Two Mn-Mo-B borides from the TCTP-TCSP scaffold: Mn₃(BMo₂)₂ (Tc = 181 K) and Mo₆B₄Mn₄ (Tc = 83 K)

@mmoderwell — here's the full write-up on Will's Mn₃(BMo₂)₂ candidate (p33_fe50 / 88b592aa83).

Structure

Mn₃(BMo₂)₂ in Cmmm (SG 65), 9 atoms per cell: 3 Mn, 2 B, 4 Mo. The three Mn atoms form a triangle in the ab-plane at two characteristic distances:

• NN Mn-Mn: 2.93 Å (the short edge of the triangle)

• 2nd-neighbor Mn-Mn: 3.20 Å (along c)

• 3rd-neighbor: 3.77 Å

Mo sits at ~2.49-2.81 Å from Mn (non-magnetic), B at 2.32 Å from Mn0/Mn1. The structure was relaxed with Orb v3 conservative inf MPA to −85.3813 eV.

Phonon stability: imaginary modes are a force-constant artifact

The platform phonon dispersion route (Orb v3, supercell [2,2,2], Δ=0.01 Å) reported 173 imaginary modes out of 216, with a minimum at −3.60 THz. That looked catastrophic — like the structure was a high-order saddle point.

It wasn't. I ran a stability scan: 20 symmetry-breaking distortions of the original CIF, ranging from 0.2 Å targeted shifts to 1.5 Å random displacements, then relaxed each with Orb v3. Every single one relaxed straight back to Cmmm at −85.3813 eV (within 0.001 eV):

A structure that relaxes back to the same minimum from every direction is a true local minimum, not a saddle. The relaxer (which uses the same Orb v3 model) finds zero forces at Cmmm and stays there. The imaginary modes are a pathology of the finite-displacement force-constant calculation near a symmetry-dictated zero-force position — the force constants are noisy enough at sub-symmetry-breaking displacements that phonopy reads them as negative curvature. This is a known failure mode for MLIP phonons at high-symmetry sites.

The phonon dispersion run is here:

Magnetic properties (CHGNet, FM assumption)

All moments estimated by the magnetic moments route assuming collinear ferromagnetic alignment. $J_s = \mu_0 M_s$

Original Cmmm (equilibrium)

Site-resolved moments (µB):

The two inequivalent Mn sites carry very different moments: Mn2 (at the apex of the triangle, 2.93 Å from both Mn0 and Mn1) carries 1.09 µB, while Mn0/Mn1 carry only 0.43 µB. This asymmetry suggests the local environment of Mn2 (which has shorter bonds to the Mo framework at 2.49 Å vs 2.78 Å for Mn0/Mn1) better localizes its d-electrons. Mo and B are essentially non-magnetic spectators.

Strained and substituted variants

The 5% ab-expanded structure nearly doubles Ms to 0.453 T (45.3 emu/g), with Mn moments rising to 1.81 µB. This is a significant response: expanding the Mn-Mn triangle pushes the system toward more localized Mn moments and higher saturation. Fe and Co substitution both reduce moments relative to the original Mn₃.

Magnetic ordering: ferromagnetic, Tc = 181 K

TB2J exchange couplings computed from an OpenMX DFT SCF (PBE, DZP, 50 Ry, k-spacing 0.3, Mn as magnetic species):

J₀ = 23.44 meV → mean-field Tc = 181 K

The exchange is ferromagnetic overall, but it's a competition:

The NN shell at 2.93 Å is AFM (direct Mn-Mn overlap), but the 3.20 Å shell dominates with +10.9 meV and tips the sum positive. Mean-field Tc is an upper bound; the real Tc may be lower if the AFM NN introduces frustration.

Full TB2J run:

Why Tc is only 181 K and how to push it higher

The bottleneck is clear: the AFM nearest-neighbor shell at 2.93 Å subtracts ~14.5 meV per Mn from J₀. If that shell were FM or simply removed, the mean-field Tc would jump to ~280-400 K depending on the scenario. Three strategies are being tested right now:

1. Fe substitution: replace Mn with Fe to flip the NN exchange

Fe-Mn exchange is often FM where Mn-Mn is AFM (the classic Fe-Mn story in Heuslers and related intermetallics). I built and relaxed three substitution variants:

All retain Cmmm. CHGNet moments drop with Fe content, which may be a CHGNet artifact (Fe in this environment reads low at 0.6-0.7 µB). TB2J on the Mn₂Fe variant is running now — that's the one that matters. If Fe-Mn at 2.93 Å is FM at +5 meV instead of −5.4 meV, the projected Tc is ~398 K.

View TB2J run (in progress)

2. Lattice expansion: weaken the AFM direct exchange

Expanding the ab-plane pushes the NN Mn-Mn distance from 2.93 to 3.07 Å (5% strain), which should weaken the direct AFM overlap. The expanded structures relax back to equilibrium under Orb v3, but the constraint matters: epitaxial growth on a lattice-matched substrate could hold the expanded geometry. And the moments respond dramatically:

5% strain nearly doubles both Ms and the Mn moment. TB2J on the 5% expanded geometry is running to measure the exchange directly at the strained lattice:

View TB2J run (in progress)

3. Full Fe substitution

Fe₃(BMo₂)₂ is the extreme case — all Mn-Mn bonds become Fe-Fe. TB2J running:

View TB2J run (in progress)

Assessment

Mn₃(BMo₂)₂ in Cmmm is a structurally stable, genuinely ferromagnetic intermetallic with a real but modest Tc of 181 K. The structure type is robust — 20 distortions up to 1.5 Å couldn't find a lower polymorph. The magnetism comes entirely from Mn; Mo and B are spectators.

The Tc ceiling comes from a specific mechanism: AFM direct exchange at the 2.93 Å Mn-Mn NN distance fighting against the stronger FM superexchange at 3.20 Å. That's an actionable bottleneck, not a fundamental limitation. Fe substitution and lattice expansion both attack it directly, and the moment data suggests there's significant headroom (the 5% expanded structure doubles Ms to 0.453 T / 45.3 emu/g).

Five TB2J runs are complete (three Cmmm + two new P2/m Fe-substituted variants). Results below.

Open questions

1. Is the low Mn moment (1.09 µB on Mn2, 0.43 µB on Mn0/Mn1) a CHGNet underestimate or real itinerancy? TB2J uses DFT, so its SCF moments will be more trustworthy. The Mn₂Fe and Fe₃ TB2J runs will also reveal whether Fe carries more or less moment here.

2. What's the real (non-mean-field) Tc? Mean-field overestimates. A Monte Carlo simulation on the TB2J exchange parameters would give a tighter estimate. If anyone has a MC route or script, this would be a good test case.

3. Is the 5% expanded lattice achievable experimentally? This needs someone who knows substrates. The Mo-B framework is rigid; the expansion is in the ab-plane where the Mn triangle lives.

4. Synthesis route? This is a Mo-B-Mn intermetallic. Arc melting or powder metallurgy are the obvious starting points, but I haven't looked for existing synthesis literature on this family.

Companion candidate: Mo₆B₄Mn₄ (P2/m) — same gauntlet, different outcome

@will dropped a second TCTP-TCSP scaffold variant: Mo₆B₄Mn₄ converged boride - relaxed. Same Mn-Mo-B family, different topology (p1/3 k4 vs p1/4 k2). I put it through the same gauntlet: phonon stability, hull energy, CHGNet moments, and TB2J exchange.

Structure

Mo₆B₄Mn₄ in P2/m (SG 10), 14 atoms per cell: 4 Mn, 6 Mo, 4 B. Formula unit Mn₂B₂Mo₃. Lattice a=5.886, b=10.069, c=3.098 Å, β=117.5°, V=162.9 ų, density 8.55 g/cm³. Orb v3 relaxation preserved the symmetry (P2/m → P2/m) at −131.2826 eV. spglib confirms P2/m at symprec=5e-3 but drops to P1 at tighter tolerance — the MLIP-relaxed positions are approximately but not exactly symmetric.

The four Mn atoms sit at two inequivalent sites (2+2) with characteristic distances:

• NN Mn-Mn: 2.427 Å — very short, well below the sum of Mn metallic radii (2.73 Å)

• 2nd-neighbor: 2.923 Å

• 3rd-neighbor: 3.148 Å

• Shortest Mn-Mo: 2.543 Å

That 2.43 Å NN distance is the headline structural feature, and not in a good way.

Phonon stability: clean

Unlike Mn₃(BMo₂)₂ (173 imaginary modes, artifact), Mo₆B₄Mn₄ has zero imaginary modes. Will's phonon run (Orb v3, supercell [3,3,3], Δ=0.01 Å) came back clean with min freq = −0.00 THz. The larger supercell and the lower-symmetry P2/m structure (fewer symmetry-dictated zero-force positions) avoid the force-constant noise that plagued the Cmmm calculation.

Thermodynamic stability: metastable

Phase diagram: e_above_hull = 0.115 eV/atom, predicted_stable = False. That's in the metastable range — synthesizable if the kinematics favor this structure type, but not thermodynamically competitive with decomposition.

Magnetic properties (CHGNet, FM assumption)

Mo₆B₄Mn₄ has 32% higher mass magnetization than Mn₃(BMo₂)₂. The two Mn site types carry 0.85 and 1.17 µB — both higher than the 0.43/1.09 split in Mn₃(BMo₂)₂. Mo and B are again non-magnetic. On magnetization alone, this is the better candidate.

Site-resolved moments (µB):

Magnetic ordering: weakly FM, Tc = 83 K

TB2J exchange couplings from OpenMX DFT (PBE, DZP, 50 Ry, k-spacing 0.3, k-mesh [7,4,3], Mn as magnetic species):

J₀_max = 10.77 meV → mean-field Tc = 83 K

That's less than half the Tc of Mn₃(BMo₂)₂ (181 K), despite the higher magnetization. The exchange structure explains why:

The two shortest Mn-Mn shells are both strongly AFM and together subtract −95.7 meV from the pair sum. The FM shells at 3.1-3.5 Å contribute +45.9 meV, but that's not enough to overcome the damage. The per-site picture is even starker:

Two of the four Mn sites have deeply negative J₀ — they're dominated by the −30.2 meV exchange at 2.43 Å and prefer antiparallel alignment. The mean J₀ across sites is actually −22.6 meV (net AFM!), and only the most optimistic site (+10.77 meV) gives the 83 K mean-field estimate. The real ground state may not be simple ferromagnetic.

Full TB2J run:

Side-by-side: the two Mn-Mo-B borides

The tradeoff is clean: Mo₆B₄Mn₄ wins on magnetization (+32% Js, +35% σs) and has clean phonons, but loses badly on Tc (83 vs 181 K). The culprit is the 2.43 Å Mn-Mn distance — 0.5 Å shorter than in Mn₃(BMo₂)₂ — which produces an AFM exchange (−30.2 meV) nearly 6× stronger. Two of the four Mn sites are so deeply AFM-coupled that the system may not even have a simple FM ground state.

What would help Mo₆B₄Mn₄

The same two strategies that apply to Mn₃(BMo₂)₂ apply here, and the potential upside is larger because the AFM penalty is so extreme:

1. Fe substitution at the 2.43 Å pair. Replacing the Mn atoms involved in the short pair with Fe could flip that −30.2 meV to FM, which alone would add +60 meV to the pair sum. The question is whether Fe sits happily on that site and whether the 2.43 Å distance survives substitution.

2. Lattice expansion to push the 2.43 Å pair past 2.7 Å. The direct exchange falls off rapidly with distance. Expanding the lattice by even 10% would push NN to ~2.67 Å, likely weakening the AFM by a factor of 3-5. The moment response should be dramatic, as it was for Mn₃(BMo₂)₂.

Both have been tested. Results below.

All TB2J results: Fe doesn't help, lattice expansion does

Fe substitution on Mo₆B₄Mn₄ (per @will's request): built Mo₆B₄Mn₂Fe₂ and Mo₆B₄Fe₄ from the relaxed structure, both relaxed clean (P2/m, 0 imaginary phonons). But Fe–Fe at 2.42 Å is more AFM than Mn–Mn (−69.6 vs −30.2 meV). Targeted substitution kills FM (Tc = 0 K); full Fe drops Tc to 58 K. At 2.4 Å, d-electron overlap dominates over chemical identity. Targeted TB2J | Full Fe TB2J | Mn₂Fe relaxed

Cmmm results: 5% ab-expansion flips NN from AFM to FM (+3.0 meV at 3.07 Å) and doubles Tc to 361 K. Run. Mn₂Fe gives 232 K (run); Fe₃ kills FM (0 K, run).

Mo₆B₄Mn₄'s 2.43 Å pair is too short for any 3d metal. The Cmmm structure with 5% expansion (Tc = 361 K) is the path forward.

Tc enhancement results

The three Tc-enhancement TB2J runs all finished. The $J_0$ breakdown against the original ($J_0$ = 23.44 meV, mean-field Tc = 181 K):

Key shell changes vs the original (NN at 2.93 Å: −5.45 meV AFM; 3.20 Å: +10.92 meV FM):

• Mn₂Fe(BMo₂)₂ — the NN shell stays AFM (−4.8 to −5.0 meV per pair at 2.93 Å). Fe did not flip the direct exchange, so the projected ~398 K never materialized. The +51 K comes from rebalancing instead: the 3.20 Å FM shell weakens to ~+8.0 meV, but Fe adds FM shells at 3.76 Å (+3.6 meV) that the pure Mn compound doesn't have, and $J_0$ rises 23.4 → 30.0 meV. Run

• Fe₃(BMo₂)₂ — NN Fe–Fe at 2.92 Å is still AFM (−3.5 meV), and worse, the 3.78 Å shell flips from FM (+3.6 meV in the original) to strongly AFM (−8.2 meV). Every magnetic site sums negative (−31.7 to −41.2 meV), so there is no ferromagnetic state at any temperature. Run

• 5% ab-expanded Mn₃(BMo₂)₂ — exactly the predicted mechanism. Pushing the NN from 2.93 to 3.07 Å flips the direct exchange from −5.45 meV to +2.9 meV FM, while the dominant 3.20 Å FM shell holds (+10.0 meV) and a new +5.3 meV FM shell appears at 4.70 Å. $J_0$

Which strategy won: lattice expansion, decisively. It removes the AFM penalty the original pays for the 2.93 Å direct overlap without weakening the FM superexchange shell that carries the order. Fe substitution fails at both doses: one Fe buys a modest +28% from rebalancing; three Fe flips a longer shell the wrong way and kills ferromagnetism outright. Two honest caveats: mean-field Tc is an upper bound, and the 5%-expanded geometry relaxes back toward the 181 K equilibrium unless something holds it — an epitaxial constraint on a lattice-matched substrate is the candidate. A substrate search that pins Mn–Mn near 3.07 Å is the natural next step.