# High-Frequency GaAs Optomechanical Bullseye Resonator

N. C. Carvalho<EMAIL_ADDRESS>Applied Physics Department and
Photonics Research Center, University of Campinas, Campinas, SP, Brazil  R.
Benevides  Applied Physics Department and Photonics Research Center,
University of Campinas, Campinas, SP, Brazil  M. Ménard  Department of
Computer Science, Université du Québec à Montréal, Montréal, Canada  G. S.
Wiederhecker  Applied Physics Department and Photonics Research Center,
University of Campinas, Campinas, SP, Brazil  N. C. Frateschi  Applied Physics
Department and Photonics Research Center, University of Campinas, Campinas,
SP, Brazil  T. P. Mayer Alegre<EMAIL_ADDRESS>Applied Physics Department
and Photonics Research Center, University of Campinas, Campinas, SP, Brazil

###### Abstract

The integration of optomechanics and optoelectronics in a single device opens
new possibilities for developing information technologies and exploring
fundamental phenomena. Gallium arsenide (GaAs) is a well-known material that
can bridge the gap between the functionalities of optomechanical devices and
optical gain media. Here, we experimentally demonstrate a high-frequency GaAs
optomechanical resonator with a ring-type bullseye geometry that is
unprecedented in this platform. We measured mechanical modes up to
$3.4\text{\,}\mathrm{GHz}$ with quality factors of 4000 (at
$77\text{\,}\mathrm{K}$) and optomechanical coupling rates up to
$39\text{\,}\mathrm{kHz}$ at telecom wavelengths. Moreover, we investigated
the material symmetry break due to elastic anisotropy and its impact on the
mechanical mode spectrum. Finally, we assessed the temperature dependence of
the mechanical losses and demonstrated the efficiency and anisotropy
resilience of the bullseye anchor loss suppression, indicating that lower
temperature operation may allow mechanical quality factors over $10^{4}$. Such
characteristics are valuable for active optomechanics, coherent microwave-to-
optics conversion via piezo-mechanics and other implementations of high-
frequency oscillators in III-V materials.

## Introduction

The engineering of light-matter interaction in optomechanical devices has
allowed the observation of very relevant fundamental phenomena such as
gravitational waves Abbott  _et al._ (2016) and ground-state cooling Chan  _et
al._ (2011); and consequently has enabled important developments in
information science technology Hill  _et al._ (2012); O’Connell  _et al._
(2010). For the past decade, silicon has been the material of choice for most
on-chip optomechanics experiments. However, as we move towards high power
efficiency and quantum-level control, device design becomes more challenging
and material properties more restrictive, which have driven intense research
into alternative materials Guha  _et al._ (2017a); Morozov  _et al._ (2019);
Zou  _et al._ (2016); Mitchell  _et al._ (2019). Therefore, gallium arsenide
(GaAs) arises as a mature platform with the potential to match or overcome
silicon in many properties, such as light confinement and optomechanical
coupling strength, due to its high-refractive-index and large photo-elastic
coefficients Balram  _et al._ (2014). Moreover, the optical losses that often
impair the performance of GaAs have been mitigated by improved etching and
surface passivation techniques, leading to optical quality factors
($Q_{\text{opt}}$) of over a million Guha  _et al._ (2017b).

Developing optomechanical resonators based on III-V materials would not only
allow for their disruptive integration with coherent light sources and single
quantum emitters, but also open routes to explore the interplay of gain and
loss in non-Hermitian physical systems Özdemir  _et al._ (2019). Besides its
advantageous optoelectronic properties, GaAs also has other valuable
characteristics, such as piezoelectricity, revealing its suitability for
wavelength conversion mediated by piezo-optomechanics Wu  _et al._ (2020),
enabling thus the ultimate integration and control of charge carriers, light,
sound, and microwave fields.

To unleash the outstanding properties of GaAs and enable an optomechanical
device operation in the resolved-sideband regime, a design supporting high
mechanical frequencies must be devised. In simple microdisks, this is limited
by the resonator radius, which turns to be unpractically small Baker  _et al._
(2014); Ding  _et al._ (2010, 2011). Also, it becomes relatively complex in
optomechanical crystals, which requires complicated designs Balram  _et al._
(2014). Despite recent efforts with nanobeam cavities showing impressive
optomechanical coupling rates, mechanical frequencies have not exceeded
$2.8\text{\,}\mathrm{GHz}$ in GaAs Balram  _et al._ (2016); Forsch  _et al._
(2020). In this work, we demonstrate GaAs optomechanical devices built using a
bullseye design that allows for very high mechanical quality factors and
measured mechanical modes up to $3.4\text{\,}\mathrm{GHz}$, with the potential
to explore even higher frequency modes, according to simulations. Our design
allows for optimizing the optical and mechanical resonances independently,
offering to couple between several high-quality factors optical modes with a
single mechanical mode, which can be interesting for the exploration of multi-
mode optomechanical experiments.

Figure 1:  (a1) Illustration of the bullseye resonator (top) and geometric
parameters (bottom). Inset: (a2) simulated displacement profile of the
mechanical modes and (a3) electric amplitude of the whispering gallery optical
mode with transverse electric polarization and fundamental radial-order
(bottom). (b) The blue (yellow) lines are the mechanical bands of the bullseye
grating simulated as a linear crystal for x-polarized or z-polarized
(y-polarized) modes. The shaded blue show the target bandgap and the insets
contain the mechanical deformation for the modes at the band-edge (X-point).
(c) 2D simulation results of the optomechanical coupling rate
($g_{0}/(2\pi)$), the gray dashed lines are drawn as a guide to the eyes. The
grating parameters in (b) and (c) are $a=$600\text{\,}\mathrm{nm}$$,
$w=$120\text{\,}\mathrm{nm}$$ and $t=$50\text{\,}\mathrm{nm}$$.

Our samples were fabricated using a technique that allows different etching
depths to be obtained in a single-step lithography process. The mechanical
performance of our device was investigated by cooling it to liquid nitrogen
temperatures and by characterizing typical sources of mechanical dissipation,
revealing that the mechanical quality factor is only limited by temperature-
dependent effects, which can be circumvented at cryogenic conditions. We also
show that our GaAs samples are strongly affected by the material elastic
anisotropy and numerically evaluate its implications, obtaining a good match
between experimental data and simulations. The devices achieved optomechanical
coupling rates and mechanical quality factors up to 39 kHz and 4000 (at
$\sim$80\text{\,}\mathrm{K}$$), respectively, operating just above the
resolved sideband limit.

## Design and fabrication

The bullseye geometry, originally devised by Santos et al. Santos  _et al._
(2017) using silicon, consists of a ring-type cavity resonator obtained by
patterning a nano-structured circular grating over a micro-disk, as shown in
the diagram in Fig. 1 (a1). In this design, the mechanical waves are confined
to the outermost ring using a radial phononic shield, which also isolates the
ring cavity from the supporting pedestal, inhibiting anchor losses and,
therefore, enabling very high mechanical quality factors. A key advantage of
the bullseye design, compared to nanobeams or optomechanical crystal devices,
is the complete decoupling between optical and mechanical resonance
frequencies. The former is mostly influenced by the radius of the disk, $R$ in
Fig. 1 (a1), whereas the latter will be defined by the external ring width
($w_{\mathrm{ring}}$). In this way, mechanical frequencies can be increased by
narrowing down the external ring, with minimal impact on the optical
frequencies, as long as $w_{\mathrm{ring}}\gtrsim$500\text{\,}\mu\mathrm{m}$$
Santos  _et al._ (2017).

Our GaAs bullseye project was based on a nominal geometry with a disk radius
of $6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and grating dimensions set to
$a=$600\text{\,}\mathrm{nm}$$, $w=$120\text{\,}\mathrm{nm}$$ and
$t=$50\text{\,}\mathrm{nm}$$, as shown in Fig. 1 (a1). This grating is
designed as a phononic shield to confine dilatational mechanical waves
propagating in the radial direction and, as long as their wavelengths are
small compared to the ring radius, one can neglect its curvature and
approximate it to a linear crystal Santos  _et al._ (2017) (Fig. 1 (b) -
inset). Using 2D cartesian Finite Element Method (FEM), we modeled a linear
grating with the geometrical parameters mentioned above, obtaining a band-
diagram as in Fig. 1 (b). Such approximation revealed a partial phononic
bandgap for longitudinal ($x$-polarized) and vertical shear waves
($z$-polarized), represented by the blue lines in Fig. 1 (b), where shear-
horizontal waves ($y$-polarized) are also shown in yellow. It resulted in two
mechanical frequency stopbands in the ranges between
$\sim$2\text{\,}\mathrm{GHz}$~{}-~{}$3\text{\,}\mathrm{GHz}$$ and
$\sim$3\text{\,}\mathrm{GHz}$~{}-~{}$4\text{\,}\mathrm{GHz}$$, able to
constrain radial dilatational mechanical waves to the edge of the disk.

The optomechanical coupling rate, $g_{0}/(2\pi)$, for the full bullseye
structure is shown in Fig. 1 (c). Our simulations (2D - axisymmetric) show
that three mechanical modes couple to the (whispering gallery-type) optical
field, as highlighted by the gray dashed lines between $2\text{\,}$ and
$5\text{\,}\mathrm{GHz}$. The mechanical displacement profiles of these modes
are shown in Fig. 1 (a2), followed by the first radial-order transverse
electric (TE) optical mode to which they are coupled (in Fig. 1 (a3)). It
reveals that C is the desired ring-type breathing mode, while A is a flexural
ring mode, and B is a grating mode that is almost independent of the external
ring size. Unlike GaAs disk resonators with similar dimensions, the $g_{0}$
for the first-order ring dilatational mechanical mode is dominated by the
photo-elastic effect; precisely, our calculations for a
$740\text{\,}\mathrm{n}\mathrm{m}$ $w_{\mathrm{ring}}$ show that the moving
boundaries Johnson  _et al._ (2002) and photo-elastic Chan  _et al._ (2012)
optomechanical coupling rates are $12\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$
and $129\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, respectively, resulting in a
total $g_{0}/(2\pi)$ of $141\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$.

The devices were fabricated from a GaAs/Al0.7Ga0.3As stack
($$250\text{\,}\mathrm{nm}$/$2000\text{\,}\mathrm{nm}$$) that was grown over a
GaAs substrate using molecular beam epitaxy (Canadian Photonics Fabrication
Centre). In contrast with the previous silicon bullseye, which was fabricated
with an optical stepper, the GaAs structure was defined using electron beam
lithography Benevides  _et al._ (2020). The fabrication steps of the GaAs
bullseye samples are summarized in Fig. 2 (a1). The devices were patterned on
the top of the GaAs wafer through a single plasma etching step. We used
positive electro-resist (ZEP-520A) and managed to define grating grooves and
remove all the material outside the microdisk region using the aspect-ratio
dependent etching, where the etching rate of narrower gaps is lower in
comparison to wider regions (Fig. 2 (a2)) Shaqfeh and Jurgensen (1989); Bailey
III  _et al._ (1995). The bullseye disks were then released from the substrate
by selective wet etching of the AlGaAs buffer layer with hydrofluoric acid
(HF), followed by standard organic cleaning. This advantageous technique
avoided the need for multiple lithography steps that would introduce complex
alignment procedures.

Figures 2 (b) and (c) present scanning microscope images of exemplar samples.
The former shows the final device surrounded by the parking lot, which was
designed to support the fiber loop. Top-view images, such as in Fig. 2 (c),
were used to characterize the dimensions of the devices. The disk radius
($R$), $w_{\mathrm{ring}}$, $a$ and $w$ were measured through this method,
whereas $t$ was only estimated from the etching rate, which was calibrated
before the fabrication of the devices. Electron beam proximity effects caused
the first outer groove ($w_{1}$ in Fig. 2 (a3)) to have a narrower width in
comparison to the inner grooves ($w$). Nevertheless, we obtained very regular
and sharp grooves that are only slightly angled. Section S1, in the
Supplementary Material, discusses in detail the impact of the verticality of
the sidewalls and the depth of the grooves in the mechanical grating
structure. Finally, in section S2, both $w_{1}$ and $t$ were found by matching
the measured and simulated values for the optical mode dispersion of our
device. All those geometrical values were then used in the FEM simulations of
Fig. 3.

## Results and discussion

The optomechanical properties of the resonators were characterized in the
setup schematically represented in Fig. 3 (a). The bullseye microcavity was
probed via evanescent coupling with a tapered fiber loop continuously fed by a
tunable C-band laser (New Focus TLB-6728). The transmitted optical signal was
then split and simultaneously measured by a slow and a fast detector. The DC
component gave information about the optical response of the cavity, while the
fast signal was measured by an electrical spectrum analyzer that read out the
mechanical mode signatures imparted on the transmitted optical signal.

Figure 2: (a1) Simplified illustration of the bullseye fabrication steps. (a2)
and (a3) contain the diagrams high-lightening the plasma etching rate contrast
for regions of different sizes, and the narrowing of the first outer groove
due to the electron beam proximity effect, respectively. (b) Scanning electron
microscope image of a typical device (bullseye resonator surrounded by the
parking lot, a structure designed to stabilize the tapered fiber loop). (c)
Top-view of a sample.

We experimentally investigated two devices with different $w_{\mathrm{ring}}$
sizes: $700\text{\,}\mathrm{nm}$ and $740\text{\,}\mathrm{nm}$, at room
temperature and atmospheric pressure. The optical spectrum of the
$740\text{\,}\mathrm{nm}$ sample is displayed in Fig. 3 (b). The inset has a
mode doublet (counter-propagating whispering gallery modes) presenting total
linewidths, $\kappa/(2\pi)$, of $\sim$6\text{\,}\mathrm{GHz}$$
($Q_{\text{opt}}\sim$3\text{\times}{10}^{4}\text{\,}$$), which was a typical
value measured in our samples. These modes provided the strongest readout of
the mechanical modes and were found to be second radial-order modes with
quasi-TE polarization (major component along the radial direction). The
identification was based on the comparison between the measured and simulated
frequency dependence of the free spectral range (due to group velocity
dispersion) Fujii  _et al._ (2020) \- see Supplementary Material, S2, for
details.

The optomechanical coupling with the second radial-order TE optical mode is
very similar to its fundamental counterpart (Fig. 1 (c)) and predicts
$g_{0}/(2\pi)=$136\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ for the sample
with a ring size of 740 nm. The complete 2D simulation for this optical mode
structure as a function of the $w_{\mathrm{ring}}$ size can be found in the
Supplementary Material, S2. Although a slightly larger $g_{0}$ is expected for
the optomechanical coupling through the first radial-order mode, our
measurement scheme was sensitive to $g_{0}^{2}/\kappa$ and, therefore, the
detection of the mechanical interaction with such optical modes was
compromised by their larger linewidths.

Figure 3:  (a) Optomechanical characterization setup (DAQ - Data Acquisition
System, PSG - Power Signal Generator, ESA - Electrical Spectrum Analyzer, PM -
Phase Modulator, HCN - Hydrogen Cyanide wavelength reference and MZI -
Mach–Zehnder Interferometer). Inset: optical microscope image of the tapered
fiber loop coupled to a device. (b) Optical transmission spectrum of the
$w_{\mathrm{ring}}~{}=~{}$740\text{\,}\mathrm{nm}$$ sample. The inset shows
the optical modes (whispering gallery-type of transverse electric polarization
and second radial-order) highlighted in light gray. The orange line is the
Lorentzian fit. (c) Power spectral density (PSD) of the selected mechanical
mode of the $740\text{\,}\mathrm{nm}$ sample (highlighted in shaded orange in
(d4)) with respective $g_{0}$ calibration peak (the orange line is the
Lorentzian fit). 3D FEM simulations of: (d1) the mechanical modes when
$\eta=0$ (fully isotropic), (d2) the optomechanical coupling rate as a
function of the material anisotropy parameter ($\eta$) and (d3) mechanical
modes when $\eta=1$ (fully anisotropic). (d4) Broad experimental PSD (raw and
digital filtered data using the Savitzky–Golay method SG ). (d5) Simulated
mechanical displacement profiles of the 2.8 GHz and the 3.2 GHz modes (highest
bars) in (d1) and (d3), respectively. The information in (d1)-(d5) corresponds
to the $w_{\mathrm{ring}}~{}=~{}$740\text{\,}\mathrm{nm}$$ and all
experimental data were acquired at room temperature and atmospheric pressure.

A 3.29 GHz mechanical resonance was measured in the $740\text{\,}\mathrm{nm}$
$w_{\mathrm{ring}}$ and shown in Fig. 3 (c). The Lorentzian fit gave a
mechanical quality factor ($Q_{\text{m}}$) of 900 and the calibration of
$g_{0}$ – through the comparison between the phase modulator RF-tone and the
optical transduction of the cavity Gorodetsky  _et al._ (2010) – resulted in
an optomechanical coupling rate of $g_{0}/(2\pi)=$34\text{\,}\mathrm{kHz}$$.
Analogously, we obtained a $Q_{\text{m}}$ of 1200 and a
$g_{0}/(2\pi)=$39\text{\,}\mathrm{kHz}$$ for a $3.13\text{\,}\mathrm{GHz}$
mode of the $700\text{\,}\mathrm{nm}$ $w_{\mathrm{ring}}$ sample. Numerical
simulations of $g_{0}$ for the $740\text{\,}\mathrm{nm}$ $w_{\mathrm{ring}}$
are shown in Fig. 3 (d1)-(d3). Also, Fig. 3 (d4) presents a
$1\text{\,}\mathrm{GHz}$-broad experimental mechanical spectrum - raw and
Savitzky–Golay filtered SG . The data used to obtain the optomechanical
coupling rates was not filtered.

The lower $g_{0}$ and the multiple peaks observed in the experimental spectra
of Fig. 3 (d4) are not consistent with axisymmetric simulations that predict a
single peak for each mechanical mode (A, B and C in Fig. 1 (a1)). Therefore,
we employed a three-dimensional FEM model to account for the well-known
elastic anisotropy of GaAs and precisely identify the mechanical modes.
Indeed, as the material anisotropy is gradually increased in the 3D
simulations, a clear degeneracy lifting of the mechanical mode frequencies is
observed, as shown in Fig. 3 (d1)-(d3), by sweeping the anisotropy parameter
$\eta$. Here, $\eta=0$ corresponds to an isotropic device and $\eta=1$ is the
full anisotropic case, according to the relation $c^{*}_{44}(\eta)=\eta
c_{44}+(1-\eta)((c_{11}+c_{22})/2)$ for the GaAs stiffness tensor component.

Figure 4:  Temperature dependence of the $3.1\text{\,}\mathrm{GHz}$ mode of
the $700\text{\,}\mathrm{nm}$ ring width sample: PSD of the mechanical
resonance measured for the lower (light blue - (a1)) and higher (light red -
(a2)) cavity input power (Lorentzian fits in orange); (b) measured mechanical
frequency shift ($\Delta f$) as a function of the cavity power (blue) and
computed frequency shift resulting from the $w_{\text{ring}}$ thermal
expansion as a function of temperature (red). (c) Comparison of the
experimental mechanical Q-factor ($Q_{\text{m}}$) power dependence to the
total model of $Q_{\text{m}}$ (Akhiezer, TLS and optomechanical anti-damping)
and the calculated optomechanical backaction effect only (narrowing of the
mechanical linewidth with increasing power). Akhiezer and TLS individual
contributions are shown. Light blue and red strips correspond to the data of
the resonant modes shown in (a1) and (a2), respectively. (d) Nanomechanical
dissipation model of the bullseye resonator. Inset: calculated $Q_{\text{m}}$
compared to the experimental data of low and room temperature measurements.

The isotropic mechanical displacement profile shown in Fig. 3 (d5) ($\eta=0$)
corresponds to the prominent bar in Fig. 3 (d1). It demonstrates that only the
C-mode is optomechanically coupled to the second radial-order optical mode.
Indeed, despite the modification of the mode structure after the loss of
symmetry in the material, the measured mechanical mode, highlighted in shaded
orange in Fig. 3 (d4), can still be related to the C-mode (external ring
breathing mode) as can be seen in Fig. 3 (d5) ($\eta=1$), which corresponds to
the highest bar in (d3). The 3D modeling also shows that the mechanical
grating isolation is resilient to the anisotropy, but predicts a reduced
$g_{0}$ when compared to the 2D simulations with isotropic material
elasticity. Thus, when $\eta=1$, it foresees a $g_{0}/(2\pi)$ of up to
$53\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$
($33\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ due photo-elastic effect and
$20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ associated to the moving
boundaries contribution - see section S2, in the Supplementary Material),
which agrees reasonably well with the calibrated data presented in Fig. 3 (c).
Moreover, such result is still above the achievable $g_{0}$ for the
fundamental radial breathing mechanical mode in GaAs disks with the same
radius and thickness
($g_{0}^{\text{(disk)}}/(2\pi)\sim(\omega_{0}/R)x_{\text{zpf}}=$12\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$,
where $\omega_{0}$ is the angular optical frequency and $x_{\text{zpf}}$ is
the zero-point amplitude fluctuation of a 230 MHz mode), with the benefit of
having mechanical GHz frequencies not yet reported in any GaAs optomechanical
structure operating at telecom wavelengths.

In order to investigate the role of the mechanical grating in inhibiting
clamping losses, thermal channels of dissipation had to be suppressed.
Therefore, our samples were cooled down to the temperature of liquid nitrogen.
Figure 4 (a1) contains the measured mechanical C-mode of the
$700\text{\,}\mathrm{nm}$ $w_{\mathrm{ring}}$. At a temperature of $\sim$
$80\text{\,}\mathrm{K}$, a $Q_{\text{m}}$ of $4\times 10^{3}$ was measured,
which is an improvement of over three times in comparison to room temperature
data. To observe the behavior of the mechanical linewidth
($\Gamma_{\text{m}}$) as a function of the cavity input power, we performed a
laser power sweep - frequency tuned for maximum optomechanical transduction,
on the blue side of the optical resonance. As the power was increased, we
noticed a simultaneous increase in the mechanical linewidth and a red-shift,
$\Delta f=f-f_{0}$, where $f_{0}$ is the initial mechanical mode frequency and
$f$ is its frequency at a given optical input power, as shown in Fig. 4 (a2).

In Fig. 4 (b) we show that $\Delta f$ goes down a few $\text{\,}\mathrm{MHz}$
when approaching $1\text{\,}\mathrm{mW}$ of optical input power. This effect
is explained by the cavity heating, which causes the material to expand,
modifying the mode frequency Benevides  _et al._ (2018). We calculated this
shift and plotted it against the data by assuming that at very low power this
heating was negligible, i. e., $\Delta f=0$, and that the initial reading of
the temperature sensor was a good approximation for the cavity temperature.
Then, it was possible to estimate a linear relationship between temperature
and the input power to the resonator, which was done by comparing the measured
power induced frequency shift to the expected shift caused by thermal
expansion.

We also measured the decrease of $Q_{\text{m}}$ with power, as displayed in
Fig. 4 (c), that contains the data of Fig. 4 (a1) and (a2) in the extreme
input power values, highlighted in light blue (low power) and light red (high
power), respectively. In order to understand this behavior, we included other
loss mechanisms to the mechanical linewidth. When neglecting anchor losses,
the linewidth broadening of nanomechanical resonators is then in general
dominated by phonon-phonon interactions and scattering by defects. The former
can introduce losses via relaxation of thermal phonons, whereas the second
dissipates mechanical energy by coupling strain waves to two-level systems
(TLS) Mohanty  _et al._ (2002); Phillips (1987).

To estimate the linewidth of our sample, we calculated the anharmonic and the
TLS mechanical attenuation through the methods described in Ref. Vacher  _et
al._ (2005) (see S3 in the Supplementary Material). From
$300\text{\,}\mathrm{K}$ to $80\text{\,}\mathrm{K}$, our bullseye resonators
falls into the Akhiezer regime Alegre  _et al._ (2011),
$\omega_{m}\tau_{ph}\lesssim 1$, where $\tau_{ph}$ is the phonon relaxation
time for this channel and $\omega_{m}$ is the mechanical resonator angular
frequency. Thermoelastic losses obtained from FEM were found to be several
orders of magnitude lower than the Akhiezer damping and are neglected in this
analysis. The double-well potentials parameters of our TLS dissipation model
were obtained from Ref. Hamoumi  _et al._ (2018), where GaAs microdisks were
investigated; as such values are highly material dependent and do not change
significantly with small variation in geometry, they serve as a reasonable
approximation for our devices. The results are displayed in Fig. 4 (d), where
the inset includes a direct comparison of the measured mechanical quality
factors ($3.1\text{\,}\mathrm{GHz}$ mode) to the theory, which predicts that
$Q_{\text{m}}\sim 12\times 10^{3}$ at 20 K. It is important to notice that, at
very low temperatures, scattering by defects is the main dissipation
mechanism, thus predictions may not be so accurate in this regime.

The measurement of $Q_{\text{m}}$ as the input laser power was increased gives
a hint about the accuracy of our mechanical dissipation model, which correctly
predicts the $Q_{\text{m}}$ optical power dependence, as shown in Fig. 4 (c).
The total model not only accounted for the thermal sources of mechanical
dissipation (Akhiezer and TLS) but also included the linewidth modification
expected from optomechanical backaction. If the measured $Q_{\text{m}}$ was
dominated by optomechanical dynamics, an opposite trend should be observed,
with increasing (decreasing) Q-factor ($\Gamma_{\text{m}}$) for higher input
power. Therefore, we can see that optomechanical anti-damping is not playing a
significant role in the mechanical linewidth, indicating that it is dominated
by temperature-dependent dissipation. Moreover, the 3D calculation including
mechanical clamping and perfectly matched layers predicted mechanical
Q-factors between $10^{4}-10^{7}$ for the ring modes (Fig. 3 (d4)), which are
at least one order of magnitude higher than the measured values, thus
suggesting that our device is not limited by anchor losses.

## Conclusion

In summary, we designed a GaAs bullseye resonator with a phononic grating
operating between $3\text{\,}\mathrm{GHz}$ and $4\text{\,}\mathrm{GHz}$. The
fabricated devices show modes ranging between
$\sim$3.0\text{\,}\mathrm{GHz}$-$3.4\text{\,}\mathrm{GHz}$$ with
optomechanical coupling rates of up to $39\text{\,}\mathrm{kHz}$ and
mechanical quality factor of $4000\text{\,}$ when cooled down to the
temperature of liquid nitrogen, rendering an optomechanical cooperativity of
$\text{C}_{\text{om}}\sim 0.05$ at $80\text{\,}\mathrm{K}$
($0.8\text{\,}\mathrm{mW}$ of cavity input power). In order to harness the
full potential of the GaAs bullseye resonators, the regime of high
cooperativity must be accessible and, thus, higher optical and mechanical
Q-factors must be achieved. The former could be obtained by minimizing
roughness Benevides  _et al._ (2020) and surface absorption Guha  _et al._
(2017b) with electroresist thermal reflow and alumina passivation
($Q_{\text{opt}}~{}\sim~{}10^{5}-10^{6}$). Additionally, mechanical
dissipation is expected to be drastically reduced at lower temperatures Forsch
_et al._ (2020) as thermal anharmonic losses are reduced
($Q_{\text{m}}~{}\sim~{}14~{}\times~{}10^{3}$). Under the above conditions,
unitary $\text{C}_{\text{om}}$ is reachable with the same
$0.8\text{\,}\mathrm{mW}$ or less power. Incorporating III-V quantum emitters
to the bullseye cavity would also enable the study of active optomechanics,
opening a plethora of possibilities, including the creation of an alternative
approach for hybrid systems that couples single emitters to mechanical strain
Yeo  _et al._ (2014) and the realization of mechanically modulated light
sources Princepe  _et al._ (2018). Microwave-to-optical conversion, on the
other hand, could take advantage of higher mechanical frequencies enabled by
narrower $w_{\mathrm{ring}}$’s or exploring higher-order mechanical modes.
Finally, the bullseye design was shown to be robust across different material
platforms and could be extended to other semiconductor materials with lower
non-linear optical losses, such as GaP Ghorbel  _et al._ (2019); Stockill  _et
al._ (2019).

##  ACKNOWLEDGMENTS

The authors would like to acknowledge CCSNano-UNICAMP for providing the micro-
fabrication infrastructure and CMC Microsystems for providing access to MBE
epitaxy and the GaAs wafers. This work was supported by São Paulo Research
Foundation (FAPESP) through grants 2017/19770-1, 2016/18308-0, 2018/15580-6,
2018/15577-5, 2018/25339-4, Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior - Brasil (CAPES) (Finance Code 001), Financiadora de Estudos e
Projetos and the National Sciences and Engineering Research Council (NSERC) of
Canada.

##  SUPPLEMENTARY MATERIAL

See Supplementary Material for details on the fabrication of the devices (S1),
the characterization of the optical modes through the frequency dependence of
the free spectral range (S2) and modeling of the mechanical dissipation
mechanisms (S3).

##  DATA AVAILABILITY

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

  * Abbott  _et al._ (2016) B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, _et al._ , Physical Review Letters 116, 061102 (2016).
  * Chan  _et al._ (2011) J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, Nature 478, 89 (2011), arXiv:1106.3614 .
  * Hill  _et al._ (2012) J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, Nature Communications 3 (2012), 10.1038/ncomms2201, arXiv:1206.0704 .
  * O’Connell  _et al._ (2010) A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, Nature 464, 697 (2010).
  * Guha  _et al._ (2017a) B. Guha, S. Mariani, A. Lemaître, S. Combrié, G. Leo, and I. Favero, Optics Express 25, 24639 (2017a), arXiv:1710.00632 .
  * Morozov  _et al._ (2019) K. M. Morozov, K. A. Ivanov, E. I. Girshova, A. S. Sokolovskii, and A. P. Mikitchuk, in  _Journal of Physics: Conference Series_, Vol. 1410 (IOP Publishing, 2019) p. 012156.
  * Zou  _et al._ (2016) C. L. Zou, X. Han, L. Jiang, and H. X. Tang, Physical Review A 94, 13812 (2016), arXiv:1604.06027 .
  * Mitchell  _et al._ (2019) M. Mitchell, D. P. Lake, and P. E. Barclay, APL Photonics 4, 016101 (2019).
  * Balram  _et al._ (2014) K. C. Balram, M. Davanço, J. Y. Lim, J. D. Song, and K. Srinivasan, Optica 1, 414 (2014), arXiv:1409.6198 .
  * Guha  _et al._ (2017b) B. Guha, F. Marsault, F. Cadiz, L. Morgenroth, V. Ulin, V. Berkovitz, A. Lemaître, C. Gomez, A. Amo, S. Combrié, B. Gérard, G. Leo, and I. Favero, Optica 4, 218 (2017b).
  * Özdemir  _et al._ (2019) K. Özdemir, S. Rotter, F. Nori, and L. Yang, “Parity–time symmetry and exceptional points in photonics,”  (2019).
  * Wu  _et al._ (2020) M. Wu, E. Zeuthen, K. C. Balram, and K. Srinivasan, Physical Review Applied 13 (2020), 10.1103/PhysRevApplied.13.014027, arXiv:1907.04830 .
  * Baker  _et al._ (2014) C. Baker, W. Hease, D.-T. Nguyen, A. Andronico, S. Ducci, G. Leo, and I. Favero, Optics Express 22, 14072 (2014), arXiv:1403.4269 .
  * Ding  _et al._ (2010) L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, Physical review letters 105, 263903 (2010).
  * Ding  _et al._ (2011) L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, Applied Physics Letters 98, 113108 (2011).
  * Balram  _et al._ (2016) K. C. Balram, M. I. Davanço, J. D. Song, and K. Srinivasan, Nature Photonics 10, 346 (2016), arXiv:1508.01486 .
  * Forsch  _et al._ (2020) M. Forsch, R. Stockill, A. Wallucks, I. Marinković, C. Gärtner, R. A. Norte, F. van Otten, A. Fiore, K. Srinivasan, and S. Gröblacher, Nature Physics 16, 69 (2020), arXiv:1812.07588 .
  * Santos  _et al._ (2017) F. G. S. Santos, Y. A. V. Espinel, G. O. Luiz, R. S. Benevides, G. S. Wiederhecker, and T. P. Mayer Alegre, Optics Express 25, 508 (2017), arXiv:1605.06318 .
  * Johnson  _et al._ (2002) S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Physical review E 65, 066611 (2002).
  * Chan  _et al._ (2012) J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, Applied Physics Letters 101, 081115 (2012).
  * Benevides  _et al._ (2020) R. Benevides, M. Ménard, G. S. Wiederhecker, and T. P. Mayer Alegre, Optical Materials Express 10, 57 (2020), arXiv:1909.06881 .
  * Shaqfeh and Jurgensen (1989) E. S. G. Shaqfeh and C. W. Jurgensen, Journal of applied physics 66, 4664 (1989).
  * Bailey III  _et al._ (1995) A. D. Bailey III, M. C. M. Van de Sanden, J. A. Gregus, and R. A. Gottscho, Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures Processing, Measurement, and Phenomena 13, 92 (1995).
  * Fujii  _et al._ (2020) L. Fujii, M. Inga, J. H. Soares, Y. A. V. Espinel, T. P. Mayer Alegre, and G. S. Wiederhecker, Optics Letters 45, 3232 (2020), arXiv:2003.11625 .
  * (25) Python-based scipy.signal.savgol_filter with window_length set to 101 and polyorder set to 2 ($\textcopyright$ Copyright 2008-2020, The SciPy community - v1.5.3).
  * Gorodetsky  _et al._ (2010) M. L. Gorodetsky, A. Schliesser, G. Anetsberger, S. Deleglise, and T. J. Kippenberg, Optics Express 18, 23236 (2010).
  * Benevides  _et al._ (2018) R. Benevides, N. C. Carvalho, M. Ménard, N. C. Frateschi, G. S. Wiederhecker, and T. P. Alegre, in  _Optics InfoBase Conference Papers_, Vol. Part F123- (Optical Society of America, 2018) pp. W4D–4.
  * Mohanty  _et al._ (2002) P. Mohanty, D. A. Harrington, K. L. Ekinci, Y. T. Yang, M. J. Murphy, and M. L. Roukes, Physical Review B - Condensed Matter and Materials Physics 66, 854161 (2002).
  * Phillips (1987) W. A. Phillips, Reports on Progress in Physics 50, 1657 (1987).
  * Vacher  _et al._ (2005) R. Vacher, E. Courtens, and M. Foret, Physical Review B 72, 214205 (2005), arXiv:0505560 [cond-mat] .
  * Alegre  _et al._ (2011) T. P. M. Alegre, A. Safavi-Naeini, M. Winger, and O. Painter, Optics Express 19, 5658 (2011), arXiv:1012.2556 .
  * Hamoumi  _et al._ (2018) M. Hamoumi, P. E. Allain, W. Hease, E. Gil-Santos, L. Morgenroth, B. Gérard, A. Lemaître, G. Leo, and I. Favero, Physical Review Letters 120 (2018), 10.1103/PhysRevLett.120.223601, arXiv:1801.09078 .
  * Yeo  _et al._ (2014) I. Yeo, P. L. De Assis, A. Gloppe, E. Dupont-Ferrier, P. Verlot, N. S. Malik, E. Dupuy, J. Claudon, J. M. Gérard, A. Auffèves, G. Nogues, S. Seidelin, J. P. Poizat, O. Arcizet, and M. Richard, Nature Nanotechnology 9, 106 (2014), arXiv:1306.4209 .
  * Princepe  _et al._ (2018) D. Princepe, G. S. Wiederhecker, I. Favero, and N. C. Frateschi, IEEE Photonics Journal 10, 1 (2018).
  * Ghorbel  _et al._ (2019) I. Ghorbel, F. Swiadek, R. Zhu, D. Dolfi, G. Lehoucq, A. Martin, G. Moille, L. Morvan, R. Braive, S. Combrié, and A. De Rossi, APL Photonics 4 (2019), 10.1063/1.5121774, arXiv:1901.05922 .
  * Stockill  _et al._ (2019) R. Stockill, M. Forsch, G. Beaudoin, K. Pantzas, I. Sagnes, R. Braive, and S. Gröblacher, Physical Review Letters 123, 163602 (2019), arXiv:1909.07850 .


# Temperature dependent equilibration of spin orthogonal quantum Hall edge
modes

Tanmay Maiti<EMAIL_ADDRESS>Saha Institute of Nuclear Physics, HBNI,
1/AF Bidhannagar, Kolkata 700064, India  Pooja Agarwal  Saha Institute of
Nuclear Physics, HBNI, 1/AF Bidhannagar, Kolkata 700064, India  Suvankar
Purkait  Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Kolkata
700064, India  G J Sreejith  Indian Institute of Science Education and
Research, Pune 411008, India  Sourin Das  Department of Physical Sciences,
IISER Kolkata, Mohanpur, West Bengal 741246, India  Giorgio Biasiol  Istituto
Officina dei Materiali CNR, Laboratorio TASC, 34149 Trieste, Italy  Lucia
Sorba  NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza San
Silvestro 12, I-56127 Pisa, Italy  Biswajit Karmakar  Saha Institute of
Nuclear Physics, HBNI, 1/AF Bidhannagar, Kolkata 700064, India

###### Abstract

Conductance of the edge modes as well as conductance across the co-propagating
edge modes around the $\nu$ = 4/3, 5/3 and 2 quantum Hall states are measured
by individually exciting the modes. Temperature dependent equilibration rates
of the outer unity conductance edge mode are presented for different filling
fractions. We find that the equilibration rate of the outer unity conductance
mode at $\nu$ = 2 is higher and more temperature sensitive compared to the
mode at fractional filling 5/3 and 4/3. At lowest temperature, equilibration
length of the outer unity conductance mode tends to saturate with lowering
filling fraction $\nu$ by increasing magnetic field $B$. We speculate this
saturating nature of equilibration length is arising from an interplay of
Coulomb correlation and spin orthogonality.

##  I INTRODUCTION

Quantum Hall (QH) systems formed the first examples of topological insulators,
where a set of gapless, topologically protected edge modes carry the current
around a bulk region that is gapped due to an interplay of the applied
magnetic field and interaction. Robustness of the QH systems has allowed
extensive theoretical and experimental investigations of the detailed chiral
edge transport revealing rich physics arising along the one dimensional
boundary [1, 2, 3, 4, 5, 6]. Though the bulk QH state fixes the total charge
conductance along the boundary, the confinement potentials and electronic
interactions can reconstruct the edge modes affecting the details of the
current distribution[3, 7, 8, 9, 10, 11, 12, 13]. Robustness and coherence of
reconstructed edge modes [14, 15] can have implications in investigation of
quantum interferrometry [16, 17, 18], braiding statistics [19, 20, 21, 22],
and in QH circuit designs for quantum electronics applications [23, 24].
Weakly equilibrating fractional conductance modes have been realized around
$\nu=1$ [25, 26] as well as around $\nu=2/3$ [5, 27, 10, 28] states, where
larger equlibration lengths are achieved in the high magnetic field limit[29].
Characterization of the equilibration processes is thus a question of active
interest.

In this work, we focus on equilibration of the spin orthogonal edge modes
occuring around states at filling fractions higher than 1 namely at
$\nu=4/3,5/3$ and $2$. The $\nu=4/3$ state is fully spin polarized in Si/SiGe
heterostructures [30]. A spin polarization transition can occur in 2D hole
gas[31, 32, 33] as well as in 2D electron gas [34] embedded in GaAs/AlGaAs
heterostructures under tilted magnetic field induced excess Zeeman splitting.
The edge of the spin unpolarized $\nu=4/3$ QH state in GaAs/AlGaAs
heterostructure has two co-propagating spin orthogonal charge modes with
conductance $1$ and $1/3$, [35, 6] with equilibration lengths measured upto of
a few hundred micrometers. Edge structure of integer QH state at $\nu=2$ is
well understood,[36, 37, 38, 39] where two unity conductance spin orthogonal
co-propagating edge modes carry the current. Scattering between the spin
orthogonal edge modes can occur through spin flip processes assisted by the
dynamics of nuclear spins [40, 41, 42, 43, 44, 45]. I-V spectroscopy of the QH
states at filling fraction $\nu$ = 4/3,5/3, 2 and 3 shows that conductance
across the copropogating edge modes is enhanced above a inter-edge mode
threshold bias $V_{th}$ and the threshold voltage $V_{th}$ increases with
decreasing filling fraction $\nu$ [46]. Above the threshold voltage,
conductance across the edge modes reaches the equilibration value for integer
filling fraction $\nu=2$ and 3. In contrast, for fractional filling $\nu=5/3$
and 4/3 the conductance across the modes is below the corresponding
equilibrium value above the threshold voltage. Above the threshold voltage
$V_{th}$ inter-edge equilibration become faster because of the flat band
scenario [46]. As a consequence, it is difficult to estimate the equilibration
length/rate of the edge modes at high imbalance. In this paper, we intend to
study the equilibration between the spin orthogonal edge modes in the linear
transport regime at filling fraction $\nu$ = 4/3, 5/3 and 2.

In this article, we study conductance of the edge modes and conductance across
the co-propagating edge modes around the $\nu=4/3,\,5/3$ and $2$ QH states by
individually exciting the modes. We measure the length scales over which the
outer unity conducatnce mode (present in all the QH states under study)
equilibrates with the inner modes of conductance 1, 2/3 and 1/3 for the bulk
filling $\nu=2,5/3$ and 4/3, respectively. We find that the equilibration rate
of the outer unity conductance mode at $\nu$ = 2 is higher and more
temperature sensitive than that of fractional filling 5/3 and 4/3. At lowest
temperature the equilibration length of the outer unity conductance mode shows
saturating behaviour with increasing magnetic field $B$ i.e. decreasing bulk
filling fraction $\nu$. The observation of significantly larger equilibration
length in the case of filling 4/3 and 5/3 relative to the $\nu$ = 2 case
potentially indicating a slower equilibration between an integer and
fractional mode.

##  II DEVICE DESCRIPTION AND MEASUREMENT PROCEDURE

Experiments are carried out on a modulation doped GaAs/AlGaAs heterostructure,
in which the two-dimensional electron gas (2DEG) resides at the GaAs/AlGaAs
heterointerface located $100\ {\rm nm}$ below the top surface. Figure 1(a)
shows the device structure, where eight Ohmic contacts
$S1,\,S2,\,D1,\,D2,\,D3,\,D4,\,D5,\,D6$ are defined for current injection and
detection and four top gates $g1,\,g2,\,g3,\,g4$ are used to tune the filling
fraction in the mesa underneath the gates. A customized pre-amplifier SR555
[29] is deployed at $S2$ contact to measure output current and for application
of ac voltage excitation simultaneously. The device is mounted in a dilution
refrigerator equipped with a 14 T superconducting magnet at a base temperature
7 mK, where electron temperature achieved is about 30 mK. Carriers are
injected by illumination with a GaAs LED in the sample at $3$ K and these
carriers are persistent at low temperatures [47]. The carrier density and
mobility of the sample become $n\sim 2.27\times 10^{11}\ {\rm cm^{-2}}$ and
$\mu\sim 4\times 10^{6}\ {\rm cm^{2}/Vs}$, respectively after light
illumination. Two terminal magnetoresistance (2TMR) [Fig. 1(b)] is measured to
find the location of the QH states namely $\nu=4/3,\,5/3,\,2$ along the
magnetic field axis at the base temperature. For our transport experiments at
filling $\nu=4/3,\,5/3,\,2$ the magnetic fields are set at $7.1,\,5.69,\,4.9$
T (as indicated in Fig. 1(b)), respectively. The top gates $g3$ and $g4$ are
kept at pinch-off condition by applying a negative voltage bias of
$Vg3=Vg4=-0.450$ V through out the experiment.

Figure 1:  (a) Optical image of the device used for transport measurement
where relevant edge mods for bulk filling fractions $\nu=4/3,5/3,2$ are shown.
Outer red channel is unity conductance mode and blue channel represents inner
mode. Edge modes are separately contacted by setting the filling fraction
beneath the two top gates at unity $\nu_{1}=\nu_{2}=1$. (b) Two terminal
magneto-resistance trace taken at base temperature. Inset shows $g$1 gate
characteristics at filling fraction $\nu=4/3$ keeping $g2$ at pinch off
condition, where blue line represents transmittance conductance
($S1\rightarrow S2$), red line represents reflected conductance
($S1\rightarrow D2$) and magenta line represents total conductance.

We set the magnetic field at $7.1$ T to carry out top gate $g1$
characteristics at a bulk filling fraction $\nu=4/3$ keeping the top gate $g2$
in pinch-off condition. Transmitted and reflected conductance are measured
between $S1\rightarrow S2$ and $S1\rightarrow D2$, respectively by varying the
top gate voltage $Vg1$ as shown in inset of Fig. 1(b).The transmitted
conductance (blue curve) shows a plateau at unit conductance, confirming the
formation of an integer QH state of filling $\nu_{1}=1$ beneath the top gate
$g1$ within a gate voltage range of $-0.168$ to $-0.099$ V. Red curve
represents reflected conductance and magenta curve shows total conductance.
The total conductance remains constant at $4/3$ throughout the gate voltage
$Vg1$ scan confirming conservation of the current. A similar gate
characteristic is also observed for the top gate $g2$ at a bulk filling
$\nu=4/3$. From the gate characteristics, we can determine the gate voltage
range in which an integer QH state of filling unity is formed beneath the
gates $g1$ and $g2$. Similarly, gate voltage ranges of top gate $g1$ and $g2$
are found to set the filling fraction unity $\nu_{1}=\nu_{2}=1$ beneath the
gates at bulk filling fraction $\nu=5/3$ and $2$.

##  III EXPERIMENTAL RESULTS AND ANALYSIS

Upon setting $\nu_{1}=\nu_{2}=1$ beneath the gates, we separately contact the
edge modes for selective current injection and detection [39, 48] in the
$\nu=4/3,\,5/3,\,2$ QH states. In this device, outer unity conductance mode
(red line) connects $S2$ to $D1$ and $S1$ is connected to $D2$ by the inner
edge mode (blue line) with conductance $1/3,2/3,1$ for filling fraction
$\nu=4/3,\,5/3,\,2$, respectively [3, 6] as shown in Fig. 1(a). To understand
the equilibration between these edge modes, we perform the temperature
dependent transport measurements from $30$ mK to only $500$ mK preventing
thermal degradation of the sample - loss of carriers and reduction of
mobility. During temperature dependence experiment, $S1$ and $S2$ are excited
with $V_{S1}=25.8\mu V$ ($17$ Hz) and $V_{S2}=25.8\mu$V ($26$ Hz) and current
at $D1$ and $D2$ are measured with the two frequency windows and back-
scattered current ($17$ Hz) is measured at $D6$. With this excitation voltage,
we work in the linear transport regime. We define the measured quantities
${}^{\nu}G_{S\rightarrow D}$ which denote two terminal conductance (TTC)
between a source $S$ and a drain $D$ at bulk filling fraction $\nu$, while
filling fractions $\nu_{1}=\nu_{2}=1$ is maintained under the gates. The TTC
${}^{\nu}G_{S\rightarrow D}$ can also be expressed in terms of transmission
probabilities of the co-propagating edge modes (see Appendix A).

Temperature dependent TTCs are presented in Fig. 2. At filling fraction
$\nu=4/3$, TTCs between contacts $S2\rightarrow D1$, and $S2\rightarrow D2$
are shown in Fig. 2(a) by the red and cyan curves, respectively. With
increasing temperature upto $500$ mK, we see that TTC
${}^{4/3}G_{S2\rightarrow D1}$ stays fixed to $1$ and zero conductance is
measured for TTC ${}^{4/3}G_{S2\rightarrow D2}$, indicating no measurable
equilibration between the outer unity conductance mode and inner $1/3$
conductance mode upto $500$ mK over a propagation length of $l=125\,\mu$m.
Similar temperature dependent experiments for filling fraction $\nu=5/3$ and
$2$ shows that with increasing temperature, ${}^{5/3}G_{S2\rightarrow D1}$ and
${}^{2}G_{S2\rightarrow D1}$ decrease [red curve of Fig. 2(b) and 2(c)] while
${}^{5/3}G_{S2\rightarrow D2}$ and ${}^{2}G_{S2\rightarrow D2}$ increase [cyan
curve of Fig. 2(b) and 2(c)]. Compensating nature of the conductances confirm
the conservation of current. At higher temperature, the outer unity
conductance mode equilibrates with inner 2/3 and 1 conductance mode for bulk
filling fraction $\nu=5/3$ and $2$, respectively. Suppression of inter-mode
scattering over propagation length of $\l=125\,\mu$m at lowest temperature is
evident in Figs. 2(a-c).

Figure 2:  Two terminal conductances as a function of $1/T$, when S2 is
excited with $25.8~{}\mu$V at $26$ Hz for bulk filling fractions (a)
$\nu=4/3$, (b) $\nu=5/3$ and (c) $\nu=2$. Two terminal conductances as a
function of $1/T$ when $S1$ is excited with $25.8~{}\mu$V at $17$ Hz for (d)
$\nu=4/3$, (e) $\nu=5/3$ and (f) $\nu=2$.

The TTC ${}^{4/3}G_{S1\rightarrow D2}$ decreases with increasing temperature
as shown in Fig. 2(d) (blue curve) while ${}^{4/3}G_{S1\rightarrow D1}$ (olive
curve) remains fixed to zero due to absence of equilibration upto $500$ mK as
also seen for ${}^{4/3}G_{S2\rightarrow D2}$ in Fig. 2(a). The TTC for the
current reaching at $D6$ from $S1$ increases with increasing temperature
[dashed orange curve of Fig. 2(d)], indicating back-scattering of the inner
$1/3$ edge mode into the oppositely moving edge channel across the bulk [Fig.
1(a)]. For filling fraction $\nu=5/3$ in the bulk, TTC
${}^{5/3}G_{S1\rightarrow D2}$ (${}^{5/3}G_{S1\rightarrow D1}$) decreases
(increases) with increasing temperature as shown in Fig. 2(e) by blue (olive)
curve and the back-scattered current at $D6$ also increases [dashed orange
curve of Fig. 2(e)] with increasing temperature. The observation indicates
simultaneous equilibration of co-propagating modes and backscattering of the
2/3 mode with increasing temperature. At a bulk filling fraction $\nu=2$,
decrease of ${}^{2}G_{S1\rightarrow D2}$ is fully compensated by increase of
${}^{2}G_{S1\rightarrow D1}$, and no current reaches at $D6$ [Fig. 2(f)],
which confirms incompressibility of QH state at $\nu=2$ within the range of
temperature variations. At filling fraction $\nu=4/3$ and $5/3$, the sub-
Kelvin bulk gaps (see Appendix B) originate from the coulomb interaction,
while the bulk gap at filling fraction $\nu=2$ is the Landau gap
$\hbar\omega_{c}$ ($\omega_{c}$ \- cyclotron frequency). Hence, break-down of
the QH state at $\nu=2$ is not observed, as the cyclotron gap
$\hbar\omega_{c}$ is much larger than the maximum applied thermal excitation
$kT$ ($T=500$ mK).

To quantify the temperature dependence of equilibration process between the
edge modes, we define equilibration length $l_{r}$ of the outer integer edge
mode, where $1/l_{r}$ is the rate of charge transfer from outer to inner mode.
Corresponding TTC of the outer mode connecting $S2$ to $D1$ can be written as
[37, 48, 29, 49, 50]

${}^{2}G_{S2\rightarrow D1}=\frac{1}{2}[1+e^{-2l/l_{r}}],$ (1)

for $\nu=2$, where the pre-factors are fixed by the boundary conditions - no
scattering into inner modes at $l=0$ and full equilibration at $l\gg l_{r}$.
Similarly, the TTC at filling fraction $\nu=4/3$ and $5/3$ can be written as
[29]

${}^{4/3}G_{S2\rightarrow D1}=\frac{1}{4}[3+e^{-4l/l_{r}}]$ (2)

and

${}^{5/3}G_{S2\rightarrow D1}=\frac{1}{5}[3+2e^{-5l/2l_{r}}],$ (3)

respectively. The above equations are utilized to estimate the value of
equilibration rate $1/l_{r}$ from the measured TTC between the contacts $S2$
to $D1$ in Figs. 2(a), 2(b) and 2(c) (red curves). It is assumed that the
small amount of current that backscatters from the inner mode do not alter the
above relations.

Equilibration rate $1/l_{r}$ of the outer integer mode is plotted as a
function of $1/T$ in Fig. 3(a) for filling fraction $\nu=4/3,\,5/3$ and $2$.
At filling fraction $\nu=1.45$ the QH state is compressible where the inner
mode does not exist; however, an effective equilibration rate can be estimated
for the outer mode using a similar exponential formulation and this shows an
intermediate value as shown in Fig. 3(a). Equilibration rate for all the
filling fractions increase monotonically with increasing temperature. The
equilibration rates for fractional filling $\nu$ = 4/3 and 5/3 are similar at
lowest temperature, but distinctly lower from that of integer filling $\nu$ =
2. To understand this observation at lowest temperature, the equilibration
length of the outer unity conductance mode is estimated $6.6\pm 0.5$ mm,
$6.3\pm 0.5$ mm, $5.1\pm 0.5$ mm and $2.3\pm 0.5$ mm for filling fraction
$\nu=4/3,1.45,5/3$ and $2$, respectively and is plotted with magnetic field in
Fig. 3(b). The equilibration length $l_{r}$ tends to saturate with increasing
magnetic field (i.e. lowering filling fraction $\nu$) at lowest temperature
and with increasing temperature the saturating trend of $l_{r}$ disappears
[Fig. 3(b)].

Figure 3:  (a) Equilibration rate of the outer unity conductance mode versus
$1/T$ at $\nu=4/3,1.45,5/3$ and $2$. (b) Plot of equilibration length versus
$B$ for different temperatures.

##  IV DISCUSION

Edge mode structures of the QH states along smooth boundaries arise from
formation of a sequence of dominant incompressible states as the electron
density changes from the bulk value to zero at the boundary [3, 51]. Here, two
spatially separated edge modes are formed by incompressibility due to spin
gap, and spin orthogonality condition prevents equilibration between these
modes at lowest temperature. Therefore equillibration process requires a spin
flip mechanism that is mediated by dynamic nuclear polarization [40, 41, 42,
43, 44].

Spin polarization of the QH system increases with lowering bulk filling
fraction $\nu$ below 2 and the QH system become fully spin polarized at
filling $\nu=1$. This change of the spin polarization from spin un-polarized
($\nu=2$) to spin polarized ($\nu=1$) results in exchange enhancement of
$g$-factor with lowering filling fraction [52, 53]. The enhanced spin gap is
reflected in the observation of increase of threshold voltage $V_{th}$ for
inter-mode transport with lowering filling fraction $\nu=2$ to $4/3$ [46]. The
exchange enhanced spin gap induces spatial separation of the opposite spin
modes. As a consequence of the increase in spatial separation, the
equilibration length $l_{r}$ should increase with increasing magnetic field
without showing saturation [29]. In contrast, the measured equilibration
length $l_{r}$ of the outer unity conductance mode tends to saturate with
increasing magnetic field $B$ as shown in Fig. 3(b) at lowest temperature.

In addition to the spin orthogonality, equilibration process may also be
suppressed due to differing character of the electron like quasiparticles in
the outer unity conductance mode and the anyon like quasiparticles in the
correlated inner mode of $\nu=5/3$ and 4/3 [54]. This could explain the
significantly larger equilibration rates in the integer QH state at $\nu=2$ as
compared to the fractional QH states at asymptotically low temperatures. A
quantitative modeling of the observed saturating nature of equilibration
length $l_{r}$ is left for further investigations.

##  V CONCLUSION

In conclusion, we study equilibration between pairs of co-propagating edge
modes of conductances $1$ on the outer side and $\nu-1$ on the inner side for
bulk filling fraction $\nu=2,5/3$ and 4/3. We observe saturating nature of
equilibration length $l_{r}$ of the outer unity conductance mode with
increasing magnetic field at the lowest temperature. We argue that the
significantly larger equilibration length for filling $\nu=5/3$ and 4/3
compared to filling $\nu=2$ is arising from suppression of equilibration due
to differing character of the electron like quasiparticles in the outer unity
conductance mode and the anyon like quasiparticles in the correlated inner
mode.

ACKNOWLEDGEMENTS

GJS acknowledges support from DST-SERB grant ECR/2018/001781. Part of the
discussions in this and related previous works were facilitated by the
International Centre for Theoretical Sciences program on Edge dynamics in
topological phases (ICTS/edytop2019/06). SD acknowledges support from DST-SERB
Grant No. MTR/2019/001043 and ARF grant from IISER Kolkata.

APPENDIX A

In our experiments two terminal conductance (TTC) ${}^{\nu}G_{S\rightarrow D}$
is measured in a multi-terminal device as shown in Fig. 1(a). The measured TTC
${}^{\nu}G_{S\rightarrow D}$ can be expressed in terms of transmission
probabilities of the co-propagating edge modes connecting the source and
detector. For this representation the outer mode and inner mode are labeled 1
and 2, respectively, where the outer mode has conductance unity and the inner
mode has conductance 1, 2/3 and 1/3 for bulk filling fraction $\nu=2,5/3$ and
4/3, respectively. Following Büttiker approach [55, 3, 29] the TTC for bulk
filling fraction $\nu$ can be expressed as

${}^{\nu}G_{\rm S\rightarrow D}=\sum_{i;j}{g_{i}}T_{ij}$ (4)

where $T_{ij}$ represents the probability of transmission from mode $i$
connected to the source $S$ into mode $j$ connected to the detector $D$. The
summation $i(j)$ is taken over all the modes connected to the source
(detector). $g_{i}$ is conductance of the $i$-th mode connected to the source
$S$. Transmission probability $T_{ij}$ between the respective modes depend on
the charge equilibration over the co-propagation length $l=125~{}\mu$m. In
this presentation we exclude back scattered current reaching to detector $D6$.

Now we focus for the bulk filling fraction $\nu=4/3$, where the TTCs can be
expressed as

${}^{4/3}G_{S2\rightarrow D1}=T_{11},$ (5) ${}^{4/3}G_{S2\rightarrow
D2}=T_{12},$ (6) ${}^{4/3}G_{S1\rightarrow D2}=\frac{1}{3}T_{22},$ (7)
${}^{4/3}G_{S1\rightarrow D1}=\frac{1}{3}T_{21}.$ (8)

At lowest temperature, the two co-propagating modes do not equilibrate, hence
the values of transmission probabilities become
$T_{11}=T_{22}=1,T_{12}=T_{21}=0$. With increasing temperature, the two co-
propagating mode starts equilibration into each other. If they fully
equilibrate, the transmission probabilities become
$T_{11}=3/4,T_{12}=1/4,T_{22}=1/4,T_{21}=3/4$. In our experiment temperature
dependence of the TTCs are plotted in Fig. 2 (a) and Fig. 2 (d) for filling
fraction $\nu=4/3$. Equilibration rate $1/l_{r}$ of the outer unity
conductance mode for $\nu=4/3$ is calculated using Eq. 2 from the measured
value of $T_{11}$ [Eq. 5]. Similarly, TTCs at other filling fraction $\nu=5/3$
and 2 can also be expressed.

APPENDIX B

In general, insulating states at the bulk gap melt at higher temperatures and
start conducting, resulting in back scattering of the edge modes as depicted
in Fig. 1 (a) (orange dashed lines). In our device, back scattered current
reaches the contact $D6$ when source $S1$ is excited and corresponding TTC
${}^{\nu}G_{S1\rightarrow D6}$ increases with increasing temperature as shown
in Fig. 2(d) for $\nu=4/3$ and in Fig. 2(e) for $\nu=5/3$ (orange curves). In
a Hall bar device, the characteristic bulk gap is estimated from the Arrhenius
plot of the finite longitudinal resistivity at elevated temperatures. A
similar activation behavior is seen in our temperature dependent measurement
[Fig. 2(d) and 2(e)], in which TTC from $S1$ to $D6$ can be expressed as [56,
57, 58]

${}^{\nu}G_{S1\rightarrow D6}\propto e^{-\frac{\Delta_{\nu}}{2T}},$ (9)

where $\Delta_{\nu}$ is the bulk energy gap (also call pair creation energy)
at filling fraction $\nu$.

Figure 4:  Arrhenius plot of the back-scattered conductance at filling $\nu$ =
4/3 and 5/3.

The back-scattered conductance ($S1$ to $D6$) for filling fraction $\nu=4/3$
and $5/3$ are presented in [Fig. 4]. The high temperature part of the data is
fitted linearly to estimate the bulk gap and is found to be
$\Delta_{4/3}=0.652\pm 0.06$K and $\Delta_{5/3}=0.534\pm 0.06$K for filling
fraction $\nu=4/3$ and $5/3$, respectively. The measured bulk gaps are
consistent with the previous measurements [56, 57].

## References

  * Chang [2003] A. M. Chang, Chiral luttinger liquids at the fractional quantum hall edge, Rev. Mod. Phys. 75, 1449 (2003).
  * MacDonald [1990] A. H. MacDonald, Edge states in the fractional-quantum-hall-effect regime, Phys. Rev. Lett. 64, 220 (1990).
  * Beenakker [1990] C. W. J. Beenakker, Edge channels for the fractional quantum hall effect, Phys. Rev. Lett. 64, 216 (1990).
  * Meir [1994] Y. Meir, Composite edge states in the $\nu$=2/3 fractional quantum hall regime, Phys. Rev. Lett. 72, 2624 (1994).
  * Sabo  _et al._ [2017] R. Sabo, I. Gurman, A. Rosenblatt, F. Lafont, D. Banitt, J. Park, M. Heiblum, Y. Gefen, V. Umansky, and D. Mahalu, Edge reconstruction in fractional quantum hall states, Nature Physics 13, 491 (2017).
  * Lin  _et al._ [2019] C. Lin, R. Eguchi, M. Hashisaka, T. Akiho, K. Muraki, and T. Fujisawa, Charge equilibration in integer and fractional quantum hall edge channels in a generalized hall-bar device, Phys. Rev. B 99, 195304 (2019).
  * Chamon and Wen [1994] C. d. C. Chamon and X. G. Wen, Sharp and smooth boundaries of quantum hall liquids, Phys. Rev. B 49, 8227 (1994).
  * Kirczenow and Johnson [1995] G. Kirczenow and B. L. Johnson, Composite fermions, edge currents, and the fractional quantum hall effect, Phys. Rev. B 51, 17579 (1995).
  * Wan  _et al._ [2002] X. Wan, K. Yang, and E. H. Rezayi, Reconstruction of fractional quantum hall edges, Phys. Rev. Lett. 88, 056802 (2002).
  * Wang  _et al._ [2013] J. Wang, Y. Meir, and Y. Gefen, Edge reconstruction in the $\nu\mathbf{=}2/3$ fractional quantum hall state, Phys. Rev. Lett. 111, 246803 (2013).
  * Wan  _et al._ [2003] X. Wan, E. H. Rezayi, and K. Yang, Edge reconstruction in the fractional quantum hall regime, Phys. Rev. B 68, 125307 (2003).
  * Joglekar  _et al._ [2003] Y. N. Joglekar, H. K. Nguyen, and G. Murthy, Edge reconstructions in fractional quantum hall systems, Phys. Rev. B 68, 035332 (2003).
  * Yang [2003] K. Yang, Field theoretical description of quantum hall edge reconstruction, Phys. Rev. Lett. 91, 036802 (2003).
  * Ronen  _et al._ [2018] Y. Ronen, Y. Cohen, D. Banitt, M. Heiblum, and V. Umansky, Robust integer and fractional helical modes in the quantum hall effect, Nature Physics 14, 411 (2018).
  * Nakamura  _et al._ [2019] J. Nakamura, S. Fallahi, H. Sahasrabudhe, R. Rahman, S. Liang, G. C. Gardner, and M. J. Manfra, Aharonov–bohm interference of fractional quantum hall edge modes, Nature Physics 15, 563 (2019).
  * McClure  _et al._ [2012] D. T. McClure, W. Chang, C. M. Marcus, L. N. Pfeiffer, and K. W. West, Fabry-perot interferometry with fractional charges, Phys. Rev. Lett. 108, 256804 (2012).
  * Ofek  _et al._ [2010] N. Ofek, A. Bid, M. Heiblum, A. Stern, V. Umansky, and D. Mahalu, Role of interactions in an electronic fabry–perot interferometer operating in the quantum hall effect regime, Proceedings of the National Academy of Sciences 107, 5276 (2010).
  * Park  _et al._ [2015] J. Park, Y. Gefen, and H.-S. Sim, Topological dephasing in the $\nu=2/3$ fractional quantum hall regime, Phys. Rev. B 92, 245437 (2015).
  * Nakamura  _et al._ [2020] J. Nakamura, S. Liang, G. Gardner, and M. Manfra, Direct observation of anyonic braiding statistics, Nature Physics 16, 931 (2020).
  * Halperin [1984] B. I. Halperin, Statistics of quasiparticles and the hierarchy of fractional quantized hall states, Phys. Rev. Lett. 52, 1583 (1984).
  * Kim [2006] E.-A. Kim, Aharanov-bohm interference and fractional statistics in a quantum hall interferometer, Phys. Rev. Lett. 97, 216404 (2006).
  * Arovas  _et al._ [1984] D. Arovas, J. R. Schrieffer, and F. Wilczek, Fractional statistics and the quantum hall effect, Phys. Rev. Lett. 53, 722 (1984).
  * Ji  _et al._ [2003] Y. Ji, Y. Chung, D. Sprinzak, M. Heiblum, D. Mahalu, and H. Shtrikman, An electronic mach–zehnder interferometer, Nature 422, 415 (2003).
  * Neder  _et al._ [2007] I. Neder, N. Ofek, Y. Chung, M. Heiblum, D. Mahalu, and V. Umansky, Interference between two indistinguishable electrons from independent sources, Nature 448, 333 (2007).
  * Kouwenhoven  _et al._ [1990] L. P. Kouwenhoven, B. J. van Wees, N. C. van der Vaart, C. J. P. M. Harmans, C. E. Timmering, and C. T. Foxon, Selective population and detection of edge channels in the fractional quantum hall regime, Phys. Rev. Lett. 64, 685 (1990).
  * Bhattacharyya  _et al._ [2019] R. Bhattacharyya, M. Banerjee, M. Heiblum, D. Mahalu, and V. Umansky, Melting of interference in the fractional quantum hall effect: Appearance of neutral modes, Phys. Rev. Lett. 122, 246801 (2019).
  * Nosiglia  _et al._ [2018] C. Nosiglia, J. Park, B. Rosenow, and Y. Gefen, Incoherent transport on the $\nu=2/3$ quantum hall edge, Phys. Rev. B 98, 115408 (2018).
  * Gross  _et al._ [2012] Y. Gross, M. Dolev, M. Heiblum, V. Umansky, and D. Mahalu, Upstream neutral modes in the fractional quantum hall effect regime: Heat waves or coherent dipoles, Phys. Rev. Lett. 108, 226801 (2012).
  * Maiti  _et al._ [2020] T. Maiti, P. Agarwal, S. Purkait, G. J. Sreejith, S. Das, G. Biasiol, L. Sorba, and B. Karmakar, Magnetic-field-dependent equilibration of fractional quantum hall edge modes, Phys. Rev. Lett. 125, 076802 (2020).
  * Lai  _et al._ [2004] K. Lai, W. Pan, D. C. Tsui, and Y.-H. Xie, Fractional quantum hall effect at $\nu=\frac{2}{3}$ and $\frac{4}{3}$ in strained si quantum wells, Phys. Rev. B 69, 125337 (2004).
  * Davies  _et al._ [1991] A. G. Davies, R. Newbury, M. Pepper, J. E. F. Frost, D. A. Ritchie, and G. A. C. Jones, Fractional quantum hall effect in high-mobility two-dimensional hole gases in tilted magnetic fields, Phys. Rev. B 44, 13128 (1991).
  * Rodgers  _et al._ [1993] P. J. Rodgers, B. L. Gallagher, M. Henini, and G. Hill, Observation of a spin polarization phase transition of the 4/3 fractional quantum hall state in a high-mobility 2d hole system, Journal of Physics: Condensed Matter 5, L565 (1993).
  * Muraki and Hirayama [1998] K. Muraki and Y. Hirayama, Re-entrant behavior of the $\nu=4/3$ fractional quantum hall effect in a front-and-back-gated 2d hole gas, Physica B: Condensed Matter 256-258, 86 (1998).
  * Clark  _et al._ [1989] R. G. Clark, S. R. Haynes, A. M. Suckling, J. R. Mallett, P. A. Wright, J. J. Harris, and C. T. Foxon, Spin configurations and quasiparticle fractional charge of fractional-quantum-hall-effect ground states in the n=0 landau level, Phys. Rev. Lett. 62, 1536 (1989).
  * Altimiras  _et al._ [2012] C. Altimiras, H. le Sueur, U. Gennser, A. Anthore, A. Cavanna, D. Mailly, and F. Pierre, Chargeless heat transport in the fractional quantum hall regime, Phys. Rev. Lett. 109, 026803 (2012).
  * Halperin [1982] B. I. Halperin, Quantized hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential, Phys. Rev. B 25, 2185 (1982).
  * Müller  _et al._ [1992] G. Müller, D. Weiss, A. V. Khaetskii, K. von Klitzing, S. Koch, H. Nickel, W. Schlapp, and R. Lösch, Equilibration length of electrons in spin-polarized edge channels, Phys. Rev. B 45, 3932 (1992).
  * Kang  _et al._ [2000] W. Kang, H. Stormer, L. Pfeiffer, K. Baldwin, and K. West, Tunnelling between the edges of two lateral quantum hall systems, Nature 403, 59 (2000).
  * Karmakar  _et al._ [2011] B. Karmakar, D. Venturelli, L. Chirolli, F. Taddei, V. Giovannetti, R. Fazio, S. Roddaro, G. Biasiol, L. Sorba, V. Pellegrini, and F. Beltram, Controlled coupling of spin-resolved quantum hall edge states, Phys. Rev. Lett. 107, 236804 (2011).
  * Machida  _et al._ [2002] T. Machida, S. Ishizuka, T. Yamazaki, S. Komiyama, K. Muraki, and Y. Hirayama, Spin polarization of fractional quantum hall edge channels studied by dynamic nuclear polarization, Phys. Rev. B 65, 233304 (2002).
  * Yang  _et al._ [2017] K. Yang, K. Nagase, Y. Hirayama, T. D. Mishima, M. B. Santos, and H. Liu, Role of chiral quantum hall edge states in nuclear spin polarization, Nature communications 8, 1 (2017).
  * Kane  _et al._ [1992] B. E. Kane, L. N. Pfeiffer, and K. W. West, Evidence for an electric-field-induced phase transition in a spin-polarized two-dimensional electron gas, Phys. Rev. B 46, 7264 (1992).
  * Wald  _et al._ [1994] K. R. Wald, L. P. Kouwenhoven, P. L. McEuen, N. C. van der Vaart, and C. T. Foxon, Local dynamic nuclear polarization using quantum point contacts, Phys. Rev. Lett. 73, 1011 (1994).
  * Dixon  _et al._ [1997] D. C. Dixon, K. R. Wald, P. L. McEuen, and M. R. Melloch, Dynamic nuclear polarization at the edge of a two-dimensional electron gas, Phys. Rev. B 56, 4743 (1997).
  * Deviatov  _et al._ [2004] E. V. Deviatov, A. Würtz, A. Lorke, M. Yu. Melnikov, V. T. Dolgopolov, D. Reuter, and A. D. Wieck, Two relaxation mechanisms observed in transport between spin-split edge states at high imbalance, Phys. Rev. B 69, 115330 (2004).
  * Deviatov  _et al._ [2008] E. V. Deviatov, A. Lorke, and W. Wegscheider, Manifestation of a complex edge excitation structure in the quantum hall regime at high fractional filling factors, Phys. Rev. B 78, 035310 (2008).
  * Nathan [1986] M. I. Nathan, Persistent photoconductivity in algaas/gaas modulation doped layers and field effect transistors: A review, Solid-State Electronics 29, 167 (1986).
  * Würtz  _et al._ [2002] A. Würtz, R. Wildfeuer, A. Lorke, E. V. Deviatov, and V. T. Dolgopolov, Separately contacted edge states: A spectroscopic tool for the investigation of the quantum hall effect, Phys. Rev. B 65, 075303 (2002).
  * Takagaki  _et al._ [1994] Y. Takagaki, K. J. Friedland, J. Herfort, H. Kostial, and K. Ploog, Inter-edge-state scattering in the spin-polarized quantum hall regime with current injection into inner states, Phys. Rev. B 50, 4456 (1994).
  * Paradiso  _et al._ [2011] N. Paradiso, S. Heun, S. Roddaro, D. Venturelli, F. Taddei, V. Giovannetti, R. Fazio, G. Biasiol, L. Sorba, and F. Beltram, Spatially resolved analysis of edge-channel equilibration in quantum hall circuits, Phys. Rev. B 83, 155305 (2011).
  * Chklovskii  _et al._ [1992] D. B. Chklovskii, B. I. Shklovskii, and L. I. Glazman, Electrostatics of edge channels, Phys. Rev. B 46, 4026 (1992).
  * Dolgopolov  _et al._ [1997] V. T. Dolgopolov, A. A. Shashkin, A. V. Aristov, D. Schmerek, W. Hansen, J. P. Kotthaus, and M. Holland, Direct measurements of the spin gap in the two-dimensional electron gas of algaas-gaas heterojunctions, Phys. Rev. Lett. 79, 729 (1997).
  * Werner and Oswald [2020] D. Werner and J. Oswald, Size scaling of the exchange interaction in the quantum hall effect regime, Phys. Rev. B 102, 235305 (2020).
  * Zülicke  _et al._ [2003] U. Zülicke, J. J. Palacios, and A. H. MacDonald, Fractional-quantum-hall edge electrons and fermi statistics, Phys. Rev. B 67, 045303 (2003).
  * Büttiker [1988] M. Büttiker, Absence of backscattering in the quantum hall effect in multiprobe conductors, Phys. Rev. B 38, 9375 (1988).
  * Boebinger  _et al._ [1987] G. S. Boebinger, H. L. Stormer, D. C. Tsui, A. M. Chang, J. C. M. Hwang, A. Y. Cho, C. W. Tu, and G. Weimann, Activation energies and localization in the fractional quantum hall effect, Phys. Rev. B 36, 7919 (1987).
  * Boebinger  _et al._ [1985] G. S. Boebinger, A. M. Chang, H. L. Stormer, and D. C. Tsui, Magnetic field dependence of activation energies in the fractional quantum hall effect, Phys. Rev. Lett. 55, 1606 (1985).
  * Willett  _et al._ [1988] R. L. Willett, H. L. Stormer, D. C. Tsui, A. C. Gossard, and J. H. English, Quantitative experimental test for the theoretical gap energies in the fractional quantum hall effect, Phys. Rev. B 37, 8476 (1988).