# Interaction-controlled transport in a two-dimensional massless-massive Dirac
system: Transition from degenerate to nondegenerate regimes

A. D. Levin,1 G. M. Gusev,1 F. G. G. Hernandez,1 E. B. Olshanetsky,2,3 V. M.
Kovalev,2,4,5 M. V. Entin2,3, and N. N. Mikhailov2,3 1Instituto de Física da
Universidade de São Paulo, 135960-170, São Paulo, SP, Brazil  2Institute of
Semiconductor Physics, Novosibirsk 630090, Russia  3Novosibirsk State
University, Novosibirsk 630090, Russia  4Novosibirsk State Technical
University, Novosibirsk 630073, Russia  5Abrikosov Center for Theoretical
Physics, Moscow Institute of Physics and Technology, Dolgoprudny, 141701,
Russia

###### Abstract

The resistivity of two-dimensional (2D) metals generally exhibits
insensitivity to electron-electron scattering. However, it’s worth noting that
Galilean invariance may not hold true in systems characterized by a spectrum
containing multiple electronic branches or in scenarios involving electron-
hole plasma. In the context of our study, we focus on 2D electrons confined
within a triple quantum well (TQW) based on HgTe. This system displays a
coexistence of energy bands featuring both linear and parabolic-like spectra
at low energy and, therefore, lacks the Galilean invariance. This research
employs a combined theoretical and experimental approach to investigate the
transport properties of this two-component system across various regimes. By
manipulating carrier density and temperature, we tune our system from a fully
degenerate regime, where resistance follows a temperature-dependent behavior
proportional to $T^{2}$, to a regime where both types of electrons adhere to
Boltzmann statistics. In the non-degenerate regime, electron interactions lead
to resistance that is weakly dependent on temperature. Notably, our
experimental observations closely align with the theoretical predictions
derived in this study. This work establishes the HgTe-based TQW as a promising
platform for exploring different interaction dominant scenarios for the
massless-massive Dirac system.

††preprint: Phys.Rev.Research

##  I Introduction

The impact of electron-electron scattering on the transport characteristics of
various two dimensional (2D) conductors, which do not adhere to Galilean
invariance, has garnered significant attention over the years pal . These
studies typically focus on systems that involve two distinct types of charge
carriers, characterized by differing charges or effective masses. The presence
of two different charge carriers within the system, each with distinct masses,
poses a challenge to the traditional concept of Galilean invariance.
Consequently, the direct proportionality between the net current and the total
particle momentum is no longer upheld nagaev1 ; nagaev2 ; murzin ; kravchenko
; hwang  .

Figure 1: (Color online) (a) The conduction and valence band edges of the
triple quantum well are schematically shown. The widths d of the HgTe wells
and the thickness $d_{b}$ of the $Hg_{1-x}Cd_{x}Te$ barriers (x=0.3) are
indicated. (b) The band structure of the triple HgTe well at $E>0$ is
calculated using the tight-binding model described in the text. The black and
blue lines represent dispersion curves for massless and massive Dirac
fermions. (c) The density of carriers is shown as a function of the chemical
potential. $n_{e}$ and $n_{d}$, representing the densities of massive and
massless electrons, $N_{s}$ is the total density. An arrow indicates the
beginning of the $E^{+}$ subband populations. (d) The resistance of the 6.4 nm
sample is plotted as a function of carrier density at T=4.2 K.

Exploring the unique scenarios of strong friction between electrons and holes,
particularly in degenerate 2D semimetals, has led to the observation of
resistivity proportional to $T^{2}$ olshanetsky ; entin . These phenomena have
been validated in various experimental settings, most notably in HgTe quantum
wells. Moreover, investigations into the nondegenerate regime have ventured
into systems like single-layer and bilayer graphene, where thermally excited
electron-hole pairs play a crucial role tan . Recent advancements have also
revealed electron-hole friction behavior in bilayer graphene systems,
showcasing resistivity scaling as $T^{2}$ with spatially separated electrons
and holes bandurin .

In a system comprising two subbands with significantly differing masses, it
can be anticipated that the resistivity will exhibit a pronounced increase
with rising temperature. This increase occurs due to the relationship between
the resistivity limits at high and low temperatures, which is proportional to
the effective mass ratio, as described in pal . In such systems, the dominance
of interactions in the transport process surpasses the Drude resistivity
resulting from impurity or phonon scattering. The behavior of transport
governed by interactions at elevated temperatures is characterized by the
principles of hydrodynamics and is often referred to as electronic fluid
behavior.

Another crucial question pertains to the investigation of interparticle
collisions and hydrodynamic conductivity within a highly adjustable system.
This system can be fine-tuned, transitioning from a non-degenerate Boltzmann
regime to a degenerate Fermi liquid regime.

As the understanding of electron hydrodynamics continues to unfold, it holds
the promise of shedding light on the intricate interplay between particle-
particle interactions, sample geometry, and charge carriers’ distinct
properties. This realm not only presents fundamental scientific inquiries but
also offers potential applications in future electronic devices and
technologies.

Previously, we offered a system that serves as a useful platform for studying
transport dominated by interaction in 2D conductors gusev . This system is a
6.3 nm HgTe quantum well, representing the spectrum as a single-valley Dirac
cone near the zero-energy state. However, it’s important to note that below
Dirac zero point energy, the spectrum represents laterally located heavy hole
valleys with a minimum at a non-zero wave vectorgusev ; buttner ; kozlov ;
gusev2 ; kristopenko ; kvon . Consequently, when the chemical potential in the
valence band reaches this lateral heavy hole band, the Dirac holes experience
strong scattering by heavy holes, and such scattering violations of Galilean
invariance lead to a strong $T^{2}$ dependence of resistance. An interesting
and important aspect of physics is explored in such conductors when heavy
particles are partially degenerate: Dirac holes follow Fermi statistics, while
heavy holes adhere to Boltzmann statistics. It’s worth noting, however, that
this system does not allow for a complete study of the transition from fully
degenerate to non-degenerate regimes for all subsystems gusev .

In this current study, we present an experimental situation that enables such
an investigation. We have introduced the HgTe-based triple quantum well (TQW)
as a convenient system featuring two subbands with both massless and massive
Dirac fermions, specifically electrons. This stands in stark contrast to the
singular HgTe well explored in a prior study gusev , where the interaction
between Dirac and heavy holes resulted in significant scattering. TQW affords
us the opportunity to investigate hydrodynamic conductivity across various
regimes, including the strongly degenerate Fermi and non-degenerate Boltzmann
regimes. Therefore, we illustrate that regardless of the carriers’ sign,
spectrum type, and other characteristics, including the confinement features
of the multilayer system, electron-electron collisions exhibit unified
properties and can emerge as the dominant mechanism in ultraclean systems.
Consequently, TQWs emerge as a promising platform for exploring novel
phenomena resulting from interactions. Notable instances encompass the
violation of the Wiedemann-Franz law, as reported in paperlucas , the
prediction of giant magnetoresistance in work levchenko , and quantum critical
conductivity, detailed in ref. muller , among other significant theoretical
predictions.

##  II ELECTRON SPECTRUM IN A TRIPLE-WELL SYSTEM

HgTe-based quantum wells have garnered significant interest due to their
capability to create unconventional two-dimensional (2D) systems, such as 2D
topological insulators konig ; hasan ; kvon2 ; gusev6 . Additionally, the
spectrum’s behavior is primarily determined by the well’s thickness, leading
to various phases characterized by insulating gaps, gapless regions, and
inverted subbands gerchikov ; kane ; bernevig ; bernevig2 . Double and triple
quantum wells are multi-layer systems comprising two or three quantum wells
separated by a tunneling-transparent barrier. Theoretical investigations
kristopenko2 ; michetti ; ferreira  propose that the phase states within these
structures are significantly altered when compared to the single quantum well
scenario, which is supported by experimental evidence gusev3 ; gusev4 . This
alteration leads to a more complex phase landscape due to the additional
degrees of freedom introduced by the increased number of 2D subbands and the
hybridization induced by tunneling between them. Here, we illustrate that a
triple HgTe-based quantum well serves as a valuable platform for studying the
influence of dominant interactions on transport behavior, primarily due to the
unique characteristics of its spectrum.

Figure 1a depicts the band structure of triple HgTe quantum wells with a
thickness of $d_{b}=3nm$ and a well width of $d=6.7nm$. The schematic
representation displays the conduction and valence band edges of the triple
quantum well.

In paper ferreira , the authors consider the confinement characteristics of
the subbands within triple HgTe/CdTe quantum wells, along with their
corresponding topological properties and edge state attributes. To accomplish
this, they employ an effective 2D Hamiltonian for the HgTe-based triple well
system.

For single HgTe quantum wells, this is achieved by projecting the Hamiltonian
onto its eigenstates at $k_{II}=0$, leading to the well-established BHZ model
bernevig ; bernevig2 . In contrast, for double HgTe quantum wells, two
intriguing approaches are explored. Firstly, similar to the derivation of the
BHZ Hamiltonian, the authors of the paper referenced as kristopenko2  project
the overall Hamiltonian onto the eigenstates at $k_{II}=0$ for the double
quantum wells (DQW).

Alternatively, in Reference michetti , the authors project the total DQW
Hamiltonian onto the subbands of the individual wells (left and right) and
introduce tunneling parameters to account for the coupling between neighboring
quantum wells.

In paper ferreira  the authors investigate the phase diagram of the HgTe
triple well and transform the 3D Kane Hamiltonian into an effective 2D 3×BHZ
model. This transformation enables them to explore the edge state
characteristics in each topological phase.

These findings reveal the existence of gapless phases, attributed to the
slight hybridization of H-like states from different quantum wells. In these
gapless phases, one or two pairs of edge states are present within the bulk.
However, there is also a phase where all E-like and H-like subbands are
inverted, leading to the formation of three sets of edge states within a bulk
gap.

Moreover, for a triple well with a well width of $d=6.7nm$, the subbands
$E_{01}$ and $H_{01}$ exhibit significant hybridization resulting in nearly
negligible mass for a linear Dirac subband. Meanwhile, the second subband
maintains a k-parabolic spectrum with massive Dirac fermions at low energies.
This effect has been substantiated through the measurement of Schubnikov de
Haas oscillations ferreira .

To obtain the analytical expression for the triple well spectrum, we employ a
tight-binding model with parameters calibrated based on Kane model
calculations, as detailed in ferreira . In this case the effective low energy
Hamiltonian is

$H=\begin{pmatrix}0&Ak&0&0&0&0\\\ Ak&0&t&0&0&0\\\ 0&t&0&Ak&0&t\\\
0&0&Ak&0&0&0\\\ 0&0&0&0&0&Ak\\\ 0&0&t&0&Ak&0\\\ \end{pmatrix}$ (1)

where $k$ is the two-dimensional momentum operator, $A=\hbar v_{F}\approx 470$
meV nm, $v_{F}$ is the Fermi velocity of the single HgTe well and the in-plane
nearest-neighbor hopping is $t\approx 24meV$.

One can see that the Hamiltonian, (1), leads to a combination of two linear
bands and four massive bands:

$\begin{gathered}E_{s}=\pm\left[t^{2}+v_{F}^{2}k^{2}+s\sqrt{t^{4}+2t^{2}v_{F}^{2}k^{2}}\right]^{1/2},\quad
s=\pm 1\\\ E_{0}=\pm v_{F}k.\end{gathered}$ (2)

In Figure 1b, we present the electronic segment of the spectrum. The Dirac-
like spectrum is represented by the blue lines, whereas the massive Dirac
subbands are denoted by the black and red lines. It’s worth noting that the
simplified tight-binding model provides a good fit for the electronic
branches, consistent with those obtained using the Kane model ferreira .
However, there is a notable difference for the hole segments of the spectrum.
Therefore, in our work, we concentrate solely on the electronic spectrum.

It’s noteworthy that in symmetric HgTe TQWs, the transition from a gapless to
a gapped phase can be achieved by disrupting the inversion symmetry of the
wells through the application of a transverse electric field, either from
asymmetric doping or external gate bias michetti ; gusev3 . This process can
induce a small gap in the energy spectrum, reminiscent of phenomena seen in
bilayer and trilayer graphene. The ability to manipulate this gap with an
external field is fascinating, offering avenues to modulate edge state
transport within HgTe double and triple well configurations. As we will show
in the next paragraph, we detected a minimal gap in our structures, indicated
by activation resistance behavior, in the order of approximately 1 meV.
However, it’s critical to underline that this gap, while present, is minuscule
relative to other system parameters and does not notably affect the spectrum
as outlined by Equation 1. Our investigation primarily concentrated on the
conduction band, which is essentially unaffected by such a negligible gap.

Due to the Dirac-like nature of the Hamiltonian, the behavior of the density
of states significantly deviates from that of conventional 2D systems. In the
spectrum of subbands denoted by $E_{s}$, we observe a nearly parabolic shape,
with particles exhibiting significant mass only in the vicinity of $k\approx
0$. As energy levels increase, the spectrum also transitions to a linear
regime. This leads to a linear dependence of the density of states on energy
and a non-monotonic change in the chemical potential with respect to gate
voltage.

In Figure 1c, the density of 2D carriers, distributed across different
subbands, is depicted as a function of the chemical potential $\mu$. Notably,
it is evident that the density of carriers in the massive Dirac branch
surpasses that of the massless carriers. Furthermore, beyond an energy
threshold of 35 meV, the second subband of the massive Dirac fermions starts
to become occupied.

The presence of various electronic branches leads to the breakdown of Galilean
invariance and a noteworthy increase in the impact of scattering between these
branches on electronic transport.

Figure 2: (Color online) Resistance as a function of the gate voltage at
different temperatures for two HgTe triple quantum wells.

##  III Experimental results

We fabricated triple quantum wells using $HgTe/Cd_{x}Hg_{1-x}$Te material with
a [013] surface orientation. The wells had equal widths, with $d_{0}$
measuring 6.7 nm and a barrier thickness of $t$ set at 3 nm. The layer
thickness was monitored during MBE growth via ellipsometry, achieving an
accuracy within ± 0.3 nm.

Figure 3: (Color online) Excess resistivity
$\Delta\rho(T)=\rho(T)-\rho(T=4.2K)$ as a function of the temperature for
various densities for sample A (a) and B (b). The red lines show $T^{2}$
dependence. The values of the densities is given in $10^{11}cm^{-2}$. Figure
4: (Color online) Excess resistivity $\Delta\rho(T)=\rho(T)-\rho(T=4.2K)$ as a
function of the temperature for two different densities for sample A. (a) The
total density is $N_{s}=5\times 10^{11}cm^{-2}$; (b) $4.1\times
10^{11}cm^{-2}$. Circles represent the experimental data, and the red lines
represent the theory of interparticle scattering between the fully degenerate
massless and massive electrons.

The devices employed in this study were multiterminal bars featuring three
consecutive segments, each 3.2 $\mu m$ wide, with varying lengths of $2\mu m$,
8 $\mu m$, and $32\mu m$. These devices were equipped with nine contacts. The
contacts were created by indium bonding to the surface of the contact pads,
which were precisely defined using lithography. Given the relatively low
growth temperature (around 180°C), the temperature during the contact
fabrication process remained low as well. Indium diffused vertically downward
on each contact pad, establishing an ohmic connection across all three quantum
wells, with contact resistance falling within the 10–50 $k\Omega$ range.

Figure 5: (Color online) Excess resistivity
$\Delta\rho(T)=\rho(T)-\rho(T=4.2K)$ as a function of the temperature for two
different densities for sample B. (a) The total density is $N_{s}=4.2\times
10^{11}cm^{-2}$; (b) $3.3\times 10^{11}cm^{-2}$. Circles represents the
experimental data, the the red lines represent the theory of interparticle
scattering between the fully degenerate massless and massive electrons.

Throughout the AC measurements, we consistently verified that the reactive
component of impedance did not exceed $5\%$ of the total impedance, confirming
the effectiveness of the ohmic contacts. Furthermore, the current-voltage
(I-V) characteristics exhibited ohmic behavior at low voltages. A 200 nm
$SiO_{2}$ dielectric layer was deposited onto the sample surface, subsequently
covered by a TiAu gate. The density variation with gate voltage was estimated
to be approximately $0.9\times 10^{11}cm^{-2}/V$, derived from the dielectric
thickness and Hall measurements, as previously reported in studies employing
similar devices. Two samples, denoted as A and B were studied. The samples
were created from a uniform substrate and were subject to identical growth
conditions. It is important to emphasize that these samples are of mesoscopic
dimensions and, as such, are expected to display all characteristics inherent
to mesoscopic physics.

sample |  $d$ (nm) |  $V_{CNP}$ (V) | $\rho_{max}(h/e^{2})$ | $\mu_{e}(V/cm^{2}s)$
---|---|---|---|---
A | 6.7 | -3.6 | 0.22 | 33.600
B | 6.7 | -6.7 | 0.2 | 39.600
Table 1:  Some of the typical parameters of the electron system in HgTe triple
quantum well at T=4.2K.

Figure 1d illustrates the variation of resistivity with charge density for
sample A at a temperature of 4.2 K. The resistance shows a prominent peak
centered at the charge neutrality point, which corresponds to the zero-energy
point. This behavior is a characteristic feature of both gapless and gapped
HgTe single well and double well devices kvon ; gusev3 . The maximum electron
density corresponds to a Fermi energy value of approximately 150 meV, as
depicted in Figure 1c. As the chemical potential approaches the range of 30-40
meV, the second subband associated with massive Dirac fermions begins to be
occupied. This is evident from a small feature in the resistivity, as
indicated by the arrow in Figure 1d. Table 1 presents the key parameters of
the gapless HgT quantum well employed in this research. These parameters
include the well width (d), the gate voltage associated with the Dirac point
position ($V_{CNP}$), the resistivity ($\rho$ value) at the charge neutrality
point (CNP), and the electron mobility ($\mu_{e}$) calculated as $1/\rho
N_{s}$, where the total electron density ($N_{s}$) is set at $2\times 10^{11}$
cm-2.

Figure 2 illustrates the variation in resistance with respect to gate voltage
across a wide range of temperatures. The plot reveals a notable increase in
resistance as temperature rises, with one notable exception: in the voltage
region close to the charge neutrality point ($-3V<V_{g}-V_{CNP}<0V$), the
resistance exhibits insulating behavior characterized by a very slight
activation gap in the range of 0.5-0.8 meV. The reduction in resistance as
temperature increases can be attributed to potential fluctuations, which may
lead to the creation of small semimetallic insulating regions or residual
gaps. It is important to acknowledge that even a small gap has the potential
to alter the energy spectrum described by Equation (2). However, for the sake
of simplicity, we have omitted such modifications in our straightforward
model.

To further investigate the temperature-dependent behavior of resistance (or
resistivity), we calculate the excess resistivity, denoted as
$\Delta\rho(T)=\rho(T)-\rho(T=4.2K)$. Subsequently, in Figure 3, we present
the excess resistivity for different electron densities across a broad
temperature range, for samples A and B. It is evident that the temperature
dependence evolves, adhering closely to a $T^{2}$ relationship at high
electron densities, while exhibiting weak temperature dependence near the
charge neutrality point. It is noteworthy that in the intermediate density
regime, the temperature dependence displays a somewhat more complex behavior.
At low temperatures, the dependence in sample A resembles a quadratic
function, whereas at higher temperatures, the power becomes significantly more
pronounced. In contrast, for sample B, the temperature dependence remains
quadratic at high temperatures, but the power is diminished at lower
temperatures. At low densities, sample A exhibits an almost plateau-like
behavior at high temperatures, while sample B demonstrates an increase in
excess resistivity. We believe that sample B displays greater levels of
disorder and inhomogeneity, and that contributions from regions of higher
density become critical at elevated temperatures. At high densities, the
experimental curve follows a quadratic dependence with a difference in the
order of amplitude for both samples. In the following discussion, we focus on
the high-density dependencies for a comparison with theoretical predictions.
Figure 4 an d 5 depicts the temperature dependence of $\Delta\rho(T)$ across
two densities for sample A and sample B respectively in more details. It is
evident that $\rho(T)$ exhibits a $T^{2}$ dependence across all depicted
densities for both samples, with some minor deviations. Notably, we observed
deviations in sample B at lower densities, which we attributed to density
inhomogeneity. It is reasonable, because we observed that the charge
neutrality point in sample B shifted towards a higher negative voltage, likely
due to charge capture by dielectrics. Furthemore, the experimental data points
at low temperatures exhibit dispersion due to their limited precision. In
sample B, we also observed a more rapid growth of excess resistivity with
increasing temperature, which deviated from the $T^{2}$ trend, possibly owing
to the broadening of the distribution function. We will delve further into the
Boltzmann statistical regime in the subsequent section.

The distinct resistance dependence of $T^{2}$ serves as an unambiguous
indicator of electron-electron scattering, as opposed to phonon scattering,
which would lead to a linear, rather than quadratic, temperature dependence,
as discussed in work melezhik . While it is a well-established fact that
electron-electron interactions do not influence the resistivity of a Galilean-
invariant Fermi liquid, there are specific scenarios where a conductivity
proportional to $T^{-2}$ can be observed, as detailed in review pal . Such
scenarios include the presence of spin-orbit interactions or the involvement
of multiple subbands, as discussed in the studies nagaev1 ; appel ; murzin ;
kravchenko ; nagaev2 ; pal .

Notably, Figure 2 reveals an extraordinary observation: the resistivity ratio
between high and low temperatures, $\rho(T=70K)/\rho(T=4.2K)$, surpasses the
4-5 limit, signifying that $\Delta\rho$ is significantly greater than
$\rho(T=4.2K)$ at elevated temperatures. This striking departure from more
conventional scenarios, where resistance is predominantly influenced by
disorder or phonon scattering, underscores the critical role of particle-
particle collisions in this context. These collisions have a pronounced
impact, far exceeding that of impurity-related scattering. This observation
validates the HgTe-based triple well as a promising experimental platform for
investigating transport phenomena dominated by interactions, opening the door
to the exploration of novel and non-trivial effects, such as violations of the
Wiedemann-Franz law sarma anomalous Coulomb drag liu  and many others (for
review, see polini ).

##  IV Comparison of theory with experiment

In the following discussion, we examine a straightforward hydrodynamic model
with two subbands, wherein electron-electron scattering is characterized by a
concept of mutual friction. The conductivity is determined through an equation
of motion for the electron featuring Dirac dispersion and massive electrons.

$-\frac{v_{d}-v_{e}}{\tau_{de}}-\frac{v_{d}}{\tau_{d}}+\frac{qE}{m_{d}}=0;$
(3)
$-\frac{v_{e}-v_{d}}{\tau_{ed}}-\frac{v_{e}}{\tau_{e}}+\frac{qE}{m_{e}}=0;$
(4)

where variables $v_{d}$ and $v_{e}$ are drift velocities of the massless
(Dirac) and massive electrons. The solution of this equations has been
performed elsewhere gusev ; entin ; bandurin . The expression for conductivity
can be obtained with some modifications, and it is given by:

$\sigma=q^{2}\frac{n_{d}n_{e}\left[\tau_{d}\left(\tau_{e}+\tau_{int}\right)\left(2+\frac{n_{d}}{n_{e}}\right)+\tau_{e}\left(\tau_{d}+\tau_{int}\right)\frac{n_{e}}{n_{d}}+\tau_{int}\left(\frac{m_{e}}{m_{d}}\tau_{d}-\frac{m_{d}}{m_{e}}\tau_{e}\right)\right]}{m_{e}n_{e}\left(\tau_{d}+\tau_{int}\right)+m_{d}n_{d}\left(\tau_{e}+\tau_{\text{int}}\right)}$
(5)

We introduce the following variables: $n_{e}$ and $n_{d}$, representing the
densities of massive and massless electrons, respectively, and $\tau_{e}$ and
$\tau_{d}$ as their respective scattering times, due to impurities and static
defects. Furthermore, we define $N_{s}$ as the total electron density. In
addition, $q$ is the elementary charge, $m_{e}=\hbar^{2}k\frac{dk}{dE_{s}}$ is
the heavy electron effective mass, $m_{d}=\mu/v_{F}^{2}$ is the Dirac electron
effective mass, $\tau_{ed(de)}$ is the collision time with heavy (Dirac)
electron per Dirac (heavy) electron. We also introduce
$1/\tau_{int}=1/\tau_{de}+1/\tau_{ed}$. The equation, considering the
constraint that interactions between massive and massless Dirac electrons
conserve the overall momentum density, is:
$n_{d}m_{d}\tau_{de}=n_{e}m_{e}\tau_{ed}$. Hence, the term in equation 5 that
remains temperature-independent is anticipated to be primarily governed by
interface roughness and impurity scattering, as discussed in work melezhik .
Conversely, the temperature-dependent component of resistivity can be
attributed to electron-electron (e-e) friction within the non-Galilean
invariant massive-massless Dirac liquid.

An insightful analysis would involve examining the limits of $T=0$ and
$T=\infty$ especially in cases where the effective masses differ
significantly. The Dirac hole’s effective scattering time can be approximated
using the relation $m_{d}=\mu/v_{F}^{2}\approx 0.006m_{0}$, with $\mu\approx
16meV$. At $T\rightarrow 0$ both bands contribute to the total conductivity
dominated by the Dirac holes:

$\sigma(T=0)=\frac{q^{2}n_{d}v^{2}\tau_{d}}{\mu}=\frac{q^{2}n_{d}\tau_{d}}{m_{d}}.$
(6)

At $T=\infty$ the conductivity becomes temperature independent and saturates
at a value approximately determined by the conductivity of the heavy hole band
:

$\sigma(T=\infty)=\frac{q^{2}(n_{d}+n_{e})^{2}\tau_{e}}{m_{e}n_{e}}$ (7)

In our case $n_{e}>>n_{d}$, the ratio of the resistivities at both temperature
limits is determined by :

$\frac{\rho(T=\infty)}{\rho(T=0)}=\frac{m_{e}n_{d}\tau_{d}}{m_{d}n_{e}\tau_{e}}$
(8)

When $n_{e}\tau_{e}\sim n_{d}\tau_{d}$, the ratio of the resistivities at
these temperature extremes is predominantly influenced by the effective mass
ratio (for Fermi energy $E_{F}=5meV$) pal :

$\frac{\rho(T=\infty)}{\rho(T=0)}\approx\frac{m_{e}}{m_{d}}\approx 4.4$ (9)

Hence, it is reasonable to anticipate that in the optimal system where
electron-electron interactions govern transport, there will be energy branches
characterized by distinct effective masses. The gapless HgTe system stands out
as a highly promising platform for conducting this type of research.

In a regime of full degeneracy, where both types of electrons adhere to Fermi
statistics, an expression for particle-particle collisions has been derived,
with slight modifications to account for the difference in their energy
spectra gusev ; kovalev :

$\frac{1}{\tau_{de}}=\alpha\frac{m_{e}(kT)^{2}U_{0}^{2}}{3\pi\hbar^{5}vv_{F}}\equiv\frac{1}{\tau_{0}}\sim
T^{2},\\\ $ (10)
$\quad\frac{1}{\tau_{ed}}=\frac{1}{\tau_{0}}\frac{\mu}{m_{e}v_{F}^{2}}\frac{n_{d}}{n_{e}}\sim
T^{2},\\\ $ (11)

where $U_{0}=\frac{2\pi e^{2}}{\epsilon q_{s}}$, $\quad
q_{s}=\frac{me^{2}}{\varepsilon\hbar^{2}}$, $\epsilon$ is the dielectric
constant of the material, v- is Fermi velocity of the massive particle,
prefactor $\alpha$ represents a numerical coefficient that varies based on the
specifics of the Coulomb interactions. Note that $1/\tau_{ed}<<1/\tau_{de}$,
and conventional $T^{2}$ behavior for $1/\tau_{int}\sim 1/\tau_{de}$,
characteristic of a particle with parabolic dispersion, becomes evident.

Remember that, in the case of finite temperature, the electron density can be
found from the equation:

$n_{d,e}=\int\limits_{0}^{\infty}D_{\varepsilon}^{d,e}(\varepsilon)(1+e^{(\varepsilon-\mu)/kT})^{-1}d\varepsilon$
(12)

where $D_{\varepsilon}^{d,e}$ is the density of the states of the Dirac and
massive electrons respectively. It is essential to emphasize that even though
the density in each subband strongly relies on the temperature at high T or in
the nondegenerate regime, the total density is constrained by the gate
voltage, expressed as $n_{e}+n_{d}=N_{s}\sim V_{g}$. As a result, the chemical
potential can be parametrically determined.

Table 2:  Fitting parameters in equations (5), (10) for 2 samples. Sample | $E_{F}$ | $N_{s}$ | $\tau_{e}$ | $\tau_{d}$ | $n_{e}$ | $n_{d}$ | $\alpha$
---|---|---|---|---|---|---|---
|  | $10^{11}$ | $10^{-13}$ | $10^{-13}$ | $10^{11}$ | $10^{11}$ |
| meV | $cm^{-2}$ | $s$ | $s$ | $cm^{-2}$ | $cm^{-2}$ |
A | 33.8 | 5 | $0.31$ | $4.7$ | $3.3$ | $1.7$ | 0.5
A | 30.4 | 4.2 | $0.3$ | $4.6$ | $2.8$ | $1.3$ | 0.7
B | 30.7 | 4.2 | $0.29$ | $4.3$ | $2.8$ | $1.3$ | 0.17
B | 26.5 | 3.3 | $0.41$ | $5$ | $2.3$ | $1$ | 0.2

To compare with the theoretical framework, we performed a fitting analysis on
the temperature-dependent data, as illustrated in Figures 4 and 5. This
analysis involved a single adjustable parameter denoted as $\alpha$, which
accounts for the interaction strength between the Dirac and massive holes as
outlined in Equations 5 -11. The scattering parameters, referred to as
$\tau_{e(d)}$, play a predominant role in determining resistivity at lower
temperatures. Importantly, varying these parameters within a reasonable range
does not impact the friction coefficient, which is the primary driver of
temperature-dependent resistivity. In Figures 4 and 5 we present the
theoretical dependencies of resistivity excess for various total density
values. Notably, the experimental data closely aligns with the expected
dependence $\Delta\rho(T)\sim T^{2}$ for the parameters specified in Table II.

It is evident that the prefactor $\alpha$, introduced as an adjustable
parameter reflecting the difference between the actual Coulomb potential and
the contact interaction potential, is close to unity and exhibits a tendency
to increase as the particle density decreases. It’s worth noting, however,
that the corresponding prefactor for sample B is smaller than that of sample
A. It is important to note that both samples exhibit a resistivity pattern
consistent with the $T^{2}$ law within one order of magnitude.

##  V Comparison of the theory with experiment in non -degenerate regime

Figure 6: (Color online) Temperature dependence of the relaxation rate for
different density ratio $n_{e}/n_{d}$. Solid lines are computed using
equations (13,15) ($1/\tau_{de}^{*}$, dashes are computed using equations
(14,16) ($1/\tau_{ed}^{*})$), dot-dashes represent $1/\tau_{int}^{*}$. Total
density $N_{s}=3\times 10^{10}cm^{-2}$, (a) ratio $n_{e}/n_{d}=0.5$ (black
lines), (b) $n_{e}/n_{d}=1.5$ (blue lines).

.

At low densities and moderately high temperatures, both massive and massless
Dirac electrons start adhering to Boltzmann statistics. In this particular
regime, an expression for electron-electron collisions has been formulated,
incorporating minor adjustments to accommodate variations in their energy
spectra kovalev :

$\frac{1}{\tau_{de}^{*}}=\beta\frac{\sqrt{\pi}}{6}\frac{n_{e}}{(kT)^{3}}\frac{v_{T}m_{e}^{4}v_{F}^{5}}{\hbar^{3}}U_{0}^{2}\mathfrak{J_{1}}\sim
T^{-1/2}\mathfrak{J_{1}}(T)$ (13)
$\frac{1}{\tau_{ed}^{*}}=\beta\frac{\sqrt{\pi}}{4}\frac{n_{d}}{(kT)^{2}}\frac{v_{T}m_{e}^{3}v_{F}^{3}}{\hbar^{3}}U_{0}^{2}\mathfrak{J_{2}}\sim
T^{1/2}\mathfrak{J_{2}}(T)$ (14)

where $U_{0}=\frac{2\pi e^{2}}{\epsilon q_{T}}$, $\quad q_{T}=\frac{2\pi
e^{2}n_{e}}{\varepsilon(kT)}$, $\epsilon$ is the dielectric constant of the
material, $v_{F}$\- is Fermi velocity of massless Dirac particles,
$v_{T}=\sqrt{\frac{2kT}{m_{e}}}$ is velocity of massive particles, The
prefactor denoted by $\beta$ corresponds to a numerical coefficient that
depends on the details of the Coulomb interactions. Integrals
$\mathfrak{J_{1,2}}$ are represented by the following expressions:

$\mathfrak{J_{1}}=\int\limits_{0}^{\infty}y^{2}dy\int\limits_{-1}^{1}\sqrt{1-x^{2}}e^{-\frac{m_{e}v_{F}^{2}}{2kT}(x+\frac{y}{2})^{2}}dx$
(15)
$\mathfrak{J_{2}}=\int\limits_{0}^{\infty}y^{2}dy\int\limits_{-1}^{1}\frac{e^{-\frac{m_{e}v_{F}^{2}}{2kT}(x+\frac{y}{2})^{2}}}{\sqrt{1-x^{2}}}dx$
(16)

Figure 7: (Color online) (a) Resistivity $\rho(T)$ as a function of
temperature for different densities for sample A. The total density is
$N_{s}\approx 0$ (charge neutrality point) (blue); $N_{s}\approx 3\times
10^{10}cm^{-2}$ (red); $N_{s}\approx 6.1\times 10^{10}cm^{-2}$ (black);
$N_{s}\approx 8\times 10^{10}cm^{-2}$ (cyan); $N_{s}\approx 10\times
10^{10}cm^{-2}$ (dark yellow). (b) Resistivity $\rho(T)$ as a function of
temperature for different densities for sample B. The total density is
$N_{s}\approx 0$ (charge neutrality point) (blue); $N_{s}\approx 3\times
10^{10}cm^{-2}$ (red); $N_{s}\approx 6.1\times 10^{10}cm^{-2}$ (black).
$N_{s}\approx 8\times 10^{10}cm^{-2}$ (cyan). Figure 8: (Color online) (a) The
resistivity ratio$\rho(T)/\rho(T=4.2K)$ as a function of temperature for
different parameters calculated from Eqs.(5,13,14). Total density
$N_{s}=3\times 10^{10}cm^{-2}$, the ratio
$\tau_{e}/\tau_{d}=5,\tau_{d}=0.4\times 10^{-12}s$, the ratio
$n_{d}/n_{e}(T=4.2K)=1.5$ (black); 1 (red); 0.5 (blue). Parameter $\beta=1$.

Figure 6 depicts the correlation between the relaxation rates $1/\tau_{de}$
and $1/\tau_{ed}$ and temperature, represented by the blue and black lines.
The computations employ Equations 13 and 14 with the parameter $\beta$ set to
1. Due to the integrals $\mathfrak{J_{1,2}}$ behaving approximately like power
functions of $T^{\delta}$ with $\delta\approx 0.5$, the relaxation rates
exhibit distinct temperature dependencies. Specifically, the rate
$1/\tau_{de}$ is a weakly temperature-dependent function, while the collision
rate $1/\tau_{ed}$ follows a quasi linear $T$ dependence.

The relationship between rates $1/\tau_{de}$ and $1/\tau_{ed}$ depends on the
ratio $n_{e}/n_{d}$ and the temperature interval. Notably, over a wide
temperature range, the interaction rate maintains the relationship
$1/\tau_{de}>1/\tau_{ed}$ for $n_{e}/n_{d}=1.5$. Moreover, the relaxation rate
$1/\tau_{de}$ surpasses $1/\tau_{de}$ at temperatures exceeding 35K for the
inverse relation $n_{e}/n_{d}=1.5$. It is essential to note that interactions
between massive and massless Dirac electrons conserve the overall momentum
density, and the total relaxation rate involved in the transport coefficient
is given by $1/\tau_{int}^{*}=1/\tau_{de}^{*}+1/\tau_{ed}^{*}$. The
temperature dependence of the relaxation rate $1/\tau_{int}^{*}(T)$ describes
an intermediate power with a temperature-dependent exponent index.

In Figure 1c, the chemical potential exhibits a range of variation from 1 to 3
meV for total electron densities in the order of $(1-3)\times 10^{10}cm^{-2}$.
Within this density range, we can assert that both massive and massless Dirac
electrons adhere to Boltzmann statistics for temperatures between $10K$ and
$100K$.

Figures 7a and 7b illustrate the temperature dependence of resistance in the
vicinity of the charge neutrality point for low electron densities. It is
evident that resistance decreases as the temperature increases near the CNP.
This trend is attributed to the presence of a small gap in the spectrum,
approximately $\Delta\sim 1meV$, leading to an activation law
$R\sim\exp(\Delta/2kT)$ for the resistance, as indicated in ferreira .
Additionally, this behavior is justified by the quantization of resistance and
nonlocal resistance resulting from the helical edge states, confirming the
topological insulator nature of the triple HgTe-based quantum well, as
demonstrated in our previous publications ferreira . Furthermore, it is
crucial to note that although the involvement of helical edge states is
significant, especially when the chemical potential resides within the gap,
their contribution becomes negligible, when $\mu$ is shifted to the
conductivity band, and the edge states become heavily intertwined with the
bulk conductivity.

Beyond the CNP, we observe weakly temperature-independent resistance, as
depicted in Figures 3a and 3b, and more comprehensively in Figures 7a and 7b.
Notably, a strong temperature dependence of resistance is discernible far from
the CNP, where the resistance increases by more than an order of magnitude
with rising temperature. This observation underscores distinct transport
regimes in the regions near and far from the CNP—specifically, for low
densities where a non-degenerate system can be anticipated, and at high
densities where the system becomes degenerate across the entire temperature
range.

To assess the agreement of our results with theory in the non-degenerate
regime, we computed the resistivity using Equations 5 and 13-16. Figure 8
illustrates these calculations for a total density $N_{s}=3\times
10^{10}cm^{-2}$ and various density ratios $n_{d}/n_{e}(T=4.2K)$.
Interestingly, there is a small ($\sim 2\%$) reduction in resistance with
increasing temperature. This finding may seem surprising given the distinct
growth observed in the scattering time $1/\tau_{int}^{*}(T)$ (see Fig. 6). It
is essential to note, however, that in the Boltzmann regime, several
parameters are expected to exhibit temperature dependence.

Primarily, the densities of massive and massless electrons strongly depend on
temperature in the non-degenerate regime (see 12). Despite this, as previously
mentioned, the total density remains fixed by the external gate voltage,
suggesting a substantial redistribution of charge carriers as temperature
increases. Additionally, the effective mass exhibits strong temperature
dependence, significantly contributing to the temperature-dependent
resistivity. These temperature-dependent parameters counterbalance the
increase in the relaxation rate $1/\tau_{int}^{*}(T)$ and ultimately result in
only a weakly temperature-dependent resistance. This effect bears some
similarity to the situation in graphene, where the non-degenerate limit has
been explored in single-layer and bilayer graphene, with electron-hole pairs
being thermally excited nam ; tan . In these cases, the electron-hole
collision rate is expected to be proportional to $T$, leading to temperature-
independent conductivity.

It’s important to acknowledge that our model, while informative, is overly
simplistic for capturing the intricate details observed in the experimental
data presented in Figures 7a and 7b, particularly in terms of exact evolution
of the resistance with T. Achieving a more accurate agreement with the
experimental results, especially with Equation 5, necessitates a precise
understanding of the behavior of the density of states within the gap.

Additionally, our tight-binding model, though useful, is too rudimentary to
accurately characterize parameters in the vicinity of the CNP, where potential
disorder may also contribute to smoothing effects. Consequently, it becomes
evident that the weak temperature dependence observed in the experiments near
the CNP aligns with our theoretical model to a certain extent. However, it is
crucial to recognize the limitations of our model and acknowledge the need for
more sophisticated approaches to capture the nuances exhibited by the
experimental data.

##  VI conclusion

In summary, our study focused on the temperature-dependent resistivity in a
triple HgTe quantum well that accommodates two branches of fermions, one with
massive and the other with massless Dirac characteristics. We observed
quadratic temperature dependencies resulting from interactions between the
Dirac fermions and massive electrons in the fully degenerate regime. In
contrast, when both type of electrons adhered to Boltzmann statistics,
resistivity remained weakly temperature-dependent.

Our findings validate that the presented model comprehensively describes the
conductivity of the triple quantum well across a broad spectrum of
temperatures and carrier densities. In our ultraclean samples, electron-
electron scattering takes precedence, exerting a significantly more notable
impact than impurity scattering. It also demonstrates the unified nature of
hydrodynamic transport across different systems, irrespective of the carriers’
sign and spectrum type.

##  VII ACKNOWLEDGMENTS

This work is supported by FAPESP (São Paulo Research Foundation) Grants No.
2019/16736-2 and No. 2021/12470- 8, CNPq (National Council for Scientific and
Technological Development), and by the Ministry of Science and Higher
Education of the Russian Federation and Foundation for the Advancement of
Theoretical Physics and Mathematics “BASIS.”

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