Screening effects of superlattice doping on the mobility of GaAs two-dimensional electron system revealed by in-situ gate control

The sample consisted of a 30-nm-wide GaAs QW sandwiched between Al_0.27Ga_0.73As barriers, grown on an $n$-type GaAs (001) substrate. The QW, with its center located 207 nm below the surface, was modulation doped on one side, with Si $\delta$-doping ([Si] = $1\times 10^{16}$ m^-2) at the center of the AlAs/GaAs/AlAs (2 nm/3 nm/2 nm) SL located 75 nm above the QW (inset of Fig. 1) Gardner et al. (2016). The Si $\delta$-doping in the thin GaAs layer provides mobile electrons not only in the QW 75-nm away but also in the neighboring AlAs layers Umansky et al. (2009); Gardner et al. (2016); Manfra (2014); Chung et al. (2020); Friedland et al. (1996). The mobile electrons in the AlAs layers provide screening of the disorder potential created by the ionized Si donors. The wafer was processed into a 120-$\mu$m-wide Hall bar with voltage-probe distance of 100 $\mu$m and fitted with a Ti/Au front gate. The $n$-type substrate was used as a back gate. Measurements were done at temperatures of $0.27$–$4.3$ K using a standard lock-in technique.

Figure 1 shows the magnetotransport of the sample measured at 1.6 K. Here, we show data taken at a positive front gate voltage ($V_{\text{FG}}$) of 0.5 V, which is supposed to increase the density of mobile electrons in the SL. Interestingly, despite the presence of mobile electrons in the SL, there are integer quantum Hall effects at Landau-level filling factor $\nu=1,2$, and $4$, where the longitudinal resistance ($R_{xx}$) drops to zero and the Hall resistance ($R_{xy}$) is quantized. From the Shubnikov-de Haas oscillations, we obtained a sheet carrier density of $1.25\times 10^{15}$ m^-2, which agrees within 3% with the value deduced from the slope of $R_{xy}$. This suggests that, even if the SL contains conduction electrons, they apparently do not contribute to transport. We confirmed similar results for $V_{\text{FG}}$ up to 0.8 V.

Figure 2 shows the $V_{\text{FG}}$ dependence of the sheet carrier density deduced from $R_{xy}$ at 0.2 T. The blue solid curve was obtained by sweeping $V_{\text{FG}}$ at 1.6 K. As shown above, at 1.6 K the measured $R_{xy}$ reflects exclusively the carrier density in the quantum well ($n_{\text{QW}}$), and not that in the SL. As we decreased $V_{\text{FG}}$ from 0.8 V, $n_{\text{QW}}$ remained almost constant for $-0.7$ $<V_{\text{FG}}<$ $0.8$ V (region I). This is consistent with the expectation that the SL contains mobile electrons, which screen the electric field from the front gate and thereby suppress its effect on $n_{\text{QW}}$. A distinct change in $n_{\text{QW}}$ occurs only at $V_{\text{FG}}<-0.7$ V (region II), indicating that the screening capability of the SL is significantly decreased in region II. Upon increasing $V_{\text{FG}}$, we observed a pronounced hysteresis in region II, where $n_{\text{QW}}$ changed at a faster rate, with an overshoot near the boundary with region I. The rate of change d$n_{\text{QW}}$/d$V_{\text{FG}}=3.5\times 10^{15}$ m^-2V^-1 for the up sweep, shown by the blue dashed line, was consistent with the geometrical capacitance between the front gate and the center of the QW calculated from the distance (207 nm) and the permittivity of AlGaAs ($\epsilon=13$). This implies that, upon increasing $V_{\text{FG}}$ in region II, electrons accumulate only in the QW, resulting in a metastable state in which the QW (SL) is overpopulated (underpopulated) with respect to its equilibrium density. We conjecture that this results from the difficulty to inject charge into the SL once it becomes close to depletion and poorly conducting. The red curve in Fig. 2, obtained by sweeping $V_{\text{FG}}$ at 4.3 K, shows similar behavior, while the boundary between regions I and II shifts to a more negative $V_{\text{FG}}$, with a smaller overshoot in the up sweep.

Turning our attention to region I, we notice that $n_{\text{QW}}$ is not constant, but varies slightly with $V_{\text{FG}}$. This indicates that part of the electric field from the gate penetrates the SL populated with electrons. This is reasonable, as the SL is not a perfect metal; it has only a finite DOS, that is, a finite screening capability. In turn, by analyzing the change in $n_{\text{QW}}$ with $V_{\text{FG}}$ as shown later, we can quantify the DOS, and hence the screening capability, of the SL. (See Ref. Eisenstein et al. (1994) for the principle of this field-penetration technique.)

Even though we used a very slow sweep rate of 0.67 mV/sec to set $V_{\text{FG}}$, in region II $n_{\text{QW}}$ gradually increased on a scale of several minutes to several tens of hours after $V_{\text{FG}}$ was set at a constant value (inset of Fig. 2). This was the case even for down sweeps for which the system is closer to equilibrium. Similar temporal behavior was reported in Ref. Rössler et al. (2010). The transient time increased with decreasing temperature and decreasing $V_{\text{FG}}$. At $V_{\text{FG}}=-1.3$ V, equilibrium was not reached even after a few days at 4.3 K. We used the following method to determine the equilibrium value of $n_{\text{QW}}$ at each $V_{\text{FG}}$, which is essential for evaluating the screening effect. First, we set $V_{\text{FG}}$ at 4.3 K to facilitate the equilibration and waited until $n_{\text{QW}}$ reached a steady value. Then, we decreased the temperature to 1.6 K and determined $n_{\text{QW}}$ from the low-field $R_{xy}$. By repeating this process for different $V_{\text{FG}}$, we obtained $n_{\text{QW}}$ as a function of $V_{\text{FG}}$, which we plot as open circles in Fig. 2. At $V_{\text{FG}}=0$ V, we obtained the same $n_{\text{QW}}$ value as that for the $V_{\text{FG}}$ sweep. However, the difference between the two methods became significant at lower $V_{\text{FG}}$. We therefore employed the data obtained by the equilibration method for $V_{\text{FG}}\leq$ 0 V and those of the $V_{\text{FG}}$ sweep for $V_{\text{FG}}>0$ V and fit them using a double exponential function (the black dashed line in Fig. 2).

To deduce the excess electron density in the SL and thereby quantitatively characterize the screening effect, we analyzed the charge equilibration among the front gate, redtwo AlAs layers [AlAs1(2)] comprising the SL, and QW using the circuit model shown in the inset of Fig. 3(a). In addition to the geometrical capacitances between the neighboring elements among these ($C_{\text{b1}}$, $C_{\text{i}}$, and $C_{\text{b2}}$), the model contains the quantum capacitances of the QW ($C_{\text{QW}}$) and the AlAs layers [$C_{\text{AlAs1(2)}}$]. The quantum capacitance is expressed as $C_{\alpha}=e^{2}D_{\alpha}$, where $D_{\alpha}=g_{\alpha}m^{\star}_{\alpha}$/$2\pi\hbar^{2}$ is the DOS ($m^{\star}_{\alpha}$ is the electron effective mass, $\alpha$ denotes QW or AlAs1(2), $e$ is the elementary charge, $g_{\alpha}$ is the degeneracy, and $\hbar=h$/2$\pi$ is the reduced Planck constant). $C_{\text{b1}}$, $C_{\text{i}}$, and $C_{\text{b2}}$ are calculated from the layer thicknesses and permittivity and $C_{\text{QW}}$ is known from the effective mass $m^{\star}_{\text{QW}}=0.067m_{0}$ of GaAs ($m_{0}$ is the electron mass in vacuum) and the twofold spin degeneracy. This leaves $C_{\text{AlAs1(2)}}$ the only unknown parameters in the model. For the model to be solvable, we need to assume that the two AlAs layers have the same density of states at the Fermi level, that is, $C_{\text{AlAs1}}=C_{\text{AlAs2}}$ ($\equiv C_{\text{SL}}$). We confirmed this assumption to be acceptable by noting that the calculated chemical potential difference between the two AlAs layers (up to 7 meV) was smaller than the disorder-broadened tail (a few tens of meV).

We calculate $\text{d}n_{\text{QW}}/\text{d}V_{\text{FG}}$ as a function of $V_{\text{FG}}$ using the equilibrium relation between $n_{\text{QW}}$ and $V_{\text{FG}}$ obtained above. By numerically solving the circuit model with the $\text{d}n_{\text{QW}}/\text{d}V_{\text{FG}}$ value at each $V_{\text{FG}}$ as an input, we can deduce $C_{\text{SL}}$ as a function of $V_{\text{FG}}$, as shown in Fig. 3(a). $C_{\text{SL}}$ decreases with decreasing $V_{\text{FG}}$, reflecting the disorder-broadened tail of the DOS. Interestingly, $C_{\text{SL}}$ is not constant even at $V_{\text{FG}}>0$ V, where it keeps increasing with $V_{\text{FG}}$. Since quantum capacitance is proportional to the DOS at the Fermi level, the obtained $C_{\text{SL}}$ can be translated into the effective mass $m^{\star}_{\text{SL}}$ that would produce the same DOS for parabolic dispersion through the relation $D_{\text{SL}}=g_{\text{SL}}m^{\star}_{\text{SL}}$/$2\pi\hbar^{2}$. In thin AlAs layers, quantum confinement and strain split the three-fold valley degeneracy in bulk into one and two, with the former becoming lower in energy for a thickness below $5.5$–$6.0$ nm Khisameeva et al. (2019); Van Kesteren et al. (1989). We therefore assumed $g_{\text{SL}}=2$, taking into account the spin degeneracy. The effective mass $m^{\star}_{\text{SL}}$ evaluated in this way is shown on the right axis of Fig. 3(a). In bulk AlAs, the effective masses in the transverse and longitudinal directions of the ellipsoid Fermi surface are $0.22m_{0}$ and $0.97m_{0}$, respectively Vurgaftman et al. (2001). For AlAs QWs thinner than 6.0 nm, the 2DES occupies the lower non-degenerate valley, where experiments report a transverse mass of ($0.2$–$0.3$)$m_{0}$ Yamada et al. (1994); Momose et al. (1999); Vakili et al. (2004). The obtained $m^{\star}_{\text{SL}}$/$m_{0}$, which approaches the expected value ($0.2$–$0.3$) with increasing $V_{\text{FG}}$, is reasonable.

Once $C_{\text{SL}}$ is obtained as a function of $V_{\text{FG}}$, one can calculate the electron density in the AlAs layers [$n_{\text{AlAs1(2)}}$] and SL ($n_{\text{SL}}=n_{\text{AlAs1}}+n_{\text{AlAs2}}$). Figure 3(b) shows the $V_{\text{FG}}$ dependence of $n_{\text{SL}}$. The right axis of the figure indicates the excess electron density normalized by the doping density ($N_{\text{Si}}=10^{16}$ m^-2). While $n_{\text{SL}}$ varies almost linearly with $V_{\text{FG}}$, the slope decreases slightly at $V_{\text{FG}}<-1.0$ V.

Now let us investigate the effect of screening on mobility. The symbols in Fig. 4(a) show the 1.6-K mobility ($\mu$) measured at the same carrier density ($n_{\text{QW}}=1.2\times 10^{15}$ m^-2) but with the sample prepared to have different $n_{\text{SL}}/N_{\text{Si}}$ values. The open circles show data obtained by re-adjusting $n_{\text{QW}}$ with the back gate after equilibrating the system at 4.3 K for $V_{\text{FG}}$ ($-1.3$–$0$ V) and cooling the sample to 1.6 K. For this $V_{\text{FG}}$ range, $n_{\text{SL}}/N_{\text{Si}}$ varied between 0.13 and 0.93 [see Fig. 3(b)]. As equilibration was not obtained for $V_{\text{FG}}<-1.3$ V at 4.3 K, we employed a different method to achieve smaller $n_{\text{SL}}/N_{\text{Si}}$. The open squares in Fig. 4(a) show data obtained by applying $V_{\text{FG}}\leq-1.7$ V at room temperature and re-adjusting $n_{\text{QW}}$ at 1.6 K with the back gate. We confirmed that the SL was already depleted (i.e., $n_{\text{SL}}/N_{\text{Si}}=0$) for $V_{\text{FG}}=-1.88$ V by noting that at 1.6 K the 2DES was depleted at zero back gate voltage. As Fig. 4(a) shows, $\mu$ decreased by 37% as $n_{\text{SL}}/N_{\text{Si}}$ decreased from 0.93 to 0.

We characterize the $n_{\text{SL}}/N_{\text{Si}}$ dependence by considering two main sources of disorder in modulation-doped GaAs 2DESs, i.e., background ionized impurities (BIs) and remote ionized impurities (RIs). For the mobility limited by RIs, we used the excess electron screening (EES) model proposed by Sammon et al. Sammon et al. (2018a). The dashed blue and green curves in Fig. 4(b) show the mobility $\mu_{\text{EES}}$ calculated using the EES model for the two limits, $n_{\text{SL}}/N_{\text{Si}}\ll 1$ and $1-n_{\text{SL}}/N_{\text{Si}}\ll 1$, respectively. The red curve is the fitting using the empirical formula derived in Ref. Sammon et al. (2018a). By adjusting the numerical parameters to connect the two limits, we obtain

Here, $k_{\text{F}}=(2\pi n_{\text{QW}})^{1/2}$ is the Fermi wave number and $d=90$ nm is the center-to-center distance between the QW and SL. The red curve in Fig. 4(a) shows the least-squares fit of the total mobility $(1$/$\mu_{\text{BI}}+1$/$\mu_{\text{EES}})^{-1}$ based on Matthiessen’s rule, where $\mu_{\text{BI}}$ is the only parameter and we assumed it to be constant. The $\mu_{\text{BI}}$ value obtained from the fit is shown by the cyan line in Fig. 4(a). The EES model well explains the overall $n_{\text{SL}}/N_{\text{Si}}$ dependence of the measured $\mu$, providing good agreement for $n_{\text{SL}}/N_{\text{Si}}\geq 0.13$. For $n_{\text{SL}}/N_{\text{Si}}<0.13$, the agreement between the experiment and model becomes less satisfactory, which is because we tried to fit both regimes of $n_{\text{SL}}/N_{\text{Si}}\ll 1$ and $1-n_{\text{SL}}/N_{\text{Si}}\ll 1$ using Eq. (1) [Fig. 4(b)]. For comparison, we also calculated the mobility using the standard model ($\mu_{\text{STD}}$) Hirakawa and Sakaki (1986), assuming independent scattering by ($N_{\text{Si}}-n_{\text{SL}}$) ionized donors. The black lines in Figs. 4(b) and  4(a) show $\mu_{\text{STD}}$ and the resultant total mobility $(1/\mu_{\text{BI}}+1/\mu_{\text{STD}})^{-1}$, respectively. The independent-scattering model predicts a mobility way below the experimental result, which in turn demonstrates the importance of the screening by excess electrons.

We also examined the possibility of excess electrons in the SL affecting $\mu_{\text{BI}}$. Sammon et al. reported that, for strong screening, the contribution of BIs to $\mu_{\text{BI}}$ is canceled out by the image-charge effect when they are located farther than $0.5d$ from the center of the QW Sammon et al. (2018b). We calculated $\mu_{\text{BI}}$ by integrating contributions from BIs over different spatial ranges, $0.5d$ and $d$. The difference between the two cases is less than 1%, thus corroborating our assumption of constant $\mu_{\text{BI}}$.

Finally, let us investigate the effect of screening on quantum lifetime ($\tau_{\text{q}}$) deduced from Shubnikov–de Haas (SdH) oscillations, a quantity often argued to be a better indicator of sample quality than mobility in terms of FQHEs. Figure 5(a) shows the SdH oscillations measured at 0.27 K at a constant carrier density ($n_{\text{QW}}=1.2\times 10^{15}$ m^-2) under different screening conditions of $n_{\text{SL}}/N_{\text{Si}}=0.93$, $0.18$, and $0$ (corresponding $V_{\text{FG}}$ of 0, $-1.20$, and $-1.88$ V, respectively). Under the well-screened condition ($n_{\text{SL}}/N_{\text{Si}}=0.93$), minima at odd filling factors due to spin splitting are more pronounced, indicating the influence of screening. We extracted the quantum lifetime $\tau_{\text{q}}$ by using the functional form of SdH oscillations, given as Coleridge (1991)

with

Here, $\Delta R$ is the amplitude of the SdH oscillations, $\omega_{\text{c}}$ is the cyclotron frequency, $R_{0}$ is the $R_{xx}$ at zero magnetic field, $\chi(T)$ is a thermal damping factor, and $k_{\text{B}}$ is the Boltzmann constant. Thus, the slope of $\Delta R$/$R_{0}\chi(T)$ vs. $1/B$, known as a Dingle plot, shown in the inset of Fig. 5(a) gives the quantum lifetime $\tau_{\text{q}}$ ¹¹1 For the data in the weak screening regime taken with $V_{\text{FG}}\leq-1.7$ V set at room temperature, analysis taking into account density inhomogeneity Qian et al. (2017) was necessary to fit the Dingle plot with the correct intercept of 4 at $B^{-1}=0$. The density inhomogeneity derived from the fit was 0.6 and 1.8% for $V_{\text{FG}}=-1.70$ and $-1.88$ V, respectively. . The obtained $\tau_{\text{q}}$ is shown by open symbols in Fig. 5(b). In contrast to $\mu$, or transport lifetime ($\tau_{\text{t}}=m_{\text{QW}}\mu/e$), $\tau_{\text{q}}$ does not show a discernible change as a function of $n_{\text{SL}}/N_{\text{Si}}$.

We compared the measured $\tau_{\text{q}}$ with the calculated quantum lifetimes limited by RIs and BIs ($\tau_{\text{qEES}}$ and $\tau_{\text{qBI}}$), which are shown in Fig. 5(b) by the green and cyan lines, respectively. Here, $\tau_{\text{qEES}}$ was calculated with the EES model Sammon et al. (2018a), whereas $\tau_{\text{qBI}}$ was calculated with the independent-scattering model using the BI concentration obtained from $\mu_{\text{BI}}$ ($=224$ m²/Vs). The expected total quantum lifetime is shown by the dashed magenta line. Although the measured $\tau_{\text{q}}$ is close to the values expected in the weak screening regime, it remains about $2.6$ ps when $n_{\text{SL}}/N_{\text{Si}}$ increases, much lower than expected in the intermediate and strong screening regimes. This discrepancy was not mitigated even when a more elaborate model was employed, such as one with different BI concentrations in the GaAs QW and AlGaAs barrier layers Sammon et al. (2018b) and remote charges on the sample surface and the back gate Chen et al. (2012); Wang et al. (2013). Extrinsic mechanisms that might reduce the apparent quantum lifetime, such as the finite density gradient Qian et al. (2017) ²²2An attempt to fit the Dingle-plot data in Fig. 5(a) using the density-gradient model in Ref. Qian et al. (2017) together with the calculated quantum lifetime resulted in a strongly nonlinear curve, which did not fit the experimental results. and the response time of the lock-in amplifier ³³3Reducing the field sweep rate from 10 mT/sec (which we normally use) to 10 $\mu$T/sec did not affect the measured value of $\mu_{\text{q}}$., did not explain the discrepancy, either. Ultra-high-quality samples with much longer $\tau_{\text{q}}$, such as those in Refs. Qian et al. (2017); Fu et al. (2018), might be necessary to observe the predicted screening effect on $\tau_{\text{q}}$. Yet, it is interesting that the visibility of the spin gap varies with $n_{\text{SL}}/N_{\text{Si}}$ even when $\tau_{\text{q}}$ remains constant, as we observed. For $n_{\text{SL}}/N_{\text{Si}}=0$, the density inhomogeneity estimated from the analysis of the Dingle plot Qian et al. (2017) is 1.8%, which may be partly responsible for the poorly developed quantum Hall effects at even as well as odd integer fillings. However, for $n_{\text{SL}}/N_{\text{Si}}=0.18$ and $0.93$, the estimated density inhomogeneity is less than 0.2% with no clear difference, which cannot account for the difference in the visibility of the spin gap. This suggests that the screening of long-range disorder becomes more important for interaction phenomena such as FQHEs. The broad $R_{xx}$ minimum around $B=8$ T seen in Fig. 1 is a precursor of the $\nu=2/3$ FQHE. By investigating FQHEs at lower temperatures under different screening conditions, it will be possible to examine how the energy gap of FQHEs is correlated with the composite fermion mobility Kang et al. (1995) deduced from the resistivity at $\nu=1/2$.