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J. Phys. Chem. Solids Pergamon Press 1969. Vol. 30, pp. 1769-1777. Printed in Great Britain.
THE JAHN-TELLER INSTABILITY WITH
ACCIDENTAL DEGENERACY
A. M. STONEHAM* and M. LANNOOt* Theoretical Physics Division, AERE Harwell, Berkshire, United Kingdom (Received 17 October 1968: in revised form 6 January 1969) Abstract-The Jahn-Teller theorem predicts that orbitally degenerate complexes in solids can lower their energy by a distortion which lowers their symmetry. Exact degeneracy is not necessary for this instability; usually it suffices if the actual separation of the energy levels involved is less than the energy reduction which would result if they were exactly degenerate. We discuss various cases of accidental degeneracy in cubic and tetrahedral systems. With accidental degeneracy it is possible to get mixed distortions which involve both trigonal and tetragonal distortions. These have probably been observed for the negative vacancy in silicon, and possibly for the neutral vacancy in diamond. The theory is applied to both these cases. and it is compared with the earlier qualitative arguments of Watkins.
1. INTRODUCTION THE JAHN-TELLER theorem asserts that orbi- tally degenerate molecules or complexes in solids can lower their energy by a distortion which lowers their symmetry[ 11. The in- stability occurs because the Hamiltonian has matrix elements linear in the distortion, whereas the elastic restoring forces are quad- ratic in this distortion. The theorem is usually applied to cases where the degeneracy of the energy levels is exact and arises from some symmetry of the system. However, exact degeneracy is not necessary for an instability [2]. Usually it is sufficient if the reduction of energy by distortion for exact degeneracy exceeds the actual separation of the levels involved. Near degeneracies can occur even when there is no underlying symmetry reason, and are known as ‘accidental’ degeneracies. Accidental degeneracy is particularly interesting because the matrix elements linear in the distortion, responsible for the instability, may connect states which transform according
_Present address:_* Coordinated Science Laboratory, University of Illinois, Urbana, Ill. 61801, U.S.A. tpermanent address: lnstitut Suptrieur d’Electronique du Nord, Lille, France. $This article is based on part of a thesis submitted by M.L. for the degree of Doctorat d’Etat en Sciences Physiques (Centre National pour les Recherches Scientifiques No. A.O. 2646).
to different representations of the appropriate symmetry group. Usually, with exact degen- eracy, matrix elements within one represen- tation alone are considered. For this reason more varied phenomena occur with accidental degeneracy, and inter-level terms are impor- tant in the examples considered here. Opik and Pryce discussed the near degen- eracy of the 2s and 2p levels of the F centre. These states are coupled by odd modes parity of T,, symmetry.^ For^ suitable^ values^ or parameters the system was unstable with respect to these distortions. The interesting feature here is that a system with full cubic symmetry can lose its inversion symmetry by a Jahn-Teller instability. As the selection rules for optical transitions depend on the parities of the states involved this particular instability may.profoundly affect the observed properties of the centre. This article is mainly concerned with nearly degenerate E and T levels in tetrahedral or cubic symmetry. The calculations show that configurations may be achieved which are of lower symmetry than those obtained ignoring the near degeneracy. Also the inter-level terms may alter the relative importance of trigonal and tetragonal distortions for relative- ly isolated levels. The results are particularly relevant to defects in the valence crystals diamond, Si and Ge. Two cases are discussed 1169
(^1770) A. M. STONEHAM and M. LANNOO
in detail. The first is the negative vacancy in Si, where Watkins[3] observed mixed trigonal and tetragonal distortions. His description in terms of a one-electron model is related to a new description, appropriate to a many- electron model of the type introduced by Coulson and Kearsley[4], for diamond. The second case discussed is the neutral vacancy in diamond. The stress-splitting of the zero phonon line of the CR 1 band shows ano- malies. and these may be related to the low- symmetry distortions discussed here [51.
In addition to these linear terms there are elastic energy terms quadratic in the distor- tions. The matrix elements of these have the form:
lKE(E’)+H’))+KT(52+r)‘)+5’))] i (2.2)
where 1 is the unit matrix.
- COMBINED TRIGONAL AND TETRAGONAL DISTORTIONS The Jahn-Teller instability arises from matrix elements of the Hamiltonian which are linear in the distortions of the environment. As an example of a system in which the matrix elements of importance connect non-degen- erate levels, we discuss in detail the case where a T, level (comprising the states I,.$>, Ir)>, I<>) lies close to an E level (comprising states IE> and (0). For zero distortion the T level lies A above the E level. Within these levels the use of symmetry simplifies the matrix elements linear in the distortion to the form:
The static distortions of this system will now be discussed, giving the configurations Q = (E, 8,(, r), 5) which minimise the total energy. As general solutions of the secular equation are so complicated as to hide the important features the results will be given for various degrees of intra-level coupling and inter-level coupling. Dynamic effects and spin- orbit coupling are ignored.
2.1 tntra-level coupling only We first consider the case when there is no coupling between the E and T, levels, i.e. Gr,,. is zero. This is the standard case, where there is only coupling within each state. Opik and Pryce concluded that the E states would distort tetragonally, with energy reduction G&/K,. The T state distorts trigonally if the reduction of energy by trigonal distortion, G&l3K,~, exceeds G&IKE; otherwise a tetra- _~ _~.. GTTV~ -GT.~S +y3G7~j
GTT
-GTx~)
The GIJ are coupling coefficients, where I = E, T labels the symmetry of the mode involved, and J = E, X, T indicates the states involved. The matrix elements within the T state have J = T and those connecting the T and E states have J = X. The notation E, 19,5,~, 4 for the normal modes of even parity is standard.*
*In tetrahedral symmetry there are, in general. two sets of T, normal modes. We will only discuss vacancy centres with this symmetry; for vacancy centres there is just one set of T2 modes, and no ambiguity arises.
~GTx< -^ G,,^ H^ +^ GEE^ E
G BE E Gm 0
____~.~
gonal distortion results. Mixed tetragonal and trigonal distortions do not provide stable distortions; this^ result^ has^ recently^ been verified with more general assumptions by Wysling and Miiller 161.
2.2 Inter-level coupling only In this section Grx is assumed to dominate, and we treat the case Gb,.E= GET = GrT = 0. The eigenvalues can be obtained, as functions of GTx.. From these configurations of minimum
1772 A. M. STONEHAM and M. LANNOO
-(G$X/KT) ($-tU’+4F’2) where^ Q^ is^ one^ of^ the^ normal^ coordinates
if IFI < (114)and+= 3/4.
0, E, f-, q or 5. With these equations we can
The other states, la), 16) and [c>, show no
express the Q in terms of the ui. Inserting
instability from interlevel coupling. They are,
these values for the Q in equation (2.9) we
of course, still affected by the coupling within
obtain a set of equations in the unknowns
the E and T levels.
E and Ui. These equations can be solved exactly. 2.3 Arbitrary intra level and inter let;el It^ is easily^ shown^ that the extrema^ obtained coupling with zero inter level coupling always exist.
We follow the method of ijpik and Pryce
Their energies were given in Section 2.1.
121, and express the wave function W as a
The first mixed solution obtained gives
linear combination of the basis functions, &, the six possible distortions with (0, E, 5, q, 4)
of the nearly degenerate states: (^) equal to (p, 0.0.0, +), c--+, -yp, 0, q, 0)
or c
--&- +$p, f 9,0,0). (2.8) The energy at these minima is:
and p and 4 are given by:
I
The coefficients Uj are solutions of the These^ formulae^ are only valid if^ F^ Lies^ between equations: -- KT GEE(GET+-@EE) Eat = 7 Xtiaj (2.9) l^ 4&^ G&f
where ZU is the sum of a Jahn-Teller matrix
and
element and an elastic energy term. The & GET(GET+@EE)__ 1. extrema of the energy are given by: 2KE:^ G&^
,
these limits exclude any divergences in the denominator of E. In the limit of zero intra
JAHN-TELLER
level coupling this state gives the minimum of Id) with += 0 discussed in Section 2.2. For sufficiently strong interlevel coupling the solution with mixed distortions is the absoluje minimum of energy. This result is important. In all cases treated previously the distortions which minimise the energy have been pure trigonal or pure tetragonal; mixed distortions do not provide absolute minima in isolated E or T levels. Other mixed solutions correspond to Id) and ]e> with # = 314; both are saddle points, and neither of these are minima. The solution for Id) has displacements (@,E, 6, q, 1;) equal to (-p’, 0, kq‘, *q’, 2~‘) and cyclic permuta- tions in the five-coordinate space. The le) solution has distortions (p”, 0, -+q”, izq”, kp”) together with cyclic permutations.
3. OTHER CASES OF NEAR DEGENERACY Certain other cases of near degeneracy lead to mixed disto~ions, and these will be discussed in the present section. Although we shall not discuss all cases of near degen- eracy,theonesweomit (A+A,A+E,E+E) are straightforward, and need no special comment. In the E + E case the secular equation can be factorised and, as for an isolated E level, the energies are functions of (e2 + P) and not of E or 13separately. For A + E, as has been observed by Elkin and Watkins[7], in an analogous case, the energy does depend on E and 0 separately; the interaction with the A state effectively produces a cubic anharmonic term which stabilises certain combinations of E and 8 distortions. In all our analysis below we refer the final energies to the centroids of the two levels
INSTABILITY (^1773)
for zero distortion. Our convention is arbit- rary, but it simplifies discussion when the order of the levels is inverted.
(a) A,+Tzor&+T, For zero distortion the T level has energy -A/4 and the A level 3A/4. The matrix giving the Jahn-Teller terms is:
There are three types of extremum. The first is the pure tetragonal distortion found for an isolated T level, with energy -A/4 - G&I&. There are three equivalent minima of this type. Secondly, there are the four equivalent trigonal distortions with IS] = 131= I<]. These correspond to the trigonal minima of an isolated T level. Although the ~sto~ions which give energy minima for an isolated T level still give extrema, they are only minima when GTX is zero. At the minima in the present case the energy has a complicated form which we give only in the limit A= 0:
E = -{ JGnl + v(G&TS. 3G$X)}2/12K,. (3.2)
The distortion has, for example:
Finally, there is a mixed distortion, with six equivalent minima. One such minimum has E= 5 = 17= 0 and finite values of 8 and 5:
JAHN-TELLER INSTABILITY 1715
in E(Q). As the solutions are complicated, we introduce the abbreviations: Iel = 3%J(^ 1-^ ($)j&
YE= G~&IKE ItI=^ Iql^
YT= GxIKT Y2 = (3YE+YT)/
[ 1
1-h ~/((2YT+YE)(YT-%)) 4y
3 47,--Y, I.
Y3 =YT(YT+2?%)i(4YT-YE). (^) To summarise, we have:
The first solution gives six equivalent minima. (^) 6) YE < YT : E, < E3 < E, These are orthorhombic, one of them having (ii) & < -)‘T< YE : E, < E, < E E finite and 19= 6 = r) = 5 = 0. Its energy is (^) (iii) yT < +yYE : E, < E2.
El=-y++(-&r} (3.13) (‘) :TT This is very similar to the example given in
and the distortion is: detail in Section 2, (E + T,). The Jahn-Teller matrix elements differ from the earlier
e=i2J(1-(&)‘).
example in that the elements involving (3’141 je) and 10) shouldbeinterchanged,andthose in 10) changed in sign. The energy reductions These are valid when 4y, > 1 Al. The second (^) and distortions are not affected, but the wave solution gives a mixed distortion, with functions are altered. The results for T, + E 8 and 5 finite, and E= t= 7 = 0. It has C, (^) can be obtained from the earlier results by symmetry, and energy: the substitution of -16) for 1~) and of Ie) for IO).
E, = -yz{ 1 t(k)‘} (3.15)
valid when 4y, > IAl. There are 12 such dis- tortions, for example:
The third solution also gives a mixed distor- tion, but with 8= {= 0, and 151= 171. The energy at this minimum is:
6, = -y3{1+ ($3 (3.17)
In this section we discuss briefly a number of systems which illustrate the Jahn-Teller effect for near, rather than exact, degeneracy. The systems described all show mixed tri- gonal and tetragonal distortions. Examples are the negative vacancy in silicon[3] and the transition ions Pd- and Pt- in silicon and Ni- in germanium[8]. The neutral vacancy in diamond may exhibit mixed distortions also, but the evidence is less direct. We will only discuss the vacancy centres in valence crystals, and will outline the reasons for expecting them to show the effects of near degeneracy. Watkins has emphasised [9, for example] that the Jahn-Teller terms are particularly large in silicon. valid when 4y, > IA/ and 4yT > YE. In this It is easy to show that, for these vacancy case the symmetry is C,, and there are 24 centres, there are likely to be energy levels equivalent sets of distortions, corresponding nearly^ degenerate^ with^ the^ ground^ state^ of to the various permutations of signs and inter- the centre. The electronic properties of change of cube axes with: vacancies in valence crystals are largely
- COMPARISON WITH EXPERIMENT
(^1776) A. M. STONEHAM and M. LANNOO
determined by the so-called ‘defect electrons’. In the neutral vacancy, for instance, four bonds were broken in foxing the centre; the four electrons on the neighbours which previously participated in the bonds are the defect electrons. New molecular orbitals can be formed by taking linear combinations of the broken bond orbitals. In particular the molecular orbital of lowest energy, Iv}, has A, symmetry; above it are three degenerate T2 orbitals, It,), jr,) and It,). These one- electron orbitals are filled successively by the defect electrons, giving electronic configura- tions v’Y~‘.However, from any one configura- tion a number of many-electron states may be constructed, and these states will, in general be close in energy[4]. For the negative vacancy in silicon three states of the same spin (2E. ‘Tt and ZT2) and one other (4A,) derive from the configuration u”P. For the neutral vacancy in diamond there are again three states of the same multiplicity (‘E, ‘T:, and IA,) and one of different multiplicity (Yr,) which all derive from v”t”. In the simplest models of the vacancy centres all states which derive from the same configuration u”P are exactly degenerate. The degeneracy will be removed by the electron-electron interaction, which admixes states of the same symmetry derived from *“-ltJl!+l. 2)ll--21,,l+Z. 1. Such configuration interaction is very’i~portant in diamond and, as we shall show, may cause large enough splittings to inhibit Jahn-Teller instabilities. On the other hand, in silicon, Watkins has found that the con~guration interaction is much smaller, and may often be ignored. For this reason he has been able to interpret his data in terms of a one-electron model, rather than in terms of Jahn-Teller coupling between many-electron states which are close in energy. The two pictures have much in common, for example, both recognise that the matrix elements which cause the instability are linear in the distortions, and indeed the one-electron model can be considered as a special case of the many-electron picture.
We shafl not compare the two viewpoints in detail here, but merely observe that Watkins’ model is particularly useful for understanding the nature of the ground states of these centres u posteriori, when the con- figuration interaction is known to be weak. The approach via the many-electron states of low energy is more appropriate when one wants to predict the ground state I( priuvi, since this approach involves a more general for- mulation of the Jahn-Teller problem and also detailed consideration of the energy separa- tions of the various many electron states. We conclude this section by discussing the neutral vacancy in diamond, since the centre has been studied extensively theoretically, and it is possible to give some idea of the relative importance of the configuration inter- action and Jahn-Teller terms. Although the Jahn-Teller terms dominate in silicon, the situation is less clear in diamond. The evidence for mixed trigonal and tetragonal distortions of the neutral vacancy in diamond comes from the stress splitting of the zero- phonon line of the GRI band, often attributed to the neutral vacancyl4,Sl. Theoretical treatments suggest that the ground state is ‘E, separated by some 0.05 eV from the “T, state; the next state, ‘TB, is about 2 eV higher in energy. The GRl band is attributed to transitions between the ‘E and ‘7’, states. The interlevel coupling which gives rise to the mixed distortions can occur in two obvious ways: coupling between ‘E and “T1, or coupl- ing between ‘E and ‘Tz. Although IE and “T, are very close in energy, their coupling depends on the spin-orbit coupling. In dia- mond the spin-orbit coupling is very weak, and the difference in multiplicity effectively reduces GTX by^ about^ lo-:‘.^ The^ interaction is then too weak; mixed distortions would only occur if the two levels were only separ- ated by about 10m7eV. The ‘E and ‘T, levels have a larger separation, and should exhibit an instability if GTX exceeds about 4 eVIA. Calculations based on the point-ion model
- I l] give 2.6 eV/A, and those based on the
