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IMPORTANCE OF SURFACE EFFECTS AND DYNAMICAL CORE-HOLE EFFECTS IN AUGER SPECTRA OF METALS AND SEMICONDUCTORS
C.-O. Almbladh, A. Morales
To cite this version:
C.-O. Almbladh, A. Morales. IMPORTANCE OF SURFACE EFFECTS AND DYNAMICAL CORE-
HOLE EFFECTS IN AUGER SPECTRA OF METALS AND SEMICONDUCTORS. Journal de
Physique Colloques, 1987, 48 (C9), pp.C9-879-C9-890. �10.1051/jphyscol:19879157�. �jpa-00227269�
JOURNAL DE PHYSIQUE
Colloque C9, supplement au n012, Tome 48, decembre 1987
IMPORTANCE OF SURFACE EFFECTS AND DYNAMICAL CORE-HOLE EFFECTS IN AUGER SPECTRA OF METALS AND SEMICONDUCTORS
C.-0. ALMBLADH and A.L. MORALES
D e p a r t m e n t o f T h e o r e t i c a l P h y s i c s , Lund U n i v e r s i t y , S d l v e g a t a n 1 4 A , 5 - 2 2 3 6 2 L u n d , Sweden
Auger C W spectra of sp-bonded materials are generally believed to be well described by one-electron theory in the bulk. We here perform detailed calculations on several simple metals and on silicon and obtain results which differ markedly from experiments. To explain these discrepancies we investigate effects of the surface and dynamical effects of the sudden disappearance of the core hole in the final state. To study core-hole effects we solve aha an-Nozieres-~e~ominicis (MND) model numerically over the entire band. The core-hole potential and other parameters in the MND model are determined by self-consistent calculations on systems with a core-hole impurity. To study surface and mean free path effects we perform slab calculations in the case of silicon, whereas we for the metals use a simpler infinite barrier model in which the valence states near the surface are constructed from Bloch states in a simple way. In all cases investigated either the effects of the surface or the effects of the core hole give important modifications and a much improved agreement with experiments.
1. INTRODUCTION
The problem of calculating Auger core-valence-valence (CVV) spectra of solids from first principles has proved to be rather delicate. Even in sp-bonded materials, naive pictures based on self-folded densities of states (DOS) fail badly, and the importance of properly including matrix element has been demonstrated by several authors[l-31. Except for the early works by Feibelman &
a[l], effects of the surface and finite Auger-electron mean free path have not been considered, and the prevailing view seems to be that one-electron bulk theory gives a rather good agreement with experiments for sp-bonded solids. However, the mean free path is quite short at typical Auger-electron energies of the order 100 eV, and an early calculation by Shulman and Dow[4] gives evidence for important core-hole effects.
In this paper we summarize our recent work on simple metals[5,6] and give new results on silicon. We present detailed one-electron spectra from bulk atoms calculated from self-consistent wavefunctions. We consider the Li KVV spectrum and the L2,3VV spectra of Na, Mg, Al, and Si and find in all cases results which deviate from experiments. To explain the data we go beyond the one-electron and bulk approximations and study the effects of the surface and mean free path, and the dynamical effects connected with the sudden disappearance of the core hole in the final state of the Auger process. As we will show, these effects give important modifications and a much better agreement with experiments. Interestingly enuugh, however, in each individual case studied here either the surface effects are important
or
the dynamical core-hole effects. Thus, the explanation is different in each case.Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19879157
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The simplest model that account for core-hole effects in a dynamical way is the independent fermion model due to Mahan, Nozihres, and DeDominicis (MND)[7]. An important aspect of our work is that the model parameters are obtained from &
initio calculations. In order to determine these parameters we perform calculations on systems with a core-hole impurity in addition to the ground-state calculations.
We then solve the MND model numerically using the finite-N method pioneered by Kotani and Toyozawa[8]. We also compared with final-state approximations based on ideas by von Barth and Grossmann[g].
To estimate the surface and mean free path effects we perform self-consistent slab calculations in the case of silicon, but use simpler approximations for the metals. In the latter case we extend an idea by Gadzuk[lO] and represent a valence orbital near the surface by a Bloch wave specularly reflected in a surface plane about one atomic radius outside the outermost nuclei. This corresponds to an infinite barrier model where the secondary reflected rays have been neglected. This is clearly a somewhat crude model for the surface electronic structure but, owing to the broad selection rules imposed by Auger matrix elements, we do not expect an Auger spectrum to probe the finer details in the local electronic structure near the surface.
2, ONE-ELECTRON THEORY OF AUGER LINESHAPES AT SURFACES AND IN THE BULK
Our computational procedure is described in detail in Refs. 5 and 6, and thus only a rather brief summary is needed here. In the one-electron approximation the angle-integrated Auger current from a bulk atom is given by
OCC
Dl(€) = 2n
I
[ ( < ~ L ~ V ~ C A > ~ ~-
< ~ ~ . J v ~ A c > * < ~ L ~ v ~ c A > ] kt kAAs we have pointed out in Ref. 5 the Coulomb matrix elements should be computed using the unscreened interaction v(?-?I) for transition energies of interest here.
To compute the spectra we use muffin-tin wavefunctions
(L is short-hand for the angular momentum label (L,m)), and keep only the contribution from the central cell, as the remaining parts are known to be very small[ll]). When we make the usual multipole expansion of the Coulomb interaction we encounter off-diagonal sums of muffin-tin amplitudes, which we approximate as follows:
Here DL is the projected density of states (PDOS) for one spin at the core-hole site. Equation ( 2 ) is exact for s and p electrons in cubic solids, and we have verified that it introduces no significant errors for hcp metals. With these approximations the Auger rate simplifies to an expression of the form
where the quantity M contains the radial matrix elements and Wigner coefficients ( U is the Fermi level).
The valence orbitals are calculated using the linear muffin-tin orbital (LMTO) method in the atomic sphere approximation[l2]. We include up to d waves. Exchange and correlation are accounted for in an approximate way via the local-density ground-state potential[l3]. We solve the Auger-electron orbitals in the same potential which is used for the core- and valence-electron orbitals, emplying two different approximations. In the first approximation we solve radial Schrodinger equations in the central sphere and match to free waves outside. In the second method we approximate the Auger-electron orbital by a single LAPW. As we have shown in Ref. 5 the latter method is applicable to the third-row materials at not too high Auger-electron energies, and in these cases the two approximations give rather similar results.
Our studies of core-hole effects also require orbitals perturbed by a core-hole impurity. We obtain these orbitals by self-consistent repeated cell calculations on a lattice where every 16th atom is a core-hole atom. Owing to the finite distance between the core holes there are some spurious effects, such as a slight increase in the bandwidth and additional but small bandgaps. We have found that these effects are fairly small, and in particular they are small compared the uncertainties in the experimental data.
To obtain the no-loss Auger current coming through the surface we should according to general photoemission theory[l4] take the final Auger orbital $A as a time-reversed LEED (low energy electron diffraction) orbital solved in the potential
Veff = VC + E;(E~) (4)
inside the solid. In Eq. (4) VC is the total Coulomb potential and C;(EA) the (time-reversed) one-electron self-energy or optical potential at the Auger energy
EA. In what follows we make the usual approximation and replace E by vxc+iy/2 where vxc is the Hohenberg-Kohn-Sham[13] ground-state potential and where the constant damping y is taken from electron-gas calculations or deduced from experimental mean free paths. In the present calculation we adopt the simplest possible approximation for $A and take it as a plane wave of the appropriate direction in vacuum, and a properly damped LAPW inside the solid with the behavior
inside the muffin-tin sphere at site-j. In Eq. ( 5 ) 2X=y/kA is the inverse mean free path, LA is short for Auger-electron angular momentum labels, and 0 is the emission angle. We have chosen our coordinate system so as to have the surface along the plane z=0. As before we approximate the valence orbitals by muffin-tin orbitals,
(kll is the momentum parallell to the surface), do a multiple expansion of the Coulomb matrix elements, and keep only the central cell part. In the presence of a
JOURNAL DE PHYSIQUE
Fig. 1. Top view of the first two silicon layers in our reconstructed 2 x 1
surface. In the first layer atom "1" is moved 0.41 a.u. upwards, and atom "2" is moved 1.45 a.u to the left and 0.38 a.u inwards to the bulk.
surface the sum over magnetic sublabels (m) can no longer be performed analytically. Averaging over the core-hole magnetic and spin sublevels and summing the contribution from all atomic layers we obtain
where
is the in general non-diagonal state density function for one spin in the unit cell at
s.,
J and wherecontains the matrix elements. (Here ck(L,L1) a Gaunt coefficient, and Rk a Slater integral.) Equations (5-9) give the general one-electron result for Auger emission from a surface in terms of muffin-tin wavefunctions and allow thus for a more realistic treatment of the Auger orbital.
The d components of the bulk spectra are quite small, and we include only s and p waves in our surface calculations. In the case of silicon, we model the valence- electron wavefunctions by selfconsistent LMTO slab calculations. We use 9 and 12 silicon layers and 16 k-points in the irrecucible surface wedges for the ideal (100) and ( 1 1 1 ) surfaces, respectively, and a sufficient number of vacuum (typically 4 ) to decouple the repeated slabs from each other. We use slab wavefunctions for layers less than half a slab thickness from the surface and take the remaining layers as bulk layers. To model the 2x1 reconstructed (100) surface we adopt the asymmetric dimer model by Chadi[l5]. However, the arrangement of atomic and empty spheres needed the LMTO treatment of open structures imposes restrictions of the possible movement of surface atoms, and therefore we have used a simplified geometry. We actually only reconstruct the top layer of silicon atoms (see Fig. I), and move neighbouring empty spheres so as to minimize the overlap between the atomic and empty spheres. The major approximation in our simple implementation is that the surface layer is not relaxed towards the bulk. The important effect we wish to model, however, is the induced splitting of the surface
states. The occupied, bonding surface states should move down in energy, and the anti-bonding ones should move up, and this is exactly what we obtain. The irrecucible surface wedge is 1/4 of the Brillouin zone including time-reversal symmetry, and we use 72 k-points in this wedge.
For the simple metals we model the valence orbitals ($k) by Bloch waves ($k) specularly reflected in a surface plane, i.e., we take
Using the transformation properties of Bloch orbitals it is not difficult to see that the muffin-tin amplitudes ($)of
Ok
are simply related to those(BE)
of theBloch waves,
for lattices where the reflection R is an allowed point-group operation. In order to evaluate the state density function FjLLI(€) we use LMTO self-consistent wavefunctions for the ground state, and use a sufficient number of k-points to converge up to the third sublayer. The fourth and the following sublayers we take as bulk layers. For the cubic metals we use about 1000 k-points in the irreducible wedge and obtain the contributions from the remaining wedges by the appropriate symmetry transformations. For Mg we use similar method with about 2000 points in 1/12 of the Brillouin zone.
3. EFFECTS OF THE CORE HOLE
The final state rule justified by von Barth and Grossmann[91 provides rather simple ways of estimating core-hole effects. According to this rule, the shaoe of a dynamical spectrum closely resembles the independent-electron result obtained with the final-state potential. In the case of Auger emission this potential contains no core hole. Auger CVV spectra involve many subchannels of comparable intensities.
The final state rule explained above pertains to each subchannel lineshape, but the subchannel yields are determined by the valence wavefunction in the completely relaxed initial state,
( 1 1 ) where l'ftQtis the yield obtained with a static core hole. This sum rule is obeyed by the MND model but follows from far more general assumptions (see e.g. Ref. 16).
Combining these two results we obtain an approximation for Auger spectra used by Ramaker[l7] in which we superimpose subchannel intensities from ground-state calculations with weights obtained with a core hole. Thus,
'a$'
D(E~) =
1 -
I ~ ~ I ( E ~ ) r (12)a>al
G a l
where the superscripts o and
*
refer to one-electron results without and with a static core hole, respectively.In order to treat core-hole effects of dynamical nature we adopt the MND independent fermion model[7]. We thus represent the valence-electron system by a one-electron Hamiltonian (H*) in the initial core-hole state, and a different one (H) in the final state without a core vacancy. As is well known the MND model can be solved analytically very close to the emission or absorption edge and gives for metals a singular edge behavior. The model can be solved also away from the edge by numerical means. Away from threshold the MND model lacks formal justification, but
C9-884 JOURNAL DE PHYSIQUE
when used with proper parameters (one-electron eigenvalues, core-hole potentials, etc) obtained from realistic calculations the model has nevertheless been shown to give a good description of x-ray emission and absorption bands in most simple metals[9,18,19]. In view of this it is not unreasonable to assume that useful insight can be gained by applying the MND model to the case of Auger CVV emission.
The MND singularity is of the form
close to the edge E = E ~ . The exponents a for different spectroscopies are simple expressions of the Fermi-surface phaseshifts of the core-hole potential. For the case of Auger CVV emission the exponents are negative (see Ref. 16 Table 61, but in general the leading edge is steeper than it would be in one-electron theory (in the latter case a = - ? ) . This effect would move the peak in the spectra to higher energies. On the other hand, the MND model also gives a broadening away from the edge originating from particle-hole shake-up, which tends to move the peak to lower energies. Thus, it is difficult to say anything definite without explicit numerical solutions over the entire spectral range.
We obtain our model parameters as follows. It has been demonstrated[9] that the shape of a dynamical MND (subchannel) spectrum (Rmo) depends essentially only on the local density of states and transition matrix elements in the final no-hole state, the number of valence electron (nR) in the central cell with and without a core hole, and the Fermi-surface phaseshifts of the core-hole potential. The shape of the total spectrum depends not only on the subchannel lineshapes but also on their relative intensities and thus on
l'iRl.
The detailed functional form of the core-hole potential in H*, on the other hand, has almost no influence on the dynamical lineshapes, and we here choose a separable Ansatz,. We require H and H* to correctly reproduce the quantities discussed above as obtained from our self- consistent field calculations. The Fermi-surface phaseshifts, however, we determine from other sources, as will be discussed in connection with our results. We also do not explicitly include the Auger electrons in our model calculations but regard the Auger process as a sudden annihilation of two valence electrons. In other words we take the transition operator aswhere the amplitudes Tvvl as chosen as separable in the orbital labels, except for the parallell-spin part for the case R = 9,' = 1 where a somewhat more complicated Ansatz is needed. (We leave out d- and higher partial waves in our MND calculations.) In each LL' channel the matrix element parameters are determined up to a constant ALL' by requiring the model to reproduce our LMTO ground-state transition densities of states D L L I ( ~ , ~ ' ) without a core hole (L is short for angular momentum and spin). The constants ALL' are finally determined by the yield sum rule in Eq. (11).
We solve the model using the finite-N method by Kotani and Toyozawa[8]. In this method the system is simulated by a finite but large number of fermions. Practical experience as well as comparisons with results by other means have shown that about 100 particles per Rmo subchannel is more than sufficient in order to obtain a good representation of the NM results. We assume that the different channels do not interact with each other, which allows us to write H and H* as
Owing to this property the transition amplitudes <slTI*> of a spectral density factorize into matrix elements from each L channel separately (s lables the possible final states), and consequently the complete spectrum is a convolution of
simpler subchannel spectra.
A passive channel L, whose electrons are not involved in the transition operator T, gives a mere overlap <N,sIN*>~ between initial and final states with N of electrons, and the corresponding spectrum just reflects the recoil to the sudden change in potential,
In the MND model the initial hole state IN*>L is made up of the N lowest orbitals ($,I of the core-hole Hamiltonian
~;f.
Similarly all final states I N , S > ~ can be taken as Slater determinants involving N orbitals (4,) of the no-hole Hamiltonian HL. Keeping N finite, the required overlaps can be done numerically and, as has been shown in detail by previous workers[9], they can be expressed in the inverse of the NxN overlap matrixS.. = <$i16 >
13 j (17)
involving the N lowest orbitals of HL and
HE.
For instance, the overlap <N,s l N*>with a final state having a particle (p) above and a hole (i) below the Fermi level is given by detS Ek <@,,I$~>S~;. Multipair excitations can be handled in an analogous way, but it is generally found that one only needs to include up to double-pair excitations for N around 100.
An active channel involving the annihilation of a valence electron with the operator
CM =
.I
MvCLv (18)v
can be handled by a similar method[9,19]. (The electron operators {cLV} refer to the no-hole Hamiltonian, and M, is a matrix element for the one-electron process in question.) The spectral profile is now
Using the expansion theorem for determinants one can show that the required overlaps <N-l,sicMIN*> can be obtained by the same procedure as above applied to the overlap matrix
where x ~ = E ~ M ~ @ ~ . In principle we have also channels where two electrons with the same angular momentum (Rm) and spin are annihilated, but LMTO calculations show that these two-electron channeds give a negligible contribution to the spectrum.
The complete spectrum thus involves only convolutions of recoil (Aa(€)) and one- electron (B&(E)) spectra. For instance, the sp part of the spectrum is obtained by convoluting together one s-wave and 5 p-wave recoil spectra with the two one- electron spectra Bs and Bp.
4. RESULTS
We first discuss the simple metals. Our one-electron bulk results for these materials are given and compared with experiments in Fig. 2 and the spectra of Li and A1 are decomposed into contributions from valence states of different angular
C9-886 JOURNAL DE PHYSIQUE
I . ' . . I . . l ~ ~ " l ' s ' ~ I ~ ' ~ ~ i ' ~ ' ~ I
H a LRTO, HND, SURF, EXP Cr> A1 LMO. I1M). SURF. E W C*>
t 0 0 -
- - -
0 . 5 0 -
-
0 . 0 0
I . . . . I . ,
- -
3 . 0 0 I . . . . I . . . . l . . . . I
AUGER ELECTRON ENERGY CRY) 3 50
3.50 4 . 0 0
AUGER ELECTR~~NSOENERGY C R Y ~ ~ O O 5.50
I I
L l LHTO. FS RULE, MNO. EXP C+>
I .OD-
-
0 . 5 0
-
--
Fig. 2. Theoretical Auger lineshapes compared with experiments. Solid curve: One- electron bulk theory; Dashed-dotted curve: Final state rule, Eq. (12);
Short-dashed curve: Full MND results; Long-dashed curve: Results for the surface model. Experimental results (crosses) are taken from Ref. 25 (Li), Ref. 26 ( M g ) , and Ref. 27 (Al).
I 1
Na LHTO. NND. SURF
- -
- -
- -
I I
I
LHTOAUGER ELECTRON ENERGY CRY) 4.00
'
AUGER ELECTRONENERGY'.!&>
AUGER ELECTRON ENERGY CRY:
I
A l L M OI
Fig. 3. One-electron Auger bulk spectra for Li and A1 decomposed according to angular momentum and spin. Parallell and anti-parallell spins of the final valence holes are denoted by p and a, respectively. Solid line:
Total spectrum; short-dashed lines ss(a); medium-dashed lines sp(a);
dashed-dotted lines sp(p); long-dashed lines pp(a); dotted lines pp(p).
momenta and spin in Fig. 3. We see that our one-electron spectra peak at too low energies. The different partial wave contributions (ss, sp, pp) peak at rather different energies. Consequently it would be rather easy to obtain good agreement with experiments by adjusting the weights in a semiempirical way, as has been done in many previous works. It is also clear that the final-state approximation, Eq.
(12), may lead to important modifications. As is seen in Table 1 , however, the relative weights rRRI/r for the third-row metals are almost unchanged when the orbitals are allowed to relax in the core-hole potential. Thus, this approximation does not improve the agreement with experiments in these cases. For Li, on the other hand, we see that the relative weights of the sp and pp channels increase when the effect of the core hole is introduced, which leads to a peak shift to higher energy and a rather good agreement with experiments (Fig. 2).
We next turn to our fully dynamical calculations. Among the model parameters the ground-state transition densities of states, the local occupancies with and without a core hole (Table 21, and the subchannel yields with a core hole present are determined by our LMTO calculations as described in the previous section. In addition we also need estimates of the Fermi-surface phaseshift of the hole potential. Experimental and theoretical estimates span a wide range, in particular for Li and Na[20]. Our values (Table 2) represent a compromise which are rather close to the values obtained by Almbladh and von Barth[21] except for Na, where we choose a smaller s-wave phaseshift. Our phaseshifts are also rather close to those which correspond to our core-hole screening charges.
Our dynamical results are also shown in Fig. 2. In both Na and Li the peak moves to higher energies, and for Li the leading edge is actually too steep. For lower energies our Li spectrum is broader than the experimental one, but the data are here less reliable because an intense plasmon satellite has been deconvoluted away. The deconvolution also takes away the particle-hole shake-up structure included in our calculation. In the case of Na there are no undifferentiated spectra avaliable in such a form that they readily allow for a detailed comparison.
Table 1. Relative subchannel yields
riR,/r
with ( * ) and without ( 0 ) a core- hole impurity.Table 2. Fermi-surface phaseshifts and the number of s and p electrons in the central cell with (n&) and without (nR) a core-hole impurity.
C9-888 JOURNAL DE PHYSIQUE
I " ~ ~ I ~ ' ~ ' I ' " ' 0 ~ ~I ' . ' . I " ' . I . . . . I . I . ~
SI C I I I ) IDEAL. BULK. EXP. BY UEISSMANN SI C100) IDEAL. 2x1. BULK. EXP. 81 1 . o o -
0 . U ) -
I , , . . I , , , , I , , , ( I , . ,
6 . 5 0 6.50
A$& ELECTRON ENERGY CRY) 6 . 5 0
AUGEGR.'&CTRON
M%R% CRY)Fig. 4. L 2 , 3 W spectra of Si from reconstructed (solid) and ideal (dashed) surfaces and bulk (dashed-dotted) compared with experiments from Ref. 28 and 29.
For the remaining cases the MND calculations mainly give an overall broadening due to particle shake-up without changing the peak position. Thus we see that the bulk results for Mg and A1 differ considerably from experiments also when dynamical core-hole effects are included.
The core hole may redistribute the relative intensities among the subchannels, and it may cause a change in shape of individual subchannel spectra. As regards the yield sum rule, it really does not rely on the MND model, and our impurity calculations should give quite reliable predictions of the relative weights. The second effect, on the other hand, relies in our calculations directly on a specific model. In Li the main effect of the core hole is the change of the relative weights, whereas the change in subchannel lineshapes is smaller but overemphasized in our calculation. In MND theory the singular edge exponents for different spectroscopies are all determined by a common set of phaseshifts[22], and this constraint seems to be violated in the case of Li. In particular the results of inelastic electron scattering[23], which show no trace of s-wave enhancement, are clearly incompatible with the large exponent deduced from x-ray photoemission spectra. It has been suggested that spin-flip scattering mediated by the core- valence exchange interaction is a likely cause for this shortcoming in the case of Li.[16,24]. In Na the entire core-hole effect is dynamical and comes from changes in the subchannel lineshapes. Both the x-ray emission and absorption spectra from Na show no doubt strong singularity effects, which supports our present finding of large dynamical effects in the Auger spectra as well.
In order to explain the remaining cases, Mg, and Al, we now discuss the effects of the surface. Our calculations model (100) surfaces for the cubic metals and a surface perpendicular to the c-axis for the hcp metals and correspond to a mean free path of 4 A in Li and A1 and 3.5 A in Na and Mg. The results for the Auger spectra at normal emission (Fig. 2) show a noticible shift of the intensity maximum for the cases of Mg L2,3 and A1 as compared to the bulk case and a rather good agreement with experiments (Fig. 2). For A1 the MND effects only give albroadening, while for Mg we notice a small MND effect that would further improve the agreement.
For the monovalent metals Li and Na we have found that the surface gives no important effects despite changes in the Local densities of states.
We finally turn to the L2,3VV spectrum of Si. Our findings above are that core- hole effects are important for mono-valent systems and that surface effects are important for polyvalent ones. Thus one would expect important surface effects in the Si case as well, and as seen in Fig. 4 this is indeed the case. Our results correspond to a mean free path of 7 A . We see in Fig. 4 that the bulk one-electron spectrum peaks at too high an energy and exhibits a shoulder from sp channels wh ch is much too strong. The surface effects move the peak to lower energies and weak.dn the shoulder, and when the surface is reconstructed the occupied surface states move down in energy and the peak at the leading edge from dangling-bond states is weakened to a shoulder. In our simple model for the reconstructed surface the bonding surface states are most probably too high in energy. Thus, one would expect the shoulder to more or less disappear in a more accurate treatment.
5. CONCLUDING REMARKS
In this work we have shown that one-electron bulk theory as a basis for parameter-free calculations does not describe Auger CVV spectra very well even in sp-bonded materials. We have investigated corrections to the usual one-electron treatment and found important effects of the interaction between the valence electrons and the core hole in monovalent metals like Li and Na. In Li the simple final state approximation, Eq. ( 1 2 ) , gives the best agreement with the data, wheras the remaining dynamical effects are overestimated by our MND calculations. In the case of Li there are similar problems also in other spectroscopies, and we have argued that they are probably inherent to the MND theory itself.
For poly-valent metals Mg and A1 we obtain only small core-hole effects but important corrections from the surface and the finite mean free path. The agreement with the data is actually better than our simple treatment of the simple-metal surfaces deserves, but we feel that our bulk and surface calculations taken as a whole show the importance of surface effects in these materials. In the case of Si we have treated the surface in a more accurate way by self-consistent slab calculations. We have obtained important surface corrections also in this case, although the agreement with experiments is not as good as for Mg and Al.
Let us finally comment on the overall reliability of the present results. The spectra considered here are rather structureless and easy to fit by parametrized theories. Thus it is of outmost importance to have accurate one-electron results in order to be able to map out various many-body effects. We feel that our LMTO wavefunctions are of sufficient accuracy, and that the remaining uncertainties mainly lie in the treatment of the outgoing Auger electron. Like previous workers we have used rather simple approximations here, but we are presently investigating more accurate schemes based on T-matrix expansions.
ACKNOWLEDGMENTS
One of us (A.L.M.) acknowledges financial support from the University of Antioquia (Medellin, Colombia). The present work has also been supported in part by the Swedish Natural Science Research Council.
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