# Phase-coherent thermoelectricity in superconducting hybrids (Brief Review)

Mikhail S. Kalenkov  I.E. Tamm Department of Theoretical Physics, P.N. Lebedev
Physical Institute, 119991 Moscow, Russia  Andrei D. Zaikin  Institute for
Quantum Materials and Technologies, Karlsruhe Institute of Technology (KIT),
76021 Karlsruhe, Germany  I.E. Tamm Department of Theoretical Physics, P.N.
Lebedev Physical Institute, 119991 Moscow, Russia

###### Abstract

We review some recent advances in studies of phase-coherent thermoelectric
effects in superconducting hybrid structures such as, e.g., Andreev
interferometers. We elucidate a number of mechanisms of electron-hole symmetry
breaking in such systems causing dramatic enhancement of thermoelectric
effects. We demonstrate that the flux-dependent thermopower exhibits periodic
dependence on the applied magnetic flux $\Phi_{x}$ which in some limits may
reduce to either odd or even functions of $\Phi_{x}$ in accordance with
experimental observations. We also show that dc Josephson current in Andreev
interferometers can be controlled and enhanced by applying a temperature
gradient which may also cause a nontrivial current-phase relation and a
transition to a $\pi$-junction state.

##  I Introduction

Applying an electric field $\bm{E}$ to a normal conductor with Drude
conductivity $\sigma_{N}$ one induces an electric current
$\bm{j}=\sigma_{N}\bm{E}$ across this conductor. Likewise, such a current can
be generated by applying a thermal gradient $\nabla T$, in which case one has
$\bm{j}=\alpha_{N}\nabla T$. This simple equation illustrates the essence of
the so-called thermoelectric effect in normal metals.

Usually the latter effect turns out to be quite small since contributions to
the thermoelectric coefficient $\alpha_{N}$ from electron-like and hole-like
excitations are of the opposite sign and almost cancel each other. As a
result, $\alpha_{N}$ is proportional to a small ratio between temperature $T$
and the Fermi energy $\varepsilon_{F}$, i.e. one has
$\alpha_{N}\sim(\sigma_{N}/e)(T/\varepsilon_{F})$.

This simple physical picture gets substantially modified as soon as a normal
metal is brought into a superconducting state. On one hand, the electric field
no longer penetrates into a superconductor and, hence, the Drude contribution
to the current vanishes. On the other hand, a supercurrent $\bm{j}_{s}$ can
now be induced in the system without applying any electric field. This current
exactly compensates for any thermoelectric current,
$\bm{j}_{s}=-\alpha_{S}\nabla T$, thus making the net electric current vanish.
For this reason the thermoelectric effect in uniform superconductors cannot be
detected in any way.

The way out has been suggested by Ginzburg Ginzburg44 ; Ginzburg91  who has
demonstrated that no such compensation takes place in spatially non-uniform
superconductors. This observation offers a possibility to experimentally
investigate thermoelectric currents in such structures as, e.g.,
superconducting hybrids. Still, similarly to normal metals the magnitude of
the thermoelectric effect in superconductors was expected Galperin73 ;
Aronov81  to be very small, i.e. $\alpha_{S}\sim\alpha_{N}$ at $T\sim T_{C}$
and $\alpha_{S}\ll\alpha_{N}$ at $T\ll T_{C}$, where $T_{C}$ is the
superconducting critical temperature.

Quite unexpectedly, already first experiments with bimetallic superconducting
rings Zavaritskii74 ; Falco76 ; Harlingen80  revealed the thermoelectric
signal which magnitude and temperature dependence were in a strong
disagreement with theoretical predictions Galperin73 ; Aronov81 . In
particular, the magnitude of the thermoelectric effect detected in these
experiments exceeded theoretical estimates by few orders of magnitude.

Later on large thermoelectric signals were also observed in multi-terminal
hybrid superconducting-normal-superconducting (SNS) structures Venkat1 ;
Venkat2 ; Petrashov03 ; Venkat3 ; Vitya  frequently called Andreev
interferometers. Furthermore, the thermopower detected in these experiments
was found to be periodic in the superconducting phase difference $\chi$ across
the corresponding SNS junction. The latter observation (i) indicates that the
thermoelectric signal in superconductors can be phase coherent and (ii) calls
for establishing a clear relation between thermoelectric, Josephson and
Aharonov-Bohm effects in systems under consideration. Both issues (i) and (ii)
– along with an experimentally observed large magnitude of the thermoelectric
effect – constitute the key subjects of our present review.

It is obvious from the above considerations that large thermoelectric effects
(not restricted by a small parameter $T/\varepsilon_{F}$) in superconducting
structures can be expected provided electron-hole symmetry in such structures
is violated in some way. In this case the contributions from electron-like and
hole-like excitations to the thermoelectric coefficient would not cancel each
other anymore and, hence, $\alpha_{S}$ can become large.

Figure 1: A simple setup illustrating electron-hole symmetry breaking due to
Andreev reflections (trajectory $b$). The setup consists of a superconducting
ring pierced by external magnetic flux $\Phi_{x}$ and attached to a piece of a
normal metal.

Electron-hole asymmetry in superconducting hybrid structures can be realized
in a variety of physical situations. As a simple example, let us consider a
superconducting ring pierced by external magnetic flux $\Phi_{x}$ and
interrupted by a normal metal as it is schematically shown in Fig. 1.
Quasiparticles propagating from a normal metal towards a superconducting ring
eventually hit either one NS interface (e.g., trajectory $a$) or both of them
(e.g., trajectory $b$). In either case an incoming electron with energy
$\varepsilon$ may be reflected back as a hole And . In the low energy limit
$\varepsilon\ll\Delta$ and in the case of trajectory $a$ the probability for
this Andreev reflection process reads

$\mathcal{R}_{a}^{e-h}(\varepsilon)=D_{1}^{2}/(1+R_{1})^{2},$ (1)

which is nothing but the standard BTK result BTK . Here and below
$D_{1,2}=1-R_{1,2}$ denote normal transmissions of the corresponding NS
interfaces.

Likewise, the Andreev reflection probability for electrons following
trajectory $b$ can be derived in the form KZ17

$\mathcal{R}_{b}^{e-h}(\varepsilon)=1-\frac{16R_{1}R_{2}}{\bigl{|}(1+R_{1})(1+R_{2})+D_{1}D_{2}e^{i\left(\chi+\frac{2\varepsilon
d}{v_{F}}\right)}\bigr{|}^{2}},$ (2)

where $d\gg v_{F}/\Delta$ is the effective distance covered by a quasiparticle
between successive scattering events at two NS interfaces and
$\chi=2\pi\Phi_{x}/\Phi_{0}$ is the superconducting phase difference across
our SNS junction. In Eq. (2) we again assume $\varepsilon\ll\Delta$.

Making use of general symmetry relations it is straightforward to verify that
the probability for an incoming hole to be reflected back as an electron
equals to $\mathcal{R}^{h-e}(\varepsilon)=\mathcal{R}^{e-h}(-\varepsilon)$.
Combining this relation with Eq. (2) we obtain

$\mathcal{R}_{b}^{e-h}(\varepsilon)\neq\mathcal{R}_{b}^{h-e}(\varepsilon),$
(3)

implying that scattering on two NS interfaces generates electron-hole symmetry
violation in our hybrid structure provided $\chi\neq\pi n$ and $0<D_{1,2}<1$.
Below we will demonstrate that this electron-hole asymmetry yields a large
thermoelectric effect in the system under consideration.

##  II Quasiclassical formalism and circuit theory

###  II.1 Eilenberger and Usadel equations

Throughout our paper we will consider superconducting hybrid structures which
can be described by means of the standard quasiclassical formalism employing
non-equilibrium Green-Keldysh matrix functions $\check{g}$ which obey the
Eilenberger equations bel

$-i\bm{v}_{F}\nabla\check{g}(\bm{p}_{F},\bm{r},\varepsilon,t)=[\check{\Omega}-\check{\Sigma},\check{g}(\bm{p}_{F},\bm{r},\varepsilon,t)],\
\check{g}^{2}=1.$ (4)

The check symbol denotes $4\times 4$ Keldysh matrices

$\check{X}=\begin{pmatrix}\hat{X}^{R}&\hat{X}^{K}\\\ 0&\hat{X}^{A}\\\
\end{pmatrix},\quad X=g,\,\Omega,\,\Sigma,$ (5)

with blocks $\hat{X}^{R,A,K}$ being $2\times 2$ matrices in the Nambu space.
The matrix $\check{\Omega}$ has the standard structure

$\hat{\Omega}^{R}=\hat{\Omega}^{A}=\begin{pmatrix}\varepsilon&\Delta\\\
-\Delta&-\varepsilon\\\ \end{pmatrix},\quad\hat{\Omega}^{K}=0,$ (6)

where $\varepsilon$ is the quasiparticle energy, $\Delta$ is the
superconducting order parameter which is chosen real further below.

Scattering of electrons on impurities is accounted for by the self-energy
matrix $\check{\Sigma}$ which can be expressed in the form

$\check{\Sigma}=-i\Gamma\left<\check{g}\right>+\check{\Sigma}_{\mathrm{m}},\quad\Gamma=v_{F}/(2\ell).$
(7)

Here the first term describes the effect of non-magnetic isotropic impurities
while the second term $\check{\Sigma}_{\mathrm{m}}$ is responsible for
electron scattering on magnetic impurities to be specified below. Averaging
over the Fermi surface is denoted by angular brackets $\left<\cdots\right>$.

Finally, the current density $\bm{j}$ is defined with the aid of the standard
relation

$\bm{j}(\bm{r},t)=-\dfrac{eN_{0}}{4}\int
d\varepsilon\left<\bm{v}_{F}\mathrm{Sp}[\hat{\tau}_{3}\hat{g}^{K}(\bm{p}_{F},\bm{r},\varepsilon,t)]\right>,$
(8)

where $N_{0}$ is the electron density of states per spin direction at the
Fermi level and $\hat{\tau}_{3}$ is the Pauli matrix.

Provided elastic mean free path $\ell$ is small enough, i.e. $\ell\ll
v_{F}/T_{C}$, Eilenberger equations (4) can be transformed into much simpler
diffusion-like Usadel equations

$\displaystyle
iD\nabla\left(\check{G}\nabla\check{G}\right)=\left[\check{\Omega},\check{G}\right],\quad\check{G}\check{G}=1,$
(9)

where $\check{G}=\left<\check{g}(\bm{p}_{F},\bm{r},\varepsilon,t)\right>$ is
isotropic part of the Eilenberger Green function and $D=v_{F}\ell/3$ is the
diffusion coefficient. The current density $\bm{j}$ in diffusion limit is
related to the matrix $\check{G}$ in the standard manner as

$\displaystyle\bm{j}=-\dfrac{\sigma}{8e}\int
d\varepsilon\operatorname{\mathrm{Sp}}(\hat{\tau}_{3}\check{G}\nabla\check{G})^{K},$
(10)

where $\sigma=2e^{2}N_{0}D$ is the normal state Drude conductivity.

###  II.2 Circuit theory for quasi-one-dimensional conductors

In the diffusive limit the above quasiclassical theory of superconductivity
can also be reformulated in the form that in some cases can be more convenient
both for quantitative calculations and for qualitative analysis of the
results. This our approach KZ2021PRB  essentially extends Nazarov’s circuit
theory Naz1 ; Naz2  which can in principle be employed for conductors of
arbitrary dimensionality. Here we focus our attention specifically on quasi-
one-dimensional conductors where substantial simplifications can be achieved.

Let us first express Keldysh component $\hat{G}^{K}$ of the Green function
matrix via retarded ($\hat{G}^{R}$) and advanced ($\hat{G}^{A}$) components of
this matrix as

$\hat{G}^{K}=\hat{G}^{R}\hat{h}-\hat{h}\hat{G}^{A},$ (11)

where $\hat{h}=h^{L}+\hat{\tau}_{3}h^{T}$ is the matrix distribution function
parameterized by two different quasiparticle distribution functions $h^{L}$
and $h^{T}$. In normal conductors the latter functions obey diffusion-like
equations

$\displaystyle D\nabla\left[D^{T}\nabla h^{T}+\mathcal{Y}\nabla
h^{L}+\bm{j}_{\varepsilon}h^{L}\right]=0,$ (12) $\displaystyle
D\nabla\left[D^{L}\nabla h^{L}-\mathcal{Y}\nabla
h^{T}+\bm{j}_{\varepsilon}h^{T}\right]=0,$ (13)

where $D^{T,L}$ and $\mathcal{Y}$ denote dimensionless kinetic coefficients
and $\bm{j}_{\varepsilon}$ represents the spectral current

$\displaystyle D^{T}=\nu^{2}+\dfrac{1}{4}|F^{R}+F^{A}|^{2},$ (14)
$\displaystyle D^{L}=\nu^{2}-\dfrac{1}{4}|F^{R}-F^{A}|^{2},$ (15)
$\displaystyle\mathcal{Y}=-\dfrac{1}{4}\left(|F^{R}|^{2}-|\tilde{F}^{R}|^{2}\right),$
(16)
$\displaystyle\bm{j}_{\varepsilon}=\dfrac{1}{2}\operatorname{\mathrm{Re}}\Bigl{(}F^{R}\nabla\tilde{F}^{R}-\tilde{F}^{R}\nabla
F^{R}\Bigr{)},$ (17)

$\nu=\operatorname{\mathrm{Re}}G^{R}$ is the local density of states and
$G^{R,A}$, $F^{R,A}$ and $\tilde{F}^{R,A}$ are components of retarded and
advanced Green functions

$\hat{G}^{R,A}=\begin{pmatrix}G^{R,A}&F^{R,A}\\\
\tilde{F}^{R,A}&-G^{R,A}\end{pmatrix}.$ (18)

In the case of quasi-one-dimensional conductors Eqs. (12) and (13) are linear
differential equations for the distribution functions $h^{T}$ and $h^{L}$
implying that there exist linear relations between these functions at
different points $x$ and $\tilde{x}$. These relations can be conveniently
represented in the Kirchhoff-like form KZ2021PRB

$\hat{G}_{x,\tilde{x}}\begin{pmatrix}h^{T}(x)\\\
h^{L}(x)\end{pmatrix}+\hat{G}_{\tilde{x},x}\begin{pmatrix}h^{T}(\tilde{x})\\\
h^{L}(\tilde{x})\end{pmatrix}=-e\begin{pmatrix}I^{T}\\\ I^{L}\end{pmatrix},$
(19)

where

$\begin{pmatrix}I^{T}\\\
I^{L}\end{pmatrix}=-\dfrac{\mathcal{A}\sigma}{4e}\begin{pmatrix}\operatorname{\mathrm{Sp}}(\check{G}\check{G}^{\prime}\hat{\tau}_{3})^{K}\\\
\operatorname{\mathrm{Sp}}(\check{G}\check{G}^{\prime})^{K}\end{pmatrix}$ (20)

is the matrix current which remains conserved along the normal wire segment,
$\hat{G}_{x,\tilde{x}}$ is $2\times 2$ conductance matrix which is in general
a complicated function of the parameters $D^{T,L}$, $\mathcal{Y}$ and
$\bm{j}_{\varepsilon}$. It obeys the following relations

$\hat{G}_{x,\tilde{x}}+\hat{\tau}_{3}\hat{G}_{\tilde{x},x}^{T}\hat{\tau}_{3}=0,\quad\hat{G}_{x,\tilde{x}}+\hat{G}_{\tilde{x},x}=\hat{G}_{j},$
(21)

where we defined $\hat{G}_{j}=\hat{\tau}_{1}G_{j}$ and $G_{j}=\sigma
j_{\varepsilon}\mathcal{A}$.

The total electric current $I$ flowing across the wire is linked to the
$I^{T}$-component of the matrix current by means of the equation

$\displaystyle I=\frac{1}{2}\int I^{T}d\varepsilon.$ (22)

In the normal wires connected to normal terminals we have
$j_{\varepsilon}\equiv 0$ and matrix conductance can be evaluated explicitly

$\hat{G}_{x,\tilde{x}}=\mathcal{A}\sigma\left[\int_{\tilde{x}}^{x}dx^{\prime}\begin{pmatrix}D^{T}(x^{\prime})&\mathcal{Y}(x^{\prime})\\\
-\mathcal{Y}(x^{\prime})&D^{L}(x^{\prime})\end{pmatrix}^{-1}\right]^{-1}.$
(23)

Under the conditions $|D^{T,L}-1|\ll 1$ and $|\mathcal{Y}|\ll 1$ the above
expression reduces to a simpler form

$\hat{G}_{x,\tilde{x}}\approx\dfrac{\mathcal{A}\sigma}{(x-\tilde{x})^{2}}\int_{\tilde{x}}^{x}dx^{\prime}\begin{pmatrix}D^{T}(x^{\prime})&\mathcal{Y}(x^{\prime})\\\
-\mathcal{Y}(x^{\prime})&D^{L}(x^{\prime})\end{pmatrix}.$ (24)

Further simplifications occur provided the energy of electrons propagating in
the normal metal remains smaller than the superconducting order parameter
$\Delta$ of the electrodes. At such energies the spectral current $I^{L}$
equals to zero inside normal wires connected to superconducting terminals and
we have

$\hat{G}_{0,x}=\begin{pmatrix}-G_{S}&0\\\
G_{j}&0\end{pmatrix},\quad\hat{G}_{x,0}=\begin{pmatrix}G_{S}&G_{j}\\\
0&0\end{pmatrix},$ (25)

where it is assumed that an SN interface is located at $x=0$, $x$ is the
coordinate inside the normal wire and $G_{S}(\varepsilon)$ is the spectral
parameter characterizing both the interface and the N-wire.

In general the matrix conductance of a quasi-one-dimensional wire can be
parametrized as

$\displaystyle\hat{G}_{x,\tilde{x}}=-\begin{pmatrix}G^{T}&G^{\mathcal{Y}}-G_{j}/2\\\
-G^{\mathcal{Y}}-G_{j}/2&G^{L}\end{pmatrix}.$ (26)

Note that diagonal elements of the matrix conductances ($G_{S}$, $G^{T,L}$)
are even functions of both energy and the superconducting phase difference
whereas their off-diagonal elements ($G^{\mathcal{Y}}$, $G_{J}$) are odd
function of these two variables.

##  III Thermoelectric effect and spin-sensitive electron scattering

In this section we will outline several examples demonstrating that spin-
sensitive electron scattering in superconductors may generate electron-hole
symmetry breaking which in turn yields dramatic enhancement of the
thermoelectric effect. Our analysis is mainly based on Refs. KZK12 ; KZ14 ;
KZ15M ; KZ15 .

###  III.1 Randomly distributed magnetic impurities

Let us first consider a superconductor doped with randomly distributed point-
like magnetic impurities with concentration $n_{\mathrm{imp}}$. Electron
scattering on such impurities can be described by the contribution
$\check{\Sigma}_{\mathrm{m}}$ to the self-energy (7) which takes the following
general form Rusinov69

$\check{\Sigma}_{\mathrm{m}}=\dfrac{n_{\mathrm{imp}}}{2\pi
N_{0}}\Bigl{\\{}\left([u_{1}+\hat{\tau}_{3}u_{2}]^{-1}+i\left<\check{g}\right>\right)^{-1}+\\\
+\left([u_{1}-\hat{\tau}_{3}u_{2}]^{-1}+i\left<\check{g}\right>\right)^{-1}\Bigr{\\}},$
(27)

where $u_{1,2}$ are dimensionless parameters characterizing the impurity
scattering potential.

It is worth pointing out that within the Born approximation the self-energy
(27) reduces to the standard Abrikosov-Gor’kov result Abrikosov60 . The latter
approximation is, however, insufficient for our purposes since it does not
allow to capture the effect of Andreev bound states with energies

$\varepsilon_{B}=\pm\beta\Delta,\quad\beta^{2}=\dfrac{(1+u_{1}^{2}-u_{2}^{2})^{2}}{(1+u_{1}^{2}-u_{2}^{2})^{2}+4u_{2}^{2}},$
(28)

which are formed and localized near each paramagnetic impurity in a
superconductor Shiba68 ; Rusinov68 . The presence of such bound states is
crucial for electron-hole asymmetry in our system. Hence, within our further
analysis it is necessary to go beyond the frequently employed Born
approximation and make use of the self-energy in the form (27).

In order to proceed we apply a temperature gradient $\nabla T$ to our
superconductor and resolve Eqs. (4) together with (7) and (27) evaluating the
linear correction to the Green-Keldysh function
$\delta\hat{g}^{K}\propto\bm{v}_{F}\nabla T$. It is straightforward to verify
that charge neutrality is explicitly maintained inside the superconductor,
since the induced voltage vanishes identically,
$V\propto\langle\operatorname{\mathrm{Sp}}\delta\hat{g}^{K}\rangle\propto\langle\bm{v}_{F}\rangle\equiv
0$. The whole calculation goes along the lines with the analysis of thermal
conductivity in unconventional superconductors Graf96 . Combining the
resulting expression for $\delta\hat{g}^{K}$ with Eq. (8) we recover the
dominating contribution to the thermoelectric coefficient $\alpha_{S}$ which
originates from electron-hole asymmetry. It reads KZK12

$\displaystyle\alpha_{S}=-\dfrac{eN_{0}v_{F}^{2}}{12T^{2}}\int_{-\infty}^{\infty}\dfrac{\mathcal{F}(\varepsilon)d\varepsilon}{\cosh^{2}(\varepsilon/2T)},$
(29)
$\displaystyle\mathcal{F}(\varepsilon)=\dfrac{\varepsilon\nu(\varepsilon)\operatorname{\mathrm{Im}}\Sigma_{0}^{R}(\varepsilon)}{\left[\operatorname{\mathrm{Im}}\mathcal{D}(\varepsilon)\right]^{2}-\left[\operatorname{\mathrm{Im}}\Sigma_{0}^{R}(\varepsilon)\right]^{2}}.$
(30)

Here $\nu(\varepsilon)$ defines the energy resolved superconducting density of
states in the presence of magnetic impurities normalized to its normal state
value and $\Sigma_{0}^{R}$ is a non-vanishing diagonal part of the retarded
self-energy matrix in Eqs. (7) and (27) which explicitly accounts for
asymmetry between electrons and holes. We obtain

$\displaystyle\Sigma_{0}^{R}(\varepsilon)=\Gamma_{0}\dfrac{\tilde{\varepsilon}^{2}-\Delta^{2}}{\tilde{\varepsilon}^{2}-\beta^{2}\Delta^{2}},\quad\nu(\varepsilon)=\operatorname{\mathrm{Re}}\dfrac{\tilde{\varepsilon}}{\sqrt{\tilde{\varepsilon}^{2}-\Delta^{2}}},$
(31)
$\displaystyle\mathcal{D}(\varepsilon)=\sqrt{\tilde{\varepsilon}^{2}-\Delta^{2}}+i\Gamma+i\Gamma_{1}\dfrac{\tilde{\varepsilon}^{2}-\Delta^{2}}{\tilde{\varepsilon}^{2}-\beta^{2}\Delta^{2}},$
(32)

where the parameter $\tilde{\varepsilon}$ is fixed by the relation Shiba68 ;
Rusinov68

$\tilde{\varepsilon}=\varepsilon+i\Gamma_{2}\dfrac{\tilde{\varepsilon}\sqrt{\tilde{\varepsilon}^{2}-\Delta^{2}}}{\tilde{\varepsilon}^{2}-\beta^{2}\Delta^{2}},$
(33)

The scattering parameters $\Gamma_{0,1,2}$ are defined as

$\displaystyle\Gamma_{0}=\dfrac{n_{\mathrm{imp}}}{\pi
N_{0}}\dfrac{u_{1}(1+u_{1}^{2}-u_{2}^{2})}{(1+u_{1}^{2}-u_{2}^{2})^{2}+4u_{2}^{2}},$
(34) $\displaystyle\Gamma_{1}=\dfrac{n_{\mathrm{imp}}}{\pi
N_{0}}\dfrac{(1+u_{1}^{2}-u_{2}^{2})(u_{1}^{2}-u_{2}^{2})}{(1+u_{1}^{2}-u_{2}^{2})^{2}+4u_{2}^{2}}$
(35) $\displaystyle\Gamma_{2}=2\dfrac{n_{\mathrm{imp}}}{\pi
N_{0}}\dfrac{u_{2}^{2}}{(1+u_{1}^{2}-u_{2}^{2})^{2}+4u_{2}^{2}}.$ (36)

Note that $\tilde{\varepsilon}$, $\Sigma_{0}^{R}$ and $\nu(\varepsilon)$ do
not depend $\Gamma$, i.e. all these parameters remain insensitive to electron
scattering on non-magnetic impurities because such kind of scattering does not
produce any pair-breaking effect in conventional superconductors. On the
contrary, scattering on magnetic impurities may strongly modify these
parameters.

In the most relevant case of diffusive superconductors with $\Gamma\gtrsim
T_{C}$ Eq. (30) reduces to
$\mathcal{F}(\varepsilon)=\nu(\varepsilon)\operatorname{\mathrm{Im}}\Sigma_{0}^{R}(\varepsilon)/\Gamma^{2}$,
i.e. $\alpha_{S}\propto 1/\Gamma^{2}$ in this limit. Further limiting
expressions for $\alpha_{S}$ are summarized elsewhere KZK12 . Here we only
provide the results of numerical evaluation of $\alpha_{S}$ as a function of
both temperature and impurity concentration. These results are displayed in
Fig. 2.

Figure 2: Thermoelectric coefficient as a function of temperature and magnetic
impurity concentration. Scattering parameters $u_{1}=u_{2}=0.5$ and the
scattering rate $\Gamma=10T_{C}$ remain the same for both panels.

We observe that the thermoelectric coefficient for a diffusive superconductor
achieves its maximum value at $T\sim T_{C}/2$ and $n_{\mathrm{imp}}$
approximately equal to one-half of the critical concentration at which
superconductivity gets fully suppressed. This maximum value takes the form

$\max_{T,n_{\mathrm{imp}}}|\alpha_{S}|\approx
0.05\dfrac{eN_{0}v_{F}^{2}T_{C}}{\Gamma^{2}}=0.2eN_{0}T_{C}\ell^{2}.$ (37)

Combining the relation $\alpha_{N}\sim(\sigma_{N}/e)(T/\varepsilon_{F})$ with
Eq. (37) we arrive at a simple estimate KZK12

$\alpha_{S}/\alpha_{N}(T_{C})\sim p_{F}\ell\gg 1,$ (38)

demonstrating that the enhancement of the thermoelectric effect is stronger in
cleaner superconductors. At the borderline of applicability of Eq. (38)
$\ell\sim v_{F}/T_{C}$ we obtain $|\alpha_{S}|\sim\sigma_{N}/e$, which defines
the absolute maximum value of $\alpha_{S}$ in conventional superconductors
doped by magnetic impurities.

Figure 3: A ring formed by two different superconductors with contacts
maintained at different temperatures $T_{a}$ and $T_{b}$.

As we already mentioned in Sec. I, one possible way to detect the
thermoelectric in superconductors is to perform an experiment with bimetallic
superconducting rings Zavaritskii74 ; Falco76 ; Harlingen80  as it is
schematically shown in Fig. 3. Provided superconducting contacts are kept at
different temperatures $T_{a}$ and $T_{b}$, thermoelectric current is
generated inside the ring and the corresponding magnetic flux $\Phi$ can be
detected. The magnitude of this flux is defined as

$\dfrac{\Phi}{\Phi_{0}}=\dfrac{4e}{c^{2}}\int_{T_{a}}^{T_{b}}[\lambda_{1}^{2}(T)\alpha_{S1}(T)-\lambda_{2}^{2}(T)\alpha_{S2}(T)]dT,$
(39)

where $\Phi_{0}=\pi c/e$ is flux quantum, $\alpha_{S1,2}$ and $\lambda_{1,2}$
denote respectively thermoelectric coefficients and the values of London
penetration depth for two superconductors. For the sake simplicity we may
assume $\alpha_{S1}\gg\alpha_{S2}$ and neglect the second term in Eq. (39).
Employing Eq. (37) together with the standard expression for the London
penetration depth in diffusive superconductors at $T=0$ we obtain a
conservative estimate for the thermally induced flux

$\dfrac{|\Phi|}{\Phi_{0}}\sim
0.01\dfrac{|T_{b}-T_{a}|}{\Gamma},\quad\Gamma\gtrsim T_{C},$ (40)

implying that for reasonably clean superconductors typical values of $\Phi$
may easily reach $\Phi\gtrsim 10^{-2}\Phi_{0}$.

###  III.2 Spin-active interfaces

Figure 4: A metallic bilayer which consists of two superconductors $S_{1}$ and
$S_{2}$ separated by a spin-active interface. The temperature $T(x)$ changes
only in the direction parallel to the interface.

As another example of a system with intrinsically induced electron-hole
asymmetry let us consider an extended bilayer consisting of two metallic
slabs, one superconducting and another one being either normal or
superconducting as well. This structure is schematically displayed in Fig. 4.
In what follows we will assume that both metals are brought into direct
contact with each other via a spin-active interface that is located in the
plane $z=0$. Such an interface can be formed, e.g., by an ultra-thin layer of
a ferromagnet.

A minimal (though sufficient) model for our spin-active interface involves
three different parameters, i.e. the transmission probabilities for opposite
spin directions $D_{\uparrow}$ and $D_{\downarrow}$ as well as the so-called
spin mixing angle $\theta$ representing the difference between the scattering
phase shifts for spin-up and spin-down electrons. These parameters are assumed
to be energy independent which can be justified for sufficiently thin
ferromagnetic layers. At the same time, the layer should not be too thin in
order to preserve its ferromagnetic state.

Finally, we will assume that the left ($x\to-\infty$) and right ($x\to\infty$)
edges of our bilayer are maintained at temperatures $T_{a}$ and $T_{b}$
respectively (see Fig. 4). Hence, quasiparticles entering our system from the
left (right) side are described by the equilibrium (Fermi) distribution
function with temperature $T_{a}$ ($T_{b}$). For the sake of simplicity we
will also assume that temperature $T$ in our bilayer structure depends only on
$x$ and does not vary along $y$\- and $z$-directions.

For the sake of definiteness let us address an SFN structure and distinguish
16 different scattering processes illustrated in Fig. 5. Depending on whether
incident electron-like or hole-like excitations arrive from the normal metal
or the superconductor one can classify all these processes into four groups
(a), (b), (c) and (d). Consider, for instance, the four scattering processes
of an electron-like excitation propagating towards the NS interface from the
normal metal side, see Fig. 5a. Provided the energy of this excitation
$\varepsilon$ does not exceed $\Delta$, it cannot penetrate deep into the
superconductor and gets reflected back into the normal metal either as an
electron (specular reflection) with probability
$\mathcal{R}^{e-e}_{NS,\sigma}$ or as a hole (Andreev reflection) with
probability $\mathcal{R}^{e-h}_{NS,\sigma}$. The probabilities for all
remaining scattering processes are defined analogously.

Figure 5: Four different electron and hole scattering processes in a
superconducting-normal bilayer.

In the case of spin-independent scattering $R_{\uparrow}=R_{\downarrow}$ and
$\theta=0$ both transmission and reflection probabilities remain symmetric
under the replacement of an electron by a hole and vice versa, i.e. we have,
e.g., $\mathcal{R}^{e-e}_{NS,\sigma}=\mathcal{R}^{h-h}_{NS,\sigma}$,
$\mathcal{R}^{e-h}_{NS,\sigma}=\mathcal{R}^{h-e}_{NS,\sigma}$ and so on. In
this case we just recover the standard BTK results BTK  (cf., e.g., Eq. (1))
and no violation of electron-hole symmetry can be expected. By contrast, in
the case of spin-sensitive scattering with $R_{\uparrow}\neq R_{\downarrow}$
and $\theta\neq 0$ it is straightforward to demonstrate KZ14  that only two
reflection probabilities remain equal,
$\mathcal{R}^{e-h}_{NS,\sigma}=\mathcal{R}^{h-e}_{NS,\sigma}$, whereas all
others differ, e.g., $\mathcal{R}^{e-e}_{NS,+}\neq\mathcal{R}^{e-e}_{NS,-}$,
$\mathcal{R}^{e-e}_{NS,\sigma}\neq\mathcal{R}^{h-h}_{NS,\sigma}$,
$\mathcal{R}^{e-h}_{NS,+}\neq\mathcal{R}^{h-e}_{NS,-}$, etc. Thus, we may
conclude that electron scattering at spin-active interfaces generates
electron-hole imbalance in SFN structures which manifests itself by different
scattering rates for electrons and holes in such systems. A similar conclusion
can also be reached in the case of bilayer structures formed by two
superconductors KZ15M .

Note that to a certain extent this situation resembles the one already
addressed in Sec. I, cf. Eq. (3). However, despite some qualitative
similarities the physical origin for this effect in the latter case is, of
course, not exactly the same. In some sense, the physical picture considered
here is closer to that addressed above in Sec. III.1 where spin-sensitive
electron on local magnetic impurities was analyzed. In this respect spin-
active interfaces can just be treated as spatially extended magnetic
impurities.

Now let us evaluate an electric current response to a temperature gradient
applied to the system along the metallic interface. In order to proceed we
need to again resolve quasiclassical Eilenberger equations (4) supplemented by
proper boundary conditions describing electron transfer across SFS-interfaces
by matching the Green function matrices $\check{g}$ for incoming and outgoing
momentum directions at both sides of this interface. In the case of spin-
active interfaces the corresponding boundary conditions have been derived in
Ref. Millis88 . An equivalent approach has been worked out in Ref. Zhao04 .

In a general case $D_{\uparrow}\neq D_{\downarrow}$, $\theta\neq 0$ and
provided two superconductors are not identical (one of them can also be a
normal metal) the thermoelectric current can be evaluated numerically or
analytically in certain limits KZ14 ; KZ15M ; KZ15 . In the presence of
particle-hole asymmetry this current can be large and strongly exceed its
values, e.g., in normal metals. Here we will not go into details of the
calculation and only present simple order-of-magnitude estimates. For
ballistic bilayers the magnitude of the thermoelectric current density can
roughly be estimated as KZ14 ; KZ15M

$|\bm{j}(z)|\sim
j_{c}(R_{\uparrow}-R_{\downarrow})\sin\theta\frac{T_{a}-T_{b}}{T_{C}},$ (41)

where $j_{c}\sim ev_{F}N_{0}T_{C}$ is the critical current of a clean
superconductor with the critical temperature $T_{C}$. We observe that, in
contrast to the situation considered in Galperin73 ; Aronov81 , Eq. (41) does
not contain the small factor $T/\varepsilon_{F}\ll 1$, i.e. the thermoelectric
effect becomes really large. For instance, by setting
$(R_{\uparrow}-R_{\downarrow})\sin\theta\sim 1$ and $T_{1}-T_{2}\sim T_{C}$,
one achieves the thermoelectric current densities of the same order as the
critical one $j_{c}$.

In the tunneling limit $D_{\uparrow},D_{\downarrow}\ll 1$ and for the case of
diffusive superconductors with very different mean free path values (i.e. for
$\ell_{1}^{2}+\ell_{2}^{2}\gg\ell_{1}\ell_{2}$) the expression for the
thermoelectric current takes a simple form KZ15

$I=\dfrac{eN_{0}}{8v_{F}^{2}}\partial_{x}T\int\dfrac{(\ell_{1}^{2}+\ell_{2}^{2})\varepsilon
d\varepsilon}{T^{2}\cosh^{2}(\varepsilon/2T)}\Bigl{<}v_{x}^{2}|v_{z}|\theta(-v_{z})\times\\\
\times(D_{\uparrow}-D_{\downarrow})\left[\nu_{1\uparrow}(0)\nu_{2\uparrow}(0)-\nu_{1\downarrow}(0)\nu_{2\downarrow}(0)\right]\Bigr{>},$
(42)

where $\nu_{i\uparrow(\downarrow)}(0)$ are the momentum and energy resolved
densities of states at the interface for the opposite electron spin
orientations. At intermediate temperatures $T\sim\Delta$ we can roughly
estimate the magnitude of the thermoelectric current as

$I\sim
eN_{0}v_{F}\ell^{2}(R_{\uparrow}-R_{\downarrow})\sin\theta\partial_{x}T.$ (43)

It is instructive to compare this result with that for the thermoelectric
current $I_{\text{norm}}$ flowing in our bilayer in its normal state. Making
use of the well known Mott relation for the thermoelectric coefficient of
normal metals, from (43) we obtain

$\dfrac{I}{I_{\text{norm}}}\sim\dfrac{\ell}{d}\dfrac{\varepsilon_{F}}{T_{C}}(R_{\uparrow}-R_{\downarrow})\sin\theta,$
(44)

where $d$ is the total thickness of our bilayer. Setting $\ell\sim d$,
$R_{\uparrow}-R_{\downarrow}\sim 1$ and $\sin\theta\sim 1$ we immediately
arrive at the conclusion that owing to electron-hole asymmetry the
thermoelectric current $I$ in the superconducting state can be enhanced by a
large factor up to $\varepsilon_{F}/T_{C}\gg 1$ as compared to that in the
normal state $I_{\text{norm}}$.

###  III.3 Related mechanisms and effects

The mechanisms of electron-hole symmetry breaking in superconducting hybrid
structures are, of course, not limited to those considered above. Electron-
hole asymmetry accompanied by large scale thermoelectric effects was also
predicted to occur in superconductor-ferromagnet hybrids with the density of
states spin split by the exchange and/or Zeeman fields Machon ; Ozaeta . These
theoretical predictions were verified in experiments with superconductor-
ferromagnet tunnel junctions in high magnetic fields Beckmann  where large
thermoelectric currents were observed. In unconventional superconductors
electron-hole symmetry breaking and enhanced thermoelectric effects can also
be caused by electron scattering on non-magnetic impurities Arfi88 ;
Lofwander04 . In this case – quite similarly to the situation considered in
Sec. III.1 – the crucial role is played by localized Andreev bound states
formed inside the system.

Finally, we mention that electron-hole asymmetry in superconducting structures
may also yield anomalously large photovoltaic effect Zaitsev86 ; KZ15a .
Despite some similarities the latter effect is rather different from the
thermoelectric one analyzed here because thermal heating of the system is
physically not equivalent to that produced by an external ac field. For more
details on this issue we refer the reader to the work KZ15a .

##  IV Phase coherence and thermoelectricity in Andreev interferometers

In the previous section we demonstrated that electron-hole asymmetry leading
to huge thermoelectric effects in superconducting structures can be generated
by spin-sensitive scattering of quasiparticles on various magnetic inclusions.
Large thermoelectric effects not restricted by a small parameter
$T/\varepsilon_{F}$ were also observed in the absence of such inclusions in
multi-terminal hybrid superconducting structures frequently called Andreev
interferometers Venkat1 ; Venkat2 ; Petrashov03 ; Venkat3 ; Vitya . Electron-
hole symmetry breaking appears to be responsible for this phenomenon also in
that case, and the key mechanism for such symmetry breaking (elucidated in
Ref. KZ17  and also in the Introduction) is directly related to Andreev
reflection at NS interfaces in the presence of magnetic flux, cf. Eqs. (2) and
(3).

Furthermore, thermoelectric signals detected in experiments Venkat1 ; Venkat2
; Petrashov03 ; Venkat3 ; Vitya  demonstrated $2\pi$-periodic dependence on
the superconducting phase difference $\chi$ across an SNS junction forming an
Andreev interferometer. The latter observation implies that in such structures
thermoelectricity is phase-coherent. Earlier this issue was addressed by a
number of authors Volkov ; V2 ; V3 ; VH ; VH2 ; VH3 . It is also remarkable
that that both odd and even in $\chi$ periodic dependencies of the thermopower
were observed in different interferometers. Several attempts to interpret this
odd-even effect were performed attributing it, e.g., to charge imbalance Titov
and mesoscopic fluctuations JW .

Figure 6: Four terminal Andreev interferometer under consideration.

In what follows we will consider Andreev interferometers schematically
displayed in Fig. 6. They consist of two normal and two superconducting
terminals interconnected by five normal metallic wires of different lengths.
Normal terminals are generally maintained at electrostatic potentials $V_{1}$
and $V_{2}$ and temperatures $T_{1}$ and $T_{2}$. The phase difference between
two superconducting terminals equals to $\chi=\chi_{2}-\chi_{1}$.

The circuit theory formalism outlined in Sec. II.2 allows one to exactly solve
the kinetic equation for the above structure expressing electric currents
inside metallic wires via quasiparticle distribution functions in the
terminals and spectral conductances of these wires. For instance, for
symmetric four-terminal Andreev interferometers the currents flowing into
normal ($I_{N}$) and superconducting ($I_{S}$) terminals read

$\displaystyle I_{N}=-\dfrac{1}{4}\int
G_{N}^{eff}(h_{N_{1}}^{T}-h_{N_{2}}^{T})d\varepsilon,$ (45) $\displaystyle
I_{S}=-\dfrac{1}{2}\int\biggl{[}G_{j}(h_{N_{1}}^{L}+h_{N_{2}}^{L})+G_{S}^{eff}(h_{N_{1}}^{T}-h_{N_{2}}^{T})\biggr{]}d\varepsilon,$
(46)

where $G_{N}^{eff}$ and $G_{S}^{eff}$ are expressed in terms of the spectral
conductances as

$\displaystyle
G_{N}^{eff}=\dfrac{(G_{S}+2G_{p}^{T})[G_{N}^{L}G_{N}^{T}+(G_{N}^{\mathcal{Y}})^{2}]}{G_{N}^{L}(G_{N}^{T}+G_{S}+2G_{p}^{T})+(G_{N}^{\mathcal{Y}})^{2}},$
(47) $\displaystyle
G_{S}^{eff}=\dfrac{G_{S}[G_{N}^{T}G_{N}^{L}+(G_{N}^{\mathcal{Y}})^{2}]-G_{j}G_{N}^{\mathcal{Y}}(G_{S}+2G_{p}^{T})}{G_{N}^{L}(G_{N}^{T}+G_{S}+2G_{p}^{T})+(G_{N}^{\mathcal{Y}})^{2}}.$
(48)

Making use of the current conservation condition one also finds

$\int\Bigl{[}G^{T}_{eff}(h_{N_{1}}^{T}+h_{N_{2}}^{T})+G_{eff}^{\mathcal{Y}}(h_{N_{1}}^{L}-h_{N_{2}}^{L})\Bigr{]}d\varepsilon=0,$
(49)

where

$\displaystyle
G^{T}_{eff}=\dfrac{G_{N}^{T}(2G_{S}G_{p}^{L}+G_{j}^{2})+G_{S}[G_{N}^{T}G_{N}^{L}+(G_{N}^{\mathcal{Y}})^{2}]}{(G_{S}+G_{N}^{T})(2G_{p}^{L}+G_{N}^{L})+(G_{j}+G_{N}^{\mathcal{Y}})^{2}},$
(50) $\displaystyle
G_{eff}^{\mathcal{Y}}=\dfrac{G_{N}^{\mathcal{Y}}(G_{S}2G_{p}^{L}+G_{j}^{2})+G_{j}[G_{N}^{T}G_{N}^{L}+(G_{N}^{\mathcal{Y}})^{2}]}{(G_{S}+G_{N}^{T})(2G_{p}^{L}+G_{N}^{L})+(G_{j}+G_{N}^{\mathcal{Y}})^{2}}.$
(51)

Eqs. (45)-(51) provide a formally exact solution describing the current
distribution in symmetric four-terminal Andreev interferometers displayed in
Fig. 6.

In what follows we will address two different types of external conditions
which could be imposed on our structure. Specifically, normal terminals could
be connected to an external battery that fixes the voltage drop
$V=V_{1}-V_{2}$ between these terminals. Alternatively, normal terminals could
be totally disconnected from any external circuit in which case one obviously
has $I_{N}\equiv 0$. In both cases the distribution functions $h^{T,L}$ in
normal terminals have the form

$h^{T/L}_{N_{i}}=\dfrac{1}{2}\left[\tanh\dfrac{\varepsilon+eV_{i}}{2T_{i}}\mp\tanh\dfrac{\varepsilon-
eV_{i}}{2T_{i}}\right],$ (52)

where $V_{i}$ and $T_{i}$ are voltage bias and temperature of the
corresponding terminal.

###  IV.1 Interplay between Josephson and Aharonov-Bohm effects

To begin with, let us ignore thermoelectric effects by setting $T_{1}=T_{2}=T$
and fix the wire lengths and cross sections respectively as
$L_{S_{1}(N_{1})}=L_{S_{2}(N_{2})}=L_{S(N)}$ and
$\mathcal{A}_{S_{1}(N_{1})}=\mathcal{A}_{S_{2}(N_{2})}=\mathcal{A}_{p}=\mathcal{A}$.
In this case the current conservation condition (49) obviously yields
$V_{1/2}=\mp V/2$ and Eqs. (46), (45) can be rewritten in a simpler form
observing the conditions $h^{T}_{N_{1}}=-h^{T}_{N_{2}}$ and
$h^{L}_{N_{1}}=h^{L}_{N_{2}}$.

In Fig. 7 we show typical energy dependence of the spectral conductances
$G_{S,N}^{eff}$ (which are even function of both energy and the
superconducting phase difference) for different values of $\chi$. In the low
and high energy limits these conductances approach the same value as a
consequence of the well known reentrance effect bel .

It is convenient to represent the spectral conductances as a sum of three
terms

$G_{S,N}^{eff}(\varepsilon,\chi)=G_{S,N}^{eff,n}+\delta
G_{S,N}^{eff}(\varepsilon)+\delta G_{S,N}^{AB}(\varepsilon,\chi),$ (53)

where $G_{S,N}^{eff,n}$ are the normal state values, $\delta
G_{S,N}^{eff}(\varepsilon)$ represent the phase independent corrections due to
the proximity effect and $\delta G_{S,N}^{AB}(\varepsilon,\chi)$ define
periodic in $\chi$ Aharonov-Bohm-type corrections due to interference effects.
The terms $\delta G_{S,N}^{eff}(\varepsilon)$ decay very slowly at high
energies ($\sim|\varepsilon|^{-1/2}$) whereas $\delta
G_{S,N}^{AB}(\varepsilon,\chi)$ decay exponentially on the energy scale of
several Thouless energies ($\sim e^{-\sqrt{|\varepsilon|/E_{\mathrm{Th}}}}$).
With this in mind we obtain $I_{N}\approx G_{S,N}^{eff,n}V$ with a small extra
excess current $\sim V^{1/2}$ at large voltages and the phase-periodic
component which amplitude does not depend on voltage in the high voltage
limit.

Analogously, we can write the current $I_{S}$ flowing between superconducting
terminals as a sum of three terms

$I_{S}=I_{0}(V)+I_{J}(V,\chi)+I_{AB}(V,\chi),$ (54)

where $I_{0}$ is the phase independent contribution, while $I_{J}$ and
$I_{AB}$ represent respectively odd and even functions of the phase difference
with zero mean value. At zero voltage $I_{J}$ simply coincides with the
equilibrium Josephson current ZZh ; GreKa . With increasing voltage the
amplitude of the phase oscillations of $I_{J}(\chi)$ decreases whereas the
amplitude of $I_{AB}(\chi)$ grows linearly $\propto V$ at low voltages and
saturates at high voltages.

Equation (54) can also be rewritten in the form

$I_{S}=I_{0}(V)+I_{1}(V,\chi-\chi_{0}(V)),\quad I_{1}(V,0)=0.$ (55)

where we introduced the phase shift $\chi_{0}$ induced by the external
voltage. This equation demonstrates that the net current $I_{S}(\chi)$ is in
general no longer an odd function of $\chi$.

Figure 7: Energy dependence of the spectral conductances for different values
of $\chi$. Solid, dotted and dashed curves correspond to the phase difference
$\chi$ equal to $0$, $\pi/2$ and $\pi$ respectively. Here we set
$L_{S}=L_{N}=L_{p}$ and $E_{\mathrm{Th}}=10^{-3}\Delta$.

The physics behind this result is transparent. In the presence of a non-zero
bias $V$ a dissipative current component, which we will further label as
$I_{d}(V)$, is induced in the normal wire segments $L_{S_{1}}$ and
$L_{S_{2}}$. At NS interfaces this current gets converted into extra
($V$-dependent) supercurrent flowing across a superconducting loop. Since at
low temperatures and energies electrons in normal wires attached to a
superconductor remain coherent (thus keeping information about the phase
$\chi$), dissipative currents in such wires also become phase (or flux)
dependent demonstrating even in $\chi$ Aharonov-Bohm-like (AB) oscillations
nakano1991quasiparticle ; SN96 ; GWZ97 ; Grenoble , i.e.
$I_{d}(V,\chi)=I_{0}(V)+I_{AB}(V,\chi)$, where $I_{0}(V)\propto V$. Combining
this contribution to the current $I_{S}$ with an (odd in $\chi$) Josephson
current $I_{J}(V,\chi)$ we immediately arrive at Eq. (54) with
$I_{1}=I_{J}+I_{AB}$.

Figure 8: Upper panel: The phase shift $\chi_{0}$ as a function of $V$. Lower
panel: The current $I_{1}(\chi)$ at $T\to 0$ for $eV=60E_{\mathrm{Th}}$ and
$eV=300E_{\mathrm{Th}}$. Here we set $L_{S}=L_{N}=L_{p}$ and
$E_{\mathrm{Th}}=10^{-3}\Delta$.

The behavior of the phase shift $\chi_{0}(V)$ displayed in Fig. 8 is
determined by a trade-off between Josephson and Aharonov-Bohm contributions to
$I_{1}$. At low voltages $I_{J}$ dominates over $I_{AB}$, and one has
$\chi_{0}\approx 0$. Increasing the bias to values $eV\sim 20E_{\mathrm{Th}}$,
in full agreement with the results WSZ  we observe the transition to a
$\pi$-junction state meaning the sigh change of $I_{J}$. Here and below
$E_{\mathrm{Th}}=D/L^{2}$ is the Thouless energy of our device and
$L=2L_{S}+L_{p}$ is the total length of three wire segments between two
S-terminals. At even higher bias voltages both terms $I_{J}$ and $I_{AB}$
become of the same order. For $v=(eV/2E_{\mathrm{Th}})^{1/2}\gg 1$ and at
$T\ll E_{\mathrm{Th}}$ we have WSZ  $I_{J}=I_{C}(V)\sin\chi$, where for our
geometry

$I_{C}(V)\simeq\frac{128(1+v^{-1})}{9(3+2\sqrt{2})}\frac{V}{R_{L}}e^{-v}\sin(v+v^{-1}).$
(56)

We also approximate $I_{AB}\approx I_{\mathrm{m}}\cos{\chi}$, where
$I_{\mathrm{m}}\approx 0.18E_{\mathrm{Th}}/eR_{L}$ and $R_{L}$ is the normal
resistance of the wire with length $L$. Then for $eV\gg E_{\mathrm{Th}}\gg T$
we get

$I_{1}\approx\sqrt{I_{C}^{2}+I_{\mathrm{m}}^{2}}\sin(\chi-\chi_{0}),\quad\chi_{0}(V)=-\arctan\frac{I_{\mathrm{m}}}{I_{C}(V)}.$

The function $\chi_{0}(V)$ demonstrates damped oscillations and saturates to
the value $\chi=\pi/2$ in the limit of large voltages, as it is also
illustrated in Fig. 8.

At higher $T>E_{\mathrm{Th}}$ the Josephson current decays exponentially with
increasing $T$ while the Aharonov-Bohm term is described by a much weaker
power-law dependence GWZ97 ; Grenoble  $I_{AB}\propto 1/T$, thus dominating
the expression for $I_{1}$ and implying that $\chi_{0}\simeq\pi/2$ at such
values of $T$.

Finally, we note that the effects similar to those discussed here have
recently been observed in non-magnetic Andreev interferometers Marg .

###  IV.2 Thermopower

Let us now turn to the thermoelectric effect. For the sake of generality we
slightly modify the setup in Fig. 6 by attaching two extra normal terminals N3
and N4 as shown in Fig. 9. These terminals are disconnected from the external
circuit and are maintained at different temperatures $T_{3}$ and $T_{4}$,
while the temperature of the remaining four terminals equals to $T$.

Figure 9: Schematics of a six-terminal Andreev interferometer consisting of
two superconducting and four normal terminals interconnected by normal
metallic wires of different lengths.

Let us define the thermopower as

$\mathcal{S}_{34}=\frac{V_{3}-V_{4}}{T_{3}-T_{4}}\Big{|}_{T_{4}\mapsto
T_{3}}.$ (57)

Making use of our general quasiclassical formalism one can evaluate the
thermopower $\mathcal{S}_{34}(V,0)$ numerically as a function of temperature.
The results of this calculation are displayed in Fig. 10. We observe a rather
non-trivial non-monotonous behavior of the thermopower as a function of $T$
which – depending on the bias voltage – can even change its sign. At high
temperatures strongly exceeding the Thouless energy $E_{\mathrm{Th}}$ the
thermopower $\mathcal{S}_{34}$ demonstrates a slow (power-law) decay with
increasing $T$ which is reminiscent of that for the Aharonov-Bohm current
component GWZ97 .

Figure 10: The thermopower $\mathcal{S}_{34}$ for $\chi=0$ as a function of
temperature at different voltage bias values. Here we set
$L_{S_{1}}=L_{S_{2}}=L_{N_{1}}=L_{N_{2}}=1/3L$, $L_{c,1}=0.1L$,
$L_{N_{3}}=L_{N_{4}}=1/2L$ and $\Delta=10^{3}E_{\mathrm{Th}}$.

In order to reconstruct the phase dependence of the thermoelectric signal it
is necessary to resolve the kinetic equations (12) and (13). Then inside the
normal wires $L_{N,3}$ and $L_{N,4}$ one finds

$\displaystyle\begin{pmatrix}I^{T}_{N_{3,4}}\\\
I^{L}_{N_{3,4}}\end{pmatrix}=\hat{G}_{N_{3,4}}\begin{pmatrix}h^{T}_{N_{3,4}}-h^{T}_{c}\\\
h^{L}_{N_{3,4}}-h^{L}_{c}\end{pmatrix}.$ (58)

Here $h^{T/L}_{c}$ are the distribution function components evaluated in the
central crossing point and $\hat{G}_{N_{3}}=\hat{G}_{N_{4}}=\hat{G}_{N}$ is
the spectral conductance matrix defined by Eq. (23) with the integration
performed either over the wire $L_{N_{3}}$ or $L_{N_{4}}$. In the absence of
superconductivity this matrix reduces to the diagonal one proportional to the
conductance of the corresponding normal wire.

Bearing in mind that no current can flow into and out of electrically isolated
terminals N3 and N4 and making use of the $T$-component of Eq. (58), we obtain

$\int
d\varepsilon\Bigl{\\{}G^{T}_{N}[h^{T}_{N_{3}}-h^{T}_{N_{4}}]+G_{N}^{\mathcal{Y}}[h^{L}_{N_{3}}-h^{L}_{N_{4}}]\Bigr{\\}}=0.$
(59)

This equation determines the relationship between the terminal temperatures
$T_{3},\ T_{4}$ and the induced thermoelectric voltages $V_{3},\ V_{4}$.
Assuming $\delta T=T_{3}-T_{4}$ to be small enough and evaluating the voltage
difference $V_{3}-V_{4}$ up to linear in $\delta T$ terms, from Eq. (57) we
get

$e\mathcal{S}_{34}\approx\dfrac{1}{4T_{N}^{2}}\dfrac{L_{N}}{\mathcal{A}\sigma_{N}}\int
d\varepsilon\dfrac{(G^{T}_{N}+G^{\mathcal{Y}}_{N})(\varepsilon+eV_{N})}{\cosh^{2}[(\varepsilon+eV_{N})/(2T_{N})]}.$
(60)

Here $V_{N}$ is the induced voltage in both terminals N3 and N4 for
$T_{3}=T_{4}=T_{N}$, i.e. in the absence of the temperature gradient. It is
also essential to keep in mind that the spectral conductance
$G^{T}_{N}(\varepsilon,\chi)$ ($G^{\mathcal{Y}}_{N}(\varepsilon,\chi)$) in Eq.
(60) is an even (odd) function of both $\varepsilon$ and $\chi$.

The first conclusion to be drawn from the above result is that the
thermoelectric effect vanishes in symmetric structures with
$L_{S_{1}}=L_{S_{2}}$, $L_{N_{1}}=L_{N_{2}}$ and with the terminals N3 and N4
attached to the wire central point. In this case one has $V_{N}=0$ and, hence,
the even term in Eq. (60) vanishes after the energy integration. In addition,
the spectral function $\mathcal{Y}$ equals to zero in the wires $L_{N_{3,4}}$,
thus providing $G^{\mathcal{Y}}=0$.

Figure 11: The oscillating parts of the thermopower (left panel) and the
current $I_{S}$ (right panel) as functions of $\chi$ at bias voltages $V$
equal to $10E_{\mathrm{Th}}/e$ (solid black lines), $50E_{\mathrm{Th}}/e$
(dotted blue lines), $100E_{\mathrm{Th}}/e$ (short dash red lines),
$200E_{\mathrm{Th}}/e$ (long dash green lines).

In the general case the result (60) can be expressed in the form DKZ18 ;
dolgirev2018topology

$\mathcal{S}_{34}(V,\chi)=\mathcal{S}_{0}(V)+\mathcal{S}_{\mathrm{odd}}(V,\chi)+\mathcal{S}_{\mathrm{even}}(V,\chi),$
(61)

where $\mathcal{S}_{0}(V)=\langle{\mathcal{S}}_{34}\rangle_{\chi}$ and the
last two terms represent respectively odd and even in $\chi$ oscillating
$2\pi$-periodic functions with zero average over $\chi$. Hence, we may
conclude that the thermopower $\mathcal{S}_{34}(V,\chi)$ (61) is, in general,
neither odd nor even function of $\chi$ taking a nonzero value at $\chi=0$.

At low enough voltages, such that $eV_{N}\lesssim E_{\mathrm{Th}}$ (which,
however, does not necessarily imply $eV\lesssim E_{\mathrm{Th}}$), Eq. (60)
implies that the odd in $\chi$ contribution dominates the thermopower. At
larger voltages $eV_{N}\gtrsim E_{\mathrm{Th}}$ the even harmonics starts to
dominate, cf. Fig. 11. The latter observation is specific to our geometry,
where at such voltages the odd harmonics becomes suppressed by the factor
$\sim(L_{c,1}-L_{c,2})/L$.

In Fig. 11 (left panel) we display the phase dependence of the thermopower at
$T=E_{\mathrm{Th}}$ and for several voltage values $V$. We observe that, while
at lower voltages the thermopower $\mathcal{S}_{34}(\chi)$ is (nearly) odd in
$\chi$, it tends to an even function of $\chi$ quite rapidly with increasing
$V$.

Let us also compare this behavior of the thermopower with that of the phase-
dependent current $I_{S}(\chi)$ between the two superconducting terminals. The
corresponding results dolgirev2018interplay  are shown in the right panel of
Fig. 11.

In contrast to the thermopower $\mathcal{S}_{34}(\chi)$, the odd in $\chi$
harmonics dominates the function $I_{S}(\chi)$ at all given values of the bias
voltage $V$. This observation allows to conclude that the thermopower
$\mathcal{S}_{34}(\chi)$ may in general have a different origin from that of
the current $I_{S}(\chi)$. This is because the current $I_{S}(\chi)$ is
defined through the spectral supercurrent $j_{E}$, while the odd harmonics of
$\mathcal{S}_{34}(\chi)$ originates from the spectral function $\mathcal{Y}$.
We further notice that the temperature dependence of the odd current harmonics
differs substantially from that of the thermopower: At sufficiently high
temperatures $T\simeq 15E_{\mathrm{Th}}$ the former is essentially suppressed
while the latter is not.

It is also worth pointing out that in the case of Andreev interferometers with
a somewhat different topology from that analyzed here it was established DKZ18
that the thermopower-phase relation closely follows the dependence
$I_{S}(\chi)$ at all values of $V$. This result seems to be in line with the
work VH  where it was suggested that the thermopower $\mathcal{S}$ is simply
proportional to ${\cal S}\propto dI_{S}/dT$. If so, one could assume that both
the supercurrent and the thermopower have the same origin both being odd
functions of $\chi$. However, numerical calculations dolgirev2018topology
clearly do not support the relation $\mathcal{S}_{34}\propto dI_{S}/dT$ for
Andreev interferometers under consideration.

Thus, we may conclude that the system topology – along with such parameters as
$V$, $T$ and $E_{\mathrm{Th}}$ – plays a crucial role for the thermopower-
phase relation. This conclusion qualitatively agrees with experimental
observations Venkat1 . Depending on the system topology the behavior of the
thermopower can be diverse: In symmetric setups it may become small or even
vanish, while in non-symmetric ones the origin of the odd-part of the
thermopower-phase relation can be determined either by the Josephson-like
effect or by the spectral function $\mathcal{Y}$ which accounts for electron-
hole asymmetry in our system.

###  IV.3 Long-range Josephson effect and $\pi$-junction states

In Sec. IV.1 we demonstrated that the current flowing between superconducting
terminals in the setup of Fig. 6 can be efficiently controlled biasing normal
terminals by an external voltage $V=V_{1}-V_{2}$. Here we consider a different
physical situation: The Andreev interferometer in Fig. 6 is exposed to a
thermal gradient, i.e. normal terminals are kept at different temperatures
$T_{1}$ and $T_{2}$ whereas no external voltage is applied $V=0$.

While this problem can be resolved in a general case KZ2021PRB , the main
physical effects are conveniently illustrated already for a simpler geometry
of Andreev interferometers with $L_{p}\to 0$ known as $X$-junctions KDZ20 ;
KZ21EPJST .

To begin with, we consider an $X$-junction configuration with
$L_{S_{1}}=L_{S_{2}}\equiv L/2$ and
${\mathcal{A}}_{S_{1}}={\mathcal{A}}_{S_{2}}\equiv{\mathcal{A}}_{S}$. In this
particular case no electron-hole asymmetry is generated KDZ20  and, hence, no
thermoelectric effect occurs, i.e. $V_{1,2}=0$. Furthermore, the distribution
function $h^{T}$ equals to zero, while the function $h^{L}$ inside the wires
is defined by

$h^{L}=r^{L}_{N_{2}}h^{L}_{N_{1}}+r^{L}_{N_{1}}h^{L}_{N_{2}},$ (62)

where $r^{L}_{N_{i}}=R^{L}_{N_{i}}/(R^{L}_{N_{1}}+R^{L}_{N_{2}})$ and

$R^{L}_{N_{i}}=\dfrac{1}{\mathcal{A}_{N_{i}}\sigma}\int\limits_{L_{N_{i}}}\dfrac{dx}{D^{L}},\quad
i=1,2.$ (63)

Let us identically rewrite Eq. (62) in the form

$h^{L}=r_{N_{2}}h^{L}_{N_{1}}+r_{N_{1}}h^{L}_{N_{2}}+W(\varepsilon)(h^{L}_{N_{1}}-h^{L}_{N_{2}}),$
(64)

where we introduced the function
$W(\varepsilon)=r^{L}_{N_{2}}r_{N_{1}}-r^{L}_{N_{1}}r_{N_{2}}$ which vanishes
identically in structures with $L_{N_{1}}=L_{N_{2}}$ and remains nonzero
otherwise decaying exponentially provided $|\varepsilon|$ exceeds the Thouless
energy of our device $E_{\mathrm{Th}}$.

Making use of Eq. (64), we immediately reconstruct the expression for the
supercurrent $I_{S}$ between the superconducting terminals S1 and S2. We find
KDZ20

$I_{S}=r_{N_{2}}I_{J}(T_{1},\chi)+r_{N_{1}}I_{J}(T_{2},\chi)+I_{S}^{\mathrm{ne}}(T_{1},T_{2},\chi),$
(65)

where

$I_{J}(T,\chi)=-\dfrac{\sigma\mathcal{A}_{S}}{2e}\int
j_{\varepsilon}\tanh\dfrac{\varepsilon}{2T}d\varepsilon$ (66)

defines the equilibrium Josephson current and

$I_{S}^{\mathrm{ne}}=\dfrac{\sigma\mathcal{A}_{S}}{2e}\int
j_{\varepsilon}W(\varepsilon)\left(\tanh\dfrac{\varepsilon}{2T_{2}}-\tanh\dfrac{\varepsilon}{2T_{1}}\right)d\varepsilon.$
(67)

Equations (65)-(67) demonstrate that provided our Andreev interferometer is
biased by a temperature gradient the supercurrent $I_{S}$ consists of two
different contributions. The first one is a weighted sum of equilibrium
Josephson currents $I_{J}$ (66) evaluated at temperatures $T_{1}$ and $T_{2}$
and the second one $I_{S}^{\mathrm{ne}}$ (67) accounts for non-equilibrium
effects.

Provided at least one of the two temperatures remains below the Thouless
energy $E_{\mathrm{Th}}$, the current $I_{S}$ (65) is dominated by the first
(quasi-equilibrium) contribution, while the non-equilibrium one (67) can be
ignored. In the opposite limit $T_{1,2}\gg E_{\mathrm{Th}}$, the equilibrium
contribution to $I_{S}$ is exponentially suppressed and reads KDZ20 :

$I_{J}=\dfrac{16\varkappa}{3+2\sqrt{2}}\dfrac{E_{\mathrm{Th}}}{eR_{n}}\left(\dfrac{2\pi
T}{E_{\mathrm{Th}}}\right)^{3/2}e^{-\sqrt{2\pi T/E_{\mathrm{Th}}}}\sin\chi$
(68)

where $R_{n}=L/(\mathcal{A}_{S}\sigma)$ and the parameter
$\varkappa=4\sqrt{\mathcal{A}_{S_{1}}\mathcal{A}_{S_{2}}}/(\mathcal{A}_{S_{1}}+\mathcal{A}_{S_{2}}+\mathcal{A}_{N_{1}}+\mathcal{A}_{N_{2}})$
is taken here at ${\mathcal{A}}_{S_{1}}={\mathcal{A}}_{S_{2}}$. Hence, the
supercurrent can now be dominated by the non-equilibrium term

$I_{S}^{\mathrm{ne}}\simeq
0.21\varkappa^{3}r_{N_{1}}r_{N_{2}}\dfrac{E^{2}_{\mathrm{Th}}}{eR_{n}}\left(\dfrac{1}{T_{1}}-\dfrac{1}{T_{2}}\right)\\\
\times\left(\dfrac{L}{L_{N_{2}}}-\dfrac{L}{L_{N_{1}}}\right)\sin\chi\cos^{2}(\chi/2).$
(69)

We observe that at temperatures strongly exceeding the Thouless energy, the
supercurrent $I_{S}\simeq I_{S}^{\mathrm{ne}}$ decays as a power-law with
increasing min$(T_{1},T_{2})$, unlike the equilibrium Josephson current in
long SNS junctions which is known to decay exponentially. This behavior is due
to driving the electron distribution function $h^{L}$ out of equilibrium by
exposing the system to a temperature gradient. In Figs. 12 and 13 we display
the critical Josephson current $I_{C}$ as a function of $T_{2}$ for fixed
$T_{1}$. At high temperatures, $T_{2}\gg E_{\mathrm{Th}}$ the current $I_{C}$
strongly exceeds both equilibrium values $I_{J}(T_{1})$ and $I_{J}(T_{2})$ and
even starts to grow for $T_{2}\gtrsim T_{1}$. This behavior implies strong
supercurrent stimulation by a temperature gradient.

Figure 12: Josephson critical current $I_{C}\equiv{\mathrm{max}}|I_{S}|$ as a
function of $T_{2}$. Inset: CPR evaluated at $T_{2}=50E_{\mathrm{Th}}$ (a),
$60E_{\mathrm{Th}}$ (b) and $75E_{\mathrm{Th}}$ (c). Solid lines correspond to
the exact numerical solution, dashed lines indicate the result (65) combined
with (68) and (69), dotted line is the quasi-equilibrium contribution
$r_{N_{2}}I_{J}(T_{1},\pi/2)+r_{N_{1}}I_{J}(T_{2},\pi/2)$ to $I_{S}$. The
parameters are: $T_{1}=70E_{\mathrm{Th}}$, $L_{S_{1,2}}=L/2$, $L_{N_{1}}=3L$,
$L_{N_{2}}=L$ and
$\mathcal{A}_{S_{1}}=\mathcal{A}_{S_{2}}=\mathcal{A}_{N_{1}}=\mathcal{A}_{N_{2}}$.

Another remarkable feature of our result (69) is the non-sinusoidal CPR that
persists even at temperatures strongly exceeding $E_{\mathrm{Th}}$. Note that
the dependence of the equilibrium Josephson current on the phase $\chi$ in SNS
junctions remains non-sinusoidal only at $T\lesssim E_{\mathrm{Th}}$ and
reduces to $I_{J}\propto\sin\chi$ at higher temperatures.

In addition, we observe that the sign of the supercurrent in Eq. (69) is
controlled by those of both length and temperature differences,
$L_{N_{1}}-L_{N_{2}}$ and $T_{1}-T_{2}$. For instance, by choosing
$L_{N_{1}}<L_{N_{2}}$ and $T_{1}<T_{2}$ we arrive at a pronounced
$\pi$-junction-like behavior, see also Fig. 13.

Figure 13: The same as in Fig. 12. The parameters are the same except
$L_{N_{1}}=L$, $L_{N_{2}}=3L$. Temperature values in the inset are
$T_{2}=65E_{\mathrm{Th}}$ (a), $75E_{\mathrm{Th}}$ (b) and $85E_{\mathrm{Th}}$
(c). Figure 14: Universal functions $P_{1}(x)$and $P_{2}(x)$. We observe that
$P_{1}(1/2)=P_{2}(1/2)=1$.

For fully asymmetric X-junction supercurrent $I_{S}$ can be also represented
in the form (65) with lengthy expression for non-equilibrium term KZ21EPJST .
Our numerical calculation demonstrates that with a good accuracy non-
equilibrium contribution can be approximately represented in the form (67)
with functions $j_{\varepsilon}$ and $W(\varepsilon)$ evaluated for particular
asymmetric X-junction.

Evaluating these functions in the high energy limit and extrapolating them to
the whole energy interval one can analytically evaluate non-equilibrium
current at sufficiently high temperatures $T_{1,2}\gg E_{\mathrm{Th}}$ which
reads

$I_{S}^{\mathrm{ne}}=\dfrac{4\varkappa^{3}}{(3+2\sqrt{2})^{2}}\dfrac{1101}{1250}r_{N_{1}}r_{N_{2}}\left(\dfrac{1}{T_{1}}-\dfrac{1}{T_{2}}\right)\dfrac{E_{\mathrm{Th}}}{eR_{n}^{a}}\times\\\
\left(\dfrac{L}{L_{N_{2}}}-\dfrac{L}{L_{N_{1}}}\right)\biggl{\\{}\dfrac{\mathcal{A}_{S_{1}}}{2\mathcal{A}_{S_{2}}}P_{1}(L_{S_{1}}/L)+\\\
\dfrac{\mathcal{A}_{S_{2}}}{2\mathcal{A}_{S_{1}}}P_{1}(L_{S_{2}}/L)+P_{2}(L_{S_{1}}/L)\cos\chi\biggr{\\}}\sin\chi,$
(70)

where the universal functions $P_{1}$ and $P_{2}$ are displayed in Fig. 14.
This result demonstrates that in a wide temperature interval
$T_{1,2}>E_{\mathrm{Th}}$ the non-equilibrium contribution to the Josephson
current is described by a universal power law dependence $\propto
1/T_{1}-1/T_{2}$, whereas the phase dependence of $I_{S}^{\mathrm{ne}}$
depends on the junction geometry only.

For partially symmetric junction with $L_{S_{1}}=L_{S_{2}}$ and
$\mathcal{A}_{S_{1}}=\mathcal{A}_{S_{2}}$ we have $P_{1}=P_{2}=1$ and the
phase dependence of $I_{S}^{\mathrm{ne}}$ reduces to the form (69). For
strongly asymmetric junctions the function $P_{1}$ increases by about an order
of magnitude, whereas $P_{2}$ varies slightly (see Fig. 14). In this case the
current-phase relation approaches a sin-like form. Numerical calculation shows
that our analytic formula (70) is in a good agreement with numerically exact
results for $I_{S}^{\mathrm{ne}}$ as long as the lengths $L_{S_{1,2}}$ remain
not very small $\min(L_{S_{1}},L_{S_{2}})\gtrsim 0.2L$.

It is easy to verify that the term $I_{S}^{\mathrm{ne}}$ dominates the
supercurrent $I_{J}$ (68) already at $\min T_{1,2}\gtrsim 30E_{\mathrm{Th}}$,
i.e. at considerably lower temperatures than in symmetric junctions considered
above, where the analogous condition reads $T_{1,2}\gtrsim 70E_{\mathrm{Th}}$.
In Figs. 15 and 16 we also observe that for asymmetric junctions with
$L_{S_{1}}\ll L_{S_{2}}$ and
$\mathcal{A}_{S_{1}}=\mathcal{A}_{S_{2}}=\mathcal{A}_{N_{1}}=\mathcal{A}_{N_{2}}$
non-equilibrium effects are visible only at $T_{1,2}/E_{\mathrm{Th}}\gtrsim
40\div 50$. Hence, we conclude that strongly asymmetric $X$-junctions with
$L_{S_{1}}\ll L_{S_{2}}$ and
$\mathcal{A}_{S_{1}}\gg\mathcal{A}_{S_{2}}+\mathcal{A}_{N_{1}}+\mathcal{A}_{N_{2}}$
appear to be most suitable candidates for observing the non-equilibrium
Josephson current (70).

Finally, we remark that for $L_{N_{1}}>L_{N_{2}}$ our $X$-junction may exhibit
two transitions between $0$\- and $\pi$-junction states. In this case the
system is in the 0-junction state as long as $T_{2}$ remains low enough to
keep the quasi-equilibrium term larger than $I_{S}^{\mathrm{ne}}$. However,
since with increasing $T_{2}$ (albeit for $T_{2}<T_{1}$) the contribution
$\propto I_{J}$ decays faster than the non-equilibrium one (now having a
negative sign), the $X$-junction eventually switches to the $\pi$-junction
state. Further increasing $T_{2}$ one reachs the point $T_{2}=T_{1}$ where
$I_{S}^{\mathrm{ne}}$ changes its sign, thus signaling the transition back to
the $0$-junction state at $T_{2}$ slightly below $T_{1}$. This behavior is
illustrated in Fig. 16.

Figure 15: Josephson critical current $I_{C}\equiv\max|I_{S}|$ as a function
of $T_{2}$. Inset: The phase dependencies of the Josephson current
$I_{S}(\chi)$ for $T_{2}=50E_{\mathrm{Th}}$ (“a” curves) and
$T_{2}=70E_{\mathrm{Th}}$ (“b” curves). Solid lines correspond to our
numerically exact solution, dashed lines indicate the result (65) combined
(70), dotted line represent the quasi-equilibrium contribution
$r_{N_{2}}I_{J}(T_{1},\pi/2)+r_{N_{1}}I_{J}(T_{2},\pi/2)$ to $I_{S}$. The
parameters are: $T_{1}=55E_{\mathrm{Th}}$, $L_{N_{1}}=L$, $L_{N_{2}}=3L$,
$L_{S_{1}}=0.2L$ and
$\mathcal{A}_{S_{1}}=\mathcal{A}_{S_{2}}=\mathcal{A}_{N_{1}}=\mathcal{A}_{N_{2}}$.
Figure 16: The same as in Fig. 15. The parameters are the same except
$L_{N_{1}}=3L$, $L_{N_{2}}=L$ and $T_{1}=65E_{\mathrm{Th}}$. Temperature
values in the inset are $T_{2}=40E_{\mathrm{Th}}$ (a), $50E_{\mathrm{Th}}$ (b)
and $70E_{\mathrm{Th}}$ (c).

##  V Concluding remarks

For decades thermoelectricity in superconductors was and remains one of the
most intriguing topics of modern condensed matter physics NL . In this review
we outlined some recent developments in the field focusing our attention on
multi-terminal superconducting-normal hybrid structures. We demonstrated that
electron-hole symmetry breaking in such systems may cause enormous enhancement
of thermoelectric effects and elucidated several physical mechanisms for such
symmetry breaking. Phase-coherent nature of thermoelectric effects manifests
itself in a periodic dependence of the thermopower on the applied magnetic
flux indicating their close relation to Josephson and Aharonov-Bohm effects.
We also demonstrated that dc Josephson current in multi-terminal hybrid
structures can be efficiently tuned by exposing the system to a temperature
gradient leading to a nontrivial current-phase relation and to a possibility
for $\pi$-junction states.

Finally, it is worth pointing out that thermoelectric effects in
superconductors give rise to a variety of applications ranging from
thermometry and refrigeration Pekola  to phase-coherent caloritronics Giazotto
that paves the way to an emerging field of thermal logic Li  operating with
information in the form of energy. We hope that theoretical results and
predictions discussed in this work not only shed light on some previously
unresolved issues but also could help to put forward these and other
applications of thermoelectric effects.

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