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.

Figure 1

Proposed real-time ab initio scheme to compute SHG and THG spectra in the $[{\Omega }_1,{\Omega }_2]$ energy range for extended systems with PBC: (a) Results from KS-DFT and $G_0{W}_0$ are input to determine the zero-field Hamiltonian. (b) The EOMs [Eq. (17)] are then integrated to obtain (c) the overlaps $S$ from which the polarization is computed as in Eq. (7). In the postprocessing step (d) the nonlinear susceptibilities are obtained by inversion of the Fourier matrix [Eq. (28)], see Sec. 3 for details.

Figure 2

Pictorial representation of the signal analysis in the postprocessing step. The signal $P\left(t\right)$ (red line) can be divided into two regions: an initial convergence region (up to $t\gg 1/{\gamma }_{\mathrm{deph}}$) in which the eigenfrequencies of the systems are filtered out by dephasing and a second region where Eq. (26) holds. In this second region the signal $P\left(t\right)$ is sampled within a period $T_L=2\pi /{\omega }_L$ to extract the $P_i^{\alpha }$ coefficients of Eq. (28). Note that $P\left(t\right)$ is not a realistic one: for illustration purposes we enhanced the second-harmonic signal that otherwise would not be visible on this scale.

Figure 3

Magnitude of ${\chi }^{\left(2\right)}(-2\omega ,\omega ,\omega )$ for bulk SiC calculated within the IPA (black triangles) and QPA (red circles). Each point corresponds to a real-time simulation at the given laser frequency (see Sec. 3). Comparison is made with results obtained ab initio by direct evaluation of the ${\chi }^{\left(2\right)}$ in Ref. 6 in IPA (gray solid line) and QPA (brown dashed line).

Figure 4

Magnitude of ${\chi }^{\left(2\right)}(-2\omega ,\omega ,\omega )$ for bulk SiC calculated within the IPA (black triangles) and TDH (red circles). Each point corresponds to a real-time simulation at the given laser frequency (see Sec. 3). Comparison is made with results obtained ab initio by direct evaluation of the ${\chi }^{\left(2\right)}$ in Ref. 6 in IPA (gray solid line) and TDH (brown dashed line).

Figure 5

Magnitude of ${\chi }^{\left(2\right)}(-2\omega ,\omega ,\omega )$ for bulk AlAs calculated within the IPA (black triangles) and QPA (red circles). Each point corresponds to a real-time simulation at the given laser frequency (see Sec. 3). Comparison is made with results obtained ab initio by direct evaluation of the ${\chi }^{\left(2\right)}$ in Refs. 6 and 38 in IPA (gray solid and dot-dashed line) and QPA (brown dashed and dotted line).

Figure 6

Magnitude of ${\chi }^{\left(2\right)}(-2\omega ,\omega ,\omega )$ for bulk AlAs calculated within the IPA (black triangles) and TDH (red circles). Each point corresponds to a real-time simulation at the given laser frequency (see Sec. 3). Comparison is made with results obtained ab initio by direct evaluation of the ${\chi }^{\left(2\right)}$ in Ref. 6 in IPA (gray solid line) and TDH (brown dashed line).

Figure 7

Magnitude of ${\chi }^{\left(2\right)}(-2\omega ,\omega ,\omega )$ for bulk CdTe calculated within the QPA (black triangles) and TDH (red circles). Each point corresponds to a real-time simulation at the given laser frequency (see Sec. 3). Comparison is made with results obtained ab initio by direct evaluation of the ${\chi }^{\left(2\right)}$ in Ref. 39 in IPA (gray solid line) and available experimental results in Refs. 40 and 41 (blue stars and diamonds).

Figure 8

Magnitude of the two independent components of the THG in bulk Si: $B=3{\chi }_{1212}^{\left(3\right)}$ (left panel) and $A={\chi }_{1111}^{\left(3\right)}$ (right panel) calculated within the QPA (black triangles). Each point corresponds to a real-time simulation at the given laser frequency (see Sec. 3). Comparison is made with results obtained by direct evaluation of the ${\chi }^{\left(3\right)}$ in Ref. 42 from empirical (gray solid line) and semi-ab initio tight binding (brown dashed line).

Figure 9

Dispersion in the magnitude of the anisotropy $\left|\sigma \right|=\left|\right(B-A)/A|$ (top panel) and in the relative phase of $B/A$ (bottom panel) calculated within the QPA (black triangles). Each point corresponds to a real-time simulation at the given laser frequency (see Sec. 3). Comparison is made with experimental results from Ref. 43 (blue stars) and results obtained by direct evaluation of the ${\chi }^{\left(3\right)}$ in Ref. 42 from empirical (gray solid line) and semi-ab initio tight binding (brown dashed line).