Loop 3 left Fe₃Ge₂Al as the one candidate with DFT support, on single points, ferromagnetic states and one decomposition. This loop tried to finish the job or kill it: relax everything with DFT, find the real competitors, and use the right magnetic states. It survived all three, with a larger margin than before.
Fe₃Ge₂Al is FeGe's kagome structure (CoSn type) with Al ordered on one of the two Ge sites in the honeycomb layer between the kagome planes.
Fe3Ge2Al, CoSn-type FeGe with Al ordered on one honeycomb Ge site. Relaxed with PBE (ABACUS, TZDP, k-spacing 0.2): a 4.918, c 4.159 A. 42-45 meV/atom below FeGe + FeAl and FeGe + Al2Fe3Ge on PBE; layer-antiferromagnetic state 8.4 meV/atom below the ferromagnet.
PBE reaction energies in meV/atom; negative means Fe₃Ge₂Al is lower than the phases it would decompose into.
Magnetic states used | vs FeGe + Al₂Fe₃Ge | vs 2 FeGe + FeAl |
|---|---|---|
Both ferromagnetic, single points at ML geometries (loop 3) | not tested | −39.3 |
Both ferromagnetic, DFT-relaxed | −40.6 | −39.3 |
Both in their lower, layer-antiferromagnetic state |
Runs behind the table: relaxations of Fe₃Ge₂Al, FeGe, Al₂Fe₃Ge, AlFeGe, FeAl, Fe₃Ge and Fe; magnetic states of FeGe, antiferromagnetic, FeGe, ferromagnetic, Fe₃Ge₂Al, antiferromagnetic and
Three things were checked.
The real competitors. The full Al–Fe–Ge phase diagram on the machine-learned surface shows that loop 3 tested the wrong decomposition. Alexandria holds two ternaries that are stable there, Al₂Fe₃Ge (P-3m1) and AlFeGe (P4/nmm), and against FeGe + Al₂Fe₃Ge the ML margin for Fe₃Ge₂Al is only 0.5 meV/atom: a tie. On PBE it is not a tie. Among FeGe, Al₂Fe₃Ge, AlFeGe, FeAl, Fe₃Ge and Fe, the lowest combination at this composition is 2 FeGe + FeAl, and Fe₃Ge₂Al is about 40 meV/atom below it.
Relaxation. Every phase relaxed in three or four steps and moved little (volumes changed by −0.8 to +2.6 % from the ML cells). The reaction energy did not move. Relaxed Fe₃Ge₂Al: a 4.918 Å, c 4.159 Å.
Magnetism. FeGe in this structure is an antiferromagnet with kagome layers alternating along c, and on PBE that state is 8.8 meV/atom below its ferromagnet (1.70 μB per Fe). Fe₃Ge₂Al does the same: 8.4 meV/atom lower, 1.82 μB per Fe. The two gains nearly cancel in the reaction, so the margin holds either way.
So the predicted ground state of Fe₃Ge₂Al is a layered antiferromagnet like FeGe, not a ferromagnet.
A free check, run after this post went up. The hull route can also build the phase diagram from the published PBE energies of Materials Project and Alexandria. That gives a reaction our code and the published data can both answer, one that does not involve the new compound: Al₂Fe₃Ge against FeGe + 2 FeAl.
Source | Al₂Fe₃Ge vs FeGe + 2 FeAl (meV/atom) |
|---|---|
Published PBE (Materials Project, Alexandria) | −1.5 |
Our PBE (ABACUS, relaxed) | +2.6 |
ML surface (Orb v3) | −10.2 |
Our PBE agrees with the published PBE to 4 meV/atom on the competitors, and the ML surface is 9 to 13 off. So the 40 meV/atom disagreement about Fe₃Ge₂Al is not our code misjudging its neighbours; it sits in the ML energy of the new cell itself, which has no published value to compare against. That makes the ML potential the likelier one to be wrong, without proving it.
The published-PBE diagram also names a competitor we had not tested: Al₅Fe₆Ge (Alexandria, Immm), which is stable there. Putting our PBE margin together with the published formation energies, Fe₃Ge₂Al comes out about 37 meV/atom below FeGe + Al₅Fe₆Ge. That number mixes two sources and is not a direct calculation; Al₅Fe₆Ge has not been run in our code.
We then ran the missing neighbours in our own code: Al₅Fe₆Ge, the published-PBE ground states of AlFeGe (I4/mmm) and Fe₃Ge
Magnetic states used | Fe₃Ge₂Al vs the PBE hull (meV/atom) |
|---|---|
All ferromagnetic | −36.8 |
FeGe and Fe₃Ge₂Al in their antiferromagnetic states | −39.9 |
Worst case: FeGe antiferromagnetic, Fe₃Ge₂Al ferromagnetic | −31.5 |
The direct number matches the −37 we had derived from mixed sources. A second cross-check against published PBE also holds: Al₅Fe₆Ge against FeGe + 5 FeAl is −5.6 meV/atom in the published data and −6.2 in ours.
One caution about our numbers. Our formation energies measured from the elements are less negative than the published ones, by an amount that grows with Al content (FeAl: −234 against −327 meV/atom). That points at the elemental Al reference in this basis set. It cancels in every comparison made here, because none of the decompositions contains an element, and the two compound-only reactions we could compare agree with published PBE to 4 and 0.6 meV/atom. It does mean our absolute formation energies should not be quoted.
The Al₅Fe₆Ge relaxation stopped at a maximum force of 0.046 eV/Å against a 0.04 target; its energy changed by 7 meV per 12-atom cell over the whole relaxation, so this does not move the result.
Before the runs we predicted −25 to +15 meV/atom against FeGe + Al₂Fe₃Ge and gave 45 % to a negative sign, because the ML surface called it a tie. PBE gives −40.6 to −44.6: outside the range, on the favourable side. The antiferromagnetic gains were predicted well (0–15 for FeGe, the two within 5 of each other).
The first antiferromagnetic run on Fe₃Ge₂Al is void. The route regroups atoms alphabetically by element, so starting moments written for an iron-first list landed on the wrong atoms and the calculation converged to a scrambled state 105 meV/atom up. Rerun with the cell written in the route's order.
Loop 3's "39 meV below FeGe + FeAl" was a true number against an incomplete list of competitors. The number survived; the claim it supported was not yet earned.
One functional (PBE), one code, one basis family. The ML potential, trained on PBE data, disagrees with these PBE numbers by about 40 meV/atom on the Al₂Fe₃Ge balance, and we have not found out which side is off. A second code or a plane-wave check would settle it.
Competitors tested on PBE are six phases. The Fe-rich and Al-rich corners (Fe₆Ge, AlFe₆Ge, Al₅Fe₆Ge, Al₁₃Fe₄) were left to the ML surface, where none enters the decomposition.
Disorder. Al prefers the honeycomb site over the in-plane site by 183 meV per Al (loop 3), but Al and Ge sharing the honeycomb layer at random was not tested, and at temperature that is the likely state.
No literature search beyond two web queries, and no synthesis assessment. Hexagonal FeGe itself is, as far as we recall from the Fe–Ge phase diagram, stable only in a limited temperature window and slow to form; we did not check that this loop, and it would matter for making an Al-substituted version.
Phonons were run on the ML surface only.
A second-opinion energy for the Fe₃Ge₂Al / FeGe / Al₂Fe₃Ge / FeAl set, to resolve the 40 meV disagreement between PBE here and the ML surface.
Al/Ge disorder on the honeycomb layer in a larger cell.
The same PBE test on the campaign's other ML-only candidates.
−41.8 |
Worst case: FeGe antiferromagnetic, Fe₃Ge₂Al ferromagnetic | −36.2 | −33.4 |
DFT-relaxed and tested against the Alexandria ternaries with the right magnetic states, Fe₃Ge₂Al stays 42–45 meV/atom below its competitors on PBE; like FeGe, it is predicted to be a layered antiferromagnet.