Zitterbewegung and its effects on electrons in semiconductors

Abstract

An analogy between the band structure of narrow gap semiconductors and the Dirac equation for relativistic electrons in vacuum is used to demonstrate that semiconductor electrons experience a Zitterbewegung (trembling motion). Its frequency is ${\ensuremath{\omega}}_{Z}\ensuremath{\approx}{\mathcal{E}}_{g}∕\ensuremath{\hbar}$ and its amplitude is ${\ensuremath{\lambda}}_{Z}$, where ${\ensuremath{\lambda}}_{Z}=\ensuremath{\hbar}∕{m}_{0}^{*}u$ corresponds to the Compton wavelength in vacuum (${\mathcal{E}}_{g}$ is the energy gap, ${m}_{0}^{*}$ is the effective mass, and $u\ensuremath{\approx}1.3\ifmmode\times\else\texttimes\fi{}{10}^{8}\phantom{\rule{0.3em}{0ex}}\mathrm{cm}∕\mathrm{s}$). Once the electrons are described by a two-component spinor for a specific energy band there is no Zitterbewegung but the electrons should be treated as extended objects of size ${\ensuremath{\lambda}}_{Z}$. The magnitude of ${\ensuremath{\lambda}}_{Z}$ in narrow gap semiconductors can be as large as $70\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$. Possible consequences of the above predictions are indicated.