Three- to two-dimensional crossover in time-dependent density-functional theory

Quasi-two-dimensional (2D) systems, such as an electron gas confined in a quantum well, are important model systems for many-body theories. Earlier studies of the crossover from 3D to 2D in ground-state density-functional theory showed that local and semilocal exchange-correlation functionals which are based on the 3D electron gas are appropriate for wide quantum wells, but eventually break down as the 2D limit is approached. We now consider the dynamical case and study the performance of various linear-response exchange kernels in time-dependent density-functional theory. We compare approximate local, semilocal, and orbital-dependent exchange kernels, and analyze their performance for inter- and intrasubband plasmons as the quantum wells approach the 2D limit. 3D (semi)local exchange functionals are found to fail for quantum well widths comparable to the 2D Wigner-Seitz radius $r_s^{2\mathrm{D}}$, which implies in practice that 3D local exchange remains valid in the quasi-2D dynamical regime for typical quantum well parameters, except for very low densities.

Figure 1

Illustrations of intersubband (top) and intrasubband (bottom) plasmon excitations with wave vector $q_{\left|\right|}$ in a quantum well with conduction band Fermi level ${\varepsilon }_F$ in the lowest subband. Intersubband plasmons involve collective transitions between two subbands, leading to density oscillations of the quasi-2D electron system perpendicular to the quantum well plane. Intrasubband plasmons (collective transitions within the lowest subband) are characterized by density oscillations and currents flowing along the plane.

Figure 2

Photoabsorption cross section for $q_{\left|\right|}=0$ intersubband excitations in quantum wells. Left panels: charge-density excitations. Right panels: spin-density excitations. Insets: density profiles at given values of $N_{\mathrm{occ}}$. The calculations were done with Eq. (15), setting $q_{\left|\right|}=0$ and using the 3D ALDA exchange kernel to obtain $n_{1\sigma }$.

Figure 3

Intersubband plasmon dispersions $\Omega \left(q_{\left|\right|}\right)$, for $N_s={10}^{12}{\mathrm{cm}}^{-2}$ and well widths 100 Å and 40 Å. The black full lines indicate the intersubband p-h continuum. The RPA only gives intersubband charge plasmons; ALDA, PBE, and PGG give both charge (full lines) and spin plasmons (dashed lines). ALDA and PBE break down when their charge plasmons fall below the p-h continuum.

Figure 4

Intersubband plasmon energies, at $q_{\left|\right|}=0$, versus well width $L$, for $N_s={10}^{12}{\mathrm{cm}}^{-2}$. The horizontal line indicates the lowest p-h transition ${\omega }_{21}$ (all energies are scaled by $L^2)$. The RPA only gives intersubband charge plasmons; ALDA, PBE, and PGG give both charge (full lines) and spin plasmons (dashed lines). ALDA and PBE break down when the charge plasmon falls below the p-h line.

Figure 5

Critical width $L_{\mathrm{crit}}^{\mathrm{inter}}$ at which the intersubband plasmon breakdown occurs, as a function of sheet density $N_s$. Full line: ALDA; dashed line: PBE.

Figure 6

Plasmon dispersions $\Omega \left(q_{\left|\right|}\right)$ for strictly 2D systems with sheet densities $N_s={10}^{10},{10}^{11}$, and ${10}^{12}{\mathrm{cm}}^{-2}$, calculated with RPA, 2D ALDA, and PGG. The full lines denote the upper boundaries of the particle-hole (p-h) continuum. Here, $\widetilde{q_{\left|\right|}}=q_{\left|\right|}/k_F^{2\mathrm{D}}$ and $\tilde{\Omega }=\Omega /{\left(k_F^{2\mathrm{D}}\right)}^2$.

Figure 7

Intrasubband plasmon dispersions for quantum wells with sheet density $N_s={10}^{10}{\mathrm{cm}}^{-2}$, for different widths $L=\lambda L_2$, where $\lambda$ takes on the values 1, 0.5, 0.2, 0.1, 0.05, 0.02, 0.01, 0.005, 0.002, and 0.001. $L_2=217$ nm is the largest width for which only the lowest subband is occupied. The individual plasmon dispersions are offset for clarity. The dashed lines are the upper boundaries of the p-h continuum. The squares indicate the wave vector ${\tilde{q}}_{\left|\right|\text{p-h}}$ where the plasmons enter the p-h continuum. Top panel: 3D ALDA. Bottom panel: PGG.

Figure 8

Wave vector ${\tilde{q}}_{\left|\right|\text{p-h}}$ at which the intrasubband plasmon merges with the p-h continuum, plotted vs well width scaling factor $\lambda$, calculated with PGG (blue) and ALDA (red). The dashed lines indicate the respective limits for the strictly 2D case. The calculations were done for sheet densities $N_s={10}^{10},{10}^{11},{10}^{12},\text{and}{10}^{13}{\mathrm{cm}}^{-2}$, as indicated. The breakdown of the 3D ALDA occurs around $\lambda =0.1$ for all $N_s$.

Figure 9

Photoabsorption cross section for $q_{\left|\right|}=0$ intersubband charge plasmons, for a quantum well with five occupied subbands, comparing PGG and exchange-only ISTLS.