co nd - m at . s up r - co n ] 4 O ct 2 00 2 Electron - Phonon Interaction in NbB 2 : A Comparison with MgB 2

We present a comparison of electron-phonon interaction in NbB2 and MgB2, calculated using full-potential, density-functional-based methods in P6/mmm crystal structure. Our results, described in terms of (i) electronic structure, (ii) phonon density of states F (ω), (iii) Eliashberg function α2F (ω), and (iv) the solutions of the isotropic Eliashberg gap equation, clearly show significant differences in the electron-phonon interaction in NbB2 and MgB2. We find that the average electronphonon coupling constant λ is equal to 0.59 for MgB2 and 0.43 for NbB2, leading to superconducting transition temperature Tc of around 22K for MgB2 and 3K for NbB2. The lack of success in finding superconductivity in other diborides with superconducting transition temperature, Tc, close to that of MgB2 [1] underscores the complex nature of interaction responsible for superconductivity in MgB2. In MgB2 the complexity is further compounded by the presence of multifaceted Fermi surface [2,3] and a highly anisotropic electron-phonon coupling, λ(k,k), over the Fermi surface [4,5]. The dependence of superconducting properties on such details has ensured that we do not know, as yet, the exact nature of interaction leading to superconductivity in MgB2. Within Eliashberg-Migdal theory [6,7] of superconductivity, a reliable description of the superconducting state requires an accurate knowledge of λ(k,k) and the renormalized electron-electron interaction, μ, which are used as input to the fully anisotropic gap equation. The present computational capability allows us to evaluate λ(k,k) accurately using density-functional-based methods but, unfortunately, μ cannot be evaluated. However, it is reasonable to assume that μ varies between 0.1 to 0.2 [4,5]. Thus, the electron-phonon coupling λ(k,k), which is a normal state function, must contain signatures of superconducting state. Preprint submitted to Elsevier Preprint 1 February 2008 In an attempt to identify some of the unique features of electron-phonon interaction in MgB2 vis-a-vis other diborides we have studied (i) the electronic structure, (ii) the phonon density of states (DOS), (iii) the Eliashbrg function, and (iv) the solutions of the isotropic Eliashberg gap equation for NbB2 and MgB2 in P6/mmm crystal structure. The choice of NbB2 has been motivated by the recent reports [8,9] of superconductivity, with Tc going up to 9.2K, under pressure in hole-doped NbxB2. Earlier experiments have shown superconductivity in stoichiometric NbB2 [9,10] as well as Boron-enriched NbB2 [11] samples. The reported Tc for stoichiometric NbB2 varies from 0.62K [10] to 5.2K [9], while for Boron-enriched NbB2.5 the Tc is found to be 6.4K [11]. We also note that, recently, Kaczorowski et al. [12] did not find any superconductivity in NbB2 down to 2K. We have calculated the electronic structure of NbB2 and MgB2 in P6/mmm crystal structure with optimized lattice constants a and c, as given in Table I. The lattice constants a and c were optimized using the ABINIT program [13] based on pseudopotentials and plane waves. For studying the electron-phonon interaction we used the full-potential linear response program of Savrasov [14,15], and calculated the dynamical matrices and the Hopfield parameter. These were then used to calculate the phonon DOS, F (ω), the electron-phonon coupling λ(k,k), and the Eliashberg function, αF (ω), for NbB2 and MgB2. Subsequently, we have numerically solved the isotropic Eliashberg gap equation [6,7,16] for a range of μ to obtain the corresponding Tc. Based on our calculations, described below, we find significant differences in the phonon DOS and the Eliashberg functions of NbB2 and MgB2. In particular, we find that the average electron-phonon coupling constant is equal to 0.59 for MgB2 and 0.43 for NbB2, leading to superconducting transition temperatures of around 22K for MgB2 and 3K for NbB2. Before describing our results in detail, we provide some of the computational details of our calculation. The structural relaxation was carried out by the molecular dynamics program ABINIT [13] with Broyden-Fletcher-GoldfarbShanno minimization technique using Troullier-Martins pseudopotential [18] for MgB2 and Hartwigsen-Goedecker-Hutter pseudopotential [19] for NbB2, 512 Monkhorst-Pack [20] k-points and Teter parameterization for exchangecorrelation. The kinetic energy cutoff for the plane waves was 110Ry forMgB2 and 140Ry for NbB2. The charge self-consistent full-potential LMTO [14] calculations were carried out with the generalized gradient approximation for exchange-correlation of Perdew et al [21] and 484 k-points in the irreducible wedge of the Brillouin zone. For MgB2, the basis set used consisted of 3κ panels and s, p, d and f orbitals at the Mg site and s, p and d orbitals at the B site. In the case of NbB2, we included 2κ panels and s, p and d orbitals at the Nb site. In all cases the potential and the wave function were expanded