《复合材料 Composites》课程教学资源（学习资料）第二章 增强体_SiC whsker10 Role of Surface Energy in the Vapor–Liquid–Solid Growth of Silicon

Original Ruasen l.l Copyrigh: 9 2003 i ebo/sin, Shrchetinited / rom Neorgonicheskie Maternal). loz. 32, No. 9,2003. pp. Role of surface energy in the vapor-Liquid-Solid growth of silicon v. A Nebolsin and A.A. shchetinin Voronezh State Technical University, Moskovskii pr: 14, Voronezh, 394026 Russia Received December 3. 2002 Abstract-The conditions of vapor-phase Si whisker growth are examined, and the role of the surface Gibbs nergy in the vapor-liquid-solid proc s evaluated The mechanism responsible for the catalytic activity of the driving force acting on the three-phase line orcoted. Experimental surface tension data are used to estimate contact upon a displacement of the liquid droplet in the course of whisker growth INTRODUCTION ture. no nutrient was fed to the reaction zone. so that The unique properties of silicon whiskers are due to whisker growth was stopped. The process was restarted the vapor-liquid-solid (VLS) growth mechanism-the only after the temperature was fully stabilized at a new only mechanism involving more than two phases in level equilibrium with one another According to the existing concepts [2], a key feature RESULTS AND DISCUSSION of the VLS mechanism is the catalytic activity of the liquid phase, that is, the large accommodation coeffi When the substrate is heated to 1300 K and the cient of atoms adsorbed on the melt surface and the nutrient gas mixture is fed to the reactor, the solid-liq- reduced activation energy for nucleation at the crystal- uid interface beneath the Si-metal melt droplet melt interface. Unfortunately, the processes underlying becomes the growth front, the volume of the etch pit this "physical catalysis" are not yet fully understood. decreases, the liquid droplet rises above the substrate The mechanisms of Si whisker growth and sha surface, its shape changes, and the wetting perimeter were considered in [3, 4] from the viewpoint of the decreases. These processes are associated with the for- thermodynamic equilibrium of a liquid droplet on the mation of interfacial regions parallel to the ill) sub- tip of the growing whisker, which can be characterized strate surface and roof-shaped protrusions at the wet whisker radius during unsteady-state growth, the for- ing crystal. The rise of the droplet over the substrate mation of a whisker pedestal, and the stability of cylin- surface is accompanied by a reduction in the area of the drical crystal growth. At the same time, a number of liquid-solid interface. As a result, the diameter of the effects could not be understood in the framework of this growing crystal decreases, and the crystal takes a coni- approach. A question of major importance is the origin cal shape. The resulting monohedral solidification front of unstable whisker growth The objective of this work was to gain detailed ng polished sections of whiskers and is also evidenced insight into the role of interfacial energy in the filamen by the sharp boundaries of impurity bands resulting from tary growth of silicon in VLS systems special vapor-phase doping of whiskers [6] In addition to the flatness of the (111 growth front, EXPERIMENTAL which points to the layer-by-layer growth mechanism the following features of experimental data warrant Silicon whiskers were grown on ( 111) single-crys- attention tal substrates in a standard chloride-hydrogen system as described in [2], using gold, copper, and other metal 1. During cylindrical crystal growth by the VLs articles as growth leaders. To examine the morphol- mechanism, Si whiskers typically have a circular cross ogy of etch pits and determine the position of the solid- section owing to the circular wetting perimeter. Lateral liquid interface at different stages of whisker growth, mechanism [4] faceting develops in later stages, by the vapor-solid we used polished axial sections of the whiskers and substrates. The axial growth rate was determined by the 2. It follows from the morphology of crystals having time marker" method [2]. During changes in tempera- a circular cross section that the liquid droplet rests on 0020-1685/03/3909-0899$2500C 2003 MAIK"Nauka/Interperiodica

0020-1685/03/3909- $25.00 © 2003 0899 MAIK “Nauka/Interperiodica” Inorganic Materials, Vol. 39, No. 9, 2003, pp. 899–903. Translated from Neorganicheskie Materialy, Vol. 39, No. 9, 2003, pp. 1050–1055. Original Russian Text Copyright © 2003 by Nebol’sin, Shchetinin. INTRODUCTION The unique properties of silicon whiskers are due to the vapor–liquid–solid (VLS) growth mechanism—the only mechanism involving more than two phases in equilibrium with one another [1]. According to the existing concepts [2], a key feature of the VLS mechanism is the catalytic activity of the liquid phase, that is, the large accommodation coeffi- cient of atoms adsorbed on the melt surface and the reduced activation energy for nucleation at the crystal– melt interface. Unfortunately, the processes underlying this “physical catalysis” are not yet fully understood. The mechanisms of Si whisker growth and shaping were considered in [3, 4] from the viewpoint of the thermodynamic equilibrium of a liquid droplet on the tip of the growing whisker, which can be characterized by the growth angle. This approach made it possible to account for many effects, such as the variation in the whisker radius during unsteady-state growth, the for￾mation of a whisker pedestal, and the stability of cylin￾drical crystal growth. At the same time, a number of effects could not be understood in the framework of this approach. A question of major importance is the origin of unstable whisker growth. The objective of this work was to gain detailed insight into the role of interfacial energy in the filamen￾tary growth of silicon in VLS systems. EXPERIMENTAL Silicon whiskers were grown on {111} single-crys￾tal substrates in a standard chloride–hydrogen system as described in [2], using gold, copper, and other metal particles as growth leaders. To examine the morphol￾ogy of etch pits and determine the position of the solid– liquid interface at different stages of whisker growth, we used polished axial sections of the whiskers and substrates. The axial growth rate was determined by the “time marker” method [2]. During changes in tempera￾ture, no nutrient was fed to the reaction zone, so that whisker growth was stopped. The process was restarted only after the temperature was fully stabilized at a new level. RESULTS AND DISCUSSION When the substrate is heated to 1300 K and the nutrient gas mixture is fed to the reactor, the solid–liq￾uid interface beneath the Si–metal melt droplet becomes the growth front, the volume of the etch pit decreases, the liquid droplet rises above the substrate surface, its shape changes, and the wetting perimeter decreases. These processes are associated with the for￾mation of interfacial regions parallel to the {111} sub￾strate surface and roof-shaped protrusions at the wet￾ting perimeter, which rapidly merge into a continuous flat front [5]. This leads to melt entrapment in the grow￾ing crystal. The rise of the droplet over the substrate surface is accompanied by a reduction in the area of the liquid–solid interface. As a result, the diameter of the growing crystal decreases, and the crystal takes a coni￾cal shape. The resulting monohedral solidification front is very flat, which can be established by directly examin￾ing polished sections of whiskers and is also evidenced by the sharp boundaries of impurity bands resulting from special vapor-phase doping of whiskers [6]. In addition to the flatness of the {111} growth front, which points to the layer-by-layer growth mechanism, the following features of experimental data warrant attention: 1. During cylindrical crystal growth by the VLS mechanism, Si whiskers typically have a circular cross section owing to the circular wetting perimeter. Lateral faceting develops in later stages, by the vapor–solid mechanism [4]. 2. It follows from the morphology of crystals having a circular cross section that the liquid droplet rests on Role of Surface Energy in the Vapor–Liquid–Solid Growth of Silicon V. A. Nebol’sin and A. A. Shchetinin Voronezh State Technical University, Moskovskii pr. 14, Voronezh, 394026 Russia Received December 3, 2002 Abstract—The conditions of vapor-phase Si whisker growth are examined, and the role of the surface Gibbs energy in the vapor–liquid–solid process is evaluated. The mechanism responsible for the catalytic activity of the liquid phase on the tip of Si whiskers is elucidated. Experimental surface tension data are used to estimate the driving force acting on the three-phase line of contact upon a displacement of the liquid droplet in the course of whisker growth

NEBOL SIN. SHCHETININ (a) Consider a liquid droplet on the tip of a whisker. Let the three-phase line of contact adjoin the fat growth front(Fig. la). In this configuration, the interfacial ten- sions a,(liquid-vapor), as (solid-vapor), and asL (solid-liquid)at point A in the three-phase line cannot be in balance because asL aL as and asL is normal a to os. Since the liquid wets only the 111) growth front and does not wet the lateral crystal surface, the droplet is in mechanical equilibrium, which can be repre sented as asL-aisin=0, where p is the angle between the liquid-vapor interface and the direction of the displacement of the three-phase Fig. 1. Geometry of whisker growth: (a)constant radius line(growth angle [7]) (8=0),(b)decreasing radius(8>0),(c)three-phase equi- Equation (1)stems from the condition that the vapor-liquid-crystal system is at equilibrium, which means that, at any variations in the droplet shape(with- the (111) growth front, without touching the lateral out volume changes), the Gibbs energy G of the system of the wh remains constant(the minimum possible G at equilib 3. Independent of the growth direction, the flat rium, dG=0) [8l rowth front of silicon whiskers is always parallel to one of the (111) singular faces, corresponding to the G=lasLCOS-+ 2a1(1 sin )TR=min, (2) minimum surface energy where the first term in square brackets represents the 64. The formation of roof-shaped protrusions always Gibbs energy of the liquid-solid interface, the second gins at the three-phase line of contact. term represents the Gibbs energy of the liquid-vapor 5. Kinetic studies [3] give grounds to believe that, at interface, and R is the droplet radius Equating the derivative of Eq. (2)to zero, we obtain ited process, which is in conflict with the layer-by-layer Eq (D),which describes the equilibrium droplet shape, growth of the atomically smooth top face corresponding to the minimum surface Gibbs energy, 6. The observed axial growth rate of whiskers -2 um/s, is almost two orders of magnitude higher asL than the growth rate of films in an analogous epitaxial (3) process[2] 7. The use of gold, nickel, platinum, and iron as where r is the whisker radius oriented)whisker growth, while the process with the surface Gibbs energy of the dro sponds to the minimum growth leaders ensures steady-state(one-dimensional In other words, angle ( p cor participation of silver, zinc, and aluminum is unstable In the absence of external perturbations, the equilib- branches and bends In the presence of tin, bismuth, unchanged in the course of whisker growth, as does the antimony, and some other elements, no whisker growth cross section of the growing crystal. For thermody- occurs namic equilibrium to be reached, the shape ofthe (111 8. Under steady-state conditions, the crystal diame- growth front must be changed so as to ensure the bal ter often varies, leading to the formation of a conical ance of forces whisker pedestal In the general case. when the lateral surface of the 9. Owing to the oxidation of the Si substrate and the whisker makes angle d with the growth axis( conical lateral surface of the whisker(in insufficiently pure whisker)(Fig. 1b), Eq (I)can be rewritten in the form hydrogen containing oxygen compounds and water vapor), either the droplet does not rise and breaks into asind+a,sin=asL (4) smaller droplets or whisker growth is unstable Let solidification occur through tangential displace 10. Na2B4O7 applied to the surface of growth leaders ment of monatomic steps of height h. Whisker growth siding on an Si substrate leads, at high temperatures thermodynamically plausible if the total gibbs in the presence of SiCl4, to the formation of through energy increment upon a displacement of the droplet by holes in the silicon substrate, which have a diameter distance h is negative and the three-phase line(point A) typical of whiskers: 1-50 um shifts in the direction corresponding to the minimum INORGANIC MATERIALS Vol 39 No 9 2003

900 INORGANIC MATERIALS Vol. 39 No. 9 2003 NEBOL’SIN, SHCHETININ the {111} growth front, without touching the lateral surface of the whisker. 3. Independent of the growth direction, the flat growth front of silicon whiskers is always parallel to one of the {111} singular faces, corresponding to the minimum surface energy. 4. The formation of roof-shaped protrusions always begins at the three-phase line of contact. 5. Kinetic studies [3] give grounds to believe that, at low temperatures, Si whisker growth is a diffusion-lim￾ited process, which is in conflict with the layer-by-layer growth of the atomically smooth top face. 6. The observed axial growth rate of whiskers, 1−2 µm/s, is almost two orders of magnitude higher than the growth rate of films in an analogous epitaxial process [2]. 7. The use of gold, nickel, platinum, and iron as growth leaders ensures steady-state (one-dimensional, oriented) whisker growth, while the process with the participation of silver, zinc, and aluminum is unstable: the liquid droplet breaks down, and the whisker branches and bends. In the presence of tin, bismuth, antimony, and some other elements, no whisker growth occurs. 8. Under steady-state conditions, the crystal diame￾ter often varies, leading to the formation of a conical whisker pedestal. 9. Owing to the oxidation of the Si substrate and the lateral surface of the whisker (in insufficiently pure hydrogen containing oxygen compounds and water vapor), either the droplet does not rise and breaks into smaller droplets or whisker growth is unstable. 10. Na2B4O7 applied to the surface of growth leaders residing on an Si substrate leads, at high temperatures in the presence of SiCl4 , to the formation of through holes in the silicon substrate, which have a diameter typical of whiskers: 1–50 µm. Consider a liquid droplet on the tip of a whisker. Let the three-phase line of contact adjoin the flat growth front (Fig. 1a). In this configuration, the interfacial ten￾sions αL (liquid–vapor), αS (solid–vapor), and αSL (solid–liquid) at point A in the three-phase line cannot be in balance because αSL 0), (c) three-phase equi￾librium

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 2003

INORGANIC 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 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 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

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 2003

902 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

ROLE OF SURFACE ENERGY IN THE VAPOR-LIQUID-SOLID GROWTH OF SILICON 903 force acting on the droplet during Si whisker growth: 4. Tatarchenko, V.A., Ustoichivyi rost kristallov(Steady Fg=0.0467J/m State Crystal Growth), Moscow: Nauka, 1988 5. Shchetinin. A.A. Kozenkov o.D. and Nebol'sin. vA a Model for the Initial Stages of Silicon Whisker CONCLUSIONS Growth, lEv. vyssh. Uchebn. Zaved, Fiz., 1989, no. 1 pp.1l7-119 The main effects of inter facial energies on the fila- 6. Bataronov I L. Shchetinin, A.A. Dunaev,A L, and Kor- mentary growth of silicon in VLS systems are the low chagin, V.V., Axial Growth Rate of Silicon Whiskers in the reductio potential barrier to crystallization, owing to ring of the the Presence of Boron Tribromide, in Tonkie plenki i tion in the surface Gibbs energy of the liquid nitevidnye kristall(thin Films and Whiskers, Voron- phase and the equilibration of the liquid droplet on the ezh: Voronezh. Pedagogich Inst., 1993, pp 92-99 whisker tip. The droplet acts as a shaper, being responsi- 7. Voronkov, VV, On the Thermodynamic Equilibrium at ble for the circular cross section of the growing crystal a Three-Phase Line of Contact, Fiz. Tverd. Tela (lenin- grad),1963,vol.5,no.2,p.570-574 8. Naidich, Yu. V, Kontaktnmye yavleniya v metallicheskikh REFERENCES rasplavakh(Contact Phenomena in Metallic Melts), Kiev: Naukova Dumka, 1972 1. Maslov, V.N., yyrashchivanie profil'nmykh poluprovodni- kovykh monokristallov(Growth of Shaped Semiconduc- 9. Voronkov, V.V., Processes at the Solidification Interface, 2. Givargizov, E. 1, Rost nmitevidnykh i plastinchatykh hr- 10. Shchetinin, AA, Bubnov, L I, Tatarenkov, A.E., and Dolgachev, A A, Dinamika izmeneniya stallov i= para(Vapor Growth of Whiskers and Platelike kapel na monokristallicheskikh podlozhkakh(variation Crystals), Moscow: Nauka, 1977 in the Contact Angle of Droplets on Single-Crystal Sub- 3. Nebolsin, V.A., Shchetinin, A.A., and Natarova, E. strates), Available from VINITI, 1987, Moscow Variation in Silicon Whisker Radius during Unsteady no.3316V87 State Growth, Neorg. Mater, 1998, vol. 34, no. 2, 11. Stepanov, A.V., Growth of Shaped Crystals, lEv. Akad. pp 135-137 [Inorg. Mater:(Engl. Transl. ) vol. 34 Nauk SSSR, Ser: Fiz. 1969 vol. 33. no. 12 2,pp.87-89] pp.1946-1948 INORGANIC MATERIALS Vol 39 No 9 2003

INORGANIC MATERIALS Vol. 39 No. 9 2003 ROLE OF SURFACE ENERGY IN THE VAPOR–LIQUID–SOLID GROWTH OF SILICON 903 force acting on the droplet during Si whisker growth: Fg = 0.0467 J/m2. CONCLUSIONS The main effects of interfacial energies on the fila￾mentary growth of silicon in VLS systems are the low￾ering of the potential barrier to crystallization, owing to the reduction in the surface Gibbs energy of the liquid phase, and the equilibration of the liquid droplet on the whisker tip. The droplet acts as a shaper, being responsi￾ble for the circular cross section of the growing crystal. REFERENCES 1. Maslov, V.N., Vyrashchivanie profil’nykh poluprovodni￾kovykh monokristallov (Growth of Shaped Semiconduc￾tor Crystals), Moscow: Metallurgiya, 1977. 2. Givargizov, E.I., Rost nitevidnykh i plastinchatykh kri￾stallov iz para (Vapor Growth of Whiskers and Platelike Crystals), Moscow: Nauka, 1977. 3. Nebol’sin, V.A., Shchetinin, A.A., and Natarova, E.I., Variation in Silicon Whisker Radius during Unsteady￾State Growth, Neorg. Mater., 1998, vol. 34, no. 2, pp. 135–137 [Inorg. Mater. (Engl. Transl.), vol. 34, no. 2, pp. 87–89]. 4. Tatarchenko, V.A., Ustoichivyi rost kristallov (Steady￾State Crystal Growth), Moscow: Nauka, 1988. 5. Shchetinin, A.A., Kozenkov, O.D., and Nebol’sin, V.A., A Model for the Initial Stages of Silicon Whisker Growth, Izv. Vyssh. Uchebn. Zaved., Fiz., 1989, no. 1, pp. 117–119. 6. Bataronov, I.L., Shchetinin, A.A., Dunaev, A.I., and Kor￾chagin, V.V., Axial Growth Rate of Silicon Whiskers in the Presence of Boron Tribromide, in Tonkie plenki i nitevidnye kristally (Thin Films and Whiskers), Voron￾ezh: Voronezh. Pedagogich. Inst., 1993, pp. 92–99. 7. Voronkov, V.V., On the Thermodynamic Equilibrium at a Three-Phase Line of Contact, Fiz. Tverd. Tela (Lenin￾grad), 1963, vol. 5, no. 2, pp. 570–574. 8. Naidich, Yu.V., Kontaktnye yavleniya v metallicheskikh rasplavakh (Contact Phenomena in Metallic Melts), Kiev: Naukova Dumka, 1972. 9. Voronkov, V.V., Processes at the Solidification Interface, Kristallografiya, 1974, vol. 19, no. 6, pp. 922–929. 10. Shchetinin, A.A., Bubnov, L.I., Tatarenkov, A.F., and Dolgachev, A.A., Dinamika izmeneniya kraevykh uglov kapel’ na monokristallicheskikh podlozhkakh (Variation in the Contact Angle of Droplets on Single-Crystal Sub￾strates), Available from VINITI, 1987, Moscow, no. 3316-V87. 11. Stepanov, A.V., Growth of Shaped Crystals, Izv. Akad. Nauk SSSR, Ser. Fiz., 1969, vol. 33, no. 12, pp. 1946−1948

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