arXiv:1911.03986v1 [cond-mat.mes-hall] 10 Nov 2019 krinsymo neato n hra os yields noise thermal and is repulsive interaction It force, skyrmion-skyrmion Magnus between skyrmions. interplay the of the that description in shown [19] point-particle Ref. of in demonstrated context is films interacting thin systematic of in The skyrmions simulations [22]. dynamics Ref. molecular in Langevin microscopic derived a is from model motion particles continuum of rigid-point equation as and with The skyrmions static interacting for [21]. the skyrmions, disorder examine quenched of simu- random to phases particle-based used collective is The driven skyrmions of particles. lations represented point-like using are 20] rigid skyrmions [19, as the Refs. where in derived approach, equation is Thiele’s The skyrmions regimes. for certain motion in a of described object be end, particle-like can this a materials To as magnetic a manipulation. in find- of skyrmion require single mechanisms and new tasks of important ing remain position their of motion the of manipulation the for cur- skyrmions. heat used and are [15–17] [18] currents rents magnetic electric The applied weak [10–14]. fields and under symmetry coupling inversion spin-orbit films broken strong thin with in materials textures bulk can spin and particle-like Skyrmions a as [9]. compact considered information of be storing development for on cells focused memory is vector study magnetization Their nontrivial the field. of topologically formed are structures Skyrmions nonlinear [5–8]. gases tum odne atrpyis[–](o ntnefrtechi- the for magnets instance cubic (for the ral [1–4] in physics important matter become it condensed later years However, physics. † ∗ lcrncades [email protected] address: Electronic lcrncades [email protected] address: Electronic ioa-rf n olcieisaiiiso krin i skyrmions of instabilities collective and Dipolar-drift h ak fmnpltn krin n controlling and skyrmions manipulating of tasks The h ocp fsymosapae ntenuclear the in appeared skyrmions of concept The ewrs rf oin aleet krin,eeti di electric skyrmions, effect, Hall motion, drift Keywords: 12.39.Dc 61.82.Ms, hydrod described. numbers: of PACS are type instability new of regimes the Three shows int fields motion. of drift magnetic gas dr and the the of electric to Hydrodynamics form leads effect. direction fie extern Hall same electric the have magneto-dipolar additional of gradients of their derivative Application but the field, moment. the by dipole to fixed the perpendicular is mag of plane gradient circulation pe the fixed of of a in quency gradients with circulates the field skyrmion in magnetic where studied nonuniform structure is The motion magnetic fields. skyrmion spiral The the moment. where insulators, tiferroic aut fpyis oooo ocwSaeUiest,Mos University, State Moscow Lomonosov physics, of Faculty krin yaisi h prxmto fpoint-particle of approximation the in dynamics Skyrmions MnSi n antcfilsi hrlmgei insulator magnetic chiral a in fields magnetic and 1) nldn h edo quan- of field the including [1]), ae .Andreev A. Pavel Dtd oebr1,2019) 12, November (Dated: ∗ n aiaI.Trukhanova Iv. Mariya and ain ewe krin yaisaddeeti be- corre- dielectric the and of dynamics development skyrmions in interest between is lations There [29–32]. a ecnrle sn antcfil rdet nthe in gradients field magnetic insulator using multiferroic controlled be can 26]. [25, law Faraday’s source the a through as field skyrmion electric moving additional of the the an seeing acquires feel force result, skyrmion Electrons transverse the the motion. In drift drift they electron as corresponding texture. uniform deviate spin and the Electrons flux skyrmion of of the effect. source through a torque pass always as transfer skyrmion [15]. is spin the trajectory the electron see its to the in leads magnetization of This local spin the the to rule parallel Hund’s that the such through skyrmion coupling, the with of electrons texture conduction spin of the interaction the is skyrmions [24]. torque using spin by Hall observed spin ob- is current- current-induced experimental effect a Hall the The skyrmion the to 23]. of servation [17, leads motion mechanism skyrmion driven torque transfer The spin relaxation. non-equilibrium during regimes different sdmntae nRf 2] oxsec feeti and electric of the Coexistence yields gradient fields [28]. field magnetic electric Ref. in the control demonstrated using to is opportunity motion, The skyrmion [28]. the orthogo- gradient of is field velocity the skyrmion to Hall The the nal average to skyrmions. the leads of skyrmion of the motion direction of the moment to interesting dipole induced parallel field, very electric pointed spatial-dependent a is the is that which shown skyrmions is field It of magnetic task. control The static field frequencies. the lower electric at in rotating lattice collectively is skyrmion gradient The of represented demonstration is [27]. experimental mechanism gradient-driven The field magnetic in gradient. collectively field rotate the skyrmions that demonstrated tally oe,mgei insulators magnetic poles, krin r icvrdi h utfrocinsulators multiferroic the in discovered are Skyrmions nadto oteeeti edgain,teskyrmions the gradient, field electric the to addition In of dynamics the control to way standard The sacmaidb nt lcrcdipole electric finite by accompanied is dwihpredclrt h magnetic the to perpendicular which ld f oino krin eutn nthe in resulting skyrmions of motion ift rcigsymosi rse nonuni- crossed in skyrmions eracting lmgei edadtemagnitude the and field magnetic al ieto fmgei ed h fre- The field. magnetic of direction krin scniee ntemul- the in considered is skyrmions o,RsinFdrto,119991. Federation, Russian cow, nmclisaiiycue ythe by caused instability ynamical iuelast h prlmotion, spiral the to leads nitude pniua antcadelectric and magnetic rpendicular rse ouiomelectric nonuniform crossed n Cu † 2 E OSeO eff × 3 B 2] ti experimen- is It [27]. drift. 2 haviors in the multiferroic materials (MFMs). The spiral frequency. The third projection of the velocity vz is a magnetic structure in insulating materials is accompa- constant in this regime. It can chosen to be equal to zero −ıΩt nied by finite electric dipole moment, which is induced in vz = 0. Hence, there is the circle motion vξ = v0e of the MFM, where the main role plays the pd-hybridization the dipolar skyrmion in x y plane with the frequency − mechanism. The unit cell of the MFM Cu2OSeO3 con- fixed by the derivative of the external magnetic field and tains 16 copper atoms and each of them is surrounded the amplitude of the dipole moment, where v0 is the ini- by several oxygen atoms. The valence electron of the tial velocity. The center of the circle orbit is motionless or oxygen atom is the p-electron, which has different orbital it can move in the z-direction creating the spiral motion symmetry from the d-electron of the copper atom. The of dipolar skyrmion. mixing of these two orbital states leads to a redistribution Include the nonuniform part of electric field directed of the electron cloud. It creates the microscopic polar- parallel to y-axis: E = E0(z)ey, which is a function ization on the copper-oxygen bond [15]. The dielectric of coordinate z with a constant value of the derivative properties of a chiral magnetic insulator Cu2OSeO3 are ∂E0(z)/∂z γ. The Newton equation (1) reduces to ≡ investigated under various magnitudes and directions of the following formv ˙ξ + ıΩvξ = ıd0γ/m. General solu- magnetic field [29] and [30], where authors found that tion appears as a superposition of solution with the zero each skyrmion carries the electric dipole moment. right-hand side and particular integral of the equation The induced dipole moment associated with a with nonzero right-hand side (vξ = d0γ/mΩ). Hence, we given skyrmionic spin configuration is found in Refs. obtain solution under simultaneous motion of the electric [29, 30]. The induced electric dipole moment can and magnetic fields be presented analytically via magnetization d = y z x z x y −ıΩt cγ λ(m m ,m m ,m m ), where λ is a measure of the vξ = v0e + . (2) β dipole moment or the magnetoelectric coupling constant [28], and m is the local normalized magnetic moment. The second term in this expression gives nonzero average This dipole moment is induced by the single unit cell. value at the averaging over time interval τ Ω−1. The The polarization of the skyrmion crystal is propor- average value of found velocity v is directed≫ parallel to h i tional to the dipole moment of single skyrmion P = ndl, the x direction v = vDex with vD = cγ/β (x-direction where l is the direction of the dipole moment and d = is chosen by properh i choosing of the time reference). λQDNskh/al, where QD is the dipolar charge, a, h are Averaging is meaningful if rotation frequency is rel- the linear dimension of the one unit cell and film thick- atively high. It corresponds to relatively large gradi- ness respectively. As a result skyrmions sensitive to the ents of the magnetic field. Overwise, there are two other gradient of electric field and effect the Hall motion [28]. regimes. Either the drift of the orbit center is small in We can use a particle-like description of skyrmion and compare with the circular motion or there is large drift assume that the skyrmion has a rigid internal structure during time t Ω−1 (some regimes are illustrated in Fig. in the regimes when the deformation of their internal 1). Obviously,≪ the drift velocity should be smaller then structure is small [22]. The overlap between different the speed of light vD c. It provides an estimation for skyrmions is assumed to be small. the value of gradients≪ of magnetic field β γ. Hence, Consider a single skyrmion motion in nonuniform elec- the magnetic field gradient should be larger≫ in compare tric and magnetic field. Single skyrmion dynamics can be with the electric field gradient. Let us point out that we described by the classic Newton equation with effective use the CGS units, where the electric and magnetic fields mass of skyrmion have same physical dimensions. The small magnetic field gradient regime can be ad- 1 mv˙ = (d )E + [v, (d )B]. (1) dressed via another particular integral of the Newton · ∇ c · ∇ equation vξ = ıd0γt/m demonstrating the acceleration of dipole. The imaginary value of v shows that this ve- Action of the electromagnetic field on the skyrmion which ξ locity is directed in y-direction (in the direction of the has the electric dipole moment d in accordance with data electric field). described above. In this model, the center of mass motion High magnetic field gradients provide drift of skyrmion of dipolar skyrmionis affected by the nonuniform electric in corresponding direction: v =2c[∂ E, B]/∂ B2 which and magnetic fields. D z z is perpendicular to the directions of the electric and mag- Consider a regime of fixed electric dipole direction dur- netic fields. Restore the contribution of the dipole mo- ing the skyrmion motion and choose z-axes along the ment in the drift velocity dipole moment direction d = d0ez. We direct the mag- netic field parallel to the dipole direction: B = B0ez. [(d )E, B] vD =2c · ∇ . (3) Next, assume that the magnitude of the magnetic field (d )B2 has constant derivative in the z direction ∂B0/∂z β. · ∇ If the electric field is equal to zero, equation (1) gives≡ It gives us the velocity of dipolar drift of skyrmion in the following equation for the velocity of the skyrmion crossed nonuniform electric and magnetic fields. motion in the magnetic fieldv ˙ξ + ıΩvξ = 0, where The dipolar drift motion in the magnetic field provides, vξ = vx + ıvy, and Ω= d0β/mc is the dipolar cyclotron let us call it, the magneto-dipolar Hall effect. It reveals 3

hydrodynamic equations allows to study waves and insta- bilities of skyrmions in crossed nonuniform electric and magnetic fields. First focus on instabilities of ideal skyrmion gas. Con- sider the electric field gradient affecting the collective be- havior of skyrmions. The external electric field creates nonzero equilibrium force field. Choose the electric field to be directed in y-direction with the gradient directed in z-direction: E = E0(z)ey, which is a nonuniform part of full external electric field Eext = Ebez + E0ey, where Eb E0. Hence, it is necessary to create an equilib- FIG. 1: Three regimes of trajectories are demonstrated. Thin rium≫ pressure gradient caused by the temperature gra- continuous blue circle shows regime of the zero electric field dient to create a macroscopically motionless equilibrium gradient. It corresponds to rotation of dipolar skyrmions with fixed center of the orbit. Second regime presented by dashed with uniform concentration. Find the following equilib- v . v v rium condition p0 = n0d0(∂E0/∂z), where n0 is the red line is calculated for D = 0 2 0, where 0 is the amplitude ∇ of rotation presented in equation (2). It corresponds to rela- constant equilibrium concentration, d0 is the equilibrium tively small drift with fast rotation. Thus, particle partially constant electric dipole moment, and p0(T ) is the equi- moves in direction opposite to the drift direction. The third librium pressure as a function of nonuniform tempera- regime is demonstrated by the thick continuous purple line. ture T (y). Consider plane wave perturbations of the de- It is plotted for vD = 1.2v0. The dipolar cyclotron frequency scribed equilibrium state obtain the following simplifica- v v v is the same in all regimes. Parameters 1, 2, 3 illustrate tion of the hydrodynamic equation appearing in the lin- the velocity of particle. ear regime ∂tδn+n0∂yδvy = 0, mn0∂tδvy = d0γδn, where −ıωt+ık y ∂Ey/∂z γ. Use the following ansatz δn = Ne y , ≡ −ıωt+ık y and δvy = Uye y , where ω is the frequency of the in shift of skyrmions in x-direction, which gives accu- wave, k is the projection of the wave vector on the y- mulation of skyrmions on the corresponding side of the y direction, N and Uy are constant amplitudes of pertur- sample. bations, to reduce the linear differential equations to the Describe collective effects in dipolar skyrmions. algebraic equations. They results in the following disper- Crossed nonuniform electric and magnetic fields lead to sion dependence the dipolar drift motion. This motion can affect the col- lective behavior of dipolar skyrmions. To study corre- 1+ ı k γd0 sponding collective behavior use the hydrodynamic equa- ω = y . (7) r tions consisting of the continuity and Euler equations √2 m with corresponding force field: Let us introduce the isotropic characteristic frequency

∂tn + (nv)=0, (4) of the electric field caused instability for the further ref- ∇ · erences ∆0 γd0k/m. ≡ and Start the smallp amplitude perturbation analysis with a simplified regime of zero electric field gradient con- p 1 m(∂t + v )v + ∇ = (d )E + [v, (d )B], (5) tribution. In this regime the system is characterized · ∇ n · ∇ c · ∇ by the equilibrium concentration n0, equilibrium pres- d e where d is the vector field of the collective dipole mo- sure p0, equilibrium collective dipole moment 0 = d0 z, ment related to the density of the electric dipole moment and the spatial derivative of the external magnetic field (the polarisability) nd, p is the thermal pressure, exist- ∂Bz/∂z = β. For small perturbations represent phys- ical parameters as follows n = n0 + δn, v = 0+ δv, ing if there is unordered part of motion of skyrmions rel- 2 2 atively to the ordered motion of collection of skyrmions. p = p0 + δp, with δp = mU δn, where U is the mean- E B square thermal velocity. It is assumed that perturbations The electric field and magnetic field are the ex- −ıωt+ıkr are proportional to e , where k = kx, ky, kz . ternal fields if the interaction between skyrmions is ne- { } glected. The account of the dipole-dipole interaction be- Linearized hydrodynamic equations lead to the tween dipolar skyrmions generalize the electric field to anisotropic spectrum of sound waves: the superposition of the external field Eext and the elec- 2 1 2 2 2 tric field caused by the dipoles Eint. The interacting 4 2 2 2 2 4 4 ω = Ω + k U Ω + 2Ω U (k⊥ kz )+ k U , electric field obeys the Poisson equation: 2 ± q −  (8) Eint = 4πd n. (6) where the anisotropy is caused by the dipolar cyclotron ∇ · − · ∇ 2 2 2 frequency Ω, and k⊥ = kx+ky. It gives the following sim- The Poisson equation is written for the dipole moments plified result for waves propagating perpendicular to the 2 2 with fixed direction. The polarization is created by corre- direction of the dipole moment (kz = 0) ω = ω⊥, where E e 2 2 2 2 sponding background electric field b = Eb z. Presented ω⊥ =Ω + k⊥U . There is single wave in this regime. It 4 is a sound-like wave spectrum shifted by the dipolar cy- growth of perturbation amplitude in accordance with the clotron frequency, which makes the gap in the spectrum. electric field instability carries described system to the The regime of propagation parallel to the direction of the nonlinear regime. dipole moment (k⊥ = 0) gives two solutions: the rotation The increment (14) can be rewritten via the character- 2 2 2 with fixed dipolar-cyclotron frequency ω+ =Ω and the istic frequency ∆0: (ky/k)(∆0/Ω). 2 2 2 sound wave ω− = kz U . If ky = 0 equation (12) shows a possibility for another Assuming that the dipolar cyclotron frequency dom- instability. It is related to the last term which propor- inates we can simplify equation (8) for arbitrary wave tional to the gradient of the electric field γ and the gra- k 2 2 2 2 2 2 2 vector direction : ω+ = Ω + k⊥U , and ω− = kz U . dient of magnetic field hidden in the dipolar cyclotron Hence, the second wave is a gapless sound wave propa- frequency Ω simultaneously. Dropping the thermal ef- 2 gating parallel to the electric dipole moment. fects U (hence ω⊥ Ω) find simplification of equation ∼ 2 ≈2 Include the external electric field directed parallel to (12) for ky = 0: ω =Ω γd0kx/m. The last term does y direction and changing its magnitude along z direc- not contain any trace of− the magnetic field, but its ap- tion include in our analysis the spatial derivative of the pearance is related to the presence of the magnetic field. external electric field γ. For waves propagating in the Hence, there is critical wave vector k0x = (β/γ)Ω/c, plane x y, perpendicular to the direction of electric where the instability appears. The dipolar drift corre- − dipole moment, find corresponding linearized hydrody- sponds to β/γ 1. ≫ namic equations δn = n0(kxδvx + kyδvy)/ω, In this regime, the structure of term causing instability is similar to described above. Hence, it can be presented 2 δn 2 ωδv k U = ıΩδv , (9) as kx∆0/k. x x y − − n0 Electric field gradient γ = 3 103 CGS units/cm 10 2 · and (γ = 10 V/m ) [28] and magnetic field gradient β = 10 G/cm (0.1 mT/mm) [27] and β = 1.8 107 G/cm (β = 2 · 2 δn ıγd0 1.8 10 T/mm) [33] are characteristic examples of the ωδvy kyU = δn ıΩδvx. (10) · − n0 mn0 − field gradients applied for the manipulation of skyrmions. However, the relation between γ and β is opposite to the This set of equations gives the following dispersion equa- condition required for the magneto-dipolar drift of dipo- tion lar skyrmions. To reach required condition one needs to generate β = 107 G/cm (102 T/mm) like in Ref. [33], at 2 2 Ω γd0 6 7 2 ω ω⊥ + kx ıky =0. (11) γ =1 10 CGS units/cm (γ = 10 10 V/m ). It gives −  ω −  m ÷ ÷ 3 the following drift velocity vD =3 10 cm/s (γ = 1 CGS 7 · The multiplier 1/ω in the last term of equation (11) in- units/cm, β = 10 G/cm). creases the degree of the dispersion equation. Assum- Including the characteristic value of the electric dipole −21 ing that the contribution of the electric field is relatively moment generate MFMs d0 = 3 10 CGS units −32 · small (as it is correspond to the drift motion condition) (d0 = 10 C m) [28], and effective mass of skyrmions −21· −24 solve this equation by the iteration method and obtain m = 3 10 g (3 10 kg) [34] estimate the dipo- lar cyclotron× frequency· for β = 107 G/cm. Hence, we −3 −1 2 2 Ω γd0 find Ω = 10 s . It is rather small frequency. There- ω = ω⊥ + ıky kx . (12)  − ω⊥  m fore, there is no rotation in this regime while particle moves in direction perpendicular to the electric field. In Solution of equation (12) demonstrates small in-plane y-direction the skyrmion accelerates up to 0.3 cm/s dur- anisotropy of the sound frequency ing 1 s at γ = 1 CGS units/cm. Hence, it moves mostly in x-direction. Ω γd0 2 2 2 2 The critical wave vector k0 0 is negligible small. ωR =Ω + k⊥U kx , (13) x − ω⊥ m Hence, the magneto-dipolar drift≈ instability happens for all wave vectors. and presence of instability with increment Role of dipole-dipole interaction in instabilities of γd0 γd0 weakly interacting skyrmion gas. It is also essential to δ = ky ky , (14) estimate the role of inter-skyrmions interactions, which mωR ≈ mΩ have the dipole-dipole nature for the MFM. The dipole- where ω = ωR +ıδ. The increment (14) shows prominent dipole interaction gives a generalization of equations (9), anisotropy. The approximate relation in equation (12) is (10), which includes the nontrivial third projection of the made for small pressure effects. Solution (12) is found in wave vector together with the z-projection of the velocity. the approximation of small instability. Hence, it shows Moreover, the following term rises on the right-hand side that as small time scale the magnetic field change the of linearized Euler equation n0d0kzkδϕint, where δϕint is instability. While the following growth of instability leads the perturbations of scalar potential of electric field obey- 2 to a transition thought intermediate regime to the electric ing the linearized Poisson equation k δϕint =4πd0ıkzδn. field instability demonstrated above (7). The following Described equations give the dispersion equation which 5 appears in the following form Conclusions.– The experimental data obtained for MFMs show that the skyrmions in MFMs have the elec- 4 2 2 2 2 ω ω (Ω + k V + ıγd0ky) tric dipole moment. We have considered the dynamics − of single dipolar skyrmions and gas of weakly interact- 2 2 2 ing dipolar skyrmions. We have applied the Newton +Ωγd0kxω +Ω kz V =0, (15) and hydrodynamic models for the point-like interacting 2 2 2 2 2 2 2 2 skyrmions in the external electromagnetic fields to study where k = k⊥ + kz and V = U +Λ kz /k , with the the drift motion of dipolar skyrmions. We have consid- characteristic frequency for the dipole-dipole interaction 2 2 2 ered the influence of the nonuniform magnetic field on Λ = 4πd0k /m. Numerical estimation of characteristic −21 the moving dipole moment. frequency of dipole-dipole interaction for d0 = 3 10 CGS units and m =3 10−21 g gives Λ 3 10−10·k s−1. We have found new way to control of skyrmion motion For the wave length of· order of 1 µm we≈ have· Λ 3 10−6 using magnetic field gradient. The large enough mag- s−1 (Λ > Ω). ≈ · netic field gradient applied together with the electric field If the contribution of the electric field is small γ 0, gradient generate the dipolar drift motion. The velocity find stable solutions of dispersion equation (15): → of dipolar drift of skyrmion in the crossed nonuniform electric and magnetic fields has been derived. The drift 1 velocity is perpendicular to the directions of the nonuni- ω2 = w2 w4 4Ω2k2(U 2 +Λ2 cos2 θ) , (16) 0 2 ± − z  form electric field and to the direction of magnetic field. p As a result we have obtained the magneto-dipolar Hall ef- 2 2 2 2 2 2 where w Ω + k (U +Λ cos θ), and kz = k cos θ. fect in crossed nonuniform electric and magnetic fields. It This is a generalization≡ of equation (8). provides an extra mechanism for control of the skyrmion The dipole-dipole interaction gives a positive non- motion and distribution of collection of skyrmions. isotropic shift of the thermal velocity square U 2. Hence, Collective dynamics of skyrmions has been addressed the shift is maximal if the wave propagate in the direction through the hydrodynamic model. We have found three of the electric dipole moment. There is no dipole-dipole regimes of collective instabilities in systems of dipolar interaction contribution for the perpendicular direction. skyrmions in the MFMs. One corresponds to accelera- Comparing this result with equations (13) and (14) we tion of dipoles by the nonuniform electric field. Other conclude that the dipole-dipole interaction decreases the two regime are related to simultaneously acting electric instability increment for the nonzero electric field gradi- and magnetic fields. One of them change the small time ent γ = 0. regime of instability happening in the direction of elec- Moreover,6 the dipole-dipole interaction increases the tric field. The another one is the instability happening value of the critical wave vector k0x for the dipolar drift in the direction of the drift motion, i.e. the dipolar drift instability increases for nonzero kz. What leads to small instability. stabilization of the system. In equation (15) the dipo- Acknowledgments.– The study of M. Trukhanova was lar drift instability is presented by the term linear on performed by a grant of the Russian Science Foundation frequency ω. (Project N 19-72-00017).

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