dinamica de vortices

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VORTEX PHYSICS INH I G H - T E M P E R AT U R E

SUPERCONDUCTORST he discovery of high-tem-peratu re superconduc-tors1 has s t imula ted d ra-matic growth in ourunderstanding of the physicsof quantized vortex lines.These superconductors ex-clude magnetic fields weakerthan a lower cr i t ica l fieldHcl < 10

2 tesla. Strongerfields penetrate as an arrayof vortices, each consisting ofexactly one quantum of flux( Therm al energy favors a vortex liquid of lines or pan-cakes.D> Vortex interac tion energy favors a perfect vortex lattice.O Pinning energy favors an amorphous or glassy solid.O Coupling energy betw een layers controls the formationof vortex lines from weakly interacting pancake vorticesin adjacent layers.

Remarkably, all four energies can be the same order

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FIGURE 1. FLEXIBLE MULTIFILAMENT SUPERCONDUCTING WIRE made from Bi2Sr2CaCu2O8 (BSCCO) superconductor in silversheathing. The wire is made by the powder-in-tube process, in which superconducting powder is put into an array of holes ina silver rod (shown in the center of the picture), which is then rolled and heat treated to produce a flat, flexible wire with hun-dreds or thousands of filaments. (Photo courtesy of American Superconductor Corp.)

of magn itude in high-tem perature superconductors, lead-ing to an unprecedented variety of liquid and solid phasesand transitio ns among them. The term vortex ma tteremphasizes that vortices are comparable in complexityand diversity to conventional atomic or molecular matter.

Vortex matter provides an excellent laboratory forexploring general phase-transformation behavior. Experi-mentally, all the relevant parameters can be varied overwide ranges: the areal vortex density over many ordersof magnitude by changing the magnetic field; thermalfluctuations through the temperature; the pinning disor-der through controlled irradiation; and the coupling energybetween CuO2 planes through the choice of material.Furthermore, vortex interactions with each other and withan external driving current are known rather precisely,allowing one to do analytical theory and numerical simu-lations with essentially no uncontrolled approximations.This combination of accessibility, diversity of phases andsimplicity of interaction is difficult to find in conventionalm atte r composed of atoms and molecules. Qu alitativelynew physics arises, moreover, because the basic degreesof freedom can be lines instead of points.

Figure 2 i s a generic phase diagram for single-crystalcuprate materials in the absence of pinning. The basiclayered struc ture of the cup rates is illustrated in the upperright. For simplicity, we assu me a field oriented alongthe c-axis, perpendicular to the CuO2 planes, which con-tain the a an d b directions. The mag netic induction B,which is proportional to the vortex density in the ab plane,is plotted because demagnetizing corrections in the usualplatelet crystals ensure that B (rather than the magneticfield H) is held fixed in most experim ents. The famousAbrikosov vortex lattice, which exists at all temperatures

below the upper critical field Bc2(T ) in mean field theory,appears here only at much lower temperatures, below aline of melting transition s Tm(B). In the presence of strongthermal fluctuations, the red curve Bc2(T ) marks the onsetof enhanced diamagnetism but is not expected to be asharp phase transition.

These two curves boun d a novel 'Vortex liquid1' regimeof flexible, entan gled line s. Although th e long-w avelengthproperties in this region are qualitatively indistinguish-able from those of a normal metal (there is a linear ohmicresistivity, for example), the quantization and entangle-ment of vortices lead nevertheless to many importantexperimental consequences.

The phase diagram of figure 2 is further subdividedby a decoupling line Td(B)y above which the discretenessof the copper oxide planes becomes significant.2 AboveTd(B), thermal fluctuations overcome the interplanar cou-pling, causing the vortex segment between adjacent planesto wander a sideways distance comparable to the averageinterlin e separation . The linelike na tur e of the vorticesis then less important: These fluxons behave more likeweakly coupled sheets of pointlike pancake vortices, al-though conservation of the total magnetic flux requiresconservation of the number of pancakes in adjacent layers.Although there are important qualitative changes at Td(B),there is no symmetry reason in the vortex liquid phasewhy the curve Td{B) must be a locus of sharp phasetransitions, just as there need not be a sharp phasetransition line separating the liquid and gas phases ofordinary matter above the critical point.

Remarkably, the extension of T d(B) into the crystallinephase is in fact associated with a sharp phase transitionto a supersolid phase due to defect proliferation.3 The

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65 70 75 80 85 90 95TEMPERATURE (kelvin)

1JLZ

R

S

N

RML

0.3

0.2

0.1

n

b

-

-

Pre-irradiation

\

S \ J

yr / ^Post-irradiation

\ .

Post-anneal

0.89 0.90 0.91 0.92 0.93NORMALIZED TEMPERATURE T/T c

0.94

358.9

358.543.15 43.20 43.25 43.30

TEMPERATURE (kelvin)

43.35

FIGURE 3. VO RTE X LATTICE MELTING, a: Resistiv ity of acrystal of strongly coupled YBa2Cu3O7 (YBCO) showing asharp drop at the melting transition and rounded behavior athigh fields, which suggests a critical point.6-7 b: Resistivity ofYBCO before and after electron irradiation to producepointlike defects, and after an nealing to remove the defects.8

The sharp pre-irradiation transition at melting is replaced by abroad transition without hysteresis in the disordered crystal.After annealing, the sharp first-order transition (and hysteresis)is recovered, c: Discontinuity at melting in the localmagnetic induction of a crystal of weakly coupled BSCCO.9

Box 1. Cage M odel of Flexible Line Meltinghe melting of a crystal of flexible vortex lines due tothermal agitation may be understood by using a simpli-

fied cage model. Intervortex interactions are described by aharmonic potential, which confines each line to the interiorof a cage defined by the surro undin g near-neighbo r lines (seethe illustration in the lower left of figure 2). Th e free energyand various other configurational averages can be estimatedby breaking the lines up into in dependen t segments just longenough to allow exploration of the interior of the cage.

Melting occurs when the root-mean-square line fluctuationsgrow to a fixed fraction of the equilibrium lattice constant,beyond which confinement is no longer effective.

We denote the trajectory of a vortex as it traverses asample with a magnetic field parallel to z by r(z) = [x(z), y(z)]and assume that the probability of a particular configurationis proportional to exp(E[r(z)]/k T)> where the energy ofthe line in its confining cage of length L is

[r(z)] = J i **W ] i (1)

The coupling constant g is a local tilt mod ulus, w hichmeasures the energy of bending the line aw ay from theexternal magnetic field direction. The curvature of theh i fii il k dharmonic confining potential k ~ d2V/dr( / ) 1 / 2 is the vortex lattice constant and

Ur , w h e r e a Q =P(r) describes

pairwise interactions between vortices. The lines interactthrough a screened Yukawa poten tial, V(r) = 2 e0 KQ(r/\),where e 0 is an interaction energy per unit length and A is th eLondon penetration depth in a plane perpendicular to z .Although K0(p) decays expo nentially to zero for p 1 , it isapproximately equal to -I n p fo r p = atf'k < 1, which is thecase for most of the experim entally relevant field range.Thus, k ~ e Q /al, where we neglect constants of order unityin this back-of-the-envelope discussion. In the same range offields, one has g = { m J m z)

o> where mjmz is the effectivemass anisotropy of the superconducting electrons. As dis-cussed in the body of the article, mjmz 0 (and a densercrystal phase) appears whenever n > 2.

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00.88 0.92 0.96

NORMALIZED TEMPERATURE 7/7.

FIGUR E 4. TRANSFORM ER EXPERIMENT, a: Geometry of theexperiment, in which voltage differences on the top andbottom surfaces probe the coherence between layers. V top andbottom count the number of vortices per unit time crossing

between the voltage taps on the top and bottom surfaces,respectively, b: Measured voltages on the top and bottomlayers of crystals of YBCO in the transformer geometry.13 Incrystals without defects, the top and bottom voltages mergeonly at the melting temperature Tm marked by the sharp dropin both voltages to zero, indicating that the liquid phase isentangled over nearly its whole range. In crystals with planarpinning defects (called twin boundaries), the top and bottomvoltages merge at 7th in the liquid phase, indicating a disentangledliquid regime between 7^ and x where the (presumably glass}'')

vortex solid forms and the voltage vanishes.

line of first-order tran sition s term ina tes.' Critical pointsare possible because pointlike disorder destroys transla-tional order in the crystal at the longest length scales,removing one sharp symmetry difference between solidand liquid phases.2

Introducing controlled point disorder (by means ofelectron irradiation, for example8) has dramatic conse-quences: The sharp transition and hysteresis disappearat all fields, giving way to a broad transition to zeroresistivity, as illustra ted in figure 3b. This behavior

qualitatively confirms the theoretical picture of first-ordertransitions to relatively perfect vortex lattices in cleanmaterials and more gradual transitions to a glassy phase

when pinning dominates. It is still unclear whether pointdisorder alone is sufficient to lead to a true second-orderphase transition.

Although transport measurements are effective forlocating the melting line in th e B-T plane, they give noinformation about the line's thermodynamic character.Therm odynam ic measu reme nts of the m agnetizat ionchange and latent heat at melting were long consideredfutile because the effects were thought to be too small toobserve. Recently, thi s pessim istic outlook has been dra-matically revised by sophisticated magnetization and cal-orimetr ic experiments showing t iny but measurablechanges in the magnetization and entropy at melting.9 12

Figure 3c shows the jump in the localmagnetic induction B on vortex melt-ing in BSCCO, implying a first-orderchange in the vortex density at themelting transition.9 From thermody-namic data like these, entropy jumpsin BSCCO and YBCO have been de-rived by using the Clausius-Clapey-ron equati on. Recently, th e first cal-orimetric experiment that directlymeasures the latent heat TA S on vor-tex melting was reported for YBCO,finding excellent agreement with theentropy derived from the magneticexperiment.12

The magnetization and calorimet-ric data are a tour de force of experi-mental technique and dramaticallyconfirm the existence of the thermo-dyna mic f i r s t - o r de r me l t i ng f i r s tsuggested by theory and transportmeasurements .

The thermodynamic measure-l me nts reveal a rem arka ble feature in

the m eltin g curv e of YBCO andBSCCO: The vortex density is higher

in the liquid th an in the solid. Thennody namicalry, this icelike me lting is equivalent to a negative slope of themelting curve, as is evident from the Clausius-Clapeyronequation, dHJdT = -AS/AM , w here AS an d AAf are theentropy and magnetization discontinuities, respectively, ata first-order mag netic trans ition. If the slope of themelting curve is negative, then the thermodynamic re-quirement of greater entropy in th e higher temperatureliquid phase requires a larger magnetization as well,equivalent to a higher density of paramagnetic vortexlines.

The physical origin of icelike melting is the flexibilityof the vortex lines combined with their long-range inter-actions (see box 1). Vortices at typical den sities interac t

with a logarithmic repulsive potential, a remarkably weakspatial dependence, which ensures tha t th e slope of themelting curve is negative and that the melting is icelike.

Entanglement and decoupling in liquid phaseAbove the melting line in figure 2, interactions (andpointlike disorder) are less important, and we must con-sider a liquid of weakly confined lines. The mean sq uaredistance that a line wanders in a distance z along thefield direction can be estima ted by using equation 1 inbox 1 (the cage potential is ignored by using a setting ofk = 0). In this approxim ation, th e line wandering is givenby a diffusion-like expressio n

< | r ( z ) - r ( 0 ) |2 > = - j - | z | .

The line diffuses as a function of the timelike variable

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Box 2 Vor tex Statistical Mechanics and the Boson M appinghe statistical mechanics of line vortices offers many oppor-tunities to exploit rich analogies between spatial configu-

rations of interacting lines in equilibrium and the non-relativis-tic quantum mechanics of interacting bosons in twodimensions.17 On e conclusion of the boson analogy is thatthe reduction in quantum zero-point energy that accompaniesthe melting of quantum Wigner crystals is represented by theincreased braiding ent rop y of the wiggling lines in the vortexliquid. Thus, the same physics that drives the melting of the

quantum Wigner crystal appears classically in the macroscopicvortex system.Space does not permit a detailed exposition of the boson

mapping. How ever, a brief qualitative outline illustrates theflavor of the argum ent. It can be shown that the classicalpartition function Z of a system of N flexible vortices withpairwise interaction V(r,j) and subject to an arbitrary z-inde-pendent pinning potential U(r) in a sample of thickness L isrelated to a quantum mechanical matrix element of an TV-par-ticle system propagating in imaginary time:

2 = J d r1 d rA J d r/1 d r

/N

w h e r e Hw = - 5 X ; X A > ) Z l i / l )

Here, g is the vortex tilt modulus discussed in box 1. Thesymmetrical integration over the vortex entry and exit points{r;} and {r';} ensures that only boson states contribute to 2.

The states in the quantum mechanical matrix element in theabove equation for 2 define the vortex positions at the twosurfaces normal to the field direction, and the operator in thematrix element is the exponential of a Ham iltonian that isidentical in form to that of correlated electrons in metals. Theboson mapping thus allows decades of work on correlatedelectron physics to be transferred to research on vortex lines,subject to the exchange of fermi for boson statistics.

The boson analogy can be exploited experimentally bymeasuring the positions of vortices on opposite surfaces of asuperconducting slab, thus defining the states of the quantummechanical matrix eleme nt discussed above. Vortex positionsare traditionally probed with decoration experiments, in whichsmall magnetic particles are allowed to settle on the samplesurface, marking the relatively high-field regions in the vortexcores. Double-sided vortex decoration experiments,18 leadingto the triangulated configurations of entry and exit pointsshown in the inset figure, allow information about vortex

DOUBLE-SIDEDTRIANGULATION o fvortex entry (a) and

exit (b) positions froma decoration ofBSCCO at low

magnetic fields.18

Shaded triangles touch defect sites, whic h

are five- orseven-coordinated.

Dislocations (clustersof red triangles) appear

as (5-7)-pairs in thisrepresentation. The

green trianglesrepresent highly

disordered regions,while the violet cluster

is a twisted bond.The two images areremarkably similar,

despite the samplethickness ofapproximately 20 /xm.

The small differencescontain information

about vortexcorrelations deep

inside the sample.

positions deep in the superconductor to be inferred from theirpositions on the entrance and exit surfaces. The fluctuatingvortex trajectories act like a spatial filter transmitting informa-tion across the sample. The decay rate of spatial correlationsin the direction of the field as a function of sample thicknessis governed by the ph on on -ro ton spectrum of the equivalentsuperfluid boson system, and it displays the same minimum atfinite wavevector in vortex systems18 as is familiar from liquidhelium-4 . Mu ch useful quantitative and qualitative informa-tion can be obtained from the double-sided decorations, includ-ing the long-wavelength elastic constants, the decay ofcrystalline and hexatic ord er across the crystal and the crossoverfrom logarithmic to exponential vortex pair interactions.

2 with diffusion constant D = kBT/gy where g is the vortextilt modulus. Once the line wanders a distance equal tothe intervortex spacing a0, the system is said to be en-tangled, in the sense th at its behavior can now be stronglyinfluenced by the barriers to cutting and reconnectingvortex lines. The entanglement length /, is th e distancealong the field required for the vortex line to wanderapproximately one intervortex spacing a0:

tzBl nBli3

where a02 ~ (f)0/B an d < f0 is the flux quantum.

Three regimes of entanglement can be defined bycomparing lz with characteristic leng ths in the sam ple.

The disentangled regime occurs for low fields such thatthe crystal thickness L is smaller than the entanglementlength: B Q ILk BT. Here, the lines are disentan-

gled and th e p roper t i e s a r e thos e of a fluid of shor tvo r tex segments that eas i ly s l ide pas t each o ther. N o tet h a t Bxl > 0 in the limit of inf in i te samp le th ickne ss .W h e n B > v l , the vor t ices entangle , and moving onevortex independen t ly of the o the r s r equ i r es f lux cutting.E n t a n g l e m e n t cont inues to increase unt i l B exceeds Bx2 =g4> 0 /sk BT, which is the field such that lz = s, the spacingbetween the double C uO2 p l anes . H ere , the flux lines a r e superentangled , or decoupled, in the sense that vor t icesw ander more than an in tervor tex spacing while pass ingfrom one s e t of C uO2 p lanes to the next . In this r eg imethe flux liquid can be viewed a s plane s of wea kly inter -act ing point l ike pancake vortices. The d isentangled , en-tangled and decoupled regimes a re il lustrated in figure 2.

E n t a n g l e m e n t of vortex lines in the flux liquid canbe probed through an e legan t pseudo - transform er exper i -m e n t .1 3 As illus trat ed in figure 4a, cu r r en t en te r s and

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F IG U R E 5. DYN AMIC PHASE DIAGRAM at constant magneticfield ol steady-state motion of vortices in a random array ofpins in two dimensions, derived from numerical molecular

dynamics simulations. 13 The vertical line marks the meltingtemperature of the perfect vortex lattice and is the upper limit

for elastic motion in the disordered solid. The dashed lineindicates a crossover from strong pinning to plastic motion.

leaves by way of the top layer of the sample, and thevoltage is monitored across two symmet rically placed p airsof taps on the top and bottom surfaces. Because of t heanisotropic geometry, th e curre nt density remai ns highe rnear the top than th e bottom, producing a stronger Lorentzforce driving th e vortices near the top. The voltage dif-ference across a pair of taps on the top or bottom directlymeasures th e number of vortices pe r unit time passingbetween them. A voltage difference betwe en the top andbottom taps implies that th e vortices on th e top and bottommove at different velocities, which requires flux cuttingand reconnectin g. Such cutt ing an d reconnection shouldbe particularly easy in the decoupled limit of independentsheets of pancake vortices. If, however, th e vortex linesare disentangled, or the barriers to flux cutting (andsubsequent reconnection) ar e sufficiently high, then th emoving lines will remain intact in the field direction an dno voltage difference between the top and bottom t apswill be observed.

Figure 4b shows th e voltages across the top andbottom taps of YBCO crystal s wit h an d without th e planarpinning defects called twin boundaries. 13 Th e sharp dropin both voltages in the untwinned samples a t T m marksthe first-order freezing transition. The two voltages ar edifferent throughout th e flux liquid phase, indicating en-tanglement and low barriers to flux cutting.

The behavior of samples with twin bound aries is evenmore interesting. In these crystals th e plana r twinboundaries run parallel to the field direction and at 45to th e cur ren t direction. Such defects pi n vortices verystrongly at low temperatures an d occur in sufficient den-sities to destroy th e vortex lattice. Both the top andbottom voltages now go smoothly to zero a t a lowertemperature 7 1 ,, suggesting a continuous transition to aglassy sta te. Remarkably , however, th e voltages at theto p an d bottom of the sample become equal at a tempera-ture T th , which is well above th e point where th e resistancevanishes. This is also th e temperature a t which dissipa-tionless supercurrents begin to flow along the c-axis. Th esimplest interpretation is that th e correlat ed di sorderembodied in the twin boundaries has combed th e hairof the flux lines sufficiently to produce a disentangled fluxliquid in the temperature interval T } T^ is consistent with a theoryof correlated pinning, which predicts a continuous transi-tion to a uBose glass phase at low temperatures in the

J0MH

Solid 1

Elastic mmotion j

Plastic motion

Liquid

/ 1TEMPERATURE

presence of twin boundaries. 2 Wande ring vortex lineshave many other important consequences that can beexplored with an analogy between vortex trajectories andthe world lines of qua ntu m mechanical bosons, as sketched

in box 2 on page 43 .

Dynamic response of vorticesThe richness of fundamental behavior in the dynamicresponse of vortices matches that of their equilibriumphase diagram and is responsible for the recent and rapidemergence of vortex dynamics as an independent subjectof investigation. Experimenters and theorists are nowdeveloping tools that are sensitive to the driven motionof vortices, as well as to the ir equilibrium configurations.It is easy to set up a steady state of driven motion in afinite sample because vortices are naturally created anddestroyed as necessary at the sample boundaries. Vorticesdisplay an impressive range of dynamical behavior, in-cluding avalanches, stick-slip dynamics, thresholds formotion, nonlinear and hysteretic dynamic response andplastic/elastic solid motion. Vortex dynamics promises tobe a major growth area for its impact on superconductingproperties and for the insight it provides into drivendynamical systems in general.

An important early step in dynamical vortex studiesis the characterization of the various steady states ofmotion in a dynamic phase diagram, 14 illustrated in figure5. In the presence of pinning and at low temperature,there is a rest phase at low driving force and a movingphase above a critical de-pinning force. The dashed linein figure 5 indicates the crossover from rest to motion.

The nature of the motion above the critical force showswide variation. In systems with random pinning, themotion is nearly always plastic at driving forces just abovethe critical force; th at is, the vortices move at differentvelocities in different par ts of the system. 15 This createsrifts along certain planes, where the velocity changesdiscontinuously. Numerical simulations show tha t theplastic motion can take the form of rivers of movingvortices sliding between stationary "river banks," with therivers sometimes only as wide as a single intervortexdistance. 15 At higher driving force, a remarkable transi-tion to elastic motion occurs, and the vortices all movewith the same average velocity 14 (There may be localelastic distortions t hat heal as the motion proceeds.) The

same elastic forces between vortices that create the equi-librium lattice stabilize this dynamic state. Intuitively,such a transition seems reasonable, because at sufficiently

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dinamica de vortices

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