Shift charge and spin photocurrents in Dirac surface states of topological insulator

The generation of photocurrent in condensed matter is of main interest for photovoltaic and optoelectronic applications. Shift current, a nonlinear photoresponse, has attracted recent intensive attention as a dominant player of bulk photovoltaic effect in ferroelectric materials. In three-dimensional topological insulators ${\text{Bi}}_2{X}_3$ ($X=\text{Te}$, Se), we find that Dirac surface states with a hexagonal warping term support shift current by linearly polarized light. Moreover, we study “shift spin current” that arises in Dirac surface states by introducing time-reversal symmetry breaking perturbation. The estimate for the magnitudes of the shift charge and spin current densities are $0.13I_0$ and $0.40I_0$ (nA/m) for ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ with the intensity of light $I_0$ measured in $(\text{W}/{\text{m}}^2)$, respectively, which can offer a useful method to generate these currents efficiently.

Figure 1

Generation of shift charge current and shift spin current are pictorially described. (a) An electronic excitation induced by light from valence band to conduction band is schematically shown. (b) When the center of charge of Bloch state is shifted upon interband transition, from the green to the red wave packet, the shift charge current is generated. (c) Two wave packets consisting of opposite spins can exhibit different shifts upon optical transition, which results in the shift spin current.

Figure 2

Warped Dirac surface state carries the shift charge current. The Fermi velocity $v_F=2.55{\hslash }^{-1}\left(\text{eV}\AA \right)$ and the warping strength $\lambda =250\left(\text{eV}{\AA }^3\right)$ are used [38]. We show the matrix elements that contribute to the nonlinear susceptibility ${\chi }_y^{yy}$. (a) The energy dispersion is drawn with electronic states satisfying the resonant condition, ${\epsilon }_c-{\epsilon }_v=\mathrm{\Omega }$, in distinct colors. (b) The coupling strength of Bloch states in the resonant condition to diamagnetic current response, $|{\left({\partial }_{k_y}{v}^y\right)}_{cv}|$, is shown in color. (c) The coupling strength of Bloch states in the resonant condition to a linearly polarized light, $|{\left(v^y\right)}_{vc}|$, is shown in color. (d) The contribution of Bloch states to the shift charge current, $|{\left(v^y\right)}_{vc}{\left({\partial }_{k_y}{v}^y\right)}_{cv}|$, is shown in color.

Figure 3

Massive Dirac surface state with band-bending carries the shift spin current. The Fermi velocity $v_F=2.84{\hslash }^{-1}\left(\text{eV}\AA \right)$, Zeeman term $m=0.03\left(\text{eV}\right)$, and band-bending strength $g=41.1{\hslash }^{-2}\left(\text{eV}{\AA }^2\right)$ are used [47] for ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. We show the matrix elements that contribute to the nonlinear susceptibility ${\chi }_{x,{\widehat{s}}_x}^{xx}$. (a) The energy dispersion is drawn with electronic states satisfying the resonant condition, ${\epsilon }_c-{\epsilon }_v=\mathrm{\Omega }$, in distinct colors. (b) The coupling strength of Bloch states in the resonant condition to diamagnetic spin current response, $|{\left(\{{\widehat{s}}_x,{\partial }_{k_x}{v}^x\}\right)}_{cv}|$, is shown in color. (c) The coupling strength of Bloch states in the resonant condition to a linearly polarized light, $|{\left(v^x\right)}_{vc}|$, is shown in color. (d) The contribution of Bloch states to the shift charge current, $|{\left(v^x\right)}_{vc}{\left(\{{\widehat{s}}_x,{\partial }_{k_x}{v}^x\}\right)}_{cv}|$, is shown in color.