Figure 1

(Color online) (a) Schematic conduction-band diagram of a typical heterostructure used in our investigation, shown as the chemical potential ${\mu }_I$ of the injector is increasing so that electrons are tunneling through the TB into vacant 2DEG states in the QW. The insulating barrier (INS) is opaque to tunneling at the values of $V_{\text{dc}}$ used in this experiment. (b) Evolution of ${\mu }_I$ over time for $V_{\text{ac}}$ of frequency $f$. If ${\tau }_{\ell }$ is much longer than the period $1/f$, then the 2DEG cannot charge or discharge fast enough for ${\mu }_{2\text{D}}$ to follow ${\mu }_I$, and the device capacitance (measured between the injector and isolated electrode) will decrease. (c) as in (a) but shown with $V_{\text{ac}}$ advanced 1/2 cycle so that electrons are tunneling out of the 2DEG into the injector.Reuse & Permissions

Figure 2

(Color online) (a) Representative capacitance-frequency curve from an impedance measurement on sample N. (b) Device equivalent circuit that we use to fit measured impedance traces, thereby allowing us to determine ${\tau }_{rc}$. Fits are made simultaneously to both the active and reactive impedance components. $C_Q$ is the capacitance of the 2DEG and $R_{\text{TB}}={\tau }_{rc}/C_Q$ is the tunneling resistance of the TB, yielding an $RC$ time constant of ${\tau }_{rc}$. $C_{\text{TB}}$ and $C_G$ are geometric capacitances of the TB and INS barrier, respectively.Reuse & Permissions

Figure 3

(Color online) Diagram depicting the energy dependence of the tunneling as a function of scaled barrier thickness $\lambda =\ell /{\ell }_0$ and scaled electron kinetic energy $\xi ={\epsilon }_K/\left(\varphi -{\epsilon }_0\right)$. For small $\lambda$ and $\xi$ (i.e., the lower left, unshaded area of the diagram), transport is predominantly momentum conserving and is approximately described by Eq. (1). In the opposite regime (shaded region), transport is dominated by scattering-assisted tunneling and varies with $\epsilon$, as in Eq. (2). Using Eq. (8) and the scaled scattering rate $\eta$, one can find values of $\xi$ and $\lambda$ for which the two transport mechanisms yield approximately the same tunneling rate (broken line, calculated for $\eta =0.1$, corresponding to an approximate sheet mobility of $5\times {10}^4{\text{cm}}^2{\text{V}}^{-1}{\text{s}}^{-1}$ for an InGaAs QW with a 30 meV ground-state confinement energy). The overlaid lines indicate the approximate regimes of operation of several different samples over a range of ${\epsilon }_K$; only samples S and S2 extend into the scattering-assisted tunneling regime. Inset: partial schematic band diagram of a sample showing parameter definitions.Reuse & Permissions

Figure 4

Schematic band diagrams of the samples used in this experiment. The thickness and alloy composition of the QW and insulating barrier are the same in all samples. (a) Sample NTU. (b) Samples N and NU (they differ only in QW $\delta$ doping, which does not significantly affect the diagram on the scale shown here). (c) Sample S. Sample S2 (not shown here) is identical except that the TB is slightly higher (0.13 eV compared to 0.1 eV).Reuse & Permissions

Figure 5

(Color online) [(a) and (b)] Capacitance-frequency data from sample N taken over a sequence of $V_{\text{dc}}$ for $T=4.2\text{K}$ and $V_{\text{ac}}=5{\text{mV}}_{\text{rms}}$. Points are data, lines are fits to data using the equivalent circuit of Fig. 2b. According to our bias convention, $n_S$ decreases with $V_{\text{dc}}$. (a) 100 mV steps in $V_{\text{dc}}$. The QW is occupied in all traces. (b) 25 mV steps in $V_{\text{dc}}$. This range of $V_{\text{dc}}$ depletes the QW, reducing the low-frequency limit of the capacitance. (c) Capacitance-voltage data from the same device taken at various values of $f$ (lines with symbols). Also shown are two charge-step simulations of the low-frequency capacitance with different QW $\delta$ dopings: $1\times {10}^{11}{\text{cm}}^{-2}$ (solid line) and $2\times {10}^{11}{\text{cm}}^{-2}$ (dashed line). The abrupt step of the simulations is not observed in the data because the added impurities make the ground-state energy nonuniform across the sample.Reuse & Permissions

Figure 6

(Color online) Capacitance versus frequency data from sample S at 5 K, $V_{\text{ac}}=5{\text{mV}}_{\text{rms}}$, and 10 mV increments of $V_{\text{dc}}$. Points are data, lines are fits using the equivalent circuit model of Figure 2. At $V_{\text{dc}}=270\text{mV}$, the QW is almost fully depleted. As $V_{\text{dc}}$ is decreased, states in the QW become available, thus increasing the low-frequency capacitance, although the roll-off frequency remains relatively constant. As $V_{\text{dc}}$ is decreased near the conditions of trace $X$ $\left(V_{\text{dc}}=210\text{mV}\right)$, the roll-off frequency of the device abruptly increases by two orders of magnitude as a result of the onset of scattering-assisted tunneling. The maximum frequency accessible with our $LCR$ meter is 1 MHz.Reuse & Permissions

Figure 7

(Color online) ${\tau }_{rc}$ versus ${\epsilon }_F$. Lines with symbols are data; lines without symbols are calculations of scattering-assisted [Eq. (2)] and momentum-conserved (Calc) tunneling. Data from samples N, NU, and NTU all show increasing ${\tau }_{rc}$ with ${\epsilon }_F$, in agreement with their respective calculations of momentum-conserved tunneling. (The calculation of NU was nearly identical to that of N and is omitted for clarity.) In contrast, measured ${\tau }_{rc}$ in sample S decreases with ${\epsilon }_F$. The calculation including scattering-assisted tunneling [S: Eq. (2)] is a much better description of the data than the calculation of momentum-conserved tunneling (S, Calc). $T=5\text{K}$ for sample S; $T=4.2\text{K}$ for all others. $V_{\text{ac}}=5{\text{mV}}_{\text{rms}}$.Reuse & Permissions

Figure 8

(Color online) Simultaneous measurements on samples N, S, and S2 of the magnetic field dependence of (a) the quasibound state lifetime $\left({\tau }_{rc}\right)$ and (b) the 2DEG thermodynamic density of states $\left(g_{2\text{D}}\right)$ at $T=1.85\text{K}$. $V_{\text{dc}}$ was held constant to give $n_S\simeq 4.78\times {10}^{11}{\text{cm}}^{-2}$ in all samples at $H=0\text{T}$. Values of $H$ corresponding to filling factors $\nu =4$, 6, and 8 are indicated with dotted lines. (a) At $\nu =4$, suppression of the scattering-assisted tunneling increases ${\tau }_{rc}$ by orders of magnitude in samples S and S2. The features at $\nu =6$ and 8 are less pronounced because the Landau-level splitting is smaller at lower $H$. (b) We also see pronounced dips in $g_{2\text{D}}$ at even integer $\nu$, indicating that all three samples show the expected formation of Landau levels. The fit represents the expected behavior of $g_{2\text{D}}$ as calculated according to Ref. 34 with a Landau-level half width of ${\Gamma }_{LL}=\left(1.22\text{meV}/{\text{T}}^{1/2}\right)\sqrt{H}$.Reuse & Permissions

Figure 9

(Color online) Measurements of sample S in magnetic field versus $V_{\text{dc}}$ at $T=5\text{K}$. (a) ${\tau }_{rc}$ and $g_{2\text{D}}$ plotted on the same scales as Fig. 8. The peaks in ${\tau }_{rc}$ are labeled with the corresponding value of $\nu$. Note that these data were acquired at higher temperature than that in Fig. 8. (b) Fan diagram of maxima in ${\tau }_{rc}$ (circles) and minima in magnetocapacitance measurements (squares). Dashed line illustrates the region of the diagram corresponding to the sweep in (a). Solid lines are least-squares fits and are labeled with their corresponding Landau-level filling factor $\nu$. All lines do not intersect at the same value of $V_{\text{dc}}$ because the carrier density $n_S$ is not strictly linear with $V_{\text{dc}}$ for nonzero $H$ (see Ref. 17 for a thorough discussion of this effect and Ref. 38 for a similar fan diagram).Reuse & Permissions

Figure 10

(Color online) Tunneling conductance calculated by two equivalent models (top panel) and the components of ${\tau }_{rc}$ (bottom panel) plotted versus $V_{\text{dc}}$ for sample NU at $H=6\text{T}$ and $T=5\text{K}$. The comparison between our analysis technique and that of Ashoori et al. (top panel) shows that the two yield almost identical results. The pronounced dips in $G_{\text{TB}}$ that occur at $\nu =2$ and 4 reflect dips in $g_s$. These appear as peaks in the plot of $G_{\text{TB}}^{-1}$ (bottom panel). The plot of ${\tau }_{rc}$ nevertheless exhibits dips at $\nu =2$ and 4 because the dips in $C_Q$ at these values of $V_{\text{dc}}$ are more pronounced than the peaks in $G_{\text{TB}}^{-1}$.Reuse & Permissions

Figure 11

(Color online) Measured capacitance and loss tangent versus frequency (symbols) and fits to both components of the data simultaneously using the equivalent circuit of Fig. 2b (lines). (a) Sample N measured at $H=0\text{T}$, $V_{\text{dc}}=0\text{V}$, and $T=4.2\text{K}$. [(b) and (c)] Sample S measured at a variety of $V_{\text{dc}}$ with $T=5\text{K}$ and $H=6\text{T}$. The bias $V_{\text{dc}}=30\text{mV}$ corresponds to $\nu =2$; $V_{\text{dc}}=-60\text{mV}$ corresponds to midway between $\nu =2$ and 4.Reuse & Permissions