Evidence of a two-stage melting of Wigner solids in two dimensions

TL;DR

This paper provides experimental evidence for a two-stage melting process of Wigner solids in two-dimensional systems, revealing an intermediate phase and advancing understanding of 2D phase transitions.

Contribution

It demonstrates the existence of a two-stage Wigner crystal melting process with an intermediate phase, supported by transport measurements in ultra-dilute 2D systems.

Findings

Evidence for a first-order WC-intermediate phase transition.

Observation of a two-stage melting process.

Identification of an intermediate phase in 2D Wigner solids.

Abstract

Two-dimensional (2D) solid-liquid transition (SLT)~\cite{Mermin1966Absence,Mermin1968Crystalline,Kosterlitz1972Long} concerns fundamental concepts of long-range correlations vital to magnetism, superconductivity, superfluidity, and topological matters. A long sought-after example is the melting of a Wigner Crystal (WC)~\cite{Wigner1934Interaction} of electrons. Detection efforts have targeted distinctive collective modes such as pinning by disorder, resonant-frequency absorption of Goldstone modes, and melting transition. However, only one-step second-order melting of softly-pinned modes was reported. Without the evidence of genuine pinning as exhibited in the charge density waves (CDWs)~\cite{PinningCDW}, these modes are likely intermediate phases which are only part of a complete SLT. To verify if there is a WC-intermediate phase transition will not only provide a solid proof of a WC, but will also unveil the nature of the SLT in relation to the two-stage Kosterlitz-Thouless (KT) model~\cite{Kosterlitz1972Long,Kosterlitz1973Ordering,Halperin1978Theory,Nelson1979Dislocation,hexatic_ceperley,hexatic_nelson, Young1979Melting}. %(or dislocation pairs) above a critical temperature $T_c$. Through transport studies of ultra-dilute high-purity 2D systems, this work presents evidence for not only a WC, but also for a two-stage WC-liquid SLT mediated by a first-order WC-intermediate phase transition.

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††preprint: AIP/123-QED

Two-dimensional (2D) solid-liquid transition (SLT) Mermin and Wagner (1966); Mermin (1968); Kosterlitz and Thouless (1972) concerns fundamental concepts of long-range correlations vital to magnetism, superconductivity, superfluidity, and topological matters. A long sought-after example is the melting of a Wigner Crystal (WC) Wigner (1934) of electrons. Detection efforts have targeted distinctive collective modes such as pinning by disorder, resonant-frequency absorption of Goldstone modes, and melting transition. However, only one-step second-order melting of softly-pinned modes was reported. Without the evidence of genuine pinning as exhibited in the charge density waves (CDWs) Lee and Rice (1979), these modes are likely intermediate phases which are only part of a complete SLT. To verify if there is a WC-intermediate phase transition will not only provide a solid proof of a WC, but will also unveil the nature of the SLT in relation to the two-stage Kosterlitz-Thouless (KT) model Kosterlitz and Thouless (1972, 1973); Halperin and Nelson (1978); Nelson and Halperin (1979); Clark et al. (2009a); Bruun and Nelson (2014); Young (1979). Through transport studies of ultra-dilute high-purity 2D systems, this work presents evidence for not only a WC, but also for a two-stage WC-liquid SLT mediated by a first-order WC-intermediate phase transition.

A WC is expected in ultra-dilute systems where the ratio of the inter-particle Coulomb energy $E_{ee}$ and the Fermi energy $E_{F}$, $r_{s}=E_{ee}/E_{F}=a/a_{B}$, surpasses 37 Tanatar and Ceperley (1989). $a=1/\sqrt{\pi n}$ is the Wigner-Seitz radius for electron density $n$ and $a_{B}=\hbar^{2}\epsilon/m^{*}e^{2}$ is the Bohr radius. The smallness of $E_{ee}=e^{2}/\epsilon a\sim 1me$V and $E_{F}$ $=n\pi\hbar^{2}/m^{*}\sim 25\mu e$V makes a WC fragile against random disorders which tend to render an Anderson insulator Anderson (1958) or a glass. More subtly, disorder fluctuations, along with the quantum fluctuations, reduce the melting temperature $T_{m}$ well below the classical melting point $T_{cm}\approx e^{2}\sqrt{\pi n}/\epsilon\Gamma$ ($\Gamma\approx 130$). As $T_{cm}$ is $\sim 100$ mK for $r_{s}=37$, $T_{m}$ can be even lower than the typical cooling capability of 10-30 mK depending on the disorder level. Disorder suppression is therefore key to keeping $T_{m}$ accessible. As conventional methods cannot provide cooling of the carriers below such a small $T_{m}$, we adopt a novel immersion cell cooling technique and demonstrate an actual $T_{m}\sim 30$ mK even in ultra-pure systems. The widely reported $T_{m}$, $\sim$150 and 300mK, is actually the liquefaction point of a part of a complete transition.

Experimental studies of WCs primarily employ transport and absorption techniques since a direct scanning method is not yet realistic. The most studied are the reentrant insulating phases (RIPs) in the fractional quantum Hall regime where AC and DC Jiang et al. (1991); Goldman et al. (1990); Williams et al. (1991), resonant absorption (rf, microwaves, and acoustic waves) Andrei et al. (1988); Chen et al. (2006); Zhu et al. (2010); Paalanen et al. (1992), and tunneling Jang et al. (2016) techniques are employed to detect the collective responses of putative WCs under pinning. Though a pinned WC should exhibit a sharp nonlinear threshold DC response, a benchmark for pinned CDWs Grüner (1988), most are reported as soft and rounded nonlinear IVs with small pinning strength ($\leq 1M\Omega$) and a large variance in the obscurely-defined pinning thresholds ($\sim$100s of mV/cm). Activated $T$ dependent pinning D’Iorio et al. (1992) hints that these putatively pinned states, with an exponentially decreasing translational correlation length $\xi$, are actually intermediate/mixed phases Nelson and Halperin (1980). This includes the systems used for resonant-absorption studies where collective modes are identified Andrei et al. (1988); Chen et al. (2006); Zhu et al. (2010); Paalanen et al. (1992); Jang et al. (2016). In contrast, $T$ dependence in a pinned WC should be moderate as quantum effects dominate.

On the other hand, the more desired $B=0$ studies require ultra-dilute carrier concentrations, i.e. $\sim 1\times 10^{9}$ cm*-2* for electrons or $\sim 4\times 10^{9}$ cm*-2* for holes in GaAs systems, in order to reach $r_{s}=37$. These have not been achieved by previous studies Kravchenko et al. (1991); Pudalov et al. (1993); Yoon et al. (1999) due to a greater challenge associated with a rising disorder effect caused by weakened inter-particle screening Huang et al. (2011). This is why localization occurs even in fairly clean systems when $a\propto n^{-1/2}$ approaches the localization length. We have overcome this barrier by realizing ultra-dilute 2D hole systems with much suppressed disorder.

Key observations include rigorously-pinned WCs and a first-order two-stage SLT. Enormous pinning with hundreds of M$\Omega$ resistivity is observed at $T<T_{m}\approx 30$ mK via sharp threshold DC-IV marked by more than three orders of magnitude drop in the differential resistance $r_{d}$ across a threshold point. Pinning collapses sharply beyond $T_{m}$, $\sim(1/4)T_{cm}$, until a liquid phase is reached. The presentation is divided into two parts: The first is a study of the RIP near filling $\nu=1/3$ using $p$-doped quantum wells, and the second is a zero-$B$-field study of ultra-dilute holes in undoped heterojunction-insulated-gate field-effect transistors (HIGFETs) at $r_{s}\geq$ 40.

The samples used for the RIP measurement are lightly doped $p$-type (100) GaAs quantum square wells. The density $p$ is $\sim 4\times 10^{10}$ cm*-2* ($r_{s}=24$), with mobility of $\mu\approx 2.5\times 10^{6}$ cm2/Vs. $a=1/\sqrt{\pi p}\sim 28n$m is half of the average carrier spacing, and $m^{*}\approx 0.45m_{e}$ is effective mass. The sample geometry is a 2.5$\times$0.5mm Hallbar realized via a wet chemical etch. Thermally deposited AuBe pads annealed at $460~{}^{\circ}$C achieve excellent Ohmic contacts to the 2D carriers, with measured contact resistances $\sim 400~{}\Omega$. Measurements are performed in a dilution refrigerator inside a fully shielded room, allowing the electronics to perform at their specified ratings.

Cooling carriers to $T_{bath}$ is critical and challenging, and it must be achieved via sufficient heat exchange which conventional methods can not accomplish. We adopt well-designed helium-3 sample immersion cells [Fig. 2(a)], which is the proven method for effective cooling down to 5 mK Huang et al. (2007a), and achieve thermalisation of the 2D holes with the mixing chamber (mc). The polycarbon cell is vacuum-tight and mounted at the lower end of a silver cold finger with its top fastened to the mc plate. The roof is a sintered silver cylindrical-block extension of the cold finger made by compressed pure silver micro-particles. During operation, helium-3 gas is continuously fed through a capillary into the cell where it condenses to fill the volume completely. Saturated sintered silver block provides $\sim$30 m2 contact area to cool the helium-3 bath. Major cooling of the 2D holes is realized via efficiently heat-sinking the metal contacts through sintered silver pillars providing 2.5 m2 surface area per lead. $T$ is monitored through a helium-3 melting curve thermometer inside an identical cell located next to the sample cell. The $T$ differential between the bath and the mc is $\leq$0.1mK at all times.

Fig. 2 (b) shows the magnetoresistance MR ($\rho_{xx}$) and the Hall resistance ($\rho_{xy}$) obtained at 10 mK via a four-terminal AC technique. The inset shows the Shubnikov de Haas (SdH) oscillations starting at 0.05 T. The RIP peak centers at $B=4.5~{}$T ($\nu=0.375$) between fillings $\nu=2/5$ and 1/3, with a dip in $\rho_{xy}$ consistent with previous studies Sajoto et al. (1993). Fixing $B=$4.5 Tesla, where the magnetic length $l_{B}=\sqrt{hc/eB}\approx a\sim 28$nm, an electrometer-level DC-IV setup is adopted to obtain the IV relationship for several $T$ from 10 to 300mK [Fig. 3(a)], with a voltage bias $V$ between $\pm$10mV (at $0.1\mu$V resolutions) to cover sufficient dynamic range. Current sensing is realized via a low-noise preamp at 50 $f$A precision. The $T$ dependence reveals three types of IV characteristics as a function of $T$: sharp threshold below some critical $T\sim$40 mK; rounded/soft nonlinear IV between 40 and 120 mK; and linear IV beyond 120 mK.

For a fixed $T$ below 40 mK, the dynamical response shows a sharp threshold IV with a striking six-thousand-times plunge in the differential resistance $r_{d}=dV/dI$, from $\sim$250 M$\Omega/\square$ down to $\sim$40k$\Omega/\square$. Joule heating of $<10^{-15}$ Watts at $V_{c}$ is negligible. The enormous sub-threshold $r_{d}$ presents a greater pinning strength than reported for pinned CDWs Grüner (1988). Sharply rising $V$ in the absence of current flow (with a current less than the threshold current $I_{c}\sim 2-3p$A) is consistent with a pinned crystal whose potential energy grows via elastic deformation on a scale of $\xi$ in response to an increasing external field $E$. Conventionally, the threshold is thought to correspond to a pinning and depinning of a WC Glasson et al. (2001). Indeed, the supra-threshold $r_{d}(V)$ becomes linear above $V=1.5$ mV, consistent with a non-pinned state. However, the values for the linear supra-threshold $r_{d}(V)$, $\sim$40k$\Omega/\square$, are in excellent agreement with that of a liquid state above the liquefaction point (shown later), indicating a pressure ($E$)-driven melting. It implies an intermediate phase between the pinned WC and a liquid. As this region is where $E_{c}$ was usually assigned in previous studies, more detailed examination is necessary.

Fig. 3(b) is a zoom-in view of the (dotted line) box of Fig. 3(a) in which IVs are shown for 10, 25, 50, 75 and 150 mK. For the two IVs for 10 and 25 mK, $r_{d}$ for pinned modes, $\sim 250$M$\Omega$, is mostly identical up to $V=0.4$ mV. It confirms only a slight $T$ dependence of $\xi$ below 40 mK. This lack of $T$ dependence is in sharp contrast to previous results Jiang et al. (1991); Goldman et al. (1990); Williams et al. (1991); Andrei et al. (1988); Yoon et al. (1999); Chen et al. (2006); Zhu et al. (2010); Paalanen et al. (1992); Jang et al. (2016); Kravchenko et al. (1991); Pudalov et al. (1993); Yoon et al. (1999) and is consistent with a quantum crystal behavior mentioned earlier. Now, deviation from pinning occurs at different points for different $T$, 0.4 mV (for 25 mK) and 0.8 mV (for 10 mK), beyond which are the intermediate phases characterized by soft/rounder IVs. Now, $E_{c}=V_{c}/L$ ($L$-length over $V_{c}$ is applied) is naturally defined as the deviation point. Lower $E_{c}$ for $T=25$ mK is qualitatively consistent with the Lindemann criterion Bruun and Nelson (2014) for higher $T$.

However, for $T$ above 40 mK, $E_{c}$ approaches zero and rigorous pinning disappears in the intermediate phase. Rounded/soft nonlinear IVs dominate even at zero-bias. Nevertheless, substantial $r_{d}$, i.e. $\sim 10$M$\Omega$ at 50 mK, indicates a finite shear modulus $\kappa$, sustained by collective modes, only expected for a first-order transitionMillis and Littlewood (1994). The disappearance of $E_{c}$ is consistent with a discontinuous $\xi$ across $T\sim$40 mK. It is now clear that the rounded nonlinear IV, which has been regarded by several previous studies  Goldman et al. (1990); Kravchenko et al. (1991); Pudalov et al. (1993); Yoon et al. (1999) as pinned WCs, is actually of an intermediate phase, which crosses over to a liquid in a second-order manner at higher $T$. The linear IV at 150 mK is beyond the liquefaction point ($T_{l}$).

$T$-driven melting transition is captured by the maximum $r_{d}(T)$ found at minimal external bias for different $T$. Fig. 3(c) shows a pronounced piece-wise dependence across a critical temperature of $\sim 38$ mK defined as $T_{m}$. A discontinuous $\xi$ acoss $T_{m}$, thus also carrier internal energy, is now apparent. Determination of $\xi$ is difficult Lee and Fukuyama (1978); Fukuyama and Lee (1978) and we provide only a rough estimate which turns out to be substantially larger than previous reportsGoldman et al. (1988); Williams et al. (1991); Jang et al. (2016). The maximum potential energy at $E_{c}$ is $U=Nw$. $w=eE_{c}a\sim 0.024\mu$V$\ll T$ is the single particle potential energy. $N=p\xi^{2}$ is the number of carriers on a scale of $\xi$. For $T=25m$K, $E_{c}=V_{c}/L\sim 8$ $mV/$cm where $L=0.5$ mm. By balancing the electrical force $NeE_{c}$ with the pinning force $\kappa a$ as shown in reference Goldman et al. (1990); Williams et al. (1991) ($\kappa$ being the shear modulus), $N\sim 1.5\times 10^{5}$ or $\xi\geq 10\mu$m is obtained. $U$ is $\sim 2.4me$V (or 30 K), comparable to $E_{ee}$. A complete phase diagram on $T$-$V$ axes is shown in Fig. 3(c).

To identify the nature of the intermediate phase is difficult because the relationship between $r_{d}$ and $\xi(T)$ is yet to be proven. Here, we show, as a minor point, that $r_{d}(T)$ can be fitted to $r_{d}=r_{0}\exp[c/(T-40mK)^{\gamma}]$ with $r_{0}\approx 23$ and $c\approx 9.5$, in the same trend as the exponentially decreasing $\xi(T)$ modeled for a hexatic phase Nelson and Halperin (1979): $\xi\sim\exp[c/(T-T_{m})^{\gamma}]$ ($\gamma\approx 0.3696$).

We now turn to the zero-field study for which undoped GaAs/AlGaAs HIGFETs Kane et al. (1995); Noh et al. (2003); Huang et al. (2007b) are adopted for the benefit of more suppressed disorder than any doped devices. A 6 mm$\times 0.8$ mm Hall bar [Fig. 4(a)] is realized with a self-align fabrication process Huang et al. (2007b). The carrier accumulation at the hetero-interface is capacitively induced through biasing a top gate beyond a turn-on voltage, $\sim-1.3$ V, at which the valance band edge meets the chemical potential [Fig. 4(b)]. The band gap of the 600nm-thick $Al_{0.3}Ga_{0.7}As$ barrier is $\sim$2 $e$V. Owning to the superior crystal quality, zero gate leakage (not measurable on a 0.05 $p$A scale) is achieved at all operating bias. $p$ is tunable from $4\times 10^{10}$ down to $7\times 10^{8}$ cm*-2*. Inset of Fig. 4(c) is a typical $R_{xx}$ for $p\sim 1\times 10^{10}$ cm*-2*.

Accessing $r_{s}\geq 37$ requires $p\leq 4.2\times 10^{8}$ cm*-2* assuming $m^{*}=0.25m_{0}$. $m^{*}$ is an estimate since a complicated dispersion associated with the light-heavy hole bands mixing and the spin-orbit coupling Winkler (2003) is not yet well-understood. On the other hand, the onset of an insulator as a function of $r_{s}$ can be obtained via the metal-to-insulator transition (MIT) Kravchenko et al. (1995). As shown in Fig. 4(c), $T$ dependence of $\rho(T)$, for $p$ from $7\times 10^{8}$ to $5.6\times 10^{9}$ cm*-2*, yields a sign change in $d\rho/dT$ around a critical density of $p_{c}\sim 4\times 10^{9}$ cm*-2*. Remarkably, $p_{c}$ corresponds to $r_{s}\sim 38$, the expected onset of a WC. Based on the RIP study, the 40-500 mK $T$ range covers from an intermediate phase to a liquid phase. $\rho(T)$ follows a nonactivated power law down to $p=7\times 10^{8}$ cm*-2*, as indicated via linear trends in the double-logarithmic scales, contrasting the hopping conduction Mott (1969); Efros and Shklovskii (1975) found in more disordered systems. For details of the nonactivated behavior and its conversion to activated transport in response to an increase of disorder, refer to references Huang et al. (2011, 2012).

Alternative to the DC voltage bias used for the RIP case, a current bias, with Keithley 6430 $f$A source, is employed with a sub-$\mu$V-resolution voltage sensing at an input impedance of $10^{16}\Omega$. Forcing a constant $I$ provides a more stringent probe to a pinned WC because it allows local effects such as melting to occur. Nonlinear IV is expected to be weakened in an inferior pinning, i.e. in an intermediate/mixed phase (or a glass). On the contrary, a pinned rigorous WC should still exhibit a sharp threshold just as for the voltage-driven scenario, even though the voltage reading above the threshold is an overestimated cordal resistance (described in Ref. Butcher et al. (2013)) due to the separation of current and voltage channels.

The following DC results are for $p=2.8\times 10^{9}$ cm*-2* (or $r_{s}\sim 45$) measured between 28 to 45 mK. The (Fermi) wavelength $\lambda$ is $\approx 400n$m, twice the carrier spacing. A pinned WC is confirmed by a sharp threshold IV obtained at 28 mK shown in Fig.5(a). This first demonstration of rigorous pinning in a zero-$B$ is marked by a 90 M$\Omega/\square$ sub-threshold $r_{d}$ below a tiny $I_{c}\sim 4p$A. The supra-threshold $r_{d}$ collapses one hundred times, slightly weaker than the RIP case, which is consistent with an overestimated $V$ mentioned above. $I_{c}$ corresponds to a smaller threshold field $E_{c}\sim 4m$V/cm (or $\sim 10^{-10}$ V$/a_{B}$), yielding a slightly larger single particle potential energy of $\sim eE_{c}a\sim 0.04\mu eV\sim 0.46$ mK due to the larger $a\sim 100$ nm. Setting $NeE_{c}=\kappa a$ as shown earlier, one obtains $N\sim 5\times 10^{5}$, corresponding to a WC on a substantial scale of $\xi\sim 100\mu$m. The dominating potential energy $U\sim 20me$V $>E_{ee}$ is consistent with a crystal. For a consistency check, the same setup is used to measure the RIP and the result is shown as the blue curve. The power dissipation is $\leq 2\times 10^{-16}$ Watts, ruling out appreciable Joule heating.

Pressure ($E$)-driven melting occurs in a similar fashion to RIP except for slight oscillations of $r_{d}$ in the intermediate phase [inset of Fig. 5(a)]. The IV curves for increasing $T$ shown in Fig. 5(b) display an involvement from rigorous to soft pinning and then to an absence of pinning, consistent with a two-stage melting. Liquefaction occurs at 42 mK, noticeably lower than the RIP. This is likely due to stronger effects of quantum fluctuations at small wavevectors and disorder fluctuations due to lack of screening in such a dilute limit. Oscillations in $r_{d}$ are stronger in the $T$-driven intermediate phase than in the $E$-driven intermediate phase. It is possibly related to the formation of stripes with long-range orientational order Nelson and Halperin (1979) that vary with increasing drive, as seen in electrons on a helium surface Glasson et al. (2001). Another possibility is that small $T_{l}$ facilitates a melting and re-crystallization of pinned WC domains, instead of or in addition to shearing, when driven across the fixed points of disorder Normand et al. (1992).

Melting transition probed by maximum $r_{d}$ (at minimal bias) is shown in Fig. 5(c) where a piecewise behavior, slight $T$ dependence below $T_{m}$ and the exponential $T$ dependence above $T_{m}$, is identical to the RIP results. Across $T_{m}\sim 30$ mK, the energy (or $\xi$) discontinuity occurs more abruptly than the RIP, in a manner agreeing with a recent quantum Monte Carlo simulation for a first order WC-intermediate phase transition mediated by a discontinuous jump in topological defects Clark et al. (2009b). The first-order nature could be a unique feature of the melting of a quantum system. A phase diagram is shown in Fig. 5(d).

To summarize, a complete melting of a quantum WC has been captured as a two-stage SLT similar to the KT model. Distinctions between a WC and an intermediate phase are evidenced by the disappearance of $E_{c}$, a drastically different $T$ dependence corresponding to a discontinuity in the internal energy (or $\xi$), and an enormous $\xi$ for a pinned WC. They strongly support a first-order WC-intermediate phase transition which is qualitatively agreeable with Ref. Clark et al. (2009a), instead of the second order KT model via the unbinding of dislocations. Effective cooling and low disorder are crucial to the formation of a WC, which is proven by rigorously pinned modes that survived stringent DC tests. $T_{m}$ is well below $T_{cm}$, indicating strong effects from quantum fluctuations and system disorders that require further understanding.

We acknowledge the support of this work from NSF under DMR-1410302. The work at Princeton was partially funded by the Gordon and Betty Moore Foundation through Grant GBMF2719, and by the National Science Foundation MRSEC-DMR-0819860 at the Princeton Center for Complex Materials.