Unified ab initio description of Fröhlich electron-phonon interactions in two-dimensional and three-dimensional materials

Sio, Weng Hong; Giustino, Feliciano

doi:10.1103/physrevb.105.115414

Unified ab initio description of Fröhlich electron-phonon interactions in two-dimensional and three-dimensional materials

Journal Article · Tue Mar 15 00:00:00 EDT 2022 · Physical Review. B

DOI:https://doi.org/10.1103/physrevb.105.115414· OSTI ID:1903341

^[1]; Giustino, Feliciano ^[2]

University of Macau (China); Univ. of Texas, Austin, TX (United States); University of Texas at Austin
Univ. of Texas, Austin, TX (United States)

Ab initio calculations of electron-phonon interactions including the polar Fröhlich coupling have advanced considerably in recent years. The Fröhlich electron-phonon matrix element is by now well understood in the case of bulk three-dimensional (3D) materials. In the case of two-dimensional (2D) materials, the standard procedure to include Fröhlich coupling is to employ Coulomb truncation, so as to eliminate artificial interactions between periodic images of the 2D layer. While these techniques are well established, the transition of the Fröhlich coupling from three to two dimensions has not been investigated. Furthermore, it remains unclear what error one makes when describing 2D systems using the standard bulk formalism in a periodic supercell geometry. In this work, we generalize previous work on the ab initio Fröhlich electron-phonon matrix element in bulk materials by investigating the electrostatic potential of atomic dipoles in a periodic supercell consisting of a 2D material and a continuum dielectric slab. We obtain a unified expression for the matrix element, which reduces to the existing formulas for three-dimensional and 2D systems when the interlayer separation tends to zero or infinity, respectively. This expression enables an accurate description of the Fröhlich matrix element in 2D systems without resorting to Coulomb truncation. We validate our approach by direct ab initio density-functional perturbation theory calculations for monolayer BN and MoS₂, and we provide a simple expression for the 2D Fröhlich matrix element that can be used in model Hamiltonian approaches. The formalism outlined in this work may find applications in calculations of polarons, quasiparticle renormalization, transport coefficients, and superconductivity, in 2D and quasi-2D materials.

View Accepted Manuscript (DOE)

Research Organization:: Univ. of Texas, Austin, TX (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES); Science and Technology Development Fund of Macau SAR (FDCT)

Grant/Contract Number:: SC0020129; AC02-05CH11231

OSTI ID:: 1903341

Journal Information:: Physical Review. B, Journal Name: Physical Review. B Journal Issue: 11 Vol. 105; ISSN 2469-9950

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (81)

Tunable Moiré Superlattice of Artificially Twisted Monolayers Chen, Po‐Yen; Zhang, Xin‐Quan; Lai, Ying‐Yu Advanced Materials, Vol. 31, Issue 37 https://doi.org/10.1002/adma.201901077	journal	July 2019
Coulomb Interaction and Excitons in a Superlattice Guseinov, R. R. physica status solidi (b), Vol. 125, Issue 1 https://doi.org/10.1002/pssb.2221250128	journal	September 1984
Optical phonon interaction effects in layered semiconductor structures Sarma, S. Das; Mason, B. A. Annals of Physics, Vol. 163, Issue 1 https://doi.org/10.1016/0003-4916(85)90351-3	journal	August 1985
A variational treatment of a two-dimensional polaron Erçelebi, A.; Süalp, G. Journal of Physics and Chemistry of Solids, Vol. 48, Issue 8 https://doi.org/10.1016/0022-3697(87)90069-2	journal	January 1987
Phononics in low-dimensional materials Balandin, Alexander A.; Nika, Denis L. Materials Today, Vol. 15, Issue 6 https://doi.org/10.1016/S1369-7021(12)70117-7	journal	June 2012
An updated version of wannier90: A tool for obtaining maximally-localised Wannier functions Mostofi, Arash A.; Yates, Jonathan R.; Pizzi, Giovanni Computer Physics Communications, Vol. 185, Issue 8 https://doi.org/10.1016/j.cpc.2014.05.003	journal	August 2014
Optimization algorithm for the generation of ONCV pseudopotentials Schlipf, Martin; Gygi, François Computer Physics Communications, Vol. 196 https://doi.org/10.1016/j.cpc.2015.05.011	journal	November 2015
EPW: Electron–phonon coupling, transport and superconducting properties using maximally localized Wannier functions Poncé, S.; Margine, E. R.; Verdi, C. Computer Physics Communications, Vol. 209 https://doi.org/10.1016/j.cpc.2016.07.028	journal	December 2016
Dielectric Genome of van der Waals Heterostructures Andersen, Kirsten; Latini, Simone; Thygesen, Kristian S. Nano Letters, Vol. 15, Issue 7 https://doi.org/10.1021/acs.nanolett.5b01251	journal	June 2015
Breakdown of Optical Phonons’ Splitting in Two-Dimensional Materials Sohier, Thibault; Gibertini, Marco; Calandra, Matteo Nano Letters, Vol. 17, Issue 6 https://doi.org/10.1021/acs.nanolett.7b01090	journal	May 2017
Dimensional Crossover in the Carrier Mobility of Two-Dimensional Semiconductors: The Case of InSe Li, Wenbin; Poncé, Samuel; Giustino, Feliciano Nano Letters, Vol. 19, Issue 3 https://doi.org/10.1021/acs.nanolett.8b04799	journal	February 2019
What Limits the Intrinsic Mobility of Electrons and Holes in Two Dimensional Metal Dichalcogenides? Cheng, Long; Liu, Yuanyue Journal of the American Chemical Society, Vol. 140, Issue 51 https://doi.org/10.1021/jacs.8b07871	journal	September 2018
Unconventional superconductivity in magic-angle graphene superlattices Cao, Yuan; Fatemi, Valla; Fang, Shiang Nature, Vol. 556, Issue 7699 https://doi.org/10.1038/nature26160	journal	March 2018
Origin of the crossover from polarons to Fermi liquids in transition metal oxides Verdi, Carla; Caruso, Fabio; Giustino, Feliciano Nature Communications, Vol. 8, Issue 1 https://doi.org/10.1038/ncomms15769	journal	June 2017
Extremely high electron mobility in a phonon-glass semimetal Ishiwata, S.; Shiomi, Y.; Lee, J. S. Nature Materials, Vol. 12, Issue 6 https://doi.org/10.1038/nmat3621	journal	April 2013
Intralayer and interlayer electron–phonon interactions in twisted graphene heterostructures Eliel, G. S. N.; Moutinho, M. V. O.; Gadelha, A. C. Nature Communications, Vol. 9, Issue 1 https://doi.org/10.1038/s41467-018-03479-3	journal	March 2018
Double-slit photoelectron interference in strong-field ionization of the neon dimer Kunitski, Maksim; Eicke, Nicolas; Huber, Pia Nature Communications, Vol. 10, Issue 1 https://doi.org/10.1038/s41467-018-07882-8	journal	January 2019
Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds Mounet, Nicolas; Gibertini, Marco; Schwaller, Philippe Nature Nanotechnology, Vol. 13, Issue 3 https://doi.org/10.1038/s41565-017-0035-5	journal	February 2018
Visualization of moiré superlattices McGilly, Leo J.; Kerelsky, Alexander; Finney, Nathan R. Nature Nanotechnology, Vol. 15, Issue 7 https://doi.org/10.1038/s41565-020-0708-3	journal	June 2020
Dielectric properties of hexagonal boron nitride and transition metal dichalcogenides: from monolayer to bulk Laturia, Akash; Van de Put, Maarten L.; Vandenberghe, William G. npj 2D Materials and Applications, Vol. 2, Issue 1 https://doi.org/10.1038/s41699-018-0050-x	journal	March 2018
Phonon-assisted processes in the ultraviolet-transient optical response of graphene Novko, Dino; Kralj, Marko npj 2D Materials and Applications, Vol. 3, Issue 1 https://doi.org/10.1038/s41699-019-0131-5	journal	December 2019
Application of Raman spectroscopy to probe fundamental properties of two-dimensional materials Cong, Xin; Liu, Xue-Lu; Lin, Miao-Ling npj 2D Materials and Applications, Vol. 4, Issue 1 https://doi.org/10.1038/s41699-020-0140-4	journal	May 2020
Electron-polaron dichotomy of charge carriers in perovskite oxides Husanu, M. -A.; Vistoli, L.; Verdi, C. Communications Physics, Vol. 3, Issue 1 https://doi.org/10.1038/s42005-020-0330-6	journal	April 2020
Electron-phonon interaction and pairing mechanism in superconducting Ca-intercalated bilayer graphene Margine, E. R.; Lambert, Henry; Giustino, Feliciano Scientific Reports, Vol. 6, Issue 1 https://doi.org/10.1038/srep21414	journal	February 2016
Interlayer interaction and related properties of bilayer hexagonal boron nitride: ab initio study Lebedev, Alexander V.; Lebedeva, Irina V.; Knizhnik, Andrey A. RSC Advances, Vol. 6, Issue 8 https://doi.org/10.1039/C5RA20882C	journal	January 2016
A first-principles study of the effects of electron–phonon coupling on the thermoelectric properties: a case study of the SiGe compound Fan, D. D.; Liu, H. J.; Cheng, L. Journal of Materials Chemistry A, Vol. 6, Issue 25 https://doi.org/10.1039/C8TA01806E	journal	January 2018
Examining two-dimensional Fröhlich model and enhancing the electron mobility of monolayer InSe by dielectric engineering Ma, Jinlong; Xu, Dongwei; Hu, Run Journal of Applied Physics, Vol. 128, Issue 3 https://doi.org/10.1063/5.0015102	journal	July 2020
Moire bands in twisted double-layer graphene Bistritzer, R.; MacDonald, A. H. Proceedings of the National Academy of Sciences, Vol. 108, Issue 30 https://doi.org/10.1073/pnas.1108174108	journal	July 2011
XX. Properties of slow electrons in polar materials Fröhlich, H.; Pelzer, H.; Zienau, S. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 41, Issue 314 https://doi.org/10.1080/14786445008521794	journal	March 1950
Fröhlich polaron and bipolaron: recent developments Devreese, Jozef T.; Alexandrov, Alexandre S. Reports on Progress in Physics, Vol. 72, Issue 6 https://doi.org/10.1088/0034-4885/72/6/066501	journal	May 2009
QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials Giannozzi, Paolo; Baroni, Stefano; Bonini, Nicola Journal of Physics: Condensed Matter, Vol. 21, Issue 39, Article No. 395502 https://doi.org/10.1088/0953-8984/21/39/395502	journal	September 2009
Advanced capabilities for materials modelling with Quantum ESPRESSO Giannozzi, P.; Andreussi, O.; Brumme, T. Journal of Physics: Condensed Matter, Vol. 29, Issue 46 https://doi.org/10.1088/1361-648X/aa8f79	journal	October 2017
First-principles calculations of charge carrier mobility and conductivity in bulk semiconductors and two-dimensional materials Poncé, Samuel; Li, Wenbin; Reichardt, Sven Reports on Progress in Physics, Vol. 83, Issue 3 https://doi.org/10.1088/1361-6633/ab6a43	journal	February 2020
k · p theory for two-dimensional transition metal dichalcogenide semiconductors Kormányos, Andor; Burkard, Guido; Gmitra, Martin 2D Materials, Vol. 2, Issue 2 https://doi.org/10.1088/2053-1583/2/2/022001	journal	April 2015
Electron–phonon interaction and transport properties of metallic bulk and monolayer transition metal dichalcogenide TaS ₂ Hinsche, Nicki Frank; Thygesen, Kristian Sommer 2D Materials, Vol. 5, Issue 1 https://doi.org/10.1088/2053-1583/aa8e6c	journal	October 2017
Hole mobility of strained GaN from first principles Poncé, Samuel; Jena, Debdeep; Giustino, Feliciano Physical Review B, Vol. 100, Issue 8 https://doi.org/10.1103/PhysRevB.100.085204	journal	August 2019
First-principles study of electron-phonon interactions and transport in anatase TiO 2 Kang, Youngho; Peelaers, Hartwin; Van de Walle, Chris G. Physical Review B, Vol. 100, Issue 12 https://doi.org/10.1103/PhysRevB.100.121113	journal	September 2019
Phonon-limited electron mobility in Si, GaAs, and GaP with exact treatment of dynamical quadrupoles Brunin, Guillaume; Miranda, Henrique Pereira Coutada; Giantomassi, Matteo Physical Review B, Vol. 102, Issue 9 https://doi.org/10.1103/PhysRevB.102.094308	journal	September 2020
Long-range quadrupole electron-phonon interaction from first principles Park, Jinsoo; Zhou, Jin-Jian; Jhalani, Vatsal A. Physical Review B, Vol. 102, Issue 12 https://doi.org/10.1103/PhysRevB.102.125203	journal	September 2020
Ab initio dipolar electron-phonon interactions in two-dimensional materials Deng, Tianqi; Wu, Gang; Shi, Wen Physical Review B, Vol. 103, Issue 7 https://doi.org/10.1103/PhysRevB.103.075410	journal	February 2021
Microscopic theory of electron-phonon interaction in insulators or semiconductors Vogl, P. Physical Review B, Vol. 13, Issue 2 https://doi.org/10.1103/PhysRevB.13.694	journal	January 1976
Self-interaction correction to density-functional approximations for many-electron systems Perdew, J. P.; Zunger, Alex Physical Review B, Vol. 23, Issue 10, p. 5048-5079 https://doi.org/10.1103/PhysRevB.23.5048	journal	May 1981
Theory of lattice-dynamical properties of solids: Application to Si and Ge Yin, M. T.; Cohen, Marvin L. Physical Review B, Vol. 26, Issue 6 https://doi.org/10.1103/PhysRevB.26.3259	journal	September 1982
Ground-state energy of a polaron in n dimensions Peeters, F. M.; Xiaoguang, Wu; Devreese, J. T. Physical Review B, Vol. 33, Issue 6 https://doi.org/10.1103/PhysRevB.33.3926	journal	March 1986
Phonon-induced shift in shallow donor levels of semiconductor quantum structures Mason, B. A.; Das Sarma, S. Physical Review B, Vol. 33, Issue 12 https://doi.org/10.1103/PhysRevB.33.8379	journal	June 1986
Electron – optical-phonon interaction in single and double heterostructures Mori, N.; Ando, T. Physical Review B, Vol. 40, Issue 9 https://doi.org/10.1103/PhysRevB.40.6175	journal	September 1989
Quasiparticle properties of a coupled two-dimensional electron-phonon system Jalabert, R.; Das Sarma, S. Physical Review B, Vol. 40, Issue 14 https://doi.org/10.1103/PhysRevB.40.9723	journal	November 1989
Many-body theory of energy relaxation in an excited-electron gas via optical-phonon emission Das Sarma, S.; Jain, J. K.; Jalabert, R. Physical Review B, Vol. 41, Issue 6 https://doi.org/10.1103/PhysRevB.41.3561	journal	February 1990
Microscopic calculation of the electron-phonon interaction in quantum wells Rücker, H.; Molinari, E.; Lugli, P. Physical Review B, Vol. 45, Issue 12 https://doi.org/10.1103/PhysRevB.45.6747	journal	March 1992
Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory Gonze, Xavier; Lee, Changyol Physical Review B, Vol. 55, Issue 16 https://doi.org/10.1103/PhysRevB.55.10355	journal	April 1997
Theory of Surface Polarons Sak, Josef Physical Review B, Vol. 6, Issue 10 https://doi.org/10.1103/PhysRevB.6.3981	journal	November 1972
Theory of atomic-scale dielectric permittivity at insulator interfaces Giustino, Feliciano; Pasquarello, Alfredo Physical Review B, Vol. 71, Issue 14 https://doi.org/10.1103/PhysRevB.71.144104	journal	April 2005
Electron-phonon interaction using Wannier functions Giustino, Feliciano; Cohen, Marvin L.; Louie, Steven G. Physical Review B, Vol. 76, Issue 16 https://doi.org/10.1103/PhysRevB.76.165108	journal	October 2007
Screening in two dimensions: G W calculations for surfaces and thin films using the repeated-slab approach Freysoldt, Christoph; Eggert, Philipp; Rinke, Patrick Physical Review B, Vol. 77, Issue 23 https://doi.org/10.1103/PhysRevB.77.235428	journal	June 2008
Phonon-limited mobility in n -type single-layer MoS 2 from first principles Kaasbjerg, Kristen; Thygesen, Kristian S.; Jacobsen, Karsten W. Physical Review B, Vol. 85, Issue 11 https://doi.org/10.1103/PhysRevB.85.115317	journal	March 2012
Optimized norm-conserving Vanderbilt pseudopotentials Hamann, D. R. Physical Review B, Vol. 88, Issue 8 https://doi.org/10.1103/PhysRevB.88.085117	journal	August 2013
Wannier interpolation of the electron-phonon matrix elements in polar semiconductors: Polar-optical coupling in GaAs Sjakste, J.; Vast, N.; Calandra, M. Physical Review B, Vol. 92, Issue 5 https://doi.org/10.1103/PhysRevB.92.054307	journal	August 2015
Two-dimensional Fröhlich interaction in transition-metal dichalcogenide monolayers: Theoretical modeling and first-principles calculations Sohier, Thibault; Calandra, Matteo; Mauri, Francesco Physical Review B, Vol. 94, Issue 8 https://doi.org/10.1103/PhysRevB.94.085415	journal	August 2016
Electron-phonon scattering effects on electronic and optical properties of orthorhombic GeS Villegas, Cesar E. P.; Rocha, A. R.; Marini, Andrea Physical Review B, Vol. 94, Issue 13 https://doi.org/10.1103/PhysRevB.94.134306	journal	October 2016
Density functional perturbation theory for gated two-dimensional heterostructures: Theoretical developments and application to flexural phonons in graphene Sohier, Thibault; Calandra, Matteo; Mauri, Francesco Physical Review B, Vol. 96, Issue 7 https://doi.org/10.1103/PhysRevB.96.075448	journal	August 2017
Diagrammatic Monte Carlo study of Fröhlich polaron dispersion in two and three dimensions Hahn, Thomas; Klimin, Sergei; Tempere, Jacques Physical Review B, Vol. 97, Issue 13 https://doi.org/10.1103/PhysRevB.97.134305	journal	April 2018
Phonon-Assisted Optical Absorption in Silicon from First Principles Noffsinger, Jesse; Kioupakis, Emmanouil; Van de Walle, Chris G. Physical Review Letters, Vol. 108, Issue 16 https://doi.org/10.1103/PhysRevLett.108.167402	journal	April 2012
Fröhlich Electron-Phonon Vertex from First Principles Verdi, Carla; Giustino, Feliciano Physical Review Letters, Vol. 115, Issue 17 https://doi.org/10.1103/PhysRevLett.115.176401	journal	October 2015
Stochastic Approach to Phonon-Assisted Optical Absorption Zacharias, Marios; Patrick, Christopher E.; Giustino, Feliciano Physical Review Letters, Vol. 115, Issue 17 https://doi.org/10.1103/PhysRevLett.115.177401	journal	October 2015
Polarons from First Principles, without Supercells Sio, Weng Hong; Verdi, Carla; Poncé, Samuel Physical Review Letters, Vol. 122, Issue 24 https://doi.org/10.1103/PhysRevLett.122.246403	journal	June 2019
Electron-Phonon beyond Fröhlich: Dynamical Quadrupoles in Polar and Covalent Solids Brunin, Guillaume; Miranda, Henrique Pereira Coutada; Giantomassi, Matteo Physical Review Letters, Vol. 125, Issue 13 https://doi.org/10.1103/PhysRevLett.125.136601	journal	September 2020
Piezoelectric Electron-Phonon Interaction from Ab Initio Dynamical Quadrupoles: Impact on Charge Transport in Wurtzite GaN Jhalani, Vatsal A.; Zhou, Jin-Jian; Park, Jinsoo Physical Review Letters, Vol. 125, Issue 13 https://doi.org/10.1103/PhysRevLett.125.136602	journal	September 2020
Ground State of the Electron Gas by a Stochastic Method Ceperley, D. M.; Alder, B. J. Physical Review Letters, Vol. 45, Issue 7, p. 566-569 https://doi.org/10.1103/PhysRevLett.45.566	journal	August 1980
Density-Functional Theory for Superconductors Oliveira, L. N.; Gross, E. K. U.; Kohn, W. Physical Review Letters, Vol. 60, Issue 23 https://doi.org/10.1103/PhysRevLett.60.2430	journal	June 1988
Generalized Gradient Approximation Made Simple Perdew, John P.; Burke, Kieron; Ernzerhof, Matthias Physical Review Letters, Vol. 77, Issue 18, p. 3865-3868 https://doi.org/10.1103/PhysRevLett.77.3865	journal	October 1996
Free Energy of the Fröhlich Polaron in Two and Three Dimensions Titantah, John T.; Pierleoni, Carlo; Ciuchi, Sergio Physical Review Letters, Vol. 87, Issue 20 https://doi.org/10.1103/PhysRevLett.87.206406	journal	October 2001
Infrared Spectra at Surfaces and Interfaces from First Principles: Evolution of the Spectra across the Si ( 100 ) − SiO 2 Interface Giustino, Feliciano; Pasquarello, Alfredo Physical Review Letters, Vol. 95, Issue 18 https://doi.org/10.1103/PhysRevLett.95.187402	journal	October 2005
Mobility of two-dimensional materials from first principles in an accurate and automated framework Sohier, Thibault; Campi, Davide; Marzari, Nicola Physical Review Materials, Vol. 2, Issue 11 https://doi.org/10.1103/PhysRevMaterials.2.114010	journal	November 2018
Structural, electronic, elastic, power, and transport properties of β − Ga 2 O 3 from first principles Poncé, Samuel; Giustino, Feliciano Physical Review Research, Vol. 2, Issue 3 https://doi.org/10.1103/PhysRevResearch.2.033102	journal	July 2020
Exact Long-Range Dielectric Screening and Interatomic Force Constants in Quasi-Two-Dimensional Crystals Royo, Miquel; Stengel, Massimiliano Physical Review X, Vol. 11, Issue 4 https://doi.org/10.1103/PhysRevX.11.041027	journal	November 2021
Phonons and related crystal properties from density-functional perturbation theory Baroni, Stefano; de Gironcoli, Stefano; Dal Corso, Andrea Reviews of Modern Physics, Vol. 73, Issue 2 https://doi.org/10.1103/RevModPhys.73.515	journal	July 2001
Nobel Lecture: Random walk to graphene Geim, Andre K. Reviews of Modern Physics, Vol. 83, Issue 3 https://doi.org/10.1103/RevModPhys.83.851	journal	August 2011
Maximally localized Wannier functions: Theory and applications Marzari, Nicola; Mostofi, Arash A.; Yates, Jonathan R. Reviews of Modern Physics, Vol. 84, Issue 4 https://doi.org/10.1103/RevModPhys.84.1419	journal	October 2012
Electron-phonon interactions from first principles Giustino, Feliciano Reviews of Modern Physics, Vol. 89, Issue 1 https://doi.org/10.1103/RevModPhys.89.015003	journal	February 2017
Erratum: Electron-phonon interactions from first principles [Rev. Mod. Phys. 89 , 15003 (2017)] Giustino, Feliciano Reviews of Modern Physics, Vol. 91, Issue 1 https://doi.org/10.1103/RevModPhys.91.019901	journal	January 2019
Excitons in hexagonal boron nitride single-layer: a new platform for polaritonics in the ultraviolet Ferreira, F.; Chaves, A. J.; Peres, N. M. R. Journal of the Optical Society of America B, Vol. 36, Issue 3 https://doi.org/10.1364/JOSAB.36.000674	journal	January 2019

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