Effects of nonlocality on second-harmonic generation in bulk semiconductors

Calculations of the second-harmonic susceptibility tensor chi(abc)(-2 omega; omega, omega) are presented for bulk semiconductors within both the v . A and the r . E gauges. The description of the semiconductor states incorporates the "scissors" Hamiltonian commonly used to obtain the correct band gap. The nonlocality of the scissors correction leads to terms in chi(abc)(-2 omega; omega, omega) not considered before within a sum-over-states approach to the v . A gauge. Using this expression, we show that the results of the two gauges give the same result for chi(abc)(-2 omega; omega, omega), within very good numerical accuracy. As part of the derivation, we clarify the well-known result for the linear optical response which states that the scissors correction rigidly shifts the spectrum along the energy axis, keeping the line-shape intact. The calculation is presented for GaAs using an all-electron scheme and a pseudopotential scheme.