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
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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
o Effect of quantum confinement
o Effect of edges
o Effect of external strain
• Application: charge transport
 Summary and conclusion
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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?
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Graphene
• A single planar layer of 𝑠𝑝2 bonded carbon atoms, densely packed in a honeycomb lattice
• Carbon to carbon bond length of 𝑎𝐶−𝐶 = 0.142 nm
• Thinnest, strongest material known, and has high electrical and thermal
conductivities
• Room temperature electron mobility of at least 10 000 𝑐𝑚2 /Vs
K. S. Novoselov, et al., Science 306, 666 (2004)
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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.
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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)
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2D Graphite (Graphene) Unit Cells
Direct Lattice
Reciprocal Lattice & BZ
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Energy Pi Bands of Graphene
R. Saito et. al, “Physical Properties of Carbon Nanotubes”
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Graphene is a zero gap semiconductor
𝐸𝐹
DOS plot: http://large.stanford.edu/courses/2008/ph373/laughlin2/
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Graphene’s Low-energy Physics: Dirac Fermions
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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
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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.
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How to solve the Band Gap Problem?
• Isoelectronic codoping with B and N: 𝐶(1−2𝑥) 𝐵𝑥 𝑁𝑥 .
• 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)
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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
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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
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AFM image of many graphene
nanoribbons parallel to each other
Cançado et al., Phys. Rev. Lett. 93, 047403 (2004)
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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)
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Effect of edge passivation with Hydrogen
• Converged geometry of a Hpassivated 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
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Passivation with other atoms or groups
A. Simbeck et al., Phys. Rev. B 88, 035413 (2013)
X. Peng, and S, Velasquez, Appl. Phys.
Letts., 98, 023112, (2011).
• Different atoms or functional groups provide
different levels of perturbations to the
nanoribbon.
• Electronic properties depend on edge
passivation
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Electronic structure: effect of quantum confinement
𝑦
𝑊
x
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Effect of strain and H-passivation: model Hamiltonian
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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).
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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).
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(a) Band Gap of Unstrained H-passivated GNR
𝐸𝑔𝑎𝑝
1.5 𝑒𝑉 𝑛𝑚
=
𝑊[𝑛𝑚]
M. Han et al.,“Energy Band-Gap
Engineering of Graphene Nanoribbons,”
PRL 98, 206805 (2007).
Benjamin O. Tayo, Mater. Focus 3, 248-254 (2014).
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(b) Effective mass of Unstrained H-passivated GNR
TB approx.
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(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).
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(a) Asymmetry of Band Gap variation with strain
• N = number of dimer lines
• Ne = number of pi electrons in
the GNR
2𝜋
• Ω=
is the length of the 1D
𝑁𝑐 𝑇
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).
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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 (𝜏𝑑𝑝 ) and mobility (𝜇𝑑𝑝 ) 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).
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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).
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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).
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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 Hpassivated 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
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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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THANK YOU FOR YOUR
ATTENTION
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