# Theory of Seebeck ratchet in noncentrosymmetric electron-phonon coupled
system

Yugo Onishi  Department of Applied Physics, University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113-8656  Hiroki Isobe  Department of Applied Physics,
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656  Naoto Nagaosa
Center for Emergent Matter Science (CEMS), RIKEN, Wako 351-0198, Japan
Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,
Tokyo 113-8656

###### Abstract

A thermoelectric effect of the noncentrosymmetric electron-phonon coupled
system is proposed, in which a temperature difference between electrons and
phonons, instead of a spatial temperature gradient, induces an electric
current. This is a realization of the Seebeck ratchet in solids with the
asymmetric energy dispersion of electrons, and the dynamical phonons act as
the fluctuation of the scalar potential and induce the dc current. Possible
realizations of this mechanism are also discussed.

##  I Introduction

The thermoelectric effect is the effect of conversion between heat and
electricity. Since they are promising for applications such as reuse of
exhaust heat and energy harvesting, thermoelectric materials with high
efficiency have been actively explored [1, 2, 3]. The performances of
thermoelectric materials are characterized by the conductivity $\sigma$, the
thermal conductivity $\kappa$, and the Seebeck coefficient $S$, and the
efficiency of the material is often measured by the dimensionless figure of
merit $ZT=\sigma S^{2}T/\kappa$ with the temperature $T$.

The thermoelectric effects most frequently discussed are the ones in metals or
semiconductors, where electrons or holes serve as carriers of charge and heat.
These systems are well described by the semiclassical Boltzmann theory and the
quantum linear response theory [4, 5, 6, 7]. It is also well known that there
exist several universal relations between thermoelectric coefficients, such as
the Wiedemann-Franz law and the Mott formula. These relations impose some
constraints on the thermoelectric effects of electronic origins in searching
for efficient thermoelectric materials. In fact, the Wiedemann-Franz law shows
that the conductivity and the thermal conductivity due to electrons are not
independent, which makes it difficult to improve $ZT$. Under these
constraints, various strategies to design efficient thermoelectric materials
have been proposed, such as band engineering and low-dimensional materials [8,
9].

While only the electrons are the carriers of charge in materials, other
degrees of freedom such as phonons can be carriers of heat. Since phonons
contribute to the heat current and the thermal conductivity, it is preferable
to reduce the thermal conductivity due to phonons in order to improve $ZT$.
Therefore, the most basic design strategy for thermoelectric materials was
“phonon-glass, electron-crystal” [9]. However, degrees of freedom such as
phonons sometimes work positively for the thermoelectric effects. For example,
in the phonon drag effect, phonons contribute to the Seebeck effect because
electrons are dragged by the heat current of phonons. The phonon drag effect
can be also described by the Boltzmann theory and the linear response theory.
This phonon drag effect can lead to a large Seebeck effect at low temperatures
in, e.g., doped semiconductors [10, 11] and, was proposed recently as an
origin of the large Seebeck effect at low temperature in FeSb2 [12]. Since
thermoelectric effects using degrees of freedom other than electrons can avoid
restrictions such as Wiedemann-Franz law, they are one of the promising ways
to achieve high-performance thermoelectric effects. Therefore, how degrees of
freedom other than electrons can contribute to thermoelectric effects is an
important issue in terms of applications.

From this viewpoint, we explore the possible role of lower symmetry of the
electron-phonon coupled system as a mechanism of the thermoelectric effect.
Especially, nonreciprocal charge transport in noncentrosymmetric systems is a
subject of recent intensive interest [13]. In addition to the broken inversion
symmetry $\cal{P}$, the time-reversal symmetry $\cal{T}$ is often required to
be broken due to the magnetic ordering or external magnetic field for the
directional transport. This can be understood from the energy dispersion of
the band structure, which satisfies
$\varepsilon_{\uparrow}(\mathbf{k})=\varepsilon_{\downarrow}(-\mathbf{k})$
when $\cal{T}$ is preserved. Hence the nonreciprocal charge transport is
represented by the formula [13]

$J=\sigma_{2}E^{2}$ (1)

where the nonlinear conductivity $\sigma_{2}$ is linear in the magnetic field
$B$ for small $B$. This indicates that the diode effect is switchable by the
magnetic field, although the rectification effect is still very small.

In the presence of electron-phonon interaction, the phonons act as the
fluctuation of the scalar potential and the consequent internal electric field
$E^{\mathrm{ph}}$. When the external electric field $E$ in Eq.(1) is naively
replaced by $E^{\mathrm{ph}}$, a dc current $J$ appears in equilibrium. Of
course, this is wrong since there is no dc current with dissipation in thermal
equilibrium, and a nonequilibrium steady state with energy flow is required to
support the dc current. This situation is analogous to a Brownian ratchet,
where the Brownian motion with different temperatures between two systems
induces the rotational motion in one direction [14].

In this work, we propose a thermoelectric effect with the asymmetric electron
dispersion driven by the phonons. Assuming that the electron temperature
$T_{\mathrm{e}}$ and that of phonons $T_{\mathrm{p}}$ are different, we find
that the correct formula is

$J\propto\expectationvalue{(E^{\mathrm{ph}})^{2}}_{T_{\mathrm{p}}}-\expectationvalue{(E^{\mathrm{ph}})^{2}}_{T_{\mathrm{e}}}.$
(2)

where the angle brackets $\expectationvalue{\dots}_{T}$ represent the thermal
average with temperature $T$. While the heat current flows spatially in the
conventional thermoelectric effects, the heat current flows between phonons
and electrons in the effect proposed here. This is the crucial difference from
previous studies. For example, in [15] a mechanism similar to the ratchet is
also proposed, but the mechanism is based on the spatial gradient of
temperature in contrast to our proposal.

Our mechanism is similar to the one proposed for the magnon-driven spin-
Seebeck effect [16]. In the magnon-driven spin-Seebeck effect, the temperature
difference between electrons and magnons induces the spin current. The results
of the present work show that a similar mechanism induces a finite electric
current in an electron-phonon coupled system. The mechanism proposed by
Bosisio et al. [17] is closely related to the present work. While they
discussed the behavior of the localized electrons in the presence of the
phonons when there is a temperature difference between electrons and phonons,
we consider itinerant electrons and thus do analysis in $k$-space. It should
be also noted that the inversion symmetry breaking (or, equivalently,
reflection symmetry breaking in one-dimensional systems) is essential to
realize the thermoelectric effects induced by the temperature difference
between electrons and phonons. While in the model proposed by Bosisio et al.
the asymmetric contacts between the system and the electrodes break the
inversion symmetry, the dispersion of the electrons breaks the symmetry in our
model. In that sense, the effect proposed in the present work is a bulk effect
in contrast to the proposal by Bosisio et al. Another closely related work is
done by Budkin et al. [18]. They proposed that the shift of the position of
electrons is induced by phonons when the effective temperatures for electrons
and phonons are different. The shift is described by the Berry connection as
in the case of shift current [19, 20]. However, this effect is also different
from our proposal because the effect proposed in [18] is an interband effect
while the effect proposed in the present work is an intraband effect.

This paper is organized as follows. In Sec.II, we present a model for an
electron-phonon coupled system considered in the present work. In Sec.III, we
solve the model perturbatively and calculate an electric current and heat
current induced by the temperature difference between electrons and phonons.
We also calculate quantities which correspond to the power factor and the
figure of merit in conventional thermoelectric materials. Sec.IV is devoted to
discussions and conclusions. We also discuss the possible realization of the
proposed mechanism such as excitation by light and a metal-insulator junction.

##  II Model

Let us consider a model which consists of spinless electrons with an
asymmetric dispersion and phonons in one dimension (Fig.1). We assume that
electrons have dispersion $\varepsilon_{k}$ and phonons have dispersion
$\omega_{q}$. The behavior of electrons can be described with the Boltzmann
equation:

$\displaystyle\partialderivative{f}{t}+\dot{r}\partialderivative{f}{r}+\dot{k}\partialderivative{f}{k}=\quantity(\partialderivative{f}{t})_{\mathrm{scatt}}$
(3)

Here, $f(r,k,t)$ is the distribution function of electrons, and the right-hand
side describes the scattering of electrons. We assume that the scattering term
is given by two terms: one corresponds to the scattering due to the electron-
phonon interaction and the other corresponds to the other sources of
scattering such as impurities. They can be expressed as follows.

$\displaystyle\quantity(\partialderivative{f_{k}}{t})_{\mathrm{scatt}}=-\frac{f_{k}-f^{\mathrm{eq}}_{k}}{\tau}+\quantity(\partialderivative{f_{k}}{t})_{\mathrm{phonon}}$
(4)
$\displaystyle\quantity(\partialderivative{f_{k}}{t})_{\mathrm{phonon}}=\sum_{k^{\prime}}\quantity(\mathcal{W}_{k^{\prime}\to
k}-\mathcal{W}_{k\to k^{\prime}})$ (5) $\displaystyle\mathcal{W}_{k+q\to
k}[f_{k},n_{q}]=\frac{2\pi}{\hbar}\frac{\absolutevalue{g_{q}}^{2}}{V}f_{k+q}\quantity(1-f_{k})$
$\displaystyle\times\quantity[n_{q}\delta(\varepsilon_{k+q}-\varepsilon_{k}+\hbar\omega_{q})+\quantity(n_{q}+1)\delta(\varepsilon_{k+q}-\varepsilon_{k}-\hbar\omega_{q})]$
(6)

where $f_{k}$ is shorthand for $f(r,k,t)$, and since we are interested only in
the spatially uniform steady state, we omit $r$ and $t$ dependencies of
$f(r,k,t)$. $f^{\mathrm{eq}}_{k}=(1+e^{\beta_{e}(\varepsilon_{k}-\mu)})^{-1}$
is the equilibrium distribution function of electron with temperature
$T_{\mathrm{e}}=1/k_{\mathrm{B}}\beta_{\mathrm{e}}$ and chemical potential
$\mu$. In Eq.(4), we use the relaxation time approximation, where $\tau$
accounts for scatterings other than the electron-phonon interaction. Here we
assume that $\tau$ is independent of $k$ for simplicity. The expression of
electron-phonon scattering term is given in Eq.(5).
$\mathcal{W}_{k^{\prime}\to k}$ is the transition rate of an electron
scattered from state $k^{\prime}$ to $k$, and is given as Eq.(6) by the
Fermi’s golden rule. $g_{q}$ is the coupling constant between phonons and
electrons, $n_{q}=n(r,q,t)$ is phonon distribution function, and $V=La^{2}$ is
the volume of the system with length $L$ and cross-section $a^{2}$ (Fig.1). We
do not consider $r$ and $t$ dependencies of $n(r,q,t)$ as in the case of
$f(r,k,t)$.

If electrons and phonon are in equilibrium with temperature
$T=1/k_{\mathrm{B}}\beta$ and chemical potential $\mu$, i.e.,

$\displaystyle f(r,k,t)$
$\displaystyle=\frac{1}{e^{\beta(\varepsilon_{k}-\mu)}+1}$ (7) $\displaystyle
n(r,q,t)$ $\displaystyle=\frac{1}{e^{\beta\hbar\omega_{q}}-1}$ (8)

then we can confirm $\mathcal{W}_{k^{\prime}\to k}=\mathcal{W}_{k\to
k^{\prime}}$. This equality comes from the following identity which holds only
in equilibrium:

$\displaystyle
f_{k^{\prime}}(1-f_{k})n_{q}\delta(\varepsilon_{k^{\prime}}-\varepsilon_{k}+\omega_{q})$
$\displaystyle=f_{k}(1-f_{k^{\prime}})(1+n_{q})\delta(\varepsilon_{k^{\prime}}-\varepsilon_{k}+\omega_{q})$
(9)

Therefore, in equilibrium, electron-phonon scattering term, Eq.(5), is zero.

In this paper, we discuss an unconventional thermoelectric effect of the
system described above. Conventionally, the source of the thermoelectric
effects is a spatial gradient of temperature. However, we consider another
type of thermoelectric effect. Instead of a spatial temperature gradient, we
consider a temperature difference between electrons and phonons as
schematically shown in Fig.2 (For the realization of such a system, see
Sec.IV). As we confirmed above, if electrons and phonons feel the same
temperature, the system relaxes into the equilibrium state and no currents
flow. However, we show that electric currents can occur when electrons and
phonons feel different temperatures.

Figure 1: One-dimensional system considered in the present work. The length of
the system is $L$ and the cross-section is $a\times a$. The electric current
flows along the wire, while the heat current flows between phonons with
temperature $T_{\mathrm{p}}$ and electrons with temperature $T_{\mathrm{e}}$.
Figure 2: The thermoelectric effect proposed in this paper. Instead of a
spatial gradient of temperature, a temperature difference between phonons and
electrons induces an electric current. The direction in which the electric
current flows is set by the inversion symmetry breaking of the electron
system.

##  III Calculation and Results

###  III.1 Solution of the Boltzmann equation

In this section, we solve the Boltzmann equation Eq.(3) up to
$\order{\absolutevalue{g_{q}}^{2}}$, when electrons and phonons feel different
temperatures, $T_{\mathrm{e}}$ for electrons and $T_{\mathrm{p}}$ for phonons.

Before proceeding to detailed calculations, we remark on the phonon scattering
term $\mathcal{W}_{k^{\prime}\to k}$. What we are interested in here is the
effect due to temperature difference $T_{\mathrm{p}}-T_{\mathrm{e}}$. In other
words, we are interested in the effect by $\delta
n^{\mathrm{eq}}_{q}=n^{\mathrm{eq}}_{q}(T_{\mathrm{p}})-n^{\mathrm{eq}}_{q}(T_{\mathrm{e}})$,
where $n^{\mathrm{eq}}_{q}(T)$ is the equilibrium distribution function at
temperature $T$ for phonon with wavevector $q$. However,
$\mathcal{W}_{k^{\prime}\to k}$ includes contribution from the deviation of
$f_{k}$ from the equilibrium distribution at $T_{\mathrm{e}}$, $\delta
f_{k}=f_{k}-f^{\mathrm{eq}}_{k}$. To clearly distinguish these two
contributions, we rewrite $\mathcal{W}_{k^{\prime}\to k}$ by expanding
$\mathcal{W}_{k^{\prime}\to k}$ with respect to the deviation from the
equilibrium distribution, $\delta f_{k}$ and $\delta n^{\mathrm{eq}}_{q}$.

$\displaystyle\mathcal{W}_{k^{\prime}\to
k}[f_{k},n^{\mathrm{eq}}_{q}(T_{\mathrm{p}})]$
$\displaystyle=W^{(0)}_{k^{\prime}\to k}+\tilde{W}_{k^{\prime}\to k}[\delta
f_{k}]+W_{k^{\prime}\to k}[\delta n^{\mathrm{eq}}_{q}]$ (10) $\displaystyle
W^{(0)}_{k^{\prime}\to k}$ $\displaystyle=\mathcal{W}_{k^{\prime}\to
k}[f^{\mathrm{eq}}_{k}(T_{\mathrm{e}}),n^{\mathrm{eq}}_{q}(T_{\mathrm{e}})]$
(11)

Here we neglect higher order terms such as $\delta f_{k}\delta
n^{\mathrm{eq}}_{q}$. To justify this expansion, $\delta f_{k}$ and $\delta
n^{\mathrm{eq}}_{q}$ need to be sufficiently small. This condition is
discussed at the end of this section. The first term is
$\mathcal{W}_{k^{\prime}\to k}$ when
$f_{k}=f^{\mathrm{eq}}_{k}(T_{\mathrm{e}})$ and
$n_{q}=n^{\mathrm{eq}}_{q}(T_{\mathrm{e}})$. Here, we explicitly write
$T_{\mathrm{e}}$ dependence of $f^{\mathrm{eq}}_{k}$ to emphasize that
$f^{\mathrm{eq}}_{k}$ is the equilibrium distribution for $T=T_{\mathrm{e}}$.
Since $W^{(0)}_{k^{\prime}\to k}=W^{(0)}_{k\to k^{\prime}}$,
$W^{(0)}_{k^{\prime}\to k}$ does not affect the Boltzmann equation, as
mentioned at the end of the last section. The second term is the contribution
from $\delta f_{k}$ and the third is the contribution from $\delta
n^{\mathrm{eq}}_{q}$. The second term can be incorporated into the relaxation
time $\tau$ in Eq.(4). The third term $W_{k^{\prime}\to k}$ can be obtained by
expanding $\mathcal{W}_{k^{\prime}\to k}$ with respect to $\delta
n^{\mathrm{eq}}_{q}$.

$\displaystyle W_{k+q\to
k}=\frac{2\pi}{\hbar}\frac{\absolutevalue{g_{q}}^{2}}{V}f^{\mathrm{eq}}_{k+q}\quantity(1-f^{\mathrm{eq}}_{k})$
$\displaystyle\times\quantity[\delta(\varepsilon_{k+q}-\varepsilon_{k}+\hbar\omega_{q})+\delta(\varepsilon_{k+q}-\varepsilon_{k}-\hbar\omega_{q})]\delta
n^{\mathrm{eq}}_{q}$ (12)

Let us now turn to solving the Boltzmann equation. We assume that there is no
spatial variation of temperature and that there is no electric field. In this
case, the Boltzmann equation for steady state is

$\displaystyle-\frac{f_{k}-f^{\mathrm{eq}}_{k}}{\tau}+\sum_{k^{\prime}}\quantity(W_{k^{\prime}\to
k}-W_{k\to k^{\prime}})=0$ (13)

Here, $f_{k}=f(r,k,t)$ does not depend on $r$ or $t$. Therefore, we consider
only $k$-dependence of the distribution function $f_{k}$. This equation can be
solved as

$\displaystyle f_{k}$
$\displaystyle=f^{\mathrm{eq}}_{k}+\tau\sum_{k^{\prime}}\quantity(W_{k^{\prime}\to
k}-W_{k\to k^{\prime}})$ (14)

Figure 3: Asymmetric one-dimensional band structure. We consider the case that
there are two Fermi points. Correspondingly we separate the Brillouin zone
into two parts, $\mathrm{R}$ and $\mathrm{L}$.

For analytic calculations, we further make three simplifications. First, we
consider a one-dimensional dispersion, i.e., $a$ is small enough so that
$\hbar^{2}/2ma^{2}\gg\mu$ where $\mu$ is chemical potential and $m$ is the
effective mass of an electron. Second, we ignore the phonon dispersion and set
$\omega_{q}=\omega$(constant). Third, we simplify the $q$-dependence of the
coupling constant $g_{q}$ as

$\displaystyle g_{q}$
$\displaystyle=\begin{cases}g_{\mathrm{F}}\quad\text{when both $k,k+q$ are on
the same side, $\mathrm{R}$ or $\mathrm{L}$}\\\ g_{\mathrm{B}}\quad\text{when
$k,k+q$ are on different side, $\mathrm{R}$ and $\mathrm{L}$}\end{cases}$ (15)

Here, R and L denote regions in the Brillouin zone of electrons near two Fermi
points (see Fig.3). Corresponding to R and L, the density of states of
electrons $N(\xi)$ are separated into two parts:

$\displaystyle N(\xi)$ $\displaystyle=N_{\mathrm{R}}(\xi)+N_{\mathrm{L}}(\xi)$
(16) $\displaystyle N_{\mathrm{R}}(\xi)$
$\displaystyle=\frac{1}{V}\sum_{k\in\mathrm{R}}\delta(\xi_{k}-\xi)$ (17)
$\displaystyle N_{\mathrm{L}}(\xi)$
$\displaystyle=\frac{1}{V}\sum_{k\in\mathrm{L}}\delta(\xi_{k}-\xi)$ (18)

where $\xi_{k}=\varepsilon_{k}-\mu$.

Under these two assumption, we can explicitly calculate the
$k^{\prime}$-summation in Eq.(14) as follows. With the functions

$\displaystyle f^{\mathrm{eq}}(\xi)$
$\displaystyle=\frac{1}{e^{\beta_{e}\xi}+1}$ (19) $\displaystyle
F(\xi_{1},\xi_{2})$
$\displaystyle=f^{\mathrm{eq}}(\xi_{1})(1-f^{\mathrm{eq}}(\xi_{2}))$ (20)
$\displaystyle n^{\mathrm{eq}}(\omega,T_{\mathrm{p}})$
$\displaystyle=\frac{1}{e^{\beta_{\mathrm{p}}\hbar\omega}-1}$ (21)

we can calculate the $k^{\prime}$ summation (see Appendix A for details) to
obtain

$k\in R$: $\displaystyle\sum_{k^{\prime}}\quantity(W_{k^{\prime}\to k}-W_{k\to
k^{\prime}})$
$\displaystyle=\frac{2\pi}{\hbar}\quantity(\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{L}}(\xi_{k}-\hbar\omega)+\absolutevalue{g_{\mathrm{F}}}^{2}N_{\mathrm{R}}(\xi_{k}-\hbar\omega))\quantity[F(\xi_{k}-\hbar\omega,\xi_{k})-F(\xi_{k},\xi_{k}-\hbar\omega)]\delta
n^{\mathrm{eq}}(\omega)$
$\displaystyle\quad+\frac{2\pi}{\hbar}\quantity(\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{L}}(\xi_{k}+\hbar\omega)+\absolutevalue{g_{\mathrm{F}}}^{2}N_{\mathrm{R}}(\xi_{k}+\hbar\omega))\quantity[F(\xi_{k}+\hbar\omega,\xi_{k})-F(\xi_{k},\xi_{k}+\hbar\omega)]\delta
n^{\mathrm{eq}}(\omega)$ (22a) $k\in L$:
$\displaystyle\sum_{k^{\prime}}\quantity(W_{k^{\prime}\to k}-W_{k\to
k^{\prime}})$
$\displaystyle=\frac{2\pi}{\hbar}\quantity(\absolutevalue{g_{\mathrm{F}}}^{2}N_{\mathrm{L}}(\xi_{k}-\hbar\omega)+\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{R}}(\xi_{k}-\hbar\omega))\quantity[F(\xi_{k}-\hbar\omega,\xi_{k})-F(\xi_{k},\xi_{k}-\hbar\omega)]\delta
n^{\mathrm{eq}}(\omega)$
$\displaystyle\quad+\frac{2\pi}{\hbar}\quantity(\absolutevalue{g_{\mathrm{F}}}^{2}N_{\mathrm{L}}(\xi_{k}+\hbar\omega)+\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{R}}(\xi_{k}+\hbar\omega))\quantity[F(\xi_{k}+\hbar\omega,\xi_{k})-F(\xi_{k},\xi_{k}+\hbar\omega)]\delta
n^{\mathrm{eq}}(\omega)$ (22b)

Here, an abbreviation $\delta
n^{\mathrm{eq}}(\omega)=n^{\mathrm{eq}}(\omega,T_{\mathrm{p}})-n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})$
is used. We calculate the electric current and the heat current from phonons
to electrons using these results in the next section. We note here that the
following calculation is valid only when the correction to the distribution
function due to scattering is sufficiently small, i.e.,

$\displaystyle\absolutevalue{\tau\sum_{k^{\prime}}(W_{k^{\prime}\to k}-W_{k\to
k^{\prime}})}\ll 1$ (23)

This condition is satisfied when
$\absolutevalue{T_{\mathrm{p}}-T_{\mathrm{e}}}$ is sufficiently small as
derived in the following. To transform this condition into a simpler form, the
following identity is useful.

$\displaystyle F(\xi-\hbar\omega,\xi)-F(\xi,\xi-\hbar\omega)$
$\displaystyle=\frac{F(\xi,\xi-\hbar\omega)}{n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})}$
(24)

By applying this identity to Eqs.(22), we can show that the inequality (23) is
satisfied when the following inequality is satisfied.

$\displaystyle
2\pi\tau\beta_{\mathrm{e}}\omega\absolutevalue{g_{\mathrm{F/B}}}^{2}N_{\mathrm{R/L}}(n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})+1)\absolutevalue{\frac{T_{\mathrm{p}}-T_{\mathrm{e}}}{T_{\mathrm{e}}}}\ll
1$ (25)

where $\mathrm{F/B}$ and $\mathrm{R/L}$ are arbitrarily chosen. If the
scattering is mainly due to electron-phonon interaction, the order of
$\hbar/2\pi\tau_{k}$ for an electron with $\xi_{k}\simeq 0$ can be estimated
as [21]

$\displaystyle\frac{\hbar}{2\pi\tau_{k}}$
$\displaystyle=\sum_{q}\frac{\absolutevalue{g_{q}}^{2}}{V}\quantity[(n_{q}+1-f_{k+q})\delta(\varepsilon_{k+q}-\varepsilon_{k}+\hbar\omega_{q})+\quantity(n_{q}+f_{k+q})\delta(\varepsilon_{k+q}-\varepsilon_{k}-\hbar\omega_{q})]$
$\displaystyle\sim\absolutevalue{g_{\mathrm{F/B}}}^{2}(n^{\mathrm{eq}}(\omega,T_{\mathrm{p}})+1)N_{\mathrm{R/L}}$
(26)

and therefore the condition Eq.(25) becomes much simpler form in this case:

$\displaystyle\frac{\hbar\omega}{k_{\mathrm{B}}T_{e}}\absolutevalue{\frac{T_{\mathrm{p}}-T_{\mathrm{e}}}{T_{\mathrm{e}}}}\ll
1$ (27)

###  III.2 Calculation of the electric current

The electric current $J$ is given by

$\displaystyle J$
$\displaystyle=\frac{e}{L}\sum_{k}\frac{1}{\hbar}\partialderivative{\xi_{k}}{k}f_{k}=\frac{e\tau}{\hbar
L}\sum_{k}\partialderivative{\xi_{k}}{k}\sum_{k^{\prime}}\quantity(W_{k^{\prime}\to
k}-W_{k\to k^{\prime}})$ (28)

where $e(<0)$ is the charge of electrons. Here we used the fact that in
equilibrium $J=e/(\hbar
L)\sum_{k}\partialderivative*{\xi_{k}}{k}f^{\mathrm{eq}}_{k}=0$.

We can take $k$-summation in Eq.(28) explicitly. After some algebra, we obtain

$\displaystyle
J=\frac{e\tau}{\hbar}\int_{-\infty}^{\infty}\frac{\differential{\xi}}{\hbar}$
$\displaystyle(-1)^{\mathrm{R}}\quantity(\absolutevalue{g_{\mathrm{F}}}^{2}-\absolutevalue{g_{\mathrm{B}}}^{2})\quantity((N_{\mathrm{L}}(\xi-\hbar\omega)-N_{\mathrm{L}}(\xi))-(N_{\mathrm{R}}(\xi-\hbar\omega)-N_{\mathrm{R}}(\xi)))$
$\displaystyle\times\quantity[F(\xi-\hbar\omega,\xi)-F(\xi,\xi-\hbar\omega)]\delta
n^{\mathrm{eq}}(\omega)$ (29)

where the factor $(-1)^{\mathrm{R}/\mathrm{L}}$ is defined as
$(-1)^{\mathrm{R}/\mathrm{L}}=\mathrm{sgn}\quantity(\evaluated{\partialderivative{\xi}{k}}_{\mathrm{R}/\mathrm{L}})$,
and we have used the assumption $(-1)^{\mathrm{L}}=-(-1)^{\mathrm{R}}$.

We can further simplify this expression as

$\displaystyle J$
$\displaystyle=(-1)^{\mathrm{R}}e\tau\omega^{2}\quantity(\absolutevalue{g_{\mathrm{F}}}^{2}-\absolutevalue{g_{\mathrm{B}}}^{2})\quantity(N_{\mathrm{R}}^{\prime}(0)-N_{\mathrm{L}}^{\prime}(0))\quantity[n^{\mathrm{eq}}(\omega,T_{\mathrm{p}})-n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})]$
(30)

Here, $N_{\mathrm{R}}^{\prime}(\xi)$ is the $\xi$-derivative of
$N_{\mathrm{R}}(\xi)$ and similar for $N_{\mathrm{L}}^{\prime}(\xi)$. For
details of the derivation of Eqs.(29),(30), see Appendix B. From Eq.(30), we
conclude that a temperature difference between phonons and electrons can
induce electric currents if
$\absolutevalue{g_{\mathrm{F}}}^{2}\neq\absolutevalue{g_{\mathrm{B}}}^{2}$ and
$N_{\mathrm{L}}^{\prime}(0)\neq N_{\mathrm{R}}^{\prime}(0)$.

###  III.3 Calculation of the heat current

We can also calculate heat current $J_{Q}$ from phonons to electrons as
follows. $J_{Q}$ can be defined as

$\displaystyle J_{Q}$
$\displaystyle=\sum_{k}\xi_{k}\quantity(\partialderivative{f_{k}}{t})_{\mathrm{phonon}}=\sum_{k}\xi_{k}\sum_{k^{\prime}}(\mathcal{W}_{k^{\prime}\to
k}-\mathcal{W}_{k\to k^{\prime}})$ (31)

This expression means that the change of energy of electrons due to phonon
scattering is equal to the heat current from phonons to electrons. Note that
we use here $\mathcal{W}_{k^{\prime}\to k}$ instead of $W_{k^{\prime}\to k}$.

Up to $\order{\absolutevalue{g_{q}}^{2}}$, $\mathcal{W}_{k^{\prime}\to
k}-\mathcal{W}_{k\to k^{\prime}}=W_{k^{\prime}\to k}-W_{k\to k^{\prime}}$ in
the absence of an electric field. By using Eqs.(22),(31), we obtain

$\displaystyle J_{Q}$ $\displaystyle=2\pi
V\hbar\omega^{2}\quantity[\absolutevalue{g_{\mathrm{F}}}^{2}(N_{\mathrm{L}}(0)^{2}+N_{\mathrm{R}}(0)^{2})+2\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{R}}(0)N_{\mathrm{L}}(0)]\quantity[n^{\mathrm{eq}}(\omega,T_{\mathrm{p}})-n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})]$
(32)

where we have used an approximation $N_{\mathrm{R}}(\xi)\simeq
N_{\mathrm{R}}(0)$ and $N_{\mathrm{L}}(\xi)\simeq N_{\mathrm{L}}(0)$ (see
Appendix B for details). Since $J_{Q}$ represents the heat current between
electrons and phonons, it is proportional to the volume $V=La^{2}$. In
particular, the ratio between $J$ and $J_{Q}$ is

$\displaystyle\frac{J}{J_{Q}}$
$\displaystyle=(-1)^{\mathrm{R}}\frac{e\tau}{2\pi
V}\frac{\quantity(\absolutevalue{g_{\mathrm{F}}}^{2}-\absolutevalue{g_{\mathrm{B}}}^{2})\quantity(N_{\mathrm{R}}^{\prime}(0)-N_{\mathrm{L}}^{\prime}(0))}{\absolutevalue{g_{\mathrm{F}}}^{2}(N_{\mathrm{L}}(0)^{2}+N_{\mathrm{R}}(0)^{2})+2\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{R}}(0)N_{\mathrm{L}}(0)}$
(33)

This ratio is independent of temperature.

###  III.4 Linear response coefficients

We can now calculate linear response coefficients. We define $2\times 2$
matrix $L$ as:

$\displaystyle j$ $\displaystyle=L_{11}E+L_{12}\quantity(-\frac{\Delta T}{T})$
(34a) $\displaystyle q$ $\displaystyle=L_{21}E+L_{22}\quantity(-\frac{\Delta
T}{T})$ (34b)

where $E$ is an external electric field, $T=T_{\mathrm{e}}$ is the temperature
of electrons, and $\Delta T$ is the temperature difference between phonons and
electrons, $\Delta T=T_{\mathrm{e}}-T_{\mathrm{p}}$. $j=J/a^{2}$ is an
electric current density, and $q=J_{Q}/V$ is a heat current per unit volume.
Note that $J_{Q}$ represents the total heat current between electrons and
phonons, and thus it is proportional to the volume $V$.

$L_{12}$ and $L_{22}$ can be calculated from Eq.(30) and (32) as

$\displaystyle L_{12}$
$\displaystyle=(-1)^{\mathrm{R}}\frac{e\tau\hbar\omega^{3}}{a^{2}k_{\mathrm{B}}T}\quantity(\absolutevalue{g_{\mathrm{F}}}^{2}-\absolutevalue{g_{\mathrm{B}}}^{2})\quantity(N_{\mathrm{R}}^{\prime}(0)-N_{\mathrm{L}}^{\prime}(0))n^{\mathrm{eq}}(\omega)(n^{\mathrm{eq}}(\omega)+1)$
(35) $\displaystyle L_{22}$
$\displaystyle=\frac{2\pi\hbar^{2}\omega^{3}}{k_{\mathrm{B}}T}\quantity[\absolutevalue{g_{\mathrm{F}}}^{2}(N_{\mathrm{L}}(0)^{2}+N_{\mathrm{R}}(0)^{2})+2\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{R}}(0)N_{\mathrm{L}}(0)]n^{\mathrm{eq}}(\omega)\quantity(n^{\mathrm{eq}}(\omega)+1)$
(36)

We can also calculate $L_{11}$ and $L_{21}$ by solving the Boltzmann equation
in the presence of an electric field.

$\displaystyle\frac{eE}{\hbar}\partialderivative{f_{k}}{k}$
$\displaystyle=-\frac{f_{k}-f^{\mathrm{eq}}_{k}}{\tau}+\sum_{k^{\prime}}\quantity(W_{k^{\prime}\to
k}-W_{k\to k^{\prime}})$ (37)

By solving the equation up to $\order{E\absolutevalue{g_{q}}^{2}}$, we can
calculate $J$ and $J_{Q}$ (see Appendix.D for details) and we obtain

$\displaystyle L_{11}$
$\displaystyle=\frac{e^{2}\tau}{4\pi^{2}\hbar^{2}a^{4}}\quantity(\frac{1}{N_{\mathrm{R}}(0)}+\frac{1}{N_{\mathrm{L}}(0)})$
(38) $\displaystyle L_{21}$
$\displaystyle=(-1)^{\mathrm{R}}\frac{e\tau\hbar\omega^{3}}{a^{2}k_{\mathrm{B}}T}(\absolutevalue{g_{\mathrm{F}}}^{2}-\absolutevalue{g_{\mathrm{B}}}^{2})(N_{\mathrm{R}}^{\prime}(0)-N_{\mathrm{L}}^{\prime}(0))n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})(n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})+1)=L_{12}$
(39)

Since $L_{12}=L_{21}$, the Onsager’s reciprocal relationship holds. From these
expressions, we define the following quantities: $\sigma$, $\lambda$, $p$, and
$ZT$.

$\displaystyle\sigma$
$\displaystyle:=L_{11}=\frac{e^{2}\tau}{4\pi^{2}\hbar^{2}a^{4}}\quantity(\frac{1}{N_{\mathrm{R}}(0)}+\frac{1}{N_{\mathrm{L}}(0)})$
(40a) $\displaystyle\lambda$ $\displaystyle:=\quantity(\frac{q}{\Delta
T})_{j=0}=\frac{L_{11}L_{22}-L_{12}L_{21}}{TL_{11}}\simeq\frac{L_{22}}{T}$
$\displaystyle=\frac{2\pi\hbar^{2}\omega^{3}}{k_{\mathrm{B}}T^{2}}\quantity[\absolutevalue{g_{\mathrm{F}}}^{2}(N_{\mathrm{L}}(0)^{2}+N_{\mathrm{R}}(0)^{2})+2\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{R}}(0)N_{\mathrm{L}}(0)]n^{\mathrm{eq}}(\omega)\quantity(n^{\mathrm{eq}}(\omega)+1)$
(40b) $\displaystyle p$ $\displaystyle:=\frac{L_{12}^{2}}{T^{2}L_{11}}$
$\displaystyle=\frac{4\pi^{2}\hbar^{4}\omega^{6}\tau}{k_{\mathrm{B}}^{2}T^{4}}\frac{1}{N_{\mathrm{R}}(0)^{-1}+N_{\mathrm{L}}(0)^{-1}}\quantity[(\absolutevalue{g_{\mathrm{F}}}^{2}-\absolutevalue{g_{\mathrm{B}}}^{2})(N_{\mathrm{R}}^{\prime}(0)-N_{\mathrm{L}}^{\prime}(0))n^{\mathrm{eq}}(\omega)(n^{\mathrm{eq}}(\omega)+1)]^{2}$
(40c) $\displaystyle ZT$
$\displaystyle:=\frac{L_{12}^{2}}{L_{11}L_{22}-L_{12}L_{21}}\simeq\frac{L_{12}^{2}}{L_{11}L_{22}}$
$\displaystyle=\frac{2\pi\tau\hbar^{2}\omega^{3}}{k_{\mathrm{B}}T}\frac{\quantity{(\absolutevalue{g_{\mathrm{F}}}^{2}-\absolutevalue{g_{\mathrm{B}}}^{2})(N_{\mathrm{R}}^{\prime}(0)-N_{\mathrm{L}}^{\prime}(0))}^{2}}{(N_{\mathrm{R}}(0)^{-1}+N_{\mathrm{L}}(0)^{-1})\quantity[\absolutevalue{g_{\mathrm{F}}}^{2}(N_{\mathrm{L}}(0)^{2}+N_{\mathrm{R}}(0)^{2})+2\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{R}}(0)N_{\mathrm{L}}(0)]}n^{\mathrm{eq}}(\omega)(n^{\mathrm{eq}}(\omega)+1)$
(40d)

Here, $\sigma$ represents the usual conductivity. $\lambda$ represents the
heat current response to the temperature difference $\Delta T$ under the
condition $j=0$. This is an analogous quantity to the thermal conductivity in
thermoelectric materials. $p$ is an analogous quantity to power factor for
thermoelectric materials, and $p(\Delta T)^{2}/4$ is the maximum of power
output per unit volume for a given $\Delta T$ [22]. $ZT$ is the figure of
merit which represents the efficiency of the thermoelectric system [22]. Note,
however, that $\lambda$ and $p$ are conceptually different from the thermal
conductivity and power factor for thermoelectric materials. In fact, their
dimensions are different from those of thermal conductivity and power factor.
In the following, we call $\lambda$ heat flow rate and $p$ power density.

##  IV Discussion

What we calculated in the previous section is the electric current induced by
the nonequilibrium distribution of phonons. As a similar phenomenon, the
phonon drag effect on the Seebeck coefficient is well known and extensively
investigated [10, 23, 12]. Although these two effects might look quite
similar, their origins are qualitatively different. In phonon drag, the
distribution of phonons which affects the electric current has nonzero net
velocity since phonons carry the heat current. These phonons with finite
velocity “drag” an electric current. On the other hand, in our calculation, we
assume dispersionless phonons. Therefore, their group velocity is zero,
resulting in no net velocity. Instead, the phonon system excites electron-hole
pairs with finite velocity. This picture become clearer when Eq.(29) is
transformed into the following form.

$\displaystyle
J=\frac{e\tau}{\hbar}\int_{-\infty}^{\infty}\frac{\differential{\xi}}{\hbar}(-1)^{\mathrm{R}}\quantity(\absolutevalue{g_{\mathrm{F}}}^{2}-\absolutevalue{g_{\mathrm{B}}}^{2})$
$\displaystyle\times\quantity((N_{\mathrm{L}}(\xi-\hbar\omega)-N_{\mathrm{L}}(\xi))-(N_{\mathrm{R}}(\xi-\hbar\omega)-N_{\mathrm{R}}(\xi)))$
$\displaystyle\times\quantity[F(\xi-\hbar\omega,\xi)n^{\mathrm{eq}}(\omega,T_{\mathrm{p}})-F(\xi,\xi-\hbar\omega)\quantity(n^{\mathrm{eq}}(\omega,T_{\mathrm{p}})+1)]$
(41)

This form can also be derived directly by using $\mathcal{W}_{k^{\prime}\to
k}$ instead of $W_{k^{\prime}\to k}$. The function $F(\xi-\hbar\omega,\xi)$ in
Eq.(41) is nothing but the probability for electron-hole pair, i.e., an
electron with energy $\xi-\hbar\omega$ and a hole with energy $-\xi$, to
exist. Therefore, Eq.(41) can be understood as an electron-hole pair
excitation (relaxation) by a phonon absorption (emission). In equilibrium, the
excitations and relaxations of electron-hole pairs balance and therefore no
net current flows, but if the phonons and the electrons have different
temperatures, this balance breaks and an electric current flows.

Another way to interpret this effect is to view the fluctuation of phonons as
an effective electric field fluctuation. The electron-phonon interaction term
we consider originally comes from the following Hamiltonian:

$\displaystyle\hat{H}_{\mathrm{e-ph}}$
$\displaystyle=\frac{1}{\sqrt{V}}\sum_{\vec{k}}g_{\vec{q}}\hat{\rho}_{\vec{q}}(\hat{a}_{\vec{q}}+\hat{a}^{\dagger}_{-\vec{q}})$
(42)

where $\hat{\rho}_{\vec{q}}$ is the Fourier component of electron density, and
$\hat{a}_{\vec{q}}$($\hat{a}^{\dagger}_{\vec{q}}$) is the
annihilation(creation) operator of phonon with wavevector $\vec{q}$.
Therefore, $g_{\vec{q}}(\hat{a}_{\vec{q}}+\hat{a}^{\dagger}_{-\vec{q}})$ can
be interpreted as an effective electric field $E_{\vec{q}}^{\mathrm{ph}}$. On
the other hand, the electric current $J$ under electric field $E$ can be
expanded with respect to $E$ as

$\displaystyle J$ $\displaystyle=\sigma_{1}E+\sigma_{2}E^{2}+\dots$ (43)

The second term represents the second order response. The expectation value
$\expectationvalue{E_{\vec{q}}^{\mathrm{ph}}}$ is always zero and thus no
current flows due to linear response. However, its fluctuation,
$\expectationvalue{(E_{\vec{q}}^{\mathrm{ph}})^{2}}\sim\absolutevalue{g_{q}}^{2}\expectationvalue{\hat{a}^{\dagger}_{\vec{q}}\hat{a}_{\vec{q}}}$,
is nonzero and can induce an electric current through the second order
response. The corresponding term in Eq.(30) is the term proportional to
$n^{\mathrm{eq}}(\omega,T_{\mathrm{p}})$. Eq.(30) also shows that this current
is cancelled by the current proportional to
$n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})$, as in Eq.(2). Therefore, no net
current flows in equilibrium, but if the phonons and electrons have different
temperatures, the net current can be finite.

As we mentioned in Sec.III.2, our calculation shows that it is necessary to
have
$\absolutevalue{g_{\mathrm{F}}}^{2}\neq\absolutevalue{g_{\mathrm{B}}}^{2}$ and
$N_{\mathrm{L}}^{\prime}(0)\neq N_{\mathrm{R}}^{\prime}(0)$ to induce finite a
thermoelectric current. The first condition requires $q$-dependence of
electron-phonon interaction parameter, $g_{q}$. The second condition requires
that an asymmetric band structure of electrons is necessary. To realize an
asymmetric band structure in one-dimensional systems, both time reversal
symmetry and spatial inversion symmetry need to be broken. Otherwise,
wavevector $k$ and $-k$ can be related to each other through the symmetries
and the two Fermi points $\mathrm{L}$ and $\mathrm{R}$ become equivalent,
resulting in $N_{\mathrm{L}}^{\prime}(0)=N_{\mathrm{R}}^{\prime}(0)$. More
physically, these symmetry breakings allow us to set a spatial direction of
the materials. When there is a temperature difference between electrons and
phonons, an electric current can flow in the direction. Because we now
consider setting the spatial direction through the band dispersion of the
electron, we need to break time reversal symmetry, but if the inversion
symmetry is broken in other ways, e.g., asymmetric contacts with leads [17],
the time reversal symmetry breaking is not necessary.

From Eq.(30) and Eq.(32), the order of magnitude estimates of the current
densities $j=J/a^{2}$ and the heat current per unit volume $q=J_{Q}/V$ are
given by

$\displaystyle j$
$\displaystyle\sim\frac{e\tau\omega^{2}}{a^{2}}(\Delta\absolutevalue{g}^{2})(\Delta
N^{\prime}(0))(n^{\mathrm{eq}}(\omega,T_{\mathrm{p}})-n^{\mathrm{eq}}(\omega,T_{\mathrm{e}}))$
(44) $\displaystyle q$
$\displaystyle\sim\hbar\omega^{2}\absolutevalue{g}^{2}N(0)^{2}(n^{\mathrm{eq}}(\omega,T_{\mathrm{p}})-n^{\mathrm{eq}}(\omega,T_{\mathrm{e}}))$
(45)

where
$\Delta\absolutevalue{g}^{2}=\absolutevalue{g_{\mathrm{F}}}^{2}-\absolutevalue{g_{\mathrm{B}}}^{2}$,
$\Delta
N^{\prime}(0)=N_{\mathrm{R}}^{\prime}(0)-N_{\mathrm{L}}^{\prime}(0),g\sim
g_{\mathrm{F/B}}$, $N(0)\sim N_{\mathrm{R/L}}$.

To estimate the order of currents, we consider GeTe doped with ferromagnetic
Mn impurities. GeTe is a noncentrosymmetric semiconductor and nonreciprocal
charge transport under magnetic field is experimentally observed at room
temperature [24]. By doping Mn, it also breaks time reversal symmetry.
Therefore, it is a promising material to realize the proposed effect. We
assume here that a temperature difference between electrons and phonons in
this system is made by, for example, laser heating (for the realization of the
temperature difference, see later discussions). Although Mn-doped GeTe is not
a one-dimensional system, we roughly estimate the order of magnitude from our
one-dimensional calculation.

We estimate the parameters for Mn-doped GeTe by the ones for GeTe [25, 24]:
$\hbar\omega\sim$10\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$,
$\tau\sim$1\text{\times}{10}^{-13}\text{\,}\mathrm{s}$$,
$N(0)\sim\varepsilon_{F}^{-1}a^{-3}$,
$N^{\prime}(0)\sim\varepsilon_{F}^{-2}a^{-3}$, Fermi energy
$\varepsilon_{F}\sim$0.1\text{\,}\mathrm{e}\mathrm{V}$$, lattice constant
$a\sim$5\text{\times}{10}^{-10}\text{\,}\mathrm{m}$$,
$\absolutevalue{g}^{2}\sim$1\text{\times}{10}^{-31}\text{\,}\mathrm{e}\mathrm{V}^{2}\mathrm{m}^{3}$$.
By using lattice constant for parameter $a$, it is equivalent to consider a 1D
wire. The value for $\tau\sim$1\text{\times}{10}^{-13}\text{\,}\mathrm{s}$$ is
calculated in [25] when considering only the scattering due to electron-phonon
coupling. Note that $\tau$ we consider in the present work represents all the
contributions to the scattering of electrons including electron-phonon
interactions as mentioned in Sec.II. Therefore, we roughly estimate the order
of $\tau$ as the one in [25]. The value for
$\absolutevalue{g}^{2}\sim$1\text{\times}{10}^{-31}\text{\,}\mathrm{e}\mathrm{V}^{2}\mathrm{m}^{3}$$
is determined so that it does not contradict to the results in [25]. Since the
relaxation time for electron-phonon scattering can be evaluated with
$\absolutevalue{g}^{2}$ as $\tau_{\mathrm{el-
ph}}\sim\hbar/(\absolutevalue{g}^{2}N(0)(n^{\mathrm{eq}}(\omega)+1))\sim$1\text{\times}{10}^{-13}\text{\,}\mathrm{s}$$
at $T=$300\text{\,}\mathrm{K}$$, the value of $\absolutevalue{g}^{2}$ is
consistent with the value of $\tau$. We estimate $\Delta N^{\prime}(0)$ as
$\Delta\varepsilon_{F}/\varepsilon_{F}^{3}a^{3}$ and
$\Delta\varepsilon_{F}\sim$0.1\text{\,}\mathrm{e}\mathrm{V}$$, which is of the
order of the Zeeman gap due to the doped Mn impurities [26]. Although the
coupling to the magnetic moment of Mn induces band splitting and does not
induce the asymmetry of the band according to [26], if one can tilt the
magnetization by applying a magnetic field, the coupling induces the asymmetry
of the band, $\Delta\varepsilon_{F}$. $\Delta\absolutevalue{g}^{2}$ is
estimated as $\Delta\absolutevalue{g}^{2}/\absolutevalue{g}^{2}\sim 0.01$.

Then

$\displaystyle
j\sim(n^{\mathrm{eq}}(\omega,T_{\mathrm{p}})-n^{\mathrm{eq}}(\omega,T_{\mathrm{e}}))\times$1\text{\times}{10}^{10}\text{\,}\mathrm{A}\mathrm{/}\mathrm{m}^{2}$$
(46) $\displaystyle
q\sim(n^{\mathrm{eq}}(\omega,T_{\mathrm{p}})-n^{\mathrm{eq}}(\omega,T_{\mathrm{e}}))\times$1\text{\times}{10}^{19}\text{\,}\mathrm{W}\mathrm{/}\mathrm{m}^{3}$$
(47)

At $T_{\mathrm{e}}=$300\text{\,}\mathrm{K}$$ and $\Delta
T=T_{\mathrm{p}}-T_{\mathrm{e}}=$0.1\text{\,}\mathrm{K}$$,
$n^{\mathrm{eq}}(\omega,T_{\mathrm{p}})-n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})\sim
0.001$, and
$j\sim$1\text{\times}{10}^{7}\text{\,}\mathrm{A}\mathrm{/}\mathrm{m}^{2}$,q\sim$1\text{\times}{10}^{16}\text{\,}\mathrm{W}\mathrm{m}^{-3}$$.
This value of the electric current density is experimentally observable. As we
discuss later, we believe that $\Delta T=$0.1\text{\,}\mathrm{K}$$ is also
experimentally realizable. Note that the inequality (27) becomes $\Delta
T/T\ll 1$ for the above parameters.

Using the same parameters, we can also estimate the quantities defined in the
previous section,

$\displaystyle\lambda$
$\displaystyle\sim\quantity(\frac{\hbar\omega}{k_{\mathrm{B}}T})^{2}n^{\mathrm{eq}}(\omega)(n^{\mathrm{eq}}(\omega)+1)\times$1\text{\times}{10}^{17}\text{\,}\mathrm{W}{K}^{-1}\mathrm{m}^{-3}$$
(48a) $\displaystyle p$
$\displaystyle\sim\quantity(\frac{\hbar\omega}{k_{\mathrm{B}}T})^{4}\quantity(n^{\mathrm{eq}}(\omega)(n^{\mathrm{eq}}(\omega)+1))^{2}\times$1\text{\times}{10}^{9}\text{\,}\mathrm{W}\mathrm{K}^{-2}\mathrm{m}^{-3}$$
(48b) $\displaystyle ZT$
$\displaystyle\sim\frac{\hbar\omega}{k_{\mathrm{B}}T}n^{\mathrm{eq}}(\omega)(n^{\mathrm{eq}}(\omega)+1)\times
10^{-5}$ (48c)

At $T=$300\text{\,}\mathrm{K}$$, $\lambda$ is of the order of
$1\text{\times}{10}^{17}\text{\,}\mathrm{W}\mathrm{K}^{-1}\mathrm{m}^{-3}$.
This corresponds to the heat current between electrons and phonons, not to the
heat current flowing spatially. A similar quantity is considered in, for
example, a model called 2T-model. 2T-model describes a relaxation process of
an electron-phonon coupled system after laser heating [27]. In this model, a
parameter for the heat current between electrons and phonons is set to
$\sim$1\text{\times}{10}^{16}\text{\,}\mathrm{W}\mathrm{K}^{-1}\mathrm{m}^{-3}$$
for gold.

If one can heat either electrons or phonons and make temperature difference
$\Delta T\sim$0.1\text{\,}\mathrm{K}$$, the corresponding power output per
volume will be $p(\Delta
T)^{2}/4\sim$1\text{\times}{10}^{7}\text{\,}\mathrm{W}\mathrm{m}^{-3}$$ at
$T=$300\text{\,}\mathrm{K}$$. This is quite large because the value
corresponds to power
factor$\sim$1\text{\times}{10}^{3}\text{\,}\mathrm{W}\mathrm{K}^{-2}\mathrm{m}^{-1}$$
for conventional thermoelectric materials when $\nabla
T\sim$1\text{\,}\mathrm{K}\mathrm{/}\mathrm{c}\mathrm{m}$$. In contrast,
$ZT\sim$1\text{\times}{10}^{-5}\text{\,}$$ are very small compared to
conventional thermoelectric materials. This is because the relaxation through
electron-phonon interaction is so fast that the heat current between electrons
and phonons is much larger than the heat current flowing spatially in the
conventional thermoelectric effects. The large heat current of the proposed
effect results in low efficiency, but the power output is still quite large.
According to Eq.(40d), $ZT$ increases monotonically as temperature increases
at high temperatures if $\tau$ is independent of temperature. However, it
should be noted that we approximate the band structure near the chemical
potential in the derivation of these expressions, Eq.(40d) is valid only when
the temperature is much smaller than the chemical potential,
$k_{\mathrm{B}}T\ll\varepsilon_{F}$. If $\tau$ is due to the electron-phonon
interaction, $\tau$ also depends on the temperature, and it decreases as the
temperature increases (Eq.(26)). In that case $ZT$ can be estimated by using
Eq.(26) as

$\displaystyle
ZT\sim\quantity(\frac{\Delta\absolutevalue{g}^{2}}{\absolutevalue{g}^{2}})^{2}\quantity(\frac{\hbar\omega\Delta
N^{\prime}(0)}{N(0)})^{2}\frac{\hbar\omega}{k_{\mathrm{B}}T}n^{\mathrm{eq}}(\omega)$
(49)

and at high temperature $\hbar\omega\ll k_{\mathrm{B}}T\ll\varepsilon_{F}$
where the expression is still valid, $ZT$ is independent of the temperature.

To realize our model experimentally, it is necessary to make the temperature
difference between the phonon system and the electron system. One way to make
such a situation is to excite only phonons (or only electrons) in a material
with light or laser. By appropriately choosing the polarization and frequency
of light, it would be possible to selectively excite either phonons or
electrons. It is discussed in literature [27] how the temperatures of
electrons and phonons change when the system is heated by ultrashort-pulsed
lasers. In this case, only electrons are excited by the laser pulse. Although
the duration of the temperature difference between electrons and phonons is
quite short ($\sim$1\text{\,}\mathrm{p}\mathrm{s}$$), the maximum of the
temperature difference reaches of the order of
$1\text{\times}{10}^{3}\text{\,}\mathrm{K}$ according to [27]. Another idea is
to use both an insulator and a metal. For example, a junction of insulator and
metal with different temperatures may be one possible realization. A similar
situation is discussed in [28], and it suggests that the temperature
difference between electrons and phonons can be $\sim$0.1\text{\,}\mathrm{K}$$
at steady state near an interface between a metal and a magnetic insulator. It
may be also possible to use a topological insulator such as Bi2Te3, instead of
a metal. In that case, the corresponding electron system is a surface state of
the topological insulator. However, as discussed above, we have to break time
reversal symmetry and inversion symmetry to set the spatial direction through
the band structure. Therefore, in the case of Bi2Te3, we also have to apply a
magnetic field. Although the surface state is not guaranteed in the presence
of the magnetic field, if the magnetic field is small so that the splitting of
the surface state bands due to the magnetic field is much smaller than the
thermal energy, the proposed effect is expected to be realized. The condition
on the field strength $B$ is given by $\sqrt{\hbar v^{2}eB}\ll
k_{\mathrm{B}}T$, where $v$ is the velocity of the Dirac electron. In the case
of Bi2Te3,
$v\sim$1\text{\times}{10}^{5}\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$$ and
thus the condition is $B\ll$100\text{\,}\mathrm{T}$$ at
$T=$300\text{\,}\mathrm{K}$$. It should be noted that, from the point of view
of symmetry, thermoelectric effects similar to the proposed effect may be
realized even without a magnetic field if the inversion symmetry is broken.
For example, in the case of Bi2Te3, the inversion symmetry is broken on the
surface, so there is a possibility that a thermoelectric effect due to the
temperature difference between electrons and phonons can be realized without a
magnetic field by a mechanism different from the proposed mechanism.

So far, we have discussed the case where phonons and electrons feel different
temperatures and show that the electric current can flow. Similarly, if there
are degrees of freedom which feel different temperatures from that of
electrons, then similar physics could occur. For example, one can replace
phonons with another electron system. In that case, an electron density
fluctuation in one system induces electric currents in the other electronic
system mediated by electron-electron interactions. Another candidate is spin
degrees of freedom. In the case of the magnon-driven spin Seebeck effect [16],
the temperature difference between electrons and magnons induces a spin
current. It is also possible to induce electric current by the temperature
difference between the electron system and the spin system if there is a
certain interaction between them, e.g., spin-orbit interaction. In any of
these systems, it should break the inversion symmetry to induce finite
thermoelectric effects.

In conclusion, we propose a new thermoelectric effect induced by the
temperature difference between phonons and electrons. This effect can be
realized in metal-insulator junctions. Similar effects are expected in
noncentrosymmetric systems if one can make a temperature difference between
electrons and other degrees of freedom such as spins.

###### Acknowledgements.

The authors are grateful to S. Maekawa for insightful discussions and useful
comments. N.N. was supported by JST CREST Grant Number JPMJCR1874 and
JPMJCR16F1, Japan, and JSPS KAKENHI Grant Number 18H03676.

##  Appendix A Solution of the Boltzmann equation

The solution of the Boltzmann equation Eq.(13) is given by

$\displaystyle f_{k}$
$\displaystyle=f^{\mathrm{eq}}_{k}+\tau\sum_{k^{\prime}}\quantity(W_{k^{\prime}\to
k}-W_{k\to k^{\prime}})$ (50)

The summation over $k^{\prime}$ can be explicitly calculated as follows. First
we consider the summation over $k^{\prime}\in\mathrm{L}$ when
$k\in\mathrm{R}$. Then, the summation becomes

$\displaystyle\sum_{k^{\prime}\in\mathrm{L}}\quantity(W_{k^{\prime}\to
k}-W_{k\to k^{\prime}})$
$\displaystyle=\sum_{k^{\prime}\in\mathrm{L}}\frac{2\pi}{\hbar}\frac{\absolutevalue{g_{\mathrm{B}}}^{2}}{V}F(\xi_{k^{\prime}},\xi_{k})\quantity[\delta(\varepsilon_{k^{\prime}}-\varepsilon_{k}+\hbar\omega)+\delta(\varepsilon_{k^{\prime}}-\varepsilon_{k}-\hbar\omega)]\delta
n^{\mathrm{eq}}(\omega)$
$\displaystyle\quad-\sum_{k^{\prime}\in\mathrm{L}}\frac{2\pi}{\hbar}\frac{\absolutevalue{g_{\mathrm{B}}}^{2}}{V}F(\xi_{k},\xi_{k^{\prime}})\quantity[\delta(\varepsilon_{k}-\varepsilon_{k^{\prime}}+\hbar\omega)+\delta(\varepsilon_{k}-\varepsilon_{k^{\prime}}-\hbar\omega)]\delta
n^{\mathrm{eq}}(\omega)$ (51)
$\displaystyle=\frac{2\pi}{\hbar}\absolutevalue{g_{\mathrm{B}}}^{2}\quantity[F(\xi_{k}-\hbar\omega,\xi_{k})-F(\xi_{k},\xi_{k}-\hbar\omega)]N_{\mathrm{L}}(\xi_{k}-\hbar\omega)\delta
n^{\mathrm{eq}}(\omega)$
$\displaystyle\quad+\frac{2\pi}{\hbar}\absolutevalue{g_{\mathrm{B}}}^{2}\quantity[F(\xi_{k}+\hbar\omega,\xi_{k})-F(\xi_{k},\xi_{k}+\hbar\omega)]N_{\mathrm{L}}(\xi_{k}+\hbar\omega)\delta
n^{\mathrm{eq}}(\omega)$ (52)

where the definitions of $f^{\mathrm{eq}}$, $F$, $\delta n^{\mathrm{eq}}$ are
given in the main text.

Similarly, we can take the summation for other cases as

$k\in\mathrm{R},k^{\prime}\in\mathrm{R}$:
$\displaystyle\sum_{k^{\prime}\in\mathrm{R}}\quantity(W_{k^{\prime}\to
k}-W_{k\to k^{\prime}})$
$\displaystyle=\frac{2\pi}{\hbar}\absolutevalue{g_{\mathrm{F}}}^{2}\quantity[F(\xi_{k}-\hbar\omega,\xi_{k})-F(\xi_{k},\xi_{k}-\hbar\omega)]N_{\mathrm{R}}(\xi_{k}-\hbar\omega)\delta
n^{\mathrm{eq}}(\omega)$
$\displaystyle\quad+\frac{2\pi}{\hbar}\absolutevalue{g_{\mathrm{F}}}^{2}\quantity[F(\xi_{k}+\hbar\omega,\xi_{k})-F(\xi_{k},\xi_{k}+\hbar\omega)]N_{\mathrm{R}}(\xi_{k}+\hbar\omega)\delta
n^{\mathrm{eq}}(\omega)$ (53) $k\in\mathrm{L},k^{\prime}\in\mathrm{L}$:
$\displaystyle\sum_{k^{\prime}\in\mathrm{L}}\quantity(W_{k^{\prime}\to
k}-W_{k\to k^{\prime}})$
$\displaystyle=\frac{2\pi}{\hbar}\absolutevalue{g_{\mathrm{F}}}^{2}\quantity[F(\xi_{k}-\hbar\omega,\xi_{k})-F(\xi_{k},\xi_{k}-\hbar\omega)]N_{\mathrm{L}}(\xi_{k}-\hbar\omega)\delta
n^{\mathrm{eq}}(\omega)$
$\displaystyle\quad+\frac{2\pi}{\hbar}\absolutevalue{g_{\mathrm{F}}}^{2}\quantity[F(\xi_{k}+\hbar\omega,\xi_{k})-F(\xi_{k},\xi_{k}+\hbar\omega)]N_{\mathrm{L}}(\xi_{k}+\hbar\omega)\delta
n^{\mathrm{eq}}(\omega)$ (54) $k\in\mathrm{L},k^{\prime}\in\mathrm{R}$:
$\displaystyle\sum_{k^{\prime}\in\mathrm{R}}\quantity(W_{k^{\prime}\to
k}-W_{k\to k^{\prime}})$
$\displaystyle=\frac{2\pi}{\hbar}\absolutevalue{g_{\mathrm{B}}}^{2}\quantity[F(\xi_{k}-\hbar\omega,\xi_{k})-F(\xi_{k},\xi_{k}-\hbar\omega)]N_{\mathrm{R}}(\xi_{k}-\hbar\omega)\delta
n^{\mathrm{eq}}(\omega)$
$\displaystyle\quad+\frac{2\pi}{\hbar}\absolutevalue{g_{\mathrm{B}}}^{2}\quantity[F(\xi_{k}+\hbar\omega,\xi_{k})-F(\xi_{k},\xi_{k}+\hbar\omega)]N_{\mathrm{R}}(\xi_{k}+\hbar\omega)\delta
n^{\mathrm{eq}}(\omega)$ (55)

Therefore, we obtain the result Eq.(22a) and Eq.(22b).

##  Appendix B Calculation of the current due to the temperature difference

The electric current and the heat current due to the temperature difference
between phonons and electrons can be calculated explicitly as follows.

The electric current can be obtained from Eq.(28) by using Eqs.(22a),(22b).
For example, we can take the summation over $k\in\mathrm{R}$ in Eq.(28)
explicitly as:

$\displaystyle\frac{1}{L}\sum_{k\in\mathrm{R},k^{\prime}}\partialderivative{\xi_{k}}{k}\quantity(W_{k^{\prime}\to
k}-W_{k\to k^{\prime}})$ (56)
$\displaystyle=\int_{\mathrm{R}}\frac{\differential{\xi}}{2\pi}(-1)^{\mathrm{R}}\left\\{\frac{2\pi}{\hbar}\quantity(\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{L}}(\xi-\hbar\omega)+\absolutevalue{g_{\mathrm{F}}}^{2}N_{\mathrm{R}}(\xi-\hbar\omega))\quantity[F(\xi-\hbar\omega,\xi)-F(\xi,\xi-\hbar\omega)]\right.$
$\displaystyle\quad\left.+\frac{2\pi}{\hbar}\quantity(\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{L}}(\xi+\hbar\omega)+\absolutevalue{g_{\mathrm{F}}}^{2}N_{\mathrm{R}}(\xi+\hbar\omega))\quantity[F(\xi+\hbar\omega,\xi)-F(\xi,\xi+\hbar\omega)]\right\\}\delta
n^{\mathrm{eq}}(\omega)$ (57)

Here we used Eq.(22a). The factor
$(-1)^{\mathrm{R}}=\mathrm{sgn}\quantity(\evaluated{\partialderivative{\xi}{k}}_{\mathrm{R}})$
comes from the conversion of $k$ summation into $\xi$ integral,

$\displaystyle\frac{1}{L}\sum_{k\in\mathrm{R}}\partialderivative{\xi_{k}}{k}$
$\displaystyle=\int_{\mathrm{R}}\frac{\differential{k}}{2\pi}\partialderivative{\xi_{k}}{k}=\int_{\mathrm{R}}\frac{\differential{\xi}}{2\pi}\absolutevalue{\partialderivative{k}{\xi_{k}}}\partialderivative{\xi_{k}}{k}=\int_{\mathrm{R}}\frac{\differential{\xi}}{2\pi}(-1)^{\mathrm{R}}$
(58)

Since the integrand is appreciable only when $\xi\simeq 0$, we can extend the
range of integration to $[-\infty,\infty]$. Then we obtain

$\displaystyle\frac{1}{L}\sum_{k\in\mathrm{R}}\partialderivative{\xi_{k}}{k}\quantity(W_{k^{\prime}\to
k}-W_{k\to k^{\prime}})$
$\displaystyle=\int_{-\infty}^{\infty}\frac{\differential{\xi}}{\hbar}(-1)^{\mathrm{R}}\left\\{\quantity(\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{L}}(\xi-\hbar\omega)+\absolutevalue{g_{\mathrm{F}}}^{2}N_{\mathrm{R}}(\xi-\hbar\omega))\quantity[F(\xi-\hbar\omega,\xi)-F(\xi,\xi-\hbar\omega)]\right.$
$\displaystyle\quad\left.+\quantity(\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{L}}(\xi)+\absolutevalue{g_{\mathrm{F}}}^{2}N_{\mathrm{R}}(\xi))\quantity[F(\xi,\xi-\hbar\omega)-F(\xi-\hbar\omega,\xi)]\right\\}\delta
n^{\mathrm{eq}}(\omega)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{\differential{\xi}}{\hbar}(-1)^{\mathrm{R}}\quantity(\absolutevalue{g_{\mathrm{B}}}^{2}(N_{\mathrm{L}}(\xi-\hbar\omega)-N_{\mathrm{L}}(\xi))+\absolutevalue{g_{\mathrm{F}}}^{2}(N_{\mathrm{R}}(\xi-\hbar\omega)-N_{\mathrm{R}}(\xi)))\quantity[F(\xi-\hbar\omega,\xi)n^{\mathrm{eq}}(\omega)-F(\xi,\xi-\hbar\omega)\quantity(n^{\mathrm{eq}}(\omega)+1)]$
(59)

and similar for summation over $k\in\mathrm{L}$. Therefore, we obtain

$\displaystyle\frac{1}{L}\sum_{k,k^{\prime}}\partialderivative{\xi_{k}}{k}\quantity(W_{k^{\prime}\to
k}-W_{k\to k^{\prime}})$
$\displaystyle=\int_{-\infty}^{\infty}\frac{\differential{\xi}}{\hbar}(-1)^{\mathrm{R}}\quantity(\absolutevalue{g_{\mathrm{F}}}^{2}(N_{\mathrm{L}}(\xi-\hbar\omega)-N_{\mathrm{L}}(\xi))+\absolutevalue{g_{\mathrm{B}}}^{2}(N_{\mathrm{R}}(\xi-\hbar\omega)-N_{\mathrm{R}}(\xi)))\quantity[F(\xi-\hbar\omega,\xi)n^{\mathrm{eq}}(\omega)-F(\xi,\xi-\hbar\omega)\quantity(n^{\mathrm{eq}}(\omega)+1)]$
$\displaystyle+\int_{-\infty}^{\infty}\frac{\differential{\xi}}{\hbar}(-1)^{\mathrm{L}}\quantity(\absolutevalue{g_{\mathrm{B}}}^{2}(N_{\mathrm{L}}(\xi-\hbar\omega)-N_{\mathrm{L}}(\xi))+\absolutevalue{g_{\mathrm{F}}}^{2}(N_{\mathrm{R}}(\xi-\hbar\omega)-N_{\mathrm{R}}(\xi)))\quantity[F(\xi-\hbar\omega,\xi)n^{\mathrm{eq}}(\omega)-F(\xi,\xi-\hbar\omega)\quantity(n^{\mathrm{eq}}(\omega)+1)]$
$\displaystyle=\int_{-\infty}^{\infty}\frac{\differential{\xi}}{\hbar}(-1)^{\mathrm{R}}\quantity(\absolutevalue{g_{\mathrm{F}}}^{2}-\absolutevalue{g_{\mathrm{B}}}^{2})\quantity((N_{\mathrm{L}}(\xi-\hbar\omega)-N_{\mathrm{L}}(\xi))-(N_{\mathrm{R}}(\xi-\hbar\omega)-N_{\mathrm{R}}(\xi)))\quantity[F(\xi-\hbar\omega,\xi)-F(\xi,\xi-\hbar\omega)]\delta
n^{\mathrm{eq}}(\omega)$ (60)

where we assumed that $(-1)^{\mathrm{L}}=-(-1)^{\mathrm{R}}$. From Eq.(60), we
obtain Eq.(29) in the main text. The following formulas and approximations are
useful.

$\displaystyle\int_{-\infty}^{\infty}\differential{\xi}F(\xi,\xi-\hbar\omega)$
$\displaystyle=\hbar\omega n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})$ (61)
$\displaystyle\int_{-\infty}^{\infty}\differential{\xi}F(\xi-\hbar\omega,\xi)$
$\displaystyle=\hbar\omega(n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})+1)$ (62)
$\displaystyle N_{\mathrm{R}}(\xi-\hbar\omega)-N_{\mathrm{R}}(\xi)$
$\displaystyle\simeq-\hbar\omega N_{\mathrm{R}}^{\prime}(0)$ (63)
$\displaystyle N_{\mathrm{L}}(\xi-\hbar\omega)-N_{\mathrm{L}}(\xi)$
$\displaystyle\simeq-\hbar\omega N_{\mathrm{L}}^{\prime}(0)$ (64)

where $n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})$ is the Bose distribution
function with temperature $T_{\mathrm{e}}$. By making use of these formulas
and approximations, we can simplify Eq.(29) as

$\displaystyle\frac{1}{L}\sum_{k,k^{\prime}}\partialderivative{\xi_{k}}{k}\quantity(W_{k^{\prime}\to
k}-W_{k\to
k^{\prime}})=\frac{(-1)^{\mathrm{R}}}{\hbar}\quantity(\absolutevalue{g_{\mathrm{F}}}^{2}-\absolutevalue{g_{\mathrm{B}}}^{2})(-\hbar^{2}\omega^{2})\quantity(N_{\mathrm{L}}^{\prime}(0)-N_{\mathrm{R}}^{\prime}(0))\delta
n^{\mathrm{eq}}(\omega)$ (65)

Therefore, from Eq.(28), we obtain the result Eq.(30).

Similarly, we can obtain the expression for the heat current by using
Eqs.(22a),(22b),(31).

$\displaystyle J_{Q}$
$\displaystyle=V\int\differential{\xi}\frac{2\pi}{\hbar}\xi\quantity[N_{\mathrm{R}}(\xi)\quantity(\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{L}}(\xi-\hbar\omega)+\absolutevalue{g_{\mathrm{F}}}^{2}N_{\mathrm{R}}(\xi-\hbar\omega))+N_{\mathrm{L}}(\xi)\quantity(\absolutevalue{g_{\mathrm{F}}}^{2}N_{\mathrm{L}}(\xi-\hbar\omega)+\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{R}}(\xi-\hbar\omega))]$
$\displaystyle\times\quantity[F(\xi-\hbar\omega,\xi)-F(\xi,\xi-\hbar\omega)]\delta
n^{\mathrm{eq}}(\omega)$
$\displaystyle+V\int\differential{\xi}\frac{2\pi}{\hbar}\xi\quantity[N_{\mathrm{R}}(\xi)\quantity(\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{L}}(\xi+\hbar\omega)+\absolutevalue{g_{\mathrm{F}}}^{2}N_{\mathrm{R}}(\xi+\hbar\omega))+N_{\mathrm{L}}(\xi)\quantity(\absolutevalue{g_{\mathrm{F}}}^{2}N_{\mathrm{L}}(\xi+\hbar\omega)+\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{R}}(\xi+\hbar\omega))]$
$\displaystyle\times\quantity[F(\xi+\hbar\omega,\xi)-F(\xi,\xi+\hbar\omega)]\delta
n^{\mathrm{eq}}(\omega)$ $\displaystyle\simeq 2\pi
V\omega\quantity[N_{\mathrm{R}}(0)\quantity(\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{L}}(0)+\absolutevalue{g_{\mathrm{F}}}^{2}N_{\mathrm{R}}(0))+N_{\mathrm{L}}(0)\quantity(\absolutevalue{g_{\mathrm{F}}}^{2}N_{\mathrm{L}}(0)+\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{R}}(0))]$
$\displaystyle\quad\times\int\differential{\xi}\quantity[F(\xi-\hbar\omega,\xi)-F(\xi,\xi-\hbar\omega)]\delta
n^{\mathrm{eq}}(\omega)$ $\displaystyle=2\pi
V\hbar\omega^{2}\quantity[\absolutevalue{g_{\mathrm{F}}}^{2}(N_{\mathrm{L}}(0)^{2}+N_{\mathrm{R}}(0)^{2})+2\absolutevalue{g_{\mathrm{B}}}^{2}N_{\mathrm{R}}(0)N_{\mathrm{L}}(0)]\quantity[n^{\mathrm{eq}}(\omega,T_{\mathrm{p}})-n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})]$
(66)

where we have used $\sum_{k}=V\int\differential{\xi}N(\xi)$ with the volume
$V$, and an approximation $N_{\mathrm{R/L}}(\xi)\simeq N_{\mathrm{R/L}}(0)$.
The last expression is Eq.(32) in the main text.

##  Appendix C Interpretation of the definition of the heat current

We can transform the definition of the heat current Eq.(31) into the following
form.

$\displaystyle J_{Q}$
$\displaystyle=\sum_{k,k^{\prime}}(\xi_{k}-\xi_{k^{\prime}})\mathcal{W}_{k^{\prime}\to
k}$
$\displaystyle=\sum_{k,k^{\prime}}\frac{2\pi}{\hbar}\frac{\absolutevalue{g_{k^{\prime}-k}}^{2}}{V}f_{k^{\prime}}\quantity(1-f_{k})\hbar\omega_{k^{\prime}-k}\quantity[n_{k^{\prime}-k}\delta(\varepsilon_{k^{\prime}}-\varepsilon_{k}+\hbar\omega_{k^{\prime}-k})-\quantity(n_{k^{\prime}-k}+1)\delta(\varepsilon_{k^{\prime}}-\varepsilon_{k}-\hbar\omega_{k^{\prime}-k})]$
(67)

Here, we have used $\mathcal{W}_{k\to k^{\prime}}$ instead of $W_{k\to
k^{\prime}}$. The form of Eq.(67) might be easier to intuitively understand
than Eq.(31). The first term in the summation in Eq.(67) represents the
probability for electrons to absorb phonon multiplied by the absorbed energy
$\hbar\omega_{k^{\prime}-k}$. Similarly, the second term corresponds to the
phonon-emitting process.

##  Appendix D Calculation of $L_{11}$ and $L_{21}$

Here we calculate $L_{11}$ and $L_{21}$. What we want to calculate is the
electric current $J$ and heat current $J_{Q}$ in the presence of an electric
field $E$ up to $\order{E\absolutevalue{g_{q}}^{2}}$. By solving the Boltzmann
equation up to $\order{E\absolutevalue{g_{q}}^{2}}$, we obtain

$\displaystyle f_{k}$ $\displaystyle=f^{\mathrm{eq}}_{k}+f_{k}^{(1)}$ (68)
$\displaystyle f_{k}^{(1)}$
$\displaystyle=-\frac{eE\tau}{\hbar}\partialderivative{f^{\mathrm{eq}}_{k}}{k}$
(69)

Here, we have used Eq.(13) based on the assumption that the scattering term
proportional to $f_{k}-f^{\mathrm{eq}}_{k}$ is incorporated into the
relaxation time term.

Using $f_{k}^{(1)}$ in Eq.(69), the currents can be written as follows.

$\displaystyle J$
$\displaystyle=\frac{1}{L}\sum_{k}\frac{e}{\hbar}\partialderivative{\xi_{k}}{k}f_{k}^{(1)}$
(70) $\displaystyle J_{Q}$
$\displaystyle=\sum_{k}\xi_{k}\quantity(\partialderivative{f_{k}}{t})_{\mathrm{phonon}}=\sum_{k}\xi_{k}\sum_{k^{\prime}}\quantity(\mathcal{W}_{k^{\prime}\to
k}-\mathcal{W}_{k\to
k^{\prime}})=\sum_{k,k^{\prime}}(\xi_{k}-\xi_{k^{\prime}})\mathcal{W}_{k^{\prime}\to
k}$
$\displaystyle=\sum_{k,k^{\prime}}(\xi_{k}-\xi_{k^{\prime}})\frac{2\pi}{\hbar}\frac{\absolutevalue{g_{k^{\prime}-k}}^{2}}{V}\quantity(f_{k^{\prime}}^{(1)}(1-f^{\mathrm{eq}}_{k})-f^{\mathrm{eq}}_{k^{\prime}}f_{k}^{(1)})\left[n(\omega_{k^{\prime}-k})\delta(\xi_{k^{\prime}}-\xi_{k}+\hbar\omega_{k^{\prime}-k})\right.\left.+(n(\omega_{k^{\prime}-k})+1)\delta(\xi_{k}-\xi_{k^{\prime}}+\hbar\omega_{k^{\prime}-k})\right]$
(71)

Note that $\mathcal{W}_{k^{\prime}\to k}$ instead of $W_{k^{\prime}\to k}$ is
used in the calculation of $J_{Q}$. Below, we calculate these quantities using
the simplification explained in the main text, i.e.,
$\omega_{k^{\prime}-k}=\omega$ and the assumption for $g_{k^{\prime}-k}$,
Eq.(15).

First, let us calculate $J$. This is the well-known contribution to the
electric current and can be calculated as follows.

$\displaystyle J$
$\displaystyle=\frac{1}{L}\sum_{\alpha=\mathrm{R},\mathrm{L}}\sum_{k\in\alpha}\frac{e}{\hbar}\partialderivative{\xi_{k}}{k}f_{k}^{(1)}=-\frac{1}{L}\frac{e^{2}E\tau}{\hbar^{2}}\sum_{\alpha=\mathrm{R},\mathrm{L}}\sum_{k\in\alpha}\quantity(\partialderivative{\xi_{k}}{k})^{2}\partialderivative{f^{\mathrm{eq}}_{k}}{\xi_{k}}$
$\displaystyle=-\frac{e^{2}E\tau}{\hbar^{2}}\sum_{\alpha=\mathrm{R},\mathrm{L}}\int_{\alpha}\frac{\differential{k}}{2\pi}\quantity(\partialderivative{\xi_{k}}{k})^{2}\partialderivative{f^{\mathrm{eq}}_{k}}{\xi_{k}}=-\frac{e^{2}E\tau}{\hbar^{2}}\sum_{\alpha=\mathrm{R},\mathrm{L}}\int_{\alpha}a^{2}N_{\alpha}(\xi)\differential{\xi}\quantity(\frac{1}{2\pi
a^{2}N_{\alpha}(\xi)})^{2}\partialderivative{f^{\mathrm{eq}}_{k}}{\xi_{k}}$
$\displaystyle=-\frac{e^{2}E\tau}{4\pi^{2}a^{2}\hbar^{2}}\sum_{\alpha=\mathrm{R},\mathrm{L}}\int_{\alpha}\differential{\xi}\frac{1}{N_{\alpha}(\xi)}\partialderivative{f^{\mathrm{eq}}_{k}}{\xi_{k}}$
(72)

Here we used a relation $N_{\alpha}(\xi)=\quantity(2\pi
a^{2}\absolutevalue{\partialderivative{\xi_{k}}{k}})^{-1}$. As in the
calculation of the currents in Appendix B, we neglect the $\xi$ dependence of
$N_{\alpha}(\xi)$ and approximate it by $N_{\alpha}(\xi=0)$. Then we obtain

$\displaystyle J$
$\displaystyle=-\frac{e^{2}E\tau}{4\pi^{2}a^{2}\hbar^{2}}\sum_{\alpha=\mathrm{R},\mathrm{L}}\int_{\alpha}\differential{\xi}\frac{1}{N_{\alpha}(0)}\partialderivative{f^{\mathrm{eq}}_{k}}{\xi_{k}}=\frac{e^{2}E\tau}{4\pi^{2}a^{2}\hbar^{2}}\quantity(\frac{1}{N_{\mathrm{R}}(0)}+\frac{1}{N_{\mathrm{L}}(0)})=\sigma_{0}E$
(73)

Similary, we can calculate $L_{21}$. Starting from (71),

$\displaystyle
J_{Q}=\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\sum_{k\in\alpha,k^{\prime}\in\beta}(\xi_{k}-\xi_{k^{\prime}})\frac{2\pi}{\hbar}\frac{\absolutevalue{g_{\beta\alpha}}^{2}}{V}\quantity(f_{k^{\prime}}^{(1)}(1-f^{\mathrm{eq}}_{k})-f^{\mathrm{eq}}_{k^{\prime}}f_{k}^{(1)})\quantity(n(\omega)\delta(\xi_{k^{\prime}}-\xi_{k}+\hbar\omega)+(n(\omega)+1)\delta(\xi_{k}-\xi_{k^{\prime}}+\hbar\omega))$
$\displaystyle=\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\sum_{k\in\alpha,k^{\prime}\in\beta}E\frac{2\pi
e\omega\tau}{\hbar}\frac{\absolutevalue{g_{\beta\alpha}}^{2}}{V}\quantity[-\partialderivative{f^{\mathrm{eq}}_{k^{\prime}}}{k^{\prime}}+\partialderivative{f^{\mathrm{eq}}_{k^{\prime}}}{k^{\prime}}f^{\mathrm{eq}}_{k}+f^{\mathrm{eq}}_{k^{\prime}}\partialderivative{f^{\mathrm{eq}}_{k}}{k}]\quantity(n(\omega)\delta(\xi_{k^{\prime}}-\xi_{k}+\hbar\omega)-(n(\omega)+1)\delta(\xi_{k}-\xi_{k^{\prime}}+\hbar\omega))$
(74)

The first term in the square bracket in (74) reads

$\displaystyle\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\sum_{k\in\alpha,k^{\prime}\in\beta}E\frac{2\pi
e\omega\tau}{\hbar}\frac{\absolutevalue{g_{\beta\alpha}}^{2}}{V}\quantity(-\partialderivative{f^{\mathrm{eq}}_{k^{\prime}}}{k^{\prime}})\quantity[n(\omega)\delta(\xi_{k^{\prime}}-\xi_{k}+\hbar\omega)-(n(\omega)+1)\delta(\xi_{k}-\xi_{k^{\prime}}+\hbar\omega)]$
$\displaystyle=\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\sum_{k^{\prime}\in\beta}E\frac{2\pi
e\omega\tau}{\hbar}\absolutevalue{g_{\beta\alpha}}^{2}\quantity(-\partialderivative{f^{\mathrm{eq}}_{k^{\prime}}}{k^{\prime}})\quantity[n(\omega)N_{\alpha}(\xi_{k^{\prime}}+\hbar\omega)-(n(\omega)+1)N_{\alpha}(\xi_{k^{\prime}}-\hbar\omega)]$
(75)

The second and the third terms in the square bracket in (74) read

$\displaystyle\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\sum_{k\in\alpha,k^{\prime}\in\beta}E\frac{2\pi
e\omega\tau}{\hbar}\frac{\absolutevalue{g_{\beta\alpha}}^{2}}{V}\quantity[\partialderivative{f^{\mathrm{eq}}_{k^{\prime}}}{k^{\prime}}f^{\mathrm{eq}}_{k}+f^{\mathrm{eq}}_{k^{\prime}}\partialderivative{f^{\mathrm{eq}}_{k}}{k}]\quantity(n(\omega)\delta(\xi_{k^{\prime}}-\xi_{k}+\hbar\omega)-(n(\omega)+1)\delta(\xi_{k}-\xi_{k^{\prime}}+\hbar\omega))$
$\displaystyle=-\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\sum_{k\in\alpha,k^{\prime}\in\beta}E\frac{2\pi
e\omega\tau}{\hbar}\frac{\absolutevalue{g_{\beta\alpha}}^{2}}{V}\partialderivative{f^{\mathrm{eq}}_{k^{\prime}}}{k^{\prime}}f^{\mathrm{eq}}_{k}\quantity(\delta(\xi_{k^{\prime}}-\xi_{k}+\hbar\omega)+\delta(\xi_{k}-\xi_{k^{\prime}}+\hbar\omega))$
$\displaystyle=-\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\sum_{k^{\prime}\in\beta}E\frac{2\pi
e\omega\tau}{\hbar}\absolutevalue{g_{\beta\alpha}}^{2}\partialderivative{f^{\mathrm{eq}}_{k^{\prime}}}{k^{\prime}}\quantity(f^{\mathrm{eq}}(\xi_{k}^{\prime}+\hbar\omega)N_{\alpha}(\xi_{k^{\prime}}+\hbar\omega)+f^{\mathrm{eq}}(\xi_{k}^{\prime}-\hbar\omega)N_{\alpha}(\xi_{k^{\prime}}-\hbar\omega))$
(76)

Therefore, $J_{Q}$ is

$\displaystyle J_{Q}$ $\displaystyle=-E\frac{2\pi
e\omega\tau}{\hbar}\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\sum_{k^{\prime}\in\beta}\left\\{\absolutevalue{g_{\beta\alpha}}^{2}\partialderivative{f^{\mathrm{eq}}_{k^{\prime}}}{k^{\prime}}\quantity[n(\omega)N_{\alpha}(\xi_{k^{\prime}}+\hbar\omega)-(n(\omega)+1)N_{\alpha}(\xi_{k^{\prime}}-\hbar\omega)]\right.$
$\displaystyle\left.+\absolutevalue{g_{\beta\alpha}}^{2}\partialderivative{f^{\mathrm{eq}}_{k^{\prime}}}{k^{\prime}}\quantity(f^{\mathrm{eq}}(\xi_{k^{\prime}}+\hbar\omega)N_{\alpha}(\xi_{k^{\prime}}+\hbar\omega)+f^{\mathrm{eq}}(\xi_{k^{\prime}}-\hbar\omega)N_{\alpha}(\xi_{k^{\prime}}-\hbar\omega))\right\\}$
$\displaystyle=-E\frac{2\pi e\omega\tau
L}{\hbar}\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\absolutevalue{g_{\beta\alpha}}^{2}\int_{\beta}\frac{\differential{\xi_{\beta}}}{2\pi}(-1)^{\beta}\quantity[{f^{\mathrm{eq}}}^{\prime}(\xi_{\beta})f^{\mathrm{eq}}(\xi_{\beta}+\hbar\omega)N_{\alpha}(\xi_{\beta}+\hbar\omega)+{f^{\mathrm{eq}}}^{\prime}(\xi_{\beta}+\hbar\omega)(f^{\mathrm{eq}}(\xi_{\beta})-1)N_{\alpha}(\xi_{\beta})]$
$\displaystyle-E\frac{2\pi e\omega\tau
L}{\hbar}\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\absolutevalue{g_{\beta\alpha}}^{2}\int_{\beta}\frac{\differential{\xi_{\beta}}}{2\pi}(-1)^{\beta}{f^{\mathrm{eq}}}^{\prime}(\xi_{\beta})n(\omega)\quantity[N_{\alpha}(\xi_{\beta}+\hbar\omega)-N_{\alpha}(\xi_{\beta}-\hbar\omega)]$
$\displaystyle=-E\frac{2\pi e\omega\tau
L}{\hbar}\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\absolutevalue{g_{\beta\alpha}}^{2}\int_{\beta}\frac{\differential{\xi_{\beta}}}{2\pi}(-1)^{\beta}[{f^{\mathrm{eq}}}^{\prime}(\xi_{\beta})f^{\mathrm{eq}}(\xi_{\beta}+\hbar\omega)N_{\alpha}(\xi_{\beta}+\hbar\omega)-{f^{\mathrm{eq}}}(\xi_{\beta}+\hbar\omega){f^{\mathrm{eq}}}^{\prime}(\xi_{\beta})N_{\alpha}(\xi_{\beta})$
$\displaystyle-{f^{\mathrm{eq}}}(\xi_{\beta}+\hbar\omega)(f^{\mathrm{eq}}(\xi_{\beta})-1)N_{\alpha}^{\prime}(\xi_{\beta})+{f^{\mathrm{eq}}}^{\prime}(\xi_{\beta})n(\omega)\quantity(N_{\alpha}(\xi_{\beta}+\hbar\omega)-N_{\alpha}(\xi_{\beta}-\hbar\omega))]\quad\text{($f^{\prime}(f-1)N$
is partially integrated)}$ $\displaystyle=-E\frac{2\pi e\omega\tau
L}{\hbar}\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\absolutevalue{g_{\beta\alpha}}^{2}\int_{\beta}\frac{\differential{\xi_{\beta}}}{2\pi}(-1)^{\beta}\quantity[{f^{\mathrm{eq}}}^{\prime}(\xi_{\beta})f^{\mathrm{eq}}(\xi_{\beta}+\hbar\omega)\quantity[N_{\alpha}(\xi_{\beta}+\hbar\omega)-N_{\alpha}(\xi_{\beta})]+{f^{\mathrm{eq}}}(\xi_{\beta}+\hbar\omega)(1-f^{\mathrm{eq}}(\xi_{\beta}))N_{\alpha}^{\prime}(\xi_{\beta})]$
$\displaystyle-E\frac{2\pi e\omega\tau
L}{\hbar}\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\absolutevalue{g_{\beta\alpha}}^{2}\int_{\beta}\frac{\differential{\xi_{\beta}}}{2\pi}(-1)^{\beta}{f^{\mathrm{eq}}}^{\prime}(\xi_{\beta})\quantity[n(\omega)(N_{\alpha}(\xi_{\beta}+\hbar\omega)-N_{\alpha}(\xi_{\beta}-\hbar\omega))]$
(77)

If we neglect $\xi$-dependence of $N_{\alpha}$ completely, $J_{Q}$ becomes 0.
Therefore, we expand $N_{\alpha}(\xi_{\beta}\pm\hbar\omega)$ up to
$\order{\hbar\omega}$ and approximate it as
$N_{\alpha}(\xi_{\beta}\pm\hbar\omega)-N_{\alpha}(\xi_{\beta})\simeq\pm\hbar\omega
N_{\alpha}^{\prime}(0)$. Then we obtain

$\displaystyle J_{Q}$ $\displaystyle\simeq-E\frac{2\pi e\omega\tau
L}{\hbar}\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\absolutevalue{g_{\beta\alpha}}^{2}N_{\alpha}^{\prime}(0)\int_{\beta}\frac{\differential{\xi_{\beta}}}{2\pi}(-1)^{\beta}[\hbar\omega{f^{\mathrm{eq}}}^{\prime}(\xi_{\beta})f^{\mathrm{eq}}(\xi_{\beta}+\hbar\omega)+F(\xi_{\beta}+\hbar\omega,\xi_{\beta})]$
$\displaystyle-E\frac{2\pi e\omega\tau
L}{\hbar}\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\absolutevalue{g_{\beta\alpha}}^{2}2\hbar\omega
N_{\alpha}^{\prime}(0)\int_{\beta}\frac{\differential{\xi_{\beta}}}{2\pi}(-1)^{\beta}{f^{\mathrm{eq}}}^{\prime}(\xi_{\beta})n(\omega)$
$\displaystyle=-E\frac{2\pi e\omega\tau
L}{\hbar}\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}\absolutevalue{g_{\beta\alpha}}^{2}\hbar\omega
N_{\alpha}^{\prime}(0)\frac{(-1)^{\beta}}{2\pi}(-\beta\hbar\omega
n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})(n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})+1))$
$\displaystyle=Ee\beta\hbar\omega^{3}\tau
Ln^{\mathrm{eq}}(\omega,T_{\mathrm{e}})(n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})+1)\sum_{\alpha,\beta\in\mathrm{R},\mathrm{L}}(-1)^{\beta}\absolutevalue{g_{\beta\alpha}}^{2}N_{\alpha}^{\prime}(0)$
$\displaystyle=E(-1)^{\mathrm{R}}e\beta\hbar\omega^{3}\tau
L(\absolutevalue{g_{\mathrm{F}}}^{2}-\absolutevalue{g_{\mathrm{B}}}^{2})(N_{\mathrm{R}}^{\prime}(0)-N_{\mathrm{L}}^{\prime}(0))n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})(n^{\mathrm{eq}}(\omega,T_{\mathrm{e}})+1)$
(78)

where we have assumed again that $(-1)^{\mathrm{R}}=-(-1)^{\mathrm{L}}$. From
Eq.(78), the expression for $L_{21}$, Eq.(39), is obtained.

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