Skip to main content
SpringerLink
Log in

Perturbation theory for weakly coupled two-dimensional layers

  • Invited Articles
  • Published:
Journal of Materials Research Aims and scope Submit manuscript

Abstract

A key issue in two-dimensional structures composed of atom-thick sheets of electronic materials is the dependence of the properties of the combined system on the features of its parts. Here, we introduce a simple framework for the study of the electronic structure of layered assemblies based on perturbation theory. Within this framework, we calculate the band structure of commensurate and twisted bilayers of graphene (Gr) and hexagonal boron nitride (h-BN), and of a Gr/h-BN heterostructure, which we compare with reference full-scale density functional theory calculations. This study presents a general methodology for computationally efficient calculations of two-dimensional materials and also demonstrates that for relatively large twist in the graphene bilayer, the perturbation of electronic states near the Fermi level is negligible.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

FIG. 1
FIG. 2
FIG. 3
FIG. 4
FIG. 5

Similar content being viewed by others

References

  1. A.K. Geim and I.V. Grigorieva: van der Waals heterostructures. Nature 499(7459), 419–425 (2013).

    Article  CAS  Google Scholar 

  2. L. Britnell, R.M. Ribeiro, A. Eckmann, R. Jalil, B.D. Belle, A. Mishchenko, Y-J. Kim, R.V. Gorbachev, T. Georgiou, S.V. Morozov, A.N. Grigorenko, A.K. Geim, C. Casiraghi, A.H. Castro Neto, and K.S. Novoselov: Strong light-matter interactions in heterostructures of atomically thin films. Science 340(6138), 1311–1314 (2013).

    Article  CAS  Google Scholar 

  3. M. Xu, T. Liang, M. Shi, and H. Chen: Graphene-like two-dimensional materials. Chem. Rev. 113(5), 3766–3798 (2013).

    Article  CAS  Google Scholar 

  4. J.H. Warner, M.H. Rümmeli, T. Gemming, B. Büchner, G. Andrew, and D. Briggs: Direct imaging of rotational stacking faults in few layer graphene. Nano Lett. 9(1), 102–106 (2009).

    Article  CAS  Google Scholar 

  5. W. Yan, M. Liu, R-F. Dou, L. Meng, L. Feng, Z-D. Chu, Y. Zhang, Z. Liu, J-C. Nie, and L. He: Angle-dependent van Hove singularities in a slightly twisted graphene bilayer. Phys. Rev. Lett. 109, 126801 (2012).

    Article  Google Scholar 

  6. L-J. Yin, J-B. Qiao, R. Xu, R-F. Dou, J-C. Nie, and L. He: Electronic structures and their Landau quantizations in twisted graphene bilayer and trilayer. arXiv e-print:1410.1621 (2014).

  7. Y. Wang, Z. Su, W. Wu, S. Nie, X. Lu, H. Wang, K. McCarty, S-S. Pei, F. Robles-Hernandez, V.G. Hadjiev, and J. Bao: Four-fold Raman enhancement of 2D band in twisted bilayer graphene: Evidence for a doubly degenerate dirac band and quantum interference. Nanotechnology 25(33), 335201 (2014).

    Article  Google Scholar 

  8. I. Brihuega, P. Mallet, H. González-Herrero, G. Trambly de Laissardière, M.M. Ugeda, L. Magaud, J.M. Gómez-Rodríguez, F. Ynduráin, and J-Y. Veuillen: Unraveling the intrinsic and robust nature of van Hove singularities in twisted bilayer graphene by scanning tunneling microscopy and theoretical analysis. Phys. Rev. Lett. 109, 196802 (2012).

    Article  CAS  Google Scholar 

  9. S. Coh, L.Z. Tan, S.G. Louie, and M.L. Cohen: Theory of the Raman spectrum of rotated double-layer graphene. Phys. Rev. B 88(16), 165431 (2013).

    Article  Google Scholar 

  10. M. Koshino: Interlayer interaction in general incommensurate atomic layers. New J. Phys. 17(1), 015014 (2015).

    Article  CAS  Google Scholar 

  11. D. Ghader, D. Szczȩśniak, and A. Khater: Theory for the electronic structure of incommensurate twisted bilayer graphene. arXiv e-print:1501.06334 (2015).

  12. H.K. Pal, S. Carter, and M. Kindermann: Theory of twisted bilayer graphene near commensuration. arXiv e-print:1409.1971 (2014).

  13. B. Cao and T. Li: Interlayer electronic coupling in arbitrarily stacked MoS2 bilayers controlled by interlayer S–S interaction. J. Phys. Chem. C 119(2), 1247–1252 (2014).

    Article  Google Scholar 

  14. M. Bokdam, T. Amlaki, G. Brocks, and P.J. Kelly: Band gaps in incommensurable graphene on hexagonal boron nitride. Phys. Rev. B 89(20), 201404 (2014).

    Article  Google Scholar 

  15. P. Hohenberg and W. Kohn: Inhomogeneous electron gas. Phys. Rev. 136(3B), B864–B871 (1964).

    Article  Google Scholar 

  16. W. Kohn and L.J. Sham: Self-consistent equations including exchange and correlation effects. Phys. Rev. 140(4A), A1133–A1138 (1965).

    Article  Google Scholar 

  17. R.M. Ribeiro and N.M.R. Peres: Stability of boron nitride bilayers: Ground-state energies, interlayer distances, and tight-binding description. Phys. Rev. B 83(23), 235312 (2011).

    Article  Google Scholar 

  18. E. Cappelluti, R. Roldán, J.A. Silva-Guillén, P. Ordejón, and F. Guinea: Tight-binding model and direct-gap/indirect-gap transition in single-layer and multilayer MoS2. Phys. Rev. B 88(7), 075409 (2013).

    Article  Google Scholar 

  19. S. Fang, R. Kuate Defo, S.N. Shirodkar, S. Lieu, G.A. Tritsaris, and E. Kaxiras: Ab initio tight-binding hamiltonian for transition metal dichalcogenides. Phys. Rev. B 92, 205108 (2015).

    Article  Google Scholar 

  20. J. Enkovaara, C. Rostgaard, J.J. Mortensen, J. Chen, M. Dulak, L. Ferrighi, J. Gavnholt, C. Glinsvad, V. Haikola, H.A. Hansen, H.H. Kristoffersen, M. Kuisma, A.H. Larsen, L. Lehtovaara, M. Ljungberg, O. Lopez-Acevedo, P.G. Moses, J. Ojanen, T. Olsen, V. Petzold, N.A. Romero, J. Stausholm-Møller, M. Strange, G.A. Tritsaris, M. Vanin, M. Walter, B. Hammer, H. Häkkinen, G.K.H. Madsen, R.M. Nieminen, J.K. Nørskov, M. Puska, T.T. Rantala, J. Schiøtz, K.S. Thygesen, and K.W. Jacobsen: Electronic structure calculations with GPAW: A real-space implementation of the projector augmented-wave method. J. Phys.: Condens. Matter 22(25), 253202 (2010).

    CAS  Google Scholar 

  21. P.E. Blöchl: Projector augmented-wave method. Phys. Rev. B 50(24), 17953–17979 (1994).

    Article  Google Scholar 

  22. G.A. Tritsaris, B.D. Malone, and E. Kaxiras: Optoelectronic properties of single-layer, double-layer, and bulk tin sulfide: A theoretical study. J. Appl. Phys. 113(23), 233507 (2013).

    Article  Google Scholar 

  23. I.V. Lebedeva, A.A. Knizhnik, A.M. Popov, Y.E. Lozovik, and B.V. Potapkin: Interlayer interaction and relative vibrations of bilayer graphene. Phys. Chem. Chem. Phys. 13(13), 5687–5695 (2011).

    Article  CAS  Google Scholar 

  24. X. Lin, Y. Xu, A.A. Hakro, T. Hasan, R. Hao, B. Zhang, and H. Chen: Ab initio optical study of graphene on hexagonal boron nitride and fluorographene substrates. J. Mater. Chem. C 1(8), 1618–1627 (2013).

    Article  CAS  Google Scholar 

  25. Y. Fan, M. Zhao, Z. Wang, X. Zhang, and H. Zhang: Tunable electronic structures of graphene/boron nitride heterobilayers. Appl. Phys. Lett. 98(8), 083103 (2011).

    Article  Google Scholar 

  26. G.A. Tritsaris, D. Vinichenko, G. Kolesov, C.M. Friend, and E. Kaxiras: Dynamics of the photogenerated hole at the rutile TiO2(110)/water interface: A nonadiabatic simulation study. J. Phys. Chem. C 118, 27393–27401 (2014).

    Article  CAS  Google Scholar 

  27. G. Kolesov, D. Vinichenko, G.A. Tritsaris, C.M. Friend, and E. Kaxiras: Anatomy of the photochemical reaction: Excited-state dynamics reveals the C–H acidity mechanism of methoxy photo-oxidation on titania. J. Phys. Chem. Lett. 6(9), 1624–1627 (2015).

    Article  CAS  Google Scholar 

Download references

ACKNOWLEDGMENTS

This work was supported in part by ARO MURI Award W911NF-14-1-0247. Mitchell Luskin was also supported in part by the Radcliffe Institute for Advanced Study at Harvard University. Calculations were performed on the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant No. ACI-1053575.

Author information

Authors and Affiliations

  1. John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts, 02138, USA

    Georgios A. Tritsaris, Sharmila N. Shirodkar & Efthimios Kaxiras

  2. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA

    Paul Cazeaux & Mitchell Luskin

  3. Department of Mathematical Sciences, University of Delaware, Newark, Delaware, 19716, USA

    Petr Plecháč

  4. Université Paris Est, Ecole des Ponts and INRIA, 77455, Marne-la-Vallée, France

    Eric Cancès

Authors
  1. Georgios A. Tritsaris
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sharmila N. Shirodkar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Efthimios Kaxiras
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Paul Cazeaux
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Mitchell Luskin
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Petr Plecháč
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Eric Cancès
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Efthimios Kaxiras.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tritsaris, G.A., Shirodkar, S.N., Kaxiras, E. et al. Perturbation theory for weakly coupled two-dimensional layers. Journal of Materials Research 31, 959–966 (2016). https://doi.org/10.1557/jmr.2016.99

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1557/jmr.2016.99

Search

Navigation