Modes of periodic domain wall motion in ultrathin ferromagnetic layers

Theory and experiments for disordered elastic manifolds, depinning, avalanches, and sandpiles

Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modeled as an elastic system subject to quenched disorder. The ensuing field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group (RG) flow involves a function, the disorder correlator Δ(w), and is therefore termed the functional RG. Δ(w) is a physical observable, the auto-correlation function of the center of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of the mappings between these systems requires specific techniques, which we develop, including modeling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional Kardar-Parisi-Zhang (KPZ) terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random-field magnets. Emphasis is given to numerical and experimental tests of the theory.

Current-induced dynamics of skyrmion strings

Dynamics of string-like objects is an important issue in a broad range of physical systems, including vortex lines in superconductors, viscoelastic polymers, and superstrings in elementary particle physics. In noncentrosymmetric magnets, string forms of magnetic skyrmions are present as topological spin objects, and their current-induced dynamics has recently attracted intense interest. We show in the chiral magnet MnSi that the current-induced deformation dynamics of skyrmion strings results in transport response associated with the real-space Berry phase. Prominent nonlinear Hall signals emerge above the threshold current only in the skyrmion phase. We clarify the mechanism for these nonlinear Hall signals by adopting spin density wave picture to describe the moving skyrmion lattice; deformation of skyrmion strings occurs in an asymmetric manner due to the Dzyaloshinskii-Moriya interaction, which leads to the nonreciprocal nonlinear Hall response originating from an emergent electromagnetic field. This finding reveals the dynamical nature of string-like objects and consequent transport outcomes in noncentrosymmetric systems.

Current-induced viscoelastic topological unwinding of metastable skyrmion strings

In the MnSi bulk chiral magnet, magnetic skyrmion strings of 17 nm in diameter appear in the form of a lattice, penetrating the sample thickness, 10–1000 μm. Although such a bundle of skyrmion strings may exhibit complex soft-matter-like dynamics when starting to move under the influence of a random pinning potential, the details remain highly elusive. Here, we show that a metastable skyrmion-string lattice is subject to topological unwinding under the application of pulsed currents of 3–5 × 10⁶ A m^–2 rather than being transported, as evidenced by measurements of the topological Hall effect. The critical current density above which the topological unwinding occurs is larger for a shorter pulse width, reminiscent of the viscoelastic characteristics accompanying the pinning-creep transition observed in domain-wall motion. Numerical simulations reveal that current-induced depinning of already segmented skyrmion strings initiates the topological unwinding. Thus, the skyrmion-string length is an element to consider when studying current-induced motion.

Understanding the dynamics of the skyrmion string lattice is the prerequisite for its potential application as next-generation information carriers. Here, the authors explore the topological unwinding of skyrmion string lattice under the application of current pulses.

Nonsteady dynamic properties of a domain wall for the creep state under an alternating driving field

With Monte Carlo simulations, the nonsteady dynamic properties of a domain wall have been systematically investigated for the thermally activated creep state under an alternating driving field. Taking the driven random-field Ising model in two dimensions as an example, two distinct growth stages of the domain interface are identified with both the correlation length and roughness function. One stage belongs to the universality class of the random depositions, and the other to that of the quenched Edwards-Wilkinson equation. In the latter case, due to the dynamic effect of overhangs, the domain interface may exhibit an intrinsic anomalous scaling behavior, different from that of the quenched Edwards-Wilkinson equation.

Transient behavior of the dynamically ordered phase in uniaxial cobalt films

We study the time dependent magnetic behavior of uniaxial cobalt films in the vicinity of the dynamic phase transition (DPT) as a function of the period P and bias H(b) of an oscillating magnetic field. In addition to the DPT at a critical period P(c), we observe the occurrence of transient behavior for P<P(c). Our data can be consistently explained by the existence of a phase line at H(b)=0 for P<P(c), which causes a first order phase transition in between two antiparallel dynamic order states, thus indicating far-reaching similarities of the DPT to equilibrium phase transitions in ferromagnets.

Magnetic soft x-ray microscopy of the domain wall depinning process in permalloy magnetic nanowires

Full-field magnetic transmission x-ray microscopy at high spatial resolution down to 20 nm is used to directly observe field-driven domain wall motion in notch-patterned permalloy nanowires. The depinning process of a domain wall around a notch exhibits a stochastic nature in most nanowires. The stochasticity of the domain wall depinning sensitively depends on the geometry of the nanowire such as the wire thickness, the wire width, and the notch depth. We propose an optimized design of the nanowire for deterministic domain wall depinning field at a notch.

Magneto-optical Kerr effect susceptometer for the analysis of magnetic domain wall dynamics

Domain wall dynamics in thin magnetic films with perpendicular and in-plane anisotropy is studied using a novel magneto-optical Kerr effect susceptometery method. The method allows for measurements of domain wall motion under ac field excitation and the analysis of dynamic modes as a function of driving frequency and magnetic field amplitude. Domain wall dynamics in the perpendicular anisotropy system, a Co/Pt multilayer, is characterized by thermally activated creep motion. For this dynamic mode, a polydispersivity exponent of β = 0.50 ± 0.03 is derived at small excitation energy, which is in excellent agreement with theoretical models. The dynamics of the other system, a Co wire with transverse uniaxial anisotropy, is dominated by viscous slide motion in a regular magnetic stripe pattern. Analytical expressions are derived for this magnetic configuration and by using these expressions, accurate values for the depinning field and the domain wall mobility are extracted from the susceptibility measurements.

Dynamic effect of overhangs and islands at the depinning transition in two-dimensional magnets

With the Monte Carlo methods, we systematically investigate the short-time dynamics of domain-wall motion in the two-dimensional random-field Ising model with a driving field (DRFIM). We accurately determine the depinning transition field and critical exponents. Through two different definitions of the domain interface, we examine the dynamics of overhangs and islands. At the depinning transition, the dynamic effect of overhangs and islands reaches maximum. We argue that this should be an important mechanism leading the DRFIM model to a different universality class from the Edwards-Wilkinson equation with quenched disorder.

Elasticity of a contact-line and avalanche-size distribution at depinning

Motivated by recent experiments, we extend the Joanny-deGennes calculation of the elasticity of a contact line to an arbitrary contact angle and an arbitrary plate inclination in presence of gravity. This requires a diagonalization of the elastic modes around the nonlinear equilibrium profile, which is carried out exactly. We then make detailed predictions for the avalanche-size distribution at quasistatic depinning: we study how the universal (i.e., short-scale independent) rescaled size distribution and the ratio of moments of local to global avalanches depend on the precise form of the elastic kernel.

Perturbation theory for ac-driven interfaces in random media

We study D-dimensional elastic manifolds driven by ac forces in a disordered environment using a perturbation expansion in the disorder strength and the mean-field approximation. We find that for D ≤ 4 , perturbation theory produces nonregular terms that grow unboundedly in time. The origin of these nonregular terms is explained. By using a graphical representation we argue that the perturbation expansion is regular to all orders for D > 4. Moreover, for the corresponding mean-field problem we prove that ill-behaved diagrams can be resummed in a way that their unbounded parts mutually cancel. Our analytical results are supported by numerical investigations. Furthermore, we conjecture the scaling of the Fourier coefficients of the mean velocity with the amplitude of the driving force h .

Direct observation of stochastic domain-wall depinning in magnetic nanowires

The stochastic field-driven depinning of a domain wall pinned at a notch in a magnetic nanowire is directly observed using magnetic x-ray microscopy with high lateral resolution down to 15 nm. The depinning-field distribution in Ni80Fe20 nanowires considerably depends on the wire width and the notch depth. The difference in the multiplicity of domain-wall types generated in the vicinity of a notch is responsible for the observed dependence of the stochastic nature of the domain-wall depinning field on the wire width and the notch depth. Thus the random nature of the domain-wall depinning process is controllable by an appropriate design of the nanowire.

Logarithmic correction to scaling in domain-wall dynamics at Kosterlitz-Thouless phase transitions

With Monte Carlo simulations, we investigate the relaxation dynamics of domain walls at the Kosterlitz-Thouless phase transition, taking the two-dimensional XY model as an example. The dynamic scaling behavior is carefully analyzed, and a domain-wall roughening process is observed. Two-time correlation functions are calculated and aging phenomena are investigated. Inside the domain interface, a strong logarithmic correction to scaling is detected.

Nonequilibrium critical dynamics with domain wall and surface

With Monte Carlo simulations, we investigate the relaxation dynamics with a domain wall for magnetic systems at the critical temperature. The dynamic scaling behavior is carefully analyzed, and a dynamic roughening process is observed. For comparison, similar analysis is applied to the relaxation dynamics with a free or disordered surface.

Creep and flow regimes of magnetic domain-wall motion in ultrathin Pt/Co/Pt films with perpendicular anisotropy

We report on magnetic domain-wall velocity measurements in ultrathin Pt/Co(0.5-0.8 nm)/Pt films with perpendicular anisotropy over a large range of applied magnetic fields. The complete velocity-field characteristics are obtained, enabling an examination of the transition between thermally activated creep and viscous flow: motion regimes predicted from general theories for driven elastic interfaces in weakly disordered media. The dissipation limited flow regime is found to be consistent with precessional domain-wall motion, analysis of which yields values for the damping parameter, alpha.