We are reporting a new type of synchronization, termed dancing synchronization, between two spin-torque nano-oscillators (STNOs) coupled through spin waves. Different from the known synchronizations in which two STNOs are locked with various fixed re
lative phases, in this new synchronized state two STNOs have the same frequency, but their relative phase varies periodically within the common period, resulting in a dynamic waving pattern. The amplitude of the oscillating relative phase depends on the coupling strength of two STNOs, as well as the driven currents. The dancing synchronization turns out to be universal, and can exist in two nonlinear Van der Pol oscillators coupled both reactively and dissipativly. Our findings open doors for new functional STNO-based devices.
Based on the findings of stripe skyrmions and the metastability of a state of an arbitrary number of skyrmions, precisely controlled manipulation of stripe skyrmions in pre-designed structures and mutual transformation between helical states and skyr
mion crystals (SkXs) are demonstrated in chiral magnetic films. As a proof of the concept, we show how to use patterned magnetic fields and spin-transfer torques (STTs) to generate nematic and smectic stripe phases, as well as UST mosaic from three curved stripe skyrmions. Cutting one stripe into many pieces and coalescing several skyrmions into one by various external fields are good ways to transform helical states and SkXs from each other.
A magnetic skyrmion is a topological object that can exist as a solitary embedded in the vast ferromagnetic phase, or coexists with a group of its siblings in various stripy phases as well as skyrmion crystals (SkXs). Isolated skyrmions and skyrmions
in an SkX are circular while a skyrmion in other phases is a stripe of various forms. Unexpectedly, the sizes of the three different types of skyrmions depend on material parameters differently. For chiral magnetic films with exchange stiffness constant $A$, the Dzyaloshinskii-Moriya interaction (DMI) strength $D$, and perpendicular magnetic anisotropy $K$, $kappaequivpi^2D^2/(16AK)=1$ separates isolated skyrmions from condensed skyrmion states. In contrast to isolated skyrmions whose size increases with $D/K$ and is insensitive to $kappall1$ and stripe skyrmions whose width increases with $A/D$ and is insensitive to $kappagg1$, the size of skyrmions in SkXs is inversely proportional to the square root of skyrmion number density and decreases with $A/D$. This finding has important implications in our search for stable smaller skyrmions at the room temperature in applications.
Skyrmions are important in topological quantum field theory for being soliton solutions of a nonlinear sigma model and in information technology for their attractive applications. Skyrmions are believed to be circular and stripy spin textures appeare
d in the vicinity of skyrmion crystals are termed spiral, helical, and cycloid spin orders, but not skyrmions. Here we present convincing evidences showing that those stripy spin textures are skyrmions, siblings of circular skyrmions in skyrmion crystals and cousins of isolated circular skyrmions. Specifically, isolated skyrmions are excitations when skyrmion formation energy is positive. The skyrmion morphologies are various stripy structures when the ground states of chiral magnetic films are skyrmions. The density of skyrmion number determines the morphology of condensed skyrmion states. At the extreme of one skyrmion in the whole sample, the skyrmion is a ramified stripe. As the skyrmion number density increases, individual skyrmion shapes gradually change from ramified stripes to rectangular stripes, and eventually to disk-like objects. At a low skyrmion number density, the natural width of stripes is proportional to the ratio between the exchange stiffness constant and Dzyaloshinskii-Moriya interaction coefficient. At a high skyrmion number density, skyrmion crystals are the preferred states. Our findings reveal the nature and properties of stripy spin texture, and open a new avenue for manipulating skyrmions, especially condensed skyrmions such as skyrmion crystals.
A generic theory of skyrmion crystal (SkX) formation in chiral magnetic films is presented. We numerically demonstrate that a chiral film can have many metastable states with an arbitrary number of skyrmions up to a maximal value. A perpendicular mag
netic field plays a crucial role in SkX formation. The energy of a film increases monotonically with skyrmion number at zero field while the film with $Q_m$ skyrmions has the lowest energy in a magnetic field. $Q_m$ first increases with the magnetic field up to an optimal value and then decreases with the field. Outside of a field window, helical states of low skyrmion number densities are thermal equilibrium phases while an SkX is metastable. Within the field window, SkXs are the thermal equilibrium states below the Curie temperature. However, the time to reach the thermal equilibrium SkX states from a helical state would be too long at a low temperature. This causes a widely spread false belief that SkXs are metastable and helical states are thermal equilibrium phase at low temperature and at the optimal field. Our findings explain well the critical role of a field in SkX formation and fascinating thermodynamic behaviours of helical states and SkXs. Our theory opens a new avenue for SkX manipulation and skyrmion-based applications.
