Critical Tunneling Currents in Quantum Hall Superfluids: Pseudospin-Transfer Torque Theory

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by  Jung-Jung Su

Critical Tunneling Currents in Quantum Hall Superfluids:
Pseudospin-Transfer Torque Theory
Jung-Jung Su1,2 and Allan H. MacDonald1
1 Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA and
2 Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545,USA
(Dated: January 18, 2010)
At total filling factor ν =1 quantum Hall bilayers can have an ordered ground state with spon-
0 taneous interlayer phase coherence. The ordered state is signaled experimentally by dramatically
1 enhanced interlayer tunnel conductances at low bias voltages; at larger bias voltages inter-layer
0 currents are similar to those of the disordered state. We associate this change in behavior with
2 theexistenceof acritical currentbeyondwhich static inter-layerphasedifferencescannot bemain-

n tained,andexaminethedependenceofthiscriticalcurrentonsamplegeometry,phasestiffness,and
a thecoherent tunnelingenergy density. Ouranalysis isbased in part on analogies between coherent
J bilayerbehaviorandspin-transfertorquephysicsinmetallic ferromagnets. Comparison withrecent

8 experimentssuggests that disorder can dramatically suppress critical currents.
1
PACSnumbers:

]
l
al I. INTRODUCTION generally depends18,21 on how the fermionic degrees of
h freedomwhichcarrychargebetweenleads areinfluenced
- by orderparameteranddisorderconfigurationsandcan-
s At Landau level filling factor ν = 1 bilayer two-
e not be described in terms of condensate dynamics alone.
dimensionalelectronsystemsinthequantumHallregime
m canhavebrokensymmetrygroundstates1–4 withsponta- In the second approach the width in voltage of the con-
t. neous inter-layer phase coherence. These ordered states ductance peak is simply equal to the product of this re-
a can be viewed as excitonic superfluids,5,6 or as XY sistance and the maximum current between source and
m pseudospin ferromagnets7 in which the pseudospin is drainatwhichthe orderparametercanmaintainatime-
- formed from the two-valued which layer quantum de- independent steady-state value.
d
n gree of freedom. The most robust experimental signa- In this article we expand on the second type of the-
o ture of these states, a vastly enhanced inter-layer tunnel ory of the conductance peak, using pseudospin-transfer
c conductance8–11 atsmallbiasvoltages,isstillpoorlyun- torque ideas22,23 borrowed from recent ferromagnetic-
[ derstood from a quantitative point of view. metalspintronicsliterature to modelthe influence ofthe

transport current on the pseudospin magnetization. We
1 Twotypesofideas,whichdiffermostessentiallyinhow
v thebiasvoltageisintroducedinthetheory,havebeenex- find that the critical current depends in general on de-
3 ploredinanefforttounderstandtheheightandwidthof tailsofthesamplegeometryandonhowdisorderandlo-
2 the tunnel conductance peak. In one approach12–15 the calization physics influence transport inside the system.
9 Generally speaking however, the critical current is pro-
bias voltage is introduced as an effective magnetic field,
2
portional to system area and to the inter-layer tunnel-
uniformacrossthebilayer,whichinducespseudospinpre-
.
1 ingamplitudewhenthecondensate’sJosephsonlengthis
cession around the zˆ axis, driving the interlayer phase
0 largerthanthesystemperimeter,andproportionaltothe
difference at a steady rate and inducing a purely oscil-
0 systemperimeterandtothesquarerootoftheinter-layer
lating inter-layer current. When the microscopic inter-
1
tunneling amplitude when it is shorter.
: layer tunneling amplitude is treated as a perturbation,
v
thermal and disorder fluctuations of the condensate are Our paper is organizedas follows. In Section II we in-
i
X then responsible for a finite dc conductance peak. We troduce the pseudospin-transfer torque theory of order-
refer to this type of theory below as the weak-coupling parameterdynamicsinabilayerquantumHallferromag-
r
a theory of the tunneling anomaly. Weak-coupling theo- net. We use this theory in Section III to discuss critical
ries predict12–15 splitting of the zero voltage tunnel con- currentvaluesfromaqualitativepointofview. InSection
ductance peak into separate finite voltage peaks in the IV we report on a series of numerical studies which take
presence of a magnetic field component parallel to the into account the two-dimensional nature of the systems
two-dimensional layers, an effect that is not16 seen ex- ofinterest andthe edge dominatedcurrent-paths typical
perimentally. The second type of transport theory17–21 ofstrongmagneticfields. Thepseudospintransfertorque
is formulated in terms of local chemical potentials of theoryenablesustoassesstheinfluence ofsamplegeom-
fermionic quasiparticles which may be altered by the or- etry on critical currents. Finally in Section V we discuss
deredstatecondensatebutarestillresponsibleforcharge thesignificanceofourfindingsinrelationtorecentexper-
conduction. In this type of theory the tunnel conduc- iments. We find that experimental critical currents are
tance is finite even in the absence of disorder and ther- severalordersofmagnitudesmallerthantheoreticalones
malfluctuationsbecausechargehastobedrivenbetween and argue that vortex-like disorder-induced pseudospin
normal-metal source and drain contacts. The resistance texturesmustbe largelyresponsibleforthisdiscrepancy.
2
We proposenew experimentalstudies whichcantestour shorter than those on which this coarse-grained contin-
ideasandtherebyachieveprogresstowardaquantitative uum theory is applied; disorder on longer length scales
theory of the spontaneous coherence tunneling anomaly. would have to be treated explicitly as we mention in the
discussionsection. The upshotis that the numericalval-
ues of the parameters in Eq.( 3) are usually not accu-
II. PSEUDOSPIN MAGNETISM AND THE rately known and likely vary substantially from sample
LANDAU-LIFSHITZ-SLONCZEWSKI EQUATION tosample. Itisworthnotingthatnmustvanishatfinite
temperatures when ∆ → 0 since two-dimensional sys-
t
In the lattice model24 of ν = 1 bilayer systems, the tems cannot support spontaneous long-range phase or-
local which layer degree of freedom can be expressed as der. Among all continuum model parameters the value
a pseudospin defined by the operator of ρs, which is typically ∼ 10−4eV, is likely the most
reliably known.
S~i = 21 a†i,σ~τσ,σ′ai,σ′ (1) troSninscpeinthheawvehicidhenlatyicearlpqsueaundtousmpinmaencdhatnhiceatlrudeescerleipc--
Xσ,σ′
tions, we can borrow from the ferromagnetic metal
spintronics literature22,23 and use the Landau-Lifshitz-
whereiis thesite index, σ is the layerindex and~τ isthe
Slonczewski(LLS)equationtodescribehowthesemiclas-
Pauli matrix vector. The pseudospin Hamiltonian has
the form24 sical pseudospin dynamics is influenced by a transport
current:
1
H = (2H −FS )SzSz−FD(SxSx+SySy) (2) dm~ (~j·∂ )m~ dm~
int 2Xi,j i,j i,j i j i,j i j i j dt =m~ ×H~eff − n~r −α(cid:18)m~ × dt (cid:19). (4)
where H = hi,σ;j,σ′|V |i,σ;jσ′i is the direct (~j is the number current density for electrons.) The sec-
i,j col
Coulomb interaction associated with the zˆ pseudospin ond term on the right-hand-side of Eq.( 4) captures the
component (i.e. with charge transfer between layers), transport current effect. Its role in these equations is
FS = hi,σ;j,σ|V |i,σ;jσi, is the exchange interac- similar to the role played by the leads in the pioneering
i,j col
tion between orbitals in the same layer and FD = analysis of coherent bilayer tunneling by Wen and Zee17
i,j
hi,σ;j,σ¯|V |i,σ;jσ¯iistheexchangeinteractionbetween who argued that a term should be added to the global
col
orbitals located in different layers. Since H is gener- condensate equation of motion to account for the con-
i,j
allylargerthanFS ,theclassicalgroundstateisaneasy- tribution oftransportcurrents to the difference in popu-
i,j
plane pseudospin ferromagnetwith a hardzˆaxis. In the lation between top and bottom layers. Eq.( 4) describes
limitofsmoothtexturesthepseudospinenergyfunctional howatransportcurrentaltersthecondensateequationof
has the form7 motion locally. Its justification for the bilayer quantum
Hallcaseisdiscussedinmoredetailbelow. Theessential
E[m~] = d2r β(m )2 validity of this expression has been verified by countless
z
Z experiments. In the first term on the right hand side
(cid:8)
1 2 2 1
+ ρs[ ∇~mx + ∇~my ]− ∆tnmx (3) H~eff =(2/~n)(δE[m~]/δm~) (5)
2 (cid:12) (cid:12) (cid:12) (cid:12) 2 (cid:27)
(cid:12) (cid:12) (cid:12) (cid:12) describes precessional pseudospin dynamics in an effec-
(cid:12) (cid:12) (cid:12) (cid:12)
where m~ = {m ,m ,m } is the local pseudospin direc- tive magnetic field which is defined by the energy func-
x y z
tion. Theparameterswhichappearinthisexpressionare tional. The third term accounts for damping of the col-
the anisotropyparameter β >0,the pseudospinstiffness lective motiondue to coupling to its environment,in the
(or equivalently the exciton superfluid density) ρ , the present case the Fermi sea of quasiparticles. The damp-
s
splitting between symmetric and antisymmetric single- ing term in Eq.( 4) has the standardisotropic form used
particlebilayerstatesduetointerlayertunneling∆ ,and in the magnetism literature. A microscopic theory26 of
t
the pseudospin density n. dampinginquantumHallbilayersmakesitclearthatthe
The mean-field-theory pseudospin ferromagnet1 con- damping is actually quite anisotropic. We return to this
sists of a full Landaulevel of electrons in identical phase point below.
coherent bilayer states. It follows that the mean-field- Magnetic order in the metallic ferromagnets to which
theory pseudospin density n is equal to the full Landau this equation is normally applied is extremely robust,
level density, (2πl2)−1. (Here l = (~c/eB)1/2, where B justifying the assumption that the spin-density magni-
is the magnetic field strength, is the magnetic length.) tude is essentially unchanged even when the system is
Mean-field-theory can also used7 to find explicit expres- driven from equilibrium by a transport current. The
sions for ρ and β. In practice the values of these three only relevantdegree of freedomis the spin-density direc-
s
parametersaremodified25 byquantumandthermalfluc- tion,whosedynamicsisdescribedbyEq.(4). We expect
tuations. The fourth parameter ∆ is exponentially sen- thatthepseudospintransfertorquedescriptionofbilayer
t
sitive to the tunnel barrier between layers. Parameters quantumHall systems willbe mostreliable when the or-
valuescanalsobeinfluencedbydisorderonlengthscales der is most firmly established, that is far away from the
3
phase boundary27 that separatesorderedand disordered larization. In terms of these variables the LLS equations
states. take the form
The transport-current(Slonczewski23) term in Eq.( 4)
2 ∆
can be understood in severaldifferent ways. In the spin- m˙ = − ρ m2 ∇~2φ+ t m sinφ
tronics literature this term is normally motivated by an z (cid:26) n~ s ⊥ ~ ⊥ (cid:27)
appeal to total spin conservation and referred to as the − ~v ·∇~)m +α m2 φ˙
spin-transfer torque. The idea is that when a ferromag- (cid:16) ps zo z ⊥
net’s spin-polarized quasiparticles carry a transport cur- 2 2 2 2 2m
φ˙ = m ρ ∇~φ + ∇~m + z ∇~2m
rent through a spatial region with a non-collinear mag- z (cid:26)n~ s(cid:18)(cid:12) (cid:12) m4⊥ (cid:12) z(cid:12) m2⊥ z(cid:19)
netization,theyviolatespin-conservation. Thecollective (cid:12) (cid:12) (cid:12) (cid:12)
4 ∆ (cid:12)1 (cid:12) (cid:12) (cid:12)
magnetizationmust therefore compensate by rotating at − β− t cosφ
n~ ~ m (cid:27)
aconstantratewhichisproportionaltothefermiondrift ⊥
α
velocity. The form we use for the pseudospin transfer − ~v ·∇~)φ − φ m˙ . (6)
torque assumes that each component of the pseudospin (cid:16) ps o m2⊥ z
current is locally equal to the number current times the
corresponding pseudospin direction cosine. This prop- InEq.(6)we havewritten~j/n,whichhasunits ofveloc-
erty does not hold locally in a microscopic theory, but is ity, as the pseudospin velocity~v .
ps
expected to be valid in the smooth pseudospin texture These equations do not on their own provide a closed
limit we address. (In metals an additional phenomeno- description of pseudospin dynamics in the presence of
logical factor is required to account for the difference in electrical bias potentials and need to be supplemented
drift velocity between majority and minority spin elec- byatheorywhichspecifies thespatialdependence ofthe
trons.) pseudospin current. In general this quantity depends21
Since pseudospin is not conserved in bilayer quantum on the order parameter configuration as well as on the
Hall systems, as we can see from Eq.( 2) or Eq.( 3), this contact geometry and external currentor voltage biases.
argumentdoesnotapplydirectly. Ifweappealtoamean- The transport theory and the pseudospin dynamics the-
fielddescriptionofthepseudospinferromagnet,however, ory are therefore not independent. In the present paper
wecanobtainthesameresultbythefollowingargument. we study voltage biased Hall bars with source and drain
Mean-field quasiparticles satisfy a single-particle Hamil- leads at opposite ends, and with a variety of shapes and
tonianforaparticleinamagneticfieldwhichexperiences sizes. We assume, as a simplification, that the current
both a scalar potential, and a pseudospin-dependent po- distributionisdefinedbyalocalconductivitytensorwith
tential that can be interpreted as a pseudospin effec- a large Hall angle. Given these simplifications, we are
tive magnetic field. Following standard textbook deriva- abletoexplicitlyevaluatethemaximumcurrentatwhich
tionsitis possible28 to deriveanexpressionforthe time- time-independent order parameters are possible. Be-
dependence of the contribution of a single quasiparticle cause collective tunneling no longer contributes strongly
to any component of the pseudospin density. The ex- to the dc interlayercurrent when the inter-layerphase is
pression contains a pseudospin precession term and an time-dependent, the interlayer conductance mechanism
additional term which is the divergence of the currentof changes qualitatively when this maximum current is ex-
that component of pseudospin. Summing over all quasi- ceeded. We therefore associate this current with the ex-
particle states we obtain a precessionterm that depends perimental critical current.
on the configuration of the order parameter, and hence
onthe pseudospin-fieldto whichit givesrise,eveninthe
absence of a current. We obtain the additional current-
III. APPROXIMATE CRITICAL CURRENTS
driven term on the right-hand side of Eq.( 4) when the
pseudospin currents are non-zero and space dependent.
In this section we discuss approximate upper bounds
Theadditionaltermintheequationofmotioncanalsobe
on the critical current which are helpful in interpreting
viewed29 as a consequence of an altered relationship be-
the numerical results described in the following section.
tweenpseudospin-dependent effective magnetic field and
We use a simplified version of the static limit of the m˙
pseudospin polarization direction for quasiparticles that z
LLS equation (Eq.( 6)) in which m is assumed to be
carry a current. Eq.( 4) should in principle also include z
small:
a currentrelateddamping term30 whichweignoreinthe
present paper.
ρ 1 ∆ n 1
0=− s ∇~2φ+ t sinφ− ~j·∇~ m . (7)
When quantum Hall bilayer pseudospin ferromagnets ~ 2 ~ 2 z
are tilted far from their easy plane, order tends31 to be
destroyed. Forthatreasonweareoftenmostinterestedin The three terms on the right-hand side can be identi-
the limit in which m is muchless than 1. It is therefore fied as contributions to the time-dependence of m (or
z z
convenient to express the pseudospin direction in terms equivalently of the exciton density) due respectively to
of the azimuthal angle φ, which is the inter-layer phase thedivergenceoftheexcitonsupercurrent,coherentcon-
difference, and m which is proportionalto the layer po- densate tunneling, and the divergence of the zˆ (layer
z
4
versionof quasiparticle counterflow current into conden-
sate counterflow current. The total counterflow current
emergingfromthe areanear the source contactis halfof
the total current flowing into the system since m =±1
z
in the contact and m → 0 far away from the contact.
z
In a tunnel geometry experiment the same counterflow
current is generated near the source and drain contacts
locatedatopposite dies ofthe sample, sothe totalcoun-
terflow supercurrent injected into the system is equal to
the total number current flowing through the system.
With this separation of length scales the quasiparticle
(pseudospintransfer torque)termcanbe droppedin the
remaining portion A˜ of the total system area A. For
FIG. 1: (Color online) Separation of transport length scales staticsolutionsoftheLLSequationsthecondensatemust
inquantumHallsuperfluidtransport. Thistheoryisintended satisfy an elliptic sine-Gordon equation inside A˜:
to apply when the ordered state is well established and Hall
angles are large because of the developing ν = 1 quantum λ2∇~2φ−sinφ=0. (9)
Hall effect. At large Hall angles current enters and leaves
the samples at the hot spot corners, even when the source When λ→∞ (∆ n→0), this equation states that ∇~2φ
t
and drain contacts (gray) fully cover the ends of a Hall bar.
iszero. ItthenfollowsfromGreen’stheoremthatnonet
Whenorderiswell established,thepseudospinorientation of counterflowsupercurrentcanflowintotheareaA˜. Inthe
the transport electrons achieves alignment with the conden-
tunnelgeometrythatmeansthattime-independentorder
satewithinarelativelysmallfractionofthesampleareaclose
parameter values cannot be maintained in the presence
to source and drain (solid yellow). In these areas pseudospin
transfer torques convert transport currents into condensate of a transport current unless ∆tn 6= 0. The maximum
counterflow supercurrents. When source and drain are con- tunneling current that can flow through the system is
nected to opposite layers a net supercurrent is injected into particularly simple to determine in the small ∆ n, large
t
the interior region of the sample. In the remaining sample λ limit. When λ is much larger than the system size the
area(shadedblue)collectiveinterlayertunnelingcanactasa angle φ cannot vary substantially over the system area.
sink for thecounterflow supercurrent. Withthissimplificationtheellipticsine-Gordonequation
can be integrated over the area A˜ to obtain
antisymmetric orcounterflow)fermionic pseudospincur- ρ A˜
rent. The last contribution would be viewed as a spin- ρsZ ∇~φ·nˆ = λs2 sin(φ) (10)
P
transfer torque in the analogous equations for an easy-
plane anisotropy ferromagnetic metal. where P is the perimeter of A˜ and nˆ is proportional to
We start our qualitative discussion of critical currents theoutwardnormal. ThelefthandsideofEq.(10)isthe
by identifying some relevant length scales. First the netsupercurrentwhichflowsoutoftheregionA˜fromits
length scale,
boundariesnearthe sourceanddraincontacts,identified
aboveasthenumbercurrentflowingthroughthesystem.
2ρ
λ= s , (8) Since the maximum value of |sin(φ)| is 1, it follows that
r∆tn themaximumcurrentconsistentwithatime-independent
order parameter in this case is
often referred to as the Josephson length because of the
similarity between these equations and those which de- eA˜ρ eA˜∆ n
scribe Josephson junctions, emerges from balancing the Ic = s = t . (11)
B ~λ2 2~
first and second terms. In this paper we assume that
the pseudospinmagnetizationdirectiondepartsfromthe Since the critical current in the small ∆ n limit is pro-
t
xˆ−yˆplaneonlyoverasmallregionclosetothesourceand portional to the area of the system we will refer to this
draincontactswhosespatialextentis smallcomparedto quantity as the bulk criticalcurrent,as suggestedby the
the Josephson length. (This issue is addressed again in notation used above.
thediscussionsection.) Ifthisiscorrect,wecanseparate For larger ∆ n, λ is no longer larger than the system
t
length scales by identifying a region close to the contact sizeanditisnotpossibletomaintainthemaximumvalue
which is small enough that we can ignore coherent tun- of sin(φ) across the system. The LLS equation critical
nelingbysetting∆ nto0,andlargeenoughthatwecan current in this regime depends on geometric details and
t
assumethatm is closeto zerointhe restofthe system. wehavenotbeenabletoobtainrigorousbounds. Wecan
z
(See Fig. (1).) When ∆ n→0, Eq.( 7) simply expresses make a rough estimate by following an argument along
t
the conservation of the sum of the excitonic and quasi- the following lines. The elliptic sine-Gordon equation is
particle counterflowcurrents. Integrationof Eq.(7) over verysimilartotheregularsine-Gordonequationinwhich
the area close to the contact then simply describes con- secondorderderivativeswithrespecttotimeandposition
5
appear with opposite signs. This 1+1 dimensional sine- where k labels the four neighboring sites, A is the pixel
Gordon equation appears as the Euler-Lagrange equa- area, and α labels components of the magnetization ori-
tion of motion of a system with a Lagrangian with a entation in the pixel of interest. I is the quasiparticle
n
kinetic-energy term proportional to ρ (∂ φ(x,t))2 and a number currentflow fromneighbor site n into this pixel.
s t
potentialenergytermproportional∆ ncos(φ(x,t). Since In this article we estimate values of I by solving the
t n
total energy (integrated over position x) is conserved by resistornetworkmodelobtainedbydiscretizingacontin-
this dynamics it follows that the variation of the typical uummodelwithalocalconductivitythatincludesaHall
value of ρ (∂ φ)2 along the space-time boundary cannot component.
s t
be larger than ∼ ∆ n. When this energy conservation
t
condition is mapped from the regular sine-Gordonequa- {I}=[G]{V} (15)
tion to the (imaginary time) elliptic sine-Gordon equa-
tionwecanconcludethatthetypicalvalueof∇~φ·nˆ along where [G] is the conductance matrix that describes the
the boundary of A˜ near the source contact cannot differ local conductivity including Hall components, {V} is a
vector of local voltages in each pixel, and {I} is a vec-
fromthe typicalvalue of∇~φ·nˆ alongthe boundarynear
tor composed of currents that flow between pixels. Us-
the drain contact by more than ∼ ∆ n. It follows that
t ing current conservation conditions the dimension of the
the current flowing through the system from source to
matrix can be reduced to the pixel number N. Given
drain should not be much larger than
the source and drain contact voltages, we can solve for
eWρ theinternalvoltageandcurrentdistributionsandforthe
IEc ∼ ~λs (12) variationofsourceanddraincurrentsacrossthecontacts.
NotethatquantumHallphysicscomesintoplaythrough
whereW isthelengthofthecontractregion,orthewidth the Hall contributions to the conductivity matrix [G].
of a Hall bar assumed to be contacted at its edges. We ThelocalHallconductivitywassettoσ =e2/h,which
xy
will refer to Ic as the edge critical current, since it is is close to the appropriate value for ν = 1 whether or
E
limited by the length of one edge. For stronger inter- nottheHallplateauisfully formed,andthe longitudinal
layer coupling then critical current is expected to vary conductivity was set to
as (∆ n)1/2 once λ is smaller than the Hall bar length.
t
Finally we note that because ofhot-spoteffects intrans- σ =0.05exp(−m2/W)(m~ ·m~ )(m~ ·m~ )e2/h (16)
xx z L R
portwithlargeHallanglesthepseudospintransfertorque
will act at the sample corners. Since the supercurrent is where m~L,R is the magnetization orientation to the left
converted into coherent pseudospin precession over the and right of a boundary separating two-pixels. The Hall
lengthscaleλ,the effectivesizeofthecontactregionwill angle used in these calculations was therefore tan−1(20)
be ∼ λ when the Hall angle is large and λ is smaller over the largest part of the sample in which the pseu-
than W. We therefore estimate that the critical current dospin magnetization is close to collinear and planar.
is close to The results we report on are not sensitive to the Hall
angle, provided that it is large.
eρ
Ic ∼ s, (13) Thecurrentwhichflowsintoapixelisassumedtohave
C ~
thesamepseudospinpolarizationasthepixelfromwhich
independentof∆ n,underthesecircumstances. Werefer it is incident. Currents entering or exiting from the con-
t
to this last critical current as the corner critical current. tacts are assumed to have mz =1 for top layer contacts
In the following section we compare numerical LLS crit- and mz = −1 for bottom layer contacts. In this way
ical currents with these rough estimates. the pseudospin transfer torque and the Landau-Lifshitz
precessionaltorqueactingoneachpixel’spseudospincan
be evaluated. Note that the pseudospin torque depends
IV. MODEL CALCULATIONS not only on current paths, but also on the pseudospin
magnetization configuration.
The numericalcalculations we describe below are sim-
ilartothosecarriedoutinmicromagneticdescriptionsof
A. Critical Current Identification
ferromagneticmetalspin-transfertorquephysics,butare
appliedhere to pseudospintransfer physicsin condensed
bilayers. We divide the system area into pixels within As mentioned in the previous section, we identify the
which the pseudospin magnetization is assumed to be critical current as the circuit current value above which
constant. Except where noted we used square 10l×10l a time independent solution no longer exists. When the
pixels where l is the magnetic length. We rewrite the external current is small, the Gilbert damping term in
spin-transfer torque term in Eq.( 6) in the discretized the LLS equation relaxes the pseudospin magnetization
form: into time-independent configurations. The behavior of
the pseudospin as the current increases is partly analo-
I
m˙ | = n [m −(m~ ·m~)m ] (14) gous to the behavior of a damped pendulum driven by
α ST Xk nApixel k,α k α an increasingly strong torque.
6
0.06 10
0.05
0.04
1
N
(cid:108)
0.03
m (cid:303) 0.02 / II0 NI ucmerical
0.1 B
Ic
0.01 E
I c
C
( L = 500 l , W = 400 l )
0.9 0.91 0.92 0.93 V 0.94 0.01
c
Voltage((cid:541)V) 1E-8 1E-7 1E-6 1E-5
2 (cid:83) l2 n (cid:39) / E
t 0
FIG. 2: Magnetization change per time step δm×N vs. ap-
plied voltage in a 500l×400l system for 2πl2n∆t =10−5E0. FIG. 3: (Color online) Critical current vs. 2πℓ2n∆t in a
This curve was obtained by changing the bias voltage by 400l×500lsystem. Thedash-dot(red),long-dash(blue)and
δV =2.5×10−5 µVateachtimestep. E0=e2/ǫl,theenergy short-dash(black)curvesplotthevaluesofthebulk(IBc),edge
unit used in all our calculations has a typical experimental (IEc)andcorner(ICc)limitedcriticalcurrentsdiscussedinthe
value∼7meV.δmdevelopslargeamplitudeoscillationwhen textforthissamplegeometry. ThesquaredotsplotLLSequa-
thevoltage exceeds thecritical voltage Vc. tioncriticalcurrentsataseriesof∆tnvalues. Allthecalcula-
tionsin thispaperwere performed usingpseudospin stiffness
(exciton superfluid density) ρs = 0.005E0 where E0 = e2/ǫl
We identify the critical current numerically by slowly is the energy unit. Currents are in units of I0 =IEc =eρs/~.
increasingthe driving voltage(by δV per time step) and For thevalueof ρs used in these calculations I0 ≃8nA.
monitoring the change in magnetization. To be more
explicit, we examine
is longer than the system size. The steady limit of the
LLS equation for the zˆ-component of the pseudospin is
δm≡ |m~ (V +δV)−m~ (V)|/N (17)
i i
Xi 1∆ nA I
t pixel
0= sinφ− (m −m ) (18)
The sum in Eq.( 17) is over all pixels. The critical cur- 2 ~ 2e z,L z,R
rent can be defined precisely as the current above which
whereI isthechargecurrentflowingthroughthesystem
δm remains finite when δV → 0. In practice we choose
andm andm arethezcomponentofpseudospinfor
z,L z,R
a suitably small value of δV and examine the current or
thesourceanddrainleadsattheleftandrightendsofthe
voltage dependence of δm. We find that δm increases
sample. Forthe draggeometry(currententeringandex-
dramatically and begins oscillating at a voltagewe iden-
iting from the same layer)m =m , there is no spin
z,L z,R
tify as the critical voltage. (See Fig.( 2).) An alter-
torqueterminthesingle-pixelcalculation,andthesteady
native but more laborious method of obtaining critical
state equation can be satisfied by setting sinφ = 0. For
currents is to sequentially examine the dynamics of the
the tunneling geometry m = −m , the maximum
z,L z,R
pseudospin magnetization at a series of fixed values of
pseudospin torque that can be compensated by the tun-
the applied voltage V. If the applied voltage is below
neling term is obtained by setting sinφ → 1. We there-
its critical value, δm will approachzero exponentially at
fore obtain
large times. If the applied voltage is above its critical
e
value δm will not approach zero and usually exhibits an Ic =Ic = ∆ nA (19)
oscillatorytime dependence. In our calculationswe used B 2~ t pixel
the first approach to determine an approximate value of
ThisgivesalineardependenceofI onthesingle-particle
c
the critical voltage (and hence the critical current) and
tunneling strength ∆ . The numerical procedures de-
t
the second method to refine accuracy.
scribed above accurately reproduce this simple result.
Asexplainedpreviouslyanddiscussedmorefullylater,
we believe that the pseudospin transfer torque in most
B. Ic vs. ∆t quantumHallsuperfluidexperimentsactsinasmallfrac-
tion of the system area. We therefore need to perform
It is useful to start by briefly discussing the single- calculationswithmanypixelsinordertorepresentatyp-
pixel limit of the calculation, which should apply ap- ical measurement. Fig. (3) shows numerical critical cur-
proximately to the case in which the Josephson length rent results for a fixed sample geometry as a function of
7

l

0
4

/
y
x / 40 l
FIG. 4: Spatial distribution of the zˆcomponent of the pseu-
dospin transfer torque in a system with 500×400 l2 area, FIG. 5: Supercurrent distribution in system with area 500×
2πl2n∆t =10−6E0, and I/Ic =0.29. The pseudospin trans- 400l2,2πl2n∆t =10−6E0,andI/Ic=0.29. Thisplotisfora
fer torque in the model studied here acts mainly in the hot tunnelinggeometryinwhichthesourceisatoplayercontact
spot pixels. E0 and l are definedas in previous figures. and the drain is a bottom layer contact. Supercurrents are
generatednearbothhotspotsandflowdiagonallytowardthe
sample center. E0 and l are definedas in previous figures.
∆ . In this figure I = Ic = eρ /~ ∼ 8nA is the unit
t 0 C s
of current and E = e2/ǫl is the unit of energy. Typi-
0
cal quantum Hall superfluid experiments are performed
at a magnetic field of roughly 2.1 Tesla for which the
magnetic length is 17.65 nm which gives E = 6.4 meV.
0
In all the calculations reported on here we used a pixel
area Apixel = 10×10l2 and the the mean-field theory l
estimate7 (ρ ≃ 0.005E ) for the pseudospin stiffness. 0
s 0 4
ThecalculationsinFig.(3)arefor40pixelsinthe width
/
W directionand50pixelsinthecurrentLdirection. For y
this system size , the Josephson length is approximately
equal to L when 2πl2∆ n∼4×10−7E . The numerical
t 0
results illustrated in Fig. (3) show the crossovers from
∆ -dependence,to∆1/2-dependence,tosaturationas∆
t t t
increases that was anticipated in our qualitative discus-
sion. At small ∆tn the critical current is reduced by a x / 40 l
small fraction compared to the single pixel result in ac-
cord with the Fig.(1).
FIG. 6: Supercurrent distribution in system with area 500×
We now examine the ingredients which enter these 400l2,2πl2n∆t =10−6E0. Thisplotisforthedraggeometry
numerical results in greater detail. According to the caseinwhichthesourceanddrainarebothtoplayercontacts,
schematic Fig. (1), the pseudospin transfer torque acts butotherparametersofthecalculation areidenticaltothose
only near the hot spots at which current enters and ex- used in the preceding tunnel-geometry figure. E0 and l are
its the sample. Fig. (4) shows a typical numerical re- definedas in previous figures.
sults for the spatialdistribution ofthe pseudospintrans-
fer torque. In the present model, the area of the region
inwhichtransportcurrentisconvertedintosupercurrent equation applies locally when the pseudospin transfer
dependsonthepixelsizeandthepseudospinstiffnessand torque is negligible, it follows from Green’s theorem and
anisotropycoefficients. In the tunneling geometry,coun- Eq.( 7) that in a steady state the total counterflow cur-
terflow supercurrent is generated near both source and rentinjectedintotheinteriorofthesample(whichforthe
drain hot spots and flows diagonally toward the center tunnel geometry equals the total charge current flowing
of the sample. In Fig. (5) we show a typical supercur- through the system) must match the area integration of
rent distribution for the tunneling geometry case. The (1/2)(∆tn/~)sinφ. For a drag geometry experiment the
corresponding distribution for an equivalent drag exper- same integral should vanish.
iment (both contacts connected to the same layer) is We now examine steady state condensate configura-
illustrated in Fig. (6). Since the elliptic sine-Gordon tions at currents near the critical current for both small
8
flow pattern spreads and because coherent tunneling is
providing the required current sink. The critical current
(cid:186)( rad) in the large n∆t limit, depends in a complex way on the
geometryofthesampleandonthespatialdistributionof
the pseudospin transfer torques. Nevertheless, the crit-
ical current saturation we find in our numerical studies
suggests that once the Josephson length is smaller com-
pared to both the width and the length of a Hall bar
samplewithalargeHallangle,itis nolongerrelevantto
the critical current value.
C. Ic vs. System Geometry
y/ l x/ l
Finally, we briefly discuss critical current dependence
on Hall bar dimensions at fixed n∆ . In Fig.( 9) we plot
t
Ic vs. Hallbarlengthinaseriesofmodelsampleswitha
FIG. 7: Pseudospin phase distribution in a system with area
fixedsinglepixelwidthW =10l andsingle-particletun-
500×400l2,2πl2n∆t =10−8E0andI/Ic =0.75. TheJoseph- neling amplitude 2πl2n∆ = 10−6E . For these param-
son length at this value of ∆t is ∼ 2500l. Note that φ is t 0
eters the W is much smaller than the Josephson length.
roughly constant through the system and that its value is
Thecriticalcurrentincreaseslinearlywithsamplelength
close toπ/2 because I is close toIc. Theunitsused here are
thesame as in previousfigures. and is therefore proportional to sample area until it sat-
urates at L ∼ 1000l. The length at which the current
saturates is a bit longer than the Josephson length and
the value of the critical current is accordingly somewhat
larger than the 1D estimate21 Ic. The large L behavior
E
(cid:186)( rad) is however consistent with the expectation that the crit-
ical current should not increase with system length once
L is substantiallylargerthan the Josephsonlengthλ. In
Fig. (10) we plot I vs. system width W with the length
c
fixedat500landthesame∆ asabove. TherangeofW
t
covered is limited somewhat by numerical practicalities
and goes from a width much smaller than the Josephson
length to a width which is somewhat larger. Over this
rangedeviations fromthe 1Dmodelin whichthe critical
current is proportional to Hall bar width are small.
y/ l x/ l
V. DISCUSSION AND CONCLUSIONS
Westartourdiscussionbycommentingbrieflyonsome
FIG. 8: Phase distribution of pseudospin in a system with
area500×400l2,2πl2n∆t =1.2×10−5E0 andI/Ic =0.97 . essential differences between the critical current of a
The Josephson length at thisvalueof ∆t is ∼70l. Note that Josephson junction and critical currents for coherent bi-
a steady state is reached even though φ varies considerably layer tunneling. In a Josephson junction, current can
across the sample and has values larger than π/2. The units flow without dissipation across a thin insulating layer
used here are thesame as in previousfigures. that separates two superconductors. The difference in
condensatephaseacrossthejunctionφ isnormallyzero
J
in equilibrium but can be driven to a non-zero steady
n∆tandlargen∆tlimits. Intheformercaseourexpecta- state value when biased by current flow IJ in the cir-
tionthatsinφshouldbe nearlyconstantexceptnearthe cuit in which the junction is placed. For thick insulating
hot spots is confirmed in Fig. (7). Collective tunneling layers the current is related to the phase difference by
actssinkstheinjectedcounterflowsupercurrentatarate
thatisnearlyconstantacrossthesamplearea. Thelarge I =Ic sin(φ ), (20)
J J J
n∆ caseismorecomplex. φvariesoveralargerangeand
t
sin(φ)changessignindifferentpartsofthesample. Near whereIc isthejunction’scriticalcurrent. Eq.(20)should
J
thesamplecornersφvariesrapidlywithpositionbecause be compared with Eq.( 7). The most obvious difference
of the large counterflow supercurrents. In the interior of is the appearance of the lateral 2D coordinate in the co-
the sample the phase changes less rapidly because the herent bilayer case. In the Josephson junction case, the
9
of the paper and
ρ m
1 IBL =e Z ~s∇~φ+ 2z ~jmz ·nˆ. (22)
P (cid:2) (cid:3)
is the injected counterflow current. The most essential
0.1 difference between tunneling in coherent quantum Hall
bilayers and Josephson junctions lies in the difference in
/ II0 Numerical pInhytshicealcacsoentoefntabJeotsweepehnsotnhejubniactsiocnurirtenitssaIJbualnkddIiBssLi-.
0.01 I c
B pationlesssupercurrentwhichflowsperpendicular to the
Ic plane of the junction. In the case of a coherent quan-
E
I c tum Hall bilayer, it is the counterflow (zˆ) component of
C
1E-3 ( (cid:79)=250.7 l , W = 10 l ) the quasiparticle current. The counterflow component
of the quasiparticle current is normally fully converted
10 100 1000 10000 to condensate counterflow supercurrents by pseudospin
L / l transfer torques, as we have discussed at length. The
voltage drop across a Josephson junction can be mea-
sured and vanishes below the critical current. Because
FIG. 9: Critical current vs. system length L for a narrow
fermionic quasiparticle transport,transversecounterflow
Hall bar with width W =10 l. The single-particle tunneling
amplitude 2πl2n∆t =10−6(E0) corresponding to λ∼250l supercurrents,andcollectiveinterlayertunneling areun-
avoidably intertwined in the coherent bilayer tunneling
case,thereisnocorrespondingmeasurablevoltagewhich
vanishes below the critical current. Instead, the critical
current is marked experimentally by an abrupt increase
in measured resistances.
Nextwedrawattentiontothesignificanceofthe qual-
1 itative difference in experiment between drag geometry
and tunnel geometry transport measurements. In our
theoretical picture is correct there should be no qualita-
I0 tive difference difference between voltage measurements
/ I Numerical inthesetwogeometriesatcurrentsbelowthecriticalcur-
I c
B rent, once coherence is well established. (Important dif-
0.1
Ic ferencesbetweenthesemeasuringgeometriescanoccurin
E
I c theinterestingregimeclosetothephaseboundarywhere
( (cid:79)=250C.7 l , L = 500 l ) interlayercoherencemaybeestablishedoverasmallfrac-
tionofthesamplearea.34)Pseudospintransfertorquesin
10 100 tunnelanddraggeometryexperimentswithsimilartotal
W / l circuitcurrentsgiveriseto similarcounterflowsupercur-
rents, as seen in Figs. (5) and (6). Excitonic conden-
sates have a maximum local (critical) supercurrent den-
FIG.10: Criticalcurrentv.s. systemwidthW forlengthL=
sitythattheycansupport,whichinthecaseofquantum
1500−06l()E.0)T,hceorsriensgploe-npdairntgictleo λtu∼nn2e5li0nlg amplitude 2πl2n∆t = Hbeal∼lb1ilAayme−r1e.xcFiotornactoynpdiceanlsastaemsphlaeswbiedetnheosfti1m0a−t4emd35thtios
corresponds to a critical current in the mA regime, or-
dersofmagnitudelargerthanthecurrentlevelstypically
lateral dependence of φ usually plays no role unless an
J
employed in quantum Hall experiments. Even account-
externalmagneticfieldispresent. Inthecoherentbilayer
ing for possible corrections due to disorder, local critical
case, on the other hand, lateral translational invariance
currents are unlikely to be approached experimentally
isalwaysbrokenbecausethepseudospintransfertorques
for either contact geometry. (If they were, we would ex-
that ultimately drive the coherent tunneling current do
pect more similarity between the two experiments.) The
not act uniformly across the system.
critical currents which are important for typical tunnel-
A closer comparison is possible in the special case in
ingexperimentsarenotlocalcriticalcurrents,butglobal
which the pseudospin stiffness is large enough to inhibit
critical currents which set an upper limit on the rate
lateralvariationofφ. IntegrationofEq.(7)overthearea
at which injected supercurrent can be sunk by collective
then yields for the coherent bilayer
tunneling. Since there is no net injected supercurrent
I =Ic sin(φ) (21) inthedraggeometryexperiment,onlylocallimitsapply.
BL B
Inthetunnel-geometryexperiment,thegloballimitmust
whereIc isthebulkcriticalcurrentdiscussedinthebody be satisfied, setting a critical current scale which can be
B
10
ordersof magnitude smaller if ∆ is small. Although the 0.1 at the lowest measurement temperatures. At these
t
relationship anticipated here between tunneling geome- low temperatures thermal fluctuations alone appear to
tryanddraggeometrytransporthasnotbeenspecifically be insufficient36 to explain the discrepancy even though
tested experimentally, it appears to us that it is clearly ∆ /k T is certainly small. The experimental finding
t B
consistent with published data. that the critical current saturates at low temperatures
Wenowturntoacomparisonofourtheorywithexper- supportsthis conclusion. Similarly, theoreticalestimates
iment, and in particular with the recent experiments of thatquantumfluctuationcorrectionstotheorderparam-
Tiemannet al.32 whohavesystematicallystudiedthede- eterarenot25 largewellawayfromthetransitionbound-
pendence of the tunneling critical currents in their sam- ary are consistent with the experimental finding of satu-
ples on temperature and layer separation. The data of ratingcriticalcurrentsinthis regime. Itappearsthatan
Tiemann et al. appear to be broadly consistent with explanationfor the smallcriticalcurrentsmust be found
that reported in earlier8,16 but focus more on tunneling in the disorder physics of quantum Hall superfluids.
anomaliesintheregionofthephasediagramfarfromthe
Theanalysisoftunnelingcriticalcurrentspresentedin
coherent state phase boundary. Tiemann et al. find i)
this paper can accommodate disorder implicitly through
that the critical tunneling currents in their samples are
its influence on the parameters ρ and ∆ n. Including
s t
proportional to system area, ii) that their typical value
disordereffects through renormalizedcoupling constants
is ∼ 10nAmm−2, and that they saturate upon moving
would be adequate if the characteristic length scales for
away from the coherent state phase boundary27 either
disorder physics are smaller than characteristic length
by lowering temperature or by decreasing the ratio of
scales like λ which are relevant for pseudospin transfer
layer separation to magnetic length.
torques. InquantumHallsuperfluidsdisordermayplaya
Tiemannetal.’sfindingthatthecriticalcurrentispro- more essentialrole by nucleating chargedmerons37 (vor-
portionaltoareawouldbeconsistentwithouranalysisif
tices). As we have explained, provided that the pseu-
the Josephson length was comparable to or longer than
dospin transfer torque acts only close to the source and
the ∼ 10mm length of their long-thin Hall bars. When
drain, the critical current is proportional to the integral
in the bulk-limited critical current regime, the critical
of ∆ n over the area of the sample. This integral is re-
t
current should be given by Ic
B duced to zero by a single undistorted meron located at
thecenterofalargesample. Ithasinfactoften12–14,40–42
Ic e∆˜
B = t =3.1B[Tesla]×∆˜ [eV]×104Amm−2 (23) beenrecognizedthatdisorderinducedvorticesmightplay
A 4πl2~ t
anessentialroleinmanyofthetransportanomaliesasso-
where ∆˜ = 2πl2∆ n is the interlayer tunneling ampli- ciated with bilayer coherence. It seems likely to us that
t t
thistypeofphysicsisverylikelyresponsibleforthesmall
tudesuitablyrenormalizedbyquantumandthermalfluc-
criticalcurrentsseeninexperiment,butthatexistingthe-
tuations and B is the magnetic field strength at ν = 1.
oryisunabletoaccountforthiseffectquantitatively. The
Inserting the magnetic field strength, experimental criti-
cal currents can be recovered by setting ∆˜ → 10−13eV. current status of the subject calls for a detailed analysis
t
Thereinliestherub. Althoughvaluesfor∆˜ onthisscale of how they influence critical currents. On the experi-
t
mental side, the importance of complex disorder-related
do imply Josephson lengths that are on the mm length
pseudospin textures for critical current values could be
scale and therefore consistent with bulk-limited critical
currents in the samples studied by Tiemann et al., they reduced and the essential physics which limits critical
currents revealed more clearly by studying samples with
arefourormoreordersofmagnitudesmallerthanvalues
much larger bare values of ∆ .
expected on the basis oftheoretical estimates. Suspicion t
that there is a fundamental discrepancy is established Although we have attributed the substantial quanti-
moreconvincingly,perhaps,byobservingthatthesecrit- tative disagreement between pseudospin transfer torque
ical currents also correspond to ∆˜ values several orders theory and experiment to disorder-induced pseudospin
t
ofmagnitude smallerthanseeminglyreliableexperimen- texturesandhavesuggestedastrategyforachievingmore
talestimates33basedontheinter-layertunnelingconduc- quantitativetests ofourtheory,itis appropriatetostep-
tanceinsimilarsamplesatzero-field. Thekeyissuethen backandreconsiderothertheoreticalpicturesthatmight
in comparing critical current theories with experiment be relevant to coherent bilayer tunneling experiments
appears to be explaining why they are so small. under some circumstances. For example, the version
We expect on general grounds that the experimental of the pseudospin-transfer torque theory applied here is
value of ∆ n ≡ ∆˜ /2πl2 should be renormalized down- based on the simplest possible assumption for the local
t t
ward by both thermal and quantum fluctuations and pseudospin-polarizationofthetransportcurrent,namely
by disorder. Indeed according to the familiar Mermin- that the pseudospin current polarization simply follows
Wagner theorem, the order parameter n must vanish for the pseudospin density polarization. This assumption
∆ →0 at finite temperatures. The importance of ther- is certainly not generically correct, but its replacement
t
mal fluctuations is strongly influenced by k T/ρ . If requires more detailed knowledge of fermionic quasipar-
B s
mean-field theory estimates7 of ρ can be trusted, the ticletransportbehavior. Oneapproachistoassumethat
s
value of ρ in typical experimental samples should be the quantum Hall effect establishes edge state transport
s
∼3×10−5eV and k T/ρ should therefore be less than and use experimental voltage probe data to infer38 the
B s