- Email: [email protected]
Scaling law for velocities of normal phase propagation over high Tc superconducting films with transport current
PII: S0011-2275(98)00034-4
Cryogenics 38 (1998) 645–647 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0011-2275/98/$19.00
Scaling law for velocities of normal phase propagation over high Tc superconducting films with transport current N.A. Buznikov and A.A. Pukhov* Scientific Center for Applied Problems in Electrodynamics, Russian Academy of Sciences, 127412, Izhorskaya 13/19, Moscow, Russia
Received 4 December 1997 The normal phase propagation over a high Tc superconducting film carrying a transport current is studied theoretically. It is shown that the values of the normal phase propagation velocity v satisfy a certain scaling law. The obtained universal formula for v takes into account the substrate effect on thermal instability development and is applicable for an arbitrary dependence of the film critical current on temperature. The scaling law allows one to put in order a number of experimental data on the normal phase propagation velocity. The results obtained may be of importance in the study of superconducting switching devices. 1998 Elsevier Science Ltd. All rights reserved Keywords: A. high Tc superconductors; A. thin films; C. propagation velocity
The transition of a high Tc superconducting (HTSC) film carrying a transport current from the superconducting to the normal state may be related to thermal instability development. The asymptotic behaviour of thermal instability development is characterized by the constant velocity v of the normal phase propagation over an HTSC film1. In Refs 2–5 the value of v has been measured for HTSC films with different dependences of the critical current Ic on the temperature T, and at distinct cooling conditions. The normal phase propagation velocity depends not only on the film parameters but also on the thermal properties and thickness of the substrate. The dependence of v on many parameters makes it difficult to analyze thermal instability development in HTSC films. In the present paper it is shown that the velocities of the normal phase propagation over an HTSC film located on a substrate, of which the reverse side is stabilized at a fixed temperature, satisfy a certain scaling law. The use of the scaling law allows for an adequate description of the experimental results2–4. The main specific features of thermal instability development in an HTSC film are related to a non-uniform temperature distribution over the substrate thickness6–8. This means that the conventional one-dimensional analysis of the normal phase propagation1,9 is inappropriate for the HTSC films, and to describe the instability development the film and substrate should be treated separately as two coupled thermal subsystems4,8.
*To whom correspondence should be addressed: Tel.: (095)3625147; fax: (095)4842633; e-mail: [email protected]
A schematic diagram of the film–substrate system is shown in Figure 1. The HTSC film of thickness Df is located on the dielectric substrate of thickness Ds, of which the reverse side is stabilized at temperature T0 ⬍ Tc (where Tc is the film critical temperature). The equations describing the temperature distribution T(X,Y) in the moving with the normal phase co-ordinate system can be written as follows8:
冉 冊
冉冊
∂T ks ∂T ∂ ∂T k + Cfv 兩 = 0 ⬍ Y ⬍ Df + Q(T) + ∂X f ∂X ∂X Df ∂Y Y = Df
(1)
Figure 1 A schematic diagram of the film–substrate system. The reverse side of the substrate is stabilized at T = T0. N and S indicate the normal and the superconducting regions of the film, respectively
Cryogenics 1998 Volume 38, Number 6 645
Scaling law for normal phase propagation velocities: N.A. Buznikov and A.A. Pukhov
冉 冊 冉 冊
∂ ∂T ∂T ∂ ∂T = 0, Df ⬍ Y ⬍ Df + Ds ks + ks + Csv ∂X ∂X ∂Y ∂Y ∂X (2) T = T0, Y = Df + Ds
(3)
where: Cf and kf are the film heat capacity and thermal conductivity; Cs and ks are the substrate heat capacity and thermal conductivity; Q(T) is the specific Joule heat release and the transport current flows along the X-axis. In Equation (1) it is taken into account that at kfDsÀksDf the film temperature is uniform over its thickness, and the last term in the left-hand side of Equation (1) corresponds to the heat removal from the film to the substrate. In the simplest approximation, the Joule heating can be expressed as1,9 Q(T) =
I2 (T − Tr ) W2D2f
Y T − T0 X ,y= ,= Ds Ds Tr − T0
(5)
(6)
Here vh = ks/CsDs is the characteristic ‘thermal’ velocity of the normal phase propagation. The value of vh depends on the substrate properties only and varies over a wide range. Let us estimate vh for different substrates of thickness Ds = 5 × 10−4 m. For MgO substrate (Cs ⬵ 5 × 105 J m−3 K−1, ks ⬵ 350 W m−1 K−1 ) or for Al2O3 substrate (Cs ⬵ 4 × 105 J m−3 K−1, ks ⬵ 650 W m−1 K−1 ) we have vh 苲 1 m s−1; for LaAlO3 substrate (Cs ⬵ 9 × 104 J m−3 K−1, ks ⬵ 20 W m−1 K−1 ) we find vh 苲 10−1 m s−1 and for SrTiO3 substrate (Cs ⬵ 2 × 106 J m−3 K−1, ks ⬵ 10 W m−1 K−1 ) we obtain vh 苲 10−2 m s−1. At typical relations between the film and substrate parameters kfDf¿ksDs and CfDf¿CsDs, the first and second terms in the left-hand side of Equation (1) can be neglected8. Let us choose the origin of the co-ordinate system in accordance with the condition T(0,0) = Tr(I), that is, (0,0) = 1. Then, Equations (1) and (3), determining the boundary conditions to Equation (6), can be written as ∂ 兩 = −1(x), 兩y = 1 = 0 ∂y y = 0
(7)
ksDfW2[Tr(I) − T0 ] I2Ds
646
Cryogenics 1998 Volume 38, Number 6
1
2 k
k=0
⬁
2− 1+
1
k=0
2 k
1−
冊 再冉 冊 冎册 冊 再冉 冊 冎
v exp 2vhk
v exp 2vhk
v + k x cos( ky), x ⬍ 0 2vh
v − k x cos( ky), 2vh
(9)
x>0
Here k = (2k + 1)/2 and k = (v2/4v2h + 2k )1/2. Taking into account that (0,0) = 1 from Equation (9) we find 1 − 2 = 2v
冘 √v +1(2 v )
(8)
2 k
2
k h
2
(10)
Equation (10) reduces the dependence of the normal phase propagation velocity v on all parameters to the universal relation between v/vh and . The value of v equals zero at = 1/2 and tends to infinity at = 0. If the dependence of the film critical current on temperature obeys the power law Ic(T)⬀(1 − T/Tc )n 2–4 for the dimensionless parameter we have
= (1 − i1/n )/␣i2
(11)
where: ␣ = I2c0Ds/ksDfW2(Tc − T0 ) is the effective Stekly parameter of the film–substrate system; i = I/Ic0 is the dimensionless current and Ic0 is the film critical current at T = T0. It is convenient to use the scaling law of Equation (10) to put in order a number of experimental data on the normal phase propagation velocity v. The results of the generalization are presented in Figure 2. Figure 2 demonstrates good agreement between the calculated dependence (Equation (10)) of v on and the experimental results2–4. It should be noted that the experimental data shown in Figure 2 have been obtained on HTSC films with different values of n (n = 12 and n = 23,4 ) by means of distinct experimental methods (direct current method3 and transient film response to current pulses2,4 ) and at different experimental conditions (a change of transport current I at fixed T03,4 and a change of T0 at fixed current amplitude I 2 ). A certain disagreement between experimental results4 and the formula of Equation (10) at low values of v (see inset in Figure 2) may be attributed to the fact that for short current pulses the transient film response does not correspond to the stationary value of v since the normal phase front has insufficient time to be formed in the time of current pulse action. At sufficiently high values of v (vÀvh ), Equation (10) may be written in the more simple form8: v = (2/ )vh−1. Taking into account Equation (11) in the case of power law Ic(T)⬀(1 − T/Tc )n we find v=
where
=
冦
⬁
k=0
and disregarding the film thickness in comparison with the substrate one (Df¿Ds ) it is convenient to rewrite Equation (2) in the following dimensionless form: ∂2 ∂2 v ∂ = 0, 0 ⬍ y ⬍ 1 + + ∂x2 ∂y2 vh ∂x
(x,y) = −1
冘 冋 冉 冘 冉
⬁
(4)
where: is the film resistivity in the normal state; I is the transport current; W is the film width; Tr(I) is the resistive transition temperature corresponding to the condition Ic(Tr ) = I; Ic(T) is the film critical current and (x) is the Heaviside stepwise function ( = 0 at x ⬍ 0 and = 1 at x ⱖ 0). Assume for simplicity that the dependences of Cf, kf, Cs, ks and on temperature can be neglected. Then, introducing the dimensionless parameters x=
is the dimensionless parameter, which is the ratio of the characteristic heat removal from the film to the substrate to the Joule heat release in the normal region. Equation (6), with boundary conditions outlined by Equation (7), can be solved through separation of variables8, which yields the following expression for the temperature distribution (x,y) in the film–substrate system:
2 I2c0 i2 2 CsDfW (Tc − T0 ) 1 − i1/n
(12)
Equation (12) presents a simplified scaling law, which is applicable if the normal phase propagation velocity is
Scaling law for normal phase propagation velocities: N.A. Buznikov and A.A. Pukhov
Figure 3 Simplified scaling law for v. Solid line, Equation (12); 쐌, 䊏, 왖, experimental data2–4. Inset: enlarged view of simplified scaling law at low values of v Figure 2 Scaling law for v. Solid line, Equation (10); 쐌, experimental data2 (a variation of T0 at fixed I = 1.33 A. Ds = 5 × 10−4 m, Cs = 9 × 104 J m−3 K−1, ks = 20 W m−1 K−1, Df = 10−6 m, W = 1.5 × 10−3 m, Tc = 85.8 K, = 4.5 × 10−6 ⍀ m); 䊏, experimental data3 (a variation of I at fixed T0 = 78 K. Ds = 5 × 10−4 m, Cs = 2 × 106 J m−3 K−1, ks = 10 W m−1K−1, Df = 4 × 10−7 m, W = 10−3 m, Tc = 87 K, = 5 × 10−7 ⍀ m, Ic0 = 1.7 A); 왖, experimental data4 (a variation of I at fixed T0 = 80 K. Ds = 5 × 10−4 m, Cs = 2 × 106 J m−3 K−1, ks = 10 W m−1K −1, Df = 4 × 10−7 m, W = 10−3 m, Tc = 87 K, = 5 × 10−6 ⍀ m, Ic0 = 1.52 A). Inset: enlarged view of scaling law at low values of v
not too low. It may be easily shown that a considerable difference between the formulae of Equations (10) and (12) may take place only in the close vicinity of = 1/2, corresponding to the minimum normal phase propagation current1,9. This fact allows one to use the simplified scaling law of Equation (12) instead of the more complicated Equation (10), with practically the same accuracy. Comparison of Equation (12) with experimental data2–4 is shown in Figure 3. Thus, in the present paper the scaling law for the velocities of the normal phase propagation over an HTSC film located on a substrate, of which the reverse side is stabilized at a fixed temperature, is found. Scaling law (Equation (10)) is obtained by means of the solution of a two-dimensional heat balance equation and hence takes into account the effects related to a non-uniform temperature distribution over the substrate thickness. Equation (10) is applicable for an arbitrary dependence of the film critical current on temperature and is convenient to put in order a number of experimental data on the normal phase propagation velocity v. It is shown also that if the dependence of the film critical current on temperature obeys the power law Ic⬀(1 − T/Tc )n, then Equation (10) reduces to the simple universal dependence of v on the transport current. In conclusion, it should be noted that quite a different situation occurs if the HTSC film is located on the substrate immersed in liquid nitrogen. Simple estimates show that in this case the temperature is approximately constant over the substrate thickness8, and experimental results5 can be described adequately in the
framework of the one-dimensional theory of the normal phase propagation1,9.
Acknowledgements The authors would like to thank V.N. Skokov for fruitful discussions. 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 Project N 96-02-18949.
References 1. Dresner, L., Stability of Superconductors. Plenum Press, New York, USA, 1995. 2. Dhali, S.K. and Wang, L., Transient response of a high Tc superconducting film. Appl. Phys. Lett., 1992, 61, 1594–1596. 3. Skokov, V.N. and Koverda, V.P., Arising and evolution of the thermal domains in current-carrying high-Tc superconducting films. Phys. Stat. Sol. (a), 1994, 142, 193–199. 4. Buznikov, N.A., Pukhov, A.A. and Skokov, V.N., Substrate effects on thermal instability development in a high Tc superconducting thin film with transport current. Cryogenics, 1998, 38, 277–282. 5. Lutset, M.O., A typical velocity for heat destruction and restoration of S-state of short sample HTSC film. Proc. ICEC 16, 1997, 2, 1341–1344. 6. Levillain, C., Manuel, P. and Therond, P.G., Effects of thermal shunt to substrate on normal zone propagation in high Tc superconducting thin films. Cryogenics, 1994, 34, 69–75. 7. Wu, J.P. and Chu, H.S., Substrate effects on intrinsic thermal stability and quench recovery for thin-film superconductors. Cryogenics, 1996, 36, 925–935. 8. Buznikov, N.A. and Pukhov, A.A., Normal-phase propagation over an HTSC film heated by microwave radiation. Supercond. Sci. Technol., 1997, 10, 318–324. 9. Wilson, M.N., Superconducting Magnets. Clarendon Press, Oxford, UK, 1983.
Cryogenics 1998 Volume 38, Number 6 647
Cryogenics 38 (1998) 645–647 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0011-2275/98/$19.00
Scaling law for velocities of normal phase propagation over high Tc superconducting films with transport current N.A. Buznikov and A.A. Pukhov* Scientific Center for Applied Problems in Electrodynamics, Russian Academy of Sciences, 127412, Izhorskaya 13/19, Moscow, Russia
Received 4 December 1997 The normal phase propagation over a high Tc superconducting film carrying a transport current is studied theoretically. It is shown that the values of the normal phase propagation velocity v satisfy a certain scaling law. The obtained universal formula for v takes into account the substrate effect on thermal instability development and is applicable for an arbitrary dependence of the film critical current on temperature. The scaling law allows one to put in order a number of experimental data on the normal phase propagation velocity. The results obtained may be of importance in the study of superconducting switching devices. 1998 Elsevier Science Ltd. All rights reserved Keywords: A. high Tc superconductors; A. thin films; C. propagation velocity
The transition of a high Tc superconducting (HTSC) film carrying a transport current from the superconducting to the normal state may be related to thermal instability development. The asymptotic behaviour of thermal instability development is characterized by the constant velocity v of the normal phase propagation over an HTSC film1. In Refs 2–5 the value of v has been measured for HTSC films with different dependences of the critical current Ic on the temperature T, and at distinct cooling conditions. The normal phase propagation velocity depends not only on the film parameters but also on the thermal properties and thickness of the substrate. The dependence of v on many parameters makes it difficult to analyze thermal instability development in HTSC films. In the present paper it is shown that the velocities of the normal phase propagation over an HTSC film located on a substrate, of which the reverse side is stabilized at a fixed temperature, satisfy a certain scaling law. The use of the scaling law allows for an adequate description of the experimental results2–4. The main specific features of thermal instability development in an HTSC film are related to a non-uniform temperature distribution over the substrate thickness6–8. This means that the conventional one-dimensional analysis of the normal phase propagation1,9 is inappropriate for the HTSC films, and to describe the instability development the film and substrate should be treated separately as two coupled thermal subsystems4,8.
*To whom correspondence should be addressed: Tel.: (095)3625147; fax: (095)4842633; e-mail: [email protected]
A schematic diagram of the film–substrate system is shown in Figure 1. The HTSC film of thickness Df is located on the dielectric substrate of thickness Ds, of which the reverse side is stabilized at temperature T0 ⬍ Tc (where Tc is the film critical temperature). The equations describing the temperature distribution T(X,Y) in the moving with the normal phase co-ordinate system can be written as follows8:
冉 冊
冉冊
∂T ks ∂T ∂ ∂T k + Cfv 兩 = 0 ⬍ Y ⬍ Df + Q(T) + ∂X f ∂X ∂X Df ∂Y Y = Df
(1)
Figure 1 A schematic diagram of the film–substrate system. The reverse side of the substrate is stabilized at T = T0. N and S indicate the normal and the superconducting regions of the film, respectively
Cryogenics 1998 Volume 38, Number 6 645
Scaling law for normal phase propagation velocities: N.A. Buznikov and A.A. Pukhov
冉 冊 冉 冊
∂ ∂T ∂T ∂ ∂T = 0, Df ⬍ Y ⬍ Df + Ds ks + ks + Csv ∂X ∂X ∂Y ∂Y ∂X (2) T = T0, Y = Df + Ds
(3)
where: Cf and kf are the film heat capacity and thermal conductivity; Cs and ks are the substrate heat capacity and thermal conductivity; Q(T) is the specific Joule heat release and the transport current flows along the X-axis. In Equation (1) it is taken into account that at kfDsÀksDf the film temperature is uniform over its thickness, and the last term in the left-hand side of Equation (1) corresponds to the heat removal from the film to the substrate. In the simplest approximation, the Joule heating can be expressed as1,9 Q(T) =
I2 (T − Tr ) W2D2f
Y T − T0 X ,y= ,= Ds Ds Tr − T0
(5)
(6)
Here vh = ks/CsDs is the characteristic ‘thermal’ velocity of the normal phase propagation. The value of vh depends on the substrate properties only and varies over a wide range. Let us estimate vh for different substrates of thickness Ds = 5 × 10−4 m. For MgO substrate (Cs ⬵ 5 × 105 J m−3 K−1, ks ⬵ 350 W m−1 K−1 ) or for Al2O3 substrate (Cs ⬵ 4 × 105 J m−3 K−1, ks ⬵ 650 W m−1 K−1 ) we have vh 苲 1 m s−1; for LaAlO3 substrate (Cs ⬵ 9 × 104 J m−3 K−1, ks ⬵ 20 W m−1 K−1 ) we find vh 苲 10−1 m s−1 and for SrTiO3 substrate (Cs ⬵ 2 × 106 J m−3 K−1, ks ⬵ 10 W m−1 K−1 ) we obtain vh 苲 10−2 m s−1. At typical relations between the film and substrate parameters kfDf¿ksDs and CfDf¿CsDs, the first and second terms in the left-hand side of Equation (1) can be neglected8. Let us choose the origin of the co-ordinate system in accordance with the condition T(0,0) = Tr(I), that is, (0,0) = 1. Then, Equations (1) and (3), determining the boundary conditions to Equation (6), can be written as ∂ 兩 = −1(x), 兩y = 1 = 0 ∂y y = 0
(7)
ksDfW2[Tr(I) − T0 ] I2Ds
646
Cryogenics 1998 Volume 38, Number 6
1
2 k
k=0
⬁
2− 1+
1
k=0
2 k
1−
冊 再冉 冊 冎册 冊 再冉 冊 冎
v exp 2vhk
v exp 2vhk
v + k x cos( ky), x ⬍ 0 2vh
v − k x cos( ky), 2vh
(9)
x>0
Here k = (2k + 1)/2 and k = (v2/4v2h + 2k )1/2. Taking into account that (0,0) = 1 from Equation (9) we find 1 − 2 = 2v
冘 √v +1(2 v )
(8)
2 k
2
k h
2
(10)
Equation (10) reduces the dependence of the normal phase propagation velocity v on all parameters to the universal relation between v/vh and . The value of v equals zero at = 1/2 and tends to infinity at = 0. If the dependence of the film critical current on temperature obeys the power law Ic(T)⬀(1 − T/Tc )n 2–4 for the dimensionless parameter we have
= (1 − i1/n )/␣i2
(11)
where: ␣ = I2c0Ds/ksDfW2(Tc − T0 ) is the effective Stekly parameter of the film–substrate system; i = I/Ic0 is the dimensionless current and Ic0 is the film critical current at T = T0. It is convenient to use the scaling law of Equation (10) to put in order a number of experimental data on the normal phase propagation velocity v. The results of the generalization are presented in Figure 2. Figure 2 demonstrates good agreement between the calculated dependence (Equation (10)) of v on and the experimental results2–4. It should be noted that the experimental data shown in Figure 2 have been obtained on HTSC films with different values of n (n = 12 and n = 23,4 ) by means of distinct experimental methods (direct current method3 and transient film response to current pulses2,4 ) and at different experimental conditions (a change of transport current I at fixed T03,4 and a change of T0 at fixed current amplitude I 2 ). A certain disagreement between experimental results4 and the formula of Equation (10) at low values of v (see inset in Figure 2) may be attributed to the fact that for short current pulses the transient film response does not correspond to the stationary value of v since the normal phase front has insufficient time to be formed in the time of current pulse action. At sufficiently high values of v (vÀvh ), Equation (10) may be written in the more simple form8: v = (2/ )vh−1. Taking into account Equation (11) in the case of power law Ic(T)⬀(1 − T/Tc )n we find v=
where
=
冦
⬁
k=0
and disregarding the film thickness in comparison with the substrate one (Df¿Ds ) it is convenient to rewrite Equation (2) in the following dimensionless form: ∂2 ∂2 v ∂ = 0, 0 ⬍ y ⬍ 1 + + ∂x2 ∂y2 vh ∂x
(x,y) = −1
冘 冋 冉 冘 冉
⬁
(4)
where: is the film resistivity in the normal state; I is the transport current; W is the film width; Tr(I) is the resistive transition temperature corresponding to the condition Ic(Tr ) = I; Ic(T) is the film critical current and (x) is the Heaviside stepwise function ( = 0 at x ⬍ 0 and = 1 at x ⱖ 0). Assume for simplicity that the dependences of Cf, kf, Cs, ks and on temperature can be neglected. Then, introducing the dimensionless parameters x=
is the dimensionless parameter, which is the ratio of the characteristic heat removal from the film to the substrate to the Joule heat release in the normal region. Equation (6), with boundary conditions outlined by Equation (7), can be solved through separation of variables8, which yields the following expression for the temperature distribution (x,y) in the film–substrate system:
2 I2c0 i2 2 CsDfW (Tc − T0 ) 1 − i1/n
(12)
Equation (12) presents a simplified scaling law, which is applicable if the normal phase propagation velocity is
Scaling law for normal phase propagation velocities: N.A. Buznikov and A.A. Pukhov
Figure 3 Simplified scaling law for v. Solid line, Equation (12); 쐌, 䊏, 왖, experimental data2–4. Inset: enlarged view of simplified scaling law at low values of v Figure 2 Scaling law for v. Solid line, Equation (10); 쐌, experimental data2 (a variation of T0 at fixed I = 1.33 A. Ds = 5 × 10−4 m, Cs = 9 × 104 J m−3 K−1, ks = 20 W m−1 K−1, Df = 10−6 m, W = 1.5 × 10−3 m, Tc = 85.8 K, = 4.5 × 10−6 ⍀ m); 䊏, experimental data3 (a variation of I at fixed T0 = 78 K. Ds = 5 × 10−4 m, Cs = 2 × 106 J m−3 K−1, ks = 10 W m−1K−1, Df = 4 × 10−7 m, W = 10−3 m, Tc = 87 K, = 5 × 10−7 ⍀ m, Ic0 = 1.7 A); 왖, experimental data4 (a variation of I at fixed T0 = 80 K. Ds = 5 × 10−4 m, Cs = 2 × 106 J m−3 K−1, ks = 10 W m−1K −1, Df = 4 × 10−7 m, W = 10−3 m, Tc = 87 K, = 5 × 10−6 ⍀ m, Ic0 = 1.52 A). Inset: enlarged view of scaling law at low values of v
not too low. It may be easily shown that a considerable difference between the formulae of Equations (10) and (12) may take place only in the close vicinity of = 1/2, corresponding to the minimum normal phase propagation current1,9. This fact allows one to use the simplified scaling law of Equation (12) instead of the more complicated Equation (10), with practically the same accuracy. Comparison of Equation (12) with experimental data2–4 is shown in Figure 3. Thus, in the present paper the scaling law for the velocities of the normal phase propagation over an HTSC film located on a substrate, of which the reverse side is stabilized at a fixed temperature, is found. Scaling law (Equation (10)) is obtained by means of the solution of a two-dimensional heat balance equation and hence takes into account the effects related to a non-uniform temperature distribution over the substrate thickness. Equation (10) is applicable for an arbitrary dependence of the film critical current on temperature and is convenient to put in order a number of experimental data on the normal phase propagation velocity v. It is shown also that if the dependence of the film critical current on temperature obeys the power law Ic⬀(1 − T/Tc )n, then Equation (10) reduces to the simple universal dependence of v on the transport current. In conclusion, it should be noted that quite a different situation occurs if the HTSC film is located on the substrate immersed in liquid nitrogen. Simple estimates show that in this case the temperature is approximately constant over the substrate thickness8, and experimental results5 can be described adequately in the
framework of the one-dimensional theory of the normal phase propagation1,9.
Acknowledgements The authors would like to thank V.N. Skokov for fruitful discussions. 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 Project N 96-02-18949.
References 1. Dresner, L., Stability of Superconductors. Plenum Press, New York, USA, 1995. 2. Dhali, S.K. and Wang, L., Transient response of a high Tc superconducting film. Appl. Phys. Lett., 1992, 61, 1594–1596. 3. Skokov, V.N. and Koverda, V.P., Arising and evolution of the thermal domains in current-carrying high-Tc superconducting films. Phys. Stat. Sol. (a), 1994, 142, 193–199. 4. Buznikov, N.A., Pukhov, A.A. and Skokov, V.N., Substrate effects on thermal instability development in a high Tc superconducting thin film with transport current. Cryogenics, 1998, 38, 277–282. 5. Lutset, M.O., A typical velocity for heat destruction and restoration of S-state of short sample HTSC film. Proc. ICEC 16, 1997, 2, 1341–1344. 6. Levillain, C., Manuel, P. and Therond, P.G., Effects of thermal shunt to substrate on normal zone propagation in high Tc superconducting thin films. Cryogenics, 1994, 34, 69–75. 7. Wu, J.P. and Chu, H.S., Substrate effects on intrinsic thermal stability and quench recovery for thin-film superconductors. Cryogenics, 1996, 36, 925–935. 8. Buznikov, N.A. and Pukhov, A.A., Normal-phase propagation over an HTSC film heated by microwave radiation. Supercond. Sci. Technol., 1997, 10, 318–324. 9. Wilson, M.N., Superconducting Magnets. Clarendon Press, Oxford, UK, 1983.
Cryogenics 1998 Volume 38, Number 6 647