Interband coupling in Bloch electron dynamics: Application to the two-band model

The theory of Bloch dynamics for a single electron in a homogeneous electric field of arbitrary time dependence is established. The complete density matrix formulation for the dynamics is developed, and the two-band model is extracted. The two-band model is studied with emphasis on the interband coupling. A standard “<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>s</a:mi><a:mtext>−</a:mtext><a:mi>p</a:mi></a:mrow></a:math>” two-band model is utilized from the nearest-neighbor tight-binding approximation to study the dynamics. In the analysis, the interband coupling paradigm is developed whereby the momentum component of the interband coupling is evaluated parametrically by using an analytically established band-matching condition. This leads to an interband coupling dependence on the band structure alone, and a significant reduction in computational complexity. The interband coupling parameter is then used to evaluate the transition probability as a function of external electric field over a wide range of field values. Results for the two-band current composites and each separate Bloch band are derived and analyzed. The temporal nature of the transition probability and current for a uniform electric field scenario is demonstrated and discussed.