Although the general instability of the iron nitride γ'-Fe4N with respect to other phases at high pressure is well established, the actual type of phase transitions and equilibrium conditions of their occurrence are, as of yet, poorly investigated. In the present study, samples of γ'-Fe4N and mixtures of α Fe and γ'-Fe4N powders have been heat-treated at temperatures between 250 and 1000 °C and pressures between 2 and 8 GPa in a multi-anvil press, in order to investigate phase equilibria involving the γ' phase. Samples heat-treated at high-pressure conditions, were quenched, subsequently decompressed, and then analysed ex situ. Microstructure analysis is used to derive implications on the phase transformations during the heat treatments. Further, it is confirmed that the Fe-N phases in the target composition range are quenchable. Thus, phase proportions and chemical composition of the phases, determined from ex situ X-ray diffraction data, allowed conclusions about the phase equilibria at high-pressure conditions. Further, evidence for the low-temperature eutectoid decomposition γ'→α+ε' is presented for the first time. From the observed equilibria, a P-T projection of the univariant equilibria in the Fe-rich portion of the Fe-N system is derived, which features a quadruple point at 5 GPa and 375 °C, above which γ'-Fe4N is thermodynamically unstable. The experimental work is supplemented by ab initio calculations in order to discuss the relative phase stability and energy landscape in the Fe-N system, from the ground state to conditions accessible in the multi-anvil experiments. It is concluded that γ'-Fe4N, which is unstable with respect to other phases at 0 K (at any pressure), has to be entropically stabilised in order to occur as stable phase in the system. In view of the frequently reported metastable retention of the γ' phase during room temperature compression experiments, energetic and kinetic aspects of the polymorphic transition γ'⇌ε' are discussed.

Department of Chemistry Faculty of Science Masaryk University Kotlářská 2 CZ 611 37 Brno Czech Republic

Freiberg High Pressure Research Centre Technische Universität Bergakademie Freiberg Lessingstraße 45 D 09599 Freiberg Germany

Institute of Inorganic Chemistry Technische Universität Bergakademie Freiberg Leipziger Straße 29 D 09599 Freiberg Germany

Institute of Materials Science Technische Universität Bergakademie Freiberg Gustav Zeuner Str 5 D 09599 Freiberg Germany

Institute of Physics of Materials Czech Academy of Sciences v v i Žižkova 22 CZ 616 62 Brno Czech Republic

Zobrazit více v PubMed

Mittemeijer E.J. Fundamentals of Nitriding and Nitrocarburizing. In: Dossett J.L., Totten G.E., editors. Steel Heat Treating Fundamentals and Processes. Volume 4A. ASM International; Materials Park, OH, USA: 2013. pp. 619–646.

Jack K.H. Binary and ternary interstitial alloys I. The iron-nitrogen system: The structures of Fe4N and Fe2N. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 1948;195:34–40. doi: 10.1098/rspa.1948.0100. DOI

Jack K.H. The iron–nitrogen system: The crystal structures of ε-phase iron nitrides. Acta Crystallogr. 1952;5:404–411. doi: 10.1107/S0365110X52001258. DOI

Leineweber A., Jacobs H., Hüning F., Lueken H., Schilder H., Kockelmann W. ε-Fe3N: Magnetic structure, magnetization and temperature dependent disorder of nitrogen. J. Alloys Compd. 1999;288:79–87. doi: 10.1016/S0925-8388(99)00150-4. DOI

Leineweber A., Jacobs H., Hüning F., Lueken H., Kockelmann W. Nitrogen ordering and ferromagnetic properties of ε-Fe3N1+x (0.10 ≤ x ≤ 0.39) and ε-Fe3(N0.80C0.20)1.38. J. Alloys Compd. 2001;316:21–38. doi: 10.1016/S0925-8388(00)01435-3. DOI

Wriedt H.A., Gokcen N.A., Nafziger R.H. The Fe-N (Iron-Nitrogen) System. Bull. Alloy Phase Diagr. 1987;8:355–377. doi: 10.1007/BF02869273. DOI

Göhring H., Fabrichnaya O., Leineweber A., Mittemeijer E.J. Thermodynamics of the Fe-N and Fe-N-C Systems: The Fe-N and Fe-N-C Phase Diagrams Revisited. Metall. Mater. Trans. A. 2016;47:6173–6186. doi: 10.1007/s11661-016-3731-0. DOI

You Z., Paek M.K., Jung I.H. Critical Evaluation and Optimization of the Fe-N, Mn-N and Fe-Mn-N Systems. J. Phase Equilibria Diffus. 2018;39:650–677. doi: 10.1007/s11669-018-0666-8. DOI

Malinov S., Böttger A.J., Mittemeijer E.J., Pekelharing M.I., Somers M.A.J. Phase transformations and phase equilibria in the Fe-N system at temperatures below 573 K. Metall. Mater. Trans. A. 2001;32:59–73. doi: 10.1007/s11661-001-0102-1. DOI

Du Marchie van Voorthuysen E.H., Boerma D.O., Chechenin N.C. Low-temperature extension of the Lehrer diagram and the iron-nitrogen phase diagram. Metall. Mater. Trans. A Phys. Metall. Mater. Sci. 2002;33:2593–2598. doi: 10.1007/s11661-002-0380-2. DOI

Andersson J.-O., Helander T., Höglund L., Shi P., Sundman B. Thermo-Calc & DICTRA, computational tools for materials science. Calphad. 2002;26:273–312. doi: 10.1016/S0364-5916(02)00037-8. DOI

Dewaele A., Denoual C., Anzellini S., Occelli F., Mezouar M., Cordier P., Merkel S., Véron M., Rausch E. Mechanism of the α-ϵ phase transformation in iron. Phys. Rev. B. 2015;91:174105. doi: 10.1103/PhysRevB.91.174105. DOI

Dewaele A., Svitlyk V., Bottin F., Bouchet J., Jacobs J. Iron under conditions close to the α-γ-ϵ triple point. Appl. Phys. Lett. 2018;112 doi: 10.1063/1.5030192. DOI

Ackermann S., Martin S., Schwarz M., Schimpf C., Kulawinski D., Lathe C., Henkel S., Rafaja D., Biemann H., Weidner A. Investigation of phase transformations in high-alloy austenitic TRIP steel under high pressure (up to 18 GPa) by in situ synchrotron X-ray diffraction and scanning electron microscopy. Metall. Mater. Trans. A. 2016;47:95–111. doi: 10.1007/s11661-015-3082-2. DOI

Dupé B., Amadon B., Pellegrini Y.P., Denoual C. Mechanism for the α → ϵ phase transition in iron. Phys. Rev. B Condens. Matter Mater. Phys. 2013;87:1–5. doi: 10.1103/PhysRevB.87.024103. DOI

Luu H.-T., Veiga R.G.A., Gunkelmann N. Atomistic Study of the Role of Defects on α → ϵ Phase Transformations in Iron under Hydrostatic Compression. Metals. 2019;9:1040. doi: 10.3390/met9101040. DOI

Mankovsky S., Polesya S., Ebert H., Bensch W., Mathon O., Pascarelli S., Minár J. Pressure-induced bcc to hcp transition in Fe: Magnetism-driven structure transformation. Phys. Rev. B Condens. Matter Mater. Phys. 2013;88:1–8. doi: 10.1103/PhysRevB.88.184108. DOI

Lord J.S., Armitage J.G.M., Riedi P.C., Matar S.F., Demazeau G. The volume dependence of the magnetization and NMR of Fe4N and Mn4N. J. Phys. Condens. Matter. 1994;6:1779–1790. doi: 10.1088/0953-8984/6/9/019. DOI

Yang C.L., Abd-Elmeguid M.M., Micklitz H., Michels G., Otto J.W., Kong Y., Xue D.S., Li F.S. Pressure effects on the electronic properties and the magnetic ground state of γ′-Fe4N. J. Magn. Magn. Mater. 1995;8853:2–6. doi: 10.1016/0304-8853(95)00530-7. DOI

Ishimatsu N., Maruyama H., Kawamura N., Suzuki M., Ohishi Y., Ito M., Nasu S., Kawakami T., Shimomura O. Pressure-Induced Magnetic Transition in Fe4N Probed by Fe K-edge XMCD Measurement. J. Phys. Soc. Jpn. 2003;72:2372–2376. doi: 10.1143/JPSJ.72.2372. DOI

Lv M., Liu J., Zhu F., Li J., Zhang D., Xiao Y., Dorfman S.M. Spin Transitions and Compressibility of ε-Fe7N3 and γ′-Fe4N: Implications for Iron Alloys in Terrestrial Planet Cores. J. Geophys. Res. Solid Earth. 2020;125:1–15. doi: 10.1029/2020JB020660. DOI

Adler J.F., Williams Q. A high-pressure X-ray diffraction study of iron nitrides: Implications for Earth’s core. J. Geophys. Res. 2005;110:B01203. doi: 10.1029/2004JB003103. DOI

Litasov K.D., Shatskiy A., Ponomarev D.S., Gavryushkin P.N. Equations of state of iron nitrides ε-Fe3Nx and γ-Fe4Ny to 30 GPa and 1200 K and implication for nitrogen in the Earth’s core. J. Geophys. Res. Solid Earth. 2017;122:3574–3584. doi: 10.1002/2017JB014059. DOI

Breton H., Komabayashi T., Thompson S., Potts N., McGuire C., Suehiro S., Anzellini S., Ohishi Y. Static compression of Fe4N to 77 GPa and its implications for nitrogen storage in the deep Earth. Am. Mineral. 2019;104:1781–1787. doi: 10.2138/am-2019-7065. DOI

Minobe S., Nakajima Y., Hirose K., Ohishi Y. Stability and compressibility of a new iron-nitride β-Fe7N3 to core pressures. Geophys. Res. Lett. 2015;42:5206–5211. doi: 10.1002/2015GL064496. DOI

Zhuang Y., Su X., Salke N.P., Cui Z., Hu Q., Zhang D., Liu J. The effect of nitrogen on the compressibility and conductivity of iron at high pressure. Geosci. Front. 2020 doi: 10.1016/j.gsf.2020.04.012. DOI

Niewa R., Rau D., Wosylus A., Meier K., Wessel M., Hanfland M., Dronskowski R., Schwarz U. High-pressure high-temperature phase transition of γ′-Fe4N. J. Alloys Compd. 2009;480:76–80. doi: 10.1016/j.jallcom.2008.09.178. DOI

Schwarz U., Wosylus A., Wessel M., Nskowski R.D., Hanfland M., Rau D., Niewa R. High-pressure-high-temperature behavior of ζ-Fe2N and phase transition to ε-Fe3N1.5. Eur. J. Inorg. Chem. 2009:1634–1639. doi: 10.1002/ejic.200801222. DOI

Clark W.P., Steinberg S., Dronskowski R., McCammon C., Kupenko I., Bykov M., Dubrovinsky L., Akselrud L.G., Schwarz U., Niewa R. High-Pressure NiAs-Type Modification of FeN. Angew. Chem. Int. Ed. 2017;56:7302–7306. doi: 10.1002/anie.201702440. PubMed DOI PMC

Laniel D., Dewaele A., Garbarino G. High Pressure and High Temperature Synthesis of the Iron Pernitride FeN2. Inorg. Chem. 2018:6245–6251. doi: 10.1021/acs.inorgchem.7b03272. PubMed DOI

Bykov M., Khandarkhaeva S., Fedotenko T., Sedmak P., Dubrovinskaia N., Dubrovinsky L. Synthesis of FeN4 at 180 GPa and its crystal structure from a submicron-sized grain. Acta Crystallogr. Sect. E Crystallogr. Commun. 2018;74:1392–1395. doi: 10.1107/S2056989018012161. PubMed DOI PMC

Wetzel M.H., Schwarz M.R., Leineweber A. High-pressure high-temperature study of the pressure induced decomposition of the iron nitride γ′-Fe4N. J. Alloys Compd. 2019;801:438–448. doi: 10.1016/j.jallcom.2019.06.078. DOI

Kresse G., Hafner J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B. 1993;47:558–561. doi: 10.1103/PhysRevB.47.558. PubMed DOI

Kresse G., Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B. 1996;54:11169–11186. doi: 10.1103/PhysRevB.54.11169. PubMed DOI

Hohenberg P., Kohn W. Inhomogeneous Electron Gas. Phys. Rev. 1964;136:B864–B871. doi: 10.1103/PhysRev.136.B864. DOI

Kohn W., Sham L.J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 1965;140:A1133–A1138. doi: 10.1103/PhysRev.140.A1133. DOI

Blöchl P.E. Projector augmented-wave method. Phys. Rev. B. 1994;50:17953–17979. doi: 10.1103/PhysRevB.50.17953. PubMed DOI

Kresse G., Joubert D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B. 1999;59:1758–1775. doi: 10.1103/PhysRevB.59.1758. DOI

Perdew J.P., Burke K., Ernzerhof M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996;77:3865–3868. doi: 10.1103/PhysRevLett.77.3865. PubMed DOI

Yuan X., Zhou Y., Huo C., Guo W., Yang Y., Li Y., Wen X. Crystal Structure Prediction Approach to Explore the Iron Carbide Phases: Novel Crystal Structures and Unexpected Magnetic Properties. J. Phys. Chem. C. 2020;124:17244–17254. doi: 10.1021/acs.jpcc.0c05129. DOI

Shatskiy A., Katsura T., Litasov K.D., Shcherbakova A.V., Borzdov Y.M., Yamazaki D., Yoneda A., Ohtani E., Ito E. High pressure generation using scaled-up Kawai-cell. Phys. Earth Planet. Int. 2011;189:92–108. doi: 10.1016/j.pepi.2011.08.001. DOI

Walker D., Carpenter M.A., Hitch C.M. Some simplifications to multianvil devices for high pressure experiments. Am. Mineral. 1990;75:1020–1028.

Rubie D.C. Characterising the sample environment in multianvil high-pressure experiments. Phase Transit. 1999;68:431–451. doi: 10.1080/01411599908224526. DOI

Bundy F.P. Phase Diagram of Bismuth to 130,000 kg/cm2, 500 °C. Phys. Rev. 1958;110:314–318. doi: 10.1103/PhysRev.110.314. DOI

Ohtaka O., Fukui H., Kunisada T., Fujisawa T., Funakoshi K., Utsumi W., Irifune T., Kuroda K., Kikegawa T. Phase relations and equations of state of ZrO2 under high temperature and high pressure. Phys. Rev. B Condens. Matter Mater. Phys. 2001;63:1741081–1741088. doi: 10.1103/PhysRevB.63.174108. DOI

Schwarz M.R. Multianvil calibration and education: A four probe method to measure the entire force-versus-pressure curve in a single run—Performed as an interdisciplinary lab-course for students. J. Phys. Conf. Ser. 2010;215 doi: 10.1088/1742-6596/215/1/012193. DOI

Nishihara Y., Doi S., Kakizawa S., Higo Y., Tange Y. Effect of pressure on temperature measurements using WRe thermocouple and its geophysical impact. Phys. Earth Planet. Int. 2020;298:106348. doi: 10.1016/j.pepi.2019.106348. DOI

Nishihara Y., Matsukage K.N., Karato S.I. Effects of metal protection coils on thermocouple EMF in multi-anvil high-pressure experiments. Am. Mineral. 2006;91:111–114. doi: 10.2138/am.2006.1883. DOI

Stokes H.T., Hatch D.M. FINDSYM: Program for identifying the space-group symmetry of a crystal. J. Appl. Crystallogr. 2005;38:237–238. doi: 10.1107/S0021889804031528. DOI

Somers M.A.J., Pers N.M., Schalkoord D., Mittemeijer E.J. Dependence of the lattice parameter of γ′ iron nitride, Fe4N1−x, on nitrogen content; Accuracy of the nitrogen absorption data. Metall. Trans. A. 1989;20:1533–1539. doi: 10.1007/BF02665509. DOI

Imai Y., Izumlyama M., Tsuchiya M. Thermodynamic Study on the Transformation of Austenite into Martensite in Iron-High Nitrogen and Iron-Carbon Binary System. Sci. Rep. Res. Inst. Tohoku Univ. Ser. A Phys. Chem. Metall. 1965;17:173–192.

Gerardin D., Morniroli J.P., Michel H., Gantois M. Microstructural study of ε-iron-carbonitrides formed by a glow discharge technique. J. Mater. Sci. 1981;16:159–169. doi: 10.1007/BF00552070. DOI

Göhring H., Kante S., Leineweber A., Mittemeijer E.J. Microstructural development and crystallographic properties of decomposing Fe-N-C compound layers. Int. J. Mater. Res. 2016;107:203–216. doi: 10.3139/146.111330. DOI

Nishiyama Z. Martensitic Transformation. 1st ed. Academic Press; London, UK: 1978.

Kral M.V. Proeutectoid ferrite and cementite transformations in steels. In: Pereloma E., Edmonds D.V., editors. Phase Transformations in Steels. Volume 1. Woodhead Publishing Limited; Cambridge, UK: 2012. pp. 225–275.

Yang J.R., Chang L.C. The effect of stress on the Widmanstätten ferrite transformation. Mater. Sci. Eng. A. 1997;223:158–167. doi: 10.1016/S0921-5093(96)10475-5. DOI

Cheng L., Böttger A., de Keijser T.H., Mittemeijer E.J. Lattice parameters of iron-carbon and iron-nitrogen martensites and austenites. Scr. Metall. Mater. 1990;24:509–514. doi: 10.1016/0956-716X(90)90192-J. DOI

Liapina T., Leineweber A., Mittemeijer E.J., Kockelmann W. The lattice parameters of ε-iron nitrides: Lattice strains due to a varying degree of nitrogen ordering. Acta Mater. 2004;52:173–180. doi: 10.1016/j.actamat.2003.09.003. DOI

Burdese A. Structural Relations and Equilibria between the ε and ζ Phases of the System Iron-Nitrogen. Metall. Ital. 1957;49:195–199.

Guo K., Rau D., von Appen J., Prots Y., Schnelle W., Dronskowski R., Niewa R., Schwarz U. High pressure high-temperature behavior and magnetic properties of Fe4N: Experiment and theory. High Press. Res. 2013;33:684–696. doi: 10.1080/08957959.2013.809715. DOI

Paranjpe V.G., Cohen M., Bever M.B., Floe C.F. The iron-nitrogen system. J. Met. 1950;2:261–267. doi: 10.1007/BF03398999. DOI

Bouchard R.J., Frederick C.G., Johnson V. Preparation and properties of submicron hexagonal FexN, 2 < x < 3. J. Appl. Phys. 1974;45:4067–4070. doi: 10.1063/1.1663913. DOI

Rauschenbach B., Kolitsch A., Hohmuth K. Iron nitride phases formed by nitrogen ion implantation and thermal treatment. Phys. Status Solidi. 1983;80:471–482. doi: 10.1002/pssa.2210800209. DOI

Somers M.A.J., Kooi B.J., Maldzinski L., Mittemeijer E.J., Van Der Horst A.A., Van Der Kraan A.M., Van Der Pers N.M. Thermodynamics and long-range order of interstitials in an h.c.p. Lattice: Nitrogen in ε-Fe2N1−z. Acta Mater. 1997;45:2013–2025. doi: 10.1016/S1359-6454(96)00307-2. DOI

Dubrovinsky L.S., Saxena S.K., Tutti F., Rekhi S., LeBehan T. In situ X-ray study of thermal expansion and phase transition of iron at multimegabar pressure. Phys. Rev. Lett. 2000;84:1720–1723. doi: 10.1103/PhysRevLett.84.1720. PubMed DOI

Dewaele A., Loubeyre P., Occelli F., Mezouar M., Dorogokupets P.I., Torrent M. Quasihydrostatic equation of state of Iron above 2 Mbar. Phys. Rev. Lett. 2006;97:29–32. doi: 10.1103/PhysRevLett.97.215504. PubMed DOI

Boehler R., Santamaría-Pérez D., Errandonea D., Mezouar M. Melting, density, and anisotropy of iron at core conditions: New X-ray measurements to 150 GPa. J. Phys. Conf. Ser. 2008;121:022018. doi: 10.1088/1742-6596/121/2/022018. DOI

Lu X.G., Selleby M., Sundman B. Implementation of a new model for pressure dependence of condensed phases in Thermo-Calc. Calphad Comput. Coupling Phase Diagr. Thermochem. 2005;29:49–55. doi: 10.1016/j.calphad.2005.04.001. DOI

Hillert M. Phase Equilibria, Phase Diagrams and Phase Transformations. 2nd ed. Cambridge University Press; Cambridge, UK: 2007.

Acet M., Gehrmann B., Wassermann E.F., Bach H., Pepperhoff W. Relevance of magnetic instabilities to the properties of interstitial solid solutions and compounds of Fe. J. Magn. Magn. Mater. 2001;232:221–230. doi: 10.1016/S0304-8853(01)00249-9. DOI

Hernlund J., Leinenweber K., Locke D., Tyburczy J.A. A numerical model for steady-state temperature distributions in solid-medium high-pressure cell assemblies. Am. Mineral. 2006;91:295–305. doi: 10.2138/am.2006.1938. DOI

Wang K., Reeber R.R. Thermal Expansion of Caesium Halides with the CsCl Structure. J. Appl. Crystallogr. 1995;28:306–313. doi: 10.1107/S0021889894014895. DOI

Köhler U., Johannsen P.G., Holzapfel W.B. Equation-of-state data for CsCl-type alkali halides. J. Phys. Condens. Matter. 1997;9:5581–5592. doi: 10.1088/0953-8984/9/26/007. DOI

Leineweber A., Hickel T., Azimi-Manavi B., Maisel S.B. Crystal structures of Fe4C vs. Fe4N analysed by DFT calculations: Fcc-based interstitial superstructures explored. Acta Mater. 2017;140:433–442. doi: 10.1016/j.actamat.2017.08.059. DOI

De Waele S., Lejaeghere K., Leunis E., Duprez L., Cottenier S. A first-principles reassessment of the Fe-N phase diagram in the low-nitrogen limit. J. Alloys Compd. 2019;775:758–768. doi: 10.1016/j.jallcom.2018.09.356. DOI

Imai Y., Sohma M., Suemasu T. Energetic stability and magnetic moment of tri-, tetra-, and octa-ferromagnetic element nitrides predicted by first-principle calculations. J. Alloys Compd. 2014;611:440–445. doi: 10.1016/j.jallcom.2014.04.171. DOI

Billingham J., Lewis M.H. Dislocation mechanisms for the nucleation of transformations in vanadium carbide. Philos. Mag. J. Theor. Exp. Appl. Phys. 1971;24:231–240. doi: 10.1080/14786437108227383. DOI

Leineweber A. Mobility of nitrogen in ε-Fe3N below 150 °C: The activation energy for reordering. Acta Mater. 2007;55:6651–6658. doi: 10.1016/j.actamat.2007.08.020. DOI

Liapina T., Leineweber A., Mittemeijer E.J., Knapp M., Baehtz C., Liu Z.Q., Mitsuishi K., Furuya K. γ′-Fe4N formation in decomposing ε-Fe3N: A powder diffraction study using synchrotron radiation. Z. Krist. Suppl. 2006;2006:449–454. doi: 10.1524/9783486992526-075. DOI

Foct J., de Figueiredo R.S. Mössbauer study of phase transformation induced by mechanosynthesis in Fe4N nitrides. Nanostruct. Mater. 1994;4:685–697. doi: 10.1016/0965-9773(94)90021-3. DOI

Foct J., de Figueiredo R.S., Richard O., Morniroli J.P. Mechanical Alloying of Interstitial Solid Solutions and Compounds. Mater. Sci. Forum. 1996;225–227:409–416. doi: 10.4028/www.scientific.net/MSF.225-227.409. DOI

De Wit L., Weber T., Custer J.S., Saris F.W. Thermodynamic stability of iron nitrides at temperatures below 350 °C. Phys. Rev. Lett. 1994;72:3835–3838. doi: 10.1103/PhysRevLett.72.3835. PubMed DOI

Pochet P., Bellon P., Chaffron L., Martin G. Phase Transformations under Ball Milling: Theory versus Experiment. Mater. Sci. Forum. 1996;225–227:207–216. doi: 10.4028/www.scientific.net/MSF.225-227.207. DOI

Zarkevich N.A., Johnson D.D. Coexistence pressure for a martensitic transformation from theory and experiment: Revisiting the bcc-hcp transition of iron under pressure. Phys. Rev. B Condens. Matter Mater. Phys. 2015;91:1–9. doi: 10.1103/PhysRevB.91.174104. DOI