  Geometric frustration is a broad phenomenon that results from an intrinsic
incompatibility between some fundamental interactions and the underlying
lattice geometry1-7. 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 glasses1-7. It has also led
to interesting findings of fractional charge quantization and magnetic
monopoles5,6. Geometric frustration related mechanisms have been proposed to
understand the origins of relaxor behavior in some multiferroics, colossal
magnetocapacitive coupling and unusual and novel mechanisms of high Tc
superconductivity1-5. Although geometric frustration has been particularly well
studied in magnetic systems in the last 20 years or so, its manifestation in
the important class formed by ferroelectric materials (that are compounds
exhibiting electric rather than magnetic dipoles) is basically unknown. Here,
we show, via the use of a first-principles-based technique, that
compositionally graded ferroelectrics possess the characteristic "fingerprints"
associated with geometric frustration. These systems have a highly degenerate
energy surface and exhibit original 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 as the
"missing" link that brings ferroelectrics into the broad category of materials
able to exhibit geometric frustration. Our ab-initio calculations allow a deep
microscopic insight into this novel geometrically frustrated system.
