repulsive force, assuming the same sign of vorticity, on each vortex of strength

F = φ0 js , (12.67)

where js is the supercurrent density associated with one vortex at the position of the

other vortex. In the presence of a transport current, j, through the superconductor

the vortices will therefore per unit length be subject to a Lorentz force of magnitude

FL = φ0 j , (12.68)

where j is the transport current density, and the direction of the force is speci¬ed by

j — B. Even a small transport current will give rise to motion of the vortex lattice

perpendicular to the current in a pure type-II superconductor in the Abrikosov“

Shubnikov phase. This motion causes dissipative processes due to the normal currents

in the core, which phenomenologically can be described, at low velocities, by a friction

force (per unit length) opposing the motion of a vortex with velocity v

Ff = ’· v . (12.69)

The friction coe¬cient is given by8

φ2

0

·= (12.70)

2πa2 ρn

where ρn is the normal resistance of the metal, and a is the size of the normal core

(approximately equal to the superconducting coherence length).

8 For a phenomenological justi¬cation of the friction term we refer to the Bardeen“Stephen model

[125], or analysis based on the time dependent Ginzburg“Landau equation [126, 127, 128]. As

proclaimed, we describe only the phenomenology of the relevant forces, no derivation based on the

microscopic theory will be done, instead we refer the reader in general to reference [129].

462 12. Classical statistical dynamics

In addition, there can also be a Hall force

FH = ± v — n

ˆ (12.71)

acting on the vortex [130].

In a real superconductor there are always imperfections, referred to as impurities,

causing the vortices to have energetically preferred positions. The pinning force is

caused by defects such as twinning or grain boundaries, or dislocation lines. These

can pin a vortex, which would otherwise move in the presence of a transport current.9

At low enough temperatures and below a critical value of the transport supercurrent,

the vortex lattice is pinned and the current carrying state dissipationless. At larger

currents or higher temperatures, the motion of the vortices occur by thermal excita-

tion of (bundles of) vortices hopping between pinning centers, the state of ¬‚ux creep.

In the regime where the pinning force, Fp , is weak compared with the driving force,

the motion of the vortex lattice is steady, characterized by a velocity, v, the super-

conductor is in the dreaded ¬‚ux ¬‚ow regime. The moving magnetic ¬eld structure

associated with the vortices, leads by induction to the presence of an electric ¬eld,

E = ’v — B. The electric ¬eld has, as a result of the friction force, a component

parallel to the current, and the work, E·j, performed by the electric ¬eld is dissipated

by the friction force. The resistance is of the order of the normal state resistance,

and the dissipation will drive the superconductor to its normal state.

There is also interaction between the vortices as discussed previously. We shall be

interested in the case where the deformation of the Abrikosov lattice is weak, leading

to a harmonic interaction between the vortices described by continuum elasticity

theory.

12.2.2 Vortex lattice dynamics

We now turn to the case of interest, the dynamics of the Abrikosov vortex lattice in

the ¬‚ux ¬‚ow regime. The formalism is identical to the previously considered case of

one particle, except the occurrence of the whole lattice of vortices with the additional

feature of their interaction.

We consider a two-dimensional (2D) description of the vortices, since we have

in mind a thin superconducting ¬lm, or a three-dimensional (3D) layered supercon-

ductor with uncorrelated disorder between the layers. We shall be interested in the

in¬‚uence of quenched disorder on the vortex dynamics in the ¬‚ux ¬‚ow regime. The

description of the vortex dynamics is, according to the previous section, described

by the Langevin equation of the form

¦RR uR t = F + ±uRt — z ’ ∇V (R + uRt ) + ξRt , (12.72)

™ ™ ˆ

m¨ Rt + · uRt +

u

R

ˆ

where uRt is the two-dimensional displacement, normal to z, at time t of the vortex

(or bundle of vortices), which initially has equilibrium position R, · is the friction

9 The existence of the Abrikosov vortex state and the pinning of vortices is, from the point

of applications using superconducting coils as magnets, the most important property. They can

produce magnetic ¬elds in the excess of tens of Tesla. Usual copper coils can not produce the stable

¬eld produced by the supercurrent, not to mention its mess of water-cooling.

12.2. Magnetic properties of type-II superconductors 463

coe¬cient, and m is a possible mass of the vortex (both per unit length). The mass

of a vortex is small and will eventually be set to zero. The interaction between the

vortices is treated in the harmonic approximation and described by the dynamic

matrix ¦RR whose relevant elasticity moduli is discussed in Section 12.6. The

force (per unit length) on the right-hand side of Eq. (12.72) consists of the Lorentz

force, F = φ0 j — ˆ, due to the transport current density j, which we eventually

z

assume constant, and the second term on the right-hand side is a possible Hall force,

characterized by the parameter ±, and V is the pinning potential due to the quenched

disorder. The pinning is described by a Gaussian distributed stochastic potential with

zero mean, V (x) = 0, and thus characterized by its correlation function

ν(x ’ x ) = V (x)V (x ) . (12.73)

The thermal noise, ξ, is the white noise stochastic process with zero mean and

correlation function speci¬ed according to the ¬‚uctuation“dissipation theorem (where

the brackets now denote averaging with respect to the thermal noise)

= 2·T δ(t ’ t ) δRR δ±±

± ±

ξRt ξR t (12.74)

and, since the forces are per unit length, the temperature T has the dimension of

energy per unit length.

Upon averaging with respect to the thermal noise and the quenched disorder, the

average restoring force of the lattice vanishes

’ ¦RR =0 (12.75)

uR t

R

since the average displacement is the same for all vortices, and a rigid translation

of the vortex lattice does not change its elastic energy, leaving the dynamic matrix

with the symmetry property

¦RR = 0 . (12.76)

R

™

Owing to dissipation, the vortex lattice reaches a steady state velocity v = uRt ,

corresponding to the average force on any vortex vanishes

F + Ff + FH + Fp = 0 , (12.77)

i.e. there will be a balance between the Lorentz force, F, the average friction force,

Ff = ’·v, the average Hall force, FH = ±v — ˆ, and the pinning force

z

Fp = ’ ∇V (R + uRt ) . (12.78)

The pinning force is determined by the relative positions of the vortices with respect

to the pinning centers and is invariant with respect to the change of the sign of ±.

The average velocity, v, is the only vector characterizing the vortex motion which

is invariant with respect to the change of the sign of ±, and the pinning force is

therefore antiparallel to the velocity. Thus, the pinning yields a renormalization of

the friction coe¬cient

’·v + Fp = ’·e¬ v . (12.79)

464 12. Classical statistical dynamics

The e¬ective friction coe¬cient depends on the average velocity of the lattice, the

disorder, the temperature, the interaction between the vortices, the Hall force, and a

possible mass of the vortex. In the absence of disorder, the e¬ective friction coe¬cient

reduces to the bare friction coe¬cient ·.

The pinning problem has no simple analytical solution. One way of attacking the

problem is a perturbation calculation in powers of the disorder potential. A second-

order perturbation calculation works well for high velocities, as we show in Section

12.5.1.10 At low enough velocities the higher-order contributions in the disorder

become important. We shall employ the self-consistent e¬ective action method of

Cornwall et al. [53] to sum up an in¬nite subset of the contributions in V . Such self-

consistent methods are uncontrolled but many times they yield surprisingly good

results. In order to apply the ¬eld theoretic methods of Cornwall et al. we need

to reformulate the stochastic problem in terms of a generating functional, which is

achieved by the ¬eld theoretical formulation of classical statistical dynamics.

In the following the in¬‚uence of pinning on vortex dynamics in type-II supercon-

ductors is investigated. The vortex dynamics is described by the Langevin equation,

and we shall employ a ¬eld-theoretic formulation of the pinning problem which al-

lows the average over the quenched disorder to be performed exactly. By using the

diagrammatic functional method for this classical statistical dynamic ¬eld theory, we

can, from the e¬ective action discussed in the previous chapter, obtain an expression

for the pinning force in terms of the Green™s function describing the motion of the

vortices.

12.3 Field theory of pinning

The average vortex motion is conveniently described by reformulating the stochastic

problem in terms of the ¬eld theory of classical statistical dynamics introduced in

Section 12.1. The probability functional for a realization {uRt }R of the motion of

the vortex lattice is expressed as a functional integral over a set of auxiliary variables

{˜ Rt }R , and we are led to consider the generating functional11

u

u

Z[F, J] = DuRt D˜ Rt J eiS[u,˜ ] , (12.80)

u

R R

where in the action

˜ ’1

u(DR u + F ’ ∇V + ξ) + Ju

˜

S[u, u] = (12.81)

the inverse free retarded Green™s function is speci¬ed by

’1

’DR uRt = m¨ Rt + · uRt + ¦RR uR t + ±ˆ — uRt ,

™ z™ (12.82)

u

R

10 Vortex pinning in the ¬‚ux ¬‚ow regime was originally considered treating the disorder in lowest

order perturbation theory [131, 132], and later by applying ¬eld theoretical methods [133, 134].

11 In the following we essentially follow reference [134].

12.3. Field theory of pinning 465

i.e.

’1

DR (R, t; R , t ) = ’¦RR δ(t ’ t ) ’ (m‚t + ·‚t )1 ’ i±σy ‚t δR,R δ(t ’ t ) ,

2

(12.83)

where matrix notation is used for its Cartesian components, i.e. 1 and σ y denote

the unit matrix (occasionally suppressed for convenience) and the Pauli matrix in

Cartesian space, respectively. The Fourier transform of the inverse free retarded

Green™s function is therefore the two by two matrix in Cartesian space given by the

expression

’i±ω

mω 2 + i·ω

’1

’ ¦q .

DR (q, ω) = (12.84)

mω 2 + i·ω

i±ω

In Eq. (12.81) we have introduced matrix notation in order to suppress the integra-

tions over time and summations over vortex positions and Cartesian indices. Thus,

˜ ’1

for example, uDR u denotes the expression

∞ ∞

˜ ’1 ’1±±

uDR u = dt dt u± (R, t) DR

˜ (R, t; R , t ) u± (R , t ) . (12.85)

RR ’∞ ’∞

±,± =x,y

The Jacobian, J = |δξRt /δ uR t |, guaranteeing the normalization of the generating

˜

functional

Z[F, J = 0] = 1 (12.86)

is given by

⎡ ¤

∞

‚ 2 V (R + uRt ) ¦

J ∝ exp ⎣’ R±±

dtDRt;Rt , (12.87)

‚x± ‚x±

R±± ’∞

where the proportionality constant is the determinant of the inverse free retarded

’1

Green™s function, |(DR )±± t |. As discussed in Section 12.1.6, in the case of a

Rt,R

nonzero mass, m = 0, the Jacobian is an irrelevant constant; and in the case of

zero mass, dropping the Jacobian from the integrand is equivalent to de¬ning the

R

retarded free Green™s function to vanish at equal times, Dtt = 0, which in turn leads

to the full retarded Green™s function satisfying the same initial condition. In terms of

diagrams, the contribution from the Jacobian exactly cancels the tadpole diagrams

as discussed in Section 12.1.6.

The average with respect to both the thermal noise and the disorder is imme-

diately performed, and we obtain the averaged functional, dropping the irrelevant

Jacobian,

Z Dφ eiS[φ]+if φ .

Z[f ] = = (12.88)

We have employed the compact notation for the ¬elds

φRt = (˜ Rt , uRt ) = (φ1 (R, t), φ2 (R, t)) (12.89)

u

466 12. Classical statistical dynamics

and for the external force and an introduced source, J(R, t),

f (R, t) = (F(R, t), J(R, t)) . (12.90)

The action obtained upon averaging, which we also denote by S, consists of two

terms

S[φ] = S0 [φ] + SV [φ] . (12.91)

The ¬rst term is quadratic in the ¬eld

1

φD’1 φ ,

S0 [φ] = (12.92)

2

where the matrix notation now in addition includes the dynamical indices, i.e. φD’1 φ

denotes the expression

∞ ∞

’1

φD’1 φ = i dt dt φ± (R, t) Dij ±± (R, t; R , t ) φ± (R , t ) . (12.93)

i j

RR ’∞ ’∞

±± ij

The inverse free matrix Green™s function in dynamical index space

’1

’1 ’1 2i·T δ(t ’ t ) δ±± δRR DR (R, t; R , t )

D11 D12

’1

D = = (12.94)

’1 ’1 ’1

D21 D22 DA (R, t; R , t ) 0

is a symmetric matrix in all indices and variables, since the inverse free advanced

Green™s function is obtained by interchanging Cartesian indices as well as position

and time variables

’1± ’1±±

DA ± (R , t ; R, t) = DR (R, t; R , t ) . (12.95)

The interaction term originating from the disorder is

∞∞

ν(uRt ’ uR t ) ±

2

i ‚

’ u±

SV [φ] = dt dt ˜Rt uR t .

˜ (12.96)

‚u± ‚u±

2 Rt Rt

’∞ ’∞

RR

±±

The source term introduced in Eq. (12.80)

∞

dt J(R, t) · u(R, t) ,

Ju = (12.97)

’∞

R

where the source, J(R, t), couples to the vortex positions, u(R, t), is added to the

action in order to generate the vortex correlation functions. For example, we have

for the average position

δZ

uRt = ’i (12.98)

δJRt J=0

12.3. Field theory of pinning 467

and the two-point unconnected Green™s function

δ2 Z

=’ . (12.99)

uRt uR t

δJRt δJR t J=0

Here and in the following we use dyadic notation, i.e. uRt uR t is the Cartesian

matrix with the components u± (R, t) u± (R , t ).

12.3.1 E¬ective action

In order to obtain self-consistent equations involving the two-point Green™s function

in a two-particle irreducible fashion, we add a two-particle source term K to the

action in the generating functional (recall Section 10.5.1)

i

Dφ exp iS[φ] + if φ + φKφ .

Z[f, K] = (12.100)

2

The generator of connected Green™s functions

iW [f, K] = ln Z[f, K] (12.101)

has accordingly derivatives

δW ±

= φi (R, t) (12.102)

δfi± (R, t)

and

δW 1± i

±

= φi (R, t) φi (R , t ) + G±± (R, t; R , t ) , (12.103)

2 ii

±± 2

δKii (R, t; R , t )

where φ is the average ¬eld, with respect to the action S[φ] + f φ + φKφ/2,

i

±

Dφ φ± (R, t) exp iS[φ] + if φ + φKφ

φi (R, t) = (12.104)

i

2

and G is the full connected two-point matrix Green™s function

⎛ ⎞

δ˜± δ˜± t δ˜± δu± t

uRt uR uRt R

δ2W

= ’i ⎝ ⎠,

Gij = ’ (12.105)

δfi δfj δu± δ˜± δu± δu±

uRt Rt Rt Rt

where

δuRt = uRt ’ uRt δ uRt = uRt ’ uRt .

˜ ˜ ˜

, (12.106)

In the physical problem of interest, the sources K and J vanish, K = 0 and

J = 0, and the full matrix Green™s function has, owing to the normalization of the

generating functional

Z[F, J = 0, K = 0] = 1 , (12.107)

468 12. Classical statistical dynamics