Two-band superconductor magnesium diboride

Two-band superconductor magnesium diboride

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2008 Rep. Prog. Phys. 71 116501

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Rep. Prog. Phys. 71 (2008) 116501 (26pp) doi:10.1088/0034-4885/71/11/116501

Two-band superconductor magnesiumdiborideX X Xi

Department of Physics, Department of Materials Science and Engineering, and Materials Research Institute,The Pennsylvania State University, University Park, PA 16802, USA

Received 1 October 2008Published 5 November 2008Online at stacks.iop.org/RoPP/71/116501

AbstractThis review focuses on the most important features of the 40 K superconductor MgB2—the weaklyinteracting multiple bands (the σ and π bands) and the distinct multiple superconducting energy gaps(the σ and π gaps). Even though the pairing mechanism of superconductor MgB2 is the conventionalelectron–phonon coupling, the prominent influence of the two bands and two gaps on its propertiessets it apart from other superconductors. It leads to markedly different behaviors in upper criticalfield, vortex structure, magnetoresistance and many other superconducting and normal-state propertiesin MgB2 from single-band superconductors. Further, it gives rise to new physics that does not exist insingle-band superconductors, such as the internal Josephson effects between the two order parameters.These unique phenomena depend sensitively on scattering inside and between the two bands, and theintraband and interband scattering can be modified by chemical substitution and irradiation. MgB2

has brought unprecedented attention to two-band superconductivity, which has been found to exist inother old and new superconductors. The legacy of MgB2 will be long lasting because of this, as wellas the lessons it teaches in terms of the search for new phonon-mediated higher Tc superconductors.

(Some figures in this article are in colour only in the electronic version)

This article was invited by Professor L H Greene.

Contents

1. Introduction 11.1. Discovery of superconductivity in MgB2 11.2. Pairing mechanism and two gaps in MgB2 21.3. Basic properties of MgB2 31.4. Potential of MgB2 for applications 31.5. Preparation of MgB2 materials 4

2. Multiple bands and multiple gaps in MgB2 52.1. Electronic structure 52.2. The E2g phonon and electron–phonon coupling 52.3. Two-gap superconductivity 52.4. Experimental evidences for two bands and two

gaps 63. Impact of multiple bands and multiple gaps on

properties of MgB2 7

3.1. Superconducting properties 83.2. Normal-state properties 113.3. New effects due to two superconducting order

parameters 134. Modification of two bands and two gaps 14

4.1. Chemical doping 154.2. Irradiation 17

5. Broader impact of MgB2 on superconductivity 175.1. Multi-band superconductivity 175.2. Search for higher Tc superconductors 19

6. Concluding remarks 19Acknowledgments 20References 20

1. Introduction

1.1. Discovery of superconductivity in MgB2

The superconductivity in MgB2 was first announced in Januaryby Akimitsu of Aoyama Gakuin University, Japan, and

reported in the 1 March 2001 issue of the journal Nature [1].MgB2 has a transition temperature Tc of about 40 K (seefigure 1), the highest in conventional superconductors andnearly two times the previous record in such superconductors.The crystal structure of MgB2 is shown in figure 2, which

0034-4885/08/116501+26$90.00 1 © 2008 IOP Publishing Ltd Printed in the UK

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Rep. Prog. Phys. 71 (2008) 116501 X X Xi

100

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Figure 1. Temperature dependence of resistivity of the MgB2

sample. (Reprinted by permission from Macmillan Publishers Ltd:Nature [1], copyright 2001.)

Figure 2. Crystal structure of MgB2. Boron atoms form stacks ofhoneycomb layers and magnesium atoms are in between the boronlayers at the center of the hexagons.

consists of honeycombed boron layers and magnesium layerslocated in between the boron layers [1]. The hexagonalunit cell has the lattice parameters a = 3.086 Å and c =3.524 Å [1]. It has been documented that the discovery was tosome extent accidental [2–5], but the excitement it generatedwas tremendous [5]. Work immediately followed the newsof the Akimitsu discovery in laboratories around the world,leading to a burst of reports on various properties of MgB2.These included the isotope effect [6], the thermodynamic andtransport properties [7], band structures [8, 9], critical currentdensity [10, 11], doping effect [12] and pressure effect [13].The initial excitement was largely based on the fact that it isa simple intermetallic compound of two inexpensive elementsand unlike high temperature cuprate superconductors (HTS), itsuperconducts via the conventional electron–phonon couplingmechanism. The prospect of practical applications for MgB2

seemed more promising than for HTS even though the Tc of40 K is much lower than 160 K in HTS. It has also becomeclear during the more than seven years since the discovery thatthe properties of MgB2 are uniquely interesting with plenty ofnew physics to be explored.

0 5 10 15 20 25 30 35 400

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Figure 3. Values of the two gaps in MgB2 extracted from thescanning tunneling spectroscopy as a function of temperature. Thelines are the BCS (T ). (Reprinted with permission from Iavaroneet al [20]. Copyright 2002 by the American Physical Society.Permission granted by Argonne National Laboratory, managed andoperated by UChicago Argonne, LLC, for the US Department ofEnergy under Contract No. DE-AC02-06CH11357.)

1.2. Pairing mechanism and two gaps in MgB2

MgB2 is a conventional BCS superconductor, i.e. it becomessuperconducting when electrons form Cooper pairs viaelectron–phonon interaction. The most direct evidence fora phonon-mediated mechanism is the isotope effect, in whicha Tc decrease when the isotopic mass of the superconductorincreases indicates the involvement of the lattice in thesuperconductivity [14]. Experiments by Bud’ko et al [6] andHinks et al showed that Tc of MgB2 increased by about 1 Kwhen 10B was used to make the compound instead of 11B.On the other hand, the isotope effect for Mg is much smaller,signaling that the B atom vibrations are more important forthe superconductivity in MgB2 [15]. The BCS mechanismwas also confirmed by photoemission spectroscopy [16],scanning tunneling microscopy [17] and neutron scatteringmeasurements [18].

MgB2, however, is by no means an ordinary supercon-ductor. Normally in a superconductor below Tc, there existsone temperature-dependent energy gap (T ) such that a min-imum energy of 2(T ) is needed to break a Cooper pair intotwo quasiparticles [14]. Two such gaps exist in MgB2, onewith (0) ∼ 2 meV and another (0) ∼ 7 meV [3, 19]. Infigure 3, the values of the two gaps extracted from a scan-ning tunneling spectroscopy are plotted as a function of tem-perature [20]. Both gaps follow the temperature dependencepredicted by the BCS theory, as indicated by the lines, andboth gaps disappear at the same Tc. According to the BCStheory, 2(0) = 3.53 kTc [14]. The two energy gaps shouldthen correspond to two Tcs of 15 K and 45 K, respectively [21].The finite coupling between the two gaps results in a single Tc

of 40 K.The two gaps arise from the existence of two bands, the

σ and π bands of the boron electrons [9]. Mazin et al havepointed out that the interband impurity scattering between theσ and π bands is exceptionally small primarily due to thedifferent symmetries of their charge density distributions [22].This small interband scattering distinguishes MgB2 from other

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Figure 4. Temperature dependence of resistivity of a MgB2 thinfilm. (From [35].)

superconductors with multiple bands at the Fermi surface,where the interband scattering normally smears out the distinctcharacteristics of the bands.

The existence of two gaps was quickly discovered andconfirmed by many experiments, including the measurementsof specific heat [23, 24], point-contact spectroscopy [25],Raman spectroscopy [26, 27], photoemission spectroscopy[28] and scanning tunneling microscopy (STM) [29].Although two-band superconductivity was first proposed inthe late 1950s [30], MgB2 is the first superconductor in whichthe two-gap effects are significant and extensive [3, 21]. Thetwo-gap superconductivity in MgB2 is the central subject ofthis review.

1.3. Basic properties of MgB2

The basic properties of MgB2 are briefly summarized here. Anearlier collection of experimental results on various propertiesof MgB2 can be found in [31]. For more detailed reading,readers are referred to two special issues of Physica C onMgB2 [32, 33].

Besides a high Tc around 40 K, MgB2 has a very lowresidual resistivity. By reacting boron fiber with Mg vapor,Canfield et al showed that the resistivity of dense MgB2 wirescan be as low as 0.38 µ cm at 42 K, leading to a residualresistivity ratio (RRR ≡ R(300 K)/R(40 K)) of over 25 [34].Figure 4 shows a resistivity versus temperature curve for a770 nm MgB2 thin film on sapphire substrate grown by a hybridphysical–chemical vapor deposition (HPCVD) technique [35].A low residual resistivity ρ0 of less than 0.1 µ cm is seen,leading to a residual resistivity ratio RRR of over 80. Fitting thedata to ρ(T ) = ρ0 + ρph(T ), where ρph(T ) is the temperature-dependent phonon scattering contribution of the generalizedBloch–Gruneisen form [36, 37], one gets ρ0 = 0.059 µ cm,the Debye temperature θ = 1232 K andρ(T ) = ρ0+const×T 3

for T θ . The electron mean free path l in this film is limitedby the scattering at the film surface and the film/substrateinterface. The clean MgB2 samples are well in the clean limit(l ξ , where ξ is the coherence length).

Figure 5. Temperature dependence of upper critical fielddetermined by different techniques for two field directions for asingle crystal of MgB2. The inset shows the temperature-dependentanisotropy coefficient γ . (Reprinted from Welp et al [38].Copyright 2003, with permission from Elsevier.)

The upper critical field Hc2 of clean MgB2 is low andanisotropic. Figure 5 shows the temperature dependence of theupper critical field for a MgB2 single crystal for two magneticfield directions H ‖ c and H ‖ ab [38]. The anisotropycoefficient, γ ≡ Hc2,ab/Hc2,c, is shown in the inset as afunction of temperature. Different measurement techniquesgive different Hc2 values, but they are around 5 T for H ‖ c and20 T for H ‖ ab at low temperature. From Hc2,c = φ0/2πξ 2

ab

and Hc2,ab = φ0/2πξabξc, where φ0 = 2.07 × 10−15 Wbis the magnetic flux quantum, this leads to ξab(0) ∼ 8 nmand ξc(0) ∼ 2 nm. A microwave measurement by a sapphireresonator technique found that the penetration depth λab(0) isabout 40 nm for clean MgB2 films [39]. Thus the Ginzburg–Landau parameter κ = λ/ξ for clean MgB2 is about 5, a rathersmall value among various type-II superconductors.

The highest self-field critical current density Jc for MgB2

has been reported in pure films deposited by HPCVD [40].Figure 6 shows the result of a transport Jc measurement of apatterned bridge of 20 µm width in an HPCVD MgB2 film.The self-field Jc is 3.4 × 107 A cm−2 at 4.2 K, which is about4% of the deparing current density (the maximum supercurrentdensity limited by the kinetic energy of the Cooper pairs) Jd ∼8.7 × 108 A cm−2 for MgB2, estimated using the Ginzburg–Landau formula Jd = φ0/[3/(

√3)πµ0λ

2(T )ξ(T )] [14]. Aself-field Jc of 1.6 × 108 A cm−2 at 2 K has been reportedby Zhuang et al from a 150 nm wide bridge in an HPCVDfilm [41]. However, Jc is suppressed quickly by the appliedmagnetic field due to the lack of pinning in the film.

1.4. Potential of MgB2 for applications

MgB2 is a promising superconductor for high-magnetic-field applications [42–44]. Unlike high-Tc superconductorswhere Jc drops sharply across the grain boundaries, grainboundaries in MgB2 do not significantly degrade Jc andeven serve as pinning centers [10, 11]. Doping or alloyingwith carbon or SiC have shown to significantly enhance

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Figure 6. Transport Jc versus temperature for a MgB2 film onSiC substrate measured under different applied magnetic fields.(From [40].)

Hc2 [45] and pinning [46, 47]. MgB2 conductors also promiselow cost [43]. Wires and tapes of MgB2 have been madeusing the so-called ‘powder-in-tube’ (PIT) technique withencouraging results [48–50]. The Tc at 40 K allows practicaloperation above 20 K using cryocoolers, allowing for cryogen-free operation of magnets such as in magnetic resonanceimaging (MRI) systems [51]. Commercial companies aroundthe world have made impressive progress producing long-length wires and magnets [51–53], and 5 years after thediscovery of superconductivity in MgB2, an MRI system usingMgB2 magnets cooled by cryocoolers was manufactured andproduced brain images [54].

The interest in MgB2 for electronic applications such as insuperconducting digital circuits is mainly due to its high Tc ascompared with the currently used Nb-based superconductors,which have to be cooled to 4.2 K [55, 56]. MgB2 devicesand circuits can be operated above 20 K, and a 20 K coolerrequires much less input power, is much lighter and of muchlower cost than a 4 K cooler with the same cooling power[57]. MgB2 Josephson junctions and SQUID have beendemonstrated to work well over 20 K [58–60]. Sandwich-type all-MgB2 junctions have been reported [61–65]. Recentlythe result by Ke et al in MgB2/MgO/MgB2 Josephson tunneljunctions showed junction Jc up to near 40 K [65]. Other MgB2

devices such as bolometers and neutron detectors have alsobeen explored [66].

The low resistivity (0.1 µ cm) shown in figure 4 incombination with an energy gap of 2 or 7 meV promises aBCS microwave surface resistance much lower than that of Nb[67], which is currently the best material for superconductingRF cavities used in accelerators [68]. The small value ofκ also leads to high values of the thermodynamic criticalfield Hc (∼800 mT as compared with 200 mT for Nb) orthe superheating field Hsh. Low BCS surface resistanceis important for high Q of a cavity, and high Hc or Hsh

is important for high ultimate RF critical field. The great

potential of MgB2 for superconducting RF cavities has beenrecognized by researchers in the field [69, 70].

1.5. Preparation of MgB2 materials

High quality bulk polycrystalline MgB2 materials can beprepared by solid-state reaction of high purity Mg and B ina sealed Ta tube [7, 34, 71]. Pure materials are characterizedby low residual resistivity near or below 1 µ cm and highRRR. The technique of reacting boron fiber with Mg vaporwas particularly successful, yielding dense MgB2 wires withresidual resistivity of 0.38 µ cm and RRR of over 25 [34].Kim et al suggested that the high RRR in these materials couldbe due to pure Mg in the samples [72]. However, thin filmswith no evidence of pure Mg in them can show even high RRRvalues [35].

Growth of MgB2 single crystals is difficult. Accordingto the thermodynamic phase diagrams of the Mg–B system,MgB2 melts congruently only under pressure higher than65 bar at the temperature of 2430 C [73], making growth ofcrystals from stoichiometric melt impossible. Reaction of Mgat high temperatures with container materials, metal fluxesand oxygen also poses serious problems [74, 75]. The mostsuccessful crystal growth technique has been the high-pressure,high temperature growth from a precursor containing Mg, Band BN, in which the formation of liquid phase eutectics inthe Mg–B–N system is helpful for crystallization of MgB2

[76, 77]. Plate-like single crystals with a size over 1 mm withgood superconducting properties have been grown by thistechnique [76, 77]. For substitution studies, single crystalswith C, Al, Mn, Fe and Li doping have been successfullygrown [77].

The most important requirement for the deposition ofMgB2 films is to provide a sufficiently high Mg vapor pressurefor the thermodynamic phase stability of MgB2 at elevatedtemperatures [73]. A clean environment for growth andpure sources of Mg and B are other key requirements. Thedeposition techniques used for MgB2 films include hightemperature ex situ annealing of B or Mg–B precursor films inMg vapor [78, 79], intermediate-temperature in situ annealingof Mg–B precursor films [80–83], low temperature in situdeposition [84–86] and high and intermediate temperaturein situ deposition [87–89]. Of all these techniques, HPCVD[35, 89] has been the most effective, producing epitaxial filmswith excellent properties. The Tc values of the pure HPCVDfilms are even higher than the bulk samples (see figure 4)due to epitaxial strain generated during the growth process[90]. The most studied doping in MgB2 films is carbon[91]. Extraordinarily high Hc2 over 60 T has been observedin carbon-alloyed HPCVD thin films [42, 45].

Noting that the resistivity of MgB2 samples reported in theliterature varies by orders of magnitude, even well beyond themetal–insulator transition, but the samples still have high Tc

values, Rowell suggested that the high resistivity is the resultof the reduction in current-carrying cross-sectional area bygrain boundaries, voids, etc [92]. The connectivity of thesuperconducting grains can be characterized by a quantityρ ≡ ρ(300 K) − ρ(50 K). A perfectly connected MgB2

4


Figure 7. The Fermi surface of MgB2 from band structurecalculation. Green and blue cylinders (hole-like) are the σ bands,and the blue (hole-like) and the red (electron-like) tubular networksare the π bands. (Reprinted with permission from Kortus et al [9].Copyright 2001 by the American Physical Society.)

sample shows ρ ∼ 7–9 µ cm, while a higher ρ indicatesreduced connectivity. A predicted correlation between ρ andtransport Jc has been proven by experiment [93].

2. Multiple bands and multiple gaps in MgB2

In the study of MgB2, theoretical investigations have achievedimpressive success and played an important role [94–96].From first-principles calculations, the electronic structure,the phononic structure and the electron–phonon coupling inMgB2 were revealed. Through Eliashberg equations, varioussuperconducting properties were calculated. Recently, thesuperconducting properties were further calculated directlyfrom first-principles using density function theory forsuperconductors, without the need for semiphenomenologicalparameters for Coulomb repulsion [97, 98]. The results ofthese theoretical works, on which the following descriptionsof the multiple bands and multiple gaps in MgB2 are based,agree with experiment remarkably well.

2.1. Electronic structure

The electrons at the Fermi surface of MgB2 are mainlyp electrons of boron, with Mg donating its s electrons to theconduction bands [8, 9]. There are four conduction bands.From the σ -bonding px,y orbitals of boron two σ bands arederived, whereas two π bands are derived from the π -bonding(hole-like) and antibonding (electron-like) pz orbitals. The σ

states are two-dimensional confined in the boron planes andthe π states are three-dimensional extending in all directions[9, 19]. The charge density distribution shows that the bondswithin the boron layers are strongly covalent, whereas thebonds between these layers are metallic [94]. The Fermisurface of MgB2 is shown in figure 7. The two cylinders aroundthe –A lines are the two σ bands and the two webbed tunnelsare due to the two π bands. The densities of states at theFermi level are 0.300 states eV−1/cell for the two σ bands and0.410 states eV−1/cell for the two π bands [94, 99].

The impurity scattering between the σ and π bands aresmall, as argued by Mazin et al [22]. This is mainly becausethe px,y (in-plane) and pz (out-of-plane) orbitals are orthogonalto each other, and the overlap integrals which determinethe interband coupling and interband impurity scattering arestrongly reduced. Further, the pz orbitals have odd parityand the px,y orbitals have even parity with respect to theB layer, leading to small hybridization between the σ andπ bands [22].

2.2. The E2g phonon and electron–phonon coupling

There are four phonon modes at the Brillouin zone centerin MgB2: the B1g, E2g, A2u and E1u modes [100, 101]. TheE2g mode involves the in-plane vibration of the boron ions inopposite directions (bond stretching). This motion changes thepx,y orbital overlap; therefore the electron–phonon couplingbetween the E2g mode and the σ band is very strong [8, 9, 101].It is this strong electron–phonon coupling that leads to the highTc in MgB2.

The E2g phonon is anharmonic: the dependence of thephonon energy on boron displacement has a large fourth-powerterm [101, 102]. The anharmonicity is confined to phononsnear the –A line, where the electron–phonon coupling is alsovery large [103]. The anharmonicity causes a hardening ofthe E2g phonon frequency and weakens the electron–phononcouplings as compared with the harmonic phonon, although theestimate of the phonon frequency shift depends on the methodsof calculation [104].

Because there are four conduction bands, the electron–phonon coupling constant becomes a 4 × 4 matrix includingphonon-mediated scattering of electron from any one of the4 bands to any other of the 4 bands [99, 102]. Since the twoσ bands are similar and the two π bands are similar (eventhough one π band is hole-like and the other is electron-like), the matrix is most often simplified to 2 × 2, and MgB2

is referred to as a two-band superconductor [102]. Theelectron–phonon coupling constant matrix elements have beencalculated by various groups: λσσ = 0.96, λππ = 0.29,λσπ = 0.23, λπσ = 0.17 and the total λ = 0.77 by Liu and co-workers [94, 102]; λσσ = 1.017, λππ = 0.448, λσπ = 0.213,λπσ = 0.155 and λ = 0.87 by Golubov et al [99]; λσσ = 0.78,λππ = 0.21, λσπ = 0.15, λπσ = 0.11 and λ = 0.61 byChoi et al (obtained from integration of λ(k, k′) over the Fermisurface [94, 103]) and λσσ = 0.83, λππ = 0.28, λσπ = 0.22,λπσ = 0.16 and λ = 0.71 by Floris et al [98]. Despite somedifferences among the results, it is clear that the electron–phonon coupling is very strong for the σ bands and muchweaker for theπ bands. The interband coupling is even weaker,although it is not negligible [105].

2.3. Two-gap superconductivity

From the information on the electron–phonon coupling con-stant matrix elements and with appropriate semiphenomeno-logical Coulomb pseudopotential matrix elements, the super-conducting energy gap and Tc were calculated [99, 102]. Theresults show 2 gaps, σ and π , that vanish together as the

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(a) (b)

Figure 8. (a) The superconducting energy gaps on the Fermi surface for MgB2 from the band structure calculation. The color scalecorresponds to the distribution of gap values shown in (b). (Reprinted by permission from Macmillan Publishers Ltd: Nature [19], copyright2002.)

temperature increases towards Tc [99], in agreement with theexperimental results such as that shown in figure 3.

The details of the two-band superconductivity have beendescribed by Choi et al [19], who employed an anisotropicEliashberg formalism and calculated the electron–phononcoupling constant for electron pairs originating from all sheetsof the Fermi surface [103]. The energy gap value for eachpoint of the Fermi surface is shown in figure 8(a). The colorscale corresponds to the distribution of gap values shown infigure 8(b). The gap on the σ bands ranges from 6.4 to 7.2 meVwith an average of 6.8 meV, and on the π bands it ranges from1.2 to 3.7 meV with an average of 1.8 meV [19]. From theBCS relation (0) = 1.764kTc, the σ gap would lead to asuperconductor with Tc of 45 K and the π gap to Tc of 15 K.Because of the finite interband coupling, superconductivityoccurs at a Tc of 39 K in MgB2 [21].

Mazin et al have argued that it is not necessary to usethe fully anisotropic Eliashberg formalism and the two-gapformalism is sufficient unless the intraband scattering is smalland the mean free path of the sample is greater than 1500 Å[106]. Indeed, the distribution of gaps shown in figure 8(b) hasbeen observed in tunnel junctions using very clean HPCVDMgB2 films [107].

2.4. Experimental evidences for two bands and two gaps

The first experimental hint to the two-band superconductivitycame from the specific heat measurements [23, 24, 108]. Asshown in figure 9 where the electronic specific heat is plottedas a function of temperature, the experimental data cannot beexplained by the one-gap BCS model with a Tc of 39.4 K [19].At lower temperatures, the experimental results become higherthan the BCS curve and then show another decrease to zerosuggestive of another transition. This shoulder is a strongindication of a second, smaller gap. Also plotted in the figureis the result of the calculation by Choi et al using two gapsshown in figure 8, and it is in excellent agreement with theexperiments [19]. The low temperature shoulder is caused bythe excitations across the π gap.

Figure 9. Electronic contribution to the specific heat as a functionof temperature for MgB2. The solid curve is the result of a two-gapmodel calculation. The dashed curve is the single-gap BCSprediction for a superconductor with Tc of 39.4 K. (Reprinted bypermission from Macmillan Publishers Ltd: Nature [19], copyright2002.)

Point-contact spectroscopy [25, 109–111] and tunnelingspectroscopy [20, 29, 112] allow direct measurement of thesuperconducting gap. Figure 10(a) shows the Cu–MgB2

point-contact spectra on several polycrystalline MgB2 samplesmeasured at T = 4.2 K by Szabo et al [25]. The dotted linesare the two-band Blonder–Tinkham–Klapwijk (BKT) modelfits with different barrier transparencies and weight factors(from 65% to 95% π -band contribution). Conductance peakscorresponding to the two gaps are clearly observed: σ =6.8 meV and π = 2.8 meV. Figure 10(b) shows tunnelingspectrum on an off-axis MgB2 epitaxial film measured at 4.2 Kby Iavarone et al clearly showing the two gaps [113]. The resultby Iavarone et al shown in figure 3 indicates σ = 7.1 meVand π = 2.3 meV at 4.2 K [20].

Figure 11 shows electronic Raman spectra, obtained bysubtracting the spectra in the normal state from that in thesuperconducting state, from a polycrystalline MgB2 sample

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Figure 10. (a) Cu–MgB2 point-contact spectra at T = 4.2 K (solidlines) for several polycrystalline MgB2 samples. The dotted linesare BTK model fitting results. (Reprinted with permission fromSzabø et al [25]. Copyright 2001 by the American PhysicalSociety.) (b) An STM spectrum tunneling into the ab plane on anoff-axis MgB2 epitaxial film at 4.2 K. (From Iavarone et al [113].)

Figure 11. Electronic Raman spectra obtained by subtracting the45 K spectra from the 15 K spectra. The thick solid lines aretheoretical fits. The HV spectra are shifted for clarity. The inset is acalculated spectrum without Gaussian broadening. (Reprinted withpermission from Chen et al [26]. Copyright 2001 by the AmericanPhysical Society.)

by Chen et al [26]. Two peaks appear as a result of thesuperconducting transition at 50 and 105 cm−1, which are thepair-breaking peaks corresponding to two binding energies ofthe Cooper pairs 2π and 2σ [26]. The gap values thusobtained are σ = 7.1 meV and π = 2.3 meV.

Figure 12. Photoemission spectra measured at 5.4 and 45 K. Theinset shows the enlarged spectrum near EF taken at 5.4 K, which canbe fitted by assuming two gaps at 1.7 meV (broken line) and5.6 meV (dotted line), respectively. (Reprinted from Uchiyamaet al [28, 117]. Copyright 2003, with permission from Elsevier.)

Photoemission spectroscopy also provided evidence forthe existence of two bands [114] and two gaps [28,115, 116]. Figure 12 (the inset shows details near EF)shows photoemission spectra of MgB2 measured in thesuperconducting and normal state by Tsuda et al [28]. Besidesa peak around 7 meV and a shift of the leading edge, indicatingthe opening of a superconducting gap, a shoulder structure at3.5 meV was observed. The spectrum can only be fitted iftwo superconducting gaps, one at 1.7 meV (broken line) andthe other at 5.6 meV (dotted line), are assumed. Later, thetwo gaps have been separately measured using angle-resolvedphotoemission spectroscopy (ARPES) [115, 116].

3. Impact of multiple bands and multiple gaps onproperties of MgB2

The characteristics of the superconducting energy gap haveoverreaching influences on the properties of a superconductor.For example, the symmetry of the gap can be s-wave as inconventional superconductors, d-wave as in HTS [118, 119],or p-wave as in Sr2RuO4 [120–122]. MgB2 is an s-wavesuperconductor in which the gap does not have nodes (the phaseof the gap does not change over the Fermi surface) [123, 124].Nontheless, MgB2 is unique in that the gap has differencemagnitudes on difference pieces of the Fermi surface.

Two-gap superconductivity has been theoretically consid-ered since the 1950s, first for transition metals in which sand d bands exist at the Fermi surface [30]. Before MgB2

was discovered, Nb-doped SrTiO3 [125] and LuNi2B2C [126]have been suggested as two-band superconductors. However,MgB2 was the first superconductor that presents a clear andstrong case for two-band superconductivity. The existence ofmultiple bands and multiple gaps has a comprehensive impacton many aspects of the properties of MgB2. It also presents tothe community of condensed matter physics opportunities to

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c2(π)

H

Figure 13. Temperature dependence of Hc2 for the dirty two-gapsuperconductor MgB2 illustrated by a bilayer toy model shown inthe inset. The dashed curves are for σ and π films in the one-gapdirty limit. The solid curve shows Hc2(T ) calculated from thetwo-gap dirty-limit BCS theory. (Reprinted from Gurevich [128].Copyright 2007, with permission from Elsevier.)

observe new physical phenomena in MgB2 that do not exist insingle-gap superconductors [21].

3.1. Superconducting properties

3.1.1. Upper critical field. For single-gap dirty-limit superconductors, the upper critical field Hc2(0) =0.69Tc(dHc2/dT )Tc and (dHc2/dT )Tc ∝ ρn [127]; thereforeHc2 increases with normal-state resistivity ρn, which canbe achieved by adding impurities and defects into thesuperconductor. Gurevich pointed out that the two-bandsuperconductor MgB2 can be understood as a weakly-coupledbilayer in which two thin films corresponding to the σ and π

bands are in contact through Josephson coupling [128] (seefigure 13). Using the dirty-limit weak-coupling multi bandBCS model taking into account both interband and intrabandscattering by nonmagnetic impurities, Gurevich showed thatthe temperature dependence of Hc2(T ) depends on whether theσ bands or π bands are dirtier and can be very different fromthat in the one-band theory [129]. As schematically shown infigure 13, the global Hc2(T ) of the bilayer is dominated bythe layer with the higher Hc2. If the π layer is dirtier, it willhave higher Hc2 at low temperature even though its Tc is muchlower. As a result, an upturn in the global Hc2(T ) occurs at lowtemperature. Because of the existence of two bands, Hc2(0) inMgB2 can exceed 0.69Tc(dHc2/dT )Tc considerably.

The Hc2 behaviors described by Gurevich have beenobserved in experiment. For example, in thin films fromvarious groups with disorder introduced in different ways,Braccini et al observed different types of temperaturedependence of Hc2, including the anomalous upturn at lowtemperature, reflecting different multi band scattering in thesesamples [45]. The value of Hc2 in carbon-doped thin filmshas reached over 60 T at low temperature (see figure 14),

0 10 20 30 400

10

20

30

40

50

60

NbTi Nb3Sn

MgB2

Fie

ld (

T)

Temperature (K)

MgB2 //

Figure 14. Hc2 versus temperature plots for a carbon-alloyedHPCVD film, Nb–Ti (bulk) and Nb3Sn (bulk). The triangle andsquare data are for the ab planes of MgB2 parallel and normal to thefield, respectively. (From [42].)

0 10 20 30 40

1

2

3

4

5

6

T [K]

γλ

γξ (a)

(b)

(c)

Tc

γλ

γξ

34 36 381.0

1.5

2.0

2.5

T [K]

Tc

1

3

5

T/Tc

γξ

γλ

Figure 15. (a) Temperature dependence of γξ (γξ = γH ) and γλ.(b) Theoretical predictions for γξ (T ) and γλ(T ). (c) Details of theexperimental data in (a) close to Tc. (Reprinted with permissionfrom Fletcher et al [135]. Copyright 2005 by the American PhysicalSociety.)

approaching the BCS paramagnetic limit of 65 T [42, 45].Considering the electron–phonon coupling effect, Gurevichargued that the strong coupling paramagnetic limit in MgB2

can be as high as 130 T; thus there is still room for furtherenhancement of Hc2 by engineering the σ - and π -bandscattering [128]. The high Hc2 in MgB2 is very attractive forhigh-magnetic-field applications.

3.1.2. Anisotropy. In anisotropic Ginzburg–Landau theory,the anisotropy parameter for the upper critical field γH ≡H

‖ab

c2 /H‖cc2 = √

m∗c/m∗

ab is not strongly temperature dependent

[130], where H‖ab

c2 and H‖cc2 are for magnetic field parallel

to ab and c axes, respectively, and m∗ is the effective mass.However, experimental results such as that shown in the insetto figure 5 and in figure 15 (γH = γξ = ξab/ξc) showed strongtemperature dependence of γH in MgB2 [131–135]. This hasbeen explained as due to the two-band effect [129, 136–138].

8


(c) (d)

Figure 16. (a) Magnitude of the order parameter, σ and π , as afunction of coordinate x on a path through the vortex center;(b) partial DOS at zero bias for the σ and π bands as a function ofdistance r from the vortex center; (c) and (d) contributions from thetwo bands to the supercurrent density around the vortex for twoξπ/ξσ values. ρ is the mixing ratio between the σ and π gaps.(Reprinted with permission from Tanaka et al [144]. Copyright2006 by the American Physical Society.)

The σ band is two-dimensional and the σ gap is veryanisotropic [19]. However, the relative contribution of theπ gap to superconductivity, which is isotropic, changes withtemperature. In the case of clean MgB2, there is more thermalmixing of the π -gap states with the σ -gap states at highertemperature, suppressing the anisotropy when temperatureincreases [130, 139]. The stronger the interband couplingbetween the σ - and π -gap states, the larger this suppressionis [138]. On the other hand, in dirty samples in which theπ band is much dirtier than the σ band, the dominant role ofπ band at low T causes the anisotropy to be smaller at lowertemperatures [129].

Kogan pointed out that although it is common to considerγH = γλ (γλ = λab/λc = √

m∗c/m∗

ab in anisotropic Ginzburg–Landau theory), in materials with anisotropic Fermi surfacesand anisotropic gaps, γH can be considerably different fromγλ away from near Tc [140]. Figure 15(b) shows the resultsfrom the two-band model calculations of γξ (T ) and γλ(T ) byGolubov et al [136] (solid lines) and Miranovic et al [137](dashed lines). Near Tc γH = γλ, but at low temperature γλ

approaches 1. It was predicted that γλ depends strongly onthe impurity scattering in the π band [136]. The theoreticalpredictions are in good agreement with the experimental resultsas shown in figure 15 [124, 135].

Both the temperature dependences of Hc2 and anisotropyillustrate that the two-band nature of superconductivityhas a large impact on the applicability of the anisotropicGinzburg–Landau theory in describing the properties of MgB2

[129, 141, 142].

3.1.3. Vortices in MgB2. Because of the two-bandsuperconductivity, the structure of vortex in MgB2 is complex.

Figure 17. Images of vortices in MgB2 from scanning tunnelingmicroscopy at T = 2 K: (a) 0.05 T and (b) 0.2 T. The color scale forthe zero bias conductance normalized to the conductance peak isshown at right. (Reprinted with permission from Eskildsenet al [112]. Copyright 2002 by the American Physical Society.)

For a single-band superconductor, the superconducting orderparameter is zero at the center of the vortex core and increasesto that of the superconductor bulk over the coherence lengthξ [14]. In MgB2, there are two coherence lengths ξσ and ξπ . Ifboth bands are dirty, ξσ = √

Dσ/2πTc and ξπ = √Dπ/2πTc,

where Dσ and Dπ are the diffusion constants (a small D

indicates large impurity scattering) [143]. If the σ band isclean, ξσ = vFσ /2πTc, where vFσ is the Fermi velocity for theσ band [144, 145]. Experiments [112, 146] show that ξπ/ξσ

is between 1 and 3. The vortex structure calculations byTanaka et al for clean σ band and dirty π band are shownin figure 16, where ρ = π/σ near Tc signifies the couplingbetween the σ and π gaps [144]. From these results, we find,among many unique features, that σ and π recover the bulkvalues over different length scales, large quasiparticle densityof state from the π bands exists where the σ band has recoveredbulk superconductivity and there are two sets of supercurrentaround the vortex core. The apparent vortex size, ξv, whichis associated with the maximum of the supercurrent densityaround the vortex center [145], depends on the ratio ξπ/ξσ andcan be larger than ξc2 extracted from Hc2, ξc2 = √

0/2πHc2

[143–145]. This effect has been observed in the scanningtunneling microscopy (STM) study of the vortex structure byEskildsen et al [112]. In figure 17, STM images of vorticesobtained by tunneling parallel to the c axis are shown for themagnetic fields of 0.05 and 0.2 T. The color scale correspondsto the the zero bias conductance, normalized to the conductancepeak, and high values represent the vortex core. The size ofthe vortices in the images leads to a ξv ∼ 50 nm, much largerthan ξc2 = 10 nm estimated from the Hc2 of the sample. Infact, it was the results of Eskildsen et al that motivated theabove-mentioned theoretical calculations.

Figure 17 shows that even a low field of 0.2 T causes asubstantial increase in the zero bias conductance between thevortices [112]. Calculation by Koshelev and Golubov [143]showed that this is primarily caused by the breaking of the π -band Cooper pairs. As shown in figure 18, the quasiparticleDOS from the π band quickly reaches its normal-state value atmagnetic fields much smaller than Hc2 (h = H/Hc2), whereasthe breaking of the σ -band Cooper pairs by magnetic fieldoccurs much more gradually [143].

The unique vortex structure and its field dependence willhave a significant influence on the vortex dynamics in MgB2.

9


N2(0)

0.2 0.4 0.6 0.8 1h

N1(0)

0

0.2

0.4

0.6

∆ max

/ T

c ∆max1

∆max2

∆max1

∆max2

N2(0)

0

0.4

0.8

0 0.2 0.4 0.6 0.8 1h

N1(0)

(a) (b)

(c) (d)

D1 = 0.2 D2 D1 = D2

Figure 18. Field dependences of maximum energy gap ((a) and (b))and averaged zero bias DOS ((c) and (d)) for D1 = 0.2D2 andD1 = D2. (Reprinted with permission from Koshelev andGolubov [143]. Copyright 2003 by the American Physical Society.)

Figure 19. I–V and dI/dV –V curves for a MgB2/I/Pb junctionmade with a MgB2 film on MgO (2 1 1) substrate measured at 4.2 K.(From [148].)

For example, the experimental finding of a finite dissipationstate at the T = 0 K limit in MgB2 has been explained by thevortex structure composed of overlapping σ -band superfluidand π -band quasiparticles [147].

3.1.4. Electron tunneling spectroscopy Electron tunnelingspectroscopy measures the quasiparticle density of statesin a superconductor; therefore tunneling experiments haveprovided the most direct evidences of the two gaps in MgB2,such as the STM result in figure 3 and the point-contactspectroscopy result in figure 10. Figure 19 shows the currentand conductance curves measured at 4.2 K for a MgB2/I/Pbjunction made with a MgB2 film on the MgO (2 1 1) substrate[148]. Two peaks were clearly seen in the dI/dV –V curve,one corresponding to σ +Pb and the other corresponding toπ +Pb, and the gap values were found to be σ = 7.4 meVand π = 2.1 meV [148].

Because the σ bands are two-dimensional and confinedto the ab plane, the tunneling results in MgB2 depend on thecrystallographical orientations of the two electrodes. If thetunneling is in the c direction, the overlap integrals with theσ orbitals are small, and the conductance is determined bythe π band [149]. The result in figure 19 was obtained on a

Figure 20. Calculated differential conductance as a function of theapplied voltage for MgB2/I/MgB2 junctions with different barriertransparencies when the angle between the orientations of the twoelectrodes γ is π/4. The barrier transparency is lower in (a) thanin (b). (Reprinted with permission from Graser and Dahm [150].Copyright 2007 by the American Physical Society.)

MgB2 film whose c axis was tilted about 19.5 away from thefilm normal, thus exposing the ab plane for tunneling. ForMgB2/I/MgB2 junctions, Graser and Dahm calculated the ab-to-ab tunneling conductance when the two MgB2 electrodeshave an angle γ between their orientations, which determinesthe effective strength of interband and intraband tunneling.[150]. The result for γ = π/4 is shown in figure 20 for(a) low- and (b) high-barrier-transparency junctions. Manyconductance peaks are expected to exist in the tunnelingconductance spectrum of MgB2/I/MgB2 junctions. The casefor MgB2/I/N junctions has been calculated by Brinkmanet al [149, 151]. The dependence of the tunneling spectrumon magnetic field is also influenced by the existence of the twogaps in MgB2 [152].

The effect of the two gaps on Cooper pair tunneling canbe seen in the temperature dependence of IcRN, where Ic is thejunction Josephson current and RN the junction normal-stateresistance. Figure 21 shows the two-band model calculationby Brinkman et al for the temperature dependence of IcRN

in MgB2/I/MgB2 Josephson junctions [153]. If there wereonly σ -to-σ tunneling, IcRN could reach about 10 mV atlow temperature. However, the π state is always presentfor tunneling due to its three-dimensional nature. Therefore,while the c-to-c junctions involve only tunneling between theπ Cooper pairs, the largest IcRN of 5.9 mV was predictedin ab-to-ab junctions for σ -to-π tunneling [153], unless theπ -gap contribution could be suppressed. Experimentally, thetemperature dependence of IcRN in the recent results by Keet al in MgB2/MgO/MgB2 tunnel junctions [65] resemblemore closely that of SINIS junctions calculated by Brinkmanet al [151].

10


Figure 21. Two-band model calculation for the temperaturedependence of IcRN in MgB2–I–MgB2 Josephson junctions.(Reprinted from Brinkman et al [56]. Copyright 2007, withpermission from Elsevier.)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

S 1210 S 1211 Nb simulation

σ 1/ σ n

t=T/Tc

Figure 22. Temperature dependence of the real part of theconductivity σ1 at 17.9 GHz for two MgB2 samples (squares andcircles) and a Nb thin film (triangles). The lines show the single-gapBCS calculation for Nb and two-gap calculation for MgB2. The datafor two MgB2 samples are shifted by 1 and 0.5 units for clarity.(Reprinted with permission from Jin et al [154]. Copyright 2003 bythe American Physical Society.)

3.1.5. Electrodynamics. The BCS theory predicts that thereal part of the complex conductivity σ1(ω) − iσ2(ω) of asuperconductor rises just below Tc and then falls exponentiallyto zero at low temperatures, displaying a ‘coherence peak’ [14].Such a coherence peak is illustrated in figure 22 for Nb,obtained from a microwave (f = 17.9 GHz) conductivitymeasurement by Jin et al [154]. The behavior of MgB2 is verydifferent. As shown in figure 22 for two MgB2 samples, thecoherence peak occurs around 0.6Tc, far below the temperaturefor the BCS coherence peak for single-band superconductors.Jin et al showed that this is due to the two band effect, inparticular, the existence of the smaller π gap, which shiftsthe coherence peak downward [154]. The simulation resultusing a two-band BCS formulism is shown in the figure bysolid lines.

Under alternating electromagnetic fields, the motion ofquasiparticles in a superconductor lags behind that of Cooperpairs, leading to a quasiparticle backflow in the total current

and nonlinear microwave response. In MgB2, the quasiparticlebackflow consists of contributions from both the σ and π bands[155]. In the clean limit, the smaller π gap causes alow temperature bump in the nonlinear microwave response.However, this bump can be suppressed when the scatteringrates in the σ and π bands are modified [155, 156].

Far-infrared spectroscopy is a powerful technique tomeasure the superconducting gap [14]. However, for a longtime a clear evidence of the two-gap structure was missingin most far-infrared measurements: only the π gap wasobserved [157]. Recently, Ortolani et al reported the two-gap feature in ultraclean MgB2 films deposited by HPCVD[158]. When the films were irradiated by neutron, the far-infrared spectra became dominated by the π gap as in mostprevious reports. They attributed the puzzling absence ofthe two-gap structure in most far-infrared measurements tothe impurity scattering in the MgB2 samples used in thoseexperiments [158].

3.2. Normal-state properties

3.2.1. Resistivity. It is widely observed that different MgB2

samples can have vastly different normal-state resistivitiesbut similar Tc. While this is partly due to the differencein connectivity among various samples [92], Mazin et alsuggested that it is also a reflection of the weak interbandimpurity scattering in MgB2 [22]. According to the two-gapsuperconductivity model, impurity interband scattering, eithermagnetic or nonmagnetic, suppresses Tc [159]. In MgB2,the interband impurity scattering rates is very small, muchsmaller than the intraband impurity scattering rates. WhileTc is determined by the maximum matrix element in λ, theconductivity is the sum of all the conducting channels. Eventhough impurities significantly impact the intraband impurityscattering, primarily in the π bands, and thus the residualresistivity, they do not have much effect on the interbandscattering and Tc. The electron–phonon coupling is muchstronger for the σ bands, which affects Tc [22].

3.2.2. de Haas–van Alphen effect. Under a strongmagnetic field B when the electron orbit quantization becomessignificant, the magnetic moment of a sample oscillatesas a function of 1/B with a period proportional to 1/A,where A is an extremal cross-sectional area of the Fermisurface in a plane normal to the magnetic field [160, 161].This effect, the de Haas–van Alphen (dHvA) effect, allowsmeasurements of the extremal Fermi surface areas and thecorresponding quasiparticle effective masses. The existenceof the four bands in MgB2 can be directly probed by the dHvAmeasurement. Figure 23 shows the Fermi surface of MgB2

plotted with the point at the Brillouin zone center and theorbits F1 . . . F6 probed by Carrington and co-workers usingdHvA measurement with the cantilever method [162–164].The experimental results agree well with the band-structurecalculations [165]. From these measurements, the band massof the quasiparticle mB, the quasiparticle effective mass m∗

and the electron–phonon coupling constant λ = m∗/mB − 1can be obtained. For example, for the σ -light-hole band

11


Figure 23. Fermi surface of MgB2 from band structurecalculation plotted with the point at the Brillouin zone center.Possible dHvA extremal orbits are indicated. (a) σ -light-hole band,(b) σ -heavy-hole band, (c) π -hole band and (d) π -electron band.(Reprinted from Carrington et al [162]. Copyright 2007, withpermission from Elsevier.)

m∗ = 0.55me for F1 and 0.61me for F2; for the σ -heavy-hole band m∗ = 1.18me for F5 and 1.2me for F6; for theπ -electron band m∗ = 0.44me for F3 and for the π -hole bandm∗ = 0.31me for F4 [164].

3.2.3. Hall effect and magnetoresistance. When themagnetic field is not so strong as to make the electron orbitquantization significant, the Hall effect and magnetoresistancein MgB2 also demonstrate unique properties due to theexistence of the multiple bands. Figure 24 shows the Hallcoefficient RH measured with the magnetic field aligned eitherparallel to the ab plane or parallel to the c axis of MgB2

single crystals by Eltsev et al [166]. They found that thein-plane RH (H‖c) is positive whereas the out-of-plane RH

(H‖ab) is negative. In materials with multiple bands, thesign of RH is determined by a sum of niµ

2i /qi over all the

bands, where n is the carrier density, µ is the mobility, q

is the charge of the carrier, which is positive for holes andnegative for electrons, and i represents different bands [161].When H‖c, all the four bands shown in figure 23 contributeto the Hall field and the hole-like carriers dominate, thusRH > 0. However, when H‖ab, the contributions of thetwo two-dimensional σ bands are substantially smaller dueto their low mobilities, and the π -electron band dominates,thus RH < 0 [166]. The experiment is in agreement with bandstructure calculations [167].

The existence of multiple bands in MgB2, in particularwhen one π band is electron-like and the other three bands

Figure 24. The in-plane (H‖c) and out-of-plane (H‖ab) Hallcoefficients as a function of temperature in the normal state ofMgB2 measured on several single crystals. (Reprinted from Masuiand co-workers [166, 168]. Copyright 2003, with permission fromElsevier.)

are hole-like, results in large magnetoresistance in cleanMgB2 samples [7, 169–171]. At 18 T and 45 K, ρ/ρ0 =130% has been observed in an HPCVD film with ρ0 =0.34 µ cm [171]. The magnetoresistance depends on a sumof σiσj (ωciτi − ωcj τj )

2 over all the bands, where σ is theconductivity, ωc is the cyclotron frequency ωc ≡ qB/m∗,which is positive for holes and negative for electrons, τ isthe relaxation time and i and j represent different bands [161].The ωciτi and ωcj τj terms add up when the i and j bands havedifferent carriers and reduce each other when the carriers areof the same type.

Li et al have shown that the magnetoresistance of MgB2

has an angular dependence which changes dramatically withtemperature [171]. In figure 25, the magnetic field dependence(left panels) and angular dependence (right panels) of themagnetoresistance for a MgB2 film are shown for threetemperatures. For H ⊥ ab, the magnetoresistance changesfrom a minimum at low temperature to a maximum at hightemperature. The temperature-dependent anisotropy can beexplained by the highly anisotropic Fermi surface of MgB2 anddifferent electron–phonon coupling from the different bands.When the applied field direction changes, the orbiting planeof the electron intercepts the Fermi surface at different angles.As a result, σi , ωci and τi of the four bands change, resultingin the angular dependence of the magnetoresistance. As thetemperature increases, σi is reduced more rapidly for theσ bands than for the π bands due to the stronger electron–phonon interaction, and the contributions of the π bandsbecome more important, leading to the observed changed inthe angular dependence [171].

In combination with band structure calculations, effortshave been made to extract the scattering information for thedifferent bands from the magnetotransport data [170, 172–175]. Figure 26 shows the temperature dependence ofelectronic lifetimes for each of the four bands in two MgB2

films with different RRR values, (a) 33.3 and (b) 20.9, obtained

12

Rep. Prog. Phys. 71 (2008) 116501 X X Xi∆ρ

/ρ0

(%)

∆ρ/ρ

0(%

)∆ρ

/ρ0

(%)

µ0H ( T) Angle (º)

(a) (b)

(c) (d)

(e)

(f)

T = 60 K

T = 100 K

T = 120 K

0

3

6

9

12

0 2 4 6 80

2

4

6

8

-180 -90 0 90 180

0

10

20

30

40

H⊥ab H//ab

3T

3T

3T

5T

5T

5T

7T

7T

7T

9T

9T

9T

H//ab H⊥ab

Figure 25. Magnetic field dependence ((a), (c) and (e)) and angulardependence ((b), (d) and ( f )) of magnetoresistance of a MgB2 film.The results are for T = 60 K ((a) and (b)), 100 K ((c) and (d)) and120 K ((e) and ( f )). A change in angular dependence is seen as thetemperature is increased. (From [171].)

by Yang et al using both the magnetoresistance and themagnetic field dependence of the Hall coefficient [175]. Inboth films, the π1 band is much dirtier than other bands at lowtemperatures, and the scattering rates become similar at hightemperatures due to the electron–phonon scattering. Disorder,which reduces RRR, mainly enhances scattering in the cleanerbands. From the significant difference in the scattering ratesin π1 and π2 bands, it is clear that the fully band-resolvedinformation on the intraband scattering is important for acorrect understanding of MgB2.

3.3. New effects due to two superconducting orderparameters

MgB2 has two superconducting order parameters, 1 = 1eiθ1

and 2 = 2eiθ2 , with an intrinsic Josephson coupling and aphase difference θ = θ1 − θ2 [176]. This feature leads to neweffects that do not exist in single-gap superconductors. In thefollowing discussion, the subscript ‘1’ stands for the π bandand ‘2’ stands for the σ band.

3.3.1. The ‘Leggett mode’. In 1966, Leggett pointed out thatin a clean two-band superconductor, there can be a collectiveexcitation corresponding to small fluctuations of the phase

Figure 26. Temperature dependence of transport scattering timesfor the four bands for two MgB2 films with RRR values of (a) 33.3and (b) 20.9. (From [175].)

difference θ between the order parameters in the two bands,the so-called Leggett mode [177]. MgB2 offers an opportunityto observe the Leggett mode experimentally [178]. Using aneffective ‘phase action’ formalism Sharapov et al derived thedispersion law for the Leggett mode [179]:

2 = ω20 + v2K2, (1)

where the Leggett mode frequency

ω20 = λ12 + λ21

λ11λ22 − λ12λ21412. (2)

The Leggett mode exists only when λ11λ22−λ12λ21 > 0, whichis satisfied in MgB2. Using the first-principles calculationresults of 1 = 1.8 meV and 2 = 6.8 meV by Choi et al [19],Sharapov et al found that ω0 = 8.9 meV using the λij valuesfrom Liu et al [102] and ω0 = 6.5 meV using the λij valuesfrom Golubov et al [99].

Leggett suggested that the Leggett mode can be detectedwhen ω0 1 [177], which is not satisfied in MgB2 [179].Ponomarev et al suggested that the Leggett mode can beobserved by its resonance coupling with the Josephson currentin a Josephson junction or by its interaction with the multipleAndreev reflection in an Andreev contact [180]. Agterberget al pointed out that in a Josephson junction between MgB2

and a single-band superconductor, there is an additionalpossibility of the resonance coupling [181]. Anishchankaet al further suggested that the application of a weak magneticfield allows one to determine the spatial dispersion of theLeggett mode using such junctions [182]. Although there aresuggestions of observations of the Leggett mode in tunnelingexperiments [149, 180], the results could be caused by artifacts.

Using polarized Raman scattering measurements, Blum-berg et al reported the observation of the Leggett mode inMgB2 single crystal [183, 184]. Figure 27 shows the Ramanspectra of an MgB2 crystal in the normal (red) and supercon-ducting (blue) states for different polarization scattering ge-ometries probing different symmetries: the top panel is for

13


0

10

20

30V.

VI.

E2g

phonon I.

2∆l

2∆0

0

2

4

10 100 1000

ω L

II.III.

IV. V.

NSC

E 2g

A 1g

χ'' (

rel.

u.)

752 nm

(a)

482 nm

(b)

Raman shift (cm-1)

χ'' (

rel.

u.)

Figure 27. The Raman response spectra of an MgB2 crystal in thenormal (red) and superconducting (blue) states for the E2g (top) andA1g (bottom) scattering channels. Decomposition intosuperconducting coherence peaks, E2g phonon and the fits are shownby solid lines. (Reprinted from Blumberg et al [183]. Copyright2007, with permission from Elsevier.)

E2g and the bottom channel for A1g symmetries [183]. InMgB2, the Leggett mode contributes only to the A1g Ramanresponse. When the sample becomes superconducting, severalnew features appear in the Raman spectra. In the E2g channel,two peaks corresponding to the pair breaking of σ (2l) andπ (20) band Cooper pairs are observed. In the A1g chan-nel, two peaks at 76 cm−1 (9.4 meV) and 106 cm−1 (13.2 meV)are seen. Blumberg et al assigned the peak at 9.4 meV tothe Leggett mode. Extending the theory on Raman scatteringin superconductors by Klein and Dierker [185] to MgB2, theauthors found good agreement between the experiment andtheory. Because the Leggett mode frequency ω0 is larger than2π , the lifetime of the mode is short and the Raman peak isbroad. If the interband electron–phonon coupling λ12 and λ21

can be reduced, ω0 could decrease below 2π and a sharperRaman peak due to the Leggett mode may be observed.

3.3.2. Interband phase textures. When the fluctuation inθ = θ1 − θ2 is not small, it cannot be treated as a harmonicoscillator and becomes a soliton, and θ between the twosuperconducting condensates slips by 2π at the soliton [186].Gurevich and Vinokur have calculated θ(x) for bridges of two-gap superconductor of several geometries and proposed thatvarious phase textures will be induced by the current flowingthrough the bridges [176, 187].

When a current equal or larger than a critical currentdensity Jt is instantaneously turned on through the bridgesshown in figures 28(a)–(c), the phase between the twosuperconducting condensates becomes unlocked (θ = 0, π )and the bridges enter into various dynamic soliton states. The

(d)

Figure 28. Geometries in which phase textures could occur:(a) a microbridge, (b) point contacts, (c) a 4-terminal geometryand (d) a bilayer model for a two-band superconductor. (Reprintedwith permission from Gurevich and Vinokur [176, 187]. Copyright2003 and 2006 by the American Physical Society.)

θ solitons appear near the normal leads and propagate into thebulk. For figures 28(a) and (b) a static phase texture eventuallyforms as shown in the figure. For figure 28(c), the solitons andantisolitons generated at the two ends annihilate in the centerand a continuous soliton motion occurs. This soliton shuttlegives rise to voltage oscillations on the bridge, which can bedetected experimentally [176].

The phase texture can also be induced by a strong enoughdc current, which breaks Cooper pairs in the π band withsmaller first, causing a redistribution of the supercurrentbetween the two bands [187]. This can be detected usingthe geometry shown in figure 28(d), where a model oftwo superconducting layers with weak interlayer Josephsoncoupling is used. The phase texture leads to a resistance acrossthe bilayer; thus the structure in figure 28(d) is in fact a current-operated switch.

3.3.3. Vortices with fractional flux quantum. Babaev hasproposed that in superconductors with two order parameters,vortices with arbitrary fractional flux quantum (analog totwo-Higgs doublet model in particle physics) can exist[188]. Vortices with fractional flux quantum in two-gapsuperconductors have been confirmed by other theoreticalapproaches [189, 190]. Such two-gap superconductors showvery unconventional magnetic properties in a so-called ‘semi-Meissner’ state and cannot be categorized into type-I or type-IIsuperconductors [191]. Such an analysis has been extendedto liquid metallic hydrogen in which two order parameters,one protonic and the other electronic, exist [192]. Babaevet al found that it cannot be categorized exclusively as asuperconductor or superfluid but a new kind of quantum fluid[192, 193]. There has been no experiment in MgB2 to directtesting these theoretical predictions, but experiment usingrings of Josephson-coupled Al layers to simulate the two-gapsuperconductor has shown states analogous to that of fractionalvortices [194].

4. Modification of two bands and two gaps

As the existence of the two bands and two gaps occupies acentral place in the physics of MgB2, one can quite naturallyexpect that their modification will have a direct impact on

14


Figure 29. Tc as a function of Al concentration in Mg1−xAlxB2

(solid symbols) and C concentration in Mg(B1−yCy)2 (opensymbols). The dotted line is the theoretical result for rigid bandfilling only, the dashed line includes the change of bands, and thesolid line further includes the E2g phonon hardening. The horizontaldotted line is for interband scattering only. (Reprinted withpermission from Kortus et al [199]. Copyright 2005 by theAmerican Physical Society.)

all aspects of MgB2 properties, lead to improvements in theproperties required for various applications and help betterunderstand the new physics that does not exist in single-gapsuperconductors.

4.1. Chemical doping

Atomic substitution shifts the Fermi level and enhancesimpurity scattering, therefore changes the properties of thetwo bands and two gaps in MgB2. Chemical doping effectshave been investigated immediately after the discovery ofsuperconductivity in MgB2 [12, 195, 196], and it was foundthat only a limited number of elements can be substituted intoMgB2 [197]. The most widely studied substitutions in MgB2

are Mg by Al and B by C [76, 77, 96, 197], and other dopantsinclude Li and Mn [76, 77, 198].

4.1.1. Al doping. Substitution of Mg by Al dopes electrons,thus reducing the number of holes in MgB2. Since the Tc

of MgB2 is determined mainly by the coupling between theσ bands and the E2g phonon, reduction in the holes in theσ bands suppresses Tc [199]. In figure 29, experimental resultsof Tc versus Al concentration in Mg1−xAlxB2 from variousgroups (solid symbols) are shown, together with the results oftwo-band Eliashberg theory taking into account various effects.The dotted line considers only band filling in rigid bands,which works well for small doping concentration. For higherdoping, additional effects—the change in the bands [200] andthe hardening of the E2g phonon [201]—need to be considered.The dashed line includes the effect of band modification, andthe solid line further includes the E2g phonon hardening. Thetheory taking into account all the factors is in good agreementwith the experimental results [96, 199].

10 15 20 25 30 35 400

2

4

6

8(a)

(b)

0.0 0.1 0.2 0.30

2

4

6

8

Gonnelli et al. Klein et al. Szabo et al.

BCS

Ene

rgy

gaps

(m

eV)

TcA (K)

Gonnelli et al.

Klein et al. Szabo et al.

Ene

rgy

gaps

(m

eV)

Al content (x)

Figure 30. (a) Andreev critical temperature and (b) Al-contentdependence of the energy gaps in Mg1−xAlxB2. The results in singlecrystals (open and solid circles [206] and gray triangles [207]) arecompared with the result in polycrystyals (gray squares [203].Dashed and solid lines were obtained by solving the two-bandEliashberg equations. (Reprinted from Gonnelli et al [198].Copyright 2007, with permission from Elsevier.)

Doping in general increases intraband and interbandscattering. While intraband scattering from nonmagneticimpurities does not affect Tc, interband scattering suppressesTc in MgB2 to about 25 K [128, 202] as shown by the horizontaldotted line in the figure. Szabo showed that Al dopingdoes not change the scattering significantly [203]. Since theσ bands are mostly confined to the B plane, the effect ofAl substitution of Mg is mainly on the intraband scatteing in thethree-dimensional π bands [96, 204]. This is reflected in thedecrease in the π -band coherence length ξπ [146]. However,the effect is weak, and as a result, Hc2 is hardly enhanced byAl substitution [205].

The effect of Al doping on the σ and π gaps is shownin figure 30 [198]. The experimental data [203, 206, 207] arecompared with the theoretical results. The dotted lines arefrom the two-band Eliashberg theory described above withoutthe interband scattering [199]. The solid lines include anadditional parameter interband scattering σπ , which dependson the Al concentration and may arise from the lattice distortiondue to Al doping [208]. The figure shows that electron dopingdue to Al substitution causes both the σ and π gaps to decrease,although there is some interband scattering that explains theinitial slight increase in the π gap. The two gaps do notmerge even when Tc is substantially reduced, indicating thatthe interband scattering is weak in Mg1−xAlxB2 [96, 198, 209].

15


Figure 31. (a) Critical temperature and (b) C-content dependence ofthe energy gaps in Mg(B1−xCx)2. The results are from [210] (solidcircles), [203] (open squares) and [211] (gray triangles). Dashedand solid lines were from the two-band Eliashberg equations.(Reprinted from Gonnelli et al [198]. Copyright 2007, withpermission from Elsevier.)

4.1.2. C doping. Substitution of B by C also dopes electronsand reduces the number of holes in MgB2. As shown infigure 29, C doping (open symbols) suppresses Tc in a similarmanner as Al doping when the doping concentration is scaledby a factor of 2 [199]. However, since the σ bands are locatedin the B plane and the π bands are also centered in the B plane,C doping has a much more significant impact on the interbandand intraband scattering. This is most clearly illustrated bythe dependence of the σ and π gaps on C doping shown infigure 31 [198], in which experimental data [203, 210, 211]are compared with the theoretical results. The decrease in theσ gap with C doping is similar to the behavior of Al doping,but for the π band, the result of Gonnelli et al in C-dopedsingle crystals shows that it is flat and eventually merges withthe σ gap at Tc = 19 K [210]. A similar result has beenobserved by Tsuda et al in C-doped single crystals [212].The merging of the σ and π bands is clear evidence for theinterband scattering, which causes the σ gap to decrease andthe π gaps to increase. The band filling effect and the interbandscattering balance each other and result in a flat dependenceof the π gap on C doping [199, 202]. On the other hand,the results of Szabo et al [203] and Tsuda et al [211] onpolycrystalline samples show no merging of the two gaps,consistent with weak interband scattering. According to bandcalculation, C doping is not expected to increase interbandscattering [208]. The discrepancy in the π gap results between

single crystal and polycrystalline samples and the mechanismof the enhanced interband scattering by C doping are not fullyunderstood [96, 198].

Unlike Al doping, C doping is very effective in enhancingHc2 [45, 205, 213] as a result of the modification of interbandand intraband scattering. In particular, this is attributed to thedirtier π bands in Mg(B1−xCx)2 samples [128, 205]. It shouldbe noted that the values of Hc2 in C-alloyed HPCVD MgB2

films (over 60 T [42, 45]) are much higher than those foundin the bulk C-doped MgB2 samples [205, 213]. Structuralanalysis shows that C-alloyed HPCVD films are full of defectsand composed of Mg(B1−xCx)2 grains and C-rich amorphousphases at the grain boundaries [91, 214]. Unlike carbon-dopedsingle crystals, where the lattice constant a decreases butc remains almost constant [215, 216], both the c and a axesin the carbon-alloyed HPCVD films expand with increasingcarbon content [91]. The origin for the much higher Hc2 inthin films is being actively investigated.

Carbon doping also enhances the irreversibility field Hirr

[217–219], indicative of stronger flux pinning. While a highHc2 can lead to a high Hirr , Hirr is also strongly influenced bystructural defects such as grain boundaries, the main source offlux pinning in MgB2 [44]. A particularly successful techniquefor achieving high Hc2, Hirr and Jc(H) values in bulk samplesis doping with SiC nanoparticles [46]. In nano-SiC doping,the C substitution is responsible for high Hc2 and defects,small grain size and nanoinclusions are responsible for theimprovement in Jc(H) [220].

4.1.3. Mn doping. Doping of MgB2 by Mn substitutesMg [76, 77]. It reduces Tc much more rapidly than Al orC doping, suppressingTc to zero at about 2% substitution [221].Rogacki et al found that Mn ions are in the divalent low-spin state; thus the substitution of Mg by Mn is isovalent(no charge doping) and the suppression of Tc is mainly bymagnetic pair breaking [221]. Both the σ and π gaps decreasewith Mn doping, and there is no merging of the σ andπ gaps, indicating that the magnetic interband scattering isnegligible [222].

4.1.4. Li doping. Replacing Mg with Li should dope holesinto MgB2 and thus increases Tc. Instead, a slight decreasein Tc has been observed experimentally [77]. In order todistinguish the roles of charge doping and lattice distortions inchemically substituted MgB2, Erwin and Mazin suggested co-doping with Na and Al in the Mg sites because it is isoelectronicand lattice distortions will be the only effect [208]. Li, havingthe same valence as Na, was chosen by Monni et al forsuch a co-doping experiment because of its more favorablepackaging effect than Na [223]. By comparing the resultsof Mg1−x(AlLi)xB2 and Mg1−xAlxB2, they showed that thesuperconductivity in Mg1−x(AlLi)xB2 is mainly affected bythe Al substitution [223]. By first-principles calculations,Bernardini and Massidda pointed out that the effect of co-doping is the sum of separate Al and Li doping. While Aldoping fills both bands, holes from Li go almost entirely tothe π bands [224]. Since Tc is determined by the σ band, itdepends only on the Al concentration. Similarly, co-doping of

16


Figure 32. (a) Corrected resistivity versus temperature of neutron irradiated thin films. (b) Tc as a function of the corrected residualresistivity of a series of MgB2 films [225, 228] and polycrystals [71, 231] irradiated with α particles and neutrons. (From Putti et al [234].)

Li with C can also be explained as the sum of separate Li andC doping [77].

4.2. Irradiation

Whereas atomic substitution dopes charge carriers andintroduces disorder at the same time, irradiation introducesdisorder only. Irradiation experiment is therefore an excellentapproach to study the effect of scattering on the propertiesof MgB2. Depending on the type, energy and fluence of theparticles, the nature of the disorder introduced is different.For example, fast neutrons induce point defects by directcollisions with nuclei, and thermal neutrons are captured bythe 10B nuclei, which decay into 7Li and energetic α particles,causing defects and displacements in a similar way as by α

particle irradiation [225]. Defect clusters with a diameter ofseveral nanometres have been observed in neutron irradiatedMgB2 samples [226]. Heavy ions such as 200 MeV Ag ions,on the other hand, generate defect clusters at high fluences[227]. The particles that have been used in the study of MgB2

include proton [11], heavy ion [227], α particle [228, 229]and neutron [225, 226, 230–233]. The effect of irradiation-induced defects on flux pinning is the focus of some of thesestudies [11, 226, 227].

Irradiation in MgB2 samples increases resistivity andsuppresses Tc, as shown in figure 32(a) for neutron irradiationof a MgB2 thin film [234]. Unlike C-doped films in whichresistivity increases dramatically due to reduction in theintergrain connectivity, neutron irradiation does not changethe connectivity significantly [225, 234]. In figure 32(b),Tc is plotted as a function of residual resistivity based onseveral reports of α particle and neutron irradiation of differentMgB2 samples [71, 225, 228, 231]. The measured resistivityvalues are corrected based on Rowell’s model on intergrainconnectivity [92] so that the corrected values reflect theintragrain resistivity. Data from the different experimentsoverlap with each other in the figure, showing a lineardependence of Tc on resistivity reaching Tc = 0 at about80 µ cm. It has been noted that this Tc − ρ0 behavior isvery similar to that in the A15 superconductors [234]. Themechanism for disorder-induced Tc suppression in MgB2 has

not been developed. Monni et al have shown that the chargescattering is significantly increased by irradiation induceddisorder [174]. In the unirradiated film, the σ band is muchcleaner than the π band. After an irradiation of 4.1×1017 cm−2

neutrons, the lifetimes of both bands are reduced but thereduction is much more dramatic for the σ -band, making itnearly as dirty as the π band [174].

The σ and π gaps in neutron irradiated MgB2 sampleswere studied by Putti et al using specific heat measurement[232]. Figure 33 shows the reduced electronic specific heatcsc/γ T for MgB2 samples without neutron irradiation (P0)and with increasing fluences of neutron irradiation (P4–P6).The transition remains sharp even for a heavily irradiatedsample, indicating a very uniform defect distribution. Whilethe result for sample P0 can only be explained by contributionsfrom both the σ and π bands, the temperature dependence ofsample P6 is characteristic of a single-gap superconductor. Theσ and π gaps versus Tc for neutron irradiated MgB2 samples,obtained from the specific heat [232] and point-contact Adreev-reflection measurement [235], are shown in figure 34 [225]. Aninitial increase in the π gap and an eventual merging of the σ

and π gaps are observed as the disorder reduces Tc. The resultsshow that neutron irradiation introduces interband scattering,changing it to a single-gap superconductor when its Tc is below∼20 K [232].

5. Broader impact of MgB2 on superconductivity

5.1. Multi-band superconductivity

Multi-band superconductivity has received unprecedentedattention because of MgB2. Since the two-band natureof superconductivity in MgB2 was well established, manyold and new superconductors are shown to be multi-bandsuperconductors. Examples include NbSe2 [236], the ternary-iron silicide Lu2Fe3Si5 [237], MgCNi3 [238], the heavyFermion PrOs4Sb12 [239, 240] and URu2Si2 [241] and thenewly-discovered iron pnictide superconductors [242].

2H -NbSe2 is a quasi-two-dimensional superconductorwith a Tc of 7.2 K and a charge density wave system below35 K. Its Fermi surface is composed of five sheets: nearly

17


0.0

0.5

1.0

1.5

2.0 P0Two-Gap

0.0

0.5

1.0

1.5

2.0P4Single-GapTwo-Gap

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

c sc(t

)/ T

c sc(t

)/ T

t

P5Single-GapTwo-Gap

0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

t

P6Single-Gap

c sc(t

)/ T

c sc(t

)/ T

Figure 33. Electronic specific heat as a function of t = T/Tc for MgB2 samples without neutron irradiation (P0) and with increasingfluences of neutron irradiation (P4–P6). The best fit curves for single-gap and two-gap models are shown. (Reprinted with permission fromPutti et al [232]. Copyright 2006 by the American Physical Society.)

Putti et al.Wang et al.

Dagheroet al.

Figure 34. σ (0) (empty symbols), π(0) (filled symbols) and(0) (half-filled symbols) as a function of Tc from [231, 232, 235].Lines are guides to the eye. (From Ferrando et al [225].)

two-dimensional hexagonal sheets derived from the Nb 4dbands, two around the point and two around the K point,and a small pancake-like sheet around the point derivedfrom the Se 4p band [236, 243]. It has long been knownthat there is a broad distribution of gaps from 0.6 to 1.4 meVin NbSe2 [244], and recent scanning tunneling spectroscopymeasurement showed features in dI/dV at 0.75 and 1.2 mV[245]. These are now believed to be the result of multi-band superconductivity analogous to MgB2 [236, 245]. Manyproperties such as the field dependence of the vortex coresize [246] and thermal conductivity [247] are consistent withmulti-band superconductivity in NbSe2.

Another example is the ternary-iron silicide R2Fe3Si5with R = Lu, Y, Sc, Tm or Er [237], an interesting systemfor the study of interaction between superconductivity andmagnetism. Its Fermi surface has two hole-like bands andone electron-like band, all disconnected from each other [237].Figure 35 shows the temperature dependence of the normalized

2.5

2.0

1.5

1.0

0.5

0.0

C e/

nT

1.00.50.0T/Tc

Lu2Fe3Si52∆1/kBTc = 4.42∆2/kBTc = 1.1

1 : 2 = 47 : 53

BCS

two-gap

-6

-4

-2

0ln

(Ce/

nTc)

151050Tc /T

~exp(-0.29Tc/T)

Figure 35. Temperature dependence of the normalized electronicspecific heat for Lu2Fe3Si5. Fitting curves using the single-bandBCS and two-gap models are also shown. (Reprinted withpermission from Nakajima et al [237]. Copyright 2008 by theAmerican Physical Society.)

electronic specific heat for Lu2Fe3Si5. It is very similar to thatshown in figure 9 for MgB2 with a shoulder at low temperatureindicative of a second, smaller gap. A fitting to the two-gap model reveals a large gap at 1.2 meV and a small gapat 0.3 meV [237].

Recently, Kamihara et al reported a layered iron-based superconductor F-doped LaOFeAs with Tc = 26 K[248]. Quickly afterwards, higher Tc was reported in relatedcompounds reaching 55 K in Sm[O1−xFx]FeAs [249]. Thesuperconductivity occurs in the FeAs layer and very close tothe magnetic ordering [250], suggesting an unconventionalpairing mechanism. There are five disconnected pieces ofthe Fermi surface—two electron cylinders around the zoneedge M–A line, two hole cylinders around the zone center

18


line and a three-dimensional hole pocket around the Z point[251], setting the stage for two-band superconductivity. Ina measurement of Hc2 in LaFeAsO0.89F0.11, Hunte et alobserved a significant upward curvature as the temperature wasdecreased, similar to that in MgB2 shown in figure 13, whichwas suggested as an evidence of two-band superconductivityin the Fe-based superconductors [242]. The Hc2(0) valuein LaFeAsO0.89F0.11 is exceptionally high, possibly above100 T, which exceeds the paramagnetic limit [242]. Thisis another example of extraordinarily high Hc2 in two-bandsuperconductors.

5.2. Search for higher Tc superconductors

Attempts to increase Tc in MgB2 have been mostlyunsuccessful. As described above, chemical substitution andirradiation have always suppressed Tc. Subjecting MgB2

to pressure also causes Tc to decrease [13]. The onlyenhancement of Tc was reported in HPCVD MgB2 films,where the epitaxial compressive strain causes the softeningof the E2g phonon and an increase in the electron–phononcoupling [90]. Theoretical calculations show that both electronand hole doping decreases electron–phonon coupling [252] andnot much increase in Tc can be expected by further increasingthe electron–phonon coupling strength [253, 254].

New superconductors with higher Tc and similar structuresto MgB2 have been proposed theoretically. Rosner et alsuggested that the Li deficient LixBC can have Tc more thantwice as high as in MgB2 due to 80% stronger electron–phononcoupling between the σ bands and the B–C bond stretchingmodes [255]. The experiment to synthesize the material hasnot been successful [254]. Another new structure predicted toshow high Tc is the metal-sandwich lithium monoboride LiBphase [256]. It is marginally stable under ambient conditionsbut stable under pressure. Liu and Mazin showed [257] thatit combines the strong electron–phonon coupling between theB σ bands and the B bond-stretching modes as in MgB2 withthe electron–phonon coupling between the free-electron-likeinterlayer states and the soft intercalant modes as in CaC6,a graphite intercalation compound with a Tc of 11.5 K [258].However, because of the absence of a π band, which wouldstrengthen the electron–phonon coupling between the σ bandand the E2g mode through screening, the coupling in LiB isactually lower. Restoring the π band by doping will lead tohigher Tc than in MgB2 [257]. Recently, Zhang et al proposeda structure with alternating MgB2 and graphene layers, whicheffectively hole-dopes MgB2 without chemical substitution[259]. The higher density of states and the in-plane latticeexpansion due to the hole doping in such a structure may leadto a higher Tc.

MgB2 offers important new clues for the search for newphonon-mediated superconductors with higher Tc, even the‘designer’ room temperature superconductors, which havebeen articulated by Pickett [254, 260]. In MgB2, strongelectron–phonon coupling only occurs for 3% of the phononmodes—in two of the nine phonon branches with wavevectorQ < 2kF. These 3% of the phonon modes have a very highmode-dependent electron–phonon coupling constant λQ ∼

20–25, leading to the high Tc in MgB2 without causing latticeinstability, and λQ is two orders of magnitude smaller forthe other modes. A further increase in λQ by 15–20% willresult in lattice instability. Pickett suggested that if one canincrease the number of phonon modes to have large λQ withouta further increase in λQ, the total electron–phonon couplingλ, which is a sum over all Q, will be further increasedwithout threatening the lattice stability. Proposing a MgB2-like material with a stiff lattice (high phonon frequency),several cylindrical Fermi surfaces and more phonon modesto strongly couple with, Pickett showed that Tc of 400–500 K is possible [254, 260]. The unexpected high Tc in thephonon-mediated superconductor MgB2 has certainly inspirednew avenues for theoretically designing room temperaturesuperconductors.

6. Concluding remarks

MgB2 is an excellent gift from nature to condensed matterphysics. Most superconductors have one order parameter:one superfluid density, one wave function and one energygap. Although the idea of two order parameters in asuperconductor was considered theoretically soon after theBCS theory [30], it took more than 40 years until thediscovery of superconductivity in MgB2 to have a clear-cuttwo-gap superconductor to study the phenomenon. The basicissues concerning the superconductivity in MgB2, such as theBCS pairing mechanism and the existence of the two gaps,have been worked out both theoretically and experimentallyrelatively quickly (as compared with the high temperaturesuperconductivity in the cuprates, whose mechanism is stillthe topic of active research 20 years after its discovery).However, there are still important questions to be answered.One significant example is the vortex dynamics in two-bandsuperconductivity. With two coherence lengths and two setsof supercurrent around the vortex core, as well as the differentresponses of the σ and π superfluids to the magnetic field, theexisting concepts of flux pinning, vortex fluctuations and phasediagram that apply to single-gap superconductors need to bemodified.

The research on new effects in MgB2 that do not exist insingle-gap superconductors is just starting, and the landscapeof superconductivity with two order parameters is yet to befully explored. While the polarized Raman measurement ofBlumberg et al is a significant advancement in the observationof the Leggett mode, confirmation by other techniques isimportant. The theoretically predicted effects such as theinterband phase textures and vortices with fractional fluxquantum have not been demonstrated. New effects due to twogaps may be discovered by further theoretical investigations.

Just as heterostructures of single-gap superconductorswith other functional materials lead to interesting physics andnovel devices, the existence of two gaps provides additionalingredients to the problem. One such example is the interactionbetween the two-gap superconductivity and magnetism. Insuperconductor (S)–ferromagnet (F) heterostructures, we nowhave two superconducting order parameters to interact withthe magnetic order parameter. The effects such as induced

19


superconductivity in the F layer [261, 262], the nonmonotonicdependence of Tc on the thickness of the F layer [263, 264] andthe π junctions in the S/F/S structures [265, 266] will have anew dimension when the superconductor is the two-gap MgB2.

A major challenge facing MgB2 researchers is the controlof the interband and intraband scattering. From the resultspresented in this paper, it is amply clear that the scatteringrates in the σ and π bands and between them are the mostcrucial parameters affecting the properties of MgB2. Althoughdefects, chemical doping and irradiation all affect the scatteringproperties, there is no effective way to modify the scatteringin each band and between the two bands at will and bydesign. For example, a record high upper critical field Hc2

has been achieved in carbon-alloyed thin films through thechange in intraband and interband scattering [42, 45], but theprocess is not yet completely under control to yield specificHc2(T ) behavior. Besides defects and dopant, other factorssuch as microstructure [214] and strain [267] may also affectscattering.

The ability to completely control intraband and interbandscattering rates will lead to significant advancement in theareas of basic science and practical applications concerningMgB2. For example, the frequency of the Leggett modedepends on the interband electron–phonon coupling, whichcould be shifted when the scattering parameters was modified,allowing a complete investigation of its behaviors. On theapplication side, the control over scattering could produceMgB2 conductors with desirable Hc2 properties or reducemicrowave nonlinearity for superconducting RF devices.

Acknowledgments

This work benefits greatly from the author’s stimulatingdiscussions with, among others, A Gurevich, D C Larbalestier,J M Rowell, Qi Li, M Putti, C Ferdeghini, R Vaglio,M Iavarone, A E Koshelev, U Welp, W E Pickett, I I Mazin,A Liu, A A Golubov, S Massidda, H H Wen, J R Shi,X G Qiu, N Newman, P C Canfield, S L Bud’ko, G Blumberg,R Flukiger, S I Lee, S X Dou and A Brinkman on the physicsof MgB2. The work is partially supported by ONR undergrant number N00014-07-1-0079 and by the ACS PetroleumResearch Fund under grant number PRF #43995-AC10.

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Two-band superconductor magnesium diboride

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