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THEORY OF COLOUR CENTRES.Cluster-Bethe
Lattice calculation for the electronic structure of
hydrogen centres in alkali-halides
H. Brandi, B. Koiller
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 7 , Tome 41, Juillet 1980, page C6-77
THEORY OF COLOUR CENTRES.
Cluster-Bethe Lattice calculation for the electronic structure
of hydrogen centres in alkali-halides
H. S. Brandi all: 13. Koiller
DeparlamenLo de Fislca, Pontificia Universidade Catolica, Cx.P. 38071, Rio de Janeiro, RJ, Brasil
Rksumk. - Nous avons utilisi: la mkthode (( Cluster-Bethe Lattice )) (CBL) dans le calcul de la structure 61ec-
tronique des centres hidrogknoi'des U, U,, U2, UJ dans les halogknures alcalins. Nos rksultats montrent que la mkthode de CBL est trks convenable pour l'interprktation de la nature des transitions optiques.
Abstract. - We have used the Cluster-Bethe Lattice (CBL) method to calculate the electronic structure of the hydrogen centres U, U,, U,, U, in alkali-halides. Our results show the adequacy of the CBL in the interpretation of the nature of the optical transitions.
1 . Introduction. - The usual theoretical treatment
of colour centres consists of molecular type calcula- tions, in which the electronic states associated to the defect are written as linear combinations of orbitals centred at atomic sites at the impurity and a few neighbours around it. This approach is often signifi- cantly improved when the number of shells of neigh- bouring sites is increased, indicating the importance of the bulk states in the electronic structure of the system. Of course molecular type methods are not able to account adequately for band states, and these states may be important in the understanding of electronic properties of the system, especially those related to excited states such as optical transitions. In this work we use the Cluster-Bethe Lattice (CBL) method to calculate the electronic structure of hydrogen centres in alkali-halides. The CBL approxi- mation consists in describing the crystal by a cluster which includes the impurity and a few neighbours, with Bethe lattices attached to each dangling bond of the boundary sites. In this way the connectivity and coordination number of the system is preserved. The Hamiltonian is written in a simple one orbital per site basis set, with correlation effects at the impu- rity treated in the Hartree-Fock approximation. We calculate the Green's function of the system self- consistently, and from it the electronic structure, including both localized and band states, is obtained. 2. Theory. - In the one orbital per-.. site basis (assumed to be complete), the Green's function matrix elements satisfy Dyson's equation :
The Hamiltonian matrix elements are written as :
f A for i = j # impurity site : +(-) sign corresponds to cation (anion) site E
+
U ( n ) for i = j = impurity hydrogen site V for i and j n.n. ; i, j # impurity site (2) V' for i and j n.n. ; i or j = impurity site 0 otherwisewhere f A and V are the perfect crystal parameters, which may be obtained from band width and gap calculations ; E is the hydrogen impurity eigenenergy, with a Madelung correction appropriate for its site and referred to the zero of energy at the middle of the gap [I, 2, 31. The term U ( n ) is the correlation energy of the doubly occupied impurity level in the Hartree-Fock approximation, where ( n ) is the occupation per spin of the level and is obtained self- consistently. The interaction of the impurity with its nearest neighbours, V', is estimated by physical arguments. In the case of an interstitial centre, two different parameters must be considered : V ; asso- ciated with the cations and V i with the anions [2]. The infinite set of coupled equations (1) may be solved exactly by introducing the transfer functions for the heteropolar Bethe lattice corresponding to the rocksalt structure [I] :
where the plus (minus) sign corresponds to i being a cation (anion) site :
E " [E'E"(E'E"-20V2)]'12
T' =
-
+
10
v
- 10 VE' ,C6-78 H. S. BRAND1 AND B. KOILLER
The sign of the square root is chosen so as to satisfy N o the proper boundary and asymptotic conditions;
we define E'
-
E+
A , E"=
E - A.The local density of states at the impurity site is given by
+
C
(res GOO(E)) -P)
(5)P E = P
where the summation is over all the P poles of Goo. E
The diagonal matrix element has been obtained Fig. 1. - Schematic representation of the local density of states
in the CBL for the U and U3 centres with a cluster at the impurity site for hydrogen centres in alkali-halides. Electronic
of 18 atoms plus hydrogen [3] and U, and U l centres excitations associated to these centres are indicated by the arrows.
with a 9 atom cluster [2].
3. Optical absorption. - To our knowledge no
previous calculation concerning the electronic struc- The nature of the electronic transition (see Fig. 1) ture of hydrogen centres in crystals has self-consistent- associated with the peak of the absorption band which ly positioned the defect states relative to both occupied emerges from our results is :
and empty bands ; this is achieved by the CBL method. i)
u
and U, centres. The peak of these absorption This information is important to interpret correctly bands is associated to a one electron transition between the nature of the optical transition associated to the doubly occupied ground staterl,
and an empty the hydrogen centres absorption band, which is not state of symmetryr4,,,
which splits-off the cationic always well understood. conduction band, in agreement with the interpretationThe perfect crystal parameter A and V, as well as Wood and ODik r41 for the U centre. A . . a
the values of 6 for the interstitial and substitutional
ii) U, and U2 centres. The absorption band, centres are obtained from reference [I]'
in this model, is associated with a transition from a The parametb V'(v;, ';) to the delocalired state arising predominantly from the interaction of the defect with its nearest neighbours valence halogen states to an H atom orbital, forming is estimated to be in the range V - 3 V. The value
an H - ion. This is in agreement with the model Vf/V is chosen according to the physical characte-
ristics of each colour centre and held fixed for all proposed by Kerkhoff et al. [5]. alkali-halides considered. In the case of a doubly
occupied localized state, the average occupation per spin, ( n ), is calculated self-consistently since the model Hamiltonian depends explicitly on this parameter.
In table I we compare the theoretical results, for typical values of the interaction parameters, with the experimental data. The value ( n ), the weight of pole, is also given in this table.
The present results are in fair numerical agree- ment with the experimental data and also reproduce the general experimental trend. Although more sophis- ticated molecular type calculations may be in good agreement with the experimental data, the nature of the peaks is sometimes difficult to be understood. The present results indicate the adequacy of the CBL in the interpretation of the type of electronic excitation associated to the optical bands of colour centres.
Table I. - Comparison of the present calculation (CBL) for the bandpeak energies with experimental data. For the U3 centre the calculated value of 3.01 eV (KCl) must be compared with the experimentalprediction 2.9 eV [8].
CLUSTER-BETHE LATTICE CALCULATION FOR THE ELECTRONIC STRUCTURE
DISCUSSION
Question. - D. SCHOEMAKER.
We have recently performed extensive Raman measurements on the U, center in many alkali- halides. Can you extend your calculations to calculate the vibrational properties of the U, centers ?
Reply. - B. KOILLER.
It would ceftainiy be possible to take the proton off-centre, since there are no high symmetry require- ments in our method. However we would need some type of parametrization for the interaction potential
between the proton and the crystal in order to perform the calculation.
Question. - J . M . SPAETH.
Did you calculate the magnetic circular dichroism of the interstitial hydrogen centres ? You gave only one optical transition energy for interstitial hydrogen centres, however, in KBr and KI these transitions are split into several bands.
Reply. - B. KOULER.
Magnetic circular dichroism was not calculated.
References
111 QUEIROZ, S. L. A,, KOELER, B., MAFFEO, B. and BRANDI, H. S., 151 KERKHOFF, F., MARTIENSSEN, W. and SANDER, W., Z. Phys. 173
Phys. Status Solidi (b) 87 (1978) 351. (1963) 184.
[2] KOILLER, B. and BRAND], H. S., Phys. Status Solidi (b) 94 [6] FISHER, F., Z. Phys. 204 (1967) 351.
(1979) K 179. [7] SCHULMAN, I. W. and COMPTON, W. D., Color Center in Solids
[3] KOILLER, B. and BRANDI, H. S., Phys. Status Solidi (b) 92 (1979) (The Mac Millan Co., New York) 1962.
279. [8] HAYES, W. and HODBY, I. W., Proc. R. SOC. A 294 (1966) 359.
