Momentum-Resolved Tunneling into Fractional Quantum Hall Edges

4 Pages·0.18 MB·English

by  U. Zuelicke

Momentum-Resolved Tunneling into Fractional Quantum Hall Edges
U. Zu¨licke,1 E. Shimshoni,2 and M. Governale1
1Institut fu¨r Theoretische Festk¨orperphysik, Universit¨at Karlsruhe, D-76128 Karlsruhe, Germany
2Department of Mathematics and Physics, University of Haifa at Oranim, Tivon 36006, Israel
(Dated: February 1, 2008)
2
0 Tunneling from a two–dimensional contact into quantum–Hall edges is considered theoretically
0
foracasewherethebarrierisextended,uniform,andparalleltotheedge. Incontrasttopreviously
2
realized tunneling geometries, details of the microscopic edge structure are exhibited directly in

n thevoltage and magnetic–field dependenceof thedifferential tunnelingconductance. Inparticular,
a it is possible to measure the dispersion of the edge–magnetoplasmon mode, and the existence of
J additional, sometimes counterpropagating, edge–excitation branchescould bedetected.

5
2 PACSnumbers: 73.43.Jn,73.43.Cd,71.10.Pm

]
l The quantum Hall (QH) effect[1] arises due to incom- 2DES
l
a z
pressibilitiesdevelopingintwo–dimensionalelectronsys-
h B
tems(2DES)atspecialvaluesoftheelectronicsheetden- y
-
s sityn andperpendicular magneticfieldB for whichthe
e 0 x
filling factor ν = 2π¯hcn /|eB| is equal to an integer p
m 0
or certain fractions. The microscopic origin of incom-
y
.
t pressibilities at fractional ν is electron–electron interac-
a
m tion. Laughlin’s trial–wave–function approach [2] suc-
cessfully explains the QH effect at ν = ν ≡ 1/(p+1)
- 1,p
d where p is a positive even integer. Our current micro-
2DES
n scopic understanding of why incompressibilities develop
o at many other fractional values of the filling factor, e.g., V
c
ν ≡ m/(mp + 1) with nonzero integer m 6= ±1, is I
[ m,p
based on hierarchical theories [3, 4, 5].

1
Theunderlyingmicroscopicmechanismresponsiblefor
v
2 creatingchargegapsatfractionalνimpliespeculiarprop- FIG. 1: Schematic picture of tunneling geometry. Two mu-
6 erties of low–energy excitation in a finite quantum–Hall tuallyperpendiculartwo-dimensionalelectronsystemsarere-
4 sample which are localized at the boundary [6]. For alized, e.g., in a semiconductor heterostructure. An external
1
ν =ν ,mbranchesofsuchedgeexcitations[7,8,9,10] magneticfieldisappliedsuchthatitisperpendiculartooneof
0 m,p
2 are predicted to exist which are realizations of strongly them(2DES⊥)butin-planefortheotherone(2DESk). When
2DES is in the quantum-Hall regime, chiral edge channels
0 correlatedchiralone–dimensionalelectronsystemscalled ⊥
form along its boundary (indicated by broken lines with ar-
/ chiralLuttingerliquids (χLL).Extensiveexperimentalef-
t rows). Wheretheyrunparallelto2DES ,electronstunnelbe-
a forts were undertaken recently to observe χLL behavior k
tween edgestates in 2DES and plane-wavestates in 2DES
m ⊥ k
because this would yield an independent confirmationof withthesame quantumnumberp ofmomentumcomponent
y
- our basic understanding of the fractional QH effect. In parallel to the barrier. Experimentally, the differential tun-
d
allofthesestudies[11,12,13,14,15,16],current–voltage neling conductancedI/dV is measured.
n
o characteristicsyieldedadirectmeasureofthe energyde-
c pendence of the tunneling density of states for the QH
v: edge. This quantity generally contains information on rˆole. New experiments are needed to test the present
i global dynamic properties as, e.g., excitation gaps and microscopic picture of fractional–QH edge excitations.
X
the orthogonality catastrophe,but lacks any momentum Here we consider a tunneling geometry that is par-
r
resolution. Power–law behavior consistent with predic- ticularly well-suited for that purpose, see Fig. 1, and
a
tions fromχLL theory wasfound[11, 12, 15]for the edge which has been realized recently for studying the in-
of QH systems at the Laughlin series of filling factors, teger QH effect in cleaved-edge overgrown semiconduc-
i.e., for ν =ν . However, at hierarchical filling factors, tor heterostructures[17]. In contrast to previous experi-
1,p
i.e.,whenν =ν with|m|>1,predictionsofχLLthe- ments, it provides a momentum–resolved spectral probe
m,p
oryare,atpresent,notsupportedbyexperiment[13,14]. of QH edge excitations[32]. With both the component
This discrepancy inspired theoretical works, too numer- of canonical momentum parallel to the barrier and en-
ous to cite here, from which, however, no generally ac- ergy being conserved in a single tunneling event, strong
ceptedresolutionemerged. Currentexperiments[16]sug- resonances appear in the differential tunneling conduc-
gestthat details of the edge potential may play a crucial tance dI/dV as a function of the transport voltage and
2
applied magnetic field. Similar resonant behavior for 1
a)
tunneling via extended uniform barriers has been used
ε+v q 2
recently[18, 19, 20, 21] to study the electronic properties q<0 c
of low–dimensional electron systems. It has also been )
ε
suggested as a tool to observe spin–charge separation in ,
q 1
Luttinger liquids[22] and the interaction–induced broad-
(
eningofelectronicspectralfunctionsatsingle-branchQH A ε−v q 2
n
edges[23]. Here we find that the number of resonantfea-
tures in dI/dV corresponds directly to the number of
chiral edge excitations present. Edge–magnetoplasmon
dispersion curves can be measured and power laws re-
v q −v q
latedtoχLLbehaviorbeobserved. Momentum–resolved n c
ε
tunnelingspectroscopyinthepresentlyconsideredgeom-
etrythusconstitutesapowerfulprobetocharacterizethe
QH edge microscopically. b) 1
To compute the tunneling conductances, we apply the ε+v q 2
general expression for the current obtained to lowest or- q>0 c
)
der in a perturbative treatment of tunneling[27]: ε

,
q
e dε
I(V) = |t |2 {n (ε)−n (ε+eV)} ( 3
¯h2 ~kk,n,X Z 2π F F A
~kX,n,X ε+v q 2
k n
×A (~k ,ε)A (n,X,ε+eV). (1)
k k ⊥
HereA andA denotesingle-electronspectralfunctions
k ⊥
−v q −v q
for2DESkand2DES⊥,respectively. (SeeFig.1). Weuse c n
a representation where electron states in the first are la- ε
beledbyatwo–dimensionalwavevector[33]k =(k ,k ),
k y z
while the quantum numbers of electrons in 2DES are
⊥
FIG. 2: Spectral functions for two–branch hierarchical
the Landau–level index n and guiding–center coordinate
fractional–QH edges at bulk filling factor 2/3 [panel a)] and
X in x direction. We assume that 2DES is located at
k 2/5 [panel b)], where the charged (edge–magnetoplasmon)
x=0. Thesimplestformofthetunnelingmatrixelement modeisassumedtobeleft–moving. a) WeshowA(0)(q,ε)≡
t reflecting translational invariance in y direction 2/3
~kk,n,X A(1)(q,ε) for a fixed value of q. Note the similarity with the
yields 2/3
spectral function of a spinless Luttinger liquid[24, 25]. The
only differenceisthat, in ourcase, velocities ofright–moving
t =t (X) δ(k −k) , (2)
~kk,n,X n y andleft–movingplasmonmodesarenotequal. b)A(0)(q,ε)≡
2/5
wherek≡X/ℓ2withthemagneticlengthℓ= ¯hc/|eB|. A(−1)(q,ε)atfixedq. Itisreminiscentofthespectralfunction
2/5
Thedependenceoftn(X)onX resultsformthpefactthat foraspinfulχLLexhibitingspin–chargeseparation[25,26]but
an electron from 2DES⊥ occupying the state with quan- differsduetotheabsenceofanyalgebraicdivergenceat−vnq.
tum number X is spatially localized on the scale of ℓ
aroundx=X. The overlapofits tailinthe barrierwith
thatofstates from2DESk willdropprecipitouslyasX/ℓ is possible to linearize the lowest–Landau–level disper-
gets large. Finally, nF(ε) = [exp(ε/kBT)+1]−1 is the sion around the Fermi point kF. At the Laughlin series
Fermi function. In the following, we use the expression ν =1/(p+1) and for short-rangeinteractions presentat
Ak(~kk,ε) = 2πδ(ε−E~k ) which is valid for a clean sys- the edge, the spectral function was found[9, 28] to be
k
tem of noninteracting electrons[34]. Here E denotes
~kk z q p
the electron dispersion in 2DES . A (q,ε)= δ(ε−r¯hv q). (4)
The spectral function of electkrons in 2DES depends p+11 p!(cid:18)2π/Ly(cid:19) e
⊥
cruciallyonthetypeofQHstateinthislayer. Atinteger Here q ≡ k − k , r = ± distinguishes the two chiral-
F
ν,whensingle–particlepropertiesdominateanddisorder ities of edge excitations, L is the edge perimeter, v
y e
broadening is neglected, it has the form the edge–magnetoplasmon velocity, and z an unknown
normalization constant. The power–law prefactor of the
A (n,X,ε)≡A (k,ε)=2πδ(ε−E ) , (3)
⊥ n nk δ–functioninEq.(4)isamanifestationofχLLbehavior.
where E is the Landau–level dispersion. Strong cor- The main focus of our work is the sharp QH edge
nk
relations present at fractional ν alter the spectral prop- at hierarchical filling factors. Here we provide explic-
erties of edge excitations. In the low–energy limit, it itly the momentum–resolved spectral functions for ν =
3
ν [35]. Microscopic theories[7, 9] predict the exis- According to χLL theory, the existence of two Fermi
±2,p
tence of two Fermi points k and k which correspond points gives rise to a discrete infinite set ofpossible elec-
Fo Fi
to outer and inner single-branch chiral edges of QH flu- tron tunneling operators at the edge. This is because an
ids at Laughlin–series filling factors ν± = 1/(p±1) and arbitrarynumberN offractional–QHquasiparticleswith
o
ν± =±1/[(2p±1)(p±1)],respectively. Thenegativesign chargeequal to eν± can be transferredto the inner edge
i o
ofν indicatesthattheinneredgemodeiscounterpropa- afteranelectronhastunneledintotheouterone[9]. Each
i−
gating. Wehaveusedstandardbosonizationmethods[29] ofthese processesgivesriseto aseparatecontributionto
applied to fractional–QH edges[9] for the calculation of the electronic spectral function at the edge which is of
the spectral functions. As these have not been obtained the general form
before, we briefly discuss their main features here.
η(N)+η(N)−1
Aν(N±2),p(q,ε) = Γ(η(N2))πΓz(η(N))(cid:18)|Lv1y/∓2πv2¯h|(cid:19) 1 2 |ε−r¯hv1q|η2(N)−1|ε∓r¯hv2q|η1(N)−1
1 2
×{Θ(r¯hv q−ε)Θ(±ε−r¯hv q)+Θ(ε−r¯hv q)Θ(r¯hv q∓ε)}. (5)
1 2 1 2
Here q ≡k−k(N), where k(N) =k −Nν±(k −k ). 2DES are denoted by ε . Eqs. (6) can be used
F F Fo o Fo Fi ⊥,k F⊥,k
The velocities v1 >v2 >0 of normal–mode edge–density to extract the lowest–Landau–level dispersion E0k from
fluctuations and the exponents η(N) depend strongly on maxima in the experimentally obtained dI/dV, thus en-
1,2
microscopic details of the edge, e.g., the self-consistent abling microscopic characterizationof real QH edges.
edge potential and inter–edge interactions. We focus
When 2DES is in a QH state at a Laughlin–series
⊥
here on the experimentally realistic case when inner
filling factor ν , it supports a single branch of edge
1,p
and outer edges are strongly coupled and the normal
excitations just like at ν = 1, and the calculation of
modes correspond to the familiar[30] charged and neu-
thedifferentialtunneling conductanceproceedsthesame
traledge-densitywaves[36]. Inthis limit, wehave[30,31]
way. The major difference is, however, the vanishing of
v = v ∼ O(log[L /ℓ]), v = v ∼ O(1) (where c and
1 c y 2 n spectral weight at the Fermi point of the edge; compare
n denote charged and neutral, respectively), and the ex-
Eqs.(3) and(4). This results in the suppressionof max-
(N)
ponents assume universal values: η1 = ηc ≡ p±1/2, ima describedby Eq.(6a),while those givenby Eq.(6b)
η(N) = η(N) ≡ (2N ±1)2/2. Note that exponents are remain. The intensity of the latter rises along the curve
2 n
generally larger than unity except for N = 0,∓1 where as a power law with exponent p.
(N)
η = 1/2. In the latter case, an algebraic singular-
2 Finally, we discuss the case of hierarchical filling fac-
ity appears in the spectral function. This is illustrated
torsν whichareexpectedtosupporttwobranchesof
±2,p
in Fig. 2. Such divergences will be visible as strong fea-
edge excitations. To be specific, we consider filling fac-
turesinthedifferentialtunnelingconductance;seebelow.
tors 2/3 and 2/5. In both cases, there are many contri-
Contributionstothespectralfunctionforallothervalues
butions to the spectral function and, hence, the differen-
ofN donotshowsuchdivergencesandwillgiveriseonly
tial tunneling conductance. However, only two of these
to a featureless background in the conductance.
exhibit algebraic singularities. It turns out that these
With spectral functions for 2DES at hand, we are
⊥ singularities give rise to either a strong maximum or a
nowableto calculatetunneling transport. We focus first
finite step in the differential tunneling conductance, de-
on the case when 2DES is in the QH state at ν = 1.
⊥ pending on the sign of voltage. (See Fig. 3). The strong
Forrealisticsituations,thedifferentialtunnelingconduc-
maximum results from a logarithmic divergence that oc-
tancedI/dV asafunctionofvoltageV andmagneticfield curs when eV = h¯v (k(N) −k ). Both the maximum
Bwillexhibittwolinesofstrongmaximawhosepositions c F Fk
and the step edge follow the dispersion of the charged
in V–B space are given by the equations
edge–magnetoplasmonmode and would therefore enable
its experimental investigation. Most importantly, how-
E = ε , (6a)
0kV F⊥ ever,thetwospectralfunctionswithsingularitiesexhibit
E = ε +eV . (6b)
0kFk F⊥ them slightly shifted in guiding–center, i.e., k direction
by an amount ν±(k −k ). Hence, two maxima and
o Fo Fi
Here kV = 2m(εFk−eV)/¯h2 and kFk, the Fermi wave a double–step feature should appear in the differential
q
vectorin2DES ,aretheextremalwavevectorsforwhich tunneling conductance whose distance in magnetic–field
k
momentum–resolvedtunneling occurs. Fermi energies in directionwillbeameasureoftheseparationofinnerand
4
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differential tunneling conductance. solve momentum even with perfect translational invari-
We thank M. Grayson and M. Huber for many useful anceparalleltotheedge.Thelatterisdestroyedanyway,
discussionsandcommentsonthe manuscript. This work in real samples, by dopant–induced disorder in the bulk
contact.Seealsoarelatedtunnelingspectroscopyofpar-
was supported by DFG Grant No. ZU 116 and the DIP
allelQHedgesbyW.Kanget al.,Nature(London)403,
projectofBMBF.U.Z.enjoyedthehospitalityofSektion
59 (2000).
Physik at LMU Mu¨nchen when finishing this work.
[33] Hereweneglect magnetic–field–induced subbandmixing
in 2DES . While this can be straightforwardly included,
k
itwilltypicallyresultinsmallquantitativechangesonly.
[34] Broadening due to scattering from disorder or interac-
tions can be straightforwardly included and does not
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