Cyclotron motion and magnetic focusing in semiconductor quantum wells with spin-orbit coupling

We investigate the ballistic motion of electrons in III-V semiconductor quantum wells with Rashba spin-orbit coupling in a perpendicular magnetic field. Taking into account the full quantum dynamics of the problem, we explore the modifications of classical cyclotron orbits due to spin-orbit interaction. As a result, for electron energies comparable to the cyclotron energy, the dynamics is particularly rich and not adequately described by semiclassical approximations. Our study is complementary to previous semiclassical approaches concentrating on the regime of weaker fields.

Figure 1

Orbital dynamics of a wave packet of initial width $d=1.0\ell$ and group wave number $k_0=2.0∕\ell$ for various initial spin states. The Rashba energy is ${\epsilon }_R=0.2\hslash {\omega }_c$, while the Zeeman energy is put to zero. In the left (right) top panel, the spin points initially along the positive (negative) $x$ direction. The middle and bottom panels show the corresponding data for the $y$ and $z$ directions, respectively. The simulation time is always $t=30∕{\omega }_c$. The strictly circular motion (dotted lines) with radius $k_0{\ell }^2$ occurring in the absence of spin-orbit coupling is always shown as a guide to the eyes.

Figure 2

Spin dynamics as expressed in terms of the time-dependent expectation values corresponding to the left column of Fig. 1. The solid lines show the quantity $\mid ⟨{\vec{\sigma }}_H\left(t\right)⟩\mid$ which is a measure of entanglement between the spin and the orbital degrees of freedom.

Figure 3

The quantities $Q_1$ (solid lines) and $Q_2$ (dashed lines) defined in the text as a function of time for the same system parameters as in Fig. 1. In the top panel, the spin points initially along the positive (negative) $x$ direction with $Q_1\left(0\right)=Q_2\left(0\right)=+1$ $[Q_1\left(0\right)=Q_2\left(0\right)=-1]$. The middle and bottom panels show the analogous data with the spin initially aligned along the $y$ and $z$ axes, respectively. Here, we always have $Q_1\left(0\right)=Q_2\left(0\right)=0$.

Figure 4

Orbital dynamics for the same system as in Fig. 1 but with a smaller initial group wave number of only $k_0=0.5∕\ell$. Again, the strictly circular motion occurring in the absence of spin-orbit coupling is always shown as a guide to the eyes.

Figure 5

Orbital dynamics for the same system as in Fig. 1 for various values of the initial group wave number $k_0$ and the spin initially always pointing along the positive $x$ axis. For a better comparison, the components of $⟨\vec{r}⟩$ are given in units of $k_0{\ell }^2$.

Figure 6

Schematic sketch of a magnetic focusing experiment.

Figure 7

The probability density $P_x(x_f-x_i)$ for a wave packet of width $d=\ell$ and totalenergy $E=2.0\hslash \omega$ at zero Zeeman coupling and different Rashba energies. The data are averaged over 500 000 randomly chosen initial spin states each. Note the different scale of the $y$ axis in the top, middle, and bottom panels.

Figure 8

The probability density $P_x(x_f-x_i)$ for the same situation as in Fig. 7 but with a Zeeman energy of ${\epsilon }_z=0.1\hslash \omega$.