CIVIL ENGINEERING 365 ALL ABOUT CIVIL ENGINEERING


Optimal dynamical control of a particle motion includes several tasks, such as acceleration, braking, and trapping. In the case of nanoparticles, ions, or atoms, the trapping problem becomes more demanding than the others, except trapping of charged particles which is relatively easy with the use of Pauli trap1,2. In the early 90-ties, it was realized that light–atom interaction allows trapping of neutral objects—cesium and sodium atoms in particular3,4. In the case of optical trapping of neutral objects, the light does two jobs: (i) it attracts the particles towards the anti-nodal points of maximum intensity of the optical lattice with the spatial period of the order of optical wavelength, and (ii) the light additionally cools down the atoms. The invention of optical tweezers in 1986 by Arthur Ashkin was a triumph for the manipulation of microparticles with laser light5. Although trapping of various particles is widely discussed in the literature, the problem of trapping of localized excited modes, especially of topological solitons (skyrmions) has not been studied yet.

The concept of skyrmion traces back to the paper of Skyrme6, and to the fundamental paper of Belavin and Polyakov7. It is now well known that skyrmion has a topological character. In particular, invariance of the topological action of the field theory, ({S}_{mathrm{top}}left({bf{n}}right)=frac{itheta }{4pi }int! {mathrm{d}}{x}_{1}{mathrm{d}}{x}_{2}{bf{n}}cdot left({partial }_{1}{bf{n}}times {partial }_{2}{bf{n}}right)), with respect to the infinitesimal transformation ({bf{n}}left({bf{x}}right)to {bf{n}}left({bf{x}}right)+{epsilon }^{a}left({bf{x}}right){R}^{a}{bf{n}}left({bf{x}}right)), (where ϵa is infinitesimal parameter and Ra stands for generators of the O(3) group) defines the specific texture of the vector field ({bf{n}}left(xright))8,9,10. The set of different textures of ({bf{n}}left(xright)), obtained from each other by means of the continuous deformation, has the same invariant topological action and the related conserved topological charge (W=frac{1}{itheta }{S}_{mathrm{top}}left({bf{n}}right)). Thus, one could argue that the topological soliton (skyrmion) is a robust object, stable with respect to small perturbations. Apart from this, skyrmions possess dual field-particle properties8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40. Skyrmions are highly mobile objects. There are several precise recipes on how to drive a skyrmion—either by a spin-polarized electron current or with a magnonic spin current that exerts a magnon pressure on the skyrmion surface. In the recent work41, an alternative mechanism of skyrmion drag was proposed, which is based on a combination of uniform temperature profile and non-uniform electric field. Nevertheless, a vital question that arises is whether the particle nature of skyrmions facilitates their trapping. In what follows, we explore trapping of a skyrmion in the laser field ezEls(x, y, z, t) (with ez being the unit polarization vector of the electric field) and the external electric field E0 = (0, 0, Ez0). There exist several methods to manipulate the polarization of the laser beam. Through these methods, the polarization of the electric field can be switched to the desired direction. For example, one can utilize ultrafast time-dependent polarization rotation in a magnetophotonic crystal42. The colloidal microspheres also can produce dominant Ez component43.

Skyrmions emerge in materials (e.g., in chiral single-phase multiferroics44,45,46) with a sizeable magnetoelectric (ME) coupling term, Eme = −E P, where P = cE[(m ) mm( m)] is the net ferroelectric polarization, with m denoting the unit vector along the magnetization and cE is the magnetoelectric coupling constant. In chiral multiferroics, the coupling of the external electric field with the ferroelectric polarization mimics the Dzyaloshinskii–Moriya (DM) term and leads to the noncolinear topological magnetic order. The mechanism of trapping of a skyrmion relies on the interaction between the electric component of the laser field and the ferroelectric polarization of the skyrmion texture. As for the specific materials, we focus on two types of materials: spin-driven single-phase multiferroics and Yttrium Iron Garnet (YIG)11,12,13,14. In particular, we present in addition to YIG, results for the multiferroic material Cu2OSeO3, which supports skyrmions. As detailed below, the emergence of the finite (but small) electric polarization due to non-collinearity of the spin allows for the movement of the skyrmions with external electric fields. YIG and single-phase multiferroic are described by free energies densities:

$$begin{array}{ccc}&&{F}_{mathrm{MF}}=int [{A}_{{rm{ex}}}{left({boldsymbol{nabla }}{bf{m}}right)}^{2}-{mu }_{0}{M}_{mathrm{s}}{m}_{z}{H}_{z}+{epsilon }_{{rm{DMI}}}+{E}_{mathrm{me}}]{mathrm{d}}{bf{r}},\ &&{F}_{mathrm{YIG}}=int [{A}_{{rm{ex}}}{left({boldsymbol{nabla }}{bf{m}}right)}^{2}-{mu }_{0}{M}_{mathrm{s}}{m}_{z}{H}_{z}+{E}_{mathrm{me}}]{mathrm{d}}{bf{r}}.end{array}$$

(1)

Here, M = Msm, where Ms is the saturation magnetization, Aex is the exchange stiffness, and Hz is the external magnetic field applied along the z-direction. The free energy of the single-phase multiferroic Cu2OSeO3 has the bulk-related DM interaction term ϵDMI = Dbm ( × m), where Db is the DMI constant. The effective magnetic field acting on the magnetization follows from the functional derivative of the free energy functional ({bf{H}}=-frac{1}{{mu }_{0}{M}_{mathrm{s}}}frac{delta {F}_{mathrm{MF/YIG}}}{delta {bf{m}}}). The interaction energy Eme = −E P between the external electric fields E and the spin-driven polarization P enters Eq. (1) as a term, which is linear in P. We note that for spin-driven multiferroics the spin-induced P is quite small (we recall that P = cE[(m ) mm ( m)], where cE is related to the spin–orbit coupling and the spatial variations in m are smooth on an atomic scale). Thus, higher-order terms (P)n and the spatial variations (P)n, which both account for the energy density of ferroelectric polarization, are negligible and therefore do not appear in the Eq. (1) above.

The laser manipulated skyrmion dynamics is governed by the stochastic Landau–Lifshitz–Gilbert (LLG) equation47,48, supplemented by the ME term

$$frac{partial {bf{M}}}{partial t}=-gamma {bf{M}}times left({{bf{H}}}_{{rm{eff}}}+{{bf{h}}}_{{l}}-frac{1}{{mu }_{0}{M}_{s}}frac{delta {E}_{mathrm{me}}}{delta {bf{m}}}right)+frac{alpha }{{M}_{mathrm{s}}}{bf{M}}times frac{partial {bf{M}}}{partial t},$$

(2)

where γ is the gyromagnetic ratio and α is the phenomenological Gilbert damping constant. The effective field Heff for the single-phase multiferroic consists of the exchange field, DM field, and of the applied external magnetic field, ({{bf{H}}}_{{rm{eff}}}=frac{2{A}_{mathrm{ex}}}{{mu }_{0}{M}_{mathrm{s}}}{nabla }^{2}{bf{m}}-frac{2{D}_{mathrm{b}}}{{mu }_{0}{M}_{mathrm{s}}}nabla times {bf{m}}+{H}_{z}{bf{z}}). The temperature in the LLG equation, is introduced through the correlation function of the thermal random magnetic field hl, (langle {h}_{l,p}(t,{bf{r}}){h}_{l,q}(t^{prime} ,{bf{r}}^{prime} )rangle =frac{2{k}_{mathrm{B}}{T}_{mathrm{sim}}alpha }{gamma {mu }_{0}{M}_{mathrm{s}}V}{delta }_{mathrm{pq}}delta ({bf{r}}-{bf{r}}^{prime} )delta (t-t^{prime} )), where p, q = x, y, z, kB is the Boltzmann constant, and V is the volume of the single cell, used in numerical simulations. The value of the temperature Tsim, we determine from the heat equation (see “Methods” section). We note that the physical temperature and the simulation temperature are related through the equation48({T}_{{rm{sim}}}=T{a}_{{rm{sim}}}/{a}_{mathrm{L}}), where aL is the lattice constant and asim is the cell length in simulation. Therefore, the physical temperature T = 50 K corresponds to the simulation temperature of ({T}_{{rm{sim}}}approx 100) K.

The z component, Ez0, of the external electric field stabilizes the skyrmion structure. Due to the Gaussian profile of the laser field, Els(x, y, z, t) has the maximum (denoted as E0) in the center of laser spot. The total z component of the electric field, Ez = Ez0 + Els(x, y, z, t), is not homogeneous in the (x, y) plane. Depending on the sign of the oscillating laser field Els(x, y, z, t), the total field Ez can be either negative or positive. We note that for an ultrashort laser pulse, the pulse compressor allows control of the spectral phase (phi (omega ),,{E}_{mathrm{ls}}(x,y,z,omega )=sqrt{| {E}_{mathrm{ls}}{| }^{2}}exp (-iphi (omega ))), where (phi (omega )=-frac{omega }{c}n(omega )d), n(ω) is the index of refraction and d is the film thickness49. In what follows, we consider both negative E0 < 0 and positive E0 > 0 values of the field. We note that modern laser technologies allow generation of ultrashort single Els(x, y, z, t) = Els(x, y, z)fscp(t) and half cycle Els(x, y, z, t) = Els(x, y, z)fhcp(t) pulses50. The temporal profiles of laser pulses are defined as follows: ({f}_{mathrm{scp}}(t)=t/{tau }_{d}exp (-{t}^{2}/{tau }_{d}^{2})), ({f}_{mathrm{hcp}}(t)=t/{tau }_{0}left[right.exp (-{t}^{2}/2{tau }_{0}^{2})-frac{1}{{b}^{{}^{2}}}exp (-{t}^{2}/b{tau }_{0})left]right.,,t>0). The ultrashort single pulse has both positive and negative Els(x, y, z, t), while the negative field part of fhcp(t) is too small. Therefore, for half-cycle pulse Els(x, y, z, t) can be viewed as positively defined.

Before presenting the numerical results, we explain the trapping mechanism. The electric field Ez is inhomogeneous only in the x direction. The functional derivative of the ME term with respect to the magnetic moment reads: (-frac{1}{{mu }_{0}{M}_{mathrm{s}}}frac{delta {E}_{mathrm{me}}({E}_{z})}{delta {bf{m}}}=frac{{c}_{mathrm{E}}}{{mu }_{0}{M}_{mathrm{s}}}[{partial }_{x}{E}_{z}({m}_{z}{{bf{e}}}_{x}-{m}_{x}{{bf{e}}}_{z})+{sum }_{j}2{E}_{z}(-{partial }_{j}{m}_{j}{{bf{e}}}_{z}+{partial }_{j}{m}_{z}{{bf{e}}}_{j})].) Here j = x, y. We focus on the first term fueled by the non-uniform electric field ∂xEz, while the second term corresponds to the effective DM interaction with a strength tunable by a constant electric field41. For tweezing, we suggest using the scanning near-field optical microscopy (SNOM) and advanced nanofabrication procedures. These two methods permit to obtain spots of light 10–20 nm in size; see recent review and references therein51. Contribution of the non-uniform electric field will be presented in the form of inhomogeneous electric torque (IET): (-gamma {bf{m}}times left(-frac{delta {E}_{mathrm{me}}({partial }_{x}{E}_{z})}{{mu }_{0}{M}_{mathrm{s}}delta {bf{m}}}right)=-frac{gamma {c}_{mathrm{E}}{partial }_{x}{E}_{z}}{{mu }_{0}{M}_{mathrm{s}}}{bf{m}}times ({bf{m}}times {{bf{p}}}_{{bf{E}}}).) The vector pE = x × ez is set by ez, which points into the direction of electric field. Obviously, the expression of IET is identical to the standard spin transfer torque—cjm × (m × p), because pE in IET mimics the spin polarization direction p. However, while cj depends on the electric current density, the amplitude of the IET depends on the gradient of the electric field ∂xEz and on the ME coupling strength cE. In the case of Gaussian laser beam (for more details, we refer to the Supplementary Note 2), the coefficient in the expression for IET, (c=frac{gamma {c}_{E}{partial }_{r}{E}_{z}}{{mu }_{0}{M}_{s}}), is determined by the gradient of electric field, while pE = er × z, where ({{bf{e}}}_{r}=({{bf{e}}}_{x}x+{{bf{e}}}_{y}y)/sqrt{{x}^{2}+{y}^{2}}) is the unit vector. The underlying mechanism of the skyrmion tweezer is as follows: depending on the direction of the laser field, the IET torque is either centripetal (drives the skyrmion to the center of the beam) or counter-centripetal (drives the skyrmion out of the beam center).

While the energy is supplied through the laser, skyrmion releases energy to the bulk and SNOM shield. Thus, for the comprehensible study of the skyrmion temperature, one needs to solve the heat equation with source and sink terms included. The maximal temperature of the skyrmion texture can be estimated analytically (see “Methods” section). In our case Tmax = 50 K and therefore the skyrmion is stable.



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