ROLE OF SURFACE ENERGY IN THE VAPOR-LIQUID-SOLID GROWTH OF SILICON urface Gibbs energy increment(maximum decrease in (sina+coso)/(cosd-sino total energy), i.e., so that the thermodynamic equilib rium condition is met on the (; face [7, 9]. The min- imum increment of the surface Gibbs energy of a step is It includes the reduction in the liquid-vapor interfacial energy(arcos()upon a displacement of the droplet by distance h(monatomic layer thickness)and the solid vapor interfacial energy increment, aswl-[(asL-a,Sin( )/as 1.2 If 0 steps may form the lateral surface of the whisker at angle 8. In such a case, whisker growth is thermody 1.0 namically plausible( the droplet wets the growth front) 90100110120130140150160170 The range of angles (p in which whisker growth +90.d thermodynamically plausible can be found by substit ing Eq. (4)in Fig 2 Plots of(sin+cos)/(cos8-sin 8) asL+aLcoS(P>ascos8 (7) tact angle of the melt droplet on the whisker tip: (1)constan adius( 8=0), (2)conical whisker(8>0).( IID) regions of We obtain whisker etching and vaporization, respectively, by the Slv chanism; (In) region of steady-state whisker growth by g sino= sin the vls mechanism responding to whisker growth with 8>0 but outside the Figure 2 shows the right-hand side of inequality (8)range of growth with 8=0 as a function of o for constant-radius(8=0)and coni- cal(8>0)whisker growth. The horizontal line shows In some M-Si systems, there is a range of angles the ratio aso,= 1.33 for silicon whiskers growing in which the reduction in the total gibbs energy upor from Au-Si liquid droplets at aL=0.900 J/m and as whisker growth is significant; that is, a in Eq. (5)is small and, hence, the difference a-OsL is negative 1. 200 J/m2[2]. In regions I and III(Fig. 2), inequality( 8) which ensures stable whisker growth. As is evident the VLs mechanism is unlikely. Clearly, in regions I and III, the reverse process takes place: silicon etching etals with relatively high values of surface energy aL and vaporization(formation of negative whiskers) Suitable solvents for Si whisker growth are Au, Ag In region that is, in a range of angles (( =31. Cu(Group I metals), Pt, Pd, and Ni(transition metals), and also for the above values of as and a, (as/a,=(table)and ay/ar ratios well below 1.4l(Fig. 2, max 1.33), the droplet shape is such that the energetically mum in curve /, representing constant-radius growth) favored process is the formation of a new solid-vapor At the same time, no silicon whiskers can be grown interface, i.e., whisker growth with the participation of Sn, Pb, Sb, Bi, or some other metals for which a/a,>1.41 Whisker growth with the Thus, there is a range of contact angles of M-Si melt roplets in which condition(8)is fulfilled and whisker use of Zn, Al, Ga(o=0.650 J/m2), or In(aL growth is thermodynamically plausible. Beyond this 0.500 J/")as the solvent yields short,, unoriented, range, whisker growth is impossible tapering crystals, often with globules on their top and contact angles suitable for whisker growth is broader. It growth perfections, which is characteristic of unstable For 8>0(conical whisker growth), the range of other imp for this reason that one often observes conical whis- The development of the ( 111) growth front is ener ker growth(pedestal ) which is, however, not followed getically favored because the energy of the Si(1l1) by a cylindrical growth stage. It seems likely that, in faces is relatively low. Reducing a,(e.g, via surfactant such cases, the contact angle falls within the range cor- adsorption on the melt surface)or increasing as(by INORGANIC MATERIALS Vol 39 No 9 2003INORGANIC MATERIALS Vol. 39 No. 9 2003 ROLE OF SURFACE ENERGY IN THE VAPOR–LIQUID–SOLID GROWTH OF SILICON 901 surface Gibbs energy increment (maximum decrease in total energy), i.e., so that the thermodynamic equilib￾rium condition is met on the {111} face [7, 9]. The min￾imum increment of the surface Gibbs energy of a step is . (5) It includes the reduction in the liquid–vapor interfacial energy (αLcosϕ) upon a displacement of the droplet by distance h (monatomic layer thickness) and the solid– vapor interfacial energy increment, If α – αSL < 0, (6) steps may form the lateral surface of the whisker at angle δ. In such a case, whisker growth is thermody￾namically plausible (the droplet wets the growth front). The range of angles ϕ in which whisker growth is thermodynamically plausible can be found by substitut￾ing Eq. (4) in αSL + αLcosϕ > αScosδ. (7) We obtain (8) Figure 2 shows the right-hand side of inequality (8) as a function of ϕ for constant-radius (δ = 0) and coni￾cal (δ > 0) whisker growth. The horizontal line shows the ratio αS/αL = 1.33 for silicon whiskers growing from Au–Si liquid droplets at αL = 0.900 J/m2 and αS = 1.200 J/m2 [2]. In regions I and III (Fig. 2), inequality (8) is not met at small and large ϕ, and whisker growth by the VLS mechanism is unlikely. Clearly, in regions I and III, the reverse process takes place: silicon etching and vaporization (formation of negative whiskers). In region II, that is, in a range of angles ϕ (ϕ ≅ 31° to 63° in curve 1 and ϕ ≅ 22° to 73° in curve 2, Fig. 2), and also for the above values of αS and αL(αS /αL = 1.33), the droplet shape is such that the energetically favored process is the formation of a new solid–vapor interface, i.e., whisker growth. Thus, there is a range of contact angles of M–Si melt droplets in which condition (8) is fulfilled and whisker growth is thermodynamically plausible. Beyond this range, whisker growth is impossible. For δ > 0 (conical whisker growth), the range of contact angles suitable for whisker growth is broader. It is for this reason that one often observes conical whis￾ker growth (pedestal), which is, however, not followed by a cylindrical growth stage. It seems likely that, in such cases, the contact angle falls within the range cor- α αS 1 αSL – αL sinϕ αS --------------------------------     2 = – – αL cosϕ αS 1 ( ) αSL – αL sinϕ /αS [ ]2 – . αS αL ----- sinϕ + cosϕ cosδ – sinδ < ------------------------------. responding to whisker growth with δ > 0 but outside the range of growth with δ = 0. In some M–Si systems, there is a range of angles ϕ in which the reduction in the total Gibbs energy upon whisker growth is significant; that is, α in Eq. (5) is small and, hence, the difference α – αSL is negative, which ensures stable whisker growth. As is evident from the table, stable growth occurs in the presence of metals with relatively high values of surface energy αL. Suitable solvents for Si whisker growth are Au, Ag, Cu (Group I metals), Pt, Pd, and Ni (transition metals), since they have relatively high surface Gibbs energies (table) and αS/αL ratios well below 1.41 (Fig. 2, maxi￾mum in curve 1, representing constant-radius growth). At the same time, no silicon whiskers can be grown with the participation of Sn, Pb, Sb, Bi, or some other metals for which αS/αL > 1.41. Whisker growth with the use of Zn, Al, Ga (αL = 0.650 J/m2 ), or In (αL = 0.500 J/m2 ) as the solvent yields short, unoriented, tapering crystals, often with globules on their top and other imperfections, which is characteristic of unstable growth. The development of the {111} growth front is ener￾getically favored because the energy of the Si {111} faces is relatively low. Reducing αL (e.g., via surfactant adsorption on the melt surface) or increasing αS (by 1.5 1.4 1.3 1.2 1.1 1.0 90 100 110 120 130 140 150 160 170 (sinϕ + cosϕ)/(cosδ – sinδ) ϕ + 90, deg 1 2 I II III Fig. 2. Plots of (sinϕ + cosϕ)/(cosδ – sinδ) against the con￾tact angle of the melt droplet on the whisker tip: (1) constant radius (δ = 0), (2) conical whisker (δ > 0). (I, III) regions of whisker etching and vaporization, respectively, by the SLV mechanism; (II) region of steady-state whisker growth by the VLS mechanism