Bond charge model of amorphous tetrahedrally coordinated solids

Journal of Non-erysta11ine Solids 35 & 36 (1980) 537-542 ~North-Holland Publishing Company

BONDCHARGEMODELOF AMORPHOUSTETRAHEDRALLY COORDINATED SOLIDS R. Dandoloff, G. DUhler and H. Bilz Max-Planck-lnstitut fur FestkUrperforschung Stuttgart, FRG

The crystalline and amorphous phases of tetrahedrally co-ordinated solids are investigated using a dynamical bond-charge model. In the Si02 systems the role of the bond charges is played by the oxygen ions. Tetrahedral solids exhibit a structural weakness against deformations of the space angle. This leads to low-frequency transverse acoustic (TA) phonon branches with anomalous anharmonic properties. I t is therefore l i kely that the amorphous phases are related to specific local distortions of ~he covalent bonds, which cause a softening of the zone boundary TAmodes. This approach suggests a new class of amorphous clusters built of 5- and ('boat'-type) 6- rings. These clusters are low-entropy or entropy-free configurations of nearly undistorted tetrahedra] units with no translational symmetry or several hundred atoms in a unit cell. The properties of amorphous solids are discussed on the basis of our model. INTRODUCTION The history of research on amorphous solids is marked by a continuous trend to an increasingly more local description of the amorphous state reflecting the direction in which most of the progress could be achieved so far. In the early work the implications of the loss of periodicity in an amorphous solid were considered in comparison with a crystalline solid by introducing statistical disorder into the strength and distribution of electronic or atomic potentials, z At this stage the reasoning of most scientists was s t i l l strongly influenced by the wellknown concepts used for the description of crystalline solids. Later on i t became obvious that a local, real-space representation of clusters or subunits was more appropriate for a realistic treatment of the problem.2 An important step was the insight into the fact that topological disorder is compatible with relatively small variations of nearest-neighbour interactions and with rigorous conservation of tetrahedral coordination. 3 The differences between amorphous and crystalline solids were no longer attributed to the loss of long-range order but rather to local variations of the topology, for example to the presence of 5and 7- fold rings in tetrahedrally coordinated structures. 4 In spite of their success, such perfect random-network models, nevertheless, failed to provide satis. factory explanations for many properties as observed in ESR or in electronic transport measurements, which appeared to be related to defects such as unsaturated dangling bonds,s As a result theactivities shifted from the investigation of larger clusters to that of these defects during the last few years. In the case of amorphous silicon or germanium there i~ by now overwhelming evidence, for exampel, that i t is rather the passivation of dangling bonds by hydrogen, than the per. fection of the'tetrahedral network which distinguishes "ideal" from "bad" samples with respect to many of their properties. 6 One might accentuate the present situation by claiming that the view of a lot of 537

538

R. Dandoloff et al. / Amorphous Tetrahedrallg Coordinated Solids

work on amorphous solids is that the topologically disordered structure plays the role of a matrix which favours the formation and s t a b i l i t y of defects, which in their turn actually determine the properties of amorphous solids. In contrast to our improved understanding of properties related to the defects, our knowledge about the amorphous structure i t s e l f is s t i l l poor. To a large extent this is due t o t h e fact that observations of properties which may be specific to the amorphous 'phase' are obscured by the influence of 'extrinsic' defect states. For instance, i t is an open question, whether the thermal anomalies and the low lying excitations are intrinsic or extrinsic properties. ~ Also we do not know how the electronic structure of an intrinsic amorphous semiconductor would look like in the bandtails. Concerning the topological structure, we ignore whether the lack of translational symmetr7 leads to a random tetrahedrally coordinated network of the Polk type, or rather favours the formation of large regular clusters compatible with 4- fold coordination and small bond distortions. Such clusters may correspond to low values of free energy. In fact there is strong evidence, that the amorphous phase of a solid qualitatively differs from a solid with very strong disorder. This was demonstrated in a persuasive way by Kalbitzer's amorphization experiments with ion bombardment, s Only ofter the transition into the 'amorphous phase' does the structure become stable against annealing. In view of our relatively poor knowledgeabout the structure, a certain amount of speculation is inevitable i f we want to proceed in our understanding of amorphicity. The purpose of this paper is to develop a new concept starting from a study of the dynamical properties of tetrahedrally-coordinated crystalline solids. We found that the dynamical bond charge model of Weber 9 provides a very appropriate description of the problem. In the f i r s t section we want to discuss the structural weakness of the diamond lattice, as indicated by the f l a t TA phonon branches and possible structural modifications in the case of strong phonon softening. In section 2 the relation between the covalent bonding and the electronic structure w i l l be considered under the aspect of variation of topology , bond angle and bond length. In section 3, f i n a l l y , consequences of the dynamical instabilities and of the relationship between electronic band structure and cluster configurations for the formation of noncrystalline structures and their properties w i l l be considered. 1. THE 'WEAKNESS' OF THE DIAMONDSTRUCTURE In Fig. I the dispersion relations for the (100) and (111) directions are shown in a combined plot for diamond, silicon germanium, and s-tin. The most remarkable features are the f l a t and low-lying TA phonon branches for Si, Ge, and e-Sn (in

"--.. "~',, ........ c

/~

| .,..

"",,~,,

,~

•

.

0t I

.

.

.

..........

/ /f - ' " ...-- ....

..__ .

.

.

-

.

.

".,_ ~X.': A

.f r

•

L

FIG. 1. Phonon dispersion relations for diamond, Si,Ge, and ~-Sn along the a and A directions. The phonon energies are scaled by the respective ion plasma frequencies. The Si and Ge curves are almost identical: thus the Si curves are shown only where they deviate perceptibly from Ge. (From Ref. 9).

R. Dandoloff et al. /Amorphous Tetrahedrally Coordinated Solids

/~

,,

'

539

x

/'

w

/

FIG.2. Transverse acoustic (TA) normal mode at the X-point with polarization vector in the (011)plane. Shown is the extreme case of the rigid rotation of tetrahedra formed by ions (small circles with their bond orbitals around the symmetry centers STA. A restoring bond bending forc~ is required in order to keep ~TA(X) finite.

contrast to diamond, see section 2) which are unexpected in view of the large shear moduli of these rigid covalent crystals. The explanation for an extremely small ratio of the effective force constants as given by

1 2Raman

/ Raman :0.04

for the TA zone boundary phonons compared with the LO phonons at the r-point, has been a major problem for a satisfactory theory of lattice dynamics in this class of crystals. 9 A particular d i f f i c u l t y was to maintain the stability of the crystal against free rotations of tetrahedra in the structure. The weakness of the diamond lattice can, in fact, be understood quite naively by looking at the normal modesnear the Brillouin zone boundary. In Fig. 2 a projection of the structure onto the(O11)-plane for the TA (lO0)-normal mode at the X-point is shown. Within the harmonic approximation the rotation of double tetrahedra around specific centers between the atoms (see Fig. 2) can be performed without affecting the bonds in the tetrahedra. An indication that this tendency towards instability is an intrinsic feature of Si, Ge, and ~-Sn and not a model artifact follows from the negative mode-GrUneisen parameters for large wave vectors. I° The adiabatic bond charge model (Fig. 3) gives a physically well-motivated and quantitatively excellent description of group IV semiconductors This ¢odel is an extension of the work by Phillips 11 and Martin 12 in which the bond charge -Z is no longer fixed at the center of gravity between two ions of charge 2Z but is allowed to move adiabatically (m~<
FIG. 3. Schematic presentation of Weber's adiabatic bond charge model. The four types of interactions (a)-(d) are discussed in the text (from Ref. 9).

540

R. Dandoloff et al. / Amorphous Tetrahedrallg Coordinated Solids

coupling (d) is of the Keating type. 13 The additional degree of freedom brings the system close enough to i n s t a b i l i t y for reproducing the observed f l a t TA branches. From this discussion i t becomes clear that the trend to i n s t a b i l i t y is enhanced by a) increasing the bond charge and b) increasing polarizability of the bond charge (see section 3). At this stage a comparison with Si02 , structures seems to be appropriate. The s-quartz crystal may be roughly described as a distorted wurtzite l a t t i c e in which the bond charges are replaced by oxygen ions. The distortion affects mainly the angles between neighboring Si ions but scarcely changes the angle between bond orbitals of the same atom. The formation of a-quartz (the most stable low temperature configuration of Si02) may be interpreted as resulting from a wurtzite struc. ture which was unstable against rotation of the double-tetrahedra (see Fig. 2) due to the large effective charges of the oxygen ions. In the elemental semiconductors oniy a structural 'weakness' was observed. In the case of SiO2 compounds the potential as a function of the bond angles at the oxygen site would correspond to a double well while the analogous function of the bond charge seems to be described by a strongly anharmonic potential. From Fig. 2 i t is obvious that the formation of the distorted phase (a-quartz requires a long-range cooperative ordering of the rotation and antirotation of neighboring double-tetrahedra which is essent i a l l y connected with the topological structure of the periodic diamond or wurtzite lattice. Even a small amount of topological disorder, however, strongly acts against such a transition into this highly distorted phase. Therefore, one is expecting smaller rotation angles in the glassy configurations. In turn this means, that the glass structure should deviate less from diamond or wurtzite with respect to second nearest Si atoms than o-quartz. A natural consequence of the smaller rotation angles would be a reduction of the depth of the double well potentials in the glasses. Moreover, because of disorder a broad distribution of potentials, including the limiting case of a f l a t ' r a t t l i n g ' potential seems plausible. Such a speculation about the origin of the low energy two-level systems as proposed by Anderson, Halperin, and Varma 14 is supported by the observation that the cubic term of the specific heat and also the atomic density of vitreous s i l i c a ~ e much closer to diamond-type B-crystobalite than to the less symmetric a- quartz. 2. ELECTRON-ION INTERACTION In thepreceeding section we pointed out the importance of the destabilizing Coulomb interaction of a movable bond charge with the l a t t i c e ions. The motion of the bond charge parallel to a bond, is directly related to the radial component ~,, of the bond polarizability. The displacement of the bond charge perpendicular to the bond (which governs the bond bending and, hence, the softening of the TA mode) is related to the lateral component of the bond polarizability a±. Is As the bond charge stands for the anisotropic part of the electronic charge distribution around the atoms, the bond p ~ r i z a b i l i t y corresponds, in terms of the inverse dielectric matrix a.-~+~ , q+G') to the off-diagonal contributions This, in turn, explains why the softening affects mainly the phonon branches close to the zone boundaries, since i t is related to the short-range part of dielect r i c screening. The bond polarizability is determined by the energetic positions of the conduction bands with r 2 , , and ris symmetry relative to the valence band edge r2s, .16 Whereas the transition from bonding sp 3 orbitals to antibonding ones is related to a polarization parallel to the bonds and scales with the inverse of E(rls ) - E(r2s,) , the polarization perpendicular to the bond a~ is related to the n-character of the r2, - band. This explains way diamond wlth its large value of E(r2,)-E(r2s,) = 10.8eV shows no flattening of the TA branches, in contrast to Si and Ge where the corresponding values are lowered dramatically (4.0 and 0.9 eV, respectively). In addition, a certain admixture of d-like states to the sp3 orbita]~ should play a role in Si, Ge, and a-Sn17 . The anharmonic contributions to the Coulomb interaction are at least of 4th order in the displacements of the bond charge in the case of the unperturbed diamond structure because of inversion symmetry. I t should be noted, however, that distor-

R. Dandoloff et al. /Amorphous Tetrahedrally Coordinated Solids

5~1

tions of the tetrahedral structur~ may ind~e static dipoles (or multipoles) in a group IV semiconductor. At the same time 3"~ order contributions to anharmonicity become possible, which then enhance the destabilizing effect of the Coulomb interactions. 3. INSTABILITY-INDUCED STRUCTURALMODIFICATIONS In order to gain some insight into 'possible' topological modifications of the diamond structure we followed the observation that a 'phas~ transition' induced by the TA(L) modes locally leads to a transformation of seat-type G-rings ('seats') into boat-type 6-rings ('boats'). Interestingly, this corresponds to the fact that the cleavage plane has (111)-orientation so that a surface which has exactly the s~nnmetry of the TA(L)-mode could be generated by a shearing between neighboring (111)-planes. The 'boats' are unstable against twisting deformations (in contrast to the stable 'seats') and therefore are interesting as molecular units to be used in alternative tetrahedral structures which are more flexible than the diamond lattice. We have found that a network of 'boats' only can be built, which contains 5-fold rings and 'boats', but s t r i c t l y avoids 'seats'. I f we also admit 'seats', this new class of regular tetrahedrally coordinated structures includes both large (but finite) clusters as well as infinite non-crystalline structures. The con~mon features of both types of configurations are the small bond angle distortions and the repetition of rather large regular subunits built of 'boats' such as dodecahedra (20 atoms) and barrel-shaped 'barrelans' (15 atoms). Fig. 4 shows a cluster containing 280 atoms with the symmetry of a dodecahedron. A continuation beyond this size with the same underlying construction principle would lead to strong distortions in this case. I t is not clear, however, whether some modifications of the systematics would allow a continuation or the incorporation of the cluster into larger units which might or might not shnw translational symmetry. As to the dynamics of such large clusters incorporated into a solid i t is at least plausible that they will contribute to the excitation spectrum with low frequency rotational modes. The actual value may vary strongly according to the unbedding of the cluster into its neighborhood. Although the low entropy and the small bond distortion should favor the formation of such regular macroclusters the question remains whether the microscopic counterparts to such models exist in the real world and, in particular, in real

FIG. 4. Regular macrocluster with tetrahedral coordination containing only boat-type 6-rings and planar 5-rings.

FIG. 5. Nucleus of an infinite noncrystalline tetrahedral structure with the full dodecahedron symmetry.

5~2

R. Dandoloff et al. /Amorphous Tetrahedrally Coordinated Solids

amorphous structures. In this context the consideration of the ~bove-mentioned i n f i n i t e clusters is of interest. Fig. 5 shows the nucleus of such an i n f i n i t e cluster with the perfect symmetry of the central dedocahedron. The rays formed by the sequence of 'barrelans' perpendicular to the dodecahedron faces are interconnected by s l i g h t l y stressed 'wurtzite' layers. The space in between is f i l l e d with a regular network of diamond structure which is only under weak uniaxial stress in the (111)-direction. The (111)-'diamond' planes represent the surface of this cluster i f i t 'grows' to macroscopic dimensions. We believe that our considerations have lead us to a point where nucleation and growth of crystalline and amorphous structures approach each other quite closely, actually much more than in the familiar picture where'only conservation of 4-fold coordinates is assumed. A very recent experimental finding which strongly encourages us to follow up these lines is the observation of regularly shaped Si 'macroclusters' up to 500 nm in diameter with 5-fold symmetry18 for which we have obtained the model counterpart. 19 In conclusion we have shown that a combination of dynamical conditions ('Instab i l i t i e s ' ) with topological arguments ('boats') versus 'seats') leads to a new concept for amorphous structures (semiconductors and glasses) providing promising aspects for a deeper understanding of the basic nature of amorphicity. ACKNOWLEDGEMENTS The authors acknowledge interesting and helpful discussions with M.J. Kelly, W. Kress,H.J.v. Schnering and W. Weber. REFERENCES (1)

(2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19)

For instance: Herman, F. and Van Dyke, J.P., Phys. Rev. Lett. 21 (1968) 1575; Maschke, K. and Thomas, P., phys. stat. sol. 39 (1970) 453; v.Heimendahl, L., in J. Stuke and W. Brenig (eds.), Amorphous and Liquid Semiconductors (Taylor and Francis, London, 1974), p. 853 Keller , J. and Ziman, J.M., J. Non-Cryst.Sol. 8-10 (1972) 111; McGill, T.C. and Klima, J., J. Phys. C 3 (1970) L 163 and Phys. Rev. B 5 (1972) 1517. Polk, D.E., J. Non. Cryst. Sol. 5 (1971) 365; Polk, D.E., Boudreaux, D.S., Phys. Rev. Lett. 31 (1973) 92 Weaire, D. and Thorpe, M.F., Phys. Rev. B 4 (1971) 2508 and 3518 Stuke, J., in: W.E. Spear (ed.), Amorphous and Liquid Semiconductors (Edinburgh, 1977), p. 406; MUller, G. and Kalbitzer, S. ibid., p. 347 Paul, W. Lewis, A.J., Connell, G.A.N., and Moustakas, T.D., Sol. State Comm. 20 (1976) 969; Brodsky, M.H., Cardona, M., and Cuomo, J.J., Phys. Rev. B 16 (1977) 3556 Dransfeld, K. and Hunklinger S., Ref. 5, p. 155 MUller G., and Kalbitzer, S., in: G.H. Frischat (ed.), Proc. 4th Int. Conf. on the Phys. of Non-Crystalline Solids (Trans.Tech. Publ., 1977), p. 278 Weber, W., Phys. Rev. B 15 (1977) 4789 Jex, H., phys. stat. sol. (b) 45, (1971) 343; Payhe, R.T., Phys. Rev. Lett. 13 (1964) 53 Phillips, J.C., Phys. Rev. 166 (1968) 832 Martin, R.M., Phys. Rev. 168 (1968) 871 Keating, P.N., 145 (1966) 637 Anderson P.W., Halperin, B.I., and Varma, C.M. Phil. Mag. 25 (1972) 1 Vukcevich, M.R., Journal of Non-Crystalline Solids 11 (1972) 25 Go, S., Bilz, H. and Cardona, M, Phys. Rev. Lett. 34 (1975) 580 and Go, S., Ph.D. Thesis Universit~t Stuttgart (1975) Kane, E.O. and A.B., Phys. Rev. B 17 (1978) 2691; Chadi, D.J., Phys. Rev. B 16 (1977) 3572 Saito, Y., Journal of Crystal Growth 47 (1979) 61 Dandoloff, R., Bilz, H., and D~hler G.H., to be published.