Improper-proper ferroelectric competition as a mechanism for multistate polarisation and ferrielectric-like behaviour

In this paper, we re-explore a simple textbook Landau model describing improper ferroelectricity and show that in the limit where both proper and improper instabilities exist and compete, improper ferroelectrics can display switching between multiple polarisation states. Using first principles calculations we highlight how the hexagonal tungsten bronze materials may be an archetypal case, with the possibility to switch between improper and proper phases. The resulting functional characteristics are akin to "ferrielectrics", with switching behaviour in the form of a triple hysteresis loop. Such functionality could be ideal for creating non-volatile multistate systems for use in memory devices or as a backbone for neuromorphic computing.