FINITE ELEMENT ANALYSIS OF EPITAXIAL
THIN FILM GROWTH
ANANDH SUBRAMANIAM
Department of Applied Mechanics
INDIAN INSTITUTE OF TECHNOLOGY DELHI
New Delhi- 110016
Ph: (+91) (11) 2659 1340, Fax: (+91) (11) 2658 1119
anandh333@rediffmail.com, anandh@am.iitd.ernet.in
http://web.iitd.ac.in/~anandh
December 2006
OUTLINE
 EPITAXIAL THIN FILMS
 FILM AND DISLOCATION ENERGETICS
 FEM SIMULATION OF FILM GROWTH
 FEM SIMULATION OF A MISFIT DISLOCATION
 CRITICAL THICKNESS
 CONCLUSIONS
EPITAXIAL THIN FILMS
EXAMPLES
FILM
~100 Å
SUBSTRATE
~100 
GaAsP
GaAs
INRTERFACIAL EDGE
DISLOCATION
InGaAs
GaP
Co
Ni
Extra-"half" plane
Metallic
Au
Ag
Semiconductor
GeSi
Si
There is a 4.2% difference in the lattice constants of Si and Ge.
Therefore when a layer of Si1-xGex is grown on top of Si, it has a
bulk relaxed lattice constant which is larger than Si.
Si1-xGex 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-xGex 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-xGex on Si
Interfacial misfit dislocation
FILM AND DISLOCATION ENERGETICS
FILM ENERGY
Eh = 2G[(1 + )/(1 - )] fm2 h
 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 parameter
as : substrate lattice parameter
DISLOCATION ENERGY
2

Gb
 0
Edl =
2  ln

4 (1   ) 
 b



 Edl – Energy per unit length of
dislocation line
 b – Modulus of the Burgers vector
 0 - size of the control volume ~ 70b
(edge dislocation)
TOTAL ENERGY
Etot = Eh + Edl (algebraic addition)
film energy per unit area & dislocation energy per unit length)
FINITE ELEMENT ANALYSIS
( Stress free strain)
FILM
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
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
y
68 Ele m ent s
x
Symmetry line
(symmetric half
of the domain taken
for analyses)
99 Elements
Region of the domain (A)
where Eshelby strain
is imposed to simulate
the strained film
Region of the domain (B)
where Eshelby strain
is imposed to simulate
the dislocation
Ge0.5Si0.5 FILM ON Si SUBSTRATE
(MISFIT STRAIN = 0.0204)
x AFTER THE GROWTH OF ONE LAYER ( ~5 Å)
50 Å
FILM
SYMMETRY LINE
EDGE 
SUBSTRATE
230 Å
Zoomed region near the edge
(MPa)
Ge0.5Si0.5 FILM ON Si SUBSTRATE
(MISFIT STRAIN = 0.0204)
x AFTER THE GROWTH OF FIVE LAYERS
145 Å
FILM
SUBSTRATE
EDGE 
SYMMETRY LINE
190 Å
(MPa)
STRESS FIELD OF AN EDGE DISLOCATION
X - PLOT OF THEORETICAL EQUATION
Gb y(3x2  y2 )
x =
2 (1 ) (x2  y2 )2
10.00
9.00
8.00
7.00
6.00
5.00
4.00
3.00
2.00
1.00
0.00
-1.00
-2.00
-3.00
-4.00
-5.00
-6.00
-7.00
-8.00
-9.00
-10.00
5.00
4.00
y (Angstroms)
3.00
2.00
1.00
0.00
-1.00
-2.00
-3.00
-4.00
-5.00
-5.00
-3.00
-1.00
1.00
x (Angstroms)
3.00
5.00
(Contour values x 104 Mpa)
STRESS FIELD OF AN EDGE DISLOCATION
X - PLOT OF THEORETICAL EQUATION
Gb y (3x2  y2 )
x 
2 (1 ) (x2  y2 )2
y (Å) →
Material → Al
b = 2.86 Å
G = 26.18 GPa
 = 0.348
(GPa)
x (Å) →
(Contour values in GPa)
STRESS FIELD OF A EDGE DISLOCATION
X – FEM SIMULATED CONTOURS
28 Å
FILM
SUBSTRATE
b
27 Å
(MPa)
(x & y original grid size = b/2 = 1.92 Å)
6
+
5
7
8
-
92.3 Å
4
5
+
1
2
4
3
(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
y – FEM SIMULATED CONTOURS
Plot of y of an edge
dislocation with
Burgers vector b:
FEM simulated contours.
(GPa)
59.7 Å
+
5
6
7
140 Å
3
5
1
+
2
6
4
3
-
140 Å
7
20.0
6
15.0
5
2.5
4
0
3
-2.5
2
-15.0
1
-20.0
(GPa)
Plot of y of an edge dislocation
with Burgers vector b: Contours
obtained from equation:
 Gb y ( x 2  y 2 )
y =
2 (1   ) ( x 2  y 2 ) 2
DISLOCATION – ENERGY/AREA OF INTERFACE
 (b) PEAK THRESHOLD
3.5E-01
3.0E-01
2.5E-01
2
(J/m )
Energy per unit area of interface
4.0E-01
2.0E-01
1.5E-01
1.0E-01
 5b THRESHOLD
5.0E-02
0.0E+00
0
2
4
6
8
10
12
14
Distance from the centre of the dislocation (in b/2 spacings)
16
Ge0.5Si0.5 FILM ON Si SUBSTRATE WITH EDGE DISLOCATION
x AFTER THE GROWTH OF FIVE LAYERS
80 Å
FILM
SUBSTRATE
SYMMETRY LINE
EDGE 
240 Å
Zoomed region near the edge
MPa
CRITICAL THICKNESS (hc) - GeSi/Ge
GLOBAL ENERGY MINIMIZATION –
EQUILIBRIUM appears suitable for metallic systems
hc (nm) =
2
1.175x10
ln8.9hc (nm)
fm
[1]
METASTABLE FILMS - 5b approach
appears suitable for semiconductor systems
hc (nm) =
1.9 x10
2
fm
3
ln2.5hc (nm)
[2]
[[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.
Ge0.5Si0.5 FILM ON Si SUBSTRATE
CONSIDERING ONLY FILM ENERGETICS ( no substrate)
Energy of film (J/m2)
3
E film
2.5
E film with
dislocation
2
1.5
Critical thickness
of 12 Å
= three film layers
1
0.5
0
1
2
3
4 5 6 7 8 9 10 11 12 13 14
Normalized thickness (with b/2)
Ge0.5Si0.5 FILM ON Si SUBSTRATE
CONSIDERING THE TOTAL ENERGY OF THE SYSTEM
(film and the substrate)
4.5
Strained layer
4
Strained layer with dislocation
Energy (J/m 2)
3.5
3
2.5
2
Critical thickness
of 22 Å = four film
layers
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9 10 11
Normalized thickness (with b/2)
12
13
14
5b 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 Ge0.5Si0.5/Si system the critical thickness corresponding to the 5b
threshold is 20 film layers (~ 110 Å)
 The experimentally determined value for the Ge0.5Si0.5/Si system is 18 film
layers (~ 100 Å)
hc (Å)
Other examples
9
8
7
6
5
4
3
2
1
0
0.65
Simulation
Theory
0.7
0.75
0.8
0.85
0.9
x in CuxAu(1-x)
Comparison between theory and finite element simulation of the critical thickness
for the onset of misfit dislocations in the CuxAu(1-x) /Ni system as a function of
copper content x in the film.
Comparison between FEM simulation and experimental results of critical thickness
(hc) for the nucleation of a dislocation in a coherently strained epitaxial film.
Film/Substrate
Co/Cu
Pt/Au
Cr/Ni
hc (experimental) (Å)
13
10
<10
hc (FEM simulated) (Å)
11
7
8
Other applications
3.5E-08
3.0E-08
2.5E-08
Theory
Simulation
2.0E-08
1.5E-08
1.0E-08
0
10
20
30
40
50
60
Total energy of a system (in J/m) of two dislocations, as a function of their
separation distance (in b).
Limitations 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
This alleviates this
Tensile stress field of
the edge dislocation
Compressive stress in the film
Tensile stress field in the substrate
This has to be present to alleviate this
Compressive stress field
of the edge dislocation
Considerable asymmetry
between the compressive
and tensile stress fields of
the edge dislocation
Free surface
10 layers
8.6
4.0
2.8
163 Å
1.6
0.5
Simulated x contours
-0.6
-1.8
-3.0
All contour values are in GPa
-4.2
272 Å
-8.2
Advantages 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 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
CONCLUSIONS
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.
Selected References
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).
References
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 METHOD
~ piecewise approximate physics
Spatio-temporal discretization of a problem
A procedure that transforms insolvable calculus problems into
approximately equivalent but solvable algebra problems
DOMAIN
SYSTEM
GOVERNING EQUATION
(Containing interior loads)
Differential
equation
Integral
equation
BOUNDARY
CONDITIONS
DOMAIN
BOUNDARY
CONDITIONS
DISCRETIZED
BOUNDARY
CONDITION
INTERELEMENT
BOUNDARY
CONDITION
GOVERNING
EQUATION
ALGEBRAIC
EQUATIONS
[K] {a} = F
Stiffness
Matrix
Load
Vector
SOLID MECHANICS
ELASTICITY
EQUILIBRIUM
EQUATIONS
CONSTITUTIVE
RELATIONSHIP
DISPLACEMENT
RELATIONS
GOVERNING
EQUATION
 x  xy

  fx
x
y
x 

E
 x   y
2
(1   )
u
x 
x
 xy

 xy 
 
E
 xy
2(1   )
u v


y x
E  2u
E
 2v
E  2u


  fx
2
2
2
2(1   ) xy 2(1   ) y
(1   ) x