Zitterbewegung of electrons and holes in III–V semiconductor quantum wells

The notion of zitterbewegung is a long-standing prediction of relativistic quantum mechanics. Here we extend earlier theoretical studies on this phenomenon for the case of III–V zinc-blende semiconductors which exhibit particularly strong spin-orbit coupling. This property makes nanostructures made of these materials very favorable systems for possible experimental observations of zitterbewegung. Our investigations include electrons in $n$-doped quantum wells under the influence of both Rashba and Dresselhaus spin-orbit interactions, and also the two-dimensional hole gas. Moreover, we give a detailed anaysis of electron zitterbewegung in quantum wires which appear to be particularly suited for experimentally observing this effect.

Figure 1

Zitterbewegung of an electron in a harmonic quantum wire perpendicular to the wire direction. The electron is initially injected into the lowest subband with its spin pointing along the $z$ direction. The wave number $k_{0y}$ for the motion along the wire is $k_{0y}\lambda =5$. The amplitude of the zitterbewegung is maximal at the resonance $2\alpha k_{0y}=\hslash \omega$ (middle panel).

Figure 2

Amplitude of the zitterbewegung perpendicular to the wire direction as a function of $\Omega ∕\omega =2\alpha k_{0y}∕\hslash \omega$ for different values of the wave number $k_{0y}$ along the wire. Again the electron spin points initially in the $z$ direction.

Figure 3

The spin expectation values $⟨{\psi }_z\mid {\vec{\sigma }}_H\left(t\right)\mid {\psi }_z⟩$ as a function of time at the resonance condition $2\alpha k_{0y}=\hslash \omega$ with $k_{0y}\lambda =5$ (cf. mid panel of Fig. 1). The time evolution of $⟨{\psi }_z\mid {\sigma }_H^x\left(t\right)\mid {\psi }_z⟩$ (solid line) is governed by a single frequency as shown in Eq. (38), while the other components $⟨{\psi }_z\mid {\sigma }_H^y\left(t\right)\mid {\psi }_z⟩$ and $⟨{\psi }_z\mid {\sigma }_H^z\left(t\right)\mid {\psi }_z⟩$ (dashed lines) show beatings between two frequencies given by ${\mu }_{\pm }$.