Thermal properties in mesoscopics: physics and ... - ResearchGate

Thermal properties in mesoscopics: physics and applications from 

thermometry to refrigeration 

Contents 

Francesco Giazotto, 1, 2, ∗ Tero T. Heikkilä, 1, 3, † Arttu Luukanen, 4 Alexander M. Savin, 1 and Jukka P. Pekola 1 

1 Low Temperature Laboratory, Helsinki University of Technology, P.O. Box 2200, FIN-02015 HUT, Finland 

2 NEST-INFM and Scuola Normale Superiore, I-56126 Pisa, Italy 

3 Department of Physics and Astronomy, University of Basel, Klingelbergstr. 82, CH-4056 Basel, Switzerland 

4 National Institute of Standards and Technology, Quantum Electrical Metrology Division, 325 Broadway, Boulder CO 

80305 USA ‡ 

(Dated: December 2, 2005) 

We present an overview of the thermal properties of mesoscopic structures. The discussion in this 

Review is based on the concept of electron energy distribution, and in particular, on controlling and 

probing it. The temperature of the electron gas is determined by this distribution: refrigeration 

is equivalent to narrowing it, and thermometry is probing its convolution with a function characterizing 

the measuring device. Temperature exists, strictly speaking, only in quasi-equilibrium, 

where the distribution is the Fermi-Dirac one. Interesting non-equilibrium deviations can also occur, 

due to slow relaxation rates of the electrons, e.g., among themselves or with lattice phonons. 

Observation and applications of non-equilibrium phenomena are also discussed. Our focus is at 

low temperatures, primarily below 4 K, where physical phenomena on mesoscopic scales and hybrid 

combinations of various types of materials, e.g., of superconductors, normal metals, insulators 

and doped semiconductors, open up a rich variety of device concepts. The present Review starts 

with the introduction to the theoretical concepts and results on thermal properties of mesoscopic 

structures. We then focus on thermometry and refrigeration with emphasis on the experimental 

status. An immediate application field of solid-state refrigeration and thermometry is in ultrasensitive 

radiation detection, which we discuss in depth. The Review also gives a summary of 

pertinent fabrication methods of the presented devices. 

I. Introduction 2 

II. Thermal properties of mesoscopic scale hybrid 

structures at sub-kelvin temperatures 2 

A. Boltzmann equation in a diffusive wire 3 

B. Boundary conditions 4 

C. Collision integrals for inelastic scattering 5 

1. Electron-electron scattering 5 

2. Electron–phonon scattering 6 

D. Quasiequilibrium limit 7 

E. Observables 8 

1. Currents 8 

2. Noise 9 

F. Examples on different systems 10 

1. Normal-metal wire between normal-metal 

reservoirs 10 

2. Superconducting tunnel structures 11 

3. Superconductor-normal-metal structures with 

transparent contacts 12 

G. Heat transport by phonons 15 

H. Heat transport in a metallic reservoir 16 

III. Thermometry on mesoscopic scale 17 

A. Hybrid junctions 18 

1. NIS thermometer 18 

2. SIS thermometer 19 

3. Proximity effect thermometry 19 

B. Coulomb blockade thermometer, CBT 20 

∗ Electronic address: F.Giazotto@sns.it 

† Electronic address: Tero.T.Heikkila@hut.fi 

‡ Present address: MilliLab, VTT Information Technology, P.O. 

Box 1202, FIN-02044 VTT, Finland 

C. Shot noise thermometer, SNT 22 

D. Thermometry based on the temperature dependent 

conductance of planar tunnel junctions 23 

E. Anderson-insulator thin film thermometry 23 

IV. Thermal detectors and their characteristics 24 

A. Effect of operating temperature on the performance of 

thermal detectors 24 

B. Bolometers: Continuous excitation 25 

1. Hot electron bolometers 27 

2. Hot phonon bolometers 28 

C. Calorimeters: Pulsed excitation 30 

D. Future directions 31 

V. Electronic refrigeration 31 

A. General principles 32 

B. Peltier refrigerators 32 

C. Superconducting electron refrigerators 33 

1. (SI)NIS structures 33 

2. S1IS2(IS1) structures 37 

3. SSmS structures 38 

4. SF systems 39 

5. HTc NIS systems 40 

6. Application of (SI)NIS structures to lattice 

refrigeration 40 

7. Josephson transistors 42 

D. Perspective types of refrigerators 43 

1. Thermionic refrigerators 44 

2. Application of low-dimensional systems to 

electronic refrigeration 44 

VI. Device fabrication 45 

A. Structure typologies and material considerations 45 

B. Semiconductor growth techniques 46 

C. Thin-film metals deposition methods 46 

1. Thermal evaporation 46 

2. Sputter deposition 47 

D. Thin film insulators 47

E. Lithography and etching techniques 48 

F. Tunnel barriers 49 

1. Oxide barriers 49 

2. Schottky barriers 49 

VII. Future prospects 50 

Acknowledgments 50 

References 51 

I. INTRODUCTION 

Solid state mesoscopic electronic systems provide a 

micro-laboratory to realize experiments on low temperature 

physics, to study quantum phenomena such 

as fundamental relaxation mechanisms in solids, and a 

way to create advanced cryogenic devices. In a broad 

sense, mesoscopic here refers to micro- and nanostructures, 

whose size falls in between atomic and macroscopic 

scales. The central concept of this Review is 

the energy distribution of mesoscopic electron systems, 

which in thermal equilibrium (Fermi-Dirac distribution) 

determines the temperature of the electron gas. The 

non-Fermi distributions are discussed in depth, since 

they are often encountered and utilized in mesoscopic 

structures and devices. This Review aims to discuss 

the progress mainly during the past decade on how 

electron distributions can be controlled, measured and 

made use of in various device concepts. When appropriate, 

earlier developments are reviewed as well. The 

central devices and concepts to be discussed are electronic 

refrigerators, thermometers, radiation detectors, 

and distribution-controlled transistors. Typically the 

working principles or resolution of these detectors rely 

on phenomena that show up only at cryogenic temperatures, 

i.e., at temperatures of the order of a few kelvin 

and below. A practical threshold in terms of temperature 

is set by liquefaction of helium at 4.2 K. This also sets 

the emphasis in this Review: the devices and principles 

working mostly at temperatures above 4.2 K are at times 

mentioned only for reference. 

Section II of this Review introduces formally the central 

ingredients; the relevant theoretical results are either 

derived or given there. We also review some of the 

new developments concerning the thermoelectric effects 

in mesoscopic systems. Although the theoretical analysis 

of the effects in the later sections is based on Sec. II, the 

main messages can be understood without reading it in 

detail. Section III explains how the electronic temperature 

is typically measured and what is required of an 

electronic thermometer. Accurate and fast thermometers 

can be utilized for thermal radiation detection as 

explained in Sec. IV, which reviews such detectors. The 

resolution of these devices is ultimately limited by the 

thermal noise, which can be lowered by refrigeration. In 

Sec. V, we show how the electron temperature can be 

lowered via electronic means, and discuss the direct applications 

of this refrigeration. Section VI explains the 

f(E) 

k B T 

L 

E 

X 

• 

Q Electrons f(x, E), T e 

G e-ph 

• 

Qe-ph Film phonons T ph 

G ph-sub 

• 

Qph-sub Substrate phonons T sub 

G 0 

• 

Q0 Heat bath (sample holder) T 0 

• 

Qe k B T 

R 

f(E) 

FIG. 1 Schematic picture of the system considered in this 

review. We describe an electron system in a diffusive wire, 

connected to two normal-metal or superconducting reservoirs 

via contacts of resistance RN . The reservoirs are further 

connected to the macroscopic measurement apparatus (see 

Subs. II.H). The heat flows and thermal conductances between 

the studied electron system and the external driving, 

the electromagnetic environment, and the phonons in the lattice 

(Subs. II.C and II.D) are indicated with the arrows. The 

description of phonons in the lattice can further be divided in 

the film phonons, substrate phonons and finally the heat bath 

on which the substrate resides (Subs. II.G). If the system is 

used as a radiation detector, it also couples to the radiation 

field, typically via some matching circuit (Sec. IV). 

main methods used in the fabrication of mesoscopic electronic 

devices, and in Sec. VII we briefly discuss some of 

the main open questions in the field and the prospects 

of practical instruments based on electronic refrigeration 

and using the peculiar out-of-equilibrium energy distributions. 

II. THERMAL PROPERTIES OF MESOSCOPIC SCALE 

HYBRID STRUCTURES AT SUB-KELVIN 

TEMPERATURES 

The schematic picture of a setup studied in typical experiments 

described in this Review is shown in Fig. 1. 

The main object is a diffusive metal or heavily-doped 

semiconductor wire connected to large electrodes acting 

as reservoirs where electrons thermalize quickly to the 

surroundings. The electrons in the wire interact between 

themselves, and are coupled to the phonons in the film 

and to the radiation and the fluctuations in the electromagnetic 

environment. The temperature Tph of the film 

phonons can, in a non-equilibrium situation, differ from 

that of the substrate phonons, Tsub and this can even differ 

from the phonon temperature T0 in the sample holder 

that is externally cooled. Under the applied voltage, the 

energy distribution function f(E) of electrons depends 

on each of these couplings, and on the state (e.g., super- 

2 

E

conducting or normal) of the various parts of the system. 

In certain cases detailed below, f(E) is a Fermi function 

feq(E; Te, µ) = 

1 

, (1) 

exp[(E − µ)/(kBTe)] + 1 

characterized by an electron temperature Te and potential 

µ. One of the main goals of this review is to explain 

how Te, and in some cases also Tph, can be driven even 

much below the lattice temperature T0, and how this low 

Te can be exploited to improve the sensitivity of applications 

relying on the electronic degrees of freedom. We 

also detail some of the out-of-equilibrium effects, where 

f(E) is not of the form of Eq. (1). In some setups, the 

specific form of f(E) can be utilized for novel physical 

phenomena. 

Throughout the Review, we concentrate on wires 

whose dimensions are large enough to fall in the quasiclassical 

diffusive limit. This means that the Fermi wavelength 

λF , elastic mean free path ℓel and the length of 

the wire L have to satisfy λF ≪ lel ≪ L. In this regime, 

the electron energy distribution function is well defined, 

and its space dependence can be described by a diffusion 

equation (Eq. (3)). In most parts of the Review, we assume 

the capacitances C of the contacts large enough, 

such that the charging energy EC = e 2 /2C is less than 

any of the relevant energy scales and can thus be ignored. 

Our approach is to describe the electron energy distribution 

function f(r, E) at a given position r of the sample 

and then relate this function to the charge and heat currents 

and their noise. In typical metal structures in the 

absence of superconductivity, phase-coherent effects are 

weak and often it is enough to rely on a semiclassical 

description. In this case, f(r, E) is described by a diffusion 

equation, as discussed in Subs. II.A. The electron 

reservoirs impose boundary conditions for the distribution 

functions, specified in Subs. II.B. The presence of 

inelastic scattering due to electron-electron interaction, 

phonons or the electromagnetic environment can be described 

by source and sink terms in the diffusion equation, 

specified by the collision integrals and discussed in 

Subs. II.C. In the limit when these scattering effects are 

strong, the distribution function tends to a Fermi function 

feq(E; Te(r), µ(r)) throughout the wire, with a position 

dependent potential µ(r) and temperature Te(r). 

In this quasiequilibrium case, detailed in Subs. II.D, it 

thus suffices to find these two quantities. Finally, with 

the knowledge of f(r; E), one can obtain the observable 

currents and their noise as described in Subs. II.E. 

In many cases, it is not enough to only describe the 

electrons inside the mesoscopic wire, assuming that the 

surroundings are totally unaffected by the changes in this 

electron system. If the phonons in the film are not well 

coupled to a large phonon bath, their temperature is influenced 

by the coupling to the electrons. In this case, it 

is important to describe the phonon heating or cooling in 

detail (see Subs. II.G). Often also the electron reservoirs 

may get heated due to an applied bias voltage, which 

has to be taken into account in the boundary conditions. 

This heating is discussed in Subs. II.H. 

At the temperature range considered in this Review, 

many metals undergo a transition to the superconducting 

state (Tinkham, 1996). This gives rise to several new 

phenomena that can be exploited, for example, for thermometry 

(see Sec. III), for radiation detection (Sec. IV) 

and for electron cooling (Sec. V). The presence of superconductivity 

modifies both the diffusion equation (inside 

normal-metal wires through the proximity effect, 

see Subs. II.A) and especially the boundary conditions 

(Subs. II.B). Also the relations between the observable 

currents and the distribution functions are modified 

(Subs. II.E). 

Once the basic equations for finding f(r, E) are outlined, 

we detail its behavior in different types of normalmetal 

– superconductor heterostructures in Subs. II.F. 

A. Boltzmann equation in a diffusive wire 

The semiclassical Boltzmann equation (Ashcroft and 

Mermin, 1976; Smith and Jensen, 1989) describes the 

average number of particles, f(r, p)drdp/(2π) 3 , in the 

element {dr, dp} around the point {r, p} in the sixdimensional 

position-momentum space. The kinetic 

equation for f(r, p) is a continuity equation for particle 

flow, 

 

∂ 

∂t + v · ∂r 

 

+ eE · ∂p f(r, p; t) = Iel[f] + Iin[f]. (2) 

Here E is the electric field driving the charged particles 

and the elastic and inelastic collision integrals Iel and 

Iin, functionals of f, act as source and sink terms. They 

illustrate the fact that scattering breaks translation symmetries 

in space and time — the total particle numbers 

expressed through the momentum integrals of f still remain 

conserved. 

In the metallic diffusive limit, Eq. (2) may be simplified 

as follows (Nagaev, 1992; Rammer, 1998; Sukhorukov 

and Loss, 1999). The electric field term can be absorbed 

in the space derivative by the substitution E = εp + µ(r) 

in the argument of the distribution function, such that 

E describes both the kinetic εp and the potential energy 

µ of the electron. Therefore, we are only left with the 

full r-dependent derivative v · ∇f = v · ∂rf + eE · ∂pf 

on the left-hand side of Eq. (2). In the diffusive regime, 

one may concentrate on length scales much larger than 

the mean free path ℓel. There, the particles quickly lose 

the memory of the direction of their initial momentum, 

and the distribution functions become nearly isotropic in 

the direction of v. Therefore, we may expand the distribution 

function f in the two lowest spherical harmonics 

in the dependence on ˆv ≡ v/v, f(ˆv) = f0 + ˆv · δf, and 

make the relaxation-time approximation to the elastic 

collision integral with the elastic scattering time τ, i.e., 

Iel = −ˆv · δf/τ. In the limit where the time dependence 

of the distribution function takes place in a much slower 

3

scale than τ, this yields the diffusion equation with a 

source term, 

(∂t − D∇ 2 r)f0(r; E, t) = Iin[f0]. (3) 

Here we assume that the particles move with the Fermi 

velocity, i.e., v = vF ˆv. As a result, their diffusive motion 

is characterized by the diffusion constant D = v2 F τ/3. In 

what follows, we will mainly concentrate on the static 

limit, i.e., assume ∂tf0(r; E, t) = 0 and lift the subscript 

0 from the angle-independent part f0 of the distribution 

function. 

Equation (3) can also be derived rigorously from the 

microscopic theory through the use of the quasiclassical 

Keldysh formalism (Rammer and Smith, 1986). With 

such an approach, one can also take into account superconducting 

effects, such as Andreev reflection (Andreev, 

1964b) and the proximity effect (Belzig et al., 1999). In 

the diffusive limit, one obtains the Usadel equation (Usadel, 

1970), which in the static case is 

D 

σA ∇ · Ǐ = −iEˇτ3 + ˇ ∆(r) + ˇ Σin(r), ˇ G(r; E) . (4) 

Here ˇ G(r; E) is the isotropic part of the Keldysh Green’s 

function in the Keldysh ⊗ Nambu space, A and σ are 

the cross section and the normal-state conductivity of the 

wire, ˇτ3 = ˆ1⊗ˆτ3 is the third Pauli matrix in Nambu space, 

ˇ∆ = ˆ1⊗ ˆ ∆ is the pair potential matrix, and ˇ Σin describes 

the inelastic scattering that is not contained in ˇ ∆. Usadel 

equation describes the matrix current Ǐ = σA ˇ G∇ ˇ G 

(Nazarov, 1999), whose components integrated over the 

energy yield the physical currents. In the Keldysh space, 

ˇG is of the form 

 

ˆG R ˆK ˇG 

G 

= 

0 Gˆ A , 

where each component is a 2 × 2 matrix in Nambu 

particle-hole space. Equation (4) has to be augmented 

with a normalization condition ˇ G 2 = 1. This implies 

( ˆ G R/A ) 2 = 1 and allows a parametrization ˆ G K = 

ˆG Rˆ h − ˆ h ˆ G A , where ˆ h is a distribution function matrix 

with two free parameters. The equations for the 

retarded/advanced functions ˆ G R/A ((1,1) and (2,2) - 

Keldysh components of Eq. (4)) describe the behavior of 

the pairing amplitude. The solutions to these equations 

yield the coefficients for the kinetic equations, i.e., the 

(1,2) or the Keldysh part of Eq. (4). This describes the 

symmetric and antisymmetric parts of the energy distribution 

function with respect to the chemical potential of 

the superconductors. The latter is assumed everywhere 

equal to allow a time-independent description. A common 

choice is a diagonal ˆ h = f L + f T ˆτ3 (Schmid and 

Schön, 1975), where f L (E) = f(−E) − f(E) is the antisymmetric 

and f T (E) = 1−f(E)−f(−E) the symmetric 

part of the energy distribution function f(E). With this 

choice, inside the normal metals where ˆ ∆ = 0, we get 

two kinetic equations of the form (Belzig et al., 1999; 

Virtanen and Heikkilä, 2004a) 

D∇ · j L = Σ L in, j L = σA(DL∇f L − T an ∇f T + jSf T ) , 

(5a) 

D∇ · j T = Σ T in, j T = σA(DT ∇f T + T an ∇f L + jSf L ) . 

(5b) 

Here jL ≡ 1 

4Tr [(τ1 ⊗ ˆ1) Ǐ ] describes the spectral energy 

current, and jT ≡ 1 

4Tr [(τ1 ⊗ ˆτ3) Ǐ ] the spectral 

charge current. The inelastic effects are described by 

the collision integrals ΣL in 

ΣT in 

1 ≡ 4Tr [(τ1 ⊗ ˆ1)[ ˇ Σin, ˇ G]] and 

1 ≡ 4Tr [(τ1 ⊗ ˆτ3)[ ˇ Σin, ˇ G]]. The kinetic coefficients are 

DL ≡ 1 

4 Tr [1− ˆ G R ˆ G A ] 

DT ≡ 1 

4 Tr [1− ˆ G R ˆτ3 ˆ G A ˆτ3] 

T an ≡ 1 

4 Tr [ ˆ G A ˆ G R ˆτ3] 

jS ≡ 1 

4 Tr [( ˆ G R ∇ ˆ G R − ˆ G A ∇ ˆ G A )ˆτ3]. 

Here, DL and DT are the spectral energy and charge 

diffusion coefficients, and jS is the spectral density of 

the supercurrent-carrying states (Heikkilä et al., 2002). 

The cross-term T an is usually small but not completely 

negligible. In a normal-metal wire in the absence of a 

proximity effect, ˆ G R = ˆτ3 and ˆ G A = − ˆτ3. Then we 

obtain DL = DT = 1, T an = jS = 0 and the kinetic 

equations (5b) reduce to Eq. (3) in the static limit. 

B. Boundary conditions 

The quasiclassical kinetic equations cannot directly describe 

constrictions whose size is of the order of the Fermi 

wavelength, such as tunnel junctions or quantum point 

contacts. However, such contacts can be described by 

the boundary conditions derived by Nazarov (1999), 

ǏL = ǏR = Z( 1 

2 { ˇ GL, ˇ GR})[ ˇ GR, ˇ GL], (6) 

where 

Z(x) = 2e2 

h 

 

n 

Tn 

2 + Tn(x − 1) . 

Here Ǐ L(R) and ˇ G L(R) are the matrix current and the 

Green’s function at the left (right) of the constriction, 

evaluated at the interface and flowing towards the right. 

The constriction is described by a set {Tn} of transmission 

eigenvalues through the function Z(x). For large 

constrictions, it is typically enough to transform the sum 

over the eigenvalues to an integral over the transmission 

probabilities T , weighted by their probability distribu- 

tion ρ(T ). In the case of a tunnel barrier, Tn ≪ 1, 

and thus Z(x) = 2e 2 /h 

n Tn ≡ GN /2. For a ballistic 

contact Tn ≡ 1 and Z(x) = GN/(x + 1). For 

4

other types of contacts, it is typically useful to find 

the observable for arbitrary T and weight it with ρ(T ), 

e.g., ρ(T ) = πGN /[(2e 2 )T √ 1 − T ] for a diffusive contact 

(Nazarov, 1994), ρ(T ) = GN /[e 2 T 3/2√ 1 − T ] for 

a dirty interface (Schep and Bauer, 1997) or ρ(T ) = 

2GN/[e 2 T (1 − T )] for a chaotic cavity (Baranger and 

Mello, 1994). This way, the observables can be related 

to the normal-state conductance GN of the junction. 

Equation (6) yields a boundary condition both for the 

”spectral” functions ˆ G R/A and for the distribution functions. 

In the absence of superconductivity, we simply 

have ˆ G R/A = ±ˆτ3, and the boundary condition for the 

distribution functions becomes independent of the type 

of the constriction, 

j L/T = GN (f L/T 

R 

L/T 

− fL ). (7) 

In this case, the two currents can be included in the same 

function by defining j(±E) = (j L (E) ± j T (E))/2. This 

yields the spectral current through the constriction 

j(E) = GN (fL(E) − fR(E)), (8) 

where f L/R is the energy distribution function on the 

left/right side of the constriction. 

Another interesting yet tractable case is the one where 

a superconducting reservoir (on the ”left” of the junction) 

is connected to a normal metal (on the ”right”) 

and the proximity effect into the latter can be ignored. 

The latter is true, for example, if we are interested in the 

distribution function at energies far exceeding the Thouless 

energy of the normal-metal wire, or in the presence 

of strong depairing. In this case, the spectral energy and 

charge currents are 

j L = 2e2 

h 

j T = 2e2 

h 

 

TnML(Tn)θ( Ē)(f L R − f L L ) (9a) 

n 

 

Tn(M 1 T (Tn)θ(−Ē) + M 2 T (Tn)θ( Ē)f T R . 

n 

(9b) 

Here Ē = |E| − ∆ and θ(E) is the Heaviside step function, 

and the energy-dependent coefficients are 

ML(T ) = 2 (2 − T )|E|Ω + T Ω 2 

((2 − T )Ω + T |E|) 2 

M 1 T (T ) = 

M 2 T (T ) = 

2T ∆2 4(T − 1)E2 + (T − 2) 2∆2 2|E| 

(2 − T )Ω + T |E| 

(10a) 

(10b) 

(10c) 

Here we defined Ω ≡ √ E 2 − ∆ 2 . In the tunneling limit 

T ≪ 1, we get 

j L/T = NS(E)(f L/T 

R 

L/T 

− fL ), (11) 

where 

 

 

 

NS(E) = 

Re 

 

 

E + iΓ 

 

Γ→0 

→ θ(Ē)|E|/Ω 

(E + iΓ) 2 − ∆2 (12) 

is the reduced superconducting density of states (DOS). 

The first form of Eq. (12) assumes a finite pair-breaking 

rate Γ, which turns out to be relevant in some cases discussed 

in Sec. V.C.1 Unless specified otherwise, we assume 

that the superconductors are of the conventional 

weak-coupling type and the superconducting energy gap 

∆ at T = 0 is related to the critical temperature Tc by 

∆ ≈ 1.764kBTc (Tinkham, 1996). 

C. Collision integrals for inelastic scattering 

The collision integral Iin in Eq. (3) is due to electron– 

electron, electron–phonon interaction and the interaction 

with the photons in the electromagnetic environment. 

1. Electron-electron scattering 

For the electron–electron interaction, the collision integral 

is of the form 

I e−e 

coll 

= κ(d) 

e−e 

 

dωdE ′ Ĩ 

α in 

ω coll(ω, E, E ′ ) − Ĩout coll(ω, E, E ′ 

) , 

5 

(13) 

where α and κ (d) 

e−e depend on the type of scattering and 

the ”in” and ”out” collisions are 

Ĩ in 

coll = [1 − f(E)][1 − f(E ′ )]f(E − ω)f(E ′ + ω) 

(14a) 

Ĩ out 

coll = f(E)f(E ′ )[1 − f(E − ω)][1 − f(E ′ + ω)]. 

(14b) 

Electron-electron scattering can be either due to the direct 

Coulomb interaction (Altshuler and Aronov, 1985), 

or mediated through magnetic impurities which can flip 

their spin in a scattering process (Kaminski and Glazman, 

2001) or other types of impurities with internal dynamics. 

In practice, both of these effects contribute to 

the energy relaxation (Anthore et al., 2003; Pierre et al., 

2000). Assuming the electron–electron interaction is local 

on the scale of the variations in the distribution function, 

the direct interaction yields (Altshuler and Aronov, 

1985) Eq. (13) with α = −2 + d/2 for a d-dimensional 

wire. In a diffusive wire, the effective dimensionality of 

the wire is determined by comparing the dimensions to 

the energy-dependent length LE ≡ D/E. The pref-

actor κ (d) 

e−e for a d-dimensional sample is 

κ (1) 

e−e = 

1 

π √ 2D2 , (Huard et al., 2004) (15a) 

νF A 

κ (2) 

e−e = 1 

, (Rammer and Smith, 1986) (15b) 

8EF τ 

κ (3) 

e−e = 

1 

8π2√22 , (Rammer, 1998) (15c) 

νF D3/2 where νF = ν(EF ) is the density of states at the Fermi 

energy EF and A is the wire cross-section. 

In the case of relaxation due to magnetic impurities, 

one expects (Kaminski and Glazman, 2001) α = −2 and 

κe−e = π 

2 

cm S(S + 1) 

νF 

 

ln 

eV 

kBTK 

−4 

. Here cm is the 

concentration, S is the spin, and TK is the Kondo temperature 

of the magnetic impurities responsible for the 

scattering. This form is valid for T > TK. For a more 

detailed account of the magnetic-impurity effects, see 

(Göppert et al., 2002; Göppert and Grabert, 2001, 2003; 

Kroha and Zawadowski, 2002; Ujsaghy et al., 2004) and 

the references therein. 

For d = 3, and for small deviations δf from 

equilibrium, the collision integral can be approxi- 

mated (Rammer, 1998) by −δf/τe−e, where τe−e = 

3 √ 3π( √ 8−1)ζ(3/2)(kBT ) 3/2 √ 

/(16kF ℓel τEF ) is the relaxation 

time (ζ(3/2) ≈ 2.612), τ = ℓel/vF is the elastic 

scattering time and kF is the Fermi momentum. In the 

case when α < −1/2, the usual relaxation-time approach 

does not work for the electron–electron interaction as 

the expression for the relaxation time has an infrared divergence 

(Altshuler and Aronov, 1985; Rammer, 1998). 

Therefore, one has to solve the full Boltzmann equation 

with the proper collision integrals. To obtain an estimate 

for the length scale at which the electron–electron interaction 

is effective, we can proceed differently. Introducing 

dimensionless position ˜x ≡ x/L and energy variables 

˜E ′ ≡ E ′ /E∗ and ˜ω ≡ ω/E∗ , we get 

∂ 2 ˜xf = −Ke−e Ĩcoll. 

Here the dimensionless integral Ĩcoll characterizes the deviation 

in the shape of the distribution function from the 

Fermi function and Ke−e depends on the specific system. 

For a quasi-1d wire with bare Coulomb interaction, 

Ke−e = 1 

√ 

2 

RD 

RK 

 

E∗ , (16) 

where RK = h/(2e 2 ), RD = L/(σA) is the resistance 

of the wire and ET = D/L 2 is the Thouless energy. 

In the case when the wire terminates in a point contact 

with resistance RT , the resistance RD in Eq. (16) should 

be replaced with the total resistance RD + RT (Pekola 

et al., 2004a). Typically the energy scale characterizing 

the deviation from (quasi)equilibrium is E ∗ = eV . At 

eV ≫ kBT , electron–electron collisions start to be effec- 

ET 

tive when Ke−e ≈ 1. This yields a length scale 

ℓ ∗ 

 

2D 

e−e = RKAσ , (17) 

eV 

where A is the cross section of the wire and σ its conductivity. 

Using a wire with resistance RD = 10 Ω and 

ET ≈ 10 µeV for L = 1 µm (close to the values in 

(Huard et al., 2004)), and a voltage V = 100 µV, we get 

ℓ ∗ e−e ≈ 24 µm. Increasing the temperature, Ĩcoll becomes 

smaller, and this effective length also decreases. The experimental 

results of Huard et al. (2004) indicate at least 

an order of magnitude larger κe−e and thus smaller ℓ ∗ e−e 

than predicted by this theory. At present, the reasons 

for this discrepancy have not been found. 

2. Electron–phonon scattering 

Another source of inelastic scattering is due to 

phonons, for which the collision integral is of the form 

(Rammer, 1998; Wellstood et al., 1994) 

I e−ph 

coll 

= 2π 

∞ 

0 

dωα 2 

in 

F (ω) Ĩcoll(E, ω) − Ĩout 

 

coll(E, ω) . 

Here 

Ĩ in 

coll(E, ω) =f(E + ω)[1 − f(E)][1 + nph(ω)] 

+ [1 − f(E)]f(E − ω)nph(ω)) 

Ĩ out 

coll(E, ω) =f(E)[1 − f(E − ω)][1 + nph(ω)] 

+ f(E)[1 − f(E + ω)]nph(ω). 

6 

(18) 

(19) 

The kernel α 2 F (ω) (the Eliashberg function) of the interaction 

depends on the type of considered phonons 

(longitudinal or transverse), on the relation between the 

phonon wavelength λph and the electron mean free path 

ℓel, on the dimensionality of the electron and phonon 

system (Sergeev et al., 2005), and on the characteristics 

of the Fermi surface (Prunnila et al., 2005). At 

sub-kelvin temperatures and low voltages, the optical 

phonons can be neglected, and one can only concentrate 

on the acoustic phonons. In what follows, we also neglect 

phonon quantization effects which may be important in 

restricted geometries. Moreover, the phonon distribution 

function nph(ω) is considered to be in (quasi)equilibrium, 

i.e., described by a Bose distribution function nph(ω) = 

neq(ω) ≡ [exp(ω/(kBT )) − 1] −1 (for phonon relaxation 

processes, see Subs. II.G). 

When the phonon temperature Tph is much lower than 

the Debye temperature TD, the phonon dispersion relation 

is linear and one can estimate the phonon wavelength 

using λph = hvS/kBTph. For typical metals, the speed 

of sound is vS ∼ 3 . . . 5 km/s, which yields a wavelength 

λph ∼ 100 . . . 200 nm at Tph = 1 K and λph ∼ 1 . . . 2 µm 

at Tph = 100 mK. In the clean limit λph ≪ ℓel, approximating 

the electron-phonon coupling with a scalar deformation 

potential, only the longitudinal phonons are coupled 

to the electrons. In this case for ω ≪ kBTD/, kF vS

Te (mK) Tph Σ (W 

(mK) m −3 K −5 Measured in 

) 

Ag 50...400 50...400 0.5 10 9 

(Steinbach et al., 1996) 

Al 35...130 35 0.2 10 9 

(Kautz et al., 1993) 

200...300 200 0.3 10 9 

(Meschke et al., 2004) 

Au 80...1200 80...1000 2.4 10 9 

(Echternach 

1992) 

et al., 

AuCu 50...120 20...120 2.4 10 9 

(Wellstood et al., 1989) 

Cu 25...800 25...320 2.0 10 9 

(Roukes et al., 1985) 

100...500 280...400 0.9...4 10 9 (Leivo et al., 1996) 

50...200 50...150 2.0 10 9 

(Meschke et al., 2004) 

Mo 980 80...980 0.9 10 9 

(Savin 

2005) 

and Pekola, 

n ++ Si 120...400 175...400 0.1 10 9 

(Savin et al., 2001) 

173...450 173 0.04 10 9 

(Prunnila et al., 2002) 

320...410 320...410 0.1 10 9 

(Buonomo et al., 2003) 

Ti 300...800 500...800 1.3 10 9 

(Manninen et al., 1999) 

TABLE I Measured electron-phonon coupling constant Σ for 

different materials. The second and third columns indicate 

the temperature ranges (electron and phonon temperatures, 

respectively) of the measurements. 

(Rammer, 1998; Wellstood et al., 1994) 

|M| 2 

α 2 F (ω) = 

4π22v 3 SνF ω 2 , (20) 

where |M| 2 is the square of the matrix element for the 

deformation potential. Generally this is inversely proportional 

to the mass density of the ions, but its precise 

microscopic form depends on the details of the lattice 

structure. Therefore it is useful to present |M| in 

terms of a separately measurable quantity, e.g., the prefactor 

Σ of the power P = ΣVT 5 dissipated to the lattice 

of volume V in the quasiequilibrium limit (see Table 

I and Subs. II.D): |M| 2 = π5v3 SΣ/(12ζ(5)k5 B ), where 

ζ(5) ≈ 1.0369. 

In the dirty limit λph ≫ ℓel, the power of ω in the 

Eliashberg function can be either 1 or 3, for the cases 

of static or vibrating disorder, respectively. For further 

details about the dirty limit, we refer to (Belitz, 1987; 

Rammer and Schmid, 1986; Sergeev and Mitin, 2000). 

The relaxation rate for electron-phonon scattering is 

given by 1/τe−ph = −{δIe−ph[f(E)]/δf(E)}| f(E)=feq(E), 

where feq(E) now is a Fermi function at the lattice temperature. 

With this definition at E = EF , (Rammer, 

1998) 

1 

τe−ph 

= 4π 

∞ 

0 

dω α2F (ω) 

. 

sinh 

(21) 

ω 

kBT 

Thus, in the clean case for kBT ≪ 2kF vS we obtain 

1/τe−ph = αT 3 , α = 7ζ(3)Σ/(24ζ(5)k 2 B νF ) ≈ 

0.34Σ/(k 2 B νF ). With typical values for Cu, Σ = 2 · 10 9 

WK −5 m −3 and νF = 1.6·10 47 J −1 m −3 , we get τe−ph = 45 

ns at T = 1 K. Assuming λF ≪ ℓel ≪ ℓe−ph, the 

electron–phonon scattering length is ℓe−ph = Dτe−ph. 

For the above values and a typical diffusion constant 

D = 0.01 m 2 /s, ℓe−ph ≈ 21 µm at T = 1 K and 

ℓe−ph ≈ 670 µm at T = 100 mK. 

In the disordered limit λph ≫ lel, the temperature dependence 

of the electron-phonon scattering rate is expected 

to follow either the T 2 or T 4 laws, depending on 

the nature of the disorder (Sergeev and Mitin, 2000). 

It seems that although most of the experiments are 

done in the limit where the phonon wavelength at least 

slightly exceeds the electron mean free path, in majority 

of the cases the results have fitted to the clean-limit expressions, 

i.e., the scattering rate ∝ T 3 e and the heat current 

flowing into the phonon system ∝ T 5 e , see Eq. (24) 

below (for an exception, see (Karvonen et al., 2005)). 

Finding the correct exponent is not straightforward, as 

the film phonons are also typically affected by the measurement, 

and because of the reduced dimensionality of 

the phonon system (see Subs. II.G). In this Review, we 

concentrate on the clean-limit expressions. 

D. Quasiequilibrium limit 

The shape of the distribution function at a given position 

of the wire strongly depends on how the inelastic 

scattering length lin compares to the length L of the 

wire. For L ≪ lin (nonequilibrium limit), we may neglect 

the inelastic scattering altogether. In this case, the 

distribution function is a solution to either Eqs. (5) or 

Eq. (3), where the collision integrals/self energies for inelastic 

scattering can be neglected. As a result, the shape 

of the electron distribution functions inside the wire at a 

finite bias voltage eV kBT may strongly deviate from a 

Fermi distribution (Giazotto et al., 2004b; Heikkilä et al., 

2003; Heslinga and Klapwijk, 1993; Pekola et al., 2004a; 

Pierre et al., 2001; Pothier et al., 1997b). The nonequilibrium 

shape shows up in most of the observable properties 

of the system, including the I − V characteristics, 

the current noise or the supercurrent. In general, it can 

only be neglected in the I −V characteristics if the charge 

transport process is energy independent as in the case of 

purely normal-metal samples. Even in this case the form 

of f(E) can be observed in the current noise. 

The kinetic equations can be greatly simplified in the 

limit where ℓin for one type of scattering is much smaller 

than L. In the quasiequilibrium limit, the energy relaxation 

length due to electron–electron scattering is much 

shorter than the wire, ℓe−e ≪ L (Nagaev, 1995). In this 

case, the local distribution function is a Fermi function 

characterized by the temperature Te(r, t) and potential 

µ(r, t). Mathematically, this can be seen by considering 

the Boltzmann equation (3) with the electron–electron 

collision integral, Eq. (13) in the limit where the prefactor 

of the latter becomes very large. As the left-hand 

side of Eq. (3) is not strongly dependent on the form of 

f(E, r) as a function of energy, the equation can only 

be satisfied if the collision integral without the prefactor 

becomes small. It can be easily shown that the latter 

7

vanishes for f(E) = feq(E). Thus, the deviations from 

the Fermi-function shape will be at most of the order of 

ℓin/L, and can be neglected in the quasiequilibrium limit. 

In this limit, we are still left with two unknowns, 

Te(r, t) and µ(r, t). Substituting f(E, r) = 

feq(E; Te(r, t), µ(r, t)) in Eq. (3) yields 

(∂t − D∇ 2 )f = (∂t − D∇ 2 )(Te∂Tef + µ∂µf)− 

D[(∇µ) 2 ∂ 2 µf) + (∇Te) 2 ∂ 2 Te f + 2∇Te∇µ∂Te∂µf] = Icoll[f], 

where Icoll[f] contains the other types of inelastic scatterings, 

e.g., those with the phonons. In the right hand side 

of the upper line, the differential operators ∂t and ∇ act 

only on Te and µ. Integrating this over the energy and 

multiplying by νF E and then integrating over E yields 

(∂t − D∇ 2 )µ(r) = 0, (22) 

Ce(r, t)∂tTe − ∇(κ(r, t)∇Te) − σ(∇µ/e) 2 = Ĩcoll. (23) 

We assumed that the energy integral over Icoll[f] vanishes. 

1 Here Ce(r, t) = π 2 νF k 2 B Te(r, t)/3 is the electron 

heat capacity, σ = DνF e 2 is the Drude conductivity, 

κ(r, t) = σL0Te(r, t) is the electron heat conductivity, 

L0 = π 2 k 2 B /(3e2 ) ≈ 2.45 · 10 −8 WΩK −2 is the Lorenz 

number and Ĩcoll(Te, µ) contains the power per unit volume 

emitted or absorbed by other excitations, such as 

phonons or the electromagnetic radiation field. The last 

term on the left hand side of Eq. (23) describes the Joule 

heating due to the applied voltage. In what follows, 

we write the volume explicitly in the collision integral 

by averaging over a small volume V around the point 

r where T (r) is approximately constant, thus defining 

Pcoll(r) ≡ V Ĩcoll. 

For the electron-phonon scattering (Wellstood et al., 

1994), Pcoll reads in the clean case (see also Table I) 

P e−ph 

coll = ΣV(Te(r) 5 − Tph(r) 5 ). (24) 

For the dirty limit specified below Eq. (20), the Eliashberg 

functions scaling with ω n translate into temperature 

dependences scaling as T n+3 , i.e., T 4 and T 6 (Sergeev 

and Mitin, 2000). 

The electrons can also be heated due to the thermal 

noise in their electromagnetic environment unless proper 

filtering is realized to prevent this heating. If one aims 

to detect the electromagnetic environment as discussed in 

Sec. IV, this discussion can of course be turned around to 

find the optimal coupling to the radiation to be observed. 

A model for such coupling in the quasiequilibrium limit 

was considered by Schmidt et al. (2004a), who obtained 

an expression for the emitted/absorbed power due to the 

external noise in the form 

P e−em 

coll 

= r k2 B π2 

6h (T 2 e − T 2 γ ). (25) 

1 In the diffusive limit where the inelastic scattering rates are lower 

than 1/τ, this is related to the particle number conservation and 

is thus generally valid. 

Here r = 4ReRγ/(Re + Rγ) 2 is the coupling constant, 

Tγ is the (noise) temperature of the environment, and 

Re and Rγ are the resistances characterizing the thermal 

noise in the electron system and the environment, 

respectively. This expression assumes a frequency independent 

environment in the relevant frequency range. 

For some examples on the frequency dependence, we refer 

to Schmidt et al. (2004a). 

E. Observables 

1. Currents 

In the nonequilibrium diffusive limit, the charge current 

in a normal-metal wire in the absence of a proximity 

effect or any such interference effects as weak localization 

is obtained from the local distribution function by 

I = −eA 

∞ 

−∞ 

dED(E)ν(E)∇f(x; E) (26) 

and the heat current from a reservoir with potential µ is 

˙Q = −A 

∞ 

−∞ 

dE(E − µ)D(E)ν(E)∇f(x; E). (27) 

Here we included the possible energy dependence of the 

diffusion constant D(E) and of the density of states ν(E), 

due to the energy dependence of the elastic scattering 

time, or due to the nonlinearities in the quasiparticle dispersion 

relation. If the Kondo effect (Vavilov et al., 2003) 

can be neglected, such effects are very small in good metals 

at temperatures of the order of 1 K or less. 

In the quasiequilibrium limit, assuming D(E) =const. 

and ν(E) = νF , Eqs. (26,27) can be simplified to 

8 

I = −σA∇µ/e (28a) 

˙Q = −κA∇T. (28b) 

When a diffusive wire of resistance RD is connected 

to a reservoir through a point contact characterized by 

the transmission eigenvalues {Tn}, the final distribution 

function is obtained after solving the Boltzmann equation 

(3) or Eqs. (5) with the boundary conditions given by 

Eq. (6), (8), or by (9). However, when RD is much less 

than the normal-state resistance 1/GN of the contact, 

we can ignore the wire and obtain the full current by a 

direct integration over Eq. (8) or Eq. (9). For example, 

for NIS or SIS tunnel junctions, the expressions for the 

charge and heat currents from the left side of the junction 

become 

I = 1 

eRT 

˙Q = 1 

e2RT 

dENL( ˜ E)NR(E)[fL(E) − fR(E)] (29) 

 

dE ˜ ENL( ˜ E)NR(E)[fL(E) − fR(E)]. (30) 

Here ˜ E = E − eV , RT = 1/GN , N L/R(E) = ν(E)/νF 

is the reduced density of states for the left/right wire,

N(E) = 1 for a normal metal and N(E) = NS(E) for a 

superconductor. Furthermore, if the two wires constitute 

reservoirs, fL/R are Fermi functions with potentials µL = 

−eV , µR = 0. The resulting NIS or SIS charge current is 

a sensitive probe of temperature and can hence be used 

for thermometry or radiation detection, as explained in 

Secs. III.A and IV, respectively. Moreover, analysis of 

the heat current Eq. (30) shows that the electrons can 

in certain situations be cooled in NIS/SIS structures, as 

discussed in Sec. V.C.1. 

In the presence of a proximity effect, the equations 

for the charge and thermal currents in the quasiclassical 

limit, i.e., ignoring the energy dependence of the diffusion 

constant and the normal-metal density of states, are 

 

I = dEj T 

(31a) 

 

˙Q = dE(Ej L − µj T ). (31b) 

Here µ is the potential of the reservoir from which the 

heat current is calculated. As in Eq. (5), these currents 

can be separated into quasiparticle, anomalous, and supercurrent 

parts. 

2. Noise 

Often one can express the zero-frequency current noise 

in terms of the local distribution function (Blanter and 

Büttiker, 2000). The noise is characterized by the correlator 

∞ 

S = 2 dt 

−∞ 

′ 〈δÎ(t + t′ )δÎ(t)〉, where δÎ = Î − 〈Î〉 and Î is the current operator. In a 

stationary system S is independent of t. In a normalmetal 

wire of length L in the nonequilibrium limit, the 

current noise S can be expressed as (Nagaev, 1992) 

S = 4eDνF A 

L 2 

L 

0 

∞ 

dx dEf(E, x)[1 − f(E, x)]. (32) 

−∞ 

In the quasiequilibrium regime, this equation simplifies 

to (Nagaev, 1995) 

S = 4eDνF AkB 

L 2 

L 

0 

dxTe(x). (33) 

When the resistance of a point contact dominates that 

of the wire, the noise power can be expressed through 

(Blanter and Büttiker, 2000) 

SNN = 2e2 

h 

 

 

dE{Tn [fL(1 − fL) + fR(1 − fR)] 

n 

+ Tn(1 − Tn)(fL − fR) 2 } 

for a normal-metal contact and (de Jong and Beenakker, 

1994) 

SNS = 2e2 

h 

 

 

n 

 

T 

dE 

2 

n 

(2 − Tn) 2 2fL(E)[1 − fL(E)] 

+ 16T 2 

n (1 − Tn) 

(2 − Tn) 4 

(f T L ) 2 

for an incoherent NS contact at E < ∆. Here f L/R 

are the distribution functions in the left/right (normal/superconducting 

in the latter case) side of the contact 

and f T L (E) = 1 − fL(E) − fL(−E), the symmetric 

part w.r.t. the S potential. If the scattering probabilities 

are independent of energy and if the two sides are 

reservoirs with the same temperature, these expressions 

simplify to 

SNN = 2e2 

h 

SNS = 2e2 

h 

+ 16T 2 

 

n 

2kBT T 2 

n + eV coth (v) Tn(1 − Tn) 

 

 

T 2 

n 

2kBT 

(2 − Tn) 2 

n 

n (1 − Tn) 

(2 − Tn) 4 (2eV coth (v) − 4kBT ) 

 

 

, 

9 

(34a) 

(34b) 

where v = eV/(2kBT ). For eV = 0 (thermal Johnson- 

Nyquist noise), S = 4kBT G, where G is the conductance 

of the point contact. In the opposite limit eV ≫ kBT 

(shot noise), one obtains S = 2eF I, where I = GV is the 

current through the junction and F is the Fano factor. 

In the presence of the superconducting proximity effect, 

the expression for the noise becomes more complicated 

(Houzet and Pistolesi, 2004). In general, it can 

be found by employing the counting-field technique developed 

by Nazarov and coworkers, see (Nazarov and 

Bagrets, 2002) and the references therein. This technique 

can also be applied to study the full counting statistics of 

the transmitted currents through a given sample within 

a given measurement time (Nazarov, 2003). 

Apart from the charge current, also the heat current 

in electric circuits fluctuates. For example, the zerofrequency 

heat current noise from the ”left” of a tunnel 

contact biased with voltage V is given by 

SQ = 2 

e2 

dEE 

RT 

2 NL(E − eV )NR(E) 

(35) 

[fR(1 − fL) + fL(1 − fR)]. 

At low voltages V ≪ kBT/e, the heat current noise obeys 

the fluctuation-dissipation result SQ = 4kBT 2 Gth, where 

Gth is the thermal conductance. This quantity is related 

to the noise equivalent power (NEP) discussed in the literature 

of thermal detectors by SQ =NEP 2 . The total 

NEP contains contributions not only from the electrical 

heat current noise, but also from other sources, such as

f(x;E) 

(a) 

1 

0.5 

0 

-1 

-0.5 0 0.5 

E /eV 

1 

1 

0.5 

x/L 

0 

1 

0.5 

f(x;E) (b) 

0 

-2 -1 0 1 

E/eV 2 

FIG. 2 Nonequilibrium quasiparticle energy distribution 

function in a diffusive normal-metal wire in the absence of 

inelastic scattering: (a) wire placed between two normalmetal 

reservoirs and (b) wire placed between a normal-metal 

(x = L) and a superconducting (x = 0) reservoir. In the 

latter picture, we assume D/L 2 ≪ eV ≪ ∆, such that 

the proximity effect and the states above the gap can be neglected. 

In both pictures, the lattice temperature was fixed 

to kBT = eV/10. 

the direct charge current noise and electron-phonon coupling. 

A detailed discussion of various NEP sources is 

presented in Sec. IV. 

Another important quantity is the cross-correlator between 

the current and heat current fluctuations. At zero 

frequency, this is given by 

SIQ = − 2 

 

dEENL(E − eV )NR(E) 

eRT 

(36) 

[fR(1 − fL) + fL(1 − fR)]. 

These types of fluctuations have to be taken into account 

for example when analyzing the NEP of bolometers (Golubev 

and Kuzmin, 2001). Recently, also the general statistics 

of the heat current fluctuations have been theoretically 

addressed by Kindermann and Pilgram (2004). 

F. Examples on different systems 

Below, we detail the solutions to the kinetic equations, 

(3) or (5) in some example systems. The aim is first to 

provide an understanding of the general behavior of the 

distribution function in these systems, but also to show 

the generic features, such as the electron cooling in NIS 

junctions. 

1. Normal-metal wire between normal-metal reservoirs 

The simplest example is a quasi-one-dimensional 

normal-metal diffusive wire of resistance RD = L/(Aσ), 

connected to two normal-metal reservoirs by clean contacts. 

In the full nonequilibrium limit, we find (see Fig. 2 

(a); the coordinate x follows Fig. 1) 

f(E, x) = fL(E) + [fR(E) − fL(E)] x 

, (37) 

L 

where fL and fR are the (Fermi) distribution functions 

in the left and right reservoirs with temperatures TL and 

1 

0.5 

x/L 

0 

f N (E) 

1 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

−1.5 −1 −0.5 0 

E/∆ 

0.5 1 1.5 

10 

FIG. 3 (Color in online edition): Quasiparticle energy distribution 

function in the center of a normal-metal wire placed 

between two normal-metal reservoirs and biased with a voltage 

eV = 10kBT . The three lines correspond to three extreme 

limits: L ≪ ℓe−e, ℓe−ph (nonequilibrium limit, black 

solid line), ℓe−e ≪ L ≪ ℓe−ph (quasiequilibrium limit, red 

dashed line), and ℓe−e, ℓe−ph ≪ L (equilibrium limit, blue 

dotted line). 

TR and potentials µL and µR = µL + eV , respectively. 

The resulting two-step form illustrated in Fig. 2a was 

first measured by Pothier et al. (1997b). 

In the quasiequilibrium limit, we get f(E, x) = 

feq(E; µ(x), Te(x)), where 

µ(x) = µL + eV x 

L 

Te(x) 2 = T 2 L + (T 2 R − T 2 L) x 2 V 

+ 

L L0 

x 

 

L 

1 − x 

L 

(38a) 

 

. (38b) 

In both cases, the current is simply given by I = V/RD 

and the heat current in the limit V = 0 by ˙ Q = L0(T 2 R − 

T 2 L )/RD. For V = 0, the resistor dissipates power and 

˙Q is not conserved. The electron distribution function in 

the center of the wire, x = L/2 is plotted in Fig. 3 for the 

nonequilibrium, quasiequilibrium and local equilibrium 

(strong electron–phonon scattering) limits. 

To obtain estimates for the thermoelectric effects due 

to the particle-hole symmetry breaking, let us lift the assumption 

of energy independent diffusion constant and 

density of states, expanding them as D(E) ≈ D0 + 

cD E−EF and ν(E) ≈ νF +cN EF 

E−EF . For linear response, 

EF 

the resulting expressions for the charge and heat currents 

are (Cutler and Mott, 1969) 

I = −GL∇µ/e + GLα∇T (39a) 

˙Q = −ΠGL∇µ/e + GthL∇T. (39b) 

Here G = e 2 νF D0A/L is the Drude conductance, 

Gth = L0GT = κA/L (Wiedemann-Franz law) is the 

heat conductance, α = eL0G ′ T/G (Mott law) is the 

Seebeck coefficient describing the thermoelectric power, 

Π = αT (Onsager relation) is the Peltier coefficient, and 

G ′ = e 2 (cDνF + D0cN)A/(LEF ) describes effects due 

to the particle-hole symmetry breaking. We see that the 

thermoelectric effects are in general of the order kBT/EF ; 

in good metals at temperatures of the order of 1 K they

E F 

eV 

(a) 

e 

E 

Δ 

Δ 

N I S 

E F 

forbidden 

levels 

Q& 

A A(nW/μm2 

) 

(b) 

20 

15 

10 

5 

0.15 

0.2 

k BT/Δ (0) = 0.3 

0 

0 3 6 9 

T (10-2 ) 

FIG. 4 (Color in online edition) (a) Sketch of the energy band 

diagram of a voltage biased NIS junction. Upon biasing the 

structure, the most energetic electrons (e) can most easily 

tunnel into the superconductor. As a result the electron gas 

in the N electrode is cooled. (b) Maximum cooling power 

surface density ˙ QA vs interface transmissivity T at different 

temperatures calculated for a NS contact. 

can hence be typically ignored. Therefore, the Peltier refrigerators 

discussed in Subs. V.B rely on materials with 

a low EF . 

2. Superconducting tunnel structures 

Consider a NIS tunnel structure, coupling a large superconducting 

reservoir with temperature Te,S to a large 

normal-metal reservoir with temperature Te,N via a tunnel 

junction with resistance RT . Let us then assume 

a voltage V applied over the system. In this case, the 

heat current (cooling power) from the normal metal is 

given by Eq. (30) with NL(E) = 1, NR(E) = NS(E), 

fL(E) ≡ feq(E − eV, Te,N ) and fR(E) ≡ feq(E, Te,S). 

For small pair breaking inside the superconductor, i.e., 

Γ ≪ ∆, ˙ QNIS is positive for eV < ∆, i.e., it cools 

the normal metal. It is straightforward to show that 

˙QNIS(V ) = ˙ QNIS(−V ). This is in contrast with Peltier 

cooling, where the sign of the current determines the direction 

of the heat current. For eV > ∆, the current 

through the junction increases strongly, resulting in Joule 

heating and making ˙ QNIS negative. The cooling power is 

maximal near eV ≈ ∆. 

In order to understand the basic mechanism for cooling 

in such systems, let us consider the simplified energy 

band diagram of a NIS tunnel junction biased at voltage 

V , as depicted in Fig. 4(a). The physical mechanism 

underlying quasiparticle cooling is rather simple: owing 

to the presence of the superconductor, in the tunneling 

process quasiparticles with energy E < ∆ cannot tunnel 

inside the forbidden energy gap, but the more energetic 

electrons (with E > ∆) are removed from the N electrode. 

As a consequence of this ”selective” tunneling of 

hot particles, the electron distribution function in the N 

region becomes sharper: the NIS junction thus behaves 

as an electron cooler. 

The role of barrier transmissivity in governing heat 

flux across the NIS structure was analyzed by Bardas 

hQ /(2 ∆ ) 

2 

. 

(a) 

0.1 

0.08 

0.06 

0.04 

0.02 

0 

- 0.02 

- 0.04 

- 0.06 

- 0.08 

- 0.1 

0 0.5 1 1.5 

eV/∆ 

(b) 

2.5 

hI /(2e ∆ ) 

2 

1.5 

1 

0.5 

11 

0 

0 0.5 

eV/∆ 

1 1.5 

FIG. 5 (Color in online edition): (a) NS point contact heat 

current from the N side as a function of voltage for different 

transparencies T (from top to bottom, tunneling limit T → 0, 

T = 0.005, T = 0.01, T = 0.05, T = 0.5 and T = 1) at 

kBT = 0.3∆. For T 0.05, the heat current is positive, corresponding 

to cooling. (b) NS point contact current-voltage 

characteristics for the same values of T and T as in (a) (now 

T increases from bottom to top). The first four curves lie essentially 

on top of each other. The corresponding NIN curves 

are shown with the dashed lines. 

and Averin (1995). They pointed out the interplay between 

single-particle tunneling and Andreev reflection 

(Andreev, 1964a) on the heat current. In the following it 

is useful to summarize their main results. 

The cooling regime requires a tunnel contact. The effect 

of transmissivity is illustrated in Fig. 4(b), which 

shows the maximum of the heat current density (i.e., the 

specific cooling power) ˙ QA versus interface transmissivity 

T at different temperatures. This can be calculated 

for a generic NS junction using Eqs. (9), (10a) and (31b). 

The quantity ˙ QA is a non-monotonic function of interface 

transmissivity, vanishing both at low and high values of 

T . In the low transparency regime, ˙ QA turns out to 

be linear in T , showing that electron transport is dominated 

by single particle tunneling. Upon increasing barrier 

transmissivity, Andreev reflection begins to dominate 

quasiparticle transport, thus suppressing heat current extraction 

from the N portion of the structure. The heat 

current ˙ QA is maximized between these two regimes at 

an optimal barrier transmissivity, which is temperature 

dependent. Furthermore, by decreasing the latter leads 

to a reduction of both the optimal T and of the transmissivity 

window where the cooling takes place. In real 

NIS contacts used for cooling applications, the average T 

is typically in the range 10 −6 ...10 −4 (Leivo et al., 1996; 

Nahum et al., 1994) corresponding to junction specific 

resistances Rc (i.e., the product of the junction normal 

state resistance and the contact area) from tens to several 

thousands Ω µm 2 . This limits the achievable ˙ QA 

to some pW/µm 2 . From the above discussion it appears 

that exploiting low-Rc tunnel contacts is an important requirement 

in order to achieve large cooling power through 

NIS junctions. However, in real low-Rc barriers, pinholes 

with a large T appear. They contribute with a large Joule 

heating (see Fig. 5, which shows the heat and charge currents 

through the NIS interface as functions of voltage for 

different T ), and therefore tend to degrade the cooling

f N (E) (a) 

1 

0.8 

0.6 

0.4 

0.2 

0 

−0.6 −0.4 −0.2 0 

E/∆ 

0.2 0.4 0.6 

f N (E) 

1 

0.8 

0.6 

0.4 

0.2 

K e−e = 

0 

1 

10 

100 

0 

−0.6 −0.4 −0.2 0 

E/∆ 

0.2 0.4 0.6 

FIG. 6 (Color in online edition): (a) Nonequilibrium distribution 

inside the SINIS island, calculated from Eq. (42) 

with Icoll = 0, T = 0.1Tc, and using a depairing strength 

Γ = 10 −4 ∆ inside the superconductors. The voltage runs 

from zero to V = 3∆/e in steps of ∆/(2e) and spans three 

regimes: (i) anomalous heating regime (eV = 0 . . . ∆, solid 

lines, widening with an increasing V ) discussed in (Pekola 

et al., 2004a), (ii) cooling regime (eV = 1.5∆ and eV = 2∆, 

dashed lines, narrowing with an increasing V ) and the strong 

nonequilibrium heating regime (eV = 2.5∆ and eV = 3∆, 

dotted lines, widening with an increasing V ). (b) Distribution 

at eV = 2.5∆ for different strengths of the electron– 

electron interaction, parametrized by the parameter Ke−e 

from Eq. (16) with RD replaced by RT and E ∗ by ∆, as 

in this case these describe the collision integral. 

performance at the lowest temperatures, and to overheat 

the superconductor at the junction region due to strong 

power injection. Barrier optimization in terms of both 

materials and technology seems to be still nowadays a 

challenging task (see also Sec. VI.F.1). 

In a SINIS system in the quasiequilibrium limit, the 

temperatures Te,N and Te,S of the N and S parts may 

differ. If the superconductors are good reservoirs, Te,S 

equals the lattice temperature. In the experimentally 

relevant case when the resistance RD of the normal-metal 

island is much lower than the resistances RL and RR of 

the tunnel junctions, the normal-metal temperature Te,N 

is determined from the heat balance equation 

(b) 

˙QSINIS(V ; Te,S, Te,N ) = Pcoll, (40) 

where ˙ QSINIS = 2 ˙ QNIS and Pcoll describes inelastic scattering 

due to phonons (Eq. (24)) and/or due to the electromagnetic 

environment (Eq. (25)). 

For further details about NIS/SINIS cooling in the 

quasiequilibrium limit, we refer to Sec. V.C.1 and 

(Anghel and Pekola, 2001). 

Let us now consider the limit of full nonequilibrium, 

neglecting the proximity effect from the superconductors 

on the normal-metal island. 2 Then the distribution function 

inside the normal metal may be obtained by solving 

the kinetic equation Eq. (3) in the static case along with 

the boundary conditions given by Eq. (8). For simplicity, 

2 This is justified in the limit RL, RR ≫ RD. 

12 

let us assume RL = RR ≡ RT ≫ RD. In this limit, the 

distribution function fN (E) in the normal-metal island 

is almost independent of the spatial coordinate. Then, 

in Eq. (3) we can use ∂ 2 xf ≈ (∂xf(L) − ∂xf(0))/L, where 

L is the length of the N wire. From Eq. (11) we then get 

RD 

RT 

(NR(fN − fR) − NL(fL − fN)) = τDIcoll, (41) 

where τD = L2 N /D is the diffusion time through the island. 

As a result (Giazotto et al., 2004b; Heslinga and 

Klapwijk, 1993), the distribution function in the central 

wire can be expressed as 

fN = fRNR + fLNL + RIVe 2νF Icoll 

. (42) 

NR + NL 

Here V = LA is the volume of the island. In the presence 

of inelastic scattering, this is an implicit equation, as Icoll 

is a functional of fN . Examples of the effect of inelastic 

scattering have been considered in (Heslinga and Klapwijk, 

1993) in the relaxation time approach, and in (Giazotto 

et al., 2004b) including the full electron–electron 

scattering collision integral. The distribution function 

fN in Eq. (42) is plotted in Fig. 6 for a few values of the 

voltage for Icoll = 0 and for a few strengths of electron– 

electron scattering at eV = 2.5∆. 

3. Superconductor-normal-metal structures with transparent 

contacts 

If a superconductor is placed in a good electric contact 

with a normal metal, a finite pairing amplitude is induced 

in the normal metal within a thermal coherence 

length ξT = D/(2πkBT ) near the interface. This 

superconducting proximity effect modifies the properties 

of the normal metal (Belzig et al., 1999; Lambert and 

Raimondi, 1998) and makes it possible to transport supercurrent 

through it. It also changes the local distribution 

functions by modifying the kinetic coefficients in 

Eq. (5). Superconducting proximity effect is generated 

by Andreev reflection (Andreev, 1964b) at the normalsuperconducting 

interface, where an electron scatters 

from the interface as a hole and vice versa. Andreev 

reflection forbids the sub-gap energy transport into the 

superconductor, and thus modifies the boundary conditions 

for the distribution functions. In certain cases, the 

proximity effect is not relevant for the physical observables 

whereas the Andreev reflection can still contribute. 

This incoherent regime is realized if one is interested in 

length scales much longer than the extent of the proximity 

effect. Below, we first study such ”pure” Andreevreflection 

effects, and then go on to explain how they are 

modified in the presence of the proximity effect. 

In the incoherent regime, for E < ∆, Andreev reflection 

can be described through the boundary conditions 

(Pierre et al., 2001) 

f(µS + E) = 1 − f(µS − E) (43a) 

ˆn · ∇[f(µS − E) − f(µS + E)] = 0, (43b)

evaluated at the normal-superconducting interface. Here 

ˆn is the unit vector normal to the interface and µS is the 

chemical potential of the superconductor. The former 

equation guarantees the absence of charge imbalance in 

the superconductor, while the latter describes the vanishing 

energy current into it. For E > ∆, the usual 

normal-metal boundary conditions are used. Note that 

these boundary conditions are valid as long as the resistance 

of the wire by far exceeds that of the interface; in 

the general case, one should apply Eqs. (9). 

Assume now a system where a normal-metal wire is 

placed between normal-metal (at x = L, potential µN 

and temperature T = Te,N) and superconducting reservoirs 

(at x = 0, µS = 0, T = Te,S). Solving the Boltzmann 

equation (3) then yields (Nagaev and Buttiker, 

2001) 

 

1 

2 

f(E) = 

 

x 1 + L 

fN (E) + 1 

 

x 

2 1 − ¯fN(E), 

L 

E < ∆ 

 

x 1 − L feq(E; µ = 0, Te,S) + x 

LfN (E), E > ∆. 

Here fN(E) = feq(E; µN , Te,N) and 

(44) 

fN(E) ¯ = 

feq(E; −µN, Te,N ). This function is plotted in Fig. 2 

(b). In the quasiequilibrium regime, the problem can 

be analytically solved in the case eV, kBT ≪ ∆, i.e., 

when Eqs. (43) apply for all relevant energies. Then the 

boundary conditions are µ = µS = 0 and ˆn · ∇Te = 0 

at the NS interface. Thus, the quasiequilibrium distribution 

function is given by f(E, x) = f(E; µ(x), Te(x)) 

with µ(x) = µN (1 − x/L) and 

Te(x) = 

 

T 2 e,N 

+ V 2 

L0 

x 

 

L 

2 − x 

L 

 

. (45) 

Note that this result is independent of the temperature 

Te,S of the superconducting terminal. 

In this incoherent regime, the electrical conductance is 

unmodified compared to its value when the superconductor 

is replaced by a normal-metal electrode: Andreev reflection 

effectively doubles both the length of the normal 

conductor and the conductance for a single transmission 

channel (Beenakker, 1992), and thus the total conductance 

is unmodified. Wiedemann-Franz law is violated 

by the Andreev reflection: there is no heat current into 

the superconductor at subgap energies. However, the 

sub-gap current induces Joule heating into the normal 

metal and this by far overcompensates any cooling effect 

from the states above ∆ (see Fig. 5). 

The SNS system in the incoherent regime has also 

been analyzed by Bezuglyi et al. (2000) and Pierre et al. 

(2001). Using the boundary conditions in Eqs. (43) at 

both NS interfaces with different potentials of the two superconductors 

leads to a set of recursion equations that 

determine the distribution functions for each energy. The 

recursion is terminated for energies above the gap, where 

the distribution functions are connected simply to those 

of the superconductors. This process is called the multiple 

Andreev reflection: in a single coherent process, a 

13 

quasiparticle with energy E < −∆ entering the normalmetal 

region from the left superconductor undergoes multiple 

Andreev reflections, and its energy is increased by 

the applied voltage during its traversal between the superconductors. 

Finally, when it has Andreev reflected 

∼ (2∆/eV ) times, its energy is increased enough to overcome 

the energy gap in the second superconductor. The 

resulting energy distribution function is a staircase pattern, 

and it is described in detail in (Pierre et al., 2001). 

The width of this distribution is approximately 2∆, and 

it thus corresponds to extremely strong heating even at 

low applied voltages. 

The superconducting proximity effect gives rise to two 

types of important contributions to the electrical and 

thermal properties of the metals in contact to the superconductors: 

it modifies the charge and energy diffusion 

constants in Eqs. (5), and allows for finite supercurrent 

to flow in the normal-metal wires. 

The simplest modification due to the proximity effect is 

a correction to the conductance in NN’S systems, where 

N’ is a phase-coherent wire of length L, connected to a 

normal and a superconducting reservoir via transparent 

contacts. In this case, jS = T an = 0 and the kinetic 

equation for the charge current reduces to the conservation 

of jT = DT ∂xf T . This can be straightforwardly 

integrated, yielding the current 

 

I = GN/e dEf T eq(E; V )DT (E), (46) 

where f T eq(E; V ) = {tanh[(E + eV )/(2kBT )] − tanh[(E − 

eV )/(2kBT )]}/2, T is the temperature in the normalmetal 

reservoir, and we assumed the normal metal in 

potential −eV . The proximity effect can be seen in the 

term DT = ( L 

0 

dx 

DT (x;E) /L)−1 . For T = 0, the differ- 

ential conductance is dI/dV = DT (eV )GN . A detailed 

investigation of DT (E) requires typically a numerical solution 

of the retarded/advanced part of the Usadel equation 

(Golubov et al., 1997). In general, it depends on 

two energy scales, the Thouless energy ET = D/L 2 of 

the N’ wire and the superconducting energy gap ∆. The 

behavior of the differential conductance as a function of 

the voltage and of the linear conductance as a function 

of the temperature are qualitatively similar, exhibiting 

the reentrance effect (Charlat et al., 1996; Golubov et al., 

1997; den Hartog et al., 1997): for eV, kBT ≪ ET and for 

max(eV, kBT ) ≫ ET , they tend to the normal-state value 

GN whereas for intermediate voltages/temperatures, the 

conductance is larger than GN , showing a maximum for 

max(eV, kBT ) of the order of ET . 

The proximity-effect modification to the conductance 

can be tuned in an Andreev interferometer, where a 

normal-metal wire is connected to two normal-metal 

reservoirs and two superconductors (Golubov et al., 1997; 

Nazarov and Stoof, 1996; Pothier et al., 1994). This system 

is schematized in the inset of Fig. 7. Due to the 

proximity effect, the conductance of the normal-metal 

wire is approximatively of the form G(φ) = GN + δG(1 + 

cos(φ))/2, where φ is the phase difference between the

two superconducting contacts and δG is a positive temperature 

and voltage-dependent correction to the conductance. 

Its magnitude for typical geometries is at maximum 

some 0.1GN. The proximity-induced conductance 

correction is widely studied in the literature and we refer 

to (Belzig et al., 1999; Lambert and Raimondi, 1998) for 

a more detailed list of references on this topic. 

The thermal conductance of Andreev interferometers 

has been studied very recently. The formulation of the 

problem is very similar as for the conductance correction. 

For E ≪ ∆, there is no energy current into the superconductors. 

Therefore, it is enough to solve for the energy 

current jL = DL∂xf L in the wires 1, 2 and 5. This yields 

˙Q = GN /e 2 

 

dEEDL(E)(f L eq(E; T2) − f L eq(E; T1)), 

where f L eq(E; T ) = tanh[E/(2kBT )] and the thermal 

conductance correction is obtained from DL(E) = 

( L dx 

0 DL(x;E) /L)−1 . Here the spatial integral runs along 

the wire between the two normal-metal reservoirs. In 

the general case, DL has to be calculated numerically. 

The thermal conductance correction has been analyzed 

by Bezuglyi and Vinokur (2003) and Jiang and Chandrasekhar 

(2005b). They found that it can be strongly 

modulated with the phase φ: in the short-junction limit 

where ET ≫ ∆, for φ = 0, DL almost vanishes, whereas 

for φ = π, DL approaches unity and thus the thermal 

conductance approaches its normal-state value. For a 

long junction with ET ∆, the effect becomes smaller, 

but still clearly observable. The first measurements 

(Jiang and Chandrasekhar, 2005a) of the proximityinduced 

correction to the thermal conductance show the 

predicted tendency of the phase-dependent decrease of κ 

compared to the normal-state (Wiedemann-Franz) value. 

Prior to the experiments on heat conductance in proximity 

structures, the thermoelectric power α was experimentally 

studied in Andreev interferometers (Dikin 

et al., 2002a,b; Eom et al., 1998; Jiang and Chandrasekhar, 

2004; Parsons et al., 2003a,b). The observed 

thermopower was surprisingly large, of the order of 100 

neV/K — at least one to two orders of magnitude larger 

than the thermopower in normal-metal samples. Also 

contrary to the Mott relation (c.f., below Eqs. (39)), 

this value depends nonmonotonically on the temperature 

(Eom et al., 1998) at the temperatures of the order of a 

few hundred mK, and even a sign change could be found 

(Parsons et al., 2003b). Moreover, the thermopower was 

found to oscillate as a function of the phase φ. The symmetry 

of the oscillations has in most cases been found to 

be antisymmetric with respect to φ = 0, i.e., α vanishes 

for φ = 0. However, in some measurements, the Chandrasekhar’s 

group (Eom et al., 1998; Jiang and Chandrasekhar, 

2004) have found symmetric oscillations, i.e., 

α had the same phase as the conductance. 

The observed behavior of the thermopower is not completely 

understood, but the major features can be explained. 

The first theoretical predictions were given by 

α SN,2 ( µV/K) 

4 

3 

2 

1 

T0 , V= 0 T0 , V= 0 

φ S S φ 

3 

5 

4 

2 

2 

1 2 

N N 

T1, V1 

T2, V2 0 

0 5 10 

k T / E 

B 2 T 

15 

14 

FIG. 7 Supercurrent-induced N-S thermopower αNS for the 

system depicted in the inset. Solid line: αNS at T1 ≈ T2. Dotted 

line: approximation (47). The correction (48) accounts 

for most of the difference. Dashed line: αNS at kBT1 = 3.6ET 

with varying T2. Dash-dotted line: the corresponding approximation. 

Inset: Andreev interferometer where thermoelectric 

effects are studied. Two superconducting reservoirs with 

phase difference φ are connected to two normal-metal reservoirs 

through diffusive normal-metal wires. Adapted from 

(Virtanen and Heikkilä, 2004b). 

Claughton and Lambert (1996), who constructed a scattering 

theory to describe the effect of Andreev reflection 

on the thermoelectric properties of proximity systems. 

Based on their work, it was shown (Heikkilä et al., 2000) 

that the presence of Andreev reflection can lead to a violation 

of the Mott relation. This means that a finite 

thermopower can arise even in the presence of electronhole 

symmetry. Seviour and Volkov (2000b), Kogan et al. 

(2002), and Virtanen and Heikkilä (2004a,b) showed that 

in Andreev interferometers carrying a supercurrent, a 

large voltage can be induced by the temperature gradient 

both between the normal-metal reservoirs and between 

the normal metals and the superconducting contacts. 

Virtanen and Heikkilä (2004b) showed that in long 

junctions, the induced voltages between the two normalmetal 

reservoirs and the superconductors can be related 

to the temperature dependent equilibrium supercurrent 

IS(T ) flowing between the two superconductors via 

V 0 

1/2 

= 1 

2 

R5(2R4/3 + R5)R3/4 [IS(T1) − IS(T2)] 

. (47) 

RNNNRSNS 

Here RNNN = R1 + R2 + R5, RSNS = R3 + R4 + R5 

and Rk are the resistances of the five wires defined in 

the inset of Fig. 7. This is in most situations the dominant 

term and it can be also phenomenologically argued 

based on the temperature dependence of the supercurrent 

and the conservation of total current (supercurrent 

plus quasiparticle current) in the circuit (Virtanen and 

Heikkilä, 2004b). A similar result can also be obtained 

in the quasiequilibrium limit for the linear-response thermopower 

(Virtanen and Heikkilä, 2004a). In addition to 

this term, the main correction in the long-junction limit 

comes from the anomalous coefficient T an , 

eV 1 

1/2 = ∓R1/2 〈T 

RNNN 

an 

1/2 〉 ∓ R3/4R5 〈T 

RNNNRSNS 

an 

5 〉. (48) 

Lk 

0 dx ∞ 

0 

Here 〈T an 

1 

an 

k 〉 ≡ dET Lk 

k (E)[f L eq(E, T1) − 

f L eq(E, T2)] and Lk is the length of wire k. One finds

that for a ”cross” system without the central wire (i.e., 

R5 = 0), this term dominates V 0 

1/2 . Further corrections 

to the result (47) are discussed by Kogan et al. (2002) 

and Virtanen and Heikkilä (2004a). 

The above theoretical results explain the observed 

magnitude and temperature dependence of the thermopower 

and also predict an induced voltage oscillating 

with the phase φ. However, the thermopower calculated 

by Seviour and Volkov (2000b), Kogan et al. (2002) and 

Virtanen and Heikkilä (2004a,b) is always an antisymmetric 

function of φ, and it vanishes for a vanishing supercurrent 

in the junction (including all the correction 

terms). Therefore, the symmetric oscillations of α cannot 

be explained with this theory. 

The presence of the supercurrent breaks the timereversal 

symmetry and hence the Onsager relation Π = 

T α (see Eqs. (39) and below) need not to be valid for 

φ = 0. Heikkilä, et al. predicted a nonequilibrium 

Peltier-type effect (Heikkilä et al., 2003) where the supercurrent 

controls the local effective temperature in an 

out-of-equilibrium normal-metal wire. However, it seems 

that the induced changes are always smaller than those 

due to Joule heating and thus no real cooling can be realized 

with this setup. 

Recently, the thermoelectric effects in coherent SNS 

Josephson point contacts have been analyzed in (Zhao 

et al., 2003, 2004). In such point contacts, the heat transport 

is strongly influenced by the Andreev bound states 

forming between the two superconductors. 

G. Heat transport by phonons 

When the electrons are thermalized by the phonons, 

they may also heat or cool the phonon system in the film. 

Therefore, it is important to know how these phonons 

further thermalize with the substrate, and ultimately 

with the heat bath on the sample holder that is typically 

cooled via external means (typically by either a dilution 

or a magnetic refrigerator). Albeit slow electron-phonon 

relaxation often poses the dominating thermal resistance 

in mesoscopic structures at low temperatures, the poor 

phonon thermal conduction itself can also prevent full 

thermal equilibration throughout the whole lattice. This 

is particularly the case when insulating geometric constrictions 

and thin films separate the electronic structure 

from the bulky phonon reservoir (see Fig. 1). In the 

present section we concentrate on the thermal transport 

in the part of the chain of Fig. 1 beyond the sub-systems 

determined by electronic properties of the structure. 

The bulky three-dimensional bodies cease to conduct 

heat at low temperatures according to the well appreciated 

κ ∝ T 3 law in crystalline solids arising from Debye 

heat capacity via 

κ = CvSℓel,ph/3, (49) 

where κ is the thermal conductivity and C is the heat capacity 

per unit volume, vS is the speed of sound and ℓel,ph 

15 

is the mean free path of phonons in the solid (Ashcroft 

and Mermin, 1976). Thermal conductivity of glasses follows 

the universal ∝ T 2 law, as was discovered by Zeller 

and Pohl (1971), which dependence is approximately followed 

by non-crystalline materials in general (Pobell, 

1996). These laws are to be contrasted to ∝ T thermal 

conductivities of pure normal metals (Wiedemann-Franz 

law, c.f., below Eq. (39)). Despite the rapid weakening 

of thermal conductivity toward low temperatures, the dielectric 

materials in crystalline bulk are relatively good 

thermal conductors. One important observation here is 

that the absolute value of the bulk thermal conductivity 

in clean crystalline insulators at low temperatures does 

not provide the full basis of thermal analysis without a 

proper knowledge of the geometry of the structure, because 

the mean free paths often exceed the dimensions 

of the structures. For example, in pure silicon crystals, 

measured at sub-kelvin temperatures (Klitsner and Pohl, 

1987), thermal conductivity κ 10 Wm −1 K −4 T 3 , heat 

capacity C 0.6 JK −4 m −3 T 3 and velocity vS 5700 

m/s imply by Eq. (49) a mean free path of 10 mm, 

which is more than an order of magnitude longer than the 

thickness of a typical silicon wafer. Therefore phonons 

tend to propagate ballistically in silicon substrates. 

What makes things even more interesting, but at the 

same time more complex, e.g., in terms of practical 

thermal design, is that at sub-kelvin temperatures the 

dominant thermal wavelength of the phonons, λph 

hvS/kBT , is of the order of 0.1 µm, and it can exceed 

1 µm at the low temperature end of a typical experiment 

(see discussion in Subs. II.C.2). A direct consequence of 

this fact is that the phonon systems in mesoscopic samples 

cannot typically be treated as three-dimensional, but 

the sub-wavelength dimensions determine the actual dimensionality 

of the phonon gas. Metallic thin films and 

narrow thin film wires, but also thin dielectric films and 

wires are to be treated with constraints due to the confinement 

of phonons in reduced dimensions. 

The issue of how thermal conductivity and heat capacity 

of thin membranes and wires get modified due to 

geometrical constraints on the scale of the thermal wavelength 

of phonons has been addressed by several authors 

experimentally (Holmes et al., 1998; Leivo and Pekola, 

1998; Woodcraft et al., 2000) and theoretically (see, e.g., 

(Anghel et al., 1998; Kuhn et al., 2004)). The main conclusion 

is that structures with one or two dimensions 

d < λph restrict the propagation of (ballistic) phonons 

into the remaining ”large” dimensions, and thereby reduce 

the magnitude of the corresponding quantities κ and 

C at typical sub-kelvin temperatures, but at the same 

time the temperature dependences get weaker. 

In the limit of narrow short wires at low temperatures 

the phonon thermal conductance gets quantized, as was 

experimentally demonstrated by Schwab et al. (2000). 

This limit had been theoretically addressed by Angelescu 

et al. (1998), Rego and Kirczenow (1998) and Blencowe 

(1999), somewhat in analogy to the well-known Landauer 

result on electrical conduction through quasi-one-

FIG. 8 The suspended silicon nitride bridge structure on the 

left, which was used by Schwab et al. (2000) to measure the 

quantum of thermal conductance shown in the graph on the 

right: thermal conductance levels off at the value 16g0 at 

temperatures well below 1 K. Adapted from (Schwab et al., 

2000). 

dimensional constrictions (Landauer, 1957). The quantum 

of thermal conductance is g0 ≡ π2k2 BT/(3h), and 

there are four phonon modes at low temperature due to 

four mechanical degrees of freedom each adding g0 to the 

thermal conductance of the quantum wire. In an experiment, 

see Fig. 8 four such wires in parallel thus carried 

heat with conductance g = 16g0. There are remarkable 

differences, however, in this result as compared to the 

electrical quantized conductance. In the thermal case, 

only one quantized level of conductance, 4g0 per wire, 

could be observed, and since the quantity transported 

is energy, the ”quantum” of thermal conductance carries 

∝ T in its expression besides the constants of nature. 

The thermal boundary resistance (Kapitza resistance, 

after the Russian physicist P. Kapitza) between two bulk 

materials is ∝ T −3 due to acoustic mismatch (Lounasmaa, 

1974). A remaining open issue is the question of 

thermal boundary resistance in a structure where at least 

one of the phonon baths facing each other is restricted, 

such that thermal phonons perpendicular to the interface 

do not exist in this particular subsystem. Classically the 

penetrating phonons need, however, to be perpendicular 

enough in order to avoid total reflection at the surface 

(Pobell, 1996). In practice though, the phonon systems 

in the two subsystems cannot be considered as independent. 

We are not aware of direct experimental investigations 

on this problem. 

On the device level reduced dimensions can be beneficial, 

e.g., in isolating thermally those parts of the devices 

to be refrigerated from those of the surrounding 

heat bath. This has been the method by which NIS 

based phonon coolers, i.e., refrigerators of the lattice 

have been realized experimentally utilizing thin silicon 

nitride films and narrow silicon nitride bridges (Clark 

et al., 2005; Fisher et al., 1999; Luukanen et al., 2000; 

Manninen et al., 1997). These devices are discussed in 

detail in Section V.C. 

Detectors utilizing phonon engineering are discussed in 

Sec. IV. 

H. Heat transport in a metallic reservoir 

16 

Energy dissipated per unit time in a biased mesoscopic 

structure is given quite generally as ˙ Q = IV , where I is 

the current through and V is the voltage across the device. 

This power is often so large that its influence on 

the thermal budget has to be carefully considered when 

designing a circuit on a chip. For instance, the NIS cooler 

of Sec. V.C.1 has a coefficient of performance given by 

Eq. (77), with a typical value in the range 0.01. . . 0.1. 

This simply means that the total power dissipated is 10 

to 100 times higher than the net power one evacuates 

from the system of normal electrons. Yet this tiny fraction 

of the dissipated power is enough to cool the electron 

system far below the lattice temperature due to the 

weakness of electron–phonon coupling. This observation 

implies that 10 to 100 times higher dissipated power outside 

the normal island tends to overheat the connecting 

electrode significantly, again because of the weakness of 

the electron–phonon coupling. Therefore it is vitally important 

to make an effort to thermalize the connecting 

reservoirs to the surrounding thermal bath efficiently. In 

the case of normal-metal reservoirs, heat can be conveniently 

conducted along the electron gas to an electrode 

with a large volume in which electrons can then cool via 

electron–phonon relaxation. In the case of a superconducting 

reservoir, e.g., in a NIS refrigerator, the situation 

is more problematic because of the very weak thermal 

conductivity at temperatures well below the transition 

temperature Tc. In this case the superconducting reservoirs 

should either be especially thick, or they should 

be attached to normal metal conductors (”quasiparticle 

traps”) as near as possible to the source of dissipation 

(see discussion in Sec. V.C.1). The latter approach is, 

however, not always welcome, because, especially in the 

case of a good metallic contact between the two conductors, 

the operation of the device itself can be harmfully 

affected by the superconducting proximity effect. 

Let us consider heat transport in a normal metal reservoir. 

In the first example we approximate the reservoir 

geometry by a semi-circle, connected to a biased sample 

with a hot spot of radius r0 at its origin (see the inset 

of Fig. 9 (b)). This hot spot can approximate, for example, 

a tunnel junction of area πr 2 0. The results depend 

only logarithmically on r0, and therefore its exact value 

is irrelevant when making estimates. We first assume 

that the electrons carry the heat away with negligible 

coupling to the lattice up to a distance r. According to 

Eq. (23), we can then write the radial flux of heat in the 

quasiequilibrium limit in the form 

˙Q(r) = −κSdT/dr, (50) 

where κ is the electronic thermal conductivity, S = πrt 

is the conduction area at distance r in a film of thickness 

t, and T is the temperature at radius r. According 

to the Wiedemann-Franz law and the temperature independent 

residual electrical resistivity in metals, one has 

κ = L0σT ≡ κ0T (see below Eq. (39)). With these as-

T/T 0 

3 

2 

(a) T(r ) = 3.0 T 0 0 0.5 (b) 50nm 

T(r ) = 2.5 T 0 0 

T(r ) = 2.0 T 0 0 

0.4 

r 0 r 

T(r ) = 1.5 T 0 0 

T(r ) = 1.2 T 0 0 

T(r ) = 1.1 T 0 0 

0.3 

200nm 

1 

0.01 0.1 

r/rS 1 

T(r 0 ) (K) 

0.2 

r 0 /r S = 0.01 800nm 

0.1 

0.1 . 1 

Q (nW) 

FIG. 9 (Color in online edition): Temperature rise according 

to the presented reservoir heating model. For details see text. 

sumptions, using Eq. (50), one finds a radial distribution 

of temperature 

 

T (r1) = 

T (r2) 2 + ˙ QR 

πL0 

ln(r2/r1), (51) 

where r1 and r2 are two distances from the hot spot, 

and we defined the square resistance as R = ρ/t. Thus 

making the reservoir thicker helps to thermalize it. The 

model above is strictly appropriate in the case where a 

thin film in form of a semi-circle is connected to a perfect 

thermal sink at its perimeter (at r = r2). A more adequate 

model in a typical experimental case is obtained 

by assuming a uniform semi-infinite film connected at 

its side to a hot spot as above, but now assuming that 

the film thermalises via electron–phonon coupling. Using 

Eq. (50) and energy conservation one then obtains (see 

Sec. II.D) 

d ˙ Q 

dr + Σπrt(T p − T p 

0 ) = 0, (52) 

where T0 is the lattice temperature, and p is the exponent 

of electron phonon relaxation, typically p = 5. We can 

write this equation into a dimensionless form 

d2u 1 du 

+ 

dρ2 ρ dρ = up/2 − 1. (53) 

Here we have defined u ≡ (T (r)/T0) 2 and ρ ≡ r/rS, 

where rS ≡ κ0/(2Σ)/T p/2−1 

0 is the length scale of temperature 

over which it relaxes towards T0. Figure 9 (a) 

shows the solutions of Eq. (53) for different values (1.1, 

1.2, 1.5, 2.0, 2.5, and 3.0) of relative temperature rise 

T (r0)/T0. We see, indeed, that rS determines the relaxation 

length. To have a concrete example let us consider 

a copper film with thickness t = 200 nm. For copper, 

using p = 5, we have κ0 1 WK−2m−1 and Σ 2 · 109 WK−5m−3 , which leads to a healing length of rS ∼ 500 

µm at the bath temperature of T0 = 100 mK. The temperature 

rise versus input power ˙ Q has been plotted in 

Fig. 9 (b) for copper films with different thicknesses. In 

particular for t = 200 nm, we obtain a linear response of 

(T (r0, ˙ Q)/T0 − 1)/ ˙ Q 7 · 10−4 /pW. 

17 

A superconducting reservoir, which is a necessity in 

some devices, poses a much more serious overheating 

problem. Heat is transported only by the unpaired 

electrons whose number is decreasing exponentially as 

∝ exp(−∆/kBT ) towards low temperatures. Therefore 

the electronic thermal transport is reduced by approximately 

the same factor, as compared to the corresponding 

normal metal reservoir. Theoretically then κ is suppressed 

by nine orders of magnitude from the normal 

state value in aluminium at T = 100 mK. It is obvious 

that in this case other thermal conduction channels, like 

electron–phonon relaxation, become relevant, but this is 

a serious problem in any case. At higher temperatures, 

say at T/TC ≥ 0.3, a significantly thicker superconducting 

reservoir can help (Clark et al., 2004). 

III. THERMOMETRY ON MESOSCOPIC SCALE 

Any quantity that changes with temperature can in 

principle be used as a thermometer. Yet the usefulness 

of a particular thermometric quantity in each application 

is determined by how well it satisfies a number of 

other criteria. These include, with a weighting factor that 

depends on the particular application: wide operation 

range with simple and monotonic dependence on temperature, 

low self-heating, fast response and measurement 

time, ease of operation, immunity to external parameters, 

in particular to magnetic field, small size and small 

thermal mass. One further important issue in thermometry 

in general terms is the classification of thermometers 

into primary thermometers, i.e., those that provide the 

absolute temperature without calibration, and into secondary 

thermometers, which need a calibration at least 

at one known temperature. Primary thermometers are 

rare, they are typically difficult to operate, but nevertheless 

they are very valuable, e.g., in calibrating the secondary 

thermometers. The latter ones are often easier to 

operate and thereby more common in research laboratories 

and in industry. 

In this review we discuss a few mesoscopic thermometers 

that can be used at cryogenic temperatures. Excellent 

and thorough reviews of general purpose cryogenic 

thermometers, other than mesoscopic ones, can be found 

in many text books and review articles, see, e.g., (Lounasmaa, 

1974; Pobell, 1996; Quinn, 1983) and many references 

therein. 

Modern micro- and nanolithography allows for new 

thermometer concepts and realizations where sensors can 

be very small, thermal relaxation times are typically 

short, but which generally do not allow except very tiny 

amounts of self-heating. The heat flux between electrons 

and phonons gets extremely weak at low temperatures 

whereby electrons decouple thermally from the lattice 

typically at sub-100 mK temperatures, unless special care 

is taken to avoid this. Therefore, especially at these low 

temperatures the lattice temperature and the electron 

temperature measured by such thermometers often de-

viate from each other. An important example of this is 

the electron temperature in NIS electron coolers to be 

discussed in Section V. 

A typical electron thermometer relies on a fairly easily 

and accurately measurable quantity M that is related to 

the electron energy distribution function f(ε) via 

M = 

∞ 

−∞ 

dεk(ε)g[f(ε)]. (54) 

Here the kernel k(ε) describes the process which is used 

to measure f(ε) and g[f] is a functional of f(ε). The 

quantity M typically refers to an average current or voltage, 

in which case g[f] = f − f0 is a linear function of 

f(ε) with some constant function f0; or it can refer to 

the noise power, in which case g[f] is quadratic in f. For 

the thermometer to be easy to calibrate, k(ε) should be a 

simple function dependent only on a few parameters that 

need to be calibrated. Moreover, if k(ε) has sufficiently 

sharp features, it can be used to measure the shape of 

f(ε) also in the nonequilibrium limit. 

A. Hybrid junctions 

Tunnelling characteristics through a barrier separating 

two conductors with non-equal densities of states (DOSs) 

are usually temperature dependent. The barrier B may 

be a solid insulating layer (I), a Schottky barrier formed 

between a semiconductor and a metal (Sc), a vacuum 

gap (I), or a normal metal weak link (N). We are going 

to discuss thermometers based on tunnelling in a CBC’ 

structure. C and C’ stand for a normal metal (N), a superconductor 

(S), or a semiconductor (Sm). As it turns 

out, the current-voltage (I − V ) characteristics of the 

simplest combination, i.e., of a NIN tunnel junction, exhibit 

no temperature dependence in the limit of a very 

high tunnel barrier. Yet NIN junctions form elements of 

presently actively investigated thermometers (Coulomb 

blockade thermometer and shot noise thermometer) to 

be discussed separately. The NIN junction based thermometers 

are suitable for general purpose thermometry, 

because their characteristics are typically not sensitive 

to external magnetic fields. Superconductor based junctions 

are, on the contrary, normally extremely sensitive 

to magnetic fields, and therefore they are suitable only 

in experiments where external fields can be avoided or at 

least accurately controlled. 

In SBS’ junctions one has to distinguish between two 

tunnelling mechanisms, Cooper pair tunnelling (Josephson 

effect) and quasiparticle tunnelling. The former occurs 

at low bias voltage and temperature, whereas the 

latter is enhanced by increased temperature and bias 

voltage. In the beginning of this section we discuss quasiparticle 

tunnelling only. 

Let us consider tunnelling between two normal-metal 

conductors through an insulating barrier. I-V characteristics 

of such a junction were given by Eq. (29). We 

assume quasi-equilibrium with temperatures Ti, i = L, R 

18 

on the two sides of the barrier. Since Ni(E) = 1 to 

high precision at all relevant energies (kBTi, eV ≪ EF = 

Fermi energy), Eq. (29) can be integrated easily to yield 

I = V/RT. Therefore the I −V characteristics are ohmic, 

and they do not depend on temperature, and a NIN junction 

appears to be unsuitable for thermometry. There is, 

however, a weak correction to this result, due to the finite 

height of the tunnel barrier (Simmons, 1963a), which will 

be discussed in subsection III.D. 

1. NIS thermometer 

As a first example of an on-chip thermometer, let us 

discuss a tunnel junction between a normal metal and a 

superconductor (NIS junction) (Rowell and Tsui, 1976). 

I-V characteristics of a NIS junction have the very important 

property that they depend on the temperature 

of the N electrode only, which is easily verified, e.g., 

by writing I(V ) of Eq. (29) with NL(E) = const. and 

NR(E) = NS(E) in a symmetric form 

∞ 

I(V ) = 1 

NS(E)[fN(E − eV ) − fN(E + eV )]dE. 

2eRT −∞ 

(55) 

This insensitivity to the temperature of the superconductor 

holds naturally only up to temperatures where 

the superconducting energy gap can be assumed to have 

its zero-temperature value. This is true in practice up to 

T/Tc 0.4. 

Employing Eq. (55) one finds that a measurement of 

voltage V at a constant current I = I0 yields a direct 

measure of fN(E), and in quasi-equilibrium, where the 

distribution follows thermal Fermi-Dirac distribution, it 

also yields temperature in principle without fit parameters. 

Figure 10 (a) shows the calculated I-V characteristics 

of a NIS junction at a few temperatures T/Tc. Figure 

10 (b) gives the corresponding thermometer calibration: 

junction voltage has been plotted against temperature at 

a few values of (constant) measuring current. 

A NIS thermometer has a number of features which 

make it attractive in applications. The sensing element 

can be made very small and thereby NIS junctions can 

probe temperature locally and detect temperature gradients. 

Junctions made by electron beam lithography can 

be much smaller than 1 µm in linear dimension (Nahum 

and Martinis, 1993). Using a scanning tunnelling microscope 

with a superconducting tip as a NIS junction one 

can most likely probe the temperature of the surface locally 

on nanometer scales in an instrument like those of 

Moussy et al. (2001); Vinet et al. (2001). Self-heating can 

be made very small by operating in the sub-gap voltage 

range e|V | < ∆ (see Fig. 10), where current is very small. 

The superconducting probe is thermally decoupled from 

the normal region whose temperature is monitored by it. 

The drawbacks of this technique include high sensitivity 

to external magnetic field, high impedance of the sensor 

especially at low temperatures, and sample-to-sample deviations 

from the ideal theoretical behaviour. Due to this

FIG. 10 (Color in online edition): Calculated I-V characteristics 

of a normal metal - superconductor tunnel junction at 

various relative temperatures T/Tc in (a). The corresponding 

voltage over the junction as a function of temperature when 

the junction is biased at a constant current is shown in (b). 

Characteristics at a few values of current are shown. 

last reason, a NIS junction can hardly be considered as a 

primary thermometer: deviations arise especially at low 

temperatures, one prominent problem is saturation due 

to subgap leakage due to non-zero DOS within the gap 

and Andreev reflection. 

A fast version of a NIS thermometer was implemented 

by Schmidt et al. (2003). They achieved sub-µs readout 

times (bandwidth about 100 MHz) by imbedding the NIS 

junction in an LC resonant circuit. This rf-NIS read-out 

is possibly very helpful in studying thermal relaxation 

rates in metals, and in fast far-infrared bolometry. 

NIS junction thermometry has been applied in x-ray 

detectors (Nahum et al., 1993), far-infrared bolometers 

(Chouvaev et al., 1999; Mees et al., 1991), in probing 

the energy distribution of electrons in a metal (Pothier 

et al., 1997a), and as a thermometer in electronic coolers 

at sub-kelvin temperatures (Leivo et al., 1996; Nahum 

et al., 1994). It has also been suggested to be used as a 

far-infrared photon counter (Anghel and Kuzmin, 2003). 

In many of the application fields of a NIS thermometer 

it is not the only choice: for example, a superconducting 

transition edge sensor (TES) can be used very conveniently 

in the bolometry and calorimetry applications 

(see Sec. IV). 

Recently Schottky contacts between silicon and superconducting 

metal have been shown to exhibit similar 

characteristics as fully metallic NIS junctions (Buonomo 

et al., 2003; Savin et al., 2001). These structures are 

discussed in subsection V.C.3. 

2. SIS thermometer 

A tunnel junction between two superconductors supports 

supercurrent, whose critical value IC has a magnitude 

which depends on temperature according to (Ambegaokar 

and Baratoff, 1963): IC = π∆ 

∆ tanh( 2eRT 2kBT ). This 

can naturally be used to indicate temperature because 

eI R T / Δ 

0.06 

0.03 

0.00 

-0.03 

-0.06 

(a) 

T/T C = 0.05 

T/T C = 0.138 

T/T C = 0.281 

T/T C = 0.396 

T/T C = 0.456 

T/T C = 0.514 

- T/T C = 0.138 

- T/T C = 0.281 

19 

- T/T C = 0.396 

- T/T C = 0.456 

- T/T C = 0.514 

-2 -1 0 1 2 -2 -1 0 1 2 

V (Δ /e) 

V ( Δ /e) 

FIG. 11 (Color in online edition): (a) Calculated and (b) 

measured I − V curves of a SIS tunnel junction at a few temperatures. 

Supercurrent has been suppressed. Experimental 

data from (Savin and Pekola, 2005). 

of the temperature dependence of the energy gap and 

the explicit hyperbolic dependence on T . Yet, these dependencies 

are exponentially weak at low temperatures. 

Another possibility is to suppress IC by magnetic field, 

e.g., in a SQUID configuration, and to work at non-zero 

bias voltage and measure the quasiparticle current, which 

can be estimated by Eq. (29) again with both DOSs given 

by Eq. (12) now. The resulting current depends approximately 

exponentially on temperature, but the favourable 

fact is that the absolute magnitude of the current is increased 

because it is proportional to the product of the 

two almost infinite DOSs matching at low bias voltages, 

and, in practical terms, the method as a probe of a quasiparticle 

distribution is more robust against magnetic 

noise. Figure 11 shows the calculated and measured dependences 

of I(V ) at a few values of temperature T/Tc. 

Although the curves corresponding to the lowest temperatures 

almost overlap here, it is straightforward to verify 

that a standard measurement of current can resolve temperatures 

down to below 0.1Tc using an ordinary tunnel 

junction. 

SIS junctions have not found much use as thermometers 

in the traditional sense, but they are extensively 

investigated and used as photon and particle detectors 

because of their high energy resolution (see, e.g., (Booth 

and Goldie, 1996) and Section IV). In this context SIS 

detectors are called STJ detectors, i.e., superconducting 

tunnel junction detectors. Another application of SIS 

structures is their use as mixers (Tinkham, 1996). While 

superficially similar to STJ detectors, their theory and 

operation differ significantly from each other. SIS junctions 

are also suitable for studies of quasiparticle dynamics 

and fluctuations in general (see, e.g., (Wilson et al., 

2001)). 

3. Proximity effect thermometry 

For applications requiring low-impedance ( 1 Ω) thermometers 

at sub-micron scale, the use of clean NS con- 

(b)

tacts may be more preferable than thermometers applying 

tunnel contacts. In an SNS system, one may again 

employ either the supercurrent or quasiparticle current as 

the thermometer. For a given phase φ between the two 

superconductors, the former can be expressed through 

(Belzig et al., 1999) 

IS(φ) = 1 

eRN 

∞ 

0 

dEjS(E; φ)(1 − 2f(E)), (56) 

where RN is the normal-state resistance of the weak link, 

jS(E; φ) is the spectral supercurrent (Heikkilä et al., 

2002), and the energies are measured from the chemical 

potential of the superconductors. Hence, the supercurrent 

has the form of Eq. (54). In practice, one does 

not necessarily measure IS(φ), but the critical current 

IC = maxφ IS(φ). In diffusive junctions, this is obtained 

typically for φ near π/2, although the maximum point 

depends slightly on temperature. 

The problem in SNS thermometry is in the fact that 

the supercurrent spectrum jS(ε) depends on the quality 

of the interface, on the specific geometry of the system, 

and most importantly, on the distance L between 

the two superconductors compared to the superconducting 

coherence length ξ0 = D/(2∆) (Heikkilä et al., 

2002). Therefore, IC(T ) dependence is not universal. 

However, the size of the junction can be tuned to meet 

the specific temperature range of interest. In the limit of 

short junctions, L ξ0 (Kulik and Omel’yanchuk, 1978), 

the temperature scale for the critical current is given by 

the superconducting energy gap ∆ and for kBT ≪ ∆, 

the supercurrent depends very weakly on the temperature. 

In a typical case ξ0 ∼ 100 . . . 200 nm, and thus 

already a weak link with L of the order of 1 µm, easily 

realisable by standard lithography techniques, lies in 

the ”long” limit. There, the critical current is IC = 

c(kBT ) 3/2 exp(− √ 

2πkBT/ET )/(eRN ET ) (Zaikin and 

Zharkov, 1981). This equation is valid for kBT 5ET . 

Here ET = D/L2 and the prefactor c depends on the 

geometry (Heikkilä et al., 2002), for example for a twoprobe 

configuration c = 64 √ 2π3 /(3 + 2 √ 2). The exponential 

temperature dependence and the crossover between 

the long- and short-junction limits were experimentally 

investigated by Dubos et al. (2001) and the 

above theoretical predictions were confirmed. 

Using a four-probe configuration with SNS junctions of 

different lengths, Jiang et al. (2003) exploited the strong 

temperature dependence of the supercurrent to measure 

the local temperature. The device worked in the regime 

where part of the junctions were in the supercurrentcarrying 

state and part of them in the dissipative state. 

In the limit where supercurrent is completely suppressed, 

there is still a weaker temperature dependence of the 

conductance, due to the proximity-effect correction (c.f., 

Eq. (46)) (Charlat et al., 1996). This can also be used for 

thermometry (Aumentado et al., 1999), but due to the 

much smaller effect of temperature on the conductance, 

it is less sensitive. 

Besides being just a thermometer of the electron gas, 

20 

FIG. 12 A typical CBT sensor for the temperature range 20 

mK - 1 K. The structure has been fabricated by electron beam 

lithography, combined with aluminium and copper vacuum 

evaporation. Both top view and a view at an oblique angle 

are shown; the scale indicated refers to the top view. Figure 

adapted from (Meschke et al., 2004). 

critical current of a SNS Josephson junction has been 

shown to probe the electron energy distribution (Baselmans 

et al., 1999, 2001b; Giazotto et al., 2004b; Huang 

et al., 2002) as indicated by Eq. (56). 

B. Coulomb blockade thermometer, CBT 

Single electron tunnelling (SET) effects were foreseen 

in micro-lithographic structures in the middle of 1980’s 

(Averin and Likharev, 1986). Such effects in granular 

structures had been known to exist already much earlier 

(Giaever and Zeller, 1968; Lambe and Jaklevic, 1969; 

Neugebauer and Webb, 1962). The first lithographic SET 

device was demonstrated by Fulton and Dolan (1987). 

Since that time SET effects have formed a strong subfield 

in mesoscopic physics. Typically SET devices operate in 

the full Coulomb blockade regime, where temperature is 

so low that its influence on electrical transport characteristics 

can be neglected; at low bias voltages current 

is blocked due to the charging energy of single electrons. 

Coulomb blockade thermometer (CBT) operates in a different 

regime, where single electron effects still play a role 

but temperature predominantly influences the electrical 

transport characteristics (Bergsten et al., 2001; Farhangfar 

et al., 1997; Meschke et al., 2004; Pekola et al., 1994). 

CBT is a primary thermometer, whose operation is based 

on competition between thermal energy kBT at temperature 

T , electrostatic energy eV at bias voltage V , and 

charging energy due to extra or missing individual electrons 

with unit of charging energy EC = e 2 /(2C ∗ ), where 

C ∗ is the effective capacitance of the system. Figure 12 

shows an SEM micrograph of a part of a typical CBT 

sensor suitable for the temperature range 0.02. . . 1 K. 

This sensor consists of four parallel one-dimensional arrays 

with 40 junctions in each. 

In the partial Coulomb blockade regime the I −V characteristics 

of a CBT array with N junctions in series do 

not display a sharp Coulomb blockade gap, but, instead, 

they are smeared over a bias range eV ∼ NkBT . The 

asymptotes of the I − V at large positive and negative 

voltages have, however, the same offsets as at low T , de-

(a) 

T CBT (mK) 

100 

10 

10 100 

T (mK) 

MCT 

(b) 

FIG. 13 (a) Typical measurement of a CBT thermometer. 

G(V )/GT is the differential conductance scaled by its asymptotic 

value at large positive and negative voltages, plotted as 

a function of bias voltage V . V1/2 indicates the full width at 

half minimum of the characteristics, which is the main thermometric 

parameter. The full depth of the line, ∆G/GT, is 

another parameter to determine temperature. In (b) temperature 

deduced by CBT has been compared to that obtained by 

a 3 He melting curve thermometer. Saturation of CBT below 

20 mK indicates typical thermal decoupling between electrons 

and phonons. Figure from Ref. (Meschke et al., 2004). 

termined by the Coulomb gap. In the partial Coulomb 

blockade regime it is convenient to measure not the I −V 

directly, but the differential conductance, i.e., the slope of 

the I−V curve, G ≡ dI/dV , vs. V . The result is a nearly 

bell shaped dip in conductance around zero bias. Figure 

13 illustrates a measured conductance curve, scaled by 

asymptotic conductance GT at large positive and negative 

voltages. The important property of this characteristic 

is that the full width of this dip at half minimum, 

V1/2, approximately proportional to T , determines the 

temperature without any fit, or any material or geometry 

dependent parameters, i.e., it is a primary measure 

of temperature. In the lowest order in EC 

kBT , one finds for 

the symmetric linear array of N junctions of capacitance 

C in series (Farhangfar et al., 1997; Pekola et al., 1994) 

G(V )/GT = 1 − EC 

kBT 

eV 

g( ). (57) 

NkBT 

For such an array C ∗ = NC/[2(N − 1)] and g(x) = 

[x sinh(x) − 4 sinh 2 (x/2)]/[8 sinh 4 (x/2)]. The full width 

at half minimum of the conductance dip has the value 

V 1/2 5.439NkBT/e, (58) 

which allows one to determine T without calibration. 

It is often convenient to make use of the secondary 

mode of CBT, in which one measures the depth of the 

dip, ∆G/GT, which is proportional to the inverse temperature, 

again in the lowest order in EC 

kBT : 

∆G/GT = EC/(6kBT ). (59) 

Measuring the inflection point of the conductance 

curve provides an alternative means to determine T without 

calibration. In (Bergsten et al., 2001) this technique 

was used for fast thermometry on two dimensional 

21 

junction arrays. A simple and fast alternative measurement 

of CBT can be achieved by detecting the third harmonic 

current in a pure AC voltage biased configuration 

(Meschke et al., 2005). 

There are corrections to results (57), (58) and (59) 

due to both next order terms in EC/kBT and due to nonuniformities 

in the array (Farhangfar et al., 1997). Higher 

order corrections do sustain the primary nature of CBT, 

which implies wider temperature range of operation. The 

primary nature of the conductance curve exhibited by Eq. 

(57) through the bias dependence g( eV 

NkBT 

) is preserved 

also irrespective of the capacitances of the array, even if 

they would not be equal. Only a non-uniform distribution 

of tunnel resistances leads to a deviation from the 

basic result of bias dependence of Eq. (57). Fortunately 

the influence of the inhomogeneity on temperature reading 

is quite weak, and it leads typically to less than 1 % 

systematic error in T . 

The useful temperature range of a CBT array is limited 

at high temperature by the vanishing signal (∆G/GT ∝ 

T −1 ). In practice the dip must be deeper than ∼ 0.1 % 

to be resolvable from the background. The low temperature 

end of the useful temperature range is set by the 

appearance of charge sensitivity of the device, i.e., the 

background charges start to influence the characteristics 

of the thermometer. This happens when ∆G/GT 0.5 

(Farhangfar et al., 1997). With these conditions it is obvious 

that the dynamic range of one CBT sensor spans 

about two decades in temperature. 

The absolute temperature range of CBT techniques 

is presently mainly limited by materials issues. At 

high temperatures, the measuring bias range gets wider, 

V 1/2 ∝ T , and the conductance becomes bias dependent 

also due to the finite (energy) height of the tunnel barrier 

(Gloos et al., 2000; Simmons, 1963a). Because of 

this, the present high temperature limit of CBTs is several 

tens of K. The (absolute) low temperature end of the 

CBT technique is determined by self-heating due to biasing 

the device. One can measure temperatures reliably 

down to about 20 mK at present with better than 1 % 

absolute accuracy in the range 0.05. . . 4 K, and about 3 

% down to 20 mK (Meschke et al., 2004). 

Immunity to magnetic field is a desired but rare property 

among thermometers. For instance, resistance thermometers 

have usually strong magnetoresistance. Therefore 

one could expect CBT to be also ”magnetoresistive”. 

On the contrary, CBT has proven to be immune to even 

the strongest magnetic fields (> 20 T) (van der Linden 

and Behnia, 2004; Pekola et al., 2002, 1998). This happens 

because CBT operation is based on electrostatic 

properties: tunnelling rates are determined by charging 

energies. Hence, CBT should be perfectly immune to 

magnetic field as long as energies of electrons with up 

and down spins do not split appreciably, which is always 

the case in experiments. 

Coulomb blockade is also a suitable probe of nonequilibrium 

energy distributions as demonstrated by Anthore 

et al. (2003).

C. Shot noise thermometer, SNT 

Noise current or voltage of a resistor (resistance R) has 

been known to yield absolute temperature since 1920’s 

(Blanter and Büttiker, 2000; Johnson, 1928; Nyquist, 

1928). Till very recently only equilibrium noise, i.e., 

noise across an unbiased resistor, was employed to measure 

temperature. In this case Johnson noise voltage vn 

and current in squared have expectation values 〈v 2 n〉 = 

R 2 〈i 2 n〉 = 4kBT R∆f, where ∆f is the frequency band of 

the measurement. In other words, the current spectral 

density is given by SI = 4kBT/R, and voltage spectral 

density by SV = 4kBT R. There are several critical issues 

in measuring temperature through Johnson noise. First, 

the bandwidth has to be known exactly. Secondly, the 

small signal has to be amplified, and the gain must be 

known to high accuracy. Thirdly, the noise signal becomes 

extremely small at cryogenic temperatures, which 

in turn means that the measurement has to be able to detect 

a very small current or voltage, and at the same time 

no other noise sources should cause comparable voltages 

or currents. 

Success in noise thermometry depends critically on the 

performance of the amplifiers used to detect the tiny 

noise current (or voltage). Constant progress in improving 

SQUIDs (Superconducting Quantum Interference Device) 

and in optimizing their operation has pushed the 

lowest temperature measurable by a Johnson noise thermometer 

down to below 1 mK (Lusher et al., 2001). 

It is not only the equilibrium noise that can be useful 

in thermometry. In the opposite limit, at eV ≫ kBT , 

the dominating noise, e.g., in tunnel junctions, is shot 

noise (Blanter and Büttiker, 2000; Schottky, 1918), whose 

current spectral density is given asymptotically by SI = 

F 2e|I|. Fano factor F equals 1 for a tunnel junction. 

The way Johnson noise transforms into shot noise upon 

increasing the bias obeys the relation (see Eq. (34a) with 

Tn → 0, and, e.g., (van der Ziel, 1986)) 

SI(T ) = 2eI coth( eV 

). (60) 

2kBT 

At |V | ≫ kBT/e, Eq. (60) obtains the Poisson expression 

above, whereas at |V | ≪ kBT/e it assumes the thermal 

Johnson form SI 4kBT/RT . 

Temperature dependent cross-over characteristics from 

Johnson to shot noise according to Eq. (60) have been 

demonstrated experimentally, e.g., in scanning tunnelling 

microscope experiments (Birk et al., 1995) as shown 

in Fig. 14. Recently, Spietz et al. (2003) employed 

this cross-over in a lithographic tunnel junction between 

metallic films for thermometry (shot noise thermometer, 

SNT) in the temperature range from few tens of mK up 

to room temperature. At both ends of this range there 

are systematic errors in the reading, which can possibly 

be corrected by a more careful design of the sensor. 

In a rather wide range around 1 K (about 0.1. . . 10 K), 

the absolute accuracy is better than 1 %. The crossover 

voltage depends only on kBT/e, which means that the 

22 

FIG. 14 Cross-over from Johnson noise to shot noise when 

increasing the bias voltage of a tunnel junction. The data 

shown by open symbols has been measured at 300 K with 

RT = 0.32 GΩ, and data shown by solid symbols at 77 K 

with RT = 2.7 GΩ. Adapted from (Birk et al., 1995). 

FIG. 15 Normalised junction noise versus normalised voltage 

at various temperatures. The residuals from the expected 

x coth(x) law are shown in the bottom half of the figure. From 

(Spietz et al., 2003). 

thermometer is indeed primary; ideally the temperature 

reading does not depend on the materials or geometry 

of the sensor. Figure 15 shows the experimental data of 

Spietz et al. (2003) at several temperatures: normalised 

(by Johnson noise) current noise has been plotted against 

normalised voltage x ≡ eV/(2kBT ), whereby all curves at 

different temperatures should lie at S norm 

I 

= x coth(x). 

As seen in the bottom half of the figure, the residuals are 

smallest around 1 K. 

As discussed above, noise measurements are difficult to 

perform especially at low temperature where signal gets 

very small. At any temperature it is critically important 

to know the frequency window of the measurement and 

the gains in the amplifiers in Johnson noise thermometry. 

Yet the SNT avoids some of these problems. It is based

on the cross-over between two noise mechanisms, both 

of which represent white noise, whereby the frequency 

window is ideally not a concern, since the same readout 

system is used in all the bias regimes. Moreover, the 

gains of the amplifiers are not that critical either because 

of the same argument. One can also use relatively high 

bandwidth which increases the absolute noise signal to be 

measured, and thereby makes the measurement faster. 

The SNT technique has a few further attractive features. 

It is likely that its operation can be easily extended 

up to higher temperatures despite the deviations 

observed in the first experiments. The sensor consists 

of just one, relatively large size tunnel junction, which 

means that it is easy to fabricate with high precision. 

Also it is likely, although not yet demonstrated, that the 

SNT is not sensitive to magnetic field, since its operation 

is based on tunnelling characteristics in a NIN tunnel 

junction as in CBT. 

Finally, noise measurements can in principle be used 

to measure the distribution function in non-equilibrium 

as well, as proposed by Pistolesi et al. (2004). 

D. Thermometry based on the temperature dependent 

conductance of planar tunnel junctions 

The effect of temperature on the current across a tunnel 

barrier with finite height is a suitable basis for thermometry 

in a wide temperature range (Gloos et al., 

2000). Simmons (1963a) showed that the tunnelling conductance 

at zero bias across a thin insulating barrier depends 

on temperature as 

G(T ) = G0[1 + (T/T0) 2 ], (61) 

where G0 is the temperature independent part of conductance 

and the scaling temperature T0 depends on the 

barrier height φ0. For a rectangular barrier of width s one 

has T 2 0 = 32 φ0 

π 2 k 2 B ms2 , where m is the effective mass of the 

electrons within the insulating barrier. Experiments over 

a temperature range from 50 K up to 400 K on Al-AlOx- 

Al tunnel junctions have demonstrated that Eq. (61) is 

obeyed remarkably well (Gloos et al., 2000; Suoknuuti 

et al., 2001). Moreover, in these measurements the scaling 

temperature was found to be T0 720 K in all samples, 

without a clear dependence on the specific (zero 

temperature) conductance of the barrier, which varied 

over three orders of magnitude from 3 µS/µm 2 up to 3000 

µS/µm 2 . This property makes the method attractive in 

wide range thermometry, and T0 can indeed be considered 

as a material specific, but geometry and thickness 

independent parameter up to a certain accuracy. 

E. Anderson-insulator thin film thermometry 

As regards to temperature read-out of microcalorimetric 

devices, resistive thin film thermometers 

near the metal-insulator transition (MIT) are relatively 

23 

FIG. 16 The suspended thermal sensor employed in (Bourgeois 

et al., 2005). The Nbx ′N1−x ′ thermometer can be seen 

in the lower part of the rectangular silicon membrane. The 

450 000 Al superconducting rings are located in the middle 

part of the membrane; examples of them are shown in (b) and 

(c). Figure from (Bourgeois et al., 2005). 

popular. Electrical resistivity properties on both sides 

of the MIT are rather well understood (Belitz and Kirkpatrick, 

1994), and in general resistance of such thin films 

shows strong temperature dependence, suitable for thermometry 

and in particular for calorimetry. On the insulator 

side resistivity ρ is determined by hopping, and 

it has typically ρ ∝ e (T0/T ) n 

temperature dependence, 

with T0 and n constants. On the metallic side, weaker 

dependence can be found. In practice, both NbxSi1−x 

(Denlinger et al., 1994; Marnieros et al., 1999, 2000) and 

Nbx ′N1−x ′ (Bourgeois et al., 2005; Fominaya et al., 1997) 

thin film based thermometers have been successfully employed. 

The suitable conduction regime can be tailored 

by adjusting x (x ′ ) in electron beam co-evaporation (Denlinger 

et al., 1994) or in dc magnetron sputtering of Nb 

in a nitrogen atmosphere (Fominaya et al., 1997). 

Bolometric and calorimetric radiation detectors are 

discussed in detail in Sec. IV. Here we briefly mention 

the application of a Nbx ′N1−x ′ thermometer in a measurement 

of the heat capacity of 450 000 superconducting 

thin film loops on a silicon membrane (Bourgeois et al., 

2005), see Fig. 16. The heat capacity of the loops is proportional 

to their total mass, which was about 80 ng in 

this case. Vortices entering simultaneously into the 450 

000 loops under application of magnetic field could be 

observed. A similar measurement (Lindell et al., 2000), 

employing a NIS thermometer could resolve the specific 

heat jump at Tc of 14 thin film titanium disks with total 

mass of 1 ng on a silicon nitride membrane.

IV. THERMAL DETECTORS AND THEIR 

CHARACTERISTICS 

The absorption of electromagnetic radiation by matter 

almost always ends in the situation where the incident 

optical power has, possibly via cascades of different 

physical processes, transformed into aggravated random 

motion of lattice ions, i.e. into heat. Here lies the fundamental 

principle behind the thermal detection of radiation: 

the transformation of the input electromagnetic 

energy to heat. In many cases, this state of maximum 

entropy has lost all the coherent properties that the incident 

radiation might have possessed, but yet there is 

information in this messy final state of the system: the 

rise in the system temperature. In this section, we give a 

short overview of thermal detectors, their theory and operation, 

and discuss some examples of thermal detectors 

and their applications. 

There is a very small difference between thermometry 

and bolometry, the thermal radiation detection. 

Yet thermometry often implies measurement of temperature 

changes over a large fractional temperature range, 

whereas in the typical bolometric application, the observed 

temperature variations are extremely small. This 

allows us in the following to concentrate on the limit 

of small temperature changes around some mean welldefined 

value, i.e., we assume the operation in the quasiequilibrium 

regime (see Sec. II). In essence, bolometry 

is really high-precision thermometry, nothing more, with 

the added ingredient of finding efficient ways of coupling 

incident radiation to the device. 

Although thermal detectors have developed enormous 

diversity since their introduction in 1880 by Samuel P. 

Langley, and major advances in the way the temperature 

rise is measured and the detectors are constructed have 

been made, the basic principle remains the same. The 

operating principle of a thermal detector can be traced 

back to the generalized thermal model, shown in Fig. 1. 

The four main parts of a thermal detector are a thermally 

isolated element, a thermal link with a thermal conductance 

Gth (here Ge−ph or Gph−sub), a thermal sensing 

element (i.e., a thermometer), and a coupling structure 

(e.g., an impedance matching structure for electromagnetic 

radiation) that serves to maximize the absorbtion 

of the incident radiation, be it in the forms of alpha particles 

or microwaves. 

A. Effect of operating temperature on the performance of 

thermal detectors 

Regardless of the exact architecture of the thermal 

detector, lowering the operating temperature will improve 

the performance significantly. This fact introduces 

the cooling techniques to the use of thermal detectors. 

Because of the diversity of the technological approaches 

it is hard to summarize the effect of the operating 

temperature in a fully universal fashion. How- 

24 

ever, some general trends can be evaluated. The figure 

of merit for thermal detectors is the noise equivalent 

power (NEP), which relates to the signal to noise ratio 

by SNR = ˙ Qopt/(NEP √ 2τint), where ˙ Qopt is the incident 

optical power and τint is the post-detection integration 

time. The limiting NEP for an optimized thermal detector 

is given by the thermal fluctuation noise (TFN) arising 

from random fluctuations of energy across the thermal 

link with a thermal conductance Gth ≡ d ˙ Q/dT which 

result in variations in the temperature of the device (see 

also discussion in Subs. II.E.2). The TFN limited noise 

equivalent power (NEP) in the linear order is given by 

the fluctuation-dissipation result (Mather, 1982) 

NEPTFN ≈ 4kBT 2 

3 ∝ T e-p decoupling 

Gth 

∝ T 5/2 lattice isolation 

(62) 

with the two cases relating the temperature dependence 

to the location of the thermal bottleneck. The expression 

for the lattice isolated case is generally valid at temperatures 

T θD/10 with θD the Debye temperature of the 

insulating material. The temperature T is the highest 

of the temperatures present in the system, i.e., the temperature 

of the thermally isolated element or that of the 

heat sink. 

Refrigeration, combined with the advances in nanolithography 

techniques have recently opened a whole new 

realm for the application of electron-phonon decoupling 

to improve the thermal isolation of bolometric detectors. 

The operation of these so-called hot electron devices 

is usually limited to low (< 1 K) temperatures 

and very small thermally active volumes V as the TFN 

between the electron gas and the lattice is given by 

5kBΣe−pV(T 6 e + T 6 ph ) (Golwala et al., 1997) where Σ 

varies between 1-4 nW/µm3 /K5 in metals (See Table I). 

The use of the lattice for the thermal isolation lends itself 

to operation over much broader temperature range as the 

geometry of the thermal link can be used to increase the 

thermal resistance. 

Typically, the bath temperatures of cryogenic thermal 

detectors are centered around four temperature ranges: 

4.2 K, the boiling temperature of liquid He at 1 atm, 

around 300 mK a temperature attainable with 3He sorption 

refrigerators, around 100 mK, easily attainable using 

a compact adiabatic demagnetization refrigerator 

(ADR), or below ∼ 50 mK, when a dilution refrigerator 

is used. In the recent years, the use of electronic refrigeration 

has become appealing with the development of the 

SINIS coolers. These coolers could enable the operation 

of thermal detectors at a much lower temperature than 

is ’apparent’ to the user. For instance, the 3He sorption 

coolers are quite attracting due to their compactness, low 

cost and simple operation. Used together with a SINIS 

cooler would allow for an affordable cryogenic detector 

system without having to sacrifice in performance. 

Since the TFN in a thermal detector is dependent on 

the higher of the temperatures in the system, it is clear

that the performance improvement is maximum when the 

temperature of the heat bath of the detector is lowered 

(Anghel et al., 2001). Modest performance increase is 

possible with direct electronic cooling, such as is the case 

in many of the SINIS bolometer experiments. 

In addition to improved noise performance, direct coupling 

of a SINIS cooler to a thermal detector could allow 

for increasing the dynamic range of for example bolometers 

based on transition edge sensors (TES): A SINIS 

cooler can be used to draw a constant power from a TES 

that would otherwise saturate due to an optical load. 

B. Bolometers: Continuous excitation 

Thermal detectors that are used to detect variations 

in the incident flux of photons or particles are called 

bolometers (from the Greek word bole - beam). This 

condition is generally met when the mean time between 

incident quanta of energy that arrive at the detector is 

much shorter than the recovery time of the bolometer 

3 . The bolometric operating principle is very simple: 

Change ∆ ˙ Qopt in the incident optical power creates a 

change in the temperature of a thermally isolated element 

by ∆T = ∆ ˙ Qopt/Gth. A sensitive thermometer is used 

to measure this temperature change. The recovery time 

τ0 is determined by the heat capacity C of the bolometer, 

and the thermal conductance Gth with τ0 = C/Gth, 

analogously to an electrical RC circuit. 

Bolometric detectors remain popular today, 125 years 

after their first introduction. Probably the most important 

advantage of thermal detectors is their versatility: 

Bolometers can detect radiation from α -particles to radio 

waves, their dynamic range can be easily adapted for 

a variety of signal or background levels. As an extreme 

example, bolometers have been used to detect infrared radiation 

from nuclear fireballs (Stubbs and Phillips, 1960), 

and the cosmic microwave background. (Lamarre et al., 

2003). 

In the early days, bolometers typically utilized handcrafted 

construction (dental floss, cigarette paper and 

balsa wood are examples of typical materials used in the 

construction) (Davis et al., 1964). If detecting electromagnetic 

waves, typically the absorber consisted of a 

metal with a suitable thickness yielding a square resistance 

of 377 Ω, i.e. matching the impedance of the vacuum. 

Further improvements on matching were achieved 

by placing the bolometer in a resonant cavity. One major 

setback with bolometers in their early days was unavoidably 

slow speed, caused by the large heat capacity 

resulting from the macroscopic size of the components 

used. The dawn of modern microfabrication techniques 

has all but eliminated this shortcoming, with bolometers 

of high sensitivity achieving time constants as short as a 

3 The opposite (calorimetric) limit will be discussed in section IV.C 

25 

few hundred nanoseconds. 

Today, the most common type of cryogenic resistive 

bolometers utilize transition edge sensors for the thermometry. 

In a TES bolometer, a superconducting film 

with a critical temperature Tc is biased within its superconductor 

- normal metal transition where small changes 

in the film temperature result to changes in the current 

through the device (or the voltage across the film). In 

most cases, the TES consists of two or more sandwiched 

superconductor-normal metal layers. The relative thicknesses 

of the S and N layers are used to tune the transition 

temperature to a desirable value by the proximity 

effect. 

Transition-edge sensors are by no means a novel type of 

a thermal detector, as first suggestions for their use came 

out already in the late thirties (Andrews, 1938; Goetz, 

1939), and first experimental results by 1941 (Andrews, 

1941). Two principal problems prohibited the wide use 

of this type of thermal detectors for some fifty years: 

Typically the normal-state resistance of the superconducting 

films was too low in order to obtain adequate 

noise matching with field effect transistor (FET) preamplifiers, 

and the lack of good transimpedance amplifiers 

usually required the films to be current biased with a voltage 

readout. This introduced a requirement to tune the 

heat bath temperature very accurately within the narrow 

range of temperatures in the superconducting transition. 

This also made the devices exceedingly sensitive to small 

variations in the bath temperature, introducing stringent 

requirements for the heat bath stability. 

These limitations can be overcome by the use of an 

external negative feedback circuit that maintains the 

film within its transition temperature and above the 

bath temperature. Such an approach was adopted by 

Clarke et al. (1977), who were able to demonstrate 

NEP=1.7 · 10 −15 W/ √ Hz at an operating temperature 

of 1.27 K, using a transformer-coupled FET as the voltage 

readout. Introducing a negative feedback has similar 

advantages as in the case of operational amplifiers: linearity 

is improved, sensitivity to internal parameters of 

the amplifier (or bolometer) is reduced, and the speed is 

increased. Interestingly, the use of an external negative 

feedback in conjunction with superconducting transition 

edge sensors never found widespread use, possibly due 

to the (slightly) more complicated read-out architecture, 

and the need for a matching transformer. 

The breakthrough of TESs came in 1995, when superconducting 

quantum interference device (SQUID) ammeters, 

which are inherently well suited in matching to low 

load impedances, were introduced as the readout devices 

for TESs (Irwin, 1995; Irwin et al., 1995a). This allowed 

for the use of voltage biasing, which introduces strong 

negative electrothermal feedback (ETF) that causes the 

thermally isolated film to self-regulate within its superconducting 

transition. The local nature of the ETF 

makes the operation of these detectors very simple as 

no external regulation is necessary. As with an external 

negative feedback, an important advantage of the voltage

iased TES is the fact that once the bath temperature 

is below ∼ Tc/2, the need for bath temperature regulation 

is significantly relaxed. Finally, the increased speed 

due to the strong negative ETF increases the bandwidth 

of the detector, allowing for either detecting faster signal 

changes in a bolometer, or higher count rates in a 

calorimeter. A comprehensive review of the theory and 

operation of voltage biased TESs has been recently published 

(Irwin and Hilton, 2005). 

The behaviour of thermal detectors is generally well 

understood. In the following discussion we shall summarize 

the main results for the theory of thermal detectors. 

The results are quite generally applicable to any resistive 

bolometers, but the treatment is geared towards voltage 

biased TES, firstly as they are currently the most popular 

type of thermal detectors, and secondly because the electrothermal 

effects are very prominent in these devices, 

having a major impact on the device speed and linearity. 

We start by writing the equation governing the thermal 

circuit (here we limit the discussion to a simple case 

of one thermal resistance and heat capacity). Generally, 

the power flow to the heat sink is given by ˙ Qout = 

K(T n −T n 0 ), where K is a constant which depends on materials 

parameters and the geometry of the link. The time 

dependence of the bolometer temperature can be solved 

from the heat equation for a bolometer with a bias point 

resistance of R = V/I absorbing a time-varying optical 

signal ˙ Qopt(t) = ˙ Qoe iωt (c.f., Eq. (23)): 

C d(δT eiωt ) 

dt 

+ K(T n − T n 0 ) + GthδT 

= ˙ Qbias + ˙ Qoe iωt + d ˙ Qbias 

δT, (63) 

dT 

where δT is used to denote the temperature change due 

to the signal power and ˙ Qbias describes the incoming heat 

flow due to the detector bias. Equating the steady state 

components of the equation yields ˙ Qbias = K(T n − T n 0 ), 

from which one can obtain the result for the average 

operating temperature of the bolometer, given by 

T = ˙ Qbias/Gth + T0 where an average thermal conductance 

Gth is defined by Gth = K(T n − T n 0 )/(T − T0) = 

˙Qbias/[( ˙ Qbias/K + T n 0 ) 1/n − T0]. 

The electrothermal feedback manifests itself through 

the fact that the change in input signal power modifies 

the bias dissipation, an effect described by the last term 

in Eq. (63). Taking a closer look at the temperature 

change we obtain 

˙Qo 

δT = 

Gth + iωC − d ˙ Qbias/dT 

(64) 

where Gth = d ˙ Qout/dT ≈ nKT n−1 is the dynamic thermal 

conductance. Now, considering the electrothermal 

term 

d ˙ Qbias 

dT = − ˙ Qbiasα 

β(ω), (65) 

T 

26 

where α = d log R/d log T describes the sensitivity of 

the detector resistance to the temperature changes and 

β(ω) ≡ [R − ZS(ω)]/[R + ZS(ω)] is the effect of the 

bias circuit (with an embedding impedance of ZS) on 

the ETF. Taking into account the thermal cut-off of the 

bolometer, the frequency-dependent loop gain is defined 

as 

β(ω) 

L(ω) ≡ L0 , (66) 

1 + ω2τ 2 

0 

where L0 ≡ ˙ Qbiasα/(GthT ) and τ0 = C/Gth is the intrinsic 

thermal time constant of the bolometer. The electrothermal 

loop gain describes the effect of varying incident 

optical power to the bias power dissipated in the detector. 

For positive bolometers with α > 0 the loop gain is 

positive for current bias (since Re[β(ω)] > 0) and negative 

for voltage bias (as Re[β(ω) < 0]). For bolometers 

with a negative temperature coefficient of resistance the 

situation is reversed. For metallic bolometers operated 

at room temperature α ∼ 1 and the loop gain is typically 

small (L0 1) so that the role of ETF is negligible. On 

the contrary, superconducting detectors with α ∼ 100 

and Gth some three orders of magnitude smaller than 

for room temperature devices can have large loop gain 

(L0 50), so that ETF plays a significant role in the detector 

characteristics. A major impact of negative ETF 

is that the bolometer time constant is reduced from τ0 

to τeff = τ0/[1 + β(0)L0]. This reduction in the time 

constant is one of the major benefits of strong negative 

ETF. 

Voltage biasing conditions are typically reached by 

driving a constant current I0 through a parallel combination 

of the bolometer and a load resistor ZS. As 

long as ZS ≪ R, the bolometer is effectively voltage biased. 

The responsivity of a voltage biased bolometer can 

be derived as follows: The current responsivity is defined 

as RI ≡ dI/d ˙ Qo. Using Eqs. (64), (65) and (66), 

V = I0ZSR/(ZS + R), and I = I0ZS/(ZS + R), the result 

for the current responsivity becomes 

RI(ω) = − 1 

V 

(1 + β)L0 1 

. (67) 

2(1 + βL0) 1 + ω2τ 2 

eff 

In the limiting case at ω = 0 with β = 1 (perfect voltage 

bias) and L0 ≫ 1, 

RI(0) = − 1 

. (68) 

V 

In order to evaluate the NEP for a bolometer, a discussion 

on the noise sources in bolometers is merited. In 

general, the NEP, noise spectral density S, and responsivity 

are related by NEP= SV,I/|RV,I|, where the subscripts 

refer to voltage or current noise and responsivity, 

respectively. The noise in bolometers is due to several 

uncorrelated sources. The most important of them is 

the thermal fluctuation noise, mentioned already above. 

Generally, this contribution is given by 

NEPTFN = 4γkBT 2 c Gth. (69)

Here γ describes the effect of the temperature gradient 

across the thermal link between the sensor and the heat 

sink (Mather, 1982) in the diffusive limit (i.e., no ballistic 

heat transport present) as 

 

(b + 1) Tc 

γ = 

b+3 − Tc −b 3+2 b 

T0 

 

2 

(3 + 2 b) Tc Tc b+1 Tc≫T0 b + 1 

→ . (70) 

− 

b+1 2b + 3 T0 

Here we assumed that the thermal conductivities obey 

κ ∝ T b . For resistive bolometers, another important contribution 

to the NEP is due to the Johnson noise, given 

by 

NEPJ = 

4kBTc 

R 

V 

L0 

 

1 + ω2τ 2 0 . (71) 

In addition to these noise sources, the current noise of 

the amplifier, Sin,amp, adds a contribution NEPamp = 

Sin,amp/RI. The total NEP of a bolometer is then 

NEP 2 tot =NEP 2 TFN + NEP 2 J 

+ NEP 2 amp + NEP 2 excess, (72) 

where the slightly ambiguous contribution NEPexcess encompasses 

various contributions from additional external 

and internal noise sources. Typical excess external noise 

contributions arise from heat bath temperature fluctuations 

and pickup in the cabling to name a few. In addition 

to the external excessive noise, it has in recent years become 

clear that there are also internal noise sources that 

are not fully accounted for. For instance, TESs with 

significant internal thermal resistance can no longer be 

treated using the simple lumped element model, and they 

suffer from internal thermal fluctuation noise (ITFN), 

adding a contribution (Hoevers et al., 2000) 

NEPITFN = 

4kBT R 

L0 

Gth 

 

1 + ω2τ 2 0 , (73) 

where L0 is the Lorenz number. 

For some devices, these noise sources are sufficient to 

explain all the observed noise. However, many groups are 

developing X-ray microcalorimeters (see below) which 

are often operated at a small fraction of the normal state 

resistance exhibit noise that increases rapidly as the bias 

point resistance is decreased, and has a significant influence 

on the performance of the detectors. Several possible 

explanations have been put forth, e.g., noise arising 

from the fluctuations in magnetic domains or phase-slip 

lines (Knoedler, 1983; Wollman et al., 1997). A systematic 

study of the excess noise in different TES geometries 

has been published recently (Ullom et al., 2004), 

showing that there exists a clear correlation between α 

and the observed excess noise, with the magnitude of 

the excess noise scaling roughly as 0.2 √ α. A quantitative 

agreement with the measured excess noise spectrum 

has so far been achieved in one experiment (Luukanen 

27 

et al., 2003), where the TES consisted of an annular (so 

called Corbino) geometry with a superconducting center 

contact, and a concentric current return at the outer 

perimeter of the annular TES. This geometry results in 

strictly radial current flow, enabling a simple analytical 

expression for the current density in the TES. The 1/r 

dependence in the current density together with a small 

radial temperature gradient causes the TES to separate 

to two annular superconducting and normal state regions. 

For any system undergoing a second order phase transition, 

order parameter fluctuations will take place. The 

excess noise arises from the thermally driven fluctuation 

of the phase boundary which manifest as resistance fluctuations. 

The volume associated with the order parameter 

fluctuation can be obtained by noting that the 

Ginzburg-Landau free energy δF associated with the 

fluctuation is δF ∼ kBTc. As the TES is biased towards 

smaller R, the relative contribution of the order parameter 

fluctuations becomes larger until it fully dominates 

over the other noise contributions. For the Corbinogeometry 

TES, the contribution due to the fluctuation 

superconductivity noise (FSN) to the NEP is 

NEPFSN = 0.24L0T 2 c G 

V 2 

Γ 1 + ω 

α 

2τ 2 0 , (74) 

where Γ ≈ 10 −8 K/ √ Hz is a constant dependent on the 

TES parameters. 

So far, this noise model has not successfully been applied 

to TESs in the more conventional square geometry, 

mainly due to the fact that the current distribution 

varies with bias point and is not easily calculable. A solution 

could be obtained by solving the full 2D Ginzburg- 

Landau equations for a square geometry. 

1. Hot electron bolometers 

In principle, modern bolometers can be artificially divided 

into two major sub-classes depending on where 

the dominant thermal bottleneck lies. The so-called hotelectron 

bolometers (HEBs) utilize the decoupling of the 

electron gas in a metal or a semiconductor from the 

phonon heat bath. The earliest HEBs were based on 

InSb, where the weak coupling of the electrons to the 

lattice at temperatures around 4 K allows the electrons 

to be heated to a temperature significantly above that 

of the lattice even in a bulk sample. Mobility in InSb 

is limited by ionized impurity scattering which results 

in decreasing resistivity with an increasing electric field. 

This is the basis of the detection mechanism (Brown, 

1984; Rollin, 1961). Often HEBs have sufficient speed 

for mixing, and in fact HEB mixers are a current topic 

of considerable interest. Most HEBs mixers operate at 4 

K, and in order to maintain our focus on phenomena and 

devices at temperatures below 1 K, we unfortunately will 

not discuss them within this review. 

HEB direct detectors (Ali et al., 2003; Karasik et al., 

2003; Richards, 1994) have been a subject of consider-

able interest, but in general a full optical demonstration 

remains to be carried out. The attractive features of 

such devices include very short time constant (well below 

1 µs), potentially very good NEP performance (below 

10 −19 W/ √ Hz when operated at or below 0.3 K), 

and simple construction that does not require surface micromachining 

steps. The architecture is essentially very 

similar to that of the HEB mixers: a small superconducting 

TES film coupled to the feed of a lithographic 

antenna. The application of the SQUID readout scheme 

utilized in typical hot phonon microbolometers and microcalorimeters 

might not be as straightforward as one 

could expect as the introduction of the SQUID input coil 

inductance to the voltage biasing circuit can make the 

system unstable due to the interaction of the poles of the 

electrical circuit and the thermal circuit. The approximate 

criterion for the stability of a voltage biased TES 

is that the effective time constant of the TES should be 

about one order of magnitude longer than the electrical 

time constant of the bias circuit (Irwin et al., 1998). 

A hot-electron bolometer that has been demonstrated 

is based on electron thermometry with NIS junctions 

(Nahum and Martinis, 1993, 1995). Here the incident optical 

power elevates the electron temperature in a small 

normal metal island weakly thermally coupled to the lattice 

phonons, and the change in the electron temperature 

can be sensed as a change in the tunneling current 

(see Subs. III.A.1). Noise equivalent powers below 

10 −19 W/ √ Hz have been predicted (Kuzmin, 2004) but 

remain to be experimentally verified. An additional attractive 

feature of the SINIS bolometer is that a DC bias 

on the device can be used to refrigerate the electrons to 

a temperature below that of the bath temperature (see 

Subs. V.C.1). Another benefit over TESs is the fact that 

the SINIS bolometer saturates much more gently compared 

to the TESs, which basically have no response at all 

once the device is overheated above Tc. The self-cooling 

property of the SINIS bolometer can also be used to compensate 

for excessive background loading, thus effectively 

giving it a larger dynamic range. The main obstacle towards 

constructing large arrays of (SI)NIS based HEBs 

is that their impedance (typically 1 kΩ - 100 kΩ) is hard 

to match to the existing cryogenic SQUID multiplexers 

(Chervenak et al., 1999; de Korte et al., 2003; Lanting 

et al., 2005; Reintsema et al., 2003; Yoon et al., 2001). In 

principle, one could apply superconducting transformers 

to match the SQUID noise, but transformers with sufficient 

impedance transformation range are quite large, 

which makes this approach unpractical. A novel readout 

method that lends itself for array readouts is a microwave 

reflectometric measurement, in which the SINIS bolometer 

is connected in series with a tuning inductor (Schmidt 

et al., 2003, 2004b). The LC resonance frequency of the 

inductor and the stray capacitance of the junctions is 

tuned to fall within the bandwidth of the cryogenic microwave 

amplifer (400-600 MHz), facilitating good impedance 

match. The dynamic resistance of the device 

is highly sensitive to the electron temperature, and thus 

28 

temperature changes cause modulation of Q of the resonance 

circuit. This modulation is sensed by sending a 

small RF signal to the resonant circuit, and measuring 

the reflected power. The electrical NEP inferred from 

noise measurements was in these experiments 1.6×10 −17 

W/ √ Hz. 

2. Hot phonon bolometers 

The second major class of bolometers are hot phonon 

bolometers (HPBs). They rely on a geometrical design 

of the heat link Ggeom so that the thermal bottleneck 

lies between two phonon populations. This approach 

is the most common, and allows for operation 

at temperatures up to and beyond room temperature. 

The earliest and most widely used of the contemporary 

HPBs are the so-called spider-web bolometers (Mauskopf 

et al., 1997), operated at 300 mK and below, where a 

free-standing Si3N4 mesh is used to support a thermal 

sensing element. Narrow Si3N4 legs provide the thermal 

isolation for the mesh. Before the introduction of 

TESs and SQUIDs the thermometer of choice was a 

small crystal of neutron-transmutated (NTD) Ge due to 

its relatively high temperature coefficient of resistance 

(d log R/d log T |T =0.3K ≈ −6) and large resistance (∼ 25 

MΩ) that allowed for good noise matching with FET 

preamplifiers. On the other hand this made the devices 

very microphonic. Efficient optical coupling was possible 

since radiation at wavelengths smaller than the mesh 

period are absorbed to a resistive film deposited on the 

Si3N4 mesh. An example of a spider-web bolometer using 

a NTD Ge is shown in Fig. 17. Later versions of the 

spider-web bolometers have adopted the use of TES as 

thermometers, coupled to a SQUID readout (Gildemeister 

et al., 1999). 

FIG. 17 (Color in online edition): A micrograph of a ”spiderweb” 

bolometer. A NTD Ge thermistor is located at the 

centre of the web. Image courtesy of NASA/JPL-Caltech. 

Alternatively to the spider-web absorber, a λ/4 resonant 

cavity can be used to maximize the optical efficiency

over a limited bandwidth. The SCUBA-2 instrument 

(Duncan et al., 2003; Holland et al., 2003) on the James 

Clerk Maxwell telescope is an ambitious overtaking with 

a pixel count of over 12 000. The bolometers consist 

of thermally isolated, (1 mm 2 )× 60 nm silicon ’bricks’ 

with the front surface of the Si degenerately doped with 

phosphorous to 377 Ω per square. The resonant cavity 

is formed by the Si in between the doped layer, and a 

Mo-Cu TES deposited to the back side of the pixel. 

Even though the superconducting TES thermometers 

are becoming increasingly popular, bolometers utilizing 

lithographic semiconducting thermistors still yield impressive 

performance and are more forgiving in terms of 

the saturation power. In the so-called pop-up bolometers, 

a lithographic, doped Si is used as the thermistor, while 

an ingenious torsional bending method of the Si3N4 legs 

is used to bend the legs and wiring layers perpendicular 

to the absorbers. This architecture enables the construction 

of arrays with a very high filling factor (Voellmer 

et al., 2003). Another promising lithographic resistive 

thermometer technology is based on thin films of NbSi 

(Camus et al., 2000). The high resistivity of these films 

allows for impedance levels that provide a good noise 

matching to room temperature JFET amplifiers. 

Instead of the surface absorbing approach, another 

method is based on the use of a lithographic antenna 

(Hwang et al., 1979; Neikirk et al., 1983), and terminating 

the induced currents to a thermally isolated bolometer, 

an approach often used in the HEB mixers discussed 

above. The attractive feature of this method is that 

the thermally sensing volume can be made much smaller 

compared to the case of the surface-absorbing bolometers, 

making these devices much faster. The low heat 

capacity often allows for lower NEP, as in many cases 

the NEPTFN is limited by the maximum time constant 

τmax = C/Gth,min allowed by the application. In this case 

NEPTFN = 4γkBT 2 C/τmax. Moreover, the small size 

of the thermally sensitive volume makes it far less sensitive 

to out of band stray light, relaxing filtering and 

baffling requirements of the incoming radiation. Antenna 

coupling also lends itself to the construction of 

integrated, on-chip filters for defining the bands for an 

array of bolometers (Hunt et al., 2003; Mees et al., 1991; 

Myers et al., 2005). Arrays of antenna-coupled bolometers 

can also utilize the inherent polarization selectivity 

of the antennas. An example of an antenna-coupled TES 

bolometer that incorporates on-chip transmission line impedance 

transformers and band-pass filters is shown in 

Fig. 18. No NEPs have yet been measured on this device, 

but the expected NEP is ∼ 10 −16 W/ √ Hz for a 

device with Tc = 450 mK. 

While not published yet, the best performance of an 

antenna-coupled HPB utilizing a similar design has been 

obtained by the JPL-Caltech bolometer group, with an 

electrical NEP=5 · 10 −19 W/ √ Hz at 230 mK (Kenyon, 

2005). The architecture is shown in Fig. 19. 

In the simplest case, the bolometer is simply a strip of 

metal placed to the feed of the lithographic antenna. Al- 

29 

FIG. 18 (Color in online edition): Micrograph of an antennacoupled 

TES bolometer. A dual-polarized double-slot antenna 

(a) is coupled via microstrip transmission lines (b) with 

band-pass and low-pass filters (c) to two Al/Ti bilayer TES 

thermometers located on suspended Si3N4 membranes (d). 

The inset shows in detail the termination of the microstrip 

line (e) to a resistor (f), and the Al/Ti TES (g). Figure courtesy 

A. T. Lee, UC Berkeley. 

FIG. 19 SEM micrograph of an isolated Si3N4 platforms with 

Mo/Au TESs. As an extreme example of thermal isolation, 

the meandering Si3N4 legs have an aspect ratio of ∼ 2800:1, 

yielding a thermal conductance Gth ≈ 100 fW/K at a bath 

temperature of 210 mK. Figure courtesy M.E. Kenyon, JPL. 

though this approach is by far the simplest, it introduces 

limitations for the performance of a HPB: Unlike in the 

case of HEB mixers, maximizing the thermal isolation of 

the bolometer is always desirable. However, the bolometer 

resistance should be matched to the impedance of the 

antenna (typically ∼ 100 Ω for broadband lithographic 

antennas on Si substrates), and thus the thermal conductance 

to the bath through the antenna,Gant , is fixed by 

the Wiedemann-Franz law, Gant = L0T/Re(Za), where 

Za is the antenna impedance. A parallel heat loss path 

is to the substrate below the bolometer film, with conductance 

Gsub ∝ LW where L and W are the length and 

width of the bolometer film, respectively. For substrate 

mounted antenna-coupled HPBs it is thus beneficial to 

minimize the size of the bolometer. This requirement 

can be relaxed if the bolometer strip is released from 

the underlying substrate to form an air-bridge (Neikirk

and Rutledge, 1984). For air-bridged devices at temperatures 

higher than 4 K, NEPTFN ∝ T 3/2 . Recently, 

an air-bridge bolometer operating at 4.2 K was demonstrated, 

showing potential for background-limited performance 

when observing 300 K blackbodies (Luukanen 

et al., 2005; Luukanen and Pekola, 2003). The potential 

applications for these devices include passive detection 

of concealed weapons under clothing, remote trace detection, 

and terrestrial submillimetre-wave imaging. 

C. Calorimeters: Pulsed excitation 

In the limit opposite to the bolometric detection, i.e. 

when the mean time between the quanta of energy arriving 

at the detector exceeds the device relaxation time, 

thermal detectors are known as calorimeters. While the 

topic of calorimetry also encompasses heat capacity measurements 

especially in mesoscopic samples, the following 

discussion concentrates on the detection of radiation 

only in order to keep the scope of our review in reasonable 

limits. For those interested in microcalorimetry in 

the sense of heat capacity measurements, we direct the 

reader to references (Bourgeois et al., 2005; Denlinger 

et al., 1994; Fominaya et al., 1997; Lindell et al., 2000) 

and (Marnieros et al., 1999). 

Cryogenic calorimeters are used today in a large variety 

of applications, from the detection of weakly interacting 

massive particles (WIMPs) in dark matter search (Akerib 

et al., 2003; Bravin et al., 1999), X-ray (de Korte 

et al., 2004; Kelley et al., 1999; Moseley et al., 1984) and 

γ-ray (van den Berg et al., 2000) astrophysics to secure 

optical communications (Miller et al., 2003; Nam et al., 

2004). As is the case with bolometers, calorimeters are 

usually operated at temperatures below 1 K. The theory 

and operation of calorimetric thermal detectors is 

very similar to bolometers, and generally the theoretical 

treatment above is valid. However, the optimization of 

calorimetric detectors can be quite different. The quantum 

of energy E deposited by either a photon, charged 

particle, WIMP etc. can be determined from the temperature 

rise ∆T = E/C, where C is the heat capacity 

of the calorimeter. This temperature rise then decays 

exponentially with a time constant τeff to its equilibrium 

value. Thermometry in the calorimeters is most often 

done either using semiconducting or TES thermometers, 

as is the case with bolometers. In addition, thermometry 

based on the change of magnetization of a paramagnetic 

sensor is appealing due to its non-dissipative nature, and 

has yielded some promising results (Fleischmann et al., 

2003; Schönefeld et al., 2000). 

The figure of merit for a calorimeter is the energy resolution, 

∆E, of full-width at half-maximum (FWHM), 

related to NEP through (Moseley et al., 1984) 

∆E =2 √ 2 ln 2 

∞ 

0 

4 

NEP 2 (f)tot 

df 

−1/2 

≈ 2 √ √ 

2 ln 2NEP(0)tot τeff, (75) 

30 

For a ’classic’ calorimeter with a white noise spectrum the 

energy resolution is in terms of the operating temperature 

and the heat capacity given by 

∆E = 2 √ 2 ln 2 kBT 2 C, (76) 

indicating that ∆E scales ∝ T 5/2 or ∝ T 3/2 depending 

whether the heat capacity of the sensor is dominated by 

the lattice or the electronic system, respectively. 

In most of the devices under development today the 

lattice temperature is being measured, as sufficient cross 

section to the incoming energy often requires a rather 

large volume of a device. An exception to the norm are 

TES calorimeters optimized for optical single photon detection 

for applications in secure quantum key distribution 

(Nam et al., 2004), shown in Fig. 20. In these 

devices, the thermal isolation is via electron-phonon decoupling. 

A trade-off is made between energy resolution 

and the speed of the detectors. Energy resolution of the 

detectors is sufficient to determine the photon-number 

state of the incoming photons, while maintaining a speed 

that is adequate for fast information transfer. 

FIG. 20 (Color in online edition): The energy spectrum of 

a pulsed 1550 nm laser, measured with an optical TES microcalorimeter. 

The peaks in the plot correspond to the 

photon-number state of the incoming pulses. The inset shows 

a micrograph of the devices. Figure courtesy of A.J. Miller, 

NIST. 

As with bolometers, the TESs currently outperform 

the competition in terms of the sensitivity. The best 

reported energy resolution for any energy dispersive detector 

was recently obtained with a Mo-Cu calorimeter 

(see Fig. 21), yielding an energy resolution of 2.38± 0.11 

eV at a photon energy of 5.89 keV (Irwin and Hilton, 

2005). The driving application in the development of Xray 

microcalorimeters are two major X-ray astrophysics 

missions planned by the European Space Agency (ESA) 

and the U.S. National Astronautics and Space Administration 

(NASA). Both missions, X-ray Evolving Universe 

Spectroscopy (XEUS) (The European Space Agency, 

2005) mission and the Constellation-X mission (NASA, 

2005) will employ X-ray microcalorimeters as their primary 

instrument. The energy resolution of the state of

FIG. 21 An energy spectrum of the 55 Mn Kα complex, obtained 

with a Mo/Cu TES microcalorimeter. The width of 

the measured characteristic x-ray lines is a convolution of the 

intrinsic x-ray linewidth and the gaussian detector contribution 

with a FWHM of 2.38 ±0.11 eV. The TES is fabricated 

on a free-standing Si3N4 membrane. The inset shows a SEM 

micrograph of the TES. The normal metal bars extending 

partially across the TES have been experimentally verified to 

improve the energy resolution of the detector (Ullom et al., 

2004). Figure courtesy of J. Ullom, NIST. 

the art microcalorimeters have already reached the requirements 

stated in the science goals for these missions. 

Thus, the current primary focus of the technical research 

is on the development of large arrays of microcalorimeters, 

and more importantly, SQUID multiplexing readouts 

for the detector arrays. A prototype 5 × 5 array of 

Ti/Au TES microcalorimeters is shown in Fig. 22. 

FIG. 22 (Color in online edition): A prototype Ti/Au microcalorimeter 

array. The dark regions within the Si3N4 membrane 

are holes in the membrane. Figure courtesy of SRON- 

MESA. 

D. Future directions 

While cryogenic thermal detectors are the most sensitive 

radiation detectors around, further significant per- 

31 

formance increases are envisioned in the future. The push 

is from single pixels to large staring focal plane arrays 

of detectors for numerous astrophysics applications. For 

this reason, novel thermal detector concepts must be able 

to be integrated into large arrays. 

The performance of hot electron bolometer mixers has 

been improving rapidly over the past years. Noise temperatures 

approaching 10×hν/(2kB) are being reported, 

and the emphasis is in pushing the operating frequency 

deeper into the THz region. In principle, frequencies in 

the infrared range are not out of the question. Lithographic 

antennas have demonstrated good performance 

up to 30 THz (Grossman et al., 1991), while it is clear 

that the fabrication becomes increasingly challenging as 

the required feature size is reduced. 

In the single pixel direct detector development, the 

quest for ever better sensitivity is still ongoing. For 

bolometers, the improvement in NEP directly translates 

to the capability of observing over less pre-detection 

bandwidth without sacrificing signal to noise ratio. This 

would enable the construction of arrays capable of yielding 

spectroscopic information. It is here where the functions 

of HEB mixers, bolometers and calorimeters conjoin 

- in the capability of performing single photon spectrometry 

at far-infrared wavelengths. 

V. ELECTRONIC REFRIGERATION 

Thermoelectric effects and, in particular, thermoelectric 

cooling have been discovered more than 170 years 

ago (Peltier, 1834). During the last 40 years considerable 

progress has been made in developing practical thermoelectric 

refrigerators for industrial and scientific applications 

(Nolas et al., 2001; Rowe and Bhandari, 1983). 

The temperature range of interest has been, however, 

far above cryogenic temperatures. Yet, during the last 

decade, solid state refrigerators for low temperature applications 

and, in particular, operating in the sub-kelvin 

temperature range have been intensively investigated. 

The motivations of this activity stem from the successful 

development and implementation of ultrasensitive radiation 

sensors and quantum circuits which require onchip 

cooling (Pekola et al., 2004b) for proper operation 

at cryogenic temperatures. Solid state refrigerators have 

typically lower efficiency as compared to more traditional 

systems (e.g., Joule-Thomson or Stirling gas-based refrigerators). 

By contrast, they are more reliable, cheaper 

and, what is more relevant, they can be easily scaled 

down to mesoscopic scale. All this gives a unique opportunity 

to combine on-chip refrigeration with different 

micro- and nanodevices. 

The aim of this part of the review is to describe the existing 

solid state electron refrigerators operating at cryogenic 

temperatures (in particular below liquid helium 

temperature), and to give an overview of some novel ideas 

and refrigeration schemes.

A. General principles 

The physical principle at the basis of thermoelectric 

cooling is that energy transfer is associated with quasiparticle 

electric current, as shown in Sec. II. Under suitable 

conditions thermal currents can be exploited for heat 

pumping, and if heat is transferred from a cold region to 

a hot region of the system, the device acts as a refrigerator. 

The term refrigeration is associated throughout 

this Review to a process of lowering the temperature of 

a system with respect to the ambient temperature, while 

cooling, in general, means just heat removal from the 

system. It is noteworthy to mention that the maximum 

cooling power of a refrigerator is achieved at a vanishing 

temperature gradient, while the maximum temperature 

difference is achieved at zero cooling power. The efficiency 

of a refrigerator is normally characterized by its 

coefficient of performance (η), i.e., the ratio between the 

refrigerator cooling power ( ˙ Qcooler) and the total input 

power (Ptotal): 

η = ˙ Qcooler 

. (77) 

Ptotal 

Irreversible processes (e.g., thermal conductivity and 

Joule heating) degrade the efficiency of refrigerators, and 

are essential elements that need to be carefully evaluated 

for the optimization of any device. 

The basic principles of the design are in general similar 

for different types of refrigerators. The increase of 

temperature gradient can be achieved by realizing a multistage 

refrigerator. In this case, the stage operating at 

a higher temperature should be designed for larger cooling 

power in order to efficiently extract the heat released 

from the lower-temperature stage (”pyramid design”). 

The enhancement of the refrigerator cooling power can 

be achieved by connecting several refrigerators in parallel. 

The parallel design is more effective both for an 

efficient heat evacuation from the hotter regions of the 

device and for the application of higher electric currents 

to the refrigerator. It also allows more freedom to properly 

bias all cooling elements. 

The temperature dependence of the electric and thermal 

properties of the active parts in the refrigerator may 

limit their exploitation at low temperatures. The reduction 

of thermal conductivity by lowering the temperature 

has both positive (better thermal insulation between cold 

and hot regions) and negative (difficulty in removing heat 

from the system) effects. 

At cryogenic temperatures different types of superconductors 

can be efficiently exploited. They can be used 

both as passive and active elements: in the former case, 

owing to their low thermal conductivity and zero electric 

resistance (e.g., as one of the two arms in Peltier 

refrigerators), in the latter as materials with an energy 

gap in the density of states for energy-dependent electron 

tunneling (e.g., in NIS coolers). 

Currently, the development of refrigerating techniques 

follows two main directions: search of new materials with 

p - type 

e + 

T cold 

e - 

Heat sink Heat sink 

I 

n - type 

FIG. 23 (Color in online edition) Basic Peltier thermoelement. 

32 

improved characteristics suitable for existing refrigeration 

schemes, and development of new refrigeration methods 

and principles. 

B. Peltier refrigerators 

Thermoelectric (Peltier) refrigeration is widely used for 

cooling different electronic devices (Nolas et al., 2001; 

Phelan et al., 2002; Rowe and Bhandari, 1983). Nowadays 

Peltier refrigerators providing temperature reduction 

down to 100...200 K and cooling power up to 100 

W are available. Peltier cooling (or heating) occurs 

when an electric current is driven through the junction of 

two different materials. The heat released or absorbed, 

˙QP eltier, depending on the direction of the electric current 

at the junction, is proportional to the electric current 

(I) driven through the circuit, ˙ QP eltier = ΠABI, 

where ΠAB = αABT , and ΠAB and αAB are the Peltier 

and Seebeck coefficients of the contact, respectively (see 

also Eq. (39)). In order to obtain enhanced values of 

the Peltier coefficient, conventional Peltier refrigerators 

generally consist of p- and n-type semiconductors with 

opposite sign of Π coefficients (see Fig. 23). The efficiency 

of a Peltier refrigerator is not determined only 

by the coefficient ΠAB = ΠA − ΠB, but also by thermal 

conductivities (κ) of both materials across the contact. 

Furthermore their electric resistances (ρ) are responsible 

of Joule heating affecting the coefficient of performance. 

The maximum temperature difference (∆Tmax) achievable 

with a Peltier refrigerator is given by (Nolas et al., 

2001) ∆Tmax = ZT 2 α2AB 

cold /2, where Z = ρκ is a figure of 

merit of the refrigerator, and Tcold is the temperature of 

the cold junction (see Fig. 23). More often, however, 

the refrigerator efficiency is characterized by the dimensionless 

figure of merit ZT . We recall that ZT ∝ ( kBT 

EF )2 . 

Most of the materials used in thermoelectric applications 

have ZT ∼ 1 at room temperature (DiSalvo, 1999; Min 

and Rowe, 2000). In general, at low temperatures only 

one single thermoelectric material is needed, because a 

superconductor can be used as one of the two arms of the 

refrigerator (Goldsmid et al., 1988; Nolas et al., 2001). 

In spite of the drastic reduction of cooling efficiency 

at low temperature, there is, however, some development 

of new materials and devices suitable for operation at

FIG. 24 Thermoelectric refrigeration at cryogenic temperatures 

using cerium hexaboride. Adapted from (Harutyunyan 

et al., 2003). 

FIG. 25 Schematic diagram of the thermoelectromechanical 

cooler, time sequences of the pulsed current applied to the 

device, and the two modes of cantilever contact: synchronized 

and optimized (Miner et al., 1999). 

cryogenic temperatures. Recently, Peltier cooling with 

∆Tmax ≈ 0.2 K was demonstrated below 10 K (Harutyunyan 

et al., 2003) (see Fig. 24). Crystals of CeB6 

were used to exploit the strong thermoelectric coefficients 

arising from the Kondo effect. The dimensionless figure 

of merit of this material is 0.1...0.25 in the temperature 

range between 4 K and 10 K. The authors claim that a 

proper optimization of the refrigerator design would allow 

more than 10% temperature reduction below 4K with 

a single-stage refrigerator. 

At millikelvin temperatures lattice specific heat and 

thermal conductivity decrease drastically, and thermoelectric 

refrigeration might become feasible. Following 

this idea and taking into account the Wiedemann-Franz 

relation, values of ZT as high as 20 at temperatures below 

10 mK and the possibility to achieve thermoelectric 

refrigeration were predicted (Goldsmid and Gray, 1979; 

Nolas et al., 2001). 

Kapitulnik (1992) proposed to exploit a metal close to 

33 

its metal-insulator transition for the implementation of 

a thermoelectric refrigerator operating below liquid-He 

temperatures. The basic concept behind this proposal 

is that near the metal-insulator transition, the transport 

coefficients acquire anomalous power laws such that their 

relevant combination describing the figure of merit for 

efficient cooling also becomes large. 

Further improvement of the efficiency of conventional 

thermoelectric coolers could be, in principle, achieved 

through a thermoelectromechanical refrigerator (Miner 

et al., 1999; Thonhauser et al., 2004). In such a device, a 

periodic variation of the electric current through a Peltier 

element combined with a synchronized mechanical thermal 

switch would allow to improve the overall cooling 

performance (see Fig. 25). The enhancement of refrigeration 

is due to the spatial separation of the Peltier cooling 

and Joule heating. 

The analysis of thermoelectric devices is usually based 

on the parameters typical of bulk materials. Significant 

progress in low temperature Peltier refrigeration might 

be achieved by using exotic materials (Goltsev et al., 

2003; Rontani and Sham, 2000) and low-dimensional 

structures (Balandin and Lazarenkova, 2003; DiSalvo, 

1999; Hicks et al., 1993), such as composite thin films, 

modulation-doped heterostructures, quantum wires, nanotubes, 

quantum dots, etc. These systems offer, in general, 

more degrees of freedom to optimize those quantities 

that affect the efficiency of thermoelectric refrigerators. 

C. Superconducting electron refrigerators 

1. (SI)NIS structures 

Although heat transport in superconducting microstructures 

originally dates back more than 40 years 

ago in SIS junctions (Chi and Clarke, 1979; Gray, 1978; 

Melton et al., 1981; Parmenter, 1961), it is instructive 

to start our description of this topic from NIS tunnel 

junction structures. Figure 26(a) shows the calculated ˙ Q 

for a NIS tunnel junction (see Sec. II.G.2) versus bias 

voltage at different temperatures (T = Te,N = Te,S). 

When ˙ Q is positive, it implies heat removal from the N 

electrode, i.e., hot excitations are transferred to the superconductor. 

For each temperature there is an optimal 

voltage that maximizes ˙ Q and, by decreasing the temperature, 

the heat current results to be peaked around 

V ∆/e. Figure 26(b) displays the heat current versus 

temperature calculated at each optimal bias voltage. The 

quantity ˙ Q(T ) is maximized at T ≈ 0.25 ∆/kB = Topt 

(as indicated by the arrow in the figure) where it reaches 

˙Q 6 × 10 −2 ∆ 2 /e 2 RT , decreasing both at lower and 

higher temperatures. Also shown in the figure is ˙ Q(T ) obtained 

assuming a temperature-independent energy gap. 

Such a comparison shows that this latter assumption is 

fully justified for T ≤ 0.2 ∆/kB. In the low temperature 

limit (Te,N ≤ Te,S ≪ ∆/kB) it is possible to give an approximate 

expression (Anghel and Pekola, 2001) for the

optimal bias voltage (Vopt), Vopt ≈ (∆ − 0.66kBTe,N )/e, 

as well as for the maximum cooling power at Vopt, 

˙Qopt ≈ ∆2 

e2 kBTe,N 

[0.59( RT ∆ 

)3/2 

 

2πkBTe,S 

− ∆ exp(− ∆ 

kBTe,S )]. 

(78) 

Equation (78) is useful for getting quantitative estimates 

on the performance of realistic coolers. In the 

same temperature limit and for V = Vopt the cur- 

rent through the NIS junction can be approximated as 

I ≈ 0.48 ∆ 

eRT 

 

kBTe,N 

∆ . The NIS junction coefficient of 

performance η is given by η(V ) = ˙ Q(V )/[I(V )V ]. For 

V ≈ ∆/e and in the low temperature limit, η thus obtains 

the approximate value 

ηopt ≈ 0.7 Te,N 

, (79) 

where we assumed ∆ = 1.764 kBTc, and Tc is the critical 

temperature of the superconductor. Equation (79) shows 

that the efficiency of a NIS junction is around or below 

20% at the typical operation temperatures. The full behavior 

of η versus temperature calculated at each optimal 

bias voltage is displayed in Fig. 26(c). The simple results 

presented above point out how the optimized operation of 

a superconducting tunnel junction as a building block of 

microrefrigerators stems from a delicate balance among 

several factors such as the contact resistance, the operation 

temperature, the superconducting gap ∆ as well as 

the bias voltage across the junction. 

The first observation of heat extraction from a normal 

metal dates back to 1994 (Nahum et al., 1994), where 

cooling of conduction electrons in Cu below the lattice 

temperature was demonstrated using an Al/Al2O3/Cu 

tunnel junction. A significant improvement was made 

two years later, still in the Al/Al2O3/Cu system, by 

Leivo et al. (1996), which recognized that using two NIS 

junctions in series and arranged in a symmetric configuration 

(i.e., in a SINIS fashion) leads to a much stronger 

cooling effect. This fact can be understood keeping in 

mind that ˙ Q is a symmetric function of V so that, at 

fixed voltage across the structure, quasi-electrons above 

∆ are extracted from the N region through one junction, 

while at the same time quasi-holes are filled in N below 

−∆ from the other junction (see Fig. 26(d)). In this configuration, 

a reduction of the electron temperature from 

300 mK to about 100 mK was obtained. Later on, several 

other experimental evidences of electron cooling in SI- 

NIS metallic structures were reported (Arutyunov et al., 

2000; Clark et al., 2004; Fisher et al., 1995, 1999; Leivo 

et al., 1997; Leoni et al., 1999, 2003; Luukanen et al., 

2000; Pekola et al., 2000a, 2004a, 2000b; Tarasov et al., 

2003; Vystavkin et al., 1999). In these experiments NIS 

junctions are used to alter the electron temperature in the 

N region as well as to measure it. In order to measure 

the temperature, the N region is normally connected to 

additional NIS contacts (i.e., ”probe” junctions) acting 

as thermometers (previously calibrated by varying the 

Tc 

34 

bath temperature of the cryostat), and operating along 

the lines described in Sec. III.A.1. Moreover, the differential 

conductance of the probe junctions gives also 

detailed information about the actual quasiparticle distribution 

function in the N region (Pekola et al., 2004a; 

Pothier et al., 1997b). 

Figure 27(a) shows the SEM micrograph of a typical 

Al/Al2O3/Cu SINIS refrigerator including the superconducting 

probe junctions. The schematic of a commonly 

used experimental setup for electron refrigeration 

and temperature measurement is shown in Fig. 27(b). 

The voltage bias Vrefr across the SINIS structure allows 

to change the electron temperature in the N region; 

at the same time, a measure of the voltage drop 

across the two probe junctions (Vth) at a constant bias 

current (I0) yields the electron temperature Te,N in the 

normal electrode (Rowell and Tsui, 1976). Figure 27(c) 

illustrates the experimental data of Leivo et al. (1996) 

of the measured electron temperature T ≡ Te,N versus 

Vrefr, taken at different bath temperatures (i.e., those 

at Vrefr = 0). As can be readily seen, the electron 

temperature rapidly decreases by increasing the voltage 

bias across the SINIS structure, reaching the lowest value 

e 2 R T /Δ 2 (0) 

Q&Q& 

-0.05 

η (%) 

0.10 

0.05 

0.00 

15 

10 

5 

(a) 

k B T/Δ(0) 

0 

0.0 0.1 0.2 0.3 0.4 0.5 

kBT/Δ(0) 0.05 

0.1 

0.2 

0.25 

0.4 

0.5 

e 2 R T /Δ 2 (0) 

Q&Q & 

T/Tc 0.0 

0.08 

0.5 1.0 

0.06 

0.04 

Δ(T) = Δ BCS (T) 

Δ(T) = Δ(0) 

NIN 

0.02 

-0.10 

0.0 0.5 1.0 1.5 

eV/Δ(0) 

(b) 

0.00 

T/T 

0.0 0.3 0.6 

c 

0.0 0.2 0.4 0.6 0.8 

kBT/Δ(0) 30 

Electric 

(c) 

25 

current 

20 

Q • 

(d) 

FIG. 26 (Color in online edition) (a) Calculated cooling power 

˙Q of a NIS junction vs bias voltage V for several temperatures 

T = Te,N = Te,S. Also shown is the behavior of a 

NIN junction. (b) ˙ Q calculated at the optimal bias voltage 

as a function of temperature, assuming both a temperatureindependent 

energy gap (dash-dotted blue line) and the real 

BCS dependence (black line). Topt indicates the temperature 

value that maximizes ˙ Q. (c) Coefficient of performance η calculated 

at the optimal bias voltage versus temperature. (d) 

Scheme of the energy band diagram of a voltage biased SINIS 

junction. The electric current flows into the normal region 

through one tunnel junction and out through the other, while 

the heat current ˙ Q flows out of the N electrode through both 

tunnel junctions. 

T opt

(b) 

R 1 

R 2 

R P 

I 0 

V refr 

V th 

R T 

R 2 

R 1 

(a) 

T 

(d) 

T e,S (K) 

(c) 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

eV/Δ(0) 

0.68 

0.51 

0.34 

0.17 

0 

T (K) e,N 

FIG. 27 (Color in online edition) (a) Scanning electron micrograph 

of a typical SINIS microrefrigerator. The structure 

was fabricated by standard electron beam lithography combined 

with Al and Cu UHV e-beam evaporation. The probe 

junction resistance usually satisfies the condition Rp ≫ RT 

in order to prevent self-cooling. (b) Sketch of a typical measurement 

setup for electron cooling and temperature measurement. 

(c) Electron temperature in the N region versus 

Vrefr measured at different bath temperatures. Dots represent 

experimental data, while solid lines are theoretical fits. 

The inset shows the cooling power (pW) against temperature 

obtained from the fits (dots). (d) Contour plot of the theoretical 

electron temperature Te,N vs bias voltage V and Te,S for 

a SINIS cooler, assuming thermal load due to the electronphonon 

interaction (see text). (a) is adapted from (Pekola 

et al., 2004a); (c) from (Leivo et al., 1996). 

around Vrefr ≈ 2∆ 360 µeV (note that now two junctions 

in series are involved in the process). Then, by 

further increasing the bias, the electron temperature rises 

due to the heat flux induced into the metal. Furthermore, 

the minimum electron temperature strongly depends on 

the starting bath temperature. The inset shows the extracted 

maximum cooling power (dots) that obtains values 

as high as 1.5 pW at T = 300 mK, corresponding 

to about 2 pW/µm 2 (in the present experiment, submicron 

NIS junctions were exploited, with barrier resistances 

RT 1 kΩ). 

The general operation of a SINIS microrefrigerator like 

that shown in Fig. 27(a) can be understood by recalling 

that the final electron temperature in the N region stems 

from the balance among several factors that tend to drive 

power into the electron system (i.e., power losses), as discussed 

in detail in Sec. II. Most of metallic SINIS refrigerators 

fabricated so far operate in a regime where strong 

inelastic electron-electron interaction tends to drive the 

system into quasiequilibrium, where the electron system 

can be described by a Fermi function at a temperature 

Te,N that differs, in general, from that of the lattice 

(Tph). At the cryogenic temperatures of interest (i.e., 

typically below 1 K), the dominant contribution comes 

35 

from electron-phonon scattering that exchanges energy 

between electrons and the lattice phonons. The refrigerator 

cooling power ( ˙ Qrefr) can be defined as the maximum 

power load the (SI)NIS device can sustain while 

keeping the N region at temperature Te,N , and can be 

generally expressed as 

˙Qrefr = 2 ˙ Q(V/2; Te,N , Te,S) − ˙ Qe−ph(Te,N , Tph), (80) 

where the factor 2 takes into account the presence of two 

identical NIS junctions, and it is often assumed that in 

the superconductors Te,S = Tph. The minimum temperature 

Te,N reached by the N metal thus fulfills the 

condition ˙ Qrefr = 0. An example of Te,N versus V behavior 

along the lines of Eq. (80) is represented by the 

solid curves plotted in Fig. 27(c). Furthermore, any additional 

term that adds thermal load into the electron 

system can be properly taken into account by including 

its specific contribution to the right-hand side of Eq. 

(80) (Fisher et al., 1999; Ullom and Fisher, 2000). Figure 

27 (d) shows an example of the calculated electron 

temperature Te,N vs voltage and Te,S as obtained from 

the solution of Eq. (80), Qrefr 

˙ = 0. Here we considered 

a typical Al/Al2O3/Cu SINIS cooler with volume 

V = 0.15 µm3 , RT = 1 kΩ, Σ = 2 nW/µm3K5 (Cu), and 

Te,S = Tph. As it can be readily seen, the Te,N value 

strongly depends on the bias voltage as well as on the 

lattice temperature. Depending on the thermal load due 

to the electron-phonon interaction and the operating V , 

the SINIS device can thus behave as a cooler or as a 

refrigerator. 

Let us now highlight some of the basic requirements 

of SINIS structures for electron cooling operation (their 

use for lattice cooling is addressed in Sec. V.C.6). In 

particular, the working temperature range of these structures 

depends mainly on the type of the superconductor 

(through its energy gap ∆ and hence its critical temperature 

Tc), on the strength of the electron-phonon interaction 

Σ, on the junction resistance RT , and on the N 

region volume V. First of all, a reduction of the active 

volume to be cooled is the most straightforward method 

to increase the efficiency of the refrigeration process, this 

being more relevant at high lattice temperatures (according 

to the assumed T 5 ph-dependence of electron-phonon 

interaction). In addition, cooling power maximization in 

the high-temperature regime (let us say 1...4.2 K) would 

require both to lower the electron-phonon coupling constant 

and to increase ∆ through a proper choice of the 

superconductor. The first issue can be solved up to some 

extent with a variety of materials with different Σ (see 

Table I). As far as the second issue is concerned, there is 

a vast choice of superconducting metals with Tc covering 

the temperature range up to about 20 K (Kittel, 1996). 

On the other hand, the question of junction resistance 

requires a careful discussion. In general, from Eq. (78), 

˙Q enhancement is expected from a reduction of RT . This 

latter issue can be accomplished both by making thinner 

barriers and by increasing the junction area. The first

(a) 

(c) 

V th (μV) 

0 

100 

200 

-0.4 -0.2 0.0 0.2 0.4 

Vrefr (mV) 

1400 

410 

320 

280 

230 

50 

T e,N (mK) 

(b) 

(d) 

0.1 

T min /T c 

0.01 

1E-4 1E-3 0.01 

FIG. 28 (Color in online edition) (a) SEM micrograph of an 

Al/Al2O3/Cu SINIS microrefrigerator exploiting large-area 

junctions (∼ 10 µm 2 ) with quasiparticle traps. (b) SEM 

image of a part of comb-like SINIS structure with 10 + 10 

junctions for cooling and a SINIS thermometer. (c) Electron 

temperature Te,N in the N region of an Al/Al2O3Cu SINIS 

refrigerator versus Vrefr measured at different bath temperatures. 

The lowest curve (red dash-dotted line) shows the 

anomalous heating effect observable at the lowest temperatures 

and attributed to the presence of quasiparticle states 

within the superconducting gap. (d) Theoretical ultimate 

minimum electron temperature of a SINIS cooler Tmin/Tc at 

V 2 ∆/e versus Γ/∆ in quasiequilibrium. (a) is adapted 

from (Pekola et al., 2000a); (b) from (Luukanen et al., 2000); 

(c) and (d) from (Pekola et al., 2004a). 

issue is not so straightforward, as already discussed in 

Subs. II.F.2, due to the intrinsic difficulty in fabricating 

high-quality low-Rc barriers, although optimized barriers 

are currently under investigation (see Sec. VI.F.1). 

Latter option was experimentally addressed by Fisher 

et al. (1999) in Al/Al2O3/Ag refrigerators, where large 

cooling powers of a few tens of pW were obtained with 

junctions of 20 × 20 µm 2 surface area. The reduction in 

electron temperature was, however, much inferior to that 

achievable with sub-micron sized junction. The problem 

intrinsic to junctions with large overlap (especially 

at the lowest temperatures) stems from the larger density 

of quasiparticles present in the superconductor, due 

to the fact that quasiparticles require a larger time to 

exit the junction region and escape from the superconductor. 

Therefore, this excess of quasiparticles alters in 

general the refrigerator performance by returning energy 

to the normal electrode, mainly due to back-tunneling 

from the superconductor as well as due to recombination 

processes, where phonons can enter and heat up the 

N region (Jochum et al., 1998; Kaplan et al., 1976; Ullom 

and Fisher, 2000). In addition, inelastic scattering 

with phonons and dynamic impurities can also lead to 

an excess of quasiparticles. These contributions can easily 

overcompensate the junction cooling power, so that 

Γ/Δ 

36 

it is crucial to remove this excess of quasiparticles from 

the superconductor. Toward this end a number of techniques 

exist among which we mention the exploitation 

of defect-free and thick S electrodes (that allow quasiparticles 

to escape ballistically from the junction area as 

well as to decrease their density near the barrier) and the 

exploitation of quasiparticle ”traps” (Parmenter, 1961), 

i.e., normal metal films connected to the superconductor 

in the junction region through a tunnel barrier or in direct 

metallic contact (Irwin et al., 1995b; Pekola et al., 

2000a) (see Fig. 28(a)). Such traps act as sinks of quasiparticles 

absorbing almost all the excess quasiparticle 

energy present in the superconductor, and have proved 

to efficiently help heat removal from the superconductor 

leading to a significant improvement of SINIS device 

cooling performance (Luukanen et al., 2000; Pekola et al., 

2000a, 2004a). All these are commonly exploited tricks 

for the thermalization of the superconducting electrodes, 

and in these conditions, it is a reasonable approximation 

to set Te,S = Tph. Furthermore, the experiments 

(Pekola et al., 2000a) demonstrated that the trap performance 

is in general superior when it is in direct metallic 

contact at short distance from the cooling junction (typically 

below 1 µm, although the optimal distance mainly 

depends on the superconductor coherence length), nevertheless 

in small-area junctions even a contact through an 

insulating barrier seems sufficient for the purpose. The 

effectiveness of trapping in SINIS structures was theoretically 

addressed in detail by Voutilainen et al. (2005) 

and Golubev and Vasenko (2002). Another possibility 

to achieve high cooling by maximizing the ratio of the 

junction area to the size of the region to be cooled is 

to use several small-area junctions (with size in the submicron 

range) connected in parallel to the N electrode 

(to limit the drawbacks typical of large junctions) (Arutyunov 

et al., 2000; Leoni et al., 1999; Luukanen et al., 

2000; Manninen et al., 1999), as shown in Fig. 28(b). 

The issue of tunnel junction asymmetry in SINIS refrigerators 

was addressed by Pekola et al. (2000b). This 

effect is fortunately weak: these authors theoretically 

showed that the maximum cooling power is reduced by 

7% in the case the junction resistances differ by a factor 

of two, as compared to a symmetric structure with 

the same total junction area. Furthermore, the reduction 

is only about 25% even when the resistance ratio 

is four. This effect stems from the ”self aligning” character 

of the double junction structure: the voltage drop 

across each junction is simultaneously close to ∆/e, thus 

corresponding to the maximum cooling power, when the 

voltage across the whole SINIS system is close to 2∆/e. 

This fact is due to the high non-linearity of the currentvoltage 

characteristics of the two junctions which carry 

the same current. The experiments have confirmed such 

a weak dependence of the cooling power on the structure 

asymmetry (Pekola et al., 2000b). 

In the low-temperature regime the situation is rather 

different. While power load from electron-phonon interaction 

becomes less and less dominant by decreasing the

temperature, and typically below 0.1 K the cooler behavior 

can be described as if the lattice would not exist 

at all, other factors can limit the achievable cooling 

power as well as the lowest attainable electron temperature. 

First of all, at the lowest temperatures the maximum 

achievable cooling power of a NIS junction is intrinsically 

limited, given by ˙ Q ∝ (e 2 RT ) −1 ∆ 1/2 (kBTe,N ) 3/2 . 

Then non-idealities in the tunnel barriers, where the possible 

presence of Andreev-like transport channels (e.g., 

pin-holes) may strongly degrade the cooling power (in 

the high temperature regime this contribution is in general 

overcome by thermal activation) (Bardas and Averin, 

1995). Furthermore, nonequilibrium effects in the N region 

as well in the S electrodes may be a limitation. 

In the N region, a suppression of the cooling power is 

expected by increasing the quasiparticle relaxation time 

(Frank and Krech, 1997), i.e., by driving the electron 

gas far from equilibrium. In the superconductors, a nonthermal 

distribution can alter the cooling response of the 

refrigeration process as well as the presence of hot quasiparticle 

excitations (like with large-area junctions) may 

be responsible of additional heat load into the N region. 

Yet, quasiparticle states within the superconducting gap 

represent a crucial problem at the lowest temperatures, 

yielding anomalous heating in the N region and limiting 

the achievable minimum temperature (Pekola et al., 

2004a). Such quasiparticle states, due mainly to inelastic 

scattering in the superconductor (Dynes et al., 1984) 

or to inverse proximity effect from the nearby N region, 

are generally taken into account by adding an imaginary 

part (Γ) to the superconductor DOS, as in Eq. 

(12). Figure 28(c) shows a representative set of cooling 

curves taken in an Al/Al2O3/Cu SINIS refrigerator at 

different lattice temperatures, where at the lowest temperature 

(red dash-dotted line) (this typically happens 

around or below T0 ≈ 100 mK) the electron gas first 

undergoes heating, then it is strongly cooled at the expected 

bias around Vrefr 2∆/e. A similar anomaly 

was reported in some experiments on SINIS refrigerators 

(Fisher et al., 1999; Pekola et al., 2004a, 2000b; Savin 

et al., 2001). The anomalous heating was attributed to 

the presence of such states within the S gap that give rise 

to a sort of dissipative channel which dominates heat current 

in a certain bias range (Pekola et al., 2004a). Pekola 

et al. (2004a) theoretically demonstrated that the minimum 

achievable electron temperature (Tmin) in SINIS 

refrigerators is set by the amount of quasiparticle states 

present within the superconducting gap, and is given by 

Tmin/Tc 2.5(Γ/∆) 2/3 at Vrefr 2∆/e in quasiequilibrium 

(see Fig. 28(d)). We note that a measure of the 

Γ/∆ value in real NIS junctions is approximately given 

by the ratio of the zero-bias to the normal state junction 

conductance at low temperature. The existence of 

quasiparticle states within the superconducting gap thus 

sets a fundamental limit to the minimum achievable temperature, 

and great care has to be devoted to get rid of 

their presence to optimize the refrigeration process at the 

lowest temperatures. 

e Q&Q& 2 2 RT /Δ1 (0) 

0.2 

(a) 

0.1 

0.0 

Δ 2/Δ 1 

0.7 

0.5 

0.3 

0.1 

SIN 

T = 0.2 Δ1(0)/kB -0.1 

0.0 0.5 1.0 

eV/Δ1 (0) 

1.5 

(b) 

(c) 

37 

FIG. 29 (Color in online edition) (a) Calculated cooling power 

˙Q of a S1IS2 junction vs bias voltage V at T = Te,S1 = 

Te,S2 = 0.2 ∆1(0)/kB for several ∆2/∆1 ratios. Red-dotted 

line represents ˙ Q when S2 is in the normal state. (b) Measured 

cooling of a Ti island to and in the superconducting 

state by quasiparticle tunneling. R0 is related to the electron 

temperature Te,S2 in Ti (see text). Dashed lines: Ti is in the 

normal state. Thin solid lines: Ti is cooled from the normal 

to the superconducting state. Thick solid line: Ti is already 

superconducting at Vrefr = 0. (c) Measured minimum electron 

temperature (Tmin) of Ti versus bath temperature T0. 

(b) and (c) are adapted from (Manninen et al., 1999). 

2. S1IS2(IS1) structures 

The enhancement of superconductivity by quasiparticles 

extraction was proposed in 1961 by Parmenter 

(1961) in the context of S1IS2 tunnel junctions, where 

S1 and S2 represent superconductors with different energy 

gaps. Later on, Melton et al. (1981) theoretically 

discussed the possibility to exploit such a system to realize 

a refrigerator (addressing both the basic features 

and performance). From an experimental point of view, 

Chi and Clarke (1979) observed in Al films an enhancement 

of the energy gap of the order of 40% due to quasiparticle 

extraction. Then, Blamire et al. (1991), using 

Nb/AlOx/Al/AlOx/Nb structures, were able to obtain 

an enhancement of the critical temperature of the 

Al layer by more than 100%. The physical mechanism 

giving rise to this effect was discussed by Heslinga and 

Klapwijk (1993) and by Zaitsev (1992). More recently, 

also Nevirkovets (1997) observed in similar structures an 

enhancement of the Al gap by quasiparticle extraction. 

Figure 29(a) shows the calculated heat current versus 

bias voltage for a S1IS2 tunnel junction at T = Te,S1 = 

Te,S2 = 0.2 ∆1(0)/kB for various ∆2/∆1 ratios (see 

Eq. (30)). The quantity ˙ Q(V ) is an even function of V , 

thus allowing connection of two junctions in a symmetric 

configuration as in the NIS case, and is positive for 

|V | < (∆1(T ) + ∆2(T ))/e where hot quasiparticle excitations 

are removed from S2. Moreover, the heat current 

is maximized at V = ±(∆1(T ) − ∆2(T ))/e, where it is

(a) (c) 

Al 

(b) 

S-Sm cooler 

junction 

V 

I 0 

A B 

Thermometer 

BOX, SiO 2 

n++ SOI film 

Al 

V th 

Si subst 

T e,N (mK) 

400 

300 

200 

100 

-0.4 0.0 

V (mV) 

0.4 

FIG. 30 (Color in online edition) (a) Optical micrograph of 

a typical Al/Si/Al microrefrigerator and schematics of the 

measurement setup. The current I0 and voltage Vth are used 

to determine the electron temperature in the n ++ -Si layer, 

and the bias voltage V is used to change its value. (b) Cross 

section of the structure along the line AB in (a). (c) Measured 

electron temperature Te,N in Si versus bias voltage across the 

Al/Si/Al cooler at different substrate temperatures. Adapted 

from (Savin et al., 2003). 

logarithmically diverging (Harris, 1974; Tinkham, 1996) 

(note that this is somewhat broadened by the smearing 

in the density of states in a realistic situation (Frank 

and Krech, 1997; Manninen et al., 1999; Pekola et al., 

2004a)). From Fig. 29(a) it follows that heat extraction 

from S2 only occurs if ∆2(T ) < ∆1(T ) holds. Then, 

at V = ±(∆1(T ) + ∆2(T ))/e a sharp transition brings 

˙Q(V ) to negative values (more details about the heat 

transport in S1IS2 junctions can be found in (Frank and 

Krech, 1997)). The dotted line represents ˙ Q(V ) when 

the electrode S2 is in the normal state (i.e., NIS case). 

Notably, when both electrodes are in the superconducting 

state, ˙ Q(V ) can exceed significantly that in the normal 

state. This peculiar characteristic of heat transport 

in S1IS2 junctions makes them promising in realizing a 

”cascade” refrigerator that might operate at bath temperatures 

higher than those for a SINIS cooler, where 

several combined superconducting stages are used to efficiently 

cool a normal or a superconducting region. 

The experimental observation of electron cooling in a 

superconductor was reported by Manninen et al. (1999) 

using Al/insulator/Ti tunnel junctions. In this experiment, 

aluminum (i.e, the larger-gap superconductor) was 

used to cool the electron system of a Ti strip from 1.02 Tc2 

to below 0.7 Tc2, where Tc2 = 0.51 K was the Ti critical 

temperature. Figure 29(b) shows a representative set 

of measurements taken at different bath temperatures 

T0. In particular, it shows the zero-bias resistance of 

the thermometer junctions R0 against the bias voltage 

across the refrigerators (Vrefr). Dashed lines indicate the 

behavior at bath temperatures where Ti is still in the normal 

state: the electron temperature in Ti decreases (i.e., 

R0 increases) by increasing Vrefr, reaching a minimum 

38 

slightly below Vrefr = 2∆1/e 420 µeV, as expected for 

a SINIS refrigerator (∆1 is the energy gap of Al). Note 

that dashed lines are the upside-down equivalent, for example, 

of the measurements shown in Fig. 27(c). Thin 

solid lines indicate the temperature behavior when Ti 

is cooled from the normal to the superconducting state. 

This happens for T0 larger than Tc2 but below 625 mK, 

at Vrefr values for which R0 has a deep minimum. Here, 

for instance, starting from T0 = 520 mK, the minimum 

electron temperature reached in Ti was Tmin ≈ 320 ± 40 

mK, thus demonstrating the effectiveness of the cooling 

mechanism also in all-superconducting refrigerators. 

Thick solid line corresponds to the case where the Ti 

strip is already superconducting, i.e., at a bath temperature 

below Tc2. The results for the minimum electron 

temperature reached in Ti against T0 is represented by 

the open circles in Fig. 29(c). Comparison to the theory 

gave good agreement with the experiment (lines in 

Fig. 29(c)) with an electron-phonon interaction constant 

ΣTi ≈ 1.3 nW/µm 3 K 5 . 

3. SSmS structures 

Superconductor-semiconductor structures have recently 

attracted interest in the field of electron cooling 

(Buonomo et al., 2003; Savin et al., 2003, 2001). The basic 

idea is to exploit, for the active part of the cooler, 

heavily-doped semiconductor layers instead of normal 

metals, with the natural Schottky barrier forming at the 

contact to the metal electrodes (Brennan, 1999; Lüth, 

2001; Rhoderick and Williams, 1988; Sze, 1981, 1985). 

The peculiar characteristic intrinsic to this scheme stems 

from the possibility to alter up to a large extent the semiconductor 

electronic properties (like, for example, the 

charge density, etc.) and to change the transmissivity 

of the Schottky barrier through proper doping or choice 

of the semiconducting layer (see also Sec. VI.F.2). 

Heavily-doped silicon is a natural choice both from a 

practical and technological point of view, mainly owing 

to the widespread and well-developed Si technology. Up 

to now, the only evidence of superconducting electron 

cooling with semiconductors comes from microrefrigerators 

realized with the silicon-on-insulator (SOI) technology 

(Buonomo et al., 2003; Savin et al., 2003, 2001). The 

first demonstration of electron refrigeration in Si was reported 

by Savin et al. (2001), where the authors obtained 

a maximum temperature reduction under hot quasiparticle 

extraction of the order of 30% at the bath temperature 

T0 = 175 mK. Figure 30(a) shows the optical micrograph 

of a typical Al/Si/Al cooler. The two bigger Al electrodes 

are used for cooling, while the two smaller for detecting 

the electronic temperature in Si. A sketch of the structure 

cross-section is displayed in Fig. 30(b), where the 

n ++ -Si region appears just on top of the silicon dioxide 

insulating layer. The structures were characterized by a 

doping level ND = 4×10 19 cm −3 . A set of cooling curves 

extracted from one of such devices is shown in Fig. 30(c)

(Savin et al., 2003). It displays the electron temperature 

Te,N in the n ++ -SOI layer versus bias voltage across the 

refrigerator at different starting bath temperatures. The 

observed electronic temperature reduction is of the order 

of 60% at T0 ≈ 150 mK. Although the device volume and 

the cooler resistances were rather high, the authors attributed 

this significant cooling to a very small electronphonon 

coupling constant in Si: as a matter of fact, the 

latter was determined to be ΣSi 0.1 nW/µm 3 K 5 (Savin 

et al., 2001), i.e., about one order of magnitude smaller 

than in Cu. The maximum achieved cooling power was 

˙Qmax ≈ 1.3 pW at T0 = 175 mK, mainly limited by the 

high value of the specific contact resistances (Rc ∼ 67.5 

kΩµm 2 ). These authors also addressed the effect of carrier 

concentration in Si on cooling performance (Savin 

et al., 2003). 

Buonomo et al. (Buonomo et al., 2003) also obtained 

a cooling effect in Al/Si/Al refrigerators, with a device 

configuration similar to that of Fig. 30(a). Their structures 

were characterized by a doping level ND = 8 × 10 18 

cm −3 and contact specific resistances Rc ∼ 100 kΩµm 2 . 

Notably, they confirmed the same value reported by 

Savin et al. (2001) for the electron-phonon coupling constant 

in Si. In addition, these authors also explored the 

Nb/Si/Nb combination for electron cooling. Their results, 

however, showed no cooling effect, owing probably 

to a too transmissive interface at the contact with 

Si. This effect was ascribed both to the slightly lower 

Schottky barrier height of Nb/Si contact (∼ 0.5...0.6 eV 

(Chang et al., 1971; Heslinga and Klapwijk, 1989)) with 

respect to the Al/Si (∼ 0.6...0.7 eV (Chang et al., 1971; 

Singh, 1994)) and to disorder-enhanced subgap conductance 

(Badolato et al., 2000; Bakker et al., 1994; Giazotto 

et al., 2001a, 2003a, 2001b; Kastalsky et al., 1991; 

Kutchinsky et al., 1997; Magnée et al., 1994; Nguyen 

et al., 1992; Nitta et al., 1994; Poirier et al., 1997; van 

Wees et al., 1992; Xiong et al., 1993). Further results 

on the electron-phonon coupling constant as well as on 

the electronic thermal conductivity in n-type Si were reported 

in (Heslinga and Klapwijk, 1992; Kivinen et al., 

2003). 

Because of the low electron-phonon coupling constant 

in heavily doped Si, one may expect to cool islands with 

larger volumes than in the normal metal case, and perhaps 

improve the high-temperature limit of cooling by 

some factor. The issue of high Rc values that limit 

the maximum achievable cooling power still represents 

a drawback of these refrigerators that has to be overcome. 

High doping of the semiconductor (in the tunneling 

regime Rc ∝ exp(N −1/2 

D ) (Sze, 1981, 1985), see 

also Sec. VI.F.2) as well as engineering of the Schottky 

barrier height through incorporation of interface layers 

at the metal-semiconductor junction (Cantile et al., 

1994a,b; Costa et al., 1991; De Franceschi et al., 1998a,b, 

2000; Grant and Waldrop, 1988; Koyanagi et al., 1993; 

Marinelli et al., 2000) can be exploited in order to tailor 

the interface transmissivity. 

4. SF systems 

39 

As discussed at the beginning of Sec. V.C.1, decreasing 

the contact resistance (RT ) is not a viable route to enhance 

heat current in NIS junctions. A possible scheme 

to surmount the problem of Andreev reflection at the 

metal-superconductor interface is to exploit, instead of 

an insulating barrier, a thin ferromagnetic (F) layer in 

good electric contact with S (Giazotto et al., 2002). The 

physical origin of the SF cooler operation stems from the 

spin-band splitting characteristic of a ferromagnet. The 

electron involved in the Andreev reflection and its phasematched 

hole belong to opposite spin bands; as a consequence, 

depending on the degree of spin polarization P 

of the F layer, strong suppression of the Andreev current 

may occur at the SF interface (de Jong and Beenakker, 

1995). In the limit of large P, the subgap current is 

thus strongly suppressed while efficient hot-carrier transfer 

leads to a sizeable heat current across the system. In 

the following we give the main results of such a proposal. 

The impact of partial spin polarization (P < 100%) in 

the F region is displayed in Fig. 31(a) where the heat current 

˙ Q versus bias voltage across the junction is plotted 

for some values of P at T = Te,F = Te,S = 0.4 ∆(0)/kB 

(Te,F is the electron temperature in F). For each P 

value there exists an optimum bias voltage (Vopt) which 

maximizes ˙ Q. In the limit of a half-metal ferromagnet 

(i.e., P = 100%) (Coey and Sanvito, 2004; Mazin, 

1999) this value is around Vopt ∆/e. Moreover, for 

P = 94% there is still a positive ˙ Q across the junction. 

The inset of Fig. 31 (a) shows ˙ Q calculated at 

each optimized bias voltage against temperature for various 

values of P. The heat current is maximized around 

T = Topt ≈ 0.4 ∆(0)/kB, rapidly decreasing both at 

higher and lower temperatures. 

The junction specific cooling power is shown in Fig. 

31(b), where ˙ QA (evaluated at each optimal bias voltage) 

is plotted versus P at different bath temperatures. 

Notably, for a half-metallic ferromagnet at T = 

0.4 ∆(0)/kB, cooling power surface densities as high as 

600 nW/µm 2 can be achieved, i.e., about a factor of 30 

larger than those achievable in NIS junctions at the optimized 

interface transmissivity (i.e., T 3 × 10 −2 at 

T 0.3∆(0)/kB, see Fig. 4(b)). This marked difference 

stems from SF specific normal-state contact resistances 

as low as some 10 −3 Ωµm 2 that are currently achieved 

in highly-transmissive junctions (Upadhyay et al., 1998). 

The inset of Fig. 31(b) shows the junction coefficient of 

performance (η) calculated at the optimal bias voltage 

versus temperature for various P values. Notably, for 

P = 100%, η reaches ∼ 23% around T = 0.3 ∆(0)/kB 

and exceeds 10% for P = 96%. In addition, in light of 

a possible exploitation of this structure in combination 

with a N region (i.e., a SFN refrigerator) it turned out 

that cooling effects comparable to the SF case can be 

achieved with a F-layer thickness of a few nm (i.e., of the 

order of the magnetic lenght). We note that the large 

˙QA typical of the SF combination makes it promising as

e 2 R /Δ T 2 (0)(×10 -2 Q&Q& 

) 

6 

k T/Δ(0) = 0.4 

B 

3 

0 

6 

-3 

3 

= 1 

0.94 

¡ = 1 

0.98 

0.96 

0.96 

0.98 

(a) 

Q& (nW/μm 

A 

2 )(×10 2 ) 

-6 

0 

0.0 

-9 

0.0 

0.94 

0.4 

kBT/Δ(0) 0.5 

0.8 

1.0 1.5 

eV/Δ(0) 

8 

7 

6 

5 

4 

3 

k T/ Δ (0)=0.4 

2 B 

1 

0 

¢ = 1 

0.98 

0.96 

0.94 

25 

20 

15 

10 

5 

0.0 0.4 0.8 0 

k T/Δ(0) 

B 

0.3 

0.2 

0.15 

0.94 0.96 0.98 1.00 

FIG. 31 (Color in online edition) (a) Calculated heat current 

˙ Q of a SF junction vs bias voltage V at T = Te,F = 

Te,S = 0.4 ∆(0)/kB for several spin polarizations P. The inset 

shows the same quantity calculated at the optimal bias 

voltage against temperature for some values of P. (b) Calculated 

maximum cooling power surface density ˙ QA versus 

P for various temperatures. The inset shows the coefficient 

of performance η calculated at the optimal bias voltage versus 

temperature for some P values. Adapted from (Giazotto 

et al., 2002, 2005). 

a possible higher-temperature first stage in cascade cooling 

(for instance, over or around 1 K), where it could 

dominate over the large thermal coupling to the lattice 

characteristic for such temperatures. 

The results given above point out the necessity of 

strongly spin-polarized ferromagnets for a proper operation 

of the SF refrigerator. Among the predicted 

half-metallic candidates it is possible to indicate CrO2 

(Brener et al., 2000; Kämper et al., 1987; Schwartz, 

1986), for which values of P in the range 85...100% have 

been reported (Coey et al., 1998; Dedkov et al., 2002; Ji 

et al., 2001; Parker et al., 2002), (Co1−xFex)S2 (Mazin, 

2000), NiMnSb (de Groot et al., 1983), Sr2FeMoO6 

(Kobayashi et al., 1998) and NiMnV2 (Weht and Pickett, 

1999). So far no experimental realizations of SF structures 

for cooling applications have been reported. 

5. HTc NIS systems 

Heat transport in high-critical temperature (HTc) NIS 

junctions was theoretically addressed by Devyatov et al. 

(2000). In these systems the cooling power depends on 

interface transmissivity as well as on orientation of the 

superconductor crystal axes and temperature. In particular, 

these authors showed that the maximum positive 

heat current in these structures can be achieved 

in junctions with zero superconducting crystallographic 

angle, at temperature T = 0.45 Tc and bias voltage 

V ≈ 0.8 ∆(0)/e. The behavior of ˙ Q(V ) turns out to 

be qualitatively similar to that of NIS junctions based 

on low-critical temperature superconductors (see Fig. 

26(a)), and with comparable values (in relative units). 

P 

η (%) 

(b) 

40 

From this it follows that the cooling power of electronic 

refrigerators based on HTc materials is approximately 

two orders of magnitude larger than in NIS junctions 

based on low-critical temperature superconductors (at 

much lower temperatures). 

A somewhat different cooling effect in HTc superconductors 

was predicted by Svidzinsky (2002), who showed 

that an adiabatic increase of the supercurrent in a ring 

(or cylinder) made from a HTc superconductor may lead 

to a cooling effect. The maximum cooling occurs if the 

supercurrent is equal to its critical value. For a clean 

HTc superconductor, the minimum achievable temperature 

(Tmin) was found to be around Tmin = T 2 0 /Tc, with 

T0 the initial temperature of the ring, thus meaning that 

substantial cooling can be achieved using large Tc values. 

Experimentally, Fee (1993) realized a Peltier refrigerator 

junction exploiting a HTc superconductor and operating 

around liquid nitrogen temperatures. In particular, 

his device consisted of a BiSb alloy and YBa2Cu3O7−δ 

superconducting rods connected by a small copper plate 

which acted as the cold junction of the device. The latter 

showed a maximum cooling of 5.35 K below the bath 

temperature T0 = 79 K. The figure of merit of the junction, 

Z, was estimated to be as large as 2.0 × 10 −3 K −1 . 

6. Application of (SI)NIS structures to lattice refrigeration 

One well established application of SINIS refrigerators 

concerns lattice cooling (Clark et al., 2005; Luukanen 

et al., 2000; Manninen et al., 1997). As a matter 

of fact, while NIS tunneling directly cools the electron 

gas of the normal electrode, the phonon system can be 

refrigerated through the electron-phonon coupling (see 

Eq. (24)). The latter, however, is typically very small at 

the lowest temperatures, thus limiting the heat transfer 

from the surroundings to the electrons. This situation 

normally happens whenever the metal to be cooled is in 

direct contact with a thick substrate, for instance, oxidized 

Si (Leivo et al., 1996; Nahum et al., 1994): only the 

electrons of the N region cool down while the metal lattice 

presumably remains at the substrate temperature. The 

metal lattice can be refrigerated considerably if the thermal 

resistance between the phonons and the substrate 

is not negligible compared to that between the electrons 

and the phonons. One effective choice to meet this requirement, 

that was indeed suggested at the beginning 

of cooling experiments (Fisher et al., 1995; Nahum et al., 

1994) as well as for detector applications (Deiker et al., 

2004; Doriese et al., 2004; Irwin et al., 1996; Nahum and 

Martinis, 1995; Pekola et al., 2004b; Ullom et al., 2004), 

is to exploit a thermally isolated thin dielectric membrane 

on which the N region of the cooler is extended. 

In this way, tunneling through the NIS junction will 

cool down the electrons of the metal, then the phonons 

of the metal (via electron-phonon coupling) that subsequently 

will refrigerate the membrane phonons (Clark 

et al., 2005; Luukanen et al., 2000; Manninen et al., 1997)

(a) 

(b) 

T min / T 0 

1.05 

1.00 

0.95 

0.90 

thermometer 

SINIS junctions 

Electron temperature at cooling junctions 

Electron temperature on the membrane 

Lattice temperature on the membrane 

200 300 400 500 600 

T 0 (mK) 

SINIS junctions 

cold fingers 

(d) 

(c) 

SIN junctions 

cold finger 

FIG. 32 (Color in online edition) (a) SEM image of a Si3N4 

membrane (in the center) with self-suspended bridges. Two 

normal-metal cold fingers extending onto the membrane are 

used to cool down the dielectric platform. The Al/Al2O3/Cu 

SINIS coolers are on the bulk (far down and top) and the 

thermometer stands in the middle of the membrane. (b) Maximum 

temperature decrease (Tmin/T0) versus bath temperature 

T0 measured at different positions in the cooler shown in 

(a). (c) Maximum lattice temperature decrease on the membrane 

versus bath temperature measured in two other samples 

similar to that shown in (a). In this case, the refrigerators exploited 

many small-area junctions arranged in parallel in a 

comb-like configuration. (d) SEM micrograph of a NIS refrigerator 

device with a neutron transmutation doped (NTD) Ge 

resistance thermometer attached on top of it. (c) is adapted 

from (Luukanen et al., 2000), (d) from (Clark et al., 2005). 

according to 

˙QSINIS(V ; Te,N , Te,S) + ˙ Qph−sub(Tph, Tsub) = 0, (81) 

where Te,N Tph, Te,S Tsub, Tph is the lattice temperature 

in the dielectric membrane, Tsub is the lattice 

temperature in the substrate, and ˙ Qph−sub is the rate 

of exchanged energy between the membrane phonons 

and substrate phonons. Eventually, if additional devices 

are standing on the same dielectric platform (for 

instance, detectors, etc.), the latter will cool down first 

the phonons of the device and then its electrons through 

the electron-phonon interaction. 

Dielectric membranes made of silicon nitride (Si3N4) 

have proved to be attractive for this purpose in light of 

their superior thermal isolation properties at low temperatures. 

Low-temperature heat transport characterization 

as well as thermal relaxation in low-stress Si3N4 membranes 

and films were quite recently addressed (Holmes 

et al., 1998; Leivo and Pekola, 1998). The first demonstration 

reported of lattice cooling (Manninen et al., 

1997) exploited such membranes in combination with 

Al/Al2O3/Cu SINIS refrigerators. In this experiment the 

authors were able to achieve a 2% temperature decrease 

in the membrane at bath temperatures T0 ≈ 200 mK. 

41 

Figure 32(a) shows a SEM image of a typical newgeneration 

lattice cooler fabricated on a Si3N4 membrane 

with self-suspended bridges. The membrane consists of 

a low-stress Si3N4 film deposited by low pressure chemical 

vapor deposition (LPCVD) on Si, and subsequently 

etched (with both wet and dry etching) in order to create 

the suspended bridge structure. In such thin membranes 

phonon propagation is essentially two-dimensional. The 

condensation of the phonon gas into lower dimensions 

in ultrathin membranes was also theoretically discussed 

(Anghel and Manninen, 1999; Anghel et al., 1998; Kuhn 

et al., 2004). The self-suspended bridges improve thermal 

isolation of the dielectric platform from the heat 

bath (Leivo and Pekola, 1998). The image also shows 

the Al/Al2O3/Cu refrigerators of the SINIS type that 

are placed on the bulk substrate (i.e., outside the membrane) 

to ensure good thermal contact with the bath. 

The N cold fingers extend onto the silicon nitride membrane, 

whose temperature is determined through an additional 

SINIS thermometer placed in the middle of the 

structure. 

The lattice refrigeration effect achieved in this SINIS 

refrigerator is shown in Fig. 32(b). Here the maximum 

temperature decrease of the membrane (Tmin/T0) against 

bath temperature T0 is displayed (red circles), and shows 

that temperature reduction as high as about 12% was 

reached in the 400...500 mK range. The electron refrigeration 

effect in the Cu region was also measured at two 

different positions in the device, i.e., nearby the cooling 

junctions and on the membrane. Notably, the Tmin/T0 

behavior is almost the same for the different sets of data; 

this basically means that: a) good thermalization was 

achieved in the cold fingers; b) the electron-phonon coupling 

was sufficiently large while Kapitza resistance between 

Cu and Si3N4 was sufficiently small to ensure the 

lattice temperature on the membrane to be nearly equal 

to the Cu electron temperature on the membrane itself. 

The best results of lattice refrigeration by SINIS coolers 

reported to date are shown in Fig. 32(c) (Luukanen 

et al., 2000) for two other devices (labeled C and D in the 

figure) similar to that of Fig. 32(a). These devices exploited 

three Cu cold fingers and several small-area junctions 

arranged in parallel in a comb-like configuration for 

the SINIS cooling stage. The junction specific resistances 

were Rc = 1.39 kΩµm 2 and Rc = 220 Ωµm 2 for device 

C and D, respectively. Lattice temperature reduction as 

high as 50% at 200 mK was achieved in the sample with 

lower Rc, thus confirming the effectiveness of small-area 

junctions in yielding larger temperature reductions. The 

achieved cooling power in these devices was estimated 

on the pW level. The reduction of the refrigeration effect 

at the lowest temperatures can be explained in terms 

of larger decoupling of electrons and phonons, but also 

the effects discussed in Sec. V.C.1 should play a role. 

Figure 32 (d) demonstrates the realization of a complete 

refrigerator device including a thermometer (Clark 

et al., 2005), where four pairs of NIS junctions are used 

to cool down a 450 × 450 µm 2 suspended Si3N4 dielectric

membrane. Each NIS junction area is 25 × 15 µm 2 , and 

the N electrode consists of Al doped with Mn to suppress 

superconductivity while Al is used for the superconducting 

reservoirs. Also shown is a neutron transmutation 

doped (NTD) Ge resistance thermometer (with volume 

250 × 250 × 250 µm 3 ) glued on the membrane. In such a 

refrigerator the authors measured with the NTD Ge sensor 

a minimum final temperature slightly below 240 mK 

starting from a bath temperature T0 = 320 mK, under 

optimal bias voltage across the cooling junctions. This 

result is promising in light of a realistic implementation 

of superconducting refrigerators, and shows the possibility 

of cooling efficiently the whole content of dielectric 

membranes through NIS junctions (Pekola, 2005). 

Possible strategies to attain enhanced lattice refrigeration 

performance in the low temperature regime (i.e., 

below 500 mK) could be a careful optimization in terms 

of the number and specific resistance of the cooling junctions 

as well as to exploit superconductors with the gap 

larger than in Al. Making the dielectric platform thinner 

and reducing thermal conduction along it should 

also increase the temperature drop across the membrane. 

The exploitation of the described method around or 

above 1 K, however, is still now not so straightforward, 

mainly due to the strong temperature dependence of the 

electron-phonon interaction that thermally shunts more 

effectively the N electrode portion standing on the bulk 

substrate to the lattice (note that also the thermal conduction 

along the membrane is strongly temperature dependent 

(Kuhn et al., 2004; Leivo and Pekola, 1998)). 

Toward this end, a reduction of the N volume placed out 

of the membrane should help; in addition, S1IS2(IS1) refrigerators 

(see Sec. V.C.2) as well as SF junctions (see 

Sec. V.C.4) might significantly improve the membrane 

cooling in the higher temperature regime. 

7. Josephson transistors 

A further interesting field of application of SINIS 

structures concerns superconducting weak links (Golubov 

et al., 2004; Likharev, 1979). In particular, in 

diffusive SNS junctions, i.e., where the junction length 

largely exceeds the elastic mean free path, coherent sequential 

Andreev scattering between the superconducting 

electrodes may give rise to a continuum spectrum of 

resonant levels (Belzig et al., 1999; Heikkilä et al., 2002; 

Yip, 1998) responsible for carrying the Josephson current 

across the structure. The supercurrent turns out to 

be given by this spectrum weighted by the occupation 

number of correlated electron-hole pairs, the latter being 

determined by the quasiparticle energy distribution 

in the N region of the weak link. In controllable Josephson 

junctions, the supercurrent is modulated by driving 

the quasiparticle distribution out of equilibrium (Heikkilä 

et al., 2002; Seviour and Volkov, 2000a; Volkov, 1995; 

Volkov and Pavlovskii, 1996; Volkov and Takayanagi, 

1997; van Wees et al., 1991; Wilhelm et al., 1998; Yip, 

(a) 

(c) 

S 

V SINIS 

L J 

S J 

N 

R T 

S J 

R T 

S 

L SINIS 

I J (μA) 

eI J R N /E Th 

5 

(b) 

4 T0 = 0.1 K 

3 

0.2 K 

2 

0.3 K 

1 0.4 K 

0.5 K 

0.6 K 

eI J R N /E Th 

42 

0 

0 1 2 

T (K) N 

0 

0 1 2 

eV /Δ SINIS Al 

3 4 

(d) 

0.20 

VSINIS (mV) 

0.2 0.3 0.4 

0.15 

0.10 

GI 

5 

4 

3 

2 

1 

20 

10 

0 

T 0 = 0 

-10 

1E-3 0.01 0.1 

ISINIS (μA) 

0.05 T0 = 72 mK 

214 mK 

0.00 

283 mK 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 

VSINIS (mV) 

FIG. 33 (Color in online edition) (a) Simplified scheme of a 

SINIS-controlled Josephson transistor. The Josephson current 

in the SJNSJ weak link (along the white dashed line) 

is controlled by applying a bias VSINIS across the SINIS line 

connected to the weak link, allowing to increase or decrease 

its magnitude with respect to equilibrium. (b) Theoretical 

normalized critical current IJ versus VSINIS calculated in the 

quasiequilibrium limit for several lattice temperatures T0 for 

a Nb/Cu/Nb long junction. The inset shows the supercurrent 

vs electron temperature characteristic calculated at φ = π/2. 

(c) SEM image of an Al/Cu/Al Josephson junction including 

the Al/Al2O3/Cu symmetric SINIS electron cooler. The 

SNS long weak link is placed in the middle of the structure. 

Also shown is a scheme of the measurement circuit. (d) Measured 

critical current IJ versus VSINIS at three different bath 

temperatures T0 for the device shown in (c). The inset displays 

the measured differential current gain GI versus ISINIS 

at T0 = 72 mK. (b) is adapted from (Giazotto et al., 2003b), 

while (c) and (d) from (Savin et al., 2004). 

1998) via dissipative current injection in the weak link 

from additional normal-metal terminals. This operation 

principle leads to a controlled supercurrent suppression 

and was successfully exploited both in all-metal (Baselmans 

et al., 2002a, 1999, 2001a,b, 2002b; Huang et al., 

2002; Morpurgo et al., 1998; Shaikhaidarov et al., 2000) 

(where a transition to a π-state was also reported) as well 

as in hybrid semiconductor-superconductor weak links 

(Kutchinsky et al., 1999; Neurohr et al., 1999; Richter, 

2000; Schäpers et al., 2003a, 1998, 2003b). The situation 

drastically changes if we allow current injection from 

additional superconducting terminals arranged in a SI- 

NIS fashion (Baselmans, 2002; Giazotto et al., 2004a,b, 

2003b). In this way, thanks to the SINIS junctions, critical 

supercurrent can be strongly increased as well as 

steeply suppressed with respect to equilibrium, leading 

to a tunable structure with large current and power gain. 

A simplified scheme of this class of transistors is displayed 

in Fig. 33(a). A diffusive SJNSJ long Josephson 

junction of length LJ (i.e., with LJ ≫ ξ0, where ξ0 is the 

SJ coherence length) shares the N region of a SINIS con-

trol line of length LSINIS. The superconductors SJ and 

S can be in general different (i.e., with gaps ∆J and ∆S, 

respectively); in addition, the SJ electrodes are kept at 

zero potential, while the SINIS line is biased with VSINIS. 

The supercurrent (IJ) behavior in response to a bias voltage 

VSINIS stems from the degree of nonequilibrium the 

SINIS line is able to generate in the weak link according 

to (Heikkilä et al., 2002; Wilhelm et al., 1998; Yip, 1998) 

∞ 

IJ(VSINIS) = 1 

dEjS(E, φ)(1 − 2f(E, VSINIS)), 

eRN 0 

(82) 

where RN is the weak link normal-state resistance, φ 

is the phase difference across the superconductors, and 

jS(E, φ) is the spectral supercurrent, obtainable from 

the solution of the Usadel equations (Usadel, 1970). The 

most straightforward situation occurs at sufficiently low 

lattice temperatures, i.e., typically below 1 K, when 

ℓe−e < LSINIS < ℓe−ph (see Sec. II). In such a case, strong 

electron-electron relaxation forces the electron system 

in the N region to retain a local thermal quasiequilibrium 

(see also Sec. V.C.1), so that 1 − 2f(E, VSINIS) = 

tanh[E/(2kBTe,N (VSINIS))]. The transistor effect in the 

structure thus simply depends on the electron temperature 

Te,N (VSINIS) established in the weak link upon biasing 

the SINIS line according to Eq. (80). The full 

calculation of the behavior of a prototype Nb/Cu/Nb 

long Josephson junction with integrated Al/Al2O3/Cu 

SINIS electron cooler is displayed in Fig. 33(b) (Giazotto 

et al., 2003b). Here the normalized IJ is plotted against 

VSINIS for several bath temperatures T0. For all chosen 

values of T0, the supercurrent value increases monotonically 

up to about VSINIS ≈ 1.8 ∆S/e, where the SINIS 

line provides the largest cooling power and allows to attain 

the lowest electron temperature. Then, by further 

increasing the voltage, an efficient suppression of IJ occurs 

due to electron heating: the device behaves as a tunable 

superconducting junction. The above given results 

can be easily understood recalling that for a long SNS 

junction at low lattice temperature (i.e., kBT0 ≪ ∆J) 

and for kBTe,N ≫ ET h = D/L2 J (D is the N-region 

diffusion coefficient), IJ depends exponentially on the effective 

electron temperature Te,N (Wilhelm et al., 1997; 

Zaikin and Zharkov, 1981) and is almost independent of 

T0. Hence, long junctions are more appropriate for the 

device to obtain large IJ changes with small Te,N variations 

(as indicated by the red hatched region in the inset 

of Fig. 33(b)). In the short junction limit (LJ ≪ ξ0), 

conversely, the IJ temperature dependence is set by the 

energy gap ∆J, thus implying a much reduced effect from 

the cooling line. Furthermore, it is expected that the 

power dissipation values from two to four order of magnitude 

smaller than with all-normal control lines can be 

obtained in these structures, thus making them promising 

as mesoscopic transistors for low dissipation cryogenic 

applications. 

So far, the only successful demonstration of this 

transistor-like operation was reported by (Savin et al., 

2004) in Al/Cu/Al SNS junctions integrated with 

43 

Al/Al2O3/Cu SINIS refrigerators. The SEM image of 

one of these samples along with a scheme of the measurement 

setup is shown in Fig. 33(c). The structure parameters 

were the following: cooler junction resistances 

RT 240 Ω, Josephson weak link normal state resistance 

RN = 11.5 Ω, and minimum SNS interelectrode 

separation LJ 0.4 µm that compared with the superconducting 

coherence length (ξ0 ≈ 62 nm) provides 

the frame of the long junction regime. The critical current 

IJ versus VSINIS at three different T0 is shown in 

Fig. 33(d). For each bath temperature, IJ increases 

around VSINIS ≈ 1.8∆S/e, as expected from the reduction 

of Te,N by electron cooling, while it is steeply suppressed 

at larger bias voltages. The resemblance of the 

experiment with the curves of Fig. 33(b) is evident. In 

the present case, IJ enhancement under hot quasiparticle 

extraction by more than a factor of two was observed 

at T0 = 283 mK. The transistor current gain 

GI = dIJ/dISINIS, shown in the inset of Fig. 33(d), 

obtained values in the range −11...20 depending on the 

control bias. As far as power dissipation is concerned, 

these authors reported low dissipated power on the 10 −13 

W level in the extraction regime while of some tens of pW 

in the regime of IJ suppression. 

The transistor behavior for arbitrary inelastic scattering 

strength in the SINIS line (or, equivalently, for 

arbitrary values of LSINIS) as well as for different SNS 

junction lengths, was theoretically addressed in detail in 

(Giazotto et al., 2004a,b). The role of geometry, materials 

combination, phase dependence, and the input noise 

power were also discussed. Notably, a marked supercurrent 

transition to a π-state under nonequilibrium (about 

two times larger than that achievable with an all-normal 

control channel (Wilhelm et al., 1998; Yip, 1998)) was 

predicted to occur for negligible or moderate electronelectron 

interaction. We recall that in the π-state (Bulaevskii 

et al., 1977) the supercurrent flows in opposite direction 

with respect to the phase difference φ between the 

two superconductors, i.e., such a sign reversal is equivalent 

to the addition of a phase factor π to the Josephson 

current-phase relation. Furthermore, current gain in the 

range 10 2 ...10 5 and power gain up to 10 3 where predicted 

to occur depending on the control voltage. 

Finally, transistor operation in a Josephson tunnel 

junction integrated with a S1IS2IS1 refrigerator was also 

theoretically addressed in the quasiequilibrium regime 

(Giazotto and Pekola, 2005). In this case, the device 

benefits from the sharp characteristics due to the presence 

of superconductors with unequal energy gaps (see 

also V.C.2), that leads to improved overall characteristics 

as compared to the SINIS-controlled SNS junction in 

the same transport regime. 

D. Perspective types of refrigerators 

Any electric current that is accompanied by the extraction 

of hot electrons (or holes) can be used, in principle,

for refrigeration purposes. For instance, this may happen 

in thermionic transport over a potential barrier as well as 

in energy-dependent tunneling through a barrier. Since 

an exhaustive analysis of all possible predicted refrigeration 

methods is far beyond the limits set to this Review, 

in the following we give a brief description of a few examples 

that we believe are relevant in the present context. 

In such devices, several different effects may contribute 

to the refrigeration process (e.g., thermionic transport, 

quantum tunneling as well as thermoelectric effects), so 

that making a proper ”classification” of the refrigeration 

principle is, strictly speaking, rather difficult. As a consequence, 

we shall try mainly to follow the definitions as 

they were introduced in the original literature. 

1. Thermionic refrigerators 

A vacuum thermionic cooling device consists of two 

electrodes separated by a vacuum gap. Cooling occurs 

when highly energetic electrons overcome the vacuum 

barrier through thermionic emission, thus reducing 

the electron temperature of one of the two electrodes. 

In such a situation, the refrigerator operation 

is mainly affected and limited by radiative heat transfer 

between the electrodes. The thermionic emission current 

density (jR) is given by the Richardson equation, 

jR = A0T 2 exp(− Φ 

kBT 

), where Φ and T are the work func- 

tion and the temperature of the emitting electrode, respectively, 

A0 = 4πemkB/h 3 is the Richardson constant, 

and m is the electron mass. From the expression above, 

a strong reduction of jR is expected upon lowering the 

temperature. Mahan (1994) developed a simple model 

for thermionic refrigeration, and demonstrated that its 

efficiency can be as high as 80% of the Carnot value. Vacuum 

thermionic refrigerators are generally characterized 

by a higher efficiency as compared to thermoelectric coolers, 

and are considered to be an attractive solution for 

future refrigeration devices (Nolas and Goldsmid, 1999). 

However, the high Φ values typical of currently available 

materials make thermionic cooling efficient, at present, 

only above 500 K. 

Several ideas on how to increase the cathode emission 

current and to improve the operation of these refrigerators 

have been proposed (Hishinuma et al., 2001, 2002; 

Korotkov and Likharev, 1999; Purcell et al., 1994). Korotkov 

and Likharev suggested to cover the emitter with a 

thin layer of a wide-gap semiconductor, and to exploit the 

resonant emission current to cool the emitter (Korotkov 

and Likharev, 1999). The analysis of such a thermionic 

cooler predicted an efficient refrigeration down to 10 K. 

The issue of high Φ values can be overcome by a reduction 

of the distance between the electrodes (in the 

submicron range) and by the application of a strong 

electric field. Following this scheme, Hishinuma et al. 

(2001) theoretically analyzed a thermionic cooler where 

the two electrodes where separated by a distance in the 

nanometer range. In such a situation, the potential 

44 

FIG. 34 (a) Schematic diagram of the potential barrier profile 

V (x) for Φ = 1 eV and with electrode separation of 60 

˚A. (b) Heat flow distribution at T = 300 K. Adapted from 

(Hishinuma et al., 2001). 

barrier is essentially lowered (see Fig. 34(a)), allowing 

both thermionic emission and energy-dependent tunneling. 

As a consequence, rather small (if compared to vacuum 

thermionic devices) external voltages (∼ 1...3 V) are 

required in order to produce significant electric currents. 

For suitable values of the applied voltage and distances 

between the electrodes, electrons above the Fermi level 

dominate the electric transport (both thermionic emission 

over the barrier and tunneling through the barrier), 

thus leading to cooling of the emitter. As it can be inferred 

from Fig. 34(b), the contribution of the energydependent 

tunneling to total cooling is essential, and this 

refrigerator could be classified as a vacuum tunneling 

device. The cooling power surface density in this combined 

thermionic-tunneling refrigerator was predicted to 

obtain values as high as 100 W/cm 2 at room temperature. 

However, in the only experimental demonstration 

of this device, a moderate emission current (below 10 

nA) was reported at room temperature, with an observed 

temperature reduction of about 1 mK (Hishinuma et al., 

2003). So far, no experimental demonstration of vacuum 

thermionic refrigeration at cryogenic temperatures has 

been reported. 

2. Application of low-dimensional systems to electronic 

refrigeration 

The exploitation of low-dimensional systems gives additional 

degrees of freedom in order to engineer materials 

that may lead to enhanced operation of thermoelectric 

and thermionic devices (Hicks et al., 1993; Hishinuma 

et al., 2002; Sales, 2002; Sofo and Mahan, 1994). 

Some of the limitations intrinsic to vacuum thermionic 

refrigerators can be overcome with solid state thermionic 

coolers (Mahan and Woods, 1998; Shakouri and Bowers, 

1997). As a matter of fact, modern growth techniques 

easily allow one to control both the barrier height 

and its width within a wide range of values. One disadvantage 

of solid state thermionic coolers stems from 

the thermal conductivity of the barrier which is essentially 

absent in vacuum devices. Nevertheless, a large

FIG. 35 (a) Scheme of a quantum-dot refrigerator. (b) 

Energy-level diagram of the structure. The reservoir R is 

cooled as its quasiparticle distribution function is sharpened 

by resonant tunneling through quantum dots DL and DR to 

the electrodes VL and VR. From (Edwards et al., 1995). 

temperature reduction can be achieved by using a multilayered 

heterostructure (Mahan and Woods, 1998; Shakouri 

and Bowers, 1997; Zhou et al., 1999). According 

to theory (Mahan and Woods, 1998; Shakouri and Bowers, 

1997), heterostructure-based thermionic refrigerators 

perform somewhat better as compared to thermoelectric 

coolers. The predicted temperature reduction of a singlestage 

device at room temperature can be as high as 40 

K, and this value can be significantly increased in a multilayer 

configuration. So far, however, the experimental 

implementations of single barrier (Shakouri et al., 1999) 

and superlattice (Fan et al., 2001; Zhang et al., 2003) 

refrigerators have reported temperature reductions of a 

few degrees at room temperature. 

Further improvement of thermionic refrigeration can, 

in principle, be achieved by combining laser cooling 

and thermionic cooling (Mal’shukov and Chao, 

2001; Shakouri and Bowers, 1997). In such an optothermionic 

device, hot electrons and holes extracted 

through thermionic emission lose their energy by emitting 

photons rather than by heating the lattice. The 

theoretical investigation of a GaAs/AlGaAs-based optothermionic 

refrigerator predicted specific cooling power 

densities of the order of several W/cm 2 at 300 K 

(Mal’shukov and Chao, 2001). 

The presence of singularities in the energy spectrum 

of low-dimensional systems can be used in solid state refrigeration. 

For instance, the discrete energy spectrum 

in quantum dots can be exploited for refrigeration at 

cryogenic temperatures (Edwards et al., 1995, 1993). In 

such a quantum-dot refrigerator (QDR) (see Fig. 35(a)) 

a reservoir (R) is coupled to two electrodes via quantum 

dots (DL and DR) whose energy levels can be tuned 

through capacitively-coupled electrodes. The QD energy 

levels can be adjusted so that resonant tunneling to the 

electrode VL is used to deplete the states in R above µ0 

and, similarly, holes below µ0 in R tunnel to VR (see 

Fig. 35(b)). As a consequence, the net result will be to 

sharpen the quasiparticle distribution function in R, thus 

leading to electron refrigeration. In spite of a rather mod- 

45 

erate achievable cooling power, the QDR was predicted 

to be effective for cooling electrons of a micrometer-sized 

two-dimensional electron gas reservoir at mK temperatures, 

and even of a macroscopic reservoir at lower temperatures 

(Edwards et al., 1995, 1993). 

VI. DEVICE FABRICATION 

A. Structure typologies and material considerations 

This section is devoted to the description of the main 

techniques and experimental procedures used for the fabrication 

of typical superconducting electronic refrigerators 

and detectors. Owing to the great advances reached 

in the last decades in micro- and nanofabrication technology 

(Bhushan, 2004; Timp, 1999), the amount of information 

related to fabrication methods is too large to 

be covered here and beyond the scope of the present review. 

Therefore, we only briefly highlight all those issues 

that we believe are strictly relevant for this research field. 

In particular we first of all focus on the two typologies 

of existing superconducting structures, namely all-metal 

and hybrid devices, and on the main differences between 

them, both in terms of materials and fabrication techniques. 

The former concern structures where the active 

parts of the device, i.e., both the superconducting elements 

and the normal regions are made of metals; in 

hybrid structures, the non superconducting active part 

of the device is made of doped semiconducting layers. 

As far as all-metal-based structures are concerned, 

they are realized with low-critical-temperature thin-film 

superconductors (normally Al, Nb, Ti and Mo), while the 

normal regions usually consist of Cu, Ag or Au. Their 

fabrication protocol includes patterning of a suitable 

radiation-sensitive masking layer through electron-beam 

or optical lithography in combination with a shadowmask 

(angle) evaporation technique (Dolan, 1977). The 

final device is thus realized in a single step in the deposition 

chamber, where additional tunnel barriers between 

different regions of the structure are in-situ created 

by suitable oxidation of the metallic layers. Although 

all this leads to an efficient way for fabricating metallic 

structures, however the electronic properties typical of 

metals are only weakly dependent on their growth conditions 

and on the specific employed technique. As a 

consequence, it is hard to tailor the metallic properties 

in order to finally match some specific requirements. 

On the other hand, the situation is rather different 

with semiconductors that offer some advantages in comparison 

to metals. The large magnitude difference of 

Fermi wave-vector between metals and semiconductors 

allows in general to observe quantum effects in structures 

much bigger than with metals. In addition, the 

availability of several techniques for growing high-purity 

crystals (e.g., molecular beam epitaxy) yields the capacity 

to tailor the semiconductor electronic properties and, 

at the same time, enables the fabrication of structures

characterized by coherent transport especially in reduced 

dimensionality (Capasso, 1990). The exploitation of lowdimensional 

electron systems as active elements of electronic 

coolers was predicted to be the next possible breakthrough 

in this research field (Edwards et al., 1995, 1993; 

Hicks and Dresselhaus, 1993a,b; Hicks et al., 1993, 1996; 

Koga et al., 2000, 1998, 1999). Moreover, the realization 

of structures where charge carriers experience arbitrarily 

chosen effective potentials is a further advantage of modern 

engineered heterostructures (Yu and Cardona, 2001). 

Last but not least, the possibility of changing the carrier 

density through electrostatic gating allows to strongly alter 

some semiconductor parameters (like the mobility as 

well as the coherence length), thus enabling to have access 

to different electronic transport regimes in the same 

structure (Ferry and Goodnick, 1997). 

Like all-metal devices, hybrid structures generally exploit 

the same superconducting materials. However, only 

heavily-doped Si layers in combination with superconductors 

were exploited up to now for the realization of superconducting 

refrigerators (Buonomo et al., 2003; Savin 

et al., 2003, 2001). The fabrication procedure of hybrid 

structures involves typically two lithographic steps, 

where the first allows to pattern the semiconductor active 

layer through various techniques (such as wet or dry 

etching). The second one the patterning of the superconducting 

electrodes. Differently from all-metal devices, at 

the metal-semiconductor interface forms the well-known 

Schottky barrier (Lüth, 2001). The latter can be tailored 

through suitable doping of the semiconductor, allowing 

to control the interface transmissivity (i.e., the junction 

specific resistance) over several orders of magnitude. 

Both all-metal and hybrid structures are normally 

fabricated on semi-insulating semiconductor substrates 

which provide both the mechanical support and rigidity 

as well as the thermalization of the device at the bath 

temperature. It is noteworthy to mention that undoped 

thermally-oxidized Si substrates are widely used toward 

this end for metallic structures, while hybrid coolers typically 

exploit the silicon-on-insulator (SOI) technique in 

order to provide the Si active layer. 

B. Semiconductor growth techniques 

Depending on the physical principle exploited, semiconductor 

growth technologies can be either physical 

or chemical, such as gas and liquid-phase chemical 

processes. In the following we give a brief survey of the 

most common semiconductor growth methods. 

The Czochralski (1918) crystal pulling method is probably 

the most common technique for growing semiconductor 

bulk single crystals and the largest amount of 

Si used in the semiconductor industry is obtained using 

this technique. This technique allows to grow large semiconductor 

single crystals. However, it is often sufficient 

and less demanding from an economical point of view, 

to grow a thin ( ∼ 1 µm) perfect crystal layer on top of 

46 

a bulk crystal of lower quality. This is accomplished by 

growing epitaxially the top layer, i.e., the atoms forming 

the latter build a crystal with the same crystallographic 

structure and orientation as the starting substrate (in 

contrast, non-epitaxial layers can be amorphous or polycrystalline). 

Gas-phase epitaxy is a widespread technique and represents 

nowadays the most important process for the industrial 

fabrication of Si and GaAs devices. In chemical 

vapour deposition (CVD) (Adams, 1983; Grove, 1967; 

Sze, 1985), the constituents of the vapor phase chemically 

react at the substrate surface and the product of the reaction 

is a film deposited on the substrate. Nowadays, CVD 

is widely employed for growing several elemental semiconductors 

and alloys and, depending on the specific application, 

it has been implemented and adapted in a number 

of different configurations (Jensen, 1989). Among 

these we mention 1) metalorganic CVD (MOCVD), exploits 

a thermally heated reactor and organometallic gas 

sources; 2) plasma-enhanced CVD (PECVD), exploits a 

plasma discharge to provide the energy for the reactions 

to occur; 3) atmospheric pressure CVD (APCVD), does 

not require a vacuum environment. 

While typical CVD processes are carried out in low 

vacuum (i.e., in the pressure range from 10 −1 to several 

Torr), molecular beam epitaxy (MBE) (Arthur, 1968; 

Chang and Ludeke, 1975; Cho, 1970; Cho and Arthur, 

1975) is an ultrahigh vacuum (UHV) growth technique, 

performed at pressures usually lower than 10 −10 Torr. In 

such vacuum conditions the amount of residual gas contaminants 

present in the growth chamber is minimized, 

thus allowing to deposit epitaxial layers of high purity 

and quality. The main features of this method are a precise 

control of both chemical and stoichiometric composition, 

and a perfect tuning of doping profiles on the scale 

of a single atomic layer (Cho, 1979). The film growth 

concept is fundamentally an evolution of UHV evaporation, 

where thermal or electron-beam sources are used 

to create a flux of molecular species. Significant MBE 

work has been achieved with Si and Ge as well as with 

IV-IV and II-VI semiconductor compounds and several 

metals, but probably the largest amount of research has 

been devoted to III-V semiconductor alloys. 

C. Thin-film metals deposition methods 

1. Thermal evaporation 

Vacuum evaporation is one of the oldest and simplest 

thin film deposition techniques (Holland, 1958; Maissel 

and Glang, 1970). It is a physical vapor deposition 

(PVD) method widely used to deposit a variety of materials, 

from elemental metals to alloys and insulators. 

Evaporation is based on the boiling off or sublimation of 

a heated source material onto a substrate surface. The 

result of the evaporant condensation is the final film. 

The rate of atoms or molecules (Ne) lost under evap-

oration from a source material per unit area per unit 

time can be expressed by the Hertz-Knudsen relation 

Ne = a[P ∗ V (T ) − P ][2πMkBT ] −1/2 , where a is the evaporation 

coefficient (a = 1 for a clean evaporant surface), P 

is the ambient hydrostatic pressure acting on the evaporant 

in the condensed phase, P ∗ V (T ) is the equilibrium 

vapor pressure of the evaporant, and M its molecular 

weight. This expression shows that the evaporation rate 

strongly depends on the evaporant vapor pressure. Most 

common metals typically deposited by thermal evaporation 

(such as Al, Au, Ga, and In) usually have vapor 

pressures in the range between 10−2 to 1 Torr in the temperature 

window of 600◦ ...2000◦ C; conversely refractory 

metals (such as Nb, Mo, Ta, W and Pt) or ceramics (such 

as BN, and Al2O3) reach such vapor pressures at much 

higher temperatures, thus making more difficult the exploitation 

of this technique for the deposition. 

Usually evaporation is performed in high or ultrahigh 

vacuum (in the range 10−5 ...10−10 Torr), where the mean 

free path ℓ for the evaporant species is much larger than 

the substrate-source distance. This translates in an almost 

line of sight evaporation which prevents covering of 

edges perpendicular to the source, the latter also referred 

to as the lack of step coverage (Madou, 1997) (note that 

this property is at the basis of lift-off processes (Moreau, 

1988) as well as of angle (shadow) evaporation technique 

using suspended masks (Dolan, 1977; Dolan and Dunsmuir, 

1988)). Furthermore, vacuum evaporation is a 

low-energy process (implying a negligible damage to the 

substrate surface), where the typical energy of the evaporant 

material impinging on the substrate is of the order 

of 0.1 eV. Nevertheless, radiative heating can be high. 

Resistive heating and electron-beam deposition are the 

two most common methods of evaporation. The former 

relies on direct thermal heating to evaporate the 

source material. This method is fairly simple, robust and 

economic but suffers from a limited maximum achievable 

temperature (of the order of 1800◦ C), which prevents 

the evaporation of refractory metals and several 

oxides. On the other side, electron-beam evaporation 

represents a crucial improvement over resistive heating. 

This method exploits a high-energy electron beam that is 

focused through a magnetic field on a localized region of 

the source material. A wide range of materials (including 

refractory metals and a wide choice of oxides) can 

be deposited owing to the generation of high temperatures 

(in excess of 3000◦ C) over a restricted area. Its 

main drawback relies on the generation of X-rays from 

the high-voltage electron beam which may damage sensitive 

substrates (such as semiconductors) (Moreau, 1988; 

Sze, 1985). Achievable deposition rates are up to several 

hundreds ˚A/sec (e.g., 0.5 µm/min for Al) (Madou, 1997). 

2. Sputter deposition 

The sputtering process has been known and used for 

over 150 years (Chapman and Mangano, 1988; Chapman, 

47 

1980; Rossnagel, 1998; Wasa and Hayakawa, 1992). It 

is a PVD method widely used nowadays for many applications, 

both in the electronic and mechanic industry 

fields as well as in the pure research environment. This 

process is based on the removal of material from a solid 

target through its bombardment caused by incident positive 

ions emitted from a (rare) gas glow discharge. The 

transferred momentum of the ions leads to the expulsion 

of atoms from the target material, thereby enabling 

the deposition (condensation) of a film on the substrate 

surface. Sputter deposition is generally performed at energies 

in the range of 0.4 to 3 keV. Furthermore, the 

average energy of emitted ions from the target source 

is in the range 10...100 eV. At these energies bombarding 

ions can penetrate up to two atomic layers in the 

substrate thus leading to a great improvement of the adhesion 

of the sputtered film (Maissel and Glang, 1970; 

Wasa and Hayakawa, 1992). Normally, relatively high 

pressures (from 10 −4 to 10 −1 Torr) are maintained in 

the growth chamber during deposition. At these pressures 

the mean free path is short (of the order of 1 mm 

at 10 −1 Torr) so that the material atoms reach the substrate 

surface with random incident angles. As a consequence, 

a very good step coverage can be achieved. Being 

essentially mechanical in nature, sputtering successfully 

allows the deposition of refractory metals (superconductors) 

like Nb, NbN, Ta, Mo and W at temperatures well 

below their melting points. 

D. Thin film insulators 

The roles of thin film insulators in solid state electronics 

are various. In particular, deposited films are often 

used as interlevel dielectrics for metals, to realize lithographic 

masking for diffusion and implantation processes, 

as well as for passivation and protective layers (Ghandhi, 

1983; Nicollian and Brews, 1983; Sze, 1985). In addition 

they can be exploited as thin amorphous membranes 

on which micro- and nanostructured devices are realized 

(Clark et al., 2005; Fisher et al., 1999; Irwin et al., 

1996; Lanting et al., 2005; Luukanen et al., 2000; Nahum 

and Martinis, 1995) in light of their specific electric and 

thermal properties (Leivo and Pekola, 1998; Leoni et al., 

2003; Manninen et al., 1997). In the following we discuss 

those insulators which are considered particularly 

relevant for microelectronic fabrication processing, i.e., 

silicon dioxide and silicon nitride. 

Silicon dioxide (SiO2) is one of the most exploited insulators 

in micro- and nanoelectronics based on Si, mainly 

due to the high quality of the SiO2/Si interface. SiO2 

films can be grown on Si substrates by thermal oxidation 

using oxygen or steam. Thermal oxidation of Si is generally 

carried out in reactors at temperatures between 900 ◦ 

C and 1200 ◦ C. The resulting SiO2 film is amorphous and 

characterized by good uniformity, lack of porosity and 

very good adhesion to the substrate. Some typical parameters 

of thermally grown silicon dioxide at 1000 ◦ C are

a refractive index of 1.46, a breakdown strength larger 

than 10 7 V/cm, and a density of 2.2 g/cm 3 (Nguyen, 

1988; Sze, 1985). An alternative way to deposit silicon 

dioxide layers is through CVD techniques (Nguyen, 1988; 

Reif, 1990). In particular PECVD is considered an effective 

technique (Adams, 1986; Hess, 1984; Kaganowicz 

et al., 1984) because of the low deposition temperature 

(SiO2 is generally deposited in the temperature range 

200 ◦ ...500 ◦ C). Silicon oxide can be deposited from silane 

(SiH4) with O2, CO2, N2O or CO (Adams et al., 1981; 

Hollahan, 1974). In addition, plasma oxide film properties 

are strongly dependent on the growth conditions such 

as reactor configuration, RF power, frequency, substrate 

temperature, pressure and gas fluxes. Typical parameters 

for plasma-deposited silicon dioxide films at 450 ◦ C 

are a refractive index of 1.44...1.50, a breakdown strength 

of 2...8 × 10 6 V/cm, and a density of 2.1 g/cm 3 . 

Silicon nitride is another insulating film that forms 

good interfaces with Si. It is nowadays successfully used 

as interlevel dielectric (Swan et al., 1967), in multilayer 

resist systems (Suzuki et al., 1982), as well as a protective 

coating as it provides an efficient barrier against moisture 

and alkali ions (e.g., Na) (Sze, 1985). PECVD is commonly 

used (Adams, 1986; Hess, 1984) because of the low 

deposition temperature (250 ◦ ...400 ◦ C) implying low mechanical 

stress. Silicon nitride is typically formed by reacting 

silane and ammonia (NH3) or nitrogen in the glow 

discharge. As for SiO2, the properties of the final film 

strongly depend on the deposition conditions (Adams, 

1986; Chow et al., 1982; Dun et al., 1981; Nguyen et al., 

1984). Typical parameters for plasma-deposited silicon 

nitride films at 300 ◦ C are a refractive index of 2.0...2.1, 

a density of 2.5...2.8 g/cm 3 , and a breakdown strength of 

6 × 10 6 V/cm. 

E. Lithography and etching techniques 

Fabrication of thin film metallic circuits as well as semiconductor 

micro- and nanodevices requires the generation 

of suitable patterns through lithographic processes 

(Campbell, 2001; Jaeger, 2002; Plummer et al., 2000). 

Lithography, indeed, is the method used to transfer such 

patterns onto a substrate (e.g., Si, GaAs, glass, etc.), thus 

defining those regions for subsequent etching removal or 

material addition. 

In photolithography (Levinson and Arnold, 1997) a 

radiation-sensitive polymeric material (called resist) is 

spun on a substrate as a thin film. The image exposure 

is then transferred to the resist through a photomask, 

consisting normally of a glass plate having the desired 

pattern of clear and opaque areas in the form of a thin 

(∼ 1000 ˚A) Cr layer. Two types of resists can be used 

in such a process, i.e., positive and negative resists (Colclaser, 

1980; Moreau, 1988; Thompson et al., 1994). In 

the former, the solubility of the exposed areas in a solvent 

called developer is enhanced, while in the latter the 

solubility is decreased. After exposure, the resist is de- 

48 

veloped and reproduces the desired pattern images for 

the subsequent processing. The radiation source for photolithography 

depends on the desired final resolution, although 

the latter is mainly limited by effects due to light 

diffraction (Levinson and Arnold, 1997). In particular, 

high pressure Hg lamps (with a wavelength λ = 365 nm 

i-line or λ = 436 nm g-line) allow a line-width larger 

than 0.250 µm, while in the range 130...250 nm deep UV 

(DUV) sources like excimer lasers are necessary, such as 

KrF (λ = 248 nm) and ArF (λ = 193 nm). 

Electron-beam lithography (Brewer, 1980; McCord 

and Rooks, 1997) represents an attractive technique 

for the fabrication of micro- and nanostructures (Kern 

et al., 1984). This method exploits a focused electron 

beam (with energy in the range 10...100 keV 

and diameter of 0.2...100 nm) to expose a polymerbased 

electron-sensitive resist (such as polymethylmethacrylate 

(PMMA)). Like in photolithography, the 

resist can be either positive or negative. Even if the wavelength 

of the impinging radiation beam can be smaller 

than 0.1 nm, the maximum achievable resolution is set 

by the electron scattering in the resist and backscattering 

from substrate (known as the ”proximity effect” (Howard 

et al., 1983; Jackel et al., 1984; Jamoto and Shimizu, 

1983; Kyser, 1983; Kyser and Viswanathan, 1975)), so 

that the resolution is generally larger than 10 nm (note, 

however, that resolutions as high as 2 nm have been 

achieved on some materials (Mochel et al., 1983)). 

In addition to lithographic procedures, etching of thin 

films or bulk substrates represents an important step for 

the fabrication of the final structure (Madou, 1997; Sze, 

1985). Toward this end, insulating or conducting thin 

films are exploited as masking layers for subsequent material 

removal. Two crucial parameters of any etching 

process are directionality and selectivity. The former 

refers to the etch profile under the masking layer. In 

particular, for an isotropic etch, the etching rate is approximately 

the same in all directions, leading to a spherical 

profile under the mask. With anisotropic etch, the 

etching rate depends on the specific direction (e.g., a particular 

crystallographic plane) thus leading to straight 

profiles and sidewalls. Selectivity instead represents how 

well the etchant can differentiate between the masking 

layer and the layer that has to be removed. Moreover, 

etching techniques can be divided into wet (Ghandhi, 

1983; Kendall and Shoultz, 1997) and dry (Madou, 1997; 

Wasa and Hayakawa, 1992) categories. In wet etching, 

the substrate is placed in a liquid solution, usually a 

strong base or acid. The advantage of wet etching stems 

from its higher selectivity in comparison to dry methods. 

Wet etching is in general isotropic for most substrates, 

and various solutions that yield anisotropic etching are 

available for some materials. In dry etching, the substrate 

is exposed to a plasma in a reactor where ions can 

etch the substrate surface. The great advantage of dry 

etching with respect to wet etching resides in its higher 

anisotropy (that allows vertical etch walls), and smaller 

undercut (that enables smaller lines to be patterned with

much higher resolution). Several materials (e.g., insulators, 

semiconductors as well as refractory metals) can be 

successfully dry etched, for which a variety of chemistries 

and recipes are available (Cotler and Elta, 1990). 

F. Tunnel barriers 

1. Oxide barriers 

Superconducting tunnel junctions represent key elements 

in a number of electronic applications (Solymar, 

1972) that span from single electron transistors (Grabert 

and Devoret, 1992) to Josephson devices (Barone and 

Paternó, 1982), just to mention two relevant examples. 

In addition, they are crucial building blocks of superconducting 

microrefrigerators (Leivo et al., 1996; Nahum 

et al., 1994) as well as of ultrasensitive microbolometers 

(Castellano et al., 1997; Nahum and Martinis, 1993). The 

simplest picture of a tunnel junction can be given assuming 

a rectangular barrier of height φI and width w. 

The electron transport across the barrier can be easily 

described within the Wentzel-Kramers-Brillouin (WKB) 

approximation. The main results of this analysis are 

(Simmons, 1963b) i) an exponential dependence of the 

zero-bias junction conductance (G0) on the barrier width, 

G0 ∝ exp[−2w √ 2m ∗ φI/], where m ∗ is the effective 

mass of electrons in the barrier; ii) a quadratic voltage 

(V ) dependence of the conductance G(V ); iii) a weak 

insulating-like quadratic dependence of G on the temperature 

T . In general, image forces acting on the electrons 

tunneling through the barrier will reduce both its height 

and its effective thickness (Simmons, 1963a). Criteria i)iii) 

require that the dominant process through the barrier 

is direct tunneling and can be used to extract the junction 

parameters in a realistic situation (Brinkman et al., 

1970; Simmons, 1963a). Among the above given criteria, 

i) seems a necessary, but not sufficient condition to establish 

that tunneling is the dominant transport mechanism 

(Rabson et al., 2001; Zhang and Rabson, 2004) due 

to the possible presence of pinholes (i.e., small regions 

where the insulator thickness vanishes) in the barrier, 

and it is generally accepted that only criterion iii) could 

be safely used to rule out the presence of such pinholes 

in the barrier (Jönsson-˚Akerman et al., 2000) and assess 

the junction quality (for instance, a reduction of junction 

conductance of about 15% on cooling from 295 K to 

4.2 K is believed to be a good practical indication of highquality 

AlOx barriers (Gloos et al., 2003, 2000); see also 

Secs. III and IV). An important figure of merit of tunnel 

contacts is the junction specific resistance Rc = RJA, 

where RJ ≡ G −1 

0 is the contact resistance and A its area. 

A route to decrease Rc is to reduce w or choose materials 

with lower effective barrier height. 

Among the available barrier materials, aluminum oxide 

(AlOx) is probably the most widespread insulator used 

to fabricate metallic tunnel junctions because it can be 

easily and reliably grown starting from an Al film. Its 

49 

main parameters are a typical barrier height φI ≈ 2 eV, 

although values in the range 0.1...8.6 eV have been reported 

(Barner and Ruggiero, 1989; Gundlach and Hölz, 

1971; Kadlec and Kadlec, 1975; Lau and Coleman, 1981) 

and a dielectric constant smaller than that of bulk Al2O3 

(4.5...8.9 at 295 K (Bolz and Tuve, 1983)). Several methods 

are currently exploited to fabricate high-quality (i.e., 

highly uniform and pinhole-free) aluminum oxide barriers 

such as in situ vacuum natural oxidation (Matsuda 

et al., 1999; Parkin et al., 1999; Sun et al., 2000; Tsuge 

and Mitsuzuka, 1997; Zhang et al., 2001), oxidation in air 

(Miyazaki and Tezuka, 1995) and plasma oxidation (Gallagher 

et al., 1997; Moodera et al., 1995; Sun et al., 1999) 

of a thin Al layer (typically below 2 nm), just to mention 

the most common techniques. The latter two methods 

lead in general to higher Rc values (of the order of several 

kΩ µm 2 or larger) with respect to natural oxidation. 

The natural oxidation allows to achieve the desired Rc by 

simply changing the oxidation pressure and time (Rc also 

depends on the original thickness of the Al layer), and almost 

any Rc value can be produced by following such a 

procedure. Aluminum oxide junctions with Rc values as 

low as some tens of Ω µm 2 or lower are currently realized 

with natural oxidation (Childress et al., 2001; Deac et al., 

2004; Fujikata et al., 2001; Liu et al., 2003; Parkin et al., 

1999; Sun et al., 2000; Zhang et al., 2001). However, 

low-Rc junctions are more prone to defects or pinholes in 

the barrier that dramatically degrade their performance. 

Promising methods and materials have been proposed 

for the fabrication of oxidized barriers with Rc as low as 

some Ω µm 2 among which we mention junctions made of 

ZrAlOx (Liu et al., 2003; Wang et al., 2002) and HfAlOx 

(Wang et al., 2003). 

2. Schottky barriers 

The metal-semiconductor (NSm) junction is an issue 

that dates back more than 60 years (Schottky, 1939), 

but still nowadays represents a relevant topic both in the 

physics of semiconductors (Brennan, 1999; Lüth, 2001; 

Rhoderick and Williams, 1988; Sze, 1981, 1985) and in 

device applications (Millman and Grabel, 1987; Singh, 

1994). The two most important types of NSm junctions 

are the Schottky barrier (SB), showing a rectifying 

diode-like I-V characteristics, and the ohmic contact, 

whose I-V is almost linear. Most of NSm junctions, with 

a few exceptions such as contacts with InAs, InSb and 

InxGa1−xAs (for x ≥ 70%) (Kajiyama et al., 1973), are 

affected by the presence of the SB that drastically affect 

their electric behavior. For most semiconductors (in the 

following we concentrate on n-type semiconductors but 

similar conclusions can be given for p-type ones), the SB 

height (φSBn) has a rather weak dependence upon the 

metal used for the contact; for instance, for metal/n-Si 

contacts φSBn = 0.7...0.85 eV, while for metal/n-GaAs 

contacts φSBn = 0.7...0.9 eV (Rhoderick and Williams, 

1988; Singh, 1994). This fact is explained in terms of the

Fermi-level pinning at the NSm interface (Bardeen, 1947; 

Heine, 1965; Mönch, 1990). 

The current across a Schottky junction depends on 

a number of different mechanisms. In the limit where 

thermionic emission dominates the electric transport, the 

rectifying action of a biased NSm junction is described 

as I(V ) = Is[exp(eV/kBT ) − 1], where the detailed expression 

for the saturation current Is depends on the 

assumptions made on carrier transport (Brennan, 1999; 

Sze, 1981, 1985). In such a case the junction specific 

resistance is Rc ∝ T −1 exp(eφSBn/kBT ), thus meaning 

that it can be lowered mainly by decreasing φSBn (typical 

Rc values at doping levels ND ≤ 10 17 cm −3 for 

metal/n-Si contacts are of the order of 10 11 ...10 13 Ωµm 2 

almost independent of ND). By contrast, tunneling 

across the SB can be the dominating transport mechanism 

if the semiconductor is heavily doped. In such a case 

I(V ) ∝ exp[−α(φSBn − V )/ √ ND], with α = 2 −1√ m ∗ ɛ 

where m ∗ is the semiconductor effective mass and ɛ the 

dielectric permittivity, i.e., the junction is not rectifying 

and the current is proportional to V for small voltages. 

The contact is thus said to be ohmic and yields 

Rc ∝ exp(αφSBn/ √ ND). This shows that Rc can be 

reduced up to a large extent by lowering the SB height 

and doping as heavily as possible (again, for metal/n- 

Si contacts and ND ≥ 10 19 cm −3 , Rc can be in the 

range 10 2 ...10 8 Ωµm 2 ). All this shows the advantage of 

using NSm contacts owing to the possibility of tuning 

the contact specific resistance over several orders of magnitude 

(from metallic-like to tunnel-like characteristics) 

through a careful choice of metal-semiconductor combinations 

and proper doping levels. This trick is commonly 

exploited to control the NSm interface resistance in current 

semiconductor technology, although heavy doping of 

the semiconductor just in proximity to the metal is often 

preferred (Giazotto et al., 2001a,b; Kastalsky et al., 

1991; Shannon, 1976; Taboryski et al., 1996). 

VII. FUTURE PROSPECTS 

Low temperature solid-state cooling is still at its infancy, 

although operation of a number of individual 

principles and techniques have been demonstrated to 

work successfully. Yet combinations of cascaded microrefrigerators 

over wider temperature ranges employing 

several stages, or combinations of different refrigeration 

principles, e.g., fluidic coolers together with electronic 

coolers, do not exist. In principle, compact low-power 

refrigerators could be fabricated using micro-machined 

helium-based fluidic refrigerators, for instance based on 

Joule-Thomson process (Little, 1984), and these devices 

could then be directly precooling NIS-refrigerators with 

niobium (Tc = 9 K) as a superconductor. A lot of engineering 

effort is, however, needed to make this approach 

work in practise as a targeted micro-refrigerator. 

The solid-state micro-circuits have already proven to 

yield new operation principles and previously unknown 

50 

concepts have been discovered in cryogenic devices, as 

demonstrated throughout this review. We believe that 

what was demonstrated here is just a presentation of a 

beginning of a new era in low temperature physics and 

instrumentation. As possible new classes of devices we 

could mention those utilizing thermodynamic Carnot cycles 

with electrons. Brownian heat engines with electrons 

are predicted to achieve efficiencies close to ideal 

(Humphrey et al., 2002). It may be possible in the future 

to make use of other types of gated cycles where 

energy selective extraction of electrons is producing the 

refrigeration effect. As a conceivable example, a combination 

of Coulomb effects and superconducting energy 

gap could form the basis of operation of a refrigerator 

where cooling power would be proportional to the operating 

frequency of the gate cycle. Such a device would 

thus be principally different from the static electronic refrigerators 

presented in this review, where a DC bias is 

in charge of the redistribution of hot electrons. 

At low temperatures, additional relaxation channels 

besides the electron-phonon scattering, such as coupling 

between electrons and photons, become important. More 

knowledge is needed on these mechanisms. 

In Subs. V.C.7, we describe how the non-equilibrium 

shape of the distribution function sometimes leads to improved 

characteristics of the device. It would be interesting 

to see if such effects could be employed to improve 

also the properties of the radiation detectors or other 

practical devices. 

The presently obvious application fields of electronic 

micro-refrigerators include astronomical detectors both 

in space as well as those based on the earth, materials 

characterization instrumentation, e.g., those devices employing 

ultra high resolution x-ray micro-analysis, and 

security instrumentation, e.g., concealed weapon search 

on the airports. It is, however, evident that once realized 

in a user-friendly and economic way, refrigeration 

becomes very important in high-tech based industry in 

a much broader perspective. Low temperature electronics 

and superconducting devices are often characterized 

by their undeniably unique possibilities, but they are often 

superior to the room temperature ones also in speed 

and power consumption. Therefore, mesoscopic on-spot 

refrigerators to attain the low temperatures forming the 

basis of these instruments are urgently needed. 

Acknowledgments 

We thank H. Courtois, R. Fazio, F. Hekking, M. Paalanen, 

F. Taddei and P. Virtanen for their insightful comments 

and for critically reading the manuscript. D. 

Anghel, A. Anthore, F. Beltram, M. Feigelman, E. Grossman, 

K. Irwin, M. Meschke, A. J. Miller, S. Nam, D. 

Schmidt, and J. Ullom are gratefully acknowledged for 

enlightening discussions. This work was supported by 

the Academy of Finland.

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