PHYSICAL REVIEW B
102
, 115432 (2020)
Microwave response of interacting oxide two-dimensional electron systems
D. Tabrea
,
1
I. A. Dmitriev
,
2
,
3
S. I. Dorozhkin,
4
B. P. Gorshunov
,
5
A. V. Boris,
1
Y. Kozuka,
6
,
7
A. Tsukazaki,
8
M. Kawasaki,
9
,
10
K. von Klitzing,
1
and J. Falson
1
,
*
1
Max-Planck-Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
2
Department of Physics, University of Regensburg, 93040 Regensburg, Germany
3
Ioffe Physical Technical Institute, 194021 St. Petersburg, Russia
4
Institute of Solid State Physics RAS, 142432 Chernogolovka, Moscow District, Russia
5
Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region 141700, Russia
6
Research Center for Magnetic and Spintronic Materials, National Institute for Materials Science, 1-2-1 Sengen,
Tsukuba 305-0047, Japan
7
JST, PRESTO, Kawaguchi, Saitama 332-0012, Japan
8
Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
9
Department of Applied Physics and Quantum-Phase Electronics Center (QPEC), University of Tokyo, Tokyo 113-8656, Japan
10
RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan
(Received 24 September 2019; revised 4 September 2020; accepted 8 September 2020; published 24 September 2020)
We present an experimental study on microwave illuminated high mobility MgZnO/ZnO based two-
dimensional electron systems with different electron densities and, hence, varying Coulomb interaction strength.
The photoresponse of the low-temperature dc resistance in perpendicular magnetic field is examined in low and
high density samples over a broad range of illumination frequencies. In low density samples a response due
to cyclotron resonance (CR) absorption dominates, while high-density samples exhibit pronounced microwave-
induced resistance oscillations (MIRO). Microwave transmission experiments serve as a complementary means
of detecting the CR over the entire range of electron densities and as a reference for the band mass unrenormal-
ized by interactions. Both CR and MIRO-associated features in the resistance permit extraction of the effective
mass of electrons but yield two distinct values. The conventional cyclotron mass representing center-of-mass
dynamics exhibits no change with density and coincides with the band electron mass of bulk ZnO, while MIRO
mass reveals a systematic increase with lowering electron density consistent with renormalization expected in
interacting Fermi liquids.
DOI:
10.1103/PhysRevB.102.115432
I. INTRODUCTION
Two-dimensional electron systems (2DES) have been the
subject of intense study as they host a remarkably rich set of
ground states depending on the strength of the interparticle
interaction. As the charge carrier density
n
is reduced, the
Coulomb energy (
E
C
∝
√
n
) becomes comparable and even-
tually even exceeds the electronic Fermi energy (
E
F
∝
n
). In
the limit of high concentration, charge carriers interact weakly
and the system’s parameters follow from band theory. At
intermediate densities, a Fermi liquid described by parameters
that undergo a renormalization due to interactions, such as
the effective mass
m
∗
and the
g
-factor, forms [
1
,
2
]. Finally,
in the dilute limit, a breakdown of the Fermi-liquid paradigm
*
falson@caltech.edu
Published by the American Physical Society under the terms of the
Creative Commons Attribution 4.0 International
license. Further
distribution of this work must maintain attribution to the author(s)
and the published article’s title, journal citation, and DOI. Open
access publication funded by the Max Planck Society.
is anticipated. This culminates either in particle localization
or, if disorder is sufficiently suppressed, in highly correlated
states such as a Wigner crystal [
3
–
5
].
The band effective mass of a 2DES is commonly measured
using cyclotron resonance (CR) since, in view of Kohn’s
theorem [
6
], the resonance frequency is insensitive to in-
terparticle correlations at the vanishingly small momentum
of the incident radiation. Estimates of the renormalized ef-
fective mass mostly rely on temperature-dependent studies
of the Shubnikov-de Haas oscillations [
1
]. Recently, oscil-
latory magnetotransport features that appear under incident
microwave radiation, referred to as microwave-induced re-
sistance oscillations or MIRO [
7
], have been advanced as
an alternative tool for obtaining the interaction-dependent
effective mass [
8
–
11
]. This method has mainly been de-
ployed in the weakly interacting regime, where
r
s
=
E
C
/
E
F
<
2[
8
–
10
]. Recently, a different region of parameter space
where Coulomb interactions prevail and
r
s
spans values
from 3 to 6 has been accessed [
11
]. This was accomplished
with Mg
x
Zn
1
−
x
O
/
ZnO heterostructures which simultane-
ously posses a low level of disorder [
12
–
14
]. Indeed,
state-of-the-art Mg
x
Zn
1
−
x
O
/
ZnO samples display electron
mobilities beyond 10
6
cm
2
/
Vs as well as quantum lifetimes
2469-9950/2020/102(11)/115432(8)
115432-1
Published by the American Physical Society
D. TABREA
et al.
PHYSICAL REVIEW B
102
, 115432 (2020)
that are comparable to what the best GaAs heterostructures
can offer [
15
,
16
]. Accordingly, exotic fractional quantum Hall
features have been reported in these samples [
17
,
18
].
Here we aim to extract the electron mass in these het-
erostructures by performing simultaneous magnetotransmis-
sion and magnetotransport measurements under microwave
illumination [
19
,
20
]. For the entire span of charge densities,
the transmission signal displays resonant features at the cy-
clotron resonance. An analysis of the density dependence of
this signal yields an electron effective mass close to the band
mass
m
b
≈
0
.
3
m
0
of bulk ZnO [
21
,
22
], where
m
0
is the free
electron mass. In contrast, the resistively detected magneto-
transport signal of the devices exhibits qualitatively different
responses depending on the charge carrier density. While low
carrier density samples (
n
<
3
.
5
×
10
11
cm
−
2
) exclusively dis-
play a conventional response due to heating of the electron
system during resonant microwave absorption at CR [
23
–
25
],
in the higher density regime the response is dominated by
the less common MIRO [
26
–
28
]. No CR related feature was
detected in the magnetoresistance of high density samples.
These two signals permit further analysis of the effective
mass. While the CR-associated feature reflects a similar band
mass
m
b
≈
0
.
3
m
0
to that obtained in transmission studies,
the value extracted from MIRO exhibits a systematic increase
with decreasing carrier concentration reflecting the renormal-
ization of the Fermi-liquid as interactions augment [
29
–
32
].
Lastly, we provide a plausible explanation for the dominance
of the CR response in the photoresistance of low-density
samples.
II. EXPERIMENT
These studies were performed on a series of
Mg
x
Zn
1
−
x
O
/
ZnO heterostructures each hosting a 2DES
at their heterointerface, with electron densities in the range
of 2
n
20
×
10
11
cm
−
2
depending on the Mg content
x
of the cap layer (0
.
01
x
0
.
15). Wafers were diced into
pieces of approximately 3
×
3mm
2
to prepare samples in
the van der Pauw geometry with four or eight contacts. The
contacts were made by evaporating Ti
/
Au and
/
or soldered
indium along the perimeter of the sample. The experimental
setup is shown in Fig.
1(a)
. Samples are mounted on ceramic
chip carriers with a drilled hole of approximately 5 mm
in diameter to allow microwave transmission through the
sample. A mylar film was glued across this opening to
provide support for the sample. Metallized mylar was
additionally placed around the perimeter of the chip to limit
the transmission of stray radiation. On the backside of the chip
carrier, a 4
×
3mm
2
carbon-covered kapton film contacted
with silver paint was placed. Its resistance
R
t
exhibited a
strong negative bolometric response
δ
R
t
(
B
)
=−
CT
s
(
B
)
P
ext
proportional to the microwave power
T
s
(
B
)
P
ext
transmitted
through the sample containing the 2DES. Since both the
sensitivity coefficient,
C
>
0, and the external microwave
power,
P
ext
,are
B
-independent, variations of
δ
R
t
(
B
) directly
reflect the
B
-dependence of the microwave transmission
coefficient
T
s
(
B
).
The experiments were carried out in a single-shot
3
He
cryostat with an axial superconducting coil. The sample is
submersed in
3
He liquid and the temperature is varied between
FIG. 1. (a) Sketch of the experimental setup. (b) Magneto-
transmission data [negated
δ
R
t
(
B
) reflecting the
B
-dependence of
the transmittance,
T
s
(
B
), solid lines] and photoresistance
δ
R
xx
(
B
)
(dashed lines) for three microwave frequencies (as marked) obtained
at
T
=
1
.
4 K on a sample with density
n
=
2
.
05
×
10
11
cm
−
2
.Curves
are shifted vertically for clarity. Linear scales are used. (c) Positions
of the minima in the magnetotransmission traces obtained for a
number of available microwave frequencies on the sample in panel
(b) (open squares). A linear fit crossing the origin for data points
corresponding to
f
75 GHz (solid line) gives the value of the
effective mass
m
∗
CR
=
(0
.
31
±
0
.
005)
m
0
associated with the CR in
transmission.
300 mK and 1.4 K by pumping on the
3
He surface. Monochro-
matic radiation with a frequency of up to 50 GHz was
generated using an Agilent 83650 B source. If needed, this
signal was additionally amplified and frequency-multiplied to
access the
f
=
75–108 GHz frequency range. The multipli-
cation leaves an inaccessible window for
f
≈
50–75 GHz.
The microwaves were delivered to the sample with the help
of an oversized rigid rectangular waveguide. Their amplitude
was modulated at a frequency
f
mod
=
1 kHz. The longitudinal
resistance
R
xx
of the 2DEG was measured using low-
frequency (
f
AC
≈
10 Hz) lock-in detection at a bias current of
I
=
500 nA. Double modulation using the dual-reference de-
tection capability of an SR860 lock-in amplifier as a fraction
of the total signal that is modulated both at
f
AC
and at
f
mod
was deployed to selectively record weak microwave-induced
changes of the low frequency resistance
δ
R
xx
. To improve
the signal-to-noise ratio we also relied on double modulation
detection of the microwave-induced changes of the carbon
resistor
δ
R
t
.
III. RESULTS AND DISCUSSION
A. Transmission measurements
Solid lines in Fig.
1(b)
display typical magnetotrans-
mission data from the carbon resistor placed below the
sample hosting a 2DES with an electron density
n
=
2
.
05
×
10
11
cm
−
2
. The change in the carbon resistor value
δ
R
t
has
115432-2
MICROWAVE RESPONSE OF INTERACTING OXIDE ...
PHYSICAL REVIEW B
102
, 115432 (2020)
been recorded at different microwave frequencies. A strong
maximum was found in each trace, corresponding to a min-
imum of the transmittance
T
s
(
B
). It is attributed to the CR.
Far from the resonance, the sample is nearly transparent to
the incoming radiation and the carbon resistor heats up and
cools down at the rate of the amplitude modulation of the
incident microwave. This translates into a negative and nearly
B
-independent off-resonant signal
δ
R
t
. Near the CR the 2DES
absorbs and reflects a larger part of microwaves, which leads
to a lower transmitted power and therefore to a lower heating
of the carbon resistor. We note that the external radiation
power reaching the sample varies significantly with the mi-
crowave wavelength. This is due to fluctuations of the incident
microwave power caused by the development of standing
waves in the waveguide and variations in the output power of
the microwave source. Therefore, we employ arbitrary units
and refrain from a quantitative comparison of the amplitude
for data recorded at different microwave frequencies. We note
that the asymmetric line shape of the CR likely originates
from interference effects within the sample that depends on
the wavelength of the radiation. Over a large frequency range
the effect is averaged out.
Figure
1(c)
demonstrates that the
B
-positions of the trans-
mission minima are proportional to the microwave frequency
in the high frequency range. The slope obtained from a linear
fit of the data for
f
75 GHz passing through the origin
(solid line) establishes that these minima match the CR condi-
tion,
f
=
eB
/
(2
π
m
∗
CR
), for an effective mass
m
∗
CR
=
(0
.
31
±
0
.
005)
m
0
close to the band mass of ZnO. Due to the finite size
of the sample, the 2DES supports a confined plasmon mode. It
hybridizes with the cyclotron resonance mode to yield a mag-
netoplasmon mode of nonzero frequency near
B
=
0. This
causes a deviation of the linear
B
-dependence of the observed
resonance frequency in the low field limit [
33
]. Therefore,
data points recorded at frequencies below 50 GHz have been
excluded from the mass analysis. The CR is an ubiquitous fea-
ture in magnetotransmission for the whole range of electron
densities
n
=
(2
−
20)
×
10
11
cm
−
2
, which we utilize below
in gauging the magnitude of mass enhancement obtained from
the analysis of photoresistance.
B. Photoresistance measurements
We now turn our attention to magnetotransport measure-
ments utilizing the double-modulation technique to extract the
microwave-induced variation of the longitudinal resistance of
the 2DES,
δ
R
xx
. In contrast to the magnetotransmission signal
that is dominated by the resonant reflection and absorption
near the CR for the whole range of electron densities, the
resistance measurements manifest more complex behavior de-
pending on the charge carrier density. In low-density samples
(
n
<
4
.
3
×
10
11
cm
−
2
), the most prominent feature in
δ
R
xx
is
a broad peak, as exemplified in Fig.
2(a)
.At
T
=
1
.
4Kthe
corresponding change in
R
xx
due to resonant absorption is of
the order of a few Ohm. Simultaneous measurement of
δ
R
xx
and of the magnetotransmission signal reveals that the peak is
aligned with the minimum in
−
δ
R
t
. A comparison between
the two sets of data is displayed in Fig.
1(b)
. It is there-
fore natural to ascribe this peak to a conventional response
due to resonant microwave absorption and associated electron
FIG. 2. Representative examples for the recorded variation of
the longitudinal dc resistance,
δ
R
xx
, induced by incident radiation
with a frequency
f
=
96 GHz at
T
=
1
.
4 K. The response differs in
samples with low [
n
=
2
.
3
×
10
11
cm
−
2
, panel (a)] and high elec-
tron density [
n
=
7
.
5
×
10
11
cm
−
2
, panel (b)] plotted on a linear
scale. (c) Position of the maxima in
δ
R
xx
for the low-density sam-
ple shown in panel (a) in the frequency vs. magnetic field plane
(open squares) together with a linear fit passing through the ori-
gin (solid line). (d) Position of selected MIRO extrema as marked
in panel (b) extracted from data obtained at different microwave
frequencies. Solid lines are linear fits to the data points using the
equation
f
=
(
N
±
1
/
4)
eB
/
2
π
m
∗
MIRO
, with
N
=
1
,
2
,
3, and 4. This
yields an average value of
m
∗
MIRO
equal to 0
.
335
m
0
. (e) Temperature
dependence of the dark resistance
R
xx
at
B
=
0 for the structures in
panels (a) and (b).
heating near the CR. In contrast, high density samples with
n
>
4
.
7
×
10
11
cm
−
2
display no detectable resonant features
in
δ
R
xx
at the CR. Instead, such samples exhibit pronounced
1
/
B
periodic magnetooscillations which can be identified as
MIRO. A typical trace for higher
n
is shown in Fig.
2(b)
.
The extracted positions of the maxima in
δ
R
xx
for the low-
density sample in Fig.
2(a)
obtained for different microwave
frequencies (open squares) are plotted in Fig.
2(c)
together
with a linear fit passing the origin (solid line). Analogous to
the transmission experiment presented above, only the high
frequency range
f
75 GHz was used in the analysis. The
slope gives the value of the CR effective mass
m
∗
CR
=
(0
.
32
±
0
.
01)
m
0
which nearly coincides with the value obtained from
the minima in the magnetotransmission data of Fig.
1(c)
.This
finding reinforces our interpretation of the peak of
δ
R
xx
as an
effect of resonant heating of the 2DES in the vicinity of the
CR. The extracted value is close to the band mass
m
b
≈
0
.
3
m
0
of bulk ZnO [
21
,
22
]. The dashed line in Fig.
2(c)
illustrates
the expected position of the lowest-order magnetoplasmon
mode in this sample for a wavelength of the dimensional
plasmon equal to twice the sample size,
λ
mp
=
6mm[
33
–
35
].
It demonstrates that finite-size effects are negligible in our
large-area samples for frequencies
f
above 75 GHz. Hence,
it is appropriate to describe the observations in terms of the
CR in an infinite 2DES.
The period, phase, as well as the damping of the 1
/
B
-
periodic MIRO oscillations observed in high-density samples
115432-3
D. TABREA
et al.
PHYSICAL REVIEW B
102
, 115432 (2020)
are all reproduced well by the conventional expression [
26
]
δ
R
xx
∝−
exp (
−
α
)sin(2
π
)
.
(1)
Here
α
describes the exponential damping at low
B
[Eq. (
1
)is
valid for
α
1]. The period of the oscillations is determined
by the quasiparticle effective mass
m
∗
MIRO
. The later enters
the ratio
=
ω/ω
c
, where
ω
=
2
π
f
is the angular microwave
frequency and
ω
c
=
eB
/
m
∗
MIRO
is the cyclotron frequency de-
termining the distance between neighboring Landau levels for
quasiparticles near the Fermi level. The “bare” cyclotron mass
m
∗
CR
extracted from the microwave transmission experiment or
the photoresistance feature represents the cyclotron dynamics
of the 2DES probed as a whole in the limit
k
→
0[
36
,
37
]. Its
value is unaffected by a renormalization of the Fermi liquid
in view of momentum conservation and Kohn’s theorem [
6
].
In contrast, MIRO involves the scattering of individual quasi-
particles at the Fermi surface. The MIRO mass
m
∗
MIRO
is
therefore expected to be modified due to renormalization by
interactions in a similar way as other transport properties such
as Shubnikov–de Haas oscillations as well as gap measure-
ments. These probe the electronic system in the opposite limit
of large
k
. When a sufficient number of MIRO harmonics
can be resolved in experiment [
33
], the MIRO mass can be
determined with high precision by fitting simultaneously the
positions of both MIRO minima and maxima to
=
N
±
1
/
4
with integer
N
(see also Refs. [
8
–
10
]). For the sample in
Fig.
2(b)
the resulting MIRO effective mass is found to be
m
∗
MIRO
=
(0
.
335
±
0
.
006)
m
0
, i.e., more than 10% larger than
the cyclotron mass. The open circles in Fig.
2(d)
shows the
positions of several selected extrema of MIRO [as marked
in Fig.
2(b)
] extracted from measurements at different mi-
crowave frequencies for illustrative purposes. Solid lines are
linear fits using the expression
f
=
(
N
±
1
/
4)
eB
/
2
π
m
∗
MIRO
,
where
N
=
1
,
2
,
3, and 4. An average over the obtained values
of the fitting parameter
m
∗
MIRO
yields 0
.
335
m
0
.
A plausible reason for the drastically different response
to microwave illumination between low [Fig.
2(a)
] and high
density samples [Fig.
2(b)
] is the much higher temperature
sensitivity of the longitudinal resistance in lower density sam-
ples. Figure
2(e)
displays this temperature dependence for
these samples in the absence of radiation and a magnetic field.
In both cases the behavior is metallic with a drop in resis-
tance as
T
is reduced. However, in the higher density sample
R
xx
bottoms out for temperatures below approximately 1 K,
whereas in the low-density sample the longitudinal resistance
continues to drop down to the lowest accessible temperature.
For low density samples, this strong
T
-dependence in the
low-temperature regime is highly reproducible [
12
]. It can
be linked to the Bloch-Grüneisen regime for acoustic phonon
scattering [
38
] as well as a higher low-
T
mobility. The Bloch-
Grüneisen regime is entered at a lower temperature in low
density samples and alloy or interfacial scattering is weaker
due to the reduced Mg-content in the Mg
x
Zn
1
−
x
O cap layer.
The response of the 2DES to microwave induced heating can
be expressed as (
δ
R
xx
/δ
T
)
T
[
23
] and is obviously enhanced
when
R
xx
shows a higher sensitivity to temperature. The pho-
toresponse is therefore prominent in low density samples, but
absent in high density samples.
Figure
3
presents
δ
R
xx
data recorded on a sample with an
intermediate density
n
=
4
.
3
×
10
11
cm
−
2
for different levels
FIG. 3. Microwave-induced change
δ
R
xx
of the longitudinal re-
sistance recorded on a sample with
n
=
4
.
3
×
10
11
cm
−
2
for different
levels of the output power
P
out
(as marked) of the
f
=
95 GHz
microwave radiation. The same data are plotted against
B
in panel
(a) and against
=
ω/ω
c
in panel (b), where
is calculated using
the MIRO mass
m
∗
MIRO
=
0
.
375. Linear scales are used.
of the output power of the microwave generator
P
out
at a
fixed frequency of 95 GHz and temperature of the surrounding
cryogenic fluid of
T
=
1
.
2 K. At the highest incident power,
P
out
=
6
.
3 mW, a strong CR peak appears at the position
corresponding to the bare cyclotron mass
m
∗
CR
, as would be
expected in a sample that still exhibits a temperature depen-
dence of
R
xx
. The CR peak is, however, accompanied by a
MIRO signal. The former decays much faster than MIRO as
the microwave power is lowered and heating is suppressed.
Indeed, at about an order of magnitude lower power,
P
out
=
0
.
78 mW, the CR feature has vanished almost entirely, while
the MIRO signal remains strong [
33
]. Samples with interme-
diate densities (4
.
3
n
4
.
7
×
10
11
cm
−
2
) therefore enable
to simultaneously extract the effective mass unaltered by in-
teractions as well as the renormalized mass from a single
δ
R
xx
trace. An additional support for this interpretation comes
from independently measured transmission signal which pro-
vides the same position of the CR as the CR feature in
δ
R
xx
. Figure
3(b)
plots
δ
R
xx
as a function of
=
ω/ω
c
using
m
∗
MIRO
=
0
.
375
m
0
obtained from an analysis of the MIRO at
T
=
600 mK. If the values of the cyclotron and MIRO mass
were the same, the CR peak would occur at
=
1. However,
we see that it coincides with the position of the first MIRO
minimum at
5
/
4. We conclude that for this particular
density the MIRO mass
m
∗
MIRO
is renormalized by interactions
and is approximately 25% larger than the bare cyclotron mass
m
∗
CR
0
.
3
m
0
.
Figure
4
is a compilation of the effective masses obtained
via four different methods for samples covering the entire
available range of carrier densities. In addition to the CR
mass obtained both from magnetotransmission (diamonds)
and from the photoresistance
δ
R
xx
(triangles) we include the
MIRO mass
m
∗
MIRO
(squares) and the mass
m
∗
SdHO
(circles) ob-
tained from the temperature dependence of the Shubnikov–de
Haas oscillations on a set of samples with similar characteris-
tics in previous studies [
14
,
15
]. Within experimental accuracy,
115432-4
MICROWAVE RESPONSE OF INTERACTING OXIDE ...
PHYSICAL REVIEW B
102
, 115432 (2020)
FIG. 4. The values of effective mass extracted using MIRO pe-
riod (squares), magnetotransmission (diamonds), CR peak in
δ
R
xx
(triangles), and SdHO (circles) versus the carrier density
n
. Dashed
line represents the band mass
m
b
≈
0
.
3
m
0
of bulk ZnO, solid lines
are guides for the eye.
the values of the CR mass extracted from the magnetotrans-
mission and from the
δ
R
xx
coincide with each other and with
the band effective mass
m
b
≈
0
.
3
m
0
(dashed line). The over-
all mean value for all samples yields
m
∗
CR
=
(0
.
3
±
0
.
01)
m
0
.
The MIRO mass
m
∗
MIRO
was obtained from the dispersion
curves
f
(
B
) of MIRO extrema, as exemplified in Fig.
2(d)
.
Its value displays an increase of 42% from 0
.
28
m
0
to 0
.
4
m
0
as the carrier density
n
is reduced from 20
×
10
11
cm
−
2
to 3.6
×
10
11
cm
−
2
. In the range of densities where both methods are
applicable,
m
∗
MIRO
agrees fairly well with
m
∗
SdHO
.
IV. CONCLUSION
In summary, we presented a combined study of mag-
netotransport and magnetotransmission on a series of
MgZnO/ZnO based 2DES under microwave illumination.
Across the entire range 2
n
20
×
10
11
cm
−
2
of charge
densities the magnetotrasmission displays the CR minima at
magnetic field positions consistent with the unrenormalized
band mass of the material. The corresponding CR-induced
features in magnetotransport were only resolved in low den-
sity devices. We identified a strong temperature dependence
of the zero-field resistance in such dilute samples, which
indicates the reason for a stronger CR response in the pho-
toresistance at low density. MIRO dominate the electrical
response in high density samples and reveal a strong renor-
malization of the quasiparticle effective mass. The reduction
at high carrier concentrations as well as the enhancement,
which augments as the electron density is diluted, agree with
the expected Fermi-liquid renormalization due to interaction
effects.
ACKNOWLEDGMENTS
We thank M. Zudov for useful comments. We acknowledge
the financial support of JST CREST Grant No. JPMJCR16F1,
Japan. J.F. is grateful for support from the Max Planck-
University of British Columbia-University of Tokyo Center
for Quantum Materials and the Deutsche Forschungsgemein-
schaft (FA 1392
/
2-1). Y.K. acknowledges JST, PRESTO
Grant No. JPMJPR1763, Japan. I.D. acknowledges support
from the Deutsche Forschungsgemeinschaft (projects DM
1/4-1 and GA501/14-1).
APPENDIX A: DETERMINATION OF THE
QUASIPARTICLE MASS FROM MIRO
The procedure of determination of the effective quasiparti-
cle mass
m
∗
MIRO
from MIRO is illustrated in Fig.
5
for a sample
with
n
=
7
.
5
×
10
11
cm
−
2
. The photoresistance
δ
R
xx
under
f
=
84 GHz microwave illumination is shown in Fig.
5(a)
as a function of magnetic field. We first extract the posi-
tions
B
e
of MIRO extrema. In Fig.
5(b)
, the inverse values
1
/
B
e
are plotted against
assuming a
∓
1
/
4offsetofthe
MIRO maxima (minima) with respect to the nodes at integer
=
N
,seeEq.(
1
) of the main text. It is seen that within
such a representation the data points fall on a straight line
going through the axes origin [
39
]. Utilizing the relation
FIG. 5. (a) Photoresponse
δ
R
xx
for a sample with
n
=
7
.
5
×
10
11
cm
−
2
at microwave frequency
f
=
84 GHz. (b) The inverted
B
-positions of MIRO maxima and minima for the data in panel
(a) plotted against
N
−
1
/
4and
N
+
1
/
4, respectively. Here,
N
is an
integer. All points fall on a straight line hitting the coordinate origin.
Fitting the slope yields the quasiparticle (MIRO) mass
m
∗
MIRO
=
(0
.
335
±
0
.
006)
m
0
. In panel (c) the measured microwave-induced
change of resistivity
δ
R
xx
[multiplied by exp(
a
/
B
) with
a
=
0
.
4T
for better visibility of high harmonics] is plotted against the inverse
of magnetic field which is rescaled to
using the obtained value of
m
∗
MIRO
. Linear scales are used.
115432-5
D. TABREA
et al.
PHYSICAL REVIEW B
102
, 115432 (2020)
FIG. 6. Power dependence of the MIRO amplitude [panel (a),
sample with
n
=
7
.
5
×
10
11
cm
−
2
] and of the amplitude of the CR
peak [panel (b), sample with
n
=
2
.
3
×
10
11
cm
−
2
]. Solid lines in
panel (a) are a guide for the eye illustrating a linear and square-root
power dependence. The analysis suggests a transition from a linear
to a sublinear regime of MIRO at the output power between 1 and 2
mW. The magnitude of the CR peak in panel (b) shows a monotonic
sublinear dependence across the entire power range within which
such a signal could be clearly identified.
=
2
π
fm
∗
MIRO
/
eB
, a linear fit with fixed zero intercept yields
m
∗
MIRO
=
(0
.
335
±
0
.
006)
m
0
. To illustrate the accuracy of the
procedure, in Fig.
5(c)
we plot full data for
δ
R
xx
against
=
2
π
fm
∗
MIRO
/
eB
calculated from the
B
values using the
obtained effective mass. For a better visibility of weak oscil-
lations at high
>
6, we multiplied
δ
R
xx
by exp(
a
/
B
) with
a
=
0
.
4 T. It is seen that all maxima and minima appear
precisely at
=
N
∓
1
/
4for
N
>
1. In this analysis we left
out the extrema around
N
=
1 where deviations are expected
due to a more complex behavior of the MIRO amplitude near
the CR [
26
].
As the above example shows, in high density samples the
MIRO effective mass can be accurately determined from a
single trace due to the large number of oscillations detectable
in the photoresponse. In the low density regime, however,
MIRO are weaker and higher harmonics (
>
4) are not visi-
ble. To improve the accuracy of extracted
m
∗
MIRO
in this case,
we processed data recorded for a larger set of microwave
frequencies. In Fig.
4
we use the average values and standard
deviations of
m
∗
MIRO
obtained from the entire collected data set
for a given sample.
APPENDIX B: POWER DEPENDENCE OF THE
MICROWAVE RESPONSE
In Fig.
6
we show the power dependence of the MIRO
amplitude [Fig.
6(a)
, sample with
n
=
7
.
5
×
10
11
cm
−
2
] and
of the amplitude of the CR peak in photoresistance [Fig.
6(b)
,
FIG. 7. The magnetic field values
B
where a photoresistance
peak is detected for different microwave frequencies
f
on a sample
with
n
=
2
.
3
×
10
11
cm
−
2
. The data points are plotted using quadratic
scales (
f
2
vs.
B
2
). The main panel presents data for the entire
frequency set, while the inset only includes data for
f
<
30 GHz.
Dashed line in the main plot is a linear fit to the data for
f
>
75 GHz
with the additional constraint that it passes through the origin. This
corresponds to the CR relationship between frequency and field,
f
=
1
/
T
c
, with
m
∗
CR
=
0
.
32
m
0
. The dashed line in the inset is a
linear fit with a nonzero offset, as in Eq. (
C1
). It yields the plasmon
frequency
f
p
=
11
.
8 GHz.
sample with
n
=
2
.
3
×
10
11
cm
−
2
]. Both measurements were
made at a temperature
T
=
1
.
2Kusing
f
=
95 GHz radia-
tion. The MIRO amplitude increases linearly in the low-power
regime
P
<
1
.
5 mW. Above this value, a sublinear be-
havior can be observed. For even higher-power radiation
(
P
>
4 mW), the MIRO amplitude saturates and eventually
starts to decrease. Importantly, no change of the MIRO phase
is observed, i.e., the minima and maxima remain shifted by
1
/
4 from integer values of
across the entire available mi-
crowave power range. This suggests that both the transition
to the sublinear growth and subsequent decay of MIRO with
increasing microwave power are due to heating [
40
], and
not due to intrinsic nonlinear effects. The later would rather
produce a significant reduction of the MIRO phase and it can
even lead to the emergence of additional oscillatory structure
around integer
[
26
,
41
,
42
]. The magnitude of the CR peak in
Fig.
6(b)
shows sublinear growth for power up to the highest
available output. At small power, it becomes difficult to isolate
the CR peak from the background signal, so, unlike MIRO
in Fig.
6(a)
, no clear transition to the linear regime could be
identified in this case.
APPENDIX C: ROLE OF CONFINED
MAGNETOPLASMONS
For large microwave frequencies
f
>
75 GHz, used for
the analysis in the main text, both the magnetic field values
where minima in the magnetotransmission [see Fig.
1(c)
] and
maxima in the magnetoresistance response [Fig.
2(c)
] appear
were found to be proportional to the microwave frequency.
115432-6