The goal was a screen for Gd-based permanent magnets: a very high moment with a workable Curie temperature. The earlier premise check explained why that is hard on paper. This post is about whether Ouro DFT can rank such compounds by Curie temperature at all.
The short answer: for Gd with nonmagnetic partners, yes, to roughly ±30%. For compounds where Fe or Co sets the ordering temperature, not with the default PBE functional. And a direct check shows the exchange couplings themselves carry an error of a quarter to a half on the shortest-wavelength mode, in opposite directions for Fe and Gd.
The exchange route used TB2J. On hcp Gd it returned a Curie temperature of 102 K against 293 K measured, and 139 K for GdZn against 270 K. Two things caused it:
A band cut. The sum stopped 5.1 eV above the Fermi level. Gd's exchange runs through its 5d states, and much of the weight sits above that.
An ill-defined atomic field. TB2J takes each atom's share of the spin splitting as a block of the Hamiltonian in the orbital basis. In the diffuse basis Gd needs (4s3p3d3f), neighboring atoms' orbitals overlap so much that the block no longer belongs to one atom.
We replaced it with our own engine, now live on the exchange and Curie temperature
Each atom's exchange field is cut out in real space on the DFT grid, so it does not depend on how diffuse the basis is. The DFT+U term is added from its projectors.
The band sum runs over every band and every pair of k and q points, with no energy cut.
The ordering temperature comes from Monte Carlo on the resulting Heisenberg model.
On bcc Fe the new engine and the old one agree (about 1000 K at a = 2.83 Å, 1043 K measured).
The first Gd result from the new engine was 477 K, far too high. The reason is physical. A Gd atom's 7 μB sits in the 4f shell, but the coupling between atoms is carried by a small 5d cloud that the 4f shell polarizes. Treating that cloud as part of a rigid spin overstates the cost of tilting neighbors against each other.
The fix: only the 4f shell is the spin. The rest of the atom's spin density, and any nonmagnetic neighbor's, is free to relax to its lowest energy for each arrangement of the 4f spins. That took hcp Gd from 477 K to 313–334 K.
Curie temperatures in kelvin, with U = 6.7 eV on Gd, at measured lattice constants:
Compound | PBE | LDA | Measured |
|---|---|---|---|
Gd | 334 | 258 | 293 |
GdZn | 297 | 240 |
Gd with nonmagnetic partners: PBE is 10–30% high, in the right order.
Fe or Co compounds: PBE is 50–80% high. PBE overstates 3d magnetism here: it gives Fe in YFe₂ a moment of 2.26 μB, and it makes YCo₂ a ferromagnet when it is a paramagnet.
LDA at the measured lattice lands within 12% on all seven. That is a small sample and it relies on knowing the lattice constant, which a screen of new compounds does not. The LDA GdFe₂ run was also less tightly converged than the rest.
We ruled out the cell, the value of U, the basis, the pseudopotential, the k-mesh, the starting moments and the Monte Carlo as causes of the PBE overestimate. Letting the Fe moment change length does not fix it either.
Calibrating against measured Tc mixes three things: the functional, the coupling formula, and the spin model. To isolate the formula we ran spin-constrained DFT (ABACUS DeltaSpin, on a 3.11 beta build). It holds each atom's moment at a chosen direction inside the self-consistent calculation, so tilting one sublattice of a two-atom cell against the other gives the summed coupling between them with no perturbation theory.
Direct | Our engine | |
|---|---|---|
bcc Fe, a = 2.83 Å | 156 meV | 104 meV |
bcc Fe, a = 2.866 Å | about 83 meV | 44 meV |
hcp Gd, slope at 20° | 34 meV |
Our engine is 25–45% low for Fe and 25–30% high for Gd on this mode. The errors go opposite ways, so this is not one scale factor.
The steep drop in Fe's coupling between 2.83 and 2.866 Å is real: the direct calculation shows it too.
The engine computes the energy of turning the exchange field, not the moment. Correcting for that closes about a third of the Fe gap at 2.83 Å and almost none at 2.866 Å. It is not in the routes.
This test only reaches the shortest-wavelength mode of each cell, where any pair-coupling formula is at its worst. The Curie temperature depends more on long wavelengths, which need larger cells we have not run. The Gd slopes also used a coarser k-mesh than the engine's number.
Even with that caveat, the errors are as large as the Tc errors we had attributed to PBE. Some of that blame may belong to the formula. And bcc Fe landing near 1043 K is weaker evidence for the engine than it looked.
Treat it as good to about ±30% for Gd compounds with nonmagnetic partners, and use it to rank, not to quote.
For compounds with Fe or Co, the default PBE number is likely 50–80% high. Setting dft_functional to LDA at a measured or carefully relaxed lattice did much better on our seven compounds.
Set hubbard_u on Gd (we used 6.7 eV). Without it the 4f states sit at the wrong energy.
Check reference_ordering_overlap in the response. Near 1 means the couplings favor the magnetic order the calculation started from. L1₀ FePt still fails this check.
For a candidate that matters, a constrained calculation of the antiparallel state and one small tilt is a cheap spot check on its couplings. It is not a route; ask if you want one run.
A long-wavelength check of the couplings in a larger Fe cell, which would say how much of the short-wavelength error reaches Tc.
Whether to add the field-versus-moment correction. It raises 3d couplings by 5–15%, which moves the PBE iron compounds further from experiment.
Couplings are computed from the ordered ground state, while Tc is where the disordered state becomes unstable.
Which atoms count as spins is still decided by element, so Co in GdCo₂ is a rigid spin by default even though YCo₂ says its moment is induced.
On the original question: nothing here changes the premise check. Gd's own ordering temperature is near room temperature, and pairing it with Fe or Co puts its moment against theirs. What we have now is a tool that can rank Gd compounds with nonmagnetic partners, and an honest error bar on it.
270 |
GdAl₂ | 222 | 158 | 170 |
GdN | 81 | 59 | 65 |
GdCo₂ | 714 | 414 | 404 |
GdFe₂ | 1337 | 886 | 796 |
YFe₂ | about 855 | 535 | 545 |
45 meV |
hcp Gd, slope at 45° | 37 meV | 45 meV |
hcp Gd, antiparallel state, per cell | 55 meV | 90 meV |
We rebuilt the exchange and Curie routes to handle gadolinium, calibrated them on seven known compounds, and then checked the couplings against a direct constrained calculation. What works, what does not, and how to read a Tc from these routes.