The g-value of the excited state associated with the GR1 line is analysed in terms of the Coulson-Kearsley defect-molecule model (1957) for the vacancy in diamond. It is shown that the experimentally observed g value is entirely consistent with the expected 1T2 level of the model provided quenching by the dynamic Jahn-Teller effect is considered. This quenching is estimated from stress splitting data for the GR1 line. The result supports strongly the view that the GR1 centre is the neutral vacancy

LOWTHER, JE

STONEHAM, AM

English

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J. Phys. C:  Solid State Phys., Vol. 11. 1978. Printed in Great Britain. @ 1978 Theoretical implications of the stress and magnetic field splitting of the GRl line in diamond J E Lowther and A M Stoneham? Department of Physics, University of the Witwatersrand, Johannesburg, South Africa Received 24 November 1977 Abstract. The g value of the excited state associated with the GR1 line is analysed in terms of the Coulson-Kearsley  defect-molecule model for the vacancy in diamond. It is shown that the experimentally observed g value is entirely consistent with the expected ’T, level of the model provided quenching by the dynamic Jahn-Teller  effect is considered. This quenching is estimated from stress splitting data for the GR1  line. The result supports strongly the view that the GR1  centre is the neutral vacancy. 1. Introduction Recently Douglas and Runciman (1977) obtained a value for the g-value of the excited state associated with the GRl line observed in diamond. The low value obtained for this excited state (1.68 x  indicates that the state has zero spin: the absence of tempera- ture dependence  further shows that the ground state has spin zero too. These results suggest the excited state is the lT, state given by the Coulson-Kearsley  ‘defect molecule’ model of  the neutral vacancy  (see Coulson and Kearsley  1957, Coulson and Larkins 1971).  On the basis of this interpretation the g-value is associated with a transferred orbi- tal moment involving the four dangling bonds surrounding the vacancy. It is evident from the stress splitting of  the GR1 doublet (Clark and Walker  1973 and Davies and Penchina 1974) that there is  a strong dynamic Jahn-Teller  effect associated with  the GR1 centre. This is in accord with earlier predictions (Lannoo and Stonehain 1968) and subsequent  theoretical  analyses  of  the  stress  splitting  and optical  properties  of  this system (Stoneham 1975, 1977 and Lowther 1975, 1976). Accordingly, not only is  the observed g value associated with a delocalised magnetic moment, but also the Zeeinan operator in the IT,  state is quenched because of the dynamic Jahn-Teller  interaction. The magnitude of  this Jahn-Teller  quenching can be evaluated from the reported stress splitting of the GR1 transitions or  from theoretical studies of the elastic properties of the centre (Larkins and Stoneham 1971). In this paper we calculate the g value of  IT2,  and show the result compares very favourably with the value reported by Douglas and Runci- man(1977). t  Permanent address: Theoretical Physics Division, AERE Harwell, Didcot. Oxon, UK 2165 D1 I 2166  J  E Lowther and A M Stoneham 2. Theory Coulson and Kearsley (1957) have described the various one-electron configurations of the neutral vacancy that can give rise to a many-electron  state IT,. Following an earlier approach (Lowther 1975) we  write the state in terms of the contributions of  three im- portant configurations : 'T,  = v/s2t2:  'T,)  + B(st3:'T2) + ?It4;  IT,)  (1) where the eigenvectors inay be obtained from the matrices (see Yainaguchi 1962) using the integrals given by  Coulson and Larkins (1971). Similarly other important many- electron states inay be written : 'E  = Pls2t2;  'E)  + Qlst3:  'E)  + Rlt4: 'E) 31'1  = L/s2t2:  3T1)  + Mist3:  3T,)  + Nit4;  3T,).  (2) Stress measurements have been performed on the ground state of the GR1 system and. taking 'E as the ground state of the GR1 transition, we obtain the following values for the stress parameters:  a L D = (O/C,,le)  = - -  (EIC/E)C('E)~ J3 a In the above expressions the coefficients C('E) and C('T2)  occur because of configuration interaction, and are given by: (4) The Jahn-Teller  interaction  introduces reduction  factors q and K(E).  In fact  q  v' 0.5 (see Ham 1972) even when there is anharmonicity in the vibronic interaction as inferred by the stress splitting of  the GR1 doublet: (see Stoneham 1975, 1977, Lowther 1975). The states given in equation (1) can also be used to give the following expression for the matrix element of the Zeeman operator: K(Tl) is a quenching  factor and, for  T  --  t  coupling, K(Tl) 1  K(E)  N exp( --9EJT/hm) (see Ham 1972). C('T2) = (v2 -  y2);  C('E) = (P2  --  R'). (1T2El-&\1T2f7)  = (YlLZI4 [1 -  2P2IK(TJ  ( 5) Figure 1. The labelling of  the sites and cube axes of the cluster is shown. Stress and magnetic  field splitting of the GR1 line in diamond  2167 3. Calculation of the g-factor We begin by calculating the g-factor without a Jahn-Teller  correction. This corresponds to the original Coulson-Kearsley  model. The Jahn-Teller  and various other corrections can then be added. The one-electron orbitals s and t (E  5, q, [) in equations (1) and (2)  can be expressed in terms of the dangling bonds shown in  figure (1). The (un-normalised) hybrids can be expressed in terms of  p orbitals along the cube axes centred on the appropriate sites: a = (sA  + xA + y, -  zA) d = (sD + xD -  yD + zD). The orbital angular momentum operator in (5) regards the centre of  the vacancy as origin. However, a local origin on an atomic neighbour  simplifies molecular  integrals. Transferring the angular momentum operator to appropriate carbon atoms we obtain the following expression for the one-electron matrix element in (5): (qlLz\E)  = N:{ [ -  (sAld/dxlxB)  + (sAld/dYlyB)  + (x,ld/dxlsB)  + (q'Ald/dylsB)18d +  4(Sf;",P"  + 3Sf;",P")}.  (7) Here N,  is the normalisation factor. The matrix elements are two-centre integrals and are to be evaluated  for two dangling bonds, centred on a and 6, separated by a distance J8d.  Numerical values  using  both Slater and Clementi wavefunctions  are shown in table 1. It is interesting to note that the terms (sAld/dx(xB)  etc. are of the same order of Table 1. Contributions to the orbital moment (a = 1.684 au). N,  is the normalisation factor associated with the overlaps among the atomic orbitals. Contributions to (7)  Slater orbitals  Clement orbitals -0.788  x  lo-' +01043 -01374 -0,633  x  lo-' +01945 +0808  x  lo-' 04532 +0.3105 magnitude as  the overlap integrals, but that these depend upon whether Slater or Cleinenti wavefunctions are used. The value of  the electronic component of  the g factor given by equation (7) is around 0.25, and is relatively insensitive to the choice of basis functions. This results from a cancellation between many of the terms appearing within the square brackets of equation (7). The Jahn-Teller  reduction factors can be calculated in two independent ways based 2168  J  E Lowther and A M Stoneham on different assumptions.  The first way follows Coulson and Kearsley in taking molecular orbitals built solely from the dangling bonds. From equation (3), taking q = 0.5, we have BID = 2K(E)  C('T,)/C('E).  (8) Since K(T,)  and K(E) are roughly equal for T-t  coupling, we can obtain K(T,) from (8) andtheDavies-Penchinavaluesforthestressparaineters(B  = -  0073 x  ineV Pa-' and D = -0.238  x  MeVPa-l). We find K(T,) LC 0.153. A  second  method is to make an assumption about the local lattice elastic constants.  The Davies-Penchina  data and the analysis of Stonehain (1977) can then be combined to give an independent esti- mate. The assumption made follows the prediction of Lidiard and Stoneham (1  967) that the elastic constant for t modes is half that for e modes. This leads to a value of the reduc- tion factor K(T,)  of 0.1  55, surprisingly close to the other prediction. The  third  factor  we  must  consider  is  configuration  admixture,  which  gives  the (1 -  2p2)  term in (5).  This has been given by Lowther (1975), and lies between 0.93 and 0.97. Collecting the three factors together (i.e.  the electronic. Jahn-Teller  and configura- tion interaction components) we find the following results : Slater (case 1)  gL  = 2.82 x Slater (case 2)  gL = 2.23 x  from experiment Cleinenti (case 1)  gL = 4.31  x  lo-,  (Douglasand Cleinenti (case 2)  g,  = 2.84 x  Runciman  1977) gL = 1.68 x Here cases  1 and 2 correspond to the cases in Lowther (1975). The predicted values are very close to observation.  It is worth noting that two further corrections have not been included. First, all the molecular integrals have been calculated for the geometry of  the perfect lattice. The atoms are moved from their sites by both the dynamic Jahn-Teller  displacements and by  a totally-symmetric distortion. Whilst one can estimate the Jahn-Teller  distortion from observed data, the symmetric contribution cannot be obtained without  special assumptions. Thus we  have been content to make restricted calculations only: these suggest that gL  will not be changed by inore than 20- 30 % by any plausible distortion. Secondly, we have followed Coulson and Kearsley in concentrating on the dangling-bond orbitals. If  the proper one-electron orbitals extend much further from the vacancy, gL  will be altered. Neither the sign nor the magnitude of the change is easy to estimate, especially given the cancellations which occur between various contributions. 4. Conclusion We have shown that standard molecular-orbital  inethods can be used to evaluate the g factor of the excited state of the GR1 centre in diamond. The results provide further evidence of the applicability of the Coulson-Kearsley  model when generalised to include the significant dynamic Jahn-Teller  quenching effects. Both the theory and the Douglas- Runciinan results suggest that the GR1 centre is the neutral vacancy, and that the princi- pal transition is from a 'E  ground state to a 'T,  excited state. Acknowledgments One of us (AMS) would like to thank the Solid State Physics Research Unit of  the Uni- Stress and magnetic  field splitting ofthe GR1 line in diamond  2169 versity of the Witwatersrand for its hospitality during the period in which this work was done. References Clark C D and Walker J 1973 Proc. R. Soc. A 334 241-57 Coulson C A and Kearsley M J 1957 Proc. R. Soc. A 241 433-54 Coulson C A and Larkins F P 1971 J.  Phys. Chem. Solids 32 2245-57 Davies G and Penchina C M  1974 Proc. R.  Soc. A 338 359-74 Douglas I N and Runciman W A 1977 J.  Phys. C:  Solid St. Phys. 10 2253-9 Ham F S  1972 Electron Paramagnetic Resonance ed S Geschwind (New York: Plenum) pp 1-119 Lannoo M and Stoneham A M  1968 J.  Phys. Chem. Solids 29 1987-2000 Larkins F P and Stoneham A M 1971 J.  Phys. C;  Solid St. Phys. 4  143-53 Lidiard A B and Stoneham A M 1967 Science and Technology oJ’Industria1  Diamonds 1 1-15 Lowther J E 1975  J.  Phys. C:  Solid St. Phys. 8 3448-54 ~ Stoneham A M  1975 Theory of Defects in Solids (Oxford: Clarendon) ch 27 Stoneham A M  1977 Solid St. Commun. 21 339-41 Yamaguchi T 1962  J.  Phys. Soc. Japan  1359-84 1976 Solid St. Commun. 20 933-5 

THEORETICAL IMPLICATIONS OF STRESS AND MAGNETIC-FIELD SPLITTING OF GR1-LINE IN DIAMOND

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