Substrate effects on thermal instability development in high Tc superconducting thin films with transport current

PII: S0011-2275(97)00159-8

Cryogenics 38 (1998) 277–282  1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0011-2275/98/$19.00

Substrate effects on thermal instability development in high Tc superconducting thin films with transport current N.A. Buznikov†, A.A. Pukhov*† and V.N. Skokov‡ †Scientific Center for Applied Problems in Electrodynamics, Russian Academy of Sciences, 127412, Izhorskaya 13/19, Moscow, Russia ‡Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, 620219, Pervomaiskaya 91, Ekaterinburg, Russia

Received 17 June 1997; revised 13 October 1997 The thermal instability development in a high Tc superconducting thin film carrying a transport current is studied both experimentally and theoretically. The normal phase propagation velocity ␯ is measured by means of both direct current and pulse current methods in a wide range of the transport currents I. It is shown that to explain the measured dependence ␯( I ) a non-linear heating over the thickness of a substrate, of which the reverse side is stabilized at a temperature T0, should be taken into account. The corresponding theory based on the solution of the non-stationary two-dimensional heat balance equation is developed and compared with experimental data. The results obtained may be of importance in the study of superconducting switching devices.  1998 Elsevier Science Ltd. All rights reserved Keywords: high Tc superconductors (A); thin films (A); propagation velocity (C)

Nomenclature Cf Cs Df Ds i I Ic kf ks L Q R t

Film heat capacity Substrate heat capacity Film thickness Substrate thickness = I/Ic, dimensionless current Transport current Critical current at T = T0 Film thermal conductivity Substrate thermal conductivity Film length Specific Joule heat release Film resistance Time

T Tc Tr T0 U v W x y z

Greek letters ␣ ␳

Introduction As is known (see, for example1,2 ), one of the possible mechanisms of the transition of a superconductor from the superconducting to normal state is the thermal instability development. The instability development results from the *To whom correspondence should be addressed.

Temperature Film critical temperature = T0 + [1 − (I/Ic )1/2 ](Tc − T0 ), resistive transition temperature Substrate reverse side temperature Voltage drop Normal phase propagation velocity Film width Longitudinal co-ordinate Transversal co-ordinate = x + vt, self-similar co-ordinate

= ␳I2c Ds/ksDfW2(Tc − T0 ), effective Stekly parameter of the film-substrate system Film resistivity in the normal state

propagation over the superconductor of the normal phase moving with constant velocity v. The normal phase propagation over a high Tc superconducting (HTSC) thin film has been studied both experimentally3–5 and theoretically6. The main peculiarities of the thermal instability development in the HTSC film are related to a non-uniform heating over the thickness of the substrate, of which the reverse side is stabilized at fixed temperature T0. In this connection, the normal phase propagation velocity v depends not on the

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Substrate effects on thermal instability development: N.A. Buznikov et al. film properties only but also on the substrate heat capacity, thermal conductivity and geometry. That is why experimental results on v obtained for HTSC films located on different substrates vary considerably3–5. In references7–10, the thermal instability development in the HTSC films temperature is assumed to vary linearly over the substrate thickness from the film temperature to the substrate reverse side temperature T0. This assumption allows the normal phase propagation to be analyzed by means of effective one-dimensional heat balance equation. However, this simple approach insufficiently describes the adequate heat removal from normal regions of the HTSC film to substrate. In general, for the correct analysis of the normal phase nucleation and propagation over the film, it is necessary to solve two-dimensional heat balance equation6,11,12, which allows one to take into account effects of the transversal heat propagation in the substrate. In this paper the normal phase propagation over an HTSC film carrying a transport current is studied both experimentally and theoretically. It is shown that the analysis based on the solution of the effective one-dimensional heat balance equation underestimates the normal phase propagation velocity v at sufficiently high transport currents I. To study the substrate effect on the thermal instability development, the theoretical model—taking into account a non-linear heating over the substrate thickness—is proposed. The dependence v(I) is found by means of the solution of the non-stationary two-dimensional heat balance equation describing the temperature distribution in the substrate. The analytical results obtained are in good agreement with experimental data.

Experimental Thin HTSC films were obtained by d.c. magnetron sputtering of a target with the stoichiometric composition YBa2Cu3O7 − x on SrTiO3 substrates4,13. The temperatures of the superconducting transition Tc ranged from 86 to 88 K, and the critical current densities were 109 ÷ 1010 A m−2 at 77 K. The temperature dependence of the film critical current Ic in the vicinity of Tc was approximated with a high accuracy by the power law Ic(T)⬀(1 − T/Tc )2. The method of lithography was used to form film bridges of width W = 1 mm and length L = 8 ÷ 10 mm. The thickness of the films Df were about 0.4 ␮m, and the substrate thickness Ds were 0.5 mm. Experiments were carried out in a gas cryostat without a direct contact of the film with liquid nitrogen. The substrate was located on the thick copper plate immersed into the liquid nitrogen (see Figure 1), that provided the stabil-

Figure 1 A sketch of the experimental arrangement: 1, copper plate; 2, SrTiO3 substrate; 3, YBa2Cu3O7 − x film; 4, current leads; 5, voltage taps.

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ization of the substrate reverse side at fixed temperature T0. Several two layer Ag-In strips deposited on the film surface by vacuum evaporation were used as potential taps. The tap arrangement is shown schematically in Figure 1. In experiments, the voltage drops U1, U2 and U3 across different parts of the film were measured. When transport current passed through the film, the normal phase nucleated at a ‘weak’ spot. Therefore, an increase of one of the values of U1, U2 and U3 was observed at first. For example, Figure 2 shows the typical dependencies of U1 and U2 on time t during the process of the superconducting state destruction in the film. It follows from Figure 2 that the transition to the normal state was sufficiently non-uniform over the film length. To determine v the pair of taps were chosen such that the normal phase nucleation took place out of these taps. The measurements of the normal phase propagation velocity were performed by means of direct current and pulse current methods. In the direct current method the normal phase propagation velocity v was estimated from the constant slope of the time dependence of the voltage drop U(t) across the potential taps. The value of v was determined from the following expression:

冉 冊 ∂U ∂t

I

=

␳I v WDf

(1)

where ␳ is the film resistivity in the normal state and (∂U/∂t)I = dU/dt is the slope of the dependence U(t) at a constant I. Equation (1) is applicable since the film length is much higher than its width, and the normal phase propagation over the film can be treated as a one-dimensional process. The dependence v(I) measured by using the direct current method is shown in Figure 3. In the pulse current method the current pulses of different shape were applied to the film. This method allows one to avoid the overheating of the film and, thus, can be used to study the thermal instability development up to almost the critical current. The rectangular pulses and the pulses with the linear current increase (triangular pulses) were used in experiments. The pulses were applied to the current leads of the bridges. Shown in Figure 4 are the time dependencies of the current (Figure 4a) and the voltage drop across the potential taps (Figure 4b and 4c) at different substrate reverse side temperatures T0 for the rectangular current pulse. Note that the pulse current method measurements differ from the direct current method measurements

Figure 2 Time dependencies of voltage drop across different parts of film. Curve 1, U1( t ); 2, U2( t ).

Substrate effects on thermal instability development: N.A. Buznikov et al.

Figure 3 Dependence of v on I. 䊏, experimental data; curve 1, Equation (7); 2, Equation (13). Ds = 5 × 10−4 m, Cs = 2 × 106 J m−3 K−1, ks = 10 W m − 1 K − 1, Df = 4 × 10 − 7 m, W = 10−3 m, Tc = 87 K, T0 = 78 K, ␳ = 5 × 10−7 ⍀m, Ic = 1.7 A.

by a short time of the pulse duration (~100 ␮s). The nonlinear increase of the voltage drop observed in the initial part of the dependence U(t) is related to the stage of the normal region formation and growth. The normal phase propagation velocity v was determined from the part of the dependence U(t) having a constant slope by using Equation (1). Shown in Figure 5 are the time dependencies of the current (Figure 5a) and the voltage drop across the potential taps (Figure 5b and 5c) at different values of T0 for the triangular current pulse. The use of such pulses allowed one to determine the dependence v(I) by means of single voltage response U(t,I). The voltage drop U depends on t due to the normal region growth and on I due to the current pulse amplitude increase. So, the value of U(t,I) satisfies the following relationship:

冉 冊 冉 冊

dU ∂U ∂U = + dI ∂I t ∂t

dt dI

I

(2)

Here (∂U/∂I)t is the partial derivative of U with respect to I at a fixed normal region length. The value of (∂U/∂I)t was estimated by means of independent experiments with extremely short triangular current pulses, when the normal phase in the film has no sufficient time to form. The duration of these pulses was about 10 ␮s, and the time dependencies of the voltage drop were similar to the ones shown in Figure 5. The partial derivative (∂U/∂t)I was determined by the film resistance increase at a constant current due to the normal phase propagation, that is, by Equation (1). Finally, from Equations (1) and (2) we have for v v=

冋 冉 冊册

WDf dU ∂U − ␳I dI ∂I

t

dI dt

(3)

The dependence v(I) measured by means of the pulse current method is shown in Figure 6. The essential increase of v in comparison with the results shown in Figure 3 is largely related to that in these measurements, the film having the much higher resistivity was used. It should be noted

Figure 4 Time dependencies of current (a) and voltage drop across potential taps at T0 = 78 K (b) and at T0 = 80 K (c) for rectangular current pulse.

that in the experiments with triangular pulses some peculiarities of the thermal instability development may be expected due to the influence of hysteresis losses and possible appearance of the thermomagnetic instability14,15. However, a remarkable difference between results of the experiments with triangular pulses and with rectangular ones was not observed (see Figure 6). Moreover, the measured normal phase propagation velocity is practically independent of the current pulse rate dI/dt up to dI/dt = 1.5 × 104 A s−1. These facts may be attributed to that in the films under investigation, the thermomagnetic disturbances in the superconducting state induced by varying current do not affect the normal phase propagation.

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Figure 6 Dependence of v on I. 䊏, experiment with triangular pulse (dI/dt = 1.1 × 104 A s−1 ); 䊐, experiments with rectangular pulses; curve 1, Equation (7); 2, Equation (13). Ds = 5 × 10−4 m, Cs = 2 × 106 J m−3 K−1, ks = 10 W m−1 K−1, Df = 4 × 10−7 m, W = 10−3 m, Tc = 87 K, T0 = 80 K, ␳ = 5 × 10−6 ⍀m, Ic = 1.52 A.

冉 冊

CsDs ∂T ∂ ∂T ks = Ds k − (T − T0 ) + Q(T)Df 2 ∂t ∂x s ∂x Ds

(4)

where Cs and ks are the substrate heat capacity and thermal conductivity, respectively; x is the longitudinal co-ordinate in the direction of transport current and Q(T) is the specific Joule heat release. In Equation (4) it is taken into account that for typical relation between parameters kfDsÀksDf (here kf is the film thermal conductivity) the film temperature is uniform over its thickness, and the factor 1/2 in the left-hand side of Equation (4) appears due to the thermal stabilization of the substrate reverse side10. The specific heat release Q(T) may be written in the form1,2 Q(T) = Figure 5 Time dependencies of current (a) and voltage drop across potential taps at T0 = 78 K (b) and at T0 = 80 K (c) for triangular current pulse.

Theory One-dimensional analysis It follows from Figure 3 and Figure 6 that the measured dependencies v(I) are similar to ones in low-temperature composite superconductors1,2. To describe adequately the experimental data it is necessary to take into account the heat removal from the film to substrate. In this section we consider a simplified approach in the framework of which the temperature is assumed to vary linearly over the substrate thickness from the film temperature T to the substrate reverse side temperature T07,8,10. Since Df¿Ds, the heat capacity and thermal conductivity of the film-substrate system are determined by the substrate properties only, and the temperature distribution along the film can be described approximately by the following effective one-dimensional heat balance equation8,10:

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␳I2 ␩(T − Tr ) W2D2f

(5)

where Tr = T0 + [1 − (I/Ic )1/2 ](Tc − T0 ) is the resistive transition temperature; Ic is the critical current at T = T0 and ␩(x) is the Heaviside stepwise function ( ␩ = 0 at x ⬍ 0 and ␩ = 1 at x ⱖ 0). It follows from Equations (4) and (5) that the temperature distribution in the moving with the normal phase coordinate system (z = x + vt) satisfies the equation Ds +

冉 冊

d dT CsDs dT ks − ks −v (T − T0 ) dz dz 2 dz Ds

(6)

␳I2 ␩(T − Tr ) = 0 W2Df

with boundary conditions T = T0 at z→ − ⬁ and T = T0 + ␳I2Ds/ksW2D2f at z→⬁. Assume for simplicity, that the dependencies of the substrate heat capacity and thermal conductivity on temperature can be neglected. Then, Equation (6) can be solved analytically1,12 which gives for the propagation velocity

Substrate effects on thermal instability development: N.A. Buznikov et al. v=2

␣i2 + 2i1/2 − 2 ks CsDs √( ␣i2 + i1/2 − 1)(1 − i1/2 )

(7)

Here ␣ = ␳I Ds/ksDfW (Tc − T0 ) is the effective Stekly parameter of the film-substrate system and i = I/Ic is the dimensionless current. In the adiabatic approximation ␣À1 Equation (7) simplifies and has the form 2 c

v=2

ks √␣i CsDs √1 − i1/2

(8)

Two-dimensional analysis To consider the fast normal phase propagation more adequately it is necessary to take into account two-dimensional heat propagation effects resulting in a non-linear heating over the substrate thickness. In this connection, film and substrate should be treated separately as two coupled thermal subsystems. At kfDsÀksDf the temperature distribution in the HTSC film satisfies the one-dimensional heat balance equation

冉 冊 冉 冊|

∂T ∂ ∂T = Df kf ∂t ∂x ∂x

+ Q(T)Df + ks

∂T ∂y

(9)

, 0 ⬍ y ⬍ Df

y = Df

where Cf is the film heat capacity and y is the transversal co-ordinate. The last term in the right-hand side of Equation (9) corresponds to the heat removal from the film to substrate. The temperature distribution in the substrate is described by the two-dimensional heat balance equation Cs

∂T ∂ = ∂t ∂x

冉 冊 冉 冊 ks

∂T ∂ + ∂x ∂y

ks

∂T , Df ⬍ y ⬍ Df + Ds ∂y (10)

Since Df¿Ds, the film thickness may be neglected, and in the self-similar co-ordinate system (z = x + vt) Equation (10) has the following form: ∂ ∂z

冉 冊 冉 冊 ks

∂T ∂y

|

=− y=0

␳I2 ␩(z), T兩y = Ds = T0 DfW2[1 − (I/Ic )1/2 ]

2

The calculated dependence (7) v on I is shown in Figure 3 and Figure 6 (curve 1). Figure 3 and Figure 6 demonstrate that the simple approach described above underestimates the values of v at sufficiently high transport currents. This fact is due to that at fast normal phase propagation the substrate is not warmed up entirely through its thickness in a time of the normal phase front passage, and the temperature distribution in the substrate differs significantly from the linear profile.

CfDf

ks

∂ ∂T + ∂z ∂y

ks

∂T ∂T = 0, 0 ⬍ y ⬍ Ds − vCs ∂y ∂z (11)

Equation (11) should be supplemented by the boundary conditions which is given by Equation (9) and by the condition of the thermal stabilization of the substrate reverse side. Taking into account that at Df¿Ds the first and second terms in Equation (9) may be neglected and using Equation (5), these boundary conditions may be written in the form

(12) The first equation in (12) means that far from the normal phase front (at ∂T/∂z = 0) the film is in the uniform superconducting (at z→ − ⬁) and normal (at z→⬁) state and the origin of co-ordinates is chosen so that T(0,0) = Tr. Thus, the normal phase propagation velocity v is determined as the eigenvalue of the hybrid Dirichlet–Neumann problem within the region 0 ⬍ y ⬍ Ds for two-dimensional Equation (11) with boundary conditions outlined by Equation (12). Assuming that the dependencies Cs and ks on T can be neglected, this problem can be solved through separation of variables12 that allows the following analytical expression for v to be obtained:

␣i2 + 2i1/2 − 2 = 2␣i2

冘 ␭ √v + (2v␭ k /C D ) ⬁

k=0

2

2 k

2

k s

s

s

2

(13)

where ␭k = ␲(2k + 1)/2. In the adiabatic approximation ␣À1 Expression (13) simplifies and by using the Euler– Maclaurin formula may be presented as v = (2/␲ )

␣i2 ks CsDs 1 − i1/2

(14)

A comparison of the dependence v(I) calculated by means of Expression (13) with experimental data is shown in Figures 3 and 6. It follows from Figures 3 and 6 that the experimental results are in good agreement with the ones obtained in the framework of two-dimensional analysis. A slight disagreement between theory and experiment with triangular current pulse at low values of I (see Figure 6) may be attributed to that in this current range, the nucleation and initial growth of the normal zone take place, and the measured propagation velocity does not correspond to its stationary value.

Conclusions In the present paper the normal phase propagation over an HTSC film carrying a transport current is studied. Specific features of the thermal instability development in the filmsubstrate system are related to a non-uniform heating over the thickness of a substrate playing the role of the ‘thermal reservoir’ for the film. The dependencies of the normal phase propagation velocity v on the transport current I are found by means of solution of both the effective onedimensional and two-dimensional heat balance equations. The dependencies obtained are shown to differ significantly at sufficiently high transport currents. This statement is especially clear from the comparison of Expressions (8) and (14) obtained in the particular case of the adiabatic approximation ␣À1. It follows from the one-dimensional analysis (Expression (8)) that at I ⬵ Ic the normal phase propagation velocity is v⬀(1 − √I/Ic )−1/2, whereas in accordance with the two-dimensional analysis (Expression (14)) the velocity increases more sharply as v⬀(1 − √I/Ic )−1. The dependence v(I) calculated by means of the two-dimensional analysis allows one to describe with a reasonable

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Substrate effects on thermal instability development: N.A. Buznikov et al. accuracy the experimental results. On the other hand, the one-dimensional analysis underestimates the values of v at sufficiently high transport currents. This fact has a clear physical sense and is due to that at fast normal phase propagation the substrate is not warmed up entirely through its thickness in a time of the normal phase front passage, and the temperature distribution in the substrate differs significantly from the linear profile. In conclusion, note that for more detailed descriptions of the normal phase propagation over an HTSC film many additional factors should be taken into account (e.g. the dependence of the film and substrate parameters on temperature, the non-uniform current distribution over the film width). Nevertheless, as shown in the present paper, the simple analytical expressions obtained are convenient to estimate the normal phase propagation velocity and allow experimental data to be described with a reasonable accuracy.

Acknowledgements This work was supported in part by the Russian State Program ‘Actual Problems of Condensed Matter Physics’, subdivision ‘Superconductivity’ under Project N 96083 and by the Russian Foundation for Basic Research under Projects N 96-02-18949 and N 96-02-16077a.

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