Intrinsic Spin-Orbit Coupling in Zigzag and Armchair Graphene Nanoribbons

Table 1

Two-center matrix elements for hopping between 𝑠 and 𝑝 orbital along a direction specified by the unit vector (𝑛_𝑥,𝑛_𝑦,𝑛_𝑧), taken from [4]. 𝑠=−8.868 eV, 𝑝=0, (𝑠𝑠𝜎)=−6.769 eV, (𝑠𝑝𝜎)=5.580 eV, (𝑝𝑝𝜎)=5.037 eV, and (𝑝𝑝𝜋)=−3.033 eV.

𝑡𝑠	𝑠	𝑡𝑝𝑥,𝑝𝑥	𝑛^2𝑥(𝑝𝑝𝜎)+(1−𝑛^2𝑥)(𝑝𝑝𝜋)
𝑡𝑝	𝑝	𝑡𝑝𝑦,𝑝𝑦	𝑛^2𝑦(𝑝𝑝𝜎)+(1−𝑛^2𝑦)(𝑝𝑝𝜋)
𝑡𝑠,𝑠	(𝑠𝑠𝜎)	𝑡𝑝𝑧,𝑝𝑧	𝑛^2𝑧(𝑝𝑝𝜎)+(1−𝑛^2𝑧)(𝑝𝑝𝜋)
𝑡𝑠,𝑝𝑥	𝑛𝑥(𝑠𝑝𝜎)	𝑡𝑝𝑥,𝑝𝑦	𝑛𝑥𝑛𝑦(𝑝𝑝𝜎)−𝑛𝑥𝑛𝑦(𝑝𝑝𝜋)
𝑡𝑠,𝑝𝑦	𝑛𝑦(𝑠𝑝𝜎)	𝑡𝑝𝑥,𝑝𝑧	𝑛𝑥𝑛𝑧(𝑝𝑝𝜎)−𝑛𝑥𝑛𝑧(𝑝𝑝𝜋)
𝑡𝑠,𝑝𝑧	𝑛𝑧(𝑠𝑝𝜎)	𝑡𝑝𝑦,𝑝𝑧	𝑛𝑦𝑛𝑧(𝑝𝑝𝜎)−𝑛𝑦𝑛𝑧(𝑝𝑝𝜋)