PHYS3531 Physics Project
Multi-valley monolayer materials
Fok Hong Ting
Under the supervision of
Dr. Wang Yao
Project outline 5
Chapter I Tight binding model 6
The secular equation 8
Chapter II Graphene 11
π energy band of graphene 15
σ energy bands of graphene 17
Chapter III Silicene 19
Tight Binding Hamiltonian of silicene without SOC 20
Constructing orthogonal basis 21
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In this project, we try to investigate the band properties of graphene and silicene with tight binding method. The principal of tight binding method is discussed in Chapter I.
In Chapter II, we will formulate a tight binding Hamiltonian for graphene and calculate its energy band structure.
In Chapter III, we will include SOC effect in the calculation of silicene energy bands and look for an analytical solution.
Finally, in Chapter IV, we focus on investigating the edge states of both graphene and silicene.
Chapter I Tight Binding Method
In this section we explain the tight binding method. In our model, we assume the material is a perfect lattice. It has no defects and can be produced by choosing a suitable unit cell.
Because of the translational symmetry of the unit cells in the direction of the lattice vectors, ai (i = 1,2,3,) , any wave function of the lattice should satisfy Bloch's theorem:
where Tai is a translational operation along the lattice vector ai, and k is the wave vector.
There are many possible functional forms of Ψ which satisfy this equation and the most commonly used form for Ψ is a linear combination of plane waves.
Plane waves are used because of the following reasons:
1. The integration of the plane wave wavefunction is easy and can be done analytically,
2. The numerical accuracy only depends on the number of the plane waves used.
However, the plane wave method also has limitations:
1. The scale of the computation is large,
2. It is difficult to relate the plane wave wavefunction to the atomic orbitals in the solid.
Another functional form which satisfies the above equation is based on the jth atomic orbital in a unit cell. This is a tight binding, Bloch function:
Where R is the position of the atom and ρj denotes the atomic wavefunction in state j.
The number of atomic wavefunctions in the unit cell is denoted by n, and we have n Bloch functions in the solid for a given k. The ρj in the N unit cells are weighted by the phase factor eik∙R and are then summed over the lattice vectors of the whole crystal.
The advantages of using orbitals in Bloch functions are as follows:
1. The number of basis functions can be small compared with the number of plane waves.
2. The formula for many physical properties can be easily derived.
The eigenfunctions in the solid Ψj are expressed by a linear combination of Bloch functions Фj' as follows:
Ψjk,r=j'=1nCjj'(k)Фj' (k, r)
where Cjj' are coefficients to be determined.
Then, the j-th eigenvalue as a function of k is given by:
Here Hjj'(k) is the transfer integral matrices and Sjj'(k) is the overlap integral matrices, respectively.
The secular equation
For a given k value, Cij is optimized so that Ej is minimized.