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Abstract
We theoretically investigate the tunable Goos-Hänchen (GH) shifts in silicene subjected to an external electric field and circularly polarized light. The prominent feature of these 2D quantum materials is the tunable bandgap that can be tuned by an external electric field or by irradiating circular polarized light beam. Using angular spectrum analysis, we obtain the analytical expressions for the spin and valley polarized spatial and angular GH shifts. We find that tuneable giant spatial and angular GH shifts exhibit extreme values near Brewster’s angles and away from the optical transition frequencies in the silicene. We demonstrate that both positive and negative giant GH shifts can be achieved in the graphene family by tuning the electric field and circularly polarized light in distinct topological regimes. Due to the topological properties of these materials, the GH shift is sensitive to the coupled spin and valley indices of the Dirac fermions as well as to the number of closed gaps. We further demonstrated that topology and spin-orbit interactions play a crucial role in beam shifts and topological quantum phase transitions of the silicene can be comprehensively and efficiently probed through GH shift at the nanoscale.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Over the last few years, two-dimensional (2D) materials have been paid more and more attention, both experimentally and theoretically due to their physical, electrical, optical, chemical, and topological properties. Among these fascinating quantum materials, one of the most celebrated 2D materials is single-layer graphene, also called a “wonder material” [1,2]. Graphene possesses extraordinary high mechanical strength, giant carrier mobility, extremely high thermal and electrical conductivities [3]. These exotic properties enable graphene to have a wide range of potential applications in electronic, spintronic [4], valleytronic [5], spin-optronic [6], optoelectronic [6], and plasmonic [7,8] devices. Graphene has triggered numerous research interests on other layered 2D quantum materials. For example, the 2D staggered materials [9, 10,11] have been identified as semiconducting 2D quantum materials having unique and exceptional topological properties [10,11]. These materials include silicene [12], germanene [13], stanene [14], and plumbene [15].
Just like graphene, these staggered 2D materials possess stable honeycomb lattice structures and outstanding electronic properties [16–18]. Unlike planar graphene, these materials have low-buckled structures and therefore have a large tunable bandgap. These materials exhibit strong spin-orbit coupling (SOC) in a bandgap near the Dirac points which provides a mass to the Dirac fermions. According to tight-binding methods, the values of the SOC ($\Delta _{so}$) in silicene [19], germanene [13] and tinene [20], are 1.55–7.9 meV, 24–93 meV, and 100 meV respectively. Using a perpendicular static external electric field ($\Delta _{z}$) and circularly polarized light ($\Lambda _{\omega }$) provides us with a tunable bandgap for each spin and valley and keep the Dirac mass controllable at the $K$ and $K'$ valleys of the staggered 2D materials. By tuning the staggered sublattice potential/laser field, the graphene family materials reveal different topological quantum phase transitions (TQPTs) [21]. For example, when $\Delta _{z}<\Delta _{so}$, then Dirac fermions in these materials are in the quantum spin Hall insulator phase but when $\Delta _{z}>\Delta _{so}$ a transition to the BI phase occurs.
Topological silicene, germanene, and stanene are key materials for the field of valleytronics [5]. Valleytronics offers an additional degree of freedom (DOF), valley pseudospin, to store and carry information. The valley states can also be utilized to encode and process information in the form of the qubit. [5,22]. In these 2D semiconductors materials valley degree of freedom for electrons arose from the honeycomb lattice structures. These materials provide an easily accessible playground for valleytronics and offer the possibility to realize novel tuneable devices [23,24].
Goos-Hanchen (GH) shift is a special phenomenon that occurs when a finite beam of light bouncing an interface undergoes longitudinal (parallel to the plane) shifts shift due to diffractive corrections under the condition of total internal reflection. The spatial GH was first confirmed experimentally by Goos and Hanchen in 1947 [25] and theoretically explained by Artmann using stationary phase method approach [26]. During the last decades a large number of theoretical and experimental works have been devoted to the study of the GH shift in many different systems, e.g. photonics [27,28], plasmonics [29,30], chiral materials [32], metamaterials [33, 34] and quantum systems [35]. The potential applications of GH shift are biosensor [30], optical measurement [30,31] and optical heterodyne sensors [36]. Meanwhile, these beam shifts have been investigated for different kinds of materials. Recently, the lateral and transverse beam shifts have been theoretically studied in the graphene-substrate system in the presence of an external magnetic field [37,38]. In addition, the GH shift on the surface of Weyl semimetals has also been predicted [41]. In the presence of a magnetic field, the mechanical steering of beams shift has been theoretically investigated in spin-orbit rich, staggered 2D materials [42,44,45]. The surface state-dependent GH has been investigated in a hybridized topological insulator thin film-substrate system by directing a circularly polarized beam with a Gaussian spatial profile on the surface [46, 47].
Motivated by the advances in topological photonics and taking advantage of the interplay between topology, topological quantum phase transitions, SOC, and Dirac physics of the graphene family materials, we attempt to explore the spin and valley polarized beam shifts on the surface of a staggered monolayer substrate system in different phases. The GH shift of the graphene family-substrate system can be tuned by an externally applied electric field as well as circularly polarized laser field. By numerical simulation, the GH shift of the graphene family in different phases is discussed. Giant positive and negative spatial spin and valley-dependent shifts can be acquired in the vicinity of the Brewster angle. We observe that the extreme values of GH shift appear away from the resonant optical transitions. Electric and laser fields modulated GH shift in different materials have potential applications in metrology and quantum information processing and for developing novel optoelectronic, valleytronic, and spintronic devices.
2. Model and theory
The low energy Hamiltonian for the graphene family materials in the presence of perpendicular electric field and circularly polarized laser field takes the form [42,48],
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. $\Delta _{so}$ is the intrinsic SOC, whereas, $\Lambda _{\omega }$ being the laser field and $\gamma =\pm 1$ sign corresponds to left and right circular polarization, respectively. Numerical diagonalization of Eq. (8) leads to the electronic bands for Dirac electron
Here, $t=\pm 1$ denotes the electron/hole band, and $\Delta _{\xi }^{s,\gamma }=\xi \Delta _{so}\hat {s}_{z}-elE_{z}-\gamma \Delta _{\omega }$ is the Dirac mass term. The Dirac mass in Eq. (2) depends on the strength of the SOC, 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 phases and is determined by the Dirac mass [21]. Here $\mathcal {C}$ is the total Chern number, $\mathcal {C}_{s}$ is the spin Chern number. The Chern numbers are insensitive to a smooth deformation of the band structure provided the bandgap is open as shown in Fig. 1. On the other hand, the Chern number changes its sign when the Dirac mass $\Delta _{\xi }^{s,\gamma }$ changes its sign [21]. For the staggered 2D materials, there exist a symmetry between the valley $K(K')$ with spin-up (down) and $K(K')$ with spin-down (up). In Fig. 1, we have shown the dispersion of the energy bands for both spin states in the $K$ valley for the graphene family material. Here, only the $K$ valley is depicted, for the $K'$ valley the spin labels only switch. The band structure evolution with electric potential $\Delta _{z}$ in the absence of $\Lambda _{\omega }$ has been shown in Fig. 1. Similar features can be shown by tuning the $\Lambda _{\omega }$ which plays an identical rule through the relationship. The tuning of $E_{z}$ (or $\Lambda _{\omega }$) achieves a TQPT.
When $\Delta _{z}=0$, the bands are spin degenerate and are separated by an insulating gap of $2\Delta _{so}$, which makes it a quantum spin Hall insulator (QSHI) as depicted in Fig. 1(a). As long as $\Delta _{z}<\Delta _{so}$, the silicenic system remains at the QSHI regime with a spin splitting and two energy gaps. For the particular case when $\Delta _{z}=0.5 \Delta _{so}$, the possible topological number is $\left (0, 1\right )$ which is non-trivial band structure. Each spin state in this phase gives rise to an independent Dirac energy gap as illustrated in Fig. 1 (b), whereas for $\Delta _{z}=0$ the spin bands overlap, as shown in Fig. 1 (a). When $\Delta _{z}$ is increased to $\Delta _{z}=\Delta _{so}$, the lower bandgap of the closes and the system hits the valley-spin polarized metal (VSPM) state. In this phase, the CN is ($1,1/2$) and the corresponding band structure is shown in Fig. 1 (c). For an even higher electric potential $\Delta _{z}>\Delta _{so}$, the system exhibits a transition from the VSPM to the band insulator (BI) phase with CN ($0,0$), and both gaps increase with $\Delta _{z}$, although a band inversion has now occurred as shown in Fig. 1(d).
Fig. 1. The electronic energy dispersion of monolayer silicene for the $K$ valley corresponding to three different topological phases: (a) QSHI (TI) ($\Delta _{z}=0$), (b) QSHI (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 (red) curves are for spin up (down).
3. Optical conductivities and Fresnel coefficients
The dynamical optical conductivity components $\sigma _{ij}$ of the silicene consists of intra-band (Drude) and inter-band conductivity for each spin and valley. Using the standard Kubo formalism, the total optical conductivity tensor of the graphene family material is obtained by adding the intra-band and interband conductivity for all spin and valley and is given by
Here $\sigma _{xx}$ and $\sigma _{xy}$ are the longitudinal and transverse Hall conductivities of the graphene family. At $T=0$ K these conductivities are given by [48,49]
We start our analysis by considering that the air-silicene monolayer-substrate interface to be in the $x,y$ plane of the laboratory Cartesian frame at $(z=0)$ as shown in Fig. 2. A static electric field is turned on along the $z$ direction which is perpendicular to a 2D material substrate. The coordinates of the incident and reflected beams are $(x_i,y_i,z_i)$ and $(x_r,y_r,z_r)$, respectively. Further, we consider that a monochromatic and well collimated Gaussian beam of light of finite width, with frequency $\omega$ and wave vector $k_{i}$ is incident on a 2D material-substrate at an incident angle $\theta _{\psi }$ from the air. The reflected amplitudes of the reflected beam can be obtained by solving Maxwell’s equations and imposing the appropriate boundary conditions, that is, $\mathbf {E}_{i}+\mathbf {E}_{r}=\mathbf {E}_{t}$ and $\mathbf {H}_{i}+\mathbf {H}_{r}-\mathbf {H}_{t}=\mathbf {J} .$ [37,38,39, 40]:
The $s$ and $p$ polarized Fresnel’s reflection coefficients are determined by the ratio of reflected and incident amplitudes:
Fresnel’s reflection coefficients of a 2D silicenic material sheet in terms of longitudinal and transverse optical conductivities are given by [38,43,44]:
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.Fig. 2. Schematic of the spatial and angular GH shifts on the surface of the graphene family materials in the presence of the electric and laser fields.
4. Goos-Hänchen shift
To illustrate the GH shifts in a general beam propagation model on the surface of silicene, we can express the incident Gaussian beam in the angular spectrum representation as
where $\Phi _{0}=\exp [-\frac {w_{0}^{2}(k_{ix}^{2}+k_{iy}^{2})}{4}]$ with the beam waist $w_{0}$. The unit complex vector $\hat {f}=(\hat {x}_{i}f_p+\hat {y}_{i}f_s)$ represents the polarization of the incident beam. The amplitudes of the incident and reflected angular spectrum are related as $\tilde {E_{r}}(k_{rx},k_{ry})=M_{R}\tilde {E_{i}}(k_{xi},k_{ri})$ which can be expressed explicitly asThe angular spectrum of the reflected field can be written as
Upon reflection, the complex amplitude for the reflected beam using the Fourier transformations, can be expressed as
For well collimated paraxial beams the transverse wavenumbers $k_{x}^{2}+k_{y}^{2}\ll 1$, so using paraxial approximation, we can write $k_{rz}=\sqrt {k_{r}^{2}-(k_{rx}^{2}+k_{ry}^{2})}=k_{r}-(k_{rx}^{2}+k_{ry}^{2})/2k_{r}$. Using the above relation Eq. (25), can be written as
In position space the general expression for the reflected angular spectrum is obtained as
The next task is to find out the relationship for the Goos-Hanchen shift in staggered 2D monolayer materials. The intensity distribution of the reflected beam $I(x_{r},y_{r},z_{r})$ is related to the longitudinal momentum current or another words to the Poynting vector. The time-averaged linear-momentum density can be written as [44]
Here the magnetic field is given by $\textrm {H}_{r}=-ik_{r}^{-1}\nabla \times \textrm {E}_{r}$. If one ignore the irrelevant proportionality factor, the intensity is directly proportional to the absolute square of the reflected angular spectrum $I(x_{r},y_{r},z_{r})\propto |E_{R}|^{2}$. The longitudinal displacements of the field centroid at any given plane $z_r$=const, is given by
Considering only the horizontal polarization with $a_{p}=1$, $a_{s}=0$ and $\eta =0$. Substituting Eqs. (27) into (29), the following expression is obtained
The superscripts identify the $K$,($K'$) and $\uparrow,(\downarrow )$ and subscripts represents polarization state ($p$).
5. Results and discussion
The phase diagram for the silicene is plotted in Fig. 3. The abbreviation of each topological phase means quantum spin Hall insulator (QSHI), band insulator (BI), polarized spin quantum Hall insulator (PS-QHI) state, and anomalous quantum Hall insulator (AQHI) state. For the sake of simplicity, we consider two different cases. In the first case the circularly polarized light field is absent $\Lambda _{\omega }/\Delta _{so}=0$ as shown by path 1 in Fig. 3. At $\Delta _{z}/\Delta _{so}=\Lambda _{\omega }/\Delta _{so}=0$, the material is characterized as a quantum spin Hall insulator. By increasing the applied electric field the $\Delta _{z}/\Delta _{so}$, the system is still in QSHI regime with Chern number ($\mathcal {C}=0,\mathcal {C}_{s}=1$). At $\Delta _{z}/\Delta _{so}=1$, the silicene is in the SVPM phase and one of the bands is closed and the silicene becomes gapless. The SPVM phase separates a QSHI phase from a BI one as shown by path $1$. In the next case, we turned on the laser field. For example, if we fix $\Lambda _{\omega }/\Delta _{so}=0.5$ and vary $\Delta _{z}/\Delta _{so}=0\rightarrow 1\rightarrow 2$ as represented by the path 2, we can see two topological quantum phase transitions. Along path 2, at $\Delta _{z}/\Delta _{so}=0.5$, the system transitions from the QSHI to the PS-QHI. Another topological phase transition occurs at $\Delta _{z}/\Delta _{so}=1.5$, in which the silicenic system crosses PS-QHI to non-trivial BI state with Chern number ($\mathcal {C}=0,\mathcal {C}_{s}=0$). When $\Lambda _{\omega }/\Delta _{so}=1$, the system is in the SPM phase. If we further increase the strength of the laser field such that $\Lambda _{\omega }/\Delta _{so}=1.5$, the silicene is in the AQHI state. In this way, the phase diagram of Fig. 3 can be fully explored by paths 3, 4, and 5 in different topological phases.
Fig. 3. 2D phase diagram of silicene 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 pink lines represent the paths used in this work to explore this diagram.
In Figs. 4(a)–(d), we show the real parts of the ${\sigma }_{xx} (\Delta _{\xi }^{\sigma,\gamma },\omega )$, ${\sigma }_{xy} (\Delta _{\xi }^{\sigma,\gamma },\omega )$ in the aforementioned phases for staggered monolayers silicene. For these simulations, we use the broadening parameter $\Gamma$=0.002$\Delta _{so}$ and assume that the chemical potential ($\mu _{F}$ =$0.2\Delta _{so}$). The resonant singularities in the longitudinal and Hall conductivities occur when the incoming light photon energy match the excitation energy gap $\hbar \omega$. In Fig. 4(a), 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 $\hbar \omega /\Delta _{so}=1$, as shown in Fig. 4(a). Table 1 summarizes different transitions excitation frequencies in different regimes. As we increase the strength of the laser 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 laser 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. In the SPM state, we can see one resonant jump and this behaviour of silicene is identical to graphene [50]. The SPM state separates two topological phases as shown in the topological phase giagram. 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 nonmetal phases with parameters $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(1,1)$, the silicene 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 topological quantum phase transition 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 I.
Fig. 4. Longitudinal conductivity and transverse Hall conductivity as a function of incident photonic energy in distinct topological regimes. (a) Real part of $\sigma _{xx}$, (b) imaginary part of $\sigma _{xx}$, (c) real part of $\sigma _{xy}$ and (d) imaginary part of $\sigma _{xy}$. The parameters used for these simulation unless otherwise specified are, $\Gamma =0.002\Delta _{so}$, $\mu _{F}=0.1\Delta _{so}$, and $\Delta _{so}=3.9$ meV.
Table 1. Allowed transitions in different topological phases.
Similarly, Figs. 5(a) and (b) illustrate the reflectivity ($R_{pp}$ and $R_{ss}$) as a function of incident angle in distinct topological regimes for the $K$ and $K'$ valleys. From Eqs. (13)–15, it is clear that the Fresnel’s reflection coefficients heavily influenced by optical conductivities. From Figs. 5(c) and (d), it is clear that $R_{pp}$ achieves a minimum value at a certain $\theta _{\psi }$ and rises again. This is called the pseudo-Brewster angle $\theta _{B}=\tan ^{-1}(n_{2}/n_{1})$, whereas $R_{ss}$ increases smoothly as the angle of incidence is increased. The phase $\phi _{pp}$ shows transition from $0$ to $-\pi$ in the vicinity of $\theta _{B}$ for different magnetic fields. A similar variation has been reported in 2D-TMDC [44]. The phase $\phi _{ss}$ shows an increasing trend with $\theta _{\psi }$ in different regimes, as shown in Fig. 5(d).
Fig. 5. Modulus and phase of the $p$ and $s$ polarized reflection coefficients for 2D staggered graphene-substrate system as a function of incident angle in distinct topological regimes (a) $R_{pp}$, (b) $R_{ss}$, (c)$\phi _{pp}$ and (d) $\phi _{pp}$. The parameters used for these simulation unless otherwise specified are, incident angle, $\Gamma =0.002\Delta _{so}$, $\mu _{F}=0.1\Delta _{so}$, $\omega =0.2\Delta _{so}$ and $\Delta _{so}=3.9$ meV.
To investigate the GH shift in the graphene family, we consider the $p$ polarized spatial and angular GH shifts as a function of the incident photonic energy and the incident angle in Figs. 6(a) and (b), in the QSHI regime with parameters $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(0,0)$. We observe a giant negative spatial GH shift in the proximity of Brewster’s angle as shown in Fig. 6(a). If we look at the $\phi _{pp}$ spectra in Fig. 5(c) together with these shifts, we observe that $\phi _{pp}$ shows a change of $-\pi$ resulting in negative spatial GH shifts. One can see the kink at the optical transition frequency $\hbar \omega /\Delta _{so}=1$. In Fig. 6(b), we have shown the spin and valley polarized angular GH shift as a function of the incident photonic energy and the incident angle. The $p$ polarized angular GH shift is positive for $\theta _{\psi }<\theta _{B}$ and negative for $\theta _{\psi }>\theta _{B}$. In Fig. 6(c), we have shown the $p$ polarized GH shift with respect to the incident photonic energy and the incident angle for the QSHI phase with parameters $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(0.5,0)$. According to the table I and Fig. 4(a), here, we can see two kinks at the optical transition frequencies. In the SPM state, $\Delta _{z}/\Delta _{so}=0$ and $\Lambda _{\omega }/\Delta _{so}=1$. The $p$ polarized spatial GH shift in this state is shown in Fig. 6(d). In this state, there is only one optical transition occurs, and the kink in the GH spectra can be seen at $\hbar \omega /\Delta _{so}=2$. Here, the spin and valley polarized spatial shift magnitude is smaller than that of the QSHI regimes near the Brewster angle. It must be noted that at the optical transition frequencies the GH shifts are less negative. In other words, the GH shifts are larger away from the optical transition frequencies.
Fig. 6. The $p$ polarized spatial and angular GH shifts as a function of the incident photonic energy and the incident angle. (a) The $p$ polarized spatial and (b) angular GH shift with parameters $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(0,0)$ in the QSHI phase. (c) The $p$ polarized spatial GH shift in QSHI and (d) SDC states.
Fig. 7(a) demonstrates the spin and valley polarized spatial GH shift as a function of the incident photonic energy and the incident angle in SDC state with parameters $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(0.5,0.5)$. In the SDC state, we have two jumps in the optical conductivity as shown in Fig. 4. Figure 7(a) indicates the giant and negative spatial GH shift near the Brewster angle $\theta _{B}$. The magnitude of the $p$ polarized spatial shift shown in Fig. 7(a) is similar to that in the SPM state. Similarly in Fig. 7(b), we also plot the spatial GH shifts as a function of the incident photonic energy and the incident angle in the PS-QHI regime. Compare to the SPM state, the distance between the optical transition is increased in the PS-QHI regime. We can see the kinks in the GH shift spectra at $\hbar \omega /\Delta _{so}=1$ and $\hbar \omega /\Delta _{so}=3$. Furthermore, in Figs. 7(c) and (d), we have exposed the spin and valley polarized spatial and angular shifts as a function of the incident photonic energy for parameters $(\Lambda _{\omega }/\Delta _{so},\Delta _{z}/\Delta _{so})=(1.5,0)$. One can note that the magnitude of the spatial GH shift decreased at the vicinity of the Brewster angle when the silicenic system is in non-metallic states AQHI. Figure 7(d) shows that the spin and valley polarized angular GH shift decrease very rapidly as the incident angle is increased, near the Brewster angle, the GH shift becomes negative and then increases again. These results are consistent with the previous studies and one can obtain a large spatial and angular shift [44,46].
Fig. 7. The $p$ polarized spatial and angular GH shifts as a function of the incident photonic energy and the incident angle. The $p$ polarized spatial angular GH shift in (a) SPM and (b) PS-QHI regimes. (c) The $p$ polarized spatial GH shifts in AQHI phase. (d) The $p$ polarized angular GH shifts in AQHI phase.
It is instructive, to examine the dependence of the spatial and angular GH shifts on the frequency (energy) of the incident THz beam. In Figs. 8 (a) and (b), we plot the spatial and angular beam shifts as a function of incident photonic energy. In Fig. 8(a), we show the response of spatial GH shift for distinct topological phases, while keeping $\mu _{F}=0.1\Delta _{so}$ and $\theta _{\psi }=54^{\circ }$ (near the Brewster angle). The beam shifts display an oscillating dependence moving gradually to higher frequencies as the excitation energy is increased. This is shown in Fig. 8(a). As the electric and laser fields are changed, rendering the system into various topological regimes, the position, as well as the magnitudes of the beam shifts, change. Our results depict that the spatial GH shifts are showing an oscillating behavior in different regimes. We observe that the resonant spectrum of the spin and valley polarized GH shift can be utilized to realize a multi-channel quantum switch. In Fig. 8(b), the valley and spin-polarized angular GH shift as a function of incident photonic energy is depicted. The amplitude of the $p$ polarized angular GH shift decreases by increasing the laser field and photonic energy.
Fig. 8. The $p$ polarized (a) spatial and (b) angular GH shifts as a function of incident photonic energy for both spins and valleys in distinct topological regimes. The parameters used for these simulation unless otherwise specified are, incident angle $\theta _{\psi }=54^{\circ }$, $\Gamma =0.002\Delta _{so}$, $\mu _{F}=0.1\Delta _{so}$, and $\Delta _{so}=3.9$ meV.
6. Conclusion
In conclusion, we presented a theoretical study on the novel spatial and angular Goos-Hänchen shifts by impinging a Gaussian light beam on the surface of the graphene family materials. We have studied the electric field/ circularly polarized light modulated valley and spin-polarized spatial and angular GH shifts due to the topological quantum phase transition in silicene. We investigated that tuneable giant spatial and angular GH shifts exhibit extreme values near Brewster’s angles and away from the optical transition frequencies in the graphene family materials. We demonstrated that both positive and negative giant spin-valley polarized spatial and angular shifts can be achieved in the graphene family by tuning the electric field and circularly polarized light in distinct topological regimes. Moreover, we observed that the interplay between topological matter and the beam shifts allows us to probe the spin and valley properties of charge carriers throughout different phase transitions. The GH shift in the graphene-family materials can be a useful way to determine the Berry curvature, topological Chern numbers, and topological quantum phase transition by direct optical measurement.
Funding
Higher Education Commission, Pakistan (10375).
Acknowledgements
The author would like to acknowledge financial support from the National Research program for Universities (NRPU), scheme number 10375 funded by the Higher Education Commission of Pakistan.
Disclosures
The author declare that there are no conflicts of interest related to this article.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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