Anisotropy of the Magnetic Phases in Cubic Helimagnets

Abstract

The magnetic phases of cubic chiral ferromagnets with the Dzyaloshinskii–Moriya interaction (MnSi, Cu₂OSeO₃) are studied in terms of microscopic and phenomenological approaches. The interaction of the magnetic moments of atoms with a local crystal field is shown to cause a cubic anisotropic term $M_{x}^{4}$ + $M_{y}^{4}$ + $M_{z}^{4}$ (M is the magnetization field) in the Landau–Lifshitz energy. This term is responsible for the existence of the helical phase at a near-zero magnetic field and can contribute to the stability of the magnetic A phase in higher fields. The A phase is simulated at various magnetic fields. When a magnetic field is turned on, the helices in the helical phase are shown to acquire elliptical and conical components.

APPENDIX

We now determine parameters θ, λ, and m of a conical–elliptical helix from Section 3.2, which are necessary to calculate the energy and the magnetic susceptibility of the helical phase.

Substituting Eq. (17) into Eq. (9) for the energy density, at θ = const we obtain

$$\mathcal{E} = {{\sin }^{2}}\theta \frac{{d\varphi }}{{dz}}\left( {\frac{1}{2}\frac{{d\varphi }}{{dz}} - 1} \right) - {{h}_{{||}}}\cos \theta - {{h}_{ \bot }}\sin \theta \cos \varphi .$$

(48)

The Euler–Lagrange equation

$$\frac{{{{d}^{2}}\varphi }}{{d{{z}^{2}}}} = \frac{{{{h}_{ \bot }}}}{{\sin \theta }}\sin \varphi $$

(49)

coincides with the equation of motion of a mathematical pendulum with the first integral

$${{\left( {\frac{{d\varphi }}{{dz}}} \right)}^{2}} = \frac{{2{{h}_{ \bot }}}}{{\sin \theta }}(a - \cos \varphi ).$$

(50)

Substituting solution (18) into Eq. (50) the, we find

$$a = \frac{2}{m} - 1,\quad {{\lambda }^{2}} = \frac{{{{h}_{ \bot }}}}{{m\sin \theta }}.$$

(51)

Substituting Eqs. (50) and (51) into Eq. (48) and averaging over z, we obtain

$$\begin{gathered} \langle \mathcal{E}\rangle = - k{{\sin }^{2}}\theta - {{h}_{{||}}}\cos \theta \\ - \,{{h}_{ \bot }}\sin \theta (2{\text{/}}m - 1 - 2\langle \cos \varphi \rangle ), \\ \end{gathered} $$

(52)

where the notation k = 〈dφ/dz〉 = 2π/p is introduced and p is the helix period. Taking into account that p = 2K(m)/λ (K(m) is the half-period of the Jacobian function), from Eq. (51) we derive the following equation for parameter m of elliptic functions:

$$m{{K}^{2}}(m) = \frac{{{{\pi }^{2}}{{h}_{ \bot }}}}{{{{k}^{2}}\sin \theta }}.$$

(53)

At low fields m ~ h_⊥, and with allowance for the expansion

$$K(m) \approx \frac{\pi }{2}\left( {1 + \frac{m}{4} + \frac{{9{{m}^{2}}}}{{64}}} \right)$$

solution (53) takes the form

$$m \approx \omega - \frac{{{{\omega }^{2}}}}{2} + \frac{{5{{\omega }^{3}}}}{{32}},\quad \omega = \frac{{4{{h}_{ \bot }}}}{{{{k}^{2}}\sin \theta }},$$

(54)

$$\langle \cos \varphi \rangle = \frac{1}{K}\int\limits_0^{2K} {{\text{s}}{{{\text{n}}}^{2}}(x;m)dx - 1 \approx \frac{m}{8}.} $$

(55)

Substituting Eqs. (54) and (55) into Eq. (52) and preserving the terms up to the second order of smallness in h inclusively, we have

$$\langle \mathcal{E}\rangle = {{\sin }^{2}}\theta \left( {\frac{{{{k}^{2}}}}{2} - k} \right) - {{h}_{{||}}}\cos \theta - \frac{{h_{ \bot }^{2}}}{4}.$$

(56)

Performing minimization with respect to k and θ, we obtain k = 1 and cosθ = h_|| and eventually have

$$\langle \mathcal{E}\rangle = - \frac{1}{2} - \frac{{h_{{||}}^{2}}}{2} - \frac{{h_{ \bot }^{2}}}{4}.$$

(57)