Curvature effects on phase transitions in chiral magnets

Periodical ground states of magnetization exist in chiral ferromagnetic films, if the constant of Dzyaloshinsii-Moriya interaction (DMI) exceeds some critical value $d_0$. Here, we demonstrate that $d_0$ can be significantly modified in curved film. As a case study, we consider tubular geometry of films with an easy axis anisotropy oriented normally to the film. For two different types of DMI we build phase diagrams of ground states for a wide region of the curvatures and DMI strengths. The periodical state of the tube can be considered as a sequence of domain walls (DW), similarly to the case of a planar film. However, in the presence of the curvature, the competition between DMI and exchange can lead to a new type of DW which is inclined with respect to the cylinder axis and possesses a structure which is intermediate between Bloch and N\'eel DW structures. The exact analytical solution for the new DW is obtained. All analytical calculations are confirmed by numerical simulations.

Periodical ground states of magnetization exist in chiral ferromagnetic films, if the constant of Dzyaloshinsii-Moriya interaction (DMI) exceeds some critical value d0. Here, we demonstrate that d0 can be significantly modified in curved film. As a case study, we consider tubular geometry of films with an easy axis anisotropy oriented normally to the film. For two different types of DMI we build phase diagrams of ground states for a wide region of the curvatures and DMI strengths. The periodical state of the tube can be considered as a sequence of domain walls (DW), similarly to the case of a planar film. However, in the presence of the curvature, the competition between DMI and exchange can lead to a new type of DW which is inclined with respect to the cylinder axis and possesses a structure which is intermediate between Bloch and Néel DW structures. The exact analytical solution for the new DW is obtained. All analytical calculations are confirmed by numerical simulations.
Introduction. -Magnetic nanostructure with arbitrary curvilinear shapes can acquire a multitude of groundstate configurations [1][2][3][4] under the twisting influence of the DMI and the effect of the curved surfaces/interfaces. Modulated states arise, if a nanomagnet has typical lengths comparable to the twisting length [5][6][7] which is determined by the material parameters and can be influenced by the curvature. The magnetic phase-diagram of curvilinear ferromagnets becomes much richer as compared to a flat specimen. Among simple curvilinear shapes, hollow cylindrical tubes or wires are very promising for a broad range of biomedical [8][9][10][11] and technological [12,13] applications, also see Review 14. Nanotubes can also be assembled into interconnected networks [15] which makes them attractive for advanced hardware concepts in neuromorphic computing [16]. It is important to note that magnetic nanotubes can be produced experimentally with different techniques [17][18][19][20][21][22].
Magnetic nanotubes belong to the simplest magnetic systems with pattern-induced chirality breaking [1]: two energetically equivalent vortex DWs with opposite chiralities possess different dynamical properties, leading to a suppression of the Walker breakdown [23] and Cherenkov-like radiation of magnons for fast DWs [24,25]. Additionally, tubular geometry results in the asymmetric spin-wave dispersion relation in azimuthally magnetized tubes [26,27], similarly to systems with intrinsic DMI [28,29]. In this context, an interrelation between effects due to intrinsic DMI and curvature-induced chirality is expected. An important question is, how the curvature modifies the critical DMI d 0 [5,7], which separates homogeneous and periodic magnetization structures. This is important for assessing the stability of skyrmions [30] and their motion [31] along the tubes and other curvilinear surfaces. Here, we present a detailed study of equilibrium states of the ferromagnetic nanotubes with intrinsic DMI of different symmetries. We show that: (i) The curvature modifies the critical DMI strength. (ii) new types of DWs appear in the periodic phase.
Model. -We consider the tubular shell as a ribbon of thickness h and width w, close-coiled upon the rod of radius R, see Fig. 1. Central line of the ribbon makes angle π/2 − ψ with the cylinder axis. The ribbon width is determined as w = 2πR sin ψ, this results in a closed cylindrical surface, i.e. without a bordering rim along the axis. The surface of the ribbon ς can be parameterized in the following way: ς (x 1 , x 2 ) = R cos (ρ s /R)x + R sin (ρ s /R)ŷ + ρ zẑ , where ρ s = x 1 cos ψ − x 2 sin ψ and ρ z = x 1 sin ψ + x 2 cos ψ, x 1 ∈ [0, L] and x 2 ∈ [−w/2, w/2] are coordinates within the ribbon surface, see Fig. 1. Such a nontrivial parametrization of the cylinder surface is useful for description of DWs which may be arbitrarily oriented along the tube axis, i.e. ψ defines the angle between the DW andẑ axis. Parametrization ς(x 1 , x 2 ) induces the natural tangential basis e α = ∂ α ς with the corresponding Euclidean metric tensor elements g αβ = e α · e β = δ αβ . Here, α, β = 1, 2 and ∂ α ≡ ∂ xα . Note that in our particular case e α are orthogonal vectors of unit length. This enables us to introduce the orthonormal basis {e 1 , e 2 , n}, where n = e 1 × e 2 is a normal vector to the surface, see Fig. 1.
Assuming small thickness of the coiled film, we consider the magnetization as a continuous function of two variables M = M (x 1 , x 2 ), which obeys the periodic boundary condition M (x 1 , w/2) = M (x 1 + T, −w/2) with T = 2πR cos ψ. The energy of the system is modelled by the functional where three contributions are taken into account. The first term in (1)   easy-normal anisotropy, where K > 0 and m n = m · n is the normal magnetization component. The competition between exchange and anisotropy results in the magnetic length = A/K, which determines a length scale of the system. The last term in (1) represents DMI contribution E d with D the DMI constant. We consider two types of DMI: is applicable for systems with T and O symmetries [29]. In the following this is called DMI of Bloch type, since for planar films it results in DWs and skyrmions of Bloch type. (ii) E n d = m n ∇ · m − m · ∇m n is valid for ultrathin films [32,33], bilayers [34] or materials belonging to C nv crystallographic group. In the following we call this DMI of Néel type. Here and below the indices b and n correspond to the Bloch and Néel DMI types, respectively.
Using a curvilinear reference frame we parametrize the magnetization in the following way m = sin θ cos φ e 1 + sin θ sin φ e 2 + cos θ n. Expressions for E x , E b d , and E n d for a general case of a local curvilinear basis were previously obtained in Refs. 35, 36, and 37, respectively. In the following we look for the equilibrium magnetization states. To this end we minimize energy (1) with respect to functions θ(x 1 , x 2 ), φ(x 1 , x 2 ) and constant ψ, see supplemental materials [38]. DMI of Bloch type. -First, we consider the case of For such kind of DMI we find two solutions [38]. The homogeneous (in the curvilinear reference frame) solution corresponds to the hedgehog state (m = ±n), its total energy normalized by E 0 = hwLK is where κ = /R is the dimensionless curvature. Additionally an inhomogeneous solution is found [38] with where d = D/ √ AK is DMI strength. It is important that angle φ b 0 , which defines orientation of the tangential magnetization component, is a coordinate independent constant. The relation (3) can be interpreted as follows: for given d and κ, there is a curvilinear frame of reference determined by the angle ψ b in which the magnetization angle φ b is constant. Angles φ b 0 and ψ b as functions of DMI strength are plotted in Fig. 2(a). For both types of DMI angle θ(x 1 ), which defines the magnitude of the normal magnetization component, depends on only one coordinate x 1 , oriented along the stripe, see  with the solution where am(•, •) is Jacobi's amplitude [39,40] and ξ = x 1 / is the dimensionless coordinate. The solution (5) describes the sequence of DWs oriented along the x 2 coordinate (perpendicularly to the ribbon, see Fig. 1). For each type of DMI, parameter λ = λ(κ, d) is a function of curvature and DMI strength. For well separated DWs, λ defines the DW width ∆ = 1/ √ λ. The integration constant C determines the period θ (ξ + T ) = θ (ξ) with K(•) is the complete elliptic integral of the first kind [39,40]. On the other hand, period T = T 0 | cos ψ|/q is predetermined by the periodical boundary conditions discussed above. Here q ∈ N determines the number of DWs N = 2q on the tube and T 0 = 2π/κ. N is even due to the periodical boundary conditions enforced by the tubular geometry. For the case of Bloch DMI constant C ≡ C b is determined by the equation (6) with ψ = ψ b taken from (3) and λ ≡ λ b = 1 + κ √ d 2 + κ 2 − κ /2. One should note that the simultaneous action of DMI and curvature decreases the width of the Bloch DW. For the corresponding period we use the notation T ≡ T b . Period T b as a function of the DMI strength is plotted in Fig. 2(b). The normalized energy of periodic states per where E(•) is the complete elliptic integral of the second kind [39,40]. For a planar film, the transition between the homogeneous and periodical state is characterized by infinite increase of period of the spiral state [7]. Although, for the cylindrical surface the period is finite in the transition point, for the limit case κ → 0 one has T → ∞. Using that C → 0 in this limit, one obtains from the equality E per b = E un b the analytical expression for the critical DMI where d 0 = 4/π is a critical DMI parameter for flat systems, which separates homogeneous and periodic magnetization distributions [5,7]. Although the expression (8) is obtained in the small curvature limit, it describes very well the existence region of the homogeneous state for the whole range of curvatures, see Fig. 3(a). The boundary (8) is also in a good agreement with results obtained by means of micromagnetic simulations in Ref. 31, see symbol ⊕ in Fig. 3(a). The equality of energies E per b (κ, d c , q) = E per b (κ, d c , q + 1) determines the boundary between states with different number of DWs. The resulting phase diagram is plotted in Fig. 3(a).
In the limit case of very small curvature (κ 1), the boundary curve (8) has the asymptotic behavior d b c ≈ ±d 0 ∓ 1 − 4/π 2 κ. Thus the curvature decreases the critical magnitude of the DMI strength. The boundary curve (8) intersects the abscissa with κ b 0 = 2/π. For κ > κ b 0 the periodical state with two DWs exists even without intrinsic DMI, see Fig. 3. This effect is analogous to the effect of spontaneous formation of the onion state in nanorings when curvature exceeds some critical value [41].
DMI of Néel type. -Let us now consider the case of Néel DMI E d = E n d . The energy of the hedgehog state (m = ±n) is reads [38] E un Similarly to the case of Bloch DMI, there is an inhomogeneous solution in form of periodical modulation. As well as in the previous case, the angle φ takes the constant value [38]: However, in contrast to the previous case, DWs are always aligned along the cylinder. This corresponds to the equilibrium value ψ n = 0 (or equivalently ψ n = π). As previously, the normal magnetization component is described by the same Eq. (5) with C ≡ C n determined by (6) with ψ = ψ n and λ ≡ λ n = 1 − κd. Note that the simultaneous action of DMI and curvature increases the width of the well separated Néel DWs. The normalized energy of the modulated state per period The equality of energies E un n (κ, d c ) = E per n (κ, d c ) determines the boundary between homogeneous and periodic states. In the small curvature limit one obtains As in the case of Bloch type DMI, the expression (12) describes the boundary of the homogeneous state in the phase diagram for a wide range of curvatures. The equality of energies E per n (κ, d c , q) = E per n (κ, d c , q + 1) determines the boundary between states with different number of DWs. The resulting phase diagram is plotted in Fig. 3(b). In the limit case of very small curvature (κ 1), the boundary curve (12) has the linear asymptotic behavior d n c ≈ ±d 0 − 2 1 + 4/π 2 κ. Thus, due to the curvature the absolute value of the critical DMI can be decreased as well as increased depending on the sign of DMI. Similarly to the case of Bloch type DMI, the boundary curve (12) intersects the abscissa with κ n 0 = κ b 0 = 2/π and for the case κ > κ n 0 the periodical state exists even without intrinsic DMI, see Fig. 3.
Conclusions. -We show that curvature modifies the value of critical DMI for curved systems, see Eqs. (8) and (12), which separates the hedgehog state with homogeneous magnetization normal to the film from the inhomogenous modulated states. For the case of Néel type of DMI this effect is approximately five times stronger (in the limit case κ 1). We found an exact solution for equilibrium states on the cylindrical surface for two different types of DMI and plotted the corresponding phase diagrams, see Fig. 3. The presence of the Néel DMI does not modify the structure of DWs, i.e. DWs are oriented along the cylinder axis (ψ n = 0) and they are of Néel type. For the case of Bloch DMI, the DWs are of a type intermediate between Bloch and Néel due to competition of intrinsic DMI and geometry-induced DMI of Néel type. These DWs are inclined by the angle ψ b ∈ (−π/4; π/4), see Eq. (3) and Fig. 3. The direction of DWs inclination (sign of the angle ψ b ) is defined by the sign of the DMI parameter. This effect is similar to the field-induced inclined DWs in flat stripes [42]. In our case the role of the external field is played by the geometry-induced easy-axial anisotropy along the cylinder axis. In both cases, the periodical boundary conditions, enforced by the closed cylindrical geometry, result in even number of domains on the cylinder.

I. THE MODEL OF FERROMAGNETIC CYLINDER
The surface parametrization ς(x 1 , x 2 ) induces the natural tangential basis g α = ∂ α ς with the corresponding metric tensor elements g αβ = g α · g β . Here, α, β = 1, 2 and ∂ α ≡ ∂ xα . As the vectors g α are orthogonal, one can introduce the orthonormal basis {e 1 , e 2 , n} with e α = g α √ g αα , n = e 1 × e 2 . (S1) Using the Gauß-Godazzi formula and Weingarten's equation [1, 2] one can obtain the following differential properties of the basis vectors Here, ∇ α ≡ (g αα ) −1/2 ∂ α (no summation over α) are components of the surface del operator and h αβ is a modified second fundamental form. The second fundamental form determines the Gauß curvature K = det h αβ and the mean curvature H = tr h αβ . Components of the spin connection vector Ω are determined by the relation Ω γ = 1 2 αβ e α · ∇ γ e β . Using curvilinear reference frame (S1), we introduce the following magnetization parametrization m = sin θ ε + cos θ n, ε = cos φ e 1 + sin φ e 2 , where θ and φ are magnetic angles, and ε is a normalized projection of the vector m on the tangential plane. We consider the ferromagnetic ribbon with the following energy functional The first term in (S4) is the exchange density E x = i=x,y,z (∂ i m) 2 with A the exchange constant. In the curvilinear reference frame exchange energy can be written as [3][4][5] E ex =∇ α m β ∇ α m β + ∇ α m n ∇ α m n +2h αβ (m β ∇ α m n − m n ∇ α m β ) + 2 αβ Ω γ m β ∇ γ m α + h αγ h γβ + Ω 2 δ αβ m α m β + H 2 − 2K m 2 n + 2 αγ h γβ Ω β m α m n . (S5a) Using the angular parametrization (S3) one can obtain [3][4][5] where Γ = h αβ · ε. The second term in (S4) corresponds to the Dzyaloshinskii-Moriya interaction (DMI) E d , with D being the DMI constant. In the curvilinear frame of reference the Néel type DMI E n d = m n ∇ · m − m · ∇m n can be written as [5] Using the angular parametrization (S3) one can obtain (up to the boundary terms) [5,6] E n d = 2 (∇θ · ε) sin 2 θ − H cos 2 θ + boundary terms, while, for the Bloch type DMI symmetry E b d = m · [∇ × m] this interaction in the curvilinear reference frame reads as [7] Substituting the angular parametrization (S3) into (S6c) results in the expression (up to the boundary terms) [7] The last term in (S4) corresponds to the uniaxial anisotropy E a = sin 2 θ, with K > 0 the easy-normal anisotropy constant.
The geometry of the curved ribbon is parameterized as ς (x 1 , x 2 ) = R cos (ρ s /R)x + R sin (ρ s /R)ŷ + ρ zẑ , where ρ s = x 1 cos ψ − x 2 sin ψ and ρ z = x 1 sin ψ + x 2 cos ψ, x 1 ∈ [0, L] and x 2 ∈ [−w/2, w/2] are coordinates within ribbon surface, see Fig. 1 (main text). This ribbon is close-coiled upon the rod with radius R. The close-coiled (without spaces) condition results in w = 2πR sin ψ and periodic conditions for the magnetization M (x 1 , w/2) = M (x 1 + 2πR cos ψ, −w/2). This parameterization results in the following first and modified second fundamental forms respectively. Tubular geometry has zero Gauß curvature K = 0, nonzero mean curvature H = −R −1 (here minus is related to the direction of the normal vector), and zero components of spin connection vector Ω = 0.
The equality of energies E per b (κ, d c , q) = E per b (κ, d c , q + 1) determines the boundary between states with different number of domain walls.
Energy as a function of DMI strength for Bloch DMI for different q is plotted in Fig. S1(a).
Energy as a function of DMI strength for Néel DMI for different q is plotted in Fig. S1(b).