FIXED SPIN MOMENT CALCULATIONS

Transition metals and their compounds display a particularly rich set of properties, many of which are related to magnetism and/ or proximity to magnetic transitions. However, despite the fact that it is possible to investigate magnetic materials using standard self-consistent LSDA electronic structure approaches, much of the richness present in the magnetic phase diagrams of even simple transition metal elements and binary compounds was not appreciated until a series of seminal papers by Moruzzi and his co-workers (Moruzzi, 1986a, Moruzzi et al., 1986b, Moruzzi and Marcus, 1988a, Moruzzi et al., 1988b, Moruzzi, 1989a, Moruzzi and Marcus, 1989b,c, Moruzzi et al., 1989d, Moruzzi, 1990a, Moruzzi and Marcus, 1990b,c). These papers used the fixed spin moment (FSM) procedure (Williams et a/., 1984, Schwarz and Mohn, 1984) which greatly simplifies the study of magnetic materials, particularly those with ferromagnetic phases. This section describes the procedure and its implementation.

The idea underlying the FSM approach is to do total energy calculations with the total moment constrained to fixed values. In this way the total energy as a function of the moment, M, can be mapped out. As Moruzzi and co-workers have shown, it is not at all unusual to have multiple minima in the E(M) curve; using the FSM these metastable states can be readily identified and studied.

When combined with the conservation of total charge, the FSM constraint amounts to separately fixing the total up and down spin charges (the sum is the total charge and the difference is the spin moment). This is straightforward in practice (Fig. 5.15).  A FSM calculation proceeds exactly as Figure 5.15 The f1xed spm moment procedure as. a standard LAPW calculation, except that separate Fermi energies are determined for the up- and down-spin channels, and these are then used in the separate construction of the up- and down-spin charge densities. The total energy calculation is exactly as in a standard calculation but with the eigenvalue sum including the eigenvalues occupied in the construction of the spin densities, i.e. up to the up- (down-) spin Fermi energy for spin-up (down) eigenvalues. This procedure is equivalent to performing calculations using constant stabilizing fields, H, and adjusting H to produce the desired value of M. Viewed in this way, H is just half the difference between the down- and up-spin Fermi energies.

Use of the FSM procedure offers several advantages in studying ferromagnetic materials. As mentioned, it allows the mapping out of E(M), and facilitates the identification of metastable ferromagnetic phases. Besides this, self-consistent iterations with the FSM method are much more stable than in standard spinpolarized calculations. In situations (e.g. Pd) where E(M) is quite flat, charge can slosh between the two spin channels with very little energy cost in a standard calculation. Although convergence accelerators, like Broyden's method do help in such situations, it can still be quite difficult to fully converge spin-polarized calculations, while FSM iterations converge as rapidly as non-spin-polarized calculations. Further, the FSM procedure is quite useful in obtaining good starting spin densities to be used in standard spin-polarized calculations. In many transition metal ferromagnets, the non-spin polarized solution is metastable, and selfconsistent iterations that do not start close enough to the magnetic solution may not find it. With the FSM procedure, self-consistent spin densities can be obtained for a moment estimated to be close to that of the magnetic solution and these can be input to an unconstrained calculation.

Although the fixed spin moment procedure was formulated for use without the inclusion of spin-orbit coupling (which mixes spin-up and spin-down states), it can be readily generalized to the case where spin-orbit is included as a perturbation (Singh and Ashkenazi, 1992). In this case, rather than fixing the fmal spin moment, the spin moment before the second variational step (see §5.14) is fixed. Since spin is a good quantum number at this point, the procedure can be followed exactly as above, i.e. two Fermi energies are determined, one for each spin, and an effective magnetic field, H, defined in terms of the difference. This effective field is then added along with the spin-orbit terms to the Hamiltonian matrix in the second variational step. Since the spins are coupled in this step, the moment although close to is not exactly equal to the enforced moment before the second variation. Although the generalized approach does not readily permit a calculation exactly at any predetermined value of M, it does provide a numerically stable prescription for calculating the energy and stabilizing field as a function of the magnetization.

Reference: Planewaves, Pseudopotentials and the LAPW Method