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Nonlinear optics from an ab-initio approach by means of the dynamical Berry phase: Application to second- and third-harmonic generation in semiconductors

We present an ab-initio real-time-based computational approach to study nonlinear optical properties in condensed matter systems that is especially suitable for crystalline solids and periodic nanostructures. The equations of motion and the coupling of the electrons with the external electric field are derived from the Berry-phase formulation of the dynamical polarization [Souza et al., Phys. Rev. B 69, 085106 (2004)]. Many-body effects are introduced by adding single-particle operators to the independent-particle Hamiltonian. We add a Hartree operator to account for crystal local effects and a scissor operator to correct the independent particle band structure for quasiparticle effects. We also discuss the possibility of accurately treating excitonic effects by adding a screened Hartree-Fock self-energy operator. The approach is validated by calculating the second-harmonic generation of SiC and AlAs bulk semiconductors: an excellent agreement is obtained with existing ab initio calculations from response theory in frequency domain [Luppi et al., Phys. Rev. B 82, 235201 (2010)]. We finally show applications to the second-harmonic generation of CdTe and the third-harmonic generation of Si.