9966
J. Phys. Chem. B 2005, 109, 9966-9969
Thermodynamic and Kinetic Size Limit of Nanowire Growth Cheng-Xin Wang, Bing Wang, Yu-Hua Yang, and Guo-Wei Yang* State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics Science and Engineering, Zhongshan UniVersity, Guangzhou 510275, P. R. China ReceiVed: December 1, 2004; In Final Form: April 1, 2005
The universal models of nucleation thermodynamics and growth kinetics were established for nanowire growth upon metal-catalyst-assisted thermal chemical vapor transport on the basis of vapor-liquid-solid (VLS) mechanism. The thermodynamic and kinetic size limit of nanowire growth was deduced from the proposed model. Theoretical predictions are in agreement with experimental data.
Introduction Recently, one-dimensional (1D) nanostructures such as wires, rods, belts, and tubes have become the focus of intensive research owing to their unique applications in mesoscope physics and fabrication of nanoscale devices.1-6 For instance, they not only provide a good system to study the electrical and thermal transport in one-dimensional confinement but also are expected to play an important role in both interconnection and functional units in fabricating electronic, optoelectronic, and magnetic storage devices.7 Up to now, most semiconducting nanowires have been synthesized using metal-catalyst-assisted thermal chemical vapor transport (CCVT) on the basis of so-called vapor-liquid-solid (VLS) mechanisms.8-11 Actually, VLS mechanisms applicable to the growth of silicon crystalline whiskers was initially proposed and demonstrated by Wagner and Ellis in 196412,13 and has subsequently been used to address the growth of semiconducting nanowires by CCVT.14-17 On the basis of these studies above, for the nanowire formation upon CCVT, the VLS mechanism reveals that (i) reactants are supplied in the vapor phase and nanowires are formed by extraction of nanowire materials from liquid droplets of metal alloy, (ii) the nanowire growth results from the difference of sticking coefficients of liquid droplets and nanowires, namely, sticking coefficients of nanowires are orders of magnitude smaller than that of liquid droplets. As a result, the droplet and the solid nanowire can capture and reject nearly all the constituents of the growing material from the vapor phase, respectively, as shown in Figure 1a. Generally, the intrinsic properties of 1D nanostructures are mainly determined by their size, shape, composition, and crystalline structure. In principle, one can control any one of these characteristics to fine-tune the properties of this nanostructure. It is, therefore, essential to pursue the basic physics and chemistry involved in the formation of nanowires. However, the size limit of nanowire growth upon CCVT still remains much less understood in theory, even though there have been a lot of relevant experimental studies to try to control the nanowire growth.17-20 For example, both theoretically and experimentally, we have not known how thin or thick nanowires could be synthesized using CCVT yet. Accordingly, in this paper, we proposed the universal nucleation thermody* Author to whom correspondence should be addressed. E-mail:
[email protected]. Telephone and Fax: +86-20-8411-3692.
Figure 1. Schematic diagram showing the VLS mechanism and XNW thermodynamic nucleation and kinetic growth processes. (a) The droplet and the solid nanowire, capturing and rejecting nearly all the constituents of the growing material from the vapor phase, respectively, because of the sticking coefficient immense difference between them. (b) The thermodynamic nucleation case. (c) The kinetic growth case.
namics and growth kinetics for nanowire growth upon CCVT on the basis of the VLS mechanism and then theoretically deduced the size limit of the nanowire growth. Importantly, taking Si nanowire (SiNW) growth as an example, we found that the theoretical predictions are in nice agreement with experimental cases. Theory Model In general, the minimal size of nanowires is both dependent on nucleation thermodynamics and growth kinetics (energy stability theory). Meanwhile, it is also dependent on the Rayleigh instability of nanowire itself (linearized stability theory).21
10.1021/jp0445268 CCC: $30.25 © 2005 American Chemical Society Published on Web 04/27/2005
Size Limit of Nanowire Growth
J. Phys. Chem. B, Vol. 109, No. 20, 2005 9967
However, when the linearized system is symmetric, the prediction of stability on the basis of the linearized stability theory is consistent with that from the energy stability theory.22,23 Namely, we could have an instability (usually called a subcritical instability) below the critical values of Rayleigh numbers determined by the linearized theory that can be in the energetically stable region. Thermodynamically, the phase transformation is promoted by the difference of Gibbs free energies. The Gibbs free energy of a phase can be expressed as a function of the pressure and temperature and determined by a general coordinate or reactive coordinate. The schematic illustration of the X nanowire (XNW) nucleation upon CCVT is shown in Figure 1b. The Gibbs free energy difference of a cluster can be expressed as24
∆G ) (σlx - σlV)S1 + σxVS2 + ∆gVV,
(1)
where σlx, σlV, and σxV are the liquid-XNW, the liquid-vapor, and the XNW-vapor interface energy density, S1 and S2 are the corresponding interface areas (see Figure 1b) expressed as: S1 ) 2πR′h and S2 ) 2πrH, respectively. When the interface between nanowires and catalyst liquid droplets is assumed to be an incoherent interface, the values of σlx can be denoted as: σlx ) (σlx + σxV)/2. V is the volume of X clusters defined as: V ) 1/6πH(H2 + 3L2) - 1/6πh(h2 + 3L2). Note that the theoretical model in our case is formulated on the basis of these assumptions of (i) the catalyst droplet and X cluster are perfectly spherical, and (ii) X nuclei are mutually noninteractive. From Figure 1b, h, H, and L2 can be defined respectively as:
h ) R′ -
( (
x
H)r 1-
R′2 - r2 +
xR′
L2 ) r2 1 -
r2(R′ cos θ - r)2 R′2 + r2 - 2R′r cos θ
R′ cos θ - r 2
+ r2 - 2R′r cos θ
(R′ cos θ - r)2 R′2 + r2 - 2R′r cos θ
)
)
(2) (3)
(4)
where R′, r, and θ denote the curvature radius of the liquid droplet, the curvature of the cluster, and the contact angle between clusters and liquid droplets, respectively. Meanwhile, one can obtain cos θ ) (σlx - σxV)/2σxV, and ∆gV is the Gibbs free energy difference per unit volume from vapor to solid, which is expressed by -RT/Vm ln(P/Pe), where P, T, R, and Vm are the pressure, the temperature, the gas constant, and the mole volume of the XNW, respectively. Pe is the X vapor phase pressure in thermal equilibrium coexistence with the liquid of composition in a flat surface. On the basis of Tan’s model,25 when the droplet is assumed to be an idea solution, ∆gV can be expressed as
∆gV ) -
( )
C RT p RT ln ) - ln eq Vm pe Vm C
(5)
where C and Ceq are X concentrations on the solid and liquid lines of the eutectic catalyst-X phase diagram, respectively. On the basis of the above deductions, the Gibbs free energy of an X cluster is expressed reasonably as:
∆G )
(
πR′(σxV - σlV) R′ -
(
2πσxVr2 1 -
((
x
R′2 - r2 +
R′ cos θ - r
r2(R′ cos θ - r)2
) (
(
C 1 RT + π ln eq 6 V C m xR′ + r - 2R′r cos θ 2
2
x ( x ( ( x ( x
R′ -
3r2 R′ -
r2(R′ cos θ - r)2
R′ - r + 2
2
R′2 + r2 - 2R′r cos θ
R′2 - r2 +
)
3
r2(R′ cos θ - r)2
R′2 + r2 - 2R′r cos θ (R′ cos θ - r)2 1- 2 R′ + r2 - 2R′r cos θ 3 R′ cos θ - r r3 1 2 2 R′ + r - 2R′r cos θ R′ cos θ - r 3r3 1 × R′2 + r2 - 2R′r cos θ
)
(
1-
) ))
R′2 + r2 - 2R′r cos θ
)
+
×
+
)
×
)
(R′ cos θ - r)2 R′2 + r2 - 2R′r cos θ
))
(6)
From eq 6, one can see that the Gibbs free energy of a cluster of XNWs is related to two factors: one is the liquid droplet composition and another one is the size of the liquid droplet. Therefore, when ∂∆G(r)/∂r ) 0, the critical size of the nuclei of XNWs can be attained.26 Distinctly, the critical size of the XNW nuclei is the thermodynamic size limit, i.e., the theoretical minimum size of the nanowire growth, from the point of view of thermodynamics. Kinetically, in the growth of semiconducting nanowires or nanowhiskers on the basis of the VLS mechanism, reactants are supplied in the vapor phase, and the diameter of nanowires is limited by the size of the metallic seed particles.19,20 Furthermore, the VLS mechanism implies that the solid wire is formed by extraction from a droplet of metal that is alloyed by the constituents of the growing material, with a melting-point depression attributable to the formation of a eutectic melt. Accordingly, we can see that after X atoms reach saturation in the liquid droplet at the given thermodynamic parameters, the number of X atoms of the droplet capturing (X1) and extracting (X2) should be equal, as shown in Figure 1c. Then, the extracting X2 atoms contribute to the growth of XNWs. Thus, the number of X2 atoms can be approximately equal to the number X1 impinging on the surface of the area whose value is equal to SN. Therefore, the numbers of X1 and X2 of X atoms is expressed reasonably as
X1 ) 2πR′(R′ + xR′2 - r′2)n ) X2 )
106 πr′2 SFNm M
(7)
where R′, r′, F, Nm, and M are the radius of droplet, the radius of XNW, the density of single crystal X, the Avogadoro constant, and the mole mass of single crystal X, respectively. The n is the number of X atoms from the vapor impinging on a unit plane surface in a unit time. For the VLS progress of the vapor growth or vapor deposition, an ideal gas should be a good approximation because of the low gas density. According to Maxwell’s velocity distribution equation, at the given temperature T and pressure +∞ +∞ P, n can be expressed as n ) ∫-∞ dux ∫-∞ duy ∫+∞ 0 nuzP(u)duz ) n′(kT/2πm)1/2 ) P/x2πmkT where P(u) is the Maxwell’s
9968 J. Phys. Chem. B, Vol. 109, No. 20, 2005
Wang et al.
velocity distribution, n′ is the molecular number of the vapor phase per unit volume, k is the Boltzmann constant, and m is the mass of X atom, respectively. In eq 7, S is the growth velocity of XNW, and can be expressed as27
S ) hν exp(-ER/RT)[1 - exp(-|∆gm|/RT)]
(8)
where h, ν, ER, R, and T are the lattice constant of the XNW in the growth direction, the thermal vibration frequency, the mole activation energy of X atoms, the gas constant, and the substrate temperature, respectively. On the basis of eq 5, the Gibbs free energy difference per mole of ∆gm can be expressed as
∆gm ) -RT ln
( ) C Ceq
(9)
According to eqs 7-9, we can obtain the relationship between the size (2R′) of the catalyst droplet and the diameter (2r′) of XNW as follows: 2r′ )
(
Figure 2. Dependence of the radius of the SiNW nucleation and the Gibbis free energy on the Si cluster size under conditions of various Au-catalyst sizes, temperature (713 K), and pressure (SiH4 partial pressure of 2 Torr).
MP ER |-RT ln(C/Ceq)| 10 x2πmkTFNmhν exp 1 - exp RT RT 2 1/2 MP (10) eq E )| |-RT ln(C/C R 106x2πmkTFNmhν exp 1 - exp RT RT 4R′
(
6
( )( ( ( )( (
))
))) )
Clearly, from eq 10, one can see that the diameter of nanowires is related to the size of the droplet. In other words, the growing size of nanowires, the theoretical possible or maximum size, depends on the droplet size from the point of view of growth kinetics. Results and Discussion
Figure 3. Relationship curve between the Au-catalyst radius and the critical energy of SiNW nucleation; the inset shows the dependence of the Au-catalyst radius and the critical nucleation of SiNW nucleation.
To validate our model, on the basis of the sufficient experiments or theoretical parameters of the Si-Au system that are presently available, we take the SiNW growth upon CCVT as an example to check its operation. From Leiber19,20 and other references, these thermodynamic parameters of our calculations by eqs 6 and 10 are as follows: the liquid droplet-gas surface energy density σlV is 0.885 J/m2,28, the solid silicon-gas surface energy density σSiV is 1.610 J/m2,21, the activation energy ER is estimated to be about 100 kJ/mol in the given temperature (713 K) and pressure (SiH4 partial pressure of ∼2 Torr) conditions on the basis of Lew’s experimental results and discussions,29 the thermal vibration frequency of Si atoms ν is about 2 × 1013 Hz,30 the lattice constant of the SiNW in the growth direction 〈111〉 h is 0.312 nm, the mole volume Vm of the single-crystal silicon is 1.2 × 10-5 kg/m3, the mole mass M and density F of single-crystal silicon are 28.09 g/mol and 2.33 g/cm3. The experimental temperature T and the SiH4 partial pressure P are 713 K and ∼2 Torr (with the total pressure of about 20 Torr).19 Note that the Au-Si phase diagram used is from ref 31. On the basis of eq 6, we give the dependence of the radius of a cluster of Si (r) and the Gibbs free energy (∆G) at the given temperature (713 K), pressure (SiH4 partial pressure of ∼2 Torr), and various Au-catalyst radii (R′), as shown in Figure 2. We can see that the Gibbs free energy (∆G) of Si clusters light changes with increase of the Au-catalyst radius. On the basis of the nucleation thermodynamics,26 the radius of Si clusters (r) corresponding to the peak value (the critical energy) should be the critical radius (r) of the SiNW nucleation. Therefore, from Figure 2, the dependent curves of the Au-catalyst radius (R′) and the critical energy (∆G), as well as the Au-catalyst radius (R′) and the critical radius of the SiNW nucleation (r)
are given in Figure 3, respectively, at the given temperature (713 K) and pressure (SiH4 partial pressure of ∼2 Torr) by numerical calculation. Definitely, the critical energy of the SiNW nucleation basically keeps little change with increasing the Aucatalyst radius when the catalyst radius is more than 10 nm, and it quickly increases with decreasing catalyst radius when the catalyst radius is less than 10 nm. These theoretical results indicate that, with the size of metallic nanoclusters decreasing, the nucleation barrier of SiNW will increase and result in increasing activation energy. In fact, this conclusion is in excellent agreement with Lieber’s prediction20 and Kikkawa’s experimental analyses.32 However, the decreasing scope of the critical energy is very little when the catalyst radius is less than 20 nm, implying that the affect of the droplet size on the critical energy of the nanowire nucleation can be actually neglected. On the other hand, in the inset of Figure 3, one can clearly see an interesting fact that the critical radius of SiNW is independent of the size of metallic catalyst droplets, suggesting that the critical radius of the nanowire nucleation is a determinate value for a certain system from the thermodynamic viewpoint. Thus, the nanowire can be grown to even smaller sizes in a certain thermodynamic system until reaching a kinetic limit. The dependence of the diameter of SiNW on the Au-catalyst size at the given temperature (713 K) and SiH4 partial pressure (2 Torr) can been attained from eq 10 as shown in Figure 4. One can see that the diameter of SiNWs increases with increase of the Au-catalyst diameter, and the SiNW diameter is nearly equal to the size of Au-catalyst droplets. Therefore, these theoretical results are consistent with experimental cases.8,19,20,33 Remarkably, these quantitative calculations are in nice agreement with Lieber’s experimental data.19,20 In addition, from Figure 4, we
Size Limit of Nanowire Growth
J. Phys. Chem. B, Vol. 109, No. 20, 2005 9969 kinetically. The validity of the proposed theory was shown when it operation in the growth of SiNWs upon CCVT, suggesting that it could be expected to be a general approach applicable to elucidating various nanowire formations upon CCVT on the basis of the VLS mechanism. Acknowledgment. The National Natural Science Foundation of Guangdong Province under Grant No.036596, the Foundation of Guangzhou under Grant No. 4205006, and the China Postdoctoral Science Foundation supported this work. References and Notes
Figure 4. Experimental data of relationship between the size of the Au catalyst and the diameter of SiNW compared to the calculated curve by using eq 10 and Au-Si phase diagram, in which the Au-Si phase diagram from ref 31 is used.
also see that the dependent relationship between the size of SiNW and Au-catalyst diameter is little different under different carrier gases and diameters of SiNW conditions. This case maybe results from the difference of the activation energy for different carrier gases (N2 in ref 19 and H2 in ref 20) and different sizes of SiNW. In detail, compared with N2 carrier gas, hydrogen can suppress the decomposition of SiH4 from the view of chemical reaction and result in the activation energy increasing SiNW growth. On the other hand, experimentally, Lieber et al.20 have found that there is a strong dependent relationship between the growth direction and the diameter of SiNWs. For instance, the smallest-diameter SiNW grows primarily along 〈110〉 (high-free-energy), and the larger-diameter SiNW grows along 〈111〉 (low-free-energy). The experimental case maybe indicates that, compared with the large-size SiNW, the small-size SiNW has large activity energies. Similarly, Buhro et al.34 simulated the relationship curve between the diameter of catalyst droplets and the size of GaAs nanowires on the basis of their empirically linear expression and experimental data. Interestingly, our theoretical predictions are consistent with Buhro’s results. Accordingly, the size of the catalyst droplet can limit the possible growing size of the nanowire synthesized by CCVT to a certain extent from the kinetic viewpoint. Conclusion In summary, we established universal models of nucleation thermodynamics and growth kinetics to address the nanowire formation upon CCVT on the basis of the VLS mechanism and found that the critical size of the nanowire nucleation is suggested to be the thermodynamic size limit, i.e., the theoretical minimum size of the nanowire grown by CCVT both thermodynamically and kinetically. Additionally, the size of the catalyst droplet could limit the growing size, i.e., the theoretical possible or maximum size of the nanowire to a certain extent in
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