  Half-metals have fully spin polarized charge carriers at the Fermi surface.
Such polarization usually occurs due to strong electron--electron correlations.
Recently [Phys. Rev. Lett. {\bf{119}}, 107601 (2017)], we have demonstrated
theoretically that adding (or removing) electrons to systems with Fermi surface
nesting also stabilizes the half-metallic states even in the weak-coupling
regime. In the absence of doping, the ground state of the system is a spin or
charge density wave, formed by four nested bands. Each of these bands is
characterized by charge (electron/hole) and spin (up/down) labels. Only two of
these bands accumulate charge carriers introduced by doping, forming a
half-metallic two-valley Fermi surface. Analysis demonstrates that two types of
such half-metallicity can be stabilized. The first type corresponds to the full
spin polarization of the electrons and holes at the Fermi surface. The second
type, with antiparallel spins in electron-like and hole-like valleys, is
referred to as a "spin-valley half-metal" and corresponds to the complete
polarization with respect to the spin-valley operator. We analyze spin and
spin-valley currents and possible superconductivity in these systems. We show
that spin or spin-valley currents can flow in both half-metallic phases.
