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New vortex state in the presence of a long Josephson junction
Physica C 316 Ž1999. 1–12
New vortex state in the presence of a long Josephson junction D. Agassi ) , J.R. Cullen Code 68, Carderock DiÕision, NaÕal Surface Warfare Center, 9500 MacArthur BlÕd. West Bethesda, MD 20817, USA Received 1 February 1999; accepted 10 February 1999
Abstract It is shown that there are two vortex states for the configuration of an Abrikosov vortex in the presence of a long Josephson junction. One state, dubbed ‘conventional’, carries one magnetic-flux quantum F 0 , as does the generating Abrikosov vortex. The second and new vortex state entails in addition to the generating vortex two Josephson vortices with opposite polarity to that of the generating-vortex and its magnetic flux is yF 0 . As a function of the vortex core and Josephson junction–plane distance, the accompanying pinning potentials are quasi-degenerate in the near proximity to the Josephson junction plane, where the conventional-vortex-state pinning potential is the lower. Conditions facilitating the experimental observation of the new vortex state and a specific signature are briefly discussed. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Josephson junction; Vortex state; Abrikosov vortex
1. Introduction The structure of a magnetic vortex threading an extended, isotropic superconductor—the Abrikosov vortex—is a standard textbook result w1,2x. The associated screening currents encircle the vortex core in trajectories of cylindrical symmetry, completely attenuate the vortex magnetic field over a London penetration depth Ž l. distance from the core, and the entire current distribution carries one flux quantum F 0 s p "cr< e <. In the special case where the vortex core is located at a Josephson junction ŽJJ., the coreencircling screening currents follow elongated-shape trajectories aligned with the JJ-plane direction w3x so that along the JJ plane the magnetic field is com-
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pletely attenuated over another characteristic length, i.e., the Josephson-penetration-depth l JJ Ž l JJ ) l.. This particular vortex—the Josephson vortex—is qualitatively distinct from the Abrikosov vortex in that the order parameter does not vanishes at its core and it involves the order-parameter phase w4,5x. In an intermediate situation where the vortex core is located at a finite distance x 0 from the JJ plane ŽFig. 1., the encircling screening currents have to screen both the field around the vortex core and that field which has reached the JJ plane and extends along it Ži.e., along the y-axis in Fig. 1.. Consequently, the screening current trajectories shape is a mix of the above two basic shapes and the resulting vortex state may be interpreted as an admixture of the Abrikosov and Josephson vortex states. It is shown here that there are only two such admixed vortex states: The first is Abrikosov-like in the sense that it carries magnetic flux F 0 as does the generating Abrikosov
0921-4534r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 1 9 8 - 7
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D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
facilitating an experimental test of the new vortex state.
2. Phase-augmented London equation The issue at hand is the calculation of the pinning potential experienced by a probe vortex located at a distance x 0 from a finite-width, infinitely long homogeneous JJ ŽFig. 1.. To keep the discussion focused and unencumbered, this section is subdivided into subsections and many mathematical details are deferred to the appendices. 2.1. The phase-augmented London equation
Fig. 1. The homogeneous finite width Josephson junction configuration and the coordinate system used in this work. The probe vortex is located at Ž x 0 , y 0 s 0. and the hatched stripe represents the JJ buffer layer. Also indicated are the left and right superconductors’ phases, f L and f R , respectively ŽEq. Ž2.3.. and the gauge-invariant phase-discontinuity x Ž y . ŽEq. Ž2.6...
vortex. The second one is a new type of vortex state which carries a magnetic flux yF 0 since its screening currents overscreening the generating vortex field along the JJ such that the ensuing magnetic field distribution entails two additional Josephson vortices with polarity opposite to that of the generating vortex. As will be shown in this work, the free energy of the exotic state is higher than that of the conventional state. However ŽFig. 6., for a vortex core located within a distance on the order of the coherence length from the JJ plane, the free energies associated with these two vortex states are quasi-degenerate. A conceivable experimental test of the exotic vortex state is looking in the immediate proximity to the JJ plane for random-telegraph flux noise. The paper is organized as follows: In Section 2, the London equation is first augmented to account for the presence of a JJ by appropriate boundary conditions. The augmented equation predicts the existence of two vortex states which are then calculated. Some of the features and consequences of the two states are presented and discussed in Section 3, as well as a brief discussion for the conditions
Without loss of generality, the probe vortex location is chosen at ™ r 0 s Ž x 0 ,0. and x 0 - 0 Žsee Fig. 1.. Assuming that the probe vortex points in the positive z direction, the London equations for the ‘L’ and ‘R’ banks are E2 B X y l2
F0s
Ex2
E2 q
E y2
BX s F 0 d Ž ™ r y™ r 0 . , X s L,R
p "c < e<
Ž 2.1 .
where BX Ž x, y . is the z-component of the magnetic field generated by the probe vortex at the X bank. In Eq. Ž2.1. and hereafter, the cgs unit system is employed and the signs correspond to ‘electron’-superconductivity. Eq. Ž2.1. is supplemented by boundary conditions as follows. The continuity of the tangential-Hz component across a ‘thin’ JJ buffer-layer Žsmall d JJ , Fig. 1. implies w6x Dw B x s0
Ž 2.2 .
where the notation for the x-discontinuity across the junction of an arbitrary function f Ž x, y . is Dw f Ž x, y .x s f Ž x s d JJ q e , y . y f Ž x s ye , y . with e ™ 0. A second boundary condition follows from the fundamental, T s 0 K, gauge-invariant constitutive relation w7,8x ™
js s y
c 4pl
™
2
Aq
F0 2p
™
=f
Ž 2.3 .
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12 ™
™
where js is the supercurrent density, A denotes the electromagnetic potential and f is the order-parameter phase. In the augmented London approximation the superconductor is assumed to be a perfect, quiescent Ž f constant. diamagnet except at its interface where f changes discontinuously.1 Correspondingly, the j y-discontinuity implied by Eq. Ž2.3. is 4p y c
D l2 j y sD A y q
F 0 Ec 2p E y
Ž 2.4 .
E AyŽ x, y. Ex
xs x M
ž
s d JJ Bz Ž x M , y . q
E AxŽ x, y. Ey
xs x M
/
Ex
s d JJ B Ž x M , y . q
x Ž y. sc Ž y. q fc Ž y. q
2
F 0 Ex Ž y . 2p
Ey
d JJ
d xA x Ž x , y .
2
`
Hy`d k
DŽ k y . s
(
k 2y q
1
1
lk y y 2
2 l D Ž k y . q d JJ
½
1
l2
ik y x Ž k y . 2p
5
s yjcJJ sin x Ž y . ,
,
`
H dk e 2p y` y
lk y y
x Žky..
Ž 2.8 .
In the particular d JJ s 0 limit, which is the only one considered hereafter, the expressions for the total fields are
Ž 2.6 .
B Ltot Ž x , y . s
where x Ž y . is the gauge-invariant phase-discontinuity across the JJ w9,10x. In the d JJ ™ 0 limit and no order-parameter-phase dependence Eq. Ž2.6. reduces to the boundary condition DŽ l2 j y . s 0. This boundary condition has been previously employed for superconductor–superconductor interfaces in the context of the London approximation w6x.
Since js is a gauge invariant construct and f is assumed to change discontinuously only at the interface, the burden of describing spatially varying screening currents is carried by the ™ electromagnetic potential A.
F0 4pl2
elk y y
`
Hy`d k
y
DŽ k y .
½
= eyD Ž k y . < xyx 0 < ye D Ž k y . x
BRtot Ž x , y . s 1
y k ye
= eyD Ž k y . < x 0 < y
x Ž y. s
,
d JJ A x Ž x M , y .
H "c 0
where denotes the Josephson critical current. The negative sign in Eq. Ž2.7. is needed to assures consistency with the Josephson current–phase relation, i.e., j2 ™ 1 s jcJJ sinŽ f 2 y f 1 . w17x. The solutions of Eq. Ž2.1. which satisfy the boundary conditions Ž2.2., Ž2.6. and Ž2.7. are obtained by the Fourier transform method Žsee Appendix A.. The ensuing equation for x is 8p
where 0 F x M F d JJ . In particular, for small d JJ values when the x-variation of BX may be neglected, insertion of Eq. Ž2.5. into Eq. Ž2.4. yields the gaugeinvariant boundary condition
Ž 2.7 .
jcJJ
iF 0 c
Ž 2.5 .
EB
j x Ž y . s j x Ž x s ye ; y . s j x Ž x s d JJ q e ; y . s yjcJJ sin x Ž y .
The discontinuity Dw A y x, in turn, can be is reex™ ™ pressed by employing curlw A x s B from the Maxwell equations. This gives D A y sd JJ
The boundary condition Ž2.6. involves the gauge invariant phase-discontinuity x Ž y . which must be determined simultaneously with the fields BX Ž x, y .. To accomplish this, one additional relation is needed. A natural choice for that relation results from imposing the Josephson current-phase relation w11–16x
,
c Ž y. sD f Ž x, y. .
D l2
3
F0 4pl2
½
ž
lk y x Ž k y .
`
Hy`
dky
2p
/5
,
x-0
e l k y yy DŽ k y . x DŽ k y .
= eyD Ž k y . < x 0 < y
ik y x Ž k y . 2p
5
,
x)0
Ž 2.9 .
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
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and Eq. Ž2.8. takes the known dimensionless form w11–16x p
p
˙
`
Hy`dh
X
E 2x Ž hX .
BJJtot Ž yrl 4 1 . s y
K 0 w p
E 2hX
s sin x Ž h . y 2 p
h
2
(J
2 2 0 qh
(
=K 1 p J 02 q h 2 .
Ž 2.10 .
The symbols in Eq. Ž2.10. denote
J0 s
p2 s
l2JJ s
x0
lJJ
hs
,
2
lJJ
ž /
4
cF 0 16p
lJJ
l jcJJ
,
Eh
s 0,
4pl
Ey
Ž 2.13 .
where BRtot Ž x s 0, yrl 4 1. s BLtot Ž x s 0, yrl 4 1. s BJJtot Ž y .. Hence, the assumed field-vanishing at yrl 4 1 implies the vanishing of x ’s derivative, as in Eq. Ž2.12.. In addition, by Ampere’s law, the magnetic field vanishing at < y
l
jdep s
jcJJ
G 1, cF 0
12'3 p 2l2j 0
Ž 2.11 .
where jdep and K nŽ z . are the depairing current w18x and modified Bessel function of order n, respectively. Eq. Ž2.10. is a generalization of the standard differential equation describing the spatial variation of x w19,20x for the case where the magnetic-field source is a Abrikosov vortex. The two parameters in Eq. Ž2.10. are the probe vortex location ŽFig. 1., given by J 0 , and a measure of the JJ-link strength given by ‘ p’. A weak link implies p 4 1. Since the JJ link-strength cannot exceed that of a perfect superconductor, p ) 1 in all cases; the p - 1 regime is not physical w21,22x. Physical solutions of the central Eq. Ž2.10. must satisfy the following two boundary conditions: Ex Ž h .
F 0 Ex Ž y .
,
3'3 j 0 jdep
s
l
2
y
evaluating Eq. Ž2.9. in the yrl 4 1 limit, see Eq. Ž3.1.. The latter yields w23x
x Ž yh . s yx Ž h . .
The method for solving Eq. Ž2.10. is based on the observation that the source term dominates the RHS for
Ž 2.12 .
< h
The h anti-symmetry condition in Eq. Ž2.12. reflects the obvious anti symmetry of the vortex encircling currents under the h ™ yh transformation. This is also apparent in the source term of Eq. Ž2.10.. The derivative boundary condition in Eq. Ž2.12. expresses the premise that at a large distance along the JJ plane from the vortex core the magnetic field vanishes. This boundary condition is derived by
Fig. 2. An example of the factors in Eq. Ž2.10. for the parameters in the header ŽEq. Ž2.11... The chosen J 0 corresponds to values ˚ 1500 A, ˚ 24. The solid and broken lines are x 0 , l, p4 s 100 A, log 10 Ž< S <., where S is the source term on the RHS of Eq. Ž2.10., and x NL Žh .rp of Eq. Ž2.18..
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
‘E 2xrEh 2 ’-factor from under the LHS-integral in Eq. Ž2.10.. Noting that w24x ` x d x K0 s pl . Ž 2.14 . l y`
H
Eq. Ž2.10. transforms into the pendulum equation w3x
E 2x ) Ž h . Eh 2
s sin x ) Ž h . ,
for h G hc .
Ž 2.15 .
The general solutions of the pendulum equation are the Elliptic functions w25x Žsee Appendix B.. The particular solutions which are physical for the problem at hand are determined by the derivative boundary-condition at h s ` of Eq. Ž2.12.. According to Eq. ŽB.1., the only such solutions are
x ) Ž h y h 0 ; w ,n .
Eh w s "1,
2w s
cosh w h y h 0 x
n s . . . , y1, 0, 1, . . .
Out of the infinity of solutions Ž2.16., the physically meaningful ones are selected by matching at h s hc to solutions of Eq. Ž2.10. in the inner interval 0 - h F hc to which we turn now. In the inner interval 0 - h F hc all terms in Eq. Ž2.10. contribute. Insight into the solution there, hereafter denoted by x - Žh ., is obtained by consid- Ž . ering the approximant x - Žh . f xsource h defined as solution of Eq. Ž2.10. where on the RHS only the source term is retained p
`
H dh p y`
X
E 2xsource Ž hX .
s y2 p 2
E 2hX h
(J
2 2 0 qh
K 0 w p
(
K 1 p J 02 q h 2 .
Ž 2.17 .
As implied by the example plotted in Fig. 2, this approximate equation is valid at least in the 0 - h 1rp interval. The general, exact solution of Eq. Ž2.17. is Žsee Appendix D and Fig. 2.
s 2 w arctg sinh w h y h 0 x q Ž 2 n q 1 . p ,
Ex ) Ž h y h 0 ; w ,n .
5
,
Ž 2.16 .
and x ) Žh s `; w , n. s 2p n where n is an integer. Each solution in this class of solutions depends on three parameters: The ‘w ’ parameter, which specifies a family of branches sharing a common derivative sign w , the ‘n’ parameter, which specifies the particular branch in a given w-family and the ‘h 0 ’ parameter which is a reference point. Fig. 3 shows two branches of each of the two w-families in Eq. Ž2.16..
Fig. 3. Representative branches of x ) Žh yh 0 ; w , n., Eq. Ž2.16.. The solid and broken lines depict w sy1 and w s1 branches, respectively, with the indicated n values.
xsource Ž h ; A,C . s x NL Ž h . q Ah q C,
h
x NL Ž h . s 2 p J 0
H0
x NL Ž ` . s p eyp J 0
dh
(
K 1 p J 02 q h 2
(J
2 2 0 qh
,
Ž 2.18 .
where the boundary condition x Žh s 0. s 0 in Eq. Ž2.12. mandates C s 0 in Eq. Ž2.18.. It is now possible to construct approximate solutions of Eq. Ž2.10. in the entire 0 F h F ` interval - Ž . by matching x ) Žh . of Eq. Ž2.16. and xsource h of Eq. Ž2.18. at h s hc . Given a choice of Ž w , n., a solution x ) Žh y h 0 ; w , n. depends only on the h 0 - Ž . parameter, while xsource h depends only on the ‘ A’ parameter. These are determined by matching x ) and x - and their derivatives at h s hc . A crude assessment which Ž w , n. are viable choices is obtained as follows. Since < E x ) rEh < F 2 ŽEq. Ž2.16.., and x NL Žh . rises and quickly flattens for h - hc ŽFig. 2., matching the x ) and x - derivatives implies that < A < F 2. Furthermore, since 0 - x NL Žh . - p ŽEq. Ž2.18.. and typically hc f 2, matching the x ) and x - values implies that
6
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
the choice hc f 1.5 the above consideration rules out all but the two Ž n s 0; w s "1. branches. Numerical solution of the matching conditions at hc confirms that there are only two viable solutions. Moreover, the above approach for constructing approximate solutions suggests a converging algorithm for the construction of exact solutions to Eq. Ž2.10. which is detailed in Appendix C. This algorithm also yields only two numerically exact solutions, in keeping with the crude consideration above and the qualitative arguments given in the Introduction section. The existence of only two solutions to the phase-discontinuity Eq. Ž2.10. which satisfy the boundary condition Ž2.12. is the central result of this work. An example of numerically exact x Žh ; w s "1, n s 0. solutions is shown in Fig. 4. Note from Fig. 4 and Eq. Ž2.13. that the in-JJ-plane magnetic field at h 4 1 is positive and negative for the w s y1 and the w s 1 solutions, respectively. A positive magnetic field conforms with the chosen direction of the probe vortex along the positive z-axis direction ŽEq. Ž2.1... In this sense the w s y1 solution has the expected polarity and hence is dubbed ‘conventional’. On the other hand, the w s 1 solution has a negative magnetic field at h 4 1, i.e., a field polarity opposite to that of the probe vortex. This unusual property implies overscreening the generating-vortex magnetic field leading to a net field in the direction opposite to that of the generating vortex. For this reason this solution is dubbed ‘exotic’. Furthermore, since for the exotic state x Ž
reverses twice its sign in the y` F h F ` interval. This feature implies that the exotic state current distribution entails two Josephson vortices. These two x-solutions are further discussed in Section 3. 2.3. The pinning potential For a finite-width JJ Ž d JJ G 0, Fig. 1. the pinning potential per unit vortex length, UP Ž 0 .rL w6x, is given by ŽAppendix D. UP Ž ™ r0 . L s
F0 8p y
B ind ™ r s™ r 0 ;r™0
/
F0
tot
ž
16p q
"jcJJ 2
` 2
Hy`d y B Ž x s x
™ M , y ;r 0
Ex Ž y .
.
Ey
`
Hy`d y Ž1 y cos
x Ž y.
.
Ž 2.19 .
where L is the vortex length in the z-direction and B tot and B ind are the total and induced magnetic fields, respectively Žsee Eq. ŽA.1... The three terms in Eq. Ž2.13. have distinct physical origins: The first term originates from the vortex’s magnetic field when reflected back off the JJ buffer layer. The other two terms correspond to the JJ kinetic and potential energies, respectively, however, with the difference that B tot is not proportional to E xrE y Žsee Eqs. Ž2.13. and Ž3.1... All terms associated with x or its
Fig. 4. Typical example of the numerically exact conventional Ž w s y1. and exotic Ž w s 1. solutions of Eq. Ž2.10., calculated for the header parameters and hc s 1.7 using the method in Appendix C. The vortex core is at h s 0 Žsee Fig. 1..
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
7
derivative are interpreted as self-field terms w3x. In the d JJ s 0 limit Eq. Ž2.19. simplifies considerably and takes the dimensionless form ŽAppendix D. UP Ž x 0 .
1
F0
2
ž /
p 4pl
L
hx Ž h .
`
H0 dh (J
s yp
1
y
(
K 1 p J 02 q h 2
`
H dhx Ž h . sin 2p 0 1
q
2 2 0 qh
x Žh .
`
H dh Ž1 y cos p 0
x Žh .
..
Ž 2.20 .
The calculation of Eq. Ž2.20. are discussed in Section 3.
3. Results and discussion To discuss the two vortex states established in Section 2, consider first their magnetic signature. Employing Eq. Ž2.9. straightforward manipulations yield B tot Ž x , y . < xs 0
Ž F 0r2pl2 .
(
s K 0 p J 02 q h 2 1
y
`
`
H d kH0 dh p 0
cos w kh x cos w kh x
(k
2
qp
2
Ex Ž h . Eh
.
Ž 3.1 . The first term on the RHS of Eq. Ž3.1. represents the contribution of the generating vortex. The choice of the generating-vortex orientation in the positive zaxis direction this term is always positive. The second term represents the self-field contribution which arises from the screening currents as they cross the non-superconducting JJ buffer-layer. In the h 4 1 limit, only k f 0 values contribute to the integralterm in Eq. Ž3.1.. Hence, it is permissible to approximate 6Ž k 2 q p 2 . f p under the integral which immediately yields Eq. Ž2.13.. Numerical evaluation of Eq. Ž3.1. with the numerically-exact solutions of Eq. Ž2.10. are presented in Fig. 5 for the parameters
Fig. 5. An example of the calculated in-JJ field ŽEq. Ž3.1... The parameters are indicated in the header. Note that the conventional state Ž w sy1. field does not change sign while there is a sign change in the exotic state Ž w s1. field, see text. Inserting the ˚ 1r 2 , l s1500 A, ˚ 1 G s 7.88=10y7 values F 0 s1630 eV 1r 2 A 1r 2 ˚ y3 r2 eV A yields for the magnetic-field unit F 0 rŽ2pl2 . s 146.3 G.
˚ 1500 A, ˚ 24. Note that for the x 0 , l, p4 s 100 A, conventional vortex state the field is positive and is monotonously decreasing with increasing h. This feature is a manifestation of the Meissner effect associated with a penetration depth O Ž lJJ .. On the other hand, for the exotic solution the field reverses sign and for most h values is negative. As pointed out above, such a sign reversal is indicative of the presence of two JJ vortices Žin the y` - h - ` interval. as part of the exotic vortex state structure w23x. The presence of JJ vortices in the exotic state and their polarity are further elucidated by considering the magnetic flux associated with each vortex state. Straightforward manipulations with Eq. Ž2.9. yield `
Fs
½
0
Hy`d y Hy`d xB
ž
sF0 1y
1 2p
tot L
`
Ž x , y . q H d xBRtot Ž x , y . 0
x Ž y s ` . y x Ž y s y` .
/
5
.
Ž 3.2 . The ‘1’-term on the RHS of Eq. Ž3.2. is the generating-vortex flux while the last two terms are associated with the JJ contribution w23x. Combining Eq. Ž3.2. with the asymptotic behavior x Žsee Fig. 4. therefore implies
F Ž w s .1 . s "F 0 .
Ž 3.3 .
8
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
In particular, for the exotic state, Eq. Ž3.1. implies that the two JJ vortices it entails carry a flux y2F 0 , hence they are directed opposite to the generatingvortex direction. Consequently, the exotic state total flux, which is the sum of the generating-vortex flux F 0 and the flux of the two Josephson vortices y2F 0 , yields a net flux yF 0 . This result can be clarified by considering the limit where the vortex core location x 0 is moved toward the JJ buffer layer. It can be shown that as x 0 tends to zero, the vortex triad entailed in the exotic-state structure Žone Abrikosov-like and two Josephson-like vortices. approach each other until they coalesce and form a Josephson vortex pointing in the negative z direction. On the other hand, as x 0 tends to zero the Abrikosov-like vortex entailed in the conventional solution tend to a Josephson vortex pointing at the positive z direction. These limiting vortex states are degenerate, as mentioned in Section 1. To assess the feasibility of observing the exotic state it is important to calculate its free energy Žor its pinning potential, which here differs from the free energy by a constant, i.e., the vortex self-energy w6x.. Fig. 6 shows the calculated pinning potentials ŽEq. Ž2.20.. for the representative parameters p s 2 Žstrong link. and p s 10 Žweak link.. The important feature borne out by the results is that for a given p value, both w s "1 pinning potentials converge to-
Fig. 6. Examples of the calculated pinning potential, Eq. Ž2.20., with the indicated parameters. The conventional and exotic state pinning potentials are indicated by broken and solid lines, respectively. Note the quasi-degeneracy of the two pinning potentials near JJ plane, Ž x 0 r l <1., and that the conventional-state Ž w s y1. potential is always lower.
ward each other with decreasing J 0 , and the conventional state Ž w s y1. potential is consistently the lower one. The latter feature can be understood by noting the overscreening in the exotic state Žsee Fig. 5.. Overscreening implies that the kinetic energy exceeds the required minimum Žas realized in the conventional state., hence the corresponding free energy is higher. The energy difference between the two vortex states is estimated as follows. Inserting F 0 s 1630 ˚ 1r2 , l s 1500 A˚ for the UP Ž x 0 .rL-unit in eV 1r2 A Eq. Ž2.20. yields ŽF 0 rŽ4pl.. 2 s 7.47 = 10y 3 ˚ Now, denoting by DU the calculated differeVrA. ence between the w s y1 and w s 1 dimensionless potentials in Fig. 6, the associated Ženergyrvortexunit-length. is DU ŽF 0rŽ4pl.. 2 s DU = 7.47y3 ˚ By comparison, the available thermal ŽeneVrA. ergyrunit-vortex-length. associated with temperature ˚ where L is T is k B TrL s Ž8.6 = 10y5 TrL. eVrA, ˚ and T in degrees. To assess if thermal given in A activation is likely to proceed between the two w s "1 vortex states, it is necessary to estimate both the free-energy difference between the two states and the barrier between them. The latter represents the energy cost to ‘flip’ a vortex state flux from a positive z to a negative z orientation Žor vice-versa. Žsee Eq. Ž3.3... This barrier is finite and depends on the geometry of the sample. Since the consideration of this barrier is beyond the scope of this work, nor are we aware of other related work, no further comment on this matter is offered at this point. By contrast, the comparison of thermal energy with the energy difference between the two vortex states can be estimated. Equating the latter two energies gives ˚ . f T Žin degrees.. Inserting in 100 = DU = L Žin A this estimate an order-of-magnitude vortex–seg˚. ment–length Žthe film’s thickness. L s O Ž1000 A and DU for the smallest x 0 and largest p values ˚ p s 10. f 0.01, it folconsidered DUŽ x 0 s 50 A; lows that for the x 0 , p4 considered here, thermal energy at T s O Ž108. is insufficient for activation between the two vortex states. Thinner films, e.g., ˚ ., and smaller x 0 seem manifestly more L s O Ž100 A favorable. Specifically, note from Fig. 6 that DU tends vanish without bounds with both increasing ‘ p’ and decreasing x 0 to zero. Consequently, with the caveat that the Žunknown. energy-barrier between the two vortex states is not too high, it is
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
conceivable that thermal-activation does proceed between the two vortex states for ‘weak’ JJs Ž p ) 10. and in the immediate proximity to the JJ buffer-layer Ž x 0 s O Žcoherence length.. 2 where the pinningpotential is the deepest ŽFig. 6., and hence is where trapped vortices spend most of the time. According to Eq. Ž3.3., such thermal rattling of vortex states implies flux to fluctuate between the values "F 0 . A hypothetical experiment to directly observe this random-telegraph flux noise may utilize a SQUID microscope w26,27x which has been recently applied to measure localized flux with spatial resolution f 4 mm. If the technique can be extended to measure localized flux time variation, it may yield a direct test of the exotic state by looking for a random-telegraph flux noise near the JJ plane. In summary, we examined the configuration of an Abrikosov vortex in the presence of a homogeneous long JJ. The key result of this work is the prediction of two vortex states which are dubbed conventional and exotic due to their distinct magnetic signature: The conventional state carries a total flux F 0 which is identical to the generating vortex flux. The exotic state, on the other hand, entails two Josephson vortices with polarity opposite to that of the generating vortex in addition to the generating vortex and its flux is yF 0 ŽEq. Ž3.3... The calculated pinning potentials associated with the two states approach quasi-degeneracy as ‘ p’ increases and ‘ x 0 ’ decreases ŽEq. Ž2.11... Combined with a thermal energy-balance estimate, these trends point to the possibility ˚ ., and weak JJs, that in thin films, O Ž L s 100 A O Ž p G O Ž10.., thermal activation between the two vortex states proceeds in the immediate proximity of the JJ plane. In that case, the different flux associated with the two states, Eq. Ž3.3. implies a localized random telegraph flux noise in the immediate proximity of JJs.
Acknowledgements We acknowledge the support of the ILIR program of the Naval Surface Warfare Center, Carderock Division and ONR. Appendix A. Solution of Eq. (2.1) Eq. Ž2.1. for the configuration of Fig. 1 is solved by the Fourier-transform method w6x. Accordingly, the B-field is decomposed into the sum of the Abrikosov vortex field, B vor, and the induced field B ind B tot Ž ™ r;™ r 0 . s B vor Ž ™ r;™ r 0 . q B ind Ž ™ r;™ r0 .
Ž A.1 .
where, without loss of generality the vortex is placed at ™ r 0 s Ž x 0 ,0., x 0 - 0 Žsee Fig. 1. and BXvor
Ž x; y.s
1 F0
e l k y yyD Ž k y . < xyx 0 <
`
2 p 2 l2
Hy`d k
y
DŽ k y .
,
X s L,R B Lind Ž x ; y . s
1
`
H dk e 2p y` y
l k y yqD Ž k y . x
BLind Ž k y . ,
for x F 0 BRtot Ž x ; y . s
1
`
H dk e 2p y` y
l k y yyD Ž k y . Ž xyd JJ .
BRtot Ž k y . ,
for x G d JJ DŽ k y . s
(
k 2y q
1
. Ž A.2 . l2 Inserting Eq. ŽA.2. into Eq. Ž2.1., imposing the boundary conditions Ž2.2., Ž2.6., Ž2.7. and assuming that B Ž x M , y . f B Ž x s 0, y . s B Ž x s d JJ , y ., i.e., d JJ is ‘small’, yields Eq. Ž2.8. and F0 B Lind Ž k y . s y 2 2 l D Ž k y . q d JJ =
2
A low-temperature-superconductor example quoted in Ref. w3x, p. 70, cites for a Nb–NbO x –Pb JJ a value lJJ s 0.019 cm Žalso Ref. w2x, p. 200 for a similar figure.. Noting that lŽNb; ˚ lŽPb; T s 08. s 370 A, ˚ these values yield ps T s 08. s 390 A, lJJ r l f 5000. Although Pb and Nb are not ‘hard’ type II superconductors, these parameters suggests that in such low-temperature-superconductor JJs thermal activation between the two vortex states is a possibility.
9
BRtot Ž k y . s
½
d JJ eyD Ž k y . < x 0 < 2
q
ik y x Ž k y . 2p
2 l DŽ k y .
5
,
F0 2
2 l D Ž k y . q d JJ
½
= eyD Ž k y . < x 0 < y
ik y x Ž k y . 2p
5
.
Ž A.3 .
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
10
The d JJ s 0 limit of Eqs. ŽA.3. and Ž2.8. are Eqs. Ž2.9. and Ž2.10., respectively.
Appendix B. Solution of Eq. (2.15) The general solutions of Eq. Ž2.15. are the Elliptic functions. In the notation of Refs. w3,25x there are two independent Elliptic functions, the main branches of which depends on the two parameters k, h 0 4 . The expression of x in terms of Elliptic functions is summarized in Table 1, where in all cases: Ex
2
ž / Eh
4 s k
2
ž
1 y k 2 cos 2
x
/
2
Ž B.1 .
and Ž2.15. implies that x Žh . is indeterminate up to an additive 2p constant. The derivative boundary condition in Eqs. Ž2.12. and ŽB.1. implies that physical solutions have k s 1 and x Žh s ` . s 2 np . Hence, constructing ‘tanw xr2x’ from Table 1 with the k s 1 Elliptic functions w25x snŽ z < k s 1. s tanhw z x, dnŽ z < k s 1. s sechw z x, then inverting the ‘tan’ function yields Eq. Ž2.16..
Appendix C. Exact solution of Eq. (2.17)
The method for solving Eq. Ž2.10. is motivated by the approximant Ž2.18.. Define a correction phase x RŽh . such that the exact x Žh . for h F hc has the decomposition
x - Ž h . s x NL Ž h . q Ah q x R Ž h . , for h F hc . Ž C.3 . Inserting Eq. ŽC.3. into Eq. Ž2.10. and employing Eq. Ž2.17. yields p
p
`
Hy`
dh X
E x NL Ž h . Eh 2
E
s y2
xŽ j . Ž h . s
½
x NL Ž h . q AŽ j.h q x RŽ j. Ž h .
/5 Ž C.1 .
/
1 2
`
Hy`
dq
e i qhyD Ž q . j DŽ q .
(
DŽ q . s q 2 q p2 .
,
Ž C.2 .
It can checked by inspection that Eq. Ž2.17. is satisfied by inserting Eqs. ŽC.1. and ŽC.2. in it. Table 1 0 F k F1 cosw x r2x s sinw x r2x s
y snwŽh yh 0 .r k < k cnwŽh yh 0 .r k < k 2 x
k G1 2x
Žj.
Ž h y h0
for 0 F h F hc
; w ;h .
for h G hc .
Ž C.5 . ‘ j’ is the iteration index and x ) Žh . is given by Eq. Ž2.16.. Inserting Eq. ŽC.2. in Eq. ŽC.4. and inverting it gives
y snwh yh 0 <1r k 2 xr k dnwh yh 0 <1r k 2 x
`
H pp 0
dq
(
sin w q h x q 2 q p 2 Q Ž jy1. Ž q . q2
,
for h F hc Q Ž jy1 . Ž q . s
and the useful representation w6x
ž (
x
)
2
0
Ž C.4 .
where
sy K 0 p J 02 q h 2
K 0 w p
xRŽ j . Ž h .
2
½ ž( E J Eh
K0 p j 2 qh2 s
E 2hX
s sin x Ž jy1 . Ž h . , for h F hc
To derive Eq. Ž2.18. note that 2
E 2xRŽ j . Ž hX .
`
H0 dh sin w qh x sin
x Ž jy1 . Ž h . . Ž C.6 .
The iterations of Eqs. ŽC.5. and ŽC.6. are carried out as follows: First a value for hc is chosen such that the RHS source term in Eq. Ž2.10. is negligible for h G hc ; typically 1.5 - hc - 2. Then for a given choice of Ž w , n. in Eq. Ž2.16., construct x Ž1. Žh . over the entire h-axis by matching Eq. ŽC.5. at h s hc . This step implies that x RŽ1. Žh . s 0 and yields the constants AŽ1., h 0Ž1.4 . From this x Ž1. Žh ., calculate x RŽ2. Žh . for h F hC from Eq. ŽC.6.. The q-integration in Eq. ŽC.6. must be carried out typically up to qmax f 50. Using this x RŽ2. Žh . and matching Eq. ŽC.5. at h s hc yields new constants AŽ2., h 0Ž2.4 . From
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
11
these x Ž2. Žh . is constructed over the entire h-axis ŽEq. ŽC.5.. which then is fed into Eq. ŽC.6. to obtain x RŽ3. Žh . for h F hC , etc. This algorithm converges in at most 5 iterations for all the studied cases.
y 2 .rlx ™ 0 for y ™ `, it yields an integral with an integrand ‘ x Ž y . EŽ K 0 w6Ž x 02 q y 2 .rlx.rŽE y .’. This integrand is recognized as the RHS source term in Eq. Ž2.10.. Therefore, expressing it in terms of the rest of Eq. Ž2.10., employing the identity for any function ‘ f ’
Appendix D. Derivation of Eq. (2.20)
E f Ž < y y yX < .
In the absence of an external magnetic field the free energy is written as the sum of contributions from the JJ banks and the magnetic energy stored in the JJ buffer layer w6,28x. In the coordinate system of Fig. 1, the corresponding expressions are 1
yields the relation:
H 8p V
Hy`d y K
™
™
™
2p c
HS
d y d z jcJJ Ž y . Ž 1 y cos x Ž y .
L
.
2
ž / 4 pl 1
sy
=
=
p
(x q y 2 0
`
Hy`d y K
ExŽ y . Ey
0
1 q 4p E yX
2
l ` 2
E x Ž y . E x Ž yX . Ey
0
2
ExŽ y .
l `
Ey
`
E x Ž y . E x Ž yX . E yX
Ey
X
K0
< y y yX <
l
JJ
where here B stands for the total magnetic induction, V banks and VJJ are the volumes of the banks and JJ, respectively, and SJJ is the JJ-plane surface. Inserting Eq. ŽA.3. into Eq. ŽD.1. and subtracting off the vortex self-energy w6x yields the pinning potential UPrL as given by Eq. Ž2.19.. Note that in general jcJJ need not be constant. In the d JJ s 0 limit the expressions simplify considerably. Inserting Eq. Ž2.9. into Eq. Ž2.19. and transforming to real-space the kinetic-energy term gives
F0
2 0
H H dydy 2p y` y` =
Ž D.1 .
UP
s
banksqVJJ
F0
q
™
(x q y
`
1
™™
dÕ BB q l2 Ž = = B .Ž = = B .
Ž D.3 .
E yX
Ey
F s Fbanks q FJJ s
sy
E f Ž < y y yX < .
`
Hy`Hy`d y d y K0
< y y yX <
l
.
X
Ž D.2 .
To simplify the double integral in Eq. ŽD.2., consider the single integral term on its RHS. When integrated by parts, and noting that K 0 w6Ž x 02 q
l q
2 l2JJ
`
Hy`d y x Ž y . sin
x Ž y. .
Ž D.4 .
Inserting Eq. ŽD.4. into Eq. ŽD.2. yields Eq. Ž2.20..
References w1x T. Van Duzer, C.W. Turner, Principles of Superconductive Devices and Circuits, Elsevier, New York, 1981, p. 310. w2x M. Tinkham, Introduction to Superconductivity, Robert E. Krieger Publishing, Malabar, FL, 1980, p. 147. w3x A. Barone, G. Paterno, Physics and Applications of the Josephson Effect, Wiley, New York, 1982, pp. 100–112. w4x M. Tinkham, Introduction to Superconductivity, Robert E. Krieger Publishing, Malabar, FL, 1980, p. 146. w5x A. Barone, G. Paterno, Physics and Applications of the Josephson Effect, Wiley, New York, 1982, p. 112. w6x D. Agassi, J.R. Cullen, Physica C 195 Ž1992. 277. w7x M. Tinkham, Introduction to Superconductivity, Robert E. Krieger Publishing, Malabar, FL, 1980, p. 117. w8x A. Barone, G. Paterno, Physics and Applications of the Josephson Effect, Wiley, New York, 1982, p. 14. w9x T. Van Duzer, C.W. Turner, Principles of Superconductive Devices and Circuits, Elsevier, New York, 1981, pp. 2, 145. w10x M. Tinkham, Introduction to Superconductivity, Robert E. Krieger Publishing, Malabar, FL, 1980, p. 198. w11x M. Tachiki, T. Koyama, S. Takahashi, Phys. Rev. B 50 Ž1994. 7065. w12x A. Gurevich, L.D. Cooley, Phys. Rev. B 50 Ž1994. 13563. w13x A. Gurevich, M. Benkraouda, J.R. Clem, Phys. Rev. 54 Ž1996. 13196.
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D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
w14x Yu.P. Denisov, Sov. Phys. Solid State 18 Ž1976. 66. w15x E.V. Gurovich, V.G. Mikhalev, Sov. Phys. JEPT 66 Ž1987. 731. w16x M.V. Fistul, G.F. Giuliani, Phys. Rev. B 58 Ž1998. 9343. w17x T. Van Duzer, C.W. Turner, Principles of Superconductive Devices and Circuits, Elsevier, New York, 1981, p. 141. w18x G. Blatter et al., Rev. Mod. Phys. 66, 1994, pp. 1125, 1356. w19x M. Tinkham, Introduction to Superconductivity, Robert E. Krieger Publishing, Malabar, FL, 1980, p. 207. w20x R.A. Ferrel, R.E. Prange, Phys. Rev. Lett. 10 Ž1963. 479. w21x A. Gurevich, Phys. Rev. B 46 Ž1992. 3187. w22x A. Gurevich, Physica C 243 Ž1995. 191.
w23x A. Barone, G. Paterno, Physics and Applications of the Josephson Effect, Wiley, New York, 1982, p. 110. w24x I.S. Gradstein, I.M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York 1965, pp. 705, 736. w25x L.M. Milne-Thompson, in: M. Abramowitz, I.A. Stegun ŽEds.., Handbook of Mathematical Functions, National Bureau of Standards, Chap. 16, 1964, p. 571. w26x J.R. Kirtly, K.A. Moler, G. Villard, A. Maignan, preprint, 1998. w27x J.A. Kirtly et al., Appl. Phys. Lett. 66 Ž1998. 1138. w28x T.P. Orlando, K.A. Delin, Foundation of Applied Superconductivity, Addison-Wesley, Reading 1991, p. 428.
New vortex state in the presence of a long Josephson junction D. Agassi ) , J.R. Cullen Code 68, Carderock DiÕision, NaÕal Surface Warfare Center, 9500 MacArthur BlÕd. West Bethesda, MD 20817, USA Received 1 February 1999; accepted 10 February 1999
Abstract It is shown that there are two vortex states for the configuration of an Abrikosov vortex in the presence of a long Josephson junction. One state, dubbed ‘conventional’, carries one magnetic-flux quantum F 0 , as does the generating Abrikosov vortex. The second and new vortex state entails in addition to the generating vortex two Josephson vortices with opposite polarity to that of the generating-vortex and its magnetic flux is yF 0 . As a function of the vortex core and Josephson junction–plane distance, the accompanying pinning potentials are quasi-degenerate in the near proximity to the Josephson junction plane, where the conventional-vortex-state pinning potential is the lower. Conditions facilitating the experimental observation of the new vortex state and a specific signature are briefly discussed. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Josephson junction; Vortex state; Abrikosov vortex
1. Introduction The structure of a magnetic vortex threading an extended, isotropic superconductor—the Abrikosov vortex—is a standard textbook result w1,2x. The associated screening currents encircle the vortex core in trajectories of cylindrical symmetry, completely attenuate the vortex magnetic field over a London penetration depth Ž l. distance from the core, and the entire current distribution carries one flux quantum F 0 s p "cr< e <. In the special case where the vortex core is located at a Josephson junction ŽJJ., the coreencircling screening currents follow elongated-shape trajectories aligned with the JJ-plane direction w3x so that along the JJ plane the magnetic field is com-
)
Corresponding author. Fax: q1-301-227-4733; E-mail: [email protected]
pletely attenuated over another characteristic length, i.e., the Josephson-penetration-depth l JJ Ž l JJ ) l.. This particular vortex—the Josephson vortex—is qualitatively distinct from the Abrikosov vortex in that the order parameter does not vanishes at its core and it involves the order-parameter phase w4,5x. In an intermediate situation where the vortex core is located at a finite distance x 0 from the JJ plane ŽFig. 1., the encircling screening currents have to screen both the field around the vortex core and that field which has reached the JJ plane and extends along it Ži.e., along the y-axis in Fig. 1.. Consequently, the screening current trajectories shape is a mix of the above two basic shapes and the resulting vortex state may be interpreted as an admixture of the Abrikosov and Josephson vortex states. It is shown here that there are only two such admixed vortex states: The first is Abrikosov-like in the sense that it carries magnetic flux F 0 as does the generating Abrikosov
0921-4534r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 1 9 8 - 7
2
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
facilitating an experimental test of the new vortex state.
2. Phase-augmented London equation The issue at hand is the calculation of the pinning potential experienced by a probe vortex located at a distance x 0 from a finite-width, infinitely long homogeneous JJ ŽFig. 1.. To keep the discussion focused and unencumbered, this section is subdivided into subsections and many mathematical details are deferred to the appendices. 2.1. The phase-augmented London equation
Fig. 1. The homogeneous finite width Josephson junction configuration and the coordinate system used in this work. The probe vortex is located at Ž x 0 , y 0 s 0. and the hatched stripe represents the JJ buffer layer. Also indicated are the left and right superconductors’ phases, f L and f R , respectively ŽEq. Ž2.3.. and the gauge-invariant phase-discontinuity x Ž y . ŽEq. Ž2.6...
vortex. The second one is a new type of vortex state which carries a magnetic flux yF 0 since its screening currents overscreening the generating vortex field along the JJ such that the ensuing magnetic field distribution entails two additional Josephson vortices with polarity opposite to that of the generating vortex. As will be shown in this work, the free energy of the exotic state is higher than that of the conventional state. However ŽFig. 6., for a vortex core located within a distance on the order of the coherence length from the JJ plane, the free energies associated with these two vortex states are quasi-degenerate. A conceivable experimental test of the exotic vortex state is looking in the immediate proximity to the JJ plane for random-telegraph flux noise. The paper is organized as follows: In Section 2, the London equation is first augmented to account for the presence of a JJ by appropriate boundary conditions. The augmented equation predicts the existence of two vortex states which are then calculated. Some of the features and consequences of the two states are presented and discussed in Section 3, as well as a brief discussion for the conditions
Without loss of generality, the probe vortex location is chosen at ™ r 0 s Ž x 0 ,0. and x 0 - 0 Žsee Fig. 1.. Assuming that the probe vortex points in the positive z direction, the London equations for the ‘L’ and ‘R’ banks are E2 B X y l2
F0s
Ex2
E2 q
E y2
BX s F 0 d Ž ™ r y™ r 0 . , X s L,R
p "c < e<
Ž 2.1 .
where BX Ž x, y . is the z-component of the magnetic field generated by the probe vortex at the X bank. In Eq. Ž2.1. and hereafter, the cgs unit system is employed and the signs correspond to ‘electron’-superconductivity. Eq. Ž2.1. is supplemented by boundary conditions as follows. The continuity of the tangential-Hz component across a ‘thin’ JJ buffer-layer Žsmall d JJ , Fig. 1. implies w6x Dw B x s0
Ž 2.2 .
where the notation for the x-discontinuity across the junction of an arbitrary function f Ž x, y . is Dw f Ž x, y .x s f Ž x s d JJ q e , y . y f Ž x s ye , y . with e ™ 0. A second boundary condition follows from the fundamental, T s 0 K, gauge-invariant constitutive relation w7,8x ™
js s y
c 4pl
™
2
Aq
F0 2p
™
=f
Ž 2.3 .
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12 ™
™
where js is the supercurrent density, A denotes the electromagnetic potential and f is the order-parameter phase. In the augmented London approximation the superconductor is assumed to be a perfect, quiescent Ž f constant. diamagnet except at its interface where f changes discontinuously.1 Correspondingly, the j y-discontinuity implied by Eq. Ž2.3. is 4p y c
D l2 j y sD A y q
F 0 Ec 2p E y
Ž 2.4 .
E AyŽ x, y. Ex
xs x M
ž
s d JJ Bz Ž x M , y . q
E AxŽ x, y. Ey
xs x M
/
Ex
s d JJ B Ž x M , y . q
x Ž y. sc Ž y. q fc Ž y. q
2
F 0 Ex Ž y . 2p
Ey
d JJ
d xA x Ž x , y .
2
`
Hy`d k
DŽ k y . s
(
k 2y q
1
1
lk y y 2
2 l D Ž k y . q d JJ
½
1
l2
ik y x Ž k y . 2p
5
s yjcJJ sin x Ž y . ,
,
`
H dk e 2p y` y
lk y y
x Žky..
Ž 2.8 .
In the particular d JJ s 0 limit, which is the only one considered hereafter, the expressions for the total fields are
Ž 2.6 .
B Ltot Ž x , y . s
where x Ž y . is the gauge-invariant phase-discontinuity across the JJ w9,10x. In the d JJ ™ 0 limit and no order-parameter-phase dependence Eq. Ž2.6. reduces to the boundary condition DŽ l2 j y . s 0. This boundary condition has been previously employed for superconductor–superconductor interfaces in the context of the London approximation w6x.
Since js is a gauge invariant construct and f is assumed to change discontinuously only at the interface, the burden of describing spatially varying screening currents is carried by the ™ electromagnetic potential A.
F0 4pl2
elk y y
`
Hy`d k
y
DŽ k y .
½
= eyD Ž k y . < xyx 0 < ye D Ž k y . x
BRtot Ž x , y . s 1
y k ye
= eyD Ž k y . < x 0 < y
x Ž y. s
,
d JJ A x Ž x M , y .
H "c 0
where denotes the Josephson critical current. The negative sign in Eq. Ž2.7. is needed to assures consistency with the Josephson current–phase relation, i.e., j2 ™ 1 s jcJJ sinŽ f 2 y f 1 . w17x. The solutions of Eq. Ž2.1. which satisfy the boundary conditions Ž2.2., Ž2.6. and Ž2.7. are obtained by the Fourier transform method Žsee Appendix A.. The ensuing equation for x is 8p
where 0 F x M F d JJ . In particular, for small d JJ values when the x-variation of BX may be neglected, insertion of Eq. Ž2.5. into Eq. Ž2.4. yields the gaugeinvariant boundary condition
Ž 2.7 .
jcJJ
iF 0 c
Ž 2.5 .
EB
j x Ž y . s j x Ž x s ye ; y . s j x Ž x s d JJ q e ; y . s yjcJJ sin x Ž y .
The discontinuity Dw A y x, in turn, can be is reex™ ™ pressed by employing curlw A x s B from the Maxwell equations. This gives D A y sd JJ
The boundary condition Ž2.6. involves the gauge invariant phase-discontinuity x Ž y . which must be determined simultaneously with the fields BX Ž x, y .. To accomplish this, one additional relation is needed. A natural choice for that relation results from imposing the Josephson current-phase relation w11–16x
,
c Ž y. sD f Ž x, y. .
D l2
3
F0 4pl2
½
ž
lk y x Ž k y .
`
Hy`
dky
2p
/5
,
x-0
e l k y yy DŽ k y . x DŽ k y .
= eyD Ž k y . < x 0 < y
ik y x Ž k y . 2p
5
,
x)0
Ž 2.9 .
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
4
and Eq. Ž2.8. takes the known dimensionless form w11–16x p
p
˙
`
Hy`dh
X
E 2x Ž hX .
BJJtot Ž yrl 4 1 . s y
K 0 w p
E 2hX
s sin x Ž h . y 2 p
h
2
(J
2 2 0 qh
(
=K 1 p J 02 q h 2 .
Ž 2.10 .
The symbols in Eq. Ž2.10. denote
J0 s
p2 s
l2JJ s
x0
lJJ
hs
,
2
lJJ
ž /
4
cF 0 16p
lJJ
l jcJJ
,
Eh
s 0,
4pl
Ey
Ž 2.13 .
where BRtot Ž x s 0, yrl 4 1. s BLtot Ž x s 0, yrl 4 1. s BJJtot Ž y .. Hence, the assumed field-vanishing at yrl 4 1 implies the vanishing of x ’s derivative, as in Eq. Ž2.12.. In addition, by Ampere’s law, the magnetic field vanishing at < y
l
jdep s
jcJJ
G 1, cF 0
12'3 p 2l2j 0
Ž 2.11 .
where jdep and K nŽ z . are the depairing current w18x and modified Bessel function of order n, respectively. Eq. Ž2.10. is a generalization of the standard differential equation describing the spatial variation of x w19,20x for the case where the magnetic-field source is a Abrikosov vortex. The two parameters in Eq. Ž2.10. are the probe vortex location ŽFig. 1., given by J 0 , and a measure of the JJ-link strength given by ‘ p’. A weak link implies p 4 1. Since the JJ link-strength cannot exceed that of a perfect superconductor, p ) 1 in all cases; the p - 1 regime is not physical w21,22x. Physical solutions of the central Eq. Ž2.10. must satisfy the following two boundary conditions: Ex Ž h .
F 0 Ex Ž y .
,
3'3 j 0 jdep
s
l
2
y
evaluating Eq. Ž2.9. in the yrl 4 1 limit, see Eq. Ž3.1.. The latter yields w23x
x Ž yh . s yx Ž h . .
The method for solving Eq. Ž2.10. is based on the observation that the source term dominates the RHS for
Ž 2.12 .
< h
The h anti-symmetry condition in Eq. Ž2.12. reflects the obvious anti symmetry of the vortex encircling currents under the h ™ yh transformation. This is also apparent in the source term of Eq. Ž2.10.. The derivative boundary condition in Eq. Ž2.12. expresses the premise that at a large distance along the JJ plane from the vortex core the magnetic field vanishes. This boundary condition is derived by
Fig. 2. An example of the factors in Eq. Ž2.10. for the parameters in the header ŽEq. Ž2.11... The chosen J 0 corresponds to values ˚ 1500 A, ˚ 24. The solid and broken lines are x 0 , l, p4 s 100 A, log 10 Ž< S <., where S is the source term on the RHS of Eq. Ž2.10., and x NL Žh .rp of Eq. Ž2.18..
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
‘E 2xrEh 2 ’-factor from under the LHS-integral in Eq. Ž2.10.. Noting that w24x ` x d x K0 s pl . Ž 2.14 . l y`
H
Eq. Ž2.10. transforms into the pendulum equation w3x
E 2x ) Ž h . Eh 2
s sin x ) Ž h . ,
for h G hc .
Ž 2.15 .
The general solutions of the pendulum equation are the Elliptic functions w25x Žsee Appendix B.. The particular solutions which are physical for the problem at hand are determined by the derivative boundary-condition at h s ` of Eq. Ž2.12.. According to Eq. ŽB.1., the only such solutions are
x ) Ž h y h 0 ; w ,n .
Eh w s "1,
2w s
cosh w h y h 0 x
n s . . . , y1, 0, 1, . . .
Out of the infinity of solutions Ž2.16., the physically meaningful ones are selected by matching at h s hc to solutions of Eq. Ž2.10. in the inner interval 0 - h F hc to which we turn now. In the inner interval 0 - h F hc all terms in Eq. Ž2.10. contribute. Insight into the solution there, hereafter denoted by x - Žh ., is obtained by consid- Ž . ering the approximant x - Žh . f xsource h defined as solution of Eq. Ž2.10. where on the RHS only the source term is retained p
`
H dh p y`
X
E 2xsource Ž hX .
s y2 p 2
E 2hX h
(J
2 2 0 qh
K 0 w p
(
K 1 p J 02 q h 2 .
Ž 2.17 .
As implied by the example plotted in Fig. 2, this approximate equation is valid at least in the 0 - h 1rp interval. The general, exact solution of Eq. Ž2.17. is Žsee Appendix D and Fig. 2.
s 2 w arctg sinh w h y h 0 x q Ž 2 n q 1 . p ,
Ex ) Ž h y h 0 ; w ,n .
5
,
Ž 2.16 .
and x ) Žh s `; w , n. s 2p n where n is an integer. Each solution in this class of solutions depends on three parameters: The ‘w ’ parameter, which specifies a family of branches sharing a common derivative sign w , the ‘n’ parameter, which specifies the particular branch in a given w-family and the ‘h 0 ’ parameter which is a reference point. Fig. 3 shows two branches of each of the two w-families in Eq. Ž2.16..
Fig. 3. Representative branches of x ) Žh yh 0 ; w , n., Eq. Ž2.16.. The solid and broken lines depict w sy1 and w s1 branches, respectively, with the indicated n values.
xsource Ž h ; A,C . s x NL Ž h . q Ah q C,
h
x NL Ž h . s 2 p J 0
H0
x NL Ž ` . s p eyp J 0
dh
(
K 1 p J 02 q h 2
(J
2 2 0 qh
,
Ž 2.18 .
where the boundary condition x Žh s 0. s 0 in Eq. Ž2.12. mandates C s 0 in Eq. Ž2.18.. It is now possible to construct approximate solutions of Eq. Ž2.10. in the entire 0 F h F ` interval - Ž . by matching x ) Žh . of Eq. Ž2.16. and xsource h of Eq. Ž2.18. at h s hc . Given a choice of Ž w , n., a solution x ) Žh y h 0 ; w , n. depends only on the h 0 - Ž . parameter, while xsource h depends only on the ‘ A’ parameter. These are determined by matching x ) and x - and their derivatives at h s hc . A crude assessment which Ž w , n. are viable choices is obtained as follows. Since < E x ) rEh < F 2 ŽEq. Ž2.16.., and x NL Žh . rises and quickly flattens for h - hc ŽFig. 2., matching the x ) and x - derivatives implies that < A < F 2. Furthermore, since 0 - x NL Žh . - p ŽEq. Ž2.18.. and typically hc f 2, matching the x ) and x - values implies that
6
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
the choice hc f 1.5 the above consideration rules out all but the two Ž n s 0; w s "1. branches. Numerical solution of the matching conditions at hc confirms that there are only two viable solutions. Moreover, the above approach for constructing approximate solutions suggests a converging algorithm for the construction of exact solutions to Eq. Ž2.10. which is detailed in Appendix C. This algorithm also yields only two numerically exact solutions, in keeping with the crude consideration above and the qualitative arguments given in the Introduction section. The existence of only two solutions to the phase-discontinuity Eq. Ž2.10. which satisfy the boundary condition Ž2.12. is the central result of this work. An example of numerically exact x Žh ; w s "1, n s 0. solutions is shown in Fig. 4. Note from Fig. 4 and Eq. Ž2.13. that the in-JJ-plane magnetic field at h 4 1 is positive and negative for the w s y1 and the w s 1 solutions, respectively. A positive magnetic field conforms with the chosen direction of the probe vortex along the positive z-axis direction ŽEq. Ž2.1... In this sense the w s y1 solution has the expected polarity and hence is dubbed ‘conventional’. On the other hand, the w s 1 solution has a negative magnetic field at h 4 1, i.e., a field polarity opposite to that of the probe vortex. This unusual property implies overscreening the generating-vortex magnetic field leading to a net field in the direction opposite to that of the generating vortex. For this reason this solution is dubbed ‘exotic’. Furthermore, since for the exotic state x Ž
reverses twice its sign in the y` F h F ` interval. This feature implies that the exotic state current distribution entails two Josephson vortices. These two x-solutions are further discussed in Section 3. 2.3. The pinning potential For a finite-width JJ Ž d JJ G 0, Fig. 1. the pinning potential per unit vortex length, UP Ž 0 .rL w6x, is given by ŽAppendix D. UP Ž ™ r0 . L s
F0 8p y
B ind ™ r s™ r 0 ;r™0
/
F0
tot
ž
16p q
"jcJJ 2
` 2
Hy`d y B Ž x s x
™ M , y ;r 0
Ex Ž y .
.
Ey
`
Hy`d y Ž1 y cos
x Ž y.
.
Ž 2.19 .
where L is the vortex length in the z-direction and B tot and B ind are the total and induced magnetic fields, respectively Žsee Eq. ŽA.1... The three terms in Eq. Ž2.13. have distinct physical origins: The first term originates from the vortex’s magnetic field when reflected back off the JJ buffer layer. The other two terms correspond to the JJ kinetic and potential energies, respectively, however, with the difference that B tot is not proportional to E xrE y Žsee Eqs. Ž2.13. and Ž3.1... All terms associated with x or its
Fig. 4. Typical example of the numerically exact conventional Ž w s y1. and exotic Ž w s 1. solutions of Eq. Ž2.10., calculated for the header parameters and hc s 1.7 using the method in Appendix C. The vortex core is at h s 0 Žsee Fig. 1..
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
7
derivative are interpreted as self-field terms w3x. In the d JJ s 0 limit Eq. Ž2.19. simplifies considerably and takes the dimensionless form ŽAppendix D. UP Ž x 0 .
1
F0
2
ž /
p 4pl
L
hx Ž h .
`
H0 dh (J
s yp
1
y
(
K 1 p J 02 q h 2
`
H dhx Ž h . sin 2p 0 1
q
2 2 0 qh
x Žh .
`
H dh Ž1 y cos p 0
x Žh .
..
Ž 2.20 .
The calculation of Eq. Ž2.20. are discussed in Section 3.
3. Results and discussion To discuss the two vortex states established in Section 2, consider first their magnetic signature. Employing Eq. Ž2.9. straightforward manipulations yield B tot Ž x , y . < xs 0
Ž F 0r2pl2 .
(
s K 0 p J 02 q h 2 1
y
`
`
H d kH0 dh p 0
cos w kh x cos w kh x
(k
2
qp
2
Ex Ž h . Eh
.
Ž 3.1 . The first term on the RHS of Eq. Ž3.1. represents the contribution of the generating vortex. The choice of the generating-vortex orientation in the positive zaxis direction this term is always positive. The second term represents the self-field contribution which arises from the screening currents as they cross the non-superconducting JJ buffer-layer. In the h 4 1 limit, only k f 0 values contribute to the integralterm in Eq. Ž3.1.. Hence, it is permissible to approximate 6Ž k 2 q p 2 . f p under the integral which immediately yields Eq. Ž2.13.. Numerical evaluation of Eq. Ž3.1. with the numerically-exact solutions of Eq. Ž2.10. are presented in Fig. 5 for the parameters
Fig. 5. An example of the calculated in-JJ field ŽEq. Ž3.1... The parameters are indicated in the header. Note that the conventional state Ž w sy1. field does not change sign while there is a sign change in the exotic state Ž w s1. field, see text. Inserting the ˚ 1r 2 , l s1500 A, ˚ 1 G s 7.88=10y7 values F 0 s1630 eV 1r 2 A 1r 2 ˚ y3 r2 eV A yields for the magnetic-field unit F 0 rŽ2pl2 . s 146.3 G.
˚ 1500 A, ˚ 24. Note that for the x 0 , l, p4 s 100 A, conventional vortex state the field is positive and is monotonously decreasing with increasing h. This feature is a manifestation of the Meissner effect associated with a penetration depth O Ž lJJ .. On the other hand, for the exotic solution the field reverses sign and for most h values is negative. As pointed out above, such a sign reversal is indicative of the presence of two JJ vortices Žin the y` - h - ` interval. as part of the exotic vortex state structure w23x. The presence of JJ vortices in the exotic state and their polarity are further elucidated by considering the magnetic flux associated with each vortex state. Straightforward manipulations with Eq. Ž2.9. yield `
Fs
½
0
Hy`d y Hy`d xB
ž
sF0 1y
1 2p
tot L
`
Ž x , y . q H d xBRtot Ž x , y . 0
x Ž y s ` . y x Ž y s y` .
/
5
.
Ž 3.2 . The ‘1’-term on the RHS of Eq. Ž3.2. is the generating-vortex flux while the last two terms are associated with the JJ contribution w23x. Combining Eq. Ž3.2. with the asymptotic behavior x Žsee Fig. 4. therefore implies
F Ž w s .1 . s "F 0 .
Ž 3.3 .
8
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
In particular, for the exotic state, Eq. Ž3.1. implies that the two JJ vortices it entails carry a flux y2F 0 , hence they are directed opposite to the generatingvortex direction. Consequently, the exotic state total flux, which is the sum of the generating-vortex flux F 0 and the flux of the two Josephson vortices y2F 0 , yields a net flux yF 0 . This result can be clarified by considering the limit where the vortex core location x 0 is moved toward the JJ buffer layer. It can be shown that as x 0 tends to zero, the vortex triad entailed in the exotic-state structure Žone Abrikosov-like and two Josephson-like vortices. approach each other until they coalesce and form a Josephson vortex pointing in the negative z direction. On the other hand, as x 0 tends to zero the Abrikosov-like vortex entailed in the conventional solution tend to a Josephson vortex pointing at the positive z direction. These limiting vortex states are degenerate, as mentioned in Section 1. To assess the feasibility of observing the exotic state it is important to calculate its free energy Žor its pinning potential, which here differs from the free energy by a constant, i.e., the vortex self-energy w6x.. Fig. 6 shows the calculated pinning potentials ŽEq. Ž2.20.. for the representative parameters p s 2 Žstrong link. and p s 10 Žweak link.. The important feature borne out by the results is that for a given p value, both w s "1 pinning potentials converge to-
Fig. 6. Examples of the calculated pinning potential, Eq. Ž2.20., with the indicated parameters. The conventional and exotic state pinning potentials are indicated by broken and solid lines, respectively. Note the quasi-degeneracy of the two pinning potentials near JJ plane, Ž x 0 r l <1., and that the conventional-state Ž w s y1. potential is always lower.
ward each other with decreasing J 0 , and the conventional state Ž w s y1. potential is consistently the lower one. The latter feature can be understood by noting the overscreening in the exotic state Žsee Fig. 5.. Overscreening implies that the kinetic energy exceeds the required minimum Žas realized in the conventional state., hence the corresponding free energy is higher. The energy difference between the two vortex states is estimated as follows. Inserting F 0 s 1630 ˚ 1r2 , l s 1500 A˚ for the UP Ž x 0 .rL-unit in eV 1r2 A Eq. Ž2.20. yields ŽF 0 rŽ4pl.. 2 s 7.47 = 10y 3 ˚ Now, denoting by DU the calculated differeVrA. ence between the w s y1 and w s 1 dimensionless potentials in Fig. 6, the associated Ženergyrvortexunit-length. is DU ŽF 0rŽ4pl.. 2 s DU = 7.47y3 ˚ By comparison, the available thermal ŽeneVrA. ergyrunit-vortex-length. associated with temperature ˚ where L is T is k B TrL s Ž8.6 = 10y5 TrL. eVrA, ˚ and T in degrees. To assess if thermal given in A activation is likely to proceed between the two w s "1 vortex states, it is necessary to estimate both the free-energy difference between the two states and the barrier between them. The latter represents the energy cost to ‘flip’ a vortex state flux from a positive z to a negative z orientation Žor vice-versa. Žsee Eq. Ž3.3... This barrier is finite and depends on the geometry of the sample. Since the consideration of this barrier is beyond the scope of this work, nor are we aware of other related work, no further comment on this matter is offered at this point. By contrast, the comparison of thermal energy with the energy difference between the two vortex states can be estimated. Equating the latter two energies gives ˚ . f T Žin degrees.. Inserting in 100 = DU = L Žin A this estimate an order-of-magnitude vortex–seg˚. ment–length Žthe film’s thickness. L s O Ž1000 A and DU for the smallest x 0 and largest p values ˚ p s 10. f 0.01, it folconsidered DUŽ x 0 s 50 A; lows that for the x 0 , p4 considered here, thermal energy at T s O Ž108. is insufficient for activation between the two vortex states. Thinner films, e.g., ˚ ., and smaller x 0 seem manifestly more L s O Ž100 A favorable. Specifically, note from Fig. 6 that DU tends vanish without bounds with both increasing ‘ p’ and decreasing x 0 to zero. Consequently, with the caveat that the Žunknown. energy-barrier between the two vortex states is not too high, it is
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
conceivable that thermal-activation does proceed between the two vortex states for ‘weak’ JJs Ž p ) 10. and in the immediate proximity to the JJ buffer-layer Ž x 0 s O Žcoherence length.. 2 where the pinningpotential is the deepest ŽFig. 6., and hence is where trapped vortices spend most of the time. According to Eq. Ž3.3., such thermal rattling of vortex states implies flux to fluctuate between the values "F 0 . A hypothetical experiment to directly observe this random-telegraph flux noise may utilize a SQUID microscope w26,27x which has been recently applied to measure localized flux with spatial resolution f 4 mm. If the technique can be extended to measure localized flux time variation, it may yield a direct test of the exotic state by looking for a random-telegraph flux noise near the JJ plane. In summary, we examined the configuration of an Abrikosov vortex in the presence of a homogeneous long JJ. The key result of this work is the prediction of two vortex states which are dubbed conventional and exotic due to their distinct magnetic signature: The conventional state carries a total flux F 0 which is identical to the generating vortex flux. The exotic state, on the other hand, entails two Josephson vortices with polarity opposite to that of the generating vortex in addition to the generating vortex and its flux is yF 0 ŽEq. Ž3.3... The calculated pinning potentials associated with the two states approach quasi-degeneracy as ‘ p’ increases and ‘ x 0 ’ decreases ŽEq. Ž2.11... Combined with a thermal energy-balance estimate, these trends point to the possibility ˚ ., and weak JJs, that in thin films, O Ž L s 100 A O Ž p G O Ž10.., thermal activation between the two vortex states proceeds in the immediate proximity of the JJ plane. In that case, the different flux associated with the two states, Eq. Ž3.3. implies a localized random telegraph flux noise in the immediate proximity of JJs.
Acknowledgements We acknowledge the support of the ILIR program of the Naval Surface Warfare Center, Carderock Division and ONR. Appendix A. Solution of Eq. (2.1) Eq. Ž2.1. for the configuration of Fig. 1 is solved by the Fourier-transform method w6x. Accordingly, the B-field is decomposed into the sum of the Abrikosov vortex field, B vor, and the induced field B ind B tot Ž ™ r;™ r 0 . s B vor Ž ™ r;™ r 0 . q B ind Ž ™ r;™ r0 .
Ž A.1 .
where, without loss of generality the vortex is placed at ™ r 0 s Ž x 0 ,0., x 0 - 0 Žsee Fig. 1. and BXvor
Ž x; y.s
1 F0
e l k y yyD Ž k y . < xyx 0 <
`
2 p 2 l2
Hy`d k
y
DŽ k y .
,
X s L,R B Lind Ž x ; y . s
1
`
H dk e 2p y` y
l k y yqD Ž k y . x
BLind Ž k y . ,
for x F 0 BRtot Ž x ; y . s
1
`
H dk e 2p y` y
l k y yyD Ž k y . Ž xyd JJ .
BRtot Ž k y . ,
for x G d JJ DŽ k y . s
(
k 2y q
1
. Ž A.2 . l2 Inserting Eq. ŽA.2. into Eq. Ž2.1., imposing the boundary conditions Ž2.2., Ž2.6., Ž2.7. and assuming that B Ž x M , y . f B Ž x s 0, y . s B Ž x s d JJ , y ., i.e., d JJ is ‘small’, yields Eq. Ž2.8. and F0 B Lind Ž k y . s y 2 2 l D Ž k y . q d JJ =
2
A low-temperature-superconductor example quoted in Ref. w3x, p. 70, cites for a Nb–NbO x –Pb JJ a value lJJ s 0.019 cm Žalso Ref. w2x, p. 200 for a similar figure.. Noting that lŽNb; ˚ lŽPb; T s 08. s 370 A, ˚ these values yield ps T s 08. s 390 A, lJJ r l f 5000. Although Pb and Nb are not ‘hard’ type II superconductors, these parameters suggests that in such low-temperature-superconductor JJs thermal activation between the two vortex states is a possibility.
9
BRtot Ž k y . s
½
d JJ eyD Ž k y . < x 0 < 2
q
ik y x Ž k y . 2p
2 l DŽ k y .
5
,
F0 2
2 l D Ž k y . q d JJ
½
= eyD Ž k y . < x 0 < y
ik y x Ž k y . 2p
5
.
Ž A.3 .
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
10
The d JJ s 0 limit of Eqs. ŽA.3. and Ž2.8. are Eqs. Ž2.9. and Ž2.10., respectively.
Appendix B. Solution of Eq. (2.15) The general solutions of Eq. Ž2.15. are the Elliptic functions. In the notation of Refs. w3,25x there are two independent Elliptic functions, the main branches of which depends on the two parameters k, h 0 4 . The expression of x in terms of Elliptic functions is summarized in Table 1, where in all cases: Ex
2
ž / Eh
4 s k
2
ž
1 y k 2 cos 2
x
/
2
Ž B.1 .
and Ž2.15. implies that x Žh . is indeterminate up to an additive 2p constant. The derivative boundary condition in Eqs. Ž2.12. and ŽB.1. implies that physical solutions have k s 1 and x Žh s ` . s 2 np . Hence, constructing ‘tanw xr2x’ from Table 1 with the k s 1 Elliptic functions w25x snŽ z < k s 1. s tanhw z x, dnŽ z < k s 1. s sechw z x, then inverting the ‘tan’ function yields Eq. Ž2.16..
Appendix C. Exact solution of Eq. (2.17)
The method for solving Eq. Ž2.10. is motivated by the approximant Ž2.18.. Define a correction phase x RŽh . such that the exact x Žh . for h F hc has the decomposition
x - Ž h . s x NL Ž h . q Ah q x R Ž h . , for h F hc . Ž C.3 . Inserting Eq. ŽC.3. into Eq. Ž2.10. and employing Eq. Ž2.17. yields p
p
`
Hy`
dh X
E x NL Ž h . Eh 2
E
s y2
xŽ j . Ž h . s
½
x NL Ž h . q AŽ j.h q x RŽ j. Ž h .
/5 Ž C.1 .
/
1 2
`
Hy`
dq
e i qhyD Ž q . j DŽ q .
(
DŽ q . s q 2 q p2 .
,
Ž C.2 .
It can checked by inspection that Eq. Ž2.17. is satisfied by inserting Eqs. ŽC.1. and ŽC.2. in it. Table 1 0 F k F1 cosw x r2x s sinw x r2x s
y snwŽh yh 0 .r k < k cnwŽh yh 0 .r k < k 2 x
k G1 2x
Žj.
Ž h y h0
for 0 F h F hc
; w ;h .
for h G hc .
Ž C.5 . ‘ j’ is the iteration index and x ) Žh . is given by Eq. Ž2.16.. Inserting Eq. ŽC.2. in Eq. ŽC.4. and inverting it gives
y snwh yh 0 <1r k 2 xr k dnwh yh 0 <1r k 2 x
`
H pp 0
dq
(
sin w q h x q 2 q p 2 Q Ž jy1. Ž q . q2
,
for h F hc Q Ž jy1 . Ž q . s
and the useful representation w6x
ž (
x
)
2
0
Ž C.4 .
where
sy K 0 p J 02 q h 2
K 0 w p
xRŽ j . Ž h .
2
½ ž( E J Eh
K0 p j 2 qh2 s
E 2hX
s sin x Ž jy1 . Ž h . , for h F hc
To derive Eq. Ž2.18. note that 2
E 2xRŽ j . Ž hX .
`
H0 dh sin w qh x sin
x Ž jy1 . Ž h . . Ž C.6 .
The iterations of Eqs. ŽC.5. and ŽC.6. are carried out as follows: First a value for hc is chosen such that the RHS source term in Eq. Ž2.10. is negligible for h G hc ; typically 1.5 - hc - 2. Then for a given choice of Ž w , n. in Eq. Ž2.16., construct x Ž1. Žh . over the entire h-axis by matching Eq. ŽC.5. at h s hc . This step implies that x RŽ1. Žh . s 0 and yields the constants AŽ1., h 0Ž1.4 . From this x Ž1. Žh ., calculate x RŽ2. Žh . for h F hC from Eq. ŽC.6.. The q-integration in Eq. ŽC.6. must be carried out typically up to qmax f 50. Using this x RŽ2. Žh . and matching Eq. ŽC.5. at h s hc yields new constants AŽ2., h 0Ž2.4 . From
D. Agassi, J.R. Cullenr Physica C 316 (1999) 1–12
11
these x Ž2. Žh . is constructed over the entire h-axis ŽEq. ŽC.5.. which then is fed into Eq. ŽC.6. to obtain x RŽ3. Žh . for h F hC , etc. This algorithm converges in at most 5 iterations for all the studied cases.
y 2 .rlx ™ 0 for y ™ `, it yields an integral with an integrand ‘ x Ž y . EŽ K 0 w6Ž x 02 q y 2 .rlx.rŽE y .’. This integrand is recognized as the RHS source term in Eq. Ž2.10.. Therefore, expressing it in terms of the rest of Eq. Ž2.10., employing the identity for any function ‘ f ’
Appendix D. Derivation of Eq. (2.20)
E f Ž < y y yX < .
In the absence of an external magnetic field the free energy is written as the sum of contributions from the JJ banks and the magnetic energy stored in the JJ buffer layer w6,28x. In the coordinate system of Fig. 1, the corresponding expressions are 1
yields the relation:
H 8p V
Hy`d y K
™
™
™
2p c
HS
d y d z jcJJ Ž y . Ž 1 y cos x Ž y .
L
.
2
ž / 4 pl 1
sy
=
=
p
(x q y 2 0
`
Hy`d y K
ExŽ y . Ey
0
1 q 4p E yX
2
l ` 2
E x Ž y . E x Ž yX . Ey
0
2
ExŽ y .
l `
Ey
`
E x Ž y . E x Ž yX . E yX
Ey
X
K0
< y y yX <
l
JJ
where here B stands for the total magnetic induction, V banks and VJJ are the volumes of the banks and JJ, respectively, and SJJ is the JJ-plane surface. Inserting Eq. ŽA.3. into Eq. ŽD.1. and subtracting off the vortex self-energy w6x yields the pinning potential UPrL as given by Eq. Ž2.19.. Note that in general jcJJ need not be constant. In the d JJ s 0 limit the expressions simplify considerably. Inserting Eq. Ž2.9. into Eq. Ž2.19. and transforming to real-space the kinetic-energy term gives
F0
2 0
H H dydy 2p y` y` =
Ž D.1 .
UP
s
banksqVJJ
F0
q
™
(x q y
`
1
™™
dÕ BB q l2 Ž = = B .Ž = = B .
Ž D.3 .
E yX
Ey
F s Fbanks q FJJ s
sy
E f Ž < y y yX < .
`
Hy`Hy`d y d y K0
< y y yX <
l
.
X
Ž D.2 .
To simplify the double integral in Eq. ŽD.2., consider the single integral term on its RHS. When integrated by parts, and noting that K 0 w6Ž x 02 q
l q
2 l2JJ
`
Hy`d y x Ž y . sin
x Ž y. .
Ž D.4 .
Inserting Eq. ŽD.4. into Eq. ŽD.2. yields Eq. Ž2.20..
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