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

We present a new specific interpretation of a previously derived general method [D. Sanjinés 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.