A defect molecule calculation for the vacancy in graphite

A defect molecule calculation for the vacancy in graphite

Carbon, 1975, Vol. 13. pp 275-282. Pergamon Press. Printed in Great Britain A DEFECT MOLECULE CALCULATION THE VACANCY IN GRAPHITE Department FOR ...

916KB Sizes 3 Downloads 29 Views

Carbon, 1975, Vol. 13. pp 275-282.

Pergamon Press.

Printed in Great Britain



A. P. P. NICHOLSON and D. J. BACON of Metallurgy and Materials Science, The University of Liverpool, P.O. Box 147, Liverpool L69 3BX, England (Recehed 2 November 1974)

Abstract-The defect-molecule calculations of Coulson et al. [2] of the electronic structure of a vacancy in graphite have been improved and extended to allow for symmetric relaxation. Various problems encountered previously[l, 21 have been shown to be due to the use of the Goeppert-Mayer and Sklar approximation. The rebonding forces on the nearest-neighbour atoms to the vacancy have been calculated, and a perfect-lattice model was used to describe lattice relaxation effects; the difficulties experienced in work on diamond and silicon[5,6] using an imperfect-lattice model were avoided. For the neutral vacancy, the stabilization energy, Ev, was found to be 1.44 eV, and the (symmetric) relaxation energy, E,, to be 3.32 eV. The neutral vacancy formation energy has been re-calculated, and using the above values for E, and E,, a value of 8.75 eV was obtained. This is somewhat higher than experimental estimates, but the difference may be explained by a high value for the formation entropy.

1.INTRODUCTION In this paper we present some results obtained from a calculation of the electronic structure of the isolated single vacancy in graphite. The calculations were performed using the defect molecule method[l], which involves treating the defect and its immediate neighbouring atoms as a molecule contained within the infinite crystal matrix, and applying standard techniques of molecular orbital theory to obtain the molecular (and hence defect) electronic structure. Similar calculations were made by Coulson et al.[2], where the various contributions to the vacancy formation energy, E,‘, were estimated. All relaxation effects were ignored, mainly on the grounds that the basic wave functions were not accurate enough. The present work had two main objectives, namely: (i) to re-calculate the stabilization energy, Ev, and (ii) to calculate the forces on nearest-neighbour atoms arising from the electronic reorganisation. These forces are then used to represent the vacancy in relaxation calculations to give, in particular, the relaxation energy, ER.The values obtained for Ev and ER have been used to re-calculate E!, using the formulation of [2]. Since the publication of [2], various modifications to the defect molecule method have been made, mostly by Larkins and his co-workers in studies of vacancies in diamond-type structures [3-81. We have followed this work in treating all relaxation effects as a first-order perturbation on the undistorted system. A simple description of the valence electrons in graphite is that, per carbon atom, three are in sp2 trigonal hybrids and the 4th is in a 2p, n-orbital. An atom is joined to its three neighbours by localized u-bonds resulting from the overlap of the sp2 hybrids; the n-electrons are in non-localized orbitals extending over the whole plane and also contribute to the binding. When a vacancy is created, the following factors need to be considered in calculating the formation energy[2] (a)Rupture of three a-bonds. (b) Loss of a-electron energy.

(c) Interaction amongst the u-electrons remaining at the vacancy site. (d)The atom reattaches itself on the crystal surface. (e) Rupture of the inter-layer Van der Waal’s bonds. (f) Relaxation of the lattice. The first four factors can be taken to be properties of the individual layer alone. The major part of the present work has involved a re-calculation of factor (c) and the inclusion of possible relaxation effects, factor (f). These calculations are described in Sections 2, 3, 5 and 6: the remaining factors are discussed in more detail in Section 4 where the formation energy is calculated. 2.THEDEFECT MOLECULE METHOD There are two main assumptions made in using the defect molecule method, namely: (i) the electrons left at the vacancy site occupy orbitals which are localized in the region of the vacancy, and (ii) the energy levels of these ‘vacancy electrons’ are not affected by exchange or polarisation with other electrons. These two assumptions allow us to regard the vacancy as a molecule which consists of the three neighbouring atoms and the vacancy electrons, and is placed in a potential representing all other atoms in the crystal. Coulson and Larkins [4] made a detailed study of both of these assumptions and concluded that the limited delocalization ( - 10%) in diamond had virtually no effect on relative energies, or on the symmetric force term. However, in studies on silicon, it was found that the inclusion of exchange with other electrons had a noticeable effect. In our calculations we have therefore followed[2] in ignoring delocalization effects, and made our calculations both with and without exchange included. As far as the undistorted vacancy is concerned, we have followed[2] closely; the main points of difference are: the effect of overlap has been included in the normalisation of the single-electron wave functions; the Goeppert-Mayer and Sklar approximation[9] was not used in calculating the potential, and exchange terms were 275

CAR Vol. 13. No. 4-B



included in 2 out of the 4 cases considered; and all the necessary basic integrals were evaluated explicitly rather than interpolating from tables. In Fig. 1 we show the 3 atomic orbitals Q, 6, c, associated with the nearest-neighbour atoms A, B, and C to the vacancy site 0. In general, we have to place n vacancy electrons in these orbitals. (n = 3 for a neutral vacancy V”, n = 4 for a negative vacancy V-, etc.) The first step is to form single-electron molecular orbitals of the correct symmetry (L&h) from a, b and c. The n-electron wave functions for the molecule are then constructed using Slater determinants. Finally configuration interaction is allowed between states of identical symmetry.

‘&is the kinetic energy operator for electron i, and V, is the potential energy for electron i due to (i) attraction to the nucleus of atom N, (ii) repulsion of K-shell electrons on N, (iii) repulsion of the electrons in the remaining trigonal hybrids around N, (iv) repulsion of the r-electron on N, and (v) an attractive exchange interaction with the bonded electrons on N. In[2], the last contribution was omitted, and the remainder evaluated using the Goeppe~-Mayer and Sklar approximation[9], writing tUlVNila)=(UlVONilU)-Q

where Q = (a(1) n(?)/~/

Fig. 1. Schematic representation of the three sp’ trigonal hybrid orbitals on the nearest neighbom atoms A&C directed towards the vacancy 0.

For the single-electron wave functions, we have: t, = N” (a t b + c) x=N;(2a-b-c)


Y = NY @ - c),

where N, = (3 + 6s))I’*, N, = (6 - 6s)“* = 3-“*NY, and S = (a 1b ) is the overlap integral. The inclusion of overlap in K, N, and NY lessens the dominance of the ground states by confi~rations with maximum occupancy of v. Following the usual procedure, the determin~~l wavefunctions are set up, and their symmetries and spin multiplicities determined. This has been carried out for n = 2,3 and 4 vacancy electrons. The results obtained are in agreement with those given in Table 2 of [2], except that they appear to have omitted the ‘A; from the (gxy) confi~ration for V”; thus only 5 levels are given in Fig. 5 of [2], instead of 6. Since the ‘Ai state is non-degenerate, and well away from the ground state, this omission has no serious effect. The spin-independent Hamiltonian may be expressed as PI

x=%f+&9’, where

and X1 = x l/fii, i
i,j = 1,2.. . . .n.


a (1) a (2)) is the one-centre

Coulomb integral, and V& is the (spherically symmetric) potential due to a neutral carbon atom with sp3 orbitals. Using the value of Q obtained from the analytic basis functions, unsatisfactory results were obtained by [ I] and[2]. A modified theory, using semi-empirical values for Q and the self-penetration integral, P = (a]T1 + Val]a) was therefore used. We have made 4 calculations, 2 for the analytic unmodified value of Q, QU = 19.415eV, and 2 for the modified value, Qnt,taken as in[2] to be 13,203eV. In each case the calc~ations were made with either exchange terms included, for which P is a constant independent of Q, or with exchange terms excluded, for which P has been chosen using a relation equivalent to eqn (3). The energy levels for a particular charge state can now be determined by evaluating the matrix elements (~~(R)~~l~i(R)), where $(R) is the determin~tal wave-function of symmetry R, belonging to configuration i. All the elements required can be expressed in terms of nine l- and 2-electron integrals, A-K, involving u, x and y, and these can each be broken down into similar integrals involving u, b and c. These integrals in turn can be expressed in terms of the basic atomic s and p orbitals. FollowingI21, we have used analytic Slater functions, with exponent 1,595. Ail the necessary basic integrals were evaluated using the MIDIAT program[lO]. The only remaining problem concerns 3-centre integrals. These were evaluated using the Muiliken approximation [ 111.

The energy levels of the undistorted positive, neutral, and negative vacancies (V’, V” and V-) have been calculated using the method outlined for the 4 different cases, as described in Section 2. Values of the 9 molecular integrals A-K required to calculate the energy levels are given for the 4 cases in Table 1, along with the values obtained in 121. The energy levels before and after configuration interaction for the 6 cases considered (See Table 1) are shown in Figs. 2-4 for V’, V0 and V- respectively. Some of the degenerate levels of V’ and V- after configuration interaction as given in[2] appear to have used the wrong normalisation factors for the confi~ration wavefunctions. The values shown are the corrected ones.


A defect molecule calculation for the vacancy in graphite Table 1. Values of the l-and 2-electron molecular integrals (in eV) @I*





























Self-Penetration Intqral, P



B= cxjtpjx>/X C = cwjH'jw>/t"


E =

F = cqlH'lxy>/X2





D =

10.906 &Ins as case (1)

10.256 8.574



same as

case (3)


case (5)

G = /TX




J = wyjH1/yx>/X2






K = /(VX~)~tJotgiven in C2f H%T1


V,, H1sl/rIpV=wjvz, and X=&>.

* Fran Cmlxln et al. C23. tkkxlifidQ, s = 13.203 ev. Urmdified Q, $1 = 19.415 ev.

Case 3

Case 2 a b

Case 1 b

Case 5

Case L



a Y



Case 6



50-K C_:/1E’ A___ 3A$




55-I&.-= __,’ IE’ -__%$ II’, 60-E -





A2 ', 55\ -

~ 3E' -*f-

3E' --_SO-'A? -\

x -.-

\5 i

3E' [email protected]‘?& \ \



Fig.2. Energy levels (in eV) for the undistorted positive vacancy V’before (a) and after(b) config~tion interaction. ~ases(l-6)~edefined inTable 1. There are several features which are common to all 3 charge states, and we discuss these first. A comparison of cases (5) and (6) shows that exchange terms have negligible effect on relative energies at least for the unmodi~ed theory. They appear to be more significant when using the modified theory. The modified theory results in much less overall spread of levels, as noted by [l] and [2]. The effect however appears to be several times smaller in our case. We believe the very large differences observed by [ I] and [2] were caused by use of the Goeppert-Mayer and Sklar approximation. A comparison of cases (1) and (3) shows that[2] predicts slightly lower and more compact energy levels, most noticeably for V. This is in agreement with the observation made in the previous section that the omission of overlap in N,, N, and NYby[2] increases the dominance of v over x and y. The effect of configuration interaction is much larger for the unmodified theory than for the modified theory. This is definitely a point in favour of using the modified theory.

Concentrating on the results of the present work, cases (3)-(6), the following specific points related to each charge centre are of interest. (i) For all cases, the ground state of the positive vacancy V” is of m~imum symmetry ‘AI, mostly from ZJ’.With this ground state at least 5 eV above the ground state of Vc or V-, it appears most unlikely that V’ will be stable. Essentially the same conclusions were reached by ]21. (ii) The ground state of V- arises from the uZe2 configuration and appears to be the triplet “Ai. In all cases however, other levels are very close to the ground state, and relaxation effects are likely to be significant. For cases (4) and (6) the ground state of V is below that of V’, suggesting that V- could be a stable centre. The conclusions of [2] were again essentially as above, at least for an u~~jsfo~ed negative vacancy. (iii) For the neutral vacancy V’, the lowest state in all cases is ‘E’, arising mainly from o’e. After configuration interaction, the next lowest level, always 4A$, is at least


A. P. P. NICHOLSON and D. J. Case 1



Case 2





















Case 6 a b



X,’ -65-



iX&L___ 2E’/-










Fig. 3. Energy levels (in eV) for the undistorted neutral vacancy V” before (a) and after(b) configuration interaction. Cases (l-6) are defined in Table 1.Level 00 is the energy of the 3 electrons in the orbitals a, b, c without interaction, and the vertical arrow denotes the stabilisation energyEv whichis 3,28,0.97,3.96,1&l,1.35and 1.21eV,for cases(1)

to (6)respectively. Case 1

Case 3 a b

Case L a b

Case 5 a


Case 6 a b

I&/’ !-55l-


‘E’ --<3


\ -\.i5 3E’ ‘-

3E’ --50+$ E ---

-z 3A&_

Fig. 4. Energy levels (in eV) for the undistortednegativevacancy V- before(a)and after(h)configurationinteraction. Cases(l-6) aredefinedin Table 1.

2 eV above the ground state. Relaxation effects are unlikely therefore to change the ground state, and we look for dipole transitions from a *E’ state. For the modified theory the transition is to another ‘E’, the results being in reasonable agreement with those of [2]. For the unmodified theory, however, the transition is to 2A i and is much larger ( - 10 eV). Since the predicted ground state of V” is doubly degenerate, there must be a Jahn-Teller effect, distorting the lattice in such a way as to destroy this degeneracy. This has not been studied in the present work. We conclude this section with the most important result for the undistorted system, the stabilization energy Ev of V’. We use the same method as [2], subsequently applied to diamond and silicon[6]. Following[2], we have disregarded multiple exchange terms, since for a 3electron system these are likely to be insignificant. We

shall give no further details here, other than to note that there appears to be a misprint in the expression given for (abc (HI abc) in[2]. We believe the factor 6 for (ab IH’l ab) should be a 3. The results have been shown on Fig. 3. It can be seen that Ev is fairly insensitive to the inclusion or otherwise of exchange terms, and also to the value of Q. However, for the modified theory at least, large changes in P result in large changes in Ev. We believe that the high values obtained in cases (1) and (3) are a result of calculating P with the Goeppert-Mayer and Sklar approximation, and that the lower values obtained here are more reasonable. 4.TIIEFORMATION ENERGY, E.’ In this section we recalculate the formation energy of the neutral vacancy, Et, using the method developed in [2]. The calculations in [2] were shown by Mayer [ 121to

A defect molecule calculation for the vacancy in graphite

imply a very large value for the inter-layer binding energy, EC.Using a value of 2~2eV for Ev, and on the basis of various other assumptions, Mayer obtained values of 64-7.5 eV for E,‘, in good agreement with experiment. Since we believe there is a large relaxation energy (- 3 eV) associated with V*, it has been necessary to reconsider Mayer’s calculation. In[2], equations were set up for the sublimation energy Es and the formation energy E! as follows: &=;E,+E,i-EC-E, E,‘=(3E,,+AE,+2E,-Evf-E,-Es

(4) (5)

where E,, = energy to break one u-bond, but without change of trigonal orbitals. (Factor (a) in Section 1). E,, = n-electron dissociation energy per atom (Factor (b)). E, = inter-layer dissociation energy per atom (Factor (e)). Ep = promotional energy, i.e. the difference in energy between the ‘P ground state of a free carbon atom and the graphite valence state (Factor (d)). AE, = change in n-electron energy on removing the atom. (Factor (b)) and Ev = stabilization energy (Factor (c)). Mayer followed[2] in using these equations, substituted for E,, from (4) into (5), and obtained:


value for p, the so-called spectroscopic p. The original calculations of E, were also made including overlap, and we then have E, = - 1.08 (p -E& where S is the overlap, and E. the separated orbital energy. We follow Coulson and Taylor [ 161in using S = 0,25 and equating EO to the work function, given most recently as 4.68 eV[ 171. Then, for /3 = -40 kcalimole, we have E,, = 3.14 eV. A valuable discussion of the points raised above concerning values of p may be found in Cotton[l8], We are now left to choose between using E,, as in121,or E, and E,, as in Mayer[ 121.In[2], the only available value for E, was 6.93 eV, derived from spectroscopic datafl91. Mayer’s major criticism of [2] was that using this value in (4) would imply E, - 5 eV which is far too large. However, Mayer assumed that E, = 5.0 eV to derive this value of E,., quoting from a remark in Coulson and Poole[ 141 that the total bond energy in graphite is 5 eV/bond. We believe Mayer’s assumption to be questionable, because the total bond energy should also include contributions E, from the n-electrons, and EC from the interlayer binding. Writing ED as the total bond dissociation energy, we should have:

Substituting this equation for ED into (4) we see that the total bond energy/atom should be equated with Es t E,,, at least so far as these calculations are concerned. It has recently been pointed out[6] that many authors equate the total bond energy~atom with ES only, and we believe this to be the origin of the value f= 2/3 Es) quoted by Coulson and Poole. In fact, Coulson and Poole used a value of 6.81 eV for E,, which was obtained from calculations on E,‘=;E,+AE,-E-i-E, -Ev. (6) CH, and CHX[20]. Mayer found that, using a reasonable value for In [2] however, (4) was used to eliminate E,, and EC E,( -0.1 eV), and Ear= 5.0eV, much lower values from (3), giving: (1.6- 3.0 eV) were obtained for E, than the theoretical value 6.93eV. He quotes Coulson * (private E:’ = (Es t I$) + AE, - 2E, - Ev. (7) communication[l2]) in saying that such low values can be justified experimentally [21], and also theoretically from The sublimation energy, Es, is well established experi- arguments of Voge[22] that the resonance of the valence mentally to be 7.40 eV[13], and following Mayer we have state with s2pz configurations will cause a reduction of E,, used this value. The loss of r-resonance energy, AE,, was by 2-3 eV. However, Larkins and Stoneham [6] could find first calculated in [2] as AE,, = (- 2.75 f 0.05)/3 where 0 is no evidence to support the Voge hypothesis, and we the resonance integral. A more sophisticated calculation follow them in using the theoretically obtained value. For by Coulson and Poole [14] confirmed this estimate, and we diamond, with an sp’ valence state, they used a recently follow Mayer in using this value for AE,. Both previous obtained value of 7.46 eV. Goldfarb and Jaffe[19] calculations used E,, = - 1.5768 [ 151. obtained 6.73 eV for this energy, 0.2eV below that The resonance integral p is usually calculated by obtained for the graphite sp’~ valence state. We could comparing the theoretical value of E, obtained from therefore assume E,, = 7.46 t 0.2 = 7.66 eV. This is very simple Hiickel theory with the empirical value of the close to the value of 7.61 eV obtained for CH1[20], which, resonance energy. This gives p = - 16 to - 20 kcal/mole as pointed out by Coulson and Poole [ 141,we should be for virtually any simple molecule. It can be argued, able to use directly for graphite. We have thus used however, that the empirical resonance energy should also E,, = 7.61 eV for the present calculations. include an electronic reorganisation contribution, namely Finally, we consider the inter-layer dissociation energy, AE,,. Then a more suitable value for p would be El.. The most recent theoretical estimate of - 40 kcal/mole, as used by Mayer and by us. 0.11 eV/atom[l3], which we have used, hardly differs As noted above, the value of E,, = - 1.576/3 was from the value used by Mayer of 0.1 eV/atom based on obtained using simple Hiickel theory, thus neglecting earlier estimates. overlap. This is usually accounted for by an even higher A summary of the calculations made is given in Table 2.


A. P. P. NICHOLSON and D. J. BACON Table 2. Calculations made of vacancy formation energy (all in eV) cwlsonetal. c2'l,case(1) Sublimation-9y,




Mayer 7.4


Present wrk

Cl23 7.4


-0.87 -1.74


n-electronener~, En





Cbnge in Elr,AE II o-b-d energy,E,





8.6* (5.0)




(0.1) 5.5*

















Inter-layereneqy, EC Pmmtionaleneqy,


Stabilizationenersy,'s Rlamtion energy, s Lmmtion


*Indicatesvaluesderives franotherdata in m

Including now the relaxation term, ER, we can write, (8) 5.RELAXATION EFFECTS We now describe the way in which relaxation effects can be taken into account. All electronic levels can couple to the totally symmetric distortion mode, and this symmetric relaxation is the only one considered in the present work. We have used a variant of the method of Larkins and Stoneham [5,6] which was applied by them to the vacancy in diamond and silicon. The complete energy level scheme for V” and V- was calculated for 3 internuclear distances Ro, R. t h, and Ro- h, where R. = Rts = Ric = R”, is the perfect lattice parameter, a = 2.46& and h = 0.1 A. The force, Fo, acting on each nearest neighbour in the unrelaxed position, is then given by Fo= [E(Ro-h)-E(Rot



By calculating the forces F+ at R. t h/2and F- at Rc,- h /2, the first derivative of the force with respect to displacements, g = (aF/au)o, can also be evaluated by a similar equation, g = ti(F+

- F-)/h.


The subscript 0 denotes perfect lattice position. In order to determine the extent of the relaxation, and the relaxation energy, ER, it is obviously necessary to know the response of the lattice to such externally applied forces representing the vacancy. In the present work a harmonic lattice mode1has been used. In keeping with the basic assumption that the forces representing the vacancy derive solely from the electronic reorganisation, we have used an equilibrium valence-force model, as in[5-81. With this model, no relaxation of the lattice occurs on removal of an atom unless external forces are applied. In our case[24], a two-dimensional mode1 of the perfect graphite layer, based on the 3-dimensional model of Young and Koppe1[25], and involving 2 independent force constants fitted to C,, and G, was used. The perfect lattice response was determined using the method of lattice statics.




We define the linear response V,,(R) of the perfect (imperfect) lattice as the displacement of the nearestneighbour atoms to the vacancy due to 3 radial forces of unit strength acting at the 3 nearest neighbours. Then, within the harmonic approximation, the total displacement of the nearest neighbour atoms, u, can be written: u = Four = FRvp


where FR is the force acting in the relaxed position, FR= Fotgu


= F&l -go,).


The corresponding relaxation energy, ER, is then given by ER = -+F,u


where the factor 3 enters because there are 3 forces acting. This relaxation energy is the sum of both the electronic and elastic contributions, and the correct value to use for the lowering of the energy levels is exactly twice this amount [5,6]. The calculation of the symmetric relaxation can be made using either a perfect-lattice or an imperfect-lattice model. In either case, the problem can be described in terms of 3 parameters, Fo, g, and u,. In a perfect-lattice model, g represents a change in force with displacement, as in (12), whereas in an imperfect-lattice mode1 g represents a change in lattice response due to the presence of the defect. This can be seen by combining (11) and (12) to give: g ,1-L v,



In most relaxation calculations, the two approaches are completely equivalent, thereby giving a single value for g. In the present work, however, this is not the case, and for given values of F. and vp, it is possible to derive 2 different values for g. One value comes from (15) and is a


A defect molecule calculatiorI for the vacancy in graphite

property of the lattice model only, and the other value comes from (lo), and is a property of the molecular model only. In previous calculations [5-s] imperfect-lattice models have been used, which have led to unreasonably large distortions. It has been found[5,6] that the use of a rigid-atom model of the lattice in which all atoms except nearest neighbours are held fixed reduces the values obtained to an acceptable size. This is surprising, since a rigid-atom model is a fairly crude approximation. A detailed analysis [24] has therefore been made of the three possible lattice models, viz. perfect, imperfect and imperfect (rigid-atom), for the earlier studies of diamond and siliconlS,6] and for graphite. It has been shown that for the lattice models used, the value obtained for DI accounts for changes in the lattice up to and including 2nd neighbours, and that the large distortions obtained are a result of incompatibility with the nearest-neighbour molecular models of the defects. This problem does not arise with either a rigid-atom model or a perfect-lattice model. There are two further points in favour of a perfect-lattice model. Firstly, it is reasonable to suppose that a change in the defect forces also arises solely from electronic reorganisation. Secondly, the perfect-lattice model allows g to be dependent on the energy level and charge state of the defect. Hence, this model has been used for the results to be presented in the following section. 6.R~U~TSFORT~DISTORTEDVACA~CY

In a perfect-lattice model formulation, the only parameter not determined by the molecular model is the linear response u,. This was found to be[24] 21,= 0.01445 A2/eV. Values for FO and g have been obtained for all energy levels of V” and V-, using eqns (9) and (10). A summary of the results obtained for the 2E’ ground state of V0 is given in Table 3. The major criticism made in Section 3 of the results for unmodified Q was that much larger configuration jnteraction occurred than for the results for modified Q. On the other hand these effects are much smaller than the



b 2E’


4A$ F\\


case (4) BM

case (6)

F,(eVi'A) g kV/@)





FR tev/n,



Ist allowxl

E$ km feV) 2 WI v



-a14 2p25 12.5&







symmetric relaxation effects, and we have shown that the values obtained for the unrelaxed FOare very insensitive

to the inclusion or otherwise of configuration interaction. This is not the case for g, and in fact the value before ~on~guration interaction for unmodified Q is positive, g = 12 eVl8’. On these grounds alone therefore, we could discount the results with unmodified Q. This is further supported by the very large value (12 eV) obtained for the first allowed transition. The overall spread of levels is also much larger now (after relaxation) than for the results with modified Q. We are therefore left with just one set of results; for modified Q including exchange and a perfect-lattice model formulation. Values of F0 and g for all the levels of V” and V- for this case only are given in Table 4. The resulting complete energy level schemes for V” and V.- are shown in Fig. 5. We consider the negative vacancy first. Symme~i~ relaxation has now changed the ground state to a ‘A i state with a 1st allowed transition of 3.0 eV to ‘E’. The ground state relaxation energy is about 1 eV more than for V’. Defect V appears to be more stable than V” with a ground state energy difference of 6.8 eV. However, such comparisons are rather suspect since the basic assumptions of the defect molecule method are less valid for a charged vacancy than for a neutral one. The only proper way of determining the stable charge state is by comparing formation energies. For the neutral vacancy V”, the ground state is still a *E’ state, but the 1st allowed transition has changed to ‘E’+ *A i :5.1 eV. Substituting for Evaand ER into eqn (8),






b \



"='JpFR 8,







\ \


-,p-, 2E' -80


Table 3. Symmetric relaxation for V”ground state *E’

v \ \\ $1 \ " \\\ \.*2/ \ Al \ \ \ \



\ \\

-\ 3A$ "4




' \ h \"v \ \ \

--go 'E'


Fig. 5. Energy levels (in eV) after configuration interaction for V’and V-before(b) and after(c) symmetric relaxation. These results are for case (4)of Table 1and a perfect lattice response model.



Table 4. Case (4) values of the unrelaxed force, F,,, and its first derivative, g, for symmetric relaxation of V” and VLevel




*E' (L) -11.01

14.3 'A; (L) 11.6 3A;






-8.05 (U)



-12.26 17.1 -9.44

6.6 lE- (L) -6.2 3E'



-10.17 13.2

1. Coulson C. A. and Kearsley M. J., Proc. Roy. Sot. A241,433

-8.31 11.3

(1957). 2. Coulson C. A. et al., Proc. Roy. Sot. A274, 461 (1963). 3. Coulson C. A. and Larkins F. P., J. Phys. Chem. Solids 30, 1963(1969). 4. Coulson C. A. and Larkins F. P., J. Phys. Chem. Solids 32, 2245 (1971). 5. Larkins F. P. and Stoneham A. M., J. Phys. C. (Solid State Phys.) 4, 143 (1971). 6. Larkins F. P. and Stoneham A. M., J. Phys. C. (Solid State Phys.) 4, 154 (1971). I. Larkins F. P., J. Phys. Chem. Solids 32, 965 (1971). 8. Larkins F. P., J. Phvs. Chem. Solids 32. 2123 (1971). 9. Goeppert-Mayer M.-and Sklar A. L., J.’ Chem.‘ Phyi. 6, 645 (1938). 10. Hopgood F. R. A., Midiaf on Atlas (Atlas Computing Laboratory publication). 11. Mu&ken R. S., J. Chim. Phys. 46, 500 (1949). 12. Mayer R. M., Carbon 7, 512 (1969). 13. Palmer H. B. and Shelef M., Chemistry and Physics ofCarbon (Edited by P. L. Walker Jr.). Vol. 4, p. 85. Marcel Dekker, New York (1968). 14. Coulson C. A. and Poole M. D., Carbon 2, 275 (1964). 15. Bradburn M.. Coulson C. A. and Rushbrooke G. S.. Proc. Rov. Sot. Edinb. A62, 336 (1948). 16. Coulson C. A. and Taylor R., Proc. Phys. Sot. 65A,815(1952). 17. Taft E. and Apker L., Phys. Rev. 99, 1831(1955). 18. Cotton F. A., Chemical ADolications of Grouo Theorv. . ” Appendix IV. John Wiley, N&v York (1983). 19. Goldfarb I. J. and JafTe,H.H., J. Chem. Phys. 30,1622(1959). 20. Jordan P. C. H. and Longuet-Higgins H. C., Mol. Phys. $21 (1962). 21. Long L. H., Experienta 7, 195 (1951). 22. yoge H. H., J. Chem. Phys. 16, 984 (1948). 23. Santos E. and Villagra A., Phys. Rev. 36, 3134 (1972). 24. Nicholson, A. P. P., Ph.D. Thesis, Liverpool (1974). 25. Young J. A. and Koppel J. U., J. Chem. Phys. 42,357 (1965). 26. Hove J. E., Proc. 1st Conj. on Industrial Carbon and Graphite, p. 501. S.C.I., London (1958). 27. Hennig G. R., J. Appl. Phys. 36, 1482(1965). 28. Henson R. W. and-Reynolds W. N., Carbon 3, 277 (1965). 29. Huntindon H. B. et al.. Phvs. Rev. 99. 1085(1955). 30. Nicholson A. P. P. and Bacoh D. J., Ph;s. Stat.‘Sol. ia) 28,613 (1975).


-10.03 10.1

6.7 'A; (U)

-8.22 11.8

*E' (M) -12.14 -17.6 k*E’


Acknowledgements-This work was undertaken with the financial support of the U.K.A.E.A., Harwell. The helpful advice of Dr. F. P. Larkins is gratefully acknowledged.

LhitsareE inev/Aardginev/y2. Values shG-3 are for levels after configuration interaction. For symnetry states with mre ttm one lwel, (L), (M), and NJ) refer to l_,middle,ana UppS levels resptively in the undistorted systm.

the vacancy formation energy is found to be 8.75 eV. This result for E! depends on 4 parameters Ev, F,, g, and v,, and is least sensitive to changes in g, and most sensitive to changes in 6. The generally accepted experimental value for E6 is about 7 eV. A possible increase in this value could be obtained since most of the estimates [2628] have used a value of zero for the formation entropy, whereas it has been suggested[29] that a lattice defect with a large relaxation energy will have a large formation entropy and vice-versa. Considering the limitations and possible sources of error in our model, we believe the value obtained to be a reasonable estimate. In conclusion, the present investigation has shown that the defect-molecule method provides an adequate model for describing the vacancy in graphite. The value of - 11 eV/A obtained for the symmetric rebonding force on the nearest-neighbour atom to the neutral vacancy is in good agreement with estimates obtained by other methods [24]. This force has been used[24,30] in conjunction with a harmonic lattice model of graphite to determine the displacements around a vacancy and the interaction energy between a vacancy and other lattice defects in graphite. In the present work, it has been used to determine the relaxation energy (3.32 eV) associated with the vacancy, and, contrary to the assumptions made in previous investigations, this energy has been shown to make a significant contribution to the vacancy formation energy (8.75 eV).