Geometric frustration is a broad phenomenon that results from an intrinsic incompatibility between some fundamental interactions and the underlying lattice geometry1, 2, 3, 4, 5, 6, 7. Geometric frustration gives rise to new fundamental phenomena and is known to yield intriguing effects such as the formation of exotic states like spin ice, spin liquids and spin glasses1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17. It has also led to interesting findings of fractional charge quantization and magnetic monopoles5, 6. Mechanisms related to geometric frustration have been proposed to understand the origins of relaxor and multiferroic behaviour, colossal magnetocapacitive coupling, and unusual and novel mechanisms of high-transition-temperature superconductivity3, 4, 5, 12, 16. Although geometric frustration has been particularly well studied in magnetic systems in the past 20 years or so, its manifestation in the important class formed by ferroelectric materials (which are compounds with electric rather than magnetic dipoles) is basically unknown. Here we show, using a technique based on first principles, that compositionally graded ferroelectrics possess the characteristic ‘fingerprints’ associated with geometric frustration. These systems have a highly degenerate energy surface and display critical phenomena. They further reveal exotic orderings with novel stripe phases involving complex spatial organization. These stripes display spiral states, topological defects and curvature. Compositionally graded ferroelectrics can thus be considered the ‘missing link’ that brings ferroelectrics into the broad category of materials able to exhibit geometric frustration. Our ab initio calculations allow deep microscopic insight into this novel geometrically frustrated system.
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