Transcript Theory of critical thickness estimation

Theory of critical thickness
estimation
B89202009
彭成毅
Outline
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SiGe Alloys
Pseudomorphic Growth and Film Relaxation
Theory of critical thickness estimation
Models : Van der Merwe (1962)
Matthhew – Blakeslee (1974)
People – Bean (1985)
Huang (2000)
SiGe Alloys
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Si and Ge
- Group Ⅳ elemental semiconductors
- Diamond lattice structure
Vegard’s rule
- a(Si1-XGeX)＝aSi＋x(a Ge－a Si)
a – lattice constant
x - Ge fraction
SiGe Allloys
Unit cell of the diamond lattice
Theoretical and experimental lattice constant of a
Si1-xGex alloy as a function of Ge fraction
Pseudomorphic Growth and
Film Relaxation
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Lattice mismatch between Si (a＝5.431A) and Ge (a＝5.658A)
- 4.17% at 300k
SiGe film on thick Si substrate
- Initial growth
- Pseudomorphic
- SiGe film is forced to adopt Si smaller
lattice constant
- Desired result for most device application
- Reach “critical thithiness”
- Relax
- Strain energy too large to maintain local equilibrium
- SiGe film relaxes via misfit dislocation formation
Pseudomorphic Growth and
Film Relaxation
Schematic 2-D representation of both
strained and relaxed SiGe on a Si substrate
Schematic representation of misfit dislocation
formed at the Si／SiGe growth interface
Theory of critical thickness
estimation
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The existence of critical thickness was first
detailed by Vand der Merwe. 「1」
Theoretically, many different models have
been established to predict the critical
thickness for strained layers.
The most celebrated ones are
Matthhew – Blakeslee (1974) 「2」
People – Bean (1985) 「3」
Huang (2000) 「4」
Theory of critical thickness
estimation
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The models of Matthhew – Blakeslee
Principle :
Minimize the total energy to get the
thermal equilibrium state
Result :
tc=5.5/x ln(tc)
Theory of critical thickness
estimation
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The models of People – Bean
Principle :
The critical thickness is determined
by the condition that the strain
energy is equal to the minimum of
dislocation energy.
Theory of critical thickness
estimation
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The dislocation energy is given by
 (
D

Gb
h
) ln( )
b
8 2a( x)
The stress energy is
1 
  2G (
) hf
1 
H
2
Theory of critical thickness
estimation
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Results
Theory of critical thickness
estimation
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The large lattice mismatch of about 4% between
germanium and silicon has limited the growth of
high-quality SiGe alloys to within a certain thickness,
the so-called critical thickness, beyond which misfit
dislocations start to generate.
To circumvent this limitation, a novel approach via
substrate engineering (i.e., tailoring the substrate to
form a finite dimension in the vertical and/or lateral
directions) has been proposed to transfer or dilute
the misfit strain.
Theory of critical thickness
estimation
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The dislocation energy in the case of an
epilayer situated on a bulk substrate is
b 2 (1  v cos2  ) h
Edis 
ln(
)
b
4 (1  )
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However the dislocation energy in the case
of a compliant structure must be
reconsidered.
Theory of critical thickness
estimation
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A compliant structure :
The dislocation is refined by the multiple
image dislocation :
2C0 ln(hr / b)  C0 ln(h1 / b)  C0 ln(h2 / b)  (image)
Theory of critical thickness
estimation
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where
 (im age)  C
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(m H  h1)  (m H  h2)
(nH  h1)(nH  h2)
)  C 0 ln(
)



(
m

1
)
H

h
1
(
m

1
)
H

h
2
h
1
h
2
m 1
n
0
ln(
When only the first-order image dislocations
are considered, the total dislocation energy
becomes


Edis  C 0 ln(h1h2 / H  h1H  h2  b )
1
2
Theory of critical thickness
estimation
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For the elastic strain energy
- in the case of a compliant structure
Es  B1 f 0 h1 /  1  B2 f 0 h2 /  2
2
where
2
2
f 0  (a1  a 2) / a 2
  [1  ( B1h1a1 ) /( B2h2a2 )]
2
1
2
2
Theory of critical thickness
estimation
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Using the PB model condition :
B1 f 0 /  1  B 2 f 0 h2 /  2
2
2
2
2
C0
(
) ln(hr / b)
2 2a ( x )
References
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[1]J. H. Van der Merwe, J. Appl. Phys,
34, 123 (1962)
[2]J. W. Mattehews and A. E. Blakeslee,
J. Cryst. Growth 27,118(1974)
[3]R. People and J. Bean, Appl. Phys.
Lett. 47, 322 (1985)
[4]F. Y. Huang, Phys. Rev. Lett. 85, 787
(2000)