8 February 2016 Classical electromagnetic model of surface states in topological insulators
Author Affiliations +
J. of Nanophotonics, 10(3), 033004 (2016). doi:10.1117/1.JNP.10.033004
Abstract
A topological insulator is classically modeled as an isotropic material with a magnetoelectric pseudoscalar Ψ existing in its bulk while its surface is charge free and current free. An alternative model is obtained by setting Ψ≡0 and incorporating surface charge and current densities characterized by an admittance γ. Analysis of planewave reflection and refraction due to a topological-insulator half space reveals that the parameters Ψ and γ arise identically in the reflection and transmission coefficients, implying that the two classical models cannot be distinguished on the basis of any scattering scenario. However, as Ψ disappears from the Maxwell equations applicable to any region occupied by the topological insulator, and because surface states exist on topological insulators as protected conducting states, the alternative model must be chosen.
Lakhtakia and Mackay: Classical electromagnetic model of surface states in topological insulators

1.

Introduction

The discovery of topological insulators1 has prompted researchers in classical optics23.4 to examine electromagnetic scattering due to bound objects made of these materials, exemplified by chalcogenides such as Bi2Se3, Bi2Te3, and Sb2Te3. As a topological insulator is considered to be an isotropic material, its frequency-domain constitutive relations are formulated to contain a magnetoelectric pseudoscalar (denoted by Ψ here) in addition to the permittivity scalar ϵ and the permeability scalar μ. The surface of the topological insulator is assumed to be charge free and current free, and the scattering problem can then be solved by following textbook techniques.5

Yet, according to condensed-matter theory, surface states exist on topological insulators as protected conducting states,2 and the characteristic electromagnetic responses of these materials are due to those surface states. Should then a topological insulator’s optical response be modeled as due solely to either

  • I. the bulk constitutive parameter Ψ with the surface of the topological insulator being charge free and current free, or

  • II. a surface parameter (denoted by γ here) that quantifies the charge density and current density on the surface of the topological insulator?

The topological insulator possesses the permittivity ϵ and permeability μ in both models. Both Ψ and γ are admittances, and whereas the magnetoelectric constitutive parameter Ψ mediates between D and B as well as between H and E throughout the topological insulator, γ is meaningful only on the surface of that material.

This communication is devoted to a comparison of models I and II, through the fundamental boundary-value problem of reflection and refraction of a plane wave. This problem is described and solved in Sec. 2 for both models. Section 3 contains a comparative discussion of the two models. Vectors are underlined. An exp(iωt) dependence on time t is implicit, with ω as the angular frequency and i=1.

2.

Fundamental Boundary-Value Problem

Suppose that all space is divided into two mutually disjoint half spaces Vout={(x,y,z):z<0} and Vin={(x,y,z):z>0} separated by the surface S={(x,y,z):z=0}. We need to solve the frequency-domain macroscopic Maxwell equations:

(1)

·B(r,ω)=0×E(r,ω)iωB(r,ω)=0·D(r,ω)=0×H(r,ω)+iωD(r,ω)=0}
in Vout and Vin separately, and impose boundary conditions on S. It is possible to do so for models I and II together.

Let the half space Vout be vacuous so that the constitutive equations

(2)

D(r,ω)=ϵ0E(r,ω),H(r,ω)=μ01B(r,ω),rVout,
hold, ϵ0 being the permittivity and μ0 being the permeability of free space. Equation (1) can then be written as

(3)

·B(r,ω)=0×E(r,ω)iωB(r,ω)=0·E(r,ω)=0×B(r,ω)+iωμ0ϵ0E(r,ω)=0},rVout
in terms of the primitive field phasors E(r,ω) and B(r,ω).

The frequency-domain constitutive relations of the material occupying Vin are

(4)

D(r,ω)=ϵ(ω)E(r,ω)+Ψ(ω)B(r,ω)H(r,ω)=μ1(ω)B(r,ω)Ψ(ω)E(r,ω)},rVin,
where ϵ, μ, and Ψ are functions of ω. Equation (4) allows us to accommodate model I. After substituting Eq. (4) into Eq. (1), we get

(5)

·B(r,ω)=0×E(r,ω)iωB(r,ω)=0·E(r,ω)=0×B(r,ω)+iωμ(ω)ϵ(ω)E(r,ω)=0},rVin.
Let us note that Ψ does not appear in the Maxwell equations applied to Vin after the convenient but inessential induction field phasors D(r,ω) and H(r,ω) have been translated into essential primitive field phasors E(r,ω) and B(r,ω).

When solving an electromagnetic boundary-value problem, it is common to use the boundary conditions:

(6)

n^(rS)·[Bout(rS,ω)Bin(rS,ω)]=0n^(rS)×[Eout(rS,ω)Ein(rS,ω)]=0n^(rS)·[Dout(rS,ω)Din(rS,ω)]=ρs(rS,ω)n^(rS)×[Hout(rS,ω)Hin(rS,ω)]=Js(rS,ω)},rSS,
with the unit normal vector n^(rS) at rSS pointing into Vout. The subscripts “in” and “out” indicate that the fields in Vin and Vout, respectively, are being evaluated on S. The quantities ρs and Js are the surface charge density and the surface current density, respectively. In order to accommodate model II, we set

(7)

ρs(rS,ω)=γ(ω)n^(rS)·Bin(rS,ω)Js(rS,ω)=γ(ω)n^(rS)×Ein(rS,ω)},rSS,
where γ describes the surface states.

Let an arbitrarily polarized plane wave in Vout be incident on S. Then the primitive field phasors in Vout can be written as

(8)

E(r,ω)={[asu^y+ap(u^xτ0+u^zκ)/k0]exp(iτ0z)+[rsu^y+rp(u^xτ0+u^zκ)/k0]exp(iτ0z)}exp(iκx)B(r,ω)=k0ω{[apu^y+as(u^xτ0+u^zκ)/k0]exp(iτ0z)+[rpu^y+rs(u^xτ0+u^zκ)/k0]exp(iτ0z)}exp(iκx)},rVout,
where k0=ωμ0ϵ0, τ0=+k02κ2, and the dependences on ω are implicit. Representing the incident plane wave, the coefficients as and ap are presumed to be known. Representing the plane wave reflected into Vout, the coefficients rs and rp are unknown. Equation (8) satisfies Eq. (3).

The primitive field phasors in Vin are given as

(9)

E(r,ω)=[tsu^y+tp(u^xτ+u^zκ)/k]exp(iτz)exp(iκx)B(r,ω)=kω[tpu^y+ts(u^xτ+u^zκ)/k]exp(iτz)exp(iκx)},rVin,
where k=ωμϵ, τ=+k2κ2, and the coefficients ts and tp are unknown. Representing the plane wave refracted into Vin, these expressions satisfy Eq. (5).

The foregoing expressions were substituted into Eqs. (2)2, (4)2, (6)2,4 and (7)2 to determine rs, rp, ts, and tp in terms of as and ap. Thus,

(10)

rs=[(ηrδr)(1+ηrδr)(Gη0)2ηr2δr]as+2  Gη0ηr2δrap(ηr+δr)(1+ηrδr)+(Gη0)2ηr2δr,

(11)

rp=[(ηr+δr)(1ηrδr)+(Gη0)2ηr2δr]ap+2  Gη0ηr2δras(ηr+δr)(1+ηrδr)+(Gη0)2ηr2δr,

(12)

ts=2ηr(1+ηrδr)as+2  Gη0ηr2δrap(ηr+δr)(1+ηrδr)+(Gη0)2ηr2δr,

(13)

tp=2ηr(ηr+δr)ap2  Gη0ηr2as(ηr+δr)(1+ηrδr)+(Gη0)2ηr2δr,
where

(14)

G=Ψ+γ,η0=μ0ϵ0,δr=τ/kτ0/k0,ηr=ϵ0μϵμ0.

We have verified that Eqs. (10) through (13) satisfy Eqs. (6)1,3 and (7)1. Moreover, Eqs. (10) through (13) simplify to the standard results:6,7

(15)

rs=asηrδrηr+δr,rp=ap1ηrδr1+ηrδrts=as2ηrηr+δr,tp=ap2ηr1+ηrδr}
for Ψ=γ=0.

3.

Discussion and Conclusion

Equations (10) through (13) can be recast in matrix form as

(16)

[rsrp]=[rssrsprpsrpp][asap],[tstp]=[tsstsptpstpp][asap].
The elements of the 2×2 matrices have either both subscripts identical or two different subscripts. The elements with both subscripts identical indicate copolarized reflection or refraction, with the remaining elements indicating cross polarization. Both cross-polarized reflection and refraction in Eqs. (10) through (13) are due to G.

Equations (10) through (13) do not contain Ψ and γ separately, but their sum G instead. Thus, measurements of the reflection coefficients rs and rp (or the transmission coefficients ts and tp, if at all possible) cannot be used to discriminate between models I (γ=0) and II (Ψ=0). Equations (6)4 and (7)2 together make it clear that measurements of the reflection and transmission coefficients of a slab made of a topological insulator cannot be used to discriminate between the two models; not only that, the solution of every scattering problem will depend on G, not on Ψ alone or γ alone.

This impasse can be resolved by realizing that surface states exist on topological insulators as protected conducting states, and the characteristic behavior of these materials is due to those surface states. Furthermore, Ψ vanishes from the Maxwell equations (5) applicable to Vin occupied by the topological insulator; indeed, Ψ would vanish even if the topological insulator were bianisotropic.8 For both of these reasons, we must choose model II, which also satisfies the Post constraint Ψ0.9

As the material occupying Vin is isotropic and achiral, cross-polarized reflection in this problem has been taken to arise from the Lorentz nonreciprocity inherent in Eq. (4).10 But now we see that surface states described by Eq. (7) by themselves are capable of yielding cross-polarized reflection, which is, therefore, not an indication of Lorentz nonreciprocity.

Acknowledgments

A.L. is grateful to the Charles Godfrey Binder Endowment at Penn State for ongoing support of his research. T.G.M. acknowledges the support of EPSRC grant EP/M018075/1.

References

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M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Modern Phys. 82(4), 3045–3067 (2010).http://dx.doi.org/10.1103/RevModPhys.82.3045Google Scholar

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M.-C. Chang and M.-F. Yang, “Optical signature of topological insulators,” Phys. Rev. B 80(11), 113304 (2009).http://dx.doi.org/10.1103/PhysRevB.80.113304Google Scholar

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F. Liu et al., “Goos–Hänchen and Imbert–Fedorov shifts at the interface of ordinary dielectric and topological insulator,” J. Opt. Soc. Am. B 30(5), 735–741 (2013).http://dx.doi.org/10.1364/JOSAB.31.000735Google Scholar

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F. Liu, J. Xu and Y. Yang, “Polarization conversion of reflected electromagnetic wave from topological insulator,” J. Opt. Soc. Am. B 31(4), 735–741 (2014).http://dx.doi.org/10.1364/JOSAB.31.000735Google Scholar

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8. 

A. Lakhtakia and T. G. Mackay, “Axions, surface states, and the Post constraint in electromagnetics,” Proc. SPIE 9558, 95580C (2015).http://dx.doi.org/10.1117/12.2190105Google Scholar

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Biography

Akhlesh Lakhtakia received his degrees from the Banaras Hindu University and the University of Utah. He is the Charles Godfrey Binder Professor of engineering science and mechanics at the Pennsylvania State University. His current research interests include nanotechnology, bioreplication, forensic science, solar-energy harvesting, surface multiplasmonics, metamaterials, mimumes, and sculptured thin films. He is a fellow of OSA, SPIE, IoP, AAAS, APS, and IEEE. He received the 2010 SPIE Technical Achievement Award.

Tom G. Mackay is a reader in applied mathematics at the University of Edinburgh and an adjunct professor in the Department of Engineering Science and Mechanics at the Pennsylvania State University. He is a graduate of the Universities of Edinburgh, Glasgow, and Strathclyde, and a fellow of the Institute of Physics (UK) and SPIE. His research interests include the electromagnetic theory of novel and complex materials, including homogenized composite materials.

Akhlesh Lakhtakia, Tom G. Mackay, "Classical electromagnetic model of surface states in topological insulators," Journal of Nanophotonics 10(3), 033004 (8 February 2016). http://dx.doi.org/10.1117/1.JNP.10.033004
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KEYWORDS
Dielectrics

Reflection

Electromagnetism

Maxwell's equations

Refraction

Scattering

Chemical elements

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