150 N. Zakharov et al Physica E 37(2007)148-152 ▲Ts=525C T=555°C 120180 Diameter in n 0.6 0.4 0.0080010001 I/R I/nm Fig. 2.(a)Size distribution of grown whiskers. re, R the smallest and largest cut off radius of whiskers (b)Whisker's length vs. growing time showing that dL/dr= 2 nm/min [13].(c)Whisker's length vs. 1/R(d)LLs ratio vs. flux I, showing that it is higher at a low flux Fig 3. (a) High-resolution TEM cross-section image of the surface area. A-distorted amorphous area is formed in an overgrown Si layer due to Si/Au eutectic intrusion from droplet E. T-twins, M-matrix, * -partial dislocation. (b) TEM cross-section image showing the very earlier stage of whisker growth Structural defects(dislocations)in an overgrown Si layer serving for stress relaxation are indicated by arrows. It should be emphasized that dislocations are absent in the vicinity of the whisker The whisker's base is always located in pits formed during the growth process. This indicates that some surrounding silicon material is definitely consumed by the whisker(see Fig. 1). Si atoms diffuse upwards to the Au droplet and are incorporated into the (1 I 1) Si /Au droplet interface. This growth process implies the presence of an ad-atom super- nm si substrate saturation Au>0 due to a gradient of the chemical potential. The balance of material can be described as follows Fig. 4. High-resolution cross-section TEM image of a surface overgrown layer. Small amorphous droplets of Si/Au eutectic are indicated by arrows. R2 dL kdr sdr 30 directly from the Si source and provides the component of with D the coefficient of Si surface diffusion, R the wh radius, u the chemical potential of the Si atoms, s vertical elongation equal to the thickness of the overgrown the square of ad-atoms diffusion, a the lattice parame Si layer Ls. The whole vertical elongation of the whisker The result is mounts to l+Ls. The visible whisker length l is full determined by flux 12, which collects the adsorbed Si ad 2Da du toms from a region with a radius Rs around the whisker. ktRdxdirectly from the Si source and provides the component of vertical elongation equal to the thickness of the overgrown Si layer Ls. The whole vertical elongation of the whisker amounts to L+Ls. The visible whisker length L is fully determined by flux I2, which collects the adsorbed Si ad￾atoms from a region with a radius Rs around the whisker. The whisker’s base is always located in pits formed during the growth process. This indicates that some surrounding silicon material is definitely consumed by the whisker (see Fig. 1). Si atoms diffuse upwards to the Au droplet and are incorporated into the (1 1 1) Si/Au droplet interface. This growth process implies the presence of an ad-atom super￾saturation Dm40 due to a gradient of the chemical potential. The balance of material can be described as follows: pR2 dL ¼ D kT dm dx S dt, (1) with D the coefficient of Si surface diffusion, R the whisker radius, m the chemical potential of the Si atoms, S ¼ 2pRa the square of ad-atom’s diffusion, a the lattice parameter. The result is dL ¼ 2Da kTR dm dx dt. (2) ARTICLE IN PRESS Fig. 2. (a) Size distribution of grown whiskers. rc, Rc the smallest and largest cut off radius of whiskers. (b) Whisker’s length vs. growing time showing that dL/dt ¼ 2 nm/min [13]. (c) Whisker’s length vs. 1/R. (d) L/Ls ratio vs. flux I1, showing that it is higher at a low flux. Fig. 3. (a) High-resolution TEM cross-section image of the surface area. A-distorted amorphous area is formed in an overgrown Si layer due to Si/Au eutectic intrusion from droplet E. T-twins, M-matrix, *-partial dislocation. (b) TEM cross-section image showing the very earlier stage of whisker growth. Structural defects (dislocations) in an overgrown Si layer serving for stress relaxation are indicated by arrows. It should be emphasized that dislocations are absent in the vicinity of the whisker. Fig. 4. High-resolution cross-section TEM image of a surface overgrown layer. Small amorphous droplets of Si/Au eutectic are indicated by arrows. 150 N. Zakharov et al. / Physica E 37 (2007) 148–152