Chiral surface twists and skyrmion stability in nanolayers of cubic helimagnets

Lorentz transmission electron microscopy (LTEM) investigations of modulated states in a FeGe wedge and detailed calculations demonstrate that chiral twists arising near the surfaces of noncentrosymmetric ferromagnets (Meynell et al. Phys. Rev. B, 90, 014406 (2014)) provide a stabilization mechanism for skyrmion lattices and helicoids in cubic helimagnet nanolayers. The calculated magnetic phase diagram for free standing cubic helimagnet nanolayers shows that magnetization processes in these compounds fundamentally differ from those in bulk cubic helimagnets and are characterized by the first-order transitions between modulated phases and the formation of specific multidomain states. The paper reports LTEM observations of multidomain patterns in FeGe free-standing nanolayers.


I. INTRODUCTION
Dzyaloshiskii-Moriya (DM) interactions 1 stabilize two-dimensional axisymmetric solitonic states (chiral skyrmions) in saturated phases of magnetic materials with broken inversion symmetry 2,3 .In uniaxial noncentrosymmetric ferromagnets chiral skyrmions condense into hexagonal lattices below a certain critical field and remain thermodynamically stable (correspond to the global minimum of the magnetic energy functional) in a broad range of applied magnetic fields 3 .This does not occur in bulk cubic helimagnets where one-dimensional modulations along the applied field (the cone phase) 4 have the lowest energy practically in the whole area of the magnetic phase diagram, and skyrmion lattices can exist only as metastable states 5,6 .
Two physical mechanisms have been proposed to date to explain the formation of skyrmion lattices in confined cubic helimagnets.One of them is based on effects imposed by induced uniaxial anisotropy 5,6 .In epilayers of cubic helimagnets on Si (111) substrates, a strong uniaxial anistropy is induced by the lattice mismatch between the B20 crystal and the substrate 6,13 .This uniaxial anisotropy suppresses the cone phase and stabilizes a number of nontrivial chiral modulated states including out-of-plane and in-plane skyrmion lattices recently observed in cubic helimagnet epilayers 6,10,12 .
The second stabilization mechanism is provided by specific modulations (chiral twists) arising near the surfaces of confined cubic helimagnets [16][17][18] .Chiral twists have been recently discovered in MnSi/Si(111) films 16,18 .However, their influence on the magnetic states arising in confined cubic helimagnets is still unclear.Also, physical mechanisms underlying the formation of skyrmionic states in free standing films of cubic helimagnets are unknown and a theoretical description of arising magnetic states in these systems is still an open question.
In this paper we report LTEM investigations of modulated states in a FeGe wedge and theoretical analysis of magnetic states in confined cubic helimagnets.Our findings show that surface twist instabilities play a decisive role in the stabilization of skyrmionic states in free standing layers of cubic helimagnets.

A. Model
The standard model for magnetic states in cubic noncentrosymmetric ferromagnets is based on the energy density functional 1,4 including the principal interactions essential to stabilize modulated states: the exchange stiffness with constant A, Dzyaloshinskii-Moriya (DM) coupling energy with con-stant D, and the Zeeman energy; m = (sin θ cos ψ; sin θ sin ψ; cos θ) is the unity vector along the magnetization vector M = mM , and H is the applied magnetic field.We investigate the functional (1) in a film of thickness L infinite in x− and y− directions and confined by parallel planes at z = ±L/2 in magnetic field H applied along z− axis (Fig. 1 a).
The equilibrium magnetic states in the film are derived by the Euler equations for energy functional (1) together with the Maxwell equations and with corresponding boundary conditions.The solutions depend on the two control parameters of the model ( 1), the confinement ratio, ν and the reduced value of the applied magnetic field, where L D is the helix period and H D is the saturation field 3,4 .

B. Modulated states in bulk cubic helimagnets
Magnetic states in bulk cubic helimagnets are commonly described by unconfined modulated states including the following three phases 1,3,4 : (i) Cones are chiral single-harmonic modulations along the applied field.The solutions for the cone phase and the equilibrium energy density are derived in analytical form 4 cos where K 0 = D 2 /(4A) = µ 0 H D M/2 is the effective easyplane anisotropy imposed by the cone modulations 5,6 .
(ii) Helicoids are one-dimensional chiral modulations with the propagation direction perpendicular to the applied field and homogeneous along the direction of the applied field 1 .Helicoids propagating along the x-axis are described by solutions (θ(x), ψ = π/2).The Euler equation for the helicoid energy density yields a set of parametrized periodic solutions θ(x, l) where the parameter l designates the period of helicoids.The equilibrium period l 0 and profile θ(x, l 0 ) are derived by minimization of the helicoid energy density with respect to l 1 .
(iii) Skyrmion lattices.The axisymmetric cores of chiral skyrmion lattice cells are described by solutions 3 where r = (ρ cos ϕ, ρ sin ϕ, z) are cylindrical coordinates of the spatial variable.The equilibrium periods and magnetization profiles θ(ρ) of the skyrmion lattice cells are derived by minimization of the energy density functional 3 ρ sin θ cos θ for different values of the core radii R, and optimization of the mean energy density of the skyrmion lattice with respect to R.
Among these solutions, the cone phase (4) corresponds to the global minimum of model (1) over the whole region where chiral modulations occur (H < H D ).The helicoids and skyrmion lattices exist as metastable states below the critical fields H h = 0.617H D and H s = 0.801H D correspondingly 1,3 .

C. Modulated states in confined cubic helimagnets
The solutions for unconfined helicoids (5) and skyrmion lattices (6) homogeneous along the film normal describe magnetic configurations in the depth of a bulk cubic helimagnet.However, the situation changes radically near the film surfaces.The gradient term, m x ∂m y /∂z − m y ∂m x /∂z in the DM energy functional (Eq.( 1)) violates transversal homogeneity of helicoids and skyrmion states and imposes chiral modulations along the z− axis that decay into the depth of the sample (surface twists) 6,16,17 .The penetration depth of these surface modulations is comparable with the characteristic length L D 6 .Mathematically, axisymmetric skyrmion cells in thin films are described by solutions of type θ = θ(ρ, z), ψ = ψ(ϕ, z), and helicoids propagating in a film along the xaxis are described by solutions of type θ(x, z), ψ(x, z).
The energy density functional for confined helicoids (w h (θ, ψ)) and skyrmion lattices (w s (θ, ψ)) can be written in the following form (8)   where the exchange (J h(s) ) and Dzyaloshinskii-Moriya (I h(s) ) energy functionals read as ρ sin θ cos θ ψ ϕ ) + sin 2 θ ψ z .The equilibrium solutions for confined helicoids and skyrmion lattices are derived by solving the Euler equations for functional (8) with free boundary conditions at the film surfaces (z = ±L/2).
Most of the investigated free standing films and epilayers of cubic helimagnets have a thickness exceeding the period of the helix (L ≥ L D ).In this paper we carry out detailed analysis of the solutions for confined chiral modulations in cubic helimagnetic films with the thickness ranging from L = L D to a bulk limit (L L D ).

III. MAGNETIC PHASE DIAGRAM A. Results of numerical simulations
The calculated ν − h phase diagram in Fig. 2 indicates the areas with the chiral modulated states corresponding to the global minimum of the energy functional and separated by the first-order transition lines.For L L D the solutions for confined helicoids and skyrmion lattices approach the solutions for the magnetic states in the unconfined case( 5), (6), which are homogeneous along the z-axis 1,3 .Surface twist instabilities arising in confined cubic helimagnets 17,18 provide a thermodynamical stability for helicoids and skyrmion lattices in a broad range of the applied fields (Fig. 2).
Another noticeable feature of the phase diagram is that the line h = 0.4 is a symmetry axis for the skyrmion lattice stability area.This follows from the fact that in bulk helimagnets this field corresponds to the minimal value of the skyrmion lattice energy compared to that of the cone phase 5,6 .This effect plays a crucial role in the formation of the A-phase pocket near the ordering temperature of bulk cubic helimagnets (for details see the Ref. 6 ).The differences between the equilibrium average energy densities of the skyrmion lattice ( ws ) and the energy density of the cone phase (w c ) ∆w ν (h) = ws (h, ν)−w c (h, ν) plotted as functions of the applied field also reach the minimum in the fields close to h = 0.4 (Inset of Fig. 2).As a result, below ν q = 7.56, the stability area of the skyrmion lattices extends around the line h = 0.4.
In the whole range of the film thickness, the helicoids with in-plane propagation directions correspond to the group state of the system.The triple point p (4.47, 0.232) and the completion point q (7.56, 0.40) split the phase diagram into three distinct areas with different types of the magnetization processes.
(I) ν > ν q = 7.56.In these comparatively thick films, the helicoids remain thermodynamically stable at low fields and transform into the cone by a first-order process at the critical line h hc (ν).The cone magnetization along the applied field increases linearly for increasing magnetic field up to the saturation at critical field H = H D .
(II) 4.47 = ν p < ν < ν q = 7.56.In this case, the magnetic-field-driven evolution of the cone is interrupted by the first-order transition in the skyrmion lattice at h hc (ν) < h q and the re-entrant transition at h hc (ν) > h q .
(III) 1 < ν < ν p = 4.47.In this thickness range, the stability area of skyrmion lattices is separated from the low field helicoid and high field cone phases by the first order transition lines.

B.
Analytical solutions for surface twists In cubic helimagnet films with L ≥ L D , twisted modulations in helicoids and skyrmions (ξ(z)) exist only in narrow regions near the film surfaces δ L. This allows us to write solutions for helicoids as θ = θ(x), ψ = π/2+ξ(z)   6)).The image size is 3000 nm × 1250 nm, the thickness varies from 90 nm (bottom) to 60 nm (top).
(the x axis is directed along the propagation direction), where θ(x) is the solution homogeneous along the z axis investigated in 1 .We write the solutions for a skyrmion lattice core as θ = θ(ρ), ψ = π/2 + ϕ + ξ(z) where and θ(ρ) is the solutions for skyrmions homogeneous along their axes 3 .The energy density of the surface twists in the helicoid (skyrmion lattice) can be reduced to the following form: e h(s) (ξ) = ∆ wh(s) (ξ) = m 2 x h(s) F h(s) (ξ) where K 0 is the effective anisotropy (4), and ρ sin θ cos θ ρdρ.The energy functional F h(s) (ξ) (9) describes surface twists ξ(z) in helicoids (skyrmion lattices) and has the same functional form as the energy functional for surface twists in a saturated helimagnet 18 .The Euler equation for ( 9) can be readily solved analytically.The equilibrium amplitude of twist modulations ξ(z) reaches the largest value on the film surface, (10)   and decays exponentially into the layer depth, tan(ξ h(s) /4) = tan(ξ Inserting (11) into the energy density (9) leads to the following expression for the negative energy density contribution imposed by surface twist modulations: The fractions of the negative surface contribution ēh(s) ∝ 1/L (12) in the total energy balance increase with decreasing film thickness, extending the stability areas of the helicoids and skyrmion lattices (2).
The magnetic phase diagram in Fig. 2 has been derived by minimization of the simplified ("isotropic") energy functional (1) with free boundary conditions.This demonstrates how a pure geometrical factor (confinement) influences the energetics of cubic helimagnet nanolayers by imposing transerverse chiral modulations (twists) in skyrmion lattices and helicoids.For the practically important thickness range L ≥ L D , the influence of chiral twists can be described by the surface energy term (12) in the cubic helimagnet energy.(color online).Contour plots show the equilibrium structures of domain boundaries between the cone and the competing skyrmion lattice (a) and helicoid phases (b) during the first-order transition.The applied magnetic field is directed perpendicular to the figures planes.The domain wall between the cone and skyrmion lattice is calculated at the critical line hsc(ν) in the vicinity of the completion point q (h = 0.40) (a), and the domain wall between the cone and helicoid at the transition line h hc (ν) for h = 0.01 (b).The bottom panel shows the reduced energy density profiles through the domain wall thickness ∆e(x) = |(w(x) − wc(h))/wc(h)| where wc(h) is defined by Eq. ( 4).
In bulk cubic helimagnets, one-dimensional singleharmonic chiral modulations (helices and cones) are observed as stable states over practically the entire region below the saturation field 20 .Contrary to bulk specimens, in free standing nanolayers of cubic helimagnets with thickness L ≤ 120 nm investigated by LTEM methods, skyrmion lattices and helicoids are observed in broad ranges of applied magnetic fields and temperatures, while the cone phase is partially or completely suppressed [7][8][9] .9]25 and bibliography in 22 ).
In our paper we use LTEM to explore first-order phase transitions into the cone phase and other specific magnetization processes imposed by the chiral surface twists (Fig. 2).For our studies, we have prepared wedge-shape single crystal FeGe(110) films.FeGe single crystals were grown by a chemical vapor transport method.A thin film specimen was made for TEM observations by using a focused ion beam technique.A series of Lorentz micrographs were taken by means of a Fresnel mode of Lorentz microscopy with a typical defocus value of 10 micrometer at T = 110 K and 250 K in a broad range of magnetic fields applied perpendicular to the film surface (Figs. 3,  4).They clearly expose the magnetic-field-driven firstorder transitions between the basic modulated states accompanied by the formation of the multidomain patterns composed of domains of the competing phases.
It should be noted that at LTEM images the domains of the cone phase appear as dark areas and cannot be distinguished from domains of the saturated state.However, in Figs. 3, 4, the dark domains arise at applied fields lower than the saturated fields (for FeGe, the saturation field µ 0 H D = 0.359 T (3) 20,21 ).Moreover, according to the theoretical results 1,3 and experimental observations 7,14 , the magnetic-field-driven transitions of the helicoid and skyrmion lattice into the saturated state advance gradually by the extension of the modulation period and formation of isolated helicoidal kinks and skyrmions.These processes exclude the formation of multidomain patterns of the competing phases characteristic of the first-order transitions 15 .
In Fig. 3, the layer thickness varies from L = 140 nm (ν = 2) at the left edge to L = 60 nm (ν = 0.86) at the right edge.In the calculated phase diagram, this thickness interval (0.86 < ν < 2) belongs to area III characterized by the first-order transitions between the helicoid and skyrmion lattice at the lower field, h hs (ν), and between the skyrmion lattice and cone at higher field, h sc (ν) (Fig. 2).Both these phase transitions are clearly observed in Fig. 3.Because the transition field h sc (ν) has lower values for larger ν, initially the cone phase nucleates at the thicker edge of the film (Figs. 3, (4)) and expands to the thinner part with an increasing applied field (Figs. 3, (5)).
The LTEM images derived at T = 110 T (Fig. 4) have been done for a wedge area belonging to the same thickness interval as that in Fig. 3 with the thickness variation from L = 90 nm (ν = 1.29) at the bottom edge to L = 60 nm (ν = 0.86) at the top edge.However, the magnetization evolution differs drastically from that observed at higher temperature.In this case, a skyrmion lattice does not arise, instead the helicoid directly transforms into the cone phase at a considerably lower field by a first-order process (Fig. 4 (2), ( 3) ).In the (ν, h) phase diagram (Fig. 2) such a magnetization evolution occurs in the area I for ν > ν q = 7.56.The suppression of skyrmion lattices and helicoids at lower temperatures is characteristic for free-standing cubic helimagnet nanolayers 8,9 .Particularly, at T = 110 K, the skyrmion lattices arise in FeGe free-standing layers only when their thickness is smaller than 35 nm 8 .This effect can be understood if we assume that the surface energy imposed by chiral twists σ h(s) (12) decreases with decreasing temperatures.As a result, at lower temperatures the existence area of skyrmion lattices in the (ν, h) phase diagram (2) would be shifted into the region of lower ν.
Finally we consider specific domain wall separating domains of the competing modulated phases during the first-order transition.The transition between the helicoid and skyrmion lattice occurs in bulk and confined chiral helimagnets 3,17 .Domain walls between the coexisting helicoids and skyrmion lattices 3 have been observed in confined cubic helimagnets 7,23,24 and FePd/Ir bilayers 26 .
In the multidomain patterns in Figs. 3, 4 the domain boundaries between the cone phase and the helicoids and skyrmion lattices represent specific transitional areas providing the compatibility of the chiral modulations along the applied field (the cone) with the in-plane modulated phases.The contour plots in Fig. 5 describe the equilibrium structure of isolated domain walls between the cone and skyrmion lattice calculated for h = 0.4 and between the cone and helicoid at h = 0.These calculations have been carried out for homogeneous along the film thickness domains of the helicoid and skyrmion lat-tice phases.The energy density profiles of the domain walls in Fig. 5 show that the potential barriers separated the equilibrium modulated phases in domain are estimated as ∆w max (h) = ∆e(0)|w c (h)| ≈ 10 −2 K 0 .

V. CONCLUSIONS
The results of micromagnetic calculations for confined chiral modulations and LTEM investigations of magnetic states in a FeGe wedge demonstrate that chiral surface twists provide the stabilization mechanism for helicoids and skyrmion lattices in free standing cubic helimagnet films.For a practically important thickness range L ≥ L D , chiral twist modulations have sizable values only near the film surfaces and can be described analytically as localized surface states exponentially decaying into the film depth (Eqs.(10), (11)).The stabilization energy in this case is described by the surface energy contributions (12).
The solutions minimizing the energy functional (1) with free boundary conditions describe chiral modulations imposed solely by the geometrical confinement and expose three basic types of the magnetization processes in cubic helimagnet nanolayers (Fig. 2).In real system, however, the confined chiral modulations arise as a result of the interplay between the stabilization mechanism imposed by the geometrical confinement and other physical factors, such as intrinsic cubic anisotropy and induced volume and surface uniaxial anisotropy, internal and surface demagnetization effects.Our findings provide a conceptional basis for detailed experimental and theoretical investigations of the complex physical processes underlying the formation of skyrmion lattices and helicoids in confined noncentrosymmetric magnets.

FIG. 1 .
FIG. 1. (color online).Magnetic structure of a helicoid with period l (a) and a skyrmion lattice cell of radius Rs (b,c) in nanolayers of cubic helimagnets.In the internal area (i) the helicoid has in-plane modulations along the x-axis, the surface areas (s) are modulated along the x and z axes.

FIG. 2 .
FIG. 2. (color online) The magnetic phase diagram of the magnetic states corresponding to the global minima for model (1) in reduced variables for the film thickness ν = L/LD and applied magnetic field h = H/HD.The existence areas of the modulated phases (cone, helicoids, and skyrmion lattice) are separated by the first-order transition lines (solid).p (4.47, 0.232) is a triple point, q (7.56, 0.40) is a completion point.Dashed line indicates the second-order transition between the cone and saturated state.Along the dotted line Ha = 0.4HD, the difference between the energy densities of the skyrmion lattice and the cone phase (∆wν ) is minimal (Inset).

FIG. 5 .
FIG. 5.(color online).Contour plots show the equilibrium structures of domain boundaries between the cone and the competing skyrmion lattice (a) and helicoid phases (b) during the first-order transition.The applied magnetic field is directed perpendicular to the figures planes.The domain wall between the cone and skyrmion lattice is calculated at the critical line hsc(ν) in the vicinity of the completion point q (h = 0.40) (a), and the domain wall between the cone and helicoid at the transition line h hc (ν) for h = 0.01 (b).The bottom panel shows the reduced energy density profiles through the domain wall thickness ∆e(x) = |(w(x) − wc(h))/wc(h)| where wc(h) is defined by Eq. (4).