The longitudinal resistance of a quantum Hall system with a density gradient

0.2 MB·English

by  R. Ilan

The longitudinal resistance of a quantum Hall system with a density gradient
R. Ilan,1 N. R. Cooper,2 and Ady Stern1
1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
2T.C.M. Group, Department of Physics, Cavendish Laboratory,
J.J. Thomson Avenue, Cambridge, CB3 0HE, U.K.
6
(Dated: February 6, 2008)
0
0 Following recent experiments, we consider current flow in two dimensional electronic systems in
2 thequantumHallregime whereagradient in theelectron densityinducesaspatial variation in the
n Hall resistivity. Describing the system in terms of a spatially varying local resistivity tensor, we
a show that in such a system the current density is generically non-uniform, with the current being
J pushed towards one side of the sample. We show that, for sufficiently large density gradient, the
6 voltagealongthatsideisdeterminedbythechangeoftheHallresistivity,andisindependentofthe
2 microscopic longitudinal resistivity, while the voltage on the opposite side is small and determined
by both longitudinal and Hall resistivities. We solve some particular models in detail, and propose

] waysby which themicroscopic longitudinal resistivity may beextracted from measurements of the
ll longitudinal voltage on both sides of thesample.
a
h
- I. INTRODUCTION rent density that leads to a difference in the longitudinal
s
e voltagedroponthetwosidesofaHallbarsample. Wede-
m Recentstrikingexperimentalobservationsoftransport scribethe conditionsunder whichthe boundarybetween
. in high mobility two-dimensional electron gases in the the quantum Hall states is “sharp” and the edge-state
t
a quantumHallregimehavebeensuccessfullyexplainedas formalism used in Ref. 1 applies; we also determine cor-
m resulting from density gradients of as small as 1%/cm.1 rectionsto this behavior. We discuss in detailthe effects
- These gradients presumably arise from inhomogeneities of a density gradient on a compressible state. In this
d in the MBE growth.1 Specifically, the longitudinal volt- case we show that, even in the presence of an unknown
n
o age drop differs on the two sides of the sample: on one density gradient, the intrinsic local resistivity ρxx can in
c side the voltage drop is close to zero; on the other the principle be extracted from resistance measurements in
[ voltagedropyields a longitudinalresistance equalto the a Hall bar geometry.

difference in the Hall resistivities corresponding to the There have been many previous studies of quantum
1
v densities at the two probes. Thus, the longitudinal re- Hall systems using this approach.3,4,5,6,7,8 These have
4 sistance of such samples does not measure an intrinsic shown that, for macroscopically homogeneous samples
1 longitudinal resistivity, ρ , but rather probes the (spa- (when the correlation length of the density inhomo-
xx
6 tially varying) Hall resistivity ρ . geneitiesissmallcomparedto thesample size),the mea-
xy
1
The explanation of these observations given in Ref. 1 sured longitudinal resistance can, depending on the ex-
0
was based on the edge-state formalism for transport in perimentalcircumstances, be determined by the fluctua-
6
quantum Hall systems. This formalism is capable of de- tionsinρ independentoftheintrinsicρ :4,5,6 notably,
0 xy xx
/ scribing inhomogeneous systems consisting of regions of the height of the peak in the longitudinal resistivity be-
t
a quantized Hall conductance that are separated by sharp tween two quantised Hall states can be equal to the dif-
m boundaries. However, there are many questions that it ferenceintheHallresistivitiesofthosestates.6 However,
cannot address, such as how “sharp” must the bound- in all macroscopically homogeneous systems, the same
-
d ary between the quantized Hall regions be for the edge- longitudinal voltage drop would appear on the two sides
n state formalism to apply? And how would the transport of a Hall bar sample. While a density gradient was in-
o
properties of a compressible system, such as the ν =1/2 voked in Ref. 8 to account for transport measurements
c
composite Fermi liquid,2 be affected by a density gradi- of certain Corbino disk samples,9 the effects of a density
:
v ent? In this case the sample is at no point in a quantum gradienthavenotbeenadequatelyexplored. Otherworks
i
X Hall state so cannot be described within the edge-state considered related effects due to inhomogeneities in the
formalism. magneticfieldandvariationsinthesamplethickness.10,11
r
a In this work we employ a more generaltheoretical ap- The paper is organizedas follows. After formulating the
proach: wemodeltheexperimentalsystemasaninhomo- mathematical problem in II, in III we provide general
geneous classical conductor,3,4 with a spatially varying arguments for the transpo§rt pro§perties of a system in
localresistivitytensorsetbytheinhomogeneouselectron which voltage probes lie in two different quantized Hall
density,with a net density gradient. Usingthis modelwe states. We show how the results of Ref. 1 emerge within
areabletoanswerthequestionsposedaboveandanalyze our theoretical approach. Our conclusions here rely on
the relation between the observed transport phenomena an assumption that current remains close to one side of
and the properties of the quantized Hall effect. As we the sample. Justification for this assumption is provided
showbelow,under quite general circumstances, the pres- by IV, V and VI which contain explicit solutions for
§ § §
ence of a density gradient results in a non-uniform cur- particular geometries: in IV and V we study a com-
§ §
2
pressible system with a uniform density gradient; in VI
§ 4 3
we calculate the effects of a uniform dissipative region
between the two quantized Hall regions. Finally, in VII
§
we provide discussion and conclusions.
I A B I
II. FORMULATION OF THE PROBLEM
y
We consider transport in a two-dimensional sample in
1 2
the x y plane. The local current density j(r) is related
to the−local electric field E(r)13 via x
E = ρ j ρ ˆz j (1)
xx xy
− × FIG. 1: Geometry of the sample considered. A total cur-
rentI flowsfrom left toright alongthelengthof thesample.
where the components of the resistivity tensor, ρ , are
ij The four voltage probes lie on thesides of thesample within
also assumed to be functions of position. These varia-
the two quantised Hall regions (A and B) at the two ends.
tionsarisefrominhomogeneitiesintheunderlyingcarrier
Thetransportpropertiesoftheinterveningshadedregionare
density in the quantum Hall samples. unimportant for the discussion of § III. The contour of in-
In a steady state, the current density and the electric tegration used for calculating the dissipation integral in the
field must satisfy Appendixis shown.
j=0 (2)
∇·
E=0 (3)
∇× any direction, or indeed could be nonuniform – we re-
as well as the appropriate boundary conditions. quire only that contacts 1 and 4 are in region A, and 2
Wecanensure(3)bydefiningascalarpotentialφ,such and 3 in B.
that Akeyresultofourworkistoshowthatwhenacurrent
I is passed along a sample of this type:
E= φ. (4) (i) the longitudinal voltage drop is larger on one side of
−∇
the samplethanthe other,the side being selectedby the
Similarly, we can ensure (2), by defining a stream func- sign of ∆ρ ρA ρB ;
xy ≡ xy− xy
tion ψ via (ii) under physically realisticconditions for the interven-
ing region, the smaller of these two voltages can be ex-
j=ˆz ψ. (5) ponentially small (in a manner to be discussed in detail
×∇
below);
Then, substituting (1) into (3), the equation for ψ is
(iii)incaseswhere(ii)applies,thenonzerovoltageyields,
to exponential accuracy, a longitudinal resistance
ρ 2ψ+ ρ ψ ˆz ψ ρ =0. (6)
xx xx xy
∇ ∇ ·∇ − ·∇ ×∇
R= ∆ρ (7)
We shall solve equation (6) using various boundary | xy|
conditions, and functional dependences of the local re-
That is, the apparent longitudinal resistance is deter-
sistivity tensor on position.
mined by the difference in the Hall resistivities at the
locations of the two contacts.
These effects are identical to those observed experi-
III. GENERAL ARGUMENTS
mentally in Ref. 1, and derived there within the edge
state transport formalism. However, we emphasize that
We begin our discussion by considering the situation our results hold in a much more general setting. For
in which the change in density along the sample is suf- example, we shall show that the two quantized Hall flu-
ficient that the two ends of the sample are in different ids may be joined by a finite region of a compressible
quantized Hall states, A and B. We assume that these quantum Hall fluid where the resistivity tensor is not
regionsexhibit goodquantized Halleffects, i.e. that dis- quantized. Furthermore, we shall be able to account for
sipation is negligible ρxx/ρxy 1. These regions are departures from the limiting behavior of results (ii) and
| | ≪
thereforecharacterizedjustby theirHallresistivitiesρAxy (iii).
and ρB . As illustrated in Fig. 1, we imagine that the First, we prove result (i) by adapting an approach of
xy
four voltage probes attached to the sample lie in these Ref. 7 to the present geometry. As described in the Ap-
regionsofquantizedHallresistivity. Fornow,the details pendix,by makinguse ofthe factthatthe regionsAand
oftheresistivityinthe interveningregion(shownshaded B havenegligible dissipationone finds that, independent
in Fig. 1) are unimportant. Moreover,the precise geom- of the spatial distribution of the resistivity between these
etry is not important: the density gradient could be in two regions, the net power dissipated within the sample
3
is A. Current distribution
Q= 21 ρAxy−ρBxy −1 V423−V122 (8) Under the conditions stated above, Eq. (6) reduces to
(cid:0) (cid:1) (cid:0) (cid:1)
where
∂ψ∂ρ ∂ψ∂ρ
ρ 2ψ xy + xy =0. (13)
x2 xx∇ − ∂x ∂y ∂y ∂x
V E dl=φ(x ) φ(x ) (9)
12 1 2
≡ · −
Zx1 The constant parameters ρxx and ρxy introduce two
x3 new lengthscales ∇
V E dl=φ(x ) φ(x ) (10)
43 4 3
≡ Zx4 · − ∂ρ −1
xy
ℓ ρ (14)
x xx
are the longitudinal voltage drops on the two sides of ≡ ∂x
(cid:18) (cid:19)
the sample. Since the dissipatedpowermustbe positive, −1
∂ρ
Q>0, Eq.(8) shows that V > V if ρA >ρB (and ℓ ρ xy (15)
| 43| | 12| xy xy y ≡ xx ∂y
viceversa). Thisprovesresult(i); italsoidentifieswhich (cid:18) (cid:19)
ofthevoltagedropsisthelargerforagivensignof∆ρxy. which characterise the spatial variation of the resistivity
The demonstration of result (ii) will be discussed in tensor: ℓ (ℓ ) is the distance over which ρ changes
x y xy
| | | |
the following sections. For now, we note that result (iii) by ρ in the x (y) direction.
xx
is a direct consequence of (ii) in the geometry described. Thereisasimple solutiontoEq.(13)thatis consistent
Let us assume that the necessary conditions have been with the boundary conditions that j =0 at y =0,w on
y
met that (ii) holds. For simplicity, we imagine that the the top and bottom sides of the Hall bar. It is sufficient
sample has been oriented such that ρA > ρB in which to set ψ(r) = ψ(y), such that j = ∂ψ = 0 everywhere.
xy xy y ∂x
case V43 > V12 . Then it is V12 that is exponentially Then Eq.(13) becomes
| | | |
smallandwhichwenowsettozero. Wecannowevaluate
∂2ψ 1 ∂ρ ∂ψ
the potential drop V43 by integrating the electric field = xy ,
from 4 to 1 to 2 to 3. Using the facts that the paths ∂y2 −ρxx ∂x ∂y
4 1 and 2 3 are in quantized Hall states, and that i.e.
→ →
V =0, one finds
12 ∂j 1 ∂ρ
x xy
= j .
V43 = I ρAxy−ρBxy (11) ∂y −ρxx ∂x x
V The solution is
R 43 = ρA(cid:0) ρB =(cid:1)∆ρ (12)
43 ≡ I xy− xy xy I e−y/ℓx
j = , j =0 (16)
This is result (iii), whichwe now see as a consequenceof x ℓx(1 e−w/ℓx) y
−
(ii). w
where I j dy is the total currentflowing along the
In this section we have asserted that the smaller volt- ≡ 0 x
sample and ℓ is the characteristiclengthscale (14). The
age drop can be exponentially small, as in result (ii). In R x
solution describes a current that is concentrated within
the following sections we describe the current distribu-
adistanceoforder ℓ ofone edgeofthe sample. Which
tions in particular (physically relevant) situations where | x|
edge this is depends on the sign of ℓ (i.e. the sign of
exact results can be obtained, and which do show that x
∂ρxy since ρ is positive). For ℓ > 0, the current is
the smaller voltage drop is, indeed, exponentially small. ∂x xx x
concentrated close to y = 0; for ℓ < 0, the current is
x
concentrated close to y =w.
IV. TRANSPORT IN A HALL BAR WITH A From this solution, it is straightforward to determine
UNIFORM GRADIENT IN THE HALL theresistancesthatwouldbeobtainedinafour-terminal
RESISTIVITY measurement on the Hall bar. We assume that the volt-
ageprobes area distance Lapart,so that, depending on
We first consider the situation of a long Hall bar the side of the sample on which these probes are placed,
aligned with the x-axis, and with edges at y =0,w. We the two longitudinal resistances are
shall discuss the current distribution in a region that is E (y =0)L L 1
x
far from the current contacts. We consider the situation R12 = I = ℓxρxx1 e−w/ℓx (17)
in which ρ is independent of position, and ρ has a −
xx xy E (y =w)L
constant gradient, ∇ρxy (which can be in an arbitrary R43 = x I =R12(0)e−w/ℓx (18)
direction). As is discussed in more detail in VII, this
§
model can be valid in a sample at a density close to a (WelabelthevoltageprobesinthesamewayasinFig.1.
◦
compressible state and with a Hall angle close to 90 Note, however, that here we do not consider an inhomo-
(ρ ρ ). For example, these conditions can be met geneous system in which there are two regions of quan-
xx xy
≪| |
inahigh-qualitysampleatadensityclosetotheν =1/2 tized Hall states: the gradient in Hall resistivity is as-
Composite Fermion liquid state.2 sumed to be uniform and constant over the whole bar.)
4
In the limit w ℓ , the current is spread uniformly
≪ | x| 4 3
over the width of the sample and we recover the results I I
for a homogeneoussystem: the longitudinal resistance is
the same on each side of the sample and set by the bulk
resistivity ρ and a geometrical factor
xx
w
L
R =R =ρ .
12 43 xx
w
In the limit w ℓ , the current is strongly confined
≫ | x| L
to one side of the sample. Assuming that ℓ < 0 so the
x 1 2
current is close to the y =w side, we find
R = 0
12 FIG.2: Illustration of thesystem describedin §V. Contacts
∂ρxy are numbered1−4, such that contact 1 is at theorigin.
R = L = ∆ρ
43 xy
∂x | |
(cid:12) (cid:12)
(cid:12) (cid:12)
(For ℓ > 0 the current (cid:12)is pus(cid:12)hed to the other side, so
x (cid:12) (cid:12)
R =0 and R = ∆ρ .) For w ℓ , on the side of Moreover,simple manipulations show that
43 12 xy x
| | ≫| |
the sample where the current is concentrated, the longi-
w R R
tudinal resistance drop between the two voltage probes ρ = 12− 43 . (20)
xx
is set by the difference ∆ρxy of the Hall resistivities at Lln[R12/R43]
| |
these two points; on the other side the longitudinal po-
Forsystemsinwhich ℓ issmallcomparedtothesample
tential vanishes. This result is identical to that for the | x|
width w, the accuracy with which ρ can be extracted
caseinwhichthecontactslieintwoquantumHallfluids, xx
becomes limited by the accuracy with which the smaller
asdiscussedin III. Weemphasize,however,thatthere-
§ resistance can be measured. In this case, in order to
sult describedhere applies to the verydifferent situation
extractρ reliably wouldrequire measurements onnar-
of a compressible system in which ρ varies smoothly in xx
xy rower samples, in which w/ℓ is of order one so that
space. | x|
R /R is not too small or too large.
Nevertheless, this solution provides an explicit exam- 12 43
The results of this section rely on the geometry of a
pleofasituationinwhichthevoltagedropononesideof
long Hall bar, such that in the vicinity of the voltage
the sample is exponentially small. If the region between
probesthecurrentdistributionisindependentofposition
the quantized Hall phases of Fig. 1 were to consist of a
alongthebar. Wenowturntodiscusstheresistancethat
regionoflinearlyvaryingHallresistivity,asdescribedby
would be measured in rectangular geometries, with the
the model studied in this section, the current would be
current contacts at the corners.
tightlyconfinedtoonesideofthesamplewithinalength-
scaleℓ . Thevoltagedropontheothersideofthesample
x
wouldbe exponentiallysmall,andthe argumentsof III
V. TRANSPORT IN RECTANGULAR
§
can be applied. In this case, we see that the parame- SAMPLES WITH A UNIFORM GRADIENT IN
ter controlling the exponential suppression is the ratio, THE HALL RESISTIVITY
w/ℓ , of the width of the Hall bar w to the lengthscale
x
| |
ℓ .
| x| Wenowconsideragainasysteminwhichthereisacon-
stantgradienttoρ ,andρ isindependentofposition.
xy xx
However, now we study a sample which is a rectangle of
B. Extracting ρxx
length L and width w, and in which current is injected
andextractedatthecorners,asdescribedinFig.2. This
It is interesting to note that even in the presence of
is representative of the geometry of the samples used in
an unknown gradient ρ , the bulk resistivity ρ can,
xy xx Ref. 1. The important new aspect, as compared to the
∇
in principle, be extracted from the two measured resis-
case of the long Hall bar, is that the current is injected
tances R ,R in this Hall bar geometry, using the re-
12 43 into the sample in a nonuniform profile and we describe
sults (17,18). (It is important that the conditions for
the current distribution in the whole sample, including
validity of these results apply. These conditions are dis-
the bending ofthe currentflowinthe vicinity ofthe cur-
cussed further in VII.) Our solution shows that the
rent contacts.
§
componentofthe gradientofthe Hallresistivityperpen-
We thereforeaimto solveEq. (13)again,withbound-
diculartotheHallbar,∂ρ /∂y,isirrelevantforthelong
xy ary conditions that correspond to two current contacts
Hall bar (but see V for finite samples). The component
at the two corners (0,w) and (L,w). These boundary
§
of the gradientfor the Hall resistivity along the Hall bar
conditions are
can be deduced from (14,17,18) to be
ψ(x,y =0)=ψ(x=0,y)=ψ(x=L,y)=C
∂ρ R R 1
xy 12 43
= − (19) ψ(x,y =w)=C , (21)
∂x L 2
5
where C ,C are constants. Without loss of generality, with these boundary conditions is
1 2
we can choose C = 0, C = I. The solution to (13)
1 2
−
ψ = ∞ I2mπ(1−(−1)me−2Lℓy)exp( x (y−w))sinh(12 (ℓ1x)2+4λmy) sin(mπx), (22)
mX=0− (2Lℓy)2+m2π2 2ℓy − 2ℓx sinh(12q(ℓ1x)2+4λmw) L
q
where correction is smaller by a factor of ℓ /L. Therefore we
x
have
m2π2 1 2
λm = L2 +(cid:18)2ℓy(cid:19) . (23) R12 ∼= 0ρ+LO(=e−∆w/ρℓx)+ (ρ ) ℓℓx ><00 . (26)
(cid:26) xxℓx xy O xx x
We note thatthere arefour lengthscales relevantto this
The voltage drop on the y =w edge is
problem: w,L, ℓ , ℓ .
y x
| | | |
In the following we examine first the case where the 8L 1 1 m2π2 1
cguraladtieenttheofcρuxryreinstinprothfielexa–nddiretchteiolnon(gℓiytu=din0a).l vWoletacgaels- V43 =−ρxxmX(odd) Im2π2 2ℓx − 2s4 L2 + ℓ2x×
on the two sides y = 0,w for the cases of positive and
1 m2π2 1
negative ℓ . When ℓ L,w, we find that the current coth( 4 + w) . (27)
flowsprimxarilyalon|gxo|n≪eofthe sides,thatisdetermined × 2s L2 ℓ2x !
by the sign of ℓ . The voltages on the two sides are
x The second term in (27) diverges. This divergence is a
again different, but the presence of the contacts require
result of the fact that by evaluating the voltage between
a certain care in analyzing the voltage drops. Then, we
x = 0 and x = L, we place our voltage probes on top of
examine the case of ℓ = 0, where the gradient of ρ is
x xy the contacts. If the probes are put at a distance ǫ from
in the y–direction. In that case, the effect of the density
the current contacts, we obtain
gradient on the current flow is more minor.
8L 1 1 m2π2 1
V = ρ I 4 +
A. Gradient in the x-direction 43 − xxmX(odd) m2π2 2ℓx − 2s L2 ℓ2x!×
mπǫ
cos( )=S S , (28)
2 1
When ∂ρ /∂y =0 Eq. (22) becomes × L −
xy
where we define
4 sinh(12 4λm+ ℓ12y) 8L 1 mπǫ
ψ =mX(odd)−Imπsinh(12q4λm+ ℓ12xxw) × (24) S1 ≡ρxxmX(odd)Im2π22ℓx cos( L ),
mπqx (y w) 8L 1 m2π2 1 mπǫ
sin( )exp − . S ρ I 4 + cos( ).
× L − 2ℓx 2 ≡ xx m2π22s L2 ℓ2x L
mX(odd)
The current components can be extracted using Eq.(5).
The first sum can be evaluated
In the limit where ℓ L,w the voltage drop along
x
≪ L ǫ
the y =0 side is S =ρ I( | |). (29)
1 xx
2ℓ − ℓ
x x
V12 =ρxxmX(odd)Im2π8L2|ℓx|s4m2π2(cid:18)ℓLx(cid:19)2+1× T1(w/hietℓhxsq,mua0rweh≡reoroeLta/sinπS|fℓo2xr|m)alwyaerbgeehaavpveparluoxe4sim(maotπfe/dL:m)f2or+wme1/≪ℓh2xamv∼=0e
| | p
w 1 ℓ 2 w 4(mπ/L)2+1/ℓ2x ∼= 2πm/L. Within that approxima-
exp( 4m2π2 x +1 ). (25) tion
× 2ℓx − 2s (cid:18)L(cid:19) |ℓx| p 1 1
V =ρ I(L 2ǫ) + (30)
Washee−nw/ℓℓxx.<W0h,etnhiℓsxv>olt0a,gtehderfiorpstisfeewxptoenrmenstiianlltyhesmsuamll, 43 xx∞ − 8L(cid:18)2|mℓxπ| − 2ℓx1(cid:19) mπǫ
(ttrhibousteiownitwhhmich≪isLp/ro|ℓpxo|)rtyioineladl atosiρgxnxifiℓLxc,anatndfintihtee cfiornst- ρxxIm(oXdd)=m0 m2π2 (cid:18) L − 2|ℓx|(cid:19)cos L
6
Thefirsttermin(30)istheonlytermthatdependson We first consider the voltage drop along y = 0. As a
the sign of ℓ : when ℓ <0 it is exactly equal to I∆ρ , first step we calculate ∂ψ :
x x xy ∂y|y=0
where∆ρ isnowthechangeinthe Hallresistivityover
xy
the length L 2ǫ; when ℓ > 0 Eq. (30) is hard to
x
−
evaluate analytically. For ǫ ≪ℓx, the second term loga- ∂ψ ∞ 2mπ(1 ( 1)me−2Lℓy)
rithmically diverges. This divergence, similar to that of = I − − (32)
equation (27), originates from the singular current den- ∂y|y=0 − ( L )2+m2π2 ×
mX=0 2ℓy
sity near a point source/sink. If ǫ > ℓ , the voltage on
x mπx x √λ
m
the y = w side is much smaller than that on the y = 0 sin( )exp .
× L 2ℓ sinh(√λ w)
edge,sincemostofthecurrentturnstoflowonthey =0 y m
side. Its precise ǫ–dependence is hard to extract from
(30).
Taking the limit of ℓ L,w we can write
To summarize the discussion of current flow along the y ≪
sinh(√λ w)=1/2exp(√λ w). An overestimate of the
ρ gradient in a finite sample, we note that for w ℓ m m
thxey side along which the current flows is determine≫d byx sum may be obtained by replacing √λmexp(−√λmw)
by √λ exp( √λ w). Since for 0<x<L
the sign of ℓx and not by the position of the contacts. 0 − 0
Along this side (which is y =0 for ℓ >0 and y =w for
x
ℓ <0),aslongasthethevoltagecontactsaresufficiently
x ∞ − L
far from the current source and sink, the voltage drop is 2mπ(1 ( 1)me 2ℓy) mπx x
− − sin =exp , (33)
dperotebrems,inaenddbisyitnhdeepdeiffnedreenntcoefinρ ρx.yAbloetnwgetehnetohtehveroletdaggee mX=0 (2Lℓy)2+m2π2 L −2ℓy
xx
the voltage drop is very small.
we get that
B. Gradient in the y-direction
∂ψ
In the case of a uniform gradient in the y direction, = I λ0exp( λ0w)=
∂y − −
the current injected at the corners propagates primarily (cid:12)y=0
(cid:12) p p
along the edge at y = w. In this case, however, the (cid:12) I/2 ℓy exp( w/2 ℓy ), (34)
(cid:12) − | | − | |
concentration of the current on the y = w is a result
of its injection at the corners of that side, and not a
result of the density gradient. Indeed, as .IVA shows, leading, when integrated over x, to an exponentially
if the current is injected uniform in y, it s§tays so. As a small voltage drop regardless of the sign of the gradient
consequence, the current profile is not exponential, and of ρxy.
the voltage drop on the y = w is not independent of
Aty =w,wecanexpecttohavethesamevoltagedrop
ρ . This qualitative picture comes out of Eq. (22). For
xx for both positive and negative gradients. We note that
∂ρ /∂x=0 we have
xy the expression for the current in the x direction remains
unchanged when replacing ℓ with ℓ , along with re-
y y
−
∞ − L placing x by L x; therefore, it is enough to consider
2mπ(1 ( 1)me 2ℓy) −
ψ = I − − (31) only the case when ℓy is positive. Integrating jx with
− ( L )2+m2π2 × respect to x and taking the limit of ℓ L,w we find
mX=0 2ℓy y ≪
the potential difference between the points x and x to
sinh(√λ y) mπx x 4 3
m
sin( )exp . be
× sinh(√λ w) L 2ℓ
m y
x4 ∞ 2mπL(1 ( 1)me−2Lℓy) 1 mπx mπ mπx x
V = ρ j dx= ρ I − − sin( ) cos( ) exp x3 . (35)
43 − xx x ∼ xx 3 2ℓ L − L L 2ℓ x4
Zx3 m=0 ( L )2+m2π2 (cid:18) y (cid:19) y
X 2ℓy (cid:12)
(cid:12)
(cid:16)q (cid:17)
We calculatethe voltagebetweenx =ǫ andx =L ǫ, tion, we obtain
4 3
−
in the limit of ǫ ℓ . Replacing summation by integra-
y
≫ 2 x x x x
V =ρ I K + K (36)
43 ∼ xx π 2ℓ 0 2ℓ 2ℓ 1 2ℓ −
(cid:18) y (cid:18) y(cid:19) y (cid:18) y(cid:19)
x x
K exp L−ǫ .
0 2ℓ 2ℓ ǫ
(cid:18) y(cid:19)(cid:19) y
(cid:12)
(cid:12)
7
The leading terms are where ψ is the stream function. The function V is ana-
lytic in z =x+ıy and hence ψ obeys the Laplace equa-
V ρ ∆ρL−ǫI ρ ∆ρǫ I (37) tion. Using the Cauchy-Reimann equations we may find
43 ∼ xx xy − xx xy the full derivative of V with respect to z
q q
where dV
= j +ıj . (42)
x y
∂ρ dz −
∆ρL−ǫ = xy(L ǫ),
xy ∂y − So the full derivative of the complex potential is a linear
∂ρ combinationofthe componentsofthe currentdensity. If
∆ρǫ = xyǫ.
xy ∂y we define
dV
This time, the voltage explicitly depends on ρxx, as the =ef(z) (43)
width of the strip within which the current flows does dz
not scale as ρ−1.
xx where f(z) is an analytic function, we get that
j = eRe(f(z))cosIm(f(z)), (44)
x
VI. UNIFORM DISSIPATIVE REGION −
BETWEEN TWO QUANTUM HALL REGIONS jy =eRe(f(z))sinIm(f(z)). (45)
The boundary conditions set for the stream function ψ
Wenowconsiderindetailasituationinwhichthereare
can now be translated into conditions on the function
quantizedHallstates atthe two ends ofthe sample asin
f(z). Therequirementonsettingj tozeroaty =(0,w)
Fig. 1, with Hall resistivities ρA and ρB and vanishing y
xy xy is translated into setting the imaginary part of f(z) to
longitudinal resistivity. In between there must be a dis-
zero there. The boundary conditions at x = (0,L) can
sipativeregion,wherethelocaldissipativeresistivityρ
xx be translated into
is not necessarily small comparedto ρ . In general,the
xy
resistivity in this region will be x–dependent. However, ρA −ρC
xy xy x=0,
wesimplifyandconsiderthecentralregionashavinglon- ρCxx
tanImf(z)= (46)
gitudinal resistivity ρC and Hall resistivity ρC that do
not vary in position (sxexe Fig. 3). xy  ρBxy−ρCxy x=L.
ρC
xx
Since the resistivity components vary along the x di-
rection only, Eq. (6) becomes Once we have solvedfor the imaginary part of f(z),
f(z)is completely determined, and therefore the electric
ρ (x) 2ψ+ρ′ (x)∂ψ +ρ′ (x)∂ψ =0 (38) field and currentdistribution are known up to a normal-
xx ∇ xy ∂y xx ∂x izationfactor,whichdeterminesthecurrentdirectionand
magnitude. We find
where ρ′ ∂ρij. Thus, in the central region where
ρ′ =0,itjhe≡stre∂axm function ψ simply obeys the Laplace ∞ 4 nπL −1
ij f(z) = sinh (47)
equation. The interfaces to the quantizedHall regionsA nπ 2w ×
and B impose boundary conditions on the current den- n(oXdd)=1 (cid:18) (cid:19)
sity (and hence on ψ). Conservation of the electric field nπz nπ(z L)
β cosh β cosh −
B A
parallel to the interface and of the current density per- × w − w
(cid:18) (cid:19)
pendicular to the interface, leads to
where β arctan[(ρA ρC )/ρC ], β arctan[(ρB
A ≡ xy− xy xx B ≡ xy−
jx −ρAxyρ−CxxρCxy x=0, ρreCxgyi)o/nρCxbxe]t.wSeeinnctehtehtewiontqeuramnetdizieadteHraelglisotnatiessaontriatnsseintidosn,
= (39)
jy  −ρBxyρ−CxxρCxy x=L, ρnCxuymibseinrsteβrAmeadnidatβeBbeistwneeegnatρiAxvyeaannddρtBxhye.oTthheurs,isonpeosoiftitvhee.
We set β < 0 and β > 0. Evaluating f(z) on the
where jx = ∂ψ/∂y,jy =∂ψ/∂x are the components of boundaryAat x=L we gBet:
−
the current density inside the central dissipative region.
∞
At the y =0 and y =w edges of the sample the current 4β nπy
B
Imf(x=L,y)= sin =β ,
must be parallel to the edge, i.e. nπ w B
n(oXdd)=1
jy(y =0)=jy(y =w)=0. (40) ∞ 4 nπy
Ref(x=L,y)= cos
nπ w ×
We simplify matters by using a conformal mapping12. n(oXdd)=1
We introduce a complex potential
nπL β
A
β coth . (48)
V =χ+ıψ (41) × B w − sinhnπL!
w
8
y = w in this situation we find the current flow to be concen-
trated along one of the edges. We showed that when
that happens, the voltage along that edge is controlled
ρxx = 0 ρxxC ρxx = 0 by the difference in Hall resistivitiesbetween the voltage
ρ A ρ C ρ B probes,whilethevoltageonthe otheredgeisverysmall.
xy xy xy
y Theseobservations,whicharelargelyindependentofthe
specific spatial dependence of both ρ and ρ in the
xx xy
y = 0 dissipative region,correspond to what is observedin the
0 L
experiment. We solved in detail two simplified models,
x
whichindeedshowedaconcentrationofthecurrentalong
one of the edges.
FIG. 3: The three different regions of the system described In one of the models we solved, the longitudinal resis-
in §VI.
tivity is assumed independent of position, and the Hall
resistivityhasauniformgradient. Insuchacase,thecur-
rentbehavioriscontrolledbythetwolengthscalesℓ and
x
ℓ . Forsampleslargecomparedtotheselengthscales,the
At the corners y = (0,w) the real part of f(z) diverges, y
currentflowsinaveryasymmetricwaythroughthesam-
going to + at y = 0, and at y = w. This diver-
∞ −∞ ple. This has been illustrated by explicit calculations in
gence implies the current density diverges at the corner
rectangular geometries, both for the ideal case of a long
(L,0) of the dissipative region, and vanishes at (L,w).
system, and for a rectangular system where the location
The same behaviour is observed at x=0, implying that
of the contacts has to be taken into account. We have
current enters and leaves the dissipative region from the
shown that in both cases, when the gradient is in the
corners. Interchangingthesignsofβ andβ movesthe
A B direction along which the current flows, the longitudinal
points of divergence to the line y =w.
resistance will be ∆ρ on one side of the sample, and
In the limit where w L, for every x in the range xy
≫ muchsmalleronthe other. Forthe longsystem,wehave
0<x<L we get
shownthatthisresultstillholds,evenifinadditionthere
∞ 4(β β )wcosnπy is a transverse component to the gradient.
Ref(x,y) B− A w . (49) For a long Hall bar geometry with a linear gradientin
≃ π2L n2
n(oXdd)=1 the localHall resistivity,we showedhow the microscopic
resistivity ρ can be extracted from the measurements
xx
This sum can be carried out explicitly to obtain of the longitudinal voltage drop on the two sides of the
sample, via Eq. (20). We believe that this model can
(βB βA) w be ofapplicationto the casesoftransportmeasurements
Ref(x,y) − y . (50)
≃ L 2 −| | on compressible states in the quantum Hall regime, for
(cid:16) (cid:17)
example the ν = 1/2 composite-fermion liquid state in
implying the currentdensity tobe largenearthe edge at
high quality samples. The conditions for application of
y = 0, and decay exponentially as a function of y. The
this model to experiment relies on the validity of the as-
decay length is ℓ = L/(β β ). In the limit of small
dissipation where ρC BρA−,BA ρC we have β sumptionsusedtoderiveEq.(13). Firstly,theseassump-
π/2 and the decay lxexng≪th|axpypro−achxeys|the limit|ℓA=,BL|/→π. tionsrequirethatthevariationinρxx overthesizeofthe
sample is small compared to its average value; typically
In the opposite limit of large dissipation where ρC
ρA,B ρC ,the decaylengthgrowslike ρC /ρA xρxB≫. this means that the fractional change in carrier density
| xy − xy| xx | xy− xy| should be small. (We note, however, that for a system
As long as this length is smaller than w, the current
with small dissipation, ρ ρ – that is with a local
is still concentrated along one side of the sample, and ◦ xx ≪| xy|
Hall angle close to 90 – even a small fractional change
the longitudinal voltage on that side is determined by
in carrier density can allow the change in the Hall resis-
ρA ρB , rather than by ρ . Again, interchanging the
sixgyn−ofxβy and β switchesxxthe edge where the current tivity ρxy to be large compared to ρxx.) Secondly, we
A B haveassumedthatwecanretainonlythe lineargradient
propagates from y =0 to y =w.
in the local Hall resistivity ρ (r). The validity of this
xy
assumptiondepends onthe functionalformofρ (r). In
xy
the case where variations in ρ (r) are all on the same
xy
VII. DISCUSSION
characteristiclength scale, which we shall refer to as the
macroscopic correlation length ξ , this assumption is
M
Inthis workwediscussedtheeffectofdensityinhomo- valid provided ξ L,min(w,ℓ ). In practice, ξ can
M x M
≫
geneities on the measurement of the longitudinal resis- beverylarge,setbyinhomogeneitiesinthegrowthofthe
tance in quantum Hall systems. samples related to characteristics of MBE machines. In
We analyzed the case in which the two ends of the realistichighmobility2DEGs,oneanticipatesthatthere
sample are in two different quantized Hall states, sep- are additional inhomogeneities on a microscopic length-
arated by a dissipative transition region. Generically, scale, ξ , oforderthe spacer-layerthickness. The model
m
9
remainsvalidprovidedthisadditionallengthscaleissuffi- We now integrate over a rectangle that encloses the cen-
cientlysmall,ξ L,min(w,ℓ ),thatthesemicroscopic tral region, as demonstrated in Fig. 1. We note that at
m x
≪
inhomogeneities simply affect the effective local resistiv- y = (0,w), the stream function ψ must be a constant,
itytensorwhichvariessmoothlyonthelongerlengthscale since we do not allow current to flow in the y direction.
ξ . We denote ψ(y =0) ψ and ψ(y =w) ψ , with
M b t
≡ ≡
Acknowledgments
I =ψ ψ (A.4)
b t
−
ThisresearchwaspartlysupportedbytheUKEPSRC
Grant No. GR/R99027/01 (N.R.C.). AS and RI thank being the total current flowing from left to right. Per-
the US-Israel Bi-National Science Foundation, the Israel forming the integration we get
Science Foundation and the Minerva foundation for fi-
nancial support. Q= ψ (φ φ ) ψ (φ φ )+
b 2 1 t 4 3
− − − −
1
(ψ2 ψ2)(ρA ρB ). (A.5)
2 b − t xy− xy
APPENDIX: CALCULATION OF THE Using (A.2,A.4) we can write
DISSIPATED POWER
1
UsingthesameapproachasinRef.7,wederiveexpres- Q = I2(ρA ρB )+I(φ φ ) (A.6)
2 xy− xy 1− 2
sionsforthepowerdissipationinasamplewithquantized
1
Hall states A and B at the two ends, as in Fig. 1. The = I2(ρB ρA )+I(φ φ ). (A.7)
2 xy− xy 4− 3
central region has arbitrary local resistivity tensor.
The power dissipation is given by
Note that in III we argue that the potential drop along
§
one side of the sample can become exponentially small.
Q= E(r) j(r)d2r.
· Clearly, from (A.6) if V12 =φ1 φ2 is to be vanishingly
Z Z small, then we must have ρA −ρB > 0 such that the
xy − xy
Since in regions A and B the resistivity tensor has only powerlossis positive;similarlyfrom(A.7)V =φ φ
43 4 3
−
off-diagonal elements which are non zero, in these two can be vanishingly small only if ρA ρB <0.
xy− xy
regionsequations (5) (4) and(1) canbe combined into a
Thegeneralresultcanbestatedmostclearlybynoting
single equation,
that,using(A.2,A.4),wecanexpressthecurrentinterms
of voltages as
ρ ψ = φ, (A.1)
∇ −∇
where ρ is a scalar, equal to ρA ,ρB in regions A and V41 V32 V43 V12
xy xy I = − = − . (A.8)
B. This equation can now be integrated to obtain the ρA ρB ρA ρB
xy− xy xy− xy
relation between φ and ψ:
Substituting this in (A.6) one then obtains
φ= ρψ+c, (A.2)
−
where c is a constant, in principle different for regionsA Q= 1(ρA ρB )−1(V2 V2), (A.9)
and B. Writing Q in terms of these scalar functions and 2 xy− xy 43− 12
using Stokes’ theorem, the dissipation integral becomes
whichindicatesthat V > V forρA >ρB ,andvice
| 43| | 12| xy xy
Q= (ψ φ) dl. (A.3) versa, as discussed in III.
− ∇ · §
Z
1 W. Pan et al.,Phys. Rev.Lett. 95, 066808 (2005). B 53, 1558 (1996).
2 B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, 8 N. R. Cooper, B. I. Halperin, C. K. Hu, and I. M. Ruzin,
7312 (1993). Phys. Rev.B 55, 4551 (1997).
3 I. M. Ruzin,Phys. Rev.B 47, 15727 (1993). 9 L. P. Rokhinson, B. Su, and V. J. Goldman, Solid State
4 N. R. Cooper and J. T. Chalker, Phys. Rev. B 48, 4530 Commun. 96, 309 (1995).
(1993). 10 G. J. C. L. Bruls et al., Phys.Rev.Lett. 46, 553 (1981).
5 S.H.SimonandB.I.Halperin,Phys.Rev.Lett.73, 3278 11 A. V.Chaplik, JETP Lett. 72, 503 (2000).
(1994). 12 R. W. Rendell and S. M. Girvin, Phys. Rev. B 23, 6610
6 A. M. Dykhne and I. M. Ruzin, Phys. Rev. B 50, 2369 (1981).
(1994). 13 Strictly speaking E is (minus) the gradient of the electro-
7 I.M.Ruzin,N.R.Cooper, andB.I.Halperin,Phys.Rev. chemical potential, and j is the transport current density.
10