Abstract

Silicene is a competitive and promising 2D material, possessing interesting topological, electronic and optical properties. The presence of strong spin orbit interaction in silicene and its analogues, germanene and tinene, leads to the opening of a gapin the energy spectrum and spin-splitting of the bands in each valley. Building upon prior work we discuss a general method to determine the magneto-optic response of silicene when a Gaussian beam is incident on silicene grown on a dielectric substrate in the presence of a static magnetic field. We use a semiclassical treatment to describe the Faraday rotation (FR) and Magneto-optical Kerr effect (MOKE). The response can be modulated both electrically and magnetically. We derive analytic expressions for valley and spin polarized FR and MOKE for arbitrary polarization of incident light in the terahertz regime. We demonstrate that large FR and MOKE can be achieved by tuning the electric field, magnetic fields and chemical potential in these fascinating 2D materials.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Monolayer graphene has garnered immense interest from a large global community of researchers. This is primarily due to its unique electronic and optical properties [1] derived from its exotic electronic structure. For example graphene possesses gapless Dirac-type band structure [2], high carrier mobilities and universal broadband optical conductivities (due to inter band transitions) [3]. Due to its fascinating optical properties, graphene is also considered to be a promising material for photonic and optoelectronic applications in the terahertz (THz) to mid-infrared ranges. For example, Faraday and Kerr rotations are non-reciprocal magneto-optic (MO) effects, in which the polarization of a plane wave is rotated when linearly polarized light is respectively transmitted or reflected from a transparent medium in the presence of static uniform perpendicular magnetic field B. Both of these effects originate from the breaking of time reversal symmetry by an external applied magnetic field. Graphene exhibits an exceptionally large Faraday and Kerr rotation in the THz region and therefore is considered a futuristic candidate for non-reciprocal tunable devices [4–6]. The magnitude of FR is about 6 in a field of strength 7 T. Unfortunately, the FR and magneto-optic Kerr effects (MOKE) observed in a single layer graphene sheet exist only at low frequencies (< 3 THz) and that too in the presence of large magnetic fields.

Graphene shares analogous properties with a large range of 2D quantum materials [7]. For example, recently, transition metal dichalcogenides (TMDC) have attracted a lot of attention due to their novelty [7, 8]. TMDC’s have the formula MX 2, where M is a transition-metal atom (Mo, W, V, etc) and X is a chalcogen atom (S, Se, or Te). TMDC’s are of particular interest because they possess a valley degree of freedom and exhibit large band gaps due to SOI [9]. These interesting spin-valley structures make TMDC’s highly attractive condidates for spintronic, valleytronic [8–11] and optoelectronic devices [12, 13].

The discovery of 2D materials has also stimulated growing interest in silicene [14], the silicon analogue of graphene. Stable silicene can be experimentally synthesized [15]. There are many electronic and physical similarities between graphene and silicene as both are found in the same group of the periodic table. The major difference is that silicene has a large SOI with an electrically tunable band gap. Just like silicene, germanene and tinene also possess stable honeycomblattice structures [15, 16]. Due to the relatively large SOI, these materials haved buckled structures, providing a mass to the otherwise massless Dirac fermions. In silicene [17], germanene [18] and tinene [19], the values of Δso have been predicted to lie in the range 1.55–7.9 meV, 24–93 meV, and 100 meV respectively. Subsequently, the interaction of an external electric field with silicene, germanene and tinene-substrate system renders the Dirac mass controllable at the K and K points, which leads to various topological phase transitions [20].

In addition to charge and spin, which are intrinsic degrees of freedom, Dirac electrons have another degrees of freedom called the valley [21–23]. The valley can also be used to encode and process information, this is the now burgeoning field of valleytronics [21]. A promising platform for valleytronics is provided by silicene. Due to spin and valley polarized responses, silicene also offers the possibility to realize novel tuneable MO devices [24, 25].

The possibility of dynamic adaptability of silicene’s electronic structure via electric and magnetic fields makes it favorable for tuneable THz applications. However, the two most important MO responses namely FR and MOKE of monolayer silicene and the wider class of Dirac materials deserves a rigorous exploration. The purpose of this work is to study FR and MOKE in these 2D lattices. Subsequently, the magnetic field dependent MO effects can be directly utilized for magnetic field sensing and optical modulation [26–28]. In addition to FR and MOKE, in this work, we also investigate the dependence of these MO effects on the incident angle, polarization state, chemical potential and temperature.

1.1. System Hamiltonian and MO conductivities

The starting point for the derivation of Faraday and Kerr rotations, and ellipticities, is the understanding of the energy levels manifold. This aspect has been thoroughly presented by several authors [29–31] and in this section, we only reproduce prior results. The low-energy physics of silicene, germanene and tinene is adequately approximated by a simple nearest-neighbor tight-binding Hamiltonian

H^ξσ=vF(ξkxτ^x+kyτ^y)12ξΔsoσ^zτ^z+12Δzτ^z

This Hamiltonian is generalized by Cysne et al. [32] to include Rashba and valley Zeeman SOI. The first term in Eq. (1) is the usual low-energy graphene-like Hamiltonian for describing massless Dirac fermions, kx,y are their crystal momentums and vF is their Fermi velocity. The parameter ξ=±1 corresponds to the valleys (K and K ) in momentum space and the vector operators τ=(τ^x,τ^y,τ^z) and σ=(σ^x,σ^y,σ^z) respectively represent Pauli matrices of the lattice pseudo spin and real spin degrees of freedom. The second term in the Hamiltonian captures intrinsic spin-orbit coupling with a band gap of Δso, whereas in the final term, Δz=aEz is responsible for breaking the A, B sublattice inversion symmetry, Ez being an electric field normal to the plane of atoms and a being the lattice constant. For Landau level (LL) quantization, we apply a static uniform magnetic field B perpendicular to this plane. Introducing the Landau gauge for the magnetic vector potential A=(yB,0,0), and diagonalizing the Hamiltonian, we obtain the eigenvalues [30],

E(ξ,σ,n,t)={t2vF2eB|n|+Δξσ2,ifn0.ξΔξσ,ifn=0.

Here, t=sgn(n) denotes the conduction/valence band, Δξσ=12ξσΔso+12Δz and n is an integer, the quantum number denoting Landau quantization and σ=±1 for spin up () and down () respectively. Note that the n=0 manifold is independent of the magnetic field and these levels can be linearly manipulated by the electric field only, whereas both the electric and magnetic fields play a role in setting the position of the n0 levels. The corresponding eigenfunctions at the K and K points are

|n¯|ξ=1=(iAn|n1Bn|n)
and
|n¯|ξ=1=(iAn|nBn|n1)
where |n is an orthonormal Fock state of the harmonic oscillator, and An and Bn are given by,
An={|E(ξ,σ,n,t)|+tΔξσ2|E(ξ,σ,n,t)|,ifn0.1ξ2,ifn=0.
and
Bn={|E(ξ,σ,n,t)|tΔξσ2|E(ξ,σ,n,t)|,ifn0.1+ξ2,ifn=0.

The next task is to calculate the magneto-optical conductivity for the 2D systems admitting the Hamiltonian in Eq. (1). As the authors in [33] describe, Kubo formula is used to derive the following general expressions for the conductivity [30, 34],

σμν(Ω)=i2πlB2σ,ξ=±1mnfnfmEnEmn¯|j^μ|m¯m¯|j^ν|n¯Ω(EnEm)+iΓ,
where fn=1/(1+e(EnμF)/kBT) is the Fermi Dirac distribution function at temperature T and chemical potential μF, j^μ=evFs^μ is the current operator, s^μ are Pauli matrices, En is the energy of the n’th landau level, Γ is the transport scattering rate responsible for the broadening of the energy levels and lB=/eB is the magnetic length. The spatial index μ can be x, y or z. The conductivity is complex whose real and imaginary parts can also be separately computed. For example, at T=0 K we have the longitudinal conductivity
ReIm}(σxx(Ω))σ0=2v2eBπξ,σm,nΘ(EnμF)Θ(EmμF)EnEm×[(AmBn)2δ|m|ξ,|n|+(BmAn)2δ|m|+ξ,|n|]{FG,
where, σ0=e2/4, F=Γ/((Ω(EnEm))2+Γ2) and G=(Ω(EnEm))/((Ω(EnEm))2+Γ2). In these expressions, the Kronecker deltas ensure the rules for electric dipole transitions between the LL’s are satisfied. The Heaviside functions Θ(EnμF) ensure that transitions across the Fermi level are possible, hence they effectively account for the so called Pauli blocking [35]. Similarly, the real and imaginary parts of the transverse conductivity are
ReIm}(σxy(Ω))σ0=2v2eBπξ,σm,nξΘ(EnμF)Θ(EmμF)EnEm×[(AmBn)2δ|m|ξ,|n|(BmAn)2δ|m|+ξ,|n|]{GF.

In the limit Δso=Δz=0, we recover graphene’s Hall conductivity [36]. For these expressions, the real (imaginary) part of σxx(σxy) is a sum of absorptive Lorentzians, each of whose FWHM depends on the scattering rate Γ, higher Γ resulting in broader and shorter peaks. Plots of these conductivities can be seen in previous works [30, 33], which set the stage and provide the formalism for computing the magneto-optic rotations discussed in the present work. Likewise, the real (imaginary) part of σxy(σxx) is a sum of dispersive Lorentzians. These peaks are positioned at Ω=(EnEm), which we call the magneto-excitation energies. The transitions obey the appropriate selection rules namely |n||m|=±1 and the conservation of real spin implying that transitions between σ=+1 and 1 levels are spin forbidden.

1.2. Magneto-optical rotations and ellipticities

We now present a general method to calculate the Faraday and Kerr rotation angles and the resulting ellipticities. Due to the rich LL structure, these MO effects are modulated by myriad stimuli such as electric and magnetic fields [4, 37, 38], chemical potential gating [37], modification through doping, optical pumping [39] as well as temperature [40] and the substrate effect [4].

Throughout this article, we consider a well-collimated, monochromatic, Gaussian beam of light with nontotal reflection impinging from one medium to the planar interface of the silicene-substrate system at an incidence angle θ1. The beam of light of frequency Ω has polarization in an arbitrary direction, and is propagating through the incident and transmitted materials with relative permittivity and permeability εn and μn respectively, where n=(1,2). The beam make an angle θ2 in the substrate which is assumed to be semi-infinite, obviating the need to consider finite substrate size effects and thin-film interference [41]. The wave vectors are k1 and k2, kn=Ωμnεn, Zn=Z0μn/εn and Z0=μ0/ε0, where μ0 and ε0 are the vacuum permeability and permittivity respectively. The Fresnel coefficients have been derived in previous work [42, 43]:

rpp=α+TαL+βα+Tα+L+β,
rss=(αTα+L+βα+Tα+L+β),
tpp=2Z2ε2Z1k1zα+Tα+Tα+L+β,
tss=2μ2k1zα+Lα+Tα+L+β,
rsp=tsp=2Z02μ0μ1μ2k1zk2z(σH+σxysym)Z1(α+Tα+L+β),
rps=k1k2zk2k1ztps=2Z02μ1μ2Z1k1zk2z(σxysymσH)α+Tα+L+β,
where,
α±L=(k1zε2±k2zε1+k1zk2zσL/(ε0Ω)),
α±T=(k2zμ1±k1zμ2+μ0μ1μ2σTΩ),
β=Z02μ1μ2k1zk2z[σH2(σxysym)2]

Here, k1z=k1cos (θ1) and k2z=k2cos (θ2). The conductivities σL(σT) are the longitudinal (transverse) components. For homogeneous, isotropic media, σL=σT=σxx=σyy. The cross conductivity of a 2D system in the presence of magnetic field is antisymmetric [44, 45] σxy=σyx. In fact, the cross conductivity σxy has symmetric σxysym and asymmetric σxyantisym parts. For anisotropic materials, such as phosphorene [46], σxysym is non-zero because the band structure of phosphorene is Dirac like (linear in k) in one direction and Schrodinger like (parabolic in k) in the other direction [47]. However, for isotropic materials such as graphene and other staggered materials (silicene, germanene, stanene, and plumbene etc.) σxysym=0. Therefore in Eqs. (14) and (15), we use σH=σxy which comprises wholly of the anti-symmetric part.

In our case, medium 1 is vacuum (ε1= 1, μ1=1) and medium 2 is nonmagnetic μ2=1. The Fresnel coefficients which are derived from the magneto-optical conductivities, subsequently determine the magneto-optic rotations and ellipticity. For incident s and p polarization, the Faraday rotation and ellipticity are computed using the expressions

ΘF,s(p)=12tan1(2Re(χF,s(p))1|χF,s(p)|2),
andηF,s(p)=12sin1(2Im(χF,s(p))1|χF,s(p)|2),
where,
χF,s=tpstss=Z0ε1μ1k1cos(θ1)σHα+L,
andχF,p=tsptpp=Z0μ2ε2μ0μ1k2cos(θ2)σHα+T

Similarly, for MOKE, the rotations and ellipticities are

ΘK,s(p)=12tan1(2Re(χK,s(p))1|χK,s(p)|2),
andηK,s(p)=12sin1(2Im(χK,s(p))1|χK,s(p)|2),
where,
χK,s=rpsrss=2Z0μ1ε1μ2k1zk2zσHαTα+L+β,
andχK,p=rsprpp=2Z0μ1ε1μ0μ2k1zk2zσHα+LαL+β

A note about the notation is in place here. The spin ( or ) or valley (K or K) will be specified in the subscripts while the superscripts identify the Faraday (F) or Kerr rotation (K) as well as the polarization state (s) or(p). If the χ’s are small, χ1, Eqs. (19) and (20) reduce to ΘF,s(p)Re(χF,s(p)) and ηF,s(p)Im(χF,s(p)) and vis-a-vis for the Kerr effect. However, for Landau quantized systems, there is no reason to believe, at the ontset, that the MO effects are small.

 figure: Fig. 1

Fig. 1 Faraday, Kerr rotation and ellipticity of silicene-substrate system as function of photon energy electric and magnetic fields. (a) The s polarized Faraday rotation and (b) ellipticity as function of incident photon energy in the K valley with modulation of the external electric field for the three distinct topological regimes, TI, VSPM and BI for a magnetic field of 1 T. The spectral peaks are labelled 1 through 6 and their origin is identified in the main text. The spectrum are vertically shifted by 15° among themselves for clearer viewing. Furthermore, in this figure we use Δz = Δso/2 (TI) and Δz = 2Δso (BI). (c) The s polarized Kerr rotation as function of incident photon energy in the Kʹ valley with modulation of the external magnetic field for the TI regime for three different values of B = 1, 3 and 5 T. (d) The maximum Faraday, Kerr rotation and ellipticity as function of magnetic field in K valley for the single transition Δ−10,K,↑. The parameters used are θ1 = 30°, Γ = 0.01Δso, refractive index n2 = 1.84 and chemical potential μF = 0.

Download Full Size | PPT Slide | PDF

2. Results and discussion

First we discuss Faraday rotation (FR) for charge neutral 2D silicene, where the inter-band transitions bridge across the valance and conduction bands. Hence μF=0. Fig. 1(a) shows the FR spectra as a function of incident photon frequency with modulation of the external electric field, landing the band structure into three distinct topological regimes [30]. The signal originating from a single spin orientation in only one of the valleys is dispersive Lorentzian, with a positive followed by a negative (or vice versa) signature. Let’s call this an anti-phase peak. This terminology is borrowed from NMR literature [48]. The anti-phase peak is centred at the magneto-optic excitation frequency EnEm with positive and negative maxima at EnEm±Γ. For the opposite spin in the same valley and an identical LL transition, we still see an absorptive anti-phase peak whose sign may be reversed, the possibility of reversal depending on the exact topological regime. The peaks corresponding to the different transitions, Em,K(K),()En,K(K),() are labelled as Δmn,K(K),(). For higher frequencies, the magnitude of the rotationis reduced in accordance with the factor of 1/(EnEm) appearing in the denominator of Eqs. (8) and (9). We now explore the three distinct topological regimes.

In the topological insulator (TI) regime (Δz<Δso), the first and second anti-phase peaks correspond to the Δ10,K, and Δ01,K, transitions for spin up and spin down respectively. In each of these transitions, one of the participating levels is an n=0 level. In a magnetic field of 1 T and Δz=Δso/2, these magneto-excitation energies are calculated as 20.3 meV (4.9 THz) and 25.1 meV (6.1 THz) respectively and are shown as 1 and 2 in the bottom spectrum of Fig. 1(a). The anti-phase peaks switch sign with spin within the same valley. The s polarized FR angles for the first two anti-phase peaks are ±6.5. The subsequent anti-phase peaks appearing at different resonant frequencies differ in magnitude for spin up and spin down cases due to spin dependent energies. The anti-phase peaks labelled 3 through 6 can also be assigned to the various transitions. For example the multiplet structure 3 originates from Δ12,K,, 4 is due to Δ21,K,, 5 is due to Δ23,K, and 6 comes from Δ32,K,.

In the valley-spin polarized metal (VSPM) instance (Δz=Δso), the gap of one of the spin-split bands closes [30] giving rise to a Dirac point. As we increase the applied electric field and begin to approach the VSPM point, the lowest frequency peaks, labeled 1 and 2 in the middle spectrum of Fig. 1(a) move apart: the Δ10,K, peak is red shifted and Δ01,K, peak is blue shifted. The excitation energies corresponding to the first two anti-phase peaks at the VSPM point are now 18.2 meV (4.4 THz) and 27.8 meV (6.7 THz). However, it is observed that at this precise electric field, the spectrum is cleaner and exhibiting fewer peaks. The peaks labeled 4 and 6 originate from the Δ21,K, and Δ32,K, transitions. The peaks that were labelled 3 and 5 in the TI regime and came from the Δ12,K, and Δ23,K, transitions are now annihilated. Eq. (2) shows that at the Dirac point in the K valley (ξ=1), the spin up (σ=1) transitions leads to Δξσ=0 which results in An=Bn=1/2 irrespective of the Landau quantum number n. For these spin up levels, therefore (AmBn)2=(BmAn)2 and from Eq. (9), the minus sign between the terms in the square brackets results in annihilation of the spectral response at the Δ12 and Δ23 frequencies. So even though, these transitions are allowed by selection rules, destructive interference between their quantum amplitudes extinguishes the response. Conversely, in the K valley (data not shown), the spin down peaks will be annihilated at the Dirac points.

For an even higher electric field (Δz>Δso), the system transitions from the VSPM to the band insulator (BI) state and the lowest band gap is opened again, resulting in sign change of some of the anti-phase peaks with respect to the TI phase. Compare the peaks 1 through 5 between the TI and BI shown in Fig. 1(a). The full range of the allowed peaks also resurfaces once the VSPM point is crossed. The separation between the anti-phase pair keeps on growing in the BI state. Consequently, all the peaks gradually shift towards higher frequencies. The magnitude of the maximum spin polarized FR angles for the first two peaks inside the anti-phase pair is ±8. If we change the polarization of the incident light, the sign of anti-phase peaks inverts with respect to the baseline. Alternatively the same effect is achieved by switching from one valley to another. If we change the valley, the spin identity of the anti-phase also changes. The juxtaposition of identities between spin up and down polarized peaks after band inversion is also observed in the K valley. Consequently the s polarized FR in the K valley will have the same form as the p polarized FR response in the K valley. The MOKE rotation spectra (data not shown) follow a similar trend. The MOKE response is also spin and valley polarized and the magnitudes of the rotation angles range between 515 for both valleys and all three topological regimes, which are in general larger than the FR angle.

 figure: Fig. 2

Fig. 2 (a) Schematic representation of the allowed transitions between LL’s for three different values of chemical potential μF = 0, 10 and 22 meV; (b) and (c) the s polarized Kerr rotation as function of incident photon frequency in K and valleys with modulation of the chemical potential in the TI regime for a magnetic field of 1 T, respectively. (d) The s polarized Faraday rotation as function of incident photon energy in K valley for different incident angles for a single transition in the TI regime. (e) The s polarized Faraday rotation as function of incident photon energy in K for different temperatures for a single transition in the TI regime. (f) The s and p polarized Kerr rotation as function of incident photon frequency in the semiclassical limit for n-type and p-type silicene (μF = 56 and -56 meV), respectively. The solid line represents the s polarized and the dashed line p polarized. The parameters used are θ = 30°, Γ = 0.01Δso and refractive index n2 = 1.84.

Download Full Size | PPT Slide | PDF

Fig. 1(b) shows the series of peaks in the ellipticity acquired by transmitted light from s polarized incident radiation originating from the K valley manifold. Faraday geometry is considered though analogous results are obtained for reflection as well. It is evident that extermely large ellipticities, of the order of 815, appear for the lowest excitations. The spectrum for ellipticity comprises absorptive Lorentzians, which are spin and valley polarized. These maxima are at the excitation energies EnEm. The rotation and ellipticity data when considered together, indicate that at the exact excitation energy EnEm, the rotation is zero while the ellipticity is maximum. Furthermore, when the rotation is maximum (Ω=EnEm±Γ), the ellipticity drops to 50% of its maximum value. The intertwined effects, although both being ultra-large, limit the use of silicene-substrate system for a pure MO rotator since significant ellipticity is also introduced.

We now demonstrate the effect of how the magnetic field modifies the magneto-optic response. The s polarized MOKE in the K valley is only one possible illustration and shown in Fig. 1(c). Here we plot the MO spectrum in the TI regime for three different values of B=1, 3 and 5 T, while keeping μF=0 and θ=30. The impact on the MOKE signal in terms of shifting magneto-excitation frequency and the amount of Kerr rotation is clear. The silicene energy levels are strongly dependent on the magnetic field B, as given by Eq. (2), and this is also true for other 2D materials including graphene [3, 30]. As we increase the strength of the applied magnetic field, the MO excitations shift towards higher frequencies with a concomitant increase in magnitude of the MOKE rotation angle. For example, the peaks labelled 1 and 2 have excitation energies 20.3 meV (4.9 THz) and 25.1 meV (6.1 THz) for B=1 T, 33.5 meV (8.1 THz) and 38 meV (9.2 THz) for B=3T and finally, 42.7 meV (10.3 THz) and 47 meV (11.32 THz) for B=5 T. The maximum value of the rotation ΘKKs exceeds ±13 at a magnetic field of 5 T, which is an exceptionally large rotation for a monolayer silicene-substrate system. Similarly the FR is also strongly field-dependent (data not shown). The primary role of the magnetic field tuning, therefore, is to shift the position of the magneto-optic excitation energies and also to modify the amount of rotation. However, unlike the electric field the magnetic field does not switch the sign of the anti-phase doublets.

Fig. 1(d) show the field dependence of ΘKFs and ΘKKs in the TI regime. Due to the dispersive MO spectrum, we chose to plot the maximum rotation. By increasing the magnetic field strength the amount of FR and Kerr rotations grows. However for stronger fields, the Kerr signal slowly decreases. At a field of 10 T, we report ΘKFs=13 and ΘKKs=5.5. The ellipticity is also strongly field dependent.

It is also instructive to discuss the effect of controlling the FR and MOKE spectra by varying the chemical potential of the silicene surface, e.g, by applying a bias voltage [37] or optical pumping [39]. For illustration purposes, we consider three different values of chemical potentials μF=0, 10 and 22 meV, while keeping the magnetic field 1 T, in the TI regime (Δz=0.5Δso) and an angle of incidence of 30. In the first case the chemical potential is at zero and lies within the n=0 manifold, for μF=10 meV the chemical potential is in between the n=0 and n=1 LL’s and for μF=22 meV the chemical potential is in between the n=1 and n=2 LL’s. These LLs are shown in Fig. 2(a). Only the K valley is depicted. For μF=0, the lowest energy excitations are also indicated on the same subfigure. They identify as Δ10,K,=20.3 meV and Δ01,K,=25.1 meV. These result in the Kerr rotations shown by anti-phase peaks 1 and 2 in the bottom spectrum of Fig. 2(b). They are interband transitions since they occur across the zero energy datum. The combination of anti-phase peaks Δ12 and Δ21 in the K valley, for both spins yield the multiplet structure 3 and the transitions Δ23, Δ32 yield the structure 4. These anti-phase are rather close in their excitation energies (e.g. Δ12,K,=44.0 meV, Δ21,K,=44.0 meV, Δ12,K,=45.5 meV and Δ21,K,=45.5 meV) and the ability to resolve this finer structure depends on the experimental capability.

As μF increases to 10 meV, certain transitions become Pauli blocked. For example the transition Δ10,K, becomes forbidden and in its stead, the intra-band transition Δ01,K,=16.0 meV emerges. The Pauli blocked transition is shown by a dashed upward pointing arrow in the middle part of Fig. 2(a) and the two lowest transitions, Δ10,K,=16.3 meV and Δ01,K,=25.1 meV are shown by solid arrows. Once again, these yield the Kerr signatures 1 and 2 shown in Fig. 2(b). The higher frequency agglomerated multiplets 3 and 4 remain unchanged. If μF is further increased to 22 meV, so that it lies between the n=1 and n=2 manifolds, both transitions starting from n=0, i.e, Δ10,K, and Δ01,K, now become Pauli blocked. These are again indicated by the dashed arrows in the rightmost part of Fig. 2(a). In their place, however, the intra-band transitions Δ12,K,=7.0 meV and Δ12,K,=7.2 meV pop up. For higher n, the LL’s are closely spaced. Hence the excitation energies also converge. These closely spaced transitions are separated by 200 μeV and are shown by the structure 1, 2 in top part of Fig. 2(b). The transitions involving the n=0 levels are completely missing from this magneto-optic spectrum. Furthermore, the transitions Δ21,K, and Δ21,K, become Pauli forbidden and hence are absent from the excitation structure labeled 3 which now comprises only Δ12,K,=44.0 meV and Δ12,k,=45.5 meV. Therefore peak 3 is a cleaner doublet of anti-phase structure when compared with the μF=0 and μF=10 meV cases. The structure 4 originates, as earlier, from rather closely spaced Δ23 and Δ32.

The magneto-optic spectrum originating from the K valley for the same values of μF is depicted in Fig. 2(c). For μF=0, the rotational peaks are coincident with the K valley as Δmn,K,=Δnm,K,(when m=0 and n0), Δ10,K,=Δ01,K, and Δ01,K,=Δ10,K,. However, these valley-specific spectrums are sign inverted with respect to each other. For μF=10 meV, the lowest energy transitions 1 and 2 occur at different positions for the two valleys. For the K valley, 1 and 2 are Δ01,K,=16 meV and Δ01,K,=25 meV respectively whereas for the K valley, the peaks 1 and 2 are Δ01,K,=13.1 meV and Δ01,K,=20.3 meV respectively.

The FR and MOKE signatures are clearly sensitive to the incident angle θ1. This is because the Fresnel coefficients are strongly dependent on the incidence angle. This dependence is shown in Fig. 2(d) for a single transition in the TI regime. An increasing incidence angle diminishes the amount of rotation until it disappears at complete grazing, θ1=π/2. A similar trend can also be seen in the MO response of graphene [19, 50].

All of the results presented so far are at 0 K but as the temperature goes up, the Fermi Dirac distribution function in Eq. (7) starts becoming significant. We can explore the temperature dependence of the FR by introducing these distributions in place of the Heaviside functions. Nevertheless, at 100 K, the FR angle is 3o, while at 300 K, the FR angle is 1. These results are shown in Fig. 2(e). The experimental value [40] of the FR angle for graphene at 1.5 K is 4 mrad which translates to 0.23. This shows that silicene has a bigger Faraday rotation than graphene in the THz range. In silicene the electrons frequently interact with scatterers. There are many scattering mechanisms including Coulomb interaction, impurities, optical phonons, acoustic phonons, and radiative decay [45]. Due to these scattering channels the peaks are additionally broadened [49]. However, in actuality, the temperature dependence of the scattering rate Γ must also be taken into account. This is ignored in the present work. The amount of MO rotation is strongly influenced by the presence of a substrate [4] (see Appendix for more details).

Tables Icon

Table 1. Table of allowed transitions in K valley in the n = −1,0,1 subspace, at a fixed Magnetic field and Chemical potential μ. Furthermore x = Δzso, y=v2eB/Δso2. and μ = μFso.

 figure: Fig. 3

Fig. 3 (a) Schematic representation of the allowed transitions between LL’s for chemical potential μ = 0.2. (b) and (c) the s polarized Faraday rotation contour plots as function of x in K valley for μ = 0 and 1.25, repectively, where x = Δzso and μ = μFso. The parameters used are θ = 30°, Γ = 0.01Δso and refractive index n2 = 1.84.

Download Full Size | PPT Slide | PDF

In studies on 2D materials placed inside magnetic fields, the semiclassical limit is valid when the LL spacing becomes unimportant and inconsequential [33]. This happens as |n| goes up and when the chemical potential is high up in the conduction band or deep down in the valance band, |μF||E0|. In this case the intra-band transitions between closely spaced levels are allowed. Suppose that μF lies between the n1 and n LL’s. Since the gap is minuscule, μFEn. In this limit we have

En+1Env2eBΔξσ2+2nv2eB=Ωc,
where, Ωc=v2eB/μF is called the classical cyclotron frequency. In this regime the Faraday and Kerr rotations can also be derived from a purely classical point of view [30]. For finite μF and n, the allowed transitions are Δ(n1)n, Δ(n1)n and Δ(n+1)n, however the latter two are large energies with diminished contributions to the magneto-optical conductivities, Eqs. (8) and (9). Hence, the allowed transition is the one that immediately across the chemical potential and results in a single large peak in all magneto-optic signatures. Furthermore in the semiclassical limit AnAn1BnBn1=1/2, and we can straightforwardly derive, using Eqs. (8) and (9), the following conductivities summed over both valleys and both spins,
Re(σxx(Ω))σ0=Im(σxy(Ω))σ0=μFπΓ((ΩΩc))2+Γ2,
Im(σxx(Ω))σ0=Re(σxy(Ω))σ0=μFπ(ΩΩc)((ΩΩc))2+Γ2

These conductivities are shaped as absorptive and dispersive Lorentzians and are directly used to compute the Fresnel coefficients Eqs. (10)(15) and subsequently the rotations. The conductivities, therefore, are modeled by classical Drude-like behavior [4, 51]. For example, in Fig. 2(f) we plot the s and p polarized Kerr rotation angles as a function of the incident photon frequency in the K valley. For n-type doping, we set μF=56 meV, which places chemical potential between the n=9 and 10 LL’s. The transitions from n=0 to higher LL’s are Pauli blocked, the selection rules dictate that only three transitions Δ9 10, Δ11 10 and Δ9 10 are allowed. The former two have negligible contributions, whereas the last mentioned transition results in a strong Drude peak, at 2.94 meV (0.71 THz). Similarly we plotted the s and p polarized Kerr rotation angles as function of the incident photon frequency for p-type silicene, with μF=56 meV. The Kerr rotation angle switches between n-type and p-type silicene indicating modulation of the rotation angle by switching the chemical potential, e.g, by switching gate bias voltage. Also note that the spin and valley information is lost in the semiclassical limit. The value of electric field Δz also becomes inconsequential at higher doping and the silicene behaves as graphene, because not only that the resonant frequency approaches that of graphene in this limit [30], but also the role of SOI becomes inconsequential.

An alternative approach to understanding the magneto-optic response is by contour plotting the rotations as a function of two variables. This method also allows one to identify topologically distinct regions and topological phase transitions [30] and may reveal discontinuations that may otherwise go unnoticed. For example, we consider transitions in the K valley within the n=1,0,1 subspace. We use dimensionless variables to simplify the analysis. We can define x=Δz/Δso, y=v2eB/Δso2 as measures of the electric and magnetic fields respectively, Ω/Δso and μ=μF/Δso as variables for photon frequency and chemical potential. In Fig. 3(a), we first plot the LL spectrum for the transitions under consideration. Table I summarizes the allowed transitions across the chemical potential. The excitation energies are also computed in the last column. It is evident that at the critical point x=12μ, the Δ10,K, transition gives way to the Δ01,K, transition, we say that the former becomes Pauli blocked. For example for a precise value of μ=0.2, this is shown by the sequence of green colored arrows that are only drawn for x12μ and the purple colored arrows drawn only for x12μ. For μ=0, this transition point is x=1.0. The contour plot in Fig. 3(b) aptly captures the scenario. For x1.0, the Δ01,K, transitions causes the Faraday rotation while for x1.0, the Δ01,K, transition kicks in yielding the Faraday rotation. Upon this transition point, the sign of the anti-phase peaks also switches. For μ=0.2, this switching now occurs at a smaller value of xc=12μ=0.6 as depicted in Fig. 3(c). Furthermore at this switching point, xc=12μ, one observes a discontinuous jump in the excitation energy. This can be computed by inserting xc into the energies Δ01,K,|xc=μ+μ2+2y2 and Δ10,K,|xc=μ+μ2+2y2 which yields a discontinuity of magnitude 2μ in the contour plot. The uninterrupted rotation which continues upward to the top right is due to Δ01,K, transition which is always switched on and is shown by thick red colored arrows in Fig. 3(a). The xc point also indicates a topological phase transition from the TI to the BI regime.

3. Conclusion

In conclusion, we have theoretically demonstrated the transitional MO effect due to the topological phase transition in silicene. We have studied the electric field modulated valley and spin polarized Faraday, Kerr rotations and ellipticities for three different topological regimes in silicene. We found that the magnitude of the maximum valley and spin polarized FR and MOKE angles for the first two anti-phase pair is 8 and 13, respectively. We also observe that if we change the polarization of the incident light or switched from one valley to another, the anti-phase peaks invert with respect to the baseline. We further investigated the magnetic field modulated MOKE for different magnetic fields and found that by increasing the magnetic field, the positions of the valley and spin polarized FR and MOKE anti-phase peaks move towards higher frequencies and the amount of FR and MOKE rotation is also enhanced. Moreover, we also note the effect of varying chemical potential on valley and spin polarized FR and MOKE.

4. Appendix

4.1. Dependence of Faraday and Kerr rotations on silicene-substrate system

For realistic applications the substrate effect cannot be ignored in Faraday and Kerr rotations. It is understood that the substrate does not introduce any magneto-optical rotation itself. Compared to free standing silicene, the substrate reduces the total rotation angle in silicene, just like in graphene [4]. If the refractive indices are small, in general the magneto-optical rotation will be large. To get higher value of the Faraday and Kerr rotations, it is recommended to use substrates with smaller refractive indices [52].

 figure: Fig. 4

Fig. 4 Faraday and Kerr rotation of silicene-substrate system as function of photon energy for three different relative permittivities. (a) The s polarized and (b) p polarized Faraday rotation as a function of incident photon energy in the K valley with modulation of relative permittivities in the TI regime for a magnetic field of 1 T. (c) The s polarized and (d) p polarized Kerr rotation as a function of incident photon energy in the K valley with modulation of relative permittivities in the TI regime for a magnetic field of 1 T, for the two transitions. The parameters used are θ1 = 30°, Γ = 0.01Δso and chemical potential μF = 0.

Download Full Size | PPT Slide | PDF

Figs. 4(a)–4(d) show the s and p polarized Faraday and Kerr rotation spectra as a function of incident photon energy in the K valley with modulation of relative permittivities in the TI regime for a magnetic field of 1 T, respectively. By increasing the relative permittivity strength the amount of FR and Kerr rotations reduces.

 figure: Fig. 5

Fig. 5 Faraday and Kerr rotation of silicene-substrate system as a function of photon energy for three different relative permeabilitis. (a) The s polarized and (b) p polarized Faraday rotation as a function of incident photon energy in the K valley with modulation of relative permeabilitis in the TI regime for a magnetic field of 1 T. (c) The s polarized and (d) p polarized Kerr rotation as a function of incident photon energy in the K valley with modulation of relative permeabilities in the TI regime for a magnetic field of 1 T, for the two transitions. The parameters used are θ1 = 30°, Γ = 0.01Δso, refractive index n2 = 1.84 and chemical potential μF = 0.

Download Full Size | PPT Slide | PDF

In Figs. 5(a)–5(d) we have shown the s and p polarized Faraday and Kerr rotation spectra as a function of incident photon energy in the K valley with modulation of relative permeabilitis in the TI regime for a magnetic field of 1 T. Here the magneto-optical effects show a different behaviour, i.e. by increasing the relative permeability, the magnitude of Faraday and Kerr rotations grows. Figs. 6(a)–6(d) show the s and p polarized ellipticity peaks for two transitions acquired by reflected and transmitted light of the silicene-substrate system as a function of photon energy for three different relative permittivities.

 figure: Fig. 6

Fig. 6 Faraday and Kerr ellipticities as a function of incident photon energy for three different relative permittivities. (a) The s polarized and (b) p polarized Faraday ellipticities as function of incident photon energy in the K valley with modulation of relative permittivities in the TI regime for a magnetic field of 1 T. (c) The s polarized and (d) p polarized Kerr ellipticities as a function of incident photon energy in the K valley with relative permittivities in the TI regime for a magnetic field of 1 T, for the two transitions. The parameters used are θ1 = 30°, Γ = 0.01Δso and chemical potential μF = 0.

Download Full Size | PPT Slide | PDF

References

1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004). [CrossRef]   [PubMed]  

2. N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, “Electronic spin transport and spin precession in single graphene layers at room temperature,” Nature , 448, 571–574 (2007). [CrossRef]   [PubMed]  

3. F. H. Koppens, D. E. Chang, and F. Javier, “Graphene plasmonics: A platform for strong light–matter interactions,” Nano Lett , 11(8), 3370–3377 (2011). [CrossRef]   [PubMed]  

4. I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single and multilayer graphene,” Nat. Phys. 7, 48–51 (2011). [CrossRef]  

5. A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. H. Castro Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011). [CrossRef]  

6. D. L. Sounas and C. Caloz, “Electromagnetic nonreciprocity and gyrotropy of graphene,” Appl. Phys. Lett. 98(2), 021911 (2011). [CrossRef]  

7. S. Ahmed and J. Yi, “Two-dimensional transition metal dichalcogenides and their charge carrier mobilities in field-effect transistors,” Nano-Micro Lett. 9, 50 (2017). [CrossRef]  

8. K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, “The valley Hall effect in MoS2 transistors,” Science 344(6191), 1489–1492 (2014). [CrossRef]   [PubMed]  

9. G. Catarina, J. Have, J. Fernández-Rossier, and N. M. R. Peres, “Optical orientation with linearly polarized light in transition metal dichalcogenides,” Phys. Rev. B 99(12), 125405 (2019). [CrossRef]  

10. R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014). [CrossRef]   [PubMed]  

11. M. A. Cazalilla, H. Ochoa, and F. Guinea, “Quantum spin Hall effect in two-dimensional crystals of transition-metal dichalcogenides,” Phys. Rev. Lett. 113(7), 077201 (2014). [CrossRef]   [PubMed]  

12. A. Pospischil, M. M. Furchi, and T. Mueller, “Solar-energy conversion and light emission in an atomic monolayer p-n diode,” Nat. Nanotechnol. 9, 257–261 (2014). [CrossRef]   [PubMed]  

13. S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015). [CrossRef]   [PubMed]  

14. G. G. Guzmán-Verri and L. C. Lew Yan Voon, “Electronic structure of silicon-based nanostructures,” Phys. Rev. B 76(7), 075131 (2007). [CrossRef]  

15. S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, “Two and one-dimensional honeycomb structures of silicon and germanium,” Phys. Rev. Lett. 102(23), 236804 (2009). [CrossRef]   [PubMed]  

16. B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015). [CrossRef]   [PubMed]  

17. M. Ezawa, “Spin-valley optical selection rule and strong circular dichroism in silicene,” Phys. Rev. B 86(16), 161407 (2012). [CrossRef]  

18. C. C. Liu, W. Feng, and Y. G. Yao, “Quantum spin Hall effect in silicene and two-dimensional germanium,” Phys. Rev. Lett. 107(7), 076802 (2011). [CrossRef]   [PubMed]  

19. Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013). [CrossRef]   [PubMed]  

20. M. Ezawa, “Photoinduced topological phase transition and a single Dirac-cone state in silicene,” Phys. Rev. Lett. 110(2), 026603 (2013). [CrossRef]   [PubMed]  

21. J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016). [CrossRef]  

22. D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: Magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007). [CrossRef]  

23. W. Yao, D. Xiao, and Q. Niu, “Valley-dependent optoelectronics from inversion symmetry breaking,” Phys. Rev. B 77(23), 235406 (2008). [CrossRef]  

24. D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108(19), 196802 (2012). [CrossRef]   [PubMed]  

25. H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotech. 7(8), 490 (2012). [CrossRef]  

26. P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for temperature-dependence of the Faraday effect by optical activity temperature shift,” IEEE Sens. J. 13(10), 832–837 (2013). [CrossRef]  

27. T. Yoshino, S. Torihata, M. Yokota, and N. Tsukada, “Faraday-effect optical current sensor with a garnet film/ring core in a transverse configuration,” Appl. Opt. 42(10), 1769–1772 (2003). [CrossRef]   [PubMed]  

28. J. W. Dawson, T. W. MacDougall, and E. Hernandez, “Verdet constant limited temperature response of a fiber-optic current sensor,” IEEE Photonics Technol. Lett. 7(12), 1468–1470 (1995). [CrossRef]  

29. M. Tahir and U. Schwingenschlögl, “Valley polarized quantum Hall effect and topological insulator phase transitions in silicene,” Sci. Rep. 3, 1075 (2013). [CrossRef]   [PubMed]  

30. C. J. Tabert and E. J. Nicol, “Magneto-optical conductivity of silicene and other buckled honeycomb lattices,” Phys. Rev. B 88(8), 085434 (2013). [CrossRef]  

31. M. Ezawa, “Valley-polarized metals and quantum anomalous Hall effect in silicene,” Phys. Rev. Lett. 109(5), 055502 (2012). [CrossRef]   [PubMed]  

32. T. P. Cysne, T. G. Rappoport, J. H. Garcia, and A. R. Rocha, “Quantum Hall effect in graphene with interface-induced spin-orbit coupling,” Phys. Rev. B 97(8), 085413 (2018). [CrossRef]  

33. C. J. Tabert and E. J. Nicol, “Valley-spin polarization in the magneto-optical response of silicene and other similar 2D crystals,” Phys. Rev. Lett. 110(19), 197402 (2013). [CrossRef]   [PubMed]  

34. M. Lasia and L. Brey, “Optical properties of magnetically doped ultrathin topological insulator slabs,” Phys. Rev. B 90(7), 075417 (2014). [CrossRef]  

35. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6 (11), 749–758 (2012). [CrossRef]  

36. V. P. Gusynin and S. G. Sharapov, “Unconventional integer quantum Hall effect in graphene,” Phys. Rev. Lett. 95(14), 146801 (2005). [CrossRef]   [PubMed]  

37. A. Dolatabady and N. Granpayeh, “Manipulation of the Faraday rotation by graphene metasurfaces,” J. Magn. Magn. Mater. 469, 231-235 (2019). [CrossRef]  

38. J. M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Moreno, J. Faist, and A. B. Kuzmenko, “Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene,” Nat. Commun. 8, 14626 (2017). [CrossRef]   [PubMed]  

39. A. N. Grebenchukov, S. E. Azbite, A. D. Zaitsev, and M. K. Khodzitsky, “Faraday effect control in graphene-dielectric structure by optical pumping,” J. Magn. Magn. Mater. 472, 25–28 (2019). [CrossRef]  

40. K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016). [CrossRef]   [PubMed]  

41. Q. Yang, X.A. Zhang, A. Bagal, W. Guo, and C.H. Chang, “Antireflection effects at nanostructured material interfaces and the suppression of thin-film interference,” Nanotechnology 24(23), 235202 (2013). [CrossRef]   [PubMed]  

42. W. J. M. Kort-Kamp, B. Amorim, G. Bastos, F. A. Pinheiro, F. S. S. Rosa, N. M. R. Peres, and C. Farina, “Active magneto-optical control of spontaneous emission in graphene,” Phys. Rev. B 92(20), 205415 (2015). [CrossRef]  

43. W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017). [CrossRef]  

44. V. P. Gusynin and S. G. Sharapov, “Transport of Dirac quasiparticles in graphene: Hall and optical conductivities,” Phys. Rev. B 73(24), 245411 (2006). [CrossRef]  

45. M. Oliva-Leyva and C. Wang, “Magneto-optical conductivity of anisotropic two-dimensional Dirac-Weyl materials,” Annals of Physics 384, 61–70 (2017). [CrossRef]  

46. T. Low, R. Roldan, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014). [CrossRef]   [PubMed]  

47. T. Low, R. Roldan, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014). [CrossRef]  

48. M. H. Levitt, “Spin dynamics: Basics of nuclear magnetic resonance” (John Wiley and Sons, New York, London, Sydney, 2008).

49. H. Funk, A. Knorr, F. Wendler, and E. Malic, “Microscopic view on Landau level broadening mechanisms in graphene,” Phys. Rev. B 92(20), 205428 (2015). [CrossRef]  

50. T. Yoshino, “Theory for oblique-incidence magneto-optical Faraday and Kerr effects in interfaced monolayer graphene and their characteristic features,” J. Opt. Soc. Am. B 30(5), 1085–1091 (2013). [CrossRef]  

51. M. Tymchenko, A. Y. Nikitin, and L. Martn-Moreno, “Faraday rotation due to excitation of magnetoplasmons in graphene microribbons,” ACS Nano 7(11), 9780–9787 (2013). [CrossRef]   [PubMed]  

52. N. Ubrig, I. Crassee, J. Levallois, I. O. Nedoliuk, F. Fromm, M. Kaiser, T. Seyller, and A. B. Kuzmenko, “Fabry-Pérot enhanced faraday rotation in graphene,” Opt. Express 21(21), 24736–24741 (2013). [CrossRef]   [PubMed]  

References

  • View by:
  • |
  • |
  • |

  1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
    [Crossref] [PubMed]
  2. N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, “Electronic spin transport and spin precession in single graphene layers at room temperature,” Nature,  448, 571–574 (2007).
    [Crossref] [PubMed]
  3. F. H. Koppens, D. E. Chang, and F. Javier, “Graphene plasmonics: A platform for strong light–matter interactions,” Nano Lett,  11(8), 3370–3377 (2011).
    [Crossref] [PubMed]
  4. I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
    [Crossref]
  5. A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. H. Castro Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
    [Crossref]
  6. D. L. Sounas and C. Caloz, “Electromagnetic nonreciprocity and gyrotropy of graphene,” Appl. Phys. Lett. 98(2), 021911 (2011).
    [Crossref]
  7. S. Ahmed and J. Yi, “Two-dimensional transition metal dichalcogenides and their charge carrier mobilities in field-effect transistors,” Nano-Micro Lett. 9, 50 (2017).
    [Crossref]
  8. K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, “The valley Hall effect in MoS2 transistors,” Science 344(6191), 1489–1492 (2014).
    [Crossref] [PubMed]
  9. G. Catarina, J. Have, J. Fernández-Rossier, and N. M. R. Peres, “Optical orientation with linearly polarized light in transition metal dichalcogenides,” Phys. Rev. B 99(12), 125405 (2019).
    [Crossref]
  10. R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
    [Crossref] [PubMed]
  11. M. A. Cazalilla, H. Ochoa, and F. Guinea, “Quantum spin Hall effect in two-dimensional crystals of transition-metal dichalcogenides,” Phys. Rev. Lett. 113(7), 077201 (2014).
    [Crossref] [PubMed]
  12. A. Pospischil, M. M. Furchi, and T. Mueller, “Solar-energy conversion and light emission in an atomic monolayer p-n diode,” Nat. Nanotechnol. 9, 257–261 (2014).
    [Crossref] [PubMed]
  13. S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015).
    [Crossref] [PubMed]
  14. G. G. Guzmán-Verri and L. C. Lew Yan Voon, “Electronic structure of silicon-based nanostructures,” Phys. Rev. B 76(7), 075131 (2007).
    [Crossref]
  15. S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, “Two and one-dimensional honeycomb structures of silicon and germanium,” Phys. Rev. Lett. 102(23), 236804 (2009).
    [Crossref] [PubMed]
  16. B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015).
    [Crossref] [PubMed]
  17. M. Ezawa, “Spin-valley optical selection rule and strong circular dichroism in silicene,” Phys. Rev. B 86(16), 161407 (2012).
    [Crossref]
  18. C. C. Liu, W. Feng, and Y. G. Yao, “Quantum spin Hall effect in silicene and two-dimensional germanium,” Phys. Rev. Lett. 107(7), 076802 (2011).
    [Crossref] [PubMed]
  19. Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013).
    [Crossref] [PubMed]
  20. M. Ezawa, “Photoinduced topological phase transition and a single Dirac-cone state in silicene,” Phys. Rev. Lett. 110(2), 026603 (2013).
    [Crossref] [PubMed]
  21. J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016).
    [Crossref]
  22. D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: Magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
    [Crossref]
  23. W. Yao, D. Xiao, and Q. Niu, “Valley-dependent optoelectronics from inversion symmetry breaking,” Phys. Rev. B 77(23), 235406 (2008).
    [Crossref]
  24. D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108(19), 196802 (2012).
    [Crossref] [PubMed]
  25. H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotech. 7(8), 490 (2012).
    [Crossref]
  26. P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for temperature-dependence of the Faraday effect by optical activity temperature shift,” IEEE Sens. J. 13(10), 832–837 (2013).
    [Crossref]
  27. T. Yoshino, S. Torihata, M. Yokota, and N. Tsukada, “Faraday-effect optical current sensor with a garnet film/ring core in a transverse configuration,” Appl. Opt. 42(10), 1769–1772 (2003).
    [Crossref] [PubMed]
  28. J. W. Dawson, T. W. MacDougall, and E. Hernandez, “Verdet constant limited temperature response of a fiber-optic current sensor,” IEEE Photonics Technol. Lett. 7(12), 1468–1470 (1995).
    [Crossref]
  29. M. Tahir and U. Schwingenschlögl, “Valley polarized quantum Hall effect and topological insulator phase transitions in silicene,” Sci. Rep. 3, 1075 (2013).
    [Crossref] [PubMed]
  30. C. J. Tabert and E. J. Nicol, “Magneto-optical conductivity of silicene and other buckled honeycomb lattices,” Phys. Rev. B 88(8), 085434 (2013).
    [Crossref]
  31. M. Ezawa, “Valley-polarized metals and quantum anomalous Hall effect in silicene,” Phys. Rev. Lett. 109(5), 055502 (2012).
    [Crossref] [PubMed]
  32. T. P. Cysne, T. G. Rappoport, J. H. Garcia, and A. R. Rocha, “Quantum Hall effect in graphene with interface-induced spin-orbit coupling,” Phys. Rev. B 97(8), 085413 (2018).
    [Crossref]
  33. C. J. Tabert and E. J. Nicol, “Valley-spin polarization in the magneto-optical response of silicene and other similar 2D crystals,” Phys. Rev. Lett. 110(19), 197402 (2013).
    [Crossref] [PubMed]
  34. M. Lasia and L. Brey, “Optical properties of magnetically doped ultrathin topological insulator slabs,” Phys. Rev. B 90(7), 075417 (2014).
    [Crossref]
  35. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6 (11), 749–758 (2012).
    [Crossref]
  36. V. P. Gusynin and S. G. Sharapov, “Unconventional integer quantum Hall effect in graphene,” Phys. Rev. Lett. 95(14), 146801 (2005).
    [Crossref] [PubMed]
  37. A. Dolatabady and N. Granpayeh, “Manipulation of the Faraday rotation by graphene metasurfaces,” J. Magn. Magn. Mater. 469, 231-235 (2019).
    [Crossref]
  38. J. M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Moreno, J. Faist, and A. B. Kuzmenko, “Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene,” Nat. Commun. 8, 14626 (2017).
    [Crossref] [PubMed]
  39. A. N. Grebenchukov, S. E. Azbite, A. D. Zaitsev, and M. K. Khodzitsky, “Faraday effect control in graphene-dielectric structure by optical pumping,” J. Magn. Magn. Mater. 472, 25–28 (2019).
    [Crossref]
  40. K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016).
    [Crossref] [PubMed]
  41. Q. Yang, X.A. Zhang, A. Bagal, W. Guo, and C.H. Chang, “Antireflection effects at nanostructured material interfaces and the suppression of thin-film interference,” Nanotechnology 24(23), 235202 (2013).
    [Crossref] [PubMed]
  42. W. J. M. Kort-Kamp, B. Amorim, G. Bastos, F. A. Pinheiro, F. S. S. Rosa, N. M. R. Peres, and C. Farina, “Active magneto-optical control of spontaneous emission in graphene,” Phys. Rev. B 92(20), 205415 (2015).
    [Crossref]
  43. W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
    [Crossref]
  44. V. P. Gusynin and S. G. Sharapov, “Transport of Dirac quasiparticles in graphene: Hall and optical conductivities,” Phys. Rev. B 73(24), 245411 (2006).
    [Crossref]
  45. M. Oliva-Leyva and C. Wang, “Magneto-optical conductivity of anisotropic two-dimensional Dirac-Weyl materials,” Annals of Physics 384, 61–70 (2017).
    [Crossref]
  46. T. Low, R. Roldan, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
    [Crossref] [PubMed]
  47. T. Low, R. Roldan, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
    [Crossref]
  48. M. H. Levitt, “Spin dynamics: Basics of nuclear magnetic resonance” (John Wiley and Sons, New York, London, Sydney, 2008).
  49. H. Funk, A. Knorr, F. Wendler, and E. Malic, “Microscopic view on Landau level broadening mechanisms in graphene,” Phys. Rev. B 92(20), 205428 (2015).
    [Crossref]
  50. T. Yoshino, “Theory for oblique-incidence magneto-optical Faraday and Kerr effects in interfaced monolayer graphene and their characteristic features,” J. Opt. Soc. Am. B 30(5), 1085–1091 (2013).
    [Crossref]
  51. M. Tymchenko, A. Y. Nikitin, and L. Martn-Moreno, “Faraday rotation due to excitation of magnetoplasmons in graphene microribbons,” ACS Nano 7(11), 9780–9787 (2013).
    [Crossref] [PubMed]
  52. N. Ubrig, I. Crassee, J. Levallois, I. O. Nedoliuk, F. Fromm, M. Kaiser, T. Seyller, and A. B. Kuzmenko, “Fabry-Pérot enhanced faraday rotation in graphene,” Opt. Express 21(21), 24736–24741 (2013).
    [Crossref] [PubMed]

2019 (3)

G. Catarina, J. Have, J. Fernández-Rossier, and N. M. R. Peres, “Optical orientation with linearly polarized light in transition metal dichalcogenides,” Phys. Rev. B 99(12), 125405 (2019).
[Crossref]

A. Dolatabady and N. Granpayeh, “Manipulation of the Faraday rotation by graphene metasurfaces,” J. Magn. Magn. Mater. 469, 231-235 (2019).
[Crossref]

A. N. Grebenchukov, S. E. Azbite, A. D. Zaitsev, and M. K. Khodzitsky, “Faraday effect control in graphene-dielectric structure by optical pumping,” J. Magn. Magn. Mater. 472, 25–28 (2019).
[Crossref]

2018 (1)

T. P. Cysne, T. G. Rappoport, J. H. Garcia, and A. R. Rocha, “Quantum Hall effect in graphene with interface-induced spin-orbit coupling,” Phys. Rev. B 97(8), 085413 (2018).
[Crossref]

2017 (4)

S. Ahmed and J. Yi, “Two-dimensional transition metal dichalcogenides and their charge carrier mobilities in field-effect transistors,” Nano-Micro Lett. 9, 50 (2017).
[Crossref]

J. M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Moreno, J. Faist, and A. B. Kuzmenko, “Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene,” Nat. Commun. 8, 14626 (2017).
[Crossref] [PubMed]

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
[Crossref]

M. Oliva-Leyva and C. Wang, “Magneto-optical conductivity of anisotropic two-dimensional Dirac-Weyl materials,” Annals of Physics 384, 61–70 (2017).
[Crossref]

2016 (2)

K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016).
[Crossref] [PubMed]

J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016).
[Crossref]

2015 (4)

S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015).
[Crossref] [PubMed]

B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015).
[Crossref] [PubMed]

H. Funk, A. Knorr, F. Wendler, and E. Malic, “Microscopic view on Landau level broadening mechanisms in graphene,” Phys. Rev. B 92(20), 205428 (2015).
[Crossref]

W. J. M. Kort-Kamp, B. Amorim, G. Bastos, F. A. Pinheiro, F. S. S. Rosa, N. M. R. Peres, and C. Farina, “Active magneto-optical control of spontaneous emission in graphene,” Phys. Rev. B 92(20), 205415 (2015).
[Crossref]

2014 (7)

T. Low, R. Roldan, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
[Crossref] [PubMed]

T. Low, R. Roldan, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

M. A. Cazalilla, H. Ochoa, and F. Guinea, “Quantum spin Hall effect in two-dimensional crystals of transition-metal dichalcogenides,” Phys. Rev. Lett. 113(7), 077201 (2014).
[Crossref] [PubMed]

A. Pospischil, M. M. Furchi, and T. Mueller, “Solar-energy conversion and light emission in an atomic monolayer p-n diode,” Nat. Nanotechnol. 9, 257–261 (2014).
[Crossref] [PubMed]

K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, “The valley Hall effect in MoS2 transistors,” Science 344(6191), 1489–1492 (2014).
[Crossref] [PubMed]

M. Lasia and L. Brey, “Optical properties of magnetically doped ultrathin topological insulator slabs,” Phys. Rev. B 90(7), 075417 (2014).
[Crossref]

2013 (10)

C. J. Tabert and E. J. Nicol, “Valley-spin polarization in the magneto-optical response of silicene and other similar 2D crystals,” Phys. Rev. Lett. 110(19), 197402 (2013).
[Crossref] [PubMed]

P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for temperature-dependence of the Faraday effect by optical activity temperature shift,” IEEE Sens. J. 13(10), 832–837 (2013).
[Crossref]

M. Tahir and U. Schwingenschlögl, “Valley polarized quantum Hall effect and topological insulator phase transitions in silicene,” Sci. Rep. 3, 1075 (2013).
[Crossref] [PubMed]

C. J. Tabert and E. J. Nicol, “Magneto-optical conductivity of silicene and other buckled honeycomb lattices,” Phys. Rev. B 88(8), 085434 (2013).
[Crossref]

Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013).
[Crossref] [PubMed]

M. Ezawa, “Photoinduced topological phase transition and a single Dirac-cone state in silicene,” Phys. Rev. Lett. 110(2), 026603 (2013).
[Crossref] [PubMed]

Q. Yang, X.A. Zhang, A. Bagal, W. Guo, and C.H. Chang, “Antireflection effects at nanostructured material interfaces and the suppression of thin-film interference,” Nanotechnology 24(23), 235202 (2013).
[Crossref] [PubMed]

T. Yoshino, “Theory for oblique-incidence magneto-optical Faraday and Kerr effects in interfaced monolayer graphene and their characteristic features,” J. Opt. Soc. Am. B 30(5), 1085–1091 (2013).
[Crossref]

M. Tymchenko, A. Y. Nikitin, and L. Martn-Moreno, “Faraday rotation due to excitation of magnetoplasmons in graphene microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

N. Ubrig, I. Crassee, J. Levallois, I. O. Nedoliuk, F. Fromm, M. Kaiser, T. Seyller, and A. B. Kuzmenko, “Fabry-Pérot enhanced faraday rotation in graphene,” Opt. Express 21(21), 24736–24741 (2013).
[Crossref] [PubMed]

2012 (5)

D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108(19), 196802 (2012).
[Crossref] [PubMed]

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotech. 7(8), 490 (2012).
[Crossref]

M. Ezawa, “Valley-polarized metals and quantum anomalous Hall effect in silicene,” Phys. Rev. Lett. 109(5), 055502 (2012).
[Crossref] [PubMed]

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6 (11), 749–758 (2012).
[Crossref]

M. Ezawa, “Spin-valley optical selection rule and strong circular dichroism in silicene,” Phys. Rev. B 86(16), 161407 (2012).
[Crossref]

2011 (5)

C. C. Liu, W. Feng, and Y. G. Yao, “Quantum spin Hall effect in silicene and two-dimensional germanium,” Phys. Rev. Lett. 107(7), 076802 (2011).
[Crossref] [PubMed]

F. H. Koppens, D. E. Chang, and F. Javier, “Graphene plasmonics: A platform for strong light–matter interactions,” Nano Lett,  11(8), 3370–3377 (2011).
[Crossref] [PubMed]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. H. Castro Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

D. L. Sounas and C. Caloz, “Electromagnetic nonreciprocity and gyrotropy of graphene,” Appl. Phys. Lett. 98(2), 021911 (2011).
[Crossref]

2009 (1)

S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, “Two and one-dimensional honeycomb structures of silicon and germanium,” Phys. Rev. Lett. 102(23), 236804 (2009).
[Crossref] [PubMed]

2008 (1)

W. Yao, D. Xiao, and Q. Niu, “Valley-dependent optoelectronics from inversion symmetry breaking,” Phys. Rev. B 77(23), 235406 (2008).
[Crossref]

2007 (3)

D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: Magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
[Crossref]

N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, “Electronic spin transport and spin precession in single graphene layers at room temperature,” Nature,  448, 571–574 (2007).
[Crossref] [PubMed]

G. G. Guzmán-Verri and L. C. Lew Yan Voon, “Electronic structure of silicon-based nanostructures,” Phys. Rev. B 76(7), 075131 (2007).
[Crossref]

2006 (1)

V. P. Gusynin and S. G. Sharapov, “Transport of Dirac quasiparticles in graphene: Hall and optical conductivities,” Phys. Rev. B 73(24), 245411 (2006).
[Crossref]

2005 (1)

V. P. Gusynin and S. G. Sharapov, “Unconventional integer quantum Hall effect in graphene,” Phys. Rev. Lett. 95(14), 146801 (2005).
[Crossref] [PubMed]

2004 (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

2003 (1)

1995 (1)

J. W. Dawson, T. W. MacDougall, and E. Hernandez, “Verdet constant limited temperature response of a fiber-optic current sensor,” IEEE Photonics Technol. Lett. 7(12), 1468–1470 (1995).
[Crossref]

Ahmed, S.

S. Ahmed and J. Yi, “Two-dimensional transition metal dichalcogenides and their charge carrier mobilities in field-effect transistors,” Nano-Micro Lett. 9, 50 (2017).
[Crossref]

Akashi, R.

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

Akturk, E.

S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, “Two and one-dimensional honeycomb structures of silicon and germanium,” Phys. Rev. Lett. 102(23), 236804 (2009).
[Crossref] [PubMed]

Amorim, B.

W. J. M. Kort-Kamp, B. Amorim, G. Bastos, F. A. Pinheiro, F. S. S. Rosa, N. M. R. Peres, and C. Farina, “Active magneto-optical control of spontaneous emission in graphene,” Phys. Rev. B 92(20), 205415 (2015).
[Crossref]

Arita, R.

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

Avouris, P.

T. Low, R. Roldan, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
[Crossref] [PubMed]

Azbite, S. E.

A. N. Grebenchukov, S. E. Azbite, A. D. Zaitsev, and M. K. Khodzitsky, “Faraday effect control in graphene-dielectric structure by optical pumping,” J. Magn. Magn. Mater. 472, 25–28 (2019).
[Crossref]

Bagal, A.

Q. Yang, X.A. Zhang, A. Bagal, W. Guo, and C.H. Chang, “Antireflection effects at nanostructured material interfaces and the suppression of thin-film interference,” Nanotechnology 24(23), 235202 (2013).
[Crossref] [PubMed]

Bastos, G.

W. J. M. Kort-Kamp, B. Amorim, G. Bastos, F. A. Pinheiro, F. S. S. Rosa, N. M. R. Peres, and C. Farina, “Active magneto-optical control of spontaneous emission in graphene,” Phys. Rev. B 92(20), 205415 (2015).
[Crossref]

Bludov, Y. V.

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. H. Castro Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

Bostwick, A.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Brey, L.

M. Lasia and L. Brey, “Optical properties of magnetically doped ultrathin topological insulator slabs,” Phys. Rev. B 90(7), 075417 (2014).
[Crossref]

Buckley, S.

S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015).
[Crossref] [PubMed]

Cahangirov, S.

S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, “Two and one-dimensional honeycomb structures of silicon and germanium,” Phys. Rev. Lett. 102(23), 236804 (2009).
[Crossref] [PubMed]

Cai, B.

B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015).
[Crossref] [PubMed]

Caloz, C.

D. L. Sounas and C. Caloz, “Electromagnetic nonreciprocity and gyrotropy of graphene,” Appl. Phys. Lett. 98(2), 021911 (2011).
[Crossref]

Carvalho, A.

T. Low, R. Roldan, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

Castro Neto, A. H.

T. Low, R. Roldan, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. H. Castro Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

Catarina, G.

G. Catarina, J. Have, J. Fernández-Rossier, and N. M. R. Peres, “Optical orientation with linearly polarized light in transition metal dichalcogenides,” Phys. Rev. B 99(12), 125405 (2019).
[Crossref]

Cazalilla, M. A.

M. A. Cazalilla, H. Ochoa, and F. Guinea, “Quantum spin Hall effect in two-dimensional crystals of transition-metal dichalcogenides,” Phys. Rev. Lett. 113(7), 077201 (2014).
[Crossref] [PubMed]

Chang, C.H.

Q. Yang, X.A. Zhang, A. Bagal, W. Guo, and C.H. Chang, “Antireflection effects at nanostructured material interfaces and the suppression of thin-film interference,” Nanotechnology 24(23), 235202 (2013).
[Crossref] [PubMed]

Chang, D. E.

F. H. Koppens, D. E. Chang, and F. Javier, “Graphene plasmonics: A platform for strong light–matter interactions,” Nano Lett,  11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Chen, S.

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
[Crossref]

Ciraci, S.

S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, “Two and one-dimensional honeycomb structures of silicon and germanium,” Phys. Rev. Lett. 102(23), 236804 (2009).
[Crossref] [PubMed]

Clark, G.

J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016).
[Crossref]

Crassee, I.

N. Ubrig, I. Crassee, J. Levallois, I. O. Nedoliuk, F. Fromm, M. Kaiser, T. Seyller, and A. B. Kuzmenko, “Fabry-Pérot enhanced faraday rotation in graphene,” Opt. Express 21(21), 24736–24741 (2013).
[Crossref] [PubMed]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Cui, X.

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotech. 7(8), 490 (2012).
[Crossref]

Cysne, T. P.

T. P. Cysne, T. G. Rappoport, J. H. Garcia, and A. R. Rocha, “Quantum Hall effect in graphene with interface-induced spin-orbit coupling,” Phys. Rev. B 97(8), 085413 (2018).
[Crossref]

Dai, J.

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotech. 7(8), 490 (2012).
[Crossref]

Dawson, J. W.

J. W. Dawson, T. W. MacDougall, and E. Hernandez, “Verdet constant limited temperature response of a fiber-optic current sensor,” IEEE Photonics Technol. Lett. 7(12), 1468–1470 (1995).
[Crossref]

Dolatabady, A.

A. Dolatabady and N. Granpayeh, “Manipulation of the Faraday rotation by graphene metasurfaces,” J. Magn. Magn. Mater. 469, 231-235 (2019).
[Crossref]

Duan, W.

Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013).
[Crossref] [PubMed]

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Ezawa, M.

M. Ezawa, “Photoinduced topological phase transition and a single Dirac-cone state in silicene,” Phys. Rev. Lett. 110(2), 026603 (2013).
[Crossref] [PubMed]

M. Ezawa, “Spin-valley optical selection rule and strong circular dichroism in silicene,” Phys. Rev. B 86(16), 161407 (2012).
[Crossref]

M. Ezawa, “Valley-polarized metals and quantum anomalous Hall effect in silicene,” Phys. Rev. Lett. 109(5), 055502 (2012).
[Crossref] [PubMed]

Faist, J.

J. M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Moreno, J. Faist, and A. B. Kuzmenko, “Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene,” Nat. Commun. 8, 14626 (2017).
[Crossref] [PubMed]

Farina, C.

W. J. M. Kort-Kamp, B. Amorim, G. Bastos, F. A. Pinheiro, F. S. S. Rosa, N. M. R. Peres, and C. Farina, “Active magneto-optical control of spontaneous emission in graphene,” Phys. Rev. B 92(20), 205415 (2015).
[Crossref]

Feng, L.

S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015).
[Crossref] [PubMed]

Feng, W.

D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108(19), 196802 (2012).
[Crossref] [PubMed]

C. C. Liu, W. Feng, and Y. G. Yao, “Quantum spin Hall effect in silicene and two-dimensional germanium,” Phys. Rev. Lett. 107(7), 076802 (2011).
[Crossref] [PubMed]

Fernández-Rossier, J.

G. Catarina, J. Have, J. Fernández-Rossier, and N. M. R. Peres, “Optical orientation with linearly polarized light in transition metal dichalcogenides,” Phys. Rev. B 99(12), 125405 (2019).
[Crossref]

Ferreira, A.

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. H. Castro Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Fromm, F.

Funk, H.

H. Funk, A. Knorr, F. Wendler, and E. Malic, “Microscopic view on Landau level broadening mechanisms in graphene,” Phys. Rev. B 92(20), 205428 (2015).
[Crossref]

Furchi, M. M.

A. Pospischil, M. M. Furchi, and T. Mueller, “Solar-energy conversion and light emission in an atomic monolayer p-n diode,” Nat. Nanotechnol. 9, 257–261 (2014).
[Crossref] [PubMed]

Garcia, J. H.

T. P. Cysne, T. G. Rappoport, J. H. Garcia, and A. R. Rocha, “Quantum Hall effect in graphene with interface-induced spin-orbit coupling,” Phys. Rev. B 97(8), 085413 (2018).
[Crossref]

Geim, A. K.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Granpayeh, N.

A. Dolatabady and N. Granpayeh, “Manipulation of the Faraday rotation by graphene metasurfaces,” J. Magn. Magn. Mater. 469, 231-235 (2019).
[Crossref]

Grebenchukov, A. N.

A. N. Grebenchukov, S. E. Azbite, A. D. Zaitsev, and M. K. Khodzitsky, “Faraday effect control in graphene-dielectric structure by optical pumping,” J. Magn. Magn. Mater. 472, 25–28 (2019).
[Crossref]

Grigorenko, A. N.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6 (11), 749–758 (2012).
[Crossref]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Guinea, F.

M. A. Cazalilla, H. Ochoa, and F. Guinea, “Quantum spin Hall effect in two-dimensional crystals of transition-metal dichalcogenides,” Phys. Rev. Lett. 113(7), 077201 (2014).
[Crossref] [PubMed]

T. Low, R. Roldan, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
[Crossref] [PubMed]

Guo, W.

Q. Yang, X.A. Zhang, A. Bagal, W. Guo, and C.H. Chang, “Antireflection effects at nanostructured material interfaces and the suppression of thin-film interference,” Nanotechnology 24(23), 235202 (2013).
[Crossref] [PubMed]

Gusynin, V. P.

V. P. Gusynin and S. G. Sharapov, “Transport of Dirac quasiparticles in graphene: Hall and optical conductivities,” Phys. Rev. B 73(24), 245411 (2006).
[Crossref]

V. P. Gusynin and S. G. Sharapov, “Unconventional integer quantum Hall effect in graphene,” Phys. Rev. Lett. 95(14), 146801 (2005).
[Crossref] [PubMed]

Guzmán-Verri, G. G.

G. G. Guzmán-Verri and L. C. Lew Yan Voon, “Electronic structure of silicon-based nanostructures,” Phys. Rev. B 76(7), 075131 (2007).
[Crossref]

Harasawa, A.

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

Hatami, F.

S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015).
[Crossref] [PubMed]

Have, J.

G. Catarina, J. Have, J. Fernández-Rossier, and N. M. R. Peres, “Optical orientation with linearly polarized light in transition metal dichalcogenides,” Phys. Rev. B 99(12), 125405 (2019).
[Crossref]

Hernandez, E.

J. W. Dawson, T. W. MacDougall, and E. Hernandez, “Verdet constant limited temperature response of a fiber-optic current sensor,” IEEE Photonics Technol. Lett. 7(12), 1468–1470 (1995).
[Crossref]

Hu, Y.

B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015).
[Crossref] [PubMed]

Hu, Z.

B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015).
[Crossref] [PubMed]

Ishizaka, K.

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

Iwasa, Y.

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

Javier, F.

F. H. Koppens, D. E. Chang, and F. Javier, “Graphene plasmonics: A platform for strong light–matter interactions,” Nano Lett,  11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Jiang, Y.

T. Low, R. Roldan, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

Jonkman, H. T.

N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, “Electronic spin transport and spin precession in single graphene layers at room temperature,” Nature,  448, 571–574 (2007).
[Crossref] [PubMed]

Jozsa, C.

N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, “Electronic spin transport and spin precession in single graphene layers at room temperature,” Nature,  448, 571–574 (2007).
[Crossref] [PubMed]

Kaiser, M.

Kawasaki, M.

K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016).
[Crossref] [PubMed]

Khodzitsky, M. K.

A. N. Grebenchukov, S. E. Azbite, A. D. Zaitsev, and M. K. Khodzitsky, “Faraday effect control in graphene-dielectric structure by optical pumping,” J. Magn. Magn. Mater. 472, 25–28 (2019).
[Crossref]

Knorr, A.

H. Funk, A. Knorr, F. Wendler, and E. Malic, “Microscopic view on Landau level broadening mechanisms in graphene,” Phys. Rev. B 92(20), 205428 (2015).
[Crossref]

Koppens, F. H.

F. H. Koppens, D. E. Chang, and F. Javier, “Graphene plasmonics: A platform for strong light–matter interactions,” Nano Lett,  11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Kort-Kamp, W. J. M.

W. J. M. Kort-Kamp, B. Amorim, G. Bastos, F. A. Pinheiro, F. S. S. Rosa, N. M. R. Peres, and C. Farina, “Active magneto-optical control of spontaneous emission in graphene,” Phys. Rev. B 92(20), 205415 (2015).
[Crossref]

Kuroda, K.

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

Kuzmenko, A. B.

J. M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Moreno, J. Faist, and A. B. Kuzmenko, “Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene,” Nat. Commun. 8, 14626 (2017).
[Crossref] [PubMed]

N. Ubrig, I. Crassee, J. Levallois, I. O. Nedoliuk, F. Fromm, M. Kaiser, T. Seyller, and A. B. Kuzmenko, “Fabry-Pérot enhanced faraday rotation in graphene,” Opt. Express 21(21), 24736–24741 (2013).
[Crossref] [PubMed]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Lasia, M.

M. Lasia and L. Brey, “Optical properties of magnetically doped ultrathin topological insulator slabs,” Phys. Rev. B 90(7), 075417 (2014).
[Crossref]

Levallois, J.

N. Ubrig, I. Crassee, J. Levallois, I. O. Nedoliuk, F. Fromm, M. Kaiser, T. Seyller, and A. B. Kuzmenko, “Fabry-Pérot enhanced faraday rotation in graphene,” Opt. Express 21(21), 24736–24741 (2013).
[Crossref] [PubMed]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Levitt, M. H.

M. H. Levitt, “Spin dynamics: Basics of nuclear magnetic resonance” (John Wiley and Sons, New York, London, Sydney, 2008).

Lew Yan Voon, L. C.

G. G. Guzmán-Verri and L. C. Lew Yan Voon, “Electronic structure of silicon-based nanostructures,” Phys. Rev. B 76(7), 075131 (2007).
[Crossref]

Liu, C. C.

C. C. Liu, W. Feng, and Y. G. Yao, “Quantum spin Hall effect in silicene and two-dimensional germanium,” Phys. Rev. Lett. 107(7), 076802 (2011).
[Crossref] [PubMed]

Liu, G. B.

D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108(19), 196802 (2012).
[Crossref] [PubMed]

Liu, P. Q.

J. M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Moreno, J. Faist, and A. B. Kuzmenko, “Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene,” Nat. Commun. 8, 14626 (2017).
[Crossref] [PubMed]

Low, T.

T. Low, R. Roldan, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

T. Low, R. Roldan, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
[Crossref] [PubMed]

Luo, H.

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
[Crossref]

MacDougall, T. W.

J. W. Dawson, T. W. MacDougall, and E. Hernandez, “Verdet constant limited temperature response of a fiber-optic current sensor,” IEEE Photonics Technol. Lett. 7(12), 1468–1470 (1995).
[Crossref]

Majumdar, A.

S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015).
[Crossref] [PubMed]

Mak, K. F.

K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, “The valley Hall effect in MoS2 transistors,” Science 344(6191), 1489–1492 (2014).
[Crossref] [PubMed]

Malic, E.

H. Funk, A. Knorr, F. Wendler, and E. Malic, “Microscopic view on Landau level broadening mechanisms in graphene,” Phys. Rev. B 92(20), 205428 (2015).
[Crossref]

Mandrus, D. G.

S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015).
[Crossref] [PubMed]

Marel, D.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Martin-Moreno, L.

J. M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Moreno, J. Faist, and A. B. Kuzmenko, “Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene,” Nat. Commun. 8, 14626 (2017).
[Crossref] [PubMed]

Martn-Moreno, L.

M. Tymchenko, A. Y. Nikitin, and L. Martn-Moreno, “Faraday rotation due to excitation of magnetoplasmons in graphene microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

McEuen, P. L.

K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, “The valley Hall effect in MoS2 transistors,” Science 344(6191), 1489–1492 (2014).
[Crossref] [PubMed]

McGill, K. L.

K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, “The valley Hall effect in MoS2 transistors,” Science 344(6191), 1489–1492 (2014).
[Crossref] [PubMed]

Mi, C.

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
[Crossref]

Mihailovic, P. M.

P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for temperature-dependence of the Faraday effect by optical activity temperature shift,” IEEE Sens. J. 13(10), 832–837 (2013).
[Crossref]

Miyamoto, K.

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

Mogi, M.

K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016).
[Crossref] [PubMed]

Moreno, L. M.

T. Low, R. Roldan, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
[Crossref] [PubMed]

Morikawa, D.

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

Morozov, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Mueller, T.

A. Pospischil, M. M. Furchi, and T. Mueller, “Solar-energy conversion and light emission in an atomic monolayer p-n diode,” Nat. Nanotechnol. 9, 257–261 (2014).
[Crossref] [PubMed]

Nedoliuk, I. O.

Nicol, E. J.

C. J. Tabert and E. J. Nicol, “Valley-spin polarization in the magneto-optical response of silicene and other similar 2D crystals,” Phys. Rev. Lett. 110(19), 197402 (2013).
[Crossref] [PubMed]

C. J. Tabert and E. J. Nicol, “Magneto-optical conductivity of silicene and other buckled honeycomb lattices,” Phys. Rev. B 88(8), 085434 (2013).
[Crossref]

Nikitin, A. Y.

J. M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Moreno, J. Faist, and A. B. Kuzmenko, “Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene,” Nat. Commun. 8, 14626 (2017).
[Crossref] [PubMed]

M. Tymchenko, A. Y. Nikitin, and L. Martn-Moreno, “Faraday rotation due to excitation of magnetoplasmons in graphene microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

Niu, Q.

W. Yao, D. Xiao, and Q. Niu, “Valley-dependent optoelectronics from inversion symmetry breaking,” Phys. Rev. B 77(23), 235406 (2008).
[Crossref]

D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: Magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
[Crossref]

Novoselov, K. S.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6 (11), 749–758 (2012).
[Crossref]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Ochoa, H.

M. A. Cazalilla, H. Ochoa, and F. Guinea, “Quantum spin Hall effect in two-dimensional crystals of transition-metal dichalcogenides,” Phys. Rev. Lett. 113(7), 077201 (2014).
[Crossref] [PubMed]

Ogawa, N.

K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016).
[Crossref] [PubMed]

Okada, K. N.

K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016).
[Crossref] [PubMed]

Okuda, T.

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

Oliva-Leyva, M.

M. Oliva-Leyva and C. Wang, “Magneto-optical conductivity of anisotropic two-dimensional Dirac-Weyl materials,” Annals of Physics 384, 61–70 (2017).
[Crossref]

Ostler, M.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Park, J.

K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, “The valley Hall effect in MoS2 transistors,” Science 344(6191), 1489–1492 (2014).
[Crossref] [PubMed]

Pereira, V.

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. H. Castro Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

Peres, N. M. R.

G. Catarina, J. Have, J. Fernández-Rossier, and N. M. R. Peres, “Optical orientation with linearly polarized light in transition metal dichalcogenides,” Phys. Rev. B 99(12), 125405 (2019).
[Crossref]

W. J. M. Kort-Kamp, B. Amorim, G. Bastos, F. A. Pinheiro, F. S. S. Rosa, N. M. R. Peres, and C. Farina, “Active magneto-optical control of spontaneous emission in graphene,” Phys. Rev. B 92(20), 205415 (2015).
[Crossref]

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. H. Castro Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

Petricevic, S. J.

P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for temperature-dependence of the Faraday effect by optical activity temperature shift,” IEEE Sens. J. 13(10), 832–837 (2013).
[Crossref]

Pinheiro, F. A.

W. J. M. Kort-Kamp, B. Amorim, G. Bastos, F. A. Pinheiro, F. S. S. Rosa, N. M. R. Peres, and C. Farina, “Active magneto-optical control of spontaneous emission in graphene,” Phys. Rev. B 92(20), 205415 (2015).
[Crossref]

Polini, M.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6 (11), 749–758 (2012).
[Crossref]

Popinciuc, M.

N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, “Electronic spin transport and spin precession in single graphene layers at room temperature,” Nature,  448, 571–574 (2007).
[Crossref] [PubMed]

Pospischil, A.

A. Pospischil, M. M. Furchi, and T. Mueller, “Solar-energy conversion and light emission in an atomic monolayer p-n diode,” Nat. Nanotechnol. 9, 257–261 (2014).
[Crossref] [PubMed]

Poumirol, J. M.

J. M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Moreno, J. Faist, and A. B. Kuzmenko, “Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene,” Nat. Commun. 8, 14626 (2017).
[Crossref] [PubMed]

Radunovic, J. B.

P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for temperature-dependence of the Faraday effect by optical activity temperature shift,” IEEE Sens. J. 13(10), 832–837 (2013).
[Crossref]

Rappoport, T. G.

T. P. Cysne, T. G. Rappoport, J. H. Garcia, and A. R. Rocha, “Quantum Hall effect in graphene with interface-induced spin-orbit coupling,” Phys. Rev. B 97(8), 085413 (2018).
[Crossref]

Rivera, P.

J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016).
[Crossref]

Rocha, A. R.

T. P. Cysne, T. G. Rappoport, J. H. Garcia, and A. R. Rocha, “Quantum Hall effect in graphene with interface-induced spin-orbit coupling,” Phys. Rev. B 97(8), 085413 (2018).
[Crossref]

Roldan, R.

T. Low, R. Roldan, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
[Crossref] [PubMed]

T. Low, R. Roldan, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

Rosa, F. S. S.

W. J. M. Kort-Kamp, B. Amorim, G. Bastos, F. A. Pinheiro, F. S. S. Rosa, N. M. R. Peres, and C. Farina, “Active magneto-optical control of spontaneous emission in graphene,” Phys. Rev. B 92(20), 205415 (2015).
[Crossref]

Ross, J. S.

J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016).
[Crossref]

Rotenberg, E.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Sahin, H.

S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, “Two and one-dimensional honeycomb structures of silicon and germanium,” Phys. Rev. Lett. 102(23), 236804 (2009).
[Crossref] [PubMed]

Sakano, M.

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

Schaibley, J. R.

J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016).
[Crossref]

S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015).
[Crossref] [PubMed]

Schwingenschlögl, U.

M. Tahir and U. Schwingenschlögl, “Valley polarized quantum Hall effect and topological insulator phase transitions in silicene,” Sci. Rep. 3, 1075 (2013).
[Crossref] [PubMed]

Seyler, K. L.

J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016).
[Crossref]

Seyller, T.

N. Ubrig, I. Crassee, J. Levallois, I. O. Nedoliuk, F. Fromm, M. Kaiser, T. Seyller, and A. B. Kuzmenko, “Fabry-Pérot enhanced faraday rotation in graphene,” Opt. Express 21(21), 24736–24741 (2013).
[Crossref] [PubMed]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Sharapov, S. G.

V. P. Gusynin and S. G. Sharapov, “Transport of Dirac quasiparticles in graphene: Hall and optical conductivities,” Phys. Rev. B 73(24), 245411 (2006).
[Crossref]

V. P. Gusynin and S. G. Sharapov, “Unconventional integer quantum Hall effect in graphene,” Phys. Rev. Lett. 95(14), 146801 (2005).
[Crossref] [PubMed]

Slipchenko, T. M.

J. M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Moreno, J. Faist, and A. B. Kuzmenko, “Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene,” Nat. Commun. 8, 14626 (2017).
[Crossref] [PubMed]

Sounas, D. L.

D. L. Sounas and C. Caloz, “Electromagnetic nonreciprocity and gyrotropy of graphene,” Appl. Phys. Lett. 98(2), 021911 (2011).
[Crossref]

Suzuki, R.

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

Tabert, C. J.

C. J. Tabert and E. J. Nicol, “Magneto-optical conductivity of silicene and other buckled honeycomb lattices,” Phys. Rev. B 88(8), 085434 (2013).
[Crossref]

C. J. Tabert and E. J. Nicol, “Valley-spin polarization in the magneto-optical response of silicene and other similar 2D crystals,” Phys. Rev. Lett. 110(19), 197402 (2013).
[Crossref] [PubMed]

Tahir, M.

M. Tahir and U. Schwingenschlögl, “Valley polarized quantum Hall effect and topological insulator phase transitions in silicene,” Sci. Rep. 3, 1075 (2013).
[Crossref] [PubMed]

Takahashi, K. S.

K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016).
[Crossref] [PubMed]

Takahashi, Y.

K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016).
[Crossref] [PubMed]

Tang, P.

Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013).
[Crossref] [PubMed]

Tokura, Y.

K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016).
[Crossref] [PubMed]

Tombros, N.

N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, “Electronic spin transport and spin precession in single graphene layers at room temperature,” Nature,  448, 571–574 (2007).
[Crossref] [PubMed]

Topsakal, M.

S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, “Two and one-dimensional honeycomb structures of silicon and germanium,” Phys. Rev. Lett. 102(23), 236804 (2009).
[Crossref] [PubMed]

Torihata, S.

Tsukada, N.

Tsukazaki, A.

K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016).
[Crossref] [PubMed]

Tymchenko, M.

M. Tymchenko, A. Y. Nikitin, and L. Martn-Moreno, “Faraday rotation due to excitation of magnetoplasmons in graphene microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

Ubrig, N.

van Wees, B. J.

N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, “Electronic spin transport and spin precession in single graphene layers at room temperature,” Nature,  448, 571–574 (2007).
[Crossref] [PubMed]

Viana-Gomes, J.

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. H. Castro Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

Walter, A. L.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Wang, C.

M. Oliva-Leyva and C. Wang, “Magneto-optical conductivity of anisotropic two-dimensional Dirac-Weyl materials,” Annals of Physics 384, 61–70 (2017).
[Crossref]

Wang, H.

T. Low, R. Roldan, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
[Crossref] [PubMed]

T. Low, R. Roldan, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

Wang, J.

Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013).
[Crossref] [PubMed]

Wen, S.

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
[Crossref]

Wendler, F.

H. Funk, A. Knorr, F. Wendler, and E. Malic, “Microscopic view on Landau level broadening mechanisms in graphene,” Phys. Rev. B 92(20), 205428 (2015).
[Crossref]

Wu, S.

S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015).
[Crossref] [PubMed]

Wu, W.

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
[Crossref]

Xia, F.

T. Low, R. Roldan, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

T. Low, R. Roldan, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
[Crossref] [PubMed]

Xiao, D.

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotech. 7(8), 490 (2012).
[Crossref]

D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108(19), 196802 (2012).
[Crossref] [PubMed]

W. Yao, D. Xiao, and Q. Niu, “Valley-dependent optoelectronics from inversion symmetry breaking,” Phys. Rev. B 77(23), 235406 (2008).
[Crossref]

D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: Magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
[Crossref]

Xu, G.

Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013).
[Crossref] [PubMed]

Xu, X.

J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016).
[Crossref]

S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015).
[Crossref] [PubMed]

D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108(19), 196802 (2012).
[Crossref] [PubMed]

Xu, Y.

Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013).
[Crossref] [PubMed]

Yaji, K.

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

Yan, B.

Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013).
[Crossref] [PubMed]

Yan, J.

S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015).
[Crossref] [PubMed]

Yang, Q.

Q. Yang, X.A. Zhang, A. Bagal, W. Guo, and C.H. Chang, “Antireflection effects at nanostructured material interfaces and the suppression of thin-film interference,” Nanotechnology 24(23), 235202 (2013).
[Crossref] [PubMed]

Yao, W.

J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016).
[Crossref]

S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015).
[Crossref] [PubMed]

D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108(19), 196802 (2012).
[Crossref] [PubMed]

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotech. 7(8), 490 (2012).
[Crossref]

W. Yao, D. Xiao, and Q. Niu, “Valley-dependent optoelectronics from inversion symmetry breaking,” Phys. Rev. B 77(23), 235406 (2008).
[Crossref]

D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: Magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
[Crossref]

Yao, Y. G.

C. C. Liu, W. Feng, and Y. G. Yao, “Quantum spin Hall effect in silicene and two-dimensional germanium,” Phys. Rev. Lett. 107(7), 076802 (2011).
[Crossref] [PubMed]

Yi, J.

S. Ahmed and J. Yi, “Two-dimensional transition metal dichalcogenides and their charge carrier mobilities in field-effect transistors,” Nano-Micro Lett. 9, 50 (2017).
[Crossref]

Yokota, M.

Yoshimi, R.

K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016).
[Crossref] [PubMed]

Yoshino, T.

Yu, H.

J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016).
[Crossref]

Zaitsev, A. D.

A. N. Grebenchukov, S. E. Azbite, A. D. Zaitsev, and M. K. Khodzitsky, “Faraday effect control in graphene-dielectric structure by optical pumping,” J. Magn. Magn. Mater. 472, 25–28 (2019).
[Crossref]

Zeng, H.

B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015).
[Crossref] [PubMed]

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotech. 7(8), 490 (2012).
[Crossref]

Zhang, H.-J.

Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013).
[Crossref] [PubMed]

Zhang, S.

B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015).
[Crossref] [PubMed]

Zhang, S. C.

Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013).
[Crossref] [PubMed]

Zhang, W.

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
[Crossref]

Zhang, X.A.

Q. Yang, X.A. Zhang, A. Bagal, W. Guo, and C.H. Chang, “Antireflection effects at nanostructured material interfaces and the suppression of thin-film interference,” Nanotechnology 24(23), 235202 (2013).
[Crossref] [PubMed]

Zhang, Y.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Zhang, Y. J.

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

Zou, Y.

B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015).
[Crossref] [PubMed]

ACS Nano (1)

M. Tymchenko, A. Y. Nikitin, and L. Martn-Moreno, “Faraday rotation due to excitation of magnetoplasmons in graphene microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

Annals of Physics (1)

M. Oliva-Leyva and C. Wang, “Magneto-optical conductivity of anisotropic two-dimensional Dirac-Weyl materials,” Annals of Physics 384, 61–70 (2017).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

D. L. Sounas and C. Caloz, “Electromagnetic nonreciprocity and gyrotropy of graphene,” Appl. Phys. Lett. 98(2), 021911 (2011).
[Crossref]

IEEE Photonics Technol. Lett. (1)

J. W. Dawson, T. W. MacDougall, and E. Hernandez, “Verdet constant limited temperature response of a fiber-optic current sensor,” IEEE Photonics Technol. Lett. 7(12), 1468–1470 (1995).
[Crossref]

IEEE Sens. J. (1)

P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for temperature-dependence of the Faraday effect by optical activity temperature shift,” IEEE Sens. J. 13(10), 832–837 (2013).
[Crossref]

J. Magn. Magn. Mater. (2)

A. Dolatabady and N. Granpayeh, “Manipulation of the Faraday rotation by graphene metasurfaces,” J. Magn. Magn. Mater. 469, 231-235 (2019).
[Crossref]

A. N. Grebenchukov, S. E. Azbite, A. D. Zaitsev, and M. K. Khodzitsky, “Faraday effect control in graphene-dielectric structure by optical pumping,” J. Magn. Magn. Mater. 472, 25–28 (2019).
[Crossref]

J. Opt. Soc. Am. B (1)

Nano Lett (1)

F. H. Koppens, D. E. Chang, and F. Javier, “Graphene plasmonics: A platform for strong light–matter interactions,” Nano Lett,  11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Nano-Micro Lett. (1)

S. Ahmed and J. Yi, “Two-dimensional transition metal dichalcogenides and their charge carrier mobilities in field-effect transistors,” Nano-Micro Lett. 9, 50 (2017).
[Crossref]

Nanotechnology (1)

Q. Yang, X.A. Zhang, A. Bagal, W. Guo, and C.H. Chang, “Antireflection effects at nanostructured material interfaces and the suppression of thin-film interference,” Nanotechnology 24(23), 235202 (2013).
[Crossref] [PubMed]

Nat. Commun. (2)

K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016).
[Crossref] [PubMed]

J. M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Moreno, J. Faist, and A. B. Kuzmenko, “Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene,” Nat. Commun. 8, 14626 (2017).
[Crossref] [PubMed]

Nat. Nanotech. (1)

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotech. 7(8), 490 (2012).
[Crossref]

Nat. Nanotechnol. (2)

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

A. Pospischil, M. M. Furchi, and T. Mueller, “Solar-energy conversion and light emission in an atomic monolayer p-n diode,” Nat. Nanotechnol. 9, 257–261 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6 (11), 749–758 (2012).
[Crossref]

Nat. Phys. (1)

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Nat. Rev. Mater. (1)

J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016).
[Crossref]

Nature (2)

N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, “Electronic spin transport and spin precession in single graphene layers at room temperature,” Nature,  448, 571–574 (2007).
[Crossref] [PubMed]

S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015).
[Crossref] [PubMed]

Opt. Express (1)

Phys. Chem. Chem. Phys. (1)

B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015).
[Crossref] [PubMed]

Phys. Rev. A (1)

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
[Crossref]

Phys. Rev. B (12)

V. P. Gusynin and S. G. Sharapov, “Transport of Dirac quasiparticles in graphene: Hall and optical conductivities,” Phys. Rev. B 73(24), 245411 (2006).
[Crossref]

W. J. M. Kort-Kamp, B. Amorim, G. Bastos, F. A. Pinheiro, F. S. S. Rosa, N. M. R. Peres, and C. Farina, “Active magneto-optical control of spontaneous emission in graphene,” Phys. Rev. B 92(20), 205415 (2015).
[Crossref]

H. Funk, A. Knorr, F. Wendler, and E. Malic, “Microscopic view on Landau level broadening mechanisms in graphene,” Phys. Rev. B 92(20), 205428 (2015).
[Crossref]

T. Low, R. Roldan, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

M. Ezawa, “Spin-valley optical selection rule and strong circular dichroism in silicene,” Phys. Rev. B 86(16), 161407 (2012).
[Crossref]

G. G. Guzmán-Verri and L. C. Lew Yan Voon, “Electronic structure of silicon-based nanostructures,” Phys. Rev. B 76(7), 075131 (2007).
[Crossref]

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. H. Castro Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

W. Yao, D. Xiao, and Q. Niu, “Valley-dependent optoelectronics from inversion symmetry breaking,” Phys. Rev. B 77(23), 235406 (2008).
[Crossref]

G. Catarina, J. Have, J. Fernández-Rossier, and N. M. R. Peres, “Optical orientation with linearly polarized light in transition metal dichalcogenides,” Phys. Rev. B 99(12), 125405 (2019).
[Crossref]

M. Lasia and L. Brey, “Optical properties of magnetically doped ultrathin topological insulator slabs,” Phys. Rev. B 90(7), 075417 (2014).
[Crossref]

C. J. Tabert and E. J. Nicol, “Magneto-optical conductivity of silicene and other buckled honeycomb lattices,” Phys. Rev. B 88(8), 085434 (2013).
[Crossref]

T. P. Cysne, T. G. Rappoport, J. H. Garcia, and A. R. Rocha, “Quantum Hall effect in graphene with interface-induced spin-orbit coupling,” Phys. Rev. B 97(8), 085413 (2018).
[Crossref]

Phys. Rev. Lett. (11)

C. J. Tabert and E. J. Nicol, “Valley-spin polarization in the magneto-optical response of silicene and other similar 2D crystals,” Phys. Rev. Lett. 110(19), 197402 (2013).
[Crossref] [PubMed]

M. Ezawa, “Valley-polarized metals and quantum anomalous Hall effect in silicene,” Phys. Rev. Lett. 109(5), 055502 (2012).
[Crossref] [PubMed]

V. P. Gusynin and S. G. Sharapov, “Unconventional integer quantum Hall effect in graphene,” Phys. Rev. Lett. 95(14), 146801 (2005).
[Crossref] [PubMed]

D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108(19), 196802 (2012).
[Crossref] [PubMed]

D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: Magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
[Crossref]

M. A. Cazalilla, H. Ochoa, and F. Guinea, “Quantum spin Hall effect in two-dimensional crystals of transition-metal dichalcogenides,” Phys. Rev. Lett. 113(7), 077201 (2014).
[Crossref] [PubMed]

S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, “Two and one-dimensional honeycomb structures of silicon and germanium,” Phys. Rev. Lett. 102(23), 236804 (2009).
[Crossref] [PubMed]

C. C. Liu, W. Feng, and Y. G. Yao, “Quantum spin Hall effect in silicene and two-dimensional germanium,” Phys. Rev. Lett. 107(7), 076802 (2011).
[Crossref] [PubMed]

Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013).
[Crossref] [PubMed]

M. Ezawa, “Photoinduced topological phase transition and a single Dirac-cone state in silicene,” Phys. Rev. Lett. 110(2), 026603 (2013).
[Crossref] [PubMed]

T. Low, R. Roldan, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
[Crossref] [PubMed]

Sci. Rep. (1)

M. Tahir and U. Schwingenschlögl, “Valley polarized quantum Hall effect and topological insulator phase transitions in silicene,” Sci. Rep. 3, 1075 (2013).
[Crossref] [PubMed]

Science (2)

K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, “The valley Hall effect in MoS2 transistors,” Science 344(6191), 1489–1492 (2014).
[Crossref] [PubMed]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Other (1)

M. H. Levitt, “Spin dynamics: Basics of nuclear magnetic resonance” (John Wiley and Sons, New York, London, Sydney, 2008).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 Faraday, Kerr rotation and ellipticity of silicene-substrate system as function of photon energy electric and magnetic fields. (a) The s polarized Faraday rotation and (b) ellipticity as function of incident photon energy in the K valley with modulation of the external electric field for the three distinct topological regimes, TI, VSPM and BI for a magnetic field of 1 T. The spectral peaks are labelled 1 through 6 and their origin is identified in the main text. The spectrum are vertically shifted by 15° among themselves for clearer viewing. Furthermore, in this figure we use Δz = Δso/2 (TI) and Δz = 2Δso (BI). (c) The s polarized Kerr rotation as function of incident photon energy in the Kʹ valley with modulation of the external magnetic field for the TI regime for three different values of B = 1, 3 and 5 T. (d) The maximum Faraday, Kerr rotation and ellipticity as function of magnetic field in K valley for the single transition Δ−10,K,↑. The parameters used are θ1 = 30°, Γ = 0.01Δso, refractive index n2 = 1.84 and chemical potential μF = 0.
Fig. 2
Fig. 2 (a) Schematic representation of the allowed transitions between LL’s for three different values of chemical potential μF = 0, 10 and 22 meV; (b) and (c) the s polarized Kerr rotation as function of incident photon frequency in K and valleys with modulation of the chemical potential in the TI regime for a magnetic field of 1 T, respectively. (d) The s polarized Faraday rotation as function of incident photon energy in K valley for different incident angles for a single transition in the TI regime. (e) The s polarized Faraday rotation as function of incident photon energy in K for different temperatures for a single transition in the TI regime. (f) The s and p polarized Kerr rotation as function of incident photon frequency in the semiclassical limit for n-type and p-type silicene (μF = 56 and -56 meV), respectively. The solid line represents the s polarized and the dashed line p polarized. The parameters used are θ = 30°, Γ = 0.01Δso and refractive index n2 = 1.84.
Fig. 3
Fig. 3 (a) Schematic representation of the allowed transitions between LL’s for chemical potential μ = 0.2. (b) and (c) the s polarized Faraday rotation contour plots as function of x in K valley for μ = 0 and 1.25, repectively, where x = Δzso and μ = μFso. The parameters used are θ = 30°, Γ = 0.01Δso and refractive index n2 = 1.84.
Fig. 4
Fig. 4 Faraday and Kerr rotation of silicene-substrate system as function of photon energy for three different relative permittivities. (a) The s polarized and (b) p polarized Faraday rotation as a function of incident photon energy in the K valley with modulation of relative permittivities in the TI regime for a magnetic field of 1 T. (c) The s polarized and (d) p polarized Kerr rotation as a function of incident photon energy in the K valley with modulation of relative permittivities in the TI regime for a magnetic field of 1 T, for the two transitions. The parameters used are θ1 = 30°, Γ = 0.01Δso and chemical potential μF = 0.
Fig. 5
Fig. 5 Faraday and Kerr rotation of silicene-substrate system as a function of photon energy for three different relative permeabilitis. (a) The s polarized and (b) p polarized Faraday rotation as a function of incident photon energy in the K valley with modulation of relative permeabilitis in the TI regime for a magnetic field of 1 T. (c) The s polarized and (d) p polarized Kerr rotation as a function of incident photon energy in the K valley with modulation of relative permeabilities in the TI regime for a magnetic field of 1 T, for the two transitions. The parameters used are θ1 = 30°, Γ = 0.01Δso, refractive index n2 = 1.84 and chemical potential μF = 0.
Fig. 6
Fig. 6 Faraday and Kerr ellipticities as a function of incident photon energy for three different relative permittivities. (a) The s polarized and (b) p polarized Faraday ellipticities as function of incident photon energy in the K valley with modulation of relative permittivities in the TI regime for a magnetic field of 1 T. (c) The s polarized and (d) p polarized Kerr ellipticities as a function of incident photon energy in the K valley with relative permittivities in the TI regime for a magnetic field of 1 T, for the two transitions. The parameters used are θ1 = 30°, Γ = 0.01Δso and chemical potential μF = 0.

Tables (1)

Tables Icon

Table 1 Table of allowed transitions in K valley in the n = −1,0,1 subspace, at a fixed Magnetic field and Chemical potential μ. Furthermore x = Δzso, y = v 2 e B / Δ s o 2 . and μ = μFso.

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

H ^ ξ σ = v F ( ξ k x τ ^ x + k y τ ^ y ) 1 2 ξ Δ s o σ ^ z τ ^ z + 1 2 Δ z τ ^ z
E ( ξ , σ , n , t ) = { t 2 v F 2 e B | n | + Δ ξ σ 2 , if n 0. ξ Δ ξ σ , if n = 0.
| n ¯ | ξ = 1 = ( i A n | n 1 B n | n )
| n ¯ | ξ = 1 = ( i A n | n B n | n 1 )
A n = { | E ( ξ , σ , n , t ) | + t Δ ξ σ 2 | E ( ξ , σ , n , t ) | , if n 0. 1 ξ 2 , if n = 0.
B n = { | E ( ξ , σ , n , t ) | t Δ ξ σ 2 | E ( ξ , σ , n , t ) | , if n 0. 1 + ξ 2 , if n = 0.
σ μ ν ( Ω ) = i 2 π l B 2 σ , ξ = ± 1 m n f n f m E n E m n ¯ | j ^ μ | m ¯ m ¯ | j ^ ν | n ¯ Ω ( E n E m ) + i Γ ,
R e I m } ( σ x x ( Ω ) ) σ 0 = 2 v 2 e B π ξ , σ m , n Θ ( E n μ F ) Θ ( E m μ F ) E n E m × [ ( A m B n ) 2 δ | m | ξ , | n | + ( B m A n ) 2 δ | m | + ξ , | n | ] { F G ,
R e I m } ( σ x y ( Ω ) ) σ 0 = 2 v 2 e B π ξ , σ m , n ξ Θ ( E n μ F ) Θ ( E m μ F ) E n E m × [ ( A m B n ) 2 δ | m | ξ , | n | ( B m A n ) 2 δ | m | + ξ , | n | ] { G F .
r p p = α + T α L + β α + T α + L + β ,
r s s = ( α T α + L + β α + T α + L + β ) ,
t p p = 2 Z 2 ε 2 Z 1 k 1 z α + T α + T α + L + β ,
t s s = 2 μ 2 k 1 z α + L α + T α + L + β ,
r s p = t s p = 2 Z 0 2 μ 0 μ 1 μ 2 k 1 z k 2 z ( σ H + σ x y s y m ) Z 1 ( α + T α + L + β ) ,
r p s = k 1 k 2 z k 2 k 1 z t p s = 2 Z 0 2 μ 1 μ 2 Z 1 k 1 z k 2 z ( σ x y s y m σ H ) α + T α + L + β ,
α ± L = ( k 1 z ε 2 ± k 2 z ε 1 + k 1 z k 2 z σ L / ( ε 0 Ω ) ) ,
α ± T = ( k 2 z μ 1 ± k 1 z μ 2 + μ 0 μ 1 μ 2 σ T Ω ) ,
β = Z 0 2 μ 1 μ 2 k 1 z k 2 z [ σ H 2 ( σ x y s y m ) 2 ]
Θ F , s ( p ) = 1 2 tan 1 ( 2 Re ( χ F , s ( p ) ) 1 | χ F , s ( p ) | 2 ) ,
and η F , s ( p ) = 1 2 sin 1 ( 2 Im ( χ F , s ( p ) ) 1 | χ F , s ( p ) | 2 ) ,
χ F , s = t p s t s s = Z 0 ε 1 μ 1 k 1 c o s ( θ 1 ) σ H α + L ,
and χ F , p = t s p t p p = Z 0 μ 2 ε 2 μ 0 μ 1 k 2 c o s ( θ 2 ) σ H α + T
Θ K , s ( p ) = 1 2 tan 1 ( 2 Re ( χ K , s ( p ) ) 1 | χ K , s ( p ) | 2 ) ,
and η K , s ( p ) = 1 2 sin 1 ( 2 Im ( χ K , s ( p ) ) 1 | χ K , s ( p ) | 2 ) ,
χ K , s = r p s r s s = 2 Z 0 μ 1 ε 1 μ 2 k 1 z k 2 z σ H α T α + L + β ,
and χ K , p = r s p r p p = 2 Z 0 μ 1 ε 1 μ 0 μ 2 k 1 z k 2 z σ H α + L α L + β
E n + 1 E n v 2 e B Δ ξ σ 2 + 2 n v 2 e B = Ω c ,
Re ( σ x x ( Ω ) ) σ 0 = Im ( σ x y ( Ω ) ) σ 0 = μ F π Γ ( ( Ω Ω c ) ) 2 + Γ 2 ,
Im ( σ x x ( Ω ) ) σ 0 = Re ( σ x y ( Ω ) ) σ 0 = μ F π ( Ω Ω c ) ( ( Ω Ω c ) ) 2 + Γ 2

Metrics