Quantum Fluctuations Stabilize Skyrmion Textures

Quantum ﬂuctuations stabilize Skyrmion textures

A. Roldán-Molina Instituto de F´ısica, Pontificia Universidad Católica de Valpara´ıso, Avenida Universidad 330, Curauma, Valpara´ıso, Chile

M. J. Santander and A.´ S. Núñez Departamento de F´ısica, Facultad de Ciencias F´ısicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile

J. Fern´andez-Rossier∗ International Iberian Nanotechnology Laboratory, Av. Mestre Jose Veiga, 4715-310 Braga, Portugal

Here we show that the zero point energy associated to the quantum spin fluctuations of a non- collinear spin texture produce Casimir-like magnetic fields. We study the effect of these Casimir fields on the topologically protected non-collinear spin textures known as skyrmions. We calculate the zero point energy, to the lowest order in the spin wave expansion, in a Heisenberg model with Dzyalonshinkii-Moriya interactions chosen so that the classical ground state displays skyrmion textures, that disappear upon application of a strong enough magnetic field. Our calculations show that the Casimir magnetic field contributes a 10 percent of the total Zeeman energy necessary to delete the skyrmion texture with an applied field, and is thereby an observable effect.

PACS numbers: 74.50.+r,03.75.Lm,75.30.Ds

The properties of many magnetic materials with bro- magnets[5–11] and in engineered surfaces[17, 18], they ken spin symmetries are correctly described in terms of have received attention for potential applications in spin- magnetic moments that have a well defined orientation tronics because it is possible to control their position with and are, in this sense, classical. Yet quantum fluctua- very low current densities[12]. In addition skyrmions dis- tions, that ultimately arise from the fact that the differ- play several features that are convenient from the view- ent projections of the spin operator do not commute with point of potential applications, such as dimensions within each other, are invariably present. The standard[1, 2] the nanometric scale, topological protection, and high description of spin excitations of these broken symmetry mobility. They have been observed in bulk magnets magnetic materials in terms of quantum spin waves im- MnSi[5–7], Fe1−xCoxSi[8, 9, 13, 14], Mn1−xFexGe[15] plies that, even in the ground state, there is a zero-point and FeGe[16] using neutron scattering and Lorentz trans- (ZP) energy ultimately associated to fluctuations of the mission electron microscopy. As it was reported, in these magnetization field[1, 2]. This ZP energy vanishes in the systems an external magnetic field can induce skyrmions case of collinear ferromagnets, but is non-zero in general. with diameters of about a few tens of nanometers. The In the case of quantum electrodynamics, the vacuum inclusion of spin transfer torques, as it is shown by nu- energy, ZP , associated to the photon zero point motion merical simulations, can be used to nucleate and manip- stored betweenE two parallel plates depends on the dis- ulate isolated skyrmions[19, 20]. In addition, two dimen- tance between L between them, resulting on an effective sional atomic-scale magnetic nanoskyrmion lattices have δEZP Casimir force[3] F = δL . In this letter we show that, also been observed by means of spin polarized scanning analogously, the vacuum− energy associated to the quan- tunneling microscopy in a monoatomic layer of Fe atoms tum spin fluctuations of a given spin texture depends on top of Ir(111) surface[17]. In this case, skyrmions on an ensemble of classical variables encoded in the lo- arrays with a length scale of only one nanometer are sta- cal spin orientation Ω~ i, resulting in an effective Casimir bilized by the interplay between Dzyaloshinskii-Moriya δEZP field ~τi = . Here we show that proper account of interaction and the breaking of inversion symmetry at δΩ~ i arXiv:1502.01950v1 [cond-mat.mes-hall] 6 Feb 2015 − those fluctuations leads to a reduction in the energy of surfaces and interfaces. skyrmion textures rendering them stable in regions where Previous work [21] has addressed the study of a purely classical analysis predicts them to be unstable. skyrmions spin excitations in a semiclassical approxima- Magnetic Skyrmions are topologically protected spin tion, for rather large systems with up to 105 localized structures[4]. Observed recently, both in chiral moments. An effective theory of magnon-skyrmion scattering in a continuum approach has been recently car- ried out by Schütte et al. [22]. Here we treat smaller ∗Permanent Address: Departamento de F´ısicaAplicada, Universi- skyrmions, using a complementary approach that per- dad de Alicante mits to study quantum fluctuations associated to the 2 excitations relevant for atomic scale skyrmions [17] for (HP) bosons[2, 23]: which the effects of quantum fluctuations of the magnetization cannot be neglected at the outset. Si Ω~ i = S ni, · p− †p S+ = 2S n a ,S− = a 2S n , (2) The non-collinear spin alignment between first neigh- i − i i i i − i bors is promoted by the competition between the stan- ~ dard Heisenberg interaction ( JS~ S~ ) and the an- where Ωi is the spin direction of the classical ground state − i · j † tisymmetric Dzyaloshinskii-Moriya interaction (DMI) on the position i, ai is a Bosonic creation operator and † term D~ S~ S~ , which is only present in non- ni = ai ai is the boson number operator. The opera- i,j · i × j centrosymmetric systems. In particular, this competi- tor ni measures the deviation of the system from the tion leads to helical spin textures with fixed chirality. classical ground state. The essence of the spin wave cal- Application of a magnetic field h along thez ˆ direction culation is to represent the spin operators in Eq. (1) (perpendicular to the system surface and then to D~ i,j for by the HP bosonic representation and to truncate the any pair i, j) breaks the up-down symmetry in the sys- Hamiltonian up to quadratic order in the bosonic oper- tem resulting, for a range of fields, in an abrupt change in ators. The linear terms in the bosonic operators vanish the ground state. In this manner, the external magnetic when the expansion is done around the correct classical field and a uniaxial anisotropy can stabilize an isolated ground state. This approach has been widely used in the Skyrmion on a ferromagnetic system with DMI. A mini- calculations of spin waves for ferromagnetic and antifer- mal model that captures the essence of the physics just romagnetic ground states[1, 2]. In the spin wave approx- − † described is given by the following Hamiltonian with four imation we thus approximate Si = √2S ai , so that the contributions, the Heisenberg and DM interactions, the creation of a HP boson is equivalent the removal of one Zeeman term and the uniaxial anisotropy: unit of spin angular momentum from the classical ground state. After some algebra[25] the quadratic Hamiltonian can X ~ ~ X ~ ~ ~ H = Ji,jSi Sj + Di,j (Si Sj) be reduced to a symmetric form in creation an annihila- − · · × tion operators written as: X~ ~ X z 2 h Si K (Si ) , (1) − · − X † ti,j τi,j aj i i H = + + a , a † (3) Ecl E0 i j τ ∗ t∗ a i,j i,j i,j j where the interactions are taken with nearest neighbors where the constant is a quantum contribution to the only. 0 ground state energyE that arises naturally in this expres- Since exact diagonalization of this Hamiltonian is not sion as a consequence of applying the bosonic conmuta- possible except for very small systems, we adopt the stan- tion relations in order to bring the Hamiltonian to its dard spin wave approach, that is implemented it two symmetric form. The second and third term in eq. (3) clearly separated steps. First, we treat eq. (1) as a clas- correspond to the called spin waves Hamiltonian HSW . sical functional H(Ω~ ), where the spin operators S~ Importantly, cl, 0, ti,j and τi,j are a functional of Ecl ≡ i E E are replaced by classical vectors Ω~ i. The classical ground the classical ground state configuration Ω~ i that satisfies δEcl state, defined as the configuration Ω~ i that minimizes the = 0. δΩ~ j functional , is determined by self-consistent iteration. Ecl The off-diagonal terms in eq. (3) do not conserve the The above Hamiltonian is studied for square systems with number of HP bosons and are in general present for non- up to 45 45 sites. We simulate hard boundary condi- × collinear classical ground state [1, 2]. In consequence, tions at the edges by the inclusion of a large magnetic the ground state of the Hamiltonian is not the vacuum of field in the border sites. Unless otherwise stated, we the a bosons and, thereby, the ground state energy at the assume [21, 27] S = 1, J = 3.0 meV, D = 0.6 meV, spin-wave approximation level is different from the classi- D~ ~r and K = 0.5(D2/J), where ~r is the unit i,j k i,j i,j cal ground state energy. The Hamiltonian HSW is solved vector between the sites i and j. We find that, for by means of a paraunitary[24, 25] transformation, anal- this choice of parameters, the skyrmion configuration is ogous to the usual Bogoliubov transformation for BCS more stable than the ferromagnetic state for fields up to Hamiltonians, that leads to: 2 hc 0.8(D /J) and skyrmion solutions can not be found ∼ 2 N at all for hc0 1.67(D /J) . Importantly, the main re- ∼ X † 1 sults discussed below do not depend qualitatively on the HSW = 0 + ~ων (αν αν + ) (4) E 2 choice of Hamiltonian parameters. ν=1

The second step permits to compute quantum fluctua- where the αν are bosonic operators formed by linear com- tions of the classical solution. For that matter, we repre- bination of both a and a† that annihilate spin waves with sent of the spin operators in terms of Holstein Primakoff energy ~ων . In fig.1 we show the evolution of the four 3

0.5 1.00

0.4 0.95

D 0.3 >

z 0.90 meV W @ Ν < nmax! E 0.2 0.85 0.1

0.80 0.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 h@D2JD h@D2JD

FIG. 1: Left panel: Spin wave spectrum as a function of magnetic field. Right panel: Average spin projection along the z axis as a function of the external magnetic field. As the field grows the size of the skyrmion is reduced. At a 0! field of h ∼ 1.67(D2/J) the system can no longer sustain the skyrmion texture and a transition towards a ferromagnetic state is induced. lowest energy modes as function of magnetic field , the abrupt change at hc0 is a consequence of the fact that for FIG. 2: Magnonic occupation of the four lowest energy ex- larger fields, the skyrmion like solution is no longer al- cited states displayed in (a), (b), (c), and (d) respectively. lowed and a transition to ferromagnetic order takes place. The plots correspond to a field h ∼ 0.75(D2/J). On each It is apparent that the skyrmion has in-gap spin wave ex- figure the dashed white line correspond to the perimeter of the skyrmion. The maximum value of the occupation is citations, in line with those obtained using the semiclassi- −3 −3 nmax = 7.1×10 for the first excited state, nmax = 5.6×10 cal approximation[21]. The first excitation has a very low −3 for the second excited state, nmax = 4.8 × 10 for the third −3 energy and can be readily associated with a gyrotropic excited state, and nmax = 2.1 × 10 for the fourth excited traslation mode. In Fig. (2) we show the magnonic oc- state. cupation, defined as

i † n = ψν a ai ψν (5) ν h | i | i point energy, . In the case of ferro- EZP ≡ EGS − Ecl † magnetic solutions it turns out that vanishes [1, 2] for the four lowest energy modes, where ψν αν GS . EZP Inspection of the magnonic occupation| in thei ≡ four| low-i because the collinear ferromagnetic ground state is also est energy excited states, shown in Fig.2, suggest that the an eigenstate of the exact Hamiltonian. second and third correspond to a breathing modes. The In general the vacuum energy ZP is not zero and, as skyrmion-ferromagnet transition is driven by a reduction it turns out, it contributes significantlyE to the relative of the energy of the lowest energy breathing mode. The stability of different spin configurations. We show this abrupt reduction of the second excitation energy near for the case of a skyrmion. In Fig. 3 we compare the the critical field signals the instability of the Skyrmion energy difference between the skyrmion and the collinear texture. Below the critical field, both internal and ex- solution, as a function of the applied field h, calculated tended spin wave states are present in the system. For at the classical level (blue line) and including (red EZP fields larger than the critical field the spin wave spectrum line). It is apparent that the critical field hc above which correspond to the usual ferromagnetic spin waves. the collinear solution is more stable increased 10% when We now discuss the main result of the manuscript. In the ZP is included: the skyrmion is more stable due contrast with the semiclassical theory, the quantum the- to quantumE fluctuations. By analogy with the Casimir ory of spin waves also yields a renormalization of the effect, variations of the vacuum energy with the local energy of the ground state whose wave function is, by moment orientation can be associated with an effective definition, the vacuum of the α operators, αν GS = 0. Casimir magnetic field. This Casimir field is the mag- | i In the case of non collinear classical order, the ground netic analogue of the Casimir force. In this manner, the state is different from the vacuum of HP bosons. The effective field on a given spin can be defined as: ground state energy reads:

N δ GS δ 1 X ~hi = E = ( cl + GS SW GS ) (7) GS = cl + 0 + ~ων cl + ZP (6) − ~ − ~ E h |H | i E E E 2 ≡ E E δΩi δΩi ν=1 which is different from the classical energy due to the The first contribution is the classical field, which vanishes contribution coming from quantum fluctuations, or zero identically by construction. Using this and the Hellman- 4

GS a†a GS , reflecting the zero point quantum fluctu- h | i i| i 20 Skyrmion ! Ferromagnetic! ations that are a consequence of a non collinear classi- phase! phase! cal state. The zero-point magnonic occupation of the skyrmion ground state is shown in figure (4)(d). Calcu- D 10 lations using different values for D, J, K show that the

meV magnitude of the ZP fluctuations increases as the size @ 0 of the skyrmion is reduced. The ZP fluctuations have FM ~ † E observable consequences. Since S Ωi = S ai ai , - both the zero point motion andh the· thermallyi − excited h i Sk -10 E spin waves will reduce the effective atomic magnetic mo- Classical difference! ments of skyrmion solutions, an effect that should be Skyrmion ZP energy! -20 Corrected difference! observable with local probes, such as STM.

0.6 0.7 0.8 0.9 1.0 1.0 0.5 a) 2 h D J ⌦z 0.0 -0.5 FIG. 3: Diﬀerences between skyrmion and ferromagnetic con- -1.0 0 10 20 30 40 ﬁguration energy. The crossing between the curves and the 2.0 z ⌦ˆ H ê L 3 i site horizontal axis sets the boundary for the region of stability b) ˆ

10 1.5 for the skyrmion in classical and corrected approaches (blue ✓ ✓ˆ

⇥ 1.0 y H L and red lines respectively). Corrected energy diﬀerence be- 0 0.5

n x tween the skyrmion and ferromagnetic textures including the 0.0 0.020 zero point fluctuations, enlarging the region of stability for ] 0 10 20 30 40 skyrmions. 0.015 c) 0.010 meV [ 0.005 ⌦ ~ h 0.000 Feynman theorem[26], the Casimir field hZP read: -0.005 0 10 20 30 40 i site ~ i δ SW hZP = GS H GS (8) −h δΩ~ i i FIG. 4: Magnetization profile on zH directionL (a), magnonic occupation (b) and Casimir field (Ω direction) (c) on a skyrmion and they can be readily calculated. Our results for the ground state for h=0.75(D2/J). The local coordinate system Casimir field are displayed in Fig. (4), together with the for spins and the corresponding Casimir fields is shown in (b) projection of the classical magnetization over the z axis inset. As consequence of the Casimir field on direction Ω, the (top panel), and the expectation value, computed with skyrmion zero point energy is reduced thereby it increase the the ground state wave function, of the magnon occupa- texture stability. tion. For reference, we sketch the spherical coordinate system as shown in the inset of Fig. (4)(b). At the cen- Importantly, the Casimir magnetic field exert torque ter of the skymion the magnetization points along zˆ, on the magnetic moments, the Casimir spin torque, de- − ~ ~ i opposite from the magnetization of the surroundings, as fined as ~τi = Ωi hZP . The ZP energy, and thereby the can be seen in Fig. (4)(a). ZP Casimir spin× torques, depend on the energy of the The stabilization of the skyrmion configuration with spin wave excitations, which are interesting on their own respect to the applied field can be associated to the right and could be observed by means of inelastic electron Casimir field using the following argument. The Zeeman tunneling spectroscopy, as recently demonstrated for spin contribution enters the energy of the system as Ωzh, in waves in short ferromagnetic chains[28]. the same manner, the contribution of the Casimir− field in In summary, we have studied the observable con- classical direction Ω~ correspond to S (~h Ω)~ = S h . sequences of quantum spin fluctuations in skyrmions. − ZP · − Ω In Fig. (4)(c) we plot hΩ, it is clear from here that the We have found that the zero-point energy stabilizes contribution of the Casimir field on the classical direction skyrmions with respect to the application of a magnetic is to lower the energy of the skyrmion state. field, increasing significantly the critical field necessary to In addition to the stabilization of the skyrmion so- ears the skyrmion. We show that this stabilization can lution, quantum fluctuations can have other observable be associated to a Casimir magnetic field driven by zero consequences that we discuss now. In order to charac- point fluctuations analogous to the Casimir effect. We terize the density of spin fluctuations of a given eigen- have also found that zero point fluctuations renormalize † state of Hamiltonian (4), ψν αν GS , we have de- the shape of the skyrmion and shrink the atomic mag- fined the magnonic occupation| i ≡ in Ec.(5).| i Importantly, netization, an effect that becomes significant for small i this quantity is non zero even in the ground state, n0 = skyrmions and could be observerved by means of spin- 5 polarized scanning tunneling microscopy. Ishizuka and K. Kimoto, Nature Nanotechnology 8, 325 The authors would like to thank funding from grants (2013). Fondecyt 1150072, ICM P10-061-F by Fondo de Inno- [11] S. Seki, X. Z. Yu, S. Ishiwata and Y. Tokura , Science vaciónpara la Competitividad-MINECON and Anillo 336, 198201 (2012). [12] A. Fert, V. Cros and J. Sampaio, Nature Nanotechnology ACT 1117. 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