Initially, our efforts were directed at understanding the fundamentals of flexoelectricity from the point of view of quantum mechanics, and at developing efficient and accurate methodologies for calculating it in real materials. This work has led to a number of breakthrough advances in the context of first-principles electronic-structure theory. One of the most powerful ideas of this proposal, i.e. combining the traditional long-wave approach with modern density-functional perturbation theory (DFPT), resulted in a bright success. We now have at our disposition a brand new method, which we call "long-wave DFPT", that elegantly solves not just flexoelectricity, but also the whole class of problems (i.e. spatial dispersion effects) that flexoelectricity belongs to. Examples include Lorentz forces (A. Zabalo, CE Dreyer and MS, PRB 2021) in extended crystals and natural rotatory power (A. Zabalo and MS, arXiv:2304.00048) in chiral systems. All these properties, including the full flexoelectric tensor for an arbitrary insulating system, can now computed in few minutes by using the publicly distributed ABINIT simulation package, where our methodological advances have been implemented and tested. Such a scenario, which we optimistically described in the "five years from now" box of the grant proposal, is now a reality.
This breakthrough did not happen in a day. We first had to address some formal issues with the treatment of the current density response to inhomogeneous fields (C. E. Dreyer, MS and D. Vanderbilt, PRB 2018), and with its representation in curvilinear coordinates (MS and D. Vanderbilt, PRB 2018; A. Schiaffino, C. E. Dreyer, D. Vanderbilt and MS, PRB 2019). Only later we could implement and test the first applications of long-wave DFPT to the clamped-ion bulk flexoelectric tensor and the dynamical quadrupoles (M. Royo and M. Stengel, PRX 2019). The lattice-mediated contributions to the flexoelectric tensor followed shortly after (M. Royo and M. Stengel, PRB 2022), together with several other spatial dispersion effects. A pioneering theory of flexomagnetism (A. Edström,... MS, PRL 2021), which was never attempted before from first principles, has also been established very recently. This activity has led to very successful (a total of four publications including two Physical Review Letters) and unforeseen collaborations on the first-principles calculation of electron-phonon interactions. The two-dimensional case, in particular, required a thorough analysis of the long-range electrostatic interactions, which led us to a theoretical and methodological milestone of its own (M. Royo and MS, PRX 2021). All these developments are a clear demonstration of the breakthrough nature of the work carried out in this project, which ended up opening opportunities that are well beyond our initial expectations.
Our work has not only been methodological. For example, we have published a highly innovative study of ferroelastic domain walls in SrTiO3 (A. Schiaffino, M. Stengel, PRL 2017), which can be regarded as our first implementation the "multiscale" part of MULTIFLEXO. This led us to the discovery of two previously overlooked coupling mechanisms that involve antiferrodistortive tilts and their gradients, which also bear important implications for the physics of SrTiO3 at low temperatures. (B. Casals et al., PRL 2018). We're currently formalizing these findings into a general approach to "first-principles macroscopic theories", with immediate relevance to the emerging research area of topological structures in ferroics (domain walls, spirals, skyrmions, etc.) Our first attempts in this direction are documented in O. Diéguez and MS, PRX 2022 and MS, arXiv:2304.06613. Meanwhile, we have successfully demonstrated (A. Zabalo and MS, PRL 2021) the relevance of flexoelectricity in the so-called ferroelectric metals, a class of materials that is attracting considerable interest lately. We have also generalized our methods to systems with lower dimensionality, which enabled us to calculate the flexoelectric properties of several two-dimensional (2D) materials (M. Springolo, M. Royo and M. Stengel, PRL 2021 and arXiv:2303.18124).
In addition to the journal publications, the above advances have been presented during invited talks at a number of prestigious international conferences, including the APS March Meeting and the Total Energy Workshop in Trieste.
