Two-fluid hydrodynamic model for semiconductors

The hydrodynamic Drude model (HDM) has been successful in describing the optical properties of metallic nanostructures, but for semiconductors where several different kinds of charge carriers are present an extended theory is required. We present a two-fluid hydrodynamic model for semiconductors containing electrons and holes (from thermal or external excitation) or light and heavy holes (in $p$-doped materials). The two-fluid model predicts the existence of two longitudinal modes, an acoustic and an optical, whereas only an optical mode is present in the HDM. By extending nonlocal Mie theory to two plasmas, we are able to simulate the optical properties of two-fluid nanospheres and predict that the acoustic mode gives rise to peaks in the extinction spectra that are absent in the HDM.

Figure 1

The dispersion relations of the optical mode (blue lines) and the acoustic mode (red lines). The full and dashed lines show the real and imaginary components, respectively, of $k_{L,j}$. The parameters are ${\omega }_b/{\omega }_a=2,{\beta }_b/{\beta }_a=4,{\gamma }_a={\gamma }_b=0.01{\omega }_a$, and ${\epsilon }_{\infty }=1$. Notice that the optical mode has a finite imaginary component (i.e., is damped) below ${\omega }_{\mathrm{eff}}/{\epsilon }_{\infty }^{1/2}$. Due to the small damping constants, the red dashed line lies almost exactly on top of the $y$ axis.

Figure 2

(a) The extinction spectrum for semiconductor A (see parameters in the main text) with $R=10\mathrm{nm}$ and ${\epsilon }_D=1$. The spectrum has been normalized with ${\sigma }_{\mathrm{geom}}=\pi R^2$. The dashed line is the local Drude model, and the full line is the two-fluid model. (b) The same spectrum plotted with the logarithmic $y$ axis. The bulk plasmon peaks are labeled with $[j,n]$, while the first acoustic peak and the LSP peak are indicated with “X” and “Y,” respectively.

Figure 3

The charge distribution in the $xz$ plane for semiconductor A at different frequencies. The damping constants have been set to ${\gamma }_a={\gamma }_b=1.0\times {10}^{11}{\mathrm{s}}^{-1}$ to make the patterns more clear. The incoming wave is directed in the $z$ direction with the electrical field polarized in the $x$ direction.

Figure 4

The spectrum of semiconductor A (parameters given in the main text) as found by the two-fluid model is shown with the solid black line. The dashed magenta line and the dash-dotted green line show the spectra found with the single-fluid HDM when including the $a$ fluid and $b$ fluid, respectively.

Figure 5

The spectrum of semiconductor A (parameters given in the main text) as found by the two-fluid model for different values of ${\beta }_b$: $0.33{\beta }_a,0.9{\beta }_a$, and $0.9999{\beta }_a$. When the $\beta$'s approach the same value, the spectrum coincides with the one predicted by the single-fluid HDM (shown with the dashed green line).

Figure 6

The spectrum of semiconductor A (parameters given in the main text) as found by the two-fluid model for different values of ${\omega }_b$: $0.5{\omega }_a,0.1{\omega }_a$, and $0.01{\omega }_a$. As ${\omega }_b$ diminishes, the spectrum becomes identical to the one obtained by the single-fluid HDM (shown with the dashed green line).

Figure 7

Extinction spectra for InSb and GaAs. In all cases is ${\epsilon }_D=1$. (a) The dashed red line and the solid blue line show the spectra for intrinsic InSb at $300\mathrm{K}$ with $R=100\mathrm{nm}$ and laser excited GaAs with $u_{\mathrm{pulse}}={10}^5\mathrm{J}{\mathrm{cm}}^{-3}$ and $R=40\mathrm{nm}$, respectively. (b) The dashed red line and the solid blue line show the spectra for intrinsic InSb at $400\mathrm{K}$ with $R=60\mathrm{nm}$ and laser excited GaAs with $u_{\mathrm{pulse}}={10}^6\mathrm{J}{\mathrm{cm}}^{-3}$ and $R=15\mathrm{nm}$, respectively. (c) The spectrum for $p$-doped GaAs with $N_A={10}^{19}{\mathrm{cm}}^{-3}$ and $R=30\mathrm{nm}$ is shown with the dashed red line. For the solid blue line the mobility of the holes is artificially set 100 times higher.

Figure 8

(a) The spectral position of the first acoustic peak as a function of ${\omega }_a$ for semiconductor A is shown with a red line, while the blue line shows the position of the LSP peak. The vertical dashed line shows the value ${\omega }_a=3.6\times {10}^{14}{\mathrm{s}}^{-1}$ which was used in previous figures with semiconductor A. The dent in the blue line around ${\omega }_a\approx {10}^{14}{\mathrm{s}}^{-1}$ is not a numerical artifact, but is caused by bulk plasmon resonances that are superimposed on the LSP peak and thereby make the definition of the peak ambiguous. (b) The amplitude of the first acoustic peak normalized with ${\sigma }_{\mathrm{geom}}$ as a function of ${\omega }_a$.

Figure 9

(a) The spectral positions of the first acoustic peak and the LSP peak as functions of $R$ for intrinsic GaAs shown with a red and blue line, respectively. GaAs was excited with a laser pulse of $u_{\mathrm{pulse}}={10}^6\mathrm{J}{\mathrm{cm}}^{-3}$. The vertical dashed line marks the value $R=15\mathrm{nm}$ which was used in Fig. 7. (b) The amplitude of the first acoustic peak as a function of $R$ (un-normalized). The number of electrons in the particle is indicated for three different sizes.