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 [email protected]

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 SE:

    2 2

m

 

U

   

E



DOS: Bulk:



V

(

E

)  1

V dN dE



V

(

E

)   2 2

m

 2 

S

(

E

)  1

dN S dE

3 / 2

E



E

0

DOS of nanostructures:



S

2

D

(

E

) 

m

  2 

n

 (

E



E n

) 

S

1

D

(

E

) 

n QW R

2

m

  

l n

,  (

E E

 

E nl E nl

) 

S

0

D

(

E

)  2

n QD n

 ,

l

,

q

 (

E



E nlq

)

Effect on optical properties:

E opt



E g



E e

1 

E h

1 

E g

 

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 (As •Ga •Al •In 4 or As •Be (p-doping) •Si (n-doping) 2 through a cracker 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 •MOCVD systems contain: – based materials and devices •For example, LED structures are grown almost exceptionally by MOCVD - 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 (AsH 3 ) and phoshpine (PH 3 ), or NH 3 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

H 2 Source of a metal-organic compound (liquid or solid state) Vapors in H 2

Radiofrequency generator (~450 kHz)

Heating up to 600-700 0 С Chemical reaction Hydrides (gaseous) Example of chemical reaction for the GaAs epitaxy: H 2 (CH 3 ) 3 Ga + AsH 3 600 0 C GaAs + 3CH 4 Growth of compound semiconductor on a crystal substrate Exhaust of gases

Modern MOCVD reactors

(1-x)Ga(CH3) 3 + NH 3 -> In x Ga + xIn(CH3) 1-x N + 3CH 3 4

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 Al 2 •Packaging O 3 •Materials (TMGa, TMAl,TMIn, gases) •Epitaxial growth of LED heterostructure • Processing and production of chips •Fabrication of final device Increasing In concentration in InGaN => larger wavelength

Direct formation of Stranski-Krastanow QDs

20 nm

Substrate Substrate Wetting layer Substrate Island growth (Volmer-Weber) Layer by layer growth (Frank – van der Merve) Combined growth (Stranski-Krastanow)

Relaxation of elastic stress in the island – main driving force for 2D-3D transition

h



L

WETTING LAYER SUBSTRATE SK growth mode

100

Direct formation of QDs (continued …)

At h=h 1c , RHEED pattern changes from strikes to spots In X Ga 1-X As/GaAs dislocations 2D-3D 2 ML InAs/GaAs 10 ε 0 >2% 1 0.0

0.2

0.4

0.6

InAs mole fraction,

x

0.8

1.0

Coherent stained islands Dislocations Critical thickness h 1c for 2D-3D transition

VLS growth of “whiskers” by Wagner & Ellis and Givargizov

Wagner & Ellis, APL 1964

Пар-жидкость-кристалл

английской литературе — или

ПЖК

(в

vapor-liquid solid

—

VLS

)) — механизм роста одномерных структур, таких как нановискеры в процессе химического осаждения из газовой фазы . High temperature (

T

~ 1000-1100 0 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 2-st stage (Vacuum or MBEchamber): Au deposition on a GaAs substrate surface GaAs wafer Au film 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

Bulk ZB GaAs becomes unstable at pressure ~ 80 GPa !!!

ABC=ccc=3C=

∞

ABA=hhh=2H=(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!

Hexagonal WZ phase in III-V NWs !!!

LPN CNRS:

APL 2005 GaAs NWs on GaAs

C. Chang-Hasnain, group:

APL 2007 InP NWs on Si TEM image InAs NWs on InAs [1 1 0 0] zone axis

0000 0002 1120

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=c 1 r 2

and perimeter

P=c 2 r, r = “radius”

Gibbs free energy of 2D island formation (fixed T, P, N):



G

 

i

  

c

2

hr

g

i



c

1

hr

2 

S

   

k B T

ln( 

c

1 s

r

2  1 ) 

F

(

i

)  2

ai



i

ln(   1 ) (1a) in

k B T

units Surface term (energetically unfavorable) Difference in chemical potentials (energetically favorable)

a



c

2 2 4

c

1 

S h

( g /

k B T

) 2 Surface energy constant

γ

– solid-vapor surface energy per unit area (J/m 2 )

Δμ – difference of chemical potentials

(J) Normally,

a

is a large parameter ~ several tens h

A=

s

i

i

g

Activation barrier for nucleation:

F

 

F

(

i c

)  ln( 

a

 1 ) Critical number of atoms:

i c

 ln

a

2 (   1 ) Half-width near maximum: 

F

 (

i c

)  ln 3 (  2

a

 1 )

Gibbs free energy

20 15 s

n

=10

-3 , a

=15  =0.75 (1), 1 (2), 1.5 (3) and 2 (4).

1 2 10 F 3 5 0 0 10 i c 20 4 30 40 Number of atoms, i 50 60

F and i

c

decrease as supersaturation increases !!!

A story about Zeldovich and nucleation theory Я.Б. Зельдович

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

???

).

Nucleation rate

f e

( Region 1: Equilibrium size distribution

i

) 

n

exp[  

F

(

i

)]

exp(F)>>1 I – nucleation rate [1/cm 2 s]

Region 2: Fluctuations [ flux

I

]

di c /dt

=0 Region 3: Growth

f(i,t) – island size distribution [1/cm 2 ]

Kinetic equation for size distribution in region II: F  

t f

  

J



i

Boundary conditions: I II

J

 

W

 (

i

)

f e

 

i

 

f f e

 

f s

/

f e

 1 ,

i

 0 i c Δi c i c i c + Δi c 

i

 2 / 

F

 (

i c

)

f s

/

f e

 0 ,

i

  III i

Nucleation rate (continued…)

Stationary solution at

J=const

with the 2 nd boundary condition:

f s

(

i

) 

J

exp[  

F

(

i

)]  

i W d i

  (

i

 ) exp[ 

F

(

i

 )]

J

=0 equilibrium

J

=const steady state To meet the 1 st boundary condition,

I

should equal:

J



n

    0 

W di

 (

i

) exp[ 

F

(

i

 )]    1 Laplace method i i+1 i-1

J



n

/ 

F

 (

i c

2  ) /

W

 (

i c

) exp( 

F

) General Zeldovich formula

J

 1  s

D

(   1 ) ln 1 / 2 (   1 ) exp    ln( 

a

 1 )   for 2D islands

Gibbs-Thomson effect and Laplacian pressure

P V

Consider liquid (L) spherical drop of radius

R

in equilibrium with vapor (V) Find

P L -P V

,

P L

and

P V

P L R γ

d

  0 at constant volume

Solution:

dV

1) System at fixed

T

,

V

and

μ

=> maximum of   

P L V L



P V V V

 g

A



dV L

 

dV V P L



P V

 g

dA

/

dV

For a cylindrical isotropic solid with

P S



P V

 g

R

For a sphere with

A

 4 

R

2

P L



P V

 2 g

R V

 4 

R

3 / 3

Laplacian surface pressure

A

 2 

RL V

 

R

2

L

yields

GT effect and Laplaciam pressure (continued …)

2) At finite

R

, equilibrium state is defined by At

R →∞

, equilibrium state is defined by 

L

(

P L

)  

V

(

P V

) 

L

(

P

 )  

V

(

P

 ) (1) (2) Subtract (1) from (2); take into account that liquid is

incompressible

and that vapor is

ideal



V



k B T

ln

P

  (

T

)  

L

(

P L

) 

V

(

P V

 

L

(

P

 ) )  

V

 (

P

 ) 

L

(

P L



k B T

 ln(

P

 )

P V

/  

L

(

P L P

 ) 

P V

)  2 

L

g

R



Liquid:

P L



P

  2 g

R



L

(

P L

)  

L

(

P

 )  2 

L

g

R

Vapor:

ln  

P V P

    2 

L

g

k B TR



V

(

P V

)  

V

(

P

 )  2 

L

g

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)

V L

= vertical growth rate of facet of radius due to 2D nucleation

R

Polynuclear growth is generally faster !

Generally,

V L =f(I,v,R) V L

  

JR

3

v V L



h



R

2  

J

1 / ,  3 ,   1  1

Kashchiev interpolation formula:

V L



h

1    

R J

/ 2

v J

 2 / 3

R

2

R R

Dependence on the nucleation barrier:

V L mono

 ( 1 /  )(

R

2 / s ) exp(  

G

* /

k B T

)

V L poly

 ( 1 /  ) exp(  

G

* / 3

k B T

)

A story about Kolmogorov-Mehl Johnson-Avrami model

A. Kolmogorov

Википедия:

Уравнение Джонсона — Мела — Аврами — Колмогорова

( англ.

Johnson — Mehl — Avrami — Kolmogorov equation

,

JMAK

) описывает процесс фазового перехода постоянной температуре. Изначально оно было получено для случая при 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.