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Electrons are expected to crystallize into a solid (so-called ‘Wigner crystal’5) when the (Coulomb) interaction energy between the electrons sufficiently dominates over the kinetic energy. One example of such an electron solid was found in a very low-density two-dimensional electron system (2DES) realized on helium surfaces4 (at zero magnetic field). Because of the low n, the zero-point motion (given by the Fermi energy Ef=n h2/2πm, where m is the electron mass) is negligibly small and at finite temperatures (T), as in the experiment4, the kinetic energy originates mainly from the classical thermal motion (kBT). The melting of such a ‘classical’ 2D electron solid is determined only by the competition between the thermal kinetic energy and Coulomb interaction (, where ε is the dielectric constant) (the International System of Units, SI, is used exclusively in this paper) and is thought to be describable by the Kosterlitz–Thouless theory of 2D melting6. Experimentally, the melting was found to occur4 at (where kB is the Boltzman constant) with Γ∼130, in excellent agreement with theoretical calculations7,8.

The 2DES as realized in the high-quality GaAs/AlGaAs structures in our experiment has relatively high n, thus (in the absence of magnetic fields) the zero-point motion (Ef) is significant and the 2DES does not solidify even at T=0. However, it is well known that a sufficiently strong perpendicular magnetic field (B) tends to suppress the kinetic energy of 2D electrons and induce the solidification9,10. On the other hand, at finite B the motion of electrons is quantized into Landau levels, and delicate many-body quantum correlations11,12 among electrons can cause the 2DES to condense into fractional quantum Hall (FQH)13 liquid states at certain rational fractional values of Landau level filling factor ν=n h/e B. Experimentally, the magnetic-field-induced electron solid (MIES)3 forms at sufficiently small ν, following the termination of FQH states at low T. It is deformed and pinned by disorder in the system, thus it behaves as an insulator (in d.c. transport), and has a characteristic resonance (see, for example, ref. 14) in its frequency (f) dependent, real diagonal conductivity (Re[σx x(f)]) due to the ‘pinning mode’ of domains of the elastic solid oscillating collectively around the disorder15,16,17.

The melting of such a MIES has been studied in various experiments18,19,20,21,22,23, and it was commonly presumed8,18,20,22 that at a fixed ν the melting should be similar to that of a classical electron solid (the expected exact ground state for a 2DES at infinite B), and thus Tm would be determined by n, as Tcm is. In this letter, we show unambiguously that this is not the case, that is, the melting of a MIES does not behave as a classical electron solid, and in fact in any given sample (whose intrinsic disorder is fixed), Tm is determined by ν, not n, and is unrelated to Tcm.

In our experiments we have studied the T-dependence of the pinning mode resonance of the MIES in its Re[σx x(f)] spectrum measured by microwave spectroscopy24. No resonance is observed when T is raised above some characteristic Tm, taken as the melting T of the electron solid. By systematically measuring Tm while varying both n and B, we found that within the experimental resolution, Tm in a given sample is only a function of ν (that is, Tm(n,B)=Tm(n/B)) down to the smallest ν∼0.05 attained in our experiments. Although Tm is generally sensitive to n at fixed B, Tm is insensitive to n at fixed ν. As ν=n h/e B=2(lB/r)2, where the magnetic length (where ħ=h/2π) is a measure of the size of the single electron wavefunction and is the mean separation between the electrons, our findings reflect the quantum nature of the 2D electron solid formed at finite B and demonstrate that its melting (Tm) is effectively controlled by the inter-electron quantum correlation, which depends on lB/r.

We have studied two different 2DES samples. Sample 1 is a GaAs/AlGaAs heterojunction. Sample 2 is a 15-nm-wide AlGaAs/GaAs/AlGaAs quantum well (QW). By backgating and/or having different cooldowns, the electron densities in both samples can be tuned to various extents (to be specified below). Typically at their respective as-cooled n, sample 1 has mobility μ∼6×106 cm2 Vs−1, and sample 2 has μ∼1×106 cm2 Vs−1.

The inset of Fig. 1b shows a schematic local cross-section of a typical sample (not to scale). A metal film coplanar waveguide (CPW), lithographically deposited on the surface, enables the measurement of Re[σx x(f)] of the 2DES. A network analyser generates a microwave signal propagating along the CPW and coupling capacitively to the 2DES (located 0.1–0.5 μm below the surface). The measured relative power absorption (P) by the 2DES can be related24 to Re[σx x] of the 2DES as P=exp((2l Z0/w)Re[σx x]), where l and w are the total length and slot width of the CPW respectively and Z0=50 Ω is the CPW characteristic impedance. CPW of meander shapes24 are commonly used to obtain larger geometric ratios (2l/w), and therefore to increase the strength of the signal (P). The sample is mounted on a metal block that is kept in good thermal equilibrium with the mixing chamber of a dilution refrigerator during the measurements. The microwave input is kept in the low power limit by reducing the power until the signal (P) no longer changes. A negative voltage between a backgate (located ∼200 μm below the surface) and the 2DES enables in situ reduction of n from the as-cooled values.

Figure 1: Temperature (T) dependence of the microwave spectra of the 2D electron solid measured from sample 1 at two different densities (n) with the same Landau filling factor (ν=n h/e B).
figure 1

a, T-dependence of the spectra at n=5.6×1010 cm−2 with ν=0.128 (magnetic field B=18 T). Spectra at a series of representative T values are shown and offset for clarity. The pinning resonance of the electron solid observed at T=50 mK is seen to weaken with increasing T and disappear at ∼250 mK, taken as the melting temperature (Tm) of the electron solid. The inset shows that the amplitude of the resonance extrapolates to zero at the similar Tm. Typical error bars for the resonance amplitude are less than 0.5 μS.b, T-dependence of the spectra at n=2.1×1010 cm−2 with ν=0.128 (B=6.86 T). The low T resonance disappears at a Tm similar to that shown in a (n=5.6×1010 cm−2), despite the fact that n has been reduced by more than a factor of two here (using a negative voltage between a backgate and the 2DES). The inset shows a schematic diagram of the sample. The dark regions on the top surface represent the coplanar waveguide (CPW). The backgate allows the in situ change of n.

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Figure 1 shows the T-dependence of the microwave resonance of the electron solid in sample 1 and the determination of Tm at two different values of n with ν fixed at a representative value of 0.128. In Fig. 1a, n=5.6×1010 cm−2 (B=18 T) and the Re[σx x(f)] spectrum shows a clear resonance with peak frequency (fpk) near 600 MHz at low T (∼50 mK). As T is increased, the resonance weakens. Its fpk at elevated T also decreases slightly from the base T (50 mK) value, but by no more than 20%, indicating that there is no significant loss of pinning of the electron solid due to the increase in the temperature. The resonance disappears into noise background at ∼250 mK, which we take as the melting T (Tm) of the electron solid. The inset of Fig. 1a shows that the resonance amplitude (obtained from a lorentzian fit) extrapolates to zero at a similar Tm. In Fig. 1b, n has been reduced to 2.1×1010 cm−2 (using a backgate voltage of −300 V) while B is also reduced to 6.86 T to keep ν the same value as in Fig. 1a. The lower n reduces the electron–electron interaction relative to the electron–disorder interaction, and the pinning resonance of the electron solid now occurs (at low T) near 1.6 GHz, which is significantly higher than that in Fig. 1a. However, the resonance disappears above a Tm∼270 mK, which is similar to the Tm in Fig. 1a, within the experimental uncertainty in determining Tm (typically ∼10%).

We have measured Tm in four different cooldowns of sample 1 for many combinations of n and B, at which a resonance from the MIES can be detected. We plot all of these Tm data as a function of ν in Fig. 2. Sample 1 has a particularly large range of tunable n, from ∼1.2–8.1×1010 cm−2, covering a ν range from ∼0.21 down to ∼0.03. Generally, we found Tm to be sensitive to n or B separately (when fixing one and changing the other). On the other hand, near similar ν (changing both n and B while fixing n/B), we have always found Tm to be insensitive to n (or B), within the experimental errors in Tm. Different cooldowns can vary Tm by up to ∼15% (at similar ν) but this does not affect our conclusion. Thus, we find Tm to be mainly determined by ν, so that Tm versus ν as plotted in Fig. 2 defines a melting curve for the electron solid in sample 1. A linear fit of Tm versus ν gives a guide to the eye, shown as the dashed line, which lies within 20% of all of the (Tm,ν) data points and within 10% of a majority (70%) of them. In the inset of Fig. 2 we plot the ‘reduced’8,18 tm=Tm/Tcm from two similar cooldowns versus ν, where is the melting T of a classical 2D electron solid defined earlier (with the value7,8 Γ=127). In contrast to Tm, tm can vary significantly (sometimes by a factor of 3) at similar ν. Thus, tm versus ν does not give a well-defined melting curve for the electron solid, confirming that Tm is not determined by n or Tcm. We have also checked that neither Tm nor tm plotted against either n or B gives any well-defined melting curve. Furthermore, we notice that our measured Tm is also unrelated to the hypothetical ‘thermal depinning’ temperature (Tdepin) of domains15,16,17 of the electron crystal. In fact, we can estimate Tdepin∼(1/kB)meω02ξ2(L/a)2∝n3/2 (where ω02=2πfpke B/me∝n−3/2 characterizes the ‘effective pinning potential’16 (we note that ω0 is also much larger than kBTm/ħ), ξ is the disorder correlation length, L∝n is the Larkin domain size15,16,17 and a is the electron crystal lattice constant), in contrast to Tm, which is determined by ν, and as seen in Fig. 2, typically decreases with increasing n if B is fixed.

Figure 2: (Tm,ν) phase diagram for the electron solid in sample 1.
figure 2

The Tm values are measured in a total of four cooldowns (shown as diamonds, circles, squares and triangles), over a wide range of densities (n=1.2–8.1×1010 cm−2) and magnetic fields. Within the experimental uncertainty, Tm versus ν gives rise to a well-defined melting curve of the electron solid. The dashed line is a guide to the eye, obtained by a linear fit through all of the data. Typical error bars in Tm are less than 10%. The inset shows the ‘reduced’ tm versus ν from two cooldowns. tm is defined as Tm normalized by the classical 2D electron-solid melting temperature (where we take Γ=127). tm versus ν does not result in a well-defined melting curve, indicating that the melting temperature Tm is not determined by n or Tcm.

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Figure 3 shows the (Tm,ν) melting curve measured on sample 2 (15-nm-wide QW). Sample 2 has a tunable n=2.7–4×1010 cm−2. Probably owing to the relatively narrow confinement of the 2DES in the QW, sample 2 enters the solid phase for ν<0.3 and the typical fpk observed is ∼6–8 GHz. In this sample we again found Tm to be controlled by ν, although the value of Tm is higher than that in sample 1 at similar ν.

Figure 3: (Tm,ν) melting curve of the electron solid in sample 2, a narrow QW of width 15 nm.
figure 3

Sample 2 has a tunable n=2.7–4×1010 cm−2 and enters an electron-solid phase at ν<0.3. Typical error bars in Tm are 10%.

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In each of the two samples, at a fixed high B, Tm typically increases with decreasing n. This is opposite to the classical behaviour, where Tcm decreases with decreasing n. At a fixed ν, our observed Tm is insensitive to n, and Tm in each sample is a well-defined function of ν. Recent theories25,26 have suggested that many-body quantum correlations between electrons can still be important in the solid phase terminating the FQH states at high B. The parameter that captures such quantum correlations is the Laudau filling factor ν=2(lB/r)2, which also determines the (single-particle) wavefunction overlap27 between neighbouring electrons. This overlap (Is), given27 by , is in fact quite small in the high-B, low-ν regime of the MIES. Our data show that Tm is quite strongly dependent on ν (for example, Tm changes by almost a factor of 2 when ν is changed by a factor of 2). Therefore the dependence of Tm on ν is unlikely to result from a Tm that is determined by the Coulomb Hartree energy EH (which is shown27 to have very weak ν-dependence in this low-ν regime, where the wavefunction overlap Is has a negligible contribution to EH). Our findings indicate that the melting of MIES is effectively controlled by the inter-electron quantum correlation27, through ν. The well-defined (Tm,ν) melting curve we obtained constitutes the phase boundary between a quantum solid and a correlated quantum liquid17 in each sample.

The Tm we measured in both samples are of a similar order of magnitude to those measured previously in other samples with various experimental techniques18,19,20,21,22,23. We have noticed that at similar ν, sample 2 (narrow QW) has a higher Tm than sample 1 (heterojunction). It has been suggested15 that the relevant disorder in semiconductor samples that pins the MIES comes mostly from the interfaces vertically confining the 2DES (this is consistent with the relatively high fpk observed in the narrow QW sample). Thus, our findings suggest that a 2D electron solid subjected to stronger pinning disorder melts at higher Tm, an effect that has been predicted previously28. Interestingly, we notice that studies on the effects of pinning disorder or geometric confinement29 on Tm in other solids usually find the opposite behaviour. For example, both the vortex solid in a high-temperature superconductor subjected to artificial pinning centres30 and a helium solid in a porous glass31 show a depression of Tm with the added disorder. Disorder is generally unavoidable in semiconductor samples, and as seen here can strongly influence the values of Tm measured, which may not be those of an ideal, clean Wigner crystal. On the other hand, the observation that in any given sample (thus with fixed intrinsic disorder), Tm is not sensitive to n (nor to B) if ν=n h/e B is fixed (but is sensitive to n if B is fixed) is difficult to explain in a classical picture of, for example, interplay of disorder and n (screening). The fact that ν is found to be the controlling variable for Tm in samples of quite different disorder (including sample 1, in which it can be inferred from the observed pinning resonance that the Wigner crystal there can possess substantial correlation length14,15,16,17) suggests that such a behaviour (Tm being controlled by ν) arises from some mechanism in which many-body quantum correlation between electrons may be important. The exact role of disorder (perhaps involving its interplay with quantum correlation) in determining the value and behaviour of Tm remains to be better understood.

It is also interesting to compare the quantum nature of our 2D electron solid with the quantum solids of helium. In a helium solid, the size of atoms is fixed by nature and the quantum parameter is the De Boer parameter1 (where M is the atomic mass, a the inter-atomic distance and v the inter-atomic potential strength), which is fixed at fixed n. The Tm of a helium solid is only determined by n. In the case of a 2D electron solid formed in high B, the size of the single electron wavefunction (lB) is readily tunable by B, independently of n. The quantum parameter here is ν= n h/e B, and we have found that Tm of such an electron solid in a given sample (with fixed disorder) is controlled by ν rather than n.