Graphene Journal Club On the Edge with Nano Graphene

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Graphene Journal ClubOn the Edge with Nano
Graphene

• Y. Hancock
• Many-Body Theory Group
• Laboratory of Physics
• COMP Centre of Excellence
• Helsinki University of Technology

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Acknowledgements

• Professor Martti Puska and Karri Saloriutta
• Professor Antti-Pekka Jauho
• Dr (Tech) Ari Harju

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Outline

• Tight-binding theory of graphene
• What is special about nano-graphene?
• Our current toolbox and research plan

www.ewels.info
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Graphene Theory

• Graphene was originally a theoreticians
  playground
• Wallace, Phys. Rev. 71, 622 (1947)

Two basis vectors, whose magnitude 2.46
angstroms
Define the wavefn
Wallace Phys. Rev. 71, 622 (1947)
Where,
and
Assume that these wavefunctions are orthogonal
Pz orbital
Next solve the Eigenproblem
where, the TB overlap integrals are,
and
Hence,
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Graphene Bandstructure
Eliminating ?, we obtain the secular equation
And the eigenenergy soln
Assuming that H11 H22, then with
?0 2.7 eV
and
Recent ARPES measurements on bulk graphene
confirm excellent agreement with bandstructure
theory A. Bostwick et al, Nature 3, 36 (2007)
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Simple Example
Atom A Atom B
Hamiltonian Equation is
where,
We find
Solving the Secular Eqn
Two atom basis has lead to the basic features of
the graphene bandstructure
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Theoretical Designing
Consider now that atoms A and B are different
The eigenproblem becomes
Solving the secular equation
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Application to Nano Graphene

• Graphene devices are thought to be most
  realizable on the nano scale.
• Two examples of nano-graphene
• In these systems, edge and strain effects,
  impurities and the reduced co-ordination number
  of the carbon atoms need to be considered.

Y-W Son et al PRL 97, 216803 (2006)
nanoribbon
A. Gerouki, J. Electrochem. Soc. 143, L262 (1996)
nanofragment
L. Brey H.A. Fertig, PRB 73, 235411 (2006)
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What is currently known - Experiments
Recently, IBM research has done measurements on
20 - 500nm wide nanoribbons and has shown that
the resistivity increases as the width of the
nanoribbon decreases. Boundary scattering and
trapped charges were found to affect the
transport properties. A confinement induced gap
of 30meV was found in the 20nm ribbon. Chen et.
al. arXivcond-mat/0701599
Graphene nanoribbons can be fabricated by
patterning mechanically exfoliated or epitaxially
grown graphene (Georga Tech, UMan and UColumbia).
Fragment fabrication has been claimed on top of
Ru 0001 surface, although this has not been
substantiated.
Conduction measurements as a function of width as
measured by Han et. al. Phys. Rev. Lett. 98,
206805 (2007). The conductance was measured using
a standard lock-in technique and showed a direct
relationship with the nanoribbon width. The
measured band-gap was found to decrease as the
width of the nanoribbon increases. The article
claims that a bandgap of 200meV was obtained for
a ribbon of width 15nm.
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The Possibility of Magnetism in Nano-Graphene

• Defects can be produced by irradiating the
  material with electrons of ions for example,
  single atom vacancies and hydrogen chemisorption
  defects. Edge effects become important. Both are
  implicated in the formation of magnetism in
  carbon based systems.
• SQUID measurements have been used to detect
  magnetism in carbon systems. In particular in
  highly oriented pyrolitic graphite samples
• - P. Esquinazi et al PRB 66, 024429 (2002).
• STM experiments can determine impurity states and
  near impurity site effects - Wehling et al PRB
  75, 125425 (2007).

There Are Many Open Questions
Features Bandbap metal or zero gap/gap
semi-conductor ? Chemical reactivity of the edge
? Magnetism and formation of local edge states ?
Possibility to Tune the bandgap. Change the
transport properties of the nanoribbons. Create
magnetism.
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Size DOES matter!

• Graphene nano-ribbons (GNRs)
• Edge effects and ribbon width in tuning the
  bandgap
• LSDA study - Son et al., PRL 97, 216803 (2006)

GNRs with armchair or zigzag shaped edges and
with hydrogen passivation
Both varieties have been shown to have nonzero
and direct bandgaps
Summary of findings Energy gap for armchair
edges is due to quantum confinement and edge
effects. For the zigzig edge this is due to a
staggered sublattice potential arising because of
edge magnetization edge states resulting from
flat bands hence magnetism in the system. The
magnetism seen in the zigzag edge has
ferromagnetic ordering at each edge, but
anti-ferromagnetic coupling between the two
edges. Energy gap is found to scale inversely
with the nanoribbon width.
LSDA pseudopotential calc with hydrogen
passivation and structural relaxation
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More on Edges, Defects Magnetism
K. Nakada et. al. PRB 54, 17954
(1996) Tight-binding calculations show the
formation of an edge state in zigzag nanoribbons.
Such a state does not appear for armchair
systems. Notice the sensitivity of the relative
magnitude of this peak as a function of the
system size
V.M. Pereira et. al. PRL 96, 036801 (2006) Uses
the tight-binding formalism to look at the effect
of vacancy concentration on the density of states
(DOS) results in graphene planes. Inclusion of
second nearest neighbour hopping term (t) breaks
the particle hole symmetry. Its effect is to
broaden the peak at the Fermi energy and shift it
by an amount, which is of order t.
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What about the graphene fragment?
Originally thought of as a possible negative
electrode component in rechargeable lithium-ion
batteries (Gerouki et. al. J. Electrochem. Soc.
143, L262 1996)
Two recent studies on the magnetism of the
graphene nanofragment D. Jian et. al.
arXiv0706. 0863 Ab-initio studies (DFT with
GGA), show that there is a minmum cluster size
for magnetism in zigzag clusters, and that the AF
phase is the ground state. As the length of the
cluster is increased the results are found to
approach that of the nanoribbons J.
Fernandex-Rossier and J.J. Palacios,
arXiv0707.2964v1 Has studied the role of
interaction effects, within a meanfield
description and cluster shapes on the magnetic
properties of the zigzag nanofragments. Magnetism
is found to arise when there is a difference
between the number of A and B atoms. The total
magnetic moment, S (NA-NB)/2 and when
NANB,then S0.
Gerouki et. Al. J. Electrochem. Soc. 143, L262
1996
Both studies were done for systems with hydrogen
passivated edges
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Unsolved Questions for the Fragment
Recent STM results, show that there is in fact a
mixture of zigzag and magnetically inactive
edges, which weaken the tendency towards magnetic
states. If the system was magnetic the LDOS at
the edge would have shown two symmetric peaks
around the Fermi level. Y. Niimi et. al. Phys.
Rev. B 73, 085421 (2006)
This is a big problem with the graphene
nanosystems, the mixture of edges has been shown
to destroy the magnetism in the system. What
is the device potential of this structure?
Are there ways to stabilize the magnetism of the
graphene nanofragment? What is the interaction
nature between the edges of the system? What is
the role of extended hopping on the magnetic
properties? How does edge relaxation affect the
properties of these systems? Can interaction
effects (such as magnetic impurity atoms) be used
to tune the magnetism of this system? How are
the transport properties affected and what is the
relationship between these, the electronic
structure and the magnetism of the nanofragment?
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Self-consistent HF method

• The system is modelled using the Hubbard
  Hamiltonian

Making the approximation
Equations are solved self consistently
Where,
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Plan

• Need to include extended hoppings into the
  problem up to third or fourth nearest neighbour
  - Phys. Rev. B. 66, 035412 (2002)
• Exploit the benefit of the tight-binding
  formalism
• Within this scheme, the structure can be directly
  and easily manipulated/
• perturbed.
• Compare with ab initio results
• To obtain parameterization for the tight-binding
  model through energy level matching.
• To understand the role of relaxation and modifiy
  the tight-binding calculation accordingly.
• To provide guidance for the ab initio work.
• To do finite temperature calculations within the
  TB formalism (already implemented) using the
  Matsubara Greens function formalism.
• Transport calculations