Phasorlike interpretation of tight-binding electronic motion in homogeneous electric fields

Abstract

We present a new specific interpretation of a previously derived general method [D. Sanjin\'es and J.-P. Gallinar, J. Phys.: Condens. Matter 11, 3729 (1999)] for studying electronic wave-packet evolution within the one-band approximation. As a result of analytical properties of Bessel functions, it is shown that in a homogeneous time-dependent electric field an electron's motion in a tight-binding band can be interpreted in terms of a phasor (polygonal) construction in the complex plane. The length of the phasors is proportional to the electronic-hopping matrix element and to the time increment of the dynamical evolution. When this time increment is infinitesimal, the directions of the phasors are expressed in terms of a time integral of the external field. Wave-packet mean position and velocity are also geometrically interpreted. Based upon our polygonal-curve construction, an interesting mathematical analogy is established between wave-packet evolution in a constant or in a linearly time-dependent electric field, and the optical phenomena of Fraunhofer or Fresnel diffraction, respectively. The first type of diffraction is related to the usual Bloch oscillation effect, while---associated with the mathematical properties of the Cornu spiral---the second one leads to ``asymptotic localization'' of the electron. Finally, for harmonically driven fields dynamical localization can also be elucidated within our complex-plane representation.