Present address: ]Department of Physics, Columbia University, New York, NY, USA

Hidden Quantum Hall Stripes in Al_xGa_1-xAs/Al_0.24Ga_0.76As Quantum Wells

X. Fu School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA Yi Huang (黄奕) School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA Q. Shi [ B. I. Shklovskii School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA M. A. Zudov zudov001@umn.edu School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA G. C. Gardner Microsoft Quantum Lab Purdue, Purdue University, West Lafayette, Indiana 47907, USA Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA M. J. Manfra Microsoft Quantum Lab Purdue, Purdue University, West Lafayette, Indiana 47907, USA Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA School of Electrical and Computer Engineering and School of Materials Engineering, Purdue University, West Lafayette, Indiana 47907, USA

(July 27, 2025)

Abstract

We report on transport signatures of hidden quantum Hall stripe (hQHS) phases in high ( $N>2$ ) half-filled Landau levels of Al_xGa_1-xAs/Al_0.24Ga_0.76As quantum wells with varying Al mole fraction $x<10^{-3}$ . Residing between the conventional stripe phases (lower $N$ ) and the isotropic liquid phases (higher $N$ ), where resistivity decreases as $1/N$ , these hQHS phases exhibit isotropic and $N$ -independent resistivity. Using the experimental phase diagram we establish that the stripe phases are more robust than theoretically predicted, calling for improved theoretical treatment. We also show that, unlike conventional stripe phases, the hQHS phases do not occur in ultrahigh mobility GaAs quantum wells, but are likely to be found in other systems.

Discovery of the integer quantum Hall effect in Si (von Klitzing et al., 1980) has paved the way to observations of many exotic phenomena in two-dimensional (2D) electron and hole systems. Two prime examples are the fractional quantum Hall effect (Tsui et al., 1982) and quantum Hall stripes (QHSs) Koulakov et al. (1996); Fogler et al. (1996); Moessner and Chalker (1996); Lilly et al. (1999a); Du et al. (1999). While fractional quantum Hall effects have been realized in many systems, including GaAs (Tsui et al., 1982), Si (Nelson et al., 1992; Kott et al., 2014), AlAs (De Poortere et al., 2002), GaN (Manfra et al., 2002), graphene (Du et al., 2009; Bolotin et al., 2009), CdTe (Piot et al., 2010), ZnO (Tsukazaki et al., 2010), Ge (Shi et al., 2015), and InAs (Ma et al., 2017), exploration of the QHS physics remains limited to GaAs (not, a).

Forming due to a peculiar boxlike screened Coulomb potential, QHSs can be viewed as charge density waves consisting of stripes with alternating integer filling factors $\nu$ , e.g., $\nu=4$ and $\nu=5$ (not, b). In experiments, QHSs are manifested by giant resistivity anisotropies ( $\rho_{xx}\gg\rho_{yy}$ ) in $N\geq 2$ half-filled Landau levels. Appearance of these anisotropies in macroscopic samples is attributed to a mysterious symmetry-breaking field Fil (2000); Koduvayur et al. (2011); Sodemann and MacDonald (2013); Pollanen et al. (2015), which nearly always aligns QHSs along the $\hat{y}\equiv\left<110\right>$ crystal axis of GaAs (not, c). While a sufficiently low disorder is necessary for the QHS formation, the absence of QHSs in systems beyond GaAs might simply be due to the lack of symmetry-breaking fields (not, d). Indeed, electron bubble phases Koulakov et al. (1996); Fogler et al. (1996); Moessner and Chalker (1996); Cooper et al. (1999); Eisenstein et al. (2002); Lewis et al. (2002); Deng et al. (2012); Rossokhaty et al. (2016); Friess et al. (2017); Bennaceur et al. (2018); Friess et al. (2018); Fu et al. (2019); Ro et al. (2019), which are close relatives of QHSs, have already been identified in graphene (Chen et al., 2019).

In this Letter, we report observation of transport signatures of the recently predicted (Huang et al., 2020) hidden QHS (hQHS) phases in a series of Al_xGa_1-xAs/Al_0.24Ga_0.76As quantum wells with $x<10^{-3}$ . In contrast to the ordinary QHS phases, the hQHS phases are characterized by isotropic resistivity ( $\rho_{xx}=\rho_{yy}=\rho$ ) that is independent of $\nu$ , unlike the isotropic liquid phases in which $\rho\propto\nu^{-1}$ . These unique properties make these phases detectable without symmetry-breaking fields, thereby opening an avenue to study stripe physics in systems beyond GaAs. The wide variation of mobilities in our samples allows us to construct an experimental phase diagram in the conductivity-filling factor plane. Its comparison to theoretical predictions (Huang et al., 2020) yields the electron quantum lifetimes and the stripe density of states. The latter turns out to be lower than predicted by original the Hartree-Fock theory (Koulakov et al., 1996; Fogler et al., 1996), calling for further theoretical input. We confirm this finding by a complementary experiment on an ultrahigh mobility GaAs quantum well, where we also show that, in this sample, the hQHS phase yields to the QHS phase in agreement with the theory.

Before presenting our experimental data, we briefly summarize the physics behind the hQHS phases (Huang et al., 2020). The resistance anisotropies in the ordinary QHS phase emerge due to different diffusion mechanisms along and perpendicular to the stripes (MacDonald and Fisher, 2000; von Oppen et al., 2000; Sammon et al., 2019). In this picture, an electron drifts a distance $L_{y}$ along the $y$ -oriented stripe edge in an $x$ -directed internal electric field until it is scattered by impurities to one of the adjacent stripe edges located at a distance $L_{x}=\Lambda/2\approx\sqrt{2}R_{\mbox{\scriptsize{c}}}$ (Koulakov et al., 1996; Fogler et al., 1996), where $\Lambda$ is the stripe period and $R_{\mbox{\scriptsize{c}}}$ is the cyclotron radius. When $L_{y}\gg L_{x}$ , the diffusion coefficient in the $\hat{y}$ direction is much larger that in the $\hat{x}$ direction, which leads to anisotropic conductivity, $\sigma_{yy}\gg\sigma_{xx}$ , and resistivity, $\rho_{xx}\gg\rho_{yy}$ . Since $L_{y}\propto\nu^{-1}$ and $L_{x}\propto\nu$ (Sammon et al., 2019), the anisotropy decreases with $\nu$ and eventually vanishes at some $\nu=\nu_{1}$ . At larger $\nu$ , the drift contribution to the diffusion along stripes can be neglected, and $L_{y}$ , like $L_{x}$ , is determined entirely by the impurity scattering. For isotropic scattering, it is easy to show (not, e) that $L_{y}=\sqrt{2}R_{\mbox{\scriptsize{c}}}$ which coincides with $L_{x}$ . As a result, the QHS phase yields to the hQHS phase in which the resistivity is isotropic and $\nu$ independent (since the stripe density of states does not vary with $\nu$ ). The hQHS phase persists until the stripe structure is destroyed by disorder at $\nu=\nu_{2}$ and the ground state becomes an isotropic liquid with $\rho_{xx}=\rho_{yy}\propto\nu^{-1}$ , as predicted by Ando and Uemura (Ando and Uemura, 1974) and experimentally confirmed by Coleridge, Zawadski, and Sachrajda (CZS) (Coleridge et al., 1994).

For the hQHS phase to exist and be detected, it should span a sizable range of the filling factors $\Delta\nu=\nu_{2}-\max\{\nu_{1},9/2\}\gg 1$ . The range $\Delta\nu$ depends sensitively on both transport $\tau^{-1}$ and quantum $\tau_{\rm q}^{-1}$ scattering rates, which control $\nu_{1}$ and $\nu_{2}$ , respectively (Huang et al., 2020). As we will see, ultrahigh mobility GaAs quantum wells do not support the hQHS phase as $\nu_{1}\approx\nu_{2}$ in these samples. On the other hand, adding the correct small amount of Al (not, f) to the GaAs well greatly expands $\Delta\nu$ , as it affects $\nu_{1}$ to a much greater extent than it does $\nu_{2}$ . This happens because Al acts as a short-range disorder, which contributes equally to transport $\tau^{-1}$ and quantum $\tau_{\rm q}^{-1}$ scattering rates, and because $\tau_{\rm q}/\tau\ll 1$ at $x=0$ .

Table 1: Sample ID, Al mole fraction

x

, electron density

n_{e}

, mobility

\mu

, and Drude conductivity, in units of

e^{2}/h

\tilde{\sigma}_{0}=hn_{e}\mu/e

at zero magnetic field (

B=0

Sample ID	$x$	$n_{e}$ ( $10^{11}$ cm^-2)	$\mu$ ( $10^{6}$ cm²/Vs)	$\tilde{\sigma}_{0}$ ( $10^{3}$ )
A	0.00057	3.0	6.5	8.0
B	0.00082	2.9	4.1	4.9
C	0.0078	2.7	1.2	1.3

Apart from different $x$ , all our Al_xGa_1-xAs quantum wells share an identical heterostructure design Gardner et al. (2013). Electrons are supplied by Si doping in narrow GaAs wells surrounded by narrow AlAs layers and placed at a setback distance of $75$ nm from each side of the 30-nm-wide Al_xGa_1-xAs well hosting the 2D electrons. Parameters of samples A, B, and C such as Al mole fraction $x$ , electron density $n_{e}$ , mobility $\mu$ , and Drude conductivity $\tilde{\sigma}_{0}=hn_{e}\mu/e$ in units of $e^{2}/h$ at zero magnetic field ( $B=0$ ) are listed in Table 1. The samples are approximately 4 mm squares with eight indium contacts positioned at the corners and at the midsides. Longitudinal resistances $R_{xx}$ and $R_{yy}$ were measured in sweeping magnetic fields using a four-terminal, low-frequency (a few Hz) lock-in technique at a temperature $T\approx 25$ mK at which the resistances are nearly temperature independent. The current was sent along either the $\hat{x}\equiv\left<1\bar{1}0\right>$ or $\hat{y}\equiv\left<110\right>$ direction using the midside contacts, and the voltage was measured between contacts along the edge. To account for anisotropies due to nonideal geometry, $R_{xx}$ or $R_{yy}$ was multiplied by a factor (typically $\lesssim 1.1$ ) that was chosen to make $R_{xx}=R_{yy}$ in the low field regime.

In Fig. 1, we present longitudinal resistances $R_{xx}$ and $R_{yy}$ as a function of filling factor $\nu$ measured in sample B. At low half-integer filling factors ( $\nu=9/2$ , $11/2$ , and $13/2$ ) the data reveal conventional QHS phases, as evidenced by $R_{xx}>R_{yy}$ . At high half-integer filling factors ( $\nu>25/2$ ) we identify the CZS phase in which $R_{xx}\approx R_{yy}\propto\nu^{-1}$ (cf. dash-dotted line). At intermediate half-integer filling factors, $\nu=15/2,...,23/2$ , one readily confirms both characteristic features of the hQHS phase; indeed, the data show that two longitudinal resistances are practically the same ( $R_{xx}\approx R_{yy}$ ) and are independent of $\nu$ (cf. dashed line). From Fig. 1, we can easily identify the characteristic filling factors $\nu_{1}\approx 7$ and $\nu_{2}\approx 12.5$ which mark the crossovers from the QHS to the hQHS phase and from the hQHS to the CZS phase, respectively.

In a similar manner, we have obtained $\nu_{1}$ and $\nu_{2}$ for sample A and $\nu_{2}$ for sample C (which does not support the QHS phase due to higher Al mole fraction $x$ ), which we then use to construct the experimental phase diagram shown in Fig. 2. We start by adding points representing the dimensionless Drude conductivity $\tilde{\sigma}_{0}$ for samples A, B, C (see Table 1) and the corresponding filling factors $\nu_{1}$ (solid circles) and $\nu_{2}$ (solid squares). To connect these data points we use the theoretical boundaries of the hQHS phase (Huang et al., 2020). The lower boundary $\nu=\nu_{1}$ , separating the QHS and the hQHS phases, is given by (Huang et al., 2020)

\nu_{1}\simeq\frac{\sqrt{\tilde{\sigma}_{0}}}{\alpha}\,,

(1)

where $\alpha$ (not, g) is the QHS density of states in units of the density of states per spin at $B=0$ , $g_{0}=m^{\star}/2\pi\hbar^{2}$ . This boundary can be obtained by either matching the parameter-free geometric average of the resistivities in the QHS phase $\sqrt{\rho_{xx}\rho_{yy}}=(h/e^{2})/(2\nu^{2}+1/2)\approx(h/e^{2})/2\nu^{2}$ (MacDonald and Fisher, 2000; Sammon et al., 2019) and the resistivity in the hQHS phase (Huang et al., 2020),

\tilde{\rho}_{\rm hQHS}\equiv\frac{h}{e^{2}}\frac{\alpha^{2}}{2\tilde{\sigma}_{0}}\,,

(2)

or, equivalently, by setting the resistivity anisotropy ratio to unity, $\rho_{xx}/\rho_{yy}\approx(\tilde{\sigma}_{0}/\alpha^{2}\nu^{2})^{2}=1$ (Sammon et al., 2019; Huang et al., 2020).

Refer to caption — Figure 1: Longitudinal resistances $R_{xx}$ (solid line) and $R_{yy}$ (dotted line) as a function of the filling factor $\nu$ measured in sample B. Gap centers between spin-resolved Landau levels are labeled by $N=2,3,...$ , at the top axis ( $\nu=2N+1$ ). The conventional QHS phase ( $R_{xx}>R_{yy}$ ) and the CZS phase ( $R_{xx}\approx R_{yy}\propto\nu^{-1}$ ) occur at half-integer $\nu=9/2,11/2,13/2$ and at $\nu=27/2,29/2,...$ , respectively. The hQHS phase is identified at intermediate half-integer filling factors, $\nu=15/2,....,25/2$ , where the resistance is isotropic *and* $\nu$ independent. The characteristic $\nu^{0}$ ( $\nu^{-1}$ ) dependence of the isotropic resistance in the hQHS (CZS) phase is marked by dashed (dash-dotted) line.

The higher boundary $\nu=\nu_{2}$ marks the crossover from the hQHS to the CZS phase and is represented by

\nu_{2}\simeq\frac{\tilde{\sigma}_{0}}{\alpha^{2}}\frac{\tau_{\rm q}}{\tau}\,.

(3)

This boundary can be obtained by equating $\alpha$ and the density of states at the center of the Landau level in CZS phase, in units of the density of states at $B=0$ , $\sqrt{\omega_{\mbox{\scriptsize{c}}}\tau_{\rm q}}$ (Raikh and Shahbazyan, 1993; Mirlin et al., 1996) or by matching $\rho_{\rm hQHS}$ and the resistivity in the CZS phase (Coleridge et al., 1994),

\rho_{\rm CZS}\equiv\frac{h}{e^{2}}\frac{1}{\nu}\frac{\tau_{\rm q}/2\tau}{(\tau_{\rm q}/2\tau)^{2}+1}\approx\frac{h}{e^{2}}\frac{1}{\nu}\frac{\tau_{\rm q}}{2\tau}\,.

(4)

We thus see that for a given carrier density, as mentioned above, $\nu_{2}$ and $\nu_{1}$ are controlled by $\tau$ and $\tau_{\rm q}$ , respectively. Strictly speaking, Eqs. (1), (3) are not sharp boundaries but rather gradual crossovers between corresponding phases.

With the help of Eq. (1) and experimental values of $\nu_{1}$ in samples A and B, we estimate $\alpha\approx 11$ , which is smaller than the theoretical estimate of $\alpha\simeq 18$ (Sammon et al., 2019; not, g). We then parameterize scattering rates $\tau^{-1}$ and $\tau_{\rm q}^{-1}$ as

\tau^{-1}=\tau_{0}^{-1}+\kappa x\,,~~~\tau_{\rm q}^{-1}=\tau_{\rm q0}^{-1}+\kappa x\,,

(5)

where $x$ is the Al mole fraction, $\kappa\approx 24$ ns^-1 per % Al (Gardner et al., 2013), and $\tau_{0}^{-1}\approx 3$ ns^-1 (Gardner et al., 2013) is the transport scattering rate in the limit of $x\rightarrow 0$ . To find the remaining parameter $\tau_{\rm q0}^{-1}$ , which is the quantum scattering rate in the limit of $x\rightarrow 0$ , we use experimental $\nu_{2}$ values and notice that Eqs. (1), (3) yield $\tau_{\rm q}/\tau\simeq\nu_{2}/\nu_{1}^{2}$ . Using Eq. (5) we then obtain an estimate for $\tau_{\rm q0}\simeq 0.05$ ns which is in good agreement with $\tau_{\rm q}$ values found from low $B$ experiments (Shi et al., 2016a, 2017a; Zudov et al., 2017) on microwave-induced (Zudov et al., 2001; Ye et al., 2001; Dmitriev et al., 2012) and Hall-field-induced (Yang et al., 2002; Zhang et al., 2007; Vavilov et al., 2007) resistance oscillations in GaAs quantum wells.

We next use $n_{e}=3\times 10^{11}$ cm^-2 and $m^{\star}=0.06\,m_{0}$ (Coleridge et al., 1996; Tan et al., 2005; Hatke et al., 2013; Shchepetilnikov et al., 2017; Fu et al., 2017) to compute the phase boundaries, Eqs. (1), (3), which are shown in Fig. 2 by solid lines. Both lines pass in close proximity to the experimentally obtained $\nu_{1}$ (solid circles) and $\nu_{2}$ (solid squares) from all samples, showing excellent agreement between theory (Huang et al., 2020) and experiment. Finally, we add data points (open circles and squares) from three other Al_xGa_1-xAs/Al_0.24Ga_0.76As quantum wells that were investigated in a different context (Shi, 2017). These points are also in agreement with the theory and the present experiment.

Having confirmed the existence of the hQHS phases in Al_xGa_1-xAs/Al_0.24Ga_0.76As quantum wells, we next examine the possibility for these phases to exist in ultrahigh mobility GaAs quantum wells (without alloy disorder). In such samples, the lower boundary $\nu_{1}$ , Eq. (1), might approach and even merge with the higher boundary $\nu_{2}$ , Eq. (3), eliminating the hQHS phase as a result. To test this scenario, we revisit the data obtained from sample A of Ref. Sammon et al., 2019 with $\tilde{\sigma}_{0}\approx 3.4\times 10^{4}$ , much higher than in samples used in the present study. As illustrated in Fig. 3, showing $\rho_{xx}$ (solid triangles) and $\rho_{yy}$ (open triangles) (not, h) as a function of the filling factor $\nu$ , the QHS anisotropy in this sample collapses at $\nu_{1}\approx 20$ . Using Eq. (1), we can then estimate $\alpha=\sqrt{\tilde{\sigma}_{0}}/\nu_{1}\approx 9$ (not, i). With $\tau_{\rm q}\simeq 0.05$ ns, Eq. (3) gives $\nu_{2}\approx 21$ , which is very close to $\nu_{1}\approx 20$ . Indeed, the data in Fig. 3 show that the QHS phase crosses over directly to the CZS phase, bypassing the intermediate hQHS phase.

In the QHS phase, the easy resistivity is $\nu$ independent and is described by $\rho_{yy}=\rho_{\rm hQHS}$ , Eq. (2), while the hard resisitivty exhibits clear $\nu^{-4}$ dependence and follows (Huang et al., 2020)

\rho_{xx}\simeq\frac{h}{e^{2}}\frac{\tilde{\sigma}_{0}}{2\alpha^{2}\nu^{4}}\,.

(6)

However, the agreement between theory and experiment breaks down at $\nu<\nu_{\rm d}\approx 8$ , where one observes significant deviations leading to the reduction of the anisotropy. While the nature of such reduction is unclear, it becomes more pronounced upon further cooling and might reflect a crossover to another competing ground state (Qian et al., 2017; Fu et al., 2020). We can account for the observed anisotropy reduction at lower filling factors assuming that the QHS phase has a finite concentration of dislocations separated by an average distance $L_{\rm d}=\beta\Lambda/2$ along stripes, where $\beta$ is a numerical factor. Scattering of drifting electrons by these dislocations limits their drift length by $L_{\rm d}\ll L_{y}$ and the resistivities calculated in Refs. Sammon et al., 2019, Huang et al., 2020 need to be modified to (not, j, k)

\rho_{xx}=\frac{h}{e^{2}}\frac{\beta}{2\nu^{2}}\,,

(7)

\rho_{yy}=\frac{h}{e^{2}}\frac{1}{2\beta\nu^{2}}\,.

(8)

Equations (7), (8) are plotted as dashed lines in Fig. 3. Equating Eq. (7) to Eq. (6) [or Eq. (8) to Eq. (2)], we find that the crossover to the dislocation limited transport happens at

\nu_{\rm d}\equiv\frac{\nu_{1}}{\sqrt{\beta}}\,.

(9)

With $\nu_{\rm d}\simeq 8$ and $\nu_{1}\simeq 20$ we find $\beta=(\nu_{1}/\nu_{\rm d})^{2}\simeq 6.3$ . This value does not seem unreasonable and correctly accounts for the saturation of the anisotropy, $\rho_{xx}/\rho_{yy}=\beta^{2}\approx 40$ (Sammon et al., 2019).

Our experimental findings in Al_xGa_1-xAs quantum wells (Fig. 2) and in a clean GaAs quantum well (Fig. 3) can be unified in a phase diagram shown in Fig. 4 which treats $\sigma_{0}/\alpha^{2}$ and $\tau_{\rm q}/\tau$ as independent parameters. Here, the QHS phase is observed above the horizontal line corresponding to $\nu_{1}=9/2$ . To detect the hQHS phase, one should satisfy both $\nu_{2}-\nu_{1}>1$ and $\nu_{2}>11/2$ , since at least two half-integer filling factors are needed to establish the $\nu$ independence of the resistance (not, l). As a result, the most favorable conditions for the hQHS phase are realized at the top-right corner of the diagram. However, as demonstrated by our experiments on Al_xGa_1-xAs quantum wells, the hQHS can be detected at modest mobilities provided that the ratio $\tau_{\rm q}/\tau$ is sufficiently high. On the other hand, this ratio is much smaller in clean GaAs quantum wells, which makes the hQHS detection difficult in such systems despite their high mobility. The phase diagram shown in Fig. 4 provides a a road map for future experiments aiming to detect the hQHS phases.

In summary, we have observed hidden quantum Hall stripe (hQHS) phases (Huang et al., 2020) forming near half-integer filling factors of Al_xGa_1-xAs/Al_0.24Ga_0.76As quantum wells with varying $x$ . These phases reside between the conventional stripe phases and the isotropic liquid phases and are characterized by isotropic resistivity that is not sensitive to the filling factor. Analysis of the experimental phase diagram reveals that the QHS density of states is smaller than predicted by the Hartree-Fock theory (Koulakov et al., 1996; Fogler et al., 1996), calling for improved theory. The unique transport characteristics of the hQHS phases should allow exploration of the stripe physics in 2D systems that, unlike GaAs, lack symmetry-breaking fields. On the other hand, ultrahigh mobility GaAs quantum wells favor conventional QHSs over hQHSs due to a shrinking filling factor range where the hQHS phases can form.

Acknowledgements.

We thank G. Jones, T. Murphy, and A. Bangura for technical support. Calculations by Y.H. were supported primarily by the National Science Foundation through the University of Minnesota MRSEC under Award Number Nos. DMR-1420013 and DMR-2011401. Experiments by X.F., Q.S., and M.Z. were supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award DE-SC0002567. Growth of quantum wells at Purdue University was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award DE-SC0006671. X.F. acknowledges the University of Minnesota Doctoral Dissertation Fellowship. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement Nos. DMR-1157490 and DMR-1644779, and the State of Florida.

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not (i) The analysis of the anisotropy ratio in this sample (Sammon et al., 2019) with theoretical value of $\alpha=18$ leads to $\gamma\approx 0.15$ (see also Ref. not, g) and a ratio of concentrations of Coulomb impurities in the spacer $N_{1}$ to that in the quantum well $N_{2}$ , $N_{1}/N_{2}\simeq 60$ , much larger than $N_{1}/N_{2}\simeq 10$ estimated in Ref. Sammon et al., 2018. Our present experiments, however, suggest lower value of $\alpha$ , leading to larger $\gamma$ and restoring the agreement with Ref. Sammon et al., 2018. Larger $\gamma$ can also result from interface roughness scattering, which was not theoretically considered in Ref. Sammon et al., 2018.
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Hidden Quantum Hall Stripes in AlxGa1-xAs/Al0.24Ga0.76As Quantum Wells

Abstract

Acknowledgements.

References

Hidden Quantum Hall Stripes in Al_xGa_1-xAs/Al_0.24Ga_0.76As Quantum Wells