Ab initio calculation of nonlinear optical properties for chiral carbon nanotubes. Second harmonic generation and dc-Pockels effect

Abstract

Although the static first hyperpolarizability $\beta$ of graphene and chiral carbon nanotubes (CNTs) vanishes by symmetry, that is no longer true for frequency-dependent fields in which case there is a unique nonzero component. We evaluate that component for the dc-Pockels effect and second harmonic generation of several CNTs by means of the coupled perturbed Kohn–Sham method with the CAM-B3LYP functional as implemented in the CRYSTAL code. Both electronic and vibrational contributions are considered. Of particular interest is the frequency dependence in the region below the first resonance. Our results for the role of band gap, type of chirality, CNT radius and effect of orbital relaxation are analyzed using the conventional sum-over-states formulation in conjunction with the calculated band structure and density of states.

Appendix

1.1 Electronic contribution to $\beta$

Equation 1 may be written as:

$$\begin{aligned} \beta _{xyz}(-~\omega _\sigma ;\omega _1,\omega _2)= & {} \sum _{n_\perp }\left( \sum _{n_\parallel } \langle 0\vert x\vert n_\parallel \rangle \langle n_\parallel \vert y\vert n_\perp \rangle \langle n_\perp \vert z\vert 0\rangle \left( \frac{1}{(\omega _{n_\parallel }-\omega _\sigma )(\omega _{n_\perp }-\omega _2)}-\frac{1}{(\omega _{n_\perp }+\omega _1)(\omega _{n_\parallel }+\omega _\sigma )}\right) \right. \nonumber \\&\quad + \left. \sum _{n_\parallel }\langle 0\vert x\vert n_\parallel \rangle \langle n_\parallel \vert z\vert n_\perp \rangle \langle n_\perp \vert y\vert 0\rangle \left( \frac{1}{(\omega _{n_\parallel }-\omega _\sigma )(\omega _{n_\perp }-\omega _1)}-\frac{1}{(\omega _{n_\perp }+\omega _2)(\omega _{n_\parallel }+\omega _\sigma )}\right) \right. \nonumber \\&\quad + \left. \sum _{n_\perp '}\langle 0\vert y\vert n_\perp \rangle \langle n_\perp \vert x\vert n_\perp '\rangle \langle n_\perp '\vert z\vert 0\rangle \left( \frac{1}{(\omega _{n_\perp }+\omega _1)(\omega _{n_\perp }'-\omega _2)}-\frac{1}{(\omega _{n_\perp }+\omega _2)(\omega _{n_\perp }'-\omega _1)}\right) \right) \end{aligned}$$

(10)

In deriving Eq. 10 we have utilized the symmetry relations:

$$\begin{aligned} \langle 0\vert x\vert n_\parallel \rangle \langle n_\parallel \vert y(z)\vert n_\perp \rangle \langle n_\perp \vert z(y)\vert 0\rangle = -~\langle 0\vert y(z)\vert n_\perp \rangle \langle n_\perp \vert z(y)\vert n_\parallel \rangle \langle n_\parallel \vert x\vert 0\rangle \end{aligned}$$

(11)

and

$$\begin{aligned} \langle 0\vert y(z)\vert n_\perp \rangle \langle n_\perp \vert x\vert n_\perp '\rangle \langle n_\perp '\vert z(y)\vert 0\rangle = -~\langle 0\vert z(y)\vert n_\perp \rangle \langle n_\perp \vert x\vert n_\perp '\rangle \langle n_\perp '\vert y(z)\vert 0\rangle \end{aligned}$$

(12)

where the wavefunctions are assumed to be real. The relations in Eqs. (4)–(5) are a consequence of the $\theta =\pi /2$ screw $T_q^w$-axis rotation operation for the $T_q^wD_n$ symmetry group of the $(n_1,n_2)$ chiral nanotube (see Ref. [49] for the definition of q and w in the $2\pi w/q$ rotation angle). This operation corresponds to the transformation $x\rightarrow x\pm 0.5a$ (a is the cell parameter), $y\rightarrow \pm ~ z$, $z\rightarrow \mp ~y$.

1.2 Vibrational contribution to $\beta$

Using Eq. 1 with vibronic states denoted by the $\vert n,i_v\rangle$, where i refers to the vibrational mode and the subscript v to the vibrational level, we have:

$$\begin{aligned} \beta _{yxz}^{\mathrm{vib}}(-~\omega _\sigma;~\omega _1,~\omega _2)&= {} P_{(y/-\omega _\sigma ,x/\omega _1,z/\omega _2)} \sum \limits _{i=1}^{3N-6} \sum \limits _{n\ne 0,i_v} \left( \frac{\langle i_0,0\vert y\vert 0,i_1\rangle \langle i_1,0\vert x\vert n,i_v\rangle \langle i_v,n\vert z\vert 0,i_0\rangle }{(\omega _i-\omega _\sigma )(\omega _{n,i_v}-\omega _2)} \right. \nonumber \\&\quad \left. +\frac{\langle i_0,0\vert y\vert n,i_v\rangle \langle i_v,n\vert x\vert 0,i_1\rangle \langle i_1,0\vert z\vert 0,i_0\rangle }{(\omega _{n,i_v}-\omega _\sigma )(\omega _i-\omega _2)}\right) \nonumber \\&= {} P_{(y/-\omega _\sigma ,x/\omega _1,z/\omega _2)} \sum \limits _{i=1}^{3N-6}\left( \frac{\left\langle i_0\left| \frac{\partial \mu _{y}}{\partial Q_i}Q_i\right| i_1\right\rangle \left\langle i_1\left| \frac{1}{2}\frac{\partial \alpha _{xz}'(\omega _2)}{\partial Q_i}Q_i\right| i_0\right\rangle }{\omega _i-\omega _\sigma } \right. \nonumber \\&\quad +\left. \frac{\left\langle i_0\left| \frac{1}{2}\frac{\partial \alpha _{yx}'(-\omega _\sigma )}{\partial Q_i}Q_i\right| i_1\right\rangle \left\langle i_1\left| \frac{\partial \mu _z}{\partial Q_i} Q_i\right| i_0\right\rangle }{\omega _i-\omega _2} \right) \end{aligned}$$

(13)

here $\omega _i$ and $Q_i$ are, respectively, the phonon frequency and normal coordinate of the i-th vibrational mode at the $\varGamma$-point, and $\mu _u$ is the dipole moment component along the Cartesian u direction. In addition, $\alpha _{vw}'(\pm ~\omega )$ is a frequency-dependent electronic polarizability obtained by assuming that vibrational energy differences are small compared to electronic energy differences so that $\omega _{n,i_v}\simeq \omega _{n,i_0}\simeq \omega _n$:

$$\begin{aligned} \alpha _{vw}'(\pm ~\omega )= & {} 2 \sum \limits _{n\ne 0}\frac{\langle 0\vert v\vert n\rangle \langle n\vert w\vert 0\rangle }{\omega _n\mp \omega } \simeq \sum \limits _{n\ne 0}\langle 0\vert v\vert n\rangle \langle n\vert w\vert 0\rangle \left( \frac{1}{\omega _n-\omega }+\frac{1}{\omega _n+\omega }\right) \nonumber \\= & {} \alpha _{vw}(\omega )\simeq \alpha _{vw}(0) \end{aligned}$$

(14)

This quantity is null at the equilibrium geometry for $v\ne w$ (in the case of chiral CNTs), but not its derivative with respect to $Q_i$. The approximation made in Eq. (14) is that we are operating in a non-resonant region where, for all excited electronic states, $\omega _n\gg \omega$. This assumption is, however, not completely valid for the electronic transitions in the IR region that were considered for some of the largest tubes: for example, the gap of (14,10)CNT is smaller than 0.2 eV, which corresponds to the edge of the IR phonon spectrum ($1600~\hbox {cm}^{-1}$).

Permuting $y/-~\omega _\sigma$ and $z/\omega _2$ in the first term of the last line of Eq. (13), we obtain:

$$\begin{aligned} \beta _{yxz}^{\mathrm{vib}}(-~\omega _\sigma;~\omega _1,~\omega _2)&\simeq {} P_{(y/-~\omega _\sigma ,x/\omega _1,z/\omega _2)} \sum \limits _{i=1}^{3N-6}\left( \frac{\left\langle i_0\left| \frac{\partial \mu _{z}}{\partial Q_i}Q_i\right| i_1\right\rangle \left\langle i_1\left| \frac{1}{2}\frac{\partial \alpha _{xy}(-~\omega _\sigma )}{\partial Q_i}Q_i\right| i_0\right\rangle }{\omega _i+\omega _2} +\frac{\left\langle i_0\left| \frac{1}{2}\frac{\partial \alpha _{yx}(-~\omega _\sigma )}{\partial Q_i}Q_i\right| i_1\right\rangle \left\langle i_1\left| \frac{\partial \mu _z}{\partial Q_i} Q_i\right| i_0\right\rangle }{\omega _i-\omega _2}\right) \nonumber \\ & \simeq {} P_{(y/-~\omega _\sigma ,x/\omega _1,z/\omega _2)} \sum \limits _{i=1}^{3N-6}\left\langle i_0\left| \frac{1}{2}\frac{\partial \alpha _{yx}(-~\omega _\sigma )}{\partial Q_i}Q_i\right| i_1\right\rangle \left\langle i_1\left| \frac{\partial \mu _z}{\partial Q_i} Q_i\right| i_0\right\rangle \left( \frac{1}{\omega _i-\omega _2}+\frac{1}{\omega _i+\omega _2}\right) \nonumber \\\simeq & {} \frac{1}{2}P_{(y/-~\omega _\sigma ,x/\omega _1,z/\omega _2)} \sum \limits _{i=1}^{3N-6} \left( \frac{\partial \alpha _{yx}(-~\omega _\sigma )}{\partial Q_i}\right) \left( \frac{\partial \mu _z}{\partial Q_i}\right) \frac{1}{\omega _i^2-\omega ^2_2} \end{aligned}$$

(15)

where we have used the integral: $\langle i_0\vert Q_i\vert i_1\rangle =\sqrt{1/2\omega _i}$.

The chiral nanotube symmetry [see discussion in connection with Eqs. (11) and (12)] leads to the following relation between numerators in the vibrational contribution:

$$\begin{aligned}&\left\langle i_0\left| \frac{1}{2}\frac{\partial \alpha _{zx}(-~\omega _\sigma )}{\partial Q_i}Q_i\right| i_1\right\rangle \left\langle i_1\left| \frac{\partial \mu _y}{\partial Q_i} Q_i\right| i_0\right\rangle \nonumber \\&\quad =-~\left\langle i_0\left| \frac{1}{2}\frac{\partial \alpha _{yx}(-~\omega _\sigma )}{\partial Q_i}Q_i\right| i_1\right\rangle \left\langle i_1\left| \frac{\partial \mu _z}{\partial Q_i} Q_i\right| i_0\right\rangle \end{aligned}$$

(16)

Due to symmetry $\frac{\partial \mu _x}{\partial Q_i}$ and $\frac{\partial \alpha _{yz}}{\partial Q_i}$ cannot simultaneously be nonzero, which leaves four remaining nonzero permutation terms that can be combined as:

$$\begin{aligned} \beta _{yxz}^{\mathrm{vib}}(-~\omega _\sigma;~\omega _1,~\omega _2) \simeq&\nonumber \\ \sum \limits _{i=1}^{3N-6} \left( \frac{\partial \alpha _{yx}}{\partial Q_i}\right) \left( \frac{\partial \mu _{z}}{\partial Q_i}\right) \left( \frac{1}{\omega _i^2-\omega _2^2}-\frac{1}{\omega _i^2-\omega _\sigma ^2}\right)&\end{aligned}$$

(17)

for frequencies in the non-resonant regime ($\omega _\sigma,~\omega _1,~\omega _2\ll \omega _n$).