Introduction to the Ginzburg-Landau Equations

Transcription

1 Intodction to the Ginzbg-Landa Eations James Chabonnea Depatment of Physics & Astonomy, Univesity of Bitish Colmbia, Vancove, B.C. Canada, V6T 1Z1 (Dated: Novembe, 005 I. INTRODUCTION In 1950 Landa and Ginzbg poposed a theoy which phenomenologically descibes mch of the behavio seen in specondctos. Not only does it encapslate the wok done by F. London and H. London in explaining the Meissne effect, bt was sed to postlate some vey emakable phenomena. The focs of this pape is Abikosov s pediction of the votex, a line defect in the specondcto which caies antized magnetic flx. It is impotant to note that Ginzbg and Landa deived this theoy phenomalogically, befoe the BCS theoy of specondctivity was intodced, and that many yeas Gokov showed that it comes fom BCS natally. We will stat with the Landa-Ginzbg fee enegy and a deivation of the eations of motion [1]. In pat III the eations of motion will then be sed to show that the theoy contains the Meissne effect []. Pat IV will discss cylindically symmetic soltions which lead to votices and the antization of magnetic flx [1] [3]. Also, the eations of motion will be solved to investigate the stcte of the condensate in a votex [4]. Using this, the enegy of a single votex [1] will be discssed in pat V and the inteaction enegy between two votices will be fond [5] in pat VI. This will give s insight into the stability of votices in type I and type II specondctos. Finally, in pat VII, the citical magnetic fields fo type I [3] and type II [1] specondctos will be fond. II. THE GINZBURG-LANDAU ENERGY The Ginzbg-Landa enegy is based on the wok of Gote and Casimi who intodced the idea of an ode paamete ψ popotional to the density of specondcting electons to descibe the state of a specondcto. They postlated a fee enegy fo a specondcto nea citical tempeate T c. E = v ψ + ψ 4 (1 Landa noticed this idea cold be expanded by consideing a complex ode field ψ (x which cold be sed to descibe flctations in the ode paamete by adding a gadient to Gote and Casimi s gess of the fee enegy. He and Ginzbg cold then wite the fee enegy of a specondcto nea the citical tempeate T c. To investigate at a specondctos in magnetic fields, simila to F. London and H. London, they added the field enegy and a gage invaiant deivative to aive at, { E = dx ( m ia (x c ψ (x } v ψ (x + ψ (x ( A (x 8π This enegy can be minimized to yield the Landa-Ginzbg eations. Minimizing with espect to the vecto potential A gives s, 1 ( A ( A = ( 1 (ψ i ( 4π m i c A ψ i c A ψ ψ (3 ( james@phas.bc.ca

2 The ight hand side can be witten in tems of a Noethe cent[6], j = 1 ( (ψ i i c A ψ ( i c A ψ ψ (4 = 1 i ( ψ ψ ψ ψ c A ψ (5 This identification of the cect is citical and will late lead to the eslt that flx is antized inside a votex. Minimization of the fee enegy with espect to the ode field ψ yields, ( i m c A ψ = ψ ψ vψ (6 This will be sed to detemine the stcte of the flx tbe. Note that the Ginzbg-Landa eations ae invaiant nde the gage tansfomation A (x A (x + ϕ (x (7 ψ (x e i c ϕ(x ψ (x (8 This tansfomation can be sed to emove the phase of the ode paamete. Now that we have established the field eations we can begin to apply them. The fist will be a demonstation of the Meissne effect and a deivation of the London pepetation depth. III. THE MEISSNER EFFECT The Meissne effect follows fom eation 3 ite nicely. Conside it in catesian coodinates fo now, whee a specondcting state exists fo x > 0 and a nomal state fo x < 0. Using cl identities and taking a ewiting the cent we can wite eation 3 as, 1 4π A = mc We will se the pola decomposition of ψ as a ansatz. ψ (x = ( 1 ( ψ ψ ψ ψ i c A ψ v ρ (x eiϕ(x (10 Fo now ρ(x = [0, 1] whee 0 indicates a nomal state and 1 indicates a specondcting state. The states between ae called mixed states. Sbstititing this ansatz into eation 9 yields, A(x = mc ϕ (x v 4π ρ(x mc A(x v ρ(x (11 Let s assme we ae looking in a egion of the specondcto withot many distbances. This is the same as setting ρ(x = 1. Taking the cl of both sides gives s the London eation, with a penetation depth λ = mc 4π v A(x = v mc ϕ(x 4π v mc A(x (1 B(x = 4π v mc B(x (13. The soltion is B(x = e x λ which indicates that the magnetic field penetates λ past the sface of the specondcto. Compaing this to the London eation we notice that v indeed matches (9

3 p with the density of specondcting electons. It is also instctive to note that = e and m = m e. This is consistent with the picte that Coope pais ae esponsible fo the condensate. Having made se that the theoy contains the fndamental eslts of the London eations we can now see what new phenomena the theoy pedicts. 3 IV. VORTEX LINES The votex line soltion comes fom solving the eations of motion in cylindical coodinates. A votex is a cylindcally symetic line defect which exists in an othewise ndistbed specondcto. It is simila to a the flid votices that ae fomed when wate goes down a dain. In a specondcto the electons otate aond a coe whee the density of specondcting electons dops to zeo. We ll fist investigate how this stcte leads to the antization of magnetic flx. A. Flx Qantization An indication that somthing inteesting is happening comes fom o definition of the cent j given by 4. If we solve eation 4 fo the vecto potential we get, A = c j ψ + hc 1 1 ( ψ i ψ ψ ψ ψ (14 In cylindical coodinates, eation 6 incidates that as 0, ρ 0. If the specondcto is not in a voltage potential then cent can only be podced by distbances in the specondcto. Fa away fom = 0, the specondcto is in an ndistbed state so j = 0. Sbstitting the ansatz fo ψ gives, A = c 1 1 ( ψ i ψ ψ ψ ψ (15 A = c ϕ (x (16 Integating on a closed conto aond the votex leads to a antization condition, A dl = c ϕ (x dl = hc πn (17 whee n is an intege. We can se Stokes theoem to see that the conto integal of A is also the flx thogh the sface. A dl = A ds = B ds = Φ (18 Eating the two expessions indicates that the flx is antized with antm nmbe n. Φ = nφ o with Φ o = πhc (19 The intege n is called the winding nmbe and is an indication of the stength of the votex. It can be shown that in cetain sitations a single votex of winding n = N will decay into N voticies each with a winding nmbe of n = 1 [7]. Also notice that the flx away fom the votex is independent of the adis of the loop we integate aond. These will be consideations in choosing the ansatz in the next section. Right now it is nclea whee the flx actally is. As it has been deived it looks like it penetates the entie plane. To find whee this might be localized we have to solve the othe eation of motion govening the density of the specondcto.

4 4 B. The Stcte of the Votex The poblem with solving the field eations in cylindical coodinates is that they ae copled, non-linea diffeential eations. We ae looking fo defects so we will no longe assme that ψ(x is constant. To make it easie to decople and lineaize these eations it is convenient to define, v ψ = ρ ( eiφ (0 A = c a ( ˆφ (1 whee ρ (, a ( 1 as and ρ (, a ( 0 as 0 and and φ the cylindical coodinates. Following the discssion in pat A the phase of ψ(x is chosen to mimic a votex with winding nmbe n = 1. We can fthe define, ρ ( = 1 + σ ( ( a ( = 1 + α ( (3 sch that σ (, α ( 0 as. Let s stat by sbsitting 0 and 1 into eation 3. ( c 4π a( ( ˆφ = π v σ mc v a( σ (4 Witing the coss podcts in cylindical coodinates and sing and 3 we get, α + 1 α 1 α = πv mc α(1 + σ (5 We can lineaize this eation by taking. We keep only the tems linea in α and σ and the lone 1 goes to zeo. This leaves s with a modified bessel eation of the fist ode. α + 1 ( α λ α = 0 (6 Whee λ is the London peetation depth we deived ealie. We want a soltion that goes to zeo as so we choose the soltion to be a modified bessel fnction of the second kind, α = 1 λ K 1 ( λ. Going back thogh all the sbstittions and then sing a gage tansfomation we find that the vecto potential is, A = c ( λ K 1 ˆφ (7 λ This is the Meissne effect in cylindical coodinates. As we move fom the coe of the votex into the specondcting mateial the magnetic field decays. We will now look at the stcte of ψ as a fnction of the adis. All we know know is that it stats at zeo in the coe and goes to v at infinity. We stat by sbstitting 0 and 3 into 6. Sbstitting in and 3 and lineaizing yields, ( v 1 m ρ ρ a ρ + aρ 1 = ( ρ + ρ 3 v 3 1 (8 σ = 4mv σ (9

5 This is a modified bessel eation of the zeoth ode. The soltion is, ( σ(x = K 0 ξ The antity ξ = mv is called the coheence length. It gives a length scale fo the change in density ψ fom the non-specondcting coe at = 0 to the ndistbed specondcto = O(ξ. We can stat to ndestand what is happening in a votex. It is a distbance which has a non-specondcting coe of adis ξ it caies anta of magnetic flx. Becase the specondcto displays the Meissne effect this flx can only be caied along the coe of the votex, whee specondctivity is destoyed. Essentially the votex is a tbe of magnetic flx allowing a magnetic field to penetate the specondcto. 5 (30 V. THE ENERGY OF A VORTEX Having solved the field eations fo ψ(x and A(x it is now possible to get a ogh estimate of the enegy of a votex. If we take the field eation 6 and sbstitte it into eation 1 we get an expession fo the fee enegy. { } v E = dx ( ρ4 A + (31 8π We know that zeoth ode Bessel fnctions have the following popety, dk 0 (µ d and, in cylindical coodinates, is a Geen s fnction fo Using these on 7, when 0, we get, A = 1 = µk 1 (µ (3 ( µ K 0 (µ = πδ ( (33 µk 1(µ = 1 K 0(µ = µ K 0 (µ (34 It can be shown that at lage the leading ode of ρ 4 is ξ E = 0 ddφ v ξ + [1]. Using all of this the enegy becomes, ( ( c λ K 0 λ 8π (35 The fist tem is easily integated, bt is nbonded at its limits. Instead of = we se a ctoff = Λ which is the size of the containe holding the specondcto. We emove the singlaity at = 0 by neglecting the coe of the votex < ξ. The second tem is evalated by sing K 0 0 ( = 1. The enegy pe nit length of the votex becomes, E = π m v log ( Λ + 1 ξ 8 ( c (36 λ The fact that a ctoff is eied indicates that a votex can only exist in a containe of finite size. Now that we have calclated the enegy fo a single votex it will be inteesting to look at two voticies and the inteaction enegy between them. VI. INTERACTION ENERGY BETWEEN TWO VORTICES The philosophy behind calclating the intevotex foce is to find the enegy of the entie system and then sbtact off the enegy of the individal voticies as oiginally otlined by Kame [8]. The technie we will se was intodced

6 by Speight [5]. The same philosophy is sed bt the actal calclation becomes mch less cmbesome. We will edce the theoy to a non-inteacting, linea one and model the voticies as point soces. The inteaction enegy is then easily calclated fom this linea theoy. Stat by lineaizing the theoy, only keeping tems that ae adatic in σ, whee σ is defined as ealie. { v E fee = dx m ( σ + 1 (( A + Aλ } 8π + v σ (37 6 And the soce tems ae, E soce = dx {τσ + j A} (38 whee τ and j ae the soces fo the fields σ and A. Minimizing this we get the eations of motion, ( ξ σ = v τ (39 ( 1λ A = 4πj (40 We want to solve fo the soces j and τ sch that they have the same asymtotic soltions we obtained ealie in 7 and 30. Using 33 and the deivative of 33 we can solve fo the soces. τ = v m πδ (x (41 j = c δ (x x ˆφ (4 The inteaction enegy is fond by sbstitting j = j 1 + j, A = A 1 + A, τ = τ 1 + τ and σ = σ 1 + σ into the total enegy E = E fee + E soce and sbtacting of the enegies of the votices, leaving only coss tems. The sbscipts 1 and efe to two sepeate voticies and positions x 1 and x espectivly. The coss tems left ove ae intepeted as the inteaction enegy. E inteaction = dx {τ 1 σ + j 1 A } (43 Using 1,, 41 and 4 the inteaction enegy can be witten, { ( E inteaction = dx v m πδ (x x 1 K 0 ξ (x x ( ( ( c 1 d = λ K 0 π v d λ m K 0 ξ ( ( ( = π v d d K 0 K 0 m λ ξ c δ (x x 1 x ( } c x λ K x 1 (44 λ whee d = x 1 x. Thee ae two tems woking to oppose each othe. The fist tem is a eplsive foce simila to the foce between two wies with cents in opposite diections. The cent in this case is cased by the electons otating aond the votex. Twovotices placed side by side will have cents nning in the opposite diection and be epelled. We can see fom eation 4 that the cent is in the ˆφ diection aond the votex. The second tem is an attactive foce cased by the specondcto pefeing to be in a state with no defects and attemping to estoe ode by making only one votex. When the fist tem is lage E inteaction is positive and the voticies epel. When the second tem (45 (46

7 7 is lage the votices attact. What govens this is the elative size of λ and ξ. Fo votices to epel, d d λ < ξ (47 eaanging yields, κ > 1 whee κ = λ ξ (48 The dimensionless antity κ is the famos Ginzbg-Landa paamete sed to detemine whethe a specondcto is type I o type II. A type II specondcto is one which allows patial penetation of a magnetic field. A type I specondcto is one which flly displays the Meissne effect. If κ > 1 then votices epel fom each othe and they will fom a tiangla lattice [9] [10], each votex caying a anta of flx Φ 0. This acconts fo the patial penetation of the magnetic field exhibited by type II specondctos. If κ < 1 then all the votices attact each othe and collapse. The specondcto now has no mechanism to cay flx and exhibits the Meissne effect, behaving like a type I specondcto. VII. CRITICAL MAGNETIC FIELDS Type I and type II specondndctos have anothe distingishing feate, the magnetic fields at which the Meissne effect is destoyed. A type I specondcto will display the Meissne effect ntil a citical extenal field B ci destoys the specondcting state. A type II specondcto will display the Meissne effect ntil a citical field B cii when votices stat to fom and allow pat of the field to penetate it. Inceasing the magnetic field stength fthe will ceate moe and moe voticies ntil thee ae so many specondctivity is destoyed. Let s fist conside a type I specondcto. The density ψ is nifom and thee is no magnetic field inside so eation 1 ickly becomes, E condensate = V ψ 4 V v ψ (49 whee V is the volme of the specondcto. This has a minima at ψ = v. Applying an extenal magnetic field changes the enegy by B 8π. If we set E condensate = 0 the condenstate has been destoyed and we get a citical magnetic field. 4πv B ci = (50 Now conside a type II specondcto. Thee ae both enegy gadients and magnetic fields inside the specondcto. We se the enegy of a votex we calclated ealie and this time the magnetic field inside the votex B int coples with the extenal filed B ext thogh the inteaction tem. E votex = π m v log ( Λ + 1 ξ 8 ( c λ dx B int B ext 8π The last tem can be simplified as this integal is the flx antization, dx B int = π c. The enegy E = 0 is when a votex will fist fom inside the specondcto. B cii = 4π v mc ( π log(λ ξ Comapaing the two citical fields we see that B cii is mch smalle than B ci. This is expected becase a type II specondcto only has to let one antm of flx thogh at B cii, whee B ci has the enegy to destoy the entie state. (51 (5

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