# Anisotropic Behavior of the Thermoelectric Power and the Thermal
Conductivity in a Unidirectional Lateral Superlattice: A Typical Anisotropic
System Exhibiting Two Distinct Nernst Coefficients

Akira Endo<EMAIL_ADDRESS>The Institute for Solid State Physics,
The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan
Shingo Katsumoto  The Institute for Solid State Physics, The University of
Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan  Yasuhiro Iye  The
Institute for Solid State Physics, The University of Tokyo, 5-1-5 Kashiwanoha,
Kashiwa, Chiba 277-8581, Japan

###### Abstract

We have calculated the thermoelectric conductivity tensor $\varepsilon_{ij}$
and the thermal conductivity tensor $\lambda_{ij}$ of a unidirectional lateral
superlattice (ULSL) ($i,j=x,y$, with the $x$-axis aligned to the principal
axis of the ULSL), based on the asymptotic analytic formulas of the electrical
conductivity tensor $\sigma_{ij}$ in the literature valid at low magnetic
fields where large numbers of Landau levels are occupied. With the resulting
analytic expressions, we clarify the conditions for the Mott formula
(Wiedemann-Franz law) to be applicable with high precision to
$\varepsilon_{ij}$ ($\lambda_{ij}$). We further present plots of the
commensurability oscillations $\delta\varepsilon_{ij}$, $\delta\lambda_{ij}$,
$\delta\kappa_{ij}$, and $\delta S_{ij}$ in $\varepsilon_{ij}$,
$\lambda_{ij}$, (an alternative, more standard definition of) the thermal
conductivity tensor $\kappa_{ij}$, and the thermopower tensor $S_{ij}$,
calculated using typical parameters for a ULSL fabricated from a GaAs/AlGaAs
two-dimensional electron gas (2DEG). Notable features of the $\delta S_{ij}$
are (i) anisotropic behavior ($\delta S_{xx}\neq\delta S_{yy}$) and (ii) the
dominance of the $xy$ component over the other components ($|\delta
S_{xy}|\gg|\delta S_{yx}|,|\delta S_{xx}|,|\delta S_{yy}|$). The latter
clearly indicates that the two Nernst coefficients, $S_{xy}$ and $S_{yx}$, can
be totally different from each other in an anisotropic system. Both (i) and
(ii) are at variance with the previous theory and are attributable to the
inclusion of a damping factor due to the small-angle scattering characteristic
of GaAs/AlGaAs 2DEGs, which have not been taken into consideration in $\delta
S_{ij}$ thus far.

##  I Introduction

A broken symmetry often plays a pivotal role in condensed matter physics. In
the present paper, we consider a system possessing arguably the simplest form
of a broken rotational symmetry, namely, a unidirectional lateral superlattice
(ULSL), in which twofold in-plane anisotropy is artificially introduced into
an otherwise isotropic two-dimensional electron gas (2DEG) by imposing a
unidirectional periodic potential modulation. We focus on the thermoelectric
and thermal transport coefficients, which are currently the subject of growing
interest as an effective tool to probe exotic electronic states difficult to
be accessed by more widely studied electrical transport coefficients [1, 2, 3,
4, 5]. Main interest in the present study is to see how the anisotropy
manifests itself in the thermoelectric/thermal transport coefficients when
placed in a magnetic field and whether or not qualitative difference is found
in the appearance of the anisotropy compared with that in the electrical
transport coefficients.

One of the best-known magnetotransport phenomena in a ULSL is the
commensurability oscillations (COs) resulting from the commensurability
between the period $a$ of the unidirectional potential modulation and the
cyclotron radius $R_{\mathrm{c}}=\hbar k_{\mathrm{F}}/(e|B|)$, where
$k_{\mathrm{F}}=\sqrt{2\pi n_{e}}$ is the Fermi wavenumber with $n_{e}$ the
areal density of the 2DEG. Here and throughout the paper, we set the 2DEG
plane as the $x$-$y$ plane, the principal axis of the modulation as the
$x$-axis, and the direction of the magnetic field $B$ as the $z$-axis.
Following the experimental discovery in the resistivities $\rho_{xx}$ and
$\rho_{yy}$ (Weiss oscillations) [6], various electrical, thermal, and
thermodynamic quantities in a ULSL were theoretically predicted [7] to
oscillate with a magnetic field $B$. Experimentally, COs have further been
observed in the capacitance [8], thermopower [9, 10, 11, 12, 13], electrical
conductivity [14], and recently also in the Hall resistance [15].

In Ref. 7, the calculated electrical, thermoelectric, and thermal transport
coefficients were mainly presented in the form of the summation of the terms
containing the (associated) Laguerre polynomials over the Landau index $N$.
Thus, numerical calculations are required to perceive the behaviors of the
transport coefficients. For the conductivities $\sigma_{xx}$ and
$\sigma_{yy}$, however, analytic formulas were also provided [7], which allows
us to readily grasp the dependence of the amplitude and the phase of the COs
on the magnetic field $B$, the temperature $T$, and the parameters of the ULSL
sample. The analytic formulas were obtained by replacing the (associated)
Laguerre polynomials by their asymptotic expressions at large $N$. Generally,
many Landau levels are occupied in a typical ULSL ($n_{e}\gtrsim 10^{15}$ m-2,
$a\gtrsim 100$ nm) in the magnetic-field range COs are observed ($B\lesssim 1$
T), validating the use of the asymptotic expressions. Following the same
strategy, the present authors deduced an approximate analytic expression also
for $\sigma_{yx}$ in a recent publication [15].

One of the purposes of the present paper is to present similar analytic
formulas for the thermoelectric power and the thermal conductivity. The
formulas are obtained by straightforward calculations of the first- and the
second-order moments performed on the analytic expressions for $\sigma_{xx}$,
$\sigma_{yy}$, and $\sigma_{yx}$ just mentioned above, with a slight
modification to include an additional damping factor accounting for the effect
of the small-angle scattering that impede the completion of a cycle of the
cyclotron orbit [16, 17]. The thermoelectric power and the thermal
conductivity are often deduced approximately from the electrical conductivity
via the Mott formula and the Wiedemann-Franz law, respectively at low
temperatures. We examine the conditions for these approximations to be
accurate using the obtained analytic expressions.

We present examples of COs calculated by these analytic formulas at
temperatures ranging from 0.02 K to 10 K, plugging in the parameters for a
typical ULSL sample fabricated from a conventional 2DEG residing at the
GaAs/AlGaAs heterointerface. The plots clearly reveal the deviation from the
Mott formula or the Wiedemann-Franz law with increasing temperatures.
Important aspects worth highlighting in the COs of the thermopower tensor are
the anisotropic behavior of the diagonal (Seebeck) components and the far
dominance of the $xy$ component, one of the off-diagonal (Nernst) component,
over the other Seebeck and Nernst components. These behaviors are at variance
with the previous theory [7]. The disagreement mainly results from the
inclusion of the additional damping factor mentioned above [16, 17], which was
shown to be indispensable for achieving good agreement between theoretical and
experimental COs in the resistivity [17] and the Hall resistance [15]. We will
also comment on the difficulties, caused by the tilting of the temperature
gradient in the magnetic field [18], in the interpretation of the
experimentally observed COs in the thermopower.

##  II Transport Coefficients

First we outline the general properties of the thermoelectric and thermal
transport coefficients to be calculated below. The electrical conductivity
tensor $\hat{\sigma}$, the thermoelectric conductivity tensor
$\hat{\varepsilon}$, the Peltier tensor $\hat{\pi}$, and the thermal
conductivity tensor $\hat{\lambda}$ are defined as the coefficients relating
the electrical current density $\bm{j}$ and the thermal current density
$\bm{j_{Q}}$ to the electric field $\bm{E}$ and the temperature gradient
$-\bm{\nabla}T$ in the linear response regime [19, 20, 21, 22]111The notations
of the coefficients here follows those of Ref. 19. $\sigma$, $\varepsilon$,
$\pi$, and $\lambda$ are denoted as $L_{EE}$, $-L_{ET}$, $L_{TE}$, and
$-L_{TT}$ in Ref. 20 and $e^{2}L_{11}/T$, $-eL_{12}/T$, $-eL_{21}/T$, and
$L_{22}/T$ in Ref. 22

$\bm{j}=\hat{\sigma}\bm{E}-\hat{\varepsilon}\bm{\nabla}T$ (1)
$\bm{j_{Q}}=\hat{\pi}\bm{E}-\hat{\lambda}\bm{\nabla}T,$ (2)

where we used the ‘hat’ $\hat{X}$ to denote the $2\times 2$ tensor containing
the components $X_{ij}$ ($i,j=x,y$). Alternative linear response equations
relating $\bm{E}$ and $\bm{j_{Q}}$ to $\bm{j}$ and $-\bm{\nabla}T$ [20],

$\bm{E}=\hat{\rho}\bm{j}+\hat{S}\bm{\nabla}T$ (3)
$\bm{j_{Q}}=\hat{\Pi}\bm{j}-\hat{\kappa}\bm{\nabla}T,$ (4)

define the resistivity tensor $\hat{\rho}$, the thermopower tensor $\hat{S}$,
and (more standard definitions of) the Peltier and the thermal conductivity
tensors, $\hat{\Pi}$ and $\hat{\kappa}$. By substituting Eq. (3) to Eqs. (1)
and (2), one can readily find the relations between the coefficients,

$\hat{\rho}=\hat{\sigma}^{-1}$ (5)
$\hat{S}=\hat{\sigma}^{-1}\hat{\varepsilon}=\hat{\rho}\hat{\varepsilon}$ (6)
$\hat{\Pi}=\hat{\pi}\hat{\rho}$ (7)
$\hat{\kappa}=\hat{\lambda}-\hat{\pi}\hat{S}=\hat{\lambda}-\hat{\pi}\hat{\rho}\hat{\varepsilon}.$
(8)

According to the Kelvin-Onsager relations [24, 25, 20, 22], we further have

$\displaystyle\hat{\sigma}(B)$ $\displaystyle=$ ${}^{t}\\!\hat{\sigma}(-B)$
(9a) $\displaystyle\hat{\lambda}(B)$ $\displaystyle=$
${}^{t}\\!\hat{\lambda}(-B)$ (9b) $\displaystyle\hat{\pi}(B)$ $\displaystyle=$
$\displaystyle T\ ^{t}\\!\hat{\varepsilon}(-B),$ (9c)

which, combined with Eqs. (5), (6), and (7), lead also to the relations

$\displaystyle\hat{\rho}(B)$ $\displaystyle=$ ${}^{t}\\!\hat{\rho}(-B)$ (10a)
$\displaystyle\hat{\kappa}(B)$ $\displaystyle=$ ${}^{t}\\!\hat{\kappa}(-B)$
(10b) $\displaystyle\hat{\Pi}(B)$ $\displaystyle=$ $\displaystyle T\
^{t}\\!\hat{S}(-B).$ (10c)

In general, the thermoelectric conductivity $\hat{\varepsilon}$ is composed of
two components: the diffusion contribution and the phonon-drag contribution
[19, 26]. In the present study, we limit ourselves to the former contribution.
The latter contribution can become the dominant contribution in GaAs/AlGaAs
2DEGs in the standard experimental setup using an external heater to introduce
the temperature gradient not only in the 2DEG but also in the lattice hosting
the 2DEG [19, 27]. We can, however, eliminate the phonon-drag contribution by
confining ourselves to very low temperatures ($\lesssim 0.2$ K) where the
electron-phonon interaction becomes negligibly small [14, 28] or by employing
the current heating technique [29, 30], which introduce the temperature
gradient only into the 2DEG without affecting the lattice temperature.
Similarly, we only consider the thermal conductivity due to the electrons and
neglect the phonon contribution. The diffusion thermoelectric conductivity and
the electronic thermal conductivity are related to the electrical conductivity
at $T=0$ as [7, 19, 20, 21, 31]

$\varepsilon_{ij}=-\frac{1}{eT}\int_{-\infty}^{\infty}dE\left[-f^{\prime}\left(E\right)\right]\left(E-E_{\mathrm{F}}\right)\sigma_{ij,{T=0}}\left(E\right)$
(11)

and

$\lambda_{ij}=\frac{1}{e^{2}T}\int_{-\infty}^{\infty}dE\left[-f^{\prime}\left(E\right)\right]\left(E-E_{\mathrm{F}}\right)^{2}\sigma_{ij,{T=0}}\left(E\right),$
(12)

respectively. Here $f^{\prime}(E)$ is the energy derivative of the Fermi-Dirac
distribution function
$f(E)=\\{1+\exp[(E-E_{\mathrm{F}})/k_{\mathrm{B}}T]\\}^{-1}$, with
$E_{\mathrm{F}}$ the Fermi energy and $k_{\mathrm{B}}$ the Boltzmann constant.
By expanding $\sigma_{ij,{T=0}}(E)$ around $E_{\mathrm{F}}$ (Sommerfeld
expansion),

$\sigma_{ij,{T=0}}(E)=\sigma_{ij,{T=0}}(E_{\mathrm{F}})+\left.\frac{\partial\sigma_{ij,{T=0}}}{\partial
E}\right|_{E_{\mathrm{F}}}\\!\\!\\!\\!\\!\\!\\!(E-E_{\mathrm{F}})+O[(E-E_{\mathrm{F}})^{2}],$
(13)

and retaining only the lowest order in $E-E_{\mathrm{F}}$, we arrive at the
Mott formula [31, 20, 21]

$\varepsilon_{ij}^{\mathrm{M}}=-L_{0}eT\left.\frac{\partial\sigma_{ij,{T=0}}(E)}{\partial
E}\right|_{E_{\mathrm{F}}}$ (14)

and the Wiedemann-Franz law [20, 21]

$\lambda_{ij}^{\mathrm{W}}=L_{0}T\sigma_{ij,{T=0}}(E_{\mathrm{F}})$ (15)

as low-temperature approximations, where
$L_{0}=\pi^{2}{k_{\mathrm{B}}}^{2}/(3e^{2})=2.44\times 10^{-8}$ V2/K2 is the
Lorenz number. We will discuss the temperature range for Eqs. (14) and (15) to
be valid for COs.

The Kelvin-Onsager relations Eqs. (9a), (9b), (10a), and (10b), along with the
antisymmetry $X_{ij}(-B)=-X_{ij}(B)$ ($i\neq j$) of the off-diagonal
components under magnetic-field reversal, yields the relations
$\sigma_{xy}=-\sigma_{yx}$, $\lambda_{xy}=-\lambda_{yx}$,
$\rho_{xy}=-\rho_{yx}$, and $\kappa_{xy}=-\kappa_{yx}$. In addition, the
diffusion thermoelectric conductivity also possesses similar relation
$\varepsilon_{xy}=-\varepsilon_{yx}$ owing to Eq. (11). For $\hat{\sigma}$,
$\hat{\varepsilon}$, $\hat{\lambda}$, and $\hat{\kappa}$, therefore, we only
need to calculate $xx$, $yy$, and $yx$ components. For the thermopower tensor
$\hat{S}$, however, the Kelvin-Onsager relation, Eq. (10c), does not place any
constraints on the relation between $S_{xy}$ and $S_{yx}$. Since
$S_{xy}=-\rho_{xx}\varepsilon_{yx}+\rho_{xy}\varepsilon_{yy}$ and
$S_{yx}=-\rho_{xy}\varepsilon_{xx}+\rho_{yy}\varepsilon_{yx}$, the relation
$S_{xy}=-S_{yx}$ holds for isotropic systems with $\rho_{xx}=\rho_{yy}$ and
$\varepsilon_{xx}=\varepsilon_{yy}$. In anisotropic systems, however, the
relation is not generally valid. A ULSL provides an archetypal example for how
anisotropy affects the relation between the two Nernst components, $S_{xy}$
and $S_{yx}$. We thus need to calculate both $S_{xy}$ and $S_{yx}$ separately,
in addition to $S_{xx}$ and $S_{yy}$. As we will see, the calculations reveal
considerable difference between the two Nernst components.

##  III Analytic Formulas

###  III.1 Electrical conductivity and resistivity

Figure 1: (Color online) Plots of thermal damping factors $A(T/T_{a})$ (solid
red line), $B(T/T_{a})$ (dashed dark-green line), and $C(T/T_{a})$ (dotted
blue line) given by Eqs. (17), (26), and (27) respectively.  Figure 2: (Color
online) Commensurability oscillations in the electrical conductivity vs.
$2R_{\mathrm{c}}/a$ calculated for various temperatures by picking out the
oscillatory parts from Eq. (21) and using typical ULSL parameters:
$V_{0}=0.26$ meV, $a=200$ nm, $\mu_{\mathrm{b}}=5.3$ m2/(Vs),
$\mu_{\mathrm{c}}=\mu_{\mathrm{h}}=\mu_{\mathrm{b}}/2$, $n_{e}=4.2\times
10^{15}$ m-2, and $\mu=72$ m2/(Vs). The top axis shows $B$. (a)
$\delta\sigma_{xx}$, (b) $\delta\sigma_{yy}$, and (c) $\delta\sigma_{yx}$.
Vertical dotted lines indicate the locations of the flat-band conditions Eq.
(19).  Figure 3: (Color online) Commensurability oscillations in the
resistivity vs. $2R_{\mathrm{c}}/a$ calculated for various temperatures by Eq.
(22) using the same ULSL parameters as in Fig. 2. The top axis shows $B$. (a)
$\delta\rho_{xx}$, (b) $\delta\rho_{yy}$, and (c) $\delta\rho_{xy}$. Vertical
dotted lines indicate the locations of the flat-band conditions Eq. (19).

We start by reviewing the well-documented analytic asymptotic expressions for
the electrical conductivity and resistivity tensors. We assume a sinusoidal
modulation $V(x)=V_{0}\cos(2\pi x/a)$ with small amplitude
$V_{0}/E_{\mathrm{F}}\ll 1$. It is well known that the COs in the electrical
conductivity arise from two separate mechanisms: the band contribution and the
collisional contribution [7]. The former originates from the oscillations of
the $y$-component of the group velocity (band dispersion) with $B$, and is
explained also by a semiclassical picture invoking the guiding center drift of
the cyclotron motion in the $y$ direction [32]. The band contribution is thus
contained only in $\sigma_{yy}$ and accounts for the major part in its COs,
$\delta\sigma_{yy}$. Here and in what follows, we denote the oscillatory part
(COs) contained in the coefficient $X_{ij}$ by $\delta X_{ij}$. The asymptotic
expression for the band contribution in the electrical conductivity is given
as [7]

$\sigma_{yy}^{\mathrm{band}}=\frac{\sigma_{0}V_{0}^{2}}{E_{\mathrm{F}}\hbar\omega_{\mathrm{c}}ak_{\mathrm{F}}}\left[1+A\left(\frac{\pi}{\mu_{\mathrm{b}}B}\right)A\left(\frac{T}{T_{a}}\right)\sin{r_{\mathrm{c}}}\right],$
(16)

where $\sigma_{0}=en_{e}\mu$ is the electrical conductivity at $B=0$ with
$\mu$ the mobility, $\omega_{\mathrm{c}}=e|B|/m^{*}$ is the cyclotron angular
frequency with $m^{*}$ the effective mass ($m^{*}=0.067m_{e}$ for GaAs with
$m_{e}$ the electron mass), and $r_{\mathrm{c}}\equiv 2\pi(2R_{\mathrm{c}}/a)$
is the reduced cyclotron radius. The temperature dependence of the CO
amplitude is dictated by the thermal damping factor $A(T/T_{a})$ (shown in
Fig. 1), with
$T_{a}\equiv\hbar\omega_{\mathrm{c}}ak_{\mathrm{F}}/(4\pi^{2}k_{\mathrm{B}})$
representing the characteristic temperature and

$A\left(x\right)\equiv\frac{x}{\sinh{x}}=1-\frac{x^{2}}{6}+O\left(x^{4}\right).$
(17)

Although absent in the original theory [7], an additional damping factor
$A(\pi/\mu_{\mathrm{b}}B)$ is included to account for the effect of the small-
angle scattering that deflects the electrons out of the cyclotron trajectory
before completing a cycle. The damping factor was theoretically deduced using
the Boltzmann equation formalism in Ref. 16. The present authors showed that
an excellent agreement with experimental COs is obtained with the value of the
parameter $\mu_{\mathrm{b}}$ close to the quantum mobility $\mu_{\mathrm{q}}$
deduced from the damping of the Shubnikov-de Haas (SdH) oscillations [17]. The
collisional contribution, on the other hand, is caused by the oscillations in
the density of states (DOS), analogous to SdH oscillations, and thus
isotropically affects the electrical conductivity. The asymptotic expression
[7] is written as

$\sigma_{xx}^{\mathrm{col}}=\frac{3\sigma_{0}V_{0}^{2}ak_{\mathrm{F}}}{8\pi^{2}E_{\mathrm{F}}\hbar\omega_{\mathrm{c}}\mu^{2}B^{2}}\left[1-A\left(\frac{\pi}{\mu_{\mathrm{c}}B}\right)A\left(\frac{T}{T_{a}}\right)\sin{r_{\mathrm{c}}}\right].$
(18)

Here, we simply assumed that the effect of small-angle scattering can be
accounted for by a damping factor $A(\pi/\mu_{\mathrm{c}}B)$ similar to that
in the band contribution, although the theory for the collisional contribution
that takes the small-angle scattering into account is still missing to the
best of our knowledge. Note, however, that, without the damping factor, the
amplitude of COs in $\sigma_{xx}$ and $\sigma_{yx}$ increases with decreasing
magnetic field and diverges with $B\rightarrow 0$, which is in clear
disagreement with the experiments [14, 15]. We allowed the parameter
describing the damping, $\mu_{\mathrm{c}}$, to differ from $\mu_{\mathrm{b}}$
in Eq. (16), admitting the possibility that the effect of the small-angle
scattering on the COs varies for a distinct mechanism. The oscillatory parts,
$\delta\sigma_{yy}^{\mathrm{band}}$ and $\delta\sigma_{xx}^{\mathrm{col}}$,
are given by the second terms in Eqs. (16) and (18), respectively. We can
readily see that $\delta\sigma_{yy}^{\mathrm{band}}$ and
$\delta\sigma_{xx}^{\mathrm{col}}$ oscillate with opposite phases, with the
former (latter) taking minima (maxima) at the flat-band conditions,

$\frac{r_{\mathrm{c}}}{2\pi}=\frac{2R_{\mathrm{c}}}{a}=n-\frac{1}{4},\hskip
14.22636pt(n=1,2,3,...),$ (19)

and that
$|\delta\sigma_{yy}^{\mathrm{band}}|\gg|\delta\sigma_{xx}^{\mathrm{col}}|$
noting that generally $\mu B\gg 1$ in the magnetic-field range COs are
observed.

The Hall conductivity also contains COs originating from the collisional
contribution. However, asymptotic expressions were not presented in the
original theory [7]. From the formula given as the summation over Landau index
$N$, Eq. (28) in Ref. 7, the present authors have recently deduced approximate
analytic formulas presented as the sum of the following three constituents
[15],

$\sigma_{yx}^{\left(1\right)}=\frac{3}{2}\nu\frac{e^{2}}{h}{\lambda_{\mathrm{c}}}^{2}\left[1-A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)A\left(\frac{T}{T_{a}}\right)\sin{\left(r_{\mathrm{c}}+\delta_{\mathrm{F}}\right)}\right],$
(20a)
$\sigma_{yx}^{\left(21\right)}=-\nu\frac{e^{2}}{h}{\lambda_{\mathrm{c}}}^{2}\frac{ak_{\mathrm{F}}}{\pi}\sin{\left(\frac{\pi}{{ak}_{\mathrm{F}}}\right)}\left[\cos{\left(\frac{\delta_{\mathrm{F}}}{2}+\frac{\pi}{{ak}_{\mathrm{F}}}\right)}+A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)A\left(\frac{T}{T_{a}}\right)\sin{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-\frac{\pi}{{ak}_{\mathrm{F}}}\right)}\right],$
(20b) and
$\sigma_{yx}^{\left(22\right)}=-\mathrm{sgn}(B)\frac{e^{2}}{h}{\lambda_{\mathrm{c}}}^{2}\frac{ak_{\mathrm{F}}}{\pi}\cos{\left(\frac{\pi}{{ak}_{\mathrm{F}}}\right)}\left[\sin{\left(\frac{\delta_{\mathrm{F}}}{2}+\frac{\pi}{{ak}_{\mathrm{F}}}\right)}-A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)A\left(\frac{T}{T_{a}}\right)\cos{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-\frac{\pi}{{ak}_{\mathrm{F}}}\right)}\right],$
(20c)

where $\nu=n_{e}h/eB$ represents the Landau-level filling fraction,
$\lambda_{\mathrm{c}}\equiv
2\sqrt{2/\pi}[V_{0}/(\hbar\omega_{c})][\pi/(ak_{\mathrm{F}})]{r_{\mathrm{c}}}^{-1/2}$,
$\delta_{\mathrm{F}}\equiv 2\cot^{-1}{r_{\mathrm{c}}}$, and $\mathrm{sgn}(x)$
represents the sign of $x$. We introduced yet another parameter
$\mu_{\mathrm{h}}$ for the damping factor due to the small-angle scattering.
We have recently found that experimentally obtained $\delta\sigma_{yx}$ is
fairly in good agreement with $\delta\sigma_{yx}$ calculated with
$\mu_{\mathrm{h}}=\mu_{b}/2$ [15], indicating that the small-angle scattering
diminishes CO amplitudes more severely in the collisional contribution than in
the band contribution. Noting that both $\delta\sigma_{xx}^{\mathrm{col}}$ and
$\delta\sigma_{yx}$ are stemming from the collisional contribution, we assume
$\mu_{\mathrm{c}}=\mu_{\mathrm{h}}=\mu_{b}/2$ in the present paper. Again, the
oscillatory parts $\delta\sigma_{yx}^{\left(1\right)}$,
$\delta\sigma_{yx}^{\left(21\right)}$, and
$\delta\sigma_{yx}^{\left(22\right)}$ are the second terms in Eqs. (20a),
(20b) and (20c). Collecting the relevant terms, the three independent
components of the electrical conductivity tensor are given as

$\displaystyle\sigma_{xx}$ $\displaystyle=$
$\displaystyle\sigma_{xx}^{\mathrm{sc}}+\sigma_{xx}^{\mathrm{col}},$ (21a)
$\displaystyle\sigma_{yy}$ $\displaystyle=$
$\displaystyle\sigma_{xx}^{\mathrm{sc}}+\sigma_{xx}^{\mathrm{col}}+\sigma_{yy}^{\mathrm{band}},$
(21b) $\displaystyle\sigma_{yx}$ $\displaystyle=$
$\displaystyle-\sigma_{xy}=\sigma_{yx}^{\mathrm{sc}}+\sigma_{yx}^{\left(1\right)}+\sigma_{yx}^{\left(21\right)}+\sigma_{yx}^{\left(22\right)},$
(21c)

where $\sigma_{xx}^{\mathrm{sc}}=\sigma_{0}/(1+\mu^{2}B^{2})$ and
$\sigma_{yx}^{\mathrm{sc}}=\sigma_{0}\mu B/(1+\mu^{2}B^{2})\simeq\nu(e^{2}/h)$
are the diagonal and the off-diagonal components, respectively, of the
semiclassical electrical conductivity for an unmodulated 2DEG. The
corresponding oscillatory parts are
$\delta\sigma_{xx}=\delta\sigma_{xx}^{\mathrm{col}}$,
$\delta\sigma_{yy}=\delta\sigma_{xx}^{\mathrm{col}}+\delta\sigma_{yy}^{\mathrm{band}}$,
and
$\delta\sigma_{yx}=\delta\sigma_{yx}^{\left(1\right)}+\delta\sigma_{yx}^{\left(21\right)}+\delta\sigma_{yx}^{\left(22\right)}$.

The resistivity tensor $\hat{\rho}$ is obtained by inverting the electrical
conductivity tensor $\hat{\sigma}$, Eq. (5). Assuming that
$|\delta\sigma_{ij}|\ll|\sigma_{ij}|$, the oscillatory parts of $\hat{\rho}$
are given, to the lowest order in $\delta\sigma_{ij}$ 222Or, to be more
precise, to the lowest order in the perturbations caused by $V(x)$. The effect
of the non-oscillatory background induced by $V(x)$ on the oscillation is the
second-order perturbation and thus neglected, as

$\displaystyle\delta\rho_{xx}=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\rho_{0}}^{2}(-\delta\sigma_{xx}+\mu^{2}B^{2}\delta\sigma_{yy}-2\mu
B\delta\sigma_{yx}),$ (22a) $\displaystyle\delta\rho_{yy}=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\rho_{0}}^{2}(-\delta\sigma_{yy}+\mu^{2}B^{2}\delta\sigma_{xx}-2\mu
B\delta\sigma_{yx}),$ (22b) $\displaystyle\delta\rho_{xy}=$
$\displaystyle-\delta\rho_{yx}={\rho_{0}}^{2}[\left(1-\mu^{2}B^{2}\right)\delta\sigma_{yx}-\mu
B\left(\delta\sigma_{xx}+\delta\sigma_{yy}\right)],$

where $\rho_{0}\equiv 1/\sigma_{0}$ is the resistivity at $B=0$.

In Figs. 2 and 3, we plot $\delta\sigma_{ij}$ and $\delta\rho_{ij}$ calculated
for the temperatures $T=0.02$, 1.5, 4.2, 10 K, employing the parameters for a
typical ULSL sample [12, 13]: $V_{0}=0.26$ meV, $a=200$ nm,
$\mu_{\mathrm{b}}=5.3$ m2/(Vs),
$\mu_{\mathrm{c}}=\mu_{\mathrm{h}}=\mu_{\mathrm{b}}/2$, $n_{e}=4.2\times
10^{15}$ m-2, and $\mu=72$ m2/(Vs). The figures exhibit well-established
behaviors of COs in the electrical conductivity and the resistivity [7, 15]:
$\delta\sigma_{yy}$ and $\delta\rho_{xx}$ arising mainly from the band
contribution oscillate with minima at the flat-band conditions and with the
amplitudes far exceeding the other components showing (nearly) anti-phase
oscillations. (Strictly speaking, the phases of $\delta\sigma_{yx}$ and
$\delta\rho_{xy}$ slightly depend on $B$. See Ref. 15 for more details.)
Compared with the original theory [7], the inclusion of the additional damping
factor $A[\pi/(\mu_{s}B)]$ ($s=\mathrm{b},\mathrm{c},\mathrm{h}$) makes the
damping of the oscillations with decreasing magnetic field more rapid in
$\delta\rho_{ij}$ and suppresses the divergence of the oscillation amplitude
at low magnetic field in $\delta\sigma_{ij}$. Furthermore, the heavier damping
of the collisional contributions compared to the band contribution, recently
found experimentally [15] and accounted for by the relation
$\mu_{\mathrm{c}}=\mu_{\mathrm{h}}=\mu_{b}/2$, enhances the dominance of
$\delta\sigma_{yy}$ and $\delta\rho_{xx}$ over the other components. As will
be shown below, this enhancement exerts a significant impact on the behaviors
of $\delta S_{ij}$.

###  III.2 Thermoelectric conductivity

Now we are ready to calculate the thermoelectric conductivity. By substituting
Eqs. (40), (41), and (42) [corresponding to Eqs. (16), (18), and (20) at
$T=0$, respectively, with the suitable $E$-dependence of the quantities
involved] into Eq. (11), we obtain the band and the collisional contributions
for the diagonal components,

$\displaystyle\varepsilon_{yy}^{\mathrm{band}}=-\frac{L_{0}eT}{E_{\mathrm{F}}}\frac{\sigma_{0}V_{0}^{2}}{E_{\mathrm{F}}\hbar\omega_{\mathrm{c}}ak_{\mathrm{F}}}\times$
$\displaystyle\left\\{\left(p-\frac{1}{2}\right)+\left[\left(p-\frac{1}{2}\right)A\left(\frac{\pi}{\mu_{\mathrm{b}}B}\right)+\frac{p_{\mathrm{b}}}{3}\left(\frac{\pi}{\mu_{\mathrm{b}}B}\right)^{2}C\left(\frac{\pi}{\mu_{\mathrm{b}}B}\right)\right]D\left(\frac{T}{T_{a}}\right)\sin{r_{\mathrm{c}}}+A\left(\frac{\pi}{\mu_{\mathrm{b}}B}\right)C\left(\frac{T}{T_{a}}\right)\frac{r_{\mathrm{c}}}{2}\cos{r_{\mathrm{c}}}\right\\}$
(23)

and

$\displaystyle\varepsilon_{xx}^{\mathrm{col}}=-\frac{eL_{0}T}{E_{\mathrm{F}}}\frac{3\sigma_{0}V_{0}^{2}ak_{\mathrm{F}}}{8\pi^{2}E_{\mathrm{F}}\hbar\omega_{\mathrm{c}}\mu^{2}B^{2}}\times$
$\displaystyle\left\\{-\left(p-\frac{1}{2}\right)-\left[-\left(p-\frac{1}{2}\right)A\left(\frac{\pi}{\mu_{\mathrm{c}}B}\right)+\frac{p_{\mathrm{c}}}{3}\left(\frac{\pi}{\mu_{\mathrm{c}}B}\right)^{2}C\left(\frac{\pi}{\mu_{\mathrm{c}}B}\right)\right]D\left(\frac{T}{T_{a}}\right)\sin{r_{\mathrm{c}}}-A\left(\frac{\pi}{\mu_{\mathrm{c}}B}\right)C\left(\frac{T}{T_{a}}\right)\frac{r_{\mathrm{c}}}{2}\cos{r_{\mathrm{c}}}\right\\}$
(24)

as well as the three constituents for the off-diagonal component,

$\displaystyle\varepsilon_{yx}^{(1)}$
$\displaystyle=-\frac{L_{0}eT}{E_{\mathrm{F}}}\frac{3}{4}\nu\frac{e^{2}}{h}{\lambda_{\mathrm{c}}}^{2}\left\\{-1+\left[A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)-\frac{2p_{\mathrm{h}}}{3}\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)^{2}C\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)\right]D\left(\frac{T}{T_{a}}\right)\sin{\left(r_{\mathrm{c}}+\delta_{\mathrm{F}}\right)}\right.$
$\displaystyle\left.-A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)C\left(\frac{T}{T_{a}}\right)r_{\mathrm{c}}\cos{\delta_{\mathrm{F}}}\cos{\left(r_{\mathrm{c}}+\delta_{\mathrm{F}}\right)}\right\\},$
(25a) $\displaystyle\varepsilon_{yx}^{(21)}$
$\displaystyle=-\frac{L_{0}eT}{E_{\mathrm{F}}}\frac{1}{2}\nu\frac{e^{2}}{h}{\lambda_{\mathrm{c}}}^{2}\left\\{\cos{\left(\frac{\delta_{\mathrm{F}}}{2}+2\frac{\pi}{ak_{\mathrm{F}}}\right)}+A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)C\left(\frac{T}{T_{a}}\right)\sin{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-2\frac{\pi}{ak_{\mathrm{F}}}\right)}\right.$
$\displaystyle-\frac{ak_{\mathrm{F}}}{\pi}\sin{\left(\frac{\pi}{ak_{\mathrm{F}}}\right)}\left[\frac{1}{2}\sin{\delta_{\mathrm{F}}}\sin{\left(\frac{\delta_{\mathrm{F}}}{2}+\frac{\pi}{ak_{\mathrm{F}}}\right)}+\frac{2p_{\mathrm{h}}}{3}\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)^{2}C\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)D\left(\frac{T}{T_{a}}\right)\sin{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-\frac{\pi}{ak_{\mathrm{F}}}\right)}\right.$
$\displaystyle\left.\left.+A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)C\left(\frac{T}{T_{a}}\right)r_{\mathrm{c}}\cos^{2}{\left(\frac{\delta_{\mathrm{F}}}{2}\right)}\cos{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-\frac{\pi}{ak_{\mathrm{F}}}\right)}\right]\right\\},$
(25b) and $\displaystyle\varepsilon_{yx}^{(22)}$
$\displaystyle=-\mathrm{sgn}(B)\frac{L_{0}eT}{E_{\mathrm{F}}}\frac{1}{2}\frac{e^{2}}{h}{\lambda_{\mathrm{c}}}^{2}\left(\cos{\left(\frac{\delta_{\mathrm{F}}}{2}+2\frac{\pi}{ak_{\mathrm{F}}}\right)}-A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)C\left(\frac{T}{T_{a}}\right)\sin{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-2\frac{\pi}{ak_{\mathrm{F}}}\right)}\right.$
$\displaystyle-\frac{ak_{\mathrm{F}}}{\pi}\cos{\left(\frac{\pi}{ak_{\mathrm{F}}}\right)}\left\\{-2\sin{\left(\frac{\delta_{\mathrm{F}}}{2}+\frac{\pi}{ak_{\mathrm{F}}}\right)}-\frac{1}{2}\sin{\delta_{\mathrm{F}}}\cos{\left(\frac{\delta_{\mathrm{F}}}{2}+\frac{\pi}{ak_{\mathrm{F}}}\right)}\right.$
$\displaystyle+\left[2A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)-\frac{2p_{\mathrm{h}}}{3}\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)^{2}C\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)\right]D\left(\frac{T}{T_{a}}\right)\cos{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-\frac{\pi}{ak_{\mathrm{F}}}\right)}$
$\displaystyle\left.\left.+A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)C\left(\frac{T}{T_{a}}\right)r_{\mathrm{c}}\cos^{2}{\left(\frac{\delta_{\mathrm{F}}}{2}\right)}\sin{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-\frac{\pi}{ak_{\mathrm{F}}}\right)}\right\\}\right).$
(25c)

Here we introduced two new functions,

$C\left(x\right)\equiv-3\frac{A^{\prime}\left(x\right)}{x}=\frac{3}{\sinh{x}}\left(\coth{x}-\frac{1}{x}\right)=1-\frac{7}{30}x^{2}+O\left(x^{4}\right)$
(26)

and

$D\left(x\right)\equiv-3A^{\prime\prime}\left(x\right)=\frac{3}{\sinh{x}}\left[2\coth{x}-\frac{2x}{\sinh^{2}{x}}-x\right]=1-\frac{7}{10}x^{2}+O\left(x^{4}\right)$
(27)

describing the damping, which is plotted in Fig. 1 along with $A(x)$ in Eq.
(17). We assumed, as is usually done [26], power-law dependence, $\mu\propto
E^{p}$ and $\mu_{s}\propto E^{p_{s}}$ ($s=\mathrm{b},\mathrm{c},\mathrm{h}$),
of the mobilities on the energy $E$. The oscillatory parts
$\delta\varepsilon_{yy}^{\mathrm{band}}$,
$\delta\varepsilon_{xx}^{\mathrm{col}}$,
$\delta\varepsilon_{yx}^{\left(1\right)}$,
$\delta\varepsilon_{yx}^{\left(21\right)}$, and
$\delta\varepsilon_{yx}^{\left(22\right)}$ are obtained by collecting the
cosine and sine terms containing $r_{\mathrm{c}}$ in the argument. The three
independent components of $\hat{\varepsilon}$ are thus given by

$\displaystyle\varepsilon_{xx}$ $\displaystyle=$
$\displaystyle\varepsilon_{xx}^{\mathrm{sc}}+\varepsilon_{xx}^{\mathrm{col}},$
(28a) $\displaystyle\varepsilon_{yy}$ $\displaystyle=$
$\displaystyle\varepsilon_{xx}^{\mathrm{sc}}+\varepsilon_{xx}^{\mathrm{col}}+\varepsilon_{yy}^{\mathrm{band}},$
(28b) $\displaystyle\varepsilon_{yx}$ $\displaystyle=$
$\displaystyle-\varepsilon_{xy}=\varepsilon_{yx}^{\mathrm{sc}}+\varepsilon_{yx}^{\left(1\right)}+\varepsilon_{yx}^{\left(21\right)}+\varepsilon_{yx}^{\left(22\right)},$
(28c)

where
$\varepsilon_{xx}^{\mathrm{sc}}=-(L_{0}eT/E_{\mathrm{F}})[1+p(1-\mu^{2}B^{2})/(1+\mu^{2}B^{2})]\sigma_{xx}^{\mathrm{sc}}$
and
$\varepsilon_{yx}^{\mathrm{sc}}=-(L_{0}eT/E_{\mathrm{F}})[1+2p/(1+\mu^{2}B^{2})]\sigma_{yx}^{\mathrm{sc}}\simeq-(L_{0}eT/E_{\mathrm{F}})\nu(e^{2}/h)$
are the diagonal and the off-diagonal components of the semiclassical
thermoelectric conductivity tensors for an unmodulated 2DEG. The oscillatory
parts are $\delta\varepsilon_{xx}=\delta\varepsilon_{xx}^{\mathrm{col}}$,
$\delta\varepsilon_{yy}=\delta\varepsilon_{xx}^{\mathrm{col}}+\delta\varepsilon_{yy}^{\mathrm{band}}$,
and
$\delta\varepsilon_{yx}=\delta\varepsilon_{yx}^{\left(1\right)}+\delta\varepsilon_{yx}^{\left(21\right)}+\delta\varepsilon_{yx}^{\left(22\right)}$.
The oscillation phase of $\delta\varepsilon_{xx}$ and $\delta\varepsilon_{yy}$
are rather complicated, since the relative weight of the terms with $\sin
r_{\mathrm{c}}$ and $\cos r_{\mathrm{c}}$ in
$\delta\varepsilon_{yy}^{\mathrm{band}}$ and
$\delta\varepsilon_{xx}^{\mathrm{col}}$ varies with both $B$ and $T$. We can
find, however, that the oscillations are dominated by the last terms in Eqs.
(23) and (24), noting that $r_{\mathrm{c}}$ generally takes a large value when
COs are observed. Roughly speaking, therefore,
$\delta\varepsilon_{yy}^{\mathrm{band}}$ and
$\delta\varepsilon_{xx}^{\mathrm{col}}$ oscillate with opposite phases and
with the amplitude
$|\delta\varepsilon_{yy}^{\mathrm{band}}|\gg|\delta\varepsilon_{xx}^{\mathrm{col}}|$,
similar to the case in $\hat{\sigma}$, leading to antiphase oscillations in
$\delta\varepsilon_{xx}$ and $\delta\varepsilon_{yy}$ with
$|\delta\varepsilon_{yy}|\gg|\delta\varepsilon_{xx}|$. The phases of
$\delta\varepsilon_{xx}$ and $\delta\varepsilon_{yy}$ are approximately
$\pi/2$ retarded compared to their counterparts in $\delta\sigma_{xx}$ and
$\delta\sigma_{yy}$. The phase of $\delta\varepsilon_{yx}$ is further
complicated by the $B$-dependent $\delta_{\mathrm{F}}$ in the argument of
cosine and sine terms. As we pointed out previously [15], however, both
$\delta_{\mathrm{F}}$ and $\pi/(ak_{\mathrm{F}})$ contained in the arguments
are expected to be small for typical ULSLs. Noting also that
$\delta\varepsilon_{yx}^{\left(22\right)}$ is much smaller than
$\delta\varepsilon_{yx}^{\left(1\right)}$ and
$\delta\varepsilon_{yx}^{\left(21\right)}$ due to the lack of the factor
$\nu$, we find that the dominant terms are the last terms in Eq. (25a) and in
Eq. (25b), and both roughly oscillate as $\cos r_{\mathrm{c}}$. Therefore, the
phase of $\delta\varepsilon_{yx}$ is close to that of
$\delta\varepsilon_{xx}$, as expected from their common origin (collisional
contribution).

Figure 4: (Color online) Commensurability oscillations in the thermoelectric
conductivity divided by the temperature vs. $2R_{\mathrm{c}}/a$ calculated for
various temperatures by picking out the oscillatory parts from Eq. (28) and
using the same ULSL parameters as in Fig. 2 and
$p=p_{\mathrm{b}}=p_{\mathrm{c}}=p_{\mathrm{h}}=3/2$. The top axis shows $B$.
(a) $\delta\varepsilon_{xx}/T$, (b) $\delta\varepsilon_{yy}/T$, and (c)
$\delta\varepsilon_{yx}/T$. $\delta\varepsilon_{ij}^{\mathrm{M}}/T$ and
$\delta\varepsilon_{ij}^{\mathrm{MB}}/T$ calculated employing the Mott formula
[Eq. (14)] and the modified Mott formula [Eq. (29)] are plotted by thick-
dotted (Mott) and thin-dotted [Mott($d/dB$)] lines, respectively. The inset to
(b) shows the close up of the part enclosed by the box in the main panel.
Vertical dotted lines indicate the locations of the flat-band conditions Eq.
(19).

We can draw out low-temperature approximations of $\hat{\varepsilon}$ by the
Mott formula Eq. (14). By performing the energy differentiation on Eqs. (16),
(18), and (20) at $T=0$, we find that the obtained approximations
$\varepsilon_{yy}^{\mathrm{M,band}}$, $\varepsilon_{xx}^{\mathrm{M,col}}$,
$\varepsilon_{yx}^{\mathrm{M(1)}}$, $\varepsilon_{yx}^{\mathrm{M(21)}}$,
$\varepsilon_{yx}^{\mathrm{M(22)}}$ are given by Eqs. (23), (24), (25a),
(25b), and (25c) with every occurrence of $C(T/T_{a})$ and $D(T/T_{a})$
replaced by unity. Therefore, the Mott formulas become a good approximation
when $C(T/T_{a})$ and $D(T/T_{a})$ are close enough to 1, or when $T\ll T_{a}$
(see Fig. 1), which guarantees $\sigma_{ij,{T=0}}\left(E\right)$ to vary
slowly within the energy window $|E-E_{\mathrm{F}}|\lesssim k_{\mathrm{B}}T$.
A general condition for the Mott formula to be applicable is
$k_{\mathrm{B}}T\ll E_{\mathrm{F}}$, which justifies discarding the higher-
order terms in the expansion Eq. (13). Since
$E_{\mathrm{F}}/(k_{\mathrm{B}}T_{a})=\pi^{2}(2R_{\mathrm{c}}/a)\gg 1$ when
COs are observed, $T\ll T_{a}$ poses a much more stringent constraint.

An alternative convenient way often employed to estimate the thermoelectric
conductivity is to use a modified Mott formula with the energy derivative
replaced by the magnetic-field derivative [30, 34]. Noting that the behaviors
of COs are governed by the variation of the ratio
$2R_{\mathrm{c}}/a\propto\sqrt{E}/B$, we replace $d/dE$ by
$(-B/2E_{\mathrm{F}})(d/dB)$ 333Note the difference from the similar formula
for the SdH effect, in which $d/dE$ is replaced with
$(-B/E_{\mathrm{F}})(d/dB)$ [30] since the oscillations are governed by
$\nu\propto E/B$, yielding an alternative approximation formula,

$\varepsilon_{ij}^{\mathrm{MB}}=L_{0}eT\frac{B}{2E_{\mathrm{F}}}\left.\frac{\partial\sigma_{ij,{T=0}}}{\partial
B}\right|_{E_{\mathrm{F}}}.$ (29)

An advantage of this formula is that it allows us to estimate
$\hat{\varepsilon}$ by numerically differentiating the data of
$\hat{\sigma}(B)$ obtained (often by inverting the resistivity
$\hat{\rho}(B)$) experimentally from the most basic magnetic-field sweeps
using a conventional ULSL sample with fixed $n_{e}$. By contrast, the
acquisition of $d/dE$ requires an additional uniform gate to vary $n_{e}$. By
carrying out the magnetic-field differentiation, we obtain

$\varepsilon_{yy}^{\mathrm{MB,band}}=-\frac{L_{0}eT}{E_{F}}\frac{\sigma_{0}V_{0}^{2}}{E_{\mathrm{F}}\hbar\omega_{\mathrm{c}}ak_{\mathrm{F}}}\left\\{\frac{1}{2}+\frac{1}{2}\left[A\left(\frac{\pi}{\mu_{\mathrm{b}}B}\right)-\frac{1}{3}\left(\frac{\pi}{\mu_{\mathrm{b}}B}\right)^{2}C\left(\frac{\pi}{\mu_{\mathrm{b}}B}\right)\right]\sin{r_{\mathrm{c}}}+A\left(\frac{\pi}{\mu_{\mathrm{b}}B}\right)\frac{r_{\mathrm{c}}}{2}\cos{r_{\mathrm{c}}}\
\right\\},$ (30)
$\varepsilon_{xx}^{\mathrm{MB,col}}=-\frac{L_{0}eT}{E_{F}}\frac{3\sigma_{0}V_{0}^{2}ak_{\mathrm{F}}}{8\pi^{2}E_{\mathrm{F}}\hbar\omega_{\mathrm{c}}\mu^{2}B^{2}}\left\\{\frac{3}{2}-\frac{1}{2}\left[3A\left(\frac{\pi}{\mu_{\mathrm{c}}B}\right)-\frac{1}{3}\left(\frac{\pi}{\mu_{\mathrm{c}}B}\right)^{2}C\left(\frac{\pi}{\mu_{\mathrm{c}}B}\right)\right]\sin{r_{\mathrm{c}}}-A\left(\frac{\pi}{\mu_{\mathrm{c}}B}\right)\frac{r_{\mathrm{c}}}{2}\cos{r_{\mathrm{c}}}\right\\},$
(31)
$\displaystyle\varepsilon_{yx}^{\mathrm{MB(1)}}=-\frac{L_{0}eT}{E_{F}}\frac{3}{4}\nu\frac{e^{2}}{h}{\lambda_{c}}^{2}\left\\{2-\left[2A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)-\frac{1}{3}\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)^{2}C\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)\right]\sin{\left(r_{\mathrm{c}}+\delta_{\mathrm{F}}\right)}\right.$
$\displaystyle\left.-A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)r_{\mathrm{c}}\cos{\delta_{\mathrm{F}}}\cos{\left(r_{\mathrm{c}}+\delta_{\mathrm{F}}\right)}\right\\},$
(32a) $\displaystyle\varepsilon_{yx}^{\mathrm{MB(21)}}$
$\displaystyle=-\frac{L_{0}eT}{E_{F}}\frac{1}{2}\nu\frac{e^{2}}{h}{\lambda_{c}}^{2}\frac{ak_{\mathrm{F}}}{\pi}\sin{\left(\frac{\pi}{ak_{\mathrm{F}}}\right)}\left\\{-2\cos{\left(\frac{\delta_{\mathrm{F}}}{2}+\frac{\pi}{ak_{\mathrm{F}}}\right)}-\frac{1}{2}\sin{\delta_{\mathrm{F}}}\sin{\left(\frac{\delta_{\mathrm{F}}}{2}+\frac{\pi}{ak_{\mathrm{F}}}\right)}\right.$
$\displaystyle\left.-\left[2A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)-\frac{1}{3}\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)^{2}C\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)\right]\sin{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-\frac{\pi}{ak_{\mathrm{F}}}\right)}-A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)r_{\mathrm{c}}\cos^{2}{\left(\frac{\delta_{\mathrm{F}}}{2}\right)}\cos{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-\frac{\pi}{ak_{\mathrm{F}}}\right)}\right\\},$
(32b) and $\displaystyle\varepsilon_{yx}^{\mathrm{MB(22)}}$
$\displaystyle=-\mathrm{sign}\left(B\right)\frac{L_{0}eT}{E_{F}}\frac{1}{2}\frac{e^{2}}{h}{\lambda_{c}}^{2}\frac{ak_{\mathrm{F}}}{\pi}\cos{\left(\frac{\pi}{ak_{\mathrm{F}}}\right)}\left\\{-\sin{\left(\frac{\delta_{\mathrm{F}}}{2}+\frac{\pi}{ak_{\mathrm{F}}}\right)}+\frac{1}{2}\sin{\delta_{\mathrm{F}}}\cos{\left(\frac{\delta_{\mathrm{F}}}{2}+\frac{\pi}{ak_{\mathrm{F}}}\right)}\right.$
$\displaystyle\left.+\left[A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)-\frac{1}{3}\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)^{2}C\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)\right]\cos{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-\frac{\pi}{ak_{\mathrm{F}}}\right)}-A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)r_{\mathrm{c}}\cos^{2}{\left(\frac{\delta_{\mathrm{F}}}{2}\right)}\sin{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-\frac{\pi}{ak_{\mathrm{F}}}\right)}\right\\}.$
(32c)

Note that the non-oscillatory parts in Eqs. (30), (31), and (32) make little
sense, since the replacement $d/dE\rightarrow-(B/2E_{\mathrm{F}})d/dB$ is
justifiable only for the COs. It can readily be found that the dominant terms
in $\delta\varepsilon_{yy}^{\mathrm{MB,band}}$,
$\delta\varepsilon_{xx}^{\mathrm{MB,col}}$,
$\delta\varepsilon_{yx}^{\mathrm{MB}\left(1\right)}$,
$\delta\varepsilon_{yx}^{\mathrm{MB}\left(21\right)}$, and
$\delta\varepsilon_{yx}^{\mathrm{MB}\left(22\right)}$ [the last terms in Eqs.
(30), (31), and (32)] coincide with the corresponding terms in
$\delta\varepsilon_{yy}^{\mathrm{M,band}}$,
$\delta\varepsilon_{xx}^{\mathrm{M,col}}$,
$\delta\varepsilon_{yx}^{\mathrm{M}\left(1\right)}$,
$\delta\varepsilon_{yx}^{\mathrm{M}\left(21\right)}$, and
$\delta\varepsilon_{yx}^{\mathrm{M}\left(22\right)}$. For completeness, we
mention that the components of the thermoelectric conductivity tensor for the
(modified) Mott formula, $\varepsilon_{xx}^{\mathrm{M(B)}}$,
$\varepsilon_{yy}^{\mathrm{M(B)}}$, and $\varepsilon_{yx}^{\mathrm{M(B)}}$,
are obtained by the same relations as Eq. (28).

Figure 5: (Color online) (a) $\delta\varepsilon_{yy}/T$ vs.
$2R_{\mathrm{c}}/a$ at $T=10$ K for various values of the exponent
$p(=p_{\mathrm{b}}=p_{\mathrm{c}}=p_{\mathrm{h}})$. The other sample
parameters are the same as in Fig. 4. The top axis shows $B$. Vertical dotted
lines indicate the locations of the flat-band conditions Eq. (19). (b) Thermal
damping factors $A(T/T_{a})$, $C(T/T_{a})$, and $D(T/T_{a})$ vs.
$2R_{\mathrm{c}}/a$ for various temperatures and
$T_{a}=(1/\pi^{2})(E_{\mathrm{F}}/k_{\mathrm{B}})(a/2R_{\mathrm{c}})$ with
$a=200$ nm and $n_{e}=4.2\times 10^{15}$ m-2 [the same parameters as in (a)],
plotted by solid, dashed, and dotted lines, respectively.

In Fig. 4, we plot $\delta\varepsilon_{ij}/T$ ($T=0.02$ K, 1.5 K, 4.2 K, and
10 K), $\delta\varepsilon_{ij}^{\mathrm{M}}/T$, and
$\delta\varepsilon_{ij}^{\mathrm{MB}}/T$ calculated using the same ULSL
parameters employed in Figs. 2 and 3 and, in addition, setting the exponents
$p=p_{\mathrm{b}}=p_{\mathrm{c}}=p_{\mathrm{h}}=3/2$, noting that the
scatterings are primarily caused by the remote ionized impurities in
GaAs/AlGaAs 2DEGs [36]. As can be seen from Eqs. (14) and (29),
$\delta\varepsilon_{ij}^{\mathrm{M}}/T$ and
$\delta\varepsilon_{ij}^{\mathrm{MB}}/T$ do not depend on the temperature. The
figures allow us to confirm the basic characteristics mentioned above, namely,
large amplitude oscillations in $\delta\varepsilon_{yy}$ with the phase
roughly $\pi/2$ shifted with respect to $\delta\sigma_{yy}$ and much smaller
antiphase oscillations in $\delta\varepsilon_{xx}$ and
$\delta\varepsilon_{yx}$. Although all the $\delta\varepsilon_{ij}$ are
virtually indistinguishable from the corresponding
$\delta\varepsilon_{ij}^{\mathrm{M}}$ at 0.02 K, small but discernible
discrepancy becomes apparent already at 1.5 K. The oscillation amplitudes
exhibit sublinear increase with $T$, thereby enhancing the discrepancy from
the Mott formula with $T$. The deviation from the Mott formula varying with
the temperature and the magnetic field are visualized by the behavior of
$C(T/T_{a})$ and $D(T/T_{a})$ plotted in Fig. 5(b). Deviation of the phase
also becomes clear at higher temperatures. The phase shift is caused by the
increase in the relative weight of the terms oscillating with $\sim\sin
r_{\mathrm{c}}$. Since the weight depends on the exponents $p$ and $p_{s}$
($s=\mathrm{b},\mathrm{c},\mathrm{h}$), the phase shift can, in principle, be
applied to the experimentally probe the values of the exponents, either to
confirm $p=p_{s}=3/2$ or to prove otherwise [see Fig. 5 (a)]. We note in
passing that $\delta\varepsilon_{ij}^{\mathrm{MB}}/T$ provides a fairly good
approximation of $\delta\varepsilon_{ij}^{\mathrm{M}}/T$, albeit with a slight
shift in the phase [see the inset to Fig. 4 (b)].

###  III.3 Thermopower

Figure 6: (Color online) Commensurability oscillations in the thermopower
divided by the temperature vs. $2R_{\mathrm{c}}/a$ calculated for various
temperatures by Eq. (33) using the same set of parameters as in Fig. 4. The
top axis shows $B$. (a) $\delta S_{xx}/T$, (b) $\delta S_{yy}/T$, (c) $\delta
S_{xy}/T$, and (d) $\delta S_{yx}/T$. $\delta S_{ij}^{\mathrm{M}}$ and $\delta
S_{ij}^{\mathrm{MB}}$ calculated by Eq. (33) with $\delta\varepsilon_{ij}$
replaced by the Mott formulas $\delta\varepsilon_{ij}^{\mathrm{M}}$ and
$\delta\varepsilon_{ij}^{\mathrm{MB}}$ are also plotted with thick-dotted
(Mott) and thin-dotted [Mott($d/dB$)] lines, respectively. Vertical dotted
lines indicate the locations of the flat-band conditions Eq. (19).

The thermopower tensor $\hat{S}$ is given by the product of the resistivity
tensor and the thermoelectric conductivity tensor, Eq. (6). The oscillatory
parts of $\hat{S}$ is written, to the lowest order in the perturbations
generated by $V(x)$, as

$\displaystyle\delta S_{xx}=$
$\displaystyle\rho_{0}\delta\varepsilon_{xx}+\rho_{\mathrm{H}}\delta\varepsilon_{yx}+\varepsilon_{xx}^{\mathrm{sc}}\delta\rho_{xx}+\varepsilon_{yx}^{\mathrm{sc}}\delta\rho_{xy},$
(33a) $\displaystyle\delta S_{yy}=$
$\displaystyle\rho_{0}\delta\varepsilon_{yy}+\rho_{\mathrm{H}}\delta\varepsilon_{yx}+\varepsilon_{xx}^{\mathrm{sc}}\delta\rho_{yy}+\varepsilon_{yx}^{\mathrm{sc}}\delta\rho_{xy},$
(33b) $\displaystyle\delta S_{xy}=$
$\displaystyle-\rho_{0}\delta\varepsilon_{yx}+\rho_{\mathrm{H}}\delta\varepsilon_{yy}-\varepsilon_{yx}^{\mathrm{sc}}\delta\rho_{xx}+\varepsilon_{xx}^{\mathrm{sc}}\delta\rho_{xy},$
(33c) $\displaystyle\delta S_{yx}=$
$\displaystyle\rho_{0}\delta\varepsilon_{yx}-\rho_{\mathrm{H}}\delta\varepsilon_{xx}+\varepsilon_{yx}^{\mathrm{sc}}\delta\rho_{yy}-\varepsilon_{xx}^{\mathrm{sc}}\delta\rho_{xy},$
(33d)

where $\rho_{\mathrm{H}}=\rho_{xy}^{\mathrm{sc}}\equiv B/(en_{e})$ is the
semiclassical Hall resistivity. In Fig. 6, we plot $\delta S_{ij}$ calculated
by Eq. (33) using the same set of sample parameters as in Fig. 4. We also plot
$\delta S_{ij}^{\mathrm{M}}$ (Mott) and $\delta S_{ij}^{\mathrm{MB}}$ [Mott
($d/dB$)] calculated by replacing $\delta\varepsilon_{ij}$ in Eq. (33) with
the Mott formulas $\delta\varepsilon_{ij}^{\mathrm{M}}$ and
$\delta\varepsilon_{ij}^{\mathrm{MB}}$, respectively. We can see that the
amplitude of $\delta S_{xy}$ exceeds those of the other components by more
than an order of magnitude. Accordingly, the relation $S_{xy}=-S_{yx}$,
expected for isotropic systems as mentioned earlier, does not hold. We can
also find that $\delta S_{xx}$ and $\delta S_{yy}$ exhibit quite dissimilar
oscillations. These behaviors are in contradiction to the previous theory [7].
In Ref. 7, only $S_{xx}$ and $S_{xy}$ were shown, and $S_{yy}$ were referred
to as “almost identical” to $S_{xx}$. No mention was made of $S_{yx}$ or its
relation to $S_{xy}$.

We can readily find, by calculating the terms separately, that the first two
terms in Eq. (33) containing $\delta\varepsilon_{ij}$ are much larger than the
last two terms with $\delta\rho_{ij}$ 444This can be qualitatively understood
by noting that, loosely speaking, $\delta\varepsilon\propto d\delta\sigma/dE$
(Mott formula) while $\delta\rho\propto\delta\sigma$. Since
$|\rho_{\mathrm{H}}/\rho_{0}|=\mu|B|\gg 1$ in the magnetic-field range COs are
observed, both Eqs. (33a) and (33b) are dominated by the identical second term
if $|\delta\varepsilon_{xx}|$, $|\delta\varepsilon_{yy}|$, and
$|\delta\varepsilon_{yx}|$ are of the same order of magnitude. We surmise that
this is the main reason behind the rather counterintuitive isotropic relation
$S_{xx}\simeq S_{yy}$ predicted in Ref. 7. As we have seen in Fig. 4, however,
$\delta\varepsilon_{yy}$ possess the amplitude more than an order of magnitude
larger than the other components, which inherits the relation
$|\delta\sigma_{yy}|\gg|\delta\sigma_{xx}|\sim|\delta\sigma_{yx}|$ in the
electrical conductivity (Fig. 2). As mentioned above, the dominance of the
$yy$ component has been intensified by the inclusion of the small-angle-
scattering damping factors, which, according to the recent experiment [15],
diminishes the oscillations of the collisional contribution more effectively
compared to those of the band contribution. As a result, while the first term
in $\delta S_{xx}$ remains negligibly small, the first term in $\delta S_{yy}$
outweighs the second term at low magnetic fields ($B\lesssim 0.3$ T), leading
to the oscillation with the phase roughly opposite to $\delta S_{xx}$. The
second term is the dominant term also in both $\delta S_{xy}$ and $\delta
S_{yx}$, as was in $\delta S_{xx}$. The second term in $\delta S_{xy}$
consists of the large amplitude oscillations $\delta\varepsilon_{yy}$
multiplied by the large $\rho_{\mathrm{H}}$, which is responsible for the
oscillation amplitude far exceeding those of the other components. $\delta
S_{yx}$ oscillates with roughly the same phase and much smaller amplitude.
This can readily be understood by recalling the relative phase and amplitude
between $\delta\varepsilon_{xx}$ and $\delta\varepsilon_{yy}$ described
earlier.

The deviation from the Mott formula with increasing temperature, the phase
shift at high temperatures, and the fairly good agreement between $\delta
S_{ij}^{\mathrm{M}}$ and $\delta S_{ij}^{\mathrm{MB}}$ (albeit with a slight
phase shift), all reflects the behaviors of $\delta\varepsilon_{ij}$ described
above.

###  III.4 Thermal conductivity

Finally, we calculate the (electronic) thermal conductivity tensors,
$\hat{\lambda}$ and $\hat{\kappa}$. Substituting Eqs. (40), (41), and (42)
into the definition Eq. (12) of $\hat{\lambda}$ and following the procedure
similar to that employed in calculating the thermoelectric conductivity, we
obtain the band and the collisional contributions for the diagonal components,

$\lambda_{yy}^{\mathrm{band}}=L_{0}T\frac{\sigma_{0}V_{0}^{2}}{E_{\mathrm{F}}\hbar\omega_{\mathrm{c}}ak_{\mathrm{F}}}\left[1+A\left(\frac{\pi}{\mu_{\mathrm{b}}B}\right)D\left(\frac{T}{T_{a}}\right)\sin{r_{\mathrm{c}}}\right],$
(34)

and

$\lambda_{xx}^{\mathrm{col}}=L_{0}T\frac{3\sigma_{0}V_{0}^{2}ak_{\mathrm{F}}}{8\pi^{2}E_{\mathrm{F}}\hbar\omega_{\mathrm{c}}(\mu
B)^{2}}\left[1-A\left(\frac{\pi}{\mu_{\mathrm{c}}B}\right)D\left(\frac{T}{T_{a}}\right)\sin{r_{\mathrm{c}}}\right],$
(35)

and the three constituents of the off-diagonal components,

$\lambda_{yx}^{\left(1\right)}=L_{0}T\frac{3}{2}\nu\frac{e^{2}}{h}{\lambda_{\mathrm{c}}}^{2}\left[1-A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)D\left(\frac{T}{T_{a}}\right)\sin{\left(r_{\mathrm{c}}+\delta_{\mathrm{F}}\right)}\right],$
(36a)
$\lambda_{yx}^{\left(21\right)}=-L_{0}T\nu\frac{e^{2}}{h}{\lambda_{\mathrm{c}}}^{2}\frac{ak_{\mathrm{F}}}{\pi}\sin{\left(\frac{\pi}{{ak}_{\mathrm{F}}}\right)}\left[\cos{\left(\frac{\delta_{\mathrm{F}}}{2}+\frac{\pi}{{ak}_{\mathrm{F}}}\right)}+A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)D\left(\frac{T}{T_{a}}\right)\sin{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-\frac{\pi}{{ak}_{\mathrm{F}}}\right)}\right],$
(36b)
$\lambda_{yx}^{\left(22\right)}=-L_{0}T\mathrm{sgn}(B)\frac{e^{2}}{h}{\lambda_{\mathrm{c}}}^{2}\frac{ak_{\mathrm{F}}}{\pi}\cos{\left(\frac{\pi}{{ak}_{\mathrm{F}}}\right)}\left[\sin{\left(\frac{\delta_{\mathrm{F}}}{2}+\frac{\pi}{{ak}_{\mathrm{F}}}\right)}-A\left(\frac{\pi}{\mu_{\mathrm{h}}B}\right)D\left(\frac{T}{T_{a}}\right)\cos{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-\frac{\pi}{{ak}_{\mathrm{F}}}\right)}\right].$
(36c)

The oscillatory parts $\delta\lambda_{yy}^{\mathrm{band}}$,
$\delta\lambda_{xx}^{\mathrm{col}}$, $\delta\lambda_{yx}^{\left(1\right)}$,
$\delta\lambda_{yx}^{\left(21\right)}$, and
$\delta\lambda_{yx}^{\left(22\right)}$ are given by the second terms in Eqs.
(34), (35), (36a), (36b), and (36c), respectively. The three independent
components of $\hat{\lambda}$ are given by

$\displaystyle\lambda_{xx}$ $\displaystyle=$
$\displaystyle\lambda_{xx}^{\mathrm{sc}}+\lambda_{xx}^{\mathrm{col}}$ (37a)
$\displaystyle\lambda_{yy}$ $\displaystyle=$
$\displaystyle\lambda_{xx}^{\mathrm{sc}}+\lambda_{xx}^{\mathrm{col}}+\lambda_{yy}^{\mathrm{band}}$
(37b) $\displaystyle\lambda_{yx}$ $\displaystyle=$
$\displaystyle\lambda_{yx}^{\mathrm{sc}}+\lambda_{yx}^{\left(1\right)}+\lambda_{yx}^{\left(21\right)}+\lambda_{yx}^{\left(22\right)},$
(37c)

where $\lambda_{xx}^{\mathrm{sc}}=L_{0}T\sigma_{xx}^{sc}$ and
$\lambda_{yx}^{\mathrm{sc}}=L_{0}T\sigma_{yx}^{sc}$ are the diagonal and off-
diagonal compoments of the semiclassical thermal conductivities for an
unmodulated 2DEG. The oscillatory parts are
$\delta\lambda_{xx}=\delta\lambda_{xx}^{\mathrm{col}}$,
$\delta\lambda_{yy}=\delta\lambda_{xx}^{\mathrm{col}}+\delta\lambda_{yy}^{\mathrm{band}}$,
and
$\delta\lambda_{yx}=\delta\lambda_{yx}^{\left(1\right)}+\delta\lambda_{yx}^{\left(21\right)}+\delta\lambda_{yx}^{\left(22\right)}$.
By employing the Wiedemann-Franz law Eq. (15) instead of Eq. (12), we obtain
$\lambda_{yy}^{\mathrm{W,band}}$, $\lambda_{xx}^{\mathrm{W,col}}$,
$\lambda_{yx}^{\mathrm{W(1)}}$, $\lambda_{yx}^{\mathrm{W(21)}}$,
$\lambda_{yx}^{\mathrm{W(22)}}$ given by Eqs. (34), (35), (36a), (36b), and
(36c), respectively, with $D(T/T_{a})$ replaced by unity. The corresponding
three components $\lambda_{ij}^{\mathrm{W}}$ ($ij=xx,yy,yx$) and their
oscillatory parts $\delta\lambda_{ij}^{\mathrm{W}}$ are obtained with similar
relation as in $\lambda_{ij}$ and $\delta\lambda_{ij}$.

Although we referred also to $\hat{\lambda}$ as the “thermal conductivity” in
this paper (following Ref. 19) for lack of a more appropriate nomenclature,
thermal conductivity is generally defined as $\hat{\kappa}$ in Eq. (8).
Recalling the relation $\varepsilon_{xy}=-\varepsilon_{yx}$ and the symmetry
of the diagonal components under magnetic-field reversal,
$X_{ii}(-B)=X_{ii}(B)$, we have
${}^{t}\\!\hat{\varepsilon}(-B)=\hat{\varepsilon}(B)$. Further with the help
of Eq. (9c), Eq. (8) can be rewritten as

$\hat{\kappa}=\hat{\lambda}-T\hat{\varepsilon}\hat{S}=\hat{\lambda}-T\hat{\varepsilon}\hat{\rho}\hat{\varepsilon}.$
(38)

The oscillatory parts are

$\delta\kappa_{xx}=\delta\lambda_{xx}-T(\varepsilon_{xx}^{\mathrm{sc}}\delta
S_{xx}-\varepsilon_{yx}^{\mathrm{sc}}\delta
S_{yx}+S_{xx}^{\mathrm{sc}}\delta\varepsilon_{xx}-S_{yx}^{\mathrm{sc}}\delta\varepsilon_{yx}),$
(39a)
$\delta\kappa_{yy}=\delta\lambda_{yy}-T(\varepsilon_{xx}^{\mathrm{sc}}\delta
S_{yy}+\varepsilon_{yx}^{\mathrm{sc}}\delta
S_{xy}+S_{xx}^{\mathrm{sc}}\delta\varepsilon_{yy}-S_{yx}^{\mathrm{sc}}\delta\varepsilon_{yx}),$
(39b)
$\delta\kappa_{yx}=\delta\lambda_{yx}-T(\varepsilon_{xx}^{\mathrm{sc}}\delta
S_{yx}+\varepsilon_{yx}^{\mathrm{sc}}\delta
S_{xx}+S_{yx}^{\mathrm{sc}}\delta\varepsilon_{yy}+S_{xx}^{\mathrm{sc}}\delta\varepsilon_{yx}).$
(39c)

Figure 7: (Color online) Commensurability oscillations in the (electronic)
thermal conductivity divided by the temperature vs. $2R_{\mathrm{c}}/a$
calculated for various temperatures. The top axis shows $B$. (a)
$\delta\lambda_{xx}/T$ and $\delta\kappa_{xx}/T$, (b) $\delta\lambda_{yx}/T$
and $\delta\kappa_{yx}/T$, (c) $\delta\lambda_{yy}/T$ and
$\delta\kappa_{yy}/T$. $\delta\lambda_{ij}/T$ and $\delta\kappa_{ij}/T$ are
plotted by dashed and solid lines, respectively. In addition,
$\delta\lambda_{ij}^{\mathrm{W}}/T$ calculated by applying the Wiedemann-Franz
law is plotted by the thick dotted-line (W-F). The inset to (b) displays
magnified view of the part enclosed by the rectangle in the main panel.

In Fig. 7, we plot $\delta\lambda_{ij}/T$ and $\delta\kappa_{ij}/T$ calculated
for various temperatures using the same set of sample parameters as has been
used so far. Again, the CO amplitude is overwhelmingly larger in the $yy$
component than in the other components, taking over similar relations in
$\hat{\sigma}$. We also plot $\delta\lambda_{ij}^{\mathrm{W}}/T$, which do not
depend on the temperature. We find that $\delta\lambda_{ij}/T$ and
$\delta\kappa_{ij}/T$ are almost indistinguishable at low temperatures, but
the latter exhibits a slight phase shift at high temperatures [see the inset
to Fig. 7(b)]. Sign reversal takes place for low magnetic fields at high
temperatures (4.2 K and 10 K), which is attributable to the sign reversal of
$D(T/T_{a})$ [see Fig. 5(b)]. Deviation of $\delta\lambda_{ij}/T$ and
$\delta\kappa_{ij}/T$ from the Wiedemann-Franz law is discernible even at 1.5
K, resulting from the deviation of $D(T/T_{a})$ from unity (see Fig. 1).

##  IV Discussion

Figure 8: (Color online) Schematic drawings depicting the experimental setup
designed to measure the four components of the thermopower tensor $\hat{S}$ in
a ULSL. The principal axis ($x$ axis) of the ULSL is aligned along and across
the Hall bar in (a) and (b), respectively. Temperature gradient
$-\bm{\nabla}T$ is introduced by heating the right end of the Hall bar
devices. At $B=0$, $-\bm{\nabla}T$ is aligned essentially parallel to the
length of the Hall bar, and thus the resulting thermoelectric voltages
$V_{xx}$, $V_{yx}$, $V_{yy}$, and $V_{xy}$ represent $S_{xx}$, $S_{yx}$,
$S_{yy}$, and $S_{xy}$, respectively, after correcting for the geometrical
factors of the devices.  Figure 9: (Color online) Drawings similar to Fig. 8
but with a finite magnetic field $B$ applied perpendicular to the 2DEG plane.
The magnetic field deflects $-\bm{\nabla}T$ away from the length of the Hall
bar roughly by the Hall angle $\theta_{\mathrm{H}}$. Owing to the tilting,
$V_{xx}$, $V_{yx}$, $V_{yy}$, and $V_{xy}$ are not simply related to $S_{xx}$,
$S_{yx}$, $S_{yy}$, and $S_{xy}$; $V_{xx}$, for example, contains
contributions from both $S_{xx}$ and $S_{xy}$.

Here we discuss the implication of the present results on the interpretation
of experiments. First we recall the practical difficulties one encounters in
the measurement involving the temperature gradient in a magnetic field for a
device fabricated from a high-mobility 2DEG. Measurement of the thermopower or
thermal conductivity using a Hall-bar geometry requires the temperature
gradient along the length of the Hall bar. In the absence of the magnetic
field, this can readily be achieved by heating one end of the Hall bar (see
Fig. 8). The situation is drastically altered, however, by the application of
the magnetic field. When the 2DEG is mostly decoupled from the lattice system,
as assumed in the present study, the thermal current is mainly carried by the
electrons and thus can readily be bent by the magnetic field. Accordingly, the
temperature gradient $-\bm{\nabla}T$ generated by heating one end is tilted
away from the longitudinal direction of the Hall bar roughly by the Hall angle
$\theta_{\mathrm{H}}=\arctan{(\sigma_{yx}/\sigma_{xx})}$ (see Fig. 9) [18]. In
a high-mobility 2DEG, the relation $|\sigma_{yx}|\gg\sigma_{xx}$ holds in the
magnetic field range COs are observed, leading to a high
$|\theta_{\mathrm{H}}|$ approaching $\pi/2$. Therefore, $-\bm{\nabla}T$ is
roughly redirected to the transverse direction across the Hall bar, with some
position-dependent variation in the direction.

At $B=0$, we can employ the pairs of voltage probes located along $V_{xx}$
($V_{yy}$) or across $V_{yx}$ ($V_{xy}$) the Hall bar to deduce the components
of the thermopower tensor $S_{xx}$ ($S_{yy}$) or $S_{yx}$ ($S_{yx}$),
respectively, employing a ULSL device with the $x$ axis aligned along (across)
the length of the Hall bar as depicted in Fig. 8 (a) [Fig. 8 (b)], and making
use of the relation $\bm{E}=\hat{S}\bm{\nabla}T$ [Eq. (3) with $\bm{j}=0$]. In
the magnetic field, however, these correspondences are lost due to the tilt of
$-\bm{\nabla}T$ (Fig. 9). To the thermovoltage $V_{xx}$ shown in Fig. 9 (a),
for instance, not only $S_{xx}$ but also $S_{xy}$ makes a contribution due to
the $y$ component $-\nabla_{y}T$ of $-\bm{\nabla}T$ introduced by the tilting.
Moreover, noting that $|\delta S_{xy}|$ is more than an order of magnitude
larger than $|\delta S_{xx}|$ (see Fig. 6) and that $|\nabla_{y}T|$ is larger
than the $x$ component $|\nabla_{x}T|$ owing to the high Hall angle, we can
conclude that the oscillatory part $\delta V_{xx}$ mostly reflects the
behavior of $\delta S_{xy}$ with minor modification attributable to $\delta
S_{xx}$. As can be seen in Fig. 9 (b), $\delta V_{xy}$ also picks up $\delta
S_{xy}$ and $\delta S_{xx}$, with the relative weight of the latter enhanced
since $|\nabla_{x}T|$ outweigh $|\nabla_{y}T|$ in this setup. In both cases,
where the pair of voltage probes are aligned along the $x$ direction, the COs
in the thermoelectric voltage are expected to mainly results from $\delta
S_{xy}$ owing to its overwhelmingly large oscillation amplitude. On the other
hand, thermoelectric voltages measured with the pairs along the $y$ direction,
$V_{yx}$ and $V_{yy}$, are composed of contributions from $\delta S_{yx}$ and
$\delta S_{yy}$. Even in these cases, however, it is possible for the
contribution of $\delta S_{xy}$ to mix into the measured thermoelectric
voltage, caused by small misalignment of the locations of the voltage arms or
of the orientation of the ULSL, which are difficult to completely eliminate in
the actual experiments. The dependence of $-\bm{\nabla}T$ on the location, as
well as its variation with $B$ [18], further complicates the interpretation of
experimentally obtained $V_{ij}$ [12, 13].

In Ref. 9, the thermoelectric voltage $V_{xx}$ was measured on a ULSL in the
configuration shown in Fig. 9 (a). The authors assumed the temperature
gradient $-\bm{\nabla}T$ along the length of the Hall bar [as in Fig. 8 (a)],
and attributed the COs observed in $V_{xx}$ to $\delta S_{xx}$. The amplitude
and the phase of the COs were found to be roughly in agreement with $\delta
S_{xx}$ in the previous theory [7]. In hindsight, however, the agreement in
the amplitude seems to have been caused partly by incorrect analysis in
deducing the amplitude $V_{0}$ of the potential modulation. 555Although
lithographically defined period of the grating gate was 400 nm, the period of
COs in $R_{xx}$ unambiguously indicates that the period responsible for the
COs was $a=200$ nm, which reveals that the second harmonic content was
dominant in the strain-induced modulation potential in this sample. [40, 41]
Accordingly, $V_{0}$ determined from the position $B_{c}$ of the maximum of
the low-field positive magnetoresistance should also become half the value
quoted in the paper ($V_{0}=0.26$ meV), which, in turn, makes the amplitude of
$\delta S_{xx}$ in Ref. 7 smaller With the correct value of $V_{0}$,
theoretical amplitude becomes smaller than the experimentally-observed
amplitude. Moreover, while Ref. 7 predicted the amplitude of the COs in the
thermopower to increase with decreasing magnetic field, the amplitude of the
experimentally observed COs in $V_{xx}$ decreased with decreasing magnetic
field in contradiction to the theory. This clearly indicates that additional
damping factors, such as $A[\pi/(\mu_{s}B)]$
($s=\mathrm{b},\mathrm{c},\mathrm{h}$) in the present study, are essential for
the theory to account for the experimentally observed COs in the thermopower.
As discussed above, we consider COs observed in this experiment are also
mainly attributable to $\delta S_{xy}$ due to the tilting of $-\bm{\nabla}T$.
In fact, we found that Eq. (33c) with the parameters for the sample in Ref. 9
666We used the values $V_{0}=0.13$ meV, $a=200$ nm, $n_{e}=2.9\times 10^{15}$
m-2, and $\mu=180$ m2/(Vs). We also note that the transverse ($y$) component
of $-\bm{\nabla}T$ changes sign by reversing the magnetic field. Therefore the
phase of $\delta S_{xy}$ changes by 180∘ depending on the direction of the
magnetic field, which was not specified in the paper. Here we assumed the
direction to be that resulting in the phase close to the experimentally-
observed phase of COs in the thermopower roughly reproduces the amplitude and
the phase of the COs presented in Fig. 4 of Ref. 9 including the magnetic-
field dependence of the amplitude, assuming the damping parameters to be
typical values for GaAs/AlGaAs 2DEGs,
$\mu_{\mathrm{b}}=2\mu_{\mathrm{c}}=2\mu_{\mathrm{h}}=6.0$ m2/(Vs).

##  V Conclusions

We have deduced asymptotic analytic expressions for the COs in
$\hat{\varepsilon}$ and $\hat{\lambda}$ with Eqs. (11) and (12), respectively,
from those of the electrical conductivity tensor $\hat{\sigma}$ in the
literature [7, 15] slightly modified by introducing an additional damping
factor $A[\pi/(\mu_{\mathrm{s}}B)]$
($\mathrm{s}=\mathrm{b},\mathrm{c},\mathrm{h}$) [Eq. (21)]. Employing
$\hat{\varepsilon}$ and $\hat{\lambda}$ thus obtained, we have further
calculated COs in the thermopower tensor $\hat{S}$ and the thermal
conductivity tensor $\hat{\kappa}$ with Eqs. (6) and (8), respectively. The
resulting analytic formulas for $\varepsilon_{ij}$, $\lambda_{ij}$, $S_{ij}$,
and $\kappa_{ij}$, namely, Eqs. (28), (37), (33), and (39), respectively,
allow us to easily find out how the amplitude and the phase of COs in these
transport coefficients vary with the temperature, the magnetic field, and the
parameters of the ULSL samples. The analytic expressions also reveal that the
temperature range for the Mott formula and the Wiedemann-Franz law to be
applicable is limited to $T\ll
T_{a}=\hbar\omega_{\mathrm{c}}ak_{\mathrm{F}}/(4\pi^{2}k_{\mathrm{B}})$. For
typical ULSL samples, this poses more stringent conditions compared to the
general condition $T\ll E_{\mathrm{F}}/k_{\mathrm{B}}$. We have also shown
that an empirical variant of the “Mott formula” obtained by replacing $d/dE$
with $(-B/2E_{\mathrm{F}})(d/dB)$ makes a fairly good approximation to the
genuine Mott formula.

For the COs in the thermopower tensor, we have found marked anisotropy between
the Seebeck components ($\delta S_{xx}\neq\delta S_{yy}$), in contradiction to
the previous theory [7]. We have also found that the amplitude of COs in one
of the Nernst components ($S_{xy}$) far exceeds those in the other components
($|\delta S_{xy}|\gg|\delta S_{xx}|,|\delta S_{yy}|,|\delta S_{yx}|$). The
origin of these properties can be traced back to the damping factor
$A[\pi/(\mu_{\mathrm{s}}B)]$, which accounts for the small-angle nature of the
scatterings in a device fabricated from a GaAs/AlGaAs 2DEG wafer and have
proven to be indispensable for achieving good agreement between the
theoretical and the experimentally obtained COs in the $\rho_{xx}$ [17] and
$\rho_{yx}$ [15]. Our recent experimental finding of the extremely small CO in
$\rho_{xy}$ [15] suggests much heavier damping of the collisional contribution
compared to the band contribution (quantified by
$\mu_{\mathrm{c}}=\mu_{\mathrm{h}}=\mu_{\mathrm{b}}/2$), which reinforces the
dominance of $|\delta X_{yy}|$ over $|\delta X_{xx}|$ and $|\delta X_{yx}|$
for $\hat{X}=\hat{\sigma}$, $\hat{\varepsilon}$, and $\hat{\lambda}$. Due to
the relation
$|\delta\varepsilon_{yy}|\gg|\delta\varepsilon_{xx}|,|\delta\varepsilon_{yx}|$,
the relative weight of the term $\rho_{\mathrm{H}}\delta\varepsilon_{yx}$,
which both $\delta S_{xx}$ and $\delta S_{yy}$ have in common [see Eqs. (33a)
and (33b)], is heavily diminished, leading to the prominent anisotropy. Noting
that $\rho_{\mathrm{H}}\gg\rho_{0}$, we can readily see that
$\rho_{\mathrm{H}}\delta\varepsilon_{yy}$ possesses by far the largest
amplitude among all the terms in Eq. (33), letting $\delta S_{xy}$, the only
component containing this term, far surpass $\delta S_{xx}$, $\delta S_{yy}$,
and $\delta S_{yx}$ in the amplitude. We stress that COs in ULSLs provide a
prototypical example of anisotropic systems lacking simple relations between
the two Nernst coefficients, $S_{xy}$ and $S_{yx}$. Finally, we have discussed
the difficulties that obstruct the attempts to experimentally determine
$\delta S_{ij}$’s and the possibility that observed COs are dominated by
$\delta S_{xy}$ regardless of the configurations of the voltage probes or the
orientation of the ULSL.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant Numbers 20K03817, 22H01805, and
19H00652.

*

##  Appendix A Electrical Conductivity at $T=0$ as a Function of the Energy

In this Appendix, we present, for completeness, the explicit expressions for
the electrical conductivity at $T=0$ as a function of the energy (spectral
conductivity), $\sigma_{ij,{T=0}}\left(E\right)$, used to calculate
$\varepsilon_{ij}$ and $\lambda_{ij}$ via Eqs. (11) and (12), respectively.
The band and collisional components for the diagonal components, corresponding
to Eqs. (16) and (18) at $T=0$, respectively, are

$\sigma_{yy,{T=0}}^{\mathrm{band}}(E)=\frac{2}{\pi}\frac{e^{2}}{h}\frac{{V_{0}}^{2}}{(\hbar\omega_{\mathrm{c}})^{2}}\tilde{\mu}|B|\alpha
E^{p-1/2}\left[1+A\left(\frac{\pi}{\tilde{\mu}_{\mathrm{b}}BE^{p_{\mathrm{b}}}}\right)\sin{\left(\frac{8\alpha\sqrt{E}}{\hbar\omega_{\mathrm{c}}}\right)}\right]$
(40)

and

$\sigma_{xx,{T=0}}^{\mathrm{col}}(E)=\frac{3}{4\pi}\frac{e^{2}}{h}\frac{{V_{0}}^{2}}{(\hbar\omega_{\mathrm{c}})^{2}}\frac{E^{-(p-1/2)}}{\tilde{\mu}|B|\alpha}\left[1-A\left(\frac{\pi}{\tilde{\mu}_{\mathrm{c}}BE^{p_{\mathrm{c}}}}\right)\sin{\left(\frac{8\alpha\sqrt{E}}{\hbar\omega_{\mathrm{c}}}\right)}\right],$
(41)

while the three constituents of the off-diagonal components corresponding to
Eqs. (20a), (20b), and (20c) are given by

$\sigma_{yx,{T=0}}^{\left(1\right)}(E)=\mathrm{sgn}(B)\frac{3}{\pi}\frac{e^{2}}{h}\frac{{V_{0}}^{2}}{(\hbar\omega_{\mathrm{c}})^{2}}\frac{\alpha}{\sqrt{E}}\left\\{1-A\left(\frac{\pi}{\tilde{\mu}_{\mathrm{h}}BE^{p_{\mathrm{h}}}}\right)\sin{\left[\frac{8\alpha\sqrt{E}}{\hbar\omega_{\mathrm{c}}}+\delta(E)\right]}\right\\},$
(42a)
$\sigma_{yx,{T=0}}^{\left(21\right)}(E)=-\mathrm{sgn}(B)\frac{2}{\pi}\frac{e^{2}}{h}\frac{{V_{0}}^{2}}{(\hbar\omega_{\mathrm{c}})^{2}}\sin{\left(\frac{\alpha}{\sqrt{E}}\right)}\left\\{\cos{\left[\frac{\delta(E)}{2}+\frac{\alpha}{\sqrt{E}}\right]}+A\left(\frac{\pi}{\tilde{\mu}_{\mathrm{h}}BE^{p_{\mathrm{h}}}}\right)\sin{\left[\frac{8\alpha\sqrt{E}}{\hbar\omega_{\mathrm{c}}}+\frac{\delta(E)}{2}-\frac{\alpha}{\sqrt{E}}\right]}\right\\},$
(42b) and
$\sigma_{yx,{T=0}}^{\left(22\right)}(E)=-\mathrm{sgn}(B)\frac{1}{\pi}\frac{e^{2}}{h}\frac{{V_{0}}^{2}}{\hbar\omega_{\mathrm{c}}E}\cos{\left(\frac{\alpha}{\sqrt{E}}\right)}\left\\{\sin{\left[\frac{\delta(E)}{2}+\frac{\alpha}{\sqrt{E}}\right]}-A\left(\frac{\pi}{\tilde{\mu}_{\mathrm{h}}BE^{p_{\mathrm{h}}}}\right)\cos{\left[\frac{8\alpha\sqrt{E}}{\hbar\omega_{\mathrm{c}}}+\frac{\delta(E)}{2}-\frac{\alpha}{\sqrt{E}}\right]}\right\\},$
(42c)

where $\alpha\equiv\pi\hbar/(a\sqrt{2m^{*}})$, $\delta(E)\equiv
2\cot^{-1}(8\alpha\sqrt{E}/\hbar\omega_{\mathrm{c}})$, and $\tilde{\mu}$ and
$\tilde{\mu}_{s}$ ($s=\mathrm{b},\mathrm{c},\mathrm{h}$) are defined by the
relations $\mu=\tilde{\mu}E^{p}$, $\mu_{s}=\tilde{\mu}_{s}E^{p_{s}}$.

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