Actual-time commentary of topological defect dynamics mediating two-dimensional skyrmion lattice melting

Magnetic multilayer stack

The Ta(5 nm)/Co₂₀Fe₆₀B₂₀(0.9 nm)/Ta(0.07 nm)/MgO(2 nm)/Ta(5 nm) multilayer stack (layer thickness with an accuracy higher than 0.01 nm) is deposited by d.c./radio-frequency magnetron sputtering utilizing a Singulus Rotaris machine with a base strain of three × 10⁻⁸ mbar. The hexagonal geometric confinement is patterned by electron-beam lithography adopted by argon-ion etching.

Interfacial Dzyaloshinskii–Moriya interplay^54,55 is especially induced on the Ta/Co₂₀Fe₆₀B₂₀ interface; the Co₂₀Fe₆₀B₂₀/MgO interface causes perpendicular magnetic anisotropy. The Ta(0.08) dusting layer is used to not solely steadiness the perpendicular magnetic anisotropy and Dzyaloshinskii–Moriya interplay^7,56 to host skyrmions but in addition to optimize the power panorama for skyrmion lattice formation and dynamics. We offer the OOP hysteresis loop in Supplementary Be aware 4 with Supplementary Fig. 4. The non-trivial topology of the noticed bubbles is experimentally confirmed by spin–orbit-torque-driven skyrmion movement and supported by micromagnetic simulations^7,31,33.

Moreover, the skyrmion interplay potential has been demonstrated to be purely repulsive within the studied materials stack²⁹; specifically, we be aware that it’s of a type wherein the KTHNY transitions are predicted to happen²⁸. In distinction, different supplies also can result in enticing skyrmion interplay potentials^57,58,59. Though basically, the existence of a Magnus drive is an extra distinctive property of skyrmions, the relative energy of the impact is, nevertheless, small in our system. It’s roughly proportional to the ratio of the domain-wall width (10–20 nm) to the skyrmion core diameter (few micrometres)^33,60 and, subsequently, negligible in our system, leading to a most skyrmion Corridor angle of some levels⁶¹. Moreover, the hopping-like skyrmion dynamics within the non-flat-energy panorama is dominated by pinning forces^7,33,34, suppressing the Magnus impact. Small skyrmions or (close-to) pinning-free diffusion techniques can, nevertheless, result in a large Magnus drive, which is of particular curiosity as topological defect dynamics in techniques with a Magnus drive and odd elasticity⁵³ is an open query.

Skyrmion stabilization and imaging

A commercially obtainable evico magnetics Kerr microscope is used to determine magnetic distinction with a decision of 300 nm in house and 62.5 ms in time, utilizing a blue light-emitting diode mild supply and a charge-coupled system digital camera with a subject of view of 200 × 150 µm². Magnetic fields might be utilized in each in-plane (IP) and OOP instructions. The alignment of coils is optimized by aligning the shift of OOP hysteresis loops with and with out an IP subject. The OOP magnet is customized made to permit subject management with sub-microtesla precision. The pattern itself is positioned onto a Peltier ingredient straight on prime of the coil for temperature management. The temperature is stored fixed at 333.5 Okay and monitored by a Pt100 sensor straight subsequent to the pattern to make sure temperature stability higher than 0.1 Okay.

Skyrmions are nucleated by making use of an IP-field pulse, which saturates the pattern within the IP path at a relentless OOP subject. The ensuing skyrmion lattice is equilibrated by an oscillating OOP magnetic subject at 100 Hz with amplitudes as much as 60 µT along with the fixed OOP-field offset earlier than measuring the obtained configuration.

We’ve got direct and exact management over the skyrmion dimension by way of the utilized OOP magnetic subject^33,39,40. The sizes of the person skyrmions are detected by a machine learning-based pixel-wise classification⁶². Moreover, we are able to repeatedly tune the skyrmion diffusivity by sinusoidal OOP-field oscillation along with the offset subject^9,34. Within the melting process introduced in Figs. 1–4, we improve the exterior OOP-field offset each 62.5 s (comparable to 1,000 frames) in steps of 6 µT. Throughout every interval of 62.5 s, the sphere is stored fixed to acquire cheap statistics for each subject worth. In a relentless exterior subject, the skyrmion ensemble is in equilibrium. Within the picosecond–nanosecond timescale of the magnetization dynamics, the intrinsic precessional dynamics of magnetization is all the time damped out on the timescales we examine, whereas the thermally activated diffusive skyrmion dynamics takes place on the millisecond–second timescale that we examine. The skyrmions additionally react quick (≲milliseconds)³⁴ to subject modifications and the steps of 6 µT are very small—the system ordering responds to dimension modifications sometimes sooner than 1 s, aside from fluctuations near the noticed transitions. Due to this fact, we are able to deal with the entire melting course of to be in quasi-equilibrium. Accordingly, we’ve got chosen the time intervals and subject steps to protect quasi-equilibrium and guarantee steady measurement situations throughout the entire melting protocol. Nevertheless, oscillating fields—that are solely used to initialize the skyrmion lattice order right here, however which can be used to destabilize the lattice order (Supplementary Fig. 4)—are anticipated to introduce non-equilibrium properties as they completely drive skyrmion dimension modifications.

Quantification of 2D order

The translational order is quantified by the translational correlation operate

averaging the hyperlink between two particle positions r_j and r_okay with respect to a reciprocal lattice vector G over the gap r. The orientational correlation operate

quantifies the orientational order primarily based on the native orientational order parameter

of a particle at place r_j with N nearest neighbours labelled okay = 1 to N. θ_jk denotes the angle of the connecting vector r_okay–r_j with respect to an arbitrary axis¹⁹.

In a 2D stable, G_T(r) decays algebraically as (propto {r}^{-{eta }_{{rm{T}}}}), signalling QLRO. When the exponent η_T reaches its crucial worth of 1/3, an exponential decay ∝exp(–r/ξ_T) with correlation size ξ_T units in; translational QLRO has disappeared¹⁹. In distinction, G₆(r) is fixed in a stable, however reveals an algebraic decay (propto {r}^{-{eta }_{6}}) when the translational order vanishes if orientational QLRO persists. Therefore, orientational order continues to be current in what’s known as the hexatic part, which is exclusive to 2D techniques^19,20. When η₆ reaches its crucial worth of 1/4, G₆(r) turns into exponential (∝exp(–r/ξ₆)) with correlation size ξ₆, leading to an isotropic liquid. On the transition from exponential to algebraic decay, the respective correlation lengths of each correlation capabilities diverge, inflicting the exponential time period to fade within the crucial (QLRO) phases¹⁹.

Just like the correlation capabilities in house, we calculate the orientational time correlation as

as a operate of time delay τ, which reveals the dynamics for each subject interval²³. The angle brackets characterize the common over all particles and all beginning instances t inside the interval of fixed subject. Principle suggests a relentless behaviour of G₆(τ) within the stable part, algebraic decay within the hexatic part and exponential decay within the liquid part²³. The hexatic and liquid phases are separated by a crucial exponent η_τ = 1/8. Our outcomes proven in Prolonged Knowledge Fig. 1 match the speculation qualitatively properly. The anticipated crucial worth of η_τ = 1/8 for an infinite system is, nevertheless, too giant to match our state of affairs. We will attribute the improved time correlation in our experiment to the results of confinement and non-flat-energy panorama.

Any lattice web site with the variety of nearest neighbours totally different from N = 6 is a topological defect. A dislocation is a pair of defects with reverse topological cost: one N = 5 and one N = 7 defect. In a stable, just a few dislocation pairs happen, that are tightly sure and of reverse orientation. The orientation of a dislocation is specified by the Burgers vector. The Burgers vector is decided because the lacking vector when encircling a dislocation counterclockwise with a set of lattice vectors, which might yield a closed path in an ideal lattice. On the transition level separating the stable from the hexatic part, the dislocation pairs unbind and proliferate. This formation of remoted free dislocations inflicting the lack of translational QLRO is measurable macroscopically as a vanishing shear modulus µ. On the transition level separating the hexatic from the liquid part, the dislocations ultimately unbind and proliferate into two remoted disclinations^18,19.

Knowledge evaluation

For the detection of skyrmions from the greyscale video and linking them to trajectories, we use the trackpy package deal⁴⁶ in Python. The obtained positions are used for each skyrmion to find out the native order parameter ψ₆ and its nearest neighbours making use of a Voronoi tessellation, which robotically determines the lattice defects. Skyrmions on the fringe of the system are uncared for for the evaluation of ψ₆ and lattice defects as their place on the edge produces artefacts within the Voronoi tessellation⁶³.

For all skyrmions that aren’t situated on the fringe of the system, we decide a worth for G_T and G₆ with respect to all different skyrmions. We bin the values of the respective correlation and carry out a median in each bin, ensuing within the distance-dependent correlation capabilities G_T(r) and G₆(r) (Fig. 1d–i). The dedication of the correlation operate works for single-frame photographs; nevertheless, we common the correlation capabilities of ten consecutive frames (over 0.625 s) to cut back noise considerably. Due to this fact, all of the plots and matches of the correlation capabilities are carried out on the averaged information. To find out the decay of the translational correlation operate, we match G_T(r) with a power-law decay (propto {left(r{/r}_{0}proper)}^{-{eta }_{{rm{T}}}}) as a operate of distance r in items of the skyrmion lattice fixed r₀. We use the preliminary power-law match to find out if the system is translationally ordered (η_T beneath a crucial worth of 1/3) or not (η_T > 1/3). In dysfunction, nevertheless, the exponent η_T is not properly outlined for the reason that decay of G_T is now solely exponential. Due to this fact, for the disordered circumstances, we match the exponential (∝exp(–r/ξ_T)) as a substitute of the facility regulation. Because the exponential time period is technically additionally current within the ordered crucial regime, we additionally match an exponential for the occurrences of η_T < 1/3, however as an extra issue to the facility regulation. We use this extra issue within the match as affirmation that the correlation size ξ_T turns into infinite within the ordered regime. For the orientational correlation operate G₆, we proceed analogously to find out the exponent η₆ in addition to the correlation size ξ₆. Nevertheless, the orientational correlation has a special crucial worth of η₆ = 1/4, which we use to find out whether or not the system is orientationally ordered (η₆ < 1/4) or not (η₆ = 1/4) and whether or not we match the exponential as an extra issue to or as a substitute of the facility regulation, respectively.

In our system, we lack the likelihood to use stress forces to straight measure the elastic moduli. As an alternative, we analyse the native deformations of the lattice in actual house to estimate the shear modulus μ (refs. ^41,42,64). Because the reference lattice, we use a central skyrmion with six completely organized nearest neighbours at positions ({{bf{X}}}_{{rm{i}}}^{{rm{ref}}}) with common lattice spacing. To this reference, we match an area deformation tensor ({{{delta }}}) for each skyrmion and its neighbours within the experimental lattice, such that the squared distance

between experimental lattice positions ({{bf{X}}}_{{rm{i}}}^{exp }) and the tweaked reference is minimized. To extract the shear part, we decompose ({delta }={epsilon}+{{{R}_{alpha }}}) to a symmetric pressure tensor ({epsilon }) and an anti-symmetric rotation ({{{R}_{alpha }}}) by an angle α. The diagonal parts of ({{epsilon }}) describe the pressure alongside x and y, whereas the off-diagonal ingredient is the shear part. In case of linear elasticity, a shear deformation is related to a shear power ({E}_{{rm{shear}}}=frac{1}{2}{(2{epsilon }_{{xy}})}^{2}Vcdot mu), the place V denotes the quantity over which the shearing takes place (space spanned by the closest neighbours in our case). Assuming a Boltzmann distribution P(E) ∝ exp(–E/okay_BT) of the shear power at temperature T, we match µ because the slope of

when calculating a histogram over the sq. bracket as a measure of the logarithm of the shear power distribution. The process requires the belief of linear elasticity, which turns into much less relevant in a much less dense system, particularly in liquid. Due to this fact, the shear modulus doesn’t vanish fully throughout melting. Additionally, the distribution of shear energies related to the decided deformations will not be completely Boltzmann like, as already noticed for colloid techniques⁴². Because the dependence will not be completely linear, we carry out a set of matches over totally different ranges and use the usual deviation as error of the imply worth.

The dedication of topological defects follows straight from the Voronoi tessellation used for calculating the native ordering. Each skyrmion with quite a lot of nearest neighbours N totally different from 6, which isn’t situated on the fringe of the system, is recognized as a lattice defect. Since defects within the stable and hexatic regimes nearly solely happen pairwise, figuring out these pairs as dislocations is trivial. Nevertheless, transitioning to a liquid, complicated clusters of defects evolve. The complicated look, together with the interactions between defect clusters, makes the identification of the formal connection between defects unimaginable. To analyse the additional evolution of defects, we set up a simplified strategy of figuring out pairs of defects. To each 5-defect i, we assign precisely one 7-defect j and take the gap between the defects as d_ij. To determine distinctive pair connections, we decrease the entire sq. distance

related to all potential connections ij utilizing the Hungarian technique⁶⁵. We determine a decided defect pair as a dislocation if the corresponding d_ij is a nearest-neighbour connection; in any other case, we determine the 2 linked defects as two disclinations. To review the dislocation dynamics, we hold solely the centre of mass of all of the recognized dislocations and hyperlink them to trajectories with trackpy⁴⁶. Be aware that the 5/7-defect pair matching in addition to the linking of dislocation trajectories work typically properly till deep within the hexatic regime as defects all the time happen in pairs and don’t absolutely dissociate. On the onset of the liquid regime, nevertheless, disclinations and sophisticated defect clusters begin occurring and make the formally appropriate matching and evolution of defect pairs inaccessible. With our strategy being purely primarily based on distance minimization, we, subsequently, anticipate a potential systematic error within the quantification of defect dynamics from the onset of the liquid regime, whereas the elevated dynamics as a direct consequence of defect fluctuations, rearrangements and dissociation continues to be mirrored.

To guage the diffusion coefficient of the skyrmions at totally different instances of the measurement, we decide the MSD as

by calculating the sq. distance of skyrmion place r at time t relative to the place on the time of preliminary prevalence t₀ and take the common over all skyrmions. The MSD is additional associated to the dimensionality d of the system (right here d = 2) and D over t within the case of regular diffusion^7,31,34. Since we need to decide D at any time t₀ with dependable statistics, we think about all trajectories current in a 10-s time window round t₀ and use the time of first prevalence as t₀. We then match the primary 1 s of the ensuing MSD to find out D. For the dislocations, we proceed analogously however use all of the trajectories occurring in a time window of 31 s round t₀ to suit D for statistical causes as a result of there are considerably fewer dislocations than skyrmions.

To correlate defect occurrences of time, for each skyrmion n at time t, we affiliate a variable

to be correlated. We calculate the Pearson correlation

for each pair of instances t₁ and t₂ by averaging over all skyrmions n (refs. ^50,51,52). Right here μ and σ characterize the imply and normal deviation of u on the respective time. The corresponding two-time correlation map is proven in Fig. 5a. The correlation decreases over time in the course of the melting and one can observe extra fast modifications within the time areas of the beforehand decided transitions (Fig. 5a, dashed pink traces).

By averaging over equal time delays τ = t₂ – t₁, we convert the two-time correlation map to a one-time correlation operate g₂(τ) for each interval of fixed magnetic subject (as the sphere is modified stepwise each 62.5 s). Though g₂(τ) (Fig. 5b) stays nearly fixed for the fields representing the stable regime, it decays notably and more and more quickly all through the melting course of. Because the decay of g₂(τ) is straight associated to the dynamics of the underlying characteristic^51,52—that’s, the topological defects on this case—this corroborates that the defect dynamics retains growing all through the melting process.