Distinct moiré textures of in-plane electric polarizations for distinguishing moiré origins in homobilayers

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Distinct moiré textures of in-plane electric polarizations for

distinguishing moiré origins in homobilayers

Hongyi Yu1,2†, Ziheng Zhou1, Wang Yao3,4

1 Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing & School of Physics and

Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China

2 State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen University (Guangzhou

Campus), Guangzhou 510275, China

3 Department of Physics, The University of Hong Kong, Hong Kong, China

4 HKU-UCAS Joint Institute of Theoretical and Computational Physics at Hong Kong, China

† yuhy33@mail.sysu.edu.cn

Abstract: In binary compound 2D insulators/semiconductors such as hexagonal boron nitride (hBN), the

different electron affinities of atoms can give rise to out-of-plane electric polarizations across inversion

asymmetric van der Waals interface of near 0° interlayer twisting. Here we show that at a general stacking

order where sliding breaks 2π/3-rotational symmetry, the interfacial charge redistribution also leads to an

in-plane electric polarization, with a magnitude comparable to that of the out-of-plane ones. The effect is

demonstrated in hBN bilayers, as well as in biased graphene bilayers with gate-controlled interlayer charge

redistributions. In long wavelength moiré patterns, the in-plane electric polarizations determined by the

local interlayer stacking registries constitute topologically nontrivial spatial textures. We show that these

textures can be used to distinguish moiré patterns of different origins from twisting, biaxial- and uniaxial-

heterostrain, where vector fields of electric polarizations feature Bloch-type merons, Néel-type merons, and

anti-merons, respectively. Combinations of twisting and heterostrain can further be exploited for

engineering various electric polarization textures including 1D quasiperiodic lattices.

Keywords: moiré patterns, van der Waals stacking, electric polarization, hexagonal boron nitride, graphene

PACS: 73.21.Cd, 73.21.Ac, 77.80.–e, 73.43.Cd

Long-wavelength moiré patterns formed in van der Waals layered insulator/semiconductor

systems have emerged as a fascinating platform for exploring novel physical phenomena [1,2],

including moiré excitons [3-8], and electron correlation phenomena [9-20] where many-body

interaction becomes significant due to the flat dispersion of minibands [21-23]. The wavelength

of the moiré pattern usually lies between several to several tens nm, much larger than the

monolayer lattice constant. In a nanoscale region small compared to the moiré supercell, the

interlayer coupling is determined by the local stacking registry, which varies smoothly and

periodically in the long-wavelength moiré landscape. This gives rise to spatial modulations in a

variety of physical quantities including the interlayer distance, local bandgap, optical transition

dipole, and magnetization [24-28].

Some less intuitive moiré modulated phenomena are recently noted in the context of binary

compound homobilayer insulators/semiconductors such as hexagonal boron nitride (hBN) and

semiconducting transition-metal dichalcogenides (TMDs). With the two layers being identical,

layer pseudospin describing the layer occupation of the carrier becomes an active quantum degree

of freedom, which is associated with an out-of-plane electric polarization. Theoretical studies

have shown that in the moiré landscape the layer pseudospin is subject to an effective Zeeman

field of topologically nontrivial spatial texture, underlying various topological phenomena in

homobilayer moiré [29-33]. In particular, the out-of-plane component of such pseudospin

Zeeman field arises from a spontaneous interfacial electric polarization that is determined by the

stacking registries between the layers [34-41]. Taking AB/BA-stacked hBN bilayers as an

example, where B atoms of one layer are vertically aligned with N atoms of the other, it has been

pointed out that different electron affinities of the atoms lead to a small amount of electron

transfer from N to nearest B atoms across the van der Waals interface [42]. The resultant out-of-

plane electric polarization has opposite signs in the AB and BA configurations, which are

connected by an interlayer sliding. The observation of the out-of-plane electric polarization and

its sliding control have been reported in commensurate and marginally twisted bilayer hBN and

TMDs [34-41], which can be exploited for ferroelectric functionalities, as well as for noninvasive

engineering of superlattice potentials in adjacent 2D materials [43].

Here we show that the nature of the stacking registry determined electric polarization

necessarily implies the presence of an in-plane component and its moiré-patterned spatial texture,

accompanying the out-of-plane one already observed. Just like the AB/BA stacking where 2π/3-

rotational (Ĉ3) symmetry only allows an out-of-plane electric polarization, there exist stacking

registries with π-rotational (Ĉ2) symmetry about an in-plane axis which only allows an in-plane

polarization [44]. Long wavelength moiré patterns formed by twisting, biaxial- or uniaxial-

heterostrain have distinct spatial distributions of the Ĉ2-symmetric locales. While this difference

has been neglected in other contexts of moiré phenomena, we show that it dictates distinct vector

fields of the polarization textures in moiré patterns of the three different origins, corresponding to

superlattices of Bloch-type merons, Néel-type merons and anti-merons, respectively.

Combinations of twisting and heterostrain can further be exploited for engineering various

polarization textures including 1D quasiperiodic lattices. The effect is demonstrated in bilayer

hBN of the R-type stacking, as well as in biased graphene bilayer and H-type hBN bilayer with

the gate-controlled interlayer charge redistribution. We find that the in-plane electric polarization

at the Ĉ2-symmetric locale can have comparable magnitude to that of the well characterized

sizable out-of-plane ones at Ĉ3-symmetric locales.

We first give an intuitive understanding to the out-of-plane and in-plane electric polarizations

in van der Waals bilayer semiconductor/insulator systems induced by the interlayer coupling. A

quantitative calculation follows. We use hBN as an illustration because of its simple band

structure, while the generalization to other systems like TMDs [37-40,44] is straightforward. In

bilayer hBN, a small amount of the electrons in the N atom can be transferred to the nearby B

atoms in the other layer due to their different electron affinities [42]. Such an interlayer charge

redistribution gives rise to local electric dipoles pointing from B to N, and the overall polarization

determined by the summation of them is generally finite (see Fig. 1(a)). We consider an R-type

bilayer hBN with a 0° interlayer twisting, characterized by an interlayer translation

. Fig. 1(b)

shows three high-symmetry stacking orders. The top and middle panels correspond to the so-

called AB and BA stackings with 0D

=ra and D

a, respectively [45,46], where

D12

a aa

º+

is the long diagonal line of the unit cell. These stackings are Ĉ3-symmetric since a B atom in one

layer horizontally overlaps with an N atom in the other layer, where the interlayer charge

redistribution results in electric dipoles along the out-of-plane direction. Because these stackings

have the minimum interlayer distances between B and N, the magnitudes of the charge

redistribution and the resultant out-of-plane electric polarization are expected to be the largest.

The lower panel of Fig. 1(b) and its Ĉ3 rotations correspond to the intermediate stackings between

AB and BA, denoted as IM. It has Ĉ2 symmetry about an axis within the 2D plane, allowing an

electric dipole in-plane but not out-of-plane. In Fig. 1(b) we use green arrows to indicate local

electric dipoles between nearest interlayer B-N pairs. For AA stacking with 0

r (not shown),

the bilayer structure has Ĉ3 and in-plane mirror symmetries, thus the total electric polarization

vanishes. For general stacking patterns with arbitrary

, the induced electric polarizations are

expected to have both the out-of-plane and in-plane components (Fig. 1(a)).

Fig. 1 (a) Top and side views of an R-type bilayer hBN with an arbitrary interlayer translation 0

r. The

large (small) blue dots denote N (B) atoms in the upper layer, and the large (small) orange dots denote N

(B) atoms in the lower layer. The green arrows indicate the local electric dipole pointing from a B atom in

one layer to a nearest N atom in the other layer, induced by the interlayer charge redistribution. (b) The

three high-symmetry stacking patterns AB, BA and IM (from up to down). AB and BA stackings are Ĉ3-

symmetric with out-of-plane electric polarizations, whereas IM stacking with in-plane Ĉ2 symmetry has an

in-plane electric polarization. D12

a aaº+ is the long diagonal line of the unit cell. (c) A schematic

illustration of the interlayer couplings (black arrows) for the conduction and valence bands. Vis the

interlayer bias. The magnitudes of kvc'

t,and kcv'

t,are much smaller than the energy separation k

D thus

can be eliminated by a second-order perturbation (see Eq. (1)). The Fermi level lies in the gap so both

valence states are occupied whereas both conduction states are empty.

To quantitively calculate the induced electric polarization, we start from a layer-decoupled

bilayer hBN. The lower-layer (upper-layer) Bloch state at wave vector k is denoted as kn

()

kn'

with an energy kn

()

kn'

. Here we consider the two π-bands of hBN [45], termed as

conduction and valence bands ()=n cv,. The interlayer coupling is treated as a perturbation,

which couples kn

and kn'

with a strength knn'

t,, see Fig. 1(c). As knn'

t, is much weaker than

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摘要：

Distinctmoirétexturesofin-planeelectricpolarizationsfordistinguishingmoiréoriginsinhomobilayersHongyiYu1,2†,ZihengZhou1,WangYao3,41GuangdongProvincialKeyLaboratoryofQuantumMetrologyandSensing&SchoolofPhysicsandAstronomy,SunYat-SenUniversity(ZhuhaiCampus),Zhuhai519082,China2StateKeyLaboratoryofOptoel...

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Distinct moiré textures of in-plane electric polarizations for distinguishing moiré origins in homobilayers

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