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Ginzburg-Landau theory of vortices in a multi-gap superconductor

M. E. Zhitomirsky and V.-H. DaoCommissariat a lEnergie Atomique, DMS/DRFMC/SPSMS,17 avenue des Martyrs, 38054 Grenoble, Cedex 9 France

(Dated: February 2, 2008)

The Ginzburg-Landau functional for a two-gap superconductor is derived within the weak-couplingBCS model. The two-gap Ginzburg-Landau theory is, then, applied to investigate various magneticproperties of MgB2 including an upturn temperature dependence of the transverse upper critical fieldand a core structure of an isolated vortex. Orientation of vortex lattice relative to crystallographicaxes is studied for magnetic fields parallel to the c-axis. A peculiar 30-rotation of the vortex latticewith increasing strength of an applied field observed by neutron scattering is attributed to themulti-gap nature of superconductivity in MgB2.

PACS numbers: 74.70.Ad, 74.20.De, 74.25.Qt

I. INTRODUCTION

Superconductivity in MgB2 discovered a few yearsago1 has attracted a lot of interest both from funda-mental and technological points of view.2 Unique phys-ical properties of MgB2 include Tc = 39 K, the high-est among s-wave phonon mediated superconductors,and the presence of two gaps 1 7 meV and 2 2.5 meV evidenced by the scanning tunneling3,4 and thepoint contact5,6 spectroscopies and by the heat capacitymeasurements.7,8,9,10 The latter property brings back theconcept of a multi-gap superconductivity11,12 formulatedmore than forty years ago for materials with large dispar-ity of the electron-phonon interaction for different piecesof the Fermi surface.Theoretical understanding of normal and supercon-

ducting properties of MgB2 has been advanced in thedirection of first-principle calculations of the electronicband structure and the electron-phonon interaction,which identified two distinct groups of bands with largeand small superconducting gaps.13,14,15,16,17,18,19 Quan-titative analysis of various thermodynamic and trans-port properties in the superconducting state of MgB2was made in the framework of the two-band BCSmodel.20,21,22,23,24,25,26,27,28 An outside observer wouldnotice, however, a certain lack of effective Ginzburg-Landau or London type theories applied to MgB2. Thisfact is explained of by quantitative essence of the dis-cussed problems, though effective theories can often givea simpler insight. Besides, new experiments constantlyraise different types of questions. For example, recentneutron diffraction study in the mixed state of MgB2has found a strange 30-reorientation of the vortex lat-tice with increasing strength of a magnetic field appliedalong the c-axis.29 Such a transition represents a markedqualitative departure from the well-known behavior ofthe Abrikosov vortex lattice in single-gap type-II super-conductors. Nature and origin of phase transitions inthe vortex lattice are most straightforwardly addressedby the Ginzburg-Landau theory.In the present work we first derive the appropriate

Ginzburg-Landau functional for a two-gap superconduc-

tor from the microscopic BCS model. We, then, in-vestigate various magnetic properties of MgB2 usingthe Ginzburg-Landau theory. Our main results includedemonstration of the upward curvature of Hc2(T ) fortransverse magnetic fields, investigation of the vortexcore structure, and explanation of the reorientationaltransition in the vortex lattice. The paper is organizedas follows. Section 2 describes the two-band BCS modeland discusses the fit of experimental data on the tempera-ture dependence of the specific heat. Section 3 is devotedto derivation of the Ginzburg-Landau functional for atwo-gap weak-coupling superconductor. In Section 4 wediscuss various magnetic properties including the uppercritical field and the structure of an isolated vortex. Sec-tion 5 considers the general problem of an orientationof the vortex lattice in a hexagonal superconductor inmagnetic field applied parallel to the c-axis and, then,demonstrates how the multi-gap nature of superconduc-tivity in MgB2 determines a reorientational transition inthe mixed state.

II. THE TWO-BAND BCS MODEL

A. General theory

In this subsection we briefly summarize the thermo-dynamics of an s-wave two-gap superconductor with theaim to extract subsequently microscopic parameters ofthe model from available experimental data for MgB2.We write the pairing interaction as

VBCS =

n,n

gnn

dxn(x)n(x)n(x)n(x) ,

(1)where n = 1, 2 is the band index. A real space represen-tation (1) is obtained from a general momentum-spaceform of the model11,12 under assumption of weak mo-mentum dependence of the scattering amplitudes gnn .We also assume that the active band has the strongestpairing interaction g11 = g1 compared to interaction inthe passive band g22 = g2 and to interband scattering

http://arXiv.org/abs/cond-mat/0309372v1

2

of the Cooper pairs g12 = g21 = g3. Defining two gapfunctions

n(x) =

n

gnnn(x)n(x) (2)

the total Hamiltonian is transformed to the mean-fieldform

HMF = Econst +

n

dx[

n(x)h(x)n(x)

+ n(x)n(x)

n(x) + h.c.

]

, (3)

h(x) being a single-particle Hamiltonian of the normalmetal. The constant term is a quadratic form of anoma-lous averages n(x)n(x). Using Eq. (2) it can beexpressed via the gap functions

Econst=1

G

dx[

g2|1|2+g1|2|2 g3(12+21)]

(4)with G = det{gnn} = g1g2 g23 . The above expres-sion has to be modified for G = 0. In this case the twoequations (2) are linearly dependent. As a result, the ra-tio of the two gaps is the same for all temperatures andmagnetic fields 2(x)/1(x) = g3/g1, while the constantterm reduces to Econst =

dx |1|2/g1.The standard Gorkovs technique can then be applied

to derive the Greens functions and energy spectra in uni-form and nonuniform states with and without impurities.In a clean superconductor in zero magnetic field the twosuperconducting gaps are related via the self-consistentgap equations

n =

n

nnn

D

0

d

2 +2ntanh

2 +2n

2T

(5)with dimensionless coupling constants nn = gnnNn ,Nn being the density of states at the Fermi level foreach band. The transition temperature is given byTc = (2De

C/)e1/, where D is the Debay frequency,C is the Euler constant and is the largest eigenvalue ofthe matrix nn :

= (11 + 22)/2 +

(11 22)2/4 + 1221 . (6)

Since > 11, the interband scattering always increasesthe superconducting transition temperature compared toan instability in a single-band case. The ratio of the twogaps at T = Tc is 2/1 = 21/( 22). At zerotemperature the gap equations (5) are reduced to

n =

n

nnn ln2Dn

. (7)

By substituting n = 2Drne1/ the above equation is

transformed to

rn =

n

nnrn

(

1

ln rn

)

. (8)

For 1/ ln rn, one can neglect logarithms on the righthand side and obtain for the ratio of the two gaps thesame equation as at T = Tc implying that 2/1 istemperature independent.30 This approximation is validonly for rn 1, i.e., if all the coupling constants nnhave the same order of magnitude. (For g23 = g1g2 theabove property is an exact one: the gap ratio does notchange neither with temperature nor in magnetic field.)However, for g3 g2 < g1, the passive gap 2 is signifi-cantly smaller than the active gap 1 and r2 1 so thatthe corresponding logarithm cannot be neglected. It fol-lows from Eq. (8) that the ratio 2/1 increases betweenT = Tc and T = 0 for small g3. Such variations becomemore pronounced in superconductors with larger valuesof , which are away from the extreme weak-couplinglimit 1. Ab-initio calculations indicate that MgB2has an intermediate strength of the electron-phonon cou-pling with 12(21) 11

3

0 0.25 0.5 0.75 1T/Tc

0

0.5

1

1.5

2C

/CN

1:0.3:0.22:0.2:0.8

FIG. 1: Theoretical dependence of the specific heat in thetwo-band BCS model. Numbers for each curve indicate valuesof g1, g2, and g3 (N1 = 0.4, N2 = 0.6). Open circles are theexperimental data.7,10

bands,15,16,17,18 the two -bands can be represented as asingle active band, which has N1 = 0.4N(0) of the totaldensity of states and drives superconducting instability,whereas a combined -band contributes N2 = 0.6N(0)to the total density of states and plays a passive role inthe superconducting instability. The above numbers areconsistent with N1 = 0.45N(0) and N1 = 0.42N(0) forthe partial density of states of the of the electrons in the-bands obtained in the other studies.14,17

The gap equations (5) have been solved self-consistently for N2/N1 = 1.5 and various values of cou-pling constants. The specific heat is calculated from

C(T ) =

nk

EnkdnF (Enk)

dT, (10)

where Enk =

k +2n is a quasiparticle energy foreach band and nF () is the Fermi distribution. Figure 1shows two theoretical fits to the experimental data ofGeneva group7,10 using a weak g1N1 = 0.4 and a moder-ate g1N1 = 0.8 strength of the coupling constant in theactive band. Constants g2 and g3 have been varied toget the best fits. In the first case the gap ratio changesin the range 1/2 =3.2.5 between T = Tc and T = 0,while in the second case 1/2 2.7. Both theoreticalcurves reproduce quite well the qualitative behavior ofC(T ). Somewhat better fits can be obtained by increas-ing the partial density of states in the -band. Quantita-tive discrepancies between various theoretical fits and theexperimental data are, however, less significant than dif-ferences between different samples.10 We, therefore, con-clude that though the specific heat data clearly agreewith the two-gap superconducting model in the regimeof weak interband interaction, a unique identification ofcoupling constants is not possible from available data.

III. THE GINZBURG-LANDAU FUNCTIONAL

We use the microscopic theory formulated in the pre-vious section to derive the Ginzburg-Landau functionalof a two-gap superconductor. In the vicinity of Tc theanomalous terms in the mean-field Hamiltonian (3) aretreated as a perturbation Va. Then, the thermodynamicpotential of the superconducting state is expressed as

s = Econst 1

ln

T exp[

0

Va()d]

, (11)

where = 1/T . Expansion of Eq. (11) in powers ofVa yields the Ginzburg-Landau functional. Since thenormal-state Greens functions are diagonal in the bandindex, the Wicks decoupling of Va in s does not pro-duce any mixing terms between the gaps. As a result, theweak-coupling Ginzburg-Landau functional has a singleJosephson-type interaction term:

FGL =

dx[

1|1|2 + 2|2|2 (12 +21)

+ 121|1|4+ 122|2|4+K1i|i1|2+K2i|i2|2]

,

i = i + i2

0Ai, 1,2 =

g2,1G

N1,2 ln2De

C

T, (12)

n =7(3)Nn162T 2c

, =g3G, Kni =

7(3)Nn162T 2c

v2Fni ,

0 being the flux quantum. For > 0, the interactionterm favors the same phase for the two gaps. For < 0,if, e.g., the Coulomb interactions dominate the interbandscattering of the Cooper pairs and g3 < 0, the smaller gapacquires a -shift relative to the larger gap.33,34

The gradient term coefficients depend in a standardway on the averages of Fermi velocities vFn over vari-ous sheets of the Fermi surface. Numerical integrationof the tight-binding fits16 yields the following results: forthe -band v2Fx = 2.13 (3.55, 1.51) and v2Fz = 0.05(0.05, 0.05); for the -band v2Fx = 1.51 (1.47, 1.55) andv2Fz = 2.96 (2.81, 3.10) in units of 1015 cm2/s2, num-bers in parentheses correspond to each of the constituentbands. The effective masses of the quasi two-dimensional-band exhibit a factor of 40 anisotropy between in-plane and out of plane directions. In contrast, the three-dimensional -band has a somewhat smaller mass alongthe c-axis. Using N2/N1 = 1.5 we find that the in-planegradient constants for the two bands are practically thesame K2/K1 1.06, while the c-axis constants differby almost two orders of magnitude K2z/K1z 90.A very simple form of the two-gap weak-coupling

Ginzburg-Landau functional is somewhat unexpected.On general symmetry grounds, there are possiblevarious types of interaction in quartic and gradientterms between two superconducting condensates of thesame symmetry, which have been considered in theliterature.35,36,37 The above form of the Ginzburg-Landau functional is, nevertheless, a straightforward

4

extension of the well-known result for unconventionalsuperconductors. For example, the quartic term fora momentum-dependent gap is |(k)|4 in the weak-coupling approximation.38,39 In the two-band model(k) assumes a step-like dependence between differentpieces of the Fermi surface, which immediately leads tothe expression (12).The Ginzburg-Landau equations for the two-gap su-

perconductor, which are identical to those obtained fromEq. (12), have been first derived by an expansion ofthe gap equations in powers of .30 Recently, a simi-lar calculation has been done for a dirty superconductor,with only intraband impurity scattering and the corre-sponding form of the Ginzburg-Landau functional hasbeen guessed, though with incorrect sign of the couplingterm.26 Here, we have directly derived the free energyof the two-gap superconductor. The derivation can bestraightforwardly generalized to obtain, e.g. higher-ordergradient terms, which are needed to find an orientation ofthe vortex lattice relative to crystal axis (see below). Wealso note that strong-coupling effects, e.g., dependence ofthe pairing interactions on the gap amplitudes, will pro-duce other, generally weaker, mixing terms of the fourthorder in . The interband scattering by impurities cangenerate a mixing gradient term as well.Finally, for G = (g1g2 g23) < 0 a number of spurious

features appears in the theory: the matrix nn and thequadratic form (4) acquire negative eigenvalues, while aformal minimization of the Ginzburg-Landau functional(12) leads to an unphysical solution at high tempera-tures. Sign of 2/1 for such a solution is opposite tothe sign of g3. The origin of this ill-behavior lies in the ap-proximation of positive integrals on the right-hand sideof Eq. (5) by logarithms, which can become negative.Therefore, negative eigenvalues of nn and Econst yieldno physical solution similar to the case when the BCStheory is applied to the Fermi gas with repulsion. Theconsequence for the Ginzburg-Landau theory (12) is thatone should keep the correct sign of 2/1 and use theGinzburg-Landau equations, i.e., look for a saddle-pointsolution rather than seeking for an absolute minimum.

IV. THE TWO-GAP GINZBURG-LANDAU

THEORY

In order to discuss various properties of a two-gap su-perconductor in the framework of the Ginzburg-Landautheory we write 1 = a1t with a1 = N1, t = ln(T1/T ) (1T/T1) and T1 = (2DeC/)eg2/GN1 for the first ac-tive band and 2 = 20 a2t with a2 = N2, 20 =(11 22)/GN1 for the passive band.

A. Zero magnetic field

For completeness, we briefly mention here the behav-ior in zero magnetic field. The transition temperature

is found from diagonalization of the quadratic form inEq. (12):

tc =202a2

2204a22

+2

a1a2. (13)

For small , one finds tc 2/(a120). Negative sign oftc means that the superconducting transition takes placeabove T1, which is an intrinsic temperature of supercon-ducting instability in the first band. The ratio of the twogaps = 2/1 = /(20 a2tc). Below the transitiontemperature, the gap ratio obeys

2 +213(a1t+ ) = 0 . (14)

For small , one can approximate /2 and due to adecrease of 2 with temperature, small to large gap ratio increases away from tc.

B. The upper critical field

1. Magnetic field parallel to the c-axis

Due to the rotational symmetry about the c-axis, thegradient terms in the a-b plane are isotropic and canbe described by two constants Kn Kn. The lin-earized Ginzburg-Landau equations describe a system oftwo coupled oscillators and have a solution in the form1 = c0f0(x) and 2 = d0f0(x), where f0(x) is a stateon the zeroth Landau level. The upper critical field isgiven by Hc2 = hc20/2

hc2 =a1t

2K1 2

2K2+

(

a1t

2K1+

22K2

)2

+2

K1K2(15)

The ratio of the two gaps = d0/c0 along the uppercritical line is

=

2 +K2hc2. (16)

The above expression indicates that an applied mag-netic field generally tends to suppress a smaller gap.Whether this effect overcomes an opposite tendency toan increase of 2/1 due to a decrease of 2 withtemperature depends on the gradient term constants.For example, in the limit 20 we find from (16) /[20 (a2 a1K2/K1)t]. If K2 is significantlylarger than K1, while a2 a1, the smaller gap is quicklysuppressed along the upper critical field line. However,for MgB2 one finds K2/K1 1 for in-plane gradientterms. Therefore, the ratio 2/1 continues to growalong Hc2(T ), though slower than in zero field.

2. Transverse magnetic field

We assume that H y and consider a homogeneoussuperconducting state along the field direction. The gra-dient terms in two transverse directions x and z have

5

0 10 20 30 40Temperature, K

0

5

10

15

20

25

Mag

netic

Fie

ld, T

0 10 20 30 40T, K

1

10

100

1/

2

FIG. 2: Temperature dependence of the upper critical fieldin MgB2 for magnetic field in the basal plane. Solid line: thetwo-gap Ginzburg-Landau theory with parameters given inthe text, squares: experimental data by Lyard et al .40 Theinset shows variation of the gap ratio along the Hc2(T ) linefor the same set of parameters.

different stiffness constants Kn and Knz, respectively.In a single band case, rescaling x x(Kx/Kz)1/4 andz z(Kz/Kx)1/4 allows to reduce an anisotropic prob-lem to the isotropic one in rescaled coordinates. A multi-gap superconductor has several different ratios Kn/Knzand the above rescaling procedure does not work. Inother words, coupled harmonic oscillators described bythe linearized Ginzburg-Landau equations have differentresonance frequencies. To solve this problem we followa variational approach, which is known to give a goodaccuracy in similar cases. The vector potential is chosenin the Landau gauge A = (Hz, 0, 0) and we look for asolution in the form

(

12

)

=

(

)1/4

ez2/2

(

cd

)

, (17)

where , c, and d are variational parameters. After spa-tial integration and substitution = h/, h = 2H/0,the quadratic terms in the Ginzburg-Landau functionalbecome

F2 = (a1t+ hK1)|c|2 + (2 + hK2)|d|2 (18)(cd+ dc) , Kn = 12 (Kn+Knz/)

The determinant of the quadratic form vanishes at thetransition into superconducting state. Transition fieldis given by the same expression as in the isotropic case(15), where Kn have to be replaced with Kn. The up-per critical field is, then, obtained from maximizing thecorresponding expression with respect to the variationalparameter . In general, maximization procedure has tobe done numerically. Analytic expressions are possible intwo temperature regimes. At low temperatures t |tc|,the upper critical field is entirely determined by the ac-

tive band and

hc2 =a1tK1K1z

. (19)

In the vicinity of Tc, an external magnetic field has asmall effect on the gap ratio = d/c /20 and aneffective single-gap Ginzburg-Landau theory can be ap-plied. The upper critical field is given by a combinationof the gradient constants Kni weighted according to thegap amplitudes:

hc2 =a1(t tc)

(K1 + 2K2)(K1z + 2K2z). (20)

Since, in MgB2 one has K1z 0.01K2z and 2 0.1, theslope of the upper critical field near Tc is determined byan effective gradient constantKeffz 2K2z > K1z (while(K1 +

2K2) K1). Thus, the upper critical field lineHc2(T ) shows a marked upturn curvature between thetwo regimes (20) and (19). Such a temperature behav-ior has been recently addressed in a number of theoret-ical works based on various forms of the two-band BCStheory.24,25,26,27 We suggest here a simpler descriptionof the above effect within the two-gap Ginzburg-Landautheory.Finally, we compare the Ginzburg-Landau theory with

the experimental data on the temperature dependence ofthe upper critical field for magnetic field parallel to thebasal plane.40 We choose ratios of the gradient term con-stants and the densities of states in accordance with theband structure calculations16 and change parameters and 20, which are known less accurately, to fit the ex-perimental data. The best fit shown in Fig. 2 is obtainedfor 20/a1 = 0.65 and /a1 = 0.4. The prominent up-ward curvature ofHc2(T ) takes place between tc = 0.18(Tc = 36 K) and t 0.2 (T = 26 K), i.e. well within therange of validity of the Ginzburg-Landau theory. Theabove values of 20 and can be related to g2/g1 andg3/g1 and they appear to be closer to the second choiceof gn used for Fig. (1). The ratio of the two gaps, as itchanges along the Hc2(T ) line, is shown on the inset inFig. 2. It varies from 1/2 2.3 near Tc = 36 Kto 1/2 45 at T = 18 K, where the Ginzburg-Landau theory breaks down. Due to a large differencein the c-axis coherence lengths between the two bands,the smaller gap is quickly suppressed by transverse mag-netic field. Also, the strong upward curvature of Hc2(T )leads to temperature variations of the anisotropy ratioan = H

c2(T )/H

cc2(T ), which changes from an = 1.7

near Tc to an = 4.3 at T = 18 K. These values are againconsistent with experimental observations,41 as well aswith theoretical studies.24,25,26

C. Structure of a single vortex

The structure of an isolated superconducting vortexparallel to the c-axis has been studied in MgB2 by the

6

0 5 10 15Distance,

0

0.5

1

Nor

mal

ized

gap

s t =0.4 / d =2

t =0.1 / d =1.64

t =0. / d =1.4

t=0.1 / d =1.17

1

2

2

2

2

2

FIG. 3: Spatial dependencies of the gaps for various temper-atures with t = lnT1/T 1 T/T1 and K2/K1 = 9.

scanning tunneling microscopy.42 Tunneling along the c-axis used in the experiment probes predominantly thethree-dimensional -band and the obtained spectra pro-vide information about a small passive gap. A largevortex core size of about 5 coherence lengths c =

0/2Hcc2 was reported and attributed to a fast sup-pression of a passive gap by magnetic field, whereas thec-axis upper critical field is controlled by a large gap inthe -band.42 The experimental observations were con-firmed within the two-band model using the Bogoliubov-de Gennes23 and the Usadel equations.28 We have, how-ever, seen in the previous Subsection that a -gap inMgB2 is not suppressed near Hc2(T ) for fields appliedalong the c-axis. To resolve this discrepancy we presenthere a systematic study of the vortex core in a two-gap su-perconductor in the framework of the Ginzburg-Landautheory.We investigate the structure of a single-quantum vor-

tex oriented parallel to the hexagonal c-axis. The twogaps are parametrized as n(r) = n(r)e

i , where isan azimuthal angle and r is a distance from the vortexaxis. Since the Ginzburg-Landau parameter for MgB2is quite large,2 25, magnetic field can be neglectedinside vortex core leading to the following system of theGinzburg-Landau equations

nnn +n3nKn(n+n/rQ2n) = 0 (21)

for n = 1, 2 (n = 2, 1) and Q 1/r. Away from the cen-ter of a vortex, the two gaps approach their asymptoticamplitudes 0n

01 =

a1t+

1, 02 =

/ 22

(22)

with obeying Eq. (14). All distances are measured inunits of a temperature-dependent coherence length de-rived from the upper critical field Eq. (15). In order to

0 5 10 15Distance,

0

0.5

1

Nor

mal

ized

gap

s 1 =0.6/d =1.8

2 =0.3/d =2.1

=0.1/d =2.6

=0.03/d =2.72

2

2

2

FIG. 4: Spatial dependencies of the gaps for various values of (given in units of a1) for t = 0.3 and K2/K1 = 9.

solve Eq. (21) numerically, a relaxation method has beenused43 on a linear array of 4000 points uniformly set ona length of 80 from the vortex center. An achieved ac-curacy is of the order of 106.The obtained results are shown in Figures 35, where

amplitudes n(r) are normalized to the asymptotic valueof the large gap 01. To quantify the size of the vortexcore for each component we determine the distance dn,where n(r) reaches a half of its maximum value 0n. Inthe case of a single-gap superconductor such a distanceis given within a few percent by the coherence length. Ina two-gap superconductor the characteristic length scalefor the large gap d1 remains close to , while d2 cansubstantially vary. Size of the vortex core is given bydv = max(2d1, 2d2).Results for temperature dependence of the vortex core

are presented in Fig. 3. The parameters 20 and aretaken the same as in the study of the upper critical field,while we choose K2/K1 = 9 in order to amplify effectfor the small gap. As was discussed above, the equilib-rium ratio of the two gaps 02/01 grows with decreas-ing temperature (increasing t). Simultaneously, the smallgap becomes less constrained with its interaction to thelarge gap and the half-amplitude distance d2 shows a no-ticeable growth. For K2 K1 such a less constrainedbehavior of 2(r) at low temperatures does not lead toan increase of the core size because both gaps have sim-ilar intrinsic coherence lengths.This trend becomes more obvious if the coupling con-

stant is changed for fixed values of all other param-eters, see Fig. 4. For vanishing , the distance d2 ap-proaches asymptotically an intrinsic coherence length inthe passive band. This length scale depends on K2(d2/d1|=0

K2/K1 = 3), but is not directly relatedto an equilibrium value of the small gap: the small gapis reduced by a factor of 7 between = 0.6 and = 0.03,while the core size increases by 50% only. Therefore, thesingle-band BCS estimate 2 = vF /(2) for the char-

7

0 5 10 15Distance,

0

0.5

1

Nor

mal

ized

gap

s

K /K =0.01 : d =0.68K /K =1 : d =1.08K /K =4 : d =1.52K /K =16 : d =2.5K /K =36 : d =3.54

1

2

2

2

2 1

2

2

1

1

1

1 2

2

2

2

2

FIG. 5: Spatial dependencies of the gaps for various values ofK2/K1 for t = 0.3.

acteristic length scale of the small gap sometimes usedfor interpretation of experimental data42 is not, in fact,applicable for a multi-gap superconductor.

Finally, Fig. 5 presents evolution of the vortex corewith varying ratio K2/K1, where again 20/a1 = 0.65and /a1 = 0.4. The apparent size of the vortex coredv 2d2 becomes about 56 coherence lengths for K2exceeding K1 by an order of magnitude. For K2/K1 1,which follows from the band structure calculations, thevortex core size does not change significantly comparedto the standard single-gap case. These results gener-ally agree with the previous study,28 though we concludethat unrealistically large values of K2/K1 are required toexplain the experiment.42 Different strength of impurityscattering in the two bands cannot explain this discrep-ancy either. It is argued that the -band is in the dirtylimit.22 The coefficient K2 in Eq. (12) is accordingly re-placed by a smaller diffusion constant. The numericalresults (Fig. 5) as well as qualitative consideration showthat in such a case the core size for 2(r) can only de-crease. Note, that the spatial ansatz (r) tanh(r/a)with a = used to fit the experimental data42 should beapplied with a = 1.8 even for a single-gap superconduc-tor in the large limit.44

V. ORIENTATION OF VORTEX LATTICE

Recent neutron scattering measurements29 in MgB2for fields along the hexagonal c-axis have discovered anew interesting phase transition in the mixed state ofthis superconductor: a triangular vortex lattice rotatesby 30 such that below the first transition field (0.5 T atT = 2 K) a nearest-neighbor direction is aligned perpen-dicular to the crystal a-axis, whereas above the secondtransition field (0.9 T) the shortest intervortex spacing isparallel to the a-axis.29 We show in this section that such

a peculiar behavior is determined by the two-gap natureof superconductivity in MgB2.

A. Single-gap superconductor

An orientation of the flux line lattice in tetragonaland cubic superconductors has been theoretically stud-ied by Takanaka.45 Recently, similar crystal field effectswere found to be responsible for the formation of thesquare vortex lattices in the borocarbides.46,47 The caseof a single-gap hexagonal superconductor is treated bya straightforward generalization of the previous works.Symmetry arguments suggest that coupling between thesuperconducting order parameter and a hexagonal crys-tal lattice appears at the sixth-order gradient terms in theGinzburg-Landau functional. For simplicity, we assumethat gap anisotropy is negligible. Then, the six-ordergradient terms derived from the BCS theory are

F6 =(7)N0326T 6c

(

1 127

)

vFivFjvFkvFlvFmvFn

(ijk)(lmn) . (23)

The above terms can be split into isotropic part andanisotropic contribution, the latter being

F an6 = (7)N0646T 6c

(

1 127

)

(

v6Fx v6Fy)

[

6x 154x2y + 152x4y 6y]

(24)

= 12K6

[

(x + iy)6 + (x iy)6]

.

In this expression x is fixed to the a-axis in the basalplane. (An alternative choice is the b-axis.) If x andy are simultaneously rotated by angle about the c-axis, (x iy)6 acquires an extra factor e6i. In thefollowing we always make such a rotation in order to havex pointing between nearest-neighbor vortices. PeriodicAbrikosov solutions with chains of vortices parallel tothe x-axis are most easily written in the Landau gaugeA = (Hy, 0, 0).48The higher order gradient terms Eq. (24) give a small

factor H2 (1 T/Tc)2 and can be treated as a per-turbation in the Ginzburg-Landau regime. The Landaulevels expansion yields (x) = c0f0(x) + c6f6(x) + ...,where the coefficient for the admixed sixth Landau levelis c6/c0 (

6!/3)h2e6iK6/K. When substituted into

the quartic Ginzburg-Landau term, this expression pro-duces the following angular dependent part of the freeenergy:

F () = 26!K63K

h2c2|c0|4|f0|2f0 f6 cos(6) , (25)

with |c0|2 = K(hc2 h)|f0|2/(|f0|4). Spatial aver-aging of the combination of the Landau levels is done in

8

a standard way

|f0|2f0 f6|f0|22

=

125

n,m

cos(2nm)e(n2+m2)

[

33(nm)6 15222(nm)4

+45

4(nm)2 15

8

]

, (26)

where summation goes over all integer n and m and pa-rameters and describe an arbitrary vortex lattice.48

For a hexagonal lattice ( = 1/2, =3/2), the nu-

merical value of the lattice factor is |f0|2f0 f6/|f0|4 =0.279. Hence, F () +K6 cos(6) and for v6Fx >v6Fy (K6 > 0) the equilibrium angle is = /2 (/6),which means that the shortest spacing between vorticesin a triangular lattice is oriented perpendicular to the a-axis, while for the other sign of anisotropy the shortestside of a vortex triangle is along the a-axis. Thus, theFermi surface anisotropy fixes uniquely the orientation ofthe flux line lattice near the upper critical field.

B. Two-gap superconductor

In a multiband superconductor effect of crystalanisotropy may vary from one sheet of the Fermi sur-face to another. We apply again the tight-bindingrepresentation16 to obtain a quantitative insight aboutsuch effects in MgB2. Explicit expressions for dispersionsof the two hole -bands are presented in the Appendix.Hexagonal anisotropy in the narrow -cylinders is en-hanced by a nonanalytic form of the hole dispersions.Combined anisotropy of the -band is v6Fx = 4.608,v6Fy = 4.601, while for the -band v6Fx = 1.514,v6Fy = 1.776 in units of 1046 (cm/s)6. According tothe choice of the coordinate system,16 the x-axis is par-allel to the b-direction and the y-axis is parallel to thea-direction in the boron plane. The above values mightbe not very accurate due to uncertainty of the LDA re-sults, however, they suggest two special qualitative fea-tures for MgB2. First, relative hexagonal anisotropy ofthe Fermi velocity vFn() differs by almost two orders ofmagnitude between the two sets of bands. Second, cor-responding hexagonal terms have different signs in thetwo bands. In Appendix, we have shown that the signdifference is a robust feature of the tight-binding approx-imation and cannot be changed by a small change of thetight-binding parameters.We investigate equilibrium orientation of the vortex

lattice in MgB2 within the two-gap Ginzburg-Landautheory. Anisotropic sixth-order gradient terms of thetype (24) have to be added to the functional (12) sepa-rately for each of the two superconducting order param-eters. As was discussed in the previous paragraph theanisotropy constants have different signs K61 > 0 andK62 < 0 and obey |K61| |K62|. In the vicinity of theupper critical field the two gaps are expanded as 1(x) =

c0f0(x) + c6f6(x) and 2(x) = d0f0(x) + d6f6(x). Solu-tion of the linearized Ginzburg-Landau equations yieldsthe following amplitudes for the sixth Landau levels:

c6 = 46!h3e6i

K612c0 +K62d012 2

,

d6 = 46!h3e6i

K621d0 +K61c012 2

(27)

with 1,2 = 1,2 + 13K1,2h. Subsequent calculations fol-low closely the single-gap case from the preceding sub-section. The angular dependent part of the free energy isobtained by substituting (27) into the fourth-order terms:

F () =[

1c30(c6+c

6)+2d

30(d6+d

6)]

|f0|2f0 f6 . (28)

The resulting expression can be greatly simplified if oneuses (2/1)

2 0.1 as a small parameter. With accu-racy O[(2/1)

4] we can neglect the angular dependentpart determined by the small gap. This yields in a closeanalogy with Eq. (25) the following anisotropy energy forthe vortex lattice near Hc2

F () = 26!

3K1h2c21|c0|4|f0|2f0 f6Keff6 cos(6) ,

Keff6 = K61 +K622

(2 +K2h)(2 + 13K2h). (29)

Despite the fact that we have omitted terms d30d6, theFermi surface anisotropy of the second band still con-tributes to the effective anisotropy constant Keff6 via lin-earized Ginzburg-Landau equations. Along the uppercritical line this contribution decreases suggesting the fol-lowing scenario for MgB2.In the region near Tc the second band makes the largest

contribution to Keff6 : a small factor 2/22 0.1 is out-

weighed by |K61/K62| < 0.1. As a result, Keff6 is neg-ative and = 0, which means that the shortest inter-vortex spacing is parallel to the b-axis. At lower tem-peratures and higher magnetic fields the second term inKeff6 decreases and the Fermi surface anisotropy of thefirst band starts to determine the (positive) sign of Keff6 .In this case, = /2 (/6) and the side of the vor-tex triangle is parallel to the a-axis. The very small|K61/K62| = 1.8 102, which follows from the bandstructure data,16 is insufficient to have such a reorienta-tion transition in the Ginzburg-Landau region. Absolutevalues of anisotropy coefficients are, however, quite sensi-tive to the precise values of the tight-binding parametersand it is reasonable to assume that experimental values ofK6n are such that the reorientation transition is allowed.The derived sequence of the orientations of the flux line

lattice in MgB2 completely agrees with the neutron scat-tering data,29 though we have used a different scan line inthe HT plane in order to to demonstrate the presence ofthe 30-orientational transformation, see Fig. 6. Condi-tion Keff6 = 0, or similar one applied to Eq. (28), definesa line H(T ) in the HT plane, which has a negative

9

H*

HH

T

c2

ab

FIG. 6: Phase diagram of MgB2 for fields parallel to the c-axis. Shaded region corresponds to intermediate orientationsof the vortex lattice separated by dotted lines of the second-order transitions. Dashed lines indicate the scans used in theexperiment (vertical) and in the presented theory.

slope at the crossing point with Hc2(T ). The six-foldanisotropy for the vortex lattice vanishes along H(T )and all orientations with different angles become de-generate in the adopted approximation. The sequence oforientational phase transition in such a case depends onweaker higher-order harmonics. One can generally write

F () = K6 cos(6) +K12 cos(12) , (30)

where the higher-order harmonics comes with a small co-efficient |K12| |K6|. Depending on the sign of K12transformation between low-field = 0 and high-field = /6 (/2) orientations, when K6 changes sign, goeseither via two second-order transitions (K12 > 0) or viasingle first-order transition (K12 < 0). In the formercase the transitions take place at K6 = 4K12, whereasin the latter case the first-order transition is at K6 = 0.These conclusions are easily obtained by comparing theenergy of a saddle-point solution cos(6) = K6/(4K12)for Eq. (30), which is Fsp = K26/(8K12), to the ener-gies of two extreme orientations.In order to determine sign of the higher-order harmon-

ics for a two-gap superconductor we expand the fourth-order terms in the Ginzburg-Landau functional (12) tothe next order

F () =1

2

[

1c20(c

26 + c

26 ) + 2d

20(d

26 + d

26 )

]

f20 f26 .(31)

These terms are responsible for a cos(12) anisotropyintroduced before. Similar angular dependence is alsoinduced by higher-order harmonics of the Fermi veloc-ity vF (), though our estimate shows that even forthe -bands corresponding modulations are very small.49

Sign of cos(12) term in Eq. (31) depends only on ageometric factor, spatial average of the Landau levelswave functions. We find for a perfect triangular lattice

f20 f26 /|f0|22 = 0.804. Thus, the twelveth-order har-monics in Eq. (30) has a positive coefficient and trans-formation between the low-field state with = 0 and thehigh-field state = /6 goes via a phase with intermedi-ate values of separated by two second order transitions.The anisotropy terms of the type (24) also produce a

six-fold modulation of the upper critical field in the basalplane. Sign of the corresponding modulations of Hc2()should also change at a certain temperature, which is de-termined by a suppression of the small gap in transversemagnetic field and is not, therefore, related to the inter-section point of Hc2(T ) and H

(T ) lines on the phasediagram for H c, Fig. 6.

VI. CONCLUSIONS

We have derived the Ginzburg-Landau functional of atwo-gap superconductor within the weak-coupling BCStheory. The functional contains only a single interac-tion term between the two superconducting gaps (con-densates). This property allows a meaningful analysisof various magnetic properties of a multi-gap supercon-ductor in the framework of the Ginzburg-Landau theory.Apart from confirming the previous results on an unusualtemperature dependence of the transverse upper criticalfield in MgB2, we have presented detailed investigationof the vortex core structure and have shown that theorientational phase transitions observed in the flux linelattice in MgB2 is a manifestation of the multi-band na-ture of superconductivity in this material. The proposedminimal model for the 30-rotation of the vortex latticeincludes only anisotropy of the Fermi surface. An addi-tional source of six-fold anisotropy for the vortex latticecan arise from angular dependence of the superconduct-ing gap. It was argued that the latter source of (four-fold) anisotropy is essential for physics of the square todistorted triangular lattice transition in the mixed stateof borocarbides.50 For phonon-mediated superconductiv-ity in MgB2, the gap modulations should be quite small,especially for the large gap on the narrow -cylindersof the Fermi surface. Experimentally, the role of gapanisotropy can be judged from the position of H(T ) linein the HT plane. H(T ) does not cross Hc2(T ) line inscenarios with significant gap anisotropy.50 A further in-sight in anisotropic properties of different Fermi surfacesheets in MgB2 can be obtained by studying experimen-tally and theoretically the hexagonal anisotropy of theupper critical field in the basal plane.

Acknowledgments

The authors would like to acknowledge useful discus-sions with R. Cubitt, M. R. Eskildsen, V. M. Gvozdikov,A. G. M. Jansen, S. M. Kazakov, K. Machida, I. I. Mazin,and V. P. Mineev. We also thank F. Bouquet and P.Samuely for providing their experimental data.

10

APPENDIX: ANISOTROPY IN -BANDS

We give here expressions for the dispersions and Fermisurface anisotropies in the two -bands, which are de-rived from the tight-binding fits of Kong et al .16 Thein-plane px,y boron orbitals in MgB2 undergo an sp

2-hybridization with s-orbitals and form three bondingbands. At k = 0 these bands are split into a non-degenerate A-symmetric band and doubly-degenerate E-symmetric band, which lies slightly above the Fermi level.Away from the k = 0-line the E-band splits into lightand heavy hole bands. Their dispersions obtained by ex-pansion of the tight-binding matrix16 in small k are

l(k) = (kz) 2t[

18k

2 + dk

4

2g(k) + 1 1/d384(1 + d)

]

,

h(k) = (kz) 2t[

38dk

2 dk4

2g(k) + 7 + 9d

384(1 + d)

]

,

where (kz) = 0 2tz cos kz and g(k) = (k6x 15k4xk2y +15k2xk

4y k6y)/k6. The tight-binding parameters pre-

sented in Ref. 16 are 0 = 0.58 eV, t = 5.69 eV,tz = 0.094 eV, and d = 0.16. The six-fold anisotropy isgiven by unusual nonanalytic terms, which are formallyof the fourth order in k. Appearance of such nonanalyticterms is a direct consequence of the degeneracy of the twobands at k = 0. For example, a nonanalytic form of (k)is known for four-fold degenerate hole bands of Si andGe,51 which have cubic anisotropy already in O(k2) or-der. Nonanalyticity of l,h(k) leads to a relative enhance-ment of the hexagonal anisotropy on two narrow Fermisurface cylinders. This anisotropy has opposite sign inlight- and heavy-hole bands. The net anisotropy of thecombined -band is determined mostly by the light-holes,which have larger in-plane Fermi velocities.

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