Symmetries in the collective excitations of an electron gas in core-shell nanowires

We study the collective excitations and inelastic light-scattering cross section of an electron gas confined in a GaAs/AlGaAs coaxial quantum well. These systems can be engineered in a core-multishell nanowire and inherit the hexagonal symmetry of the underlying nanowire substrate. As a result, the electron gas forms both quasi-one-dimensional channels and quasi-two-dimensional channels at the quantum-well bents and facets, respectively. Calculations are performed within the random-phase approximation and time-dependent density functional theory approaches. We derive symmetry arguments which allow one to enumerate and classify charge and spin excitations and determine whether excitations may survive to Landau damping. We also derive inelastic light-scattering selection rules for different scattering geometries. Computational issues stemming from the need to use a symmetry-compliant grid are also investigated systematically.

Figure 1

(a) Schematics of the studied core-multishell NW, (b) normalized single-particle DOS in arbitrary units, and (c) envelope functions of the 12 lowest-lying subbands. The principal quantum number $n$ and the irrep of the $D_{6h}$ group of each state is indicated.

Figure 2

CDE dispersion in $q_z/k_F$ for a NW with one occupied subband calculated within the RPA. The color map is plotted with a logarithmic scale to emphasize the electron-hole single-particle excitation continua. The bounds of the latter are also delimited by blue solid lines that are analytically calculated. The lateral labels show the subbands involved in the excitation and the symmetry of the excited state.

Figure 3

CDE dispersion with three occupied subbands calculated within the (a) RPA and (b) TDLDA approaches. The bounds of the single-particle excitation continua for excitations from $n=1$ ($n=2,3$) to higher subbands are delimited by blue solid (dashed) lines, which can be calculated analytically from energy and momentum conservation. Dark blue rectangles highlight selected CDEs which are Landau damped. Panels (c) and (d) show low-energy CDEs between the occupied subbands calculated within the RPA and TDLDA, respectively. Black dots: CDEs; gray (red) area: single-particle continuum for transitions from $n=1$ ($n=2,3$). The labels show the subbands involved in the excitation and the symmetry of the excited state.

Figure 4

CDE dispersion for a NW with five occupied subbands calculated within the the TDLDA. The bounds of the single-particle excitation continua from subband 1, (2,3), and (4,5) are delimited by blue solid, dashed, and dotted lines, respectively, and calculated analytically. The labels show the subbands involved in three selected excitations and the symmetry of the excited states. The dark blue rectangles illustrate selected Landau damping of these CDEs.

Figure 5

(a) SDE dispersion in $q_z/k_F$ for a NW with three occupied subbands calculated within the TDLDA. The bounds of the single-particle excitation continua from subband 1, (2,3) are delimited by blue solid (dashed) lines that are calculated analytically. The labels show the subbands involved in the excitations and the symmetry of the excited states. The dark blue rectangles illustrate selected Landau damping of SDEs. (b) Low-energy SDE spectrum for excitations between the occupied subbands calculated within the TDLDA. The black dots show the SDE dispersion, whereas the gray (red) area is the single-particle continuum for transitions from subband 1 (2,3).

Figure 6

(a) Imaginary part of the density-density response function given by Eq. (5) for a system with three occupied subbands and $q_z=0$. (b),(c) ILS spectra for the same system when photons are applied perpendicular to the top facet and along the maximal diameter, respectively, as illustrated in the left insets. The 2D color-map insets show the IDDs calculated at selected peaks and labeled with the excitation irreps of the $D_{2h}$ group. The vertical dashed lines are guides to the eye showing the energy position of selected excitations indicated by the top labels.

Figure 7

(a) Schematic position space domain, (b) periodic replication with hexagonal pattern of the domain used for the HDFT, and (c) schematic momentum-space domain.