Photogalvanic currents from first-principles real-time density-matrix dynamics

Abstract

The photogalvanic effect is the generation of a second-order direct current by illumination of a non-centrosymmetric material. In this work, we develop a first-principles real-time density matrix (FPDMD) formalism enabling the calculations of the photogalvanic current in all time regimes: transient and steady. Unlike past \textit{ab-initio} studies which focused only on the photo-excitation process, our first-principles theory framework encodes all quantum scatterings (intra/interband relaxation and electron-hole recombination) mediated by bosons (photons and phonons), and is thus predictive of photogalvanic currents in realistic materials. In particular, for the linear photogalvanic effect, we find electron scatterings mediated by phonons contribute significantly to the shift current for prototypical piezoelectrics like BaTiO$_3$. For the circular photogalvanic effect, we develop a self-consistent theory of a steady injection current that incorporates realistic scattering mediated by phonons. Our formulation developed for photogalvanic current elucidates its connection with fundamental quantum-geometric quantities such as the Berry curvature and the quantum metric. A phonon-based explanation is proposed for the bipolar transient photogalvanic current observed by the THz emission spectroscopy.

Figures (4)

• Figure 1: Kinetic process and electron occupation under an external light illumination. (a) Kinetic processes including light excitation, scattering and recombination. (b) Excited carrier occupation changes with time in conduction bands within 200 ps; the inset zooms in the first 30 fs. The excitation peak is formed within 10 fs, but conduction electrons are then quickly scattered to conduction band edge. Within $25$ fs (the energy relaxation time), the precursor of a quasi-Fermi-Dirac distribution forms, with a quasi temperature 315K. (c) For the conduction band of BaTiO$_3$, the intra-band Berry curvature $\mathbf{\Omega}^\mathbf{k}$ is represented as a purple vector field enclosed by the excitation surface at $\hbar\omega-E_g=0.32$ eV. Under electron-phonon scattering, the blue arrows ($p$) illustrate one representative electron trajectory from $\mathbf{k}$ to band bottom, while the green arrows ($T\circ p$) is the trajectory that is the time-reverse of $p$. For a small-momentum scattering event from $\mathbf{k}$ to $\mathbf{k}'$, the electron is displaced in real space by the phononic shift vector $\mathbf{S}^\nu_{\mathbf{k}'\leftarrow\mathbf{k}}=\mathbf{\Omega}^{\bar{\mathbf{k}}}\times (\mathbf{k}'-\mathbf{k})$ with $\bar{\mathbf{k}}=(\mathbf{k}+\mathbf{k}')/2$. This cross product of a purple arrow and a blue/green arrow is represented as a red circle and points in the z direction, which is the polar axis of BaTiO$_3$. The shift contributions from $p$ and $T\circ p$ add up instead of canceling out, because under time-reversal, $\mathbf{\Omega}^{-\bar{\mathbf{k}}}=-\mathbf{\Omega}^{\bar{\mathbf{k}}}$ and $(\mathbf{k}-\mathbf{k}')=(-\mathbf{k}+\mathbf{k}')$, leaving the shift vector invariant.
• Figure 2: Different DC photocurrent conductivities of BaTiO$_3$. (a) Excitation shift current conductivity in $zxx$ direction. Real time result is from evaluating the steady-state off-diagonal part of the density matrix, with eliminating other contributions except excitation current. Reference data is from Ref.fei2020shift. (b) Phonon shift current conductivity from perturbation theory, FPDMD, and Berry curvature formula at RT in $zxx$ direction. FPDMD is obtained by eliminating other contributions except phonon shift current. Perturbation refers to results calculated by Eq.\ref{['eqn:shift_ph']}. The Berry curvature results are obtained by replacing $\mathbf{S}^\nu_{\mathbf{k}'s\leftarrow \mathbf{k} s}$ with $\mathbf{q}\times\mathbf{\Omega}^{\bar{\mathbf{k}}}_{ss}$, and only considering intra-band electron-phonon scattering in Eq.\ref{['eqn:shift_ph']}. (c) Injection current conductivity of BaTiO$_3$ from perturbation theory (Eq. \ref{['eqn:Jinj']}) and FPDMD. The perturbation result uses state-dependent electron-phonon relaxation rate (Eq. \ref{['statedepGamma']}).
• Figure 3: Photogalvanic current and conductivity of BaTiO$_3$ in $zxx$ direction. (a) Comparison of photocurrent (Eq. \ref{['dccurrent']}) from real-time FPDMD, experiment result koch1975bulk and reference work dai2021phonon. (b) The total shift current conductivity and separate current components from FPDMD.
• Figure 4: FPDMD simulation of $J(t)$ and their kernel-modeled results for BaTiO$_3$, using a Gaussian light pulse centered at $t=0$ with width 20 fs. The modeled results are fitted against FPDMD. Whether $J(t)=J_{tot}(t)$ is bipolar [as in panel (a) with photon energy at $0.545eV+E_g$] or unipolar [panel (b) with photon energy at $0.445eV+E_g$] is determined by whether the steady-state excitation and phonon shift currents have opposite sign [the second dashed line in Fig. \ref{['fig:3']}(b)] or the same sign [the first dashed line in Fig. \ref{['fig:3']}(b)]. Panel (c) shows the fitting of real time dynamics to the convolution of $I(t)$ and $K(t)$ (kernel-models), where three different time scales are obtained, roughly in agreement with the electron-phonon scattering time $\tau_{eph}$, energy relaxation time $\tau_{er}$, and recombination time $\tau_{rec}$ of this system Note7 (Details in SI Sec. IV).