Skyrmions are (meta-)stable for different magnetic field regimes depending on the magnetic properties of the system. By appropriately tuning the applied field, the skyrmion lattice phase can become the lowest energy state of the system. This then means that skyrmions spontaneously form, either because other states, such as stripe domains, become unstable or because thermal activation overcomes the energy barrier, leading to the formation of a skyrmion lattice that is lower in energy.

As previously shown experimentally, we can additionally use AC field excitations to switch the magnetic configuration from stripe domains to a lower energy skyrmion lattice.

Finally, localized fields can be used to locally write a skyrmion. This has been demonstrated using magnetic force microscopy tips as localized field sources and a scheme based on the magnetic field from ultrashort electron pulses has been additionally suggested.

For a system with a skyrmion lattice state being the ground state, it was shown that one can generate skyrmions from stripe domains by current injection or field pulses that leads to heating which in turn generates strong magnetic excitations that allow the system to transform to its ground state configuration. This effect is currently investigated intensively experimentally as one finds a strong temperature dependence of the necessary current densities to nucleate skyrmions, but no complementary theoretical calculations are available so far.

Given that spin torques and in particular spin orbit torques efficiently manipulate skyrmions, different mechanisms have been put forward to use spin torques to write skyrmions. Converting DWs into skyrmions is one option and using spin torques in wires with geometrical variations to generate skyrmions is another possibility. Also, by using local spin currents flowing perpendicular to a wire, the magnetization can be locally reversed forming skyrmions.

One can influence magnetic systems by electric fields including via resulting strain in order to tune the magnetic properties (DMI, anisotropy, etc.) and thereby enable key operations such as the generation (writing) and annihilation (deleting) of chiral particles. The voltage controlled reversal of the core of a skyrmion was predicted by micromagnetic simulations using electric-field modulation of the anisotropy. The complete operation of a memory device based on skyrmions was analyzed theoretically and a modulation of the DMI with an electric field was found to be extremely efficient for writing and erasing skyrmions. Electric field-driven switching of single skyrmions was shown by spin-polarized STM, where both writing and deleting of skyrmions were demonstrated with electric fields with one polarity creating and the other polarity removing skyrmions.

Fundamentally, the presence or absence of a skyrmion can be efficiently tuned by the electric field modulation of the magnetic properties as this effect is not limited by the time reversal symmetry that prevents easy electric-field switching of the magnetization from up to down. Typically, the formation of skyrmions in external magnetic fields is explained by a balance between Heisenberg exchange interactions, DMI interaction and magneto-crystalline anisotropy, and thus it is clear that a change of any of these parameters on the application of an electric field will modify the energy landscape. It was claimed that the dominating effect of the electric field is on the exchange, where thus one electric field direction favors the ferromagnetic state, and the other stabilizes the skyrmionic state. In contrast, Ma et al. suggest that electric fields could write skyrmions if both the anisotropy profile generated by a thickness gradient and the electric field-induced anisotropy would change. A large voltage induced DMI modulation has also been recently reported, which can modify the states.

The simplest approach to manipulate skyrmion positions is to change the energy of the skyrmion when it moves laterally by a field gradient. This was proposed where it was found that in confined geometries, the resulting dynamics is more complex due to the confining potential of the edges. Moreover, a gradient has been predicted to accelerate the skyrmion. Experimentally, skyrmion dynamics due to field gradients was then realized in where a field pulse initiated the nucleation of a domain whose spatially strongly varying stray field displaced a skyrmion. By measuring the relaxation dynamics, a gyrotropic spiraling trajectory was determined that allowed for the identification of a topologically non-trivial ?=1W=1 skyrmion. More recently, this approach has been proposed to position and trap skyrmions and to guide skyrmions on a track.

In addition to creating skyrmions by electric fields, very recently a number of schemes have been proposed for electric field driven skyrmion dynamics. One clear option to achieve this is via e-field induced changes in the anisotropy, as shown in micromagnetic simulations by way of anisotropy gradients from a varying thickness dielectric or stepped magnetic anisotropy generated by multiple surface electrodes. Ma et al. were also able to demonstrate experimentally the electric field driven creation and directional motion of skyrmions in tracks with thickness and resulting anisotropy gradients via the e-field induced changes in the anisotropy. Another suggestion is based on electric-field generated surface acoustic waves for ferromagnetic materials grown on piezoelectrics, with analytical studies of both propagation along tracks and for oscillators. Yuan et al. proposed an alternative scheme based on parametric pumping of the system via alternating out-of-plane electric fields in the presence of a symmetry breaking in-plane magnetic field. Their simulations revealed a resulting so-called “rock-and-roll” motion of the skyrmion with combined excitations of breathing mode excitations and net motion along a set trajectory. The breathing mode excitations result in spin-wave emission which acts to propagate the skyrmion, resulting in the largest velocities being found theoretically for the resonance of the internal mode.¹⁷⁷

Spin torques induced by an effective acting spin-polarized current in the system affect the magnetic textures and allow one to move them. For magnetic skyrmions, it turns out that the current-induced dynamics is special as in addition to forces collinear with the current; also, forces perpendicular to the current direction can occur, so called Magnus forces which originate from the skyrmion topology. This becomes evident in the quasiparticle Thiele description where a perpendicular force arises from the interplay of the current and the gyrotropic tensor. The transverse motion of skyrmions to the current direction is called the skyrmion Hall effect and recently has been experimentally observed. Overall, the details of the skyrmion dynamics depend to a large extent on the symmetries of the systems and the kind of torque that is acting. Below, we discuss explicitly the two main type of torques, namely, spin transfer torques and spin-orbit torques.

Magnons, the quanta of spin wave excitations, have been shown to displace spin structures, such as DWs, and recently skyrmions have moved to the focus. The interaction between magnons and skyrmions was studied, where it was found that magnons can excite the eigenmodes of skyrmions and that a magnon current can induce skyrmion motion. As the scattering of magnons from skyrmions is not isotropic due to the topology of the skyrmion, there is an associated skyrmion Hall effect due to the incoming magnon propagation direction. The theory of magnon-skyrmion interaction was further developed and it was found that the interaction depends strongly on the magnon wave number and there is a finite momentum associated with the skyrmion. Finally, the impact of skyrmions on magnon properties was also ascertained.

One particular approach to displace spin textures that involves magnons is the use of temperature gradients. Here, thermal spin currents can transfer angular momentum and additionally entropic effects can also lead to a motion of spin structures as well. Overall, the physics is quite complex and both effects can displace magnetic textures toward the colder as well as the hotter part in certain regimes. This approach has been extensively discussed for DWs theoretically and experimentally there are reports of thermal effects on DWs. This approach was also proposed for skyrmions, the dynamics of a skyrmion due to a temperature gradient was calculated.

Even without any external drives, skyrmions can change their positions randomly due to thermal fluctuations.  it was seen that a skyrmion can move by thermal fluctuations from one position to another in the absence of any external driving force. Recently, this was studied in detail. In low pinning materials, strong thermal skyrmion diffusion was identified by measuring the mean square displacement with time. While the diffusion is in qualitative agreement with analytical predictions, it was found that the skyrmion diffusion depends on the skyrmion size and exhibits an exponential scaling with temperature. This surprising behavior can be explained by diffusion in a non-flat potential energy landscape leading to locally different pinning properties. The average energy barriers were extracted quantitatively from the experiments. Analogous to the diffusion in solids, this behavior shows that in the optimized systems, skyrmions exhibit thermally activated dynamics that can be uniquely used to quantify key unknown properties, including energy landscapes, ascertain interaction potentials, etc. Finally, short timescale fluctuations have recently been probed by free electron laser measurements.

Shortly after the theoretical prediction of skyrmion lattices, the clockwise, the counterclockwise, and the breathing mode were observed by means of microwave absorption spectra in Cu22OSeO33 and optical pump-probe techniques. Since then, by means of magnetic resonance techniques or inelastic neutron scattering, skyrmion lattice excitations were observed in several compounds including metals, semiconductors, and insulators. It has been shown that these excitations have a universal character and can be quantitatively modeled by only two material specific quantities characterizing the DMI energy and the critical field value for the field polarized phase. In addition to the thus far observed excitations, which symmetry wise couple to homogeneous excitations, there is an ongoing effort to experimentally access new theoretically predicted modes.

Numerical simulations and analytical calculations have identified a number of different internal modes, such as the gyrotropic, uniform breathing, and polygon-like distortion modes. For a thin film, below the magnon gap, there exist a few magnon-skyrmion bound states which are labeled according to their angular momentum quantum number mm. Note that due to the translational invariance, the translational mode has zero frequency. The breathing mode, parametrized by the angular momentum m=0m=0, can, for example, be activated by fields that are directed out-of-plane and which do not affect the radial symmetry of the skyrmions. Imaging of skyrmion breathing modes induced by spin-orbit torques has also been achieved experimentally and the gyrotropic resonance has also been measured. In-plane magnetic fields, on the other hand, do break the radial symmetry and can potentially excite modes that affect the shape of the skyrmion. The quadrupolar mode with m=2m=2 indicates the elliptical instability of the skyrmion and also a sextupolar mode with m=3m=3 is realized only in a small range above the elliptical instability. For more bubble-like skyrmions, also more modes are expected to exist. Experimentally those modes are interesting for microwave generation. So far, the polygon-like distortion modes are still to be observed.

In confined samples, e.g., in ferromagnetic nanodiscs, the translational mode turns into a gyrotropic mode with a non-zero frequency. Furthermore, in asymmetric confining potentials, new modes emerge for increased aspect ratios of the elements.¹⁶⁰