Not only is the gap size much smaller than small molecules like compar-He and H2, but the delocalized π electron cloud also works to repel molecules:16 thismakes graphene ideal to be use
Trang 1This part consists of three introductory chapters to give the reader sufficient backgroundmaterial to understand the Ph.D work that is reported in the later chapters.
In Chap.1, a review of the physical and chemical properties of graphene and hexagonalboron nitride is given in order to provide motivation for studying these materials A re-view of the fabrication methods of these materials is also presented Chap.2is devoted
to the discussion of density-functional theory, which is the workhorse for the tional simulations done in this thesis In Chap.3, the Landauer approach for electronicand thermal transport is discussed, and we show the derivation for the equations that weuse to calculate the thermoelectric properties of graphene-related materials in the laterchapters
computa-4
Trang 2Introduction to graphene-related
materials
Abstract: In this chapter, we briefly introduce the physical and chemical properties of graphene, graphene nanoribbons and quantum dots, and hexagonal boron nitride.
1.1 Physical and chemical properties of graphene
Unit cell of graphene – Graphene is composed of a two-dimensional hexagonal work of C atoms, as shown in Fig.1.1(a) The primitive unit cell (PUC) of graphene ismade up of two C basis atoms which are labeled A and B Graphene is a bipartite latticebecause the atoms A and B can be grouped into two different sublattices where atoms
net-A are only bonded to atoms B, and vice versa.10The lattice vectors of the primitive unitcell (PUC) are
a1=√
3acc
√3
2 ˆx +
1
2ˆy
!and a2=√
3acc
√3
√
3ˆx + ˆy
and b2= √2π
3acc
1
Trang 3a2
X Y
b2
Γ
Κ'ΜΚ
a1
a2
Figure 1.1: (a) The atomistic model of the hexagonal network of graphene; the greyspheres represent C atoms The primitive unit cell (PUC) of graphene is denoted bylattice vectors a1 and a2 and is composed of two C basis atoms A and B Assuming
a C–C bond length of 1.42 Å, |a1| = |a2| = 2.46 Å (b) The corresponding reciprocallattice vectors b1 and b2 of the real-space unit cell denoted in (a) The corners of thelight blue hexagon represent all the reciprocal lattice points closest to origin, and the1st Brillouin zone in dark blue is constructed from the lines that bisect the vectorsconnecting the origin to the nearest reciprocal lattice points The letters Γ, M, K, andK’ mark the high-symmetry points of the 1st Brillouin zone
which obey the relation ai· bj= 2πδi j, where δi j is the Kronecker delta Similarly,the entire reciprocal lattice is obtained by translating the reciprocal PUC in any generaldirection G = m1b1+ m2b2, mi∈ Z
Bonding in graphene – The 2s, 2px, and 2py orbitals of each C atom in graphene bridize to form sp2orbitals that lie in the x − y plane and are separated from each other
hy-by 120◦ Within the x − y plane of graphene, the C atoms are connected via σ bondsdue to the overlap of these sp2 orbitals The remaining 2pz orbitals of the C atomsare oriented perpendicular to the x − y plane, and overlap to form π bonds When oneattempts to connect the C atoms in graphene by double bonds to fulfill the octet rule,three equivalent Clar structures11 can be constructed, as shown in Fig 1.2 Since abenzenoid ring appears in each hexagon ring of graphene when all three Clar structuresare superimposed on each other, the π electrons from the 2pz orbitals are fully delo-calized over the entire graphene structure The highly delocalized nature of the π elec-trons lead to high electronic conductance in graphene, as will be discussed later Thepartial double bond character of the C–C bonds makes graphene mechanically strong:theoretical calculations showed that the Young’s modulus of graphene is 1.05 TPa foruniaxial tensile strain applied along the plane of graphene;12 experimentally, a break-
Trang 4Figure 1.2: The three equivalent Clar structures of graphene
ing strength of 42 Nm-1 and Young’s modulus of 1.0 TPa was measured for graphene
in the out-of-plane direction.13This great mechanical strength makes graphene capable
of withstanding high electron current densities without structural failure.14 In ison,15 Cu has a Young’s modulus of 130 GPa The geometric size of the gap in themiddle of each hexagon ring of graphene is estimated to be ∼ 0.064 nm, when the C–Cbond length and van der Waals’ radius of the C atoms are assumed to be 0.142 nm and0.11 nm, respectively Not only is the gap size much smaller than small molecules like
compar-He and H2, but the delocalized π electron cloud also works to repel molecules:16 thismakes graphene ideal to be used as an impermeable gas membrane.17
Electronic band structure and conductance of graphene – The many interestingproperties of graphene are due to the unique characteristics of its electronic band struc-ture, which will be derived here using the tight-binding approach.7,18 Since the inter-esting electronic properties of graphene are due to the electrons in the π-band, and the2pz orbital does not overlap significantly with the 2s, 2px, and 2py orbitals, we shalluse pz atomic orbital functions centered on the A and B basis atoms in the nth PUC of
a graphene supercell as the minimal basis set; i.e.,
pz(r − rA− Rn) and pz(r − rB− Rn), (1.3)
where Rnis the position vector of the nth PUC in the graphene supercell consisting of
N PUCs, and rX is the position vector of basis atom X from the origin of the nth PUC
Trang 5The electronic wavefunction of the π electrons in graphene can be expressed as
Hˆ
pRn0 ,A z
Hˆ
pRn0 ,B z
Hˆ ... surfaces like Cu,11 9
Ni,11 8,11 9 Rh,12 0 Pd,12 1 and Ru12 2 have been used for the CVD growth of h-BN pending on the... deposition growth process of h-BN on the stepped surface of Ru(00 01)
1. 3 .1. 1 Exfoliation of h-BN crystal
Similar to graphene, we can cleave multiple layers of h-BN from a h-BN crystal... catalyticdecomposition of C60 molecules using the Ru(00 01) metal surface
1. 2 .1. 1 Exfoliation of graphite
Since graphite is composed of layers of graphene held together