Bloch-electron dynamics in homogeneous electric fields: Application to multiphoton absorption in semiconductors and insulators

Abstract

The theory of Bloch-electron dynamics for carriers in a homogeneous electric field of arbitrary time dependence is developed in consideration of the properties of multiphoton absorption (MPA) in semiconductors and insulators. The general approach is to utilize the accelerated Bloch-state representation (ABR) as a basis thereby treating the electric field exactly; also, the electric field is described in the vector potential gauge. In developing the ABR, the instantaneous eigenstates for the central Hamiltonian are obtained as the accelerated Bloch states and utilized to find the time-dependent solution to the Schr\"odinger equation. In introducing the Wigner-Weisskopf approximation (WWA), the transition probability amplitude between the states of the system is obtained and used in the application to MPA in semiconductors and insulators. The probability amplitude for MPA is derived for a plane-polarized radiation field. The periodicity of the time-dependent electric field serves as a principle time constant in developing the probability amplitude per period. Utilizing the WWA, the general MPA transition probability is derived for transitions between an arbitrary set of valence and conduction bands for a time-varying electric field in the $x$ direction. Within the WWA, the MPA transition probability, equivalent to the well-known Keldysh transition amplitude, is derived and expressed in terms of infinite-variable generalized Bessel functions and modified Bessel functions. The exact MPA result is analyzed in the small ($\ensuremath{\lesssim}{10}^{6}$ V/cm) and large ($\ensuremath{\gtrsim}{10}^{8}$ V/cm) electric field amplitude limits.