tion on the central Si. The largest contribution to near-neighbor hopping in the Al-Si plane between p$_{z}$ and p$_{x}$ or p$_{z}$ is most likely due to this ring structure, as the p lobes are confined to the inside of a square of near-neighbor Al atoms, which are only edge shar- ing with the nearest Si atoms along b vectors. This is the likely reason that all the hoppings along b are approximately of the same magnitude. The disper- sion which creates the FSs along Γ-Z (seen in Fig. 4) is composed only of the p$_{x}$ and p$_{y}$ WFs. This is not caused by the large hoppings, but by smaller hoppings along b ∗ between p$_{x}$ and p$_{y}$ WFs. With- out these small hoppings, the band just above ε$_{F}$ is dispersionless along Γ-Z. A deformation potential D is the shift in an en- ergy band with respect to sublattice atomic displace- ment. One can freeze in phonon modes to calculate deformation potentials, which at the FS are directly connected to electron-phonon matrix elements. 30 Moving the Si atoms in the z direction by ± 1% of the experimental parameter, such that the tetra- hedra surrounding the Al atoms stretch or flatten (while remaining centered on Al), gives an average deformation potential of ∼ 0.8 eV/ ˚ A over five band positions near the Fermi level. The largest shift is for the ellipsoidal electron pockets, with D ∼ 1 . 2 eV/ ˚ A. These ellipsoids disappear when the Si atoms are displaced toward the Al plane (see Fig. 6). Flattening the Na bilayer, so as to remove the buckling of the Na atoms, requires a (very large) 12% change in the z component of the Na atoms. We chose such a large displacement because we do not expect a substantial deformation potential for Na movement. Even this large displacement does not alter very much the valence bands, as expected, and the hole FSs remain virtually unchanged. The con- duction bands at the Fermi level however shift appre- ciably, resulting in a modulation of the electron el- lipsoids along (1,1) near M. In addition, the acciden- tal four-band near-degeneracy that is 0.5 eV above ε$_{F}$ at M splits the two separate doubly-degenerate states, opening up a gap of ∼ 0.7 eV, which is equiv- alent to a deformation potential of ∼ 0.8 eV/ ˚ A (but 6 TABLE I: Selected hopping integrals in meV for the Si 3 p WFs along the vectors a = ( a, 0 , 0) (hopping within a Si layer), b = ( a/ 2 , a/ 2 , d ) (hopping across an Al layer), and b ∗ = ( a/ 2 , a/ 2 , c − d ) (hopping across a Na layer). d is the distance in the z direction between Si atoms above and below Al planes. FIG. 6: (Color online.) Comparison of band structures near ε F for different z Si values. FIG. 5: (Color online.) Isosurface of the WFs for (a) Si 3 p x and (b) Si 3 p z . Na atoms are large and yellow (light) colored, Si atoms are small and blue (dark) col- ored. The two colors of the isosurface represent different signs. (c) The tight-binding fatbands band structure de- scribed in the text for the WFs, compared to the DFT band structure (black lines). IV. RESPONSE TO CHANGES A. Electron-ion coupling p x p y p z a p x 761 60 p y -62 60 p z 60 60 -40 2a p x 128 27 b p x 361 300 360 p y 300 361 360 p z 360 360 360 b ∗ p x 12 5 50 p y 5 12 50 p z 50 50 185