Zero differential resistance in two-dimensional electron systems at large filling factors

Abstract

We report on a state characterized by a zero differential resistance observed
in very high Landau levels of a high-mobility two-dimensional electron system.
Emerging from a minimum of Hall field-induced resistance oscillations at low
temperatures, this state exists over a continuous range of magnetic fields
extending well below the onset of the Shubnikov-de Haas effect. The minimum
current required to support this state is largely independent on the magnetic
field, while the maximum current increases with the magnetic field tracing the
onset of inter-Landau level scattering.

Full text

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arXiv:1007.2832v1 [cond-mat.mes-hall] 16 Jul 2010
Zero differential resistance in two-dimensional electron systems at large filling factors
A.T. Hatke,1H.-S. Chiang,1M.A. Zudov,1, ∗L.N. Pfeiffer,2and K.W. West2
1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA
2Princeton University, Department of Electrical Engineering, Princeton, NJ 08544, USA
(Received 10 February 2010)
We report on a state characterized by a zero differential resistance observed in very high Landau levels of
a high-mobility two-dimensional electron system. Emerging from a minimum of Hall field-induced resistance
oscillations at low temperatures, this state exists over a continuous range of magnetic fields extending well
below the onset of the Shubnikov-de Haas effect. The minimum current required to support this state is largely
independent on the magnetic field, while the maximum current increases with the magnetic field tracing the
onset of inter-Landau level scattering.
PACS numbers: 73.43.Qt, 73.63.Hs, 73.43.-f, 73.21.-b, 73.40.-c
Over the past decade it was realized that high mobil-
ity two-dimensional electron systems (2DESs) exhibit an ar-
ray of fascinating phenomena occurring in very high Lan-
dau levels where the Shubnikov-de Haas oscillations (Sd-
HOs) are not yet resolved. Among these are three classes of
magneto-oscillations, namely microwave-,1–16phonon-,17–20
and Hall field-21–24induced resistance oscillations (HIROs).
Remarkably, the minima of microwave-induced oscillations
can evolve into states with zero resistance.25–32These exotic
states are currently understood in terms of the absolute neg-
ative resistance which leads to an instability with respect to
formation of current domains.33–36Unfortunately, direct ex-
perimental confirmation of the domain structure has proven
difficult in irradiated 2DES and awaits future studies.
is therefore of great interest to explore if other classes of
oscillations give rise to phenomenologically similar states.
Recently, experiments revealed states with zero differential
resistance which emerged from the maxima of microwave-
induced resistance oscillations37,38and from the maxima of
the SdHOs.39–42Such states are analogous to the radiation-
induced zero-resistance states in a sense that they can also be
explained by the domain model.39
In this Rapid Communication we report on another state
characterized by a zero differential resistance which require
neither microwave irradiation nor the Shubnikov-de Haas ef-
fect. This state emerges from a minimum of HIROs in a high
mobility 2DES at low temperatures. Appearing in very high
Landau levels, this state is observed over a continuous mag-
netic field range extending well below the onset of the Sd-
HOs. The minimum current required to support such a state is
largelyindependentonthemagneticfield,whilethemaximum
currentincreases roughlylinearlywith the magneticfield trac-
ing the onset of inter-Landau level scattering. According to
the domain model,39these currents should be associated with
currents inside the domains.
The data presented in this Rapid Communication were ob-
tained on a Hall bar (width w = 100 µm) etched from a sym-
metrically doped GaAs/AlGaAs quantum well. After a brief
low-temperature illumination with visible light, density and
mobility were ne≃ 3.8×1011cm−2and µ ≃ 1.0×107cm2/Vs,
respectively. Differential resistivity, rxx≡ dVxx/dI, was mea-
sured using a quasi-dc (a few hertz) lock-in technique at tem-
peratures ranging from T = 1.5 to 3.0 K.
It
Since the zero differential resistance state (ZdRS) reported
here originates from the minimum of HIROs, we first dis-
cuss the basic physical picture behind this effect. According
to the “displacement” model,43–46HIROs originate from the
impurity-mediatedtransitions between Landaulevels tilted by
theHallelectricfield, Edc= ρHj, whereρHis theHall resistiv-
ity and j = I/w is the currentdensity. In this scenario, a domi-
nantscatteringprocessinvolvesanelectronwhichis backscat-
tered off an impurity. The guiding center of such an elec-
tron is displaced by a distance equal to the cyclotron diameter
2Rc. When 2Rcmatches an integral multiple of the real-space
Landau level separation, the probability of such events is en-
hanced. Thisenhancementmanifestsas amaximuminthedif-
ferential resistivity occurring whenever ǫdc≡ eEdc(2Rc)/?ωc
(ωcis the cyclotron frequency) is equal to an integer.23,44As
we will show, disappearanceof the ZdRS is directly related to
the fundamental HIRO peak at ǫdc= 1.
At 2πǫdc? 1, the theory44predicts another source of non-
linearities known as the “inelastic”47mechanism.
model a dc field creates a nonequilibriumdistribution of elec-
tron states which, in turn, leads to a resistance drop. We note
that both models were developed in the limit of strongly over-
lappedLandaulevels,aconditionwhichisnotalwayssatisfied
in experiments.23,39,40,48As a result, the relative importanceof
thesemechanismsat 2πǫdc? 1remainspoorlyunderstoodand
calls for further investigations. At the same time this regime
is directly relevant to the formation of states with zero differ-
ential resistance, as we show below.
We now present our experimental results. In Fig.1 (a) and
1 (b) we plot the longitudinal differential magnetoresistivity
rxx(B) acquired at T = 1.5 K and currents of I = 10 µA and
I = 20 µA, respectively. For comparison, each panel also in-
cludes the linear response (I = 0) longitudinal magnetoresis-
tivity ρxx(B),which is essentially featureless and exhibits only
the SdHOs starting to develop at B ? 2.5 kG. At low mag-
netic fields, the 2DES remains in the linear response regime
as manifested by the overlapping curves obtained at zero and
finite currents. However at higher magnetic fields, the data
obtained at finite currents show several distinct characteristics
signaling strong nonlinearities. First, the differential resistiv-
ity at 10 and 20 µA reveals a pronounced peak at B ≃ 0.2
kG and B ≃ 0.4 kG, respectively (cf.,↓). This peak occurs at
ǫdc≃ 1and,asdiscussedabove,originatesfromtheresonantly
In this

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2
2
1
0
rxx [?]
T = 1.5 K
I = 10 ?A
ZdRS
a
?
SdHO
I = 0 ?A
2
1
0
rxx [?]
3210
B [kG]
T = 1.5 K
I = 20 ?A
ZdRS
b
?
SdHO
I = 0 ?A
FIG. 1: (Color online) (a)[(b)] Differential magnetoresistivity rxx(B)
measured at I = 10 µA [20 µA] and T ≃ 1.5 K. Magnetoresistivity
ρxx(B) at I = 0 is shown for comparison.
enhancedscatteringdueto electrontransitionsbetweenneigh-
boring Hall field-tilted Landau levels. However, the most re-
markable feature of Fig.1 is the dramatic drop of rxxat higher
B (smaller ǫdc) which extends all the way to zero at B ? 1
kG. This drop marks a transition to the ZdRS whose generic
characteristics are the focus of this Rapid Communication.
We now show that the ZdRS presented in Fig.1 is quali-
tatively different from those reported earlier.37,39Indeed, the
ZdRS in Ref.37 emerged from a maximum of microwave-
induced resistance oscillations whereas our experiments are
performedwithoutmicrowaves. TheZdRS reportedin Ref.37
occurred in high magnetic fields where the linear response
resistivity is dominated by the SdHOs, and the ZdRS were
formedat thediscrete valuesofthe magneticfield correspond-
ing to the SdHO maxima (odd filling factors). Clearly, the
ZdRSshowninFig.1extendsoveracontinuousrangeofmag-
netic fields and does not rely on the existence of the SdHOs at
all; similar to the microwave-inducedzero-resistance states, it
persists to magnetic fields much lower than the onset of the
SdHOs. Finally, we note that our data reveal neither nega-
tive spikes in differential resistance preceding the ZdRS nor
temporal fluctuations reported in Ref.39.
In the context of the domain model, the range of currents
supporting the ZdRS is of particular interest. To investigate
this range it is convenient to employ an alternative measure-
ment technique in which the magnetic field B is held constant
and the current I is varied. This approach readily reveals both
the minimum and the maximum currents for a given mag-
netic field. One example of such a measurement performed at
B = 1.3 kG is presented in Fig.2(a) showing the differential
resistivity rxxas a function of applied current I. We observe
that rxxexhibitsa dramaticdropwith increasingcurrentwhich
eventually evolves into a state with zero differential resistance
(cf.,“ZdRS”). Formation of the ZdRS can therefore be char-
acterizedby a current I1≃ 10µA (cf., left line). Once formed,
the ZdRS persists up to a current I2≃ 23 µA (cf., right line)
-10
0
10
Vxx [?V]
-50050
I [?A]
ZdRS
b
2
0
rxx [?]
B = 1.3 kG
ZdRS
a
I2
I1
FIG. 2: (Color online) (a) Differential magnetoresistivity rxx(I) and
(b) voltage Vxxat B = 1.3 kG and T ≃ 1.5 K. Dashed line in (b)
represents Ohm’s law which holds at small I. Vertical lines mark the
ZdRS critical currents, I1and I2.
above which the differential resistivity starts to increase.
The drop in the rxxpreceding the ZdRS can be examined
quantitatively by fitting the data with a Gaussian rxx(I) =
rxx(0)exp(−I2/∆2
tivity and ∆1is the characteristic current which can be related
to I1. Anexampleofsuchafit overthecurrentrangefrom−20
to +20 µA is shown in Fig.2(a) by a dark line. It describes
the experimental data remarkably well yielding ∆1≃ 4.5 µA
from which I1can be estimated as I1≃ 2∆1.
The phenomenon can also be illustrated by a current-
voltage characteristic, shown in Fig.2(b), which is obtained
by integrating the data shown in Fig.2(a). Concurrent with
the drop in the rxxobserved in Fig.2(a), the longitudinal volt-
age Vxxdeparts from Ohm’s law (cf.,dashed line) and satu-
rates to a plateau which extends over a finite current range.
Within this range (cf., vertical lines), the voltage is inde-
pendent of the applied current and, as we show next, is also
largely insensitive to the magnetic field.
In Fig.3(a)-3(c)we present the differentialresistivity rxx(I)
obtained at higher magnetic fields, i.e. (a) B = 1.7 kG, (b) 2.1
kG, and (c) 2.5 kG, each measured at three different temper-
atures T = 1.5 K (solid line), 2.0 K (dotted line), and 3.0
K (dashed line). At T = 1.5 K all data show well devel-
oped ZdRS. At T = 2.0 K the ZdRS becomes narrower and
at T = 3.0 K are totally destroyed. While recent experiments
suggest electron-electron interactions as the origin of HIROs
temperature dependence,24this issue has not yet been theo-
retically considered43,44and awaits future studies. In what
follows we thus limit our discussion to the T = 1.5 K data
showing well developed ZdRS.
Examination of the data in Fig.3 reveals that the lin-
ear response resistivity rxx(0) and the lower critical current
(cf.,vertical lines), I1, bothhaveveryweak dependenceon the
magnetic field. As a result, the voltage at the plateau, which is
equal to the area underthe zero bias peak, is also largely inde-
1). Here rxx(0) is the linear response resis-

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3
4
2
0
rxx [?]
B = 2.1 kG
?
b
4
2
0
rxx [?]
B = 1.7 kG
?
a
4
2
0
rxx [?]
100 50
I [?A]
0
B = 2.5 kG
?
c
FIG. 3: (Color online) Differential resistivity rxxvs I at (a) 1.7 kG,
(b) 2.1 kG, and (c) 2.5 kG at T = 1.5 K (solid), 2.0 K (dotted), and
3.0 K (dashed). IHis marked by ↑ (see text).
pendent on B. As illustrated in Fig.3(a)-3(c) all the T = 1.5
K data at I ? I1are well described by exp(−I2/∆2
lines) with ∆1 ≃ 4.0 µA. We note that this value is slightly
lower than the one obtained at B = 1.3 kG and that the ZdRS
is not developed in our 2DES at B ? 1 kG. This behavior can
be linked to the crossover from separated to overlapped Lan-
dau level regime. Using the quantum scattering time τq≃ 19
ps extracted from the Dingle analysis of HIROs we find23,24
that ωcτq≃ 5 at B ≃ 1 kG which suggests that the ZdRS form
in separated Landau levels.
Further examination of T = 1.5 K data in Fig.3 reveals
that the ZdRS becomes wider at higher magnetic fields as
its higher critical current I2increases with B. At I > I2the
differential resistivity grows and then shows a fundamental
(ǫdc ≃ 1) HIRO peak (cf., ↑) which occurs at IH = jH· w,
where jH = ene(ωc/2kF) ∝ B and kF =
wave number. Therefore, the increase of I2is largely deter-
mined by the increase of IHand thus is related to the onset
of inter-Landau level scattering. We proceed by fitting the ex-
perimental data with rxx(I) = rxx(IH)exp[−(I − IH)2/∆2
find that, similar to the width of the zero-bias peak ∆1, ∆2
is roughly B independent. It is, however, noticeably larger,
ranging from ≃ 18 to ≃ 22 µA. A rough estimate for I2can be
obtained as I2≃ IH− 2∆2.
To summarize our experimental observations we construct
a “phase diagram” in the (I, B) plane which is presented in
Fig.4. We observethattheexperimentalpositionofthefunda-
mental HIRO peak IH(cf., open circles) is well described by a
1) (cf., dark
√2πneis the Fermi
2] and
120
100
80
60
40
20
0
0.0
I [?A]
2.52.01.51.00.5
B [kG]
ZdRS
T = 1.5 K
j1
j2
w1
w
j
(a)
j1
j2 j
Edc
(b)
FIG. 4: (Color online) Current at the fundamental HIRO peak IH
(open circles), ∆1(squares), and IH− ∆2(solid circles) vs B. Shaded
area marks the region I1(B) ? I ? I2(B) where the ZdRS is formed.
Inset (a) shows the simplest domain structure containing domains
of width w1, current j1(top) and of width w2 = w − w1, current j2
(bottom) separated by a wall (dashed line). Inset (b) depicts a generic
Edcvs j dependence with domain currents j1and j2.
linearrelation(cf.,solidline)computedusingǫdc= 1. Theex-
tracted ∆1and IH−∆2are shown by solid squares and circles,
respectively. The shaded area roughly marks the phase space,
I1(B) ? I ? I2(B) where the ZdRS is formed. In the sim-
plest case of two domains39these currents should be associ-
ated with the currents inside the domains, Ii= ji·wi(i = 1,2),
where j1is the domain current density and wiis the domain
width [see insets (a) and (b)]. The position of the domain
wall can be found from the boundary condition I = I1+ I2
as w1/w = (I2− I)/(I2− I1). For B = 1.5 kG, we esti-
mate I1 ≃ 10 µA, I2 ≃ 43 µA, and for I = 20 µA obtain
w1/w ≃ 23/33 ≃ 0.7, thesituation depictedin the inset (a). As
the current approaches either I1or I2, the domain wall moves
to the sample boundary and the ZdRS is destroyed.39
In summary, we reported on a state with a zero differential
resistance in a dc-driven high-mobility 2DES subject to weak
magnetic fields and low temperatures. This state emerges
from a minimum of Hall field-induced resistance oscillations
in the absence of microwave radiation and disappears in the
regimeofstronglyoverlappedLandaulevelsandwithincreas-
ing temperature. Occurring in very high Landau levels, the
state extends over a continuousrange of electric and magnetic
fields persisting far below the onset of the Shubnikov-deHaas
oscillations. The minimum current required to support this
state is largelyindependentonthe magneticfield andthe max-
imum current traces the onset of inter-Landau level scattering
increasing linearly with the magnetic field. According to the
domain model39these currents should be associated with cur-
rents inside the domains formed in a dc-driven high-mobility

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4
2DES. To explain the temperature dependence and the mech-
anism leading to the ZdRS, theories might need to consider
the effects of electron-electron and electron-phonon scatter-
ing, and be extendedto the regime of separatedLandaulevels.
Since the state under study is similar to a microwave-induced
zero-resistance state in a sense that it can also be explainedby
the domain model, it offers exciting experimental opportuni-
ties. In particular, it might allow one to explore instabilities
leading to domain formation. Such studies have proven diffi-
cult in irradiated 2DES and no direct experimental confirma-
tion of domains is currently available.
We thank I. A. Dmitriev and B. I. Shklovskii for useful dis-
cussions and remarks. The work at Minnesota was supported
by the NSF Grant No. DMR-0548014.
∗Corresponding author: zudov@physics.umn.edu
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