SISIS Josephson transistor in the nonequilibrium regime - Low ...

Helsinki University of Technology Special assignment
Department of Engineering Physics Tfy-3.393 Computational Physics
and Mathematics 30th August 2006
SISIS Josephson transistor in the nonequilibrium regime
Matti Laakso
62975L

Contents
1 Introduction 3
2 Formalism 4
2.1 General concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Superconductivity . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Energy nonequilibrium . . . . . . . . . . . . . . . . . . . . 5
2.2 SISIS transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Quasiclassical Green’s functions . . . . . . . . . . . . . . . . . . . 6
3 Equations 8
3.1 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Green functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Order parameter and observables . . . . . . . . . . . . . . . . . . 12
4 Results 13
4.1 Full nonequlibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Relaxation-time approximation . . . . . . . . . . . . . . . . . . . 19
4.3 Collision integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4 Asymmetric structure . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4.1 Spin-flip scattering . . . . . . . . . . . . . . . . . . . . . . 26
5 Discussion 27
A Tunneling rate 29
B Ambegaokar-Baratoff formula for the critical current 30
2

1 Introduction
Mesoscopic physics deals with phenomena in systems which are small enough
in size to exhibit quantum mechanical properties, but sufficiently large that the
need to take individual atoms into account does not arise. In general this means
structures with spatial dimensions of nanometer and micrometer scale. Mesoscopic
experiments usually aim to study electron transport in small metallic
wires and metal-to-metal contacts, quantum mechanics of relaxation and lowtemperature
phenomena. The central concept is often the energy distribution
of mesoscopic electron systems, which in thermal equilibrium defines the temperature
of the electron gas in the sample. Also nonequilibrium distributions
are often encountered and utilized in mesoscopic systems. The ability to control
and measure electron distributions can be used in various device concepts, such
as electronic refrigerators, thermometers, radiation detectors and novel types of
transistors. Typically the resolution of these devices requires the minimization
of thermal noise and fluctuations which can be achieved at cryogenic temperatures.
A practical threshold of temperature is given by the liquefaction of
helium at 4.2 K.
Many metals turn superconducting at these temperatures, an example of
quantum mechanical phenomenon which gives rise to macroscopic effects. The
most famous property of superconductors is probably its ability to carry electric
current without dissipation, i.e., without inducing a voltage drop. This is also
known as supercurrent. Another property is the Meissner effect, in which an
external magnetic field creates eddy currents on the surface of the superconductor.
These circulating currents create an opposing magnetic field with the
result that the total magnetic field is unable to penetrate into the superconductor.
The most important superconducting effect in mesoscopic systems is the
Josephson effect which allows the supercurrent to flow through a weak link, e.g.
an insulating barrier, between two superconductors. These Josephson junctions
are very versatile objects due to their nonlinear current-voltage characteristics.
A particular application of Josephson junctions is the Josephson transistor,
which has been a subject of recent interest in nanoelectronics. Its latest type is
composed of superconducting (S) and normal (N) metals separated with insulating
oxide barriers (I). By arranging these constituents in a SINIS structure
the magnitude of the supercurrent flowing through the system can be increased
as well as suppressed by driving the energy distribution in the normal metal out
of equilibrium [1]. This transistorlike operation with large current and power
gain has also been experimentally demonstrated [2]. Also an all superconducting
device has been proposed, where the supercurrent is controlled by voltage
biasing the middle electrode of an SISIS line. This has been theoretically addressed
in the quasiequilibrium regime in Ref. [3]. The SISIS transistor benefits
from sharp characteristics brought by the superconducting interelectrode which
leads to improved operational properties.
In this work we study the SISIS transistor in full nonequilibrium, as well
as with different strengths of energy relaxation. We also examine the effects of
asymmetry in the SISIS line driving the energy distribution out of equilibrium.
General concepts and the formalism used is introduced in Sec. 2. Relevant equations
are derived in Sec. 3. We finish by presenting the results and discussion
in Secs. 4 and 5.
3

2 Formalism
2.1 General concepts
2.1.1 Superconductivity
The microscopic theory of superconductivity was established in 1957 by Bardeen,
Cooper and Schrieffer (BCS). According to it even a weak attractive interaction
can bind electrons into a bound state called the Cooper pair. This attractive
interaction arises from the polarization of the medium by a first electron attracting
positive lattice ions, which in turn attract the second electron. Since these
lattice deformations are phonons, the strength of the attraction is heavily dependent
on temperature and vanishes completely above a material specific critical
temperature TC (typically a few Kelvin for conventional superconductors, such
as Al, Nb, Pb or Sn). This collective behaviour results in a condensate of Cooper
pairs, which may be described with a single macroscopic wavefunction. Central
to this model is the assumption that the overall interaction between the electrons
is attractive up to energy differences of the order of Debye frequency, which
characterizes the cutoff of the phonon frequency spectrum in solids. Therefore
only electrons within the Debye frequency around the Fermi level will take part
in forming the condensate. This is also called the BCS cutoff energy.
A certain amount of energy, 2∆, is required to break a pair and create two
“normal-state” electron excitations. This leaves two holes in the Fermi sea
which can also be interpreted as two excitations of positively charged particles.
These excitations are collectively called quasiparticles. Because the minimum
excitation energy of a quasiparticle from the BCS ground state is ∆, the energy
spectrum has an energy gap in which there are no states. Electrons with energies
falling inside the gap prefer to form a Cooper pair. The density of states in BCStheory
is
|E|
g(E) = g0
θ(|E| − |∆|),
E2 − |∆| 2
where g0 is the density of states at the Fermi level when the sample is in the
normal state and θ is the Heaviside step function. The magnitude of the energy
gap at T = 0 is found to be related to the critical temperature through ∆0 =
1.764 TC. 1
In 1962 Josephson predicted that an electric current carried by Cooper pairs
(i.e., supercurrent) should flow between two superconductors in a weak contact,
usually separated with a thin insulating barrier which only allows the
Cooper pairs to tunnel across it. Such a junction is called an SIS junction
(Superconductor-Insulator-Superconductor). The magnitude of the supercurrent
is given by the dc Josephson relation
I = IC sin(∆χ), (1)
where ∆χ is the difference across the junction in the phase of the order parameter
to be defined later and IC is the maximum supercurrent that the junction can
support. If the current driven through the junction exceeds IC, a voltage starts
to appear, and the phase difference evolves in time according to ∂t∆χ = 2eV .
1 Throughout this text we use units in which = kB = c = 1.
4

This leads to the ac Josephson relation
I = IC sin(∆χ(0) + 2eV t), (2)
i.e. the current through a Josephson junction with an applied dc voltage oscillates
in time. For a more thorough review of superconductivity and related
effects, see for example the book by Tinkham [4].
2.1.2 Energy nonequilibrium
Macroscopic electron systems in thermal equilibrium can be described with the
Fermi-Dirac distribution function
f 0 1
(E) =
exp((E − µ)/T) + 1 ,
where µ is the chemical potential and T is the temperature. Consider a sample
connected to large leads acting as reservoirs of electrons. The sample assumes an
equilibrium distribution with a temperature equal to the reservoirs and a chemical
potential profile that depends on the voltage applied across it. The charge
carriers that enter the sample due to the potential difference scatter from the
impurities, lattice dislocations, phonons and other electrons, thereby thermalizing
with the underlying lattice and maintaining thermal equilibrium. The mean
free path that the electron travels before scattering is characterized by scattering
lengths lel for elastic scattering and le−ph and le−e for inelastic electronphonon
and electron-electron scattering, respectively. In mesoscopic systems
typical orders of magnitude are lel ≈ 10 . . .100 nm and le−e ≈ 1 . . .20 µm. The
electron-phonon scattering length depends strongly on temperature. At 1 K
le−ph ≈ 21 µm but at 100 mK we already have le−ph ≈ 670 µm [5].
If the spatial dimensions of the sample satisfy lel ≪ L, the distribution function
is described by the semiclassical Boltzmann equation in a diffusive wire
with source and sink terms called the collision integrals describing the various
scattering processes. In the so called quasiequilibrium limit, le−e ≪ L ≪ le−ph,
the electron-phonon interaction is nearly absent and the sample can be considered
as detached from the phonon bath. The high frequency of electron-electron
collisions still serves as a method of relaxation and the electrons assume a Fermi
distribution but with a temperature that in general differs from the temperature
of the phonon bath [5]. If the sample is small enough with L ≪ le−e, le−ph,
the injected electrons are not able to relax even through collisions with other
electrons in the sample, and the distribution may deviate strongly from the
Fermi function. This is called the nonequilibrium limit and we may neglect the
inelastic scattering altogether. In the nonequilibrium limit the concept of temperature
is rendered obsolete because it is defined through the Fermi equilibrium
distribution. However, it is still possible to define some effective temperature
with the shape of the distribution function or the magnitude of the energy gap,
for example, but these are by no means unambiguous. The nonequilibrium form
of the distribution shows up in many observable properties of the system, e.g.,
current-voltage (I-V) characteristics and current noise.
2.2 SISIS transistor
The superconducting structure under study is schematically depicted in Fig. 1.
It consists of two crossing SISIS lines with a common central superconductor.
5

R1
µ = 0
5 R5
1 3
2
µ = 0
4
Figure 1: The SISIS structure studied in this work. The superconducting island
in the middle (2) is connected with tunnel contacts to four large superconducting
leads (1,3,4 and 5). The control line is biased with voltage V , which controls
the energy distribution in the island. A supercurrent IS is driven across the
island from lead 4 to 5, and its magnitude depends on the distribution function
in the island.
The superconducting island in the middle is assumed to have small dimensions
so that L ≪ le−e, le−ph, i.e., the energy relaxation via inelastic scattering is in
practice very weak. Each of the SIS-junctions is characterized by a junction
resistance Ri, which is at least hundreds of Ohm. In contrast, the normal-state
resistances of the superconductors are typically of the order 1Ω. We bias the first
SISIS line with an adjustable voltage V , thus creating an energy nonequilibrium
in the island, and drive a supercurrent through the second SISIS line, which
is kept at zero chemical potential. The magnitude of the supercurrent may
be controlled with the external voltage, so this structure works basically as a
transistor. Studying this dependence is the main objective of this work.
2.3 Quasiclassical Green’s functions
In quantum mechanics Green’s function describes the propagation of disturbances
in which a single particle is added to a many-particle equilibrium system
at x1 and removed at x2. Here x = (r, t) represents both space and time coordinates.
The retarded Green function describes the propagation from past to
present (t1 < t2) and the advanced Green function describes the propagation
from future to present time (t1 > t2). If the retarded and advanced Green
function for a system is known, it is possible to calculate all energy-dependent
quantities of the system, e.g., density of states. These can then be related to
the equilibrium properties of the system, such as supercurrent. In BCS-theory
6
IS
R4
R3
V

the Green function is a 2 × 2 matrix in the particle-hole (Nambu) space of the
form
ˆG(x1,
G(x1, x2)
x2) =
−F
F(x1, x2)
† (x1, x2)
G(x1, ¯ .
x2)
(3)
Here G is the Green function of the propagating electron, ¯ G is the Green function
of the propagating hole and F is the pair amplitude, which is coupled to the
pair potential ∆ with
∆(x1) = λ lim F(x2, x1). (4)
x2→x1
Here λ is a parameter characterizing the strength of the attractive interaction.
The pair potential measures the amount of correlations between the pairs of
electrons and it is also called the order parameter of the superconductor. The
pair amplitude and pair potential are complex functions, and the magnitude
of the pair potential |∆| coincides with the energy gap of the superconductor.
In bulk superconductors |∆| is often spatially constant, but the phase χ may
vary in space. This phase difference gives rise to the supercurrent. In normal
metals λ = 0 and the pair potential vanishes, but it is possible for a normal
metal to have a nonzero pair amplitude in a small region in good contact with
a superconductor. This is called the proximity effect.
Green’s functions can be found as the solutions of the Gor’kov equations.
In nonstationary processes the physical system may be out of equilibrium and
we need to know the exact distribution of the excitations in addition to their
spectrum. For this we need the real-time Green functions, which describe the
evolution of a system in nonequilibrium. There are a few methods for finding
the real-time Green functions, most notably one by Keldysh which is also used
in this work. Keldysh technique is described in detail in [6]. In the following
the retarded and advanced Green functions are denoted with ˆ GR and ˆ GA and
the Keldysh Green function with ˆ GK .
The full double-coordinate Green functions work well for homogeneous superconductors,
but when there are spatial inhomogeneities, such as the insulating
layer in a Josephson junction, they become very cumbersome. The Green
function in Eq. (3) oscillates as a function of the relative coordinate |r1−r2| with
a magnitude of the Fermi wavelength λF[7]. In conventional low-temperature
superconductors this is much shorter than the characteristic coherence length
ξ = vF /∆. Moreover, usually of interest are the effects which depend on the
phase of the Cooper pair wavefunction, which in turn depends only on the center
of mass coordinates. For these reasons it is possible to integrate out the
dependence of the Green function on the relative coordinate. Fourier transformation
over the relative coordinate to momentum space produces a sharp peak
at |p| = pF, therefore integration over the relative coordinate corresponds to
integration of the transformed function over ξp = p2 /2m − EF , which depends
on the magnitude of the momentum. The quasiclassical Green function is then
defined by
dξp
ˆg =
πi ˆ G. (5)
We may also perform a Fourier transformation over the temporal coordinates
resulting in a Green function that depends on the direction of the momentum,
as well as energies E1 and E2.
In the system under study we assume that the resistance of the superconducting
island is negligible compared to the resistances of the tunnel contacts,
7

which implies that no potential drop appears inside the island. We also assume
the superconducting reservoirs to have spatially large dimensions compared to
the impurity mean free path. These combined allow us to use the tunnel Hamiltonian
approach, in which each region has spatially constant, separate energy
distributions. In momentum representation this allows us to simplify the Green
functions even further by averaging out the direction of the momentum.
3 Equations
Central to this work is to find the nonequilibrium quasiparticle distribution
function for the superconducting island. It is advantageous to separate the
distribution function to a symmetric and an antisymmetric part with respect to
E = 0, which allows us to work solely with positive energies. This can be done
by introducing the odd-in-E and even-in-E distribution functions
fL(E) = −f(E) + f(−E), fT(E) = 1 − f(−E) − f(E), (6)
where f is the full quasiparticle distribution function. We can recover the full
distribution function with 2f(E) = 1 − fL(E) − fT(E). In equilibrium
f 0
1 E − µ E + µ
L/T (E, µ) = tanh ± tanh ,
2 2T 2T
where the symmetries fL(E, −µ) = fL(E, µ) and fT(E, −µ) = −fT(E, µ) are
evident. Once we have found the distribution function and the Green functions,
we are able to calculate the energy gap for the superconductor and relevant
physical observables such as electrical currents and potentials.
3.1 Kinetic equations
Here we derive the kinetic equations that are required to solve the fL and
fT components of the nonequilibrium distribution function. From the BCS
Hamiltonian and the definition of the Green function it is possible to derive the
inverse matrix Green function
where
ˆG −1 ∂
(x1) = −iˆτ3
∂t1
ˆH
0 −|∆|e
=
i2µt1
|∆|e−i2µt1
,
0
− ∇2 r1
2m − µ + ˆ H, (7)
and ˆτ3 is the third Pauli spin matrix. The Gor’kov equations for the retarded,
advanced and Keldysh Green functions now read [8]
( ˆ G −1 − ˆ Σ R(A) ) ◦ ˆ G R(A) = ˆ1δ(x1 − x2),
( ˆ G −1 − ˆ Σ R ) ◦ ˆ G K − ˆ Σ K ◦ ˆ G A = 0, (8)
ˆG R(A) ◦ ( ˆ G −1 − ˆ Σ R(A) ) = ˆ1δ(x1 − x2),
ˆG K ◦ ( ˆ G −1 − ˆ Σ A ) − ˆ G R ◦ ˆ Σ K = 0, (9)
8

where the product ˆ ΣK ◦ ˆ GA is a convolution
ˆΣ K ◦ ˆ G A
=
ˆΣ K E1,E ˆ G A E,E2
dE
2π .
Here and in the following we demote the energy parameters to subscript for
clarity. Here we also introduce the self-energy Σ, which modifies the dynamics of
the system due to interactions between the particle and the surrounding system,
e.g. tunneling and collisions with other particles. Next we subtract Eqs. (8)
from Eqs. (9) and substitute the quasiclassical Green functions averaged over
directions of the momenta yielding the transport-like equations
−E1ˆτ3ˆg K E1,E2 + ˆgK E1,E2E2ˆτ3
+ ˆH ◦ ˆg
, K
= ÎK , (10)
E1,E2
−E1ˆτ3ˆg R(A)
+ ˆgR(A)
E1,E2 E1,E2E2ˆτ3 +
where the collision integrals are
ˆH ◦ , ˆg R(A)
Î K E1,E2 = ˆ Σ R ◦ ˆg K − ˆg K ◦ ˆ Σ A − ˆg R ◦ ˆ Σ K + ˆ Σ K ◦ ˆg A ,
ˆΣ R(A) ◦ ˆg
, R(A)
.
Î R(A)
E1,E2 =
Here
A ◦ B
,
= A ◦ B − B ◦ A.
= ÎR(A) , (11)
E1,E2
Equations (10) and (11) are known as Eliashberg and Usadel equations, respectively.
The Keldysh Green function can be parametrized as [8]
ˆg K E1,E2 = ˆgR E1,E2 (fL,E2 + ˆτ3fT,E2) − (fL,E1 + ˆτ3fT,E1)ˆg A E1,E2 ≡
K g fK −f †K ¯g K
.
(12)
By combining the Eliashberg and Usadel equations together with the parametrization
above we obtain two kinetic equations for diagonal components of the distribution
matrix ˆ1fL + ˆτ3fT
Tr ˆME,E K R
= Tr ÎE,E − ÎE,E − ÎA
E,E fL,E , (13)
Tr ˆτ3 ˆ
K R
ME,E = Tr ˆτ3 ÎE,E − ÎE,E − ÎA
E,E fL,E , (14)
where the matrix ˆ M is
ˆM =
ˆH ◦ ˆg
, K
− fL ˆH ◦ ˆg
, R − ˆg A
.
Once we know the exact forms of the retarded and advanced Green functions
and self-energies, we are able to solve fL and fT from the kinetic equations
above.
9

3.2 Green functions
Next we need expressions for the self-energies and Green functions. In a superconductor
that has tunnel contacts to other superconductors the tunneling
self-energy is [8]
ˆΣT =
iηjˆgj, (15)
j
where the sum goes over all the indices of the contacting superconductors and
the tunneling rate is (see Appendix A)
ηj = (4νe 2 ΩRj) −1 .
Here ν is the normal state density of states at the Fermi level, Ω is the volume
and Rj is the tunnel resistance to superconductor j. Elastic processes are
dropped out of self-energies after averaging out momentum directions. The selfenergies
for the inelastic processes are more complex and we will not present
them here. They are derived in for example [9].
The retarded and advanced Green functions are obtained from the Usadel
equation Eq. (11) supplemented with the normalization condition [10]
ˆg ◦ ˆg = ˆ1. (16)
If a constant potential difference is maintained over a hybrid structure with
more than one superconductor, at least one of the superconductors will have
a nonzero chemical potential µ. This leads to a time-dependent phase of the
order parameter evolving according to the ac Josephson relation χ = 2µt, while
the magnitude of the order parameter stays constant. We may choose one of
the superconductors at zero chemical potential to have a phase χ = 0 without
losing generality. The order parameter may be written in terms of the absolute
temporal coordinates t1 and t2 as ∆ = |∆|ei2µ(t1−t2) , which may be Fourier
transformed to
∆E1,E2 = |∆|2πδ(E1 − E2 + 2µ), ∆ ∗ E1,E2 = |∆|2πδ(E1 − E2 − 2µ).
Therefore the Usadel equation Eq. (11) is satisfied by [8]
g R(A)
= gR(A)
E1,E2 E1+µ 2πδ(E1 − E2), ¯g R(A)
= ¯gR(A)
E1,E2 E1−µ 2πδ(E1 − E2)
f R(A)
= fR(A)
E1,E2 E1+µ 2πδ(E1 − E2 + 2µ), f †R(A) †R(A)
= f E1,E2 E1−µ 2πδ(E1 − E2 − 2µ),
(17)
where the functions g R(A)
E = −¯g R(A)
E
Usadel equation
−Eˆτ3 + ˆ H, ˆg R(A)
E
and f R(A)
E
= f †R(A)
E
satisfy the stedy-state
= ÎR(A)
E . (18)
The normalization condition for the steady-state Green functions simplifies to
to 2
−
2 = 1. (19)
g R(A)
E
f R(A)
E
In our system the superconducting island has tunnel contacts to four other
superconductors. Once we insert the self energies to the collision integral in Eq.
(18) we get
−Eˆτ3 + ˆ H, ˆg R(A)
2
=
iηj
10
j
ˆg R(A)
j
, ˆg R(A)
. (20)
2

10
8
6
4
2
0
−2
−4
−6
−8
−60
f (−)
f (+)
−10
−5 −4 −3 −2 −1 0
E
1 2 3 4 5
(a)
60
−60
Figure 2: Functions f (±) (a) and g (±) (b).
10
8
6
4
2
0
−2
−4
−6
−8
60
g (−)
g (+)
−10
−5 −4 −3 −2 −1 0
E
1 2 3 4 5
The steady state Green functions may be conveniently written as
ˆg R(A) = g R(A) ˆτ3 + f R(A) iˆτ2,
simplifying the equation to
⎛
⎝E +
iηjg R(A)
⎞
⎠ = g R(A)
⎛
⎝|∆| +
iηjf R(A)
⎞
⎠,
f R(A)
2
j=2
j
which has the solution
g R(A) E ± iγ
2 = ±
(E ± iγ) 2 − (|∆| ± iδ) 2 ×
1, |E| < |∆|
sgn(E), |E| > |∆|
f R(A)
|∆| ± iδ
2 = ±
(E ± iγ) 2 − (|∆| ± iδ) 2 ×
1, |E| < |∆|
sgn(E), |E| > |∆|
2
j=2
(b)
j
60
−60
,
, (21)
where γ = R(A)
j ηjgj and δ = R(A)
j ηjfj . In the tunneling limit η ≪ ∆,
therefore we neglect the exact forms of γ and δ and instead use a constant
γ = δ = 10−4 in the numerical calculations. We also define
g (−) = Re g R = 1
2 (gR − g A ), g (+) = Im g R = 1
2i (gR + g A ),
f (−) = Re f R = 1
2 (fR − f A ), f (+) = Im f R = 1
2i (fR + f A ). (22)
These are plotted in Fig. 2. For energies inside the energy gap g (−) and f (−)
are of the order of γ. The same applies for g (+) and f (+) outside the gap. The
normalized superconducting density of states is just g (−) . These functions are
extensively used below.
11

3.3 Order parameter and observables
The magnitude of the order parameter in the nonequilibrium Keldysh formalism
is given by [6]
|∆|
λ =
EC
dE
−EC 8 Tr (ˆτ1 − iˆτ2) ˆg K EC
dE
=
−EC 4 fK , (23)
where EC is the BCS cutoff energy. Inserting f K for our SISIS system explicitely
yields
|∆| = λ
EC
dE fL2f
2 −EC
(−)
E
(+)
− ifT2f
E = λ
EC
2 −EC
dEfL2f (−)
E , (24)
where the last equality follows because the magnitude of the order parameter
is real. Because f (−) in the integrand depends on the magnitude of the order
parameter, this is a self-consistency equation.
As shown in the previous subsection, gR(A) = −¯g R(A) in a stationary nonequilibrium
state. This is equivalent to a particle-hole symmetry in the quasiclassical
approximation, i.e., there are always as many particle excitations as there are
hole excitations. Therefore the net charge density N is unchanged after a transition
to a superconducting state and the chemical potential of a Cooper pair (µ)
is the same as the chemical potential of an excited electron (eϕ). However, this
equilibrium may be broken by charged perturbations, e.g., electron injection,
or conversion of normal current to supercurrent near an interface. This leads
to a situation known as charge imbalance in which there are differing amounts
of electrons and holes on opposite sides of the Fermi level. To compensate this
effect the chemical potential of the excitations must be adjusted in order to
maintain overall electrical neutrality. The charge density is given by [9]
dE
N = N0 + 2ν(eϕ + µ) − ν
2 Tr(ˆgK ), (25)
where N0 is the charge density in normal state. Because N = N0, the chemical
potential of the excitations is coupled to the chemical potential of the condensate
through
dE
eϕ = −µ +
4 Tr(ˆgK ). (26)
The net current flowing into an island is given by the time derivative of the total
charge, which leads to [8]
dE
I = ieνΩ
4 Tr(ˆτ3 ÎK T ). (27)
Here we have included only the tunneling part of the collision integral, because
elastic and inelastic processes conserve the particle number. In the absence of
a potential difference this results in an equation for the supercurrent. For a
finite supercurrent to appear, a phase gradient must exist across the junction
according to the dc Josephson relation. Therefore we will not put χ = 0 as
earlier, rather we write the Green functions as
ˆg R(A) = g R(A) ˆτ3 + f R(A) i (cosχˆτ2 + sin χˆτ1).
12

We obtain for the expression of the supercurrent across junction 2-4
I 2→4
S = − 1
dE fL2f
2eR4
(−)
2 f (+) (−)
4 + fL4f 4 f (+)
2 sin(χ4 − χ2)
+(fT2 − fT4) g (−)
2 g(−) 4 + f(+) 2 f (+)
4 cos(χ4 − χ2) . (28)
Because also the supercurrent is conserved, I2→4 = −I2→5 . If a voltage is
applied across the junction, Eq. (27) gives the quasiparticle current. It reads
I 1→2 = − 1
dE g
4eR1
(−)
1,E+µ g(−)
2,E (fL2 + fT2 − fL1 − fT1)
+g (−)
1,E−µ g(−)
2,E (−fL2
+ fT2 + fL1 − fT1) . (29)
In a similar manner it is possible to obtain the the energy current, which is
given by [8]
dE
Iε = −iνΩ
4 Tr
(E + eϕˆτ3) ÎK
T . (30)
These are the basic observables required to describe the behaviour of our system.
4 Results
4.1 Full nonequlibrium
We begin by presenting the calculated distribution function along with the order
parameter and electric currents for the simplest, namely left-right symmetric
case, where the tunnel junction resistances are the same and reservoirs 1 and
3 are similar superconductors, i.e., R1 = R3 = R and |∆1| = |∆3| = |∆L|.
Moreover, we assume the energy relaxation to be completely absent by setting
the inelastic contributions to the self-energies in the previous section to zero.
We fix the chemical potential of the island to zero and bias the structure with
a voltage V . Current conservation forces the chemical potentials of reservoirs 1
and 3 to µ1 = eV/2 and µ3 = −eV/2, respectively. This implies fL1 = fL3 = fL,
. We also set the phase of the
fT1 = −fT3 = fT and g (−)
E±µ1
= g(−)
E∓µ3
= g(−)
E±µ
order parameter in the island to zero. This leads to ˆ H = −|∆|iˆτ2 simplifying
the kinetic equations (13) and (14) to
with solutions
g (−)
E+µ (fL2 − fL − fT) + g (−)
g (−)
E+µ + g(−)
E−µ
g (−)
2,E R−1 fT2
fL2 = g(−)
E+µ (fL + fT) + g (−)
E−µ (fL2 − fL + fT) = 0, (31)
= 4i|∆2|fT2f (+)
2,E ν2e 2 Ω2, (32)
g (−)
E+µ + g(−)
E−µ
E−µ (fL − fT)
,
fT2 = 0. (33)
This may also be written in terms of the full distribution functions as
f2 = g(−)
E+µ f1 + g (−)
E−µ f3
g (−)
E+µ + g(−)
E−µ
13
.

f 2
1
0.8
0.6
0.4
0.2
0
T=0.1 T C
E=−eV/2+∆ L
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
E/∆ 0
E=eV/2−∆ L
v=0
v=0.5
v=1.5
v=2.0
v=2.5
v=3.0
Figure 3: Nonequilibrium distribution function for the superconducting island
at T = 0.1 TC. The cooling effect reducing the number of excited quasiparticles
as the voltage is increased is evident. Here and below we denote v = eV/∆0
and ∆0 = ∆L(T = 0).
The full nonequilibrium distribution function for the superconducting island at
a bath temperature, i.e., electron temperature in the large superconductors, of
T = 0.1 TC 2 for various bias voltages is shown in Fig. 3. Upon increasing the
voltage above eV = ∆L, the number of excited quasiparticles in the island is
clearly decreased with respect to equilibrium. This can be interpreted as a lower
effective electron temperature. This cooling effect in SINIS and SISIS structures
and its applications have been reviewed in [5]. The shape of the distribution
function can be qualitatively explained with a simple semiconductor model for
the tunneling of quasiparticles, which is depicted in Fig. 4. In the semiconductor
model the superconductors are modelled as ordinary semiconductors with
BCS density of states and energy gaps of 2∆. The transitions between metals
are all horizontal, i.e. they occur at constant energy levels after adjusting the
relative levels of the chemical potentials to account for the potential difference.
This model also presumes the symmetry between states outside and inside the
Fermi surface and it is inadequate for dealing with processes in which Cooper
pairs tunnel between metals. The states above the Fermi level (E > 0) are
depopulated because there is no injection of particles from the energy gap in
reservoir 3, but extraction to states above the gap in reservoir 1 is possible.
The states below the Fermi level (E < 0) are overpopulated because injection
is possible but extraction to gap is not. This can also be seen as excited holes
tunneling out of the island. Maximum cooling effect is achieved with a voltage
of eV = 2∆L. If the voltage is further increased, hole injection from above
2 Here and in the following TC denotes the critical temperature of the leads, i.e. reservoirs
1 and 3.
14

Reservoir 1 Island 2 Reservoir 3
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Figure 4: A semiconductor model for the tunneling of quasiparticles in the case
eV > 2∆L. The dashed lines represent the Fermi level in each part. The shape
of the distribution in the middle electrode can be understood by noting that the
tunneling of electrons (e − ) or holes (h + ) to the energy gap is forbidden. The
divergence of the BCS density of states enhances the tunneling near gap edges
because the quasiparticles have more available states to tunnel to. The electric
current through the SISIS control line flows from left to right.
the gap and particle injection from below the gap in reservoirs 1 and 3, respectively,
becomes possible. This leads to the peculiar shape of f2 for energies
|E| < eV/2 − ∆L.
In Fig. 5 the distribution function is plotted at higher bath temperatures for
bias voltages eV/∆0 = 1 and eV/∆0 = 3. At higher temperatures the reservoirs
have more excited quasiparticles above and below the gap, and the small notches
at |E| = eV/2+∆L are a result of their injection. The shape of the distribution
for |E| < eV/2 − ∆L is retained at high voltages and it should be noted that
the edges stay very sharp regardless of the temperature.
We also need to know the density of states in addition to their occupation
within the superconductor. This is given by g (−) , and we only need to find out
the magnitude of the energy gap. This is given by Eq. (23). The interaction
constant λ can be excluded in favor of the zero-temperature order parameter
by setting fL2 = fL2(T → 0) = sgn(E) and taking the limit γ → 0 in f (−) . We
have
EC
|∆0|
|∆0| =λ dE
∆0 E2 − |∆0| 2 = λ|∆0|
EC +
ln
E2
C − |∆0| 2
|∆0|
2EC
≈λ|∆0| ln ,
|∆0|
15

f 2
1
0.8
0.6
0.4
0.2
0
v=1.0
E=−eV/2−∆ L
E=eV/2+∆ L
T=0.5 T C
T=0.7 T C
T=0.9 T C
−3 −2 −1 0 1 2 3
E/∆
0
(a) eV/∆0 = 1
f 2
1
0.8
0.6
0.4
0.2
0
v=3.0
E=−eV/2−∆ L
E=−eV/2+∆ L
E=eV/2−∆ L
T=0.5 T C
T=0.7 T C
T=0.9 T C
E=eV/2+∆ L
−3 −2 −1 0 1 2 3
E/∆
0
(b) eV/∆0 = 3
Figure 5: Nonequilibrium distribution function for the superconducting island
at various bath temperatures for voltages eV/∆0 = 1 (a) and eV/∆0 = 3 (b).
valid when EC ≫ |∆0|. Therefore
λ =
ln
1
2EC
|∆0|
For the numerics we adopt a method presented in [11]. We may approximate
that fL2 ≈ 1 for energies above a certain scale E∗ and thus get
∗
E
1
|∆| ≈ dE(fL2 − 1)f
ln 0
(−)
EC
|∆|
+ dE
∆ E2 − |∆| 2
2EC
|∆0|
E ∗
0
.
2EC
|∆0|
|∆|
1
= dE(fL2 − 1)f
2EC
ln 0
|∆0|
(−)
2EC
+ |∆| ln
|∆|
ln
·
2EC 2EC
ln =ln +
|∆0| |∆|
1
∗
E
dE(fL2 − 1)f
|∆| 0
(−)
|∆|
ln =
|∆0|
1
∗
E
dE(fL2 − 1)f
|∆| 0
(−)
|∆| =∆0 exp − 1
∗
E
dE(1 − fL2)f
|∆|
(−)
. (34)
The magnitude of the order parameter of the island as a function of voltage at
various bath temperatures is shown in Fig. 6(a). At T = 0.1 TC the odd-in-
E part of the distribution is effectively unchanged outside the gap giving the
same result as for equilibrium. However, above eV = 2∆L the peculiar shape
of the distribution makes it possible to have a lower value solution for the order
parameter as well, giving rise to a hysteretic behavior with three solutions. Once
voltage reaches eV = 2(∆2 + ∆L) the only possible solution is ∆2 = 0, because
16

∆/∆ 0
1.1
1
0.9
0.8
0.7
0.6
0.5
eV=2(∆ L +∆ 2 )
0.4
0.3
0.2
0.1
T=0.1 T
C
T=0.5 T
C
T=0.7 T
C
T=0.9 T
C
0
0 0.5 1 1.5 2 2.5
eV/∆
0
3 3.5 4 4.5 5
(a)
∆/∆ 0
1.4
1.2
1
0.8
0.6
0.4
0.2
T=0.7 T C
eV=2(∆ 2 −∆ L )
∆ 2 /∆ L =0.2
∆ 2 /∆ L =0.5
∆ 2 /∆ L =0.7
∆ 2 /∆ L =1
∆ 2 /∆ L =1.3
0
0 1 2 3 4
eV/∆
0
Figure 6: The order parameter as a function of bias voltage at various bath
temperatures (a) and ratios ∆2/∆L (b).
the order parameter can never exceed its zero-temperature value, which would
be needed in order to move the gap above eV/2 − ∆L (this stems from the fact
that fL can never exceed 1). Equation (23) determines a situation in which the
free energy of the system is minimized, i.e., dF/d∆ = 0. This condition can
also describe a maximum in F [12]. Therefore, the multivalued behavior of the
order parameter can be interpreted as different minima and maxima in the free
energy. The values of the order parameter corresponding to minima in the free
energy are the solutions of Eq. (34) with a positive derivative with respect to
∆
d
d∆
∆=∆2
|∆| − ∆0 exp
− 1
|∆|
E ∗
0
(b)
dE(1 − fL2)f (−)
> 0.
In this case the largest and smallest values represent minima and the middle
value represents a maximum. If we increase the voltage from zero, the system
stays in the free-energy minimum corresponding to a superconducting state.
Once we enter the hysteretic region, thermal fluctuations may cause the system
to jump to normal state, which is the other free-energy minimum. In the absence
of fluctuations, the system finally jumps to normal state at eV = 2(∆2 + ∆L).
If we now proceed by decreasing the voltage, the jump to the superconducting
state may again occur somewhere in the hysteretic region. Once the voltage is
decreased enough, only the superconducting state is possible.
At higher bath temperatures the order parameter is initially in its equilibrium
value, but increases with increasing voltage as the island cools. In Fig.
6(b) the order parameter is shown for single temperature of T = 0.7 TC but
for different zero-temperature ratios ∆2/∆L = TC2/TC. For lower ratios, the
island is initally in normal state because the bath temperature is above its critical
temperature. Upon increasing the voltage, the island turns superconducting
once the electron distribution is able to support an energy gap.
Finally, we calculate the supercurrent through the island using Eq. (28).
In this case fT2 = fT4 = 0, so the latter term does not contribute, and the
17

eR 4 I/∆ 0
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
∆ 2 /∆ L =1
T=0.1 T C
T=0.5 T C
T=0.7 T C
T=0.9 T C
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
eV/∆
0
(a)
eR 4 I/∆ 0
1.4
1.2
1
0.8
0.6
0.4
0.2
∆ 2 /∆ L =0.7
T=0.7 T C
∆ 4 /∆ L =0.8
∆ 4 /∆ L =0.9
∆ 4 /∆ L =1.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
eV/∆
0
Figure 7: The supercurrent as a function of bias voltage at different bath temperatures
(a) and ratios ∆4/∆L (b). The rising and falling edges of the hysteresis
have been marked with arrows. The small arrows in (b) indicate voltages above
which ∆2 > ∆4.
expression for the supercurrent reduces to the form of the dc Josephson equation
Eq. (1), with a critical current of
IC = − 1
dE fL2f
2eR4
(−)
2 f (+) (−)
4 + fL4f 4 f (+)
2 . (35)
For γ → 0, the first part of the integrand above is finite between ∆2 < E < ∆4
and the second part is finite between ∆4 < E < ∆2. The critical current versus
control voltage has been plotted in Fig. 7(a) for various bath temperatures.
As can be seen, the supercurrent through the junction depends linearily on the
magnitude of the order parameter in the island and the hysteretic behaviour of
∆2 carries over to the supercurrent as well. Once the island jumps into normal
state, the supercurrent vanishes. In Fig. 7(b) the more interesting characteristics
of the supercurrent are shown. For ∆2/∆L = 0.7 the non-hysteretic rising
edge at low voltage gives a well-tunable supercurrent. The lower ratios ∆4/∆L
also exhibit a current drop at voltage above which ∆2 > ∆4 where the second
part of the integrand contributes. For higher ratios ∆2 is always smaller than
∆4 and changing the ratio merely scales the magnitude of the supercurrent.
The electric current through the SISIS control line may be calculated with Eq.
(29). The resulting current is plotted in Fig. 8. It has the familiar form of
a current through a voltage biased SIS-junction. At low bath temperatures
with few excited quasiparticles there is no current until the voltage exceeds
eV = 2(∆L + ∆2) and the potential difference gives enough energy to create
a particle on one side and a hole on the other side. At higher temperatures a
small current exists below that due to tunneling of thermally excited particles
above the gap.
18
(b)

eR 1 I/∆ 0
2.5
2
1.5
1
0.5
T=0.1 T C
T=0.7 T C
T=0.9 T C
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
eV/∆ 0
eV=2(∆ L +∆ 2 )
Figure 8: The electric current through the SISIS control line as a function of
bias voltage.
4.2 Relaxation-time approximation
The quasiparticles entering the island tend to relax towards energy equilibrium
as discussed in Subs. 2.1.2. This is achieved via inelastic collisions with impurities,
lattice vibrations (phonons) and other electrons. At low temperatures the
most relevant scattering mechanism is electron-electron scattering. The simplest
method to include the energy relaxation of the injected particles is the
relaxation-time approach. Typically one compares the interaction times measured
from experiments to some characteristic scale of the problem. We assume
that the distribution function has relaxed to the form
f(E) = f 0 (E) + δf(E),
where f0 is a Fermi function and δf is a small deviation from equilibrium.
Therefore we may expand the collision integral as
J (ee)
1 [f] ≈
δf ≡ − 1 0
f − f , (36)
∂J (ee)
1
∂f
f=f 0
where τee is the electron-electron scattering time which is assumed independent
of the energy of the excitations. We further assume that f 0 is the Fermi
function in quasiequilibrium, i.e., steady state where electric and energy currents
are conserved. These define the chemical potential and temperature of
the quasiequilibrium, respectively, and are given by Eqs. (27) and (30). Inserting
these into kinetic equations yields the solution, presented in terms of the
19
τee

f 2
1
0.8
0.6
0.4
0.2
0
v=1.0 T=0.7 T C
E=−∆ 2
E=∆ 2
Γτ ee =10 5
Γτ ee =100
Γτ ee =1
Γτ ee =0.01
−3 −2 −1 0 1 2 3
E/∆
0
(a) eV/∆0 = 1
f 2
1
0.8
0.6
0.4
0.2
0
v=3.0 T=0.7 T C
E=−∆ 2
E=∆ 2
Γτ ee =10 5
Γτ ee =100
Γτ ee =1
Γτ ee =0.01
−3 −2 −1 0 1 2 3
E/∆
0
(b) eV/∆0 = 3
Figure 9: Nonequilibrium distribution function for the superconducting island
for various values of Γτee and voltages eV/∆0 = 1 (a) and eV/∆0 = 3 (b).
full distribution functions,
f2 =
g (−)
E+µ f1 + g (−)
E−µ f3 +
e2
ν2Ω2R/ g (−)
g (−)
E+µ + g(−)
E−µ + e2ν2Ω2R/ 2,E τee
g (−)
2,E τee
f 0 2
. (37)
The strength of the relaxation may be adjusted by varying the dimensionless
parameter (Γτee) −1 ≡ e 2 ν2Ω2R/τee, where Γ is the tunneling injection rate into
the island. The full distribution function for various strengths of the relaxation
is shown in Fig. 9. As Γτee → ∞, the full nonequilibrium is retained. The
limit Γτee → 0 corresponds to quasiequilibrium, where f2 → f 0 2. The density of
states for the island appearing in Eq. (37) enhances the relaxation inside the
energy gap and creates sharp drops at |E| = ∆2.
The effect of energy relaxation to the magnitude of the order parameter
and supercurrent is shown in Fig. 10. With large values of Γτee no changes
with respect to the full nonequilibrium can be seen. This is because the part of
the distribution function affecting the magnitude of the order parameter, i.e.,
outside the gap, is mostly unchanged. The ehancement of the order parameter
is suppressed with Γτee = 1 and vanishes completely when Γτee = 0.01. Also
the hysteretic behaviour is lost in that limit.
4.3 Collision integral
The more accurate method to include the energy relaxation is to use the full
electron-electron collision integrals. Here we use the collision integral for the
simplest form of electron-electron interaction, namely point interaction. If a
superconductor is devoid of all impurities, the potential of a distant electron
is completely screened by all other electrons. Therefore the potential may be
20

∆/∆ 0
1
T=0.7 T
C
0.8
0.6
0.4
0.2
Γτ ee =10 5
Γτ ee =100
Γτ ee =1
Γτ ee =0.01
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
eV/∆
0
(a) Order parameter
eR 4 I/∆ 0
1.4
1.2
1
0.8
0.6
0.4
0.2
Γτ ee =10 5
Γτ ee =100
Γτ ee =1
Γτ ee =0.01
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
eV/∆
0
(b) Supercurrent
Figure 10: The magnitude of the order parameter (a) and supercurrent (b) for
various values of Γτee at T = 0.7TC. Here all the superconductors were assumed
similar, i.e., ∆2/∆L = ∆4/∆L = 1. The curves for Γτee = 10 5 and Γτee = 100
coincide.
modelled with a delta function V (r) = V δ(r) and the collision integral is [9]
J (ee)
1 =4λ2 ee π
pFvF
×
dE1dE2
g (−)
E1 g(−) − f(−)
E2 E1 f(−)
g E2
(−)
E g(−)
E3
(1 − fE)fE1fE2fE3 − fE (1 − fE1)(1 − fE2) (1 − fE3)
+ f(−)
E f(−)
E3
, (38)
where pF and vF are Fermi momentum and velocity, respectively, λee is the
electron-electron interaction constant and energies satisfy the conservation law
E = E1+E2+E3. We note that because of the terms g (−)
E3
and f(−)
E3 multiplying
the kernel of the integral, J (ee)
1 ≈ γ when |E3| < ∆ and therefore it has very
little effect on excitations inside the gap. We also note that the relaxation is
strongest for excitations at |E3| = ∆. This is in stark contrast to the assumed
energy independent scattering time of the relaxation-time approximation. Moreover,
the collision integral at a given energy is in general not proportional to
the difference between the true distribution function and the quasiequilibrium
function f 0 evaluated at the same energy. The collision integral vanishes for the
quasiequilibrium function however, as can be seen by inserting f = f 0 . This
time it is not possible to solve f2 explicitly, rather we must solve the resulting
self-consistent equation
f2 =
g (−)
E+µ f1 + g (−)
E−µ f3 +
g (−)
E+µ + g(−)
E−µ
e 2 ν2Ω2R/g (−)
2,E
J (ee)
(39)
by numerical methods. The strength of the relaxation can be adjusted by varying
the parameter κ ≡ 4πλ 2 eee 2 ν2Ω2R/pFvF. The distribution function for
various κ with T = 0.7 TC is shown in Fig. 11. With the proper collision integral
the distribution at energies inside the gap is practically unaffected. At the
21

f 2
1
0.8
0.6
0.4
0.2
0
T=0.7 T C
v=1
E=−∆ 2
E=∆ 2
κ=0
κ=0.01
κ=0.1
κ=1
−3 −2 −1 0 1 2 3
E/∆
0
(a) eV/∆0 = 1
f 2
1
0.8
0.6
0.4
0.2
T=0.7 T C
v=3
E=−∆ 2
κ=0
κ=0.01
κ=0.1
κ=1
0
E=∆
2
−5 −4 −3 −2 −1 0
E/∆
0
1 2 3 4 5
(b) eV/∆0 = 3
Figure 11: The full distribution function for the island for different stregths of
relaxation and voltages eV/∆0 = 1 (a) and eV/∆0 = 3 (b). Only quasiparticles
with energies outside the energy gap ∆2 are affected.
lower voltage the island stays in the superconducting state with finite energy
gap even with very high relaxation, and the quasiequilibrium limit cannot be
reached. At the higher voltage the difference to relaxation-time approximation
is most notable for small values of κ or (Γτee) −1 . Once the strength of the
relaxation is increased, the gap goes to zero and the difference is lost. The
order parameter and supercurrent for various values of κ are shown in Fig. 12.
The magnitude of the order parameter is hysteretic even with strong relaxation
contrary to the behaviour in the quasiequilibrium limit, where the hysteretic
behaviour vanished.
4.4 Asymmetric structure
Let us now examine an asymmetric situation, where R1 = R3 or ∆1 = ∆3. By
solving Eqs. (13) and (14) with J (inel)
1 = J (inel)
2 = 0 we obtain quite lengthy
expressions for the odd and even parts of the distribution function
DfL2 =4e 2 f (+)
2 ν2Ω2|∆2| G1 (fL1 − fT1)g (−)
1,E−µ1 + (fL1 + fT1)g (−)
+ 1,E+µ1
G3
− g (−)
2,E
+G3
DfT2 = − g (−)
+G3
(fL3 − fT3)g (−)
3,E−µ3 + (fL3 + fT3)g (−)
3,E+µ3
2,E
G1
2fL1G1g (−)
1,E−µ1 + (fL1 + fL3 + fT1 − fT3)G3g (−)
3,E−µ3
2fL3G3g (−)
3,E−µ3 + (fL1 + fL3 − fT1 + fT3)G1g (−)
1,E−µ1
G1
g (−)
3,E+µ3
2fT1G1g (−)
1,E−µ1 + (fL1 − fL3 + fT1 + fT3) G3g (−)
3,E−µ3
2fT3G3g (−)
3,E−µ3 + (−fL1 + fL3 + fT1 + fT3)G1g (−)
1,E−µ1
22
g (−)
1,E+µ1
,
g (−)
1,E+µ1
g (−)
3,E+µ3
(40)
,

∆/∆ 0
1
0.8
0.6
0.4
T=0.7 T C
0.2
κ=0
κ=0.01
κ=0.1
κ=1
0
0 1 2
eV/∆
0
3 4
(a) Order parameter
eR 4 I/∆ 0
1.2
1
0.8
0.6
0.4
T=0.7 T C
0.2
κ=0
κ=0.01
κ=0.1
κ=1
0
0 1 2
eV/∆
0
3 4
(b) Supercurrent
Figure 12: The magnitude of the order parameter (a) and supercurrent (b) for
various strength of the relaxation and T = 0.7TC. Again all the superconductors
were assumed similar. The small notches in the supercurrent with κ = 1 and
κ = 0.1 are again the result of different parts of the integrand in Eq. (28) being
finite.
where
D =4e 2 f (+)
2 ν2Ω2|∆2| G1 g (−)
+ g(−) + G3 g 1,E−µ1 1,E+µ1
(−)
3,E−µ3
G1g (−)
(−)
+ G3g G1g 1,E−µ1 3,E−µ3
(−)
(−)
+ G3g ,
1,E+µ1 3,E+µ3
− 2g (−)
2,E
+ g(−)
3,E+µ3
and Gi = 1/Ri are the conductances of the tunnel contacts. The distribution
functions depend on the volume, energy gap and normal state density of states
at the Fermi level, but these can be included in dimensionless constants of the
type G/|∆|νΩe 2 . In the asymmetric case the potentials µ1 and µ3 must be
chosen such that the electrical current is conserved. This implies the vanishing
of the total net current I into the island in Eq. (27). The conservation of electric
current also forces the imaginary part in Eq. (24) to vanish regardless of fT.
The current through the junction and therefore also the potentials depend on
the order parameter of the island. Since the correct potentials are in turn needed
for calculating the order parameter, the equations must be solved iteratively.
However, at low temperatures and low bias voltages the order parameter stays
reasonably close to ∆0, and we can get an idea of the behaviour of the potentials
by choosing the order parameter to be a constant. The potential of reservoir 1
has been plotted in Fig. 13 for different degrees of asymmetry at eV/∆0 = 1
with a fixed order parameter. At low bath temperature the current through
the control line is very small and very high resistance or energy gap ratios are
required in order to significantly change the potential from eV/2. At higher
temperatures the thermally excited quasiparticles are able to tunnel above the
gaps leading to a higher current and a larger effect to the potentials. In the
limit ∆1,2,3 → 0 the condition on current conservation results in the Kirchoff’s
voltage law.
The full distribution function at T = 0.7 TC1 has been plotted for various
23

eµ 1 /∆ 0
1
0.9
0.8
0.7
0.6
0.5
10 0
v=1
10 1
R 1 /R 3
(a) ∆1 = ∆3
T=0.1 T C
T=0.7 T C
10 2
eµ 1 /∆ 0
1
0.9
0.8
0.7
0.6
v=1
T=0.1 T C
0.5
T=0.7 T
C
1 1.5 2 2.5 3 3.5
∆ /∆
1 3
4 4.5 5 5.5 6
(b) R1 = R3
Figure 13: The potential µ1 as a function of the ratio of resistances (a) or order
parameters (b). The order parameter of the island has been fixed at ∆2 = ∆1.
In the asymmetric situation we have introduced a slightly different notation
with ∆0 = ∆1(T = 0) and TC = TC1. Also note the logarithmic scale in (a).
zero-temperature ratios ∆3/∆1 with a voltage eV/∆0 = 2.0 and ratios R1/R3
with a voltage eV/∆0 = 1.5 in Fig. 14. Qualitatively the distribution function
takes a form that can be deduced from the simple model presented earlier. The
small notches at |E| = |µi| + ∆i result from the injection of thermally excited
quasiparticles above the gaps of the reservoirs. The larger peaks at |E| =
|µi| − ∆i result from the injection of quasiparticles below the gaps. From the
small notches we may also observe that the distribution is no longer symmetric
around E = 0 because fT is finite and there is charge imbalance.
The order parameter and supercurrent for χ4 − χ2 = π/2 is shown in Fig.
15. In the asymmetric structure the magnitude of the order parameter seems
to take a value close to its value in the symmetric case with a voltage of eV =
2 max(|µ1|, |µ3|). This is reasonable because the distribution function in the
region |E| > max(∆1 + |µ1|, ∆3 + |µ3|) is similar to the distribution in the
symmetric structure. The superconductivity is lost once |µ1| > ∆2 + ∆1 or
|µ3| > ∆2 + ∆3. With high asymmetry ratios the potentials differ very much
from ±eV/2 and supercondictivity is lost at a lower bias voltage compared to
the symmetric structure. Also the hysteretic region is evident. Once the voltage
is increased over the hysteretic region, no solution to Eq. (24) together with the
conservation of current can be found. This may lead to some time-dependent
dynamic behaviour to which our formalism is unsuitable.
Because fT is finite, also the latter part of Eq. (28) may contribute depending
on the phase difference between the superconductors. Its magnitude can be
investigated by setting the phase difference to zero. In this case the supercurrent
is significantly smaller. The cos(∆χ) deviation from the dc Josephson relation
is negligible, because it is at least three orders of magnitude smaller than the
product g (−)
2 g(−)
4 . This is illustrated in Fig. 16. The small magnitude of the
cos(∆χ) deviation results either from the small values of fT2 or small values of
24

eR 4 I/∆ 0
f 2
1.2
T=0.7 T
C
1
0.8
0.6
0.4
0.2
1
0.8
0.6
0.4
0.2
0
R 1 /R 3 =1.1
R 1 /R 3 =2
R 1 /R 3 =5
−3 −2 −1 0 1 2 3
E/∆
0
(a) ∆1 = ∆3
f 2
T=0.7 T
C
1
0.8
0.6
0.4
0.2
0
∆ 3 /∆ 1 =1.1
∆ 3 /∆ 1 =2
∆ 3 /∆ 1 =3
−3 −2 −1 0 1 2 3
E/∆
0
(b) R1 = R3
Figure 14: The distribution function for different ratios R1/R3 at eV/∆0 = 1.5
(a) and ∆3/∆1 (b) at eV/∆0 = 2.0. The arrows are located at energies −µ1−∆1,
µ3 − ∆3, −µ1 + ∆1 and µ3 + ∆3 (from left to right). The corresponding peaks
can also be seen above the Fermi level at exactly opposite energies.
R 1 /R 3 =1.1
R 1 /R 3 =2
R 1 /R 3 =5
0
0 0.5 1 1.5 2 2.5
eV/∆
0
(a) ∆1 = ∆3
1.2
1
0.8
0.6
0.4
0.2
0
∆/∆ 0
eR 4 I/∆ 0
1.2
1
0.8
0.6
0.4
0.2
∆ 3 /∆ 1 =1.1
∆ 3 /∆ 1 =2
∆ 3 /∆ 1 =3
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
eV/∆
0
0
(b) R1 = R3
Figure 15: The magnitude of the order parameter and supercurrent for different
ratios R1/R3 (a) and ∆3/∆1 (b).
25
1.2
1
0.8
0.6
0.4
0.2
∆/∆ 0

eR 4 I/∆ 0
x 10−3
1
χ −χ =0
4 2
0
−1
−2
−3
−4
−5
R 1 /R 3 =1.1
R 1 /R 3 =2
R 1 /R 3 =5
−6
0 0.5 1 1.5 2 2.5
eV/∆
0
(a) Supercurrent
eR 4 I/∆ 0
x 10−7
0.5
0
−0.5
−1
−1.5
−2
cos(χ) prefactor
R 1 /R 3 =1.1
R 1 /R 3 =2
R 1 /R 3 =5
−2.5
0 0.5 1 1.5 2 2.5
eV/∆
0
(b) Prefactor of cos(∆χ)
Figure 16: The magnitude of the supercurrent for χ4 − χ2 = 0.
f (+)
2 f (+)
4 . The latter may be increased by smearing out the density of states.
This can be achieved by introducing magnetic impurities to the sample which
is addressed in the next subsection.
In principle, the energy relaxation in the asymmetric case could be included
by solving the kinetic equations with the collision integrals (either proper or
approximated). However, we do not have the proper collision integrals for the
case of finite charge imbalance and their application would be computationally
unwieldy, therefore we do not pursue the energy relaxation further this time.
4.4.1 Spin-flip scattering
If magnetic impurities are introduced to the central island, the injected electrons
may interact with them and scatter. In spin-flip scattering the incident electron
changes the direction of its spin in the scattering event. The spin-flip scattering
smears out the edges in the quasiparticle density of states and creates available
states also inside the gap. The self-energy of the spin-flip scattering is [9]
ˆΣS = i
ˆτ3ˆg ˆτ3, (41)
2τS
where 1/τS ≡ γ is the spin-flip rate. In this case the steady-state Usadel equation
can be written [13]
f R(A)
2 E + iγf R(A)
2 g R(A)
2 = g R(A)
2 |∆|. (42)
The solution to this equation is presented with a slightly different notation in
[13]. Even with the additional states inside the gap, the prefactor to cos(∆χ)
is still two orders of magnitude smaller than the complete supercurrent with
χ4 − χ2 = 0 as can be seen from Fig. 17. We can conclude that fT2 is too
small where the rest of the integrand is finite for the cosine-part to contribute.
Indeed, as can be seen from Fig. 14 fT2 is finite only near the small notches at
energies much larger than the gap, whereas f (+)
2 f (+)
4 is sufficiently large only
inside the gap irregardless of the magnitude of γ.
26

eR 4 I/∆ 0
1
0
−1
−2
−3
x 10−4
2
−4
0 0.5 1 1.5 2
eV/∆
0
(a)
I C
cos(χ) prefactor
× 10 −2
eR 4 I/∆ 0
x 10−4
0.5
0
−0.5
−1
−1.5
−2
−2.5
−3
× 10 −2
−3.5
I
C
cos(χ) prefactor
−4
0 0.1 0.2
γ
0.3 0.4 0.5
Figure 17: The magnitude of the supercurrent and prefactor of cos(∆χ) for
χ4 − χ2 = 0 with a spin-flip rate of γ = 0.0125 (a) and with various spin-flip
rates at eV/∆0 = 2 (b).
5 Discussion
We have studied the SISIS Josephson transistor in full nonequilibrium regime
with the quasiclassical Keldysh-Green function formalism. The equations derived
here are mostly the same as in Ref. [8], with the exception of the expression
for the supercurrent in Eq. (28) and the exact form of the Green functions in
a superconductor with tunnel junctions in Eq. (21). The results show that
the self-consistent calculation of the superconducting order parameter is important,
because its hysteretic behaviour in nonequilibrium prevents the use of this
structure as a transistor with a large differential current gain. The rising edge
of the supercurrent with unequal energy gaps in Fig. 7(b) can still be utilized,
as well as the non-hysteretic falling edge where ∆2 and ∆4 cross.
Another approach would be to probe the degree of nonequilibrium in the
superconducting island by measuring the supercurrent flowing through it. However,
the deviations from the Fermi distribution are strongest inside the energy
gap, whereas the supercurrent is only affected by the distribution outside the
gap. Only the changing magnitude of the order parameter and its effect to
the supercurrent can produce observable characteristics that would enable us to
extract information about the degree of nonequilibrium.
The energy relaxation inside the energy gap with the proper collision integral
is qualitatively very different from the relaxation-time approximation. As
a result the quasiequilibrium limit is unreachable as long as the island is superconducting.
Outside the gap their difference is much smaller. The magnitude
of the order parameter and therefore also the supercurrent drop down to low
values more quickly with the proper energy relaxation and most notably the
sharp characteristics of the supercurrent in the quasiequilibrium limit become
hysteretic. To our knowledge this calculation has not been done before for
superconductors.
To further pursue the study of this topic one could take into account the
27
(b)

operation of the Josephson junctions in the dissipative regime, which is required
in order to calculate the real values of power and current gain. The actual
behaviour of the asymmetric system at large voltages can also present interesting
results. The possibility of a supercurrent with a cos(∆χ) dependence in the
presence of a charge imbalance might warrant some additional study as well,
and it might be realizable with a different structure.
References
[1] F. Giazotto, T. T. Heikkilä, F. Taddei, R. Fazio, J. P. Pekola, and F. Beltram.
Phys. Rev. Lett., 92, 137001 (2004).
[2] A. M. Savin, J. P. Pekola, J. T. Flyktman, A. Anthore, and F. Giazotto.
Appl. Phys. Lett., 84, 4179 (2004).
[3] F. Giazotto and J. P. Pekola. J. Appl. Phys., 97, 023908 (2004).
[4] M. Tinkham. Introduction to superconductivity. Dover publications Inc.,
Mineola, New York, 2nd edition (1996).
[5] F. Giazotto, T. T. Heikkilä, A. Luukanen, A. M. Savin, and J. P. Pekola.
Rev. Mod. Phys., 78, 217 (2006).
[6] J. Rammer and H. Smith. Rev. Mod. Phys, 58, 323 (1986).
[7] W. Belzig, F. K. Wilhelm, C. Bruder, G. Schön, and A. D. Zaikin. Superlattices
and Microstructures, 25, 1251 (1999).
[8] J. Voutilainen, T. T. Heikkilä, and N. B. Kopnin. Phys. Rev. B, 72, 054505
(2005).
[9] N. B. Kopnin. Theory of nonequilibrium superconductivity. International
series of monographs on physics. Oxford university press, Oxford (2001).
[10] U. Eckern and A. Schmid. J. Low Temp. Phys., 45, 137 (1981).
[11] T. T. Heikkilä. Simple model for nonequilibrium heating of a superconductor
(2004). Unpublished.
[12] D. R. Heslinga and T. M. Klapwijk. Phys. Rev. B, 47, 5157 (1993).
[13] A. Anthore. Decoherence mechanisms in mesoscopic conductors. Ph.D.
thesis, University of Paris (2003).
[14] G.-L. Ingold and Y. V. Nazarov. Single charge tunneling, volume 294 of
NATO ASI Series B: Physics, chapter 2, pages 21–107. Plenum Press, New
York (1992).
[15] T. T. Heikkilä. Private communication.
[16] V. Ambegaokar and A. Baratoff. Phys. Rev. Lett., 10, 486 (1963).
28

A Tunneling rate
A standard way of calculating the tunneling rate through a junction is to apply
perturbation theory. The Hamiltonian of the system is written
H = Hqp + HT, (43)
where Hqp describes the quasiparticle dynamics and HT their tunneling through
the junction. Written with the help of the creation and annihilation operators
c † and c, the tunneling Hamiltonian has the form [4]
HT =
σkq
Tkqc †
kσ cqσ + Herm. conj., (44)
where the subscript k refers to one metal and q to the other and T is a tunneling
matrix element. Therefore the first part of the tunneling Hamiltonian transfers
an electron from the first metal with wave vector k to the second with wave
vector q, while the Hermitian conjugate does the opposite. If the tunneling
Hamiltonian is small, we may apply Fermi’s Golden Rule, i.e., use the leading
order approximation in the perturbation to calculate the tunneling rate (see for
example [14]). According to it, the transition rate from some initial state i to a
final state f is given by
Γf←i = 2π |〈f|HT |i〉| 2 δ(Ef − Ei). (45)
To obtain the total tunneling rate from metal 1 to metal 2 we need to sum over
all initial states weighed with their probabilities and over all final states:
Γ12(V ) = 2π
pk |〈ǫq|HT |ǫk〉| 2 δ(ǫq − ǫk − eV ), (46)
kq
where pk is the probability of an initial state and eV is the difference in chemical
potentials. We note that the matrix element 〈ǫq|c †
kσ cqσ|ǫk〉 is nonvanishing only
for initial states that have a quasiparticle of wave vector k and spin σ present and
final states that have a state with wave vector q and spin σ unoccupied. Now
we may write the occupation probabilities with Fermi-functions and transform
the sum to an integral
∞
Γ12(V ) =2π dǫkdǫqf 0 (ǫk)(1 − f 0 (ǫq))
|Tkq| 2 ν1(ǫk)ν2(ǫq)Ω1Ω2δ(ǫq − ǫk − eV )
= 1
e 2 RT
−∞
σkq
∞
dEf 0 (E)(1 − f 0 (E + eV ))
−∞
≈ eV
e2 , (47)
RT
where we have introduced the tunnel resistance RT ≡ (4πe 2 |T | 2 ν1ν2Ω1Ω2) −1
and assumed that the densities of state are effectively constant in the range
where the integrand is finite. We have further assumed a tunneling probability
T independent of the initial or final states. The last approximation is valid at
low temperatures. The total current through the junction is given by
I(V ) = e (Γ12(V ) − Γ21(V )), (48)
29

which results in the same expression as Eq. (29). For the tunneling rate from
one initial state (which also gives the lifetime of the state) we have [15]
η12 = Γ12
ν1Ω1eV = (4ν1e 2 Ω1RT) −1 , (49)
which is also used in the tunneling self energy.
B Ambegaokar-Baratoff formula for the critical
current
A formula for the critical current between two similar superconductors (same
energy gaps) in equilibrium was derived by Ambegaokar and Baratoff in [16].
The same formula can be derived analytically from Eq. (35) and we present
it here for completeness. At zero bias voltage fL2 = fL4 = tanh(E/2T) and
f (−)
2 f (+)
4
IC = − 1
eR
= f(−) 4 f (+)
2 = 1/4i(fR − fA )(fR + fA ) resulting in
∞
E 1 |∆|
dE tanh
2T 4i
2
(E + iγ) 2 |∆|
−
− |∆| 2 2
(E − iγ) 2 − |∆| 2
.
−∞
This has poles at zj = ±|∆| ±iγ and zj = 2iTπ 1
2 + m , where m is an integer.
Integration along a contour around the poles in the upper half plane and the
use of residue theorem yields
IC = 1
⎧
⎨
π
z
Res tanh
eR ⎩ 2 z=zj 2T
|∆| 2
(z + iγ) 2 |∆|
−
− |∆| 2 2
(z − iγ) 2 − |∆| 2
⎫ ⎬
⎭ .
j
Calculating the residues and letting γ → 0 results in
IC = π|∆|
2eR tanh
which is the Ambegaokar-Baratoff-relation.
30
|∆|
2T
, (50)