# Strong magnetoresistance in a graphene Corbino disk at low magnetic fields

Masahiro Kamada  Low Temperature Laboratory, Department of Applied Physics,
Aalto University School of Science, P.O. Box 15100, 00076 Aalto, Finland
Vanessa Gall  Institute for Quantum Materials and Technologies, Karlsruhe
Institute of Technology, 76021 Karlsruhe, Germany Institut für Theorie der
Kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe,
Germany Jayanta Sarkar  Low Temperature Laboratory, Department of Applied
Physics, Aalto University School of Science, P.O. Box 15100, 00076 Aalto,
Finland  Manohar Kumar Low Temperature Laboratory, Department of Applied
Physics, Aalto University School of Science, P.O. Box 15100, 00076 Aalto,
Finland  QTF Centre of Excellence, Department of Applied Physics, Aalto
University, P.O. Box 15100, 00076 Aalto, Finland  Antti Laitinen  Low
Temperature Laboratory, Department of Applied Physics, Aalto University School
of Science, P.O. Box 15100, 00076 Aalto, Finland  Igor Gornyi  Institute for
Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76021
Karlsruhe, Germany Institut für Theorie der Kondensierten Materie, Karlsruhe
Institute of Technology, 76128 Karlsruhe, Germany Ioffe Institute, 194021 St.
Petersburg, Russia  Pertti Hakonen<EMAIL_ADDRESS>Low Temperature
Laboratory, Department of Applied Physics, Aalto University School of Science,
P.O. Box 15100, 00076 Aalto, Finland  QTF Centre of Excellence, Department of
Applied Physics, Aalto University, P.O. Box 15100, 00076 Aalto, Finland

###### Abstract

We have measured magnetoresistance of suspended graphene in the Corbino
geometry at magnetic fields up to $B=0.15$ T, i.e., in a regime uninfluenced
by Shubnikov-de Haas oscillations. The low-temperature relative
magnetotoresistance $[R(B)-R(0)]/R(0)$ amounts to $4000B^{2}\%$ at the Dirac
point ($B$ in Tesla), with a quite weak temperature dependence below 30 K. A
decrease in the relative magnetoresistance by a factor of two is found when
charge carrier density is increased to $|n|\simeq 3\times 10^{-10}$ cm-2. The
gate dependence of the magnetoresistance allows us to characterize the role of
scattering on long-range (Coulomb impurities, ripples) and short-range
potential, as well as to separate the bulk resistance from the contact one.
Furthermore, we find a shift in the position of the charge neutrality point
with increasing magnetic field, which suggests that magnetic field changes the
screening of Coulomb impurities around the Dirac point. The current noise of
our device amounts to $10^{-23}$ A2/$\sqrt{\textrm{Hz}}$ at 1 kHz at 4 K,
which corresponds to a magnetic field sensitivity of 60
nT/$\sqrt{\textrm{Hz}}$ in a background field of 0.15 T.

##  I Introduction

Besides extraordinary physical characteristics, graphene exhibits superb
electrical transport properties [1, 2]. Charge carrier conduction in monolayer
graphene can display ballistic behavior over several microns, though the mean
free path is often limited by Coulomb scattering and short-range scatterers
[3]. Typically Coulomb scatterers, embedded in the substrate or caused by
fabrication residues, dominate the transport and short-range scattering
becomes important only at large carrier densities. Using freely suspended
graphene flakes and current annealing [4], however, impurity scattering can be
minimized and intrinsic properties of graphene can be reached. Many of the
basic transport properties of graphene have been revealed using suspended
devices. Suspended graphene in Corbino disk geometry, for example, has turned
out to be valuable in sensitive investigations of fractional quantum Hall
states in graphene [5].

Magnetoconductance is a powerful tool for studying basic quantum transport in
monolayer graphene [6, 7, 2, 8]. Typically, magnetoconductance at low magnetic
fields and low temperatures is governed by impurity scattering which leads to
quantum corrections to the conductance and universal conductance fluctuation
contributions [9]. For classical magnetoresistance, both linear and quadratic
behavior is expected according to effective medium theory [10, 11]. Owing to
strong demand for magnetic field sensors based on magnetotransport, various
ways to generate large magnetoresistance in graphene have been developed in
monolayer [8, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] and multilayer graphene
[22, 23]. In this work, we demonstrate that intrinsic behavior of suspended
graphene in the Corbino ring geometry, already as such, yields a huge
magnetoresistance. This magnetoresistance can serve as an efficient tool for
sample characterization as well as it can be used for sensing purposes. Our
work demonstrates a magnetic field sensitivity on the order of 60
nT/$\sqrt{\textrm{Hz}}$ at 4 K in a background field of 0.15 T.

In the Hall bar geometry, both the transverse and longitudinal bulk
conductivities – $\sigma_{xy}(B)$ and $\sigma_{xx}(B)$, respectively –
determine the resistivity $\rho_{xx}(B)$. As a result, $\rho_{xx}(B)$ turns
out to be independent of the applied magnetic field $B$ in the simplest one-
band model. The Corbino-ring measurement setting for magnetoresistance is
special already for this simplest case, because the Hall conductivity
$\sigma_{xy}$ (or the Hall voltage) drops out from the resistivity. The latter
is then obtained just as the inverse of the longitudinal conductivity
$\rho_{xx}(B)=1/\sigma_{xx}(B)$. In a way, a Corbino disk is equivalent to an
infinitely wide sample, in which the effect of the side walls can be
neglected. The magnetoresistance for a generic anisotropic Corbino samples is
calculated in Ref. [24]; in the isotropic case, relevant to our setup, the
resistance of the Corbino sample is expressed through to the bulk resistivity
as

$R(B)=\frac{1}{2\pi}\ \rho_{xx}(B)\ \ln\frac{r_{\text{out}}}{r_{\text{in}}}.$
(1)

Here $r_{\text{in}}$ and $r_{\text{out}}$ are, respectively, the inner and
outer radii of the disk. The logarithmic geometrical factor in Eq. (1)
reflects the total current conservation in the Corbino disk.

Bulk magnetoresistance measured in disordered graphene in the Hall-bar
geometry may display complex magnetic field dependence [25, 26, 27, 28]; for
example, indications of $\sqrt{B}$ dependence at small fields have been
reported [29, 30]. Our results, on the contrary, display a strong parabolic
($B^{2}$) magnetoresistance for arbitrary disorder, while only small
corrections to the $B^{2}$ dependence are found for magnetic fields up to 0.1
T.

##  II Sample fabrication and characterization

Our graphene samples were fabricated using a technique based on lift off
resist (LOR) sacrificial layer [31]; details of the employed process can be
found in Ref. [5]. The current annealing of the samples at low temperature
before measurements guaranteed a high maximal field-effect mobility
$\mu_{\text{FE}}^{\text{max}}\simeq 1-2\times 10^{5}$ cm2/Vs. Most of the data
was measured on a Corbino disk with inner and outer radii of 0.9 and 2.25
$\mu$m, respectively. A false-color scanning electron microscope picture of a
Corbino sample is displayed in Fig. 1, together with its connections to the
employed measurement devices. The gate capacitance $C_{g}=1.5\times 10^{-5}$
F/m2 was obtained using the Landau level fan diagram [5].

Figure 1: Schematics of our measurement configuration and a scanning electron
microscope image of a suspended graphene Corbino disk (green part in the
center) with 4.5 $\mu$m diameter outer Au contact and 1.8 $\mu$m diameter
inner Au contact.

In our experiments, we employed standard voltage-biased measurements and
recorded current through the sample over the bias range of $1-13$ mV. These
rather large voltages were chosen in order to measure the low-frequency noise
spectra simultaneously and the resistance noise of our samples was in the
range of $\delta R/R\lesssim 2\times 10^{-5}$ at 1 Hz. The current was
amplified using a transimpedance amplifier at gain $10^{5}$ V/A. Bias-T
components facilitated insertion of rf-signals to the sample. For details of
the measurement system, see Ref. [32]. In the measurements, both positive and
negative bias voltages $V$ were used. A weak $V$ dependence was removed by
extrapolating data at $V<0$ and at $V>0$ down to zero bias: the two
extrapolations differed less than 3%.

First, we characterized the gate voltage dependence of the resistance
$R(V_{g})$ of our sample at $B=0$ (see lowest curves in Fig. 2) at two
temperatures, $T=4$ K and $T=27$ K. The offset of the Dirac point from zero
amounted to $V_{g}^{D}(0)\simeq 0.2$ V at $T=27$ K, and the corresponding
residual charge density was found to be $n_{0}\simeq 8\times 10^{9}$ cm-2. For
$T=4$ K, the zero-$B$ offset of the Dirac point had the opposite sign:
$V_{g}^{D}(0)\simeq-0.2$ V. The temperature dependence of the offset (and, in
particular, the change of its sign) can be related to different concentration
of (quasi)resonant adsorbed atoms, see below.

Figure 2:  Resistance vs. $V_{g}$ at various magnetic fields. (a) $4$ K for
magnetic fields from 0 to 0.15 T with 0.005 T step (from bottom to top); the
broken curves are cubic spline interpolations to the data. (b) $27$ K for
magnetic fields from 0 to 0.15 T with 0.01 T step (from bottom to top); the
broken curves are cubic spline interpolations.

Figure 2 includes also our data at small magnetic fields, $B<0.15$ T. For both
temperature values, the resistance grows strongly with $B$ and,
simultaneously, there appears a shift of the maximum value to slightly higher
gate voltage $V_{g}^{D}(B)$. The data points were joined using cubic spline
fits in order to make observed changes with $B$ easier to distinguish.
According to the spline fits, the shift of the Dirac point for $T=27$ K
amounts to $\Delta V_{g}=V_{g}^{D}(B)-V_{g}^{D}(0)=5B$ V, where $B$ is
measured in Tesla. Even stronger shift is seen for $T=4$ K.

The observed shift of the Dirac point with increasing $B$ can be attributed to
changes in screening of charged impurities in a magnetic field near charge
neutrality (see Appendix A for more details). In this case, the total charge
density induced on the membrane is less at the Dirac point at finite $B$,
which would indicate generation of more positive charge on graphene by the
screening. Thus the screening should take place by positive carriers and the
impurities are negatively charged. In addition, there was a slow shift of the
Dirac point position towards positive $V_{g}$ over time (on the order of 0.2 V
in one month). It is worth noting that the effect of magnetic field on
screening is suppressed at higher densities and higher temperatures. This is
in line with the stronger shift of the resistance maximum at $T=4$ K.

One more peculiarity seen in the resistance curves plotted over the gate
voltage is a feature close to $-2$ V for both temperatures shown in Fig. 2.
The resistance around this voltage is somewhat enhanced compared to the
resistance away from this voltage. There is no comparable feature at the
electron side of the resistance curve and the feature is stronger for lower
temperatures. We attribute this feature to a broadened resonance level
associated with adsorbed local impurities. When the chemical potential is
moved by the gate voltage into the vicinity of this quasi-resonance, the
scattering amplitude for such impurities is enhanced, leading to shorter
transport scattering time and, hence, to the increase in resistance. At the
same time, the broadening of this resonance is sufficiently strong to avoid
truly resonant scattering (as, e.g., in the case of vacancies); in contrast to
infinitely strong impurities (vacancies), the position of the quasi-resonance
is shifted away from the Dirac point. Away from the resonant energy, these
impurities produce weak short-ranged disorder.

For higher temperatures, some adsorbed dirt is thermally removed form the
sample, leading to a less pronounced feature. This suggests that the role of
scattering off short-range disorder at higher temperatures could be decreased.
Below, based on the analysis of the magnetoresistance curves, we will discuss
this issue in more detail. The dependence of the concentration of local quasi-
resonant impurities on temperature can also explain the $T$ dependence of
zero-$B$ shift of the Dirac point mentioned above. Indeed, at higher
temperature ($T=27$ K in Fig. 2), the shift of the chemical potential is
smaller, which is consistent with the above picture of lower concentration of
adsorbed impurities.

In Sec. IV, we deduce charge carrier mobility from the measured geometric
magnetoresistance. For comparison, we display in Fig. 3 the field-effect
mobility defined by $\mu_{\text{FE}}=e^{-1}d\sigma_{xx}/dn$, obtained from
measurements of differential resistance $R(V_{g})$ at the end of the
experiments. These data measured at $T=4$ K indicate that, for our sample at a
tiny bias voltage, the maximum mobility for holes $\mu_{\text{FE}}\simeq 13$
m2/Vs is clearly larger than that for electrons $\mu_{\text{FE}}\simeq 7$
m2/Vs. However, at the employed bias voltages $1\dots 10$ mV, the influence of
the pn interface at positive gate voltages appears to be reduced and
$\mu_{\text{FE}}$ for electrons and holes becomes almost equal.

Figure 3:  Field effect mobility $\mu_{\text{FE}}$ at $T=4$ K determined from
the measured differential resistance $dV/dI$ at $B=0$. The Dirac point shift
$V_{g}^{D}(0)$ has been subtracted off from the gate voltage before
calculating the charge carrier density $n$. The dip in $\mu_{\text{FE}}$
around $n=0$, indicated by a grey shadow, corresponds to the density range
governed by disorder broadening of the Dirac point. The extend of this range
is consistent with the value $n_{*}$ given in Table 1

##  III Theoretical background

Before analyzing the obtained magnetoresistance data, we present in this
Section the basic facts about disorder-dominated transport in graphene (for
the hydrodynamic—collision-dominated—transport, higher temperatures are
typically required than those in our experiment, $50$ K $<T<150$ K, while
phonons become important at yet higher $T$ [33]). Distinct from conventional
electron gases in 2D semiconductor heterostructures, graphene displays a
linear energy dispersion relation of the carriers,
$\varepsilon_{\mathrm{k}}=\pm v\hbar k$. This leads to the linear-in-energy
density of states in clean graphene:

$\displaystyle\nu_{0}(\varepsilon)=\frac{N|\varepsilon|}{2\pi
v_{F}^{2}\hbar^{2}},$ (2)

where $v_{F}=10^{6}~{}\text{m/s}$ is the Fermi velocity, $\varepsilon$ is the
energy counted from the Dirac point, and $N=4$ is the degeneracy due to spin
and valley degrees of freedom. Following from this density of states, the
charge density of carriers is given by

$\displaystyle n=N\frac{\varepsilon_{F}^{2}}{4\pi\hbar^{2}v_{F}^{2}},$ (3)

where $\varepsilon_{F}$ is the Fermi energy. A consequence of the linear
dispersion relation is that the cyclotron frequency $\omega_{c}$ becomes
energy-dependent [2] :

$\displaystyle\omega_{c}(\varepsilon)=\frac{eB}{m_{c}(\varepsilon)}=\frac{\hbar}{m_{c}(\varepsilon)\ell_{B}^{2}},$
(4)

where $B$ is the magnetic field and
$m_{c}(\varepsilon)={\varepsilon}/{v_{F}^{2}}$ is the cyclotron mass, also
dependent on the energy, and where we have defined a magnetic length by
$\ell_{B}=\sqrt{\hbar/eB}$.

The linear dispersion relation also influences the scattering and relaxation
rates of charge carriers. We assume that the major contributions to resistance
arise from short-range scattering (s) and Coulomb scattering (C) and adopt
this mixed-disorder model [34] to describe the magnetoresistance. For the
interpretation of the effect of charged scatterers, one should keep in mind,
that graphene is not perfectly flat, but has ripples [35, 36, 37]. The
scattering off ripples is similar to that for charged impurities [33], so that
their contribution is effectively included in our treatment of Coulomb
scattering.

We first address scattering on the short-range impurity potential. We limit
our discussion to not too high carrier densities for which the length scale
$d$ for random potential variations is smaller than the carriers wave-length
and larger than the interatomic spacing $a$, i.e., $a\ll d\ll\lambda$. For
such disorder the intervalley scattering can be disregarded and the quantum
$\tau_{q}$ and transport $\tau_{\text{tr}}$ scattering times can be estimated
using Fermi’s golden rule [38, 2]:

$\tau_{{q}}^{s}(\varepsilon)=\frac{\hbar\gamma_{s}}{|\varepsilon|}\>,\;\;\;\;\tau_{\text{tr}}^{s}(\varepsilon)=2\tau_{{q}}^{s}(\varepsilon)\>,$
(5)

where

$\gamma_{s}=\frac{2\hbar^{2}v_{F}^{2}}{n_{\text{imp}}^{s}U_{0}^{2}},$ (6)

$n_{\text{imp}}^{s}$ is the concentration of short-range impurities, and
$U_{0}$ denotes the magnitude of the impurity potential. In what follows, we
will characterize the strength of short-range disorder in the samples by the
parameter $\gamma_{s}$ which is energy independent. The difference between
$\tau_{\text{tr}}$ and $\tau_{q}$ is caused by the weak scattering anisotropy
which originates from the spinor nature of the wave functions.

The scattering times for charged impurities can be brought to similar form
(see Appendix B), although the effective parameter $\gamma_{C}$ is then, in
general, a function of the energy and the Fermi energy, as well as temperature
and magnetic field (through the corresponding dependence of screening of
Coulomb impurities by charged carriers). Below, we mainly consider
sufficiently low temperatures and magnetic fields, keeping only the energy
dependence of $\gamma_{C}$:

$\tau_{{q}}^{C}(\varepsilon)=\frac{\hbar\gamma_{C}^{\prime}(\varepsilon,\varepsilon_{F})}{|\varepsilon|}\>,\;\;\;\;\tau_{\text{tr}}^{C}(\varepsilon)=2\frac{\hbar\gamma_{C}(\varepsilon,\varepsilon_{F})}{|\varepsilon|}\>,$
(7)

Here $\gamma_{C}(\varepsilon,\varepsilon_{F})$ and
$\gamma_{C}^{\prime}(\varepsilon,\varepsilon_{F})$ are functions of the
effective coupling (graphene “fine-structure constant”) $\alpha=e^{2}/(\hbar
v_{F}\epsilon_{\infty})$, with $\epsilon_{\infty}$ the background dielectric
constant, see Appendix B. In the absence of a screening environment and
neglecting the renormalization of velocity by Coulomb interaction, the nominal
value of this constant is $\alpha_{0}=2.2$. However, both the screening and
renormalization [2] effects reduce this value. For intermediate values of
$\alpha$, the relation $\tau_{\text{tr}}^{C}(\varepsilon)\approx
2\tau_{{q}}^{C}(\varepsilon)$, similar to Eq. (5), holds, but it is no longer
exact.

The conductivity in zero magnetic field in the presence of only short-range
scatterers is given by the Drude formula:

$\sigma_{xx}^{D}=\sigma_{0}=\frac{2e^{2}\gamma_{s}}{\pi\hbar}.$ (8)

By comparing this with the typical conductivity of high-quality graphene
samples, we observe that $\gamma_{s}$ should be of the order of unity if
impurities in high-mobility samples, such as ours, were short-ranged (this
estimate corresponds to measured quantum scattering time at $V_{g}=10$ V and
scaled to our measurement regime).

An important parameter for the magnetoresistance is the product
$\omega_{{c}}\tau_{q}^{s}$ which describes the broadening of Landau levels.
Since for short-range scatterers both $\omega_{{c}}$ and $\tau_{q}^{s}$ depend
on energy [see Eqs.(4) and (5)], the parameter

$x=\omega_{{c}}\tau_{q}^{s}=\frac{\gamma_{s}\hbar^{2}v_{F}^{2}}{\varepsilon^{2}\ell_{B}^{2}},$
(9)

can be either small or large, depending on the energy $\varepsilon$ [28]. For
scattering on Coulomb impurities, the quantum scattering time decreases
linearly with energy $\varepsilon\propto\sqrt{n}$, which means that the
parameter $x=\omega_{{c}}\tau_{q}^{C}$ becomes energy independent. The
parameter $x$ determines the dependence of the density of states
$\nu(\varepsilon)$ of disordered graphene on magnetic field. Since the
transport scattering time $\tau_{\text{tr}}$ has the same energy dependence as
$\tau_{q}$, the same parameter $x$ governs the quasiclassical bending of
particle trajectories in magnetic field.

The general result for the longitudinal conductivity
$\sigma_{xx}(\varepsilon)$ is given by Eq. (4.13) of Ref. [38]. Introducing
the relative density of states
$\tilde{\nu}(\varepsilon)=\nu(\varepsilon)/\nu_{0}(\varepsilon)$, where
$\nu_{0}(\varepsilon)$ is the zero-field density of states, we write the
conductivity kernel (conductivity of particles at energy $\varepsilon$) in
terms of $\tilde{\nu}(\varepsilon)$ as follows:

$\sigma_{xx}(\varepsilon)=\sigma_{0}\frac{\tilde{\nu}(\varepsilon)^{2}}{\tilde{\nu}(\varepsilon)^{2}+[\omega_{\mathrm{c}}(\varepsilon)\tau_{\text{tr}}(\varepsilon)]^{2}},$
(10)

where

$\displaystyle\sigma_{0}$
$\displaystyle=\frac{e^{2}v^{2}}{2}\tau_{\text{tr}}(\varepsilon_{F})\nu(\varepsilon_{F})\equiv\frac{e^{2}\gamma
N}{2\pi\hbar}.$ (11)

Here, we have introduced the dimensionless disorder strength $\gamma$ that has
a meaning of a dimensionless conductance per spin per valley. For the short-
range disorder, $\gamma=\gamma^{s}$. In the presence of both short-range and
Coulomb scatterers, the total transport time is determined by

$\displaystyle\frac{1}{\tau_{\text{tr}}(\varepsilon)}=\frac{1}{\tau_{\text{tr}}^{s}(\varepsilon)}+\frac{1}{\tau_{\text{tr}}^{C}(\varepsilon)},$
(12)

and $\gamma$ is related to the total transport scattering time, as given by
Eq. (11).

The finite-temperature conductivity is given by the kernel (10) integrated
with the derivative of the Fermi function $n_{F}(\varepsilon)$:

$\displaystyle\sigma_{xx}=\int_{-\infty}^{\infty}\mathrm{d}\varepsilon\left(-\frac{\partial
n_{F}(\varepsilon)}{\partial\varepsilon}\right)\sigma_{xx}(\varepsilon)$ (13)

At zero temperature, the derivative gives the delta-function and the
conductivity reduces to Eq. (10) with $\varepsilon\to\varepsilon_{F}$. At zero
magnetic field it is given by $\sigma_{0}$ from Eq. (11). The temperature
dependence of the Drude conductivity arises from the energy dependence in the
kernel (10) when the thermal broadening of the delta-function is taken into
account. At low temperatures, $k_{B}T\ll\varepsilon_{F}$, the finite-$T$
corrections to the zero-$T$ result are small, and the measured conductivity is
given approximately by $\sigma_{xx}\approx\sigma_{xx}(\varepsilon_{F},T=0)$.
Under these conditions, if $\tilde{\nu}_{0}$ is independent of the magnetic
field, the conductivity in a finite magnetic field can be written in the
conventional Drude form:

$\displaystyle\sigma_{xx}(B)=\frac{en\mu_{0}}{1+(\mu_{0}B)^{2}},$ (14)

where $\mu_{0}$ is the mobility at $B=0$, i.e,

$\displaystyle\mu_{0}=\frac{\sigma_{xx}(B=0)}{ne}.$ (15)

As can be seen from the calculation of $\tilde{\nu}_{0}(\varepsilon)$ in Refs.
[39, 28], corrections to the density of states arising from the finite
magnetic field can be neglected, as long as $x\ll 1$, which we will show is
true over a large range of data in the present experiment.

Comparison with Eq. (10) shows that
$\omega_{c}(\varepsilon)\tau_{\text{tr}}(\varepsilon)/\tilde{\nu}_{0}(\varepsilon)$
corresponds to $\mu_{0}B$. This means that the Drude conductivity is given by

$\displaystyle\sigma_{xx}\approx\frac{e^{2}\gamma
N}{2\pi\hbar}\frac{1}{1+\left(\frac{2e\gamma}{\pi\hbar n}\right)^{2}B^{2}},$
(16)

which has finite temperature corrections that are detailed in the Appendix C.
There are also temperature-dependent quantum corrections to the Drude
conductivity, in particular, those arising from the electron-electron
interaction (EEI), as discussed in Appendix C.1.

For both short-range and Coulomb impurities, Eq. (10) then yields a parabolic
magnetoresistance in the Corbino geometry. The Drude resistivity in the
Corbino geometry takes a simple form

$\displaystyle\rho_{xx}(B)=\frac{1}{\sigma_{0}}\left[1+(\mu_{0}B)^{2}\right].$
(17)

According to the Mathiessen rule, the inverse mobility can be written as a sum
of the contributions of different momentum-relaxing scattering processes

$\displaystyle\mu_{0}^{-1}$ $\displaystyle=\mu_{C}^{-1}+\mu_{s}^{-1},$ (18)

which yields in the zero-$T$ limit:

$\displaystyle\frac{1}{\mu_{0}}$ $\displaystyle=\frac{\pi\hbar
n}{2e\gamma}=\frac{\pi\hbar
n}{2e}\left(\frac{1}{\gamma_{s}}+\frac{1}{\gamma_{C}(\varepsilon=\varepsilon_{F},\varepsilon_{F})}\right),$
(19) $\displaystyle\gamma_{C}$
$\displaystyle(\varepsilon=\varepsilon_{F},\varepsilon_{F})=\frac{n}{c(\alpha)n_{\text{imp}}^{C}},$
(20)

with $c(\alpha)$ defined in Appendix B. One sees that the contribution of
Coulomb scatterers to the inverse mobility is density independent. On the
other hand, the mobility governed by short-range impurities decreases with
charge carrier density as $\mu_{s}\propto 1/n$. Thus, the density dependence
of the total mobility allows one to characterize the role of short-range and
Coulomb impurities in transport.

Below a certain chemical potential or the corresponding density $n_{*}$,
disorder-induced broadening smears the single-particles energy and the density
of states saturates. This energy scale is given by the self-consistent
equation for $\varepsilon$:

$\displaystyle\frac{\hbar}{\tau_{q}(\varepsilon_{*})}\sim\varepsilon_{*}$ (21)

For the mixed disorder model with $\gamma_{s}\gg 1$, we get for the
corresponding density

$\displaystyle n_{*}\sim d(\alpha)n_{\text{imp}}^{C},$ (22)

where $d(\alpha)$ is given in Appendix B. The value of $n_{*}$ depends on the
density of charged impurities, fine-structure constant $\alpha$. We model this
saturation effect by performing the replacement
$n\rightarrow\sqrt{n^{2}+n_{*}^{2}}$, which effectively interpolates between
$n$ at high densities and $n^{*}$ at the neutrality point, in all formulas,
when used for plotting or fitting. In order to keep the notation clear, we do
not explicitly write down this replacement in the main text. Since this
replacement is an approximate interpolation, it describes the behavior of the
density of states (and other observables) at $n\sim n_{*}$ only qualitatively,
see Appendix D for details. Nevertheless, this simple interpolation function
allow us to confidently extract the system parameters, when the range of
densities $n\gg n_{*}$ is included in the fit.

##  IV Analysis of data and Results

The relative magnetoresistance $\Delta R(B)/R(0)=R(B)/R(0)-1$ of our sample at
$B<0.15$ T is illustrated in Fig. 4 which depicts the relative resistance
$R(B)/R(0)$ as a function of $B^{2}$ measured at $T=4$ K (Fig. 4a) and at
$T=27$ K (Fig. 4b). In both data sets, the magnetoresistance is found to be
the strongest at the Dirac point, which is in agreement with the weakest
effect of scattering when $|n|$ is smallest. Both data sets are influenced by
the growing shift of the Dirac point $\Delta V_{g}$ as $B$ increases.

The strength of the measured magnetoresistance depends only weakly on
temperature up to 27 K. However, when comparing the data at 4 K and 27 K, one
observes that the $B^{2}$ dependence is followed better at 27 K than at 4 K in
small magnetic fields. Qualitatively, this could be a signature of increased
role of electron-electron scattering [27, 28] and macroscopic inhomogeneities
[11, 30, 40]. In our suspended graphene sample such inhomogeneities can be due
to static ripples.

Figure 4: (a) Scaled resistance $R/R(0)$ vs. $B^{2}$ at $T=4$ K measured at
various gate voltage values between $-3\dots+3$ V ($0.8\times 10^{9}$ cm-2
$<|n|<3\times 10^{10}$ cm-2). The Dirac point corresponds to $V_{g}\simeq 1$
V, the data at which is denoted by black symbols in the figure. The solid
curves are guides for the eye, emphasizing an overall parabolic
magnetoresistance and slight deviations from parabolicity. (b) Scaled
resistance $R/R(0)$ vs. $B^{2}$ at $T=27$ K measured at various gate voltage
values between $-3\dots+3$ V. The magnetoresistance at low fields grows faster
at 27 K than at 4 K.

The nature of scattering does not appear to play a large role in the measured
magnetoresistance. The parabolic field dependence is followed for both Coulomb
and short-range impurities in the range of parameters covered: $|n|\sim
0.8-3\times 10^{10}$ cm-2 and $T=4-27$ K. In general, the $B^{2}$ dependence
at small magnetic fields is more closely followed in the 27 K data. The 4 K
data displays deviations from $B^{2}$ behavior at $B<30$ mT, which may be a
sign of coherent behavior and quantum interference effects, either regular
weak localization type or Corbino-geometry related as predicted for graphene
in Ref. [41]. At the largest magnetic fields around $0.1-0.15$ T, small
deviations from $B^{2}$ dependence become obvious, in particular near the
Dirac point. One can interpret this deviation as the onset of the Shubnikov -
de Haas (SdH) oscillation regime in the sample [42] that corresponds to $x\sim
1$.

On top of this one has to consider additional contribution $R_{\text{cont}}$
to the measured resistance related to the contact effects. These contributions
are the resistance of the metal-graphene contacts and the interface resistance
of the contact-doped graphene region. The former contribution is a microscopic
material property, which we take to be constant. The latter, discussed in Ref.
[43] is of the type of the $pn$-junction resistance. This contribution to the
total resistance depends on the density of charge carriers in the bulk of the
sample and is the main cause for the usually observed electron-hole asymmetry
in transport measurements. In low magnetic fields the cyclotron radius is
larger than the geometrical length scales characterizing the contact region
and, hence, the overall contact resistance should not show a pronounced
magnetic-field dependence.

The parabolic magnetoresistance is associated with the bulk contribution,
whereas the total resistance includes the contact resistance:
$R=R_{\text{bulk}}+R_{\text{cont}}$, where $R_{\text{bulk}}$ describes the
disorder-induced bulk resistance and $R_{\text{cont}}$ the contact
contribution. Since $R_{\text{cont}}$ depends on the gate voltage, the
normalized magnetoresistance shown in Fig. 4 is not particularly convenient
for extracting the density dependence of the mobility. Indeed, the value of
$R(B=0)$ in the denominator of the scaled magnetoresistance is not equal to
$R_{\text{bulk}}(B=0)$ in front of the $B$-dependent term coming from Eq.
(17), so that the coefficient in front of the $B^{2}$ term in the scaled
magnetoresistance is not equal to $\mu_{0}^{2}$.

To overcome this complication caused by the contact resistance, we have
employed a fitting function of the form

$R(B)-R(0)=MB^{2},$ (23)

for the total resistance, where we have a single fitting parameter $M$ fully
determined by the properties of the bulk of the sample. According to Eq. (17)
we have $M=R_{\text{bulk}}(0)\mu_{0}^{2}$, where $R_{\text{bulk}}(0)$
describes bulk resistance at zero field and $\mu_{0}$ is the mobility. We
recall that $R_{\text{bulk}}(0)$ is different from the measured $R(0)$ because
the latter includes the contact contribution.

Since magnetoresistance is related to mobility, the data can be employed to
derive information on the impurity scattering in our sample. From the obtained
fitting parameter $M$ which is given by

$\displaystyle M$
$\displaystyle=R_{\text{bulk}}(0)\mu_{0}^{2}=\frac{\gamma}{\pi^{2}\hbar
n^{2}}\ln\frac{r_{\text{out}}}{r_{\text{in}}}$
$\displaystyle=\frac{1}{\pi^{2}\hbar
n^{2}}\left(\frac{1}{\gamma_{s}}+\frac{c(\alpha)n_{\text{imp}}^{c}}{n}\right)^{-1}\ln\frac{r_{\text{out}}}{r_{\text{in}}}.$
(24)

In order to include the disorder-induced saturation of the density of states,
we replace $n$ with $\sqrt{n^{2}+n_{*}^{2}}$ in the fitting function, and then
extract $\gamma_{s}$, $n_{*}$, and the effective concentration of Coulomb
impurities $c(\alpha)n_{\text{imp}}^{C}$. The values we extract from these
fits are shown in Table 1.

|  $T=4$ K |  $T=27$ K
---|---|---
$\gamma_{s}$ | 9.6 | 38.2
$n_{*}$ [m-2] | 1.7$\times 10^{14}$ | 1.8$\times 10^{14}$
$c(\alpha)n_{\text{imp}}^{C}$ [m-2] | 1.1$\times 10^{14}$ | 0.88$\times 10^{14}$
Table 1: Parameters extracted from the fit of the data. These parameters are
used in all following plots.

Notably, the energy corresponding to $n_{*}$ is $\varepsilon_{*}\approx 9$
meV, which is larger than the energies corresponding to 4 K and 27 K, which
are $0.3$ meV and $2.3$ meV, respectively, so that finite-$T$ corrections are
small, even for $27$ K. Moreover the value of $n_{*}$ is consistent with
region corresponding to disorder broadening of the Dirac point in the field
effect mobility shown in Fig. 3. In Fig. 5, we show a comparison of the
shifted magnetoresistance and the corresponding fit by the theoretical curves
obtained from Eq. (17) with the fitting parameters from Table 1.

Figure 5: Non-normalized magnetoresistance $R(B)-R(0)$ for 4 K in (a) and 27 K
in (b). The points are obtained after shifting the gate voltage by
$V_{g}^{D}$. The dashed lines correspond to the fitted function Eq. (23) and
solid lines to theoretical zero-temperature magnetoresistance, Eq. (17),
calculated using the parameters from Table 1. Since these parameters are
obtained at finite temperature and are electron-hole averaged, the dashed and
solid lines do not exactly coincide for the same values of the gate voltage.

Figure 6: (a) Inverse mobility, Eq. (19), derived using the parameters
extracted from the $4$ K magnetoresistance data. (b) Inverse mobility derived
from the $27$ K data. The parameters are given in Table 1.

With the parameters obtained from fitting the magnetoresistance curves, we get
the mobility from Eq. (19), see Fig. 6, where the inverse mobility
$\mu_{0}^{-1}$ is displayed as a function of gate voltage difference
$U=V_{g}-V_{g}^{D}$ relative to the gate voltage $V_{g}^{D}$ corresponding to
the Dirac point. A clear minimum is found at the Dirac point (maximum for the
mobility). The slope of the inverse mobility away from the Dirac point is
determined by the strength of short-range scatterers $\gamma_{s}$. We observe
that at $T=27$ K, the mobility varies only very slightly as a function of the
gate voltage. This indicates that the role of short-range impurities is
suppressed at higher temperature. Possibly, with increasing temperature
residual dirt (adsorbed atoms) is removed from the sample.

In order to summarize the effect of the two different types of impurities, in
Fig. 7 we show the zero-field conductivity $\sigma_{0}$ determined by Eq. (11)
for both our mixed disorder model and a Coulomb-impurity model. Already at
experimentally accessible density ratios $n/n_{\text{imp}}^{C}\approx 2$ we
observe the sublinear conductivity due to short-range scatterers discussed in
Ref. [34].

Figure 7: (a) Conductivity at $B=0$T, Eq. (11), derived using the parameters
extracted from the $4$ K magnetoresistance data. (b) Zero-field conductivity
derived from the $27$ K data. The blue curve is the result for our mixed
disorder model, the orange one for $\gamma_{s}=\infty$, i.e., no short-range
scatterers. The parameters are given in Table 1, additionally we used
$\alpha=1.3$ to determine $n_{\text{imp}}^{C}$.

Thus, looking at the deduced impurity scattering strengths from the obtained
mobilities, our magnetoresistance data should reflect effects related to both
Coulomb and short-range scatterers. Note that our Hall mobility is slightly
smaller than the field-effect mobility obtained using $R(V_{g})$ which yields
approximately $10^{5}$ cm2/Vs near Dirac point for the average mobility of
electrons and holes. The value of mobility extracted from the analysis of the
magnetoresistance is consistent with the field-effect mobility.

Subtracting the disorder-induced bulk resistance with the parameters obtained
from the measured magnetoresistance, we get access to the overall contact
resistance, which is shown in Fig. 8.

Figure 8: Zero-$B$ resistance at $T=4$ K in (a) and $T=27$ K in (b). Blue
curves: the measured resistance. Orange curves: the zero-$T$ bulk resistance
calculated from Eq. (1) with the parameters from the fit (Table 1). Green
curves: the contact resistance which is obtained as the difference of the
measured and bulk resistances. Note that the contact resistance is actually
larger than the bulk contribution, as expected for high-mobility samples.

The obtained value of the contact resistance is somewhat higher than the one
reported for similar samples in the Corbino geometry, which can be related to
the fact that the sample has been cooled down several times.

##  V Discussion and conclusions

In summary, we have investigated geometric magnetotoresistance in suspended
graphene Corbino ring at low temperatures. The magnetoresistance, $\Delta
R(B)/R(0)\propto B^{2}$, is ultrastrong: it amounts to $4000B^{2}$ % at the
Dirac point ($B$ in Tesla), with quite small temperature dependence below 30
K. This is comparable with the “extraordinary magnetoresistance” in
encapsulated graphene in a disk geometry observed at room temperature in Ref.
[8] (although the physical mechanism behind the low-temperature ultrastrong
magnetoresistance is different). The relative magnetoresistance decreases with
charge density and, at $|n|\simeq 3\times 10^{-10}$ cm-2, it is already
reduced by a factor of two.

The simple zero-temperature analysis appears to be sufficient to explain the
main features of the measured magnetoresistance. The quadratic magnetic field
dependence is followed for both short-range and charged scatterers. The gate
dependence of the magnetoresistance allows one to estimate the partial
contributions of the short-range and long-range impurity scattering to the
mobility. In particular, away from the Dirac point, the gate-voltage
dependence of the mobility is entirely determined by the short-range component
of the impurity potential. At the same time, we see that transport around the
neutrality point is dominated by scattering on long-range disorder (Coulomb
impurities or ripples). However, no fundamental difference is observed between
Coulomb and short-range in terms of the magnetic-field dependence of the
resistance.

We find that the mobility extracted from the parabolic magnetoresistance is
sufficiently high (of the order of $10^{5}$ cm2/Vs), which is in agreement
with previous estimates for similar (slightly cleaner) samples used for
studying the quantum Hall effect, including the fractional quantum Hall
effect. It is worth noting that, somewhat counterintuitively, the mobility is
found to be higher at higher $T$. This can be related to the technological
process of preparing the sample, where the concentration of adsorbed atoms
depends on temperature. The total measured resistance is given by the sum of
the bulk and contact contributions. We see that in our geometry, the contact
resistance is even higher than the bulk resistance, which is consistent with
the high quality of the sample.

We emphasize that knowledge on the contact resistance is essential in the
analysis of our geometric magnetoresistance. According to Eq. (24), the
measured magnetoresistance is proportional to
$R_{\textrm{bulk}}\mu_{0}^{2}B^{2}$ where
$R_{\textrm{bulk}}=R-R_{\textrm{cont}}$. If we employed $R$ instead of
$R_{\textrm{bulk}}$, we would overestimate the reduction of $\mu_{0}$ obtained
from the magnetoresistance data as a function of gate voltage. For example,
naively fitting the normalized data in Fig. 4b we would deduce a reduction in
$\mu^{2}$ by a factor of two, while from Fig. 6b we obtain only $<13$%
reduction in $\mu^{2}$. Thus, in the former case the strength of the short-
range scattering would appear almost three times larger than in the correct
analysis.

In the vicinity of the Dirac point at 4 K, we find a high sensitivity of
resistance variation with respect to magnetic field $dR\big{/}dB=12.5$
k$\Omega$/T at $B=0.15$ T. According to our current noise measurements
yielding $S_{I}=10^{-23}$ A2/Hz at 1kHz for the same sample at 10 $\mu$A, we
may estimate a magnetic field sensitivity of 60 nT/$\sqrt{\textrm{Hz}}$ for
our device at 4 K. This sensitivity is excellent when compared with graphene
Hall magnetometers, since our result is on par with magnetic field
sensitivities of devices working in 20 times larger magnetic fields [44].

To conclude, the analysis of graphene magnetoresistance at different values of
the gate voltage in the Corbino geometry allowed us to extract information
about scattering mechanisms in the sample, and to separate the bulk and
contact contributions to the resistance. The strong magnetoresistance of
Corbino geometry at low magnetic fields appears to be a very powerful tool of
characterization of the graphene samples. As an outlook, it will be very
interesting to investigate experimentally and theoretically the
magnetoresistance of graphene in this geometry at elevated temperatures, when
electronic hydrodynamic effect become pronounced.

## Acknowledgements

We thank Pavel Alekseev, Dmitry Golubev, and Sheng-Shiuan Yeh for fruitful
discussions and comments. This work was supported by the Academy of Finland
projects 314448 (BOLOSE), 310086 (LTnoise) and 312295 (CoE, Quantum Technology
Finland), by ERC (grant no. 670743), as well as by the DFG within FLAG-ERA
Joint Transnational Call (Project GRANSPORT) and RFBR (grant no. 20-02-00490).
This research project utilized the Aalto University OtaNano/LTL infrastructure
which is part of European Microkelvin Platform EMP (funded by European Union’s
Horizon 2020 Research and Innovation Programme Grant No. 824109).

##  Appendix A Dirac point shift

In the resistance vs. gate voltage data, Fig. 2, we observe a shift of the
resistance maximum (associated with the position of the Dirac point) with
increasing magnetic field. We attribute this shift to the dependence of the
screening length on the magnetic field, which is only relevant for small
densities. In order to be able to fit the quadratic magnetoresistance, one
first has to get rid of this shift of the Dirac point.

Within the picture based on the effect of magnetic field on screening, one
should only shift data points close to the Dirac point, while not affecting
those farther away from it. Since we have data only for a discrete set of
values of the gate voltage, in order to determine the gate voltage
corresponding to the Dirac point, we find the maximum of the $R(V)$ curve
given by a cubic spline interpolation of the measured data points. For each
value of magnetic field, this maximum $V_{0}$ is shifted to the same voltage
$V_{1}$. For all other data points, we adopt a phenomenological Ansatz, where,
away from the Dirac point, the shift reduces exponentially, i.e., a measured
point on the curve corresponding to $B$ is shifted by

$\displaystyle V\Rightarrow V-(V_{0}-V_{1})e^{-\frac{|V-V_{0}|}{V_{2}}}.$ (25)

The voltage $V_{1}$ is the same for all curves and is determined by the
maximum of the most symmetric curve in the unshifted case, and $V_{2}$
describes the characteristic voltage window where the effect of magnetic field
on screening is seen. Since the shift for different magnetic fields will
generically be different, one also has to use the spline interpolation to get
access to the resistance at the same voltages for all magnetic fields. We take
this voltages to be the ones on the shifted zero magnetic field curve. A
result for thus shifted and unshifted data points is shown in Fig. 9.

Figure 9: Resistance at $4$ K (a) and $27$ K (b), shifted according to Eq.
(25). Colored dots are the shifted measured points and black stars show the
data obtained from the spline. (a): $V_{2}=1$V and $V_{1}=0$; (b): $V_{2}=1$V
and $V_{1}=0.5$V.

Comparing Fig. 9 with Fig. 2 of the main text, we see that the shifted curves
are indeed more symmetric, which supports the idea of stronger shift around
the neutrality point as a result of magnetic-field effect on screening. We
note, however, that a homogeneous (independent of the distance to the Dirac
point) shift, used for the actual fitting procedure, captures correctly the
shift near the neutrality point. Since this is the main manifestation of the
screening-induced shift in magnetoresistance curves (the difference in the
tails of magnetoresistance is not that important for the fitting procedure),
we adopt the simplest shifting in our analysis. Note that, in all cases, the
small feature close to $-2$V (attributed to a quasi-resonance in scattering by
adsorbed local impurities) is preserved after shifting the curves.

##  Appendix B Charged impurities

In the main text, we gave the transport and quantum scattering times for
short-range impurities, Eq. (5), and Coulomb impurities, Eq. (7). Here we show
how to derive these expressions for large energies. The behaviour close to the
neutrality point is discussed in Sec. D.

Very generally, the scattering rates can be written as

$\displaystyle\frac{1}{\tau_{i}(\varepsilon)}$
$\displaystyle=\frac{n_{\text{imp}}\nu(\varepsilon)}{8\hbar}\int_{0}^{2\pi}\mathrm{d}\theta\,\left|V\left(q(\theta)\right)\right|^{2}$
$\displaystyle\times\begin{cases}\sin^{2}\theta,&i=\text{tr}\\\
(1+\cos\theta),&i=q\end{cases},$ (26)

where $\nu(\varepsilon)$ is the density of states and we express the
transferred momentum as

$q(\theta)=2(\varepsilon/\hbar v)|\sin(\theta/2)|$

with $\theta$ being the scattering angle. For short-range impurities,
$V(q)=U_{0}$ and $n_{\text{imp}}=n_{\text{imp}}^{s}$; the integration over
$\theta$ then leads exactly to Eq. (5). In the case of charged impurities, we
consider the screened interaction potential in the random-phase approximation:

$\displaystyle V(q)=\frac{2\pi e^{2}/q\epsilon_{\infty}}{1+(2\pi
e^{2}/q\epsilon_{\infty})\Pi(q,0)}.$ (27)

Here $\epsilon_{\infty}$ is the background dielectric constant and $\Pi(q)$ is
the static polarization operator which is given by the thermodynamic density
of states ($\upmu$ is the chemical potential):

$\displaystyle\lim_{q\rightarrow 0}\Pi(q)=\frac{\partial n}{\partial\upmu}.$
(28)

This quantity can be connected to the single-particle density of states via

$\displaystyle n(\upmu)$
$\displaystyle=\int\mathrm{d}\varepsilon\,n_{F}(\varepsilon)\nu(\varepsilon)$
$\displaystyle\Rightarrow\ \frac{\partial n}{\partial\upmu}$
$\displaystyle=\int\mathrm{d}\varepsilon\left(-\frac{\partial
n_{F}(\varepsilon)}{\partial\varepsilon}\right)\nu(\varepsilon).$

As mentioned in the main text,the density of states can in general be deduced
from Eq. (36) in Ref. [28]. From there we see that away from the neutrality
point, the oscillatory magnetic-field dependent corrections to the
compressibility are exponentially suppressed and hence not seen in the
experiment. Moreover, since we are at temperatures below than the chemical
potential away from the neutrality point, we can use a zero $T$ approximation
and get

$\displaystyle\lim_{q\rightarrow 0}\Pi(q,0)\approx\frac{2\upmu}{\pi
v_{F}^{2}\hbar^{2}}=\nu(\upmu).$ (29)

By introducing the effective interaction strength

$\alpha=\frac{e^{2}}{\hbar v_{F}\epsilon_{\infty}},$

we can bring the screened Coulomb interaction to the form

$\displaystyle V(q)$ $\displaystyle=\frac{2\pi
e^{2}}{\epsilon_{\infty}(q+K)},$ (30) $\displaystyle K$
$\displaystyle=\frac{2\pi e^{2}N}{\epsilon_{\infty}}\lim_{q\rightarrow
0}\Pi(q,0)\approx 2\alpha\pi\hbar v_{F}\nu(\upmu).$ (31)

We get thus get for the scattering rates:

$\displaystyle\frac{1}{\tau_{i}^{C}(\varepsilon)}$
$\displaystyle=\frac{\pi^{2}}{4}\hbar^{3}v_{F}^{4}\alpha^{2}n_{\text{imp}}^{C}\nu(\varepsilon)$
$\displaystyle\times\int_{0}^{\pi}\frac{\mathrm{d}\theta}{\left(\varepsilon\sin\frac{\theta}{2}+\alpha\nu(\upmu)\pi\hbar^{2}v_{F}^{2}\right)^{2}}$
$\displaystyle\times\begin{cases}\sin^{2}\theta,&i=\text{tr}\\\
1+\cos\theta,&i=q\end{cases}.$ (32)

At low temperatures, the typical energies are very close to the chemical
potential and we can set $\varepsilon\to\upmu$ in the transferred momentum $q$
in the interaction matrix element:

$\displaystyle\frac{1}{\tau_{\text{tr}}^{C}(\varepsilon)}$
$\displaystyle\approx\frac{\pi}{4}\alpha^{2}v_{F}^{2}\hbar
n_{\text{imp}}^{C}\varepsilon\int_{0}^{2\pi}\frac{\mathrm{d}\theta\,\sin^{2}\theta}{[\upmu|\sin(\theta/2)|+\alpha
N\upmu/2]^{2}}$ $\displaystyle=\frac{\pi}{2}v^{2}\hbar
n_{\text{imp}}^{C}\frac{\varepsilon}{\upmu^{2}}c(\alpha),$ (33) $\displaystyle
c(\alpha)$
$\displaystyle=\alpha^{2}\int_{0}^{\pi}\frac{\mathrm{d}\theta\sin^{2}\theta}{(\sin(\theta/2)+\alpha
N/2)^{2}}.$ (34)

This can be brought into the form

$\displaystyle\tau_{\text{tr}}^{C}(\varepsilon)=\frac{2\gamma^{C}\hbar}{\varepsilon},\quad\gamma^{C}=\frac{\upmu^{2}}{\pi
v_{F}^{2}\hbar^{2}n_{\text{imp}}c(\alpha)},$ (35)

which is exactly Eq. (7) for zero temperature, where $\upmu=\varepsilon_{F}$.
Using relation (3), this is also exactly $\gamma_{C}$ as introduced in Eq.
(20). Similarly, we get for the quantum scattering rate

$\displaystyle\frac{1}{\tau_{q}^{C}}$
$\displaystyle\approx\frac{\pi}{2}\alpha^{2}v_{F}^{2}\hbar
n_{\text{imp}}^{C}\frac{\varepsilon}{\upmu^{2}}\int_{0}^{\pi}\frac{\mathrm{d}\theta\,(1+\cos\theta)}{[\sin(\theta/2)+\alpha
N/2]^{2}}$ $\displaystyle=\frac{\pi}{2}v_{F}^{2}\hbar
n_{\text{imp}}^{C}\frac{\varepsilon}{\upmu^{2}}d(\alpha),$ (36) $\displaystyle
d(\alpha)$
$\displaystyle=\alpha^{2}\int_{0}^{\pi}\frac{\mathrm{d}\theta(1+\cos\theta)}{(\sin(\theta/2)+\alpha
N/2)^{2}}$ (37)

which again is of the form

$\displaystyle\tau_{q}^{C}(\varepsilon)=\frac{\gamma_{C}^{\prime}\hbar}{\varepsilon},\quad\gamma_{C}^{\prime}=\frac{2\upmu^{2}}{\pi
v_{F}^{2}\hbar^{2}n_{\text{imp}}^{C}d(\alpha)},$ (38)

as used in Eq. (7). The values of $c(\alpha)$ and $d(\alpha)$ for some
realistic $\alpha$ are given by

$\displaystyle c(\alpha)$ $\displaystyle=\begin{cases}0.14,&\alpha=0.5\\\
0.22,&\alpha=1.0\\\ 0.26,&\alpha=1.5\\\ 0.29,&\alpha=2\end{cases}\ ,$
$\displaystyle d(\alpha)$ $\displaystyle=\begin{cases}0.43,&\alpha=0.5\\\
0.55,&\alpha=1.0\\\ 0.61,&\alpha=1.5\\\ 0.65,&\alpha=2\end{cases}\ .$ (39)

Thus, for intermediate $\alpha$, the relation between the two scattering
rates, given by $\frac{d(\alpha)}{2}\approx c(\alpha)$, is similar to that for
short-range scattering. Whenever we use a specific value of $\alpha$, we chose
$\alpha=1.3$, thus accounting for the renormalization of velocity, as well as
for the screening by the metallic parts of the setup.

In a general setup, both short-range and Coulomb scatterers are present. Here
we discuss how this mixture affects the scattering times for energies away
from the Dirac point, where the density of states is not affected by disorder.
We will stick with the assumption of no inter-valley scattering, diagonality
in the sublattice space, and no correlations between different kinds of
scattering. This corresponds to summing up the self-energies, where we would
not consider mixed diagrams, and translates into a sum rule for transport
times, Eq. (12). We will further assume that in the relevant limit, the
density of states is not modified by the magnetic field, since corrections are
exponentially suppressed. Then we can use the transport times as written in
Eq. (5) and Eq. (7) to write the total transport time $\tau_{\text{tr}}$ as

$\displaystyle\tau_{\text{tr}}(\varepsilon)=\frac{2\gamma\hbar}{|\varepsilon|},\quad\gamma=\frac{\gamma_{C}\gamma_{s}}{\gamma_{C}+\gamma_{s}}.$
(40)

The conductance kernel, as found from the Boltzmann equation, is then given by

$\displaystyle\sigma_{xx}(\varepsilon)$
$\displaystyle=\frac{e^{2}v_{F}^{2}}{2}\frac{\tau_{\text{tr}}(\varepsilon)\nu(\varepsilon)}{1+\left[\omega_{c}(\varepsilon)\tau_{\text{tr}}(\varepsilon)\right]^{2}}=\frac{\sigma_{0}}{1+[\omega_{c}(\varepsilon)\tau_{\text{tr}}(\varepsilon)]^{2}}$
$\displaystyle=\frac{2e^{2}\gamma}{\pi\hbar}\frac{1}{1+\left(\frac{2\gamma\hbar^{2}\Omega^{2}}{\varepsilon^{2}}\right)^{2}},$
(41)

where $\Omega=v_{F}/\ell_{B}$, leading to Eq. (16) of the main text.

##  Appendix C Finite-temperature effects

In the main text, we restricted ourselves to the zero temperature limit of Eq.
(13). Here we discuss finite-temperature corrections in the Drude formula, as
well as effects due to electron-electron interaction (EEI) which also
introduce finite-temperature corrections to Eq. (16) and thus Eq. (17). Let us
consider, for the illustrative purpose, the case of only short-range
potential.

The problem generally has two energy scales, the first one is the temperature
$k_{B}T$, and the second one is the energy $\varepsilon_{m}$ introduced by the
product $\omega_{c}(\varepsilon)\tau_{\text{tr}}(\varepsilon)$ which we write
as

$\displaystyle\omega_{c}(\varepsilon)\tau_{\text{tr}}(\varepsilon)=\frac{2eB\hbar
v_{F}^{2}}{\varepsilon^{2}}\equiv\frac{\varepsilon_{m}^{2}}{\varepsilon^{2}}$
(42)

We will restrict ourselves to low temperatures $k_{B}T\ll\upmu$ and small
magnetic fields $,\varepsilon_{m}\ll\upmu$ For finite but low temperatures and
small fields, the integral of the conductance kernel Eq. (10) over energies
can be brought into the form [28]:

$\displaystyle\sigma_{xx}$
$\displaystyle=\sigma_{0}\int_{-\infty}^{\infty}\mathrm{d}\varepsilon\left(-\frac{\partial
n_{F}(\varepsilon)}{\partial\varepsilon}\right)\left(1-\frac{\varepsilon_{m}^{4}}{\varepsilon^{4}+\varepsilon_{m}^{4}}\right)$
$\displaystyle=\sigma_{0}\left[1-\int_{-\infty}^{\infty}\mathrm{d}\varepsilon\left(-\frac{\partial
n_{F}(\varepsilon)}{\partial\varepsilon}\right)\frac{\varepsilon_{m}^{4}}{\varepsilon^{4}+\varepsilon_{m}^{4}}\right],$
(43)

we used Eq. (5) and Eq. (7) for the transport scattering time.

One notices that both the derivative of the Fermi function and the fraction in
the second expression are peaked. The derivative of the Fermi function is
peaked around $\varepsilon=\upmu$ with a width given by the temperature, while
the term $\varepsilon_{m}^{4}/(\varepsilon^{4}+\varepsilon_{m}^{4})$ is peaked
around $\varepsilon=0$ and its width is determined by $\varepsilon_{m}$. Since
we assume that both $T,\varepsilon_{m}\ll\upmu$, these peaks are well
separated, and the integral can be written as a sum of the contributions of
the two peaks, i.e.,
$\sigma_{xx}=\sigma_{0}-\left(\sigma_{xx}^{(T)}+\sigma_{xx}^{(\varepsilon_{m})}\right).$

For low temperatures, the derivative of the Fermi function has a finite width
of the order of $T$ around $\varepsilon=\upmu$. To incorporate finite-
temperature corrections to the conductivity, we thus expand the fraction in
powers of $(\varepsilon-\upmu)$ and after evaluating the integral with get

$\displaystyle\sigma_{xx}^{(T)}$
$\displaystyle\approx\sigma_{0}\left[\frac{\varepsilon_{m}^{4}}{\upmu^{4}+{\varepsilon_{m}}^{4}}-\frac{2\pi^{2}T^{2}\upmu^{2}{\varepsilon_{m}}^{4}}{3}\frac{\left(3{\varepsilon_{m}}^{4}-5\upmu^{4}\right)}{\left(\upmu^{4}+{\varepsilon_{m}}^{4}\right)^{3}}\right].$
(44)

The first term of this expression is the only non-vanishing contribution at
zero $T$ and leads exactly to Eq. (16).

The contribution of the second peak is found by fixing the value of the Fermi
function at its value at $\varepsilon=0$ and then evaluating the integral:

$\displaystyle\sigma_{xx}^{(\varepsilon_{m})}\simeq\sigma_{0}\varepsilon_{m}\frac{e^{-\upmu/T}}{\sqrt{2}T}.$
(45)

Thus the total Drude resistivity for short-range impurities is given, in the
regime of low temperatures and low magnetic fields, by

$\displaystyle\rho_{xx}=\frac{1}{\sigma_{xx}}\approx\frac{1}{\sigma_{0}}$ (46)
$\displaystyle\times\left[1+\frac{{\varepsilon_{m}}^{4}}{\upmu^{4}}\left(1+\frac{10\pi^{2}T^{2}}{3\upmu^{2}}\right)-\frac{{\varepsilon_{m}}^{8}}{\upmu^{8}}\frac{16\pi^{2}T^{2}}{3\upmu^{2}}+\frac{\pi{\varepsilon_{m}}}{\sqrt{2}T}e^{-\frac{\upmu}{T}}\right].$

From this expressions we see, that the mobility $\mu_{0}$ itself does not
acquire finite temperature corrections, since they all require finite magnetic
field. However the $B$ dependence of the magnetoresistance does acquire
additional terms (in particular, the $B^{4}$ term which is absent at $T=0$, as
well as the $\sqrt{B}$ term [28] which is, however, exponentially suppressed
at low $T\ll\upmu$). Furthermore, the $B^{2}$ dependence is also slightly
modified by a finite temperature. Finite-$T$ corrections would have a similar
structure if one includes Coulomb impurities.

###  C.1 Effects of electron-electron interaction

The effect of electron-electron interaction (EEI) on the magnetoresistance of
graphene was explored in Refs. [45, 27]. Since EEI does not influence
$\sigma_{xy}$, we can directly employ their main result, which is the EEI
correction to magnetoresistivity:

$\displaystyle\Delta\rho_{\text{EEI}}$
$\displaystyle=\left[(\omega_{c}\tau_{\text{tr}})^{2}-1\right]\frac{e^{2}\rho_{0}^{2}}{2\pi^{2}\hbar}A\,\ln\frac{k_{B}T\tau_{\text{tr}}}{\hbar},$
(47)

where

$\displaystyle A$
$\displaystyle=1+c\left[1-\frac{\ln(1+F_{0}^{\sigma})}{F_{0}^{\sigma}}\right],$
(48) $\displaystyle F_{0}^{\sigma}$
$\displaystyle=-\alpha\int_{0}^{2\pi}\frac{\mathrm{d}\theta}{2\pi}\frac{\cos^{2}\theta/2}{\sin\theta/2+2\alpha},$
(49)

and $c$ is the number of multiplets. Depending on the temperature, $c=3$, 7,
or 15 for very low, moderately low, and high temperatures, respectively. It
describes the number of ungapped non-singlet two-particle states contributing
to Hartree-type correction to the conductivity. Since each electron has a well
defined spin and valley quantum number, which can both take two values, there
are in total 16 possibilities for two particle states, one of which will
always be a spin and valley singlet. In fact, since there may be inter- or
intra-valley scattering, valley is not necessarily a good quantum number in
this sense, depending of the hierarchy of the temperature, the intra-valley
phase breaking time $\tau_{*}$, and the inter-valley scattering time
$\tau_{\text{iv}}$. For $k_{B}T<\hbar/\tau_{*}$, channels mixing different
spins do not contribute, thus there remain $2\times 4=8$ channels, of which
one is a singlet, i.e., $c=7$. If $k_{B}T<\hbar/\tau_{\text{iv}}$, valley is
not a good quantum number anymore, thus we get $4$ states, of which one is a
multiplet, i.e., $c=3$. According to Ref. [27], the relation
$\tau_{*}<\tau_{\text{iv}}$ is usually fulfilled. Below we give the numerical
values of the Fermi liquid constant $F_{0}^{\sigma}$ for some values of
$\alpha$:

$\displaystyle F_{0}^{\sigma}=\begin{cases}-0.18,&\alpha=0.5\\\
-0.21,&\alpha=1\\\ -0.22,&\alpha=1.5\\\ -0.23,&\alpha=2\end{cases}\ .$ (50)

Correction (47) does influence the mobility and it also influences the
prefactor of the $B^{2}$ dependence. For low temperatures, this is a negative
correction. The effect of EEI alone on the resistance at $T=4$ K is shown in
Fig. 10. By plugging the value of $\gamma$ obtained from the fit of
magnetoresistance in the analytical expression for zero $T$, Eq. (17), we
obtain the left panel of Fig. 11. For comparison, in the right panel of Fig.
11, we show the result of numerical evaluation of Eq. (13) at $T=27$ K, with
the EEI correction included. A comparison with the zero-$T$ plot in Fig. 11
shows that the EEI correction at $T=27$ K is almost negligible. At the same
time, the EEI produces the leading finite-$T$ correction at $T=4$ K. Overall,
we observe that the effect of finite temperature is rather weak in the
considered range of parameters.

Figure 10: Magnetoresistance at $T=4$ K with parameters from the fit (Table 1)
without the EEI correction (a) and including it (b). The EEI correction leads
to a small suppression of the magnetoresistance, without changing the
functional form. This negative correction is shown in inset (c). Here
$\alpha=1.3$ and $c=3$ were chosen.

Figure 11: (a) Magnetoresistance obtained from the zero-temperature solution
(17) with the parameters from the fit (Table 1) for $T=27$ K. (b)
Magnetoresistance obtained by using the parameters from the fit and solving
Eq. (13) numerically, with the EEI correction included according to Eq. (47)
($\alpha=1.3$ and $c=7$). The inset (c) shows the finite temperature
corrections to the magnetoresistance. Depending on the magnitude of the
magnetic field this correction is either positive or negative and it also
changes the functional form very slightly.

In combination, finite temperature does change the mobility and it also
influences the magnetic field behaviour of the resistance, so that formula
(17) is no longer exact and the relation (24) does not, strictly speaking,
yield the true parameters. However, as we see from Figs. 10 and 11, these
effects are rather small in the experimentally accessed range, which justifies
the neglect of these effects.

##  Appendix D Vicinity of the Dirac point

As seen in Fig. 2, the resistance does not diverge at the Dirac point, as
would be expected from combining the resistivity $\rho_{0}=1/\sigma_{0}$ from
Eq. (11) with the effective $\gamma$ from Appendix B using $\gamma_{C}$ as
expressed in Eq. (20). The reason for this is the saturation of the density of
states close to the Dirac point due to disorder [2, 39]. Below a certain
chemical potential $\upmu_{*}$, the quasiparticle pole in the Green’s function
is effectively absent, and all quantities should be fixed below this value.
The relevant scale for this behaviour is given by Eq. (21). In order to
determine the changes this induces and find Eq. (22), we discuss here how to
find the relevant scale $\mu_{*}$ from the condition

$\displaystyle\frac{\hbar}{2\tau_{q}(\upmu_{*})}=\upmu_{*}.$ (51)

Below $\upmu_{*}$, the density of states saturates, while it is not affected
for larger energies, i.e.

$\displaystyle\nu(\varepsilon)$
$\displaystyle=\begin{cases}\nu_{*},&|\varepsilon|\ll\upmu_{*},\\\
\nu_{0}(\varepsilon),&|\varepsilon|\gg\upmu_{*}\end{cases}.$ (52)

This value enters directly into the calculation of all scattering rates Eq.
(26); it also directly determines the screening radius in the Coulomb impurity
case Eq. (31). While it is clear, that Eq. (26) is only true if the density of
states is not strongly broadened, the idea is to approach the crossover from
the side of large energies, where this is the case. We can then calculate the
Coulomb scattering rates as follows:

$\displaystyle\frac{1}{\tau_{\text{tr}}^{C}(\upmu)}$
$\displaystyle=\frac{\pi^{2}}{4}\hbar^{3}v_{F}^{4}n_{\text{imp}}^{C}c(\alpha)\begin{cases}\dfrac{\nu_{*}}{\upmu_{*}^{2}},&\upmu\ll\upmu_{*}\\\\[5.69046pt]
\dfrac{\nu_{0}(\upmu)}{\upmu^{2}},&\upmu\gg\upmu_{*}\end{cases}\ ,$ (53)
$\displaystyle\frac{1}{\tau_{q}^{C}(\upmu)}$
$\displaystyle=\frac{\pi^{2}}{4}\hbar^{3}v_{F}^{4}n_{\text{imp}}^{C}d(\alpha)\begin{cases}\dfrac{\nu_{*}}{\upmu_{*}^{2}},&\upmu\ll\upmu_{*}\\\\[5.69046pt]
\dfrac{\nu_{0}(\upmu)}{\upmu^{2}},&\upmu\gg\upmu_{*}\end{cases}\ ,$ (54)

and, in a similar fashion, we get for short-range scatterers:

$\displaystyle\frac{1}{\tau_{\text{tr}}^{s}(\upmu)}$
$\displaystyle=\frac{1}{\gamma_{s}}\frac{\pi\hbar
v_{F}^{2}}{4}\begin{cases}\nu_{*},&\upmu\ll\upmu_{*}\\\
\nu_{0}(\upmu),&\upmu\gg\upmu_{*}\end{cases}\ ,$ (55)
$\displaystyle\frac{1}{\tau_{q}^{s}(\upmu)}$
$\displaystyle=\frac{1}{\gamma_{s}}\frac{\pi\hbar
v_{F}^{2}}{2}\begin{cases}\nu_{*},&\upmu\ll\upmu_{*}\\\
\nu_{0}(\upmu),&\upmu\gg\upmu_{*}\end{cases}\ .$ (56)

By assuming that the expression for $\upmu\gg\upmu_{*}$ is still reasonably
close for $\upmu\sim\upmu_{*}$ and using both short-range and Coulomb
impurities in Eq. (51) we find

$\displaystyle\upmu_{*}$
$\displaystyle=\sqrt{\frac{\pi}{2}v^{2}\hbar^{2}d(\alpha)\frac{n_{\text{imp}}^{C}}{2-1/\gamma_{s}}},$
$\displaystyle n_{*}$
$\displaystyle=\frac{d(\alpha)n_{\text{imp}}^{C}}{4-{2/\gamma_{s}}}.$ (57)

which leads to the approximation Eq. (22). This value depends on the density
of charged impurities and $\gamma_{s}$. The effective $\gamma$ can then be
extracted from the definition of $\gamma$, Eq. (11). We see that the density
of states actually drops out, only the influence of $\nu(\varepsilon)$ on the
electronic density (via $\upmu^{2}$) is relevant. We find the required
asymptotics

$\displaystyle\frac{1}{\gamma}=\frac{1}{\gamma_{s}}+\begin{cases}\dfrac{c(\alpha)n_{\text{imp}}^{C}}{n_{*}},&n\ll
n_{*}\\\\[5.69046pt] \dfrac{c(\alpha)n_{\text{imp}}^{C}}{n},&n\gg
n_{*}\end{cases},$ (58)

which we phenomenologically fulfill by tweaking the relation between density
and chemical potential in graphene to

$\displaystyle\sqrt{n^{2}+n_{*}^{2}}=N\frac{\upmu^{2}}{4\pi\hbar^{2}v^{2}},$
(59)

and consequently we replace $n$ by $\sqrt{n^{2}+n_{*}^{2}}$ in all fits and
plots, as mentioned in the main text.

Figure 12: Effective disorder parameter $1/\gamma$ defined by Eq. (61) and its
two parts $1/\gamma_{C}$ and $1/\gamma_{s}$ for $T=4$ K (a) and $T=27$ K (b).
The parameter $\gamma$ is smaller for 4 K, which means that disorder is
stronger at lower temperatures. The parameters are given in Table 1.

Using the general conductivity formula (16) with the finite disorder
broadening, Eq. (59), we get the broadened form of Eq. (19) and Eq. (20):

$\displaystyle\frac{1}{\mu_{0}}$
$\displaystyle\approx\frac{\pi\hbar}{2e}\left(\frac{\sqrt{n^{2}+n_{*}^{2}}}{\gamma_{s}}+c(\alpha)n_{\text{imp}}^{C}\right)$
(60)

and

$\displaystyle\frac{1}{\gamma}=\frac{1}{\gamma_{s}}+\frac{c(\alpha)n_{\text{imp}}^{C}}{\sqrt{n^{2}+n_{*}^{2}}}.$
(61)

Figure 13: The product $\omega_{c}\tau_{\text{tr}}$ at $B=0.1$ T with the
parameters from the fit (Table 1) for $4$ K in (a) and $27$ K in (b). The
transport time is larger for the higher temperature, which is consistent with
the reduction of the contribution of short-range impurities to scattering
processes.

##  Appendix E Details of the fitting procedure

Here we discuss how the fits are performed and obtain the parameters for Table
1. The first step is to remove the Dirac point shift as discussed in Appendix
A. In practice, we do not employ the inhomogeneous shift introduced there, but
rather just shift the curves as a whole, such that the maximum is at zero
voltage. From the cubic spline through the measured data points, we read off
the resistance values at the original voltages and additionally all half
integer ones for $27$ K.

We then fit the obtained magnetoresitances $R(B)-R(0)$ over the whole measured
range of $B$ with the fit function Eq. (23) for all gate voltages. From the
thus obtained parameter $M$ we extract all parameters of the theory according
to Eq. (24). We fit the magnetoresistance with three parameters $\gamma_{s}$,
$c(\alpha)n_{\text{imp}}^{C}$, and $n_{*}$, expressing $M$ as follows:

$\displaystyle\frac{1}{M}=\pi^{2}\hbar\left(n^{2}+n_{*}^{2}\right)\left[\frac{1}{\gamma_{s}}+\frac{c(\alpha)n_{\text{imp}}^{C}}{\sqrt{n^{2}+n_{*}^{2}}}\right].$
(62)

A plot of the resulting effective disorder parameters $\gamma$ for the two
temperatures is shown in Fig. 12.

In Fig. 13, we also show the product $\omega_{c}\tau_{\text{tr}}$ for $B=0.1$T
and the two temperatures using the obtained parameters to calculate
$\tau_{\text{tr}}$. These plots, demonstrating the dependence of mobility
$\mu_{0}$ on the gate voltage, are in agreement with Fig. 6. We see that for
$B=0.1$ T the bending of cyclotron trajectories should be already substantial.
Since the quantum scattering time in graphene is smaller by about a factor of
two compared to the transport scattering time, the parameter
$x=\omega_{c}\tau_{q}$ is still smaller than one, and hence Landau levels
overlap strong enough to not lead to any more intricate effects in the range
magnetic fields $B<0.15$ T addressed here.

## References

  * Novoselov et al. [2004] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004), ISSN 0036-8075, eprint https://science.sciencemag.org/content/306/5696/666.full.pdf, URL https://science.sciencemag.org/content/306/5696/666.
  * Castro Neto et al. [2009] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009), URL https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.81.109.
  * Peres [2010] N. M. R. Peres, Rev. Mod. Phys. 82, 2673 (2010), URL https://link.aps.org/doi/10.1103/RevModPhys.82.2673.
  * Bolotin et al. [2008] K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Communications 146, 351 (2008), ISSN 0038-1098, URL https://www.sciencedirect.com/science/article/pii/S0038109808001178.
  * Kumar et al. [2018] M. Kumar, A. Laitinen, and P. Hakonen, Nature Communications 9, 2776 (2018), ISSN 20411723, eprint 1611.02742, URL www.nature.com/naturecommunications.
  * Novoselov et al. [2005] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005), ISSN 0028-0836, URL http://www.nature.com/articles/nature04233.
  * Zhang et al. [2005] Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature 438, 201 (2005), ISSN 0028-0836, URL http://www.nature.com/articles/nature04235.
  * Zhou et al. [2020] B. Zhou, K. Watanabe, T. Taniguchi, and E. A. Henriksen, Applied Physics Letters 116, 053102 (2020), ISSN 0003-6951, URL https://doi.org/10.1063/1.5142021.
  * Das Sarma et al. [2011] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev. Mod. Phys. 83, 407 (2011), ISSN 0034-6861, URL https://link.aps.org/doi/10.1103/RevModPhys.83.407.
  * Tiwari and Stroud [2009] R. P. Tiwari and D. Stroud, Phys. Rev. B 79, 165408 (2009), URL https://link.aps.org/doi/10.1103/PhysRevB.79.165408.
  * Ping et al. [2014] J. Ping, I. Yudhistira, N. Ramakrishnan, S. Cho, S. Adam, and M. S. Fuhrer, Physical Review Letters 113, 047206 (2014), ISSN 0031-9007, URL https://link.aps.org/doi/10.1103/PhysRevLett.113.047206.
  * Cho and Fuhrer [2008] S. Cho and M. S. Fuhrer, Phys. Rev. B 77, 081402(R) (2008), URL https://link.aps.org/doi/10.1103/PhysRevB.77.081402.
  * Friedman et al. [2010] A. L. Friedman, J. L. Tedesco, P. M. Campbell, J. C. Culbertson, E. Aifer, F. K. Perkins, R. L. Myers-Ward, J. K. Hite, C. R. Eddy, G. G. Jernigan, et al., Nano Letters 10, 3962 (2010), ISSN 1530-6984, URL https://pubs.acs.org/doi/10.1021/nl101797d.
  * Bai et al. [2010] J. Bai, R. Cheng, F. Xiu, L. Liao, M. Wang, A. Shailos, K. L. Wang, Y. Huang, and X. Duan, Nature Nanotechnology 5, 655 (2010), ISSN 17483395.
  * Gopinadhan et al. [2013] K. Gopinadhan, Y. J. Shin, I. Yudhistira, J. Niu, and H. Yang, Physical Review B 88, 195429 (2013), ISSN 10980121\.
  * Wang et al. [2014] W. J. Wang, K. H. Gao, Z. Q. Li, T. Lin, J. Li, C. Yu, and Z. H. Feng, Applied Physics Letters 105, 182102 (2014), ISSN 0003-6951, URL http://aip.scitation.org/doi/10.1063/1.4901175.
  * Chuang et al. [2016] C. Chuang, Y. Yang, R. E. Elmquist, and C.-T. Liang, Materials Letters 174, 118 (2016), ISSN 0167-577X, URL https://www.sciencedirect.com/science/article/pii/S0167577X16303573.
  * Laitinen et al. [2018a] A. Laitinen, M. Kumar, P. Hakonen, and E. Sonin, Scientific Reports 8, 594 (2018a), ISSN 2045-2322, URL http://www.nature.com/articles/s41598-017-18959-7.
  * Chuang et al. [2018] C. Chuang, C. T. Liang, G.-H. Kim, R. E. Elmquist, Y. Yang, Y. P. Hsieh, D. K. Patel, K. Watanabe, T. Taniguchi, and N. Aoki, Carbon 136, 211 (2018), ISSN 0008-6223, URL https://www.sciencedirect.com/science/article/pii/S0008622318304287.
  * Song et al. [2019] G. Song, M. Ranjbar, and R. A. Kiehl, Communications Physics 2, 65 (2019), ISSN 23993650, URL http://dx.doi.org/10.1038/s42005-019-0161-5.
  * Hu et al. [2020] J. Hu, J. Gou, M. Yang, G. J. Omar, J. Tan, S. Zeng, Y. Liu, K. Han, Z. Lim, Z. Huang, et al., Advanced Materials 32, 2002201 (2020), ISSN 0935-9648, URL https://onlinelibrary.wiley.com/doi/10.1002/adma.202002201.
  * Gopinadhan et al. [2015] K. Gopinadhan, Y. J. Shin, R. Jalil, T. Venkatesan, A. K. Geim, A. H. C. Neto, and H. Yang, Nature Communications 6, 8337 (2015), ISSN 2041-1723, URL http://www.nature.com/articles/ncomms9337.
  * Li et al. [2017] H. Li, Y. J. Zeng, X. J. Hu, H. H. Zhang, S. C. Ruan, M. J. Van Bael, and C. Van Haesendonck, Carbon 124, 193 (2017), ISSN 00086223.
  * Nomokonov et al. [2019] D. V. Nomokonov, A. V. Goran, and A. A. Bykov, Journal of Applied Physics 125, 164301 (2019), eprint https://doi.org/10.1063/1.5085932, URL https://doi.org/10.1063/1.5085932.
  * Zheng and Ando [2002] Y. Zheng and T. Ando, Phys. Rev. B 65, 245420 (2002), URL https://link.aps.org/doi/10.1103/PhysRevB.65.245420.
  * Müller et al. [2008] M. Müller, L. Fritz, and S. Sachdev, Phys. Rev. B 78, 115406 (2008), URL https://link.aps.org/doi/10.1103/PhysRevB.78.115406.
  * Jobst et al. [2012] J. Jobst, D. Waldmann, I. V. Gornyi, A. D. Mirlin, and H. B. Weber, Phys. Rev. Lett. 108, 106601 (2012), ISSN 00319007\.
  * Alekseev et al. [2013] P. S. Alekseev, A. P. Dmitriev, I. V. Gornyi, and V. Y. Kachorovskii, Physical Review B 87, 165432 (2013), ISSN 1550235X, eprint 1210.6081.
  * Vasil’eva et al. [2012] G. Y. Vasil’eva, P. S. Alekseev, Y. L. Ivanov, Y. B. Vasil’ev, D. Smirnov, H. Schmidt, R. J. Haug, F. Gouider, and G. Nachtwei, JETP Letters 96, 471 (2012), ISSN 1090-6487, URL https://doi.org/10.1134/S0021364012190137.
  * Vasileva et al. [2019a] G. Vasileva, P. Alekseev, Y. Vasilyev, A. Dmitriev, V. Kachorovskii, D. Smirnov, H. Schmidt, and R. Haug, physica status solidi (b) 256, 1800525 (2019a), eprint https://onlinelibrary.wiley.com/doi/pdf/10.1002/pssb.201800525, URL https://onlinelibrary.wiley.com/doi/abs/10.1002/pssb.201800525.
  * Tombros et al. [2011] N. Tombros, A. Veligura, J. Junesch, J. Jasper van den Berg, P. J. Zomer, M. Wojtaszek, I. J. Vera Marun, H. T. Jonkman, and B. J. van Wees, Journal of Applied Physics 109, 093702 (2011), eprint https://doi.org/10.1063/1.3579997, URL https://doi.org/10.1063/1.3579997.
  * Laitinen et al. [2018b] A. Laitinen, M. Kumar, T. Elo, Y. Liu, T. S. Abhilash, and P. J. Hakonen, Journal of Low Temperature Physics 191, 272 (2018b), ISSN 0022-2291, URL http://link.springer.com/10.1007/s10909-018-1855-x.
  * Gornyi et al. [2012] I. V. Gornyi, V. Y. Kachorovskii, and A. D. Mirlin, Phys. Rev. B 86, 165413 (2012), URL https://link.aps.org/doi/10.1103/PhysRevB.86.165413.
  * Hwang et al. [2007] E. H. Hwang, S. Adam, and S. Das Sarma, Phys. Rev. Lett. 98, 186806 (2007), URL https://link.aps.org/doi/10.1103/PhysRevLett.98.186806.
  * Meyer et al. [2007] J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. J. Booth, and S. Roth, Nature 446, 60 (2007), ISSN 1476-4687, URL https://doi.org/10.1038/nature05545.
  * Lau et al. [2012] C. N. Lau, W. Bao, and J. Velasco, Materials Today 15, 238 (2012), ISSN 1369-7021, URL https://www.sciencedirect.com/science/article/pii/S1369702112701141.
  * Gornyi et al. [2015] I. V. Gornyi, V. Y. Kachorovskii, and A. D. Mirlin, Phys. Rev. B 92, 155428 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.92.155428.
  * Shon and Ando [1998] N. Shon and T. Ando, Journal of the Physical Society of Japan 67, 2421 (1998), eprint https://doi.org/10.1143/JPSJ.67.2421, URL https://doi.org/10.1143/JPSJ.67.2421.
  * Ostrovsky et al. [2006] P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, Physical Review B 74, 235443 (2006), ISSN 10980121.
  * Vasileva et al. [2019b] G. Y. Vasileva, P. S. Alekseev, Y. B. Vasil’ev, A. P. Dmitriev, V. Y. Kachorovskii, D. Smirnov, H. Schmidt, and R. Haug, Journal of Experimental and Theoretical Physics 129, 438 (2019b), ISSN 10906509.
  * Rycerz [2010] A. Rycerz, Physical Review B 81, 121404(R) (2010), ISSN 1098-0121, URL https://link.aps.org/doi/10.1103/PhysRevB.81.121404.
  * Laitinen et al. [2018c] A. Laitinen, M. Kumar, and P. J. Hakonen, Physical Review B 97, 075113 (2018c), ISSN 2469-9950, URL https://link.aps.org/doi/10.1103/PhysRevB.97.075113.
  * Laitinen et al. [2016] A. Laitinen, G. S. Paraoanu, M. Oksanen, M. F. Craciun, S. Russo, E. Sonin, and P. Hakonen, Physical Review B 93, 115413 (2016), ISSN 1550235X, eprint 1502.04330v2.
  * Schaefer et al. [2020] B. T. Schaefer, L. Wang, A. Jarjour, K. Watanabe, T. Taniguchi, P. L. McEuen, and K. C. Nowack, Nature Communications 11, 4163 (2020), ISSN 2041-1723, URL https://doi.org/10.1038/s41467-020-18007-5.
  * Kozikov et al. [2010] A. A. Kozikov, A. K. Savchenko, B. N. Narozhny, and A. V. Shytov, Phys. Rev. B 82, 075424 (2010), URL https://link.aps.org/doi/10.1103/PhysRevB.82.075424.