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
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 inter-
action 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 unrenormalized 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.
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
inter-particle interaction. As the charge carrier density
n
is reduced, the Coulomb energy (
E
C
∝
√
n
) becomes
comparable and eventually even exceeds the electronic
Fermi energy (
E
F
∝
n
). In the limit of high concentra-
tion, charge carriers interact weakly and the system’s pa-
rameters follow from band theory. At intermediate densi-
ties, a Fermi liquid described by parameters that undergo
a renormalization due to interactions, such as the effec-
tive mass
m
∗
and the
g
-factor, forms.
1,2
Finally, in the
dilute limit, a breakdown of the Fermi-liquid paradigm
is anticipated. This culminates either in particle local-
ization 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 mea-
sured using cyclotron resonance (CR), since in view
of Kohn’s theorem
6
the resonance frequency is insen-
sitive to inter-particle correlations at the vanishingly
small momentum of the incident radiation. Estimates
of the renormalized effective mass mostly rely on tem-
perature dependent studies of the Shubnikov-de Haas
oscillations.
1
Recently, oscillatory magnetotransport fea-
tures that appear under incident microwave radiation,
referred to as microwave induced resistance oscilla-
tions or MIRO,
7
have been advanced as an alterna-
tive tool for obtaining the interaction-dependent effective
mass.
8–11
This method has mainly been deployed 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 simultaneously
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 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
heterostructures by performing simultaneous magneto-
transmission and magnetotransport measurements under
microwave illumination.
19
For the entire span of charge
densities, the transmission signal displays resonant fea-
tures at the cyclotron resonance. An analysis of the
density dependence of this signal yields an electron ef-
fective mass close to the band mass
m
b
≈
0
.
3
m
0
of bulk
ZnO,
20,21
where
m
0
is the free electron mass. In contrast,
the resistively detected magnetotransport signal of the
devices exhibits qualitatively different responses depend-
ing on the charge carrier density. While low carrier den-
sity samples (
n <
3
.
5
×
10
11
cm
−
2
) exclusively display a
conventional response due to heating of the electron sys-
tem during resonant microwave absorption at CR,
22–24
in
the higher density regime the response is dominated by
the less common MIRO.
25–27
No CR related feature was
arXiv:2006.13627v1 [cond-mat.mes-hall] 24 Jun 2020
2
detected in the magnetoresistance of high density sam-
ples. These two signals permit further analysis of the ef-
fective mass. While the CR-associated feature reflects a
similar band mass
m
b
≈
0
.
3
m
0
to that obtained in trans-
mission studies, the value extracted from MIRO exhibits
a systematic increase with decreasing carrier concentra-
tion reflecting the renormalization of the Fermi-liquid as
interactions augment.
28–31
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
×
3 mm
2
in order
to prepare samples in the van der Pauw geometry with
four or eight contacts. The contacts were made by evap-
orating Ti/Au and/or soldered indium along the perime-
ter 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 my-
lar film was glued across this opening to provide support
for the sample. Metalized 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
×
3 mm
2
carbon-covered kapton film contacted with sil-
ver paint was placed. Its resistance
R
t
exhibited a strong
negative bolometric response
δR
t
(
B
) =
−
C T
s
(
B
)
P
ext
proportional to the microwave power
T
s
(
B
)
P
ext
trans-
mitted through the sample containing the 2DES. Since
both the sensitivity coefficient,
C >
0, and the exter-
nal microwave power,
P
ext
, are
B
-independent, variations
of
δR
t
(
B
) directly reflect the
B
-dependence of the mi-
crowave 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 var-
ied between 300 mK and 1.4 K by pumping on the
3
He
surface. Monochromatic 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 multiplication leaves an inacces-
sible window for
f
≈
50
−
75 GHz. The microwaves
were delivered to the sample with the help of an over-
sized rigid rectangular waveguide. Their amplitude was
modulated at a frequency
f
mod
= 1 kHz. The longitu-
dinal 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 detection capability of an SR860 lock-in
amplifier as a fraction of the total signal that is modu-
FIG. 1: (a) Sketch of the experimental setup. (b) Mag-
netotransmission data (negated
δR
t
(
B
) reflecting the
B
-
dependence of the transmittance,
T
s
(
B
), solid lines) and pho-
toresistance
δR
xx
(
B
) (dashed lines) for three microwave fre-
quencies (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.
lated both at
f
AC
and at
f
mod
was deployed to selectively
record weak microwave-induced changes of the low fre-
quency resistance,
δR
xx
. To improve the signal-to-noise
ratio we also relied on double modulation detection of the
microwave induced changes of the carbon resistor,
δR
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
δR
t
has been recorded at different microwave fre-
quencies. A strong maximum was found in each trace,
corresponding to a minimum 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 mi-
crowave. This translates into a negative and nearly
B
-
independent off-resonant signal
δR
t
. Near the CR the
2DES absorbs and reflects a larger part of microwaves,
which leads to a lower transmitted power and therefore
3
to a lower heating of the carbon resistor. We note that
the external radiation power reaching the sample varies
significantly with the microwave 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 re-
frain 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 orig-
inates from interference effects within the sample that de-
pends on the wavelength of the radiation. Over a large
frequency range the effect is averaged out.
Panel (c) demonstrates that the
B
-positions of the
transmission minima are proportional to the microwave
frequency in the high frequency range. The slope ob-
tained from a linear fit of the data for
f
≥
75 GHz pass-
ing through the origin (solid line) establishes that these
minima match the CR condition,
f
=
eB/
(2
πm
∗
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 sam-
ple, the 2DES supports a confined plasmon mode. It
hybridizes with the cyclotron resonance mode to yield a
magnetoplasmon mode of non-zero frequency near
B
= 0.
This causes a deviation of the linear
B
-dependence of
the observed resonance frequency in the low field limit.
32
Therefore, data points recorded at frequencies below 50
GHz have been excluded from the mass analysis. The
CR is an ubiquitous feature 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 pho-
toresistance.
B. Photoresistance measurements
We now turn our attention to magnetotransport mea-
surements utilizing the double-modulation technique to
extract the microwave-induced variation of the longitu-
dinal resistance of the 2DES,
δR
xx
. In contrast to the
magnetotransmission signal that is dominated by the res-
onant reflection and absorption near the CR for the whole
range of electron densities, the resistance measurements
manifest more complex behavior depending on the charge
carrier density. In low-density samples (
n <
4
.
3
×
10
11
cm
−
2
), the most prominent feature in
δR
xx
is a broad
peak, as exemplified in Fig. 2(a). At
T
= 1
.
4 K the cor-
responding change in
R
xx
due to resonant absorption is
of the order of a few Ohm. Simultaneous measurement
of
δR
xx
and of the magnetotransmission signal reveals
that the peak is aligned with the minimum in
−
δR
t
. A
comparison between the two sets of data is displayed in
Fig. 1(b). It is therefore natural to ascribe this peak to
a conventional response due to resonant microwave ab-
sorption and associated electron heating near the CR.
In contrast, high density samples with
n >
4
.
7
×
10
11
cm
−
2
display no detectable resonant features in
δR
xx
at
FIG. 2: Representative examples for the recorded variation
of the longitudinal dc resistance,
δR
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 electron density [
n
= 7
.
5
×
10
11
cm
−
2
,
panel (b)] plotted on a linear scale. (c) Position of the max-
ima in
δR
xx
for the low-density sample shown in panel (a)
in the frequency vs. magnetic field plane (open squares) to-
gether with a linear fit passing through the origin (solid line).
(d) Position of selected MIRO extrema as marked in panel (b)
extracted from data obtained at different microwave frequen-
cies. Solid lines are linear fits to the data points using the
equation
f
= (
N
±
1
/
4)
eB/
2
πm
∗
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).
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
δR
xx
for
the low-density sample in Fig. 2(a) obtained for differ-
ent 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 min-
ima in the magnetotransmission data of Fig. 1(c). This
finding reinforces our interpretation of the peak of
δR
xx
as an effect of resonant heating of the 2DES in the vicin-
ity of the CR. The extracted value is close to the band
mass
m
b
≈
0
.
3
m
0
of bulk ZnO.
20,21
The dashed line in
Fig. 2(c) illustrates the expected position of the lowest-
order magnetoplasmon mode in this sample for a wave-
length of the dimensional plasmon equal to twice the
sample size,
λ
mp
= 6 mm.
32–34
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
4
1
/B
-periodic MIRO oscillations observed in high-density
samples are all reproduced well by the conventional
expression
25
δR
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 os-
cillations is determined by the quasi-particle effective
mass
m
∗
MIRO
. The latter enters the ratio
=
ω/ω
c
,
where
ω
= 2
πf
is the angular microwave frequency and
ω
c
=
eB/m
∗
MIRO
is the cyclotron frequency determining
the distance between neighboring Landau levels for quasi-
particles near the Fermi level. The “bare” cyclotron mass
m
∗
CR
extracted from the microwave transmission exper-
iment or the photoresistance feature represents the cy-
clotron dynamics of the 2DES probed as a whole in the
limit
k
→
0.
35,36
Its value is unaffected by a renormal-
ization of the Fermi liquid in view of momentum conser-
vation 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 har-
monics can be resolved in experiment,
32
the MIRO mass
can be determined with high precision by fitting simulta-
neously the positions of both MIRO minima and maxima
to
=
N
±
1
/
4 with integer
N
(see also Refs. [8], [9], and
[10]). For the sample in Fig. 2(b) the resulting MIRO ef-
fective 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)] ex-
tracted from measurements at different microwave fre-
quencies for illustrative purposes. Solid lines are linear
fits using the expression
f
= (
N
±
1
/
4)
eB/
2
πm
∗
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 tem-
perature sensitivity of the longitudinal resistance in lower
density samples.
Fig. 2(e) displays this temperature
dependence for these samples in the absence of radia-
tion and a magnetic field. In both cases the behavior
is metallic with a drop in resistance 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 con-
tinues to drop down to the lowest accessible tempera-
ture. For low density samples, this strong
T
-dependence
in the low temperature regime is highly reproducible.
12
It can be linked to the Bloch-Gr ̈uneisen regime for acous-
tic phonon scattering
37
as well as a higher low-
T
mobil-
ity. The Bloch-Gr ̈uneisen regime is entered at a lower
temperature in low density samples and alloy or in-
terfacial 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 (
δR
xx
/δT
)∆
T
,
22
and is obviously enhanced when
R
xx
shows a higher sensitivity to temperature. The photore-
sponse is therefore prominent in low density samples, but
absent in high density samples.
Figure 3 presents
δR
xx
data recorded on a sample with
an intermediate density
n
= 4
.
3
×
10
11
cm
−
2
for differ-
ent levels 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 high-
est incident power,
P
out
= 6
.
3 mW, a strong CR peak ap-
pears at the position corresponding to the bare cyclotron
mass
m
∗
CR
, as would be expected in a sample that still
exhibits a temperature dependence 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.
32
Samples with intermediate den-
sities (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
δR
xx
trace. An additional support for this interpretation
comes from independently measured transmission signal
which provides the same position of the CR as the CR
feature in
δR
xx
. Figure 3(b) plots
δR
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 co-
incides 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 ob-
tained via four different methods for samples covering
the entire available range of carrier densities. In addi-
tion to the CR mass obtained both from magnetotrans-
mission (diamonds) and from the photoresistance
δR
xx
(triangles) we include the MIRO mass
m
∗
MIRO
(squares)
and the mass
m
∗
SdHO
(circles) obtained from the tem-
perature dependence of the Shubnikov-de Haas oscilla-
tions on a set of samples with similar characteristics
in previous studies
14,15
. Within experimental accuracy,
the values of the CR mass extracted from the magne-
totransmission and from the
δR
xx
coincide with each
other and with the band effective mass
m
b
≈
0
.
3
m
0
(dashed line). The overall 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
5
FIG. 3: Microwave-induced change
δR
xx
of the longitudinal
resistance 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
calculated using
the MIRO mass
m
∗
MIRO
= 0
.
375. Linear scales are used.
FIG. 4: The values of effective mass extracted using MIRO
period (squares), magnetotransmission (diamonds), CR peak
in
δR
xx
(triangles), and SdHO (circles) versus the carrier den-
sity
n
. Dashed line represents the band mass
m
b
≈
0
.
3
m
0
of
bulk ZnO, solid lines are guides for the eye.
applicable,
m
∗
MIRO
agrees fairly well with
m
∗
SdHO
.
IV. CONCLUSIONS
In summary, we have presented a combined study of
magnetotransport and magnetotransmission on a series
of MgZnO/ZnO based 2DES under microwave illumina-
tion. 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 corre-
sponding CR-induced features in magnetotransport were
only resolved in low density devices. We identified a
strong temperature dependence of the zero-field resis-
tance in such dilute samples, which indicates the reason
for a stronger CR response in the photoresistance at low
density. MIRO dominate the electrical response in high
density samples and reveal a strong renormalization of
the quasi-particle 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 inter-
action effects.
Acknowledgements
We thank M. Zudov for useful comments. We acknowl-
edge the financial support of JST CREST Grant Num-
ber 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 Forschungsgemeinschaft (FA 1392/2-1).
Y.K. acknowledges JST, PRESTO Grant Number JP-
MJPR1763, Japan. I.D. acknowledges support from the
Deutsche Forschungsgemeinschaft (DM 1/4-1).
Appendix A: Determination of the quasiparticle
mass from MIRO
The procedure of determination of the effective quasi-
particle mass
m
∗
MIRO
from MIRO is illustrated in Fig. 5
for a sample with
n
= 7
.
5
×
10
11
cm
−
2
. The photore-
sistance
δR
xx
under
f
= 84 GHz microwave illumina-
tion is shown in Fig. 5(a) as a function of magnetic field.
We first extract the positions
B
e
of MIRO extrema. In
Fig. 5(b), the inverse values 1
/B
e
are plotted against
assuming a
∓
1
/
4 offset of the MIRO maxima (min-
ima) with respect to the nodes at integer
=
N
, see
Eq. (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.
38
Utilizing the relation
= 2
πfm
∗
MIRO
/eB
, a linear fit with fixed zero intercept
yields
m
∗
MIRO
= (0
.
335
±
0
.
006)
m
0
. To illustrate the ac-
curacy of the procedure, in panel (c) we plot full data
for
δR
xx
against
= 2
πfm
∗
MIRO
/eB
calculated from the
B
values using the obtained effective mass. For a better
visibility of weak oscillations at high
>
6, we multiplied
δR
xx
by exp(
a/B
) with
a
= 0
.
4 T. It is seen that all
maxima and minima appear precisely at
=
N
∓
1
/
4 for
N >
1. In this analysis we left out the extrema around
N
= 1 where deviations are expected due to a more com-
plex behavior of the MIRO amplitude near the CR.
25
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 oscilla-
tions detectable in the photoresponse. In the low-density
regime, however, MIRO are weaker and higher harmon-
6
FIG. 5: Panel (a): Photoresponse
δR
xx
for a sample with
n
= 7
.
5
×
10
11
cm
−
2
at microwave frequency
f
= 84 GHz.
Panel (b): The inverted
B
-positions of MIRO maxima and
minima for the data in panel (a) plotted against
N
−
1
/
4
and
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
δR
xx
(multiplied by exp(
a/B
)
with
a
= 0
.
4 T for better visibility of high harmonics) is plot-
ted against the inverse of magnetic field which is rescaled to
using the obtained value of
m
∗
MIRO
. Linear scales are used.
ics (
>
4) are not visible. In order to improve the accu-
racy 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 [panel (a), sample with
n
= 7
.
5
×
10
11
cm
−
2
]
and of the amplitude of the CR peak in photoresistance
[panel (b), sample with
n
= 2
.
3
×
10
11
cm
−
2
]. Both
measurements were made at a temperature
T
= 1
.
2 K
using
f
= 95 GHz radiation. The MIRO amplitude in-
creases linearly in the low-power regime
P <
1
.
5 mW.
Above this value, a sublinear behavior 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
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.
integer values of
across the entire available microwave
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,
39
and not
due to intrinsic nonlinear effects. The latter would rather
produce a significant reduction of the MIRO phase and it
can even lead to the emergence of additional oscillatory
structure around integer
.
25,40,41
The magnitude of the
CR peak in panel (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 back-
ground signal, so, unlike MIRO in panel (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. From the CR condition
f
= 1
/T
c
,
with
T
c
= 2
πm
∗
CR
/eB
, it is possible to extract the CR
7
effective mass
m
∗
CR
. It is found to be close to the band
mass of ZnO. At low microwave frequencies
f <
50 GHz
we observe a systematic deviation from the linear rela-
tionship between the
B
-field at extrema and microwave
frequency. Below we show that this deviation is due to a
coupling of the cyclotron motion with plasma oscillations
in a finite-size 2DES.
Within the simplest model considering a clean 2DES
and neglecting electrodynamic retardation effects, the
spectum of magnetoplasmons is given by
42
f
2
mp
=
f
2
p
+
T
−
2
c
.
(C1)
Here the square of the 2DES plasmon frequency,
f
2
p
=
ne
2
8
π
2
m
∗
̄
q,
(C2)
is proportional to the magnetoplasmon wave vector
q
and
inversely proportional to the effective electron mass
m
∗
.
In our square-shaped samples with side length
L
, the low-
est wave vector corresponding to the fundamental mag-
netoplasmon mode is
q
=
π/L
. Taking into account that
the lateral size
L
= 3 mm of the samples significantly
exceeds their thickness
w
= 0
.
3 mm (determined by the
thickness of the ZnO substrate having dielectric constant
1
= 8
.
5
0
in units of the vacuum permittivity
0
), the
effective dielectric constant ̄
entering Eq. (C2) can be
approximated as
43
̄
=
0
2
+
1
2
·
1
tanh(
qw
) +
0
1
+
0
tanh(
qw
)
'
2
.
23
0
.
(C3)
With the help of Eqs. (C1)-(C3) the experimental data
can be reproduced well, as illustrated in Fig. 7 for a sam-
ple with density
n
= 2
.
3
×
10
11
cm
−
2
. Here we take
the magnetic field values
B
corresponding to the resis-
tance peaks detected for different microwave frequencies
f
, and plot
f
2
against
B
2
(open circles). In accordance
with Eq. (C1), the data points are found to closely follow
a straight line. Within the scale of the main panel (show-
ing data for the entire frequency set) the offset of this line
away from the origin [given by
f
2
p
according to Eq. (C1)]
is barely visible. This justifies the application of the sim-
ple CR linear relationship between field and microwave
frequence,
f
= 1
/T
c
, for
f >
75 GHz to determine the
cyclotron mass in the main text. The corresponding lin-
ear fit (obtained for
f >
75 GHz data with
f
p
set to zero)
is shown as a dashed line in the main panel. The slope
corresponds to the cyclotron mass
m
∗
CR
= 0
.
32
m
0
from
the main text.
In the inset to Fig. 7 we present a magnified portion of
data for
f <
30 GHz. In the chosen representation,
f
2
vs.
B
2
, the data points still lie on a straight line, but a posi-
tive offset becomes evident. A linear fit of this portion of
the data (with slope fixed by
m
∗
CR
= 0
.
32
m
0
) yields the
offset value
f
2
p
, corresponding to
f
p
= 11
.
8 GHz. The
same value is obtained from calculation using Eqs. (C2)
and (C3). This good agreement supports the validity of
our interpretation of the data in terms of the dimensional
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 non-zero offset, as in Eq. (C1). It yields the plasmon
frequency
f
p
= 11
.
8 GHz.
magnetoplasmon resonance, although the precise coinci-
dence between calculated and extracted values of
f
p
can
also be accidential, in particular, in view of the simpli-
fied description of magnetoplasmons used for the above
estimates.
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