Berry curvature spin Hall eect and nonlinear optical response in moir e transition metal dichalcogenide heterobilayers Jin-Xin Hu1Ying-Ming Xie1and K. T. Law1y

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Berry curvature, spin Hall eﬀect and nonlinear optical response in moir´e transition

metal dichalcogenide heterobilayers

Jin-Xin Hu,1Ying-Ming Xie,1, ∗and K. T. Law1, †

1Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

Recently, topological ﬂat bands and the spin Hall eﬀect have been experimentally observed in AB-

stacked MoTe2/WSe2heterostructures. In this work, we systematically study the Berry curvature

eﬀects in moir´e transition metal dichalcogenide (TMD) heterobilayers. We point out that the moir´e

potential of the remote conduction bands would induce a sizable periodic pseudo-magnetic ﬁeld

(PMF) on the valence band. This periodic PMF creates net Berry curvature ﬂux in each valley

of the moir´e Brillouin zone. The combination of the eﬀect of the Berry curvature and the spin-

valley locking can induce the spin Hall eﬀect being observed in the experiment. Interestingly, the

valley-contrasting Berry curvature distribution generated by the PMF can be probed through shift

currents, which are DC currents induced by linearly polarized lights through nonlinear responses.

Our work sheds light on the novel quantum phenomena induced by Berry curvatures in moir´e TMD

heterobilayers.

I. INTRODUCTION

The discovery of two-dimensional moir´e materials leads

to the engineering of new platforms for the study of

novel topological, superconducting, and magnetic prop-

erties of electrons in recent years [1–10]. For example,

magneto-electric and nonlinear Hall eﬀects have been

demonstrated in twisted graphene superlattice [11–15]

and twisted transition metal dichalcogenide (TMD) ho-

mobilayers [16].

Notably, moir´e TMD heterobilayers, in which moir´e

pattern mainly originated from the lattice mismatching

between two distinct TMD layers, have been observed to

exhibit nontrivial topological and correlated properties

[17–27]. The study showed that a quantum anomalous

Hall state at ﬁlling with ν= 1 (one hole per moir´e unit

cell) was observed in AB stacked moir´e MoTe2/WSe2het-

erobilayers [28–35]. Very recently, the spin Hall torque

has been demonstrated near ν= 1 and ν= 2 stem-

ming from the large Berry curvature in this AB-stacked

2L-MoTe2/WSe2heterostructures [36]. However, unlike

the graphene moir´e superlattice or twisted TMD ho-

mobilayers, the novel responses induced by the Berry

curvature in TMD heterobilayers remain unknown the-

oretically. Moreover, in previous works [37–40], the

model for TMD heterobilayers is simply described by

H=−ˆp2/(2m) + V(r), where ˆpis the crystal momen-

tum operator, mis an electron eﬀective mass and V(r)

is the moir´e potential. As Hsimply represents a valence

band free Fermion moving in a periodic potential, the

discovery of Berry curvature induced spin Hall eﬀect in

the experiment is quite surprising.

In this work, we describe the moir´e TMD heterobi-

layers as a massive Dirac Fermion moving in a periodic

moir´e potential, in which the moir´e potential of both con-

∗Corresponding author: yxieai@connect.ust.hk

†Corresponding author: phlaw@ust.hk

duction band and valence band is taken into account.

Given that the low energy states are near the valence

band edge, we project out the freedom of the conduc-

tion band by using the quantum commutation relation

of crystal momentum ˆ

pand position ˆ

r. Remarkably, we

ﬁnd that the moir´e potential on the conduction band,

which although being 1 ∼2 eV away, contributes a pe-

riodic pseudo-magnetic ﬁeld (PMF) to the valence band

in the low energy state. We next show that the peri-

odic PMF results in a moir´e valley-contrasting Berry cur-

vature distribution, which exhibits net Berry curvature

ﬂux in each valley. Being consistent with the experiment

in [36], we ﬁnd a large spin Hall eﬀect in this case. It

arises from a combination of the giant Ising spin-orbit

coupling and the net Berry curvature ﬂux induced by

PMF. Finally, we show that the predicted moir´e valley-

contrasting Berry curvature distribution induced by the

periodic PMF could exhibit a salient feature in the shift

current response, which is a second-order DC response

by applying a linear polarized light. The shift current re-

sponse is tied to the quantum geometric properties of the

system and varies microscopically due to changes in prop-

erties of the Bloch wavefunction upon excitation between

bands [41–43]. Due to the presence of valley-contrasting

Berry curvature distribution, we ﬁnd that the photocur-

rent as a function of photon energy exhibits two peaks

and the peak separation is proportional to the strength

of PMF. Our theory highlights that the periodic PMF

plays an important role in the novel responses induced

by Berry curvature in moir´e heterobilayer TMDs.

II. MODEL HAMILTONIAN

Due to a large band oﬀset (hundreds of meV) between

the two layers in moir´e TMD heterobilayers, we assume

that the low energy states are arisen from one layer, while

the other layer contributes to a periodic moir´e potential.

It is known that the 2H-TMD monolayer is described by

massive Dirac Fermions [44]. For MoTe2/WSe2heterobi-

arXiv:2210.00759v4 [cond-mat.mes-hall] 18 Feb 2023

-200

-150

-100

-50

E (meV)

(a)

(b) (c)

FIG. 1: (a) The schematic picture of TMD heterobilayers,

with a top layer (red and blue atoms) and bottom layer (yel-

low and green atoms). The low energy physics of the top

layer is described by a massive Dirac model with a modiﬁed

moir´e potential. (b) The landscape of a C3symmetric peri-

odic PMF indicated by Eq.(4). We set Uc= 20 meV, φc= 0.4

π,B0= 30 T. (c) The calculated moir´e bands with B0=30

T, φc= 0.4π,Uv= 12 meV, φv= 0.3π. The zero energy is

shifted to the band edge.

layers, the valence band maximum of MoTe2is about 300

meV higher than WSe2[45]. We thus model the MoTe2

layer with a massive Dirac Hamiltonian including slow-

varying moir´e potential on both conduction and valence

band

H=vF0π†

π0+Uc(ˆr) 0

0Uv(ˆr)+∆

2σz,(1)

where ˆπis the momentum operator with ˆπ=τˆpx+iˆpy,

vFis the Fermi velocity, ∆ is the energy gap between

the conduction band and the valence band, τ=±de-

note Kand K0valleys. See Fig. 1(a) for an illustra-

tion of this model. Ucand Uvrepresent the moir´e po-

tential of conduction and valence band with Uc(r) =

2UcP3

i=1 cos(Gi·r+φc), Uv(r) = 2UvP3

i=1 cos(Gi·r+

φv), which is dedicated by the D3point group symmetry.

Gj=G0(sin(4(j−1)π

3),cos(4(j−1)π

3)), G0=4π

√3LM. To be

speciﬁc, we set the moir´e lattice constant LM≈5 nm,

vF= 4 ×105m/s and ∆ =1 eV, which are estimated

from the MoTe2/WSe2moir´e heterobilayers [46].

We next project out the conduction band and obtain

a low energy eﬀective Hamiltonian to describe the states

near the valence band edge in moir´e TMD heterobilayers.

To the ﬁrst order, we get the eﬀective Hamiltonian

Heff =−1

2m∗ˆπ(1 −Uc(ˆr)

∆)ˆπ†+Uv(ˆr)−∆

2,(2)

where m∗is the eﬀective mass with m∗= ∆/(2v2

F). By

using the commutation relation [ˆr,ˆp] = i~, we ﬁnd the

eﬀective Hamiltonian becomes

Heff =−1

2m∗(p2

x+p2

y+ 2eτp·A) + Uv(r)−∆

2,(3)

where the vector potential A(r) = −A0[a2sin(G1·r+

φc)−a1sin(G2·r+φc)−a3sin(G3·r+φc)] with

A0=~UcG0

e∆,a1= (1/2,−√3/2),a2= (1,0),a3=

a2−a1. The vector potential A(r) obeys Coulomb gauge

∇ · A(r) = 0. The details of deriving the continuum

model are shown in Appendix A.

Notably, we ﬁnd besides the kinetic energy part, the

eﬀective Hamiltonian includes a p·Aterm. This term

arises from the conduction band’s moir´e potential and the

momentum-dependent mixing induced by the momentum

operator ˆπ. One can regard Aas a gauge potential so

that we deﬁne the PMF Bps(r) as

Bps(r) = ∂xAy−∂yAx=τ B0

i=1

cos(Gi·r+φc),(4)

with the strength of PMF B0=~UcG2

0/(e∆). The

strength of PMF is mainly determined by the energy

gap ∆ and the conduction band moir´e potential Uc. It

is worth noting that the moir´e potential on the valence

band has no inﬂuence on the PMF though it plays an

important role in the band structure.

The topography of this PMF Bps(r) is shown in

Fig.1 (b), which displays the same period as the moir´e

superlattice. By using a conduction band moir´e poten-

tial Uc= 20 meV and energy gap ∆ = 1 eV, we ﬁnd the

PMF strength B0is as sizable as 30 T. Naively, it seems

one can completely neglect the conduction band and its

moir´e potential as ∆ is very large in this case. However,

our ﬁnding points out that the conduction band’s moir´e

potential would enable the states at the valence band to

experience an eﬀective PMF.

To see how the PMF aﬀects the moir´e band structure

of the TMD heterobilayers, we then diagonalize the ef-

fective Hamiltonian with plane wave basis. The resulting

moir´e bands of K-valley are plotted in Fig.1 (c), whereas

the K0-valley is related by the time-reversal symmetry

operation. To verify the accuracy of our projected ef-

fective continuum model, in Fig.2 we compare the cal-

culated Berry curvature of the top moir´e band (purple

band in Fig.1 (c)) between the full Dirac Hamiltonian in

Eq.1 (Fig.2 (a),(b)) and the eﬀective Hamiltonian in Eq.3

(Fig.2 (c),(d)), which shows a good agreement. It can be

seen that there is a Berry curvature centering around Km

and −Kmpockets within the moir´e Brillouin zone. The

PMF enables a distinct gap between these two pockets.

By further tuning the conduction band’s moir´e poten-

tial Ucto change the PMF, the top two moir´e bands can

further exchange Berry curvature by gap closing and re-

opening and undergo a topological phase transition. Fol-

lowing Ref. [29] using the three-band continuum model

near ±Kmpoint, we can obtain the topological phase

transition boundary lines analytically

B0sin(φc+π

6) = ±4√3m∗

e~Uvcos(φv+π

6).(5)

In Fig.3 (a) we numerically calculate the topological

phase diagram with various B0and φcby using the con-

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Berrycurvature,spinHalleectandnonlinearopticalresponseinmoiretransitionmetaldichalcogenideheterobilayersJin-XinHu,1Ying-MingXie,1,andK.T.Law1,y1DepartmentofPhysics,HongKongUniversityofScienceandTechnology,ClearWaterBay,HongKong,ChinaRecently,topologicalatbandsandthespinHalleecthavebeenexperiment...

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Berry curvature spin Hall eect and nonlinear optical response in moir e transition metal dichalcogenide heterobilayers Jin-Xin Hu1Ying-Ming Xie1and K. T. Law1y

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