Coexistence of Composite-Bosons and Composite-Fermions in nu=1/2 + 1/2 Quantum Hall Bilayers

4 Pages·0.14 MB·English

by  Steven H. Simon

Coexistence of Composite-Bosons and Composite-Fermions in ν = 1 + 1 Quantum Hall
2 2
Bilayers
Steven H. Simona, E. H. Rezayib, and Milica Milovanovicc
a Lucent Technologies, Bell Labs, Murray Hill, NJ, 07974
bDepartment of Physics, California State University, Los Angeles California 90032
3
0 c Institute of Physics, P.O. Box 68, 11080, Belgrade, Yugoslavia
0 (February 2, 2008)
2
InbilayerquantumHallsystemsatfillingfractionsnearν =1/2+1/2, asthespacingdbetween

n the layers is continuously decreased, intra-layer correlations must be replaced by inter-layer corre-
a
lations, and the composite fermion (CF) Fermi seas at large d must eventually be replaced by a
J
composite boson(CB) condensateor“111 state”atsmall d. WeproposeascenariowhereCBsand
3
CFs coexist in two interpenetrating fluids in the transition. Trial wavefunctions describing these
1
mixed CB-CF states compare very favorably with exact diagonalization results. A Chern-Simons

transport theory is constructed that is compatible with experiment.
]
l
l
a Bilayer quantum Hall systems show a remarkable va- state is specified by the number of CFs and the number
h
riety ofphenomena [1]. Perhapsthe moststudied case is ofCBswiththetotalnumberofCBsplusCFsremaining
-
s whentheelectrondensityineachofthetwolayersissuch fixed. A first order transition could also be described as
e that ν =nφ /B = 1, where n is density, φ =2π¯hc/e is a mixture of CBs and CFs but with phase separation of
m 0 2 0
the flux quantum and B is the magnetic field perpendic- thetwofluids. Hereweinsteadconsiderstateswherethe
t. ular to the sample. At this filling fraction, it is known CFandCBfluidsinterpenetrate. Sincethewavefunction
a
that at least two types of states can occur depending on mustbefullyantisymmetricintermsoftheoriginalelec-
m
thespacingdbetweenthelayers. Forlargedthetwolay- trons, one might think of the electron having some CF
d- ers must be essentially independent ν = 12 states, which characterandsomeCBcharacter. Thisinterpolationbe-
n are thought to be well described as compressible com- tween CF and CB is advantageous since it allows us (by
o posite fermion (CF) Fermi seas with strong intralayer varyingtheratioofCFstoCBs)toconstructstateswith
c correlations and no interlayer correlations [2]. For small varying degrees of inter- versus intra-layer correlations.
[
enoughvaluesofdoneshouldhaveaninterlayercoherent We find that such mixed CF-CB states agree well with

1 “111state”whichcanbedescribedasacompositeboson exact diagonalizations. Further, we find that a Chern-
v
(CB) condensate with strong interlayer correlations and Simons version of our mixed Bose-Fermi theory is con-
3
intralayercorrelationswhich areweakerthan that of the sistent with experimental observation, and in particular
0
2 CF Fermi sea [1]. The nature of the transition between also predicts the above mentioned semicircle relation.
1 CFsandCBsisthefocusofthispaper. (Throughoutthis CF Fermi Sea and 111 State: Near ν = 1 in a
0 paper we will assume zero interlayer tunnelling, assume 2
single layer, the Jain CF [2] picture is given by at-
3
the spins arefully polarized,andset the in-plane field to
0 taching 2 zeros of the wavefunction to each parti-
zero.)
/ cle. Thus we write the electron wavefunction as Ψ =
at Previous numerical work [4] suggested that the tran- Φ [z ,z ,...,z ,z ] (z z )2 where repre-
m sition between the CF Fermi sea and the CB 111 state Ph f 1 1 N N Qi<j i− j i P
sents projection onto the lowest Landau level, where
may be first order. (A number of more exotic mecha-
-
for convenience here and elsewhere the gaussian factors
d nisms have also been proposed [5]). However, experi-
n mentsclearlyshowthatinterlayercorrelationsandcoher- exp[−Pi|zi|2/(4ℓ2B)] will not be written explicitly, and
o z = x +iy is the complex representation of the posi-
ence turn on somewhat continuously as d/ℓ is reduced i i i
c B tion. In the above equation Φ represents the fermionic
v: [6–8] (ℓB = (h¯e/Bc)1/2 is the magnetic length). Based wavefunction of the CFs in thfe effective magnetic field
on a picture of a first order transition and percolating
i = B 2nφ where n is the density. Generally, we
X puddles of one phase within the other, Ref. [3] predicts B − 0
will assume the CFs are weakly interacting so that the
r the drag resistivity tensor ρD should roughly obey the
a semicircle relation (ρD )2 +(ρD +π¯h/e2)2 = (π¯h/e2)2 wavefunction Φf can be written as a single Slater de-
xx xy terminantappropriatefor noninteractingfermions inthe
which agrees reasonably well with experiment [8]. On
effectivefield . Atν =1/2, is zeroandΦ represents
theotherhand,thefirstordertransitionmodelhastrou- B B f
a filled Fermi sea wavefunction.
ble accounting for the strong interlayer correlations that
appearto occur evendeepinto the putative Fermiliquid In Chern-Simons fermion theory [9,2], each electron
state [8]. We are thus motivated to look for a more con- is exactly transformed into a fermion bound to 2 flux
tinuous transition from the CF Fermi liquid to the CB quanta. At mean field level, the fermions see magnetic
111 state. The picture we have in mind is a family of field =B 2nφ which is zero at ν = 1. The effective
B − 0 2
states interpolating between these endpoints where each (mean) electric field seen by a fermion is given analo-
1
gously by E = E 2ǫJ where E is the actual electric zero within its own layer (forming a CF dipole [2]).
field,Jis the fermi−oncurrent(whichis equaltothe elec-
One can also write a Chern-Simons boson theory for
trical current) and ǫ = (2π¯h/e2)τ where τ = iσ is the
y the111state. Here,eachelectronisexactlymodeledasa
2 by 2 antisymmetric unit tensor (here σ is the Pauli
y bosonboundto1fluxquantumwherethebosonsseethe
matrix). Defining a transport equation for the weakly
Chern-Simons flux from both layers. Thus, the effective
interacting fermions E =ρ J (where ρ is approximated
f f magnetic(mean)fieldseenbya bosoninlayerα isgiven
as simple Drude or Boltzmann transport for fermions in
by α =B φ (n1+n2)withnibeingthedensityinlayer
0
zero magnetic field), we obtain the RPA expression for B −
α. Here,attotalfillingfractionofν +ν =1,thebosons
the electrical resistivity ρ=ρ +2ǫ. 1 2
f ineachlayerseeaneffectivefieldofzeroandcancondense
We nowturnto the double layersystemsandfocus on
to form a superfluid (or quantum Hall) state. The effec-
filling fraction ν = ν = 1. If the two layers are very
1 2 2 tiveelectricfieldseenbythebosonsinlayerαissimilarly
far apart, then we should have two independent ν = 21 given by Eα = Eα ǫ(J1 +J2) where Eα is the actual
systems. Thus, we would have a simple CF liquid state −
electric field in layer α. This can be supplemented by a
in each layer with the total wavefunction being just a
transport equation for the bosons in zero magnetic field
product of the wavefunctions for each of the two layers. Eα = ραJα. If the bosons are indeed condensed, then
It should be noted that such a state completely neglects b
we can set ρα = 0. This results in the perfect Hall drag
the correlation effects of the interlayer interaction. indicative ofbthe 111 state where E1 =E2 =ǫ(J1+J2).
On the other hand, if the two layers are brought very
closetogether,the intra-andinter-layerinteractionswill Transition Wavefunctions: At intermediate d we need
be roughly the same stength. In this case, it is known to ask what the energetic price is for binding within the
thatthesystemwillinsteadbedescribedbytheso-called layer (to form a CF) versus out of the layer (to form a
111 state. The wavefunction for this state is written as CB). The important thing to note is that (as we might
Ψ= (z z ) (w w ) (z w )wherethe expect for fermions) each fermion put into the Fermi
z ’s rQepir<ejsenit−thje Qeleic<tjroni−coorjdiQnai,tjesiin−thje first layer sea costs successively more energy (each having a higher
i
andthe w ’srepresentelectroncoordinatesinthe second wavevector than the last). Thus, it might be advanta-
i
layer. Here each electron is bound to a single zero of geousforsomeofthefermionsatthetopoftheFermisea
the wavefunction within its own layer as well as being tobecomeunboundfromtheirzeroswithinthelayerand
bound to a single zero of the wavefunction in the oppo- bind to the other layer — falling into the boson conden-
site layer. Of course by Fermi statistics, each electron sate. We imagine having some number Nfα of electrons
must be bound to at least one zero within its own layer in layer α that act like CFs (filling a Fermi sea) and Nα
b
so the only additional binding here is interlayer. Thus, that act like CBs (which can condense). Of course we
whendissmall,the electronbindsazerointhe opposite should have Nα +Nα = Nα the total number of elec-
f b
layer (to form a CB inter-layer dipole) whereas when d trons in layer α. We can write down mixed Fermi-Bose
is largethe electronminimizes its energybybinding to a wavefunctions for double layer systems as follows:

Ψ=APΦ1f[z1,z1,...,zNf1,zNf1]Φ2f[w1,w1,...,wNf2,wNf2] Y (zi−zj)2 Y (wi−wj)2× (1)
 i<j≤Nf1 i<j≤Nf2

(z z ) (w w ) (z w ) (w z )
Y i− j Y i− j Y i− j Y i− j
j<i;N1<i j<i;N2<i i,j;N1<i i,j;N2<i;j≤N1 
f f f f f .
We have chosen to order the particles so that parti- jection). Thesecondlineofthewavefunctionbindszeros
cles i = 1,...Nα are fermions and i = Nα+1,...,Nα to each boson such that the wavefunction vanishes as z
f f
are bosons. The antisymmetrization operator anti- asanyparticle(bosonorfermion)approachesthatboson
A
symmetrizes only over particle coordinates within each fromeitherlayer. (Thebosonsareassumedcondensedso
layer,and is the lowestLandaulevelprojection. Here, the explicit boson wavefunction is unity).
P
Φα is the CF wavefunction of Nα fermions in layer α. ItisperhapseasiertodescribetheseJastrowfactorsin
f f
The first line of the wavefunction is thus the fermionic terms of an effective Chern-Simons description. Within
part, including the CF wavefunctions and also the Jas- such a description, we write expressions for the effective
trow factors. Here the Jastrow factors bind two zeros to magnetic field α seen by either species in layer α as
B
each fermion within a layer in the sense that the wave-
functionvanishesasz2 astwofermionsinthesamelayer Bfα =B−2φ0nα−φ0(n1b +n2b) (2)
approacheachother(beforeantisymmetrizationandpro- α =B φ (n1+n2+n1 +n2) (3)
Bb − 0 b b f f .
2
We note, however, that unlike in the prior 111 case and method of numerical generation is involved and will be
Fermi liquid case,the transformationto a Chern-Simons discussed in a forthcoming paper. We note that in the
theoryhereisnotexact(evenbeforemakingameanfield Jastrow factors of Eq. 1 the fermions (z ,w N ) ex-
i i f
≤
approximation)becauseofcomplicationsassociatedwith perience oneless flux quantumthanthe bosons(z ,w >
i i
arbitrarilychoosingwhichparticlesarebosonsandwhich N ). Thus, when the CBs are in zero effective magnetic
f
are fermions. It seems plausible that as a low energy de- field (so they can condense), the CFs experience a single
scription, to the extent that zeros are bound tightly to fluxquantum. Thus,Φ ismodifiedtorepresentfermions
f
electrons, such a mixed Bose-Fermi theory will be sensi- on a sphere with a monopole of charge φ in the center.
0
ble. However, by making such an approximation we ex- We generate wavefunctions with 0,1,2,3,4,5 fermions per
plicitlyneglectprocessesbywhichazeroofthewavefunc- layer (and correspondingly 5,4,3,2,1,0 bosons per layer).
tionistransferredfromoneelectrontoanother(changing By construction the 1 fermion state is identical to the 0
a boson into a fermion or vice versa). fermion state. However all of the remaining states are
Since at ν = 1 the effective magnetic field seen by linearly independent. Thus we have generated 5 states
2
bosons or fermions is zero, we will take Φα to be a filled which we label 0 Fermions , 2 Fermions , 3 Fermions ,
f | i | i | i
FermiseaofNαCFsinlayerα. SimilarlytheCBsinzero 4 Fermions , and 5 Fermions . Of course 0 Fermions
f | i | i | i
field condense into a k = 0 ground state (as assumed in is the 111 state, and 5 Fermions is two separate layers
| i
thewavefunction). Ifwehavetwolayersofmatchedden- of composite fermions in the presence of 1 flux quanta.
sity we must have N1 = N2 in the ground state. How- (Inmostofthewavefunctions,thefermionicgroundstate
f f
ever, the overall number of CFs versus CBs is a matter in a single layer [10] is at maximal angular momentum
of energetics and presumably varies continuously as d L = 0. We always couple these to produce a bilayer
6
changes. When N =0 in both layers,this wavefunction L=0 wavefunction).
f
is the 111state, whereasN =0 (or N =N) consistsof We next perform an exact lowest Landau level diag-
b f
2 uncorrelated layers of CFs. onalization on the bilayer sphere using a pure coulomb
interaction v (r) = e2/r within a layer and v (r) =
11 12
1 e2/√r2+d2 betweenthe two layerswhere r is the chord
2> 0 Fermions distance between two points. We vary the spacing d
d 0.8 2 Fermions between the two layers and obtain exact ground states
groun 0.6 34 FFeerrmmiioonnss at each spacing d. In Figure 1, we show the overlap
Ψ 5 Fermions (squared) of each of our trial wavefunctions with the ex-
Ψ | trial 0.4 sahctowgrtohuenrdelsattaivteeeanteeragcyhofofthtehevavraioluuessgorofudn/dℓbs.taWteesaalssoa
< 0.2 functionofd/ℓ . ItisclearthatthemixedCB-CFstates
B
0 have very high overlap with the exact ground state and
very low energy in the transition region. This supports
d 0.15
un the picture of the transition from CF to CB occurring
o
gr throughasetofgroundstateswithinterpenetratingCF-
E 0.1
- CB mixtures.
E trial 0.05 InFigure 2 we show the inter- andintra-layerelectron
pair correlationfunctions (g and g ). We see that the
12 11
0 mixed CB-CF states allow us to interpolate between the
0 0.5 1 1.5 2 2.5 3
types of correlations that exist in the limiting 111 and
d / l
B Fermi liquid states.
FIG.1. (Top) Overlap squared of trial wavefunctions with
Chern-Simons RPA: We can calculate the resistivities
exact groundstateasafunctionof layerspacing d/ℓB. (Bot-
anddragresitivities ofthese mixedBose-Fermistatesby
tom)Energyoftrialstatesminusenergyoftheofgroundstate
using the Chern-Simons RPA approach. Analogous to
asa function ofd/ℓB. Calculations aredonewith 5electrons
Eqs. 2 and 3 we canwrite effective electric (mean) fields
per layer on a bilayer sphere with flux 9φ0. At low spac-
seen by the bosons or fermions in layer α:
ing the 111 state (|0 Fermionsi) has the highest overlap and
the lowest energy. However as the spacing increases, we go
Eα =Eα 2ǫJα ǫ(J1+J2) (4)
through a sequence of states |2 Fermionsi, |3 Fermionsi, and f − − b b
|4 Fermionsiandfinallyatlargespacing|5 Fermionsi(theCF Eα =Eα ǫ(J1+J2+J1 +J2) (5)
Fermi sea) has thehighest overlap and lowest energy. b − b b f f
where Jα is the Fermi (Bose) current in layer α with
f(b)
Numerical Calculations: We have considered a finite the total current in layer α given by Jα =Jαb +Jαf.
sized bilayer sphere with 5 electrons per layer and a We supplement Eqs. 4, 5 with transport equations for
monopole of flux 9φ0 at its center. We first generate the fermions (bosons) in each layer Eαf(b) = ραf(b)Jαf(b).
mixed CB-CF wavefunctions of the form of Eq. 1. The Finally, for a drag experiment we fix J2 = 0 and fix
3
J1 finite. We then solve for E1 and E2 yielding the tionedsemicirclelaw(ρD )2+(ρD +π¯h/e2)2 (π¯h/e2)2
in-layer resistivity (E1 = ρ11J1) and the drag resistiv- from our above expressxixon for ρxDy. In additi≈on, in this
ity (E2 = ρDJ1). Assuming layer symmetry so ρα limit one can derive ρ11 ρD which is also reasonably
b(f) xx ≈ xx
is not a function of α, the results of such a calculation consistent with published data [8].
yield ρ11 = (G + H)/2 and ρD = (G H)/2 where
Our mixed CB-CF theory can be generalized for un-
G=(ρ−1+ρ−1)−1+2ǫ, and H−1 =ρ−1−+(ρ +2ǫ)−1.
b f b f equal densities as well as for filling fractions away from
At filling fraction ν1 = ν2 = 21 both ρb and ρf are di- ν1 =ν2 = 21. Further,onemaybeabletotreattheeffects
agonal (since CBs and CFs are in zero effective field). of an in-plane magnetic field. Indeed, such an approach
Thus,thereareonlytwofreeparameters(ρb,xxandρf,xx) was already used in Ref. [12] to understand bilayer tun-
and four measurable quantities (ρxx,ρxy,ρDxx,ρDxy) which nelling experiments in tilted fields.
enables us to derive experimental predictions, such as
In summary, we have constructed a theory describing
ρD +ρ11 = 4π¯h/e2 [11]. Other more complicated rela-
xy xy the crossover between a CF liquid at large d to the 111
tionships can also be derived.
CB state at small d which can be thought of as a two-
fluidmodel. Comparisonsoftrialwavefunctionstoexact
5 Fermions diagonalizationareveryfavorable,andthecorresponding
11 Chern-Simons transport theory appears to be in reason-
g(r)12 34 F Feermrmioionnss abAlecakgnroewemledengtemweitnhtse:xpTehriemeanuttahlodrsataw.ould like to ac-
00..55
2 Fermions
knowledge helpful conversations with J. Eisenstein and
0 Fermions
B. I. Halperin. MM acknowledges support from Grant
00
number 1899 of the Serbian Ministry of Science, Tech-
nology, and Development. EHR acknowledges support
1 0 Fermions
from DOE under contract DE-FG03-02ER45981.
g(r)11 2 Fermions
0.5 3 Fermions
4 Fermions
5 Fermions
0
0 1 2 3 4 5 6 7
d/l
B
FIG. 2. Interlayer(top) and Intralayer (bottom) electron [1] For an introduction to bilayer quantum Hall see “Per-
paircorrelationfunctionsg12andg11asafunctionof(arc)dis- spectives in Quantum Hall Effects”, ed. S. Das Sarma
tanceforeachofthetrialwavefunctions(5electronsperlayer and A.Pinczuk, Wiley, 1997; and therein.
again). As we go from the Fermi liquid state(|5 Fermionsi) [2] For a review of CFs see “Composite Fermions”, ed O.
to the 111 state (|0 Fermionsi), replacing CFs with CBs, the Heinonen, World Scientific,1998; and therein.
short range intralayercorrelations are reduced and theinter- [3] A. Sternand B. I.Halperin Phys.Rev.Lett. 88, 106801
layer correlations build up. The deviation of g12 from unity (2002); see experimental data in [8]
intheFermiliquidstateiscausedbythefact thattwoL6=0 [4] John Schliemann, S.M. Girvin, A.H. MacDonald, Phys.
single layer states are combined to form an L = 0 bilayer Rev. Lett.86, 1849 (2001).
state. [5] Y.-BKimetal.,Phys.Rev.B.63,2053152001;E.Dem-
ler et al., Phys. Rev.Lett. 86 1853 (2001).
When d is large, we assume that there are very few [6] I.B.Spielman,J.P.Eisenstein,L.N.Pfeiffer,andK.W.
CBs and thus ρ should be large (at least at finite WestPhys.Rev.Lett.84,5808(2000); I.B.Spielman,J.
b,xx
temperature or with disorder). Furthermore, from the P.Eisenstein,L.N.Pfeiffer,andK.W.West,Phys.Rev.
Lett. 87, 036803 (2001).
experimentally measured CF resistivity, it is clear that
ρ is small ( 2π¯h/e2) at least at large d. As d is [7] M. Kellogg, I. B. Spielman, J. P. Eisenstein, L.N. Pfeif-
f,xx ≪ fer, and K. W.West Phys.Rev.Lett. 88, 126804 (2002)
reduced, presumably the density of CBs increases and
[8] M. Kellogg, J.P. Eisenstein, L.N. Pfeiffer, and K.W.
ρ drops until the CBs condense at some critical den-
b,xx West; cond-mat/021150.
sity and ρb,xx = 0 (yielding perfect Hall drag as in the [9] B.I.Halperin,P.A.Lee,andN.Read,Phys.Rev.B47,
111 state ). Simultaneously, as d decreases, ρf,xx pre- 7312 (1993).
sumably increases, but only mildly [9] and may remain [10] E. Rezayiand N. Read, Phys.Rev.Lett. 72, 900 (1994)
small even when the CBs condense. Only when the den- [11] We thank B. I. Halperin for pointing out that this rela-
sityofCFsisnearzeroshouldρ diverge. Inthe limit tion should hold in any Galilean invariant theory.
f,xx
that ρ 2π¯h/e2 it is easy to derive the above men- [12] M. Milovanovic, submitted to Phys.Rev.B.
f,xx
≪
4