Transitional Goos-H&#x000E4;nchen effect due to the topological phase transitions

Weijie Wu; Wenshuai Zhang; Shizhen Chen; Xiaohui Ling; Weixing Shu; Hailu Luo; Shuangchun Wen; Xiaobo Yin

doi:10.1364/OE.26.023705

1. 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 [3–6]. 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. [15–17]. 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 [19–22], 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 the time-averaged Poynting vector of the evanescent wave is directed along the interface. In fact, the angular and spatial GH effects are caused by the angular gradient of the amplitude and phase of complex reflection coefficients [26,27]. The transitional GH effect manifests as a sharp jump 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.

2. Topological phase transitions

Figure 1 illustrates that a Gaussian beam of frequency ω₀ impinges at an angle θ on a silicene-substrate 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) = (A₀ sin ωt, A₀ cos ωt), where ω ≫ ω₀ is the frequency of circularly polarized light with ω < 0 (ω > 0) specifying left (right) circular polarization and A₀ is the amplitude. The incident beam generating GH effect can be written as ${\tilde{E}}_{i} \propto ({\hat{x}}_{i} f_{p} + {\hat{y}}_{i} f_{s}) exp [- Z_{R} (k_{i x}^{2} + k_{i y}^{2}) / 2 k_{i}]$ [28]. Here $Z_{R} = π w_{0}^{2} / λ$ is the Rayleigh range, k_i is the wave number, w₀ is the beam waist, the vectors x̂, 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 vector f̂ = (f_px̂_i + f_sŷ_i)/(|f_p|² + |f_s|²)^1/2.

Fig. 1 Schematic representation of the wave reflection at a freestanding silicene sheet. The freestanding silicene is subjected to a static electric field E_z and a circularly polarized lasers. The lattice constant and staggering length values are a = 3.86, and ℓ = 0.23, and the effective spin-orbit coupling λ_SO = 3.9meV. For the freestanding silicene, we can choose the refractive index of the substrate as n = 1.

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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]

H_{η} = ℏ ν_{F} (η k_{x} τ_{x} + k_{y} τ_{y}) + λ_{S O} σ_{z} η τ_{z} - ℓ E_{z} τ_{z} - Λ η τ_{z} .

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. ℓ 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_{s}^{η} |

with the Dirac mass

m_{s}^{η} = η s λ_{SO} - ℓ E_{z} - η Λ,

where Λ = (eaA₀ν_F)²ħ/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 $𝒜_{i} (\vec{k}) = i 〈 ψ (\vec{k}) | \frac{\partial}{\partial k_{i}} | ψ (\vec{k}) 〉$ , where ψ(k⃗) is eigenvector of Hamiltonian, and vector 𝒜_i(k⃗) is called the Berry connection. Then we may define the Berry curvature, ℱ(k⃗) = ∂𝒜_y(k⃗)/(∂k_x) − ∂𝒜_x(k⃗)/(∂k_y). And the Chern number is the integral of the Berry curvature ℱ(k⃗) over the first Brillouin zone. Each topological phase is indexed by the Chern number 𝒞 and the Z₂ index. The Z₂ 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, $𝒞_{s}^{η} = η sgn (m_{s}^{η}) / 2$ for the K_η valley. Then, the Chern and spin-Chern numbers are obtained by $𝒞 = \sum_{η = \pm} (𝒞_{+}^{η} + 𝒞_{-}^{η})$ and $𝒞_{s} = \sum_{η = \pm} (𝒞_{+}^{η} - 𝒞_{-}^{η}) / 2$ . Therefore, the topological number is (𝒞, 𝒞_s), as shown in the Fig. 2(d).

Fig. 2 Phase diagram of (a) longitudinal conductivity and (d) transverse Hall conductivity in the (E_z, Λ)/λ_SO plane, and the Chern number and spin-Chern number (𝒞, 𝒞_s) are also indicated. The solid line and the dashed line together outline the phase boundaries indexed by K_η, where the solid line represents the spin up (s = +1) and the dashed line represents the spin down (s = −1). (b) Longitudinal conductivity, (e) transverse Hall conductivity, (c) horizontally polarized reflection coefficient, and (f) cross-polarized reflection coefficient is a function of the photon energy of circularly polarized light in the states of QSHI, PS-QHI, and QAHI. The parameters are Λ = 0λ_SO(blue), 1λ_SO(green), and 2λ_SO(red), E_zl = 0.5λ_SO, ħΓ = 0.002λ_SO, μ = 0.1λ_SO. And the solid (dashed) line represents the real (imaginary) part.

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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]

\begin{array}{l} \frac{{\tilde{σ}}_{x x}^{η s}}{σ_{0} / 2 π} & = \frac{4 μ^{2} - {| m_{s}^{η} |}^{2}}{2 ℏ μ Ω} Θ (2 μ - | m_{s}^{η} |) + (1 - \frac{{| m_{s}^{η} |}^{2}}{ℏ^{2} Ω^{2}}) \tan^{- 1} (\frac{ℏ Ω}{M}) + \frac{{| m_{s}^{η} |}^{2}}{ℏ Ω M}, \\ \frac{{\tilde{σ}}_{x y}^{η s}}{σ_{0} / 2 π} & = \frac{2 η m_{s}^{η}}{ℏ Ω} \tan^{- 1} (\frac{ℏ Ω}{M}) . \end{array}

Here, σ₀ = e²/4ħ,

{\tilde{σ}}_{x x}^{η s} = {\tilde{σ}}_{y y}^{η s}

,

{\tilde{σ}}_{x y}^{η s} = - {\tilde{σ}}_{y x}^{η s}

,

M = max (| m_{s}^{η} |, 2 | μ |)

, T is temperature, μ is doping, and Ω = −iω + Γ, where Γ is the scattering rate.

The reflected amplitudes are obtained by solving Maxwell’s equations and imposing the appropriate boundary conditions [35, 36], that is, E_i + E_r = E_t and H_i + H_r − H_t = J. 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_ij for incoming i-polarized and outgoing j-polarized plane waves, are obtained [37–39],

\begin{array}{l} r_{p p} = \frac{α_{-} β_{+} + σ_{p s} σ_{s p}}{α_{+} β_{+} + σ_{p s} σ_{s p}}, & r_{s s} = \frac{α_{+} β_{-} - σ_{p s} σ_{s p}}{α_{+} β_{+} + σ_{p s} σ_{s p}}, \\ r_{p s} = - \frac{2}{Z_{i}} \frac{σ_{p s}}{α_{+} β_{+} + σ_{p s} σ_{s p}}, & r_{s p} = \frac{2}{Z_{i}} \frac{σ_{s p}}{α_{+} β_{+} + σ_{p s} σ_{s p}} . \end{array}

Here, α₊ = σ_ppk_iz/k_i + k_izk_t/(Z_tk_ik_tz) + 1/Z_i, α₋ = σ_ppk_iz/k_i + k_izk_t/(Z_tk_ik_tz) − 1/Z_i, β₊ = −σ_ssk_i/k_iz − k_ik_tz/(Z_tk_izk_t)−1/Z_i, β₋ = σ_ssk_i/k_iz + k_ik_tz/(Z_tk_izk_t)−1/Z_i,

Z_{i} = 1 / \sqrt{∊_{i}}

, and

Z_{t} = 1 / \sqrt{∊_{t}}

.

From 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 $| m_{s}^{η} | > 2 | μ |$ , we could get $M = | m_{s}^{η} |$ . When $ℏ Ω / | m_{s}^{η} |$ is small enough, the conductivity can be simplified to ${\tilde{σ}}_{x y}^{η s} = 2 σ_{0} η m_{s}^{η} ℏ Ω / (2 π ℏ Ω | m_{s}^{η} |) = e^{2} 𝒞_{s}^{η} / (2 π ℏ)$ . And the Hall conductivity is $σ_{x y} = \sum_{η, s} {\tilde{σ}}_{x y}^{η s} = e^{2} 𝒞 / (2 π ℏ)$ , which is in good agreement with the description of Haldane [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) $ℏ Ω / | m_{s}^{η} |$ 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.

3. 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]

\begin{array}{l} E_{r} \propto exp (i k_{i} z_{r} - \frac{k_{i}}{2} \frac{x_{r}^{2} + y_{r}^{2}}{Z_{R} + i z_{r}}) \\ \times {{\hat{x}}_{r} [f_{p} r_{p p} (1 - \frac{i x_{r}}{Z_{R} + i z_{r}} \frac{\partial ln r_{p p}}{\partial θ_{i}}) + f_{s} r_{p s} (1 - \frac{i x_{r}}{Z_{R} + i z_{r}} \frac{\partial ln r_{p s}}{\partial θ_{i}})] \\ + {\hat{y}}_{r} [f_{s} r_{s s} (1 - \frac{i x_{r}}{Z_{R} + i z_{r}} \frac{\partial ln r_{s s}}{\partial θ_{i}}) + f_{p} r_{s p} (1 - \frac{i x_{r}}{Z_{R} + i z_{r}} \frac{\partial ln r_{p s}}{\partial θ_{i}})]}, \end{array}

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,

D_{GH} = \frac{\int \int x_{r} I (x_{r}, y_{r}, z_{r}) d x_{r} d y_{r}}{\int \int I (x_{r}, y_{r}, z_{r}) d x_{r} d y_{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) ∝ S̄ · ẑ_r. Then the Poynting vector related to the electromagnetic field can be obtained by

\bar{S} \propto Re (E_{r} \times H_{r}^{*})

.

After substituting Eq. (5) into Eq. (6), we get the following expression

D_{GH} = \frac{R_{p p}^{2} φ_{p p} + R_{s p}^{2} φ_{s p}}{k_{i} (R_{s p}^{2} + R_{p p}^{2})} - z_{r} \frac{R_{p p}^{2} ρ_{p p} + R_{s p}^{2} ρ_{s p}}{k_{i} (R_{s p}^{2} + R_{p p}^{2}) Z_{R}} .

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

Δ_{GH}^{P} = \frac{R_{p p}^{2} φ_{p p} + R_{s p}^{2} φ_{s p}}{k_{i} (R_{p p}^{2} + R_{s p}^{2})},

Similarly, if just consider the cross-polarized component, we obtain

Δ_{GH}^{SP} = \frac{1}{k_{i}} φ_{s p} .

If s-polarization is considered, only the positions of s and p need to be interchanged. We have

Δ_{GH}^{S} = \frac{R_{s s}^{2} φ_{s s} + R_{p s}^{2} φ_{p s}}{k_{i} (R_{s s}^{2} + R_{p s}^{2})},

Δ_{GH}^{PS} = \frac{1}{k_{i}} φ_{p s} .

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)].

Fig. 3 Transitional spatial GH effect in the (E_z, Λ)/λ_SO plane. The spatial shift, which is generated by (a) horizontally and (b) vertically polarized incident light, is a function of external electric field and coupling constants on the surface of silicene. The wavelength of incident light is λ_i = hc/E, incident angle θ_i = 60°, ħω = 0.1λ_SO, and the beam waist is chosen as w₀ = 1mm. The (c) horizontally and (d) vertically polarized spatial shifts are modulated.

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We next consider the angular deviations in GH effect. For p-polarization, the expression of angular deviation is obtained as

Θ_{GH}^{P} = - \frac{1}{k_{i} Z_{R}} \frac{R_{p p}^{2} ρ_{p p} + R_{s p}^{2} ρ_{s p}}{R_{s p}^{2} + R_{p p}^{2}} .

If we block the primary polarization with only the cross-polarized component being taken into consideration, we get

Θ_{GH}^{SP} = - \frac{1}{k_{i} Z_{R}} ρ_{s p} .

If s-polarization is considered, only the positions of s and p need to be interchanged. So, we have

Θ_{GH}^{S} = - \frac{1}{k_{i} Z_{R}} \frac{R_{s s}^{2} ρ_{s s} + R_{p s}^{2} ρ_{p s}}{R_{p s}^{2} + R_{s s}^{2}},

Θ_{GH}^{PS} = - \frac{1}{k_{i} Z_{R}} ρ_{p s} .

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.

Fig. 4 Transitional angular GH effect in the (E_z, Λ)/λ_SO plane. The angular deviation, which is generated by (a) horizontally and (b) vertically polarized incident light, is a function of external electric field and coupling constants on the surface of silicene. The (c) horizontally and (d) vertically polarized angular deviations are modulated. The parameters are the same as in Fig. 3.

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4. 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.

Funding

National Natural Science Foundation of China (NSFC) (61835004); National Natural Science Foundation of China (NSFC) (11474089).

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Transitional Goos-Hänchen effect due to the topological phase transitions

Abstract

1. Introduction

2. Topological phase transitions

3. Transitional Goos-Hänchen effect

4. Conclusion

Funding

References

Cited By

Figures (4)

Equations (15)

Optics Express