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  • Open Access

In Search of the Electron-Phonon Contribution to Total Energy

Samuel Poncé1,2,* and Xavier Gonze1

  • *Contact author: samuel.ponce@uclouvain.be
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PRX Energy 5, 043004 – Published 22 October, 2026

DOI: https://doi.org/10.1103/zx5f-cw7t

Abstract

The total energy is a fundamental characteristic of solids, molecules, and nanostructures. In most first-principles calculations of the total energy, the nuclear kinetic operator is decoupled from the many-body electronic Hamiltonian and nuclear potential, and the dynamics of the nuclei is reintroduced afterward. This two-step procedure introduced by Born and Oppenheimer (BO) is approximate. Energies beyond the electronic and vibrational (or phononic) main contributions might be relevant when small energy differences are important, such as when predicting stable polymorphs or describing the magnetic energy landscape. We clarify the different flavors of BO decoupling and give an exact formulation for the total energy in the basis of BO electronic wave functions. Then, we list contributions beyond the main ones that appear in a perturbative expansion in powers of M0−1/4, where M0 is a typical nuclear mass, up to sixth order. Some of these might be grouped and denoted as the electron-phonon contribution to total energy, Eelph, which first appears at fourth order. The electronic inertial mass contributes at sixth order. We clarify that the sum of the Allen-Heine-Cardona zero-point renormalization of eigenvalues over occupied states is not the electron-phonon contribution to the total energy but rather part of the phononic contribution. The computation of the lowest-order Eelph is implemented and shown to be small but non-negligible in the case of diamond (3.8 meV per atom) and GaN (9.7 meV per atom). We also estimate the electronic inertial mass contribution and the quasiharmonic one for diamond. We confirm the size consistency of all computed terms.

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Popular Summary

Computer simulations guide the search for better batteries, solar cells, and energy-efficient electronics, yet their verdicts hinge on tiny energy differences of a few hundredths of an electronvolt per atom. Almost all such simulations split the problem in two: first the electrons, with the atomic nuclei frozen in place, then the vibrations of the nuclei added afterwards. This near-century-old shortcut, introduced by Born and Oppenheimer, quietly discards small energy terms, among them the one produced by electrons and vibrations influencing each other. This work sorts out which missing terms deserve the name “electron-phonon contribution to the total energy,” corrects a recent proposal in the literature, and implements the leading term in the Abinit software. It amounts to 2% of the vibrational energy in diamond, but up to 18% in gallium nitride, the workhorse of power electronics and LEDs. Including it makes predicted energies of competing crystal structures more trustworthy.

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References (111)

  1. R. Jones, Density functional theory: Its origins, rise to prominence, and future, Rev. Mod. Phys. 87, 897 (2015).
  2. R. M. Wentzcovitch, Invariant molecular-dynamics approach to structural phase transitions, Phys. Rev. B 44, 2358 (1991).
  3. M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, Iterative minimization techniques for ab initio total-energy calculations: Molecular dynamics and conjugate gradients, Rev. Mod. Phys. 64, 1045 (1992).
  4. C. Freysoldt, B. Grabowski, T. Hickel, J. Neugebauer, G. Kresse, A. Janotti, and C. G. Van de Walle, First-principles calculations for point defects in solids, Rev. Mod. Phys. 86, 253 (2014).
  5. G. Ceder, Y.-M. Chiang, D. R. Sadoway, M. K. Aydinol, Y.-I. Jang, and B. Huang, Identification of cathode materials for lithium batteries guided by first-principles calculations, Nature (London) 392, 694 (1998).
  6. G. J. Snyder and E. S. Toberer, Complex thermoelectric materials, Nat. Mater. 7, 105 (2008).
  7. W. A. Saidi, S. Poncé, and B. Monserrat, Temperature dependence of the energy levels of methylammonium lead iodide perovskite from first-principles, J. Phys. Chem. Lett. 7, 5247 (2016).
  8. S. Pimputkar, J. S. Speck, S. P. DenBaars, and S. Nakamura, Prospects for LED lighting, Nat. Photonics 3, 180 (2009).
  9. A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, and K. A. Persson, Commentary: The materials project: A materials genome approach to accelerating materials innovation, APL Mater. 1, 011002 (2013).
  10. W. Sun, S. T. Dacek, S. P. Ong, G. Hautier, A. Jain, W. D. Richards, A. C. Gamst, K. A. Persson, and G. Ceder, The thermodynamic scale of inorganic crystalline metastability, Sci. Adv. 2, e1600225 (2016).
  11. P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 (1964).
  12. W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140, A1133 (1965).
  13. M. Born and R. Oppenheimer, Zur quantentheorie der molekeln, Ann. Phys. 389, 457 (1927).
  14. M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford, 1954).
  15. S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys. 73, 515 (2001).
  16. G.-M. Rignanese, J.-P. Michenaud, and X. Gonze, Ab-initio study of the volume dependence of dynamical and thermodynamical properties of silicon, Phys. Rev. B 53, 4488 (1996).
  17. S. Rostami and X. Gonze, Approximations in first-principles volumetric thermal expansion determination, Phys. Rev. B 110, 014103 (2024).
  18. S. Rostami, M. Giantomassi, and X. Gonze, Anisotropic temperature-dependent lattice parameters and elastic constants from first principles, npj Comput. Mater. 11, 271 (2025).
  19. O. Hellman, I. A. Abrikosov, and S. I. Simak, Lattice dynamics of anharmonic solids from first principles, Phys. Rev. B 84, 180301 (2011).
  20. L. Monacelli, R. Bianco, M. Cherubini, M. Calandra, I. Errea, and F. Mauri, The stochastic self-consistent harmonic approximation: Calculating vibrational properties of materials with full quantum and anharmonic effects, J. Phys. Condens. Matter 33, 363001 (2021).
  21. F. Knoop, N. Shulumba, A. Castellano, J. P. A. Batista, R. Farris, M. J. Verstraete, M. Heine, D. Broido, D. S. Kim, J. Klarbring, I. A. Abrikosov, S. I. Simak, and O. Hellman, TDEP: Temperature dependent effective potentials, J. Open Source Software 9, 6150 (2024).
  22. N. C. Handy, Y. Yamaguchi, and H. F. Schaefer, The diagonal correction to the Born–Oppenheimer approximation: Its effect on the singlet–triplet splitting of CH2 and other molecular effects, J. Chem. Phys. 84, 4481 (1986).
  23. E. F. Valeev and C. D. Sherrill, The diagonal Born–Oppenheimer correction beyond the Hartree–Fock approximation, J. Chem. Phys. 118, 3921 (2003).
  24. F. Giustino, Electron-phonon interactions from first principles, Rev. Mod. Phys. 89, 015003 (2017).
  25. S. Baroni, P. Giannozzi, and A. Testa, Green’s-function approach to linear response in solids, Phys. Rev. Lett. 58, 1861 (1987).
  26. X. Gonze, First-principles responses of solids to atomic displacements and homogeneous electric fields: Implementation of a conjugate-gradient algorithm, Phys. Rev. B 55, 10337 (1997).
  27. X. Gonze and C. Lee, Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory, Phys. Rev. B 55, 10355 (1997).
  28. X. Gonze, G.-M. Rignanese, and R. Caracas, First-principle studies of the lattice dynamics of crystals, and related properties, Z. Kristallogr. Cryst. Mater. 220, 458 (2005).
  29. X. Gonze, S. Rostami, and C. Tantardini, Variational density functional perturbation theory for metals, Phys. Rev. B 109, 014317 (2024).
  30. S. Poncé, W. Li, S. Reichardt, and F. Giustino, First-principles calculations of charge carrier mobility and conductivity in bulk semiconductors and two-dimensional materials, Rep. Progress Phys. 83, 036501 (2020).
  31. J.-M. Lihm and S. Poncé, Beyond-quasiparticle transport with vertex correction: Self-consistent ladder formalism for electron-phonon interactions, Phys. Rev. X 16, 011008 (2026).
  32. P. B. Allen and J. L. Feldman, Thermal conductivity of disordered harmonic solids, Phys. Rev. B 48, 12581 (1993).
  33. L. Lindsay, First principles Peierls-Boltzmann phonon thermal transport: A topical review, Nanoscale Microsc. Thermophys. Eng. 20, 67 (2016).
  34. E. R. Margine and F. Giustino, Anisotropic Migdal-Eliashberg theory using Wannier functions, Phys. Rev. B 87, 024505 (2013).
  35. W. H. Sio, C. Verdi, S. Poncé, and F. Giustino, Polarons from first principles, without supercells, Phys. Rev. Lett. 122, 246403 (2019).
  36. V. Vasilchenko, M. Giantomassi, S. Poncé, and X. Gonze, Variational first-principles approach to self-trapped polarons, Phys. Rev. B 112, 014314 (2025).
  37. S. Poncé, Y. Gillet, J. Laflamme Janssen, A. Marini, M. Verstraete, and X. Gonze, Temperature dependence of the electronic structure of semiconductors and insulators, J. Chem. Phys. 143, 102813 (2015).
  38. S. Poncé, J.-M. Lihm, and C.-H. Park, Verification and validation of zero-point electron-phonon renormalization of the bandgap, mass enhancement, and spectral functions, npj Comput. Mater. 11, 117 (2025).
  39. C. Freysoldt and J. Neugebauer, First-principles calculations for charged defects at surfaces, interfaces, and two-dimensional materials in the presence of electric fields, Phys. Rev. B 97, 205425 (2018).
  40. M. Černý, J. Pokluda, M. Šob, M. Friák, and P. Šandera, Ab initio calculations of elastic and magnetic properties of Fe, Co, Ni, and Cr crystals under isotropic deformation, Phys. Rev. B 67, 035116 (2003).
  41. J. Xu, H. Huang, and P. Hu, An approach to calculate the free energy changes of surface reactions using free energy decomposition on ab initio brute-force molecular dynamics trajectories, Phys. Chem. Chem. Phys. 22, 21340 (2020).
  42. J. Bouchet, F. Bottin, V. Recoules, F. Remus, G. Morard, R. M. Bolis, and A. Benuzzi-Mounaix, Ab initio calculations of the B1-B2 phase transition in MgO, Phys. Rev. B 99, 094113 (2019).
  43. P. B. Allen, Theory of thermal expansion: Quasi-harmonic approximation and corrections from quasi-particle renormalization, Mod. Phys. Lett. B 34, 2050025 (2020).
  44. P. B. Allen and V. Heine, Theory of the temperature dependence of electronic band structures, J. Phys. C 9, 2305 (1976).
  45. S. Paul, M. P. Gururajan, A. Bhattacharya, and T. R. S. Prasanna, Theoretical issues in the accurate computation of the electron-phonon interaction contribution to the total energy, arXiv:2309.04279.
  46. See Supplemental Material at http://link.aps.org/supplemental/10.1103/zx5f-cw7t for (S1) hybrid Born-Oppenheimer-based approximations; (S2) the proof that the total wave function factorizes up to second order in λ; (S3) the eigenenergies of the auxiliary Hamiltonian, including all sixth-order energy terms; (S4) the Born-Oppenheimer perturbation expansion in normal-mode coordinates; (S5) software improvements and numerical details; (S6) the zero-point renormalization at q=Γ; and (S7) size-consistency tests, which includes Refs. [14,37,49,57,63,78–80,82,87,89,90,103].
  47. P. B. Allen and M. Cardona, Theory of the temperature dependence of the direct gap of germanium, Phys. Rev. B 23, 1495 (1981).
  48. X. Gonze et al., Recent developments in the ABINIT software package, Comput. Phys. Commun. 205, 106 (2016).
  49. X. Gonze et al., The Abinit project: Impact, environment and recent developments, Comput. Phys. Commun. 248, 107042 (2020).
  50. A. H. Romero et al., ABINIT: Overview and focus on selected capabilities, J. Chem. Phys. 152, 124102 (2020).
  51. M. J. Verstraete et al., Abinit 2025: New capabilities for the predictive modeling of solids and nanomaterials, J. Chem. Phys. 163, 164126 (2025).
  52. G. Brunin, H. P. C. Miranda, M. Giantomassi, M. Royo, M. Stengel, M. J. Verstraete, X. Gonze, G.-M. Rignanese, and G. Hautier, Phonon-limited electron mobility in Si, GaAs, and GaP with exact treatment of dynamical quadrupoles, Phys. Rev. B 102, 094308 (2020).
  53. G. Brunin, H. P. C. Miranda, M. Giantomassi, M. Royo, M. Stengel, M. J. Verstraete, X. Gonze, G.-M. Rignanese, and G. Hautier, Electron-phonon beyond Fröhlich: Dynamical quadrupoles in polar and covalent solids, Phys. Rev. Lett. 125, 136601 (2020).
  54. J. Y. Tsao et al., Ultrawide-bandgap semiconductors: Research opportunities and challenges, Adv. Electron. Mater. 4, 1600501 (2018).
  55. M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, New York, NY, 1996).
  56. C. J. Ballhausen and A. E. Hansen, Electronic spectra, Annu. Rev. Phys. Chem. 23, 15 (1972).
  57. L. A. Nafie, Adiabatic molecular properties beyond the Born–Oppenheimer approximation. Complete adiabatic wave functions and vibrationally induced electronic current density, J. Chem. Phys. 79, 4950 (1983).
  58. G. A. Hagedorn, High order corrections to the time-dependent Born-Oppenheimer approximation I: Smooth potentials, Ann. Math. 124, 571 (1986).
  59. J. C. Tully, Perspective on “Zur Quantentheorie der Molekeln”, Theor. Chem. Acc. Theor. Comput. Model. 103, 173 (2000).
  60. A. Scherrer, F. Agostini, D. Sebastiani, E. Gross, and R. Vuilleumier, On the mass of atoms in molecules: Beyond the Born-Oppenheimer approximation, Phys. Rev. X 7, 031035 (2017).
  61. G. Cohen, R. Steinitz-Eliyahu, E. K. U. Gross, S. Refaely-Abramson, and R. Requist, Nonadiabaticity from first principles: Exact-factorization approach for solids, Phys. Rev. B 112, 075102 (2025).
  62. A. Abedi, N. T. Maitra, and E. K. U. Gross, Exact factorization of the time-dependent electron-nuclear wave function, Phys. Rev. Lett. 105, 123002 (2010).
  63. C. A. Mead and D. G. Truhlar, On the determination of Born–Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei, J. Chem. Phys. 70, 2284 (1979).
  64. T. Qin, J. Zhou, and J. Shi, Berry curvature and the phonon Hall effect, Phys. Rev. B 86, 104305 (2012).
  65. T. Saito, K. Misaki, H. Ishizuka, and N. Nagaosa, Berry phase of phonons and thermal Hall effect in nonmagnetic insulators, Phys. Rev. Lett. 123, 255901 (2019).
  66. O. Bistoni, F. Mauri, and M. Calandra, Intrinsic vibrational angular momentum from nonadiabatic effects in noncollinear magnetic molecules, Phys. Rev. Lett. 126, 225703 (2021).
  67. D. Saparov, B. Xiong, Y. Ren, and Q. Niu, Lattice dynamics with molecular Berry curvature: Chiral optical phonons, Phys. Rev. B 105, 064303 (2022).
  68. J. Bonini, S. Ren, D. Vanderbilt, M. Stengel, C. E. Dreyer, and S. Coh, Frequency splitting of chiral phonons from broken time-reversal symmetry in CrI3, Phys. Rev. Lett. 130, 086701 (2023).
  69. S. Ren, J. Bonini, M. Stengel, C. E. Dreyer, and D. Vanderbilt, Adiabatic dynamics of coupled spins and phonons in magnetic insulators, Phys. Rev. X 14, 011041 (2024).
  70. M. Royo and M. Stengel, Dynamical response of noncollinear spin systems at constrained magnetic moments, Phys. Rev. X 16, 011049 (2026).
  71. G. D. Mahan, Many-Particle Physics (Springer, Boston, MA, 1990), 10.1007/978-1-4613-1469-1.
  72. R. Requist, C. R. Proetto, and E. K. U. Gross, Exact factorization-based density functional theory of electron-phonon systems, Phys. Rev. B 99, 165136 (2019).
  73. G. Stefanucci, R. van Leeuwen, and E. Perfetto, In and out-of-equilibrium ab initio theory of electrons and phonons, Phys. Rev. X 13, 031026 (2023).
  74. R. M. Martin, Electronic Structure: Basic Theory and Practical Methods (Cambridge University Press, Cambridge, 2004).
  75. D. R. Hamann, M. Schlüter, and C. Chiang, Norm-conserving pseudopotentials, Phys. Rev. Lett. 43, 1494 (1979).
  76. P. B. Allen, Second erratum—theory of thermal expansion: Quasi-harmonic approximation and corrections from quasi-particle renormalization, Mod. Phys. Lett. B 36, 2292002 (2022).
  77. A. V. Ramasimha Varma, S. Paul, A. Itale, P. Pable, R. Tibrewala, S. Dodal, H. Yerunkar, S. Bhaumik, V. Shah, M. P. Gururajan, and T. R. S. Prasanna, Electron–phonon interaction contribution to the total energy of group IV semiconductor polymorphs: Evaluation and implications, ACS Omega 8, 11251 (2023).
  78. A. Miglio, V. Brousseau-Couture, E. Godbout, G. Antonius, Y.-H. Chan, S. G. Louie, M. Côté, M. Giantomassi, and X. Gonze, Predominance of non-adiabatic effects in zero-point renormalization of the electronic band gap, npj Comput. Mater. 6, 167 (2020).
  79. S. Poncé, G. Antonius, Y. Gillet, P. Boulanger, J. L. Janssen, A. Marini, M. Côté, and X. Gonze, Temperature dependence of electronic eigenenergies in the adiabatic harmonic approximation, Phys. Rev. B 90, 214304 (2014).
  80. X. Gonze, P. Boulanger, and M. Côté, Theoretical approaches to the temperature and zero-point motion effects on the electronic band structure, Ann. Phys. 523, 168 (2011).
  81. J.-M. Lihm, S. Poncé, and C.-H. Park, Self-consistent electron lifetimes for electron-phonon scattering, Phys. Rev. B 110, L121106 (2024).
  82. S. Poncé, G. Antonius, P. Boulanger, E. Cannuccia, A. Marini, M. Côté, and X. Gonze, Verification of first-principles codes: Comparison of total energies, phonon frequencies, electron–phonon coupling and zero-point motion correction to the gap between ABINIT and QE/Yambo, Comput. Mater. Sci. 83, 341 (2014).
  83. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  84. M. J. van Setten, M. Giantomassi, E. Bousquet, M. J. Verstraete, D. R. Hamann, X. Gonze, and G.-M. Rignanese, The pseudodojo: Training and grading a 85 element optimized norm-conserving pseudopotential table, Comput. Phys. Commun. 226, 39 (2018).
  85. G. T. Surratt, R. N. Euwema, and D. L. Wilhite, Hartree-Fock lattice constant and bulk modulus of diamond, Phys. Rev. B 8, 4019 (1973).
  86. S. Poncé, F. Macheda, E. R. Margine, N. Marzari, N. Bonini, and F. Giustino, First-principles predictions of Hall and drift mobilities in semiconductors, Phys. Rev. Res. 3, 043022 (2021).
  87. M. Royo and M. Stengel, First-principles theory of spatial dispersion: Dynamical quadrupoles and flexoelectricity, Phys. Rev. X 9, 021050 (2019).
  88. S. G. Louie, S. Froyen, and M. L. Cohen, Nonlinear ionic pseudopotentials in spin-density-functional calculations, Phys. Rev. B 26, 1738 (1982).
  89. A. Eiguren and C. Ambrosch-Draxl, Wannier interpolation scheme for phonon-induced potentials: Application to bulk MgB2, W, and the (1×1) H-covered W(110) surface, Phys. Rev. B 78, 045124 (2008).
  90. J.-M. Lihm and C.-H. Park, Phonon-induced renormalization of electron wave functions, Phys. Rev. B 101, 121102 (2020).
  91. F. Giustino, M. L. Cohen, and S. G. Louie, Electron-phonon interaction using Wannier functions, Phys. Rev. B 76, 165108 (2007).
  92. S. Poncé, E. R. Margine, C. Verdi, and F. Giustino, EPW: Electron–phonon coupling, transport and superconducting properties using maximally localized Wannier functions, Comput. Phys. Commun. 209, 116 (2016).
  93. C. Verdi and F. Giustino, Fröhlich electron-phonon vertex from first principles, Phys. Rev. Lett. 115, 176401 (2015).
  94. J. Sjakste, N. Vast, M. Calandra, and F. Mauri, Wannier interpolation of the electron-phonon matrix elements in polar semiconductors: Polar-optical coupling in GaAs, Phys. Rev. B 92, 054307 (2015).
  95. M. Stengel, Unified ab initio formulation of flexoelectricity and strain-gradient elasticity, Phys. Rev. B 93, 245107 (2016).
  96. M. Royo, K. R. Hahn, and M. Stengel, Using high multipolar orders to reconstruct the sound velocity in piezoelectrics from lattice dynamics, Phys. Rev. Lett. 125, 217602 (2020).
  97. V. A. Jhalani, J.-J. Zhou, J. Park, C. E. Dreyer, and M. Bernardi, Piezoelectric electron-phonon interaction from ab initio dynamical quadrupoles: Impact on charge transport in Wurtzite GaN, Phys. Rev. Lett. 125, 136602 (2020).
  98. M. Royo and M. Stengel, Exact long-range dielectric screening and interatomic force constants in quasi-two-dimensional crystals, Phys. Rev. X 11, 041027 (2021).
  99. S. Poncé, M. Royo, M. Stengel, N. Marzari, and M. Gibertini, Long-range electrostatic contribution to electron-phonon couplings and mobilities of two-dimensional and bulk materials, Phys. Rev. B 107, 155424 (2023).
  100. S. Poncé, M. Royo, M. Gibertini, N. Marzari, and M. Stengel, Accurate prediction of Hall mobilities in two-dimensional materials through Gauge-covariant quadrupolar contributions, Phys. Rev. Lett. 130, 166301 (2023).
  101. F. P. Bundy and J. S. Kasper, Hexagonal diamond—a new form of carbon, J. Chem. Phys. 46, 3437 (1967).
  102. A. P. Jones, P. F. McMillan, C. G. Salzmann, M. Alvaro, F. Nestola, M. Prencipe, D. Dobson, R. Hazael, and M. Moore, Structural characterization of natural diamond shocked to 60 GPa; Implications for Earth and planetary systems, Lithos 265, 214 (2016).
  103. V. Brousseau-Couture, É. Godbout, M. Côté, and X. Gonze, Zero-point lattice expansion and band gap renormalization: Grüneisen approach versus free energy minimization, Phys. Rev. B 106, 085137 (2022).
  104. N. Ikeda, Y. Niiyama, H. Kambayashi, Y. Sato, T. Nomura, S. Kato, and S. Yoshida, GaN power transistors on Si substrates for switching applications, Proc. IEEE 98, 1151 (2010).
  105. T. J. Flack, B. N. Pushpakaran, and S. B. Bayne, GaN technology for power electronic applications: A review, J. Electron. Mater. 45, 2673 (2016).
  106. S. Poncé, D. Jena, and F. Giustino, Hole mobility of strained GaN from first principles, Phys. Rev. B 100, 085204 (2019).
  107. B. N. Pantha, R. Dahal, J. Li, J. Y. Lin, H. X. Jiang, and G. Pomrenke, Thermoelectric properties of InxGa1−xN alloys, Appl. Phys. Lett. 92, 042112 (2008).
  108. K. Atsumi, Y. Inoue, H. Mimura, T. Aoki, and T. Nakano, Neutron detection using boron gallium nitride semiconductor material, APL Mater. 2, 032106 (2014).
  109. O. Brandt, H. Yang, B. Jenichen, Y. Suzuki, L. Däweritz, and K. H. Ploog, Surface reconstructions of zinc-blende GaN/GaAs(001) in plasma-assisted molecular-beam epitaxy, Phys. Rev. B 52, R2253(R) (1995).
  110. H. Xia, Q. Xia, and A. L. Ruoff, High-pressure structure of gallium nitride: Wurtzite-to-rocksalt phase transition, Phys. Rev. B 47, 12925 (1993).
  111. S. Poncé and X. Gonze, In search of the electron-phonon contribution to total energy, Materials Cloud Archive, 10.24435/materialscloud:4d-rw (2026).

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PRX Energy 5, 043004– Published 22 October, 2026
  • Received 4 December 2025
  • Revised 26 July 2026
  • Accepted 18 September 2026
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