Geometrical frustration and the Magnetic ordering

Geometrical frustration is an important feature in magnetism, where it stems from the relative arrangement of spins. A simple 2D example is shown in Figure 1. Three magnetic ions reside on the corners of a triangle with antiferromagneticinteractions between them; the energy is minimized when each spin is aligned opposite to neighbors. Once the first two spins align anti-parallel, the third one is frustrated because its two possible orientations, up and down, give the same energy. The third spin cannot simultaneously minimize its interactions with both of the other two. Since this effect occurs for each spin, the ground state is sixfold degenerate. Only the two states where all spins are up or down have more energy.

Figure 1

Similarly in three dimensions, four spins arranged in a tetrahedron (Figure 2) may experience geometric frustration. If there is an antiferromagnetic interaction between spins, then it is not possible to arrange the spins so that all interactions between spins are antiparallel. There are six nearest-neighbor interactions, four of which are antiparallel and thus favourable, but two of which (between 1 and 2, and between 3 and 4) are unfavourable. It is impossible to have all interactions favourable, and the system is frustrated.

Figure 2

Geometrical frustration is also possible if the spins are arranged in a non-collinear way. If we consider a tetrahedron with a spin on each vertex pointing along the easy axis (that is, directly towards or away from the centre of the tetrahedron), then it is possible to arrange the four spins so that there is no net spin (Figure 3). This is exactly equivalent to having an antiferromagnetic interaction between each pair of spins, so in this case there is no geometrical frustration. With these axes, geometric frustration arises if there is a ferromagnetic interaction between neighbours, where energy is minimized by parallel spins. The best possible arrangement is shown in Figure 4, with two spins pointing towards the centre and two pointing away. The net magnetic moment points upwards, maximising ferromagnetic interactions in this direction, but left and right vectors cancel out (i.e. are antiferromagnetically aligned), as do forwards and backwards. There are three different equivalent arrangements with two spins out and two in, so the ground state is three-fold degenerate.

Figure 3

Figure 4