Coulomb-driven terahertz-frequency intrinsic current oscillations in a double-barrier tunneling structure

We investigate time-dependent, room-temperature quantum electronic transport in GaAs/AlGaAs double-barrier tunneling structures (DBTSs). The open-boundary Wigner-Boltzmann transport equation is solved by the stochastic ensemble Monte Carlo technique, coupled with Poisson's equation and including electron scattering with phonons and ionized dopants. We observe well-resolved and persistent terahertz-frequency current-density oscillations in uniformly doped, dc-biased DBTSs at room temperature. We show that the origin of these intrinsic current oscillations is not consistent with previously proposed models, which predicted an oscillation frequency given by the average energy difference between the quasibound states localized in the emitter and main quantum wells. Instead, the current oscillations are driven by the long-range Coulomb interactions, with the oscillation frequency determined by the ratio of the charges stored in the emitter and main quantum wells. We discuss the tunability of the frequency by varying the doping density and profile.

Figure 1

A schematic of the double-barrier GaAs/AlGaAs tunneling structure. The barrier and quantum-well widths are denoted by $d$ and $w$, respectively, and $L$ is the system length. We assume that the device dimensions in the $x$ and $y$ directions are much greater than its length $L$.

Figure 2

(a) Time-averaged potential profile (solid) and charge density (dashed) as a function of position for the uniformly doped DBTS at an applied bias of 96 mV (bias for which intrinsic current oscillations are observed). The potential profile is obtained from a self-consistent solution of the coupled WBTE and Poisson's equation. (b) Time-averaged current density vs applied bias for DBTSs with 6-nm-wide (solid red) and 7-nm-wide (dashed blue) quantum wells. The time average is obtained over 1-ps intervals. Arrows in panel (b) are guides for Fig. 3. Arrow 2 also corresponds to the bias value of 96 mV, which is used in (a).

Figure 3

Current density vs time (left column) and the corresponding Fourier transform amplitudes (right column) for the uniformly doped DBTS from Sec. 3a at an applied bias of 1:50 mV (top row), 2:96 mV (middle row), and 3:130 mV (bottom row). Panel numbers 1–3 refer to the bias values marked in Fig. 2.

Figure 4

(a) Time-averaged potential profile (solid line) and charge density (short dashed line) as a function of position for the traditionally doped RTD at an applied bias of 180 mV. Long dashed line shows the doping profile of the device. The potential profile is obtained from a self-consistent coupled solution of the WBTE and Poisson's equation. (b) Time-averaged current density vs applied bias. The time average is obtained over 1-ps intervals. The arrow marks the bias value 180 mV, which is used in (a).

Figure 5

(a) Current density as a function of time for a bias of 180 mV (peak current) for the traditionally doped RTD (lower blue curve) and a comparison with the uniformly doped DBTS from Sec. 3a for a bias of $0.096$ mV (upper red curve). (b) Fourier spectrum of the current-density oscillations for the traditionally doped (blue curve) and uniformly doped DBTS (red curve). Fourier transform amplitudes are normalized so the maximum equals one.

Figure 6

Fourier transform amplitude of the time-dependent current density for the uniformly doped DBTS from Sec. 3a with the full scattering rates (blue squares), $50\%$ of the full scattering rates (green triangles), and $25\%$ of the full scattering rates (red circles). The inset shows the corresponding time-averaged potential profiles for a part of the device.

Figure 7

(a) Snapshot of the potential profile (thick black curve) for the uniformly doped DBTS from Sec. 3a at an applied bias of 96 mV (bias for which intrinsic oscillations are observed). The solid red and the dashed blue curves correspond to the ground-state probability densities in the well and emitter regions, respectively. (b) Snapshot of the potential profile (thick black curve) of the traditionally doped RTD from Sec. 3b in a steady state for an applied bias of 180 mV (bias for which intrinsic oscillations are observed). The solid red curve corresponds to the ground-state probability density of the main quantum well bound state. The emitter quantum well is very shallow and contains no bound states. In both (a) and (b), the two arrows define the emitter and well regions (the computational domain for the wave functions). The two domains overlap in the region of the left barrier.

Figure 8

Fourier transforms of the potential on the left side of the first barrier at $z_b=6$ nm (red open circles), the middle of the well at $z_w=0$ (green closed circles), and the potential difference $\Delta \left(t\right)=V(z_b,t)-V(z_w,t)$ (blue squares) for the uniformly doped DBTS from Sec. 3a. In the legend, the tilde denotes a Fourier transform from the time to frequency $\left(\nu \right)$ domain. The inset shows a snapshot of the potential profile where the positions $z_b$ (red downward-pointing triangle) and $z_w$ (green upward-pointing triangle) are marked. The figure shows that the largest Fourier components of the emitter and well potential oscillations are for frequencies above the intrinsic current oscillations. However, these high-frequency plasmonic oscillations [Eq. (9)] are in phase, as are the low-frequency oscillations around $0.4$ THz. In contrast, the potential difference between the barrier and well region oscillates with the same frequency as the current density ($4.5$ THz).

Figure 9

(a) Time evolution of $\Delta$, the energy difference between the MQW ground state and the EQW ground state (red circles) and current density (blue triangles) for the uniformly doped DBTS from Sec. 3a. From the figure we see that the energy difference between the ground states in the EQW and the MQW is oscillating with the same frequency as the current density and is close to zero when $\Delta$ is at a minimum. The current density leads $\Delta$ in phase by ${49}^{\circ }$. (b) Probability density for the emitter ground state at a few points in time (the corresponding values of $\Delta$ are denoted in the inset). We see that the EQW wave function varies weakly over time.

Figure 10

(a) Frequency of current oscillation $f_{\max }$ vs doping density. (b) Amplitude of current oscillations $J_A$ vs doping density. The amplitude is calculated using $J_A=2\sqrt{2}\sigma$, where $\sigma$ is the standard deviation in current density. (c) Relative amplitude of current oscillations $J_A/J_{\mathrm{av}}$, where $J_{\mathrm{av}}$ is the time-averaged current density. (d) Operating bias $V_{\mathrm{op}}$ (see main text for definition) as a function of doping density. (e) Frequency of current oscillations vs the ratio of time-averaged charge in the MQW and the EQW (see main text for details). All panels are for the uniformly doped DBTS from Sec. 3a.

Figure 11

Time-averaged potential profile in the uniformly doped DBTS from Sec. 3a at a doping density of $1.0\times {10}^{18} {\mathrm{cm}}^{-3}$ (solid red) and $2.0\times {10}^{18} {\mathrm{cm}}^{-3}$ (dashed blue) for applied bias of 96 and 77 mV, respectively. For the higher doping density, the potential well on the collector side is deep enough to form a bound state, which disrupts the current-density oscillations that are governed by charge transfer between the emitter and main quantum wells.

Figure 12

Fourier transform of the time-dependent potential value at the center of the MQW region, corresponding to $V(z_w,t)$ in Fig. 8, for doping densities of $1.0\times {10}^{18} {\mathrm{cm}}^{-3}$ (blue squares), $1.5\times {10}^{18} {\mathrm{cm}}^{-3}$ (filled circles), and $1.8\times {10}^{18} {\mathrm{cm}}^{-3}$ (red circles). Vertical lines mark the corresponding electron plasma frequencies, $9.65,11.81$, and $12.94$ THz, respectively. Plasma frequencies are calculated using Eq. (9), assuming an electron density equal to doping density. We see that as the electron density is increased, plasma oscillations increase both in frequency and amplitude. The lower-frequency Fourier component, corresponding to the frequency of intrinsic current oscillations, varies more slowly and remains between 4 and 5 THz.

Figure 13

Solid black curve shows the potential profile for the uniformly doped DBTS described in Sec. 3a. Long-dashed red curve shows the bias potential $V_{\Delta }\left(z\right)$, with parameters $\beta =0.035 \mathrm{nm}{}^{-1},z_0=0$, and $V_0=0.096$ eV. Short-dashed blue curve shows $V_{\delta }\left(z\right)$, which is the potential remaining when the bias and barriers have been subtracted from the total potential $V\left(z\right)$.