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

anandh333@rediffmail.com, anandh@am.iitd.ernet.inhttp://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