(c11 ’ c66 )qx qy

2 2

φ0 c11 qx + c66 qy

¦q = , (12.168)

(c11 ’ c66 )qx qy 2 2

c66 qx + c11 qy

B

488 12. Classical statistical dynamics

where q belongs to the ¬rst Brillouin zone, and B is the magnitude of the external

magnetic ¬eld, and φ0 /B is therefore equal to the area, a2 , of the unit cell of the

vortex lattice. In the continuum limit we obtain a longitudinal branch, el (q) · q = 1,

ˆ

with corresponding eigenvalues Kl (q) = c11 a q , and a transverse branch, et (q) · q =

22 ˆ

22

0, with corresponding eigenvalues Kt (q) = c66 a q .

12.6.1 High-velocity limit

√ 2

At high velocities, v ν0 /(·rp ), where lowest-order perturbation theory in the

disorder is valid, we can neglect the self-energies in the self-consistent expression for

the pinning force, Eq. (12.133), i.e. we can insert the free retarded Green™s function

for the lattice and, assuming v T /(·rp ), neglect the ¬‚uctuation exponent, and

obtain for the pinning force

·k · v (k · eb (k))2

dk

Fp = ’ k ν(k) . (12.169)

(·k · v)2 + (Kb (k))2

(2π)2

b=l,t

The maximum values, attained at the boundaries of the Brillouin zones, of the trans-

verse and longitudinal eigenvalues are speci¬ed by the compression and shear moduli,

Kt ∼ c66 and Kl ∼ c11 . The compression modulus is much greater than the shear

modulus, c11 c66 , in thin ¬lms and high-temperature superconductors (see for ex-

ample reference [137]). The order of magnitude of the ¬rst term in the denominator

’2

of Eq. (12.169) is ·v 2 rp , since the range of the impurity correlator is rp , and at

intermediate velocities, c66 rp /· v c11 rp /·, only the transverse mode therefore

contributes to the pinning force, and we obtain

(k · et (k))2

dk

=’ k ν(k) . (12.170)

Fp

·k · v

(2π)2

The eigenvalues et (k) are periodic in the reciprocal lattice and, assuming short-range

disorder, rp a, the rest of the integrand is slowly varying, and we obtain for the

pinning force

dk ν(k)k 2

1 ν0

Fp = ’ =’ v. (12.171)

k

(2π)2 ·k · v 8πrp ·v 2

4

2

At very high velocities, v c11 rp /·, the eigenvalues of the dynamic matrix in

Eq. (12.169) can be neglected compared with the velocity-dependent term in the

denominator, and the longitudinal and transverse parts of the free retarded Green™s

function give equal contributions to the pinning force, and we obtain

ν0

Fp = ’ v. (12.172)

4πrp ·v 2

4

12.6. Vortex lattice 489

This result is identical to the expression for the pinning force on a single vortex,

√ 2

Eq. (12.142), in the high velocity regime, v ν0 /(·rp ), since the in¬‚uence of the

elastic interaction is negligible.

12.6.2 Numerical results

In this section we consider the pinning force on the vortex lattice obtained from

the self-consistent theory. For any given average velocity of the lattice, the coupled

equations of Green™s functions and self-energies, Eq. (12.124) and Eq. (12.125), may

be solved numerically by iteration. In order to simplify the numerical calculation,

the self-consistent equations are brought on dimensionless form by introducing the

1/2 1/2

following units for length, time, and mass, a, ·a3 /ν0 , and · 2 a4 /ν0 . Starting

by neglecting the self-energies, we obtain numerically the response and correlation

functions. From Eq. (12.133) we can then determine the pinning force as a function

of the velocity. We have calculated the velocity dependence of the pinning force for

vortex lattices of sizes 4 — 4, 8 — 8, and 16 — 16 using the self-consistent theory, and

the results are shown in Figure 12.12.

5.5

5

4.5

Fp

4

3.5

3

2.5

10 20 30 40 50

v

Figure 12.12 Pinning force (in units of ν0 a’2 ) as a function of velocity (in units of

1/2

· ’1 ν0 a’2 ) obtained from the self-consistent theory for three di¬erent lattice sizes.

1/2

The stars correspond to a 4 — 4 lattice, and the two curves correspond to 8 — 8 and

16 — 16 lattices, respectively. The mass and temperature are chosen to be zero, and

1/2 1/2

the elastic constants are given by c66 a3 = 100ν0 and c11 a3 = 104 ν0 , and the

range of the disorder correlator is chosen to be rp = 0.1a.

490 12. Classical statistical dynamics

The di¬erence between the results obtained for the 8 — 8 and the 16 — 16 lattice

is small, and we conclude that the pinning force is fairly insensitive to the size of the

lattice.

In Figure 12.13 we compare the pinning force as a function of the velocity for

lattices of di¬erent sti¬nesses, and we ¬nd that the pinning force decreases with

increasing sti¬ness of the lattice.

6

5

4

Fp

3

2

1

0 10 20 30 40 50

v

Figure 12.13 Pinning force (in units of ν0 a’2 ) on a vortex lattice of size 16 — 16

1/2

as a function of velocity (in units of · ’1 ν0 a’2 ) obtained from the self-consistent

1/2

1/2

theory for the compression modulus given by c11 a3 = 104 ν0 and three di¬erent

1/2 1/2

shear moduli: c66 a3 = 50ν0 (upper dashed line), c66 a3 = 100ν0 (solid line) and

1/2

c66 a3 = 200ν0 (lower dashed line). The mass and temperature are both chosen to

be zero, and rp = 0.1a.

Generally, the interaction between the vortices lowers the pinning force, since the

neighboring vortices in a moving lattice drag a vortex over the potential barriers.

This can be inferred from the self-consistent theory by comparing the pinning forces

depicted in Figures 12.3 and 12.12, and in perturbation theory by noting the extra

term originating from the elastic interaction in the denominator of the expression for

12.6. Vortex lattice 491

the pinning force, Eq. (12.169).

When the temperature is increased, the pinning force decreases, except at very

high velocity, as apparent from Figure 12.14. This feature is common to the single

vortex case, and simply re¬‚ects that thermal noise helps a vortex over the potential

barriers.

3.7

3.6

3.5

3.4

Fp

3.3

3.2

3.1

3

2.9

10 20 30 40 50

v

Figure 12.14 Pinning force (in units of ν0 a’2 ) on a vortex lattice of size 16—16 as

1/2

a function of velocity (in units of · ’1 ν0 a’2 ) obtained from the self-consistent theory

1/2

1/2

for two di¬erent temperatures. The elastic constants are given by c66 a3 = 100ν0

’1/2

and c11 a3 = 104 ν0 , and rp = 0.1a and m = 1.0 · 10’4 · 2 a3 ν0

1/2

. The dashed line

1/2 ’1

corresponds to T = 0, and the solid line to T = 0.5ν0 a .

The convergence of the iterative procedure is monitored by checking that energy

conservation is ful¬lled. The energy conservation relation for a vortex lattice is

obtained as in Section 12.5.5, and since the term originating from the harmonic

interaction between the vortices disappears owing to the symmetry property of the

dynamic matrix, Eq. (12.76), we obtain for a vortex lattice the energy conservation

relation

· ‚t tr ’i‚t GK (R, t; R, t ) + 2 T GR (R, t; R, t ) |t =t = ’v · Fp . (12.173)

492 12. Classical statistical dynamics

The convergence of the iteration procedure, employed when solving the self-consistent

equations, has been checked by numerically calculating the terms in Eq. (12.173). We

¬nd that the right- and left-hand sides of the energy conservation relation di¬er by

no more than a few percent after twenty iterations.

12.6.3 Hall force

We now consider the in¬‚uence of a Hall force on the dynamics of a vortex lattice. The

motion of the vortex lattice, with its associated magnetic ¬eld, induces an average

electric ¬eld. The relationship between the average vortex velocity and the induced

electric ¬eld, E = v — B, and the expression for the Lorentz force, yields for the

resistivity tensor of a superconducting ¬lm

φ0 B ·e¬ ±

ρ= , (12.174)

’±

2 ·e¬

·e¬ + ±2

where the e¬ective friction coe¬cient, ·e¬ , was introduced in Eq. (12.79).13 Accord-

ing to Eq. (12.174), the following relationship between the transverse, ρxy , and the

longitudinal resistivities, ρxx , is obtained

±2

±

ρxy = ρ2 1+ . (12.175)

xx 2

Bφ0 ·e¬

If the Hall force is small, ± ·e¬ , the scaling relation

±

ρxy = ρ2 (12.176)

xx

Bφ0

is seen to be obeyed. This scaling law is valid for all velocities of the vortex, provided

the Hall force is small compared with the friction force, ± ·. We note that the

scaling law is also valid at small vortex velocities for arbitrary values of the Hall

force, if the e¬ective friction coe¬cient diverges at small velocities. This occurs if

the pinning force decreases slower than linearly in the vortex velocity. This is indeed

the case, according to the self-consistent theory, at temperatures lower than the

√

average barrier height, T ν0 /rp , as indicated by the low velocity behavior of the

pinning force in Figure 12.15. This behavior of the pinning force is also obtained for

non-interacting vortices as apparent from Figure 12.8.

In Figure 12.15 is shown the pinning force obtained from the self-consistent theory

as a function of velocity for the case of zero temperature. As expected there is

no in¬‚uence of the Hall force on the pinning force at low velocities, but we ¬nd

a suppression at intermediate velocities, and at very high velocities, v c11 a/·,

we recover the high velocity limit of the single vortex result, i.e. Eq. (12.165). By

comparison of Figures 12.8 and 12.15, we ¬nd that the Hall force has a much weaker

in¬‚uence at intermediate velocities on the pinning of an interacting vortex lattice

than on a system of non-interacting vortices. Furthermore, the in¬‚uence of the Hall

force on the pinning force is more pronounced for a sti¬ lattice than a soft lattice, as

seen from the inset in Figure 12.15.

13 The e¬ective friction coe¬cient was determined to lowest order in the disorder in reference [138].

12.7. Dynamic melting 493

3.6

3.5

3.4

3.3

3.2

Fp 1.6

1.5

3.1 Fp

1.4

3 1.3

2.9 1.2

0 50

2.8

v

2.7

0 10 20 30 40 50

v

Figure 12.15 Pinning force (in units of ν0 a’2 ) on a vortex lattice of size 16 — 16

1/2

as a function of velocity (in units of · ’1 ν0 a’2 ) obtained from the self-consistent

1/2

theory. The solid and dashed lines correspond to ± = 0 and ± = ·, respectively.

The temperature and mass are both chosen to be zero, and rp = 0.1a. The elastic

1/2 1/2

constants are given by c11 a3 = 104 ν0 and c66 a3 = 100ν0 . Inset: pinning force as

1/2

a function of velocity for ± = 0 and ± = ·, respectively. Here c66 a3 = 300ν0 and

the other parameters are unchanged.

In Figure 12.16 the dependence of the Hall angle on the velocity is presented for

various sti¬nesses of the vortex lattice; the sti¬est lattice has the greatest Hall angle.

Since the pinning force is reduced by the interaction between the vortices, the Hall

angle for a lattice is larger than for an independent vortex, except at high velocities

where they saturate at the same value. A similar behavior of the Hall angle at zero

velocity, as observed for a single vortex in Section 12.5.5, pertains to a vortex lattice.

12.7 Dynamic melting

In this section we consider the in¬‚uence of quenched disorder on the dynamic melting

of a vortex lattice. This non-equilibrium phase transition has been studied experi-

mentally [139, 140, 141, 142, 143, 144], as well as through numerical simulation and

a phenomenological theory and perturbation theory [145, 146, 147]. The notion of

dynamic melting refers to the melting of a moving vortex lattice where, in addition to

the thermal ¬‚uctuations, ¬‚uctuations in vortex positions are induced by the disorder.

A temperature-dependent critical velocity distinguishes a transition between a phase

494 12. Classical statistical dynamics

0.8

0.7

0.6

θ

0.5

0.4

0.3

0 10 20 30 40 50

v

Figure 12.16 Hall angle obtained from the self-consistent theory for a vortex lattice

of size 16 — 16 as a function of velocity (in units of · ’1 ν0 a’1 ) for a moderately

1/2

1/2

strong Hall force, ± = ·. The compression modulus is given by c11 a3 = 104 ν0 ,

and the three curves correspond to decreasing values of the shear modulus c66 a3 =

1/2 1/2 1/2

200ν0 , 100ν0 , 50ν0 . The mass and temperature are both chosen to be zero, and

rp = 0.1a.

where the vortices form a moving lattice, the solid phase, and a vortex liquid phase.

Before solving the self-consistent equations by numerical iteration in order to

obtain the phase diagram, we consider the heuristic argument for determining the

phase diagram for dynamic melting of a vortex lattice presented in reference [145].

There, the disorder induced ¬‚uctuations were estimated by considering the correlation

function

(p)

(p)

κ±± (x, t) = f± (x, t) f± (0, 0) (12.177)

of the pinning force density

f (p) (x, t) = ’ δ(x ’ R ’ uRt ) ∇V (x ’ vt) . (12.178)

R

Neglecting the interdependence of the ¬‚uctuations of the vortex positions and the

¬‚uctuations in the disorder potential, the pinning force correlation function factorizes

δ(x ’ R ’ uRt ) δ(R ’ uR 0 ) ∇±∇± V (x ’ vt)V (0) .

κ±± (x, t)

RR

(12.179)

Introducing the Fourier transform (A is the area of the ¬lm)

CRR (q, t) = A’1 e’iq·(R+uRt ’R ’uR 0 ) (12.180)

12.7. Dynamic melting 495

of the vortex density-density correlation function

δ(x ’ R ’ uRt ) δ(R ’ uR 0 )

CRR (x, t) = (12.181)

and employing the translational invariance yields

’nV δ(x ’ R ’ uRt ’ R ’ uR 0 ) ∇± ∇± ν(x ’ vt) ,

κ±± (x, t) =

RR

(12.182)

where nV is the density of vortices. In the ¬‚uidlike phase the motion of di¬erent

vortices is incoherent and the o¬-diagonal terms, R = R , can be neglected yielding

κ±± (x, t) = ’nV δ(x) ∇± ∇± ν(vt) . (12.183)

In analogy with the noise correlator, the e¬ect of disorder-induced ¬‚uctuations is