High-frequency impact ionization and nonlinearity of photocurrent induced by intense terahertz radiation in HgTe-based quantum well structures

We report on a strong nonlinear behavior of the photogalvanics and photoconductivity under excitation of HgTe quantum wells (QWs) by intense terahertz (THz) radiation. The increasing radiation intensity causes an inversion of the sign of the photocurrent and transition to its superlinear dependence on the intensity. The photoconductivity also shows a superlinear raise with the intensity. We show that the observed photoresponse nonlinearities are caused by the band-to-band light impact ionization under conditions of a photon energy less than the forbidden gap. The signature of this kind of impact ionization is that the angular radiation frequency $\omega =2\pi f$ is much higher than the reciprocal momentum relaxation time. Thus the impact ionization takes place solely because of collisions in the presence of a high-frequency electric field. The effect has been measured on narrow HgTe/CdTe QWs of 5.7 nm width; the nonlinearity is detected for linearly and circularly polarized THz radiation with different frequencies ranging from $f=0.6$ to 1.07 THz and intensities up to hundreds of kW/${\mathrm{cm}}^2$. We demonstrate that the probability of the impact ionization is proportional to the exponential function, $exp(-E_0^2/E^2)$, of the radiation electric field amplitude $E$ and the characteristic field parameter $E_0$. The effect is observable in a wide temperature range from 4.2 to 90 K, with the characteristic field increasing with rising temperature.

Figure 1

(a) Setup scheme for a photocurrent measurement between contacts 2 and 5. The current is measured as a voltage drop $U$ across a load resistor. (b) Setup scheme for a photoconductivity measurement between contacts 2 and 5. The signal is measured as a voltage drop $U$ along a load resistor $R_L$, while a bias voltage $V$ is applied.

Figure 2

(a) Reference laser pulse shape over time in arbitrary units. The signal is measured by the fast room temperature photon drag detector [30]. [(b)–(f)] Time variation of the photocurrent response excited by the illumination of the sample bulk with right-handed circularly polarized radiation at different intensities with a frequency of $f=0.6$ THz and measured diagonally between contacts 3 and 6. At low intensities, the current response follows the reference pulse with a negative amplitude, while at the higher intensities a nonlinear positive contribution is observed, which results in a change of sign of the photocurrent at approximately 10 kW/${\mathrm{cm}}^2$.

Figure 3

(a) The intensity dependences of the edge photocurrent excited by right-handed (open circles) and left-handed (full circles) circularly polarized radiation with a frequency of $f=0.6$ THz. The signal was picked up from contacts 1 and 6 under illumination of the corresponding sample edge. The change of sign and the nonlinear raise of the photocurrent is clearly seen, while there is no difference within the measuring accuracy between signals for the left- and right-handed excitation polarizations. (b) Dependencies of the photocurrent at the edge (blue, picked up from contacts 1 and 6) and in the sample bulk (red, picked up from contacts 2 and 5) excited by linearly polarized radiation with a frequency of $f=0.6$ THz on the intensity $I$. (c) Dependence of the photocurrent induced in the sample bulk on the quarter-wave plate angle $\phi$ measured at the intensity of $I=6.5$ kW/${\mathrm{cm}}^2$ for the radiation frequency of $f=0.6$ THz. The solid line shows a fit after Eq. (3) with the fitting parameters $J_0=-1.0\mu \mathrm{A}$, $J_1=-1.1\mu \mathrm{A}$, $J_2=-0.3\mu \mathrm{A}$, and $J_3=0.1\mu \mathrm{A}$.

Figure 4

Intensity dependencies of the photocurrent excited by linearly polarized radiation with a frequency of $f=0.6$ THz for the temperatures of $T=4.2$, 60, and 90 K. The signal was picked up from contacts 2 and 5 under illumination of the sample bulk. The inset shows the temperature dependence of the inversion intensity $I_{\text{inv}}^{\text{L}}$, which shifts to higher intensities at higher temperatures.

Figure 5

Dependencies of the normalized photoconductivity $\mathrm{\Delta }\sigma /\sigma$ excited by linearly polarized radiation in the bulk of the QW for the radiation frequencies of 0.6 (blue triangles), 0.77 (red triangles), and 1.07 THz (black triangles) on the intensity $I$. The signal is measured for contacts 2 and 5 using the setup shown in Fig. 1. Solid lines present the corresponding fits after Eq. (4) [see also theoretical Eq. (10)] with the fitting parameters $A$ and $I_0$. A nonlinear raise of the signal is clearly seen for all frequencies. Inset shows the kinetics for an exemplary photoconductivity pulse (solid line) measured in the bulk of the samples for linearly polarized radiation with a frequency of 0.6 THz and an intensity of 57 kW/${\mathrm{cm}}^2$. The dashed line shows the reference laser pulse.

Figure 6

Intensity dependences of the normalized photoconductivity $\mathrm{\Delta }\sigma /\sigma$ excited by linearly polarized radiation with $f=0.6$ THz in the bulk of the QW. The signal was measured for three temperatures of $T=4.2$, 60, and 90 K for the contacts 2 and 5 using the setup shown in Fig. 1. Solid lines show corresponding fits after Eq. (4) [see also theoretical Eq. (10)] with the fitting parameters $A$ and $I_0$. A nonlinear raise of the signal is clearly seen for all frequencies. Inset shows the temperature dependence of the fitting parameter $E_0^2\propto I_0$.

Figure 7

Logarithmic plot of the relative photoconductivity $\mathrm{\Delta }\sigma /\sigma$ excited by linearly polarized radiation in the bulk of the sample on the inverse squared electric field $E^{-2}$. Data are shown for the radiation frequencies of 0.6 (blue triangles), 0.77 (red triangles), and 1.07 THz (black triangles) and have been measured between the contacts 2 and 5. Solid lines present the corresponding fits after Eq. (4) [see also theoretical Eqs. (10) and (11).] with fitting parameters $A$ and $E_0^2$. Inset shows a dependence of the fitting parameter $E_0^2$ extracted from the corresponding fits from panel (a) on the squared radiation frequency $f^2$.

Figure 8

Intensity dependencies of the normalized photoconductivity $\mathrm{\Delta }\sigma /\sigma$ excited by linearly polarized radiation with $f=0.6$ THz in the sample with semitransparent gate. Solid lines show corresponding fits after Eq. (4) [see also theoretical Eq. (10)] with the fitting parameters $A$ and $I_0\propto E_0^2$. Inset shows the dependence of the fitting parameter $E_0^2$ on the Fermi energy. The latter has been obtained from the corresponding magnetotransport measurements. Note that due to unknown intensity acting on 2DEG after transmission through the gate all values are given in arbitrary units.

Figure 9

Calculated energy dispersion for a quantum well with a thickness of 5.7 nm using the $\mathit{k}·\mathit{p}$ Hamiltonian from Ref. [38]. Here, the band gap between the electron subband E1 (red) and the hole subband H1 (blue) is ${\varepsilon }_{\text{g}}=17.6$ meV. The dashed line indicates the Fermi level.

Figure 10

Dependencies of the photoconductivity (black triangles) and the photocurrent (red squares) excited by linearly polarized radiation with a frequency of $f=0.6$ THz on the intensity. Signals were measured between contacts 2 and 5. Dashed black line shows fit after Eq. (4). Here the fitting parameters $A=0.15$ and $I_0=41$ kW/${\mathrm{cm}}^2$ obtained from the photoconductivity measurements, see Figs. 5 and 7, are used. Solid red line shows a fit according to Eq. (25) [also see Eqs. (23) and (24)]. Here, $A$ and $I_0$ are taken from the photoconductivity fit, $n=3\times {10}^{11}{\mathrm{cm}}^{-2}$ from transport measurements and the parameters ${\chi }_c=3.4\times {10}^{-13}\mu \mathrm{A}{\mathrm{cm}}^4$ ${\mathrm{kW}}^{-1}$ and ${\chi }_v=8.2\times {10}^{-12}\mu \mathrm{A}{\mathrm{cm}}^4$ ${\mathrm{kW}}^{-1}$ are obtained from fitting. At low intensities, the photocurrent varies linearly with the intensity, while the photoconductivity signal is negligibly small. At higher intensities the photocurrent starts to show nonlinear behavior as the photoconductivity increases superlinearly. The inset shows the dependence of the diagonal photocurrent between contacts 3 and 6 excited by linearly polarized radiation with a frequency of $f=0.6$ THz on the intensity. The solid line shows a fit according to Eq. (25) with the fitting parameters ${\chi }_c=2.7\times {10}^{-13}\mu \mathrm{A}{\mathrm{cm}}^4$ ${\mathrm{kW}}^{-1}$ and ${\chi }_v=6\times {10}^{-12}\mu \mathrm{A}{\mathrm{cm}}^4$ ${\mathrm{kW}}^{-1}$. It is seen that the inversion intensity $I_{\text{inv}}$ increases in this measurement direction.