Computation of Large Systems with Economic Basis Set: Simulation of

J. Phys. Chem. B 2002, 106, 625-631

625

Computation of Large Systems with Economic Basis Set: Simulation of Diamond Metallization Using Titanium R. Q. Zhang,* W. C. Lu, H. F. Cheung,† and S. T. Lee Center Of Super-Diamond and AdVanced Films (COSDAF) & Department of Physics and Materials Science, City UniVersity of Hong Kong, Hong Kong S. A. R., China ReceiVed: June 25, 2001; In Final Form: October 12, 2001

Economic basis sets for ab initio calculations of systems involving titanium (Ti) atom have been studied. The basis set in which the polarization and diffuse functions are applied only to Ti and its nearest neighboring atoms is shown to give reliable binding energies and geometries for various carbon-based compounds. Such a basis set is used as the economic basis set for calculations of medium-sized clusters simulating Ti and diamond interactions. For large clusters composed of Ti and C atoms, the more economic basis set 6-31G is considered for obtaining reliable energetics with the correction using high-level single-point energy calculations with 6-31+G*. The approach to determine the economic basis set is expected to be useful for a general system containing transition metal element. With the economic basis sets, simulations of diamond metallization have revealed reasonable behavior of the Ti atomic interaction with diamond film. On the basis of the determined energetics of Ti deposition, the high Ti concentration in CVD diamond after metallization is conjectured to occupy the grain boundaries rather than the bulk of diamond grain. For a Ti atom with high kinetic energy, the Ti atom may penetrate through a diamond (001) surface more easily than through a diamond (111) surface. In the diamond bulk, the Ti atom may energetically favor to substitute a C site more than diffusion.

1. Introduction Diamond has high potential in advanced electronic devices such as diodes, radiation sensors, thermistors, and transistors which could operate at elevated temperatures. Owing to the success in the past decade in achieving heteroepitaxial diamond films,1-3 such desirable applications appear to be realistic. Because metallization is an indispensable step in device processing, the study on formation of low-resistant Ohmic contact for diamond with metals such as Ti and Al of electrode is important. There has been a few reports on metallization of single crystal diamond film, whereas polycrystalline CVD diamond have been frequently used as substrates in experiments.4-9 It has been observed that the formation of good Ohmic or rectifying contacts is not always easily accomplished on diamond films grown by CVD.4 The performance of Ohmic contacts on diamond are found to depend on the choice of metals, doping concentration, surface pretreatment, and so on. A deeper understanding of the various related problems such as metal diffusion into diamond and metal substitution for carbon atom in diamond should be important to realize such a key technology. Nowadays, satisfactory agreements with experiments have been achieved using ab initio calculations for small- or mediumsize molecular systems in terms of their energetic, spectroscopic and other properties. The metal-diamond interaction could also be expected for simulation using a cluster-model approach. However, this may lead to the computation of large systems in order to include relatively realistic environment of diamond. Although semiempirical molecular orbital theory could be used * To whom correspondence should be addressed. E-mail: aprqz@ cityu.edu.hk. † E-mail: [email protected].

for the study of large systems as we did for some issues in CVD diamond nucleations and interfacial structures,10-13 it cannot be used for the present problem due to the involvement of transition metal. To perform ab initio calculation for a reliable simulation, huge CPU time is required if using a medium level computation resource, indicating that a practical approach has to be found. Because the use of computation resource in ab initio calculation depends largely on the basis set used, it would be desirable that an economic scheme be developed for selecting basis sets with which the reliability of the result is maintained, whereas the computational resource used is minimized. Recently, the principle for choosing such a basis set for a nonmetal system has been studied by the authors,14,15 and Truhlar and coworkers.16,17 However, there is no previous work on the required economic basis set for a system involving transition metal elements. In this work, various schemes for selecting basis set have been examined in calculations for Ti-containing systems using density functional theory (DFT). The ionic TiCHx+ (x ) 1-3) and TiC2H2+,18-20 the neutral TiCHx (x ) 1-3) and TiC2H2, as well as the TiC(CH3)x (x ) 1-3) clusters simulating the Ti-C bonding on diamond surfaces, were selected for studying the basis set effects in TiC systems. The determined economic basis sets were then used to study the interactions between transition metal Ti and diamond clusters in order to evaluate the energetics and bond formation in metalization of diamond films with titanium. 2. Theoretical Approach and Computation Accurate ab initio calculations for transition metal complexes usually confront with the large computational cost. To solve this problem, several methods have been developed by treating

10.1021/jp0123942 CCC: $22.00 © 2002 American Chemical Society Published on Web 12/14/2001

626 J. Phys. Chem. B, Vol. 106, No. 3, 2002 the active site of the system and the remainder differently. For example, the hybrid quantum mechanics/molecule mechanics (QM/MM) method, where the active site is calculated with QM and the rest with MM, has been found to be useful in the field of computational organometallic chemistry.21 The ONIUM calculation22 is another technique dividing the molecular system into two or three layers which are treated with different model chemistries. As an alternative approach to reduce the computational cost, the economic basis set has been introduced by considering the different role of different basis functions adopted in basis set used in ab initio calculations for a heteroatomic system.14-17 It has been noticed that even for the same kind of basis functions a different role may be found when they are used in different systems. Therefore, the amount and type of the basis functions for describing an atom may be optimized according to the real occupation and the behavior of valence electrons in the molecule. In general, the more electrons it possesses, the higher level (the angular moment type) of the basis functions it may require. Furthermore, if the atom is negatively charged, a higher-level basis function should be used. In contrast, if the atom is positively charged, the use of the basis function could be reduced. On the other hand, the polarization and diffuse functions are the correction functions, which can describe the deformation and diffusion of the electronic cloud. When atoms form bond, electron transfer occurs, leading to complicated distribution of the electron cloud and the requirement of the polarization function and/or the diffuse function. When an atom loses electron, its electron cloud would be contracted. Such a kind of deformation of electron cloud can be described possibly without the usage of polarization function and/or diffusion functions. To examine the basis set selection scheme for transition metal Ti, we considered the following basis sets, 6-311+G(2d,p), 6-31+G*, 6-31G*, 6-31G, LanL2DZ,23 as well as the designed composite basis sets based on 6-31G. These composite basis sets were constructed by adding the polarization and diffuse functions to (1) the Ti atom; (2) atoms directly linked to the Ti atom; and (3) the Ti atom and the atoms directly bound with it. We use {Ti}6-31G* to indicate that only the Ti atom is described using 6-31G* while using 6-31G for other atoms in the considered system. {Ti,C}6-31+G* or {Ti,C}#6-31+G* represents that only Ti and C atoms directly bound to Ti are described using the 6-31+G* basis set while using the 6-31G or 3-21G basis set for all other atoms in the system. In a similar way, several other composite basis sets have also been designated in this study. Geometric optimizations were performed with the various basis sets using B3LYP approach of DFT. Binding energies were obtained with these basis sets and their corrections were done by doing single-point energy calculations using the standard 6-31+G* basis set. All calculations were carried out with the Gaussian 98 package.24 3. Results and Discussion 3.1 Economic Basis Sets for Ti-Containing Systems. Due to their fundamental interest in surface chemistry, organometallic chemistry, and catalysis, the interactions of ligands with metal ions have been intensively studied and accurate bond strengths have been provided for many transition-metal-ligand ions.18-20,25-27 These works thus form the basis to evaluate the economic basis sets studied in this work. From the experimental study of small Ti-containing systems,18-20 the binding energies obtained for TiCH+, TiCH2+, TiCH3+, and TiC2H2+ are 114 ( 1, 91 ( 2, 51 ( 1, and 61 ( 5 kcal/mol, respectively. Theoretical calculations on these ions

Zhang et al. using various methods and basis sets have also been reported. The study of the transition-metal carbyne cations with CH using a MCSCF/SCF method with basis sets of 14s11p6d for Ti, 9s5p1d for C, and 4s1p for H has yielded the binding energy of 113.0 kcal/mol for TiCH+.25 The bonding of transition-metal ions to acetylene has been obtained with a MCPF/HF theory using basis sets of 14s11p6d3f for Ti, valence double-ζ and natural orbital sets for C and H and the binding energy for Ti+C2H2 was revealed to be 39.9 kcal/mol.26 In addition, the binding energy for TiCH+, TiCH2+, and TiCH3+, reported by Li and co-workers at a PMP3/HF level of theory with the LANL2DZ basis set, were 144.7, 99.7, and 75.7 kcal/mol, respectively.27 It can be found that the calculated Ti+-C binding energies are sensitive to the methods and basis sets adopted. Unfortunately, little theoretical information is reported on the neutral Ti-C bonding systems. Table 1 lists the calculated binding energies using 6-311+G(2d,p) and 6-31+G* basis sets. The deviations of binding energies determined using the designed basis sets from the values calculated with 6-31+G* are also tabulated in Table 1. The binding energies were determined by the total energy differences respectively between 1Tit CH+ and 4Ti+ + 4CH, 2TidCH + and 4Ti+ + 3CH , 3Ti-CH + and 4Ti+ + 2CH 2 2 3 3, 2TiC H + and 2Ti+ + C H , 2TitCH and 3Ti + 4CH, 2 2 2 2 3TidCH and 3Ti + 3CH 4Ti-CH and 3Ti + 2CH , 3TiC H 2 2, 3 3 2 2 and 3Ti + C2H2, 2TiCCH3 and 3Ti + 4CCH3, 3TiC(CH3)2 and 3Ti + 3C(CH ) , as well as 4TiC(CH ) and 3Ti + 2C(CH3) . 3 2 3 3 3 For the ions, TiCH+, TiCH2+, TiCH3+, and TiC2H2+, the derived binding energies at the B3LYP/6-31+G* level are 131.68, 89.66, 64.85, and 63.76 kcal/mol, respectively, which agree well with the corresponding experimental values. Table 2 shows the corresponding geometrical parameters. It can be seen that the B3LYP/6-31+G* binding energies and geometries are very close to those of B3LYP/6-311+G(2d,p), suggesting that the former basis set is good enough for predicting reliable energetics and geometries. The binding energies at B3LYP/6-31+G* level of the neutrals of TiCH, TiCH2, TiCH3, and TiC2H2 are 125.66, 84.98, 57.18, and 48.48 kcal/mol, respectively. Thus, the neutrals are energetically less favorable than the corresponding ions; a Ti-C bond is about 5∼8 kcal/mol weaker than its corresponding Ti+-C bond, following the trend indicated by their bond lengths (see Table 2). The binding energies of TiC(CH3)x (x ) 1-3) at the same level of calculation are 124.6, 72.39, and 41.80 kcal/mol, respectively, which are smaller than the corresponding values for TiCHx, possibly due to the smaller electronegativity of C(CH3)x than CHx.28 Furthermore, it can be seen from Table 1 that the calculated binding energies using the {Ti}6-31+G* basis set are larger, whereas that using {C}6-31+G* smaller than those from 6-31+G*. The magnitudes of these positive and negative deviations are different from one system to another. However, when the polarization and diffuse functions of the Ti atom and its nearest neighboring C atoms are all introduced to the basis set, i.e., {Ti,C}6-31+G* or {Ti,C}#6-31+G*, the calculated binding energies for the Ti-containing systems are considerably improved. Comparing with the other composite basis sets considered here, {Ti,C}6-31+G* and {Ti,C}#6-31+G* are generally superior in determining an accurate Ti-C bond strength. It is noted that for the considered Ti-containing positive ions, the 6-31G binding energies are in good agreement with those of 6-31+G*, and the effect of polarization and diffusion functions is small. In such cases, the accuracy of such reduced basis set as 6-31G may be due to the positive charges of these

Simulation of Diamond Metallization Using Titanium

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TABLE 1: Binding Energies (in kcal/mol) Calculated at B3LYP Level of Theorya basis set

TiCH+

TiCH2+

TiCH3+

TiC2H2+

6-311+G(2d,p) 6-31+G* 6-31G* 6-31G LANL2DZ {Ti}6-31G* {Ti}6-31+G* {C}6-31G* {C}6-31+G* {Ti,C}6-31G* {Ti,C}6-31+G* exp[18-20]

130.10 131.68

88.93 89.66

64.42 64.85

60.62 63.76

-0.06 -0.17

0.60 -1.23 -0.72 -0.77 1.95 0.26 -0.03 0.61 -0.04

-0.14 -0.13 -0.05 -0.02 -0.06 0.00

0.05 -0.35 -0.33 -0.32 0.03 -0.06 0.03 -0.02

-0.34 -0.49 5.34 -0.21 3.06 -0.55 -1.48 -0.09 0.16

-0.05 -0.12 -0.12 -0.14 -0.05 0.01 -0.03 0.00

-0.16 -0.04 -0.03 -0.01 -0.12 0.01 -0.20 -0.01

114 ( 1

91 ( 2

basis set

TiCH

TiCH2

TiCH3

TiC2H2

6-311+G(2d,p) 6-31+G* 6-31G* 6-31G LANL2DZ {Ti}6-31G* {Ti}6-31+G* {C}6-31G* {C}6-31+G* {Ti,C}#6-31G* {Ti,C}#6-31+G*

124.73 125.66

84.84 84.98

57.03 57.18

45.40 48.48

-0.12 -0.12 -0.13 -0.07 -0.08 -0.04 -0.14 0.00

-1.76 -3.29 -2.17 -2.79 1.70 -2.13 0.88 -1.85 -0.07

0.11 -0.31 -0.30 -0.28 0.08 -0.05 0.08 -0.02

-3.73 -3.70 3.31 -3.48 2.30 -3.91 -2.01 -3.76 0.07

-0.15 -0.14 -0.12 -0.15 -0.17 -0.05 -0.15 -0.01

basis set

TiCCH3

TiC(CH3)2

6-311+G(2d,p) 6-31+G* 6-31G* 6-31G LANL2DZ {Ti}6-31G* {Ti}6-31+G* {C}6-31G* {C}6-31+G* {Ti,C}6-31G* {Ti,C}6-31+G* {Ti,C}#6-31G* {Ti,C}#6-31+G*

123.66 124.61

71.91 72.39

-0.15 -0.10 -0.12 -0.09 -0.11 -0.07 -0.14 0.02 -0.14 0.02

-2.11 -2.85 -0.36 -2.45 2.13 -1.94 0.22 -1.63 0.69 -1.41 1.08

-0.22 0.03 -0.01 -0.02 0.01 0.04 -0.05 0.11 -0.14 0.04

51 ( 1

0.17 0.42 7.19 0.63 2.68 -0.03 -1.68 0.60 0.23

-4.41 -4.22 0.69 -4.15 1.66 -4.47 -4.14 -4.43 0.10

61 ( 5

-0.20 -0.06 -0.04 0.01 -0.18 -0.01 -0.26 -0.01

2.86 -0.58 0.54 0.39 1.59 1.99 -0.35 3.14 0.19

-0.52 -3.32 3.23 -2.64 0.81 -1.14 -0.44 -0.48 0.10

TiC(CH3)3

-3.30 -3.83 4.19 -3.51 1.79 -4.06 -3.53 -3.78 0.12 -3.36 0.75

41.17 41.80 0.08 0.16 0.17 0.10 0.18 0.34 0.19 0.17 0.04 0.11

-4.32 -4.47 5.21 -4.40 1.00 -4.90 -4.06 -4.84 0.21 -3.89 1.36

a For each compound, the left column represents the differences of binding energies calculated with B3LYP/6-31+G*//B3LYP/Basis set from those of B3LYP/6-31+G*, whereas the right column shows the difference between the values of B3LYP/Basis set and those of B3LYP/6-31+G*. The basis set {Ti}6-31G** indicates that only Ti atom is described using 6-31G*, while using 6-31G for other atoms. {Ti,C}6-31+G* or {Ti,C}#631+G* indicates that only Ti and C atoms directly bound to Ti are described using the 6-31+G* basis set while using the 6-31G or 3-21G basis set for all other atoms. Similar designations are applied to the other composite basis sets.

ions which result in contraction of the electronic clouds. In contrast, the effect of polarization and diffusion functions is considerably increased for Ti-containing neutrals, as can be seen from Table 1. For the study of large Ti-C systems with 6-31G, the single-point energy calculation with a high level of basis set such as 6-31+G* is necessary for giving an accurate Ti-C binding energy. For all the considered Ti-C species, the errors of the corrected binding energies at B3LYP/6-31+G*//B3LYP/ 6-31G from those at B3LYP/6-31+G* are within 1 kcal/mol. Accordingly, when the TiC systems get too large to be calculated with {Ti,C}6-31+G* or {Ti,C}#6-31+G* basis set, 6-31G may be considered as the more economic basis set aided by corrections made using the single-point energy calculations with a higher-level basis set. The feasibility that the single-point energies are used for binding energy corrections are supported by the fact that the determined geometric parameters with these economic basis sets are very close to those from 6-31+G*. From Table 2, the Ti-C bond lengths of 6-31G are obviously very close to the values of 6-31+G*, and the further improvements of geometries due to the introduction of polarization and diffuse functions on Ti or C are not obvious. The {Ti,C}6-31+G* and {Ti,C}#6-31+G* basis sets can generally give a more reliable

Ti-C bond lengths. Therefore, with these similar geometric structures the same basis set 6-31+G* would result in similar energetics. In contrast, the deviations of binding energies using LANL2DZ are frequently the largest, as seen from the results for TiCH2+, TiCH3+, TiC(CH3)2, and TiC(CH3)2. And the binding energies obtained with these different geometric structures cannot be corrected by single-point energy calculations with 6-31+G*, due to the deteriorated structures from optimization using LANL2DZ as seen in Table 2. The above analysis of basis set effects indicates that the addition of the polarization and diffuse functions to only the key atoms in the systems involving Ti-C interaction can yield much accurate results of geometric parameters and thus the binding energies. The composite basis set {Ti,C}6-31+G* or {Ti,C}#6-31+G* may be a reliable economic basis set suitable for calculations of interactions of medium-sized ligands with the Ti atom. However, for much larger systems, such as the one simulating Ti substitution in diamond as illustrated in the latter part of this paper, the significantly increased requirement for computation resource forces the use of more efficient and economic basis sets such as 6-31G, in conjunction with the correction using 6-31+G* single-point energies. The reduction

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TABLE 2: Geometrical Parameters Optimized in B3LYP Calculations Using 6-311+G(2d,p) and 6-31+G* and the Deviations of Geometrical Parameters Determined with Various Smaller Basis Sets from Those Using 6-31+G* Basis Set a

6-311+G(2d,p) 6-31+G* 6-31G* 6-31G LANL2DZ {Ti}6-31G* {Ti}6-31+G* {C}6-31G* {C}6-31+G* {Ti,C}#6-31G* {Ti,C}#6-31+G*

TiCH+

TiCH2+

TiCH3+

TiC2H2+

TiCH

TiCH2

TiCH3

TiC2H2

Ti-C

Ti-C

Ti-C

Ti-C

Ti-C

Ti-C

Ti-C

Ti-C

1.673 1.672 -0.012 0.002 0.021 -0.005 0.008 -0.008 -0.003 -0.013 0.000

1.845 1.843 -0.008 -0.001 0.016 -0.004 0.002 -0.006 -0.004 -0.010 -0.001

2.067 2.065 0.007 0.007 -0.011 0.005 -0.003 0.009 0.010 0.005 -0.001

1.942 1.933 -0.009 0.008 0.013 0.002 0.010 -0.005 -0.001 -0.009 0.000

1.686 1.686 -0.016 -0.003 0.012 -0.010 0.005 -0.012 -0.004 -0.017 0.000

1.871 1.871 -0.017 -0.012 0.005 -0.016 -0.003 -0.014 -0.005 -0.019 -0.001

2.126 2.121 0.007 0.004 0.004 0.002 -0.009 0.010 0.016 0.005 -0.001

1.972 1.965 -0.007 0.008 0.005 0.003 0.006 -0.003 0.000 -0.007 0.000

TiCCH3 6-311+G(2d,p) 6-31+G* 6-31G* 6-31G LANL2DZ {Ti}6-31G* {Ti}6-31+G* {C}6-31G* {C}6-31+G* {Ti,C}6-31G* {Ti,C}6-31+G* {Ti,C}#6-31G* {Ti,C}#6-31+G* a

TiC(CH3)2

TiC(CH3)3

Ti-C1

Ti-C1

C2C1Ti

C3C1Ti

Ti-C1

1.701 1.698 -0.018 -0.005 0.009 -0.011 0.008 -0.013 -0.006 -0.017 0.000 -0.017 0.000

1.875 1.872 -0.024 -0.009 0.033 -0.015 0.012 -0.015 -0.007 -0.020 0.006 -0.021 0.004

143.2 142.0 3.4 0.9 -17.3 1.6 -4.5 2.3 2.2 3.1 -1.8 2.7 -2.0

99.9 101.6 -3.7 -1.0 18.7 -1.8 4.9 -2.1 -1.6 -2.9 2.3 -2.5 2.6

2.151 2.146 0.000 -0.002 -0.015 -0.005 -0.010 0.002 0.004 -0.001 -0.004 -0.006 -0.008

Bond lengths in angstroms and bond angles in degrees.

TABLE 3: CPU Time and the Number of Basis Functions for TiC(CH3)3 in B3LYP Calculations with Various Basis Sets basis set

6-31+G*

{Ti,C}#6-31+G*

6-31G

no. of basis functions CPU time (min) for single-point calculation CPU time (min) for frequency calculation

136 64

109 35

81 28

191

92

55

of the total number of basis functions and the improvement of the computational efficiency thus make it possible to calculate medium and large systems with good accuracy. The decrease of the basis-set size would reduce the limitation of the computer disk space and may greatly save CPU time. Table 3 lists the CPU time with different basis sets using SGI Origin200 (4xR10000) server machine for single-point energy and frequency calculations for TiC(CH3)3, along with the number of basis functions. It shows that the designed composite basis sets largely reduce the CPU time compared with those using the standard basis set 6-31+G*. In comparison with the singlepoint calculation using 6-31+G*, the calculations with {Ti,C}#631+G* and 6-31G save about 45% and 66% of CPU time. The reduction in CPU time would be greatly enlarged when a larger TiC system such as that simulating Ti substitution in diamond is considered. 3.2 Ti Interactions with Diamond Surfaces and Ti Substitution in Bulk Diamond. To reveal the mechanism of interdiffusion and the Ti-C bond formation in the metallization of diamond, potential energy surface (PES) for Ti atom depositing onto the diamond surface and substitution in diamond phase were performed with DFT calculation using the economic basis sets studied above. In the present calculations, the PESs are calculated with all C and boundary H atoms being fixed while the orientation of the surface H atoms are allowed to relax

Figure 1. Models for Ti-C bond formation (a) and Ti diffusion (b) on diamond (001) surface (numbers represent Z coordinates).

due to the consideration that the surface H atoms can move much more freely than the diamond C atoms which would be constrained in a crystal environment in experiment. In addition to the studies of TiC(CH3)2 and TiC(CH3)3 as illustrated in the previous section, the characteristics of Ti-C bonding on diamond (001) or diamond (111) surface may further be examined by PES calculations. The substrate models for Ti-C bond formation and Ti diffusion on diamond (001) surface are shown in Figure 1. The energetic behaviors of the deposited Ti atom calculated with the economic {Ti,C}#6-31+G* basis set are shown in Figure 2. The deposition sites are assumed to be (1) the ontop site, and (2) the bridge site. In the former case, the Ti atom can form TidC bond with surface carbon atom with a bond distance of about 1.9 Å. The TidC bond strength is shown to be about 3.5 eV (i.e., 80.57 kcal/mol), which is larger than the 72.39 kcal/mol for TiC(CH3)2 (relative to the Ti atom and the most stable C(CH3)2 radical, see Table 1). For the case of Ti depositing onto the bridge site, the C9H16 cluster

Simulation of Diamond Metallization Using Titanium

Figure 2. PES curves for Ti diffusions on the saturated (] and the unsaturated (O) diamond (001) surface calculated at the B3LYP/ {Ti,C}#6-31+G* level of theory.

Figure 3. Models for Ti-C bond formation (a) and Ti diffusion (b) on diamond (111) surface (numbers represent Z coordinates).

model was adopted. Because of the consideration as noted above, the surface hydrogen atoms are allowed to relax in orientation while the other atoms are fixed in the PES calculation. In this case, the Ti atom (in its ground state of triplet) encounter for a small energy barrier to enter the bulk, as shown in Figure 2. In addition, the PES calculation is also performed using an unsaturated diamond (001) surface, with the two facing hydrogen atoms on the surface (Figure 1(b)) being taken away. In such a case, there is a complex stabilized by about 4.2 eV when Ti is at about Z ) -2.4 Å (see Figure 2), which corresponds to a bridge site of deposition. There is also a small energy barrier when the Ti further goes inside the bulk. The PES results show that through diamond (001) surface the Ti atom can easily be inserted into the space between the first and the second surface carbon layers. For Ti-C bonding and Ti diffusion onto diamond (111) surface, the C4H9 and C10H16 cluster models (Figure 3) were used as the substrate. PES result of Ti-C bonding calculated with the {Ti,C}#6-31+G* basis set indicates that there is a bond formation at the Ti-C distance of 2.1 Å, consistent with the Ti-C bond length (2.1464 Å) of the TiC(CH3)3 cluster (see Table 2). The binding energy here is about 2.3 eV (53.04 kcal/ mol), which is larger than the 41.80 kcal/mol for TiC(CH3)3 (see Table 1). For the Ti atom depositing onto the diamond (111) surface along the symmetrical line of the C10H16 model

J. Phys. Chem. B, Vol. 106, No. 3, 2002 629

Figure 4. PES curves for Ti diffusions on the saturated (]) and unsaturated (O) diamond (111) surfaces calculated at the B3LYP/ {Ti,C}#6-31+G* level of theory.

(Figure 3(b)), the corresponding potential energy curve is illustrated in Figure 4, showing a quite large energy barrier of about 41.9 eV for a Ti atom crossing the surface layer. The relaxation of surface hydrogen atoms shows little effect on the barrier of diffusion into diamond (111) surface because it is mainly determined by the Ti-C interactions. When the Ti atom approaches nearer the bulk, there appears a minimum stabilized by 12.4 eV. This diffusion process is shown to be endothermic by as much as 29.5 eV. Figure 4 also shows the energetics of a similar diffusion process of the Ti atom but on an unsaturated surface with the three surface hydrogen atoms (Figure 3(b)) being taken away. Comparing to the hydrogen-saturated surface, the energy barrier of 29.8 eV is greatly reduced but still too high for the Ti diffusing into the bulk diamond. The stabilization energy for Ti atom after entering the bulk diamond is 5.5 eV. Although the relaxation of the diamond structure in experimental Ti deposition can lower the energy barrier to some extent for the Ti diffusion, it can still be estimated that the Ti diffusion through a diamond (111) surface is much more difficult to occur than through a diamond (001) surface. The behavior of Ti in the diamond bulk is further studied for the case that the deposited Ti atom possesses enough kinetic energy to cross the diamond surface (probably through diamond (001) surface). The C14H20 cluster model shown in Figure 5 is used as a diamond substrate for calculation. The results show that the interstitial minimum for a Ti atom is located at about Z ) ( 0.75 Å. The Ti diffusion from one cell of interstitial vacancy to another requires about 12.6 eV (Figure 6(a)), which is much smaller than the energy barriers for Ti passing through the saturated and unsaturated diamond (111) surfaces. However, such a barrier is still too high for thermal activation. Compared with the Ti diffusion in the diamond bulk, a C atom, as also studied in this work (see Figure 6(b)), shows much small energy barrier (about only 3.2 eV) and a quite different feature of potential energy profile for diffusing in the diamond bulk. There is an interstitial minimum for a C atom located at the interface of the two cells (Z ) 0 Å). The C atom should be much easier to move among the interfacial minimum and the two cells than a Ti atom moving from one cell to another. The energetics for a Ti atom taking up a substitution site has also been investigated in this work. Two cluster models of C29H36 and TiC28H36, as shown in Figure 7, were chosen for the study. Because these two clusters are too large to perform

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Figure 5. Model for Ti diffusion in the bulk of diamond.

Figure 7. C29H36 (a) and TiC28H36 (b) clusters for studying the substitution of a Ti atom for a C atom in diamond. The calculated Ti-C bond length of TiC28H36 is 1.964 Å at B3LYP/6-31G.

and C atoms, due to the influence of the surrounding carbon atoms. The substitution of Ti + C29H36 f C + TiC28H36 is found to be exothermic by about 7.2 eV by considering the energies of Ti and C atom at their interstitial minima (see Figure 6). In view of the various energetics of Ti depositing onto the diamond surfaces, it is clear that the Ti diffusion through diamond (001) surface is much easier than through diamond (111) surface. For the further Ti diffusion in the bulk of the diamond, the energy barrier (12.6 kcal/mol) is still too high for thermal activation as in a pure thermal annealing process. This indicates that a Ti deposition through thermal evaporation may not cause Ti diffusion into the single crystal of diamond, but may favor bond formation with the surface C atoms, especially for those on diamond (001) surface. For a CVD diamond film, which is normally polycrystalline, the Ti atom may go into the boundary. However, Ti atoms with higher kinetic energy would likely pass through diamond (001) surface and go into the bulk of the diamond. Comparing the energetics of Ti diffusion in diamond bulk and Ti taking up a substitution site, the Ti atom of high kinetic energy may favor substitution from the thermodynamic viewpoint, suggesting the facilitation of Ti-C formation. 4. Conclusions Figure 6. PES curves for Ti (a) and C (b) diffusions in the bulk of diamond calculated at the B3LYP/{Ti,C}#6-31+G* level of theory.

high-level calculations, their structures were optimized at the B3LYP/6-31G level, followed by the single-point energy calculations at the B3LYP/6-31+G* level. According to the basis set effect discussed above, such a treatment can also give reliable binding energies for carbon systems interacting with Ti. For Ti and C atoms located in the diamond bulk, their energy difference is decreased compared to that between the free Ti

The basis set {Ti,C}6-31+G* or {Ti,C}#6-31+G* which applies polarization and diffuse functions only to Ti and its nearest neighboring C atoms is shown to be advantageous and economic for predicting reliable geometric parameters and binding energies for medium-sized Ti-containing clusters. For large Ti-containing systems or mixed TiC systems, the 6-31G basis set could be used as the more efficient and more economic basis sets aided by the single-point energy calculations using 6-31+G* basis set. This principle for basis set selection can be extended to general ab initio calculations for large systems

Simulation of Diamond Metallization Using Titanium involving transition metal-organic interaction. The application of the economic basis sets in simulation of metallization of diamond film has revealed reasonable behavior of Ti atomic interactions with diamond and indicates that the high concentration of Ti in CVD diamond after metallization would occupy the grain boundaries rather than the bulk of diamond grain. Depositions with high kinetic energy using sputtering or ion implantation may result in Ti penetrating the diamond (001) surface to the bulk and favor substitution for C atom by Ti. Acknowledgment. The work described in this paper was fully supported by a grant from City University of Hong Kong (Project No. 7000764). References and Notes (1) Jiang, X.; Klages, C. P.; Zachai, R.; Hartweg, M.; Fu¨sser, H.-J. Appl. Phys. Lett. 1993, 62, 3438. (2) Wolter, S. D.; Stoner, B. R.; Glass, J. T.; Ellis, P. L.; Buhaenko, D. S.; Jenkins, C. E.; Southworth, P. Appl. Phys. Lett. 1993, 62, 1215. (3) Lee, S. T.; Lin, Z. D.; Jiang, X. Mater. Sci. Eng. R 1999, 25, 123. (4) Das, K.; Venkatesan, V.; Miyata, K.; Dreifus, D. L.; Glass, J. T. Thin Solid Films 1992, 212, 19. (5) Zhu, Y.; Zheng, B.; Yao, W.; Cao, L. Diamond Relat. Mater. 1999, 8, 1073. (6) Viljoen, P. E.; Lambers, E. S.; Holloway, P. H. J. Vacu. Sci., & Technol. B 1994, 12, 2997. (7) Wong, N. B.; Li, G. Q.; Zhu, S. M.; Tjong, S. C.; Lee, S. T. Phys. Stat. Sol. A 1998, 169, 5. (8) Monterio, O. R.; Salvadori, M. C.; Cattani, M.; Mammana, V.; Brown, I. G. Thin Solid Films 1997, 308, 215. (9) Meng, W. J.; Curtis, T. J.; Rehn, L. E.; Baldo, P. M. J. Appl. Phys. 1998, 83, 6076. (10) Zhang, R. Q.; Wang, W. L.; Esteve, J.; Bertran, E. Appl. Phys. Lett. 1996, 69, 1086. (11) Zhang, R. Q.; Bertran, E.; Wang, W. L.; Esteve, J.; Lee, S. T. Diamond Relat. Mater. 2000, 9, 146.

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