Physical Review B
American Physical Society
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.
Chao Gao, A. L. Ritter, JR Dennison and N. A. W. Holzwarth, "Spectral Momentum Density from Graphite from (e,2e)Spectroscopy: Comparison with First Principles Calculations," Phys. Rev. B, 37, 3914-3923 (1988).