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Appendix A
AGNR Optical Matrix Elements

To obtain the transition rules in Sec. 3.2, the gradient approximation Eq.3.12 and the wavefunction Eq. 3.22 are used. The momentum matrix elements Eq. 3.23 are obtained as

1 im N∑ ∑N pn,m (kx) = -----------0 [ (N + 1) ℏ p=1q=1 + (xB - xA )eikx(xBq -xAp)sin(nθp )sin (m θq)⟨Ap|H |Bq⟩e-iφm(kx) q p - (xA - xB )eikx(xAq -xBp)sin(nθp )sin (m θq)⟨B |H |A ⟩e+iφn(kx)] , q p p q
(A.1)

where ⟨Ap|H|Bq⟩ = ⟨Bp|H|Aq⟩ = t for p = q and p = q ± 1, otherwise the matrix elements are zero. Therefore, Eq. A.1 can be written as

N ---1----im0- ∑ pn,m (kx) = (N + 1) ℏ tacc sin (nθp)[ p=1 ( ) + e-iφm (kx) +e+ikxacc sin (m θp) - 1e- ikxacc∕2[sin (m θ(p - 1)) + sin (m θ(p + 1))] 2 ( ) - e+iφn(kx) - e- ikxacc sin (m θp) + 1e+ikxacc∕2[sin (mθ (p - 1)) + sin (m θ (p + 1))] ] . 2 [ N ] ---1----im0- ∑ = (N + 1) ℏ tacc sin (nθp) sin (m θp) × p=1 -iφm (kx)( +ikxacc -ikxacc∕2 ) +iφn(kx)( - ikxacc +ikxacc∕2 ) ( + e e - e cos(m θ) + e e - e cos(m θ) ) .
(A.2)

Here the relation sin (x) + sin (y) = 2 sin ((x + y)∕2) cos ((x - y) ∕2) is employed. Using Eq. 3.21, Eq. A.2 can be written as

[ N ] ---1----im0- ∑ pn,m (kx) = (N + 1) ℏ tacc sin (nθp) sin (m θp) × p=1 1 ( ) ( + -------- 1 - 2 cos2(m θ) + 2e+i3kxacc∕2cos (m θ) - e- i3kxacc∕2cos(m θ) |fm (kx)| 1 ( ) + -------- 1 - 2 cos(m θ)sin(n θ) + 2e- i3kxacc∕2cos(n θ) - e+i3kxacc∕2cos (m θ) ) . ◟-|fn(kx)|---------------------------◝◜-------------------------------------◞ Fn,m (kx ) [ ] 1 im0 ∑N = ------------tacc sin (nθp) sin (m θp) Fn,m (kx) . (N + 1) ℏ p=1
(A.3)

The summation over the sine functions in Eq. A.3 determines the transition rules. Using some trigonometric identities one can write this summation as

∑N 1 (n - m )π (n - m )πN ( (n - m )π ) -1 sin (nθp)sin(m θp ) = -[ + cos----------sin ------------ sin ---------- p=1 2 2 2(N + 1) 2(N + 1) ( ) (n + m )π (n + m )πN (n + m )π -1 - cos ----------sin ------------ sin ---------- ] . 2 2(N + 1) 2(N + 1)
(A.4)

If n ± m = 2k + 1, where k is a non-zero integer, both terms in the bracket of Eq. A.4 will be zero. In the case of n±m = 2k, both terms in the bracket will be equal to -1, therefore, the summation will be again zero. However, if n = m, the fist term in will be equal to N and the second term will be equal to -1. Therefore, only transitions between valence and conduction subbands with the same band-index are allowed

{ N∑ N--+-1 , n = m sin(n θp)sin(m θp) = 2 p=1 0 , n ⁄= m
(A.5)

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