Sifting for substitutional elements that decrease thermal conductivity of a thermal barrier material

We derive a relation between phonon relaxation time and the differences in masses and volumes of substitutional ions from host ions in the doped crystal using perturbation methods and apply it to perovskite SrZrO_3 doped with rare earth elements, a potential material for thermal barrier coating (TBC). In contrast to previously reported Hamiltonians for doped crystals, the Hamiltonian for this study was derived by simultaneously considering the volume and mass of a unit cell as random variables with respect to the position of the unit cell. Within the perturbation framework for perfect crystals, substitutional ions only introduce the phonon relaxation time rather than change the phonon dispersion relation, and therefore the thermal conductivity only decreases with decreasing phonon relaxation time. To decrease thermal conductivities, researchers experimentally dope crystals with chemical elements different from the host elements. However, if researchers were to adopt enumeration methods, numerous possible dopants and their combinations exist that require significant experimentation and resources. To reduce the possibilities, a relation is developed that sifts out those elements that cannot decrease the thermal conductivity of a given crystal as efficiently as others. The remaining elements can then be studied through experiments and computational analysis, thereby saving experimentation time and resources. Moreover, the relation accounts for the lower thermal conductivity with higher mass and size differences between the host and substitutional ions. Finally, the relation is applied to obtain the minimum thermal conductivities of SrZrO_3 doped with rare earth elements. We find that lutetium provides the optimal solution and, in decreasing thermal conductivity, mass rather than volume difference has the stronger effect.