ISSN 10637842, Technical Physics, 2014, Vol. 59, No. 1, pp. 26–35. © Pleiades Publishing, Ltd., 2014. Original Russian Text © M.A. Zelikman, K.A. Potseluev, 2014, published in Zhurnal Tekhnicheskoi Fiziki, 2014, Vol. 84, No. 1, pp. 28–37.
THEORETICAL AND MATHEMATICAL PHYSICS
Pinning of Linear Vortices and Possible Distances between Them in a 3D Ordered Josephson Medium with a Nonzero Structural Factor M. A. Zelikman* and K. A. Potseluev St. Petersburg State Technical University, Politekhnicheskaya ul. 29, St. Petersburg, 195251 Russia *email:
[email protected] Received February 11, 2013
Abstract—On the basis of the fluxoid quantization conditions, we derive a system of equations describing the current configuration of two interacting linear vortices in a 3D ordered Josephson medium in the entire range of possible values of structural factor b. The axes of these vortices are located in the middle row of an infinite strip with a width comprising 13 meshes. We propose a method for solving this system, which makes it possible to calculate the current configurations exactly. The critical values of pinning parameter Id are calculated, for which two linear vortices can still be kept at a distance of d meshes between their centers in the entire range of possible values of parameter b. The formula describing the Id(b) dependences for various values of d is derived. The dependences of the maximal pinning force F on parameter I for various values of b are analyzed. It is shown that for the same value of I, larger values of b correspond to larger maximal pinning forces. DOI: 10.1134/S1063784214010228
such a model, which is usually referred to as an ordered 3D Josephson medium, exhibits all properties typical of superconductors in an external magnetic field (Meissner screening currents, interacting vorti ces, a set of characteristic fields, etc.); even the quan titative relations are analogous to those for traditional and highTc superconductors. The mathematical description based on the equations of fluxoid quanti zation in meshes makes it possible to analyze all details of current configurations. Therefore, such an approach is expedient for analyzing processes occur ring in real HTSCs.
INTRODUCTION Analysis of the structure, motion, and pinning of vortices emerging in a sample placed in a magnetic field is one of the most important problems in super conductor physics. The theory of vortices based on the Ginzburg–Landau equations for traditional super conductors had been constructed even before the development of the BCS theory [1–3]. In ceramic hightemperature superconductors (HTSCs), the the oretical description of vortices encounters certain dif ficulties associated with the grain structure of the sam ple. These difficulties are primarily associated with the mesh structure of the medium in which contacting superconducting grains are separated by insulating regions. In the regions where granules touch one another, a large number of Josephson junctions are formed so that such media are sometimes referred to as Josephson media. All these Josephson junctions are nonlinear elements, which strongly complicates anal ysis of such media. The current states (both screening and vortex states) differ in their structure from the states existing in traditional superconductors. The Ginzburg–Landau equations are inapplicable in this situation, and another mathematical basis is required for an analytic description of grained superconduc tors. In the model of the grained HTSC proposed in [4], a system of equations for quantization of a fluxoid in meshes is used as the mathematical basis [5]. This model is a cubic grid with period h, consisting of superconducting wires of radius δ in which each link contains a Josephson junction. Calculations show that
The set of equations derived in [4] contains two dimensionless parameters I and b the meaning of which will be explained later. The approach proposed in [4] was used in [6] for analyzing the Meissner cur rent configurations in the entire range of parameters I and b and in [7] for studying possible structures and the energy of a solitary linear vortex. Important infor mation on the pinning of vortices can be obtained from analysis of possible distances between two soli tary vortices. Upon a decrease in the pinning force and an increase in the repulsive forces acting between the vortices, the minimal possible distance between their centers increases. The critical values of pinning parameter Id for which two vortices can still be kept at a distance of d meshes between their centers were cal culated in [8], but calculations were made only for b = 0. The present study aims at determining the depen dence of critical values of Id as functions of b in the entire range of possible values of parameter b. 26
PINNING OF LINEAR VORTICES AND POSSIBLE DISTANCES
1. METHOD OF CALCULATIONS AND BASIC EQUATIONS Following [8], we will carry out our analysis using the model in the form of a cubic grid with period h, consisting of superconducting wires of radius δ, in which each link contains a Josephson junction; all junctions are small in size and are characterized by the same critical current JC. The current distributions have a planar structure (i.e., the currents in all parallel planes perpendicular to the vortex axis and separated by distance h are distributed identically). Let us consider two parallel linear vortices whose axes pass through the center of a 2D plate with a thick ness of (2N – 1) meshes, which is infinitely large in two directions, which are separated from each other by d meshes. The cross section of this pattern by a plane perpendicular to the axes of the vortices is an infinitely long strip with a width of (2N – 1) meshes the middle row of which contains the centers of the two vortices under investigation. Figure 1 shows a quadrant of such a configuration for a strip of 13 meshes (N = 7). The central mesh of one of the two vortices is marked by a dot. The center of the other vortex lies below (outside Fig. 1) and is symmetric either to the central line of the lower row of meshes, or to the lower current line. In the former case, distance d between the centers of the vortices is equal to an even number of meshes (2M), while in the latter case, it is equal to an odd number of meshes (2M + 1). This means that there are (2M – 1) and 2M meshes between the central meshes of the vor tices, respectively. In each mesh, the fluxoid quantization condition holds, 2πΦ m /Φ 0 +
∑ϕ
(m) k
= 2πK m ,
(1)
k
∑
(m) ϕ k k
where is the sum of phase jump at the Joseph son junctions of the mth mesh, Φm is the total magnetic flux through the mth mesh, Φ0 is the magnetic flux quantum, and Km is an integer equal to unity for the central mesh of the vortex and zero for the remaining meshes. With increasing distance from the centers of the vortices, Josephson currents Jk = JC sinϕk decrease, the rate of this decrease increasing with crit ical current JC. As a result of calculations, we see that phase jumps at all junctions (except the largest phase jumps ϕ1–ϕ7) can be treated as small (so that sinϕk ≈ ϕk). Further, we assume that the condition of the smallness is satisfied when ϕk < 0.5. The regions in which phase jumps cannot be treated as small are outlined in Fig. 1 by bold segments. To avoid writing the conditions of current balance at junctions, it is convenient to use the meshcurrent method. Let us suppose that the mesh current equal to the product of JC by the corresponding “mesh” phase jump passes anticlockwise in each mesh. In Fig. 1, mesh phase jumps in each mesh are indicated. Then, TECHNICAL PHYSICS
Vol. 59
No. 1
2014
ϕ5 ϕ3
27
ψ2
ξ2
η2
ρ2
ζ2
μ2
χ2
ψ1 ϕ4
ξ1
η1
ρ1
ζ1
μ1
χ1
η0
ρ0
ζ0
μ0
χ0
γ0
θ0
ω0
ν0
π0
ψ0 ϕ1 α0 ϕ2
ϕ5
ξ0
ϕ3 β0
ϕ7 ϕ6
α1
β1
γ1
θ1
ω1
ν1
π1
α2
β2
γ2
θ2
ω2
ν2
π2
αM
βM
γM
θM
ωM
νM
πM
Fig. 1. Distribution of phase jumps for two vortices in an infinitely long strip with a width of 13 meshes in a plane perpendicular to the vortices. The mesh phase jump is indicated at the center of each mesh. All mesh currents are directed counterclockwise. The pattern can be continued symmetrically to the left and downwards.
the values of phase jumps ϕk at the junctions (except ϕ1–ϕ7) are determined as the differences of corre sponding mesh values. The magnetic flux through the mth mesh can be written in the form [1] ⎛ Φ m = μ 0 S/h ⎜ i m + b ⎝
m⎞ k
∑ J ⎟⎠ ,
(2)
k
where b is the factor of field nonuniformity due to the discrete current distribution along the vortex axis, S is (m) the mesh area, im is the mesh current, and k J k is the algebraic sum of the currents at the junctions of the mth mesh. We will further assume that the structural factor is defined, as in [7], by the expression
∑
1 1 – exp ( – 2π ( 1 – δ/h )) . b = ln 2π ( 1 – 2δ/h ) 1 – exp ( – 2πδ/h ) Figure 2a shows the dependence of parameter b on δ/h; it can be seen that significant values of structural factor b corresponding to 0.01 ≤ δ/h ≤ 0.50 lie in the range from 0.454 to 0.049. Substituting expression (2) into (1), we obtain the following system of equations for all cells with indices of mesh currents greater than or equal to unity above the central row of vortices (for k ≥ 1):
28
ZELIKMAN, POTSELUEV 0.5
Considering that sin ϕ4 = Ψ – ψ1, where Ψ stands for the mesh phase jump in the mesh under ψ1 (since notation ψ0 will be used for another purpose), we can write Eq. (5) in the form ( 4 + g )ψ 1 – ψ 2 (6) – ( ψ 1 + IbΨ + ϕ 4 )/ ( Ib + 1 ) – 2ξ 1 = 0. Introducing the notation (7) ψ 0 = ( ψ 1 + IbΨ + ϕ 4 )/ ( Ib + 1 ), we obtain Eq. (6) in the form exactly corresponding to system (3). Thus, variable ψ0 is not the mesh phase jump of the corresponding mesh. Its introduction allows us to unify the system of equations. An increase in the number of variables by unity leads to the emer gence of additional equation (7). We seek the solution to the system of linear differ ence equations (3) in the form
(a)
b, a.u.
0.4 0.3 0.2 0.1 0.045 0.1
0.2
0.3 δ/h, a.u.
0.4
0.5
(b)
g, a.u.
0
k
k
ψ k = Aλ ,
k
ξ k = Bλ ,
η k = Cλ ,
k
ρ k = Dλ , k
k
k
ζ k = Eλ , μ k = Fλ , χ k = Gλ . Substituting these relations in system (3), we trans form it to
I
−1/I
(8)
0 0.045
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
b, a.u.
Fig. 2. (a) Dependence of parameter b on δ/h; (b) depen dence of parameter g on b.
( 4 + g )ψ k – ψ k + 1 – ψ k – 1 – 2ξ k = 0, ( 4 + g )ξ k – ξ k + 1 – ξ k – 1 – ψ k – η k = 0, ……………………………………… ( 4 + g )χ k – χ k + 1 – χ k – 1 – μ k = 0,
a –λ . 0 0
– 2λ a . 0 0
0 –λ . 0 0
0 0 . 0 0
0 0 . –λ 0
0 0 . a –λ
⎞⎛ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎠⎝
0 0 . –λ a
A B · F G
⎞ ⎟ ⎟ ⎟ = ( 0 ), ⎟ ⎟ ⎟ ⎠
(9)
where the following notation has been introduced: 2
(3)
(10) a = ( 4 + I )λ – λ – 1. Expanding the nth order determinant of type (9) in the first row, we obtain the following recurrent rela tion:
where g = I/(Ib + 1) and pinning parameter I is defined as before by the expression
2
Δ n = af n – 2λ f n – 1 ,
(11)
2
2
I = 2πμ 0 ( h – 2δ ) J C /hΦ 0 .
(4)
Quantity g is the “effective” value of the pinning parameter, which determines the behavior of the vor tex in the range of small phase jumps. Figure 2b shows the dependence of g on b. Upon a change in parameter I, the hyperbole should be moved horizontally so that the curve intersects the g axis at point I. Note that Eqs. (3) are valid for all meshes above the central row of vortex except for the mesh with current JCψ1 (third mesh in the left row). For this mesh, the fluxoid quantization condition assumes the form 3ψ 1 – ψ 2 – 2ξ 1 – ϕ 4 + Iψ 1 + Ib ( 3ψ 1 – ψ 2 – 2ξ 1 – sin ϕ 4 ) = 0.
(5)
(12) f n = af n – 1 – λ f n – 2 , where fn is the nthorder determinant of the form
fn =
a –λ . 0
–λ a . 0
0 –λ . 0
. . . .
. . . .
. . . .
0 0 . –λ
0 0 . . a
(13)
Using formulas (11)–(13), we can find the determi nant of matrix (9): 7
6
4
2
(14) Δ 7 = λ k ( k – 7k + 14k – 7 ), where k = a/λ. Equating relation (14) to zero, we obtain the conditions for the existence of a nonzero solution to system (3): TECHNICAL PHYSICS
Vol. 59
No. 1
2014
PINNING OF LINEAR VORTICES AND POSSIBLE DISTANCES
k 2, 3 = ± 0.867767,
k 1 = 0,
k 4, 5 = ± 1.56366,
m
(15)
k 6.7 = ± 1.94985. Substituting relations (15) into (10), we obtain eigenvalues λi (i = 1, …, 7): 2
λ i = ( 4 + I – k i – ( 4 + I – k i ) – 4 )/2.
(16)
We have discarded the versions with the plus sign of radical (16) because expressions (8) in the given geom etry should not increase indefinitely with k (i.e., all values of |λi| must be smaller than unity). Solving system (9), we express coefficients B, C, …, G in Eq. (8) in terms of A: B = kA,
2
2
C = ( k – 2 )A,
4
2
D = k ( k – 3 )A,
4 2 F = k ( k – 5k + 5 )A, (17)
E = ( k – 4k + 2 )A, 6
4
2
G = ( k – 6k + 9k – 2 )A. The general solution to system (3) has the form (for m ≥ 0) 7
ψm =
∑
ξ m = 0.5
i=1
∑A k λ i i
m i ,
∑
m
A i ( k i – 2 )λ i ,
∑ A k (k i i
2 i
i=1
∑
(18) –
2 4k i
+
4
2
∑
4
2
m
A i ( k i – 6k i + 9k i – 2 )λ i .
i=1
Writing the quantization conditions for fluxoid (1) for all meshes with mesh current indices greater than or equal to unity below the central row of the vertex and introducing analogously to (7) the additional vari able (19) α 0 = ( α 1 + Ibα + ϕ 2 )/ ( Ib + 1 ) (α0, like ψ0, is not the mesh phase jump), we obtain a system of equations identical to (3) for quantities αk, βk, γk, θk, ωk, νk, and πk (for k ≥ 0). For these quanti ties, all formulas (8)–(17) hold. The only change is associated with the fact that there are no grounds for rejecting in expressions (16) the solutions with the plus sign of the radical. Instead of this, the symmetry con ditions for configuration relative to the horizontal symmetry axis appear: αm, βm, γm, θm, ωm, νm, and πm are described by expressions analogous to (18) with i TECHNICAL PHYSICS
Vol. 59
No. 1
+ Ib ( 4η 0 – η 1 – ρ 0 – γ 0 – ξ 0 ) = 0,
(21)
( 4 + g )π 0 – ν 0 – χ 0 – π 1 = 0,
7
6
+ Ib ( 4β 0 – ξ 0 – α – γ 0 – β 1 ) = 0,
( 4 + g )ν 0 – π 0 – ω 0 – μ 0 – ν 1 = 0,
m
A i k i ( k i – 5k i + 5 )λ i ,
i=1
χ m = 0.5
ϕ 6 – ϕ 3 + 2β 0 – β 1 – ξ 0 + Iβ 0
( 4 + g )ω 0 – θ 0 – ν 0 – ζ 0 – ω 1 = 0,
7
∑
sin ϕ 6 = β 0 – γ 0 , sin ϕ 7 = ξ 0 – η 0 . As well as 14 conditions (1) for meshes of two rows with zero indices in Fig. 1: ϕ 1 + ϕ 2 + 2ϕ 3 + Iα + Ib ( 4α – 2β 0 – Ψ – α 1 ) = 2π,
( 4 + g )θ 0 – θ 1 – γ 0 – ω 0 – ρ 0 = 0,
m 2 )λ i ,
i=1
μ m = 0.5
(20)
– ϕ 7 + 3η 0 – η 1 – ρ 0 – γ 0 + Iη 0
m
– 3 )λ i ,
7
ζ m = 0.5
sin ϕ 5 = Ψ – ξ 0 ,
+ Ib ( 4ξ 0 – β 0 – ξ 1 – Ψ – η 0 ) = 0,
7
4 Ai ( ki
sin ϕ 4 = Ψ – ψ 1 ,
ϕ 7 – ϕ 5 + 2ξ 0 – β 0 – ξ 1 + Iξ 0
i=1
ρ m = 0.5
d–m
ϕ 4 – ϕ 1 + 2ϕ 5 + IΨ + Ib ( 4Ψ – α – 2ξ 0 – ψ 1 ) = 0,
7
2
m
by ( λ i + λ i ), where d is the separation between the centers of the vortices and the set of ki and λi for i = 8–14 is the same as for the interval from 1 to 7. To determine 14 values of Ai (i = 1, 2, …, 14), seven phase jumps ϕ1–ϕ7 and two mesh currents Ψ and α, we write two additional conditions (7) and (19), seven conditions for the current in bold regions in Fig. 1, sin ϕ 1 = α – Ψ, sin ϕ 2 = α – α 1 , sin ϕ 3 = α – β 0 ,
+ Ib ( 4γ 0 – β 0 – γ 1 – η 0 – θ 0 ) = 0,
i=1
η m = 0.5
varying from 8 to 14, in which factors λ i are replaced
– ϕ 6 + 3γ 0 – γ 1 – η 0 – θ 0 + Iγ 0
7
m
Ai λi ,
29
2014
( 4 + g )ρ 0 – η 0 – ζ 0 – θ 0 – ρ 1 = 0, ( 4 + g )ζ 0 – ρ 0 – ω 0 – μ 0 – ζ 1 = 0, ( 4 + g )μ 0 – ν 0 – ζ 0 – χ 0 – μ 1 = 0, ( 4 + g )χ 0 – π 0 – μ 0 – χ 1 = 0. This gives 23 equations in 23 unknowns. A method for solving such a system based on its linearization (with the exception of two equations which were solved numerically) was proposed in [8]. In contrast to [8], we will solve this system using the Newton method (i.e., linearization of all equations followed by iterative approximation to the solution). Let us repre sent all significant phase jumps ϕ1–ϕ7 and their sines 0
0
0
in the form ϕk = ϕ k + δk, sinϕk = sin ϕ k + cos ϕ k δk 0
(k = 1, 2, 3, …, 7), where we assume that ϕ k are known quantities and δk are new unknowns (instead of ϕk). The resultant system of 23 equations is linear in all 23 unknowns. Solving this system, we can find all δk and
30
ZELIKMAN, POTSELUEV
Table 1 b=0 d
kd
1 2 3 4 5 6
1.570 0.960 0.593 0.397 0.288 –
b = 0.1
b = 0.45
Id
Fh/E0π
Id
Fh/E0π
Id
Fh/E0π
Id
Fh/E0π
0.934 0.339 0.140 0.0587 0.0196 0
0.575 0.0907 0.0256 0.00886 0.00266 –
0.805 0.309 0.132 0.0564 0.0191 0
0.492 0.0843 0.0246 0.00858 0.00260 –
0.635 0.263 0.119 0.0524 0.0181 0
0.387 0.0740 0.0227 0.00809 0.00248 –
0.550 0.238 0.111 0.0498 0.0174 0
0.331 0.0680 0.0215 0.00776 0.00239 –
0
0
calculate new quantities ϕ k using the formula ϕ k = 0 ϕk
b = 0.3
+ δk. Then, we analyze the system with the new 0
variables ϕ k . The applicability of this method is con firmed by the results. The iterative procedure con verges (i.e., with each new iteration, the value of δk becomes smaller and ultimately tends to zero), which justifies the use of the expansion of the sine function. In this way, the initial system can be solved to any degree of accuracy using just a few steps. Testing of this method revealed that for the first iteration, we can set 0 all ϕ k equal to unity as initial conditions. The results obtained using this method completely coincides with the results obtained by the method proposed in [8]. For values of pinning parameter I smaller than a certain critical value Id (subscript d corresponds to the number of meshes between the centers of the vortices), the system has no solution, which means that for such
Fig. 3. Calculated distribution of phase jumps over junc tions for separation d = 5 between the centers of the vorti ces for b = 0.45 and critical value of I5 = 0.0174. The pat tern can be continued symmetrically to the left and down wards.
values of I, the vortices cannot be kept at such short distances. For I > Id, the system has two solutions cor responding to two possible configurations of the vorti ces (both “a” or both “b” [7]). Upon a decrease in I, these solutions become closer and coincide for I = Id. It should be noted that one more variant can also exist, when one of the vortices has configuration a and the other has configuration b. In this case, the symmetry of the current configurations is not observed, and analysis in this case would be cumbersome. However, since we are interested in critical values of Id, and con figurations a and b coincide for I = Id (see above), the critical value in this case will be the same. 2. RESULTS OF CALCULATIONS, THEIR INTERPRETATION, AND ANALYSIS Table 1 contains the results of calculation of the critical values of parameter Id for various distances d between the centers of the vortices and for different values of b. Calculations were performed for a strip width of 13 meshes (N = 7). To verify the validity of the initial assumptions, let us consider the resultant structure of the typical pat tern. By way of example, Fig. 3 shows the calculated distribution of the phase jump over junctions for d = 5 and b = 0.45 and for critical value of I5 = 0.0174. It can be seen from the figure that the assumption concern ing the smallness of all ϕk except ϕ1–ϕ7 is justified. Figure 4 shows the dependences of Id on b for vari ous separations d between the centers of the vortices. With increasing b, the values of Id decrease for all val ues of d. Note that the values of Id for b = 0 coincide with those obtained earlier [8]. At first glance, these results appear as astonishing. Indeed, the initial system of equations contains parameter b only in combination Ib, whose values are small for all Id. Therefore, it could be expected that the value of Id for Ib Ⰶ 1 is independent of b; however, it can be seen from Fig. 4 that it depends on b quite noticeably. To explain this phenomenon, we will find the analytic expression describing the curves in Fig. 4. For this purpose, we confine our analysis to the first two terms in the expansion Id(b) = Id(0) – kdId(b)b + …. This gives TECHNICAL PHYSICS
Vol. 59
No. 1
2014
PINNING OF LINEAR VORTICES AND POSSIBLE DISTANCES
I d ( b ) = I d ( 0 )/ ( 1 + k d b ).
(22)
The values of kd for various separations d are given in the second column of Table 1. Expression (22) describes the curves in Fig. 4 very exactly for all values of d. The errors in the description do not exceed sev eral fractions of a percent in the entire range of b (from 0 to 0.45). For d > 5, the critical state is not observed (i.e., the system has solutions for any indefinitely small values of I). This result confirms the conclusion drawn in [9]: when the value of I tends to zero, the minimal possible separation between linear vortices does not increase indefinitely as in the case of 2D vortices, but attains a certain finite value d0 and then remains unchanged. True, the value of this constant predicted in [9] is larger (d0 = 7–8). Let us determine the maximal pinning force for a vortex (i.e., the force that should be applied to the vor tex to set it in motion). It is equal to the force acting on the vortex for the critical value of pinning parameter Id. For calculating this force, we will use the continuous medium approximation. Each vortex has a central part (core) with a size of a few meshes, in which the dis creteness of the medium is significant, while the dependence of ϕ on the coordinates observed in the outer regions can be assumed to be quasicontinuous. Since the interaction between vortices is determined by their outer regions, we can use in analysis of the equilibrium conditions the expressions for the forces of interaction of vortices in a continuous medium. The transition from the discrete to continuous case for small values of pinning parameter I for b = 0 was con sidered in [10]. It was shown that a vortex in a contin uous medium can be described by the same equations as those for a conventional Abrikosov vortex [3], but the role of the Londons penetration depth is played by quantity λ = h/ I . It can easily be seen that for nonzero values of b, in the range of small phase jumps, the equations of sys tems (3) and (21) have the same form as for b = 0, but parameter I is replaced by g = I(Ib + 1). Performing the corresponding substitution in the formulas [10], we obtain the following expression for the energy of interaction of two vortices per vortex: U int = ± E 0 πgK 0 ( r g/h ),
(23)
where the plus sign corresponds to repulsion and the minus sign to attraction; K0 is the zerothorder Bessel (Hankel) function of the imaginary argument, r is the separation between the centers of the vortices, and 2 E0 ≡ Φ 0 /4π2μ0h2. Taking advantage of the fact that dK0/dx = –K1(x), we can find the force exerted on a vortex by another vortex: F int = – ∂U int /∂r (24) 3/2 = ± E 0 π/hg K 1 ( r g/h ). TECHNICAL PHYSICS
Vol. 59
No. 1
2014
31
Id 1.0 0.9 0.8 0.7 0.6 d=1 0.5 0.4 0.3 d=2 0.2 0.1 0
0.1
0.2
b
0.3
0.4
d=3 d=4 d=5 0.5
Fig. 4. Dependences of Id on b for various separations d between the centers of the vortices. Crosses correspond to calculated values.
This formula makes it possible to find the force of interaction between two solitary vortices in a sample infinitely large in all directions; however, the geometry considered here generally describes a different situa tion. At distances between the vortices comparable with the strip width, the vortices cannot be treated as solitary, and the boundaries cannot be ignored. The effect of boundaries can be estimated using the follow ing considerations [8]. It can easily be seen that the above configuration is equivalent to a periodic sequence of pairs of vortices with alternating orienta tions. To verify this, let us consider such a sequence in which the separation between the central lines of adja cent pairs is 2N meshes (i.e., (2N – 1) meshes fit between their central meshes). By way of example, Fig. 5a shows such a configuration with N = 3 and d = 2. The central meshes of the vortices will be marked by dots or crosses depending on their orientation. It is clear from the symmetry of the pattern that all hori zontal currents in the row of meshes located in the middle between the central lines of adjacent pairs (in Fig. 5a, this row is crossed by line AA) are zero. For this reason, the structure of the pattern between two such adjacent “axial” vertical rows (between line AA and the line symmetric to it and passing to the left of the vorti ces) is identical to the above case of a pair of vortices located in the central row of a layer of width (2N – 1) meshes. Thus, the effect of the strip boundary can be taken into account and estimated as the action of the entire system of periodically arranged extra vortices on the two vortices under investigation. The situation resembles the image method in electrostatics. The above analysis is valid only in the case when the separation between the central lines of adjacent pairs is
32
ZELIKMAN, POTSELUEV (a)
A 0 0
deform it, but not strive to move it. Therefore, analyz ing the equilibrium conditions for this vortex, we must take into account only the forces exerted on it by the vortices from another chain. Taking into account the corresponding cosines, we obtain the following expression for the total force: E 0 πg F = g K1 ( d g ) h
0 0 A (b)
∑K ( 1
m=1
(c)
(d) 0
Fig. 5. Infinite periodic sequences of vortices for the paral lel or alternate configuration, which are equivalent to a single vortex in a 2D wafer with a thickness of 13 meshes and corresponding to different boundary conditions: (a) alternate even, (b) alternate odd, (c) parallel even, and (d) parallel odd. Bullets indicate the central meshes of the vortices.
equal to the even number of meshes. The configura tion shown in Fig. 5a will be referred to as the alternate even configuration in accordance with the orientation of adjacent pairs. Let us also consider the following configuration: alternate odd (Fig. 5b), parallel even (Fig. 5c), and parallel odd (Fig. 5d). In each of these configurations, the last equation in system (3) (and, hence, matrix (9)) has its own form; as a result, rela tions (11)–(18) will change. In particular, in the con figuration shown in Fig. 5b, we must replace extreme right element a in the lower row of matrix (6) by (a + λ); for Fig. 5c, it must be replaced by (a – λ), and the sec ond right element –λ of the lower row for Fig. 5d should be replaced by –2λ. Let us consider one of the vortices of the initial pair for the above alternate configuration. This vortex interacts with all vortices in both infinitely long chains. However, the vortices from its own chain can only
(25)
∞
+2
∫
2
d + ( mL )
2
m d g ) ( – 1 ) , 2 2 d + ( mL )
where d is the separation between the centers of a vor tex pair and L is the period of the chain (the values of d and L are expressed in terms of the number of meshes). In the case when the pinning parameter is equal to its critical value for a given distance d between the vortices, the latter cannot be held by pinning; i.e., formula (25) determines the maximal pinning force. The first term in the parentheses in Eq. (25) corre sponds to mutual repulsion of the two vortices under investigation, while the remaining terms describe the force exerted on the sample by the sample boundaries. In the alternate configuration, this force is directed against the force of interaction of the pair of vortices (i.e., reduces it and thus helps to keep the vortices in their positions). For calculations in the parallel con figuration, we must exclude (–1)m from the terms of series (25). Therefore, the interaction of a vortex with the boundaries facilitates displacement. These argu ments lead to the conclusion that the maximal pinning force must be stronger than for a solitary vortex in the alternate configuration and weaker in the parallel con figuration. Let us suppose that the value of I is smaller than the critical value for the parallel configuration (i.e., two interacting vortices cannot be kept at a given distance). Owing to the interaction with the bound aries, this is still possible in the alternate configura tion. It follows hence that the critical values of Id for the alternate configurations are lower than for the par allel configurations; the value of Id for a solitary vortex lies between these values. We have considered earlier the alternate configura tion with N = 7 and L = 14 (see Table 1). Let us now investigate the parallel configuration for N = 7 and L = 13. Table 2 contains the calculated critical values of parameter Id for this case. As expected, these values are larger than in the case of the alternate configura tion (Table 1). It should be noted that for arbitrarily large values of d, the values of Id differ from zero. This fact determines the importance of analysis of parallel configurations because it is these configurations that make it possible to calculate the pinning forces for very small values of I. The Id(b) dependences for all values of d are successfully described by formula (24); coeffi cients kd are given in the second column of Table 2. With increasing d, the values of coefficients kd TECHNICAL PHYSICS
Vol. 59
No. 1
2014
PINNING OF LINEAR VORTICES AND POSSIBLE DISTANCES
33
Table 2 b=0 d
kd
1 2 3 4 5 51
1.570 0.926 0.536 0.331 0.211 0
b = 0.1
b = 0.3
b = 0.45
Id
Fh/E0π
Id
Fh/E0π
Id
Fh/E0π
Id
Fh/E0π
0.934 0.340 0.144 0.0702 0.0392 0.000364
0.575 0.0909 0.0263 0.0103 0.00498 0.0000332
0.805 0.311 0.137 0.0680 0.384 0.000364
0.492 0.0847 0.0252 0.0100 0.00491 0.0000332
0.635 0.265 0.124 0.0639 0.0369 0.000364
0.387 0.0744 0.0234 0.00959 0.00477 0.0000332
0.550 0.240 0.116 0.0611 0.0358 0.000364
0.331 0.0684 0.0223 0.00934 0.00466 0.0000332
decrease monotonically and vanish for quite large val ues of d (i.e., Id become independent of b). Figure 6 shows for comparison the curves describ ing the dependence of Id on b for alternate and parallel configurations for various separations d between the vortices. The curves for d = 1 and 2 for both types of configurations almost coincide and are not shown in Fig. 6. Maximal pinning force F is a function of the struc tural factor b that (as mentioned above) appears in the initial set of equations only in combination Ib whose values are small for all values of Id. Expanding F into a series in Ib and retaining only the firstorder terms, we obtain F ( I, b ) = F ( I, 0 ) + kIb.
Figure 7 shows the curves describing the depen dences of the maximal pinning force F normalized to E0π/h on parameter I for alternate configuration with L = 14 for various values of b (see Table 1). It can easily be seen that for the same value of I higher values of b correspond to larger maximal values of the pinning force (i.e., k > 0 in Eq. (26)); the distances from the lower curve (b = 0) to the curves with nonzero values of b are approximately proportional to b. Crosses cor respond to the results of calculation and the curves are drawn through them approximately. Therefore, the value of k for Eq. (26) cannot be calculated rigorously, but the qualitative agreement between the curves and expression (26) can be traced.
(26) Fh/πE0 0.6
1
Id 0.14
2
0.5
0.13 0.12 d=3
0.11
3
0.4
0.10
4
0.09 0.3
0.08 0.07 0.06
d=4
0.2
0.05 0.04
0.1
0.03
d=5
0.02 0.01 0
0 0.1
0.2
0.3
0.4 0.45
b
Fig. 6. Dependences of Id on b for alternate (lower curves) and parallel (upper curves) configurations for various sep arations d between vortices. TECHNICAL PHYSICS
Vol. 59
No. 1
2014
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 l
Fig. 7. Dependences of the maximal pinning force on parameter I for alternate configurations with L = 14 for various values of b: 0 (1), 0.1 (2), 0.3 (3), and 0.45 (4). Crosses denote calculated points.
34
ZELIKMAN, POTSELUEV
Fh/πE0 0.028 1
1'
0.024 2
2'
0.020 0.016 0.012 0.008 0.004
0
0.02 0.04 0.06
0.08 0.10 0.12 0.14 I
Fig. 8. Dependences of the maximal pinning force on parameter I for alternate configurations with L = 14 and parallel configuration with L = 13 for b = 0 (curves 1, 1') and 0.45 (curves 2, 2 '); curves 1 and 2 correspond to alter nate configurations and 1' and 2 ' for parallel configura tions. Crosses denote calculated points.
Figure 8 shows the F(I) dependences for the alter nate and parallel orientations considered here for b = 0.45. As expected, the curve for the alternate configu ration passes above that corresponding to the parallel configuration. The curve for a solitary vortex in an unbounded medium must pass between these curves. It can be seen from Table 2 that for large d, the val ues of Id become independent of quantity b. There fore, these values can be calculated for b = 0. The cal culations performed in [8] show that the condition for the equality of the maximal pinning force to the force of interaction between two solitary vortices assumes the form E 0 πI π (27) 0.1E 0 I = IK 1 ( d I ). h h Since K1(x) < 1/x, it follows from Eq. (27) that the minimal separation d0 between the vortices is smaller than 10 meshes. An analogous result was obtained in [9], where the minimal separation was 7–8 meshes. Thus, when I tends to zero, the minimal separation between two solitary linear vortices in a discrete medium does not increase indefinitely, but attains a certain finite value d0 and then remains unchanged. This fact indicates that when I 0, the pinning of linear vortices cannot be disregarded. These arguments are valid only for a solitary pair of vortices at an infinitely large distance from the bound ary, where both real and “imaginary” extra vortices are absent. The presence of extra vortices changes the sit uation. In the case of alternate configuration, they
help the vortices to stay at their site; therefore, the value of d0 becomes even smaller, which is confirmed by numerical calculations (see Table 1). In parallel configurations, extra vortices increase the repulsive force; therefore, it is more difficult for a vortex to remain at its site. Let us suppose that for a small value of I, the conditions that force F from Eq. (27) (without π factors (–1)m) is equal to pinning force 0.1E0 I holds h at distant d between the vortices in the pair. On the verge of equilibrium, the ratio of the pinning force to I is equal to constant 0.1E0π/h (i.e., is independent of I and d). Let us estimate the behavior of ratio F/I from Eq. (27) upon a decrease in I (g = I for b = 0). Even if d I Ⰶ 1 (as a result of which the first terms in the parentheses in Eq. (27) are proportional to 1/d I ), the argument of function K1(x) with increasing m is no longer small, and for not very small x, function K1(x) increases upon a decrease in x faster than 1/x. There fore, with decreasing I, ratio F/I increases and the pin ning force cannot keep a vortex at the previous dis tance. The value of d increases, as a result of which F/I decreases and the vortex can be in equilibrium in its new position. These arguments do not depend on period L. Thus, for any indefinitely large period L (i.e., plate thickness), for I 0, the minimal possible separation between the vortices of a pair increases indefinitely (i.e., the value of d0 is infinitely large). In other words, in the case of the parallel configuration, the critical value of pinning parameter Id differs from zero for indefinitely large separations d between the vortices. The validity of this statement is confirmed by the results of calculations (see Table 2). CONCLUSIONS On the basis of the fluxoid quantization conditions, we have derived a system of equations describing the current configuration of two interacting linear vortices in a 3D ordered Josephson medium in the entire range of possible values of structural factor b. The method proposed for solving this system is based on the fact that the phase jumps at the junctions can be treated as small even at a small distance from the centers of the vortices; this allows us to pass to a linear system of finitedifference equations. This sys tem has different solutions in the region between the vortices under investigation and in two symmetric regions from the centers of the vortices to infinity. Joining these solutions in the meshes close to the cen ter of a vortex, in which the linearity of the equations is violated, we can obtain the exact solution. In addi tion, we are using the idea of approaching the exact solution by successive iterations in the values of phase jumps, which cannot be treated as small. The conver gence of the method in all cases is confirmed by verifi TECHNICAL PHYSICS
Vol. 59
No. 1
2014
PINNING OF LINEAR VORTICES AND POSSIBLE DISTANCES
cation of the correctness of the initial equations of the system. This method makes it possible to calculate quite exactly the current configuration whose cross section by a plane perpendicular to its axis is an infinitely long strip with the centers of the vortices in the middle row. The width of the strip in the cases considered here is 13 meshes. It is shown that for a given separation d between the central meshes of the vortices and a fixed value of structural factor b, this system has a solution only for values of the pinning parameter exceeding a certain critical value Id(b). The critical values of pinning parameter Id for which two linear vortices can still be kept at a distance of d meshes between their centers are calculated in the entire range of possible values of parameter b. The for mula is derived, which describes the Id(b) dependence for various values of d. For large d, the values of Id become independent of b. Therefore, these values can be calculated for b = 0. The dependence of the maximal pinning force F on parameter I for various values of b are analyzed. It is shown that for the same value of I, larger values of b correspond to larger maximal pinning forces. When the value of I tends to zero, the minimal sep aration between two solitary vortices in a discrete
TECHNICAL PHYSICS
Vol. 59
No. 1
2014
35
medium does not increase indefinitely (as in the case of Abrikosov vortices in a continuous medium), but attains a certain finite value d0 and then remains unchanged. This fact indicates that for I 0, the pinning of linear vortices cannot be disregarded. REFERENCES 1. A. A. Abrikosov, Sov. Phys. JETP 5, 1174 (1957). 2. B. B. Goodman, Rev. Mod.Phys. 36, 12 (1964). 3. P. De Gennes, Superconductivity of Metal and Alloys (Benjamin, New York, 1966). 4. M. A. Zelikman, Supercond. Sci. Technol. 10, 469 (1997). 5. I. O. Kulik and I. K. Yanson, Josephson Effect in Super conducting Tunnel Structures (Nauka, Moscow, 1970). 6. M. A. Zelikman and K. A. Potseluev, Tech. Phys. 57, 579 (2012). 7. M. A. Zelikman and K. A. Potseluev, Tech. Phys. 58, 316 (2013). 8. M. A. Zelikman, Tech. Phys. 51, 1174 (2006). 9. M. A. Zelikman, Tech. Phys. 50, 967 (2005). 10. M. A. Zelikman, Supercond. Sci. Technol. 10, 795 (1997).
Translated by N. Wadhwa
