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Nonlinear response theory for orbital photocurrent in semiconductors

Kakeru Tanaka, Hiroaki Ishizuka

Abstract

Recent theoretical studies on the nonlinear response of spin and orbital degrees of freedom have discovered spin and orbital analogs of the photocurrent, with potential for characterizing topological materials and for applications. In this paper, we develop a general theory for calculating spin and orbital currents in semiconductors and study the properties of optical responses in the Bernevig-Hughes-Zhang and Luttinger models, where nonlinear orbital responses and a topological phase transition occur. We study the evolution of optical responses at the topological phase transition and how they manifest. In addition, we find that the relaxation time dependence of the orbital conductivity is somewhat distinct from that of the photocurrent. The theory is straightforwardly applicable to complex models of real materials, allowing quantitative predictions of the nonlinear responses of orbital and spin.

Nonlinear response theory for orbital photocurrent in semiconductors

Abstract

Recent theoretical studies on the nonlinear response of spin and orbital degrees of freedom have discovered spin and orbital analogs of the photocurrent, with potential for characterizing topological materials and for applications. In this paper, we develop a general theory for calculating spin and orbital currents in semiconductors and study the properties of optical responses in the Bernevig-Hughes-Zhang and Luttinger models, where nonlinear orbital responses and a topological phase transition occur. We study the evolution of optical responses at the topological phase transition and how they manifest. In addition, we find that the relaxation time dependence of the orbital conductivity is somewhat distinct from that of the photocurrent. The theory is straightforwardly applicable to complex models of real materials, allowing quantitative predictions of the nonlinear responses of orbital and spin.

Paper Structure

This paper contains 14 sections, 40 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Nonlinear response theory
  3. Nonlinear conductivity
  4. Shift current
  5. Injection current
  6. Bernevig-Hughes-Zhang model
  7. Model
  8. Orbital shift and injection currents
  9. Four-band effect
  10. Luttinger model
  11. Model
  12. Optical conductivity
  13. Nonlinear orbital current
  14. Summary and Discussion

Figures (10)

  • Figure 1: The dispersion of the BHZ model with $\epsilon_s=3.5\mathrm{eV},t_{ss}=t_{sp}=1\mathrm{eV},$ and $\epsilon_p=t_{pp}=0\mathrm{eV}$, which corresponds to $\lambda=3.5$.
  • Figure 2: Nonlinear orbital conductivity for the BHZ model. The frequency dependence of (a) ${\rm Re}(\sigma_{xxy}^{(2A)})$, (b) ${\rm Im}(\sigma_{xxy}^{(2B)})$, and (c) ${\rm Re}(\sigma_{yyy}^{(2A)})$, and (d) joint density of states.
  • Figure 3: The $\omega$ dependence of (a) ${\rm Re}(\sigma_{xxy}^{(2A)})$ and (b) ${\rm Im}(\sigma_{xxy}^{(2B)})$ with different relaxation time $\tau$.
  • Figure 4: The imaginary part of nonlinear (a),(c),(e) spin and (b),(d),(f) orbital conductivities for the BHZ model with the Rashba term. (a),(b) Nonlinear conductivities calculated with Eq. \ref{['2nd']}, (c),(d) their shift-current part in Eq. \ref{['eq:shift']} and (e),(f) their injection-current part in Eq. \ref{['eq:inj']}. The results are for $\epsilon_s=3.5\mathrm{eV},t_{ss}=t_{sp}=1\mathrm{eV}$, $\epsilon_p=t_{pp}=0\mathrm{eV}$, and $E=0.1\mathrm{eV}$.
  • Figure 5: (a) spin and (b) orbital conductivities $\mathrm{Im}\sigma^{(2B)}_{xxy}$, where $\tau=0.4$.
  • ...and 5 more figures