Mixed state in high temperature superconductors

~

Solid State Communications, Vol. 77, No. 5, pp. 389-396, 1991. Printed in Great Britain.

0038-1098/9153.00+.00 Pergamon Press plc

MIXED STATE IN HIGH TEMPERATURE SUPERCONDUCTORS V.V.Moshchalkov* Centre d'Etude Nucleaires de Grenoble, DRF/SBT-LCP, 85 X 38041 Grenoble Cedex, France (Received 11 October 1990 by P.Burlet)

In high T c superconductors the macroscopic mixed state (MACMS), corresponding to a mixture of macroscopically large superconducting and normal domains, may be realized in magnetic field H. On the H-T phase diagram MACMS lies between high temperature normal state and low temperature Abrikosov vortex state. The irreversibility line corresponds in this case to the topological cross-over from MACMS with isolated singly-connected superconducting "droplets", surrounded by the normal phase, to the sponge-like MACMS with multiply-connected superconducting "body", surrounding isolated normal-state domains, which finally decay into Abrikosov vortices, as temperature goes down. The irreversibility appears due to the change of connectivity caused by the transition from superconducting droplets to superconducting sponge. The posibility of the multiple ~o normal cores existence has been demonstrated. Related experimental data on resistivity, magnetization, critical currents of high T c superconductors in magnetic field are discussed. The new experiments have been proposed to check predictions of the MACMS model.

INTRODUCTION

divided into superconducting and normal regions. This is the origin of the mixed state. Nowdays the mixed state is usually associated with the Abrikosov vortex state in type II superconductors, while less and less cited are the classical papers by Schubnikov [1] , Mendelssohn [2] , Shoenberg [3] and Shalnikov [4] , who studied the mixed state in type I superconductors. In fact, in any sample of a type I superconductor, except a long cylinder in a uniform parallel magnetic field, the formation of the mixed (Schubnikov) state is inevitable. As a result, a superconducting sample in magnetic field is divided into macroscopic superconducting and normal domains. This effect is stronger in thin specimens and, as it has been shown by Landau [5], thin plate in transverse field consists of macroscopic domains, branching towards the surface. The domains in superconducting samples are analogous to well known magnetic domains. The dimensions of superconducting (ds) and

The superconducting samples in magnetic field are characterized by the formation of a mixture of the superconducting and normal state areas. This effect is a result of the interplay between two contributions into free energy H2c AF = ~ V s

H2 ~ VN

(1)

The term with the thermodynamical critical field Hc corresponds to the energy increase due to the distruction of superconductivity in volume Vs, while the second term gives the drop in energy due to the penetration of magnetic field H into volume V,. If AF < 0 then the sample, to minimize AF, is spontaneously * Permanent address: Laboratory of High T c Superconductivity, Physics Department, Moscow State University, Moscow 117234, USSR.

389

390

MIXED STATE IN HIGH TEMPERATURE SUPERCONDUCTORS

normal (dN) domains and their distribution over the sample depend upon the sample shape and on the penetration of magnetic field into superconductor, 1. The superconducting domains have macroscopic dimensions which tend to zero at H=H c , where dN/d s diverges as Hc/(Hc-H) . Measurements carried out on tin [4} give ds~ 0.2 mm for H/He=0.8 and T=3 K. Therefore, the existence of the macroscopic mixed state (Schubnikov phase) in type I superconductors has been proved long before studies of type II superconductors. Though the origin of the mixed state in both type I and type II superconductors is similar, it is mainly the tendency to optimize the field penetration into superconducting areas, the important difference between the two should be emphasized here. The vortex state predicted by Abrikosov [6] is a novel quantized ultimate realization of such field penetration into a superconductor, which, optimizing ~F (Eq.1) gives to each flux line the single flux quantum ~o This is a microscopic mixed state with dN~K , where is the coherence length. The Abrikosov vortex state may be observed even in long cylinders in parallel field. On the other hand, the Schubnikov mixed state in type I superconductors is a macroscopic state with the normal domain size increasing with field. This state is not found in long cylinders in parallel field. THE MODEL OF MIXED STATE

THE

possibility of the superconducting phase neucleation below Tc(H) with superconducting "droplets" which are quite large and are also characterized by an ability to carry multiple flux quanta N~ o with N >> I. These three arguments seem to be very important for layered extreme type II (A >> ~) superconductors, such as high T c oxides. Let us consider a type II superconductor in regime 1/12<< nL<< I/~ 2 , when the Abrikosov vortex lattice iS quite dense and the number of flux lines nt, each carrying ~o , is large. In this case the free energy is given by the formula [9] : nL~

~F@o=

1 1 / ~ o ] nL ( H ~ - H 2 ) ~ o in - + 16~ 2 A 2 ~ ~ ) 8~

(2)

Here H=nL~o=(nL/2)(2~ o) and we may try to create 2~ o vortices by the substitutions: ~o ~ 2~ o and nt~ nL/2 in Eq.2 (nt /2)4q%o2 AF~

=

11/~

ol

in

+

0

n L (Hc2-H 2 ) 2 Introducing

8","

~

coefficients

(3)

o

A=Hqbo/(16~ 212 );

C=4~7H and D=H(Hc2-H 2)/(B~) , we have to minimize the following expressions to find the optimum (~@o)°pt and ( ~ o)°pt

MACROSCOPIC

After the introduction, it is proper to formulate the main hypothesis of the present paper: the macroscopic mixed state (MACMS) exists in high T c superconductors (and, probably, in any extreme type II superconductor) between the low temperature Abrikosov vortex state and high temperature normal state (T > To) There are direct magnetic decoration experiments [7,8], proving the existence of the low temperature Abrikosov vortex state. But what are the arguments in favour of MACMS ? These are the following: (i) possibility of the multiple- ~o vortex state (MVS) with normal cores carrying multiple flux quanta N~ o ; (ii) divergence of the dN/ds~ ~/A ratio for H > H c ; (iii)

Vo|. 77, No. 5

AFro

AF~

= A in

= 2A in °

The

(4)

[~o } + D~o

+ ~ ~

(5)

~ ~o)

corresponding

= A/(2D) and ( ~ the AF minima

°

values

are

)opt = 2A/D, which give O

+

mln

(AF~°),,In

(K~o)opt

(6)

-

2

= 2A in ( ~ D D I

+ A

(7)

The energy ~F is lower for 2@ o than for ~o vortices if (~F@o)mi n / ( A F ~ o)min > I which leads to H~ o

~212 (He2 _H 2)

> e = 2.78

(8)

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MIXED STATE IN HIGH TEMPERATURE SUPERCONDUCTORS

Using the a p p r o x i m a t i o n o b t a i n (Fig.l)

4 ~ 2 He1 = ¢ o '

4 Hcl

- -

"~

we

H

391

3¢0

~Hc2

(9)

H > H 2_ - H 2 e

If H2~ He2 then H ~ (~e/4)Hc2 = I-Ic2 and, of cause, the single 4Po flux lattice is realized. On the o t h e r hand, c l o s e r to H c in fields

4.rr 4

A4 e2

+ Hc

2 ~ 2 A2

<

H

<

(10)

Hc

the free energy AF is lower for 2~ o flux lines. Within the framework of the approximation 4wA2Hcl = ~o Eq.10 may be r e w r i t t e n in the f o l l o w i n g way

0

To

T

Fig.2

[~2e 2

+

H~ -

~re

(11)

< H < Hc

The optimization procedure is easily g e n e r a l i z e d to the m u l t i p l e v o r t e x (N~ o ) states and corresponding fields H ~ , H ~ o , etc lie c l o s e r to the He(T) line, as it is c l e a r l y seen from the g r a p h i c solution of Eq.9 (Fig.l). The interval b e t w e e n two s u c c e s s i v e values H~ and • O H(~I~ b e c o m e s shorter as N zncreases. It m a y ° b e c o n c l u d e d at this point that the single ~o vortex lattice b e c o m e s unstable towards multiple ~o vortex state formation, as the Hc(T) line is approached (Fig.2). The s i t u a t i o n for single ~o state gets even worse, when H > H c , b e c a u s e in this case the saddle point singularity of the free energy

fl ,f2

0

H2@o~I

LHo

H

H3~ -~ Flg.1 The g r a p h i c i l l u s t r a t i o n to the s o l u t i o n of Eg.9. M u l t i p l e - ~ o flux lines may be formed b e t w e e n H ~ , H~ ,...etc O O and H c .

extreme

The H-T type II

phase diagram superconductor:

Abrikosov vortex state, multiple-~ o vortex state, m a c r o s c o p i c m i x e d state.

of an AVS -

MVS MACMS

-

(Eq.3) is crossed and m i n i m i z a t i o n procedure becomes divergent, giving formally (~ )opt ~ ~" Therefore, in the v i c i n i t y of the He(T) line we may expect the transition from the single ~o Abrikosov v o r t e x state to the m u l t i p l e ~o vortex state (see the MVS area in Fig.2). The divergence (~q)opt /A h T in the v i c i n i t y of H c indlcates t a between Hc and Hc2 in type II superconductors the m i x e d state may be m a c r o s c o p i c (MACMS). The MVS formation below He(T) may be even further enhanced, if to t a k e into account in Eq.3 and Eq.2 a s t r o n g e r repulsion b e t w e e n flux lines in layered superconductors, where the 3d r e p u l s i v e energy Uij N in(A/rlj ) should be m o d i f i e d in the d i r e c t i o n of a larger r e p u l s i o n for 2d systems which have Uij ~ I/rij [9]. The increase of the vortex r e p u l s i v e energy in l a m e l l a r s u p e r c o n d u c t o r s may be caused by a n o t i c e a b l y stronger vortex i n t e r a c t i o n through the "empty" space between superconducting planes. So, there are two main driving forces for the transition into MVS: v o r t i c e s repulsion and the (Hc2-H 2 ) sign inversion. If the MVS below He(T) is really possible, it is interesting then to investigate neucleation

in more in the bulk,

details the trying to find

392

Vol. 77, No. 5

MIXED STATE IN HIGH TEMPERATURE SUPERCONDUCTORS

additional arguments in favour of MVS and MACMS coming this time into a superconducting area on the H-T plane from high fields and temperatures. The s o l u t i o n of the l i n e a r i z e d Ginzburg-Landau equation

1 ( . 2eA~ 2 2--~ [-1~V- c ) ~ = -me

V

.~.L) 112

o

(a)

(12)

gives the well k n o w n r e s u l t -~ = ~1

m

v z2

+

n+~

(13)

mc

a r i s i n g from the formal identity between Eq.12 and the S c h r ~ d i n g e r e q u a t i o n for a particle with charge 2e in a u n i f o r m magnetic field. For n=0 and Vz=0 in Eq.13 one obtains the r e l a t i o n ~2 (T c (0) -~

T c (H))

=

i

~,

~eH =

2m T c (0) 52o

vi

(14)

:

h

~i~i

R

(b)

mc I

which determines the He2 (T) line. Two things are not c o m p l e t e l y clear here: (i) why the solutions of Eq.13 with n > 0 and Vz> 0 are not c o n s i d e r e d at all ? and (ii) is it possible, at least in the limit of e x t r e m e l y large A and full field penetration, to find the solution of the nonlinearized Ginzburg-Landau equation ? For this purpose we shall use the a p p r o a c h similar to T i n k h a m m e t h o d [10] d e v e l o p e d for the e x p l a n a t i o n of the Little and Parks [11] oscillations. This method assumes that the a m p l i t u d e I~I of the o r d e r p a r a m e t e r is constant, which might be a realistic s i t u a t i o n for e x t r e m e l y large A and full field penetration:

&F = l e 1 2 ( ~

+ ~le12+

21 m V 2 ) + 'h~2

(15)

For a cylinder which does n o t carry c u r r e n t along its axis and is in a fully penetrating p a r a l l e l field V = c o n s t and a f t e r c o u n t u r i n t e g r a t i o n along a circle with radius R Eq.15 is m i n i m i z e d w h e n [10] V = minf~_~_in_~ iI = & minl n

kmR

~o)

~RH

(16)

V

.

Cc)

I

Fig.3 The illustration of the optimized V(R) function (Eq.16) for different m a g n e t i c fields (a,b,c). The peak positions are given by Eq.17. Superconductivity is possible for those R which c o r r e s p o n d to ~ > V(R).

of peaks on the saw-like V(R) d e p e n d e n c e are easely found: n~ o R2

=

(17)

The distance between two successive peaks and their height d e c r e a s e with R J

The

~F (Eg.15) m i n i m u m c o r r e s p o n d s to 1412 =(-~-mV2/2)8 -I and the solution exists if -~ > my2/2 , where V is given by Eq.16. The o p t i m i z e d V(R) function (Eq.16) is shown in Fig.3. The p o s i t i o n s

Rn÷1

-Rn =

~H 4Po (n+l)

(18)

and the h i g h e s t peak is the first one w i t h V ( R = R I )=~wR IH/(m~ o ) = ( ~ / m ) ~ 7 ~ o . If the first peak lies outside the

Vol. 77, No. 5

MIXED STATE IN HIGH TEMPERATURE SUPERCONDUCTORS

superconducting cylinder ( Re> RI, weak field limit, R o is the sample radius), then the superconducting area I~1 > my2/2 (see Fig.3a) carries ~o and expands when Tc-T ~ I~I increases during the field-cooled process. Finally the superconducting area, growing towards the cylinder boundary, occupies the whole sample (full Meissner effect), w h i c h takes place when

H

[

AVS

393

1.

L

\l

~2 ~2 R~ H2 I~1 >

&T ~

2m

(19)

~

The t r a n s i t i o n width ~T is p r o p o r t i o n a l to H 2 . If applied field increases, then the first peak at R=R I enters the sample and R ~ = ~ o / ( ~ H ) = R ~ (Fig.3b). After that the s u p e r c o n d u c t i v i t y c r i t e r i u m I~I > mV2/2 should be treated separately for two d i f f e r e n t cases: (i) either 0 < I ~ 1 < ~2~H/(2m~o) and horizontal line l~l=const intersects Saw-like peaks (Fig.3c) or (ii) I~t ~ ~2~H/(2m~o) and the l~l=const line lies above the saw-like V(R) dependence. In the first case concentric-rings-like multiple #o solution may be possible (Fig.3) with normal state rings area being d e c r e a s e d as T goes down. But, strictly speaking, this solution is in a c o n t r a d i c t i o n with the c o n d i t i o n that l~l=const e v e r y w h e r e .in the sample. Nevertheless, as unstable or fluctuating superconductivity this c o n c e n t r i c - r i n g s solution still might be realized. In the second case l~l=const and I~I > my2/2 everywhere in the sample. As a result, superconducting area is m a c r o s c o p i c a l l y large and again it is c h a r a c t e r i z e d by the m u l t i p l e - ~ o field penetration. It is interesting to note here that the superconductivity criterium I~I > mV2/2 gives in the second case the I~I value which is 2 times smaller than that used earlier for the He2 d e t e r m i n a t i o n (Eq.14): Tc(H)-Tc (0) I~1

=

--

2

~eH =

To~

(20)

2mc

For a given fixed I~I, the field H in Eq.20 is two times larger than H ~ in Eq.14. If to take the concentric rings like solution seriously, then the n e u c l e a t i o n is d e v e l o p e d at temperatures below the vertical line T=Te(0) on the H-T phase d i a g r a m (Fig.4), while Eq.20 determines the possibility of the

,l%4"'inel,

H(Ro=R1)........ \

.........

\/ I aeissner phase I

T,

To

T

Fig. 4 obtained

The tentative H-T phase diagram, from the analysis of Eq.15.

divergence, c o r r e s p o n d i n g to the MACMS existence. It is not completely clear what are Hcl and He2 in this model. Formally, the full Meissner effect is to be observed only if H < ~o/(~R~) , whereas the onset of the diamagnetic response should correspond to the vertical line T=Tc(0) (Fig.4), because the concentric rings like solution appears immediately below Tc(0) for any small I~I (Fig.3). On the other hand, Eq.20 in the l~l=const model gives not the n e u c l e a t i 0 n onset but K divergence and therefore the H vs T line found from Eq.20 should be close to the MVS line, d i s c u s s e d above in this paper. Of cause, the real K value is limited by the full field penetration requirement and it should follow the penetration depth variations. Finally, the l~l=const approach also leads to conclusions, similar to those obtained in the section on the MVS. At this stage it is possible to give the following tentative picture of the field-cooling development of s u p e r c o n d u c t i v i t y (Fig.5). After crossing the vertical line T=Tc(0) , the "problematic" c o n c e n t r i c - r i n g s - l i k e solution may appear at many n e u c l e a t i o n centers (Fig.5a). Fluctuations may transform rings into stripes, while the solution I=I > ~2~H/(2m~o) is more stable. As temperature goes down, the dimension of the superconducting droplets grows, but they are still mostly singly connected ones and for

394

MIXED STATE IN HIGH TEMPERATURE SUPERCONDUCTORS

O Normal phase © Superconducting phase

O

(a) T=T4, MACMS

(b) T=T3, MACMS

(c) T=T2, MVS

Vol. 77, No. 5

singly connected superconductor it is impossible to obtain an irreversible behaviour. Closer to the H vs T line, where ~ ~ ~ , a topological cross-over from singly-connected macroscopic superconducting droplets surrounded by the normal phase to a multiply-connected sponge-like superconducting body (Fig. Sb) with normal state inclusions surrounded by the superconducting phase takes place. And this is the irreversibility line, because irreversibility immediately appears, if one creates a hole in a superconductor, even in type I superconductor. The instructive and simple example here is a ring made of the type I superconductor. This ring demonstrates strongly irreversible magnetization behaviour [12] . Therefore, the irreversibility line is of the common origin for all superconductors: this line corresponds to a topological cross-over associated with the change of the connectivity. If, below the irreversibility line, temperature goes further down, than the flux lines, trapped into pores of a sponge-like superconducting body, first decay into the multiple- ~o vortex structure (Fig. Sc) and finally into the A b r i k o s o v vortex state (Fig.5). DISCUSSION

(d) T=T1 , AVS Fig.5 A schematic illustration of the superconducting phase formation during the field-cooling process, when macroscopic mixed state (MACMS), m u l t i p l e - ~ o vortex state (MVS) and A b r i k o s o v vortex state (AVS) are crossed successively: T 4 > T 3 > T 2 > T I . The irreversibility temperature Tirr corresponds to the topological transition from isolated singly connected s u p e r c o n d u c t i n g droplets (a), surrounded by the normal phase to a m u l t l p l y - c o n n e c t e d sponge-like s u p e r c o n d u c t i n g body (b) which surrounds n o r m a l - s t a t e areas.

Arguments given above indicate the possibility of the MVS and MACMS formation in extreme type II superconductors between the low temperature Abrikosov vortex state and high temperature normal state on the H-T plane. Irreversibility line Hitr (T), as a line of the topological cross-over, may have the following features: (i) Due to its proximity to the Hc(T) line, it should not be much dependent on the N orientation, if the anisotropy of l.llc =Hfl~ /~ci ±c Hcl and Hc2 is the same: Hc½C,.c2 and therefore H~C=H~ c , because in this case the anisotropy is completely cancelled. This seems to explain the independence of the Hitr (T) line in Bi2Sr2CaCu208 upon field orientation [13] .(ii) The "misterious" independence of the irreversibility line on the irradiation [14] is easely understood, because the Hitr (T) should not be changed till H c itself is not changed. The latter seems to be true almost up to

Vol. 77, No. 5

MIXED-STATE IN HIGH TEMPERATURE SUPERCONDUCTORS

the strong sample damage under a huge irradiation doses which evidently have not been reached in Ref. 14. But if the sample was not much damaged and still looked more or less like YBaCuO, then both H e and Hitr should remain almost the same as corresponding values in YBaCuO. Similar arguments may be used to interpret the universal scaling of the irreversibility line in LiTi204 samples with different critical temperatures, found in Ref.15. (iii) The MVS and MACMS, introduced above, may be dependent upon the sample size and sample geometry due to a strong influence of these two factors on the domain formation. Critical currents for the MACMS should be very sensitive to inclusions of the normal state areas which have macroscopically large sizes, comparable with those of the normal domains in the MACMS. This explains the critical current enhancement obtained by Murakami [16] on YBaCuO with macroscopically large inclusions of the insulating phase. The domain formation in the MACMS will be stronger in thin films than in bulk single crystals, like domain formation in magnetic films. The higher stability of the domain distribution in thin films may lead to higher critical currents. Flux creep [17,18] should be strongly suppressed in the MACMS above Hitr (T), because in this case flux lines are mostly outside the singly connected superconducting droplets, whereas field penetrates into macroscopically large normal areas which surround isolated superconducting droplets. The melting of

395

the flux lattice, in fact, corresponds to the AVS instability and reflects the onset of the transition into MVS and MACMS. The scatter of the characteristic sizes of the normal domains in the MACMS may be related to the existence of the distribution of the pinning potentials in high T c superconductors. The enhanced noise below T c in oxide superconductors [19] is probably caused by the domain formation and by the variation of their characteristics with temperature (see Fig.5). Similar processes in magnetic materials have been known for a long time. A sharp peak in the temperature dependence of the noise power measured in magnetic field [19] seems to correspond to the AVS~MVS, MACMS transition. In the regime of the full field penetration the onset of the diamagnetic response may correspond to the vertical line T=Tc(0)=const, rather than to Hc2 , which may be associated with the ~ ~ line (Fig.3,4). From this point of view, the latter may be related to the resistivity zero p(T) ~ 0 and the saturation of the diamagnetic response x(T), but not to the onset of the p(T) and x(T) superconducting transitions. To verify directly the MVS and MACMS existence all types of decoration and visualization experiments in the H-T area between He(T) and normal phase are extremely promising. ACKNOWLEDGEMENTS Useful discussions with J.Rossat-Mignod, C.Ayache and A r e acknowledged.

G.Uimin

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396

MIXED STATE IN HIGH TEMPERATURE SUPERCONDUCTORS

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VO|.77, NO. 5

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