# A Computational Study of Superconducting Materials: A Case Study in Carbon-Doped MgB<sub>2</sub>

Open Access

- Graduate Program:
- Materials Science and Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- June 10, 2011
- Committee Members:
- Zi Kui Liu, Dissertation Advisor
- Zi Kui Liu, Committee Chair
- Joan Marie Redwing, Committee Member
- Vincent Henry Crespi, Committee Member
- Qi Li, Committee Member

- Keywords:
- thermodynamic modeling
- density functional theory
- superconductor
- CALPHAD
- MgB<sub>2</sub>

- Abstract:
- <p>Properties of superconductivity materials below their critical temperature such as critical magnetic field (H<sub>c2</sub>) or critical current density (J<sub>c</sub>) are greatly affected by the structures/compositions determined at high temperature during the synthesis processes. It is well known that different processing conditions can result in different superconducting properties at low temperatures. Therefore, the ability to control the structures/compositions of the superconducting materials made at high temperatures is crucial for the development of the superconducting materials. Thermodynamics is a very useful tool to achieve control over the structures/compositions of the synthesis processes. With the improvement in the computational power and advancement in the field of computational thermodynamic and density functional theory, combining the powerful and robust CALculation of PHAse Diagram technique with quantum mechanical first-principles calculations has rapidly become the preferred approach for materials design especially for metallic alloys system in the past few years.</p> <p>In this work, high temperature thermodynamics of the MgB<sub>2</sub> superconductor will be investigated. With the discovery of the unusually high superconducting temperature in MgB<sub>2</sub>, countless investigations have been carried out on the superconducting properties of single crystal, polycrystal, thin films, or doped MgB<sub>2</sub>. As mentioned earlier, properties of superconductors are affected by the structures/compositions of the superconducting materials processed at temperature much higher than superconductivity transition temperature. It is very important to understand the high temperature properties, so the superconducting properties at low temperatures can be easily controlled. First-principles calculations based on DFT are employed to obtain various thermodynamic and structural properties such as enthalpy of formation, heat capacity, mixing enthalpy, bulk modulus, elastic constants, and thermal expansion for the endmembers of the B<sub>4+x</sub>C sublattice model and various phases present in the Mg-B-C system. The calculated thermochemical data are then used to optimize the Gibbs energy description of each phase in the system during the CALPHAD modeling. Temperature-dependent properties of those structures can be obtained using phonon calculations under the harmonic approximation via the supercell method. In some instances, the Debye-Grüneisen model is used instead of the supercell phonon approach to obtain the temperature-dependent thermodynamic properties that are necessary to the CALPHAD modeling because the phase has imaginary phonon frequencies present from dynamically unstable structures. The CALPHAD modeling is then applied to construct the B-C system using the five sublattice model, (B)<sub>11</sub>(B,C)(B,C,Va)(B,Va)(B,C,Va), to represent the solubility in the B<sub>4+x</sub>C phase. The Gibbs energy description of the ternary compound, MgB<sub>2</sub>C<sub>2</sub>, will be obtained purely from first-principles calculations since there is no experimental data available. By combining three binary systems, Mg-B, Mg-C, and B-C together, the Mg-B-C thermodynamic database including MgB<sub>2</sub>C<sub>2</sub> and the solubility of carbon in MgB<sub>2</sub> can be created. This dissertation focuses on the coupling of first-principles calculations with CALPHAD modeling to construct a thermodynamic database for the system which has limited experimental data. Then, properties such as the carbon solubility in MgB<sub>2</sub> can be predicted. The thermodynamic properties in the region that experimental data are rarely available such as decomposition temperature of MgB<sub>2</sub>C<sub>2</sub> can be estimated. With the availability of the Mg-B-C thermodynamic database, the carbon solubility limit in MgB<sub>2</sub> as a function of temperature or pressure can be calculated. The knowledge obtained from CALPHAD modeling can be very useful for the experimentalist since it provide a guideline on how external variables can effect the solubility of carbon in MgB<sub>2</sub>. The synthesis processes can be modified to obtain the maximum carbon solubility or to avoid the presence of second phase.</p> <p>By using the predicted results from CALPHAD modeling and the properties calculated from first-principles, the model that predicts the unusual behavior of lattice parameters in carbon-doped MgB<sub>2</sub> thin films, based on the difference in coefficient of thermal expansion and elastic properties in MgB<sub>2</sub> and other phases, is proposed. First-principles calculations on X-doped MgB<sub>2</sub>, with X being 49 elements substituting the Mg atom, have been performed in order to suggest potential dopants for the MgB<sub>2</sub> superconductor system. Based on the calculations, the solubility in MgB<sub>2</sub> is possible for 24 elements. In those 24 elements, 8 of them introduce positive strain on the MgB<sub>2</sub> lattice, as predicted by first-principles calculations.</p>