Nucleation theory in growth modeling of nanostructures

1
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_at_mail.ioffe.ru
Repino, 13- July 2013, Lecture 1
2
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
3
Size-dependent quantum effects in nanostructures
SE
DOS
Bulk
DOS of nanostructures
Effect on optical properties
4
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 !!!
5
Alfred Cho the father of MBE
6
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

7
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
8
In situ monitoring by RHEED
9
In situ monitoring by RHEED (continued )
Physical nature of RHHED oscillations
10
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
11
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

12
Chemistry of MOCVD growth process for GaAs
Radiofrequency generator (450 kHz)
Source of a metal-organic compound (liquid or
solid state)
H2
Vapors in H2
Heating up to 600-7000?
Chemical reaction
Hydrides (gaseous)
Example of chemical reaction for the GaAs
epitaxy
Growth of compound semiconductor on a crystal
substrate
H2
(CH3)3Ga AsH3
GaAs 3CH4
6000C
Exhaust of gases
13
Modern MOCVD reactors
(1-x)Ga(CH3)3 xIn(CH3)3 NH3 -gt InxGa1-x N
3CH4 Reactor Aixtron 2000/HT (2003) GaN growth
6 x 2-inch substrates Productivity gt 500 blue
LED structures monthly Each wafer contains 10
000 LED chips 0.350.35 mm
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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 gt larger
wavelength
15
Direct formation of Stranski-Krastanow QDs
20 nm
Relaxation of elastic stress in the island main
driving force for 2D-3D transition
SK growth mode
16
Direct formation of QDs (continued )
At hh1c, RHEED pattern changes from strikes to
spots
2 ML InAs/GaAs
Coherent stained islands
Dislocations
e0gt2
Critical thickness h1c for 2D-3D transition
17
VLS growth of whiskers by Wagner Ellis and
Givargizov
Wagner Ellis, APL 1964
???-????????-???????? ??? ??? (? ??????????
??????????  vapor-liquid-solid  VLS)) 
???????? ????? ?????????? ????????, ????? ???
??????????? ? ???????? ??????????? ????????? ??
??????? ????.
High temperature (T 1000-11000 C) CVD
experiments of 1960-70s with micrometer diameters
18
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
19
Typical RHEED patterns during the wire growth
200 nm GaAs/Si(100)
20
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
Bulk ZB GaAs becomes unstable at pressure 80
GPa !!!
ABCccc3C8
ABAhhh2H(11)
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!
21
Hexagonal WZ phase in III-V NWs !!!
LPN CNRS
APL 2005
InAs NWs on InAs
GaAs NWs on GaAs
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
22
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
23
Nucleation
Consider 2D island of ML height h, area Ac1r2
and perimeter Pc2r, r radius
Gibbs free energy of 2D island formation (fixed
T, P, N)
(1a)
in kBT units
Difference in chemical potentials (energetically
favorable)
Surface term (energetically unfavorable)
i
Surface energy constant
As i
? solid-vapor surface energy per unit area
(J/m2) ?µ difference of chemical potentials
(J) Normally, a is a large parameter several
tens
g
h
24
Gibbs free energy
?n10-3 , a15 ?0.75 (1), 1 (2), 1.5 (3) and 2
(4).
Activation barrier for nucleation
Critical number of atoms
F
ic
Half-width near maximum
F and ic decrease as supersaturation increases
!!!
25
A story about Zeldovich and nucleation theory
?????????? ?????? ?????? ??????? ???????????
???????? (???? ?.?. , 1996), ?????? ??????????????
???? ??????????? ?????? ??????? ???????????,
??????? ??????? ? ?????????? ? ????????????
????? ????????, ??????????? ????????? ??????????
???? ? ?????? ??????????????, ????????????-??????
???????? ????????????? ? ?????? ????????? ??? ?
??????? ???????????? ?????????????? ????????
???????????. ??????? ????????? ???????????
???????? ??????????? (????????? ?????????? ???).
?.?. ?????????
26
Nucleation rate
exp(F)gtgt1
Region 1 Equilibrium size distribution
I nucleation rate 1/cm2s
F
Region 2 Fluctuations flux I
dic/dt0
Region 3 Growth
f(i,t) island size distribution 1/cm2
I
III
II
Kinetic equation for size distribution in region
II
i
ic-?ic
ic?ic
ic
Boundary conditions
27
Nucleation rate (continued)
Stationary solution at Jconst with the 2nd
boundary condition
J0 equilibrium
Jconst steady state
To meet the 1st boundary condition, I should
equal
i1
i
Laplace method
i-1
General Zeldovich formula
for 2D islands
28
Gibbs-Thomson effect and Laplacian pressure
Consider liquid (L) spherical drop of radius R
in equilibrium with vapor (V)
PV
Find PL-PV, PL and PV
R
PL
Solution
1) System at fixed T, V and µ gt maximum of
?
at constant volume
For a sphere with
Laplacian surface pressure
yields
For a cylindrical isotropic solid with
29
GT effect and Laplaciam pressure (continued )
2) At finite R, equilibrium state is defined by
(1)
(2)
At R?8, equilibrium state is defined by
Subtract (1) from (2) take into account that
liquid is incompressible and that vapor is ideal
Vapor
Liquid
30
Mononuclear and polynuclear growth
I nucleation rate, vdr/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, VLf(I,v,R)
R
R
Kashchiev interpolation formula
Dependence on the nucleation barrier
31
A story about Kolmogorov-Mehl-Johnson-Avrami
model
?????????
????????? ????????  ????  ??????  ???????????
(????. Johnson  Mehl  Avrami  Kolmogorov
equation, JMAK) ????????? ??????? ????????
???????? ??? ?????????? ???????????. ??????????
??? ???? ???????? ??? ?????? ??????????????
????????? ? 1937 ???? ?. ?. ????????????, ?
??????????? ??????? ? 1939 ???? ?. ?. ????? ? ?.
?????????, ? ????? ???? ???????????????? ? ?????
?????? ?. ?????? ? 19391941 ?????.

A. Kolmogorov