Nonlinear Terahertz Response of $n$-Type GaAs

Excitation of an $n$-type GaAs layer by intense ultrashort terahertz pulses causes coherent emission at 2 THz. Phase-resolved nonlinear propagation experiments show a picosecond decay of the emitted field, despite the ultrafast carrier-carrier scattering at a sample temperature of 300 K. While the linear THz response is in agreement with the Drude response of free electrons, the nonlinear response is dominated by the super-radiant decay of optically inverted impurity transitions. A quantum mechanical discrete state model using the potential of the disordered impurities accounts for all experimental observations.

Figure 1

(a) Linear propagation of THz pulses through an $n$-type GaAs layer. The electric field is plotted as a function of time for the incident pulse $E_{\mathrm{in}}(t)$ (dashed line) and for the transmitted pulse $E_{\mathrm{out}}(t)$ (dots). The expected $E_{\mathrm{out}}(t)$ according to the Drude model including reflection losses at the surfaces is shown for comparison (solid line). Inset: Experimental setup for the linear regime. (b) Corresponding power spectra. Diamonds, measured difference $|E_{\mathrm{out}}(\nu ){|}^2-|E_{\mathrm{in}}(\nu ){|}^2$. Solid line, the expected Drude result.

Figure 2

Nonlinear propagation of THz pulses through the $n$-type GaAs layer without the substrate. (a) Dashed line, incident pulse $E_{\mathrm{in}}(t)$; solid line, transmitted pulse $E_{\mathrm{out}}(t)$. (b) Corresponding power spectra. Inset: experimental setup. (c) The difference signal $E_{\mathrm{out}}(t)-E_{\mathrm{in}}(t)$ shows weakly damped oscillations at a frequency of $\nu =2\mathrm{THz}$. (d) The difference of the power spectra (diamonds) exhibits a pronounced, positive peak at 2 THz—in distinct contrast to the expected result from Drude theory (solid line).

Figure 3

(a) Difference signals $E_{\mathrm{out}}(t)-E_{\mathrm{in}}(t)$ for several amplitudes $E_{\max }$ of the incident THz pulse. The coherent oscillations after the main pulse (solid lines) occur above a threshold of $E_{\max }\approx 40\mathrm{kV}/\mathrm{cm}$. (b) Fourier transforms of $E_{\mathrm{out}}(t)-E_{\mathrm{in}}(t)$ after the main pulse. (c) Dots: Spectral power at 2 THz as a function of $E_{\max }^2$. For comparison, the dashed line shows a linear dependence.

Figure 4

Results of solving the single particle Schrödinger equation with the Coulomb potentials of randomly distributed impurities. (a) The solid line shows the joint density of states as a function of the energy difference between the ground state and excited states. (b) Dots, calculated transmission spectrum of the $n$-type GaAs layer at room temperature from our quantum mechanical model, which reproduces very well the result from the Drude model (solid line). Diamonds, ionization of half the neutral donors results in a pronounced gain around 2 THz. (c) Sketch of typical wave functions in the potential of disordered impurities.