Measurement of characteristic currents in stabilized superconductors

Measurement of characteristic currents in stabilized superconductors V.A. Altov, A.A. Akhmetov and V.V. Sytchev All-Union Research Institute of Metrological Service, Moscow 117334, USSR * Moscow Power Engineering Institute, Moscow 105835, USSR

Received 14 September 1987 The superconductivity recovery current,/r, in a short composite superconductor with a hot spot is calculated. The relation between/r and the minimum normal zone propagation current, /p, in a long composite is described.

Keywords: superconductors; characteristic currents; mathematical models

The minimum propagation current, lp, of the normal zone in a superconducting composite is the most important feature of the conductor in superconductive devices. Measurements of lp by well known methods are usually carried out on samples which may be many times longer than the thermal length of the composite 1. However, in a number of cases, the use of long samples in small testing coils is difficult due to conductor rigidity. In the present Paper, we investigate the possibility of determining the minimum current of the normal zone propagation by I - V characteristic measurement of a short composite superconductor. The current distribution between the superconductor and the stabilizing normal metal is taken into account in the calculations. It is known that in a narrow, short superconductor bounded normal zone (hot spot) can exist supported by the Joule heating 2'3. Current reduction in a conductor with a hot spot results in a jump-like recovery of superconductivity at l = I~. The value I~ and the I - V characteristic of the sample with the hot spot are calculated in Reference 2 for the case where the superconductor resistivity changes from zero to some constant value at the critical temperature, To. However, a similar approach cannot be used for the calculation of short composite I - V characteristics, as an increase in the composite effective resistivity occurs over a certain temperature interval 1. This leads to a noticeable change in the results. Let us assume that the temperature changes only along the conductor (x axis). Thus, the dimensionless co-ordinate X = x/l,, (where: 1z = xA/hP; x is the specific thermal conductivity of a normal metal; A is its transverse section; P is the cooled perimeter of the conductor; and h is the heat transfer coefficient into a cooler with the temperature To) can be used to give the heat flow equation in the following form 1 (~20 Ox2

- -

i= I/Ic and

pl 2 APh(T~ - To)

where: ct is the Stekly stability parameter; Ic is the critical current of the superconductor at T = To; p is the specific resistivity of a normal metal; r is the dimensionless effective resistance of the composite 1, depending on dimensionless temperature, 0, and current, i. We first consider a model with a jump-like change of parameter r; r = 0 at 0 < 1 - i and r = 1 at 0/> 1 - i. Let us assume that the normal zone exists on the 2l length and that at the ends of the 2L length sample the temperature 0 = 0. This approach, as shown in Reference 2, is valid at L>>l where L - / > I due to a sharp temperature decrease inside the 'shores'. Then, from the solution of Equation (1) and using the condition 0(/)= 1 - i, we obtain a relation between I and i i = D [ ( 1 + ~2)~ ' _ 1]

~iZr

= 0

Cryogenics 1988 Vol 28 June

(1)

(2)

where 1

D = 2ct { 1 + cth/[cth(L --/)3 } This dependence at i > it. x has two branches l =/(i): stable increasing and unstable decreasing. The current of superconductivity recovery, it, l, is easily determined from the condition of the minimum D value at l = L/2, then !

0 +

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370

where T-To OT ~ - To

D = 2~ (1 + cth L/2)

Measurement of characteristic currents in stabilized superconductors. V.A. Altov et al. At L - - * ~ , the current is ir,1 --* ip, l = (2/~ + l/~2)l/g

where ip, 1 is the m i n i m u m p r o p a g a t i o n current in the simple m o d e P . It should be noted that the superconductivity recovery current obtained in Reference 2, under the assumption r = 0 at 0 < 1 and r = 1 at 0 >> 1, with bridge length prolongation tends to the value which is also the m i n i m u m p r o p a g a t i o n current of the normal zone for the given dependence r = r(i, 0). For the model with linear dependence of critical current on temperature, the effective resistance, r, in the temperature interval 1 -- i ~< 0 ~< 1 is 1 0;

0<1

O+i

r=

1.

i

1;

i

1--i<~O<~ l

(3)

0>1

F r o m Equation (1) we have a system of equations to determine the dependence of the normal and resistive state length on current: 11 = ll(i) and 12 = I2(i) cos[(ai - I)I(12 - 11)] - ( o d - l)~th I1 sin[(7i - 1)~'(12 - 10] i-1 -- ~ i 2 -- 1

( ~ i - 1)~th I~ cos[(~i - 1){(12 - I1) ] + sin[(~i - 1)~(12 - 11)] = (1 - i)(~i 0~i2-

where O(x) > 1

1)~ c t h ( L - 12)

(4)

1

at Ixt < l~

l - i <~ O(x) <~ l

at l~ <<.]xl <~12 at Ixl > 12

O(x) < 1 - i

The solution of Equation (4) was achieved numerically and the calculated results for ~ = 4, L = 4 are shown in Figure 1, where 0(11) = 1 and O(lz) = 1 - i. It is evident from the figure that the hot spot exists for i~>i,. At i > i~ > i, on the lower branch of the curve, length 11 = 0

which is expected for a resistive state without a normal zone. The temperature profiles calculated at i = 0.65 > il for the upper and lower branches of dependences Ii, 2 = 11,2(0 are shown in Figure 2. The dependence 0 = O(x) determined for i = ir is also shown. F r o m the calculations it follows that at i = ir the length is L = ll + 12 (see Figure 1). This condition enables us to evaluate easily the limit to which the superconductivity recovery current tends with bridge length prolongation. In fact, by solving Equation (4) at th(/0 = 1 we obtain the condition z ~i 2 - 2 + i, = 0. From this we can obtain the following expression i,=

(1 2)'1 4~2+ -~=ip

(5)

which coincides with the m i n i m u m p r o p a g a t i o n current of the normal zone under the chosen dependence r = r(i, O) (see Reference 1). T o find the analytical dependence of the superconductivity recovery current, ir, on the bridge length, L, we used a formal analogy of Equation (1) with that of the single mass particle m o v e m e n t under the potential energy U(O), where variable X is time and 0 is a co-ordinate. For U(O) we have

U(O)

_If (~i2r(i'

0) dO

0)-

(6)

In Figure 3 the dependence U(O) is shown for the effective resistance r, given by Equation (3) at three i values: i = 0.6, 0.62 and 0.65 (for ~ = 4 ) . The chosen current values exceed the m i n i m u m p r o p a g a t i o n current. Thus, the calculated dependence U(O) intersects the abscissa three times 4. The ' t i m e ' of particle m o v e m e n t in the given potential from point 0 = 0 o to 0 = 0 is determined from

ff"

=

dO {2[U(0o)- u(0)]}~

(7)

Dependence z = ~(0o), constructed from Equation (7), for the four increasing values of i current is given in Figure 4. It is evident that under some value 0 o = Oo(i ), the ' t i m e ' of particle m o v e m e n t is minimum. However, ~ ~ ov at

3

2 2

I

I

I

I

0

0

0.6 ~I

/I

0.65

/"

Figure

1

Dependences/1,2 =/1,2(i) at ~ = 4 and L = 4 . 1,/2; 2, / 1

i I1

I

I II

/2

2

/2 /i

/2

4

g

Figu re 2 Temperature profiles calculated at ~ = 4 and L = 4 for: 1, i = 0 . 6 5 , a lower branch; 2, i = 0.65, an upper branch

i=i,; 3,

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Measurement of characteristic currents in stabilized superconductors." V.A. Altov et al. from the initial 'co-ordinate' 0 o for i = 0.6. That is, in the sample with bridge length L = 4, a hot spot at i = 0.6 does not exist; this can also be clearly seen in Figure 1. If i = i r in the 2L length specimen, curve r = Z(0o) touches the straight line L = c o n s t a n t at one point (00.b = 0O,t = 00,,); that is, the condition

!

I3

0.2

0z(0 o, i) -

0.1

-

--0

00o

0

-0.1

Figure 3

Dependences U = U ( 0 )

= 0.62; 3, i = 0.64. - - - ,

at ¢ = 4 for: 1, i = 0 . 6 0 ; Solution o f T(00) = 4

2, i

and 0o = 0a, where 0a corresponds to the point of the U(O) dependence intersecting the abscissa [U(0~) = 0]. It should be noted that for i = 0.65, 0~ < 1 (see Figures 3 and 4); that is, at 0~ ~< 0 o ~< 1 the particle starts to move under the potential value found from the resistive state. In a short conductor of length 2L (with the chosen boundary conditions 0(_+ L ) = 0), the hot spot existence is determined by the identity L = ~. F o r comparison, the straight lines L = 4 and L = 6 are shown in Figure 4. It is evident from the figure that the straight line L = 6 intersects the curves ~ = ~(0o) constructed using given values of current i at points 0o, b < 0o,t. Consequently, in this case, the hot spot can exist and 0O,b and 0o.t are the temperatures at X = 0 determined for the lower and upper branches of the dependences 11, 2 = l 1,2(i), calculated using Equation (4) for L = 6. However, the straight line L = 4 is not intersected by the 'time movement' dependence

is fulfilled (see Figure 4). This condition can be used to determine the analytical dependence i~ = i,(L). In fact, in Figure 3 a dotted line connects the points corresponding to the solution of equation r(0o)= 4. It is seen here that at i = ir the potential is U(Oo,,) << 1. If the bridge length is enlarged, a further decrease in the value U(Oo,, i,) occurs; simultaneously 0o,r (the temperature in the hot spot centre at i = i,) tends to the value 0ti2. By putting Or/O0o = 0 at i = constant = ir, U ( 0 o , i~) << 1 and 0~i2 - - 0o,r << 1, from Equation (7) it is easy to obtain

Oo,~.~ ~i2 -- (°ti~z ) ½

(8)

0 o ~ cti 2

I0

I

I

I

2

3

where z = cti2 - 2 + i,. By substituting the expression obtained above into Equation (7) and neglecting the small components, we obtain the ratio between L and i, 2 1 L ~ (~ir_ 1)½ arcsin ~

8(1 + In

i~)(cti2 - I) + ~i2z ~i2rz (9)

Thus, it can be seen that z-~ 0 when L ~ oo. The L = L(ir) curve calculated using Equation (9) for 0t = 4 is shown in Figure 5. The points obtained numerically using Equation (4) are also shown. A good agreement of the calculated results is evident. It should be noted that the accuracy in the determination of the superconductivity recovery current using only an approximate equation [Equation (9)] somewhat decreases, when under conductor intensive cooling, the values ir ~ 1 and ip--* 1.

4 I0

L=6 ,S

5-

L=4

I

I I

I I I

I I I

_t

I I 1

I [

I i

1.0

eo,b %,z

%,t 1.5

Figure 4 Particle movement "time" from the point % for = = 4 and four increasing current values: 1, i = 0.60; 2, i = i r at L = 4; 3, i = 0.62; 4, i = 0.65

372

Cryogenics 1988 Vol 28 June

0

%

/p

I

I

0.60

O.61 ir

Figure 5 =4

Dependence L = L(ir) calculated using Equation (9) at

Measurement of characteristic currents in stabilized superconductors. V.A. Altov et al. 0.9

model. At the same time, the corrections for ip and ipl , induced by the sample finite length, are sufficiently small. In this case, Equation (9) or in a more general case Equation (4), can be used together with Equation (5) to determine the minimum propagation current of the normal zone according to the superconductivity recovery current in a short sample.

1 2

0.8

Conclusions

0.7

0.6

",,

m

\.

0.5-

0

\\

\\ --,,..., -%

I

I

I

2

4 (~

6

Figure 6

Dependence of the characteristic currents on the c( parameter at L = 4: 1, i2; 2, ip; 3, ir,1 ; 4, ip,1

The dependence of the characteristic currents on the stability parameter, ~, at L = 4 is illustrated in Figure 6. It is evident from the figure that the current values for normal zone propagation at the parameters chosen are noticeably different for the most simple and more realistic

The superconductivity recovery current in a limited conductor with an increase in sample length, tends to the minimum propagation current of the normal zone. The values of the characteristic currents are sufficiently close already at 2L ~ 10 to enable us to utilize the measurements made on a comparatively short sample to determine the minimum propagation current of the normal zone.

References 1 Altov, V.A., Zenkevich, V.B., Kremlev, M.G. and Sytchev, V.V. Stabilization of superconducting magnetic systems Plenum Press, New York, USA (1977) 338 2 Skocool, W.J., Bensley, M.R. and Tinkham, M. Self-heating hot spots in superconducting thin film microbridges J Appl Phys (1974) 45 4054-4066 3 Freytag, L. and Huebener, R.P. Temperature structures in superconductor as a non-equilibrium phase transition J Low Temp Phys (1985) 60 377-393 4 Gurevitch, ANI. and Mints, R.G. Localized waves in inhomogeneous media Usp Fiz Nauk (1984) 142 61-98 (in Russian)

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