# Origin of pinning disorder in magnetic-field-induced Wigner solids

Matthew L. Freeman  National High Magnetic Field Laboratory, Florida State
University, Tallahassee, Florida 32310, USA  P. T. Madathil  L. N. Pfeiffer
K. W. Baldwin  Y. J. Chung  Department of Electrical and Computer Engineering,
Princeton University, Princeton, New Jersey 08544, USA  R. Winkler  Northern
Illinois University, DeKalb, Illinois 60115 USA  M. Shayegan  Department of
Electrical and Computer Engineering, Princeton University, Princeton, New
Jersey 08544, USA  L. W. Engel  National High Magnetic Field Laboratory,
Florida State University, Tallahassee, Florida 32310, USA

###### Abstract

At low Landau level filling factors ($\nu$), Wigner solid phases of two-
dimensional electron systems in GaAs are pinned by disorder, and exhibit a
pinning mode, whose frequency is a measure of the disorder that pins the
Wigner solid. Despite numerous studies spanning the last three decades, the
origin of the disorder that causes the pinning and determines the pinning mode
frequency remains unknown. Here we present a study of the pinning mode
resonance in the low-$\nu$ Wigner solid phases of a series of ultralow-
disorder GaAs quantum wells which are similar except for their varying well
widths, $d$. The pinning mode frequencies, $f_{\rm p}$, decrease strongly as
$d$ increases, with the widest well exhibiting $f_{\rm p}$ as low as
$\simeq$35 MHz. The amount of reduction of $f_{\rm p}$ with increasing $d$ can
be explained remarkably well by tails of the wave function impinging into the
alloy-disordered AlxGa1-xAs barriers that contain the electrons. However, it
is imperative that the model for the confinement and wave function includes
the Coulomb repulsion in the growth direction between the electrons as they
occupy the quantum well.

When the Coulomb energy of a system of electrons dominates the kinetic energy,
and disorder is sufficiently small, minimization of the Coulomb energy drives
the formation of a Wigner solid (WS) [1], which is invariably pinned by
interaction with the host system. Because of pinning by disorder, unlike the
essentially uniform pinning of WS in some moiré systems [2, 3], a finite
correlation length of crystalline order is produced. The pinning makes the WS
an insulator, and also causes the spectrum of a WS to exhibit a pinning mode,
a small, collective oscillation of electrons of the solid about their pinned
positions. The frequency of the pinning mode is a measure of the strength of
the pinning disorder. In this Letter we utilize the pinning mode to identify
the disorder pinning a WS in state-of-the-art, ultralow-disorder GaAs quantum
wells [4, 5] in high magnetic fields.

There is a great deal of evidence for the existence of WS in low-disorder,
GaAs-hosted two-dimensional electron systems (2DESs) at high magnetic fields,
which effectively freeze out the kinetic energy [6, 7, 8, 9, 10]. These
include pinning-mode studies [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23], $I$-$V$ curves [24]higher-temperature transport [25] and
photoluminescence [26, 27, 28]. Most recently, evidence for the WS has also
come from studies of the tunneling density of states [29], from the geometric
resonance of nearby mobile carriers in a composite fermion metal by a WS in a
nearby plane [30], and from NMR relaxation [31]. In low-disorder GaAs 2DESs, a
WS is expected in high magnetic fields, starting near the fractional quantum
Hall effect (FQHE) at Landau level filling ($\nu=1/5$). Theoretical work [32,
33] has characterized the high magnetic field WS to be composed of composite
fermions [34], which can be thought of as electrons bound to an even number of
flux quanta. The present study of WS pinning disorder then has relevance to
the disorder that produces vanishing diagonal conductivity in FQHE states, in
which composite fermions are localized, either as single particles or as a
pinned solid [21]. A more detailed understanding of disorder effects on
carriers in high magnetic field may also be of value to the development of
quantum computation schemes utilizing FQHE quasiparticles [35].

Here we present a study of four GaAs quantum well (QW) samples, grown by
molecular beam epitaxy to be similar to each other except for their well
thickness, $d$. The samples are grown according the state of the art described
in Ref. 4. The QWs have $d=30,40,50$ and $70$ nm, and respectively mobilities
10, 17, 16 and 26 $\times 10^{6}$ cm2/Vs and carrier densities of
$n=4.5,4.5,4.2$ and $4.7\times 10^{10}$ cm-2. While mobilities depend on $n$,
the overall trend is of mobility increasing with $d$. According to the
calculations of Ref. 5, with realistic estimates of the interfacial asperity
characteristics, interface roughness disorder reduces the mobility for the
narrower wafers. Flanking the QWs on each side are 280-nm-thick barriers of
Al0.12Ga0.88As. In all four samples only the lowest QW electric subband is
populated. The QW thicknesses and carrier densities of our samples are not
compatible with a low $\nu$ bilayer (two-component) WS [36, 37, 38].

We find a strong decrease in the pinning mode frequency, $f_{\rm p}$, on
increasing $d$. The measured $f_{\rm p}$ decreases by nearly an order of
magnitude as $d$ increases from 30 to 70 nm. The 70-nm-wide QW has an
extremely low $\mbox{$f_{\rm p}$}\simeq 35$ MHz, indicating the extremely low
disorder of the sample. $f_{\rm p}$ vs $d$ can be fit very well using a model
of alloy disorder arising from penetration of the tails of the wave functions
into the AlxGa1-xAs barriers that flank the well. To explain the data, the
model must use a Poisson-Schrödinger approach that self-consistently
incorporates the effect of the populated well on the charge distribution along
the growth direction.

To measure the radio-frequency (rf) conductivity we used coplanar waveguide
(CPW) transmission lines patterned onto the top surfaces of the samples as
shown schematically in the inset of Fig. 1. The CPW couples capacitively to
the 2DES, about 900 nm below. The driven center conductor of the CPW was 75
$\mu$m wide, and was separated from the grounded side planes by
$50$-$\mu$m-wide slots. The transmission coefficients were measured with a
room-temperature network analyzer. We calculated the 2DES conductivity,
$\sigma_{\rm xx}$, from a distributed model of the coupled system of the CPW
and 2DES. The model, described in Ref. 19, was suggested originally in Ref.
39, and takes the 2DES to be in the small-wave-vector limit. The model
corrects for the tendency of the rf field in the 2DES to spread out from the
region under the CPW slots near the resonance condition, particularly at low
frequencies. Pinning-mode frequencies, from which the conclusions of this
study are drawn, are essentially the same whether assessed from the raw
transmission coefficient spectra or from the Re($\sigma_{\rm xx}$) data shown
below. The measurements were carried out in the low-power limit, in which the
measured conductivity is not sensitive to the excitation power, with the 2DES
at $\simeq 35\,$mK, estimated from the dependence of the resonance on cryostat
temperature.

The experimentally deduced 2DES conductivity, Re($\sigma_{\rm xx}$), vs
magnetic field, $B$, is shown in Fig. 1 for the 30-nm QW, at a frequency of
250 MHz. Minima in Re($\sigma_{\rm xx}$) due to FQHEs are clearly visible. The
Re($\sigma_{\rm xx}$) behavior between the 2/9 and 1/5 FQHE minima is an
effect of the varying proximity of the pinning resonance frequency to the
measurement frequency in that $\nu$-range. The apparent increase of
Re($\sigma_{\rm xx}$) as $B$ goes below 4 T comes from rf parallel conduction
in the sample, which freezes out as $B$ increases.

Figure 2 shows color-scale Re($\sigma_{\rm xx}$) spectra taken at many
fillings, $0.14<\nu<0.22$, for the four QWs. For $\nu\lesssim 0.19$, all the
QWs develop a striking, sharp resonance whose frequency, $f_{\rm p}$, depends
strongly on the QW thickness. Low $f_{\rm p}$, observed particularly for the
70-nm QW, signifies a long correlation length of crystalline order. For the
70-nm QW the number of electrons per (Larkin) domain can be estimated [40,
39], as $2\pi\mu_{c}/eB\mbox{$f_{\rm p}$}\simeq 1000$, using the classical WS
shear modulus [41], $\mu_{c}=0.245e^{2}n^{3/2}/4\pi\epsilon_{0}\epsilon$.
Consistent with the interpretation of the resonance as a pinning mode, the
resonance is absent in the range of the 1/5 FQHE, which is clearly visible as
a low-conductivity band that appears as dark blue. As 1/5 is approached from
either the higher or lower $\nu$ side, the resonance frequency, $f_{\rm p}$,
is reduced sharply as the resonance fades away [42]. For $\nu$ above the 1/5
FQHE, in the reentrant range [43] of the WS, up to $\nu=0.22$, a resonance is
also present, although much weaker and broader. The 40-nm QW data show a weak
but sharp secondary resonance at higher frequency than that of the main peak,
although such a resonance is absent in the spectra of the other samples. While
likely related to the main peak, this secondary peak does not appear to be a
harmonic.

Figure 1:  Real diagonal conductivity, Re($\sigma_{\rm xx}$), vs magnetic
field, $B$ for the 30-nm QW. Inset: Schematic of measurement setup, with the
metal film of coplanar waveguide shown as black.  Figure 2: Color-scale plots
of real diagonal conductivity, Re($\sigma_{\rm xx}$), on frequency-filling
factor ($f$-$\nu$) axes, for the four QWs, marked with well widths, $d$. The
graphs are shown with larger $\nu$ at the bottom, so magnetic field, $B$,
increases going upward. The color scale is in percentages of full scales,
which are 50, 210, 120 and 100 $\mu$S respectively for (a) through (d). For
the 70-nm QW of panel (d) only, Re($\sigma_{\rm xx}$) is multiplied by a
factor of five for $\nu>0.20$, to enhance visibility in that range, which has
small Re($\sigma_{\rm xx}$).

The central result of this work is the pronounced decrease of $f_{\rm p}$ with
increasing QW thickness. Though $f_{\rm p}$ is determined by the shear modulus
($\mu$) of the WS as well as by the pinning disorder [40, 44, 39], we rule out
the shear modulus effect as an explanation for the observed tenfold decrease
of $f_{\rm p}$ as $d$ is increased. $\mu$ is expected to decrease as $d$
increases, because in a thicker well the electron-electron interaction is
softer at short range. However, in the context of weak pinning [40, 44, 39] a
softer WS results in a larger $f_{\rm p}$. The inverse relationship of $f_{\rm
p}$ and $\mu$ follows from the tendency of carriers, when the WS is softer, to
be more closely associated with disorder potential minima and hence to
experience larger restoring force upon displacement.

The densities, $n$, of our QWs are not identical, and $n$ can affect $f_{\rm
p}$, because $\mu$ increases with $n$. For well-developed resonances in the
$\nu$ ranges shown here, the expected relation is $\mbox{$f_{\rm p}$}\propto
n^{-3/2}$. This relationship is predicted in theory [44, 40, 39] and is also
borne out by studies of $f_{\rm p}$ vs $n$ [15]. To capture the trends solely
due to changing $d$ in our $f_{\rm p}$ vs $d$ data we multiply the measured
$f_{\rm p}$ by a near-unity factor $\alpha=(n/4.5\times 10^{10}{\rm\
cm}^{-2})^{3/2}$ to correct all the frequencies to their equivalents for
$n=4.5\times 10^{10}{\rm\ cm}^{-2}$. We emphasize that the corrections are
small: for $d=30,40,50$ and $70$ nm QWs respectively, $\alpha=1.0,1.0,0.90$
and $1.07$. It is clear that the variation of $n$ from sample to sample cannot
explain the large variation of $f_{\rm p}$ with $d$.

Figure 3 shows $\alpha\mbox{$f_{\rm p}$}$ at $\nu=0.15$ vs $d$. The graph is
plotted in log-log format to account for the wide range of $f_{\rm p}$ as $d$
is varied. The change of $f_{\rm p}$ with $\nu$ where the resonance is well-
developed ($0.14\leq\nu\leq 0.19$) is shown as the error bars on the figure.
The other curves in Fig. 3 show different model calculations for $f_{\rm p}$
vs $d$, and are discussed below.

In the remainder of this Letter we consider different sources of disorder in
our samples and show how the variation of $\alpha\mbox{$f_{\rm p}$}$ with $d$
points to the origin of the disorder that pins the WS. The types of disorder
affecting the 2DES in QWs are well-known: (1) Charged impurities (usually C)
which are dispersed through the growth, and which limit [5] the possible
mobilities at zero magnetic field for dilute samples with very thick spacer
layers; (2) Smooth disorder, stemming from the granularity of the remote
donors grown into the sample, on a long length scale given by the spacing of
the donors; (3) Interface roughness at the edges of the QW; and finally (4)
Alloy disorder [20], from penetration of the tails of the wave functions into
the AlxGa1-xAs barriers, whose Al is randomly distributed. The dispersed
impurities (1) would not be expected to produce the observed decreasing
$f_{\rm p}$ with $d$. Reference 5 shows that the effect of this disorder on
mobility is independent of $d$.

The remote ionized donors (2) are about the same distance away from the WS for
all the QWs, and also would not be expected to produce the strong thickness
dependence. This leaves interface roughness or alloy scattering in the
barriers as candidates to explain the data.

Figure 3:  Density-corrected resonance frequency, $\alpha\mbox{$f_{\rm p}$}$
(see text), vs QW width, $d$. Measured $\alpha\mbox{$f_{\rm p}$}$ for the four
QWs at $\nu=0.15$ are shown as black dots. For $d=30,40,50$ and $70$ nm
respectively, $\alpha=1.0,1.0,0.90$ and $1.07$. Curves for the models
discussed in the text are as shown in the legend. The one-particle infinite QW
(interface roughness) and finite QW (roughness and alloy) curves are
multiplied by scale factors to intercept $(d,\alpha\mbox{$f_{\rm p}$})=(45.3$
nm$,0.125$ GHz$)$, the centroid of the measured $\alpha\mbox{$f_{\rm p}$}$
data points on the log-log plot. The curves marked “alloy” are the calculated
$P_{B}^{2}$, and all “roughness” curves are the calculated $(\partial
E_{s}/\partial d)^{2}$; see text for details. The $P_{B}^{2}$ curves generated
for the Hartree and LDA self-consistent calculations are scaled to least
squares fit the $\alpha\mbox{$f_{\rm p}$}$ data. The fits are done on the
logged $\alpha\mbox{$f_{\rm p}$}$ data.  Figure 4:  (a) Normalized charge
density, $\rho_{c}$, (red), and self-consistent potential, $V$ (black) for QWs
of widths, $d=30,50$ and $70$ nm. These are from Poisson-Schrödinger, self-
consistent (Hartree) calculations for B=12.4 T. (b) Hartree (solid) and LDA
(dashed) calculated charge distribution tails within the barriers, for the
four QWs. The edge of the barrier is the origin of the horizontal axis.

To identify the pinning disorder, we make use of an effective disorder
potential, $V({\bf r})$, to estimate the pinning energy. The effective
potential takes into account the probability that the electron wave function
in the WS will couple to a source of pinning disorder. In weak-pinning theory
[40, 39], $f_{\rm p}$ goes as a correlation function of this effective
potential, $\langle V(0)V({\bf r})\rangle=V_{0}^{2}f(r)$, where $f(r)$ is
normalized to unity and decays with length much shorter than the WS lattice
constant, so $\mbox{$f_{\rm p}$}\propto V_{0}^{2}$ for a given type of
disorder. We model $\mbox{$f_{\rm p}$}(d)$ by considering the dependence of
$V_{0}^{2}$ on $d$ to within a constant of proportionality, which we treat as
an adjustable parameter.

Interface roughness disorder has been modeled based on the spatial
fluctuations of the QW subband energy [45, 46, 47]. In this picture,
asperities in the interface locally alter the well thickness and hence the
subband energy. For an infinitely-deep QW, the subband energy, $E_{s}$, goes
as $d^{-2}$, and a fluctuation in $d$ of $\xi\ll d$ produces a change in
$E_{s}$ of $\sim-\xi d^{-3}$. The interface roughness fluctuation size, $\xi$,
should be the same for all the samples, because they were grown in the same
system and under similar conditions. So in considering the sample series, and
taking the fluctuation in subband energy to be $V_{0}$, we find
$V_{0}^{2}\propto(\partial E_{s}/\partial d)^{2}\propto d^{-6}$. We also
obtained $E_{s}$ and $(\partial E_{s}/\partial d)^{2}$ from a simple model of
a finite square well, whose depth is the conduction band discontinuity, 0.095
eV [48], for the Al0.12Ga0.88As barriers closest to the QW, and effective mass
of $0.067m_{0}$, both inside and outside the QW, where $m_{0}$ is the free-
electron mass. The $(\partial E_{s}/\partial d)^{2}$ curves for the infinite
and finite square wells are shown in Fig. 3 as a dotted gray line and a
dashed-dotted blue line. For comparison with the data the curves are
multiplied by prefactors to intersect the centroid of the data on the log-log
plot. The slopes of the two curves are similar, and are both clearly too large
compared to the data.

For effective alloy disorder from the evanescent penetration of the WS charge
density into the barriers, $V_{0}$ is proportional to the probability $P_{B}$,
that a carrier is within the barriers, so $f_{\rm p}$ is modeled to be
proportional to $P_{B}^{2}$. In the simple, finite square QW model of the
previous paragraph, $P_{B}$ is obtained by integrating the tails of the charge
distribution in the barriers. The resulting $P_{B}^{2}$, again multiplied by a
factor to intersect the centroid of the data, is shown in Fig. 3 as a solid
blue curve. Clearly, this line also decreases with $d$ more strongly than the
experimental $f_{\rm p}$ data.

The simple “textbook”, one-particle, square well models neglect the effect of
Coulomb repulsion between the electrons along the growth direction. This
repulsion tends to push charge to the edges of a QW as it is widened [49, 50,
51]. We performed self-consistent calculations of the charge distribution and
potential, using the Poisson and Schrödinger equations, at magnetic field
$B=12.4$ T, and $n=4.5\times 10^{10}$ cm-2 to match $\nu=0.15$ as in Fig. 3.
Figure 4(a) shows potential and charge distributions calculated for the 30, 50
and 70 nm QWs in the self-consistent Hartree approximation [52]. As $d$
increases, the charge distributions flatten and eventually become bimodal.

We carried out the self-consistent calculations for high magnetic field, with
the charge density coming from the occupied Landau level only. Because of the
Landau level degeneracy, the second electric subband, occupied for $d>70$ nm
at $B=0$, is not occupied in high field. We evaluated the wave functions of
the Landau levels via an $8\times 8$ multiband Hamiltonian [52]. Besides the
self-consistent Hartree approximation, we also used spin density-functional
theory in the local-density approximation (LDA) based on the spin-dependent
exchange-correlation potential parameterized in Ref. [53].

Estimates of the effect of interface roughness in the context of the self-
consistent calculations come from taking $\mbox{$f_{\rm p}$}\propto(\partial
E_{s}/\partial d)^{2}$, with $E_{s}(d)$ (measured from the well center) coming
from the self-consistent calculations. The Hartree and LDA calculations give
the red and green dotted curves in Fig. 3, respectively. The curves are again
multiplied by scale factors to intercept the centroid of the measured data.
The curves are nearly identical and decrease much more strongly than the
experimental data.

In Fig. 3, the red and green solid curves show $P_{B}^{2}$ vs $d$ obtained
from the Hartree and LDA calculations respectively, multiplied by scale factor
parameters to least-squares fit the density-adjusted experimental data,
$\alpha\mbox{$f_{\rm p}$}$. The fits are remarkably good. $P_{B}^{2}$ is
proportional to the contribution of alloy disorder to $f_{\rm p}$, so the good
fits suggest that the charge density in the barriers is indeed the origin of
the pinning disorder, and that the self-consistent modeling of the populated
well is required to reasonably estimate the $P_{B}$ associated with the charge
density in the barriers. In comparison to the one-particle square-well models,
the self-consistent models lessen the effect of the widening well, because the
pushing outward of the charge distributions from Coulomb repulsion preserves
more charge density in the barriers as the well gets wider. In the theoretical
curves, this effect continues at larger $d$, so that the curves increase for
$d>70$ nm.

The self-consistent models are only approximations because they take the
charge as uniform sheets in the 2D plane, which is not the case in a WS. For
our samples with $n\simeq 4.5\times 10^{10}$ cm-2, the effective in-plane size
of an electron is the magnetic length $l_{B}\simeq 7$ nm at $\nu=0.15$, while
the WS lattice constant $a=2^{1/2}\cdot 3^{-1/4}n^{-1/2}\simeq 50$ nm. In
between WS lattice sites, because of the smaller charge density, the
penetration of charge into the barriers will be somewhat less than the model
estimates. Likewise more penetration than predicted will occur at lattice
sites. These effects would not vary much with $d$, explaining the strikingly
good fits we see.

In summary, we have studied the pinning mode of the magnetic-field-induced WS
in a series of ultralow-disorder GaAs QW samples, of varying thickness. An
extremely low $f_{\rm p}$ down to 35 MHz is found in the 70 nm QW. We identify
the disorder responsible for the WS pinning as Al alloy disorder in the QW
barriers. For this disorder, the decreasing curve of $f_{\rm p}$ vs well
thickness is quantitatively explained by the reduction of charge density in
the barriers for thicker wells, if this reduction is self-consistently
evaluated to account for the QW population.

###### Acknowledgements.

A portion of this work was performed at the National High Magnetic Field
Laboratory (NHMFL), which is supported by National Science Foundation (NSF)
Cooperative Agreement No. DMR-1644779 and the state of Florida. The rf
measurements at NHMFL were supported by Department of Energy (Grant No. DE-
FG02-05-ER46212). Work at Princeton University was supported by the NSF Grant
No. DMR 2104771 for measurements, by the U.S. Department of Energy Basic
Energy Office of Science, Basic Energy Sciences (Grant No. DEFG02-00-ER45841)
for sample characterization, and the NSF Grant No. ECCS 1906253 and the Gordon
and Betty Moore Foundation’s EPiQS Initiative (Grant No. GBMF9615 to L.N.P.)
for sample synthesis. We also thank David Huse, J. K. Jain and Adbhut Gupta
for illuminating discussions.

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