Dielectrics in a time-dependent electric field: A real-time approach based on density-polarization functional theory

In the presence of a (time-dependent) macroscopic electric field the electron dynamics of dielectrics cannot be described by the time-dependent density only. We present a real-time formalism that has the density and the macroscopic polarization $\mathcal{P}$ as key quantities. We show that a simple local function of $\mathcal{P}$ already captures long-range correlation in linear and nonlinear optical response functions. Specifically, after detailing the numerical implementation, we examine the optical absorption, the second- and third-harmonic generation of bulk Si, GaAs, AlAs, and CdTe, at different levels of approximation. We highlight links with ultranonlocal exchange-correlation functional approximations proposed within a linear response time-dependent density functional theory framework.

Figure 1

Optical absorption in bulk Si (top left), GaAs (top right), CdTe (bottom left), and AlAs (bottom right). Experimental optical absorption spectra (open circles) are compared with real-time simulations at different levels of approximation: TD-LDA (continuous orange line), RPA (green dash-dotted line), both without the scissor correction, and the IPA (violet dotted line) and RPA (green dashed line) with scissor correction.

Figure 2

Optical absorption in bulk Si (top left), GaAs (top right), CdTe (bottom left), and AlAs (bottom right). Experimental optical absorption spectra (open circles) are compared with real-time simulations at different levels of approximation: opt-PF (blue dashed line), JGM-PF (pink continuous line), RPA (gray dotted line). All approximations include the scissor operator correction.

Figure 3

Effect of microscopic components in the JGM-PF on the optical absorption (right panel) and SHG (left panel) of AlAs. The plots compare JGM-PF spectra with (green dashed line) and without microscopic effects (magenta continuous line) and the opt-PF (blue dotted line).

Figure 4

Upper panel: Contribution of the macroscopic xc field to the optical spectrum of Si calculated as the difference between the the opt-PF and the RPA optical absorption spectra (pink continuous line) compared with $-\alpha \text{Re}\left({\chi }_{ii}^0\right)$ (green dashed line). Bottom panel: Phase delay $\varphi$ between the polarization and the applied electric field as a function of the applied field frequency at the RPA (dotted line) and opt-PF (continuous line) level of approximation. The horizontal line highlights the $\varphi =\pi /2$ delay. See text.

Figure 5

SHG spectra of GaAs (top panels), AlAs (middle panels), and CdTe (bottom panels) obtained from real-time simulations at different levels of approximation. Left panels: IPA (dotted violet), RPA (dashed green), and TD-LDA (continuous orange)—all without scissor operator correction. For comparison we included the RPA spectrum of GaAs and AlAs calculated by Luppi et al. [57] (open triangles). Right panels: opt-PF (dashed blue) and JGM-PF (continuous pink) are compared with IPA (dotted gray) and RPA for CdTe and GaAs. Available experimental results are shown for GaAs (open circles) [58] and CdTe (open circles [59] and stars [60]).

Figure 6

THG of Si: $|{\chi }_{1111}^{\left(3\right)}|$ and $|3{\chi }_{1212}^{\left(3\right)}|$ components (see text). Spectra obtained from real-time simulations at different levels of approximation. Left panels: TD-LDA (continuous orange line), RPA (green dashed line), IPA (dotted violet line) without scissor operator correction are compared with and IPA (gray dotted line) with scissor operator correction. Right panels: JGM-PF (continuous pink line), opt-PF (blue dashed line), and RPA (gray dotted line) with scissor operator correction.

Figure 7

THG of Si: Anisotropy parameters $\sigma$ and $\varphi$ (see text). Experimental data (open circles) [62] compared with results obtained from real-time simulations at different levels of approximation. Left panels: TD-LDA (continuous orange line), RPA (green dashed line), IPA (dotted violet line) without scissor operator correction are compared with the IPA (gray dotted line) with scissor operator correction. Right panels: JGM-PF (continuous pink line), opt-PF (blue dashed line), and RPA (gray dotted line) with scissor operator correction.