Table Of ContentPhase Transitions in Isolated Vortex Chains
Matthew J. W. Dodgson
Theory of Condensed Matter Group, Cavendish Laboratory, Cambridge, CB3 0HE, UK.
(Dated: January 11, 2002)
2 In very anisotropic layered superconductors (e.g. Bi2Sr2CaCu2Ox) a tilted magnetic field can
0 penetrate as two co-existing lattices of vortices parallel and perpendicular to the layers. At low
0 out-of-plane fields the perpendicular vortices form a set of isolated vortex chains, which have re-
2 cently been observed in detail with scanning Hall-probe measurements. We present calculations
thatshowaverydelicatestabilityofthisisolated-chainstate. Asthevortexdensityincreasesalong
n
the chain there is a first-order transition to a buckled chain, and then the chain will expel vortices
a
J in a continuous transition to a composite-chain state. At low densities there is an instability to-
wardsclustering, duetoalong-range attraction between thevorticeson thechain, and at verylow
3
densities it becomes energetically favorable to form a tilted chain, which may explain the sudden
1
disappearance of vortices along thechains seen in recent experiments.
]
n PACSnumbers: 74.60.Ec,74.60.Ge
o
c
- I. INTRODUCTION ture of the vortex lattice as a function of magnetic field
r
p angle,14,15 which is consistent with the crossing-lattice
u state rather than a single tilted flux-line lattice.13 In
The vortexsystem in layeredsuperconductors,1 which
s addition, the work of Koshelev explained quantitatively
t. includes the high-Tc cuprates, displays a rich set of the attraction of perpendicular flux lines that leads to
a physical phenomena, such as a thermodynamic melting
m transition,2 a pinning-induced disordering transition,3 the vortex-chain state in a certain field range. Apart
from more detailed work on the melting transition,4 and
- and various structural transitions at different angles.4,5
d on magnetization curves,5 these concepts have also in-
The average density and orientation of the vortices are
n spiredrecentscanningHall-probemeasurements16onthe
controlledbythemagneticfield,becauseeachvortexcar-
o vortex-chain state. These experiments are similar to
ries one quantum of flux, Φ = hc/2e. In this paper we
c 0 the Bitter decoration technique in that they only probe
[ areconcernedwiththevortexchainsthatappearinacer-
the field distribution emanating from the surface of the
tainregimeoftiltedmagneticfields. Thesechains,which
1 superconductor. However, they have the advantage of
consist of a high density of flux lines perpendicular to
v speed and resolution, and are not a “one-off” measure-
the layers,werefirst observedwith the Bitter decoration
7 ment, so that the system can be finely tuned to observe
9 technique by Bolle et al.6 A qualitative explanation fol-
different effects. These new experiments have shownun-
1 lowed shortly7 in terms of the proposed crossing-lattice
expected properties within the chain state. In particu-
1 state in tilted fields.8,9 This state consists of a lattice of
0 flux lines perpendicular to the layers crossed by a lat- lar, it is possible to tune to a field range where all of the
2 out-of-plane flux lines are arranged in chains. It is this
tice of flux lines along the layers. The in-plane lattice
0 “isolated-chainstate” that we will study in this paper.
is strongly distorted due to the anisotropy,10 and has a
/
at largespacingalongthelayers. Husesurmisedthatapos- We will use the following geometry. The z-axis is per-
m sibleattractiveinteractionbetweenthetwospeciesofflux pendicular to the layers, and the component of mag-
line would lead to a higher density of out-of-plane flux netic induction B determines the density of “pancake”
- z
d lines along chains, with an inter-chain separation equal vortices17 in every superconducting layer. The in-plane
n to the in-plane flux-line spacing. This picture seems to field is along the x-axis, and B gives the density of the
x
o be consistent with the experimental observations.11 The flux linesinthis direction. Thesein-plane flux lineshave
c
energy of the crossing-lattice state was considered by their centers in the spacing between layers, and so are
:
v Benkraouda and Ledvij,12 who found a transition from commonly called Josephson vortices.18 A z-directed flux
Xi a single lattice of tilted flux lines to the crossing-lattice line is made from a stack of pancake vortices, and con-
state as the tilt angle is increased for sufficiently large tainscirculatingcurrentsinthelayersuptoadistanceof
r
a anisotropy. Their work, however, neglected interactions the penetration depth, λ λ, from the vortex center.
ab
between the two crossing lattices. A Josephson vortex in the x≡-direction has an ellipsoidal
Interest in the crossing-lattice state has been revital- currentpattern,withthefluxandcurrentsdecayingover
ized in the last couple of years. Koshelev13 has shown the much larger distance λ = γλ in the y-direction.
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that the regime for crossing lattices is larger than previ- Here, γ is the anisotropy ratio, which is large for weakly
ouslyexpected. Thisisbecauseacorrecttreatmentmust coupled layers. The phase singularity of the Josephson
include the interactions between the perpendicular flux vortex is confined between two neighboring layers, with
lines, and a distortion away from ideal crossing lattices separationd,wherethephasedifferenceacrossthelayers
has a lower energy. Experimental evidence comes in the changes by 2π over a distance of the Josephson length
form of the unusual dependence of the melting tempera- γd.
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a / dJz23450000 buckled chainplus ejected stacks mixed buckled chain solated straight chain
ch i clusters of chain
FIG. 1: Optimal displacements of the vortex chain of pan-
10
cake stacks due to the crossing of Josephson vortices, using
the parameters appropriate for BiSCCO quoted in the text.
tilted chain
The figure takes the results from calculations for ach = 10λ, 0
and aJz =20d which for theisolated-chain state corresponds 2 4 6 8 10 12
toBx =53GandBz =0.8G.Thedisplacementsaremagni- ach / λ
fied by two for clarity, and thez-scale is not thesame as the
x-scale. FIG. 2: Phase diagram of the isolated vortex chain, show-
ingfourphases: Isolatedchainoftiltedstacks;Isolatedchain
of straight stacks crossing Josephson vortices; Isolated buck-
led chain; state with chains plus ejected vortices. Axes show
The most simple structure of the crossing-lattice
thespacingofpancakevorticesach andtheJosephson vortex
state is when an ideal triangular lattice (with spac- spacing aJz within the chain.
ing a = [(2/√3)Φ /B ]1/2) of pancake-vortex stacks
P 0 z
crosses a stretched triangular lattice of Josephson vor-
100
tices, with separation in the z-direction of a = n
[(2/√3)Φ0/Bx]1/2/√γ. Koshelev has shown how tJhze in- iahc n hain n
ttdmtpeireacirennaeaassnccniagtltseikiuoeatelsnhdassratwttpaobecateakhntanswecrraeleeeokffecienewna-cvttitpleeoilrdavrebtnesoeectaxvaelteadkltrarretgatitnhecsittcedaiehocic,esnekwrtn.seo1it3rtea(htrAnBisoadtznohtsfJih<gftoerhhsoleeΦeomprw0Jhd/otseλpshonea2ensp)niihtcdvtysahoeokoarines-fl B [G]x6800 detlit clusters of chai solated straight cmixed buckled chai
i
vortices. Thisleadstothehigh-densitychainsofpancake 40
stacks that are observed by Bitter decoration,6,11 with
an inter-chain spacing of aJy = √γ[(√3/2)Φ0/Bx]1/2. buckled chain
20
Eventually, for small Bz Φ0/λ2, all of the pan- plus ejected stacks
≪
cakeslieoverthe Josephsonvortices. Thisisolated-chain
state has a pancake-stack separation along the chains of
0
a =(Φ /B )/a , see Fig. 1. 0 1 2 3
ch 0 z Jy
B [G]
The crossing lattices must compete energetically with z
the more conventional tilted lattice of vortices. This is
FIG. 3: Same phase diagram as in Fig. 2 but with axes
quite similar to the vortex lattice in a continuous super- convertedtothethein-planeandout-ofplanecomponentsof
conductor, only there is a kinked structure along each magnetic field,assuming theJosephson vortices form aregu-
vortexwithaperiodicityL=d(Bx/Bz). Theanisotropy lar stretched triangular lattice.
of the currents mean that there is a stretched aspect ra-
tio of this lattice. In addition, this distortion may be
different from that expected from simple rescaling19 due recentscanningHallprobemeasurements.16Wefindthat
to the attraction of tilted vortices along the direction of the stability of the isolated-chain state is quite delicate.
tilt. This may lead to a “tilted chain state”,20 which We first note in Section II that the existence of a sta-
is distinct from the crossing chains mentioned above. ble crossing configuration in the isolated chains is only
Koshelev showed that for intermediate pancake densi- possible for large enough anisotropy, layer spacing, and
ties Bz > Φ0/λ2 there should be a transition at a small pancake separation. We also find that the energy of the
v(Balxue<oBfx∗B)xan=d Bthx∗e ≈cro0s.s0in1gBzlatbteictwesee(nBx∗th<eBtixlt)e.d lattice ihsaoslaatemdinchimaiunma,si.ae.f,utnhcetrieonisoafnpoapntcimakuem-stdaecnksisteypaorfaptaionn-
In this work we will concentrate on very small out-of- cakestacksalongachain,andtherewillbeaninstability
plane fields and B B < Φ /λ2 when all of the pan- towards clustering when the total density is low. As the
z x 0
≪
cakestacksbecomeattachedtothecentersoftheJoseph- density of pancake stacks along the chain increases (at a
son vortices, giving the isolated-chain state observed in fixeddensityofJosephsonvortices),thechainwillbuckle
2
(Section III), and then eject vortices at a critical mini- felt by a given pancake. We can therefore reduce this
mumseparation(SectionIV). Moresurprisingly,atvery many-body optimization problem to a one-dimensional
low pancake density the chain state may be replaced by problem, considering the displacement of a row of pan-
a chain of tilted stacks (Section V). These results are cakes in one layer under the potential due to the ideal
summarized in Figs. 2 and 3, where we plot the calcu- chain of pancake stacks (the corrections due to the dis-
lated phase boundaries for three first-order transitions: placements in other layers is small when the number of
clustering, buckling and tilting as well as a continuous pancakeswithlargedisplacementsismuchlessthanλ/d).
ejection transition. Finally, in Section VI we discuss the Within this scheme, the energy profile (per stack) for
extent to which these transitions have been observedex- displacingbyu arowofpancakesinthe nthlayerwhen
n
perimentally, and consider the effect of fluctuations due thereisa Josephsonvortexbetweenlayers0and1cross-
to finite temperature or quenched pinning disorder. ing the chain is,
We conclude this introduction with a note on pa-
rameters. The experiments we have referred to,6,11,16 ∆E (u )= Φ0dJyu +Vrow(u ). (1)
were performed on the extremely anisotropic cuprate n n − c n n em n
Bi Sr CaCu O (BiSCCO).Wherewemakeexplicitcal-
2 2 2 x
The first term here comes from the Lorentz force on
culations we will take the following appropriate param-
eters γ = 500, λ = 2000 ˚A, and d = 15 ˚A. This gives the pancakes due to the in-plane current density from
the Josephson vortex, Jy, and tends to pull the pan-
a Josephson length (the size of the non-linear core of a n
Josephson vortex) of γd=7500 ˚A λ. cakes away from their stacks. The form of this current
≫ will be discussed more in Section III, but the numeri-
cal value is taken from Ref. 28, e.g., the current in the
n = 1 layer (immediately above the Josephson vortex)
II. STRUCTURE OF THE ISOLATED VORTEX
is Jy = 2.28ε c/Φ γd. The second term is the attrac-
CHAIN 1 0 0
tive magnetic interaction of the pancake row with the
remainder of all the stacks in the chain. Clem showed
Koshelev has estimated the structure of a pancake for a single pancake,24 that this has the same interac-
stack that crosses the center of a single Josephson tion energy as the sum of a pancake with a full stack,
vortex.13 In the crossingconfigurationthere is a Lorentz Vstack(R) = 2ε dK (R/λ) plus a pancake with its anti-
em 0 0
force on the pancakes in a direction parallel to the imageVpc−pair =2ε dln(R/L),sothatfortheentirerow
em 0
Josephson vortex, which must balance the attraction we find
of each pancake to its stack. This distorts the pan-
cake stack, with the largest displacements by the two Vrow(u) = 2ε d[K (u/λ)+ln(u/2λ)+γ ] (2)
em 0 0 E
pancakes immediately above and below the Josephson
ja +u ja ja +u
co8r.e4.ε0Td(hλe/γddis)t2o/rltnio(3n.s5γgdiv/eλ)a.nHeenreerεg0ydg=ai(nΦ0o/f4∆πλE)×2d≈is +2ε0dXj6=0K0(cid:20) chλ (cid:21)−K0(cid:18) λch(cid:19)+ln(cid:20) jchach (cid:21).
−
the typical energy scale for pancake interactions. This
causes the pancake stack to be attracted to a stack [Euler’sconstantγ isneededtofix thezeroofenergyto
E
of Josephson vortices within the crossing-lattice state. that of fully aligned stacks, c.f. the small x expansionof
Koshelev’s resultuses a quadratic approximationfor the the modified Bessel function K (x)].
0
pancake-to-stack attraction. Recent work21 using the Such energy profiles for the n = 1 pancake row are
correct potential has shown that, while Koshelev’s es- shown in Fig. 4 for different values of a with the
ch
timate for the crossing energy is close to the full result, choice of the ratio γd/λ = 3.75 (a reasonable choice for
a stable crossing configuration only exists for extreme BiSCCO). For dilute chains there is a stable minimum
enough anisotropy, γ > γmin = 2.86λ/d. In this section (as expected from the results of Ref. 21), which deter-
we include the interaction between pancakes in different mines the optimal displacement. However, this stabil-
stacks separated by ach along an isolated chain. For fi- ity disappears once the separation is below the critical
nite ach we find a further reduction of the stability limit value of amchin = 5.3λ. There can be no stable isolated-
of crossing found in Ref. 21. We also find a minimum in chainstate for densities higher than this critical value.22
the energyofthe chainas a function ofstackseparation, Fig.4alsosuggeststhatthecrossingconfigurationisonly
which leads to a clustering instability when ach ≫λ. metastable,withalowerenergyatu1 =0.5ach,butthisis
Tocalculatethecrossingconfigurationofthechain,we notreliableasthesimpleLorentzforceargumentdoesnot
usethefollowingassumptions: First,weneglecttheeffect holdforpancakedisplacementsofsuchalargefractionof
ofinducedJosephsoncurrentsduetodisplacingpancakes thestack-separation. Forsuchdisplacementstheoriginal
fromtheirstacks(reasonableforlargevaluesofγd/λand Josephsonvortexbecomescompletely fragmented,anda
small enough displacements u < γd). Second, we utilize differentapproachisneeded. Infact,thecompetingstate
the long rangeof the remainingelectromagneticpancake here is a “tilted chain”, and in Section V we will calcu-
interactions,17 of order λ in the z-direction. There are late the energy of the tilted chain, to compare to the
many( λ/d 102)pancakesthatcontributetothecur- isolated chain state. (A third possibility is a soliton-like
rent dis∼tributi∼on in one layer, determining the potential structure23 that we will not consider here.)
3
3 a /λ
0 5 ch 10 15 20
-8
ach/λ= ∞ ] (a)
d0 -9
2 ach/λ=10.0 -3ε10 -10 attraction
[
h repulsion
εd0 u1 ach/λ=7.0 Ec-11
E(u) /111 ach/λ=5.3 2] -0.05 unstable
∆ ach/λ= ∞ -ελ0 -1 locally stable
5
0 -10
[ -1.5 globally stable
a /λ=10.0 )
Instability point for ach/λ=5.3 ch (Bhz -2
-10 1 2 u /λ 3ach/λ=7.0 4 5 fc-2.5 (b) B*z1
1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
B [G]
FIG. 4: Energy profiles for displacing a pancake row within z
a chain of pancake stacks, Vermow(u1), (dashed lines) and the FIG. 5: (a) The energy per pancake within the isolated
total energy change ∆E1(u1) with a Josephson vortex be- chain as a function of separation ach when aJz = 20d. Note
low the pancake row (full lines), for different values of the the presence of a minimum energy at ach = 8.2λ. (b) The
mpaunmcadkeis-asptapcekarsspfaocrinsgpaaccihn.gsNsmotaellherowthathneammchienta=st5a.b3λle.mEaincih- fsuanmcetiroensuolftt,hbeuotuetx-opfr-epslsaendeaflsutxhdeeennseitrygyBdze=nsΦity0/facchh(aBJyz.)Tashae
curve is only plotted up to u1 =ach/2. The fact that, for all thick line shows where the function has positive curvature,
values of ach the lowest energy is for large displacements, is required to ensure local stability. The entire function must
notaphysicalresult,astheLorentzforceargumentwillbreak be convex, and so for densities less than B∗1 the chain will
z
down when u1 is a significant fraction of ach. These results phase separate into clusters with ach =a∗ch1 and regions with
are for a fixedvalue of γd/λ=3.75. nostacks.
The displacements of the entire stacks are shown in homogeneous chain at very low densities.
Fig. 1 for a = 10λ, and a Josephson-vortex separation Toformallydescribethisinstabilityweshouldconsider
ch
of a = 20d (this corresponds to B = 53 G for our thefreeenergydensityasafunctionofout-of-planefield,
Jz x
parameters). All of the displacements contribute to the f(B ) = f (B )+ε B /Φ . Here ε is the line
z ch z stack z 0 stack
total crossing energy gain. We find that the crossing energy of a pancake stack, f (B ) = E (a )/a a d
ch z ch ch ch Jy
energy per pancake stack only lowers from ∆E ( ) and B = Φ /a a . In Fig. 5b we plot f (B ) using
× z 0 ch Jy ch z
0.21ε dforanisolatedstackto∆E (amin) 0.∞26ε≈d the same data in Fig. 5a. We also note that the thermo-
− 0 × ch ≈− 0
asthestacksreachthecriticalseparationamin. Therefore dynamically stable phase is determined by the minimum
ch
within the region of stable crossing, there is a pinning Gibbs free energy g(H ) = f[B (H )] B H /4π with
z z z z z
−
energytotheJosephsonvortexcentersperunitlengthof H = 4π∂f/∂B . By a geometric construction we see
z z
pancake stack, thatthepointB =B∗1 onf (B )whereastraightline
z z ch z
from the origin connects with the same gradient must
εJv−pin =∆E×/aJz 0.2ε0(d/aJz), (3) have g = 0, and there is a first-order phase transition
≈
between a Meissner phase with no pancake stacks and a
wherethelastresultiswithourparametersforBiSCCO. finite density of stacks with a = a∗1. All the points
ch ch
In Fig. 5a we plot the total energy per pancake at lower density B < B∗1 have g > 0 and so are ther-
z z
of the isolated chain Ech = 21 j6=0Vesmtack(jach) + modynamically unstable. In fact, similar arguments de-
(d/aJz)∆E×(ach). Note that therPe is a minimum at termine that fch(Bz) must always be a convex function.
a = 8.2λ, reflecting the fact that at large separations Thedottedlinerepresentstheregionwherethecurvature
ch
the pancake stacks attract each other. This unusual fea- of f (B ) is positive, meaning that the solution here is
ch z
turehasbeenexplainedbyBuzdinandBaladie26 bycon- always unstable.
sidering the distorted pancake stacks of Fig. 1 as the su- The straight line that determines B∗1 is given by
z
perposition of a straight stack plus a series of pancake– f = BδHz /4π where δHz is the change in the lower
c1 c1
anti-pancakedipoles. Thestraightstacksgivearepulsive critical field due to the attraction of pancake stacks
term, but this is exponentially small for a λ, while to the Josephson vortices. In Fig. 5b it is given by
ch
the dipoles give a weak attractive term that≫only falls δHz = 2.5 10−3Hz which is hardly measurable. In
off like 1/a2 (note there is some similarity to the at- reacl1expe−rimen·ts howec1ver, the geometry of the samples
ch
tractionbetweentilted flux lines20). Thereforethere is a oftenmakesdemagnetizationeffectsimportantsuchthat
net attractionat largedistances, whichwill destabilize a the average B becomes fixed to the external field H ,
z ext
4
rather than to H. This means that small values of B
z
are accessible, and that for B < B∗1 we may expect E uy
a coexistence of B = 0 and Bz = Bz∗1 phases (c.f. the stack rep
z z z 0.005 repulsion
intermediate state in type-I superconductors27). Alter-
natively we could describe this mixed regime in terms of
“clusters” of pancake stacks with separation a∗1. The d
ch ε0
rdeeltaetrimveinperdopboyrttihoenvoafluspeaocfeBtaz,kebnutupthbeysitzheeoseficnludsivteidrsuaisl yE(u)/ 0 total enEerrepg+y∆E×
clusters depends on the energy of the “domain wall” be-
tween the cluster of stacks and the region of no stacks,
compared to the magnetic energy cost of large clusters -0.005 crossing optimal
with the wrong flux density. Also, in the experimen- energy ∆E× displacement
tal situation one can have different pancake densities on
thedifferentchains,sotheinhomogenousstatemayhave
0 2 4 6 8
coexisting empty and filled chains. Our results for the
uy/λ
critical field separating the clustered phase from the ho-
mogeneous isolated straight chains are shown in Figs. 2 FIG. 6: The energy profile when we pull one pancake stack
and 3. away a distance uy from the isolated chain state, for ach =
5.5λ, and aJz = 25d. The geometry is shown in the inset.
Thedashedlineshowstheenergyfromthecrossingeventson
III. BUCKLING INSTABILITY WITHIN thedisplacedstack. Thedottedlineistheinteraction energy
VORTEX CHAIN ofdisplacedstackwiththeotherstacksonthechain. Thefull
line is thesum of these two contributions. Note thepresence
of two minima, one for uy = 0, and a lower minimum at
Itisimportanttorealizethat, whilethe isolated-chain uy =4.9λ.
stategainsenergyduetotheLorentzforceoftheJoseph-
son vortex currents, there is an energy penalty for the
pancake stacks to be close to each other. One conse- 8.81(n 1)y˜[y˜2 (n 1)2+2.77]
+ − 2 − − 2 . (5)
quence of this is a maximum density above which pan- [(n 1)2+y˜2+2.02]4
cake stacks will be ejected from the chain. This criti- − 2
cal density will be derived in the next section. Below When y = 0, this gives the results used in Section II,
this density, however, the chain can already react to the with J (0) = (2cε /Φ γd)C /n 1 , and C = 0.57,
n 0 0 n | − 2| 1
stack repulsion by buckling. Note that we will assume C =0.86, C =0.99, and C 1 for n>3.
2 3 n
≈
that the Josephsonvortices remainstraight,as in exper- We can now recalculate the crossing energy for an ar-
iments they are held in place by the interactions with bitrary distance between a pancake stack and the cen-
Josephson vortices in neighboring chains, and the chain ter of a Josephson vortex. Within the quadratic ap-
separationismuchsmallerthantheinteractionrangefor proximation used in Refs.13 and 28 one can easily solve
Josephson vortices, aJy γλ. In contrast, the interac- the crossing configuration for a single stack. Writing
≪
tions between pancake stacks on different chains are in ∆E (u ) = (Φ d/c)J (y)u + 1αu2, where we take
n n − 0 n n 2 n
the dilute limit, aJy ≫λ, and so are easily displaced. α = (ε0d/λ2)ln(λ/rw) with rw a short distance cut-off,
Tocalculatethisbuckling,weneedtoknowtheenergy we find the relaxation energy,
gain from a crossing event when the pancake stack is
displaced away from the center of a Josephson vortex. λ 2 1 ∞
∆E (y)= 2 [p (y/γd)]2. (6)
We therefore need the full current profile of a Josephson × n
− (cid:18)γd(cid:19) ln(λ/r )
vortex, which is a solution of the non-linear equations w n=X−∞
thatarisefromtheLondon-Lawrence-Doniachmodel.9,18
Koshelev takes r =u ( 0.29λ2/γd for y =0), and in
AnaccuratenumericalsolutionforaJosephsonvortexis w 1 ≈
Ref. 21 we have shown this to be a good approximation
describedinAppendixBofRef.25. Foravortexdirected
alongxˆcenteredaty =0andbetweenthen=0,1layers, to the result when the full form of the pancake-stack
interaction is used.
the current in the y direction can be written28 (ignoring
This result for the crossing energy as a function of
screening, i.e. n<λ/d and y <λ ),
c stack displacement uy is shown as the dashed line in
2cε Fig. 6, where there is one Josephson vortex for every
0
J (y)= p (y/γd), (4)
n n 25 pancakes along a stack. As might be expected, the
Φ γd
0
energy increases quadratically at small uy. More inter-
with p (y˜) = φ′ (y˜) the reduced superfluid momentum, esting is the fact that the crossing energy vanishes only
n n
where the phase has the form,29 as ∆E λ2/uy2 for uy γd (note that an ex-
×
≈ − ≫
ponential suppression will occur at extremely long dis-
φ (y˜) = tan−1 (n− 21) + 0.35(n− 12)y˜ tances uy > λc). In contrast, also shown in Fig. 6
n (cid:20) y˜ (cid:21) [(n 1)2+y˜2+0.38]2 (the dotted line) for a = 5.5λ is the repulsive energy
− 2 ch
5
0
(a)
y
-3 u
-0.5
d
ε) / 0-4 -2ελ]0 -1 straight chain buckled chain
3b-cy10E(u -5 decreasing ach -5f(B) [10chz-1-.25
-2.5
-6
B*z2- B*z2+
-3
0 2 4 6 8 0 0.5 1 1.5 2 2.5 3 3.5
uy/ λ Bz [G]
8 FIG. 8: Energy density of the buckled-chain state with
aJz =20dasafunctionofout-of-planefieldBz. Alsoshownis
λ 6 theresultforthestraightchainfromFig.5b. Thetwopoints
y / 4 joined by the straight line have the same Gibbs free energy,
u and those densities between these points are thermodynami-
2 callyunstabletowardsamixedphaseofstraight andbuckled
(b)
chains.
0
2 3 4 5 6 7
a / λ
ch
becomes unstable if a < 2.65λ. Finally, to calculate
ch
FIG.7: (a)Energyofthebuckledchain(seeinset)asafunc- the energygainfromcrossingthe Josephsonvortices,we
tion of the buckle parameter uy, for stack separation along
use the result (6) giving the total energy of the buckled
the chain ach/λ=5.2, 5.4, 5.6, 5.8, and 6.0. For the separa- chain,
tion ach/λ = 6.0 there is only one minimum at uy = 0. For
ach/λ=5.8thereisasecondminimumatuy =3.3λ. There- d
mainingcaseshavetheirlowestminimumatfiniteuy. (b)The Eb−c(uy)=Eb−c(uy)+ [∆E (0)+∆E (uy)]. (8)
size of the optimum buckling distortion uy as a function of 0 2aJz × ×
opt
ach.
In Fig. 7a we plot this energy for a /λ from 5.2 to 6.0,
ch
with a = 25d, showing how the minimum energy of a
Jz
cost for displacing one pancake stack awayfroma chain, buckledchaincrossestheenergyofastraightchainasach
E = Vstack (ja )2+uy2/λ .Thiscontribu- decreases. Notethattheminimumforafinite uy isquite
tiorenpvanPishj6=es0exepmone(cid:16)nptiallycwhhenuy >√(cid:17)achλ,andsothe sehxatellnotwo,fabnudcksolinwgeamloignhgttehxepcehcatinlardgueefltuocrtaunadtioomnsvinortthexe
totalenergyis alwaysnegativefor largeenoughuy. This pinning, which is discussed in Section VI. Also shown in
gives the possibility of two minima in the total energy Fig. 7b is the optimal displacement uy as a function of
opt
(see the full line in Fig. 6), and a first-order transition a ,whichhasasmallestvalueofuy =2.3λatthefirst
wherethedisplacementjumps toafinitevalueasa de- ch opt
ch appearance of (meta)stable buckling at a = 5.8λ and
ch
creases. While we havestartedby consideringdisplacing
then increases at higher densities.
a single stack from the chain, it is this instability that
It has been shown in Section II that one should plot
drives a buckling of the whole chain.
the energy density of the chain as a function of the out-
Toestimatethebucklingtransitionweuseavariational
of-planefluxdensityB ,inordertodeterminethestable
z
approach for the buckled configuration (See the inset of
thermodynamic phases. Such a plot is shown in Fig. 8
Fig. 7a). This configuration is only characterized by a
comparing the straight and buckled chains. It shows
single displacement parameteruy, and so we can look at that the isolated straight chains at B = B∗2− 1.3 G
the energy profiles in uy for different a /λ. First, the z z ≈
ch has the same Gibbs free energy as the buckled-chain
energy per pancake is calculated before crossing relax- state at B = B∗2+ 2.5 G. Therefore at fields with
ations, B∗2− < Bz < Bz∗2+ ≈the thermodynamic phase will be
z z z
a coexistence of straight and buckled regions, with the
ε d
Eb−c(uy)= 0 K (i j)2a2 +(y y )2/λ relative proportion linearly dependent on B . Note that
0 N Xi6=j 0(cid:18)q − ch i− j (cid:19) Bz∗2− is only slightly higher than Bz∗1 so tzhat there is
(7) only a small regime where pure isolated straight chains
with y = 0 for i even, and y = ( 1)(i+1)/2uy for i are the stable phase. The same procedure has been fol-
i i
−
odd. Next we checked for the stability of crossing, as lowed for a range of a /d, and the resulting boundaries
Jz
in Section II, and found that the crossing configuration to the mixed buckled chains are shown in Figs. 2 and 3.
6
IV. EJECTION OF PANCAKE STACKS FROM L
catpBohorfozeemWs.trinhtpeTeeoeoihtnsssdeiotltaoaiweolctucreascotdtteonaeeantrndtlisadenpiotduatwtnerIoihcraShuteemhOesarieseopLesotfchiAemlhaaarpTastpafoieeErlnnfdeatsDct-cr.machtahkiTiCnoosaeshdnHitiniestrsAltoiaosafmnftIncoasNtkorihtbtsdeteieeohntalnpweoswateieasnienltcnltferauhrsatkghnepheyocpidtwsseeoitocndfatnoshtcmiohkatonyets-f 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(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)aJz
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)a(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
chains. The simple model assumes that the chain spac- ch
inga ,andthe spacingbetweenpancakestacksis much
Jy .
greater than the penetration depth λ. In this case we FIG.9: Configuration ofthetiltedchain,wherepancakesin
can separate the total interaction energy density of pan- a stack are displaced from layer to layer byL=ach/aJz.
cake stacks in the formf =f (ν )+f (ν ), where
tot ch ch dil dil
ν and ν are the (2D) densities of pancake stacks on
ch dil
and off the chains respectively, and the total density is V. CROSSING-TO-TILTED TRANSITION
ν +ν = ν = B /Φ . The separation of pancakes WITHIN ISOLATED CHAINS
ch dil tot z 0
along the chain is a = 1/a ν . The chain energy
ch Jy ch
fch(νch) is to be calculated as in Sections II and III for In this section we calculate the energy of a chain of
straightandbuckledchainsrespectively. Wewillseethat tilted vortices,andcompare to the chainof crossingvor-
the phase transition to the composite state occurs when tices. The tilted chain is an alternative configuration
there is a minimum in fch(νch), and so near this transi- to the crossed chain with the same density of pancake
tion we will expand, and Josephson vortices (see Fig. 9) where the tilt angle
is determined by tanθ = L/d = a /a . These tilted
f (ν )=ǫ′λ2(ν νc )2. (9) ch Jz
ch ch ch− ch stacks have an increased energy of pancake interactions
The density of pancake stacks not on a chain is small but there may be an energy gain from replacing fully
nearthetransition,sothattheinteractionenergydensity developed Josephson vortices with many short Joseph-
is simply (using the limit K (x) π/2xe−x and only son segments of length L < γd. To calculate the energy
0
≈
including nearest neighbors), p of the tilted chain, we use the linear approximation of
the London-Lawrence-Doniach model,9,25 where we ig-
fdil(νdil)=3ε0√2πλνd−il5/4e−1/λνd1i/l2. (10) nore the non-linear effects from regions with large phase
differences across neighboring layers. This approxima-
Note that in the small ν limit, all terms in a power
dil tion is justified as long as the length L is much smaller
series expansion of f (ν ) are zero, i.e. the function is
dil dil than the Josephson length γd and the pancake separa-
extremely flat. For this reason the critical total density
tion a . Note that the condition L<γd corresponds to
ch
at the transition is only determined by the minimum in
B /B < γ, which is the case we are interested in. The
f (ν ) at νc . For ν < νc , all of the pancake stacks x z
ch ch ch tot ch energy due to pancake interactions in the linear approx-
are on chains. At densities just above νc the energy is
ch imation is,
minimized with a dilute off-chain density,
1 d2k dq 1+(γλ)2(k2+q˜2) S 2
νdil =δνtot− 3√42πεε0′ λ5/21δν1/4e−1/λδνt1o/t2. (11) Epticlt = 8π Z (2π)22π[1+λ(cid:2)2(k2+q˜2)][1+(γλ(cid:3))2|k2z+| λ2q˜2]
tot (12)
The second term is extremely small for the fields we are with Sz(k,q) = Φ0d j,ne−iqnde−ikx(jach+nL) and q˜2 =
interested in B Φ /λ2, so that we can say in the (2/d)2sin2(qd/2). ThPe integral should be cut off due to
z 0
composite state a≪t ν > νc there is a fixed density the pancake cores at k > π/ξ and due to the layered
tot ch ab
on the chains ν = νc and the remaining density is structure at q >π/d. Similarly, the Josephson segments
ch ch
ν = ν νc . The phase boundary νc between the interact with an energy contribution,
dil tot − ch ch
compositestateandtheisolated-chainstateiscalculated
from the minimum of fch. In Figs. 2 and 3 this bound- Etilt = 1 d2k dq |Sx|2 (13)
ary is plotted, but is not distinguishable by eye from Jv 8π Z (2π)22π[1+(γλ)2k2+λ2q˜2]
the transition between mixed buckled and pure buckled
chain. Therefore the pure buckled chain has a very nar- withSx = 2kΦx0 sin kx2L j,ne−iq(n+21)de−ikx[jach+(n+12)L].
row range of existence. It is worth stating that an ejec- The integrals ca(cid:0)n al(cid:1)lPbe done exactly, leaving a sum
tion transition has been observed experimentally16 with over the values of k = 2πm/a . We have evaluated
x ch
results that seem to be consistent with the above, al- these sums numerically to find the energy of the tilted
though a quantitative analysis of the transition has not chain. This is shown for a = 25d as the dotted line
Jz
yet been published. in Fig. 10. Also shown as the full line is the energy per
7
30 VI. DISCUSSION
pancake dominated J-vortex dominated
25
We now discuss the extent to which these transitions
have been observed experimentally. In the recent scan-
20 Tilted chain
ningHallprobeexperimentsofGrigorenkoetal.,16 there
d
ε015 does seem to be a sharp transition as a function of Bz
between a composite state of chains with a dilute lattice
/
E and the isolated chain state, as derived in Section IV.
10 Isolated chain At lower fields there is some evidence for buckling and
clustering (Sections II and III) although the influence
5 5 × ∆E of pinning disorder may be contributing to this. Finally,
Grigorenko et al. report a strange transition at very low
0
B where the chains are replaced by faint, homogeneous
z
“stripes” of flux. These stripes would be consistent with
0 10 20 30 40 50 60
a / λ the flux distribution from a tilted chain, and it is likely
ch
thatatransitionfromtheisolated-chainstatetoatilted-
FIG.10: Energyperpancakeofthetiltedchain(dottedline) chain state (as found in Section V) takes place. Experi-
and the chain of straight pancake stacks crossing Josephson mentally,alargejumpinB (i.e. inthepancakedensity)
z
vortices,foraJz =25d,plottedagainstthepancakeseparation isseenatthetransitiontotiltedchains;unfortunatelywe
ach/λ. Also shown (dashed line) is the difference of the two were not able to calculate the jump in this paper, as it
energies, which crosses zero at ach =45λ. requiresanaccuratetreatmentofnon-lineareffectsinthe
smallB limit. Other experimentalfeatures arethat the
z
tiltedchainsaremuchstraighterthanthecrossingchains,
pancake of the crossing chain with the same density of and less pinned. This is consistent with the fact that
pancakes and Josephson vortices, Ecross = Ech+EJcvross, the tilted chains are really part of a 3D line lattice with
where Ech is defined in Section II and stronger interactions that dominate over pinning disor-
der, whereas in the crossing case the pancake stacks are
in the extreme dilute limit, and therefore easily pinned
ε da λ ja
EJcvross = γ0 ach 1.55+ln(cid:18)d(cid:19)+ K0(cid:18) λJz(cid:19). in random low-energy sites.
Jz Xj6=0 Finally, we briefly discuss the effect of fluctuations.
(14) The calculations in this paper have found the lowest-
The tilted chain has a lower energy from Josephson energy vortex configurations of the chains. In reality
vortex contributions (dominant at low density) but a there will be distortions of these states due to thermal
higher energy from pancake vortex contributions (dom- fluctuations and random pinning to inhomogeneities in
inant at high density), and there is a first-order tran- the underlying crystal. It is well established that ther-
sition between the isolated chain of straight stacks and mal fluctuations can lead to a melting of the vortex
the chain of tilted stacks. In fig. 10 it is shown that crystal.1 A melting transition to completely decoupled
for ach < 45λ (higher density) the preferred state is the layers is studied for Bx = 0 in Ref. 31, involving short-
crossing chain, while for ach > 45λ (lower density), the wavelengthfluctuationsofpancakestackswithkz >π/λ.
lowest energy is for the tilted chain. Note that Savelev While there will be some modification in the presence of
etal.,30 alsofinda“reentrant”transitionbacktoatilted Josephsonvorticesintiltedfields,inthelowout-of-plane
lattice for fields close to the ab-plane, when λ>γd. fields studied here, B Φ /λ2, this melting will take
z 0
≪
AstricttreatmentshouldfollowSectionsIIandIIIand place at a high temperature, close to the Berezinskii-
consider the points of the same Gibbs free energy den- Kosterlitz-Thouless transition32 at T = ε d/2. If we
BKT 0
sity. This is not possible within our level of treatment, consider long wavelength fluctuations, k 0, then we
z
→
however, as we would need accurate results at very low mayexpectsomekindofmeltingofthechainatverylow
B < B /γ where non-linear effects in the tilted chain fields due to the exponentially small interaction of pan-
z x
are important. Here, we can only show that there must cakestacks. However,themechanismofsuchatransition
be a low-density transition to the tilted chain, but we must be different from the “entanglement” proposed in
cannotplotthecoexistenceboundaries. Evenso,weplot the original derivation of low-field vortex lattice melting
the line where the energies cross in Figs. 2 and 3 sepa- by Nelson.33 Considering the nature of the crystalline
rating the clustered chain from a tilted chain phase. In order of the isolated vortex chain, we should recognize
realexperimentsthereisafinite2DdensityofJosephson the two-dimensional nature of this state. A simple con-
vortices,whichcannotbeconsideredisolatedinthesense sequence is that thermal fluctuations at long wavelength
of the pancake vortex chains (the interaction rangeλ is willleadtoaquasi-long-rangeorderedstate. Itmaythen
c
greaterthan the chainspacinga ). Thereforeafull cal- bepossibletohavea2Dcontinuousmeltingtransitionvia
Jy
culation of the 3D tilted lattice requires a minimization the unbinding ofdislocations. However,a dislocationfor
over the aspect ratio, which we leave for a future work. this system of stacks corresponds to a stack that termi-
8
nates in the middle of the chain, and this should cost an thechainssothatthereisonlyashortrangeorder. There
energy linear in the system size, rather than the usual will also be significanttransversedisplacements, and the
logarithmic energy of a dislocation. This will suppress a buckling effect should be enhanced by disorder.
dislocation unbinding transition.
In the experiments of Grigorenko et al. the pinning It is a pleasure to thank Simon Bending and Sasha
disorder seems to be more important in disturbing the Grigorenko for enthusiastically introducing me to the
chainstatesthanthermalfluctuations. Thefactthatthe world of chains and crossing lattices, as well as Dima
pancake stacks are observed in fixed positions is due to Geshkenbein for useful discussions and Alex Koshelev
pinning(otherwisethermalfluctuationswouldsmearout for demonstrating how to calculate the currents near a
theaveragedensityinthechainstate),andthisalsotends Josephsonvortex. TheauthorissupportedbyanEPSRC
to disorder the chains. On general grounds we expect Advanced Fellowship AF/99/0725. Work at the ETH-
pinning-induced wandering of the pancake stacks within Zu¨rich was financially supported by the Swiss National
Foundation.
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9
