Summary
The theory of the RPA optical response of a solid has been generalized in order to take into account also the possible presence of spatially nonlocal potentials in the Hamiltonian. Explicit expressions for first- and second-order susceptibilities are given in the new framework. The expressions obtained depend on the matrix elements of operators of the form of a commutator of a component of the position operatorr and an operator that commutes with the lattice translations. The problem of the evaluation of these matrix elements is solved in a simple manner by introducing an auxiliary, periodic position operator,XXXr. In such a way a general formulation is obtained that preserves the gauge invariance. As an application of the new theory, the second harmonic generation (SHG) from a semiconductor in a simple two-band model has been studied. The differences between our correct gauge-invariant results and those obtained in the usual local approximation is an indication of a slow convergence of the expressions obtained in the local approximation.
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Piparo, E., Girlanda, R. Nonlocal potential and optical properties of solids. Il Nuovo Cimento D 14, 679–694 (1992). https://doi.org/10.1007/BF02451715
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DOI: https://doi.org/10.1007/BF02451715