ent structure superconduct in the 5–8 K range. 13,14 Linear-response and frozen-mode calculations indi- cate electron-phonon coupling is the likely mech- anism; in particular, an ultra-soft phonon mode appears and is suggested to play a role in the superconductivity. 15–20 It is curious that in this compound, where divalent Ca (comparing it with NaAlSi) contributes one additional electron into the Si-Al sp bands, the preferred structure is that of AlB$_{2}$ ( i.e. , MgB$_{2}$) with sp 2 planar bonding 15,16,18 rather than the more sp $^{3}$-like bonding in NaAlSi. Electronic structure calculations show that CaAlSi has one electron in the conduction band above a bonding-antibonding band separation at the NaAlSi band-filling level, a situation which would not ap- pear to be particularly favorable for sp 2 bonding. In this paper we analyze first-principles elec- tronic structure calculations that reveal that NaAlSi is a naturally self-doped semimetal, with doping occurring—thus charge transfer occurring—between covalent bands within the Al-Si substructure, and two-dimensional (2D) free-electron-like bands within the Al layer. The resulting small Fermi surfaces (FSs) are unusual, complicated by the small but seemingly important interlayer coupling along the crystalline c -axis. 92 approximation 22 was applied for the exchange- correlation potential. The experimental lattice con- stants obtained by Kuroiwa et al. 7 ( a = 4 . 119 ˚ A and c = 7 . 632 ˚ A) and internal coordinates pub- lished by Westerhaus and Schuster 23 ( z$_{Na}$ = 0 . 622, z$_{Si}$ = 0 . 223) were used in our calculations. First-principles, local density approximation (LDA) calculations were carried out using the full- potential local-orbital (FPLO) scheme. 21 A k -point mesh of 20 × 20 × 12 was used, and the Perdew-Wang Above ε$_{F}$ lie non-bonding and antibonding bands, and the Na s bands. Among these there are a pair of distinctive bands, which can be identified most eas- ily by their Al s character in the top panel of Fig. 2. These bands are nearly free-electron-like with large dispersions, and cross many other bands with little mixing. Along Γ-M they are degenerate and easily identifiable in Fig. 2, as they disperse up through the Fermi level to nearly 10 eV at the M point. Along Γ- X, and similarly at the top of the zone Z-R, they are distinct: one again disperses upward rapidly, cutting through many other bands, also to nearly 10 eV at X; the other disperses much more weakly to X, with a bandwidth of about 2 eV. Their Al s character and nearly vanishing Si character identify these as free-electron states, in which electrons move down channels of Al atoms separately in x and y direc- tions. (Note that their lack of k$_{z}$ dispersion identi- fies them as planar bands.) There is some coupling to the states in a parallel channel of Al atoms, giv- ing rise to the 2 eV transverse dispersion. These bands lie 0.5 eV below ε$_{F}$ at Γ and contain elec- trons. Without interference with other bands near the Fermi level and supposing them to be isotropic in the plane (but see below), such FSs might include 3–4% of the area of the zone, which would equate to an intrinsic electron doping for two bands, both spins of around 0.12–0.16 carriers per unit cell, and the concentration of hole carriers would be equal. The anisotropy, discussed below, makes the actual The calculated band structure of NaAlSi is shown in Fig. 2. As expected, the Na ion gives up its elec- tron to the Si-Al-Si trilayer (see Fig. 1), which may have some ionic character, though it is not easy to quantify (the valence bands have much more Si char- acter than Al character, seemingly more than sug- gested by their number of valence electrons). There are several readily identifiable classes of bands. Two primarily Si 3 s bands, with a small amount of Al 3 s character, are centered 9 eV below the Fermi level ε$_{F}$ and have a width of 2.5 eV. Above them there is a six-band complex of Al-Si s - p bands (heavily Si) that are very nearly filled, the band maximum only slightly overlapping ε$_{F}$ . III. ELECTRONIC STRUCTURE A. Discussion of the band structure II. COMPUTATIONAL METHODS FIG. 1: (Color online.) Crystal structure 24 of NaAlSi. Four Si atoms tetrahedrally surround an Al atom, and these Al-Si networks sandwich the Na atoms. The unit cell is outlined in black. 2