Availableonlineatwww.sciencedirect.co ° ScienceDirect PHYSICAE ELSEVIER Physica E37(2007)148-152 www.elsevier.com/locate/phy Growth of Si whiskers by MBE: Mechanism and peculiarities n. Zakharov.p. Werner. L. Sokolov u. gisele Max Planck Institute of Microstructure Physics, Weinberg 2. D-06120 Halle, Germany Available online 7 September 2006 We analyzed the stress-driven mechanism of MBE Si whisker growth. It is shown that the driving force for mBe whisker growth is determined by the relaxation of elastic energy stored in the overgrown layer L, due to gold intrusion. In this case the supersaturation is determined by the interplay between elastic stresses and surface energy. The latter is considerably decreased due to decoration of the Si surface by gold resulting in formation of thin liquid Si/Au eutectic layer. This suggests that in our case the Si supersaturation is not an ndependent growth parameter as it is in the chemical vapor deposition growth method. Instead it is determined by stress in the ergrown Si layer. This approach allows us to explain quite well the growth kinetic and the relationship between the radius and the length of the whiskers. The whisker growth in our case can be considered as a stress relaxation mechanism, where the stress relaxation occurs due to transition from the two-dimensional system to the three-dimensional one. C 2006 Elsevier B.v. All rights reserved PACS:6146.-w;81.07.Vb;8l.15.Hi Keywords: Semiconductor nanostructures: Silicon nano wires: Molecular beam epitaxy 1. Introduction tion of the vapor-phase on the liquid surface of the Si/M eutectic, where M is catalytic metal. The necessary super The renaissance of interest in whisker growth is saturation of Si ad-atoms is determined by the vapor supported by the possibility to employ one-dimensional pressure in the reaction chamber and can easily be varied in quantum confinement in electronic and photonic devices wide region. The Si whiskers grow from the supersaturated [1-4]. However, the first whiskers boom occurred more liquid solution of Si in Si /M eutectic. A variety of method than four decades ago. At that time, scientists found out such as metal organic chemical vapor deposition that the strength of crystalline materials is about three (MOCVD)[7-10], pulse laser deposition(PLD)[ll],gas- orders of magnitude lower than it should be according to source molecular epitaxy (GS-MBE)[12, 13] and molecular the theoretical prediction [5]. The main reason is the beam epitaxy (MBe)[14] are nowadays used to grow Si formation of structural defects during the plastic deforma- wires on a Si(l 1 1) substrate. Each of them has its tion. The strength of individual whiskers approaches the advantages and disadvantages. In this paper we would like heoretical limit because they can be grown practically to pay attention to the peculiarities of MBE Si whisker defect-free. Although all attempts failed to develop a bulk growth. Advantageous are the possibilities of well-con- composite material on their basis with a strength close to trolled growing conditions(high vacuum, a clean substrate the theoretical predictions, a plenty of experience in and a well-controlled environment in the growth chamber, whisker growth was gained at that time, and so-called the precise control of atomic fluxes which is extremely vacuum liquid solid (VLS) growth mechanism was devel- important for doping, temperature control, etc. ) The goal oped [6]. According to this mechanism growth of whiskers of this work is to cast light on the growth mechanism and occurs on metallic droplets because of a high accommoda- kinetic of whisker growth by MBE. To simplify the problem as much as possible we investigate the growth of *Corresponding author. Tel: +49 345 5582669 fax: +493455511223. elemental semiconductor Si whiskers on a Si substrate E-mail address: zakharov(@ mpi-halle de(N. Zakharov) using gold droplets as a catalyst 1386-9477/- see front matter c 2006 Elsevier B V. All rights reserved doi:10.1016/ physe.2006.07018
Physica E 37 (2007) 148–152 Growth of Si whiskers by MBE: Mechanism and peculiarities N. Zakharov, P. Werner, L. Sokolov, U. Go¨sele Max Planck Institute of Microstructure Physics, Weinberg 2, D-06120 Halle, Germany Available online 7 September 2006 Abstract We analyzed the stress-driven mechanism of MBE Si whisker growth. It is shown that the driving force for MBE whisker growth is determined by the relaxation of elastic energy stored in the overgrown layer Ls due to gold intrusion. In this case the supersaturation is determined by the interplay between elastic stresses and surface energy. The latter is considerably decreased due to decoration of the Si surface by gold resulting in formation of thin liquid Si/Au eutectic layer. This suggests that in our case the Si supersaturation is not an independent growth parameter as it is in the chemical vapor deposition growth method. Instead it is determined by stress in the overgrown Si layer. This approach allows us to explain quite well the growth kinetic and the relationship between the radius and the length of the whiskers. The whisker growth in our case can be considered as a stress relaxation mechanism, where the stress relaxation occurs due to transition from the two-dimensional system to the three-dimensional one. r 2006 Elsevier B.V. All rights reserved. PACS: 61.46.w; 81.07.Vb; 81.15.Hi Keywords: Semiconductor nanostructures; Silicon nano wires; Molecular beam epitaxy 1. Introduction The renaissance of interest in whisker growth is supported by the possibility to employ one-dimensional quantum confinement in electronic and photonic devices [1–4]. However, the first whiskers boom occurred more than four decades ago. At that time, scientists found out that the strength of crystalline materials is about three orders of magnitude lower than it should be according to the theoretical prediction [5]. The main reason is the formation of structural defects during the plastic deformation. The strength of individual whiskers approaches the theoretical limit because they can be grown practically defect-free. Although all attempts failed to develop a bulk composite material on their basis with a strength close to the theoretical predictions, a plenty of experience in whisker growth was gained at that time, and so-called vacuum liquid solid (VLS) growth mechanism was developed [6]. According to this mechanism growth of whiskers occurs on metallic droplets because of a high accommodation of the vapor-phase on the liquid surface of the Si/M eutectic, where M is catalytic metal. The necessary supersaturation of Si ad-atoms is determined by the vapor pressure in the reaction chamber and can easily be varied in wide region. The Si whiskers grow from the supersaturated liquid solution of Si in Si/M eutectic. A variety of methods such as metal organic chemical vapor deposition (MOCVD) [7–10], pulse laser deposition (PLD) [11], gassource molecular epitaxy (GS-MBE) [12,13] and molecular beam epitaxy (MBE) [14] are nowadays used to grow Si wires on a Si(1 1 1) substrate. Each of them has its advantages and disadvantages. In this paper we would like to pay attention to the peculiarities of MBE Si whisker growth. Advantageous are the possibilities of well-controlled growing conditions (high vacuum, a clean substrate and a well-controlled environment in the growth chamber, the precise control of atomic fluxes which is extremely important for doping, temperature control, etc.). The goal of this work is to cast light on the growth mechanism and kinetic of whisker growth by MBE. To simplify the problem as much as possible we investigate the growth of elemental semiconductor Si whiskers on a Si substrate using gold droplets as a catalyst. ARTICLE IN PRESS www.elsevier.com/locate/physe 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.07.018 Corresponding author. Tel.: +49 345 5582669; fax: +49 345 5511223. E-mail address: zakharov@mpi-halle.de (N. Zakharov).
N. Zakharov et al/ Physica E 37(2007)148-152 2. Experiment The base of the whiskers was always located in triangular pits formed during the growth process. This We used oriented 5" Si wafers cleaned by the clearly indicates that some surrounding silicon material conventional RCA (Radio Corporation of America) was definitely consumed by the growing whisker(see Fig procedure were used as substrates. Our MBE system la, b). During the growth Si atoms diffused upward to the includes three electron-beam guns for the evaporation of Au droplet and were then incorporated into the( 11) Au and Si as well as of Ge [14]. a thin Au film with a Si/Au droplet interface on the top This growth process ominal thickness of 2 nm was deposited on the substrate implies the presence of an ad-atom supersaturation at a substrate temperature Ts= 525C. During the nano Au kT>o due to a difference in the chemical potential wires (Nw) growth the constant Si flux II ranged in between the overgrown layer and the top of the whisker terval 0.013-0.108 nm/s. Two growth temperatures Normally grown whiskers are defect-free(see Fig. 5) Ts=525 and 545C were chosen. The vacuum in the however, they sometimes contain single inclined stacking chamber during the growth was 2 x 10 Pa. The samples fault coming from the interface between the substrate and were investigated by transmission electron microscopy the overgrown Si layer (TEM) and high-resolution scanning electron microscopy The intrusion of Au into the Si overgrown layer during (SEM). Reflection high energy electron diffraction whisker growth(see Fig. 3a)leads to an increase of the (RHEED)was used to monitor the evolution of the elastic energy and the formation of structural defects such surface structure during the growth process as twins, partial dislocations and small amorphous Si/Au eutectic particles inside the overgrown layer. The presence of an Au-enriched surface layer was also demonstrated by [15]. The very early stage of whisker growth is shown in 3. Results Fig. 3b, where the structural defects serving for elastic energy relaxation are indicated by arrows. It should be The gold deposition at T=525C resulted in the noted that the region underneath the growing whisker is formation of Au droplets on the Si surface. The diameters defect-free. This suggests that whisker growth results from of the droplets ranged from 20 to 400 nm. During the the relaxation of elastic stresses in overgrown layer subsequent Si deposition, NWs were formed on the Au In the cross-section TEM images at room temperature droplets(Fig. 1). Under these conditions their diameter d one can clearly see the presence of tiny amorphous droplets lated to the size of the droplet and ranged from 70 of Si/Au eutectic approximately 5nm in diameter(see to 200 nm(2re<d<2Rc)(see Fig. 2a). Smaller as well as Fig. 4). We assume that at a growth temperature of 525C larger droplets outside of this range did not initiate whisker they turn into a thin liquid layer on the Si surface. Thi growth at all. The length of the grown whiskers L seems to be the case in Fig. 5, where RHEED patterns was proportional to the growth time Lot and varied in taken at temperatures 150C (a), and 360C(b)are shown inverse proportion to the radius L1/R(Fig. 2b and c espectively) Even though the growth rate of the whiskers dL/dt 4. Discussion decreased with the Si flux I1, the ratio L/Ls increased as can be seen in Fig. 2d (Ls the thickness of overgrown Si layer The growth process of whiskers by MBE is shown ig. 6). This suggests that longer whiskers can be grown at schematically in Fig. 6. Two fluxes of the Si ad-atoms 11 low Si flux I1 and I2 can be distinguished. The uniform flux I, comes 100nm 400nn Fig 1. SEM (a)and TEM(b) images of Si whiskers. Each whisker is placed in the pit. They are defect-free. The surface of the whiskers is decorated by tiny Au droplets(see also Fig. 4)
2. Experiment We used /111S oriented 500 Si wafers cleaned by the conventional RCA (Radio Corporation of America) procedure were used as substrates. Our MBE system includes three electron-beam guns for the evaporation of Au and Si as well as of Ge [14]. A thin Au film with a nominal thickness of 2 nm was deposited on the substrate at a substrate temperature TS ¼ 525 1C. During the nano wires (NW) growth the constant Si flux I1 ranged in interval 0.013–0.108 nm/s. Two growth temperatures TS ¼ 525 and 545 1C were chosen. The vacuum in the chamber during the growth was 2 107 Pa. The samples were investigated by transmission electron microscopy (TEM) and high-resolution scanning electron microscopy (SEM). Reflection high energy electron diffraction (RHEED) was used to monitor the evolution of the surface structure during the growth process. 3. Results The gold deposition at T ¼ 525 1C resulted in the formation of Au droplets on the Si surface. The diameters of the droplets ranged from 20 to 400 nm. During the subsequent Si deposition, NWs were formed on the Au droplets (Fig. 1). Under these conditions their diameter d related to the size of the droplet and ranged from 70 to 200 nm (2rcodo2Rc) (see Fig. 2a). Smaller as well as larger droplets outside of this range did not initiate whisker growth at all. The length of the grown whiskers L was proportional to the growth time Lt and varied in inverse proportion to the radius L1/R (Fig. 2b and c respectively). Even though the growth rate of the whiskers dL/dt decreased with the Si flux I1, the ratio L/Ls increased as can be seen in Fig. 2d (Ls the thickness of overgrown Si layer, Fig. 6). This suggests that longer whiskers can be grown at low Si flux I1. The base of the whiskers was always located in triangular pits formed during the growth process. This clearly indicates that some surrounding silicon material was definitely consumed by the growing whisker (see Fig. 1a,b). During the growth Si atoms diffused upward to the Au droplet and were then incorporated into the (1 1 1) Si/Au droplet interface on the top. This growth process implies the presence of an ad-atom supersaturation Dm/kT40 due to a difference in the chemical potential between the overgrown layer and the top of the whisker. Normally grown whiskers are defect-free (see Fig. 5); however, they sometimes contain single inclined stacking fault coming from the interface between the substrate and the overgrown Si layer. The intrusion of Au into the Si overgrown layer during whisker growth (see Fig. 3a) leads to an increase of the elastic energy and the formation of structural defects such as twins, partial dislocations and small amorphous Si/Au eutectic particles inside the overgrown layer. The presence of an Au-enriched surface layer was also demonstrated by [15]. The very early stage of whisker growth is shown in Fig. 3b, where the structural defects serving for elastic energy relaxation are indicated by arrows. It should be noted that the region underneath the growing whisker is defect-free. This suggests that whisker growth results from the relaxation of elastic stresses in overgrown layer. In the cross-section TEM images at room temperature one can clearly see the presence of tiny amorphous droplets of Si/Au eutectic approximately 5 nm in diameter (see Fig. 4). We assume that at a growth temperature of 525 1C they turn into a thin liquid layer on the Si surface. This seems to be the case in Fig. 5, where RHEED patterns taken at temperatures 150 1C (a), and 360 1C (b) are shown. 4. Discussion The growth process of whiskers by MBE is shown schematically in Fig. 6. Two fluxes of the Si ad-atoms I1 and I2 can be distinguished. The uniform flux I1 comes ARTICLE IN PRESS Fig. 1. SEM (a) and TEM (b) images of Si whiskers. Each whisker is placed in the pit. They are defect-free. The surface of the whiskers is decorated by tiny Au droplets (see also Fig. 4). N. Zakharov et al. / Physica E 37 (2007) 148–152 149
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. ktRdx
directly 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 adatoms 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 supersaturation 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
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 10-3which
The 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
N. Zakharov et al Physica E 37(2007)148-152 corresponds to the supersaturation Aoo/kTa002 Such a The minimal radius of the whiskers is determined by low supersaturation will not be sufficient to support interplay between the gain in elastic and the loses in surface whisker growth by classical gas apitaxy. However, the energies rc=2y/Ag ( Gibbs-Tomson effect). The stress presence of the liquid Si/Au phase makes the growth relaxation efficiency decreases with the radius of the possible due to a low nucleation energy in the liquid phase. whisker [18]. This results in the reduction of supersatura Thus, the role of Au droplets in our case is the formation of tion for thicker whiskers with r>R and their growth i/Au liquid phase. The difference between (220) inter- interruption. This also explains the experimental fact that planar distances near the top of the whisker and inin the case of mbe thinner whiskers grow faster. The vergrown layer measured in this investigation in HRTEM whisker growth in our case can be considered as a stress images is AE= AEyya5 x 10. This should be sufficient relaxation mechanism similar to the Stranski-Krastanow to drive the whisker growth. A very similar mechanism of mechanism [19], where stress relaxation occurs by transi- n whisker growth on a stressed Sn surface of Cu-Sn tion from two-dimensional system to three-dimensional bimetallic film was experimentally observed by Tu [17]. In one specimen maintained at room temperature, the Sn and Cu yers were both in compression and tension. The whiskers grew only on compressed Sn film. The driving force of this Acknowledgments process was attributed to the interdiffusion and reaction that occurs in the cu-Sn film The authors would like to thank l. schubert and a The tiny amorphous particles, about 5nm in size, are Frommfeld for the MBE experiments, F. Syrowatka for present on the Si surface at room temperature as can be seen in Fig. 4. They are most probably Si/Au eutectic SEM analysis, S. Hopfe for TEM specimen preparation, phase. The RHEED in situ control of the surface structure and M. Werner for specific TEM analysis. The work was also partially supported by European project NODE (FP6 during the heating of Si wafer deposited by 2 nm Au points /015783)and Russian foundation of basic research(04-02 Above this transition temperature instead of sharp reflec- 16747) tions from the crystalline Si, the diffuse halo appears(s Fig. 5). This is a strong indication that at a growth temperature of 525C the whole specimen is covered by a References thin Si/Au liquid eutectic layer. Using Eq. (5)and remembering that Ls= l,t we can write [Y. Nakajiama, Y. Takahashi, S Horiguchi, K. Iwadate, H. Namatsu, K. Kurihara, M. Tabe, Appl. Phys. Lett. 65(1994)2833 L2Da△ 2 N. Usami, T. Mine, S. Fukatsu, Y. Shiraki, Appl. Phys. Lett. 64 (1994)1126 3]J. L. Liu, Y. Shi, F. Wang, Y. Lu, R. Zhang, P. Han, S.L. Gu,Y D Thus, Eq.(6)and Fig. 2d show that the elongation of Zheng, Appl. Phys. Lett. 68(1996)352. whisker L is controlled by the diffusion of Si ad-atoms and 4] C.M. Lieber, MRS Bull. 28(2003)128 stress in the overgrown layer L, [A.H. Cottrell, in: Dislocations and Plastic Flow in Crystals. Clarendon Press, Oxford, 1953, p. Il 5. Conclusions [6R.S. Wagner, w.C. Ellis, Appl. Phys. Lett. 4(1964)89 964)2907. w.C. Ellis, K. Jackson, S M. Arnold, J. Appl. Phys. 35 It has been shown that the driving force for MBE [8R.s.Wagner, W.C.Ellis, Trans. Metall. Soc. AIME 233(1965)1053 energy stored in an overgrown layer L, due to gold 0 R S Wagner. W.C. Ellis, Appl.Phys. Lett. 4(1964)89 intrusion. The supersaturation is determined by the inter- [12] J.L. Liu, S.J. Gai, G L. Jin, S.G. Tomas, K.L. Wang, J. Cryst play between elastic stresses and surface energy(Eq (3)) Growth200(1999)106. The last one could be considerably decreased due to the [3]Q Tang, X Liu, T.I. Kamins, G.S. Salomon, J.S. Harris, Appl. Phys ett.81(2002)2452 surface. This suggests that in our case the supersaturation [4 L, Schubert Werne N.D lakharov, g Gerth K oIb, L. Long. is not an independent growth parameter. This, however, is [15)S. Takeda, K Ueda, N Ozaki, Y Ohno, Appl. Phys. Lett. 82(2003) lot the case for the CVd growth technique where the supersaturation is determined by vapor pressure and can be []C. Messmer, J.C. Bilello, J Appl. Phys. 52(1981)4623 sily varied as an independent parameter. This approach (7KN. Tu, Acta MetalL. 21(1973)347 allows us to explain quite satisfactorily the growth kinetics (8 N.D. akharov, P: Werre G, Gert, L Schubert, L. Sokolov and the relationship between the radius and the length of [19 I.N. Stranski, L. Krastanow, Sitzungsberichte d. Akad. D. wi senschaften in Wien, Abt IIb, Band 146(1937)797
corresponds to the supersaturation Dso/kTE0.02. Such a low supersaturation will not be sufficient to support whisker growth by classical gas apitaxy. However, the presence of the liquid Si/Au phase makes the growth possible due to a low nucleation energy in the liquid phase. Thus, the role of Au droplets in our case is the formation of Si/Au liquid phase. The difference between {2 2 0} interplanar distances near the top of the whisker and in overgrown layer measured in this investigation in HRTEM images is Dezz ¼ DeyyE5 103 . This should be sufficient to drive the whisker growth. A very similar mechanism of Sn whisker growth on a stressed Sn surface of Cu–Sn bimetallic film was experimentally observed by Tu [17]. In specimen maintained at room temperature, the Sn and Cu layers were both in compression and tension. The whiskers grew only on compressed Sn film. The driving force of this process was attributed to the interdiffusion and reaction that occurs in the Cu–Sn film. The tiny amorphous particles, about 5 nm in size, are present on the Si surface at room temperature as can be seen in Fig. 4. They are most probably Si/Au eutectic phase. The RHEED in situ control of the surface structure during the heating of Si wafer deposited by 2 nm Au points to a phase transition which occurred at (360720)1C. Above this transition temperature instead of sharp reflections from the crystalline Si, the diffuse halo appears (see Fig. 5). This is a strong indication that at a growth temperature of 525 1C the whole specimen is covered by a thin Si/Au liquid eutectic layer. Using Eq. (5) and remembering that Ls ¼ I1t we can write L Ls ¼ 2Da kT Dso lRI 1 1 2g DsR . (6) Thus, Eq. (6) and Fig. 2d show that the elongation of whisker L is controlled by the diffusion of Si ad-atoms and stress in the overgrown layer Ls. 5. Conclusions It has been shown that the driving force for MBE whisker growth is supported by the relaxation of elastic energy stored in an overgrown layer Ls due to gold intrusion. The supersaturation is determined by the interplay between elastic stresses and surface energy (Eq. (3)). The last one could be considerably decreased due to the formation of a thin liquid Si/Au eutectic layer on the Si surface. This suggests that in our case the supersaturation is not an independent growth parameter. This, however, is not the case for the CVD growth technique where the supersaturation is determined by vapor pressure and can be easily varied as an independent parameter. This approach allows us to explain quite satisfactorily the growth kinetics and the relationship between the radius and the length of the whiskers. The minimal radius of the whiskers is determined by an interplay between the gain in elastic and the loses in surface energies rc ¼ 2g/Ds (Gibbs–Tomson effect). The stress relaxation efficiency decreases with the radius of the whisker [18]. This results in the reduction of supersaturation for thicker whiskers with R4Rc and their growth interruption. This also explains the experimental fact that in the case of MBE thinner whiskers grow faster. The whisker growth in our case can be considered as a stress relaxation mechanism similar to the Stranski–Krastanow mechanism [19], where stress relaxation occurs by transition from two-dimensional system to three-dimensional one. Acknowledgments The authors would like to thank L. Schubert and A. Frommfeld for the MBE experiments, F. Syrowatka for SEM analysis, S. Hopfe for TEM specimen preparation, and M. Werner for specific TEM analysis. The work was also partially supported by European project NODE (FP6 /015783) and Russian foundation of basic research (04-02- 16747). References [1] Y. Nakajiama, Y. Takahashi, S. Horiguchi, K. Iwadate, H. Namatsu, K. Kurihara, M. Tabe, Appl. Phys. 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