The Wigner solid (WS), an ordered array of electrons, favored when the Coulomb repulsion energy dominates over the thermal and Fermi energies, is one of the longest-anticipated and most exotic correlated phases of a strongly-interacting electron system [1]. In a lowdisorder, two-dimensional electron system (2DES) under a large perpendicular magnetic field (B), the Fermi energy is quenched and the electrons condense into the lowest Landau level (LL). If the separation between the LLs is large compared to the Coulomb energy so that LL mixing (LLM) can be ignored, a magnetic-field-induced, 2D quantum WS is expected at very small LL filling factors (ν 1/5) [2][3][4][5]. There is, however, a close competition with interacting liquid phases, such as the fractional quantum Hall states (FQHSs) [6]. In very high mobility 2DESs confined to GaAs quantum wells where LLM is small, insulating phases are seen near the FQHS at ν = 1/5, and are generally believed to signal the formation of a WS, pinned by the small but ubiquitous disorder [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. Many properties of these insulating phases support the pinned WS picture [17]; these include the nonlinear current-voltage and noise characteristics [10][11][12][13], microwave resonances [7,11,19,20], photoluminescence [15,16], nuclear magnetic resonance features [21], tunneling resonances [23], and screening characteristics [24]. There is also a recent experiment in a GaAs bilayer electron system with very imbalanced densities where one layer is near ν = 1/2 and contains composite fermions while the other layer is at very low fillings (ν 1/5) and hosts a WS [22]. The commensurability oscillations of the composite fermions induced by the periodic potential of the WS layer are used to directly probe the lattice constant of the WS.
The 2D hole systems (2DHSs) in low-disorder GaAs quantum wells provide a particularly exciting platform for studies of the quantum WS phases, both at B = 0 [25][26][27] and at high B [23,[27][28][29][30][31][32][33][34][35][36][37]. The effective mass for holes in GaAs is m * 0.5 (in units of the free electron mass) [38], much larger than m * 0.067 for GaAs 2D electrons, rendering the 2DHS effectively more dilute and therefore more interacting; note that the r s parameter, the inter-particle distance in units of the effective Bohr radius, scales with m * . Signatures of a quantum WS at B = 0 have indeed been reported in dilute GaAs 2DHSs with very large r s [25][26][27]. At high B, the larger m * means that the LL separation is small so that there is a significant mixing of the higher LLs into the collective states of the 2D system. (For our samples reported here, the LLM parameter κ, defined as the ratio of the Coulomb to cyclotron energies ranges between ∼ 5 and 16.) Such LLM generally weakens the FQHSs, whose stability relies on short-range correlations, and favors the ground states with long-range order, such as the WS [39][40][41][42][43][44][45]. Consistent with this expectation, experiments on GaAs 2DHSs have indeed shown that the onset of the magnetic-field-induced WS moves to higher fillings (ν 1/3, compared to ν 1/5 for 2D electrons) [17,[28][29][30][31][32][33][34][35][36][37]. A recent study on ZnO 2DESs with parameters similar to GaAs 2DHSs also shows the onset of the WS at ν 1/3 [46]. Here we present experiments on very low disorder 2DHSs confined to modulation-doped GaAs quantum wells, and probe two fundamental WS-liquid phase diagrams: a temperature vs. ν phase diagram for the thermal melting of the WS, and a κ vs. ν diagram for its quantum melting.
We studied 2DHSs confined to modulation-doped, 30nm-wide GaAs quantum wells (QWs) grown on GaAs (100) substrates. The details of the sample parameters are provided in the Supplemental Material (SM) [47]. The samples have 2DHS densities (p) ranging from 2.0 to 7.9, in units of 10 10 cm -2 which we will use throughout this paper, and their low-temperature mobility is 1.5 × 10 6 cm 2 /Vs. We present data in the main text for two samples with densities p = 3.8 and 7.9; data for other densities are shown in the SM [47]. We performed all our measurements on 4 mm × 4 mm van der Pauw geometry samples, which are fitted with gate electrodes deposited on their top and bottom surfaces. The density in a given sample is tuned using both the front and back gates while keeping the charge distribution in the QW symmetric. We made measurements primarily in a dilution refrigerator with a base temperature of 40 mK.
Figure 1 shows the temperature dependence of longitudinal resistivity ρ xx vs. B at p = 3.8. The expanded (grey) trace at 40 mK shows a series of FQHSs attesting to the good quality of the sample. At the highest temperature, there is even a hint of a developing ν = 1/5 FQHS. The ν = 1/3 FQHS is fully developed and has a vanishing ρ xx minimum at the lowest temperatures. On the other hand, on its flanks (e.g., at ν = 0.30 and 0.37), ρ xx has very high values, which decrease rapidly as temperature is raised. This insulating behavior is generally believed to signal a disorder-pinned WS state [17,28,29,[31][32][33][34][35][36][37], and can be seen more conveniently in the Arrhenius plot shown in Fig. 1 inset. Also shown in this inset are the temperature dependence of ρ xx at ν = 2/5 and 1/3. In contrast to the insulating behavior at ν = 0.30 and 0.37, ρ xx at ν = 1/3 and 2/5 decreases as temperature is lowered, and is activated at ν = 1/3 with an energy gap of 1.76 K.
We probe the thermal melting of the WS by monitoring the screening efficiency [24,[54][55][56] of the 2DHS. This technique was used recently [24] to study the magneticfield-induced WS in GaAs 2DESs near ν 1/5, and the deduced melting phase diagram was found to be in good agreement with previous measurements. The measurement setup is shown schematically in Fig. 2 inset. The top and bottom yellow plates represent the front and back gates. The blue layer in the middle represents the 2DHS we are probing. We apply an AC excitation voltage V AC of 1 mV between the back and front gates at 22 kHz as shown in the inset. This AC voltage generates an electric field E P penetrating through the 2DHS. The magnitude of E P depends on the screening efficiency of the 2DHS. The magnitude of the penetrating current I P is then probed in response to E P .
The trace in Fig. 2 shows I P vs. B at our base temperature ( 40 mK) and p = 3.8. At fillings where the 2DHS is in an integer or FQHS, its bulk is incompressible and the screening is minimal. As a result, I P shows a maximum. When the bulk is compressible between the QHSs, I P comes down as a result of the increasing screening efficiency of the 2DHS. At ν = 0.30, where the WS phase develops, I P shows a local maximum, consistent with the WS phase being insulating and having, therefore, lower screening efficiency. I P at ν = 0.37 shows a "shoulder" at this density, but develops into a well-defined local maximum at lower densities [47]. At very high B, beyond 8 T, the 2DHS becomes strongly insulating and I P approaches the same value it has at the strongest QHSs, consistent with the screening efficiency being minimal.
The right inset in Fig. 2 shows the temperature depen-dence of I P at different ν. At ν = 0.120, I P starts with high value at the lowest temperature, consistent with an insulating WS. At the highest temperatures, where we expect the WS to have melted, I P saturates at a value which is lower than its maximum value. This is consistent with a compressible liquid phase which has a higher screening efficiency than the WS. However, as temperature is raised, instead of decreasing monotonically from its low-temperature value and saturating at the hightemperature limit, I P shows a well-defined minimum at a critical temperature T C . This temperature dependence is generic for all the traces shown in Fig. 1 inset except for ν = 0.370 and ν = 0.270, where I P at the lowest temperature is lower than its high-temperature limit. This is because the lowest temperature achieved in our measurements (T 40 mK) is close to T C for these two fillings; we expect I P to increase if lower temperatures were accessible.
The data shown in Fig. 2 inset suggest that the 2DHS becomes particularly efficient at screening near T C . A qualitatively similar behavior was recently seen in lowdensity GaAs 2D electron systems [24]. Associating T C with the melting temperature of the WS, Ref. [24] found the measured dependence of T C on ν to be consistent with the WS melting phase diagrams reported previously for the magnetic-field-induced WS in GaAs 2DESs. It is not clear why a WS should become particularly efficient at screening as it melts. It is possible that the minimum in I P signals the presence of an intermediate phase near the melting temperature, as has been suggested in a recent report [37]. Alternatively, very recent calculations [57] suggest that dissipation from mobile dislocations and uncondensed charge carriers become especially important near the melting of the WS phase. It is possible that they contribute to the extra screening at the melting.
Associating T C with the melting temperature of the WS, a plot of our measured T C vs. ν, as shown in Fig. 3, provides the WS thermal melting phase diagram of a 2DHS at p = 3.8. As ν increases from small values, T C decreases until the WS phase is "interrupted" by the well-developed ν = 1/3 FQHS. When ν is higher than 1/3, there is a reentrant WS phase between the 1/3 and 2/5 FQHSs, around ν 0.37. We note that our T C 50 mK at ν = 0.37 is consistent with the WS melting temperature reported in Ref. [37] for a 2DHS with a similar density at ν = 0.375.
The competition between the WS and FQHS liquid phases depends on the mixing between the LLs [28,29,[31][32][33][34][35][36][37][39][40][41][42][43][44][45]. This is often quantified in terms of the LLM parameter κ, defined as the ratio between the Coulomb energy and the LL separation: κ = (e 2 /4π 0 l B )/( eB/m * ), where l B = /eB is the magnetic length. Note that κ ∝ m * . When κ is large, the mixing with the higher LLs reduces the FQHS energy gaps and favors the formation of a WS at filling factors higher than 1/5 [28,29,[31][32][33][34][35][36][37][39][40][41][42][43][44][45]. Recent theoreti- cal work by Zhao et al. [45] directly mapped out a zerotemperature phase diagram for the quantum melting of the WS in the κ-ν space. The calculated phase diagram is reproduced in Fig. 4 for a direct comparison with our experimental results.
Our GaAs 2DHSs allow us to test the role of LLM. Compared to the GaAs 2DES, the 2DHSs have large m * . Because of the non-parabolicity of the valence bands and spin-orbit coupling, however, m * for 2DHS is intrinsically complex and depends on the specific sample parameters such as the QW width and the symmetry of the charge distribution [58]. For a systematic study, it is therefore essential to use both the front and back gates to keep the hole charge distribution in the QW symmetric while changing the density. Cyclotron resonance experiments [38] on 2DHSs confined to symmetric, 30-nm-wide QWs grown on GaAs (100) substrates yield a weakly densitydependent m * 0.48 in the density range we studied here. We use this value of m * to determine values of κ at a set of representative filling factors ν = 0.30, 1/3, 0.37 and 2/5 for our samples, and show these in Fig. 4 using symbols described in the inset.
For p = 3.8, the experimental data are represented by blue circles in Fig. 4. Data at all four fillings are consistent with the calculation results: as ν decreases, the 2DHS ground state changes from a FQHS at ν = 2/5 to a WS at 0.37, then to a FQHS at 1/3, and finally back to a WS at 0.30. In order to lower κ, we made measurements on a higher density 2DHS. The ρ xx vs. B data for this sample are shown in Fig. 5. At this density, well-developed FQHSs are seen at ν = 1/3, 2/5, and 2/7. Moreover, in contrast to the trace at p = 3.8 (Fig. 1), ρ xx at ν = 0.37 has comparable value to ρ xx at higher fillings, and depends only very weekly on temperature. This implies that the ground state at ν = 0.37 is not a WS at p = 7.9. On the other hand, similar to the data for p = 3.8, the trace in Fig. 5 shows a very large and strongly temperature dependent ρ xx peak at ν = 0.30, consistent with a pinned WS. We show the four experimental points for p = 7.9 at ν = 0.30, 1/3, 0.37, and 2/5, in Fig. 4 by purple circles. The data are again consistent with the theoretical phase diagram: as κ is reduced, the WS phase at ν = 0.37 disappears but it is still present at ν = 0.30.
We also performed measurements at three other densities, p = 6.2, 2.9, and 2.0; the results are presented in the SM [47], and are summarized in Fig. 4. For p = 6.2, the results are consistent with the theoretical phase diagram. For the lowest two densities, p = 2.9 and 2.0, however, there is a hint of a FQHS at ν = 2/5, but the data suggest a competition with an insulating phase, signaled by a rise in ρ xx as the temperature is lowered. This might indicate an apparent discrepancy between the experimental data and the theoretical phase diagram, which predicts that the ground state should be a FQHS (liquid) phase at ν = 2/5 in the entire range of κ in Fig. 4. We believe that disorder, whose role certainly increases at very low densities but is neglected in theory of Ref. [45], is at least partly responsible for the discrepancy [47]. It is worth remembering that, in early studies of GaAs 2DES, qualitatively similar observations were made. Early samples, which had lower quality, showed a competition between a FQHS and an insulating phase at ν = 1/5 [8], and a FIG. 5. Magneto-resistivity data for a 2DHS with p = 7.9 at 30 and 60 mK. The y-scale for the grey trace is expanded by factor of 50 to show the numerous observed FQHSs, attesting to the high quality of the 2DHS.
clear FQHS with a vanishing ρ xx at the lowest temperature was only seen when samples of much better quality were available [9].
In conclusion, we report a thermal melting phase diagram for the magnetic-field-induced WS in GaAs 2DHSs deduced from its screening efficiency. The phase diagram shows the clear reentrant behavior of the WS around the FQHS at ν = 1/3, and provides data for a quantitative comparison with future theoretical calculations. We also systematically study the quantum melting of the WS as a function of LLM, varied by changing the 2DHS density. While we find good overall agreement with the results of calculations, we would like to emphasize the complexity of the 2DHS LL diagram [58]. As discussed in more detail in the SM [47], the 2DHS LLs are non-linear and also can cross as a function of magnetic field. Moreover, the interaction between holes is subtle because of the multicomponent and mixed (spin and orbital) nature of the hole states. These make a quantitative assessment of the role of LLM challenging. We hope that our experimental data provide incentive for a more precise theoretical evaluation of the role of LLM, as well as disorder, in the competition between the WS and FQHS phases in GaAs 2DHSs. We performed all our measurements on 4 mm × 4 mm van der Pauw geometry samples, with InZn on the corners and sides annealed at 380 °C to make eight contacts to the 2DHS. For samples A and C, Ti/Au was deposited on top and used as front gates, and back gates were made by placing the samples on top of melted In. Both front and back gates are used to tune the density while keeping the charge distribution in the QW symmetric. We studied two densities p = 6.2 and 3.8 ×10 10 cm -2 for sample A, p = 7.9 × 10 10 cm -2 for sample B, and three densities p = 3.7, 2.9, and 2.0 × 10 10 cm -2 for sample C. The results for sample A at p = 3.8 × 10 10 cm -2 and sample B at p = 7.9 × 10 10 cm -2 are presented in the main text. Samples A and C were measured in a cryogen-free dilution refrigerator with a base temperature of 40 mK, while sample B was measured in a wet dilution refrigerator with a base temperature of 30 mK. All in-plane, magneto-transport measurements in this study were done using a low-frequency lock-in technique at 3 to 7 Hz frequency with excitation currents ranging from 1 to 10 nA. For the screening-efficiency (capacitance) measurements, we applied 1 mV AC excitation voltage V AC to the back gate at various frequencies around 20 kHz while keeping the 2DHS grounded, and measured the penetrating current from the front gate. Note that, because of the large distance between the back gate and the 2DES ( 500 µm), the density modulation due to V AC is negligible.
We have performed self-consistent calculations of the hole subband structure using the multiband envelope function approximation based on the 8×8 Kane Hamiltonian [58] and using the Hartree approximation. Figure 6 shows the confinement potential, charge distribution and LL diagram for 2DHSs confined in a 30-nm-wide GaAs QW with densities p = 2.0, 3.8, and 7.9 × 10 10 cm -2 . Figures 6(a) to (c) indicate that, as the density increases, the separation between the first and the second subband energies decreases from 2.49 meV at p = 2.0 × 10 10 cm -2 to 1.99 meV at p = 7.9 × 10 10 cm -2 . For all three densities, the Fermi energy, as indicated by the red dashdotted line, lies well above the second subband, leading to a single-layer-like charge distribution as shown in Figs.
The calculated LL diagrams are shown in Figs. 6(g) to (i). The LLs are non-linear and show multiple crossings as a function of magnetic field, which is very different from what one would expect based on a constant effective mass. Such complex LL diagrams make a quantitative assessment of the role of LLM very challenging. We note that in the analysis of the experimental data, we calculate the LLM parameter κ based on the effective mass measured via cyclotron resonance at low magnetic fields [38]. As described in the main text (e.g., see Fig. 4), we find good agreement between our experimental data and the theoretical quantum melting phase diagram calculated by Zhao et al. [45]. Note that the parameter κ used in the calculations of Ref. [45] is based on a single value for the effective mass, and assuming LLs which have a simple, linear dependence on magnetic field. However, considering the LL diagrams shown in Fig. 6, one should be cautious about the implications of such agreement.
For all three densities in Fig. 6, the calculations indicate a crossing of the two highest LLs at some fractional filling factor. Naïvely, this corresponds to a vanishing cyclotron frequency (i.e., an infinite effective mass) and thus a diverging LLM parameter κ. However, we argue that due to the intricate nature of hole LLs, the LLM is greatly suppressed for this pair of LLs. To show this we consider the effective 2D interaction that is frequently used to approximate the Coulomb interaction between charge carriers in quasi-2D systems. In a one-band model appropriate for electrons, the effective 2D interac-
6. Panels (a)-(c) show the calculated, self-consistent, valence-band confinement potential (black curves) and the relevant subband and Fermi energies for 2DHSs confined in a 30-nm-wide GaAs QW with densities p = 2.0, 3.8, and 7.9 × 10 10 cm -2 . The blue dashed lines in each figure show the energies of the first two subbands, and the red dash-dotted line the Fermi energy. Panels (d)-(f) show the corresponding self-consistent charge distributions in the QW. Panels (g)-(i) show the Landau level diagrams in the relevant magnetic field range. Black and green lines indicate Landau levels of opposite parity. The grey dashed lines mark the magnetic field positions of ν = 1/3 for different densities, and the red dash-dotted line traces the Fermi energy.
where ρ = (x, y) are the in-plane coordinates and ξ(z) is the wave function for the out-of-plane motion. At short distances ρ w (w is the width of the quasi-2D system), V * 2D (ρ) is weaker than the bare 2D Coulomb interaction V 2D (ρ) = 1/ρ; and V * 2D (ρ) approaches V 2D (ρ) for large distances ρ w.
We want to generalize V * 2D (ρ) to the case appropriate for holes where the charge carriers are characterized by multicomponent envelope functions [58]:
where φ i α (ρ) and ξ i α (z) denote the in-plane and out-ofplane parts of the ith spinor component of Ψ α in the basis of the bulk band-edge Bloch functions u i (ρ, z). The symbol α represents a generic index for the wave functions (C2). In the present case α stands for the LL index. Ignoring the i-dependence of φ i α (ρ), we obtain, similar to Eq. (C1), the direct Coulomb interaction:
and the exchange Coulomb interaction:
(C3b) In a one-band model, both V D 2D (ρ) and V X 2D (ρ) reduce to Eq. (C1). We may expect that in a 2D jellium model with a homogenous background ensuring charge neutrality, the direct Coulomb term (C3a) can be ignored, i.e., the Coulomb interaction between the charge carriers is represented by the exchange term (C3b) [50].
For the 2D hole systems studied here, spin-orbit coupling is large, so that spin is not a good quantum number for the states (C2). However, for symmetric QWs as in the present experiments, the envelope functions Ψ α (ρ, z) characterizing the LLs are eigenstates of parity [51], as illustrated in Figs. 6(g)-(i), where black and green lines indicate LLs of opposite parity. In particular, it turns out that the envelope functions Ψ α (ρ, z) and Ψ β (ρ, z) for the two lowest LLs have opposite parity, i.e., for each i the spinors ξ i α (z) and ξ i β (z) have opposite parity. The exchange interaction V X 2D (ρ) between these LLs is thus greatly reduced in magnitude, in particular for large distances ρ, when each z integral is effectively an integral over an odd function (Fig. 7). These theoretical considerations are consistent with our interpretation of the experiments using a large, but finite cyclotron effective mass, as cyclotron absorption is forbidden between states of opposite parity.
The suppression of the exchange interaction and thus the suppression of LLM between the two highest LLs is similar to the well-known fact that, if spin is a good quantum number, the exchange interaction acts only between charge carriers with the same spin orientation. It is also closely related to the fact that the enhancement of the Coulomb interaction in low-density quasi-2D hole systems can be greatly reduced in magnitude compared with the well-known enhancement of the Coulomb interaction in low-density quasi-2D electron systems, in particular when the quasi-2D hole system is spin-polarized [50,52]. A more detailed theoretical study of these aspects will be published elsewhere.
Figure 8 shows a summary of the frequency dependence measurements for sample A at p = 3.8 × 10 10 cm -2 . Figure 8(a) shows the penetrating current I P vs. magnetic field B at 22 and 1 kHz respectively. Despite the quantitative difference in the magnitude of I P , the traces show qualitatively the same behavior. Figure 8(b) shows the temperature dependence of I P at 22 and 1 kHz, measured at ν = 0.21. The critical temperature T C at which I P shows a minimum is the same even though the measurement frequencies differ by more than an order of magnitude, indicating that the measured critical temperature is independent of the measurement frequency.
Figure 9 compares the temperature dependence of the screening efficiency (I P ) and the transport (ρ xx ) measurements at ν = 0.21 for sample A at p = 3.8 × 10 10 cm -2 . In sharp contrast to the monotonic decrease of ρ xx as the temperature is increased, I P decreases first and then increases, showing a well-defined minimum at a critical temperature T C . A qualitatively similar behavior was reported in Ref. [24] for the magnetic-field-induced WS states near ν = 1/5 in GaAs 2D electrons. The contrast between the I P and ρ xx traces implies that the I P measurements provide additional information which is not discernible in transport measurements. While we do not have a clear explanation for why this is so, we speculate that the I P measurements might be more sensitive to the presence of a possible intermediate phase [37] or the additional dissipation from mobile dislocations and uncondensed charge carriers near the WS melting temperature [57], as we mentioned in the main text. Figure 10 shows the longitudinal resistivity ρ xx vs. perpendicular magnetic field B for sample A at p = 6.2 × 10 10 cm -2 . The trace was taken at a base temperature of 40 mK. The vertical marks indicate the expected positions of the filling factors as labeled. At this density, the ν = 1/3, 2/5, and 2/7 fractional quantum Hall states (FQHSs) are fully developed but the values of ρ xx at fillings between the FQHSs, namely at ν = 0.37 and 0.30, are quite high, consistent with disorder-pinned WS phases. In particular, ρ xx at ν = 0.30 is 507 kΩ/ . At ν = 0.37 ρ xx is 50 kΩ/ , about an order of magnitude larger than the ρ xx peaks at lower magnetic fields. When we raise the current from 1 nA to 5 nA, ρ xx at ν = 0.37 drops by a factor of two, but stays nearly constant at lower fields. We therefore conclude that at this density, the 2DHS is likely to be a pinned WS at ν = 0.37 and ν = 0.30.
2. Data for p = 3.7 × 10 10 cm -2
In this section we present additional data for sample C at p = 3.7×10 10 cm -2 . Figure 11 shows the temperature dependence of ρ xx vs. B, measured at 50, 85, and 210 mK. Similar to the data measured at p = 3.8 × 10 10 cm -2 , shown in Fig. 2 of the main text, the ν = 1/3 FQHS is fully developed at the lowest temperature while on its flanks ρ xx shows high resistivity consistent with an insulating phase. The developing 2/7 FQHS is also prominent here. The inset shows the Arrhenius plot of ρ xx at filling factors ν = 0.30, 0.37 and 2/5. The ρ xx values are comparable to those of Fig. 2 in the main text at similar temperatures; also, ρ xx at ν = 0.30 and 0.37 increases with decreasing temperature, while at ν = 2/5, it decreases.
Figure 12 shows the penetrating current I P vs. B measured at various temperatures. The trace at the lowest temperature of 50 mK shows features qualitatively similar to those at p = 3.8 × 10 10 cm -2 at 40 mK, as shown in Fig. 2 of the main text.
The six red triangles in Fig. 12 indicate the magnetic field positions at which I P shows local minima. As described in Ref. [24], this is an alternative way for probing the melting temperature of the WS.
Figure 13 provides a summary of the WS melting phase diagram at this density. Figure 13(a) shows the temperature sweeps at four filling factors ranging from ν = 0.140 to 0.296. The positions at which I P shows minimum values are shown by black squares in Fig. 13(b). The open squares correspond to the magnetic field positions indicated by the red triangles in Fig. 12. The yellow region indicates the WS phase and the white region the liquid phase. The vertical marks indicate the positions of ν = 1/3 and 2/7. In our measurements on this sample, we could only achieve temperatures down to ≈ 50 mK, and the WS phase on the higher filling side of 1/3 is already melted at this temperature: At ν = 0.37, we found I P to be monotonically increasing as a function of temperature for T ≥ 50 mK.
3. Data for p = 2.9 × 10 10 cm -2
In this section we present additional measurement results for sample C at p = 2.9 × 10 10 cm -2 . Figure 14 shows the temperature dependence of ρ xx vs. B measured at 50, 95, 210, and 410 mK. At this density, the ν = 1/3 FQHS is fully developed. There is a sign of an insulating background around ν = 2/5, and the 2/7 FQHS is no longer present. The inset shows the Arrhenius plot of ρ xx at ν = 0.30, 0.37, 2/5, and 1/3. ρ xx at ν = 0.30, 0.37, and 2/5 shows an insulating behavior, in stark contrast to the behavior at ν = 1/3, which is activated with a FQHS energy gap of 0.86 K.
Figure 15 shows the I P vs. B measured at different temperatures. From the lowest temperature trace we can see that the quantum Hall states are all weaker compared The penetrating current IP , normalized to its maximum value, vs. B measured at various temperatures for sample C at p = 2.9 × 10 10 cm -2 . The horizontal dashed line shows the maximum of IP when the 2DHS screening is minimal. The vertical marks indicate the field positions of fillings ν = 2, 1, 2/3, 2/5, 0.37 and 1/3, where IP shows local maxima at the lowest temperatures.
to those seen in Fig. 12. Furthermore, the local maximum at ν = 0.37 at the lowest temperatures is stronger. At high magnetic fields, I P starts to rise right after the 1/3 FQHS to the saturating value, and the local max- imum at ν = 0.30 is washed out by the rapidly rising background.
Figure 16 shows the measured WS melting phase diagram at this density. Figure 16(a) shows the temperature sweeps of I P at filling factors ranging from ν = 0.090 to 0.371. The critical temperatures at which I P shows minimum values are shown as black squares in Fig. 16(b). The yellow region indicates the WS phase and the white region the liquid phase. The vertical line indicates the position of the ν = 1/3 FQHS, and as shown here, the WS phase on the higher filling side of 1/3 is now present above 50 mK. Figure 17 shows the temperature dependence of ρ xx vs. B for sample C at p = 2.0 × 10 10 cm -2 . There is a lifting of ρ xx background at low temperatures around ν = 2/5, which is now so strong that there is no longer a minimum at ν = 2/5. At ν = 1/3, ρ xx increases with decreasing temperature and starts to decrease only at the lowest temperature, consistent with previous low-density measurements [33]. The inset shows the Arrhenius plot of ρ xx at ν = 1/3, 2/5, 0.37, and 0.30. At this low density the insulating behavior at ν = 0.30, 0.37, and 2/5 is much stronger, meaning that ρ xx decreases more as temperature rises compared to higher densities, and ρ xx at ν = 1/3 becomes non-monotonic with temperature.
Figure 18 shows the I P vs. B measured at various temperatures. At the lowest temperature, the ν = 1/3 FQHS is much weaker at this density. On the other hand, the I P peak corresponding to the insulating phase at ν = 0.37 becomes stronger. When the temperature is raised, the peak at ν = 0.37 vanishes quickly, indicating the melting of the WS while the peak at ν = 1/3 lasts even up to 300 mK.
Figure 19 shows the deduced melting phase diagram. The two insets show the temperature dependence of I P at a number of filling factors on the left and right sides of ν = 1/3 respectively, color coded to match the data points in the main figure. Similar to Fig. 16, the WS phase exists on both sides of the ν = 1/3 FQHS. On the higher-filling side of 1/3, the phase boundary exhibits a "dome" shape whose maximum is at ν 0.37. The dome shape appears to be asymmetric. Also, it goes through ν = 2/5 instead of coming down and approaching zero at ν = 2/5. We attribute this asymmetry to the developing insulating background near ν = 2/5 mentioned earlier. Figure 20 shows a summary of the WS melting phase diagrams plotted together for comparison. The data points shown by stars are from sample A and those shown by circles are from sample C. The vertical mark indicates the position of ν = 1/3. On the higher filling side of 1/3, T C increases as the density decreases. This general trend is also seen on the lower filling side of 1/3 down to ν ∼ 0.15. The increase in melting temperature at lower density is unexpected in the clean limit because the Coulomb energy is lower at lower density, and should bring down the melting temperature. This was indeed observed in Ref. [24] for very high quality GaAs two-dimensional electron systems (2DESs). However, mi- crowave resonance studies on 2DESs at very low densities have reported WS melting temperatures that are nearly independent of density [20], and have attributed it to the increasing role of the disorder pinning potential at lower densities [53]. Theoretical work in Ref. [53] in fact predicts that a stronger disorder (impurity) potential raises the melting temperature of the WS. It is possible that in our 2DHSs, the disorder potential at very low densities plays a strong role, leading to an increase of the melting temperature as the density is lowered. This is also consistent with a previous study of a lower quality 2DHS having a slightly higher melting temperature compared with our data [30].