CRYSTAL GROWTH & DESIGN
Dependence of the Growth Rate of Nanowires on the Nanowire Diameter
2006 VOL. 6, NO. 5 1154-1156
Dimo Kashchiev* Institute of Physical Chemistry, Bulgarian Academy of Sciences, ul. Acad. G. BoncheV 11, Sofia 1113, Bulgaria ReceiVed NoVember 18, 2005; ReVised Manuscript ReceiVed February 27, 2006
ABSTRACT: The dependence of the rate G of nucleation-mediated growth of nanowires on the nanowire diameter D is considered. It is shown that an existing general formula for this dependence allows experimental determination of the nucleation rate and growth constant of the two-dimensional islands on the nanowire tip. Comparison with available G(D) data for GaAs nanowires demonstrates agreement between theory and experiment. Introduction Nanowires (or nanowhiskers) are nanoscale materials with applicability to nanoelectronics, nanooptics, and other areas of nanoscience and nanotechnology. Different techniques are used to grow nanowires by the vapor-liquid-solid (VLS) mechanism.1-11 This mechanism is rather complex, and the theoretical description of the nanowire growth rate G (m/s) is not a simple problem. Finding G as a function of the nanowire diameter D (m) and the supersaturation ∆µ (J) of the system is of great importance for a better control over the production of nanowires for various purposes. In 1973, Givargizov and Chernov1 proposed a formula for the dependence of G on D and ∆µ, and recently, Dubrovskii and coauthors12-14 developed a detailed theory that allows numerical determination of G for nanowires having different diameters and growing at constant supersaturation. The objective of the present study is to demonstrate how an existing general formula15 for G(D) dependence can be used for experimental determination of two basic kinetic parameters characterizing the nanowire growth when the rate-limiting step in the growth is nucleation and spreading of two-dimensional (2D) islands on the nanowire tip.
Figure 1 schematizes a nanowire (e.g., of Si) on the surface of a crystal (e.g., of Si). The nanowire growth is catalyzed by an alloy droplet (e.g., of Si and Au) on the tip of the nanowire, because the liquid alloy phase is supersaturated. In VLS growth, the growth species (e.g., Si) are supplied by direct impingement on the surface of the alloy droplet and by surface diffusion along the nanowire lateral surface toward the periphery of the alloy droplet. These species are first dissolved into the droplet, then transported to the surface of the nanowire tip, and finally attached to available growth sites on this surface. When the tip surface is atomically smooth and free of screw-dislocation emergence points, these sites are generated, and, hence, nanowire growth is mediated by nucleation and lateral growth of 2D islands on the nanowire tip. The nanowire growth rate is defined by G ≡ dL/dt, where L (m) is the nanowire length, and t is time. Givargizov and Fax:
Chernov’s formula for G is:1,2
G ) k1(∆µ/kT - k2/D)2
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(1)
where k is the Boltzmann constant, T (K) is the absolute temperature, k1 (m/s) is the kinetic coefficient of growth of the solid/liquid interface between the nanowire tip and the alloy droplet, and k2 (m) is a thermodynamic parameter. Both k1 and k2 are considered as independent of ∆µ and D, but while k1 is theoretically undetermined k2 is given by1,2
k2 ) 2σSVυ0/kT
General
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Figure 1. Cross section of a nanowire having an alloy droplet on its tip and growing on a crystal surface.
(2)
Here σSV (J m-2) is the appropriately averaged specific surface energy of the solid/vapor interface between the nanowire and the vapors around it, and υ0 (m3) is the volume of a growth species in the nanowire. The D term in eq 1 accounts for the Gibbs-Thomson effect of elevation of the chemical potential of a cylindrically or prismatically shaped phase with diameter D and infinite length (in the latter case D is the diameter of the circle inscribed in the prism cross section provided this section is a regular polygon). This elevation of the chemical potential brings about the effective decrease of the supersaturation ∆µ in eq 1. The factor 2 in eq 2 corrects for the factor 4 in the original Givargizov-Chernov G(D) formula, because 4 applies to the chemical potential of spherically shaped phases.15 Eq 1 was not obtained theoretically. Rather, it was established empirically by finding that it described satisfactorily the experimental G(D) data1,2 for VLS growth of Si whiskers with D from 0.05 to 5 µm in vapors of SiCl4 and H2 at T ) 1273 K (the catalyst in the alloy droplets was Au). In their comprehensive analysis, Dubrovskii and coauthors12,13 also allowed for the Gibbs-Thomson effect on the supersatu-
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Growth Rate of Nanowires
Crystal Growth & Design, Vol. 6, No. 5, 2006 1155
ration ∆µ that drives the nanowire growth. In addition, they took into account that the change of the area bD2 of the nanowire cross section can affect the number of 2D islands mediating the nanowire growth (b is a numerical shape factor, e.g., b ) π/4 for cylindrical nanowires, b ) 1 for nanowires as square prisms, and b ) 31/2/2 for nanowires as regular hexagonal prisms if D is the diameter of the circle inscribed in the nanowire cross section). Indeed, when a growing crystal face is large enough, many 2D islands can be nucleated on it so that the filling of one layer and, hence, crystal growth occurs by the polynuclear mechanism.15 When, however, this face is sufficiently small, under otherwise equal conditions only one 2D island succeeds in nucleating within the time necessary for the filling of one layer: growth then proceeds by the mononuclear mechanism.15 Since in polynuclear growth G is independent of D, and in mononuclear growth it is proportional to D2, differently sized nanowires are bound to grow at a D-dependent rate. Although the theory of Dubrovskii and coauthors12,13 does not provide explicitly for G(D) dependence, it allows numerical determination of this dependence by solving a set of several equations. Using such a determination, Dubrovskii and coauthors12,13 have shown that their theory describes well experimental G(D) data for VLS growth of GaAs nanowires with D from 30 to 110 nm at T ) 823 K (the catalyst in the alloy droplets was Au). We can now recall the general formula for nucleationmediated crystal growth,15 which is directly applicable to growth of nanowires whose tips are free of screw dislocations. This formula is valid for progressive nucleation at a time-independent rate J (m-2 s-1) of 2D islands having a given height h (m) and growing laterally according to the power law
r ) (υt)n
(3)
Here r (m) is the island effective radius, υ (m1/n s-1) is the island growth constant, and n is a number typically between 1/2 and 1. For instance, n ) 1 for direct attachment of growth species to the periphery of a growing island (see, e.g., ref 15), and then υ is merely the island growth rate, i.e., the propagation velocity of the step bordering the island. Under the above restrictions for the nucleation and growth of the 2D islands on the nanowire tip, we can use the corresponding general formula for nucleation-mediated crystal growth15 and represent the dependence of the nanowire growth rate on the nanowire diameter in the form
G ) a1D2/(1 + a2D2)
(4)
where the kinetic parameters a1 (m-1 s-1) and a2 (m-2) are given by
a1 ) bhJ a2 ) [b/ψc
1/(2n+1)
](J/υ)
(5) 2n/(2n+1)
(6)
In eq 6, ψ is a number for which ψ ) 1 is a good approximation, and c is the island shape factor (e.g., c ) π for circles, c ) 4 for squares, c ) 2 × 31/2 for hexagons, etc.). Also, the parameters a1 and a2 depend on ∆µ through J and υ, and in this way they dictate the supersaturation dependence of G from eq 4. Being functions of ∆µ, a1 and a2 can therefore depend on D, because the Gibbs-Thomson effect discussed above results in effectively lower supersaturations for the nanowires with smaller diameters. It seems, however, that the possible D dependence of a1 and a2 will be felt in eq 4 only for nanowire diameters smaller than a few tens of nanometers and could often
be neglected. For example, for VLS growth of Si nanowires, with υ0 ) 0.021 nm3 and σSV ) 1.4 J m-2 (ref 1), from eq 2 it follows that at T ) 1273 K and ∆µ/kT ) 1 the Gibbs-Thomson D term in eq 1 is more than 10% of ∆µ/kT only for D < 33 nm. For a greater ∆µ/kT value the corresponding D value is smaller. The above ∆µ/kT value is relevant, because nucleation on crystal faces free of nucleation-active centers usually occurs at supersaturations ∆µ/kT > 1. The general eq 4 for nucleation-mediated nanowire growth has two limiting forms. For D < < a-1/2 it reduces to the 2 equation16,17
G ) a1D2 ) bhJD2
(7)
which describes growth by the mononuclear mechanism. In the opposite limiting case of D > > a-1/2 , eq 4 converts into the 2 equation15
G ) a1/a2 ) ψh(cυ2nJ)1/(2n+1)
(8)
applicable to growth by the polynuclear mechanism. In the particular case of n ) 1 (linear growth of the 2D islands) for a2 from eq 6 we have a2 ) (b/ψc1/3)(J/υ)2/3. Then eq 4 takes the form of the equation derived by Obretenov et al.,18 and eq 8 turns into that given by Hillig.19 As to the dependences of J, υ, and G on the supersaturation ∆µ, theoretical expressions for them are presented, e.g., in refs 12, 13, 15, and 20. When the kinetic parameters a1 and a2 in eq 4 are treated as D-independent, this equation describes in a simple way the G(D) function for nanowires growing by the nucleation mechanism. The attractive feature of eq 4 is that it offers the possibility for a reliable determination of the nucleation rate J and growth constant υ of the 2D islands on the nanowire tip. Indeed, using eq 4 for a fit to experimental G(D) data obtained at constant supersaturation ∆µ allows calculation of the two free parameters a1 and a2 and evaluation of J and υ from the expressions
J ) a1/bh
(9)
υ ) (a1/h)(b/ca22n+1)1/2n
(10)
which follow from eqs 5 and 6 with ψ ) 1. Doing this evaluation with the help of G(D) data obtained at different ∆µ values would lead to an experimental determination of the J(∆µ) and υ(∆µ) dependences, which can then be compared with available theoretical ones such as, e.g., in refs 12, 13, 15, and 20. A limitation with respect to the determination of υ is the necessity of independent knowledge or assumption about the value of n. Comparison with Experiment Let us now see how eq 4 describes the above-mentioned experimental G(D) data for GaAs nanowires12,13 provided a1 and a2 are treated as D-independent free parameters. The circles in Figure 2 represent these data, and the solid curve illustrates the best fit to them resulting from eq 4. As seen, there is a reasonable agreement between theory and experiment. The fit is obtained with a1 ) (3.35 ( 0.25) × 10-5 ML nm-2 s-1 and a2 ) (7.9 ( 1.5) × 10-5 nm-2. With these a1 and a2 values and with h ) 1 monolayer (ML) and assumed b ) π/4 (cylindrical nanowires), c ) π (circular 2D islands) and n ) 1 (linear island growth), from eqs 9 and 10 we find that J ) 42 µm-2 s-1 and υ ) 34 nm/s under the experimental conditions. These values for the nucleation and growth rates of the 2D
1156 Crystal Growth & Design, Vol. 6, No. 5, 2006
Kashchiev
and assumed σSL ) 0.2 J m-2, from eq 11 it follows that i* ) 198 GaAs molecules. This number is too large: nucleation typically occurs when i* < 50 (ref 15). The inference is therefore that despite its reasonable agreement with the experimental data in Figure 2, eq 1 can hardly be considered as adequate for their description.
Figure 2. Dependence of the nanowire growth rate on the nanowire diameter: circles - experimental data12,13 for GaAs nanowires grown by the VLS mechanism at T ) 823 K; solid curve - eq 4; dashed curve - numerically obtained G(D) dependence;12,13 dotted curve eq 1.
islands on the tip of a GaAs nanowire growing by the VLS mechanism seem reasonable. For comparison, the dashed curve in Figure 2 depicts the G(D) dependence obtained numerically by Dubrovskii and coauthors12,13 with the use of parameter values leading to a fit with the experimental G(D) data. Unlike the solid curve, the dashed curve takes into account the GibbsThomson effect but does not result in a fit that is considerably different from that corresponding to the solid curve. This is an indication that under the conditions of the experiment the D dependence of the parameters a1 and a2 in eq 4 is negligible. We note as well that no direct evaluation of J and υ with the help of the dashed curve is possible. Also for comparison, the dotted curve in Figure 2 exhibits the best-fit G(D) dependence from eq 1. This fit is obtained with k11/2∆µ/kT ) 0.590 ( 0.016 ML1/2 s-1/2 and k11/2k2 ) 16.2 ( 1.2 ML1/2 nm s-1/2, and, as seen, it is comparable with the fits corresponding to the solid and dashed curves. From eq 2, with T ) 823 K, υ0 ) 0.05 nm3 and assumed σSV ) 1.2 J m-2, the parameter k2 is estimated to be k2 ) 10.4 nm. Using this k2 value and the values of k11/2∆µ/kT and k11/2k2 given above yields k1 ) 2.5 ML/s and ∆µ ) 0.37kT. While nothing can be said about the former value, because k1 is merely an empirical parameter, the value of ∆µ seems rather small. This can be seen if it is used for calculation of the number i* of molecules in the 2D GaAs nucleus on the surface of the nanowire tip. For a circular nucleus with monolayer thickness this number is given by15
i* ) πa0κ2/∆µ2
(11)
where a0 (m2) is the area occupied by a molecule on the surface of the nanowire tip, and κ (J m-1) is the specific peripheral (or edge) energy of the nucleus. The product a0κ2 can be approximated by hυ0σSL2, since a0h ≈ υ0 and κ ≈ σSLh, where σSL (J m-2) is the specific surface energy of the solid/liquid interface between the nanowire tip and the alloy droplet. With the above T, υ0 and ∆µ values and with h ) 1 ML ) 0.57 nm
Conclusion In conclusion, it should be noted that since the general eq 4 for the G(D) dependence of nanowires applies only when the rate-controlling step in their growth is the nucleation and lateral growth of 2D islands on the surface of the nanowire tip, it is valid irrespectively of whether the growth species are transported to the alloy droplet directly from the ambient phase or by surface diffusion along the nanowire lateral surface. When the GibbsThomson effect is negligible, the parameters a1 and a2 in eq 4 are D-independent, and using this equation for analysis of experimental G(D) data allows a reliable determination of the nucleation rate J and growth constant υ of the 2D islands on the nanowire tip. Evaluating J and υ from eqs 9 and 10 with parameters a1 and a2 found from G(D) data at different fixed supersaturations ∆µ makes it possible to reveal the J(∆µ) and υ(∆µ) dependences and compare them with theoretical ones. It should be noted as well that eq 4 is directly applicable also to nucleation-mediated growth of nanowires when rather than by a gas phase, they are surrounded by a liquid phase. References (1) Givargizov, E. I.; Chernov, A. A. Kristallografiya 1973, 18, 147153. (2) Givargizov, E. I. J. Cryst. Growth 1975, 31, 20-30. (3) Duan, X.; Wang, J.; Lieber, C. M. Appl. Phys. Lett. 2000, 76, 11161118. (4) Ohlsson, B. J.; Bjork, M. T.; Magnusson, M. H.; Deppert, K.; Samuelson, L.; Wallenberg, L. R. Appl. Phys. Lett. 2001, 79, 33353337. (5) Wu, Y.; Fan, R.; Yang, P. Nano Lett. 2002, 2, 83-86. (6) Lew, K.-K.; Redwing, J. M. J. Cryst. Growth 2003, 254, 14-22. (7) Dailey, J. W.; Taraci, J.; Clement, T.; Smith, D. J.; Drucker, J.; Picraux, S. T. J. Appl. Phys. 2004, 96, 7556-7567. (8) Mcllroy, D. N.; Alkhateeb, A.; Zhang, D.; Aston, D. E.; Marcy, A. C.; Norton, M. G. J. Phys.: Condens. Matter 2004, 16, R415-R440. (9) Tonkikh, A. A.; Cirlin, G. E.; Samsonenko, Yu. B.; Soshnikov, I. P.; Ustinov, V. M. Semiconductors 2004, 38, 1217-1220. (10) Kikkawa, J.; Ohno, Y.; Takeda, S. Appl. Phys. Lett. 2005, 86, 123109(1-3). (11) Plante, M. C.; LaPierre, R. R. J. Cryst. Growth 2006, 286, 394399. (12) Dubrovskii, V. G.; Sibirev, N. V. Phys. ReV. E 2004, 70, 031604(1-7). (13) Dubrovskii, V. G.; Sibirev, N. V.; Cirlin, G. E. Tech. Phys. Lett. 2004, 30, 682-686. (14) Dubrovskii, V. G.; Cirlin, G. E.; Soshnikov, I. P.; Tonkikh, A. A.; Sibirev, N. V.; Samsonenko, Yu. B.; Ustinov, V. M. Phys. ReV. B 2005, 71, 205325(1-6). (15) Kashchiev, D. Nucleation: Basic Theory with Applications; Butterworth-Heinemann: Oxford, 2000. (16) Volmer, M.; Marder, M. Z. Phys. Chem. A 1931, 154, 97-112. (17) Kaischew, R.; Stranski, I. N. Z. Phys. Chem. A 1934, 170, 295299. (18) Obretenov, W.; Kashchiev, D.; Bostanov, V. J. Cryst. Growth 1989, 96, 843-848. (19) Hillig, W. B. Acta Metall. 1966, 14, 1868-1869. (20) Kashchiev, D. J. Cryst. Growth 2004, 267, 685-702.
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