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4.2 Wave Functions

In the following, the relation between θ and kx is obtained. Setting n = N + 1 in Eq. 4.1 gives

( √ -- ) E ϕB = tϕA + 2t cos --3k a ϕA . 0 N 2 x cc N+1
(4.19)

On the other hand the wave functions for n = N and n = N + 1 are given by Eq. 4.10

ϕA = sin(N-θ)-ϕA, N sin(θ) 1 sin((N + 1)θ) ϕAN+1 = -------------ϕA1. sin(θ)
(4.20)

By imposing a hardwall boundary condition, ϕ0B = 0, Eq. 4.19 can be rewritten as

( √ -- ) tsin(N-θ)ϕA + 2tcos --3-k a sin-((N--+-1)θ)ϕA = 0, sin(θ) 1 2 x cc sin(θ) 1
(4.21)

and we obtain the quantization condition,

( ) sin (N θ) √3-- ---------------= - 2cos ---kxacc . sin ((N + 1) θ) 2
(4.22)

From Eq. 4.22, θ can be extracted and used in the analytical derivation of the wave functions and energy dispersion relation of ZGNRs. Figure 4.2 shows the variation of θ as a function of kx for different subbands.

As can be seen, a weak dependency exists between θ and kx. However, we assume a constant θ with respect to kx, so the right hand side of Eq. 4.22 becomes a constant. At this point, two approaches exist to simplify Eq. 4.22 based on choosing kx. In general, using all kx in the range of [0,π∕√ -- 3acc] except kx = 2π∕3√3--acc results in non analytical solution and curve fitting is required to extract θ. This approach is first discussed. For kx = 0, Eq. 4.22 can be written as

---sin-(N-θ)--- sin ((N + 1)θ) = - 2.
(4.23)

Using curve fitting one finds

{ Q Q = p1υ + p2 θ = -------= P = p′υ + p′, N + P 1 2
(4.24)

where υ is related to the subband number q (q = 2υ) and N is the index of ZGNR. p1, p2, p1′ and p 2′ are fitting factors. Performing a fit for the range of N ∈ [6, 100],

3.31 υ - 0.5208 θ = ----------------------. N + 0.07003 υ + 0.5216
(4.25)

Figure 4.3 compares θ obtained from Eq. 4.25 with exact numerical results.

PIC


Figure 4.3: The wavenumber θ as a function of the ribbon index, N, for different subband numbers, q. Dashed lines show the first approximation (Eq. 4.25) and circles show the second approximation (Eq. 4.28).

Another approach for approximating θ is selecting kx = 2π∕3√ -- 3acc which makes the right side of Eq. 4.22 equal to -1

sin(N θ) + sin ((N + 1)θ) = 0, 2sin((2N + 1)θ ∕2)cos(θ∕2 ) = 0,
(4.26)

and has the following solution

υ π θ = --------. N + 1 ∕2
(4.27)

By representing υ in terms of subband index q, θ is obtained as a function of q and N,

qπ θ = --------. 2N + 1
(4.28)

Figure 4.3 compares the discussed approximations with exact numerical results. The value of θ for Eq. 4.28 shows a good agreement with that obtained from numerical calculations. The approximation is valid for a wide range of ZGNR indices. Using the analytical expression for θ, analytical wave functions and energy dispersion are evaluated from Eq. 4.12 and Eq. 4.18. The results for a 6-ZGNR and a 19-ZGNR are shown in Fig. 4.4 and Fig. 4.5, respectively.

PIC


Figure 4.4: The electronic band structure of 6-ZGNR ((a), (b), and (c)) and 19-ZGNR ((d), (e), and (f)). The analytical model (black dashed lines) is compared against the numerical results (red solid lines).

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