Excitonic fine structure in TlCl

Solid State Communications,Vol. 17, pp. 685—687, 1975.

Pergamon Press.

Printed in Great Britain

EXCITONIC FINE STRUCTURE IN T1C1 Ch. Uth]ein and J. Treusch Institut für Physik, Universität Dortmund, 46 Dortmund, Germany (Received 2 May 1975 by M. Cirlona)

The doublet structure of the is-exciton of simple cubic thallous chloride is quantitatively explained as due to coulombic and exchange interaction, which leads to intra- and intervalley scattering between K =0 excitons formed of electron—hole pairs at non-equivalentX.points of the Brillouin.zone.

THE EXCITONIC doublet structure of single cubic T1C11 has been a puzzle until it was qualitatively explained2 in terms of intervalley scattering between K =0 excitons centered at the inequivalent X-points of the Brillouin zone, where the direct gap of 110 occurs.3 The question was left open whether the nature of this scattering mechanism is purely coulombic (including exchange) or whether phonons play a significant role. In this paper a quantitative calculation of intra- and intervalley scattering matrix elements is performed on the basis of LCAO-wave functions taken 4 which has been adjusted to theanexisting KKR-band structure.3 from empirical band model

atomic orbitals of the atom indicated in the superscript. If we set up an exciton Hamiltonian in the usual sense5 using a Kohn—Luttinger basis set,we have no longer a simple two band model, because of the existence of three nonequivalent X-points. If we confine ourselves from the beginning to excitons with vanishing translational momentum, the Hamiltonian reads Hjf~ 4L’(k,k’) = + (c~Ic N~EG*O

IVJMXVJM’ 1e 1G12

iGr

Ic,.~~’i

4ire2

The wave functions cj~and z~constituting the bottom of the conduction band and the top of the valence band are according to4 given by

—

c 3, +

~

=

c

3,

—

=

v3, + ~ = v3, + ~ =

—

ap~’It)+ ~/s/2(p~

—

7s’I~~f)mp~It

1 +KJf)r IG+Kj,’ +k—k’12

(Civ Ie’’i’~lcj’~’)(vj’~’ Ic_

VIM)

In deriving equation 2 we have made use of the plane

(1)

wave expansion of the Coulomb interaction. G is a reciprocal wave vecto~and i11’ the vector connecting the valleys / and/’.

—

7S1’hI,I~)+

(2)

—

G

ipy)ThI,I~)

p~”I.I~) + j3/y’2(p~+ ip~)”It)

{1 ~ }~

i6p~’R>,

where / denotes the various X-valleys, v, p are spin indices the X~-conduction X~-valence 2 of = 0.85, ~32 = 0.15,72 and = 0.3 and 62 = band, 0.7. and a It s understood that valley 3 points in [001]-direction and that the wave functions corresponding to the

Theener~’ first term, of equation 2 contains kinetic of theH0, electron and hole and the the. intravalley parts of the Coulomb interaction. The Is-like eigenstates of H 0, in the following denoted by I/vp), are 12-fold degenerate, since H0 does not depend on

valleys 1 and 2 are obtainedby cyclic permutations of the cartesian coordinates. p~,P~,P~and s are the

/, v, p. In accordance with group theory this high 685

686

EXCITONIC FINE STRUCTURE IN TiC!

:::~_:::__~:--~“

2

0

-

-1

-

_M

0.6

0.7

0.8

Vol. 17, No.6

e~p

-

-1

0.9

1.0

1.1

07

0.6

0,8

09

tO

1.1

Locolisotuon 2/g

FIG. 1. Calculated matrix elements (dashed lines) and resulting energies for the A- and B-excitons (solid lines) versus degree ~oflocalization p/pa. Arrows indicate the experimental splitting (reference 2). degeneracy is partly lifted by the remaining perturbation terms of equation (2). These terms are intra- and intervalley contributions to the analytic part of the exchange interaction6 and the intervalley parts of the Coulomb interaction. In the latter case the wave vector k k’ may be neglected as compared to i~’. Thus both perturbation terms represent a contact potential between electron and hole. —

All electron—hole interaction terms have to be screened in principle. But in contrast to the intravalley parts of the Coulomb interaction the perturbation terms

The linear combinations belonging to valleys 1 and 2 may again be obtained by cyclic permutations. The dipole active states t,Li$~~~ transform according to the representation X~ and X~, the pure triplet states t/i7 according to Xj. These states will be mixed by the intervalley parts of the Hamiltoman (2) thus finally leading to exciton states which transform according to , , 1’~,and r’~ (point group Oh). The dipole active representation f’~occurs twice. For ls-exciton states the perturbation terms of equation (2) are given by the following matrix2 ~n 0 E

~i

of (2) have to be screened by = 4q), where q refers to the involved scattering wave vector. For TICI a 7 type calculation on the basis of our Penn model empirical LCAO band structure4 yields c = 2 for the analytic exchange and e = 2.4 for the intervalley

—

I~

E~

11

‘I4~

M

—f-— {13, 4,—4) + 13,—4, 4))

—

—

0

E3 272McT(l, 2)

E1 = 2(32 72 MAT —4(3 E 272McT(1, 2) E 2 = 4(32-y2MAT 3 = 2cx 2 2 ~ 2a,&y MAT c43’y MCT 1, 3 where ‘I~= l/\/2(~Li~ + ~14)and cI~= iJ4 transform according to r~ , and ~ = I /~J2(~ ~4)according to r~.The pure triplet states ~ [‘i) are not affected by the perturbation terms of equation (2). — —

=

4

2

1~ Coulomb terms. In both cases the phonon screening is negligible. In the following we give linear combinations of the unperturbed exciton states which are diagonal with respect to the intravalley parts of our Hamiltonian (2).

~rn M 0 1~

—

—

—

1 ~ ~J2 ~

~

—

‘

2~

—

2

3

—

2~2

(3) =

—~-

{

3, 4-

4)—

3,— 4, —4))

The quantities MAT and MCT(JI’) are defined by —

=

~{I3,4,4)+l3,—4,—4)}.

AT

ire

2

(s

2 G*0

Ti iGr

e

Ti

Ti -iG~ TI

p~)(p~le

1G12

Is )

Vol. 17, No.6

EXCITONIC FINE STRUCTURE IN TiC!

4ire2 McT(j,J~)=

XE

—

The matrix elements E 1 , E2 , E3 and ~ which result from this calcualtion are plotted as functions of p/po in Fig. 1. In addition we depicted the energies

10(0)12

~ IG + K11 12

G

687

(5)

a, (3 and 7 are the mixing coefficients of equation (1) and 0(r) is the envelope function of the Is-exciton. This result was obtained by use of the LCAO wave functions of equation (1), neglecting all two-center integrals occuring in the matrix elements of equation (2). We have calculated the exchange- and Coulomb-8 like terms MAT and MCTfunctions using Hermann—Skillmann 6s- and 6p-atomic wave of thallium. The degree of localization of these functions was varied by coordinate-stretching, P/Po varying between 0.6 and l.l.~10(0)12 has been adjusted to reproduce the experimentally 2 measured total oscillator strength of the doublet: f = 4~h ______72 ~ 0(0)2 ~ 373 x iO~. mw

(6)

of the two dipole active modes, which in accordance with reference 2 are labelled A and B. Their energetic distance is in good agreement with experiment. All energies refer to the pure triplet states having zero energy. The right part of Fig. 1 shows corresponding results for (32 = 030, which seems to be the maximum value in accordance with the measured ratio of oscillator Asbetween may be easily seen, neither thestrengths energeticph/1~B• distance the two dipole active modes 1~Aand 1~B,nor the strength of the intervalley matrix element M~,change drastically, when the degree of localization or the composition of the wave functions is changed over the rather wide range discussed here. In all cases agreement with experiment is satisfactory.’° So, in conclusion, we feel, that the mechanism which splits the 1 s-exciton of TlCl is proven to be of Coulombic nature (including exchange) and that phonons play, if at all, a minor role. Acknowledgement The authors are grateful to E. Mohier, Frankfurt, for a very helpful discussion. —

REFERENCES 1.

BACHRACH R.Z. & BROWN FC.,Phys. Rev. BI, 818 (1970).

2.

MOHLER E., SCHLOGL G. & TREUSCH J.,Phys. Rev. Lett. 27,424(1971).

3.

OVERHOF H. & TREUSCH L, Solid State Commun. 9, 53 (1971).

4.

TREUSCH J. Phys Rev. Lett. 34, 1343 (1975).

5.

KNOX RS., Theory ofExcitons, Solid State Physics, Suppl. 5. Academic Press, NY (1963).

6.

The nonanalytic exchange vanishes for transverse excitons, which show up in optical spectra.

7.

PENN D.R., Phys. Rev. 128, 2093 (1962).

8.

HERMAN F. & SKILLMAN S., Atomic Structure Calculations, Prentice Hall, NY (1963).

9.

A reasonable approximation to the partially ionic character of the Ti-wave functions seems to be obtained by taking plpo 0.8.

10.

During the preparation of this manuscript the authors became aware of as yet unpublished work by KURITA S., KOBAYASHI K. & ONODERA Y. describing magneto-optical measurements on T1C1. Their experimental findings including the energetic position of the pure triplet states confirm our results. Their theoretical attack is similar to ours, but no quantitative calculations are performed. —

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