ABSTRACT
Here, e kn is the energy and Hab (a, b = 1, 2) are the interaction matrix elements. The latter can be readily obtained by taking into account the nearest neighbor interactions:
H H e p r t H p r t R
H
= = - - - =
=
◊Âc c e c
A A A A A( ) ( ) k k
H e p r t H p r t R
H H
c e
c c
= - - - =
=
e p r t H p r t R
e e
B A B= - - - = - - -
( ) ( ) g g g0 0 01 - ◊
◊= = - - -
=
H H e p r t H p r t R
2 21 c cB A B A( ) ( )
Here, ezA (ezB) is the energy of pz orbital for atom A (B) after the hybridization but without the formation of bonds with neighboring atoms and g0 is the hopping energy between nearest neighbor atoms. As we are only interested in the excitation spectrum, we may set ezA(ezB) = 0. The electronic band structure can be obtained by letting the determinant be zero, i.e.,
11 12 21 22 0--ÈÎÍÍ ˘˚˙˙ =e e (2.11)
Solving Eq. (2.11) gives the energy dispersion:
E k k a k a k ax y y( ) cos cos cos = ± + ÊËÁ ˆ¯˜ ÊËÁ ˆ¯˜ + ÊËÁ ˆ¯˜g 0 1 4 32 32 4 322 . (2.12)
Note that, for clarity, we have replaced e kn by E(k) since we have only two bands which are corresponding to the conduction and the valence band of electrons. In Eq. (2.12), kx and ky are the components of k in the (kx, ky) plane, g 0 2 75= . eV is the nearest-neighbor hopping energy, and plus (minus) sign refers to the upper (p*) and lower (p) band. Figure 2.1c shows the three-dimensional electronic dispersion (left) and energy contour lines (right) in the k-space. Near the K and K’ points, the energy dispersion has a circular cone shape which, to a first order approximation, is given by E k kF( ) | | = ± n . (2.13) Here v aF ms= ª -32 100 6 1g is the Fermi velocity. Note that in Eq. (4) the wave vector k is measured from the K and K’ points, respectively. This kind of energy dispersion is distinct from that of non-relativistic electrons, i.e., E k k
m ( )= 2 22 , where m is the mass of electrons. The linear dispersion becomes “distorted” with increasing
k away from the K and K’ points due to a second order term with a threefold symmetry; this is known as the trigonal warping of the electronic spectrum in literature [3−5]. The peculiarity of electrons in graphene near the K (K’) points can be intuitively understood as follows. The 2pz orbital of each carbon atom in the A sub-lattice interacts with the three nearest neighboring atoms in the B sub-lattice (and vice versa) to form energy bands. Although the interaction between the two atoms is strong (as manifested by the large hopping energy), the net interaction with the three nearest neighboring atoms diminishes as k approaches the K (K’) points. This can be readily verified by substituting the K a( , / , )0 4 3 3 0p and K a a¢( / , / , )2 3 2 3 3 0p p points into Eqs. (2.10) and (2.12). The strong interaction with individual neighboring atoms makes it possible for electrons to move at a fast speed in graphene and the diminishing net interaction at Fermi level leads to a zero band gap. This result indicates that any honeycomb lattice consisting of same atoms will exhibit similar energy dispersion curves, and it is not necessary that one must have a carbon lattice. 2.2.2 Low-Energy Electronic SpectrumAlthough the electronic band structure of graphene can be calculated by the tight-binding model, the salient features of low-
energy electron dynamics in graphene are better understood by modeling the electrons as relativistic Weyl fermions (within the k p◊ approximation), which satisfy the 2D Dirac equations [2, 6, 7]. - ◊— =
- ◊— = i v E
i v E
FF around K point)around K s y ys y y ( (* ′ ′ ′ point) (2.14)
where s s s= ( , )x y , s s s* ( , )= -x y , s sx y ii= ÈÎÍ ˘˚˙ = -ÈÎÍ ˘˚˙0 11 0 0 0, , y y y= ( , )A B , and y y y′ ′ ′= ( , )A B . Equation (2.14) can be solved to obtain the eigenvalues and eigenfunctions (envelope functions) as follows:
E v k k k
e
e
a
y a
= +
= Ê
Ë ÁÁ
ˆ
¯ ˜˜
( ) ( )
b = 1 (−1) refers to the K and K¢ valley, and qk y xk k= -tan ( / )1 is determined by the direction of the wave vector in the k-space. Therefore, for both the valleys, the rotation of k in the ( , )k kx y plane (surrounding K or K‘ point) by 2p will result in a phase change of p of the wave function (so-called Berry phase) [8, 9]. The Berry phase of p has important implications to electron transport properties, which will be discussed shortly. The eigenfunctions are two-component spinors; therefore, low-energy electrons in graphene possess a pseudospin (with a = +(−) 1 corresponding to the up (down) pseudospin) [10]. It is worth stressing that the pseudospin has nothing to do with the real electronic spin; the latter is an intrinsic property of electron with quantum mechanical origin, while the former is a mathematical convenience to deal with A and B atoms in graphene, which represent two intervened triangular lattices. The spinors are also the eigenfunctions of the helicity operator h p
p
= ◊12s | | . It is straightforward to show that hy ab yab ab= 12 . Taking n as the unit
vector in the momentum direction, one has n ◊ =s 1 for electrons and n ◊ = -s 1 for holes, for the K valley, and the opposite applies to the K’ valley [11]. The unique band structure near the K point is also accompanied by a unique energy-dependence of density of states (DOS). For a 2D system with dimension L L¥ , each electron state occupies an area of 2 2p / L in the k-space. Therefore, the low-energy DOS of graphene can readily be found as g g E v
s v | |2 2 2p F , where gs and gv are the spin and valley degeneracy, respectively [1, 7, 11]. The linear energy dependence of DOS holds up to E ª0 3 0. g , beyond which the DOS increases sharply due to trigonal warping of the band structure at higher energy [11]. Figure 2.1 compares the basic features of the electronic band structure of graphene with that of conventional 2D electron gas system [12]. In the latter case, the electron is confined in the z direction by electrostatic potentials, leading to the quantization of kzand thus discrete energy steps. As kx and ky still remain as continuous, associated with each energy step is a sub-band with a parabolic energy dispersion curve. Due to energy quantization, the DOS is now given by a sum of step functions, and between the neighboring steps the DOS is constant. In contrast, graphene is a “perfect” 2D system; therefore, there are no sub-bands emerged from the confinement in the z direction. Furthermore, the single band has a linear energy dispersion in the (kx, ky) plane instead of a parabolic shape as it is in the case of conventional 2D system. Note that quantum wells with a well thickness of one atomic layer have been realized in several material systems; but these systems are fundamentally different from graphene. In addition to single-layer quantum wells, ultrathin 2D sheets have also been realized in many other material systems [13]. However, these nanosheets are fundamentally different from graphene either in lattice structure or in the constituent elements. Although the linear energy dispersion or Dirac points are also found to exist in some bulk materials, in most cases, they do not play a dominant role in electrical transport; therefore, it is difficult to study electron behavior in these materials directly through electrical transport measurements.
