Characteristics of $ta-C$ and $a-C$

The macroscopic properties enable us to distinguish between the $ta-C$ structure and the $a-C$ structure. The former has higher density, is transparent and is much harder than the latter. From the microscopic point of view, the ratio of fourfold, diamondlike bonds to threefold, graphite-like bonds ($sp^3/sp^2$) will determine the kind of structure that is obtained. This ratio is strongly affected by the way the amorphous solid is prepared and depends on temperature and pressure. By evaporation in an electron beam or carbon arc, one can produce an $a-C$ solid with a percentage of $sp^2$ bonds ranging from $\sim 70 \%$ to $\sim 98 \%$ [20,22,23]. A $ta-C$ solid can be prepared by ion beam deposition with 90 % of $sp^3$ bonds [21,23], with ion energies between several ten and several hundred eV.

During the last few years, many investigations of conditions for formation of the $ta-C$ and $a-C$ structures were carried out by means of computer simulation. The Monte Carlo method with an empirical potential was used by J. Tersoff [24] and P.C.Kelires [23] to generate amorphous carbon in three different ways. By simulations of homogeneous condensation of vapor [24] and by ultrafast quenching of liquid carbon [24,23], the resulting samples were rather graphitic $a-C$, with a bond angle distribution peaked around 120$^{\circ}$, as expected. Indeed, experimentally, preparation of amorphous solids includes a cooling process whose nature determines the final structure. It has to be fast enough to avoid a creation of a more ordered structure (crystallization), and must be done until a temperature that corresponds to the amorphization transition is reached [18]. These two factors can be managed by the Monte Carlo method.

The third way was a simulation of molten carbon quenched under a pressure of 1 Mbar [24,23], producing a $ta-C$ lattice, identical to that produced by D. R. McKenzie in the laboratory [21], though the Tersoff sample was poorer in fourfold $sp^3$ bonds and the bond angle distribution showed two overlapping peaks centered at 110$^{\circ}$ and 120$^{\circ}$. This procedure is not directly related to the kinetics of the actual growth process, but still based on the concept of applied pressure. This high applied pressure attempts to reproduce the high compressive stress generated by energetic atoms during deposition [21].

There is presently considerable interest in the formation of amorphous diamond-like carbon films (DLC). This material has a high fraction of tetrahedrally coordinated atoms. The process of diamondlike film growth itself, by deposition of energetic carbon atoms, has been investigated by molecular dynamics simulation, via the Tersoff potential [25]. The sample created was very similar to that of McKenzie et al. But P.C. Kelires showed that the diamondlike lattice he generated was not stable under further annealing at higher temperature. Many of the $sp^3$ bonds disappeared (the weaker ones, with large angle distortion), the sample created remaining quite dense (2.8 gr/cm$^3$) but with a majority of threefold $sp^2$ bonds and a mean bond length of 1.48 Å.

Jäger and Albe [27] also used the Tersoff and Brenner potentials to simulate the thin-film deposition of amorphous carbon. Unfortunately, their attempts have not been overly successful, the fraction of sp$^3$ bonding (52% for the C$^{+}$ ion energies $E=30-80 eV$) was in poor agreement with experiment, as would be anticipated from the liquid-quench calculations.

With the same empirical potential, U.Stephan and M.Haase [26] generated amorphous structures via molecular dynamics calculations, at three different densities, which were fixed at the beginning. The mean bond angle obtained was close to that of graphite. Statistical data for the models above are listed in the Table 2.2. The values of the diamond and graphite energy, and their first nearest neighbor distance (the bond length) are derived from the Tersoff potential [24]. $z$ is the coordination number and $r_{\circ}$ the nearest neighbor distance.

Table 2.2: Statistical data for amorphous samples obtained from computer simulations.

	potential	% $sp^2$	$z$	$r_{\circ}$ (Å)	density (gr/cm$^3$)	ref.
Graphite		100	3.00	1.46	2.27	[24]
Tersoff (condensed)	Tersoff	91	3.08	1.47	2.18	[24]
Tersoff (quenched)	Tersoff	91	3.09	1.47	2.39	[24]
Kelires (quenched)	Tersoff	88	3.1	1.47	2.00	[23]
Galli	ab initio	85	3.20	1.44	2.00	[28]
Wang (1)	TB	80.6	2.96		2.20	[15]
Wang (2)	TB	80	3.08		2.44	[15]
Stephan (1)	Brenner	80	2.93	1.50	2.00	[26]
Stephan (2)	Brenner	79	3.05	1.49	2.50	[26]
Kelires (annealed )	Tersoff	75	3.2	1.48	2.8	[23]
Wang (3)	TB	71	3.16		2.69	[15]
Stephan (3)	Brenner	66	3.33	1.51	3.00	[26]
Tersoff (1Mbar)	Tersoff	50	3.40	1.51	3.02	[24]
Wang	TB	26		1.52	3.5	[16]
Kelires (1Mbar)	Tersoff	10	3.70	1.53	3.2	[23]
McKenzie	Tersoff	8	3.7	1.53	2.9	[21]
Diamond		0	4.00	1.54	3.51	[24]

Another computational technique used to ''create'' amorphous lattices was performed by G. Galli, R. Martin, R. Car and M.Parinello [28]. They used an ab initio molecular dynamics where the motion of the atomic core is treated classically, while the electron wave functions are represented in terms of large basis set of plane waves, keeping the energy of the whole system closed to a minimum with respect to the wave functions. This technique is more accurate than the classical ones presented above, but numerically intensive and unable to describe the dynamics of thousands of atoms in a reasonable simulation time. The 54 atoms $a-C$ lattice formed in ref. [28] by quenching from a liquid consisted of 85% $sp^2$ sites at 300 K, with an average bond angle of 117$^{\circ}$, starting from a density of 2 gr/cm$^3$.

N.A. Marks, D.R. McKenzie, B.A. Pailthorpe with M. Bernasconi and M. Parinello [29] repeated this simulations with a 64 carbon atoms at a density 2.9 gr/cm$^3$. The simulated structure of amorphous carbon contains 65% fourfold and 35% threefold- coordinated carbon sites. Three- and four membered rings are present in the structure and give the network an unusual topology. An important parameter identified in this work is the rate at which the liquid cools to amorphous solid. The structure produced using the instant cooling profile had the highest sp$^3$ fraction, with 69% of the atoms fourfold coordinates. Another cooling rate, named ''slow'', is more physically realistic, the temperature fell linearly from 5000K to 300K over 0.5 ps. The slow cooling profile produced a structure containing just 57% sp$^3$ atoms.

Alvarez at al. [30] used the slightly modified ab initio molecular dynamics method based on the Harris [31] functional to generate an amorphous carbon network. The authors slowly heated the crystalline diamond structure with 64 carbon atoms and two different densities: 1.8 and 2.6 g/cm$^3$, from 300 to 4000K and immediately cooled them to 0 K by cooling rate 9.25*10$^{15}$ K/s. Then the structures was subjected to annealing cycles at 700 K with an intermediate quenching process. The amorphous carbon samples contain 75% and 51% fourfold coordinated atoms respectively. This way to build the amorphous carbon network is not designed to reproduce the way an amorphous material is grown, but the generated amorphous carbon sample adequately represents those obtained experimentally.

Finally, a tight binding molecular dynamics method was applied to investigate $ta-C$ and $a-C$ structures [15,16,32]. Here the electron wave functions are expanded in terms of a basis set of valence electrons wave functions, rather than plane waves, controlling the attractive part of the potential, while the repulsive one is treated empirically. $ta-C$ and $a-C$ models were obtained by quenching 216 atoms from the liquid phase, at four different densities in order to understand the effects of density on the macroscopic structures [15,16]. Statistical analysis of these samples is shown in Table 2.2. To investigate the size effects, the 64 atoms samples of amorphous carbon at a density 2.2 gr/cm$^3$ were generated by the same way (C.Z. Wang, K.M. Ho [1]). The authors showed that the size effects have a very slight influence on the amorphous carbon structures obtained by tight-binding molecular dynamics.

From the measurements and theoretical calculations made in the studies presented above, the following conclusions about structural characteristics of $ta-C$ and $a-C$ solids emerge. $ta-C$ is a hard and dense material, mostly made of distorted $sp^3$ sites. A considerable amount of strain exists, due to localized melting and rapid quenching during its formation, that leaves the lattice in a stressed state. Part of the internal strain energy is relieved by the presence of $sp^2$ sites. These threefold atoms tend to form small clusters, owing to the delocalization of the $\pi$ states [16], and perhaps $\pi$-bonded pairs [23]. They control the band gap which is assumed to be of order of 2 eV. Moreover, an annealing process at high temperature may relieve stresses and bring the sample to a still dense but $sp^2$ phase. This is explained by the fact that threefold sites are the energetically favorable geometry. Temperature will supply the necessary energy to remove weakly bonded $sp^3$ atoms from their sites. The bond is then transformed to a threefold bond, relieving the local strain and lowering the energy.

The $a-C$ lattice has a less dense structure and is mainly threefold bonded. In a sample made of $\sim 80 \%$ $sp^2$, threefold coordinated atoms tend to form large cluster of diameter $\sim 15$ Å embedded in a matrix of fourfold atoms. These $sp^2$ sites are distorted and nonplanar, arranged in thick layers with an average thickness of 1.0-1.2 Å and spaced more closely than the sheets of graphite. They are composed of mainly fivefold and sevenfold rings, with a small amount of sixfold rings [23,28,20,25], leading to an unsignificant medium range order. These observations are not supported by the work of J. Robertson and E.P. O'Reilly [33] who found a much more ordered configuration of sheets with mostly sixfold rings.

Moreover, increasing temperature sensitively promotes the creation of graphitelike bonds [28], making the $sp^2$ fraction range from 60% at 30${^\circ} C$ to 90% at 1050 ${^\circ} C$ [23]. The process involved in that phenomenon is the same as that seen above where a $sp^3$ to $sp^2$ sites transition occurs during thermal annealing.