Physics Department, Pittsburg State University Pittsburg, KS Electronic and transport properties of graphene nanoribbons: influence of edge passivation and uniaxial strain Benjamin O. Tayo Physics Department, Pittsburg State University Pittsburg, KS WSU Physics Seminar Wichita, KS November 12, 2014

Outline Part I: Graphene Review (Tutorials) Introduction to graphene Structural properties of graphene Electronic properties Graphene’s band structure and the band gap problem Part II: Graphene nanoribbons Structural properties, edge passivation Electronic structure Effect of quantum confinement Effect of edges Effect of external strain Application: charge transport Summary and conclusion

Part I: Graphene Review 1. Introduction, structural properties 2. Electronic structure Uniqueness of Graphene’s band structure Band gap problem How to solve the band gap problem?

Graphene K. S. Novoselov, et al., Science 306, 666 (2004) A single planar layer of 𝑠𝑝 2 bonded carbon atoms, densely packed in a honey-comb lattice Carbon to carbon bond length of 𝑎 𝐶−𝐶 =0.142 nm Thinnest, strongest material known, and has high electrical and thermal conductivities Room temperature electron mobility of at least 10 000 𝑐𝑚 2 /Vs K. S. Novoselov, et al., Science 306, 666 (2004)

Graphene: building block of other carbon materials Castro Neto et al. Peres, 2006a, Phys. World 19, 33 Graphene (top left) is a honeycomb lattice of carbon atoms. Graphite (top right) can be viewed as a stack of graphene layers. Carbon nanotubes are rolled-up cylinders of graphene (bottom left). Fullerenes C60 are molecules consisting of wrapped graphene by the introduction of pentagons on the hexagonal lattice.

Graphene Isolation of graphene in 2004 by Manchester group headed by Andre Geim 2010 Nobel prize in physics awarded to Andre Geim and Konstantin Novoselov “for groundbreaking experiments regarding the two-dimensional material graphene” Country Rankings in Graphene Publications to Date (source: Thomson Reuters ISI Web of Science; search dated 15 June 2012 using “Topic=graphene”; 19,017 records)

2D Graphite (Graphene) Unit Cells Direct Lattice Reciprocal Lattice & BZ

Energy Pi Bands of Graphene R. Saito et. al, “Physical Properties of Carbon Nanotubes”

Graphene is a zero gap semiconductor 𝐸 𝐹 DOS plot: http://large.stanford.edu/courses/2008/ph373/laughlin2/

Graphene’s Low-energy Physics: Dirac Fermions

Experimental evidence of massless Dirac fermions in graphene: Cyclotron mass A = Area in k space enclosed by electron’s orbit n = carrier concentration Fitting the theoretical result with experimental data yields: vF = 106 m/s, t = 3.0 eV Solid State Physics, Ashcroft and Mermin, 1976 The electronic properties of graphene, Rev. Mod. Phys. Vol. 81, 2009

The Band Gap problem in graphene Graphene’s electrical charge carriers (electrons and holes) move through a solid with effectively zero mass and constant velocity, like photons. Graphene's intrinsically low scattering rate from defects implies the possibility of almost ballistic transport. The primary technical difficulty has been controlling the transport of electrical charge carriers through the sheet.

How to solve the Band Gap Problem? Isoelectronic codoping with B and N: 𝐶 (1−2𝑥) 𝐵 𝑥 𝑁 𝑥 . Graphene substrate interaction, e.g. epitaxial graphene on h-BN or SiC. N-type doping by Potassium deposition onto the graphene sheet. Try other materials (2D materials revolution): MoS2, MoSe2, WS2, WSe2, etc. Lithographic patterning into graphene nanoribbons. L. Liu & Z. Shen, Appl. Phys. Letts 95, 252104, (2009) T. Ohta et al., "Controlling the electronic structure of bilayer graphene," Science 313, 951 (2006). K. S. Novoselov et. al, “Two-dimensional atomic crystals,” PNAS, vol. 102 no. 30, 10451, (2005)

Part II: Graphene Nanoribbons 1. Structural properties, edge passivation 2. Electronic structure Effect of quantum confinement Effect of edges Effect of external strain 3. Application: charge transport

Graphene Nanoribbons (GNRs) 𝑦 x W GNRs are elongated stripes of single layered graphene with a finite width Electronic properties depend on edge geometry and width Structurally very similar to carbon nanotubes

AFM image of many graphene nanoribbons parallel to each other Cançado et al., Phys. Rev. Lett. 93, 047403 (2004)

Graphene Nanoribbon structural parameters N = Number of dimer lines N-AGNR = GNR with armchair edges and N-dimer lines N-ZGNR = GNR with zig-zag edges and N-dimer lines N= 3p, 3p+1, 3p+2, where p is a positive integer (family pattern). Benjamin O. Tayo, Mater. Focus 3, 248-254 (2014)

Effect of edge passivation with Hydrogen Converged geometry of a H-passivated 7-AGNR Edge C-C bond lengths are shortened by 3 to 5% compared to those in the middle of the ribbon *Optimization was performed using DFT with the B3YPL XC potential and the 6-31 G(d) basis set, with the Gaussian 09 code

X. Peng, and S, Velasquez, Appl. Phys. Letts., 98, 023112, (2011). Passivation with other atoms or groups A. Simbeck et al., Phys. Rev. B 88, 035413 (2013) Different atoms or functional groups provide different levels of perturbations to the nanoribbon. Electronic properties depend on edge passivation X. Peng, and S, Velasquez, Appl. Phys. Letts., 98, 023112, (2011).

Electronic structure: effect of quantum confinement 𝑦 𝑊 x

Effect of strain and H-passivation: model Hamiltonian

Model Hamiltonian At k = 0, Hamiltonian is: Y.W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. 97, 216803 (2006). Benjamin O. Tayo, Mater. Focus 3, 248-254 (2014).

Tight-Binding Parameters Hopping integrals are calculated using analytic expressions for TB matrix elements between C atoms For edge carbon atoms, additional strain due to H passivation has to be taken into account D. Porezag, et al., Phys. Rev. B 51, 12947 (1995).

(a) Band Gap of Unstrained H-passivated GNR M. Han et al.,“Energy Band-Gap Engineering of Graphene Nanoribbons,” PRL 98, 206805 (2007). 𝐸 𝑔𝑎𝑝 = 1.5 𝑒𝑉 𝑛𝑚 𝑊[𝑛𝑚] Benjamin O. Tayo, Mater. Focus 3, 248-254 (2014).

(b) Effective mass of Unstrained H-passivated GNR TB approx.

(c) Band Gap and Effective mass of strained H-passivated GNR X. Peng & S. Velasquez, “Strain modulated band gap of edge passivated armchair GNRs,” APL 98, 023112 (2011).

(a) Asymmetry of Band Gap variation with strain N = number of dimer lines Ne = number of pi electrons in the GNR Ω = 2 𝜋 𝑁 𝑐 𝑇 is the length of the 1D BZ per allowed state Line width = 0.01 eV Energy range -2 eV < E < 2 eV EF = 0 (Fermi Energy) B. Tayo, Mater. Focus 3, 248-254 (2014).

Application: Charge transport in AGNR Carrier scattering by longitudinal acoustic phonons plays a significant role in charge transport in intrinsic semiconductors. Within the deformation potential theory (dp), the carrier relaxation time ( 𝜏 𝑑𝑝 ) and mobility ( 𝜇 𝑑𝑝 ) are given by: C = stretching modulus; E1 = Deformation potential constant J. Bardeen and W. Shockley, Phys. Rev. 80, 72 (1950). F. B. Beleznay, F. Bogr, and J. Ladik, J. Chem. Phys. 119, 5690 (2003).

Application: Charge transport in AGNR The advantage of gapless graphene is its high carrier mobility. When a non-zero gap is engineered by patterning graphene into nanoribbons, the mobility has been shown to decrease dramatically The hardness to achieve high mobility and large on/off ratio simultaneously limits the development of graphene electronics. Suitable choice of strain and edge passivation could be used to open the band gap while maintaining a low effective mass. X. R. Wang, Y. J. Ouyang, X. L. Li, H. L. Wang, J. Guo, and H. J. Dai, Phys. Rev. Lett. 100, 206803 (2008). J. Wang, R. Zhao, M. Yang, Z. Liu, and Z. Liu, Chem. Phys. 138, 084701 (2013).

Future of Graphene Electronics Walt A. de Heer: “Researchers should stop trying to use graphene like silicon, and instead use its unique electron transport properties to design new types of electronic devices that could allow ultra-fast computing” “Exceptional ballistic transport in epitaxial graphene Nanoribbons,” J. Baringhaus, M. Ruan, F. Edler, A. Tejeda, M. Sicot, A. Taleb-Ibrahimi, A. Li, Z. Jiang, E. H. Conrad, C. Berger, C. Tegenkamp, and Walt A. de Heer, Nature Physics 10, 182, (2014).

Summary Edge passivation and strain can both be described within the TB approx. by simply renormalizing the C-C hopping integral. Studied relationship between carrier mass and band gap energy for strained H-passivated AGNRs belonging to different families: N = 3p, 3p+1, 3p+2 For unstrained H-passivated AGNRs, the effective mass exhibits a linear dependence on band gap energy for small energy gaps or large ribbon width. However for ribbons with small width or larger band gaps, the effective mass dependence on energy gap is parabolic. In the presence of strain, both band gap and effective mass displays a nearly zigzag periodic pattern, indicating that the effective mass remains proportionate to the band gap even in the presence of applied strain. Finally, we discussed the implications of non-zero band gap on carrier mobility

Acknowledgement Pittsburg State University summer faculty fellowship Supercomputer core time from NERSC @ LBNL Use of Gaussian 09 software for DFT calculations

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