Physics of the Solid State, Vol. 45, No. 3, 2003, pp. 409–413. Translated from Fizika Tverdogo Tela, Vol. 45, No. 3, 2003, pp. 385–389. Original Russian Text Copyright © 2003 by Artemov, Medvedev. METALS AND SUPERCONDUCTORS Phase Diagram of a Wide Current-Carrying Superconducting Film A. N. Artemov and Yu. V. Medvedev Donetsk Physicotechnical Institute, National Academy of Sciences of Ukraine, Donetsk, 83114 Ukraine e-mail: [email protected] Received December 14, 2001; in final form, May 18, 2002 Abstract—A model of current heating of a wide superconducting film is suggested assuming nonlinearity of the film conductance. Within this model, the parameters of the film can be characterized by a single dimension- less coefficient of thermal balance which includes both thermal and resistive parameters of the system. The sta- bility of the state of the current-carrying film is analyzed. A phase diagram of the film is constructed in terms of the coefficient of thermal balance and the average current density. The propagation velocity of a stationary nonuniform temperature distribution in the film is calculated as a function of current density at various values of the coefficient of thermal balance. © 2003 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION resistive region in order to characterize the power of A current-carrying type-II superconductor is a heat source Q˙ (J, T) in the conductor. For composite rather complex object. Depending on temperature and superconductors and narrow superconductive films, current density, it can exhibit a superconducting, resis- this analysis is commonly simplified by using a one- tive, or normal state. Its states can also be homogeneous dimensional heat conduction equation under the and inhomogeneous, as well as stationary and nonsta- assumption of linear surface heat removal [W(T) = tionary. Experimental and theoretical studies of the h(T – T0), where h is the coefficient of heat transfer and states of the superconductor, their stability, and the T is the temperature of the thermostat] and of the resis- physical processes responsible for these states are 0 tivity being a step function of temperature [1]. How- widely discussed in the literature [1]. However, many aspects of the behavior of a superconductor with a cur- ever, in high-temperature superconductors with a lay- rent remain unstudied. One of such topical problems is ered structure, a topological phase transition can occur, related to current-carrying wide superconductive films. which is inherent in two-dimensional systems. In this case, it is important to take into account the nonlinear Thermal or electromagnetic influences on type-II dependence of ρ on current density J. Nonlinearity of superconductors can stimulate the occurrence of self- the function ρ(J) in highly anisotropic layered systems maintained nonuniform dissipative regions in them [1– is caused by internal physical processes, namely, by 6]. The evolution of these regions manifests itself in the variety of current–voltage (I–V) characteristics (see, dissociation of fluctuation-induced pairs of the vortex– e.g., [4]) and may result in the transition of the whole antivortex type under the action of a current. Nonlinear- sample into resistive and normal states or in the restor- ity determines the condition for the existence and sta- ing of the dissipative state. Such transitions are accom- bility of uniform and nonuniform stationary states [11, panied by the propagation of “switching waves” 12]. through the sample. The calculation of the velocities of A similar mechanism of nonlinearity can be these waves allows one to describe, under particular observed in wide superconducting films carrying a cur- simplifying assumptions, the development of excita- tions in superconductors [7–10] in both the presence rent. The resistivity in such systems is caused by the and absence of a magnetic field. In the latter case, the motion of magnetic vortices penetrating through the film edge under the action of the Lorentz force. The flux instability of the superconducting state is related to the ρ nonisothermal dynamics of magnetic vortices penetrat- flow resistivity f, responsible for the resistive losses, ing into the sample due to the magnetic field created by depends on the concentration of vortices nv in the ρ ρ πξ2 the current. superconductor: f ~ n nv. The concentration of vor- In the general case of nonisothermal dynamics of tices surmounting the edge barrier and entering the magnetic flux, analysis of the destruction (restoration) sample is a function of the current density flowing in of the superconducting properties of a current-carrying the sample. This dependence is the main cause of the element requires the use of the equation of thermal bal- nonlinearity of resistive losses in wide superconducting ance and the model description of the resistivity ρ of a films. 1063-7834/03/4503-0409 $24.00 © 2003 MAIK “Nauka/Interperiodica” 410 ARTEMOV, MEDVEDEV The authors of [2] considered the kinetics of flux All calculations of thermal processes will be made flow in wide films and their I–V characteristics and by ignoring the heating nonuniformity across the sam- found that the time-averaged longitudinal electric field ple width caused by the nonuniformity of current distri- E in a resistive region is a nonlinear function of the bution [13]. This means that the current density in average current density in the film (in the absence of Eq. (3) is averaged over the cross section of the film. vortex pinning) and has the form For a large resistive region, this approximation is quite ρ (), justified, because the current has enough time to be E = n Jf J T , (1) redistributed and become uniform over the region; the fJT(), = ()1 – J /J /1[]– J /J + AJT(), , (2) heat production, being a function of the current, also c c becomes almost constant along the vortex path. where ρ is the resistivity of the film in a normal state, n ˙ Jc is the average density of the critical current Ic for the The function Q (J, T), describing the heat produc- α onset of dissipation in the film, A(J, T) = (Jc/J)[(Tc – tion, is a local function and depends on the current den- 3/2 α sity and temperature at a given point of the film. We are T)/Tc] , is a dimensionless constant of the order of unity, and T is the critical temperature of the supercon- interested in solutions to Eq. (3) that describe inhomo- c geneous states. In this case, superconducting, resistive, ductor. and normal states can simultaneously exist in the film. The present article is devoted to the study of the evo- ˙ lution of stationary states of a wide superconducting The function Q (J, T) accounts for these possible states; film associated with the process of heat production due it is a piecewise continuous function and can be written ˙ ρ 2 to dissipative effects of viscous motion of magnetic as Q (J, T) = nJ f(J, T), where the function f(J, T) has flux. We use a model expression for the power of the the form heat source Q˙ (J, T) which generalizes the results obtained in [2]. fJT(), Within the model proposed, the phase diagram of ≥ 1, TTn the film is constructed to show the dependence of the (5) ()[](), ≥≥ film state on the average current density and on the = 1 – Jc/J /1– Jc/J + AJT , T n TTr coefficient of thermal balance, which includes both the 0, TT≤ . thermal characteristics of the substrate and the super- r conductor parameters determining the heat production power. In addition, nonuniform stationary solutions are Our definition of the nonlinear function f(J, T) is found, namely, superconducting–normal and supercon- somewhat different from Eq. (2) used in [2]. We take ducting–resistive phase switching waves (S–N and S–R into account that the film transforms into the normal waves, respectively), similar to those considered in state not at the temperature Tc, as assumed in [2], but at [11]. The propagation velocities of these waves as a Tn, where the average current density J becomes equal function of current density in the film are also found. to the dispairing current density JGL. For this reason, we redefine the quantity α in f(J, T) and assume it to be not 2. MODEL a constant but rather dependent on the current density. This quantity becomes equal to zero at the temperature If the vortices that have penetrated into the film Tn (which is always lower than Tc) and has the form α move perpendicular to the direction of the current and = 1 – J/JLG(T). The temperature of the transition of the thermal length is considerably larger than all char- the film to the resistive state Tr is determined by the acteristic lengths of the sample, then the state of the relation J = J (T ). superconducting film can be studied by using a one- c r dimensional equation of heat transfer: The model proposed contains many parameters describing the resistive properties of the superconduc- ∂T ∂2T D C ------ = D k --------- + d Q˙()JT, – WT(). (3) tor and the thermal characteristics of the substrate. In s s ∂ s s 2 f t ∂x order to derive universal dependences of the properties of our model on these parameters, we conveniently Here, Ds is the substrate thickness; Cs and ks are the spe- write the equations in a dimensionless form. For this cific heat and heat conductivity of the substrate, respec- reason, we introduce the dimensionless variables tively; df is the film thickness; and the x axis coincides with the current direction. The rate of heat removal is x τ ks assumed to be a linear function: l ==------, ------------2t, Ds C D () () s s WT = hT– T0 , (4) where h = k /D is the effective coefficient of heat θ T J s s ==----- , j ------------()- removal.