Core excitons in the soft X-ray region: Solid argon

Solid State Communications,Vol. 16, pp. 143—146, 1975.

Pergamon Press.

Printed in Great Britain

CORE EXCITONS IN THE SOFT X-RAY REGION: SOLID ARGON M. Altarelli* Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, U.S.A. and W. Andreoni and F. Bassani Istituto di Fisica, Università di Roma, Rome, Italy (Received 23 September 1974 by F. Bassani)

A theoretical calculation of core excitons is performed for the L11,.~1soft X-ray threshold of solid argon at 245 eV. The binding energies and the relative transition amplitudes for the lowest allowed exciton states are computed by formulating the problem in terms of Wannier functions and solving the resulting integral equation in the one-site approximation.The results obtained allow to locate the onset of interband transitions at an energy a few eV above previous theoretical determinations. Therefore, sharp structure previously interpreted in terms of conduction band density of states is attributed to discrete excitonic transitions, as strongly suggested by the close analogy with the atomic absorption spectrum. A comparison with the fundamental excitonic absorption in the vacuum ultra violet region is carried out in terms of the ratio of the electron—hole exchange interaction to the spin—orbit splitting of the hole states. 3 IN RECENT years, theinvestigated soft X-ray spectroscopy of solids has been widely using synchrotron radiation as a continuum source.1 The new high resolution data made available by this technique have stimulated considerable theoretical interest on photoabsorption transitions from core levels. In non-metals, this interest has focused on the question of the relative importance of excitonic effects and of one-electron interband transitions in the photoabsorption process.2 In fact, if excitonic effects could be neglected, the soft X-ray spectrum would provide a direct experimental determination of the density of final states, to be compared with theoretical band structure calculations. The experimental results for solid rare gases and alkali-halides, however, show an abundance of sharp *

structure near the onset transitions of from inner shells, which is indicative of theofimportance excitonic effects. It is the purpose of the present communication to carry out a detailed calculation of core exciton binding energies for the ~ soft X-ray threshold in solid argon. This will allow us to locate the energy position of the interband edge and to assign in detail the observed structure. The results obtained are strongly supported by the very close similarity of the experimental spectra of gaseous and solid argon4 for a few eV above the onset of transitions from the 2p-shell (Fig. 1). This similarity, clearly indicative of important excitonic effects, seems to have been overlooked in earlier theoretical investigations.5’6 In order to compute the binding energy of core

Work supported in part by the U.S. Army Research Office, Durham, North Carolma, under Contract ARO DA-HCO4-74-C-0005.

excitons, we take advantage of a procedure already 143

144

CORE EXCITONS IN THE SOFT X-RAY REGION

5~

Vol. 16, No. 1

~

112

B

b/

~i/

~ItH

V

~0.8

2~—

~0.6~

i~ C I

244 246 248 250 252 254 256 258 260 262 264 266 268 270 272 E(eV)

FIG. I. Experimental photoabsorption cross section (left scale) or absorption coefficient (right scale) of solid (solid line) and gaseous (dashed line) argon, after reference 4. Corresponding peaks in the two spectra are labelled with the same letter. Assignment of the excitonic peaks in the crystal to specific band edges (see text) is also shown. applied successfully to the fundaniental exciton spectrum of Ar, and described in detail elsewhere.7’8 We take the integral equation for exciton states as our starting point, and retain only the dominant terms in the expressions of the Coulomb and exchange integrals between core and conduction states in terms of Wannjer functions. We than assume that the electron and the hole are confined to the same unit cell (onesite approximation), in analogy with our treatment of valence excitons;7’8 this assumption is even more justifled for the strongly localized core excitons. With this approximation and taking into account the spin—orbit splitting of the 2p-core level in the r

8

-

and F6 -states, the solution of the integral equation is reduced to the simple 2 X 2 determinantal equation: + [_Q+~(J+D)]

G6+,8-(E)

The computations of the parameters Q, J and D requires explicitly the knowledge of Wannier functions for the core and conduction bands. The core functions are for all purposes coincident with the 2p atomic functions, whereas the conduction band Wannier function is replaced by a summation over the Brillouin zone of a plane wave orthogonalized to valence and core states on the origin and on the nearest neighbor sites.8 Another important parameter is the spin—orbit splitting of the core levels, which is identical to the atomic value, and which enters equation (1) through the Green functions. The value of the parameters used in the calculations is given in Table I, where the contributions to Q and J from the various angular momentum components of the conduction band

—~(J+D)G6÷8.(E) =0(1)

—~~(J+D)G6+6..(E)

1 + [—Q+~(J+D)]

where Q is the Coulomb integral, J and D are the short 8 and long-range components of the exchange interaction, G 6+~(E)is the Green function for the F6+—F,. edge, which includes the detailed structure of the conduction band throughout the Brillouin zone.

G6+8-(E)

Wannier function (which is of cubic but not spherical symmetry), are also shown. It is important to notice in Table 1 that the polarization term D is negligible, and that the exchange J

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CORE EXCITONS IN THE SOFT X-RAY REGION

145

Table 1. Values (in eV) of the Coulomb and exchange integrals Q and J of the polarization term D and ofthe spin— orbit splitting ~ of the 2p core level, for solidAr. The contribution to Q and J ofdifferent angular momentum components of the conduction band Wannier function is also shown

Q J D

1=0

1=4

1=6

1=8

6.073 0.060

0.117 0.4 X i0~

0.286 0.1 X l06

0.082 0.3 X i0~

is much smaller than the spin—orbit splitting, as a consequence of the localized nature of the core states. We therefore expect, in contrast to the situation for valence excitons, a very small admixture of the two spin—orbit split states. Table 2. Binding energies (with respect to the I’~—F~ band edge) Eb and relative intensities ofthe two lowest ris core exciton states F Eb

1=10 0.005 0.1 X

Total 6.563 0.060 —l.5XlfY5 2.03

10_b

exciton states, rather than to features of the joint density of states. This conclusion is strongly suggested by the close similarity of the atomic and solid state absorption spectra in this region, which is also of great help in assigning the observed structure to specific exciton states. The lowest peak A is the exciton state computed above, and associated with the s-like conduction band. It corresponds indeed to the atomic transition 2P3/2 4s, labelled a in Fig. I. The spin—orbit -~

15(l) 3200 37

F15 (2) 11~ 37

partners, the crystal and theintense atom, are labelled A’ and a’,inrespectively. The in very B peak is closely related to the atomic 2p3/2 3d tiansition, —~

Finally, in order to compute the Green function G(E), we interpolate the conduction band structure, as Calculated by Lipari,9 with a three term tight. binding formula. We can then proceed to the solution

labelled b, and is therefore an exciton associated with the higher conduction band of d-like symmetry, which Is very close in energy to the lowest band at the X point of the Brillouin zone. Its spin—orbit partner B’ is degenerate with the 2p 3 ~ conduction band continuum and appears as a sharp resonant feature above threshold. The weak feature C, finally, is a higher exciton (n = 2, in the effective mass notation) and corresponds to the peak labelled c in the atomic spectrum. In conclusion, it is important to remark that the -~

of equation (1), which yields the results shown in Table 2. The two n = 1 exciton states turn out to be separated by the spin—orbit splitting the relative intensity being in the ratio 8 2 : 1. This is because the effective exchange term = 2(J+D) (2) ~,

is much smaller than the spin—orbit splitting already pointed out,

~,

as

Our result for the binding of the lowest exciton allows to locate the onset of the continuum about 3.4 eV above the first peak, i;e. at 248.5 eV, immediately above the shoulder labelled C in Fig. 1. All the structure observed below this energy must therefore be associated with transitions to bound

.

.

ab inztio calculation presented here and based on the integral equation approach allows to understand the difference between the fundamental and the soft X-ray absorption threshold in terms of the ratio of the exchange interaction to the spin—orbit splitting of the hole. Furthermore, it allows an appropriate definition of the continuum edge and a classification of the excitation spectrum of the crystal which is based on the band structure, but explains the correspondence with atomic transitions.

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CORE EXCITONS IN THE SOFT X-RAY REGION

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REFERENCES 1.

For recent reviews see CODLING K.,Rep. Progr. Phys. 36, 541 (1973); HAENSEL R. and SONNTAG B. in Computational Solid State Physics (Edited by HERMAN F., DALTON N.W. and KOEHLER T.R.) p. 43, Plenum, New York (1972).

2.

See for instance ALTARELLI M. and DEXTER D.L.,Phys. Rev. Lett. 29, 1100(1972) and references therein; PANTELIDES S.T. and BROWN F.C.,Phys. Rev. Lett. 33, 298 (1974) and references therein.

3.

CODLING K., reference I and references therein.

4.

HAENSEL R., KEITEL G., KOSUCH N., NIELSEN U. and SCHREIBER P.,J. dePhysique 32, C4, 236 (1971).

5. 6.

ROSSLER U.,Phys. Status Solidi (b), 45,483(1971). KUNZ A.B. and MICKISH D.J.,Phys. Rev. B 8,779(1973).

7.

ALTARELLI M. and BASSANI F., J. Phys. C 4, L328 (1971); Proc. p. 196, Polish Scientific Publishers, Warsaw (1972).

8.

ANDREONI W., ALTARELLI M. and BASSANI F. (to be published).

9.

LIPARI N.O.,Phys. Rev. B6, 4071 (1972).

mt. Conf Phys. Semicond.

Warsaw, 1972,