Influence of elastic strains on LPE growth kinetics in the Cd-Hg-Te System
By comparing the results of calculations concerning the dependence of the parameters of a layer on the growth conditions with and without regard for mechanical strains in the growing system, we have analyzed the influence of the elastic energy of the strained solid phase on the phase formation in...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2007
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| Цитувати: | Influence of elastic strains on LPE growth kinetics in the Cd-Hg-Te System / P.P. Moskvin, L.V. Rashkovetsky, V.V. Khodakovsky // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 3. — С. 70-74. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1181262017-05-29T03:04:00Z Influence of elastic strains on LPE growth kinetics in the Cd-Hg-Te System Moskvin, P.P. Rashkovetsky, L.V. Khodakovsky, V.V. By comparing the results of calculations concerning the dependence of the parameters of a layer on the growth conditions with and without regard for mechanical strains in the growing system, we have analyzed the influence of the elastic energy of the strained solid phase on the phase formation in the Cd–Hg–Te system. It is shown that the occurrence of elastic strains in a layer results in an insignificant reduction of the growth rate and has almost no influence on the composition of a growing layer. The ideas of coherently matched phases in the presence of elastic deformations in the system, as well as the assumption about the existence of the chemical equilibrium of phases on the interface, give rather close results as for the crystallization of the material. Both approaches describe the experimental data on the growth of layers in various temperature-time regimes quite satisfactorily 2007 Article Influence of elastic strains on LPE growth kinetics in the Cd-Hg-Te System / P.P. Moskvin, L.V. Rashkovetsky, V.V. Khodakovsky // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 3. — С. 70-74. — Бібліогр.: 14 назв. — англ. 1560-8034 PACS 64.90.+b http://dspace.nbuv.gov.ua/handle/123456789/118126 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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By comparing the results of calculations concerning the dependence of the
parameters of a layer on the growth conditions with and without regard for mechanical
strains in the growing system, we have analyzed the influence of the elastic energy of the
strained solid phase on the phase formation in the Cd–Hg–Te system. It is shown that the
occurrence of elastic strains in a layer results in an insignificant reduction of the growth
rate and has almost no influence on the composition of a growing layer. The ideas of
coherently matched phases in the presence of elastic deformations in the system, as well
as the assumption about the existence of the chemical equilibrium of phases on the interface,
give rather close results as for the crystallization of the material. Both approaches
describe the experimental data on the growth of layers in various temperature-time
regimes quite satisfactorily |
| format |
Article |
| author |
Moskvin, P.P. Rashkovetsky, L.V. Khodakovsky, V.V. |
| spellingShingle |
Moskvin, P.P. Rashkovetsky, L.V. Khodakovsky, V.V. Influence of elastic strains on LPE growth kinetics in the Cd-Hg-Te System Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Moskvin, P.P. Rashkovetsky, L.V. Khodakovsky, V.V. |
| author_sort |
Moskvin, P.P. |
| title |
Influence of elastic strains on LPE growth kinetics in the Cd-Hg-Te System |
| title_short |
Influence of elastic strains on LPE growth kinetics in the Cd-Hg-Te System |
| title_full |
Influence of elastic strains on LPE growth kinetics in the Cd-Hg-Te System |
| title_fullStr |
Influence of elastic strains on LPE growth kinetics in the Cd-Hg-Te System |
| title_full_unstemmed |
Influence of elastic strains on LPE growth kinetics in the Cd-Hg-Te System |
| title_sort |
influence of elastic strains on lpe growth kinetics in the cd-hg-te system |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118126 |
| citation_txt |
Influence of elastic strains on LPE growth kinetics in the Cd-Hg-Te System / P.P. Moskvin, L.V. Rashkovetsky, V.V. Khodakovsky // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 3. — С. 70-74. — Бібліогр.: 14 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT moskvinpp influenceofelasticstrainsonlpegrowthkineticsinthecdhgtesystem AT rashkovetskylv influenceofelasticstrainsonlpegrowthkineticsinthecdhgtesystem AT khodakovskyvv influenceofelasticstrainsonlpegrowthkineticsinthecdhgtesystem |
| first_indexed |
2025-07-08T13:24:25Z |
| last_indexed |
2025-07-08T13:24:25Z |
| _version_ |
1837085298202247168 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 70-74.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
70
PACS 64.90.+b
Influence of elastic strains on LPE growth kinetics
in the Cd–Hg–Te system
P.P. Moskvin1, L.V. Rashkovetsky2, V.V. Khodakovsky1
1Zhytomyr State Technological University
103, Chernyakhovsky str., 10005 Zhytomyr, Ukraine
E-mail: moskvin@us.ztu.edu.ua
2V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine
45, prospect Nauky, 03028 Kyiv, Ukraine
Abstract. By comparing the results of calculations concerning the dependence of the
parameters of a layer on the growth conditions with and without regard for mechanical
strains in the growing system, we have analyzed the influence of the elastic energy of the
strained solid phase on the phase formation in the Cd–Hg–Te system. It is shown that the
occurrence of elastic strains in a layer results in an insignificant reduction of the growth
rate and has almost no influence on the composition of a growing layer. The ideas of
coherently matched phases in the presence of elastic deformations in the system, as well
as the assumption about the existence of the chemical equilibrium of phases on the inter-
face, give rather close results as for the crystallization of the material. Both approaches
describe the experimental data on the growth of layers in various temperature-time
regimes quite satisfactorily.
Keywords: A2B6 semiconductors, LPE, elastic strains.
Manuscript received 12.07.07; accepted for publication 27.09.07; published online 30.11.07.
Cd–Hg–Te systems including solid-solution layers with
thickness of several micrometers are widely used as
materials for infrared detectors. They are obtained by the
growing from the own melt on the substrate made of a
binary compound (mostly it is cadmium telluride). In
works [1, 2], the thermodynamic information allowing
one to analyze the system’s phase diagram is
theoretically obtained with high reliability. In [3, 4], the
influence of the growth kinetics on the composition of
layers is considered in terms of a diffusion-limited
model. According to this model, the thermodynamic
equilibrium was supposed to exist on the interface. Thus,
the equations of phase equilibrium [1, 2] are valid for the
concentrations of components on a growing surface, and
the diffusion mass transfer problem can be posed. At the
same time, the difference (although insignificant)
between the crystal lattice constants of initial mixed
compounds, CdTe and HgTe, occurs in the Cd–Hg–Te
system. During the solid solution crystallization on the
substrate from a binary compound, this factor is
responsible for the occurrence of elastic strain fields in
the growing system. Just this excess mixing energy
component responds for the formation of misfit
dislocations near the “substrate-layer” interface and it,
probably, governs complex diffusion phenomena in the
solid phase, by determining the shape of a hetero-
junction.
Thus, it is possible to consider the grown epitaxial
layer to be elastically strained, and the description of
such crystallization process should involve the presence
of the elastic energy in the system. This means that, in
the description of the material growth kinetics, the
boundary concentrations of components on the “strained
solid phase–melt” interface should be connected by the
coherent diagram equations [5, 6], instead of equations
which take into account only the chemical phase
equilibrium.
Therefore, the purpose of the present work is to
analyze the influence of a shift of phase equilibria due to
the absent or incomplete relaxation of elastic strains in
CdxHg1-xTe solid solutions on the heteroepitaxy kinetics
for a layer grown from binary compounds on a substrate.
In view of important practical applications of the posed
problem, it is necessary to theoretically estimate the
possibility to rule the strains in a heterojunction by
changing the dependence of a composition of the layer
on its thickness while obtaining the layers under noniso-
thermal conditions.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 70-74.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
71
The equations of phase equilibria between the
elastically strained solid phase and the own melt for
ternary systems are obtained in [5, 6]. On their
derivation, the thermodynamic functions describing the
liquid phase were supposed to be the same as those in
the description of general phase equilibria, whereas the
calculation of component activities in the solid state was
performed, by considering the existence of elastic strain
fields occurring in a thin layer of the solid solution
coherently matched with a massive substrate. For the
Cd–Hg–Te system, these equations look as
( )
;/])1()1[(ln
ln/ln
/)(
22
CdTeCdTe
TeCd
TeCd
CdTe
CdTeTeCd
CdTeCdTe
RTxxBx
xx
xx
RT
T
T
TTTC
RTTTS
S
StStF
FF
FF
−−−⋅+γ=
=
⋅
⋅
+
−−∆+
+−∆
−
( )
;/)(ln
ln/ln
/)(
22
HgTeHgTe
TeHg
TeHg
HgTe
HgTe
TeHgHgTeHgTe
RTxxBx
xx
xx
RT
T
T
TTT
CRTTTS
S
StStF
F
FFF
−⋅+γ=
=
⋅
⋅
+
−−×
×∆+−∆ −
(1)
ATeTeATeA CCCC LLF −+=∆ − ; ;)1(ln 2
j
S
j xRT −α=γ⋅
;1CdTeHgTe =+xx ;HgCd,A = ,HgTeCdTe,=j
where FF ST ∆, − melting temperature and entropy of
fusion of CdTe, HgTe compounds. “St” stands for the
stoichiometric liquid phase, −iC molar heat capacities
of the initial elements of the liquid phase and ATe
compounds; jγ , Sα - the activity coefficient of the j-th
component and the solid phase interaction parameter;
Sj xx , – compositions of the solid solution and
substrate; and B – elastic constant of the solid solution
dependent on the crystallographic orientation of a layer.
It is known [5, 6] that the value of parameter B for
a (111)-oriented layer is a little more than the respective
value for a (100)-oriented one. Taking into account that,
in practice, (111)-oriented layers are grown more often
and considering that, as a whole, the elastic energy
contribution to the total energy balance of the solid
phase is expected to be not so significant, the following
calculations are carried out just for the latter layer
orientation. Then, according to the data [5, 6] for (111)-
oriented layers of crystals with cubic symmetry, this
parameter is the following combination of elastic
constants:
( )
( )441211
441211
2
A
42
26
4 ccc
ccc
a
dx
daN
B
++
+
= .
Here, kmca, – crystal lattice parameter of an
unstrained solid solution and its elastic constants, and
−AN Avogadro number.
Elastic constants, as well as the crystal lattice
parameter for the current solid solution composition,
were calculated by the method of linear interpolation, by
using the data on the appropriate properties of initial
binary compounds as the initial information. Data
necessary for the calculation of the mentioned para-
meters are borrowed from work [7] and submitted in
Table.
Table. Elastic constants of CdTe and HgTe compounds at
various temperatures [7].
Compound K,T 11C ,
GPa
12C ,
GPa
44C ,
GPa
298 53.8 37.4 20.18
CdTe
77 56.2 39.4 20.61
300 53.6 33.6 22.30
HgTe
77 58.7 40.5 21.10
As follows from Table, the elastic constants of
compounds are dependent on temperature.
While the experimental data concerning these
parameters are absent for the temperatures of growth,
their values were corrected by diminution by 10 %
relatively to respective ones for room temperature. Such
a degree of correction corresponded to their reduction up
to values at the typical temperatures of crystallization in
the system.
The thermodynamic functions needed to calculate
the liquid phase parameters within the model of
polyassociated solutions are taken from our works [1, 2].
The shift of phase equilibria due to the presence of
elastic strains in the multicomponent system is
characterized by the contact supercooling which is
defined as the difference between the liquidus
temperature and the temperature of phase equilibrium
between the strained solid phase and the melt of the
same composition [5, 6]. The calculated values of
contact supercooling on the solid solution growth on
CdTe and HgTe (111)-oriented substrates, which are
obtained by solving Eqs. (1), are shown in Fig. 1. As
follows from these data, the contact supercooling for
typical growth temperatures is not greater than 1 K. Such
a result is not unexpected, taking into account the above-
mentioned properties of the solid solutions under study.
Really, the direct calculations of the elastic component
in Eq. (1) show that the contribution of external elastic
strains into the solid-phase total energy is not greater
than 10 % relative to the excess mixing energy of the
solid phase (parameter Sα ). The last data concern the
(111)-oriented layer growth. For (100)-oriented layers,
the calculated contact supercooling is even less.
The occurrence of elastic energy in the growing
system results not only in a change of the phase
equilibrium temperature, but also in a change of the
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 70-74.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
72
Fig. 1. Contact supercooling with solid solution composition
at the liquidus temperature TL = 550 ºC for СdTe (1) and
HgTe (2) (111)-oriented substrates.
composition of the growing strained solid phase. This
effect is known as the lattice-pulling effect [5]. The
calculations of this effect with the use of the equations of
the coherent diagram (1) have shown that the
compositions of strained and equilibrium solid phases
differ only in the third digit after a point, i.e. the
considered effect is rather insignificant. Thus, the
obtained data allow us to assume that the mechanical
strains in a growing solid solution should result in an
insignificant diminution of the growth rate of layers due
to the contact supercooling at an insignificant change of
the growing material’s composition relative to the
composition predicted by the equilibrium phase diagram.
Such a distinction in the characteristic curves of both
coherent and equilibrium Cd–Hg–Te system’s phase
diagrams should be taken into account in the growth
kinetics analysis.
To describe the growth kinetics of a ternary
solution strained layer, we should solve the system of
differential equations of diffusion. The assumption about
a coherent match between the growing layer and the
substrate should be taken as boundary conditions for
them, instead of the assumption about the chemical
equilibrium on the interface. In such a situation, the
coherent diagram equations (1) are valid for the
concentrations of components on the interface, while the
equations of mass conservation remain constant [5, 8].
Thus, the mathematical description of the process of
growth of the coherently matched epilayer and a massive
substrate is given by the equations
z
x
V
z
x
D
t
x isi
i
i
∂
∂
ρ
ρ
+
∂
∂
=
∂
∂
2
2
,
( )
z
tx
DxxV i
iiiS
l
s
∂
∂
=−
ρ
ρ ),0(00
,
Growth time, min1/2
Fig. 2. Layer thickness with square root of time for CdxHg1-xTe
solid solution isothermal growth with x ≈ 2 mol. frac. without
elastic strains taken into account (dashed curve) and with them
(solid curve). Growth temperatures: 1 − 501; 2 − 503; 3 − 505,
4 − 507 ºC. Liquidus temperature TL = 513 ºC.
where 0
,isx , 0
ix – concentrations of solid and liquid
phase components on the interface, respectively,
corresponding to the coherent diagram equations (1),
z
x
D i
i ∂
∂
− component fluxes nearby the surface
boundary to the interface; −)(tV growth rate; −
ρ
ρ
l
s
ratio of the solid- and liquid-phase molar densities; i =
Cd, Hg – under the solid solution growth from a
tellurium enriched liquid phase, z − coordinate directed
deep into the melt.
The mass transfer problem formulated in the
mentioned way was solved by numerical methods both
for isothermal growth conditions and for the layers
grown by the equilibrium and supercooling techniques.
In the last case, the process of growth was realized by
the uniform cooling of a liquid phase supercooled by ∆T.
So Fig. 2 shows the calculated and experimental data [9]
on the solid-solution layer thickness grown on the CdTe
(111)B substrate with isothermal growth time for
different supercoolings and with mechanical strains,
being taken into account or not. The partial diffusion
coefficients necessary for the calculations were supposed
to be independent of temperature in the considered
interval and equal to: s/cm105.2 25
Hg
−⋅=D and
s/сm108.2 25
Cd
−⋅=D [9]. As follows from the
calculations, elastic strains and the contact supercooling
in the system reduce the total liquid phase supercooling
degree and slow down the material’s growth rate, which
is reflected in a reduction of the layer thickness.
However, in general, the influence of contact super-
cooling on the crystallization process is not significant.
This means that the kinetic calculations carried out by
C
on
ta
ct
c
up
er
co
ol
in
g,
°C
La
ye
r t
hi
ck
ne
ss
, µ
m
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 70-74.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
73
Fig. 3. CdxHg1−xTe layer thickness with x ≈ 0.8 mol. frac.
versus growth time. Liquidus temperature Тl = 462 ºC. 1 –
initial growth temperature Т = 455 ºC (∆Т = 7 ºC), cooling
rate Vc = 0.25 ºC /min; 2 – Т = 456 ºC (∆Т = 8 ºC), Vc =
0.54 ºC /min, 3 − Т = 452 ºC (∆Т = 10 ºC), Vc = 0.56 ºC /min,
4 – Т = 450 ºC, (∆Т = 12 ºC), Vc =0.53 ºC /min. Experimental
data [14].
Fig. 4. Solid solution composition versus layer thickness. Initial
growth temperature Т = 552 ºC. Liquidus temperature Тl =
= 555 ºC. Cooling rate Vc = 0.25 ºC/min. Experimental data [11].
using the phase equilibrium relations and the coherent
ones as boundary conditions, give close results, and both
these approaches describe the isothermal crystallization
quite satisfactorily.
Influence of mechanical strains must be the most
essential while growing the layers of a variable compo-
sition. Layer composition changes in such a situation
occur, first of all, due to a temperature decrease in the
system. For the quantitative description of the layer
growth at a temperature decrease in the system, the
problem of diffusion was solved in such a way that, on
each iterative step of integration, the growth temperature
was reduced linearly with the given rate. The growth rate
of a layer, its composition, and thickness at each time
moment were obtained. The comparison of calculated
and experimental results [10] for the thickness of layers
grown by the supercooling technique at various degrees
of supercooling and cooling rates is given in Fig. 3.
Quite satisfactory conformity of the results of calcu-
lations to experimental data is observed. Such a result
confirms the efficiency of the diffusion-limited model in
the description of the solid solution LPE process, when
the model of polyassociative solutions is used for the
description of a phase equilibrium in the system.
To estimate the influence of mechanical strains on
the growth process, the comparative analysis of the
results of calculations of the solid solution composition
distribution in the cases where layers are grown under
various time-temperature conditions both with and
without the account of elastic energy in the total
system’s energy balance should be carried out. Such data
for a solid solution grown from the liquid phase with the
composition xCd = 0.006 at. frac., xHg = 0.217 at. frac. at
an initial supercooling of 2 ºC and various cooling rates
are shown in Fig. 5. It is obvious from Fig. 5 that the
results of calculations of the composition distribution
carried out with and without account of mechanical
strains differ poorly. The calculated dependences carried
out with the coherent diagram application give some
reduction of a composition gradient relatively to those
calculated with the use of the phase equilibrium
conditions. Naturally, these data reflect the well-known
effect of solid solution composition pulling to a
substrate’s one (lattice-pulling effect) [6]. However, in
case of CdxHg1-xTe solid solution, for which the
variation of a lattice constant with the composition
change is very insignificant, this effect is noticed to be
poor. We note that, with decrease in the initial growth
temperature, the considered effect should be manifested
less significantly. It is illustrated by the calculated data
in Fig. 5 when the initial growth temperature is reduced
down to 450 ºC.
It is worth to note the observed satisfactory
correlation between the calculated values of composition
gradients over the layer’s thickness and the experimental
data [13] and a little worse correspondence for the solid
phase composition.
We note that, under the typical growth conditions
of an epitaxial layer in the considered system, the
experimental data on the composition variation versus
the layer thickness are close to linear. This means that, in
the considered temperature intervals, the liquidus and
solidus slopes of the system can be plotted as straight
lines with high degree of accuracy, i.e., the linear model
of diffusion growth earlier developed for the description
of a similar process in systems of 53BA solid solutions
[6] can be also applied to this system of semiconductor
Layer thickness, µm
La
ye
r t
hi
ck
ne
ss
, µ
m
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 70-74.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
74
Fig. 5. CdxHg1-xTe solid solution composition versus layer
thickness for the initial growth temperature T = 500 ºC,
degree of supercooling ∆Т = 2 ºC, different cooling rates, and
elastic strains taken or not into account: Vc = 0.2 ºC/min (1),
0.4 (2), 0.6 (3).
solid solutions. It is also necessary to note that the
quantitative application of the diffusion-limited model
with linear forms of the phase diagram surface will
result inevitably in mathematical difficulties in
calculations of the model’s characteristic parameters (for
example, the partial liquidus slopes T
xi
∂
∂ ). The
model’s software offered in this work has no such lacks.
Thus, the achieved good correspondence of the
results of calculations of the growth kinetics of
CdxHg1−xTe solid solutions in various time-temperature
regimes to experiments confirms the applicability of the
diffusion-limited model to the description of the growth
process of layers. The solution of the problem of growth
under mechanical strains in the system shows that the
elastic energy reduces the material’s growth rate by
10 % on the average. Thus, a variation in the composition
of an elastically strained layer is insignificant and not
greater than several percents in comparison with that
predicted by the equilibrium phase diagram.
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Layer thickness, µm
|