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Structure and chemical bonding in amorphous diamond
fB[AMOND RE TED ndATERIALS
ELSEVIER
Diamond and Related Materials 5 (1996) 175-185
Structure and chemical bonding in amorphous diamond G. Jungnickel,
Th. Kiihler, Th. Frauenheim, Department of Physics, Technical
M. Haase, P. Blaudeck,
U. Stephan
University, D 09107 Chemnitz, Germanq
Received 11 August 1995; accepted in revised form 22 November 1995
Abstract The atomic structures of relaxed amorphous carbon models with diamond density are investigated, and the related scattering, electronic, and vibrational properties are analysed. Purely tetrahedrally coordinated models were generated using a bond switch Monte Carlo algorithm similar to the Wooten-Winer-Weaire method for amorphous silicon. These defect-free models are compared with models produced by density-functional-based tight-binding molecular dynamics as well as by molecular dynamics using the classical Tersoff potential. The stability and properties of “amorphous diamond” are discussed, and the consequences for network strain and gap states of frequently appearing undercoordinated atoms are deduced. Keywords: Amorphous
diamond; Atomic structure; Chemical bonding; Molecular dynamics
1. Introduction The production of highly tetrahedral forms of amorphous carbon (ta-C) [l-6] and the theoretical confirmation of corresponding high density metastable structures [ 7-91 have considerably stimulated contemporary research in amorphous semiconductors. These materials have been shown to behave like semiconductors with relatively large bandgaps [l] and they have been successfully doped [lo]. Whereas experimentally the highest fraction of fourfold coordinated sp3-like carbon atoms is determined to be about 85%, related to a density that is about 14% below the diamond density of 3.5 g cmm3 [11,12], the simulations of corresponding structures give lower sp3 fractions of about 65% at 3.0 gcmm3 and 75% at 3.3 g cmd3 [ 13,141. In a previous paper we examined the structure-related electronic properties of the corresponding model structures generated by a density-functional-based tightbinding molecular dynamical (MDDF) scheme and discussed the bandgap properties on the level of n-bonded cluster formation versus electronic defect generation [14]. Moreover, a model containing 85% four-fold coordinated atoms at diamond density was examined for the first time. The main result was the explanation of the occurrence of large TI-~* band gaps (2 3 eV) in these structures by the formation of a few small K clusters, predominantly 0925-9635/96/$15.000 1996Elsevier Science S.A. All rights reserved SSDZ 0925-9635(95)00498-X
pairs of three-fold coordinated carbon atoms embedded within a rigid bonding matrix of four-fold coordinated diamond-like atoms, although embedding effects generally tend to reduce the gap width [15]. The n-bonded atom pairs yield a further gain in binding energy, stabilizing the amorphous network by simultaneously removing internal strain. Most of all, the investigations clearly showed that the presence of three-fold coordinated atoms does not necessarily produce electronic defects in amorphous carbon, as is the case for amorphous silicon. This is a major challenge for bandgap engineering using carbon-based materials. It is still not known whether the fraction of four-fold coordinated atoms can be further increased to create finally a stable completely tetrahedrally bonded amorphous diamond-like structure with a defect-free bandgap. In the following we shall focus on the simulation of “amorphous diamond”, i.e. on the modelling of lowenergy structures with the maximum fraction of fourfold coordinated carbon atoms at diamond density. To do this, we generated various high-density models of amorphous carbon by applying different methods as described in Section 2. In particular, we use the WootenWiner-Weaire technique [ 161 to obtain completely fourfold coordinated continuous distorted carbon networks (CDNs) which are investigated in Section 3 with respect to their network statistics and their scattering properties in comparison with models obtained by molecular dynamics applying both classical potential and density-
G. Jungnickel et al./Diamond and Related Materials 5 (1996) 175-185
176
functional-based tight-binding molecular dynamics. Finally, in Sections 4 and 5 we concentrate on the electronic and vibrational properties of the models, and we discuss the stability of amorphous diamond in particular by further relaxing one of the completely four-fold coordinated CDN models using the MDDF method, taking into account the ability of three-fold coordinated carbon atoms to form 7cbonds.
2. Theoretical modelling 2.1, Monte Carlo relaxation (MCK
using Keating’s potential
models)
The empirical Keating potential [ 171 for modelling the atomic interaction was used to generate two models with 512 and 128 carbon atoms respectively via an extended Monte Carlo algorithm similar to that described by Wooten et al. [ 161 and Wejchert et al. [ 181. A sequence of geometrically relaxed bond transpositions are introduced in an initially crystalline (cubic) diamond supercell to generate a completely four-fold coordinated disordered random network. These bond rearrangements mainly affect the ring structure of the network, creating rings of several sizes and in particular odd-membered rings. To guarantee the applicability of the potential that was originally developed to describe elasticity in diamond crystals by considering only small local deviations from the crystalline short-range order the bond switching process is controlled by the following rules. (1) The initial bond rearrangement process is stopped at a switch density of one bond transposition per carbon atom. Although the network energy seems to be approaching a saturation point when the number of switches introduced exceeds 0.2 per atom, there is a clear tendency to increase the Keating potential further when continuing this process. (2) If rings with less than five carbon atoms are introduced into the network, the bond transposition is forbidden to avoid local distortions that are too large. This condition need not necessarily be applied, as was demonstrated in the case of silicon [19] where the lowest-energy structures have been obtained by considering four-membered rings. However, as will be discussed in Section 3, this contradicts our MDDF results. (3) The length of anyObond after a bond transposition should not exceed 1.85 A. This guarantees that the local coordination number for each atom, counted in spheres at this radius, remains at 4 after each transposition. (4) Bond rearrangements are allowed only for bonds which lie nearly parallel. (5) An additional purely repulsive potential term prevents non-bonded neighbours from approaching to within bonding distance.
The initially amorphized disordered models handled within periodic boundary conditions are characterized by a bond length of 1.56 f 0.12 A and a bond angle of 109.3” + 15.6” for the 512-atom model, and by a bond length of 1.55 + 0.12 A and a bond angle of 109.3” + 15.4” for the 128-atom model. These parameters correspond to high-temperature states. Therefore a Monte Carlo relaxation including bond switches and atomic vibrations was also applied to remove the excessive strain from the network by reswitching thermally unstable bond rearrangements and/or introducing additional switches which influenced their neighbourhood to reach low energy states. During the Monte Carlo bond transpositions the initialization constraints were no longer applied, except that four-membered rings were still excluded. After an equilibration at 10000 K the models were further relaxed by a Monte Carlo simulated annealing technique (cooling rate, 0.5 K per Monte Carlo step), which cooled the structure down to room temperature. Finally, the two low-energy structures MCKl with 512 atoms and MCK2 with 128 atoms were obtained. The rms deviations of the bond lengths and the bond angles were reduced to about two-thirds of the values after the initial amorphization. Furthermore, as is shown in Table 1, the mean bond lengths were corrected during the relaxation so that they approached the diamond value. 2.2. Molecular dynamics relaxation functional-based
using the densitytight-binding method (MDDF models)
To evaluate the structure in amorphous carbon modifications more precisely, we have carried out molecular dynamics (MD) simulations based on quantummechanical calculation of the total energy as a function of atomic coordinates. The gradients of the total energy, evaluated at the given atom positions, provide the interatomic forces. Within these simulations, the Newtonian equations of motion for all atoms within the structural configuration are integrated numerically using the Verlet algorithm, and the configurations are adjusted deterministically at finite temperature. On the basis of the carbon-carbon interaction, a time step of 80 atu Table 1 Structural properties of the theoretical models examined Model structure
Number of atoms
c& (%)
Kc,
R, (A)
Q (de&
MCKl MCK2 MCKMD MDDFl MDDFZ MDT
512 128 128 128 128 512
100 100 95 88 92 52
4.00 4.00 3.95 3.88 3.92 3.51
1.540 + 0.089 1.540 &-0.087 1.537 & 0.059 1.535 * 0.062 1.538 f 0.060 1.491 i 0.068
109.21 + 11.43 109.08 * 11.31 109.05 + 11.76 109.63 * 11.07 109.48 & 10.46 112.04 _+ 10.65
100
4.00
1.545 * 0.000 109.47 f 0.00
Diamond
G. Jungnickel et al. JDiamond and Related Materials 5 (1996) 175.-185
atu =
x 10 l7 s) found to sufficient for the Verlet and to conservation energy during simulation run. method which outlined previously a density-functional-based tight-binding (DFTB) approach [ 201 includes first-principle ideas. The Kohn-Sham orbitals of the atomic configuration are expanded in a minimal basis of localized atomic valence orbitals of the free atoms, which, together with the single atomic potentials, are determined self-consistently within the local density approximation (LDA). By making use of a simplified non-self-consistent scheme for the manyatom configuration, the effective one-electron potential in the Kohn-Sham Hamiltonian of the solid is approximated as a sum of contracted potentials of neutral atoms. The contraction improves the transferability of the attractive part of the potential from the free to the solid environment. The repulsive energy, which includes the core-core interactions, double counting and exchange correlation terms, is modelled by a short-range repulsive pair potential. This is fitted to reproduce the potential energy curve of the C, molecule obtained from a self-consistent field (SCF) calculation as a function of a wide range of interatomic separations. The method has already been used to investigate amorphous carbon structures over a wide range of mass densities and including hydrogen atoms [7,21,22]. The cohesive energies per carbon atom as a function of structure and coordination closely approximate results obtained by more sophisticated SCF methods [20]. An important property of the method is that it accounts well for TC bonding, which is the source of the complex carbon chemistry. The MDDFl structure was generated using simulated annealing. In particular, we performed a rapid quench (cooling rate, 1Ol5 K s-l over a total annealing time of 2 ps) of a liquid structure that was partly equilibrated at 10000 K. The simulation has been performed under the constraint of a fixed volume at the given diamond density with 128 atoms within a periodic simple cubic supercell. The MDDF2 structure was obtained by applying a more extended simulation run including a stochastic annealing process [23]. For this, the total relaxation time was increased to about 8 ps. In order to investigate the stability of the completely four-fold coordinated MCK structures, the MCK2 model containing 128 atoms was post-relaxed under the DF-based potential, performing a MD run at room temperature for 1 ps. 2.3. Molecular dynamics relaxation using Tersoff ‘s potential (MDTmodel)
Another model was constructed via MD by describing the interatomic forces on the basis of the classical Tersoff
117
potential [24] which is a sum of two-body interactions similar to the well-known Morse potential, except that the force constant of its attractive part is a function of the coordination of the pair of atoms. In this way manyparticle interactions are introduced, which is necessary to obtain sensible bond angles. The model containing 512 atoms was generated starting from a random arrangement of carbon atoms within a periodic supercell. It was initially equilibrated at about 5000 K over a total time of 1.5 ps. Finally, it was cooled down in a simulated annealing relaxation over a total annealing time of 1.5 ps and equilibrated again at room temperature for a further 1.25 ps. The method has been used to study amorphous carbon models at three different densities 1251.
3. Static structural properties Table 1 summarizes the structural data of the generated models derived from the composition, atomic coordinates and bond maps of the models. The bond maps of the molecular dynamically generated models MCKMD, MDDFl, MDDF2 and MDT were set up by searching all atoms in a spherical neighbourhood of an atom within a radius of 1.85 A. The bond maps of the MCK models are defined by construction from rearranging a bond map of diamond. Table 1 gives the fraction csP3 of four-fold coordinated sp3-like bonded carbon atoms, the coordination numbers kcc and the mean values and deviations of the bond lengths and bond angles. Additionally, in Fig. 1 we provide the bond length, bond angle and dihedral angle statistics necessary for a full characterization of the models investigated (except for the less interesting MDT model). On analysing the data, we found large differences in the most interesting property, i.e. the fraction of fourfold coordinated atoms. In the MCK models the coordination number of all atoms is 4 by construction. However, it should be noticed, that we generated their bond maps as a test for their relaxation state in a similar way as for the other models. Under this condition we found that eight atoms in the 512-atom model MCKl are surrounded by five rather than four nearest neighbours. This means that some atom pairs that have been treated as non-bonded during the bond transposition technique had finally reached a distance less than 1.85 A. Although an additional repulsive potential (see Section 2, MC method) was applied during the relaxation to reduce such effects, the local strain in some network regions was high enough to force the atoms into these positions. Therefore the fraction of sp3-like atoms for MCKl would be slightly smaller than 100% if one does not count these over-coordinated species because of their deviations from sp3-like hybridization. The MCK2 model does not contain similar atoms or extremely
178
G. Jungnickel et al. JDiamond and Related Materials 5 i 1996) 175-185
BondLengthHistograms II - "__,,,,,,,,
I.15 A
1.45 A
1.75 A
1.15 A
I .45 A
1.75 A
1.15 A
I .45 A
I .75 A
I ISA
1.45 A
I .75 A
I.I~A
I 45A
I .75 A
Bond Angle Histograms
80”
120
160"
80
120
160"
Fig. 1. Bond length, bond angle and dihedral angle statistics for the DFTB relaxed models than for the CDN;.
80"
120"
160"
for the amorphous
strained bonds. Nevertheless, by construction the MCK models are completely four-fold coordinated CDNs [ 261 with a short-range order similar to the diamond crystal. Hence they can be characterized as models of “amorphous diamond”. Also, the electronic structure is very similar to the crystalline counterpart (see Section 4). The two density-functional-based structures MDDFl and MDDF2 differ in their composition because of the different relaxation times. They contain significant fractions of three-fold coordinated carbon atoms (12% and 8% respectively). The longer the relaxation the lower is the number of under-coordinated atoms, since the atoms have more time to scan their neighbourhood to capture additional neighbours. In contrast, after an additional MD relaxation of the four-fold coordinated network MCK2 at room temperature for only 1 ps with the DFTB method, three bonds are strongly elongated and finally broken. The local network strain is reduced by separating the associated atoms from each other and cutting their former bonds. The network then has more degrees of freedom to relax further. The previously bonded atoms become under-coordinated and create defect states. The creation of defect states that cause the
diamond
80"
120"
160"
80"
120"
models. Note that the bond length deviations
160"
are smaller
total energy to increase is in competition with the reduction of elastic strain energy. These effects are the most important here; however, in general, rehybridization into sp2-like states can also occur. This may result in 7cbonds which could also lower the total energy. The Tersoff potential seems to be inappropriate for describing the high-density structures in the constantvolume regime considered, since the MDT model contains only 52% of carbon atoms in the sp3 state even at the diamond density. By using a different MC technique this potential was applied more successfully to studies of amorphous carbon with high mass densities [S]. However, this technique involved changes of the supercell volume that are not considered in this study. Recent progress in filtered vacuum arc deposition confirms that, at lower densities of about 3.0 g cme3, the fraction of four-fold coordinated atoms, the MDT model fails to predict a mean bond length and a mean bond angle close to the diamond values expected. It is worth noting that the bond-angle deviation of the MCK models is about lo%, which is approximately the same as that obtained in studies of a-Si for the sillium model [ 261. However, a major difference between
G. Jungnickel et al./Diamond and Related Materials 5 (1996) 175-185
a-C and a-Si models involves the bond-length deviations. For sillium it is half the result obtained with the carbon CDN. This in turn gives a Keating energy for sillium that is 3.6 times less than that of amorphous diamond. Moreover, the inclusion of a few three-fold coordinated carbon atoms reduces the bond-length deviation substantially (by about 30%) in the MCKMD model which was post-relaxed using the DFTB method. Fig. 1 shows the relevant statistical data for the full characterization of the models with a vast majority of sp3-like atoms. The individual statistics are normalized to unity for comparison. As is expected from the values given in Table 1, the short-range order differs most with respect to bond length statistics. The most stable structure MDDF2 is characterized by almost symmetric bond-length and bond-angle histograms. Differences are found in the dihedral angle statistics that describe the relative orientation of attached network units (distorted tetrahedrons and trigonal planes). Note that the dihedral angles related to sp*-sp3 bonds are equally distributed. The few sp’-sp* bonds found in the MDDF models originate from isolated n-bonded atom pairs; this is supported by their small bond lengths, typical of double bonds, and the tendency to graphite-like dihedral torsions at these adjacent sites. The latter effect is much stronger for the MDDF2 model, where the double bonds are much better relaxed with respect to their bond length. This is a secondary consequence of the extended relaxation time, since the double bonds have more time to influence their neighbourhood in such a way that they can improve their n-orbital overlap. The sinusoidal curvature of the dihedral angle distribution for adjacent tetrahedral sites in amorphous networks is a topological effect [28]; the deviation of the distribution from diamond behaviour (delta functions at 60” and 180”) is smallest for MDDF2. Fig. 2 summarizes the ring statistics of the models. The rings were counted using the shortest path criterion [29]. Only rapid quenching yields a considerable number of four-membered rings for the DF-based models. Therefore the general prohibition from closing network loops with fewer than five atoms applied in the construction of the MCK models does not alter the quality of the CDN models. We find almost equal numbers of five-, six- and seven-membered rings in the MCK and MDDF models and only a few larger rings. There is a tendency to prefer six-membered over fiveand seven-membered rings. The Tersoff potential clearly suppresses the generation of five-membered rings. An interesting feature is that in MCK2 the number of fivemembered rings is higher than the number of sevenmembered rings. This is not changed by post-relaxation with the DFTB method. In the MDDF models, however, seven-membered rings are preferred over five-membered rings. In summing all rings up to ring size 10 we could
179
not reach the ring density of the diamond crystal for either the MCK or the MDDF models. The static structure of the models yields the total reduced structure factors shown in Fig. 3 as thick lines. They are given by F(Q) = Q [S(Q) - 11, where Q represents the momentum transfer associated with a scattering process and S(Q) is the structure factor defined in the usual way. F(Q) is determined via Fourier transformation of the atomic pair correlation function that can easily be obtained from the histogram of pair distances for a model under consideration. The most striking feature is the central peak at Q z 10 A, which changes its shape from a single asymmetric peak to a clear double-peak feature with an increasing fraction of three-fold coordinated carbon atoms. One reason for this is the underlying sp2-sp3 contribution which causes a relative minimum to occur at Q= 10 A. However, the strong bond-length deviations in the CDN models smear out any small subpeak on the right flank. Note also that the doublepeak contour obtained for the MDDF models including 12% sp2-like bonded atoms is clearly different from what is measured in the 3.0 g cm-- 3 case [ 11,271, thus supporting the previously published result that the fraction of three-fold coordinated atoms incorporated in filtered vacuum-arc-deposited samples exceeds 12% [13]. Comparing F(Q) for the post-relaxed model MCKMD with that for MDDF2 shows that the central peak must also be influenced by a structural difference, since the composition and short-range order of these two models are almost equal. The only difference that goes beyond short-range correlations in these models is the relationship between the number of five- and sevenmembered rings, which is reversed. Finally, attention should be paid to the first sharp diffraction peak at Q 2 1.5 A, the occurrence of which seems to be an artefact of the smaller models. It vanishes for the 512-atom models.
4. Electronic properties The electronic properties of the models are evaluated according to the method described elsewhere [ 151 using n-cluster statistics. The related data are summarized in Table 2. They include the ratio np+&, of the number of p and 71:states not involved in CTbonds to the number of G bonds, the ratio rip/n,, of p defect states to the bonding and antibonding 7cstates in the gap region and the number np_w_,k~ of weak (5 bonds. We also give the number of n-bonded atom clusters per atom and the (T gap width. The latter is rather approximate since, mainly because of the minimal basis approach, the gap width is not very well defined. Therefore we provide the width of the gap relative to the direct gap of diamond as calculated in the DFTB method. This is done by removing the defect states due to isolated p orbitals or weak
180
G. Jungnickel et uh/Diamond and Related Materials 5 (1996) 175-185
-2.5
2.5
Fig. 2. The ring statistics of the models the CDNs with the MDDF models. Table 2 Electronic Model structure
defect and n-cluster np+n In,
MCKl MCK2 MCKMD MDDFl MDDF2 MDT
0.001 0.000 0.016 0.032 0.032 0.139
Diamond
0.000
statistics
show a difference
in the relation
between
the number
of five- and seven-membered
rings when comparing
of the models n,ln,
0.333 1.000 0.333 1.119
np-weak D per atom
?i clusters
Gap width (%)
per atom
334
5-6
7kx
0.002 0.000 0.023 0.420 0.006 0.870
0.000 0.000 0.000 0.023 0.008 0.033
0.000 0.000 0.000 0.000 0.000 0.008
0.000 0.000 0.000 0.000 0.000 0.008
39 64 72 47 27 67
0.000
0.000
0.000
0.000
100
o orbitals from the gap region prior to the determination of the gap. The width of the latter is then given by the highest occupied molecular levels and the lowest unoccupied levels that are not identified to be defects. This method yields the width of a o-o* gap for a model that does not contain z-bonded atom clusters. In contrast, if such clusters are formed by sp2-like atoms, the width of the related z--71*gap is determined in this way. The total densities of states (DOS) for the models are
shown in Fig. 4. The MCK models contain no threefold coordinated atoms and therefore should not have rc states next to the Fermi level E,. Large o bandgaps are obtained for both MCKl and MCKZ. However, in MCKl there are a few strained bonds and a weak cr bond yielding isolated states in the gap region. MCK2 is a true amorphous diamond with a pure gap, since the model is free from any defect states (n,+,/n,=O.O). The MDDF models and the MDT model all contain sp2-
G. Jungnickel et al./Diamond and Related Materials 5 (1996) 175-185
181
lbng Statistics 0.8 MCK2
MCKl 0.6
0.0 MCKh4D
MDDF
MDDFZ
MDT
I
0.6
0.6
0.4
0.2
0.0 3
5
6
I
8
9
10
4
5
6
7
8
9
IO
Ring Size Fig. 3. The total The contributions
interference functions F(Q) (thick lines) of the models differ, particularly from the different bond types are marked by thin lines.
like carbon atoms, of which some are isolated species but others are n-bonded atom pairs. These double-bondlike atomic arrangements yield z and 7c* states in the o gap region of the DOS, thus reducing the effective gap width. We note that, owing to the extended relaxation time, the sp2-like atoms in MDDFZ have a greater chance of forming x-bonded atom pairs than during the relaxation of MDDFl. The Tersoff potential fails to handle the three-fold coordinated atoms correctly, and this is indicated by the relatively high value for the number of weak c~ bonds in agreement with the largest number of p defects comparing all models. This is the source of the highest DOS close to the Fermi level (see also Ref. [ 151).
5. Vibrational properties The vibrational density of states (VDOS) has been calculated within the harmonic approximation by diagonalization of the dynamical matrix to obtain the normalmode frequencies. For each network we set up the dynamic matrix dependent on the potential used to
regarding
the central
(double)
peak at about
10
A-‘.
generate the model. To analyse the chemical nature of vibrations we have decomposed the total VDOS into the contributions from the different carbon hybrids, applying the projection technique described in Ref. [ 231. We use the inverse participation ratio P-‘(o) [30] as the measure for the localization or extension of each eigenmode. The details of the theoretical methods are discussed elsewhere [ 231. All spectra have been convoluted by a constant Lorenzian resolution function of width 40 cm-’ in order to simulate the VDOS of an extended system. In Fig. 5 we present the vibrational density of states (VDOS) of the CDNs and the models relaxed with the DFTB method. The most prominent difference between them is the shape of the VDOS. For models MCKl and MCK2 based on Keating forces we obtain a diamondlike shape with a clear peak centred at about 1300 cm-‘. These VDOSs are similar to CDN models of amorphous silicon [26,30], although the low-frequency bands here are fused to one broad band between 250 and about llOOcm_‘. In contrast, structures MCKMD, MDDFl and MDDF2 develop a characteristic hemispherical shape over the whole frequency range under the influence
182
G. Jungnickel et aLlDiamond and Related Materials 5 (1996) 175-185
DOS [ 1/eV/atom]
MCKl
ID
;
Energy [eV] Fig. 4. The under-coordinated
atoms yield defect and II states in the gap region of the electronic DOS. Energies are given relative to the Fermi level.
of the tight-binding forces. This is particularly obvious for the vibrational density of states of the sp3 sites only. The cause of this difference is not yet clear. It is possible that the particular reversed relationship between the five- and the seven-membered rings could have some influence in dropping high-frequency modes and enhancing the low- and medium-frequency modes. Then the VDOS for MCKMD would be clearly different from the VDOS of model MDDFl. Another factor that could reduce the number of vibrations with higher frequencies relative to the others in the MDDF models is the reduced stress in the systems. Also, it should be noted that the force constants in the potentials are not exactly equal and that the range for interactions is larger for the tight-binding than for the Keating potential. However, the main band is found to lie between 250 and about 1700 cm- ’ for all models. Considering the high-frequency limit for diamond at about 1350 cm-‘, we find a growing localization of transversal vibrations above this value in all of our models. This is shown in Fig. 6, where the defined inverse participation ratios for the model eigenmodes are compared, and again is similar to the results obtained for
amorphous silicon models [30] where the localization of modes at the high-frequency edge was found to be enhanced by the presence of five-coordinated sites. Comparison of the intensities for MCKl and MCK2 in Fig. 6 shows exactly the same result. The occurrence of the regions with atoms that are effectively overcoordinated enhances the high-frequency localization for MCKl. For the molecular dynamically generated structures which contain under-coordinated sp2-like atoms rather than over-coordinated sites, we also find defect bands beyond 1700 cm-’ with strongly localized excitations. These are mostly due to stretching motions of isolated sp’ pairs within the rigid sp3 network. The width of these defect bands in the CDN structures is much smaller than obtained for the other models. Additionally, we find localized low-frequency modes in the models relaxed with the DFTB method and assign them to the isolated under-coordinated atoms. This is consistent with the findings in Ref. [30]. Furthermore, we identified a strongly localized highfrequency mode at 2780 cm-’ in MCKl (Fig. 6). A stretch vibration of a five-membered ring found in the region of high local stress mentioned above is responsible
G. Jungnickel et al./Diamond and Related Materials 5 (1996) 175-185
INTENSITY
[ arb.
units
183
] MCKl
MCKZ
MCKMD
MDDFl
MDDF2
0
2000
1000
3
w[cn-l] Fig. 5. VDOSs of the CDNs and the models relaxed with the DFTB method. The models relaxed with the Keating potential show a pronounced peak at about 1300 cm-l. Dotted lines show the contribution of sp3 atoms, dashed lines that of sp3-like species.
for this very high-frequency mode. The distance between the two diametral atoms taking part in this vibration is only about 1.77 A. However, they are not recognized as a bonded pair of atoms in the bond transposition modelling procedure. Counting this pair of atoms as a bond would cause a three-membered ring to appear. Hence the two atoms are under compressive stress in their local environment and vibrate at a particularly high frequency.
6. Conclusions We have generated two completely four-fold coordinated continuous distorted networks (CDNs) to simulate the properties of “amorphous diamond” by applying the Wooten-Winer-Weaire bond transposition technique. Although the models containing 512 and 128 carbon atoms were relaxed with respect to a Keating potential, they are unstable against the creation of under-
G. Jungnickel et al./Diamond and Related Materials 5 (1996) 175-185
184
INTENSITY
[ arb. units ]
MCKI
MCK2
MCKMD
P-‘(w)
MDDFl
I
MDDFB
3ci(IO
Go0 w [cm-l]
Fig. 6. The inverse participation ratios P-‘(w) reveal localized high-frequency modes in all models, but only a few localized low-frequency vibrations in the models containing isolated sp2-like atoms.
coordinated atoms. This was tested in an additional molecular dynamics simulation where the interatomic forces were described with a more realistic densityfunctional-based tight-binding potential. The smaller model continued to relax under these forces by breaking some bonds and creating isolated three-fold coordinated sites that gave rise to defect states in the gap region close to the Fermi level. Additionally, we generated models by means of molecular dynamics and the simulated annealing technique describing the interatomic forces with the tight-binding
as well as with the semi-empirical Tersoff potential. When Tersoff forces were used we were unable to obtain a realistic number of sp3-like atoms within the constant volume regime considered. In contrast, the models based on the tight-binding forces contain about 90% four-fold coordinated sites. The under-coordinated atoms in these models try to create pairs in order to make up double bonds that would yield a z-rr* gap. However, we also find defect states due to isolated p orbitals in these models. By extending the relaxation time and using a stochastic cooling technique the total energy has been
G. Jungnickel et al,/Diamond and Related Materials 5 (1996) 175-185
reduced owing to the improvements in the formation of n-bonded atom pairs. However, this causes the effective gap width to decrease. The analysis of the vibrational modes of these models shows that there is a difference in the shape of the vibrational density of states between the models where the atoms interact with Keating forces and the models where they interact via tight-binding forces. The typical diamond-like peak found in the former does not emerge in the latter. All the models are characterized by localized high-frequency modes. Localized low-frequency modes appear only in the networks containing isolated sp2like atoms. It appears that isolated three-fold coordinated atoms and n-bonded pairs of under-coordinated atoms remove excessive strain from completely tetrahedrally bonded amorphous carbon networks with diamond density. The resulting gain in total energy stabilizes these systems, although defect and n states in the gap region with energies above the valence o-band edge appear. If the n-bonded atom pairs can relax properly in their rigid sp3-bonded neighbourhood, they create WTC*gaps which are smaller than the expected o--(5* gaps for completely four-fold coordinated amorphous networks. The gap width for the latter models in the absence of overcoordinated species is roughly estimated to be about 64% of that of crystalline diamond.
Acknowledgement We gratefully acknowledge support by the Deutsche Forschungsgemeinschaft.
References [l]
[2]
D.R. McKenzie, D. Muller, E. Kravtchinskaia, D. Segal, P.B. G. Amaratunga, P.H. Gaskell Mater., 1 (1991) 51. D.R. McKenzie, D.A. Muller,
B.A. Pailthorpe, Z.H. Wang, Lukins, P.D. Swift, P.J. Martin, and A. Saeed, Diamond Relat. E. Kravtchinskaia,
D. Segal,
185
D.J.H. Cockayne, G. Amaratunga and R. Silva, Thin Solid Films, 206 (1992) 198. c31 Y. Lifshitz, G.D. Lempert, S. Rotter, I. Avigal, C. Uzan-Saguy and R. Kalish, Diamond Relat. Mater., 2 (1993) 285. c41 J. Ishikawa, Y. Takeiri, K. Ogawa and T. Tagaki, .I. Appl. Phys., it61 (1991) 773. c51 B. Rother, J. Siegel, K. Breuer, I. Mtihling, S. Deutschmann, J. Vetter, G. Trommer and B. Rau, J. Mater. Rex, 6 (1991) 101. S. Misawa, S. Yoshida and S. Gonda, J. Appl. C61 T. Miyazawa, Phys., 55 (1984) 188. G. Jungnickel, Th. Kbhler and U. Stephan, c71 Th. Frauenheim, J. Non-Cryst. Solids, 182 (1995) 186, and references cited therein. PI P.C. Kelires, Phys. Rev. B, 47 (1993) 1829. R. Wang and S. Klemm, Phys. Rev. B, 43 c91 D.A. Drabold, (1991) 5132. and CA. Davis, Diamond Relat. Mater., 4 Cl01 J. Robertson (1994) 441. Cl11 P.H. Gaskell, A. Saeed, PC. Chieux and D.R. McKenzie, Phys. Rev. Lett., 67 (1991) 773. r-121 S.D. Berger, D.R. McKenzie and P.J. Martin, Philos. Mag. Lett., 57 (1988) 285. M. Ktihn, S. Deutschmann, F. Richter, Cl31 G. Jungnickel, U. Stephan, P. Blaudeck and Th. Frauenheim, Diamond Relat. Mater., 3 (1994) 1056. P. Blaudeck and G. Jungnickel. Cl41 U. Stephan, Th. Frauenheim, Phys. Reo. B, 50 (1994) 1489. Lee, W.R.L. Lambrecht, B. Segall, P.C. Kelires, Cl51 C.H. Th. Frauenheim and U. Stephan, Phys. Rev. B, 49 (1994) 11448. Cl61 F. Wooten, K. Winer and D. Weaire, Phys. Reu. Lett., 54 (1985) 1392. Cl71 P.N. Keating, Phys. Rev., 145 (1966) 637. Cl81 J. Wejchert, D. Weaire and F. Wooten, J. Non-tryst. Solids., 122 (1990) 241. Cl91 F. Wooten and D. Weaire, J. Non-tryst. Solids, 97-98 (1987) 349. Th. Frauenheim, Th. KBhler, R. Kaschner and c201 D. Porezag, G. Seifert, Phys. Rev. B, 51 (1995) 12947. Th. Kbhler, M. Haase, S. Deutschmann, c211 G. Jungnickel, U. Stephan, P. Blaudeck and Th. Frauenheim, J. Non-cry& Solids, 183 (1995) 161. Th. Frauenheim, D. Porezag, P. Blaudeck, c221 G. Jungnickel, U. Stephan and R.J. Newport, Phys. Reu. B, 50 (1994) 6709. and G. Jungnickel, Phys. Rev. B, ~231 Th. KBhler, Th. Frauenheim 52 (1995) 118. c241 J. Tersoff, Phys. Rea. B, 37 (1988) 6991. and M. Haase, J. Phys. Condens. Matter, 5 ~251 U. Stephan (1993) 9157. C261 K. Winer, Phys. Rev. B, 35 (1987) 2366. ~271 P.H. Gaskell, A. Saeed, P. Chieux and D.R. McKenzie, Philos. Mag. B, 66 (1992) 155. C281 R.J. Temkin, J. Non-tryst. Solids, 28 (1978) 23. Phys. Rev. B, 44 (1991) 4925. ~291 D.S. Franzblau, G.S. Grest and c301 R. Biswas, A.M. Bouchard, W.A. Kamitakahara, C.M. Soukoulis, Phys. Rev. Lett.. 60 (1988) 2280.
ELSEVIER
Diamond and Related Materials 5 (1996) 175-185
Structure and chemical bonding in amorphous diamond G. Jungnickel,
Th. Kiihler, Th. Frauenheim, Department of Physics, Technical
M. Haase, P. Blaudeck,
U. Stephan
University, D 09107 Chemnitz, Germanq
Received 11 August 1995; accepted in revised form 22 November 1995
Abstract The atomic structures of relaxed amorphous carbon models with diamond density are investigated, and the related scattering, electronic, and vibrational properties are analysed. Purely tetrahedrally coordinated models were generated using a bond switch Monte Carlo algorithm similar to the Wooten-Winer-Weaire method for amorphous silicon. These defect-free models are compared with models produced by density-functional-based tight-binding molecular dynamics as well as by molecular dynamics using the classical Tersoff potential. The stability and properties of “amorphous diamond” are discussed, and the consequences for network strain and gap states of frequently appearing undercoordinated atoms are deduced. Keywords: Amorphous
diamond; Atomic structure; Chemical bonding; Molecular dynamics
1. Introduction The production of highly tetrahedral forms of amorphous carbon (ta-C) [l-6] and the theoretical confirmation of corresponding high density metastable structures [ 7-91 have considerably stimulated contemporary research in amorphous semiconductors. These materials have been shown to behave like semiconductors with relatively large bandgaps [l] and they have been successfully doped [lo]. Whereas experimentally the highest fraction of fourfold coordinated sp3-like carbon atoms is determined to be about 85%, related to a density that is about 14% below the diamond density of 3.5 g cmm3 [11,12], the simulations of corresponding structures give lower sp3 fractions of about 65% at 3.0 gcmm3 and 75% at 3.3 g cmd3 [ 13,141. In a previous paper we examined the structure-related electronic properties of the corresponding model structures generated by a density-functional-based tightbinding molecular dynamical (MDDF) scheme and discussed the bandgap properties on the level of n-bonded cluster formation versus electronic defect generation [14]. Moreover, a model containing 85% four-fold coordinated atoms at diamond density was examined for the first time. The main result was the explanation of the occurrence of large TI-~* band gaps (2 3 eV) in these structures by the formation of a few small K clusters, predominantly 0925-9635/96/$15.000 1996Elsevier Science S.A. All rights reserved SSDZ 0925-9635(95)00498-X
pairs of three-fold coordinated carbon atoms embedded within a rigid bonding matrix of four-fold coordinated diamond-like atoms, although embedding effects generally tend to reduce the gap width [15]. The n-bonded atom pairs yield a further gain in binding energy, stabilizing the amorphous network by simultaneously removing internal strain. Most of all, the investigations clearly showed that the presence of three-fold coordinated atoms does not necessarily produce electronic defects in amorphous carbon, as is the case for amorphous silicon. This is a major challenge for bandgap engineering using carbon-based materials. It is still not known whether the fraction of four-fold coordinated atoms can be further increased to create finally a stable completely tetrahedrally bonded amorphous diamond-like structure with a defect-free bandgap. In the following we shall focus on the simulation of “amorphous diamond”, i.e. on the modelling of lowenergy structures with the maximum fraction of fourfold coordinated carbon atoms at diamond density. To do this, we generated various high-density models of amorphous carbon by applying different methods as described in Section 2. In particular, we use the WootenWiner-Weaire technique [ 161 to obtain completely fourfold coordinated continuous distorted carbon networks (CDNs) which are investigated in Section 3 with respect to their network statistics and their scattering properties in comparison with models obtained by molecular dynamics applying both classical potential and density-
G. Jungnickel et al./Diamond and Related Materials 5 (1996) 175-185
176
functional-based tight-binding molecular dynamics. Finally, in Sections 4 and 5 we concentrate on the electronic and vibrational properties of the models, and we discuss the stability of amorphous diamond in particular by further relaxing one of the completely four-fold coordinated CDN models using the MDDF method, taking into account the ability of three-fold coordinated carbon atoms to form 7cbonds.
2. Theoretical modelling 2.1, Monte Carlo relaxation (MCK
using Keating’s potential
models)
The empirical Keating potential [ 171 for modelling the atomic interaction was used to generate two models with 512 and 128 carbon atoms respectively via an extended Monte Carlo algorithm similar to that described by Wooten et al. [ 161 and Wejchert et al. [ 181. A sequence of geometrically relaxed bond transpositions are introduced in an initially crystalline (cubic) diamond supercell to generate a completely four-fold coordinated disordered random network. These bond rearrangements mainly affect the ring structure of the network, creating rings of several sizes and in particular odd-membered rings. To guarantee the applicability of the potential that was originally developed to describe elasticity in diamond crystals by considering only small local deviations from the crystalline short-range order the bond switching process is controlled by the following rules. (1) The initial bond rearrangement process is stopped at a switch density of one bond transposition per carbon atom. Although the network energy seems to be approaching a saturation point when the number of switches introduced exceeds 0.2 per atom, there is a clear tendency to increase the Keating potential further when continuing this process. (2) If rings with less than five carbon atoms are introduced into the network, the bond transposition is forbidden to avoid local distortions that are too large. This condition need not necessarily be applied, as was demonstrated in the case of silicon [19] where the lowest-energy structures have been obtained by considering four-membered rings. However, as will be discussed in Section 3, this contradicts our MDDF results. (3) The length of anyObond after a bond transposition should not exceed 1.85 A. This guarantees that the local coordination number for each atom, counted in spheres at this radius, remains at 4 after each transposition. (4) Bond rearrangements are allowed only for bonds which lie nearly parallel. (5) An additional purely repulsive potential term prevents non-bonded neighbours from approaching to within bonding distance.
The initially amorphized disordered models handled within periodic boundary conditions are characterized by a bond length of 1.56 f 0.12 A and a bond angle of 109.3” + 15.6” for the 512-atom model, and by a bond length of 1.55 + 0.12 A and a bond angle of 109.3” + 15.4” for the 128-atom model. These parameters correspond to high-temperature states. Therefore a Monte Carlo relaxation including bond switches and atomic vibrations was also applied to remove the excessive strain from the network by reswitching thermally unstable bond rearrangements and/or introducing additional switches which influenced their neighbourhood to reach low energy states. During the Monte Carlo bond transpositions the initialization constraints were no longer applied, except that four-membered rings were still excluded. After an equilibration at 10000 K the models were further relaxed by a Monte Carlo simulated annealing technique (cooling rate, 0.5 K per Monte Carlo step), which cooled the structure down to room temperature. Finally, the two low-energy structures MCKl with 512 atoms and MCK2 with 128 atoms were obtained. The rms deviations of the bond lengths and the bond angles were reduced to about two-thirds of the values after the initial amorphization. Furthermore, as is shown in Table 1, the mean bond lengths were corrected during the relaxation so that they approached the diamond value. 2.2. Molecular dynamics relaxation functional-based
using the densitytight-binding method (MDDF models)
To evaluate the structure in amorphous carbon modifications more precisely, we have carried out molecular dynamics (MD) simulations based on quantummechanical calculation of the total energy as a function of atomic coordinates. The gradients of the total energy, evaluated at the given atom positions, provide the interatomic forces. Within these simulations, the Newtonian equations of motion for all atoms within the structural configuration are integrated numerically using the Verlet algorithm, and the configurations are adjusted deterministically at finite temperature. On the basis of the carbon-carbon interaction, a time step of 80 atu Table 1 Structural properties of the theoretical models examined Model structure
Number of atoms
c& (%)
Kc,
R, (A)
Q (de&
MCKl MCK2 MCKMD MDDFl MDDFZ MDT
512 128 128 128 128 512
100 100 95 88 92 52
4.00 4.00 3.95 3.88 3.92 3.51
1.540 + 0.089 1.540 &-0.087 1.537 & 0.059 1.535 * 0.062 1.538 f 0.060 1.491 i 0.068
109.21 + 11.43 109.08 * 11.31 109.05 + 11.76 109.63 * 11.07 109.48 & 10.46 112.04 _+ 10.65
100
4.00
1.545 * 0.000 109.47 f 0.00
Diamond
G. Jungnickel et al. JDiamond and Related Materials 5 (1996) 175.-185
atu =
x 10 l7 s) found to sufficient for the Verlet and to conservation energy during simulation run. method which outlined previously a density-functional-based tight-binding (DFTB) approach [ 201 includes first-principle ideas. The Kohn-Sham orbitals of the atomic configuration are expanded in a minimal basis of localized atomic valence orbitals of the free atoms, which, together with the single atomic potentials, are determined self-consistently within the local density approximation (LDA). By making use of a simplified non-self-consistent scheme for the manyatom configuration, the effective one-electron potential in the Kohn-Sham Hamiltonian of the solid is approximated as a sum of contracted potentials of neutral atoms. The contraction improves the transferability of the attractive part of the potential from the free to the solid environment. The repulsive energy, which includes the core-core interactions, double counting and exchange correlation terms, is modelled by a short-range repulsive pair potential. This is fitted to reproduce the potential energy curve of the C, molecule obtained from a self-consistent field (SCF) calculation as a function of a wide range of interatomic separations. The method has already been used to investigate amorphous carbon structures over a wide range of mass densities and including hydrogen atoms [7,21,22]. The cohesive energies per carbon atom as a function of structure and coordination closely approximate results obtained by more sophisticated SCF methods [20]. An important property of the method is that it accounts well for TC bonding, which is the source of the complex carbon chemistry. The MDDFl structure was generated using simulated annealing. In particular, we performed a rapid quench (cooling rate, 1Ol5 K s-l over a total annealing time of 2 ps) of a liquid structure that was partly equilibrated at 10000 K. The simulation has been performed under the constraint of a fixed volume at the given diamond density with 128 atoms within a periodic simple cubic supercell. The MDDF2 structure was obtained by applying a more extended simulation run including a stochastic annealing process [23]. For this, the total relaxation time was increased to about 8 ps. In order to investigate the stability of the completely four-fold coordinated MCK structures, the MCK2 model containing 128 atoms was post-relaxed under the DF-based potential, performing a MD run at room temperature for 1 ps. 2.3. Molecular dynamics relaxation using Tersoff ‘s potential (MDTmodel)
Another model was constructed via MD by describing the interatomic forces on the basis of the classical Tersoff
117
potential [24] which is a sum of two-body interactions similar to the well-known Morse potential, except that the force constant of its attractive part is a function of the coordination of the pair of atoms. In this way manyparticle interactions are introduced, which is necessary to obtain sensible bond angles. The model containing 512 atoms was generated starting from a random arrangement of carbon atoms within a periodic supercell. It was initially equilibrated at about 5000 K over a total time of 1.5 ps. Finally, it was cooled down in a simulated annealing relaxation over a total annealing time of 1.5 ps and equilibrated again at room temperature for a further 1.25 ps. The method has been used to study amorphous carbon models at three different densities 1251.
3. Static structural properties Table 1 summarizes the structural data of the generated models derived from the composition, atomic coordinates and bond maps of the models. The bond maps of the molecular dynamically generated models MCKMD, MDDFl, MDDF2 and MDT were set up by searching all atoms in a spherical neighbourhood of an atom within a radius of 1.85 A. The bond maps of the MCK models are defined by construction from rearranging a bond map of diamond. Table 1 gives the fraction csP3 of four-fold coordinated sp3-like bonded carbon atoms, the coordination numbers kcc and the mean values and deviations of the bond lengths and bond angles. Additionally, in Fig. 1 we provide the bond length, bond angle and dihedral angle statistics necessary for a full characterization of the models investigated (except for the less interesting MDT model). On analysing the data, we found large differences in the most interesting property, i.e. the fraction of fourfold coordinated atoms. In the MCK models the coordination number of all atoms is 4 by construction. However, it should be noticed, that we generated their bond maps as a test for their relaxation state in a similar way as for the other models. Under this condition we found that eight atoms in the 512-atom model MCKl are surrounded by five rather than four nearest neighbours. This means that some atom pairs that have been treated as non-bonded during the bond transposition technique had finally reached a distance less than 1.85 A. Although an additional repulsive potential (see Section 2, MC method) was applied during the relaxation to reduce such effects, the local strain in some network regions was high enough to force the atoms into these positions. Therefore the fraction of sp3-like atoms for MCKl would be slightly smaller than 100% if one does not count these over-coordinated species because of their deviations from sp3-like hybridization. The MCK2 model does not contain similar atoms or extremely
178
G. Jungnickel et al. JDiamond and Related Materials 5 i 1996) 175-185
BondLengthHistograms II - "__,,,,,,,,
I.15 A
1.45 A
1.75 A
1.15 A
I .45 A
1.75 A
1.15 A
I .45 A
I .75 A
I ISA
1.45 A
I .75 A
I.I~A
I 45A
I .75 A
Bond Angle Histograms
80”
120
160"
80
120
160"
Fig. 1. Bond length, bond angle and dihedral angle statistics for the DFTB relaxed models than for the CDN;.
80"
120"
160"
for the amorphous
strained bonds. Nevertheless, by construction the MCK models are completely four-fold coordinated CDNs [ 261 with a short-range order similar to the diamond crystal. Hence they can be characterized as models of “amorphous diamond”. Also, the electronic structure is very similar to the crystalline counterpart (see Section 4). The two density-functional-based structures MDDFl and MDDF2 differ in their composition because of the different relaxation times. They contain significant fractions of three-fold coordinated carbon atoms (12% and 8% respectively). The longer the relaxation the lower is the number of under-coordinated atoms, since the atoms have more time to scan their neighbourhood to capture additional neighbours. In contrast, after an additional MD relaxation of the four-fold coordinated network MCK2 at room temperature for only 1 ps with the DFTB method, three bonds are strongly elongated and finally broken. The local network strain is reduced by separating the associated atoms from each other and cutting their former bonds. The network then has more degrees of freedom to relax further. The previously bonded atoms become under-coordinated and create defect states. The creation of defect states that cause the
diamond
80"
120"
160"
80"
120"
models. Note that the bond length deviations
160"
are smaller
total energy to increase is in competition with the reduction of elastic strain energy. These effects are the most important here; however, in general, rehybridization into sp2-like states can also occur. This may result in 7cbonds which could also lower the total energy. The Tersoff potential seems to be inappropriate for describing the high-density structures in the constantvolume regime considered, since the MDT model contains only 52% of carbon atoms in the sp3 state even at the diamond density. By using a different MC technique this potential was applied more successfully to studies of amorphous carbon with high mass densities [S]. However, this technique involved changes of the supercell volume that are not considered in this study. Recent progress in filtered vacuum arc deposition confirms that, at lower densities of about 3.0 g cme3, the fraction of four-fold coordinated atoms, the MDT model fails to predict a mean bond length and a mean bond angle close to the diamond values expected. It is worth noting that the bond-angle deviation of the MCK models is about lo%, which is approximately the same as that obtained in studies of a-Si for the sillium model [ 261. However, a major difference between
G. Jungnickel et al./Diamond and Related Materials 5 (1996) 175-185
a-C and a-Si models involves the bond-length deviations. For sillium it is half the result obtained with the carbon CDN. This in turn gives a Keating energy for sillium that is 3.6 times less than that of amorphous diamond. Moreover, the inclusion of a few three-fold coordinated carbon atoms reduces the bond-length deviation substantially (by about 30%) in the MCKMD model which was post-relaxed using the DFTB method. Fig. 1 shows the relevant statistical data for the full characterization of the models with a vast majority of sp3-like atoms. The individual statistics are normalized to unity for comparison. As is expected from the values given in Table 1, the short-range order differs most with respect to bond length statistics. The most stable structure MDDF2 is characterized by almost symmetric bond-length and bond-angle histograms. Differences are found in the dihedral angle statistics that describe the relative orientation of attached network units (distorted tetrahedrons and trigonal planes). Note that the dihedral angles related to sp*-sp3 bonds are equally distributed. The few sp’-sp* bonds found in the MDDF models originate from isolated n-bonded atom pairs; this is supported by their small bond lengths, typical of double bonds, and the tendency to graphite-like dihedral torsions at these adjacent sites. The latter effect is much stronger for the MDDF2 model, where the double bonds are much better relaxed with respect to their bond length. This is a secondary consequence of the extended relaxation time, since the double bonds have more time to influence their neighbourhood in such a way that they can improve their n-orbital overlap. The sinusoidal curvature of the dihedral angle distribution for adjacent tetrahedral sites in amorphous networks is a topological effect [28]; the deviation of the distribution from diamond behaviour (delta functions at 60” and 180”) is smallest for MDDF2. Fig. 2 summarizes the ring statistics of the models. The rings were counted using the shortest path criterion [29]. Only rapid quenching yields a considerable number of four-membered rings for the DF-based models. Therefore the general prohibition from closing network loops with fewer than five atoms applied in the construction of the MCK models does not alter the quality of the CDN models. We find almost equal numbers of five-, six- and seven-membered rings in the MCK and MDDF models and only a few larger rings. There is a tendency to prefer six-membered over fiveand seven-membered rings. The Tersoff potential clearly suppresses the generation of five-membered rings. An interesting feature is that in MCK2 the number of fivemembered rings is higher than the number of sevenmembered rings. This is not changed by post-relaxation with the DFTB method. In the MDDF models, however, seven-membered rings are preferred over five-membered rings. In summing all rings up to ring size 10 we could
179
not reach the ring density of the diamond crystal for either the MCK or the MDDF models. The static structure of the models yields the total reduced structure factors shown in Fig. 3 as thick lines. They are given by F(Q) = Q [S(Q) - 11, where Q represents the momentum transfer associated with a scattering process and S(Q) is the structure factor defined in the usual way. F(Q) is determined via Fourier transformation of the atomic pair correlation function that can easily be obtained from the histogram of pair distances for a model under consideration. The most striking feature is the central peak at Q z 10 A, which changes its shape from a single asymmetric peak to a clear double-peak feature with an increasing fraction of three-fold coordinated carbon atoms. One reason for this is the underlying sp2-sp3 contribution which causes a relative minimum to occur at Q= 10 A. However, the strong bond-length deviations in the CDN models smear out any small subpeak on the right flank. Note also that the doublepeak contour obtained for the MDDF models including 12% sp2-like bonded atoms is clearly different from what is measured in the 3.0 g cm-- 3 case [ 11,271, thus supporting the previously published result that the fraction of three-fold coordinated atoms incorporated in filtered vacuum-arc-deposited samples exceeds 12% [13]. Comparing F(Q) for the post-relaxed model MCKMD with that for MDDF2 shows that the central peak must also be influenced by a structural difference, since the composition and short-range order of these two models are almost equal. The only difference that goes beyond short-range correlations in these models is the relationship between the number of five- and sevenmembered rings, which is reversed. Finally, attention should be paid to the first sharp diffraction peak at Q 2 1.5 A, the occurrence of which seems to be an artefact of the smaller models. It vanishes for the 512-atom models.
4. Electronic properties The electronic properties of the models are evaluated according to the method described elsewhere [ 151 using n-cluster statistics. The related data are summarized in Table 2. They include the ratio np+&, of the number of p and 71:states not involved in CTbonds to the number of G bonds, the ratio rip/n,, of p defect states to the bonding and antibonding 7cstates in the gap region and the number np_w_,k~ of weak (5 bonds. We also give the number of n-bonded atom clusters per atom and the (T gap width. The latter is rather approximate since, mainly because of the minimal basis approach, the gap width is not very well defined. Therefore we provide the width of the gap relative to the direct gap of diamond as calculated in the DFTB method. This is done by removing the defect states due to isolated p orbitals or weak
180
G. Jungnickel et uh/Diamond and Related Materials 5 (1996) 175-185
-2.5
2.5
Fig. 2. The ring statistics of the models the CDNs with the MDDF models. Table 2 Electronic Model structure
defect and n-cluster np+n In,
MCKl MCK2 MCKMD MDDFl MDDF2 MDT
0.001 0.000 0.016 0.032 0.032 0.139
Diamond
0.000
statistics
show a difference
in the relation
between
the number
of five- and seven-membered
rings when comparing
of the models n,ln,
0.333 1.000 0.333 1.119
np-weak D per atom
?i clusters
Gap width (%)
per atom
334
5-6
7kx
0.002 0.000 0.023 0.420 0.006 0.870
0.000 0.000 0.000 0.023 0.008 0.033
0.000 0.000 0.000 0.000 0.000 0.008
0.000 0.000 0.000 0.000 0.000 0.008
39 64 72 47 27 67
0.000
0.000
0.000
0.000
100
o orbitals from the gap region prior to the determination of the gap. The width of the latter is then given by the highest occupied molecular levels and the lowest unoccupied levels that are not identified to be defects. This method yields the width of a o-o* gap for a model that does not contain z-bonded atom clusters. In contrast, if such clusters are formed by sp2-like atoms, the width of the related z--71*gap is determined in this way. The total densities of states (DOS) for the models are
shown in Fig. 4. The MCK models contain no threefold coordinated atoms and therefore should not have rc states next to the Fermi level E,. Large o bandgaps are obtained for both MCKl and MCKZ. However, in MCKl there are a few strained bonds and a weak cr bond yielding isolated states in the gap region. MCK2 is a true amorphous diamond with a pure gap, since the model is free from any defect states (n,+,/n,=O.O). The MDDF models and the MDT model all contain sp2-
G. Jungnickel et al./Diamond and Related Materials 5 (1996) 175-185
181
lbng Statistics 0.8 MCK2
MCKl 0.6
0.0 MCKh4D
MDDF
MDDFZ
MDT
I
0.6
0.6
0.4
0.2
0.0 3
5
6
I
8
9
10
4
5
6
7
8
9
IO
Ring Size Fig. 3. The total The contributions
interference functions F(Q) (thick lines) of the models differ, particularly from the different bond types are marked by thin lines.
like carbon atoms, of which some are isolated species but others are n-bonded atom pairs. These double-bondlike atomic arrangements yield z and 7c* states in the o gap region of the DOS, thus reducing the effective gap width. We note that, owing to the extended relaxation time, the sp2-like atoms in MDDFZ have a greater chance of forming x-bonded atom pairs than during the relaxation of MDDFl. The Tersoff potential fails to handle the three-fold coordinated atoms correctly, and this is indicated by the relatively high value for the number of weak c~ bonds in agreement with the largest number of p defects comparing all models. This is the source of the highest DOS close to the Fermi level (see also Ref. [ 151).
5. Vibrational properties The vibrational density of states (VDOS) has been calculated within the harmonic approximation by diagonalization of the dynamical matrix to obtain the normalmode frequencies. For each network we set up the dynamic matrix dependent on the potential used to
regarding
the central
(double)
peak at about
10
A-‘.
generate the model. To analyse the chemical nature of vibrations we have decomposed the total VDOS into the contributions from the different carbon hybrids, applying the projection technique described in Ref. [ 231. We use the inverse participation ratio P-‘(o) [30] as the measure for the localization or extension of each eigenmode. The details of the theoretical methods are discussed elsewhere [ 231. All spectra have been convoluted by a constant Lorenzian resolution function of width 40 cm-’ in order to simulate the VDOS of an extended system. In Fig. 5 we present the vibrational density of states (VDOS) of the CDNs and the models relaxed with the DFTB method. The most prominent difference between them is the shape of the VDOS. For models MCKl and MCK2 based on Keating forces we obtain a diamondlike shape with a clear peak centred at about 1300 cm-‘. These VDOSs are similar to CDN models of amorphous silicon [26,30], although the low-frequency bands here are fused to one broad band between 250 and about llOOcm_‘. In contrast, structures MCKMD, MDDFl and MDDF2 develop a characteristic hemispherical shape over the whole frequency range under the influence
182
G. Jungnickel et aLlDiamond and Related Materials 5 (1996) 175-185
DOS [ 1/eV/atom]
MCKl
ID
;
Energy [eV] Fig. 4. The under-coordinated
atoms yield defect and II states in the gap region of the electronic DOS. Energies are given relative to the Fermi level.
of the tight-binding forces. This is particularly obvious for the vibrational density of states of the sp3 sites only. The cause of this difference is not yet clear. It is possible that the particular reversed relationship between the five- and the seven-membered rings could have some influence in dropping high-frequency modes and enhancing the low- and medium-frequency modes. Then the VDOS for MCKMD would be clearly different from the VDOS of model MDDFl. Another factor that could reduce the number of vibrations with higher frequencies relative to the others in the MDDF models is the reduced stress in the systems. Also, it should be noted that the force constants in the potentials are not exactly equal and that the range for interactions is larger for the tight-binding than for the Keating potential. However, the main band is found to lie between 250 and about 1700 cm- ’ for all models. Considering the high-frequency limit for diamond at about 1350 cm-‘, we find a growing localization of transversal vibrations above this value in all of our models. This is shown in Fig. 6, where the defined inverse participation ratios for the model eigenmodes are compared, and again is similar to the results obtained for
amorphous silicon models [30] where the localization of modes at the high-frequency edge was found to be enhanced by the presence of five-coordinated sites. Comparison of the intensities for MCKl and MCK2 in Fig. 6 shows exactly the same result. The occurrence of the regions with atoms that are effectively overcoordinated enhances the high-frequency localization for MCKl. For the molecular dynamically generated structures which contain under-coordinated sp2-like atoms rather than over-coordinated sites, we also find defect bands beyond 1700 cm-’ with strongly localized excitations. These are mostly due to stretching motions of isolated sp’ pairs within the rigid sp3 network. The width of these defect bands in the CDN structures is much smaller than obtained for the other models. Additionally, we find localized low-frequency modes in the models relaxed with the DFTB method and assign them to the isolated under-coordinated atoms. This is consistent with the findings in Ref. [30]. Furthermore, we identified a strongly localized highfrequency mode at 2780 cm-’ in MCKl (Fig. 6). A stretch vibration of a five-membered ring found in the region of high local stress mentioned above is responsible
G. Jungnickel et al./Diamond and Related Materials 5 (1996) 175-185
INTENSITY
[ arb.
units
183
] MCKl
MCKZ
MCKMD
MDDFl
MDDF2
0
2000
1000
3
w[cn-l] Fig. 5. VDOSs of the CDNs and the models relaxed with the DFTB method. The models relaxed with the Keating potential show a pronounced peak at about 1300 cm-l. Dotted lines show the contribution of sp3 atoms, dashed lines that of sp3-like species.
for this very high-frequency mode. The distance between the two diametral atoms taking part in this vibration is only about 1.77 A. However, they are not recognized as a bonded pair of atoms in the bond transposition modelling procedure. Counting this pair of atoms as a bond would cause a three-membered ring to appear. Hence the two atoms are under compressive stress in their local environment and vibrate at a particularly high frequency.
6. Conclusions We have generated two completely four-fold coordinated continuous distorted networks (CDNs) to simulate the properties of “amorphous diamond” by applying the Wooten-Winer-Weaire bond transposition technique. Although the models containing 512 and 128 carbon atoms were relaxed with respect to a Keating potential, they are unstable against the creation of under-
G. Jungnickel et al./Diamond and Related Materials 5 (1996) 175-185
184
INTENSITY
[ arb. units ]
MCKI
MCK2
MCKMD
P-‘(w)
MDDFl
I
MDDFB
3ci(IO
Go0 w [cm-l]
Fig. 6. The inverse participation ratios P-‘(w) reveal localized high-frequency modes in all models, but only a few localized low-frequency vibrations in the models containing isolated sp2-like atoms.
coordinated atoms. This was tested in an additional molecular dynamics simulation where the interatomic forces were described with a more realistic densityfunctional-based tight-binding potential. The smaller model continued to relax under these forces by breaking some bonds and creating isolated three-fold coordinated sites that gave rise to defect states in the gap region close to the Fermi level. Additionally, we generated models by means of molecular dynamics and the simulated annealing technique describing the interatomic forces with the tight-binding
as well as with the semi-empirical Tersoff potential. When Tersoff forces were used we were unable to obtain a realistic number of sp3-like atoms within the constant volume regime considered. In contrast, the models based on the tight-binding forces contain about 90% four-fold coordinated sites. The under-coordinated atoms in these models try to create pairs in order to make up double bonds that would yield a z-rr* gap. However, we also find defect states due to isolated p orbitals in these models. By extending the relaxation time and using a stochastic cooling technique the total energy has been
G. Jungnickel et al,/Diamond and Related Materials 5 (1996) 175-185
reduced owing to the improvements in the formation of n-bonded atom pairs. However, this causes the effective gap width to decrease. The analysis of the vibrational modes of these models shows that there is a difference in the shape of the vibrational density of states between the models where the atoms interact with Keating forces and the models where they interact via tight-binding forces. The typical diamond-like peak found in the former does not emerge in the latter. All the models are characterized by localized high-frequency modes. Localized low-frequency modes appear only in the networks containing isolated sp2like atoms. It appears that isolated three-fold coordinated atoms and n-bonded pairs of under-coordinated atoms remove excessive strain from completely tetrahedrally bonded amorphous carbon networks with diamond density. The resulting gain in total energy stabilizes these systems, although defect and n states in the gap region with energies above the valence o-band edge appear. If the n-bonded atom pairs can relax properly in their rigid sp3-bonded neighbourhood, they create WTC*gaps which are smaller than the expected o--(5* gaps for completely four-fold coordinated amorphous networks. The gap width for the latter models in the absence of overcoordinated species is roughly estimated to be about 64% of that of crystalline diamond.
Acknowledgement We gratefully acknowledge support by the Deutsche Forschungsgemeinschaft.
References [l]
[2]
D.R. McKenzie, D. Muller, E. Kravtchinskaia, D. Segal, P.B. G. Amaratunga, P.H. Gaskell Mater., 1 (1991) 51. D.R. McKenzie, D.A. Muller,
B.A. Pailthorpe, Z.H. Wang, Lukins, P.D. Swift, P.J. Martin, and A. Saeed, Diamond Relat. E. Kravtchinskaia,
D. Segal,
185
D.J.H. Cockayne, G. Amaratunga and R. Silva, Thin Solid Films, 206 (1992) 198. c31 Y. Lifshitz, G.D. Lempert, S. Rotter, I. Avigal, C. Uzan-Saguy and R. Kalish, Diamond Relat. Mater., 2 (1993) 285. c41 J. Ishikawa, Y. Takeiri, K. Ogawa and T. Tagaki, .I. Appl. Phys., it61 (1991) 773. c51 B. Rother, J. Siegel, K. Breuer, I. Mtihling, S. Deutschmann, J. Vetter, G. Trommer and B. Rau, J. Mater. Rex, 6 (1991) 101. S. Misawa, S. Yoshida and S. Gonda, J. Appl. C61 T. Miyazawa, Phys., 55 (1984) 188. G. Jungnickel, Th. Kbhler and U. Stephan, c71 Th. Frauenheim, J. Non-Cryst. Solids, 182 (1995) 186, and references cited therein. PI P.C. Kelires, Phys. Rev. B, 47 (1993) 1829. R. Wang and S. Klemm, Phys. Rev. B, 43 c91 D.A. Drabold, (1991) 5132. and CA. Davis, Diamond Relat. Mater., 4 Cl01 J. Robertson (1994) 441. Cl11 P.H. Gaskell, A. Saeed, PC. Chieux and D.R. McKenzie, Phys. Rev. Lett., 67 (1991) 773. r-121 S.D. Berger, D.R. McKenzie and P.J. Martin, Philos. Mag. Lett., 57 (1988) 285. M. Ktihn, S. Deutschmann, F. Richter, Cl31 G. Jungnickel, U. Stephan, P. Blaudeck and Th. Frauenheim, Diamond Relat. Mater., 3 (1994) 1056. P. Blaudeck and G. Jungnickel. Cl41 U. Stephan, Th. Frauenheim, Phys. Reo. B, 50 (1994) 1489. Lee, W.R.L. Lambrecht, B. Segall, P.C. Kelires, Cl51 C.H. Th. Frauenheim and U. Stephan, Phys. Rev. B, 49 (1994) 11448. Cl61 F. Wooten, K. Winer and D. Weaire, Phys. Reu. Lett., 54 (1985) 1392. Cl71 P.N. Keating, Phys. Rev., 145 (1966) 637. Cl81 J. Wejchert, D. Weaire and F. Wooten, J. Non-tryst. Solids., 122 (1990) 241. Cl91 F. Wooten and D. Weaire, J. Non-tryst. Solids, 97-98 (1987) 349. Th. Frauenheim, Th. KBhler, R. Kaschner and c201 D. Porezag, G. Seifert, Phys. Rev. B, 51 (1995) 12947. Th. Kbhler, M. Haase, S. Deutschmann, c211 G. Jungnickel, U. Stephan, P. Blaudeck and Th. Frauenheim, J. Non-cry& Solids, 183 (1995) 161. Th. Frauenheim, D. Porezag, P. Blaudeck, c221 G. Jungnickel, U. Stephan and R.J. Newport, Phys. Reu. B, 50 (1994) 6709. and G. Jungnickel, Phys. Rev. B, ~231 Th. KBhler, Th. Frauenheim 52 (1995) 118. c241 J. Tersoff, Phys. Rea. B, 37 (1988) 6991. and M. Haase, J. Phys. Condens. Matter, 5 ~251 U. Stephan (1993) 9157. C261 K. Winer, Phys. Rev. B, 35 (1987) 2366. ~271 P.H. Gaskell, A. Saeed, P. Chieux and D.R. McKenzie, Philos. Mag. B, 66 (1992) 155. C281 R.J. Temkin, J. Non-tryst. Solids, 28 (1978) 23. Phys. Rev. B, 44 (1991) 4925. ~291 D.S. Franzblau, G.S. Grest and c301 R. Biswas, A.M. Bouchard, W.A. Kamitakahara, C.M. Soukoulis, Phys. Rev. Lett.. 60 (1988) 2280.