# Weak Localization and Antilocalization in Twisted Bilayer Graphene

Hongyi Yan  Center for Advanced Quantum studies, Department of Physics,
Beijing Normal University, Beijing 100875, China  Haiwen Liu
<EMAIL_ADDRESS>Center for Advanced Quantum studies, Department of
Physics, Beijing Normal University, Beijing 100875, China

###### Abstract

We theoretically study the weak localization (WL) and weak antilocalization
(WAL) effects in twisted bilayer graphene that placed in alignment on a
hexagonal boron nitride substrate. The low-energy band of the top layer
exhibits a Dirac cone with a negligible gap, while the bottom layer has a
relatively large band gap. The system features a low concentration of
impurities, and the quantum correction to the conductivity arises from the
quantum interference between two time-reversed impurity scattering
trajectories. Through bias voltage tuning, we find that inter-layer scattering
significantly contributes to the conductivity correction when the Fermi
surface areas of the two valleys at low energy are comparable. We observe a
double crossover from WL to WAL and back to WL at a specific range of Fermi
energy, which is a particularly intriguing phenomenon.

##  I Introduction

Within the framework of classical mechanics, the Drude model can be used to
describe electron motion and obtain the transport properties of materials.
However, when the coherence length is sufficiently long at low temperature
such that the quantum effect cannot be ignored, the Drude model needs to be
corrected. Electrons in the material are scattered by impurities and form an
infinite number of trajectories. Quantum interference between two closed time-
reversed trajectories of electrons tends to increase the probability of
electrons staying in original place, which usually leads to a decrease in
conductivityHikami  _et al._ (1980); Altshuler  _et al._ (1980). This quantum
interference can be suppressed by applying a magnetic field that breaks time-
reversal symmetry and thereby breaks quantum coherence. Unlike the behavior of
conventional electrons, relativistic Dirac electrons in monolayer graphene are
characterized by their linear energy band, large Fermi velocity, and
chiralityNovoselov  _et al._ (2004). Quantum conductivity correction to the
Drude model in monolayerAndo  _et al._ (1998); Suzuura and Ando (2002); McCann
_et al._ (2006); Morozov  _et al._ (2006); Wu  _et al._ (2007); Kechedzhi  _et
al._ (2007a); Fal’ko  _et al._ (2007); Yan and Ting (2008); Tikhonenko  _et
al._ (2008, 2009); Hilke  _et al._ (2014) and bilayer grapheneGiovannetti  _et
al._ (2007); Kechedzhi  _et al._ (2007a, b); Horsell  _et al._ (2008); Liao
_et al._ (2010) has been studied extensively both theoretically and
experimentally over the past decade. In general, monolayer graphene exhibits
weak antilocalization for long-ranged scattering impurities, while bilayer
graphene manifests weak localization. In addition to graphene systems,
heterostructuresAmann  _et al._ (2022); Wang  _et al._ (2015, 2016); Yang  _et
al._ (2016); Völkl  _et al._ (2017); Zihlmann  _et al._ (2018); Wakamura  _et
al._ (2018), semiconductorsLu  _et al._ (2013) and topological insulatorsImura
_et al._ (2009); Shan  _et al._ (2012); Lu and Shen (2011); Lu  _et al._
(2011) are also used as platforms to study the phenomenon of weak localization
and weak antilocalization. Researchers have taken into account the influence
of electron-electron interactionYan and Ting (2008) spin-orbit couplingHikami
_et al._ (1980); Shan  _et al._ (2012); Lu  _et al._ (2013); Amann  _et al._
(2022); Wang  _et al._ (2015, 2016); Yang  _et al._ (2016); Völkl  _et al._
(2017); Zihlmann  _et al._ (2018); Wakamura  _et al._ (2018), and
magnetismShan  _et al._ (2012); Lu  _et al._ (2011) and found that they all
have a significant and complicate impact on the transport properties.

In recent years, the discovery of the correlated insulating state and
superconducting state in twisted bilayer graphene (TBG) near the magic
angleCao  _et al._ (2018a, b) has made the twisted moiré system a hot topic of
researchLu  _et al._ (2019); Yankowitz  _et al._ (2019). TBG is composed of
two graphene layers rotated at a specific angle $\theta$ to each other.
Through twisting, the original monolayer Brillouin zone’s Dirac point
duplicates into two equal Dirac points with the same chirality Bistritzer and
MacDonald (2011); Lopes dos Santos  _et al._ (2007); Moon and Koshino (2013);
Koshino  _et al._ (2018). The interlayer hopping renormalizes the Fermi
velocity at low energy, which decreases as the twist angle decreases. The
crucial feature of TBG is that when the twist angle decreases to the so-called
“magic angle”, the bands near the charge neutral point are further compressed
into flat bands. The kinetic energy of electrons on the bands dramatically
reduces and becomes comparable to the strength of the Coulomb interaction.
Thus, the electron correlation in TBG near the magic angle becomes so
significant that it has to be considered. Currently, much attention has been
focused on the study of strong correlation properties at the flat band, while
weak (anti-)localization of TBG has not been extensively studied, and only a
few experiments conducted on this subjectMeng  _et al._ (2012); Kashiwagi  _et
al._ (2022). It remains unclear whether quantum correction of TBG or some
artificial structure based on it that has exotic characters compared to
monolayer and bilayer graphene.

Figure 1: (a) Schematic diagram of the twisted bilayer graphene placed on hBN
substrate system. (b) Blue (red) line is the band structure of the twisted
bilayer graphene with (without) inter-layer bias, the twist angle
$\theta=3^{\circ}$ . The band gap of the bottom layer is denoted as
$\it\Delta$, and the bias-voltage induced bottom layer energy shift is
represented by $\it U$. (c) The low-energy band structure extracted from (b).
Valley $\textbf{K}_{1}$ features a gapless and massless Dirac cone while near
valley $\textbf{K}_{2}$ is a massive energy band with a gap. The Fermi energy
$E_{F}$ value is relative to the charge neutral point.

In this work, we present a theoretical investigation of the quantum
conductivity correction in twisted bilayer graphene placed on a hexagonal
boron nitride (hBN) substrate. The bottom layer of twisted bilayer graphene
aligns with hBN. We consider a low concentration of long-range non-magnetic
impurities as the main cause of electron scattering at low temperature and
neglect electron-phonon and electron-electron scattering effects in the
system. We particularly focus on the conductivity correction induced by inter-
valley scattering between two valleys with different Fermi surface sizes
(valley in this context refers to a mini-valley as described in previous
workPhinney  _et al._ (2021)), which has not been studied previously. The
paper is structured as follows. In Sec. II, we introduce an effective
Hamiltonian model for the system and the type of impurities considered. We
then calculate the scattering time and velocity correction for intra- and
inter-valley scattering. In Sec. III, we carefully calculate the Cooperons for
the two kinds of scattering. In Sec. IV, we present our calculations for the
quantum conductivity correction and magneto-conductivity. In Sec. V, we
summarize our findings and provide a discussion of our study. Finally in Sec.
VI we give our acknowledgement.

##  II Model

###  II.1 Effective Hamiltonian

The system of our study is twisted bilayer graphene that placed in alignment
on a hexagonal boron nitride substrate, as depicted in Fig. 1(a). Near the
original K point are two equal Dirac points derived from the top and bottom
layers respectivelyMeyer  _et al._ (2007). Since impurity scattering in the
system is long-ranged, $\textbf{K}-\textbf{K}^{\prime}$ scattering is
suppressed, we only need to consider scattering within one valley. For
simplicity, we label the two Dirac points $\textbf{K}_{1}$ and
$\textbf{K}_{2}$ and refer to them as “valleys” instead of “mini-valleys” in
this article. The interaction between the hBN substrate and the bottom layer
of graphene breaks the sublattice symmetry of the latter and results in a band
gap opening whose magnitude depends on the on-site energy difference of the A
and B sublattices of the bottom layer Giovannetti  _et al._ (2007); Skomski
_et al._ (2014). We employ the modified Bistritzer-MacDonald continuum model
Bistritzer and MacDonald (2011) in which we set the on-site energy of the
sublattices A and B in the bottom layer to be different to induce a band gap
whose magnitude is $\Delta$. If we neglect the corrugation between the two
layers, the band in the top layer remains gapless and masslessBistritzer and
MacDonald (2011). while if we take account into it, a small and negligible gap
opens in the band of the top layer. Moreover, by tuning the bias voltage, one
can control the relative energy shift $U$ between the two Dirac cones. The
energy band of the system is depicted in Fig. 1(b), and the simplified bands
at low energy is extracted in Fig. 1(c). The Van Hove singularity is located
at the midpoint of the valleys $\textbf{K}_{1}$ and $\textbf{K}_{2}$, its
energy is positively correlated with the twist angle Moon and Koshino (2013),
i.e, a larger twist angle corresponds to a higher Van Hove energy. When the
Fermi energy approaches the Van Hove energy, the shape of the cone deforms
such that we cannot use a simple Dirac equation to fit the band. Therefore, to
obtain a relatively good cone shape, the Fermi energy should be far away from
the Van Hove energy. Under these conditions, we can describe the low-energy
band structure of the system using a simple effective Hamiltonian in the
continuum limit, which is given by:

$H_{eff}=\hat{\tau}_{0}\otimes\hbar
v_{F}\textbf{k}\cdot\hat{\bm{\sigma}}+\frac{\left(\hat{\tau}_{0}-\hat{\tau}_{z}\right)}{2}\otimes(\frac{\it\Delta}{2}\hat{\sigma}_{z}+U\hat{\sigma}_{0}),$
(1)

$v_{F}$ is renormalized Fermi velocity, $\hat{\bm{\sigma}}$,
$\hat{\sigma}_{0}$ and $\hat{\sigma}_{z}$ are Pauli operators act in
sublattice (A and B) space and $\hat{\tau}_{0}$ and $\hat{\tau}_{z}$ act in
valley ($\textbf{K}_{1}$ and $\textbf{K}_{2}$) space. We then obtain the
conduction band energy $E(\textbf{k}_{1})$ ($E(\textbf{k}_{2})$) and the
corresponding eigenstates $\psi_{1}$ ($\psi_{2}$) near the valley
$\textbf{K}_{1}$ ($\textbf{K}_{2}$). The wave function of the propagating
electron is $\psi_{j}=\left\langle r\mid
k_{j}\right\rangle=\frac{1}{\sqrt{S}}\left(a_{j},b_{j}e^{i\phi_{\textbf{k}}}\right)e^{i\textbf{k}_{j}\cdot
r}$. Here, $\phi_{\textbf{k}}$ is the phase angle of the wave vector, $j$ is
the valley index. For two valleys $a_{1}=1/2$, $b_{1}=1/2$,
$a_{2}=\sqrt{(E_{F}+U+\it\Delta/2)/2(E_{F}+U)}$,
$b_{2}=\sqrt{(E_{F}+U-\it\Delta/2)/2(E_{F}+U)}$.

###  II.2 Impurity type and scattering time

We consider the case of weak non-magnetic impurity scattering, where
perturbation treatment is applicable. Electron spin doesn’t flip in this
process, and we investigate the transport properties of the system through
standard weak (anti-)localization calculations. When the twist angle is small,
the two valleys are in close proximity, the impurities can scatter electrons
within the same valley or between the two valleys. The intra-valley scattering
potential is given by the following expression:

$U_{0}(\textbf{r})=\sum_{i}u_{0}^{i}\tau_{0}\otimes\sigma_{0}F\left(\textbf{r}-\textbf{R}_{i}\right).$
(2)

$F(\textbf{r}-\textbf{R}_{i})$ is the distribution function of the long-ranged
impurity scattering, it can be simplified to the form of the delta function
$\delta(\textbf{r}-\textbf{R}_{i})$Lu  _et al._ (2013). Although it seems
contradictory at first sight because the delta function is short-ranged, the
feasibility has been verified. Under the impurity average, $U_{0}(\textbf{r})$
satisfies $\left\langle U_{0}(\textbf{r})\right\rangle_{imp}=0$, $\left\langle
U_{0}(\textbf{r})U_{0}(\textbf{r}^{\prime})\right\rangle_{imp}\sim\delta(\textbf{r}-\textbf{r}^{\prime})$.
The intra-valley scattering amplitude reads:

$\left\langle
j,\mathbf{k}\left|U_{0}\right|j,\mathbf{k}^{\prime}\right\rangle_{intra}=\frac{1}{S}\sum_{i}u_{0}^{i}e^{i\left(\mathbf{k}^{\prime}-\mathbf{k}\right)\cdot\mathbf{R}_{i}}\left(a^{2}_{j}+b^{2}_{j}e^{i\left(\phi_{\mathbf{k}^{\prime}}-\phi_{\mathbf{k}}\right)}\right).$
(3)

The intra-valley scattering time can be evaluated with the first Born
approximation:

$\frac{1}{\tau}=\frac{2\pi}{\hbar}\sum_{\mathbf{k}^{\prime}}\left\langle
U_{\mathbf{k},\mathbf{k}^{\prime}}U_{\mathbf{k}^{\prime},\mathbf{k}}\right\rangle_{imp}\delta\left(E_{F}-\varepsilon_{\mathbf{k}^{\prime}}\right),$
(4)

substitute Eq. (3) into Eq. (4) one can get:

$\frac{1}{\tau_{jj}}=\frac{2\pi
N_{j,F}}{\hbar}n_{0}u_{0}^{2}\left(a_{j}^{4}+b_{j}^{4}\right),$ (5)

with $n_{0}u_{0}^{2}=\sum_{i}(u_{0}^{i})^{2}/S$. $n_{0}$ represents the
concentration of non-magnetic impurities, $u_{0}$ is the average scattering
potential. $N_{j,F}$ is the density of states (DOS) of valley $j$ at the Fermi
energy per unit area of the system. Recent experimental studies have confirmed
the existence of electron scattering between valleys in twisted bilayer
graphene and twisted double bilayer graphene (TDBG)Tomić  _et al._ (2022);
Phinney  _et al._ (2021); Kashiwagi  _et al._ (2022). Kashiwagi discovered
that the ratio between intra-valley and inter-valley scattering time is
proportional to $1/(\it\Delta K_{\theta})^{2}$ in twisted double bilayer
grapheneKashiwagi  _et al._ (2022), where $\it\Delta K_{\theta}$ =
$\left|\textbf{K}_{2}-\textbf{K}_{1}\right|$. We speculate same result can be
found in TBG. Under this condition, the inter-valley scattering amplitude is
inversely proportional to $\it\Delta K_{\theta}$. We assume that the inter-
valley scattering potential has a similar form as in Eq.(2),

$U^{\prime}_{0}(\textbf{r})=\sum_{i}(u^{\prime})_{0}^{i}\tau_{x}\otimes\sigma_{0}F\left(\textbf{r}-\textbf{R}_{i}\right).$
(6)

$u^{\prime}$ is the scattering strength of the inter-valley which is usually
smaller than that of the intra-valley. Phinney found when the twist angle
$\theta=1.65^{\circ}$, the ratio between the intra-valley and inter-valley
scattering time approximates 3/4Phinney  _et al._ (2021). Therefore, We use
the factor $\beta/\it\Delta K_{\theta}$ to express the ratio
$(u^{\prime})_{0}^{i}/u_{0}^{i}$, so that the amplitude of the inter-valley
scattering is inversely proportional to $\it\Delta K_{\theta}$,

$\displaystyle\left\langle
1,\textbf{k}\left|U^{\prime}_{0}\right|2,\textbf{k}^{\prime}\right\rangle_{\text{inter
}}=$ $\displaystyle\frac{\beta}{\it\Delta
K_{\theta}}\frac{1}{S}\sum_{i}u_{0}^{i}e^{i(\textbf{K}_{2}-\textbf{K}_{1})\cdot\textbf{R}_{i}}e^{i\left(\textbf{k}^{\prime}-\textbf{k}\right)\cdot\textbf{R}_{i}}$
(7)
$\displaystyle\times\left(a_{1}a_{2}+b_{1}b_{2}e^{i\left(\phi_{\textbf{k}^{\prime}}-\phi_{\textbf{k}}\right)}\right).$

Using Eq. (4) the inter-valley scattering time can be calculated,

$\frac{1}{\tau_{j\bar{j}}}=\frac{2\pi
N_{\bar{j},F}}{\hbar}\frac{\beta^{2}}{\left(\it\Delta
K_{\theta}\right)^{2}}n_{0}u_{0}^{2}\left(a_{j}^{2}a_{\bar{j}}^{2}+b_{j}^{2}b_{\bar{j}}^{2}\right).$
(8)

here $j$ = 1, 2, $\bar{j}$ is another valley relative to $j$.
$\tau_{j\bar{j}}$ denotes the scattering time of electron scattered from
valley $j$ to valley $\bar{j}$. The scattering time is inversely proportional
to the DOS of the destination. The total scattering time of the two valleys
are:

$\displaystyle\frac{1}{\tau_{1}}=\frac{1}{\tau_{11}}+\frac{1}{\tau_{12}},$ (9)
$\displaystyle\frac{1}{\tau_{2}}=\frac{1}{\tau_{22}}+\frac{1}{\tau_{21}}.$

###  II.3 Velocity correction

Firstly, we consider the Diffuson approximation, which is a vertex correction
that describes the multiple scattering of electrons on the impurity potential
Akkermans and Montambaux (2007); Datta (1997). The Diffuson correction
corresponds to a series of ladder diagrams and contributes to the velocity
correction through an iterative equationAltshuler  _et al._ (1980); Shan  _et
al._ (2012):

$\widetilde{v}_{j,\textbf{k}}^{x}=v_{j,\textbf{k}}^{x}+\sum_{j^{\prime},\textbf{k}^{\prime}}G_{j^{\prime},\textbf{k}^{\prime}}^{R}G_{j^{\prime},\textbf{k}^{\prime}}^{A}\left\langle
U_{\textbf{k},\textbf{k}^{\prime}}U_{\textbf{k}^{\prime},\textbf{k}}\right\rangle_{imp}\widetilde{v}_{j^{\prime},\textbf{k}^{\prime}}^{x},$
(10)

$G_{j,\textbf{k}}^{R/A}=\frac{1}{\omega-\varepsilon_{j,\textbf{k}}\pm
i\hbar/2\tau_{j}}.$ (11)

${v}_{j,\textbf{k}}^{x}$, $\widetilde{v}_{j,\textbf{k}}^{x}$ respectively
denote bare velocity and corrected velocity of valley $j$ in the $x$
direction, $v^{x}=vcos\phi_{\textbf{k}}$. Eq. (11) is the expression of
retarded and advanced Green’s functions, corresponding to the plus and minus
signs in the denominator on the right-hand side of the equation. Eq. (10) can
be further expanded and written as a set of equations:

$\displaystyle v_{1,\textbf{k}}^{x}=$
$\displaystyle\left(1-\frac{\tau_{1}}{\tau_{11}}\frac{a_{1}^{2}b_{1}^{2}}{a_{1}^{4}+b_{1}^{4}}\right)\widetilde{v}_{1,\textbf{k}}^{x}-\frac{\tau_{2}}{\tau_{12}}\frac{a_{1}a_{2}b_{1}b_{2}}{a_{1}^{2}a_{2}^{2}+b_{1}^{2}b_{2}^{2}}\widetilde{v}_{2,\textbf{k}_{//}}^{x},$
(12) $\displaystyle v_{2,\textbf{k}}^{x}=$
$\displaystyle\left(1-\frac{\tau_{2}}{\tau_{22}}\frac{a_{2}^{2}b_{2}^{2}}{a_{2}^{4}+b_{2}^{4}}\right)\widetilde{v}_{2,\textbf{k}}^{x}-\frac{\tau_{1}}{\tau_{21}}\frac{a_{1}a_{2}b_{1}b_{2}}{a_{1}^{2}a_{2}^{2}+b_{1}^{2}b_{2}^{2}}\widetilde{v}_{1,\textbf{k}_{//}}^{x}.$

$\textbf{k}_{//}$ denotes the wave vector in the valley $\textbf{K}_{2}$
($\textbf{K}_{1}$) on the Fermi surface that is parallel to the wave vector k
in the valley $\textbf{K}_{1}$ ($\textbf{K}_{2}$). Note that the two bands are
isotropic, the Fermi velocities on their own Fermi surface are of equal
magnitudes and independent of directions. By numerically solving Eq. (12), we
can obtain the velocity correction factor $\eta_{j}$:

$\displaystyle\eta_{1}=$
$\displaystyle\widetilde{v}_{1,\textbf{k}}^{x}/v_{1,\textbf{k}}^{x},$ (13)
$\displaystyle\eta_{2}=$
$\displaystyle\widetilde{v}_{2,\textbf{k}}^{x}/v_{2,\textbf{k}}^{x}.$

##  III Cooperon correction

###  III.1 Cooperon

Next, we consider the Cooperon correction that corresponds to the so-called
“maximum-crossed diagrams” Akkermans and Montambaux (2007); Datta (1997),
which describes the quantum interference between two time-reversal electron
wave trajectories of oppositely ordered collision sequences. Although for
intra-valley scattering, the wave vectors k and $-\textbf{k}$ are not strictly
time-reversal because they are relative to the Dirac point K rather than
$\Gamma$ in the Brillouin zone of monolayer graphene, we still consider them
satisfy generalized time reversion. The Cooperon directly contributes to the
conductivity correction as follows:

$\displaystyle\it\Delta\sigma=$ $\displaystyle\frac{e^{2}\hbar}{2\pi
S}\sum_{j,j^{\prime},\textbf{k}}\sum_{\textbf{q}}\Gamma_{jj^{\prime}}(\textbf{Q}+\textbf{q})\widetilde{v}_{j,\textbf{k}}^{x}\widetilde{v}_{j^{\prime},\textbf{Q}+\textbf{q}-\textbf{k}}^{x}$
(14) $\displaystyle\times
G_{j,\textbf{k}}^{R}G_{j,\textbf{k}}^{A}G_{j^{\prime},\textbf{Q}+\textbf{q}-\textbf{k}}^{R}G_{j^{\prime},\textbf{Q}+\textbf{q}-\textbf{k}}^{A}.$

$\Gamma_{jj^{\prime}}(\textbf{Q}+\textbf{q})$ is the Cooperon corresponding to
the scattering from valley $j$, vector k to valley $j^{\prime}$, vector
$\textbf{k}^{\prime}$. $j$, $j^{\prime}$ = 1, 2 and
$\textbf{k}+\textbf{k}^{\prime}=\textbf{Q}+\textbf{q}$. Here, Q is defined as
a “nesting vector” for inter-valley scattering. Its modulus equals the radius
difference between the two Fermi circles,
${\left|\textbf{Q}\right|=\left|\textbf{k}_{1F}\right|-\left|\textbf{k}_{2F}\right|}$,
as shown in the Fig. 2(a). For intra-valley scattering, the nesting vector is
$\textbf{Q}=0$. To obtain the Cooperon $\Gamma(\textbf{Q}+\textbf{q})$, we
need to evaluate the bare vertex function $\gamma$. The bare vertex function
can be calculated by:

$\gamma_{j,\textbf{k};j^{\prime},\textbf{k}^{\prime}}\equiv\left\langle
U_{j,\textbf{k};j^{\prime},\textbf{k}^{\prime}}U_{j^{\prime},\textbf{Q}+\textbf{q}-\textbf{k};j,\textbf{Q}+\textbf{q}-\textbf{k}^{\prime}}\right\rangle_{imp}.\\\
$ (15)

In the following, we will provide a detailed expression of the bare vertex
function $\gamma$ for intra-valley and inter-valley scattering, and then
calculate the corresponding Cooperons.

To begin with, we calculate the Cooperon for intra-valley scattering
($j=j^{\prime}$). By combining Eqs. (3) and (15), and setting the nesting wave
vector $\textbf{Q}=0$, we calculate the bare vertex function for intra-valley
scattering as follows:

$\displaystyle\begin{aligned}
\gamma_{j,\textbf{k};j,\textbf{k}^{\prime}}&\equiv\left\langle
U_{j,\mathbf{k};j,\mathbf{k}^{\prime}}U_{j,\mathbf{q}-\mathbf{k};j,\mathbf{q}-\mathbf{k}^{\prime}}\right\rangle_{imp}\\\
&=\frac{n_{0}u_{0}^{2}}{S}\\{a_{j}^{4}+2a_{j}^{2}b_{j}^{2}e^{i\left(\phi_{\mathbf{k}^{\prime}}-\phi_{\mathbf{k}}\right)}+b_{j}^{4}e^{2i\left(\phi_{\mathbf{k}^{\prime}}-\phi_{\mathbf{k}}\right)}\\}\end{aligned}$
(16)

In Eq. (16), the bare vertex function has three channels $L$ = 0, 1, 2,
corresponding to the three terms separately. Here, $L$ is the coefficient of
$\phi_{\textbf{k}^{\prime}}-\phi_{\textbf{k}}$ in the exponent. To calculate
the quantum correction to the conductivity, we set
$\phi_{\textbf{k}^{\prime}}-\phi_{\textbf{k}}=\pi$ and
$\textbf{k}+\textbf{k}^{\prime}=\textbf{q}$. The Cooperon is related to the
bare vertex function through the Bethe-Salpeter equation Akkermans and
Montambaux (2007):

$\Gamma_{j,\textbf{k};j,\textbf{k}^{\prime}}=\gamma_{j,\textbf{k};j,\textbf{k}^{\prime}}+\sum_{\textbf{k}^{\prime\prime}}\gamma_{j,\textbf{k};j,\textbf{k}^{\prime\prime}}G_{j,\textbf{k}^{\prime\prime}}^{R}G_{j,\textbf{q}-\textbf{k}^{\prime\prime}}^{A}\Gamma_{j,\textbf{k}^{\prime\prime};j,\textbf{k}^{\prime}}.$
(17)

The Feynman diagram representation of this equation is depicted in Fig. 2(b).
Similar to the bare vertex function $\gamma$, the Cooperon $\Gamma$ also has
three channels denoted by $L$ = 0, 1, 2:

$\displaystyle\Gamma_{j,\textbf{k};j,\textbf{k}^{\prime}}$
$\displaystyle=\Gamma_{jj}(\textbf{q})$ (18)
$\displaystyle=\Gamma_{j}^{(0)}+\Gamma_{jj}^{(1)}e^{i\left(\phi_{\textbf{k}^{\prime}}-\phi_{\textbf{k}}\right)}+\Gamma_{jj}^{(2)}e^{2i\left(\phi_{\textbf{k}^{\prime}}-\phi_{\textbf{k}}\right)}.$

The intra-valley scattering Cooperon is written in a simple form as
$\Gamma_{jj}(\textbf{q})$. We assume that the three channels obey the Bethe-
Salpeter equation separately and do not couple with each other Imura  _et al._
(2009). In Eq.(17), the product of
$\gamma_{j,\textbf{k};j,\textbf{k}^{\prime\prime}}\Gamma_{j,\textbf{k}^{\prime\prime};j,\textbf{k}^{\prime}}$
does not contain the parameter $\phi_{\textbf{k}^{\prime\prime}}$ as it is
canceled out in the exponent. Therefore, this term can be taken out of the
summation of $\textbf{k}^{\prime\prime}$. We can then define the summation as
$\Pi$, transform the summation of $\textbf{k}^{\prime\prime}$ into an
integral, and evaluate it to obtain:

$\Pi_{jj}=\sum_{\textbf{k}^{\prime\prime}}G_{j,\textbf{k}^{\prime\prime}}^{R}G_{j,\textbf{q}-\textbf{k}^{\prime\prime}}^{A}=\frac{2\pi
N_{j,F}}{\hbar}\tau_{j}\left(1-D_{j}q^{2}\tau_{j}\right),$ (19)

here $\tau_{j}$ is the total scattering time of the valley $j$ in the Eq. (9),
$D_{j}$ is diffusion constant and $D_{j}={v}_{j}\tau_{j}^{2}/2$. The Eq. (17)
can be transformed into a concise form:

$\Gamma_{jj}=\gamma_{jj}+\Pi_{jj}\gamma_{jj}\Gamma_{jj}.$ (20)

$\gamma_{jj}$ is a concise form of the bare vertex function. Solve the Eq.
(20) and the Cooperon $\Gamma_{jj}$ can be obtained:

$\Gamma_{jj}=\frac{\gamma_{jj}}{1-\prod_{jj}\gamma_{jj}}.$ (21)

Combining Eqs. (18)-(23), we can obtain an explicit expression of the
Cooperon:

$\Gamma_{jj}^{(L)}=\frac{1}{Z_{j}D_{j}\tau_{j}^{2}}\frac{1}{l_{L}^{-2}+q^{2}},$
(22)

$l_{L}^{-2}=\frac{1-Z_{j}\tau_{j}\gamma_{jj}^{(L)}}{Z_{j}\tau_{j}^{2}D_{j}\gamma_{jj}^{(L)}}.$
(23)

In Eqs. (22) and (23), $Z_{j}=2\pi N_{j,F}/\hbar$, where $N_{j,F}$ is the
density of states of the $j$ valley at the Fermi level, and $l_{L}$ is the
scattering length of channel $L$. Since there are always three channels $L$ =
0, 1, 2 for Cooperon, we omit the channel index $L$ for brevity in the
following. It should be noted that all six channels from the two valleys have
a finite scattering length. The gap in the Cooperon of channel $L$ = 1 of
valley $\textbf{K}_{1}$ results from the inter-valley scattering that changes
the scattering time in Eq. (9). This differs from the situation in monolayer
graphene, where the $L$ = 1 channel corresponds to a Goldstone mode when only
intra-valley scattering occurs Imura  _et al._ (2009).

When considering the inter-layer scattering Cooperon, the situation becomes
more complicated because the Fermi surface of the two valleys are not equal in
size. For any given wave vector k in valley $\textbf{K}_{1}$, we cannot find
$-\textbf{k}$ in valley $\textbf{K}_{2}$. Thus, we introduce a non-zero
nesting vector Q. The primary contribution to the conductivity correction
comes from the portion of wave vectors k on the Fermi surface of one valley
that scatter to the wave vector $\textbf{Q}+\textbf{q}-\textbf{k}$ of the
other valley. This proportion is represented by the red arcs in Fig. 2(a). The
bare vertex function for inter-valley scattering is expressed as:

$\displaystyle\gamma_{12,21}\equiv$ $\displaystyle\left\langle
U_{1,\textbf{k};2,\textbf{k}^{\prime}}U_{2,\textbf{Q}+\textbf{q}-\textbf{k};1,\textbf{Q}+\textbf{q}-\textbf{k}^{\prime}}\right\rangle_{imp}$
(24) $\displaystyle=$ $\displaystyle\frac{\beta^{2}}{(\it\Delta
K_{\theta})^{2}}\frac{n_{0}u_{0}^{2}}{S}\\{a_{1}^{2}a_{2}^{2}+2a_{1}a_{2}b_{1}b_{2}e^{i\left(\phi_{\textbf{k}^{\prime}}-\phi_{\textbf{k}}\right)}$
$\displaystyle+b_{1}^{2}b_{2}^{2}e^{2i\left(\phi_{\textbf{k}^{\prime}}-\phi_{\textbf{k}}\right)}\\}.$

The subscripts 12,21 of $\gamma$ represent the correlation between the
scattering events of electrons from valley $\textbf{K}_{1}$ to valley
$\textbf{K}_{2}$ and its time-reversed process. “12” denotes $1,\textbf{k}$
and $2,\textbf{Q}+\textbf{q}-\textbf{k}$ on the upper (blue) and lower (red)
electron line respectively before scattering event happens as seen in Fig.
2(c). To distinguish the inter-valley scattering bare vertex function and
Cooperon from those of intra-valley in Eqs. (16) and (18), we use four indices
to represent them as shown in Eq. (24). To calculate the Cooperon
$\Gamma_{12,21}$, we need another Cooperon $\Gamma_{12,12}$ seen in the second
line of Fig. 2(c). The bare vertex function of $\gamma_{12,12}$ are defined
as:

$\displaystyle\gamma_{12,12}\equiv$ $\displaystyle\left\langle
U_{1,\textbf{k};1,\textbf{k}^{\prime}}U_{2,\textbf{Q}+\textbf{q}-\textbf{k};2,\textbf{Q}+\textbf{q}-\textbf{k}^{\prime}}\right\rangle_{imp}$
(25) $\displaystyle=$
$\displaystyle\frac{n_{0}u_{0}^{2}}{S}\\{a_{1}^{2}a_{2}^{2}+(a_{1}^{2}b_{2}^{2}+a_{2}^{2}b_{1}^{2})e^{i\left(\phi_{\textbf{k}^{\prime}}-\phi_{\textbf{k}}\right)}$
$\displaystyle+b_{1}^{2}b_{2}^{2}e^{2i\left(\phi_{\textbf{k}^{\prime}}-\phi_{\textbf{k}}\right)}\\}.$

The form of $\gamma_{12,12}$ is similar to that of intra-valley scattering in
Eq. (16). However, it represents the correlation between intra-valley
scattering events in valley $\textbf{K}_{1}$ and $\textbf{K}_{2}$, and its
meaning is completely different from that of Eq. (16). Additionally, the
Cooperon of this type, $\Gamma_{12,12}$, does not contribute to the
conductivity correction because the two wave vectors on the diagonal of the
Cooperon, $1,\textbf{k}$ and $2,\textbf{Q}+\textbf{q}-\textbf{k}^{\prime}$,
are not equal. As a result, the Bethe-Salpeter equation takes a coupled form:

$\displaystyle\Gamma_{12,12}=$
$\displaystyle\gamma_{12,12}+\sum_{\textbf{k}^{\prime\prime}}^{\prime}\gamma_{12,12}G_{1,\textbf{k}^{\prime\prime}}^{R}G_{2,\textbf{Q}+\textbf{q}-\textbf{k}^{\prime\prime}}^{A}\Gamma_{12,12}$
(26)
$\displaystyle+\sum_{\textbf{k}^{\prime\prime}}^{\prime}\gamma_{12,21}G_{2,\textbf{k}^{\prime\prime}}^{R}G_{1,\textbf{Q}+\textbf{q}-\textbf{k}^{\prime\prime}}^{A}\Gamma_{21,12}.$
$\displaystyle\Gamma_{21,12}=$
$\displaystyle\gamma_{21,12}+\sum_{\textbf{k}^{\prime\prime}}^{\prime}\gamma_{21,12}G_{1,\textbf{k}^{\prime\prime}}^{R}G_{2,\textbf{Q}+\textbf{q}-\textbf{k}^{\prime\prime}}^{A}\Gamma_{12,12}$
$\displaystyle+\sum_{\textbf{k}^{\prime\prime}}^{\prime}\gamma_{21,21}G_{2,\textbf{k}^{\prime\prime}}^{R}G_{1,\textbf{Q}+\textbf{q}-\textbf{k}^{\prime\prime}}^{A}\Gamma_{21,12}.$

A prime on the summation sign in Eq. (26) indicates that the summation over
the wave vector $\textbf{k}^{\prime\prime}$ is not over whole points on the
Fermi surface but over the points on the arc. The Feynman diagram
representation of this set of equations is shown in Fig. 2(c). The product of
$\gamma\Gamma$ does not contain $\phi_{\textbf{k}^{\prime\prime}}$, so it can
be factored out of the summation, leaving the summation similar to Eq. (19),
which we define as the summation $\Pi_{j\bar{j}}$:

$\Pi_{j\bar{j}}=\sum_{\textbf{k}^{\prime\prime}}^{\prime}G_{j,\textbf{k}^{\prime\prime}}^{R}G_{\bar{j},\textbf{Q}+\textbf{q}-\textbf{k}^{\prime\prime}}^{A},$
(27)

To calculate $\Pi_{j\bar{j}}$ exactly, two things need to be clarified.
Firstly, only the $\textbf{k}^{\prime\prime}$ points near the tangent point
and on the red arcs in Fig. 2(a) are considered in the calculation. By
introducing the nesting wave vector Q, we match as many wave vectors k of one
valley and $\textbf{k}^{\prime}$ of another valley as possible, such that they
satisfy $\textbf{k}=\textbf{Q}+\textbf{q}-\textbf{k}^{\prime\prime}$ and all
wave vectors on the two arcs correspond one to one. Due to the isotropy of the
two bands, we suppose to approximate the partial summation by summing over all
points on the Fermi surface multiplied by a hyperbolic tangent function
$f=\tanh(\lambda k_{min}/Q)$, where $k_{min}$ is the modulus of the smaller
Fermi wave vectors of the two valleys. Secondly, the summations $\Pi_{12}$ and
$\Pi_{21}$ are equal. Since they correspond to the same wave vector
$\textbf{k}^{\prime\prime}$ summation, and they are conjugate to each other.
Furthermore, considering the fact that they are real numbers, we can assume
that $\Pi_{12}$ and $\Pi_{21}$ are equal. Let $\Pi_{12}=\Pi_{21}=\Pi$. With
the above analysis, the form of $\Pi$ can be approximately given:

$\Pi=tanh(\lambda\frac{k_{min}}{Q})\frac{2\pi
N}{\hbar}\tau\left(1-Dq^{2}\tau\right).$ (28)

In Eq. (28), we have already used the condition
$\epsilon_{j,\textbf{k}}=\epsilon_{\bar{j},\textbf{Q}-\textbf{k}}$. Here, $N$
represents the average density of states of the two valleys at Fermi energy,
given by $N=(N_{1,F}+N_{2,F})/2$. The scattering time $\tau$ satisfies
$2/\tau=1/\tau_{1}+1/\tau_{2}$, where $\tau_{1}$ and $\tau_{2}$ are the
scattering times in valleys $\textbf{K}_{1}$ and $\textbf{K}_{2}$
respectively. The diffuson constant $D$ is defined as $D=v^{2}\tau/2$, where
the velocity $v$ satisfies
$v^{2}=(N_{1,F}v_{1}^{2}+N_{2,F}v_{2}^{2})/(N_{1,F}+N_{2,F})$. This treatment
mainly takes the average of the two valleys in order to obtain the equal
$\Pi_{12}$ and $\Pi_{21}$. The Bethe-Salpeter equation in Eq. (26) can be
modified into a set of equations:

$\left(\begin{array}[]{l}\Gamma_{12,12}\\\
\Gamma_{21,12}\end{array}\right)=\left(\begin{array}[]{l}\gamma_{12,12}\\\
\gamma_{21,12}\end{array}\right)+\left(\begin{array}[]{ll}\gamma_{12,12}&\gamma_{12,21}\\\
\gamma_{21,12}&\gamma_{21,21}\end{array}\right)\Pi\left(\begin{array}[]{l}\Gamma_{12,12}\\\
\Gamma_{21,12}\end{array}\right).$ (29)

Furthermore, the bare vertex functions satisfy the following relations:
$\gamma_{12,12}=\gamma_{21,21}=\gamma_{c}$,
$\gamma_{12,21}=\gamma_{21,12}=\gamma_{t}$, where $\gamma_{c}$ and
$\gamma_{t}$ represent the “cis” and “trans” forms, respectivelyImura  _et
al._ (2009). We can rewrite $\Gamma_{12,12}$ as $\Gamma_{c}$ and
$\Gamma_{21,12}$ as $\Gamma_{t}$, and simplify Eq. (29) as:

$\left(\begin{array}[]{c}\Gamma_{c}\\\
\Gamma_{t}\end{array}\right)=\left(\begin{array}[]{l}\gamma_{c}\\\
\gamma_{t}\end{array}\right)+\left(\begin{array}[]{ll}\gamma_{c}&\gamma_{t}\\\
\gamma_{t}&\gamma_{c}\end{array}\right)\Pi\left(\begin{array}[]{c}\Gamma_{c}\\\
\Gamma_{t}\end{array}\right)$ (30)

Only $\Gamma_{t}$ contributes to the conductivity correction and it can be
solved with the form:

$\displaystyle\Gamma_{t}=\frac{\gamma_{t}}{1-2\Pi\gamma_{c}+\Pi^{2}\left(\gamma_{c}^{2}-\gamma_{t}^{2}\right)}.$
(31)

Different from the Cooperon solved in Eq. (22), the Cooperons in Eq. (31)
cannot be directly transformed to a standard form of $1/(l^{-2}+q^{2})$. At
this point, we can neglect the $q^{4}$ term in the denominator, which is a
small quantity of higher order. As a result, the Cooperon $\Gamma_{t}$ can be
expressed as:

$\displaystyle\Gamma_{t}=\alpha_{t}\frac{1}{ZD\tau^{2}}\frac{1}{l_{t}^{-2}+q^{2}}$
(32)
$\displaystyle\alpha_{t}^{-1}=2\frac{\gamma_{c}}{\gamma_{t}}-2Z\tau\frac{\gamma_{c}^{2}-\gamma_{t}^{2}}{\gamma_{t}}$
$\displaystyle
l_{t}^{-2}=\frac{1}{D}(\frac{1}{Z\tau^{2}\gamma_{t}}-{2\frac{\gamma_{c}}{\tau_{c}\gamma_{t}}}+Z\frac{\gamma_{c}^{2}-\gamma_{t}^{2}}{\gamma_{t}}).$

$l_{t}$ is the scattering length of the inter-valley scattering Cooperon.
$Z=2\pi N\cdot\tanh(\lambda k_{min}/Q)/\hbar$. The form of the Cooperon is
analogous to that of intra-valley scattering, as shown in Eq. (22). Notice
that each expression contains three channels.

,

Figure 2: (a) The Fermi surface of the system at a specific energy. To aid
visualization, we shift the Fermi circles of the two valleys to the same
center. The larger (smaller) circle corresponds to the Fermi surface of valley
$\textbf{K}_{1}$ ($\textbf{K}_{2}$). $\textbf{k}_{1F}$ and $\textbf{k}_{2F}$
denote the Fermi wave vectors on the Fermi circle of the two valleys. The
nesting wave vector Q has a modulus of
$\left|\textbf{Q}\right|=\left|\textbf{k}_{1F}\right|-\left|\textbf{k}_{2F}\right|$.
The red arcs on the two Fermi circles indicate the part of wave vector k on
the larger circle and $\textbf{k}^{\prime}$ on the smaller circle that
satisfies $\textbf{k}+\textbf{k}^{\prime}=\textbf{Q}+\textbf{q}$ and
contribute to the conductivity correction. (b) The Bethe-Salpeter equation
that the intra-valley scattering Cooperon $\Gamma_{11}$ satisfies. The blue
(red) line represents the retarded (advanced) Green function. The pentacle
denotes an impurity, and impurity scattering is shown by dotted lines. (c) The
Bethe-Salpeter equations that the “trans” and “cis” Cooperons satisfy. (d) The
Feynman diagram depicts the intra- and inter- valley scattering contribution
of conductivity correction.

##  IV Conductivity correction versus temperature and magnetic field

The conductivity correction is directly connected with the Cooperon through:

$\displaystyle\it\Delta\sigma_{intra}=$
$\displaystyle\frac{e^{2}\hbar}{2\pi}\sum_{j}\sum_{\textbf{q}}\Gamma_{jj}(\textbf{q})\sum_{\textbf{k}}\widetilde{v}_{j,\textbf{k}}^{x}\widetilde{v}_{j,\textbf{q}-\textbf{k}}^{x}$
(33) $\displaystyle\times
G_{j,\textbf{k}}^{R}G_{j,\textbf{k}}^{A}G_{j,\textbf{q}-\textbf{k}}^{R}G_{j,\textbf{q}-\textbf{k}}^{A},$

$\displaystyle\it\Delta\sigma_{inter}=$
$\displaystyle\frac{e^{2}\hbar}{2\pi}\sum_{j}\sum_{\textbf{q}}\Gamma_{j\bar{j}}(\textbf{Q+q})\sum_{\textbf{k}}^{\prime}\widetilde{v}_{j,\textbf{k}}^{x}\widetilde{v}_{\bar{j},\textbf{Q+q}-\textbf{k}}^{x}$
(34) $\displaystyle\times
G_{j,\textbf{k}}^{R}G_{j,\textbf{k}}^{A}G_{\bar{j},\textbf{Q+q}-\textbf{k}}^{R}G_{\bar{j},\textbf{Q+q}-\textbf{k}}^{A}.$

Eqs. (33) and (34) represent the conductivity correction contributions from
intra- and inter- valley scattering. To begin with, we calculate the summation
of k:

$\sum_{\textbf{k}}\widetilde{v}_{j,\textbf{k}}^{x}\widetilde{v}_{j,\textbf{q}-\textbf{k}}^{x}G_{j,\textbf{k}}^{R}G_{j,\textbf{k}}^{A}G_{j,\textbf{q}-\textbf{k}}^{R}G_{j,\textbf{q}-\textbf{k}}^{A}=-\frac{2\pi\eta_{j}^{2}v_{j}^{2}N_{j,F}\tau_{j}^{3}}{\hbar^{3}},$
(35)

$\displaystyle\sum_{\textbf{k}}^{\prime}\widetilde{v}_{j,\textbf{k}}^{x}\widetilde{v}_{\bar{j},\textbf{Q+q}-\textbf{k}}^{x}G_{j,\textbf{k}}^{R}G_{j,\textbf{k}}^{A}G_{\bar{j},\textbf{Q+q}-\textbf{k}}^{R}G_{\bar{j},\textbf{Q+q}-\textbf{k}}^{A}$
(36) $\displaystyle=$
$\displaystyle-\frac{2\pi\eta_{j}\eta_{\bar{j}}v_{j}v_{\bar{j}}N\tau_{j}\tau_{\bar{j}}\tau}{\hbar^{3}}tanh(\lambda\frac{k_{min}}{Q}).$

In the limit $\textbf{q}\rightarrow 0$. The condition
$\epsilon_{j,\textbf{k}}=\epsilon_{\bar{j},\textbf{Q}-\textbf{k}}$ is used to
obtain Eq. (36). Next, we need to calculate the summation of Cooperon. By
using the explicit expressions of the Cooperons given in Eqs. (22) and (32),
and transforming the summation over q into an integral, the conductivity
correction can be written in the following form Datta (1997); Shan  _et al._
(2012):

$\displaystyle\it\Delta\sigma_{intra}$
$\displaystyle=-\sum_{j}\frac{\eta_{j}^{2}}{2\pi}\frac{e^{2}}{h}\sum_{L=0,1,2}(-1)^{L}\int_{l_{\varphi}^{-2}}^{l_{j,e}^{-2}}\frac{1}{l_{j,L}^{-2}+q^{2}}dq^{2}$
(37)
$\displaystyle=-\sum_{j}\frac{\eta_{j}^{2}}{2\pi}\frac{e^{2}}{h}\sum_{L=0,1,2}(-1)^{L}\ln\frac{l_{j,L}^{-2}+l_{j,e}^{-2}}{l_{j,L}^{-2}+l_{\varphi}^{-2}}.$

$\displaystyle\it\Delta\sigma_{inter}=$
$\displaystyle-\sum_{j}\frac{\eta_{j}\eta_{\bar{j}}}{2\pi}\frac{e^{2}}{h}\frac{\tau_{j}\tau_{\bar{j}}}{\tau^{2}}\frac{v_{j}v_{\bar{j}}}{v^{2}}\sum_{L=0,1,2}$
(38)
$\displaystyle\times(-1)^{L}\alpha_{t,L}\int_{l_{\varphi}^{-2}}^{l_{t,e}^{-2}}\frac{1}{l_{t,L}^{-2}+q^{2}}dq^{2}$
$\displaystyle=$
$\displaystyle-\sum_{j}\frac{\eta_{j}\eta_{\bar{j}}}{2\pi}\frac{e^{2}}{h}\frac{\tau_{j}\tau_{\bar{j}}}{\tau^{2}}\frac{v_{j}v_{\bar{j}}}{v^{2}}\sum_{L=0,1,2}$
$\displaystyle\times(-1)^{L}\alpha_{t,L}\ln\frac{l_{t,L}^{-2}+l_{t,e}^{-2}}{l_{t,L}^{-2}+l_{\varphi}^{-2}}.$

Here, $l_{j,e}$ represents the mean free path of valley $j$, where the inverse
square is taken as the upper limit of the integral with respect to
$\textbf{q}^{2}$. For inter-valley scattering, the effective mean free path is
denoted as $l_{t,e}$ and is calculated as $l_{t,e}=\sqrt{D\tau}$. The
coherence length $l_{\varphi}$ has a strong temperature dependence which is
given by $l_{\varphi}\propto T^{-1/2}$. We provide an explicit expression for
coherence length as $l_{\varphi}=l_{1,e}(T/T^{*})^{-1/2}$, where $T^{*}$ is
the temperature at which coherence length equals the mean free length of the
electron in valley $\textbf{K}_{1}$. The coefficient $\alpha_{t,L}$ has been
given in Eq. (32). The total conductivity correction is the sum of the
contributions from intra- and inter-scattering:

$\it\Delta\sigma=\it\Delta\sigma_{intra}+\it\Delta\sigma_{inter}.$ (39)

We investigate the conductivity correction of the system with twist angle
equals $3^{\circ}$, the Fermi energy $E_{F}=42$ meV, the gap of valley
$\textbf{K}_{2}$ $\it\Delta$ = 53 meV, $\lambda$ = 0.1,
$\beta=\sqrt{3/4}\it\Delta K_{\theta=1.65^{\circ}}$Phinney  _et al._ (2021),
where $\it\Delta K_{\theta=1.65^{\circ}}$ is the wave vector difference
between two Dirac points for a twist angle of $1.65^{\circ}$. The results are
presented in Figs. 3(a) and 3(b). In Fig. 3(a), the Fermi surface areas of the
two valleys are unequal when the bias voltage is turned off. We observe that
the contribution from inter-valley scattering denoted by green line is
negligible and the total correction is negative across the entire temperature
range, indicating a weak localization regime. When the bias voltage is turned
on and the energy shift $U$ = 8 meV, the bottom energy band is lifted, the
Fermi surfaces of the two valleys become close in size as is shown in Fig.
1(b). In Fig. 3(b), we find that inter-valley scattering plays a significant
role at this time, and the sign of the total conductivity correction changes
to positive value at low temperatures, indicating a crossover from weak
localization to weak anti-localization by tuning the bias voltage.

Figure 3: Conductivity correction and magnetoconductivity plots. (a)
Conductivity correction of the system at the Fermi energy $E_{F}$ = 42 meV
without bias. The gap $\it\Delta$ is set to be 53 meV. The red solid line
denotes the total correction, while the black, blue, and green dotted lines
represent the contribution of the corresponding parts. (b) Same parameter used
as (a) except for an energy shift $U$ = 8 meV. (c) The magneto-conductivity
plot corresponds to (a). (d) The magneto-conductivity plot corresponds to (b).
The mean free path $l_{1,e}$ is set to be 40 nm, and the temperature $T/T^{*}$
= 0.01. We observe a crossover from weak localization to weak anti-
localization at low temperatures.

Next, we turn to the calculation of magneto-conductivity. The external
vertical magnetic field induces the appearance of Landau levels, which
discretizes the wave vector q into a series of discrete numbers:
$q_{n}=\sqrt{(n+1/2)/l_{B}^{2}}$, where $l_{B}$ is the magnetic length of
Cooperon defined as $l_{B}=\sqrt{\hbar/4e\left|B\right|}$Datta (1997), The
summation over wave vector q can then be transformed into a summation over the
energy level index $n$. The finite-field conductivity correction is expressed
in terms of digamma functions $\Psi$,

Figure 4: (a) The phase diagram of the system with a twist angle of
$\theta=3^{\circ}$. The dotted box marks the double crossover region. (b) The
phase diagram of the system with a twist angle of $\theta=5^{\circ}$. The
temperature is set to be $T/T^{*}$ = 0.01.

$\displaystyle\it\Delta\sigma_{intra}(B)=$
$\displaystyle-\sum_{j,L}\frac{\eta^{2}}{2\pi}\frac{e^{2}}{h}(-1)^{L}\\{\Psi(\frac{1}{2}+\frac{l_{B}^{2}}{l_{j,L}^{2}}+\frac{l_{B}^{2}}{l_{j,e}^{2}}).$
(40)
$\displaystyle-\Psi(\frac{1}{2}+\frac{l_{B}^{2}}{l_{j,L}^{2}}+\frac{l_{B}^{2}}{l_{\phi}^{2}})\\}.$

$\displaystyle\it\Delta\sigma_{inter}(B)=$
$\displaystyle-\sum_{j,L}\frac{\eta_{j}\eta_{\bar{j}}}{2\pi}\frac{e^{2}}{h}\frac{\tau_{j}\tau_{\bar{j}}}{\tau^{2}}\frac{v_{j}v_{\bar{j}}}{v^{2}}(-1)^{L}$
(41)
$\displaystyle\times\\{\Psi(\frac{1}{2}+\frac{l_{B}^{2}}{l_{t,L}^{2}}+\frac{l_{B}^{2}}{l_{t,e}^{2}})-\Psi(\frac{1}{2}+\frac{l_{B}^{2}}{l_{t,L}^{2}}+\frac{l_{B}^{2}}{l_{\phi}^{2}})\\}.$

The magneto-conductivity is defined as the change of the conductivity
correction under a finite magnetic field:

$\it\Delta\sigma_{B}=\it\Delta\sigma(B)-\it\Delta\sigma(0).$ (42)

Figs. 3(c) and 3(d) present the magneto-conductivity curves. In the presence
of a magnetic field, time reversal symmetry is broken, suppressing quantum
interference in the system. The total magneto-conductivity curve shown in Fig.
3(c) exhibits an upward trend upon increasing the magnetic field beyond a
certain threshold, suggesting the existence of the original weak localization
state. On the other hand, the curve in Fig. 3(d) demonstrates an obvious
downward trend with an increasing magnetic field, indicating the system’s
original state is weak antilocalization. Experimental measurement of the
magneto-conductivity is feasible, and our results provide clear evidence of
the WL-WAL crossover via tuning of the bias voltage.

Furthermore, we present phase diagrams of WL and WAL in the Fermi energy
$E_{F}$ and bias voltage $U$ plane for different twist angles in Figs. 4(a)
and 4(b). The system with a small twist angle corresponds to stronger inter-
valley scattering, as evidenced by Eq. (7). In Fig. 4(a), for the system with
a twist angle of $\theta=3^{\circ}$, we observe that the red WAL region
penetrates deep into the blue WL region, resulting in a double crossover of
WL-WAL-WL. The energy shift $U$ and Fermi energy are within the region bounded
by the dotted box, with the Fermi energy ranging from 40 meV to 43 meV. This
double crossover can be explained by the relation between the inter-valley
scattering and the bias voltage. As we tune the bias voltage, the energy band
of valley $\textbf{K}_{2}$ shifts, and when the Fermi surface area of the two
valleys approaches each other, inter-valley scattering dramatically
contributes to the conductivity correction. As one continue to adjust the bias
voltage, the Fermi surface size of the two valleys gets further apart, and
only intra-valley scattering contributes to the conductivity correction. In
contrast, in Fig. 4(b), for a twist angle of $\theta=5^{\circ}$, the WAL and
WL regions are clearly separated, and the double crossover disappears due to
the weakening of the inter-valley scattering strength. Therefore, we conclude
from Figs. (3) and (4) that inter-valley scattering plays a critical role in
the double crossover. However, the conditions for detecting the double
crossover are quite stringent; the twist angle of the system cannot be too
large, and the energy region in which this crossover can be detected is
narrow.

##  V Summary and discussions

To sum up, we explore the quantum conductivity corrections about weak
(anti-)localization effects in the system with adjacent gapless and gapped
Dirac cones. We find that the inter-valley scattering has a significant impact
on conductivity correction, especially when the Fermi surface area of the two
valleys are equal. Through standard calculation, we observe a double crossover
in a specific range of Fermi energy by tuning bias voltage when the twist
angle is not too large. Meanwhile, we note that a tiny gap appears at the top
layer which is around 4 meV for twist angle $\theta=3^{\circ}$ when
considering lattice relaxation or corrugation. It may slightly weaken the
strength of WAL or WL, resulting in a less drastic double crossover.
Nevertheless it doesn’t influence our quantitative result. Our empirical
method for handling the inter-valley scattering with unequal Fermi surface
areas may be applicable to other multi-valley systems, and our calculations
may be beneficial for future experimental investigations on the quantum
transport properties of two-dimensional twisted moiré materials.

##  VI Acknowledgements

This work was financially supported by the National Basic Research Program of
China (Grants No. 2017YFA0303301), the National Natural Science Foundation of
China (Grants No. 12022407).

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