MgB : superconductivity and pressure effects 2 V. A. Ivanov, J. J. Betouras, and F. M. Peeters Departement Natuurkunde, Universiteit Antwerpen (UIA), Universiteitsplein 1, B-2610 Antwerpen, Belgium 3 0 0 2 WepresentaGinzburg-Landautheoryforatwo-bandsuperconductorwithempha- n sison MgB2. Experimentsareproposed whichlead toidentification ofthepossible a scenarios: whetherbothσ-andπ-bandssuperconductorσ-alone. Accordingtothe J second scenario a microscopic theory of superconducting MgB2 is proposed based 4 onthestronglyinteractingσ-electronsandnon-correlatedπ-electronsofboronions. 2 The kinematic and Coulomb interactions of σ-electrons provide the superconduct- ] ingstatewithananisotropicgapofs*-wavesymmetry. ThecriticaltemperatureTc n has a non-monotonic dependence on the distance r between the centers of gravity o ofσ-andπ-bands. ThepositionofMgB2 onabell-shapedcurveTc (r)isidentified c intheoverdopedregion. Thederivedsuperconductingdensityofelectronicstatesis - r inagreementwithavailableexperimentalandtheoreticaldata. Itisarguedthatthe p effectsofpressurearecrucialtoidentifythemicroscopicoriginofsuperconductivity u in MgB2. Possibilities for Tc increase are discussed. s . ThediscoveryofsuperconductivityinMgB2 [1]posesmanyinterestingandfundamentalquestions t a regarding the nature of the superconducting state as well as the possibility of multi-band super- m conductivity. The crystal belongs to the space group P6/mmm or AlB -structure where borons 2 - are packed in honeycomb layers alternating with hexagonal layers of magnesium ions. The ions d Mg2+ are positioned above the centers of hexagons formed by boron sites and donate their elec- n trons to the boron planes. The electronic structure is organized by the narrow energy bands with o c near two-fold degenerate σ-electrons and the wide-band π-electrons. Without any of the lattice [ strain,theσ dispersionrelationsareslightlysplittedduetothetwoboronatomsperunitcell. The correspondingportions of the Fermi surface consist of coaxialcylinders along the Γ - A symmetry 3 v direction of the Brillouin zone (BZ), whereas the π- bands are strongly dispersive. In the follow- 5 ing sections we present a Ginzburg-Landau (GL) analysis of the two-band superconductor with 8 emphasisonMgB ,we apply itin pressureexperiments whichpotentially distinguishthe different 2 0 superconducting bands [2], then we provide a microscopic model. 4 0 2 I. GINZBURG-LANDAU ANALYSIS AND PRESSURE EFFECTS 0 / t a We distinguish different possibilities: m Case I: Both σ and π bands superconduct. The GL free energy functional for MgB in this case 2 - can be written as : d n 1 1 F = d3r Π~ψ 2+α ψ 2+β ψ 4+ Π~ψ 2+α ψ 2+β ψ 4 o Z {2mσ| σ| σ| σ| σ| σ| 2mπ| π| π| π| π| π| c : (~ A~)2 v +r(ψ∗ψ +ψ ψ∗)+β(ψ 2 ψ 2) + ∇× , (1) i σ π σ π | σ| | π| 8π } X where Π~ = i~~ 2e/cA~ and A~ is the vector potential, α =α0 (T T0 ). r − ∇− σ,π σ,π − cσ,π a (i) If r = 0 then Eq. 1 is the free energy for two bands without pairing transfer (Josephson coupling) between them. The quartic term which mixes the two order parameters reflects the 1 fact that although the two bands are different, there is a constraint coming from the common chemical potential which connects them in equilibrium. It is also the realizationand a measure of the strength of the interband interaction of quasiparticles and it affects the critical temperatures. The absence of a bilinear in the two order parameters term facilitates the observation of the Leggett’s mode in the Josephson tunneling between a two-band superconductor and an ordinary superconductor [3]. The onset of the superconducting state in one band does not imply the onset in the other. To investigate this possibility in MgB an experiment which checks the splitting of 2 the transitionsdue to the lattice deformationbythe strainfields canbe decisive. Inthe MgB the 2 compression due to pressure is anisotropic [4–6]. According to Ref. [4] the compressibility along thec-axisisalmosttwicelargerthatoftheplanecompressibility. Thereforeapplicationofuniaxial pressure on single crystals will affect differently the two gaps and the result will be a splitting of the critical temperatures, even if at ambient pressure they appear to be the same. This can be measured through a specific heat experiment under pressure. Following Ozaki’s formulation [7,8], we addtothe GL functionalEq.(1)the termwhichcouples the orderparameters,insecondorder, with the strain tensor ǫ to first order. F = C (Γ )[δ(ǫ +ǫ )+ǫ ]ψ 2 C (Γ )(ǫ +ǫ +ǫ )ψ 2, (2) strain 1 1 xx yy zz σ 2 1 xx yy zz π − | | − | | where C (Γ ) are coupling constants, δ is given in terms of the elastic constants of the material 1,2 1 δ = (c +c c )/(c c ). The effect of the pressure on these two order parameters, each 11 12 13 13 33 − − one belonging to an one-dimensional representation is to shift the critical temperatures. The new critical temperatures can be found, by solving the coupled equations for ψ 2 and then setting σ,π | | the coefficients of ψ 2 to zero. The result is : T = T0 η p, where for uniaxial | σ,π| cσ,π cσ,π − σ,π × pressure η >η >0. In order to detect the splitting, the pressure must be above a certain value π σ p which depends on the resolution of the experiment, so that the difference in the two critical min temperatures ∆T = T0 T0 +η η p is experimentally detectable. c | cσ− cπ σ − π| min (ii)Ifr=0thenthepairtransferingtermispresentanditmeansthattheonsetofsuperconductivity 6 in one band implies automatically the appearance of superconductivity in the other. There is a single observed T with a different pressure dependence. Analyzing the equations which result c from the minimization of the GL free energy we get in the regime T0 >T >T0 : cσ cπ α α =r2 (3) π σ This gives the pressure dependence of the T : c T (p)= 1 T0 +T0 (η +η )p + 1 (T0 T0 +(η η )p)2+a2 1/2 (4) c 2 cσ cπ− π σ 2 cσ− cπ π − σ (cid:2) (cid:3) (cid:2) (cid:3) and a2 = 4r2/(α0α0). Deviations from a straight line at moderate values of pressure can be π σ attributed to the two bands. Case II: Only the σ- bands superconduct. In this case there is one order parameter initially and the GL free energy functional is the usual one. The dominant physical situation is an important change in the electronic properties of the material under pressure, because the band (σ ) which is 2 below the Fermi level at ambient pressure [9] can, partially, overcome the barrier and get above the Fermi level at a certain value of pressure (crossover pressure), restoring the degeneracy of the twoσ bands atpoint Γ. Thenthere will be a crossoverfroma superconducting state ψ to a state 1 ψ +ψ . To understand this effect we need to write the GL free energy taking into account the 1 2 second order parameter of the same symmetry as well as the hexagonal symmetry of the boron layers. The irreducible representation of the D group are four one-dimensional ones (three of 6h them have line nodes) and two two-dimensionalones [8]. There are experimental data compatible with s-wave or an anisotropic s∗-wave order parameter. and a penetration depth experiment [10] whichsuggests nodes oranisotropics∗-wave. Thereforewe consider the gapto be a function ofk x and k with a smooth modulation in the k direction as in Ref. [15]. As a consequence the most y z 2 promisingcandidatesfortheinitialorderparameterarethebasisfunctionsoftheΓ representation 1 oftheD group. Toconstructthe functionalofthefreeenergyfortwoorderparameters,weneed 6h to take into accountthe decompositionof the terms containing the derivatives Π~, which belong to the Γ representation. Then the decomposition 5 Γ ∗ Γ ∗ Γ Γ =Γ +Γ , (5) 5 1 5 1 1 2 ⊗ ⊗ ⊗ containstheΓ representationonceandsuggeststhatthereispreciselyonetermquadraticinΠ 1 x,y which mixes the two order parameters (to first order each). There is also a quadratic term which mixes the two components, according to the trivial decomposition Γ Γ and a quartic term 1 1 ⊗ as well. Taking into accounts all the usual terms, we obtain the following expression for the free energy: 1 1 F = d3r Π~ψ 2+α ψ 2+β ψ 4+ Π~ψ 2+α ψ 2+β ψ 4+ 1 1 1 1 1 2 2 2 2 2 Z {2m | | | | | | 2m | | | | | | 1 2 (~ A~)2 γ (ψ∗ψ +c.c.)+γ (Π ψ Π∗ψ∗+Π ψ Π∗ψ∗+c.c.)+β (ψ 2 ψ 2) + ∇× . (6) 1 1 2 2 x 1 x 2 y 1 y 2 3 | 1| | 2| 8π } We have used the fact that the γ -term favors the coupling of linear combination of the two 1 order parameters with phase difference 0 or π between them [11], therefore a term ψ2∗ψ2 +c.c. 1 2 is incorporated into the β -term of the free energy. The resemblence between Eq.(1) and Eq.(7) 3 is apparent; in the first case the mixing terms are a consequence of the interactions between the two bands, in the second case it is due to the modification of the existing gap in the electronic spectrum. With the present approach we are in the position to describe the crossover and to consider the possibilityofdifferenteffectivemassesandparameters(theinclusionoftheeffectatp ). Theeffect c of pressure can be taken into account as usual [12] with the modification due to the particular physicalsituationbywritingα =α0(T T (p))Θ(p p ),γ =γ0(p 1)Θ(p p ). Theadditional 2 2 − c − c 1 1 − − c term which couples the order parameter with the strain tensor ǫ, in the regime where p>p is : c F = C(Γ )[(1+δ)(ǫ +ǫ )+2ǫ ] ψ +ψ 2. strain 1 xx yy zz 1 2 − ×| | The coupling to strain affects both order parameters in the same way. Theapproximatevalueofthecrossoverpressurecanbeestimatedasfollows. Theenergydifference between the two subbands is approximatedby (1 n )2∆, where the fractionof superconducting σ − electronsis1 n 0.03,i.e. thecarrierdensityperboronatominMgB . Theapproximatevalue σ 2 − ∼ ofthe crossoverpressurep whichsuppressesthe deformationpotential∆canbe estimatedbythe c expression p Ω (1 n )√∆ where Ω = 30˚A3 is a unit cell volume of MgB and ∆ = 0.04eV2 c σ 2 ∼ − is the deformation potential for a boron displacement u 0.03˚A [13]. Using these parameters, ∼ we get a crossover pressure of p 30GPa. This estimation shows, that an applied pressure at c ∼ a realistic value influences drastically the electronic structure, producing the degeneracy of two bands whichareinitially splitted. Superconductivityinthe secondbandwilloccur atlowervalues of the estimated p . The requested band overlapping around the Fermi energy will also occur at c lower values due to the corrugation of the Fermi surfaces and the already existing strains in the material and the anisotropic compressibility. These considerations make the above estimation an upperlimitofp . Thedegreetowhichshearstressesofthesampleandthesurroundingfluidunder c pressure affect the data of pressure measurements is still under investigation [14]. The physics of this crossoverissimilarin spiritto anelectronictopologicaltransitionof5/2kind[16]. InFig.1in amodelcalculation,weillustratetheexpectedbehaviorofT andtheformoftheorderparameter c asafunctionofpressureatfixedtemperatureandschematicallythekinkatp ofT . Theanomaly c c close to p can be detected directly in a penetration depth experiment under pressure. c 3 1.0 0) 0.9 1.0 ( yp)/ 0.8 ( 0.7 y ) 0.6 0 (c 0.5 p)/T 0.5 ssu1 perconducts 0.0 0.5 1p.0/p 1.5 2.0 (c c T s & s 1 2 superconduct 0.0 0.0 0.5 1.0 1.5 p/p c FIG.1. TheformofTc asafunctionofpressureforcaseII,pcisthecrossoverpressurewherethesecond band becomes superconducting as well. Inset : The normalized absolute value of the order parameter ψ(p)/ψ(0)=|ψ1(p)+ψ2(p)|/|ψ1(0)|atT =0.7Tc1. Thechosenparametersare: α1 =2(T/Tc1−1−0.1p/pc), α2 =(T/Tc1−0.8−0.1p/pc),β1 =β2 =1, β3 =Θ(p/pc−1), γ1=0.4(p/pc−1)Θ(p/pc−1). Thepictureweprovideforthesecondcaseiscomplimentarytothefirst. Bothcasesmayberealized since the effects can be detected at different values of pressures. In case I, the value of pressure necessary to see the effects is limited either by the experimental resolution or by the validity of the Eq. 4 at low pressure. The specific symmetry of the order parametersis only used to produce the additional terms due to strain. In case II there is a larger predicted value for the crossover pressure and there are more specific assumptions regarding the symmetry of the order parameter. Early pressure experiments [17] demonstrated the overall decrease of T with pressure which was c attributedtothe lossofholes. Intwoofthe samplesthereisalineardependence ofT onpressure c andintwoothersaweakquadraticdependence. Westressthatthesamplesarepolycrystallineand the experiment is effectively under hydrostatic pressure. Also the degree of nonstoichiometry was not known. The almost linear dependence for a wide range of pressures,makes the GL functional as presented, valid for the MgB . Experiments on single crystals will be able to verify the effects 2 which are described. We do not attempt at the moment any fitting of experimental data because there is no experimental consensus on the different values of key parameters of the theory (e.g. there is a wide range of published data on the value of dT /dp [14]). c II. MICROSCOPIC MODEL The presented microscopic model of superconductivity in MgB , is based on the correlated σ- 2 and the non-correlated π-band. The negatively charged boron layers and the positively charged magnesium layers provide a lowering of the π-band with respect to the σ-band which was also noticedfirstinbandcalculations[18,19]. Inthis microscopictreatmentthe σ-bands aretreatedas degenarate. Thequasi-2Dσ-electronsaremorelocalizedthanthe3Dπ-electrons. Thisleadstoan enhancement of the on-site electron correlations in the system of degenerated σ-electrons. They are takento be infinite, while the intersite Coulombinteractionsand electron-phononinteractions simply shift the on-site electron energies. After carrying out a fermion mapping to X-operators the Hamiltonian becomes: 4 H = t(σ)(p)Xso(p)Xos(p)+V n (i)n (j) r n (i)+ σ σ π − Xp,s hXi,ji Xi + ε(π)(p) π+(p)π (p)+H.c. µ [n (i)+n (i)]. (7) 0 s s − σ π Xp,s (cid:2) (cid:3) Xi Here, the X-operators describe the on-site transitions of correlated σ-electrons between the one- particle ground states (with spin projection s = ) and the empty polar electronic states (0) ± of the boron sites and µ is the chemical potential. The wide π-band is shifted with respect to the σ-band by an energy r, comprising the mean-field π-electron interactions and the electron- phonon interactions. We included in Eq. (7) also the nearest neighbour Coulomb repulsion V of σ-electrons, which is essential for the low density of hole-carriers in MgB . The mutual hopping 2 between σ- and π-electrons is assumed to be negligible due to the characteristic space symmetry of the orbitals involved. A schematic view of the model is presented in Fig. 2. FIG.2. A schematic view of the energy band diagram. The degenerate σ-electrons are represented by a lower correlated band. The diagonal X-operators (X· X··) satisfy the completeness relation, X0+4Xs =1, and their ≡ thermodynamic averages ( X0 , Xs ) are the Boltzmann populations of the energy levels of the h i unperturbed on-site Hamil(cid:10)toni(cid:11)an in (7). Due to the orbital and spin degeneracy the σ-electrons occupy their one-particle ground state with density n = 4 Xs per boron site. The correlation σ h i factor for the degenerated σ-electrons is f = X0+Xs = 1 3 Xs = 1 3n . It plays an − h i − 4 σ important role in all numerics starting from th(cid:10)eir unper(cid:11)turbed (zeroth order) Green’s function, D(0)(ω)=f/( iω µ),whereasforπ-electronsD(0)(ω)=1/( iω µ r). Notethatforthe σ n π n − − − − − conventional Hubbard model the correlationfactor in the paramagnetic phase is 1 n/2 (see [23] − and Refs. therein). The energy dispersion of both bands, ξ(p) = ft(p) µ and ε(p) = ε (p) r µ, are gov- 0 − − − erned by the zeros of the inverse Green’s function D−1(ω,p) = diag D−1(ω,p);D−1(ω,p) = σ π diag −iωn+ξ(p); iω +ε(p) , which follows from the Dyson equation(cid:8)D−1(ω,p)=D(0)−1(ω(cid:9))+ n f − n o t(p), to first order with respect to the tunneling matrix t(p)=diag t(p);ε (p) . 0 { } bThe chemical potential µ for the MgB2 =Mg++B− p2 b2=Mg++B−(pnσpnπ)2-system (Eq. (1)) obeys the equation for the total electron density pe(cid:0)r b(cid:1)oron site nσ +nπ = 2, with the partial electron densities: n =2T eiωδD (ω,p) n (r,µ). (8) σ,π σ,π σ,π ≡ Xn,p Theelectrondensitiessatisfytherequirements0<n <1(duetothecorrelationfactorf,providing σ the quarter-foldnarrowingof the degenerateσ-band) and 0<n <2. For any energy difference r π 5 betweentheσ-and π-bandsthechemicalpotentialµhastobesuchthattheconstraintn +n =2 σ π is satisfied. From the system of Eqs. of the constraint and (9), for a flat density of electronic states (DOS hereafter)ρ (ε)=(1/2w )θ w2 ε2 withhalf-bandwidthsw andw forσ-andπ-electrons, σ,π 1,2 1,2− 1 2 respectively) we derive the chem(cid:0)ical poten(cid:1)tial of the A2+B(pnσpnπ) systems as 2 w 5r 2 µ= − w (9) 1. 5w +4w 1 2 The non-correlated π-electrons play the role of a reservoir for the σ-electrons. In the energy dispersionξ(p)fortheσ-electrons,thecorrelationfactorf canbeexpressedalsoviatheparameters w ,w and r as f =(2w +w +3r)/(5w +4w ). 1 2 1 2 1 2 The anomalous self-energy for the σ-electrons (Fig. 3) is written self-consistently as ∨ ∨ Σ(q) Σ(p)=T Γ (p,q) , (10) 0 ∨ Xn,q ω2 +ξ2(q)+Σ(q) n where the vertex Γ is determined by the amplitudes of the kinematic and Coulomb interactions 0 suchthatΓ (p,q)= 2t +V (p q). We do notinclude the otherkinematic verticesinΓ which 0 q 0 − − are essential at a moderate concentration of carriers [23]. ∨ FIG.3. Theanomalousself-energyΣ(p)forσ-electrons. ThesolidlineisananomalousGreen’sfunction. In momentum space the Coulomb repulsion between the nearest neighbours (Eq. (7)) reflects the tight-binding symmetry of the boron honeycomb lattice [22]. Near the Γ A line of the Brillouin − zone the Coulomb vertex can be factorized as V (p q) = 2βt(p)t(q), where the parameter β = − V/6t2 expresses the Coulomb repulsion for the nearest σ-electrons and the energy dispersion is p2+p2 t(p) = 3t 1 x y . Considering the explicit form of the vertex Γ in Eq. (10), and after (cid:16) − 12 (cid:17) 0 summation over the Matsubara frequencies ω =(2n+1)πT one obtains n ∨ ∨ ∨ tanh ξ2(q)+Σ2(q)/2T Σ(p)= t(q)(1 βt(p))Σ(q) q . (11) − ∨ Xq ξ2(q)+Σ2(q) q ∨ ThesearchforasolutionintheformΣ(p)=Σ +t(p)Σ convertsEq. (11)forthesuperconducting 0 1 critical temperature and the gap to tanh ξ2(p)+Σ2(p)/2T 1= t(p)(1 βt(p)) , (12) − pξ2(p)+Σ2(p) Xp p with the gap function Σ(p)=[1 βt(p)]Σ . 0 − Setting the gap function in Eq. (12) equal to zero, one can derive analytically T in a logarithmic c approximation: 6 T = w1 (w +w r)(w +4r)exp 1 , c 5w1+4w2 1 2− 1 −λ p (cid:0) (cid:1) λ= (5w1+4w2)(3+5βw1)(w 5r) r βw1w2−2w1−w2 . (13) (2w1+w2+3r)3 2− h − 3+5βw1 i Underthe prefactorinthe squarerootthe restrictionsforanenergyshiftr guaranteethe assumed volume of the correlated σ-band, namely n 0 (r w +w ) and n 1 (r w /4= t). σ 1 2 σ 1 ≥ ≤ ≤ ≥− − T (r) is plotted in Fig. 4 for different values of the parameter β, reflecting the suppression of c superconductivity with an increase of the Coulomb repulsion. In the MgB case, T = 40 K, 2 c corresponds to r =0.085 eV and a dimensionless value βw =8 V/3w . 1 1 FIG. 4. The non-monotonic dependence of Tc(r) at different magnitudes of the Coulomb repulsion (parameterβw1). Herew2/w1 =8. MgB2 ismarkedintheinsetwithasolidcirclepositionedatr=0.085 eV and βw1 =1.2 for w1 =1 eV and w2 =8 eV. The superconducting gap equation follows from Eq. (12) taken for T=0 dε 1= ρ (ε)ε(1 βε) . (14) σ Z − ξ2(ε)+Σ2(1 βε)2 q 0 − It defines the anisotropic superconducting order parameter of s∗-wave symmetry. The near-cylindricalhole-likeσ-FermisurfacesinMgB givesroomto calculatethe superconduct- 2 ing DOS ρ(E) = δ E ε2(p)+Σ2(p) , where the gap function is Σ = Σ (1 βt(p)) = b 1+acos2ϑ , whPerpe a(cid:16)=−βwp/(12(1 βw )(cid:17)), b = Σ (1 βw ) and ϑ is the az0imu−thal angle. 1 1 0 1 − − T(cid:0)hen the DOS(cid:1) normalized with respect to the normal DOS is 1 ρ(E) dz =E . (15) ρ0(E) Z E2 b2(1+az2)2 0 q − ForaCoulombrepulsionsuchthatβw <1the parametersatisfiesa>0andthe superconducting 1 DOS becomes ρ(b<E <(1+a)b) = E K(q), ρ0(E) q2ab ρ(E >(1+a)b) = E F sin−1 ab ;q , (16) ρ0(E) q2ab (cid:16) q(E+(1+a)b)q2 (cid:17) whichisexpressedintermsofthecompleteandincompleteellipticintegralsK andF,respectively with modulus q = (E b)/2E. − p 7 FIG. 5. The superconducting DOS, normalized with respect to the normal DOS, for the Coulomb parameter range βw1 >1. At βw1 =0.92, theresult by Haas and Maki [15] is reproduced (inset). The DOS (16) has cusps at E= (1+a)b. A similar result was obtained in Ref. [15] for a non- ± specified parameter a > 0. In our case the parameter a is controlled by the in-plane Coulomb repulsion V as the authors of Ref. [20] noted. At βw = 0.92 the DOS of Eq. (16) (see Fig. 5, 1 inset) reproduces Fig. 1(b) of Ref. [15]. For our case of an enhanced Coulomb repulsion βw >1, we have to take the parameter a<0 in 1 the gap function and the superconducting DOS (Fig. 4) is then given by ρ((1 a)b<E <b) −| | = E F sin−1 1; 2E , ρ0(E) √(E+b)|a|b (cid:16) q qE+b(cid:17) ρ(E >b) = E F sin−1q; E+b . (17) ρ0(E) q2|a|b (cid:18) q 2E (cid:19) ItcontainstwologarithmicdivergenciesatE = bandagapintheenergyrange E <(1 a)b. ± | | −| | The two-gapratio is 1/(1 a). −| | Measurements on MgB with scanning tunneling spectroscopy [24,25] and with high-resolution 2 photo-emission spectroscopy [26] revealed the presence of these two gap sizes. From the ratio 3.3 betweenthetwogapsinRef.[24]wecanextracttheparameters a 2/3andβw 1.14,whereas 1 | |≈ ≈ from data of Ref. [25] one can derive βw = 1.21. A value βw = 1.14 can be estimated from 1 1 Ref. [26]. Point-contact spectroscopy [27] shows gaps at 2.8 and 7 meV, from which we estimate a Coulomb repulsion parameter βw =1.16. The recent study of energy gaps in superconducting 1 MgB by specific-heat measurements revealed two gaps at 2.0 meV and 7.3 meV [28] for which 2 βw = 1.13, and a gap ratio 3 2.2 [29], for which βw = 1.14 1.15. Measurements of the 1 1 − − specific heatofMg11B alsogiveevidence forasecondenergygap[30]. Ramanmeasurements[31] 2 establishedpronouncedpeaks,correspondingwithgapsat100cm−1and44cm−1. Fromthesedata one can extract a = 0.56 and βw = 1.18. At βw = 1 we have gapless like superconductivity. 1 1 − In this case the superconducting DOS is linear with respect to the energy near the nodes of the superconducting order parameter. Then the superconducting specific heat C T2 and the NMR e ∼ boron relaxationrate T3 at low temperatures. From this point of view it is interesting that the ∼ data of Ref. [32]shows a C T2 behaviour anda deviationfrom the exponentialBCS behaviour e in T−1(T) of 11B [33] visual∼ized in MgB . 1 2 III. CONCLUSIONS We have done a GL analysis of the superconductivity of a two-band superconductor with par- ticular attention to MgB . In that framework, we provide the different possibilities for a T-P 2 8 phase diagram, make predictions for the role of the bands and discuss different experiments from which crucial information can be extracted. Then we have analyzed the superconducting prop- erties of the material MgB within the framework of a correlated model Eq. (7). The existing 2 electron-phonon and non-phonon approaches to the superconducting mechanism in MgB can be 2 separated in two groups: one pays attention to the σ-electrons and the other to the π-electron subsystem. We have taken into account both the correlated σ- and noncorrelated π-electrons. Analysis of our results leads to the conclusion that superconductivity occurs in the subsystem of σ-electrons with degenerate narrowenergy bands whereas the wide-band π-electrons play the role of a reservoir. Superconductivity is driven by a non-phonon kinematic interaction in the σ-band. A lot of evidences in favour oftwo different superconducting gaps canbe explained by anisotropic superconductivitywithanorderparameterofs∗-wavesymmetry,inducedbythein-planeCoulomb repulsion. For an enhanced interboron Coulomb repulsion (βw >1) the logarithmic divergencies 1 in the superconducting DOS (Eq. 17, Fig. 5) are manifested by a second gap in the experiments. Inourapproachtheelectron-phononcouplingishiddenintheparameterr. Thereforethepressure and isotope effects can be explained by the dependence of r(ω) on the phonon modes. From the non-monotonicT dependence on r it followsthat the MgB materialis inthe underdoped regime c 2 (around r & w /4). For a fictitious system A2+B , where the two electrons are contributed by 1 2 − atomA,thesuperconductingcriticaltemperatureincreaseswithanrincrease. Apressureincrease lowerstheσ-bandwithanr decreaseresultinginanegativepressurederivativeofT inagreement c with experiment [34]. The bell shaped curve T (r), with the MgB position in the underdoped c 2 regime, shows a possibility to reach higher T ’s in diboride materials with an AlB crystal struc- c 2 ture. We suggest the synthesis of materials with increased r-values (e.g. with ”negative chemical pressure”) and optimized smaller interatomic B-B distances in the honeycomb plane. ACKNOWLEDGMENTS This work was supported by the Flemish Science Foundation (FWO-Vl), the Concerted Action program(GOA), the Inter-University Attraction Poles researchprogram(IUAP-IV) and the Uni- versity of Antwerp (UIA). [1] J. Nagamatsu et al.,Nature410, 63 (2001). [2] J. J. Betouras, V.A. Ivanovand F. M. Peeters, submitted (2002). [3] D.F. Agterberg, E. Demler, and B. Janko, cond-mat/0201376. [4] A.F. Goncharov et al., cond-mat/0104042; Phys. Rev.B 64, 100509(R) (2001). [5] V.G. Tissen et al., cond-mat/0105475. [6] T. Vogt et al.,Phys. Rev.B 63, 220505(R) (2001). [7] M.Ozaki,Prog. Theor.Phys.75,442(1986); M.Sigrist, R.Joynt,andT.M.Rice,Phys.Rev.B36, 5186 (1987). [8] M. Sigrist and K.Ueda, Rev.Mod. Phys. 63, 239 (1991) [9] J. M. An and W.E. Pickett,Phys. Rev.Lett. 86, 4366 (2001). [10] C. Panagopoulos et al.,Phys. Rev.B 64, 4514 (2001) [11] J. Betouras and R. Joynt, Europhys. Lett., 31, 119 (1995); J. J. Betouras and R. Joynt, Phys. Rev. B 57, 11752 (1998) [12] R.Joynt, Phys.Rev.Lett. 71, 3015 (1993) [13] T. Yildirim et al., Phys.Rev.Lett. 87, 037001 (2001). 9 [14] J. S. Schilling et al., to appear in: Studies of High-Temperature Superconductors (ed. A. Narlikar), Vol. 38, (NovaScience, NY,2002); cond-mat/0110268. [15] S.Haas and K. Maki, Phys. Rev.B 65, 020502 (2002). [16] I.M. Lifshits, Sov.Phys. JETP 11, 1130 (1960). [17] M. Monteverdeet al.,Science 292, 75 (2001). [18] D.R. Armstrong, P. G. Perkins, J. Chem. Soc. Faraday II 75, 12 (1979). [19] N.I. Medvedevaet al.,JETP Lett. 73, 336 (2001) [20] K.Voelker, V. I.Anisimov, T. M. Rice, cond-mat/0103082. [21] Y.Kong et al.,Phys. Rev.B 64, R020501 (2001). [22] V.A. Ivanov,M. van den Broek and F. M. Peeters, Solid State Commun. 120, 53 (2001). [23] V.Ivanov,Europhys.Lett. 52, 351 (2000). [24] C.-T. Chen et al.,cond-mat/0104285. [25] F. Giubileo et al.,cond-mat/0105592. [26] S.Tsuda et al.,cond-mat/0104489. [27] P.Szabo et al.,cond-mat/0105598. [28] R.A. Fisher et al.,cond-mat/0107072. [29] A.Junodetal.,toappearin: StudiesofHigh-Temperature Superconductors (ed.A.Narlikar),Vol.38, (NovaScience, NY,2002); cond-mat/01106394. [30] F. Bouquet et al, Phys.Rev. Lett.87, 047001 (2001) [31] X.K. Chen et al.,cond-mat/0104005 version 2 (21 May 2001) [32] Y.Wang,T. Plackowski, A. Junod, Physica C 355, 179 (2001). [33] A.Gerashenko et al., cond-mat/0102421. [34] H.Tou et al.,J. Phys.: Condens. Matter 13, L267 (2001). 10