Method for Detecting Superconducting Stripes in High-Temperature SuperconductorsBased on Nonlinear Resistivity Measurements

Rodrigo A. Muniz1,2 and Ivar Martin2

1Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

(Received 25 April 2011; published 13 September 2011)

We theoretically study the effect that stripelike superconducting inclusions would have on the nonlinear

resistivity in single crystals. Even if the stripe orientation varies throughout the sample between two

orthogonal directions due to twinning, we predict that there should be a universal dependence of the

nonlinear resistivity on the angle between the applied current and the crystal axes. This prediction can be

used to test the existence of superconducting stripes at and above the superconducting transition

temperature in cuprate superconductors.

DOI: 10.1103/PhysRevLett.107.127001 PACS numbers: 74.81.�g, 74.25.F�, 74.72.Kf

The study of the high-temperature superconductivityproblem in cuprates has been complicated by the proximityof the superconducting state to nonsuperconducting or-dered phases. Among them, only the antiferromagneticinsulating state has been unambiguously identified as al-ways present in the undoped parent compounds across allfamilies of cuprates. The nature of the pseudogap regime[1] that emerges upon doping, has remained elusive, eventhough there are many theoretical proposals for its origin[2–6]. One of the likely candidates for at least a part of thepseudogap regime is the spin and/or charge density waves(SDW/CDW), or their strongly-coupled cousin, the stripestate [3,4]. Indeed, neutron scattering commonly detectselastic and inelastic responses characteristic of static orslowly fluctuating incommensurate SDWs in a majority ofcuprate families [7,8]. Charge modulation is also observed[9–11], albeit less commonly [12]. Naturally, charge andspin modulations can locally modify the conditions for theonset of superconductivity. Moreover, it is possible that inthe anharmonic regime that characterizes the stripe state,local pairing mechanisms, which are not present in themore conventional CDW and SDW setting, may becomeoperational [13–15]. In this case, stripes would be animportant contributor to the high values of the supercon-ducting transition temperature Tc in cuprates. The in-planespatial inhomogeneity of superconductivity would be con-sistent with several anomalies commonly observed in theunderdoped cuprates, including suppressed superfluid den-sity [16] and the spectral weight transfer in the opticalconductivity [17].

Identification of the stripe states, in contrast to harmonicspin or charge modulations has been notoriously difficult.Direct experiments that would measure the transportanisotropy induced by the rotational symmetry breakingcaused by stripes are often hindered by sample twin-ning or stripe ordering patterns that restore rotationalsymmetry on the measurement length scale [18]. In addi-tion, it is believed that in order to be compatible with

superconductivity, the stripes have to be dynamical, i.e.,fluctuating on some time scale.In this Letter we propose a method capable of detecting

signatures of superconducting stripes diluted in a normalmatrix despite all these complications. It is based on prob-ing the spatial anisotropy of the nonlinear transport, whichshould be induced by superconducting stripes in singlecrystal samples, even when linear transport is completelyisotropic. The method takes advantage of the fact thatunlike the linear response conductivity tensor, which hasto be rotationally invariant in tetragonal systems, the non-linear response in general is not. Superconductors haveinherently nonlinear I-V characteristics, which makesthem ideally suited for the proposed method. Therefore,nonlinear transport measurements can help both to estab-lish the temperature range within which superconductinginclusions persist above Tc, as well as to determine thelocal spatial structure of these inclusions. The method isnot limited to static stripe configurations; it applies as longas the fluctuation time scale is sufficiently slower than thesuperconducting pairing time scale.We qualitatively demonstrate how anisotropic nonlinear

resistivity can arise in a tetragonal system by considering aschematic model in Fig. 1(a). Suppose that within the

0 1 2 3 40

1

2

3

4

Js

Vs(

J s)

Ic = 0.5

Ic = 1.0

Ic = 1.5

Ic = 2.0

Ic = 2.5

(b)(a)

FIG. 1 (color online). (a) System with stripe domains alignedalong directions [100] and [010]. (b) I-V relations used forthe superconducting stripes for different values of the criticalcurrent Ic.

PRL 107, 127001 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending

16 SEPTEMBER 2011

0031-9007=11=107(12)=127001(5) 127001-1 � 2011 American Physical Society

CuO2 planes there are domains of superconducting stripes,which are primarily oriented along [100] and [010] direc-tions. Each superconducting segment has a nonlinearI-V characteristic, as shown schematically in Fig. 1(b).If the characteristic domain size is much smaller thanthe crystal size, the tetragonal symmetry of the system isnot broken—the [100] direction is still equivalent to[010]. Then, we are immediately forced to conclude thatlinear resistivity must remain isotropic. In particular�0½110� ¼ �0

½100� (superscript ‘‘0’’ indicates that the resistiv-ity is taken at zero current).

Now, let us consider the nonlinear resistivity of thesystem. Suppose we apply current jx and measure electricfield Ex, with direction x either parallel to [100] or [110](two high-symmetry directions). For small applied current,the superconducting segments have negligible resistivity,and hence the average resistivity of the system is thesmallest. As jx increases, some of the segments quench(when the local critical current Ic is reached), and theoverall resistivity increases. We therefore anticipate thatin both cases, the resistivity will increase nonlinearly withjx. However, the nonlinear resistivity in two cases willnot be the same, as can be easily seen from the followingargument, which is particularly simple in the dilute stripeconcentration limit. Suppose that in addition to the currentin the direction x, there is also a current in the y direction.Qualitatively, it is clear that it should make little differencefor Ex in case x ¼ ½100�, but a big difference whenx ¼ ½110�. That is because, in the latter case, jy current

will directly contribute to reaching the critical current Ic onthe superconductor (SC) links involved in the transportalong x, while in the former, jy primarily saturates the

horizontal superconducting links, which are largely irrele-vant for the transport along x. Thus we see that the non-linear resistivity has to be anisotropic, and that the I-Vdependence in the coordinate system where x ¼ ½100� andy ¼ ½010� with good accuracy is given by

Ex ¼ �ðjxÞjx; Ey ¼ �ðjyÞjy: (1)

That is, the electric field along the stripe pinning directiondepends only on the current in the same direction.

In order to obtain the relation between theE and j in anyother planar coordinate system we only need to performa rotation of Eqs. (1) around z axis by angle � with the

matrix Rz�,

E0 ¼ Rz��ðJÞJ ¼ Rz

��ðRz��J

0ÞRz��J

0 (2)

where

�ðJÞ � �ðJxÞ 00 �ðJyÞ

" #: (3)

The resistivity tensor in the rotated basis is

�0ðJ0Þ ¼ Rz��ðRz

��J0ÞRz

��: (4)

Naturally, in the limit of vanishing current we see that theresistivity tensors coincide, �0 ¼ �, as expected from lin-ear response of a tetragonal system. However, at a finitecurrent, the relationship is more complex. For instance, if acurrent j0 is applied along direction x0, the electric fieldresponse becomes

Ex0 ¼ ½�ðj0 cos�Þcos2�þ �ðj0 sin�Þsin2��j0Ey0 ¼ ½�ðj0 cos�Þ � �ðj0 sin�Þ� sin� cos�j0: (5)

In particular, for the high-symmetry direction � ¼ �=4the resistivity is obtained from resistivity along � ¼ 0

by dilation of the current axis by affiffiffi2

pfactor, �0ðj0Þ ¼

�ðj0= ffiffiffi2

p Þ. Another notable feature is the appearance of thetransverse electric field if current is applied away from thehigh-symmetry directions, i.e., when � is not an integermultiple of �=4. The induction of transverse electric fieldin tetragonal systems is only possible in the presence ofnonlinear response.To test the above reasoning, we consider an explicit

resistor network model that incorporates the describedqualitative features. The resistors, connecting the nearestneighbor sites of a square lattice, Rij, are chosen at random

to be either ‘‘normal,’’ with constant resistivity Rn, or‘‘superconducting,’’ with the current-dependent resistivityRsðJsÞ. The probability of superconducting bonds is p. TheKirchhoff’s equations

Xj¼n:n:

Vi � Vj

RijðVi � VjÞ ¼ 0; (6)

combined with current or voltage boundary conditions givea system of nonlinear equations that can be used to deter-mine the local voltages fVig. In our model calculation,for the superconducting resistors we take the I-V depen-dence as

Vs ¼ RsðJsÞJs ¼�R0s þ Rn � R0

s

e�4ðJs�IcÞ þ 1

�Js (7)

which at small currents has resistivity of approximatelyR0s < Rn and at large currents Js � Ic saturates to the

resistance of the normal links Rn as illustrated inFig. 1(b). We will take Rn ¼ 1 and R0

s ¼ 0:05 unless statedotherwise. In the simulations we used a system of size80� 80 sites with fixed voltage boundary conditionsin the direction of applied current, and periodic boundaryconditions in the transverse direction. Simulationswere performed for current applied along the bond anddiagonal directions (see Supplemental Material for illus-trations [19]).In Fig. 2(a) we present the result of lattice simulation

when current is applied along the bond direction (squares)for various values of superconducting link concentrationsp and for Ic ¼ 2. As expected, for large values of appliedcurrent Jm, the resistivity saturates to the normal value 1,while at low currents it drops by an amount that increases

PRL 107, 127001 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending

16 SEPTEMBER 2011

127001-2

with p. The width of the intermediate nonlinear regiondecreases with increasing p.

Our lattice simulation results can be very accuratelyreproduced by the effective medium theory (EMT). EMTis typically applied to study linear random resistor net-works, where it well captures, e.g., percolation phenomena[20]. Within EMT, one solves exactly the problem of aparticular random resistor R0 embedded in a perfect matrixcomposed of identical effective medium resistors Rm. Thevalue of Rm is determined self-consistently from the con-dition that the voltage drop V0 on resistor R0, averaged overthe distribution, be equal to the average voltage drop Vm

per link of the network. Here we apply this procedure to thebinary network that contains nonlinear resistors. Withthe average current parallel to the bond direction, it leadsto the following equations

Js ¼ 2gsðJsÞgmðJsÞ þ gsðJsÞ Jm; (8)

gmðJsÞ ¼�p� 1

2

�½gsðJsÞ � gn�

þ��

p� 1

2

�2½gsðJsÞ � gn�2 þ gngsðJsÞ

�1=2

; (9)

where Jm is the externally applied uniform current per unitcell, Js is the current through a superconducting link,which needs to be determined self-consistently, and forconvenience we introduced conductances, gi � 1=Ri.These coupled nonlinear equations can be solved numeri-cally for any specified form of RsðJÞ. As can be seen fromFig. 2(a), nonlinear EMT (solid lines) very well approx-imates our lattice simulation results, with the largest de-viation occurring at the percolation threshold, p ¼ 0:5.

In Fig. 2(b) we present the comparison of lattice simu-lations for the current applied along the bond (circles) anddiagonal (squares) directions. After rescaling the current

axis by a factorffiffiffi2

p, the diagonal resistivity (dashed lines)

matches the resistivity in the bond direction, as was

anticipated from Eq. (5). Even at the percolation threshold,p ¼ 0:5, the scaling works remarkably well. The deviationoccurs due to the cross talk between the horizontal andvertical superconducting bonds, which was neglected inEq. (1). We have also performed simulations assumingthat superconducting links have random length, randomcritical current, or internal structure [‘‘patches’’ similar to

Fig. 1(a)]. All these simulations [19] show the same ‘‘ffiffiffi2

p’’

scaling.When current is applied in an arbitrary direction relative

to the crystal axes, in addition to the longitudinal resistiv-ity, there is also a finite transverse resistivity. The fullangular dependence of the longitudinal and transverseresistivity obtained using Eq. (5) for Ic ¼ 2 and forp ¼ 0:2, 0.8 is presented in Fig. 3. As expected, the biggestanisotropy effects are concentrated in the range of currentswhere the resistivity of the superconducting links is non-linear. This transverse response would be completely ab-sent in the linear resistivity of a tetragonal system.Therefore, it represents a ‘‘null-point measurement’’ thatcan be a very sensitive diagnostic of the presence of localrotational symmetry breaking induced by superconductingstripes.We now discuss the expected range of applicability of

our model to real materials. Our central assumption is thatonly special links are superconducting, i.e., that theJosephson coupling between them is relatively negligible.Therefore, our model applies for the current densitiesexceeding the interstripe critical current density. On theother hand, the nonlinear behavior is expected to occurfor current densities up to the intrastripe critical current.This current can be estimated as a fraction of the criticalcurrent density of the optimally doped cuprates, which atlow temperatures is about 107 A=cm2, decreasing with

0 1 2 3

0.2

0.4

0.6

0.8

1

Jm

Rm p = 0.1

p = 0.3p = 0.5p = 0.7p = 0.9

(a)

0 1 2 3 4 5

0.2

0.4

0.6

0.8

1

Jm

Rm

p = 0.1p = 0.5p = 0.9

(b)

FIG. 2 (color online). (a) Comparison between EMT and thenumerical lattice simulation. The lines correspond to Eq. (9) andthe squares to the numerical simulation. (b) Verification of the‘‘

ffiffiffi2

pscaling.’’ Circles and squares correspond to numerical

lattice simulations for the [100] and [110] directions, respec-tively. The dashed lines correspond to the numerical data fordirection [110] with the current axis divided by

ffiffiffi2

p.

Ix / I

c

Iy /

I c

0 0.5 1 1.5 20

0.5

1

1.5

2

0.65

0.75

0.85

0.95

(a)

Ix / I

c

Iy /

I c

0 1 2 30

1

2

3

−0.08

−0.04

0

0.04

0.08(b)

Ix / I

c

Iy /

I c

0 0.5 1 1.5 20

0.5

1

1.5

2

0.1

0.3

0.5

0.7

0.9

(c)

Ix / I

c

Iy /

I c

0 1 2 30

1

2

3

−0.2

−0.1

0

0.1

0.2

(d)

FIG. 3 (color online). Longitudinal (a),(c) and transverse (b),(d) resistivity as a function of applied current (Jx, Jy) for

p ¼ 0:2 (a),(b) and p ¼ 0:8 (c),(d).

PRL 107, 127001 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending

16 SEPTEMBER 2011

127001-3

temperature. For underdoped cuprates, due to large inter-stripe separation this leaves a large window of applicabil-ity; however, in the optimal and overdoped materials, thewindow will shrink. Finally, we note that the predictednonlinear transport signatures, are distinctly different fromanother potential source of nonlinearity—the Joule heat-ing. Namely, the window of currents where nonlineareffects due to local superconductivity should exist, shrinksas the temperature increases above Tc, vanishing at someT�c , while Joule heating should increase with increasing

current indefinitely and should be sensitive to the details ofthe thermal coupling to environment. Moreover, Jouleheating is not expected to cause any spatial anisotropy inresistivity.

Another issue that we can comment on within our super-conducting percolation model is the apparent discrepancybetween the superconducting fluctuation regime, as ex-tracted from the bulk probes [21] and from the ac transportmeasurements [17,22,23]. In the ac transport, one attemptsto detect features characteristic of superconductivity, suchas the superfluid density, and tracks their disappearance asa function of increasing temperature. Experiments consis-tently find that the superconducting fluctuation rangeabove Tc, extracted this way, is much more narrow thanthe one obtained from the bulk measurements, such asdiamagnetic response [21]. Within the uniform fluctuatingsuperconductor scenario these differences are difficult toreconcile [23]. However, if we assume that the supercon-ductor is intrinsically inhomogeneous, as in the modelconsidered above, the rapid disappearance of the transportsignatures of superconductivity can be attributed to reduc-tion of the superconducting fraction p below the percola-tion threshold above Tc. Indeed, let us assume that theconductivity of the normal links is gn ¼ ��n and of thesuperconducting links is gs ¼ ��s=ð1þ i!�sÞ in the rele-vant frequency range, !�n � 1, where �s is the relaxationtime in the superconducting links and �n is the normal staterelaxation rate (�s � �n). The effective medium conduc-tivity gm can be obtained from EMT, Eq. (9). We find,above percolation threshold, gm � ð2p� 1Þgs, while be-low—gm � gn=ð1� 2pÞ, the latter having only a very

small imaginary part. The crossover occurs in a narrowrange of �p ffiffiffiffiffiffiffiffiffi

!�np � 1; i.e., the superconducting con-

tribution disappears very rapidly as the fraction of thesuperconducting links goes below the percolation thresh-old pc ¼ 1=2 near Tc, even though disconnected super-conducting inclusions may still persist to much highertemperatures and contribute to other superconductivity-sensitive measures. In Fig. 4 we show schematically thebehavior of gmð!Þ expected from the EMT, which agreesqualitatively with the experimental trends.The lattice percolation model presented here, while

crude, provides a transport-based method to test the exis-tence of local superconducting inclusions—superconduct-ing stripes—in the cuprates at and above Tc. Incombination with the bulk probes, which are sensitive tothe local superfluid density ( / Ic) and the volume fractionp, it may help to shed light on the nature of the pseudogapregime, from which the high-temperature superconductiv-ity emerges.We acknowledge useful discussions with P. Armitage, I.

Bozovic, H. Lee, C. Panagopoulos, and T. Park. This workwas carried out under the auspices of the National NuclearSecurity Administration of the U.S. Department of Energyat Los Alamos National Laboratory under ContractNo. DE-AC52-06NA25396 and supported by the LANL/LDRD Program.

[1] T. Timusk and B. Statt, Rep. Prog. Phys. 62, 61 (1999).[2] P.W. Anderson, Science 235, 1196 (1987).[3] J. Zaanen and O. Gunnarsson, Phys. Rev. B 40, 7391

(1989); D. Poilblanc and T.M. Rice, Phys. Rev. B 39,9749 (1989); K. Machida, Physica (Amsterdam) 158C ,192 (1989).

[4] S. A. Kivelson, E. Fradkin, and V. Emery, Nature(London) 393, 550 (1998).

[5] S. Chakravarty, R. B. Laughlin, D.K. Morr, and C. Nayak,Phys. Rev. B 63, 094503 (2001).

[6] C.M. Varma, Phys. Rev. B 73, 155113 (2006).[7] J.M. Tranquada et al., Nature (London) 375, 561

(1995).[8] V. Hinkov et al., Science 319, 597 (2008).[9] C. Howald, H. Eisaki, N. Kaneko, and A. Kapitulnik, Proc.

Natl. Acad. Sci. U.S.A. 100, 9705 (2003); Y. Kohsakaet al., Science 315, 1380 (2007).

[10] M. Huecker et al., arXiv:1005.5191.[11] C. V. Parker et al., Nature (London) 468, 677 (2010).[12] P. Abbamonte et al., Science 297, 581 (2002).[13] V. J. Emery, S. A. Kivelson, and O. Zachar, Phys. Rev. B

56, 6120 (1997).[14] A. H. Castro Neto, Phys. Rev. B 64, 104509 (2001).[15] I. Martin, D. Podolsky, and S. A. Kivelson, Phys. Rev. B

72, 060502 (2005); I. Martin, G. Ortiz, A.V. Balatsky, andA. R. Bishop, Int. J. Mod. Phys. B 14, 3567 (2000).

[16] Y. J. Uemura et al., Phys. Rev. Lett. 62, 2317 (1989).[17] J. Orenstein, Physica (Amsterdam) 390C, 243 (2003);

J. Orenstein, in Handbook of High-Temperature

0.05 0.15 0.25 0.35

0.01

0.02

0.03

ω τn

Re

g m(ω

)/g s(ω

= 0

) p = 0.6p = 0.5p = 0.4p = 0.3p = 0.2p = 0.1

(a)

0.05 0.15 0.25 0.350

0.02

0.04

ω τn

−Im

gm

(ω)/

g s(ω =

0) p = 0.6

p = 0.5p = 0.4p = 0.3p = 0.2p = 0.1

(b)

FIG. 4 (color online). Frequency dependence of (a) real and(b) imaginary parts of conductivity obtained from effectivemedium theory for systems with different stripe concentrationsp. �s=�n ¼ 100.

PRL 107, 127001 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending

16 SEPTEMBER 2011

127001-4

Superconductivity, edited by J. R. Schrieffer and J. S.Brooks (Springer, New York, 2007), p. 299.

[18] Y. Ando, K. Segawa, S. Komiya, and A. Lavrov, Phys.Rev. Lett. 88, 137005 (2002).

[19] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.107.127001 foradditional numerical simulations confirming the

universality of the ‘‘ffiffiffi2

p’’ scaling in the angular depen-

dence of nonlinear resistivity.[20] S. Kirkpatrick, Rev. Mod. Phys. 45, 574 (1973).[21] L. Li et al., Phys. Rev. B 81, 054510 (2010).[22] J. Corson et al., Nature (London) 398, 221 (1999).[23] L. S. Bilbro et al., Nature Phys. 7, 298 (2011); L. S. Bilbro

et al., arXiv:1103.2402.

PRL 107, 127001 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending

16 SEPTEMBER 2011

127001-5