Ab initio nonlinear optics in solids: linear electro-optic effect and electric-field induced second-harmonic generation

Abstract

Second-harmonic generation (SHG), linear electro-optic effect (LEO) and electric-field induced second-harmonic generation (EFISH) are nonlinear optical processes with important applications in optoelectronics and photovoltaics. SHG and LEO are second-order nonlinear optical processes described by second-order susceptibility. Instead, EFISH is a third-order nonlinear optical process described by third-order susceptibility. LEO and EFISH are only observed in the presence of a static electric field. These nonlinear processes are very sensitive to the symmetry of the systems. In particular, LEO is usually observed through a change in the dielectric properties of the material while EFISH can be used to generate a “second harmonic” response in centrosymmetric material. In this work, we present a first-principle formalism to calculate second- and third-order susceptibility for LEO and EFISH. LEO is studied for GaAs semiconductor and compared with the dielectric properties of this material. We also present how it is possible for LEO to include the ionic contribution to the second-order macroscopic susceptibility. Concerning EFISH we present for the first time the theory we developed in the framework of TDDFT to calculate this nonlinear optical process. Our approach permits to obtain an expression for EFISH which does not contain the mathematical divergences in the frequency-dependent second-order susceptibility that caused until now many difficulties for numerical calculations.

Appendices

Appendix 1

Writing a time-dependent electric field as

$$\begin{aligned} {\mathbf {E}}(t)={\mathbf {E}}_0\left( e^{-i\omega t}+e^{i\omega t}\right) \end{aligned}$$

(23)

the time-dependent second-order polarization induced by ${\mathbf {E}}(t)$ is

$$\begin{aligned} {\mathbf {P}}^{(2)}(t) &= \chi ^{(2)}{\mathbf {E}} {\mathbf {E}}=\chi ^{(2)}_{\mathrm{SHG}} E_0^2\left( e^{-2i\omega t}+ e^{2i\omega t}\right) \nonumber \\&\quad +2\chi ^{(2)}_{\mathrm{OR}} E_0^2 \end{aligned}$$

(24)

where the second-order susceptibility $\chi ^{(2)}$ relates the polarization to the electric field. The first term corresponds to the second harmonic generation (SHG) and the second term to the optical rectification (OR).One can define a frequency-dependent SHG an OR polarization as ${\mathbf {P}}^{(2)}_{\mathrm{SHG}}(2\omega )=\chi ^{(2)}_{\mathrm{SHG}} E_0^2$ and ${\mathbf {P}}^{(2)}_{\mathrm{OR}}=2\chi ^{(2)}_{OR} E_0^2$.

In the general case, where the electric field contains two different frequencies,

$$\begin{aligned} {\mathbf {E}}(t)={\mathbf {E}}_{0,1}\left( e^{-i\omega _1t}+e^{i\omega _1t}\right) +{\mathbf {E}}_{0,2}\left( e^{-i\omega _2t}+e^{i\omega _2t}\right) \end{aligned}$$

(25)

the time-dependent second-order polarization becomes

$$\begin{aligned} {\mathbf {P}}^{(2)}(t) &= \chi ^{(2)}_{\mathrm{SHG}} E_{0,1}^2\left( e^{-2i\omega _1t}+e^{2i\omega _1t}\right) \nonumber \\&\quad +\chi ^{(2)}_{\mathrm{SHG}} E_{0,2}^2\left( e^{-2i\omega _2t}+e^{2i\omega _2t}\right) \nonumber \\&\quad +2\chi ^{(2)}_{\mathrm{OR}} E_{0,1}^2+2\chi ^{(2)}_{OR} E_{0,2}^2 \nonumber \\&\quad +2\chi ^{(2)}_{\mathrm{SFG}} E_{0,1}E_{0,2}\left( e^{-i(\omega _1+\omega _2)t}+e^{i(\omega _1+\omega _2)t}\right) \nonumber \\&\quad +2\chi ^{(2)}_{\mathrm{DFG}} E_{0,1}E_{0,2}\left( e^{-i(\omega _1-\omega _2)t}+e^{i(\omega _1-\omega _2)t}\right) \nonumber \\ \end{aligned}$$

(26)

where the last two terms correspond respectively to the sum frequency generation (SFG) and the difference frequency generation (DFG).

For the linear electro-optic effect (LEO), the total field is

$$\begin{aligned} E(t)=E_{0}\left( e^{-i\omega _1t}+e^{i\omega _1t}\right) +E_{dc} \end{aligned}$$

(27)

where $E_{\mathrm{dc}}=\lim \limits _{\omega \rightarrow 0}E_2(t)=2E_{0,2}$. The polarization corresponding to the linear electro-optic effect is then

$$\begin{aligned} \begin{aligned} P^{(2)}_{\mathrm{LEO}}(t)=2\chi ^{(2)}_{\mathrm{LEO}} E_{0}E_{\mathrm{dc}}\left( e^{-i\omega _1t}+e^{i\omega _1t}\right) \end{aligned} \end{aligned}$$

(28)

One can generalize these conventions to the third order case and we get for the EFISH process

$$\begin{aligned} P^{(3)}_{\mathrm{EFISH}}(t)= 3\chi ^3_{\mathrm{EFISH}} E_{\mathrm{dc}} E_{0}^2\left( e^{-2i\omega t}+e^{2i\omega t}\right) \end{aligned}$$

(29)

Appendix 2

In this Appendix, we present the detailed derivation of the ionic contribution to the LEO susceptibility. The frequency-dependent dielectric tensor depends on the frequency $\omega$, on the amplitude of the static electric field ${\mathcal {E}}$ and on the ionic displacement induced by the static field ${\mathbf {R}}(\pmb{\mathcal{E}})$. The ionic contribution is given by

$$\begin{aligned}&\frac{{\text {d}} \epsilon ^{\mathrm{ionic}}_{ij}({\mathbf {R}}(\pmb{\mathcal{E}}),\pmb{\mathcal{E}},\omega ) }{d {\mathcal {E}}_{k}}\vert _{{\mathbf {R}}_0,{\mathcal {E}}_{k}=0}\nonumber \\&\quad = \sum _{n\alpha }\frac{\partial \epsilon _{ij}({\mathbf {R}},0,\omega )}{\partial \tau _{n \alpha }}\vert _{{\mathbf {R}}_0} \frac{\partial \tau _{n \alpha } }{\partial {\mathcal {E}}_{k}}\vert _{{\mathcal {E}}_{k}=0} \end{aligned}$$

(30)

The evaluation of $\frac{\partial \tau _{n \alpha } }{\partial {\mathcal {E}}_{k}}\vert _{{\mathcal {E}}_{k}=0}$ has been proposed in [32]; it is based on the fact that the electric enthalpy $F(\pmb{\mathcal{E}})$ of a solid in an electric field is obtained by the minimization

$$\begin{aligned} F(\pmb{\mathcal{E}})= \min _{{\mathbf {R}}}F(\pmb{\mathcal{E}}, {\mathbf {R}}(\pmb{\mathcal{E}})) \end{aligned}$$

(31)

Expanding the electric enthalpy in terms of the static field, we get

$$\begin{aligned}&F({\mathbf {R}}(\pmb{\mathcal{E}}), \pmb{\mathcal{E}})=F({\mathbf {R}}(\pmb{\mathcal{E}}), \pmb{\mathcal{E}}=0)-\Omega _0\sum _{k}{\mathcal {P}}_k({\mathbf {R}}(\pmb{\mathcal{E}})){\mathcal {E}}_k \nonumber \\&\quad -\frac{\Omega _0}{2}\sum _{kj}\chi ^{(1)}_{kj}({\mathbf {R}}(\pmb{\mathcal{E}})){\mathcal {E}}_k{\mathcal {E}}_j + \cdots \end{aligned}$$

(32)

and, to first order in terms of $\pmb{\mathcal{E}}$, the minimization leads to

$$\begin{aligned} \sum _{n'\alpha '}\frac{\partial ^2 F({\mathbf {R}}, {\mathcal {E}}= 0)}{\partial \tau _{n\alpha }\partial \tau _{n'\alpha '}} \vert _{R_0}\frac{\partial \tau _{n'\alpha '}}{\partial {\mathcal {E}}_k}\vert _{{\mathcal {E}}_k=0}= \Omega _0\frac{\partial {\mathcal {P}}_k({\mathbf {R}})}{\partial \tau _{n\alpha }}\vert _{R_0} \end{aligned}$$

(33)

By decomposing $\tau _{n'\alpha '}$ in the basis of the zone-center phonon-mode eigendisplacements(${\mathbf {q}}=0$):

$$\begin{aligned} \tau _{n'\alpha ' }=\sum _m \tau _{m }U_m(n'\alpha ') \end{aligned}$$

(34)

we get

$$\begin{aligned}&\sum _m \frac{\partial \tau _{m } }{\partial {\mathcal {E}}_k} \vert _{{\mathcal {E}}_k=0} \sum _{n'\alpha '}\frac{\partial ^2 F({\mathbf {R}}, {\mathcal {E}}= 0)}{\partial \tau _{n\alpha }\partial \tau _{n'\alpha '}} \vert _{R_0}U_m(n'\alpha ')\nonumber \\&\quad = \Omega _0\frac{\partial {\mathcal {P}}_k({\mathbf {R}})}{\partial \tau _{n\alpha }}\vert _{R_0} \end{aligned}$$

(35)

Multiplying by $U_p(n\alpha )$ and summing over $n\alpha$

$$\begin{aligned}&\sum _m \sum _{n\alpha } \frac{\partial \tau _{m } }{\partial {\mathcal {E}}_k} \vert _{{\mathcal {E}}_k=0} \sum _{n'\alpha '}\frac{\partial ^2 F({\mathbf {R}}, {\mathcal {E}}= 0)}{\partial \tau _{n\alpha }\partial \tau _{n'\alpha '}} \vert _{R_0}U_m(n'\alpha ')U_p(n\alpha )\nonumber \\&\quad = \Omega _0\sum _{n\alpha }\frac{\partial {\mathcal {P}}_k({\mathbf {R}})}{\partial \tau _{n\alpha }}\vert _{R_0}U_p(n\alpha ) \end{aligned}$$

(36)

we finally obtain

$$\begin{aligned} \frac{\partial \tau _{m } }{\partial {\mathcal {E}}_k} \vert _{{\mathcal {E}}_k=0} =\frac{\Omega _0}{\omega ^2_{m}} \sum _{n\alpha }\frac{\partial {\mathcal {P}}_k({\mathbf {R}})}{\partial \tau _{n\alpha }}\vert _{R_0}U_m(n\alpha ) \end{aligned}$$

(37)

where we have used the fact that the second derivative of the enthalpy in Eq. (36) corresponds to the inter-atomic forces $M_n$ and the normalisation relation for the phonon modes $\sum _{n\alpha } M_{n} U_{m}(n \alpha )U_p(n\alpha )=\delta _{mp}$.

From Eqs. (30) and (37) and with the definition of the mode polarity, $p_{mk}=\Omega _0 \sum _{n\alpha }\frac{\partial {\mathcal {P}}_k({\mathbf {R}})}{\partial \tau _{n\alpha }}U_m(n\alpha )$, the ionic contribution becomes

$$\begin{aligned} \frac{{\text {d}} \epsilon ^{ionic}_{ij}({\mathbf {R}}(\pmb{\mathcal{E}}),\pmb{\mathcal{E}},\omega ) }{{\text {d}} {\mathcal {E}}_{k}}\vert _{{\mathbf {R}}_0,{\mathcal {E}}_{k}=0}= \sum _m \frac{p_{mk}}{\omega ^2_{m}} \frac{\partial \epsilon _{ij}({\mathbf {R}},0,\omega )}{\partial \tau _{m}}\vert _{{\mathbf {R}}_0} \end{aligned}$$

(38)