vortices in superconductors: modelling and computer

Vortices in superconductors: modellingand computer simulations

B y Jennifer Deang1, Qiang D u2, M a x Gunzburger2

and Janet Peterson2

1Department of Mathematics, Virginia Polytechnic Institute,Blacksburg, VA 24061-0123, USA

2Department of Mathematics, Iowa State University, Ames, IA 50011-2064, USA

Vortices in superconductors are tubes of magnetic flux, or equivalently, cylindricalcurrent loops, that penetrate into a material sample. Knowledge about the structureand dynamics of collections of vortices is of importance both to the understanding ofthe basic physics of superconductors and to the design of devices. We first discuss ho-mogeneous isotropic superconductors that can be modelled by the Ginzburg–Landautheory. We then discuss variants of this model that can account for inhomogenietiesand anisotropies due to impurities, thickness variations, and thermal fluctuations.These all effect changes in the vortex state, as do changes in the applied magneticfield and current strengths and directions. Through computational simulations, weuse the various models to illustrate these changes. In particular, we examine thepinning of vortices by thickness variations in thin-films, by impurities and by grainboundaries, the effects that changes in the thickness of a simple Josephson junctionhave on the structure of the vortex state, transitions that occur in the vortex state asthe applied magnetic field is increased, and distortions of that state due to thermalfluctuations.

1. Introduction

The fascination of superconductivity centres on its two hallmarks. Perfect conduc-tivity is the flow of current without resistance; in fact, the resistivity of the super-conducting state is below the detection capability of any past or current measuringdevice. Perfect diagmagnetism is the expulsion of a magnetic field from a sample asit is cooled below a critical temperature at which it becomes superconducting. Thisdescription of superconductivity applies to metal superconductors in the bulk, i.e.away from boundaries.

The situation for other superconductors, including the recently developed high crit-ical temperature superconductors, is more complicated. In particular, mixed statesoccur in which both the normal, i.e. non-superconducting, and superconductingphases coexist. In this mixed state, the magnetic field can penetrate the samplein a tube-like configuration as depicted in the sketch in figure 1. These tubes ofmagnetic flux are referred to as (superconducting) vortices. In this paper, we explorevarious phenomena associated with these vortices.

Phil. Trans. R. Soc. Lond. A (1997) 355, 1957–1968 c© 1997 The Royal SocietyPrinted in Great Britain 1957 TEX Paper

1958 J. Deang and others

SUPERCURRENT LOOPS

MAGNETIC FLUX LINES

Meissner

Vortex

Normal

Hc1

Hc2

Temperature

App

lied

fie

ld

(a) (b)

Figure 1. (a) Depicts magnetic flux tubes, i.e. vortices in a superconductor. (b) Depicts statesin an ideal type-II superconductor.

2. Ginzburg–Landau vortices

A satisfactory theoretical explanation of superconductivity eluded physicists fromthe time of its discovery (1911) until 1957 when Bardeen, Cooper and Shrieffer (BCS)developed their microscopic theory. Before that time, phenomenological continuumtheories had been developed, e.g. by London & London in the 1930s and by Ginzburg& Landau (GL) in 1950. The latter theory was connected, in 1959, to the BCS theoryby Gor’kov. Also in 1959, the GL theory was used by Abrikosov to predict the vortex-like structures mentioned in §1; this represents a remarkable triumph for the GLtheory since, at that time, these structures had not been observed in experimentsand were thus not known to exist. Only a few years later was Abrikosov’s predictionverified in the laboratory. We now summarize some important points concerning theGL theory; more detailed accounts of the material of this section can be found in,for example, Tinkham (1975).

The Ginzburg–Landau theory of superconductivity applies to homogeneous andisotropic materials in the steady state. The primary dependent variables employed bythe theory are the vector-valued magnetic potential, A, and a complex-valued orderparameter ψ. Based on Landau’s theory of second-order phase transitions, Ginzburg& Landau wrote down that the Gibbs free energy associated with a superconductoris of the following form:

G(ψ,A) =∫

Ω

(fn + α|ψ|2 + 1

2β|ψ|4)

dΩ +∫

Ω

1ms

∣∣∣(ih∇+es

cA)ψ∣∣∣2 dΩ

+∫

Ω

( | curlA|28π

− ∇×A ·H4π

)dΩ .

The three terms on the right-hand side correspond to the condensation, kinetic andmagnetic energies; here, H denotes a given applied field. The coefficient α is tem-perature dependent and changes sign (from positive to negative) as the temperaturedecreases from above to below the critical temperature Tc. For T > Tc, the sample isin a normal (non-superconducting) state while for T < Tc, the sample is supercon-ducting. The other coefficients are roughly temperature independent. In particular,ms and es are the mass and charge of the superconducting charge carriers; theseare often chosen to be twice the corresponding quantities for an electron. The basic

Phil. Trans. R. Soc. Lond. A (1997)

Vortices in superconductors 1959

thermodynamic postulate of the GL theory is that the superconducting sample is ina state such that its Gibbs free energy is a minimum.

Physically interesting variables may be recovered from ψ and A through the rela-tions

Ns = |ψ|2 = density of available of superconducting charge carriers,

j =es

ms(h∇φ− es

cA)|ψ|2 = supercurrent,

h = ∇×A = magnetic field,

where φ denotes the phase of the order parameter, i.e. ψ = |ψ|eiφ.There are two length scales in the GL theory, the penetration depth and coherence

length given by

λ =

√−βmsc2

4παe2s

and ξ =√− h

2msα,

respectively. These give the scale for variations in the magnetic field and order pa-rameter, respectively. Their ratio, κ = λ/ξ, is a material parameter that is referredto as the Ginzburg–Landau parameter.

Superconductors can be divided into two classes according to the value of κ. Type-Isuperconductors are characterized by κ < 1/

√2; in this case the surface energy as-

sociated with phase transitions (from the superconducting to the normal phase) ispositive so that phase transitions are avoided. The description of superconductivitygiven in the first paragraph of the introduction is largely that of a type-I supercon-ductor.

For type-II superconductors, κ > 1/√

2 and the surface energy associated with aphase transition is positive so that phase transitions are favoured. Note that mostsuperconductors of practical use, including high-temperature superconductors, areof type-II; in fact, the latter are strongly type-II with, typically, κ > 100. Onemay naturally ask if the surface energy associated with phase transitions in type-II superconductors is negative, what stops phase transitions from occurring overarbitrarily small length scales? The answer lies in fluxoid quantization. Let ∂Σ denotea closed curve lying in the material sample such that |ψ| 6= 0 everywhere on the curve,i.e. the curve ∂Σ nowhere intersects a normal region. Let Σ denote a surface boundedby the closed curve ∂Σ . We define the fluxoid to be

Φ′ =∫

Σh · dΣ +

msc

e2s

∫∂Σ

j

|ψ|2 · d(∂Σ ) =hc

es

∫∂Σ∇φ · d(∂Σ ).

Then, we have that Φ′ = (2πhc/es)n, where n is an integer, i.e. Φ′ is quantized.Fluxoid quantization and the negative surface energies associated with phase tran-

sitions results in the existence of a mixed normal-superconducting state in type-IIsuperconductors. In this state, we have that 0 < |ψ| < ψ0 =

√−α/β, i.e. |ψ| variesbetween the normal and superconducting values, and h 6= 0 or H, i.e. the field pen-etrates the sample. This penetration takes the form of tubes of flux, i.e. supercon-ducting vortices, as depicted in figure 1; for clean low-temperature superconductors,the vortices arrange themselves in a regular hexagonal lattice. The phase diagramfor such superconductors, at least in the bulk, is sketched in figure 1; for low enoughfields and temperatures, we have a Meissner region in which the field is excludedfrom the sample and |ψ| attains its full superconducting value ψ0. For high enough



Figure 2. Vortex lattice in a two-dimensional type-II superconductor. On the left are the levelcurves of the magnitude of the order parameter; on the right is the supercurrent distribution.

fields or temperatures, superconductivity is destroyed. For intermediate fields andtemperatures, we have the mixed vortex state. Thus, we have the two critical fieldsHc1(T ) and Hc2(T ) which separate the three regions in the phase diagram.

If we introduce appropriate non-dimensionalizations, e.g. lengths by λ, ψ by ψ0,H by

√2Hc =

√8πα2/β, and A by λHc, the Gibbs free energy takes the form

G(ψ,A) =∫

Ω(fn − |ψ|2 + 1

2 |ψ|4) dΩ +∫

Ω

∣∣∣∣( iκ∇+A

)ψ

∣∣∣∣2 dΩ

+∫

Ω(|∇ ×A|2 − 2∇×A ·H) dΩ .

We now can see the importance of the Ginzburg–Landau parameter, κ; indeed,it is the sole parameter appearing in the Ginzburg–Landau model. In this non-dimensionalization, we have that Hc2 = κ and Hc1 ≈ lnκ/(2κ).

Using standard techniques from the calculus of variations, one may derive theEuler–Lagrange conditions that correspond to the minimization on the Gibbs free en-ergy; this system of partial differential equations are the celebrated Ginzburg–Landauequations. We do not list them here since they are merely the time-independent ver-sions of the equations given in §2 a.

A computational simulation of the vortex state in a square two-dimensional sampleof side 100λ is given in figure 2. We have that κ = 5 and the applied field H = 0.5κand is directed in the z-direction whereas the sample cross-section lies in the (x, y)-plane. The level curves of the magnitude of the order parameter and a vector plotof the supercurrent are provided. Notice the hexagonal lattice of vortices near thecentre of the sample. The flux tubes sketched in figure 1 are perpendicular to thepaper in figure 2. At the centre, or core, of the vortices, the order parameter goes tozero. Note the supercurrent circulating around the core of each vortex and aroundthe boundary of the sample. The latter acts as a shielding current that prevents themagnetic field from penetrating into the sample, except as flux tubes through thevortex cores.

(a ) The time-dependent Ginzburg–Landau equationsDue to the resulting Lorentz force, the motion of vortices, i.e. of the magnetic flux

tubes, induces resistance and thus a loss of perfect conductivity. Time-dependent ver-sions of the Ginzburg–Landau model have been developed, by among others, Gor’kov



& Eliashberg (1968). As a result of its ability to account for transient phenomena,this model cannot only be used to study the equilibrium structure of the vortex state,but also how this structure is achieved. Furthermore, it also can account for effectsdue to applied currents.

In addition to ψ and A, we must now add the scalar potential Φ as a primaryvariable. In non-dimensional form, the time-dependent Ginzburg–Landau equations(TDGL) are given by

−(∂ψ

∂t+ iκΦψ

)=(

iκ∇+A

)2

ψ − ψ + |ψ|2ψ,

−σ(∂A

∂t+∇Φ

)= ∇× (∇×A−H) + |ψ|2A+

i2κ

(ψ∗∇ψ − ψ∇ψ∗).The normal ‘conductance’, σ, is a second parameter appearing in the TDGL. Justas κ is a ratio of the relaxation length scales for A and ψ, σ is the (inverse) ratio oftheir relaxation time scales. The physically interesting variables Ns and h are givenas before; we also have that

E = −∂A∂t−∇Φ = electric field,

j =(

1κ∇φ−A

)|ψ|2 − σ

(∂A

∂t+∇Φ

)= current.

Note that the current not only has, as before, a resistanceless supercurrent com-ponent, but also has a resistive, Ohmic, normal current component, i.e. the secondterm in the above equation. In general, one should append to the TDGL equationsanother equation for Φ. However, in many situations, Φ may be determined as agauge choice (see, for example, Du 1994a).

In the Ginzburg–Landau model, the vortices are at equilibrium and a gauge mayalways be chosen so that Φ = 0. Thus, the Ginzburg–Landau equations are obtainedby merely setting the left-hand side of the TDGL equations to zero.

We give the results of two simulations using the TDGL equations. Figure 3, on theleft, shows, through a series of snapshots in increasing time, the nucleation of vorticesat the boundaries of a sample and the development of the vortex lattice of figure 2.Figure 3, on the right, shows, through a series of snapshots in increasing time, themotion of vortices induced by an applied current. The applied field is perpendicularto the paper; the applied current is in the vertical direction and the vortices movefrom left to right.

For both the TDGL and its steady state predecessor, a variety of analytical resultshave been derived. For example, the global existence and uniqueness of appropriatelydefined weak solutions and the continuous dependence of the solution on the initialdata have been proved. An invariant integral has been identified and convergenceresults for semi-discrete and fully discrete Galerkin finite dimensional approximationshave been obtained (see, for example, Du 1994a, b).

3. The vortex state in non-ideal superconductors

The models and resulting behaviour discussed so far are complicated by a varietyof effects that can affect the vortex structure in type-II superconductors. We nowexplore some of these complications and, based on variants of the Ginzburg–Landau



Figure 3. On the left, the nucleation of vortices and the subsequent formation of the vortexlattice in a two-dimensional type-II superconductor. On the right, vortex motion in the presenceof a constant applied current in the vertical direction.

model, we use numerical simulations to illustrate some of the resulting changes thatoccur in the vortex structure.

(a ) Pinning of vorticesAs was already mentioned, the motion of vortices causes resistance. Further, in

a clean superconductor, any applied current will cause vortices to move. Thus, theidentification of mechanisms that can ‘pin’ vortices, i.e. stop them from moving,are of great interest. Here, we discuss three such mechanisms and how they can beeffectively modelled through variations in the Ginzburg–Landau equations.

Vortices can be pinned by normal impurities within the superconductor. The ap-pearance of such inclusions can be modelled by allowing α and ms to vary acrossthe sample. In regions occupied by the superconducting material, α < 0 and ms =twice the electron mass. In regions occupied by the normal, non-superconductingmaterial, we choose α > 0 and allow the mass to be a second material parameter. Innon-dimensional form, the parameters that define the model are |αn/αs| and ms/mn,where the subscripts s and n refer to the superconducting and normal materials, re-spectively. It was shown in Chapman et al. (1995a) that this model correctly accountsfor the physical behaviour at superconducting/normal and superconducting/vacuuminterfaces.

Another pinning mechanism is variation in thickness in thin films. Such materialscan be modelled by the full, three-dimensional Ginzburg-Landau model; however, ifone takes advantage of the thinness of the film, one can derive simplified models thatare much less costly to compute with. A variable thickness, thin film model is derivedin Chapman et al. (1996) where it is shown that, to leading order, the magneticfield is unaffected by the superconductor. Thus, if the applied field is a constant, toleading order, the magnetic field is constant throughout the sample. As a result, thevector potential can be determined independently of the order parameter. The lattercan be determined, to leading order, from a variable-coefficient version of the firstGinzburg-Landau equation using the known vector potential. This equation is givenby (

iκ∇+A0

)· d(

iκ∇+A0

)ψ + d(|ψ|2 − 1)ψ = 0 ,

where d(x, y) denotes a function describing the thickness variation of the film andA0 the known vector potential. Higher-order corrections can also be determined. In



Figure 4. On the left, the pinning of vortices by normal inclusions. In the centre, the pinning ofvortices by thinner regions in a variable thickness thin film. On the right, the pinning of vorticesby a grain boundary in an anisotropic superconductor.

addition, it has been shown, using this simplified model, that all superconductingmaterials, whether type-I or type-II in bulk, behave as type-II superconductors whenmade into sufficiently thin films. Furthermore, the rigorous connection between solu-tions of the thin film model and solutions of the three-dimensional Ginzburg-Landauequations has been established.

A third pinning mechanism involves grain boundaries in anisotropic superconduc-tors. Anisotropies can be accounted for within the Ginzburg-Landau formalism byassuming that the mass in the Gibbs free energy is a tensor, i.e. we replace ms witha tensor ms. If the lattice is aligned with the coordinate axes, ms is diagonal, i.e.

ms =

m1 0 00 m2 00 0 m3

For details, see, e.g. Chapman et al. (1995b). This model accounts for anisotropicbut homogeneous materials. It can be combined with the other models discussed inthis subsection to treat anisotropic, inhomogeneous materials as well as anisotropicthin films.

We now use some numerical simulations to see, in figure 4, how the vortex latticeis affected by the three pinning mechanisms. The parameters in these models are asfor figure 2, so that that figure gives the lattice structure in the absence of any ofthe pinning mechanisms. In the left of figure 4, we have pinning by the inclusion ofsome normal impurities within a two-dimensional superconductor. These impuritiesare located at the small squares evident in the figure; in these squares we have chosenαn = −αs and mn = ms. Note that one vortex is attracted to each of the ten smallimpurities. In the centre figure we have pinning by thin regions in a thin film. Thethin regions are again located at the small squares evident in the figure; in thesesquares, the thickness of the film is half of the thickness of the rest of the film.Again, note that one vortex is attracted to each of the ten small thin regions. Onthe right of figure 4, we have pinning by a grain boundary in a two-dimensionalsuperconductor. The position of that boundary is indicated by the diagonal line inthe figure. Above that line, we have that m1/m2 = 0.5 while below we have thereverse, i.e. m1/m2 = 2. Note that vortices are attracted to the grain boundary.Thus, the simulations of figure 4 show how each of the three pinning mechanismsdistort the vortex lattice of figure 2 by attracting vortices toward the pinning sites.



(b ) Josephson junctionsA simple Josephson junction consists of a thin slice of normal, non-superconducting

material sandwiched between slabs of superconducting material. An important prop-erty of such junctions is that, if the normal slice is thin enough, a resistancelesssupercurrent tunnels across the junction; this is known as the DC–Josephson effect .For junctions that can be modelled as one-dimensional objects, e.g. the normal sliceis of finite thickness but of infinite extent in the other two directions and the super-conducting slabs are of semi-infinite extent, De Gennes developed a theory relatingproperties in the superconductor on one side of the junction to the correspondingproperties in the superconductor on the other side (see, for example, De Gennes1966). If we let (·)+ and (·)− denote evaluation, at the interfaces with the normalmaterial, of a quantity in the superconductors on the two sides of the simple one-dimensional junction, the De Gennes relation is given by ψ(

1κ∇− iA

)ψ

+(M11 M12

M21 M22

) ψ(1κ∇− iA

)ψ

− ,where the Mij are real and are determined by the particular junction and depend onits thickness, the type of material, etc. De Gennes also gave the expression

J =1M12|ψ+ψ−| sin(φ+ − φ−),

for the tunnelling supercurrent flowing across the one-dimensional junction in termsof quantities in the superconductor on each side of the junction. The De Gennestheory does not given any detailed information about the vortex state near or at thejunction.

The model used in §3 a for treating normal impurities can also be used for Joseph-son junctions. In fact, it was shown in Chapman et al. (1995a) that all De Gennestheory for Josephson junctions can be recovered from the ‘variable α, variable m’model described in §3 a. For example, if d denotes the thickness of the normal slice,one finds that the above De Gennes relations hold with

M11 = cosh(2κd√α), M12 =

mn√α

sinh(2κd√α),

M21 =√α

mn

sinh(2κd√α), M22 = cosh(2κd

√α).

Here, we have assumed that the same superconducting material is used on both sidesof the junction; the model of Chapman et al. (1995a) can account for different super-conductors on each side of the simple junction and remains in complete agreementwith the De Gennes theory for that case as well. If one examines the expressionsfor the tunnelling current, J , and for M12, one sees that the model of Chapman etal. (1995a) predicts that, as expected, J decreases exponentially as d increases andas α increases. Furthermore, J also decreases as mn increases. Since large mn cor-responds to a highly insulating material, this is in agreement with the experimentalobservation that junctions made from insulating materials need to be thinner thanjunctions made from metals in order to obtain the same tunnelling current.

The model of Chapman et al. (1995a) not only recovers the De Gennes theoryfor simple Josephson junctions, but also can give detailed information about thevortex state in and near the junction. Of course, it can also treat bounded samples



Figure 5. Magnitude of the order parameter (top) and current (bottom) in a simple Josephsonjunction of thickness 1ξ, 3ξ, and 8ξ (left to right).

Figure 6. Vortex state in increasing fields; magnetic field strength = 0.9κ, 1.0κ, 1.2κ, 1.4κ,1.6κ, 1.7κ, 1.8κ and 1.9κ.

in multi-dimensions. Here, in figure 5, we give the structure of the vortex state inthree junctions having differing thicknesses. We now non-dimensionalize lengths byξ so that sample size is now 20ξ by 20ξ. For the thinnest junction of thickness,one coherence length, we see that vortices are attracted to the normal material inthe junction; the corresponding current plot shows that supercurrent is crossingthe junction, both as a shielding current near the boundary and as current loopscirculating around the vortices in the junction. For the junction of thickness 4ξ,the vortices have become much ‘weaker’, and much less supercurrent is crossingthe junction. For the 8ξ junction, the normal material is already thick enough thatno supercurrent is tunnelling across the junction; the two superconductors on each



side of the normal slice have become essentially uncoupled. Note that there is somesupercurrent flowing in the normal material parallel and near to the interfaces withthe superconducting materials; this effect, known as the proximity effect , is alsoeffectively treated by the model of Chapman et al. (1995a).

(c ) Surface superconductivity in high fieldsThe phase diagram of figure 1 holds in the idealized case of pure homogeneous

isotropic superconductors of infinite extent. For samples that have interfaces withnormal materials, e.g. bounded samples, the sharp transition from the vortex state forfields below Hc2 to the non-superconducting, normal state for fields above Hc2 doesnot occur as indicated by that phase diagram. In fact, superconductivity can persistfor fields higher than Hc2. For semi-infinite superconductors, it can be shown thatsuperconductivity persists up to a field Hc3 ≈ 1.69Hc2 (see, for example, Tinkham1975). Moreover, near Hc3, it is thought that the bulk of the superconductor isin the normal state and the superconducting state exists only near the surface ofthe sample which, in the semi-inifinite case, is just the planar interface betweenthe superconductor and the normal material. However, these simple one-dimensionalanalytical theories cannot account for bounded samples in predicting the structureof the superconducting state nor can they account for how, say, corners can affect thevalue of the highest applied field for which superconductivity persists. These theoriescannot say much, if anything, about the structure of the superconducting state forfields between Hc2 and Hc3 and not near either.

The Ginzburg–Landau model can be used, in principle, to examine the structureof the superconducting state for any value of the applied field and for boundedsamples. Thus, one can examine this state as one increases the field from a valuebelow Hc2 at which the superconductivity appears as a vortex lattice all the way upto a high enough field that superconductivity is destroyed. In figure 6, we presentsome preliminary computational results of such a study. For a sample of size 5ξ by 5ξ,we see that the vortex state (in this case a single vortex) persists relatively unchangedas one crosses the value Hc2 = κ, which is the value at which an infinitely largesuperconductor would lose superconductivity. At a value of the field above 1.4κ, thevortex state changes and four vortices appear; this state persists for fields above thevalue Hc3 = 1.69κ at which one-dimensional theories predict that superconductivityis totally destroyed. In fact, superconductivity persists for fields up to 2.09κ. Thisindicates that corners can aid in preserving superconductivity at higher fields.

(d ) Thermal fluctuationsThe idealized phase diagram of figure 1 is further complicated by thermal fluctu-

ations, i.e. the motion of vortices due to the vibrations of molecules in the super-conductor. This effect becomes especially pronounced as one approaches the criticaltemperature, Tc, above which an ideal superconductor loses it superconductivity.Indeed, thermal fluctuations can induce a (local) loss of superconductivity at tem-peratures below Tc and, conversely, induce (local) superconductivity at temperaturesabove Tc. Fluctuations can also cause depinning, especially near Tc. Indeed, shouldthe vibratory motion of vortices due to thermal fluctuations be of sufficiently largeamplitude, vortices may wander far enough away from a pinning site that they be-come permanently dislodged and, for example, continue moving in a resistive fashiondue to the application of a current.

Thermal fluctuations can be modelled (see Hohenberg & Halperin 1977) by intro-



Figure 7. Nucleation of vortices and development of the vortex lattice in the presence ofthermal fluctuations having variance two.

Figure 8. The effect of increasing temperature, modelled through the variance of the randomthermal fluctuations, on the vortex lattice; the variance ranges from 1 to 10.

ducing a random Langevin forcing term in the first TDGL equation, i.e.(∂ψ

∂t+ iκΦψ

)+(

iκ∇+A

)2

ψ − ψ + |ψ|2ψ = η.

Here η is a Gaussian random variable in time and space having zero mean. Thevariance increases as the temperature approaches Tc, i.e. as the T approaches Tc theprobability of having larger fluctuations increases.

We now present some preliminary computational results for thermal fluctuations.First, in figure 7, we show the nucleation of vortices and the development of thevortex lattice for the same configuration as in figure 2, but now we add a Langevinforcing term having variance two. Note that, although things look somewhat chaoticat the beginning, eventually we arrive at a lattice as in figure 2, but now distortedby the the effects of the thermal fluctuations. Of course, a true steady-state is neverachieved, but instead the individual vortices eventually reach a quasi-steady-statein which they vibrate about some fixed location. In figure 8, we show the effects of



increasing the temperature, which is modelled through increasing the variance in therandom forcing term. We give snapshots, taken at the same time, of computationalsimulations for different values of the variance. The time chosen is large enoughthat the quasi-steady-state described above has been reached. We see that as thevariance, i.e. the temperature, is increased, the effects of thermal fluctuations becomemore pronounced, and that for large enough variances, the vortex lattice becomesunrecognizable. This may model the transition to the vortex liquid state, i.e. thedestruction of the Abrikosov lattice at fields below Hc2.

ReferencesChapman, S. J., Du, Q. & Gunzburger, M. 1995a A Ginzburg–Landau type model of supercon-

ducting/normal junctions including Josephson junctions. Europ. Jl Appl. Math. 6, 97–114.Chapman, S. J., Du, Q. & Gunzburger, M. 1995b On the Lawrence–Doniach and anisotropic

Ginzburg–Landau models for layered superconductors. SIAM Jl Appl. Math. 55, 156–174.Chapman, S. J., Du, Q. & Gunzburger, M. 1996 A model for variable thickness superconducting

thin films. J. Appl. Math. Phys. 47, 410–431.De Gennes, P. 1966 Superconductivity in metals and alloys. New York: Benjamin.Du, Q. 1994a Global existence and uniqueness of solutions of the time-dependent Ginzburg–

Landau model for superconductivity. Applicable Analys. 52, 1–17.Du, Q. 1994b Finite element methods for the time-dependent Ginzburg–Landau model of su-

perconductivity. Comp. Math. Appl. 27, 119–133.

Gor’kov, L. & Eliashberg, G. 1968 Generalization of the Ginzburg–Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities. Sov. Phys. JETP 27,328–334.

Hohenberg, P. & Halperin, B. 1977 Theory of dynamic critical phenomena. Rev. Mod. Phys. 49,435–479.

Tinkham, M. 1975 Introduction to superconductivity. New York: McGraw-Hill.


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