Modeling of semiconductor nanowires
V.G. Dubrovskii
St. Petersburg Academic University &
Ioffe Physical Technical Institute RAS, St.-Petersburg, Russia
Plan:
• Introduction
• Growth modeling
• Crystal structure of III-V nanowires
• Strain induced by lattice mismatch
• Self-induced GaN nanowires
• Self-regulated pulsed nucleation in VLS nanowires
dubrovskii@mail.ioffe.ru
Repino, 14 July 2013, Lecture # 2
Books
Monograph
“Theory of formation of
epitaxial nanostructures”
By V.G. Dubrovskii
Moscow, Fizmatlit 2009
352 p.
New book (2013):
V.G. Dubrovskii
“Nucleation theory and growth
of nanostructures”
Springer
Selected papers on NWs
4
13
5
8
12
Papers on nucleation theory
Nucleation and growth:
3
Ostwald ripening:
Linear peptide chains:
Modern NWs and their applications
InAs, MOCVD,
nanoimprint (Lund U)
Nanoelectronics
GaAs, MBE, e-beam
(Ioffe & LPN CNRS)
InAs/InP, Lund U
Nanophotonics
NW
GaN/AlN,
Ioffe & LPN CNRS
Nanosensors
Modern nanowires and their importance
Nano Lett. 10, 1529 (2010)
Where is the
killer application?
Exponential increase in the number of
publications:
1) Nanowire based single cell endoscopy
Biological probe for endoscopy, spot delivery and
sensing within a single living cell
2) Nanowires for direct solar to fuel conversion
3) Integrated nanophotonics
1 – solar cells, 2 – LEDs/lasers, 3 – nanoribbons,
4 – photonic bandgap NW arrays, 5 – sample analysis
chambers, 7- photodetectors, - microfluidic systems
4) fundamental physics: growth and properties
Advantages of nanowire based optoelectronics
 Easy to fabricate uniform arrays by organizing seeds before growth
Smallest LEDs / lasers of any kind (10s nm in diameter, a few microns in
length)
 (Potentially) high efficiency (electronic active medium and
optical waveguide being identical: large confinement factor)
 Vertical cavity and surface emitting
 Easy to realize single photon emission
 Much less restricted by lattice mismatch => III-Vs on Si substrates,
coherent strained heterostructures in NWs
 Wurtzite phase of ZB III-Vs
(C. Chang-Hasnain group, UC Berkeley APL 2007)
Nanowire heterostructures
Au-assisted VLS growth: the first wires
Au-assisted CVD of Si “whiskers” on Si(111) at T~1000 0C (Wagner & Ellis, 1964)
Fundamental aspects of VLS growth: Givargizov, in “Highly anisotropic crystals”, 1975
Alloy at
equilibrium
with solid
Liquid
Vapor
Au catalyst
Si wires
Si is transferred from vapor to solid through liquid drop on the
wire top (Tm=363 0C)
Liquid drop acts as a chemical catalyst: pyrolysis rate > 0 at
the drop surface and = 0 at the substrate surface
 Simple phase diagram of Au-Si alloy: no Au in the wire?
Au-assisted VLS growth of III-V nanowires by
MBE
Kinetic processes driving nanowire growth

2
1 – direct impingement
2 – desorption from the drop
3 – diffusion from the sidewalls
4 – desorption from the sidewalls
5 – diffusion from the substrate
to the sidewalls,
6 – diffusion from the substrate to
the drop
7 – surface nucleation
VL
1

3
6
2R
L0

L
4
Wire
5
Island
7
Hs
Surface layer
Substrate
  V / Vdes 1  V / Vdes
eq
  C / Ceq 1
surf
1
Nucleation-mediated wire growth
resulting in the vertical
growth rate
Supersaturation of gaseous phase to the solid
(= to equilibrium alloy with concentration Ceq)
Supersaturation of (liquid) alloy in the drop to the solid
V.G.Dubrovskii et al., PRE 2004, PRB 2005, PRE 2006, PRB 2008, PRB 2009; PRB 2010, APL 2011
W.Seifert et al., JCG 272, 211 (2004), L.Schubert et al., APL 84, 4968 (2004)….
Model of diffusion-induced NW growth
• Stationary growth with R = const
J
β
α
• Direct impingement
• Adatom diffusion, substrate and sidewalls
• GT effect in the drop
γ
l
2R
Surface adatoms (s):
z
θf
Ds ns   s J cos 
L
λf
r
θs
λs
0
r 
dn
Ds s
dr
Df
 D f
rR
dz
z 0
0
d 2n f
dz
2
  f J sin  
nf
f
0
ω = 1 in MOCVD and 1/π in MBE
Constant concentration
far away from the wire
dn f
s
Sidewall adatoms (f):
Four boundary conditions:
dns
dr
ns
 s ns ( R)   f n f (0)
Continuity of flux
at the wire base
V.G.Dubrovskii et al., PRB 2005, 2009, PRE 2006


Continuity of chemical
potential at the wire base

k BT ln  f n f ( L)  l 
2l
R
Continuity of chemical
potential at the wire top
Growth kinetics
l  L/ f
Due to GT effect, coefficients
A, B and C can be of either signs !
h  H / f
Sidewall
adatoms
Surface
adatoms
dl
BU (l )  C

A

dh
U (l )
Direct impingement,
Surface growth
J
α
Sa
β
l (h  0)  l0
U (l )  sinh(l )  cosh(l )  1
B=0, C=0 (no diffusion): dl/dh=A, Classical Givargizov-Chernov case
Generally:
 dl 
CA
 
dh
  l 0
 dl 
 B A
 
 dh  l 
DI growth:
s / R  1
dU
 BU  C
dh
U (h  0)  U (l0 )
( f / R) tan  1
h(l ) 
1  BU (l )  C 
ln 

B  BU (l 0 )  C 
Theoretical L(t) curves
@ RGT=3.5 nm
1200
R=50 nm
1000
R=30 nm
Au-assisted MBE
of GaAs NWs
L [nm]
800
R=20 nm
600
L(t) curves are essentially
non-linear !!!
400
R=10 nm
200
0
1000
2000
3000
t [s]
4000
5000
6000
V.G. Dubrovskii et al., PRB 2009
Narrowing size distribution of <110> Ge NWs
2500
70 min
Length (nm)
2000
1500
50 min
1000
40 min
 s ~ 100 nm
30 min
25 min
15 min
500
0
60
80
100
120
140
160
180
Diameter (nm)
L  s
2gs
abgf
  2ag f H  
  1
exp
  R  
Initial stage: L  2 s V (t  t0 ) / R
0
Infinite growth:  s0  l  f  l
Limited growth: g s  0, g f  0
Dubrovskii et al., PRL 2012
Role of surface energies in NW polytypism
Hexagonal
cross-section:
 (2 1 1) 
8
 10.23  nm  2
2
0.7819  nm
 (1 1 00 ) 
6
 7.67  nm  2
2
0.7822  nm
 (1120 ) 
4
 8.86  nm  2
2
0.4516  nm
1-st approximation
for lateral
surface energy:
 wv
n

S
i
i
i
i
 (1 1 0 )
4

 8.86  nm  2
2
0.4514  nm
Surface energies: summary
Surface energy ratio WZ to ZB
Facet type
Surface energy,
J/m2
(2 1 1 )
1.73
(1 1 0)
1.50
(1 1 00)
1.30
(1120)
1.50
Transition
(2 1 1 )
(1 1 00)
(2 1 1 )
(1120)
(1 1 0)
(1 1 00)
(1 1 0)
(1120)
Dubrovskii et al., PRB 2008; Phys. Solid State 2010
   WZ /  ZB
0.75
0.867
0.867
1
Role of nucleation
F.Glas et al., Phys. Rev. Lett. 2007, V.G. Dubrovskii et al., PRB 2008, J. Johansson et al,
Cryst. Growth & design 2009 …
 At lower surface energy of NW sidewalls, WZ phase can form only when nucleation
takes place at the triple phase line (TL)
 In a mononuclear mode, the structure is dictated by the monolayer island orientation
Two conditions of WZ phase formation:
LV
 Condition for TL nucleation (straight sidewalls):
lSL

SL
WV
C nucleation
TL nucleation
(a)
(b)
l
 WV   LV sin    SL
G 22<G 1
*
G1
LV surface energy should not be too high!
*
G2
 High enough supersaturation to create a
stacking fault
G *j 
G
0
r
* *
r2 r1

G j2
 
  c 
Nucleation barriers
 WZ
1  (GWZ / GZB ) 2
Theoretical conclusions
 Surface energy of relevant WZ sidewalls is indeed lower than of ZB ones
 TL nucleation can be suppressed by a lower surface energy catalyst
 Structure retains to bulk ZB at large R because the ring of critical size
dissapears (Dubrovskii et al., PRB 2008)



3ρ/2
C nucleation:
ρ*max = 3ρ/2
Growth and phase diagrams:
TL nucleation:
ρ*max = ρ
CUB – blue curves
0.83 0.875 HEX – red curves
1.0
fmax
7
6
HEX
5
fCR
CUB
4
fminHEX,TL
Probabilities pCUB, pHEX
Normalized chemical potential f
8
0.8
τ=0.95
0.6
0.91
0.4
0.2
fminCUB,TL
3
0
0.25
0.5
0.75
1
Normalized wire radius 
1.25
1.5
0.0
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0
Liquid chemical potential f
GaAs, R=20 nm
Two step growth with temperature ramping
 1 nm Au layer deposited on GaAs(111)B surface
 Sample A grown at 6300C from the beginning => no NW growth
 Growth at 5300C for tLT; growth temperature ramped
from 530 to 6300C within 2 min, Ga and As4 fluxes maintained;
growth at 6300C, V=0.2 nm/s
 Sample B: tLT=1.5 min, NO NW growth
Sample C: tLT=15 min, NW GROW longer than 2 nm
V.G. Dubrovskii et al.,
PRB 2009
Riber 32 (LPN)
Complex NW shape:
Branching, tapering
Continuing growth
Sample A: tLT=0
Sample B: tLT=1.5 min
Sample C: tLT=15 min
Two step growth with temperature ramping
- 0.4 nm Au layer deposited on GaAs(111)B surface
- Samples 1 grown at 6300 C from the beginning => no NW growth
Growth at 5500 C with V=0.3 nm/s for tLT
- Growth temperature ramped from 530 to 630 0C,
- Growth at 6300 C for 48 min, V=0.15 nm/s, V/III=4.
-Sample 2: tLT =2 min, NO NW growth - Sample 3: tLT =12 min, NW GROW longer than 10 microns
EP1203 (Ioffe)
More regular shape
Sample 3 before and after high temperature growth step
Control of crystal structure: stacking fault free
GaAs NWs grown with two T steps via scenario IV
High resolution TEM studies of a NW detached from sample C:
200 nm
40 nm
Pure WZ
Pure ZB
Transition region
Optical properties of WZ and WZ/ZB GaAs NWs:
Pure WZ NWs
B.V. Novikov et al., PSS RRL 2010:
WZ/ZB heterostructures
D. Spikoska et al., PRB 2009:
Type II band structure:
Predominantly ZB NWs
Pure WZ NWs
Band alignment and the first
e and h levels v thickness
EZB-EWZ=41 meV, redshift opposite to InP
PL spectra: 1.51 to
1.43 eV shift for different
proportions of WZ
Control of crystal phase by growth catalyst: Ga-catalyzed
GaAs NWs
Dubrovskii et al., PRB 2010, Nano Letters 2011
TPL nucleation condition:
   WV   SL   LV sin   0
 WV  1.3 J/m2
 SL  0.59 J/m2
 LV  1.0 J/m2 for Au-Ga
(at 40% Ga percentage)

= -0.23 to -0.11 J/m2
for contact angles from 110 to 1250
A=drop
B=WZ
C=WZ-ZB
mix-up
D=ZB all the way
For pure liquid Ga at the
growth temperature:
 LV  0.67

= 0.08 to 0.16 J/m2
Strain relaxation and critical dimensions
in freestanding nanowires

a
a 1   0 
a
• Because of free lateral surfaces
strain relaxation is expected to be
much more efficient
than in 2D layers and even QDs
• Model
Elastic modulus
- linear isotropic elasticity
- same elastic parameters E, 
Poisson ratio
Barton J. Appl. Mech. (1941)

Strain maps
zz /0
E = 90 GPa, ε0 = 0.46, ν = 0.3
axis
outer surface
Heterostructured nanowires (QDs in NW)
Because of free lateral surfaces
strain relaxation much more efficient
than in 2D layers and even QDs !!!
InAs QDs in InP NWs:
Axial or radial heterostructures
Lund University
Relaxation of elastic stress in NS grown on a lattice
mismatched substrate: existing models
w( )  w2 D z( )
w2 D  (E 0 ) /(1  )
2
Major asymptotic properties:

z( ) 
 0 / 
0
z( ) 
1
Simple:
Elastic energy of 2D layer
(per atom)
z( ) 
1
1   / 0
Ratsch-Zangwill:
z( ) 
Aspect ratio:
  H /(2 R)
1  exp( /  0 )
 / 0
Glas:
z( ) 
p1
 (1  p1 ) exp( p3  )
1  p2 
Gill-Cocks:
1  6 2
z( ) 
1  2b (1  6 2 )  (16  10k ) 2
Results for elastic energy relaxation
z( )  W ( ) / W2 D
Elastic constants of a cubic material
Solid lines – calculations for different geometries
Relative strain energy for cylinders
Dashed lines - fits
A
Z ( )  1 /(1  A )
5.5 (cone), 8 (truncated cone 700), 15
(cylinder) and 50 (reverse cone 1100)
Critical thickness for plastic relaxation
Energy per of a dislocation pair (Glas, PRB 2007):
 _ 
E (1  v cos  )b  h 
Wd  4 R
ln  1
2

8 (1  v )
 b 


2
_
h  h if h  R
z
r
2
_
h  R if
h  R
  2/
h
b is the core cutoff parameter for elastic stress, θ is the angle
between the Burgers vector and dislocation line
2R
Elastic energy:
E
E R 2 h 02 a 2  a  1
2
We 
Z (  )V 0 
1 v
1  v 1  A
3
a = 1 for cylinder and 0 for cone
Critical thickness for plastic relaxation
The excess energy of dislocation pair with respect to a fully coherent state is:
_

2 2


2
  beff beff  a  a  1
Er0   h 
 h( R, h) 

W ( R, h) 
Z



C
ln
 1
Rh
0
2


1  v  2R 
R
3
b
4R







C  (1  v cos2  )b2 /[2 (1  v)]
Pure edge dislocations:
600 dislocations:
W  0
W ( R, h)  0
  4/
  /2
  /3
beff  b
beff  b / 2
Coherent state is stable
W  0
Dislocations
Critical thickness for dislocation formation
hc (R)
Critical thickness for plastic relaxation
 0  0.03 600 dislocations in cylinder geometry:
Critical thickness tends
to infinity at certain
critical radius which
depends on lattice
mismatch and NS
geometry!
4% - GaAs/Si, 8.1% - InP/Si, 11.6% - InAs/Si
Critical dimension for plastic deformation
Z (  )  1 / A
at    , therefore the equation for critical dimension is given by
2 2

2 (a 2  a  1)   beff
 R

 beff  0 Rc   C  ln c  1  0
 4

3A
b




Critical radius v mismatch for different
geometries:
Dots showing MOCVD
and MBE experimental data
III-V NWs on Si substrates: MOCVD
a – InAs with 20 nm Au on Si(111)
b – InP with 20 nm Au
c – InP with 60 nm Au
d – InP with 120 nm Au
e – TEM of 17 nm diameter InAs NW
WZ phase !!!
Critical diameter for the growth of epitaxial NWs
on the lattice mismatched substrates
(C. Chang-Hasnain group, APL 2007)
III-V NWs on Si substrates: MBE
Cirlin et al., PSS RRL 2010
Problems with VLS nanowires
• Unwanted Au contamination
• Uncontrolled zincblende-wurtzite polytypism
Use catalyst-free NN formation (GaAs on Si or sapphire)
Use self-catalyzed growth (Ga instead of Au
in the case of GaAs NWs)
Au distribution in Au-seeded Si NWs
(by P. Pareige, Rouen University,
France)
Au contamination of Si and Ge NWs grown by MBE
Nanoscale RL
Self-induced GaN NWs on Si: new growth mechanism
• No Ga drops are detected on top => not VLS mechanism
• GaN never nucleates as NW, nanoislands of different shapes are
formed in the beginning (different shapes on an amorphous SixN
interlayer or on mismatching AlN layer)
• Even on AlN, misfit dislocations are formed before NW formation; NWs
are relaxed from the very beginning
• MBE of self-induced GaN NWs employs specific growth conditions:
high N flux and high temperature are required
• Surface diffusion plays a crucial role in NW growth
• GaN NWs usually grow in both vertical and radial directions
• GaN NWs are hexahedral, restricted by 6 equivalent low energy m-planes
Self-induced GaN NWs on Si(111): radial growth !
Histograms showing
diameter distributions:
Growth mechanism:
Length-diameter dependence:
 No drops are seen on NW tops
 NWs growing in vertical and radial direction
Nucleation on lattice mismatched AlN layer
•
•
•
MBE on Si(111) substrates
5 nm thick AlN buffer layer
GaN growth at T=800 C, N/Ga fluxes ratio =10
HR TEM images:
RHEED patterns:
2 min, AlN buffer
10 min, GaN islands
17 min, GaN NWs
RHEED and HRTEM studies show misfit
dislocations in islands!
a – SC islands; b – truncated pyramids
c – full pyramids, d – NWs,
island to NW transition at ~ 13-14 nm radius
Role of misfit dislocations
Height v radius for different structures:
dislocation
GaN NWs are relaxed from the beginning!
Model suggesting a series of shape transformations
to relax elastic stress, NW is already relaxed:
Plastic relaxation in islands is also shown in:
O. Landre, C. Bougerol, H. Renevier, and
B. Daudin. Nanotechnology 20, 415602 (2009)
Nucleation on an amorphous interlayer
•
•
•
•
Si(111) substrate
5 min exposure to active N to form SixNy amorphous
layer
GaN growth at T=780 C, N/Ga fluxes ratio =6.2
Epitaxial constraint should be weak!
HR TEM:
RHEED patterns:
Incubation SC Transition
r0=5 nm
NWs
Scaling model for nucleation and growth of GaN NWs
J. Tersoff, R.M. Tromp, Phys. Rev. Lett. 70, 2782 (1993)
Assumptions:
•
•
•
•
No strain-induced contributions, directly applicable on an amorphous interlayer
Anisotropy of surface (and edge) energy as the dominant driving force
Growth anisotropy: superlinear length-radius dependence of GaN NWs !
Compare surface energy of isotropic island and anisotropic NW at given volume
Illustration of the model:
Surface energy of isotropic island:


GISL    kn n  ki ( i   s ) r02  k  ISL r0
 n

In SC geometry:
k 
n n
n
 2 SC /(1  cos ) ki  
k  2
Island volume: VISL  kV r03
In SC geometry:
kV  [f ( )] / 3
f ( )  [(1  cos )(2  cos )] /[(1  cos ) sin  ]
NW volume:
Surface energy of NW: GNW  6 SW rh 
3 3
( TOP   i   S )r 2  6 NW r
2
VNW  (3 3 / 2)r 2h
Scaling model for h(r)
h   r
Superlinear dependence of NW length on radius with >1 for all t
With this dependence,
from VISL  VNW
 3 3 
r0  
 2kV
1/ 3
 23
 r


Using this in previous equations, the
driving force for island to NW shape
transformation is obtained in the form
g (r )  (GNW  GISL ) /A r 2( 2 ) / 3 
 1
g (r )  b r
3
sidewalls
 c r
c  C / A
 3 3 
A  
 2kV




3 3
( TOP
2
2( 1)
3
 d r

( 2 1)
3
 e r

(  2)
3
1
edges
in-plane
b  B / A
C

e  E / A
d  D / A
2/3


  k n n  ki ( i   s )  B  6   SW
 n

1/ 3
 3 3  
 k  ISL
E  
  i   S ) D  6 NW
2k 

V

Results of statistical analysis of TEM and
SEM data remarkably follows the scaling
dependence at:
  2.46 and    0.088
General condition for anisotropic growth
g (r )  0
g (r )  0
NW anisotropic growth is energetically preferred
NW growth is suppressed
No edge contributions
where
g (r )  0 between r1,2  x13,/(2 1)
d  e  0
x1, 2 are positive roots of cubic equation b x3  x 2  c  0


b2c  4 / 27
0,50
b=0.34
0,25
Interesting NW case relates to
r2  1
g(r)
r1 ~ 1
b  1 c ~ 1
0,00
-0,25
NW sidewall energy should be much
smaller and in-plane energy compared
to surface energy of the island !
b=0.12
c=0.7
-0,50
0
10
20
  2.46    0.088
d  14
30
40
r (nm)
e  4.5
50
60
70
c  0.7
Parameters of GaN spherical caps and NWs
Boxy hexahedral islands with
constant aspect ratio h/r = 0.088
r0  5 nm from experimental data
0,50
r1  3.4 nm from growth law
0,25
c  0.7
r1
0,00
d  14 e  4.5
d  e In view of small prefactor
g(r)
b  0.14
r2
Edge terms
included
-0,25
g(r)
and larger contact angle of NWs
 TOP  130 meV/A2
 S  137 meV/A2
g1(r)
-0,50
known
Assume  SW  100 meV/A2
No edge terms
-0,75
0
10
20
30
40
50
60
70
NW radius r (nm)
 i  109meV/A2 (was 40 meV/A2 by analogy with Si/SiO2)
 SC  230 meV/A2 (was 130-176 meV/A2 from Young’s eq.)
  2.46 b  0.14 c  0.7
   0.088 d  14 e  4.5
Scaling in GaN NW growth: kinetic model
L  L0 ~  ~ 40nm
Elongation:

R 2 dL   f J sin 
 
 J top 2R  ( top J cos  J des  J surf )R 2
 dt 


Tip SW surface
Top facet
Radial growth:
Schematics of possible growth scenarios:
Yellow – NW surface contributing to elongation
Magenta – desorption area
Grey – NW surface contributing to radial growth
Blue – overgrown shells
a – no radial growth, R=const
b – R~t
c – tapered shape
d – cylindrical shape, SCALING!
Neglect c, adopt model d with

2RL dR   f J sin 
 
 J SW 2RL
 dt 


SW collection
1 dL a
 c
V dt R
1 dR
 B
V dt
L(t  t0 )  L0
R(t  t0 )  R0
  const/ L
Scaling in GaN NW growth: L(t), R(t)
1 dL a

V dt R
a  (2 f g f  tan ) / 
1 dR b

V dt L
b  (2 f gSW  tan ) / 
g f  1  (Jtop ) /( f J sin  )
gSW  1  (J SW ) /( f J sin  )
V=0.045 nm/s; a=65 nm
R(t0)=17 nm, L(t0)= 140 nm
=2.46:
 /( 1)
60
2000
1 /( 1)
 a (  1) V (t  t0 ) 
R  L0 1 


L
R
0 0



R 
L  L0  
 R0 
 f J sin   Jtop
a gf
 

b gSW  f J sin   J SW
1500
40
30
1000
20
NW radius (nm)
50
NW length (nm)
 a(  1) V (t  t0 ) 
L  L0 1 


L
R
0 0


500
10
0
0
5000
10000
15000
Growth duration (s)
20000
0
25000
Condition for super-linear NW growth:
J step  Jtop
Timescale hierarchy and self-regulated pulsed
nucleation in catalyzed nanowire growth
V.G. Dubrovskii
St. Petersburg Academic University &
Ioffe Institute RAS, St. Petersburg, Russia
Plan:
• Nucleation statistics
• Oscillating morphology of growth interface
• Sharp nucleation probability: impact on length uniformity
• Nucleation theory applied to monolayer growth cycle
• Timescale hierarchy
• Au-catalyzed GaAs nanowires
• Conclusions
dubrovskii@mail.ioffe.ru
V. G. Dubrovskii Phys. Rev. B. 87, 195426 (2013)
Lecture 3, Repino , 14 July
Usual assumptions
• Droplet is liquid
• Supersaturation in the droplet is constant during growth.
• Liquid-solid growth interface is planar
From Dubrovskii & Sibirev JCG 2007:
From Glas et al. PRL 2007:
Nucleation statistics in InPAs nanowires
Post-growth study of compositional modulated InPxAs1-x wires:
Au-catalyzed MBE
(a) – HAADF STEM image showing composition
oscillations, related to a given time interval
(b) – measured L(t) fitted by the diffusion
growth model
Std deviation
v length:
Experimental determination of nucleation statistics
(a) – Length of successive nucleations, dashed line
is the mean height and solid line is the mean length
(b) – Histogram of nucleation events per osilattion
Blue line – Poissonian; Red line – model of Glas
Periodically changing morphology of the growth interface in
catalyzed Si, Ge and GaP nanowires
If a truncated facet is stable:
y0
y
1  
Sawtooth
y
 (t )
1
a0  a1 (t / t ML )
Cyclic supersaturation in Au-catalyzed Ge nanowire growth
2011
Impact on the length distribution of nanowires
Regular NW arrays with L=const: if droplets are organized before growth,
then the wires have a narrow distribution over L
L=const for R=const!
pm (t )
Au-seeded InAs,
MOCVD, nanoimprint
Au-seeded GaAs,
MBE, e-beam
Hypothetical growth from identical droplets,
starting simultaneously at t=0, with average
growth rate V (in ML/s), and RANDOM nucleation:
Probability to observe a
NW with m MLs at time t
dp0
 Vp 0
dt
dpm
 V ( p m 1  p m ) m  1,2,3...
dt
Poissonian length distribution
p m (t )  e
Vt
(Vt ) m
m!
Evolution of length distribution with growth time
0.125
Vt  10
Probability pm(t)
0.100
0.075
Vt  50
0.050
Vt  100
0.025
0.000
0
20
40
60
80
100
120
Number of monolayers in nanowire
Why this unwanted Poissonian broadening is not observed experimentally?
Material balance within 1 ML growth cycle
Consider an element that limits nucleation (Si in Au-Si or As in Au-Ga-As).
  k BT ln(  1)
Supersaturation:
c  N / N0  1
  c / ceq  1
Atomic concentration of As in the droplet
N0  f ( )R3  const
Droplet volume
Linear scaling:   N /( N 0 ceq )
Formation of 1 ML removes
N  R 2  / h
Refill time:
V  I / h
  jdiff /(Vh)
Perfect alloy approx.
As atoms from the droplet
 
h
ceq Rf (  )
t r 
1
2V [(1  cos  ) 1   / R]
Deposition rate
Effective diffusion length on NW sidewalls
Model system:
Nucleation and growth of 2D island
The probability density of island nucleation: (hR2 / ) P(t )
P  exp[a / ln(  1)]
a  h( / kBT )2  1
Re-normalized Zeldovich nucleation rate
Island growth rate: dr / dt  r0 / 
Maximum
suoersaturation
Material balance (in absence of desorption):
t
 2R I

hR
 (t ) 




N (0)  
 2Rjdiff t  N (t ) 
dt P(t )  dt


1

cos





0
 t

2 t
2
Atoms dissolved
Total number of atoms arrived in the droplet
to the droplet by time t
Analytical solutions:
 t  t*
 t  t * 

(t )  P( * ) exp
 exp

t

t
 n 
 n
Probability density
Number of atoms
In 2D island
2
 (0)  *  
P(t  0)  0
tn  (*t ) / G
G ~ ic (* )  1 !!!

 t  t * 



Q(t )  1  exp exp
a
 *  exp

1

 t n 

3
ln(
t
/

)



Nucleation probability
Time scale hierarchy
t g
Island growth time = R /( * r0 )
ceq f (  )
R2


2hV R /(1  cos  )  
t
Nucleation to refill:
t n  * ceq f (  ) R

t r
G
h
Maximum supersaturation  *
Growth to nucleation:
Analysis for Au-catalyzed GaAs:
  0.045 nm3 h  0.326 nm
t g
G  R
 2
t n  * t  r0
t g  2t n  t r
450 to 600 C,   0.35 J/m2
a  40
ceq  0.005
  /2
 0
Island growth << Nucleation interval << Refill
V  5ML / s
R=25 nm
Radius dependence
Same parameters of GaAs NWs
Non-overlapping probabilities: narrow nucleation pulses, anti-correlation, uniform L
Overlapping probabilities: random Poissonian
1.0
40
R=25 nm
R=150 nm
0.8
30
20
10
0
0.00
V  2  104
0.05
0.10
0.15
0.20
Time t (s)
0.25
0.30
Probability Q(t)
Probability density P(t)
V  5  105
R=25 nm
R=150 nm
0.6
0.4
0.2
0.0
0.00
0.05
0.10
0.15
Time t (s)
Pulsed nucleation requires (i) modest growth rate; (ii) fast diffusion in the liquid;
(iii) small enough radius
0.20
Impact of truncated edge on crystal structure
When main facet does not meet the trijunction, islands do not nucleate at the trijunction!
Wetting VLS growth predicted in
Experimental verification of wetting:
C. García Núñez et al. J. Cryst. Growth 372, 205 (2013)
Wetting without or with a truncated
facet has a very similar impact on
the crystal structure: