Strongly Correlated Fractional Quantum Hall Line Junctions

4 Pages·0.15 MB·English

by  U. Zuelicke

Strongly Correlated Fractional Quantum Hall Line Junctions
U. Zu¨licke1 and E. Shimshoni2,3
1Institut fu¨r Theoretische Festk¨orperphysik, Universit¨at Karlsruhe, D-76128 Karlsruhe, Germany
2Department of Mathematics–Physics, University of Haifa at Oranim, Tivon 36006, Israel
3Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854–8019
3
(Dated: February 1, 2008)
0
0
Wehavestudiedacleanfinite–lengthlinejunctionbetweeninteractingcounterpropagatingsingle–
2
branchfractional–quantum–Halledgechannels. Exactsolutionsforlow–lyingexcitationsandtrans-

n port properties are obtained when the two edges belong to quantum Hall systems with different
a filling factors and interact via the long–range Coulomb interaction. Charging effects due to the
J couplingtoexternaledge–channelleadsarefullytakenintoaccount. Conductancesandpowerlaws

3 in the current–voltage characteristics of tunnelingare strongly affected by inter–edge correlations.
1

] Two–dimensional(2D)electronsystemsexhibitincom- low–lying excitations are represented by two decoupled
l
l pressibilitieswhentheirsheetdensityn iscommensurate bosonicmodesthathaveoppositechirality. Oneofthem,
a 0
with the value B of perpendicular magnetic field such the chargedmode, corresponds to fluctuations in the to-
h
- that the filling factor ν =2π¯hn0/eB is integer or equal talelectrondensity inthe twoedges andis free. The dy-
s | |
to certain fractions [1, 2]. Such an incompressible phase namics of its orthogonal complement, the neutral mode,
e
m is characterized by a quantized value of the Hall resis- turns out to be governed by realizations of chiral sine–
tance and low–lying excitations that are localized at the Gordon models [27, 28] that can be solved exactly for
.
at boundary of the 2D system [3, 4, 5, 6, 7]. When ν is certain values of the parameter ν˜=νRνL/νR νL . We
| − |
m the inverse of an odd integer, these quantum–Hall (QH) use these exact solutions to calculate transport through
edge excitations have been shown [8] to be isomorphous the line junctionandfindthatthe stronginter–edgecor-
-
d to those of a single–branch chiral one–dimensional (1D) relationssignificantlyaffectthe conductanceandtunnel-
n
electron system [9]. Similar to their nonchiral counter- ingresonancesarisingfromtranslationalinvariancealong
o
parts that are realized, e.g., in semiconductor quantum the barrier.
c
[ wires [10] or carbon nanotubes [11], QH edges are ex- Starting point of our calculations is the Hamiltonian

pected [8] to exhibit power laws in electronic correlation for our model of the line junction. (See Fig. 1.) It reads
2
v functions that are the hallmark of Luttinger–liquid be- H = L/2 dx ( + + + ) where
7 havior[12]. Virtuallynootherquasi–1Delectronsystem, J −L/2 HR HL Hint Htun HR/L
describeRisolated edge channels, the inter–edge in-
4 however,matchesthe versatilityofQHedgesintailoring Hint
teraction,and uniformtunnelingacrossthebarrier:
3 their electronic properties which can be achieved, e.g., Htun
0
simply by adjusting the magnetic field and/or appropri-
1
ate nanostructuring techniques. Recent applications of a
2
0 the cleaved–edge overgrowth method [13, 14] have suc- 2 3
/ ceededincreatingextendeduniformtunneljunctionsbe-
t ν ν
a tweentwo integerQHedges [15]or a 2Delectronsystem
m R L
andaQHedge[16]. Besidesopeningupnewpossibilities
- for electron tunneling spectroscopy [17], these new sam-
d 1 4
ple geometries enable the controlled experimental study
n
o oftheinterplaybetweentunnelingandinteractioneffects b
1 2
c in low dimensions [18, 19, 20, 21] that cannot be per-
: formed in conventional systems.
v
i A diverse set of electron–correlation effects has been 4 3
X
proposed for QH line junctions where both edge chan-
r nels belong to QH systems having the same filling −L/2 0 L/2 x
a
factor[22, 23, 24, 25, 26]. In our work presented here,
we consider the case of a clean junction where the filling FIG.1: Schematiclayoutofaquantum–Halllinejunction. An
extended uniform tunnel barrier couples counterpropagating
factors ν and ν , characterizing the respective right–
R L single–branch edge channels from two 2D electron systems
moving and left–moving edges, are different. Such a sit-
that have different fractional filling factors νR and νL. (See
uationcouldbe realized,e.g.,insampleswithtwomutu-
panel a.) We model this situation by two parallel infinite
ally perpendicular 2D electron systems [16] and a mag- chiral fractional edge channels that are coupled via uniform
netic field properlyadjustedinmagnitude anddirection. tunnelingand Coulomb interactions along the finitejunction
Intheexperimentallyrealisticlimitwhereelectronsfrom region where |x|≤L/2. Outside the junction, edge branches
thetwoedgesinteractstronglyviaCoulombinteractions, correspond to leads connecting to external reservoirs.
2
∆ := Y Y . We see from Eqs. (3) that the charged
R L
−
mode is free while the neutral–mode dynamics is that of
HR/L = 41π n¯hvR(0/)L+ ν2Rπ/L Uo (cid:2)∂xφR/L(cid:3)2 , (1a) aitschsoirlaultisoinn,e–wGeoirndtoenrjemctodaenl e[2la7b,o2r8a]t.ionBeofnoreobosbetravianbilnegs
√νRνL
= λU ∂ φ ∂ φ , (1b) related to electron transport.
Hint 4π2 x R x L
To discuss transport properties,it is necessary to con-
Htun = tnψR†(x)ψL(x)+H.c.o . (1c) sidercontinuityequationsandchemicalpotentialsforthe
various edge densities. Denoting time by τ, the operator
Here we denote the second–quantized annihilation op- forparticlecurrent(perunitlength)acrossthe barrieris
erators for electrons from the two edges as ψ . The
R/L
chiral bosonic phase fields φ (x) are related to the
R/L
d d
respective edge densities via ̺R/L = ψR†/LψR/L = IJ(x) = −dτ ̺R(x)= dτ ̺L(x) , (5a)
√νR/L∂xφR/L/(2π) and obey the commutation rela- t
tions φR/L(x),φR/L(x′) = ±iπsgn(x−x′). The bare = i¯h nψR†(x)ψL(x)−H.c.o , (5b)
(confin(cid:2)ement–induced)e(cid:3)lectronvelocitiesinthetwoedge
channels are given by v(0) , U is the matrix element In our limit of interest, its bosonizedexpressiondepends
R/L on the neutral mode only,
of screened Coulomb interactions, and we have taken
into account the generally different interactionstrengths
t φ x∆
n
within and between the edges by a factor 0 < λ ≤ IJ(x)=2¯h√zRzL sin(cid:18)√ν˜ + ℓ2 (cid:19) . (6)
1. It is straightforward to diagonalize the part H
J
−
L/2
dx , and using the bosonization identity [29] Itisstraightforwardtofindacontinuityequationforden-
−L/2 Htun
R sity ̺ := ∂ φ /(2π) associated with the charged mode,
c x c
ixYR/L±i φR/L(x)
ψR/L(x)=√zR/L R/L e ℓ2 (νR/L)1/2, (2)
F d ̺ +̺ d
R L
0 = ̺ , (7a)
thetunnelingtermHtun canbeexpressedsolelyinterms dτ νR νL ≡ dτ c
of bosonic phase fields as well. [In Eq. (2), z are | − |
R/L = ∂ p+sgn(ν ν )v ∂ ̺ , (7b)
normalization constants, R/L Klein factors, YR/L the { τ R− L c x} c
F
guiding–centercoordinatesofedgeelectronsinthedirec-
and the corresponding one for ̺ :=∂ φ /(2π),
n x n
tion perpendicular to the junction, and ℓ = ¯hc/eB
| |
the magnetic length.] However, explicit exprepssions for d sgn(ν ν )
R L
0 = ̺ − , (8a)
normal modes in the most general case are unilluminat- n J
dτ − √ν˜ I
ing. We therefore proceed immediately to the special
= ∂ sgn(ν ν )v ∂ ̺ . (8b)
case of interest to us where strong Coulomb interactions { τ − R− L n x} n
dominatethebareelectronvelocitiesineachbranch,i.e.,
In the stationary regime, densities have no explicit time
¯hv(0) U, and the distance between the two edges is dependence. The above continuity equations then imply
R/L ≪
of the order of or smaller than ℓ such that λ 1. It is thattheyassumeconstantvalues̺¯ alongthejunction.
→ c/n
thenpossibletorewritetheQH–junctionHamiltonianas We can also derive expressions for local chemical poten-
HJ = −LL/2/2dx (Hc+Hn) with the contributions tials µc/n :=δHJ/δ̺c/n ≡∓isgn(νR−νL) HJ,φc/n as-
R sociated with the charged and neutral mod(cid:2)es: (cid:3)
¯hv
= c [∂ φ ]2 , (3a)
Hc 4π x c µc(x) = 2π¯hvc̺c(x) , (9a)
= ¯hvn [∂ φ ]2+2t√z z cos φn + x∆ (,3b) µn(x) = 2π¯hvn̺n(x)
Hn 4π x n R L (cid:18)√ν˜ ℓ2 (cid:19) L
π¯h 2
+ dx′ (x′) sgn(x x′). (9b)
J
for independent c(harged) and n(eutral) modes [36]. √ν˜Z−L I −
2
Theircorrespondingphasefieldsobeycommutationrela-
tions φ (x),φ (x′) = sgn(ν ν )iπsgn(x x′) Specializing these to the stationary limit, we find
c/n c/n R L
± − −
and a(cid:2)re related to the(cid:3) densities of the original right–
µ (x) = 2π¯hv ̺¯ =const. , (10a)
moving and left–moving edge electrons via c c c
π¯h
µ ( L/2) = 2π¯hv ̺¯ I , (10b)
̺ = ±sgn(νR−νL) ν ∂ φ √ν ν ∂ φ . n ± n n± √ν˜ J
R/L R/L x c R L x n
2π ν ν −
p| R− L| (cid:0) (cid:1) (4) where I := L/2 dx (x) is the total current flowing
J −L/2 IJ
The velocities are vc = |νR − νL|U/(2π¯h) and vn = through the jRunction. The values of ̺¯c/n and IJ can be
ν v(0)+ν v(0) /ν ν ,andweusedtheabbreviation related to chemical potentials in the external leads that
R L L R | R− L|
(cid:0) (cid:1)
3
connectthe QHjunctiontoreservoirs(seeFig.1),which It has been shown [28] that the cosine term constitutes
we proceed to show now. only an irrelevant perturbation to the dynamics of the
In a real setup as sketched in Fig. 1a, edge channels chiralbosonicfieldwhen m m′ >2. Moreinteresting,
| − |
away from the junction are not coupled via tunneling or and easier to realize experimentally, are the two cases
interactions anymore. However, for each of these chi- 1/ν˜ 2,4 where the cosine term turns out to be rele-
∈{ }
ral 1D leads, Coulomb interactions between electrons at vantormarginal,respectively,andexactsolutionsforits
the same edge stillgivethe dominantcontributiontothe dynamics can be found [27, 28].
edge–magnetoplasmonvelocity [30]. We therefore model Lets first consider the case ν˜ = 1/2 realized, e.g.,
the dynamics of electrons in regions x > L/2 by the at a line junction between QH systems having filling
| |
leadHamiltonianH = −L/2+ ∞ dx ( + ). factors 1 and 1/3, respectively. Introducing an auxil-
E n −∞ L/2o HR HL iary chiral boson field η(x) that has the same chiral-
While H is not diagonaRl in theRcharged and neutral
E ity and velocity as φ , we can define fictitious chiral
modes, their corresponding chemical potentials are still n
fermionoperatorsΨ via aninversebosonizationiden-
well–defined in the lead regions. As the densities in ↑/↓
tity Ψ exp isgn(ν ν )[η+σφ ]/√2 . Note
eachofthefourchiraledge–channelleadscannotchange, σ R L n
∝ − −
chemicalpotentialsremainconstantaswell[31]. Assum- that the pseudo(cid:8)spin degree of freedom indexed(cid:9)by σ is
ing again U ¯hv(0) , we find for these (in a compact not related to the spin of the original electrons in the
≫ R/L sample. Inthenewfictitious–fermiondegreesoffreedom,
notation where µ means µ or µ , respectively)
i/j i j Eqs. (3b) and (6) are quadratic:
ν µ ν µ
µ = R 1/2− L 4/3 sgn(ν ν ), (11a) ′ = Ψ† [isgn(ν ν )h¯v ∂ ]Ψ
c|x><∓L ν ν R− L Hn σ R− L n x σ
2 | R− L| Xσ
µn|x><∓L = −√pν˜ µ1/2−µ4/3 sgn(νR−νL).(11b) +t Ψ†↑Ψ↓eixℓ∆2 sgn(νR−νL)+H.c. (,13a)
2 (cid:0) (cid:1) (cid:16) (cid:17)
t
Dlineemaarnrdeilnagtiocnosntiinnvuoiltvyinogf tµhce/nchaetmxic=al±poLt/e2ntyiaielsldisnfothuer IJ′ = −i¯h (cid:16)Ψ†↑Ψ↓eixℓ∆2 sgn(νR−νL)−H.c.(cid:17) .(13b)
four leads, the constants ̺¯ , and I . After some In addition, density and chemical potential of the neu-
c/n J
straightforwardalgebra,we obtain the expressions tralmodearerelatedtocorrespondingfictitious–fermion
quantities via Ψ†Ψ Ψ†Ψ √2̺ and µ µ
2π¯h ↑ ↑ − ↓ ↓ ≡ n ↑ − ↓ ≡
µ = µ I , (12a) √2µ . Note that only the pseudospin sector of the new
2/4 1/3± ν J n
R/L fermionic theory has any bearing for real observations,
νRµ1 νLµ3+2π¯hIJ while the pseudocharge sector describedby the auxiliary
v ̺¯ = − sgn(ν ν ), (12b)
c c R L
2π¯h ν ν − field η is hidden. We see that the problem of the orig-
R L
| − |
p inal line junction between two strongly interacting edge
µ µ + πh¯ + πh¯ I
v ̺¯ = 1− 3 (cid:16)νR νL(cid:17) Jsgn(ν ν )(,12c) branches with opposite chirality has been mapped onto
n n 2π¯h/√ν˜ R− L that of tunneling between two branches of noninteract-
− ing andchiral fermions. Thelatterisreadilysolvedusing
relating the ‘output’ chemical potentials µ2/4 as well as standardmethods. Forexample,whentunneling isweak
the constants ̺¯c/n to the experimentally adjustable ‘in- enoughsuchthatitispossibletoneglectthecontribution
put’ chemical potentials µ1/3 and the a priori unknown IJ in Eq. (12c), we find e2IJ = GJ(µ1 µ3) with the
current IJ [37]. To obtain the full solution for the trans- ∝linear electric conductance −
port problem of our finite QH junction that is attached
tdoeteedrmgei–nceh̺a¯nnnaenldleIaJdsse,lfw–ceoanrseistneonwtlyleuftsiwngiththtehdeytnaasmkitcos GJ = 2eπ2¯h sin2hπL1Lt+pξ12+ξ2i . (14)
oftheneutralmodeexpressedinEqs.(3b),(6),(9b),and
(12c). We now show how this can be achieved. Here Lt = π¯hvn/t is a fundamental length scale set by
| |
The Hamiltonian of the neutral mode [given by tunneling, and ξ = h¯vn∆/(2tℓ2) a resonance parame-
| |
Eq.(3b)]isarealizationofrecentlydiscussed[27,28]chi- ter: GJ is maximal for ξ 0 which corresponds to the
→
ral sine–Gordon models. In contrast to the well–known resonance condition where both energy and 1D momen-
nonchiral sine–Gordon model [32], the quantum charac- tum are conserved in a tunneling event [38]. The ob-
terofits dynamicsarisesbecausethe bosonicfieldenter- tainedlinear dependence ofIJ onµ1 µ3 has to be con-
ing the cosine term does not commute with itself. Pos- trastedwiththepowerlawIJ (µ1 −µ3)νR−1+νL−1−1 that
∝ −
sible values of the parameter ν˜ for the QH line junction is to be expected for momentum–resolved tunneling at
discussedherecanbeparameterizedbytwonon–negative aline junction betweennoninteracting chiral–Luttinger–
integers m,m′ such that 1/ν˜ = 2m m′ . These val- liquid [8] edge channels. Furthermore, chirality of the
| − |
ues correspondto QH junctions between systems having effective tunneling problem described by Eqs. (13) re-
filling factors 1/(2m+1) and 1/(2m′+1), respectively. sults in charge oscillations along the junction similar to
4
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[36] ThisactuallyrequiresalsotheconditionU ≫|t|ℓtohold
tunneling currents oscillate in time. which is typically satisfied. No Klein factors appear in
We thank N. Andrei, C. Chamon and V. Tripathi for Eqs.(3)astheygiveriseonlytoanoverallfactor±1that
useful discussions. This work was supported in part by can be absorbed into thetunnelingmatrix element [28].
the German Science Foundation (DFG) through Grant [37] The result that external voltages enter as uniform con-
straintsonchiralchargedensitiesinthe tunneling region
No. ZU 116/1 and the German Federal Ministry of Ed-
is nontrivial. Similar relations arise for a homogeneous
ucation and Research (BMBF) in the framework of the
interacting quantum wire in the absence of backscatter-
DIP program. ing.See,e.g.,R.EggerandH.Grabert,Phys.Rev.B58,
10761 (1998); I. Safi, Eur.Phys. J. B 12, 451 (1999).
[38] Note that the condition of weak tunneling used here is
valid for GJ ≪ e2/2π¯h, yet it does not restrict the tun-
neling strength to theperturbativelimit L≪Lt.
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