Two-photon absorption in semiconductors: A multiband length gauge analysis

The simplest approach to deal with light excitations in direct-gap semiconductors is to model them as a two-band system: one conduction and one valence band. For such models, particularly simple analytical expressions are known to exist for the optical response such as multiphoton absorption coefficients. Here we show that generic multiband models do not require much more complicated expressions. Our length-gauge analysis is based on the semiconductors Bloch equations in the absence of all scattering processes. In the evaluation, we focus on two-photon excitation by a pump-probe scheme with possibly nondegenerate and arbitrarily polarized configurations. The theory is validated by application to graphene and its bilayer, described by a tight-binding model, as well as bulk zinc-blende semiconductors described by $\mathit{k}·\mathit{p}$ theory.

Figure 1

Example path (dark blue) in the algorithm for determining an ensemble of $\mathbf{k}$ points on a resonance contour line (red numbered points connected by dashed lines).

Figure 2

Local gauge restoration scheme to determine band-structure properties at each resonant $\mathbf{k}$ point (red circles). Eigenfunctions are required for all points of the “neighbor grid” and fixed in gauge, as shown by arrows. Connection elements ${\mathit{\xi }}_{\mathbf{k}}$ are determined at the nearest neighbors (black circles) in order to obtain ${\mathbf{\nabla }}_{\mathbf{k}}{\mathit{\xi }}_{\mathbf{k}}$ at the central point.

Figure 3

(a) Energy contours for the monolayer graphene VB-CB transition, $2{\gamma }_0\left|s_{\mathbf{k}}\right|$; cf. Eq. (50). (b) The CB spectrum of monolayer (dashed) and bilayer (solid) graphene along the $k_y=0$ line, indicated by the dashed line in (a). The inset shows a zoom close to the $K$ region. The spectrum is electron-hole symmetric.

Figure 4

Linear optical conductivity of mono- and bilayer graphene. The monolayer result corresponds to Eq. (65). The dotted line is the analytical low-frequency result for the bilayer given in Eq. (6) of Ref. [29].

Figure 5

Degenerate 2PA coefficients of mono- and bilayer graphene for co- and cross-polarized excitation. These are isotropic, i.e., independent of the light polarization with respect to the crystallographic axes. In (a), $\beta$ is shown as a function of photon wavelength, while in (b) the same data are shown using a different scaling and as a function of energy. In (a), the dotted line is $\beta =(\overline{K}/L_{\mathrm{c}}){\omega }^{-4}$, the dash-dotted line is $\sim {\omega }^{-3}$. In (b), the 2D absorption coefficient (i.e., not divided by $L_{\mathrm{c}}$), normalized by the long-wavelength result for the monolayer, is shown. The horizontal lines correspond to the values 1 and 2.

Figure 6

2PA coefficient of zinc-blende semiconductors as a function of the probe polarization direction (${\theta }_p$). The pump is copolarized (${\theta }_e={\theta }_p$) or cross polarized (${\theta }_e={\theta }_p+\pi /2$); the angle $\theta$ may be interpreted here as either ${\theta }_p$ or, to match with Ref. [11], ${\theta }_e$.

Figure 7

Dependence of bulk ZnSe 2PA coefficients ${\beta }_{\perp },{\beta }_{\parallel }$ on the nondegeneracy parameter. The sum frequency corresponds to $\hslash {\omega }_{\mathrm{\Sigma }}=3.098\mathrm{eV}$. The experimental data labeled “data” are those shown in Ref. [20]; no error bars are available. The polarization settings for each sample are also defined therein. The curves labeled “cw” are obtained from the analytical approach presented in this work, using the same eight-band $\mathit{k}·\mathit{p}$ model as in Ref. [20]. The curves labeled “pulses” are the results from dynamical simulations already shown in Ref. [20], but corrected by a small postprocessing mistake, a constant factor $\pi /3$. All curves and data are for room temperature. The curve labeled “2-band” is a fit of ${\beta }_{\parallel }\sim n_{0p}^{-1}{n}_{0e}^{-1}{\omega }_p^{-3}{\omega }_e^{-4}$ to the “cw” curve.

Figure 8

Same data as in Fig. 7, but plotting the ratio of cross- and copolarized 2PA coefficients ${\beta }_{\perp }/{\beta }_{\parallel }$. See the caption of Fig. 7 for details.

Figure 9

Comparison of the analytical approach with dynamical simulations as in Fig. 7, for bulk GaAs with $\hslash {\omega }_{\mathrm{\Sigma }}=1.5694\mathrm{eV}$. The “pulses” curves have some uncertainty and are spline fits to only three discrete data points per curve, since the computational expense is large.