Semiconductor optics in length gauge: A general numerical approach

Abstract

We present a numerical method for studying electron dynamics in semiconductor crystals interacting with electromagnetic fields. The approach employs the length gauge and is consequently free of the unphysical divergences that arise in the velocity gauge as → 0. The topology of the electronic structure of the Brillouin zone is taken into account by the discretization employed, and the computational method is invariant under local gauge symmetry of the Bloch functions. Arbitrary electronic structures and any temporal dependence of the fields can be handled within this approach. We give some illustrative calculations to compare and contrast numerical results of velocity and length gauges for subgap excitations of semiconductors.within the dipole approximation, where H 0 is the full Hamiltonian of the crystal and E͑t͒ is the electric field. When ex-PHYSICAL REVIEW B 76, 035213 ͑2007͒

Theory of strong-field injection and control of photocurrent in dielectrics and wide band gap semiconductors

“…In contrast, the corresponding length-gauge equations have a term proportional to ∇ k δ(k ′ − k), which couples different k-points and introduces singularities, and matrix elements ξ f i (k), which are not uniquely defined. These problems with the length gauge in the description of optical phenomena in solids are well-known, and have been solved in recent theoretical treatments 22,23 . Nevertheless, the treatment of the crystal polarization in the velocity gauge has a few important drawbacks, discussed, for example, in Ref.…”

“…Nevertheless, the treatment of the crystal polarization in the velocity gauge has a few important drawbacks, discussed, for example, in Ref. 23. First, the solution of dynamic equations requires a large number of bands, only a few of which contain a significant carrier population at the end of a simulation.…”

Ultrafast Control of Strong-Field Electron Dynamics in Solids

“…Also, when periodic boundary conditions are applied in the length gauge, the interaction potential becomes discontinuous at the boundaries of the unit cell. Nevertheless, once these difficulties are addressed, the length gauge becomes an appropriate choice for numerical simulations [12][13][14][15][16].…”

“…The localization length of a Wannier-Stark state is given by 15) where ∆ i is the energy interval covered by band i. Our introduction to the Wannier-Stark states has so far followed the one given by Wannier [19].…”

“…The case of degenerate bands presents additional mathematical challenges [27,35,36]; in particular, the transition matrix elements ξ ξ ξ i j (k) are singular at degeneracies [36]. For reference, we also give the relation between ξ ξ ξ i j (k) and the matrix elements of the position operator between the Bloch functions [15,37,38]:…”