LT 21 Proceedings of the 21st International Conference on Low Temperature Physics Prague, August 8-14, 1996

Part $2 - Superconductivity 1: HTS - Theory

P R O P A G A T I O N O F H E A T E D D O M A I N S IN H T S C F I L M S

J. Le6n', E. Holguin ~ and J. F. Loude b

"Departamento de Fisica, Universidad del Cauca, Popayfin, Colombia

blnstitut de Physique Nucldaire, Universitd de Lausanne, 1015 Lausanne, Switzerland

A beam of laser radiation incident on the surface of a metallic fihn which is in thermal contact with an undercritieal current-carrying HTSC film, produces a localized destruction of superconductivity in the interface, which then propagates along all the CuO2-planes of the superconductor. Using a quasi-stationary approximation of the heat equation, we have developped a theoretical model which shows that the corresponding I-V characteristics are due to the flow of the current through a large number of time-varying resistors in parallel.

1. INTRODUCTION The High-Tc superconductors have a

common distinctive feature: a layered structure composed of CuO2-planes, sandwiched between insulating phases (for instance, of antiferromagnetic type). Here we investigate theoretically the propaszfion of heat along thin films made of these materials.

ourselves to the case of quasi-stationary processes at low regimes [1], the conductive heat loss between a CuO2-plane and the adjacent insulating planes will follow the law of Newton; furthermore, since d>>do, we may assume, as a first approximation, that the insulating planes remain at the temperature "lb.

2. MODEL Fig. 1 shows the geometry of a HTSC film

deposited on an isolating substrate, at both extremities in thermal contact with metallic films (MF) used as electric contacts, and immersed in a bath at temperature Tb. A small undercritical current l<<]dTd flows through the HTSC film, where I~ is the critical current. The film of width w has a layered structure, where CuO2-planes alternate with insulating planes, respectively of thickness do~lnm and d=10nm. A beam of laser radiation is applied on the surface of left MF, producing a step-like localized increase of the temperature. Since the thickness is small, the temperature reaches almost simultaneously the same value To in all the CuO2-planes of the interface. If To>T~, superconductivity is thermally destroyed and a normal phase (N) begins to expand along each CuO2-plane of the HTSC fihn.

We may hence write the heat equation in one dimension, with x along the length of the film and the origin in the interface MF-HTSC. Limiting

........................... .ili!!iiiiiV ,

/J/ o

. , ~

i~i~ ~:iEiEiE::i~E~ ~:':'.:.:':':::'::ii~!i i ! i i i I N S U L A I O l l T b / / / /

Fig. 1 Geometry ofa HTSC film used to study the propagation of heat along the CuO2-planes

Under these conditions, we may consider that inside an expanding N-phase, the temperature is given by

KN ~ 2 2-~o[TN(X,t)-Tb]=O ( l )

where KN and k are respectively the coefficient of

Czechoslovak Journal of Physics, Vol. 46 (1996), Suppl. $2 999

thermal conductivity in the CuO2-plane and tile heat transfer coefficient between the CuOrplanes and the adjacent insulating planes. If f(O is the position at time t of the front of propagation at temperature T~, eq. (1) admits a quasi-stationary solution of the form

_,,, . , ~ . ~.~ sinh(anx)

sinh{mv [x - f( t)] } x (2)

sinh[a ̂ , f(t)]

where a~=(2"k/Kl~lofl/e. A similar equation gives the spatio-temporal distribution of the temperature in the remaining superconducting phase (S), which extends for x>f(t)with Tb_<Ts(x,O<Tr Then

Ts(x , t )= T b + ( T c - T b ) e x p { - a s [ x - f( t)]} (3)

--/v'K-- .t/2 where a,=tz ,ao/ . On the other hand [2], the velocity v(t) of thermal expansion is given by the absorption rate of latent heat L at the N-S interface

c~Ttr (x,t) tgTs(x,t) - Krr + Ks = Lv(t) (4)

Then, using eqs. (2) to (4), we get

(Tb - To) cosh[ax f( t ) ] + (T0- Tb) v(t) = {a~ K~,

sinh[aN f( t ) ]

- a s K s ( T c - Tt,)} L -l (5)

Assuming the same thermal properties (i.e. KN~Ks, thus implying aN-~as,) for tile N and S regions, we integrate the position f(O from eq. (5). Hence

exp(ctN X0 ) - 1 2sinh(txN Xo ) In{ exp(a^, x o) - exp[atr f ( t ) ] }

a N f ( t ) exp[aNf ( t ) ] - 1 =T

exp(a ~r x o) exp[a N f( t)] (6)

where r=2(otg)2KM'l'o-Tb)t/L is a reduced time and Xo the position at which stops the thermal expansion with velocity v(t)=O. We calculate

1 ~ - ~ xo= (7)

It must pointed out that this point also belongs to the temperature distribution curve T(x) of a non superconducting material with same thermal properties, when one of its extremities is heated at To. Eq. (6) shows that the time necessary to reach the stationary situation, i.e. f(t)=xo, is infinite.

To measure the expansion of these normal phases, we place two widely separated potential probes P~ and P2 along the film with P~, for instance, in the MF-HTSC interface. Since all the CuO2-planes of the HTSC film form an electric network of n resistors in parallel, the difference of potential u(t) is

f ( t ) ] u(t) (s)

where p~ is the normal-state resistivity of a CuO2- plane. The quantity between brackets represents the time variation of a single CuO2-plane resistance during the thermal expansion.

~.~f(tl

~N X 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

, T

Fig.2 Position at time t of the front f(t) of heat propagation at Tc expressed in reduced quantities

For typical HTSC films of thickness about lgm, we have n=100. In Fig. 2, we show the reduced quantities az~f(t) vs. r.

3. CONCLUSIONS According to the present theoretical work,

the layered structure of the High-To superconductors exerts a determinant influence on their thermal properties. We are designing experiments to test this hypothesis.

REFERENCES [I] N. Amaioua, E. Holguin and J.F. Loude, J. Low

Temp. Phys. 102(1996)157. [2] W. C. Overton, J. Low Temp. Phys. 5(1971)

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1000 Czech. J. Phys. 46 (1996), Suppl. $2