Optical Matrix Elements

B.3 Optical Matrix Elements

Using Eq. 4.12 and Eq. 3.13 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

P ′ = (x ′ - x ) im0-⟨θ|H |θ′⟩ θ,θ θ θ ℏ

(B.19)

N N im0-∑ ∑ ik(xBm- xAn) ′ B A Pθ,θ′ = ℏ Ω [ e sin (nθ)sin(m θ)⟨An |H |Bm ⟩(x m - xn) n=1 m=1 - eik(xAm- xBn )sin(n θ′) sin(m θ)⟨Bn |H |Am ⟩(xAm - xBn)].

(B.20)

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

( ) ( √ -- ) N ( ) im0-- i-3acct- ∑ i√3kxa∕2 - i√3kxa∕2 ′ Pθ,θ′ = ℏΩ 2 [ e - e sin(nθ) sin((N - n + 1)θ ) ( √ - √ -n=1 ) - e-i 3kxa∕2 - ei 3kxa∕2 sin(nθ ′)sin((N - n + 1)θ)],

(B.21)

after some algebra and replacing Ω from Eq. 4.16, the optical matrix elements are

√ -- ( √ -- ) - 2 3m0acct 3 N∑ ′ Pθ,θ′ = -------------sin ---kacc [sin(nθ) sin((N - n + 1)θ )- ℏ (2N + 1) 2 n=1 sin (n θ′) sin((N - n + 1)θ)].

(B.22)

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