Extended Uniform Ginzburg-Landau Theory for Novel Multiband Superconductors
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Extended Uniform Ginzburg-Landau Theory for Novel Multiband Superconductors
Brendan J. Wilson ∗ and Mukunda P. Das † Department of Theoretical Physics, Research School of Physics and Engineering,The Australian National University, Canberra ACT 0200, Australia
The recently discovered multiband superconductors have created a new class of novel supercon-ductors. In these materials multiple superconducting gaps arise due to the formation of Cooperpairs on different sheets of the Fermi surfaces. An important feature of these superconductors isthe interband couplings, which not only change the individual gap properties, but also create newcollective modes. Here we investigate the effect of the interband couplings in the Ginzburg-Landautheory. We produce a general τ (2 n +1) / expansion ( τ = 1 − T /T c ) and show that this expansionhas unexpected behaviour for n ≥
2. This point emphasises the weaker validity of the GL theoryfor lower temperatures and gives credence to the existence of hidden criticality near the criticaltemperature of the uncoupled subdominant band.
PACS numbers: 74.20.De, 74.20.Fg, 74.50.+r
I. INTRODUCTION
The BCS theory is considered as a conventional the-ory of superconductivity at a microscopic level, wherebosonic excitations like phonons or spin fluctuations playthe role of mediators in the formation of Cooper pairs andhence superconductivity occurs in a metallic state. In re-cent years many varieties of superconductors have beendiscovered which are unconventional. Some examples are1. cuprates, where anisotropic d-wave gaps occur withnodes (vanishing gap) at some symmetry points onthe Fermi surface ,2. co-existence of superconductivity with anti-ferromagnetism (CeCuSi ) or with ferromagnetism(UGe ) predominantly in heavy fermion systems ,3. coexistence of superconductivity with charge/spinordering (NbSe , Cuprates) ,4. non-centrosymmetric heavy fermion systems(CePt Si, CeInSi ) , where lack of inversionsymmetry gives rise to spin-orbit interaction withno definite parity in the ground state. In this casesinglet and triplet pairings coexist,5. strong electronic correlations dominated so-callednon-fermi liquid state, believed to be found in high T c oxides and in heavy fermion systems .In addition to these, there is another class of uncon-ventional superconducting systems, the novel multibandsuperconductors. In these systems two or more energybands are cut by the fermi energy giving rise to multi-ple energy gaps with different magnitudes in the differentFermi sheets. Recent measurements of tunnelling, point-contact spectroscopy, angle-resolved photoemission andspecific heat provide clear evidence of multiple gap struc-tures. Examples of such systems are MgB , RNi B C(R= Lu,Y), 2H-NbSe and many in the pnictide FeAsfamily. The BCS theory of superconductivity has been gener-alised to multiband systems. In one of our recent publi-cations we have given a brief appraisal of the history ofmultiband BCS theory and have presented a theory ofthe time-reversal symmetry broken state in the BCS for-malism. We are reminded that a phenomenological the-ory of superconductivity by Ginzburg and Landau [GL]was developed before the proposal of microscopic BCStheory. The GL approach is a successful theory of phasetransitions with many practical applications. The basisof this theory rests on two approximations correct aroundthe critical temperature ( T c ) (i) the order parameter, Ψ,is small near T c and (ii) Ψ ∼ τ / , where τ = 1 − ( T /T c ).Despite these limitations there is a myth that the GLtheory applies not only around T c , rather it is useful formuch lower temperatures. In early years soon after theappearance of BCS theory, Gor’kov established theequivalence of the BCS energy gap, ∆, with the orderparameter of the GL theory for single band supercon-ductors with the above conditions (i) and (ii).In multiband superconductors the equivalence hasbeen investigated by many authors (see a brief reviewin ref. ). In both theories (BCS and GL) an additionalinteraction term appears due to interband interaction,which is recognised as the Josephson term. This term isthe lowest order coupling between the gaps (in BCS) andorder parameter (in GL) in the different bands. The pres-ence of Josephson terms in multiband superconductorscauses several problems in the Gor’kov type derivations.Recently in a series of papers Vagov and coworkers have made detailed analysis and established a generalisa-tion of the standard GL theory (which is correct to τ / )by retaining additional terms in the expansion up to or-der τ (2 n +1) / . In practice they have analysed the n = 1corrections to the order parameter for 1, 2 and 3 bands. They call this formalism extended GL theory. Thisextended version with τ / corrections seems to have im-proved the validity of the GL expansion to some lowertemperatures away from T c in one- and two-band sys-tems.In this paper we adopt the microscopic approach ofGor’kov generally for uniform multiband systems withisotropic (spherical) Fermi surfaces. Other types of Fermisurfaces for dirty superconductors and with anisotropycan be done appropriately with more complications. Wepresent here our detailed calculations of BCS gaps andGL order parameters for superconductors with one andtwo bands.In Sec.2 we extend the Gor’kov technique to multi-band superconductors in the absence of an external mag-netic field, going beyond the standard/traditional modelof GL. The coefficients for all terms in the series expan-sion of the self-consistent gap equation are given explic-itly, and we show how to solve the resulting equations forthe gap functions.In Sec.3 results for the one band superconductor arepresented showing clearly the departure of the standardGL order parameter (with τ / ) while comparing withthe BCS result. Higher order corrections are reportedwith impressive agreement with the BCS. We see thateach additional term increases the range of τ for whichthe expansion is accurate.In Sec.4 similar results for the two band superconduc-tors are presented. These calculations are done for differ-ent interband couplings. In contrast with the single bandresults, additional terms in the two-band GL expansiononly improve the agreement with the BCS result up toa certain value for τ . Pushing beyond this point, theagreement becomes worse as additional terms are added.This disagreement is associated with the appearance ofa second critical temperature in the weak coupling limit.In Sec.5 we present the conclusions and summary ofthis work. II. DERIVATION OF EXTENDEDGINZBURG-LANDAU THEORY
The BCS theory is generalised to a multibandtheory by allowing multiple fermion operators, whichare identified by a band index, ν , and including a Joseph-son interband term in the interaction. This term allowsfor Cooper pairs to tunnel from band to band. With thisgeneralisation, the effective multiband BCS Hamiltonianin real space is given byˆ H eff = X σ X ν Z d x ˆ ψ † σν ( x ) (Π ν ( x ) − µ ν ) ˆ ψ σν ( x )+ X ν Z d x (cid:16) ∆ ∗ ν ( x ) ˆ ψ ↓ ν ( x ) ˆ ψ ↑ ν ( x ) + h . c . (cid:17) , (1)where Π ν ( x ) = m ν (cid:16) − i ~ ∇ − e A ( x ) c (cid:17) , A ( x ) is the vec-tor potential, ˆ ψ σν ( ˆ ψ † σν ) are fermionic annihilation (cre-ation) operators, ν , ν ′ are band indices, σ are spin in-dices, m ν is the electron mass, µ ν is the chemical po-tential, g νν ′ are the interband coupling parameters, and ∆ ν ( x ) = P ν ′ g νν ′ D ˆ ψ ν ′ ↑ ( x ) ˆ ψ ν ′ ↓ ( x ) E is the superconduct-ing gap.Following the Gor’kov technique, the Green function G ν,ω n ( x , x ′ ) and anomalous Green function F † ν,ω n ( x , x ′ )can be written as a pair of coupled integral equations : G ν,ω n ( x , x ′ ) = G (0) ν,ω n ( x , x ′ ) − ~ − Z d y G (0) ν,ω n ( x , y )∆ ν ( y ) F † ν,ω n ( y , x ′ )(2) F † ν,ω n ( x , x ′ ) = ~ − Z d y e G (0) ν,ω n ( x , y )∆ ∗ ν ( y ) G ν,ω n ( y , x ′ ) , (3)where G (0) ν,ω n ( x , y ) is the normal Green function and e G (0) ν,ω n ( x , y ) = G (0) ν, − ω n ( y , x ), ∆ ν ( x ) is the superconduct-ing gap function in band ν , and the fermionic Matsubarafrequency ω n = (2 n +1) πβ ~ , with β = 1 /k B T . The normalGreen functions satisfy the equations " i ~ ω n + ~ m ν (cid:18) ∇ + ie A ( x ) ~ c (cid:19) + µ ν G (0) ν,ω n ( x , x ′ )= ~ δ ( x − x ′ )(4) " − i ~ ω n + ~ m ν (cid:18) ∇ − ie A ( x ) ~ c (cid:19) + µ ν G (0) ν,ω n ( x , x ′ )= ~ δ ( x − x ′ ) . (5)Using substitution, we can transform equations (2) and(3) into decoupled nonlinear integral equations, and bycontinued substitution we can write the anomalous Greenfunction as a series expansion in the gap and the normalGreen function F † ν,ω n ( x , x ′ ) = ∞ X m =0 ( − m ~ m +1 m +1 Y j =1 Z d y i e G (0) ν,ω n ( x , y )∆ ∗ ν ( y ) × m Y j =1 G (0) ν,ω n ( y − , y )∆ ν ( y ) e G (0) ν,ω n ( y , y + )∆ ∗ ν ( y + ) × G (0) ν,ω n ( y + , x ′ ) . (6)The gap is defined in terms of the anomalous Greenfunction by X ν ′ (cid:2) g − (cid:3) νν ′ ∆ ∗ ν ′ ( x ) = lim η → + X n e − iω n η β ~ F † ν,ω n ( x , x ) . (7) A. Uniform field free case
In this paper we are interested in finding the meanvalue for the gaps, so we will consider the case where themagnetic field is zero and the gap does not depend on x ,so the superconductor is uniform. By requiring the gapto satisfy equation 7, we obtain the self-consistent gapequation in matrix formˇ g − .~ ∆ = ~R, (8)where ˇ g is the interband coupling matrix with elements g ν,ν ′ , ~ ∆ is a column vector with elements ∆ ν , and ~R is acolumn vector with elements given by R ν = ∞ X m =0 ∆ ν | ∆ ν | m P ν,m , (9) P ν,m = lim y + → y m +1 Y j =1 Z d y j ! Q ν,m ( { y } m +2 ) , (10) Q ν,m ( { y } m +2 ) =( − m β ~ m +1) X n m +1 Y j =1 G (0) ν,ω n ( y − , y − ) e G (0) ν,ω n ( y − , y ) , (11)with { y } m = { y , y . . . , y m } . Equation 8 is a coupledequation involving the gaps from all bands, ν . Theseequations must be solved simultaneously.The normal Green function can be solved in Fourierspace to find G (0) ν,ω n ( y , x ) = 1(2 π ) Z d k ~ e i k . ( y − x ) i ~ ω n − ξ ν,k , (12)with ξ ν,k = ~ k m ν − µ ν . Performing each of the real spaceintegrals in equation 10 produces a delta function, andthese can be used to compute all but one of the k spaceintegrals, resulting in the simplified expression P ν,m = ( − m β ∞ X n = −∞ Z d k ~ ω n ) + ξ ν,k ) m +1 = ( − m β N ν (0) ∞ X n = −∞ Z d ξ ~ ω n ) + ξ ) m +1 , (13)with N ν (0) is the density of states in band ν . When m = 0 this integral diverges logarithmically, and so mustbe cut off at the Debye energy, ~ ω D . In this case we find P ν, = β − N ν (0) ∞ X n = −∞ Z ~ ω D − ~ ω D d ξ ~ ω n ) + ξ ≈ N ν (0) A − a ν ln (cid:18) − τ (cid:19) , (14) A = ln (cid:18) ~ ω D e γ πk B T c (cid:19) , (15) a ν = − N ν (0) , (16) where γ ≈ . τ = 1 − T /T c with T c to be defined later. The remainingterms with m ≥ P ν,m = ( − m β N ν (0) ∞ X n = −∞ Z ∞−∞ d ξ ~ ω n ) + ξ ) m +1 = − b ν,m − τ ) m , (17) b ν,m = − N ν (0) ( − m (cid:0) m +1 − (cid:1) (2 m )! ζ (2 m + 1)(4 π ) m ( m !) ( k B T c ) m , (18)where ζ ( z ) is the Riemann zeta function. Putting thisback together we find R ν = N ν (0) A ∆ ν − a ν ln (cid:18) − τ (cid:19) ∆ ν − ∞ X m =1 b ν,m − τ ) m | ∆ ν | m ∆ ν . (19)We then regroup terms to rewrite equation 8 in theform0 = ˇ L.~ ∆ + ~W , (20) W ν = a ν ln (cid:18) − τ (cid:19) ∆ ν + ∞ X m =1 b ν,m − τ ) m | ∆ ν | m ∆ ν , (21)with ˇ L = ˇ g − − ˇ N (0) A , and ˇ N (0) is a diagonal matrixwith elements N ν (0) on the diagonal. B. Expansion in small τ Near the transition temperature, τ is a small param-eter, so we will expand equation 20 in powers of τ .To truncate this expansion, keeping only terms up to O (cid:0) τ (2 n +1) / (cid:1) , we first make the scaling∆ ν = τ / ¯∆ ν . (22)After scaling and then dividing through by τ / we find0 = ˇ L.~ ¯∆ + ~ ¯ W, (23)¯ W ν = a ν ln (cid:18) − τ (cid:19) ¯∆ ν + ∞ X m =1 b ν,m τ m (1 − τ ) m (cid:12)(cid:12) ¯∆ ν (cid:12)(cid:12) m ¯∆ ν . (24)Then the gap is expanded in powers of τ , as is all theother dependence on τ in ¯ W ν . The ¯∆ ν and ¯ W ν expan-sions are given by ¯∆ ν ( τ ) = ∞ X n =0 ¯∆ ( n ) ν τ n , (25)¯ W ν = ∞ X p =1 ¯ W ( p ) ν τ p . (26)We recover a set of equations for ¯∆ ( n ) ν by collectingpowers of τ in equation 23 and requiring that the equalityholds for all τ . The leading order behaviour is a constant.Collecting these constant terms leads to the lowest orderequation 0 = ˇ L.~ ¯∆ (0) . (27)This has a non-trivial solution if det ˇ L = 0. We choose T c to be the largest value such that this equation is satisfied.We note that the T c of the combined system depends onthe interband coupling. In the one band case, the wellknown solution for T c is T c = 2 e γ ~ ω D π exp (cid:18) − g N (0) (cid:19) , (28)while for the two-band case, the solution for T c is T c = 2 e γ ~ ω D π exp − g N (0) + g N (0) − p ( g N (0) − g N (0)) + 4 g N (0) N (0)2( g g − g ) N (0) N (0) ! (29) ≈ T c (cid:18) g N (0) g N (0)( g N (0) − g N (0)) + O (cid:0) g (cid:1)(cid:19) , (30)when g N (0) > g N (0), where T c is the critical tem-perature of the uncoupled first band, which is assumedto be the dominant band. We note that the critical tem-perature is enhanced over that of the dominant band dueto the interband coupling, regardless of sign.Now, since det ˇ L = 0, there is at least one eigenvectorof ˇ L with a zero eigenvalue. We shall assume that this isnon-degenerate, so that there is only one zero eigenvalue.We choose the base eigenvector to have the form ~η =[ ρ , ρ , . . . , ρ N ] T , (31) ρ i = c ,i c , , (32) c ijk...,lmn... =( − i + j + k + ... + l + m + n + ... M ijk...,lmn... , (33)where c ijk...,lmn... is the cofactor of the matrix ˇ L , and M ijk...,lmn... is the minor of ˇ L , defined as the determinantof the matrix obtained by removing the rows i, j, k, . . . and columns l, m, n, . . . from ˇ L . Assuming all ρ i are finiteand nonzero, we can then obtain a complete basis withthe remaining vectors ~η i = [ ρ , ρ , . . . , − ρ i , . . . , ρ N ] T . (34)The superconducting gaps can be written with this basisas ~ ¯∆ ( n ) = X j ψ ( n ) j ~η j . (35)Putting this back into equation 27 and using the fact thatˇ L.~η = 0 and ˇ L.~η j = 0, j = 1, we find ψ (0) j = 0 , j = 1 , (36) ~ ¯∆ (0) = ψ (0)1 ~η . (37) where ψ (0)1 is yet to be determined. The term linear in τ gives the equation0 = ˇ L.~ ¯∆ (1) + ~ ¯ W (1) , (38)¯ W (1) ν = a ν ¯∆ (0) ν + b ν, ¯∆ (0) ν (cid:12)(cid:12)(cid:12) ¯∆ (0) ν (cid:12)(cid:12)(cid:12) . (39)This mixes ~ ¯∆ (0) with ~ ¯∆ (1) , however, as pointed out inref 14, we can remove the ~ ¯∆ (1) dependence using the factthat ~η T . ˇ L = 0. Projecting this equation on to ~η andusing the solution for ~ ¯∆ (0) we find0 = X ν a ν η ,ν ψ (0)1 + b ν, η ,ν ψ (0)1 (cid:12)(cid:12)(cid:12) ψ (0)1 (cid:12)(cid:12)(cid:12) (40)= aψ (0)1 + b ψ (0)1 (cid:12)(cid:12)(cid:12) ψ (0)1 (cid:12)(cid:12)(cid:12) , (41)with a = P ν a ν η ,ν and b = P ν b ν, η ,ν . This has thesame form as the one band uniform G-L equation. Koganand Schmalian pointed out that the gradient term isalso the same as the one band G-L equation, and thusthere is only one coherence length near T c , and the orderparameters are proportional to each other.Projecting equation 38 onto the other basis vectors, ~η i , results in a further set of equations for the highercomponents, ψ (1) j .0 = X j =1 ~η Ti . ˇ L.~η j ψ (1) j + X ν a ν η i,ν η ,ν ψ (0)1 + b ν, η i,ν η ,ν ψ (0)1 (cid:12)(cid:12)(cid:12) ψ (0)1 (cid:12)(cid:12)(cid:12) (42)= X j =1 γ ij ψ (1) j + α i ψ (0)1 + β i, ψ (0)1 (cid:12)(cid:12)(cid:12) ψ (0)1 (cid:12)(cid:12)(cid:12) , (43)with γ ij = ~η i . ˇ L.~η j , α i = P ν a ν η i,ν η ,ν = a − a i ρ i and β i, = P ν b ν, η i,ν η ,ν = b − b i, ρ i . The indices i and j refer to the basis vectors, ~η j , not the band indices, ν .This process can be continued recursively to find theG-L approximation to any order. We provide the formfor the terms ¯ W (2) ν and ¯ W (3) ν .¯ W (2) ν = a ν ∆ (1) ν + b ν, (cid:18) (1) ν (cid:12)(cid:12)(cid:12) ∆ (0) ν (cid:12)(cid:12)(cid:12) + ∆ (0)2 ν ∆ (1) ∗ ν (cid:19) + 12 a ν ∆ (0) ν + 2 b ν, ∆ (0) ν (cid:12)(cid:12)(cid:12) ∆ (0) ν (cid:12)(cid:12)(cid:12) + b ν, ∆ (0) ν (cid:12)(cid:12)(cid:12) ∆ (0) ν (cid:12)(cid:12)(cid:12) , (44)¯ W (3) ν = a ν ∆ (2) ν + b ν, (cid:18) (2) ν (cid:12)(cid:12)(cid:12) ∆ (0) ν (cid:12)(cid:12)(cid:12) + ∆ (0)2 ν ∆ (2) ∗ ν (cid:19) + b ν, (cid:18) (0) ν (cid:12)(cid:12)(cid:12) ∆ (1) ν (cid:12)(cid:12)(cid:12) + ∆ (1)2 ν ∆ (0) ∗ ν (cid:19) + 12 a ν ∆ (1) ν + 2 b ν, (cid:18) (1) ν (cid:12)(cid:12)(cid:12) ∆ (0) ν (cid:12)(cid:12)(cid:12) + ∆ (0)2 ν ∆ (1) ∗ ν (cid:19) + b ν, (cid:18) (1) ν (cid:12)(cid:12)(cid:12) ∆ (0) ν (cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12) ∆ (0) ν (cid:12)(cid:12)(cid:12) ∆ (0)2 ν ∆ (1) ∗ ν (cid:19) + 13 a ν ∆ (0) ν + 3 b ν, ∆ (0) ν (cid:12)(cid:12)(cid:12) ∆ (0) ν (cid:12)(cid:12)(cid:12) + 4 b ν, ∆ (0) ν (cid:12)(cid:12)(cid:12) ∆ (0) ν (cid:12)(cid:12)(cid:12) + b ν, ∆ (0) ν (cid:12)(cid:12)(cid:12) ∆ (0) ν (cid:12)(cid:12)(cid:12) . (45)All higher order terms can similarly be produced fromthe full definition of ¯ W ν . III. SINGLE-BAND GINZBURG-LANDAUTHEORY
Applying this procedure to a single band superconduc-tor is fairly straight forward. The matrix ˇ L becomes anumber, and the equation for T c becomes trivial to solve.The basis vector η = 1 so that ¯∆ ( n ) = ψ ( n )1 in equation35.This procedure has been performed for the one bandcase to high order, with the results shown in figure 1.The BCS solution is given by the bold black dots inthe top plot. The thin red line that overshoots this isthe conventional τ / GL theory, while a selection ofplots with higher order corrections up to τ (2 n +1) / with n = 50 are also shown. The first correction, τ / is seenas the dashed line just above the BCS solution , whilehigher order corrections are almost indistinguishable ex-cept near τ = 1. Including a larger number of correctionsincreases the range of convergence, and it is presumedthat the infinite sum will converge for all τ <
1. How-ever for any large finite sum, the deviation near τ = 1 isexpected to remain large.On the bottom plot of figure 1 we plot the magnitudeof each term in the sum. The error of any finite sum is ap-proximately given by the magnitude of the next term inthe sum, and so this plot can be viewed as an estimation H a L
0. 0.25 0.5 0.75 1.0.1.2.3. 1 - Τ Ú j = i ¡ D H j L ¥ Τ j + (cid:144) (cid:144) k B T c
0. 0.05 0.11.51.61.71.81.9
0. 0.25 0.5 0.75 1.10 - - - Τ ¡ D H i L ¥ Τ i + (cid:144) (cid:144) k B T c H b L i BCS01251020304050 FIG. 1: (Colour online) (a) The extended GL expansion iscompared to a numerical calculation of the full BCS result.The extended G-L converges to the true solution on the region τ < τ it converges quickly to the BCSsolution. Inset: A close up of the region near τ = 1. Thereare singularities in the BCS function infinitesimally close to τ = 1 which prevent the extended G-L from converging atthis point. (b) The magnitude of the lowest terms in the G-L expansion are shown on a Log plot. The magnitude of thehigher terms decays quickly except near the point τ = 1 whereit remains finite. This shows that the expansion is convergingon the region τ < of the error in any given finite sum. The magnitude ofeach term decreases in general except near τ = 1, where,after the first few terms, it remains approximately con-stant.For the single band case, an exact form for each termin the expansion can be computed, though the numberof terms needed increases rapidly. We report the resultfor the first three terms in the expansion.∆ (0)1 = k B T c s π ζ (3)∆ (1)1 =∆ (0)1 (cid:18) −
34 + 93 ζ (5)196 ζ (3) (cid:19) ∆ (2)1 =∆ (0)1 (cid:18) − − ζ (5)784 ζ (3) + 8649 ζ (5) ζ (3) − ζ (7)1372 ζ (3) (cid:19) . (46) IV. TWO-BAND GINZBURG-LANDAUTHEORY
In two band GL, things progress in much the same way.However, there is now more a larger range of possibilities È D Ν BC S È (cid:144) T c H a L H b L -Τ È D Ν G L È (cid:144) T c H c L -Τ H d L g FIG. 2: (Colour online) Numerical calculations of the BCS gap is compared to the high expansion in the extended GL theory.We use the parameters g = 0 . g = 0 . N (0) = N (0) = 0 . ~ ω D = 0 .
09. a) BCS solution band 1. b) BCS Solution band2. c) GL solution band 1. d) GL solution band 2. The GL plots are calculated to order τ n +1 / where n = 50. due to three parameters in the interband coupling matrix, g νν ′ , especially the role of the interband interaction, g .We know from BCS theory that in the limit that theinterband coupling goes to zero, the two gaps are inde-pendent and each has their own critical temperatures,which we label T c and T c respectively. When the in-terband coupling is small but nonzero, there is still alarge change in the behaviour of the smaller gap near thetemperature T c . However the critical temperature ofthe combined system is an enhancement of the dominantband’s critical temperature.The exact lowest order solution can easily be calcu-lated, with the result∆ (0)1 = k B T c p L N (0) + L N (0) q L N (0) + L L N (0) s π ζ (3) (47)∆ (0)2 = k B T c L p L N (0) + L N (0) L q L N (0) + L L N (0) s π ζ (3) (48)The higher order terms become increasingly complicated,however the results for specific parameters are calculatednumerically to high order.In figure 2 we show plots of the BCS solution for arange of values for the interband coupling, g . In a) thefirst band is plotted, and it is seen that the interbandcoupling only has a weak effect on the behaviour of this band, while in b), the second gap shows a drastic changeas g increases, especially near T c , the critical tempera-ture of the second band in the noninteracting limit. Withthe increase of the coupling strength, the large up-swellof the second band near this critical temperature getswashed out, so that at large coupling the plot looks rem-iniscent of a one band BCS plot.Plots c) and d) depict the order parameters of band 1and 2 respectively as calculated using the extended GLformalism derived earlier. For 1 − τ & . − τ . . g . The pointwhere the solutions begin to disagree is very close to thelocation of T c , which in the small coupling limit is T c ≈ . T c . While this finite summation approach does notprove that the series is divergent, it is clear that the sumhas not converged in this range for the large number ofterms computed. We expect that in general the sumwill converge for all T & T c , but converge very slowlyor diverge for T . T c . Komendova et al. argue thatthere is a possibility of hidden criticality near T c whichbecomes critical in the limit that the coupling goes tozero. This feature is likely to be associated with theanomalous behavior of the GL gaps near this point, andis expected to prevent the series from converging belowthis point. Ú j = i ¡ D Ν H j L ¥ Τ j + (cid:144) (cid:144) k B T c Ν= H a L Ν= H b L H c L
0. 0.25 0.5 0.75 1.0.1.2.3. 1 -Τ H d L
0. 0.25 0.5 0.75 1.1 -Τ ¡ D Ν H i L ¥ Τ i + (cid:144) (cid:144) k B T c - - Ν= Ν= - - - -
0. 0.25 0.5 0.75 1.10 - - -Τ
0. 0.25 0.5 0.75 1.1 -Τ iBCS 1 5 20 400 2 10 30 50 FIG. 3: (Colour online) The extended GL expansion is compared to a numerical calculation of the full BCS result. In all plotswe use the parameters g = 0 . g = 0 . N (0) = N (0) = 0 . ~ ω D = 0 .
09. a) g = 0 . g = 0 .
01, c) g = 0 . g = 0 .
55. Columns one and three correspond to band 1 while columns two and four correspond to band 2. The firsttwo columns compare a finite sum of terms in the G-L expansion to the full BCS solution. The second two columns show themagnitude of each additional term on a log plot. In all plots the vertical black lines are located at what would be the criticalpoint of the second band in the uncoupled limit, T c /T c . We see that for 1 − τ & T c /T c the trend is for additional terms todecrease in magnitude, and the series seems to be converging, while for 1 − τ . T c /T c , the terms tend to grow and the seriesseems to be diverging. It is seen that the G-L expansion only converges to the BCS result in the region τ . − T c /T c < Surprisingly, while the BCS solution for the first bandshowed only a weak perturbation with the interband cou-pling, the non-convergent behaviour seen in the GL solu-tion of the smaller band also affects the dominant band.This occurs for any small non-zero interband coupling,even though the solution converges for all τ if the inter-band coupling is zero.In figure 3 the first two columns show the extended GL of band 1 and band 2 respectively as a function of1 − τ for various g . We can see that as the numberof terms included in the expansion is increased, the GLsolution departs from the BCS solution, shown as dots, inthe region T . T c , and increasing the number of termsincreases this difference. Therefore, with this number ofterms, the solution is not converging to the true solutionin this range. Ú j = i ¡ D Ν H j L ¥ Τ j + (cid:144) (cid:144) k B T c Ν= H a L Ν= H b L H c L
0. 0.25 0.5 0.75 1.0.1.2.3. 1 -Τ H d L
0. 0.25 0.5 0.75 1.1 -Τ ¡ D Ν H i L ¥ Τ i + (cid:144) (cid:144) k B T c - - Ν= Ν= - - - -
0. 0.25 0.5 0.75 1.10 - - -Τ
0. 0.25 0.5 0.75 1.1 -Τ iBCS 1 5 20 400 2 10 30 50 FIG. 4: (Colour online) The extended GL expansion is compared to a numerical calculation of the full BCS result. In all plotswe use the parameters g = 0 . g = 0 . N (0) = N (0) = 0 . ~ ω D = 0 .
09. a) g = 0 . g = 0 .
01, c) g = 0 . g = 0 .
55. Columns one and three correspond to band 1 while columns two and four correspond to band 2. The firsttwo columns compare a finite sum of terms in the G-L expansion to the full BCS solution. The second two columns show themagnitude of a selection of individual terms on a log plot. In all plots the vertical black lines are located at what would be thecritical point of the second band in the uncoupled limit, T c /T c . Because the critical temperatures are close in the uncoupledlimit, the extended GL solution at small coupling only has a very small region of validity. We also see that at larger interbandcoupling, the location of non-convergent is much lower than the point T c /T c . Thus, while the non-convergent behaviour isassociated with this point at small interband coupling, the location of this point is also a function of g . The second two columns show the magnitude of eachof the terms in the sum on a log plot. In these plots itis shown that there is approximately a pivot point abovewhich the magnitude of the terms decrease, while belowthis point the magnitude of the terms increase. At thepivot point the magnitude of the terms remains approxi-mately constant. The location of the pivot point is closeto the point T c /T c , especially for small interband cou-pling. The location of the point T c /T c is shown as avertical black line in the figure. As g increases the location of the pivot point seemsto move towards T = 0. However we know that T c is a constant. A possible reason for this behaviour ofthe pivot point is that as g increases, T c does indeedremain constant, but T c increases, so that T c /T c shouldmove towards 0 as g increases. It is this increase in T c that makes the non-convergent point move towards zeroas g increases.In figure 4 we have produced a similar plot to figure 3but where the gaps in the noninteracting limit have sim-ilar critical temperatures. With these parameters, theBCS solution shows that the dominant band is almostunperturbed by the interband interaction. At small in-terband interaction, the second band is weakly perturbedexcept near T c , however with increasing interband inter-action, the second band quickly becomes indistinguish-able from the first band. This is expected since the in-terband interaction causes the two bands to behave asa single band. Since the properties of the two gaps arealready similar in the uncoupled limit, only a reason-ably small interband interaction is required before thetwo bands behave like a single band.When we look at the GL solution, we see that withthese parameters and small interband coupling the re-gion of validity of the solution is very tiny. Even afterthe inclusion of a very large number of terms, the regionwhere the GL solution has converged to the BCS solu-tion is only in the range τ . .
1. When the interbandcoupling is increased, this range of convergence increasessignificantly. At large coupling, the solution convergesover almost the complete temperature range. In this casethe two gaps are almost identical.We see that in the two band case where the two gapsare close to degenerate and the interband coupling is veryweak, the GL approximation is only valid in a very smalltemperature region near T c , and the theory should beapplied with care. However, for the case where one bandis very dominant, or where the interband coupling is verylarge, the GL theory performs very well, and convergesquickly to the BCS result over a fairly large temperaturerange. V. CONCLUSION
In this paper we have reconstructed the relationshipof the BCS theory with the GL theory with the limita-tions developed by Gor’kov in his ground-breaking work.The theory has been restricted to the case of a uniformsystem, but has been extended to allow multiple bandsand large order in τ . This extends on the work of ref 14where the authors calculated a similar expansion keepingterms of order τ / in the presence of a magnetic field.We have shown that in a one band superconductor the τ / correction improves the magnitude of order param- eter closer to the BCS value. Higher order correctionsfor n ≥ τ (2 n +1) / improve the agreement with theBCS result except at T = 0, where the series for the gapappears to be nonconvergent.In the two band situation, the interband coupling playsa pivotal role in enhancing the smaller order parameterabove the T c value in the BCS model. As the interbandcoupling increases the point of inflection around T c healsgradually. At large interband coupling both gaps looksimilar to a one band solution. The critical temperatureof the system evolves smoothly out of the largest criti-cal temperature, T c , and is enhanced by the interbandcoupling.In the GL model there are significant differences forboth the gaps below T . T c . The large deviationpersists for weaker interband couplings despite includ-ing larger τ (2 n +1) / corrections. This issue is significantwhen T c is close to the critical temperature T c . In thiscase the range of validity of the GL solution can be ex-tremely small. The GL solution to the gaps below T c is unreliable, and therefore care must be taken when ap-plying the GL model to multiband superconductors.When the interband coupling is larger or when oneof the gaps is very dominant, the GL solution performsmuch better and including higher order terms can makethe solution close to the BCS value over a large tempera-ture range. Similar to the one band case, the point T = 0is nonconvergent in the multiband solution regardless ofinterband coupling.In summary we have clearly demonstrated the impor-tance of τ (2 n +1) / expansion for large n for multibandGL superconductors. This point emphasises the weakervalidity of the GL theory for lower temperatures, and es-pecially for applications with small interband coupling.We are of the opinion that any use or misuse of GL the-ory has to be carefully examined considering its domainof applicability. Acknowledgements
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