Nucleation theory in growth modeling of
nanostructures
V.G. Dubrovskii
St. Petersburg Academic University &
Ioffe Physical Technical Institute RAS, St.-Petersburg, Russia
Plan:
• Introduction
• Epitaxy techniques
• Semiconductor quantum dots and nanowires
• Elements of nucleation theory
• Zeldovich nucleation rate
• Gibbs-Thomson effect and Laplacian pressure
• Nucleation on laterally confined facets
dubrovskii@mail.ioffe.ru
Repino, 13- July 2013, Lecture # 1
Modeling of nanostructure formation
•
•
•
•
•
•
Growth theory
Nucleation
Theory of nanostructure formation
Quantum dots
Nanowires
Epitaxial techniques (MBE, MOCVD…)
InAs/GaAs(100) QDs
Main goals of modeling:
 Understanding
 Prediction
 Optimization
 New morphology
 New structure
 New materials
GaAs/GaAs(111)B-Au NWs
Size-dependent quantum effects in nanostructures
 2




U

   E
 2m

SE:
DOS: V ( E ) 
Bulk:
1 dN
V dE
 S (E) 
2 m
V ( E )  2  2 
  
1 dN
S dE
3/ 2
E  E0
DOS of nanostructures:
 S2 D ( E ) 
m
 2
S1D ( E)  nQWR
 ( E  E
n
)
n
2m
( E  Enl )

 n,l E  Enl
 S0 D ( E)  2nQD  ( E  Enlq )
n ,l , q
Effect on optical properties:
Eopt  Eg  Ee1  Eh1  Eg  E(L)
Transformation of QD distribution function into
DOS
Required properties of NS
ensembles:
• High uniformity
• High density (?)
• Controlled composition
• Controlled morphology
• Controlled crystal structure
Morphology of nanostructure
ensembles depends on growth
process !!!
Alfred Cho – the father of MBE
Technologies of nanostructure
formation: MBE and CVD
1. Molecular beam epitaxy = MBE
•Developed in early 70s
•Now widely used to produce high-quality layers of different
compound semiconductors with very abrupt interfaces and good control
of thickness, doping and composition
•Materials are deposited in a form of molecular beams on a heated substrate
•Molecular beams are originated from thermally evaporated elemental sources
(effusion cells)
•Growth rates are typically of order of several angstroms per second
•MBE system consists of 3 main vacuum chambers:
-Growth chamber
-Buffer chamber (preparation and storage of samples)
-Load lock (to bring samples in and out of the vacuum environment)
•Rotating samples (manipulator)
•Pressure gauge (ion gauge)
•Nitrogen cooler
•Cryo-pumps, ion pump, turbo pumps to remove gases, residual pressure is
typically less than 10-11 Torr
•Substrates holders made from Ta, Mo or pyrolytic boron nitride
Scheme of typical MBE system
Monitor residual
gases, source beams
In situ growth control
Deposition
Example for GaAs:
•As (As4 or As2
through a cracker
•Ga
•Al
•In
•Be (p-doping)
•Si (n-doping)
Sample rotation
In situ monitoring by RHEED
In situ monitoring by RHEED (continued …)
Physical nature of RHHED
oscillations
Modern MBE reactors
•GaAs growth
•6 x 3 inch substrates
•Growth rate 1-3 A/s
•10 sources
•As cracking
•Two parallel loading
systems
•RHEED
•QMA
•Cryo-panel
•4 standard HEMT
processes daily
Riber 49
MOCVD
•Metal organic chemical vapor deposition (MOCVD) = MOVPE is being used for
crystal growth from 1960 and in 1980s was applied for the fabrication of
compound semiconductor – based materials and devices
•For example, LED structures are grown almost exceptionally by MOCVD
•MOCVD systems contain:
- the gas handling system to meter and mix reactants
- the reactor (vertical or horizontal in design)
- the pressure control system
-the exhaust facilities
•Basic principle is the deposition of the required growth species with precursors
at ~ atmospheric pressure of a carrier gas and chemical reaction in the
temperature field of a heated substrate
•Group III sources are trimethylgallium (TMGa), TMAl, TMIn
•Group V species are typically hydride gases such as arsine (AsH3) and phoshpine
(PH3), or NH3 for GaN
•Very high V/III ratios (50-100) because the incorporation of group V elements
Is self-limited (very high partial pressure of group V species)
•Growth rate and composition is controlled by partial pressures of the species and
by the substrate temperature
Chemistry of MOCVD growth process for GaAs
H2
Source of a
metal-organic
compound
(liquid or solid state)
Radiofrequency
generator (~450 kHz)
Vapors in H2
Hydrides
(gaseous)
Example of chemical reaction
for the GaAs epitaxy:
H2
(CH3)3Ga + AsH3
GaAs + 3CH4
0
600 C
Heating up to 600-7000С
Chemical
reaction
Growth of compound semiconductor
on a crystal substrate
Exhaust of
gases
Modern MOCVD reactors
(1-x)Ga(CH3)3 + xIn(CH3)3
+ NH3 -> InxGa1-x N + 3CH4
Reactor Aixtron 2000/HT
(2003): GaN growth
6 x 2-inch substrates
Productivity > 500 blue LED
structures monthly
Each wafer contains ~ 10 000
LED chips 0.35*0.35 mm
Heterostructres for blue-green and white LEDs
Main technological stages:
•Wafers Al2O3
•Materials (TMGa, TMAl,TMIn, gases)
•Epitaxial growth of LED heterostructure
• Processing and production of chips
•Packaging
•Fabrication of final device
Increasing In concentration in InGaN =>
larger wavelength
Direct formation of Stranski-Krastanow QDs
20 nm
Island growth
(Volmer-Weber)
Substrate
Layer by layer growth
(Frank – van der Merve)
Substrate
Wetting layer
Combined growth
(Stranski-Krastanow)
Substrate

h
L
WETTING LAYER
SUBSTRATE
SK growth mode
Relaxation of elastic stress
in the island – main driving
force for 2D-3D transition
Direct formation of QDs (continued …)
At h=h1c, RHEED pattern
changes from strikes to spots
InXGa1-XAs/GaAs
dislocations
2D-3D
Critical thickness, ML
100
2 ML InAs/GaAs
10
Coherent stained islands
Dislocations
ε0>2%
1
0.0
0.2
0.4
0.6
0.8
InAs mole fraction, x
1.0
Critical thickness h1c
for 2D-3D transition
VLS growth of “whiskers” by Wagner & Ellis and
Givargizov
Wagner & Ellis, APL 1964
Пар-жидкость-кристалл или ПЖК (в
английской литературе — vapor-liquidsolid — VLS)) — механизм роста одномерных
структур, таких как нановискеры в процессе
химического осаждения из газовой фазы.
High temperature (T ~ 1000-11000 C) CVD experiments of 1960-70s with
micrometer diameters
Formation of vertical nanowires on activated
surfaces by MBE
GaAs/GaAs(111)B-Au
1-st stage (MBE chamber):
oxide desorption from GaAs
substrate and buffer layer growth
GaAs wafer
Au film
2-st stage (Vacuum or
MBEchamber):
Au deposition on a GaAs
substrate surface
GaAs wafer
GaAs NW
3-st stage (MBE chamber):
formation of Au-Ga alloy droplets;
deposition of GaAs – growth of NW
GaAs wafer
Typical RHEED patterns during the wire growth
200 nm
GaAs/Si(100)
200 nm
GaAs/GaAs(111)B
ZB and WZ phase of III-Vs
All III-V NWs, except nitrides, have STABLE ZB cubic phase in BULK FORM
In GaAs:
Difference in cohesive energies
= 16. 6 – 24 meV per pair
at zero ambient pressure.
T.Akiyama et al, Jpn.J.Appl.Phys,
2006;
M.I.McMahon and R.J.Nelmes,
PRL, 2005
ABC=ccc=3C=∞
ABA=hhh=2H=(11)
Bulk ZB GaAs becomes
unstable at pressure ~ 80 GPa !!!
Most of ZB III-V nanowires contain WZ phase:
A.I.Person et al., Nature Materials 2004, Au-assisted MOVPE of III-V/III-V
J.C.Harmand et al., APL 2005, Au-assisted MBE of GaAs/GaAs
I.P.Soshnikov et al., Phys. Sol. State 2006, Au-assisted MBE of GaAs/GaAs
P.Mohan et al., Nanotechnology 2005, selective area catalyst free growth of III-Vs
C.Chang-Hasnain group, Au-assisted MOCVD of III-V/Si
AND MANY OTHERS!
Hexagonal WZ phase in III-V NWs !!!
LPN CNRS:
APL 2005
GaAs NWs on GaAs
InAs NWs on InAs
[1 1 0 0] zone axis
C. Chang-Hasnain,
group:
0002
0000
APL 2007
1120
InP NWs on Si
TEM image
FFT of TEM image
ZB-WZ transition in GaAs NWs (Ioffe & LPN)
Au-assisted MBE of GaAs
on the GaAs(111)B substrate
ZB
Switching from WZ
to ZB at the end of
growth
WZ
I.P.Soshnikov et al,
Phys. Sol. State 2005
Switching from ZB
to WZ at the
beginning
of growth
ZB phase systematically
appears at low
supersaturation !
F.Glas et al., Phys. Rev. Lett 2007
Nucleation
Consider 2D island of ML height h, area A=c1r2 and perimeter P=c2r, r = “radius”
Gibbs free energy of 2D island formation (fixed T, P, N):
G  i  c2 hrg
hr 2 c1r 2
i  c1

S
s
  k BT ln(  1)
F (i)  2 ai  i ln( 1)
(1a)
in kBT units
Difference in
Surface term
chemical potentials
(energetically
(energetically
unfavorable)
favorable)
2
c2
a
 S h(g / k BT ) 2
4c1
i
Surface energy
constant
γ – solid-vapor surface energy per unit area (J/m2)
Δμ – difference of chemical potentials (J)
Normally, a is a large parameter ~ several tens
A=s i
h
g
Gibbs free energy
sn=10-3 , a=15
=0.75 (1), 1 (2), 1.5 (3) and 2 (4).
Activation barrier for nucleation:
ln(  1)
Critical number of atoms:
ic 
a
ln (  1)
2
Half-width near maximum:
ln (  1)
F (ic ) 
2a
3
20
Free energy of island formation, F
F  F (ic ) 
a
1
15
10
2
F
3
5
ic
0
0
10
20
4
30
40
50
60
Number of atoms, i
F and ic decrease as supersaturation
increases !!!
A story about Zeldovich and nucleation theory
ФИЗИЧЕСКИЕ ОСНОВЫ ТЕОРИИ ФАЗОВЫХ
ПРЕВРАЩЕНИЙ ВЕЩЕСТВА (КУНИ Ф.М. , 1996), ФИЗИКА
Сформулированы цели современной теории фазовых превращений,
введены понятия о стабильных и нестабильных фазах вещества,
образовании зародышей стабильной фазы в недрах
метастабильной, вероятностно-статистическое представление
о потоке зародышей как о ведущей кинетической характеристике
фазового превращения.
Описана временная зависимость фазового превращения
(уравнение Зельдовича ???).
Я.Б. Зельдович
Nucleation rate
Region 1: Equilibrium size distribution
exp(F)>>1
f e (i)  n exp[F (i)]
I – nucleation
rate [1/cm2s]
Region 2: Fluctuations [ flux I]
F
dic/dt=0
Region 3: Growth
f(i,t) – island size distribution [1/cm2]
I
III
II
Kinetic equation for size distribution in region II:
i
f
J

t
i
Boundary
conditions:
 f 
J  W (i) f e  
i  f e 

f s / f e  1, i  0
ic-Δic
ic
ic+Δic
i  2 / F (ic )
f s / f e  0, i  
Nucleation rate (continued…)
Stationary solution at J=const with the 2nd boundary condition:

di
f s (i)  J exp[F (i)]  exp[F (i)]
W (i)
i
J=0
equilibrium
J=const
steady state
To meet the 1st boundary condition, I should equal:
 di

J  n    exp[F (i )]
 0 W (i )


i+1
1
i
Laplace method
i-1
/ F (ic ) / 
J n
W (ic ) exp( F )
2
General Zeldovich formula


a
J
(  1) ln (  1) exp

ln(


1
)
 s D


1
1/ 2
for 2D islands
Gibbs-Thomson effect and Laplacian pressure
PV
Consider liquid (L) spherical drop of radius R
in equilibrium with vapor (V)
Find PL-PV, PL and PV
PL
R
Solution:
γ
1) System at fixed T, V and μ => maximum of
  PLVL  PVVV  gA
d  0 at constant volume dV  dVL  dVV
PL  PV  gdA/ dV
For a sphere with
2g
PL  PV 
R
A  4R 2 V  4R 3 / 3
Laplacian surface
pressure
2
For a cylindrical isotropic solid with A  2RL V  R L yields
PS  PV 
g
R
GT effect and Laplaciam pressure (continued …)
2) At finite R, equilibrium state is defined by
 L ( PL )  V ( PV )
(1)
At R→∞, equilibrium state is defined by
 L ( P )  V ( P )
(2)
Subtract (1) from (2); take into account that liquid is incompressible and that
vapor is ideal
V  kBT ln P   (T )
2 Lg
 L ( PL )   L ( P )   L ( PL  P )   L ( PL  PV ) 

R
 V ( PV )  V ( P )  k BT ln(PV / P )
Vapor:
Liquid:
 PV
ln
 P
2g
PL  P 
R
 L ( PL )   L ( P ) 
2 L g
R
 2 L g
 
 k BTR
2 L g
V ( PV )  V ( P ) 
R
Mononuclear and polynuclear growth
I – nucleation rate, v=dr/dt – 2D island growth rate, R – face radius
I and v are time-independent during growth (constant supersaturation)
Polynuclear growth
is generally faster !
VL = vertical growth rate of facet of radius R
due to 2D nucleation
VL
Generally, VL=f(I,v,R)

JR
v
3
R 2 J ,   1
VL  h
1/ 3
 v 2 J ,   1
 
R
Kashchiev interpolation formula:
VL  h
R
Dependence on the nucleation barrier:
R J
2
1   J / v  R 2
2/3
VL
mono
 (1/  )(R2 / s ) exp(G* / kBT )
VL
poly
 (1/  ) exp(G* / 3k BT )
A story about Kolmogorov-MehlJohnson-Avrami model
Википедия:
A. Kolmogorov
Уравнение Джонсона — Мела — Аврами —
Колмогорова (англ. Johnson — Mehl —
Avrami — Kolmogorov equation, JMAK)
описывает процесс фазового перехода при
постоянной температуре. Изначально оно
было получено для случая кристаллизации
расплавов в 1937 году А. Н. Колмогоровым, и
независимым образом в 1939 году Р. Ф.
Мелом и У. Джонсоном, а также было
популяризировано в серии статей М. Аврами
в 1939—1941 годах.