Understanding the physical properties of magnetic skyrmions is important for fundamental research with the aim to develop new spintronic device paradigms where both logic and memory can be integrated at the same level. Here, we show a universal model based on the micromagnetic formalism that can be used to study skyrmion stability as a function of magnetic field and temperature. We consider ultrathin, circular ferromagnetic magnetic dots. Our results show that magnetic skyrmions with a small radius - compared to the dot radius - are always metastable, while large radius skyrmions form a stable ground state. The change of energy profile determines the weak (strong) size dependence of the metastable (stable) skyrmion as a function of temperature and/or field.
ASJC Scopus subject areas
- Electronic, Optical and Magnetic Materials
- Condensed Matter Physics
Access to Document
Other files and links
FingerprintDive into the research topics of 'Origin of temperature and field dependence of magnetic skyrmion size in ultrathin nanodots'. Together they form a unique fingerprint.
Origin of temperature and field dependence of magnetic skyrmion size in ultrathin nanodots. / Tomasello, R.; Guslienko, K. Y.; Ricci, M.; Giordano, A.; Barker, J.; Carpentieri, M.; Chubykalo-Fesenko, O.; Finocchio, G.In: Physical Review B, Vol. 97, No. 6, 060402, 09.02.2018.
Research output: Contribution to journal › Article › peer-review
TY - JOUR
T1 - Origin of temperature and field dependence of magnetic skyrmion size in ultrathin nanodots
AU - Tomasello, R.
AU - Guslienko, K. Y.
AU - Ricci, M.
AU - Giordano, A.
AU - Barker, J.
AU - Carpentieri, M.
AU - Chubykalo-Fesenko, O.
AU - Finocchio, G.
N1 - Funding Information: et al. Tomasello R. 1 Guslienko K. Y. 2,3 Ricci M. 4 Giordano A. 5 Barker J. 6 Carpentieri M. 7 Chubykalo-Fesenko O. 8 Finocchio G. 5 * Department of Engineering, Polo Scientifico Didattico di Terni, 1 University of Perugia , 50100 Terni, Italy Departamento Física de Materiales, 2 Universidad del País Vasco , UPV/EHU, 20018 San Sebastián, Spain 3 IKERBASQUE , the Basque Foundation for Science, 48013 Bilbao, Spain Department of Computer Science, Modeling, Electronics and System Science, 4 University of Calabria , I-87036 Rende, Italy Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, 5 University of Messina , 98166 Messina, Italy Institute for Materials Research, 6 Tohoku University , 980-8577 Sendai, Japan 7 Department of Electrical and Information Engineering , Politecnico di Bari, 70125 Bari, Italy 8 Instituto de Ciencia de Materiales de Madrid , CSIC, Cantoblanco, 28049 Madrid, Spain * Corresponding author: email@example.com 9 February 2018 February 2018 97 6 060402 22 June 2017 ©2018 American Physical Society 2018 American Physical Society Understanding the physical properties of magnetic skyrmions is important for fundamental research with the aim to develop new spintronic device paradigms where both logic and memory can be integrated at the same level. Here, we show a universal model based on the micromagnetic formalism that can be used to study skyrmion stability as a function of magnetic field and temperature. We consider ultrathin, circular ferromagnetic magnetic dots. Our results show that magnetic skyrmions with a small radius—compared to the dot radius—are always metastable, while large radius skyrmions form a stable ground state. The change of energy profile determines the weak (strong) size dependence of the metastable (stable) skyrmion as a function of temperature and/or field. Consiglio Nazionale delle Ricerche 10.13039/501100004462 B52I14002910005 Türkiye Bilimsel ve Teknolojik Araştirma Kurumu 10.13039/501100004410 113F378 Ministerio de Economía y Competitividad 10.13039/501100003329 MAT2013-47078-C2-1-P MAT2013-47078-C2-2-P FIS2016-78591-C3-3-R Tohoku University 10.13039/501100006004 Nonlinear localized excitations have attracted the attention of physicists for a long time. Such excitations, including solitary waves or solitons, play an important role in optics, quantum field theory, condensed matter, and other fields. It is sometimes possible to associate integer numbers (topological charges) to the solitons, which are preserved in their dynamics. Topologically nontrivial magnetization configurations in ferromagnets, such as domain walls, vortices, and skyrmions are currently the focus of a lot of research activity. These spin textures are also candidates for nanoscale device applications—computational paradigms, magnetic storage, and programmable logic—due to their small size [1–13] . Skyrmion solutions were obtained first by Skyrme in the nonlinear field theory  . Subsequently chiral skyrmions were predicted  , and discovered experimentally in noncentrosymmetric cubic B20 compounds [16–20] which permit an antisymmetric anisotropic interaction, namely, the Dzyaloshinskii-Moriya interaction (DMI). This arises from a relativistic correction and relies on spin-orbit interactions [21,22] . Recent efforts have focused on materials with interfacial DMI—especially ultrathin transition-metal/heavy-metal multilayers with large spin-orbit coupling such as Co/Pt and Co/Ir [8,23,24] . The DMI is necessary to yield axisymmetric skyrmions in ultrathin magnetic elements and the chiral skyrmions can be further stabilized by an external magnetic field [8,23,24] . Temperature is usually considered to be detrimental to skyrmion stability, leading to either the transformation of the skyrmion state into a more energetically favorable state  or to nucleation of multiple skyrmions and labyrinth domains [8,23,24] . In addition, recent room-temperature experiments with an external out-of-plane magnetic field [8,23] , showed a strong nonlinear dependence of the skyrmion radius on the external field strength pointing out the key role of the external field. Rohart and Thiaville  developed a domain-wall model of Néel skyrmions at zero temperature T and zero magnetic field H ext , and identified a critical DMI parameter D c quantifying the influence of DMI that is independent of the dot size. They found that the skyrmion radius R sk diverges as the DMI parameter D → D c . In a finite magnetic dot, however, R sk is determined by the confining potential and cannot exceed the dot radius R d  . This analytical model yields incorrect results for D > D c and should be improved. The temperature dependence of the skyrmion radius in an infinite film was calculated numerically by Barker and Tretiakov  who attempted to use Rohart's expression for R sk with the temperature dependence of the magnetic properties of the host material to explain the behavior. In that paper  , no temperature dependence of D was considered. The authors concluded that in ferromagnets the skyrmion radius has a weak quasilinear temperature dependence. This is rather a result of using the equation by Rohart et al. for small radius (∼10 nm) skyrmions which have a size comparable with the domain-wall width. Our work is motivated by the lack of understanding regarding the skyrmion size and stability as a function of control parameters such as temperature and external field in a finite-size sample. Here, we provide a simple analytic tool for identifying materials and geometries which can host skyrmions in a target temperature and field window. Particularly, the analytical model predicts weak/strong dependence of skyrmion size on control parameters which is further confirmed by numerical modeling. More in detail, we present a theoretical approach to skyrmion stability based on minimization of the magnetic energy using a skyrmion ansatz in confined geometries. We show that the thermal evolution of skyrmion size in ultrathin dots is best expressed in a Q-d phase space of the skyrmion quality factor Q = 2 K u / 2 K u μ 0 M s 2 μ 0 M s 2 ( K u is the uniaxial anisotropy constant, M s is the saturation magnetization, μ 0 is the vacuum permeability), and the reduced DMI strength d = | D | l ex / A ( l ex = 2 A / μ 0 M s 2 is the exchange length, with A being the exchange constant). The proximity of the point describing the skyrmion state in this Q-d phase space, to the border of the skyrmion stability, is the crucial parameter in understanding the linear or nonlinear change in radius with temperature or field as observed in micromagnetic simulations. The temperature dependence of the skyrmion stability is expressed as scaling relations of both Q and d with changing the reduced magnetization m ( T ) = M s ( T ) / M s ( T ) M s ( 0 ) M s ( 0 ) . We find the equilibrium skyrmion size increases rapidly with temperature only when approaching a critical line d c ( Q ) = ( 4 / π ) Q − 1 . The size becomes comparable with the dot radius at d ( T ) > d c ( T ) . The metastable skyrmions have small radius ( R sk ≪ R d ), whereas the stable (ground state) ones have large radius. The out-of-plane external magnetic field can be used to bring metastable skyrmions closer to or farther from the critical line d c ( Q ) where the metastable skyrmion becomes a ground-state skyrmion. The magnetic skyrmion can be described by the energy functional E [ m ] = ∫ d V ɛ ( m ) , with the energy density (1) ɛ ( m ) = A ( ∇ m ) 2 + ɛ DMI + K u m z 2 − 0.5 M s m · H m − M s m · H ext , where m = M / M s is the unit magnetization vector ɛ DMI = D [ m z ( ∇ · m ) − ( m · ∇ ) m z ] is the interface DMI energy density, m z is the magnetization z component, H m is the magnetostatic field, and H ext is the external magnetic field. We assume that in an ultrathin circular dot the magnetization m does not depend on the thickness ( z coordinate). We introduce polar coordinates in the dot plane ρ = ( ρ , ϕ ) , and define spherical angles of the magnetization vector ( Θ , Φ ) as functions of ρ . We also assume that the static skyrmion centered in the dot is axially symmetric and use the equations Θ 0 ( ρ ) = Θ 0 ( ρ ) , Φ 0 ( ρ ) = ϕ + ϕ 0 . In seeking a general expression for the dependence of the skyrmion size on both temperature and external magnetic field, and to overcome drawbacks of previous approaches in estimating the skyrmion radius  (see note 1 in the Supplemental Material), we developed an analytical approach using a different skyrmion ansatz from Refs. [26,29] . To minimize the energy functional (1) , we propose the trial function (2) tan Θ 0 ( r ) 2 = r sk r e ξ ( r sk − r ) , where r = ρ / l ex , ξ 2 = Q − 1 , and the polar angle Θ 0 describes the static skyrmion profile. The accuracy of the ansatz (2) was numerically verified in the case of two-dimensional easy axis infinite ferromagnets ( Q >1) in Ref.  . Equation (2) recovers the Belavin-Polyakov solution for the isotropic case ξ = 0 and leads to the finite exchange energy at r ≈ 0 . Equation (2) is also very similar (in the area of its applicability, at r ≈ r sk ≫ 1 ) to the ansatz used in the theory of bubble domains  in highly anisotropic ferromagnetic films  (see approach 3 in note 1 in the Supplemental Material). The energy E and the equilibrium skyrmion radius r sk = R sk / l ex can now be expressed as functions E [ r sk , Q , d , H ext ] and r sk [ Q , d , H ext ] . The conditions of the skyrmion stability can be found using a standard variational procedure, solving the equations ∂ E / ∂ r sk = 0 , ∂ 2 E / ∂ r sk 2 = 0 . To account for the temperature dependence of the skyrmion radius, we use the scaling approach for macroscopic (micromagnetic) parameters which usually decrease with temperature increasing due to the magnetization fluctuation effects. Including thermal effects into a micromagnetic approach is known to largely overestimate the Curie temperature as compared to the more accurate atomistic approach  . However, the largest temperatures considered in the present study are very far from the Curie temperature ( T / T c < 1 / 2 ). In this case, the micromagnetic approach produces accurate results in terms of the thermodynamically averaged quantities as a function of the magnetization m ( T ) . We calculated the temperature dependence of uniaxial magnetic anisotropy K u ( T ) = K u ( 0 ) m ( T ) γ achieving γ = 3.03 that is close to the Callen-Callen relation  . The scaling of the exchange A ( m ) = A ( 0 ) m α and DMI D ( T ) = D ( 0 ) m ( T ) β parameters is found by atomistic simulations of an infinite thin film calculating the thermal spin-wave spectrum [34,35] and fitting the long-wavelength regime (small k vectors) with a linear spin-wave theory  (see note 2 in the Supplemental Material). We obtain α = β = 1.5 , consistent with other recent results [36,37] . To benchmark the theory, we consider a circular nanodot (e.g., Co) of diameter 2 R d = 400 nm and thickness of 0.8 nm assumed to be in contact with a thin layer of heavy metal (e.g., Pt) giving an appreciable interfacial DMI. We performed systematic micromagnetic simulations to calculate the skyrmion size as a function of the out-of-plane external field H ext (from 0 to 50 mT) and temperature (from 0 to 300 K) integrating the Landau-Lifshitz-Gilbert equation of motion for the reduced magnetization m = M / M s [28,38–42] (see note 3 in the Supplemental Material). At T = 0 K , we used the following material parameters: M s = 600 kA / m  , A = 20 pJ / m  , D = 3.0 mJ / m 2 [44,45] , K u = 0.60 MJ / m 3  , and Gilbert damping α G = 0.1  . As a reference, Rohart's critical DMI value D C = 3.48 mJ / m 2 for our parameters at T = 0 K  . Temperature causes the skyrmion to diffuse and leads to fluctuations of the diameter and deformations of the skyrmion shape [27,47,48] , losing the circular symmetry. We therefore calculated the effective diameter by assuming that the area of the skyrmion core (here it is the region where the z component of the magnetization is negative) is equivalent to a circle  . The skyrmion diameter and perimeter display approximately Gaussian statistical distributions  (see note 3 in the Supplemental Material) with increasing widths with temperature. Figure 1(a) compares the magnetic field dependence of the skyrmion diameter calculated by micromagnetic simulations with the analytic skyrmion ansatz, Eq. (2) , at T = 0 K . There is a reasonably good agreement considering there are no fitting parameters (maximum difference around 8% at zero field). Figure 1(b) shows the temperature dependence of the skyrmion diameter calculated with micromagnetic simulations for three values of the external field (open symbols). At zero field, the skyrmion diameter rapidly expands with increasing temperature. The increasing out-of-plane field causes a weak quasilinear increasing of the diameter up to T = 300 K [red circles and green triangles in Fig. 1(b) ]. Our computations show that two thermal/field regimes exist: at high temperature ( T > 200 K ) and low field ( H ext < 10 mT ), the skyrmion size is strongly influenced by thermal fluctuations, whereas at low temperature ( T ≤ 200 K ) independent of the field, or at high temperature and high field ( H ext ≥ 10 mT ), the skyrmion is weakly affected by thermal fluctuations. These different behaviors were not observed in Ref.  , because with the parameters used, the skyrmions were always in the metastable region. When calculating the skyrmion diameter as a function of the external field for T = 300 K [Fig. 1(c) ], there is a large, nonlinear expansion of the diameter and larger variance as the external field is reduced. This is consistent with the experimental results in Refs. [8,23] . 10.1103/PhysRevB.97.060402.f1 1 FIG. 1. (a) Skyrmion diameter as a function of the perpendicular external field as computed by micromagnetic simulations (red circles) and by analytical computations based on Eq. (2) (blue squares) at T = 0 K . (b) Skyrmion diameter as a function of temperature for three values of the perpendicular external field. The symbols represent the mean value of the skyrmion diameter as obtained by micromagnetic simulations including thermal fluctuations, where the error bar corresponds to the standard deviation, while the dashed curves are related to the analytical results with scaled values of the macroscopic parameters, using α = 1.50 , β = 1.50 , and γ = 3.585 . γ is approximately 15 % larger than the one derived from atomistic simulations  . (c) Mean value of the skyrmion diameter and corresponding standard deviation as error bar as a function of applied external field for T = 300 K , calculated from micromagnetic simulations. By including the scaling relations in the analytical approach, the results are in agreement with micromagnetic simulations until the skyrmion is in the metastable region far from the boundary of the stable region  (e.g., when T < 200 K for H ext = 0 mT ; see note 3 in the Supplemental Material). When approaching the ground-state region, the confining potential—due to the magnetostatic field and the DMI boundary conditions—starts to play a significant role in fixing the skyrmion size [11,26] . To account for this, we consider the scaling exponent γ of the uniaxial perpendicular anisotropy K u ( T ) = K u ( 0 ) m ( T ) γ as a fitting parameter. This is because the analytical model is developed within the thin-film approximation where the effective anisotropy is computed as K eff ( T ) = K u ( T ) − 0.5 μ 0 M s ( T ) 2 , but the numerical micromagnetic solution includes the full magnetostatic calculation. By performing athermal (deterministic) micromagnetic simulations  (see note 3 in the Supplemental Material), an excellent agreement is found for γ = 3.585 . With this value, we can calculate the skyrmion diameter dependence on temperature and field by Eq. (2) [dashed lines in Fig. 1(b) ]. The analytical [ansatz Eq. (2) ] and micromagnetic results agree well and we conclude that our analytical expression is accurate in predicting the skyrmion size for arbitrary temperature and external field combinations. Figure 1(c) shows how even a small external field strength H ext = 5 mT significantly reduces the skyrmion size at room temperature. This occurs because the magnetization region outside of the skyrmion core expands—the field direction is opposite to the skyrmion core magnetization—leading to a shrinking of the skyrmion. The nonlinear field–skyrmion size dependence is in qualitative agreement with recent experimental results (see, for comparison, Fig. 4a in Ref.  and Fig. S8 in the Supplemental Material of Ref.  ). To understand the origin of the two thermal/field regimes, we focus on the data obtained at zero external field. We calculate the critical DMI parameter for each set of the scaled parameters by the expression D c ( T ) = 4 A ( T ) K eff ( T ) / 4 A ( T ) K eff ( T ) π π  . When D approaches D c (as temperature increases; see Fig. 2 ), a sharp increase of the skyrmion size R sk ( T ) occurs  . This could explain why, at room temperature, the skyrmion size increases for H ext < 5 mT exhibiting a nonlinear dependence on the external field [8,23] [see Fig. 1(c) ]. 10.1103/PhysRevB.97.060402.f2 2 FIG. 2. Comparison of temperature dependences of the scaled DMI parameter D and the scaled critical DMI parameter D c when H ext = 0 mT . The thermal scaling of the macroscopic parameters leads to the temperature dependence of the quality factor Q ( T ) = Q ( 0 ) m ( T ) γ . Taking into account the definition d ( T ) = D ( T ) l ex ( T ) / D ( T ) l ex ( T ) A ( T ) A ( T ) , we derive the temperature dependence of the reduced DMI parameter as d ( T ) = d ( 0 ) [ m ( T ) ] β − α / 2 − 1 . Note that temperature dependence of the reduced critical DMI parameter d c ( m ) ∝ m − 1 K eff ( m ) ∝ Q ( m ) − 1 includes neither the exchange stiffness α nor the DMI exponent β . This justifies the use of the skyrmion stability diagram expressed in the reduced coordinates, Q and d, presented in Fig. 3(a) . The parameters at T = 0 K are Q ( 0 ) = 2.65 , d ( 0 ) = 1.41 , and l ex ( 0 ) = 9.4 nm [the scaling law for the reduced DMI parameter is d ( m ) ∝ m β − α / 2 − 1 , i.e., d ( m ) ∝ m − 0.84 ). The dependence d ( m ) determines the evolution of the point describing the skyrmion configuration in the Q-d plane accounting for the change of temperature via the m ( T ) law [Fig. 3(a) ]. The increase of parameter d leads to the stabilization of the skyrmion state and a rapid expansion of the skyrmion radius. This effect is especially strong at H ext = 0 mT . A finite value of the field H ext , opposing the skyrmion core magnetization, results in a contraction of the skyrmion radius and the skyrmion radius weakly increases with T . In other words, the skyrmion radius increase is suppressed by the finite magnetic field which does not occur at zero field. 10.1103/PhysRevB.97.060402.f3 3 FIG. 3. (a) Q-d skyrmion stability diagram for H ext = 0 mT . The green squares indicate the metastable skyrmion states at T = 0 K and T = 300 K , while the magenta square indicates the stable skyrmion state at T = 400 K obtained with our parameters. The dashed red line represents the critical DMI parameter D c as defined in Ref.  . The blue arrows indicate how the state evolves as a function of temperature. Insets: example of spatial distributions of the magnetization at T = 0 , 300, and 400 K. Sketch of the energy profile of the dot when the skyrmion corresponds to (b) a metastable state and (c) when it occupies an absolute energy minimum (ground state). Two spatial distributions of the magnetization shown in (b) and (c) refer to deterministic full micromagnetic simulations with scaled values of the parameters for T = 300 K and T = 400 K , respectively. For all the spatial distributions of the magnetization, a color scale linked to the normalized z component of the magnetization is also illustrated. There are two qualitatively different behaviors of how the skyrmion radius R sk ( T ) changes with temperature at H ext = 0 mT according to the parameters Q and d : (1) The skyrmion radius R sk ( T ) is an almost constant function of temperature when the skyrmion configuration remains in the metastable region. (2) The skyrmion radius R sk ( T ) increases sharply with increasing temperature when approaching the region of skyrmion ground-state stability. The first scenario is realized when the T = 0 K parameters Q (0), d (0), and l ex ( 0 ) correspond to the skyrmion metastable state [Figs. 3(a) and 3(c) ]. In this case, the skyrmion radius at T = 0 K is small and has only a weak temperature dependence R sk ( T ) up to 200 K because the skyrmion is deep in the region of metastability in the Q-d phase diagram. The decreasing function d ( m ) guarantees that the skyrmion state stays in the ( Q , d ) area of the skyrmion metastability for increasing temperature. The second behavior occurs when a skyrmion, initially in the metastable state with parameters Q (0), d (0), and l ex ( 0 ) [see Figs. 3(a) and 3(b) ], moves toward the area of global stability in the Q-d space. This is realized for our dot magnetic parameters from T = 200 K to T = 350 K [green points in Fig. 3(a) ]. In this case, the skyrmion radius at T = 0 K is small, but it shows a strong dependence with increasing temperature R sk ( T ) . For further increasing of T up to 400 K, the skyrmion reaches the area of its stability [magenta point in Fig. 3(a) ] crossing the uniform state–skyrmion equilibrium line [red dashed line in Fig. 3(a) ]. At this line the skyrmion energy is equal to the perpendicular uniform state energy, d = d c ; the skyrmion radius is large and depends on the confining potential. In particular, the skyrmion radius calculated from deterministic micromagnetic simulations with scaled values of the parameters (corresponding to T = 400 K ) is much larger than the skyrmion radius for T = 300 K [compare the spatial distribution of the magnetization in Fig. 3(b) with that in Fig. 3(c) ]. In the region of stability where the skyrmion is the ground state of the ferromagnet, the ratio R sk / R is a weak function of all parameters. On the other hand, when considering in the micromagnetic simulations thermal fluctuations at T = 400 K , we observe large deformations of the skyrmion which turn into a “horseshoe”-like configuration as already observed in experiments  , while it remains quasicircular for T = 300 K [see insets in Fig. 3(a) ]. Here we do not consider stabilization of multiple skyrmions or labyrinth domain textures, assuming that the parameter d is not very large and we are always in the area of the single skyrmion stability. From a technological point of view, understanding how to move skyrmions between the metastability and the confined ground state can be exploited to enhance the electrical skyrmion detection  (see note 4 in the Supplemental Material and Supplemental Material Video 1  ). In summary, we developed a theoretical framework describing the skyrmion stability in ultrathin dots as a function of the external magnetic field and temperature by combining a proper ansatz with thermal scaling relations of the micromagnetic parameters A , K u , and D obtained by atomistic spin dynamics (although this direct estimation can be time-consuming, the scaling exponents can be alternatively retrieved from experimental measurements). We showed that the strong temperature dependence of the skyrmion diameter occurs because the thermal evolution of an initially metastable skyrmion brings it toward the region where the skyrmion is the ground state due to an increase of the effective DMI strength in comparison to the anisotropy. Our results, corroborated by extensive micromagnetic simulations, provide a tool, the Q - d phase diagram, to determine the transition from the metastable to ground-state skyrmion configuration, as well as the skyrmion size in the presence of both out-of-plane external field and temperature. Our achievements could be beneficial for racetrack memories where localized manipulation of magnetic parameters can be used to vary the skyrmion size and stability, improving the ability for skyrmion electrical detection. This work was supported by the program of scientific and technological cooperation between Italy and China (code CN16GR09) funded by Ministero degli Affari Esteri e della Cooperazione Internazionale, and by the bilateral Italy-Turkey project (CNR Grant No. B52I14002910005, TUBITAK Grant No. 113F378). R.T. and M.R. acknowledge support from Fondazione Carit Projects “Sistemi Phased-Array Ultrasonori” and “Sensori Spintronici.” K.G. acknowledges support by IKERBASQUE (the Basque Foundation for Science). The work of K.G. and O.C.-F. was supported by the Spanish Ministry of Economy and Competitiveness under Projects MAT2013-47078-C2-1-P, MAT2013-47078-C2-2-P, and FIS2016-78591-C3-3-R. J.B. acknowledges support from the Graduate Program in Spintronics, Tohoku University. Publisher Copyright: © 2018 American Physical Society.
PY - 2018/2/9
Y1 - 2018/2/9
N2 - Understanding the physical properties of magnetic skyrmions is important for fundamental research with the aim to develop new spintronic device paradigms where both logic and memory can be integrated at the same level. Here, we show a universal model based on the micromagnetic formalism that can be used to study skyrmion stability as a function of magnetic field and temperature. We consider ultrathin, circular ferromagnetic magnetic dots. Our results show that magnetic skyrmions with a small radius - compared to the dot radius - are always metastable, while large radius skyrmions form a stable ground state. The change of energy profile determines the weak (strong) size dependence of the metastable (stable) skyrmion as a function of temperature and/or field.
AB - Understanding the physical properties of magnetic skyrmions is important for fundamental research with the aim to develop new spintronic device paradigms where both logic and memory can be integrated at the same level. Here, we show a universal model based on the micromagnetic formalism that can be used to study skyrmion stability as a function of magnetic field and temperature. We consider ultrathin, circular ferromagnetic magnetic dots. Our results show that magnetic skyrmions with a small radius - compared to the dot radius - are always metastable, while large radius skyrmions form a stable ground state. The change of energy profile determines the weak (strong) size dependence of the metastable (stable) skyrmion as a function of temperature and/or field.
UR - http://www.scopus.com/inward/record.url?scp=85042145328&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85042145328&partnerID=8YFLogxK
U2 - 10.1103/PhysRevB.97.060402
DO - 10.1103/PhysRevB.97.060402
M3 - Article
AN - SCOPUS:85042145328
VL - 97
JO - Physical Review B
JF - Physical Review B
SN - 2469-9950
IS - 6
M1 - 060402