finite element analysis of epitaxial thin film growth
TRANSCRIPT
FINITE ELEMENT ANALYSIS OF EPITAXIAL FINITE ELEMENT ANALYSIS OF EPITAXIAL THIN FILM GROWTHTHIN FILM GROWTH
ANANDH SUBRAMANIAMDepartment of Applied Mechanics
INDIAN INSTITUTE OF TECHNOLOGY DELHINew Delhi-
110016Ph: (+91) (11) 2659 1340, Fax: (+91) (11) 2658 1119
[email protected], [email protected]://web.iitd.ac.in/~anandh
December 2006
EPITAXIAL THIN FILMSEPITAXIAL THIN FILMS
FILM AND DISLOCATION ENERGETICSFILM AND DISLOCATION ENERGETICS
FEM SIMULATION OF FILM GROWTHFEM SIMULATION OF FILM GROWTH
FEM SIMULATION OF A MISFIT DISLOCATIONFEM SIMULATION OF A MISFIT DISLOCATION
CRITICAL THICKNESSCRITICAL THICKNESS
CONCLUSIONSCONCLUSIONS
OUTLINEOUTLINE
EPITAXIAL THIN FILMSEPITAXIAL THIN FILMS
FILM
SUBSTRATE INRTERFACIAL EDGE DISLOCATION
~100 Å
~100
EXAMPLES
GeSiSi
GaAsPGaAs
InGaAsGaP
AuAg
CoNi
Semiconductor
Metallic
Extra-"half" plane
There is a 4.2% difference in the lattice constants of Si
and Ge. Therefore when a layer of Si1-x
Gex
is grown on top of Si, it has a bulk relaxed lattice constant which is larger than Si.
Si1-x Gex on Si
If layers are grown below the critical thickness then they become strained with the lattice symmetry changing from cubic to tetragonal.
Strained
Si1-x Gex on
Si
Above the critical thickness, it costs too much energy to strain additional layers of material into coherence with the substrate. Instead misfit dislocations ‘form’, which act to partly relieve the
strain in the epitaxial film.
Partly relaxed
Si1-x Gex on Si
Interfacial misfit dislocation
FILM AND DISLOCATION ENERGETICSFILM AND DISLOCATION ENERGETICS
Eh
= 2G[(1 + )/(1 -
)] fm2 h
FILM ENERGY
DISLOCATION ENERGY
TOTAL ENERGY
Edl
=
bGb 0
2
ln2)1(4
Etot = Eh + Edl (algebraic addition)film energy per unit area & dislocation energy per unit length)
Eh – Energy of film per unit area of interface Film parallel to (001), (111) or (011) G – Shear modulus
– Poisson’s ratio fm – Misfit strain = (af - as )/af
af : film lattice parameteras : substrate lattice parameter
Edl – Energy per unit length of dislocation line
b – Modulus of the Burgers vector 0 - size of the control volume ~ 70b
(edge dislocation)
FINITE ELEMENT ANALYSISFINITE ELEMENT ANALYSIS( Stress free strain)
1. Constructing a strain-free layer of GeSi on the Si substrate
2. Imposing the coherency at the interface through a lattice misfit strain
3. Simulation is repeated for successive build-up of the layers to model the growth of the film
Elastic constants for the GeSi alloy calculated by linear interpolation of values
Anisotropic conditions
Lattice constants at 550 0C – the growth temperature
FILM
DISLOCATION
Edge dislocation is modelled by feeding the strain (Tdl ) corresponding to the introduction of an extra plane of atoms
b = as /2 [110]
Tdl = ((as [110] + bs ) - as [110]) / (as [110] + bs ) = bs /3bs =1/3
yx
S y m m e t r y l i n e(symmetric half of the domain taken for analyses)
Region of the domain (B)where Eshelby strain is imposed to simulate the dislocation
Region of the domain (A)where Eshelby strain is imposed to simulate the strained film
99 Elements
68 E
lem
ents
GeGe 0.50.5 SiSi 0.50.5 FILM ON FILM ON SiSi SUBSTRATESUBSTRATE(MISFIT STRAIN = 0.0204)
x AFTER THE GROWTH OF ONE LAYER ( ~5 Å)
50 Å
230 Å
(MPa)
Zoomed region near the edge
SUBSTRATE
FILM
EDGE
SYMMETRY LINE
GeGe 0.50.5 SiSi 0.50.5 FILM ON FILM ON SiSi SUBSTRATESUBSTRATE(MISFIT STRAIN = 0.0204)
x AFTER THE GROWTH OF FIVE LAYERS14
5 Å
190 Å
(MPa)
SYMMETRY LINE
SUBSTRATE
FILM
EDGE
STRESS FIELD OF AN EDGE DISLOCATIONSTRESS FIELD OF AN EDGE DISLOCATIONX - PLOT OF THEORETICAL EQUATION
-5.00 -3.00 -1.00 1.00 3.00 5.00
x (Angstroms)
-5.00-4.00-3.00-2.00-1.000.001.002.003.004.005.00
y (A
ngst
rom
s)
-10.00-9.00-8.00-7.00-6.00-5.00-4.00-3.00-2.00-1.000.001.002.003.004.005.006.007.008.009.0010.00
(Contour values x 104 Mpa)
x
= 222
22
)()3(
)1(2 yxyxyGb
STRESS FIELD OF AN EDGE DISLOCATIONSTRESS FIELD OF AN EDGE DISLOCATIONX - PLOT OF THEORETICAL EQUATION
(Contour values in GPa)
Material → Alb = 2.86
Å
G
=
26.18
GPa
=
0.348
(GPa)
x (Å) →
y (Å
) →
2 2
2 2 2
(3 )2 (1 ) ( )x
Gb y x yx y
STRESS FIELD OF A EDGE DISLOCATIONSTRESS FIELD OF A EDGE DISLOCATIONX – FEM SIMULATED CONTOURS
(MPa)(x & y original grid size = b/2 = 1.92 Å)
27 Å
28 Å
FILM
SUBSTRATEb
92.3
Å
(x & y original grid size = b/2 =
2.72 Å)
8 2.82
7 2.03
6 1.24
5 0.44
4 1.15
3 -1.94
2 -2.74
1 -3.54
(GPa)
59.7 Å
-
-
+
+1
2
3
7
6
8
5
5
4
4
y – FEM SIMULATED CONTOURS
Plot of y
of an edge dislocation with
Burgers vector b: FEM simulated contours.
140
Å
7 20.0
6 15.0
5 2.5
4 0
3 -2.5
2 -15.0
1 -20.0
(GPa)
140 Å
2 6
3
1
6 7
3
4
5
-
+
+
-
5
Plot of y
of an edge dislocation with Burgers vector b: Contours
obtained from equation:
y = 222
22
)()(
)1(2 yxyxyGb
DISLOCATION DISLOCATION –– ENERGY/AREA OF INTERFACEENERGY/AREA OF INTERFACE
0.0E+00
5.0E-02
1.0E-01
1.5E-01
2.0E-01
2.5E-01
3.0E-01
3.5E-01
4.0E-01
0 2 4 6 8 10 12 14 16
Distance from the centre of the dislocation (in b/2 spacings)
Ener
gy p
er u
nit a
rea
of in
terfa
ce
(J/m
2 )
(b) PEAK THRESHOLD
5b THRESHOLD
GeGe 0.50.5 SiSi 0.50.5 FILM ON FILM ON SiSi SUBSTRATE WITH EDGE DISLOCATIONSUBSTRATE WITH EDGE DISLOCATIONx AFTER THE GROWTH OF FIVE LAYERS
MPa
Zoomed region near the edge
80 Å
240 Å
FILM
SUBSTRATE
SYMMETRY LINE EDGE
CRITICAL THICKNESS (CRITICAL THICKNESS (hhcc ) ) -- GeSi/GeGeSi/Ge
GLOBAL ENERGY MINIMIZATION – EQUILIBRIUM appears suitable for metallic systems
METASTABLE FILMS - 5b approach appears suitable for semiconductor systems
hc
(nm) = )(9.8ln10175.1 2
nmhfx
cm
hc
(nm) = )(5.2ln109.12
3
nmhfx
cm
[[1] F.C. Frank, J. Van der
Merve, Proc. Roy. Soc. A 198 (1949) 216-225.[[2] R. People, J.C. Bean, Appl. Phys. Lett. 47 (1985) 322-324.
[1]
[2]
GeGe 0.50.5 SiSi 0.50.5 FILM ON FILM ON SiSi SUBSTRATESUBSTRATE
CONSIDERING ONLY FILM ENERGETICS ( no substrate)
0
0.5
1
1.5
2
2.5
3
1 2 3 4 5 6 7 8 9 10 11 12 13 14Normalized thickness (with b/2)
Ene
rgy
of fi
lm (
J/m
2)
E film
E film withdislocation
Critical thickness of 12 Å
= three film layers
CONSIDERING THE TOTAL ENERGY OF THE SYSTEM(film and the substrate)
GeGe 0.50.5 SiSi 0.50.5 FILM ON FILM ON SiSi SUBSTRATESUBSTRATE
00.5
11.5
22.5
33.5
44.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14Normalized thickness (with b/2)
Ener
gy (J
/m2 )
Strained layerStrained layer with dislocation
Critical thickness of 22 Å = four film
layers
5b THRESHOLD APPROACH5b THRESHOLD APPROACH
Energy per unit interfacial area of the dislocation is taken at
a distance of 5b/2
from the centre of the dislocation (corresponding to an interfacial width of 5b)
When the energy of the growing film (per unit area of the interface) exceeds
this threshold value dislocation is nucleated
For the Ge0.5
Si0.5
/Si system the critical thickness corresponding to the 5b
threshold is 20 film layers (~ 110 Å)
The experimentally determined value for the Ge0.5
Si0.5
/Si system is 18 film
layers (~ 100 Å)
Other examplesOther examples
0123456789
0.65 0.7 0.75 0.8 0.85 0.9
x in CuxAu(1-x)
h c (Å
)
Simulation
Theory
Comparison between theory and finite element simulation of the critical thickness for the onset of misfit dislocations in the Cux Au(1-x) /Ni system as a function of copper content x in the film.
Film/Substrate Co/Cu Pt/Au Cr/Nihc
(experimental) (Å) 13 10 <10hc
(FEM simulated) (Å) 11 7 8
Comparison between FEM simulation and experimental results of critical thickness (hc
) for the nucleation of a dislocation in a coherently strained epitaxial film.
Other applicationsOther applications
1.0E-08
1.5E-08
2.0E-08
2.5E-08
3.0E-08
3.5E-08
0 10 20 30 40 50 60
TheorySimulation
Total energy of a system (in J/m) of two dislocations, as a function of their separation distance (in b).
Limitations of the current theoriesLimitations of the current theories
The energy of a growing film is assumed to be a linear function of the
thickness
The substrate is assumed to be rigid and only the energy of the film is taken
into account (when the film is just a few monolayers thick the energy stored in the substrate is about 10% of the total energy but this value goes to about 30% for growth of 20 layers)
Even though the energetics
of the substrate is ignored the energy of the whole
dislocation is taken into account
physically this is unacceptable as the tensile part of the coherency stresses relieve the compressive part of the dislocation stresses and vice-versa.
The full dislocation energy is used in calculation even though the dislocation is not the one present in an infinitum with ‘antisymmetry’
[x
(x,y) = x
(x,y), y
(x,y) = y
(x,y)] between compressive and tensile stress fields
The edge dislocation is an interfacial dislocation with different material properties above and below the interface
this aspect is ignored in standard calculations
Compressive stress in the filmTensile stress field of the edge dislocation
Compressive stress field of the edge dislocation
This alleviates this
Tensile stress field in the substrate
This has to be present to alleviate this
8.6
4.0
2.8
1.6
0.5
-0.6
-1.8
-3.0
-4.2
-8.2
All contour values are in GPa
272 Å
10 layers
163
Å
Free surface8.6
4.0
2.8
1.6
0.5
-0.6
-1.8
-3.0
-4.2
-8.2
All contour values are in GPa
272 Å
10 layers
163
Å
Free surfaceFree surface
Simulated x contours
Considerable asymmetry between the compressive and tensile stress fields of
the edge dislocation
Advantages of the current simulationsAdvantages of the current simulations
(i)
As growth progresses, the upper layers are expected to be more relaxed energetically as compared to the layers closer to the substrate and this aspect is captured in the simulation
(ii)
The simulation calculates the energy of the interfacial dislocation in a film/substrate system (with separate material properties for the
film and substrate), wherein there is considerable asymmetry between the tensile and compressive stress fields of the dislocation and hence the energy of a interfacial dislocation is different from that of a dislocation in a bulk crystal
(iii) The methodology adopted automatically takes into account the interaction between the film coherency and dislocation strain fields
(iv) Equilibrium critical thickness is calculated taking into account the energy of the entire system and not just the film as in many models
Limitations of the current simulationLimitations of the current simulation
(i)
E and
values calculated from single crystal data (bulk values) have been used for the thin films
future work will try to take thin film effects into account(ii)
Linear interpolation is used to calculate the lattice parameter and the material properties of the alloy films
(iii)
For computational convenience the thickness and width of the substrate considered is small as compared to the real physical dimensions
(iv)
For computational convenience and for comparison with available experimental data highly strained films have been considered in the current analysis and the model will have to be tested for low strain systems wherein the critical thickness values are very large
(v)
Core structure & energy of the dislocation are ignored in the simulation
ways will be sought to meaningfully incorporate core energy into calculations in a simple way (initially this would be attempted without actually simulating the core structure)
(vi)
The convergence of the solution cannot be checked by mesh refinement is not possible as the mesh dimension is already the interatomic spacing
CONCLUSIONSCONCLUSIONS
1)
Misfit strain is fed as the stress-free Eshelby
strain in the finite element model to effectively simulate lattice mismatch strain in
an
epitaxial layer
2)
Feeding the stress-free strain corresponding to the introduction of an extra plane of atoms can simulate an edge dislocation
3)
The equilibrium critical thickness for epitaxial films can be determined by a combined simulation of a growing film with a edge dislocation (The results obtained show a close correspondence with the standard theoretical expressions and experimental results for epitaxial metallic films)
4)
For the GeSi/Si
system, the experimental values match satisfactorily with that of the 'threshold approach', when the energy per unit area of the simulated dislocation is taken at a characteristic distance (xch
) of 5b.
1.
F.C. Frank and J. Van der
Merve, Proc. Roy. Soc. A 198, 216 (1949).
2.
S.C. Jain, A.H. Harker
and R.A. Cowley, Philos. Mag. A
75, 1461 (1997).
3.
J.P. Hirth
and J. Lothe, Theory of Dislocations (McGraw-Hill, New York, 1968).
4.
W. Bollmann, Crystal Defects and Crystalline Interfaces (Springer-Verlag, Berlin, 1970).
5.
J.W. Matthews, Misfit Dislocations in Dislocations in Solids, edited by F.R.N. Nabarro, (North-Holland, Amsterdam, 1979).
6.
R. People and J.C. Bean, Appl. Phys. Lett. 47, 322 (1985).
7.
T. Mura, Micromechanics of Defects in Solids (Martinus
Nijhoff, Dordrecht, 1987).
8.
J.W. Cahn, Acta
Metall. 10, 179 (1962).
9.
J.W. Matthews and A.E. Blakeslee, J. Cryst. Growth, 27, 118 (1974).
Selected ReferencesSelected References
ReferencesReferences
1.
"Critical thickness of equilibrium epitaxial thin films using finite element method"
Anandh Subramaniam
Journal of Applied Physics, 95, p.8472, 2004.
2.
"Analysis of thin film growth using finite element method"
Anandh Subramaniam
and N. Ramakrishnan
Surface and Coatings Technology, 167, p.249, 2003.
3.
"FEM Simulation of Dislocations"
Anandh Subramaniam and N. Ramakrishnan
Proceedings of the 8th International Symposium on Plasticity and Impact Mechanics (IMPLAST-2003) (Ed.: N.K. Gupta), (Refereed paper), Phoenix Publishing House Pvt. Ltd., New Delhi, p.291, 2003.
FINITE ELEMENT METHODFINITE ELEMENT METHOD ~ piecewise approximate physics~ piecewise approximate physics
SpatioSpatio--temporal temporal discretizationdiscretization
of a problemof a problem
A procedure that transforms insolvable calculus problems into A procedure that transforms insolvable calculus problems into approximately equivalent but solvable algebra problemsapproximately equivalent but solvable algebra problems
GOVERNING EQUATION (Containing interior loads)
Differential equation
Integral equation
BOUNDARY CONDITIONS
DOMAIN
SYSTEM
DOMAIN
BOUNDARY CONDITIONS
GOVERNING EQUATION
DISCRETIZED
BOUNDARY CONDITION
INTERELEMENT BOUNDARY CONDITION
ALGEBRAIC EQUATIONS
[K] {a} = F
Stiffness Matrix
Load Vector
SOLID MECHANICSSOLID MECHANICS
EQUILIBRIUM EQUATIONS
DISPLACEMENT RELATIONS
CONSTITUTIVE RELATIONSHIP
ELASTICITY
xxyx fyx
yxxE
)1( 2 xyxy
E
)1(2
xu
x
xv
yu
xy
xfyuE
yxvE
xuE
2
22
2
2
2 )1(2)1(2)1( GOVERNING EQUATION