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Journal articles on the topic 'Wannier function'

Author: Grafiati

Published: 4 June 2021

Last updated: 23 February 2023

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1

Panayotaros, Panayotis. "Discrete Nonlinear Schrödinger Systems for Periodic Media with Nonlocal Nonlinearity: The Case of Nematic Liquid Crystals." Applied Sciences 11, no. 10 (May 13, 2021): 4420. http://dx.doi.org/10.3390/app11104420.

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We study properties of an infinite system of discrete nonlinear Schrödinger equations that is equivalent to a coupled Schrödinger-elliptic differential equation with periodic coefficients. The differential equation was derived as a model for laser beam propagation in optical waveguide arrays in a nematic liquid crystal substrate and can be relevant to related systems with nonlocal nonlinearities. The infinite system is obtained by expanding the relevant physical quantities in a Wannier function basis associated to a periodic Schrödinger operator appearing in the problem. We show that the model can describe stable beams, and we estimate the optical power at different length scales. The main result of the paper is the Hamiltonian structure of the infinite system, assuming that the Wannier functions are real. We also give an explicit construction of real Wannier functions, and examine translation invariance properties of the linear part of the system in the Wannier basis.

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2

Karabulut, Hasan. "A Wannier function made from distributed Gaussians." Journal of Mathematical Physics 46, no. 7 (July 2005): 073504. http://dx.doi.org/10.1063/1.1946529.

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3

Albert, J. P., C. Jouanin, D. Cassagne, and D. Bertho. "Generalized Wannier function method for photonic crystals." Physical Review B 61, no. 7 (February 15, 2000): 4381–84. http://dx.doi.org/10.1103/physrevb.61.4381.

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4

Fitzhenry, P., M. M. M. Bilek, N. A. Marks, N. C. Cooper, and D. R. McKenzie. "Wannier function analysis of silicon carbon alloys." Journal of Physics: Condensed Matter 15, no. 2 (December 18, 2002): 165–73. http://dx.doi.org/10.1088/0953-8984/15/2/316.

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5

McCulloch, D. G., A. R. Merchant, N. A. Marks, N. C. Cooper, P. Fitzhenry, M. M. M. Bilek, and D. R. McKenzie. "Wannier function analysis of tetrahedral amorphous networks." Diamond and Related Materials 12, no. 10-11 (October 2003): 2026–31. http://dx.doi.org/10.1016/s0925-9635(03)00196-1.

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6

Wilkinson, M. "Generalized Wannier function and renormalization of Harper's equation." Journal of Physics A: Mathematical and General 27, no. 24 (December 21, 1994): 8123–48. http://dx.doi.org/10.1088/0305-4470/27/24/021.

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7

Mizel, A., and M. L. Cohen. "Wannier function analysis of InP nanocrystals under pressure." Solid State Communications 113, no. 4 (December 1999): 189–93. http://dx.doi.org/10.1016/s0038-1098(99)00466-4.

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8

Busch, Kurt, Sergei F. Mingaleev, Antonio Garcia-Martin, Matthias Schillinger, and Daniel Hermann. "The Wannier function approach to photonic crystal circuits." Journal of Physics: Condensed Matter 15, no. 30 (July 18, 2003): R1233—R1256. http://dx.doi.org/10.1088/0953-8984/15/30/201.

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9

NAKAMURA, Kazuma. "First-principles Calculations for Polarization and Wannier Function." Hyomen Kagaku 29, no. 7 (2008): 432–36. http://dx.doi.org/10.1380/jsssj.29.432.

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10

Bu, Xiangtian, and Shudong Wang. "Electron–phonon scattering and mean free paths in D-carbon." Physical Chemistry Chemical Physics 22, no. 7 (2020): 4010–14. http://dx.doi.org/10.1039/c9cp06504k.

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11

Rui, Xue, Liang Zhao-Xin, and Li Wei-Dong. "The nonlinear Wannier function in square Krönig-Penney model." Chinese Physics B 18, no. 10 (September 29, 2009): 4130–35. http://dx.doi.org/10.1088/1674-1056/18/10/011.

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12

Šášik, R., and D. Stroud. "Wannier-function approach to phase transitions in superconducting films." Physical Review B 50, no. 5 (August 1, 1994): 3294–301. http://dx.doi.org/10.1103/physrevb.50.3294.

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13

Mizel, Ari, and Marvin L. Cohen. "Electronic transitions in InAs nanocrystals using Wannier function method." Solid State Communications 104, no. 7 (November 1997): 401–5. http://dx.doi.org/10.1016/s0038-1098(97)00339-6.

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14

Mizel, Ari, and Marvin L. Cohen. "Electronic energy levels in semiconductor nanocrystals: A Wannier function approach." Physical Review B 56, no. 11 (September 15, 1997): 6737–41. http://dx.doi.org/10.1103/physrevb.56.6737.

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15

Mizel, A., and M. L. Cohen. "Erratum to “Wannier function analysis of InP nanocrystals under pressure”." Solid State Communications 117, no. 7 (January 2001): 449. http://dx.doi.org/10.1016/s0038-1098(00)00503-2.

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16

Hermann, Daniel, Matthias Schillinger, Sergei F. Mingaleev, and Kurt Busch. "Wannier-function based scattering-matrix formalism for photonic crystal circuitry." Journal of the Optical Society of America B 25, no. 2 (January 29, 2008): 202. http://dx.doi.org/10.1364/josab.25.000202.

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17

Leung, K. M. "Defect modes in photonic band structures: a Green’s function approach using vector Wannier functions." Journal of the Optical Society of America B 10, no. 2 (February 1, 1993): 303. http://dx.doi.org/10.1364/josab.10.000303.

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18

Smirnov, V. P., and D. E. Usvyat. "Change of the Wannier-function symmetry point by choice of Bloch-function phase factors." Physical Review B 59, no. 15 (April 15, 1999): 9695–98. http://dx.doi.org/10.1103/physrevb.59.9695.

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19

Pavarini, E., A. Yamasaki, J. Nuss, and O. K. Andersen. "How chemistry controls electron localization in 3d1perovskites: a Wannier-function study." New Journal of Physics 7 (September 5, 2005): 188. http://dx.doi.org/10.1088/1367-2630/7/1/188.

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20

Shukla, Alok, Michael Dolg, and Hermann Stoll. "A Wannier-function-based ab initio Hartree–Fock study of polyethylene." Chemical Physics Letters 294, no. 1-3 (September 1998): 126–34. http://dx.doi.org/10.1016/s0009-2614(98)00850-1.

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21

Kohn, Walter. "Density functional/Wannier function theory for systems of very many atoms." Chemical Physics Letters 208, no. 3-4 (June 1993): 167–72. http://dx.doi.org/10.1016/0009-2614(93)89056-n.

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22

SCHWEIGERT, CHRISTOPH, and EFROSSINI TSOUCHNIKA. "KRAMERS–WANNIER DUALITIES FOR WZW THEORIES AND MINIMAL MODELS." Communications in Contemporary Mathematics 10, no. 05 (October 2008): 773–89. http://dx.doi.org/10.1142/s0219199708002983.

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We study Kramers–Wannier dualities for Wess–Zumino–Witten theories and (super-) minimal models in the Cardy case, i.e. the case with bulk partition function given by charge conjugation. Using the TFT approach to full rational conformal field theories, we classify those dualities that preserve all chiral symmetries. Dualities turn out to exist for small levels only.

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23

Rai, D. P., and C. E. Ekuma. "Origin of strong Coulomb interactions in borophene: First-principles Wannier function analysis." Journal of Applied Physics 131, no. 14 (April 14, 2022): 145105. http://dx.doi.org/10.1063/5.0088860.

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We report the first-principles Wannier function study of the electronic structure of two polymorphs of borophene: 8-[Formula: see text] and 6-[Formula: see text] (henceforth denoted as 6-[Formula: see text]) borophene, where 8 and 6 depict the number of nonequivalent atoms per unit cell. Both structures are found to be anisotropic metals with electronic structures dominated by weak inter- and intra-hopping physics, suggesting strongly correlated metallic ground states. Our findings could aid in explaining the recently observed strong Coulomb interaction in related materials such as graphene bilayer.

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24

MISIRPASHAEV, T. SH. "TO THE TOPOLOGY OF KRAMERS-WANNIER DUALITY." International Journal of Modern Physics A 09, no. 16 (June 30, 1994): 2755–72. http://dx.doi.org/10.1142/s0217751x94001126.

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We show that for an even-dimensional hypercubic lattice one can modify the construction of a dual lattice to have the correspondence edge-edge instead of the conventional correspondence edge-(d−1)-dimensional face. This gives a straightforward generalization of Kramers-Wannier duality for an even-dimensional Ising model. In the same way as the partition function for the 2D Ising model is related to a sum over paths on a torus, higher-dimensional models involve sums over paths on Riemannian surfaces of higher genus. The critical temperature can be located only in the d=2 case in which all topological effects disappear from the thermodynamic limit. The duality in higher dimensions, however, being weak, leads nevertheless to some interesting relations for sums over paths on Riemannian surfaces, which can be considered as a topological characteristic of a critical point.

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25

Postorino, Sara, Jianbo Sun, Saskia Fiedler, Laurent O. Lee Cheong Lem, Maurizia Palummo, and Luca Camilli. "Interlayer Bound Wannier Excitons in Germanium Sulfide." Materials 13, no. 16 (August 12, 2020): 3568. http://dx.doi.org/10.3390/ma13163568.

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We report a cathodoluminescence (CL) study of layered germanium sulfide (GeS) where we observe a sharp emission peak from flakes covered with a thin hexagonal boron nitride film. GeS is a material that has recently attracted considerable interest due to its emission in the visible region and its strong anisotropy. The measured CL peak is at ~1.69 eV for samples ranging in thickness from 97 nm to 45 nm, where quantum-confinement effects can be excluded. By performing ab initio ground- and excited-state simulations for the bulk compound, we show that the measured optical peak can be unambiguously explained by radiative recombination of the first free bright bound exciton, which is due to a mixing of direct transitions near the Γ-point of the Brillouin Zone and it is associated to a very large optical anisotropy. The analysis of the corresponding excitonic wave function shows a Wannier–Mott interlayer character, being spread not only in-plane but also out-of-plane.

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26

Zarubin, Alexander V., Felix Kassan-Ogly, and Alexey I. Proshkin. "Diffuse Scattering on Ising Chain with Competing Interactions." Materials Science Forum 845 (March 2016): 122–25. http://dx.doi.org/10.4028/www.scientific.net/msf.845.122.

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We considered the Ising 1D chain in an external magnetic field taking into account the nearest and next-nearest interactions. By the method of Kramers–Wannier transfer-matrix, the rigorous analytical expression for Fourier-transform of pair spin-spin correlation function was obtained, and the temperature evolution of the scattering was analyzed for various relations of exchange parameters.

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27

Kazansky, A. K., and V. N. Ostrovsky. "Green-function approach to electron angular correlations in the Wannier threshold law." Physical Review A 48, no. 2 (August 1, 1993): R871—R874. http://dx.doi.org/10.1103/physreva.48.r871.

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28

Busch, Kurt, Christian Blum, Alexandra M. Graham, Daniel Hermann, Martin Köhl, Patrick Mack, and Christian Wolff. "The photonic Wannier function approach to photonic crystal simulations: status and perspectives." Journal of Modern Optics 58, no. 5-6 (March 10, 2011): 365–83. http://dx.doi.org/10.1080/09500340.2010.526256.

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29

Davison, S. G., R. A. English, Z. L. Miskovic, F. O. Goodman, A. T. Amos, and B. L. Burrows. "Recursive Green-function study of Wannier - Stark effect in tight-binding systems." Journal of Physics: Condensed Matter 9, no. 30 (July 28, 1997): 6371–82. http://dx.doi.org/10.1088/0953-8984/9/30/006.

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30

Alfonso, Alexys Bruno. "Wannier-function approach to electron states in superlattices under an electric field." Microelectronics Journal 35, no. 1 (January 2004): 63–64. http://dx.doi.org/10.1016/s0026-2692(03)00235-0.

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31

Liang, G. C., Y. A. Lin, D. Z. Y. Ting, and Y. C. Chang. "Multiband Quantum Transmitting Boundary Method for Non-orthogonal Basis." VLSI Design 8, no. 1-4 (January 1, 1998): 507–13. http://dx.doi.org/10.1155/1998/90280.

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We generalize the Multiband Quantum Transmitting Boundary Method (MQTBM) for computing transmission coefficients in heterostructure to tight-binding-like band structure models with non-orthogonal basis and multiple neighbor interactions. We implement this method based on the newly developed planar-basis pseudopotential method which uses the generalized planar Wannier function basis. We demonstrate the method by computing transmission coefficients for a GaAs/AlAs double barrier structure.

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32

Tanaue, Helena B., and Alexys Bruno-Alfonso. "Wannier-function expansion of localized modes in 1D photonic crystals without inversion symmetry." Journal of the Optical Society of America B 37, no. 12 (November 9, 2020): 3698. http://dx.doi.org/10.1364/josab.401754.

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33

Lim, S. H. N., D. G. McCulloch, A. R. Merchant, N. A. Marks, M. M. M. Bilek, and D. R. McKenzie. "Wannier function analysis for understanding disordered structures generated using Car-Parrinello molecular dynamics." Molecular Simulation 28, no. 10-11 (October 2002): 971–79. http://dx.doi.org/10.1080/089270204000002601.

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34

Silvestrelli, Pier Luigi, and Alberto Ambrosetti. "Inclusion of Van der Waals Interactions in DFT using Wannier Functions without empirical parameters." EPJ Web of Conferences 230 (2020): 00010. http://dx.doi.org/10.1051/epjconf/202023000010.

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We describe a method for including van der Waals (vdW) interactions in Density Functional Theory (DFT) using the Maximally-Localized Wannier functions (MLWFs), which is free from empirical parameters. With respect to the previous DFT/vdW-WF2 version, in the present DFT/vdW-WF2-x approach, the empirical, short-range, damping function is replaced by an estimate of the Pauli exchange repulsion, also obtained by the MLWFs properties. Applications to systems contained in the popular S22 molecular database and to the case of adsorption of Ar on graphite, and Xe and water on graphene, indicate that the new method, besides being more physically founded, also leads to a systematic improvement in the description of systems where vdW interactions play a significant role.

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35

Souza, Ivo, Richard M. Martin, Nicola Marzari, Xinyuan Zhao, and David Vanderbilt. "Wannier-function description of the electronic polarization and infrared absorption of high-pressure hydrogen." Physical Review B 62, no. 23 (December 15, 2000): 15505–20. http://dx.doi.org/10.1103/physrevb.62.15505.

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36

Dong, Lu, and Yang Guang. "The use of Wannier function in the calculations of band structure of covalent crystals." Solid State Communications 58, no. 11 (June 1986): 785–88. http://dx.doi.org/10.1016/0038-1098(86)90769-6.

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37

Ekuma, Chinedu E., Chia-Hui Lin, Juana Moreno, Wei Ku, and Mark Jarrell. "First-principles Wannier function analysis of the electronic structure of PdTe: weaker magnetism and superconductivity." Journal of Physics: Condensed Matter 25, no. 40 (September 11, 2013): 405601. http://dx.doi.org/10.1088/0953-8984/25/40/405601.

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38

Sit, Patrick H. L., Federico Zipoli, Jia Chen, Roberto Car, Morrel H. Cohen, and Annabella Selloni. "Oxidation State Changes and Electron Flow in Enzymatic Catalysis and Electrocatalysis through Wannier-Function Analysis." Chemistry - A European Journal 17, no. 43 (September 9, 2011): 12136–43. http://dx.doi.org/10.1002/chem.201101916.

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39

THONHAUSER, T. "THEORY OF ORBITAL MAGNETIZATION IN SOLIDS." International Journal of Modern Physics B 25, no. 11 (April 30, 2011): 1429–58. http://dx.doi.org/10.1142/s0217979211058912.

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In this review article, we survey the relatively new theory of orbital magnetization in solids — often referred to as the "modern theory of orbital magnetization" — and its applications. Surprisingly, while the calculation of the orbital magnetization in finite systems such as atoms and molecules is straight forward, in extended systems or solids it has long eluded calculations owing to the fact that the position operator is ill-defined in such a context. Approaches that overcome this problem were first developed in 2005 and in the first part of this review we present the main ideas reaching from a Wannier function approach to semi-classical and finite-temperature formalisms. In the second part, we describe practical aspects of calculating the orbital magnetization, such as taking k-space derivatives, a formalism for pseudopotentials, a single k-point derivation, a Wannier interpolation scheme, and DFT specific aspects. We then show results of recent calculations on Fe, Co, and Ni. In the last part of this review, we focus on direct applications of the orbital magnetization. In particular, we will review how properties such as the nuclear magnetic resonance shielding tensor and the electron paramagnetic resonance g-tensor can be elegantly calculated in terms of a derivative of the orbital magnetization.

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40

Shukla, Alok, Michael Dolg, and Hermann Stoll. "Wannier-function-basedab initioHartree-Fock approach extended to polymers: Applications to the LiH chain andtrans-polyacetylene." Physical Review B 58, no. 8 (August 15, 1998): 4325–34. http://dx.doi.org/10.1103/physrevb.58.4325.

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41

Shimojo, Fuyuki, Kozo Hoshino, and Y. Zempo. "Bonding Properties of Liquid Tellurium under Pressure: A Maximally Localized Wannier Function Approach with Ultrasoft Pseudopotentials." Journal of the Physical Society of Japan 72, no. 10 (October 15, 2003): 2417–20. http://dx.doi.org/10.1143/jpsj.72.2417.

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42

Do, Thi-Nga, Danhong Huang, Po-Hsin Shih, Hsin Lin, and Godfrey Gumbs. "Atomistic Band-Structure Computation for Investigating Coulomb Dephasing and Impurity Scattering Rates of Electrons in Graphene." Nanomaterials 11, no. 5 (May 1, 2021): 1194. http://dx.doi.org/10.3390/nano11051194.

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In this paper, by introducing a generalized quantum-kinetic model which is coupled self-consistently with Maxwell and Boltzmann transport equations, we elucidate the significance of using input from first-principles band-structure computations for an accurate description of ultra-fast dephasing and scattering dynamics of electrons in graphene. In particular, we start with the tight-binding model (TBM) for calculating band structures of solid covalent crystals based on localized Wannier orbital functions, where the employed hopping integrals in TBM have been parameterized for various covalent bonds. After that, the general TBM formalism has been applied to graphene to obtain both band structures and wave functions of electrons beyond the regime of effective low-energy theory. As a specific example, these calculated eigenvalues and eigen vectors have been further utilized to compute the Bloch-function form factors and intrinsic Coulomb diagonal-dephasing rates for induced optical coherence of electron-hole pairs in spectral and polarization functions, as well as the energy-relaxation time from extrinsic impurity scattering of electrons for non-equilibrium occupation in band transport.

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43

Demmouche, Kamel, and José Coutinho. "Electronic exchange-correlation, many-body effect issues on first-principles calculations of bulk SiC polytypes." International Journal of Modern Physics B 32, no. 29 (November 20, 2018): 1850328. http://dx.doi.org/10.1142/s0217979218503289.

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The first-principles Projector-Augmented Wave method (PAW) is used to investigate the electronic, phonon band structure and dielectric properties of four bulk silicon carbide (SiC) polytypes. We employ PAW pseudopotential density functional theory with Perdew, Burke and Ernzerhof (PBE) and hybrid HSE06 approximations of the exchange-correlation functional. Many-body effects are incorporated using the GW approximation of the self-interaction to study SiC properties. GW method in its single-shot variant, which is based on the many-body perturbation theory (MBPT), is used to calculate the quasi-particle (QP) energies of the band structure and the dielectric properties for different polytypes. The electronic band structure determination within GW method uses the Wannier procedure where a basis set of maximally localized Wannier function (MLWF) is constructed to interpolate the QP energies of few regular mesh k-points to the high-symmetry lines in Brillouin zone. As a consequence of QP correction to the Kohn–Sham energies, bandgap is increased by upto 3 eV in case of 4H–SiC, as compared to PBE bandgap. GW results are comparable to those of hybrid functionals and are in good agreement with the experimental results. The optical properties are then studied within PBE, HSE06 and include many-body effects. In addition, the phonon band structure has been investigated within HSE06 and compared to previous PBE results. We found good agreement with the previous theoretical results and the experimental available data.

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44

Zhang, Haiwu, B. Yu Yavorsky, and R. E. Cohen. "Polar Metallocenes." Molecules 24, no. 3 (January 29, 2019): 486. http://dx.doi.org/10.3390/molecules24030486.

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Crystalline polar metallocenes are potentially useful active materials as piezoelectrics, ferroelectrics, and multiferroics. Within density functional theory (DFT), we computed structural properties, energy differences for various phases, molecular configurations, and magnetic states, computed polarizations for different polar crystal structures, and computed dipole moments for the constituent molecules with a Wannier function analysis. Of the systems studied, Mn2(C9H9N)2 is the most promising as a multiferroic material, since the ground state is both polar and ferromagnetic. We found that the predicted crystalline polarizations are 30–40% higher than the values that would be obtained from the dipole moments of the isolated constituent molecules, due to the local effects of the self-consistent internal electric field, indicating high polarizabilities.

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45

Mączka, Mariusz, and Stanisław Pawłowski. "A Polynomial Approximation to Self Consistent Solution for Schrödinger–Poisson Equations in Superlattice Structures." Energies 15, no. 3 (January 20, 2022): 760. http://dx.doi.org/10.3390/en15030760.

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The paper deals with a new approach to iterative solving the Schrödinger and Poisson equations in the first type of semiconductor superlattice. Assumptions of the transfer matrix method are incorporated into the approach, which allows to take into account the potential varying within each single layer of bias voltage superlattice. The key process of the method is to approximate the charge density and wave functions with polynomials. It allows to obtain semi-analytical solutions for the Schrödinger and Poisson equations, which in turn have significant impact on the accuracy and speed of superlattice simulations. The presented procedure is also suifihue for finding eigenstates extended over relatively large superlattice area, and it can be used as an effective pro-gram module for a superlattice finite model. The obtained quantum states are very similar to the Wannier-Stark functions, and they can serve as the base under non-equilibrium Green’s function formalism (NEGF). Exemplary results for Schrödinger and Poisson solutions for superlattices based on the GaAs/AlGaAs heterostructure are presented to prove all the above.

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46

KLEINERT, P., and V. V. BRYKSIN. "QUANTUM TRANSPORT IN SEMICONDUCTOR SUPERLATTICES BEYOND THE KADANOFF–BAYM ANSATZ." International Journal of Modern Physics B 15, no. 31 (December 20, 2001): 4123–43. http://dx.doi.org/10.1142/s0217979201008123.

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Quantum transport in semiconductor superlattices subject to quantizing parallel electric and magnetic fields is studied based on the Kadanoff–Baym–Keldysh double-time Green function approach. Exploiting the symmetry properties of the underlying Hamiltonian, coupled kinetic equations are derived and analytically solved for the density-of-states and the carrier distribution function. Scattering giving rise to collisional broadening plays an important role in our transport model, whose unperturbed eigenstates are completely discrete due to Wannier–Stark and Landau quantization. It is shown that a correct description of the stationary quantum transport in superlattices with field-induced localized eigenstates requires the determination of a time-dependent distribution function from a kinetic equation, which emerges beyond the Kadanoff–Baym Ansatz. Depending on the scattering strength, gaps are predicted to occur in the electric and magnetic field dependence of the current density. The rigorous quantum-mechanical approach reveals the hopping nature of the nonlinear transport in narrow miniband superlattices. This is compared with results obtained recently within the density-matrix approach.

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47

Mączka, Mariusz. "Effective Simulations of Electronic Transport in 2D Structures Based on Semiconductor Superlattice Infinite Model." Electronics 9, no. 11 (November 4, 2020): 1845. http://dx.doi.org/10.3390/electronics9111845.

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Effective simulations of semiconductor superlattices are presented in the paper. The simulations have been based on the Wannier function method approach where a new algorithm, inspired by Büttiker probes, has been incorporated into determining the Green function procedure. The program is of a modular structure, and its modules can either work independently, or interact with each other following a predefined algorithm. Such structuring not only accelerates simulations and makes the transport parameters possible to initially assess, but also enables accurate analysis of quantum phenomena occurring in semiconductor superlattices. In this paper, the capabilities of type I superlattice simulator, developed earlier, are presented, with particular emphasis on the new block where the Fermi levels are determined by applying Büttiker probes. The algorithms and methods used in the program are briefly described in the further chapters of our work, where we also provide graphics illustrating the results obtained for the simulated structures known from the literature.

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48

Okuno, Yukihiro, and Yukio Sakashita. "Estimation of Piezoelectric Response Using Maximally Localized Wannier Function and Its Application to the Bismuth-Based Ferroelectric Materials." Japanese Journal of Applied Physics 50, no. 10R (October 1, 2011): 101503. http://dx.doi.org/10.7567/jjap.50.101503.

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49

Okuno, Yukihiro, and Yukio Sakashita. "Estimation of Piezoelectric Response Using Maximally Localized Wannier Function and Its Application to the Bismuth-Based Ferroelectric Materials." Japanese Journal of Applied Physics 50, no. 10 (October 20, 2011): 101503. http://dx.doi.org/10.1143/jjap.50.101503.

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50

Mostofi, Arash A., Jonathan R. Yates, Young-Su Lee, Ivo Souza, David Vanderbilt, and Nicola Marzari. "wannier90: A tool for obtaining maximally-localised Wannier functions." Computer Physics Communications 178, no. 9 (May 2008): 685–99. http://dx.doi.org/10.1016/j.cpc.2007.11.016.

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