Unconventional phonon-mediated superconductivity in MgB_{2}

We have evaluated the total carrier mass enhancement factor f_{t} for MgB_{2} from two independent experiments (specific heat and upper critical field). These experiments consistently show that f_{t} = 3.1\pm0.1. The unusually large f_{t} is incompatible with the measured reduced gap (2\Delta (0)/k_{B}T_{c} = 4.1) and the total isotope-effect exponent (\alpha = 0.28\pm0.04) within the conventional phonon-mediated model. We propose an unconventional phonon-mediated mechanism, which is able to quantitatively explain the values of T_{c}, f_{t}, \alpha, and the reduced energy gap in a consistent way.

The recent discovery of superconductivity near 40 K in MgB 2 [1] has led to a remarkable excitement in the solid-state physics community. Such a high-T c superconductivity in this simple intermetallic compound immediately raises a question of whether mechanisms other than the conventional electron-phonon interaction are responsible for the superconductivity. A significant boron isotope effect (α B = −d ln T c /d ln M B = 0.26±0.03, where M B is the mass of boron) [2][3][4] and nearly zero magnesium isotope effect (α Mg = 0.02±0.01) [3] suggest that electron-phonon coupling plays an important role in the pairing mechanism. To explain the 40 K superconductivity, an electron-phonon coupling constant of about 1 is needed. First-principle calculations give the coupling constant of 0.7-0.9 (Refs. [5][6][7][8]), which appears to be sufficient to explain the observed high-T c superconductivity. On the other hand, recent specific heat data [9] indicate a very strong electron-phonon coupling with a total mass enhancement factor of about 3.2. This would suggest a coupling constant of 2.2 and 2∆(0)/k B T c ≈ 5 within the conventional phonon-mediated mechanism [10], in contradiction with the bulk-sensitive Raman scattering experiments which show 2∆(0)/k B T c = 4.1 (Ref. [11]).
Here we evaluate the carrier mass enhancement factor f t from the measured upper critical field. The deduced f t is 3.1±0.1, in remarkably good agreement with that deduced from the independent specific heat data [9]. The large mass enhancement factor is not compatible with the measured gap amplitude and the reduced total isotopeeffect exponent within the conventional phonon-mediated model. We thus propose an unconventional phononmediated mechanism where both the long-range Fröhlichtype and short-range electron-lattice interactions are considered and treated distinctively. Within this scenario, we are able to quantitatively explain the T c value, the mass enhancement factor, the total isotope-effect exponent, and the reduced energy gap.
It is clear that the large mass enhancement factor is not compatible with the measured energy gap (2∆(0)/k B T c = 4.1) [11] and the reduced isotope exponent (α = 0.28±0.04) [4] within the conventional phonon-mediated model. In order to resolve the above controversy, one needs to consider a long-range Fröhlich-type electronphonon interaction that results from the Coulomb interactions of electronic charge carriers with the ions of ionic (polar) materials [15]. This interaction is distinct from the short-range electron-lattice interaction (e.g., deformation-potential interaction, Holstein interaction) that has a carrier's energy only depending on the positions of the atoms with which it overlaps. The Fröhlich electron-phonon interaction is particularly strong in an ionic solid or a perovskite oxide where its static dielectric constant ǫ • is always much larger than its high-frequency dielectric constant ǫ ∞ , whereas the short-range electronlattice interaction (including the electron-lattice interaction with acoustic phonons) is present in all the materials [15]. Recent Quantum Monto Carlo simulation [16] has shown that the Fröhlich-type electron-phonon interaction is nonretarded and can always lead to a mass enhancement factor of f p = exp(g 2 ) no matter how weak this interaction is. Here g 2 = A/ω • , A is a constant, and ω • is the characteristic frequency of optical phonons. It is apparent that this mass enhancement factor f p will strongly depend on the isotope mass if g 2 is substantial, in contrast to the conventional retarded electronphonon coupling model where the mass enhancement factor is essentially isotope-mass independent [10]. On the other hand, the short-range component of the electronlattice interaction is retarded and can be modeled within the conventional Eliashberg equations when the coupling constant is less than 1 and the phonon energies are much lower than the bare hopping integral [16,17]. When only a short-range electron-lattice interaction is present and the coupling constant is far larger than 1, the interaction becomes nonretarded and small polarons can be formed [15,17]. In this case, the Migdal adiabatic approximation breaks down [17] and one cannot use the Eliashberg equations to describe superconductivity [15,17].
Since the Fröhlich-type electron-phonon interaction is nonretarded, we can assume that the role of this interaction is simply to enhance the density of states [17] and reduce the direct Coulomb interaction between two carriers [15,17]. In other words, we can model the retarded short-range electron-phonon coupling within the conventional Eliashberg equations, but the effective density of states and the coupling constant λ are enhanced by a factor of f p due to the presence of the Fröhlichtype electron-phonon interaction. The effective Coulomb pseudopotential µ * in the Eliashberg equations will not change significantly when the Fröhlich electron-phonon interaction sets in. This is because this interaction enhances the density of states, which tends to increase µ * , while it produces an attractive potential that effectively reduces the direct Coulombic repulsion by a factor of about ǫ • /ǫ ∞ [15] and thus tends to decrease µ * (see Eq. 4 below). Therefore, the Fröhlich-type electron-phonon interaction can enhance superconductivity mainly through increasing the effective coupling constant for the retarded electron-phonon interaction. For ionic materials (e.g., MgB 2 ), this type of electron-phonon coupling should be rather strong, so that the superconductivity can be enhanced substantially.
Within this simplified approach, the effective coupling constant for the retarded short-range electron-phonon interaction is λ = λ b f p , where λ b is the bare short-range retarded electron-phonon coupling constant in the absence of the Fröhlich-type electron-phonon interaction. The value of λ b for MgB 2 has been calculated to be 0.7-0.9 (Ref. [5][6][7][8]). The total carrier mass enhancement factor is then given by The factor 1 + λ b f p is the mass enhancement factor due to the retarded electron-phonon interaction with the enhanced coupling constant λ = λ b f p . The Coulomb pseudopotential µ * is Here µ ∝ U f p (U is the effective Coulomb interaction between two carriers, which is renormalized by the Fröhlich interaction ); ; ω ln is the logarithmically averaged frequency, which is normally lower than the Debye frequency ω D by a factor of about 1.2. Indeed, the calculatedhω ln is 53.8 meV (Ref. [19]), which is a factor of 1.2 smaller than the measuredhω D (64.3 meV) [2,9].
Since f p = exp(g 2 ), which depends on the isotope mass [16], the quantities λ, µ * , and f t are all isotope-mass dependent. One can easily deduce that the total exponents of the isotope effects on λ (α λ = − d ln λ/d ln M j , where M j is the mass of the jth atom in the unit cell) and on µ * (α λ = − d ln µ * /d ln M j ) are given by and Furthermore, from the McMillian formula [10], we can also determine the total exponent of the isotope effect on T c (α = − d ln T c /d ln M j ), From the above equations, we can calculate f p , λ,hω ln , α λ , α µ * , and α as a function of λ b using fixed parameters f t = 3.2, T c = 40 K, E F = 0.57 eV (Ref. [18]), and µ * = 0.1. In Fig. 1, we plot the calculated α,hω ln , and f p as a function of λ b . One can see that the calculated α is 0.3 for λ b = 0.8, a value lying within the first-principle calculations (λ b = 0.7-0.9) [5][6][7][8]. The calculated isotope exponent α is in quantitative agreement with the measured one (0.28±0.04) [4]. The reduction in the isotope exponent is due to the fact that the coupling constant λ has a negative isotope effect which partially cancels out the positive isotope effect on the prefactor of Eq. 7. As λ b increases, f p must decrease to keep f t a constant (see Eq. 3 and Fig. 1c). The decrease of f p reduces the magnitude of the negative isotope effect on the coupling constant, and thus increase the isotope effect on T c , as seen clearly in Fig. 1a.
Meanwhile, the calculatedhω ln is about 40 meV for λ b = 0.8 (see Fig. 1b), which appears to be lower than the one (53.8 meV) determined from the phonon density of states [19]. However, inelastic neutron scattering experiments [20] show that a low-energy phonon mode at about 17 meV is strongly coupled to conduction electrons; the intensity of the 17 meV peak increases with decreasing temperature for T > T c whereas below T c it starts decreasing. The strong coupling to the low energy mode is not expected from the theoretical calculations [5][6][7][8], and may be related to some kind of structural instability [20]. If this is true, the calculatedhω ln in Ref. [19] should be substantially overestimated. Indeed, from the measured 2∆(0)/k B T c = 4.1, one can determine the magnitude of hω ln using a formula [10] Substituting T c = 40 K and 2∆(0)/k B T c = 4.1 into Eq. 9, we gethω ln = 40 meV, in quantitative agreement with the above independent calculation. From Fig. 1c, one can see that f p is about 1.5 at λ b = 0.8, that is, the Fröhlich-type electron-phonon interaction enhances the density of states and the coupling constant by a factor of 1.5. Without this interaction, the material would have a transition temperature of about 22 K. Therefore, the Fröhlich-type electron-phonon interaction in ionic materials can indeed enhance superconductivity.
Although the conventional phonon-mediated model could account for the observed T c value [5][6][7][8], it cannot consistently explain the T c value, the reduced isotope exponent (≈0.3), the large carrier mass enhancement factor (3.1), and the reduced energy gap (2∆(0)/k B T c = 4.1). The accurate determination of 2∆(0)/k B T c = 4.1 places a strong constraint onhω ln , that is,hω ln ≃ 40 meV. With hω ln = 40 meV, and T c = 40 K, one cannot find any parameters that could lead to α ≃ 0.28. Further, one would never get a carrier mass enhancement factor larger than 3 within the conventional model. On the other hand, the proposed unconventional phonon-mediated mechanism naturally resolves these controversies and is able to quantitatively explain these experiments in a consistent way. In addition, an important prediction of this scenario is that both electronic specific heat γ and London penetration depth λ L (0) depend on the isotope mass of boron. One can easily show that, upon replacing 10 B with 11 B, both γ and λ 2 L (0) will decrease by about 4%. The isotope effect on γ should be observable if one could measure the specific heat of the isotope samples down to a low temperature (< 1 K) under a magnetic field higher than H c2 (0). On the other hand, a special cation must be taken in determination of the intrinsic London penetration depth since the extrinsic contribution to the measured penetration depth due to defects [21] would mimic the isotope effect if two isotope samples have different densities of defects.
In summary, the carrier mass enhancement factor f t has been determined for MgB 2 from the measured upper critical field and the calculated bare Fermi velocity. It is found that f t is 3.1±0.1, in remarkably good agreement with that deduced from the independent specific heat data. The unusually large f t is inconsistent with the measured reduced gap and the total isotope-effect exponent (α ≈ 0.3) within the conventional phonon-mediated model. We thus propose an unconventional phononmediated mechanism where long-range Fröhlich electronphonon interaction and short-range retarded electronphonon interaction are modeled separately. Within this scenario, we are able to quantitatively explain the values of T c , f t , α and 2∆(0)/k B T c .