Transcript
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: 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 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
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).
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