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Zhu Guo-Bao, Yang Hui-Min. Transport properties of the topological Kondo insulator SmB6 under the irradiation of light. Chinese Physics B, 2016, 25(10): 107303  
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Transport properties of the topological Kondo insulator SmB6 under the irradiation of light
Zhu Guo-Bao1, 2, Yang Hui-Min1, †,
Department of Physics and Electronic Engineering, Heze University, Heze 274015, China
Institute of Theoretical Physics, Heze University, Heze 274015, China

 

† Corresponding author. E-mail: yangyhm2016@163.com

Project supported by the National Natural Science Foundation of China (Grant Nos. 11504095 and 11447145), the Foundation of Heze University (Grant Nos. XY14B002 and XYPY01), and the Project funded by the Higher Educational Science and Technology Program of Shandong Province, China (Grant No. J15LJ55).

Abstract
Abstract

In this paper, we study transport properties of the X point in the Brillouin zone of the topological Kondo insulator SmB6 under the application of a circularly polarized light. The transport properties at high-frequency regime and low-frequency regime as a function of the ratio (κ) of the Dresselhaus-like and Rashba-like spin–orbit parameter are studied based on the Floquet theory and Boltzmann equation respectively. The sign of Hall conductivity at high-frequency regime can be reversed by the ratio κ and the amplitude of the light. The amplitude of the current can be enhanced by the ratio κ. Our findings provide a way to control the transport properties of the Dirac materials at low-frequency regime.

PACS: 73.43.–f;78.67.–n
Keyword:SmB6;light;Flouqet theory;Boltzmann equation
1. Introduction

In the past few years, topological insulators[1,2] have attracted significant interest in condensed matter physics both in theory and experiment. A weakly interacting system with strong spin–orbit coupling and insulators driven by strong electron correlations can result in a topological insulating state. The narrow gap insulator in which the gap is created by electron correlations are called topological Kondo insulators. The first well-known topological Kondo insulator material is SmB6.[3–5] The topological properties of SmB6 have been studied extensively both theoretically and experimentally.[5–13] Lu et al.[5] have found non-trivial Z2 topology in typical mixed valence SmB6 based on first-principles calculations. Kim et al.[11] have investigated the surface in-gap states in a potential mixed-valent SmB6 based on first-principles density functional theory. In Refs. [12] and [13], the authors studied the properties of topologically protected surface states in SmB6, and found that there are four different topological phases in SmB6, which are distinguished by the sign of mirror Chern numbers. Transport properties of surface stats in the material that are poorly understood.

Different experiments have shown that transport is indeed dominated by the surface contributions at low temperature.[14–16] In Ref. [17], the authors provided conclusive evidence for surface-dominated transport in SmB6 at low temperature. The spin- and angle-resolved photoemission spectroscopy measurements showed that the surface states of SmB6 around the X point are spin-polarized.[18] The surface states of SmB6 around the X point can be described by a combination of Rashba- and Dresselhaus-like spin–orbit Hamiltonian.[19,20] The interplay between the Rashba and Dresselhaus parameters on transport properties is studied in our paper.

Generally speaking, the effect of light irradiation on electron systems are photon-dressing of band structures through the mixing of different bands and redistribution of electron occupation numbers through the absorptions/emissions of photons leading to non equilibrium distributions.[21] In this paper, we use the Floquet theory and linearized Boltzmann equation to analyze the influence of light on transport properties of SmB6 in low-frequency and high-frequency limit, respectively.

The structure of this paper is organized as follows. In Section 2 we present a brief account of the effective Hamiltonian with a circularly polarized light. Based on the Floquet theory, we study the transport properties of our model at high frequency in Section 3. In Section 4, we study the transport properties of our model at low frequency by solving the Boltzmann equation. In Section 5, we summarize our results.

2. Model Hamiltonian

Based on k · p theory and density functional theory, the low energy physics of the surface states of an ultrathin slab of SmB6 at the X point can be captured by the following Hamiltonian[19,20]

where αR,D are the Rashba- and Dresselhaus-like coupling coefficients respectively. are Pauli matrices. k = (kx,ky,0) are the momentum measured from the X point. Iz is 2×2 unit matrix vector in the z direction. Δ is the surface-hybridization induced band gap of the surface. The band dispersions relationship of the free Hamiltonian are

and the the corresponding wave functions are

where

and tan φ = (kx + κky)/(ky + κ kx). κ = αD/αR is the ratio between the Dresselhaus and Rashba coupling coefficients. When the Dresselhaus component is ignored (κ = 0), the model is reduced to massive Dirac model of the magnetically doped surface states of a three-dimensional (3D) topological insulator,[22] and the corresponding energy contours for E+ are shown in Fig. 1(a). If the Rashba component is ignored, the model is reduced to the effective Hamiltonian for the electrons in graphene in the presence of sublattice potential.[23] When both αD and αR are nonzero, three different energy contours for E+ are shown in Figs. 1(b)–1(d) as increasing of κ for small κ. The energy in Fig. 1 is plotted in units of αR. The similar energy contours can be obtained when the unit of energy is αD.

Fig. 1. Band structure at the X point and contour plot of the corresponding conduction band for fixed Δ = 0 and different κ.

When the system under illumination by circularly polarized light represented by the time-dependent vector potential

the time-dependent Hamiltonian is

where A = E0/ω, E0 and ω are the field strength and frequency respectively. The varied frequency and amplitude of the light can be used to control the dynamics of the system. In the high-frequency regime, the emission and absorption of a single virtual photon are enough to capture the energy dispersion and Floquet states based on the Floquet theory. Therefore, the dynamics of the system can be described by a time-independent effective Hamiltonian. In the low-frequency regime, there are more emissions and absorptions of photons to contribute to the transport. The photons lead the mixing of Floquet states and redistribution of electron occupation numbers. We use the Boltzmann transport equation to study the time evolution of the system. The two regimes are due to the two important physical effects of the application light. We focused on the analyses of the two effects in two regimes respectively.

3. Transport properties in the high-frequency regime

Previous works[24,25] proposed that light can dress the band structures of the system in the high frequency regime, and the system can be described by static Hamiltonian. Based on the Floquet theory,[26–29] the static time-independent Hamiltonian is written as

where

in which T = 2π/ω is the time period of the light. The processes only include the emission and absorption of a single virtual photon. Then the effective Hamiltonian in the presence of light can be written as

The last term in the above equation denotes that a photo-induced modulation of the gap would be possible.

In the clean limit, the intrinsic Hall conductivity can be obtained from[30–32]

where Ωn is the momentum-space Berry curvature for the n-th band

The summation is over all occupied bands below the bulk gap, and v̂x(y) is the velocity operator along x (y) direction. The band dispersions relationship of the effective Hamiltonian and the the corresponding wave functions are obtained from Eqs. (2)–(4) by replacing Δ with

When the chemical potential Ef is in positive energy or negative energy, we find

When the chemical potential is in the gap, we find

The direct correspondence between the Chern number and the Hall conductance for two-dimensional (2D) insulators is characterized by: σxy = Ce2/h. In Fig. 2(a), we show the band Chern numbers as functions of different externally tunable parameters (κ and A). Figure 2(b) shows that Chern numbers are plotted as a function of A for κ = 0.5 (dashed blue line) and κ = 2 (solid red line). A is measured in units of . Based on Eq. (13), the amplitude of Chern numbers is independent of the frequency. However, the sign of Chern numbers is dependent on the frequency, and the frequency follows the condition of high-frequency limit ω ≫ α. κ and A allow us to tune the Chern number of the lower spin-degenerate bands of the effective Floquet Hamiltonian (9) between C = 1/2 and C = −1/2. The direction of Hall current can be controlled by illuminating with light and choosing the Rashba and Dresselhaus parameters due to the fact that they lead the change of sign of the Berry curvature of occupied states. The Berry curvature of occupied states is

Fig. 2. (a) Phase diagram of periodically driven SmB6 at a high-frequency regime for circularly polarized light in the (A,κ) plane. (b) Chern number as a function of A for κ = 0.5 and κ = 2. A is measured in units of , and ω ≫ α.

The circularly polarized light breaks the time reversal symmetry and induces a gap term (see Eq. (9)). The band gap ) is closed and reopened along the change of Chern number between C = 1/2 and C = −1/2 for fixed κ when varying A, and the direction of the Hall current changed. For fixed A and varied κ, the varied κ changed the sign of Berry curvature, therefore the direction of Hall current changed with unchanged gap.

4. Transport properties in low-frequency regime

In the low-frequency regime, Flouqet theory cannot capture the effect of redistribution of electrons in the band structure. Within the relaxation time approximation, the Boltzmann transport equation for the electronic distribution function reads

where f0(En) is the Fermi–Dirac distribution for the n-th band energy of the system, E(t) = −∂ A(t)/∂ t = E0(sinωt,− cosωt), and τ is the relaxation time. We consider that the system is subjected to a weak finite ac field. Then the distribution to the linearized term has the form

where gn(k,t) follows the differential equation

After straightforward calculation, we obtain

where .

The light-induced current in the system is expressed through

Substituting Eq. (16) into Eq. (19) and performing integration, we obtain

where Θ(x) is the Heaviside function, and

The current can be divided into the linear and nonlinear on Ef. The linear term is nothing but Drude’s-like result. The nonlinear term originates from the band gap Δ. The current is dependent on the parameter of κ. In Fig. 3, we show the current for fixed Ef in units of e2E0/8ħπ2 for κ = 0,0.5,2. The amplitude of current with κ ≠ 0 is greatly enhanced compared to the current with κ = 0. The locations of nodal point and antinodal point are shifted. The current is determined by the group velocity

We can find that the κ affected the group velocity, and the energy band dispersion (see Fig. 1) is also changed. Therefore, the light-induced current can be controlled by κ.

Fig. 3. Light-induced current plotted for κ = 0,0.5,2 in units of .
5. Conclusion and summary

To conclude, we have analyzed transport properties of the X point in the Brillouin zone of the topological Kondo insulator SmB6 under the application of a circularly polarized light. We have employed Floquet theory and Boltzmann equation to compute the Hall conductivity at high-frequency regime and the current at low-frequency regime as a function of the ratio (κ) of the Dresselhaus-like and Rashba-like spin–orbit parameters respectively. At high-frequency regime, Berry curvature can be reversed by the ratio κ and the amplitude of the light. Therefore, the direction of Hall current can be controlled by the choice of the Rashba and Dresselhaus parameters and the amplitude of the light. At low-frequency regime, the group velocity can be modified by the ratio κ. The amplitude of the light-induced current is enhanced due to the ratio κ. The locations of nodal point and antinodal point are also shifted. Our findings provide a way to control the transport properties of the Dirac materials. It is also important for studying the effect of spin–orbit coupling in the materials.

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