Nonlinear Electromagnetic Response of a Uniform Electron Gas

The linear electromagnetic response of a uniform electron gas to a longitudinal electric field is determined, within the self-consistent-field theory, by the linear polarizability and the Lindhard dielectric function. Using the same approach, we derive analytical expressions for the second- and third-order nonlinear polarizabilities of the three-, two-, and one-dimensional homogeneous electron gases with the parabolic electron energy dispersion. The results are valid both for degenerate (Fermi) and nondegenerate (Boltzmann) electron gases. A resonant enhancement of the second- and third-harmonics generation due to a combination of the single-particle and collective (plasma) resonances is predicted.

Figure 1

The real (black solid curves) and imaginary (red dashed curves) parts of the first-order polarizability of the 2D electron gas (9) as a function of the wave vector $Q$ at (a) $\mathrm{\Omega }=0.5$ and (b) $\mathrm{\Omega }=1.5$, Ref. [2]. The inset in (b) shows the single-particle absorption continuum (shaded area) bounded by the curves (13) with $o=1$. The red dash-dotted lines in the inset show the values of $\mathrm{\Omega }$ for which the polarizability ${\pi }_{\mathit{q}\omega ;\mathit{q}\omega }^{(1),2}$ is plotted on the main plots.

Figure 2

The second-order polarizability of the 2D electron gas (11) at $k_F{a}_B=1$. The inset in (b) shows the single-particle absorption (shaded) areas bounded by the curves (13) with $o=1$ and $o=2$. The range of $Q$ shown in (a) and (b) is encircled in the inset.

Figure 3

The third-order polarizability (12) of the 1D electron gas at $k_F{a}_B=1$. The shaded area in the inset in (b) is the single-particle absorption region, which has a different form in the 1D case, as compared to the 2D and 3D cases. The range of $Q$ shown in (a) and (b) is encircled in the inset.

Figure 4

The spectrum of 2D plasmons (thick red curve) at $k_F{a}_B=1.2$. The numbers 1, 2, 3 on the curves mark the single-particle absorption areas (13) with $o=1,2,3$. The maximum of the second- (third-) order response function is achieved at the point $A$ ($B$) corresponding to the intersection of the 2D plasmon curve with the second- and third-order boundary curves (13).