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The recursive Green’s function method for graphene

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Abstract

We describe how to apply the recursive Green’s function method to the computation of electronic transport properties of graphene sheets and nanoribbons in the linear response regime. This method allows for an amenable inclusion of several disorder mechanisms at the microscopic level, as well as inhomogeneous gating, finite temperature, and, to some extend, dephasing. We present algorithms for computing the conductance, density of states, and current densities for armchair and zigzag atomic edge alignments. Several numerical results are presented to illustrate the usefulness of the method.

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Notes

  1. The number of sites per slice does not need to be equal for all slices.

  2. There are several good textbooks, such as Refs. [32, 49, 50], that discuss nonequilibrium Green’s functions. In particular, Ref. [32] concisely covers all required background material. We refer the reader to these books for the derivation of expressions involving G < and related functions and further insight into the subject.

  3. One set of random numbers for each α-component of the gauge field, such that \(\langle c_{k}^{\alpha}\rangle= 0\) and \(\langle c_{k}^{\alpha}c_{k^{\prime}}^{\alpha^{\prime}} \rangle= \delta_{kk^{\prime}} \delta_{\alpha\alpha^{\prime}}\).

  4. The discussion and results that follow complement the material presented in Ref. [2].

References

  1. Castro Neto, A.H.F., Guinea, F., Peres, N.M.R., Novoselov, K.S., Geim, A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81, 109 (2009)

    Article  Google Scholar 

  2. Mucciolo, E.R., Lewenkopf, C.H.: Disorder and electronic transport in graphene. J. Phys. Condens. Matter 22, 273201 (2010)

    Article  Google Scholar 

  3. Shon, N.H., Ando, T.: Quantum transport in two-dimensional graphite system. J. Phys. Soc. Jpn. 67, 2421 (1998)

    Article  Google Scholar 

  4. Ostrovsky, P.M., Gornyi, I.V., Mirlin, A.D.: Electron transport in disordered graphene. Phys. Rev. B 74, 235443 (2006)

    Article  Google Scholar 

  5. Nomura, K., MacDonald, A.H.: Quantum transport of massless Dirac fermions. Phys. Rev. Lett. 98, 076602 (2007)

    Article  Google Scholar 

  6. Nomura, K., Koshino, M., Ryu, S.: Topological delocalization of two-dimensional massless Dirac fermions. Phys. Rev. Lett. 99, 146806 (2007)

    Article  Google Scholar 

  7. Tworzydło, J., Groth, C.W., Beenakker, C.W.J.: Finite difference method for transport properties of massless Dirac fermions. Phys. Rev. B 78, 235438 (2008)

    Article  Google Scholar 

  8. Hernández, A.R., Lewenkopf, C.H.: Finite-difference method for transport of two-dimensional massless Dirac fermions in a ribbon geometry. Phys. Rev. B 86, 155439 (2012)

    Article  Google Scholar 

  9. Thouless, D.J., Kirkpatrick, S.: Conductivity of the disordered linear chain. J. Phys. C 14, 235 (1981)

    Article  Google Scholar 

  10. Drouvelis, P.S., Schmelcher, P., Bastian, P.: Parallel implementation of the recursive Green’s function method. J. Comput. Phys. 215, 741 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. MacKinnon, A.: The calculation of transport properties and density of states of disordered solids. Z. Phys. B 59, 385 (1985)

    Article  Google Scholar 

  12. Sols, F., Macucci, M., Ravaioli, U., Hess, K.: Theory for a quantum modulated transistor. J. Appl. Phys. 66, 3892 (1989)

    Article  Google Scholar 

  13. Baranger, H.U., DiVincenzo, D.P., Jalabert, R.A., Stone, A.D.: Classical and quantum ballistic-transport anomalies in microjunctions. Phys. Rev. B 44, 10637 (1991)

    Article  Google Scholar 

  14. Kazymyrenko, K., Waintal, X.: Knitting algorithm for calculating Green functions in quantum systems. Phys. Rev. B 77, 115119 (2008)

    Article  Google Scholar 

  15. Kramer, T., Kreisbeck, C., Krueckl, V.: Wave-packet dynamics approach to transport in mesoscopic systems. Phys. Scr. 82, 038101 (2010)

    Article  Google Scholar 

  16. Yuan, S., De Raedt, H., Katsnelson, M.I.: Modeling electronic structure and transport properties of graphene with resonant scattering centers. Phys. Rev. B 82, 115448 (2010)

    Article  Google Scholar 

  17. Weiße, A., Wellein, G., Alvermann, A., Fehske, H.: The kernel polynomial method. Rev. Mod. Phys. 78, 275 (2006)

    Article  MATH  Google Scholar 

  18. Ferreira, A., Viana-Gomes, J., Nilsson, J., Mucciolo, E.R., Peres, N.M.R., Castro Neto, A.H.: Unified description of the dc conductivity of monolayer and bilayer graphene at finite densities based on resonant scatterers. Phys. Rev. B 83, 165402 (2011)

    Article  Google Scholar 

  19. Triozon, F., Roche, S.: Efficient linear scaling method for computing the Landauer-Büttiker conductance. Eur. Phys. J. B 46, 427 (2005)

    Article  Google Scholar 

  20. Liu, M.-H., Richter, K.: Efficient quantum transport simulation for bulk graphene heterojunctions. Phys. Rev. B 86, 115445 (2012)

    Article  Google Scholar 

  21. Datta, S.: Electronic Transport in Mesoscopic Systems. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  22. Ferry, D.K., Goodnick, S.M.: Transport in Nanostructures. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  23. Pastawski, H.M., Medina, E.: Tight-Binding methods in quantum transport through molecules and small devices: from the coherent to the decoherent description. Rev. Mex. Fis. 47, 1 (2001)

    Google Scholar 

  24. Landauer, R.: Spatial variation of currents and fields due to localized scatterers in metallic conduction. IBM J. Res. Dev. 32, 306 (1988)

    Article  MathSciNet  Google Scholar 

  25. Caroli, C., Combescot, R., Nozieres, P., Saint-James, D.: Direct calculation of the tunneling current. J. Phys. C 4, 916 (1971)

    Article  Google Scholar 

  26. Lee, P.A., Fisher, D.S.: Anderson localization in two dimensions. Phys. Rev. Lett. 47, 882 (1981)

    Article  MathSciNet  Google Scholar 

  27. Fisher, D.S., Lee, P.A.: Relation between conductivity and transmission matrix. Phys. Rev. B 23, 6851 (1981)

    Article  MathSciNet  Google Scholar 

  28. Sajjad, R.N., Polanco, C., Ghosh, A.W.: Atomistic deconstruction of current flow in graphene based hetero-junctions (2013). arXiv:1302.4473

  29. Meir, Y., Wingreen, N.S.: Landauer formula for the current through an interacting electron region. Phys. Rev. Lett. 68, 2512 (1992)

    Article  Google Scholar 

  30. Hernández, A., Apel, V.M., Pinheiro, F.A., Lewenkopf, C.H.: Quantum electronic transport: linear and nonlinear conductance from the Keldysh approach. Physica A 385, 148 (2007)

    Article  Google Scholar 

  31. Lewenkopf, C.H., Mucciolo, E.R., Castro Neto, A.H.: Numerical studies of conductivity and Fano factor in disordered graphene. Phys. Rev. B 77, 081410R (2008)

    Article  Google Scholar 

  32. Haug, H., Jauho, A.-P.: Quantum Kinetics in Transport and Optics of Semiconductors, 2nd edn. Springer, Heidelberg (2008)

    Google Scholar 

  33. Mucciolo, E.R.: Unpublished

  34. Schomerus, H.: Effective contact model for transport through weakly-doped graphene. Phys. Rev. B 76, 045433 (2007)

    Article  Google Scholar 

  35. Areshkin, D.A., Nikolić, B.K.: I-V curve signatures of nonequilibrium-driven band gap collapse in magnetically ordered zigzag graphene nanoribbon two-terminal devices. Phys. Rev. B 79, 205430 (2009)

    Article  Google Scholar 

  36. Lopez Sancho, M.P., Lopez Sancho, J.M., Rubio, J.: Highly convergent schemes for the calculation of bulk and surface Green functions. J. Phys. F, Met. Phys. 15, 851 (1985)

    Article  Google Scholar 

  37. Umerski, A.: Closed-form solutions to surface Green’s functions. Phys. Rev. B 55, 5266 (1997)

    Article  Google Scholar 

  38. Rocha, A.R., García-Suárez, V.M., Bailey, S., Lambert, C., Ferrer, J., Sanvito, S.: Spin and molecular electronics in atomically generated orbital landscapes. Phys. Rev. B 73, 085414 (2006)

    Article  Google Scholar 

  39. Wimmer, M.: Quantum transport in nanostructures: From computational concepts to spintronics in graphene and magnetic tunnel junctions. Dissertation, University Regensburg (2009). http://epub.uni-regensburg.de/12142/

  40. Tworzydło, J., Trauzettel, B., Titov, M., Rycerz, A., Beenakker, C.W.J.: Sub-poissonian shot noise in graphene. Phys. Rev. Lett. 96, 246802 (2006). Supplementary material is found in arXiv:cond-mat/0603315

    Article  Google Scholar 

  41. Rycerz, A., Tworzydło, J., Beenakker, C.W.J.: Anomalously large conductance fluctuations in weakly disordered graphene. Europhys. Lett. 79, 57003 (2007)

    Article  Google Scholar 

  42. Metalidis, G., Bruno, P.: Green’s function technique for studying electron flow in two-dimensional mesoscopic samples. Phys. Rev. B 72, 235304 (2005)

    Article  Google Scholar 

  43. Todorov, T.N.: Tight-binding simulation of current-carrying nanostructures. J. Phys. Condens. Matter 14, 3049 (2002)

    Article  Google Scholar 

  44. Cresti, A., Farchioni, R., Grosso, G., Parravicini, G.P.: Keldysh-Green function formalism for current profiles in mesoscopic systems. Phys. Rev. B 68, 075306 (2003)

    Article  Google Scholar 

  45. Yazyev, O.: Emergence of magnetism in graphene materials and nanostructures. Rep. Prog. Phys. 73, 056501 (2010)

    Article  Google Scholar 

  46. Son, Y.W., Cohen, M.L., Louie, S.G.: Energy gaps in graphene nanoribbons. Phys. Rev. Lett. 97, 216803 (2006)

    Article  Google Scholar 

  47. Xue, Y., Datta, S., Ratner, M.A.: First-principles based matrix Green’s function approach to molecular electronic devices: general formalism. Chem. Phys. 281, 151 (2002)

    Article  Google Scholar 

  48. Areshkin, D.A., Nikolić, B.K.: Electron density and transport in top-gated graphene nanoribbon devices: first-principles Green function algorithms for systems containing a large number of atoms. Phys. Rev. B 81, 155540 (2010)

    Article  Google Scholar 

  49. Bruus, H., Flensberg, K.: Many-Body Quantum Theory in Condensed Matter Physics: An Introduction. Oxford University Press, New York (2004)

    Google Scholar 

  50. Nazarov, Y.V., Blanter, Y.M.: Quantum Transport: Introduction to Nanoscience. Cambridge University Press, New York (2009)

    Book  Google Scholar 

  51. Büttiker, M.: Coherent and sequential tunneling in series barriers. IBM J. Res. Dev. 32, 63 (1988)

    Article  Google Scholar 

  52. D’Amato, J.L., Pastawski, H.M.: Conductance of a disordered linear chain including inelastic scattering events. Phys. Rev. B 41, 7411 (1990)

    Article  Google Scholar 

  53. Martin, J., Akerman, N., Ulbricht, G., Lohmann, T.: Observation of electron-hole puddles in graphene using a scanning single electron transistor. Nat. Phys. 4, 144 (2008)

    Article  Google Scholar 

  54. Zhang, Y., Brar, V.W., Girit, C., Zettl, A., Crommie, M.F.: Origin of spatial charge inhomogeneity in graphene. Nat. Phys. 5, 722 (2009)

    Article  Google Scholar 

  55. Ishigami, M., Chen, J.H., Cullen, W.G., Fuhrer, M.S., Williams, E.D.: Atomic structure of graphene on SiO2. Nano Lett. 7, 1643 (2007)

    Article  Google Scholar 

  56. Meyer, J.C., Geim, A.K., Katsnelson, M.I., Novoselov, K.S., Obergfell, D., Roth, S., Girit, Ç., Zettl, A.: On the roughness of single- and bi-layer graphene membranes. Solid State Commun. 143, 101 (2007)

    Article  Google Scholar 

  57. Levy, N., Burke, K.L., Meaker, S.A., Panlasigui, M., Zettl, A., Guinea, F., Castro Neto, A.H., Crommie, M.F.: Strain-induced pseudo-magnetic fields greater than 300 Tesla in graphene nanobubbles. Science 329, 544 (2010)

    Article  Google Scholar 

  58. Chen, J.H., Jang, C., Adam, S., Fuhrer, M.S., Williams, E.D., Ishigami, M.: Charged-impurity scattering in graphene. Nat. Phys. 4, 377 (2008)

    Article  Google Scholar 

  59. Chen, J.-H., Cullen, W.G., Jang, C., Fuhrer, M.S., Williams, E.D.: Defect scattering in graphene. Phys. Rev. Lett. 102, 236805 (2009)

    Article  Google Scholar 

  60. Han, M., Brant, J.C., Kim, P.: Electron transport in disordered graphene nanoribbons. Phys. Rev. Lett. 104, 056801 (2010)

    Article  Google Scholar 

  61. Peres, N.M.R.: Colloquium: the transport properties of graphene: an introduction. Rev. Mod. Phys. 82, 2673 (2010)

    Article  Google Scholar 

  62. Das Sarma, S., Adam, S., Hwang, E.H., Rossi, E.: Electronic transport in two-dimensional graphene. Rev. Mod. Phys. 83, 407 (2011)

    Article  Google Scholar 

  63. Adam, S., Brouwer, P.W., Das Sarma, S.: Crossover from quantum to Boltzmann transport in graphene. Phys. Rev. B 79, 201404 (2009)

    Article  Google Scholar 

  64. Kłos, J.W., Zozoulenko, I.V.: Effect of short- and long range scattering on the conductivity of graphene: Boltzmann approach vs tight-binding calculations. Phys. Rev. B 82, 081414(R) (2010)

    Google Scholar 

  65. Pereira, V.M., Castro Neto, A.H., Peres, N.M.R.: Tight-binding approach to uniaxial strain in graphene. Phys. Rev. B 80, 045401 (2009)

    Article  Google Scholar 

  66. Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, 3rd edn. Pergamon, London (1986)

    Google Scholar 

  67. Kim, E.-A., Castro Neto, A.H.: Graphene as an electronic membrane. Europhys. Lett. 84, 57007 (2008)

    Article  Google Scholar 

  68. Vozmediano, M.A.H., Katsnelson, M.I., Guinea, F.: Gauge fields in graphene. Phys. Rep. 496, 109 (2010)

    Article  MathSciNet  Google Scholar 

  69. Peierls, R.E.: Zur Theorie des Diamagnetismus von Leitungselektronen. Z. Phys. 80, 763 (1933)

    Article  Google Scholar 

  70. Lundeberg, M.B., Folk, J.A.: Rippled graphene in an in-plane magnetic field: effects of a random vector potential. Phys. Rev. Lett. 105, 146804 (2010)

    Article  Google Scholar 

  71. Sols, F., Guinea, F., Castro Neto, A.H.: Coulomb blockade in graphene nanoribbons. Phys. Rev. Lett. 99, 166803 (2007)

    Article  Google Scholar 

  72. Vérges, J.A., Guinea, F., Chiappe, G., Louis, E.: Transport regimes in surface disordered graphene sheets. Phys. Rev. B 75, 085440 (2007)

    Article  Google Scholar 

  73. Evaldsson, M., Zozoulenko, I.V., Xu, H., Heinzel, T.: Edge disorder induced Anderson localization and conduction gap in graphene nanoribbons. Phys. Rev. B 78, 161407(R) (2008)

    Article  Google Scholar 

  74. Mucciolo, E.R., Castro Neto, A.H., Lewenkopf, C.H.: Conductance quantization and transport gaps in disordered graphene nanoribbons. Phys. Rev. B 79, 075407 (2009)

    Article  Google Scholar 

  75. Brey, L., Fertig, H.A.: Electronic states of graphene nanoribbons studied with the Dirac equation. Phys. Rev. B 73, 235411 (2006)

    Article  Google Scholar 

  76. Zârbo, L.P., Nikolić, B.K.: Spatial distribution of local currents of massless Dirac fermions in quantum transport through graphene nanoribbons. Europhys. Lett. 80, 47001 (2007)

    Article  Google Scholar 

  77. Chang, P.-H., Nikolić, B.K.: Edge currents and nanopore arrays in zigzag and chiral graphene nanoribbons as a route toward high-ZT thermoelectrics. Phys. Rev. B 86, 041406(R) (2012)

    Google Scholar 

  78. Tao, C., Jiao, L., Yazyev, O.V., Chen, Y.-C., Feng, J., Zhang, X., Capaz, R.B., Tour, J.M., Zettl, A., Louie, S.G., Dai, H., Crommie, M.F.: Spatially resolving edge states of chiral graphene nanoribbons. Nat. Phys. 7, 616 (2011)

    Article  Google Scholar 

  79. Wakabayashi, K., Takane, Y., Sigrist, M.: Perfectly conducting channel and universality crossover in disordered graphene nanoribbons. Phys. Rev. Lett. 99, 036601 (2007)

    Article  Google Scholar 

  80. Wakabayashi, K., Takane, Y., Yamamoto, M., Sigrist, M.: Electronic transport properties of graphene nanoribbons. New J. Phys. 11, 095016 (2009)

    Article  Google Scholar 

  81. Lima, L.R.F., Pinheiro, F.A., Capaz, R.B., Lewenkopf, C.H., Mucciolo, E.R.: Effects of disorder range and electronic energy on the perfect transmission in graphene nanoribbons. Phys. Rev. B 86, 205111 (2012)

    Article  Google Scholar 

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Acknowledgements

Financial support by the Brazilian funding agencies FAPERJ and CNPq is gratefully acknowledged.

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Authors and Affiliations

  1. Instituto de Física, Universidade Federal Fluminense, 24210-346, Niteroi, RJ, Brazil

    Caio H. Lewenkopf

  2. Department of Physics, University of Central Florida, Orlando, FL, 32816-2385, USA

    Eduardo R. Mucciolo

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Correspondence to Eduardo R. Mucciolo.

Appendices

Appendix A: Steps in the linear conductance

Let us show that the surface Green’s function in Eq. (67) leads to the expect steps in the linear conductance. For this purpose, let us begin by noticing that, in the case of a square lattice lead, only propagating modes yield a finite level width: For |E−ε ν |<2t x ,

$$ \tilde{\varGamma}_\nu= -2 \operatorname{Im} [ \tilde{ \varSigma}_\nu ] = 2 \operatorname{Im} \bigl[ ( \tilde{g}_\nu )^{-1} \bigr] = 2 t_x \sin\phi_\nu, $$
(210)

where \(\sin\phi_{\nu}= \sqrt{1 - (E-\varepsilon_{\nu})^{2}/4t^{2}}\), in which case we can write \(\tilde{g}_{\nu}= e^{-i\phi_{\nu}}/t_{x}\).

In order to obtain the retarded Green’s function across the system, we add one slice between the left and right contacts and use the following expression, easily derivable from Eqs. (32), (33), (41) and (42):

$$ G_{0,2} = t_x^2 g_L^2 \bigl( g_L^{-1} - t_x^2 g_L \bigr)^{-1} $$
(211)

Since G 0,2 depends solely on g L , we can rewrite in the propagation mode basis, in which case the Landauer formula is reduced to (see Eq. (15))

$$ \mathcal{T} = {\sum_{\nu}}^\prime \tilde{ \varGamma}_\nu ( \tilde{G}_{0,2} )_\nu \tilde{ \varGamma}_\nu ( \tilde{G}_{0,2} )_\nu^\ast, $$
(212)

where the prime indicates that the sum runs only over states such that |E−ε ν |<2t x and

$$ ( \tilde{G}_{0,2} )_\nu= t_x^2 \tilde{g}_\nu^2 \bigl( \tilde{g}_\nu^{-1} - t_x^2 \tilde{g}_\nu \bigr)^{-1} = \frac{e^{-2i \phi_\nu}}{2i t_x \sin\phi_\nu}. $$
(213)

Putting all together, we find that

(214)

which is the expected result for a clean ballistic system.

Appendix B: Peierls hopping phases

Here we evaluate the phase of the hopping matrix elements between any two arbitrary sites due to the presence of a perpendicular magnetic field. We pick the vector potential in the generic Landau gauge A x =(α−1)By and A y =αBx, with 0≤α≤1. The (directional) Peierls phase between two neighboring sites k and k′ is given by [69]

(215)

where θ kk′ is the angle that the segment k–k′ makes with the x axis.

Notice that φ k,k′=−φ k′,k . If we sum over all the bond phases around the perimeter of a hexagon, we obtain \(\sum\varphi= \sqrt{3}ecBa_{0}^{2}/2\hbar= 2\pi (\varPhi/\varPhi_{0})\), where Φ 0=h/ec (flux quantum), and \(\varPhi= B A_{\rm hex}\), with \(A_{\rm hex} = \sqrt{3} a_{0}^{2}/2\) being the area of the hexagon.

Appendix C: Random flux estimate

Let us estimate the rms value of the random magnetic field produced by the random vector potential:

(216)

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Lewenkopf, C.H., Mucciolo, E.R. The recursive Green’s function method for graphene. J Comput Electron 12, 203–231 (2013). https://doi.org/10.1007/s10825-013-0458-7

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