# Wigner crystallization and metal-insulator transition of two-dimensional holes in GaAs at B=0

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It is well known that the ground state of a two-dimensional (2D) electron or hole system realized in semiconductors at B=0 is expected to be a Wigner crystal (WC) in the low density limit when carrier-carrier interaction becomes dominant over the kinetic energy of the carriers. Tanatar and Ceperley (1)from their Monte Carlo simulation for ideally clean systems find that the critical density, expressed in terms of the dimensionless parameter $r_{s}$ , is $r_{s}=37\pm 5$ . We note that $r_{s}$ is the Coulomb energy to Fermi energy ratio, given by $r_{s}=(\mbox{p}\pi)^{-1/2}m^{*}e^{2}/4\pi\hbar^{2}\epsilon\epsilon_{o}$ , where $m^{*}$ is the effective mass, p the charge carrier density, and $\epsilon$ the dielectric constant. Disorder, which is always present in any experimental system, also plays important roles. In the clean limit, it is expected to pin the WC, rendering the system into an insulator. In the highly disordered dirty limit, the Coulomb interaction is negligible, or can be treated as a small perturbation; the system is better described by single particle localization. In between, when both interaction and localization are equally important, neither the electronic processes nor the nature of the phase is known and there is no clear physical picture for the system at the present time. Also, it is believed that if the disorder is not too strong, it can enhance Wigner crystallization. Chui and Tanatar (2)find from their Monte Carlo studies that the disorder they put in their model can stabilize the WC phase at $r_{s}$ as low as 7.5.

On the experimental side, a number of publications (3); (4); (5); (6); (7); (8); (9); (10), since the discovery by Kravchenko et al. (3)of a 2D metal-insulator transition (MIT) in high mobility Si metal- oxide-semiconductor field-effect transistors (Si MOSFETβs) at $r_{s}\approx 10$ , have reported transport studies in the large $r_{s}$ regime of 2D electron or hole systems in several different semiconductor heterostructures. These studies generally do not address the properties in the insulating phase. They focus mostly on the nature of the metallic phase and the scaling behavior near the MIT. Their results agree with the conclusion of Kravchenko et al. (3)that there is a MIT in high mobility samples and the critical density for the transition varies from $r_{s}\approx 5$ in $\mbox{Si}_{0.88}\mbox{Ge}_{0.12}$ to $r_{s}\approx 28$ in p-GaAs. The origin and nature of the insulating phase, however, can be inferred from earlier Si-MOSFET experiments, where the low mobility samples show a cross-over from weak localization to strong localization as the carrier density is decreased. In high mobility samples, which show MITβs at $r_{s}\approx 10$ , Pudalov et al. (11)first studied the transport characteristics in the insulating phase and attributed them to those of a pinned WC. However, Shashkin et al. (12)and Mason et al. (13)subsequently find that single particle localization gives a better description and the transport is via hopping in the presence of a Coulomb gap. It thus appears that in the regime where disorder and localization must be treated on equal footing, transport and I-V characteristics do not allow us to identify whether the insulating phase is a WC. A more desirable approach to the problem has to come from clear identification of a 2D system in the clean limit, where the physics is expected to be simpler and theoretical predictions more reliable.

In this letter, we report transport measurements on a 2D hole system (2DHS) in GaAs/Al ${}_{x}$ Ga ${}_{1-x}$ As, which has an unprecedentedly high peak mobility, $\mu_{p}=7\times 10^{5}\mbox{cm}^{2}$ /Vs. This mobility, approximately twice time higher than previously studied 2DHS, allowed us to reach p as low as $4.8\times 10^{9}\mbox{cm}^{-2}$ . From a systematic study of density, temperature and bias dependences of the hole transport, we find a MIT at $r_{s}=35.1\pm 0.9$ , which is in good agreement with $r_{s}=37\pm 5$ predicted for the WC transition by Tanatar and Ceperley. We also examine the dependence of $r_{s}$ at the MIT, from all 2D electron and hole systems that have been reported to show MITβs, on the disorder in the sample and show that our sample is indeed in the clean limit. In the rest of this paper, we present this result and also describe in detail related new transport characteristics from our 2DHS.

The 2DHS is created in a modulation doped GaAs/Al ${}_{x}$ Ga ${}_{1-x}$ As heterostructure grown on the (311) A surface of GaAs substrate by molecular-beam epitaxy. A backgate, used to change the density, is situated 250 $\mu$ m away from the 2DHS, and does not screen the carrier-carrier interaction. Since the holes are produced by modulation doping, the background neutralizing charges are immobile. In contrast, the 2D carriers in the insulated gate field-effect transistors used in other experiments (3); (4); (5); (6); (7)are capacitively induced by applying a voltage to a metallic gate located close to the 2D systems. In such transistors, the background neutralizing charges are mobile and the carrier-carrier interaction is screened. The sample was a rectangle of $4.2\times 0.4\mbox{mm}^{2}$ cut along the $[\overline{2}33]$ direction, after the wafer was thinned to 250 $\mu$ m. Resistivity ( $\rho$ ) was measured by a four probe method. To avoid sample heating, current was limited to below 1nA, and voltage was measured across two contacts separated by 0.7mm located near the center. An ac technique using a lock-in at a frequency $<$ 10Hz was used whenever possible. At low densities where capacitive coupling became significant, a dc method was employed. The sample was cooled in the mixing chamber of a dilution refrigerator. Three samples cut from the same wafer were studied, and all showed MITβs with the same characteristics. However, the data reported in this Letter were collected over three cool-downs on the same sample. The sample was warmed up to room temperature between cool-downs.

Fig. 1 shows the T dependence of $\rho$ , measured in the zero current limit, in the range 50mK $<$ T $<$ 1.3K, for $4.8\times 10^{9}\leq p\leq 3.72\times 10^{10}\mbox{ cm}^{-2}$ or $16.0\leq r_{s}\leq 44.4$ . For p $>$ 2 $\times$ 10 ${}^{10}$ cm ${}^{-2}$ (the two bottom traces), the sign of d $\rho$ /dT, which has been usually used to distinguish metallic phase from the insulating phase, is positive in the entire T range, showing metallic behavior. For p $\leq$ 7.2 $\times$ 10 ${}^{9}$ cm ${}^{-2}$ (the four top traces), d $\rho$ /dT is always negative exhibiting the characteristic of an insulator. In the intermediate range of p, however, there is a peak in $\rho$ resulting from d $\rho$ /dT $<$ 0 at high T and d $\rho$ /dT $>$ 0 at low T. This peak starts to appear at p=1.98 $\times$ 10 ${}^{10}$ cm ${}^{-2}$ ; it becomes more pronounced and shifts to a lower temperature as p decreases. The maximum is at T $\approx$ 0.7K at p= $1.98\times 10^{10}\mbox{cm}^{-2}$ , and $\approx$ 0.18K at p= $9.0\times 10^{9}\mbox{cm}^{-2}$ . The presence of such a peak makes it ambiguous to tell whether the 2DHS is in the metallic phase or insulating phase using the sign of d $\rho$ /dT as the criterion. For example, at p= $9.0\times 10^{9}\mbox{cm}^{-2}$ , if the measurements are made only down to 0.3K, one would conclude from the fact d $\rho$ /dT is negative that the 2DHS is in the insulating phase, while the data below 0.18K indicate that the 2DHS is in fact in the metallic phase.

However, this ambiguity can be avoided if we define the phase as metallic when the I-V is superlinear (d ${}^{2}$ I/dV ${}^{2}$ $<$ 0) and insulating when it is sublinear (d ${}^{2}$ I/dV ${}^{2}$ $>$ 0). Fig. 2 is the differential conductance (dI/dV) plotted against V at p= $9.0\times 10^{9}\mbox{cm}^{-2}$ , taken at several temperatures below and above the $\rho$ maximum, as marked by the arrows in the inset. In sharp contrast to d $\rho$ /dT, which shows a metallic behavior below and an insulating behavior above T=0.18K, the dI/dV vs. V taken at all four temperatures show superlinear I-Vβs, i.e., dI/dV decreases with increasing V. Such a superlinear I-V is observed down to p= $8.1\times 10^{9}\mbox{cm}^{-2}$ . When p is reduced below $7.3\times 10^{9}\mbox{cm}^{-2}$ , where $\rho$ shows monotonic increase with decreasing T, the I-V becomes sublinear, indicative of an insulating phase. It should also be pointed out that identifying the metallic phase and the insulating phase based on the sign of d ${}^{2}$ I/dV ${}^{2}$ gives exactly the same result as that obtained using the sign of d $\rho$ /dT at our lowest T.

Using the sign of d ${}^{2}$ I/dV ${}^{2}$ , we locate the MIT at p= $7.7\pm 0.4\times 10^{9}\mbox{cm}^{-2}$ , or equivalently at $r_{s}=35.1\pm 0.9$ . At the MIT, $\rho$ is approximately 35k $\Omega$ at T $\rightarrow$ 0, and $E_{F}$ is calculated to be 0.58K. The critical $r_{s}$ for the MIT in our sample is approximately four times that in high mobility Si MOSFETβs and 25% larger than that in previously studied highest mobility 2DHS in GaAs/AlGaAs, of which $\mu_{p}\approx 4\times 10^{5}\mbox{cm}^{2}$ /Vs is approximately half of that in our 2DHS. It is clear from this result that the $r_{s}$ , at which the 2D MIT is observed, depends strongly on the disorder in the 2D system. In the inset of Fig. 1, we plot ${r_{s}}^{c}$ , the 2D MIT critical $r_{s}$ , from all samples reported in the literature as a function of the scattering rate of the 2D carriers, defined by $1/\tau=e/(m^{*}\mu_{p})$ , which gives a simple measure of the disorder and also takes into account of the differences in effective mass, $m^{*}$ , of the charge carriers in different material systems. It is clear that for 1/ $\tau$ $>$ 1 $\times$ 10 ${}^{11}$ sec ${}^{-1}$ , where all the high mobility Si MOSFET data lie, ${r_{s}}^{c}$ is approximately constant. The data from the 2D systems in GaAs/AlGaAs heterostructures, which are known to have low disorder and show fractional quantum Hall effect characteristics in high magnetic fields, are all in the 1/ $\tau$ $<$ 1 $\times$ 10 ${}^{11}$ sec ${}^{-1}$ regime. In this low scattering rate region, ${r_{s}}^{c}$ shows a steep increase with decreasing 1/ $\tau$ and extrapolates to ${r_{s}}^{c}$ $\approx$ 40 as 1/ $\tau$ $\rightarrow$ 0. Our data, shown as the filled circle in the figure is in the clean limit and the observed ${r_{s}}^{c}=35.1\pm 0.9$ is in good agreement with the $r_{s}=37\pm 5$ for the WC transition predicted by Tanatar and Ceperley.

In the clean limit, a WC is expected to be pinned, giving rise to a depinning threshold voltage, which is characterized by a sudden increase in current as the applied voltage exceeds the threshold. In Fig. 3, the dV/dI taken at $r_{s}$ =44.4 is plotted against V. At 50mK, the dV/dI is nearly constant until V reaches about 1.5mV, beyond which it drops rapidly by an order of magnitude at V $\approx$ 3mV. This is suggestive of the depinning threshold expected for a pinned WC. In the WC conduction model proposed by Chui (14), where the WC is weakly pinned by the potential of remote dopants, transport is expected to be mediated by the creation of dislocation pairs through quantum tunneling. The energy needed to create a dislocation pair leads to a conduction threshold voltage. The observed threshold voltage of 1.5mV, corresponding to a field of 2.1V/m, is in reasonable agreement with Chuiβs model, according to which, the depinning threshold field estimated for our sample is $E_{d}=0.09{n_{i}}a^{2}e^{2}/\epsilon d^{3}=2.5$ V/m. Here, $n_{i}=8\times 10^{11}\mbox{cm}^{-2}$ is the remote doping impurity density, $a=(\pi\mbox{p})^{-1/2}$ , and the doping impurity setback distance d=1200 $\AA$ . However, there are caveats. The pinning gap deduced from $E_{d}$ , $\Delta_{p}=\hbar(eE_{d}/m^{*}a)^{1/2}\approx$ 2.7K, is not consistent with the observation that no well defined depinning threshold feature is seen when the thermal energy is still much smaller than the gap. In Fig. 3, the data show no clear threshold at T as low as 145mK. Moreover, as shown in the inset of Fig. 3, the I-V taken at T as low as 50mK is linear within the experimental resolution at voltages below 1.5mV. This ohmic conduction below the depinning threshold voltage cannot be explained by possible backgate leakage current, which is much less than 1pA.

Finally, we turn to the overall symmetry in the nonlinear transport properties of our low disorder 2DHS as its density is varied across the MIT. Shahar et al. (15)recently reported a new symmetry across the quantum Hall liquid to Hall insulator transition. They find that the I-V curves in the quantum Hall liquid phase can be mapped into the I-V curves in the Hall insulator phase by simply interchanging I with V. This I-V inversion symmetry can be viewed as reflecting the charge-flux duality symmetry in the composite boson description of the fractional quantum Hall effect. More recently, Kravchenko et al. (4)and Simonian et al. (16)reported a similar reflection symmetry across the 2D MIT, though its origin is still unclear at the present time. We have made a detailed comparison of the I-V characteristics in the metallic and insulating phases in our sample, and find that they are qualitatively different. Instead of comparing I-V with V-I in the two phases, we compare their derivatives which are more sensitive to changes in the curvature. In Fig. 4(a) and 4(b), dI/dV in the insulating phase and dV/dI in the metallic phase are shown against I and V, respectively. In the metallic phase, as V increases, dV/dI initially increases, but stops increasing at V $\approx$ 0.15mV. Above this characteristic voltage, dV/dI stays constant to I as high as 100nA, the highest current used in our measurements. If we assume a reflection symmetry with respect to a straight line in I-V corresponding to $\rho$ =35k $\Omega$ /sqr. at the MIT, the characteristic voltage of 0.15mV in the metallic phase implies a characteristic current of 2.5nA in the insulating phase, above which dI/dV should remain constant. It is clear from Fig. 4(a) that such a characteristic current is absent in all traces of the dI/dV vs. I in the insulating phase. We conclude that the I-V reflection symmetry across the 2D MIT is not observed in our data in the ranges of I and V shown in Fig. 4.

To summarize, we presented transport properties of a 2DHS that undergoes a MIT at ${r_{s}}^{c}=35.1\pm 0.9$ , which is unambiguously determined using the sign of d ${}^{2}$ I/dV ${}^{2}$ rather than d $\rho$ /dT. The dependence of ${r_{s}}^{c}$ on disorder, from all 2D electron and hole systems that have been reported to show MITβs, indicates that our 2DHS is in the clean limit, and the value of ${r_{s}}^{c}$ in our sample is in good agreement with $r_{s}=37\pm 5$ expected for the WC transition by Tanatar and Ceperley. In the insulating phase, a conduction threshold that is quantitatively consistent with the WC conduction model of Chui is found, but the pinning gap estimated from the observed threshold field is too large compared with our observations. We also find that the I-V reflection symmetry across the MIT is absent in our low disorder 2DHS.

We thank R. Bhatt, S. Chakravarty, M. Hilke, Y. Hanein, and S. Papadakis for fruitful discussions. This work was supported by the NSF.

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Fig. 1. T dependence of $\rho$ , from the top curve, at p=0.48, 0.55, 0.64, 0.72, 0.90, 1.02, 1.27, 1.98, 2.72, and 3.72 in units of 10 ${}^{10}$ cm ${}^{-2}$ . The inset shows the dependence of ${r_{s}}^{c}$ on 1/ $\tau$ , from various 2D systems; this work (filled circle), 2DHS in GaAs/AlGaAs (open circles. (7); (8); (9)), 2DES AlAs/GaAlAs (filled triangle, (6)), 2DHS in Si ${}_{0.88}$ Ge ${}_{0.12}$ /Si (filled square, (10)), and 2DES in Si-MOSFET (open triangles, (3); (4); (5)). $m^{*}$ / $m_{e}$ =0.37, 0.46, 0.18, and 0.19 for p-GaAs (17), n-AlAs (18), p-Si ${}_{0.88}$ Ge ${}_{0.12}$ (19), and n-Si-MOSFET.

Fig. 2. dI/dV in the metallic phase near the MIT (p=9.0 $\times$ $10^{9}$ cm ${}^{-2}$ ) at four different temperatures marked by the arrows in the inset. Qualitatively same nonlinear I-V characteristics are seen at all four temperatures. In contrast, d $\rho$ /dT shows a metallic behavior below and an insulating behavior above T=0.18K.

Fig. 3. dV/dI at p=4.8 $\times$ 10 ${}^{9}$ cm ${}^{-2}$ is plotted against V at T=50mK (open circles), 145mK (filled circles), and 284mK (triangles). The inset shows the I-V curve at T=50mK.

Fig. 4. (a) dI/dV vs. I in the insulating phase and (b) dV/dI vs. V in the metallic phase at 50mK. The numbers shown next to the symbols are the densities in units of 10 ${}^{10}$ cm ${}^{-2}$ . The corresponding rs values are in parenthesis.