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Tunable THz surface polariton based on a topological /layered superconductor hybrid structure

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Citation Li, Mingda, Zuyang Dai, Wenping Cui, Zhe Wang, Ferhat Katmis, Jiayue Wang, Peisi Le, Lijun Wu, and Yimei Zhu. “Tunable THz Polariton Based on a /layered Superconductor Hybrid Structure.” Phys. Rev. B 89, no. 23 (June 2014). © 2014 American Physical Society

As Published http://dx.doi.org/10.1103/PhysRevB.89.235432

Publisher American Physical Society

Version Final published version

Citable link http://hdl.handle.net/1721.1/88644

Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW B 89, 235432 (2014)

Tunable THz surface plasmon polariton based on a topological insulator/layered superconductor hybrid structure

Mingda Li,1,* Zuyang Dai,2 Wenping Cui,3 Zhe Wang,1 Ferhat Katmis,4 Jiayue Wang,5 Peisi Le,1 Lijun Wu,6 and Yimei Zhu6 1Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA 2Department of , Tsinghua University, Beijing 100084, China 3Institut fur¨ Theoretische Physik, Universitat¨ zu Koln,¨ Zulpicher¨ Straße 77, D-50937 Koln,¨ Germany 4Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA 5Department of Engineering Physics, Tsinghua University, Beijing 10084, China 6Department of Condensed Physics, Brookhaven National Lab, Upton, New York 11973, USA (Received 6 March 2014; revised manuscript received 5 May 2014; published 25 June 2014)

We theoretically investigate the surface plasmon polariton (SPP) at the interface between a three-dimensional strong topological insulator (TI) and a layered superconductor/magnetic insulator structure, within the random approximation. The tunability of the SPP through electronic doping can be enhanced when the magnetic permeability of the layered structure becomes higher. When the interface is gapped by or perpendicular magnetism, the SPP is further distorted, accompanied by a shift of group velocity and penetration depth. Such a shift of the SPP reaches a maximum when the magnitude of the Fermi level approaches the gap value, and may lead to observable effects. The tunable SPP at the interface between layered- superconductor and magnetic materials in proximity to the TI surface may provide new insight in the detection of Majorana .

DOI: 10.1103/PhysRevB.89.235432 PACS number(s): 73.20.Mf, 73.25.+i

I. INTRODUCTION backscattering probability due to topological protected surface states [33–35]. However, unlike the booming studies in Surface plasmon polariton (SPP) is the collective excitation graphene plasmonics, plasmon hybrid devices in TIs have been of at the interface between a conductor and dielectrics seldom reported, even with comparable performance in the driven by electromagnetic (EM) waves. [1,2] Despite its wide THz range as well as other promising features such as net spin applications in nanophotonics [3], near-field optics and tip- polarization, i.e., “spin plasmon” [24–26,31] and spin-charge enhanced Raman spectroscopy [4,5], and biological sensors separation [30]. and antennas [6,7], the SPP in general suffers from problem Therefore, in this paper, we propose a plasmonic hybrid of huge nonradiative loss due to the strong absorption of structure composed of a 3D TI in close contact with a the metal [1–4] accompanied by additional radiative loss [8], layered superconductor. This structure provides a new platform which limits SPP lifetime and propagation length for further where SPP waves are supported. The tunability of the SPP application in integrated devices. propagation can be achieved independently through either gate In order to solve the SPP loss problem, low-loss plasmonics voltage or external magnetic field. Since the Majorana bound based on graphene [9–23] and topological insulators (TIs) states, which are non-Abelian in superconductors and [24–31] has attracted much recent attention. In far- have great significance in topological computation and THz range, the major loss mechanism in graphene lies in [34], are predicted to exist at the boundary between a 3D TI the scattering between electrons and optical [11]. and a superconductor, this plasmonic structure may provide a A number of studies in graphene plasmonics have been new perspective in search of Majorana bound states. conducted utilizing the properties of low loss and tunability. Yan et al. [32] have reported enhanced plasmon in a patterned graphene-insulator stack structure when compared II. THEORY with single-layered graphene, while Ju et al. [12] demonstrated an enhanced tunability in the THz range in microribbon A. of an anisotropic SPP wave graphene metamaterials. These efforts target manipulation of Since the SPP wave is well localized at the interface, while and miniaturization of optical devices, and could the Dirac electrons only exist on the surface of a TI, we could be further integrated and hybridized toward applications in apply anisotropic dielectric functions to model the dielectric detectors, modulators or other integrated devices. function of a 3D TI in order to capture both the dielectric On the other hand, in a doped three-dimensional (3D) function of surface 2D chiral Dirac electrons (Fig. 1, xy plane, TI, the -impurity scattering becomes dominant due z = 0) and the bulk dielectric constant (Fig. 1, z>0 region). to weak electron- coupling [28], with a further reduced In order to describe wave propagation in the layered structure (Fig. 1, z<0 region), we adopt the method proposed by Averkov et al. [36] using an anisotropic dielectric function. * Author to whom correspondence should be addressed: This is valid when λSP D, where λSP is the SPP wavelength, [email protected] and D is the spatial periodicity of the layered structure. Since

1098-0121/2014/89(23)/235432(6) 235432-1 ©2014 American Physical Society LI, DAI, CUI, WANG, KATMIS, WANG, LE, WU, AND ZHU PHYSICAL REVIEW B 89, 235432 (2014)

speed in vacuum c = 1 and μ is the magnetic permeability of the layered superconductor material. Substituting the EM field components back into Eq. (1), we obtain the localization constants, i.e., the inverse of penetration depth away from the interface.  o = 2 + 2 − 2 κ1 qx qy ε2D(q,ω)ω ,    q2 + q2 e = x y − 2 κ1 ε2D(q,ω) ω , εd FIG. 1. (Color online) The schematic configuration of a 3D  (3) o = 2 + 2 − 2 TI/layered superconductor hybrid structure. The z<0regionconsists κ2 qx qy μεab(q,ω)ω , of alternating layers of superconductor and insulator. The SPP wave  propagates along the interface. A back gate is present to tune the ε q,ω e c( ) 2 2 2 Fermi level of the interfacial electrons, which leads to a change of κ = q + q − μεc(q,ω)ω . 2 ε (q,ω) x y dielectric functions and furthermore a change of SPP propagation ab properties. The dispersion relation of the resulting surface wave can be written in a determinant form:  2   2 q2−(κo)  we are interested in the long-wavelength THz range, the − o iω ε1 − y 2  iκ1 κe iμκo 0 condition λSP D is guaranteed to be met. In this approach,  1 2   2 2 2 2  the anisotropy leads to the existence of an optical axis where  −q q 0 −q + μω εab  y x x  = 0(4) the EM wave suffers no birefringence [37]. Thus the electric  2 2 q2q2   o 2 ω qx ε1 x y 2 e  field and magnetic field at the interface can be written as a κ1 qy κe μκo ω εabκ2   1 2  superposition of an ordinary wave and an extraordinary wave: 11 −1 −1   o −κo|z| e −κe|z| i q x+q y−ωt E = E e j + E e j e ( x y ), In this paper, we only consider the wave propagating along the j  j j  − o| | − e| | + − (1) x direction, neglecting the oblique excitation. The dispersion = o κj z + e κj z i(qx x qy y ωt) Hj Hj e Hj e e , relation can finally be simplified as  where j = 1,2 denote the TI and layered superconductor sides, ε ε (ω) [ε (ω) − με (q,ω)] respectively. Based on our model, the TI dielectric function q = ω d ab c 2D (5) εc(ω)εab(ω) − ε D(q,ω)εd is defined as ε1 = (ε2D(q,ω),ε2D(q,ω),εd ) with optical axis 2 along the z direction, while the dielectric function of the lay- which is the main analytical result. In this expression, μ is the = ered superconductor is defined as ε2 (εc(ω),εab(ω),εab(ω)), effective permeability of the layered superconductor structure which is anisotropic along the x direction. and q is the wave number along the x direction, and can be a Noticing the different directions of optical axis in the complex number. When both the upper and lower materials are upper TI region and lower superconductor region, then for isotropic, i.e., ε2D = εd = ε1 and εab =εc = ε2, it is further ordinary wave and extraordinary wave we have Eo = 0, 1z = ε1ε2 o = e = e = reduced to the well known result q ω ε +ε . E2x 0 and H1z 0, H2x 0, respectively. These equalities 1 2 are valid approximately, since, when the gap is opened, the electromagnetic constitutive relation is modified to incorporate B. Dielectric function of the layered superconductor the topological magnetoelectric effect [38], Defining dimensionless = ω/ωJ ,withωJ the

0cαθ Josephson plasmon frequency of the layered superconductor, D = E − B the dielectric function of the layered superconductor can be π (2) written as [36] 1 αθ H = B + E     μ cμ π 1 γ 2 0 ε ( ) = ε 1 − ,ε( ) = ε 1 − , (6) c s 2 ab s 2 However, despite the O(α) correction to the electromagnetic field by this constitutive relation, it actually contributes where the imaginary parts are neglected. Throughout this only an order of O(α2) ∼ 10−4 correction to the plasmon article, we take the value reported in [36] and set the interlayer energy [38]. Moreover, since we are interested in the relative dielectric constant εs = 16, current-anisotropy parameter γ = SPP energy shift upon gating or gapping, but not the absolute 200, and ωJ = 4meV. magnitude of SPP energy, the contribution from the axion term will be canceled out when calculating the energy difference, C. Dielectric function of the gapless topological insulator since the ungated or gapless SPP energy already contains The dielectric function of the chiral gapless topological the contribution from the axion term [39]. Thus, compared insulator surface has been reported in [24,25,27], where, for with the huge energy shift (∼1% originated from gapping or the Hamiltonian for a 2D helical Dirac electron H0, doping) in all following calculations, we could safely neglect the axion term and set α = 0. Within this approximation, the H =  + ×  · 0 vF k (zˆ k) σ k, (7) components of the EM fields are shown in Appendix A, with k

235432-2 TUNABLE THz SURFACE PLASMON POLARITON BASED . . . PHYSICAL REVIEW B 89, 235432 (2014) the Lindhard Dielectric function ε2D(q,ω) can be written as e2 ε2D(q,ω) = 1 − V2D(q)(q,ω) = 1 − (q,ω)(8) 2ε0q with the polarization operator

g n − n +  (q,ω) = dk k,γ k q,γ 4π 2 ω + E − E +  + iδ γ,γ k,γ k q,γ 2 ×|fk,γ |fk+q,γ  | . (9) In this expression, g is the spin/valley degeneracy, for the chiral states we have g = 1 due to spin-momentum locking, nk,γ is the Fermi occupation value at the energy eigenvalue, and Ek,γ = γ vF k, with γ = 1 for conduction band and γ =−1 | = for valence band, respectively.√ The spinor eigenstates fk,γ − iθk/2 iθk/2 = (e ,iγ e )/ 2, with θk defined as tan θk ky /kx .We FIG. 2. (Color online) The dispersion relations (a),(b) and gap- = × 5 take the value of Fermi velocity vF 6.2 10 m/s[40]in induced group velocity changes (c),(d) of the SPP at various Fermi all calculations. levels with respect to gap opening. The Fermi levels are taken at The random-phase approximation (RPA) relaxation time three different values with 80 meV intervals. In (c), at μ = 1, even (RT) approximation [41] can be applied to describe the SPP if the tuning of the Fermi level does not change much relative to the damping along the propagation direction, which leads to a dispersion relation shown in (a), it still shows a shift to the SPP group finite propagation length. Defining χ(q,ω) ≡ velocity in the high-Q range. e2 q2 (q,ω), the renormalized polarizability, taking the conser- vation of electron number into account, can then be written quantity, and moreover we are interested in the frequency and as the tunability of the SPP but not the exact propagation length, (1 + i/ωτ)χ(q,ω + i/τ) for simplicity we take τ =∞in all following calculations, χτ (q,ω) = , (10) using Eq. (9) instead of Eq. (10). 1 + (i/ωτ)χ(q,ω + i/τ)χ(q,0) Equation (9) can be calculated either analytically [10,27] where τ is the relaxation time, which is in general frequency or numerically, thanks to the identical expression of the dependent, and mainly originates from the electron-phonon polarization operator within the RPA for a simple Dirac gas coupling, scattering by superconductor , as well and a helical Dirac gas. In our approach, we choose T = 10 K as impurities scattering. In this situation, τ can be calculated for numerical integration to avoid discontinuity, and compare using electron self-energy [11]: with analytical result with at least six-digit agreement. Then the numerical result for the gapless TI is applied to solve for the = + + (E) e−ph(E) sc(E) imp(E) (11) dielectric function of the TI when the surface state is gapped. When the SPP frequency is higher than the gap of the superconductor or above the optical frequency of the D. Dielectric function of the gapped topological insulator topological insulator, the self-energy in Eq. (11) contains a Either the magnetic field perpendicular to the interface prominent imaginary part, which leads to a drastic reduction (z direction in Fig. 1) or the superconductivity would open of propagation length. However, since we are interested in up a gap to the gapless Dirac cone, and lift out the degeneracy low-frequency SPPs (ω<ωJ = 4 meV, shown in Fig. 2), at k = 0, and further alter the dielectric function as well as SPP where only impurity scattering dominates,√ the relaxation time dispersions. In order to take into account the effect of gaps, we can be directly written as τ = μ nπ/evF , without the need adopt the Bogoliubov–de Gennes Hamiltonian [33,42], which of implementing Eq. (11). Since τ ∼ 10−12s is a very large can be regarded as a generalization of Eq. (7):

− + − | | 1 kx σy ky σx Mσz EF i  σy H = + , (12) BdG 2 k − | | − − − − k k i  σy kx σy ky σx Mσz EF

= + + | | where spinor k (ck↑,ck↓,c−k↑,c−k↓), M and  denote the magnetic gap and superconductivity gap, respectively, and EF is the chemical potential. Both Eqs. (8) and (9) still hold, but the eigenvalues and eigenvectors are changed. The eigenvalues can now be written as    = | |2 + | |2 2 + 2 2 + 2 2 1/2 + 2 + 2 + 2 E1  2  M EF M EF k M k EF ,    = | |2 − | |2 2 + 2 2 + 2 2 1/2 + 2 + 2 + 2 E2  2  M EF M EF k M k EF ,

235432-3 LI, DAI, CUI, WANG, KATMIS, WANG, LE, WU, AND ZHU PHYSICAL REVIEW B 89, 235432 (2014)    =− | |2 − | |2 2 + 2 2 + 2 2 1/2 + 2 + 2 + 2 E3  2  M EF M EF k M k EF ,    =− | |2 + | |2 2 + 2 2 + 2 2 1/2 + 2 + 2 + 2 E4  2  M EF M EF k M k EF .

The corresponding unnormalized eigenvectors are shown in level EF is shown in Fig. 3, at different values of gap. It can Appendix B. be seen directly that the shift is increased at larger gap value, and it increases as a function of Q. Most importantly, the shift III. RESULTS AND DISCUSSIONS reaches a peak value when the Fermi level approaches the gap value. This feature can be seen more clearly in a line plot with In order to conveniently express the dielectric functions fixed Q value [Fig. 3(d)]. We also see that SPP group velocity for both the TI and layered superconductor, we define dimen- vg does not shift when Fermi level EF = 0. In contrast to the Q sionless wave number Q = v q /(cω ), and all energies are F x J and EF dependent shift of group velocity caused by the energy dimensionless and expressed in units of ωJ . gap, the localization constants show a qualitatively different The SPP dispersion relations are obtained by solving Eq. behavior. The localization constant in the TI side κo is sensitive (5). The typical dispersion relations at various gate voltages, 1 neither to EF nor to Q [Figs. 4(a), 4(c), and 4(e)], while in the i.e., Fermi levels EF , are shown in Figs. 2(a) and 2(b).We o layered superconductor side κ2 is tunable by EF but still Q see an enhanced tunability, i.e., a shift of dispersion relation independent. The importance of the localization constant shift when EF varies, when the lower layer has increased effective with Q can hardly be overestimated, in that it indicates that magnetic permeability μ. By defining the SPP group velocity the different propagation properties of the SPP are almost fully = vg d /dQ, we see an additional change of propagation coming from the group velocity, other than the difference in | | properties induced by either M or  [Figs. 2(c) and 2(d)]by SPP wavelength. Fewer controllable variables would definitely plotting the percentage SPP group velocity shift, make the experimental results easier to explain. v v (gapped) − v (gapless) g = g g . vg vg(gapless) Here both the magnetic gap M and superconductivity gap || are dimensionless in units of the Josephson plasmon frequency ωJ . In order to further investigate the effect of the gap on the SPP propagation, the relative shift of group velocity vg/vg as a function of dimensionless wave number Q and Fermi

FIG. 4. (Color online) Relative shift of the o-light component of o FIG. 3. (Color online) The percentage shift of SPP group velocity localization constants for TI κ1 [panels (a), (c), (e)], and layered = o as a function of Q and EF , at three different gap values M 5, superconductor κ2 [panels (b), (d), (f)]. The localization constants | |= = | |= = | |= e e  0.5(a);M 10,  1.0 (b); and M 15,  1.5(c). for e-light κ1 and κ2 have different magnitudes but features similar o o The change is negative for EF > 0 and positive for EF < 0, and to κ1 and κ2 , respectively. We see that the shift is not sensitive to reaches a maximum when the Fermi level is close to the gap values the Fermi level EF of topological insulators, but always reaches peak (d). Notice all gap values are expressed in units of ωJ (ωJ = 4meV values when EF is near the value of the gap. Unlike the shift of group throughout the calculation); for instance the green line M = 5, velocity, which is Q dependent, the localization constants are almost ||=0.5 corresponds to M = 20 meV, ||=2meV. independent of Q value.

235432-4 TUNABLE THz SURFACE PLASMON POLARITON BASED . . . PHYSICAL REVIEW B 89, 235432 (2014)

The reason that the maximum vg shift occurs when the ACKNOWLEDGMENTS gap matches the Fermi level can be understood in the light Author Mingda Li would thank Prof. Ju Li for his generous of electronic transition and available electron density for support and helpful discussions. plasmonic oscillation. Electrons in the conduction band are mostly extended and dominate the surface plasmon excitation. When the Fermi level is lower than the energy gap, charge carriers need additional energy to fulfill the transition to APPENDIX A: COMPONENTS OF EM FIELDS the conduction band in order to contribute to the collective excitation. On the other hand, since the SPP is excited by κo q iκoq EM waves, when the Fermi level is higher than the energy o = 1 o o =− y o o =− 1 y o H1x E1y ,E1x E1y ,H1y E1y , gap, the electronic transition by optical excitation is limited iω qx ωqx by occupied electrons. The increasing forbidden transition 2 2 q + q iωε (q,ω) o = x y o e = 2D e leads to a reduced shift of dielectric function, and moreover H1z E1y ,H1x e E1y , ωqx κ the vg shift is reduced accordingly. Thus, only when the 1 q Fermi level reaches the value of energy gap is the number e = x e E1x E1y , of available electrons participating in the electronic transition qy and collective excitation maximized, and leads to a shift of iωq ε (q,ω) e =− x 2D e SPP frequency and group velocity. H1y e E1y , qy κ  1  iκe iω2ε (q,ω) IV. CONCLUSIONS e = 1 + 2D e E1z e E1y , We provide a generic theoretical framework to study the qy qy κ1 surface plasmon polariton (SPP) at the interface between a q iq q iq H o = x Eo ,Ho = x y Eo ,Eo =− y Eo , topological insulator (TI) and layered superconductors. The 2z 2y 2y o 2y 2z o 2y ωμ ωμκ2 κ2 SPP in this hybrid structure may be widely applied to study   2 novel optical and transport phenomena at the interface, through q2 − κ‘o κe H o = y 2 Eo ,Ee = 2 Ee , the tunability of SPP propagation by gating or gapping the 2x o 2y 2z 2y iωμκ2 iqy surface states of the TI. It can also be generalized to a larger ωε (q,ω) category of materials in proximity to a TI surface. For instance, e = ab e H2z E2y , Wei et al. [43,44] have shown that when the ferromagnetic qx insulator EuS is on the top of topological insulator Bi2Se3,it e 2 − 2 e ωεab(q,ω)κ e e qx μω εab(q,ω) e induces significant magnetic moment in Bi Se thin films and H =− 2 E ,E= E , 2 3 2y iq q 2y 2x q q 2y induces breaking of T -reversal symmetry. The SPP in such a x y x y ferromagnetic/TI hybrid structure can also be studied within this approach. Furthermore, the Majorana zero mode is predicted to exist APPENDIX B: (UNNORMALIZED) EIGENVECTORS [35,42,45] as a domain wall state at the interface between OF THE BdG HAMILTONIAN TI and ferromagnetic-superconductor boundaries. Therefore, 2 = 2 + 2 − =| |− + =| |+ at the TI/layered superconductor interface, propagating SPPs Defining k kx ky, δ  M, and δ  may interact with the Majorana domain wall state, and lead to M, the four eigenvectors can be written as the shift of the SPP propagation properties, including group − − − − T velocity, reflectivity and transmissivity, etc. In this regard, the δ + (δ )2+k2 δ + (δ )2 + k2 |fk,1= −1, , − ,1 , change of optical properties of the SPP can be considered ikx + ky ikx + ky a semiclassical manifestation of the existence of Majorana T fermions, which is a pure quantum phenomenon with non- + − + 2 + 2 + − + 2 + 2 | = δ (δ ) k δ (δ ) k Abelian statistics. The existence of the zero-mode level may fk,2 1, , ,1 , ikx + ky ikx + ky contribute to electronic transition, and leads to a further change T of dielectric functions. Unlike transport measurements, which δ− − (δ−)2 + k2 −δ− + (δ−)2 + k2 involve only a single Majorana domain wall state, this |f = − , , , , k,3 1 + + 1 hybrid structure can be regarded as the interaction between ikx ky ikx ky the SPP and a series of domain wall states, coming from each T + + + 2 + 2 + + + 2 + 2 domain wall state along the interface. In a nutshell, the SPP δ (δ ) k δ (δ ) k |fk,4= 1, , ,1 . on the TI surface may provide insights to the detection of ikx + ky ikx + ky Majorana Fermions as a conceptually novel platform.

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