# Probing quantum phases in ultra-high-mobility two-dimensional electron
systems using surface acoustic waves

Mengmeng Wu  International Center for Quantum Materials, Peking University,
Haidian, Beijing, 100871, China  Xiao Liu  International Center for Quantum
Materials, Peking University, Haidian, Beijing, 100871, China  Renfei Wang
International Center for Quantum Materials, Peking University, Haidian,
Beijing, 100871, China  Yoon Jang Chung  Department of Electrical Engineering,
Princeton University, Princeton, New Jersey, 08544, USA  Adbhut Gupta
Department of Electrical Engineering, Princeton University, Princeton, New
Jersey, 08544, USA  Kirk W. Baldwin  Department of Electrical Engineering,
Princeton University, Princeton, New Jersey, 08544, USA  Loren Pfeiffer
Department of Electrical Engineering, Princeton University, Princeton, New
Jersey, 08544, USA  Xi Lin<EMAIL_ADDRESS>International Center for Quantum
Materials, Peking University, Haidian, Beijing, 100871, China
Interdisciplinary Institute of Light-Element Quantum Materials and Research
Center for Light-Element Advanced Materials, Peking University, Haidian,
Beijing, 100871, China  Yang Liu<EMAIL_ADDRESS>International Center
for Quantum Materials, Peking University, Haidian, Beijing, 100871, China

###### Abstract

Transport measurement, which applies an electric field and studies the
migration of charged particles, i.e. the current, is the most widely used
technique in condensed matter studies. It is generally assumed that the
quantum phase remains unchanged when it hosts a sufficiently small probing
current, which is, surprisingly, rarely examined experimentally. In this work,
we study the ultra-high mobility two-dimensional electron system using a
propagating surface acoustic wave, whose traveling speed is affected by the
electrons’ compressibility. The acoustic power used in our study is several
orders of magnitude lower than previous reports, and its induced perturbation
to the system is smaller than the transport current. Therefore we are able to
observe the quantum phases become more incompressible when hosting a
perturbative current.

Two-dimensional electron systems (2DES) with extremely low disorder host a
plethora of exotic quantum many-body states Tsui  _et al._ (1982); Prange and
Girvin (1987); Jain (2007). The quantum Hall state (QHS) is incompressible
signaled by vanishing longitudinal resistance and quantized Hall resistance
Jain (2007). At high Landau level fillings factors $\nu>4$, various charge
density waves are stabilized by the large extent of the electron wavefunction
Lilly  _et al._ (1999); Du  _et al._ (1999); Koulakov  _et al._ (1996). The
enigmatic 5/2 fractional quantum Hall state (FQHS) attracts tremendous
interest Willett  _et al._ (1987); Pan  _et al._ (1999); Dean  _et al._
(2008); Kumar  _et al._ (2010); Rezayi and Haldane (2000); Peterson  _et al._
(2008); Wójs  _et al._ (2010); Willett (2013); Lin  _et al._ (2014); Wan  _et
al._ (2008) because its quasi-particles might obey non-Abelian statistics
Moore and Read (1991); Kitaev (2003); Stern (2010); Nayak  _et al._ (2008).
Varies of experimental studies are employed to study its topological
properties and quasi-particle statistics, such as weak tunnelingRoddaro  _et
al._ (2003); Miller  _et al._ (2007); Radu  _et al._ (2008); Fu  _et al._
(2016), interferometryWillett  _et al._ (2009); Zhang  _et al._ (2009);
Willett  _et al._ (2010); Nakamura  _et al._ (2020), shot noiseSaminadayar
_et al._ (1997); de Picciotto  _et al._ (1997); Bid  _et al._ (2010); Dolev
_et al._ (2010) and thermal transportChickering  _et al._ (2013); Banerjee
_et al._ (2017, 2018). Most of these studies rely upon the hypothesis that a
quantum state is unperturbed by the tiny probing current passing through the
$\mu$m size device.

Surface acoustic wave (SAW) is a useful current-free technique to investigate
the property of 2DES Wixforth  _et al._ (1986); Paalanen  _et al._ (1992);
Willett  _et al._ (1993, 2002); Friess  _et al._ (2018, 2020, 2017); Drichko
_et al._ (2011, 2016). The propagating piezo-electric field accompanying the
SAW interacts with the charge carriers, which in turn affects its velocity
($v$) and attenuation. Qualitatively, this interaction is related to the
compressibility of 2DES: $v$ increases when the 2DES becomes incompressible
and thus unable to respond to the SAW 111Previous work in Ref. Wixforth  _et
al._ (1986) explains the velocity shift with the conductivity. Such an
analysis may not be suitable here because our ultra-high mobility 2DES has a
very long transport scattering time $\tau_{tr}\simeq 0.7$ ns comparable to the
SAW frequency.. In this work, we probe the 2DES using a pW-level, continuous-
wave SAW and discover that the $\sim$ 100 nA current flowing through the
$\sim$ 1 mm size sample causes a $\sim$ 0.1 ppm (parts per million, $10^{-6}$)
increase of the SAW velocity at very low $T\lesssim 250$ mK. Such a current-
induced SAW velocity shift illustrates that a close and careful examination on
the charge transport mechanism is essential and imperative.

Figure 1: (a) A photo of our device. The zoom-in plot shows the structure of
the aluminum IDT. Two pairs of orthometric IDTs are evaporated on the sample
labeled by the alphabets A to D, and four contacts are made at the four
corners labeled by numbers 1 to 4. IDT C is used to excite the SAW and IDT A
is used for receiving SAW. Unless otherwise specified, the current is injected
into contact 1 and flows out from contact 2 while the contact 3 and 4 are
float. (b) The measured amplitude ($|S_{\text{21}}|$) and phase delay
($\mit\Phi$) of the transmission coefficient as a function of frequency at
base temperature $T$ and magnetic field $B\sim 50$ mT.  Figure 2:  (a) The
longitudinal ($R_{\text{xx}}$) and Hall ($R_{\text{xy}}$) resistance vs. $B$.
(b) The measured SAW velocity increase $\eta(B)=\Delta v(B)/v_{0}$. A
$f_{0}=0.125$ Hz, $I_{\text{p}}$ = 707 nA AC current passes through the sample
(contact 1 $\rightarrow$ 2) during the measurement, imposing a 4-s-period
oscillation to $\eta$, see the enlarged plot in the red dashed box. Red solid
box shows $\eta$ near $\nu=3/2$. (c) The extracted oscillation using a digital
bandpass filter centered at 0.25 Hz (pink curve). Its amplitude can be
measured using a lock-in amplifier (black curve). Inset: power spectrum
density of the oscillation. The Y coordinate $\kappa$ is defined as
$\eta_{\text{m}}^{-1}\cdot(\partial\eta/\partial|I|)$. The meaning of $\kappa$
is explained in Fig3.

Our sample is made from a GaAs/AlGaAs wafer grown by molecular beam epitaxy.
The 2DES is confined in a 30-nm-wide quantum well, whose electron density is
$2.91\times 10^{11}$ cm-2 and low-temperature mobility is about $2\times
10^{7}$cm2/(V$\cdot$s). We make a Van der Pauw mesa (length of side
$d_{\text{m}}$ = $1.2$ mm) by wet etching, and then evaporate
5-$\mu\text{m}$-period interdigital transducers (IDTs) on each side of the
mesa. 50 $\Omega$ resistance is connected in parallel to each IDT for
broadband impedance matching. When applied with an AC voltage whose frequency
matches the resonance condition, the IDT generates a propagating SAW. The SAW
will be captured by the IDT on the opposite side of the sample as a voltage
output through the piezoelectric effect White and Voltmer (1965), see Fig.
1(a).

We use a custom-built superheterodyne demodulation system Wu  _et al._ (2023)
to analyze the attenuation $|S_{21}|$ and phase delay $\mit\Phi$ of the output
signal 222See supplementary material for a detailed description of our setup,
which includes Refs. Sullivan  _et al._ (1990); Durdaut  _et al._ (2019)..
From the measured $|S_{21}|$ and $\mit\Phi$ vs. frequency $f$ shown in Fig.
1(b), we can calculate the reference SAW velocity $v_{\text{0}}$ at low field
($\simeq 2950$ m/s) from the IDT period (5 $\mu$m) and the resonant frequency
$f_{\text{c}}$ (589.5 MHz). We can also derive the delay time
$\partial\mit\Phi/\partial(2\pi f)$ = 1.1 $\mu s$ and 54 ns near and away from
the SAW resonance peak, consistent with the $\sim 3$ mm SAW travel distance
and $\sim 11$-meter-long coaxial cable (5.5 m each way). The experiment is
carried out in a dilution refrigerator whose base temperature is $\lesssim$ 10
mK.

Figure. 2(a) and (b) show the magneto-resistance ($R_{\text{xx}}$,
$R_{\text{xy}}$) and the measured relative SAW velocity increase
$\eta(B)=\Delta v(B)/v_{0}$, where $\Delta v(B)=v(B)-v_{0}$. The positive
(negative) velocity shift results in the decrease (increase) in the delay
time. We can directly deduce $\eta$ from the measured SAW phase shift
$\mit\Phi$ through $\eta\simeq-\Delta\tau/\tau=-\mit\Phi/(2\pi
f_{\text{c}}\tau)$ , where $\tau=d_{m}/v_{0}=407$ ns is the SAW’s traveling
time through the 2DES. At high $B$ fields, $\eta$ exhibits maxima
(corresponding to enhanced SAW velocity) when the 2DES forms an incompressible
quantum Hall state and its screening capability vanishes Drichko  _et al._
(2011), see Fig. 2(b). The SAW interacts with 2DES by inducing a screening
charge distribution. Therefore $\eta$ is related to the 2DES compressibility
$\frac{dn}{d\mu}$ 333See supplementary material for more information.. $\eta$
at integer fillings increases monochromatically with decreasing $\nu$. We
linearly fit the data at $\nu$ $\lesssim$ 10 and define the intercept at
$\nu=0$ as $\eta_{\text{m}}=124$ ppm Note (3).

Figure 3: (a) $\delta\eta$ vs. DC Current $I$ at $B$ = 4.62 T. (b)
$\delta\eta$ vs. time measured with different ranges of sweeping current
(black and red). (c-d) $\delta\eta_{\text{p}}$ vs Current peak $I_{\text{p}}$
at transition states and QHSs. Inset: Black curve is the sinusoidal input
current with peak amplitude $I_{\text{p}}$ and frequency $f_{0}$. Red curve is
the current induced velocity shift ($\delta\eta$) with amplitude
$\delta\eta_{\text{p}}$. Blue dashed curve is the second harmonic component of
$\delta\eta$. (e) $\kappa$ vs. $B$ when current ($I_{\text{p}}$ = 707 nA)
flows perpendicular (between contacts 1 $\&$ 2) and parallel (4 $\&$ 1) to the
SAW propagation direction. Figure 4: (a) $\kappa$ vs. $B$ at different $T$.
(b) $\kappa$ vs. $T$ at different $B$. (c-d) Schematic explanation of the CIVS
at $\nu=11/8$. Solid and open dots represent negative-charged quasiparticles
and positive-charged quasiholes respectively. (c) and (d) depict situations
when currents flow on different directions. The dashed box represents the
phase boundary. The grey (white) domain is the region with $\nu_{CF}>1.5$
($\nu_{CF}<1.5$ ).

Unlike the vanishing plateau seen in $R_{xx}$, we observe “V”-shape maxima in
$\eta$. At the vicinity of integer filling factors $\nu=N+\nu^{*}$, $N$ is an
integer, the 2DES consists of an incompressible quantum Hall liquid and
additional quasiparticles/quasiholes whose filling factor $|\nu^{*}|<1$. The
fact that $\eta$ has a linear dependence on the quasiparticles/quasiholes
density $n^{*}=n|\nu^{*}|/\nu$ suggests that the quantum phase formed by these
dilute quasiparticles/quasiholes is compressible Note (3); Chen  _et al._
(2003). The SAW velocity enhancement is also seen as clear “V”-shape maximum
at $\nu=4/3$, 5/3, 6/5, etc., as well as developing maxima at $\nu=5/2$, 7/3,
and 11/5 where FQHS develop. The SAW velocity enhancement is seen when the SAW
propagates along the hard axis of the stripe phase formed at $\nu=9/2$, 11/2,
etc., consistent with previous reports Friess  _et al._ (2017). Interestingly,
$\eta$ is large near $\nu=3/2$ where the 2DES forms compressible composite
Fermion Fermi sea, possibly because the composite Fermions with extremely
large effective mass are inert to the SAW-induced field 444Such behavior is
surprising but not inconsistent with the previous report, since our frequency
is much lower than the geometric resonance condition Willett  _et al._ (1993,
2002)..

We are able to reach $\sim$ 0.1 ppm resolution in $\eta$ while using
excitation that is orders of magnitude smaller than previous reports Friess
_et al._ (2017). The input RF power in Fig. 2(b) is 1 nW (-61 dBmW) and only a
tenth of it turns into SAW considering the attenuation of cables and the
efficiency of the IDT. The SAW induced potential on the 2DES is only $\sim$ 10
$\mu$eV, leading to $\lesssim 10^{4}$ cm-2 electron density fluctuation Note
(3). Therefore, we can resolve very delicate response of 2DES while preserving
the fragile many-body states. One of the most surprising discovery is a
velocity shift $\delta\eta=\eta(B,I)-\eta(B,I=0)$ induced by a current passing
through the 2DES, see Fig. 3(a). $\delta\eta$ increases nearly linearly by 8
ppm when $I$ increases from 0 to 1 $\mu$A. $\delta\eta$ is an even function of
$I$, so that if we sweep the current from -0.5 to 0.5 $\mu$A, $\eta$ displays
a triangle waveform, see Fig. 3(b). We define a parameter
$\kappa=\eta_{\text{m}}^{-1}\cdot(\partial\eta/\partial|I|)$ to describe this
current induced velocity shift (CIVS) effect.

We note if the input current is sinusoidal at frequency $f_{0}$, the leading
component of $\delta\eta$ would be the second harmonic at frequency 2$f_{0}$,
see the Fig. 3(c) inset. Therefore, we can use lock-in technique to measure
amplitude ($\delta\eta_{\text{p}}$) of the $\delta\eta$ oscillation from its
second harmonic Note (3) and deduce
$\kappa\simeq\eta_{\text{m}}^{-1}\cdot(\delta\eta_{\text{p}}/\delta
I_{\text{p}})$ even when the CIVS effect is small. We measure
$\delta\eta_{\text{p}}$ at different filling factors as a function of the AC
current amplitude $I_{\text{p}}$ in Fig. 3(c) and (d). In the left panel where
the 2DES is compressible, $\delta\eta_{\text{p}}$ increases linearly and then
saturates at large current amplitudes. In the right panel, the tiny
$\delta\eta_{\text{p}}$ value does not rise obviously with the increase of
current at integer fillings. At fractional fillings, we discover a clear
threshold behavior where $\delta\eta_{\text{p}}$ remains almost zero until
$I_{\text{p}}$ reaches about 600 nA.

The Fig 2(b) data is taken when a $f_{0}=$ 0.125 Hz, $I_{\text{p}}$ = 707 nA
current passes through the 2DES. In the expanded plot of $\eta$ in Fig. 2(b)
and the power spectrum of the $\eta$ in the Fig. 2(c), we can clearly observe
a 4-s (2$f_{0}$) period oscillation in $\eta$. We apply a digital band-pass
filter to the Fig. 2(b) data to extract this oscillation (pink shade) and
deduce $\kappa$ from its amplitude (red trace) in Fig. 2(c). Alternatively, we
can use a lock-in amplifier to measure this oscillation amplitude (black
trace). The Fig. 2(c) data clearly evidences the dependance of CIVS effect on
the quantum phases of 2DES. At strong quantum Hall effects, unlike the
“V”-shape maxima in the $\eta$ trace and the plateau in the $R_{\text{xx}}$
trace, $\kappa$ presents a “W”-shape minimum – it has a positive peak at exact
integer $\nu$ = 1, 2, 3, etc. and reduces to zero on both sides before
increasing. Between $\nu=1$ and 2, $\kappa$ exhibits clear minima at
$\nu=4/3$, 5/3, 7/5, 8/5 and 6/5 when FQHS form, similar to the $\eta$ and
$R_{\text{xx}}$ traces. Surprisingly, clear minimum can be seen in the
$\kappa$ trace corresponding to the fragile FQHS at $\nu=5/2$, 7/3, 8/3, 11/5
and 14/5 while the $\eta$ trace only shows a glimmer of maxima.

We can rule out the possibility that finite $\kappa$ is caused by the heating
effect. As discussed in the supplementary, $\eta$ has little temperature
dependence when the sample temperature is below 100 mK. Although there’s no
reliable approach to detect the electron temperature of the 2DES, we are quite
confident that it is well below 100 mK when taking Fig. 2 data. We can see
clear features in $\eta$ and $\kappa$ in Fig. 2(b) and (c) for several fragile
quantum phases such as the FQHS at $\nu=5/2$, 7/3 and 8/3 as well as the
unidirectinal charge density waves at $\nu=9/2$ and 11/2, which are only
stable at temperatures well below 100 mK. The transport measurements taken
using 500 nArms current also shows clear difference when temperature raises to
100 mK Note (3). Lastly, $\kappa$ dip at a fragile QHS such as $\nu=5/2$ is
much more obvious than the composite Fermion Fermi sea at $\nu=3/2$, although
the former is more sensitive to the temperature.

Figure. 4(a) shows that at all fields $\kappa$ decreases as $T$ increases, and
eventually vanishes when $T\simeq$ 250 mK. The summarized $\kappa$ vs. $T$
data at different fields in Fig. 4(b) suggests an exponential dependence
$\kappa\propto\exp(-T/T_{\text{C}})$ where the characteristic temperature
$T_{\text{C}}$ is about 50 mK at $2<\nu<3$ and 70 mK at $1<\nu<2$. More data
show that the $T_{\text{C}}$ is insensitive to the probing SAW
frequencies/wavelengths Note (3). It is important to mention that the
vanishing of $\kappa$ is unlikely a direct result of reduced quantum Hall
stability, since the QHS around $3/2$ remains quite strong at $T\simeq 250$ mK
when $\kappa$ vanishes.

The measured $\kappa$ is almost always positive. The increased SAW velocity
suggests that the 2DES becomes more incompressible when carrying current.
Intuitively, the current cripples the incompressible phases by introducing
more defects/inhomogeneities and broadening the domain walls, so that the 2DES
are expected to be more compressible. Also, we observe no changed in $\kappa$
when we rotate the current direction to be parallel to SAW, see Fig. 3(e),
indicating that the CIVS has no dependence on which direction the current
flows. Unfortunately, there’s very little investigation on the morphing of the
quantum phase when carrying a non-destructive current. Meanwhile, the large
$\kappa$ is seen at the transition between two neighboring QHS, where a
rigorous description of charge transport mechanism must involve quasiparticle
localization and percolation and is particularly hard.

We propose a simple hand-waving mechanism to understand the positive $\kappa$
in Fig. 4(c). At $\nu=4/3$ and 7/5, the electrons in the partially filled
Landau level form $\nu=1/3$ and 2/5 fractional qauntum Hall states,
respectively, if the 2DES is fully-spin-polarized. These two states can be
explained as the $\nu_{\text{CF}}=1$ and 2 integer quantum Hall states of
composite Fermions, and the phase transition happens at $\nu=11/8$ when the
average composite Fermion filling factor
$\textless\nu_{\text{CF}}\textgreater$ = 1.5. Because of the density
fluctuation Chung  _et al._ (2019), the regions with $\nu_{\text{CF}}<1.5$
($\nu_{\text{CF}}>1.5$) consist of an incompressible $\nu=4/3$ ($\nu=7/5$) QHS
and additional movable negative-charged quasiparticles (positive-charged
quasiholes), see Fig. 4(c). When a current passes through the sample, e.g.
from left to right (red arrow), quasiparticles move leftward and quasiholes
move rightward. The effective magnetic field poses a Lorentz force, leading to
the accumulation and depletion of quasiparticles/quasiholes at the phase
boundary. The depletion (accumulation) of quasiholes and accumulation
(depletion) of quasiparticles occur at the same boundary, leading to an
increase (decrease) in the local density and the formation of incompressible
QHS with $\nu_{\text{CF}}=2$ ($\nu_{\text{CF}}=1$). If we rotate the current
to the vertical direction (blue arrow), the incompressible regions rotate as
well. In short, the current passing through the disordered 2DES induces
incompressible phases at domain boundaries which are always parallel to the
current direction. This explains why we measure the same $\kappa$ in different
current directions as shown in Fig. 3(e). Similar discussion can be easily
extended to QHS, where the flowing current drives the sparsely distributed,
disorder-pinned quasiparticles/quasiholes out of their equilibrium positions
and piles them at boundaries of the incompressible liquid phase.

In conclusion, we use the interaction between SAW and electrons to study the
morphing of quantum phases in ultra-high mobility 2DES. We discover that the
SAW velocity increases, suggesting that the 2DES becomes more incompressible
when a non-destructive current flows through the 2DES. This effect is only
seen with a revolutionarily enhanced sound velocity resolution at very low
temperatures and disappears at $T\gtrsim$ 250 mK.

###### Acknowledgements.

We acknowledge support by the National Key Research and Development Program of
China (Grant No. 2021YFA1401900 and 2019YFA0308403) and the National Natural
Science Foundation of China (Grant No. 92065104 and 12074010) for sample
fabrication and measurement. The Princeton University portion of this research
is funded in part by the Gordon and Betty Moore Foundation’s EPiQS Initiative,
Grant GBMF9615.01 to Loren Pfeiffer. We thank Xin Wan, Zhao Liu and Bo Yang
for valuable discussions.

## References

  * Tsui  _et al._ (1982) D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982).
  * Prange and Girvin (1987) R. E. Prange and S. M. Girvin, eds.,  _The Quantum Hall Effect_ (Springer, New York, 1987).
  * Jain (2007) J. K. Jain,  _Composite Fermions_ (Cambridge University Press, Cambridge, UK, 2007).
  * Lilly  _et al._ (1999) M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 82, 394 (1999).
  * Du  _et al._ (1999) R. R. Du, D. C. Tsui, H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, Solid State Communications 109, 389 (1999).
  * Koulakov  _et al._ (1996) A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, Phys. Rev. Lett. 76, 499 (1996).
  * Willett  _et al._ (1987) R. L. Willett, J. P. Eisenstein, H. L. Störmer, D. C. Tsui, A. C. Gossard, and J. H. English, Phys. Rev. Lett. 59, 1776 (1987).
  * Pan  _et al._ (1999) W. Pan, J.-S. Xia, V. Shvarts, D. E. Adams, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, Phys. Rev. Lett. 83, 3530 (1999).
  * Dean  _et al._ (2008) C. R. Dean, B. A. Piot, P. Hayden, S. Das Sarma, G. Gervais, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 101, 186806 (2008).
  * Kumar  _et al._ (2010) A. Kumar, G. A. Csáthy, M. J. Manfra, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 105, 246808 (2010).
  * Rezayi and Haldane (2000) E. H. Rezayi and F. D. M. Haldane, Phys. Rev. Lett. 84, 4685 (2000).
  * Peterson  _et al._ (2008) M. R. Peterson, T. Jolicoeur, and S. Das Sarma, Phys. Rev. Lett. 101, 016807 (2008).
  * Wójs  _et al._ (2010) A. Wójs, C. Tőke, and J. K. Jain, Phys. Rev. Lett. 105, 096802 (2010).
  * Willett (2013) R. L. Willett, Reports on Progress in Physics 76, 076501 (2013).
  * Lin  _et al._ (2014) X. Lin, R. Du, and X. Xie, National Science Review 1, 564 (2014).
  * Wan  _et al._ (2008) X. Wan, Z.-X. Hu, E. H. Rezayi, and K. Yang, Phys. Rev. B 77, 165316 (2008).
  * Moore and Read (1991) G. Moore and N. Read, Nuclear Physics B 360, 362 (1991).
  * Kitaev (2003) A. Kitaev, Annals of Physics 303, 2 (2003).
  * Stern (2010) A. Stern, Nature 464, 187 (2010).
  * Nayak  _et al._ (2008) C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).
  * Roddaro  _et al._ (2003) S. Roddaro, V. Pellegrini, F. Beltram, G. Biasiol, L. Sorba, R. Raimondi, and G. Vignale, Phys. Rev. Lett. 90, 046805 (2003).
  * Miller  _et al._ (2007) J. B. Miller, I. P. Radu, D. M. Zumbühl, E. M. Levenson-Falk, M. A. Kastner, C. M. Marcus, L. N. Pfeiffer, and K. W. West, Nature Physics 3, 561 (2007).
  * Radu  _et al._ (2008) I. P. Radu, J. B. Miller, C. M. Marcus, M. A. Kastner, L. N. Pfeiffer, and K. W. West, Science 320, 899 (2008).
  * Fu  _et al._ (2016) H. Fu, P. Wang, P. Shan, L. Xiong, L. N. Pfeiffer, K. West, M. A. Kastner, and X. Lin, Proc. Natl. Acad. Sci. 113, 12386 (2016).
  * Willett  _et al._ (2009) R. L. Willett, L. N. Pfeiffer, and K. W. West, Proc. Natl. Acad. Sci. 106, 8853 (2009).
  * Zhang  _et al._ (2009) Y. Zhang, D. T. McClure, E. M. Levenson-Falk, C. M. Marcus, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 79, 241304 (2009).
  * Willett  _et al._ (2010) R. L. Willett, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 82, 205301 (2010).
  * Nakamura  _et al._ (2020) J. Nakamura, S. Liang, G. C. Gardner, and M. J. Manfra, Nature Physics 16, 931 (2020).
  * Saminadayar  _et al._ (1997) L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne, Phys. Rev. Lett. 79, 2526 (1997).
  * de Picciotto  _et al._ (1997) R. de Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, and D. Mahalu, Nature 389, 162 (1997).
  * Bid  _et al._ (2010) A. Bid, N. Ofek, H. Inoue, M. Heiblum, C. L. Kane, V. Umansky, and D. Mahalu, Nature 466, 585 (2010).
  * Dolev  _et al._ (2010) M. Dolev, Y. Gross, Y. C. Chung, M. Heiblum, V. Umansky, and D. Mahalu, Phys. Rev. B 81, 161303 (2010).
  * Chickering  _et al._ (2013) W. E. Chickering, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 87, 075302 (2013).
  * Banerjee  _et al._ (2017) M. Banerjee, M. Heiblum, A. Rosenblatt, Y. Oreg, D. E. Feldman, A. Stern, and V. Umansky, Nature 545, 75 (2017).
  * Banerjee  _et al._ (2018) M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman, Y. Oreg, and A. Stern, Nature 559, 205 (2018).
  * Wixforth  _et al._ (1986) A. Wixforth, J. P. Kotthaus, and G. Weimann, Phys. Rev. Lett. 56, 2104 (1986).
  * Paalanen  _et al._ (1992) M. A. Paalanen, R. L. Willett, P. B. Littlewood, R. R. Ruel, K. W. West, L. N. Pfeiffer, and D. J. Bishop, Phys. Rev. B 45, 11342 (1992).
  * Willett  _et al._ (1993) R. L. Willett, R. R. Ruel, K. W. West, and L. N. Pfeiffer, Phys. Rev. Lett. 71, 3846 (1993).
  * Willett  _et al._ (2002) R. L. Willett, K. W. West, and L. N. Pfeiffer, Phys Rev Lett 88, 066801 (2002).
  * Friess  _et al._ (2018) B. Friess, V. Umansky, K. von Klitzing, and J. H. Smet, Phys Rev Lett 120, 137603 (2018).
  * Friess  _et al._ (2020) B. Friess, I. A. Dmitriev, V. Umansky, L. Pfeiffer, K. West, K. von Klitzing, and J. H. Smet, Phys. Rev. Lett. 124, 117601 (2020).
  * Friess  _et al._ (2017) B. Friess, Y. Peng, B. Rosenow, F. von Oppen, V. Umansky, K. von Klitzing, and J. H. Smet, Nature Physics 13, 1124 (2017).
  * Drichko  _et al._ (2011) I. L. Drichko, I. Y. Smirnov, A. V. Suslov, and D. R. Leadley, Phys. Rev. B 83, 235318 (2011).
  * Drichko  _et al._ (2016) I. L. Drichko, I. Y. Smirnov, A. V. Suslov, Y. M. Galperin, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 94, 075420 (2016).
  * Note (1) Previous work in Ref. Wixforth  _et al._ (1986) explains the velocity shift with the conductivity. Such an analysis may not be suitable here because our ultra-high mobility 2DES has a very long transport scattering time $\tau_{tr}\simeq 0.7$ ns comparable to the SAW frequency.
  * White and Voltmer (1965) R. M. White and F. W. Voltmer, Applied physics letters 7, 314 (1965).
  * Wu  _et al._ (2023) M. Wu, X. Liu, R. Wang, X. Lin, and Y. Liu, “A high-performance surface acoustic wave sensing technique,” (2023), arXiv:2311.01718 [physics.app-ph] .
  * Note (2) See supplementary material for a detailed description of our setup, which includes Refs. Sullivan  _et al._ (1990); Durdaut  _et al._ (2019).
  * Note (3) See supplementary material for more information.
  * Chen  _et al._ (2003) Y. Chen, R. M. Lewis, L. W. Engel, D. C. Tsui, P. D. Ye, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 91, 016801 (2003).
  * Note (4) Such behavior is surprising but not inconsistent with the previous report, since our frequency is much lower than the geometric resonance condition Willett  _et al._ (1993, 2002).
  * Chung  _et al._ (2019) Y. J. Chung, K. W. Baldwin, K. W. West, N. Haug, J. van de Wetering, M. Shayegan, and L. N. Pfeiffer, Nano Letters 19, 1908 (2019).
  * Sullivan  _et al._ (1990) D. Sullivan, D. Allan, D. Howe, and F. Walls, Inst. Stand. Technol. Technical Note 1337 (1990).
  * Durdaut  _et al._ (2019) P. Durdaut, M. Höft, J.-M. Friedt, and E. Rubiola, Sensors 19, 185 (2019).

##  I Supplementary Materials

###  I.1 I. Calibrating our SAW measurement

Figure S1: (a) A simplified diagram of our superheterodyne quadrature phase
demodulation system. $V_{\text{in}}$ and $V_{\text{out}}$ are the input and
output signal of the device. (b) The $|S_{21}|$ of the testing LiNbO3 device
measured by our setup as a function of frequency. The device has similar in-
band loss compared to the GaAs device used in our main manuscript. (c) The I/Q
demodulation output of our setup when the input signal amplitude is -155 dBm,
from which we can calibrate the electrical noise floor of our setup is about
-170 dBm. (d) The phase noise spectrum of our demodulation output.

We introduce a superheterodyne-scheme radio frequency lock-in technique which
can measure the phaseshift ($\mit\Phi$) of the delay-line device. The whole
setup is optimized for the best phase noise background and stability; see ref.
Wu  _et al._ (2023) for detailed discussion. A schematic diagram of the system
is shown in Fig. S1(a). We use fractional phase-locked-loop (PLL) to generate
two single-frequency signals $V_{\text{in}}$ and $V_{\text{LO}}$ whose
frequencies are linked to the reference clock frequency $f_{\text{CLK}}$ by
$f_{\text{in}}=\frac{n}{m}\cdot f_{\text{CLK}}$ and
$f_{\text{LO}}=\frac{n+k}{m}\cdot f_{\text{CLK}}$ respectively; $n$, $m$ and
$k$ are integer values. $V_{\text{in}}$ is attenuated and sent to the SAW
device which attenuates the amplitude by $\mit\Gamma$ and introduce a phase
delay $\mit\Phi$. The output signal of the device $V_{\text{out}}$ will be
magnified by a broadband low noise amplifier (LNA) and then multiply with
$V_{\text{LO}}$ at the mixer. We then use a FPGA-based digital lock-in
technique to digitize and analyze the differential frequency component
$V_{\text{d}}$. The I/Q component obtained from the quadrature demodulation is
able to deduce the amplitude and phase of $V_{\text{out}}$.

We calibrate the measurement system using a delay-line device made from a
LiNbO3 (64∘-Y cut) substrate whose center frequency $f_{\text{c}}$ is 837MHz,
see the $S_{21}$ spectrum in Fig. S1(b). In order to analyze the electronic
white noise background, we send a 1 fW (-120 dBm), 837-MHz signal with an on-
off modulation using a mechanical switch (10mHz, 70$\%$ duty cycle). The
demodulation output ($I$ and $Q$) using resolution bandwidth of about 2 Hz is
plotted in Fig. S1(c). When the switch is off, no signal is sent into the
device and the demodulation output origins from the leaking signal, resulting
in the right cluster of points in Fig. S1(c). When the switch is on, the
detected signal is expected to be -155 dBm after attenuated by the cables
(about -5dB ) and the device (about -30dB), corresponding to the left cluster
of points in Fig. S1(c). The two clusters are separated by a distance
$d\simeq$ 1.29 $\mu$V and their standard deviation $\sigma$ is 0.24 $\mu$V,
consistent with a equivalent -170 dBm electronic noise background. The phase
resolution limited by such electrical noise background is
$\delta\theta=\frac{\sigma}{d}=0.18$ rad at -120 dBm input power, and becomes
better than 0.2 mrad when the input signal is about -60 dBm.

We can increase the signal amplitude to increase the phase resolution.
However, this enhance has a limit at large signal amplitude when the phase
noise of $V_{\text{in}}$ and $V_{\text{LO}}$ dominates over the electronic
noise Sullivan  _et al._ (1990); Durdaut  _et al._ (2019). Fig. S1(d) shows
the phase noise spectrum at -61 dBm (about 1 nW) input RF power. The measured
phase noise spectral density $S_{\varphi}$(f) is proportional to $1/f^{0.5}$
at frequencies below $\sim$ 1 Hz (the red solid line), consistent with the
frequency jittering of the signal source. It equals about 0.2 mrad/Hz 0.5 at 1
Hz, leading to bounded random phase drift of about 0.3 mrad (i.e. constant
Allan deviation).

###  I.2 II. The temperature response of $\eta$

In this section, we discuss two different mechanism of temperature induced SAW
velocity shift.

Usually, the acoustic velocity increases when temperature reduces, and a
$\eta\propto-T^{3}$ relation is expected in regular materials, e.g. see ref.
Wu  _et al._ (2023). At very low temperatures $T\lesssim$ 4 K, we observe a
surprising decrease of the SAW velocity in a delay-line sample made from an
intrinsic GaAs substrate without any 2DES. The temperature dependence of the
SAW velocity shift $\eta$ of this sample is shown in Fig. S2(a) and (b).
Although we have not figure out the mechanism behind this peculiar behavior,
we find that $\eta$ remains basically unchanged at temperatures below 200mK.
The electron-phonon coupling becomes extremely weak at temperature below 1 K
and the 2DES is believed to be cooled by contacts. It is unlikely that the
heat generated by the 500 nA current passing through the high mobility 2DES,
which is less than 1 nW, can heat up the lattice to above 50 mK. Furthermore,
the heating power of this current is roughly proportional to magnetic field,
because the 2-point resistance of these ultra-high mobility 2DES is
approximately $R_{xy}$ at magnetic fields $B\gtrsim 0.5$ T. This is also
inconsistent with the oscillating behavior of our measured
$\kappa=\eta_{\text{m}}^{-1}\cdot(\partial\eta/\partial|I|)$.

Besides the lattice effect, the quantum phase in 2DES itself is also sensitive
to temperature so that $\eta$ can also be influenced. In Fig. S2(c), we show
the $\eta$ vs. $B$ measured at different temperatures. The traces taken at
different temperature are offset so that the $\eta$ peak at strong integer
quantum Hall effects such as $\nu=1$, 2 and 3 as well as at low magnetic field
$B\simeq 0$ T, which is assumed to be insensitive to temperatures, are
aligned. The Fig. S2(d) shows $\eta$ at $B=10.2$ T where the largest
temperature dependence is seen. $\eta$ has a linear dependence on $T$ and
$d\eta/dT$ is about 0.2 ppm/mK, and saturates below about 70 mK. Therefore, it
is only possible to achieve the $\simeq$ 10 ppm amplitude of
$\delta\eta_{\text{P}}$ if the 2DES is heated up to about 120 mK. On the other
hand, we show $R_{xx}$ measured using 500 nA current in Fig. S2(e). At
$\nu=9/2$ and $\nu=11/2$, where the spontaneous unidirectional charge density
wave phase is stabilized, a clear difference in $R_{xx}$ is seen when the
mixing chamber temperature raises from 64 mK to 100 mK, suggesting that the
2DES temperature is below 100 mK if the 500 nA passing through the device at
fridge base temperature. Therefore the heating effect of 2DES is unlikely the
leading cause of the large current-induced increase in the sound velocity.

Figure S2: (a) SAW velocity shift $\eta$ of an intrinsic GaAs sample as a
function of the temperature $T$. The SAW velocity increases as temperature
decreases at relative high temperatures $T\gtrsim 4$ K. (b) The SAW velocity
decreases i.e. $\eta$ reduces, when temperature decreases from 4K to 200mK,
and remains unchanged at temperatures below about 200 mK. (c) $\eta$ vs. $B$
at different Temperatures. (d) The $\eta$ vs. Temperature $T$ at $B=10.2$ T
shows a linear increasing of $\eta$ at temperatures above 70 mK. (e) $R_{xx}$
taken near filling factors $\nu=9/2$ and 11/2 using 500 nA measurement current
at different temperatures. The $R_{xx}$ minima at $\nu=9/2$ and 11/2 signal
the formation of unidirectional charge density waves, whose temperature
dependence suggests that the 2DES remains below 100 mK when 500 nA current
passing through the device.  Figure S3: SAW velocity shift $\eta$ measured
under different input SAW powers. Inset: Schematic diagram of the measurement
setup. The IDT is connected in parallel with a 50 $\Omega$ resistance ($R$).
The coaxial cable ($L$ $\sim$ 5.5 m) is connected from the base temperature to
room temperature.

###  I.3 III. Electron density fluctuation

In this section, we provide a rough estimation of the electron density
fluctuation of 2DES induced by SAW. The SAW speed $v_{0}\simeq 2950$ m/s is
much smaller than the Fermi velocity of charged quasiparticles
($v_{\text{F}}\simeq 3\times 10^{5}$ m/s), and the transport scattering time
is of the same order as the SAW period. We can assume that the response of
2DES to the SAW-induced electric field is fast enough so that the 2DES is
almost always at equilibrium and the problem can be solved statically. The
propagating piezoelectric field introduced by SAW at a given time (e. g.
$t=0$) is

$\vec{E}_{\text{ext}}=\vec{E}\cdot\cos(-\omega
t+kx)|_{\text{t=0}}=\vec{E}\cdot\cos(kx)$ (1)

where $k$ denotes the wavevector of the SAW. We then have the SAW-induced
external potential is

$\mit\Phi_{\text{ext}}=-\int\vec{E}_{\text{ext}}(x)dx=-\frac{E}{k\sin(kx)}$
(2)

Using Thomas-Fermi screening model, the induced charge is

$\rho_{\text{ind}}=-e^{2}\cdot\frac{dn}{d\mu}\mit\Phi_{\text{scr}}/D$ (3)

where $\frac{dn}{d\mu}$ is the 2DES’s compressibility, i.e. the density of
state at Fermi energy for Fermi sea. And we assume the 2DES as a flat plate of
uniform thickness $D=20$ nm and neglect the stray field. The screened
potential $\mit\Phi_{\text{scr}}$ is the sum of the external potential and the
induced potential

$\mit\Phi_{\text{scr}}=(\mit\Phi_{\text{ind}}+\mit\Phi_{\text{ext}})$ (4)

$\mit\Phi_{\text{ind}}$ can be calculated from $\rho_{\text{ind}}$ using the
Gaussian law

$\nabla^{2}\mit\Phi_{\text{ind}}=-\rho_{\text{ind}}/\epsilon_{0}\epsilon_{\text{r}}$
(5)

Combining the above relations, the induced potential can be deduced by the
following equation self-consistently.

$\nabla^{2}\mit\Phi_{\text{ind}}=e^{2}\frac{dn}{d\mu}\frac{1}{D\epsilon_{0}\epsilon_{\text{r}}}(\mit\Phi_{\text{ind}}+\mit\Phi_{\text{ext}})$
(6)

It is easy to see that

$\mit\Phi_{\text{ind}}=-\frac{\mit\Phi_{\text{ext}}}{1+\frac{D\epsilon_{0}\epsilon_{\text{r}}k^{2}}{e^{2}\frac{dn}{d\mu}}}$
(7)

In GaAs $\epsilon_{\text{r}}=13.1$, the density of state in 2DES at zero field
is $\frac{dn}{d\mu}=\frac{m_{\text{e}}}{2\pi\hbar^{2}}$. The induced potential
cancels the external potential if
$\frac{D\epsilon_{0}\epsilon_{\text{r}}k^{2}}{e^{2}\frac{dn}{d\mu}}\ll 1$. We
can deduce the induced electron density.

$n_{\text{ind}}=-\rho_{\text{ind}}\cdot
D/e=\epsilon_{0}\epsilon_{\text{r}}\nabla^{2}\mit\Phi_{\text{ind}}\cdot
D/e=\frac{D\epsilon_{0}\epsilon_{\text{r}}k^{2}\mit\Phi_{\text{ext}}}{e}$ (8)

If the 2DES is incompressible, i.e. $dn/d\mu=0$, $\mit\Phi_{\text{ind}}$ and
$n_{\text{ind}}$ are both zero and the interaction between 2DES and SAW
vanishes. The induced electrons with $n_{\text{ind}}$ propagate along with the
SAW. The power dissipation and phase delay of SAW occur when induced electrons
are scattered by impurities or the quasiparticle velocity is comparable or
less than the SAW velocity.

$\mit\Phi_{\text{ext}}$ can be estimated from the signal amplitude received by
the output IDT. In our experiment as shown by the inset of Fig. S3, the
typical input RF power is -61 dBm (about 1 nW) and the output is about -97
dBm. The cable attenuation is about -5 dB at 600 MHz so that the power at the
output IDT is -92 dBm. The IDT capacitance $C_{\text{IF}}$ $\sim$ 20 pF
corresponds to $\sim$ 10 $\Omega$ impedance at 600 MHz, sufficiently small to
be neglected. We can estimate that the voltage at the receiving IDT is about
10 $\mu$V, leading to $\sim 10^{4}$ cm-2 electron density fluctuation, orders
of magnitude smaller than the density fluctuation of the 2DES itself ($\sim
10^{9}$ cm-2 even in very high-mobility samples Chung  _et al._ (2019)).

In the experiment, the SAW excitation power (-61 dBm) is a trade-off between
the principle of “non-perturbative” measurement and the signal-to-noise ratio
(SNR). Namely, the power should be sufficiently low so that $n_{\text{ind}}$
is negligible, while it should be large enough to resolve interesting
phenomena. The proper input power is chosen by the following rationale. We
first measure $\eta$ using the lowest possible power, e.g. -91 dBm which
corresponds to $n_{\text{ind}}\sim 300$ cm-2. We then increase the power to
-61 dBm. We find that the measured two $\eta$ are almost identical, see Fig.
S3, suggesting that -61 dBm is sufficiently low. The power received by our
lock-in amplifier is about -92 dBm. Our setup has -170 dBm noise background
when using 300 ms time constant (corresponds to about 0.3 Hz effective noise
bandwidth). The 80 dB SNR ratio leads to about 0.1 ppm resolution in $\eta$,
sufficiently high for this study.

###  I.4 IV. $\eta$ near integer fillings

Near the integer Landau level filling factors $\nu=N$, we observe “V”-shape
maxima instead of plateaus in $\eta$. This might be related to the finite
compressibility of Wigner crystal formed by the dilute
quasiparticles/quasiholes whose effective filling factor $\nu^{*}=|\nu-N|$ is
small Chen  _et al._ (2003). In Fig. S4(a), we observe a linear dependence
between $\eta$ and the effective quasi-particle density
$n^{*}=n|\nu^{*}|/\nu$.

$\eta$ at exact integer fillings $\nu=N$ increases when the quantum Hall state
becomes stronger. We summarize $\eta$ at different integer filling in Fig.
S4(b). $\eta$ has a rather linear dependence on $\nu$ at $\nu\textless 10$,
whose intercept at $\nu=0$ is about $\eta_{\text{m}}=124.2$ ppm.
$\eta_{\text{m}}$ marks the maximum effect the 2DES poses on the SAW and can
be used to normalization $\kappa$.

###  I.5 V. The origin of the second harmonics

As shown in Fig. 3 of the manuscript, the SAW velocity shift is proportional
to the amplitude of current. Therefore, an AC current passing through the
sample

$I=I_{\text{p}}sin(\omega t)$ (9)

induces a SAW velocity shift

$\delta\eta=\delta\eta_{\text{p}}|sin(\omega t)|=\eta_{\text{m}}\kappa
I_{\text{p}}|sin(\omega t)|$ (10)

The Fourier series of $\delta\eta$ can be expanded as

$\delta\eta=\delta\eta_{\text{p}}(\frac{4}{\pi}-\frac{4}{3\pi}\cos(2\omega
t)-\cdots-\frac{2(1+(-1)^{n})}{(n^{2}-1)\pi}\cos(n\omega t))$ (11)

where $n$ is a non-negative even integer. The dominate component of the signal
is second harmonics, whose amplitude is

$\delta\eta_{\text{2f}}=\delta\eta_{\text{p}}\cdot\frac{4}{3\pi}$ (12)

Therefore, we can measure $\delta\eta_{\text{2f}}$ and deduce
$\delta\eta_{\text{p}}$. We can also deduce $\kappa$ by

$\kappa=\frac{1}{\eta_{\text{m}}}\frac{\partial\eta_{\text{p}}}{\partial
I_{\text{p}}}=\frac{3\pi}{4\eta_{\text{m}}}\frac{\partial\eta_{\text{2f}}}{\partial
I_{\text{p}}}$ (13)

Figure S4: (a) $\eta$ vs effective electron density $n^{*}$ at different
integer fillings. (b) The linear relationship between $\eta$ and $\nu$.

###  I.6 VI. Data using lower SAW frequency

We repeat the acoustic study in another sample. The IDT period $\lambda$ is 12
$\mu$m and the SAW resonance frequency is $f_{\text{c}}=243.1$ MHz. The
primary feature of $\kappa$ in this device (see Fig. S5) is similar to the
data reported in the main text. Similarly, $\kappa$ decreases with increasing
temperature and finally vanishes when $T\simeq$ 250 mK. We conclude that the
phenomena we reported in the manuscript have no dependence on the SAW
frequency and wavelength.

Figure S5: $\kappa$ vs magnetic field at different temperatures of another
sample with lower resonance frequency (243.1 MHz). The IDT period $\lambda$ is
12 $\mu$m .

###  I.7 VII. Data from a lower mobility sample

For comparison, we perform similar acoustic study on a lower mobility sample.
The electron density of the 2DES is $1.12\times 10^{11}$ cm-2 and its low-
temperature mobility is about $1.14\times 10^{6}$ cm2/(V$\cdot$s). $\eta$
$\approx$ 700 ppm at $\nu$ = 1 is larger in this sample, because the 2DES is
shallower and the interaction between 2DES and SAW is stronger. Thanks to this
large $\eta$, we can measure $\delta\eta$ directly with DC current, similar to
the Fig. 3(a) data in the main text. The threshold current is about 200 nA for
both the $\nu$ = 1 QHS and the $\nu$ = 2/3 FQHS, see Fig. S6(a). Therefore,
integer and fractional QHS exhibit similar CIVS if the quasiparticles are
randomly localized and no Wigner crystal exists.

Figure S6: In the lower mobility sample, (a) $\delta\eta$ vs. DC Current $I$
at $\nu$ = 1 (black curve) and $\nu$ = 2/3 (red curve). (b) $\delta\eta$
(black curve) and normalized differential longitudinal resistance
($R_{\text{xx}}/R_{\text{$\nu$ =1/2}}$) (red curve) vs. large DC Current $I$
at $\nu$ = 1. The differential longitudinal resistance is measured under small
AC current and normalized by the longitudinal resistance at $\nu$ = 1/2.

Fig. S6(b) shows $\delta\eta$ as a function of the DC current $I$ at $\nu$ =
1. $\delta\eta$ increases when the current increases above the threshold. It
saturates at large current $I$ $\textgreater$ 2 $\mu$A. Meanwhile, the
differential longitudinal resistance increases by less than a percent,
signaling that the integer QHS is still far below breakdown.

###  I.8 VIII. Device fabrication

The samples are 5 × 5 mm square cleaved directly from a GaAs/AlGaAs wafer
grown via molecular beam epitaxy. The 30-nm-wide quantum well locates at 390
nm below the surface. We use three $\delta$-doping layers, the deepest one is
about 80 nm below the quantum well. The low temperature mobility of the 2DES
is about $2\times 10^{7}$ cm-2/(V $\cdot$ s).

We make ohmic contacts at the four corners of the sample by depositing
Ge/Au/Ni/Au alloy and annealing at 440 °C for 30 minutes. A $d_{\text{m}}$ =
1.2 mm square Van der Pauw mesa is then created by wet etching using a ${\rm
H_{2}O}$:${\rm H_{2}O_{2}}$:${\rm H_{2}SO_{4}}$ solution (240:8:1) for 4
minutes. The etching depth is approximately 800nm. We pattern the interdigital
transducers (IDTs) with evaporated 400 ÅAl using a maskless laser lithography
system and lift-off process. Each IDT has 170 pairs of fingers with a
periodicity of $\lambda$ = 5 $\mu$m. The center-to-center spacing of opposite
pair of IDTs is 2450 $\mu$m.