# Electron-hole collision-limited resistance of gapped graphene

Arseny Gribachov  Moscow Institute of Physics and Technology, Dolgoprudny
141700, Russia  Vladimir Vyurkov  Moscow Institute of Physics and Technology,
Dolgoprudny 141700, Russia  Valiev Institute of Physics and Technology RAS,
36/1 Nahimovsky Ave., 117218, Moscow, Russia  Dmitry Svintsov
<EMAIL_ADDRESS>Moscow Institute of Physics and Technology, Dolgoprudny
141700, Russia

###### Abstract

Collisions between electrons and holes can dominate the carrier scattering in
clean graphene samples in the vicinity of charge neutrality point. While
electron-hole limited resistance in pristine gapless graphene is well-studied,
its evolution with induction of band gap $E_{g}$ is less explored. Here, we
derive the functional dependence of electron-hole limited resistance of gapped
graphene $\rho_{eh}$ on the ratio of gap and thermal energy $E_{g}/kT$. At low
temperatures and large band gaps, the resistance grows linearly with
$E_{g}/kT$, and possesses a minimum at $E_{g}\approx 2.5kT$. This contrast to
the Arrhenius activation-type behaviour for intrinsic semiconductors.
Introduction of impurities restores the Arrhenius law for resistivity at low
temperatures and/or high doping densities. The hallmark of electron-hole
collision effects in graphene resistivity at charge neutrality is the
crossover between exponential and power-law resistivity scalings with
temperature.

##  I Introduction

The problem of minimal graphene conductivity $\sigma_{\min}$ observed at
charge neutrality has been a subject of long debate since the discovery of
graphene. Experimental studies have shown that $\sigma_{\min}$ varies slightly
between samples and with changing the temperature Novoselov  _et al._ (2005);
Tan  _et al._ (2007); Bolotin  _et al._ (2008), which posed questions about
the universality of this quantity. Numerous theoretical works attempted to
derive $\sigma_{\min}$ from Kubo’s formula for clean graphene at zero
frequency of electric field $\omega$, zero temperature $T$ and doping
$\varepsilon_{F}$ Herbut  _et al._ (2008); Mishchenko (2008); Gorbar  _et al._
(2002). The apparent ’universal’ result appeared to depend on the order of
taking the limits of zero frequency, temperature and doping Ziegler (2007).
The latter fact indicated on the deficiency of model of ’clean graphene’ for
derivation of universal minimal conductivity.

The first resolution of the minimal conductivity puzzle appeared by realizing
that graphene at charge neutrality has some residual doping. This doping comes
from random impurities in the sample, arranged in positively and negatively
charged clusters. The free carriers of graphene tend to screen these impurity
charges, forming the electron-hole puddles Martin  _et al._ (2008). The root-
mean-square density of charge carriers in graphene appears to be non-zero.
Instead, it is proportional to the impurity density $n_{i}$, and so is the
carrier scattering rate. These two proportionalities lead to a very weak
dependence of $\sigma_{\min}$ on the density of residual impurities and
temperature, which can be considered as ’approximate universality’ Adam  _et
al._ (2007); Cheianov  _et al._ (2007).

With the current level of graphene technology, the density of residual
impurities can readily be lower than the density of thermally activated
electrons and holes $n_{\rm th}\approx 8\times 10^{10}$ cm-2 Cao  _et al._
(2015). In such a situation, electron-hole puddles and impurity scattering
make a minor contribution to the resistivity. The scattering between electrons
and holes now governs the experimentally measured value of minimum
conductivity Nam  _et al._ (2017); Berdyugin  _et al._ (2022); Bandurin  _et
al._ (2022); Gallagher  _et al._ (2019). A strong violation of Wiedemann-
Frantz relation between electrical and thermal conductivity at charge
neutrality Crossno  _et al._ (2016) and the appearance of new electron-hole
sound waves Zhao  _et al._ (2023) also indicate on the dominant role of e-h
scattering in clean samples. A scaling estimate of e-h limited conductivity
was presented in Ref. Vyurkov and Ryzhii, 2008 and resulted in
$\sigma_{\min}=C\alpha_{c}^{-2}e^{2}/h$, where $\alpha_{c}=e^{2}/\hbar v_{0}$
is the Coulomb coupling constant, $v_{0}$ is the velocity of massless
electrons in graphene, and $C$ is the numerical prefactor. A rigorous solution
of the kinetic equation using the variational principle confirmed the result
and established $C\approx 0.76$ Kashuba (2008); Fritz  _et al._ (2008).

While most theoretical and experimental studies were devoted to the electron-
hole scattering in pristine gapless graphene, very little attention have been
paid to the same process in gapped systems. The gap induction in single
graphene layer is possible under lattice reconstruction on boron nitride
substrates Jung  _et al._ (2015). The gap is readily induced in graphene
bilayer under the action of transverse electric fields McCann  _et al._
(2007). The derivatives of graphene are not the only examples of 2d electron
systems with small energy gap. Another family is represented by quantum wells
based on mercury cadmium telluride of sub-critical thickness König  _et al._
(2007); Marcinkiewicz  _et al._ (2017). Given the large variety of clean 2d
systems with small band gap and their potential applications in nano- and
optoelectronics, it is natural to study the factors limiting their electrical
resistivity, particularly, the inevitable electron-hole scattering. An attempt
to derive and measure the electron-hole limited resistivity was presented in
Tan  _et al._ , 2022. Its results cannot be considered as satisfactory because
the electron-hole scattering times $\tau_{eh}$ in the presence of band gap
$E_{g}$ were not derived, but rather guessed. The authors of have proposed a
universal function $f(E_{g}/kT)$ governing the scaling of e-h limited
conductivity with band gap; the function possessed a quadratic maximum at
$E_{g}=0$ and dropped exponentially at $E_{g}/kT\gg 1$. Such dependence was
seemingly confirmed by the experiment.

The present paper is aimed at a rigorous derivation of electron-hole
scattering time and conductivity at neutrality point in gapped graphene. Our
formalism is based on kinetic equation with carrier-carrier collision
integral; the carriers are assumed interacting via unscreened Coulomb
potential. The kinetic equation is solved with a variational principle which
yields good results for the estimates of conductivity Fritz  _et al._ (2008).
We find that at large band gaps, the conductivity scales as
$\sigma_{\min}\propto kT/E_{g}$. This behaviour differs essentially from
conventional Arrhenius-type activation. Such non-Arrhenius behaviour of
minimal conductivity can be explained with simple gas kinetics arguments. The
free path time of a trial electron against a dilute hole background is
inversely proportional to the hole density $\tau_{eh}\propto n^{-1}_{h}\propto
e^{E_{g}/2kT}$, i.e. grows exponentially with gap induction. The Drude
conductivity is proportional to the product of electron density and free-path
time, $\sigma_{\min}\propto n_{e}\tau_{eh}/m^{*}$. As electron and hole
components are balanced at charge neutrality, $n_{e}=n_{h}$, the leading
Arrhenius exponents are cancelled in the expression for conductivity.
Eventually, the conductivity depends on the gap only via effective mass,
$m^{*}=E_{g}/2v_{0}^{2}$. This justifies the hyperbolic dependence of
$\sigma_{\min}$ on $E_{g}$.

The above intuitive explanation is missing the long-range character of Coulomb
interaction. In classical plasmas, the latter led to log-divergent collision
integral Landau (1936). We show that no such divergences appear during the
evaluation of conductivity. The collision integral converges both at small
momentum transfers $q\rightarrow 0$ as such momenta do not change the electric
current, and at large momenta $q\rightarrow\infty$ due to small quantum-
mechanical overlap between scattered states. All in all, our variational
derivation results in following expression for the conductivity at charge
neutrality valid at $E_{g}\gg kT$

$\sigma_{\min}=\frac{8}{\pi}\frac{e^{2}}{h}\alpha_{C}^{-2}\frac{kT}{E_{g}}.$
(1)

##  II Variational approach to kinetic equation with carrier-carrier
collisions

Electron states in the gapped graphene are described by a ’massive’ Dirac
Hamiltonian

$\hat{\mathcal{H}}_{D}=\left[\begin{array}[]{cc}E_{g}/2&v_{0}\left(\hat{p}_{x}-i\hat{p}_{y}\right)\\\
v_{0}\left({\hat{p}}_{x}+i{\hat{p}}_{y}\right)&-E_{g}/2\\\ \end{array}\right]$

Such Hamiltonian is applicable to the single layer graphene aligned to boron
nitride substrate, and to 2d electron system in CdHgTe quantum wells. Its
applicability to graphene bilayer is limited, as the latter has a quadratic
band touching at $E_{g}=0$. We will further argue that such Hamiltonian is
applicable for bilayer at large induced gaps under proper replacement of
parameters.

Figure 1: Scattering diagrams for the processes contributing to graphene
resistivity at charge neutrality point: electron-impurity, electron-electron,
and electron-hole scattering (normal and annihilation-type processes)

The dc conductivity is obtained by solving the kinetic equation with respect
to distribution function which we linearize as $f_{p}=f_{0}-\Phi_{e/h,\bf
p}\partial f_{0}/\partial\varepsilon$, here $\Phi_{e/h,\bf p}$ is the linear
function of electric field ${\bf E}$, the subscripts $e$ and $h$ distinguish
between electrons and holes. We restrict our consideration to the carrier-
carrier and carrier-impurity collisions. The former involve electron-electron
(e-e), hole-hole (h-h), and electron-hole (e-h) collisions (Fig. 1). The
kinetic equation for electrons takes on the form

$-e{\bf E}{\bf v_{p}}\frac{\partial
f_{0}}{\partial\varepsilon}=-{\mathcal{C}}_{ei}\\{\Phi_{e,\bf
p}\\}-{\mathcal{C}}_{ee}\\{\Phi_{e,\bf p}\\}-{\mathcal{C}}_{eh}\\{\Phi_{e,\bf
p},\Phi_{h,\bf p}\\},$ (2)

and similarly for holes with apparent change of signs. At charge neutrality
electrons and holes move in the opposite directions, thus $\Phi_{e,\bf
p}=-\Phi_{h,\bf p}\equiv\Phi_{\bf p}$. The complex structure of carrier-
carrier collision integrals makes the exact solution of (2) impossible, at
least in the non-degenerate case Combescot and Combescot (1987). For this
reason, we use the variational approach with a reasonable trial form of
distribution function to get an estimate of the resistivity Ziman (2001);
Maldague and Kukkonen (1979). The problem of resistivity is weakly sensitive
to the specific form of $\Phi_{\bf p}$ (unlike the problems of thermoelectric
coefficients Takahashi  _et al._ (2023)), thus the variational approach is
efficient for predicting its functional dependence.

Within this approach, one maximizes the entropy generation rate being a
quadratic functional of distribution function $\Phi$:

$\displaystyle\mathcal{Q}[\Phi]=\mathcal{Q}_{0}[\Phi]+\mathcal{Q}_{ei}[\Phi]+\mathcal{Q}_{ee}[\Phi]+\mathcal{Q}_{eh}[\Phi],$
(3) $\displaystyle\mathcal{Q}_{0}[\Phi]=-eN\sum_{\bf p}({\bf v_{p}}{\bf
E})\frac{\partial f_{0}}{\partial\varepsilon}\Phi_{\bf p},$ (4)
$\displaystyle{\mathcal{Q}}_{ei}[\Phi]=\frac{N}{2}\sum\limits_{\bf
pp^{\prime}}{\frac{\partial
f_{0}}{\partial\varepsilon}(\Phi_{p}-\Phi_{p^{\prime}})^{2}W^{ei}_{\bf
pp^{\prime}}},$ (5)

$\displaystyle\mathcal{Q}_{ee}[\Phi]=\frac{N}{8}\frac{1}{kT}\sum\limits_{\bf{pkq}}{{{f}_{0}}\left(\bf{p}\right){{f}_{0}}\left(\bf{k}\right)[1-{{f}_{0}}\left({\bf{{p}^{\prime}}}\right)][1-{{f}_{0}}\left({\bf{{k}^{\prime}}}\right)]W_{\bf{pk},\bf{{p}^{\prime}{k}^{\prime}}}^{ee}{{\left[\left({{\Phi}_{p}}+{{\Phi}_{k}}\right)-\left({{\Phi}_{{{p}^{\prime}}}}+{{\Phi}_{{{k}^{\prime}}}}\right)\right]}^{2}}},$
(6)
$\displaystyle\mathcal{Q}_{eh}[\Phi]=\frac{N}{8}\frac{1}{kT}\sum\limits_{\bf{pkq}}{{{f}_{0}}\left(\bf{p}\right){{f}_{0}}\left(\bf{k}\right)[1-{{f}_{0}}\left({\bf{{p}^{\prime}}}\right)][1-{{f}_{0}}\left({\bf{{k}^{\prime}}}\right)]W_{\bf{pk},\bf{{p}^{\prime}{k}^{\prime}}}^{eh}{{\left[\left({{\Phi}_{p}}-{{\Phi}_{k}}\right)-\left({{\Phi}_{{{p}^{\prime}}}}-{{\Phi}_{{{k}^{\prime}}}}\right)\right]}^{2}}}$
(7)

Above, we have introduced the Fermi golden rule scattering probabilities in
unit time, given by:

$\displaystyle W_{\bf{pp^{\prime}}}^{ei}=\frac{2\pi}{\hbar}n_{\rm
imp}{{\left|V\left(\bf{q}\right)\right|}^{2}}{{\left|M_{\bf{p{p}^{\prime}}}^{++}\right|}^{2}}\delta\left({{\varepsilon}_{\bf{p}}}-{{\varepsilon}_{\bf{p^{\prime}}}}\right),$
(8) $\displaystyle
W_{\bf{pk},\bf{{p}^{\prime}{k}^{\prime}}}^{ee}=\frac{2\pi}{\hbar}N{{\left|V\left(\bf{q}\right)\right|}^{2}}{{\left|M_{\bf{p{p}^{\prime}}}^{++}\right|}^{2}}{{\left|M_{\bf{k{k}^{\prime}}}^{++}\right|}^{2}}\delta\left({{\varepsilon}_{\bf{p}}}+{{\varepsilon}_{\bf{k}}}-{{\varepsilon}_{{\bf{{p}^{\prime}}}}}-{{\varepsilon}_{{\bf{{k}^{\prime}}}}}\right),$
(9) $\displaystyle
W_{\bf{pk},\bf{{p}^{\prime}{k}^{\prime}}}^{eh}=\frac{2\pi}{\hbar}N\left[{\left|V\left(\bf{q}\right)\right|}^{2}{{\left|M_{\bf{p{p}^{\prime}}}^{--}\right|}^{2}}{{\left|M_{\bf{k{k}^{\prime}}}^{++}\right|}^{2}}+{\left|V\left(\bf{p}+\bf{k}\right)\right|}^{2}{{\left|M_{\bf{pk}}^{+-}\right|}^{2}}{{\left|M_{\bf{p^{\prime}k^{\prime}}}^{+-}\right|}^{2}}\right]\delta\left({{\varepsilon}_{\bf{p}}}+{{\varepsilon}_{\bf{k}}}-{{\varepsilon}_{{\bf{{p}^{\prime}}}}}-{{\varepsilon}_{{\bf{{k}^{\prime}}}}}\right)$
(10)

where $N=4$ is the degeneracy factor,
$\varepsilon_{p}=(E_{g}^{2}/4+p^{2}v_{0}^{2})^{1/2}$ is the energy spectrum in
the gapped graphene, $V({\bf q})=2\pi e^{2}/\kappa|{\bf q}|$ is the Fourier-
transformed Coulomb potential, $\kappa$ is the background dielectric constant,
$M^{ss^{\prime}}_{\bf pp^{\prime}}$ is the overlap of chiral wave functions
between bands $s$ and $s^{\prime}$ ($s=+1$ for the conduction and $s=-1$ for
the valence band, respectively)

$|M^{ss^{\prime}}_{\bf
pp^{\prime}}|^{2}=\frac{\varepsilon_{p}\varepsilon_{p^{\prime}}+(E_{g}/2)^{2}+ss^{\prime}(v_{0}{\bf
p}\cdot v_{0}{\bf p^{\prime}})}{2\varepsilon_{p}\varepsilon_{p^{\prime}}}.$
(11)

Two fundamental differences between effects of e-e and e-h collision integrals
should be mentioned at that stage. First, there’s a difference in signs with
which the function $\Phi_{\bf p}$ enters the expressions (6) and (7). It stems
from the fact that the entropy production rate ${\mathcal{Q}}$ is finite only
when collisions change the electric current; and the quadratic-in-$\Phi$
expression in square brackets of (6) and (7) can be associated with the
collision-induced change in electric current. Naturally, the current carried
by two particles with momenta ${\bf p}$ and ${\bf k}$ depends on their charge,
which explains the difference of collision integrals for e-e and e-h
scattering.

Another difference between e-e and e-h collisions lies in the presence of an
’annihilation scattering’, where electron and hole collide with virtual photon
production, and subsequently yield yet another pair. Such scattering is
represented by the second term in square brackets of (10). In all other
aspect, electron and hole collisions become identical if we neglect the
exchange effects. The latter appear to be numerically small for quite a large
number of particle ’sorts’ $N=4$.

Our trial distribution function is selected as

$\Phi_{\bf p}=\tau e({\bf v_{p}}\cdot{\bf E})$ (12)

where $\tau$ is the parameter subjected to the optimization having the meaning
of transport relaxation time, and ${\bf v_{p}}=\partial\varepsilon_{\bf
p}/\partial{\bf p}$ is the electron velocity. Such choice of $\Phi_{\bf p}$ is
important to reproduce the finite resistivity of gapless graphene. Other forms
lead to the log-divergent collision integral due to the prolonged interaction
of carriers with collinear momenta Fritz  _et al._ (2008). In the gapped case,
$\Phi_{\bf p}$ given by (12) is not the only possible choice, but we stick to
it for traceability of our result to the preceding studies.

The optimization of the entropy functional with respect to the scattering time
yields the following result:

$\tau^{*}=\frac{D}{C},\qquad\sigma_{eh}=D\tau^{*},$ (13)

where $D$ is the Drude weight and $C=C_{ei}+C_{ee}+C_{eh}$ is net collision
rate. Expressions for $D$ and $C$ are readily obtained from the entropy
functional:

$\displaystyle C_{ei}=\frac{Ne^{2}}{2}\sum\limits_{\bf
pp^{\prime}}{\frac{\partial f_{0}}{\partial\varepsilon}({\bf v_{p}}-{\bf
v_{p^{\prime}}})^{2}W^{ei}_{\bf pp^{\prime}}},$ (14) $\displaystyle
C_{ee}=\frac{Ne^{2}}{8kT}\sum\limits_{\bf{pkq}}{{{f}_{0}}\left(\bf{p}\right){{f}_{0}}\left(\bf{k}\right)[1-{{f}_{0}}\left({\bf{{p}^{\prime}}}\right)][1-{{f}_{0}}\left({\bf{{k}^{\prime}}}\right)]W_{\bf{pk},\bf{{p}^{\prime}{k}^{\prime}}}^{ee}{{\left[\left({{\bf{v}}_{\bf{p}}}+{{\bf{v}}_{\bf{k}}}\right)-\left({{\bf{v}}_{{\bf{{p}^{\prime}}}}}+{{\bf{v}}_{{\bf{{k}^{\prime}}}}}\right)\right]}^{2}}},$
(15) $\displaystyle
C_{eh}=\frac{Ne^{2}}{8kT}\sum\limits_{\bf{pkq}}{{{f}_{0}}\left(\bf{p}\right){{f}_{0}}\left(\bf{k}\right)[1-{{f}_{0}}\left({\bf{{p}^{\prime}}}\right)][1-{{f}_{0}}\left({\bf{{k}^{\prime}}}\right)]W_{\bf{pk},\bf{{p}^{\prime}{k}^{\prime}}}^{eh}{{\left[\left({{\bf{v}}_{\bf{p}}}-{{\bf{v}}_{\bf{k}}}\right)-\left({{\bf{v}}_{{\bf{{p}^{\prime}}}}}-{{\bf{v}}_{{\bf{{k}^{\prime}}}}}\right)\right]}^{2}}},$
(16) $\displaystyle
D=-\frac{Ne^{2}}{2}\sum\limits_{\mathbf{p}}{v_{p}^{2}}\frac{\partial{{f}_{0}}}{\partial\varepsilon}.$
(17)

Expressions for the electron-electron and electron-hole collision rates look
very similar, and seem formally of the same order of magnitude. However, in
parabolic gap case (realized at $E_{g}\gg kT$), the momentum conservation upon
e-e collisions implies the conservation of total current by the virtue of
proportionality ${\bf p}=m^{*}{\bf v_{p}}$. This feature is captured by Eq.
(15), where the velocity factor in the square brackets is exactly zero if
${\bf p}=m^{*}{\bf v_{p}}$. Even in the gapless case, where proportionality
between velocity and momentum does not hold, e-e collisions make a numerically
small contribution to resistivity, compared to the e-h processes Svintsov  _et
al._ (2014).

Expressions (12 \- 16) are the central results of our paper and, in principle,
enable the direct numerical evaluation of conductivity for the arbitrary value
of the gap. For numerical purposes, it is convenient to eliminate the energy
delta-functions by introducing the extra transferred energy variable $\omega$

$\delta\left({{\varepsilon}_{\bf{p}}}+{{\varepsilon}_{\bf{k}}}-{{\varepsilon}_{{\bf{{p}^{\prime}}}}}-{\varepsilon_{{\bf{k^{\prime}}}}}\right)=\\\
\int{d\omega\delta\left(\omega-{{\varepsilon}_{\bf{p}}}+{\varepsilon_{\bf
p-\bf q}}\right)\delta\left(\omega-{{\varepsilon}_{\bf k+\bf
q}}+\varepsilon_{\bf{k}}\right)},$ (18)

and integrate over the angles $\theta_{\bf pq}$ and $\theta_{\bf kq}$
analytically. Further simplifications are possible only in the limit of large
gaps $E_{g}\gg kT$ and are described in Appendix A.

##  III Gap-dependent conductivity limited by carrier-carrier scattering

We start our inspection of collision frequencies $\tau^{-1}$ and resistivity
$\rho=\sigma^{-1}$ from the case of pristine graphene, i.e. neglecting the
electron-impurity collisions. Such problem has only two dimensionless
parameters: the coupling constant $\alpha_{c}=e^{2}/\kappa\hbar v_{0}$ and the
normalized gap $E_{g}/kT$. Within the Born approximation to carrier-carrier
scattering, the collision frequency and resistivity appear proportional to the
coupling constant squared. Restoring the dimensionality of the collision rate
and resistivity, we are always able to present them in the form:

$\displaystyle\tau_{eh}^{-1}=\alpha_{c}^{2}\frac{kT}{\hbar}\tilde{\nu}\left(\frac{E_{g}}{kT}\right),$
(19)
$\displaystyle\sigma_{eh}=\frac{e^{2}}{\hbar}\alpha_{c}^{-2}\tilde{\sigma}\left(\frac{E_{g}}{kT}\right),$
(20)

where $\tilde{\nu}$ and $\tilde{\sigma}$ are the dimensionless collision
frequency and the dimensionless conductivity depending only on the normalized
gap $E_{g}/kT$.

Figure 2: Electron-hole limited resistivity $\rho_{eh}$ (a) and associated
collision frequency $\tau_{eh}$ (b) of pristine gapped graphene at charge
neutrality point. The collision frequency is scaled by ’thermal frequency’
$\alpha_{c}^{2}kT/\hbar$, while the resisitivity is scaled by
$\rho_{0}=\alpha_{c}^{2}h/2e^{2}$. The partial contributions of normal and
annihilation-type electron-hole scattering are shown with red and orange
curves, respectively

The resulting dependences of collision rate and resistivity on $E_{g}/kT$ are
shown in Fig. 2. The collision frequency $\tilde{\nu}$ decays monotonically
with increasing the gap, which reflects the lack of thermally excited carriers
to collide with in a non-degenerate semiconductor. It is possible to show that
$\tilde{\nu}\propto e^{-E_{g}/2kT}$ at $E_{g}/kT\gg 1$. Still, the
conductivity $\tilde{\sigma}_{eh}$ is scaling with gap non-exponentially. The
reason is that Drude weight $D$ has a compensating exponential $D\propto
e^{-E_{g}/2kT}$, which again reflects the exponentially small number of
carriers in the band. The resulting dependence of $\tilde{\sigma}_{eh}$ on
$E_{g}/kT$ appears to be hyperbolic at large values of gap, while the
resistivity scales linearly, $\rho_{eh}\propto E_{g}/kT$. The latter scaling
can be ascribed to the enhancement of effective mass in the Drude formula
$\sigma_{eh}=ne^{2}\tau_{eh}/m^{*}$, provided the combination $n\tau$ is gap-
independent.

Analyzing the intermediate-gap region, $E_{g}\sim kT$, it is instructive to
split the electron-hole collision frequency into the contributions of ’normal’
and ’annihilation-type’ scattering. The dependences of partial collision
frequencies on $E_{g}/kT$ are both decaying, yet the annihilation collision
frequency decays faster than the normal collision frequency. This fact is
explained by approximate orthogonality of conduction and valence band states
located in $kT$-window near the edges of conduction and valence bands. It is
the states orthogonality which reduces the probability of annihilation,
$|M^{+-}_{\bf pk}|^{2}\ll 1$, and makes the annihilation-type collisions
irrelevant to the conductivity of the large-gap semiconductor. Yet, the
annihilation scattering is by no means small in the zero-gap state, and even
appears stronger than the conventional scattering.

Large frequency of annihilation-type collisions at $E_{g}=0$ and their rapid
decay at $E_{g}\gg kT$ result in a non-trivial dependence of resistivity on
band gap $E_{g}/kT$. At small induced gaps, the resistivity decays due to the
rapid cancellation of annihilation-type scattering, and reaches a minimum at
$E_{g}\approx 2.5kT$. At larger values of $E_{g}$, the resistivity grows
linearly due to the enhancement of effective mass. The presence of such
annihilation minimum on the gap dependence of resistance is a natural hallmark
of the carrier-carrier collision-limited transport.

To conclude the study of resitivity in pristine gapped graphene, we point to
the main steps in deriving an approximate expression for resistivity at large
band gaps. In that case, exchange electron-hole and electron-electron
contributions to $C$ can be neglected. The Fermi distribution functions are
reduced to the Boltzmann exponents, which simplifies the energy integration.
The natural cutoff for the transferred momentum $q$ appears order of
$E_{g}/v_{0}$ – otherwise, the scattered electron cannot reside on the
dispersion curve. The latter fact disables the appearance of the Coulomb
logarithm Landau (1936) in the expression for resistivity. Performing these
steps, described in detail in Appendix A, we arrive at the conductivity of
graphene with large induced gap in the form:

$\sigma_{eh}(E_{g}\gg
kT)=\frac{8}{\pi}\frac{e^{2}}{h}\alpha_{c}^{-2}\frac{kT}{E_{g}}.$ (21)

This value is very different numerically from the e-h limited conductivity of
gapless graphene, obtained in Ref. Fritz  _et al._ , 2008 and reproduced in
our calculations:

$\sigma_{eh}(E_{g}=0)=0.76\frac{e^{2}}{h}\alpha_{c}^{-2}.$ (22)

Despite numerical and functional differences between small- and large-gap
asymptotics of $\sigma_{eh}$, both (21) and (22) can be presented in a similar
form. This is achieved by introducing the ’running’ coupling constant
depending on the average thermal carrier velocity

$\overline{\alpha}_{c}^{2}=\frac{e^{4}}{\kappa^{2}\hbar^{2}\langle
v_{p}^{2}\rangle}.$ (23)

Here $\langle v_{p}^{2}\rangle=v_{0}^{2}$ in the gapless state
($\overline{\alpha}_{c}\rightarrow\alpha_{c}$), and $\langle
v_{p}^{2}\rangle=2kT/m^{*}$ in the limit of large gap. With the notation (23),
we present the large-gap conductivity as

$\sigma_{eh}(E_{g}\gg
kT)=\frac{2}{\pi}\frac{e^{2}}{h}\overline{\alpha}_{c}^{-2}\approx
0.6\frac{e^{2}}{h}\overline{\alpha}_{c}^{-2}.$ (24)

We may now speculate that the gap-dependent electron-hole collision-limited
conductivity of variable-gap semiconductor is more universal than it was
assumed previously Tan  _et al._ (2022). First, the scaling of $\sigma_{eh}$
with gap is non-exponential and much slower. Second, the conductivity can be
expressed only via conductance quantum $e^{2}/h$ and the running coupling
constant $\overline{\alpha}_{c}$, with a numerical prefactor very weakly
depending on the gap.

##  IV Observability of the electron-hole conductivity in disordered samples

It is now tempting to compare the magnitudes of the carrier-carrier and
carrier-impurity contributions to the resistivity in realistic disordered
samples. Within the adopted variational approach, the contributions of these
scattering channels to the collision frequency and resistivity are simply
additive. From scaling considerations, the e-i collision frequency should be
of the form:

$\tau^{-1}_{ei}=\alpha_{c}^{2}\frac{n_{\rm imp}}{(kT/\hbar
v_{0})^{2}}\frac{kT}{\hbar}{\tilde{\nu}}_{ei}\left(\frac{E_{g}}{kT}\right).$
(25)

The normalized electron-impurity collision frequency depends weakly (non-
exponentially) on the scaled band gap $E_{g}/kT$.

It becomes clear now that the electron-hole scattering-limited resistivity is
dominant over impurity-limited if the hole density exceeds the impurity
density. Of course, this conclusion is valid only at the charge neutrality
point. With increasing the band gap, the number of thermally-excited carriers
is reduced, while the density of impurities remains approximately constant.
This makes us conclude that e-h collisions always become irrelevant at
$E_{g}/kT\gg 1$.

Figure 3: Temperature-dependent resistivity $\rho$ of gapped graphene under
the combined action of electron-hole and impurity scattering for various
values of band gap: $E_{g}=20$ meV (a), $E_{g}=50$ meV (b), and $E_{g}=100$
meV (c). The resistivity is scaled by $\rho_{0}=\alpha_{c}^{2}h/2e^{2}$.
Curves of different color correspond to different impurity densities $n_{\rm
imp}$, red curve in each panel corresponds to the pristin graphene, gray
dashed line is a guide for eye showing the Arrhenius behaviour

The latter conclusion is illustrated in Fig. 3, where the total resistivity is
shown as a function of inverse temperature at various impurity densities and
band gaps. At the smallest band gap $E_{g}=20$ meV (Fig. 3a), the total
resistivity is dominated by e-h collisions in a wide temperature range
starting from the highest temperatures to $(kT)^{-1}\approx 60$ eV-1
($T\approx 200$ K). With increasing the band gap to $E_{g}=50$ meV and then to
$E_{g}=100$ meV (Fig. 3 b,c), we observe quite a rapid increase in total
resistivity. The total resistivity deviates from electron-hole limited one
already at nearly-room temperature, and becomes dominated by impurities. The
Arrhenius law for impurity scattering is expectantly fulfilled: the logarithm
of resistivity vs $T^{-1}$ becomes linear (see the gray dashed curve in Fig.
3). We suggest that the seeming agreement between the exponential e-h
conductivity scaling function $f(E_{g}/kT)$ of Ref. Tan  _et al._ (2022) and
experiment was due to the presence of residual impurities, which guaranteed
the restoration of Arrhenius law.

The density of impurities in gapped graphene should be pretty low to observe
the effects of electron-hole scattering below the room temperature. For
$E_{g}=20$ meV and $n_{\rm imp}=5\times 10^{8}$ cm-2, the e-h and e-i
contributions to resistivity become equal to each other at $T\approx 90$ K.
Such density of impurities is achievable in high-quality van der Waals
heterostructures.

##  V Discussion and conclusions

We have performed a systematic variational evaluation of resistivity due to
electron-hole and electron-impurity scattering in gapped graphene. We have
found that for pristine graphene, the resistivity at charge neutrality point
scales as $\rho_{eh}\propto E_{g}/kT$. The scaling is non-exponential and
violates the empirical Arrhenius law. This fact is related to simultaneous
reduction in carrier density and collision frequency with increasing the band
gap. A weak residual dependence of resistivity on the gap is due to the gap-
dependent effective mass within the Dirac model. We expect that $\rho_{eh}$
would not depend on $E_{g}$ at all for semiconductors described by other
Hamiltonians, where $E_{g}$ and $m^{*}$ are decoupled.

In this aspect, it’s worth noting that gapped Dirac Hamiltonian does not
provide an adequate description of electronic bands in graphene bilayer. Its
spectrum remains parabolic at $E_{g}=0$, and possesses a van Hove singularity
at both band edges for finite gap Zhang  _et al._ (2009). Evaluation of
$\rho_{eh}$ for this kind of spectrum should be performed independently.
Still, the cancellation of Arrhenius exponents in the expression for
resistivity $\rho_{eh}=m^{*}\nu_{eh}/ne^{2}$ is a generic phenomenon
independent of these spectral details.

Another important feature of gap-dependent resistivity $\rho_{eh}(E_{g}/kT)$
is the presence of minimum achieved at $E_{g}\approx 2.5kT$. This minimum is
provided by the competition of two opposite effects occurring upon gap
induction: the enhancement of effective mass and reduction in probability of
electron-hole annihilation scattering. Such an annihilation minimum is quite
sensitive to the numerical values of the normal and annihilation scattering
probabilities. Rigorously speaking, the minimum may not appear for other model
potentials of electron-hole scattering, e.g. statically-screened or
dynamically-screened Coulomb interaction. Still, the effects of screening
should be small in the gapped case $E_{g}/kT\gg 1$, where the number of free
carriers is low.

Our study was primarily devoted to the 2d systems with massive Dirac-type
spectrum, where the maximum resistivity is achieved for equal and non-
degenerate densities of electrons and holes. A different situation can emerge
in semimetals with overlapping conduction and valence bands, where CNP is
achieved for a degenerate electron-hole system. Such situation is realized in
HgTe quantum wells above the critical thcikness, where e-h scattering-limited
resistivity scales as $T^{2}$ Kvon  _et al._ (2011); Entin  _et al._ (2013).
Scattering between degenerate electrons and degenerate holes can also be
realized in drag experiments Bandurin  _et al._ (2022) and in 2d
semiconductors with interband population inversion Svintsov  _et al._ (2014).

All the above discussion concerned the resistivity of gapped graphene exactly
at the charge neutrality point. Away from the CNP, the electron-hole
collisions make a minor contribution to the resistivity due to the effect of
Coulomb drag Vyurkov and Ryzhii (2008); Zarenia  _et al._ (2019). More
precisely, the majority carriers tend to drag the minority ones in the same
direction, in spite of the opposite charges. The co-directional motion of
electrons and holes reduces the strength of their mutual scattering. One may
speculate that e-h scattering tends to nullify its own magnitude even for
small density imbalance. An estimate of this imbalance can be obtained from
the two-fluid hydrodynamic model Svintsov  _et al._ (2012) and reads as
$\delta n/n_{T}\gtrsim\tau_{eh}/\tau_{ei}$. Definitely, a similar estimate
could be obtained from a variational method using two different variational
parameters $\tau_{e}$ and $\tau_{h}$. We refrain from these lengthy
calculations due to the expected minor effect of e-h collisions away from the
CNP.

Potential extensions of this work lie in the field of thermoelectric
coefficients of gapped graphene in the electron-hole collision regime. The
interest to thermoelectricity in such a system stems from (1) enhancement of
Seebeck coefficient with gap induction even for impurity- and phonon-limited
scattering (2) further enhancement of thermopower and reduction in thermal
conduction in the hydrodynamic regime Andersen  _et al._ (2019); Principi and
Vignale (2015). The combination of this factors can make clean gapped graphene
a very strong candidate for thermoelectric photodetection Titova  _et al._
(2023).

The work was supported by the grant # 22-29-01034 of the Russian Science
Foundation.

##  Appendix A Resistivity at large band gaps

Throughout this section, we work in the units $\hbar=k=v_{0}=1$. The
resistivity of gapped graphene at $E_{g}\gg T$ is limited by normal electron-
hole scattering. Annihilation-type process can be neglected due to its small
matrix element, while electron-electron scattering can be neglected due to the
approximate conservation of current if carriers reside at the parabolic part
of the spectrum. The contribution of normal e-h scattering to average
collision rate $C$, which we denote by $C^{++}_{eh}$, can be presented in the
form Zarenia  _et al._ (2019); Zheng and MacDonald (1993)

$C_{eh}^{++}=\frac{e^{2}\pi^{2}}{T}\int
qdqd\omega{|V(q)|^{2}n_{\omega}[1+n_{\omega}]\left[{\Im}\Pi_{0}{\Im}\Pi_{2}+{\Im}{\bf\Pi}_{1}{\Im}{\bf\Pi}_{1}\right],}$
(26)

where $n_{\omega}=[e^{\omega/T}-1]^{-1}$ are the Bose functions, and the
’generalized polarizabilities’ $\Pi_{n}(q,\omega)$ are defined by

$\Pi_{n}({\bf q},\omega)=N\sum_{\bf
p}{\frac{f_{0}(\varepsilon_{p})-f_{0}(\varepsilon_{p-q})}{\varepsilon_{p}-\varepsilon_{p-q}-\hbar\omega+i\delta}|M^{++}_{{\bf
p},{\bf p}-{\bf q}}|^{2}({\bf v_{p}}-{\bf v_{p-q}})^{n}.}$ (27)

Representation (26) is achieved by introducing an extra energy variable
$\omega$ with the aid of (18) and developing the square of four particle
velocities:

$[{\bf v}_{\bf p}-{\bf v}_{\bf k}-{\bf{v}}_{{\bf{p-q}}}+{\bf
v}_{\bf{k+q}}]^{2}=({\bf v_{p}}-{\bf v_{p-q}})^{2}+({\bf v_{k}}-{\bf
v_{k+q}})^{2}-2({\bf v_{p}}-{\bf v_{p-q}})({\bf v_{k}}-{\bf v_{k+q}}).$ (28)

Evaluation of imaginary part of polarizability is achieved with Sokhotski rule

$\Im\Pi_{0}({\bf q},\omega)=-\pi N\sum_{\bf p}{\left[f_{0}(\varepsilon_{\bf
p})-f_{0}(\varepsilon_{\bf p-q})\right]\delta[\varepsilon_{\bf
p}-\varepsilon_{\bf p-q}-\omega]|M^{++}_{{\bf p},{\bf p}-{\bf q}}|^{2}.}$ (29)

The angular integration is performed with the aid of delta function, which
results in

$\Im\Pi_{0}({\bf
q},\omega)=-\frac{N}{2\pi}\frac{1}{\sqrt{q^{2}-\omega^{2}}}\int\limits_{E_{\min}}^{\infty}d\varepsilon{\left[f_{0}(\varepsilon+\frac{\omega}{2})-f_{0}(\varepsilon-\frac{\omega}{2})\right]\frac{\varepsilon^{2}-\frac{q^{2}}{4}}{\sqrt{\varepsilon^{2}-E_{\min}^{2}}}.}$
(30)

The minimum carrier energy at which the quantum $({\bf q},\omega)$ can be
absorbed is denoted by $E_{\min}$:

$E_{\min}=\frac{q}{2}\sqrt{1+\frac{E_{g}^{2}}{q^{2}-\omega^{2}}}.$ (31)

Further integration over energies can be performed in the Boltzmann limit
$E_{g}\gg T$. All slowly varying functions of energy $\varepsilon$ can be
evaluated at $\varepsilon=E_{\min}$, while the remainder is evaluated exactly:

$\Im\Pi_{0}({\bf q},\omega)=-\frac{\pi
N}{(2\pi)^{2}}\frac{q^{2}\sinh(\frac{\beta\omega}{2})}{\sqrt{q^{2}-\omega^{2}}}\sqrt{\pi}(B^{2}-1)\frac{e^{-\beta
Bq/2}}{\sqrt{\beta Bq}},$ (32)

where we have introduced the inverse temperature $\beta=T^{-1}$ and the
dimensionless parameters

$B(q,\omega)=\sqrt{1+\frac{E_{g}^{2}}{q^{2}-\omega^{2}}},\quad
l(q,\omega)=\frac{\omega}{q}$ (33)

Similar steps can be done to evaluate $\Im\Pi_{1,2}$ in the Boltzmann limit,
which results in:

$\displaystyle{\Im}{\bf\Pi}_{1}=2\pi\frac{N}{(2\pi)^{2}}\sinh(\frac{\beta\omega}{2})\sqrt{q^{2}-\omega^{2}}B\sqrt{\pi}\frac{B^{2}-1}{B^{2}-l^{2}}\frac{e^{-\beta
Bq/2}}{\sqrt{\beta Bq}}\frac{{\bf q}}{q},$ (34)
$\displaystyle{\Im}\Pi_{2}=4\pi\frac{N}{(2\pi)^{2}}\sinh(\frac{\beta\omega}{2})\sqrt{q^{2}-\omega^{2}}B^{2}\sqrt{\pi}\frac{B^{2}-1}{B^{2}-l^{2}}\frac{1-l^{2}}{B^{2}-l^{2}}\frac{e^{-\beta
Bq/2}}{\sqrt{\beta Bq}}.$ (35)

Considerable cancellations appear after collecting the expressions (32 \- 35)
into $C^{++}_{eh}$:

$C_{eh}^{++}=\frac{\pi^{3}N^{2}e^{2}\alpha_{c}^{2}}{8}\int
dqd\omega\exp\left[-\beta
E_{g}qB(q,\omega)\right]\frac{q^{2}B(q,\omega)(q^{2}-\omega^{2})}{\left[q^{2}+\frac{(q^{2}-\omega^{2})^{2}}{E_{g}^{2}}\right]^{2}}$
(36)

where $q$ and $\omega$ were normalized to the band gap $E_{g}$.

Let us inspect the function under the exponent $f(q)=qB(q,\omega)$. Considered
as a function of $q$, it has a minimum value $f_{\min}=1+\omega$. Instead of
$q$, we pass to the new variable

$\tau=qB(q,\omega).$ (37)

The dimensionless integral in the expression for $C^{++}_{eh}$ is now recast
in the form

$I\equiv\int dqd\omega\exp\left[-\beta
E_{g}qB(q,\omega)\right]\frac{q^{2}B(q,\omega)(q^{2}-\omega^{2})}{\left[q^{2}+\frac{(q^{2}-\omega^{2})^{2}}{E_{g}^{2}}\right]^{2}}=2\int\limits_{0}^{\infty}d{\omega}\int\limits_{1+\omega}^{\infty}d\tau\frac{\tau^{2}}{(\tau^{2}-{\omega}^{2})^{2}}\frac{e^{-\beta
E_{g}\tau}}{\sqrt{(\tau^{2}-{\omega}^{2}-1)^{2}-4{\omega}^{2}}}.$ (38)

Integrating over $\tau$, we again proceed with the steepest descend method,
i.e. we set $\tau=1+\omega$ in all smooth functions under the integration
sign. This results in

$I\approx\frac{(\beta
E_{g})^{2}}{2}\int\limits_{0}^{\infty}d{\omega}\frac{(1+\omega)^{2}}{((1+\omega)^{2}-{\omega}^{2})^{2}}\frac{1}{2\sqrt{{\omega}+1}}\int\limits_{1+\omega}^{\infty}d\tau\frac{e^{-\beta
E_{g}\tau}}{\sqrt{\tau^{2}-(1+\omega)^{2}}}=\\\ =\frac{(\beta
E_{g})^{2}}{4}e^{-\beta
E_{g}}\int\limits_{0}^{\infty}d\omega\frac{({\omega}+1)^{3/2}}{(1+2{\omega})^{2}}K_{0}({\omega}\beta
E_{g})\approx\frac{\pi\beta E_{g}}{8}e^{-\beta E_{g}}.$ (39)

Hence,

$C_{eh}^{++}=\frac{N^{2}e^{2}\alpha^{2}}{8}(\beta E_{g})\frac{\pi}{8}e^{-\beta
E_{g}}.$ (40)

The Drude weight $D$ is also simply evaluated with Boltzmann statistics

$D\approx\frac{Ne^{2}}{2\pi\hbar^{2}}kTe^{-\beta E_{g}/2}.$ (41)

Substituting $C$ and $D$ in the Boltzmann limit [Eqs. (40) and (41)] into the
general expression for the conductivity (13), we find the linear-in-$T$
scaling of conductivity (21).

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