# Cooperative Nernst Effect of Multilayer Systems: Parallel Circuit Model
Study

Hiroyasu Matsuura<EMAIL_ADDRESS>Department of Physics,
University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  Alexander
Riss  Fabian Garmroudi  Michael Parzer  Ernst Bauer  Institute of Solid State
Physics, Technische Universität Wien, Vienna, Austria

###### Abstract

Transverse thermoelectric power generation has emerged as a topic of immense
interest in recent years owing to the orthogonal geometry which enables better
scalability and fabrication of devices. Here, we investigate the thickness
dependence of longitudinal and transverse responses in film-substrate systems
i.e., the Seebeck coefficient, Hall coefficient, Nernst coefficient and
anomalous Nernst coefficient in a unified and general manner based on the
circuit model, which describes the system as the parallel setup. By solving
the parallel circuit model, we show that the transverse responses exhibit a
significant peak, indicating the importance of a cooperative effect between
the film and the substrate, arising from circulating currents that occur in
these multilayer systems in the presence of a temperature gradient. Finally,
on the basis of realistic material parameters, we predict that the Nernst
effect in bismuth thin films on doped silicon substrates is boosted to
unprecedented values if the thickness ratio is tuned accordingly, motivating
experimental validation.

##  I Introduction

The Seebeck and Nernst effect, which convert a temperature gradient into a
longitudinal and transverse voltage, respectively, have attracted much
attention in recent years, not only in terms of fundamental science but also
from the perspective of effective utilization of thermal energy [1, 2].
Heavily doped semiconductors have been studied since the middle of the 20th
century, while other classes of materials such as semimetallic systems
including Heusler compounds [3, 4, 5, 6] or Dirac and Weyl semimetals [7, 9,
8] as well as transition metal systems with strong interband scattering [10]
have also been the focus of recent attention. In particular, research on
thermoelectric materials focusing on drag effects such as magnon drag [11, 12]
and paramagnon drag effects [13, 14] for longitudinal thermoelectric response
and the anomalous Nernst effect using exotic magnetic states for transverse
thermoelectric effect have been in a specific focus of research [2].

With the development of microfabrication technology, thermoelectric effects in
film-substrate systems have been also studied extensively [15, 16, 17, 18,
19]. For example, regarding the Seebeck effect, it was discovered that FeSe
films on a SrTiO3 substrate exhibit huge power factors [17, 20]. Furthermore,
it has been reported that MoO2+x shows a much larger power factor on silicon
(Si) than on quarz substrate [18]. There have also been various studies on the
transverse thermoelectric effect in film-substrate systems [21, 22, 23, 26,
24, 25, 26, 27, 28, 29, 30, 31]. The ordinary Nernst effect in such systems
has been investigated as an example of transport properties in semiconducting
films with a perspective of application [21, 22, 23]. On the other hand, a
large anomalous Nernst effect has been discovered in ferromagnetic films,
resulting in active research on the anomalous Nernst effect in film-substrate
systems [24, 25, 26, 27]. In full-Heusler Co2MnGa films, a very large value of
about -3$\upmu$V/K has been observed at a thickness of 40 nm [26]. It has also
been reported that the anomalous Nernst coefficient increases with decreasing
thickness in ferromagnetic Fe or Ni films on Si substrates [24].

The above-indicated studies infer the importance of a substrate contribution
to measurements in the film-substrate system. Presently, phenomenological
theoretical methods for discussing the influence of the substrate on the
Seebeck coefficient of film-substrate systems [32, 33] and those for two-
material composite [34, 35, 36] have been discussed.

More recently, in bilayer systems of Fe-Ga films on n-type silicon substrate,
it was found that the anomalous Nernst effect unexpectedly increases
drastically as the thickness of substrate increases, reaching a peak of 15
$\upmu$V/K, being much larger than that of bulk Fe-Ga and much larger than the
current record observed in Weyl semimetal Co2MnGa [37]. The thickness
dependence of the bilayer system has been also studied on the basis of a
circuit model, and theoretical results are in good agreement with experiments
[37].

Clearly, the film-substrate system has the potential to originate a huge
thermoelectric effect by tuning the thickness. Therefore, in this paper, we
discuss the Hall coefficient, Seebeck effect and transverse thermoelectric
effects such as normal and anomalous Nernst effects in film-substrate systems
in a unified and general manner based on the parallel circuit model in matrix
form. In conclusion, the Seebeck coefficients vary gradually with parameters
characterizing the ratio of the thickness of film and substrate, while it is
found that both the anomalous Nernst effect (studied in ref. [37]) as well as
the ordinary Nernst effect exhibit a significant peak. Finally, as an
realistic example of unsuspected features, we study the Nernst effect in
bismuth film on doped silicon substrate system.

The paper is organized as follows. In Sec. II, we introduce the linear
response for the electrical and thermoelectric effects. In Sec. III, we derive
the linear response in the film-substrate system based on the circuit model,
which is used in Sec. IV to perform model calculations. In Sec. V, we
illustrate the results with bismuth films on silicon substrate.

##  II Linear response for the electrical and thermoelectric effects

In this section, we study the linear response for electrical and
thermoelectric effects with/without magnetic field [38]. When applied fields
in the system A comprise an electric field (${\bm{E}}_{\rm A}$), a thermal
gradient ($({\nabla}T)_{\rm A}$), and a magnetic field along the $z$-axis
($B_{z,{\rm A}}$), the linear response in the $x$,$y$ directions of system A
is

$\displaystyle\begin{pmatrix}j^{x}_{\rm e,A}\\\ j^{y}_{\rm e,A}\\\ j^{x}_{\rm
Q,A}\\\ j^{y}_{\rm Q,A}\\\ \end{pmatrix}=\bar{L}_{\rm
A}\begin{pmatrix}E_{x,{\rm A}}\\\ E_{y,{\rm A}}\\\ -(\nabla T)_{x,{\rm
A}}/T\\\ -(\nabla T)_{y,{\rm A}}/T\\\ \end{pmatrix},$ (1)

where ${\bm{j}}_{\rm e}=(j^{x}_{\rm e,A},j^{y}_{\rm e,A})$ and ${\bm{j}}_{\rm
Q}=(j^{x}_{\rm Q,A},j^{y}_{\rm Q,A})$ are electrical and thermal current
densities in system A, respectively. $\bar{L}_{\rm A}$ is given by

$\displaystyle\bar{L}_{\rm A}=\begin{pmatrix}\bar{L}_{11,{\rm
A}}&\bar{L}_{12,{\rm A}}\\\ \bar{L}_{21,{\rm A}}&\bar{L}_{22,{\rm A}}\\\
\end{pmatrix},$ (2)

where $\bar{L}_{ij,{\rm A}}$ is a $2\times 2$ matrix of system A represented
by

$\displaystyle\bar{L}_{ij,{\rm A}}=\begin{pmatrix}L_{ij,{\rm
A}}^{xx}&L_{ij,{\rm A}}^{xy}\\\ L_{ij,{\rm A}}^{yx}&L_{ij,{\rm A}}^{yy}\\\
\end{pmatrix}.$ (3)

In this paper $L_{ij,{\rm A}}^{kl}$ is called the linear response coefficient
of system A, and $L_{ij,{\rm A}}^{xy}=-L_{ij,{\rm A}}^{yx}$. In particular,
$L_{11}^{xx(yy)}$ and $L_{11}^{xy}$ are the same as $\sigma^{xx(yy)}$ and
$\sigma^{xy}$, which are the electrical conductivity and the Hall
conductivity, respectively. As shown next, the Hall coefficient, Seebeck
coefficient, Nernst coefficient and anomalous Nernst coefficient, which are
transverse thermoelectric effects with/without a magnetic field, can be
expressed in terms of the linear response coefficients.

###  II.1 Hall coefficient

The Hall effect measures the electric field that appears in the $y$ direction
when an electrical current is applied in the $x$ direction in the presence of
a perpendicular magnetic field $B_{z,{\rm A}}$ and in the absence of a thermal
gradient. The Hall coefficient of system A ($R_{\rm H,A}$) is defined as

$\displaystyle R_{\rm H,A}=\frac{E_{y,{\rm A}}}{j_{e,{\rm A}}^{x}B_{z,{\rm
A}}}.$ (4)

From Eqs. (1)-(3), we obtain

$\displaystyle\begin{pmatrix}j^{x}_{\rm e,A}\\\ j^{y}_{\rm e,A}\\\
\end{pmatrix}=\bar{L}_{11,{\rm A}}\begin{pmatrix}E_{x,{\rm A}}\\\ E_{y,{\rm
A}}\\\ \end{pmatrix}.$ (5)

By solving Eq. (5) in the condition of $j^{y}_{\rm e,A}=0$, the Hall
coefficient is given as

$\displaystyle R_{\rm H,A}=\frac{L_{11,{\rm A}}^{xy}}{L_{11,{\rm
A}}^{xx}L_{11,{\rm A}}^{yy}B_{z,{\rm A}}}.$ (6)

###  II.2 Seebeck and Nernst coefficients

The Seebeck and Nernst coefficients are obtained by measuring the electric
field that appears in the $x$ and $y$ direction, respectively, when a thermal
gradient is applied in the $x$ direction, and, for the Nernst coefficient, a
magnetic field is present. The Seebeck coefficient $S_{xx,\text{A}}$ and
Nernst coefficient $\nu_{yx,\text{A}}$ are defined as

$\displaystyle S_{xx,{\rm A}}$ $\displaystyle=$ $\displaystyle\frac{E_{x,{\rm
A}}}{(\nabla T)_{x,{\rm A}}},$ (7) $\displaystyle\nu_{yx,{\rm A}}$
$\displaystyle=$ $\displaystyle\frac{E_{y,{\rm A}}}{(\nabla T)_{x,{\rm
A}}B_{z,{\rm A}}}.$ (8)

If ${\bm{j}}_{e}$ is zero, the linear response equation reads

$\displaystyle 0=\bar{L}_{11,{\rm A}}\begin{pmatrix}E_{x,{\rm A}}\\\ E_{y,{\rm
A}}\\\ \end{pmatrix}+\bar{L}_{12,{\rm A}}\begin{pmatrix}-(\nabla T)_{x,{\rm
A}}/T\\\ -(\nabla T)_{y,{\rm A}}/T\\\ \end{pmatrix}.$ (9)

Since the thermal gradient in the $y$ direction can be neglected,

$\displaystyle\begin{pmatrix}E_{x,{\rm A}}\\\ E_{y,{\rm A}}\\\
\end{pmatrix}=-\bar{L}_{11,{\rm A}}^{-1}\bar{L}_{12,{\rm
A}}\begin{pmatrix}-(\nabla T)_{x,{\rm A}}/T\\\ 0\\\ \end{pmatrix},$ (10)

where $\bar{L}_{11,{\rm A}}^{-1}$is the inverse matrix of $\bar{L}_{11,{\rm
A}}$.

Ignoring the term containing $L_{11,{\rm A}}^{xy}L_{11,{\rm A}}^{yx}$, a
second-order effect of the magnetic field, yields the Seebeck and Nernst
coefficients as

$\displaystyle S_{xx,{\rm A}}$ $\displaystyle=$ $\displaystyle\frac{L_{12,{\rm
A}}^{xx}}{TL_{11,{\rm A}}^{xx}},$ (11) $\displaystyle\nu_{yx,{\rm A}}$
$\displaystyle=$ $\displaystyle\frac{L_{11,{\rm A}}^{xy}L_{12,{\rm
A}}^{xx}-L_{11,{\rm A}}^{xx}L_{12,{\rm A}}^{xy}}{TL_{11,{\rm
A}}^{xx}L_{11,{\rm A}}^{yy}B_{z,{\rm A}}}$ (12) $\displaystyle=$
$\displaystyle(S_{xx,{\rm A}}\theta_{{\rm H},{\rm A}}-S_{yy,A}\alpha_{{\rm
N},{\rm A}})/B_{z,{\rm A}},$

where $\nu_{yx,{\rm A}}=-\nu_{xy,{\rm A}}$. $\theta_{{\rm H},{\rm A}}$ and
$\alpha_{{\rm N},{\rm A}}$ are the Hall and Peltier angles [25] defined as
(See Appendix A)

$\displaystyle\tan{\theta_{{\rm H},{\rm A}}}\simeq\theta_{{\rm H},{\rm A}}$
$\displaystyle=$ $\displaystyle\frac{L_{11,{\rm A}}^{xy}}{L_{11,{\rm
A}}^{yy}},$ (13) $\displaystyle\tan{\alpha_{{\rm N},{\rm
A}}}\simeq\alpha_{{\rm N},{\rm A}}$ $\displaystyle=$
$\displaystyle\frac{L_{12,{\rm A}}^{xy}}{L_{12,{\rm A}}^{yy}}.$ (14)

In the Nernst effect, $L_{11,A}^{xy}$ is proportional to the magnetic field,
i.e., $L_{11,A}^{xy}\propto B_{z,{\rm A}}$. Thus, using Eq. (6), the Hall
angle is given by

$\displaystyle\theta_{{\rm H},{\rm A}}=R_{\rm H,A}L_{11,{\rm A}}^{xx}B_{z,{\rm
A}}.$ (15)

###  II.3 Thermoelectric transports with anomalous components

In the above discussion, we studied the Hall, Seebeck and Nernst coefficients.
On the other hand, $L^{xy}_{ij}$ can be finite in ferromagnetic materials even
without a magnetic field, yielding to anomalous effects.

The anomalous Nernst coefficient is defined as

$\displaystyle N_{yx,A}=\frac{E_{y,A}}{(\nabla T)_{x,A}}.$ (16)

Using Eqs. (7), (10) and (16), the Seebeck coefficient with anomalous
components and the anomalous Nernst coefficient are

$\displaystyle S_{xx,A}^{AN}$ $\displaystyle=$
$\displaystyle\frac{L_{11,A}^{yy}L_{12,A}^{xx}+L_{11,A}^{xy}L_{12,A}^{xy}}{T(L_{11,A}^{xx}L_{11,A}^{yy}+(L_{11,A}^{xy})^{2})}$
(17) $\displaystyle=$ $\displaystyle\frac{S_{xx,A}+\theta_{\rm H,A}\alpha_{\rm
N,A}r_{12}S_{yy,A}}{1+\theta_{\rm H,A}^{2}r_{11}},$ $\displaystyle N_{yx,A}$
$\displaystyle=$
$\displaystyle\frac{L_{11,A}^{xy}L_{12,A}^{xx}-L_{11,A}^{xx}L_{12,A}^{xy}}{T(L_{11,A}^{xx}L_{11,A}^{yy}+(L_{11,A}^{xy})^{2})}$
(18) $\displaystyle=$ $\displaystyle\frac{S_{xx,A}\theta_{\rm
H,A}-S_{yy,A}\alpha_{\rm N,A}}{1+\theta_{\rm H,A}^{2}r_{11}},$

where $r_{ij}=L_{ij}^{yy}/L_{ij}^{xx}$ and $N_{yx,A}=-N_{xy,A}$.

##  III Linear response of a film-substrate system based on the circuit model

In this section, we derive a linear response coefficient in the film-substrate
system based on circuit model which treats the film and substrate as a
parallel setup. Fig. 1(a) shows a sketch of a film-substrate systems. As
discussed in [33], this system can be mapped to the parallel circuit as shown
in Fig. 1(b).

Figure 1: Schematic image of a) film-substrate system, b) parallel circuit
consisting of $\bar{L}_{f}$ and $\bar{L}_{s}$, and c) the circuit organized by
Kirchhoff’s law. $V_{x,i}=E_{x,i}a$, and $(\Delta T)_{y,i}=(\nabla T)_{x,i}a$
for $i=s,f$, with $s$ and $f$ denoting the substrate and film, respectively.

Here, the linear response coefficients of the film (system A $=f$) and
substrate (A $=s$) are defined as $\bar{L}_{f}$ and $\bar{L}_{s}$,
respectively. We assume that $B_{z,f}=B_{z,s}\equiv B_{z}$.

In this paper, the volume of film and substrate are $v_{f}=a_{x}\times
a_{y}\times d_{f}$ and $v_{s}=a_{x}\times a_{y}\times d_{s}$, respectively,
where $a_{x(y)}$ is the length in $x$ $(y)$ direction, and $d_{f}$ and $d_{s}$
are thicknesses of film and substrate, respectively. From Kirchhoff’s law in
parallel circuits (see Fig. 1(c)), the electrical potentials of the film and
substrate ($V_{x,f}=E_{x,f}a_{x}$ and $V_{x,s}=E_{x,s}a_{x}$) satisfy the
following equations:

$\displaystyle V_{x,f}=V_{x,s}=V_{x},$ (19) $\displaystyle
V_{y,f}=V_{y,s}=V_{y},$ (20)

where $V_{x}$ and $V_{y}$ are electrical potentials in the $x$ and $y$
directions in the system of the combined circuit, respectively. Thus, the
electric fields in the $x$\- and $y$\- directions in the combined circuit are
defined as $E_{x}=V_{x}/a_{x}$ and $E_{y}=V_{y}/a_{y}$, respectively.

The temperature gradient is considered similarly as for in the electrical
circuit. When the temperature differences in the $x$ ($y$) direction at film
and substrate are $(\Delta T)_{x(y),f}=(\nabla T)_{x(y),f}a_{x(y)}$ and
$(\Delta T)_{x(y),s}=(\nabla T)_{x(y),s}a_{x(y)}$, we obtain

$\displaystyle(\Delta T)_{x,f}=(\Delta T)_{x,s}=(\Delta T)_{x},$ (21)
$\displaystyle(\Delta T)_{y,f}=(\Delta T)_{y,s}=(\Delta T)_{y},$ (22)

where $(\Delta T)_{x}$ and $(\Delta T)_{y}$ are the temperature differences in
the combined circuit.

In the parallel system formed by the film and the substrate, the total
electrical ($\beta=e$) and thermal ($\beta=Q$) currents in $x$ (y) direction
are given as

$\displaystyle j_{\beta,{\rm
tot}}^{x(y)}a_{y(x)}(d_{s}+d_{f})=j_{\beta,f}^{x(y)}a_{y(x)}d_{f}+j_{\beta,s}^{x(y)}a_{y(x)}d_{s}.$
(23)

From Eqs. (1), (19)-(22), and (23), the total linear response coefficient of
film-substrate system is given as

$\displaystyle\bar{L}_{\rm
tot}\equiv\frac{d_{f}}{d_{f}+d_{s}}\bar{L}_{f}+\frac{d_{s}}{d_{f}+d_{s}}\bar{L}_{s}.$
(24)

For simplicity, we assume below that the response is isotropic in the $x$\-
and $y$-directions. Applying Eq. (24) to Eqs. (11), (13) and (14), the Seebeck
coefficient ($S_{\rm tot}$), the Hall angle ($\theta_{\rm H,\rm tot}$), and
the Peltier angle ($\alpha_{\rm N,\rm tot}$) are

$\displaystyle S_{\rm tot}$ $\displaystyle=$
$\displaystyle\frac{S_{f}+\epsilon_{\sigma}S_{s}}{1+\epsilon_{\sigma}},$ (25)
$\displaystyle\theta_{\rm H,\rm tot}$ $\displaystyle=$
$\displaystyle\frac{\theta_{\rm H,f}+\epsilon_{\sigma}\theta_{\rm
H,s}}{1+\epsilon_{\sigma}},$ (26) $\displaystyle\alpha_{\rm N,\rm tot}$
$\displaystyle=$ $\displaystyle\frac{S_{f}\alpha_{\rm
N,f}+\epsilon_{\sigma}S_{s}\alpha_{\rm N,s}}{S_{f}+\epsilon_{\sigma}S_{s}},$
(27)

and

$\displaystyle\epsilon_{\sigma}=\frac{L_{11,s}^{xx}d_{s}}{L_{11,f}^{xx}d_{f}}=\frac{L_{11,s}^{yy}d_{s}}{L_{11,f}^{yy}d_{f}}=\frac{R_{f}}{R_{s}},$
(28)

where $R_{A}$ is the resistance of film ($A=f$) or substrate ($A=s$) [33]. The
expression of the Seebeck coefficient in Eq. (25) is the same as in a previous
study [33]. The Hall angle (Eq. (26)) and the Peltier angle (Eq. (27)) are
functions of the same form as the Seebeck coefficient (Eq. (25)), when
$\epsilon_{\sigma}S_{s}/S_{f}$ is replaced by $\tilde{\epsilon}_{\sigma}$ in
Eq. (27).

Next, applying Eq. (24) to Eq. (6), the Hall coefficient ($R_{\rm H,\rm tot}$)
in the film-substrate systems is

$\displaystyle R_{\rm H,\rm tot}=\frac{R_{{\rm H},f}+\epsilon_{\sigma}R_{{\rm
H},s}}{1+\epsilon_{\sigma}}+\frac{\epsilon_{\sigma}}{(1+\epsilon_{\sigma})^{2}}R_{{\rm
H},s}\delta R_{{\rm H}},$ (29)

where

$\displaystyle\delta R_{{\rm H}}$ $\displaystyle=$
$\displaystyle(\rho_{f}-\rho_{s})(\theta_{\rm H,s}-\theta_{\rm H,f})/R_{{\rm
H},s}B.$ (30)

Here, $\rho_{f}$ and $\rho_{s}$ are the electrical resistivities of film and
substrate, respectively.

Finally, applying Eq. (24) to Eqs. (12), (17) and (18), the Nernst coefficient
($\nu_{\rm tot}$), the Seebeck coefficient with the anomalous components
($S_{\rm tot}^{AN}$), and the anomalous Nernst coefficient ($N_{\rm tot}$) are

$\displaystyle\nu_{\rm tot}$ $\displaystyle=$ $\displaystyle\frac{\nu_{\rm
f}+\epsilon_{\sigma}\nu_{\rm
s}}{1+\epsilon_{\sigma}}+\frac{\epsilon_{\sigma}}{(1+\epsilon_{\sigma})^{2}}\nu_{\rm
s}\delta\nu,$ (31) $\displaystyle S_{\rm tot}^{AN}$ $\displaystyle=$
$\displaystyle\frac{1}{1+\theta_{\rm H,tot}^{2}}\bigr{[}\frac{(1+\theta_{\rm
H,f}^{2})S_{\rm f}^{AN}+\epsilon_{\sigma}(1+\theta_{\rm H,s}^{2})S_{\rm
s}^{AN}}{1+\epsilon_{\sigma}}+\frac{\epsilon_{\sigma}}{(1+\epsilon_{\sigma})^{2}}S_{\rm
s}^{AN}\delta S^{AN}\bigr{]}$ (32) $\displaystyle\simeq$
$\displaystyle\frac{S_{\rm f}^{AN}+\epsilon_{\sigma}S_{\rm
s}^{AN}}{1+\epsilon_{\sigma}}+\frac{\epsilon_{\sigma}}{(1+\epsilon_{\sigma})^{2}}S_{\rm
s}^{AN}\delta S^{AN},$ $\displaystyle N_{\rm tot}$ $\displaystyle=$
$\displaystyle\frac{1}{1+\theta_{\rm H,tot}^{2}}\bigr{[}\frac{(1+\theta_{\rm
H,f}^{2})N_{\rm f}+\epsilon_{\sigma}(1+\theta_{\rm H,s}^{2})N_{\rm
s}}{1+\epsilon_{\sigma}}+\frac{\epsilon_{\sigma}}{(1+\epsilon_{\sigma})^{2}}N_{\rm
s}\delta N\bigr{]}$ (33) $\displaystyle\simeq$ $\displaystyle\frac{N_{\rm
f}+\epsilon_{\sigma}N_{\rm
s}}{1+\epsilon_{\sigma}}+\frac{\epsilon_{\sigma}}{(1+\epsilon_{\sigma})^{2}}N_{\rm
s}\delta N,$

where

$\displaystyle\delta\nu$ $\displaystyle=$
$\displaystyle(S_{f}-S_{s})(\theta_{\rm H,s}-\theta_{\rm H,f})/\nu_{\rm
s}B_{z},$ (34) $\displaystyle\delta S^{AN}$ $\displaystyle=$
$\displaystyle(S_{f}\alpha_{N,f}-S_{s}\alpha_{N,s})(\theta_{\rm
H,s}-\theta_{\rm H,f})/S_{\rm s}^{AN},$ (35) $\displaystyle\delta N$
$\displaystyle=$ $\displaystyle(S_{f}-S_{s})(\theta_{\rm H,s}-\theta_{\rm
H,f})/N_{\rm s}.$ (36)

In Eqs. (32) and (33), we used $1+\theta_{H,i}^{2}\simeq 1$ for $i={\rm tot}$,
$s$, and $f$. Therefore, Eqs. (29), and (31) - (33) show the same
$\epsilon_{\sigma}$ dependence. The anomalous Nernst coefficient of Eq. (33)
is the same as in ref. [37], when we neglect the Hall angle of substrate.

In the Appendix B, we derive a $\epsilon_{\sigma}$ dependence of Hall
resistivity. Although the linear responses of a single film-substrate system
is explained in this section, it is easy to extend to the n-layer systems. In
appendix C, the respective formulation is described.

##  IV Model Calculations

In this section, we further elaborate the differences between two types:
type-I (Eqs. (25)-(27)), and type-II (Eq. (29)) and (Eqs. (31)-(33)).
Particularly, the $\epsilon_{\sigma}$ dependencies, i.e. the weighted
contribution characterizing the ratio of the film to substrate resistances,
are studied for the Seebeck coefficient (type-I) and the Nernst coefficient
(type-II).

### Type-I

Figure 2 shows the $\epsilon_{\sigma}$ dependence of the Seebeck coefficient.
The Seebeck coefficient gradually changes and there is no extremum. The result
is consistent with a previous study [33].

Figure 2: $\epsilon_{\sigma}$ dependence of the Seebeck coefficient (Type-I).
The Hall angle and the Peltier angle are the same forms of function as the
Seebeck coefficient.

The origin of this dependence is understood based on the circulating current
shown in Fig. 4(a) [33]. If a temperature difference is applied in the $x$
direction ($\Delta T$), a voltage is caused by both the Seebeck effect in the
film and the substrate, respectively. The circulating current $I_{circ}^{x}$
is flowing around the film and the substrate. Employing Ohm’s law, the
circulating current reads

$\displaystyle I_{circ}^{x}=\frac{(S_{f}-S_{s})\Delta T}{R_{f}+R_{s}}.$ (37)

Thus, the measured voltage in the $x$ direction is given by

$\displaystyle V_{x}$ $\displaystyle=$ $\displaystyle S_{f}\Delta
T+I_{circ}^{x}R_{f},$ (38)

and the measured Seebeck coefficient becomes

$\displaystyle S_{tot}=\frac{V_{x}}{\Delta
T}=\frac{S_{s}R_{s}+S_{f}R_{f}}{R_{s}+R_{f}}.$ (39)

This equation is consistent with Eq. (25). The circulating current flowing
around the film and substrate is thus essential to understand the
$\epsilon_{\sigma}$ dependence of Eq. (25).

### Type-II

Here, we study the type-II equations, such as the Nernst coefficient of Eq.
(31). If $\delta\nu=0.0$ (see Eq. (34)), $\nu_{tot}$ behaves the same as the
Seebeck coefficient in Fig. 2. On the other hand, when $\delta\nu$ is finite,
an additional contribution appears. As shown in Fig. 3, it is found that the
Nernst coefficient exhibits distinct maxima/minima for
$\epsilon_{\sigma}\simeq 1.0$ as $|\delta\nu|$ increases.

Figure 3: $\epsilon_{\sigma}$ dependence of the Nernst coefficient $\nu_{\rm
tot}$ (Type-II). We fix $\nu_{\rm tot}/\nu_{\rm s}=10$. The Hall coefficient
($R_{\rm H,\rm tot}$), the Seebeck coefficient with the anomalous components
($S_{\rm tot}^{AN}$), and the anomalous Nernst coefficient ($N_{\rm tot}$)
also show the same $\epsilon_{\sigma}$ dependence with the Nernst coefficient.

To understand the origin of the $\epsilon_{\sigma}$ dependence, Eq. (31) is
divided into two terms as

$\displaystyle\nu_{\rm tot}$ $\displaystyle=$ $\displaystyle\nu_{1}+\nu_{2},$
(40)

where

$\displaystyle\nu_{1}$ $\displaystyle=$ $\displaystyle\frac{\nu_{\rm
f}+\epsilon_{\sigma}\nu_{\rm s}}{1+\epsilon_{\sigma}},$ (41)
$\displaystyle\nu_{2}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{\sigma}}{(1+\epsilon_{\sigma})^{2}}(S_{f}-S_{s})(\theta_{\rm
H,s}-\theta_{\rm H,f})/B_{z}.$ (42)

If a temperature difference is applied in the $x$ direction and the magnetic
field perpendicular to film-substrate systems, we conclude, in analogy to the
Seebeck case (Fig. 4(a)), for $I_{circ}^{y}$ (Fig. 4(b))

$\displaystyle I^{y}_{circ}=\frac{(\nu_{f}-\nu_{s})B_{z}\Delta
T}{R_{f}+R_{s}};$ (43)

the measured Nernst coefficient is then

$\displaystyle\frac{V_{y}}{B_{z}\Delta
T}=\frac{\nu_{f}R_{f}+\nu_{s}R_{s}}{R_{f}+R_{s}},$ (44)

which is the same as $\nu_{1}$ (Eq. (41)). Thus, the origin of $\nu_{1}$ is
caused by the circulating current in the y-direction, flowing the film and the
substrate.

Next, let us discuss the origin of $\nu_{2}$. Using the circulating current of
Eq. (37) ($I_{circ}^{x}$) and the resistance of the total system $R_{tot}$
defined by $1/R_{tot}=1/R_{f}+1/R_{s}$, $\nu_{2}$ follows from

$\displaystyle\nu_{2}B_{z}\Delta T=V_{A}+V_{B},$ (45)

where

$\displaystyle V_{A}$ $\displaystyle=$ $\displaystyle
I_{circ}^{x}R_{tot}\theta_{\rm H,s},$ (46) $\displaystyle V_{B}$
$\displaystyle=$ $\displaystyle-I_{circ}^{x}R_{tot}\theta_{\rm H,f}.$ (47)

Thus, the Nernst voltages are derived from the circulating current of
$I_{circ}^{x}$; the total voltage is the series circuit. It should be noted
that for the additional voltage (Eq. (45)) to appear, both a difference in the
Seebeck coefficient ($S_{f}-S_{s}$) and a difference in the Hall angle
($\theta_{\rm H,s}-\theta_{\rm H,f}$) of the film and the substrate are
necessary.

Figure 4: Schematic picture of the film-substrate system for describing the
Seebeck effect (a) and Nernst effect (b)-(c), where $\Delta T$ is a
temperature gradient in the x-direction. $I_{circ}^{x}$ is the circulating
current flowing around the film and the substrate along x direction, and it is
proportional to the difference of the Seebeck coefficients in the film and
substrate. The total voltage is obtained by $V_{x}=I_{circ}^{x}(R_{f}+R_{s})$,
where $R_{f}$ and $R_{s}$ are resistances in the film and the substrate,
respectively. In the same way, the circulating current along y direction
($I_{circ}^{y}$), which is proportional to the difference of the nernst
coefficients in the film and the substrate, is flowing around the film and the
substrate. As shown in (b), the nernst voltage is
$V_{y}=I_{circ}^{y}(R_{f}+R_{s})$. The Cooperative effect is the cross term of
$I_{circ}^{x}$ and Hall angles in the film and the substrate ($\theta_{\rm
H,s}$ and $\theta_{\rm H,f}$). The sign different between $V_{A}$ and $V_{B}$
shown in (c) is due to the direction of $I_{circ}^{x}$ in the film and the
substrate. The voltage due to the cooperative effect is
$V_{y}=V_{A}+V_{B}=I_{circ}^{x}R_{tot}(\theta_{\rm H,s}-\theta_{\rm H,f})$
where $R_{tot}$ is a resistance of the total system as defined by
$1/R_{tot}=1/R_{f}+1/R_{s}$.

Fig. 4(c) shows a scheme of the circuit, corresponding to Eqs. (45)-(47).
Since $\nu_{2}$ is the contribution from the series circuit of Nernst voltages
derived from the circulating current, which is essentially different from
$\nu_{1}$, we call this contribution the cooperative effect. Here, the sign
difference with Eqs. (46) and (47) is due to the difference that the direction
of the circulating current is positive in the $x$ direction for the ﬁlm but
negative for the substrate, as shown in Fig. 4(a). Since the amplitude and the
sign of these contributions ($\nu_{1}$ and $\nu_{2}$) depend on the systems,
we discuss specific examples in the next section.

In this section, only the Nernst coefficient was considered, however, we
unambiguously obtained that this cooperate effect is present in the Hall
coefficient ($R_{\rm H,\rm tot}$), the Seebeck coefficient with the anomalous
components ($S_{\rm tot}^{AN}$), and the anomalous Nernst coefficient ($N_{\rm
tot}$).

Finally we shortly comment on the anomalous Nernst effect in the bilayer
system of ref. [37]. When one neglects the Hall angle of the substrate, the
second term of Eq. (33) is proportional to $(S_{f}-S_{s})\rho^{xy}_{f}$, which
is proportional to the circulating current. Thus, this term includes the
cooperative effect. This effect is the central part in the field of the
anomalous Nernst effect based on the closed circuit, consisting of a magnetic
material and a thermoelectric material [28, 29, 30, 31].

Figure 5:  Ratio of the thickness ($d_{s}/d_{f}$) dependence on the Hall
coefficient, Seebeck coefficient and Nernst coefficient in bismuth film-
Lightly doped silicon substrate system (Bi@L-Si) and bismuth film-heavily
doped silicon substrate (Bi@H-Si). $d_{f}$ and $d_{s}$ indicates the thickness
of film (bismuth) and substrate (silicon).

Figure 6:  $\epsilon_{\sigma}$ dependence of $\nu_{1}$ (black line), and
$\nu_{2}$ (pink line) in bismuth film-lightly doped silicon substrate system
(Bi@L-Si) and bismuth film-heavily doped silicon substrate system (Bi@H-Si),
respectively. The blue and red lines indicate the total of Nernst coefficient
($\nu_{\rm tot}$).

##  V Prediction of cooperative effect

To verify realistic amplitudes of this effect, the case of bismuth films on
doped silicon substrates is discussed. Here, we estimate the Hall coefficient,
Seebeck coefficient, and Nernst coefficient from the experimental results.
Table 1 shows experimental data at room temperature, extracted from refs. [39,
40, 41, 42] for doped silicon and bismuth.

Table 1: Typical values of electrical conductivity, Hall coefficient, Hall mobility, Seebeck coefficient and Nernst coefficient at room temperatures. The experimental data are extracted from refs. [39, 40, 41, 42].  | lightly doped silicon | heavily doped silicon | bismuth
---|---|---|---
Impurity concentration (cm-3) | 3 $\times$1014 | 1 $\times$ 1017 |
| (Sample num. 131 [39]) | ( Sample num. 139 [39]) |
| (Sample num. 2 [41]) | (Sample num. 13 [41]) |
Electrical conductivity ($\mu\Omega^{-1}$m-1) | 5 $\times$ 10-6 [39] | 0.005 [39] | 0.88 [42]
Hall coefficient (m3/C) | -0.05 [39] | -10-4 [39] | -3.6 $\times$ 10-7 [42]
Seebeck coefficient ($\upmu$V/K) | -1600 [40] | -1000 [40] | -62 [42]
Nernst coefficient ($\upmu$V/KT) | -40 [41] | -10 [41] | -14 [42]

We set $d_{f}$ as the thickness of bismuth film, and $d_{s}$ as the thickness
of lightly (heavily) doped silicon. Figure 5 depicts the $d_{s}/d_{f}$
dependence of the Hall coefficient, Seebeck coefficient and Nernst coefficient
in the bismuth film-lightly doped silicon substrate system (blue line) and
bismuth film-heavily doped silicon substrate system (red line), respectively.
From here, we refer the bismuth film-lightly doped silicon substrate system as
Bi@L-Si and bismuth film-heavily doped silicon substrate system as Bi@H-Si,
respectively. Although the Hall coefficient has a potential to have a peak due
to the cooperative effect, the Hall and Seebeck coefficients change gradually
in both Bi@L-Si and Bi@H-Si, as the ratio of thickness increases. On the other
hand, as already demonstrated in Fig. 3, the Nernst coefficient exhibits a
large peak of -55 $\upmu$V/KT unexpectedly at $d_{s}/d_{f}\simeq 100$ in
Bi@H-Si, and in the case of Bi@L-Si, the Nernst coefficient decreases
drastically at $d_{s}/d_{f}\simeq 10^{5}$.

In order to understand the origin and sign of the peak in Bi@L-Si and Bi@H-Si,
$\nu_{1}$ and $\nu_{2}$ (Eq. (40)), Figure 6 shows the $\epsilon_{\sigma}$
dependence of $\nu_{1}$, $\nu_{2}$, and $\nu_{\rm tot}$ in Bi@L-Si and
Bi@H-Si. The cooperative effect ($\nu_{2}$) exhibits a negative peak at
$d_{s}/d_{f}\simeq 100$ in Bi@H-Si, and a positive one at $d_{s}/d_{f}\simeq
10^{5}$ in Bi@L-Si. The sign of the cooperative effect depends on the material
parameters.

As discussed in the introduction, a huge anomalous Nernst effect was observed
in the bilayer system of Fe-Ga and n-type silicon substrate [37]. Although we
did not consider the anisotropy explicitly and neglected interfacial
properties, we obtained significant peaks in the system of bismuth thin films
on different silicon substrates, depending on the film thickness.

##  VI Conclusion

In conclusion, we studied the thickness dependence of longitudinal and
transverse responses in film-substrate systems in a unified and general manner
based on the parallel circuit model in matrix form. We found that in
transverse responses such as the Hall coefficient, the Nernst effect and the
anomalous Nernst effect, there is a potential to obtain huge responses by
adjusting the thickness of the film or the substrate, thereby fine-tuning the
conductance ratio $\epsilon_{\sigma}$. Our work emphasizes the importance of
the cooperative effect in film and substrate systems. Finally, in order to
verify the role of the cooperative effect in realistic scenarios, bismuth
film-doped silicon substrate systems were considered. Using experimental
parameters for the transport properties of bismuth and silicon, the Nernst
effect of the system is drastically enhanced, originating from the interplay
of film and substrate via the cooperative effect.

###### Acknowledgements.

H. M. is grateful to M. Ogata for the discussion. This work is supported by
Grants-in-Aid for Scientific Research from the Japan Society for the Promotion
of Science (No. JP20K03802, No. JP22K18954, and No. JP22KK0228), and JST-Mirai
Program Grant (No. JPMJMI19A1).

##  Appendix A On the definitions of Hall and Nernst angles

In the absence of a temperature gradient, zero heat current, and also zero
electrical current in the $y$ direction, the electric field arising in the $y$
direction, when an electric field is applied in the $x$ direction, is obtained
from

$\displaystyle\begin{pmatrix}j^{x}_{\rm e,A}\\\ 0\\\
\end{pmatrix}=\bar{L}_{11,{\rm A}}\begin{pmatrix}E_{x,{\rm A}}\\\ E_{y,{\rm
A}}\\\ \end{pmatrix}.$ (48)

The ratio of $E_{y}$ and $E_{x}$, namely Hall angle, is

$\displaystyle\tan{\theta_{H,{\rm A}}}\simeq\theta_{H,{\rm
A}}=\frac{E_{y}}{E_{x}}=-\frac{L_{11,{\rm A}}^{yx}}{L_{11,{\rm
A}}^{yy}}=\frac{L_{11,{\rm A}}^{xy}}{L_{11,{\rm A}}^{yy}},$ (49)

where we assumed that the Hall angle is much smaller than $\pi/2$. The Hall
mobility is defined as

$\displaystyle\mu_{H,{\rm A}}=\frac{L_{11,{\rm A}}^{xy}}{L_{11,{\rm
A}}^{yy}B_{z,{\rm A}}}=\frac{\sigma_{{\rm A}}^{xy}}{\sigma_{{\rm
A}}^{yy}B_{z,{\rm A}}}.$ (50)

The Peltier angle is also obtained by the ratio of $(\nabla T)_{x}$ and
$(\nabla T)_{y}$ as

$\displaystyle\tan{\alpha_{N,{\rm A}}}\simeq\alpha_{N,{\rm A}}=\frac{(\nabla
T)_{y}}{(\nabla T)_{x}}=\frac{L_{12,{\rm A}}^{xy}}{L_{12,{\rm A}}^{yy}}.$ (51)

where we also assumed that the Peltier angle is much smaller than $\pi/2$.

##  Appendix B Hall resistivity in film-substrate system

The electrical resistivity in system A is defined as

$\displaystyle\begin{pmatrix}E_{x,A}\\\ E_{y,A}\\\
\end{pmatrix}=\begin{pmatrix}\rho^{xx}_{A}&\rho^{xy}_{A}\\\
\rho^{yx}_{A}&\rho^{yy}_{A}\\\ \end{pmatrix}\begin{pmatrix}j_{x}^{e,A}\\\
j_{y}^{e,A}\\\ \end{pmatrix}.$ (52)

By comparison with eq. (5), the Hall resistivity $\rho^{xy}_{A}$ is given as

$\displaystyle\rho^{xy}_{A}=\frac{-L^{xy}_{11,A}}{L^{xx}_{11,A}L^{yy}_{11,A}+(L^{xy}_{11,A})^{2}}.$
(53)

The Hall resistivity in the film-substrate system becomes

$\displaystyle\rho^{xy}_{\rm tot}$ $\displaystyle=$
$\displaystyle\frac{-L^{xy}_{11,\rm tot}}{L^{xx}_{11,\rm tot}L^{yy}_{11,\rm
tot}+(L^{xy}_{11,\rm tot})^{2}}$ (54) $\displaystyle=$
$\displaystyle\frac{1}{\frac{d_{f}}{d_{f}+d_{s}}L_{11,f}^{xx}}\frac{\theta_{H,\rm
tot}}{(1+\epsilon_{\sigma})(1+\theta_{H,\rm tot}^{2})}.$

Substituting eq. (26) to eq. (54), and approximating with $1+\theta_{H,\rm
tot}^{2}\simeq 1$, the Hall resistivity is given as

$\displaystyle\rho^{xy}_{\rm
tot}\simeq\frac{1}{\frac{d_{f}}{d_{f}+d_{s}}L_{11,f}^{xx}}\frac{\theta_{\rm
H,f}+\epsilon_{\sigma}\theta_{\rm H,s}}{(1+\epsilon_{\sigma})^{2}}.$ (55)

##  Appendix C n-layer system

As discussed in Eq. (24), the total response coefficient in the n-layer
systems is given as

$\displaystyle\bar{L}_{\rm tot}^{{\rm
n-layer}}\equiv\sum_{i=1}^{n}\frac{d_{i}}{d_{\rm tot}}\bar{L}_{i}.$ (56)

where $d_{i}$ indicates the thickness of the i-th layer, and $d_{\rm tot}$ is
the total thickness defined as $d_{\rm tot}=\sum_{i=1}^{n}d_{i}$. Here we
define the ratio corresponding to Eq. (28) as

$\displaystyle\epsilon_{i}=\frac{L_{11,i}^{xx}d_{i}}{L_{11,1}^{xx}d_{1}}=\frac{R_{1}}{R_{i}}.$
(57)

where $R_{i}$ and $L_{11,i}^{xx}$ are the electrical resistivity in the i-th
layer, and the electrical conductivity in the $i$-th layer, respectively.
Applying Eq. (56) to Eqs. (6), (11), (13) and (14), the Hall coefficient, the
Seebeck coefficient, the Hall angle and the Peltier angle become

$\displaystyle R_{H,\rm tot}^{{\rm n-layer}}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{n}\frac{d_{\rm
tot}}{d_{i}}\epsilon_{i}^{2}R_{H,i}}{(\sum_{i}\epsilon_{i})^{2}},$ (58)
$\displaystyle S_{\rm tot}^{{\rm n-layer}}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{n}\epsilon_{i}S_{i}}{\sum_{i=1}^{n}\epsilon_{i}},$
(59) $\displaystyle\theta_{H,{\rm tot}}^{{\rm n-layer}}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{n}\epsilon_{i}\Theta_{H,i}}{\sum_{i=1}^{n}\epsilon_{i}},$
(60) $\displaystyle\alpha_{N,{\rm tot}}^{{\rm n-layer}}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{n}\epsilon_{i}S_{i}\alpha_{n,i}}{\sum_{i=1}^{n}S_{i}\epsilon_{i}},$
(61)

Using these equations, the Nernst coefficient, the anomalous Nernst
coefficient, and the Seebeck coefficient with the anomalous components are
given as

$\displaystyle\nu_{\rm tot}^{{\rm n-layer}}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{n}\epsilon_{i}(\nu_{i}-S_{i}\theta_{H,i}/B_{z})}{\sum_{i=1}^{n}\epsilon_{i}}+S_{\rm
tot}^{{\rm n-layer}}\theta_{H,{\rm tot}}^{{\rm n-layer}}/B_{z},$ (62)
$\displaystyle S_{\rm tot}^{AN,{\rm n-layer}}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{n}\epsilon_{i}(S_{i}^{AN}-S_{i}\alpha_{{\rm
n},i})}{\sum_{i=1}^{n}\epsilon_{i}}+S_{\rm tot}^{{\rm n-layer}}\alpha_{{\rm
n},{\rm tot}}^{{\rm n-layer}},$ (63) $\displaystyle N_{\rm tot}^{{\rm
n-layer}}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{n}\epsilon_{i}(\nu_{i}-S_{i}\theta_{H,i})}{\sum_{i=1}^{n}\epsilon_{i}}+S_{\rm
tot}^{{\rm n-layer}}\theta_{H,{\rm tot}}^{{\rm n-layer}}.$ (64)

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