Calculated Changes in the Elastic Properties of MgCNi₃ at the Superconducting Transition

Abstract

We calculated the elastic properties of MgCNi₃ at the superconducting transition (𝑇_𝑐) using various thermodynamic and acoustic data. From the calculations, a step discontinuity of 8 ppm in the bulk modulus, 7 ppm in the Young’s modulus, and 3 ppm in the longitudinal sound velocity (𝑣_𝑙) is expected at 𝑇_𝑐. The step discontinuities at the transition temperature indicated the importance of lattice changes to the superconducting mechanism of MgCNi₃. The Debye temperature was calculated to be 460 K. The electron-phonon coupling constants calculated in the weak and strong coupling limits of the BCS theory and the van Hove scenario showed that MgCNi₃ is a moderately strong coupled superconductor.

1. Introduction

Superconductivity in MgCNi₃ with transition temperature (𝑇_𝑐) near 8 K is of interest especially from the pairing mechanism point of view. The electron-phonon coupling constant of MgCNi₃ has been determined using specific heat measurements [1]. Evidence of strong electron-phonon coupling and the van Hove type singularity in the electronic density of states near the Fermi energy has also been suggested [2].

The nature of superconductivity in this material is still controversial. A step discontinuity in the longitudinal sound velocity indicating lattice softening has been observed in the conventional superconductors. Thus, information on the lattice properties at 𝑇_𝑐 is very useful in understanding the MgCNi₃ superconductor. The acoustic method is a sensitive probe of phonon states and electron-phonon coupling in materials. The elastic properties of MgCNi₃ have also been reported [3, 4].

The objectives of this work were to estimate the electron-phonon coupling constant, changes in the elastic properties, and sound velocity at the transition temperature of MgCNi₃. The elastic properties of MgCNi₃ including the change in the bulk and Young’s moduli and discontinuity in the longitudinal sound velocity at 𝑇_𝑐 were calculated using various sound velocity and thermodynamic data. The acoustic Debye temperature was also calculated and compared to reported values determined by other methods. The electron-phonon coupling constant in various scenarios is also reported.

2. Basic Formulation

The standard isotropic elastic medium approximation can be applied to the polycrystalline samples. In this case, two independent elastic moduli, namely, the shear modulus 𝐺 can be written as 𝐺=𝜌𝜈²_𝑠, and the bulk modulus 𝐵 can be written as 𝐵=𝜌𝜈²_𝑙−(4/3)𝜌𝜈²_𝑠, where 𝜌 is the mass density, 𝜈_𝑠 is the shear velocity, and 𝜈_𝑙 is the longitudinal velocity [5]. The longitudinal modulus can be written as 𝐿=𝜌𝜈²_𝑙, and the Young’s modulus is 𝑌=𝐺(3𝐿−4𝐺)/(𝐿−𝐺).

For the superconducting transition in zero fields, the thermodynamics of second-order phase transition [6, 7] gives the following anomalies: Δ𝐵𝐵=−Δ𝐶_𝑝𝑇_𝑐𝐵𝜕𝑇_𝑐𝜕𝑃²,Δ𝑌𝑌=−Δ𝐶_𝑝𝑇_𝑐𝑌𝜕𝑇_𝑐𝜕𝜎_𝑖², (1a)(1b)where 𝜎_𝑖 is the 𝑖th component of the stress, Δ𝐶_𝑝 is the discontinuity in specific heat at 𝑇_𝑐, and 𝑃 is the pressure. 𝐵 is expected to show a step discontinuity, and 𝐺 is supposed to show a change in the slope at 𝑇_𝑐 from the mean-field transition [6]. The change in 𝐵 can be written as Δ𝐵𝐵≈𝜈²_𝑙2Δ𝜈_𝑙/𝜈_𝑙−(4/3)𝜈²_𝑠2Δ𝜈_𝑠/𝜈_𝑠𝜈²_𝑙−(4/3)𝜈²_𝑠. (2)

3. Results and Discussion

Several papers have reported the elastic moduli of MgCNi₃ [3, 4, 8] with a considerable spread in values. Because this material is usually prepared in the polycrystalline form, in this paper, we have used values for polycrystalline form as reported by Shein et al. [3]. Using bulk modulus 𝐵=171.05 GPa, shear modulus of 𝐺=61.46, and Young’s modulus 𝑌=168.16 GPa as published previously [3], the longitudinal sound velocity was calculated to be 𝜈_𝑙=6270 m/s and shear sound velocity 𝜈_𝑠=3100 m/s.

The sound velocities can be used to calculate the acoustic Debye temperature by using the standard formula 𝜃_𝐷=(ℎ/𝑘_𝐵)(3𝑁/4𝜋𝑉)^1/3𝜈_𝑚, where ℎ is the Planck’s constant, 𝑘_𝐵 is the Boltzmann’s constant, 𝑁 is the number of mass-point, 𝑉 is the atomic volume, and 𝜈_𝑚 is the mean velocity given by 3/𝜈³_𝑚=1/𝜈³_𝑙+2/𝜈³_𝑠. Using this expression, the acoustic Debye temperature of MgCNi₃ is calculated to be 460 K. This is near to the 𝜃_𝐷 values calculated theoretically [4, 8, 9] but much higher than the value determined from specific heat measurement [10, 11].

The BCS theory for weak coupling gives 𝑇_𝑐=1.13𝜃_𝐷𝑒^−1/𝜆, where 𝜆 is the electron-phonon coupling. The electron-phonon coupling constant for the weak coupling case calculated using the previous acoustic Debye temperature is 𝜆=0.25.

The relevance of van Hove density of states has been suggested for MgCNi₃ [2]. The singularity in the density of states at the Fermi level was taken into account in the van Hove scenario of a two-dimensional system. In this scenario, 𝑇_𝑐 is given as 𝑇_𝑐=2.72𝑇_𝐹𝑒^{√(−1/𝜆)}, where 𝑇_𝐹=𝐸_𝐹/𝑘≈10𝜃_𝐷 (𝐸_𝐹 is the Fermi energy) [12]. The electron-phonon coupling constant calculated using this formula is 𝜆=0.018.

The McMillan expression for the strong coupling BCS case is given as [13] 𝑇_𝑐=(𝜃_𝐷/1.45)exp[−1.04(1+𝜆)/(𝜆−𝜇^∗(1+0.62𝜆))], where 𝜇^∗ (with conventional values of 0.05−0.1) is the screened Coulomb repulsion between electrons in a Cooper pair. The electron-phonon coupling constant derived from this expression ranged from 0.49 to 0.59. This is consistent with most measured values for MgCNi₃ [1, 10]. Thus, this material is a moderately strong coupled superconductor.

The following is a discussion on the elastic properties of MgCNi₃ at 𝑇_𝑐. Using (1a), Δ𝐶_𝑝/𝑇_𝑐=66 mJ/mol K² [10], and 𝜕𝑇_𝑐/𝜕𝑃=0.15 K/GPa [14], the calculated step discontinuity in the bulk modulus is 8 ppm. Using (2), the discontinuity in the longitudinal velocity is 3 ppm. By assuming that 𝜕𝑇_𝑐/𝜕𝜎_𝑖≈𝜕𝑇_𝑐/𝜕𝑃, the expected discontinuity in the Young’s modulus calculated using (1b) is 7 ppm. These discontinuities are similar to Pb [15] and LiFeAsOF [16] superconductors but are much smaller than the calculated [17] and observed [18] values in the cuprate-based high 𝑇_𝑐 superconductors and MgB₂ [19]. Table 1 shows the sound velocity and changes in bulk and Young’s modulus at 𝑇_𝑐 of MgCNi₃ and other superconductors.

Table 1 

The second-order phase transition does not allow step discontinuity in the shear velocity at 𝑇_𝑐 in zero fields; hence, the shear velocity was assumed continuous at 𝑇_𝑐. In these calculations, the thermodynamic data used are representative of the various reported values, which are of the same order of magnitude. The expected step discontinuity in the longitudinal velocity in MgCNi₃ is similar to the measured value of Pb which is a BCS-type superconductors and are within the resolution range of most ultrasonic apparatus. Our calculations showed that MgCNi₃ is a moderately strong coupled BCS-type superconductor.

Acknowledgments

The author thanks Dr. W. Kong (KLIUC) for the assistance. This work has been supported by the Ministry of Higher Education of Malaysia (Grant no. ERGS/1/2011/STG/UKM/01/25) and Universiti Kebangsaan Malaysia under Grant no. DIP-2012-32.