Higher-order bulk photovoltaic effects, quantum geometry and application to $p$-wave magnets

Abstract

The injection and shift currents are generalized to the $\ell $th-order injection and shift currents for the longitudinal conductivities in the two-band model, where $\ell $ is the power of the applied electric field. In addition, the formulas for the higher-order injection current are expressed in terms of the quantum metric and the higher-order shift current in terms of the higher-order quantum connection. Then, they are applied to $p$-wave magnets. It is shown that the injection and shift currents are zero. On the other hand, the $\ell $th-order injection and shift currents with odd $\ell $ are nonzero when the direction of the Néel vector of the $p$-wave magnet points to an in-plane direction.

Figures (4)

• Figure 1: Energy spectrum in units of $\varepsilon _{0}$. (a) The Néel vector is along the $x$ direction ($\Theta =\pi /2;\Phi =0$). (b) The Néel vector is along the $z$ direction ($\Theta =0$). The horizontal axis is $k_x$. We have set $J=\varepsilon _{0}/k_{0}$, $\hbar ^{2}k_{0}^{2}/\left( 2m\right) =\varepsilon _{0}/2$ and $B=0.5\varepsilon _{0}$.
• Figure 2: (a) Higher-order injection currents log$_{10}\left\vert \sigma _{\text{inject}}^{x;x^{\ell }}/\sigma _{\text{inject}}^{\left( \ell \right) }\right\vert$ in units of $\sigma _{\text{inject}}^{\left( \ell \right) }$. (b) Higher-order shift currents log$_{10}\left\vert \sigma _{\text{shift}}^{x;x^{\ell }}/\sigma _{\text{shift}}^{\left( \ell \right) }\right\vert$ in units of $\sigma _{\text{shift}}^{\left( \ell \right) }$. Blue curves indicate $\ell =3$, orange curves indicates $\ell =5$, green curves indicate $\ell =7$ and red curves indicate $\ell =9$. Dashed curves indicate asymptotic behaviors for large $\omega$. We have set $J=\varepsilon _{0}/k_{0}$, $\hbar ^{2}k_{0}^{2}/\left( 2m\right) =\varepsilon _{0}/2$ and $B=0.5\varepsilon _{0}$.
• Figure 3: (a) Jerk current at finite temperature. (b) Third-order shift current at finite temperature. Red dashed curves indicate the absolute zero temperature $T=0$, blue dashed curves indicates $k_{\text{B}}T=0.1\varepsilon _{0}$ and green curves indicate $k_{\text{B}}T=\varepsilon _{0}$. We have set $\Theta =\pi /2,$$J=\varepsilon _{0}/k_{0}$, $\hbar ^{2}k_{0}^{2}/\left( 2m\right) =\varepsilon _{0}/2$ and $B=0.5\varepsilon _{0}$.
• Figure 4: (a) $\omega$ dependence of the Fermi distribution difference $f_{-+}$. (b) Temperature dependence of the Fermi distribution difference $f_{-+}$ at the optical band edge $\omega =2B$. We have set $J=\varepsilon _{0}/k_{0}$, $\hbar ^{2}k_{0}^{2}/\left( 2m\right) =\varepsilon _{0}/2$ and $B=0.5\varepsilon _{0}$.