Formation energy on the convex hull is the standard criterion for thermodynamic stability: a material is stable if no linear combination of competing phases has lower formation energy. In the materials informatics community, this has become the dominant screening criterion — primarily because the Materials Project database makes convex hull distances accessible for millions of compositions without running new calculations.
The problem is that thermodynamic stability is necessary but not sufficient. A material can sit on the convex hull — be thermodynamically stable at 0 K — and still be dynamically unstable. Dynamic instability means the structure is at a saddle point on the potential energy surface: it has no net force at the equilibrium geometry, but the second derivative of energy with respect to certain atomic displacements is negative. Any perturbation from equilibrium in the direction of those displacements lowers the energy, driving spontaneous distortion to a different structure.
In cathode materials, dynamic instability is not a rare edge case. It is a routine occurrence in high-throughput screening results, and it is a common cause of synthesis failures in programs that rely only on formation energy filtering.
The phonon dispersion as a stability indicator
Phonons are the quantized vibrational modes of a crystal lattice. The phonon dispersion relation ω(q) describes how phonon frequencies vary as a function of wavevector q across the Brillouin zone. For a dynamically stable structure, all phonon frequencies are real — the squared frequency ω²(q) is positive everywhere. For a dynamically unstable structure, imaginary frequencies appear at one or more wavevectors: ω²(q) < 0, which in a dispersion plot appears as a curve dipping below zero on the frequency axis.
The physical interpretation of an imaginary frequency at wavevector q is that the structure has a soft mode at that q: an atomic displacement pattern that lowers the energy. If the imaginary mode is at q = Γ (the zone center), the structure will distort without a change in periodicity — a zone-center soft mode. If the imaginary mode is at a zone-boundary q-point, the distortion involves a supercell-doubling: the soft mode has a wavelength that is a multiple of the unit cell, and the resulting distorted structure has a lower symmetry and larger unit cell than the parent phase.
The practical consequence for cathode screening: a structure with imaginary phonon modes will not form as the target phase under synthesis conditions. It will distort to a different, lower-symmetry structure — which may or may not be electrochemically useful, but is almost certainly not what the synthesis protocol was optimized for.
The finite displacement method
Computing phonon dispersion from first principles requires the interatomic force constant (IFC) matrix: the matrix of second derivatives of energy with respect to atomic displacements. Two methods are standard for extracting the IFC matrix from DFT: density functional perturbation theory (DFPT), which computes the IFCs analytically through linear response, and the finite displacement (frozen phonon) method, which approximates the second derivatives numerically from a set of DFT calculations with small atomic displacements.
MaterSynq uses the finite displacement method, implemented via phonopy interfaced to VASP. The procedure: starting from the equilibrium geometry, small displacements (typically 0.01–0.02 Å) are applied to each inequivalent atom in each independent Cartesian direction in a supercell. The resulting forces on all atoms are used to construct the IFC matrix. The dynamical matrix at each q-point is obtained by Fourier transformation of the IFCs, and phonon frequencies are the eigenvalues of the dynamical matrix at that q-point.
Supercell size matters for phonon accuracy: the IFC matrix elements decay with interatomic distance, and the supercell must be large enough that the IFCs have decayed to negligible values at the supercell boundaries. For layered oxide cathodes with the R3̄m structure, 2×2×2 supercells (24 atoms × 8 = 192-atom supercell for Ni-rich NMC) provide adequate IFC convergence while remaining tractable for DFT calculation. For spinel and olivine structures, different supercell choices are appropriate.
What the phonon screen catches that formation energy misses
In a representative 100-candidate screening campaign across Ni-rich NMC compositions, a typical distribution looks like this: approximately 60–70% of candidates pass the hull distance threshold (≤50 meV/atom above the hull). Of those, the phonon screen eliminates an additional 15–30% as dynamically unstable. The most common failure modes are zone-boundary soft modes along the high-symmetry Γ-M path in the Brillouin zone, associated with in-plane Li layer distortions that arise from unfavorable Ni²⁺/Mn⁴⁺ ordering at certain compositions.
The compositions that fail phonon screening despite passing hull distance filtering tend to cluster around specific stoichiometries where cation ordering frustration is high — sites where the energetic preferences of Ni and Mn for particular local coordination environments conflict. These compositions are not random noise in the screening output; they are physically interpretable as frustrated orderings, and the phonon calculation provides the mechanistic explanation for why the thermodynamic stability metric failed to predict instability.
Phonon accuracy with MLIP vs DFT
MLIP-based phonon calculations are faster than DFT by the same factor as energy calculations, but phonon accuracy is more sensitive to the quality of force predictions than formation energy accuracy. The IFC matrix is constructed from small-displacement forces, and errors in force prediction at those displacements propagate into phonon frequencies. For well-trained MLIPs on in-distribution chemistries, the phonon stability classifier accuracy (stable vs. dynamically unstable) exceeds 92% on held-out test sets in our validation. But the absolute phonon frequencies can deviate from DFT reference by 2–5% — which is sufficient for stability classification but not for computing thermodynamic properties from phonon density of states (heat capacity, entropy).
For the purpose of screening — binary classification of stable vs. dynamically unstable — MLIP-based phonon calculations are adequate and reduce the cost of phonon screening to something that can be run on all hull-passing candidates without impractical compute overhead. For thermodynamic property prediction from phonons, DFT-level force constants are required.
Integrating phonon screening into the funnel
The right placement of phonon screening in the candidate funnel depends on the compute budget and the tolerance for false negatives. The approach MaterSynq uses: phonon screening is applied to all candidates that pass the hull distance threshold, using MLIP-derived force constants for the initial classification. Candidates with clearly imaginary modes (imaginary frequencies more negative than −5 meV at zone-boundary q-points) are filtered before DFT single-points. Borderline cases (soft modes within ±5 meV of zero) are passed to DFT for single-point energy confirmation, with a note flagging the borderline dynamic stability.
This two-tier approach — MLIP phonon pre-screen, followed by DFT confirmation for borderline cases — adds approximately 20% to the total compute cost of the campaign while eliminating 15–30% of candidates that would otherwise consume experimental resources. The expected value of this compute expenditure, measured in synthesis experiments avoided, is consistently positive for any program where a synthesis attempt costs more than a few hundred dollars of researcher time.
Formation energy screening tells you which phases exist. Phonon screening tells you which of those phases will form as intended. Both are required for a defensible candidate list.