Bound small polaron optical absorption in tetrahedral symmetry

Solid State Communications, Vol. 18, pp. 1345—1348, 1976.

Pergamon Press.

Printed in Great Britain

BOUND SMALL POLARON OPTICAL ABSORPTION IN TETRAHEDRAL SYMMETRY O.F. Schirmer Institut für Angewandte Festkorperphysik der FraunhoferGesellschaft, D 78 Freiburg, West Germany and R. Schnadt IBM Laboratorien, D 703 Boblingen, West Germany (Received 23 October 1975 by E. Burstein) The distortion caused by capture of a charge carrier at site-degenerate positions near a defect in tetrahedral symmetry is shown to lead to broad, intense optical absorption due to charge transfer between equivalent sites. The optical absorption of BeO : Li, shown here, exhibits the predicted features. IN ORDER to understand the structure of deep centers in semiconductors, it is necessary to take into account the interaction between trapped carrier and lattice. In fact, generally a sizeable part of the carrier binding energy is caused by this coupling. The important role played by this carrier—lattice interaction for radiationless electron—hole recombination at deep centers has recently been demonstrated by Henry eta!.1 Previously we have shown2 that the lattice interaction of holes bound to negative defects on cation sites in alkaline earth oxides can be directly derived from their polaronic optical absorption. There the sixfold cubic symmetry of the lattice around the defect is broken by the hole—lattice interaction in such a way that the hole is fixed at one of the six 02- ions around the defect, and the optical absorption consists of light induced hole transfer from one to an equivalent 02- site, the distorted nuclear frame being fixed during the transition. In explaining the experimental fmdings, the language of the Pseudo-Jahn—Teller effect3 was cornbined with that of small polaron theory,4 connecting these fields, which so far had developed independently.

however, is a little more favorable energetically by 34meV5 because of the polarity of the wurtzite structure. This intrinsic alignment easily allows to test the symmetry properties of the optical transitions by their polansation. Since the hole lattice interaction will turn out to be much larger than this wurtzite stabiisation energy, we shall neglect the latter in the following and treat the site symmetry as tetrahedral. The axial orbital, stabiised by the wurztite field, will be called Ia) in the following, the nonaxial ones l13~),1132), 1133), Fig. 1(a). In this letter we shall deal only with the new features introduced by tetrahedral symmetry and refer to the previous work2’6 for additional information. In describing the charge transfer optical properties of this system we start with the fictitious situation that the hole can tunnel freely among the four eligible sites. The corresponding states would be ~‘1

=

f~) =

ir’

5~)

=

In this communication we extend this previous treatment to the case of tetrahedral symmetry, present in most technologically used semiconductors. As a good prototype for a suitable defect in such systems 2~ wein BeO. choose bound to U~ substituting Bestudy the The higha hole bandgap of this material allows for us to entire broad optical absorption band, permitting us to clarify the absorption mechanism, whereas in lower gap materials only the low energy tails of the bands might be observable. The center itself offers the advantage that its groundstate is very well known from ESR,5 showing

=

~ + i13~+ ~(_

132>

~>+ (3~)+

+ 133)) 132>—

1133>)

~(— a) + I13~> 132) + 133)) ~ + I13~) 1132>— 1133)) —

—

neglecting overlap corrections to the normalisation. The IF1)resonance and IF5) states would separated fourthe times the integral J. Asbe observed by by ESR, high 7’d symmetry, however, is broken to C~,by hole—lattice interaction. We use the generally employed simplification that among the available lattice phonons of suitable symmetry only one is assumed to be coupling. In the present case this vibration mode must have 1~5symmetry in order to reduce the symmetry from 7’d to C 3~, Thus the Hamiltonian describing the resonance and hole lattice interactions wifi appear in the above tunneling basis as: .~

that the hole can be localised at each of the four 02_ ions surrounding U~in p-orbitals pointing towards this site [Fig. 1(a)]. The 02_ ion lying along the c-axis,

1345

1346

OPTICAL ABSORPTION IN TETRAHEDRAL SYMMETRY

r/~

Vol. 18, Nos. 9/10

‘~

I~3i> 1p2>

3

1P3> b

a

C

Fig. 1. (a) Arrangement of hole orbitals in BeO : Li. The thick arrows indicate 02_ p-orbitals. At low temperatures, only Ia) is populated. (b) Tetrahedral arrangement of equipotential surfaces of the hole—lattice system in the space of the F5 vibration mode configuration coordinates. The designations mean that the system is stabilised in well a, if the hole is trapped in orbital a), etc. (c) Hole energy level scheme and transitions of the system in 1(a) under C3~, distortion. —3J

VQ~

~

.~

VQ~

~

~

~Y

~Z

J

V’QX ~

~Z

VQ~

This condition (i~Q~ 8/3EJT, = 0) means the energy that the distance optical absorption between theisminimum peaked atof one potential sheet and those of the three other wells at the position of the minimum.

VQ2 ~

~Y

V’Q~ ~

~

The hole lattice coupling strength is described by the reduced matrix elements V and V’, the ratio of the offdiagonal matrix elements otherwise being given by the relevant vector coupling coefficients.8 Transforming this to the single ion basis a), 13~),we obtain

/v(Qx

=

(

Given is thethus same EJT, the in tetrahedral syrnmetry peaked at aabsorption higher energy, 8/3EJT, than in octahedral symmetry, 2EJT.2’6 Since in such a transition the vibrational state of the lattice, called p) in the following, is not changed, immediately after the transition the lattice will not vibrate in the eigenstates of the new wells, Iz’~),but

~QY+QZ)

J

J

V(—Q~+ Q~—Q~) J J

J

J

V(—Q~—Q~+Q~) J J V(Q~—Q~—Q~)

where it has been assumed that V = V’. (Otherwise cornplete localisation at one site is not described.) Adding to this matrix the potential of the F5 vibration mode, which is diagonal in anyenergy hole basis, I x (pw~/2)Q2,(1 is the 4 x 4 unit matrix), we see that the system has potential minima at Q~= = = — V/pw~along the [111] direction in the space of the F5-configuration coordinates and also along the three equivalent directions, Fig. 1(b). These minima are lower than that of the undistorted configuration, = 0, 2/pc~.TheQ~ saddle by the stabilisation energy E,,~ = 3/2V between neighboring minima lies at 2/3EJT above the niiriiina. For strong coupling, the localisation of the systern in one of the wells means that the nuclear framework is permanently trigonally distorted. All optical transitions will take place in this fixed trigonal symmetry.

rather in the non-stationary phonon state ~ v

1)(p~p) This leads to2’6 the following distribution of transition probabilities: 2 = (,i~ lP~~ >1 ~(E~ 1— EM — hw) where ~p2), v,~)are the vibrational functions extending along the direction connecting wells a and J3, and E0. and E,2 are the corresponding vibrational energies. This is a Poisson distribution, which for large hole lattice coupling can be approximated by the Gaussian :~ I(hc~.,) = exp {— w [hw — (8/3)EJTI 2} (1) with w~= l6/3E~,.hw 0. This is valid at T = 0. At .

higher T, w

—i

.

is given to good accuracy by

w~(T) = w_i(O) coth (hwo/2kT).

Vol. 18, Nos. 9/10

OPTICAL ABSORPTION IN TETRAHEDRAL SYMMETRY absorption 1) (cm

1347

______

~uc

295K

exP<

£

theory

—

\

3

—

3.0

2.0

1.0

eV

Fig. 2. Optical absorption of holes trapped near Li in BeO. Except for frequency independent factors, the absorption curve is given by I(hw)- 11w. The hole can be excited to the three sites l13~), i = 1, 2, 3, equivalently. Resonance interaction between them leadsstates to the(now formation tunneling in C of the following excited 3~symmetry): 3(I13i) + 132) + 133)) — I— ri)

IF

31)

—

liv

=

l/~~J~2 I13i) — 1132)— 133))

—

,

,—,.

II’32) — energy l,v2d132) —1133)).are — 2J and J, The corresponding eigenvalues respectively, Fig. 1(c). The r 1(F3) levels can by light with its E parallel (perpendicular) to be theexcited c-axis. It should that thebyabove excited tunneling states be arekept all toinbemind multiplied the non-stationary phonon states corresponding to the vibrational groundstate p> of the system. The absorption will thus be given by the superposition of two Gaussian shapefunctions, one of them belonging to parallel, the other one to perpendicular polarisation. A transition from a) to any of the I13~)individually is possible by the resonance admixture2’6 Ia’)

=

a) —(3J/8EJT) ~ I13~)

and is polarised along the vector 1 connecting these two sites. Its intensity is proportional to (3J/8EJTE .1)2. Generalising this to the excited tunneling states, one finds that the parallel proportional to the perpendicular one intensity to 3/2l~2,iswhere I~and 1~ are3l~f, the projections of I parallel and perpendicular to c. In ideal

tetrahedral surroundings i~= 4/3 a, 11 = 2\/~7~a, a being the bond length. Under this condition, one thus expects an intensity ratio ofA 11/A1 = 4. If the tetrahedron is distorted, so that the nonaxial bonds form an angle 2/sin2~. For ~ with c, this ratio becomes 2(1 + cos ~) = 90°,this is 2. Let us now compare results, these predictions experimental absorption shown u~with ig. 2.the They .

were obtained with a BeO crystal, which after X-irradiation5exclusively ESRatof holes Among theshowed spectrathe taken 6K, the trapped parallel at Li sites. absorption follows the derived absorption line shape 0 This fit was obtained with EJT = I(hw)-hw 1.08 eV, 11wvery well.’ 0 = 0.11 eV. The first value is a typical 2’6 stabilisation energy for of holes in oxideoscillator lattices, whereas the frequency the trapped representative matches the zone boundary LO frequency of BeO.1’ As predicted by the model, the absorption is smaller for El c than for E II c. The lineshape, however, is less well reproduced. We suppose that this is due to a superposition with another absorption band of unknown origin, since no homogeneous phonon broadened absorption band can have a larger width than small polaron absorptions, where the entire excitation energy appears in the width (see the expression for w). Assuming that the perpendicular absorption is predominantly due to the described center, one concludes that 4 is smaller than the tetrahedral bond angle, because A 11 IAI <4. This is in accord 5with the relaxation of 0 and Li~away fromThe eachdifference other. between parallel and perpendicular peak positions can be identified with 3J, giving

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OPTICAL ABSORPTION IN TETRAHEDRALSYMMETRY

Vol. 18, Nos. 9/10

J 0.2 eV. At elevated temperatures the spectra become more isotropic (peak heights and positions moving to-

It remains to discuss the sign of / (>0). It is such that the nodeless Fi(C3~)state lies lowest for holes. This

ward each other, see Fig. I), as expected, because the hole then is thermally distributed also among the nonaxial 02_ ions lying higher in energy by 34meV. The spectra become broader, (3), and also second because light first withon E IIaccount c or £ Iofcequation is polarised at oblique angles with respect to the axes of the nonaxial centers and thus leads to a superposition of the two bands. Since the resonance integral J, 0.2 eV, is larger than the representative phonon energy, 0.11 eV, the hole will tunnel from one il,) to a neighboring one before the lattice can adjust to its presence in Ill,). This will reduce the probability that the hole is trapped at 13, after having been transferred there from a and might explain, why a population increase of the nonaxial sites under irradiation has not been observed yet. The descnbed situation is similar to the well-known stabilisation of systems with high spin-orbit coupling against Jahn—Teller distortion.

means that positive charge density is attracted towards the region between the equivalent 02_ ions. This situation has also been encountered by 2Watkins hole It couldfor be aattritrappedtoata repulsion a zinc vacancy in ZnSe.’ buted of electrons out of the region between the twofold negatively charged 02_ and Se2 ions, respectively. In the alkaline earth oxides, where it was concluded, that J had the opposite sign,2 electron density could be stabilised in the corresponding region by the presence of twofold positive alkaline ions near the overlap region, present there in the rocksalt but not in the zincblende or wurtzite structure.

I.

Acknowledgements We thank Dr. S.B. Austerman for lending the BeO specimen, Dr. F.S. Ham and Dr. G.D. Watkins for a discussion, and U. Kaufmann and Dr. P. Koidl for comments on the manuscript. —

REFERENCES HENRY C.H. & LANG D.V., Proc. 12th mt. Conf. Phys. Semicond. Stuttgart 1974 (Edited by PILKUHN M.H.), p. 411. Stuttgart (1974).

2.

SCHIRMER 0.F., KOIDL P. & REIK H.G., Phys. Status Solidi(b) 62, 385 (1974).

3. 4.

See e.g.: HAM F.S. in Paramagnetic Resonance (Edited by GESCHWIND S.), p. 106. NY (1972); WATKINS G.D., in LatticeDefects in Semiconductors, 1974 (Edited by HUNTLEY F.A.), p. 1. London (1975). See e.g.: APPELJ.,Solid State Phys. 21,193(1968).

5.

SCHIRMER 0.F.,J. Phys. Chem. Solids 29, 1407 (1968).

6.

SCHIRMER 0.F., BLAZEY K.W., BERLINGER W. & DIEHL R., Phys. Rev. Bi 1,4201(1975).

7.

KOSTER G.F., DIMMOCK J.O., WHEELER R.G. & STATZ H., Properties of the Thirty-Two Point Groups, p. 100. Cambridge, MA (1963).

8.

Reference 7, p. 94.

9. 10. 11. 12.

KEILT.,Phys.Rev. 140, A601 (1965). Deviations at energies above the peak energy are usually observed (References 2 and 6) and can be attributed to hole transfer into higher lying nonequivalent orbitals (to be published). LOH E.,Phys. Rev. 166,673 (1968). WATKINS G.D., in Radiation Effects in Semiconductors, (Edited by WHITEHOUSE J.E.), p.228. London (1973).