CRITICAL STATE MODEL FOR CUPRATE SUPERCONDUCTORS

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CRITICAL STATE MODEL FOR CUPRATE

SUPERCONDUCTORS

A. Portis, M. Stalder, G. Stefanicki, F. Waldner, M. Warden

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C8, Supplement au no 12, Tome 49, dkcembre 1988

CRITICAL STATE MODEL FOR CUPRATE SUPERCONDUCTORS

A. M. Portis1, M. Stalder, G. Stefanicki, F. Waldner and M. Warden

Physics Institute, University of Ziirich, 8001 Zurich, Switzerland

Abstract. - Microwave absorption in granular superconductors results from the motion of fluxons within Josephson junctions. Critical state development accounts for the sign reversals and absorption minima that have been observed when the magnetic field is modulated or scanned with the assumption thai flux flow is proportional t o the gradient of the flux density.

Introduction

One of the many interesting features of modulated microwave absorption in the planar cuprate superconductors [I, 21 is the unusual dependence of the absorption signal on the amplitude but not the frequency of the modulation field and on the direction but not the rate of sweep of the applied magnetic field.

The observed absorption has been associated with the damped motion of fluxons driven by microwave current [3] and is characterized by an effective-medium microwave surface resistance

where f B

/

cpo is the concentration of mobile fluxons within a microwave penetration depth XL =

(me2

/

4 npne2)1/2 of the surface with n the super- conducting carrier concentration and p the permeabil- ity. Bo = 8 .rrwprlX;

/

( P O is a characteristic field with q the fluxon viscosity, cpo = h e / 2e the quantum of flux and Xo = 4 nwpX~

/

c2 the surface reactance in zero magnetic field. For f a function of depth, an average weighted by the microwave intensity fac- tor exp (-22

/

XL) is required to compute the surface resistance.

The obtained modulation signals are interpreted as arising from a critical state 141 in which a magnetic field above a threshold H,*l penetrates linearly into Joseph- son junctions

where Jc is the critical current density for flux flow. Fields smaller than the threshold field Hz1 penetrate exponentially with a characteristic relaxation distance A'

B (2) = Hext e - x / A' (3)

where X1 is related to J, and H,*l by (4 n

/

c ) J,X1 = HA.

The concentration of mobile fluxons a t depth x has been taken [4] to be proportional to the absolute value of the gradient of the magnetic field

where l z 0.1 pm is a characteristic length.

We demonstrate here that the absorption that has been observed is a consequence of reversible and ir- reversible processes associated with the pinning and depinning of fluxons and with the dependence of the concentration of mobile fluxons on the critical current density J,, which may be a function of magnetic field. M e a n fields a n d gradients

For a field given by (3) the average over a microwave penetration depth is

( B ) = (2

/

XL) /OL B

(z)

exp {-20

/

X L ) dx

From (3) the field gradient at the surface is Hext

/

A'

and the maximum gradient that the medium will support is (4 n

/

c) Jc. Thus, for Hex*

>

Hz1 = (4 .rr

/

c) X'J,, the field drops linearly for x

<

x, =

A' (Hext

-

H,*l)

/

H> and exponentially for x

>

x,. The mean flux density for Hext

>

H,*l is

( B ) = Hext

-

(XL

/

2 ~ ' ) H,*1

x

(1 - [(XL

/

2X1)

/

(1

+

XL

/

2 4 1

x

exp {-2xc

/

XL))

.

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Mobile fluxons can be expected for both increasing and decreasing fields. When the direction of field scan is changed, however, the density of mobile fluxons first decreases and then increases again, independent of the

l ~ l s o at IBM Rcsearch Division, Ziirich Research Laboratory, 8803 Riischlikon, Switzcrland and ETH ZiiricJ~ Institute for Intcrmcdiate Encrgy Physics, Paul Schcrrcr Institute, 5234 Villigcn, Switzcrland. Permanent address: Dcpartli~ent of Physics, University of California, Berkcley, CA 04720, U.S.A.

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C8 - 2232 JOURNAL DE PHYSIQUE

initial direction. With the field penetration given by

( 3 ) the mean concentration of free fluxons is

(n) =

( e /

X'PO) Hext

1

( I

+

X L

/

A')

.

( 7 )

For Hext increasing, the field within the superconduc-

tor is given by ( 2 ) , which leads for He,t

>>

H A t o a

maximum mean concentration of mobile fluxons

If the field scan is reversed a t H,,, the internal field for Hext

>

Hmax

-

2H; is given by

B (x) = Hm,

-

(Hmax - Hext) e-"'"

-

(2

/

A') H L (9)

leading to a reduced mean density of free fluxons for

Hmax

>>

Hz1

(n) =

( e l

X ' P O ) {(Hmax

-

Hext)

x [I-2exp{-xo ( ~ / x ' + ~ / x L ) ) ] / ( ~ + x L / ~ x ' ) -Hzl [ I - 2 exp (-220

/

X L ) ] )

.

₍₁₀₎

For Hmax - H:l

<

Hext, we take xo = 0. For Hm, - 2H;

<

Hext

<

Hm, - Hzl, we take xo =

A' 1n [(Hmax

-

Hext)

/

H L ] .

Had a decreasing field been reversed, (n) would have been reduced t o the same extent. This is precisely what is observed in the planar cuprates [5] and in con-

ventional type I1 superconductors [6], supporting the

assumption that the driving force for fluxon depinning is the critical current. .

Critical field d i s t r i b u t i o n

The concentration of mobile fluxons may be computed from (10) as a function of (H,,

-

Hext)

/

H>

for various values of A'

/

X L . In the limit A'

>>

X L the

computed trough has a triangular shape with a discon- tinuity in slope a t Hext = Hm,

-

Hzl. The observed

troughs [5] initially decrease linearly and then slope

gradually back t o f = 1. The likely explanation is that

the relaxation distance A' is not much larger than the

penetration depth X L . An additional possibility is that

there exists a distribution in Hzl, which can in princi-

ple be extracted from the experimental results. With a distribution of threshold fields f ( H z l ) , (8)

becomes

For a unique threshold field, f ( H z l ) is a triangular function with a second derivative that is a delta function a t Hext = Hmax

-

Hzl

The second derivative of (11) is then

d2 (n)

/

d~~ =

( e l

X'VO)

f

( H A ) . (13)

Curves off (H;l) show a threshold field that decreases

with increasing Hext. At moderate fields, A' is expected

t o decrease t o a fixed granular distance in the micron range with any further decrease in H: a consequence

of a reduction in the critical current density J, a t higher fields.

The observed field dependence of f ( H z ) is sugges- tive of the spin-glass structure expected of weakly cou- pled granular superconductors [7]. The field dependence of f ( H z ) for a bulk superconductor should be

quite different since the critical current density is rel- atively fixed in this field range while the relaxation distance A' may be expected t o increase because of

increased rigidity a t higher fluxon densities [8].

Acknowledgments

We gratefully acknowledge helpful conversations with our colleagues, K. W. Blazey, I. Morgenstern, K. A. Miiller and Ch. Rossel.

[ I ] Blazey, K. W., Miiller, K. A,, Bednorz, J . G.,

Berlinger, W., .4moretti, G., Buluggiu, E., Vera, A. and Matacotta, F. C., Phys. Rev. B 36 (1987) 7241.

[2] Khachaturyan, K., Weber, E. R., Tejedor, P., Stacy, A. M. and Portis, A. M., Phys. Rev. B 36 (1987) 8309.

[3] Portis, A. M. Blazey, K. W., Miiller, K. A. and

Bednorz, J. G., Europhys. Lett. 5 (1988) 467. [4] Blazey, K. W., Portis, A. M. and Bednorz, J. G.,

Solid State Commun. 65 (1988) 1153.

[5] Warden, M., Portis, A. M., Stalder, M., Stefan-

icki, G. and Waldner, F., this conf.

[6] Walton, B. L., Rosenblum, B. and Bridges, F., Phys. Rev. Lett. 32 (1974) 1047;

Walton, V. L. and Rosenblum, B., Low Tempera- ture Physics, LT-19, Eds. W. J. OISullivan, K. D.

Timmerhaus and E. F. Hammel (Plenum, New York) 1973.

[7] John, S. and Lubensky, T. C., Phys. Rev. B 34

(1986) 4815;

Ebner, C. and Stroud, D., Phys. Rev. B 31

(1985) 165.

[8] Campbell, A. M. and Everts, J. E., Critical Cur- rents in Superconductors, Adv. Phys. 2 1 (1972) 199. Reprinted in Taylor and Francis Monographs