Table Of ContentEvidence for zero-differential resistance states in electronic bilayers
G. M. Gusev,1 S. Wiedmann,2,3 O. E. Raichev,4 A. K. Bakarov,5 and J. C. Portal2,3,6
1Instituto de F´ısica da Universidade de S˜ao Paulo, CP 66318, S˜ao Paulo, SP,Brazil
2Laboratoire National des Champs Magn´etiques Intenses, CNRS-UJF-UPS-INSA, 38042 Grenoble, France
3INSA Toulouse, 31077 Toulouse Cedex 4, France
4Institute of Semiconductor Physics, NAS of Ukraine, Prospekt Nauki 41, 03028, Kiev, Ukraine
5Institute of Semiconductor Physics, Novosibirsk 630090, Russia and
6Institut Universitaire de France, 75005 Paris, France
1 (Dated: January 27, 2011)
1
We observe zero-differential resistance states at low temperatures and moderate direct currents
0
inabilayerelectron systemformedbyawidequantumwell. Severalregionsofvanishingresistance
2
evolve from the inverted peaks of magneto-intersubband oscillations as the current increases. The
n experiment,supportedbyatheoreticalanalysis,suggeststhattheoriginofthisphenomenonisbased
a oninstabilityofhomogeneouscurrentflowunderconditionsofnegativedifferentialresistivitywhich
J
leads to formation of current domains in our sample, similar to thecase of single-layer systems.
6
2 PACSnumbers: 73.43.Qt,73.63.Hs,73.21.-b,73.40.-c
]
l
l Studies of non-linear transport in high-quality two- explained within the same model assuming formation of
a
dimensionalelectronsystem(2DES)haverevealedmany currentdomains when the negative resistance conditions
h
interesting phenomena which occur in a perpendicular are reached and homogeneous current picture becomes
-
s magnetic field at large filling factors. In the presence unstable [18]. In order to learn more about the origin
e
of AC excitation by microwaves, there exist microwave- of ZdRS, clear understanding of the domain model is re-
m
induced resistance oscillations (MIROs) [1] which obey quired. In this connection, studies of the systems which
.
t the periodicity ω/ωc, where ω and ωc are the radiation differ from standard 2DES are of crucial interest.
a
m frequencyandthe cyclotronfrequency,respectively. The In this Rapid Communication, we report observation
minima of these oscillations evolve into zero-resistance ofZdRSinahigh-qualitybilayerelectronsystemformed
-
d states (ZRS) for high electron mobility and elevated mi- by a wide quantum well (WQW). Owing to scattering
n crowave power [2]. The MIROs have been found also in of electrons between two populated 2D subbands, ZdRS
o bilayer and trilayer electron systems [3], where they in- in bilayer electron systems evolve from inverted MIS os-
c terfere with magneto-intersubband(MIS) oscillations [4] cillations for relatively low DC and obey MIS oscilla-
[
becauseofthe presenceofmorethanone populatedsub- tion periodicity. In contrast to both regimes of ZdRS in
1 band. Recently, it has been demonstrated [5] that ZRS single-layersystems [16, 17], ZdRS occur in the interme-
v exist in bilayer systems despite of additional intersub- diate regime from overlapping to separated LLs (0.1 T<
4 band scattering. Stimulated by experimental findings, B <0.3 T). A theoretical consideration of magnetoresis-
0 theorists have proposed several microscopic mechanisms tance in DC-driven electronic bilayers shows that nega-
1
whichreasonablyexplainnon-lineartransportcausedby tive differential resistance (NDR) can be reached in our
5
. microwave excitation [6–9]. system, thereby supporting the domain model.
1
0 Adirectcurrent(DC) excitationofhigh-quality2DES Our experiments are carried out on high-quality
1 leads to another group of non-linear phenomena caused WQWs with a well width of w=45 nm, high electron
1 by Landau quantization and, therefore, distinct from density ns 9.1 1011 cm−2 and a mobility of µ
v: electronheatingeffectsobservedundersimilarconditions 1.9 106 cm≃2/V s×after a brief illumination with a re≃d
×
i in samples with lower mobilities. The Hall-field induced light-emitting diode. Several samples in Hall bar geom-
X
resistance oscillations (HIROs) are found in numerous etry (length L = 500 µm and width W = 200 µm) have
ar experiments [10–13] and are explained [10, 14] in terms been studied, while we focus here on two samples (A
of large-angle elastic scattering between Landau levels and B, with a slightly higher electron density for sam-
(LLs) tilted by the Hall field. Further, even a moderate ple B). Measurements have been performed at mK tem-
DC causes a considerable decrease of the resistance [15], peratures in a dilution refridgerator (base temperature
whichinhigh-qualitysamplesmayleadtothezerodiffer- 50mK)(sampleA)andupto4.2Kinacryostatewitha
entialresistancephenomenon. Thezero-differentialresis- variabletemperatureinsert(sampleB).We recordlongi-
tancestates(ZdRS)foundinsingle-layersystemsemerge tudinalresistanceusingacurrentof0.5µAatafrequency
either from the inverted maxima of Shubnikov-de Haas of13Hz. DirectcurrentIDC wasappliedsimultaneously
(SdH) oscillations at relatively high magnetic fields [16] through the same current leads to measure the differen-
or from a minimum of HIROs [17]. In the second case tial resistance rxx =dVxx/dIDC.
ZdRS appears atlow magnetic fields, before the onsetof In Fig. 1 we show differential resistance r as a
xx
SdH oscillations, and extends overa continuous range of function of the magnetic field for I =0, I =5 µA
DC DC
fields. The two seeminglydifferentregimes,however,are and I =15 µA. Dark magnetoresistance exhibits well-
DC
2
developed MIS oscillations and confirms the existence of
two populated subbands [4]. The inversion of MIS os-
cillations for B < 0.17 T is caused by the alternating
current of 0.5 µA. The position of the inversion field is 10
T=50 mK
in a reasonableagreementwith our theoreticalestimates
9
[19]. At this low temperature, we also observe SdH os-
cillationswhicharesuperimposedonMISoscillationsfor 8 IDC= 0 A
B >0.15 T. If we apply a constant DC, the MIS oscilla-
7
tionsareinvertedinthewholerangeofmagneticfieldsfor
I =5 µA ata temperature of 50mK. ForI =15µA,
DC DC 6
we observe ZdRS for B > 0.14 T which are developed )
IDC= 6 A
from the inverted MIS oscillations. Having a closer look (x5
x
at magnetic fields 0.1 T< B <0.3 T, we see that the r
4
maximabetweenZdRSaresplit, whichalsooccurswhen IDC= 10 A
applying a strong AC or DC excitation separately [19]
3
and can be explained by the influence of Landauquanti- IDC= 16 A
zationoninelasticscatteringofelectronsintwo-subband 2
systems (see also Fig. 4).
1 IDC= 19.2 A
0
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
B (T)
T= 50 mK
30 VAC IAC IDC FIG. 2: (Color online) Evolution of differential magne-
toresistance rxx with increasing DC bias from IDC=0 to
IDC=19.2 µA (step ∆IDC=0.4 µA).For thehighest current,
) we findfour ZdRSfor B >0.12 T.
r (xx 20
IDC= 0
IDC= 5 A from IDC=16 µA the widths of ZdRS remain constant.
10
AfurtherincreaseinI leadstoheatingoftheelectron
DC
gasinthe sample,thereforeourstudies atT=50mK are
IDC= 15 A limited to a maximal current of Imax=20 µA.
0 DC
Since ZdRS in electronic bilayers occur neither at
0.0 0.1 0.2 0.3 nearly discrete values of B [16] nor in a wide region of
B (T) magneticfields[17],andareseparatedbymaximainr ,
xx
it is important to illustrate this phenomenon by sweep-
ing I at a constant magnetic field, see Fig. 3. We
FIG.1: (Coloronline)Differentialmagnetoresistancerxxasa DC
fix the magnetic field at several positions (split maxima
function of themagnetic fieldat IDC=5µAand IDC=15µA
for sample A. The top trace, which is shifted up for clarity, and ZdRS in rxx), mark them by arrows with numbers
shows magnetoresistance (IDC=0) which exhibits MIS oscil- (1) to (5) in Fig. 3(a), and plot differential magnetore-
lationstogetherwith SdHoscillations. Experimentalsetupis sistance rxx(IDC) in Fig. 3(b). For B = 0.254 T and
shown in theinset. B =0.189T,wearesituatedinthecenterofZdRS.Both
ZdRSaredevelopedatI >10µA. ForB=0.254T we
DC
Asconcernstemperaturedependence,wehavetoapply findsmallkinksinr at 5.8µAbeforer isstabilized
xx xx
±
ahigherDCtomaintainZdRSathighertemperature,see near zero. Beyond ZdRS, we show similar plots for the
Fig 4(a) later. The reason for this is rooted in the fact two maxima in r at B = 0.204 T and B = 0.224 T
xx
that the nonlinear response to a DC bias weakens with which occur between the ZdRS, and in the transition
increasing temperature due to an increase in electron- from these maxima to ZdRS at B =0.2 T. The longitu-
electron scattering [9, 14]. dinal voltage V as a function of I is plotted in Fig.
xx DC
In orderto give a more detailed overviewof the evolu- 3(c) for the ZdRS around B=0.25 T. This trace is ob-
tion from MIS oscillations (I =0) to the regime where tainedbyintegratingthedataatB =0.254Tandshows
DC
inversionofallMISpeakstakesplaceuntilwereachzero- atypicalvoltage-currentcharacteristicwithsaturationof
differentialresistance,weplotinFig. 2 rxx as afunction Vxx corresponding to the onset of ZdRS.
of the magnetic field starting from I =0 (top trace) Wenowanalyzeanddiscussourexperimentalobserva-
DC
to I =19.2 µA (bottom trace) at a constant tempera- tion. First, we point out that the occurrence of ZdRS in
DC
ture of 50 mK for both negative and positive magnetic electronicbilayersisdistinctfromthecaseofsingle-layer
field orientation in steps of ∆I = 0.4 µA. Starting systemswhereZdRSemergeinawide rangeofmagnetic
DC
3
anism). A more rigorous theory, which includes another
(displacement) mechanism of nonlinear response and re-
mains valid at higher currents, has been developed in
Ref. [14]. To investigate the possibility of NDR for our
(a) T= 50 mK
) 10 (3)(2) samplesandtoexplainothercharacteristicfeaturesofthe
( observedmagnetoresistance,wehavegeneralizedthethe-
rxx (4) (1) oryofRef. [14]beyondtheregimeofoverlappingLandau
5
IDC= 20 A (5) levels and for the case of multisubband occupation. Be-
low we present the results valid both for single-subband
0 systemsandforthesystemswithtwocloselyspacedsub-
0.00 0.05 0.10 0.15 0.20 0.25 0.30 bands (our samples). In the regime of classically strong
B ( T) magnetic fields, the magnetoresistance of the system of
electrons interacting with the static potential of impu-
30 (b) B= 0.254 T (1)
rities is found from the following relation between the
()rxx 20 B= 0.204 T (3) ddeinnasliteyleocftrtihcefiaelpdplEie|d| =cuVrxrxen/tL:j = IDC/W and longitu-
B= 0.224 T (2)
10 B= 0.2 T (4)
E|| =jρ0(cid:20)1− a2kγk′′− akak−k′Ak′ γk′ −γk′−k′ (cid:21),
0 k(Xk6=0) Xkk′ (cid:0) (cid:1)
B= 0.189 T (5)
(1)
V) 5 (c) where ρ0 = m/e2nsτtr is the classical Drude resistiv-
B= 0.254 T (1)
(Vxx 0 ity (e and m are the electron charge and the effec-
tive mass). The contribution of SdH oscillations is ne-
-5
glected. The sums are taken over the cyclotron har-
-20 -10 0 10 20
monics, k are integers, a are the coefficients of expan-
IDC( A) k
sion of the density of electron states in such harmonics,
D(ε) = a exp(ik2πε), where D(ε) is the density of
FIG. 3: (Color online) (a) Differential resistance rxx for states inPunkitskof m/π~~ω2c. For two-subband systems, one
IDC=20 µA. (b) Dependence of rxx on the DC at several should use D(ε)=(D +D )/2, where D is the den-
chosenmagneticfieldsaslabeled atatemperatureof50mK. 1ε 2ε iε
sity of states in the subband i. In both cases a are
(c) Corresponding voltage Vxx at B=0.254 T. k
taken real, which is attainable by a proper choice of the
reference point for energy. The coefficients Ak′, which
describe harmonicsof non-equilibriumoscillatingcorrec-
fields well below the onset of SdH oscillations. In a two-
tion to the distribution function, are found from the lin-
subband system several ZdRS emerge from the peaks of ear system of equations k′Ckk′Ak′ =akγk′ with
inverted MIS oscillations and follow the 1/B periodic-
P
iatryatoiofnth∆es.eIonscoiltlhaetironwodredtse,rmeaicnhedZdbyRSthaepspuebabrsanadrosuenpd- Ckk′ = ττitnr[a3k−k′+a2k(ak+k′−2ak−k′)]+ak−k′(γk−k′−γk).
the magnetic fields given by the set of cyclotron ener- (2)
gies ~ωc = ∆/k, where k is an integer. Next, the other The first part of the matrix Ckk′ is determined by tak-
important properties of ZdRS, such as the dependence ing into account the Landau quantization in the lin-
ofr onthe DC andthe voltage-currentcharacteristics, earized electron-electron collision integral (see details in
xx
areverysimilartothosereportedforsingle-layersystems Ref. [19]). The coefficients γk describe the effect of
[16, 17]. the current and are defined by averaging over the scat-
tering angle θ: γ γ(ζk) = J [2ζksin(θ/2)]τ /τ(θ),
From the theoretical point of view, it is quite pos- k 0 tr
≡
sible that the differential dissipative resistance rxx = where J0(x) is the Bessel function, ζ = 4π3j2/e2nsωc2
dV /dI (or even absolute resistance R ) in high- (for single-subband systems one shouldpmultiply ζ by
xx DC xx
mobility samples approaches zero with increasing cur- √2),and1/τ(θ)=(m/~3)w[2kF sin(θ/2)]istheangular-
rent, since an application of the current leads to a con- dependent scattering rate (kF is the Fermi wavenum-
siderable decrease in the resistivity. Such a decrease, ber, w(q) is the correlator of random scattering poten-
observed both in single-layer [15] and double-layer [19] tial). The transport time τtr is given in the usual way,
systems, is reasonably explained by the theory of Ref. 1/τ = (1 cosθ)/τ(θ). Finally, γ′ and γ′′ denote, re-
tr − k k
[9] based on a modification of the electron distribution spectively,thefirstandthesecondderivativesofthefunc-
function due to current-induced diffusion of electrons in tion γ(ζk) over its argument.
the energy space in the presence of Landau quantiza- To carry out the calculations, we numerically com-
tion. This non-equilibrium distribution function is sta- puted the density of states in the self-consistent Born
bilized by inelastic electron-electron collisions described approximation, using the quantum lifetime of electrons
throughthe inelastic scatteringtime τ (inelastic mech- τ = 6.6 ps determined from linear low-temperature
in 0
4
and γ(ζ) (τ /τ )/ 1+χζ2 [14].
tr 0
≃
In Fig. 4 the calcuplated differential magnetoresistance
plots are compared with experimental data recorded on
0)1.0 (a) T= 4.2 K T= 2.5 K sample B with a subband separation energy ∆ = 1.4
( meV. Notice that this separation is slightly larger than
x
rx insampleA,sothe positionsofinvertedMISpeaks(and
)/ sample B
B of the corresponding ZdRS) are slightly shifted towards
(0.5
rxx experiment higher magnetic fields with respect to sample A. Apply-
ing I = 50 µA, we observe ZdRS around B = 0.27 T
IDC= 50 A DC
0.0 at low temperatures (see the plot for T = 1.4 K) and
T= 1.4 K
its disappearance at T >2.5 K. The calculation shows a
0.1 0.2 0.3 similar behavior with a NDR regionaroundB =0.27 T.
0)1.0 (b) T= 4.2 K The other important feature of magnetoresistance, the
( peak splitting leading to minima at 0.23 T and 0.32 T,
x
x
r is also reproduced in the calculations, see Ref. [19] for
)/
B a detailed description of this phenomenon. Therefore,
(0.5
xx theory a direct calculation of magnetoresistance demonstrates
r
that NDR is attainable for our samples. In the NDR re-
T= 2.5 K
0.0 gions, according to Ref. [18], the homogeneous flow of
T= 1.4 K
the current becomes unstable and the system exhibits a
0.1 0.2 0.3 transition to current domains. The simplest two-domain
structure discussed in Refs. [16, 17] can as well describe
B (T)
ZdRS in our experiment.
To summarize, we have found evidence for zero-
FIG. 4: (Color online) (a) Differential magnetoresistance rxx
differential resistance in a bilayer (two-subband) elec-
in experiment and (b) theory for three chosen temperatures
tronsystemwhereZdRSdevelopfrominvertedmagento-
(sample B).
intersubband oscillations under a relatively small direct
current. This experimental result, together with a the-
magnetoresistance measurements. Next, we applied the oretical consideration, suggests that the ZdRS occurs as
theoretical [9] estimate τ ~ε /T2, where ε is the a result of current instability under negative differential
Fermi energy and T is thine e≃lectrFon teemperatureF. Heat- resistance conditions. The domain model proposed for
e
ing of electron gas by the current is taken into account explanation of ZdRS in single-subband 2DEG [16, 17]
by considering energy loss due to electron-phonon in- very likely applies to multisubband systems.
teraction [3, 20] (for I = 50 µA the heating is ap- Note added in proof. We have recently become aware
DC
preciable only at low temperatures, at T = 1.4 K we ofrelatedexperimentalworkonasimilarsystem,byRef.
find T 1.6 K). To describe elastic collisions, we used 21.
e
≃
the model of long-range scattering potential, w(q) We acknowledge support from COFECUB-USP
∝
exp( l q) (where l 1/k is the correlation length), (project number U Ph 109/08), FAPESP and CNPq
c c F c
whic−h produces τ =≫τ (1+χ)/χ with χ = 1/(k l )2 (Brazilian agencies).
tr 0 F c
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