Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors

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520 G. Blatter and V.B. Geshkenbein

with ∇·A = 0; the expressions ∇

(2)

, 

(2)

,andA

(2)

de-

note the planar components of ∇,,andA).Equation

(12.101)is equivalent to the current conservation law

∇·j =0.

12.5.2 Josephson Vortices

We considera magnetic fieldin the ab-planepointing

along the y-directionand investigate the structureof

an individual Josephson vortex aligned with the y-

axis [61–63,194].We choose a gauge with A

=0and

, A

depending only on x and z. Subtracting the

phase equations (12.101) for n +1andforn from

each other provides a coupled set of equations for

the gauge invariant phase differences ¥

n+1,n

[78],



n+1,n

=2sin¥

n+1,n

−sin¥

n+2,n+1

−sin¥

n,n−1



sin ¥

n+1,n

. (12.103)

Outside a core region with dimensions  and d the

non-linearity and discreteness of the above equa-

tionsareirrelevantand(12.103)canbeapproximated

by the continuum differential equation



+ "

−





¥ (r)=0. (12.104)

Indeed, it is easy to see that the screening current

density along thez-axisin thecontinuumanisotropic

description,j

=−(2e/m)|¦ |

˜' ≈ c¥

"/8

becomes equal to the Josephson current density j

a distance x ∼ , hence for x >  the linearized

continuum approximation provides a good descrip-

tion of the situation. On the other hand, for small

distances x <  a further increase in the current

density j

is cut off by the condition |j

| < j

and the

discreteness of the problem becomes relevant.

Equation (12.104) tells us that the driving (gauge

invariant) phase differences vanish due to screen-

ing on a length /" along the x-axis and on a scale 

along the z direction.Since the phase difference is the

quantity driving the currents we obtain a magnetic

size /" (along x)and (along z) for the Joseph-

son vortex, see Fig. 12.5. On these magnetic length

scales the Josephson and Abrikosov vortices in an

anisotropic superconductor are roughly equivalent.

Within the phasecore [63] with dimensions  (along

x)andd (along z) we have to take the non-linearity

and the discreteness of (12.103) into account. On

these scales the phase is changing rapidly and the

current density j

reaches its maximum value j

at a

distance ∼  away from the axis of the vortex [194].

Furthermore, within this region the order parame-

ter is weakly suppressed in the layers adjacent to the

axis of the Josephson vortex: Ignoring thecoupling to

other layers, the Ginzburg–Landau equation for the

order parameter ¦

(¦

∞

=1)inthen -th layer reads,



−i'

∇

(|¦

)+|¦

| − |¦

= 0 ; (12.105)

separating into real and imaginary parts we obtain

for the real part



[

(2)

−(∇

(2)

)

]+1−|¦

|¦

| =0. (12.106)

If a Josephson vortex directed along the y-axis is

present between the n-th and the n +1-th layer

Fig. 12.5. Josephson vortex in a strongly layered superconductor. The usual normal core of the Abrikosov vortex (di-

mensions " and  along z and x , respectively) is replaced by the phase core (dimensions d and  = d/" along z and

x, respectively) within which the non-linearity and the discreteness of the problem is relevant. The region outside the

phase core is roughly equivalent to the corresponding regime in an Abrikosov vortex with screening currents extending

adistance along z and /" along x

12 Vortex Matter 521

the phase '

(x) will rapidly change on the scale 

and we can approximate the derivative of the phase

≈ 1/. Combining this estimate with (12.106)

we obtain a suppression of the order parameter ¦

the order of ı|¦

|≈(/)

at the center (x =0)of

the Josephson vortex: contrary to the Abrikosov vor-

tex where the large current ﬂow near the core leads to

a complete suppression of the order parameter, the

corresponding order parameter suppression in the

superconducting layers is only weak in the presence

of a Josephson vortex.

Note that the set of equations (12.103) for the

gauge-invariant phase differences ¥

n+1,n

can be gen-

eralized to an arbitrary field direction by substituting

the derivative @

with the planar Laplacian @

and

completing the resulting equations with appropriate

boundary conditions describing the singularities in

the phase field produced by vortices penetrating the

layers [195]

∇

(2)

∧ (∇

(2)

n+1,n

) ≡ @

n+1,n

− @

n+1,n

=2







ı(r − s

n+1 

)−ı(r − s

n

)



, (12.107)

where s

n

denotes the position of the singularity in

the phase '

corresponding to the  -th vortex in the

n-th layer.Notethat a straight vortex with s

n

= s

n+1 

produces no driving singularity for the phase differ-

ences ¥

n+1,n

, i.e., no screening currents crossing the

layers are set up. It is quite remarkable that such a

closed set of equations involving only the gauge in-

variant phase differences ¥

n+1,n

can be found.

The Josephson vortex in a layered superconduc-

tor involves two length scales along the x-axis, the

core size  = d/" and the magnetic size 

= /".

This has to be contrasted with the Josephson vor-

tex in the junction between two weakly coupled bulk

superconductors, for which there is only one charac-

teristic length scale 

=[c¥

/16

j

]

1/2

,with the

penetration depth of thebulksuperconductorsand j

the coupling current density of the junction.The dif-

ference can be understood in the following way: Con-

sider two bulk superconductorsjoined via a junction

with a coupling j

.Weplaceavortexlineintothe

junction and analyze its evolution from anAbrikosov

vortex at strong coupling j

≈ j

to a Josephson vor-

tex at weak coupling j

 j

[196,197].As soon as the

translational symmetry is broken by the junction,the

structure of the Abrikosov vortex is modified as the

zero in the order parameter disappears and is re-

placed by a finite jump across the junction (this is

easily understood by solving the GL equation ¦ ≈

0 near the singularity with the boundary condition

− ¦

−

= l@

−

, ¦

=(x +iy) ± l/2; the

length l is related to the transparency of the junction,

here chosen along the y-axis). Decreasing the cou-

pling j

belowj

the normalcoreof theAbrikosovvor-

tex is transformed into a phase core extending over a

distance 

≈ (j

) >  along the junction where

the phase of the order parameter changes by ∼ 2.

As the current across the junction is limited to a value

< j

the order parameter suppression is reduced,

ı¦ ∼ (j

)

, as compared to the complete order

parameter suppression in the Abrikosov vortex core.

With decreasing coupling j

the phase core expands

and the suppression of the order parameter becomes

small. Decreasing the coupling j

below the critical

value j

≈ j

/ the phase core expands beyond the

magnetic screening length  and the transformation

to the Josephson vortex with only one length scale



describing the phase, the currents, and the field

across the junction has been completed. Upon a fur-

ther decrease in j

the order parameter suppression

then is reduced to the value ı¦ ∼ (j

)/.

The analogous discussion for a layered supercon-

ductor produces quite a different result: In a layered

material, the screening currents ﬂowing perpendic-

ular to the planes have to cross not only one sin-

gle junction but have to overcome the large num-

ber of /d junctions. Upon decreasing the coupling

strength between the layers the screening current

density along the c-axis is also reduced such that

the current pattern of the original Abrikosov vortex

immediately starts to expand along the junction.As

a result, the ratio between the magnetic extent of the

vortex and the size of its phase core remains always

the same, /

= d/ = const, independent of the

coupling strength,and we always keep the two length

scales  and 

describing the core and the magnetic

size of the Josephson vortex.

From the abovediscussion we can understand that

a Josephson vortex is very similar to an Abrikosov

522 G. Blatter and V.B. Geshkenbein

vortex, however, with a different core size and struc-

ture.Thisconclusionisalsoconfirmedbythecal-

culation of the Josephson vortex line energy [61,62]

which is obtained through an integration of the ki-

netic energy of the currents



≈ ""





+1.55



, (12.108)

where the inner and outer cutoff lengths in the log-

arithm are provided by the magnetic lengths (

, )

and the dimensions of the phase core (, d)ofthe

Josephson vortex (the constant 1.55 is found in a

more accurate analysis, see [198]). The correspond-

ing result for an Abrikosov vortex in an anisotropic

superconductor and directed along the planes is

[125] 

= ""

[ln(/")+0.497]; note the different

temperature dependence for these two results aris-

ing from the substitution of 

(T)="(T)bythe

layer distance d under the logarithm. The line en-

ergy (12.108) defines the lower critical fieldalong the

planes H

(# =0)=4

/¥

andthe (in-plane)elas-

tic tension of a Josephson vortex, "



= 

.Finally,the

viscous drag coefficient depends on the core struc-

ture and differs only by a numerical factor from the

corresponding (Bardeen–Stephen) expression for an

Abrikosov vortex: substituting the appropriate di-

mensions for the phase core the result reads [199]

(see also [194])



≈ 1.8

2

"

, (12.109)

with 

the normal-state resistivity along the c-axis

and where we have assumed the relation 

= "



between thein-plane andc-axisresistivity (thecorre-

sponding Bardeen–Stephen result for the Abrikosov

vortex is 



= ¥

/2

"

). In high-T

super-

conductors the resistivity–anisotropy relation is vi-

olated below T

and 

/

< "

; in this case the

in-plane dissipation is enhanced and we should re-

place 

in (12.109) by the expression 

/(0.79 +

0.21"



/

) ≈ 4.76

12.5.3 Pancake Vortices

We consider a magnetic field directed along the c-

axis and discuss the structure of a vortex and its

planar constituents, the pancake vortices [65]. In

this context it is interesting to understand the sim-

ilarities and differences in the vortex structure for

the three cases of an isolated thin film [200], an un-

coupled stack of superconducting layers with j

[62, 64, 65, 201,202], and a layered superconductor

with j

> 0 [64,195,201,203]. We start with the sim-

plest case of a 2D thin film of thickness d and choose

 to denote the penetration depth of the correspond-

ing bulk material; the effective penetration depth of

the film is 

eff

=2

/d. The solution for a vortex

positioned at the origin of the coordinatesystem has

been given by Pearl [200,204]: The London equation

(cf. (12.99); we write A

(2)

= A)



A = dı(z)(a + A) , (12.110)

with the source term

a =

2

∇' =−

2

ˆz ∧ R

, a

=i¥

ˆz ∧ K

(12.111)

is solved with the Fourier AnsatzA(r)=



k/(2)

× A

exp(ik · r),

=−

d(a

+ A

)



, with



R A(R, z =0)e

−iK ·R

, (12.112)

and an integration over k

provides the result A

/(1+

eff

K).Inserting this result back into (12.112)

we obtain

=−2i¥

ˆz ∧ K

(1 + 

eff

(12.113)

and transforming back to real space making use of

the cylindrical symmetry we find the vector potential

(R, z)=¥

∞



2

(KR)e

−K|z|

1+

eff

, (12.114)

with J

(x) the Bessel function of integer order. The

associated magnetic field asymptotically resembles

that of a positively/negatively charged magnetic

monopole generating a magnetic ﬂux ¥

in the up-

per/lower half space, see Fig. 12.6,

B(r  

eff

) ≈

2

|z|

. (12.115)

12 Vortex Matter 523

Fig. 12.6. Pearl solution for a vortex in a superconducting

film with a magnetic field taking asymptotically the form

of a positively/negatively charged magnetic monopole in

the upper/lower half space

The magnetic ﬂux crossing the film within a circle

of radius R is

¥ (R) ≈ ¥

⎧

⎪

⎨

⎪

⎩



eff

, R  

eff

1−



eff

, 

eff

 R ,

(12.116)

andapproachesthe unit ﬂux quantum ¥

at large dis-

tances. The circular current density J

(R)(perunit

length) decays algebraically with distance and gen-

erates (via the Lorentz force) a screened logarithmic

interaction V

int

(R) between a pair of Pearl vortices,

(R) ≈

4



eff

⎧

⎪

⎨

⎪

⎩



eff



eff

int

(R) ≈ 2"

⎧

⎪

⎨

⎪

⎩



eff

, R  

eff



eff

, 

eff

 R .

Next, we discuss the case of an array of uncou-

pled (i.e., j

= 0) parallel superconducting layers

where the interaction is of electromagnetic origin

[62,64,65,201,202].Ignoring small variations in the

vector potential on the scale d we rewrite (12.99) in

the form (we use that A

(2)

= A)

[

 −1]A =

d¥

2



ı(z − nd)∇'

. (12.117)

An individual vortex line threading the stack is char-

acterized by the positions S

of the vortex cores

within each layer. The gradients of the phase fields

∇'

(R)=−ˆz ∧ (R − S

)/|R − S

drive the screen-

ing currents in the individual layers, which in turn

are coupled electromagnetically through (12.117);

hence, screening currents in response to a pancake

vortex in the n-th layer will be set up within the other

layers of the stack.The linearity of (12.117) allows us

to reduce the problem to the analysis of the elemen-

tary building block of the vortex line, the pancake

vortex [62,64,65,201,202].We consider a single pan-

cakevortexwithitsdrivingfield∇'

(R)=−ˆz∧R/R

placed at the origin of the n-th layer (positioned at

z = 0) and with all the other driving terms vanishing.

Transforming (12.117) to Fourier space we find the

vector potential

=−id¥

ˆz ∧ K

(1 + 

)

(12.118)

and transforming back to real space using cylindrical

symmetry we obtain the solution

(R, z)=¥

2

∞



2

(12.119)

(KR)e

−

√

1+

|z|/

√

1+

The associated magnetic field (see Fig. 12.7)

(r)=

4

−r/

(2)

(r)=

4

d R

signz (12.120)



−|z|/

−

|z|

−r/



differs quite drastically from the monopole-likePearl

solution: The screening effect due to the other su-

perconducting planes in the stack squeezes the field

524 G. Blatter and V.B. Geshkenbein

into a layer of width  along the z -axis and the mag-

netic field has to escape parallel to the layers rather

than spread out uniformly over the entire solid an-

gle. This field redistribution due to screening has

important consequences: First, the squeezing of the

field reduces the ﬂux threading the central layer n to

¥ (R)=¥

(d/2)[1 − exp(−R/)] hence the current

ﬂow is not able to screen the driving phase-field in

the central layer. Indeed, the current density of the

central layer decays like 1/R to all length scales,

(R, z =0)=

4



eff



eff



1−

2



1−e

−R/





(12.121)

and therefore the logarithmic dependence of the in-

teraction potential V

int

between two vortices placed

in the same layer persists to infinity,

int

(R, z =0)=±2"

d ln

+ O(d/) ,

 < R < L , (12.122)

with L the system-size cutoff (the − (+) sign refers to

equally (oppositely) charged vortices; note that the

total energy includes an additional self-energy term

d ln(L/), with the bulk planar coherence length

 serving as a short distance cutoff).

Fig. 12.7. Pancake vortex in a layered superconductor with

vanishing interlayer Josephson coupling. The screening

currents present in the neighboring layers squeeze the

magnetic field into the planar direction. The screening cur-

rent density in the central layer decays like j ∝ 1/R to all

scales

In order to find the general expression for the

interaction energy between pancake vortices (po-

sitioned at S

n

in layer z

= nd)weexpress

the current and magnetic field terms in the free

energy (12.97) through the current density j =

−(c/4

)[(¥

/2)d



ı(z − nd)∇'

+A ] and the

vector potential A, in Fourier space,

=−

4

id¥

ˆz ∧ K



1+



n,

−iK ·S

n

−ik

, (12.123)

=−

id¥

ˆz ∧ K

1+



n,

−iK ·S

n

−ik

; (12.124)

combining terms we arrive at the free energy

F =

8



(2)

1+

(12.125)



n, ,m,

−iK ·(S

n

−S

m

)

−ik

(n−m)d

In order to make contact with the expressions (12.62)

and (12.80) describing interacting vortex segments

in the continuous anisotropic situation one has to

take the appropriate limit " → 0 and account for the

continuity of vortex lines via the relation ∇·s =0

(we remind the definition s



= r



+ u



of the vor-

tex positions), in Fourier space, K · S

=−k

and hence S

˛,k

⊥˛

⊥ˇ

ˇ,−k

= K

− |K · S

−|k

.Theterms∝|s

in the freeenergy

(12.62) then combineinto a term |s

(1+

)

(cf.(12.126)),while those∝ (|s

+|s

)comewith

a prefactor 1/(1+

+

) which vanishes in the

limit " → 0.

An individual pancake vortex involves the energy

8



(2)

(1 + 

)

= "





−

2





, (12.126)

where the first term is due to the driving term ∝∇'

and involves the core size  as a short scale cutoff,

12 Vortex Matter 525

while the second term is due to screening. A vortex–

anti-vortex pair in the same layer (n) and separated

adistanceR costs an energy

+−

(R)=

4



(2)

[1 − cos(K · R)]

(1 + 

)

=2"



1−J

(KR)



1−

d/2

√

1+



≈ 2"

d ln



, (12.127)

reproducing the result (12.122) above. Placing a sec-

ond pancake vortex (with equal phase turn) in layer

m we obtain the expression

(R, z)="





(KR)

−

√

1+

|z|/

√

1+

, (12.128)

with R = |R

− R

| and z = z

− z

=(m − n)d.

In order to handle the (logarithmic) divergence at

small K (cf. the correction term in (12.126)) we de-

termine the (attractive) force f (R, z)=−@

(R, z)

between the two pancake vortices,

f = "





dKJ

(KR)

−

√

1+

|z|/

√

1+

. (12.129)

At very small distances R < |z| <  the inte-

gral is cut by the exponential and we have f ≈

/2

)(R/|z|);at intermediatescales|z| < R < 

we find the constant force f ≈ "

/

;forlargedis-

tances  < R the integral is cut by the J

factor and

f ≈ "

/R). In the end we obtain an attractive in-

teraction between vortices which is small (by a factor

d/), but which is long ranged (extending over /d

layers),

int

(R, z = nd =0)≈ "



⎧

⎪

⎨

⎪

⎩

4|z|

, R |z| ,



, |z|R   ,



,   R .

(12.130)

For separations |z| >  the interaction decays ex-

ponentially ∝ exp(−|z|/) (the factor 1/2quoted

in [150] is incorrect).Finally,constructinga full vor-

tex line witha phase singularity present in each layer,

the calculation of the field distribution, the line en-

ergy, and the lower critical field H

[65] gives re-

sults in agreement with the continuous anisotropic

description,

2









4

= "

[ln  +const.] . (12.131)

Similarly, the line energy 

(# ) for a vortex tilted

by an angle  = /2−# away from the c-axis

agrees (to logarithmic accuracy) with the continu-

ous anisotropic result (12.25) in the limit " → 0,



(# )="

sin # ln





sin #

1+sin#



(12.132)

(note that "

=sin# for the present case of perfectly

decoupled layers with " =0).

A finite Josephson coupling between the super-

conducting layers renders the introduction of a sin-

gle pancake vortex into the layered superconductor

impossible [64,195,201,203]: in addition to the elec-

tromagnetic interaction between the layers, the in-

terlayer Josephson coupling contributes the energy











1−cos¥

n+1,n



)



. (12.133)

The phase difference ¥

n+1,n

= 0 between the plane

containing the vortex and its two neighboringplanes

and the coupling energy F

to these two layers be-

comes infinite: ignoring a possible relaxation of the

phase pattern, equation (12.133) predicts an energy

of an individual pancake vortex which grows with

526 G. Blatter and V.B. Geshkenbein

the sample area. However, the phase pattern due to a

single pancake vortex can relax to a state with an en-

ergy growing only linearly in the sample size,the lin-

ear energy arising from the Josephson strings which

take the ﬂux to and away from the pancake vortex.

Introducing a pair of oppositely “charged” pancake

vortices at a distance R into the same plane creates

two Josephson strings which contribute the energy

(12.133) with the phase differences ¥

n+1,n



) deter-

mined by the coupled set of equations (12.101). In

ordertotruncatethissystemofequationsweassume

the phases '

n±1

to be undisturbed, while the phase

is distorted due to the presence of the pair in layer

n. The phase difference ¥ = '

n+1

− '

= '

n−1

− '

then is a solution of





(2)

¥ =−2sin¥ , (12.134)

where we have neglected the additional screening

term due to the vector potential which becomes rel-

evant on large scales R > /". For small phase dif-

ferences ¥ we can expand the sine in (12.134) and

we obtain the natural length scale (screening length)

 = d/" of the problem. On scales R <  the

Josephson currents cannot build up and the system

essentially behaves like an uncoupled layered mate-

rial with its ideal 2D behavior. On the other hand, for

distances R >  the currents along z are no longer

hampered by the discreteness of the material and a

continuous anisotropic 3D behavior results.

At small separation R   thesolutionof (12.134)

behavesas¥ (R  R



 ) ≈ R·∇' =(R/R



)cos',

with ' = arctan(y/x) the azimuthal angle in the

plane,and vanishes ∝ exp(−R



/)atlargerdistances



 .For small phase differences ¥ we can expand

the cosine in (12.133) and a trivial integration sup-

plies us with the Josephson interaction energy of the

pair,

int

(R) ≈ "









, R   . (12.135)

For large separations R  , two Josephson strings

connecting the pair are created between the layers n

and n ± 1. The interaction between the two pancake

vortices then is dominated by the energies of the two

strings and takes the form (see (12.108))

int

(R) ≈ 2"





−





,   R  /" ,

(12.136)

with ˛ a numerical of order unity arising from cut-

ting off the logarithm in (12.108) at a distance of

the order of the separation d between the Josephson

strings [70,205].The last term in (12.136) is a higher

order correction to the leading string term (arising

again from the Josephson coupling[203]) and decays

exponentially at distances R > /".Thecomparison

of the Josephson interaction energies (12.135) and

(12.136) with the electromagnetic energy (12.122) of

the pair shows that the electromagnetic contribution

is larger at small distances R <  and the Josephson

term dominates for distances R > .

In summary,we arrive at the following picturefor

the interaction between two pancake vortices placed

inthesamelayerofaJosephsoncoupledsystem:At

small distances R <  the Josephson currents have

not yet built up, the electromagnetic energy is dom-

inant and we obtain a logarithmic interaction. As R

grows beyond , the Josephson currents have been

established, the magnetic field between the two vor-

tices has been redirected into two Josephson strings

and the interaction grows linearly, i.e., the pair be-

comes confined.

Two further pancake-vortex configurations are

the kink–anti-kink excitation along one Josephson

vortex [203] with the energy

K,−K

int

(R) ≈ 2"

⎧

⎪

⎨

⎪

⎩



,  < R <  ,





−



,  < R < /" ,





, /" < R ,

(12.137)

and the double-kink configuration providing the

building block of a tilted vortex with an energy

[201] (we assume  < ; otherwise, the linear term

∝ dR/

saturates at R ∼ , cf. (12.130))

K,K

int

(R) ≈ "

⎧

⎪

⎨

⎪

⎩











,  < R <  <  ,



,  < R < /" .

(12.138)

12 Vortex Matter 527

These basic pancake-vortex configurations allow to

study more complicated vortex configurations and

their energies in terms of point-like interacting par-

ticles. An example is the calculation of the line en-

ergy for a tilted vortex from which we can find the

elastic line tensions for the in-plane and for the out-

of-plane tilt modes: Summing up the energies of the

individual pancake vortices including their interac-

tion as given by (12.138) and the contributions from

the Josephson strings (12.135) and (12.136) we find

the line energy of a vortex tilted by an angle # with

respect to the layers.For large angles # > " the sep-

aration between neighboring pancake vorticesis less

than  and the line energy becomes [195,206]



(# > ")="

sin # (12.139)



˛

 sin #

2tan



 tan #



where ˛ and ˛



are numericals of order unity. For

small angles 0 < # < ", the Josephson string con-

tributesa large energy (12.136) and the result reads



(0 < # < ")="

sin # (12.140)







tan #



tan #



Using the definitions (12.37) for "

⊥

and for "



above

we easily find the expressions for the out-of-plane

and the in-plane line tensions in a layered mate-

rial [150] (we ignore logarithmic corrections and

the divergence of "

⊥

due to the lock-in transition

[207–209] at # =0):

⊥

(# )=

⎧

⎪

⎨

⎪

⎩

cos

, 0 < # < " ,

sin

, " < # <





(# )=""

⎧

⎪

⎨

⎪

⎩

cos # , 0 < # < " ,

sin #

, " < # <



These equations strongly resemble the results for the

anisotropic superconductor, a consequence of the

similarity in the current ﬂow set up by a vortex in

an anisotropic and in a layered material.

12.5.4 Renormalized Superﬂuid Stiffness

The presence of pancake vortices leads to a down-

ward renormalization of the superﬂuid density in

the superconducting layers [76,210]; this effect has

various consequences for the structural and statis-

tical properties of the vortex system in layered su-

perconductors, such as a reduction in the effective

anisotropy, a modification of the characteristics of

the Josephson vortices,andimplicationsfor the layer

decoupling transition. Strictly speaking, in the ab-

sence of pinning the presence of pancake vorticesre-

moves the (homogeneous) superﬂuid stiffness alto-

gether. However,a finite superﬂuid stiffness survives

at finite distances: we can define a short-wavelength

≈ /d) superﬂuid stiffness J in an individual

superconducting layer. This stiffness J(K)isdisper-

sive, where the (in-plane) long-wavelength limit J(0)

is determined by the pinning of the pancake vortices

within the pancake-vortex stacks produced by the

vortices in the other layers; its dispersive component

∝ K

is due to the finiteshear energy associated with

the deformation of the pancake-vortex lattice in the

plane.

In order to gain a better understanding of this

renormalized in-plane dispersive phase stiffness we

search for a free energy functional ıF[ıA, u]de-

scribing the stiffness of the gauge-invariant vector

potential ıA in the presence of mobile vortices de-

scribedby their displacement field u (wefirst concen-

trate on a 2D (translation invariant) situation with

the field B ˆz); note that ıF and ıA describe the

deviations in the free energy and in the gauge field

with respect to the equilibrium vortex lattice state.

In particular, the gauge field ıA contains a term

−j/

, 

≡ c/4

= e

/mc, originating from the

(macroscopic) current density j and a second term

u ∧ B due to the vortex displacement u, originating,

e.g., in response to an applied external current den-

sity j

ext

ıA =−



j + u ∧ B . (12.141)

Here, the field B ˆz generates the vortices re-

ducing the superﬂuid stiffness. The vortex-induced

gauge potential is easily understood in the dynam-

ical situation: taking the derivative of ıA

= u ∧ B

528 G. Blatter and V.B. Geshkenbein

with respect to time we find the Josephson relation

ıE =−@

ıA

/c = B ∧ v /c with the vortex velocity

v = @

u. Alternatively, the relation ıA

= u ∧ B fol-

lows from explicit calculation of the distortionin the

vortex phase fieldı'

= '

(R−u)−'

(R) ≈ −u·∇'

and suitable average over areas ıA > a

: using the

definition∇'

=−





ˆz ∧(R − S



)/|R − S



,where



denote the vortex positions, and transforming to

Fourier space one finds the useful relations

2

(ˆz ∧ iK)

ı'

2

∧ iK) ·ˆz

. (12.142)

Substituting iK

ı'

→∇ı'

→ (2/¥

)ıA

one easily reproduces the relation ıA

= u ∧ B.

The free energy functional we are searching for

then should contain terms describing the kinetic and

magnetic energies associated with ıA and terms ac-

counting for the elastic energy following from the

displacement field u; indeed, the free energy density

ıf [ıA, u]=

8



(∇∧ıA)





ıA − u ∧ B





[∇⊗u]

(12.143)

exhibits all the correct features (the tensor product

in (12.143) stands for [∇⊗u]

≡ @

): i)

Variation with respect to the vector potential ıA re-

produces the relation (12.141). ii) “Integrating” over

the gauge field ıA viaminimizingwith respect to ıA,

ıA

u ∧ B + 

K [K · (u ∧ B)]

1+

, (12.144)

and inserting the result back into (12.143), we find

the elastic energy density ıf

=(1/2){[c

(K)−

] |K · u

+ c

} with the correct modu-

lus c

(K)−c

=(B

/4)/(1 + 

) (note that

we should identify (B

/4)/(1 + 

)withthe

combination c

(K)−c

, cf. [13] and the remark

below (12.68); the tensor product in (12.143) then

transforms to c

/2 and the combination

− |K · u

]=c

|K ∧ u

). And iii),

variation with respect to u produces the force equa-

tion

j ∧B =−c

∇

u , (12.145)

where j accounts both for current densities applied

externally as well as those induced internally via

compression. Note that the kinetic energy of the

smooth macroscopic currents (external and from

vortex lattice compression) enter the free energy

density (12.143) via the term ∝ 

−2

,whereasthe

small microscopic currents due to the distortion u

of the vortex system are accounted for in the shear

term.

Further insight into the nature of the superﬂuid

stiffness in the presence of vortices is gained when

the force equation (12.145) is modified to account

for vortex pinning: using the Labusch [6] parame-

ter ˛

we introduce an effective pinning force den-

sity f

=−˛

u balancing the Lorentz force density

ext

∧ B/c (note that the elastic forces due to shear

and compression are taken care of in the effective

pinning parameter ˛

). Inserting the displacement

field u = j

ext

∧B/c˛

back into (12.141) we find that

the response j

ext

=−

eff

ıA involves the renormal-

ized superﬂuid density



eff

4



+ B

/4˛

; (12.146)

as expected, in the absence of pinning, ˛

→ 0, the

effective superﬂuid density 

eff

→ 0 vanishes. The

renormalized superﬂuid density (12.146)defines the

Campbell penetration depth 





+ B

/4˛

describing the penetration of a static external signal

into the superconductor in the presence of (pinned)

vortex lines [211].On the other hand, accounting for

the finiteshear forcein (12.145) the same calculation

provides the dispersive response



eff

(K)=

4

1+16/a

≈

4

16

(12.147)

Next, we modify (12.143) to describe our pancake

vortex system in a layered superconductor (we as-

sume that B ˆz). We separate the term describing

the smooth macroscopic current ﬂow in the planes

and complete with the elastic and Josephson cou-

pling terms,

12 Vortex Matter 529

ıF =

8





(∇∧ıA)





ıA

(2)

− u ∧ B











[∇⊗u

]

d



1−cos¥

n+1,n





, (12.148)

where the shear energy is determined by the density

of pancake vortices n

= B/¥

= a

−2

, c

= "

d/4a

The electromagnetic component of the tilt energy

(12.85) is strongly dispersive ∝ k

−2

and turns the

elastic tilt energy into a potential term with a pref-

actor v

≈ ("

d/2a



)ln[a

/(d

+ u

)], where we

have assumed k

∼ 1/d in the logarithm.Finally,a fi-

nite Josephson coupling between layers is accounted

for within a Lawrence–Doniach type description (cf.

(12.97)) and involves the gauge invariant phase dif-

ference ¥

n+1,n

Inordertofindtheeffectivefree energy describing

the gauge invariant vector potential ıA we integrate

over the displacement field u in (12.148); the rele-

vant terms involving the displacement field define

the partial free energy density

ıf

2"





(2)

− u

∧ B



(12.149)

16

+ g |u



The first term describes the kinetic energy density

of the smooth macroscopic current ﬂow. The second

term describes the shear stiffness and the third con-

tributionoriginatesfrom the electromagnetictilt en-

ergy which “binds” the pancake vortices into stacks;

here, we have rewritten the electromagnetic tilt en-

ergy in the form g (2"

d/a

)|u

with the reduc-

tion factor

g =

8

+ u

; (12.150)

the displacement u

in (12.150) is determined by the

pancake displacement typical for the actual physical

situation (e.g.,see (12.159) below). Note that it is this

“pinning” via the pancake vortex stack which will

establish a finite short-wavelength superﬂuid den-

sity in the individual layers. Finally, we “integrate”

over the displacement field u: minimizing (12.150)

we find

B ∧ ıA

(2)

1+a

/16 + g

(12.151)

and inserting this solution back into (12.150) we ob-

tain the free energy density

ıf

[ıA]=

8

g + a

/16

1+g + a

/16

|ıA

(2)

(12.152)

from whichwefindtherenormalizedsuperﬂuidstiff-

ness



eff

(K)=

4

g + a

/16

1+g + a

/16

. (12.153)

Comparing with (12.146) we can trace back the fi-

nite long-wavelength superﬂuid stiffness ∝ g/(1+g)

to the out-of-plane pinning action of the pancake

vortex stacks with the effective Labusch parame-

ter ˛

=4g"

derived from the “pinning” po-

tential V

= ˛

/2 due to the electromagnetic

interaction defining the stacks. Similarly, compar-

ison with (12.147) shows that the dispersive con-

tribution ∝ a

/16 is due to the in-plane vor-

tex shear energy. The full result (12.153) then fol-

lows from the combination of the force equation

j ∧ B d/c =(v

+ c

) u with (12.141).

Finally,wederive an expression for the free energy

in terms of the phase variable

ı' through the substi-

tution ıA

(2)

→ (¥

/2)∇ı' (the overline reminds

us that ıA

(2)

and hence also ∇ı' are smooth fields

averaged over distances larger than the vortex lat-

tice spacing); expanding (12.152) in a

/16 and

combining with (12.148) we arrive at