Spiral spin state with open boundary conditions in a magnetic field

In order to model a spiral spin state in a thin film, we study a classical Heisenberg model with open boundary conditions. With magnetic field applied in the plane of the film, the spin state becomes ferromagnetic above a critical field that increases with thickness $N$. For a given $N$, the spiral passes through states with $n=n_0$ up to 0 complete periods in steps of 1. These numerical results agree with earlier analytic results in the continuum limit and help explain the susceptibility jumps observed in thin films.

Figure 1

A spiral spin state propagating along $\mathbf{z}$. The thin film has dimensions $M\times M\times N$, where $N$ is the thickness and $M\gg N$.

Figure 2

Phase diagram of spiral spin state with dimensionless DM interaction $d=0.16$. Plots show dimensionless magnetic field $b$ versus thickness $N$. The number of complete periods of the spiral is $n$. (a) Green squares were evaluated by sweeping the thickness, and blue circles by sweeping the magnetic field. The horizontal dashed line shows $b_c\approx {(\pi /4)}^2{d}^2=0.0157$ [12]. (b) Square points separate regions with different values of $N$ at the phase boundaries.

Figure 3

Spin along the field direction as a function of site $i$ for different thicknesses and $b=0.01$. In (a), $N=137$ and 139 correspond to $n=2$ and 3, respectively. In (b), $N=179$ and 181 correspond to $n=3$ and 4, respectively.

Figure 4

Spin along the field direction as a function of site $i$ for $N=119$ and (a) $b=0$ and 0.0005, and (b) $b=0.001$. Starting with $n=2$ complete periods at $b=0$, the spin configuration changes to $n=3$ at $b=0.0005$ and back to $n=2$ at $b=0.001$. The increase in $n$ in (a) is produced by the downturn in $cos{\theta }_i$ at the end sites, as shown in the inset.

Figure 5

Average magnetization as a function of magnetic field for thicknesses $N=159$–163 (a) and $N=211$ (b). The vertical dashed line shows $b_c=0.0157$ [12]. Circles in (b) are the numerical results with the fixed boundary condition taken from Ref. [14]. The inset of (a) shows the jump from $n=4$ to 3 for each thickness $N$. The inset of (b) shows the jumps from $n=2$ to 0.