Coherence and stiffness of spin waves in diluted ferromagnets

We present the results of a numerical analysis of magnon spectra in supercells simulating two-dimensional and bulk random diluted ferromagnets with long-range pair exchange interactions. We show that low-energy spectral regions for these strongly disordered systems contain a coherent component leading to interference phenomena manifested by a pronounced sensitivity of the lowest excitation energies to the adopted boundary conditions. The dependence of configuration averages of these excitation energies on the supercell size can be used for an efficient determination of the spin-wave stiffness $D$. The developed formalism is applied to the ferromagnetic Mn-doped GaAs semiconductor with optional incorporation of phosphorus; the obtained concentration trends of $D$ are found to be in reasonable agreement with recent experiments. Moreover, a relation of the spin stiffness to the Curie temperature $T_{\mathrm{C}}$ has been studied for Mn-doped GaAs and GaN semiconductors. It is found that the ratio $T_{\mathrm{C}}/D$ exhibits qualitatively the same dependence on Mn concentration in both systems.

Figure 1

The number of states $N\left(E\right)$ for the diluted 2D ferromagnet with 3% of magnetic sites simulated by a square supercell with $200\times 200$ sites (solid line) and by a rectangular supercell with $150\times 300$ sites (dashed line).

Figure 2

The configurationally averaged lowest excitation energies $E_1$ for the diluted 2D ferromagnet as functions of the $\mathbf{q}$-vector magnitude derived from the supercell size: for square supercells (open squares) and for rectangular supercells (solid diamonds). The straight dotted line results from a fit, and the dash-dotted line refers to the VCA dependence.

Figure 3

The number of states $N\left(E\right)$ for GaAs doped by 5% Mn simulated by a cubic supercell with $4\times {28}^3$ sites (solid line) and by a rhombohedral supercell with ${44}^3$ sites (dashed line).

Figure 4

The configurationally averaged lowest excitation energies $E_1$ for Mn-doped GaAs as functions of the $\mathbf{q}$-vector magnitude derived from the supercell size for cubic supercells (open squares) and for rhombohedral supercells (solid diamonds). The straight dotted line results from a fit, and the dash-dotted line refers to the VCA dependence.

Figure 5

The calculated spin-wave stiffness $D$ as a function of Mn concentration in (Ga,Mn)As (open circles). The experimental values (solid circles) [13] are shown as well.

Figure 6

Correlations between the Curie temperature $T_{\mathrm{C}}$ and the spin-wave stiffness $D$ as calculated for (Ga,Mn)As (open circles) and (Ga,Mn)N (solid diamonds).

Figure 7

The ratio $T_{\mathrm{C}}/D$ as a function of Mn concentration for (Ga,Mn)As (open circles) and (Ga,Mn)N (solid diamonds).