PHYSICAL REVIEW B
VOLUME 57, NUMBER 6
1 FEBRUARY 1998-II
Thermally activated Hall creep of flux lines from a columnar defect
D. A. Gorokhov and G. Blatter
Theoretische Physik, ETH-Hönggerberg, CH-8093 Zürich, Switzerland
~Received 28 April 1997; revised manuscript received 22 August 1997!
We analyze the thermally activated depinning of an elastic string ~line tension e ) governed by Hall dynamics
from a columnar defect modeled as a cylindrical potential well of depth V 0 for the case of a small external
force F. An effective one-dimensional-field Hamiltonian is derived in order to describe the two-dimensional
string motion. At high temperatures the decay rate is proportional to F 5/2T 21/2exp@F0 /F2U(F)/T#, with F 0 a
constant of order of the critical force and U(F);( e V 0 ) 1/2V 0 /F the activation energy. The results are applied
to vortices pinned by columnar defects in superclean superconductors. @S0163-1829~98!07105-7#
I. INTRODUCTION
The search for mechanisms of pinning is both scientifically challenging as well as an important problem to be studied in view of technological applications of high-T c superconductors. Even if the transport current is less than the
critical one, energy dissipation takes place due to quantum or
thermally activated creep of vortices.1 Recent measurements
of the critical current and of the magnetization relaxation rate
in layered high-temperature superconductors show that columnar defects produced by irradiation with heavy ions can
strongly suppress the vortex motion.2 In this paper we
present a theoretical study of the thermally activated depinning of a vortex from a columnar defect in the presence of a
small transport current ~classical creep!.
The inverse lifetime G of a metastable state can be written
in the form G5Ae 2S/\ , where S is the Euclidean action
along the extremal trajectory and A is the prefactor determined by the fluctuations around the saddle-point solution
~see Ref. 3 for a general review concerning the decay of
metastable states!. At low temperatures the saddle-point solution is time dependent and, consequently, S depends on the
dynamics of the system ~quantum creep!, whereas at high
temperatures the calculation of S alone does not involve the
dynamics ~classical creep!. Here, we concentrate on high
temperatures but go beyond the usual exponential accuracy
by calculating the prefactor A, a task which does involve the
dynamics of the flux lines as well.
In high-T c superconductors the dynamics of vortices may
be dominated by either the dissipative or Hall term in the
equation of motion. Microscopic calculations of the dynamic
constants4 show that the ratio of Hall and dissipative coefficients a / h is approximately equal to v 0 t , with v 0 and t the
level spacing between localized Caroli–de Gennes–Matricon
states in the core and the relaxation time, respectively. Recent experimental studies on 90 K crystals of Y-Ba-Cu-O
~see Ref. 5! have been interpreted as providing evidence for
the superclean limit, with v 0 t ;15 below 15 K, in which
case the contribution of dissipative forces can be neglected in
this regime.
Quantum depinning of flux lines governed by Hall dynamics from a columnar defect has been considered by Sonin
and Horovitz6 for the case of a small external force. For
pancake vortices this problem has been studied by Bulae0163-1829/98/57~6!/3577~9!/$15.00
57
vskii, Larkin, Maley, and Vinokur.7 The case j c 2 j! j c has
been investigated by Chudnovsky, Ferrera, and Vilenkin8
and by Morais-Smith, Caldeira, and Blatter.9 On the other
hand, the problem of the depinning of a massive string from
a linear object has been solved by Skvortsov in the whole
temperature range10 and the thermal depinning of a flux line
governed by dissipative dynamics has been considered by
Krämer and Kulić.11
The present paper is organized as follows: In Sec. II we
reduce the two-dimensional problem of the string motion to
an effective one-dimensional ~1D! problem. In Sec. III the
decay rate of a trapped string is calculated. In Sec. IV the
results are applied to vortices in superclean superconductors.
II. 1D EFFECTIVE HAMILTONIAN
Let us consider a string which is pinned by a columnar
defect in the presence of an external force. Both the cylindrical defect as well as the vortex are directed along the c
axis of the anisotropic superconductor and the external magnetic field is supposed to be sufficiently small, such that the
interaction between the vortices can be neglected. Furthermore, we consider the situation where each vortex is pinned
by an individual defect, i.e., the concentration of defects is
assumed to be larger than that of vortices. The free-energy
density of the string describing this situation is
S D
~1!
U ~ u! 5V cyl~ Au 2x 1u 2y ! 1V ext~ u x ! .
~2!
G ~ u! 5
e ]u 2
1U ~ u! ,
2 ]z
where
Here, e is the elasticity of the string and u x and u y describe
its displacement along the x and y directions, with the z axis
chosen parallel to the defect. In Eq. ~2!, V cyl( Au x 2 1u y 2 )
denotes the cylindrical pinning potential and V ext52Fu x is
the forcing potential. The function V cyl(r) is supposed to be
monotonously increasing and restricted from below and
above. We assume boundary conditions u(6L/2)50.
If FÞ0, the state of the vortex becomes metastable. Our
main goal is to investigate the decay rate as a function of
temperature T and force F, where the external force is as3577
© 1998 The American Physical Society
3578
D. A. GOROKHOV AND G. BLATTER
sumed to be small. The Lagrangian of the vortex ~in real
time dynamics! can be written as
L @ u x ,u y # 5
E
1L/2
2L/2
S
dz a u y
D
]ux
2G @ u x ,u y # ,
]t
~3!
where a is the Hall coefficient. The corresponding equations
of motion take the form
]ux ]U
] 2u y
5
2e 2 ,
]t ]uy
]z
~4!
]uy
]U
] 2u x
52
1e 2 .
]t
]ux
]z
~5!
a
a
Equations ~4! and ~5! can be formulated as the equations of
motion of the 1D Hamiltonian density
F S D S D S DG
e ]x
p
H5 U x, 1
a
2 ]z
e
2
1
2a2
]p
]z
2
,
]H
] ] H ] U ~ x, p/ a ! e ] 2 p
2
2 2 2,
5
]p ]z ]pz
]p
a ]z
ṗ52
where v 0 is the unstable mode growth rate and Z is the
partition function of the system under consideration. Equation ~9! is applicable if the temperature T satisfies the condition T@\ v 0 , otherwise quantum effects become relevant.
If the transition from quantum to classical behavior in the
decay of the metastable state is of second order, the parameter T 0 5\ v 0 /2p determines the temperature of the crossover. However, in general the transition can be of first order,
in which case the crossover temperature differs from
\ v 0 /2p ~see Ref. 15!. Note that at low temperatures the
equation for the decay rate can be written as \G
52T ImZ/Z. We emphasize that Eq. ~9! can be applied only
if the system is properly equilibrized, i.e., if its characteristic
relaxation time is much smaller than G 21 .
We assume that the quasiclassical approximation is applicable. With the partition function written as a path integral,
Z5
~6!
where we have used the definitions x[u x and p[ a u y . Using the variational procedure for the Hamiltonian density H
we obtain
ẋ5
57
]H
] ]H
] U ~ x, p/ a !
] 2x
1
1e 2 ,
52
]x ]z ]xz
]x
]z
~7!
~8!
and one can easily see that Eqs. ~7! and ~8! are equivalent to
Eqs. ~4! and ~5! with x5u x and p5 a u y . Thus we have
reduced the 2D problem of the motion of a vortex governed
by Hall dynamics with an action given by Eq. ~3! to the 1D
problem of the motion of a vortex described by the effective
action * dz dt(pẋ2H), with H given by Eq. ~6!. The above
idea has been introduced by Volovik12 for the tunneling of
2D vortices in a liquid-helium film, the motion of which is
equivalent to that of a massless particle in a magnetic field.
Later, this idea has been used by Feigel’man, Geshkenbein,
Larkin, and Levit13 within the context of vortex tunneling in
superclean high-T c superconductors. Chudnovsky, Ferrera,
and Vilenkin8 have generalized this idea for the description
of the depinning of a flux line governed by Hall dynamics
from a columnar defect near criticality. In their case
U(u x ,u y )5au 2x /22bu 3x /31cu 2y /2 and the elasticity along
the y direction can be neglected if j c 2 j! j c , i.e., the problem can be reduced to the 1D massive string problem. Above
we have generalized the problem for the case of an arbitrary
potential and nonzero elasticity in the y direction.
E$
S
Du% exp 2
D
S Eucl@ u#
,
\
~10!
we can use the steepest descent method for its calculation. In
this approximation the partition function is determined by its
stationary points. The most significant contribution to Z
arises from the trajectory u(z,t)50—the position of the
minimum of the potential, whereas the imaginary part of the
partition function is determined by the neighborhood of the
saddle-point solution u(z,t)5u0 (z) which is time independent at high temperatures. Hence, the problem is reduced to
the calculation of the unstable mode growth rate v 0 , and of
the real and imaginary parts of the statistical sum. Below we
shall calculate all these quantities.
A. Unstable mode growth rate
At high temperatures the saddle-point solution does not
depend on time. The stationary extremal trajectory u0 (z) for
the Euclidean action
S Eucl@ u x ,u y # 5
E
1L/2
2L/2
dz
E
1\/2T
2\/2T
S
d t G @ u x ,u y # 2i a u y
D
]ux
]t
~11!
corresponding to the real-time Lagrangian ~3! satisfies the
Euler equation
e
] 2u
]z
2
5
] U ~ u!
,
]u
u5 ~ u x ,u y ! .
~12!
The unstable mode growth rate then is determined by the
negative eigenvalue of the operator d 2uS Euclu u0(z) . Near the
saddle-point solution u0 (z) the perturbed solution can be expanded in the form
u x ~ z, t ! 5u x0 ~ z ! 1 c ~ z ! exp~ i2 p T t /\ ! ,
~13!
III. DECAY RATE
u y ~ z, t ! 5 w ~ z ! exp~ i2 p T t /\ ! ,
~14!
The decay rate G of an arbitrary metastable Hamiltonian
system at high temperatures is given by the expression14
with small distortion amplitudes c (z) and w (z) for T&T 0 .
Substituting Eqs. ~13! and ~14! into the equations of motion
for the Euclidean action ~11! and expanding, we obtain the
system of equations determining the crossover temperature
T 0.
\G5
\ v 0 Im Z
,
p
Z
~9!
THERMALLY ACTIVATED HALL CREEP OF FLUX . . .
57
3579
The unstable mode growth rate v 0 is determined by the
negative eigenvalue l̃21 of Eq. ~17!, l̃21 52( a v 0 ) 2 . In
Appendix A we derive a variational principle for the present
problem and show that the negative eigenvalue of Eq. ~17!
indeed exists and is unique. ~We exploit that the operator
1/2
Ĥ 1/2
y Ĥ x Ĥ y is Hermitian and has the same eigenvalues as
Ĥ y Ĥ x ). Furthermore, we derive both upper and lower bounds
FIG. 1. The ‘‘potential’’ U x (z) for the 1D Schrödinger operator
Ĥ x @see Eq. ~15!#. The characteristic size of the potential is given by
D5 A2 e V 0 /F and its height is equal to k .
d c
2
2e
2e
dz
2
d 2w
dz 2
on the eigenvalue l̃21 of the operator Ĥ x Ĥ y in Appendix B,
such that we finally arrive at the estimate
T 05
2
1 a v 0w 1
2 a v 0c 1
d V cyl
dr 2
u r5u x0 ~ z ! [Ĥ x c 1 a v 0 w 50,
~15!
1 dV cyl
u
[Ĥ y w 2 a v 0 c 50,
r dr r5u x0 ~ z !
~16!
subject to the boundary conditions c (6L/2), w (6L/2)50.
The ~lowest! eigenvalue v 0 is related to the crossover temperature T 0 through the condition v 0 52 p T 0 /\.
The ‘‘potentials’’ U x (z) and U y (z) of the Schrödinger
operators defined in Eqs. ~15! and ~16! are shown in Figs. 1
and 2, respectively.
B. Ratio ImZ/Z
The most significant contribution to ReZ arises from the
neighborhood of the minimum of the potential V cyl . In this
region we can write @see Eq. ~2!#
U ~ x,p/ a ! >
zero eigenvalue. The operator Ĥ y is positive and has only
positive eigenvalues.
Eliminating the function w from the system of Eqs. ~15!
and ~16! we obtain ~we denote eigenvalues of Ĥ x Ĥ y by l̃
and those of Ĥ x by l)
Ĥ y Ĥ x c 5l̃c .
~17!
k 2 k 2
x 1 2p ,
2
2a
~19!
where k 5d 2 V cyl /du 2x u u50 . The equations of motion ~7! and
~8! take the form
S
a 2 ẋ5 k 2 e
function of the 1D Schrödinger operator Ĥ x with one node
and there must be another function corresponding to the
ground-state ‘‘wave function’’ with a negative eigenvalue.
We conclude that the operator Ĥ x has one negative and one
~18!
with j P @ 1.1530,2.7314# a universal constant and V 0 denotes
the depth of the potential V cyl . The result ~18! holds independently of the details of the pinning potential V cyl as long
as the driving force F is small, F!F c . We note that T 0
;F 2 ; the same dependence holds for a string with dissipative dynamics. For a massive string depinning from a linear
defect T 0 ;F, see Ref. 10.
The operator Ĥ x has only one negative eigenvalue: it can
be easily seen that in the limit L→` the function du x0 /dz is
an eigenfunction of the operator Ĥ x with zero eigenvalue. Its
derivative du x0 /dz can be understood as the ‘‘velocity of a
particle’’ moving in the potential U(u x0 ,u y0 ) with the ‘‘velocity’’ changing its sign once. Hence, du x0 /dz is an eigen-
j \F 2
\ Au l̃21 u
5
,
2pa
2p aV0
]2
]z2
D
~20!
p[Ĥ 0 p,
~21!
ṗ52Ĥ 0 x.
Hence, p5 a 2 Ĥ 21
0 ẋ and the Euclidean action of the vortex
near the equilibrium position can be written in the form @we
make use of Eq. ~6!#
S 0Eucl5
1
2
E
1\/2T
2\/2T
dt
E
1L/2
2L/2
S
dz x 2 a 2
]2
]t
2
D
Ĥ 21
0 1Ĥ 0 x.
~22!
The same procedure as above provides the variation of the
Euclidean action near the thermal saddle-point solution
S SEucl5
1
2
E
1\/2T
2\/2T
dt
E
1L/2
2L/2
S
dz d x 2 a 2
]2
]t2
D
Ĥ 21
y 1Ĥ x d x,
~23!
where the operators Ĥ x and Ĥ y are given by Eqs. ~15! and
~16!. Using the usual measures for harmonic oscillators in
the path integrals for S 0Eucl and S SEucl ~see Ref. 16!,
FIG. 2. The ‘‘potential’’ U y (z) for the 1D Schrödinger operator
Ĥ y @see Eq. ~16!#. The characteristic size of the potential is given by
D5 A2 e V 0 /F and its height is equal to k .
F S
d m 0 5 det 2 a 2
]2
]t
Ĥ 21
2 0
DG
1/2
dC 0a
)a A2 p \
~24!
3580
D. A. GOROKHOV AND G. BLATTER
and
F S
d m S 5 det 2 a 2
]2
]t2
Ĥ 21
y
DG
The double-prime sign reminds us that we excluded the
1/2
1/2
negative and zero eigenvalues of the operator ĥ 1/2
y ĥ x ĥ y .
dC Sb
)b A2 p \ ,
~25!
~here
and
are expansion coefficients over the systems
of normalized functions! and defining the dimensionless opC 0a
C Sb
erators ĥ 0 , ĥ x , and ĥ y as Ĥ 0 / k , Ĥ x / k , and Ĥ y / k , respectively @ k 5V 9cyl(0)#, we obtain for ImZ/Z
ImZ L
5
Z
2
A FE
S DG
S DF G
k
2pT
]u0
dz
]z
2L/2
3
U
2 1/2
1L/2
m x,0~ L !
m xy,0~ L !
U
3exp 2
T
2
2
det@ 2 ~ a / k !~ ] / ] t
2
2
2
2
2
! 1ĥ 20 #
U
1L/2
u0
z
2L/2
of ĥ 20 is continuous, m 0 (k)5(11 e k 2 / k ) 2 with the spectral
density r 0 (k). Using
2
dz.
V
4
A2 e V 0 0 ,
3
F
G5
~27!
erators ĥ x and
defined on the interval @ 2L/2,
1L/2# , L→`.
The eigenvalues of the operators 2( a 2 / k 2 )( ] 2 / ] t 2 )
1/2
ĥ 1/2
y ĥ x ĥ y
1/2
1ĥ 20 and 2( a 2 / k 2 )( ] 2 / ] t 2 )1ĥ 1/2
y ĥ x ĥ y are equal to
D
2paT
n
k\
1 m 0,a
and
S
2paT
n
k\
D
2
1 m xy,b ,
n50,61,62 . . . , ~28!
with m 0,a and m xy,b the eigenvalues of the operators ĥ 20 and
1/2
ĥ 1/2
y ĥ x ĥ y , respectively. Substituting these eigenvalues into
Eq. ~26! and calculating the product over n using ) `n51 (1
1x 2 /n 2 )5sinh(px)/px we obtain
A k S D
FE S D G
ImZ a L
5
Z
4\
3
2L/2
dz
]u0
]z
F S(
3exp
3
F
T
1
m x,0~ L !
U
exp 2
2p
T sin~ p T 0 /T ! m xy,0~ L !
1L/2
\k
2aT
a
G
1/2
Am 0,a 2 ( 9 Am xy,b
b
) 9 @ 12exp„2 ~ \ k / a T ! Am xy,b …#
DG
.
FE
5exp
G
dk d r ~ k ! lnf „m ~ k ! … ,
A k S *D
F E S D G F mm
HE d F
aT0
T
U
exp 2
2p
T
~30!
2\ 2
1L/2
2L/2
M
3
)
b51
dz
`
0
F
]u0
]z
L
sin~ p T 0 /T !
2 1/2
x,0~ L !
xy,0~ L !
G
1/2
X S
dk r ~ k ! ln 12exp 2
S
12exp 2
\k
Am ,
a T xy,b
DG
e
\k
11 k 2
aT
k
D CG J
21
~31!
,
with
\k
U * 5U2
2a
FE
`
0
S
D
M
e
dk d r ~ k ! 11 k 2 2 (
k
b51
Am xy,b
G
~32!
the quantum renormalized activation energy. Let us investigate the correction to the activation energy ~32! arising from
large k values. We will see that the large-k modes lead to an
ultraviolet divergence which we have to cut off by some
physical length scale. It is convenient to rewrite the integral
* k0 dk 8 d r (k 8 )(11 e k 8 2 / k ) as
E
k
0
S
dk 8 d r ~ k 8 ! 11
D
e 2
k8 5
k
N 0~ k !
(n
~ 11 e k 2n / k 2 !
N S~ k !
2
2 1/2
)a @ 12exp„2 ~ \ k / a T ! Am 0,a …#
b
)b
f ~ m xy,b !
3exp
2
2
zero eigenvalue of the operator 2 a 2 Ĥ 21
y ( ] / ] t )1Ĥ x .
m x,0(L) and m xy,0(L) are the ‘‘zero’’ eigenvalues of the op-
S
f ~ m 0,a !
3
is the activation barrier and L is the length of the string ~we
performed standard integration over the shift mode, see Refs.
17,18!. The prime in Eq. ~26! indicates that we exclude the
2
)a
with d r (k)5 r 0 (k)2 r S (k) we arrive at the final expression
for the decay rate.
,
1/2
! 1ĥ 1/2
y ĥ x ĥ y #
where (L→`)
E S ]] D
1/2
The positive part of the spectrum of ĥ 1/2
x ĥ y ĥ x consists of
M discrete eigenvalues m xy,b and a continuous part m xy (k)
5(11 e k 2 / k ) 2 with a spectral density r S (k). The spectrum
1/2
~26!
U5 e
C. Result for G
1/2
det@ 2 ~ a / k !~ ] / ] t
2
57
(n
~ 11 e q 2n / k 2 ! , ~33!
where k n and q n are trivially related to the eigenvalues of the
~29!
1/2
operators ĥ 20 and ĥ 1/2
y ĥ x ĥ y and N 0 (k) and N S (k) denote the
number of eigenvalues inside the interval @ 0,k # . Note that
1/2
N 0 (`)2N S (`)5M 12. The two operators ĥ 20 and ĥ 1/2
y ĥ x ĥ y
define two scattering problems. We define appropriate scattering states c k (z) and c q (z) with asymptotics e ikz and e iqz
(z→2`) and e ikz ,e iqz1 d (q) (z→1`), with d (q) the appropriate scattering phase shift. The general solutions of the
‘‘Schrödinger’’ equations can be written as C k (z)
57
THERMALLY ACTIVATED HALL CREEP OF FLUX . . .
5C1ck(z)1C2c2k(z) and C q (z)5C 81 c q (z)1C 82 c 2q (z).
Since we require that the ‘‘wave functions’’ satisfy the condition C p (6L/2)50, L@ Ae V 0 /F, we obtain the discrete
levels through the phase equations ~see also Ref. 19!
q n L1 d ~ q ! 5 p n,
k n L5 p n,
S D S
pn
L
.1
2
2pn
L
p n2 d ~ q !
L
2
2
D
2
2q
d~ q !.
L
d ~ q ! .1
Cutting off the integral in Eq. ~32! at large momentum k *
we obtain
U * .U2
~34!
and hence
k 2n 2q 2n 5
~35!
For q@ Ak / e we can make use of the semiclassical approxi1/2
mation for the operator ĥ 1/2
y ĥ x ĥ y and calculate the phase
shift directly: With the usual ansatz C(z);e i * q(z)dz the
function q(z) can be found from the equation
@ e q 2 ~ z ! 1U x #@ e q 2 ~ z ! 1U y # 5 k 2 E q ,
2
2 4
~36!
G5
2
1/2 1/2
continuous spectrum of ĥ 1/2
y ĥ x ĥ y . At large q we obtain
q~ z !.
Ake
S
E 1/4
12
q
~ U x 1U y !
4 k AE q
Similarly, for the operator ĥ 20 we obtain
k~ z !.
S
Ake
E 1/4
12
k
1
2 AE k
D
D
d~ q !.
Ake E
E 1/4
q
2L/2
dz
~38!
,
~ 2 k 2U x 2U y !
4 k AE q
.
Ae
2V 0 k
,
qF
E
0
S
D
e k 82
e
.
k
k
.
~40!
N 0~ k !
E
(n
~ k 2n 2q 2n !
k
A2 e V 0
0
dk 8
pF
G
F
U
T
~42!
1L/2
dz
2L/2
det~ ĥ 20 !
1/2
1/2
det9 ~ ĥ 1/2
y ĥ x ĥ y !
G5
kL
2pa
A k S DF E
2pT
exp 2
U
T
U
~39!
A2V 0 / e /FL. At large k the number of states
i.e.,
in the interval dk is equal to L dk/2p and the integral
dk 8 d r ~ k 8 ! 11
F
2pT
exp 2
L→`
k 2n 2q 2n 52 k
k
AV 0 e .
A k S DF E
3 Au m xy,21 u lim Am x,0~ L !
Taking into account that at large q, E q . e 2 q 4 / k 2 and that
U x (z),U y (z)! k for u z u , A2 e V 0 /F, we obtain
d~ q !.
A2 p a F
G
S DG
]u0
]z
2 1/2
1/2
~43!
.
Using the same technique as before, see Eq. ~26!, we arrive
at the more suitable form
and we arrive at the difference in the phase shift
1L/2
kL
2pa
m x,0~ L !
3
m xy,0~ L !
~37!
.
\kk*
The theory is self-consistent if the correction to the activation energy is small as compared to U ~here we have assumed the cutoff k * to be large such that we can neglect the
contribution from the bound states!.
In case of a continuous transition from quantum to classical behavior the result Eq. ~31! with Eq. ~32! is applicable
for any T.T 0 except for a narrow temperature interval
;\ 3/2 around T 0 ~see Ref. 20!. An explicit expression for the
decay rate can be obtained only for the case \→0, i.e., in the
classical limit. In this limit we can expand the exponents in
Eq. ~31!, 12exp(2\k/aT).\k/aT and the quantum correction to the activation energy ~32! tends to zero. Performing
these steps we obtain the result
where E q 5(11 e q / k ) . e q / k is the eigenvalue of the
2
3581
~41!
diverges linearly at large k ~for the case of a massive string
the correction to the activation diverges as lnk, see Refs.
10,18!. The integral then has to be cut off at some wave
vector k * . For a string the natural cutoff is p /r, with r the
radius of the string. For a vortex parallel to the c axis of an
anisotropic superconductor k * ; p /max(jc ,d), with j c the
coherence length in the c direction and d being the distance
between the superconducting layers.
1L/2
2L/2
dz
S DG
]u0
]z
det~ ĥ 0 !
2 1/2
UF G
1/2
det„ĥ x ~ L ! …
det~ ĥ 0 !
det~ ĥ y !
1/2
,
~44!
where m x,0(L) is the zero eigenvalue of the operator ĥ x (L).
The calculation of the instanton determinant ratios
~44! can be elegantly performed using the Gelfand-Yaglom
formula.21 The renormalized21 determinant det Ĥ of the
Schrödinger operator Ĥ52 ] 2 / ] z 2 1 p(z) is the solution of
the differential equation Ĥ f (z)50 with the initial conditions
f u z52L/250 and d f /dz u z52L/251. The value of the
function f (z5L/2) provides the renormalized determinant. As has been shown in Ref. 11, in the limit
F→0,
m x,0(L);(Fa/V 0 )exp@2Ak / e (L22 A2 e V 0 /F) # ,
det(ĥ 0 )/ u det„ĥ x (L)…u ;exp(Ak / e L), and det(ĥ 0 )/det(ĥ y )
;(Fa/V 0 )exp(2A2 k V 0 /F), where the coefficients of proportionality depend on the detailed form of the pinning potential and a is the characteristic radius of the pinning well.
The eigenvalue m xy,21 52 j 2 F 4 / k 2 V 20 , see Eq. ~18!, and we
make use of Eq. ~27!. Substituting these dependences into
Eq. ~44! we obtain
G5g
V 0L
aa
3
A S D
V0 F
e Fc
5/2
SD S
T̃
T
1/2
exp 2
D
U 2 A2 k V 0
1
,
T
F
~45!
3582
D. A. GOROKHOV AND G. BLATTER
where g is a dimensionless proportionality coefficient which
depends on the detailed form of the pinning potential, T̃
5a Ae V 0 , F c is the critical force, and the activation energy
U is given by Eq. ~27!. As an illustration we have explicitly
solved the differential equations for a pinning potential given
by the equation
F
V cyl5V 0 12cos2
S
D
G
p
Au 2 1u 2y u ~ a 2 2u 2x 2u 2y ! , ~46!
2a x
where u (t) is the step-function @ u (t)51 if t.0 and u (t)
50 otherwise#. The external force has the form F(u x )
5F u ( u u x u 2a).
We
find
m x,0(L)5(24Fa/ p V 0 )exp@2Ak / e (L
22 A2 e V 0 /F) # , det(ĥ 0 )/ u det(ĥ x ) u 5(1/2)exp(Ak / e L), and
det(ĥ 0 )/det(ĥ y )54Fa/ p V 0 exp(2A2 k V 0 /F), and substituting these expressions into Eq. ~44! we obtain g5 pj /25/4
.1.32j .
A real physical system will involve a finite cutoff k * ,
which we can account for in Eq. ~31! by a restricted integration over the modes k,k * . However, in making use of the
Gelfand-Yaglom formula we actually account for unphysical
fluctuations with k.k * . In order for the classical result ~45!
to be valid we then must require the cutoff k * to be sufficiently large such as to validate the Gelfand-Yaglom approach. On the other hand, k * has to be small enough in
order to arrive at a small quantum correction in Eq. ~32!.
Let us first specify the regime of applicability of our results. The zero-temperature Euclidean action takes the value
S Eucl(T50)5 a V, where V is the volume encircled by the
string in the course of the tunneling motion. In the x and y
directions the characteristic size of the loop is V 0 /F and the
length of the string segment along the z direction is of order
Ae V 0 /F, hence S Eucl(T50); a Ae /V 0 (V 0 /F) 3 . On the other
hand, S Eucl(T5T 0 )5\U/T 0 ;S Eucl(T50), i.e., even if the
transition from quantum to classical behavior at T c .T 0 is
first-order like, its temperature will still be of order T c ;T 0
and our results are applicable in the regime T.T 0 .
Next, we estimate the temperature T 0 using parameters
for
the
moderately
anisotropic
superonductor
YBa2 Cu3 O72 d . We choose our columnar defect in the form
of a cylindrical cavity of radius a. j ab (0), where j ab is the
coherence length in the ab plane. The expression for the
pinning potential then can be written in the form22
U~ r !5n
r2
e0
,
4 r 2 12 j 2ab
Substituting j ab 516 Å, l ab 51400 Å, l c /l ab 55, a
5 p \n, n5231021 cm23 , e 5 e 0 l 2ab /l 2c , and j .2, we obtain T 0 50.6 K• n (F/F c ) 2 , and \ k / a 515 K• n . Taking
into account that n might be ;0.1, see Ref. 23, we see that at
T;10215 K when the Hall force appears to be large,5 the
conditions T@T 0 ,\ k / a are well satisfied.
The ratio a / h .15 for T&15 K ~see Ref. 5! was obtained
from indirect measurements. In Ref. 24 direct measurements
of the Hall angle in 60 K YBa2 Cu3 O72 d are reported to yield
a ratio a / h .1. We wish to point out that even if a / h &1 it
is still possible to use Eq. ~45!: We have shown that the
inverse lifetime of a vortex pinned by a columnar defect
behaves at high temperatures as
S
G;F 5/2T 21/2exp 2
~47!
where e 0 5(F 0 /4p l ab ) 2 is the relevant energy scale and the
factor n (0, n ,1) describes the pinning efficiency factor,23
F 0 and l ab are the flux quantum and the penetration length
in the ab plane, respectively. For the model potential ~47!
the temperature T 0 , the critical force F c , and the temperature
\ k / a can be easily calculated ~the Hall coefficient a is related to the electron density n as a 5 p \n at T50 in the
superclean
limit
v 0 t @1):
T 0 5( nj /64p 2 )( e 0 /
2
2
n j ab )(F/F c ) , F c 5( A2 n /16) e 0 / j ab , \ k / a 5 n \ e 0 /4a j 2ab .
D
U 2 A2 k V 0
1
.
T
F
~48!
It is interesting to note that the same dependence of the decay rate on the external force and temperature has been obtained in Refs. 11 for a flux line governed by dissipative
dynamics. Moreover, the equation for the decay rate from
Refs. 11 can be obtained up to a numerical prefactor from
Eq. ~45! by the substitution a → h , indicating that in the
regimes a & h or a ! h it is possible to use Eq. ~45! with
a →max(a,h).
The scaling law ~48! for the decay rate leads then to the
resistivity scaling
r~ F !;
IV. CONCLUSION
57
FA
G S D
F
U
Fexp~ 2 A2 k V 0 /F ! exp 2
,
T
T
~49!
where the factor in square brackets represents the correction
to the standard result, which arises from the inclusion of
classical fluctuations around the saddle point ~prefactor!.
We note that Eq. ~48! is not valid for T.FU/2 A2 k V 0 as
the quasiclassical approximation is not applicable at these
large temperatures. It is well known that the problem of a
vortex pinned by a cylindrical potential is equivalent to that
of a 2D quantum particle trapped in a radially symmetric 2D
potential. The latter always has a bound state in 2D, and thus
the vortex is pinned by the defect at any temperature. However, the thermal fluctuations lead to a large downward
renormalization of the pinning energy at sufficiently high
temperatures.22,23
Finally, let us make an estimate of the cutoff momentum
k * . The theory developed above works well if k * @ Ak / e ;
with k * 5 p /d, where d512 Å is the distance between superconducting layers, we obtain k * / Ak / e .5, such that this
condition is marginally satisfied.
Briefly summarizing, we have considered the problem of
the thermally activated depinning of a flux line governed by
Hall dynamics from a columnar defect in the presence of a
small (F!F c ) external force. We have shown how to reduce
the 2D problem of the string motion to a 1D effective problem. The expression for the decay rate has been obtained for
the whole temperature region where the saddle-point solution
is time independent @see Eqs. ~31!, ~44!, and ~45! for high
temperatures#. An analytical expression for the classical asymptotics of G has been calculated for a model potential as
given by Eq. ~46!, and we have discussed possible applications of these results to high-T c superconductors.
57
THERMALLY ACTIVATED HALL CREEP OF FLUX . . .
3583
ACKNOWLEDGMENT
1/2
The image of the operator B̂5Ĥ 1/2
is
y l 21 u 21 &^ 21 u Ĥ y
We thank A. Yu Alekseev, V. V. Cheienov, and P. B.
Rodin for discussions and gratefully acknowledge financial
support from the Swiss National Foundation.
one-dimensional, i.e., B̂ is a self-adjoint projector of rank 1.
The operator
Â5Ĥ 1/2
y
APPENDIX A: EXISTENCE AND UNIQUENESS
We prove that a solution of Eq. ~17! with a negative eigenvalue exists. Second, we show the uniqueness of this solution. We mention that in the limit L→` the boundary condition c →0 for u z u →` is asymptotically satisfied.
S(
a Þ21
D
l a u a &^ a u 10• u 21 &^ 21 u Ĥ 1/2
~A6!
y
is a non-negative Hermitian operator with two zero eigenvalues. We perturb the operator  with a projector of rank 1 and
make use of the following theorem.25
Existence. As we already know, the operator Ĥ x has one
Theorem. Let  be a self-adjoint operator and B̂ a selfadjoint operator of rank 1. Take an interval D on the real
negative eigenvalue. We show that the operator Ĥ y Ĥ x has a
negative eigenvalue, too. It is possible to rewrite the equation
axis and denote by n(D) the number of eigenvalues of  in
Ĥ y Ĥ x c 5l̃c in the form
1B̂ within D satisfies the estimate:
1/2
21/2
Ĥ 1/2
c ! 5l̃~ Ĥ 21/2
c!
y Ĥ x Ĥ y ~ Ĥ y
y
~A1!
~the operator Ĥ y has only positive eigenvalues, hence, operators like Ĥ 11/2
or Ĥ 21/2
are well defined and Hermitian!. The
y
y
1/2
1/2
operator Ĥ y Ĥ x Ĥ y is also Hermitian and its lowest eigenvalue can be obtained by minimization of the functional
1/2
^ x u Ĥ 1/2
y Ĥ x Ĥ y u x &
F@x#5
^xux&
~A2!
in the space of functions with integrable modulus squared.
is self-conjugate and defining f
Using the fact that H 1/2
y
5Ĥ 1/2
y x , the functional can be rewritten as
F@f#5
this interval. Then the number m(D) of eigenvalues of Â
^ f u Ĥ x u f &
f u Ĥ 21/2
f&
^ Ĥ 21/2
y
y
5
^ f u Ĥ x u f &
^ f u Ĥ 21
y uf&
.
~A3!
The operator Ĥ x has a negative eigenvalue, i.e., there is a
function u f & such that ^ f u Ĥ x u f & ,0. The operator Ĥ y has
only positive eigenvalues, i.e., the form ^ f u Ĥ 21
y u f & is always positive. Consequently we constructed the Rayleigh-
n ~ D ! 21<m ~ D ! <n ~ D ! 11.
Applying this theorem to the interval (2`,0) ~we exclude
1/2
zero! one can easily see that the operator Ĥ 1/2
y Ĥ x Ĥ y has not
more than one negative eigenvalue. But the eigenvalues of
this operator are the same as those of the operator Ĥ y Ĥ x so
that our problem has only one negative eigenvalue.
APPENDIX B: NEGATIVE EIGENVALUE PROBLEM
We construct upper and lower bounds for the negative
eigenvalue l̃21 of the operator Ĥ x Ĥ y . In the limit F→0 the
lowest eigenvalues of the operators Ĥ x and Ĥ y do not depend
on the detailed form of the pinning potential.11 This is due to
the rapid decay of the eigenfunctions in the region u z u .D,
where the form of the potential is irrelevant. The lowest eigenvalue of the operator Ĥ x is equal to l 21 52 m 2 F 2 /2V 0 ,
with m the root of m tanhm51 ~see Refs. 11!. In the region
u z u ,D the normalized ‘‘ground-state’’ wave function of the
operator Ĥ x is
Ritz principle for the equation Ĥ y Ĥ x c 5l̃c and proved that
there is a function with a negative average value. The eigenvalue l̃21 is even lower, thus we have demonstrated the
existence of a negative eigenvalue for the problem ~17!.
Uniqueness. Let us rewrite the operator Ĥ x in the following form:
Ĥ x 5
(a l au a &^ a u 5 a Þ21
( laua& ^au1l21u21& ^21u,
f 21 ~ z ! 5
cosh~ Au l 21 u / e z !
ADcosh~ Au l 21 u / e D !
functions of the operator Ĥ y by those of the infinitely deep
quantum well
l 8n 5
p 2F 2 2
n ,
8V 0
f 8n ~ z ! 5
1
AD
cos
genvalue of the operator Ĥ x , the vector u 21 & denoting the
1/2
‘‘ground state’’of the operator Ĥ x . The operator Ĥ 1/2
y Ĥ x Ĥ y
can be rewritten in the form
1/2
1/2
Ĥ 1/2
y Ĥ x Ĥ y 5Ĥ y
(
a Þ21
l a u a &^ a u Ĥ 1/2
y
1/2
1Ĥ 1/2
y l 21 u 21 &^ 21 u Ĥ y [Â1B̂.
S
D
pn
p
z1 ~ n21 ! ,
2D
2
n51,2 . . . . ~B2!
Using the variational principle ~A3! we can write
l̃21 <
~A5!
~B1!
.
As the characteristic depth k of the ‘‘potential well’’ U y (z)
is much larger than the ‘‘size quantization energy’’ e /D 2
~see Fig. 2!, we can approximate the eigenvalues and eigen-
~A4!
where l a are the eigenvalues of Ĥ x , l 21 is the lowest ei-
~A7!
^ f 21 u Ĥ x u f 21 &
^ f 21 u Ĥ 21
y u f 21 &
52
m 2 F 2 /2V 0
(m
.
8 & u 2 /l m8 !
~ u ^ f 21 u f m
~B3!
D. A. GOROKHOV AND G. BLATTER
3584
Substituting the eigenvalues and eigenfunctions ~B2! into
Eq. ~B3! we obtain
l̃21 <2
m 2 F 4 /2V 20
`
(
m50
$ 8/@ m 2 1 p 2 ~ 2m11 ! 2 /4# %
As ^ f u Ĥ x u f & ,0, 2c 221 u l 21 u 1c 21 l 1 1c 22 l 2 1 •••,0, and
taking into account that ( n c 2n 51 we obtain
F4
521.3292 2 .
V0
2
1. u c 21 u .
~B4!
In order to obtain a lower estimate l̃21 >B.2` @see Eq.
~A3!#, we use the following inequality:
l̃21 5minf PL2 F @ f # >minCPD
l 21
^ f u Ĥ 21
y uf&
,
~B5!
where the functions belonging to D satisfy the conditions ~i!
^ f u f & 51, ~ii! ^ f u Ĥ x u f & ,0, and ~iii! f (z)5 f (2z). The
last condition is a consequence of the parity symmetry of the
operator Ĥ x Ĥ y . Following the discussion in Sec. III A, the
operator Ĥ x has one negative eigenvalue with an even eigenfunction. For any odd function f , ^ f u Ĥ x u f & >0, thus an odd
function cannot minimize the functional ~A3!, hence the
eigenfunction corresponding to the lowest eigenvalue of the
57
A
l1
50.9191,
u l 21 u 1l 1
~B8!
where we used the fact that l 21 521.4392e /D 2 and l 1
57.8309e /D 2 ~see Refs. 10,11!. As u ^ f̃ u f 81 & u < uu f̃ uuuu f 81 uu
< A12c 221 50.3941, we obtain u ^ f u f 81 & u 2 > f ( u c 21 u ,K),
where
f ~ u c 21 u ,K ! 5c 221 u ^ f 21 u f 18 & u 2 1K 2 22 u c 21 uu ^ f 21 u f 18 & u K.
~B9!
We note that u c 21 u P @ 0.9191,1 # and KP @ 0,0.3941# . A
simple analysis shows that the minimal value of f ( u c 21 u ,K)
is equal to f (0.9191,0.3941)50.119 and substituting this
value into Eqs. ~B5! and ~B6! we obtain
operator Ĥ x Ĥ y has to be even. Our goal is to find some
~large! constant b such that ^ C u Ĥ 21
y u C & .b.0 for any C
PD. We rewrite
^ f u Ĥ 21
y uf&5
(m
8 &u2
u^fufm
8
lm
>
u ^ f u f 81 & u 2
l 18
~B6!
and expanding f in the eigenfunctions of the operator Ĥ x ,
f 5 ( n521 c n f n [c 21 f 21 1 f̃ , we can write
u ^ f u f 81 & u
F4
l̃21 >27.4603 2 .
V0
Hence, we have shown that in the limit F→0 the lowest
eigenvalue l̃21 of the operator Ĥ x Ĥ y , l̃21 behaves as
2 g F 4 /V 20 with g P @ 1.3292,7.4603# . The temperature T 0 and
the eigenvalue l̃21 are connected by the equation T 0
5\ Au l̃21 u /2p a and we obtain the estimate
2
8 u f 18 & u 2 1 u ^ f̃ u f 18 & u 2 12c 21 ^ f 21 u f 18 &^ f̃ u f 18 &
5c 221 u ^ f 21
T 05
>c 221u^f21uf81&u21u^f̃uf81&u222uc 21u u^f21uf81&u u^f̃uf81&u.
~B7!
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