Terahertz-induced interband tunneling of electrons in GaAs

Ultrafast high-field transport is studied in $n$-type GaAs with ultrashort terahertz (THz) pulses of an electric field amplitude of up to 300 kV/cm. At lattice temperatures between $T=300$ and 80 K, we observe coherent ballistic transport of electrons over a major part of the first Brillouin zone. At $T=300\text{K}$, ballistic transport occurs at a constant electron density whereas at lower temperatures, the THz pulses generate additional electron-hole pairs by field-induced tunneling between valence and conduction bands. We show that the ultrashort decoherence time of superpositions of valence- and conduction-band states plays a crucial role for the efficiency of the tunneling process. The extremely fast interband decoherence at room temperature results in a negligible tunneling rate.

Figure 1

(Color online) (a) Electronic band structure of GaAs along the [100] direction calculated by the local pseudopotential method without spin-orbit interaction (Refs. 3, 4, 5). (b) Corresponding velocity of electrons in the lowest conduction band (cb) and in the light-hole band (lh), Eq. (2). (c) Interband matrix element squared $X_{\text{lh-cb}}^2$ for the transition between the light-hole band and the lowest conduction band.

Figure 2

(Color online) The Ti:sapphire oscillator-amplifier system generates 25 fs pulses at a wavelength of 800 nm with a repetition rate of 1 kHz and a pulse energy of 0.5 mJ. THz pulses, obtained by optical rectification in a GaSe crystal, are focused onto the sample, which is mounted on the cold finger of a cryostat. Electro-optic sampling in a [110] ZnTe crystal allows the measurement of the electric field as a function of delay time. The polarization state of the probe pulses from the oscillator is read out by the quarter-wave plate $\left(\lambda /4\right)$, a Wollaston polarizer (WP), and the two photodiodes PD1 and PD2. The whole THz path and the sample are in a vacuum chamber.

Figure 3

(Color online) Transients of the incident [$E_{\text{in}}\left(t\right)$, dotted lines, right ordinate scale] and of the emitted [$E_{\text{em}}\left(t\right)$, solid lines, left ordinate scale] electric fields at temperatures of 200 K [(a) and (b)] and 80 K [(c) and (d)] for different field amplitudes.

Figure 4

(Color online) (a) Measured emitted field transients $E_{\text{em}}\left(t\right)=E_{\text{tr}}\left(t\right)-E_{\text{in}}\left(t\right)$ for lattice temperatures of $T=80\text{K}$ [solid line, same data as in Fig. 3d] and of 300 K (dots). Although the amplitude of $E_{\text{in}}\left(t\right)$ is 300 kV/cm in both cases, the low-temperature curve has an amplitude corresponding to a ten times higher carrier density, i.e., to $n_{\text{e}}\approx 10\times N_D=2\times {10}^{17}{\text{cm}}^{-3}$. (b) Normalized transiently absorbed energy [Eq. (6)] for $T=80\text{K}$ (solid line). After the THz pulse the absorbed energy of 15 eV corresponds to $\approx 2\times {10}^{17}{\text{cm}}^{-3}$ field-generated electron-hole pairs. The dashed lines in (a) and (b) are calculated from the theoretical model explained in Sec. 4.

Figure 5

(Color online) Maximum energy $W_{\text{max}}$ (open symbols) and irreversibly absorbed energy $W_{\text{irr}}$ (closed symbols) as a function of the incident electric field amplitude, calculated according to Eq. (6), for temperatures of 80, 200, and 300 K. For $T=80\text{K}$ and for $T=200\text{K}$, the dashed lines are fits to $W_{\text{max}}=\text{const}{E}_{\text{in}}^2$. The hatched area shows the possible values for $W_{\text{max}}$ if only the electrons present by doping participate, i.e., if no tunneling occurs.

Figure 6

(Color online) Population of the upper level (conduction band) after the end of the exciting THz pulse calculated for a two-level system as a function of the constant decoherence time $\tau$ (dots). The diamonds are estimates for the decoherence times at the temperatures of our experiment.

Figure 7

(Color online) Dynamics of the electron wave function during interband tunneling: (a) valence-band electron wave packet (schematic) before THz excitation. (b) The THz field drives an off-resonant interband Rabi oscillation resulting in a coherent superposition of valence- and conduction-band wave packets. The amount of conduction-band character is determined by the parameter $\Theta$. (c) Further acceleration of the superposition will spatially separate the conduction-bandlike part from the valence-bandlike part of the same wave function with a pronounced nonclassical character.

Figure 8

(Color online) (a) Incident electric field $E_{\text{in}}\left(t\right)$ (field amplitude 300 kV/cm, dashed line), corresponding vector potential $A\left(t\right)$ (dashed-dotted line), and time derivative of the conduction-band population $d{p}_{\text{cb}}/dt$ calculated for ${\gamma }_m={\left(1\text{ps}\right)}^{-1}$ and $T=80\text{K}$. The arrows mark the zeros of the vector potential. (b) Time dependence of the conduction-band population calculated for different momentum relaxation rates ${\gamma }_m$.

Figure 9

(Color online) Two possible geometries for the transfer of the electric field from the sample to the electro-optic crystal (EOX). (a) In our setup, the radiation from the sample is collimated (in the actual experiment instead of lenses parabolic mirrors are used), propagates to the second lens and is focused onto the electro-optic crystal. This results in the on-axis $\left(\rho =0\right)$ electric field at the detector ${\vec{E}}_{\text{det}}\left(0,t\right)=-{\vec{E}}_{\text{sample}}\left(0,t\right)$. Also shown is our sample with the etched substrate (gray). (b) In another often used setup (Refs. 47, 48, 49), the THz emission is focused by one lens onto the electro-optic crystal. This results in ${\vec{E}}_{\text{det}}\left(0,t\right)\propto \partial {\vec{E}}_{\text{sample}}\left(0,t\right)/\partial t$.