Fully Frustrated Face-Centered Cubic Antiferromagnet
We present here some results of the spin resistivity in a film of fully frustrated antiferromagnetic face-centered cubic lattice with Ising spins . We show that the spin resistivity versus temperature exhibits a discontinuity at the phase transition temperature: an upward jump or a downward fall, depending on how many parallel and antiparallel localized spins interacting with a given itinerant spin. The surface effects as well as the difference of two degenerate states on the resistivity are analyzed. Comparison with non- frustrated antiferromagnets is shown to highlight the frustration effect. We also show and discuss the results of the Heisenberg spin model on the same lattice.
We consider a thin film of FCC lattice structure where each lattice site is occupied by an Ising spin whose values are ±1.
Figure 17.7 Spin resistivity versus T for |/2| = 0.26|/i| for several values of £>i: from up to down Di = 0.7a, 0.8a, 0.94a, a, 1.2a. Other parameters are Nx = Ny = 20, Nz = 6,10 = K0 = 0.5, D2 = a, D = 1, e = 1.
The Heisenberg spin model is considered in Section 17.5.4. The interaction between the lattice spins is limited to nearest-neighbor (NN) pairs with the following Hamiltonian:
where S, is an Ising spin, ]ifj the exchange integral between the NN spin pair S, and Sj. Hereafter we take /,-ry = Js for surface spins and /,,y = J for other NN spin pairs. We consider here the antiferromagnetic interaction )s, J <0 for the rest of this chapter. The system size is Nx x Ny x Nz where Nx is the number of FCC cells in the x direction, etc. Periodic boundary conditions (PBC) are used in the x and у directions while the surfaces perpendicular to the z axis are free. The film thickness is Nz.
The FCC antiferromagnet is a fully frustrated system which is composed of tetrahedra each of which has four equilateral triangles. We know that it is impossible to fully satisfy simultaneously the three antiferromagnetic bond interactions on each triangle. As a consequence, the bulk lattice has an infinite ground-state
Figure 17.8 Ground state spin configuration of the FCC cell at the film surface (basal xy plane). The horizontal (vertical) axis is the x (z) axis. Top: ground state when |/s| < 0.51/ |. Middle and bottom: first and second degenerate ground states when |/s| > 0.5|/ |.
degeneracy . In the case of a thin film, the surface spin configuration depends on ]s as shown in Fig. 17.8 .
For |/s| < 0.51/ I, the ground state is composed of ferromagnetic xy planes antiferromagnetically stacked in the z direction a shown in the upper figure of Fig. 17.8. For |/s| > 0.5|/ |, the ground state is two-fold degeneracy as shown in the middle and lower figures of Fig. 17.8. The difference of these two configurations is that the middle figure is an alternate stacking of up- and down-spin planes in the у direction while the lower figure is an alternate stacking of up- and down-spin planes in the x direction. These degenerate states are not equivalent in the spin transport in the x direction as seen below: In the first degenerate state, the itinerant spins move in the x direction between an up-spin plane and a down-spin plane, while in the second degenerate state the itinerant spins meet successively an up-spin plane and a down-spin plane perpendicular to their trajectories. We will present our results for these two cases separately.
In our simulations, we use the lattice size Nx = Ny = 20 and Nz = 8.
For studying the spin transport, we consider N0 = (Nx x Ny x Nz)/2 itinerant spins (one electron per two FCC unit cells). Except otherwise stated, we choose interactions /0 = K0 = 0.5, D e [0.6a; 2a], D2 = a, D = 0.35, e = 1, N0 = 1600, and r0 = 0.05a. A discussion on the effect of a variation of each of these parameters was given in 17.2.2.
Note, however, that due to the form of the interaction given by Eq. (17.7), the itinerant spins have a tendency to form compact clusters to gain energy. This tendency is neutralized more or less by the concentration gradient term, or chemical potential, given by Eq. (17.9). The value of D has to be chosen so as to avoid a collapse of itinerant spins. We show in Fig. 17.9 the phase diagram in the space (K0, D). The limit depends, of course, on the values of D and D2.
Figure 17.9 Collapse phase diagram in the space (K0, D). The black zone is the collapse region. D, = D2 = a. See text for comments.
Figure 17.10 Staggered magnetization of antiferromagnetic FCC thin film of thickness Nz = 8 versus T. The transition temperature Tc ~ 1.79.
Results for the Ising case
We show in Fig. 17.10 the staggered magnetization of the lattice as a function of T. As seen here the transition is of first order with a discontinuity at Tc ~ 1.79. Note that the Ising antiferromagnetic FCC thin film shows a first-order transition down to a thickness of about four atomic layers .
We consider the first degenerate configuration shown in the middle figure of Fig. 17.8 with Js = J = -1. In Fig. 17.11, we show the spin resistivity p versus T for two typical values of D. In all cases resistivity p is small for low T then increases with increasing T. At Tc, it undergoes a discontinuity upward jump. After transition, the resistivity decreases slowly to the same value for all Di in paramagnetic phase.
In Fig. 17.12 we show the spin resistivity p versus T for two typical values of D using the second degenerate ground state (bottom figure of Fig. 17.8). Here we see that depending on Dj, the jump at the transition can be upward or downward.
In Fig. 17.13 we show the resistivity of the first and second degenerate states at a given D = a, for comparison. The upward and downward jumps are seen. This difference is due to the positions of down spins on the trajectory of the itinerant spins [see the discussion below Eq. (17.13)].
Figure 17.11 Resistivity of thin film of size Nx = Ny = 20 and Nz = 8 for N0 = 1600 itinerant spins versus T for = a (black circles) and Di = 1.25a (white circles), a being the lattice constant. Case of the first degenerate state (middle figure of Fig. 17.8). Js = J = —1.0, /„ = K0 = 0.5, D = 0.35.
Figure 17.12 Resistivity versus temperature in the case of second degenerate state (bottom figure of Fig. 17.8) for Di = a (black circles) and Di = 1.25a (white circles) with Nz = 8, N0 = 1600,JS = ] = —1.0, I0 = K0 = 0.5, D = 0.35.
Figure 17.13 Resistivity versus T for first (black points) and second degenerate (white points) configurations for = a with Nz = 8, N0 = 1600, Js =J= -1.0,10 = K0 = 0.5, D = 0.35.
In order to enhance the surface effect, in addition to a small value of Js we allow the exchange interaction between a surface spin and its neighbors in the beneath layer to be Jr which will be taken to be small in magnitude. We show in Fig. 17.14 the surface magnetization and the magnetizations of the interior layers as functions of T for Js = Jr = —0.5 and J = -1. As seen here, the surface transition takes place at a lower temperature Ti ~ 1.2 while interior layers become disordered at T2 — 1.8. As a consequence, one expects that the surface fluctuations at T will induce an anomaly in p in addition to that at T2. This is shown in Fig. 17.15. Note that the increase of p at low T is an effect of a freezing of itinerant spins at low T as discussed above.
Results for the Heisenberg Case
In this section, we presently briefly the results on the same lattice with the Heisenberg spin model. Itinerant spins are the same as used above, namely polarized Ising spins. This assumption allows to outline only the effect of the continuous nature of the Heisenberg lattice spin on the resistivity. The full Hamiltonian with different
Figure 17.14 Layer magnetizations versus T for Js = ]r = —0.5 and ] = — 1. Other parameters: Dy = a,Nz = 8, N0 = 1600,10 = K0 = 0.5, D = 0.35. The surface transition is at Г] ~ 1.2. The vertical dotted line is a guide to the eye indicating the discontinuous fall of interior layer magnetization.
Figure 17.15 Resistivity versus temperature T in the case shown in Fig. 17.14. There are two anomalies occurring, respectively, at the surface and bulk transition temperatures.
kinds of interaction is assumed as above except the exchange interaction between lattice spins. This is given by
where S, is the Heisenberg spin at the site / and A an Ising-like anisotropy which is assumed to be negative to favor an antiparallel spin ordering on the z axis. When A is zero, one has the isotropic Heisenberg model. In order to have at phase transition at a non-zero T, we should take a non-zero value for A because it is known, by the theorem of Mermin-Wagner , that for vector spin models there is no long-range ordering at finite temperatures in two dimensions. The small thickness considered here is, in a phase-transition point of view, equivalent to a two-dimensional system. Except A, note that we use the same assumptions as in Eq. (17.13).
Let us show now in Fig. 17.16 the resistivity as a function of T for two typical values of D. As seen, depending on the value of D, p undergoes a sharp increase or decrease at Tc. At some values such as that corresponding to the upper curve of the upper figure, the resistivity can go across a large region of fluctuations without a sharp jump. So in experiments, care should be taken to interpret similar behavior if any. Note that the second degenerate configuration yields always a larger resistivity than in the first one, as observed in the Ising case in the previous section.
The effect of A on the resistivity is not veiy important in the reasonable range [0.1, 1.5]: Except the fact that Tc varies with A, for instance Tc ~ 0.65 for A = 0.5 and Tc — 0.55 for A = 0.1, the discontinuity of p at Tc diminishes only slightly with decreasing A.
We have shown in this section that the spin resistivity p of the fully frustrated FCC antiferromagnet is quite different from that of ferromagnets  and non-frustrated antiferromagnets  shown in previous section, p does not show a peak at the magnetic phase transition temperature. It shows instead a discontinuous jump at the transition temperature Tc. The jump depends on the numbers of parallel and antiparallel localized spins which interact with an
Figure 17.16 Heisenberg case. Resistivity of thin film of size Nx = Ny = 20 and Nz = 8 for N0 = 1600 itinerant spins versus T for Dt = a (black circles) and D^ = 1.25a (white circles) in unit of the lattice constant a for first (upper) and second (lower) degenerate states. A = —l,/s = J = —1.0, I0 = K0 = 0.5, D = 0.35.
itinerant spin. After transition, the resistivity tends to a saturation value independent of D. The abrupt behavior of p at Tc in the antiferromagnetic FCC Ising lattice is an effect of the frustration which causes a first-order transition of the lattice magnetic ordering leading to a discontinuity of p at Tc.
We are not aware of experiments performed on spin transport in materials with first-order magnetic transition. Our result is thus a prediction which would be useful for future experiments. Note, however, that for electrical transport, the electrical resistivity shows a discontinuity at a metal-insulator "first-order" transition in РгЫЮз  and NdNi03 . The magnetic resistivity found in this chapter has also a discontinuity behavior at a magnetic "first-order” transition. This similarity shows that the resistivity is closely related to the nature of the phase transition, whatever its origin (magnetic, insulator-metal, ...) may be. The mapping between the two cases, however, is not the scope of this chapter.
We have also shown that the surface disordering causes a peak of the resistivity at the surface transition temperature.
In the Heisenberg model, the spin continuous degrees of freedom weaken the first-order transition, yielding in general a reduction of the critical temperature and a less abrupt change of the resistivity at the transition.
As a last remark, let us emphasize that the behavior of the spin resistivity at Tc is quite different from one antiferromagnet to another. It depends on many factors such as the lattice structure, the interaction range, the spin model and the instability (in particular due to frustration) of the spin ordering. We have studied here the effects of some of them, but a throughout understanding needs much more investigations and analysis.