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Tight-Binding Model for Electrons on the Honeycomb Lattice 25 of the system because it simply leads to a redefinition of the weights ak and bk which aquire a different relative phase [18]. With the help of these wavefunctions, we may now search the solutions of the Schr ̈odinger equation Hψk = ǫkψk. Here, we have chosen an arbitrary representation, which is not necessarily that in real space.3 Multiplication of the Schr ̈odinger equation by ψk∗ from the left yields the equation ψk∗Hψk = ǫkψk∗ψk, which may be rewritten in matrix form with the help of Eqs. (2.5) and (2.6) (a∗k,b∗k)Hk ak = ǫk (a∗k,b∗k)Sk ak . bk bk Here, the Hamiltonian matrix is defined as (2.7) (2.8) (2.9) accounts for the non-orthogonality of the trial wavefunctions. The eigen- values ǫk of the Schr ̈odinger equation are the energy dispersions or energy bands, and they may be obtained from the secular equation d e t H k − ǫ λk S k = 0 , ( 2 . 1 0 ) which needs to be satisfied for a non-zero solution of the wavefunctions, i.e. for ak ̸= 0 and bk ̸= 0. The label λ denotes the energy bands, and it is clear that there are as many energy bands as solutions of the secular equation (2.10), i.e. two bands for the case of two atoms per unit cell. Notice that the generalisation to n atoms per unit cell is straight-forwards – in this case, the wavefunction (2.5) needs to be written as n ψk = X a(j)ψ(j), kk j=1 3The wavefunction ψk(r) is, thus, the real space representation of the Hilbert vector ψk. and the overlap matrix ψ(A)∗Hψ(A) ψ(A)∗Hψ(B) ! † Hk≡ k k k k =H, ψ(B)∗Hψ(A) ψ(B)∗Hψ(B) k kkkk ψ(A)∗ψ(A) ψ(A)∗ψ(B) ! † S≡kkkk=S k ψ(B)∗ψ(A) ψ(B)∗ψ(B) k kkkkPDF Image | Physical Properties of Graphene
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