Phonon Calculations With PHIN

Summary

PHIN OS and PHIN Atomic engine provide a powerful platform for performing high-throughput phonon calculations. Its modular design, active-learning based engine, and efficient task orchestration make it possible to explore phonon properties across diverse material classes with accuracy and speed beyond what is possible with a DFT-only approach. The computed phonon band structures and DOS are in good agreement with DFT-based reports, demonstrating that PHIN Atomic and PHIN OS are powerful tools for reliable high-throughput phonon calculations. The results presented here capture the essential vibrational features of MAX phases, NiTi shape memory alloys, and Mg₃Bi₂ thermoelectric, underscoring the utility of PHIN OS for high-throughput materials discovery, characterization, and design. 

Background

Thermal properties of materials are a crucial aspect in designing and utilizing materials across aerospace, automotive, and energy industries. A key component in understanding thermal properties are the phonon band structure and density of states (DOS). These provide insight into how the crystal structure governs thermal transport behavior. The ability to rapidly perform digital experiments to calculate phonon band structures, DOS, and related thermal properties opens new opportunities for discovery and design of advanced materials.

As a case study, we present the phonon band structures and DOS of three classes of materials: NiTi shape memory alloys, MAX phases and thermoelectric materials. All require knowledge of phonon properties to understand their performance in thermal management and thermoelectricity. PHIN OS enables setting up these calculations in minutes with subsequent execution on cloud resources for obtaining results quickly. The results computed with PHIN’s fine-tuned models reproduce DFT and experimental results with state of the art accuracy at a fraction of the cost.

Phonon spectra can be computed using several approaches (e.g., finite-displacement “frozen phonon”, density-functional perturbation theory). A widely used approach is the finite-displacement method in the harmonic approximation: atoms are displaced infinitesimally about equilibrium, and the resulting forces define the second-order interatomic force constants. This method does not include intrinsic anharmonic interactions. The quasi-harmonic approximation (QHA) can be layered on top by evaluating volumes so that phonon frequencies are dependent on volume, capturing thermal expansion but not phonon–phonon scattering. For many crystalline solids, the harmonic approximation (and QHA when thermal expansion is important) is accurate at low to moderate temperatures; strongly anharmonic materials may require higher-order force constants or molecular-dynamics–based treatments.

Phonon Workflow

Calculating phonon spectra requires constructing a supercell, applying symmetry-inequivalent displacements, and computing forces to assemble the dynamical matrix and obtain dispersions. When forces are computed with DFT, the cost is significant due to the number of configurations and supercell sizes. PHIN Atomic reduces this cost by using fine-tuned machine learning models, accelerating evaluations while obtaining near-DFT accuracy.

The phonon workflow is organized into modular tasks that are orchestrated as a digital experiment. PHIN OS composes tasks into grouped sub-experiment tasks to streamline experiment development. As seen in Figure 1, the phonon experiment contains two sub-experiments, the first of which finds the minimum energy configuration of an input structure and the second calculates the phonon band structure.

Figure 1: PHIN OS geometry optimization and phonon workflow

Within the phonon experiment, three sequential tasks displace the atomic coordinates, calculate the forces, and construct the second-order force constants and the dynamical matrix. These are composed within the Phonon Analysis Experiment in Figure 1, and seen in more detail in Figure 2. Moreover, because phonon calculations are often needed for many different structures, PHIN OS  supports high-throughput sweeps allowing users to easily vary input structures, supercell sizes, or displacement magnitudes, etc. and if any parameter is modified PHIN OS automatically handles rerunning the necessary steps. This makes the process both efficient and reproducible.

Figure 2: Workflow for Mg3Bi2 phonon calculation

Case Studies

We ran the phonon workflow on three material classes using the PHIN's proprietary foundation models available on PHIN OS. The systems studied include Ti₂AlX (X = B, N) MAX phases, and Mg₃Bi₂ thermoelectrics, all which require knowledge of phonon properties due to their interesting applications in superconductivity, shape memory effects, and ultralow thermal conductivity [1–3]. 

Phonon Digital Experimental Method

The structures were obtained from the Materials Project, with geometry relaxations performed via the conjugate gradient method while allowing for isotropic changes to the unit cell. The phonon workflow requires specifying the supercell size and the atomic displacement magnitude [1]. The displaced structures are simulated with PHIN Atomic to compute atomic forces, which are used to assemble the dynamical matrix using the phonopy package. From this, the phonon band structures, DOS, and partial DOS were obtained.

Ni-Ti Shape Memory Alloys

The first class of materials studied are NiTi shape memory alloys. The phonon band structure for the B2 and B19 phases are simulated through the phonon workflow. The B2 structure shows the well-known phonon instability at the M-point along the Γ–M direction, which is the characteristic signature of its martensitic transformation pathway [2], with the complete mode softening characteristics of the B2 structure captured in the bandstructures in Figure 4. The B19 band structures are computed in Figure 5. The results demonstrate that the PHIN OS workflow and PHIN Atomic engine are able to capture the essential phonon characteristics of NiTi systems [4].

Figure 3: NiTi B2 phase phonon band structure and DOS

Figure 4: NiTi B19 phase phonon band structure and DOS

Mg-Bi Thermoelectrics

The thermoelectric Mg₃Bi₂ material is dynamically stable with no imaginary branches. The partial DOS shows that the heavier pnictogen atom dominate the low-frequency spectrum, while Mg atoms contribute mainly to the mid- and high-frequency modes. The Mg₃Bi₂ exhibits a softened transverse acoustic branch along Γ–M, with cutoffs near 0.8–1.2 THz. This feature produces shoulders in the DOS and is attributed to weak Mg–Bi bonds that suppress group velocities, leading to ultralow lattice thermal conductivity [5-6]. At higher frequencies, the optical manifold near 5.8–6.2 THz appears nearly flat, further enhancing phonon scattering and reducing conductivity, as demonstrated in Figure 5.

Figure 5: Mg3Bi2 phonon bandstructure and DOS

Ti-Al-X MAX Phases

The Ti₂AlX MAX phases are accurately calculated to be dynamically stable as indicated by the absence of imaginary frequencies. The computed spectra can be separated into two regimes: a low-frequency region dominated by Ti–Al modes below about 12 THz, and a high-frequency region dominated by the lighter X atoms (B,N), with optical branches extending to 15–21 THz. This division of vibrational modes is clearly reflected in the partial DOS, as illustrated in Figures 6 and 7, and reproduces previous DFT and experimental results [7].

Figure 6: Ti2AlB MAX phase phonon bandstructure and DOS

Figure 7: Ti2AlN phase phonon bandstructure and DOS

References

[1] E. Karaca, P.J.P. Byrne, P.J. Hasnip, M.I.J. Probert, Prediction of phonon-mediated superconductivity in new ti-based MAX phases, Scientific Reports 12 (2022) 13198. https://doi.org/10.1038/s41598-022-17539-8.

[2] Z. Wu, J.W. Lawson, Theoretical investigation of phase transitions in the shape memory alloy NiTi, Phys. Rev. B 106 (2022) L140102. https://doi.org/10.1103/PhysRevB.106.L140102.

[3] H. Yang, B. Jia, L. Xie, D. Mao, J. Xia, J. Yang, M. Yuan, Q. Gan, X. Liu, M. Hu, J. Shuai, J. He, Achieving high thermoelectric performance through ultra-low lattice thermal conductivity based on phonon localization, Joule 8 (2024) 2667–2680. https://doi.org/10.1016/j.joule.2024.06.020.

[4] P. Kumar, U.V. Waghmare, First-principles phonon-based model and theory of martensitic phase transformation in NiTi shape memory alloy, Materialia 9 (2020) 100602. https://doi.org/https://doi.org/10.1016/j.mtla.2020.100602.

[5] J. Ding, T. Lanigan-Atkins, M. Calderón-Cueva, A. Banerjee, D.L. Abernathy, A. Said, A. Zevalkink, O. Delaire, Soft anharmonic phonons and ultralow thermal conductivity in Mg3 (sb, bi)2 thermoelectrics, Science Advances 7 (2021) eabg1449. https://doi.org/10.1126/sciadv.abg1449.

[6] Y. Chen, Z. Ma, N. Zhao, Y. Li, X. Yao, X. Dou, Unraveling the lattice thermal conductivity and thermoelectric properties of monolayer Mg3 Bi2, Physical Chemistry Chemical Physics 27 (2025) 12919–12928. https://doi.org/10.1039/D5CP01499A.

[7] M.I. Naher, M. Montasir, M.Y.H. Khan, M.A. Ali, M.M. Hossain, M.M. Uddin, M.Z. Hasan, M.A. Hadi, S.H. Naqib, Investigation of the Pressure Dependent Physical Properties of MAX Phase Ti2AlX (X = B, C, and N) Compounds: A First-Principles Study, (2024). https://doi.org/10.48550/arXiv.2412.19431.