Thermal Mechanisms of Stable Macroscopic Penetration of Applied Currents in High Temperature Superconductors and their Instability Conditions

Transcription

1 1 The Open Applied Physis Journal, 212, 5, 1-33 Open Aess Thermal Mehanisms of Stable Marosopi Penetration of Applied Currents in High Temperature Superondutors and their Instability Conditions V R Romanovskii * National Researh Center Kurhatov Institute, NBIK- Center, Mosow, Russia Abstrat: Marosopi mehanisms of applied urrent penetration in high-t superondutors are disussed to understand the basi physial trends, whih are harateristi for the stable and unstable formation of the thermo-eletrodynamis states of high-t superondutors plaed in DC external magneti field The performed analysis shows that the definition of the urrent stability onditions of high-t superondutors must take into onsideration the development of the interonneted thermal and eletrodynamis states when the temperature of superondutor may stably rise before instability As a result, the thermal degradation effet of the urrent-arrying apaity of superondutor exists The boundary values of the eletri field and the urrent above whih the applied urrent is unstable are defined taking into aount the size effet, ooling onditions, non-linear temperature dependenes of the ritial urrent density of superondutor It is proved that the allowable stable values of eletri field and urrent an be both below and above those determined by a priori hosen ritial values of the eletri field and urrent of the superondutor The violation features of the stable urrent distribution in high-t superondutors ooled by liquid oolant are studied The neessary riteria allowing one to determine the influene of the properties of superondutor and oolant on the urrent instability onset are written Keywords: High-T superondutors, I-V harateristis, marosopi behavior, ritial urrent, urrent instability 1 INTRODUCTION The eletrodynamis of superondutors is one of the most fundamental issues of superondutivity Suh investigations made in the marosopi approximation are of importane for both understanding the basi mehanisms underlying the stable superonduting state formation and many appliations of superondutivity, in partiular, when the pratial use of superondutors requires the analysis of stability onditions of superonduting state As known, they depend on the marosopi phenomena ourring in the superondutors and may lead to the so-alled magneti or urrent instabilities [1, 2] The numerous previous investigations of the magneti instabilities in high-t superondutors have been made [3-12] In the meantime, basi features of the stable harging of applied urrent into high-t superondutors, espeially, influene of thermal mehanisms on the eletrodynamis state formation have still not disussed This is partly explained by the fat that the limiting urrent-arrying apaity of low-t superondutors, whih have steep voltage-urrent harateristis, is desribed by their ritial urrents [1, 2] This onept is also widely used to determine the urrentarrying apaity of high-t superondutors as their basi property However, the finite voltage indued in high-t superondutors by applied urrent appears long before the urrent instability onset In this ase, the marosopi eletrodynamis of superondutor is a onsequene of the non- *Address orrespondene to this author at the National Researh Center Kurhatov Institute, NBIK- enter, Mosow, Russia; Fax: 7 ( ; Tel: 7 ( ; -mail: vromanovskii@netsapenet loal dynamis of vortex lattie that is gradually oming to the motion [13] The onept of the resistive states signifiantly expands the range of permissible stable operating mode of high-t superonduting materials [14-23] However, they are haraterized by non-trivial temperature hange of superondutor, whih depends on the eletrodynamis proesses in superondutor For the first time, this feature, whih takes plae even in low-t superondutors, was formulated in [24, 25] The heating issues of high-t superondutors are also important for the -axis intrinsi Josephson juntion geometries There is a big disussion on the pseudogap in high-t superondutors inferred from the V-I urves during the intrinsi Josephson juntion tunneling spetrosopy [26, 27] The intrinsi tunneling spetrosopy of bismuth uprate superondutors has reeived onsiderable attention This tehnique utilizes intrinsi Josephson juntions, whih are unique in probing bulk eletroni properties, among other important methods suh as sanning tunneling spetrosopy and the break-juntion tehnique However, intrinsi tunneling spetra an be seriously affeted by self-heating due to poor thermal ondutivity of the uprate materials and relatively large urrent densities required to reah the gap voltage Therefore, experimentally, intrinsi tunneling spetra are usually different from those obtained in sanning tunneling spetrosopy and break-juntion experiments As a result, it is important to eliminate the heating in order to obtain the genuine intrinsi tunneling spetra Another important stability problem onerns thermally ativated phase slips and quantum phase slips in a long hannel or thin superonduting wire and films They are treated as so-alled one dimensional (1D system that has / Bentham Open

2 Stable Marosopi Penetration of Applied Currents in High Temperature Superondutors The Open Applied Physis Journal, 212, Volume 5 11 width smaller than the oherene length During a phase slip, the superonduting order parameter flutuates to zero at some point along the superondutor resulting in a voltage pulse The sum of these pulses gives the resistive voltage By applying a urrent, the 1D superondutor an transform from the superonduting state to normal One of the most striking features of this transition is the multiple steps in the voltage-urrent urve indued by phase slip These resistive states are nonstationary and nonequilibrium (see, for example, [28, 29] and referenes therein The stability of their formation an be investigated using numerial integration of the time-dependent Ginzburg - Landau equations However, the disussion of the instability of these metastable mirosopi states is beyond the sope of given paper beause our main purpose is the marosopi phenomena in high-t superondutors when magneti penetration depth is not mirosopi and dynamis of whih an be desribed by models based on the Maxwell equations Thus, in this paper, the role of the heat transfer mehanisms in the formation of the marosopi eletrodynamis states of bulk high-t superondutors during urrent harging is disussed in detail The models proposed permit: - to investigate the existene of the non-isothermal (J harateristis of a high-t superondutor that may take plae during experiments; - to study the thermal peuliarities of the eletrodynamis state formation of high-t superondutor aounting for non-uniform temperature distribution in its ross setion and to get the appropriate urrent instability onditions defining the maximum stable value of applied urrent; - to onsider thermal features of violation of the stable urrent mode of high-t superondutors under different ooling regimes, and formulate the riteria defining the possible mehanisms of urrent instability in high-t superondutors Aordingly, the results obtained allow to formulate the marosopi features of the thermo-eletrodynamis proesses in high-t superondutors, whih are important for the analysis of applied urrent experiments 2 GOVRNING THRMO-LCTRODYNAMICS MODLS To investigate the basi features of the thermo-eletrodynamis phenomena in superondutors, let us onsider simple approximation believing that a superondutor with a slab geometry (-a<x<a, - <y<, -b<z<b, b>>a plaed in a onstant external magneti field parallel to its surfae in the Z-diretion that is penetrated over its ross setion (S=4ab Suppose that the applied urrent is harged in the Y-diretion inreasing linearly from zero at the onstant sweep rate di/dt and its self magneti field is negligibly lower than the external magneti field [2] Let us desribe a onstitutive relation (J of superondutor by a power law and approximate the dependene of the ritial urrent on the temperature by the linear relationship As known, power law orresponds to a logarithmi urrent dependene of the potential barrier This dependene is observed in many experiments that deal with both low- and high-temperature superondutors [3-33] Assume also that the superondutor has the transverse size in the X-diretion, whih does not lead to the magneti instability As it was shown in [34, 35], the flux penetration phenomena during reep are haraterized by a finite veloity Therefore, taking into onsideration the existene of the moving boundary of the urrent penetration region, the transient equations desribing the evolution of the temperature and eletri field inside the superonduting slab is independent of z and y oordinates and may be written as follows # T # % # T " $, < x < xp C( T = (!( T # t # x * # x $, J, xp < x < a 2 µ! J,, =! t > " x 2 p < x < a! t! x Here, the eletri field (x,t, urrent density J(x,t and ritial urrent density J (T,B onform the following relationships [ / (, ] n = J J T B (3 J ( T, B = J ( B T ( B! T / T ( B! T (4 B B For the problem under onsideration, the initial and boundary thermo-eletrodynamis onditions are given by T ( x, = T, ( x, = (5 " T " T (, t =,! ( a, t h[ T ( a, t # T ] = " x " x ( x, t =, x > p! (, t =, xp =! x! µ di ( a, t =! x 4b dt p Here, C and λ are the speifi heat apaity and thermal ondutivity of the superondutor, respetively; h is the heat transfer oeffiient; T is the ooling bath temperature; n is the power law exponent of the -J relation; is the voltage riterion defining the ritial urrent density of the superondutor; J and T B are the known ritial parameters of superondutor at the given external magneti field B; x p is the moving boundary of the urrent penetration area following from the integral relation a di 4 b! J ( x, t dx = t (8 dt xp defining the onservation law of harged urrent [34, 35] Note that approah (4 is reasonable for operating states when the temperature variation of a superondutor is not notieable However, the huge flux reep of high-t superondutors leads to the strong nonlinear temperature dependene of their ritial urrents in the high magneti field Therefore, the relationship (4 is not universal In these ases, the various approximations are used [36-42] to desribe the temperature-dependent ritial urrent of high- (1 (2 (6 (7

3 12 The Open Applied Physis Journal, 212, Volume 5 V R Romanovskii Fig (1 Critial urrent of Ag/Bi2212 versus temperature (a, magneti indution (b and temperature dependene of di /dt (: ( - after [4], ( - alulations aording to (9, ( linear approximation T superondutors In partiular, this quantity may be desribed by equation % T J ( T, B = J ( 1 * T!, B % " B - (1 # # exp( / B B B exp( $ T / T / * 1 proposed in [42] It is good approximation for Bi-based superondutors Fig (1 shows the possible temperature variation of the ritial urrent I =ηj S saled for the Bi 2 Sr 2 CaCu 2 O 8 (Bi2212 omposite (η is the volume fration of a superondutor in a omposite The alulations presented were made at T=42 K using the following onstants T =871 K, B =4655 T, α=133, β=676, γ=173, χ=27, B =1T, J = A/m 2 (1 whih were reported in [42] and were also adopted aording to the measured data published in [43] for the Ag/Bi2212 superondutor having S=1862 m 2 and η=2 The diffusion one-dimensional model defined by equations (1 - (8 may be simplified in the ases when the temperature distribution inside the superondutor is pratially uniform Under the onsidered slab geometry, the uniform temperature distribution exists when the ondition ha/λ << 1 takes plae (Below this ondition will be stritly proved Integrating equation (1 with respet to x from to (9 a and onsidering the boundary onditions (6, it is easy to get the following transient zero-dimensional heat equation dt h C( T =! ( T! T ( t J ( t (11 dt a This equation with the relations (3 and (4 desribes the uniform time variation of the temperature and eletri field as a funtion of the applied urrent I(t=J(tS=dI/dt t The limiting transition di/dt, applying to the model desribed by equation (11, leads to the stati zerodimensional heat balane equation In this ase, i e at ha/λ << 1, uniform stati thermal state of the superonduting slab is the solution of the following equation J = h( T! T / a (12 The models formulated allow one to investigate the influene of the thermal mehanisms on the marosopi formation of the eletrodynamis states of high-t superondutors during urrent harging and find the orresponding stability riteria 3 POTNTIAL THRMO-LCTRODYNAMICS STATS OF HIGH - T C SUPRCONDUCTORS To appreiate the temperature influene on the eletri field dynamis inside superondutor, let us redued the set of equations (1 - (3, eliminating the urrent density Omitting the intermediate algebra, it is easy to get the following equation

4 Stable Marosopi Penetration of Applied Currents in High Temperature Superondutors The Open Applied Physis Journal, 212, Volume J " # $ $ dj " # = % 2 $ $ x µ µ n ( t C T dt ( * " # $ " $ T #, J! -, $ x ( $ x - / whih is valid in the urrent penetration region To simplify the analysis, the existene only the temperature dependene of J was assumed during this transformation The term in left part and the first term in right part of this equation desribe the eletri field evolution in the isothermal approximation In this ase, the inequality 2 / x 2 > always takes plae in aordane with equation (2 when J / B is negligible Besides, as it follows from [34, 35], the isothermal sweep dynamis of the eletri field on the surfae of superondutor satisfies the estimator (a,t t n/(n1 Therefore, the next inequalities (a,t/ t > and 2 (a,t/ t 2 < will exist in the urrent penetration region when eletri field is small ( << and temperature distribution is uniform However, the temperature rise of the superondutor hanges these regularities This effet depends on the third term in right part of equation written Using equation (3, it is easy to prove that the finite value of dj /dt, whih is negative for pratiable superondutors, will lead to other relations when the temperature of superondutor indued by applied urrent beome differ from its initial value Namely, one an obtain (a,t/ t > and 2 (a,t/ t 2 > In other words, the dynamis of the eletri field will beome more intensive when temperature inreases Physially, this behavior takes plae beause the applied urrent diffusion is aompanied with the energy dissipation Indeed, the temperature rise inreases the eletri field and hanges the indued urrent density of the applied urrent, whih penetrates more deeply into the superondutor In turn, this will lead to new dissipation produing the orresponding rise of temperature and so on As a result, the sharp rise of the eletri field and temperature may our when the urrent penetration front exeeds some harater boundary The latter defines the stability margin of eletrodynamis states [1, 2] To prove the existene of the stability margin, let us rewrite the latter equation in the form 2 1 " " =! q 2 D " t " x m where D m 2/ n n $ %! ( T $ % =, ", 2 µ J ( T = a ( $ % q = *( T # ( T ( Here, D m is the magneti diffusion oeffiient, µ! ( T = C( T 2 J T a " J " T is the so-alled magneti stability parameter [1, 2], (13 2 2!( T " J 1 d! $ # T % # T # T = µ, " ( T = 2 C( T " T!( T dt ( # x # x quation (13 is diffusion type equation desribing the fission-hain-reation phenomena, in whih the quantity γ is the like-fission oeffiient and q is the volume soure In the ase under onsideration, the oeffiient γ is positive due to the negative value of dj /dt for pratially used superondutors Thus, the evolution of the thermo-eletrodynamis state of superondutor indued by applied urrent has the fission-hain-reation nature As known, suh proesses beome an irreversible when γ exeeds some harater value γ q in aordane with the fission quality obeying the timedependent law depending on [exp(γ-γ q t] - terms As a result, the eletri field evolution in superondutor has the instability nature leading to the avalanhe modes The latter happens as the spontaneous rise of the eletri field and temperature Note that the quantity D m inreases with inreasing n- value Therefore, the instability develops more intensively when (J relation of superondutor is steeper Consequently, it has sharp harater in low-t superondutors and gets more smoothed nature in high-t superondutors This feature shows experiments However, subsequent temperature rise may lead to the harater peuliarities that underlie the basis of the marosopi phenomena in superondutors The quantity γ has essential temperature dependene First of all, it varies in inverse proportion to the heat apaity of a superondutor So, the fission-hain evolution of the indued eletri field in high-t superondutors may deay after a ertain temperature inrease In other words, high-t superondutors have the thermal self-stabilization modes even after instability onset In partiular, the swept eletri field may ome lose to the steady modes without the transition of the high-t superondutor to the normal state due to its large ritial temperature There exist other reasons of thermal effet on the eletri field evolution They are due to the temperature dependene of the ritial urrent density and the value dj /dt Indeed, Fig (1 shows that not only J may be small but the dj /dtvalues beome very small in aordane with the huge fluxreep degradation in high temperature range, whih, however, is not lose to the ritial temperature Thereby, the thermal effet on the eletri field evolution depends on the quantity dj /dt after relatively high temperature rise of superondutor Besides, the effet of the temperature on the urrent diffusion depends also on the value and sign of dλ/dτ( T/ x 2 - term The latter may be both positive and negative over the ross setion of a superondutor This depends on the ooling onditions, ross setion of superondutor, its thermal ondutivity oeffiient and ritial parameters Thus, the evolution of thermo-eletrodynamis states of high-t superondutors is the phenomenon having the fission-hain-reation harater There exist the harateristi thermal effets aording to whih stable eletrodynamis state formation of high-t superondutors depends on the joint non-trivial temperature variation of the temperature dependenes of superondutor s properties, whih are C(T, λ(t, J (T and dj /dt The importane of

5 14 The Open Applied Physis Journal, 212, Volume 5 V R Romanovskii this onlusion should be emphasized To understand the stability mehanism of high-t superondutors, one usually allows for only the heat apaity of high-t superondutor believing that it just dereases the probability of the instability onset However, marosopi eletrodynamis phenomena in high-t superondutors depend on the thermal interonnetion between above-mentioned properties of high-t superondutors This feature plays essential role in the marosopi eletrodynamis of high-t superondutors, as it will be shown below 4 ZRO-DIMNSIONAL APPROXIMATION OF APPLID CURRNT INSTABILITY The boundary after whih the unstable eletrodynamis states exist may be determined, first of all, from the equilibrium analysis of the stati states Within the framework of the mathematial models under onsideration, the simplified study may be made using stati zero-dimensional model desribed by equations (3, (4 and (12, as the first step allowing to get the riteria of instability Aordingly, the following relations aj! " T = T # $ J n 1! J a " n h 1 % # $ h( TB T % J! " = n 1 # $ J a n 1 % h( TB T, (14 (15 desribe the T( and J( harateristis of a superonduting slab These formulae demonstrate the features, whih are rooted in a formation of the thermo-eletrodynamis states of superondutors Namely, the term J a h( T! T B ( n 1/ n desribes the influene of temperature on the eletrodynamis state In partiular, the limiting ase h (perfet ooling orresponds to the isothermal states (T=T In general, the nearly isothermal eletrodynamis states are defined by the ondition n/( n 1! h( TB # T " << $ % aj (16 If this ondition does not satisfy, then the thermal dissipation ours in a superondutor and its overheating influenes on the eletrodynamis states In this ase, the temperature hange of superondutor depends not only on the ross setion and heat transfer oeffiient but its ritial parameters, as follows from (16 In partiular, the higher the ritial urrent, the higher the temperature influene Formulae (14 and (15 are onvenient for the validity analysis of the different eletri field riteria used for the desription of the ritial urrent measurements Indeed, using (14 one an obtain the temperature of the sample during an experiment and estimate of the temperature influene on the stable penetration of an applied urrent However, formulae (14 and (15 do not desribe the boundary of the stable states To define it, let us analyze the differential resistivity of a superondutor, whih determines the slope of its (J harateristi Aording to (15, it is written as " aj # 1 ( n 1/ n $ % h( TB! T = ( 2 J 1 J J * a!, n - h T T (1! n/ n 2/ n ( B! 2 (17 This formula indiates that the differential resistivity of the superondutor may have both positive ( Ε/ J> and negative ( Ε/ J< branhes The positive derivative of Ε/ J orresponds to the stable states in the ase under onsideration Correspondingly, the negative value determines unstable states Aording to these peuliarities, the slope of Τ/ J has a similar harater of the hange Therefore, analyzing Ε/ J (or Τ/ J it is easy to find the stable and unstable operating thermo-eletrodynamis states under the applied onditions Aordingly, the boundary between them is defined by ondition Ε/ J (18 For the first time, this ondition was formulated in [14] for low-t superondutors ooled by a oolant with the onstant oeffiient of heat transfer This ooling mode is harateristi of the ondution-ooled or gas-ooled superondutors Under the equality (17 and ondition (18, the boundary of the instability is determined by the so-alled quenh values of the eletri field q,, urrent I q, and temperature T q, They are equal to! h " q, = $ ( TB # T % nj a n/( n 1 n n! J h " Iq, = S $ ( TB # T % n 1 n a q, q, q, /, 1/( n 1 (19 (2 T = T J a h (21 in the stati zero-dimensional approximation under onsideration These formulae demonstrate the following peuliarities of the stable eletrodynamis state formation of high-t superondutors with the power (J-harateristi First, it is seen that the stability parameters are invariant to the oupled quantities {J, } when J and orrespond to the values defined in the isothermal states Indeed, in these ases, the ratio J / 1/n is onstant for the given value n Therefore, the quenh values do not depend on the ratio J / 1/n within the framework of the linear temperature dependene of the ritial urrent density desribed by equation (4 Consequently, the quantities J and may be arbitrarily hosen on the isothermal part of the (J- harateristi of a superondutor This transformation does not hange the stability boundary defined by { q,, I q,, T q, }

6 Stable Marosopi Penetration of Applied Currents in High Temperature Superondutors The Open Applied Physis Journal, 212, Volume 5 15 Seond, many experiments show [see, for example, 21-23] that the stable over-ritial regimes exist when applied urrent may be higher than the ritial urrent determined by arbitrarily hosen value ( > This feature is the result of an a priori set of oupled isothermal values {J, } The onditions of the existene of over-ritial regimes are formulated below Besides the ondition (18, one an use another method of the determination of urrent instability onset analyzing the removal effiieny of the Joule heating into the oolant [21] Aordingly, the stable regimes must satisfy the following onditions G(T=W(T, G(T/ T= W(T/ T (22 in the stati homogeneous states Here, G is the power of the Joule heating, W is the power of the heat removal In the ase under onsideration, G=(TJ(Ta and W=h(T-T Then using stability ondition (22 it is easy to find that the quenh values are also desribed by relations (19 - (21 The results presented allow one to investigate the physial features of the formation of stable thermo-eletrodynamis states of high-t superondutors ooled by a ooler with the onstant heat transfer oeffiient, ie without the infringement of the heat exhange onditions with a oolant on a surfae of a superondutor Let us find the instability boundary of steady urrent modes of superondutors ooled by oolant when the heat transfer oeffiient depends on temperature The latter exist, for example, when liquid oolant is used As above, let us use zero-dimensional approximation Then the stati uniform distribution of the eletri field and temperature in superonduting slab for a given value of the urrent density may be defined solving the following system of equations n(t =!" J / J (T # $, (23 J = W ( T (24 Here, W(T=q(T/a and q(t is the heat flux to the oolant Assume that it is defined as follows q( T = h( T ( T " T! (25 In partiular, this formula may be used to desribe the heat transfer onditions during nuleate and film boiling regimes [1] Note also that the temperature dependene of n- value as well as arbitrary dependene J (T are taken into aount within the model formulated To find the temperature of the superondutor before the urrent instability let use the inequality! J /! T < (26 that is a onsequene of the fat that the ondition (18 divides the temperature-urrent harateristi of a superondutor into two areas xepting the eletri field from equations (23 and (24, one gets relation between the urrent density and temperature The latter may be written as 1/( n 1 n J ( T =! # W ( T J T / " $ (27 Then onsidering that the ritial urrent density of real superondutor and its n-value derease with temperature, the urrent distribution in the superondutor is unstable, if the inequality n dj dn J 1 dw ln > (28 J dt dt J W dt is realized Aordingly, the stable states are determined by the boundary values, whih follow from the solution of equation n dj 1 dn J 1 dw ln = J dt n 1 dt W W dt (29 Criteria (28 or (29 allow one to determine diretly the temperature of superondutor T q before applied urrent instability In partiular, let us find this value assuming for simpliity that the ritial urrent density is desribed by the linear relationship (4 Then the value of T q satisfies the solution of equation 1 n(t q 1 = " " dn ln a J T q dt T =Tq h T q! T n! T q! T T B! T q In the simplest approximation n=onst, it equals T q = T T B! T 1 n / " (3 The written above analytial expressions make it possible to define the instability boundary in the lose form They are onvenient to evaluate the stability onditions of many pratially important modes without the large volume of omputations taking into onsiderations different mehanisms of the instability onset 5 TH INSTABILITY CONDITIONS OF APPLID CURRNT IN ON-DIMNSIONAL APPROXIMA- TION As it was shown above, the urrent after whih unstable states exist an be determined by the analysis of the stati modes of a superondutor and an be obtained from the onditions (18, (22 or (29 However, these riteria were formulated under the assumption that the distributions of the temperature, eletri field and urrent inside superondutor are uniform As follows from the heat ondution theory [44], suh thermal modes take plae at ha/λ << 1 Under this ondition, the uniform temperature distribution in the ross setion of superondutor is the most probable in the ases of low thikness, non-intensive ooling onditions or high thermal ondutivity For example, operating regimes with uniform temperature distribution may be observed during urrent harging into a superonduting omposite (superondutor with a high ondutivity matrix At the same time, the thermal ondutivity of superonduting materials having no stabilizing matrix is low Therefore, the temperature distribution inside the superondutor is not, stritly speaking, uniform [1, 2, 45] In addition, the riterion of the spatial heterogeneity of the thermal states, ie ondition ha/λ >> 1, does not desribe the influene of the ritial pro-

7 16 The Open Applied Physis Journal, 212, Volume 5 V R Romanovskii perties of superondutor on the spatial features of the development of stable thermo-eletrodynamis states Therefore, let us find the onditions desribing the instability boundary of the eletrodynamis state of high-t superondutor aounting for non-uniform temperature distribution in its ross setion Let us investigate the stati heterogeneity operating mode of the slab, whih takes plae in the fully penetrated mode at di/dt, using the following set of equations " # T %!( T " $ J =, " x " x (!! x 2 = 2 under the boundary onditions (31 (32!T!T (,t =, "!x!x (a,t h $% T (a,t # T =, (33!! (, t =, ( a, t =! x! x taking into aount the relationships (3 and (4 (34 As follows from (32 and (34, =onst at di/dt The existene of states when the distribution of the eletri field indued by the applied urrent is pratially uniform at a small rate will be shown below In general, the solution of the problem desribed by equation (31 and ondition (33 requires the use of numerial methods However, assuming that the thermal ondutivity oeffiient is onstant (λ=λ =onst, it an be solved analytially Indeed, let us introdue the dimensionless variables! = ( T B " T / ( T B " T, X = x / a Then the problem (3, (4, (31 and (33 an be rewritten as follows d 2! dx " #! =, d! 2 dx =, d! ( dx 1 H! ( 1 = (35 (n1/ n Here, H=ha/λ is the thermal resistane of superon- J dutor,! = a 2 $ " ( T B # T % is the dimensionless ( parameter that desribes the influene of the material properties of the superondutor on the heterogeneous harater of the thermo-eletrodynamis state formation This parameter an be rewritten in the form of! = Ja " T B # T / a Therefore, the physial meaning of the parameter γ is equal to the ratio of the power of the Joule heating in the superondutor to the power of the heat flux transferred by thermal ondutivity Thus, it desribes diretly the influene of the ondutive heat transfer mehanism on the eletrodynamis state formation in superondutor In partiular, the boundary problem (35 leads to a model with a uniform temperature distribution in the superondutor at γ << 1, ie, Ja 2 <<! T " T at B Parameter γ also allows one to estimate the legitimay of the usage of the zero-dimensional model taking into onsideration the ritial properties of superondutor In partiular, it is seen that the influene of the non-uniform distribution of temperature on the eletrodynamis state of superondutor will be more notieable when its ritial urrent density is higher or the so-alled temperature margin of superondutor, whih is equal to (T B - T, is lower with other parameters being equal Their influene the more essential, the more intensive the ooling The analytial solution of the boundary problem (35 is given by! ( X =! osh " X,! = H " sinh " H osh " Then, in the dimensional form, the stati non-uniform distribution of temperature in the superonduting slab is desribed by the expression $ T (x = T B! ( T B! T " osh # x % a( (36 during fully penetrated mode In this ase, the relationship between the urrent per width and eletri field determined a I ( = I / (2 b = 2! J ( dx is written in the form of as * " # * I = 2Ja$ %! H!! H / tanh (37 aording to the symmetry of the examined problem This formula strongly proves that the boundary of the urrent instability in the stati non-uniform approximation follows from the ondition! /! I " # (38 Its formal differene from the ondition (18 shows that the urrent instability ondition depends on the harater of the urrent redistribution in the superonduting slab when the temperature distribution in the superondutor is nonuniform Moreover, it will be shown below that the urrents in the internal part of superondutor are more unstable in omparison with the ones flowing on its surfae due to the orresponding non-uniform temperature distribution As a result, the ondition (18 loses its physial meaning Aordingly, the non-uniform distribution of the temperature in the superondutor also hanges the formulation of the stability riterion (22 In order to formulate this ondition taking into aount the non-uniform distribution of temperature, let us integrate the heat equation (31 with respet to x from to a Then taking into aount the boundary onditions at x= and x=a it is not diffiult to obtain the following relationship a! J dx = h T " T x=a (39

8 Stable Marosopi Penetration of Applied Currents in High Temperature Superondutors The Open Applied Physis Journal, 212, Volume 5 17 This relation desribes the first equality of the onditions (22 in the integral form Obviously, the seond equality is modified aording to the given formulation of the G and W Using the urrent-voltage harateristi (37 and the ondition (38, it is easy to find the quenh eletri field and quenh urrent per width of the slab defining the urrent instability onset in the non-uniform distribution of temperature in the slab Aordingly, in terms of the dimensionless variables used, the value q is determined by the solution of equation H = 2n n 1! q tanh! q 2! q " n " 1 n 1 sinh 2! q Then the value of I * q is equal to " # * q Iq = 2Ja $ %! q H! q / tanh! q Here, H J! q = a 2 $ " T B # T % q ( 1 1 n (4 (41 Note that the dependene γ q (H in the framework of the zero-dimensional approximation is alulated by formula! q, = H / n (42 whih follows from (19 or an be obtained from (4 under the assumption that γ q << 1 The alulations presented below show that the value γ q does not exeed unit within a wide range of the thermal resistane variations This onlusion allows one to find the approximate solution of equation (4 After some transformation, it is easy to get! " H q n H ( n # 1 / 3 (43 In the dimensional variables, the equality (43 is written as follows # 1 q! % $ % n n " 1 H / 3 J a h T B " T ( ( n/(n1 (44 In this ase, the limiting urrent stably flowing in superondutor is equal to I q! I # 1 % $ % n H n " 1 1 / 3 J a h T B " T n H(n " 1 / 3 H n H(n " 1 / 3 / tanh ( ( 1/(n1 H n H(n " 1 / 3 (45 Here, I is the ritial urrent of a superondutor (I =J S Comparing this value of quenh eletri field with the value desribed by the formula (19, it is not diffiult to find differene between the estimated boundary values of the eletri field alulated in the framework of the zerodimensional and one-dimensional approximations The orresponding relationship an be written as q, n! 1 q " # = $ 1 H % 3n n/( n 1 (46 Therefore, differene between the orresponding values of the urrent of instability is defined as! H " * # $ Iq, n 1 n H ( n % 1 / 3 = # $ * Iq n 1 # n H ( n % 1 / 3 H $ # tanh n H ( n % 1 / 3 $! n % 1 " # 1 H $ 3n 1/( n 1 (47 These expressions allow one to estimate quantitatively the influene of the spatial heterogeneity of thermo-eletrodynamis states on the urrent stability onditions In partiular, the formulae (46 and (47 show that differene between the quenh values of eletri field and urrent, whih follow from the zero-dimensional and one-dimensional alulations, always takes plae for any finite value of γ Moreover, differene inreases with inreasing γ As a result, the one-dimensional approximation orretly desribing the urrent stability onditions at intensive ooling onditions (H >> 1 allows one to find the limiting value γ q,i at H Aording to (4, it is the solution of equation 2! q,i = n " 1 n 1 sinh 2! q,i It is essential that the value γ q,i annot be found in terms of the zero-dimensional model beause the spatially heterogeneous harater of the thermo-eletrodynamis proesses is ritially dependent on its thermal resistane at H > 1 In these states, the inrease in the limiting values of the eletri field and urrent beomes signifiantly nonlinear with inreasing H and the following estimate I n 1 n tanh * * q q " Iq, 1/( n 1! $ n # 1 %! q 1 H ( 3n may be used aording to (47 This relation shows that the urrent of instability will not inrease proportionally to an inrease in the thikness or the ritial urrent density of superondutor in the intensive ooling onditions unlike the orresponding proportional hange of the ritial urrent of the superondutor with inreasing quantities a and J In other words, the thermal size effet will lead to unavoidable thermal degradation of the urrent-arrying apaity of superondutor In this ase, the inrease in the quantities a

9 18 The Open Applied Physis Journal, 212, Volume 5 V R Romanovskii and J leading to inrease in the ritial urrent of the superondutor is not aompanied by a orresponding inrease in the urrents of instability This thermal degradation effet, whih ours due to stable temperature rise of superondutor, will be disussed below 6 FFCT OF HAT CAPACITY ON TRANSINT LCTRODYNAMICS STATS OF HIGH-T C SUPRCONDUCTORS Using the results obtained above, let us disuss the physial features of stable and unstable formation of thermoeletrodynamis states in the urrent harging into a Bi2212- superondutor under non-intensive ooling onditions, whih take plae in ondution-ooled magnets [46-48] Under this mode, the temperature distribution in a superondutor is pratially uniform if ha/λ << 1 The typial temperature and voltage dependenes of high-t superondutors as a funtion of a urrent are shown in Fig (2 The results presented were based on the numerial solution of the problem defined by equations (1 - (8 For onveniene of the performed analysis, it was done using the urrent normalized by the slab width (I * =5I/b In this ase, normalized urrent sweep rate di * /dt=5b -1 di/dt was single variable quantity The simulation was made for a Bi 2 Sr 2 CaCu 2 O 8 superonduting slab (a=5 1-2 m for two values of di*/dt initially ooled at B=1 T and T =42 K The following onstants h=1-3 W/(m 2 K, n=1, = 1-6 V/m were used Aording to the linear fitting presented in Fig (1, the ritial parameters of the superondutor in the framework of approximation (4 are equal to T B =261 K, J = A/m 2 (48 The speifi heat apaity and thermal ondutivity of the superondutor were defined as follows " J # C $ m 3 K % ( * 3! 6 585T 22 T, T 1K = 1, -!, 4, 4 > T, T 1K (49 # W $!( T % = m K (" 1234* * 1 " 5 " 4 * " * * T " 6 2 " 7 3 " T T T aording to [49, 5], respetively (5 As it follows from Fig (2, the temperature variation of the superondutor under onsideration has a minor effet on the dynamis of eletrodynamis states in the stage of partially penetrated mode Therefore, it is easy to find the eletri field f on the surfae of the superondutor when its ross setion is ompletely filled with urrent Aording to the saling approximation formulated in [34], this value is aµ di equal to f = The orresponding values are shown 4b dt in Fig (2 Using it, let us estimate the influene of the urrent harging rate on the non-uniform redistribution of the eletri field inside superondutor during initial stage of the fully penetrated mode when I > I f Here, I f = n µ J a n 1 n1 n! µ di $ # " 4b dt % 1n n di dt Curves 2, 2, 3 and 3 in Fig (2 depit the evolution of eletri field and temperature in the superondutor during both partially and fully penetrated states on the surfae of a slab (2, 3, 4 and in the enter (2, 3 Curve 1 orresponds to a stati uniform distribution indued in the slab by infinitely slow urrent harging and follows from approximation (14 and (15 As a whole, the urves in Fig (2 indiate the existene of the harateristi dynamis of the fully penetrated states that are proper for the stable and unstable urrent harging modes of high-t superondutors and depend on di/dt and must be onsidered during experimental definition of the ritial urrent of superondutors First, omparing urves 1-3 in the sub-ritial eletri fields ( <, it is easy to understand that the eletrodynamis state formation will be pratially uniform in the Fig (2 Sweep rate dependene of the temperature-urrent (a and voltage-urrent (b harateristis of a high-t superondutor: 1 - di * /dt ; 2, 2 - di * /dt=1 2 A/(s m; 3, 3 - di * /dt=1 3 A/(s m, 4 - di * /dt=1 2 A/(s m, C=C(T

10 Stable Marosopi Penetration of Applied Currents in High Temperature Superondutors The Open Applied Physis Journal, 212, Volume 5 19 fully penetrated mode when f <<, for example, at di / dt << 4b / aµ Seond, there exists the transient period after whih the fully penetrated eletri fields on the surfae of the slab and in its entre beome pratially equal To estimate the influene of the harging rate on the time during whih the non-uniform distribution of the eletri field exists, let us use the estimator! /!t! ( / a 2! /!J (51 following from equation (2 It shows that the redistribution of the eletri field in the ross-setion of the slab beomes more intensive with inreasing differential resistivity of a superondutor / J and value of the indued eletri field Taking into onsideration this onlusion and evident fat that indued eletri field inreases with inreasing di/dt (Fig 2, it is easy to understand that the time window of the eletri field redistribution between non-uniform and pratially uniform fully penetrated states will derease with inreasing sweep rate, as it is depited in Fig (2 Third, Fig (2 also reveals that in the over-ritial eletri field region (> the fully penetrated dependenes (I *, di * /dt and T(I *, di * /dt, whih follows from transient model, do not identially with the orresponding stati dependenes defined by the stati model Namely, the transient differential resistivity of the superondutor not only dereases with inreasing di * /dt but has only positive values Consequently, the transient voltage-urrent harateristi of a high-t superondutor does not allow one to find the boundary of the urrent instability onset in the ontinuous urrent harging experiments Really, the limiting urrentarrying apaity is defined by the ondition (18 in the stati zero-dimensional approximation The orresponding quenh values q, and I * q, defined aording to the (19 - (21 are depited in Fig (2 by the dotted lines Note that the alulated boundary of stability is loated in the over-ritial eletri field region for the superondutor under onsideration The onditions desribing the existene of the stable over-ritial stati states are formulated below To explain why the ondition (18 is not observed in the transient thermo-eletrodynamis states and the transient (I * relation has only positive slope, let us use the transient zero-dimensional approximation Assuming that the ritial urrent of a superondutor does not depend on the magneti field, it is easy to get dt dj! " 1 # $ d dj dt = % dj J ( T! " # $ n % (52 using equation (3 To find the term dt/dj, let us utilize equation (12 taking into onsideration that dt 1 dt di = dt S dj dt Then dependene dt/dj is given by dt dj J! ( T! T h / a C( T di dt = S (53 As dt/dt> during urrent harging then dt/dj> Therefore, as follows from equations (52 and (53, the differential resistivity of a superondutor is always positive in the monotonously urrent harging and the slope of the transient (I *, di * /dt and T(I *, di * /dt dependenes will beome smaller when the heat apaity of a superondutor is higher, ie, will derease with inreasing temperature of a superondutor during both stable and unstable states At the same time, as follows from Fig (2, there exists stable temperature rise of a superondutor before urrent instability This unavoidable temperature rise of a superondutor will hange its heat apaity and the effet of heat apaity of superondutor on the eletrodynamis state is inevitable during urrent harging To illustrate this feature, Fig (2 also presents the dependenies (I * and T(I * obtained under the assumption that the heat apaity of the superondutor alulated at the temperature of the oolant does not hange in the temperature (urve 4 Fig (2 learly depits that the temperature of the superondutor before the urrent instability onset is not equal to the temperature of the oolant In partiular, allowable temperature rise of a superondutor before urrent instability onset depends on the urrent harging rate: the higher the di/dt, the higher the stable overheating of a superondutor, as shown in the inset of Fig (2 In these ases, the effet of the heat apaity of a superondutor on the eletrodynamis states of a superondutor beomes more notieable The results obtained demonstrate the reason aording to whih the transient voltage-urrent harateristis of high-t superondutor do not allow to find the boundary of the urrent instability onset To avoid this omplexity, the urrent harging with break (method of the fixed urrent is used in the experiments [16-18, 21-23] Aording to this method, there are two final states: temperature and voltage either are stabilized or begin to grow spontaneously In the first ase, the stable urrent distribution is a diret onesquene of the existene of stable stati states that underlie the formation of a stable branh of the voltage-urrent harateristi of a superondutor However, if the harged urrent exeeds the orresponding quenh urrent I q, then the formed states are unstably even in spite of the fat that the orresponding values of the eletri field and the temperature may be below than the boundary values q, and T q, on the transient voltage-urrent or temperature-urrent harateristis of a superondutor Therefore, all urrents in Fig (2 exeeding the relevant value I * q orrespond to unstable urrents Note that this method of the urrent instability determination was proposed for the first time in [24, 25] analyzing the onditions of the urrent instability onset of the low-t superondutors To prove these peuliarities, the transient states of the superondutor that will be observed near the instability urrent boundary are shown in Fig (3 The alulations were arried out both in the ontinuous urrent harging and during urrent harging to a fixed value I * after whih di/dt is equal to zero Here, the orresponding values of q, and

11 2 The Open Applied Physis Journal, 212, Volume 5 V R Romanovskii Fig (3 Time variation of the eletri field (a and temperature (b on the surfae of superondutor during ontinuous harging at di * /dt=1 3 A/(s m (urve 1 and urrent harging with break (urve 2 - I * =1524 A/m; urve 3 - I * =1523 A/m T q, that are the same as in Fig (2 are also given It is seen that the values q, and T q, are the boundary values between transient stable and unstable states during fully penetrated mode when the ondition ha/λ <<1 takes plae regardless of the finite value of urrent harging rate and the temperature dependene of heat apaity of a superondutor Note important feature that is harateristi of modern high-t superondutors The results presented show that the urrent instability ours during fully penetrated modes In this ase, the instability onset does not depend on the urrent harging rate and temperature dependene of heat apaity As known, the urrent instability onset in the low-t superondutors an also our in the partially penetrated modes [2, 24, 25] This fat is due to differenes in ritial properties of low-t and high-t superondutors Let us write the ondition under whih the urrent instability onset in superondutors will our only in the fully penetrated mode This would be in the ase when q, > f, ie at " $ # $ nj h a T B! T % n n1 > aµ 4b di dt (54 If this ondition is not valid then limiting harged urrent will depend on the sweep rate and temperature dependene of heat apaity [2, 24, 25] In this ase, the permissible overheating of the superondutor before the instability may essentially differ from T q, and inreases with inreasing di/dt [24, 25] The ondition (54 shows that, in general, the urrent instability onset depends on dj /dt as dj /dt~j / (T -T Therefore, the probability of the urrent instability onset in the partially penetrated mode will inrease with inreasing ritial urrent density of high-t superondutors In other words, there may be suh urrent harging modes when the urrent-arrying properties of the high-t superondutor with high ritial urrent density will be not fully used due to the instability of the harged urrents that may our in the partially penetrated mode In addition, the partially or fully penetrated regimes of instability onset depend on the n-value of (J relation of a superondutor In partiular, the probability of instability onset dereases with dereasing n Thus, the heat apaity of high-t superondutor may play essential role in the stable and unstable formation of its eletrodynamis states in the over-ritial modes It appears that it is impossible to see in an experiment a steep transition from the stable state to the unstable one even at helium temperature level Stritly speaking, this is possible only at very low urrent harging rates Note that the heat apaity of low-t superondutors also influenes on the formation of their voltage-urrent and temperature-urrent haraterristis However, its role is insignifiant beause of the small temperature rise in the stable states Curves 4 in Fig (2 learly show the harateristi type of voltage-urrent and temperature-urrent harateristis of superondutors with low heat apaity The role of the heat apaity will inrease when the operating temperature inreases Fig (4 shows the influene of the urrent harging rate on the thermo-eletrodynamis states of the superonduting slab under onsideration ooled at operating temperature T = 2 K The numerial simulation was performed under the assumption of the nonintensive ooling ondition (h=1-3 W/(m 2 K at B=1 T and the ritial urrent density is desribed by equation (9 with parameters (1 This allows one to use the transient and stati zero-dimensional models (11 and (12 Aordingly, solid lines 2-4 desribe the transient thermo-eletrody-

12 Stable Marosopi Penetration of Applied Currents in High Temperature Superondutors The Open Applied Physis Journal, 212, Volume 5 21 Fig (4 letri field (a and temperature (b versus applied urrent at T =2 K and various urrent harging rate namis states of the superondutor and line 1 shows the stati state Dash-dotted urve 5 orresponds to the isothermal (J harateristi alulated at T = 2 K Comparing results shown in Fig (2 and Fig (4, it is easy to see the following operating temperature effet: the higher the temperature of the oolant or the urrent harging rate, the more lose the non-isothermal dependenes to the orresponding isothermal ones in the high eletri field range This peuliarity follows from the relationships (52 and (53 at the limiting transition dt/dj if C(TdI/dt It allows one to make unexpeted onlusion that is important for pratial appliations: an isothermal voltage-urrent harateristis in the over-ritial eletri field range an be obtained harging the urrent into the non-intensive ooled superondutor at high value of the urrent harging rate It is obvious that in this ase the ondition ha/λ <<1 must take plae Thus, in the ontinuous urrent harging modes, transient voltage-urrent and temperature-urrent harateristis of high-t superondutors during fully penetrated states have only one branh with the positive slope, whih depends on the temperature dependene of heat apaity That is why their slope dereases when the value di/dt inreases This thermal effet exists both in the stable and unstable thermoeletrodynamis states It is due to the finite temperature rise of a high-t superondutor both before and after instability onset that inreases the heat apaity of a superondutor Its role is the most signifiant in the high eletri fields or high operating temperature Therefore, the transient (J harateristis of high-t superondutors do not allow to find the urrent instability onset in the experiments and the possible stable inrease in temperature of high-t superondutors should be taken into aount for the orret determination of their ritial urrents 7 THRMAL PCULIARITIS OF LCTRODYNA- MICS STAT FORMATION IN TH NON-UNIFORM TMPRATUR DISTRIBUTION Let us investigate the formation peuliarities of the eletrodynamis states of high-t superondutor aounting for non-uniform temperature distribution in its ross setion using the appropriate urrent instability onditions formulated above To disuss the basi physial features, let us use, first of all, the one-dimensional diffusion model desribed by equations (1 - (8 onsidering the non-isothermal penetration of applied urrent into examined Bi 2 Sr 2 CaCu 2 O 8 superondutor at B=1 and T =42 K under parameters (48 - (5 assuming that intensive ooling ondition takes plae (h=1w/(m 2 K As above, the performed analysis will be done using the urrent and urrent sweep rate normalized by the slab width (I * =5I/b, di * /dt=5b -1 di/dt Fig (5 shows the eletri field and the temperature as a funtion of applied urrent, whih will exist on the surfae and in the enter of the superonduting slab at di * /dt=1 2 A/ (s m at different urrent sweep rate To understand the peuliarities of the non-isothermal development of eletrodynamis states in the superondutor that take plae both in stable and unstable regimes, the isothermal (I * harateristi of a superondutor (urve 1 alulated aording to the formulae (3 and (4 at T=42 K is also presented in Fig (5 Besides, the urve 2 desribes the non-isothermal stati (I * harateristi determined under the assumption of the uniform temperature and urrent distributions that takes plae over the whole range of the transport urrent harging In this ase, the stati thermo-eletrodynamis states of the superondutor are desribed by equations (3, (4 and (12 in whih the density of the transport urrent is equal to J=I * /(2a Fig (5a also depits the eletri field q, and the urrent I * q, defining the stability boundary of spatially