NEBOL SIN. SHCHETININ Surface Gibbs energy aL, contact angle +900 on( 111; faces, as/a, ratio, and stability of whisker growth in metal-silicon systems(as=1.200 J/m2[2D) Metal TK[8] αx,J/m2[8] aS/CL 9+ 90, deg [10] Stability of whisker growth 1.300 135 0.910 110 0.900 1.33 0.400 3.15 No whisker growth Pt 210 1.740 0.72 High 0.820 0.360 3.5 No whisker growth 1.750 0.72 800 0.355 3.38 No whisker growth removing impurities from the surface layer) may lead accordingly, the cylindrical shape of the growing whis- to the fulfillment of the condition ker. In the course of whisker growth, the three-phase as>a,cos(p asL (9) system acts as a natural shaper, determining the diame The total Gibbs energy increment is then positive, and is no contact between the crystallizing material and the reverse process is thermodynamically favored, i.e., shaper walls, in contrast to many other growth tech silicon vaporization(etching) through the liquid drop- niques [4, 11], the resulting whiskers have a more per- let. In other words, if a, is low compared to as, a single fect structure (solid-liquid) interface is energetically more favorable than two(solid-vapor and liquid-vapor) interfaces The rise of the liquid droplet during whisker growth (Figs. la, 1b)is due to the contact angle hysteresis at Indeed, this process may take place and leads to the the three-phase line. If the contact angle of the lateral formation of either negative whiskers by the solid-liq whisker surface()is smaller than its equilibrium con- uid-vapor(SLV)mechanism [2]or cylindrical through tact angle( 0)(Fig. Ic), satisfying the Young equation holes in the silicon substrate, as in the presence of a borax film, which eliminates adsorbed particles and arcoS 0+ aSL =as oxides from the silicon surface a driving force(Fg) appears, applied to the three-phase The formation of a thin SiO2 film on the substrate or line of contact. The magnitude of this force can be teral whisker surface when the reactant gases are found by jointly solving Eq(10)and inequality(7 insufficiently pure increases the surface energy as and leads to the fulfillment of condition (9). Whisker Fr=a (cos -cos Acos8)-as(cos8-1)>0.(11) able, and the process is unstable or does not occur at all. At 8=0, inequality(11) takes the form The energy gain responsible for the accelerated rate Fg=al(cos (p-cos0)>0 (12) of whisker growth omparison with epitaxial film Condition(12) is fulfilled for <0. For this reason, the growth in an analogous chemical process is obviously advance of the droplet along the lateral whisker surface liquid phase, a, upon the displacement of the three- can be thought of as the reverse of liquid spreading over phase line during whisker growth. For this reason, whisker growth at relatively low temperatures(150- Taking 0=60 for the silicon-gold system [3] and 200 K below the Si epitaxy temperature)is diffusion- using the above values of az and as, we obtain from limited, and the axial growth rate, 1-2 um/s, is notably Eq(10)as=0.750 J/m2. For constant-diameter whis higher than the rate of epitaxial growth ker growth(6=0), we find p≡56.5° from Eq(1)and The mechanical equilibrium of a liquid droplet, rep- a=0.703 J/m- from Eq (5). Since a< asu, the forma- resented by Eq. (1), is due to surface tension forces and tion of a new solid-vapor interface is energetically results in a circular cross section of the crystal, which more favorable than the formation of a solid-liquid reflects the shape of the three-phase line of contact and, interface. Finally, from Eq. (12)we obtain the driving INORGANIC MATERIALS Vol. 39 No 9 2003902 INORGANIC MATERIALS Vol. 39 No. 9 2003 NEBOL’SIN, SHCHETININ removing impurities from the surface layer) may lead to the fulfillment of the condition αS > αLcosϕ + αSL. (9) The total Gibbs energy increment is then positive, and the reverse process is thermodynamically favored, i.e., silicon vaporization (etching) through the liquid drop￾let. In other words, if αL is low compared to αS , a single (solid–liquid) interface is energetically more favorable than two (solid–vapor and liquid–vapor) interfaces. Indeed, this process may take place and leads to the formation of either negative whiskers by the solid–liq￾uid–vapor (SLV) mechanism [2] or cylindrical through holes in the silicon substrate, as in the presence of a borax film, which eliminates adsorbed particles and oxides from the silicon surface. The formation of a thin SiO2 film on the substrate or lateral whisker surface when the reactant gases are insufficiently pure increases the surface energy αS and leads to the fulfillment of condition (9). Whisker growth may then become thermodynamically unfavor￾able, and the process is unstable or does not occur at all. The energy gain responsible for the accelerated rate of whisker growth in comparison with epitaxial film growth in an analogous chemical process is obviously due to the reduction in the surface Gibbs energy of the liquid phase, αL, upon the displacement of the three￾phase line during whisker growth. For this reason, whisker growth at relatively low temperatures (150– 200 K below the Si epitaxy temperature) is diffusion￾limited, and the axial growth rate, 1–2 µm/s, is notably higher than the rate of epitaxial growth. The mechanical equilibrium of a liquid droplet, rep￾resented by Eq. (1), is due to surface tension forces and results in a circular cross section of the crystal, which reflects the shape of the three-phase line of contact and, accordingly, the cylindrical shape of the growing whis￾ker. In the course of whisker growth, the three-phase system acts as a natural shaper, determining the diame￾ter and circular cross section of the crystal. Since there is no contact between the crystallizing material and shaper walls, in contrast to many other growth tech￾niques [4, 11], the resulting whiskers have a more per￾fect structure. The rise of the liquid droplet during whisker growth (Figs. 1a, 1b) is due to the contact angle hysteresis at the three-phase line. If the contact angle of the lateral whisker surface (ϕ) is smaller than its equilibrium con￾tact angle (θ) (Fig. 1c), satisfying the Young equation αLcosθ + αSL = αS, (10) a driving force (Fg) appears, applied to the three-phase line of contact. The magnitude of this force can be found by jointly solving Eq. (10) and inequality (7): Fg = αL(cosϕ – cosθcosδ) – αSL(cosδ – 1) > 0. (11) At δ = 0, inequality (11) takes the form Fg = αL(cosϕ – cosθ) > 0. (12) Condition (12) is fulfilled for ϕ < θ. For this reason, the advance of the droplet along the lateral whisker surface can be thought of as the reverse of liquid spreading over a solid surface, i.e., droplet contraction. Taking θ = 60° for the silicon–gold system [3] and using the above values of αL and αS , we obtain from Eq. (10) αSL = 0.750 J/m2 . For constant-diameter whis￾ker growth (δ = 0), we find ϕ ≅ 56.5° from Eq. (1) and α = 0.703 J/m2 from Eq. (5). Since α < αSL, the forma￾tion of a new solid–vapor interface is energetically more favorable than the formation of a solid–liquid interface. Finally, from Eq. (12) we obtain the driving Surface Gibbs energy αL, contact angle ϕ + 90° on {111} faces, αS/αLratio, and stability of whisker growth in metal–silicon systems (αS = 1.200 J/m2 [2]) Metal T, K [8] αL, J/m2 [8] αS/αL ϕ + 90, deg [10] Stability of whisker growth Cu 1400 1.300 0.97 135 High Au 1400 0.910 1.37 110 High Ag 1300 0.900 1.33 – Intermediate Sn 1300 0.575 2.8 – No whisker growth Pb 1300 0.400 3.15 – No whisker growth Pt 2100 1.740 0.72 120 High Zn – – – Low Al 1300 0.820 1.46 – Low Sb 900 0.360 3.5 – No whisker growth Ni 1850 1.750 0.72 120 High Bi 800 0.355 3.38 – No whisker growth