Core excitons in solid rare gases: Kr-3d and Xe-4d transitions

Solid State Communications, Vol. 25, pp. 835—837, 1978.

Perganion Press.

Printed in Great Britain

CORE EXCITONS IN SOLID RARE GASES: Kr-3d AND Xe-4d TRANSITIONS G. Grosso, L. Martinelli and G. Pastori Parravicini Istituto di Fisica, Università di Pisa and Gruppo Nazionale di Struttura della Materia del CNR, Pisa, Italy (Received 17 October 1977 by F. Bassani) Core excitons from 3d levels of Kr and 4d levels of Xe are examined within the integral equation formalism both in the isolated atoms and in the crystals. Earlier interpretation that resonances at the interband edge Kr-3d and Xe-4d are due to is, 2p, 3p excitons is well supported by the present calculation. OPTICAL PROPERTIES of rare gas atoms and solids have been widely investigated in the extreme ultraviolet region [I]. On the theoretical side the intermediately

(~90 eV), superimposed on

bound exciton theory [21 has brought a renewed interest on valence and core excitons in rare gas solids. More recently several semi-empirical approaches have been developed to relate atomic excitations with crystal excitons. In paper [3] a simple expression for central cell corrections has been derived within the effective mass approximation. The quantum defect method has been extended to exciton problem and successfully applied to solid neon [4]. Valence excitons in rare gas solids have been studied within the envelope function formalism [5] combined with an appropriate use of atomic exciton energies. In this paper we examine core excitons originated from 3d levels of Kr and 4d levels of Xe using the envelope function formalism. With respect to valence excitons we have here two significant differences: (i) the much higher energy at which exciton resonances appear; (ii) the fact that at k = 0 core band to conduction band dipole matrix elements are forbidden. At a forbidden interband edge in the electric-dipole approximation optical transitions are possible only to excitons with p-like envelope functions [6]. Transitions to s-excitons may occur due to electric-quadrupole or magneticdipole terms [7] ; in solid Kr and Xe is excitons are in fact experimentally detected as a very weak shoulder at the onset of 3d and 4d transitions [8,9]. We are thus interested in excitons with is and np (n > 2) symmetry. In the following we consider specifically only krypton, the case of xenon being completely similar. All parameters of interest used throughout this paper (atomic radius r0, lattice constant a0, Mott—Littleton radius RML, static dielectric constant e8 and effective mass m* of conduction electrons) are the same of [5]. Near the onset of 3d transitions in gaseous Kr

the spin—orbit energy (~ 1.22 eV) of the 3d shell. For convenience exciton energies of the series 3d512 np and the corresponding series limit E1 are reported in Table 1. These data are used for semi-empirical determination of the short range potential VR (r) in the effective mass equation ~ 2 + V~(r)+ VR(r) F(r) = (E—E V 1)F(r) (1) i 2~ In equation (I) Vc(r) is the Coulomb potential corresponding to 3d wavefunction [111. The short range potential is taken in the form of a step function A S(r r0) and the value of A is adjusted in such a way that the computed eigenvalues of the Schrodinger equation (1) for np (n ~ 2) envelope functions fit the corresponding experimental values of the exciton series 342 ~(3 + n)p (n > 2). With A = 1.55 Rydberg, the agreement between experimental and computed atomic exciton energies is within 0.01 eV (see Table 1). An interesting aspect of the exciton effective equation (1) stands in the possibility of calculating atomic exciton series with different angular symmetry, once experimental data are available for a given symmetry only. For instance, in neon, the same potential which fits s-type valence excitons was able to reproduce the experimental binding energies [12] of p-type, d-type etc. valence excitons within ~ 2%; refinements should further reduce residual differences. We notice explicity that there is a simple correspondence between usual spectroscopic notations for atomic excitons (in which quantum numbers of initial and final states are given) and envelope function notations (in which quantum numbers of the relative electron—hole motion are given). This correspondence can be worked

the photoionization continuum there are two series of resonances [10] (342 np and 3d312 np with n> 5) separate by -+

-~

835

—~

—

.

—

836

CORE EXCITONS IN SOLID RARE GASES

Vol.25, No. 10

Table 1. Experimental and computed energies (in e V) for core excitons originated from 3d512 levels of Kr. Both atomic spectroscopic notations and envelope function notations are given for convenience Gaseouskrypton Experiment [10] Spectroscopic notations 3d—’- 5s 3d—’- 6s 3d—’- 5p 3d—~6p 3d—’-7p Series limit

Envelope function notations is 2s 2p ‘4p

Solid krypton Experiments Solid state effects —1.61 —0.20 —0.78

Binding energy Ref. [8] Ref. [9] —2.02 —1.90

Exciton energy Ref. [8] Ref.[9] 90.28 90.40

-

-

-

91.25

Model Binding energy potential —3.61 —1.56 —2.58 —2.59

—0.69

—0.57

91.61

91.73

92.60

—1.23

—1.23

—0.12

—0.31

—0.22

91.99

92.08

93.10 93.83

—0.73

—0.73

—0.06

-

-

-

Exciton energy -

92.3

Table 2. Experimental and computed energies (in e V) for core excitons originated from 4d512 levels of Xe. Both atomic spectroscopic notations and envelope function notations are given for convenience Gaseous xenon Spectroscopic notations

Envelope function notations

4d—’-6s 4d—+ is

is 2s

4d—’-6p 4d—7p 4d-. 8p Series limit

3p ‘ip

Solid xenon

Experiment [10] Exciton energy

Model Binding energy potential

.

-

-

-

65.12 66.37 66.85 67.55

—2.43 —1.18 —0.70

out using Elliott’s concept [6] of first-class excitons, second-class, etc., andcontains the general property that at small kthird-class a scattering state an s-wave (with coefficient independent on k), a p-wave (with coefficient linear in k), a d-wave (with coefficient quadratic in k) etc. To label excitons we will use indifferently spectroscopic or envelope function notations, For core excitons, the above mentioned aspect of the exciton effective equation (1) is of major interest because of the limited amount of experimental information presently available; in particular it allows us to calculate the energy of ls exciton in Kr, which is unknown both experimentally and theoretically. We solve in fact for s-states the Schrodinger equation (1) with potential which fits the p-exciton In Krthe thesame computed binding energy (3.61 eV) of series. is core exciton is close to that of is valence exciton [12] (3.966 eV). A similar situation occurs in Xe. Also in Ar, where experimental data [13] are available, the binding energies of 3p 4s and 2p -+45 excitons (4.135 eV and 4.02 eV) are very near. It seems a general feature that gas binding energies Is different, valence and core the excitons in rare atoms are notoftoo despite drastic change in the hole localization, -+

In the crystal the electron—hole interaction is primarily modified by screening effects; for valence excitons, the polarization potential was taken in the simplified form [5]

Experiments Binding energy Exciton energy Ref. [8] Ref. [9] Ref. [8] Ref. [91

Solid state effects

—3.32 —1.49 —2.44 —1.18 —0.70

—1.16 —0.09 —0.49 —0.05 —0.02

Vp(r)

e ~ RML 2 /

2 (1 =

i_ ji r where RML

=

—1.24

—1.30

64.36

64.30

-

-

-

-

—0.32 —0.12

—0.25 -

65.28 65.48

65.35 65.6

-

-

-

-

65.6

._-L\

‘f
— ~

I

r

ML

if r > RML

(2)

e~/

ira

0/6 3446 is the Mott—Littleton radius at the lowest order of approximation. The same expression is assumed here for core excitons; to which extent such an approximation is justified requires further investigation. Exciton binding energies in the crystal are obtained solving numerically the equation 2 + Vc(r) + VR (r) + Vp(r) F(r) —

2m

v

—

‘E

—

~

—

E ~

(3

GI

for is and np (n ~ 2) envelope functions. For r > a 0 the free electron mass m has been replaced by the conduction band effective mass. This simplication could be 2 dk, checked by evaluating the transform quantity fofF(r), E~(k)IF(k)I where F~)is the Fourier which is known numerically; the present poor knowledge of correlated conduction band energies E~(k)makes premature a similar refinement. Dipole—dipole interactions, responsible for transverse-longitudinal splitting could be included in the calculations; but they are safely negligible for a forbidden band-to-band transition.

Vol. 25, No. 10

CORE EXCITONS IN SOLID RARE GASES

837

In Tables 1 and 2 we report the computed energies for Kr-3d and Xe-4d crystal excitons, together with experimental data for comparison. Despite some uncertamty in expenmental values of bmdmg energies and the adopted simplications in treating solid state

assignment [8] that resonances at the interband edge are due to is, 2p, 3p excitons. Acknowledgements We are mdebted to Prof. F. Bassani and Dr. R. Resta for many stimulating discussions. Correspondence with Dr. L. Resca is also gratefully

effects, our present calculation well supports earlier

acknowledged.

—

REFERENCES 1.

For a review see for instance SONNTAG B.,Rare Gas Solids (Edited by KLEIN M.L. &VENABLES J.A.) Vol. 2. Academic Press, New York (1977).

2.

3.

ALTARELLI M. & BASSANI F.,J. Phys. C4, L328 (1971); ALTARELLI M., ANDREONI W. & BASSANI F., Solid State Commun. 16, 143 (1975);Phys. Rev. B! 1, 2352 (1975); ANDREONI W., PERROT F. & BASSANI F.,Phys. Rev. B14, 3589 (1976). RESCA L. & RODRIGUEZ S. (to be published).

4.

RESTA R. (to be published).

5.

7.

MARTINELLI L. & PASTORI PARRAVICINI G.J. Phys. dO, L687 (1977); GROSSO G., MARTINELLI L. & PASTOR! PARRAVICINI G., Solid State Commun. 25,435 (1978). ELLIOTT RJ.,Phys. Rev. 108,1384(1957). See, for instance, RUSTAGI K.C., Solid State Commun. 12,607 (1973).

8.

HAENSEL R., KEITEL G., SCHREIBER P. & KUNZ C.,Phys. Rev. 188, 1375 (1969).

6.

9. 10.

HAENSEL R., KOSUCH N., NIELSEN U., ROSSLER U. & SONNTAG B.,Phys. Rev. B7, 1577 (1973). CODLING K. & MADDEN R.P.,Phys. Rev. Lett. 12, 106 (1964).

11. 12.

CLEMENTI E. & RAIMONDI D.,J. Chem. Phys. 38,2686 (1963); 47, 1300 (1967). MOORE C.E., Nat. Bur. Standards 1, Circular No. 467 (1949).

13.

WATSON W.S. & MORGAN F.J.,J. Phys. B2, 277 (1969).