B.3 Optical Matrix Elements

Using Eq. 4.12 and Eq. 3.13 the matrix elements pθ,θ(kx) ≡⟨+,θ,kx|px|-,k x for an interband transition from a valence band state |-,θ,kxto a conduction band state |+,k xare 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 An|H|Bm= 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)