Energy spectra for quantum wires and two-dimensional electron gases in magnetic fields with Rashba and Dresselhaus spin-orbit interactions

We introduce an analytical approximation scheme to diagonalize parabolically confined two-dimensional (2D) electron systems with both the Rashba and Dresselhaus spin-orbit interactions. The starting point of our perturbative expansion is a zeroth-order Hamiltonian for an electron confined in a quantum wire with an effective spin-orbit induced magnetic field along the wire, obtained by properly rotating the usual spin-orbit Hamiltonian. We find that the spin-orbit-related transverse coupling terms can be recast into two parts $W$ and $V$, which couple crossing and noncrossing adjacent transverse modes, respectively. Interestingly, the zeroth-order Hamiltonian together with $W$ can be solved exactly, as it maps onto the Jaynes-Cummings model of quantum optics. We treat the $V$ coupling by performing a Schrieffer-Wolff transformation. This allows us to obtain an effective Hamiltonian to third order in the coupling strength $k_{\text{R}}\ell$ of $V$, which can be straightforwardly diagonalized via an additional unitary transformation. We also apply our approach to other types of effective parabolic confinement, e.g., 2D electrons in a perpendicular magnetic field. To demonstrate the usefulness of our approximate eigensolutions, we obtain analytical expressions for the $n\text{th}$ Landau-level $g_n$ factors in the presence of both Rashba and Dresselhaus couplings. For small values of the bulk $g$ factors, we find that spin-orbit effects cancel out entirely for particular values of the spin-orbit couplings. By solving simple transcendental equations we also obtain the band minima of a Rashba-coupled quantum wire as a function of an external magnetic field. These can be used to describe Shubnikov-de Haas oscillations. This procedure makes it easier to extract the strength of the spin-orbit interaction in these systems via proper fitting of the data.

Figure 1

Energy spectrum of the zeroth-order Hamiltonian ${\tilde{H}}_0-\frac{1}{2}{k}^2$ [Eq. (6)] for $k_{\text{R}}=0.25$ (first ten wire modes shown). The circles and squares indicate the crossings between adjacent and next-to-adjacent, respectively, opposite-spin oscillator states. Here $k^{\ast }=\frac{1}{2k_{\text{R}}}$ defines the crossing point of the energy dispersions of the states $|k,n,-1⟩$ and $|k,n+1,+1⟩$.

Figure 2

Eigenenergies of the first ten Rashba wire modes for $k_{\text{R}}=0.25$ (which yields $k^{\ast }=2$). The solid lines are obtained within our analytical approximation scheme [Eqs. (19, 20)] while the empty circles correspond to the (exact) eigenenergies from a numerical diagonalization of $H_{\text{R}}$ in Eq. (4). The deviation between the numerical and the approximate analytical spectra cannot be seen on scale of the plot. The dashed line denotes the Fermi energy $E_F$ and the empty squares show the crossings of the states separated by $3\hslash {\omega }_0$.

Figure 3

Similar to Fig. 2 but for the transverse states corresponding to $n=50–59$. Here the (Rashba) spin-orbit coupling strength $k_{\text{R}}=0.10\Rightarrow k^{\ast }=5$. Note that even for such large $n$’s the deviation of the analytical solution (lines) from the numerical (circles) can hardly be distinguished on the scale of the plot.

Figure 4

Eigenenergies of the first ten transverse modes of the Rashba-Dresselhaus wire. Here the spin-orbit coupling strength $k_{\text{RD}}=0.25$ and $\theta =0.126$ $\left(\alpha ⪢\beta \right)$. The solid lines denote the approximate analytical solution in Eq. (37) while the empty circles correspond to the numerical diagonalization of the corresponding Hamiltonian, Eq. (26). The inset zooms in on the $n=9$ dispersion, showing more clearly the deviation between the analytical and numerical solutions. The maximum deviation is about 4% near $k=0$.

Figure 5

Similar to Fig. 4 with $k_{\text{RD}}=0.10$ and $\theta =0.126$ $\left(\alpha ⪢\beta \right)$. The deviation of the analytical solution (lines) from the numerical (circles) is on the order of a few percents, and on the scale of the plot can hardly be distinguished.

Figure 6

Similar to Fig. 5 but for the wire modes corresponding to $n=50–59$. Here $k_{\text{RD}}=0.10$ and $\theta =0.126$ $\left(\alpha ⪢\beta \right)$. The inset shows more clearly the deviation of the analytical solution (solid line) from the numerical one (circles) for the $n=59$ mode. The maximum deviation is less than 4% near $k=0$.

Figure 7

Here we plot the numerical and analytical solution for $k_{\text{RD}}=0.25$ with $\alpha ⪡\beta$ (solid line and circles) and $\alpha \approx \beta$ (dashed line and squares) for levels $\left(5,\uparrow \right)$ and $\left(9,\uparrow \right)$. The analytical solution is the same as that derived for the $\alpha ⪡\beta$ case. Note that the $\alpha \approx \beta$ solution is accurate near $k=0$ but deviates more for $k\approx 2k^{\ast }$, see text for details.

Figure 8

$B=0$ eigenenergies of a Rashba wire [Eq. (37)] as a function of $k$ for a Rashba coupling strength $k_{\text{R}}=0.135$ and $E_F=30\hslash {\omega }_0$.

Figure 9

Eigenenergies [Eqs. (19, 20)] as a function of the magnetic field with $k=0$, $k_{\text{RD}}=0.135$, $g^{\ast }=-4$, $m/m_0=0.037$, and $E_F=30\hslash {\omega }_0$. The inset shows a zoom of the intersections of the dispersion curves with $E_F$.

Figure 10

Separation between subsequent roots $\Delta r_i=r_{i+1}-r_i$, plotted as a function of $r_i$. The separation of the node points, e.g., $r\approx 1.0$ and $r\approx 2.5$ for the uppermost curve, is determined by the value $\alpha /\left({\ell }_0\hslash \omega \right)$.

Figure 11

$n\text{th}$-Landau-level $g_n$ factors for two different values of $\left|g^{\ast }\right|=4$ and $\left|g^{\ast }\right|=1$, as a function of Rashba coupling. Other parameter values are $B=2\text{T}$, $\frac{m}{m_0}=0.037$, and $\beta =6\text{meV}\text{nm}$.