Theory of Optical Activity in Doped Systems with Application to Twisted Bilayer Graphene K. Chang1Z. Zheng1 2J. E. Sipe3and J. L. Cheng1 2y

2025-05-06 0 0 925.39KB 23 页 10玖币

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Theory of Optical Activity in Doped Systems with Application to

Twisted Bilayer Graphene

K. Chang,1, ∗Z. Zheng,1, 2 J. E. Sipe,3and J. L. Cheng1, 2, †

1GPL Photonics Lab, State Key Laboratory of Applied Optics,

Changchun Institute of Optics, Fine Mechanics and Physics,

Chinese Academy of Sciences, Changchun 130033, China.

2University of Chinese Academy of Science, Beijing 100039, China.

3Department of Physics, University of Toronto,

Toronto, Ontario M5S 1A7, Canada

(Dated: October 11, 2022)

arXiv:2210.03960v1 [cond-mat.mes-hall] 8 Oct 2022

Abstract

We theoretically study the optical activity in a doped system and derive the optical activity

tensor from a light wavevector-dependent linear optical conductivity. Although the light-matter

interaction is introduced through the velocity gauge from a minimal coupling Hamiltonian, we ﬁnd

that the well-known “false divergences” problem can be avoided in practice if the electronic states

are described by a ﬁnite band eﬀective Hamiltonian, such as a few-band tight-binding model. The

expression we obtain for the optical activity tensor is in good numerical agreement with a recent

theory derived for an undoped topologically trivial gapped system. We apply our theory to the

optical activity of a gated twisted bilayer graphene, with a detailed discussion of the dependence of

the results on twist angle, chemical potential, gate voltage, and location of rotation center forming

the twisted bilayer graphene.

I. INTRODUCTION

Optical activity, also known as optical rotatory power, describes the rotation of the

polarization direction as light propagates through an optically active medium [1, 2]. This

phenomenon arises from the diﬀerent responses to left and right circularly polarized light;

the diﬀerence in absorption of the two polarizations is referred to as circular dichroism [3].

Despite the broad application of circular dichroism in detecting the chirality of molecules

[4, 5], the relevant research in crystals is limited [6, 7]. The quantum treatment of optical

activity tensor is usually obtained from the charge-current density response [1, 8–10] with

the light-matter interaction included via the minimal coupling Hamiltonian [9]. However,

this method can lead to “false divergences” when the set of bands involved in the calculation

is inevitably truncated [11, 12], and appropriate “sum rules” must be applied to show that

the prefactors multiplying the divergent terms in fact vanish. To avoid this diﬃculty in

calculating the optical activity, Mahon and Sipe [1] proposed a multipole moment expansion

method for optical conductivity, where the macroscopic ﬁelds are introduced through the

interactions with electric dipole, magnetic dipole, and electric quadrupole moments associ-

ated with Wannier functions at the lattice sites. Despite the link this approach identiﬁes

∗knchang@ciomp.ac.cn

†jlcheng@ciomp.ac.cn

with treating the optical response of isolated molecules, the derivation is complicated and

the treatment is at present limited to undoped, topologically trivial insulators.

Alternate derivations — particularly if they are simpler — can often stimulate new re-

search. Starting from a minimal coupling Hamiltonian, in this paper we derive the expres-

sions for the optical activity from the linear conductivity tensor at ﬁnite light wavevector.

We then apply this method to study the optical activity of twisted bilayer graphene (TBG)

using a simple tight-binding model [13–15] to describe its electronic states. The optical

response of TBG has been extensively studied [16–19], with the optical activity of undoped

TBG investigated both experimentally [20] and theoretically [13]. In the undoped limit we

ﬁnd agreement with the results of Mahon and Sipe [1], except for an extra term that indeed

seems to exhibit a “false divergence”; however, while we cannot conﬁrm analytically that the

prefactor of this term vanishes, we can verify that it has a negligible value in our ﬁnite band

tight-binding model. And our approach is more general than that of Mahon and Sipe [1]

in that it can be extended to doped systems; as well, it does not explicitly involve Wannier

functions and the topological considerations necessary to construct them as localized func-

tions. With our results in hand, we explore the dependence of the optical activity tensor on

twist angle, chemical potential, gate voltage, and location of rotation center forming twisted

bilayer graphene.

II. MODELS

A. Optical activity tensor

An optically active material has diﬀerent responses to left and right circularly polarized

light, which are described by its linear optical conductivity σda(q, ω). For an electric ﬁeld

E(r, t) = E(q, ω)ei(q·r−ωt)+ c.c., the induced optical current is J(r, t) = J(q, ω)ei(q·r−ωt)+

c.c.with Jd(q, ω) = σda(q, ω)Ea(q, ω). The Roman letters in the superscript denote Carte-

sian directions x/y/z, and the repeated superscripts are summed over. Without losing

generality, considering an incident light propagating along the z-direction, the response to

the circularly polarized light can be written as [21]

Jδ1(qˆ

z, ω) = σδ1δ2(qˆ

z, ω)Eδ2(qˆ

z, ω),(1)

where Jδ= (Jx+iδJy)/√2 with δ=±gives the left (+)/right (−) circular component of

the current; a similar deﬁnition applies to Eδ. Then the diagonal response coeﬃcients are

σδδ(qˆ

z, ω) = σxx +σyy −iδ(σxy −σyx)

2.(2)

For nonzero σxy −σyx, the responses to the left and right circularly polarized lights are

not the same, and the circular dichroism can be characterized by the ellipticity spectra

Ψ=(α+−α−)/(2(α++α−)), where αδis the absorption of the δcircularly polarized light

[22]. For two-dimensional materials, the absorption is proportional to Re[σδδ], and we can

get

Ψ = Im[σxy −σyx]

2Re[σxx +σyy].(3)

The wave vector qappearing in the linear conductivity σda(q, ω) is very small compared

to most electron wave vectors, and up to the ﬁrst order in qthe conductivity can be expanded

σda(q, ω) = σda(ω) + Sdac(ω)qc+··· ,(4)

with σda(ω)≡σda(0, ω) giving the long wavelength limit, and

Sdac(ω)≡∂σda(q, ω)

∂qcq=0

.(5)

The latter arises from eﬀects of magnetic dipole and electric quadrupole, and modiﬁed

electric dipole eﬀects [21]. From macroscopic optics, it gives the response of the current to

the spatial derivative of the electric ﬁeld, including contributions following from Faraday’s

law.

For non-magnetic materials, the conductivity tensor σda(ω) has either no oﬀ-diagonal

components or equal oﬀ-diagonal components σda(ω) = σad(ω) [23], and the optical activity

mainly comes from the terms involving Sdac(ω), which is our focus in this paper. Because

Sdac(ω) is a third-order tensor, it is nonzero only for crystals breaking inversion symmetry;

more speciﬁcally, the nonzero Im[σxy −σyx] indicates a chiral structure.

B. A microscopic response theory for Sdac(ω)

For very weak electromagnetic ﬁelds, the Hamiltonian with the inclusion of light-matter

interaction is taken from the minimal coupling as

H(t) = ˆ

H0+−e

2[ˆvaAa(r, t) + Aa(r, t)ˆva] + (−e)2

4hˆ

MdaAd(r, t)Aa(r, t)

+Ad(r, t)Aa(r, t)ˆ

Mdai+··· ,(6)

with the electronic charge e,ˆ

va= [ˆra,ˆ

H0]/(i~), and ˆ

Mda = [ˆrd,ˆva]/(i~). Here ˆ

H0is the

crystal Hamiltonian without external ﬁeld, ˆvais the velocity operator, ˆ

Mda is an operator as-

sociated with the mass term, and Aa(ˆ

r, t) is the ath component of the vector potential of the

electromagnetic ﬁeld. The detailed explanation of this Hamiltonian is listed in Appendix A.

The ﬁeld operator can be expanded as

Ψ(r, t) = X

nZdkˆank(t)φnk(r).(7)

Here φnk(r)=1/p(2π)3eik·runk(r) are the band eigenstates of ˆ

H0with band eigenenergies

εnkand periodic functions unk(r), and ˆank(t) is an annihilation operator of this state. The

second quantization form of the Hamiltonian ˆ

H(t) is

H(t) = X

nZBZ

dkεnkˆa†

nkˆank+Zdq

(2π)3(−e)Aa

q(t)X

nm ZBZ

dkV(1);a

nk+q,mkˆa†

nk+qˆamk

2Zdq1dq2

(2π)6(−e)2Aa

q1(t)Ab

q2(t)X

nm ZBZ

dkV(2):ab

nk+q1+q2,mkˆa†

nk+q1+q2ˆamk.(8)

Here Aa

q(t) = RdrAa(r, t)e−iq·ris the Fourier component of the vector potential, and the

matrix elements are given by

V(1);a

nk,mk1=1

lva

nlkUlk,mk1+Unk,lk1va

lmk1,(9)

V(2);ab

nk,mk1=1

lMab

nlkUlk,mk1+Unk,lk1Mab

lmk1,(10)

Unk,mk1=Zuc

Ωu∗

nk(r)umk1(r),(11)

where va

nmkand Mab

nmkare the matrix elements of single particle operators ˆvaand ˆ

Mab,

respectively; Ω is the volume of the unit cell. The current density operator is given as

Ja(q, t) = Zdrˆ

Ja(r, t)eiq·r=−Zdrδˆ

H(t)

δAa(r, t)eiq·r=−(2π)3δˆ

H(t)

δAa

−q(t),(12)

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摘要：

TheoryofOpticalActivityinDopedSystemswithApplicationtoTwistedBilayerGrapheneK.Chang,1,Z.Zheng,1,2J.E.Sipe,3andJ.L.Cheng1,2,y1GPLPhotonicsLab,StateKeyLaboratoryofAppliedOptics,ChangchunInstituteofOptics,FineMechanicsandPhysics,ChineseAcademyofSciences,Changchun130033,China.2UniversityofChineseAcadem...

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Theory of Optical Activity in Doped Systems with Application to Twisted Bilayer Graphene K. Chang1Z. Zheng1 2J. E. Sipe3and J. L. Cheng1 2y

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