Journal of Crystal Growth 240 (2002) 6–13
Calculated strain energy of hexagonal epitaxial thin films
Jianyun Shena,b,*, Steven Johnstona, Shunli Shangb, Timothy Andersona
a
University of Florida, Department of Chemical Engineering, Gainesville, FL 32611-6005, USA
b
Research Institute for Non-ferrous Metals of Beijing, 100088 Beijing, China
Received 21 June 2001; accepted 21 November 2001
Communicated by C.R. Abernathy
Abstract
Generalized formulae for the strain energy of hexagonal thin films on both hexagonal and rhombohedral substrates
have been developed. These formulae require knowledge of the elastic stiffness coefficients and the lattice parameters of
the film and only the lattice parameters of the substrate. Example calculations of the strain energy present in the
strained film-substrate material combinations GaN/Al2O3 and GaN/LiGaO2 are presented for different film
crystallographic directions and rotation with respect to the substrate. Finally, phase equilibrium calculations are
performed for the Ga–N binary system which show the substantial influence of strain energy on equilibrium in the
system. r 2002 Elsevier Science B.V. All rights reserved.
PACS: 82.60.Hc; 82.60.s; 64.; 82.20.Wt
Keywords: A1. Phase diagrams; B1. Gallium compounds; B1. Nitrides
1. Introduction
The increased interest in heteroepitaxy of
semiconductor films has motivated the development of a reliable method to calculate the strain
energy produced by the mismatch between the
epitaxial growth layer and the underlying substrate. Lattice mismatch as well as the subsequent
strain can have significant influence on the extent
of dislocation generation, the morphological,
electrical, and optical properties, and the phase
equilibrium behavior of the growth layer. While
*Corresponding author. Department of Chemical Engineering, University of Florida, Gainesville, FL 32611-6005, USA.
Tel.: +1-352-392-2420; fax: +1-352-392-9513.
E-mail address: [email protected]fl.edu (J. Shen).
cubic film/cubic substrate combinations have been
analyzed previously [1], systems involving hexagonally oriented material as either the film or
substrate have not been thoroughly investigated to
date. Examples of important semiconductor materials that exist in the hexagonal crystal structure
include the wide band gap compound semiconductors GaN, SiC, BN and many II–VI semiconductors such as CdS. These materials are
promising candidates for use in optoelectronic
applications including visible and ultraviolet emitters, high power–high temperature electronics, and
in the case of cBN as an abrasive [2,3]. The lack of
large area bulk crystals for most of these
hexagonal compounds has necessitated the use of
alternative substrates with different lattice parameters and thermal expansion coefficients.
0022-0248/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 2 2 0 9 - 6
J. Shen et al. / Journal of Crystal Growth 240 (2002) 6–13
Furthermore, device applications often require
alloying that introduces additional stress in the
film. As an example, alloying GaN with InN to
lower the band gap energy increases the value of
the lattice parameter.
The strain energy stored in a film grown on a
mismatched substrate increases the Gibbs energy
of the material. This strain energy contribution
modifies such equilibrium quantities as melting
temperature, vapor pressure, and miscibility limits.
While the growth of hexagonal materials has been
extensively studied experimentally, quantitative
calculation of the inherent strain energy has not
been fully performed [4]. Furthermore, the effect
of the strain energy on the resulting equilibrium
has not been addressed. In this work elastic
compliance equations are developed and their
relationship to the overall strain energy of a
hexagonally oriented film and substrate are presented. These general relations are then applied to
the growth of GaN on different substrates.
2. Analysis
The development of the general equations
necessary to calculate the strain energy of a
hexagonal film on either a hexagonal or rhombohedral substrate is presented in this section.
The general form of strain energy for an elastic
body can be written as
U¼
1
2ð sx ex
þ sy ey þ sz ez þ txy gxy
þ tyz gyz þ tzx gzx Þ;
ð1Þ
where U is the strain energy per unit volume, sx ;
sy and sz are the normal stresses, txy tyz and tzx are
the shear stresses, ex ; ey and ez are the normal
strains, and gxy ; gyz and gzx are the shear strains.
Generally only the normal strain and stress in two
perpendicular axes are considered to be produced
as a result of lattice mismatch in epitaxial growth.
Hook’s law can be written as
ex ¼ s011 sx þ s012 sy ;
ð2Þ
ey ¼ s022 sy þ s012 sx ;
ð3Þ
s0ij
where are the elastic compliances of the plane of
interest. These values can be obtained from the
7
fourth-rank transform of the sij values in the
principle plane. The fourth-rank transform of the
sij values in the principle plane is
3 X
3 X
3 X
3
X
s0ijkl ¼
aim ajn akp alq smnpq ;
ð4Þ
m¼1 n¼1 p¼1 q¼1
where aim ; ajn ; akp ; and alq are the direction cosines
of the axes of the plane studied with respect to
those of the principle plane. It should be noted
that sij is actually the reduced representation of the
fourth tensor sijkl according to Voigt’s notation [5].
In the case where the elastic compliances are
unknown but the elastic stiffness coefficients, cij ;
are available, the sij values can be calculated from
cij since the matrix of sij and that cij are mutually
inverted.
The matrix of sij for the hexagonal crystal lattice
is given by Lekhnitskii [5] as:
3
2
s11 s12 s13 0
0
0
7
6
7
6 s12 s11 s13 0
0
0
7
6
7
6s
0
0
7
6 13 s13 s33 0
7
6
7
6 0
0
0
s
0
0
44
7
6
7
6 0
0
0
0 s44
0
5
4
0
0
0
0
0 2ðs11 s12 Þ
For the hexagonal lattice, the elastic compliances
in the plane of interest can be written as
s011 ¼ ð1 l32 Þ2 s11 þ l34 s33 þ l32 ð1 l32 Þð2s13 þ s44 Þ;
ð5Þ
s012 ¼ s11 ðl1 m1 l2 m2 Þ2 þ s12 ðl1 m2 l2 m1 Þ2
þ s13 ½ð1 l32 Þm23 þ l32 ð1 m23 Þ þ s33 l32 m23
þ s44 l3 m3 ðl2 m2 l1 m1 Þ:
ð6Þ
s022 ¼ ð1 m23 Þ2 s11 þ m43 s33
þ m23 ð1 m23 Þð2s13 þ s44 Þ;
ð7Þ
where l1 ; l2 and l3 are the direction cosines of the x0
axis of the plane studied with respect to the x; y;
and z axes of the principle plane. Similarly m1 ; m2 ;
and m3 are that of y0 axis. The relationship
between sij and cij in the principle plane can be
written as
s11 ¼
c11 c33 c213
;
ðc11 c12 Þðc11 c33 þ c12 c33 2c213 Þ
ð8Þ
J. Shen et al. / Journal of Crystal Growth 240 (2002) 6–13
8
s12 ¼
c213 c12 c33
;
ðc11 c12 Þðc11 c33 þ c12 c33 2c213 Þ
s13 ¼
c13
;
c11 c33 þ c12 c33 2c213
ð10Þ
s33 ¼
c11 þ c12
;
c11 c33 þ c12 c33 2c213
ð11Þ
s44 ¼
1
:
c44
ð12Þ
ð9Þ
Once these relationships are known calculation
of a meaningful elastic energy term can made. Two
different planes of film growth were considered,
the (0 0 0 1) basal plane and the (1 0 1% 0) prism
plane (see Fig. 1). Since the (0 0 0 1) basal plane of
a hexagonal crystal is isotropic, the x and y axes
can be chosen coincident with those of the
principle plane. Therefore,
ex ¼ s11 sx þ s12 sy ;
ð13Þ
ey ¼ s22 sy þ s12 sx :
ð14Þ
For a growth on the (0 0 0 1) basal plane there
are two possible film/substrate crystallographic
relationships. The first one occurs when the planes
of film growth and the underlying substrate are
both hexagonal. The second one is when either the
film growth plane or the substrate is not hexagonal
(e.g., zincblende, rhombohedral). Each case was
analyzed.
5.185 Å
[0001]
a3
[1010]
(1010)
-a1
-a2
a2
[1210]
[1210]
a1
where a and a0 are the lattice parameters of the
growth layer and the substrate in the basal plane,
respectively. Using Eqs. (13) and (14), the normal
stresses to the film/substrate interface can then be
written as
1
a a0
s x ¼ sy ¼
:
ð16Þ
s11 þ s12 a0
Thus from Eq. (1) the strain energy per unit
volume is simply
1
a a0 2
:
ð17Þ
U¼
s11 þ s12
a0
In the second case, when either the growth layer
or substrate is not hexagonal, the strain in the x
and y directions will not be equal (ex aey ) as a
result of the departure from symmetry. From
Eq. (13) and (14), the following relations are
evident:
s11 ex s12 ey
rx ¼ 2
;
ð18Þ
s11 s212
ry ¼
-a3
Fig. 1. Basal plane (0 0 0 1) and prism plane (1 0 1% 0) of a
hexagonal crystal. The [1% 0 1 0] direction is parallel to the lower
basal plane.
s11 ey s12 ex
s211 s212
ð19Þ
with a resulting strain energy expression of
U¼
3.189 Å
(0001)
When both the film and substrate are hexagonal
and the basal plane of each is used in deposition,
the strain in the x and y directions are equal as
required by symmetry and can be written as
a a0
ex ¼ ey ¼
;
ð15Þ
a0
s11 ðe2x þ e2y Þ 2ex ey s12
2ðs211 s212 Þ
:
ð20Þ
Deposition on the [1 0 1% 0] prism plane in the
[1% 2 1% 0] and [0 0 0 1] directions (defined as x0 and y0
direction of the plane studied as shown in Fig. 1)
was also considered. On this plane the direction
cosines of the x0 and y0 axes are l1 ¼ 0; l2 ¼ 1; l3 ¼
0; m1 ¼ 0; m2 ¼ 0; and m3 ¼ 1: Therefore the
elastic compliances can be transformed from the
plane of interest to the principle plane using
s011 ¼ s11 ;
ð21Þ
s012 ¼ s13 ;
ð22Þ
s022 ¼ s33 ;
ð23Þ
J. Shen et al. / Journal of Crystal Growth 240 (2002) 6–13
9
The strains in the [1% 2 1% 0] and [0 0 0 1] directions
are thus
Table 1
( of GaN, Al2O3, and LiGaO2
Lattice constant (A)
ex0 ¼ sx0 s011 þ sy0 s012 ¼ sx0 s11 þ sy0 s13 ;
ð24Þ
Material
a
b
c
ey0 ¼ sy0 s022 þ sx0 s012 ¼ sy0 s33 þ sx0 s13
ð25Þ
and the normal stresses are
ex0 s33 ey0 s13
rx 0 ¼
;
s11 s33 s213
GaN (wurtzite)
Al2O3 (wurtzite)
LiGaO2 (orthorhombic)
3.189
4.758
5.402
F
F
6.372
5.185
12.991
5.007
ð26Þ
ry 0 ¼
ey0 s11 ex0 s13
:
s11 s33 s213
ð27Þ
cij (Gpa)
It directly follows as before that the strain
energy per unit volume is:
U¼
e2x0 s33 þ e2y0 s11 2ex0 ey0 s13
2ðs11 s33 s213 Þ
:
ð28Þ
If a hexagonal lattice structured film with lattice
parameters a and c is grown with its prism plane
on a rectangular face of a substrate with lattice
parameters a0 and c0 ; then the system’s strain is
ex0 ¼ ða a0 Þ=a0 ; ey0 ¼ ðc c0 Þ=c0 ; and its resulting
strain energy is
U¼
Table 2
Elastic stiffness coefficients cij and compliances sij of GaN
ða a0 =a0 Þ2 s33 þ ðc c0 =c0 Þ2 s11
2ðs11 s33 s213 Þ
2ða a0 =a0 Þðc c0 =c0 Þs13
:
2ðs11 s33 s213 Þ
ð29Þ
Eq. (29) is general and can thus be used for all the
other prism planes as well as ‘‘internal’’ planes
such as (1 2% 1 0) because of the isotropic nature of
the basal plane.
3. Lattice mismatch and strain energy calculation
Three independent sets of calculations were
performed to demonstrate the strain energy
calculation of a GaN film grown on Al2O3 and
LiGaO2 substrates, using the fundamental data
provided in Tables 1 and 2 [6]. Table 3 summarizes
the strain energy calculated for three different film/
substrate orientations and their corresponding
lattice mismatches. From both Eqs. (17), (20) and
(29) and the data in Table 3, it is clear that an
increase in the lattice mismatch increases the
system’s strain energy.
c11
374
c12
106
c13
70
c33
379
c44
101
sij (1/GPa) s11
s12
s13
s33
s44
0.00297 0.00077 0.00041 0.00279 0.00990
The minimum lattice mismatch in the GaN/
Al2O3 system occurs when the GaN film is rotated
301 with respect to the sapphire substrate so that
the (0 0 0 1)8(0 0 0 1), [1% 0 1 0]8[1 2% 1 0] GaN/Al2O3
interface is formed and a 3:1 film/substrate unit
cell ratio is used. For instance, the lattice mismatch
for no rotation and 301 rotation with a 3:1 GaN/
Al2O3 unit cell ratio are 33% and 16.1%,
respectively. Rotation of GaN films on Al2O3
has been observed experimentally [7] and is
illustrated in Fig. 2. The strain energy of the
16.1% mismatched system is 11.755 GPa or
161.49 kJ/mol. Of course these values of lattice
mismatch exceeds the limits to maintain elastic
deformation in the growth layer. Therefore,
experimentally, a relaxation of the strain energy
by large-scale dislocation formation would be
inevitable without the use of a buffer layer
(typically GaN or AlN). Consistent with experimental results and a recent set of calculations we
have performed it is energetically favorable for the
majority of the stress in the GaN/Al2O3 system to
be compensated by dislocation formation with
approximately a 2% residual strain remaining at
the interface [8].
Deposition of the (0 0 0 1)8(0 0 1), [1% 2 1% 0]8[0 1 0]
GaN/LiGaO2 system (Fig. 3) is an attractive
alternative due to its lower mismatch [9]. LiGaO2
has an orthorhombic structure consisting of an
alternate stacking of a two-dimensional array
J. Shen et al. / Journal of Crystal Growth 240 (2002) 6–13
10
Table 3
Strain energy produced by lattice mismatch
Growth layer/substrate
Growth orientation
(0 0 0 1)8(0 0 0 1) [1% 0 1 0]8[1 2% 1 0]
(0 0 0 1)8(0 0 1), [1% 2 1% 0]8[0 1 0]
(1 2% 1 0)8(0 1 0) [1 0 1% 0]8[1 0 0]
w-GaN / w-Al2O3
w-GaN /o-LiGaO2
w-GaN /o-LiGaO2
GaN
Al2O3
[1010] of GaN
Lattice mismatch (%)
Strain energy
ex
ey
GPa
kJ/mol
16.1
2.25
2.25
F
0.09
3.56
11.76
0.094
0.36
161.5
1.3
4.9
pffiffiffi
tion, the normal strains are ex ¼ ð 3 3:189 5:402Þ=5:402 ¼ 2:25%
and
ey ¼ ð2 3:189 6:372Þ=6:372 ¼ 0:09%: Thus the strain energy is
reduced by Eq. (20) to 0.094 GPa or 1.29 kJ/mol.
Another possible film/substrate orientation
for prism plane growth is the (1 2% 1 0)8(0 1 0),
[1 0 1% 0]8[1 0 0] GaN/LiGaO2 system (Fig. 4). The
normal strains
pffiffiffi for this material combination are
then ex ¼ ð 3 3:189 5:402Þ=5:402 ¼ 2:25% and
ey ¼ ð5:185 5:007Þ=5:007 ¼ 3:56%: The strain
energy for this orientation is then calculated using
Eq. (29) to be 0.359 GPa or 4.94 kJ/mol.
[1210] of Al2O3
4. Results and discussion
Fig. 2. Scheme for the deposition of the (0 0 0 1)8(0 0 0 1),
[1% 0 1 0]8[1 2% 1 0] GaN/Al2O3 system.
[1210] of GaN
[010] of LiGaO2
Interfacial bond of Ga (or Li)-N;
• Ga and Li from LiGaO 2,
• N from GaN
3.136 Å
6.372 Å
3.189 Å
GaN
3.186 Å
5.402 Å
after deformation
LiGaO2
before deformation
Fig. 3. Scheme for the deposition of the (0 0 0 1)8(0 0 1),
[1% 2 1% 0]8[0 1 0] GaN/LiGaO2 system.
consisting of oxygen tetrahedra centered Ga and
Li ions. Therefore the material has a wurtzitic
superstructure with a small departure from hexagonal symmetry as a result of the need to
accommodate metallic atoms of different size.
When GaN is grown on LiGaO2 in this orienta-
Thermodynamic equilibrium calculations were
carried out on the strained Ga–N system to
demonstrate the importance of strain energy on a
system’s final equilibrium state. Thermochemical
data for all species except for GaN(s) were taken
from the ThermoCalc version K sub94 database
[10]. The data for GaN(s) were reassessed by
Davydov and Anderson [11]. Their results were
incorporated. A strain energy term calculated from
Eqs. (16), (21) or (29) was added to the value of
G–Hser for GaN(s).
The effect of strain energy on the strained Ga–N
equilibrium binary system is shown in Figs. 5 and
6. Fig. 5 depicts equilibrium in the unstrained Ga–
N heterogeneous system at 0.1 MPa and will be
used as a reference for the strained GaN/LiGaO2
and GaN/Al2O3 equilibrium systems. Fig. 6 illustrates the effect of strain energy on the resulting
Ga–N equilibrium. A decrease in the GaN
thermodynamic stability (i.e., its melting point
temperature) results from the addition of a
positive term to the Gibbs energy expression of
J. Shen et al. / Journal of Crystal Growth 240 (2002) 6–13
(1210)
GaN
5.185 Å
(1210) GaN / (010)
LiGaO2 interface
11
3.189 Å
a3
-a1
a2
-a2
LiGaO2
a1
[1010] -a3
Fig. 4. Scheme for the deposition of the (1 2% 1 0)8(0 1 0), [1 0 1% 0]8[1 0 0] GaN/LiGaO2 system.
3000
3000
Gas
Gas
2500
Temperature (K)
Temperature (K)
2500
2000
Liquid + Gas
1500
1055 K
1000
500
0
0.0
0.2
0.4
Liquid + Gas
1500
0
0.0
0.6
0.8
1.0
GaN + Gas
GaN + Liquid
303K
Ga + GaN
0.2
(a)
N Mole Fraction
0.4
0.6
0.8
1.0
N Mole Fraction
1100
Fig. 5. T x diagram of the unstrained heterogeneous system
Ga–N as a function of temperature and N mole fraction at
0.1 MPa.
No Strain
U = 1.29 kJ/mol
U = 4.94 kJ/mol
1080
Temperature (K)
GaN. As Fig. 6 shows, at a system pressure of
0.1 MPa the melting temperature of GaN decreases by 11 and 43 K with the addition of the
strain energies associated with the (0 0 0 1)8(0 0 1),
[1% 2 1% 0]8[0 1 0] GaN/LiGaO2 and (1 2% 1 0)8(0 1 0),
[1 0 1% 0]8[1 0 0] GaN/LiGaO2 growth systems, respectively. The smaller change in the melting
temperature corresponds to less strained GaN
basal plane deposition (Fig. 3) while the larger
melting temperature is represents GaN prism
plane growth (Fig. 4). Due to the decrease in the
GaN melting point temperature the liquid+gas
1055K
1000
500
GaN + Gas
GaN + Liquid
303K
Ga + GaN
2000
Liquid + Gas
1060
1055K
1044K
1040
GaN + Liquid
GaN + Gas
1020
1012K
1000
0.0
(b)
0.2
0.4
0.6
0.8
1.0
N Mole Fraction
Fig. 6. T x diagram of the strained heterogeneous system
Ga–N on LiGaO2 at 0.1 MPa system pressure. (a) Overall
diagram; (b) Zoomed view of change of melting point.
J. Shen et al. / Journal of Crystal Growth 240 (2002) 6–13
12
stability domain will increase as the film/substrate
strain energy is increased. However, since the
strain energy only affects solid GaN there is no
change in the dew point line. Furthermore, the
GaN+liquid/GaN+gas as well as the GaN+
liquid/Ga+GaN equilibrium lines remain unchanged because the strain energy affects the
GaN in both phases equally. Thus its impact on
each phase cancels.
It is desirable in most semiconductor deposition
techniques including CVD, MOCVD, and MBE to
have only solid and gas phases present. Thus, from
a thermodynamic viewpoint, deposition of GaN
on LiGaO2 at near atmospheric conditions must
occur below 1012 or 1044 K (1055 K in the unstrained
case) and Ga:N molar ratios less than unity.
Fig. 7 shows the thermodynamic equilibrium
deposition of GaN on Al2O3. A striking difference
between Figs. 5 and 7 can be seen immediately.
The strain energy of the GaN film deposited on
Al2O3, 161.5 kJ/mol, is so large that it prevents
GaN from being a stable solid equilibrium phase
at any temperature or mole fraction. Other
calculations revealed that the maximum allowable
strain energy that would still result in a region of
GaN stability is E89 kJ/mol. Hence the strain
energy present when GaN is deposited on sapphire
is over 80% too large for solid GaN stability. The
phase diagrams shown in Fig. 7 are experimentally
inaccurate because a high dislocation density
would occur at this large of a lattice misfit which
would significantly relax the strained growth layer.
Therefore the phase diagram should be calculated
by combining the strain and the dislocation
energies with the Gibbs energy of GaN. This
calculation will be published in our next paper.
A general relationship between the decrease of
the GaN melting point temperature and the lattice
mismatch at 0.1 MPa system pressure is presented
in Fig. 8. As shown in Fig. 5, the maximum
possible change in the melting temperature, and
thus the limit of GaN stability, is 752 K (1055–
303 K). Therefore, the maximum symmetric strain
the Ga–N system can possess and still have solid
GaN stability is when ex and ey equal 0.12. Elastic
incorporation of this much lattice mismatch
induced strain energy to enable epitaxial growth
is unreasonable and would experimentally result in
dislocation formation and strain relief. Changes in
the GaN melting point temperature are evident
even at lower film strain energy values characteristic of high quality films.
3000
Gas
Temperature (K)
2500
2000
Liquid + Gas
1500
1000
500
303K
Ga + Gas
0
0.0
0.2
0.4
0.6
0.8
1.0
N Mole Fraction
Fig. 7. T x diagram of the strained heterogeneous system
Ga–N on Al2O3 at 0.1 MPa system pressure.
Fig. 8. Generalized T x diagram of the strained heterogeneous system Ga–N at 0.1 MPa system pressure.
J. Shen et al. / Journal of Crystal Growth 240 (2002) 6–13
5. Conclusions
A series of generalized equations have been
developed from fundamental stress/strain principles to calculate the lattice mismatch induced
strain energy of a hexagonal film on hexagonal
and rhombohedral substrates. Addition of strain
energies to the thermodynamic equilibrium calculations of the Ga–N phase diagram shows
dramatic differences between the strained and
unstrained cases. At 0.1 MPa system pressure the
GaN melting point decreased 11 and 43 K
depending on the film/substrate orientation when
GaN was deposited on LiGaO2. Even more
interesting was the disappearance of a thermodynamically stable GaN region when the strain
energy arising from its deposition on Al2O3 was
included. This contradicts what is observed
experimentally. The dew point line and Ga melting
point temperature lines were unchanged due to the
canceling effect of the GaN in each adjacent
deposition region. At equilibrium N2 partial
pressures, deposition on LiGaO2 again decreased
the GaN melting point temperature though a large
GaN+liquid phase remained. The profound effect
of strain energy in a lattice mismatched system
indicates that the predicted equilibrium behavior
of epitaxial thin films may differ greatly from that
13
of unstrained thermodynamic systems. These
calculations suggest that strain energy has significant influence on both the experimental and
equilibrium deposition processes of semiconductor
materials. Further refinement of this model by the
inclusion of dislocation and surface energy effects
will be presented in a future paper.
References
[1] W.A. Brantly, J. Appl. Phys. 44 (1973) 534.
[2] H. Amano, T. Asahi, I. Akasaki, Jpn. J. App. Phys. Lett.
28 (1990) L150.
[3] S.N. Mohammad, A.A. Salvador, H. Morkoc, Proc. IEEE
83 (1995) 1306.
[4] R. Thokala, J. Chaudhuri, Thin Solid Films 266 (1995)
189.
[5] S.G. Lekhnitskii, Theory of Elasticity of an Anisotropic
Elastic Body, Holden-Day, San Francisco, CA, 1963.
[6] Y. Takagi, M. Ahart, T. Azuhata, T. Sota, K. Suzuki, S.
Nakamura, Physica B 219–220 (1996) 547.
[7] I. Akasaki, H. Amano, Y. Koide, K. Hiramatsu, N.
Sawaki, J. Crystal Growth 98 (1989) 209.
[8] O. Ambacher, J. Phys. D: Appl. Phys. 31 (1998) 2653.
[9] Y. Tazou, T. Ishii, S. Miyazawa, Jpn. J. Appl. Phys. 36
(1997) L746.
[10] B. Sundman, B. Jansson, J.O. Andersson, CALPHAD 9
(1985) 153.
[11] A.V. Davydov, T.J. Anderson, Electrochem. Soc. Proc.
98–18 (1998) 38.