Lifshitz Transition Including Many-Body Effects in Bi-Layer Graphene and Change in Stacking Order

Partha Goswami

Physics Department, D.B. College, University of Delhi, New Delhi, India.

DOI: 10.4236/graphene.2013.22013                

Abstract

We consider the AB-(Bernal) stacking for the bi-layer graphene (BLG) system and assume that a perpendicular electric field is created by the external gates deposited on the BLG surface. In the basis (A₁, B₂, A₂, B₁) for the valleyKand the basis (B₂, A₁, B₁, A₂) for the valley K′, we show the occurrence of trigonal warping [1], that is, splitting of the energy bands or the density of states on the k_x - k_y plane into four pockets comprising of the central part and three legs due to a (skew) interlayer hopping between A₁ and B₂. The hopping between A₁ - B₂ leads to a concurrent velocity v₃ in addition to the Fermi velocity v_F. Our noteworthy outcome is that the above-mentioned topological change, referred to as the Lifshitz transition [2, 3], is entirely bias-tunable. Furthermore, the many-body effects, which is known to yield logarithmic renormalizations [4] in the band dispersions of monolayer graphene, is found to have significant effect on the bias-tunability of this transition. We also consider a variant of the system where the A atoms of the two layers are over each other and the B atoms of the layers are displaced with respect to each other. The Fermi energy density of statesfor zero bias corresponds to the inverted sombrero-like structure. The structure is found to get deformed due to the increase in the bias.

Keywords

AB-(Bernal) Stacking; Trigonal Warping; Lifshitz Transition; Logarithmic Renormalizations; Inverted Sombrero

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. S. Nunez, E. Suarez Morell, and P. Vargas, “Trigonal Distortion of Topologically Confined Channels in Bilayer Grapheme,” Applied Physics Letters, Vol. 98, No. 26, 2011, Article ID: 262107. doi:10.1063/1.3605568
[2]	A. A. Abrikosov, “Fundamentals of the Theory of Metals,” Elsevier, Amsterdam, 1988.
[3]	Y. Lemonik, I. L. Aleiner and V. I. Fal’ko, “Competing Nematic, Antiferromagnetic, and Spin-Flux Orders in the Ground State of Bilayer Grapheme,” Physical Review B, Vol. 85, No. 24, 2012, Article ID: 245451. doi:10.1103/PhysRevB.85.245451
[4]	C. Hwang, D. A. Siegel, S.-K. Mo, W. Regan, A. Ismach, Y. Zhang, A. Zettl and A. Lanzara, “Fermi Velocity Engineering in Graphene by Substrate Modification,” Scientific Reports, Vol. 2, 2012, p. 590.
[5]	A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov and A. K. Geim, “The Electronic Properties of Grapheme,” Reviews of Modern Physics, Vol. 87, No. 1, 2009, pp. 109-162. doi:10.1103/RevModPhys.81.109
[6]	V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Reviews of Modern Physics, Vol. 84, No. 3, 2012, pp. 1067-1125. doi:10.1103/RevModPhys.84.1067
[7]	J. B. Oostinga, H. B. Heersche, X. Liu, A. F. Morpurgo, L. M. K. Vandersypen, “Gate-Induced Insulating State in Bilayer Graphene Devices,” Nature Materials, Vol. 7, 2008, pp. 151-157. doi:10.1038/nmat2082
[8]	Y. Zhang, T.-T. Tang, C. Girit, Z. Hao, M. C. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, F. Wang, “Direct Observation of a Widely Tunable Bandgap in Bilayer Graphene,” Nature, Vol. 459, No. 7248, 2009, pp. 820-823. doi:10.1038/nature08105
[9]	K. Kechedzhi, V. I. Fal’ko, E. McCann, and B. L, “Influence of Trigonal Warping on Interference Effects in Bilayer Graphene,” Physical Review Letters, Vol. 98, No. 7, 2007, Article ID: 176806. doi:10.1103/PhysRevLett.98.176806
[10]	E. McCann and V. I. Fal’ko, “Landau-Level Degeneracy and Quantum Hall Effect in a Graphite Bilayer,” Physical Review B, Vol. 96, 2006, Article ID: 086805.
[11]	F. Guinea, A. H. Castro Neto, and N. M. R, Peres, “Electronic States and Landau Levels in Graphene Stacks,” Physical Review B, Vol. 73, No. 24, 2006, Article ID: 245426. doi:10.1103/PhysRevB.73.245426
[12]	A. Ramasubramaniam, D. Naveh, and E. Towe, “Tunable Band Gaps in Bilayer Graphene-BN Heterostructures,” Nano Letters, Vol. 11, No. 3, 2006, pp. 1070-1075. doi:10.1021/nl1039499
[13]	M. S. Dresselhaus and G. Dresselhaus, “Intercalation Compounds of Graphite,” Advances in Physics, Vol. 51, No. 1, 2002, pp. 1-186. doi:10.1080/00018730110113644
[14]	Y. Barlas, R. C?té, J. Lambert and A. H. MacDonald, “Anomalous Exciton Condensation in Graphene Bilayers,” Physical Review Letters, Vol. 104, No. 9, 2010, Article ID: 096802. doi:10.1103/PhysRevLett.104.096802
[15]	K. S. Kim, T.-H. Kim, A. L. Walter, Th. Seyller, H. W. Yeom, E. Rotenberg and A. Bostwick, “Visualizing Atomic-Scale Negative Differential Resistance in Bilayer Graphene,” Physical Review Letters, Vol. 110, No. 3, 2013, Article ID: 036804.
[16]	C. Toke and V. I. Fal’ko, “The Effect of the Electron- Electron Interaction on the Lifshitz Transition Density in Bilayer Graphene,” Unpublished.
[17]	H. Min, R. Bistritzer, J. J. Su and A. H. MacDonald, “Room-Temperature Superfluidity in Graphene Bilayers,” Physical Review B, Vol. 78, No. 12, 2008, Article ID: 121401. doi:10.1103/PhysRevB.78.121401
[18]	B. Seradjeh, J. E. Moore and M. Franz, “Exciton Condensation and Charge Fractionalization in a Topologic,” Physical Review Letters, Vol. 103, No. 6, 2009, Article ID: 06642.
[19]	R. Nandkishore, and L. Levitov, “Quantum Anomalous Hall State in Bilayer Grapheme,” Physical Review B, Vol. 82, No. 11, 2010, Article ID: 115124.
[20]	E. Akkermans and G. Montambaux, “Mesoscopic Physics of Electrons and Photons,” Cambridge University Press, New York, 2007.
[21]	M. Maggiore, “A Modern Introduction to Quantum Field Theory,” Oxford University Press, Oxford, 2005.
[22]	J. Cserti, “Minimal Longitudinal dc Conductivity of Perfect Bilayer Grapheme,” Physical Review B, Vol. 75, No. 3, 2007, Article ID: 033405.
[23]	I. Snyman and C. W. J. Beenakker, “Ballistic Transmission through a Graphene Bilayer,” Physical Review B, Vol. 75, No. 4, 2007, Article ID: 045322.
[24]	J. Cserti, A. Csordas and G. David, “Role of the Trigonal Warping on the Minimal Conductivity of Bilayer Graphene,” Physical Review Letters, Vol. 99, 2007, Article ID: 066802.