Crystalline phases in chiral ferromagnets: Destabilization of helical order

In chiral ferromagnets, weak spin-orbit interactions twist the ferromagnetic order into spirals, leading to helical order. We investigate an extended Ginzburg-Landau theory of such systems where the helical order is destabilized in favor of crystalline phases. These crystalline phases are based on periodic arrangements of double-twist cylinders and are strongly reminiscent of blue phases in liquid crystals. We discuss the relevance of such blue phases for the phase diagram of the chiral ferromagnet MnSi.

Figure 1

(Color online) Cut through a double-twist cylinder: Double-twist-configuration of the magnetization. Sheets of constant magnetization are wrapped as cylinders around a common axis.

Figure 2

Amplitude function $\lambda \left(r\right)$ which minimizes Eq. (7) using the directional dependence [Eq. (6)], the stabilizing term $h_1$ (with $\xi =-6.5$, $\eta =5$), and the boundary condition $\lambda \left(r\right)=0$ for $r>R$. Here, $q=0.38q_0$ and $R=3.98$ are free variational parameters where $q_0=1∕2$ is the helix wave vector. The resulting free energy gain per length is given by $f^{\mathrm{cyl}}=-0.21$.

Figure 3

(Color online) Crystalline structures built from double-twist cylinders: (a) Unit cell of the square lattice of double-twist cylinders in the $x\text{−}y$ plane. The magnetization on the cylinder axes, as well as at the points where cylinders touch, is shown explicitly. (b) Unit cell of the cubic lattice of double-twist cylinders. At the points where cylinders touch, the magnetization has twisted to an angle of 45° from the cylinder axis.

Figure 4

Phase diagram for $\eta =0.05$. In addition to the helical and the isotropic phases, there is a parameter range where the cubic lattice of double-twist cylinders minimizes the free energy.

Figure 5

Ratio of double-twist wave vector $q$ [see Eq. (6)] versus helix wave vector $q_0$ [see Eq. (4)] for $\eta =0.05$: $q∕q_0$ varies considerably but remains of order 1 in the parameter region where the cubic crystal is stable.