We report on magnetotransport properties of a MgZnO/ZnO heterostructure
subjected to weak direct currents. We find that in the regime of overlapping
Landau levels, the differential resistivity acquires a quantum correction
proportional to both the square of the current and the Dingle factor. The
analysis shows that the correction to the differential resistivity is dominated
by a current-induced modification of the electron distribution function and
allows us to access both quantum and inelastic scattering rates.Comment: 4 pages, 3 figure

Falson, J.

Kawasaki, M.

Kozuka, Y.

Shi, Q.

Smet, J.

Tsukazaki, A.

Zudov, M. A.

English

arXiv

We report on magnetotransport properties of a MgZnO/ZnO heterostructure subjected to weak direct currents. We find that in the regime of overlapping Landau levels, the differential resistivity acquires a quantum correction proportional to both the square of the current and the Dingle factor. The analysis shows that the correction to the differential resistivity is dominated by a current-induced modification of the electron distribution function and allows us to access both quantum and inelastic scattering rates

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Nonlinear response of a MgZnO/ZnO heterostructure close to zero bias
Q. Shi,1 J. Falson,2 M. A. Zudov,1, ∗ Y. Kozuka,3 A. Tsukazaki,4 M. Kawasaki,3, 5 and J. Smet2
1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA
2Max-Planck-Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
3Department of Applied Physics and Quantum-Phase Electronics Center, University of Tokyo, Tokyo 113-8656, Japan
4Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
5RIKEN Center for Emergent Matter Science, Wako 351-0198, Japan
(Received 26 June 2017)
We report on magnetotransport properties of a MgZnO/ZnO heterostructure subjected to weak direct currents.
We find that in the regime of overlapping Landau levels, the differential resistivity acquires a quantum correction
proportional to both the square of the current and the Dingle factor. The analysis shows that the correction to
the differential resistivity is dominated by a current-induced modification of the electron distribution function
and allows us to access both quantum and inelastic scattering rates.
Nonlinear magnetotransport in high Landau levels of two-
dimensional electron systems (2DESs) offers a unique ap-
proach to obtain information on both electron-impurity and
electron-electron scattering. For example, at high direct cur-
rents the differential resistance exhibits Hall field-induced
resistance oscillations (HIRO) [1–11] which originate from
electron (or hole [10]) backscattering off impurities leading
to transitions between Landau levels. Since such transitions
are accompanied by a displacement of the electron guiding
center by a cyclotron diameter 2Rc, applied current density
j translates to an energy scale eρHj(2Rc), where ρH is the
Hall resistivity. HIRO then result from the commensurabil-
ity between this energy and the inter-Landau level spacing
~ωc, where ωc is the cyclotron frequency of a charge carrier.
In overlapping Landau levels, the corresponding correction to
the differential resistance r is given by [12]
δr
R0
≈ 16
π
τ
τπ
λ2 cos 2πǫj , πǫj ≫ 1 , (1)
where ǫj = eρHj(2Rc)/~ωc, R0 is the low-temperature,
linear-response resistance at zero magnetic field (B = 0),
τ is the disorder-limited transport scattering time, τπ is the
backscattering time, λ = exp(−π/ωcτq) is the Dingle factor,
and τq is the quantum lifetime.
The disorder in a 2DES can often be conveniently separated
into a short-range (e.g., background impurities) and a long-
range (e.g., remote ionized donors) component, characterized
by “sharp” and “smooth” scattering rates (τ−1
sh
and τ−1sm ), re-
spectively. When τ ≫ τq, as in a conventional high-mobility
modulation-doped 2DES, τsh ≈ τπ and τsm ≈ τq. Therefore,
the analysis of the HIRO amplitude using Eq. (1) can yield in-
formation on both sharp and smooth disorder components in
a 2DES under study.
In the regime of weak electric fields, the differential re-
sistance acquires a negative quantum correction which scales
with j2, as has been observed in GaAs heterostructures [3, 4,
6, 8, 13–19]. In contrast to Eq. (1), this current-induced cor-
rection originates both from the low ǫj counterpart of Eq. (1)
[12] (displacement mechanism) and from the oscillatory mod-
ification of the energy distribution function (inelastic mecha-
nism) [20]. More specifically, in overlapping Landau levels,
δr can be written as [12]
δr
R0
≈ −αǫ2j , πǫj ≪ min{1, (2τ/τin)1/2} , (2)
where
α = α0λ
2 , α0 = 12π
2
(
3τ
16τ⋆
+
τin
τ
)
. (3)
Here, τ−1⋆ entering the first (displacement) term can be ex-
pressed as τ−1⋆ = 3τ
−1
0 − 4τ−11 + τ−12 , where τ−1n represents
n-th angular harmonic of the rate of scattering on angle θ,
τ−1θ =
∑
τ−1n e
inθ [21]. This displacement term can never ex-
ceed 9/16 (sharp-disorder limit). In contrast, the second (in-
elastic) term in Eq. (3), given by τin/τ ∼ ~EF /τ(kBT )2 (EF
is the Fermi energy, τin is the inelastic relaxation time), can be
significantly larger than unity, especially in high density and
low mobility 2DESs [8, 13–19]. In such systems, nonlinear
transport at small j offers a convenient way to obtain τin and
thus access the strength of electron-electron interactions in the
2DES under study.
In this paper the capability of nonlinear transport to re-
veal information about scattering sources is exploited on a
MgxZn1−xO/ZnO heterostructure [11, 22–27]. We find that
the correction to the differential resistivity can be well de-
scribed by Eq. (2). The analysis of the curvatureα reveals that
the observed nonlinear response is governed by the current-
induced modification of the electron distribution function,
consistent with the theoretical estimates. From the Din-
gle analysis we obtain τq ≈ 2 ps, in good agreement with
the values found from recent measurements of Shubnikov-de
Haas [24], microwave-induced [27], and Hall field-induced
[11] resistance oscillations, confirming the applicability of
Eqs. (2), (3). More importantly, our experiments allow us to
estimate the inelastic relaxation time τin ≈ 40 ps, which could
not be accessed in previous studies [11, 24, 27]. While a sim-
ilar approach has been previously employed to study inelas-
tic relaxation in GaAs/AlGaAs quantum wells, a much lower
mobility and higher density of our MgZnO/ZnO heterostruc-
ture suggest its potential applicability to a diverse variety of
2DESs.
2




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FIG. 1. (Color online) (a) r(B) at different I from 0 (bottom curve)
to 0.75 mA (top curve), in steps of 0.25 mA. HIRO maxima are
marked by 1+ and 2+. The curves are not shifted. (b) r(I) at dif-
ferent B from 1 to 2 T, in steps of 0.25 T. The dashed line represents
r(I) at B = 0.
Our sample was fabricated from a MgxZn1−xO/ZnO het-
erostructure grown using liquid ozone-based molecular beam
epitaxy [26, 28]. A Hall bar of width of ≈ 0.09 mm and
distance between voltage probes of ≈ 0.8 mm was defined
by scratching the wafer with a tungsten needle [11]. Electri-
cal contacts were made by soldered indium. At T ≈ 1.35
K, our 2DES has density ne ≈ 2.0 × 1012 cm−2 and mo-
bility µ ≈ 2.3 × 104 cm2/Vs. The differential resistance
r was recorded using a standard four-terminal lock-in tech-
nique at a constant coolant temperature T ≈ 1.35 K either
while sweeping magnetic field B at constant direct current I
or while sweeping I at a constant B.
To facilitate the discussion of our results in the regime of
weak currents, we first briefly summarize the main findings
of related HIRO experiments. In Fig. 1(a) we present r as a
function of B recorded at different I from 0 (bottom curve)
to 0.75 mA (top curve), in steps of 0.25 mA. The trace at
I = 0 is rather featureless, showing only weak Shubnikov-
de Haas oscillations at B & 1 T. The data at I = 0.25
mA and higher currents, however, reveal HIRO (cf. 1+, 2+)
which spread over a wider B range with increasing I . The
positions of the HIRO maxima are well described by B+N ≈
(m⋆
√
8π/ne/e
2)(j/N) ∝ j/N [11], where N = 1, 2, 3, ... .
However, in contrast to what one might expect, the increase
in I does not lead to observation of more oscillations. This
is in line with the recent study [11] which found that τq de-
creases rapidly with I . Also, like Ref. 11, we observe that
the zero-field differential resistance r0 also increases with I ,
suggesting a decrease of τ . Both observations are consistent
with a scenario that Joule heating leads to elevated electron
temperature which, in turn, causes enhanced electron-electron
and electron-phonon scattering [11]. Owing to this uninten-
tional heating, our HIRO experiments never revealed more
than three oscillations (regardless of I) which precluded ex-
tracting τq from a conventional Dingle analysis. Instead, we
had to resort [11] to fitting experimental curves with the HIRO
expression whose applicability, in contrast to Eq. (1), is not
limited to πǫj ≫ 1 [12]:
δr
R0
= − 2τ
τsh
λ2
(
ζ
[
J20 (ζ)
]′′)′
. (4)
Here, J0 is the Bessel function and the prime denotes a deriva-
tive with respect ζ = πǫj . Not surprisingly, the obtained τq
value showed significant dependence on I , reflecting a sizable
electron-electron scattering contribution [5, 29]. As a result,
the impurity-limited τq could only be estimated by extrapolat-
ing the I-dependence of the quantum scattering rate to zero
current [11].
An alternative way to study HIRO is to sweep I while keep-
ing B fixed [3]. In Fig. 1(b) we present r(I) at different B
from 1 to 2 T in steps of 0.25 T (solid lines) and at B = 0
(dashed line). At B = 1 T and higher, oscillations in r with
I are superimposed on a smooth, monotonically increasing
background which closely mimics r(I) at B = 0. Concur-
rently, the oscillations move to higher I with increasing B
while growing in amplitude. This increase originates from
enhanced modulation of the density of states at higherB.
At the focus of the present work is a maximum centered
at I = 0 which becomes more pronounced with increas-
ing B, see Fig. 1(b). As we show below, this maximum ap-
pears due to Landau quantization, primarily, as a result of
current-inducedmodification of the electron distribution func-
tion [12, 20]. We further illustrate how this feature can be used
to obtain both quantum and inelastic lifetimes.
In Fig. 2(a) we replot r shown in Fig. 1(b) as a function of
ǫj = 2eρHRcj/~ωc, where ωc = eB/m
⋆,m⋆ = 0.3m0 [11],
and m0 is the mass of a free electron. Oscillations at all B
are lined up and display maxima at ǫj ≈ 1, 2 and a minimum
at ǫj ≈ 1.5, indicating that ǫj was obtained consistently. The
reduction of the oscillation amplitude at smaller B is as result
of increased Landau level overlap manifested in the decay of
the Dingle factor. We also notice that the oscillation amplitude
decreases with ǫj ∝ j which can be attributed to Joule heating
discussed above.
At small ǫ2j ≪ 1, the differential resistance is suppressed
and this suppression becomes more pronounced with increas-
3

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FIG. 2. (Color online) (a) r(ǫj) at different B from 1 to 2 T, in steps
of 0.25 T. (b) δr/R0(0) vs. ǫj at different B from 0.6 to 1.5 T, in
steps of 0.1 T. The dashed line is the fit to the data atB = 1.5 T with
Eq. (2).
ing B. To examine this regime in detail, we present in
Fig. 2(b) δr/R0(0) ≡ [r − r(0)]/R0(0) as a function of ǫj
at differentB from 0.6 (top) to 1.5 T (bottom), in steps of 0.1
T. While at B = 0.6 T the trace is virtually flat, the curva-
ture rapidly increases with B. An example of a fit to the data
at B = 1.5 T with Eq. (2) (dashed line) demonstrates excel-
lent overlap with the experimental data at |ǫj | . 0.02. As we
show below, the applicability condition of Eq. (2) is well sat-
isfied since (2τ/τin)
1/2 ≈ 0.5, which far exceeds our fitting
range of |πǫj | . 0.06
We next fit our data at all otherB and obtain the curvatureα
which is the only fitting parameter. Being guided by Eq. (3),
we construct a Dingle plot, presented in Fig. 3, showing the
reduced curvatureα/12π2 as a function of 1/B on a log-linear
scale. At 1/B & 0.7 T, the curvature exhibits exponential
dependence from which we obtain τq ≈ 2.0 ps using Eq. (3).
The deviation of the data from the exponential dependence at
lower 1/B has been previously observed in experiments on
GaAs quantumwells in the regime of separated Landau levels
[8]. In our MgZnO/ZnO sample, the Landau levels start to
separate atB ≈ 1.3 T, as estimated from ωcτq = π/2 [30, 31].




a


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c
FIG. 3. (Color online) Reduced curvature α/12π2, obtained from
the fits in Fig. 2(b) as a function of 1/B on a log-linear scale. The
fit to the lower B part of the data with Eq. (3) (solid line) generates
τq ≈ 2.0 ps and 3τ/16τ⋆ + τin/τ ≈ 11.
We now examine the intercept of the Dingle plot in Fig. 3
and extract the inelastic scattering time. We first recall that
the first term in Eq. (3), representing the displacement con-
tribution, has its maximal value of 3τ/16τ⋆ ≈ 9/16 (sharp
disorder limit), a condition which was established in a recent
HIRO study [11]. Since the intercept of the fit in Fig. 3 yields
α0/12π
2 ≈ 11≫ 9/16, we conclude that the inelastic mech-
anism dominates the nonlinear response at small direct cur-
rents in our MgZnO/ZnO heterostructure. From this value,
we then find τin ≈ 40 ps.
It is interesting to compare the obtained value of τin to the
one expected from theoretical considerations [20, 32]. In the
regime of our experiment [33], theory predicts
τin ≈ 0.82τee , ~
τee
≈ πk
2
BT
2
4EF
ln
(
2vF
aBωc
√
ωcτ
)
, (5)
where τ−1ee is the electron-electron scattering rate for a test
particle at the Fermi energy, vF is the Fermi velocity, and
aB ≈ 1.5 nm is the Bohr radius. At B = 0.7 T, the loga-
rithmic factor is about 5.9 and Eq. (5) yields τin ≈ 140 ps,
a few times larger than the value obtained from our Dingle
analysis.
Since we have limited our analysis to currents consider-
ably smaller than those needed to induce noticeable change
in r at B = 0, we believe that any residual Joule heating
is unlikely to be responsible for this discrepancy. On the
other hand, Eqs. (2), (3) were derived assuming ωcτ ≫ 1 and
ωcτq . 1. Since in high-mobility GaAs samples τ ≫ τq,
both of these conditions can be simultaneously met. The situ-
ation is markedly different in our MgZnO/ZnO sample where
τq . τ ≈ 3.8 ps and the above constraints are only marginally
satisfied. Indeed, at B = 0.7 T we estimate ωcτ ≈ 1.5 and
ωcτq ≈ 0.8.
In summary, we have studied nonlinear magnetotransport
in a MgxZn1−xO/ZnO heterostructure in the regime of weak
4direct currents. We have found that the differential resistiv-
ity acquires a correction δr which is quadratic in direct cur-
rent and decays exponentially with 1/B. The analysis of
the B-dependence of the curvature suggests that the nonlin-
ear response is governed by a dc field-induced modification
of the electron distribution function. The Dingle analysis in
the regime of overlapping Landau levels reveals the quan-
tum lifetime τq ≈ 2.0 ps, consistent with existing magneto-
transport studies in similar MgxZn1−xO/ZnO heterostructures
[11, 24, 27]. However, the regime of small direct currents ex-
plored in the present work also allowed us to obtain the in-
elastic relaxation time of τin ≈ 40 ps, which was not pre-
viously measured in MgxZn1−xO/ZnO heterostructures. Our
study demonstrates that the nonlinear magnetotransport is a
powerful technique to access scattering times in 2D systems
not necessarily having high mobility. As such, the technique
could be useful to investigate electron-impurity and electron-
electron scattering in a broad variety of materials.
We thank I. A. Dmitriev for discussions and P. Herlinger
for assistance with the microscope. This work was supported
by the US Department of Energy, Office of Basic Energy Sci-
ences, under Grant No. DE-SC002567 (University of Min-
nesota) and by Grant-in-Aids for Scientific Research (S) No.
24226002 from MEXT, Japan (University of Tokyo).
∗ Corresponding author: zudov@physics.umn.edu
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Nonlinear response of a MgZnO/ZnO heterostructure close to zero bias

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Nonlinear response of a MgZnO/ZnO heterostructure close to zero bias

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