Spectral momentum density of graphite from (e,2e) spectroscopy: Comparison with first-principles calculation

Abstract

We have measured the spectral momentum density ρ(E,q) of graphite by (e,2e) spectroscopy for momentum parallel and perpendicular to the crystal c axis. In the independent-electron approximation, ρ(E,q)=ΣG‖Uk(G)‖2 δ(q-k-G)δ(E-E(k)) where the one-electron wave function is Ψk(r)=eik⋅rΣGUk(G)eiG⋅r) and G is a reciprocal-lattice vector. The measurements covered a range of momentum parallel to the c axis equal to 0≤‖q‖≤1.84 Å-1 and a range of momentum perpendicular to the c axis equal to 0≤‖q‖≤2.35 Å-1. The energy range spanned the valence band of graphite from 4.4 eV above the Fermi energy to 27.6 eV below the Fermi energy. The momentum resolution was 0.47 and 0.73 Å-1 (full width at half maximum) for momentum parallel and perpendicular to the c axis, respectively. The energy resolution was 8.6 eV. The maximum coincidence rate was ∼0.02 counts/sec. The band structure E(k) and spectral density ‖Uk(G)‖2 have been calculated from first principles using a self-consistent density-functional theory in the local-density approximation with a mixed-basis pseudopotential technique. The agreement within experimental uncertainties between measurement and theory is excellent.