Low-Temperature Formation of Thermal and Electrodynamic States in Bi 2 Sr 2 CaCu 2 O 8

Transcription

1 ISSN , Technical Physics, 008, Vol. 53, No., pp Pleiades Publishing, Ltd., 008. Original Russian Text N.A. Lavrov, V.K. Ozhogina, V.R. Romanovskii, 008, published in Zhurnal Tekhnicheskoœ Fiziki, 008, Vol. 78, No., pp SOLIDS Low-Temperature Formation of Thermal and lectrodynamic States in Bi Sr CaCu O 8 N. A. Lavrov, V. K. Ozhogina, and V. R. Romanovskii Russian Research Centre Kurchatov Institute, pl. Kurchatova, Moscow, 38 Russia Received une 5, 007 Abstract The formation of thermal and electrodynamic states in Bi Sr CaCu O 8 under the condition of current input is studied. The analysis is carried out for partial and complete current penetration under the assumption that the superconductor is cooled down to liquid helium temperature at the zero time. When the current input is continuous, the temperature dependence of the Bi Sr CaCu O 8 specific heat influences the form of the I V and I T characteristics of the superconductor. This effect is observed at high electric fields when both stable and unstable states form. As a result, the nonstationary I V characteristic of Bi Sr CaCu O 8 has the only branch the slope of which is positive and decreases with increasing temperature. Therefore, the higher the rate of current input, the more pronounced the decrease in the slope. It is concluded that one cannot find the current above which instability develops from the Bi Sr CaCu O 8 I V characteristic if the current input is continuous. PACS numbers: 7.60.Ge, 7.60.g, 85.5.Kx, 85.5.L DOI: 0.3/S As is known, the critical properties of Bi Sr CaCu O 8 (Bi) in high magnetic fields at a temperature of. K are much more pronounced than those of ordinary (low-temperature) superconductors. This offers considerable scope for producing superconducting magnets generating high-induction fields. In particular, a promising approach to designing high-field magnetic systems is development of cryocooler-cooled superconducting magnets with the central section made of Bi-superconductor [, ]. As in the case of lowtemperature superconducting magnets, the designers engaged in the above experiments must know conditions under which the current-carrying properties of Bi superconductor remain good when a very high current is introduced. It is therefore clear that analysis of mechanisms behind the current instability in Bi superconductor at helium temperatures of the coolant and development of techniques to determine stability conditions are vital in creating new current-carrying elements. In this work, we study the physics of stable and unstable thermal and electrodynamic states forming when a current is introduced into a Bi superconductor. The analysis is performed under the assumption that cooling conditions are similar to cooling by a cryocooler [3]. Let us investigate the dynamics of thermal and electrodynamic states in a Bi superconductor having the form of a plane-parallel plate with half-thickness a. The plate is placed in constant external magnetic field B and is cooled by a coolant with constant heat-transfer coefficient h. Let the current be absent in the superconductor at the initial time instant and then increase with constant rate di/dt. We assume that the I V characteristic of the superconductor is described by a power-type function, while its critical current decreases linearly with increasing temperature. In the presence of magnetic flux creep, the current was shown [, 5] to penetrate into a superconductor with a finite rate. Therefore, with regard to the moving boundary of the region into which the current penetrates, the D space time distributions of the temperature, T(x, t); electrical field, (x, t); and current density, (x, t), in the superconductor can be determined by solving the set of equations CT ( ) T = t µ t = λ ( T ) T 0, 0 < x< x p, x p < x< a, --, t > 0, 0 x p < x< a. () The I V characteristic of the superconductor and the critical current density satisfy the relationships = c - n, c ( T, B) c ( T, B) c0 ( B) T cb( B) T = ----, T cb ( B) T 0 () and the initial and boundary conditions are expressed (according to the statement of the problem) as T T( x, 0) = T 0, ( 0, t) = 0; λ T ( at, ) htat [ (, ) T 0 ] = 0, x0 (, ) = 0; 66

2 LOW-TMPRATUR FORMATION OF THRMAL AND LCTRODYNAMIC STATS 67 x ( p, t) = 0, x p > 0 ( 0, t) = 0, x p = 0; ( at, ) = (3) Here, n,, c0, and T cb are given parameters of the superconductor; C and λ are its specific heat and thermal conductivity, respectively; T 0 is the coolant temperature; b is the half-width of the plate; and x p is the moving coordinate of current penetration region. Coordinate x p is determined from the integral relationship [, 5] a x p () If the temperature and electric field are uniformly distributed over the superconductor s cross section, the analysis of thermal and electrodynamic state formation based on qs. () () can be simplified. Such states arise when di/dt 0 and ha/λ 0. In these limits, the temperature, current density, and electric field intensity in the superconducting plate are related as [6] (5) When the cross section of the superconductor is filled with the current only partially, one can make use of a self-similar approximation [, 5] to simplify the description of electrodynamic states. In the rectangular coordinate system, the electric field intensity inside the area where the current flows and penetration depth at weak creep (n 0) can then be determined as (6) To solve qs. () (), the finite difference method was used. The input parameters were B = 0 T, T 0 =. K, n = 0, h = 0 3 W/(cm K), = 0 6 V/cm, T cb = 6. K, µ 0 b ( x, t) dx = di b dt di ----t. dt a T T c0 = /n, h -- c0 a ht --- ( n )/n ( cb T 0 ) /n /n c0 = /n. -- c0 a ht --- ( n )/n ( cb T 0 ) xt (, ) 0, 0 x x p = µ di ---- ( x x b dt p ), x p x a, n x p () t a -----t n/ n = n µ di ---- n b dt ( )/( n ) ( ) / n µ n n 0 c0 ( )., V/cm '' f 0 7 (b) x p, cm ' f 0 8 c0 =.5 0 A/cm, and a = 0.05 cm. The temperatures dependences of the specific heat and thermal conductivity were calculated by the formulas [7, 8] CT ( ) = 58.5T T , T 0K; m 3 K λ( T) = T T T -- W. mk T The current was normalized to the plate width (I* = 0.5I/b) for convenience. Therefore, the reduced rate of current input (di*/dt = 0.5b di/dt) was taken as a variable parameter in the calculations with regard to the symmetry of the problem. The variations of the current penetration depth (the current gradually fills the cross section) and electric field intensity on the surface of the plate with time are shown in Fig.. The results were obtained with models () () and (6) for two values of di*/dt. It follows from Fig. that, for the input parameters listed above, the temperature of the superconductor has a minor effect on the dynamics of electrodynamic states at the stage of current partial penetration. Keeping this in mind, we will find the electric field intensity on the surface of the superconductor when its cross section is completely filled with current. According to the self-similar approximation, the fill field is given by f = µ 0 a/(b)di/dt. The corresponding values are shown in Fig.. Using them, one can easily estimate how the rate of current input influ- (a) Fig.. Time variation of the (a) current penetration depth (coordinate of the moving boundary) and (b) electric field intensity on the surface of the plate calculated by model () () (solid lines) and model (6) (dashed lines): di*/dt = () 0 and () 0 3 A/(s cm). Values f ' and f '' are the electric field intensities on the superconductor surface calculated in the self-similar approximation for given values of di*/dt.

3 68 LAVROV et al., V/cm 0 5, ', V/cm , 600 q '' f ' f 000 ' 3' I* q 00 I*, A/cm 3, 3' Fig.. lectric field intensity ( ) on the surface and (', 3') at the center of the superconductor vs. reduced current I* = I/b. () di*/dt 0; (, ') di*/dt = 0 A/(s cm), C = C(T); (3, 3') di*/dt = 0 3 A/(s cm), C = C(T); and () di*/dt = 0 A/(s cm), C = C(T 0 ) x = x = a , A/cm Fig. 3. lectric field intensity at the center of the plate (x = 0) and on its surface (x = a) vs. the current density. di*/dt = 0 (solid lines) and 0 3 A/(s cm) (dotted lines). ences the nonuniformity of the electric field distribution in the superconductor at complete current penetration. Curves, ', 3, and 3' in Fig. show the variation of the electric field intensity on the surface and at the center of the superconducting plate throughout the current input process, and curve is the I V characteristic calculated by static approximant (5). From the run of curves 3 in the range of subcritical electric field intensities ( < ), it follows that, for f (i.e., for di/dt b /µ 0 a, electrodynamic states in the subcritical range of complete filling asymptotically tend to become steady states satisfying q. (5). Let us see how the rate of current input influences the duration of the transition to steady states at current complete penetration. It follows from () that the following estimate is valid for these states, ----, (7) t a i.e., the rate of diffusion processes under the total-current conditions depends on the differential resistance of the superconductor and on respective values of the electric field intensity. It follows from expression (7) and Fig. 3, which shows the dependences () on the surface and at the center of the superconductor for different values of di*/dt, that the transient period over which the states become almost steady shrinks as the rate of current input rises. In fact, for the input parameters used, an increase in the rate of current input influences the differential resistance of the superconductor only slightly (Fig. 3), but the electric field intensity grows for evident reasons. As a result, with an increase in di/dt, the rate of electric field redistribution over the cross section of the superconductor increases, which, in turn, shortens the transient (Fig. ). It also follows from Fig. that, in the range of supercritical electric field intensities ( > ), the curves (I*) calculated in the nonstationary, relationships () (), and stationary, relationship (5), approximations ascend in a different manner, the divergence between the curves increasing with rate of current input. As a result, the nonstationary I V characteristics of Bi superconductor (curves, ', 3, and 3') have branches the slope of which is always positive and decreases with increasing di/dt. At the same time, the stationary I V characteristic of Bi superconductor (curve ) has branches with positive and negative slopes. In the range / > 0, the current stationary distribution is stable, while for / < 0, it is unstable. Hence, the boundary of stable states in the stationary approximation is found from the condition / [9]. The corresponding boundary values of the electric field intensity, q, and current, I q *, are given by /n q = --- h ā - ( T cb T 0 ) n c0 n c0 n/ ( n ), I q * an = a q = ( T n n a cb T 0 ) / ( n ), (see Fig. ). It follows from these expressions that, unlike the case of low-temperature superconductors, one cannot find the boundary value of the stable-state range from the I V characteristic of Bi superconductor observed at continuous input of current, because its differential resistance is always positive. To take in this result, let us turn to the equation for the nonstationary I V characteristic of the supercon-

4 LOW-TMPRATUR FORMATION OF THRMAL AND LCTRODYNAMIC STATS 69 ductor under the assumption that the current is uniformly distributed over the cross section. In this case, the thermal balance for the plate is expressed by CT ( ) dt h = -- ( T T (8) dt a 0 ). Neglecting the influence of the magnetic field on the critical current density for simplicity, we find from the equation for the I V characteristic that d d d c /n dt dt d = ---. c ( T) /n n (9) The derivative dt/d can be found from (8) with regard to dt dt where S = ab is the cross-sectional area of the superconductor. Then we come to dt d (0) When the current is applied continuously, the temperature of the superconductor rises permanently; that is, dt/dt > 0 and, hence, dt/d > 0. It then follows from qs. (9) and (0) that the differential resistance will decrease with increasing the specific heat of the superconductor. Consequently, an increase in the Bi temperature should be taken into account in analysis of the electrodynamic state formation in Bi. The variation of the thermal state of the Bi superconductor with the rate of current input is shown in Fig. in the form of I T characteristics corresponding to the I V curves shown in Fig.. Shown also are the dependence T(I*) calculated in stationary approximation (5) (curve ) and corresponding boundary temperature T q above which the superconducting state is unstable. To illustrate how the temperature dependence of the specific heat influences the formation of thermal and electrodynamic states, Figs. and show the calculated dependences (I*) and T(I*), which were obtained under the assumption that the specific heat of the superconductor is independent of the temperature (curve ). As follows from Fig., the temperature of the superconductor preceding instability is not equal to the coolant temperature. The primary reason is that the I V characteristic of the superconductor admits of some (allowable) overheating [9]. The rate of current input also influences the running temperature of the material: the higher is di/dt, the higher is the overheating and, hence, the more pronounced is the influence of the specific heat on the I V and I T characteristics, which will only increase (have a positive slope) when the current = -- dt di ----, S d dt ( T T 0 )h/a = ---- S. CT ( ) di/dt T, K T q I*, A/cm Fig.. Temperature of the Bi superconductor vs. the reduced current for different rates of current input and different calculated temperature dependences of its specific heat C = C(T), continuous variation of the specific heat with temperature; C = C(T 0 ), constant specific heat calculated at the coolant temperature): () di*/dt 0; () di*/dt = 0 A/(s cm), C = C(T); (3) di*/dt = 0 3 A/(s cm), C = C(T); and () di*/dt = 0 A/(s cm), C = C(T 0 ). Curve is the dependence T(I*) calculated in stationary approximation (5); T q is the boundary value of an allowed increase in the superconductor temperature calculated in stationary approximation (5). is introduced continuously. This all is demonstrated in Figs. and. Thus, one cannot find the boundary value of the stable-state range from the nonstationary I V characteristics of the superconductor. To overcome this uncertainty, the following method is used in experiments: the current of a given value is introduced into the superconductor and its thermoelectrodynamic state is determined as a long time passes [0 6]. Two final states may arise in this case: the temperature and electric field intensity either stabilize or begin to increase spontaneously. The appearance of a stable voltage with the superconductivity retained is a direct consequence of the existence of stable stationary states, which are the basis for the formation of the I V characteristic of the superconductor. In the simplest case, these states are described by model (5). Therefore, for currents higher than current I q *, which is determined from the stationary condition of stability in the limit /, forming states are unstable even despite the fact that the respective running values are lower than boundary values q and T q in the nonstationary I V and I T characteristics of the superconductor (Figs. and ). To illustrate the aforesaid, the time variation of the temperature of the superconductor is shown in Figs. 5 and 6 for different rates of current input. The calculations were carried out for continuous input of current and also for the case when the current reached some given value I 0 * and then di/dt was set equal to zero. The I* q 3

5 70 LAVROV et al. T, K 6.0 T q T, K 6.0 T q T(0, t) T(a, t) Fig. 5. Time variation of the Bi surface temperature according to the current input conditions and specific heat of the superconductor: (, 5) T(a, t) calculated at continuous input of current for di*/dt = 0 A/(s cm) and C = C(T) and C = C(T 0 ), respectively. ( ) T(a, t) calculated at C = C(T) and fixed current I 0 = () 55, (3) 5, and () 53 A/cm and (6 8) T(a, t) calculated at C = C(T 0 ) and current I 0 = (6) 55, (7) 5, and (8) 53 A/cm. T q is the same as in Fig Fig. 6. Time variation of the temperature at the center (solid lines, T(0, t)) and on the surface of the superconducting plate (dashed lines, T(a, t)) calculated at continuous input of current (curves ) and at fixed values of the current that are close to the instability-initiating current (curves, 3): () di*/dt = 0 3 A/(s cm), () I 0 * = 5, and (3) I 0 * = 53 A/cm. corresponding boundary value of T q is also shown here as in Fig.. Figure 5 also shows the results of calculation carried out for a constant specific heat of the superconductor, and Fig. 6 substantiates the uniform temperature distribution over the cross-sectional area of the superconductor in the course of current diffusion. Obviously, the temperature distribution is almost uniform when ha/λ the condition that is satisfied for the initial parameters used. At the same time, it follows from Fig. 6 that whether or not the temperature distribution is uniform depends on the current input conditions. However, the discussion of how the time variation of the temperature influences current instability is beyond the scope of this paper. Figures 5 and 6 clearly demonstrate that static value T q delineates the boundary between the stable and unstable states regardless of the rate of current input and temperature dependence of the specific heat. Therefore, if the temperature during the transient observed after the current reaches a preset value does not exceed T q, the state is stable. Otherwise, the introduced current is unstable and cause spontaneous heating of the superconductor. Thus, an increase in the temperature of Bi Sr CaCu O 8 in both the stable and unstable states changes the form of its I V characteristic. When the current is introduced continuously, the characteristic has the only branch with a positive slope decreasing as the temperature rises. This effect following from the temperature dependence of the specific heat is the most pronounced in high electric fields, when the temperature of the superconductor may appreciably and steadily grow (for example, if the rate of current input is relatively high). As a result, when the current is applied continuously, the running values of the currentinduced electric field intensity and temperature are lower than those calculated in the framework of stationary model (5). On the whole, our study indicates the need to take into account an allowed increase in the temperature in analysis of electrodynamic state formation in high-temperature superconductors, especially in the range of high operating temperatures. ACKNOWLDGMNTS This work was supported by the Russian Foundation for Basic Research, project no a. RFRNCS. M. S. Newson, D. T. Ryan, M. N. Wilson, and H. ones, I Trans. Appl. Supercond., 75 (00).. K. Watanabe, S. Awaji, and M. Motokawa, Physica B , 87 (003). 3. R. H. Bellis and Y. Iwasa, Cryogenics 3, 9 (99).. V. R. Romanovskii, Zh. Tekh. Fiz. 70 (5), 7 (000) [Tech. Phys. 5, 557 (000)]. 5. V. R. Romanovskii, Cryogenics, 9 (00). 6. V. R. Romanovskii, Zh. Tekh. Fiz. 73 (), 55 (003) [Tech. Phys. 8, 5 (003)]. 7. A. unod, K. O. Wang, T. Tsukamoto, et al., Physica C 9, 09 (99). 8. P. F. Herrmann, C. Albrecht,. Bock, et al., I Trans. Appl. Supercond. 3, 876 (993).

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