Understanding the interplay of lattice and magnetic degrees of freedom in structurally complex insulators from first principles

Two themes recurring throughout nature are spontaneous symmetry breaking and the emergence of new phenomena through collective behavior of seemingly simpler, well-understood parts. These themes are especially prevalent in condensed matter systems where the interplay between diverse microscopic degrees of freedom, such as spin, charge, and lattice excitations, produces new and unusual macroscopic properties. For example, ferromagnets spontaneously break time-reversal symmetry, while ferroelectrics break space inversion symmetry. Materials displaying either spontaneous spin polarization or electrical polarization are ubiquitous and many possess simple crystalline structures, yet materials breaking both time and space inversion, appropriately called multiferroic, are quite rare and prevalently found in structurally and chemically complex materials. The study of the emergence of new phenomena in multiferroic materials holds much promise to further our fundamental understanding of how spin and lattice degrees of freedom interact with one another. Much effort has already been invested in identifying new multiferroic materials; attention has now turned to the question of how to produce a strong coupling between the two distinct order parameters.; In this thesis we investigate the interplay of lattice and spin degrees of freedom in complex solids using first-principles density-functional methods. Specifically, we study (1) lattice-lattice coupling in ferroelectrics and (2) spin-lattice coupling in both simple and geometrically frustrated magnets. Building upon these two studies we developed a unique approach in which the interplay of phonons and spins leads to a strong coupling between the ferroelectric and ferromagnetic order parameters in multiferroics.