nucleation theory in growth modeling of nanostructures v.g. dubrovskii st. petersburg academic...
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Nucleation theory in growth modeling of nanostructures
V.G. Dubrovskii
St. Petersburg Academic University & Ioffe Physical Technical Institute RAS, St.-Petersburg, Russia
Repino, 13- July 2013, Lecture # 1
dubrovskii@mail.ioffe.ru
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
Modeling of nanostructure formation
• Growth theory • Nucleation• Theory of nanostructure formation• Quantum dots• Nanowires• Epitaxial techniques (MBE, MOCVD…)
InAs/GaAs(100) QDs
GaAs/GaAs(111)B-Au NWs
Main goals of modeling:
Understanding Prediction Optimization New morphology New structure New materials
Size-dependent quantum effects in nanostructures
DOS of nanostructures:
EU
m2
2
Effect on optical properties:
)(11 LEEEEEE ghegopt
dE
dN
VEV
1)(
SE:
DOS:dE
dN
SES
1)(
0
2/3
22
2)( EE
mEV
Bulk:
n
nD
S EEm
E )()(2
2
ln nl
nlQWR
DS EE
EEmnE
,
1 )(2)(
qln
nlqQDD
S EEnE,,
0 )(2)(
Transformation of QD distribution function into DOS
• High uniformity• High density (?)• Controlled composition• Controlled morphology• Controlled crystal structure
Required properties of NSensembles:
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 controlof 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 istypically less than 10-11 Torr•Substrates holders made from Ta, Mo or pyrolytic boron nitride
Scheme of typical MBE system
Monitor residualgases, source beams
In situ growth control
Sample rotation
Deposition
Example for GaAs:•As (As4 or As2 through a cracker•Ga•Al•In•Be (p-doping)•Si (n-doping)
Modern MBE reactors
Riber 49
•GaAs growth•6 x 3 inch substrates•Growth rate 1-3 A/s•10 sources•As cracking•Two parallel loadingsystems•RHEED•QMA•Cryo-panel•4 standard HEMTprocesses daily
MOCVD•Metal organic chemical vapor deposition (MOCVD) = MOVPE is being used forcrystal 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 precursorsat ~ 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 elementsIs self-limited (very high partial pressure of group V species)•Growth rate and composition is controlled by partial pressures of the species andby the substrate temperature
Chemistry of MOCVD growth process for GaAs
Source of ametal-organic
compound(liquid or solid state)
H2
Hydrides (gaseous)
Chemical reaction
Radiofrequencygenerator (~450 kHz)
Heating up to 600-7000С
Growth of compound semiconductoron a crystal substrate
Example of chemical reaction for the GaAs epitaxy:
(CH3)3Ga + AsH36000C
H2GaAs + 3CH4
Vapors in H2
Exhaust ofgases
Modern MOCVD reactors
(1-x)Ga(CH3)(1-x)Ga(CH3)33 + xIn(CH3) + xIn(CH3)33
+ NH+ NH33 -> In -> InxxGaGa1-x1-x N + 3CH N + 3CH44
ReactorReactor Aixtron 2000/HT Aixtron 2000/HT (2003):(2003): GaN growth GaN growth 6 x 26 x 2--inchinch substratessubstratesProductivity > 500 blue LEDProductivity > 500 blue LEDstructures monthlystructures monthlyEach wafer contains ~ 10 000Each wafer contains ~ 10 000LED chips 0.35*0.35 mmLED 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
Substrate
Wetting layer
Substrate
SubstrateIsland growth
(Volmer-Weber)
Layer by layer growth(Frank – van der Merve)
Combined growth(Stranski-Krastanow)
SUBSTRATE
WETTING LAYER
L
h
Relaxation of elastic stressin the island – main drivingforce for 2D-3D transition
SK growth mode
20 nm
Direct formation of QDs (continued …)
Critical thickness h1c for 2D-3D transition
Coherent stained islands
At h=h1c, RHEED patternchanges from strikes to spots
ε0>2% Dislocations
2 ML InAs/GaAs
VLS growth of “whiskers” by Wagner & Ellis and Givargizov
High temperature (T ~ 1000-11000 C) CVD experiments of 1960-70s withmicrometer diameters
Пар-жидкость-кристалл или ПЖК (в английской литературе — vapor-liquid-solid — VLS)) — механизм роста одномерных структур, таких как нановискеры в процессе химического осаждения из газовой фазы.
Wagner & Ellis, APL 1964
Formation of vertical nanowires on activated surfaces by MBE
1-st stage (MBE chamber): oxide desorption from GaAs substrate and buffer layer growth
GaAs wafer
3-st stage (MBE chamber):formation of Au-Ga alloy droplets;deposition of GaAs – growth of NW GaAs wafer
GaAs NW
2-st stage (Vacuum or MBEchamber): Au deposition on a GaAssubstrate surface
GaAs wafer
Au film
GaAs/GaAs(111)B-Au
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 pairat 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 !!!
Most of ZB III-V nanowires contain WZ phase: A.I.Person et al., Nature Materials 2004, Au-assisted MOVPE of III-V/III-VJ.C.Harmand et al., APL 2005, Au-assisted MBE of GaAs/GaAsI.P.Soshnikov et al., Phys. Sol. State 2006, Au-assisted MBE of GaAs/GaAsP.Mohan et al., Nanotechnology 2005, selective area catalyst free growth of III-VsC.Chang-Hasnain group, Au-assisted MOCVD of III-V/Si AND MANY OTHERS!
ABC=ccc=3C=∞ ABA=hhh=2H=(11)
Hexagonal WZ phase in III-V NWs !!!
LPN CNRS:
GaAs NWs on GaAs InAs NWs on InAs
C. Chang-Hasnain,group:
TEM image
[1 1 0 0] zone axis
0002
0000
1120
FFT of TEM image
InP NWs on Si
APL 2005
APL 2007
ZB-WZ transition in GaAs NWs (Ioffe & LPN)
Au-assisted MBE of GaAson the GaAs(111)B substrate
WZ
ZB
Switching from WZto ZB at the end ofgrowth
I.P.Soshnikov et al,Phys. Sol. State 2005
Switching from ZBto WZ at the beginningof growth
F.Glas et al., Phys. Rev. Lett 2007
ZB phase systematically appears at low supersaturation !
Nucleation
Gibbs free energy of 2D island formation (fixed T, P, N):
(1a)
Difference in chemical potentials
(energetically favorable)
Surface term(energeticallyunfavorable)
i
γ – solid-vapor surface energy per unit area (J/m2)Δμ – difference of chemical potentials (J)Normally, a is a large parameter ~ several tens
h
)1ln(2)( iaiiF
A=i
Consider 2D island of ML height h, area A=c1r2 and perimeter P=c2r, r = “radius”
hrciG 2
in kBT units
21
2
1
rchrci
S
)1ln( TkB
Surface energy constant
2
1
22 )/(
4Tkh
c
ca BS
Gibbs free energy
0 10 20 30 40 50 600
5
10
15
20
4
3
2
1
Number of atoms, i
Fre
e e
nerg
y o
f is
land
fo
rma
tion
, F
n=10-3 , a=15 =0.75 (1), 1 (2), 1.5 (3) and 2 (4).
)1ln()(
a
iFF c
Activation barrier for nucleation:
)1(ln 2
a
ic
Critical number of atoms:
aiF c 2
)1(ln)(
3
Half-width near maximum:
F
ic
F and ic decrease as supersaturationincreases !!!
A story about Zeldovich and nucleation theory
Я.Б. Зельдович
ФИЗИЧЕСКИЕ ОСНОВЫ ТЕОРИИ ФАЗОВЫХ ПРЕВРАЩЕНИЙ ВЕЩЕСТВА (КУНИ Ф.М. , 1996), ФИЗИКАСформулированы 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 ???).
Nucleation rate
)(/2 ciFi
F
iic-Δic ic+Δicic
exp(F)>>1
I II III
Region 1: Equilibrium size distribution
)](exp[)( iFnife
Region 2: Fluctuations [ flux I]
I – nucleationrate [1/cm2s]
Region 3: Growthdic/dt=0
f(i,t) – island size distribution [1/cm2]
Kinetic equation for size distribution in region II:
iJ
tf
ee f
fi
fiWJ )(
0,1/ iff esBoundary conditions:
iff es ,0/
Nucleation rate (continued…)
)](exp[)(
)](exp[)(
i
s iFiWid
iFJif
Stationary solution at J=const with the 2nd boundary condition:
i-1
i
i+1
J=0equilibrium
J=conststeady state
To meet the 1st boundary condition, I should equal:
1
0
)](exp[)(
iF
iWdi
nJLaplace method
)exp()(2
/)(/FiW
iFnJ c
c
)1ln(
exp)1(ln)1(1 2/1
a
JD
General Zeldovich formula
for 2D islands
Gibbs-Thomson effect and Laplacian pressure
RPL
PV
Consider liquid (L) spherical drop of radius R in equilibrium with vapor (V)
Find PL-PV, PL and PV
Solution:
1) System at fixed T, V and μ => maximum of
AVPVP VVLL
0d at constant volume VL dVdVdV
dVdAPP VL /For a sphere with 24 RA 3/4 3RV
RPP VL
2
For a cylindrical isotropic solid with RLA 2 LRV 2 yields
RPP VS
γ
Laplacian surfacepressure
GT effect and Laplaciam pressure (continued …)
2) At finite R, equilibrium state is defined by )()( VVLL PP
At R→∞, equilibrium state is defined by )()( PP VL
Subtract (1) from (2); take into account that liquid is incompressible and thatvapor is ideal )(ln TPTkBV
)/ln()()(
2)()()()(
PPTkPPR
PPPPPP
VBVVV
LVLLLLLLL
Liquid:
RPPL
2
RPP L
LLL
2)()(
Vapor:
RPP L
VVV
2)()(
TRkP
P
B
LV
2ln
(1)
(2)
Mononuclear and polynuclear growth
I – nucleation rate, v=dr/dt – 2D island growth rate, R – face radiusI and v are time-independent during growth (constant supersaturation)
vJR3
1,
1,3/12
2
Jv
JRhVL
Kashchiev interpolation formula:
23/2
2
/1 RvJ
JRhVL
VL = vertical growth rate of facet of radius R due to 2D nucleation
Generally, VL=f(I,v,R)
R R
Polynuclear growth is generally faster !
Dependence on the nucleation barrier:
)/exp()/)(/1( *2 TkGRV B
monoL
)3/exp()/1( * TkGV Bpoly
L
VL
A story about Kolmogorov-Mehl-Johnson-Avrami model
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.
Википедия:
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