Second-Harmonic Generation Spectroscopy from Time-Dependent Density-Functional Theory
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Second-Harmonic Generation Spectroscopy from Time-Dependent DensityFunctional Theory E. Luppi1,2, H. Hübener1,3, M. Bertocchi1,4, E. Degoli4 , S. Ossicini4 and V. Véniard1 1
Laboratoire des Solides Irradiés, Ecole Polytechnique, 91128 Palaiseau, France. Department of Chemistry, University of California, Berkeley CA, 94720, U.S.A. (current affiliation). 3 Department of Materials, University of Oxford, Parks Road, Oxford, OX (current affiliation). 4 Department of Physics, University of Modena and Reggio Emilia, 41125 Modena, Italy. 2
ABSTRACT We developed an ab initio formalism based on Time-Dependent Density-Functional Theory for the calculation of the second-order susceptibility χ (2) (Luppi et al. J. Chem. Phys. 132, 241104 (2010)). We apply this formalism to the calculation of second-harmonic generation spectra of hexagonal SiC polytypes, ZnGeP2 (ZGP) and GaP. Starting from the independent-particle approximation, we include manybody effects, such as quasiparticle via the scissors operator, crystal local fields and excitons. We consider two different types of kernels: the ALDA and the “long-range” kernel. We analyze the effects of the different electron-electron descriptions in the spectra, finding good agreement with experiments. INTRODUCTION Nonlinear optics studies the interaction of light with matter where the response of the material to the applied electromagnetic field is nonlinear in the amplitude of the field [1]. Among the nonlinear phenomena existing in nature, the main role is played by second-harmonic generation (SHG). In SHG, a pump wave with a frequency of ω generates a signal at the frequency 2ω which propagates through a medium with a quadratic nonlinearity proportional to the macroscopic second-order susceptibility χ (2) . In this work, we calculate SHG spectra for hexagonal SiC polytype, ZnGeP2 (ZGP) and GaP. We use the ab initio theory we developed [2] based on Time-Dependent Density-Functional Theory (TDDFT). We compare the effects of many-body effects (crystal local fields and excitons). For ZGP and GaP, we also show the real and imaginary part of the dielectric function. THEORY The SHG spectra presented in this work is calculated using the ab-initio theory in Ref. [2]. We formulate the macroscopic second-order polarization P (2) M in terms of the total electric field E M , which reads as (2) P (2) (q1 + q 2 , q1, q 2 , ω )E M (q1 )E M (q 2 ) , M (q1 + q 2 , ω ) = d
where the vector q indicate the polarization of the electric field. Here, we use d (2) instead of
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(1)
χ (2) because it keeps track of the symmetry properties of the system. We recall that if the last two indices of the tensors are the same, we have dijj(2) = χ ijj(2) , while if the indices are different we (2) [1]. Later, when discussing results, we show χ (2) quantities, since they are more have 2dijk(2) = χ ijk
commonly used. As we are interested in the low energy part of the SHG spectrum we will consider only the optical limit, i.e. q → 0 . It has thus been natural to develop our approach in TDDFT. We obtain that the m
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