Open Access
Add to BibSonomy
Abstract
Materials belonging to the graphene family are two-dimensional staggered monolayers that undergo topological phase transitions under the influence of an external electric field or off-resonant optical field. Inspired by the interplay between topological matter and the helicity of photons, we investigate various topological quantum phases of the graphene family materials (GFMs), when subject to an external electric field and irradiated by off-resonant light. Using the Kubo formalism, we derive analytic expressions of the valley and spin-resolved conductivities of silicene. We then show that the topological quantum phase transitions can be modulated by an external electric field or irradiating circularly polarized light on the surface. Based on a general beam propagation model, we theoretically investigate the transitional Kerr rotations in silicene in different phases. Our results identify topological phases where Kerr rotations and ellipticity can be maximized. We believe that our results are helpful for developing novel practical devices based on the Kerr effect of silicene.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In the last few years, topological photonics has grown exponentially in which topological ideas have been applied to both electronic and photonic systems, opening up new opportunities to realize exotic topological novel effects [1,2]. Topological quantum materials [3–5] have emerged as a unifying concept in modern material science with potential applications in novel electronic, spintronic, valleytronic, spintronic, optoelectronic, and plasmonic devices [3,6–8]. For example, staggered two-dimensional semiconductor atomic crystals (monolayer of silicon, germanium, and tin atoms arranged in a honeycomb lattice) hold great promise due to their exceptional electronic and optical properties [9–12]. Unlike 2D graphene, these materials are non-planar and possess stronger spin-orbit interaction (SOI) resulting in a gap in the low-energy Dirac-like band structure in each of the $K$ and $K'$ valleys [13,14]. The strengths of the intrinsic SOI in silicene [13], germanene [15,28] and tinene [16], are of the order of 1.55–7.9, 24–93, and 100 meV, respectively. By inducing an external electric field or off-resonant circularly polarized light, the four Dirac gaps, in general, become non-degenerate possibly triggering a topological quantum phase transition (TQPT) between a robust quantum spin Hall insulator (QSHI) and a band insulator (BI). This transition is subject to the spin-valley optical selection rules [12,17].
This article delineates the interplay of topology and Kerr rotations (KR) originating in these topological materials due to broken time-reversal symmetry [18,19]. It is well known that graphene exhibits an exceptionally large Faraday and Kerr rotation in the THz region which is attributed to the cyclotron resonance and Landau level (LLs) transitions. The magnitude of FR is about $6^{\circ }$ in a field of strength 7 T [20]. In the presence of an external magnetic field, the graphene is considered a futuristic candidate for non-reciprocal tunable devices [20–22]. 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. On the other hand, in topological insulator W. K. Tse et. al. theoretically studied the Faraday and Kerr effects and predicted that MOKE may exhibit a giant $\pi /2$ rotation [23,24].
The FR and MOKE have been further studied in various prior works, in the presence of electric and magnetic fields in 2D lattices with spin-orbit interaction [25]. Széchenyi et al. theoretically predicted that large Faraday and Kerr rotations can be achieved for bilayer graphene in a quantum anomalous Hall state in the absence of the magnetic field [26]. Recently, FR and MOKE in the presence of an external magnetic field have been investigated in hybridized three-dimensional topological insulator thin films for the top and bottom Dirac surface states [27]. It is, therefore pertinent to determine the Kerr rotations without the magnetic field in the graphene family by exclusively driving TQPTs between different quantum phases and investigating the role of topological invariants and their connection with the Kerr signal. This is the intent of the current article. Therefore the article explores the quantitative predictions of the Kerr effect in various topological phases, driven by the electric field and the off-resonant circularly polarized light.
In this work, we investigate the topological quantum phase transitions, distinct topological phases, and spin-valley polarized optical response of monolayer silicene, which is exposed to an electric field and irradiated by an off-resonant light. The graphene 2D materials reveal several electronic phases resulting from changes in electric field and circularly polarized laser light. In the process of the electronic energy dispersion evolution by external stimuli, a configurable topological phase diagram is presented. There exist generous topological phases in these 2D quantum materials, such as quantum spin Hall insulator (QSHI), spin-valley polarized metal (SVPM), spin-polarized metal (SPM), spin half-metal (SHM), anomalous quantum Hall insulator (AQHI), single Dirac cone (SDC) phase and band insulator (BI). By using the famous Kubo formula, we determine the longitudinal and transverse optical conductivities to obtain Fresnel’s reflection coefficients. The Fresnel’s coefficients for circularly polarized Gaussian light beam are calculated by solving the Maxwell equations and applying appropriate boundary conditions at the surface of the GFMs.
2. General theoretical model
2.1 System Hamiltonain
The crystal structure of the GFM has been shown in Fig. 1. The low-energy physics of the GFMs which include silicene, germanene, and tinene in the presence of a perpendicular uniform electric field is described by a simple low energy Hamiltonian as [28–30]
The parameter $\xi =\pm 1$ corresponds to the valleys ($K$ and $K^{'}$) in momentum space and the vector operators $\vec {\tau }=(\hat {\tau }_{x},\hat {\tau }_{y},\hat {\tau }_{z})$ and $\vec {s}=(\hat {s}_{x},\hat {s}_{y},\hat {s}_{z})$ respectively represent Pauli matrices of the lattice pseudo spin and real spin degrees of freedom. The intrinsic spin-orbit coupling is denoted by $\Delta _{so}$, whereas, $E_{z}$ being an electric field perpendicular to the plane of atoms. Additionally, we consider a circularly polarized wave of intensity $I_{0}$ in our model with frequency $\omega _{0}\gg \omega$ propagating along the $z$ direction, where $\omega$ is the frequency of light. Based on Floquet formulism [31] the electromagnetic (EM) light source can be treated as an external perturbation that is periodic in time. The corresponding EM vector potential $A(t)$ is given by
Under the approximation that the light frequency $\omega$ is much higher than for any electron transition $\omega _{0}$, the photon dressed effective Floquet Hamiltonian can be written as [32]
where $H_{0}$ is unperturbed Hamiltonian of the system and $H_{\pm 1}$ represent the Fourier components of the Hamiltonian and are given byAfter some straightforward calculations we obtain
Here,
is the the spin- and valley-dependent mass which is tunable by $E_{z}$ and polarization potential of the off-resonant light. Using Eqs. (6) and (7), the energy dispersion is obtained as where $t=\pm 1$ denotes the electron/hole band. The Dirac mass gap in Eq. (7) depends on the strength of the SOI, the spin and valley quantum numbers of the carriers, and the helicity of photons. Throughout this article, we consider the topological number $(\mathcal {C},\mathcal {C}_{s})$ to differentiate different topological regimes. Figure 2 shows the dispersion of the energy bands for both spin states in the $K$ valley and the impact of varying the externally applied electric potential $\Delta _{z}$. Similar features can be achieved by adjusting the input high intensity $\Lambda _{\omega }$ which plays an identical rule through the relationship. The tuning of $\Delta _{z}$ (or $\Lambda _{\omega }$) achieves a TQPT [33].
Fig. 1. (a) The un-buckled structure of the staggered 2D semiconductor monolayer. (b) Illustration of the buckled honeycomb lattice of of the staggered 2D semiconductors which is distorted due to large ionic radius of a silicon atom and generates a staggered sublattice potential $E_{z}$.
Fig. 2. The electronic energy dispersion of the staggered 2D monolayer for the $K$ valley corresponding to three different topological regimes: (a) TI ($\Delta _{z}=0$), (b) TI ($\Delta _{z}=0.5 \Delta _{so}$), (c) VSPM ($\Delta _{z}=\Delta _{so}$) and (d) BI ($\Delta _{z}=2 \Delta _{so}$) respectively. The solid blue (dashed red) curves are for spin up (down).
In Fig. 2, we have shown the low-energy bands of silicene as a function of the dimensionless parameter in different topological phases. The dashed red (solid blue) curves correspond to spin-down (spin-up) bands. The GFMs can be further controlled by a staggered electric potential $elE_{z}=\Delta _{z}=0$, which generates an electrostatic potential $2lE_{z}$ between the two different atoms in the silicone unit cell. When $\Delta _{z}=0$, then the electronic state is the QSHI phase as shown in Fig. 2(a). For the particular case when $\Delta _{z}=0.5 \Delta _{so}$, the possible topological number is $\left (0, 1\right )$ and the system is still in the QSHI also called the topological insulator (TI) state which is non-trivial band structure. Each spin state in this phase gives rise to an independent Dirac energy gap as shown in Fig. 2(b), whereas for $\Delta _{z}=0$ the spin bands overlap, as illustrated in Fig. 2(a). Fig. 2(c), demonstrates the case when $\Delta _{z}$ is increased to $\Delta _{z}=\Delta _{so}$. At this point, the lower bandgap of spin-up closes, and the system hits the valley-spin polarized metal (VSPM) state in the $K$ valley, while at the $K'$ valley the corresponding mass gap remains and it is the spin-down gap that closes. In this state, the topological Chern number is ($1,1/2$). For an even higher electric potential $\Delta _{z}>\Delta _{so}$, the spectrum becomes gapped again and the system transition from the VSPM to the band insulator (BI) phase as shown in Fig. 2(d). The topological Chern number for the BI phase is ($0,0$).
2.2 Phase diagram, topological quantum phase transitions and optical signature
The phase diagram is a mosaic of regions corresponding to the quantum phases QSHI ($0, 1$), QAHI ($-2, 0$), the photoinduced polarized-spin quantum Hall insulator (PS-QHI) ($-1, 1/2$), BI ($0, 0$) and the interaction points SPVM ($1, 1/2$). To understand the TQPTs in the GFMs, let us first look into topological signatures in the phase diagram summarized in Fig. 3. For the sake of simplicity, we consider two different cases. In the first case, we fix the circularly polarized light $\Lambda _{\omega }/\Delta _{so}=0$ as shown by path $1$ in Fig. 3. At $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(0,0)$, the silicenic layer can be characterized as a QSHI. Now the electric potential is tuned from $\Delta _{z}/\Delta _{so}=0{\rightarrow }\;1\;{\rightarrow }\;2$. We note that as $\Delta _{z}$ increases, then the phase boundaries come closer to each other. For parameters $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(0,1)$, the phase boundaries cross at a single point SVPM. In the SVPM state. We observe that along this path, we have two topological phases. Further increasing $\Delta _{z}$, the magnitude of all four Dirac masses increases, and the GFM becomes a regular BI. The SVPM phase separates a QSHI phase from BI one with Chern numbers $(\mathcal {C},\mathcal {C}_{s})$ changing as ($0, 1$) $\rightarrow$ ($1, 1/2$) $\rightarrow$ ($0, 0$).
Fig. 3. 2D Phase diagram of the staggered 2D materials in the $elE_{z}/\Delta _{so}$ and $\Lambda _{\omega }/\Delta _{so}$ plane. The electronic phases, Chern number, and spin-Chern number are also indicated. The dashed lines represent the paths used in this work to explore this diagram.
An alternative approach to understanding the phase diagram of the Fig. 3 is to consider the presence of circularly polarized light, i.e. turning on the off-resonant optical field. By fixing $\Delta _{z}=0$ and increasing $\Lambda _{\omega }$, the system goes through a topological quantum phase transition from the QSHI phase to an SPM phase at the critical value $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(1,0)$, where the energy gap of one of the spins closes. Further increasing $\Lambda _{\omega }/\Delta _{so}$ beyond the SPM, the AQHI phase is reached. If we fix the circularly polarized light and modulate the staggering potential correspondingly, the phase diagrams are maintained. For example, we consider that $\Lambda _{\omega }/\Delta _{so}=0.5$ and modulate $\Delta _{z}/\Delta _{so}$ as expressed by path $2$. Here, we can see two topological quantum phase transitions. For example, at the first phase transition point $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(0.5,0.5)$, the system transitions from the QSHI to the PS-QHI phase, which is the combination of the AQHI and QSHI phases. We can see the second topological quantum phase transition at $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(1.5,0.5)$. In this way, the first quadrant of the phase diagram of Fig. 3 can be fully explored by paths $3$, $4$, and $5$ in different topological phases.
2.3 Optical response of the graphene family materials
Let us first look into the standard Kubo formalism [35–37] to calculate the dynamical optical response for the GFMs by taking into account its dependence on frequency and mass gap. The dynamical optical conductivity components $\sigma _{ij}$ consists of intra-band (Drude) and inter-band conductivity. Within the linear response regime, the spin and valley polarized 2D conductivity tensor of the GFMs is given by the sum of the intra-band and inter-band terms as
Here $\sigma _{xx}$ and $\sigma _{xy}$ are the longitudinal and transverse Hall conductivities of the GFMs. At $T=0$ K, the explicit analytical expressions of the conductivities are given by [34,35]
Lets spell out the nomenclature here. $\Theta (2\mu _{F}-|\Delta _{\xi }^{s,\gamma }|)$ is the Heaviside function which ensure that transitions across the Fermi level are possible and $\mu _{F}$ is the chemical potential. Here, $\tilde {\sigma }_{xx}^{\xi,s}(\Delta _{\xi }^{s,\gamma },\omega )=\tilde {\sigma }_{yy}^{\xi,s}(\Omega )$, $\tilde {\sigma }_{yx}^{\xi,s}(\Delta _{\xi }^{s,\gamma },\omega )=-\tilde {\sigma }_{xy}^{\xi,s}(\Delta _{\xi }^{s,\gamma },\omega )$, $\sigma _{0}=e^2/4\hbar$, $\Omega =-i\omega +\Gamma$, $\Gamma$ is the scattering due to impurity which is $\Gamma =\tau ^{-1}$ accounts for the relaxation time and $M=\max (|\Delta _{\xi }^{s,\gamma }|,2|\mu _{F}|)$.
In Figs. 4(a) and (b), we show the real part of the longitudinal and the transverse dynamical conductivities in the aforementioned phases of the GFMs in the $\Delta _{z}/\Delta _{so}$ and $\Lambda _{\omega }/\Delta _{so}$ plane. The parameters we use for these simulations are $\Gamma$=0.002$\Delta _{so}$ and the chemical potential $\mu _{F}$ =$0.1\Delta _{so}$. The phase diagram of longitudinal and transverse Hall conductivities are illustrated in Figs. 4(a) and (b), respectively. We observe that both conductivities present signatures of the topological quantum phase transitions taking place in the GFM. Due to the valley and spin-resolved nature of the conductivities, the different Chern numbers are also indicated. To fully explore the phase diagram in Figs. 4(a) and (b), we have shown the real part of the longitudinal and transverse conductivity as a function of the incident normalized photonic energy for different topological phases in Figs. 4(c) and (d). In Fig. 4(c), we capture some representative phases of the topological phase diagram. For parameters $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(0,0)$, there is a single jump in the conductivity at the excitation frequency $\hbar \omega /\Delta _{so}=1$. Table 1 summarizes different transitions excitation frequencies in different phases. As we increase the strength of the off-resonant optical field, e.g., $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(0.5,0)$, the single feature splits into two jumps. The silicenic system still behaves as QSHI. For a stronger optical field $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(1,0)$ the system makes transitions from the QSHI phase to the SPM state as shown in Fig. 3, by path 3.
Fig. 4. Phase diagram of (a) longitudinal conductivity, (b) transverse Hall conductivity, for staggered monolayers in the $E_{z}/\Delta _{so}$ and $\Lambda _{\omega }/\Delta _{so}$ plane. The electronic phases, Chern number and spin-Chern number are also indicated. The solid line represents the spin up ($\uparrow$) and the dashed line represents the spin down ($\downarrow$). (c) Longitudinal conductivity, and (d) transverse Hall conductivity as a function of incident photonic energy. (e) Modulus of $r_{ss}$ and (f) $r_{pp}$ as a function of incident photonic energy. The parameters used for these simulation unless otherwise specified are, $\Gamma =0.002\Delta _{so}$, $\mu _{F}=0.1\Delta _{so}$, $\omega =0.2\Delta _{so}$ and $\Delta _{so}=3.9$ meV.
Table 1. Allowed transitions in different topological phases.
In the SPM state, we can see one resonant jump and this behavior of silicene is identical to graphene [38]. The SPM state separates two topological phases as shown in the topological phase diagram. For parameters $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(0.5,0.5)$, the silicenic system is in the SDC state and the jumps in the conductivity occur at $\hbar \omega /\Delta _{so}=1$ and $\hbar \omega /\Delta _{so}=2$. In non-metal phases with parameters $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(1,1)$, the GFM is in the PS-QHI state and again we can see two jumps in the conductivity at transition frequencies $\hbar \omega /\Delta _{so}=1$ and $\hbar \omega /\Delta _{so}=3$. Lastly, to analyze the TQPT between the AQHI and PS-QHI phases, we tune the parameters as $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(1.5,0)$. Now the optical transitions occur at lower resonant frequencies as shown in table 1. We show the real part of the Hall conductivity as a function of the dimensionless photonic energy in Fig. 4(d), in distinct topological regimes. In the QSHI phase, we can see the two sharp features in the conductivity spectra. The first positive feature occurs at lower photonic energy and is associated with the excitation of the spin-up electron while the negative one is due to the excitation of the spin-down electrons. In other phases, the Hall conductivity is always negative for both spin-up and spin-down electrons.
2.4 Polarization re-orientation effects in the graphene family materials
The GFM is placed on a dielectric media in the $xy$ plane at $z = 0$, while the $xz$ plane is the plane of incidence for the electromagnetic light beam as shown in Fig. 5. The relative permittivity and permeability of the medium for the incident (reflected) light is denoted by $\epsilon _{i}$ ($\epsilon _{r}$) and $\mu _{i}$ ($\mu _{r}$), respectively. The off-resonant circularly polarized light beam comes with the angle of incidence $\theta _{\psi }$ and is refracted by angle $\theta _{\chi }$. The $s$ and $p$ polarized Fresnel’s reflection coefficients are determined by the ratio of reflected and incident amplitudes [35]:
Fresnel’s reflection coefficients of a 2D quantum material sheet in terms of longitudinal and transverse optical conductivities are given by [35–37]:
Here, $Z_{i}=1/\sqrt {\epsilon _{i}}$. For the convenience of calculation of reflection coefficients, the coordinate transformation is performed for the conductivities, which are given as
where $\varphi$ is the angle between $x$ axis and the crystal axis. The Fresnel’s reflection coefficients as a function of normalized photonic energy are shown in Figs. 4(e) and (f), for distinct topological phases. From Eqs. (15)–(17), it is obvious that the optical conductivities have an impact on Fresnel’s reflection coefficients. Figures 4(e) and (f) present the $s$ and $p$ polarized Fresnel coefficients spectra with respect to the normalized photonic energy for different phases. The reflection peaks are clearly observed at the excitation photonic energies.2.5 Phenomenological description of the topological Kerr effects
Optical signatures can be an important and convenient experimental indicator for probing the topological regimes of such a semiconductor material. In this article, we exemplify the correlation between the Kerr rotation and the Chern numbers. This means that the Kerr effect becomes a topological signature. Kerr rotations are based on the Fresnel coefficients which are derived from optical conductivities. Their detailed derivation can be found elsewhere [39]. The Kerr and Faraday rotation geometries are shown in Fig. 6. The Fresnel coefficients help to determine the Kerr rotations $\Theta ^{\textrm {K},\textrm {K'},\uparrow,\downarrow }_{K,\textrm {s}(\textrm {p})}$ and ellipticity $\eta ^{\textrm {K},\textrm {K'},\uparrow,\downarrow }_{K,\textrm {s}(\textrm {p})}$. For incident $s$ and $p$ polarization, the Kerr rotation and ellipticity are computed using the expressions [25,39]
To confirm the topological quantum phase transitions in the GFMs, the $s$ and $p$ polarized Kerr rotation and ellipticity are plotted in Fig. 7. It is shown that the Kerr rotation and ellipticity are very sensitive to the Chern numbers. The magnitude of Kerr rotation and ellipticity in the SPM, and SVPM phases presents a sharp jump when a topological quantum phase transition occurs. The same trend in the magnitude of Kerr rotation and ellipticity can be seen in the SDC state, which is only one closed gap with a linear dispersion in $K$ or $K'$ valleys. The impact on the Kerr signal in terms of electric field and the off-resonant circularly polarized light is clear. The GFM energy dispersion is strongly dependent on these stimuli, as given by Eq. (8). The primary role of the off-resonant laser field tuning is that it controls the band structure of the 2D semiconductor materials and is responsible for spin and valley polarized responses. It also shifts the position of the singularities (excitation energies) and modifies the amount of rotation. As we increase the strength of the applied electric field and laser field, the excitation resonant frequencies also increase with a concomitant increase in the magnitude of the Kerr rotation angle. The magnitude of the Kerr angles and ellipticities for the $s$ polarized light is very small in distinct topological phases as shown in Fig. 7(a) and (c). Extremely large Kerr angles and ellipticities appear for different resonant excitations for the $p$ polarized off-resonant light beam. For example, the magnitude of the maximum $p$ polarized Kerr angles in the QSHI and BI regimes is $\approx \pm 6^\circ$ as shown in Fig. 7(b). In the AQHI regime, we report $\Theta _{K,p}^{K, K',\uparrow,\downarrow }\approx \pm 23^{\circ }$ and $\eta _{K,p}^{K, K',\uparrow,\downarrow }\approx \pm 30^{\circ }$. The incident angle for these simulations is $\theta _{\psi }=55^{\circ }$.
Fig. 7. Density plots of the (a) $s$ and (b) $p$ polarized Kerr rotation in $\Delta _{z}/\Delta _{so}$ and $\Lambda _{\omega }/\Delta _{so}$ plane. Density plots of the (c) $s$ and (d) $p$ polarized ellipticity in $\Delta _{z}/\Delta _{so}$ and $\Lambda _{\omega }/\Delta _{so}$ plane. Heavy pink lines represent phase boundaries indexed by $K$ and $K'$ and spin up $\uparrow$ and spin down $\downarrow$. Parameters are $\Gamma =0.002\Delta _{so}$, $\mu _{F}=0.1\Delta _{so}$, $\omega =0.2\Delta _{so}$, $\theta _{\psi }=55^\circ$ and $\Delta _{so}=3.9$ meV.
The Kerr rotation angles are sensitive to the incident angle $\theta _{\psi }$. This is because the Fresnel coefficients are strongly dependent on the incidence angle. An increasing incidence angle diminishes the amount of rotation until it disappears at complete grazing, $\theta _{\psi }=\pi /2$. A similar trend can also be seen in the optical response of other 2D materials [27,36]. In Figs. 8(a)–(d), we examine the $p$ polarized Kerr rotations as a function of normalized photonic energy and the incident angle in distinct topological regimes. We can observe that in different phases, we have positive and negative $p$ polarized Kerr rotation. In Fig. 8(a), we show the Kerr rotation in the QSHI phase. According to table 1, for $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(0.5,0)$, we have transitions at $0.5\hbar \omega /\Delta _{so}$ and $1.5\hbar \omega /\Delta _{so}$, respectively. These transitions can be seen in the Kerr rotation spectra. At incident angles, $\theta _{\psi }=22.5^\circ$ and $45^\circ$, the magnitudes of the Kerr rotation angles range between $15^\circ$–$20^\circ$ for both valleys as shown in Fig. 8(a). In the SPM state, we have a giant Kerr rotation peak at $2.0\hbar \omega /\Delta _{so}$ excitation photonic energy as shown in Fig. 8(b), which indicates a giant Kerr rotation of $10^\circ$–$23^\circ$ in this topological phase.
Fig. 8. $p$ polarized Kerr rotation as a function of incident angle and normalized photonic energy in (a) the QSHI, (b) SPM, (c) PS-QHI, and (d) AQHI phases. Parameters are the same as in Fig. 7.
Now, we want to investigate the Kerr rotation of topological GFMs in nonmetal phases. Figure 8(c) demonstrates the Kerr rotation peaks when the parameters $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(1.0,1.0)$, i.e, when the system is in the PS-QHI phase. In this phase, the optical resonant transitions occur at $1.0\hbar \omega /\Delta _{so}$ and $3.0\hbar \omega /\Delta _{so}$ photonic energies, respectively. From Fig. 8(c), one can see that the Kerr rotation is $\approx 15^\circ$, when the GFM is in a non-metallic state. Lastly, we analyze the Kerr rotation in the AQHI phase with parameters $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(1.5,0.5)$. The resonant excitation energies are clear in Fig. 8(d) in the AQHI phase. Our numerical simulation shows that we have a giant Kerr rotation of $\approx 23^\circ$ when the GFM is in the non-metal phase.
3. Conclusion
In summary, we have theoretically investigated the topological quantum phase transitions in the Kerr effect by impinging a Gaussian beam on the surface of a GFM. The Kerr rotation is modulated by off-resonant circularly polarized optical and electric fields in different topological regimes and is analyzed in the THz frequency range. We found that the magnitude of the maximum valley and spin-polarized Kerr angles and ellipticity are $\approx 23^{\circ }$ and $\approx 30^{\circ }$, respectively. We also observe that if we change the polarization of the incident light or switch from one valley to another, the Kerr rotation invert. We demonstrated that in addition to the SVPM phase, the GFM exhibits large Kerr rotation in the AQHI phase. We believe that the transitional Kerr effect provides an extraordinary routeway to determine the Berry curvature, Chern numbers, and topological quantum phase transition by direct optical measurement and to further investigate spintronics, spinoptics, valleytronics, and next-generation electronic devices. Our treatment of the Kerr rotations can be easily extended to other 2D quantum materials such as transition metal dichalcogenides (TMDs), 2D topological insulators, etc., with Hall conductivities.
Funding
Higher Education Commission, Pakistan (10375).
Acknowledgement
We thank Dr. Ali Akbar for discussions. The author would like to acknowledge financial support from the National Research pro-gram for Universities (NRPU), scheme number 10375 funded by the Higher Education Commission of Pakistan.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014). [CrossRef]
2. M. Segev and M. A. Bandres, “Topological photonics: Where do we go from here?” Nanophotonics 10(1), 425–434 (2020). [CrossRef]
3. M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010). [CrossRef]
4. X. L. Qi and S. C Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011). [CrossRef]
5. Y. Ando, “Topological insulator materials,” J. Phys. Soc. Jpn. 82(10), 102001 (2013). [CrossRef]
6. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012). [CrossRef]
7. W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015). [CrossRef]
8. H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, “Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface,” Nat. Phys. 5(6), 438–442 (2009). [CrossRef]
9. C. L. Kane and E. J. Mele, “Evidence for Dirac Fermions in a Honeycomb Lattice Based on Silicon,” Phys. Rev. Lett. 95(22), 226801 (2005). [CrossRef]
10. A. Castellanos-Gomez, “Why all the fuss about 2D semiconductors?” Nat. Photonics 10(4), 202–204 (2016). [CrossRef]
11. J. Zhao, H. Liu, Z. Yu, R. Quhe, S. Zhou, Y. Wang, C. C. Liu, H. Zhong, N. Han, J. Lu, Y. Yao, and K. Wu, “Rise of silicene: A competitive 2D material,” Prog. Mater. Sci. 83, 24–151 (2016). [CrossRef]
12. C. Grazianetti, E. Cinquanta, and A. Molle, “Two-dimensional silicon: the advent of silicene,” 2D Mater. 3(1), 012001 (2016). [CrossRef]
13. M. Ezawa, “Spin-valley optical selection rule and strong circular dichroism in silicene,” Phys. Rev. B 86(16), 161407 (2012). [CrossRef]
14. S. Sattar, N. Singh, and U. Schwingenschlögl, “Proximity-induced spin-valley polarization in silicene or germanene on F-doped WS2,” Phys. Rev. B 94(20), 205415 (2016). [CrossRef]
15. P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet, and G. Le Lay, “Silicene: Compelling Experimental Evidence for Graphenelike Two-Dimensional Silicon,” Phys. Rev. Lett. 108(15), 155501 (2012). [CrossRef]
16. 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]
17. M. Ezawa, “Photoinduced Topological Phase Transition and a Single Dirac-Cone State in Silicene,” Phys. Rev. Lett. 110(2), 026603 (2013). [CrossRef]
18. D. Vanderbilt, “Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators,” (Cambridge University Press, Cambridge, UK, 2018).
19. N. R. Cooper, J. Dalibard, and I. B. Spielman, “Topological bands for ultracold atoms,” Rev. Mod. Phys. 91(1), 015005 (2019). [CrossRef]
20. 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(1), 48–51 (2011). [CrossRef]
21. 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]
22. D. L. Sounas and C. Caloz, “Electromagnetic nonreciprocity and gyrotropy of graphene,” Appl. Phys. Lett. 98(2), 021911 (2011). [CrossRef]
23. W. K. Tse and A. H. MacDonald, “Giant magneto-optical Kerr effect and universal Faraday effect in thin-film topological insulators,” Phys. Rev. Lett. 105(5), 057401 (2010). [CrossRef]
24. W. K. Tse and A. H. MacDonald, “Magneto-optical Faraday and Kerr effects in topological insulator films and in other layered quantized Hall systems,” Phys. Rev. B 84(20), 205327 (2011). [CrossRef]
25. M. Shah and M. S. Anwar, “Magneto-optical effects in the Landau level manifold of 2D lattices with spin-orbit interaction,” Opt. Express 27(16), 23217–23233 (2019). [CrossRef]
26. A. Kormányos, G. Szechenyi, M. Vigh, and J. Cserti, “Transfer matrix approach for the Kerr and Faraday rotation in layered nanostructures,” J. Phys.: Condens. Matter 28(37), 375802 (2016). [CrossRef]
27. M. Shah, A. Akbar, M. Sajid, and M. S. Anwar, “Transitional Faraday and Kerr effect in hybridized topological insulator thin films,” Opt. Mater. Express 11(2), 525–538 (2021). [CrossRef]
28. C. Liu, H. Jiang, and Y. Yao, “Low-energy effective Hamiltonian involving spin-orbit coupling in silicene and two-dimensional germanium and tin,” Phys. Rev. B 84(19), 195430 (2011). [CrossRef]
29. M. Shah, “Probing topological quantum phase transitions via photonic spin Hall effects in spin-orbit coupled 2D quantum materials,” J. Phys. D: Appl. Phys. 55(10), 105105 (2022). [CrossRef]
30. M. Shah, “Electrically tunable Goos-Hänchen shift in two-dimensional quantum materials,” Opt. Mater. Express 12(2), 421–435 (2022). [CrossRef]
31. T. Kitagawa, T. Oka, A. Brataas, L. Fu, and E. Demler, “Transport properties of nonequilibrium systems under the application of light: Photoinduced quantum Hall insulators without Landau levels,” Phys. Rev. B 84(23), 235108 (2011). [CrossRef]
32. A. López, A. Scholz, B. Santos, and J. Schliemann, “Photoinduced pseudospin effects in silicene beyond the off-resonant condition,” Phys. Rev. B 91(12), 125105 (2015). [CrossRef]
33. M. Ezawa, “Monolayer Topological Insulators: Silicene, Germanene and Stanene,” J. Phys. Soc. Jpn. 84(12), 121003 (2015). [CrossRef]
34. L. Stille, C. J. Tabert, and E. J. Nicol, “Optical signatures of the tunable band gap and valley-spin coupling in silicene,” Phys. Rev. B 86(19), 195405 (2012). [CrossRef]
35. A. Akbar, M. Shah, and M. Shah, “Topological Imbert–Fedorov shifts in silicene,” J. Opt. Soc. Am. B 38, 2736–2741 (2022).
36. M. Shah and M. S. Anwar, “Magneto-optic modulation of lateral and angular shifts in spin-orbit coupled members of the graphene family,” OSA Continuum 3(4), 878–892 (2020). [CrossRef]
37. M. Shah and M. S. Anwar, “Valley and spin polarized quantized spin dependent photonic shifts in topological silicene,” Phys. Lett. A 401, 127342 (2021). [CrossRef]
38. A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009). [CrossRef]
39. A. Akbar, M. W. Khalid, and M. S. Anwar, “Low temperature Voigt effect in the terbium gallium garnet crystal,” Opt. Express 25(24), 30550–30559 (2017). [CrossRef]
