A topological insulator is a state of quantum matter that, while being an insulator in the bulk, hosts topologically protected electronic states at the surface. These states open the opportunity to realize a number of new applications in spintronics and quantum computing. To take advantage of their peculiar properties, topological insulators should be tuned in such a way that ideal and isolated Dirac cones are located within the topological transport regime without any scattering channels. Here we report ab-initio calculations, spin-resolved photoemission and scanning tunnelling microscopy experiments that demonstrate that the conducting states can effectively tuned within the concept of a homologous series that is formed by the binary chalcogenides (Bi 2 Te 3 , Bi 2 se 3 and sb 2 Te 3 ), with the addition of a third element of the group IV.

The switching between topologically distinct skyrmionic and ferromagnetic states has been proposed as a bit operation for information storage. While long lifetimes of the bits are required for data storage devices, the lifetimes of skyrmions have not been addressed so far. Here we show by means of atomistic Monte Carlo simulations that the field-dependent mean lifetimes of the skyrmionic and ferromagnetic states have a high asymmetry with respect to the critical magnetic field, at which these lifetimes are identical. According to our calculations, the main reason for the enhanced stability of skyrmions is a different field dependence of skyrmionic and ferromagnetic activation energies and a lower attempt frequency of skyrmions rather than the height of energy barriers. We use this knowledge to propose a procedure for the determination of effective material parameters and the quantification of the Monte Carlo timescale from the comparison of theoretical and experimental data.

In a broad range of applied magnetic fields and material parameters isolated magnetic skyrmions condense into skyrmion lattices. While the geometry of isolated skyrmions and their lattice counterparts strongly depend on field and Dzyaloshinski-Moriya interaction, this issue has not been adequately addressed in previous studies. Meanwhile, this information is extremely important for applications, because the skyrmion size and the interskyrmion distance have to be tuned for skyrmion based memory and logic devices. In this investigation we elucidate the size and density-dependent phase diagram showing traditional phases in field versus material parameters space by means of Monte-Carlo simulations on a discrete lattice. The obtained diagram permits us to establish that, in contrast to the continuum limit, skyrmions on a discrete lattice cannot be smaller than some critical size and have a very specific shape. These minimal skyrmions correspond to the micromagnetic configuration at the energy barrier between the ferromagnetic and the skyrmionic states. Furthermore, we use atomistic Landau-Lifshitz-Gilbert simulations to study dynamics of the skyrmion annihilation. It is shown that this procees consists of two stages: the continuous skyrmion contraction and its discontinuous annihilation. The detailed analysis of this dynamical process is given.

The localized magnon modes of isolated kπ skyrmions on a field-polarized background are analyzed based on the Landau-Lifshitz-Gilbert equation within the terms of an atomistic classical spin model, with system parameters based on the Pd/Fe biatomic layer on Ir(111). For increasing skyrmion order k a higher number of excitation modes are found, including modes with nodes in the radial eigenfunctions. It is shown that at low fields 2π and 3π skyrmions are destroyed via a burst instability connected to a breathing mode, while 1π skyrmions undergo an elliptic instability. At high fields all kπ skyrmions collapse due to the instability of a breathing mode. The effective damping parameters of the spin waves are calculated in the low Gilbert damping limit, and they are found to diverge in the case of the lowest-lying modes at the burst and collapse instabilities, but not at the elliptic instability. It is shown that the breathing modes of kπ skyrmions may become overdamped at higher Gilbert damping values.

We determine sizes and activation energies of kπ-skyrmions on a discrete lattice using the Landau-Lifshitz-Gilbert equation and the geodesic nudged elastic band method. The employed atomic material parameters are based on the skyrmionic material system Pd/Fe/Ir(111). We find that the critical magnetic fields for collapse of the 2π-skyrmion and 3π-skyrmion are very close to each other and considerably lower than the critical field of the 1π-skyrmion. The activation energy protecting the structures does not strictly decrease with increasing k as it can be larger for the 3π-skyrmion than for the 2π-skyrmion depending on the applied magnetic field. Furthermore, we propose a method of switching the skyrmion order k by a reversion of the magnetic field direction in samples of finite size.