Modeling the momentum distributions of annihilating electron-positron pairs in solids

Measuring the Doppler broadening of the positron annihilation radiation or the angular correlation between the two annihilation gamma quanta reflects the momentum distribution of electrons seen by positrons in the material. Vacancy-type defects in solids localize positrons and the measured spectra are sensitive to the detailed chemical and geometric environments of the defects. However, the measured information is indirect and when using it in defect identification comparisons with theoretically predicted spectra is indispensable. In this article we present a computational scheme for calculating momentum distributions of electron-positron pairs annihilating in solids. Valence electron states and their interaction with ion cores are described using the all-electron projector augmented-wave method, and atomic orbitals are used to describe the core states. We apply our numerical scheme to selected systems and compare three different enhancement (electron-positron correlation) schemes previously used in the calculation of momentum distributions of annihilating electron-positron pairs within the density-functional theory. We show that the use of a state-dependent enhancement scheme leads to better results than a position-dependent enhancement factor in the case of ratios of Doppler spectra between different systems. Further, we demonstrate the applicability of our scheme for studying vacancy-type defects in metals and semiconductors. Especially we study the effect of forces due to a positron localized at a vacancy-type defect on the ionic relaxations.

Figure 1

(Color online) (a) Bulk Cu/bulk Al and (b) bulk Ag/bulk Al ratio curves of momentum distributions of annihilating electron-positron pairs calculated using the state-dependent scheme. The experimental data (Ref. 35) is shown with circles. The theoretical curves are convoluted with a Gaussian function with a FWHM of $4.3\times {10}^{-3}{m}_{0}c$.

Figure 2

(Color online) Bulk Cu/bulk Al ratio curve of momentum distributions of annihilating electron-positron pairs. The comparison between the state-dependent and the state-independent LDA schemes is shown. The experimental data is from Ref. 35. The theoretical curves are convoluted with a Gaussian function with a FWHM of $4.3\times {10}^{-3}{m}_{0}c$.

Figure 3

(Color online) Bulk Ag/bulk Al ratio curve of momentum distributions of annihilating electron-positron pairs. The comparison between the state-dependent and the state-independent LDA schemes is shown. The experimental data is from Ref. 35. The theoretical curves are convoluted with a Gaussian function with a FWHM of $4.3\times {10}^{-3}{m}_{0}c$.

Figure 4

(Color online) Ratio curves of momentum distributions of annihilating electron-positron pairs calculated using different approximations. The experimental data is from Ref. 36. The theoretical curves are convoluted with a Gaussian function with a FWHM of $4.7\times {10}^{-3}{m}_{0}c$.

Figure 5

(Color online) Bulk Cu/bulk Fe ratio curve of momentum distributions of annihilating electron-positron pairs calculated using different approximations. The experimental data is from Ref. 36. The theoretical curves are convoluted with a Gaussian function with a FWHM of $4.7\times {10}^{-3}{m}_{0}c$.

Figure 6

(Color online) Bulk Cu/bulk Al ratio curve of momentum distributions of annihilating electron-positron pairs. The comparison between the BN-LDA and the GGA for annihilation rates within the state-dependent scheme is shown. The experimental data is from Ref. 35. The theoretical curves are convoluted with a Gaussian function with a FWHM of $4.3\times {10}^{-3}{m}_{0}c$.

Figure 7

(Color online) Bulk Ag/bulk Al ratio curve of momentum distributions of annihilating electron-positron pairs. The comparison between the BN-LDA and the GGA for annihilation rates within the state-dependent scheme is shown. The experimental data is from Ref. 35. The theoretical curves are convoluted with a Gaussian function with a FWHM of $4.3\times {10}^{-3}{m}_{0}c$.

Figure 8

(Color online) Different energy components as functions of vacancy relaxation in % of the nearest-neighbor distance (relative to the relaxed geometry, only nearest neighbor ions are moved) for ${\mathrm{V}}_{\mathrm{Si}}$ in Si and ${\mathrm{V}}_{\mathrm{Al}}$ in Al. The solid lines denote the total energy, the dashed lines the energy of the electron-ion system, and the dash-dotted line the positron energy eigenvalue. The figures show results in which the forces are calculated using a positron state solved in the PAW potential (엯) and in the ATSUP potential (◇).

Figure 9

(Color online) Theoretical ratio curves for the Ga vacancy in GaAs. The experimental data, measured from electron-irradiated GaAs, is from Ref. 40. The theoretical curves are convoluted with a Gaussian function with a FWHM of $5.5\times {10}^{-3}{m}_{0}c$.

Figure 10

Theoretical ratio curve for the neutral Si vacancy in Si. The the data is convoluted with a Gaussian function with a FWHM of $3.7\times {10}^{-3}{m}_{0}c$