High temperature resistivity measured at {\nu} = 5/2 as a predictor of 2DEG quality in the N=1 Landau level

TL;DR

This study demonstrates that the resistivity at filling factor 5/2 at high temperature can predict the excitation gap of the fractional quantum Hall state, providing a new metric for assessing 2DEG quality beyond mobility.

Contribution

It introduces resistivity at ν=5/2 as a novel high-temperature predictor of the excitation gap, highlighting its correlation with long-range disorder effects.

Findings

Resistivity at ν=5/2 correlates with the excitation gap Δ_{5/2}.

Resistivity reflects long-range disorder, unlike mobility.

Resistivity at ν=5/2 is a reliable predictor across different samples.

Abstract

We report a high temperature (T = 0.3K) indicator of the excitation gap $\Delta_{5/2}$ at the filling factor $ \nu=5/2$ fractional quantum Hall state in ultra-high quality AlGaAs/GaAs two-dimensional electron gases. As the lack of correlation between mobility $\mu$ and $\Delta_{5/2}$ has been well established in previous experiments, we define, analyze and discuss the utility of a different metric $\rho_{5/2}$, the resistivity at $\nu=5/2$, as a high temperature predictor of $\Delta_{5/2}$. This high-field resistivity reflects the scattering rate of composite fermions. Good correlation between $\rho_{5/2}$ and $\Delta_{5/2}$ is observed in both a density tunable device and in a series of identically structured wafers with similar density but vastly different mobility. This correlation can be explained by the fact that both $\rho_{5/2}$ and $\Delta_{5/2}$ are sensitive to long-range disorder from remote impurities, while $\mu$ is sensitive primarily to disorder localized near the quantum well.

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The two-dimensional electron gas (2DEG) in GaAs/AlGaAs heterostructures remains the preeminent semiconductor platform for the study of strong electron-electron correlations in reduced dimensions. As the design of GaAs/AlGaAs 2DEG structures becomes more sophisticated and ultra-low temperature experiments become more complicated, the question of how best to make a preliminary assessment of sample quality becomes increasingly important DasSarma2014 . This is especially true for heterostructures designed to explore the most fragile fractional quantum Hall states (FQHSs) in the N=1 Landau level (LL), a regime in which many distinct phases are separated by small intervals in filling factor $\nu=hn/eB$ (h is Planck’s constant, n is 2DEG density, e is the charge of the electron and B is magnetic field) that must be examined at temperatures below 50mK. For example, transport signatures of the putative non-Abelian $\nu=5/2$ and $\nu=12/5$ FQHS and reentrant integer quantum Hall effect (RIQHE) states are strongest at temperatures T$\leq$20mK. Traditionally, zero-magnetic-field mobility measured at much higher temperatures (0.3K$\leq$T$\leq$2K) has been used as a primary metric of 2DEG quality, but a large body of experimental and theoretical evidence has now shown that mobility does not necessarily encode all information needed to predict high-field behavior in the fractional quantum Hall regimeNeubler2010 ; Gamez ; Manfrareview ; DasSarma2014 ; Pan2011 . Evidently additional methods must be employed to predict behavior at lower temperatures and high magnetic field in the highest quality samples Manfrareview . In the context of the present work, quality is quantified by the size of the energy gap of the FQHS at $\nu=5/2$, $\Delta_{5/2}$. We note that samples with high $\Delta_{5/2}$ generally show high-quality transport throughout the N=1 LL.

Composite fermion (CF) theory replaces a system of highly interacting electrons with a system of weakly-interacting composite fermions by vortex attachment Halperin1993 ; Jain1 ; Jain2 and explains the physics around filling factor $\nu=1/2$ in the N=0 LL. Extending this theory to other half-fillings in higher LLs, we expect that at T=0.3K, the state at $\nu=5/2$ is also described by composite fermions experiencing zero effective magnetic field $B_{eff}=0$. Indeed, Willett and collaborators BoB demonstrated the existence of a Fermi surface at $\nu=5/2$ at T=0.3K using surface acoustic wave techniques. The CF model has also been successfully used to analyze energy gaps of odd denominator FQHSs around $\nu=5/2$ by using the CF cyclotron energy Kumar ; Samkharadze . In the CF picture, the fractional quantum Hall state that emerges at T$\leq$100mK is viewed as a condensation of CFs driven by a BCS-like p-wave pairing instability Moore1991 ; JainBook . In this work, we assume that at T=0.3K a Fermi sea of CFs forms at $\nu=5/2$, and measure the resistivity of this metallic phase, $\rho_{5/2}$, analogously to the zero field resistivity. We investigate this high field resistivity as a high temperature (T = 0.3K) indicator of the strength of the $\nu=5/2$ FQHS at low temperatures.

The longitudinal resistance as a function of magnetic field B measured at T = 0.3K in the vicinity of $\nu=5/2$ is plotted in the inset of Fig. 1. A resistance minimum is observed at $\nu=5/2$. It is noteworthy that the positive magnetoresistance near $\nu=5/2$ resembles the transport behavior near $\nu=3/2$ and $\nu=1/2$ Du1993 ; Rokhinson1995 ; Rokhinson1997 , but contrasts with the negative magnetoresistance often observed near zero field. The temperature dependence of resistance at $\nu=5/2$ is shown in Fig. 1. In this exemplary Arrhenius plot, $R_{xx}$ at $\nu=5/2$ shows activated behavior below 100mK: it increases as temperature increases following $R_{xx}\varpropto e^{-\frac{\Delta}{2k_{B}T}}$. A linear fit through the activation region yields an energy gap $\Delta_{5/2}$=570mK. However, $R_{xx}$ at $\nu=5/2$ starts to saturate when temperature exceeds 150mK, and at T=300mK it is insensitive to temperature. The very weak temperature dependence observed around T=0.3K is an important attribute; it indicates that a description based on a gapped FQHS with a dilute population of thermally activated FQHS quasiparticles is no longer justified as it is at significantly lower temperatures. It is appealing to consider this change a temperature-driven transition to a CF Fermi sea. As a purely practical matter, the temperature insensitivity of $\rho_{5/2}$ suggests the utility of our method of characterization in much the same way that B=0 mobility is insensitive to temperature below T$\sim$1K in ultra-high quality samples. The resistivity measured at $\nu=1/2$ is equal to the CF resistivity. It can be shown that the resistivity measured at $\nu=5/2$ is equivalent to CF resistivity up to a numerical scale factor Rokhinson1995 ; Halperin1993 .

We have studied two types of samples. The first sample is a density tunable device: an in situ back-gated 2DEG. The 2DEG is grown by molecular beam epitaxy (MBE) and resides in a 30nm GaAs quantum well bounded by Al0.24Ga0.76As barriers with Si $\delta$-doping in a narrow GaAs layer flanked by pure AlAs layers placed 66nm above the principal 30nm GaAs quantum well. This design has been shown to yield the largest gap energy for the $\nu=5/2$ FQHS Manfrareview ; Deng1 ; Deng2 ; Josh . It is believed Neubler2010 that low conductivity electrons confined in the X-band of the AlAs barriers surrounding the narrow GaAs doping wells screen potential fluctuations caused by remote donor impurities, promoting expression of large gap FQHS in the N=1 LL. This particular sample displays the largest excitation gap for the $\nu=5/2$ FQHS reported in the literature, attesting to its high quality Waston . The in situ gate consists of an $n^{+}$ GaAs layer situated 850 nm below the bottom interface of the quantum well. There is an 830nm GaAs/AlAs superlattice between the 2DEG and back gate to minimize leakage current. The device is a 1mm by 1mm lithographically-defined Van der Pauw (VdP) square with eight Ni$\setminus$Ge$\setminus$Au$\setminus$Ni stack contacts diffused along the sample edges; processing details have been described in reference Waston .

In a second set of experiments we examine a series of samples each sharing the same heterostructure design: a 30nm GaAs quantum well with 2DEG density $n=3.0\times$ $10^{11}/cm^{2}$. The quantum well is flanked by symmetric Si $\delta$-doping in GaAs doping wells as described previously Manfrareview . These samples are grown in a single MBE growth campaign; sample mobility improves as the unintentional impurities emanating from the source material decrease with increasing growth number. Each specimen consists of a 4mm by 4mm VdP square with 8 diffused indium contacts on the edges. We perform standard low frequency (13-85 Hz) lock-in measurements. The excitation currents for resistivity measurement and gap measurement are 10nA and 2nA, respectively. We use the Van der Pauw method for measuring resistivites; the quoted resistivity is the usual average of four distinct resistance measurements. The samples are homogeneous such that the resistances measured along different crystallographic directions are similar, both at $\nu=5/2$ and zero field.

In Fig. 2(a) we present the dependence of $\rho_{5/2}$ (left axis) and $\Delta_{5/2}$ (right axis) for various densities of the in situ back gated GaAs 2DEG. The typical uncertainty for gap and resistivity measurement is $\pm 5\%$. As $n$ increases, $\Delta_{5/2}$ increases and $\rho_{5/2}$ decreases. A clear correlation between $\rho_{5/2}$ and $\Delta_{5/2}$ is observed in this density tunable device.

In the density-tunable device, we expect that as density increases the resistivity $\rho_{5/2}$ should decrease due to the increasing carrier concentration, and the energy gap $\Delta_{5/2}$ should increase due to the increase of the Coulomb energy scale. It is also possible that the scattering rate may change with changing density Halperin1993 ; DasSarma2014 , so the change of $\rho_{5/2}$ and $\Delta_{5/2}$ with changing density likely reflects both their explicit density dependence and the effects of scattering. The strong correlation we observe between $\rho_{5/2}$ and $\Delta_{5/2}$ in this device indicates that $\rho_{5/2}$ captures both the density dependence and the effects of scattering on the energy gap.

We note that at high densities, $\rho_{5/2}$ begins to increase and $\Delta_{5/2}$ plateaus and decreases slightly. This has been explained by occupation of the first excited subband of the quantum well Waston , and may be responsible for the slight mismatch between the minimum of $\rho_{5/2}$ and the peak of $\Delta_{5/2}$; evidently the correlation between $\rho_{5/2}$ and $\Delta_{5/2}$ is not as strong when there is parallel conduction through an excited subband.

We then measured $\rho_{5/2}$ for the series of wafers with the identical heterostructure design and fixed 2DEG density. The relationship between $\rho_{5/2}$ and $\mu$ is illustrated in Fig. 3: $\rho_{5/2}$ initially drops monotonically as mobility increases, but it saturates at $\mu\sim 15\times 10^{6}cm^{2}/Vs$ even though mobility keeps increasing over the course of the MBE growth campaign. In this high mobility range $\rho_{5/2}$ and $\mu$ appear to have no relationship to one another; samples with the same $\rho_{5/2}$ may have vastly different $\mu$.

A few samples with various combinations of $\rho_{5/2}$ and $\mu$ are chosen from the sample set in Fig. 3 to measure energy gaps ($\Delta_{5/2}$) of the fractional quantum Hall state that forms at $\nu=5/2$ at lower temperatures. Fig. 4 (a) displays the relation between $\Delta_{5/2}$ and $1/\rho_{5/2}$ for those samples: $\Delta_{5/2}$ increases monotonically as $1/\rho_{5/2}$ increases. For the largest $1/\rho_{5/2}$, where $\rho_{5/2}$ is $39\Omega$, $\Delta_{5/2}$ reaches 570mK, among the largest gaps at this density ever reported in literature Deng1 ; Waston ; Kumar . Here, in this comparison of different samples with the same density but different levels of disorder, we also observe a strong correlation between $\Delta_{5/2}$ and $\rho_{5/2}$, indicating that $\rho_{5/2}$ is sensitive to the scattering mechanisms that limit $\Delta_{5/2}$. We also plot $\Delta_{5/2}$ vs. $\mu$ for these samples. As it is shown in Fig. 4 (b), in the low mobility range where $\mu<10\times 10^{6}cm^{2}/Vs$, $\Delta_{5/2}$ increases as $\mu$ increases. However, no correlation exists in the high mobility range where $\mu>10\times 10^{6}cm^{2}/Vs$.

We briefly discuss why the CF resistivity measured by $\rho_{5/2}$ may contain different information than the zero-field mobility and show a stronger correlation with the low-temperature $\nu=5/2$ FQHS. Zero-field resistivity is dominated by large-angle scattering [1]. Because of this, the zero-magnetic-field mobility is primarily limited by Coulomb scattering from impurities located directly in the quantum well DasSarma2008 ; DasSarma2014 ; Deng1 ; Gardner2016 , while remote impurities primarily cause small-angle scattering which has significantly less impact on mobility. Composite fermions at half-filling are also scattered by impurities; however, remote charged impurities also induce spatial variations in the effective magnetic field $B_{eff}$ due to variations in the 2DEG density Halperin1993 ; Zhang1992 . These effective magnetic field variations cause increased large-angle scattering of CFs, resulting in an enhanced contribution by remote impurities to the CF resistivity Halperin1993 ; Rokhinson1995 ; Mirlin1995 . Because of this, we expect that $\rho_{5/2}$ is more sensitive to remote impurities than the zero-field mobility and thus contains different information about the distribution of impurities in the system. This increase in large-angle scattering has been observed experimentally at $\nu=1/2$ Du1993 . However, the connection between CF resistivity at $\nu=5/2$ and FQHS gap strength has not been previously studied. Our data suggests that variations of $B_{eff}$ from remote impurities dominate the measured $\rho_{5/2}$.

Other experiments Rokhinson1995 ; Rokhinson1997 ; Kang1995 and theory Khveshchenko1996 ; Mirlin1997 at $\nu=1/2$ have shown that short-range CF-CF interactions via gauge field fluctuations result in an additional correction to the CF scattering rate and resistivity which is not present at zero field. This may be another reason that $\rho_{5/2}$ provides information about the disorder potential from impurities that $\mu$ does not.

Next, we address the sensitivity of the strength the $\nu=5/2$ FQHS to disorder. It has been shown through computational methods that the size of quasiparticles and quasiholes in the $\nu=5/2$ state is unusually large, on the order of at least 12 times the magnetic length Neubler2010 ; this large size is expected to make $\Delta_{5/2}$ sensitive to long-range disorder from remote charged impurities Neubler2010 ; Pan2011 . Experiments studying the effects of remote doping schemes have confirmed that remote impurities from ionized donors do indeed have a large impact on $\Delta_{5/2}$, but minimal effect on $\mu$ Gamez ; Umansky2009 . A particularly illuminating experiment is detailed in Ref. Deng1 : it was found that intentionally placing short-range disorder directly in the quantum well drastically reduced mobility but had a comparatively small effect on $\Delta_{5/2}$, confirming that $\mu$ is limited by short-range disorder while $\Delta_{5/2}$ is more sensitive to long-range disorder from remote impurities. The fact that both $\Delta_{5/2}$ and $\rho_{5/2}$ are sensitive to long-range disorder from remote impurities explains the strong correlation we observe between the two quantities and the lack of correlation with $\mu$.

Additionally, if the $\nu=5/2$ FQHS is considered to be a p-wave Cooper-paired state of composite fermions as in the Moore-Read Pfaffian state Moore1991 ; JainBook , then it is natural to compare our results to what is observed in p-wave superconductors. In p-wave superconductors, as the normal-state resistivity increases due to impurity scattering, the superconducting transition temperature $T_{c}$ is expected to decrease Abrikosov1960 ; Millis1988 ; Levin1993 ; Mackenzie2003 , and strong suppression of $T_{c}$ with increasing normal-state resistivity has been observed experimentally in the putative p-wave superconductor Sr2RuO4 Mackenzie1998 ; Mackenzie2003 . Therefore, the direct correlation we observe between the normal-state CF resistivity at T=0.3K and the low-temperature $\nu=5/2$ FQHS energy gap is in qualitative agreement with the strong dependence of $T_{c}$ on normal-state resistivity in p-wave superconductors.

We mention that the quantum scattering time $\tau_{q}$ measured by low-field Shubnikov-de Hass oscillations Coleridge1989 ; Coleridge1991 is also expected to be sensitive to long range disorder, and thus might be expected to be a predictor of $\Delta_{5/2}$ DasSarma2014 . However, a previous experiment in a density-tunable device Neubler2010 showed no correlation between $\tau_{q}$ and the strength of the $\nu=5/2$ FQHS. We also measured $\tau_{q}$ in our back-gated device and found no correlation with $\Delta_{5/2}$ (data not shown here); a detailed analysis of quantum scattering time in our samples is presented in a forthcoming publication.

In conclusion, we observe a strong correlation between $\rho_{5/2}$ and $\Delta_{5/2}$ in both a density-tunable device and in a series of samples with fixed 2DEG density. Therefore, we propose the use of $\rho_{5/2}$ measured at T = 0.3K as a metric to predict the strength of $\nu=5/2$ FQHS at low temperatures. The fact that we observe this correlation both when the electron density is varied (in the back-gated device) and when the level of disorder is varied (in the series of fixed-density samples) makes our method a robust tool for predicting $\Delta_{5/2}$. A possible physical origin for the correlation is that $\rho_{5/2}$ is sensitive to large-angle scattering by remote impurities due to the variation of the $B_{eff}$ and to short-range CF-CF interactions, neither of which are reflected in the zero-field mobility.

This work was supported by the Department of Energy, Office of Basic Energy Sciences, under Award number DE-SC0006671. Additional support for sample growth from the W. M. Keck Foundation and Microsoft Station Q is gratefully acknowledged. We thank S. H. Simon for important comments on this work.