zbMATH — the first resource for mathematics

Improved recursive Green’s function formalism for quasi one-dimensional systems with realistic defects. (English) Zbl 1375.81093
Summary: We derive an improved version of the recursive Green’s function formalism (RGF), which is a standard tool in the quantum transport theory. We consider the case of disordered quasi one-dimensional materials where the disorder is applied in form of randomly distributed realistic defects, leading to partly periodic Hamiltonian matrices. The algorithm accelerates the common RGF in the recursive decimation scheme, using the iteration steps of the renormalization decimation algorithm. This leads to a smaller effective system, which is treated using the common forward iteration scheme. The computational complexity scales linearly with the number of defects, instead of linearly with the total system length for the conventional approach. We show that the scaling of the calculation time of the Green’s function depends on the defect density of a random test system. Furthermore, we discuss the calculation time and the memory requirement of the whole transport formalism applied to defective carbon nanotubes.
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81-08 Computational methods for problems pertaining to quantum theory
81V70 Many-body theory; quantum Hall effect
SIESTA; transiesta
Full Text: DOI
[1] Büttiker, M.; Imry, Y.; Landauer, R.; Pinhas, S., Generalized many-channel conductance formula with application to small rings, Phys. Rev. B, 31, 10, 6207-6215, (1985)
[2] Datta, S., Quantum transport: atom to transistor, (2005), Cambridge University Press Cambridge
[3] Capelle, K., A Bird’s-eye view of density-functional theory, Braz. J. Phys., 36, 4A, 1318-1343, (2006)
[4] Haydock, R.; Heine, V.; Kelly, M. J., Electronic structure based on the local atomic environment for tight-binding bands, J. Phys. C, Solid State Phys., 5, 20, 2845-2858, (1972)
[5] Haydock, R., The recursive solution of the Schrödinger equation, Comput. Phys. Commun., 20, 1, 11-16, (1980)
[6] Thouless, D. J.; Kirkpatrick, S., Conductivity of the disordered linear chain, J. Phys. C, Solid State Phys., 14, 3, 235-245, (1981)
[7] MacKinnon, A., The calculation of transport properties and density of states of disordered solids, Z. Phys. B, Condens. Matter, 59, 4, 385-390, (1985)
[8] Wimmer, M.; Richter, K., Optimal block-tridiagonalization of matrices for coherent charge transport, J. Comput. Phys., 228, 23, 8548-8565, (2009) · Zbl 1194.82087
[9] Mason, D. J.; Prendergast, D.; Neaton, J. B.; Heller, E. J., Algorithm for efficient elastic transport calculations for arbitrary device geometries, Phys. Rev. B, 84, 15, (2011)
[10] Cuthill, E.; McKee, J., Reducing the bandwidth of sparse symmetric matrices, (Proceedings of the 1969 24th National Conference, ACM ’69, (1969), ACM New York, NY, USA), 157-172
[11] Kazymyrenko, K.; Waintal, X., Knitting algorithm for calculating Green functions in quantum systems, Phys. Rev. B, 77, 11, (2008)
[12] Tsukamoto, S.; Hirose, K.; Blügel, S., Real-space finite-difference calculation method of generalized Bloch wave functions and complex band structures with reduced computational cost, Phys. Rev. E, 90, 1, (2014)
[13] George, A., Nested dissection of a regular finite element mesh, SIAM J. Numer. Anal., 10, 2, 345-363, (1973) · Zbl 0259.65087
[14] Li, S.; Wu, W.; Darve, E., A fast algorithm for sparse matrix computations related to inversion, J. Comput. Phys., 242, 915-945, (2013) · Zbl 1297.65030
[15] Petersen, D. E.; Li, S.; Stokbro, K.; Sørensen, H. H.B.; Hansen, P. C.; Skelboe, S.; Darvez, E., A hybrid method for the parallel computation of Green’s functions, J. Comput. Phys., 228, 14, 5020-5039, (2009) · Zbl 1280.82010
[16] Kuzmin, A.; Luisier, M.; Schenk, O., Fast methods for computing selected elements of the Green’s function in massively parallel nanoelectronic device simulations, (Wolf, F.; Mohr, B.; an Mey, D., Euro-Par 2013 Parallel Processing, Lecture Notes in Computer Science, vol. 8097, (2013), Springer-Verlag Berlin, Heidelberg), 533-544
[17] Fujimoto, Y.; Hirose, K., First-principles treatments of electron transport properties for nanoscale junctions, Phys. Rev. B, 67, 19, (2003)
[18] Takayama, R.; Hoshi, T.; Sogabe, T.; Zhang, S.-L.; Fujiwara, T., Linear algebraic calculation of the Green’s function for large-scale electronic structure theory, Phys. Rev. B, 73, 16, (2006)
[19] Ono, T.; Egami, Y.; Hirose, K., First-principles transport calculation method based on real-space finite-difference nonequilibrium Green’s function scheme, Phys. Rev. B, 86, 19, (2012)
[20] Iwase, S.; Hoshi, T.; Ono, T., Numerical solver for first-principles transport calculation based on real-space finite-difference method, Phys. Rev. E, 91, 6, (2015)
[21] Hestenes, M. R.; Stiefel, E., Methods of conjugate gradients for solving linear systems, J. Res. Natl. Bur. Stand., 49, 6, 409-436, (1952) · Zbl 0048.09901
[22] Anderson, P. W., Absence of diffusion in certain random lattices, Phys. Rev., 109, 5, 1492-1505, (1958)
[23] MacKinnon, A.; Kramer, B., One-parameter scaling of localization length and conductance in disordered systems, Phys. Rev. Lett., 47, 21, 1546-1549, (1981)
[24] Abrahams, E.; Anderson, P. W.; Licciardello, D. C.; Ramakrishnan, T. V., Scaling theory of localization: absence of quantum diffusion in two dimensions, Phys. Rev. Lett., 42, 10, 673-676, (1979)
[25] López Sancho, M. P.; López Sancho, J. M.; Rubio, J., Quick iterative scheme for the calculation of transfer matrices: application to mo (100), J. Phys. F, Met. Phys., 14, 5, 1205-1215, (1984)
[26] López Sancho, M. P.; López Sancho, J. M.; Rubio, J., Highly convergent schemes for the calculation of bulk and surface Green functions, J. Phys. F, Met. Phys., 15, 4, 851-858, (1985)
[27] Zienert, A.; Schuster, J.; Gessner, T., Metallic carbon nanotubes with metal contacts: electronic structure and transport, Nanotechnology, 25, 42, (2014)
[28] Fuchs, F.; Zienert, A.; Wagner, C.; Schuster, J.; Schulz, S. E., Interaction between carbon nanotubes and metals: electronic properties, stability, and sensing, Materials for Advanced Metallization 2014, Microelectron. Eng., 137, 124-129, (2015)
[29] Fediai, A.; Ryndyk, D. A.; Cuniberti, G., Electron transport in extended carbon-nanotube/metal contacts: ab initio based Green function method, Phys. Rev. B, 91, 16, (2015)
[30] Biel, B.; García-Vidal, F. J.; Rubio, A.; Flores, F., Ab initio study of transport properties in defected carbon nanotubes: an O(N) approach, J. Phys. Condens. Matter, 20, 29, (2008)
[31] Flores, F.; Biel, B.; Rubio, A.; García-Vidal, F. J.; Gómez-Navarro, C.; de Pablo, P. J.; Gómez-Herrero, J., Anderson localization regime in carbon nanotubes: size dependent properties, J. Phys. Condens. Matter, 20, 8, (2008)
[32] Biel, B.; García-Vidal, F. J.; Rubio, A.; Flores, F., Anderson localization in carbon nanotubes: defect density and temperature effects, Phys. Rev. Lett., 95, 26, (2005)
[33] Greene-Diniz, G.; Jones, S. L.T.; Fagas, G.; Haverty, M.; Lacambra, C. M.; Shankar, S.; Greer, J. C., Divacancies in carbon nanotubes and their influence on electron scattering, J. Phys. Condens. Matter, 26, 4, (2014)
[34] Lee, A. T.; Kang, Y.-J.; Chang, K. J., Transport properties of carbon nanotubes: effects of vacancy clusters and disorder, J. Phys. Chem. C, 116, 1, 1179-1184, (2012)
[35] Khoeini, F.; Shokri, A. A.; Farman, H., Electronic transport through superlattice-like disordered carbon nanotubes, Solid State Commun., 149, 21-22, 874-879, (2009)
[36] Blase, X.; Adessi, C.; Biel, B.; López-Bezanilla, A.; Fernández-Serra, M.-V.; Margine, E. R.; Triozon, F.; Roche, S., Conductance of functionalized nanotubes, graphene and nanowires: from ab initio to mesoscopic physics, Phys. Status Solidi B, 247, 11-12, 2962-2967, (2010)
[37] López-Bezanilla, A.; Blase, X.; Roche, S., Quantum transport properties of chemically functionalized long semiconducting carbon nanotubes, Nano Res., 3, 4, 288-295, (2010)
[38] López-Bezanilla, A.; Triozon, F.; Latil, S.; Blase, X.; Roche, S., Effect of the chemical functionalization on charge transport in carbon nanotubes at the mesoscopic scale, Nano Lett., 9, 3, 940-944, (2009)
[39] Anantram, M. P.; Govindan, T. R., Conductance of carbon nanotubes with disorder: a numerical study, Phys. Rev. B, 58, 6, 4882-4887, (1998)
[40] Jiang, J.; Dong, J.; Yang, H. T.; Xing, D. Y., Universal expression for localization length in metallic carbon nanotubes, Phys. Rev. B, 64, 4, (2001)
[41] Teichert, F.; Zienert, A.; Schuster, J.; Schreiber, M., Strong localization in defective carbon nanotubes: a recursive Green’s function study, New J. Phys., 16, 12, (2014)
[42] Krasheninnikov, A. V.; Nordlund, K.; Sirviö, M.; Salonen, E.; Keinonen, J., Formation of ion-irradiation-induced atomic-scale defects on walls of carbon nanotubes, Phys. Rev. B, 63, 24, (2001)
[43] Rodriguez-Manzo, J. A.; Banhart, F., Creation of individual vacancies in carbon nanotubes by using an electron beam of 1 å diameter, Nano Lett., 9, 6, 2285-2289, (2009), pMID: 19413339
[44] Kim, S.; Kim, H.-J.; Lee, H. R.; Song, J.-H.; Yi, S. N.; Ha, D. H., Oxygen plasma effects on the electrical conductance of single-walled carbon nanotube bundles, J. Phys. D, Appl. Phys., 43, 10, (2010)
[45] Atomistix toolkit version 12.8.2, Quantum Wise A/S
[46] Brandbyge, M.; Mozos, J.-L.; Ordejón, P.; Taylor, J.; Stokbro, K., Density-functional method for nonequilibrium electron transport, Phys. Rev. B, 65, 16, (2002)
[47] Perdew, J. P.; Zunger, A., Self-interaction correction to density-functional approximations for many-electron systems, Phys. Rev. B, 23, 10, 5048-5079, (1981)
[48] Troullier, N.; Martins, J. L., Efficient pseudopotentials for plane-wave calculations, Phys. Rev. B, 43, 3, 1993-2006, (1991)
[49] Soler, J. M.; Artacho, E.; Gale, J. D.; García, A.; Junquera, J.; Ordejón, P.; Sánchez-Portal, D., The SIESTA method for ab initio order-N materials simulation, J. Phys. Condens. Matter, 14, 11, 2745-2779, (2002)
[50] Porezag, D.; Frauenheim, T.; Köhler, T.; Seifert, G.; Kaschner, R., Construction of tight-binding-like potentials on the basis of density-functional theory: application to carbon, Phys. Rev. B, 51, 19, 12947-12957, (1995)
[51] Seifert, G.; Porezag, D.; Frauenheim, T., Calculations of molecules, clusters, and solids with a simplified LCAO-DFT-LDA scheme, Int. J. Quant. Chem., 58, 2, 185-192, (1996)
[52] Gaus, M.; Goez, A.; Elstner, M., Parametrization and benchmark of DFTB3 for organic molecules, J. Chem. Theory Comput., 9, 1, 338-354, (2013)
[53] Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, T.; Suhai, S.; Seifert, G., Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties, Phys. Rev. B, 58, 11, 7260-7268, (1998)
[54] Thorgilsson, G.; Viktorsson, G.; Erlingsson, S. I., Recursive Green’s function method for multi-terminal nanostructures, J. Comput. Phys., 261, 256-266, (2014) · Zbl 1349.82124
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.