Superconducting-normal interface propagation speed in superconducting thin films
aa r X i v : . [ c ond - m a t . s up r- c on ] J un Superconducting-normal interface propagation speed in superconducting samples
Artorix de la Cruz de O˜na ∗ A. Center of theoretical Physics and Applied Mathematics,Dynamical System Project, Montr´eal, H3G 1M8, Canada.District Scolaire 9, 3376 rue Principale C.P. 3668, NB E1X 1G5, Canada. (Dated: October 22, 2018)In this paper a new approach to obtain the interface propagation speed in superconductors bymeans of a variational method is introduced. The results of the approach proposed coincide withthe numerical simulations. The hyperbolic differential equations are introduced as an extension ofthe model in order to take into account delay effects in the front propagation due to the pinning.
PACS numbers: 05.45.-a, 82.40.Ck, 74.40.+k, 03.40.Kf
I. INTRODUCTION
The study of interface propagation is one of the mostfundamental problems in nonequilibrium physics. Theunderstanding of the magnetic field penetration or ex-pulsion in Superconducting samples has been a majorchallenge. An important problem to be solved is the de-termination of the speed at which the interface movesfrom a superconducting to a normal region.In Ref.1, Di Bartolo and Dorsey have obtained thefront speed by using heuristic methods such as Marginalstability hypothesis(MSH) and Reduction order.In general, the nonlinear equations have been employedto model fronts propagation in different areas such aspopulation growth and chemical reactions. Our startpoint is the nonlinear diffusion equations(ND) of the form u t = u xx + f ( u ) obtained from the Ginzburg-Landauexpressions (GL). The GL comprise a coupled equationsfor the density of superconducting electrons and the localmagnetic field.Benguria and Depassier have developed a varia-tional speed selection method(BD) to compute the frontspeed in ND equations. In the BD method a trail func-tion g ( x ) is defined a priori and one may find accuratelower and upper bounds for the speed c . Only if the lowerand the upper bounds coincide, then the value of c canbe determined without any uncertainty. To eliminate theambiguity in the speed determination, Vincent and Fortin Ref.6 have proposed a more accurate approach basedon the BD method. The approach assumes some approx-imative considerations from where the function g ( x ) isdetermined.The purpose of this paper is to develop further the in-sights into the front propagation afforded by the work inRef.1. We aboard the determination of the propagationspeed from a variational point of view. We use an alter-native approach to the one developed by Vincent-Fort .In order to describe the evolution of the system be-tween two homogeneous steady state, we assume aSC sample embedded in a stationary applied mag-netic field equal to the critical H = H c . The mag-netic field is rapidly removed, so the unstable normal-superconducting planar interface propagates toward thenormal phase so as to expel any trapped magnetic flux, leaving the sample in Meissner state. We have consideredthat the interface remains planar during all the process.The existence of a delay time in the interface propaga-tion systems is an important aspect that can be modeledby hyperbolic diffusion equations(HD) which generalizethe ND. The HD has been recently applied in biophysicsto model the spread of humans , bistable systems , for-est fires and in population dynamics . With the goal totake into account the delay effect on the interface prop-agation speed in superconductors, due to, for example,imperfections, vortex-vortex interactions, the presence ofpinning , we have included the relaxation time τ forthe front, and indee introduce the hyperbolic differentialequations. Traveling wave solutions . In this paper, we are inter-ested in the one-dimensional time-dependent Ginzburg-Landau equations, which in dimensionless units are: ∂ t f = (1 /κ ) ∂ x f − q f + f − f and ¯ σ∂ t q = ∂ x q − f q .Here, the quantity f is the magnitude of the supercon-ducting order parameter, q is the gauge-invariant vec-tor potential (such that h = ∂ t q is the magnetic field),¯ σ is the dimensionless normal state conductivity (theratio of the order parameter diffusion constant to themagnetic field diffusion constant) and κ is a parameterwhich determines the type of superconducting material; κ < / √ κ > / √ f ( x, t ) = s ( x − c t ) and q ( x, t ) = n ( x − c t ), where z = x − c t with c > κ s zz + c s z − n s + s − s = 0 ,n zz + ¯ σc n z − s n = 0 , (1) II. VARIATIONAL ANALYSIS
Vector potential q = 0. In this section, we assume q = 0 for the GL equations, ∂ t f = 1 κ ∂ x f + f − f . (2)Then, there exists a front f = s ( x − ct ) joining f = 1,the state corresponding to the whole superconductingphase to f = 0 the state corresponding to the normalphase. Both states may be connected by a traveling frontwith speed c . The front satisfies the boundary conditionslim s →−∞ f = 1 , lim s →∞ f = 0. Then Eq.(2) can be writ-ten as, s zz + c κ s z + F k ( s ) = 0 , (3)where F k = κ s (1 − s ) and F = (1 /κ ) F k .We define p ( s ) = − ds/dz > g such that h = − dg/ds >
0. Taking into account hp + ( g F k /p ) ≥ √ g h F k and following the BD method we arrive to c ≥ (2 /κ ) Z ( g h F ) ds/ Z g ds. (4)Now, the asymptotic speed of the front for sufficientlylocalized initial conditions may be determined in the limit t → ∞ . In the limit one has s → z → −∞ , and s is aslowly varying function of z . Therefore one has s zz ≪ s z ,and from Eq.(3) we have that κ c s z + κ F ( s ) ≃ p ≃ − s z ≃ F /c .Assuming p = F ( s ) /α >
0, where α is a positive con-stant to determine, we can write in general form the trialfunction as, g ( s ) = exp (cid:18) − α Z F − ( s ) ds (cid:19) . (5)Multiplying in both sides by the function h in the ex-pression F k g/p = hp , we have that, h F k g = h p (6)By using Eq.(6), the relation h F k g = h F /α is ob-tained. Then, the following relation is valid,2 p h F k g = 2 α h F . (7)Substituting Eq.(7) in Eq.(4), the general expression forthe speed is given by, c ≃ max α ∈ (0 , (cid:18) α κ Z F ( s ) h ( s ) ds/ Z g ( s ) ds (cid:19) . (8)Taking into account the form of F and Eq.(5), the trialfunction can be written as g ( s ) = (cid:2) ( s − /s (cid:3) α / , (9) c New approachNumericalBD method
FIG. 1: Illustration of the front speed obtained by differentmethods in the q = 0 case versus the GL parameter. and the function h ( s ), h ( s ) = α s − (1+ α ) (1 − s ) ( α − / . (10)The integrals in Eq.(8) are given by, Z g ( s ) ds = 1 √ π Γ (cid:0) (1 − α ) / (cid:1) Γ (cid:0) α / (cid:1) , (11)which is valid for α ≥ Z f ( s ) h ( s ) ds = 1 √ π [ Γ (cid:0) (1 − α ) / (cid:1) (12) − (cid:0) (3 − α ) / (cid:1) ] Γ (cid:0) α / (cid:1) , which is valid for 0 < α < c ≃ max α ∈ (0 , κ α − (cid:2) (3 − α ) (cid:3) Γ (cid:2) (1 − α ) (cid:3) ! . (13)Notice that for α = 1, we obtain the maximum forEq.(13), c = 2 /κ which is the result obtained by usingthe MSH method.In Fig.1 the front speed versus the time delay is shown.The continuous line represents the results of the approachproposed in this paper following Eq.(13) and the numer-ical simulation by Eq.(2). The results coincides. Also,the dashed line represents the bound from the varia-tional(BD) method . Vector potential q = 1 − f . For a set of parameters κ = 1 / √ σ = 1 /
2, we have that s ( z ) + n ( z ) = 1,then Eq.(1) takes the form s zz + ( c/ s z + F ( s ) = 0,where F ( s ) = s (1 − s ) is the reaction term. Proceedingas in Eq.(5) we have that, g ( s ) = [(1 − s ) /s ] α exp (cid:0) α /s (cid:1) , (14)and the velocity is given by, c ≃ max α ∈ D √ α Z F ( s ) h ( s ) ds/ Z g ( s ) ds ! . (15)The interface speed is given by, c ≃ max α ∈ (0 , (cid:18) √ α Γ( α )Γ(1 + α ) (cid:19) , (16)for α = 1 / c = √ III. FRONT FLUX EXPULSION WITH DELAY
It is well known the existence of pinning produces a de-lay time in the magnetic field penetration o expulsion.This can be taken into account by resorting to hyperbolicdifferential equations seen in Section I, which generalizethe parabolic equation. The aim of this section is tostudy of the interface speed problem in superconductingsamples by means of the HD equations, which can bewritten as τ ∂ u∂ t + ∂ u∂ t = ∂ u∂ x + f ( u ) + τ ∂ f ( u ) ∂ t , (17)In the absence of a delay time ( τ = 0), this reduces tothe classical equation u t = u xx + f ( u ). Vector potential q = 0. Taking into account the Eqs.(2)and (17) we can write the following expression, κ τ ∂ f∂ t + κ ∂ f∂ t = ∂ f∂ x + κ F + κ τ ∂ F ∂ t , (18)where F = s (1 − s ).It has been proved that Eq.(17) has traveling wavefronts with profile s ( x − ct ) and moving with speed c > − a c ) s zz + c [ κ − a F ′ ( s )] s z + F k ( s ) = 0 , (19)where z = x − ct , a = κ τ , F k = κ F , and with boundaryconditions lim z →∞ s = 0, lim z →−∞ s = 1, and s z < , s z vanishes for z → ±∞ .We define p ( s ) = − ds/dz > g such that h = − dg/ds >
0. Taking into account (1 − a c ) hp +( g F k /p ) ≥ √ − ac √ g h F k and following the BD method we ar-rive to c √ − a c ≥ κ R ( g h F ) / ds R g ( κ − a F ′ ) ds . (20)In order to obtain the trial function g ( s ), we take inconsideration that in the lim s → − c [ κ − a F ′ ( s )] p + F κ ( s ) ≃ , (21)since s zz ≪ s z . Then, we write an expression for p interms of F , p = F κ /α [ κ − a F ′ ( s )] . (22) c NumericalNew approachUpper boundLower bound
FIG. 2: Time-delayed interface propagation speed for the caseof q = 0 versus the time delay τ . The expression for the speed is given by, c − a c ≃ max α ∈ (0 , κ α R (cid:2) h F / ( κ − a F ′ ) (cid:3) ds R g ( κ − a F ′ ) ds , (23)where the integrals can be only solved by numerical meth-ods. Taking into account Eq.(22) and the expression (1 − ac ) hp = F κ g/p , we have obtained the relation forthe trial function, g = exp (cid:20) − α (1 − ac ) Z ( κ − a F ′ ) F κ ds (cid:21) . (24)from where we have for our case, g = exp " α κ (1 − ac ) (3 as ) + lg " ( s − (2 a + κ ) s a − κ ) , (25)and for h , h ( s, α, a ) = α (cid:0) κ + a (3 s − (cid:1) κ s ( s −
1) ( a c − g ( s, α, κ ) (26)In Fig.2 the front speed versus the time delay is shown.The continuous line represents the result of the approachproposed in this paper following Eq.(23) which coincideswith the numerical simulation done using Eq.(18). Also,we have represented the lower and upper bounds fromthe variational(BD) method . Vector potential q = 1 − f . Taking into account theEqs.(17) and (18) we can write the following expression, τ ∂ f∂ t + 12 ∂ f∂ t = ∂ f∂ x + 12 F + τ ∂ F ∂ t , (27)where F = s (1 − s ).Then we can write Eq.(27) as follows,(1 − a c ) s zz + c [ κ − a F ′ ( s )] s z + F k ( s ) = 0 , (28)where we have assumed F k = (1 / F and a = τ / c NumericalNew approachLin. StabilityUpper boundLower bound
FIG. 3: Time-delayed interface propagation speed for q =1 − f versus the time delay. The expression for the velocity is given by c √ − ac ≥ √ R ( g h F ) / ds R g (1 − a F ′ ) ds . (29)In order to obtain an expression for the trial function g ( s ), we take in consideration that in the lim s → − c [(1 / − a F ′ ( s )] p + F κ ( s ) ≃ , (30)since s zz ≪ s z . Then, we write an expression for p interms of F , p = F κ /α [(1 / − a F ′ ] . (31)The expression for the speed is given by, c − a c ≃ max α ∈ (0 , κ α R [ h F / ( (1 / − a F ′ )] ds R g ( (1 / − a F ′ ) ds . (32) Taking into account Eq.(31) and the expression (1 − ac ) hp = F κ g/p , we have obtained the relation for thetrial function, g = exp (cid:20) − α (1 − ac ) Z ((1 / − a F ′ ) F κ ds (cid:21) . (33)from where we have for our case, g ( s, α, a ) = exp (cid:20) α s (1 − ac ) ( g + g ) (cid:21) , (34)where g = 1 + 24 a s (3 s − g = lg ( s − s (4 a +1) s s (16 a − , h ( s, α, a ) = α [1 + 4 a s (3 s − s (1 − s ) (1 − a c ) g ( s, α, a ) (35)The integrals in Eq.(32) can be only solved by numericalmethods.In Fig.3 the front speed versus the time delay isshown. The continuous line represents the results basedon the approach proposed in this paper following Eq.(32)which coincides with the numerical simulation done usingEq.(27). Also, we have represented the lower and upperbounds from the variational(BD) method . Conclusion . We have computed for the Ginzburg-Landau equations in the form of parabolic and hyperbolicequations the superconducting-normal interface propaga-tion speed by a new approach. This approach is basedin the method proposed by Vincent and Fort in Ref.6.We have obtained the expressions for the trial function g in each case developed. The results of our methodol-ogy coincide with the numerical results for the examplesanalyzed. ∗ Electronic address: [email protected] J.Bartolo and A.Dorsey, Phys.Rev.Lett. , 4442(1996) A.T. Dorsey, Ann. Phys. (N.Y.) , 248(1994) R.Benguria and M.Depassier,Phys.Rev.Lett. ,2272(1994) R.Benguria and M.Depassier, Phys.Rev.E. , 6493(1998) R.Benguria and M.Depassier, Phys.Rev.E. , 3258(1995) V. Mendez and J. Fort, Phys.Rev.E. , 011105(1997) J. Fort and V. Mendez, Phys. Rev. Lett. , 867(1999) V. Mendez and A. Compte, Physica A , 90(1998) V. Mendez and J.E. Lebot, Phys. Rev. E. , 6557(1997) V. Mendez and J. Camacho, Phys. Rev. E. , 6476(1997) E.Altshuler and T.Johansen, Rev.Mod.Phys. , 471(2004) E.H. Brandt, Rep. Prog. Phys. , 1465(2002)13