(a)
(b) 79/1
7.9/1
(c)
7.9/10
(d) 79/10
1.3
ss
C /C
o
1.4
1.2
1.1
1
1
1.2 P/Po 1.4
1
1.2 P/Po 1.4
1
R=LV Eq.
R= 25 nmLV Eq.
R= 10 nmLV Eq.
R=10 nm
R=25 nm
30/1
(f)
100/1
(g)
1
1.2 P/P0 1.4
R=75 nm
R=200 nm
R=500 nm
R=1500 nm
ML
(h)
30/10
100/10
ss
C /C
o
1.8 (e)
1.2 P/P0 1.4
1.4
1
1
o
1.4 P/P 1.8 1
1.4 P/Po 1.8 1
R= LV Eq.
R=25 nmLV Eq.
R=10 nmLV Eq.
R=10 nm
R=25 nm
1.4 P/P
o
1.8 1
o
1.4 P/P 1.8
R=50 nm
R=75 nm
R=100 nm
R=200 nm
ML
Fig. 6. (a)-(h) Variation of CSS/Co with P/Po for LL growth by IF nucleation using SiCl4
(top row) and SiH4 (bottom row). Pair of numbers at the top of every plot represent the
values of QR/QD used to calculate the value of CSS/Co in that plot. L-V Eq. represents
liquid-vapor equilibrium. Fig. 6(a) and (b) are also included in the main paper and have
been provided here again for ease of comparison.
(i) 7.9/1 (TPB)
(j) 79/1 (TPB)
(k)
(l) 79/10 (TPB)
7.9/10 (TPB)
ss
C /C
0
1.4
1
1
1.2 P/P0 1.4
o
1
1.2 P/P 1.4
1
1.2 P/Po 1.4
R=10 nm
R=25 nm
R=75 nm
R=LV Eq.
R=25 nmLV Eq.
R=10 nmLV Eq.
30/1 (TPB)
(n)
100/1 (TPB)
(o)
30/10 (TPB)
R=200 nm
R=500 nm
R=1500 nm
(p)
100/10 (TPB)
ss
C /C
o
1.8 (m)
1.2 P/Po 1.4
1
1.4
1
1
o
1.4 P/P 1.8 1
1.4 P/Po 1.8 1
1.4 P/Po 1.8 1
1.4 P/Po 1.8
R= LV Eq.
R=10 nm
R=75 nm
R=25 nmLV Eq.
R=25 nm
R= 100 nm
R=10 nmLV Eq.
R=50 nm
R=200 nm
Fig. 6. (i)-(p) Variation of CSS/Co with P/Po for LL growth by TPB nucleation using
SiCl4 (top row) and SiH4 (bottom row). Pair of numbers at the top of every plot represent
the values of QR/QD used to calculate the value of CSS/Co in that plot. L-V Eq. represents
liquid-vapor equilibrium.
30/1
100/1
(e)
(f)
30/10
100/10
(g)
(h)
ss
C /C
o
2
1.5
1
0
0
100
200
R (nm)
7.9/1
(i)
100
200 0
R (nm)
7.9/10
(k)
100
200
R (nm)
79/10
(l)
600
1200
R (nm)
600
1200
R (nm)
ss
C /C
0
1.4
100
200 0
R (nm)
79/1
(j)
1.2
1
0
600
1200
R (nm)
0
100
200
R (nm)
100/1
0
30/10
(n)
0
100/10
(o)
(p)
ss
C /C
0
2
0
600
1200
R (nm)
30/1
(m)
1.5
1
0
100
200
R (nm)
0
100
200
R (nm)
0
100
200
R (nm)
Fig. 7. Variation of CSS/Co with wire radius R. (a)-(d) In main body of manuscript. (e)-(h)
LL growth by SiH4 and interface nucleation and ML growth. (i)-(l) LL growth by TPB
nucleation using SiCl4 at 1000C. (m)-(p) LL growth by SiH4 at 480C. The CSS/Co
values were calculated from the equality representing the steady state, Eq. 30, by using a
Matlab code. In all rows, the solid symbols are for growth by the radius independent ML
mode. The solid lines give the transition radius calculated using Eq. 25. Pair of numbers
at the top of every figure give the combination of QR and QD used to calculate the data in
that figure. Symbols o, □, ◊, ∆, and are for P/Po = 1.1, 1.2, 1.3, 1.4 and 1.5 respectively
in the middle row and P/Po= 1.1, 1.3, 1.5, 1.7 and 1.9 in the top and bottom rows. The
solid symbols represent growth rate using the ML mode.
Growht Rate (nm/sec)
30
30/1
(e)
0.003
20
0.002
10
0.001
100/1
0.03
30/10
(g)
0.0008
100/10
(h)
0.02
0.0004
0
0
0.01
0
0
100
200
R (nm)
1.1 LL
1.6 LL
Growth Rate (nm/sec)
(f)
600
7.9/1 (TPB) (i)
0.8
100
200
R (nm)
0
0
1.7 LL
1.8 LL
1.9 LL
1.1 ML
79/1 (TPB)
(j)
0.6
400
200
0
0
100
200
R(nm)
1.6 ML
1.7 ML
7.9/10 (TPB)
(k)
100
200
0
0
600 1200
R (nm)
Growht Rate (nm/sec)
1.1 LL
1.2 LL
30/1 (TPB)
4
(m)
0.002
2
79/10 (TPB)
0.6
(l)
0.2
0.2
600 1200
R (nm)
1.8 ML
1.9 ML
0.4
0.4
0
0
100
200
R (nm)
0
0
1.3 LL
1.4 LL
0
0
600 1200
R (nm)
1.5 LL
1.1 ML
100/1 (TPB) (n) 0.002
600 1200
R (nm)
1.2 ML
1.3 ML
30/10 TPB
(o)
1.4 ML
1.5 ML
0.0008
100/10 (TPB) (p)
0.0006
0.001
0.001
0.0004
0.0002
0
0
100
200
R (nm)
0
0
1.1 LL
1.6 LL
100
200
R (nm)
1.7 LL
1.8 LL
0
0
1.9 LL
1.1 ML
100
200
R (nm)
1.6 ML
1.7 ML
0
0
100
200
R (nm)
1.8 ML
1.9 ML
Fig. 8. Variation of growth rate Radius R . (a)-(d) In main body of manuscript. (e)-(h) LL
growth by IF nucleation using SiH4. (i)-(l) LL growth by TPB nucleation using SiCl4.
(m)-(p) LL growth by TPB nucleation using SiH4. Pairs of numbers at the top of each
graph represents the combination of QR and QD used to calculate the growth rates. Solid
symbols represent the growth rates due to the radius independent ML mode.
S. 1: Calculated growth rates for TPB nucleation by SiCl4(middle row in Fig. 8
above) and IF and TPB nucleation using SiH4 (top and bottom rows in Fig. 8
above):
1. The trends in the calculated growth rates for LL mode by TPB nucleation for
SiCl4 are the same as that for LL mode by IF nucleation in the main body of the
text.
2. As in the case of growth using SiCl4, experimental data of growth from SiH4
displays a rising trend with R up to up to about 30 nm and then saturates to a
constant value.14 Typical growth rates are about 2-3 nm/sec. In good agreement
with these experimental trends, in three of the calculated plots 8(e), (g) and (h),
the same observation is made, that is an initial increase followed by a saturation
Figure 8(e) or decrease Figure 8(g) and (h). Such a decrease in growth rates at
large radii has been experimentally observed6 for growth by SiCl4, by some
groups. Reasons for the same have been discussed in the paper in the section on
“Wire growth rates.”
3. In one of the data sets, Figure 8(j) the growth rate has not saturated or reached its
maximum. While this might seem as an anomaly, it really is not. Even in the case
of SiCl4 the non-saturating behavior as mentioned is present but not easily visible
because of the scale used. This particular data set therefore brings out the fact that
the rise and saturation behavior observed is not a general feature of the set of
equations representing the steady state kinetics of LL growth, but occurs when the
right combination of activation parameters and supersaturation are present.
S.2: Reconciling the radius dependence of the growth mechanism and the
experimentally observed radius independent growth rate:
The experimental observation of a radius independent growth rate in spite of LL mode
growth can only be possible if and only if the nucleation rate (frequency per unit area) J,
eq 18, itself decreases with an increase in R, to compensate for the larger area available
for nucleation. As can be seen from eq 18 this is possible only if C and hence the
chemical potential of Si in the droplet decreases with an increase in R. Indeed as seen
from eqs 4, 5, 15, 16(a) and 18 we get, for the steady state condition at constant pressure,
K8
R
2

 (C / C o ) ln( C / C o )

1/ 2
  K9 

exp 
o 
ln(
C
/
C
)


(S1)
using equations representing IF nucleation. In deriving eq S1, the second term in eq 1 has
been neglected based on the justification given previously. A similar analysis can be
performed for TPB nucleation.
K8 and K9 are constants that can be evaluated from the respective equations and do not
have any radius or concentration dependence. For the range of C/Co involved for the AuSi system, 1 to 5, the change in the non-exponential term is negligible compared to that of
the exponential one. Hence, if the logarithmic and linear dependence on C/Co is neglected
(and incorporated into K8) with respect to the exponential dependence on the RHS of eq
S1 then we get
  K9 
K9
1
  2 or ln( C / C o ) 
exp 
o 
ln( R 2 / K 8 )
 ln( C / C )  R
(S2)
eq S2 shows that as the radius increases the concentration C and hence chemical
potential of Si in the droplet should decrease under these conditions as required by the
observed radius independence of the growth rate. Substituting eq S2 back in eq 18 (main
text), it can be easily verified that the nucleation rate varies as 1/R 2 and hence the growth
rate, eq 26a as required becomes independent of R. A similar exercise can be performed
for TPB nucleation. injection, the reduction in C with R, leads to the saturation and also a
dip of the growth rates at larger radii.
S.3: Dependence of growth rates on C/Co by the LL and ML modes.
100
60
8
6
40
LL, R= 10 nm, 1
LL, R= 10 nm, 10
LL, R= 200 nm, 1
LL, R= 200 nm, 10
ML, 1
ML, 10
(b)
4
20
0
1
10
Growth Rate (nm/sec)
Growth Rate (nm/sec)
80
(a)
LL, R=10 nm, 1
LL, R=100 nm, 1
LL, R=1000 nm, 1
LL, R=10 nm, 10
LL, R=100 nm, 10
LL, R=1000 nm, 10
ML, 1
ML, 10
2
1.1
1.2 C/Co 1.3
1.4
1.5
0
1
1.5
2 C/Co 2.5
Variation in growth rate with C/Co as would be expected from eqs 26 and 28 for LL
growth by IF nucleation and ML growth at (a) 1000C using SiCl4 and (b) 480C using
SiH4. Plots for TPB nucleation are included in the supporting material as Figure 5(c) and
(d). Points shown are NOT the results of calculations using the steady state equations but
rather meant to help differentiate between the various curves. The shaded boxes represent
the range of typical growth rates reported in the literature and hence the expected range
of C/Co values. Numbers next to the values of radius, R, represent QD/kT values used.
3
S. 4 List of Equations
 LS  kT ln(
C
2 Au / Si
)

  Si
o
R
C

 LS   VS  kT ln( P / P o ) 

(1)
2 Si

R
(2)

2 Au / Si  
VL  kT ln( P / P 0 )  kT ln( C / C 0 ) 

R


(3)
 dC 
 dC 
 dC   dC   dC 
 
     
 dt 
 dt 1or 3  dt  5  dt  6  dt  7
1or 2
(4)
Adroplet  K 1 P s 
 K1 P s 
K3
 dC 






s
s
 dt 
  2 1  K 2 P  droplet Vdroplet 1  K 2 P  droplet R
(5)
S
2R  K1P S 
K 4
 dC   K1P 


S 
2
 dt  1  K P S 
 3 
2
 diffusion Vdroplet 1  K 2 P  diffusion R
(6)
  a exp(
Qdesorption  Qsurfacediffusion
2kT
Eq
 dC   P  P


 
1/ 2
 dt  5  (2mkT )
)
 K3

 R
o

 2 Au / Si   K 3
 dC   CP



exp 

 
1/ 2 
RkT
 dt  5  (2mkT) 

 R
(7)
(8)
(9)
(SiCl4)Gas or Adsorbed  (Si)Gas or Adsorbed + 2(Cl2)gas or adsorbed
(10a)
(Si)Gas or Adsorbed  (Si)Droplet
(10b)
 Q R  VL 
 dC 
C [Cl ] 2

  K 5 exp  
kT
 dt  6


(11)
 VL 
 dC 
C

  K 6 exp  
 dt  6
 kT 
K 6 C o ( P / P o ) exp( 
KK
 dC 

  1 3
R
 dt  6
J IF
K K PS
2 Au / Si 
) 1 3
RkT
R
 C /Co

o
 P/P



2
(13)
2

2 Au / Si   S
 exp(
)  P
RkT


(14)
1 R 2 dh K 7 dh
 dC 
=

 
 dt  7 Vdroplet  dt R dt
(15)
 dh 
 ( JR 2 )a
 
 dt  LL  IF
(16a)
 dh 
 ( J 2R)a
 
 dt  LL TPB
(16b)
 dh 
2
   ( J .r )a
dt
  ML
(17)
Q  C

  N oC o exp(  D ) o
kT  C

J TPB
(12)
C o
Q  C

exp(  D ) o
kT  C
 a
1/ 2
  C 
 ln  o  
  C 
  C
 ln  o
 C

 

1/ 2


K9

exp  
o 
 ln( C / C ) 
(18a)


K9

exp  
o 
 ln( C / C ) 
(18b)
r  (v/2J )1 / 3
(19)


1
 exp( QD / kT )C o  ln( C / C o )
v  a 
 1  1 / 2 exp( / 2kT ) 
3
2RTransition
J 1
v
dh  K 1 P S

dt  1  K 2 P S
 K3

K 
 7
(20)
(21)
(22a)
dh K 1 K 3 P S

dt
K7
(22b)
 2622 exp( QR / RT ) 
K 3 P S


kT
dh 


dt
K7
 K P 
 
 C ss / C o
1

K 3    K1 K 3 

o
 1  K 2 P  droplet  
 P/P
C / C o 
K1 K 3 P  K1 K 3 
o 
P/P 
2



2
(22c)
2

2 Au / Si    K 7 dh
 exp(
)  P 
RkT    dt


(23)
2

K
2 Au / Si  
 exp
 P  7 (R 2 )aJ
RKT 


(24)
1
2
  C / C o  2 
K9
2 Au / Si   
C 2
IF
2 C 


K 1 K 3 P 1  
exp

K
K
R
(
)
ln(
) exp( 
) (25)

7
10

o
o 
o 


RKT  
C  C 
ln C / C o
  P / P  

SS
Q  C SS

 dh 
 aR 2  N oC o exp(  D ) o
 
kT  C
 dt  LL  IF

  C SS
 ln  o
 C
 

 


1/ 2


K9
 (26)
exp  
SS
o 
 ln( C / C ) 
2
2

   K 7
 C / C o  
K1 K 3 P 1  
exp
 .av 2/3 J 1 / 3
 
o  
RKT   
P/P  

 

S
dh
= ( / 4)1 / 3 .av 2/3 J 1 / 3
dt
A V
3
a

or R 
3
SC 
(Cos   3Cos  2)
(27)
(28)
(29)