Heat Capacity of MgB2: Evidence for Moderately Strong Coupling Behavior

We characterize the superconducting state of a phase pure polycrystalline sample of the new layered high-temperature superconductor MgB2 by specific heat measurements in magnetic fields up to 9 Tesla. The characteristic jump at the superconducting transition is observed and compared with the predictions of weak coupling BCS-theory and the α-model. Our analysis shows excellent agreement with the predictions for α = ∆/kBTC =2.1(1) with a Sommerfeld term γ of 1.1(1) mJ/mol K indicating that MgB2 is a superconductor in the moderately strong electron-phonon coupling regime. 74.25B, 74.80Dm, 74.72.-h Typeset using REVTEX 1 Recently, Akimitsu and co-workers reported superconductivity at TC ≈ 39K in the layered diboride MgB2 [1]. This discovery on the one hand again focuses attention on the borides as possible candidates for high-TC superconductivity. On the other hand, TC of MgB2 exceeds the generally agreed theoretical limit for phonon mediated superconductivity and raises the question as to another possible coupling mechanism [2]. However, first measurements of the thermodynamic properties and in particular of a sizeable partial B isotope effect by Bud’ko et al. strongly hint at the importance of phonons for superconductivity in MgB2 [3]. Kortus and co-workers, based on electronic band structure calculations conclude that the metallic character is due to covalent B-B bonds and that TC particularly benefits from strong electron-phonon coupling in concert with high frequency vibrations associated with the light mass of the boron atoms [4]. Evidence for strong-coupling s-wave superconductivity was indeed found from B nuclear spin lattice relaxation measurements from which a rather large gap ratio of 2∆/kBTC ≈ 5 was derived [6]. Strong electron-lattice coupling also seems to be able to explain Al doping experiments carried out by Slusky et al. that show a decrease of TC with increasing Al content and finally a loss of bulk superconductivity at ≈ 10% Al doping [5]. These experiments, in particular, reveal MgB2 to be close to a structural instability involving a boron interlayer alternation rather than bond alternation in the B layers. All presently available tunneling spectroscopy experiments consistently fit very well to an s-wave BCS quasi-particle density of states and consistently exclude d-wave symmetry of the order parameter. However, the gap values resulting from these studies span a wide range and currently leave a somewhat inconclusive situation. The very first experiments by Rubio-Bollinger et al. [7] gave a surprisingly small value of 2 meV (weak-coupling BCS value is 5.9 meV). More recent work by Schmidt et al. [8] (∆=4.3 meV) and Sharoni et al. [9] (∆=5 7 meV) put ∆ closer to the weak-coupling value or even into an intermediate coupling regime. This wide range of gap values seems to result from defects or minor non-superconducting impurity phases, or chemical reactions at the surface of the polycrystalline specimen. In this Letter, we report a heat capacity study in zero field and a magnetic field of 2 9 Tesla of a phase pure sample of MgB2. In the zero-field measurements we observe the characteristic jump in CP at the superconducting transition temperature TC=38.67(5)K associated with the formation of the superconducting condensate. The relative specific heat jump ∆CP/TC is significantly larger than expected for weak-coupling BCS theory indicating a moderately strong coupling scenario as an appropriate description. The shape of the heat capacity jump very well fits to a model assuming a BCS-like temperature dependence of the gap and a gap ratio α=∆(0)/kB TC ≈ 2.1(1) and assuming a Sommerfeld coefficient γ=1.1(1) mJ/molK. The heat capacity jump is shifted to temperatures below 20 K and almost completely smeared out when a magnetic field of 9 Tesla is applied. A polycrystalline sample of MgB2 was prepared from stoichiometric mixtures according to the procedure described in ref. [1]. Phase purity was checked with a STOE powder diffractometer. The superconducting transition temperature was determined with a Quantum Design MPMS7 magnetometer and conventional 4-point resistivity measurements and amounted to 38.5K. For the heat capacity measurements a pellet of 3 mm diameter and ≈14 mg was pressed, sealed in a Ta tube under Ar atmosphere and sintered at 950 C for 24 h. The heat capacity was measured with a Quantum Design PPMS relaxation calorimeter in the temperature range 1.8 to 100K in fields up to 9 T. The addenda were determined in a separate run and subtracted. Fig. 1 displays the zero-field heat capacity of MgB2 in comparison with the data taken with in external field of 9 Tesla. A sharp anomaly centered at≈38 K is visible in the zero-field data which is suppressed by a magnetic field of 9 Tesla. The height of the anomaly amounts to ∆CP/TC ≈ 2 mJ/mol K . The difference of the zero-field and the heat capacity at 9 Tesla is shown in Fig. 2 together with a theoretical curve (solid line in Fig. 2). The theory (α-model by Padamsee et al. [10]) allows for a variable ratio α = ∆(0)/kBTC with αBCS =1.76 corresponding to the weak-coupling limit. This simplistic approach was successfully used to model the specific heat jumps e.g. of layered organic superconductors as well as of the layered rare earth carbide halides [11,12]. To calculate the model heat capacities we used a polynomial representation of the tabu3 lated temperature dependence of the BCS gap [13]. To fit the experimental data we varied as adjustable parameters α, the Sommerfeld coefficient γ, TC and a parameter that allowed for a gaussian broadening of TC [12]. In addition, we included a weak linear slope of the background, the origin of which is not clear at present. The fits converge rapidly to α = 2.1(1) with TC=38.67(5) and δTC=0.55(5) (σ in a gaussian distribution of TC) corresponding to a smearing of TC of 1.4% emphasizing a good sample homogeneity. The fits consistently converged to γ=1.1(1) mJ/mol K. A reliable independent determination of the Sommerfeld coefficient e.g. from low temperature heat capacity data turned out to be difficult since a field of 9 T is not sufficient to suppress superconductivity completely. In fact, we observe sizeable deviations from linear behavior in a CP/T vs. T 2 representation below about 15 K in agreement with measurements of Hc2 by Takano et al. and Bud’ko et al. [14,15]. Assuming normal state behavior above ≈15 K a fit using CP/T=γ + β T 2 yields γ=1.17(3) mJ/mol K in best agreement with the finding from the fits of the heat capacity jump and β=1.04(1)×10 J/mol K corresponding to a low temperature Debye temperature ΘD of ≈ 800K in good agreement with the results of the previous 1957 heat capacity study by Swift and White [16]. Our results clearly prove bulk superconductivity in MgB2 and substantiate the scenario of MgB2 being in the ’intermediate’ or moderately strong coupling regime as conjectured from tunneling spectroscopy experiments by Sharoni et al. [9]. Our method does not suffer from the inherent problems of tunneling techniques, namely defects or surface deterioration and indicate a gap value of 7.0(3) meV for MgB2. The ratio (TC/ΘD) 2 amounts to 2.34×10 and is close to classical elemental strongly-coupled superconductors, e.g. Hg. Using a typical value μ∗ ≈ 0.15 for the Coulomb pseudopotential we estimate an empirical electron-phonon coupling constant λ ≈ 1 which is also close to the results found for elemental stronglycoupled superconductors. The Sommerfeld coefficient γ obtained independently from the fits to the heat capacity jump and the low temperature normal state heat capacity match reasonably well and are in good agreement with the results of band structure calculations by Kortus et al. [4] and An et al. [17] where a value of 1.7 mJ/mol K was consistently found. 4 In summary, we present and analyze heat capacity measurements which show that the new high-TC superconductor MgB2 is a BCS superconductor in the intermediate or ’moderately’ strong electron-phonon coupling regime. From fits of the heat capacity jump we conclude a gap of 7.0(3) meV. We would like to thank E. Brücher and G. Siegle for experimental assistance. We gratefully acknowledge useful discussions with A. Bussmann-Holder, U. Habermeier, O. Jepsen, J. Köhler and W. Schnelle.