Time-Dependent Kohn-Sham Theory with Memory

In time-dependent density-functional theory, exchange and correlation (xc) beyond the adiabatic approximation can be described by viscoelastic stresses in the electron liquid. In the time domain, the resulting velocity-dependent xc vector potential has a memory containing short- and long-range components, leading to decoherence and energy relaxation. We solve the associated time-dependent Kohn-Sham equations, including the dependence on densities and currents at previous times, for the case of charge-density oscillations in a quantum well. We illustrate xc memory effects, clarify the dissipation mechanism, and extract intersubband relaxation rates for weak and strong excitations.

Figure 1

Memory kernel $Y(n,t-t^{\prime })$ for $r_s=3$, using the QV and GK parametrizations [3, 20] for $f_{\mathrm{xc}}^L(\omega )$. Inset: $Y^{\mathrm{GK}}$ for $r_s$ between 1 and 15, indicating exponential memory loss, with a longer-ranged memory for lower densities.

Figure 2

Memory part of the xc potential [Eq. (8)], evaluated for $n(z,t)$ of Eq. (13) in GK and QV parametrization, shown in a stroboscopic plot during the 4th cycle after switch-on. The heavy lines indicate equidistant snapshots. Compared to the ALDA fluctuations (bottom panel), $v_{\mathrm{xc}}^{\mathrm{M},\mathrm{QV}}$ and $v_{\mathrm{xc}}^{\mathrm{M},\mathrm{GK}}$ have a phase lag a little over $\pi /2$. The density oscillation in the inset is drawn with enhanced amplitude for clarity.

Figure 3

Dipole moment $d(t)$ of a 40 nm $\mathrm{GaAs}/{\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}$ quantum well with electron density $1\times {10}^{11}{\mathrm{cm}}^{-2}$, initially in a uniform electric field ${\mathcal{E}}_1=0.01\mathrm{mV}/\mathrm{nm}$ (top) and ${\mathcal{E}}_2=0.5\mathrm{mV}/\mathrm{nm}$ (bottom), which is abruptly switched off at $t=0$. Dashed lines: ALDA. Solid lines: $\mathrm{ALDA}+\mathrm{M}$ (using QV).

Figure 4

Dissipation of excitation energy $E(t)$, for the quantum well of Fig. 3 and initial fields $\mathcal{E}$ between 0.01 and $1\mathrm{mV}/\mathrm{nm}$ as indicated, calculated with $\mathrm{ALDA}+\mathrm{M}$ (using QV). Inset: initial occupation probabilities of the first three subbands.