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Vibrational properties of amorphous GaN
Journal of Non-Crystalline Solids 283 (2001) 203±210
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Vibrational properties of amorphous GaN William Pollard * Department of Physics, Astronomy and Geology, Valdosta State University, Valdosta, GA 31698, USA Received 21 September 2000
Abstract The vibrational densities-of-states (DOS) of crystalline hexagonal GaN and a simulated amorphous GaN (a-GaN) structure are calculated. Neutron scattering spectra of simulated amorphous GaN structures are also calculated. Our results for hexagonal GaN yield energies of zone-centered phonons which are consistent with experimental Raman and infrared (IR) absorption studies, and a density of states spectrum, which is in accord with neutron scattering measurements.Our results for amorphous GaN show that the vibrational properties of a-GaN may be attributed to the nearly crystalline nearest-neighbor geometry of the amorphous network. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 61.43.D; 78.30.L
1. Introduction The addition of crystalline GaN to the family of device quality semiconductors has opened new opportunities in short-wavelength photonic devices for display and data-storage applications, and high-temperature/high-power electronics. A great deal of experimental and theoretical eort has been expended to understand the bulk properties of the crystalline phases of these materials for various applications [1±7]. Although amorphous GaN (a-GaN) may be viewed as a byproduct of these eorts, it has been the subject of few theoretical and experimental investigations [8± 12]. This is probably due primarily to a lack of
*
Tel.: +1-912 333 7171; fax: +1-912 333 7389. E-mail address: [email protected] (W. Pollard).
technological interest. Comparatively, a-GaN is characterized by a shorter bond length and higher relative ionicity than amorphous GaAs (a-GaAs). Like amorphous silicon dioxide (a-SiO2 ), a-GaN has a large band gap (3.39 eV), but unlike a-SiO2 , a-GaN is semiconducting. Twenty years ago, amorphous gallium nitride was ®rst synthesized by reactive sputtering of gallium in nitrogen ambient [9]. More recently, thin ®lm transistors have been made using a-GaN ®lms [8]. The formation of a layer of a-GaN by implanting gallium into hydrogenated amorphous silicon nitride (a-SiNx :H) has also been reported [10]. Theoretically, based on approximate ab initio molecular dynamics studies, structural models of a-GaN at dierent densities have been proposed. These models are described as `highly disordered', possessing no like-atom bonds and exhibiting a large state-free optical gap [11,12]. Although this study focused on the electronic states, another manifestation of bulk
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 4 6 1 - 6
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topography and bonding structure of a-GaN may be found in its vibrational properties. Numerous studies of the vibrational properties of amorphous semiconductors and insulators suggest that various structural aspects of the crystalline GaN phases will be retained in the amorphous phase and will be evidenced by the presence of particular Raman and infrared (IR) active vibrational modes [13±19]. In contrast to the situation for a-GaN, the vibrational structure of crystalline GaN has received considerable experimental and theoretical attention [1±7]. Crystalline GaN is observed to nucleate in two tetrahedrally bonded structures. It is most commonly observed in the equilibrium hexagonal (wurtzite) structure, but it can also crystallize in a metastable cubic (zincblende) structure. Hexagonal and cubic GaN are dierentiated by the stacking sequence of the tetrahedrally bonded Ga±N bilayer, such that the individual bond lengths and local atomic environments are nearly identical, while the overall symmetry of the crystal is determined by the stacking periodicity. Depending on the stacking order, the bonding between Ga and N atoms in adjacent bilayer planes is of either zincblende or wurtzite nature. Zincblende bonds are rotated 60° with respect to nearest neighbors while hexagonal bonds are mirror images. Each type of bonding provides a slightly altered atomic environment making some lattice sites inequivalent and thereby reducing the overall crystal symmetry. However, the hexagonal structure may be viewed as having a slightly distorted zincblende structure with a small change in nearest neighbor distances. Consequently, there is not a substantial energy dierence between the vibrational modes of the hexagonal and cubic structures. This conclusion is supported by the relatively small energy dierences observed between the infrared and Raman active vibrational modes reported for crystalline hexagonal and cubic GaN. A similar conclusion should also hold for the vibrational modes of a-GaN. To date, there have been few experimental or theoretical studies of the vibrational states of a-GaN. Although several issues concerning a-GaN are of fundamental interest, a major issue involves the nature of its
vibrational states, and the eects of chemical, short-range and intermediate-range disorders upon them. As a ®rst step in probing the vibrational structure of a-GaN, we report the results of a systematic and quantitative investigation of its vibrational states. To avoid the diculties inherent in other computational methodologies, we have employed a theoretical approach which has proven successful in elucidating the vibrational properties of a variety of amorphous elemental (e.g., amorphous Si, Ge, As) and compound (e.g., amorphous GaAs, SiO2 ; As2 O3 , SiC) amorphous materials [13±19]. In this approach, the actual bulk amorphous structure is simulated by an in®nite eective network or lattice which preserves various elements of short- and intermediate-range orders found in the crystalline systems. 1 Due to the breakdown of selection rules in amorphous materials, spectroscopic features are generally interpreted in terms of the vibrational density-ofstates (DOS) spectra which are excellent measures of the vibrational excitations of a material. To establish how the atoms are moving, we have calculated the vibrational densities of states, and use them to determine how the displacements of nearby atoms correlate. From an analysis of the DOS, the symmetry, character, and degree of localization of vibrational states of interest may be determined. More importantly, the infrared and Raman spectra may be viewed as the DOS modulated by frequency-dependent matrix elements. Neutron scattering spectra, which are excellent experimental probes of the vibrational density-of-states, are also calculated. Our results suggest that similar to a-SiO2; the vibrational
1
This approach is based on the cluster Bethe lattice method (CBLM). The basic idea behind the CBLM is that the electronic or vibrational behavior of an atom or cluster of atoms in an amorphous structure can be simulated by attaching appropriate Bethe lattices to the surface bonds of the cluster. The Bethe lattice is a non-periodic in®nite network of atoms with the same atomic coordination as the real system, but which contains no closed rings-of-bond. Hence, by bonding Bethe lattices to the surface bonds of a cluster of atoms, one makes a surrogate system which like the real system is in®nite, which maintains the local atomic environment of the atom(s) of interest, and whose vibrational states can be determined exactly.
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properties of a-GaN may be attributed to the nearly crystalline nearest- neighbor geometry of the amorphous network and to the eects of bond-angle and dihedral angle disorders. Our results also show that the eects of structural disorder are to cause the density of states spectrum of a-GaN to be `smooth' and the absence of the sharp features, which characterize the DOS spectrum of the crystal. The a-GaN DOS spectrum is characterized by three vibrational `bands' (centered near 87, 71 and 22 meV), and exhibits four dominant optic peaks (69, 74, 86 and 87 meV), which should be infrared as well as Raman active. 2. Constraints on theory The absence of periodicity is one of the major obstacles to studying the vibrational properties of systems, such as a-GaN. Any realistic theory must take this lack of periodicity into account, which precludes the straightforward use of pseudopotential, or other methods that are applicable only to periodic systems. Attempts to use these methods by arti®cially inducing periodicity into the system (e.g., supercell approximation) can lead to a distorted view of the structure itself, the vibrational states, and other properties of physical importance [11,12,20±26]. Similarly, cluster calculations, in which the atomic sites of interest and their immediate atomic environment are extracted from the real system and are treated as a large molecule, have several disadvantages including limitations on the size of the cluster and the diculty of discriminating between the states of interest and those associated with the ®nite size of the cluster. Recently, some advances have been made in this area with real-space methods, which retain the advantages of a pseudopotential approach without arti®cially inducing periodicity into the system [24]. In order to avoid the diculties inherent in other computational methodologies, we have employed a theoretical approach in which the actual aperiodic a-GaN structure is simulated by an in®nite eective network or lattice which preserves various elements of short- and intermediate-range orders found in the actual structure but without
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closed `rings-of-bond' 2; 3; 4 (see also Footnote 1). A similar approach has been used as a ®rst step in delineating the vibration states in a variety of amorphous materials [13±19]. In our application of this method, the short-range order, de®ned by the coordination of the Ga and N atoms, the symmetries at their bonding sites, bond-lengths and bond-angles, is basically the same in our simulated a-GaN as in the crystal. This means that we implicitly assume that the crystal (equilibrium hexagonal phase) and amorphous structure share the same short-range order. Variations in shortrange and intermediate-range orders are taken into account by including a distribution of bond angles (109:47° 10°) and dihedral angles (between 60° and 180°) determined by the bonding chemistry of the crystalline GaN structures. The vibrational interactions between atoms in both the crystalline and amorphous structures are described by the same valence-force-®eld (VFF) model of the elastic energy and include both twoand three-body forces (cf. Footnote 4). The VFF parameters were obtained by scaling those for SiC and GaAs taking into account ionicity and bond length dierences. The force constants used in this study are given in Footnote 4. The anisotropy, which characterizes the crystalline hexagonal GaN structure, is taken into account by introducing two
2 The vibrational Green's function, G x, is related to the DOS, n w, by n x 2mx=pIm fTrG xg and to the projected DOS on site a by na x 2ma x=pIm fTrGaa xg. 3 In the VFF, the change in the potential energy per atom associated with displacements from equilibrium can be written schematically as V Rj Kr Drj 2 =2 Rj>K Kh r0 Dhjik 2 =2 Rj>k Krr0 Drj Drk ; where Drj is the scalar change in the length of the nearestneighbor bond joining atoms i and j; Dhjik is the change in the angle formed by bonds connecting atoms i and j, and atoms i and k, and r0 is the equilibrium bond length. Kr ; Kh , and Krr0 are the bond-stretching, bond-bending, and simultaneous bondstretching, force constants, respectively. 4 The valence-force-®eld parameters for the vibrational states of GaN lattice, as de®ned in Footnote 3, are (in 105 dyn/cm): Kr Ga±N 1:25; Kh N±Ga±N 0:092; Kh Ga±N±Ga 0:05; Krr0 Ga±N±Ga 0:017; Krr0 Ga±N±Ga 0:0067. Along c-direction: Kr Ga±N 2:4; Kh N±Ga±N 0:17; Kh N± Ga±N 0:11; Krr0 Ga±N±Ga 0:032; Krr0 Ga±N±Ga 0:011.
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sets of parameters: one describing interactions along the c-axis and one for interactions perpendicular to the c-axis [1,3,6]. Our approach is developed in a Green's function formalism that allows for the calculation of vibrational DOS spectra in Footnote 2. More importantly, from an analysis of the DOS, the symmetry, character and nature of vibrational states of interest may be determined. 3. Results 3.1. Vibrational states of hexagonal GaN It is instructive to begin our analysis by examining the vibrational states of the bulk crystalline structure. Hexagonal GaN has a wurtzite structure possessing two formula units (two inequivalent nitrogen atoms and two inequivalent gallium atoms) per primitive unit cell. The four atoms in the unit cell give rise to 12 phonon branches for a general wavevector. This fact makes a detailed interpretation of the phonon dispersion curves for the hexagonal GaN structure more complicated than for the zincblende GaN structure. Hence, most theoretical and experimental studies have focused only upon the zone-center optic phonons. The question whether these data adequately represent the vibrational properties remains. The onephonon DOS spectrum provides insight into the vibrational properties without the aforementioned complications. Using a VFF model of the elastic energy, the DOS spectrum of hexagonal GaN is calculated. Our results are shown in Fig. 1. There are three distinctive regions evident in the DOS spectrum. The highest energy region may be labeled the longitudinal optic band. This band extends from 82 to 91 meV and is centered at about 87 meV. The transverse optic band is separated from the longitudinal optic band by an approximately 4-meV phonon gap and extends from 61 to 78 meV. Vibrational states of lowest energy comprise the region of the spectrum termed the acoustic bands that are centered at about 21 meV. These states are separated from the states of the transverse optic band by a phonon gap of about 25 meV. Signi®cant peaks characterize each band.
Fig. 1. The vibrational densities-of-states (DOS) of crystalline hexagonal GaN. The spectrum is calculated using the VFF parameters of Footnote 4 that take into account the anisotropy, which characterizes the crystalline hexagonal GaN structure.
Our examination of the local DOS spectra reveals that gallium atom motions dominate the acoustic bands, while nitrogen atom motions dominate the optic bands. The large mass dierence between the gallium atoms and nitrogen atoms is re¯ected in the density of states spectrum by the relatively smaller widths of the longitudinal and transverse optical bands as compared to the corresponding acoustic band. Moreover, the large mass mismatch also accounts for the large phonon gap separating the acoustic and optical bands. The higher energy of the longitudinal optic band, and the gap between it, and the transverse optic band (LO±TO splitting) is related to the ionicity of the bonding between Ga and N atoms [3,6,7]. Numerous spectroscopic measurements have been performed on hexagonal crystalline GaN [1± 7]. Most of these studies have focused on the Raman and IR active zone-centered phonons. There is a consensus that the A1 (TO) mode is observed at 66 meV (533 cm 1 ), the E1 (TO) is observed at 70 meV (561 cm 1 ), and the A1 (LO) and E1 (LO) modes at 91 meV (735 cm 1 ) and 92 meV (740 cm 1 ), respectively. From Raman spectroscopy, the E2 modes were observed at 18 meV (144 cm 1 ) and 71 meV (569 cm 1 ). The high- and low-frequency B modes are not optically active. The positions in energy of the zone-centered phonons, in general, do not coincide with the peaks in
W. Pollard / Journal of Non-Crystalline Solids 283 (2001) 203±210
the DOS. This is due to the dispersion of the phonon branches across the zone ± the peaks in the DOS include many unresolved critical energies at the zone boundaries. However, we have calculated the frequencies of zone-centered phonons for comparison using our VFF parameters. In the optic region, we ®nd the energies of the A1 (TO) and the E1 (TO) modes at 67 meV (543 cm 1 ) and 69 meV (557 cm 1 ), respectively. The corresponding energies of the A1 (LO) and E1 (LO) modes are 86 meV (695 cm 1 ), and 87 meV (705 cm 1 ), respectively. The high and low energy E2 modes occur at 72 meV (580 cm 1 ), and 17 meV (140 cm 1 ), while high and low energy B modes are found at energies of 85 meV (689 cm 1 ) and 32 meV (250 cm 1 ). Here, the agreement between theory and experiment is fair, and perhaps would be improved if long-range Coulomb contributions to the dynamical force were explicitly included in our model of the vibrational interactions. It is interesting to note that the agreement is comparable to that achieved using quantum chemical approaches [3,5±7]. Neutron scattering is also an excellent probe of phonon modes. Unlike Raman and IR measurements, neutron scattering is not restricted by selection rules, and probes phonon modes throughout the Brillouin zone. The reduced neutron dynamical structure factor is extremely sensitive to the structural characteristics of a solid. Moreover, in the limit of large momentum transfers (incoherent limit) it approaches the vibrational densities of states. Recently, Nipko et al. [2] measured the phonon density of states of hexagonal GaN using time-of-¯ight neutron spectroscopy. Their measured (open circles) and calculated (solid line) neutron-weighted vibrational density of state spectra are shown in Fig. 2 (taken from [2]). The overall agreement between the theoretical DOS spectrum of Fig. 1 and the neutron weighted DOS spectra in Fig. 2 is fairly good, but as the comparison with IR and Raman measurements indicated that the longitudinal optic modes of the theory are about 4.8 meV too low in energy. The major dierence between the spectra is probably the relative weights of the optic and acoustic bands. The nitrogen local densities of states are weighted more in the neutron spectrum than they
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Fig. 2. Measured (open circles) and calculated (lines) neutronweighted DOS for GaN (from [2]).
are in the theoretical DOS. This results from a dierence in the neutron cross-sections for nitrogen and gallium nuclei, the net result being to overemphasize the contribution of the nitrogen atoms to the density of states. 3.2. Vibrational states of amorphous GaN To calculate the DOS spectrum of the simulated a-GaN structure we have used the same VFF parameters used to calculate the crystal DOS. Our results are shown in Fig. 3. Vibrational states are observed to occur in three identi®able energy re-
Fig. 3. The vibrational DOS spectrum of the simulated a-GaN structure. The spectrum is characterized by the same gross features as those found in the crystalline DOS spectrum of Fig. 1.
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gions, which are centered near 87, 71 and 22 meV, respectively. Several prominent peaks characterize each region. The vibrational modes corresponding to the peak at lowest energy (25 meV) arises mainly from motions in which second neighbor Ga (N) atoms move out-of-phase with each other along directions perpendicular to the Ga±N bond. Gallium atom motion is also responsible for the peak at 33 meV. In this mode, the Ga atoms move out-of-phase along the Ga±N bond while the N atoms are at rest. The vibrational density of states between 60 and 77 meV is mainly given by inphase (second neighbor) N atoms motion perpendicular to the Ga±N bonds. The mode corresponding to the peak at 69 meV involves less in-phase motion of (second neighbor) Ga atoms than that at 74 meV. The highest energy region (83±94 meV) of the DOS also involves in-phase nitrogen (90%) and in-phase gallium (10%) stretching motions, but in directions parallel to the Ga±N bond. Due to the absence of periodicity (i.e., longrange order), the simulated GaN lattice's DOS spectrum is found to be `smooth' and shows the absence of the sharp features, which characterize the crystal's DOS spectrum (Fig. 1). Clearly, the gross features of the crystalline spectrum are retained in the DOS spectrum of the simulated aGaN lattice. The retention of these features, and the gross similarity of the crystal's DOS spectrum and simulated GaN lattice's DOS spectrum follow from the fact that these aspects of the DOS spectra are determined, for the most part, by the mass dierence between gallium and nitrogen atoms and the presence of the same short-range order (tetrahedral local bonding con®gurations) in both structures. More importantly, the vibrational modes corresponding to the peaks in the higher energy region of the simulated a-GaN's DOS spectrum should be infrared as well as Raman active. There have been few reported experimental measurements of the vibrational states of a-GaN. However, recently, Fourier transform infrared spectroscopy measurements have been reported for Ga-implanted a-SiNx :H matrix [10]. These measurements show the presence of an a-GaN layer and a Ga±N stretching vibrations at 91 meV
Fig. 4. Calculated reduced neutron structure factor for the simulated a-GaN network. In the same way as in the experiment, the calculated S E is obtained at a temperature of 8 K and by performing a spherical average over the directions of the wavevector Q.
which is consistent with the predictions of the DOS spectrum of the simulated a-GaN structure shown in Fig. 3. Because neutron scattering has proven useful in elucidating the vibrational states of other amorphous materials, it would be informative to calculate the inelastic neutron structure factor of the simulated a-GaN network. The theoretical computational approach employed is outlined in some detail in [16,17,20,21]. 5 Using the same energies and performing a spherical average over the momentum transfer Q, we have calculated the reduced neutron structure factor for the simulated a-GaN network. Our results are shown
5 The coherent double dierential neutron cross-section for one-phonon annihilation is ``proportional'' to the reduced dynamical structure for S 0 Q; x. This factor determines the way in which the scattering intensity varies for dierent momentum transfers. The scattering process itself is characterized by the energy change x and the momentum transfer Q, which are interrelated by the energy and momentum conservation conditions. The reduced structure factor for a monatomic solid can be written as S 0 Q; x e 2W h xRQ ImGij x2 Q expiQ Ri Rj i; where the summation is over atoms i and j in the solid; Q is the unit vector along Ri and Rj denoting the position of atoms i and j, respectively; 2W is the Debye±Waller factor; and h i denotes a spherical average over the direction of Q. Note that the reduced dynamical structure factor is related to the normal structure factor with the Bose factors and the magnitude of the momentum transfer divided out.
W. Pollard / Journal of Non-Crystalline Solids 283 (2001) 203±210
in Fig. 4. The spectrum exhibits activity in three main regions. There are several prominent features evident in these spectra. As in the experimental spectra for crystalline GaN (Fig. 2), the regions determined more by the motion of nitrogen atoms are weighted more than those determined by gallium atom motions. Compared to DOS spectrum, the optical region of the calculated structure factor is characterized by large changes in the overall intensity. There is a shift of intensity from the lower portion of this region to the upper region. The acoustical region, on the other hand, shows an overall reduction in intensity. 4. Discussion Previous studies of compound amorphous materials have shown that there are four types of disorder: (1) like-atom bonds; (2) bond-angle variations; (3) dihedral angle variations; and (4) variations in topology (rings-of-bond) [16±25]. In reality, these are not independent of each other. Moreover, the possibility of native defects such as under- or over-coordinated atoms should be added to the above list. In contrast to a-GaAs, the relatively large electronegativity dierence between gallium and nitrogen atoms would argue against the presence of homopolar Ga±Ga or N±N bonds in a-GaN. The absence of like-atom bonds clearly indicates that any variations in topology would not include the occurrence of an appreciable fraction of odd-membered rings-of-bond. Since the types of rings-of-bond most likely to form in the actual amorphous solid are those that form in the crystal, we expect a predominance of sixfold rings-of-bond in the actual a-GaN network. Moreover, vibrational modes associated with the sixfold rings-of-bond should be evident as peaks in the region between 9 and 31 meV of the vibrational DOS spectrum of an actual a-GaN network. In the actual a-GaN DOS spectra, however, one would expect these topologically induced features to be less distinct than those induced in the crystalline GaN DOS spectrum. Most of the disparity between the crystalline GaN DOS (Fig. 1) and the DOS of the simulated a-GaN DOS (Fig. 3) may be attributable to the predominance of sixfold rings-
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of-bond in the crystalline structure. The net eect of the topological disorder is to cause the density of states of an actual a-GaN network to closely resemble that of our simulated a-GaN structure. Bond and dihedral angle variations should be the principal sources of disorder in the actual aGaN network. We have investigated the eects of dihedral and bond-angle variations on the simulated a-GaN structure. The net eect on the simulated a-GaN density of states of altering bond angles by small amounts (less than about 10°) is to cause a small increase in the width of the TO-like band (60 and 77 meV). This is accompanied by a slight shift of the center of the TO-like band to higher energies. Similar behavior is observed in the lowest energy region of the DOS spectrum. The displacement in energy of the vibration states by bond-angle ¯uctuations should result in a smearing of the density of states in the actual amorphous network. On the other hand, the distortion energies associated with dihedral angles ¯uctuations are smaller than those associated with bond-angle ¯uctuations and are consequently more likely to ¯uctuate in the amorphous structure, but the density of states is relatively insensitive to these ¯uctuations. Note that both bond and dihedral variations in an actual a-GaN network should have the net eect of broadening the peaks induced in the DOS by topology, so that they disappear almost completely, causing the DOS to be almost indistinguishable from that of the simulated aGaN structure's density of states. Finally, under- and over-coordinated atoms play an important role in the determination of the optical, electronic, and vibrational properties of many amorphous materials. The dangling bond defects in amorphous silicon and valence alternation pairs in amorphous selenium are prototypical examples. Molecular-dynamic studies of a-GaN suggest that strong chemical short-range order and the presence of a large percentage of threefold coordinated gallium and nitrogen atoms characterize the amorphous network [11,12]. Similar molecular-dynamic studies of a-GaAs also predict the presence of a large percentage (about 20%) of threefold coordinated As and Ga sites [22,25]. Yet, vibrational modes (which should be optically active) associated with under-coordinate sites have not been identi®ed
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in the measured infrared and Raman spectra of aGaAs. This may suggest that the large percentage of threefold sites is an artifact of the computational procedure, and that the actual percentage of the threefold coordinated sites is probably much smaller than that predicted by molecular-dynamic studies. Given the large phonon gap between the optic-like and acoustic-like bands in the vibrational DOS of a-GaN, and the expected energies and optical activities of vibrational modes associated with threefold coordinated nitrogen and gallium sites, experimental infrared and Raman measurements should be able to easily determine the densities of defective threefold coordinated nitrogen and gallium sites in a-GaN. 5. Conclusions As a ®rst step in probing the vibrational structure of a-GaN, we report the results of a systematic and quantitative investigation of its vibrational states. Using a VFF model of the elastic energy, the vibrational DOS spectra of crystalline hexagonal GaN and a simulated a-GaN structure were calculated. The large mass mismatch accounts for the large phonon gap separating the acoustic-like and optical-like bands in the DOS spectra of both the crystalline and amorphous structures. Our results for hexagonal GaN yield energies of zone-centered phonons, which are consistent with the Raman and IR studies, and a DOS, which is in accord with the neutron scattering measurements. There are few spectroscopic measurements of the vibrational properties of a-GaN to which our calculated DOS can be compared. Our results show that similar to a-SiO2 , the vibrational properties of a-GaN may be attributed to the nearly crystalline nearestneighbor geometry of the amorphous network and to the eects of bond-angle and dihedral angle disorders. Our results also show that the eects of structural disorder are to cause the density of states spectrum of a-GaN to be `smooth' and the absence of the sharp features, which characterize the spectrum of the crystal. The a-GaN spectrum is charactized by three vibrational `bands' (centered near 87, 71 and 22 meV), and exhibits four
dominant optic peaks (68, 74, 86, and 87 meV) which should correspond to the vibrational mode which are infrared as well as Raman active.
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Vibrational properties of amorphous GaN William Pollard * Department of Physics, Astronomy and Geology, Valdosta State University, Valdosta, GA 31698, USA Received 21 September 2000
Abstract The vibrational densities-of-states (DOS) of crystalline hexagonal GaN and a simulated amorphous GaN (a-GaN) structure are calculated. Neutron scattering spectra of simulated amorphous GaN structures are also calculated. Our results for hexagonal GaN yield energies of zone-centered phonons which are consistent with experimental Raman and infrared (IR) absorption studies, and a density of states spectrum, which is in accord with neutron scattering measurements.Our results for amorphous GaN show that the vibrational properties of a-GaN may be attributed to the nearly crystalline nearest-neighbor geometry of the amorphous network. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 61.43.D; 78.30.L
1. Introduction The addition of crystalline GaN to the family of device quality semiconductors has opened new opportunities in short-wavelength photonic devices for display and data-storage applications, and high-temperature/high-power electronics. A great deal of experimental and theoretical eort has been expended to understand the bulk properties of the crystalline phases of these materials for various applications [1±7]. Although amorphous GaN (a-GaN) may be viewed as a byproduct of these eorts, it has been the subject of few theoretical and experimental investigations [8± 12]. This is probably due primarily to a lack of
*
Tel.: +1-912 333 7171; fax: +1-912 333 7389. E-mail address: [email protected] (W. Pollard).
technological interest. Comparatively, a-GaN is characterized by a shorter bond length and higher relative ionicity than amorphous GaAs (a-GaAs). Like amorphous silicon dioxide (a-SiO2 ), a-GaN has a large band gap (3.39 eV), but unlike a-SiO2 , a-GaN is semiconducting. Twenty years ago, amorphous gallium nitride was ®rst synthesized by reactive sputtering of gallium in nitrogen ambient [9]. More recently, thin ®lm transistors have been made using a-GaN ®lms [8]. The formation of a layer of a-GaN by implanting gallium into hydrogenated amorphous silicon nitride (a-SiNx :H) has also been reported [10]. Theoretically, based on approximate ab initio molecular dynamics studies, structural models of a-GaN at dierent densities have been proposed. These models are described as `highly disordered', possessing no like-atom bonds and exhibiting a large state-free optical gap [11,12]. Although this study focused on the electronic states, another manifestation of bulk
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 4 6 1 - 6
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topography and bonding structure of a-GaN may be found in its vibrational properties. Numerous studies of the vibrational properties of amorphous semiconductors and insulators suggest that various structural aspects of the crystalline GaN phases will be retained in the amorphous phase and will be evidenced by the presence of particular Raman and infrared (IR) active vibrational modes [13±19]. In contrast to the situation for a-GaN, the vibrational structure of crystalline GaN has received considerable experimental and theoretical attention [1±7]. Crystalline GaN is observed to nucleate in two tetrahedrally bonded structures. It is most commonly observed in the equilibrium hexagonal (wurtzite) structure, but it can also crystallize in a metastable cubic (zincblende) structure. Hexagonal and cubic GaN are dierentiated by the stacking sequence of the tetrahedrally bonded Ga±N bilayer, such that the individual bond lengths and local atomic environments are nearly identical, while the overall symmetry of the crystal is determined by the stacking periodicity. Depending on the stacking order, the bonding between Ga and N atoms in adjacent bilayer planes is of either zincblende or wurtzite nature. Zincblende bonds are rotated 60° with respect to nearest neighbors while hexagonal bonds are mirror images. Each type of bonding provides a slightly altered atomic environment making some lattice sites inequivalent and thereby reducing the overall crystal symmetry. However, the hexagonal structure may be viewed as having a slightly distorted zincblende structure with a small change in nearest neighbor distances. Consequently, there is not a substantial energy dierence between the vibrational modes of the hexagonal and cubic structures. This conclusion is supported by the relatively small energy dierences observed between the infrared and Raman active vibrational modes reported for crystalline hexagonal and cubic GaN. A similar conclusion should also hold for the vibrational modes of a-GaN. To date, there have been few experimental or theoretical studies of the vibrational states of a-GaN. Although several issues concerning a-GaN are of fundamental interest, a major issue involves the nature of its
vibrational states, and the eects of chemical, short-range and intermediate-range disorders upon them. As a ®rst step in probing the vibrational structure of a-GaN, we report the results of a systematic and quantitative investigation of its vibrational states. To avoid the diculties inherent in other computational methodologies, we have employed a theoretical approach which has proven successful in elucidating the vibrational properties of a variety of amorphous elemental (e.g., amorphous Si, Ge, As) and compound (e.g., amorphous GaAs, SiO2 ; As2 O3 , SiC) amorphous materials [13±19]. In this approach, the actual bulk amorphous structure is simulated by an in®nite eective network or lattice which preserves various elements of short- and intermediate-range orders found in the crystalline systems. 1 Due to the breakdown of selection rules in amorphous materials, spectroscopic features are generally interpreted in terms of the vibrational density-ofstates (DOS) spectra which are excellent measures of the vibrational excitations of a material. To establish how the atoms are moving, we have calculated the vibrational densities of states, and use them to determine how the displacements of nearby atoms correlate. From an analysis of the DOS, the symmetry, character, and degree of localization of vibrational states of interest may be determined. More importantly, the infrared and Raman spectra may be viewed as the DOS modulated by frequency-dependent matrix elements. Neutron scattering spectra, which are excellent experimental probes of the vibrational density-of-states, are also calculated. Our results suggest that similar to a-SiO2; the vibrational
1
This approach is based on the cluster Bethe lattice method (CBLM). The basic idea behind the CBLM is that the electronic or vibrational behavior of an atom or cluster of atoms in an amorphous structure can be simulated by attaching appropriate Bethe lattices to the surface bonds of the cluster. The Bethe lattice is a non-periodic in®nite network of atoms with the same atomic coordination as the real system, but which contains no closed rings-of-bond. Hence, by bonding Bethe lattices to the surface bonds of a cluster of atoms, one makes a surrogate system which like the real system is in®nite, which maintains the local atomic environment of the atom(s) of interest, and whose vibrational states can be determined exactly.
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properties of a-GaN may be attributed to the nearly crystalline nearest- neighbor geometry of the amorphous network and to the eects of bond-angle and dihedral angle disorders. Our results also show that the eects of structural disorder are to cause the density of states spectrum of a-GaN to be `smooth' and the absence of the sharp features, which characterize the DOS spectrum of the crystal. The a-GaN DOS spectrum is characterized by three vibrational `bands' (centered near 87, 71 and 22 meV), and exhibits four dominant optic peaks (69, 74, 86 and 87 meV), which should be infrared as well as Raman active. 2. Constraints on theory The absence of periodicity is one of the major obstacles to studying the vibrational properties of systems, such as a-GaN. Any realistic theory must take this lack of periodicity into account, which precludes the straightforward use of pseudopotential, or other methods that are applicable only to periodic systems. Attempts to use these methods by arti®cially inducing periodicity into the system (e.g., supercell approximation) can lead to a distorted view of the structure itself, the vibrational states, and other properties of physical importance [11,12,20±26]. Similarly, cluster calculations, in which the atomic sites of interest and their immediate atomic environment are extracted from the real system and are treated as a large molecule, have several disadvantages including limitations on the size of the cluster and the diculty of discriminating between the states of interest and those associated with the ®nite size of the cluster. Recently, some advances have been made in this area with real-space methods, which retain the advantages of a pseudopotential approach without arti®cially inducing periodicity into the system [24]. In order to avoid the diculties inherent in other computational methodologies, we have employed a theoretical approach in which the actual aperiodic a-GaN structure is simulated by an in®nite eective network or lattice which preserves various elements of short- and intermediate-range orders found in the actual structure but without
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closed `rings-of-bond' 2; 3; 4 (see also Footnote 1). A similar approach has been used as a ®rst step in delineating the vibration states in a variety of amorphous materials [13±19]. In our application of this method, the short-range order, de®ned by the coordination of the Ga and N atoms, the symmetries at their bonding sites, bond-lengths and bond-angles, is basically the same in our simulated a-GaN as in the crystal. This means that we implicitly assume that the crystal (equilibrium hexagonal phase) and amorphous structure share the same short-range order. Variations in shortrange and intermediate-range orders are taken into account by including a distribution of bond angles (109:47° 10°) and dihedral angles (between 60° and 180°) determined by the bonding chemistry of the crystalline GaN structures. The vibrational interactions between atoms in both the crystalline and amorphous structures are described by the same valence-force-®eld (VFF) model of the elastic energy and include both twoand three-body forces (cf. Footnote 4). The VFF parameters were obtained by scaling those for SiC and GaAs taking into account ionicity and bond length dierences. The force constants used in this study are given in Footnote 4. The anisotropy, which characterizes the crystalline hexagonal GaN structure, is taken into account by introducing two
2 The vibrational Green's function, G x, is related to the DOS, n w, by n x 2mx=pIm fTrG xg and to the projected DOS on site a by na x 2ma x=pIm fTrGaa xg. 3 In the VFF, the change in the potential energy per atom associated with displacements from equilibrium can be written schematically as V Rj Kr Drj 2 =2 Rj>K Kh r0 Dhjik 2 =2 Rj>k Krr0 Drj Drk ; where Drj is the scalar change in the length of the nearestneighbor bond joining atoms i and j; Dhjik is the change in the angle formed by bonds connecting atoms i and j, and atoms i and k, and r0 is the equilibrium bond length. Kr ; Kh , and Krr0 are the bond-stretching, bond-bending, and simultaneous bondstretching, force constants, respectively. 4 The valence-force-®eld parameters for the vibrational states of GaN lattice, as de®ned in Footnote 3, are (in 105 dyn/cm): Kr Ga±N 1:25; Kh N±Ga±N 0:092; Kh Ga±N±Ga 0:05; Krr0 Ga±N±Ga 0:017; Krr0 Ga±N±Ga 0:0067. Along c-direction: Kr Ga±N 2:4; Kh N±Ga±N 0:17; Kh N± Ga±N 0:11; Krr0 Ga±N±Ga 0:032; Krr0 Ga±N±Ga 0:011.
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sets of parameters: one describing interactions along the c-axis and one for interactions perpendicular to the c-axis [1,3,6]. Our approach is developed in a Green's function formalism that allows for the calculation of vibrational DOS spectra in Footnote 2. More importantly, from an analysis of the DOS, the symmetry, character and nature of vibrational states of interest may be determined. 3. Results 3.1. Vibrational states of hexagonal GaN It is instructive to begin our analysis by examining the vibrational states of the bulk crystalline structure. Hexagonal GaN has a wurtzite structure possessing two formula units (two inequivalent nitrogen atoms and two inequivalent gallium atoms) per primitive unit cell. The four atoms in the unit cell give rise to 12 phonon branches for a general wavevector. This fact makes a detailed interpretation of the phonon dispersion curves for the hexagonal GaN structure more complicated than for the zincblende GaN structure. Hence, most theoretical and experimental studies have focused only upon the zone-center optic phonons. The question whether these data adequately represent the vibrational properties remains. The onephonon DOS spectrum provides insight into the vibrational properties without the aforementioned complications. Using a VFF model of the elastic energy, the DOS spectrum of hexagonal GaN is calculated. Our results are shown in Fig. 1. There are three distinctive regions evident in the DOS spectrum. The highest energy region may be labeled the longitudinal optic band. This band extends from 82 to 91 meV and is centered at about 87 meV. The transverse optic band is separated from the longitudinal optic band by an approximately 4-meV phonon gap and extends from 61 to 78 meV. Vibrational states of lowest energy comprise the region of the spectrum termed the acoustic bands that are centered at about 21 meV. These states are separated from the states of the transverse optic band by a phonon gap of about 25 meV. Signi®cant peaks characterize each band.
Fig. 1. The vibrational densities-of-states (DOS) of crystalline hexagonal GaN. The spectrum is calculated using the VFF parameters of Footnote 4 that take into account the anisotropy, which characterizes the crystalline hexagonal GaN structure.
Our examination of the local DOS spectra reveals that gallium atom motions dominate the acoustic bands, while nitrogen atom motions dominate the optic bands. The large mass dierence between the gallium atoms and nitrogen atoms is re¯ected in the density of states spectrum by the relatively smaller widths of the longitudinal and transverse optical bands as compared to the corresponding acoustic band. Moreover, the large mass mismatch also accounts for the large phonon gap separating the acoustic and optical bands. The higher energy of the longitudinal optic band, and the gap between it, and the transverse optic band (LO±TO splitting) is related to the ionicity of the bonding between Ga and N atoms [3,6,7]. Numerous spectroscopic measurements have been performed on hexagonal crystalline GaN [1± 7]. Most of these studies have focused on the Raman and IR active zone-centered phonons. There is a consensus that the A1 (TO) mode is observed at 66 meV (533 cm 1 ), the E1 (TO) is observed at 70 meV (561 cm 1 ), and the A1 (LO) and E1 (LO) modes at 91 meV (735 cm 1 ) and 92 meV (740 cm 1 ), respectively. From Raman spectroscopy, the E2 modes were observed at 18 meV (144 cm 1 ) and 71 meV (569 cm 1 ). The high- and low-frequency B modes are not optically active. The positions in energy of the zone-centered phonons, in general, do not coincide with the peaks in
W. Pollard / Journal of Non-Crystalline Solids 283 (2001) 203±210
the DOS. This is due to the dispersion of the phonon branches across the zone ± the peaks in the DOS include many unresolved critical energies at the zone boundaries. However, we have calculated the frequencies of zone-centered phonons for comparison using our VFF parameters. In the optic region, we ®nd the energies of the A1 (TO) and the E1 (TO) modes at 67 meV (543 cm 1 ) and 69 meV (557 cm 1 ), respectively. The corresponding energies of the A1 (LO) and E1 (LO) modes are 86 meV (695 cm 1 ), and 87 meV (705 cm 1 ), respectively. The high and low energy E2 modes occur at 72 meV (580 cm 1 ), and 17 meV (140 cm 1 ), while high and low energy B modes are found at energies of 85 meV (689 cm 1 ) and 32 meV (250 cm 1 ). Here, the agreement between theory and experiment is fair, and perhaps would be improved if long-range Coulomb contributions to the dynamical force were explicitly included in our model of the vibrational interactions. It is interesting to note that the agreement is comparable to that achieved using quantum chemical approaches [3,5±7]. Neutron scattering is also an excellent probe of phonon modes. Unlike Raman and IR measurements, neutron scattering is not restricted by selection rules, and probes phonon modes throughout the Brillouin zone. The reduced neutron dynamical structure factor is extremely sensitive to the structural characteristics of a solid. Moreover, in the limit of large momentum transfers (incoherent limit) it approaches the vibrational densities of states. Recently, Nipko et al. [2] measured the phonon density of states of hexagonal GaN using time-of-¯ight neutron spectroscopy. Their measured (open circles) and calculated (solid line) neutron-weighted vibrational density of state spectra are shown in Fig. 2 (taken from [2]). The overall agreement between the theoretical DOS spectrum of Fig. 1 and the neutron weighted DOS spectra in Fig. 2 is fairly good, but as the comparison with IR and Raman measurements indicated that the longitudinal optic modes of the theory are about 4.8 meV too low in energy. The major dierence between the spectra is probably the relative weights of the optic and acoustic bands. The nitrogen local densities of states are weighted more in the neutron spectrum than they
207
Fig. 2. Measured (open circles) and calculated (lines) neutronweighted DOS for GaN (from [2]).
are in the theoretical DOS. This results from a dierence in the neutron cross-sections for nitrogen and gallium nuclei, the net result being to overemphasize the contribution of the nitrogen atoms to the density of states. 3.2. Vibrational states of amorphous GaN To calculate the DOS spectrum of the simulated a-GaN structure we have used the same VFF parameters used to calculate the crystal DOS. Our results are shown in Fig. 3. Vibrational states are observed to occur in three identi®able energy re-
Fig. 3. The vibrational DOS spectrum of the simulated a-GaN structure. The spectrum is characterized by the same gross features as those found in the crystalline DOS spectrum of Fig. 1.
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gions, which are centered near 87, 71 and 22 meV, respectively. Several prominent peaks characterize each region. The vibrational modes corresponding to the peak at lowest energy (25 meV) arises mainly from motions in which second neighbor Ga (N) atoms move out-of-phase with each other along directions perpendicular to the Ga±N bond. Gallium atom motion is also responsible for the peak at 33 meV. In this mode, the Ga atoms move out-of-phase along the Ga±N bond while the N atoms are at rest. The vibrational density of states between 60 and 77 meV is mainly given by inphase (second neighbor) N atoms motion perpendicular to the Ga±N bonds. The mode corresponding to the peak at 69 meV involves less in-phase motion of (second neighbor) Ga atoms than that at 74 meV. The highest energy region (83±94 meV) of the DOS also involves in-phase nitrogen (90%) and in-phase gallium (10%) stretching motions, but in directions parallel to the Ga±N bond. Due to the absence of periodicity (i.e., longrange order), the simulated GaN lattice's DOS spectrum is found to be `smooth' and shows the absence of the sharp features, which characterize the crystal's DOS spectrum (Fig. 1). Clearly, the gross features of the crystalline spectrum are retained in the DOS spectrum of the simulated aGaN lattice. The retention of these features, and the gross similarity of the crystal's DOS spectrum and simulated GaN lattice's DOS spectrum follow from the fact that these aspects of the DOS spectra are determined, for the most part, by the mass dierence between gallium and nitrogen atoms and the presence of the same short-range order (tetrahedral local bonding con®gurations) in both structures. More importantly, the vibrational modes corresponding to the peaks in the higher energy region of the simulated a-GaN's DOS spectrum should be infrared as well as Raman active. There have been few reported experimental measurements of the vibrational states of a-GaN. However, recently, Fourier transform infrared spectroscopy measurements have been reported for Ga-implanted a-SiNx :H matrix [10]. These measurements show the presence of an a-GaN layer and a Ga±N stretching vibrations at 91 meV
Fig. 4. Calculated reduced neutron structure factor for the simulated a-GaN network. In the same way as in the experiment, the calculated S E is obtained at a temperature of 8 K and by performing a spherical average over the directions of the wavevector Q.
which is consistent with the predictions of the DOS spectrum of the simulated a-GaN structure shown in Fig. 3. Because neutron scattering has proven useful in elucidating the vibrational states of other amorphous materials, it would be informative to calculate the inelastic neutron structure factor of the simulated a-GaN network. The theoretical computational approach employed is outlined in some detail in [16,17,20,21]. 5 Using the same energies and performing a spherical average over the momentum transfer Q, we have calculated the reduced neutron structure factor for the simulated a-GaN network. Our results are shown
5 The coherent double dierential neutron cross-section for one-phonon annihilation is ``proportional'' to the reduced dynamical structure for S 0 Q; x. This factor determines the way in which the scattering intensity varies for dierent momentum transfers. The scattering process itself is characterized by the energy change x and the momentum transfer Q, which are interrelated by the energy and momentum conservation conditions. The reduced structure factor for a monatomic solid can be written as S 0 Q; x e 2W h xRQ ImGij x2 Q expiQ Ri Rj i; where the summation is over atoms i and j in the solid; Q is the unit vector along Ri and Rj denoting the position of atoms i and j, respectively; 2W is the Debye±Waller factor; and h i denotes a spherical average over the direction of Q. Note that the reduced dynamical structure factor is related to the normal structure factor with the Bose factors and the magnitude of the momentum transfer divided out.
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in Fig. 4. The spectrum exhibits activity in three main regions. There are several prominent features evident in these spectra. As in the experimental spectra for crystalline GaN (Fig. 2), the regions determined more by the motion of nitrogen atoms are weighted more than those determined by gallium atom motions. Compared to DOS spectrum, the optical region of the calculated structure factor is characterized by large changes in the overall intensity. There is a shift of intensity from the lower portion of this region to the upper region. The acoustical region, on the other hand, shows an overall reduction in intensity. 4. Discussion Previous studies of compound amorphous materials have shown that there are four types of disorder: (1) like-atom bonds; (2) bond-angle variations; (3) dihedral angle variations; and (4) variations in topology (rings-of-bond) [16±25]. In reality, these are not independent of each other. Moreover, the possibility of native defects such as under- or over-coordinated atoms should be added to the above list. In contrast to a-GaAs, the relatively large electronegativity dierence between gallium and nitrogen atoms would argue against the presence of homopolar Ga±Ga or N±N bonds in a-GaN. The absence of like-atom bonds clearly indicates that any variations in topology would not include the occurrence of an appreciable fraction of odd-membered rings-of-bond. Since the types of rings-of-bond most likely to form in the actual amorphous solid are those that form in the crystal, we expect a predominance of sixfold rings-of-bond in the actual a-GaN network. Moreover, vibrational modes associated with the sixfold rings-of-bond should be evident as peaks in the region between 9 and 31 meV of the vibrational DOS spectrum of an actual a-GaN network. In the actual a-GaN DOS spectra, however, one would expect these topologically induced features to be less distinct than those induced in the crystalline GaN DOS spectrum. Most of the disparity between the crystalline GaN DOS (Fig. 1) and the DOS of the simulated a-GaN DOS (Fig. 3) may be attributable to the predominance of sixfold rings-
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of-bond in the crystalline structure. The net eect of the topological disorder is to cause the density of states of an actual a-GaN network to closely resemble that of our simulated a-GaN structure. Bond and dihedral angle variations should be the principal sources of disorder in the actual aGaN network. We have investigated the eects of dihedral and bond-angle variations on the simulated a-GaN structure. The net eect on the simulated a-GaN density of states of altering bond angles by small amounts (less than about 10°) is to cause a small increase in the width of the TO-like band (60 and 77 meV). This is accompanied by a slight shift of the center of the TO-like band to higher energies. Similar behavior is observed in the lowest energy region of the DOS spectrum. The displacement in energy of the vibration states by bond-angle ¯uctuations should result in a smearing of the density of states in the actual amorphous network. On the other hand, the distortion energies associated with dihedral angles ¯uctuations are smaller than those associated with bond-angle ¯uctuations and are consequently more likely to ¯uctuate in the amorphous structure, but the density of states is relatively insensitive to these ¯uctuations. Note that both bond and dihedral variations in an actual a-GaN network should have the net eect of broadening the peaks induced in the DOS by topology, so that they disappear almost completely, causing the DOS to be almost indistinguishable from that of the simulated aGaN structure's density of states. Finally, under- and over-coordinated atoms play an important role in the determination of the optical, electronic, and vibrational properties of many amorphous materials. The dangling bond defects in amorphous silicon and valence alternation pairs in amorphous selenium are prototypical examples. Molecular-dynamic studies of a-GaN suggest that strong chemical short-range order and the presence of a large percentage of threefold coordinated gallium and nitrogen atoms characterize the amorphous network [11,12]. Similar molecular-dynamic studies of a-GaAs also predict the presence of a large percentage (about 20%) of threefold coordinated As and Ga sites [22,25]. Yet, vibrational modes (which should be optically active) associated with under-coordinate sites have not been identi®ed
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in the measured infrared and Raman spectra of aGaAs. This may suggest that the large percentage of threefold sites is an artifact of the computational procedure, and that the actual percentage of the threefold coordinated sites is probably much smaller than that predicted by molecular-dynamic studies. Given the large phonon gap between the optic-like and acoustic-like bands in the vibrational DOS of a-GaN, and the expected energies and optical activities of vibrational modes associated with threefold coordinated nitrogen and gallium sites, experimental infrared and Raman measurements should be able to easily determine the densities of defective threefold coordinated nitrogen and gallium sites in a-GaN. 5. Conclusions As a ®rst step in probing the vibrational structure of a-GaN, we report the results of a systematic and quantitative investigation of its vibrational states. Using a VFF model of the elastic energy, the vibrational DOS spectra of crystalline hexagonal GaN and a simulated a-GaN structure were calculated. The large mass mismatch accounts for the large phonon gap separating the acoustic-like and optical-like bands in the DOS spectra of both the crystalline and amorphous structures. Our results for hexagonal GaN yield energies of zone-centered phonons, which are consistent with the Raman and IR studies, and a DOS, which is in accord with the neutron scattering measurements. There are few spectroscopic measurements of the vibrational properties of a-GaN to which our calculated DOS can be compared. Our results show that similar to a-SiO2 , the vibrational properties of a-GaN may be attributed to the nearly crystalline nearestneighbor geometry of the amorphous network and to the eects of bond-angle and dihedral angle disorders. Our results also show that the eects of structural disorder are to cause the density of states spectrum of a-GaN to be `smooth' and the absence of the sharp features, which characterize the spectrum of the crystal. The a-GaN spectrum is charactized by three vibrational `bands' (centered near 87, 71 and 22 meV), and exhibits four
dominant optic peaks (68, 74, 86, and 87 meV) which should correspond to the vibrational mode which are infrared as well as Raman active.
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