Transitional Goos-Hänchen effect due to the topological phase transitions

We examine the Goos-Hänchen (GH) effect for a Gaussian beam impinging on the surface of silicene whose topological phase transitions can be modulated by external electric field and/or irradiating circular polarized light. It is shown that both the spatial and angular shifts in the GH effect present a sharp jump due to the topological phase transitions. The transitional GH effect can be attributed to transitional optical conductivity, which relates to Berry curvature and Chern numbers. These results can be broadly extended to other two-dimensional atomic crystals in the graphene family. We believe that the transitional GH effect may offer a possible way to determine the Berry curvature, Chern numbers, and topological phase transition by a direct optical measurement. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
The topological phase transition has attracted great attention, not only for its fundamental scientific importance but also for its potential applications [1,2]. Silicene is a monolayer of silicon atoms with buckled sublattices [3][4][5][6]. As a two-dimensional topological insulator, topological phase transitions in silicene can be modulated by external electric field and/or irradiating circular polarized light. Recently reported that graphene-like electronic band structure, supporting charge carriers behaving as massless Dirac fermions [7], exist in the silicene. The strong spin-orbit coupling provides a mass to Dirac electrons and favors an experimentally accessible Kane-Mele type quantum spin Hall effect [8,9]. The interaction of external electric field and/or circularly polarized light with silicene makes Dirac mass controllable, which leads to topological phase transition. However, how to measure the topological phase transition in the experiment is still a challenging problem [10,11].
The Goos-Hänchen (GH) effect [12] have exhibited remarkable potential for optical measurement and sensor [13,14]. Recently, the GH effect on the surface of graphene has been investigated. [15][16][17]. It has been demonstrated that the beam shifts in GH effect are very sensitive to the quantized Hall conductivity of graphene in quantum Hall regime [18]. By incorporating the quantum weak measurement techniques [19][20][21][22], the GH shifts have been applied as pointer to detect the layers of graphene [23]. Therefore, we expect that GH effect would be sensitive to topological phase transitions in silicene. In a simple explanation, the GH effect is a position shift of the beam intensity distribution centroid relative to its geometrical optics position [24,25]. The GH effect is attributed to the penetration of an evanescent field, and on the boundary of topological phase transitions, but this does not occur in the same regime of topological phase.
In this paper, the physical mechanism between different topological phases and transitional GH effect has been revealed through boundary conditions. Firstly, the optical conductivity and Dirac mass are used to depict topological phase diagram. Chern number and spin-Chern number have been calculated. Both optical conductivity and reflection coefficient have a very close relationship with Chern number. By regulating external electric field and/or irradiating circular polarized light, topological phase transition will occur. Then, the transitional GH effect appears. Since the transitional GH effect has a corresponding relationship with different topological states, we believe that this work provides a convenient scheme for directly optical measurement of the different topological features. Figure 1 illustrates that a Gaussian beam of frequency ω 0 impinges at an angle θ on a silicenesubstrate system. A local coordinate system needs to be established to determine the position of the beam split. Since a static electric field E z is applied along the z axis, an electrostatic potential 2lE z among the two inequivalent sublattices of the monolayer can be generated. Furthermore, the properties of silicene can also be modified by irradiating circularly polarized light. The corresponding electromagnetic potential is given by A(t) = (A 0 sin ωt, A 0 cos ωt), where ω ω 0 is the frequency of circularly polarized light with ω < 0 (ω > 0) specifying left (right) circular polarization and A 0 is the amplitude. The incident beam generating GH effect can be written [28]. Here Z R = πw 2 0 /λ is the Rayleigh range, k i is the wave number, w 0 is the beam waist, the vectorsx,ŷ represent the directions of parallel and perpendicular to the incidence plane, respectively. And the polarization of the beam is determined by the complex-valued unit vectorf

Topological phase transitions
Near the Fermi energy, the Rashba terms are ignored since the Rashba couplings is week. And these results still holds under the adiabatic condition [29,30]. Therefore, the effective Dirac Hamiltonian can be written as [10,11] Here σ = (σ x , σ y , σ z ) and τ = (τ x , τ y , τ z ) are the Pauli matrices of the spin and the sublattice pseudospin, respectively. η = ±1 is the valley index, ν F is the Fermi velocity, λ SO is the effective spin-orbit (SO) coupling. is the buckled height, Λ is the coupling constant between the monolayer and the circularly polarized light. The first term is the Hamiltonian for Dirac electrons. The second term is the Kane-Mele spin-orbit coupling term. The third term is related to the perpendicular electric field E z . And the last term is associated with the irradiating circular polarized light. By diagonalizing the Hamiltonian in Eq.(1), the band gap is given by 2|m where Λ = (eaA 0 ν F ) 2 /ω, a is the lattice constant. That is to say, the Dirac mass can be controlled by applying electric field E z and coherent laser beam.
In the momentum space, the gauge potential can be defined as where ψ( ì k) is eigenvector of Hamiltonian, and vector A i ( ì k) is called the Berry connection. Then we may define the Berry curvature, And the Chern number is the integral of the Berry curvature F( ì k) over the first Brillouin zone. Each topological phase is indexed by the Chern number C and the Z 2 index. The Z 2 index is identical to the spin-Chern number C s modulo 2, when the spin s is a good quantum number. It is determined uniquely by the Dirac mass and the valley index, that is, C Therefore, the topological number is (C, C s ), as shown in the Fig. 2(d). Using the Kubo's formula [31,32], the expressions for the longitudinal and transverse Hall conductivities with zero temperature can be written as [33,34] σ ηs xx Here, The reflected amplitudes are obtained by solving Maxwell's equations and imposing the appropriate boundary conditions [35,36], that is, To calculate the reflection of an incoming beams plane-polarized in an arbitrary direction, we decompose the incoming field into its p and s components. Sequentially, the four Fresnel's reflection coefficients, r i j for incoming i-polarized and outgoing j-polarized plane waves, are obtained [37][38][39], Here,  Figs. 2(a) and 2(d), the Dirac mass plays an important role in the topological phase transition. That is, the phase boundaries appear when Dirac mass is zero. The occurrence of the topological phase transition is caused by the change of the sign of Dirac mass. For instance, if Λ is gradually increased from 0 and E z = 0 remains unchanged, the entire system has undergone topological phase transition from quantum spin Hall insulator (QSHI) to spin-polarized metal (SPM) to quantum anomalous Hall insulator (QAHI). In other words, the band gap is also closed from the beginning of the open state and then reopening. In addition, when the topological numbers of the band insulator (BI) is (0, 0), the QSHI being (0, 1), the QAHI being (−2, 0), and the photoinduced polarized-spin quantum Hall insulator (PS-QHI) being (−1, 1/2), these states are insulators with the band gap open. All processes are adiabatically performed. Then, the SPM, spin-valley-polarized metal (SVPM), and single Dirac cone state appear at the intersection of different topological phase, and these gaps is closed.
From Figs. 2(b), 2(c), 2(e) and 2(f), the behavior of longitudinal conductivity, horizontally reflection coefficient, Hall conductivity, and cross-polarized reflection coefficient are illustrated as a function of the photon energy of circularly polarized light in the aforementioned topological phases. From Eq. (4), the optical conductivities have an impact on Fresnel's reflection coefficients and thus influence transitional GH effect. Next, note that cross-polarized reflection coefficient and transverse Hall conductivity are proportional to Chern number at low frequencies and small dissipation. The main reason is that the Hall conductivity can be reduced to be proportional to Chern number.
If |m  [40]. Hence, the Hall conductivity can perfectly describe the topological phase diagram determined by Dirac mass. From Eq. (3), due to the relationship between the longitudinal conductivity and Dirac mass, the boundaries in phase diagram can still be described by longitudinal conductivity, but the different topological states cannot be depicted. In addition, the Hall conductivity is proportional to Chern number, the following two conditions need to be satisfied: (i) the absolute value of Dirac mass is greater than the absolute value of doping; (ii) Ω/|m η s | should be small enough. The Hall conductivity plays a important role in cross-polarized reflection coefficient. Considering the peaks caused by different photon energy, we find that the number of peaks of conductivity and the reflection coefficient is related to the Chern number as shown in Fig. 2.

Transitional Goos-Hänchen effect
To reveal the relation between the transitional GH effect and topological phase transitions, we give a derivation of the GH effect. Considering a incident Gaussian beam, the reflected light field can be obtained based on the boundary condition [18] where f p = a p ∈ R, f s = a s exp(iη). By calculating the centroid of the reflected light field, the general GH effect expression is derived. That is, x r I(x r , y r , z r )dx r dy r ∫ ∫ I(x r , y r , z r )dx r dy r , where I is beam intensity, and (x r , y r , z r ) is the reflected beam's coordinate frame. The beam intensity is closely linked to flux of the time averaged Poynting vector I(x r , y r , z r ) ∝S ·ẑ r . Then the Poynting vector related to the electromagnetic field can be obtained byS ∝ Re(E r × H * r ). After substituting Eq. (5) into Eq. (6), we get the following expression Here, r A = R A exp(iφ A ), A ∈ {pp, ss, ps, sp}, ρ A = Re(∂ ln r A /∂θ i ), and ϕ A = Im(∂ ln r A /∂θ i ).
The first term is the spatial shift, which does not changes with z r . The second term is the angular deviation, and it is z r -dependent [25]. We first consider the spatial shifts in GH effect. For p-polarization we have Similarly, if just consider the cross-polarized component, we obtain If s-polarization is considered, only the positions of s and p need to be interchanged. We have From Eqs. (8)- (11), we find that the spatial shifts are related to the derivation of the phase of the complex Fresnel's reflection coefficients.
To confirms the topological phase transitions in silicene, the spatial shifts are plotted in Fig. 3. It is shown that the the original spatial shifts are very sensitive to the Chern numbers [Figs. 3(a) and 3(b)]. The magnitude of spatial shifts in the states of SPM, SVPM presents a sharp jump when a topological phase transition occurs. And the magnitude of spatial shifts in the single Dirac cone state, which is only one closed gap with a linear dispersion in K or K points, has the same feature. The more clear jumping boundary in spatial shifts can be found if we only consider the beam shifts of the crossed components of reflected field [Figs. 3(c) and 3(d)].
We next consider the angular deviations in GH effect. For p-polarization, the expression of angular deviation is obtained as If we block the primary polarization with only the cross-polarized component being taken into consideration, we get If s-polarization is considered, only the positions of s and p need to be interchanged. So, we have From Eqs. (12)-(15), we find that the angular deviations are related to the derivation of the amplitude of the complex Fresnel's reflection coefficients.
To further confirms the topological phase transitions in silicene, the angular deviations are plotted in Fig. 4. It is shown that the large angular deviations in the states of QSHI and BI are presented, where the Chern number is zero. For some states including QAHI and PS-QHI, the angular deviations are small where the Chern number is non-zero [Figs. 4(a) and 4(b)]. Hence, the GH effect provides a possible scheme to determine the Chern number. Similar to the case of spatial shifts, the clear jumping boundary in angular deviations can be found if we only consider the beam shifts of the crossed components of reflected field [Figs. 4(c) and 4(d)]. These interesting phenomenon is attributed to the Hall conductivity directly related to the topological phase transitions.

Conclusion
In conclusion, we have theoretically shown transitional GH effect due to the topological phase transition. By analyzing the Hamiltonian, we have demonstrated that the topological phase diagram, which is described by GH effect, is consistent with the one described by Dirac mass. The topological number, including Chern number and spin-Chern number, can characterize different topological phases, and they are indicated in topological phase diagram. We have revealed that the transitional GH effect is sensitive to Chern number, topological phase transition. As a consequence, we believe that the experimental detection of topological phase transition by directly optical measurement is possible. Our method can be extended to entire graphene family including silicene, germanene, and stanene. Our results can also be extend to topological photonics [41,42] due to the similar topological root.