Barrier noise and flux creep in high-Tc superconductors

Barrier noise and flux creep in high-Tc superconductors

Volume 139, number 3,4 PHYSICS LETTERS A 31 July 1989 BARRIER NOISE AND FLUX CREEP IN HIGH-T. SUPERCONDUCTORS E. ~IMANEK Department of Physics, Uni...

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Volume 139, number 3,4

PHYSICS LETTERS A

31 July 1989

BARRIER NOISE AND FLUX CREEP IN HIGH-T. SUPERCONDUCTORS E. ~IMANEK Department of Physics, University of California, Riverside, CA 92521, USA Received 29 March 1989; revised manuscript received 24 May 1989; accepted for publication 25 May 1989 Communicated by A.A. Maradudin

We calculate the rate ofthermal activation offlux lines over a barrier exhibiting time-dependent fluctuations of the activation energy. Strong deviations from the linear T-dependence of the rate are predicted in the low temperature region, where the barrier fluctuations are of quantum nature.

Shortly after the discoveryofthe new class of highT~superconductors [1], MUller et a!. [2] reported glass-like time and history dependent phenomena in the Ba—La—Cu—O ceramics. Subsequently, Mota et al. [3,4] have observed logarithmic time decays of the magnetization in Sr—La—Cu—U and Ba—La—Cu— o specimens in sintered and powder form. The logarithmic rate of the decay is found to exhibit interesting behaviour as a function of temperature: For 2 K < T< 10 K the rate is approximately linear with temperature, but for T< 1 K a much weaker T-dependence is observed [3,4]. Recently, we have pointed out a possible explanation of this low-ternperature anomaly [5]. Following ref. [21, we assume that the glassy features result from the formation of a hierarchy of current loops, consisting of weakly coupled superconducting grains [61. Furthermore, we assume that there are local fluctuations of the normal conductance, which in turn produce critical current noise of the weak links. If the intensity and correlation time of this noise is nearly T-independent, the calculation of ref. [5] shows that the rateof the phase-slips in the Josephson junctions deviates from the Arrhenius law at low temperatures in qualitative agreement with the experiment [3,4]. The time-logarithmic relaxation of the permanent magnetization has also been observed in the 1:2:3 Y— Ba—Cu--O single crystals [7—10].Yeshurun and Malozemoff [10] have interpreted the relaxation rate results in terms of the classical flux creep mechanism [11,12] applied to an extended version of the crit-

ical state model [131, involving field dependent critical current. Measurements ofthe relaxation ratehave been done only for T> 4 K [91 but the data do not exclude the possibility ofa low temperature anomaly similar to that seen in refs. [3,4]. In the present note we describe a calculation of the magnetization decay which also exhibits at low ternperatures a deviation from the linear T-dependence. The idea is to incorporate into the flux creep model of ref. [101 time-dependent fluctuations of the energy barriers. For a granular superconductor the origin of such fluctuations is easy to understand. If we consider, for example, a two-dimensional array of junctions, the energy barrier which a vortex has to overcome, as it moves between the elementary cells, is a fraction of the Josephson coupling energy [141. Then the barrier fluctuations result from the normal conductance noise for which there is a convincing experimental evidence [151.In a single crystal, the data of ref. [161 suggest an intrinsic flux pinning to microscopic defects such as impurities or oxygen Vacancies. These defects presumably contain localized electron levels (traps) with fluctuating occupation number [171. Since the vortex core is only 10—20 A large the pinning energy can be sensitive even to single electron changes at the trapping sites. We consider the diffusive motion of a flux-bundle of cross-section L2 and length L jumping over the barriers between pinningcenters separated by L [181. The phenornenological equation of motion for the flux-bundle in the direction x is

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Volume 139, number 3,4

PHYSICS LETTERS A

M~+i~5c=2aL3—(mU0/L)sin(2xx/L),

where (1)

where M and ~ are the mass and viscosity of the flux bundle and (]~/2is the amplitude of the pinning potential, which of is assumed periodic. Thegiven coefficient is the gradient the magnetic pressure by [181a ä(h/4it)

a=B

BJ =—

(2)

~,

where h is the local field and J is the current density. Assuming that the flux-lines are overdamped and introducing a dimensionless bundle-coordinate y= 2itx/L, we obtain from eq. (1) the following Langevin equation, 3—(itU

2it

v= —F

[2aL



3— (irU [aL 0/L) sin 2v+ sin

2it K(y) =

Q=J

+

~Qsin 2, (8a)

dt.

(8b)

Following the high-barrier approximation of Kramers [201 (valid for ~7~>> k 11T), we calculate from eq. (7) the “down-hill” and “up-hill” jump rates r~ and r_, respectively [5], 2v —

(+Qsin

1)

4it

2e”~’”~,

x [IH” (Yt) IH” (y,~)]“

(3)

viscousF(t) where dissipation, is the random presumably force associated due to thewith normal the currents. Its correlation function is assumed in the form

y]

,

K2(y) = Q

r~=

0/L) sin y]+F(t)

1(t) siny,

31 July 1989

r_ =

4it —

(9a)

2v)

(+Q sin

X [I H”

(Yt)

H” (y,,,)]

~

(9b)

j~,,,) ,

where Yi is the initial bundle-position at a minimum

=ó(t1 —t2) where

(4)

2kBT

8lt

~

,

of the pinning potential, ym andym are positions of the adjacent maxima in the “down-hill” and “up-hill” directions, fined by respectively. The function H(y) is de-

(5)

.

K

The last term in eq. (3) comes from the barrier fluctuations u0(t)= U0(t) U0, where U0 is the time average of the barrier energy. The correlator of the random force F1(t) is

H(y) =2

J

1(y)

dYKU.

(10)



4~

‘~

(6) Eqs. (3 )—(6) are formally similar to those describing an overdamped Josephson junction in the presence of the critical current noise [5]. If the correlation time of F1 (t) is much larger than the bundle relaxation time M/ij, one can apply the Stratonovich method [19] to derive a Fokker—Planck equation for the probability distribution P(y, t): öP(y, t)

at

[K1(y)P(y, t) I

— — —

(9b) a perturbation expansion in r~_~waexp{(Uo/2knT) x [A+(p) — (Q/)N±(p)

Q [5],

]}

,

(1 la)

( 1 0/2kBT) (Q/ )N_ (p) 1 }

,

(I lb)

7

r_

W~,exp{

x [A (p)



where Wa is the attempt frequency w~/2i~~, W0 being the oscillation frequency about the minimum of the 4/irU tilted The p=2aL functions A±(p) and N+ (p),pinning of the potential. “tilting” ratio 0, are defined as follows,

2 +~~—~[K 1 d

2)~2,

2(y)P(y,t)],

184

Approximate, but transparent results for the flux creep rate can be obtained by applying to eqs.(9a).

(7)

A±(p)=p(~—2sin~p)—2(l—p

:

Volume 139, number 3,4

—2 sin ‘p) (2~~2)(l~~2) l/2

N~(p) —

PHYSICS LETTERS A

= ~p(it



+

~(1 p2)

3/2

(12)

and

31 July 1989

critical state model the current density is spatially constant and proportional to the critical current J~ [13]. Flux creep produces a time-decay ofJ~,which according to eq. (17) is given (for large t and small T) by [11,12]

A_(p)=A÷(p)—27t,

Jc(t)Jeff[1(1CBT/t~eff) ln(t/1

0)]

(13)

N_(p)=N÷(p)—ir.

,

(19)

The expressions (12) can be8) approximated overPa by a simple linear large region of p (O~p~O. dependence,

where lit0 Wa. Inserting eq. (19) into eq. (1) of ref. calculate (from andmagnetiza(3) of ref. [10])[10] thewe logarithmic decayeqs. rate(2) of the tion in a cylinder of radius r,

A÷(p)~tp—2,

dM R(T)= dlnt

=

~5(H”~2—H~2)kBT

(14)

Using eqs. (13) and (14) in eqs. (11) we obtain r~ Wa exp{

x [—2+4Q/3c+ itp(l —Q/2c)]}, Wa

<~H
(20)

(U 0/2k~T)

r_

for ~

(15a)

where JJ* is the field for which the currents first penetrate the entire sample, H~1is the lower critical field and n~1 is a phenomenological power in the h-de-

(15b)

pendence of J~.Inserting the expressions (18a), (18b) into eq. (20) we see that the barrier noise modifies the simple linear T-dependence ofRQo(T)

exp{( Uo/2kB T)

x [—2+4Q/3— ~tp(l Q/2)]}. —

In keeping with the standard flux creep theory [11,18] we assume that gradient of the magnetic

by a factor R(T)

pressure a is so large that the following inequality holds for the p-proportional terms in expressions

RQO

(15):

The explicit expression for Q/2 follows from eqs. (5), (6) and (8b), Q it2S~ (22) 2 4i~L2k~T’

4 aL kBT

itU0p 2kBT

(16)

0= v=L(r÷—r_)~Lr÷ LWaexp[Ueff/kBT(l_Jeff)]

,

(17)

where eqs. (2) and (16) are used to express p in terms of J. The parameters Ueff and jeff are the energy barrier and critical current renormalized by the barrier noise, respectively. On comparing eqs. (1 5a) and (17) we have (l8a)

UeffUO(12Q/3f),

(1 —2Q/3) i—Q/2



2Q/ 3) 2

(21)

where S

Jco

(1

—=

This implies that r~>> r_ and the net creep velocity in the direction of the magnetic pressure gradient can be written as

jeff

( T)

l—Q,/2



~8hi~

where J~0is the critical current in the absence of thermal activation and barrier noise as well. In the

5

dt

(23)

is the intensity of the barrier noise. We now apply these results to the relaxation ofthe zero-field-cooled magnetization in an Y—Ba—Cu—O crystal for field parallel to the orthorhombic c-axis [9]. The electron number at the pinning site is assumed to fluctuate between 0 and 1, resulting in a local barrier potential which can be approximated by a random telegraphic signal. The correlator of the “global” pinning energy of the whole bundle of n fluxlines is according to eq. (23)

s0 = 2 u~ r, = 2 U~t’n

,

(24)

1=l

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Volume 139, number 3,4

PHYSICS LETTERS A

where t, is the correlation time of the local pinning barrier acting on the ith flux line and f is its bundleaverage. U~is the time-averaged pinning energy per site. For the viscosity coefficient we use the expression [21] nLçb~ ‘1 21tc2~hpa/,

(25)

where øo is the flux quantum, ~ab and Pab are the coherence length and resistivity in the ab-plane, respectively. From eqs. (22)—(25) we have

Q



2



21t~a2bPahC2U?

Ø~L3kBT

(26)

For sufficiently low temperatures the local charge fluctuations at the pinning sites enter into the quanturn regime and the spectral density S 0 of eq. (24) becomes T-independent [17]. Consequently, in this region, the expression (26) can be written as T0/ T where T0 is a constant proportional to the zero temperature limit of the product ~bPab U~.Then the Tdependence of the decay rate of eqs. (20) and (21) can be described at low T by the expression R ( T)

T—T0 2 (1—4T0/3 T)

(27)

______________

As the temperature is lowered, R ( T) develops a minimum at T~3.1 T 0 followed by a divergence at T=4T0/3 and a rapid drop to zero taking place at T= T0. For T< T0 the rate assumes negative values. This unphysical negative rate and the above divergence need to be taken seriously because of the breakdown of the perturbation theory for temperatures of the order of T0. Nevertheless, the rate is expected to show considerable departure from the linear T-dependence at low temperatures. The data of ref. [9]show a qualitative agreement with this prediction: A distinct change of the slope of the rate R ( T) is observed near T= 10 K. For an orientational estimate ofT0 The we use ~ab= dis16 0)25 j.t~ cm [22]. mean A andofPab(T tance the pinning centers L is not well known and we take L= 300 A, which is probably an overestimate, in view of the fact that there may be several percent of randomly distributed vacancies in the copper—oxygen planes [23]. The pinning energy per defect is given by the condensation energy in the vol186

31 July 1989

ume of ~ which has been estimated in ref. [9] to be about 0.15 eV. Inserting these parameters into eq. (24) we obtain T0=QT/2=3xlO’°t(K/s)

(28) 1°s which is With T0~3K, eq. (28) t=10 shorter cornot easy to reconcile withimplies considerably relation time, expected for the purely electronic hybridization model of ref. [17]. Presumably the defect trapping processes involve configurational ionic transitions, which at low temperatures are driven .

predominantly by quantum tunneling. Such processes have been recently identified in the conductance noise studies in submicron metal—insulator— metal junctions [241. They also have been invoked [5] to explain the low-temperature anomaly in the magnetization decay rate of high-Tv superconducting ceramics [3,4]. The latter materials are best described by the model of a superconducting glass, in which different metastable flux states are separated by energy barriers, which have to be jumped before the system reaches thermodynamic equilibrium [25]. Thus we expect that barrier noise will modify the magnetization rate sults derived here. It glass is interesting the to logarithin a superconducting in a way that similar the remic decay rate obtained recently by Mota et al. [4] in the Sr—La—Cu—O powder actually exhibits a mmimum at T—~1 K, predicted by eq. (21) of this paper. I wish to thank Elga Pakulis for a valuable discussion. I am also indebted to the referee of this paper for important suggestions, especially for pointing out the relevance of ref. [101 for this work.

References [1] J.G. Bednorz and K.A. MUller, Z. Phys. B 64 (1986) 1989. [21 K.A. M. 1143. Takashige and J.G. Bednorz, Phys. Rev. Lett. Muller, 58 (1987) [3]A.C. Motta, A. Pollini, P. Visani, K.A. MUller and J.G. Bednorz, Physica C 153—155 (1988) 67. [4]A.C. Motta, A. Pollini, P. Visani, K.A. Muller and J.G. Bednora, Phys. Rev. B 36 (1987) 4011. [5]E. ~imánek, Phys. Rev. B 39 (1989), in press. [6]C. Ebner and D. Stroud, Phys. Rev. B 31(1985) 165. [7] T.K. Worthington, W.J. Gallagher and T.R. Dinger, Phys. Rev. Lett. 59 (1987) 1160.

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[8] W.J. Gallagher, T.K. Worthington, T.R. Dinger, F. Holtzberg, D.L. Kaiser and R.L. Sandstorm, Physica B 14 (1987) 228. [9] Y. Yeshurun and A.P. Malozemoff, Phys. Rev. Lett. 60 (1988) 2202. [10] Y. Yeshurun, A.P. Malozemoff, F. Holtzberg and T.R. Dinger, Phys. Rev. B 38 (1988)11828. [11] P.W. Anderson, Phys. Rev. Lett. 9 (1962) 309. [12] A.M. Campbell and J.E. Evetts, Adv. Phys. 21(1972)199. [13] C.P. Bean, Phys. Rev. Lett. 8 (1962) 250. [14] C.J. Lobb, D.W. Abraham and M. Tinkham, Phys. Rev. B 27 (1983) 150. [15] B.W. Ricketts, R. Driver and H.K. Welsh, Solid State Commun. 67 (1988) 133. [16] G.J. Dolan, G.V. Chandrasekhar, T.R. Dinger, C. Field and F. Holtzberg, Phys. Rev. Lett. 62 (1989) 827.

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[171 G.M. Eliashberg, Pis’ma Zh. Eksp. Teor. Fiz. 46 (1987) 94 [JETP Lett. 46 (1988) S81]. [18] M. Tinkham, Introduction to superconductivity (McGrawHill, New York, 1975). [19] R.L. Stratonovich, Topics in the theory of random noise (Gordon and Breach, New York, 1963). [20] H.A. Kramers, Physica 7 (1940) 284. [21] J. Bardeen and M.J. Stephen, Phys. Rev. 140 (1965) A1197. [22] S.J. Hagen, T.W. Jing, Z.Z. Wang, J. Horvath and N.P. Ong, Phys. Rev. B 37 (1988) 7928. [23] J.W. Halley and H.B. Shore, Phys. Rev. B 37 (1988) 525. [24] C.T. Rogers and R.A. Buhrman, Phys. Rev. Lett. 53 (1984) 1272. [25] J. Choi and J.V. José, Phys. Rev. Lett. 62 (1989) 320.

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