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Solid-Vapor Equilibrium Phase Diagram of Hydrogen-Adsorbed Carbon K. H. Lee,* S. J. Noh, B. S. Lee, and S. J. Yoon Department of Applied Physics, Dankook University, Seoul, Korea Received December 31, 1999. In Final Form: June 27, 2000 We studied the phase equilibrium of carbon which holds the hydrogen-adsorbed surface at low pressure. All of the molecules in a chamber of a finite volume are assumed to be fully decomposed so that the system is considered to be a mixture of atomic hydrogen and carbon vapor with a crystalline carbon. The equilibrium conditions are characterized by the same chemical potentials for both phases of each ingredient. The phase boundary of a two-dimensional solid with a smaller bond strength is separable from that of a threedimensional solid with a larger bond strength. The covering ratio of hydrogen atoms on the surface was also calculated and turns out to be larger for a two-dimensional solid.
Introduction Diamond has been expected to be useful for electronic devices because of its large energy gap, high breakdown voltage, and chemical stability.1,2 Recent progress in the synthesis of diamond and diamond-like films by various kinds of chemical vapor deposition (CVD) methods has led to a great deal of interest in many possible applications.3-11 Such techniques mainly use activated plasma of carbon species from a mixture of hydrocarbons, such as methane (CH4) and hydrogen (H2) gases. The film growth involves a highly nonequilibrium process accompanied by a series of the complex chemical reactions. Although much effort has been concentrated on enhancing the growth rate and finding the optimum condition,12,13 the solid-vapor equilibrium phase diagram of carbon at low pressure has hardly been studied since the phase diagram of carbon was described by Leipunsky.14 In this paper, we report the thermal equilibrium phase diagram of carbon in a condition that may be related to the diamond film growth from the vapor phase. Our calculation includes hydrogen atoms adsorbed on the solid carbon surface, which are allowed to have simple vibrational degrees of freedom. The atomic hydrogen is believed to provide the preferential etching of graphite over diamond.4-6,15,16 In this sense, it is also of interest to investigate the hydrogen-adsorption covering ratio which * Corresponding author (E-mail:
[email protected]). (1) Pan, L. S.; Kania, D. R., Eds. Diamond: Electronic Properties and Applications; Kluwer Academic Publishers: Norwell, MA, 1995. (2) Spear, K. E.; Dismukes, J. P., Eds. Synthetic Diamond: Emerging CVD Science and Technology; John Wiley & Sons Inc.: New York, 1994. (3) Cappelli, M.; Paul, P. H. J. Appl. Phys. 1990, 67, 2596. (4) Angus, J. C.; Mayman, C. C. Science 1988, 241, 913. (5) Angus, J. C.; Jansen, F. J. Vac. Sci. Technol. 1988, A6(3), 1778. (6) Yarbrough, W. A.; Messier, R. Science 1990, 247, 688. (7) Okano, K.; Naruki, N.; Akiba, Y.; Kurosu, T.; Iida, M.; Hirose, Y. Jpn. J. Appl. Phys. 1988, 27, L173. (8) Shimada, Y.; Mutsukura, N.; Machi, Y. Jpn. J. Appl. Phys. 1992, 31, 1958. (9) Suzuki, K.; Sawabe, A.; Inuzuka, T. Jpn. J. Appl. Phys. 1990, 29, 153. (10) Jubber M. G.; Milne, D. K. Phys. Status Solidi 1996, A154, 185. (11) Harris, S. J.; Shin, H. S.; Goodwin, D. G. Appl. Phys. Lett. 1995, 66, 891. (12) Battaile, C. C.; Srolovitz, D. J.; Butler, J. E. J. Appl. Phys. 1997, 82, 6293. (13) Battaile, C. C.; Srolovitz, D. J.; Butler, J. E. J. Mater. Res. 1999, 14, 3439. (14) Leipunsky, O. Usp. Khim. 1939, 8, 1519. (15) Frenklach, M.; Wang, H. Phys. Rev. 1991, B43, 1520. (16) Spear, K. E. J. Am. Ceram. Soc. 1989, 72, 171.
is defined as a fractional number of hydrogen-terminated sites to the total adsorption centers on the surface. A simple kind of the gas-solid adsorption is known as Langmuir’s theory of adsorption.17,18 We do not seek any growth mechanism for diamond in a CVD process, which actually involves kinetics of many carbon allotropes and radicals. We assume that all of the molecules in a chamber of a finite volume are fully activated and decomposed so that the system may be considered as a mixture of atomic hydrogen and carbon vapor with a solid phase of carbon. The thermal equilibrium can be sustained locally although the spatial region is very limited in a short temporal period. This assumption is fairly appropriate when the growth rate of the solid phase is extremely low. Because of this highly simplified situation, the resulting sublimation curve turns out to be a lower limit, but it qualitatively gives a possible explanation for the growth of a three-dimensional (3D) solid over a two-dimensional (2D) solid at low pressure. Phase Boundaries and Adsorption The atomic hydrogen can be adsorbed on the solid surface. The solid phase of carbon is represented by a 3D bulk or a quasi-2D planar sample. Because a graphite which is stable under normal condition has an sp2 hexagonal structure and its π-bond along the c-axis is relatively weak compared to the basal plane, it can be considered more likely as a planar structure. The graphite actually has a small shear modulus along the basal plane. In sharp contrast, a diamond forms a typical covalent sp3 hybrid bond and all carbon atoms are strongly coupled to construct tetrahedral units. In this respect, we regard the diamond as a 3D harmonic oscillator and the graphite as a 2D harmonic oscillator. Equilibrium conditions for the hydrogen-adsorbed carbon in the presence of hydrogen and carbon vapors are the same chemical potentials of both phases:
µgC ) µsC
(1)
µgH ) µaH
(2)
where µgC, µsC, µgH, and µaH are the chemical potentials of (17) Langmuir, I.; Villars, D. S. J. Am. Chem. Soc. 1931, 51, 486. (18) Pilling, M. J.; Seakins, P. W. Reaction Kinetics; Oxford University Press: Oxford, 1995.
10.1021/la9916878 CCC: $19.00 © 2000 American Chemical Society Published on Web 12/01/2000
Equilibrium Phase Diagram of Carbon
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vapor carbon, solid carbon, vapor hydrogen, and adsorbed hydrogen, respectively. We assume that all the molecules including H2 and many hydrocarbons such as CH4 and CH3 are fully decomposed into individual atoms. In this case, the chamber of a fixed volume V holds a mixture of atomic hydrogen and carbon gases, which is thermally equilibrated with a partially hydrogen-terminated solid carbon. We used such an assumption that all the molecules are fully decomposed only for calculating convenience. We are aware that there are many hydrocarbon species still remaining as gaseous molecules inside the container. However, it is difficult to obtain the chemical potentials for H2 and CH3, which must satisfy equilibrium conditions for atomic and molecular vapor and crystalline harmonic oscillator. To consider such molecules, it is necessary to have several additional parameters such as equilibrium constants and molecular constants for rotation and vibration. To avoid this complicated situation, we include only atomic hydrogen, which is known to be one of key elements that make the diamond film growth possible. Vapor Phase. First, we examine the chemical potentials of the vapor phase, which can be obtained by assuming that the container is filled with classical ideal gases. Because the volume occupied by the solid phase is negligible compared to the total volume of the chamber, the volume occupied by the vapor phase, Vg, is reasonably approximated by the total volume, Vg ≈ V. The chemical potentials for the vapor phase are
µgH ) -kBT ln g
(
) ( ) ) ( )
2πmCkBT
µgC ) -kBT
(
h
2
2πmHkBT 2
h
3/2
3/2
V f Ng
V (1 - f)Ng
(3)
has narrow IR peaks at 3107 and 1405 cm-1, corresponding to the C-H stretching and bending vibration, respectively.19,20 The chemical potentials for the solid phase can be readily calculated from the partition function.
µsC ) -kBT ln q1
(
µaH ) kBT ln
N0 - NaH
)
- ln z1
(6)
where NaH is the number of adsorbed hydrogen atoms out of the total N0 adsorption centers, which is roughly estimated by N0 ≈ (NsC)(5-d)/3, where d is the dimensionality for the solid phase. In eqs 5 and 6, q1 and z1 are the partition function of a single d-dimensional harmonic oscillator and that of a single adsorption center, respectively. Both are, using β ) 1/kBT,
q1(T) ) (2 sinh βpωD/2)-d eβc z1(T) ) 2eβa (2 sinh βhν|/2)-2(2 sinh βhν⊥/2)-1
(7) (8)
where c is the cohesive energy for the harmonic oscillator, a is the adsorption energy relative to the energy of a free hydrogen atom, and ν| and ν⊥ are the bending and stretching vibrational frequencies, respectively. The adsorption energy, like the cohesive energy, is defined to be positive. The prefactor 2 in eq 8 comes from the spin degeneracy of the hydrogen nucleus. Vapor-Solid Phase. When the thermal equilibrium is established on the surface through the bulk, the solidvapor phase boundary can be obtained from eqs 3 and 5:
(4)
where N is the total number of vapor atoms, and mC and mH are the masses of constituent elements. In eqs 3 and 4, we have used the fractional number of the vapor carbon atoms, f ) NgC/Ng ) NgC/(NgC + NgH), where NgC and NgH are the number of carbon and hydrogen atoms in vapor phase, respectively. Such expressions of the chemical potentials are relatively simple because there is no contribution from the internal degrees of freedom which molecular gases must hold. For this reason, we allow only atomic gases in this work. Solid Phase. We turn to the solid phase which is composed of NsC carbon atoms with NaH adsorbed hydrogen atoms on the surface. The solid phase is considered to be a classical d-dimensional harmonic oscillator, which is assumed to have a delta function-like phonon spectrum at a certain single frequency. Such an Einstein model is reasonable because the actual phonon density of states, especially for diamond, is extremely high at the Debye frequency, ωD. Another solid phase we have to take into account is the hydrogen adsorption on the carbon surface. We also suppose a single bond between the adsorbed hydrogen and the surface carbon atom, which has simple vibrational degrees of freedom both vertically and horizontally. This monohydride bond hypothesis is only for calculational compactness, although there should be several possible bonding ways along different surface orientations. The monohydride allows two degrees of vibrational freedom: a stretching mode characterized by a frequency ν⊥ and two bending modes characterized by ν|. It has been reported that a natural type-Ia diamond
NaH
(5)
P)
(
)
kBT 2πmCkBT f h2
3/2
1 q1
(9)
With more convenient units, eq 9 can be written as follows:
(
)
θD d -11600c/T T5/2 2 sinh e P ) 0.108 f 2T
(10)
where the unit of pressure is mega-Pascals, θD is the Debye temperature in kelvin, and c is in electronvolts per atom. In a similar way, we can calculate the hydrogen covering ratio, defined as
θ≡
NaH ) N0
1 1+
a
-βµH z-1 1 e
(11)
This expression is reminiscent of the particle number distribution of fermions. It is actually obtained from an equilibrium condition that the adsorption rate is exactly matched with the desorption rate. From eqs 4 and 6, we obtain
θ)
A0P(1 - f) 1 + A0P(1 - f)
(12)
where P is the total vapor pressure shown in eq 9 and A0 (19) Hamza, A. V.; Kubiak, G. D.; Stulen, R. H. Surface Sci. Lett. 1988, 206, L833. (20) Runciman, W. A.; Carter, T. Solid State Commun. 1971, 9, 315.
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Lee et al.
Figure 1. The equilibrium phase boundaries of carbon at low pressure. Parameters used are c ) 7 eV (solid and dotted curves) for d ) 3 and c ) 5 eV (dashed and dash-dotted curves) for d ) 2. The same Debye temperature, θD ) 1860 K, is used for both samples. The normal condition of a CVD diamond film growth is located in the upper-left corner around a pressure range of 10-4-10-2 MPa and temperature range of 1000-1500 K.
is the adsorption constant given by
A0 )
(
)
h2 2πmHkBT
z1 kBT
3/2
(13)
For a practical usage, eq 12 can be written as follows:
(
1 - θ ) 1 + 41.1
)
1 - f z1 f q1
-1
(14)
where
z1 ) 2e-11600(c-a)/T (2 sinh θD/2T)d × q1 (2 sinh 0.719k⊥/T)-1 (2 sinh 0.719k|/T)-2 (15) In eq 15, the energies are in units of electronvolts per atom and the wavenumbers, k⊥ and k|, are in centimeters-1. Results Inside a CVD chamber, the fraction of vapor carbon atoms is believed to be approximately the same order as the volume fraction of methane gas in hydrogen molecules. Hence, the values of the fraction f should be in the range of 10-4-10-2. Because the f values are so small, the sublimation curve in the pressure-temperature phase diagram is significantly increased to higher values of pressure by order of 100-10 000. As shown in Figure 1, the phase boundaries are shifted to lower temperatures at a fixed pressure value as f decreases. The cohesive energy used for a 3D sample, d ) 3, is c ) 7 eV (solid and dotted curves), which may be appropriate for diamond at 0 K and 1 atm. We also use c ) 5 eV (dashed and dashdotted curves) for a 2D harmonic oscillator, d ) 2, which may represent graphite at a normal CVD situation.21,22 The same Debye temperature, θD ) 1860 K, is used for both samples. Smaller f values mean relatively larger values of the partial pressure by the atomic hydrogen gas. The total vapor pressure is mostly governed by the partial pressure
Figure 2. The temperature-dependent hydrogen covering ratio on the surface of 3D (solid and dotted curves) and 2D (dashed and dash-dotted curves) samples. See text for the parameters used. The covering ratio increases as f decreases, as expected.
of the atomic hydrogen. The partial pressure of the vapor carbon is almost f times smaller than that of the hydrogen gas in this system. Note that in the phase diagram there is a fairly large region which corresponds to a stable 3D solid phase between two phase boundaries at a fixed f value. This indicates that such a region may be the right place where diamond is in its solid phase while graphite is still in vapor phase. We turn to the temperature-dependent hydrogen covering ratio on the surface of either a 2D or 3D harmonic oscillator, shown in Figure 2. Parameters used are c ) 7 eV for d ) 3 (solid and dotted curves) and c ) 5 eV for d ) 2 (dashed and dash-dotted curves). Both samples are assumed to have the same Debye temperature of θD ) 1860 K and the adsorption energy of a ) 4 eV. The bending and stretching vibrational wavenumbers for a C-H bond on the surface are assigned as k| ) 1405 cm-1 and k⊥ ) 3107 cm-1, respectively.19,20 The values of energies used in this work have been chosen according to refs 21 and 22. Although these energies should depend on temperature, we use fixed values for simplicity and we believe that the general trend of our results should not be changed much. Our result of increasing θ as temperature grows is consistent with other reports.12,13,21 Such an increase is not surprising because in our approach the chamber of a fixed volume is assumed to be in a thermal equilibrium state with two phases. The increasing temperature results in higher vapor pressure inside the chamber so that the atomic hydrogens are more likely to sit on the adsorption centers. Note that in a real CVD chamber, however, the hydrogen covering ratio may decrease with the increasing temperature because the vapor pressure is kept at a constant optimal value. For a temperature-independent chemical potential, which is definitely not our case, it also decreases as the temperature increases, as shown in eq 11. The covering ratio increases as f decreases, as expected. An important fact, as in the equilibrium phase boundaries, is that the covering ratio of a 2D sample with a small cohesive energy is much higher than that of a 3D sample. This observation probably means that for the hydrogen (21) Pierson, H. O. Handbook of Carbons, Diamond and Fullerene; Noyes Publications: New Jersey, 1993. (22) Kittel, C. Introduction to Solid State Physics, 7th ed.; John Wiley & Sons: New York, 1996.
Equilibrium Phase Diagram of Carbon
Figure 3. The covering ratio as a function of f at fixed temperatures of T ) 1600 and 1900 K. The solid and the dotted curves are for a 3D harmonic oscillator while the dashed and the dot-dashed ones are for a two-dimensional one. The parameters used are the same as in Figure 2.
adsorption, the diamond surface is less active than the graphite surface. Such an important result may lead to the well-known preferential etching scenario in the diamond film growth. The atomic hydrogen helps to suppress the formation of nondiamond phase to ensure the formation of diamond phase. In our calculation, the higher hydrogen covering ratio on the 2D sample may provide one of the correct explanations to the possible growth of diamond over graphite. The small values of the fraction f also act to enhance the hydrogen covering ratio. The covering ratio θ at a fixed temperature increases with the decreasing value of f, as expected. Figure 3 shows the covering ratio as a function of f at typical temperatures of 1600 K (solid and dashed curves) and 1900 K (dotted and dot-dashed curves). We used c ) 7 eV for d ) 3 (solid and dotted curves), and c ) 5 eV for d ) 2 (dashed and dot-dashed curves). The rest of parameters used are the same as in Figure 2. As f decreases, θ increases, indicating that higher hydrogen vapor pressure is required to cover the surface with more hydrogen atoms. It is easy to see that the 2D solid compared to the 3D solid has a much higher hydrogen covering ratio for all values of f. Higher temperature is also found to improve the extent of the hydrogen adsorption. Using different values of the hydrogen adsorption energy, a, does change the covering ratio. However, the general trend such as the higher covering ratio for d ) 2 and for small f, is unchanged. Conclusions We calculated both the thermal equilibrium phase boundaries of a crystalline carbon and the hydrogen covering ratio on its surface. Most experiments on diamond film growth from the vapor phase are in a regime which maintains the substrate in a temperature range between 800 and 1000 °C in a decomposed hydrocarbon environment of pressure 1-100 Torr. Because such a regime is located on the upper-left corner of the phase diagram shown in Figure 1, the calculated phase boundaries turn out to be a lower limit of the experimental studies. This discrepancy is thought to be due to the highly simplified model which does not consider the complex kinetics. The contribution from the partial pressure of gaseous molecules such as H2 and methyl radicals (CH3), which we have neglected in the model, may also explain the difference.
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Despite such a low limit, however, the general trend for the phase boundaries of the monotonic increasing vapor pressure as a function of temperature is very similar to the phase diagram in ref 21 which considers carbon species only. The partial pressure by the hydrogen vapor can significantly shift the phase boundaries to the left-hand side of the phase diagram. Accordingly, the experimental regime for the CVD diamond film is attained by reducing the value of the vapor carbon fraction f sufficiently, but it may reduce the growth rate. Such a statement is plausible because the small value of the fraction f leads to less chance that any form of the carbon film will stack. Thus, our model describes qualitatively (but does not quantitatively) the diamond-related film growth from the vapor phase with an extremely low deposition rate. One of the main points that we can obtain is that the phase boundary of the 3D solid with higher cohesive energy is located on the right side of that of the 2D solid with smaller cohesive energy. In other words, the phase boundary of a 2D solid with a small bond strength is separable from that of a 3D solid with a large bond strength. This is encouraging in a sense that such a region between two phase boundaries is where the 3D phase is in its solid phase while the 2D phase is still in vapor phase. This of course is purely based on equilibrium conditions. Such a phase separation is mainly due to the different values of cohesive energies. The change in the total vapor pressure at a given temperature is governed by changes in the fraction f and the cohesive energy. By varying the fraction from f1 to f2 and the cohesive energy from c1 to c2, the ratio for both pressures is P2/P1 ≈ (f1/f2) exp[-5.8(c2 c1)] at a temperature of T ) 2000 K. For example, if the fraction f decreases by a factor of 100 at a certain fixed cohesive energy, the vapor pressure increases by the same factor. However, the pressure ratio at a given fraction f reaches up to 105 times by reducing only 2 eV in cohesive energy. It leads to the fairly large region between two phase boundaries. The hydrogen covering ratio also turns out to be larger for a 2D solid. It is generally believed that the formation of surface hydrides in a diamond is far less than that in a graphite. However, this belief should not be confused with the ratio of sp3 and sp2 hybrid bonds in a diamondlike carbon (DLC) which has been reported to be larger than unity for most values of atomic hydrogen content.5 Although the experiment on DLC is based on the real situation that two hybrid bonds coexist in a reaction chamber, our model deals with the equilibrium vapor phases of 3D and 2D solid phases independently. We treat the separate phase diagrams only with different values of dimensionality and cohesive energy of two solid phases. The hydrogen covering ratio on the hydrogenated surface may be experimentally observed, but it should be carefully compared because our model is performed in a fixed volume so that pressure and temperature are linked together in an equilibrium state. One may think that most of the surface hydrogen might be desorbed when the sample is subject to very high temperature. This is true only when the sample is in a vanishingly small pressure. The increasing H sites in a CVD diamond with increasing temperature have also been reported by a kinetic Monte Carlo study.12,13 Note that because we used the same hydrogen adsorption energy and the same vibrational wavenumbers for both 3D and 2D solid phase, there is plenty of room for different shapes of the curves. Such a choice for the above parameters is only for calculating simplicity, so it is more suitable to understand that an adsorption in this study is just a simple C-H bond on the
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surface of a cubic (3D) bulk and a square (2D) plane rather than that on a diamond (sp3) and a graphite (sp2), respectively. Our results are undoubtedly based on a highly simplified equilibrium thermodynamic calculation. If the thermal equilibrium can be sustained locally for a short temporal period, however, our result may provide one aspect of the possible explanation for the growth of diamond over graphite. For further studies, one may try more realistic model calculation including several dominant hydrocarbon
Lee et al.
species. In this case the possible dynamical degrees of freedom for methyl radicals and hydrogen molecules should also be considered with complex surface reconstructions. Acknowledgment. This work was supported by Korean Ministry of Education through the research grant BSRI 98-2452. LA9916878
