Electromodulation of the bilayer $ν$=2 quantum Hall phase diagram

4 Pages·0.19 MB·English

by  L. Brey

Electromodulation of the bilayer ν=2 quantum Hall phase diagram.
L.Brey1,2, E.Demler1, and S.Das Sarma1,3
1 Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106
2 Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049, Madrid, Spain.
3 Department of Physics, University of Maryland, College Park, MD 20742
9
9 Wemakeanumberofpreciseexperimentalpredictionsforobservingthevariousmagneticphases
9 and the quantum phase transitions between them in the ν=2 bilayer quantum Hall system. In
1
particular, weanalyzetheeffect ofan externalbiasvoltage onthequantumphasediagram, finding

n thatafinitebiasshould readilyenabletheexperimentalobservation oftherecentlypredictednovel
a canted antiferromagnetic phase in transport and spin polarization measurements.
J

6 PACS numbers: 73.40.Hm; 75.20.Kz
2

Recent theoretical work [1–3] predicts the existence of take ¯h=1 throughout); the interlayer tunneling energy
]
l a novel canted antiferromagnetic (C) phase in the ν=2 characterizedby∆ ,thesymmetric-antisymmetricen-
l SAS
a bilayer quantum Hall system under quite general exper- ergy gap; the Zeeman energy or the spin-splitting ∆ ;
z
h
imental conditions, and encouraging experimental evi- the intralayer Coulomb interaction energy and the in-
-
s dence in its support has recently emerged through in- terlayer Coulomb interaction energy. The application of
e
elastic light scattering spectroscopy [4,5] and transport the external electric field adds another independent en-
m
measurements [6]. Very recent theoretical works have ergy scale, the bias voltage, to the problem. Neglecting
t. shown that such a C-phase may exist [7] in a multilayer the largest (ωc, which we take to be very large) energy
a superlatticesystem(withν=1perlayer),andthatinthe scale,oneisstillleftwithfourindependentdimensionless
m
presence of disorder-induced-interlayer tunneling fluctu- energy variables to consider in constructing the ν=2 bi-
-
d ations the C-phase may break up into a rather exotic layer quantum phase diagram in the presence of finite
n spin Bose glass phase [8] with the quantum phase tran- bias. In the absence of any bias the quantum phase
o sition between the C-phase and the Bose glass phase be- diagram is surprisingly simple, allowing for only three
c
ing in the sameuniversalityclass asthe two-dimensional qualitatively different quantum magnetic phases, as es-
[
superconductor-insulator transition in the dirty boson tablishedbyamicroscopicHartree-Focktheory[1–3,7],a

1 system. In this Letter we consider the effect of an exter- longwavelengthfieldtheory basedonthe quantumO(3)
v
nal electric field induced electromodulation (through an nonlinear sigma model [2,3], and a bosonic spin theory
6
applied gate bias voltage) of the ν=2 bilayer quantum [8]. These three magnetic phases are the fully spin po-
9
2 phase diagram. Our goal is to provide precise experi- larized ferromagnetic phase (F), which is stabilized for
1 mentalpredictionswhichwillfacilitatedirectandunam- large values of ∆ (or for strong intralayer Coulomb in-
z
0 biguousobservationsofthevariousmagneticphases,and teraction), the paramagnetic symmetric or spin singlet
9
more importantly the quantum phase transitions among (S)phase,whichisstabilizedforlargevaluesof∆ (or
9 SAS
them. We find the effect of a gate bias to be quite dra- forstronginterlayerCoulombinteraction),andtheinter-
/
t matic on the ν=2 bilayer quantum phase diagram. In mediate C phase, where the electron spins in each layer
a
m particular, a finite gate bias makes the C-phase more aretiltedawayfromtheexternalmagneticfielddirection
stable which could now exist even in the absence of any due to the competition between ferromagnetic and sin-
-
d interlayer tunneling in contrast to the situations consid- glet ordering. Note that the S phase is fully pseudospin
n
eredinreferences[1–3]wheretheinterlayertunnelingin- polarizedwith ∆ , the symmetric-antisymmetricgap,
SAS
o
duced finite symmetric-antisymmetricgapwascrucialin actingasthe effective pseudospinsplitting. The C phase
c
: the stability of the C-phase. Thus, a finite gate bias, ac- is a true many-body symmetry-broken phase not exist-
v
cordingtoourtheoreticalcalculationspresentedhere,has ing inthe single particlepicture (andis stabilizedby the
i
X aqualitativeeffectontheν=2bilayerquantumphasedi- interlayer antiferromagnetic exchange interaction). The
r agram – it produces a spontaneously interlayer-coherent single particle theory predicts a level crossing and a di-
a
canted antiferromagnetic phase which exists even in the rectfirst ordertransitionbetween the S phase and the F
absence of any inter-layer tunneling. The prediction of phase(nominallyat∆ =∆ )astheZeemansplitting
z SAS
this spontaneously coherent canted (CC) phase is one of increases. Coulomb interaction creates the new symme-
the new theoreticalresults ofthis paper. Thetheoretical try broken C phase, which prevents any level crossing
construction of the bilayer ν=2 quantum phase diagram (and maintains an energy gap throughout so that there
andpredictingitsexperimentalconsequencesinthepres- isalwaysaquantizedHalleffect)betweenFandSphases,
ence of the bias voltage are our main results. andmakesallphasetransitionsinthe systemcontinuous
The bilayer ν=2 system is characterized by five in- second order transitions. The canted phase is canted in
dependent energy scales: the cyclotron energy, ω (we both the spin and the pseudospin space.
c
1
One key experimental difficulty in observing the pre- tween them. Note that the S-phase, which is the singlet
dicted phase transitions (in the absence of any external or the symmetric phase, is also stabilized for ∆ =0
SAS
bias voltage) is that a given sample (with a fixed value by finite bias effects. This phase (i.e. the S-phase along
of ∆ , determined by the system parameters such as the∆ =0)isthespontaneousinterlayercoherentsym-
SAS SAS
well widths, separations, etc.) is always at a fixed point metric or singlet phase (the CS-phase) and is analogous
in the quantum phase diagram calculated in references to the corresponding ν=1 spontaneous interlayer coher-
[1–3,7,8] because ∆ , ∆ and the Coulomb energies ent phase studied extensively [13] in the context of the
z SAS
are all fixed by the requirement ν=2 and the sample pa- ν=1 bilayer quantum phase diagram. There is, however,
rameters. Therefore,agivenexperimentalsampleinthis a fundamental difference between the coherentCS phase
so-called balanced condition (i.e. no external bias, equal for our ν=2 bilayer system and the corresponding [13]
electrondensitiesinthetwolayersontheaverage)iscon- ν=1 spontaneous interlayer coherent phase; our ν=2 bi-
strainedtolieintheForCorSphase,andtheonlyway layerCSphasecanonlyexistunderafinite external bias
to seeanyphasetransitionsisto makeanumber ofsam- (the same as our CC phase). Unlike the corresponding
ples with different parameters lying in different parts of ν=1bilayersystem[13]ortherecentlystudiedzeromag-
the phasediagramandto investigateandcompareprop- netic field bilayer system [14] there is no spontaneous
erties, as was done in the light scattering experiments breaking of the pseudospin U(1) symmetry (generated
of references [4,5]. This is obviously an undesirable sit- by the interlayer electron density difference) in our ν=2
uation because what one really wants is to vary an ex- coherentbilayer phases which can only exist in the pres-
perimental control parameter (e.g. an external electric enceofanexternalvoltagebias. Weemphasizethatthere
field) to tune the system through the phase boundaries isnoanalogytoourcantedphase(CorCCphase)inthe
andstudythequantumphasetransitioninsteadofstudy- corresponding ν=1 bilayer quantum phase diagram [13].
ing different samples. (Theoreticallythis tuning is easily We note that the main difference (cf. Figs.1 and 2)
achievedbymaking∆ ,∆ andtheCoulombenergies betweenthe Hartree-Fock[1–3,7]theoryandthe bosonic
z SAS
continuous variables in the phase diagram [1–3,7,8], but spin [8] theory is that the Hartree-Fock theory underes-
experimentally, of course, this cannot be done.) In this timates the stability of the S phase (compared with the
Letter we show that an externally applied electric field bosonic spin theory) at small values of ∆ . This is a
z
through a gate bias, which takes one away (off-balance) realeffectandarisesfromthe neglectofquantumfluctu-
from the balanced condition and introduces [9] unequal ations in the Hartree-Focktheory which treats the inter-
layer electron densities is potentially an extremely pow- layertunnelingasafirstorderperturbationcorrectionin
erfulexperimentaltoolinstudyingtheν=2bilayerquan- the S-phase. The bosonic spintheory is essentially exact
tumphasetransitions. Ourresultsindicatethatusingan for the S-phase and is therefore more reliable near the
externalgatebias as a tuning parameter,a technique al- C-S phase boundary, particularly for small values of ∆
z
ready extensively used [6,10,11] in experimental studies where tunneling effects are important.
of bilayer structures, should lead to direct experimen- In Fig.3 we show our calculated quantum phase dia-
talobservationsofthe predictedquantumphasesinν=2 grams in the gate voltage (V+) -tunneling (∆SAS) space
bilayer systems and the continuous transitions between for fixed values of the Zeeman energy ∆ (and the
z
them in both transport measurements [6,10,11] and in Coulomb energies) using both the Hartree-Fock and the
spin polarization measurements through NMR Knight bosonicspintheory. Thephasediagramsinthetwotheo-
shift experiments [12]. ries are qualitatively similar, and the interlayercoherent
We have used two complementary techniques, the di- phases (CC and CS phases) are manifestly obvious in
rectHartree-Focktheory[1–3,7]andtheeffectivebosonic Fig.3becausethe C andthe Sphasesnowclearlyextend
spin [8] theory, to evaluate the bilayer ν=2 quantum to the ∆ =0 line (the ordinate) for finite bias volt-
SAS
phase diagram including the effect of a finite bias volt- age. In general, the presence of bias therefore allows for
age. Theresultingbiasdependentphasediagrams(inthe sixdifferentquantummagneticphasesintheν=2bilayer
∆ −∆ space) for the Hartree-Fock theory and the system: theusualF,C,andSphasesofreferences[1–3]as
z SAS
bosonic spin theory are shown in Figs. 1 and 2, respec- well as the purely N´eel (N) phase [1–3] along the ∆ =0
z
tively. Although there are some quantitative differences line in Fig.1 (the F,C,S,N phases are all allowed in the
between the phase diagrams in the two models (to be balanced V+=0 situation), and two new (bias-induced)
discussed below), the main qualitative features are the coherent phases (CC and CS) along the ∆ =0 line
SAS
same: increasing bias voltage enhances the phase space in Figs. 1-3. The most important effect of the external
ofthe C phase mostlyatthe costofthe Fphase,andfor bias,whichisanimportantnewpredictionofthecurrent
large enough bias the C phase becomes stable even for paper, is that it allows for a continuous tuning of the
∆ =0, this CC-phase is spontaneously coherent. We quantum phase of a ν=2 bilayer system within a single
SAS
note that the CC-phase (i.e. the bias induced C-phase gatedsample,as is obviousfromFigs.1-3. The predicted
alongthe∆ =0line)andtheC-phasearecontinuously quantum phase transitions can now be studied in light
SAS
connected and there is no quantum phase transition be- scattering [4,5], transport [6,10], and NMR [12] experi-
2
ments in single gatedsamples by tuning the bias voltage US-ONR.
to sweep through various phases as shown in Figs.1-3.
The last issue we address here is what one expects
to see experimentally in transport and spin polarization
measurements, in sweeping through the phase diagram
ofFigs.1-3underanexternalgatebias. InFig.4weshow
our calculated results for the variation in the spin polar-
[1] L. Zheng, R. J. Radtke and S. Das Sarma, Phys. Rev.
ization of the system as a function of the bias V+ with Lett., 78, 2453 (1997).
alltheothersystemparametersbeingfixed. Asexpected
[2] S.DasSarma,S.SachdevandL.Zheng,Phys.Rev.Lett.,
the spin polarization is complete in the F phase and re- 79, 917 (1997).
mains a constant as a function of V+ until it hits the F- [3] S. Das Sarma, S. Sachdev and L.Zheng , Phys. Rev. B,
C phase boundary where it starts to drop continuously 58, 4672 (1998).
through the C phase, essentially dropping to zero at the [4] V.Pellegrini et al.,Phys.Rev.Lett. 79, 310 (1997).
the C-S phase boundary, remaining zero in the S-phase. [5] V.Pellegrini et al.,Science 281, 799 (1998).
At zero temperature the two phase transitions (i.e. F-C [6] A.Sawada et al.,Phys.Rev.Lett.80, 4534 (1998).
[7] L.Brey, Phys.Rev.Lett. 81, 4692 (1998).
and C-S) are characterized by cusps in the spin polar-
[8] E.Demler and S.Das Sarma, cond-mat/9811047.
ization (Fig.4) which perhaps will not be observable in
[9] L.Brey and C.Tejedor, Phys.Rev.B 44, 10676 (1991);
finite temperature experiments. The main features of
T.Jungwirth and A.H.MacDonald, Phys.Rev.B53, 9943
the calculated spin polarization as a function of bias, as
(1996).
shown in Fig.4, should, however be readily observable in [10] Y.W.Suen et al.,Phys.Rev.Lett.72, 3405 (1994).
NMRKnightshiftmeasurements[12],includingpossibly [11] A.G.Davies et al.,Phys.Rev.B 54, R17331 (1996).
the Knight shift difference in the two layers (Fig.4). We [12] S. E. Barret et al., Phys. Rev. Lett.,74, 5112 (1995);
havealsocarriedoutcalculationsoftheinterlayercharge N.N.Kuzma et al.,Science 281, 686 (1998).
imbalance (which is zero in the F phase and then rises [13] K.Yang et al.,Phys.Rev.Lett. 72, 732 (1994).
continuously throughout the C and the S phases reach- [14] L.Zheng et al., Phys.Rev.B55, 4506 (1997).
ing full charge polarization for large V+ in the S phase) [15] OurnumericalHartree-Fockcalculations(notshownhere
forthesakeofbrevity)forthesamplesofref.[6]takeinto
as a function of the bias voltage. There are two cusps
in the calculated imbalance as a function of V+, corre- account theactual well widthsandothersample param-
eters given in ref. [6], and also include realistic Landau
sponding to the F→C and the C→S phase transitions,
level mixing effects. Wefindthat the ν=2sample (Fig.4
which should be experimentally observable. The calcu-
of ref. [6]) at the balanced density point to be in the S
lated imbalance therefore looks almost exactly comple- phase for density <∼ 0.5×10cm−2 and in the F phase
mentary to the spin polarization results shown in Fig.4. for density >∼2.8×10cm−2, and to be in theC-phase in
Finallywehavealsocalculatedthechargedexcitationen- between.
ergies within the simple Hartree-Fock and bosonic theo-
ries (assuming no textural excitations such as skyrmions
or merons), which lead to weak cusps in the activation
energies at the phase boundaries. Using the parameters
ofthesamplesinref.[6],weconcludefromournumerical
calculations[15]thatthephasetransitionbeingobserved
in the ν = 2 bilayer transport experiments of ref. [6] is
thetransitionfromtheC-phasetotheS-phaseasafunc-
tion of the density (and not from the F -phase to the C
-phase as implied in ref. [6]) Neither phase in ref. [6] is
spontaneous interlayer coherent phase (because ∆ is
SAS
finite in the experiment) in contrast to the claims of ref.
[6]. Ourresultsindicate,however,thatitshouldbepossi-
bletoseeallthreeν=2quantumphases(F,CandS)ina
singlegatedsamplebyvaryingthebiasvoltage. Wehope
that the detailed results presented in this paper will en-
courage future bilayers ν=2 experiments under external
gate bias to explore the predicted rich phase diagram.
We aregratefulto A.H.MacDonaldandZ.F.Ezawafor
usefuldiscussion. ThisworkissupportedbytheNational
Science Foundation (at ITP, UCSB). LB also acknowl-
edge financial support from grants PB96-0085 and from
the Fundaci´on Ramo´n Areces. SDS is supported by the
3
0.1
1.4
F
S
1.2
0.075 C V
+
∆
Z
0 ∆ 0.6
0.05 0.1 0.8 SAS
C
V
C +
Z S
∆
S
0.025 0.4
0 ∆ 0.08 F
SAS
0
0 0.2 0.4 0.6
∆ 0 0.1 0.2
∆
SAS
SAS
FIG. 1. Calculated Hartree-Fock ν=2 bilayer phase
(∆SAS −∆z) diagrams for different bias V+. Dot, dashed forFfiIGxe.d3∆. Cz.atlchuelamteadinpfihgausreed:iBagorsaomniscisnptinhepVh+as−e∆diSaAgrSasmpafocer
and continuous lines corresponds to V+=0, 0.5 and 0.65 re- ∆z=0.01 (all other parameters correspond to Fig.2). Insert:
spectively. InsetcorrespondstotheV+=1.42case. Thelength Hartree-Fock phase diagram for ∆z=0.01 (all other parame-
and the energy units are the magnetic length, ℓ, and the in-
ters correspond to Fig.1)
tralayerCoulombenergye2/(ǫℓ). Theinterlayerseparationis
1.
0.5
V = 0.2
0.2 + 0.25
0.1 V = 0.4 S
+ z
∆
0.7
z
0.15 0.25
F
0.1 0.15
C V = 0
0.1 +
S
0
0.4 V 0.8 1.2
0 0.15 0.3 +
∆
FIG.4. Calculated z-componentofthetotalspinpolariza-
SAS tion in each of the layers hSzi as a function of the bias V+
FIG.2. Phase diagram in thebosonic spin theory for dif-
for fixed ∆z and ∆SAS. This quantity is proportional to the
ferent bias voltages. All the units and parameters are the
Knight shift[12]. Main figure: solid line Hartree-Fock the-
same as in Figure 1. (See ref.[8] for details on the bosonic
oryfor∆SAS=0.05anddashedlinefor∆SAS=0.1(∆z=0.01
spin model parameters).
throughout);allotherparameterscorrespondtoFig.1. Insert:
Bosonic spin theory for the same parameters.
4