Comparison of calculated photoionization cross-sections of diamond and silicon in the approximations of plane wave and orthogonalized plane wave

Abstract

In the approximation of the orthogonalized plane wave the shape of X-ray photoelectron spectra of the valence band in diamond and silicon has been calculated. It is shown that consideration of orthogonality terms influences greatly the value of the ratio between photoionization cross-sections of s- and p-electrons. X-ray emission and photoelectron spectroscopy allow us to define the density of states for silicon valence electrons. X-ray photoelectron spectroscopy is at present widely used for the study of the electronic structure of solids. The photoelectron spectra of crystals clearly reveal all the variations in the density of states. However the curves for the photoelectron energy distribution may be different from the calculated density of states for valence electrons. This indicates the importance of the transition probability for the formation of a photoelectron spectrum. In refs. 1 and 2 the X-ray photoelectron spectra obtained with high resolution for diamond and silicon crystals are compared with the density of states and the conclusion made that the s-states lie higher than the p-states. Densities of states and X-ray photoelectron spectra for diamond and silicon have been calculated 3 . The wave functions of the valence electrons were found by the tight-binding method 4 while plane waves were used as wave functions for the excited electrons. From a theoretical point of view it is more reasonable to use orthogonalized plane waves to describe the electron states in the conduction band. Both the core and valence states are to be orthogonalized. The present work reports calculation of the shape expected for X-ray photo-electron spectra of diamond and silicon valence electrons in the approximation of orthogonalized plane wave. Investigation was also made of the influence caused by the orthogonality terms on the ratio of photoionization cross-sections of s- and p-electrons. The electronic structure of diamond and silicon was calculated also by the tight-binding method with the use of the parameters from ref. 3. X-ray photo-electron spectra of valence electrons were calculated over 5230 points in 1/48 part of the Brillouin zone. For simplicity it was assumed that the polarization vector of the electromagnetic wave A is directed along the crystal Z-axis. As was done in ref. 3, the X-ray photoemission intensity was averaged over angular variables. As an example we shall give the formula used for calculation of the X-ray photoelectron spectrum for diamond I (ω, E ) ∼ ∝ E σ n,k [( 1 3 ET 2s 2 + 1 3 T 2p 2 U 2p−2s 2 - 2 3 ∝ ET 2s T 2p U 2p−2s )| C s n ( K | 2 + 1 5 ET 2p 2 (| C x n ( K | 2 + 3 5 ET 2p 2 + 1 3 T 1s U 1s-2p + T 2s U 2s-2p ) 2 + 2 3 √ ET 2p ( T 1s U 1s-2p + T 2s U 2s-2p ) √ C z n ( k )| 2 ] where T nl ( E ) = fx O ∞ r 2 R nl (tr) jl (∝ Er dr and U ij = ( E i - E j fx 8 r 3 R i R j ( r ) R j ( r )dr R i ( r ) is the radial part of the atomic wave function, C i n ( k ), is found from the Schrödinger equation by the tight-binding method. A similar formula is valid for silicon but the number of integrals T nl and U ij will be larger owing to the fact that there are more electron states in the silicon atomic core. In eqn. (1) the terms | C x n | 2 ,| C y n | 2 and | C z n | 2 have different factors because the A vector is directed along the crystal Z-axis. These factors will be the same when the A vector is directed along (111). Therefore the contribution of p-electrons to the photoelectron spectrum will be proportional to the partial density of p-states. The formula (1) is simplified in the approximation of a plane wave I (ω, E ∼ E 3 2 Σ n,k [ T s 2 1 3 | C s n ( k )| 2 + T p 2 ( 1 5 | C x n ( k )| 2 + C y n ( k )| 2 + C z n ( k )| 2 )] (2) Figures 1 and 2 show the results obtained from eqns. (1), (2). Here both the spectra calculated in the plane wave approximation and those found experimentally are given. The ratio between maximum III (p-states prevail) and maximum I (s-states dominate) is given in Table 1. As can be seen from Table 1 and Figs. 1 and 2, the spectra calculated in the 4 Relationship between maximum III and maximum I and between photoionization cross-sections of s- and p- electrons in diamond and silicon Diamond Silicon I III /I I Density of states of valence electrons 2.64 2.29 XPS (in PW-approximation) 0.27 1.65 XPS (in OPW-approximation) 0.41 1.12 XPS (experimental) 0.44 1.40 σ s /σ p PW-approximation 25.0 1.7 OPW-approximation 13.6 2.4 Free atom 12.0 3.4 approximation of the orthogonalized plane wave fit the experimental results better than the spectra obtained in the approximation of the plane wave. Important effects of orthogonality terms on X-ray spectrum shape were observed. Contributions of these terms may be different at different electron configurations. Thus when orthogonality is taken into account this increases p-states in the case of diamond and s-states in the case of silicon. Some information about the relation between photoionization cross-sections of s- and p-electrons may be obtained from the expression σ s σ p ≈ I s N p I p N s where I s , I p , and N s , N p are the integral contributions of s- and p-states to the spectrum, and the density of these states, respectively. It may be seen from Table 1 that when one allows for orthogonalization this greatly affects the σ s /σ p value. σ s , and σ p are averaged characteristics depicting the magnitude of photoionization cross-sections of s- and p-electrons in the crystal. McFeely et al. have estimated the number of s- and p-electrons for diamond crystals. These authors used the X-ray photoelectron spectrum, the X-ray emission K-spectrum, the calculated density of electron states and the ratio of atomic photoionization cross-sections σ s /σ p . We have determined the density of states of valence electrons from the X-ray photoelectron spectrum, as well as from the emission K- and L-spectrum. Use was also made of the photoionization cross-sections ratio for s- and p-electrons in the approximation of an orthogonalized plane wave. It is known that in crystals of light elements the K- and L-spectra are proportional to the partial densities of states of p- and s-electrons J K ( E ) = P p N p ( E ) and J L ( E ) = P s N s ( E ) where, neglecting the crossover transitions, one may treat P s and P p as constants independent of energy. On the other hand the X-ray photoelectron spectrum intensity and the electron states density can be written as I ( E ) = σ s N s ( E ) +σ p N p ( E ) and N ( E ) = N s ( E ) + N p ( E ) Figure 3 shows that satisfactory approximation of the photoelectron spectrum is achieved by superposition of the emission K- and L-spectrum with allowance for excitation probabilities of s- and p-electrons. The density of states obtained for the silicon valence electrons is given in Fig. 4. This is in agreement with the density of states calculated in ref. 3 which is broadened with the dispersion curve (0.4 eV at half-width) --corresponding to the resolution with which the X-ray emission L-spectra have been taken for silicon. Thus one may conclude from the above results that in diamond and silicon crystals the orthogonality terms make an important contribution. The use of X-ray emission and photoelectron spectra together with the data on photoionization cross-sections for valence electrons allowed us to determine the density of states in silicon. The tight-binding method provides good agreement with experiment. How- ever the simple theoretical model does not take into account all the physical processes that occur in electron photoemission. This perhaps is partly the cause of the discrepancy between theory and experiment. Further advances in this field need a theoretical study of the effects which accompany electron emission from the sample (the loss of excitation energy by electrons, consideration of the existence of a hole in the valence band, surface effects). The resulting information may be used for development of a more accurate model. Special attention should be paid to angular dependence of X-ray photoemission.