J. Phys. Chem. C 2010, 114, 19941–19945
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Prediction of Formation of Cubic Boron Nitride Nanowires inside Silicon Nanotubes Shengliang Hu,*,†,‡ Xiaochao Lu,‡ Jinlong Yang,‡ Wei Liu,‡ Yingge Dong,‡ and Shirui Cao‡ Key Laboratory of Instrumentation Science & Dynamic Measurement, Ministry of Education; Science and Technology on Electronic Test and Measurement Laboratory; and School of Materials Science and Engineering, North UniVersity of China, Taiyuan 030051, P. R. China ReceiVed: August 25, 2010; ReVised Manuscript ReceiVed: October 12, 2010
A model that predicts the formation of cBN nanowires inside Si nanotubes was developed by taking the effect of surface tension induced by the nanosize curvature of critical nuclei and the Si nanotubes into account. The radius and formation energy of the critical nuclei of cBN, the phase transition probability, and the growth velocity of the cBN nuclei were calculated with the proposed model, and the effect of the radii of Si nanotubes on the nuclei was also investigated. The results show that the temperature and supersaturation limited to a range of 1300-1500 K and 2-5, respectively, and radii of Si nanotubes of less than 5 nm are favorable for the formation of cBN nanowires inside Si nanotubes. 1. Introduction One-dimensional nanostructures such as nanowires, nanorods, nanobelts, and nanotubes have become the focus of intensive research owing to their unique applications in microscopic physics and the fabrication of nanoscale devices.1-3 Although carbon nanotubes have high mechanical strength, good chemical stability, and unique thermal properties, large-scale controlled fabrication of carbon nanotubes with semiconductor properties is difficult.3 However, Si nanotubes not only have some properties similar to carbon nanotubes but also have better photoluminescence and good compatibility with the commonly used complementary metal oxide semiconductor (CMOS) chip, etc.4-6 Therefore, Si nanotubes have greater potential for applications in photoelectric devices.5,6 Similar to that for carbon nanotubes, Si nanotubes can be filled with selected materials to develop a composite with new physical and chemical properties. Cubic boron nitride (cBN) has high thermal conductivity, the second hardest, a wide band gap, and chemical inertness and can be easily doped to p- and n-type semiconductors.7 If cBN fill the Si nanotubes, we predict that the resulting composite would have novel properties, such as for photics and electrics, that could be applied in the fabrication of nanoscale photoelectric devices, for example. Presently, there are many composite structures successfully prepared by filling nanotubes with selected materials. For example, BN nanotubes were filled with Ni, NiSi2, and R-Al2O3 nanowires,8,9 carbon nanotubes were filled with Fe, Ni, and Co nanowires, etc.10,11 However, to our best knowledge, the successful placement of cBN nanowires in Si nanotubes has not yet been reported in the literature. In this paper, we propose a simple thermodynamic and dynamical model of cBN nanowires grown inside Si nanotubes. Our calculations show the nucleation and growth conditions of cBN inside Si nanotubes, which provides an important basis for experimental preparation. * Corresponding author. +86-351-3557519; e-mail:
[email protected]. † Key Laboratory of Instrumentation Science & Dynamic Measurement, and Science and Technology on Electronic Test and Measurement Laboratory. ‡ School of Materials Science and Engineering.
Figure 1. Schematic illustration of nucleation and growth inside a Si nanotube with a radius of R.
2. Theoretical Model 2.1. Nucleation Stability. When the reaction gases (such as BF3, N2)12 flow along the Si nanotubes, the BN clusters are condensed on the inner wall of the Si nanotubes by a series of surface reactions and diffusions. Sequentially, cBN nucleation and growth will occur inside the Si nanotubes by phase transition (as shown in Figure 1). On the basis of the thermodynamic nucleation theory, we will discuss the nucleation behavior of the cBN clusters inside the Si nanotubes. The theory is formulated based on the following assumptions:13 nanonuclei are perfect spherical caps and are mutually noninteracting. The formation energy of a nucleus with radius r on the substrate (i.e., the inner wall of the Si nanotube) with radius R is:14
∆G ) (γsc - γsv)Ssc + γcvScv + ∆gV
(1)
where γsc, γsv, and γcv are the interface energy of the substratenucleus, the substrate-vapor, and the nucleus-vapor, respectively, and Ssc, Scv are corresponding interface areas. Generally, γcv and γsv can be estimated by15 γ ) d[(SvibHm)/(2kVmR)]0.5, where R is the ideal gas constant, Hm is the bulk melting enthalpy of crystals, Svib is the vibrational part of the overall melting entropy, k is the compressibility, and d is the atomic diameter. ∆g is the Gibbs free energy difference per unit volume. Taking into account the effect of nanosize-induced additional
10.1021/jp108045w 2010 American Chemical Society Published on Web 11/04/2010
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Hu et al.
pressure on the Gibbs free energy of critical nuclei inside the Si nanotubes, ∆g can be expressed by:16
∆g ) -
[
]
2γsc 2γcv 1 RT ln(P/Pe) + ) + 2 Vm R r 2γsc 2γcv 1 RT ln(1 + σ) + + 2 Vm R r
[
]
(2)
where R, β can be obtained from the following Figure 1. When ∂∆G/∂r ) 0, the radius and the free energy of the critical nucleus can be obtained as follows:
2γcv ∆g
(4)
3 16π γcv f(m, x) 3 (∆g)2
(5)
where14,17
f(m, x) )
1 1 (x + m) 1 1 + mx 3 - x3 2 - 3 + 2 2 g 2 g 3 x+m x+m 3 + mx2 -1 g 2 g
(
[
)
(
)]
(
)
( )[
J ) hν exp -
1 V ) πr3(2 - 3 cos β + cos3 β) + 3 1 3 πR (2 - 3 cos R + cos3 R) (3) 3
∆G* )
]
[ ]
Ea - ∆g Ea - exp RT RT
(8)
2.3. Nucleus Growth. Based on the Wilson-Frenkel growth law, generally, the growth velocity J of the crystalline nucleus can be expressed as:19
where P and T are the pressure and temperature, respectively, Pe is the equilibrium-vapor pressure of cBN, σ is the supersaturation,17 R is the gas constant, and Vm is the mole volume of cBN, which can be generally measured by experiment or estimated based on the crystal structure, the mole mass, and density of cBN. V is the volume of the cBN nucleus and can be expressed as:
r* ) -
[
f ) exp -
(6)
and m ) cos θ ) (γsv - γsc)/γcvx ) R/r*g ) (1 + x2 + 2mx)0.5
(7) 2.2. Phase Transformation. Generally, the Gibbs free energy is an adaptable measure of the energy of a state in phase transformation among competing phases. At the given thermodynamic condition, both cBN and hexagonal BN (hBN) phases can coexist, but only one of the two phases is stable and the other must be metastable and may transform into the stable state. Here, we studied the probability of the phase transition from hBN to cBN inside Si nanotubes. The probability of the phase transformation from the metastable phase to the stable phase is related not only to the Gibbs free energy difference ∆g but also to an activation energy (Ea - ∆g) that is necessary for the phase transformation. The general expression of the probability f of the phase transformation from the initial states to final states is:18
Ea |∆g| 1 - exp RT RT
(
)]
(9)
where h and ν are the lattice constants of cBN nuclei in the growth direction and the thermal vibration frequency, respectively. 3. Results and Discussion According to eqs 2 and 4, Vm ) 7.123 × 10-6 m3/mol, γcv ) 4.72 J/m2, γsv ) 1.24 J/m2, and γsc ) (γcv + γsv)/2 ) 2.98 J/m2.20 We calculated the critical nucleation radii under the different supersaturations and temperatures when the radius of the Si nanotubes is 10 nm, and the results are shown in Figure 2a. Since Si nanotubes can exist only below 2200 K,21 all of the calculations in the above theoretical model must be considered below this temperature. It can be seen from Figure 2a that the critical radii of the cBN nuclei decrease as the temperature and supersaturation increase. Figure 2b shows the relationship curve between the formation energy of the critical nuclei and the supersaturation at various temperatures. Clearly, the formation energy of the cBN nuclei decreases with an increase in superaturation and increases with a decrease in temperature. These results show that the high temperature and high pressure are propitious to the formation of the cBN nucleus. When ln P/Pe ) 0.8,13 the dependence of the radius and formation energy of cBN critical nuclei on the radii of the Si nanotubes can be obtained in Figure 3 when the temperature is in the range of 1300-1700 K. It can be seen from Figure 3a and 3b that both the radius and formation energy of cBN critical nuclei decrease as the radii of Si nanotubes decrease. Therefore, these results indicate that the radii of Si nanotubes can greatly affect the cBN formation inside the Si nanotubes. The smaller the radius of Si nanotubes, the easier the formation of cBN nuclei. According to experimental data, Ea ) 207 kJ/mol,18 using eqs 2 and 8, we calculated the probability curves of the phase transformation inside the Si nanotubes when r ) 5 nm and R ) 10 nm as shown in Figure 4a. Clearly, the probability of the phase transformation increases as the supersaturation and temperature increase. Additionally, the probability of the phase transformation hardly changes in the temperature range from 1000 to 1900 K when the supersaturation is close to zero. When ln P/Pe ) 0.8, the dependence of the phase transformation probability on the radii of Si nanotubes can be obtained in Figure 4b. The probability of the phase transformation increases as the radius of the Si nanotubes decreases at a given temperature. These results indicate that the smaller the radius of the Si nanotube, the higher the probability of the phase transformation, i.e., the phase transformation would take place more easily inside Si nanotubes with a smaller radius. Additionally, when the radii of Si nanotubes are beyond 10 nm, the probability of the phase transformation changes much slowly. According to eq 9, when h ) 2.09 nm, and V ) 2.5 × 1013 Hz,19 we calculated the relationship curves between the growth velocity of cBN nuclei inside the Si nanotubes and both the temperature and supersaturation when r ) 5 nm and R ) 10 nm. The results are shown in Figure 5a and 5b. Clearly, one
Formation of cBN Nanowires inside Silicon Nanotubes
Figure 2. (a) The relationship curves between the radii of cBN critical nuclei inside the Si nanotubes and both temperature and supersaturation. (b) The dependence of the supersaturation on the formation energy of cBN critical nuclei inside Si nanotubes at a given temperature.
J. Phys. Chem. C, Vol. 114, No. 47, 2010 19943
Figure 4. (a) The relationship curves between the phase transition probability and both temperature and supersaturation. (b) The dependence of the phase transition probability on the radii of the Si nanotubes.
Figure 3. The dependence of the radii (a) and the formation energy (b) of cBN critical nuclei on the radii of the Si nanotubes under conditions of variable temperature.
can see that the growth velocity of cBN nuclei inside the Si nanotubes increases as both the temperature and the supersaturation increase. The growth velocity changes much faster when the temperature is over 1600 K and the supersaturation is below 5. When ln P/Pe ) 0.8, we calculated the relationship curve between the growth velocity of cBN nuclei inside the Si nanotubes and the radii of Si nanotubes at a given temperature, and the results are shown in Figure 5c. Figure 5c clearly shows that the growth velocity of cBN nuclei inside the Si nanotubes decreases as the radii of Si nanotubes increase at a given temperature. These results indicate that the cBN nuclei exhibit more favorable growth inside Si nanotubes of smaller radii at a given temperature and pressure.
Figure 5. The dependence of the temperature (a), the supersaturation (b), and the radii of Si nanotubes (c) on the growth velocity of cBN nuclei inside Si nanotubes.
It is well-known that cBN, which is the simplest and purely artificial III-V compound, can be synthesized under high pressure and high temperature similar to that for the synthesis of diamonds.22 However, it has been observed recently that cBN in nanoscale is also obtained at low temperature and pressure.12,19 Many studies show that the formation of nanosized cBN is attributed to nanosize-induced interior pressure.18-20,22 Generally,
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Figure 6. The dependence of the difference of Gibbs free energy between hBN and cBN phases on the size under various temperatures.
in the case of the nucleation of a cluster from gases, the phase stability is quite different from that of the phase diagram that is determined at atmospheric pressure, i.e., the nuclei are under high pressure arising from so-called “capillarity” that is expressed by the Laplace-Young equation.20 Many researchers have studied the nucleation stability of both nanodiamond and nanographite, the formation of quantum dots, and the other topics by employing the Laplace-Young equation.23-26 Especially, Yang et al. have successfully developed a new nanothermodynamics model to analyze the metastable phase nucleation on nanoscale.20 Using their nanothermodynamics model, Yang, et al. investigated the formation mechanism of cBN nuclei under chemical vapor deposition (CVD) and concluded that the effect of surface tension induced by the nanometer-size curvature of the critical nuclei could drive the metastable phase region of cBN nucleation into the stable phase region of the BN phase diagram. In our theoretical model, we considered the additional pressure contribution not only due to the existence of surface stress induced by the curvature of the critical nuclei mentioned above but also from the surface tension induced by the Si nanotube curvature. Since cBN is a high pressure phase, it is clear that the smaller the radius of the Si nanotube, the more the addition pressure induced by nanotube curvature increases, which makes the formation of cBN nuclei easier. Therefore, our calculated results above show that a decrease in the radius of the Si nanotubes promotes the nucleation and growth of cBN inside Si nanotubes. According to Yang’s nanothermodynamic theory, the Gibbs free energy difference of both hBN and cBN phases can be expressed as:27
∆Gh-c ) Gh - Gc )
(
)
2γcv 4πr3 ∆V P - Pe + + 4πr2γcv 3Vm r (10)
Here, Gh and Gc represent the Gibbs free energy of hBN and cBN phases, respectively, which is a function of the pressure and temperature. Pe is the equilibrium phase boundary between hBN and cBN and can be expressed approximately by27 Pe ) 1.833 × 106T + 1.117 × 109. On the basis of eq 10, the dependence of the difference of Gibbs free energy between hBN and cBN phases on the size can be determined, as shown in Figure 6. Gh is larger than Gc when the size of the crystalline grains is below a threshold size at a given temperature. This result suggests that the Gibbs free energy of formation of the cBN phase will be less than that of the hBN phase if the size of the formed critical nuclei is lower than the threshold size. Since the nucleation of a phase is confined in the Si nanotubes, it is favorable to form critical nuclei with small size. Thus, under such small size, cBN could have a lower formation energy for
Hu et al. the critical nuclei than hBN, which would be preferable for nucleation inside the Si nanotubes. However, could the Si atoms react with N and B atoms during cBN nucleation by condensation? Although an answer cannot be obtained presently, the previous experimental and calculated results show that the stable temperature of the existence of Si nanotuobes can be around 1700-2200 K. cBN are more easily formed on Si substrates with a “mirror-like” surface than other noncovalent or soft substrates, and cBN can be formed over a wide range of temperatures (from about 100 °C to over 800 °C).21,28 Furthermore, the growth velocity of cBN nuclei is quite fast based on the calculated results above, suggesting that cBN nanowires can be obtained by fast growth once cBN nuclei are formed. Generally, B and N atoms and other molecules will be first absorbed on the surface of Si and then the interaction between Si and other atoms takes place.29,30 Note that their interaction depends on the temperature. When Si nanotubes have a low temperature, they may not react with B and N atoms, but a higher temperature may improve the reactivity of the B and N atoms, which not only strengthens their interaction with Si but also leads to a structural change in the BN nuclei that allows formation of a stable phase.28 Since cBN is a stable phase on nanoscale and can be formed over a wide range of temperatures, cBN nuclei can also possibly be formed inside Si nanotubes on the basis of the considerations mentioned above. 4. Summary A model that predicts the formation of cBN nanowires inside Si nanotubes was developed by taking the effect of surface tension induced by the nanosize curvature of critical nuclei and Si nanotubes into account. The following results were obtained by our calculations: (i) The smaller the radii R of the Si nanotubes, the easier the nucleation and growth of cBN inside the Si nanotubes. (ii) A higher temperature T can decrease the formation free energy and the radius of critical nuclei of cBN inside the Si nanotubes, which increases the probability of phase transformation and growth velocity of cBN nuclei inside the Si nanotubes. (iii) The effect of supersaturation σ on the formation of cBN nanowires is larger at first and then becomes smaller. Therefore, the most favorable conditions for the formation of cBN nanowires inside Si nanotubes are T ) 1300-1800 K, σ ) 2-5, and R e 5 nm. The proposed model is expected to be a general approach for the elucidation of nucleation and growth of materials conducted on a nanometer scale. Acknowledgment. This work was financially supported by the National Natural Science Foundation of China (no. 50902126), Shanxi Province Science Foundation for Youths (no. 2009021027), Program for the Top Young Academic Leaders of Higher Learning Institutions of Shanxi, and North University of China Science and Talent Startup (no. 20080302) Foundation for Youths. References and Notes (1) Xia, Y.; Yang, P. AdV. Mater. 2003, 15, 351. (2) Liu, Q. X.; Wang, C. X.; Yang, Y. H.; Yang, G. W. Appl. Phys. Lett. 2004, 84, 4568. (3) Kuchibhatla, S. V. N. T.; Karakoti, A. S.; Bera, D.; Seal, S. Prog. Mater. Sci. 2007, 52, 699. (4) Jeong, S. Y.; Kim, J. Y.; Yang, H. D.; Yoon, B. N.; Choi, S.-H.; Kang, H. K.; Yang, C. W.; Lee, Y. H. AdV. Mater. 2003, 15, 1172. (5) Hu, J.; Bando, Y.; Liu, Z.; Zhan, J.; Golberg, D.; Sekiguchi, T. Angew. Chem. 2003, 116, 65. (6) Fagan, S.; Baierle, R. J.; Mota, R.; da Silva, A. J. R.; Fazzio, A. Phys. ReV. B 2000, 61, 9994. (7) Sekkal, W.; Bouhafs, B.; Aourag, H.; Certier, M. J. Phys.: Condens. Matter 1998, 10, 4975.
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J. Phys. Chem. C, Vol. 114, No. 47, 2010 19945 (20) Wang, C. X.; Yang, G. W. Mater. Sci. Eng. R. 2005, 49, 157. (21) Fagan, S. B.; Mota, R.; Baierle, R. J.; Paiva, G.; da Silva, A. J. R.; Fazzio, A. J. Mol. Struct. 2001, 539, 101. (22) Wang, C. X.; Yang, Y. H.; Liu, Q. X.; Yang, G. W. J. Phys. Chem. B 2004, 108, 728. (23) Hwang, N. W.; Hahn, J. H.; Poon, D. Y. J. Cryst. Growth 1996, 160, 87. (24) Tolbert, S. H.; Alivisatos, A. P. Annu. ReV. Phys. Chem. 1995, 46, 595. (25) Gao, Y. H.; Bando, Y. Nature 2002, 415, 599. (26) Gao, Y. H.; Bando, Y. Appl. Phys. Lett. 2002, 81, 3966. (27) Wang, C. X.; Liu, Q. X.; Yang, G. W. Chem. Vap. Deposit. 2004, 10, 280. (28) Mirkarimi, P. B.; McCarty, K. F.; Medlin, D. L. Mater. Sci. Eng. R. 1997, 21, 47. (29) Rangelov, G.; Stober, J.; Eisenhut, B.; Fauster, T. Phys. ReV. B 1991, 44, 1954. (30) Mui, C.; Widjaja, Y.; Kang, J. K.; Musgrave, C. B. Surf. Sci. 2004, 557, 159.
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