Magnetosubbands of semiconductor quantum wires with Rashba spin-orbit coupling

We have investigated the electronic structure of Rashba spin-split quantum wires in a magnetic field. For our numerical calculations, a harmonic confinement was assumed. We find that wire structures with several occupied one-dimensional subbands still exhibit a beating pattern in the magnetoresistance. The wire width turns out to strongly affect the magnetic field values at which nodes occur in the beating pattern. In the limit of narrow wires, the beating pattern would vanish altogether because spin-split subbands become populated equally.

Figure 1

(Color online) Energy dispersion of the spin-split subbands at $B=0$ for weak spin-orbit coupling: ${\Delta }_{\mathit{so}}∕\hslash {\omega }_0=0.01$ (a) and for strong spin-orbit coupling for ${\Delta }_{\mathit{so}}∕\hslash {\omega }_0=1$ (b). The anticrossing between the second and third subband is marked by a circle. The quantity $b_0=\sqrt{\hslash ∕m*{\omega }_0}$ is the characteristic length scale of the harmonic confinement potential. The inset shows the geometry of the wire structure.

Figure 2

(Color online) (a) Energy dispersion at zero magnetic field. The inset shows a detail of the anticrossing of the lower subbands. The relative strength of the Rashba spin-orbit coupling ${\Delta }_{\mathit{so}}∕\hslash {\omega }_0$ was taken to be 0.1. (b) Energy spectrum at $k_y=0$ as a function of ${\omega }_c∕{\omega }_0$. For the Zeeman energy splitting $g{\mu }_{B}B$, we assumed $-0.072\hslash {\omega }_c$. (c) Energy dispersion at a finite magnetic field $(\hslash {\omega }_c=1.5\hslash {\omega }_0)$.

Figure 3

(Color online) Fermi energy as a function of the magnetic field for a $300\mathrm{nm}$ wide wire. For the Rashba coupling parameter a value of $8\times {10}^{-12}\mathrm{eV}\mathrm{m}$ was assumed.

Figure 4

(Color online) (a) One-dimensional density of states of a $500\mathrm{nm}$ wide wire as a function of a magnetic field at a fixed Fermi energy of $39\mathrm{meV}$. For the Rashba coupling parameter a value of $7.5\times {10}^{-12}\mathrm{eV}\mathrm{m}$ was assumed. (b) Density of states for a lifetime broadened energy spectrum. For the scattering parameter $\Gamma$ a value of $0.4\mathrm{meV}$ was assumed.

Figure 5

(Color online) (a) and (c) subband spectra at $k_y=0$ as a function of the magnetic field for a 600 and $200\mathrm{nm}$ wide wire, respectively. Here, a Rashba coupling parameter of $7.5\times {10}^{-12}\mathrm{eV}\mathrm{m}$ was assumed. (b) The position of the nodes in the magnetoresistance as a function of the wire width.

Figure 6

(Color online) (a)–(c) Expected node positions as a function of wire width with $E_{F,1D}=39\mathrm{meV}$ for ${\alpha }_R=5\times {10}^{-12}\mathrm{eV}\mathrm{m}$, $7.5\times {10}^{-12}\mathrm{eV}\mathrm{m}$, and $10\times {10}^{-12}\mathrm{eV}\mathrm{m}$, respectively. (d)–(f) Node positions for $E_{F,1D}=26\mathrm{meV}$, $32.5\mathrm{meV}$, and $39\mathrm{meV}$, respectively. The Rashba coupling parameter was kept at $7.5\times {10}^{-12}\mathrm{eV}\mathrm{m}$.