AlN and GaN Nanostructures: Analytical Estimations of the Characteristics of the Electronic Spectrum

Abstract

Analytical expressions for the band gaps and characteristic velocities and effective masses of carriers are obtained for infinite flat sheets, free and decorated nanoribbons with zigzag edges, and chains of aluminum and gallium nitrides. The values obtained are compared with the characteristics for silicon carbide and carbon nanostructures calculated within the same models. The role of the substrate is also briefly discussed.

APPENDIX

In accordance with the Harrison theory [17, 18], at a = 1.42 Å, we obtain t = 2.38 eV for the π bond of p orbitals. For an infinite graphene (Gr) sheet, the Fermi velocity (characteristic velocity) is c = $3at{\text{/}}2\hbar$.

For a carbon ZNR, within the same double-chain nanoribbon model but at A = B, the dispersion relation can be written as E(k) = 0 and E(k) = $\pm t(1 \pm \sqrt {1 + 4\Phi } ){\text{/}}2$, where the energy of the p state of carbon atom is considered to be the reference point [22]. Thus, for low-energy branches at the Brillouin zone boundary, we have E_g = 0. At q = π/2 – k → 0, for a positive low-energy spectral branch, we obtain E(q) ≈ tΦ(q); hence, m* = ${{\hbar }^{2}}{\text{/}}8{{m}_{0}}{{a}^{2}}t$. In the case of DZNR, the situation is more complex: low-energy branches of E(k) have extrema at k₀a ≈ 0.84, so that the band gap E_g ≈ 4t/3 arises in the spectrum (see Fig. 2a in [22] for the case of ε_a = 0, τ = 1). On the assumption that E(k) ≈ ${{\hbar }^{2}}(k$$- \;{{k}_{0}}{{)}^{2}}{\text{/}}2{{m}_{0}}m{\text{*}}{{a}^{2}}$ near the minimum, we obtain m* ≈ 0.05.

In the case of one-dimensional chain of carbon atoms (carbyne), one should distinguish metallic carbyne (or cumulene (c)) having double bonds (…=C=C=…) and semiconductor carbyne (or polyyne) with alternating single and triple bonds (…≡D–C≡C–C≡…) [21–23]. In the absence of decoration, we have E_g = 0 and c = at/$\hbar$ for cumulene, where a = 1.28 Å and t = 2.92 eV. For polyyne, we obtain E_g = 2Δt and m* = ${{\hbar }^{2}}{{\bar {\tau }}^{2}}{\text{/}}{{m}_{0}}{{a}^{2}}\Delta t$, where $\bar {\tau }$ = Δt/t, Δt = t₁ – t₂, and t₁ = 3.00 eV and t₂ = 2.84 eV are the transition energies for the single and triple bonds, respectively [21].

Decoration of cumulene was considered in [22]. If cumulene is decorated by carbon atoms, we consider (as previously) the position of the p level of carbon atom as the energy reference point and obtain the dispersion in the form E(k) = 0 and E_±(k) = $\pm t\sqrt {\Phi + 2}$. Therefore, the band gap at the Brillouin zone boundary is E_g = $2t\sqrt 2$ and the effective mass is m* = ${{\hbar }^{2}}{\text{/}}2\sqrt 2 {{m}_{0}}{{a}^{2}}t$.

Here, we present additionally the results of [23], where free and decorated zigzag graphene edges were considered within the chain model. In the absence of decoration, E(k) = $\pm (t{\text{/}}\sqrt 2 )(1$ + Φ ± $\sqrt {1 + 2\Phi + {{\Phi }^{2}}} {{)}^{{1/2}}}$, so that E_g = 0. At k → π/2a, low-energy bands E(k) = $\pm (t{\text{/}}\sqrt 2 )(1$ + Φ – $\sqrt {1 + 2\Phi + {{\Phi }^{2}}} {{)}^{{1/2}}}$ are flattened, and the concept of effective mass cannot be introduced (as in the case of DZNR).

In the case of decoration with carbon atoms, the electron dispersion relation takes the form

$$\begin{gathered} E(k) = \pm (1 + 2{{\cos }^{2}}(ka) \\ \pm \;\cos (ka)\sqrt {1 + {{{\cos }}^{2}}(ka)} {{)}^{{1/2}}}, \\ \end{gathered} $$

so that we have E_g = 2t = 4.76 eV and c = $at{\text{/}}\hbar$ = 0.52 × 10⁶ m/s at the Brillouin zone boundary.