Fig. 4: Effective 2nd neighbor hopping \(\tilde{t}\) via “stepping stone” sites.
From: Tunning the tilt of the Dirac cone by atomic manipulations in 8Pmmn borophene
a Formation of bonding \(\left\vert {\phi }_{1}\right\rangle\) and anti-bonding \(\left\vert {\phi }_{2}\right\rangle\) orbitals between the ridge atoms 5 and 6. b Effective hopping \(\tilde{t}\) between second neighbor inner sites 3 and \({3}^{{\prime} }\) can be formed by virtual hopping via the anti-bonding orbital \(\left\vert {\phi }_{3}\right\rangle\) of two ridge atoms. 2Δ is difference of atomic on-site energy. E0 is atomic on-site energy of \(\left\vert 3\right\rangle\) and \(\left\vert {3}^{{\prime} }\right\rangle\) and it is set to zero. c Energy diagram after the formation of molecular states which connect B3 to one of its successive third nearest neighbors through antibonding orbitals.
