B. Density of states C. Unusual Fermi surfaces Fig. 3 shows the total, partial, and projected den- sities of states (DOS) of NaAlSi. The Na contri- bution near the Fermi level is negligible and thus not shown. Except for a strong dip (“pseudogap”) near the Fermi level and a less severe dip in the − 3 to − 4 eV range, the DOS hovers around 3 states/eV throughout both valence and conduction bands. Within the pseudogap encompassing the Fermi energy, there is an anomalous sharp and nar- row peak with ε$_{F}$ lying on its upper slope, as noted previously by Kuroiwa et al. 7 The value of N ( ε$_{F}$ ) is 1.1 states/eV. We discuss below the FSs of both hole (Si) and electron (Al) character. It seems clear that the transport properties and low-energy properties (which have not yet been re- ported), and in particular the superconductivity of NaAlSi, are intimately associated with this sharp and narrow peak in the DOS, which includes the Fermi level. The projected DOS shows the flat bands that give rise to this peak are very strongly Si- derived. There is Al 3 s character that turns on just below ε$_{F}$ , but it is relatively small compared to the Si character at ε$_{F}$ , and its magnitude remains low and nearly constant through the peak. There is Al 3 p character of the same magnitude in the vicinity of the Fermi level. The top edge of the peak coin- cides with the flat band along Γ-X at 0.13 eV. The width of the peak, about 0.35 eV, must be due to dispersion and anticrossings that are mostly not visi- ble along symmetry directions and arise from mixing away from symmetry lines of the valence and con- duction bands. Nonetheless, the slope of the DOS at ε$_{F}$ is rather steep, and this may give rise to high thermopower for the material. The standard low-temperature limit of thermopower (the Seebeck coefficient tensor) S ( T ) in semiclassical Bloch-Boltzmann theory is The conductivity tensor σ ( E ) can be written in terms of the average velocity ( ⃗v ( E )) product, DOS, and scattering time τ ( E ) over the constant energy ( E ) surface: The thermopower thus picks up contributions from the energy variation of three quantities: the dyadic product 〈 ⃗v⃗v 〉 , N ( E ), and τ ( E ). Often the energy dependence of τ is neglected, out of lack of detailed knowledge, though it also can be argued to follow roughly 1 /τ ( E ) ∝ N ( E ) for elastic scattering. The Fig. 4 depicts the calculated FSs. In spite of the generally 2D band structure, the small k$_{z}$ dispersion of bands at ε$_{F}$ make some of the FSs surprisingly three-dimensional. Electron pockets and hole pock- ets coexist in the Brillouin zone, with electron and hole concentrations necessarily being equal. Hole surfaces. Four hole “fan-blade” surfaces lie oriented in the xz - and yz -planes. At the center, extending from Γ half way to Z, lies a long and nar- row surface with square cross section. The top view allows the origin of these surfaces to be understood. The cross sections in the xy -plane are of two ellipses that are very anisotropic (in the xy -plane) and at right angels to each other. Each corresponds to a dispersion that is weak in one direction (the long major axis) and strong in the other (minor axis). These bands would intersect, but in fact are inter- sected by the electron band that cuts a squarish hole (rotated by 45 ◦ ), within which the elongated hole surface inside re-emerges. where M $^{$_{−}$1}$( E ) is (a diagonal element of) the in- verse mass tensor (second derivative of ε$_{k}$ ) averaged over the constant energy surface. This form makes it clear that the expressions should, for any quan- titative estimate, be generalized to two-band form, since the valence and conduction bands have differ- ing signs of their effective masses, and the scattering time—and its energy variation—is likely to be very different for Si-derived covalent valence bands and Al-derived free-electron conduction bands. Measure- ment of the thermopower, and a quantitative theo- retical treatment, would be very useful in extend- ing the understanding of the transport properties of NaAlSi. energy dependence of v $^{2}$( E ) also counteracts the en- ergy dependence of N ( E ). Nevertheless it is ob- served that materials with large slope in N ( E ) fre- quently have large thermopower. For NaAlSi we cal- culate d ln N ( E ) /dE |$_{ε}$$_{F}$ = − 4 . 0 eV $^{$_{−}$1}$. This value can be compared with other materials that have fine structure near the Fermi level: TiBe$_{2}$, where d ln N ( E ) /dE |$_{ε}$$_{F}$ = 10–12 eV $_{−}$1 and N ( ε$_{F}$ ) also is much larger; 25 and MgCNi$_{3}$ with its very impres- sive peak very near ε$_{F}$ , for which d ln N ( E ) /dE |$_{ε}$$_{F}$ ∼ -15-20/eV $^{$_{−}$1}$. 26 The energy derivative of diagonal elements of σ that occurs in Eq. 1 can also be expressed as 1 σ dσ dE = d ln τ ( E ) dE + 1 2 π 2 M $^{$_{−}$1}$( E ) v $^{2}$( E ) , (3) S ( T ) → − π $^{2}$k$_{B}$ 3 e d ln σ ( E ) dE ∣ ∣ ε$_{F}$ k$_{B}$T. (1) σ ( E ) = 4 πe $^{2}$〈 ⃗v ( E ) ⃗v ( E ) 〉 N ( E ) τ ( E ) . (2) 4