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)

In situ monitoring by RHEED

In situ monitoring by RHEED (continued …)

Physical nature of RHHED oscillations

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

Typical RHEED patterns during the wire growth

200 nm GaAs/GaAs(111)B

200 nm GaAs/Si(100)

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: