PHYSICAL REVIEW B
VOLUME 58, NUMBER 21
1 DECEMBER 1998-I
Nonadiabatic distortion in the current distribution around a moving vortex
D. M. Gaitonde
Mehta Research Institute, Chatnaag Road, Jhusi, Allahabad 211019, India
~Received 19 May 1998!
We study the phase distribution around a vortex in uniform motion. We consider both the cases of neutral
and charged superfluids. The motion of the vortex causes the density of the system to fluctuate. This in turn
produces a compensating current, thus ensuring current conservation locally. We explicitly calculate this
current. @S0163-1829~98!00942-4#
In recent years, there has been a great deal of interest ~see
Ref. 1 for a recent review! in the static and dynamic properties of vortices in superfluids and superconductors. A study
of high-temperature and other superconductors has led to the
discovery of several new phenomena such as flux-lattice
melting,2 quantum flux creep,3 anomalous sign change of the
Hall coefficient,4 and anomalous ac electromagnetic
response.5 This has led to a renewed upsurge in efforts to
understand the dynamics of moving vortices. Issues such as
the size and sign of the Magnus force6 and the vortex inertial
mass7 have been intensely studied. Theoretical studies of
these phenomena have been carried out, both within an effective Ginzburg-Landau theory8 and microscopically9
within the BCS theory.
However, most studies of vortex dynamics have tended to
assume the vortex motion to be adiabatic, i.e., the instantaneous distribution of the superconducting order parameter
around the vortex is assumed to be identical with that of a
vortex at rest. In this paper we address the question of
nonadiabatic distortions in the phase distribution around a
moving vortex. Our calculations are performed using a
Ginzburg-Landau phase-only functional. We solve the equations of motion to second order in the vortex velocity (u)
and thus obtain corrections to the phase distribution.
The vortex motion induces fluctuations in the instantaneous density distribution around the vortex. Current conservation therefore causes a compensating current to flow, thus
conserving the density locally. We explicitly calculate the
induced current, to second order in u for neutral as well as
charged superfluids. The constant current contours are found
to be distorted from their circular shape. However, there is
no change in the energy, to second order in u, coming from
the nonadiabatic distortions in the case of a neutral superfluid. For a charged superfluid the displacement current, associated with a time-dependent vector potential, contributes
to a change in the energy. This contribution is suppressed by
a factor of u 2 /c 2 and is therefore negligibly small. These
effects, thus have no consequences either for the Magnus
force or for the vortex mass.
The superfluid is described by the action functional per
unit length S 8 5S/L5 * dt * drW @ L u 1L em # , where
S
D
S
a1
2eAW
2eA 0 2 a 2
L u5
u̇ 2
2
¹u2
2
\
2
\c
D
2
1 gu̇
~1a!
and
0163-1829/98/58~21!/14183~4!/$15.00
PRB 58
L em 5
W / ] t !# 2 2 ~ ¹3AW ! 2
@ ¹A 0 1 ~ 1/c !~ ] A
.
8p
~1b!
Here u is the phase of the superconducting order parameter
W are the scalar and vector potentials associated
and A 0 and A
with the electromagnetic field. This functional has been derived microscopically10 and the coefficients a 1 and a 2 can
be related to appropriate polarizabilities of the underlying
fermionic system. The size and sign of the coefficient g,
which determines the Magnus force, remains controversial.
For the purposes of this paper, we treat the Ginzburg-Landau
coefficients as phenomenological parameters.
We will first consider the case of a neutral superfluid.
Accordingly, we put e50 in Eq. ~1! and switch off the electromagnetic fields. We then have
L ns 5
a1 2 a2
u̇ 2 ~ ¹ u ! 2 1 gu̇ .
2
2
~2!
The Euler-Lagrange equation of motion is easily found to be
a 1 ü 5 a 2 ¹ 2 u .
~3!
Notice that the term linear in u̇ corresponds to a topological
phase and makes no contribution to the equation of motion.
For a uniformly moving vortex we have u (rW ,t)
5 u (rW 2uW t). Substituting this in Eq. ~3! we get
a 1 uW •¹ ~ uW •¹ u ! 5 a 2 ¹• ~ ¹ u ! .
~4!
We seek a solution to Eq. ~4! of the form
u ~ rW ,t ! 5 u ~ 0 ! ~ rW 2uW t ! 1 u ~ 1 ! ~ rW 2uW t ! 1 u ~ 2 ! ~ rW 2uW t ! 1¯ ,
~5!
where u (m) (rW 2uW t) is of mth order in uW . Here ¹ u (0) (rW 8 )
5 f̂ 8 /r 8 ~where we define the coordinate rW 8 5rW 2uW t for convenience! is the phase distribution of a static vortex while the
remaining terms are nonadiabatic phase distortions induced
by the vortex motion. The quantization of the vorticity,
which stems from the single valuedness of the superconducting order parameter, ensures that for mÞ0 u (m) (rW 8 ) is nonsingular and ¹ u (m) is purely longitudinal.
Substituting Eq. ~5! in Eq. ~4! and equating terms of the
same order in u, we arrive at the result
14 183
©1998 The American Physical Society
14 184
BRIEF REPORTS
¹ 8 2 u ~ 1 ! ~ rW 8 ! 50
~6a!
and
a 1 uW •¹ 8 @ uW •¹ 8 u ~ 0 ! ~ rW 8 !# 5 a 2 ¹ 8 2 u ~ 2 ! ~ rW 8 ! .
~6b!
Since ¹ u (1) is constrained to be purely longitudinal, Eq. ~6a!
implies that it is zero. We therefore set u (1) 50.
Fourier transforming Eq. ~6b! with respect to rW 8 we solve
for u (2)
q to get
u ~q2 ! 5
22 p a 1
uW •ẑ3qW
uW •qW
a2
q4
~7a!
and find the corresponding current ¹ 8 u (2)
q to be
~7b!
It is easy to see from Eq. ~2! that this current, which is purely
longitudinal, does not contribute to the action to second order in u. This implies that both the Magnus force and the
vortex mass are unaffected by the existence of this current.
To get a better idea of the induced current distortion, we
Fourier transform ¹ 8 u q ~the details are outlined in the Appendix! to get
F
f̂ 8
a 1u 2
12
cos 2f 8
r8
2a2
G
~8!
for the x axis chosen to be directed along uW . It is easily seen
from Eq. ~8! that the circular symmetry of a static vortex is
destroyed by the extra current.
We now turn our attention to the charged case. The EulerLagrange equations of motion are now given by
a1
S
D
S
W ~ qW ! 5
A
~ 2uW •qW ! qW \uW •¹ u q
2
W W 2
2ec ~ q 2 l TF
11 !@ q 2 1l 22
L 2 ~ u •q /c ! #
1
~ \c/2e ! ¹ u q
,
2 2
q l L 112 ~ l L uW •qW /c ! 2
D
]
2e
2e
A 0 5 a 2 ¹• ¹ u 2
AW ,
u̇ 2
]t
\
\c
S
W / ] t !#
2e a 1
2e
¹• @ ¹A 0 1 ~ 1/c !~ ] A
52
u̇ 2 A 0
4p
\
\
~9a!
D
~9b!
S
S
W
1 ]A
¹A 0 1
c ]t
D
2e
2e
AW .
5a2
¹u2
\c
\c
\ uW •¹ 8 u qW
,
2
2e q 2 l TF
11
2
q 2 l TF
11
5 a 2 qW •¹ 8 u q .
~12!
u ~q2 ! 52
2
uW •ẑ3qW q 2 l TF
2pa1
Wu •qW
.
2
a2
q 4 q 2 l TF
11
~13!
Notice that as e→0 and l TF→`, Eq. ~13! recovers the earlier result of Eq. ~7a! for a neutral superfluid.
Substituting Eq. ~13! in Eq. ~11! we find, to second order
in u, the vector potential to be
AW ~ qW ! 5
F
G
l 2L ~ uW •qW ! 2
\c 2 p ẑ3qW
11 2 2 2
.
2e iq 2 q 2 l 2L 11
c q l L 11
~14!
As expected, the vector potential is purely transverse as the
longitudinal terms exactly cancel, consistent with our gauge
condition.
Making use of Eqs. ~13! and ~14! we find that the current
density Wj (rW 8 )5 @ ¹ 8 u 2 (2e/\c)AW # is given by Wj 5 Wj a 1 Wj b
1 Wj c where
Wj a ~ qW ! 5
2 p ẑ3qW q 2 l 2L
iq 2 q 2 l 2L 11
~15a!
is the current density of a vortex at rest,
D
Wj b ~ qW ! 5
~9c!
W 50. Then, on Fourier transWe choose the gauge ¹•A
forming Eq. ~9b! we find
A 0 ~ qW ! 5
2
q 2 l TF
We put u 5 u (0) 1 u (1) 1 u (2) 1¯ as before. We once again
find u (1) to be zero and get
and
1 ]
¹3 ~ ¹3AW !
1
4p
4pc ]t
~11!
where the penetration depth l L is given by 4 p a 2 (2e/
\c) 2 5l 22
L . In writing Eq. ~11!, we have made use of the
relations AW (rW ,t)5AW (rW 2uW t) and A 0 (rW ,t)5A 0 (rW 2uW t). All
Fourier transforms have been performed with respect to the
co-ordinate rW 8 5rW 2uW t as before.
We now substitute our result for A 0 @Eq. ~10!# in Eq. ~9a!
and make use of our gauge condition to arrive at
a 1 uW •qW uW •¹ 8 u q
uW •ẑ3qW
2pa1
qW ~ uW •qW !
.
¹ 8 u ~q2 ! 5
a2
iq 4
¹ 8 u ~ rW 8 ! 5
PRB 58
~10!
where the Thomas-Fermi screening length is given by
22
. On Fourier transforming Eq. ~9c! and
4 p a 1 (2e/\) 2 5l TF
making use of Eq. ~10! we have
2 p l 2L ẑ3qW q 2 l 2L
iq 2 c 2 ~ q 2 l 2L 11 ! 2
~ uW •qW ! 2
~15b!
corresponds to the ‘‘displacement current’’ of the moving
flux, and
Wj c ~ qW ! 5
2 p l 2L
iq 2 c 2
qW ~ uW •qW !
uW •ẑ3qW
2
q 2 l TF
11
~15c!
is the induced current necessary for local charge conservation.
We now proceed to evaluate Wj (rW 8 ). Fourier transforming
Wj a (qW ) we find ~see the Appendix for details!
Wj a ~ rW 8 ! 5
f̂ 8
K ~ r 8 /l L ! .
lL 1
~16!
PRB 58
BRIEF REPORTS
which is the usual result for a vortex at rest. Here K m (x) are
the modified Bessel functions. In a similar manner, we evaluate Wj b (rW 8 ) and Wj c (rW 8 ), details of which have been provided in
the Appendix.
We find
Wj b ~ rW 8 ! 5
u2
4c
2
nal. The occurrence of the factor u 2 /c 2 is indicative of the
electromagnetic origin of this effect. These currents are,
however, strongly suppressed by the factors of u 2 /c 2 as well
as efficient Coulomb screening in the case of j c , making
their observation a difficult task.
APPENDIX
$ @~ 21cos 2f 8 ! I 1 2cos 2f 8 I 2 # f̂ 8
1sin 2f 8 ~ I 1 1I 2 ! r̂ 8 % ,
14 185
~17!
In this appendix, we outline the derivation of the results
stated above for the current Wj (rW 8 ). We will first take up the
case of a neutral superfluid. Then we have
where
F
1
r8
I 15
3K 1 ~ r 8 /l L ! 1 K 81 ~ r 8 /l L !
2l L
lL
G
~18a!
and
I 25
r8
2l 2L
K 2 ~ r 8 /l L ! .
~18b!
2 p ẑ3qW
iq 2
u 2 l 2L
4c 2
@~ I 4 2I 3 ! sin 2f 8 r̂ 8 2 ~ I 3 1I 4 ! cos 2f 8 f̂ 8 # .
2pa1
uW •ẑ3qW
qW ~ uW •qW !
,
a2
iq 4
E
dqW
~ 2p !2
exp~ iqW •rW 8 ! ¹ 8 u q .
~19!
¹ 8 u ~ rW 8 ! a 5
I 3 '0
~20a!
and
8
r 83
@ 12J 0 ~ r 8 / j !# 2
8
r 82j
J 1~ r 8/ j ! 1
4
r 8j 2
J 18 ~ r 8 / j !
~20b!
and J m (x) are Bessel functions. The smallness of I 3 and I 4
stems from the small screening length (l TF) associated with
j c @see Eq. ~15c!#. This length is much smaller than the coherence length j, which is the coarse-graining scale beyond
which a description in terms of a phase-only action functional is applicable. Notice also that both j b and j c are
strongly suppressed by factors of u 2 /c 2 making their observation a difficult task.
We now consider the drawbacks of our calculation. The
phase-only action functional considered by us is applicable
only for T!T c as normal currents which are important near
T c have been ignored. However, the solution of the corresponding nonlinear problem is beyond the scope of our
work. We have also ignored effects arising from the vortex
core, whose proper inclusion would require a use of microscopic theory.
Finally, we summarize the main results of this paper. We
have calculated the nonadiabatic phase distortion of a moving vortex. Vortex motion induces time-dependent density
fluctuations, which in turn gives rise to additional currents
necessary to ensure local charge conservation. For the neutral superfluid, the extra supercurrent is purely longitudinal.
In the charged case there are two distinct contributions. One
coming from the ‘‘displacement current’’ ~j b above! is
purely transverse whereas the other ~j c above! is longitudi-
~A1!
~A2!
The first term in Eq. ~A1! is easily Fourier transformed to
yield the standard result
Here
I 4'
1
where the first term on the right-hand side is the usual current of a vortex at rest and the second term is the distortion
induced by the vortex motion. Thus, the current in real space
is given by
¹ 8 u ~ rW 8 ! 5
Similarly,
Wj c ~ rW 8 ! 5
¹ 8 u q5
f̂ 8
r8
~A3!
.
We find, on performing the angular integrals in Eq. ~A2!, the
second term in the current to be
¹ 8 u ~ rW 8 ! b 5
2 a 1u 2
4a2
E
dq $ r̂ 8 sin 2u @ J 3 ~ qr 8 ! 2J 1 ~ qr 8 !#
1 f̂ 8 cos 2u @ J 3 ~ qr 8 ! 1J 1 ~ qr 8 !# % ,
~A4!
where u is the angle uW makes with respect to rW 8 . The integrals in Eq. ~A4! can now be performed to yield
¹ 8 u ~ rW 8 ! b 5
2 a 1u 2
f̂ 8
.
cos 2u
2a2
r8
~A5!
On adding the results of Eqs. ~A3! and ~A5! and choosing uW
to lie along the x axis we finally arrive at Eq. ~8! above.
We now consider the charged case. In this case, the current Wj (qW ) is the sum of three different contributions Wj a , Wj b ,
and Wj c . The corresponding Fourier transforms Wj a (qW ), Wj b (qW ),
and Wj c (qW ) have been obtained earlier @Eq. ~15!#. We will
evaluate each piece separately:
Wj a ~ rW 8 ! 5
E
dqW
~2p!
exp~ iqW •rW 8 !
2
2 p ẑ3qW q 2 l 2L
iq 2 q 2 l 2L 11
~A6!
is the usual current associated with a static vortex. On performing the angular integration we get
Wj a ~ rW 8 ! 5 f̂ 8
E
dq
q 2 J 1 ~ qr 8 !
q 2 1l 22
L
5
f̂ 8
K ~ r 8 /l L ! .
lL 1
~A7!
BRIEF REPORTS
14 186
PRB 58
We now consider Wj b (rW 8 ):
Wj b ~ rW 8 ! 5
E
dqW
~ 2p !2
exp~ iqW •rW 8 !
2 p l 2L ẑ3qW q 2 l 2L
iq 2 c 2 ~ q 2 l 2L 11 ! 2
I 45
~ uW •qW ! 2 .
E
dq
E
dq
I 15
~A9a!
~ q 2 l 2L 11 ! 2
and
I 25
q 4 l 4L J 3 ~ qr !
~ q 2 l 2L 11 ! 2
I 1 can be rewritten as
2l 3L ]
I 15
2 ]lL
E
~A9b!
.
F
G
2l 3L ] K 1 ~ r 8 /l L !
dq 2 2
5
2 ]lL
q l L 11
l 3L
~A10!
q 2 J 1 ~ qr 8 !
I 35
Wj c ~ rW 8 ! 5
E
dqW
~ 2p !2
exp~ iqW •rW 8 !
iq 2 c 2
qW ~ uW •qW !
I 45
E
dq
On performing the angular integration in Eq. ~A11! we arrive
at the result of Eq. ~19! where
I 35
E
dq
q 2 J 1 ~ qr 8 !
2
q 2 l TF
11
1
G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin,
and V. M. Vinokur, Rev. Mod. Phys. 66, 1125 ~1994!.
2
A. Oral et al., Phys. Rev. Lett. 80, 3610 ~1998!, and references
therein.
3
A. F. Th. Hoekstra et al., Phys. Rev. Lett. 80, 4293 ~1998!.
4
T. Nagaoka et al., Phys. Rev. Lett. 80, 3594 ~1998!; S. Bhattacharya, M. J. Higgins, and T. V. Ramakrishnan, ibid. 73, 1699
~1994!.
5
S. Spielman et al., Phys. Rev. Lett. 73, 1537 ~1994!.
q 2 J 1 ~ qr 8 !
2
q 2 l TF
11
I 45
1
3
l TF
14
~A12b!
.
1
3
l TF
K 1 ~ r 8 /l TF! .
~A13!
22
E
] 8
]
r
dq
q @ J 0 ~ qr 8 ! 22J 18 ~ qr 8 !#
2
q 2 l TF
11
K 1 ~ r 8 /l TF! 22
]2
]r8
2
E
dq
]
]r8
F
J 1 ~ qr 8 !
2
q 2 l TF
11
K 0 ~ r 8 /l TF!
2
l TF
.
~A14!
G
~A15!
.
We ignore the first two terms on the right-hand side of Eq.
~A15! as they are exponentially small and in view of the
small screening length, approximate the last integral by
E
~A12a!
and
2
q 2 l TF
11
This expression can be further simplified to yield
uW •ẑ3qW
.
2
q 2 l TF
11
~A11!
q 2 J 3 ~ qr 8 !
The Thomas-Fermi screening length l TF! j where j is the
superconducting coherence length beyond which our coarsegrained picture based on a phase-only functional is valid.
Thus K 1 (r 8 /l TF)'exp(2r8/lTF)→0 in this (r. j ) regime.
To evaluate I 4 we make use of the relation J m21 (x)
8 (x) which is obeyed by Bessel functions.
2J m11 (x)52J m
Using this relation, we find that I 4 can be rewritten as
which reduces to the result of Eq. ~18a!. I 2 is directly evaluated to yield the result stated earlier in Eq. ~18b!.
Finally we consider Wj c (rW 8 ):
2 p l 2L
dq
I 3 is easily evaluated to yield
~A8!
On performing the angular integral we arrive at Eq. ~17!
above where
q 4 l 4L J 1 ~ qr !
E
dq
J 1 ~ qr 8 !
2
11 !
~ q 2 l TF
'
E
j 21
0
dqJ 1 ~ qr 8 ! 5
1
r8
@ 12J 0 ~ r 8 / j !# .
~A16!
Substituting these approximations in Eq. ~A15! we finally
arrive at the result of Eq. ~20b!.
6
D. J. Thouless, P. Ao, and Q. Niu, Phys. Rev. Lett. 76, 3758
~1996!; G. E. Volovick, JETP Lett. 62, 66 ~1995!.
7
D. M. Gaitonde and T. V. Ramakrishnan, Phys. Rev. B 56, 11
951 ~1997!, and references therein.
8
J-M. Duan, Phys. Rev. B 48, 333 ~1993!.
9
A. van Otterlo, M. V. Feigelman, V. B. Geshkenbein, and G.
Blatter, Phys. Rev. Lett. 75, 3736 ~1995!.
10
T. V. Ramakrishnan, Phys. Scr. T27, 24 ~1989!; I. J. R. Aitchison
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