Effect of spin-orbit interaction on the plasma excitations in a quantum wire

Incorporating the spin-orbit interaction into the Hamiltonian of a quantum wire, we have calculated the plasma excitation energies. We include both the Rashba term ($\alpha$ coupling) arising from the asymmetry of the heterostructure forming the two-dimensional electron gas, and the $\beta$ coupling due to the quantum wire confinement which we model with a harmonic potential. The $\alpha$ coupling lifts the degeneracy of the spin states and the $\beta$ coupling causes the quantized transverse single-particle energy to have negative dispersion. Our model yields several interesting features which may be observed with the use of inelastic light scattering and electron energy loss spectroscopy (EELS). The collective excitations for the quantum wire are determined by the allowed transitions between subbands. The collective plasma excitations split off from each branch of allowed particle-hole modes. The subband structure gives rise to a plasmon mode with a negative group velocity.

Figure 1

The transverse energy eigenvalues ${ϵ}_x=2{ϵ}_{k_y,n}∕\hslash \Omega -k_y^2{\lambda }_0^2$ as a function of $k_y{\lambda }_0$ for ${\lambda }_0∕l_{\alpha }=0.1$ and ${\lambda }_0∕l_{\beta }=0.1$. The degeneracy of the ↑- and ↓-spin levels is is lifted for finite $k_y$ by the Rashba SO term in the Hamiltonian in Eq. (1).

Figure 2

For $\alpha =0$ and $\beta =0$, the normalized frequencies $\overline{\omega }=2\omega ∕\Omega$ of the plasmon excitations and particle hole modes are plotted as a function of $q_y{\lambda }_0$. The values for $m*$, ${ϵ}_b$, and $E_F$ are given in the text. The shaded regions denote the single-particle excitations giving rise to Landau damping.

Figure 3

For $\alpha =0$ and $\beta =0$, the form factor is plotted as a function of $k_y{\lambda }_0$.

Figure 4

The plasmon excitations and particle hole mode frequencies $\omega$, in units of $\Omega ∕2$, are plotted as a function of $q_y{\lambda }_0$. The SPE’s are shown as the shaded regions. The values for $\alpha$, $\beta$, $m*$, ${ϵ}_b$, and $E_F$ are given in the text.