We have measured the spectral momentum density p(E,q) of graphite by (e,2e) spectroscopy for momentum parallel and perpendicular to the crystal c axis. In the independent-electron approximation, p(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| ≤ 1.84Å-1 and a range of momentum perpendicular to the c axis equal to 0 ≤ |q| ≤ 2.35Å-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.
The spectral momentum density p(E,q) of graphite has been measured by (e,2e) spectroscopy for one direction of momentum in the basal plane (0 ≤ |q| ≤ 2.35Å-1) and for momentum parallel to the c axis (0 ≤ |q| ≤ 1.84Å-1). This is the first study of a crystal by this experimental technique. In the single-particle approximation the spectral density is equal to |Φk(q=k+G)|2 when E=E(k) (G is a reciprocal-lattice vector). The spectral density of graphite has been calculated from first principles using self-consistent, density-functional theory in the local-density approximation with a mixed-basis pseudo-potential technique. Excellent agreement is found within the experimental uncertainties between theory and the measurements reported here. The same theoretical approach has been used to calculate the band structure and charge density of graphite.1 The band structure has been measured by several groups using angle-resolved photo-electron spectroscopy (ARPES) for electron momentum in the basal plane of the crystal.2-11 The difference between theory and these measurements is less than or of order 1 eV. There is also good agreement between the calculated and measured charge densities.1 The density-functional technique has been used to calculate the ground-state properties of many solids and the agreement with measurements is often within a few percent. Thus, from our perspective, a mature theory of graphite exists and has been well confirmed by several independent experiments. We view the calculated spectral density as a benchmark for evaluating this new experimental technique, (e,2e) spectroscopy. The agreement between our measurements and theory is strong evidence that the analysis12,13 relating the (e,2e) coincidence rate to the spectral momentum density is correct.
The spectral momentum density provides very detailed information about the electronic structure of solids and can be measured only by (e,2e) spectroscopy. The band structure of crystalline solids can be measured by ARPES, but at this time information regarding the one-electron wave functions cannot be obtained from the intensity of the photoemission peaks. The integral of p(E,q) over momentum is the density of states, which, of course, can be measured by angle-integrated photoemission. The integral of p(E,q) over energy, the momentum density, can be obtained from Compton scattering and positron annihilation. The price one pays to measure the full spectral momentum density by (e,2e) spectroscopy is low count rate. The technique is a coincidence measurement, explained in more detail in the next section, and our maximum coincidence rate was approximately 1 event per minute. The experiment required four months of data taking. In order to obtain this great rate, the energy resolution was ΔE=8.6 eV (FWHM) for momentum parallel and perpendicular to the crystal c axis, respectively. The coincidence rate is proportional to ΔE Δq4. Possible ways to increase the data rate are being explored, but it is clear that in general most information regarding the electronic structure of crystalline solids can be obtained more expeditiously and with higher resolution by other techniques. Where (e,2e) spectroscopy provides fundamental insights is in the investigation of disordered solids. For example, in an initial experiment on amorphous carbon,14 two well-defined bands were observed which did not broaden (within the experimental resolution) even well out beyond the momentum of the crystalline Brillouin zone boundary. This counterintuitive result is being investigated thoroughly.15,16
The rest of this paper is organized as follows. The theory of (e,2e) scattering from solids is discussed in the first section. The details of calculating the spectral momentum density of graphite are also described in that section. The experimental results are given in the second section and analyzed in the third section. Finally, the results are summarized.
Gao, Chao; Ritter, A. L.; Dennison, JR; and Holzwarth, N. A. W., "Spectral Momentum Density from Graphite from (e,2e) Spectroscopy: Comparison with First Principles Calculations" (1988). Journal Articles. Paper 24.