Repetition: Nucleation Rates

Repetition: Nucleation RatesSimple nucleation theory of the isolated nucleus as well as rate equations yieldnucleation rates I of the form:

The exponent p can have integer or non-integervalues.

SBTkE

p12 eRA]sm[I ⋅≅−−

Droplet-model: E=E(∆∆∆∆G*) => unambigousParticle-model: E=E(i, Ei) => ambigous

Repetition: Rate Equations General

Incomplete condensation Complete condensation

log t

log n

n1

nx

t a Coalescence log t

log n

n1

nx

t c

Coalescence

dndt

R n

a

1 10= = −τ

d n wdtx x( ) ≅ 0

dndt

R d n wdtx x1 = − ( )

Repetition: KMC Simulation

Kinetic Monte Carlo (KMC) simulations allow the determination of

the shape of the stable islandsthe size distribution of stable islandsthe influence of defects on nucleation

they only use the elemetary processes of film growth (deposition, surface diffusion, desorption, particle bonding) as input for the simulation of film growth.

Repetition: KMC - Principle

Definition of event typesDefinition of relative event probabilities("conditional probabilities")Choice of a particle for the execution of the eventDetermination of the time interval betweenspecific events

Advantage: each chosen event changes the system

Disadvantage: not all event types are known a priori; algorithm is memory consuming

Repetition: KMC – Results and Trends Variation of substrate temperature TS

Variation of deposition rate R

R=1ML/sEDiff=0.5 eVEDes=1 eVEb= 0.5 eV

TS=700 KEDiff=0.5 eVEDes=1 eVEb= 0.5 eV

TS=10K 300K 700K600K

R=0.5 ML/s 1ML/s 10 ML/s5 Ml/s

Epitaxy

Epitaxy is the deposition of layers, which are monocrystalline in large regions

Homoepitaxy:Substrate meterial = film material

Heteroepitaxy:Substrate material ≠≠≠≠ film material

Heteroepitaxy IEpitactic relation:

Van-der-Waals epitaxy:The interaxction between substrate material and film material is so weak, that film atoms can arrange themselkves in a crystallographically favorable manner.

Substrate materialFilm material

Heteroepitaxy IIHigh temperature epitaxy:

The crystallographically favorable arrangement of the atoms is reached by a high substrate temperature.

Low temperature epitaxy :The crystallographically favorable arrangement of the atoms is reached by local defect structures

Vicinal surfacesDendritic Islands

Film Growth: Nucleus Shape/Wetting

γγγγ…Surface energyO…SurfaceD…VaporK…Nucleus

Film Growth: Growth ModesGrowth modes:

A: Substrate materialB: Film material

Frank-Van der Merwe:layer by layer W >WAB BB

Volmer-Weber:islands, W <WAB BB

Stranski-Krastanov:layer/island, W >W

stress relief by 3d islandsAB BB

Characterization: e. g.: Auger-electron spectroscopy

q[ML]0 1 2 3 4

IB

IA

q[ML]0 1 2 3 4

IB

IA

0 1 2 3 4

IB

IA

q[ML]

Stress and Film Growth I

Growth modes:

While the Frank-van der Merwe and Volmer-WeberGrowth modes lead to mostly stress free films, in the Stranski-Krastanov-mode significant stresses are in-duced in the first growth phases.

A: Substrate materialB: Film material

Frank-Van der Merwe:layer by layer W >WAB BB

Volmer-Weber:islands, W <WAB BB

Stranski-Krastanov:layer/island, W >W

stress relief by 3d islandsAB BB

Stress and Film Growth II

Stranski-Krastanov-growth:+ Lattice mismatch (misfit)+ Misfit-dislocations+ Islands

Substrate

Wetting layerIslands

Stress and Film Growth IIIDetailed mechanism:

a

b

Substrate, lattice constant a

Film, lattice constant b

Pseudomorphic transition zone

[%]100a

ba ⋅−=∆

Lattice Mismatch ∆∆∆∆:

Film Growth: Experimental I

Film thickness

Substrate-temperature

Au on NaCl, R=0.1 nm/s

Film Growth: Experimental II

Coalescence:(a) d=10 nm(b) d=10.5 nm(c) d=11 nm(d) d=11.5 nm

Ag on NaCl, R=0.1 nm/s, T=100°C

Film Growth: Further Steps

Further Growth: Roughness/Film Structure

The film structure is determided by the roughness of the film growth front in the different growth phases to a high degree.

For very thin layers the roughness is in the order of the film thickness and can therefore be more important than the mean layer thickness.

d

r D

Roughness Types I

Stochastic roughness – Solid on Solid (SOS) model

ax

h(x)h max

minh

+ Particles have to have a below nearestneighbor (NN)

+ Particles stick where they land

Roughness Types IISelf similar surfaces – SOS model

+ Particles can reach energetically favorable positions (e. g. high coordination number)

+ Particle migration e. g. by surface diffusion

R

x

h(x)

R=f(L), R''>R'>R

R'R''

h

L''L'L

Roughness Types III

Ballistic aggregation – pore formation

+ Particles stick where they land+ Particles do not have to have a below NN

h(x)

x

ShadowingGiven initial profile and impungement angle distribution

+ Narrow impingement angle distribution:peaks see the same particle flow as valleys (a)

+ Narrow impingement angle distribution:peaks see larger particle flow than valleys (a)

vϕ

n( )ϕ

v ϕ

n( )ϕ

θ vn( )ϕ

v

θn( )ϕ

(a) (b)

g

gn

n n

nh(x)

Shadowing Dominated GrowthPeaks grow faster than valleys

+ Formation of columnar structures (a)+ Pore formation in combination with surface

diffusion (b)

(a) (b)

Roughness Measurement in Real Space IConversion of a continous heigth function to a set of discrete heigth values due to the finite lateral resolution of the measurement device.

x

h(x)

h

x∆x L=N x∆

i

i

Roughness values (vertical) may depend on the lateralresolution of the measurement device.

Roughness Measurement in Real Space II

+ Stylus profilometer: 1d+ Scanning tunneling microscope, STM: 2d + Scanning force microscope, AFM: 2d+ Optical near field microscopy, SNOM: 2d

The Feedback PrincipleExample: STM

(a) absolute tip position constant(b) relative tip position constant

(a) (b)

H = const

h(x)

(x) (x)I U

h(x)STM-tip

Contakttip/surface

STM-tip

UPiezo-element

d(x)d(x)=const

x x

x x

ref

tunnel Piezo

Piezo

Roughness Measurement in Fourier Space IScattering of visible light, X-rays or particles at outeror inner interfaces

(a) Specular reflexion(b) Diffuse reflexion(c) Signal combined from (a) and (b)

Roughness Measurement in Fourier Space II

Advantages:+ Damage free+ not necessarily vacuum based

Disadvantage:+ Suzrface profile not unambigously reconstructible

LightElectronsIons

Outer interfaces Inner interfaces

X-raysSynchrotron radiation

Loss of Phase InformationScattering basically yields the Fourier Transform of a surface ⇒⇒⇒⇒ loss of phase information⇒⇒⇒⇒ no unambigous reconstruction of the profile possible

h(x)=sin(x)+0.5sin(2x)+0.25sin(3x)

h(x)

0 2πx

0 2π

h(x)

x

h(x)=sin(x)+0.5sin(2x+ /2)+0.25sin(3x+ )π π

A =1 A =1ϕ =0 ϕ =0A =0.5 A =0.5ϕ =0 ϕ π = /2A =0.25 A =0.25ϕ =0 ϕ π =

1 11 12 22 2

3 33 3

∑=

ϕ+=N

0kkk )kxsin(A)x(h

knownunknown

Quantification of Roughness ILinear profile, Sampling Interval ∆∆∆∆x

x

h(x)

h

x∆x L=N x∆

i

i

xLN

∆=

Quadratic scan

xyx

LLLx

LN

yx

2

∆≡∆=∆

≡=

∆=

Mean film thickness

∑=

=N

1iih

N1h

Quantification of Roughness II

Ra-value: mean absolute deviation

∑=

−=N

1iia hh

N1R

Rq- or RMS-value: mean quadratic deviation

( )∑=

−===N

1i

2iRMSq hh

N1RMSRR

Quantification of Roughness IIIDifferent profiles may have the same Ra or. RMS-values:

h(x)R R

R R

h(x)

h h

hh

h(x)

h(x)

x

x x

x

different periodicities

different symmetries

Shape Specific ParametersAllow limited statements about profile shape:

Skewness Sk:

Kurtosis K:

( )∑=

−=N

1i

3i3

q

hhNR

1Sk

( )∑=

−=N

1i

4i4

q

hhNR

1K

Sk<0: many heigth values < hSk>0: many heigth values > h

K: Measure of mean flank steepness

Correlation FunctionsAllow detailed statements about vertical and lateral profile properties:Point-point correlations for a discretized profile:

∑−

=+ −⋅−

−=∆⋅=

nN

1inii )hh()hh(

nN1)xn(R)X(Rz. B.:

x

h(x)

hh

xn=0 => N point pairs

n=1 => N-1 point pairs

n=2 => N-2 point pairsTherefore always N-n point pairs can be correlated within the interval

x L=N xD

ii+n

i i+n

Auto Covariance Function

∑−

=+ −⋅−

−=∆⋅=

nN

1inii )hh()hh(

nN1)xn(R)X(R

Discrete

Continuum

dx)h)x(h()h)x(h(L

1)(RL

0

−τ+⋅−τ−

=τ ∫τ−

Structure Function

Discrete

Continuum

∑−

=+ −−−

−=∆⋅=

nN

1i

2nii )]hh()hh[(

nN1)xn(S)X(S

dx)]h)x(h()h)x(h[(L

1)(S 2L

0

−τ+−−τ−

=τ ∫τ−

Connection Between R(τ) and S(τ)

Normalized Autocovariance function (Autocorrelation function):

It is:

)0(R/)X(R)X( =ρ )0(R/)(R)( τ=τρor

2qR)0(R =

)](1[R2)(S 2q τρ−=τ

Summary Correlation Functions

Non normalized quantities Normalized quantities

Autocovariance function

Structure function

dx)x(h)x(h)(R τ+⋅=τ

2)]x(h)x(h[)(S τ+−=τ

Autocorrelation function

)0(R/)(R)( τ=τρ

)](1[R2)(S 2q τρ−=τ

Note: All heigth values are measured from the mean heigth h.

Correlation Length ξSurface profile

Autocovariance function

Within ξξξξ the profile exhibits similar heigth values.

Periodicities are present, if R(ττττ) exhibits maxima at ττττ≠≠≠≠0.

Correlation Functions and Fourier Spectra

"Power Spectral Density"

P(k) ist the fourier transformed of the Autocovariance function R(ττττ).

Result of a scattering experiment:

2

ikr2l

dre)r(h)2(L

1lim)k(P ∫∞

∞−∞→ π⋅

=λ

π= 2kλλλλ ... Wavelength of a characteristic surface feature

∫∞

∞−π= dre)r(R

)2(1)k(P ikr

2

A scattering experiment therefore basically yields the Autocovariance function with all related statistical quantities (ξξξξ, Rq).