Decagonal Quasiferromagnetic Microstructure on the Penrose Tiling

The stable magnetization configurations of a ferromagnet on a quasiperiodic tiling have been derived theoretically. The magnetization configuration is investigated as a function of the ratio of the exchange to the dipolar energy. The exchange coupling is assumed to decrease exponentially with the distance between magnetic moments. It is demonstrated that for a weak exchange interaction the new structure, the quasiferromagnetic decagonal configuration, corresponds to the minimum of the free energy. The decagonal state represents a new class of frustrated systems where the degenerated ground state is aperiodic and consists of two parts: ordered decagon rings and disordered spin-glass-like phase inside the decagons.

Figure 1

(a) A section of the Penrose tiling. The original Penrose rhombic tiles and the decagonal tiles are indicated. Two allowed overlapping of decagonal clusters are shown as $A$ and $B$. (b) The original Penrose rhombic tiles. Five nearest neighbor distances (the sides and the diagonals of the rhombuses) and their lengths are given. $\tau$ is the golden mean. The two strongest exchange bonds according to two shortest nearest neighbor distances are denoted as $J$ and $J^{\prime }$.

Figure 2

(a) Monte Carlo simulations. Top view of the portion of the magnetic microstructure in a sample of finite size for pure dipolar interaction, i.e., $R=J/D=0$. The microstructure has been obtained for a square sample of about 10 500 vector spins on the Penrose tiling for $D/k_{B}T=100$. The spins belonging to the perimeter of decagons (marked) form closed chains. The chains overlap according to rules given in Fig. 1. (b) Experimental model. The perspective view of the magnetic microstructure. The red arrows represent the orientation of dipolar moments of magnets fixed onto the nodes of the Penrose tiling (rhombuses). The magnets can rotate in the horizontal plane.

Figure 3

(a) Top view of the portion of the quasiferromagnetic spin configuration in a sample of finite size for $\rho =1.176a\tau$ and $R=J/D=5$. The magnetic moments are nearly coplanar to the sides of the decagons. The $X$ component of the average magnetization is $M_X=0.85$. (b) An example of a planar spin configuration in the region of transition from the single domain to the decagonal structure for $\rho =1.176a\tau$ and $R=J/D=0.4$. The microstructures have been obtained for square and disk-shaped samples of 400 and 10 500 magnetic moments at $J/(k_{B}T)=100$. The magnetic moments at the edges are oriented mainly parallel to the boundary as in a conventional vortex structure. However, only local vortices inside the decagons exist.