Appendix B  ZGNR Optical Matrix Elements

B.1  Bloch Wave Functions Prefactors

To obtain C_A and C_B in Eq. 18, one can substitute Eq. 6 and Eq. 19 into the Shrödinger equation H|ψ⟩=E|ψ⟩. Considering an A-type carbon atom at some atomic site n, and its three nearest neighbors, the Hamiltonian can be written as:

Using Eq. 1 along with the wave functions obtained in Eq. 12, one obtains:

Therefore, the relation between C_A and C_B can be written as:

By employing the relation sin(x)sin(y)=(1/2)[cos(x−y)−cos(x+y)] and using Eq. 22,

Analogously, for the N−n+1th B-type carbon atom one can obtain the following relation:

which gives

From Eq. 4 and Eq. 6, one can find that C_A=± C_B.
Also, the dispersion relation can be found by multiplying Eq. 3 by Eq. 5,

With the help of trigonometric identities and Eq. 22, this expression can be reformatted as

B.2  Transverse Wave Functions Amplitude

To solve the recursive formula,

one can consider the ansatz φ_n=tⁿ and follow similar equation,

This equation is the generating polynomial of the recursive formula 9.
The roots of 10 are

The general solution of the difference equation is

since t₁ is a root of the equation, the other root t₂ can be written as: t₂=t₁⁻¹.
By substituting those two roots in 12 one obtains

Imposing the initial condition φ₀=0 results in

and from the 13,

We obtain

By substituting 11 and 16 in 15, one obtains

17 can be rewritten as

B.3  Optical Matrix Elements

Using Eq. 12 and Eq. 13 the matrix elements p_θ,θ^′(k_x) ≡ ⟨ +, θ, k_x| p_x | −, θ^′, k_x⟩ for an interband transition from a valence band state | −, θ, k_x⟩ to a conduction band state | +, θ^′, k_x⟩ are obtained as

Considering only the nearest neighbors, each atom with some index n has two neighbors with index N−n+1 and one neighbor with index N−n, see Fig. 4.1. Therefore, the index m has only three values with ⟨ A_n | H | B_m ⟩ = t. So we have

after some algebra and replacing Ω from Eq. 16, the optical matrix elements are