# Magneto-Seebeck coefficient of Fermi-liquid in three-dimensional Dirac/Weyl
semimetal

F. R. Pratama1<EMAIL_ADDRESS>Riichiro Saito2 Nguyen T. Hung2,3
<EMAIL_ADDRESS>1Mathematics for Advanced Materials-OIL, AIST,
2-1-1 Katahira, Aoba, 980-8577 Sendai, Japan
2Department of Physics, Tohoku University, Sendai 980-8578, Japan
3Frontier Research Institute for Interdisciplinary Sciences, Tohoku
University, Sendai 980-8578, Japan

###### Abstract

We investigate dissipationless magneto-Seebeck effect in three-dimensional
Dirac/Weyl semimetal. The Hall resistivity $\rho_{yx}$ and thermoelectric Hall
coefficient $\alpha_{xy}$ exhibit plateaus at the quantum limit, where
electrons occupy only the zeroth Landau level. In this condition, quantum
oscillation in the Seebeck coefficient $S_{xx}\approx\rho_{yx}\alpha_{xy}$ is
suppressed, and the massless fermions are transformed into a Fermi liquid
system. We show that the Seebeck coefficient at the quantum limit is expressed
by the harmonic sum of Fermi wavelength and thermal de Broglie wavelength
scaled by magnetic length.

###### pacs:

72.20.Pa,72.10.-d,73.50.Lw

Research on thermoelectricity in low-dimensional materials is pioneered by the
theoretical works of Hicks and Dresselhaus [1, 2, 3]. They found that by
reducing the dimension of materials (essentially confining charge carriers in
one- or two-directions), larger Seebeck coefficient $S_{xx}$ (or thermopower)
and power factor can be obtained due to enhancement of electronic density of
states at the Fermi energy. Such quantum confinement effect takes place only
if the typical size of the material is smaller than thermal de Broglie (TDB)
wavelength [4, 5, 6]. In this Letter, we show that an analogous phenomenon
also occurs in three-dimensional (3D) Dirac/Weyl semimetal when we apply
magnetic field $B$. The confinement of massless fermions in a 3D Fermi pocket
by $B$ will induce the Fermi liquid behavior in the material.

An advantage of using a 3D Dirac/Weyl material in magneto-thermoelectric
devices was pointed out by Skinner and Fu [7]. They showed that a large, non-
saturating $S_{xx}$ as a function of the magnetic field is achieved in the
dissipationless limit. By keeping the number of carriers $n_{0}$ constant, the
thermoelectric Hall coefficient over temperature $\alpha_{xy}/T$ is
approaching a finite value that depends only on fundamental constants and
Fermi velocity [8, 9, 10]. The Hall resistivity $\rho_{yx}$ is expressed by
the classical formula $\rho_{yx}=B/(n_{0}e)$ ($e$ is the elementary charge),
and as consequence the $S_{xx}$ is linearly proportional to $B$,
$S_{xx}\approx\rho_{yx}\alpha_{xy}\propto B$ [11]. However, Galeski et al.
[12] recently observed saturating $S_{xx}$ in a nearly massless [13], 3D Dirac
semimetal ZrTe5 in magnetic field $B\approx 2~{}\mathrm{T}$, in which 3D
quantum Hall effect (QHE) occurs. They showed that the plateau of the Hall
resistivity is proportional to the half of the Fermi wavelength in the
direction parallel to the magnetic field, $\lambda_{F}^{\parallel}$.

Similar experimental results are reported by Tang et al. [14] in ZrTe5, and by
Wang et al. [15] in HfTe5. It is important to note that the QHE appears only
in two-dimensional (2D) electron systems when the Fermi level is located
between two Landau levels (LLs) [16, 17]. Thus, 3D materials normally do not
exhibit the QHE because of the continuous dispersion along the direction of
$B$. To explain the QHE in the 3D Dirac/Weyl semimetal, it was argued [14]
that the formation of charge density wave (CDW) [18, 19] due to the Fermi
surface instability plays a decisive role in discrete LL near the Fermi
energy. In other words, the CDW transforms the 3D system into a stack of
decoupled 2D layers, and each layer conducts the Hall current on the edges.
However, spectroscopic and transport measurements [12] do not show any signs
of CDW. Thus, it is necessary to confirm analytically whether the Hall plateau
arises from the intrinsic bulk properties of the 3D Dirac/Weyl fermion,
especially since recent calculations on the magneto-transport coefficients of
the Weyl semimetal with finite thickness [20, 21, 22, 23] do not show such
feature that $\rho_{yx}\propto\lambda_{F}^{\parallel}/2$.

In this Letter, we show that the saturated value of $S_{xx}$ becomes the
Seebeck coefficient of 3D Fermi liquid $\mathcal{S}$ [24]. In particular, we
show that the condition for $S_{xx}\equiv\mathcal{S}$ follows a scaling law
between three fundamental lengths: the magnetic length, the Fermi wavelength,
and the TDB wavelength in the direction perpendicular to $B$, where charge
carriers are magnetically confined. Further, we discuss that the condition
originates from the existence of the zeroth LL in the 3D Dirac/Weyl fermion,
which is independent of $B$, and thus can not be satisfied in electron gas in
conventional 3D metals or semiconductors.

The calculation of the Hall conductivity and thermoelectric Hall coefficient
is carried out within quantum edge formalism [25, 26, 27, 7, 8], where
transport scattering time $\tau$ is assumed to be small compared with the
inverse of cyclotron frequency $1/\omega_{c}$, i.e. $\omega_{c}\tau\gg 1$. In
this condition, the magnitude of the longitudinal conductivity $\sigma_{xx}$
is much smaller than $\sigma_{xy}$, which implies that the Hall resistivity is
approximately given by $\rho_{yx}\approx 1/\sigma_{xy}$, and
$S_{xx}\approx\rho_{yx}\alpha_{xy}$ [28]. Let us consider a 3D Weyl/Dirac
semimetal with a volume $L_{x}L_{y}L_{z}$ , where $L_{i}$ with $i=x,y,z$ is
the length of the material in the $i$-th direction. In the presence of an
external magnetic field in the $z$-direction $\mathbf{B}=B\hat{\mathbf{z}}$,
an electrons is confined to move on an orbit in the $xy$-plane, but move
freely in the $z$-direction with the Fermi velocity $v_{F}^{z}$. A potential
difference $V_{x}$ is applied between the right ($x=+L_{x}/2$) and left
($x=-L_{x}/2$) edges of the sample, which gives rise to the electric field
$\mathbf{E}=E_{x}\hat{\mathbf{x}}=-\partial V_{x}/\partial x$. In the presence
of both $B$ and $E_{x}$, the Landau level (LL) spectrum without the Zeeman
term is given by (see Sec. A in the Supplemental Material for the derivation):

$\displaystyle
E_{n}(B,k_{y},k_{z})=\varepsilon_{d}(B,k_{y})+\varepsilon_{n}(B,k_{z}),$ (1)

where $\varepsilon_{d}(k_{y})\equiv\hbar k_{y}v_{d}$ and
$\varepsilon_{n}(k_{z})\equiv\mathrm{sgn}(n)\sqrt{{\mathcal{E}_{B}}^{2}|n|/\gamma^{3}+\left(\hbar
v_{F}^{z}k_{z}/\gamma\right)^{2}}$. Because of $\varepsilon_{d}(k_{y})$, the
LL is dispersive along $k_{y}$, which allows charge carriers to move in the
$y$-direction with velocity
$v_{y}=\hbar^{-1}\partial\epsilon_{n}(k_{y},k_{z})/\partial k_{y}=v_{d}$,
where $v_{d}\equiv E_{x}/B$ is the drift velocity. $\varepsilon_{n}(k_{z})$
corresponds to the cyclotron energy, where $\mathcal{E}_{B}\equiv\sqrt{2}\hbar
v_{F}^{\perp}/\ell_{B}$, $\ell_{B}\equiv\sqrt{\hbar/(eB)}$ is the magnetic
length, and $v_{F}^{\perp}$ is the Fermi velocity in the direction
perpendicular to the magnetic field. Due to the relativistic nature of the
Dirac fermion, the effect of $E_{x}$ is also included in the Lorentz factor,
$\gamma\equiv 1/\sqrt{1-(v_{d}/v_{F}^{\perp})^{2}}$, which induces the
shrinking of the LL for $E_{x}\neq 0$ [29, 30]. Nevertheless, since we focus
on the case for strong $B$, we can take $\gamma\approx 1$. It is noted that
for $|k_{z}|\neq 0$, there are two possible values of the zeroth LL, i.e.
$E_{\pm 0}=\varepsilon_{d}(k_{y})\pm\hbar v_{F}^{z}k_{z}$.

By using the fact that $\ell_{B}\ll L_{z}$, $k_{z}=2\pi n_{z}/L_{z}$
($n_{z}\in\mathbb{Z}$) can be treated as a continuous variable
$\sum_{k_{z}}/L_{z}\rightarrow\int_{0}^{\infty}dk_{z}/\pi$, the density of
states per unit volume (DOS) for each spin direction is given by

$\displaystyle\mathcal{D}(E)=\frac{eB}{h}\sum_{k_{y}}\int\limits_{0}^{\infty}\frac{dk_{z}}{\pi}\sum_{n=-\infty}^{\infty}N_{n}\delta\left[E-E_{n}(k_{y},k_{z})\right],$
(2)

where we define $N_{n}\equiv 1/2~{}(1)$ for $n=0~{}(n\neq 0)$ to avoid double
counting of the zeroth LL that includes both $E_{+0}$ and $E_{-0}$. The number
of charge carriers $n_{0}$ is given by the difference between the number of
electrons and holes, as follows:

$\displaystyle
n_{0}=\int\limits_{0}^{\infty}\mathrm{d}E\mathcal{D}(E)f(E-\mu^{*})-\int\limits_{-\infty}^{0}\mathrm{d}E\mathcal{D}(E)[1-f(E-\mu^{*})],$
(3)

where $f(E-\mu^{*})\equiv 1/[e^{\beta(E-\mu^{*})}+1]$ is the Fermi-Dirac
distribution function, $\beta\equiv 1/(k_{B}T)$. The chemical potential
$\mu^{*}$ is given as a function of $x$ due to $V_{x}$:

$\displaystyle\mu^{*}(x)=\mu+{eV_{x}x}/{L_{x}},$ (4)

where $\mu$ is a constant which determines the average carrier density [31] in
the sample. Hereafter, we adopt the condition that $\mu$ in Eq. (4) is kept
constant with respect to magnetic field.

For a given $n_{0}$ in Eq. (3), the electrical $J_{y}$ and heat $J_{y}^{Q}$
currents are given by [8]

$\displaystyle\binom{J_{y}}{J_{y}^{Q}}=v_{y}n_{0}\binom{-e}{E-\mu^{*}}=-E_{x}\binom{\sigma_{xy}}{T\alpha_{xy}}.$
(5)

By considering only the currents along the edges $(x=\pm L_{x}/2)$,
$\sigma_{xy}$ and $\alpha_{xy}$ in Eq. (5) are given as a function of $\mu$
and $T$, as follows (see Secs. C and D in the Supplemental Material for the
derivation):

$\displaystyle\sigma_{xy}$ $\displaystyle=\frac{2e^{2}}{h}\sum\limits_{j=\pm
1}\sum\limits_{n=0}^{\infty}j{N}_{n}\int\limits_{0}^{\infty}\frac{dk_{z}}{\pi}f\left[\varepsilon_{n}(B,k_{z})-j\mu\right]$
$\displaystyle\equiv\sigma_{xy}^{(0)}+\sum\limits_{n=1}^{\infty}\sigma_{xy}^{(n)},$
(6)

and

$\displaystyle\alpha_{xy}=\frac{2ek_{B}}{h}\sum\limits_{j=\pm
1}\sum\limits_{n=0}^{\infty}{N}_{n}\int\limits_{0}^{\infty}\frac{dk_{z}}{\pi}\left\\{\ln\left[1+e^{-\beta\left\\{\varepsilon_{n}(B,k_{z})-j\mu\right\\}}\right]+\frac{\beta\left[\varepsilon_{n}(B,k_{z})-j\mu\right]}{1+e^{\beta\left[\varepsilon_{n}(B,k_{z})-j\mu\right]}}\right\\}\equiv\alpha_{xy}^{(0)}+\sum\limits_{n=1}^{\infty}\alpha_{xy}^{(n)},$
(7)

where $\sigma_{xy}^{(n)}$ and $\alpha_{xy}^{(n)}$ denote, respectively, the
Hall conductivity and thermoelectric Hall coefficient for $n$-th LL. The
prefactor of 2 in Eqs. (6) and (7) is given by assuming that filling factor of
the LL at $x=\pm L_{x}/2$ are the same [32, 33], where almost symmetrical
currents at the both edges flow in the same direction [31, 34]. The index
$j=+1$ and $-1$ indicate the contributions of electron and hole, respectively.
$\sigma_{xy}^{(0)}$ and $\alpha_{xy}^{(0)}$ are the Hall conductivity and
thermoelectric Hall coefficient for the zeroth LL, which are obtained
analytically as follows:

$\displaystyle\sigma_{xy}^{(0)}=\frac{2e^{2}}{h^{2}v_{F}^{z}\beta}\sum\limits_{j=\pm
1}j\mathrm{ln}\left(1+e^{j\beta\mu}\right),$ (8)

and

$\begin{split}\alpha_{xy}^{(0)}=\frac{-2ek_{B}}{h^{2}v_{F}^{z}}\sum\limits_{j=\pm
1}\Bigg{[}\frac{2}{\beta}\mathrm{Li}_{2}(-e^{j\beta\mu})+j\mu\ln{(1+e^{j\beta\mu})}\Bigg{]},\end{split}$
(9)

where $\mathrm{Li}_{2}(z)\equiv\sum_{k=1}^{\infty}z^{k}/k^{2}$ is the
dilogarithm function.

Hereafter, we adopt the parameters for calculating $\rho_{yx}=1/\sigma_{xy}$
and $\alpha_{xy}$ from the experimentally determined values of 3D Dirac
material ZrTe5 [12]. The material has an orthorombic crystal structure and
anisotropic Fermi velocities $v_{F}^{i}$ along $i=a,~{}b,$ and $c$-axis, where
$v_{F}^{a}=1.164\times 10^{5}$ m/s, $v_{F}^{b}=1.534\times 10^{4}$ m/s, and
$v_{F}^{c}=3.489\times 10^{5}$ m/s. By aligning $B$ parallel to the $b$-axis,
the Fermi wavevector $k_{F}^{b}=72.9\times 10^{-3}\mathrm{\AA}^{-1}$ is
extracted from the Hall plateau in the quantum limit, where only the zeroth LL
is occupied. In order to reproduce experimental measurement of $\rho_{yx}$
with our model, we use $k_{F}^{z}=k_{F}^{b}$, $v_{F}^{z}=v_{F}^{b}$,
$v_{F}^{\perp}=\sqrt{v_{F}^{a}v_{F}^{c}}=2.015\times 10^{5}$ m/s, and
$\mu=\hbar v_{F}^{z}k_{F}^{z}=7.36$ meV.

Figure 1: Log-log plot of $\rho_{yx}$ as a function of $B$ for a constant
$\mu=7.36~{}\mathrm{meV}$, for several values of $T$. Inset shows the linear
plot of $\rho_{yx}$ with $B=0-2~{}\mathrm{T}$.

In Fig. 1, we show a log-log plot of $\rho_{yx}$ as a function of $B$ for
$T=0.6,2,4.2,7$ and $10~{}\mathrm{K}$. For all temperatures, the quantum Hall
regime occurs at $B\sim~{}1~{}T$, as shown in the inset of Fig. 1. It is noted
that the quantization of $\rho_{yx}$ is not as pronounced as in the two-
dimensional case, due to the logarithmic singularity of the DOS upon the
integration on $k_{z}$ (see Sec. B in the Supplemental Material). As we
further increase $B$, $\rho_{yx}$ becomes independent of $B$ and $T$, which
indicates that all $n>0$ LLs are depopulated. As shown in the inset of Fig. 1,
the Hall plateau at $\rho_{xy}\approx 11.6~{}\mathrm{m}\Omega\cdot\mathrm{cm}$
occurs for $B\geq 1.2~{}\mathrm{T}$. The present calculation reproduces recent
experiments by Tang[14] and Galseki [12]. The good agreement can be attributed
to small effective mass in $\mathrm{ZrTe_{5}}$ [12, 35, 14, 36, 37] (in the
order of $\sim 0.01m_{e}$, where $m_{e}$ is the mass of free electron), and
because there is only one spin degeneracy for each zeroth LL [13, 38, 39].

Figure 2:  Log-log plot of (a) $\alpha_{xy}$ and (b) $S_{xx}$ as a function of
$B$ for several values of $T$. The chemical potential is set by
$\mu=7.36~{}\mathrm{meV}$. Grey circles indicates the field
$B=\mathcal{B}(\mu,T)$ where $\alpha_{xy}$ and $S_{xx}$ begin to saturate.

Let us discuss the quantum limit case, in which
$\rho_{yx}(\mu,T)=1/\sigma_{xy}^{(0)}(\mu,T)$. From Eq. (8), the term at
$j=-1$ vanishes for low temperature ($|\mu\beta|\gg 1$), which means that only
electrons contribute to the constant $\rho_{yx}$ for $\mu>0$ (and holes for
$\mu<0$ but with opposite sign of $\rho_{yx}$). By using
$\mathrm{ln}(1+e^{\beta\mu})\approx\beta\mu$, $\rho_{yx}$ is independent of
$T$ as follows:

$\displaystyle\rho_{xy}=\frac{2e^{2}\mu}{h^{2}v_{F}^{z}}=\frac{\pi
h}{e^{2}k_{F}^{z}}=\frac{h\lambda_{F}^{z}}{2e^{2}},$ (10)

where $\lambda_{F}^{z}=2\pi/k_{F}^{z}\equiv\lambda_{F}^{\parallel}$. It is
noted that the magnitude of the Hall plateau shown by Eq. (10) has been
confirmed experimentally in the samples of $\mathrm{ZrTe_{5}}$ with thickness
$L_{z}\approx 100~{}\mathrm{\mu m}$, while an experiment on the 2D Dirac
semimetal $\mathrm{Cd_{3}As_{2}}$ [40] with $L_{z}\approx 60-70~{}\mathrm{nm}$
does not indicate such behaviour. Thus, we show that Eq. (10) is indeed an
intrinsic property of 3D Dirac semimetal.

In Figs. 2 (a) and (b), we show log-log plots of $\alpha_{xy}$ and
$S_{xx}=\alpha_{xy}/\sigma_{xy}=\rho_{yx}\alpha_{xy}$ as a function of $B$ for
$T=0.6,2,4.2,7$ and $10$ K, respectively. We can see that for $B\leqslant
1~{}\mathrm{T}$, $\alpha_{xy}$ decreases linearly with increasing $B$, which
means that $\alpha_{xy}$ is inversely proportional to $B$ in the linear scale.
On the other hand, $S_{xx}$ does not show significant dependence on $B$. As
$B$ becomes larger, oscillations appears in both $\alpha_{xy}$ and $S_{xx}$
for $T\leqslant 4.2~{}\mathrm{K}$ since a few LLs touch the chemical
potential. Due to the smearing of the Fermi-Dirac distribution function by
increasing $T$, the individual peaks in the oscillation become
indistinguishable. As the quantum limit takes place, $\alpha_{xy}$ and
$S_{xx}$ become constants for all temperatures. By applying identity
$\mathrm{Li}_{2}(z)+\mathrm{Li}_{2}(1/z)=-\pi^{2}/6-[\mathrm{ln}(-z)]^{2}/2$
on Eq. (9), the explicit formula for thermoelectric Hall plateau
$\alpha_{xy}=\alpha_{xy}^{(0)}$ is given as follows:

$\displaystyle\alpha_{xy}=\frac{2\pi^{2}}{3}\frac{{ek_{B}}^{2}}{h^{2}v_{F}^{z}}T.$
(11)

Eq. (11) was first obtained by Kozii et al. [8] as the saturating limit for
$\alpha_{xy}$ of an ideal 3D Dirac material at a large $B$, with a fixed
number of carriers. We complement the result by showing that exactly the same
formula also prevails for a fixed chemical potential. By combining Eqs. (10)
and (11), $S_{xx}$ at the quantum limit is given by

$\displaystyle
S_{xx}=\frac{\pi^{2}}{3}\frac{{k_{B}}^{2}}{e}\frac{T}{\mu}\equiv\mathcal{S},$
(12)

where $\mathcal{S}$ is the expression for the Seebeck coefficient of the Fermi
liquid system [24].

Figure 3:  Plot of the scaled Seebeck coefficient $S_{xx}/\mathcal{S}$ as a
function of $\lambda_{H}/\ell_{B}$ for constant $\mu=7.36~{}\mathrm{meV}$ for
several values of $T$ (straight lines). Inset: $S_{xx}/\mathcal{S}$ at a
constant $T=3~{}\mathrm{K}$ for several values of $\mu$.

Now, let us discuss the onset of magnetic field, $\mathcal{B}$, in which the
3D Dirac/Weyl material behaves as the Fermi liquid system at
$\mathcal{B}(\mu,T)$ (open circles in Figs. 2). It is noted that in the
quantum limit, the uncertainty principle determines that the radius of
electron’s orbit is given by the magnetic length $\ell_{B}$. Therefore, it is
useful to express the dependence of $\mathcal{B}$ in term of the length scales
associated with $\mu$ and $T$, in the direction perpendicular to $B$ where
magnetic confinement of electrons occurs. At $T=0~{}\mathrm{K}$, the plateau
is formed when the magnetic energy is larger than the chemical potential, i.e.
$\mathcal{E}_{B}\geqslant\mu$ with $\mu=\hbar
v_{F}^{\perp}k_{F}^{\perp}=hv_{F}^{\perp}/\lambda_{F}^{\perp}$. Thus, near the
absolute zero, $S_{xx}$ becomes $\mathcal{S}$ when the ratio of the two length
scales satisfies $\ell_{B}/\lambda_{F}^{\perp}\leqslant 1/(\sqrt{2}\pi)\approx
0.225$.

For given a finite $T$, $\sigma_{xy}$ and $\alpha_{xy}$ are obtained by a
convolution of $-\partial f(E-\mu,T)/\partial
E=(\beta/4)\mathrm{sech}^{2}[\beta(E-\mu)/2]$ with those quantities at
$T=0~{}\mathrm{K}$ (see Supplemental material). Thus, the shift from
$\ell_{B}/\lambda_{F}^{\perp}$ is given by $W=2k_{B}T/\mathcal{E}_{B}$,
resulting in Eq. (12) holds if following condition is satisfied:

$\displaystyle\frac{\ell_{B}}{\lambda_{F}^{\perp}}+W\leqslant\frac{1}{\sqrt{2}\pi}.$
(13)

The length scale associated with the temperature is thermal de Broglie
wavelength [6]. We adopt a definition of the thermal de Broglie wavelength
[41] which is suitable for massless particle $\Lambda^{\perp}\equiv
hv_{F}^{\perp}/(2\pi^{1/3}k_{B}T)$. Equivalently,
$W=\sqrt{2}\pi^{2/3}\ell_{B}/\Lambda^{\perp}$. By defining $1/\lambda_{H}$ as
a weighted harmonic sum of $\lambda_{F}^{\perp}$ and $\Lambda^{\perp}$:

$\displaystyle\frac{1}{\lambda_{H}}=\frac{\sqrt{2}\pi}{\lambda_{F}^{\perp}}+\frac{2\pi^{5/3}}{\Lambda^{\perp}},$
(14)

the condition for obtaining $S_{xx}=\mathcal{S}$ is given by
${\lambda_{H}}/{\ell_{B}}\geqslant 1$. In Fig 3, we plot $S_{xx}/\mathcal{S}$
with $\mu=7.36~{}\mathrm{meV}$ for $T=2,~{}4.2$, and $10~{}\mathrm{K}$. Here,
we can see the suppression of quantum oscillation for
${\lambda_{H}}/{\ell_{B}}\geqslant 1$. In Fig 3, $\Lambda^{\perp}$ is varying
while $\lambda_{F}^{\perp}$ remains constant. For the opposite condition, we
plot $S_{xx}/S$ for $\mu=6,~{}9$, and $12~{}\mathrm{K}$ at a constant
$T=3~{}\mathrm{K}$ (see inset plot in Fig 3), where similar behaviour is
observed. This phenomenon shows that the confinement within the length of
$\ell_{B}$ will transform the 3D Dirac fermions into a Fermi liquid system. In
other words, the states of the Dirac fermions at the zeroth LL become
localized in the $k$-space within a small Fermi surface (due to low carrier
density $n_{0}$), which does not change with keeping chemical potential
constant. In contrast, for an ordinary massive 3D electron gas with a fixed
$\mu$, we show in the Section E of Supplemental material that $\rho_{yx}$
tends to increase after $n=1$ LL is emptied, while $\alpha_{xy}$ becomes zero.
This phenomenon is unlikely to occur because of large $n_{0}$, which tends to
be preserved in order to minimize the change of energy [12].

Finally, we have to consider the ranges of $B$ and $T$ in which the Fermi
liquid phase can survive. For $\lambda_{H}/\ell_{B}\leqslant 1.2$ in Fig. 3,
the maximum value of $B$ is around $6~{}\mathrm{T}$ for $T\leqslant
10~{}\mathrm{K}$ and $\mu\leqslant 12~{}\mathrm{meV}$. Around this $B$ and
higher, the scaling law may be broken if the Zeeman term becomes more
noticeable, especially if the material has a large $g$-factors. This may
explain the increase of $\rho_{xy}$ in $\mathrm{ZrTe_{5}}$ [12, 14] and
$\mathrm{HfTe_{5}}$ [15] for $B\geqslant 3$ T though the Hall plateaus are
observed for $B<3$ T. In these materials, the value of $g$ is in the order
$\sim 10$ [12, 39, 35, 42]). It also has been observed that large $B$ can
drive phase transitions such as band inversion [38, 43], anomalous QHE [42],
as well as fractional QHE [14] and mass generation [44] due to many body
interactions. Also at higher temperature, the transport becomes dissipative
due to scattering effects [9], where $\rho_{xx}$ increases, and as a
consequence, the relation $S_{xx}\approx\rho_{yx}\alpha_{xy}$ is no longer
valid. Nevertheless, our proposed scaling law for an ideal 3d Dirac material
may be observed experimentally within proper ranges of $B$ and $T$.

In summary, we have analytically shown that at a fixed chemical potential,
both the Hall resistivity and thermoelectric Hall coefficient show plateau
structures. In this condition, magnetic confinement of carriers occupying the
zeroth LL transform the 3D Dirac/Weyl fermion into the Fermi liquid.
Particularly, the threshold of this phenomenon can be parameterized by
magnetic length , the Fermi wavelength, and the thermal de Broglie wavelength.
Our ﬁndings establish a relationship between the three fundamental lengths in
the quantum Hall and thermoelectric Hall effects in the 3D Dirac/Weyl
semimetals.

## Acknowledgments

N.T.H. acknowledges JSPS KAKENHI (Grant No. JP20K15178) and the financial
support from the Frontier Research Institute for Interdisciplinary Sciences,
Tohoku University. R.S. acknowledges JSPS KAKENHI Grant No JP18H01810 and
CSIS, Tohoku University.

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