Dirac-like Hamiltonians, linear in momentum k, describe the low-energy physics of a large set of novel materials, including graphene, topological insulators, and Weyl Fermions. We show here that the inclusion of a minimal k 2 Wilson's mass correction improves the models and allows for systematic derivations of appropriate boundary conditions for the envelope functions on finite systems. Considering only Wilson's masses allowed by symmetry, we show that the k 2 corrections are equivalent to Berry-Mondragon's discontinuous boundary conditions. This allows for simple numerical implementations of regularized Dirac models on a lattice, while properly accounting for the desired boundary condition. We apply our results on graphene nanoribbons (zigzag and armchair), and on a PbSe monolayer (topological crystalline insulator). For graphene, we find generalized Brey-Fertig boundary conditions, which correctly describes the small gap seen on ab initio data for the metallic armchair nanoribbon. On PbSe, we show how our approach can be used to find spin-orbital coupled boundary conditions. Overall, our discussions are set on a generic model that can be easily generalized for any Dirac-like Hamiltonian. arXiv:1908.00145v1 [cond-mat.mes-hall]

The demand to produce clean energy has been increasing over the last years due to the various climate changes, which are strongly afflicting the world. Therefore, it is necessary to implement new and alternative energy sources that contribute for the environment preservation. That is the case, for instance, of novel energy-storage devices, which could contribute for the current demand of clean energies. In fact, most of the system used nowadays are those lead-based materials, which are strongly pollutant and contribute for the environment contamination. In this way, the interest in the development and study of lead-free materials has become a real priority in the scientific community. The objective of the present work is to synthesize and investigate the physical properties of lead-free ferroelectric systems based on BaTiO3 (BT) with technological interest. In particular, the structural and dielectric properties have been investigated as a function of the Sn 4+ content, used as a doping element in the BT hosting crystalline structure. The phase transition characteristics have been also analysed in a wide temperature and frequency range.