Impurity effects in highly frustrated diamond-lattice antiferromagnets

We consider the effects of local impurities in highly frustrated diamond-lattice antiferromagnets, which exhibit large but nonextensive ground-state degeneracies. Such models are appropriate to many A-site magnetic spinels. We argue very generally that sufficiently dilute impurities induce an ordered magnetic ground state and provide a mechanism of degeneracy breaking. The states that are selected can be determined by a “swiss cheese model” analysis, which we demonstrate numerically for a particular impurity model in this case. Moreover, we present criteria for estimating the stability of the resulting ordered phase to a competing frozen (spin glass) one. The results may explain the contrasting finding of frozen and ordered ground states in CoAl${}_2$O${}_4$ and MnSc${}_2$S${}_4$, respectively.

Figure 1

Cubic cell of an AB${}_2$X${}_4$ spinel. The sublattice of A sites (large blue spheres) is a diamond lattice, while the sublattice of B sites (smaller yellow spheres) is a pyrochlore lattice.

Figure 2

Impurity model. A nonmagnetic impurity resides on a B site indicated by the black sphere. The six nearest-neighbor A sites form a distorted hexagon around the impurity (larger and darker blue spheres).

Figure 3

“Spiral surfaces” comprising the degenerate spiral ground-state vectors for varying coupling strengths $J_2/J_1$. The surfaces are grayscale coded according to the energies $E_1\left(\mathbf{q}\right)$ and $E\left(\mathbf{q}\right)$, respectively. Dark and gray indicate high and low values, respectively. The larger dark orange spheres denote the absolute minima. The edges of the first Brillouin zone are shown for orientation. The top row shows results for a single impurity $E_1\left(\mathbf{q}\right)$, while the bottom row shows results averaged over the four possible impurity sites $E\left(\mathbf{q}\right)$. Starred directions, e.g., ${100}^{*}$, denote sets of points on the spiral surface located “around” the corresponding (unstarred) direction, e.g.,100.

Figure 4

Deviation of the local spiral wave vector ${\mathbf{q}}_i$ from the one of the spiral state at the boundary $\mathbf{q}$. The left panel is for boundary spiral states with $\mathbf{q}$ pointing along the $1\overline{1}1$ direction in the coupling range $J_2/\left|J_1\right|<1/4$. The right panel is for boundary spiral states with $\mathbf{q}$ pointing along the 100 direction in the coupling range $J_2/\left|J_1\right|<1/4$. The impurity is embedded into a system of $N=2744=8\times 7^3$ spins. The symbols correspond to two different sets of neighboring spins, $P$ (circles) and $Q$ (boxes), used to calculate the local spiral wave vectors; details are given in the text. Both plots are dominated by very short distance decay; the amplitude of the distortion at distances $d\ge 2$ is not meaningful here.

Figure 5

Plot of ${\left[E\left(q\right)\right]}_S-E\left({\mathbf{q}}_0\right)$ versus frustration $J_2/J_1$. The shift of $T_c$ for the order-by-disorder phase per impurity is proportional to this quantity [see Eq. (37)].

Figure 6

Finite size effects. Deviation of the local spiral wave vector ${\mathbf{q}}_i$ from the one of the spiral state at the boundary $\mathbf{q}$ for systems of varying size $N=8\times L^3$. For the chosen couplings $J_2/\left|J_1\right|=0.2$ the spiral wave vector $\mathbf{q}$ points along the $1\overline{1}1$ direction with length $\left|\mathbf{q}\right|=0.71\pi$. The symbols correspond to two different sets of neighboring spins, $P$ (circles) and $Q$ (boxes), used to calculate the local spiral wave vectors; details are given in the text.