# Extraordinary Thermoelectric Properties of Topological Surface States in
Quantum-Confined Cd3As2 Thin Films

Wenkai Ouyang  Department of Mechanical Engineering, University of California,
Santa Barbara, CA 93106, USA  Alexander C. Lygo  Materials Department,
University of California, Santa Barbara, CA 93106, USA  Yubi Chen  Department
of Mechanical Engineering, University of California, Santa Barbara, CA 93106,
USA  Department of Physics, University of California, Santa Barbara, CA 93106,
USA  Huiyuan Zheng  Department of Physics, the University of Hong Kong, Hong
Kong, 999077, China  Dung Vu  Department of Mechanical and Aerospace
Engineering, the Ohio State University, Columbus, OH 43210, USA  Brandi L.
Wooten  Department of Mechanical and Aerospace Engineering, the Ohio State
University, Columbus, OH 43210, USA  Xichen Liang  Department of Chemical
Engineering, University of California, Santa Barbara, CA 93106, USA  Wang Yao
Department of Physics, the University of Hong Kong, Hong Kong, 999077, China
Joseph P. Heremans<EMAIL_ADDRESS>Department of Mechanical and Aerospace
Engineering, the Ohio State University, Columbus, OH 43210, USA  Susanne
Stemmer<EMAIL_ADDRESS>Materials Department, University of California,
Santa Barbara, CA 93106, USA  Bolin Liao<EMAIL_ADDRESS>Department of
Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA

(August 28, 2024)

###### Abstract

Topological insulators and semimetals have been shown to possess intriguing
thermoelectric properties promising for energy harvesting and cooling
applications. However, thermoelectric transport associated with the Fermi arc
topological surface states on topological Dirac semimetals remains less
explored. In this work, we systematically examine thermoelectric transport in
a series of topological Dirac semimetal Cd3As2 thin films grown by molecular
beam epitaxy. Surprisingly, we find significantly enhanced Seebeck effect and
anomalous Nernst effect at cryogenic temperatures when the Cd3As2 layer is
thin. Combining angle-dependent quantum oscillation analysis,
magnetothermoelectric measurement, transport modelling and first-principles
simulation, we isolate the contributions from bulk and surface conducting
channels and attribute the unusual thermoeletric properties to the topological
surface states. Our analysis showcases the rich thermoelectric transport
physics in quantum-confined topological Dirac semimetal thin films and
suggests new routes to achieving high thermoelectric performance at cryogenic
temperatures.

Topological Dirac Semimetal, Thermoelectric Effect, Topological Surface State,
Quantum Confinement

##  I Introduction

Topological insulators and semimetals have recently attracted intense research
interests due to their exotic bulk transport properties, such as the chiral
anomaly and topologically protected surface states [1, 2]. Additionally, their
unusual thermal and thermoelectric transport properties have also offered
opportunities for potential thermoelectric energy conversion applications [3,
4]. For example, topological insulators and topological semimetals typically
feature linear or close-to-linear electronic bands near the Fermi level,
leading to a high charge carrier mobility [5] that is beneficial for
thermoelectrics. It has been experimentally shown that the thermoelectric
performance of lead selenide (PbSe) can be enhanced by a pressure-driven
topological phase transition, where the formation of linear Dirac bands
improves the electrical conductivity [6]. Furthermore, both theory [7] and
experiments [8, 9] have established that the Seebeck coefficient of
topological semimetals can increase with the magnetic field in the quantized
regime without saturation. On the thermal transport side, first-principles
simulations and inelastic neutron and x-ray scattering measurements have
suggested significant phonon softening in topological semimetals due to the
Kohn anomaly, leading to reduced thermal conductivity [10, 11, 12]. These
properties are all promising features for improving thermoelectric energy
conversion efficiency. So far, however, the explored thermoelectric properties
of topological insulators and semimetals are mostly limited to those in
intrinsic bulk crystals, while epitaxial thin films remain less examined.
Compared with bulk crystals, epitaxial thin films offer more degrees of
control, such as strain and quantum confinement, which can be further tuned to
optimize their thermoelectric properties. In addition, topological surface
states also contribute more prominently in thin films. In this work, we focus
on detailed measurements and modeling of thermoelectric properties of
epitaxial thin films of a topological Dirac semimetal Cd3As2, aiming to
isolate the impacts of surface states and quantum confinement.

Cd3As2 is an archetypal topological Dirac semimetal that exhibits a three-
dimensional linear dispersion relation and topologically protected surface
states [13, 14, 15, 16]. The linear dispersion relation and a large Fermi
velocity give rise to an electron mobility as high as 20,000
$\mathrm{cm^{2}}$/V-s at room temperature [5] in thin films grown by molecular
beam epitaxy (MBE). In addition, Cd3As2 has a lattice thermal conductivity
[17, 18, 19] around 0.3 $\sim$ 0.7 W/m-K at 300 K, which is anomalously low
for single crystals. Using first-principles lattice dynamics calculations, Yue
et al. [10] uncovered the presence of a cluster of soft optical phonon modes
in Cd3As2, which significantly enhance the scattering phase space of acoustic
phonons leading to a low lattice thermal conductivity. The combination of a
high electron mobility and a low thermal conductivity makes it a highly
promising candidate for thermoelectric applications. However, as a topological
Dirac semimetal, Cd3As2 has no bandgap in its bulk form, which is typically
detrimental to the Seebeck coefficient. As a result, a moderate Seebeck
coefficient ranging from 50 $\mu\mathrm{V/K}$ to 150 $\mu\mathrm{V/K}$ [20,
21, 19, 22, 23] was recorded experimentally in Cd3As2 bulk crystals.

To overcome this limitation, a bandgap can be opened in Cd3As2 through either
crystal symmetry breaking or quantum confinement that can be realized in
epitaxially grown thin films or heterostructures [14, 24, 25, 26, 27].
Specifically, it has been shown experimentally that in Cd3As2 thin films grown
by MBE, a bulk band gap emerges below a certain film thickness due to quantum
confinement, where the low temperature electrical transport properties are
dominated by mobile carriers restricted to two-dimensional surface states
[28]. The surface states in topological Dirac semimetal Cd3As2 consists of two
sets of Fermi arcs [29] [except for the (001) surface where the projection of
the bulk Dirac nodes becomes a point] that are capable of producing quantum
oscillations and the quantum Hall effect [30, 31], which has been
experimentally observed in MBE-grown films with a thickness below 70 nm [28,
32]. These surface states have been theoretically predicted to persist even in
thin films with a few nm thickness [29]. Furthermore, the Dirac dispersion of
surface states on (112) surfaces in MBE-grown Cd3As2 has been verified through
gating experiments [33]. However, thermoelectric transport properties of these
topological surface states are less explored and the impact of quantum
confinement on thermoelectric properties has not been examined experimentally.

In this work, we systematically investigate the thermoelectric transport in
high-quality Cd3As2 thin films grown by MBE. To resolve the contribution from
different conducting channels, we studied samples with varying Cd3As2
thicknesses: 950 nm in the bulk limit, 95 nm in the transition regime, and 25
nm with quantum confinement and prominent contributions from the surface
states. Surprisingly, we observed anomalously enhanced Seebeck and Nernst
coefficients in the 25 nm sample at cryogenic temperatures. Combining
extensive magneto-Seebeck and Nernst effect measurements, detailed angle-
dependent quantum oscillation characterization, analytical transport models
and first-principles density functional theory (DFT) simulations, we attribute
the observed thermoelectric properties to the interplay of bulk and surface
conducting channels in parallel. Our work provides a comprehensive
understanding of thermoelectric transport in confined topological Dirac
semimetal thin films and reveals the rich physics in these systems that can be
potentially employed to engineer composite materials for thermoelectric
applications.

##  II Results and Discussion

###  II.1 Temperature dependence of resistivity and Seebeck coefficient

The MBE technique was employed to achieve the epitaxial growth of
(112)-oriented Cd3As2 thin films on (111)-oriented GaAs substrates with a
150-nm (111) GaSb buffer layer [5]. More details of the growth method and the
measurement technique can be found in the Methods section. Characterization
results of the bare GaSb buffer layer is provided in Supplementary Note 1 in
the Supplementary Information (SI). Fig. 1a presents a schematic illustration
of the sample structures studied in this work as well as a picture of the
measurement device. In our experiment, three different thicknesses of Cd3As2
were prepared: 950 nm, 95 nm and 25 nm, with the same GaSb buffer layer and
substrate conditions. We refer to them as “950 nm sample”, “95 nm sample”, and
“25 nm sample”, respectively, in the following discussion.

The measured electrical resistivity of the three samples as a function of
temperature is shown in Fig. 1b. The resistivity of the 950-nm sample exhibits
a typical semimetallic behavior, where it increases with temperature. This
behavior reflects the bulk Cd3As2 property. In contrast, the 95-nm sample
shows a more pronounced semiconductor-like behavior with a decreasing
resistivity with temperature. Low-field Hall results of the two samples are
provided in the Supplementary Note 2 in the SI, indicating n-type conduction
that originates from the bulk Cd3As2 states. A very high Hall mobility up to
Therefore, the semiconductor-like behavior in the 95-nm sample suggests the
onset of bulk bandgap opening due to either weak quantum confinement or strain
that breaks the four-fold rotation symmetry around (001) in Cd3As2 protecting
the Dirac nodes [29], where the thermal activation of carriers results in a
decrease in the resistivity with an increasing temperature [34, 21]. For the
25-nm sample, a modest variation in resistivity is noted with an increasing
temperature, suggesting a delicate interplay between the semiconductor-like
behavior of the bulk states and, potentially, the metallic surface states
[34].

In Fig. 1c, we present the temperature-dependent Seebeck coefficient for the
three samples. Specifically, for the 950-nm sample, we observed an increase in
the Seebeck coefficient with temperature, where the negative sign suggests the
prevalence of n-type carriers [23, 21]. Meanwhile, for the 95-nm sample, the
Seebeck coefficient exhibits a similar temperature dependence above 50 K. In
the temperature range above 50 K, the Seebeck coefficient of both samples can
be well described by our DFT simulation using the bulk band structure of
Cd3As2 assuming a Fermi level slightly into the conduction band (details of
the DFT simulation are given in Methods and Supplementary Note 3 in the SI).
However, around 50 K, a change in the sign of the Seebeck coefficient occurs,
suggesting a transition of the dominant carriers from n-type to p-type. At
even lower temperatures, we also observed an increase of the positive Seebeck
coefficient. Both features cannot be explained by our DFT simulation
considering only the Cd3As2 bulk states. More interestingly, for the 25-nm
sample, a positive Seebeck coefficient was observed in the entire temperature
range, and the Seebeck coefficient showed a metallic feature with a negative
temperature dependence above 50 K, contrary to the behavior observed in the
other two samples. Strikingly, the Seebeck coefficient displayed a sharp
increase below 50 K, peaked at 5 K with a value of nearly 500
$\mu\mathrm{V/K}$. This surprising behavior was in sharp contrast to the other
two samples with thicker Cd3As2 thin films. Furthermore, the distinct
behaviors of the Seebeck coefficient in the three samples suggest the
existence of contributions from more than one conducting channels in the
thinner samples, thereby requiring further investigations as detailed in
subsequent sections.

###  II.2 Angle-dependent quantum oscillations

In order to comprehensively investigate the underlying mechanisms responsible
for the distinct thermoelectric behaviors of the three heterostructure samples
with different Cd3As2 thicknesses, we performed systematic measurements of
angle-dependent quantum oscillations in both the electrical resistivity (the
Shubnikov-de-Haas, or SdH, oscillation) and the Seebeck coefficient of the
three samples at 2 K. Quantum oscillations of electrical transport properties
as a function of the magnetic field are manifestations of the Fermi surface
quantization, where the oscillation frequency as a function of the angle of
the magnetic field can provide detailed information of the Fermi surface [35,
36, 37]. Figure 2a presents the Seebeck voltage oscillation of the three
samples when the magnetic field is perpendicular to the sample surface (“0
degree” direction) at 2 K. The Seebeck voltage is plotted against the inverse
magnetic field ($1/B$), from which the oscillation frequencies are obtained
through a Fourier transform (FT). The three samples exhibit distinct
oscillation frequencies, signaling different origins of the observed
thermoelectric transport [38, 36, 39]. Specifically, for the 950 nm sample, a
single prominent oscillation frequency is detected at approximately 20 T, as
indicated by peaks A-B labeled in Fig. 2a. The 95 nm sample shows two
oscillation frequencies: one is around 20 T, also denoted by peaks A-B, while
the other is around 55 T, corresponding to peaks C-D in Fig. 2a. Similarly,
the 25-nm sample also displays these two oscillation frequencies.

To further pinpoint the conducting channels corresponding to these quantum
oscillation frequencies, we conducted a systematic investigation of the SdH
and Seebeck coefficient quantum oscillation frequencies as a function of the
angle of the applied magnetic field in all three samples. Figures 2b and 2d
show the results and a comparison of these results with the DFT calculation
considering only the bulk Cd3As2 states. According to the Onsager relation
[40], the oscillation frequencies are proportional to extremal areas of the
Fermi surface perpendicular to the applied magnetic field. Figure 2b
illustrates the variation of the SdH oscillation frequency as a function of
the magnetic field angle in 950-nm and 95-nm samples, respectively. The
oscillation frequency is observed to undergo slight variations with angle in
both samples, with the 950 nm sample exhibiting a concave shape centered
around 90∘ and the 95 nm sample a convex shape. From Fig. 2b, our DFT
calculation predicts an oscillation frequency around 16 T at zero degree field
angle, assuming bulk Cd3As2 states and a Fermi level $\sim 12$ meV above the
charge neutral point (CNP). The Fermi level position is deduced from the Hall
carrier concentration measurement of the 950 nm sample (see Supplementary Note
3 in the SI). Additionally, the DFT simulation also reproduces the concave
trend shown in Fig. 2b for the 950 nm sample. Specifically, at the predicted
Fermi level position, a Lifshitz transition occurs and the two spherical Fermi
surfaces surrounding the two Dirac points along the $\Gamma$-Z direction merge
to form a peanut-shaped Fermi surface, as depicted in Fig. 2c. This Fermi
surface shape accounts for the concave trend observed in the 950 nm sample.
The agreement between the 950 nm sample data and the DFT simulation further
suggests that the thermoelectric transport in the 950 nm sample mainly arises
from the n-type Cd3As2 bulk states. Similarly, the 95 nm sample shows a SdH
frequency at the 0∘ field angle in agreement with the DFT calculation,
indicating the contribution from the Cd3As2 bulk states. However, the angle-
dependent trend observed in the 95 nm sample deviates from that in the 950 nm
sample, which can be potentially attributed to the onset of the quantum
confinement effect that can vary the shape of the Fermi surface as well as the
position of the Fermi level. Unfortunately, in the 95 nm sample, the
additional high-frequency SdH oscillation at 55 T is weak and we could not map
its angle-dependence reliably.

For the 25 nm sample, it turned out difficult to obtain reliable angle-
dependent SdH data with multiple oscillation frequencies. Instead, we found
that the quantum oscillation observed in the Seebeck coefficient as a function
of the magnetic field is much clearer and provides more reliable extraction of
the oscillation frequencies. Therefore, we conducted systematic angle-
dependent Seebeck oscillation measurements on the 25 nm sample and the results
are shown in Fig. 2d. One of the observed oscillation frequencies near 20 T is
nearly independent of the magnetic field direction, signaling its origin in
the Cd3As2 bulk states, which are expected to form closely spaced subbands due
to quantum confinement. In addition, the 25 nm sample exhibits another high
oscillation frequency around 55 T at 0∘ field direction, which also appears in
the 95 nm sample. The angle-dependent reduction in this oscillation frequency
culminating at 90∘ field direction implies that this frequency is associated
with two-dimensional (2D) surface states, since the Fermi surface associated
with the 2D surface states has zero intersection with the magnetic field at
90∘ field direction, as illustrated in Fig. 2e. The observation of this
higher-frequency oscillation and its field-angle dependence is consistent with
previous quantum oscillation measurements of Cd3As2 thin films [41]. Based on
this observation, the behavior of the Seebeck coefficient of the 95 nm sample
shown in Fig. 1c can be understood. Above 50 K, the main contribution to the
Seebeck coefficient in the 95 nm sample comes from the bulk Cd3As2 states,
similarly as the 950 nm sample. Below 50 K, the bulk states in the 95 nm
sample start to freeze out due to the onset of gap opening, while the
contribution from the surface states emerges and is responsible for the sign
change of the Seebeck coefficient. This view is also consistent with the
p-type character of the surface states observed in previous measurements of
Cd3As2 thin films when the surface Fermi level was low, due to either surface
depletion or electrostatic gating [34, 33]. In contrast, the Seebeck
coefficient of the 25 nm sample is positive throughout the entire temperature
range, suggesting that the bulk Cd3As2 states, presumably n-type, play a minor
role in thermoelectric transport even up to 300 K. Instead, the contribution
of the p-type surface states is largely amplified in the 25 nm sample compared
with the 95 nm sample due to its reduced thickness and stronger quantum
confinement that gaps out the bulk states. In the following, we provide
detailed magnetothermoelectric transport measurements and model analysis to
further quantify the contribution from surface states.

###  II.3 Magnetothermoelectric transport measurements and modeling

Based on the quantum oscillation analysis, we have established that the
thermoelectric transport in the 950-nm sample is dominated by the n-type bulk
Cd3As2 states, while additional contribution from the surface states play a
role below 50 K in the 95-nm sample. In the 25-nm sample, the contribution
from the surface states is significant even up to 300 K. To further verify and
quantify our interpretation, we discuss detailed magnetotransport
measurements, including Hall, Seebeck and Nernst coefficients, and the
associated transport models in this section. We focus on analyzing the 25-nm
sample here, while data of the 950-nm sample and the 95-nm sample are provided
in the Supplementary Note 2 in the SI. All measurements discussed in this
section were done with a magnetic field perpendicular to the sample surface.
Firstly, the magnetoresistance and the Hall resistivity data (Supplementary
Note 2 in the SI) of the 25 nm sample exhibit a typical two-carrier behavior
with contributions from two groups of carriers with opposite charge. In
particular, a group of higher-mobility n-type carriers dominate the small-
field Hall resistivity, while a group of higher-density p-type carriers
dominate the large-field Hall resistivity. To simultaneously incorporate the
magnetoresistance and the Hall resistivity, we used a two-band Drude model to
fit the Hall resistivity $\rho_{xy}$ data as a function of temperature and
magnetic field, as shown in Fig. 3a, where we presented results at three
representative temperatures. Details of the model and additional data are
provided in Supplementary Note 2 of the SI. The semiclassical Drude model is
valid for low magnetic fields before quantum oscillations appear, so we
focused on analyzing the magnetotransport data below 4 T. We used the carrier
concentration and the carrier mobility of both groups of carriers as the
fitting parameters here. To increase the fitting sensitivity, we used the same
group of parameters to simultaneously fit the low-field ($<$ 1 T) data of the
longitudinal resistivity. As shown in Fig. 3a, the two-band Drude model fits
the experimental data satisfactorily. The fitted mobilities and concentrations
for the electron and hole conduction channels are shown in Fig. 3b and c. As
expected, the Hall data can be explained by a high-mobility, low-density
electron channel representing the bulk Cd3As2 states and a low-mobility, high-
density hole channel that can be attributed to the surface states [34]. The
low concentration of the n-type Cd3As2 bulk states suggests a strong quantum
confinement. The high density of the surface state carriers is in qualitative
agreement with the observed high-frequency quantum oscillation as well as
previous reports on similar samples [34].

Figure 4 shows the magnetic-field-dependent Seebeck coefficient $S_{xx}$ and
Nernst coefficient $S_{xy}$ in the 25-nm sample measured at different
temperatures in a magnetic field up to 9 T. In the semiclassical regime, the
Seebeck coefficient and the Nernst coefficient can be calculated as [42]:

$S_{xx}(B)=A(\frac{\sigma_{xx}^{2}}{\sigma_{xx}^{2}+\sigma_{xy}^{2}}D+\frac{\sigma_{xy}^{2}}{\sigma_{xx}^{2}+\sigma_{xy}^{2}}D_{H}),$
(1)
$S_{xy}(B)=A\frac{\sigma_{xx}\sigma_{xy}}{\sigma_{xx}^{2}+\sigma_{xy}^{2}}(D_{H}-D),$
(2)

where $A=\frac{\pi^{2}k_{B}^{2}T}{3e}$, and $\sigma_{xx}$ and $\sigma_{xy}$
are longitudinal and Hall conductivity, respectively. Additional parameters
$D$ and $D_{H}$ are from the Mott relations:
$D=\frac{\partial\ln\sigma_{xx}}{\partial\zeta}$,
$D_{H}=\frac{\partial\ln\sigma_{xy}}{\partial\zeta}$, where $\zeta$ is the
Fermi level. As shown in Fig. 4a, the measured Seebeck coefficient follows the
semiclassical increase and saturation trend with the magnetic field in a
semimetal [23]. In Fig. 4b, we used the two-band semiclassical model (Eqn. 1)
with parameters extracted from the Hall resistivity fitting (Fig. 3) to fit
the low-field ($<$ 1 T) Seebeck data [42]. For consistency, we also used the
experimental $\sigma_{xx}$ and $\sigma_{xy}$ data and the fitting parameters
$D$ and $D_{H}$ to fit the low-field Seebeck data, where similar values of $D$
and $D_{H}$ are obtained (see Fig. S6 in the SI). The fitted values of $D$ and
$D_{H}$ are provided in Fig. 4c. The extracted $D$ and $D_{H}$ values steeply
rise as the temperature decreases, reaching $>1000$ eV-1, suggesting unusually
sharp changes of $\sigma_{xx}$ as a function of the Fermi level. At cryogenic
temperatures where the electronic scattering rates are largely independent of
the Fermi level, the sharp changes of $\sigma_{xx}$ most likely stem from the
large effective mass and, thus, large density of states of the p-type carriers
residing in the surface states. This interpretation is consistent with the
Hall resistivity data and in agreement with previous Landau level spectroscopy
analysis of similar samples [33].

The unusual thermoelectric properties of the topological surface states are
further manifested in the Nernst coefficient of the 25 nm sample, as shown in
Figs. 4d-f, where we provide the measured Nernst coefficient up to 9 T and the
semiclassical (Eqn. 2) prediction for the low-field Nernst data with
parameters extracted from fitting the low-field Seebeck data. The low-field
Nernst coefficient above 40 K can be reasonably explained using the
semiclassical model (Fig 4e). Below 40 K, as shown in Fig. 4f, however, there
is significant discrepancy between the experimental values and the model
predictions, suggesting the failure of the semiclassical model. For
comparison, the Nernst coefficient measured in our 950 nm sample at cryogenic
temperatures (Supplementary Note 2 in the SI) is in good agreement with
previous measurements of bulk Cd3As2 crystals [23], where the anomalous Nernst
effect (ANE) contribution due to the Berry curvature of bulk Cd3As2 states is
considered. The Nernst coefficient in the 95 nm sample is slightly increased
from that in the 950 nm sample, while the Nernst coefficient in the 25 nm
sample is about one order of magnitude higher than that in the 950 nm sample.
This experimental trend signals a significantly enhanced ANE associated with
the topological surface states in the 25 nm sample. Due to the still scarce
understanding of the Fermi arc surface states in Cd3As2 thin films [29], it is
difficult to quantitatively verify the large ANE associated with them from a
modeling point of view. One potential explanation is that the surface states
are gapped in the 25 nm sample due to hybridization of the top and bottom
surfaces and/or strain [29], which can lead to a large Berry curvature [43,
44] that contributes to an enhanced ANE. Our result exemplifies the complexity
of thermoelectric transport in the quantum-confined thin films of topological
semimetals and suggests a new route to achieving large thermoelectric
responses at cryogenic temperatures via topological surface states.

In passing, we briefly discuss the possible contribution to the measured
thermoelectric response from the GaSb buffer layer, which is a semiconductor
and can potentially possess a high Seebeck coefficient. Measurements of a bare
GaSb buffer layer are provided in the SI, where it is shown that GaSb becomes
insulating below 50 K. In principle, however, the band alignment between GaSb
and Cd3As2 at the interface can lead to charge transfer and “doping” of GaSb,
increasing its conductivity. However, in extensive previous studies of this
system down to very thin Cd3As2 layers, there is no experimental evidence that
the GaSb buffer layer affects transport measurements in any way, especially at
cryogenic temperatures [28, 33, 34, 45]. Moreover, the charge carriers in GaSb
do not carry any nontrivial Berry curvature and, thus, are not expected to
lead to any contribution to the observed ANE. On the basis of these
observations, we attribute the observed unusual thermoelectric properties to
the topological surface states, rather than the GaSb buffer layer.

##  III Conclusion

In summary, we systematically studied the thermoelectric transport properties
of epitaxially grown Cd3As2 thin films with different thicknesses. Through
detailed quantum oscillation analysis and magnetotransport measurement
complemented by DFT simulation and a two-band Drude transport model, we
identified contributions from both bulk states and topological surface states.
In particular, we revealed unusually large Seebeck and Nernst effect
associated with the topological surface states. Our results showcase the
complexity of thermoelectric transport in topological semimetal thin films and
signal new opportunities to achieve remarkable thermoelectric performances in
these emerging material systems.

##  IV Methods

###  IV.1 Molecular Beam Epitaxy Growth

(112)-oriented Cd3As2 films with thicknesses of 25 nm, 95 nm, and 950 nm were
grown on 150-nm GaSb buffer layers on GaAs (111)B substrates by molecular beam
epitaxy, with similar parameters as described previously [5]. The GaSb buffer
layer was grown with a III/V beam equivalent pressure (BEP) ratio of 4.5 at a
substrate temperature of 420°C. Following the GaSb growth, the substrate
temperature is reduced to 135°C for the Cd3As2 growth. Cd3As2 flux is supplied
from molecular sources at a BEP between $2\times 10^{-7}$ and $5\times
10^{-6}$ Torr.

###  IV.2 Thermoelectric Transport Measurement

In this study, we utilized a steady-state transport measurement system to
measure the electrical and thermoelectric properties. Specifically, a
commercial 9 T Dynacool Physical Property Measurement System (PPMS, Quantum
Design) was used. For the resistivity and the Hall measurements, we conducted
six-point probe measurements with the magnetic field perpendicular to the
sample (device pictures are shown in the SI). Voltage measurements were
carried out using Keithley 2182A nanovoltmeters, while the power supply was
provided by a Keithley 6221 current source. The formation of electrical
contact was facilitated using silver epoxy. For thermoelectric measurements,
temperature readings were obtained using 25 $\mu$m T-type thermocouples
fabricated through spot welding to minimize parasitic heat loss. It should be
noted that the thermocouples’ sensitivity deminishes below 60 K, where Cernox
thermistors were employed instead. To detect the Seebeck and Nernst voltage,
longitudinal and transverse copper voltage probes were used. A 120-$\Omega$
strain gauge served as the heater, and the heating power was adjusted
accordingly to achieve a temperature rise of 1 $\sim$ 3 $\%$. Cd3As2 samples
were placed on sapphire heat sinks. The magnetic field was swept from -9 T to
9 T with a sweeping rate of 5 mT/s and a customized LabVIEW control code was
used to effectively manage experimental control and the data acquisition
process.

###  IV.3 Angle-dependent Quantum Oscillation Measurement

A picture of the experimental setup using a rotation sample stage is provided
in the SI. At a rotation angle of 0 degree, the heat flow or current flow was
applied along the sample, while the magnetic field was perpendicular to the
sample. Angle-dependent measurements taken at 15-degree intervals between 0
and 180 degrees. The magnetic field was swept from -9 T to 9 T, and
measurements included SdH oscillations, as well as thermoelectric oscillations
(i.e. oscillations in Seebeck and Nernst coefficients at higher magnetic
fields). The raw data was fitted using polynomial equations to establish a
baseline, which were subsequently subtracted to isolate the oscillations.
Following the subtraction, Fast Fourier Transform (FFT) analysis were
performed to extract the oscillation frequencies. The FFT range selected for
analysis was from 2 T to 8.5 T, as lower field values lead to no discernible
oscillations and high noise was observed near 9Ṫ. For the 25-nm Cd3As2 sample,
a comparative analysis was conducted on both Seebeck and Nernst oscillation
signals to ensure the presence of the same oscillation frequencies in both
measurements.

###  IV.4 Density Functional Theory Simulation

We employed the Vienna ab initio simulation package (VASP) [46, 47] version
6.3.1. for DFT-based first principles calculations. The projector augmented
wave (PAW) [48] potentials were utilized. The valence electrons for Cd
included $5s^{2}3d^{10}$, while for As they were $4s^{2}3p^{3}$ electrons. The
Perdew-Burke-Ernzerhof (PBE) [49] exchange correlation functional was adopted,
including the spin-orbit coupling. The cutoff energy for the plane-wave
expansion was 400 eV. The lattice structure was optimized with a Hellmann-
Feynman force tolerance of 0.001 eV/Å. The Monkhorst-Pack $6\times 6\times 6$
$k$-mesh to sample the whole Brillouin zone was critical to obtain the correct
Fermi level. This $k$ mesh has a $k$-point ($\frac{1}{12}$, $\frac{1}{12}$,
-$\frac{1}{12}$) close to the Dirac point located at ($\frac{1}{12.5}$,
$\frac{1}{12.5}$, -$\frac{1}{12.5}$) in the 80-atom primitive-cell reciprocal
lattice basis or at (0, 0, $\pm$0.0386
$$\mathrm{\SIUnitSymbolAngstrom}$^{-1}$)=(0,0,
$\pm\frac{1}{6.25}\frac{2\pi}{c}$) in the conventional-cell reciprocal
lattice, close to the previous reports [50, 51, 13, 52]. The symmetry of
Cd3As2 is $\mathrm{I4_{1}/acd}$ and fully exploited to minimize the
computational expense. The relaxed lattice constants of the conventional cell
are $a$=$b$=12.850Å, $c$=25.905Å. The calculated electronic structure is shown
in the SI using the 80-atom primitive cell.

In addition, we compared the experimental angle-dependent quantum oscillation
frequencies with DFT calculations. With the Fourier interpolated Fermi
surfaces on a $\Gamma-65\times 65\times 83$ k-point gird, we used the SKEAF
package [53] to divide the Fermi surfaces into slices perpendicular to the
external magnetic field and obtain the slices with local maximum or minimum
areas, from which the quantum oscillation frequencies can be calculated using
the Onsager relation. For evaluating thermoelectric transport coefficients, we
applied the BoltzTraP2 [54, 55] software to calculate transport properties by
solving the linearized Boltzmann transport equation for electrons under the
constant relaxation time approximation.

## Data Availability

The authors declare that the data supporting the findings of this study are
available within the paper, its supplementary information files, and the Dryad
Data Repository (https://doi.org/10.5061/dryad.z08kprrjv).

Figure 1: Sample structures and temperature dependence of resistivity and
Seebeck coefficient. (a) Schematic of the sample structure and potential
conducting channels and a picture of the measurement device. (112) Cd3As2
samples with different thicknesses were epitaxially grown on a 150-nm (111)
GaSb buffer layer on (111) GaAs substrates. (b) Temperature dependence of the
measured resistivity of 950 nm, 95 nm and 25 nm Cd3As2 samples, respectively.
(c) Temperature dependence of the measured Seebeck coefficients in the three
samples, as compared to DFT calculation considering only Cd3As2 bulk states.
Error bars are not shown here for clarify and are less than 10$\%$ for data
shown here. Figure 2: Angle-dependent quantum oscillations. (a) Measured
thermoelectric voltage as a function of the inverse magnetic field for the
three samples. Peaks A-B correspond to a 20 T oscillation frequency and peaks
C-D, observed only in the 95 nm and 25 nm samples, correspond to a 55 T
frequency. (b) Angular dependence of the quantum oscillation frequency in 950
nm and 95 nm samples. The oscillation frequency of these two samples is
consistent with the DFT calculation considering Cd3As2 bulk states. (c) The
Fermi surface of Cd3As2 bulk states responsible for the observed quantum
oscillation frequency near 20 T. (d) Angular dependence of the quantum
oscillation frequency in the 25-nm sample. The oscillation frequency of the
25-nm sample exhibits two distinct frequencies. The lower one near 20 T
corresponds to the Cd3As2 bulk states. Additionally, there is a high-frequency
peak near 55 T at 0 degree, which decreases with an increasing field angle,
indicating its origin in the topological surface states. (e) Illustration of
the angular dependence of the oscillation frequency corresponding to
topological surface states. The Fermi surface of the surface states has the
maximum intersection area with the magnetic field and the highest oscillation
frequency at zero degree field angle, while a 90-degree field angle parallel
to the sample surface leads to a vanishing oscillation frequency.  Figure 3:
Magnetic-field-dependent Hall measurements of the 25-nm sample. (a) Measured
Hall resistivity (solid lines) of the 25-nm sample fitted by a two-band Drude
model (dashed lines) at three representative temperatures. (b) Fitted
mobilities of the electron and hole conducting channels, indicating a highly
mobile n-type channel and a low-mobility p-type channel. (c) Fitted carrier
concentrations of the n-type bulk states and the p-type surface states. Figure
4: Magnetic-field-dependent Seebeck and Nernst measurements of the 25-nm
sample. (a) Measured Seebeck coefficient of the 25-nm sample at different
temperatures with a magnetic field up to 9 T. (b) Low-field Seebeck data
(solid lines) fitted by a two-band semiclassical model (dashed lines). (c) The
fitted Mott relation parameters $D$ and $D_{H}$ as a function of temperature.
(d) Measured Nernst coefficient of the 25-nm sample at different temperatures
with a magnetic field up to 9 T. (e) Low-field Nernst data (solid lines)
fitted by a two-band semiclassical model (dashed lines) above 40 K, where
reasonable agreement between experiment and model is observed. (e) Measured
Nernst coefficient (solid lines) of the 25-nm sample fitted by a two-band
model (dashed lines) below 20 K, where large deviation between experiment and
model potentially due to an enhanced anomalous Nernst contribution from the
topological surface states is observed.

## References

  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Colloquium: topological insulators, Reviews of Modern Physics 82, 3045 (2010).
  * Lv  _et al._ [2021] B. Lv, T. Qian, and H. Ding, Experimental perspective on three-dimensional topological semimetals, Reviews of Modern Physics 93, 025002 (2021).
  * Fu  _et al._ [2020] C. Fu, Y. Sun, and C. Felser, Topological thermoelectrics, APL Materials 8, 040913 (2020).
  * Vu  _et al._ [2021] D. Vu, W. Zhang, C. Şahin, M. E. Flatté, N. Trivedi, and J. P. Heremans, Thermal chiral anomaly in the magnetic-field-induced ideal Weyl phase of Bi1-xSbx, Nat. Mater. 20, 1525 (2021).
  * Schumann  _et al._ [2016] T. Schumann, M. Goyal, H. Kim, and S. Stemmer, Molecular beam epitaxy of Cd3As2 on a III-V substrate, APL Materials 4, 126110 (2016).
  * Chen  _et al._ [2019] L.-C. Chen, P.-Q. Chen, W.-J. Li, Q. Zhang, V. V. Struzhkin, A. F. Goncharov, Z. Ren, and X.-J. Chen, Enhancement of thermoelectric performance across the topological phase transition in dense lead selenide, Nature Materials 18, 1321 (2019).
  * Skinner and Fu [2018] B. Skinner and L. Fu, Large, nonsaturating thermopower in a quantizing magnetic field, Science Advances 4, eaat2621 (2018).
  * Han  _et al._ [2020] F. Han, N. Andrejevic, T. Nguyen, V. Kozii, Q. T. Nguyen, T. Hogan, Z. Ding, R. Pablo-Pedro, S. Parjan, B. Skinner, _et al._ , Quantized thermoelectric Hall effect induces giant power factor in a topological semimetal, Nature Communications 11, 6167 (2020).
  * Zhang  _et al._ [2020] W. Zhang, P. Wang, B. Skinner, R. Bi, V. Kozii, C.-W. Cho, R. Zhong, J. Schneeloch, D. Yu, G. Gu, _et al._ , Observation of a thermoelectric Hall plateau in the extreme quantum limit, Nature Communications 11, 1046 (2020).
  * Yue  _et al._ [2019] S. Yue, H. T. Chorsi, M. Goyal, T. Schumann, R. Yang, T. Xu, B. Deng, S. Stemmer, J. A. Schuller, and B. Liao, Soft phonons and ultralow lattice thermal conductivity in the Dirac semimetal Cd3As2, Phys. Rev. Research 1, 033101 (2019).
  * Yue  _et al._ [2020] S. Yue, B. Deng, Y. Liu, Y. Quan, R. Yang, and B. Liao, Phonon softening near topological phase transitions, Physical Review B 102, 235428 (2020).
  * Nguyen  _et al._ [2020] T. Nguyen, F. Han, N. Andrejevic, R. Pablo-Pedro, A. Apte, Y. Tsurimaki, Z. Ding, K. Zhang, A. Alatas, E. E. Alp, _et al._ , Topological singularity induced chiral Kohn anomaly in a Weyl semimetal, Physical Review Letters 124, 236401 (2020).
  * Crassee  _et al._ [2018] I. Crassee, R. Sankar, W.-L. Lee, A. Akrap, and M. Orlita, 3D Dirac semimetal Cd3As2: A review of material properties, Phys. Rev. Materials 2, 120302 (2018).
  * Wang  _et al._ [2013] Z. Wang, H. Weng, Q. Wu, X. Dai, and Z. Fang, Three-dimensional Dirac semimetal and quantum transport in Cd3As2, Phys. Rev. B 88, 125427 (2013).
  * Yang and Nagaosa [2014] B.-J. Yang and N. Nagaosa, Classification of stable three-dimensional Dirac semimetals with nontrivial topology, Nature Communications 5, 4898 (2014).
  * Chorsi  _et al._ [2020] H. T. Chorsi, S. Yue, P. P. Iyer, M. Goyal, T. Schumann, S. Stemmer, B. Liao, and J. A. Schuller, Widely tunable optical and thermal properties of Dirac semimetal Cd3As2, Advanced Optical Materials 8, 1901192 (2020).
  * Spitzer  _et al._ [1966] D. P. Spitzer, G. A. Castellion, and G. Haacke, Anomalous thermal conductivity of Cd3As2 and the Cd3As2-Zn3As2 alloys, Journal of Applied Physics 37, 3795 (1966).
  * Armitage and Goldsmid [1969] D. Armitage and H. J. Goldsmid, The thermal conductivity of cadmium arsenide, Journal of Physics C: Solid State Physics 2, 2138 (1969).
  * Wang  _et al._ [2018] H. Wang, X. Luo, W. Chen, N. Wang, B. Lei, F. Meng, C. Shang, L. Ma, T. Wu, X. Dai, Z. Wang, and X. Chen, Magnetic-field enhanced high-thermoelectric performance in topological Dirac semimetal Cd3As2 crystal, Science Bulletin 63, 411 (2018).
  * Pariari  _et al._ [2016] A. Pariari, N. Khan, R. Singha, B. Satpati, and P. Mandal, Tuning the scattering mechanism in the three-dimensional Dirac semimetal Cd3As2, Phys. Rev. B 94, 165139 (2016).
  * Jia  _et al._ [2016] Z. Jia, C. Li, X. Li, J. Shi, Z. Liao, D. Yu, and X. Wu, Thermoelectric signature of the chiral anomaly in Cd3As2, Nat Commun 7, 13013 (2016).
  * Amarnath  _et al._ [2020] R. Amarnath, K. S. Bhargavi, and S. S. Kubakaddi, Thermoelectric transport properties in 3D Dirac semimetal Cd3As2, J. Phys.: Condens. Matter 32, 225704 (2020).
  * Liang  _et al._ [2017] T. Liang, J. Lin, Q. Gibson, T. Gao, M. Hirschberger, M. Liu, R. J. Cava, and N. P. Ong, Anomalous Nernst effect in the Dirac semimetal Cd3As2, Phys. Rev. Lett. 118, 136601 (2017).
  * Young  _et al._ [2012] S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, Dirac semimetal in three dimensions, Physical Review Letters 108, 140405 (2012).
  * Pan  _et al._ [2015] H. Pan, M. Wu, Y. Liu, and S. A. Yang, Electric control of topological phase transitions in Dirac semimetal thin films, Scientific Reports 5, 14639 (2015).
  * Xiao  _et al._ [2015] X. Xiao, S. A. Yang, Z. Liu, H. Li, and G. Zhou, Anisotropic quantum confinement effect and electric control of surface states in dirac semimetal nanostructures, Scientific Reports 5, 7898 (2015).
  * Narayan  _et al._ [2014] A. Narayan, D. Di Sante, S. Picozzi, and S. Sanvito, Topological tuning in three-dimensional Dirac semimetals, Physical Review Letters 113, 256403 (2014).
  * Schumann  _et al._ [2018] T. Schumann, L. Galletti, D. A. Kealhofer, H. Kim, M. Goyal, and S. Stemmer, Observation of the quantum Hall effect in confined films of the three-dimensional Dirac semimetal Cd3As2, Phys. Rev. Lett. 120, 016801 (2018).
  * Arribi  _et al._ [2020] P. V. Arribi, J.-X. Zhu, T. Schumann, S. Stemmer, A. A. Burkov, and O. Heinonen, Topological surface states in strained Dirac semimetal thin films, Physical Review B 102, 155141 (2020).
  * Potter  _et al._ [2014] A. C. Potter, I. Kimchi, and A. Vishwanath, Quantum oscillations from surface Fermi arcs in Weyl and Dirac semimetals, Nature Communications 5, 5161 (2014).
  * Wang  _et al._ [2017] C. M. Wang, H.-P. Sun, H.-Z. Lu, and X. C. Xie, 3D quantum Hall effect of fermi arcs in topological semimetals, Physical Review Letters 119, 136806 (2017).
  * Goyal  _et al._ [2018] M. Goyal, L. Galletti, S. Salmani-Rezaie, T. Schumann, D. A. Kealhofer, and S. Stemmer, Thickness dependence of the quantum Hall effect in films of the three-dimensional Dirac semimetal Cd3As2, APL Materials 6, 026105 (2018).
  * Galletti  _et al._ [2018a] L. Galletti, T. Schumann, O. F. Shoron, M. Goyal, D. A. Kealhofer, H. Kim, and S. Stemmer, Two-dimensional Dirac fermions in thin films of Cd3As2, Physical Review B 97, 115132 (2018a).
  * Galletti  _et al._ [2018b] L. Galletti, T. Schumann, T. E. Mates, and S. Stemmer, Nitrogen surface passivation of the Dirac semimetal Cd3As2, Phys. Rev. Materials 2, 124202 (2018b).
  * Liang  _et al._ [2015] T. Liang, Q. Gibson, M. N. Ali, M. Liu, R. Cava, and N. Ong, Ultrahigh mobility and giant magnetoresistance in the Dirac semimetal Cd3As2, Nature Materials 14, 280 (2015).
  * He and Li [2016] L.-P. He and S.-Y. Li, Quantum transport properties of the three-dimensional Dirac semimetal Cd3As2 single crystals, Chinese Phys. B 25, 117105 (2016).
  * Ashcroft and Mermin [1976] N. W. Ashcroft and D. N. Mermin,  _Solid State Physics_ (Cengage Learning, New York, 1976).
  * Xiang  _et al._ [2015] Z. J. Xiang, D. Zhao, Z. Jin, C. Shang, L. K. Ma, G. J. Ye, B. Lei, T. Wu, Z. C. Xia, and X. H. Chen, Angular-dependent phase factor of Shubnikov–de Haas oscillations in the Dirac semimetal Cd3As2, Physical Review Letters 115, 226401 (2015).
  * Goyal  _et al._ [2019] M. Goyal, H. Kim, T. Schumann, L. Galletti, A. A. Burkov, and S. Stemmer, Surface states of strained thin films of the Dirac semimetal Cd3As2, Phys. Rev. Materials 3, 064204 (2019).
  * Eto  _et al._ [2010] K. Eto, Z. Ren, A. A. Taskin, K. Segawa, and Y. Ando, Angular-dependent oscillations of the magnetoresistance in Bi2Se3 due to the three-dimensional bulk Fermi surface, Physical Review B 81, 195309 (2010).
  * Nishihaya  _et al._ [2019] S. Nishihaya, M. Uchida, Y. Nakazawa, R. Kurihara, K. Akiba, M. Kriener, A. Miyake, Y. Taguchi, M. Tokunaga, and M. Kawasaki, Quantized surface transport in topological Dirac semimetal films, Nature Communications 10, 2564 (2019).
  * Liang  _et al._ [2013] T. Liang, Q. Gibson, J. Xiong, M. Hirschberger, S. P. Koduvayur, R. Cava, and N. Ong, Evidence for massive bulk Dirac fermions in Pb1-xSnxSe from Nernst and thermopower experiments, Nature Communications 4, 2696 (2013).
  * Chen  _et al._ [2022] D. Chen, B. He, M. Yao, Y. Pan, H. Lin, W. Schnelle, Y. Sun, J. Gooth, L. Taillefer, and C. Felser, Anomalous thermoelectric effects and quantum oscillations in the kagome metal CsV3Sb5, Physical Review B 105, L201109 (2022).
  * Hosur and Qi [2013] P. Hosur and X. Qi, Recent developments in transport phenomena in Weyl semimetals, Comptes Rendus Physique 14, 857 (2013).
  * Lygo  _et al._ [2023] A. C. Lygo, B. Guo, A. Rashidi, V. Huang, P. Cuadros-Romero, and S. Stemmer, Two-dimensional topological insulator state in cadmium arsenide thin films, Physical Review Letters 130, 046201 (2023).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6, 15 (1996).
  * Blöchl [1994] P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  * Ali  _et al._ [2014] M. N. Ali, Q. Gibson, S. Jeon, B. B. Zhou, A. Yazdani, and R. J. Cava, The crystal and electronic structures of Cd3As2, the three-dimensional electronic analogue of graphene, Inorg. Chem. 53, 4062 (2014).
  * Conte  _et al._ [2017] A. M. Conte, O. Pulci, and F. Bechstedt, Electronic and optical properties of topological semimetal Cd3 As2, Sci. Rep. 7, 1 (2017).
  * Baidya and Vanderbilt [2020] S. Baidya and D. Vanderbilt, First-principles theory of the Dirac semimetal Cd3As2 under Zeeman magnetic field, Phys. Rev. B 102, 1 (2020), 2008.10703 .
  * Rourke and Julian [2012] P. M. Rourke and S. R. Julian, Numerical extraction of de Haas-van Alphen frequencies from calculated band energies, Comput. Phys. Commun. 183, 324 (2012), 0803.1895 .
  * Madsen and Singh [2006] G. K. Madsen and D. J. Singh, BoltzTraP. A code for calculating band-structure dependent quantities, Comput. Phys. Commun. 175, 67 (2006), 0602203 [cond-mat] .
  * Madsen  _et al._ [2018] G. K. Madsen, J. Carrete, and M. J. Verstraete, BoltzTraP2, a program for interpolating band structures and calculating semi-classical transport coefficients, Comput. Phys. Commun. 231, 140 (2018), 1712.07946 .

###### Acknowledgements.

This work is based on research supported by the National Science Foundation
(NSF) via the DMREF program under award number DMR-2118523. Transport
measurements were done at UCSB Materials Research Laboratory (MRL) Shared
Experimental Facilities, which are supported by the NSF MRSEC Program under
award number DMR-2308708. The computational work used Stampede2 at Texas
Advanced Computing Center (TACC) and Expanse at San Diego Supercomputer Center
(SDSC) through allocation MAT200011 from the Advanced Cyberinfrastructure
Coordination Ecosystem: Services & Support (ACCESS) program, which is
supported by NSF grants 2138259, 2138286, 2138307, 2137603, and 2138296. Use
was also made of computational facilities purchased with funds from NSF (award
number CNS-1725797) and administered by the Center for Scientific Computing
(CSC) at UCSB. The CSC is supported by the California NanoSystems Institute
and the UCSB MRL.

## Author Contributions

B.L., S.S. and J.P.H. conceived and supervised the project. A.C.L. prepared
the samples. W.O., B.L.W., D.V. and X.L. conducted the transport measurements.
Y.C. conducted the DFT simulations. H.Z. and Y.W. constructed the two-band
Drude model. W.O., Y.C. and B.L. drafted the manuscript. All authors have
commented and edited the manuscript.

## Competing Interests

The authors declare no competing interests.