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Configurational instabilities at isoelectronic centres in silicon
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
Configurational Instabilities at Isoelectronic Centres in Silicon Gordon Davies
Physics Department, King’s College London, Strand, London WC2R 2LS, UK
Received May 9,1994; accepted June 15,1994
Abstract When an exciton is bound on an isoelectronic centre, one particle (electron or hole) may be severely localised on the centre. In this paper attention is drawn to the importance of the lattice relaxations stimulated by this local- ised charge density. Two examples are discussed in detail for centres where the exciton binding is produced primarily by the relaxation. It is shown that Jahn-Teller theory applied to the tightly bound particle allows a very precise understanding of the effects of external perturbations on the bound exciton.
1. Introduction Luminescence from isoelectronic centres has been investi- gated for over three decades [l]. One motivation has been the very high quantum efficiency of the centres, which results from the centres having the same valence properties as the host atoms - recombination of an exciton leaves the centre in a “particle-free” state, in contrast to the recombi- nation of an exciton at a charged centre where the addi- tional charge may be excited in a non-radiative (e.g. Auger) process. However, despite the considerable studies of these centres, it is still not clear how their excited states are bound on the centres [2]. The prevalent view is that one particle (electron or hole) is bound by a short-range potential and the second (hole or electron) is trapped in the Coulomb field of the first [3]. The second particle may then have a quasi- Rydberg series of excited states, and such a series has been observed at some centres [4]. It has long been recognised that modifications to the exciton structure occur when the symmetry of the ‘centre is lower than tetrahedral, and the axial properties may be described by a local strain field, as discussed by Gislason et al. [SI. In this paper we introduce a further concept where the local strain is produced by the presence of the exciton. We will discuss two examples of isoelectronic centres in silicon which illustrate some of the consequences of the lattice relaxations.
2. A simple description
The Hopfield-Thomas-Lynch model of bound excitons [3] assumes that one particle (say the electron) is spatially located on the core of the isoelectronic centre, the binding being derived possibly from the change in electron affinity at the centre. The second particle (the hole) is then bound by the Coulomb field of the electron. ,Since the electron is highly localised, its charge density is approximately point- like on the scale of the hole orbital, and the hole orbits in an approximately effective-mass-like state.
At this stage we introduce an ingredient which does not seem to have been explicitly discussed before in the context
of binding the exciton. The high charge density of the local- ised electron will force a local lattice relaxation. For a simple general discussion we consider first only hydrostatic deformations. A uniform hydrostatic deformation (V - Vo)/Vo of the correct sign reduces the energy of the elec-
tron by A(cll + 2c12)(V - Vo)/3Vo where A is the change in electron energy per unit hydrostatic pressure. Consequently the energy of the electron is reduced if the lattice deforms over the volume Vo occupied by the electron orbital. In the lattice-continuum approximation, the elastic energy required for this deformation is (cll + 2c12)(V - V0)2/6Vo, assuming it is a uniform deformation. The total energy E of the electron plus the deformed lattice is
E = (cll + 2c12)C(~ - v ~ ) ~ / ~ V ~ + A(V - V0)/3V01,
E,, = - ( ~ 1 1 + 2C12)A2/6Vo.
(1)
(2)
and is minimised (for a given Vo) at V - Vo = -A, when
The energy saved by the deformation increases as Vo decreases, so that the lattice relaxation tends to increase the localisation of the electron. However, the limit V, + 0 is prohibited by the increasing kinetic energy of the localised electron.
A useful qualitative picture is that the energy reduction is produced by the local strain acting on the one electron, but to generate that strain involves deforming n chemical bonds within the extent of the electron orbital; the smaller n and the larger the strain that can be produced. Consequently self-binding effects can only occur when at least one particle is in a small orbital. To illustrate the potential importance of lattice relax-
ations for a real centre, suppose that the isoelectronic centre has one substitutional impurity atom, and that the electron is suf€iciently localised that the lattice deformation is limited to the four nearest neighbour atoms. We label these atoms as a, b, c, d at coordinates i l l , 111, 111, 111. The hydro- static breathing mode with a coordinate Q defined by
Q =(-% + x b + xc - xd + Y a - Y b + Yc - Y d
- -
+ z a + z b - z, - z d ) / f i (3) has an effective mass equal to the mass M of one atom. A hydrostatic stress s produces a movement Q = 21s/(cll + 2c,,) where 1 is the interatomic spacing. This movement increases the elastic energy by ~Mw’Q’ , but decreases the electron energy by 3A(c1, + 2cl,)Q/21. Stability is reached when the total energy is reduced by AE = 9A2(cll + 2c12)’/8M1202, (4) where co is the angular frequency of the breathing mode. We take A = 3meV/GPa and co = 3.2 x 1013s-’ (both values derived below from experiment), and use the perfect lattice values for the interatomic spacing I (0.234nm) and the
Physica Scripta T54
8 G. Davies
elastic constants c l l + 2c12 (298 GPa). The calculated value of AE = 55meV shows that the lattice relaxation produced by the electron can be a considerable contribution to its binding energy, even for the relatively small value of the deformation potential A used here, as long as the electron is sufficiently localised.
All this discussion can be rephrased for the alternative case of a hole-attractive centre, for which the electron is the effective-mass-like particle, and we will consider examples of both types of centre. We note that when the electron and hole recombine, the high charge density of the localised par- ticle is removed, so that the source of the lattice relaxation disappears. The properties of the vibronic sideband produc- ed by the luminescence transition give us information on the lattice relaxation, as we see next.
3. The “ABC” centre - an electron trap
One well-known centre in silicon, refered to as the ABC centre, produces luminescence with zero-phonon lines at 1122.3 meV at low temperature, corresponding to an exciton binding energy (measured from the free exciton level) of 32.9meV [a. Its chemical origin is not certain, except that the centre contains nitrogen [A. The luminescence band, Fig. 1, has a vibronic sideband with a centroid 35 f 3meV below the zero-phonon line - from standard vibronic theory [SI this energy is the lattice relaxation AE of eq. (4). The vibronic sideband has a dominant peak at w = 3.2 x ~ O ” S - ~ (the value used in Section 2) which is not a critical point in the phonon distribution [9], and phonons with a wide range of wavevectors are present in the phonon sideband (Fig. 1). This spread in wavevectors implies that we are dealing with a transition involving a highly localised state.
Since we are interested in the effects of lattice relaxations on the excited states, we consider the effects of externally applied, controlled uniaxial stresses on the zero-phonon optical transitions. Sample data are shown in Fig. 2 for the effects of (001) compressions. The points [lo] show the effects of stress as measured at 20 K (to give adequate popu- lation of the higher excited states), and the lines are a calcu- lated fit (taking into account the effects also of (111) and (1 10) compressions). The fit is based on a model in which an electron is tightly bound in a non-degenerate orbital state, and the hole is weakly bound. The 6 conduction band minima, denoted X, 8 . . . 2, form states transforming as Al, E, and & in the Td point group of a substitutional atom. For an electron-attractive centre we expect the A, state to be lowest in energy (as for a substitutional donor), and the fit in Fig. 2 (made with this assumption) shows this to be the case for the ABC centre. Zeeman measurements [6] on the luminescence emitted by the bound exciton show that it is trapped on the centre in a trigonal symmetry. In trigonal symmetry the A, electron state may be perturbed in first order by the hydrostatic component (sxx + syY + szz) of the stress. Group theory also allows perturbation by a symmetry-maintaining compression along the trigonal axis, (sXy + syz + szJ. However, sheer stresses have no effect on the conduction band minima of silicon, and so the electron cannot be perturbed by any symmetry lowering stresses. The result is that no information can be derived about the trigonal distortion from the effects of stress on the tightly-
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R
A
~
1048 1120 960
Photon energy (mev) Fig. 1. Photoluminescence recorded at 20K from the “ABC“ isoelectronic centre in silicon. Zero-phonon lines A and C produce the uniaxial stress splittings shown in Fig. 2 Superimposed on the acoustic phonon sideband is emission from the free excitons (with the involvement of one transverse optic phonon), and the baseline for this emission is indicated. (This baseline is confirmed by the bandshape of the ABC system at low temperature). The extent of the one-phonon sideband is shown by the sharp cutoff at the Raman phonon, indicated by ‘R”. Experimental data by M. Zafar Iqbd cm
bound electron, even though it may be orbitting in a highly deformed region of the crystal.
The trigonal symmetry splits the mj = 3 valence band states [SI, and is primarily responsible for the 3meV split- ting observed at zero stress. All the stress-induced splitting on Fig. 2 derives from the perturbation of the hole by the stresses. The fit is achieved with the deformation potentials of the hole being within 25% of those of the valence band states [ll]; the small change would be consistent with the hole being relatively de-localised. Small deviations could occur simply from local changes in the elastic constants as well as from the modifications of the hole’s properties pro- duced by its being bound. The fit on Fig. 2 uses a hydro- static response for all the states of 3meV/GPa, and the fact
Configurational Instabilities at Isoelectronic Centres in Silicon 9
1130 r I n
1120
1110
0 700 Compressive stress (MPa)
Fig. 2. Effects of uniaxial stresses on the A (1122meV) and C (ll2SmeV) zero-phonon lines, as recorded at 20K for compressions along the (001) axis. The lines are the calculated effects of stresses on those transitions which are predicted to be optically allowed. The B line, with zero-stress value near 1121 meV, is induced by the stresses. Experimental data by M. Zafar Iqbal [lo].
that the same value can be used for all the transitions is consistent with the common parentage of all the states.
The 3meV splitting at zero applied stress is equivalent to a compression along (111) of about 94MPa, and presum- ably reflects the (1 11) axis of the core of the centre, as seen by the relative delocalised hole. Paradoxically, in this example the tightly bound particle (the electron) gives no symmetry information about the core of the centre, because it has no response to symmetry-lowering stresses. However, in some cases the tightly bound particle can indicate a sym- metry which is quite different from the symmetry of the centre in its relaxed ground state, as is shown in the second example.
4. The 4Li "Q" centre - a hole trap
Luminescence with zero-phonon lines at 1044 and 1045 meV is observed in Lidoped silicon after radiation damage, Fig. 3 [12]. Isotope doping studies have shown that the centre contains 4 Li atoms [13]. First-principles calculations suggest that the ground state of the complex has T, sym- metry [14], but uniaxial stress measurements on the optical transitions indicate unambiguously a trigonal symmetry [l5]. Relevant data are reproduced in Fig. 4. Before dis- cussing them we note that the binding energy relative to the free exciton is llOmeV, and the centroid of the lumines- cence band is 100meV below the zero-phonon lines - as for the ABC centre, the binding appears to originate mainly from the vibronic relaxation.
The vibronic relaxation may involve atomic motion which is totally symmetric in the original T, point group, and it may involve deformations along a (111) axis to
Q
QH
970 1000 1030 1060 Photon energy (meV)
Fig. 3. Vibronic bandshape of the Li-related "Q" transition, recorded at 6 K, after Ref. [ 131. The zero-phonon lines are labelled Q ("S = 0" line) Qr ("S = 1" line) and QH for lumincsccnce from the valley-orbit split-off elec- tron state. The extent of the one-phonon sideband is shown by the sharp cutoff at the Raman phonon, indicated by 'R'. Peaks A, B, C are resonance modes produced by the Li atoms.
produce the observed trigonal symmetry. We can establish that it is predominantly trigonal. &man measurements show that the lowest energy transition at 1044meV is from an excited state which appears, at the spectral resolution, to be a spin triplet (S = 1) to a spin zero ground state, while the 1045meV line is from an "S = 0" excited state to the
lo50 r
0 100 0 100
Compressive stress (MPa) Fig. 4. The points show the effects of uniaxial stresses on the Li-related Q line (1045meQ QL line (1044meV) and the valley-orbit state QH (1048meV) for (a) (001) and (b) (111) compressions [lq. The lines are calculated as described in Seaion 4. Thicker lines are for transitions origin- ating (at low stresses) in the "S = 0" states, and thinner lines for "S = 1" states.
Physica Scripta T54
10 G. Davies
same spin zero ground state [16]. The spin states are h e - diate evidence that the orbital angular momentum of the hole has been quenched, that is, that the hole is in an axial field (with (111) orientation) at the centre [SI. To estimate the strength of the axial field we represent the valence band of silicon by the three degenerate orbital states P, , P, , Pz states; these states are mixed by the spin-orbit interaction to produce the j = $ and 8 valence band maxima. For example, the j = 3, mi = state has the form p z ( P x + iPy)J - 4 P z t and the j = 3, mi = state is Jf ( P , + iPy)J + ,& P z t . We can recover a pure Pz t state by taking the combination p~(3, 3) - A(+, *). For signifi- cant quenching of the orbital angular momentum, the axial field at the centre must be large compared to the spin-orbit interaction (44meV for the valence band of Si). Using a per- turbation V which is diagonal in the P, , P , , Pz set, so that (Px , VPX, = (P, , VP,) = - Q, (Pz , V P 3 = 2Q, we can cal- culate the mixture of the j , mi states by this perturbation, and hence calculate the strength of the optical transitions from an electron in a totally symmetric orbital. The result is shown in Fig. 5, where allowance has been made for the Werent trigonal orientations in the crystal. The points are the observed relative strengths of the “S = 1” to “S = 0” transitions for a series of Li-related and S-related bands, with the perturbation taken to be equal to the observed binding energy, i.e. the energy difference of the free exciton and the zero-phonon line. For the 4-Li centre (labelled Q on Fig. 5 ) the axial perturbation required to produce the observed transition ratio of 1/70 [16] is closely equal to the total binding energy, which we have seen is equal to the relaxation energy - the atomic relaxation produced by binding the hole is essentially all axial.
We now have a picture for the bound exciton as consist- ing of a hole, which is tightly bound by about 100meV in
r
0 \ G 0 cl
-2.0 1 -2.5
0 40 80 120 Binding energy (mev)
Fig. 5. Points show the ratios of the “S = 1” and “S = 0” zero-phonon lines for a series of similar Li-related vibronic bands in silicon [Zl], with the Li-related band discussed in section 4 labelled Q [20]. The exciton binding energy is the difference between the free-cxciton energy and the observed zero-phonon line. The line shows the calculated ratio assuming a 44meV valencsband spinabit coupling, and equating all the binding energy to the axial component.
Physiccr Scripta T54
the (111) axial field of the centre, plus an electron in rela- tively shallow states, bound by the Coulomb field of the trapped hole. As for the ABC centre, the 6 lowest conduc- tion band states of the electron transform as A,, E, and T2 in the T, symmetry of the unrelaxed centre. It turns out that the ordering is that the E state is lowest in energy and the A, state next. These states are not affected by the trigonal field of the centre (except for a possible shift in their energy centroid). However, a (001) compression produces a shift As from the hydrostatic component of the stress and addi- tionally the E and A, electron states are perturbed accord- ing to the matrix [lq ’[ -2; Bel% -$Belsc
Here the origin of the energy is the E state and the A, state lies E,, above it at zero stress. In terms of the stress tensors Sri dehed with respect to the crystal‘s cube axes,
(6) Only one parameter, Bel, controls the splitting of the E
state and its coupling to the A, state. The lines on Fig. 4(a) have been drawn with Be, = 13meV/GPa, similar to the value of 11 meV/GPa for the conduction band states [ll], confirming that the shallow-electron approach is correct, and rationalising the existence of the 1048 meV optical tran- sition as the A, electron state. The hydrostatic term is A = 4.25meV/GPa similar to the value for the exciton bound to the ABC centre, Section 3.
Under (111) stress, the electron states are not split, and all the splittings on Fig. 4(b) arise from the hole. Because the axial perturbation (- 100meV) considerably exceeds the spin-orbit coupling (of 44meV) we can use a very simple description of the stress effects in which we ignore the spin- orbit interaction. The valence band is then represented by three degenerate orbitals, p , , p,,, pz where we use the con- ventional cubic axes x, y, z of the crystal. The non- degenerate hole state for the [lll] centre is the linear combination (p, + py + p J d . An external compressive stress s along [lll] perturbs the centre by As + 2Cs/3 where the first term is for the hydrostatic component and C dehes the coupling of pi and pj by a sheer stress sij as Cs,. Similarly the hole at a centre oriented along [lil] is expected to be perturbed by As - 2Cs/9. To fit the points on Fig. 4(b) requires C = 32.6meV/GPa, considerably reduced from the valence band value of C, = 52meV/GPa
To understand the reduction we introduce the language of the Jahn-Teller effect. A triply degenerate T, state (the p orbitals) in T, symmetry can couple to a mode of vibration which transforms as T, . The coupling leaves a T, lowest in energy, but this is now a vibronic level, rather than a purely electronic level. An A, level from the first vibrational level at zerocoupling lies an energy 48 above the lowest level, where [ 181
Eo E, A ,
Ai -$Belse -$Bels, E,
] (5) Bels, -$Belse
se = 2s,, - s, - sn, se = d ( S z . - Sn).
c111.
48 - 1.32E,r exp (- l.24EIT/ho). (7) In the limit of strong coupling, when the relaxation energy EjT + CO, a four-fold degenerate ground state is obtained,
Configurational Instabilities at Isoelectronic Centres in Silicon 1 1
corresponding to distortions of the centre along the four (111) axes. For the Li-related centre, E,, - llOmeV, and taking the effective vibrational mode to have a quantum h o = 25meV (as a rough mean of the one-phonon quanta in Fig. 3), EIT/hu = 4.4, and the strong-coupling limit has almost been reached. Because the electronic ground state is replaced by a vibronic ground state spread out over the four distortion axes, the effect of a given perturbation on the lowest vibronic level is reduced relative to its effect on the purely electronic states. The reduction factor for a sheer stress is k(T,) = 213 for strong coupling (and is correct to within a few percent for the finite coupling here, Ref. [19]). The effect of sheer stresses on the valence band, C, = 52meV/GPa, will therefore be reduced to C = 34.7meV/ GPa for the trapped hole, very close to the value derived in the last paragraph. Both the electron and hole behaviour can thus be understood with remarkable precision simply in terms of the deformation potentials of the band edges.
Evaluating eq. (7, the tunneling splitting 48 - 0.62 meV. There are no data on this value. (It is not observed optically, and will be too close to the zerephonon line to be readily detected in a stress experiment). Using this value as a guide, we can estimate the extent of the trapping in each distortion in terms of the time 7 required to tunnel from one distortion to another:
(8)
7 is in any event substantially smaller than the radiative decay time of 9 . 5 ~ ~ for the S = 0 transition [20], and the Merent distortions can communicate with each other. This allows us to observe an unusual phenomenon. Under [lll] stress a repulsion occurs between the S = 1 state of the Cl113 centres and the S = 0 states of the other orientations (such as the [lil] centres). If the trigonal axis of the centre was an intrinsic property, produced by its molecular struc- ture, then this interaction would not occur, since the centres would be spatially separated in the crystal.
The situation here is that, in contrast to the usual cases considered in Jahn-Teller theory, there are two adiabatic surfaces, one for the “S = 1” states and one for the “S = 0” state, separated by AE = 1 meV. Under [111] stress, tran- sitions from the [lll] centres decrease in energy and those from the [lil] centres increase, bringing the two adiabatic surfaces into coincidence near s = 9AE/8C = 35 MPa. At this stage a quantum-mechanical repulsion can occur between the lowest energy vibronic states associated with the two surfaces. If we were dealing with true S = 1 and S = 0 states there would be no observed repulsion, since the coupling is primarily through the orbital effects associated with the lattice strain and so would be zero between orthog- onal spin states. However, because these are not pure spin states there will be a repulsion. The curves on Fig. 4(b) are calculated with a coupling of &2.5meV/GPa between the different distortions, about an order of magnitude decreased from the first-order perturbation of the states. This reduction reflects the admixture of the valence band states in the ”S = 1” and “S = 0” states. The problem is analogous to the ratio R of the optical transition intensities of the singlet and triplet lines, except that the transition intensity depends on the square of the wavefunctions. We expect the coupling to be reduced by ,h? - 10 from C, in agreement with the observed effect.
7 - h/b - 4 x 10“2s.
5. Summary
This paper has drawn attention to the role of lattice relax- ations in binding excitons at isoelectronic centres in silicon. When an electron is tightly bound, an A, orbital is likely, and the consequent lack of orbital degeneracy implies that it is difficult to obtain data about the symmetry of the core of the centre. The hole moves in a relatively diffuse orbital, and so the symmetry observed for it may be the residual symmetry-lowering of the lattice observed at this large radius, or it may be a symmetry-lowering relaxation pro- duced by the hole states. The symmetry of the relaxed ground state is difficult to obtain. This point has been emphasised for a tightly bound hole centre, where the tri- gonal symmetry has been shown to derive from lattice relax- ation effects. However, it is then possible to use the well-established results of Jahn-Teller theory as developed for the limit of strong coupling to obtain a very precise understanding of the bound exciton states of these centres.
Acknowledgements This work was supported by the Science and Engineering Research Council.
References 1.
2
3.
4.
5.
6. 7.
8.
9. 10. 11.
12
13.
1 4. 15.
16.
17. 18. 19. 20.
21.
See the review by Dean, P. J. and Herbert, D. C, in: ‘Topic in Current Physics” (Edited by K. Cho) (SpMget, Berlin 1979), vol. 14,
Discussions of the binding mechanisms are given by, e.g. Faulkner, R A., Phys Rev. 175,991 (1968), and are reviewed extensively in Ref. 1. Hopfield, J. J, Thomas, D. G. and Lynch, R T, Phys. Rev. Lett. 17, 312 (1966). Thewalt, M. L. W, Watkins, S. P, Ziemelis, U. 0, Lightowlem, E. C. and Henry, M. 0, Solid State Commun. 44, 573 (1982); Labric, D., Timusk, T. and Thewalt, M. L. W, Phys. Rev. Lett. 52,81(1984). Gislason, H. P., Monemar, B, Dean, P. J. and Herbert, D. C, Physica 117B & 118B, 269 (1983). Weber, J, Schmid, W. and Saw, R, Phys. Rev. B21,2401(1980). Alt, H. Ch. and Tapfer, L, “Roc. of the 13th Int. Cod. on Defects in Semiconductors” (Edited by L. C. Kimerling and J. M. Parsey Jr.) (The Metallurgical Society of AIME, Pennsylvania 1985), p. 833; Tahjima, M, Japanese J. Appl. Phys 21-1, 113 (1981); Sauer, R., Weber, J. and Zulehner, W, Appl. Phys. Lett. 44,440 (1984). Pryce, M. H. L, in: “Phonons in Perfect Lattices and in Lattices with Point Imperfections” (Edited by R W. H. Stephenson) (Oliver and Boyd, Edinburgh 1966), pp. 414425. Zdetsis, A. D, Chem. Phys. 40,345 (1979). Iqbal, M. Z., private communication (1993). Laude, L. D, Pollack, F. H. and Cardona, M, Phys. Rev. Lett. B3, 2623 (1971). Johnson, E. S., Compton, W. D, Noonan, J. R. and Streetman, B. C, J. Appl. Phys. 44, 511 (1973). Canham, L. T, Davies, G. and Lightowlers, E. C., J. Phys. C13, L757 (1980). Tarnow, E, J. Phys. Condensed Matter 4,1459 (1992). Davies, G, Canham, L. T. and Lightowlers, E. C, J. Phys. C17, L173
Canham, L. T, Davies, G. and Lightowlem, E. C, Inst. Phys. Con. Ser. 59,211 (1981). Wilson, D. K. and Feher, F, Phys Rev. 124,1068 (1961). Caner, M. and Englman, R, J. Chem. Phys. 44,4054 (1966). Sakamoto, N., J. Phys. C17,4791(1984). Lightawlers, E. C., Canham, L. T, Davies, G, Thewalt, M. L. W. and Watkins, S. P., Phys. Rev. BD, 4517 (1984). Lightowlers, E. C. and Davies, G, Solid State Commun. 53, 1055 (1985).
