arov et al./ Physica E 37(2007)148-1 (b) Fig. 5. RHEED Pattern taken in situ during growth: (a)Ts= 150C reflections from crystalline Au are observed: (b) Ts=(360+20)C. The halo from liquid Si/Au eutectic layer appears Substrate Fig. 7. Schematic diagram of elastic stress relaxation on the top of Fig. 6. Schematic diagram of whisker growth. Ir-flux from Si source, Ir-Si hiker. i being the relaxation length. flux from substrate: L, Ls the length of whisker and thickness of vergrown Si layer, respectively; Rs the radius of pit. The effective difference between the chemical potentials of part of the whisker on the length i(see Fig. 7). Thus we can Si atoms in the overgrown layer and on the top of the write whisker can be written as Au=△ao with i being the relaxation length. Finally where△a=(Aox+△ay+△=)/3 being the stress created in the overgrown layer Ls due to gold intrusion; o the L= 2daAoo atomic volume: y being the surface energy of the cylindrical kT R surface formed mainly by (1 10) and (1 12) planes which Thus, the whisker growth rate dL/dt is constant. This is in are parallel to the growth direction [11 1]. Ao is a complex good agreement with the experimental results(see Fig 2b) function of gold concentration in overgrown layer Ls, The radius dependence of Ll/R is also agrees quite well whisker radius R and II. The first term is the gain in elastic with the experimental data(see Fig 2c). Of course, L also energy per atom due to the strain relaxation on the top of depends implicitly on the flux from the Si source I, through whisker, while the second term is the loss of energy per Ao, because Ao =0 at I1=0. When Ao= 2y/re, the growth atom due to the increase of the side surface of the whisker. stops(Gibbs-Tomson effect). In our case rc 35 nm; the This clearly shows that the supersaturation is determined surface energy of side surface of the whisker ?2000erg/ by both the surface energy and the elastic energy stored in cm2[l; It gives△a≈14×10°erg/cm32=14×10-3Ea, the overgrown Si layer due to Au intrusion. It is not an where Esi the Young modulus. Thus, the difference of the independent variable as the gas pressure in the case of Cvd lattice parameters in the substrate and on the top of growth. The stress relaxation occurs mainly in the upper the whisker should be Ae= Aep,=1.4x 10-3whichThe effective difference between the chemical potentials of Si atoms in the overgrown layer and on the top of the whisker can be written as Dm ¼ Dso 2go R , (3) where Ds ¼ (Dsxx+Dsyy+Dszz)/3 being the stress created in the overgrown layer Ls due to gold intrusion; o the atomic volume; g being the surface energy of the cylindrical surface formed mainly by {1 1 0} and {1 1 2} planes which are parallel to the growth direction [1 1 1]. Ds is a complex function of gold concentration in overgrown layer Ls, whisker radius R and I1. The first term is the gain in elastic energy per atom due to the strain relaxation on the top of whisker, while the second term is the loss of energy per atom due to the increase of the side surface of the whisker. This clearly shows that the supersaturation is determined by both the surface energy and the elastic energy stored in the overgrown Si layer due to Au intrusion. It is not an independent variable as the gas pressure in the case of CVD growth. The stress relaxation occurs mainly in the upper part of the whisker on the length l (see Fig. 7). Thus we can write dm dx  Dm l (4) with l being the relaxation length. Finally L ¼ 2Da kT Dso lR 1 2g DsR t. (5) Thus, the whisker growth rate dL/dt is constant. This is in good agreement with the experimental results (see Fig. 2b). The radius dependence of L1/R is also agrees quite well with the experimental data (see Fig. 2c). Of course, L also depends implicitly on the flux from the Si source I1 through Ds, because Ds ¼ 0 at I1 ¼ 0. When Ds ¼ 2g/rc, the growth stops (Gibbs–Tomson effect). In our case rcE35 nm; the surface energy of side surface of the whisker gE2000 erg/ cm2 [16]; it gives DsE1.4  109 erg/cm3 ¼ 1.4  103 Esi, where Esi the Young modulus. Thus, the difference of the lattice parameters in the substrate and on the top of the whisker should be Dezz ¼ Deyy ¼ 1.4  103 which ARTICLE IN PRESS Fig. 5. RHEED pattern taken in situ during growth: (a) Ts ¼ 150 1C reflections from crystalline Au are observed; (b) Ts ¼ (360720)1C. The halo from liquid Si/Au eutectic layer appears. Fig. 6. Schematic diagram of whisker growth. I1-flux from Si source, I2–Si flux from substrate; L, Ls the length of whisker and thickness of overgrown Si layer, respectively; Rs the radius of pit. Fig. 7. Schematic diagram of elastic stress relaxation on the top of whisker, l being the relaxation length. N. Zakharov et al. / Physica E 37 (2007) 148–152 151