Dynamics of helical vortices in a superconducting wire

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Physica C 264 (1996) 204-212

Dynamics of helical vortices in a superconducting wire M. Shvartser, M. Gitterman, B.Ya. Shapiro * Institute of Superconductivity, Department of Physics, Bar-llan University, 52100, Ramat Gan, Israel Received 22 March 1996

Abstract

The dynamic theory of helical vortices in a superconducting wire subjected to a longitudinal bias current and axial magnetic field has been developed. Two distinct regimes are identified: vanishing and spreading. The helix radius and the voltage induced by the helix motion are obtained as functions of time. In the presence of an AC current the jumps of the voltage are caused by the helix motion.

1. Introduction A magnetic field penetrates a type-II superconductor in the form of magnetic flux [1]. The motion of the magnetic flux defines the resistivity and magnetic properties of superconductors. In the absence of an external magnetic field the resistivity is described by the motion of linear flux lines (or magnetic vortex rings in a cylindrical wire [2]). However, as was shown in the pioneering work of Clem [3], in the presence of an external magnetic field parallel to a resistive current, a linear flux line lying on a cylinder axis becomes unstable against the growth of helical perturbations. For higher transport currents the analogous instability occurs also in a flux-line lattice [4]. The origin of these instabilities is in a Lorentz force [j × tho]/C exerted by the applied current density j upon the quantum of flux ~b0 which makes the

* Corresponding author. Fax: +972 3 5353298; e-mail: f67410@ barilvm.bitnet.

left-hand spirals expand and the right-hand spirals contract. The Lorentz force is proportional to j, and for large enough currents this force is capable of overcoming the force due to the surface tension which has a tendency to move the vortex to the cylinder axis where it has a shortest length. This helical instability is similar to the Onsager-Feynman mechanism of viscosity in superfluids [5] and to the magnetohydrodynamic cork-screw instability in a plasma [6]. Clem considered [3] the stability conditions of a linear flux taking into account the self-energy associated with the surface tension and the energies of interactions of the flux with the sources of extemal current and magnetic field. Among other things, the following non-trivial dissipative regime was predicted: for some values of an external field H a and the current field Hj the right-hand spiral flux with the pitch angle arctan(Hj/H a) is nucleated at the surface of a cylinder. This spiral, then, contracts to the axis, becomes unstable, continues to move as a left-hand spiral, and, finally, leaves the sample. Afterwards, this " e n t r y - e x i t " regime repeats itself.

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M. Shvartser et aL / Physica C 264 (1996) 204-212

An important generalization of Ciem's work has been performed recently by Genenko [7]. Firstly, he added to the stability analyses an additional energy associated with the interaction of the flux with a surface. Secondly, he solved the London equation for the magnetic field inside a sample, and matched its solutions on the surface with those of Maxwell's equations. The latter result allows one to draw the current-voltage phase diagram describing the different regimes of the resistive states of superconductors. Some limitations of such an approach should be mentioned. For a bulk sample the current density is exponentially small everywhere but in the immediate vicinity of a surface. Therefore, the helical instability effects are of no importance. Another restriction occurs in samples in which one or two dimensions are smaller than the London penetration length A, i.e., in strips or filaments, respectively. In this case one can no longer consider the vortices as lines, and one has to solve the non-linear Ginzburg-Landau equations (GLE's) instead of the linear London equations, making the problem much more complicated. However, the problem is not hopeless. It was shown [8,9] that for strips by some rigorous procedure one can reduce the GLE to the equation of motion for a single vortex. Such a program has been realized [9] for linear vortices in a strip where we considered the motion of a single vortex, a vortexantivortex pair, their interactions with a surface, etc. A similar approach is used here for a helical vortex. We assume that a helical vortex already exists in a sample. It can be characterized initially by the radius r 0 and the pitch 2-rrL (for simplicity, we henceforth call L the pitch). Our aim will be to describe the subsequent motion of a helical vortex in a superconducting cylinder. In our previous work [10] we have shown, both analytically and numerically, the onset of a helical vortex on the surface of a superconducting wire. The present work is a continuation of Ref. [10] in the sense that we consider here the subsequent behavior of a helical vortex which has appeared on a surface, moved away from the surface and passed the cylinder axis to the outside of the wire. It rums out that the only motion allowed which preserves the helical form of the vortex is one with constant pitch (in contrast to different cases suggested in Ref. [11]).

2. Basic equations We start from the Lagrangian of a single vortex line in the form

L=-fo'ds=-o-f(dtz+u ( ,1)l "at-[dUx( Z, t)] 2 "at-[agy( z, t)12) '/2 =-try

l+--~-Z)+()

+(--~-Z

d

(1) where o- is the line tension per unit length of the vortex, ux(z, t), Uy(Z, t) and u:(z, t) are the components of deviations of the vortex from the straight line (z-axis) and z is the coordinate of an arbitrary point of the vortex on the straight line. Using the Lagrangian density

~(

du: 2

du~

2

dUy 2

(2)

one can write the equation of motion:

---az

~ a~ z

=f,

(3)

where u - {ux(z, t), Uy(Z, t), u:(z, t)} is the vector of deviations and f is the external force density. We assume that the vortex line preserves its helical form and we choose the helix axis to be the z-axis. This condition requires

u x = r(t) cos

z+.:(z, t)

L(t) ' Z + u z ( z , t) uy = r( t) sin L(t)

(4)

We assume that there is no pinning force acting on a vortex inside a wire. Hence, there are only two external forces in our problem: the viscous drag force density L and the (Lorentz) force fj. The viscous drag force density has the form fv = -"O ti, where rl is a phenomenological viscous drag coefficient per unit length of vortex [3] while the density of the Lorentz force is fj = [ j × d~o]/C.

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M. Shvartser et a l . / Physica C 264 (1996) 204-212

Both these force densities act on a unit length of the vortex, so that the force acting on a section of vortex placed between points 1 and 2 is equal to:

2

r2

ds

Therefore, in order to pass from the integration over s to the integration over z we need to use the following coefficient:

(±/2

as a=--=

1+

dz

=

(

/,/ dz : V

1 + duz

dz ]

+

I dz ]

acting on the vortex and to derive its equation of motion. The length of the line element is d s = [(d x) 2 + ( d y ) 2 + (dz)2] 1/2, i.e., the length of element of the helical vortex will be ds=

ds=

(6)

V'

so that f = a ( f v + f j ) . On substituting the latter formula for f as well as Eqs. (2), (4) and (5) into Eq. (3), projecting Eq. (3) on the z-axis and performing some calculations we obtain the extremely simple result: dUz/dz = 0, or uz(t) = const. This means that every point of the helix moves only in the x - y plane and its z component does not change. Designating from now on the coordinates of the helix points as x(z, t), y(z, t) and z where z does not depend on time, one can rewrite Eqs. (4) as

Z x( z, t) = r(t) cos "L(t----y'

(.): 1 +-~

(8)

dz

(9)

Then, the viscous force can be written as

r2 / ]/2

Fv= - f ~ . d,= -fn. = f~:v

1 + ~-~]

dz (10)

dz

with projections on the x- and y-axes:

( r2/''dz

Fv.x=f,: I+-ZI

z 4

r

= - "r/[kL sin - L

x

1+

z z

Z)l

sin -- + -- cos -L L L

,

(11)

r2 ) I/2

L(t) " ,y = - s r / ~ [

=-r/

x of homogeneous

dz.

(7)

Later on we shall prove that L does not depend on time. Note that we have changed our notations, namely the z coordinate of a point on the helix z + u z is replaced now by z.

3. A p p r o x i m a t i o n

~Z

1/2

Z y( Z, t) = r ( t ) sin

+

which in the case of our helix can be rewritten, according to Eqs. (7) as

+~dz]

1+

1-1- -~Z

current

At first, we assume that the current is homogeneously distributed across the wire. This approximation is valid when helixes constantly enter a wire and form the resistive state. In this simplest case the problem can be solved analytically. Let us assume that a left-hand helical vortex already exists inside a wire. Such a vortex is obtained as a result of an instability of a linear flux line lying on a cylinder axis [3]. Our aim now is to calculate all the forces

[

1 + ~-T

-~Lcos

dz

z z --z r~( L + cos - - + - - sin L L

(1+

(12)

where Eqs. (7) have been used. The Lorentz force acting on the left-hand helical vortex has a tendency to expand the vortex towards the outside of the wire. This force has the following form:

Fj= f [jx 4~°] ds= f [jx rk°l 1+ c

= f~:j dz

~

r2 / 1/2 FI

dz

(13)

207

M. Shvartser et al. / Physica C 264 (1996) 204-212

with components

jqbor

Fj = f -

z ~/

r2

+ L2 cos Z _ l+

j
cL

sin

where Eq. (18) contains the single dimensionless parameter

z

dz

k(14)

7'

j&o L

Eq. (18) can be easily solved with respect to t', and one finds that

j~bor Z V ~1 r 2 Fj.y = f c~r2 7 L 2 s i n - ~ _ _ + - ~ d z jqbor --cos cL

--

i 1

t' --

z L

--.

(15)

2(k-

j~bo r

r2 + L 2

c~r2 + L 2

+ rli - "qrL 1 - - c o t L

= 0.

In

]

jqbor

r2 + L2 + c~r2 + L2

(17)

Hence, we have shown that the only possible motion of a helical vortex which preserves its form is that with constant pitch L. This result can be understood in the following way: both directions along the z-axis are equivalent for the vortex with non-fixed boundary conditions, i.e., all the points of the vortex are moving only radially which obviously maintains a constant pitch. One may easily see from Eq. (17) that the positive helix velocity as a function of L has a maximum at some critical value Lcr while the negative velocity reaches its maximum at L = 0. It is convenient to transform Eq. (17) to a dimensionless form by the substitution x = r/L; t'=

(~/nL2)': x x X2 ~_-""'~-~- k ~x2 b----~ ,

(18)

-1

r02 1+-~ -1

i

1 2 ( k + 1)

In

/.2

+-~-f + 1

E

1

(16)

The y component of the equation of motion (3) differs from Eq. (16) only by replacing cot(z/L) in the last term by tan(z/L). One immediately sees that Eq. (16) is satisfied only if L = 0 and

-~=

1)

V'

tr r

"r/tt =

r2 1+-~

V

The transport current j in Eqs. (13-15) was assumed to be uniform due to the narrowness of the wire. Integrating both sides of Eq. (3) over z and substituting Eqs. (9-15), one obtains for the x component of this equation:

o'r

(19)

co

k2(k 2 - 1)

k In

r2 +-~-y + 1

V/1 + - ~r22 /

-1

%2

kVl+-~--f - 1

r0)

+-/

,

where r o = r(0) is the initial radius. These solutions are shown in Fig. 1. For k > 1, only a monotonical increase of radius occurs, independently of r 0. However, if k < 1 there are two possible regimes. For

ro>L

- 1

the radius increases monotonically (spreading regime), while in the opposite case

ro
- 1,

the radius decreases monotonically (vanishing regime). Alternatively, for each r 0 one has some critical value kcr such, that for k > kcr the spreading regime occurs, while for k < kcr the radius decreases.

M. Shvartser et a l . / Physica C 264 (1996) 204-212

208 200

15.0

X

10.0 6

7

8

9

5.0

I

oo 0.0

10.0

20.0 t'

30.0

40.0

Fig. 1. The dimensionless radius x as a function of the dimensionless time t' for different values of k. x o = v/J-, ker = 0.5. Vanishing regime: 1. k = 0.4; 2. k = 0.45; 3. k = 0.49; 4. k = 0.495; 5. k = 0.499. Spreading regime: 6. k = 0.6; 7. k = 0.55; 8. k = 0.51; 9. k = 0.505; 10.k = 0.501.

For k < kcr the radius decreases monotonically. There are three stages in this regime: a slow decrease, a fast decrease and again a slow decrease. The last slow decrease takes an infinite time, which means that our solution does not vanish in a finite time. The time of the first slow stage is defined as 2 that at which the radius reaches the value (-5)r0. The time of the fast stage is defined as the time from when the radius equals ( } ) r 0 until it equals (½)r 0. The dependence of the durations of the first and the second stages on k is shown in Fig. 2. From the experimental point of view it is possible to distinguish the spreading and vanishing regimes by the dependence of the emf V on time. This emf is connected with the power W of the current source by W = IV, where I is the full current. On the other hand, the power of the Lorentz forces can be written as

The spreading regime has two stages, a slow one and a fast one. The time of the slow stage is defined ,t as that at which the radius reaches the value (-5)r0. The time of the slow stage depends on k for every r 0. This dependence is represented on Fig. 2 for x o = r o l l = v~-. For k < 0 only the vanishing regime is available since both the line tension and the Lorentz force are squeezing the helix.

w=ftJ×*o>ds=fJ

oi

r

C

×

lf~r2 + L 2 lj( r ) 49o i.( d z - - ,)r(t), L cL

(21)

where l is the length of the considered element of a wire. Then,

lj( r) 490 i'( t) r(t) = lj( x) 490or x(t') x(t'), cLl cLhl

V= - -

(22) 20.0

or, in the case of a uniform current,

149°tr~( t,) x( t, ) V = cLrlS

15.0

t'

10.0

5.0

°°0.0

i

02

0.4

k

&

018

10

Fig. 2. The duration of the different stages (in units of the dimensionless time t') of the spreading and vanishing regimes as a function of the dimensionless parameter k for x o = v/'3-, kcr = 0.5; (a) the first stage of the vanishing regime; (b) the second stage of the vanishing regime; (c) the first stage of the spreading regime.

(23)

where S is the cross-section of a wire. The function V(t') in the two regimes are shown in Figs. 3(a) and (b). It is possible to distinguish between these two regimes in experiment by analyzing the dependence of the voltage on time. In the vanishing regime the voltage goes to zero as time progresses and in the spreading regime it goes to infinity. If the helical vortex is created at a wire surface by the external current and a longitudinal magnetic field H a, its dynamics is described by Eq. 0 8 ) with k < 0. In this case the voltage is negative and monotonically approaches zero as a function on time. Its

M. Shvartser et al. / Physica C 264 (1996) 204-212 2.0

4. Helicalvortexdrivenbyinhomogeneouscurrent

ia)

1.5

Let us now turn to the situation where one has to take into account that the current density JT is not uniform with radius. Following Ref. [3] we suppose that

10

x,V

0.5

v

0.0

\

I

lo(r/a)

Jr = ~ 27rR------~I,( R / A ) '

-0.5

-I 0

209

2d.0

,~0

0.0

3go

f

,0.o

I0,0

(26)

where I is the current and lo(r) and I](R) are the modified Bessel functions. In this case we also have to take into account the force with which the screening current Ja acts on the vortex [3]. We have

8.0

^ cna I i ( r / h ) Ja = - q~ 4-rrA 1 o ( R / A ) '

(27)

60

x~

4.0

where H a is the longitudinal magnetic field. Then the equation of motion has the form

2.0

rli"

x.V

.

.

.

.

.

.

0.0 O0

.

.

.

.

.

10.0

J.<" 20.0 t'

3o.o

]

40.0

Fig. 3. (a) The dimensionless voltage V (in units of (lcho~r)/(cLTlS)) and the radius x as functions of the dimensionless time t' in the vanishing regime for xo = v~, k = 0.49. (b) The dimensionless voltage V (in units of (ldPo(r)/(cL~S)) and the radius x as functions of the dimensionless time t' in the spreading regime for x o = v/3-, k = 0.51.

O-T r2 + L2

jT4'O r cCr 2 + L 2

c

j~4'O L ~

' (28)

or in dimensionless form:

x

~=

x

x 2+1

k l ~

k2

Io(xq) lo(xoq )

11(xq)

+1

(29)

ll( xoq) '

where absolute value is maximal when the helix arises at the wire boundary. Substituting Eq. (18) in Eq. (23) we obtain for maximal voltage Vmax = ' ~ ( J / J a ) a,

T

c,rrr3.0

1 + H~l ],

for j <
(25) a = 2,

14,o °. 2Ha T = crr r3rl Hcl m

Xo

q=A I

(24)

where Ja = cHa/2rrro is the surface current induced by the longitudinal magnetic field, and

ol = 3,

L

B

for j >> A .

It seems that the voltage magnitude Vmax is a most convenient function to be measured.

kl

R

JT(R)

L

ja(R) '

4'0 I o ( R / A )

~--" m

2 r r x o A co" I I ( R / A )

and cH a

4'o R 1 1 ( R / A )

k 2 = 47rAx----~ co" I o ( R / A ) "

(30)

All the terms in the right sides o f Eqs. (28) and (29) are negative because that is the situation when the vortex enters from the surface of the wire. The expression for the e m f in this case will be obtained from Eq. (22) by replacing j with Jv.

M. Shuartser et a l . / Physica C 264 (1996) 204-212

210 1.o

.

,

.

,

.

,

.

,

Eq. (31) is a small perturbation. We shall write Eq. (31) in the dimensionless form (similar to Eq. (18))

0.8

X

X

x2 +'~ + ~

=

sin( to' t')],

[ k + k,

0,6

(32)

X

where k 1 = (jldpoL)/(ctr) and to' = to('oL2)/o" and look for a solution of this equation in the form

0.4

x(r) = X ( / ' ) + ~:(t'),

0.2

~ ( 0 ) = 0, 0.0 0.0

1.0

2.0

3.0

4.0

(33)

5.0

t' Fig. 4. The dimensionless radius x as a function o f the d i m e n s i o n less time t' for a n o n - u n i f o r m current and different radii o f the wire for kj = k 2 = 0.4 a n d x o = 1. (a) R = ( I / 3 0 ) A (the c u r v e is completely similar to that for a u n i f o r m current); (b) R = 3A; (c) R = 10A; (d) R = 30A.

2.0

(a)

1.0 x

The results of a numeric solution are represented in Fig. 4.

x,V

0.5 0.0

5. Helical vortex driven by AC bias current -1.0

"Of =

o'r L--~ +

r2+

I

,

0.0

Let us consider now the case when a homogeneous current is composed of AC and DC components. The equation of motion in this case is

10.0

20.0

30.0

40.0

t' 15.0

,oo

jqbor

(b)l

c~r2 + L 2

jlqbor -~ c r21~------+ L2 sin(tot).

(31)

This equation is rather complicated and has to be solved numerically. The results of numerical solution for both regimes when Jl <> tr/("OL2) are represented in Figs. 5(a) and (b). This experimental set up may be used to distinguish between the two regimes, due to the strong difference between voltage oscillations at large frequencies to. In the vanishing regime the voltage oscillations decrease with time and then disappear when r--* 0. In the spreading regime the oscillations grow with time, approaching some constant value. Let us now use a perturbative procedure to solve Eq. (31). If j~ <
X, V

5.0 X

~

~

0.0

i

"5"00.0 Fig.

5.

(a)

10.0 The

dimensionless

i

20.0 t' voltage

i

30.0 V

(in

40.0 units

of

(ldPo~r)/(cLrlS)) and the radius x as functions of the dimensionless time t' in the vanishing regime for x 0 = v ~ , k = 0.49, k= = 0.05 when the current is composed of A C and DC components. (b) The dimensionless voltage V (in units o f (IdpoO)/(cLr/S)) and the radius x as functions of the dimensionless time t' in the spreading regime for x o =~/'3, k = 0.51, k I = 0.05 when the current is composed of AC and DC components.

M. Shvartser et a l. / P hy sic a C 264 (1996) 204-212

where X is the solution to the unperturbed Eq. (18). By substituting Eq. (33) into Eq. (32) and taking into account the terms of the first order in Taylor's expansion we derive the following equation for ~::

~=~

( X 2 + 1)2

( X 2 + 1) 3/2

X +

k I sin(to'f).

¢X2+ 1

(34)

Here we assumed that k t << k, so that the terms with ~:k t are small compared to those with ~k and can be neglected. If we assume that ~: changes with time much faster than X, the solution of Eq. (34) has the form sc(t ') - bto'e~" +

b[a s i n ( t o ' f )

+ to' cos( t o ' f ) ]

=

a 2 + of 2

(35) where X2-1

k

a~___

(X2+l) X b = ~--~ - +- 1

2

(X2+l)

3/2'

k I.

(36)

60,0 50.0 40.0 30.0 X

a

20.0

s~ ~

10.0 0.0 -I0.0

2000 0

i

20.0

t

40.0

i

i

60.o

80.0

loo.o

t' Fig. 6. The dimensionless radius x as a function o f the d i m e n s i o n less time t' for (a) D C current ( x o = 2, k = 0.5) a n d (b) D C a n d A C current ( x o = 2, k = 0.5, k I = - 0 . 1 1 8 5 2 , to' = 0.298).

211

It can be seen that there are some "resonance" values of to' and t' at which ¢(to', t')changes more sensitively. When the values k and k I are of the same order an effect of "dynamic pinning" may be observed. As is shown in Fig. 6 the helix may spend quite a long time fluctuating around the point of unstable equilibrium before starting to spread or vanish. Such behavior will cause a jump in the voltage.

6. Conclusion

Let us summarize the most important features of the helix dynamics in the wire. (1) The only possible type of vortex which retains its helical form is that with constant pitch. Only the radius of a helical vortex is changed during its motion. (2) For some values of the currents, a long time delay occurs, i.e., the vortex spends a long time inside the wire. (3) The voltage induced by the motion of a vortex depends on its pitch, i.e. on the initial conditions when the vortex was formed. (4) When an AC current is added to the DC current some peculiarities appear in the vortex motion, namely the voltage show both maxims at some "resonance" frequencies of the AC current and jumps under some conditions. (5) The radius of a helical vortex increases or decreases monotonically with time. It is possible to distinguish between the two regimes experimentally by observation of the voltage behavior in the wire or by observing the voltage noise behavior in the presence of a small AC current at large frequencies. (6) High-temperature superconductors are well suited for detecting the voltage Vmax which accompany the helix entrance to the wire. Indeed, in these materials the viscosity is -q ~ 1 0 - 6 g s / c m , which is two orders of magnitude smaller than the viscosity of conventional superconductors. In particular, for the superconducting wire with parameters ! = 1 cm, r 0 = 1 0 . 3 cm, o - = 2 × 1 0 -5 G 2 cm 2 for j ~ j a = 105 A / c m z, we obtain Wmax = 1 0 - 6 V.

212

M. Shvartser et aL / Physica C 264 (1996) 204-212

Acknowledgements The authors are grateful to V. V i n o k u r for fruitful discussions and critical c o m m e n t s . W e thank the M i n e r v a Center, the A c a d e m y of Science and Arts of Israel, G I F and the I s r a e l - F r a n c e F o u n d a t i o n for financial support.

References [1] A.A. Abrikosov, Fundamentals of the Theory of Metals (North-Holland, Amsterdam, 1988).

[2] A.M. Campbell and J.E. Evets, Critical Currents in Superconductors (Taylor & Francis, London, 1972). [3] J.R. Clem, Phys. Rev. Lett. 38 (1977) 1425. [4] E.H. Brandt, Phys. Let. A 79 (1980) 207; idem, Phys. Rev B 25 (1982) 5756. [5] R.P. Feynman, Progress in Low Temperature Physics, vol. 1 (North-Holland, Amsterdam, 1964). [6] A. Dattuer, Ark. Fys. 21 (1962) 71. [7] Yu.A. Genenko, Phys. Rev. B 51 (1995) 3686. [8] I. Aranson, L. Kramer and A. Weber, J. Low. Temp Phys. 89 (1992) 859. [9] !. Aranson, M. Gitterman and B.Ya. Shapiro, J. Low Temp. Phys. 97 (1994) 215. [10] I. Aranson, M. Gitterman and B.Ya Shapiro, Phys. Rev. B 51 (1995) 3092. [1 l] W.E. Timms and D.G. Walmsley, J. Phys. F 5 (1975) 287.