Department of Applied Physics and Astronomy, University of Sharjah, Sharjah, United Arab Emirates
Abstract:
A comparative analysis of three different time‐independent approaches to studying open quantum structures in a uniform electric field was performed using the example of a one‐dimensional attractive or repulsive δ‐potential and the surface that supports the Robin boundary condition. The three considered methods exploit different properties of the scattering matrix as a function of energy E: its poles, real values, and zeros of the second derivative of its phase. The essential feature of the method of zeroing the resolvent, which produces complex energies, is the unlimited growth of the wave function at infinity, which is, however, eliminated by the time‐dependent interpretation. The real energies at which the unitary scattering matrix becomes real correspond to the largest possible distortion, , or its absence at which in either case leads to the formation of quasibound states. Depending on their response to the increasing electric intensity, two types of field‐induced positive energy quasibound levels are identified: electron‐ and hole‐like states. Their evolution and interaction in the enlarging field lead ultimately to the coalescence of pairs of opposite states, with concomitant divergence of the associated dipole moments in what is construed as an electric breakdown of the structure. The characteristic features of the coalescence fields and energies are calculated and the behavior of the levels in their vicinity is analyzed. Similarities between the different approaches and their peculiarities are highlighted; in particular, for the zero‐field bound state in the limit of the vanishing , all three methods produce the same results, with their outcomes deviating from each other according to growing electric intensity. The significance of the zero‐field spatial symmetry for the formation, number, and evolution of the electron‐ and hole‐like states, and the interaction between them, is underlined by comparing outcomes for the symmetric δ geometry and asymmetric Robin wall.
was performed using the example of a one‐dimensional attractive or repulsive δ‐potential and the surface that supports the Robin boundary condition. The three considered methods exploit different properties of the scattering matrix 
as a function of energy E: its poles, real values, and zeros of the second derivative of its phase. The essential feature of the method of zeroing the resolvent, which produces complex energies, is the unlimited growth of the wave function at infinity, which is, however, eliminated by the time‐dependent interpretation. The real energies at which the unitary scattering matrix becomes real correspond to the largest possible distortion, 
, or its absence at 
which in either case leads to the formation of quasibound states. Depending on their response to the increasing electric intensity, two types of field‐induced positive energy quasibound levels are identified: electron‐ and hole‐like states. Their evolution and interaction in the enlarging field lead ultimately to the coalescence of pairs of opposite states, with concomitant divergence of the associated dipole moments in what is construed as an electric breakdown of the structure. The characteristic features of the coalescence fields and energies are calculated and the behavior of the levels in their vicinity is analyzed. Similarities between the different approaches and their peculiarities are highlighted; in particular, for the zero‐field bound state in the limit of the vanishing 
, all three methods produce the same results, with their outcomes deviating from each other according to growing electric intensity. The significance of the zero‐field spatial symmetry for the formation, number, and evolution of the electron‐ and hole‐like states, and the interaction between them, is underlined by comparing outcomes for the symmetric δ geometry and asymmetric Robin wall. 