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Workshop on thermodynamic modelling of solutions and alloys
Calphad Vol. 21, NO. 2, pp. 265-285, 1997 (D 1997 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 0364-5916/97 $17.00 + 0.00
Pergamon
00024-2
PII SO364-5916(97)
Workshop on THERMODYNAMIC
MODELLING OF SOLUTIONS AND ALLOYS
SchloB Ringberg, March lO-16,1996
GROUP 5: Solution Thermodynamics
of Electronic Materials
Group Members : T Andersson University of Florida, Gainesville, USA D de Fontaine University of California, Berkeley, USA S G Fries LTH, RWTH, Aachen, Germany
B Legendre Ecole de Pharmacie, Paris, France W A Oates (Chairman) Jiilich, Germany R Schmid-Fetzer TU Clausthal, Clausthal, Germany H-J Seifert Max Planck Institute, Stuttgart, Germany Q Chen Royal Institute of Technology, Stockholm, Sweden 265
266
T. ANDERSSON
SOLUTION
et al.
THERMODYNAMICS OF
ELECTRONIC Report
of Group V, Ringberg,
MATERIALS March 1996
Group Members T. Andersson D. de Fontaine S. G. Fries B. Legendre W. A. Oates (Chairman) R. Schmid-Fetzer H-J. Seifert Q. Chen (Assistant)
A description ‘Empirical and
of CALPHAD
methods
rely on global
use ad hoc approximate the
various
curve-fitting
models
contributions
measurements to deconvolute
to the free
energy.’
Didier de Fontaine,
Ringberg,
March 1996
Introduction Electronic
materials
ramics and polymers. these materials feature
comprise
various classes of materials:
In many commercial
are present
of microelectronics
in intimate systems
electronic
contact
semiconductors,
devices or components
with one another.
is the small dimensions
Another
metals,
ce-
often all of noteworthy
in the final product.
It is
THERMODYNAMIC obvious
that the stable and metastable
pertinent
materials
processing
are of fundamental
of electronic
phase equilibria importance
and the thermodynamics
in the design, selection,
of the
synthesis
and
materials.
Let us briefly examine tronics
267
MODELLING OF SOLUTIONS AND ALLOYS
the different
classes of materials
which are used by the elec-
industry:
1. Metals:
Of specific
The application Solution
interest
are solder alloys and metals
of phase diagrams
phase modelling
for interconnections
in soldering are detailed
in an excellent
has been applied in only a few of the systems
[l].
book [2]. of interest
to date [3]. Metallic magnetic for electronic materials,
materials
applications.
including
for electronic magnetic
thermodynamics
applications.
Some examples
Some examples
of structural
ing or masking
layers in devices.
includes
examples optical
[5].
the electronic
3. Polymers:
ceramics:
properties
phase equilibria
and interconnection
and structural
ceramics:
and packaging
ceramics
Electroceramics, optics (LiNbOs). ceramics,
insulat-
thermodynamics
thermodynamic
modelling
which
is desirable.
polymers:
photorefractive
materials.
of functional
ceramics,
group relevant
[4].
Mostly generic ceramic solution
Again, a division into functional
effects,
another
ceramics for integrated Substrates
For functional
of functional
packaging
are applied
glasses - may be divided into functional
oxides, ceramic superconductors,
has been applied
constitute
As in the case of metallic joining
metals solution
2. Ceramics,
and superconductors
and structural
optoelectronic
polymers.
polymers
or photonic
Some examples
Little in the way of the coupling
is useful: Some
devices,
nonlinear
of structural
Polymers:
of thermodynamics
has been carried out for this group of materials
although
with a start
has been made [6,7]. 4. Compound
semiconductors
are the most relevant
application
of solution
Compound
semiconductor/metal-interactions
and stability
of metal
contacts
been made in III-V-metals XXIII [8] or in Cd-Te-metal thermodynamic
group of semiconductors
for any
thermodynamics.
calculations
(Schottky
phase diagram systems
are also important or ohmic).
Substantial
investigations
[9]. Experiments
for the formation
as reported combined
were found to be useful [lo].
progress
has
at Calphad
with very simple
268
T. ANDERSSON The CALPHAD
areas associated
approach
to thermodynamic
with the design,
bulk crystal growth
patibility
in electronic
formation
of ohmic and Schottky
in general
semiconductors in its application
although
Modelling
Previous
metallization,
performance
soldering
technologically
important,
com-
interfaces,
of barrier materials,
[2]).
the Group’s
attention
Other challenges
to such systems.
dopants
and lead-free solder systems,
and the way in which the CALPHAD
was focussed approach
not addressed
could at the
are strain energy and surface effects.
Semiconductors
by the CAL-
Approach modelling
of compound
as have been so successful
been an assessed
temperature-composition
More specifically,
using a Redlich-Kister
ionic, twosublattice
polynomial
been taken as perfectly
(Al,Ga)As
or (Ga,In)(As,P)
phase diagram
[14].
(like Ga-As
description
stoichiometric
solution
have been modelled
the II-VI
liquids
model [15,16] or the
semiconductor
compounds.
the has
etc.) have usually been
[14], whereas
using the associated
has followed
The end product
More complex
by using for regular solution
phases have phases
like
intermixing
binary compounds.
This kind of modelling, ignored the technogically properties.
systems
systems.
model [17]. The solid compound
invariably
of the component
semiconductor for metallic
liquid phases in III-V systems
(like Cd-Te etc.) have been described
ducting
e.g.,
materials
at semiconductor/metal/ceramic
at the Workshop,
of Compound
CALPHAD
partially
of intentional
semiconductors;
layers, low temperature
same procedures
modelled
materials,
thin film preparation
doping and solubility
(die bonding,
time available
be improved
PHAD
contacts,
of insulating
on compound
Workshop,
could be of value in several of electronic
ceramics,
doping in compound
devices and reactions
and stability
In the short
[l l] or functional
semiconductor
native and intrinsic
joining and packaging
modelling
and processing
(LPE) (12) or f rom the vapor phase (CVD, MOVPE etc) [13]. au
nealing and recrystallization;
deposition
synthesis
of semiconductors
by liquid phase epitaxy
and impurities,
et al.
leading
to a temperature-composition
most important
feature
of these materials,
In other words, the usual temperature-composition
phase diagram,
has
viz., their semiconphase diagram
is
THERMODYNAMIC of very limited important
MODELLING OF SOLUTIONS AND ALLOYS
value to the semiconductor
functional
properties
free energy minimisation
Wagner
between
Until now, the ability to model the
has not been incorporated
into the commercially
defects
the defect structure
in dilute solution
of solids are not new, chemical
having been introduced
over 60 years ago (181. Their mass action law approach
and his classic text semiconductor
[19] includes
systems.
Gal-.As,
Pbi_,Te,
The major purpose
using results from ‘first principles’ using experimental
In the following theoretical
importance
calculations.
was to give consideration emphasis
calculations
as well as the more conventional
We then show how the results
of
route of
are likely to
and defect calculations. methods
used in the
from such calculations approximate
Lastly, we show how standard
can be used in order to arrive at better
using optimisation
investigations
defect model as well as providing
and entropies.
as to how this
was placed on the importance
in the future for both defect-free
at a satisfactory
[20]. There community.
out - see, for example,
we first give a brief review of the different
for the various defect energies techniques
Particular
results since results from such theoretical
become of increasing
used in arriving
of the Calphad
[23-251.
of the Group’s discussion
could be extended.
and
for several binary
by this approach
‘bulb’ have been been carried and Sni_,Te,
by Schottky
was refined by Krijger
of phase diagrams
in this field by some members
of the solidus
[11,21,22],
kind of approach
the construction
The Cd-Te system has been treated
has also been a little activity Some calculations
available
packages.
Models which take into account equilibrium
industry.
269
can be values
CALPHAD
values for the defect parameters
by
procedures.
Ab Initio Thermodynamics Applied to Electronic Materials 1. Definition By ‘ab initio’ or first principles input only the atomic numbers circumstances
calculations
of the constituents
the most stable crystal structure
pound semiconductor
materials,
we mean calculations
which require
and the crystal structure.
can be predicted
the basic structural
framework
as
In favorable
as well. In the usual comis always the same:
the
270 zincblende
(diamond)
structure,
apart, or the related Energies
electronic
formation)
calculations
methods
of solution
and by the approximations Methods
are obtained
according
the FPLMTO potential
(full potential)
wave methods important FPLMTO
sensitive
are ‘all electron’
are possessed
method
by the FLAPW
small number
computer
whereas
pseudopotential
( a t omit sphere approximation).
rather
conveniently.
if local atomic displacements
Full
and plane These two
and ab initio pseudo-potentials.
The
are not an essential
is the most convenient
and
atomic forces and in the case of alloys is
codes, though fully developed,
of specialists
is not universal
quantum
agreement
concerning
the definition
via the density functional most fundamental
approach
FLAPW,
and is used widely.
of ‘first-principles’(or
ab
self-consistency
are needed to perform is performed
the electronic
as part of the calculation
in the local density approximation
(LDA). This is the
level; it is that of the APW, LMTO, and ab initio pseudo-potentials.
One can also use semi-empirical to experimental
in
let us define levels of ‘ab initio character’:
(a) 1st level: only the Z values (atomic number) and electronic
training
Hence, a relatively
of these codes:
The ASA is more accessible
To clarify the situation,
calculation,
require intensive mechanics.
can handle the most sophisticated
ab initio pseudopotentials.
initio) calculations.
structure
methods
plane wave) and
to the choice of atomic sphere radii.
The associated
There
augmented
The ASA version of the LMTO
to use, but cannot calculate
All
(LDA).
seen by an electron.
(also the most computer-intensive);
their use and general knowledge of the underlying
FPLMTO,
methods.
Two main version of the LMTO exist:
and the LMTO-ASA
can also be used, particularly
economical
and at
i.e., by performing
by the choice of basis functions
linear,
allow forces on an atom to be calculated
of the problem.
zero of temperature equation,
the potential
(full potential,
are the most reliable
properties
(a, f, f)
in the local density approximation
core and valence electron.
methods
translated
to one of several approximation
method
used in describing
muffin tin orbital)
distinguish
at absolute
differ from one to another
such as the FLAPW
(linear,
methods
fee lattices
by solving the Schrijdinger
rest on the density functional
Practical
feature
i.e., two interpenetrating structure.
composition
structure
methods
Wurtzite
(cohesion,
the stoichiometric
LMTO
et a/.
T. ANDERSSON
data.
pseudopotentials
Let us label this level ‘1 l/2’.
, the parameters
of which are fitted
THERMODYNAMIC (b) 2nd level: consistency.
electronic
equation
site energies), level-one
‘band structure’
must be constructed the ‘LMTO-Tight
by empirical Binding‘.
(c) 3rd level: empirical formation
in principle
moment
potentials
methods.
integrals, calculated
terms’
Thus, a typical level 2 is
to experimentally
Example
ones
can be derived
obtained
(EAM) [41] and the tight binding
(cohesive,
measured
At level 3, the ‘potentials’
itself.
by a
is also in this class [27].
directly by fitting properties
equation
on-
of atoms,
for which ‘repulsive
but are in fact (partially)
by fitting.
No
of such approaches
method
in the second
[28].
(d) 4th level: here the interatomic for example,
for example
Total energies,
are obtained
is made to solve the Schrodinger
approximation
(hopping
can treat thousands
TB ‘bond order’ method
structure,
atom method
band structure,
lattice parameters)
where
the eigenvalues
from a level-one formulation, directly
means are not easy to obtain.
Pettifor’s
by level-one
from electronic
are the embedded
separately
self-
which must
(TB) method
is diagonalized,
is very convenient, energies.
energies, elastic constants,
or to those calculated
is the Tight Binding
are calculated
electronic
in terms of parameters
which gives the TB parameters
The TB formulation
often gives valuable
attempt
An example
or by fitting to the electronic
theory.
which do not feature
is solved (the Hamiltonian
and the TB parameters
either from LMTO-ASA
calculations
must be expressed
‘from the outside’.
indeed the Schriidinger summed),
structure
Then the Hamiltonian
be introduced
271
MODELLING OF SOLUTIONS AND ALLOYS
is obtained
potentials
by summing
are strictly empirical
up pair energies
and the total energy,
(Lennard-Jones
potentials,
for
example). Molecular the exception
dynamics
ular dynamics, concerned
(MD) simulations
combining
we shall look for analytical perature/composition
cists must consider.
techniques
methods
obtained
for deriving
ab initio calculations
dependence,
ab-initio
with molec-
In what follows, we shall not be by brute force simulation.
thermodynamic
functions,
are rarely concerned
the two most important
variables
about
Instead, hence tem-
first principles
The introduction
electronic
of the Cluster
temperature
which thermodynami-
Until about 10 years ago, outside of brute force simulation,
be done about combining namics of alloys.
[29] which performs
phase diagrams.
performing
and concentration
with 4th or 3rd level approaches,
method
MD with level 1 approach.
with thermodynamic
Physicists
operate
of the famous Car-Parrinello
structure
calculations
Expansion
Method
little could
and thermody[30] has provided
T. ANDERSSON et a/.
272 a way to combine principles
quantum
and statistical
Though predicting
the aim of the empirical properties
of materials,
rely on global measurements
available
theoretical
Empirical individual
procedures,
functions
can produce,
thermodynamic
quantities:
i.e.,
methods
to deconvolute
parts treated
the
no data
by the best
enthalpies,
of ab initio methods
methods
are based on electronic for electronic
configurational,
vibrational,
properties,
entropies,
require no fitting
temperature-dependent
is improving
of supplanting
materials
properties.
results but often unreliable
ones) which can
ones.
are in the process
with electronic
models
very accurate
rely on sound physical
but often produce results (especially
widely from experimental
meaningful
are identical, Empirical
require (in principle)
from their component
by fitting,
ab initio methods
The accuracy
methods
are opposite.
and use ad hoc approximate
methods
structure
calculations,
applications
For example,
continuously
older empirical
so that ‘first principle’
ones.
Also, since ab initio
such methods
are particularly
which, almost by definition,
band gaps, charge densities
are concerned
are obtained
as by-
of the calculations.
A significant hoc expressions
advantage
of starting
hence can be calculated
itself can also be expressed Method
of thermodynamic These tools:
with sound physical
to be fitted, is that the coefficients
physical meaning,
Variation
a new field: ‘first
formulations.
methods
etc. By contrast,
and first principles
the methodologies
to the free energy. Ab initio methods
base and build thermodynamic
entropy
creating
of ab initio methods
various contributions
products
thereby
thermodynamics’.
2. Advantages
deviate
mechanics,
from ‘first principles’.
by cluster methods.
(CVM) which naturally properties first-principle
are now well developed
rather
than ad
acquire clear
Then the configurational
Such is the spirit of the Cluster
includes short range order in the calculation
of solid solutions. electronic
structure
calculations/cluster
and some general packages are being created
‘user friendly’ than what has been available up to now. 3. Application
formalisms,
of the cluster expansion
to semiconductor
systems
expansion/CVM which will be more
THERMODYNAMIC By and large, physicists have appealed
directly
(CALPHAD).
However,
at absolute defects)
interested
most computations
zero (total energies,
Very little thermodynamic pansion
formation
properties
method.
Lambrechts calculation
systems
energies
surface reconstruction, by molecular
‘potentials’.
to the rule is the remarkable important
by the cluster and abundant
exwork
effort along these lines comes
of Ito and Mohri [32] pl us sporadic
by these groups vary depending shortly.
of semiconductor
work in this field has been performed
(331 and de Fontaine
273
than that of the empiricists
they are often obtained
using semi-empirical
The main exception
and co-workers
be explained
rather
energies of compounds,
by the group of Zunger [31]. Another
from the collaboration
properties
to date have focused on structural
are required,
or Monte Carlo simulation
produced
in predicting
to the ab initio community
and when thermal
dynamics
MODELLING OF SOLUTIONS AND ALLOYS
cluster
expansion
(341. The nature
on the computational
First, let us see what are the problems
work by
and reliability
tools available,
of
as will
to be solved.
4. Tasks to be performed Let us consider temperature distinguish
strictly
and concentration,
(b) Solid-phase
words,
i.e., those
phase diagrams
involving
specifically
in all generality.
We
In principle, solid phase
of liquid phases
quasi-chemical
can be treated
have been explored
has been performed
one.
Following
lattice
of bond strengths,
these
(CE), but only
no thorough
parameters.
angles and coordination
of an
the liquid as
The CVM approximation Pelton
and others
was
[36], it would
the CVM pair, which is equivalent
approximations
study
exception
[35] w h o simply considered
the work of Blander,
Of course,
Actually,
by CE to date, with the possible
to try a simpler approximation, model.
by cluster expansions
systematically.
and Pasture1
with slightly expanded
the tetrahedron
ordering
defects
of Al-Ni by Colinet
an fee crystal
be interesting
phase separation,
all three problems
reactions
investigation
determination
reactions:
(c) Off-stoichiometry
phases.
in other
problems,
the following problems:
(a) Liquidus-Solidus
distribution
thermodynamic
to the
do not take into account which are characteristic
the
of liquid
274
T. ANDERSSON et al.
Solid-phase
equilibria
for quasi-binary contributions
have been investigated
sections
in ternary
in detail for semiconductors,
and higher systems.
have come from the group of Zunger (311. The technical
group’s successes can be understood are generally
dominated
by elastic effects, due to the often large disparity distortions:
cell-external
[37]. The first type can be handled by minimization to lattice parameter,
but the second requires,
Hence, first-principle
required.
As mentioned
pseudopotential
electronic
that in the Mohri-Ito to be neglected Other
collaborative
dynamics
studies
deviations
methods
tend not to use
and CE remains
local atomic
relaxation
are unfamiliar
from stoichiometry
with CE (note
calculations,
as top so
effects tend
also be taken into account. chemical reactions
of point defects in electronic In oxides,
defects,
these equilibrium
with definite equilibrium
to choice of defect reactions
constants
ab initio
materials.
semiconductor
vacancies
Such effects are traditionally
associated
in addition
a recent
[38]), and tend to
lines [29].
is no problem. create antisite
however
unless one includes very elaborate
In
compounds,
and defect clusters.
these defects may be ionized in different ways. Electron-hole
The problem,
are
codes are not ‘linear-response-ready’,
along Car-Parrinello
alloys, off-stoichiometry
thermore,
in FLAPW
included
codes and ab initio
of the latter technique
by the group of Ceder and Joannopoulos
We do not know of any CE investigations metallic
is met by FLAPW
work, important
specialists
on oxide systems
stay away from thermodynamic molecular
forces on individual
[32].
pseudopotential
collaboration
relaxations
codes with ‘force theorem’
above, that requirement
Ito’s pseudopotential
systems
in atomic ‘sizes’.
and cell-internal
in general, calculating
‘linear response’ codes. Specialists
Apparently,
for that
of total energy (at zero K) with respect
structure
the CE, thus only Zunger’s group, proficient practitioners.
reasons
as follows: mixing energies in semiconductor
There result two types of structural
atoms.
particularly
In this field, the most important
equilibrium
Furmust
studied by writing down symbolic constants
(Kroger-Vink
to consider,
or the AG“ for the reactions.
approach).
is to obtain
values of
Up to now, empirical
methods
have been used. Impressive performed authors
-
‘brute force’ ab initio calculations
point out some of the difficulties
of calculations,
of defect formation
see the review by Scherz and Scheffler [39] is particularly
for example
gaps for which tractable
problems and reliable
which one encounters
relative to electronic approaches
energies
in performing
excitation
have been
interesting.
these kinds
energies
are not yet available.
These
and band
Furthermore,
THERMODYNAMIC as stated
by these
a reliable
description
MODELLING OF SOLUTIONS AND ALLOYS
authors
‘no general
of all defect
theoretical
candidates’.
approach
exists
(of more than two partners)
and highly distorted
been investigated
Little is known about corresponding
although lattice
in detail.”
Scherz and Scheffler do describe vibrations
through
be of considerable
to ab initio calculation
to investigate
linear combination
expresses
is suited
defect
vibrational
of dynamical
have not entropies,
matrix elements methods.
the CE technique Moreover,
for
complexes
metal impurities
valence force-field
whether
of defect thermodynamics.
at least one of its implementations,
‘Compound
transition
calculations
the use of approximate
interest
that
As a consequence,
275
and
It would
could be applied
since the CE method,
or
the energy of a given ‘configuration’
of the energies of simple structures,
this approach
is reminiscent
Energy Model‘ [40], but in a more general way, not requiring
as a of the
the mean field
simplification. 5. Caveat The Cluster expansion
Expansion
in cluster
in addition,
method
functions
many components
minimization
becomes
prohibitively
in particular
The convergence especially
rapidly.
are present,
in the past with short expansions, experiment,
is no panacea:
converges
large.
the numbers Practical
small clusters,
leading to transition
of cluster expansions
Its application
calculations
often resulting
temperatures
only if the
must be used.
of variables
and the incorporation
where elastic and Madelung contributions
is practical
If not, large clusters
in the free energy
have been performed in poor agreement
hundreds
with
of degrees too high.
of more distant
are important,
If,
interactions,
need to be investigated
further. One important scopic properties structure
remark:
it is generally
(thermodynamic
calculations
not good strategy
ones, for example)
to attempt
directly
since the length scales involved are just too different.
to solve practical
technological
the computation
receives input from lower levels and produces
It is generally techniques culations produce
problems by a hierarchy of approaches,
too much to ask for any scientist
of more than perhaps themselves,
a hierarchy
TB parameters,
MD simulations.
to derive macro
from ab initio electronic
two successive
output
to be an expert
for the next level.
in the computational
levels. Even for electronic
of levels may be used: for example,
a density-of-states
is obtained
It is preferable
where, at each level,
structure
cal-
LMTO calculations
and its second moment
is used in
276
T. ANDERSSON et al. For thermodynamic
the interface coefficients
It follows that
may be obtained
structure
parameters
‘scale’ of the end product
required,
with what
theories.
Or a
based on a level-two
derived from level-one calculation. in this field is dictated
by the numbers
of levels in the hierarchy
in turn, dictate
types of specialists, language.
of a linear system
by level-one averaging)
provides
For example,
must be organized
Such considerations,
find a common
methods.
by inversion
of which may be obtained
the way research
must be considered.
another,
mechanical
may be used (direct configurational
with electronic
organized,
to us here), the cluster expansion
and statistical
expansion
the parameters
more direct method
(of interest
the quantum
of the cluster
of equations,
approach
properties
between
how research
teams
and how they shall communicate
This is a non-trivial
organizational
by the which must be with one
problem,
one of
‘vertical’ integration.
Thermodynamic Assessment of a Compound Semiconductor System -
the Cd-Te System
Introduction In order to illustrate of a compound the Cd-Te
the kind of procedures
semiconductor
system
was chosen.
possible optoelectronic infrared
system,
involved in the thermodynamic
including
Crystals
optimisation
based on this system
devices with adjustable
assessment
of the defect parameters, have attracted
band gap and as a substrate
interest
as
material
for
equilibria
the
detectors.
In the standard solid semiconductor
Calphad approach phase is modelled
described
in the next section
approach
can be extended
solid phase.
for the calculation as a stoichiometric
for the Cd-Te
by considering
system.
compound.
This procedure
is
We then go on to show how this
the lattice and electronic
Lastly, we show how some of the important
can be optimised.
of liquid--solid-gas
defect structure
defect parameters
of the
of the model
THERMODYNAMIC
MODELLING
OF SOLUTIONS
AND ALLOYS
277
1400-c
A 200 II0
0.2
0.4
0.6
0.8
1.0
MOLE_FFWCTION TE Figure 1: The assessed phase diagram for the Cd-Te system (after [15]).
Standard CALPHAD
Assessment
The Cd-Te binary system has been reviewed recently (151. The liquid phase was modelled by the associated to describe
solution model having CdTe as an associate.
the Gibbs energy of formation
and with Te. The composition in a Redlich-Kister was modelled dependent
dependence
polynomial
than the CdTe-Cd
as a stoichiometric
formation
enthalpy.
of the associate of the CdTe-Te
Eight coefficients
were used
and its interaction
with Cd
interation
interaction.
The CdTe solid phase
phase with an experimentally
Fig. (1) s h ows the calculated
is one order higher
adjusted
temperature
binary phase diagram.
Inclusion of the Defect Model Defects are introduced important fortunately,
during the growth and processing
role to play in determining the identification
the functional
and the determination
defects is a difficult task so that theoretical There are two aspects (a) obtaining
of semiconductors
properties
of a semiconductor.
of the concentration
calculations
and have an Un-
of the different
take on a particular
importance.
to these calculations: the isolated defect free energies (energies
tum mechanical
calculations
of the type described
and entropies)
from quan-
in an earlier section
278
T. ANDERSSON et a/. (b) using the defect
free energies
to calculate
the concentrations
of the various
defects under given conditions.
A theoretical
group at SRI International
(Cd,%n,Hg)-Te model
systems
which
contains
substitutional
eight
sublattices
interstitial
sublattice).
consider
although
native
point
In addition
of formation
(a) vibrational
are equally as important
two contributions -
model for the Coulombic (b) degeneracy
-
distortion
of the principal
It can be seen that
The other ionisation
the Cd interstitial important
as the energies of formation.
due to the introduction
the elastic contributions
and using a point charge
a direction
degeneracy
due
has been considered.
neutral
defects
is calculated
to dominate
at the highest
It also appears
with the important
energies are consistent
calculated
in CdTe at 700°C as a function
defect at low and medium
defect is the Te antisite.
ionisation
of
contributions.
energies that the Cd vacancy is a double acceptor
The calculated
The SRI
by fitting the total energy results to a valence force
the principal
Teed and Cdi, are donors,
[46].
[45]:
of Cd
Cd pressures
from calculations
interstitial
is
Cd pressures. of the
with the first ionisation
close to the valence band edge and the second being close to mid-gap. defects,
to i.e.,
at high temperatures,
in Fig (2) the results from Ref. [46] which shows the theoretically
concentrations
Vao, whilst
of defects
as well as the usual spin degeneracy,
to Jahn-Teller
to be isolated,
as being of importance
the change in the phonon spectrum
field model for calculating
on the two
and an anionic
there are ionised defects
are considered
to this entropy
for the
a quite complex
and vacancies
defects
The defects
the concentrations
the defect can be calculated
pressure.
(antisites
a bound defect pair has been suggested
group have considered
We reproduce
out this type of analysis
defects on both a cationic
to these neutral
and holes.
In order to be able to calculate the entropies
defects
plus two interstitial
as well as electrons
unbound,
have carried
in recent years [42-471. They have considered
level
The other
two
being a double donor.
with there being a p 3
n transition
in
CdTe at high Cd pressures. The model adopted
here is a slight modification
of that used by the SRI group.
have used only one interstitial
sublattice
degeneracy.
in order to be able to illustrate
This is sufficient
We
and have not allowed for any spin or Jahn-Teller the principles
involved
in
THERMODYNAMIC
Figure
2:
Calculated
MODELLING OF SOLUTIONS AND ALLOYS
neutral defect concentrations
279
for CdTe at 700% as a
function of Cd pressure (after [46]). this type of calculation.
The three-sublattice
model may be written:
(Cd,TeX,Te~,VaX,Va~,VaZ~)(Te,CdY,VaX)(Va,CdX,Cdf,CdZ+,TeX) where we have used the standard (X refers to a neutral The software permit
defect)
packages
the inclusion
Kriiger notation
[e,h]
for the effective charges on the defects
(191.
ChemSage
of electronic
and ThermwCalc
have been extended
defects as well as charged
lattice
defects
in order to [22,49].
The
input files require the following information:
1. the standard
chemical potentials
with those for electrons 2. the site exchange been considered
of Cd and Te in the zincblende
energies for the lattice species. in the present
potential
The only interactions
model are those between
the majority
which have components
sublattices.
the models in these packages are restricted
holes only, i.e., the chemical
together
and holes and the lattice defects.
Cd and Te on the two substitutional
At present,
structure,
of the electrons
to non-degenerate is given by:
electrons
and
280
T. ANDERSSON et al.
800 I
500 ! I 4998 E-4 4998
A
I
I 5002
5000
5004
MOLE_FFtACTION TE
( [48]) solidus in the
Figure 3: Comparison of the calculated and experimental
Cd-Te system. The calculated solidus results from the optimisation of the energies of formation of three defect energies. where y, is the concentration per sublattice
site so that the electron
lattice defect concentrations. the number
of electrons
and is conveniently
concentrations
This concentration
expressed
may be directly
as the fraction
compared
may, of course, be directly
per unit volume, which is the concentration
with the
converted
to
of choice by the semiconductor
industry.
Optimisation of the Defect Model Parameters As a starting
point for the defect free energies, those given by the SRI group for HgTe were
used. The most important solidus by Greenberg measurements
(48). Optimisation
is possible
Calc). In order to illustrate the energies, The calculated
of these were then optimised
of defect parameters
by using ChemOpt the possibilities
but not the entropies, solidus is compared
to the recent measurements to the experimental
(for ChemSage)
of extending
the standard
of the three most important with the experimental
or Parrot
of the solidus
(for Therm*
Calphad
approach,
defects were optimised.
data in Fig. (3).
THERMODYNAMIC F’ut
MODELLING OF SOLUTIONS AND ALLOYS
281
ure Development s/Improvements/Recommendations
It should
be emphasised
that
been used as the starting such calculated future
results
calculations
Calphad
theoretically
point
will be the best
on compound
community
their optimisation
source of thermodynamic An important
results,
particularly
have
It is likely that
data
for defects
contribution
in
from the
values through
for the solidus but also for
properties.
the Group have identified
species,
parameters
in CdTe.
can be to assist in the refining of these calculated by using experimental
and have shown how a system charged
thermodynamic
calculations
semiconductors.
the single phase thermodynamic Although
calculated
for the defect
an important
of interest
area for CALPHAD
can be modelled
development
to give the concentrations
of
there are several areas in which there is room for improvement:
1. There is a need for better values for defect entropies recent ab intio calculations the entropy
in semiconductors.
for defect energy formation
values are of equal importance
Although
the
are a welcome development,
in the calculation
of defect equilibria
at
high temperatures. 2. There are two ways in which the currently (a) by the incorporation
of degeneracy
sion of spin degeneracy distortions
in the electron/hole
the charged
3. Semiconductor
Cd-Zn-Te
defects
approximation modelling
The defect interaction
can be quite large (AE/LT
>>
energies between
0) so that
should be incorporated
into all the free energy minimisa-
community.
carrying out more detailed
calculations
These will consider not only the temperature-composition
grams of the kind shown in this report but will also include consideration and a calculation
of the functional
results will be published
the Bragg-
used in the present model is insufficient.
of the Group are currently
system.
The inclu-
is not a major hurdle [23] but the effects of Jahn-Teller
tion packages used by the CALPHAD
Some members
equilibria.
will be a more difficult problem.
(b) by the inclusion of defect interactions.
Williams
used defect models could be improved:
(semiconducting)
in due course.
properties
on the
phase diaof gasequilibria
of the materials.
These
282
T. ANDERSSON
Although combining
outside the terms of reference of the Working Group, there is also a need for
the thermochemical
tor industry
et a/.
modelling
can relate their processing
the final crystal. is in operation
with reactor modelling
variables to the charged species concentrations
The software interface, on both ThermoCalc
so that the semiconduc-
developed
by Sundman
and ChemSage,
of
and Eriksson and which
is an ideal tool for this purpose.
References [l] Papers presented and “Materials
at the symposia Science
“Materials
of Contacts
Annual Meeting of TMS, Anaheim, [2] G. Humpston,
D.M. Jacobson,
tional, Materials [3] A. Bolcavage,
and Metallization California,
Principles
and Interconnects”
February
of Soldering
125th
4-8 , , (1996) and Brazing
ASM Interna-
Park, OH, , (1993)
and Processing
of Materials
of Thermodynamics
(Eds.: P. Nash and B. Sundman),
TMS,
PA, ~171 (1995)
[4] B. Hallemans, [5] A. Boudene,
P. Wollants, J.R. Roos, Journal of Phase Equilibria, K. Hack, A. Mohammad,
S. Fries, H.-L. Lukas, R.A. Konetzki, C. Bernard,
C. Colinet,
J.C. Mathieu, in print,
for Microelectronics”
C.R. Kao, S.-L. Chen, Y.A. Chang, Applications
in the Synthesis Warrendale,
and Technologies
A. Pasturel,
J. Rogez, B.B. Argent,
D. Neuschiitz,
E. Zimmermann,
R. Schmid-Fetzer, A. Pisch,
16, 137-149 (1995)
W. Huang,
G. Effenberg, B. Sundman,
F. Weiss, A. Rais, M. Ganteaume,
A.T. Dinsdale,
A. Watson,
High Temp. Sci.,
(1996)
[6] D.O. Lopez, J. van Braak,
J.LL. Tamarit,
H.A.J.
Oonk, CALPHAD,
(1994) [7] H.A.J. Oonk, CALPHAD, [8] A.T. Dinsdale, [9] H. Cordes, [lo] J. Klingbeil,
16, 37-46 (1992)
CALPHAD,
R. Schmid-Fetzer, R. Schmid-Fetzer,
18, 337-368 (1994) Z. Metallkde.,
86, 304-312 (1995)
CALPHAD,
18, 429-440 (1994)
18, 387-396
THERMODYNAMIC [ll]
H. Wenzl, W.A. Oates, Elsevier
283
MODELLING OF SOLUTIONS AND ALLOYS K. Mika, Handbook
of Crystal
Growth,
Ed.: D.T.J.
Hurle,
Science Publ., Vol. lA, ~103-186 (1993)
(121 M. Astles, Materials,
H. Hill, A.J. Williams,
P.J. Wright,
M.L. Young, Journal
of Electronic
15, 41-49 (1986)
(131 S. Lourdudossa, [14] I. Ansara,
S. Du, 0. Kjebona,
C. Chatillon,
B. Sundman,
H.L.Lukas,
B.B. Argent,
CALPHAD,
18, 397-407 (1994)
T. Nishizawa,
A. Watson,
H. Ohtani,
T.G. Chart,
K. Ishida, M. Hillert,
T. Andersson,
CALPHAD,
18,
177-222 (1994) [15] J. Yang, N.J. Silk, A. Watson,
A.W. Bryant,
B.B. Argent,
CALPHAD,
19, 399-414
(1995) [16] T.-C. Yu, R.F. Brebrick, [17] M. Hillert, B. Jansson, [18] C. Wagner,
J. Phase Equilibria,
B. Sundman,
W. Schottky,
[19] F.A. Kroeger,
13, 476-496 (1992)
J. eAgren,
Metall. Trans. A, 16, 261-266 (1985)
Z. phys. Chem. B, 11, 163-173 (1930)
The Chemistry
of Imperfect
Crystals,
North Holland, 2, Chaps. 14-16,
(1974) (201 F.A. Kroeger,
Rev. Phys. .4ppl., 12, 205-210 (1977)
(211 H. Wenzl, A. Dahlen, A. Fattah, Growth,
S. Petersen,
K. Mika, D. Henkel, Journal of Crystal
109, 191-204 (1991)
[22] W.A. Oates,
G. Eriksson,
H. Wenzl, Journal
of Alloys and Compounds,
220, 48-52
(1995) [23] Y.G. Sha, R.F. Brebrick,
Journal
of Electronic
Materials,
18, 421-443 (1989)
[24] J.-C. Lin, T.L. Ngai, Y.A. Chang, Metall. Trans., A 17, 1241 (1986) [25] T.L. Ngai, D. Marshall,
R.C. Sharma, Y.A. Chang, Monatsh.
[26] W.A. Oates, H. Wenzl, CALPHAD, [27] David Pettifor,
for example
Chemie, 118,277 (1987)
17, 35-46 (1993)
Ch. 4 in “Bonding
and Solids”, Oxford Science Publications
(1995).
and Crystal
Structure
of Molecules
284
T. ANDERSSON et al.
[28] see for example
F. Ducastelle,
Ch. 6 in “Order and Phase Stability
in Alloys”, North-
Holland (1991). [29] R. Car, M. Parrinello, [30] J. M. Sanchez,
Phys. Rev. Lett., 55, 2471-2473 (1985)
F. Ducastelle
and D. Gratias,
Physica
128A, 334 (1984)
[31] S.-H. Wei, L.G. Ferreira and A. Zunger, Phys. Rev. B 41, 8240 (1990). [32] T. Ito, J. Appl. Phys. 77, 4845 (1995). [33] E.A. Albanesi,
W.R.L. Lambrecht,
[34] R. McCormack
and D. de Fontaine
[35] A. Pasture& C. Colinet, Matter)
and BSegail,
A.T. Paxton
Phys. Rev. B 15 ,48, 17841 (1993).
(unpublished
work at U.C. Berkeley).
and M. van Schilfgaarde,
J. Phys. (Condensed
4, 945 (1992).
[36] P. Wu, G. E ri k sson, A.D. Pelton and M. Blander, [37] D.B. Lax, L.G. Ferreira, [38] P.D. Tepesch,
ISIJ International,
33, 26 (1993).
S. Froyen and A. Zunger, Phys. Rev. B 46, 12587 (1992).
A.F. Kohan,
G.D. Garbulsky,
Boyer, M.J. Mehl, B.P. Burton,
G. Ceder, C. Coley, H.T. Stokes,
K. Cho and J. Joannopoulos,
L.L.
J. Amer. Ceram. Sot.
(to be published). [39] U. Scherz, M. Scheffler, in ‘Defects in III-V Compounds’, Press, Boston, [40] J.-O. Andersson,
ed. E. R. Weber, Academic
1993. A. Fernandez-Guillermet,
M. Hillert, B. Jansson,
B. Sundman,
Acta
Met., 34, 437-446 (1986) [41] MS. Daw and M.I. Baskes, Phys. Rev. Lett. 50, 1285 (1983); Phys. Rev. B 29, 6443 (1984). [42] M. A. Berding,
M. van Schilfgaarde,
A. T. Paxton,
A. Sher, J. Vat. Sci. Tehnol. A ,
8, 1103-1107 (1990) [43] M. A. Berding,
M. van Schilfgaarde,
A. Sher, J. Vat. Sci. Tehnol. B , 10, 1471-1475
M. van Schilfgaarde,
A. Sher, J. Electronic
(1992) [44] M. A. Berding, (1993)
Materials,
22, 1005-1110
THERMODYNAMIC
MODELLING OF SOLUTIONS AND ALLOYS
[45] M. A. Berding, M. van Schilfgaarde, [46] M. A. Berding,
A. Sher, Phys. Rev. B, 50, 15191534
A. Sher, M. van Schilfgaarde,
J. Electronic
Materials,
285
(1994)
24, 1127-1135
(1995) [47] S. Krishnamurthy,
A. B. Chen, A. Sher, M. van Schilfgaarde,
J. Electronic
24, 1121-1125 (1995) [48] J. H. Greenberg,
J. Crystal
Growth,
161, l-11 (1996)
[49] Qing Chen, M. Hillert, to be published
in J. Alloys and Compounds
Materials,
Pergamon
00024-2
PII SO364-5916(97)
Workshop on THERMODYNAMIC
MODELLING OF SOLUTIONS AND ALLOYS
SchloB Ringberg, March lO-16,1996
GROUP 5: Solution Thermodynamics
of Electronic Materials
Group Members : T Andersson University of Florida, Gainesville, USA D de Fontaine University of California, Berkeley, USA S G Fries LTH, RWTH, Aachen, Germany
B Legendre Ecole de Pharmacie, Paris, France W A Oates (Chairman) Jiilich, Germany R Schmid-Fetzer TU Clausthal, Clausthal, Germany H-J Seifert Max Planck Institute, Stuttgart, Germany Q Chen Royal Institute of Technology, Stockholm, Sweden 265
266
T. ANDERSSON
SOLUTION
et al.
THERMODYNAMICS OF
ELECTRONIC Report
of Group V, Ringberg,
MATERIALS March 1996
Group Members T. Andersson D. de Fontaine S. G. Fries B. Legendre W. A. Oates (Chairman) R. Schmid-Fetzer H-J. Seifert Q. Chen (Assistant)
A description ‘Empirical and
of CALPHAD
methods
rely on global
use ad hoc approximate the
various
curve-fitting
models
contributions
measurements to deconvolute
to the free
energy.’
Didier de Fontaine,
Ringberg,
March 1996
Introduction Electronic
materials
ramics and polymers. these materials feature
comprise
various classes of materials:
In many commercial
are present
of microelectronics
in intimate systems
electronic
contact
semiconductors,
devices or components
with one another.
is the small dimensions
Another
metals,
ce-
often all of noteworthy
in the final product.
It is
THERMODYNAMIC obvious
that the stable and metastable
pertinent
materials
processing
are of fundamental
of electronic
phase equilibria importance
and the thermodynamics
in the design, selection,
of the
synthesis
and
materials.
Let us briefly examine tronics
267
MODELLING OF SOLUTIONS AND ALLOYS
the different
classes of materials
which are used by the elec-
industry:
1. Metals:
Of specific
The application Solution
interest
are solder alloys and metals
of phase diagrams
phase modelling
for interconnections
in soldering are detailed
in an excellent
has been applied in only a few of the systems
[l].
book [2]. of interest
to date [3]. Metallic magnetic for electronic materials,
materials
applications.
including
for electronic magnetic
thermodynamics
applications.
Some examples
Some examples
of structural
ing or masking
layers in devices.
includes
examples optical
[5].
the electronic
3. Polymers:
ceramics:
properties
phase equilibria
and interconnection
and structural
ceramics:
and packaging
ceramics
Electroceramics, optics (LiNbOs). ceramics,
insulat-
thermodynamics
thermodynamic
modelling
which
is desirable.
polymers:
photorefractive
materials.
of functional
ceramics,
group relevant
[4].
Mostly generic ceramic solution
Again, a division into functional
effects,
another
ceramics for integrated Substrates
For functional
of functional
packaging
are applied
glasses - may be divided into functional
oxides, ceramic superconductors,
has been applied
constitute
As in the case of metallic joining
metals solution
2. Ceramics,
and superconductors
and structural
optoelectronic
polymers.
polymers
or photonic
Some examples
Little in the way of the coupling
is useful: Some
devices,
nonlinear
of structural
Polymers:
of thermodynamics
has been carried out for this group of materials
although
with a start
has been made [6,7]. 4. Compound
semiconductors
are the most relevant
application
of solution
Compound
semiconductor/metal-interactions
and stability
of metal
contacts
been made in III-V-metals XXIII [8] or in Cd-Te-metal thermodynamic
group of semiconductors
for any
thermodynamics.
calculations
(Schottky
phase diagram systems
are also important or ohmic).
Substantial
investigations
[9]. Experiments
for the formation
as reported combined
were found to be useful [lo].
progress
has
at Calphad
with very simple
268
T. ANDERSSON The CALPHAD
areas associated
approach
to thermodynamic
with the design,
bulk crystal growth
patibility
in electronic
formation
of ohmic and Schottky
in general
semiconductors in its application
although
Modelling
Previous
metallization,
performance
soldering
technologically
important,
com-
interfaces,
of barrier materials,
[2]).
the Group’s
attention
Other challenges
to such systems.
dopants
and lead-free solder systems,
and the way in which the CALPHAD
was focussed approach
not addressed
could at the
are strain energy and surface effects.
Semiconductors
by the CAL-
Approach modelling
of compound
as have been so successful
been an assessed
temperature-composition
More specifically,
using a Redlich-Kister
ionic, twosublattice
polynomial
been taken as perfectly
(Al,Ga)As
or (Ga,In)(As,P)
phase diagram
[14].
(like Ga-As
description
stoichiometric
solution
have been modelled
the II-VI
liquids
model [15,16] or the
semiconductor
compounds.
the has
etc.) have usually been
[14], whereas
using the associated
has followed
The end product
More complex
by using for regular solution
phases have phases
like
intermixing
binary compounds.
This kind of modelling, ignored the technogically properties.
systems
systems.
model [17]. The solid compound
invariably
of the component
semiconductor for metallic
liquid phases in III-V systems
(like Cd-Te etc.) have been described
ducting
e.g.,
materials
at semiconductor/metal/ceramic
at the Workshop,
of Compound
CALPHAD
partially
of intentional
semiconductors;
layers, low temperature
same procedures
modelled
materials,
thin film preparation
doping and solubility
(die bonding,
time available
be improved
PHAD
contacts,
of insulating
on compound
Workshop,
could be of value in several of electronic
ceramics,
doping in compound
devices and reactions
and stability
In the short
[l l] or functional
semiconductor
native and intrinsic
joining and packaging
modelling
and processing
(LPE) (12) or f rom the vapor phase (CVD, MOVPE etc) [13]. au
nealing and recrystallization;
deposition
synthesis
of semiconductors
by liquid phase epitaxy
and impurities,
et al.
leading
to a temperature-composition
most important
feature
of these materials,
In other words, the usual temperature-composition
phase diagram,
has
viz., their semiconphase diagram
is
THERMODYNAMIC of very limited important
MODELLING OF SOLUTIONS AND ALLOYS
value to the semiconductor
functional
properties
free energy minimisation
Wagner
between
Until now, the ability to model the
has not been incorporated
into the commercially
defects
the defect structure
in dilute solution
of solids are not new, chemical
having been introduced
over 60 years ago (181. Their mass action law approach
and his classic text semiconductor
[19] includes
systems.
Gal-.As,
Pbi_,Te,
The major purpose
using results from ‘first principles’ using experimental
In the following theoretical
importance
calculations.
was to give consideration emphasis
calculations
as well as the more conventional
We then show how the results
of
route of
are likely to
and defect calculations. methods
used in the
from such calculations approximate
Lastly, we show how standard
can be used in order to arrive at better
using optimisation
investigations
defect model as well as providing
and entropies.
as to how this
was placed on the importance
in the future for both defect-free
at a satisfactory
[20]. There community.
out - see, for example,
we first give a brief review of the different
for the various defect energies techniques
Particular
results since results from such theoretical
become of increasing
used in arriving
of the Calphad
[23-251.
of the Group’s discussion
could be extended.
and
for several binary
by this approach
‘bulb’ have been been carried and Sni_,Te,
by Schottky
was refined by Krijger
of phase diagrams
in this field by some members
of the solidus
[11,21,22],
kind of approach
the construction
The Cd-Te system has been treated
has also been a little activity Some calculations
available
packages.
Models which take into account equilibrium
industry.
269
can be values
CALPHAD
values for the defect parameters
by
procedures.
Ab Initio Thermodynamics Applied to Electronic Materials 1. Definition By ‘ab initio’ or first principles input only the atomic numbers circumstances
calculations
of the constituents
the most stable crystal structure
pound semiconductor
materials,
we mean calculations
which require
and the crystal structure.
can be predicted
the basic structural
framework
as
In favorable
as well. In the usual comis always the same:
the
270 zincblende
(diamond)
structure,
apart, or the related Energies
electronic
formation)
calculations
methods
of solution
and by the approximations Methods
are obtained
according
the FPLMTO potential
(full potential)
wave methods important FPLMTO
sensitive
are ‘all electron’
are possessed
method
by the FLAPW
small number
computer
whereas
pseudopotential
( a t omit sphere approximation).
rather
conveniently.
if local atomic displacements
Full
and plane These two
and ab initio pseudo-potentials.
The
are not an essential
is the most convenient
and
atomic forces and in the case of alloys is
codes, though fully developed,
of specialists
is not universal
quantum
agreement
concerning
the definition
via the density functional most fundamental
approach
FLAPW,
and is used widely.
of ‘first-principles’(or
ab
self-consistency
are needed to perform is performed
the electronic
as part of the calculation
in the local density approximation
(LDA). This is the
level; it is that of the APW, LMTO, and ab initio pseudo-potentials.
One can also use semi-empirical to experimental
in
let us define levels of ‘ab initio character’:
(a) 1st level: only the Z values (atomic number) and electronic
training
Hence, a relatively
of these codes:
The ASA is more accessible
To clarify the situation,
calculation,
require intensive mechanics.
can handle the most sophisticated
ab initio pseudopotentials.
initio) calculations.
structure
methods
plane wave) and
to the choice of atomic sphere radii.
The associated
There
augmented
The ASA version of the LMTO
to use, but cannot calculate
All
(LDA).
seen by an electron.
(also the most computer-intensive);
their use and general knowledge of the underlying
FPLMTO,
methods.
Two main version of the LMTO exist:
and the LMTO-ASA
can also be used, particularly
economical
and at
i.e., by performing
by the choice of basis functions
linear,
allow forces on an atom to be calculated
of the problem.
zero of temperature equation,
the potential
(full potential,
are the most reliable
properties
(a, f, f)
in the local density approximation
core and valence electron.
methods
translated
to one of several approximation
method
used in describing
muffin tin orbital)
distinguish
at absolute
differ from one to another
such as the FLAPW
(linear,
methods
fee lattices
by solving the Schrijdinger
rest on the density functional
Practical
feature
i.e., two interpenetrating structure.
composition
structure
methods
Wurtzite
(cohesion,
the stoichiometric
LMTO
et a/.
T. ANDERSSON
data.
pseudopotentials
Let us label this level ‘1 l/2’.
, the parameters
of which are fitted
THERMODYNAMIC (b) 2nd level: consistency.
electronic
equation
site energies), level-one
‘band structure’
must be constructed the ‘LMTO-Tight
by empirical Binding‘.
(c) 3rd level: empirical formation
in principle
moment
potentials
methods.
integrals, calculated
terms’
Thus, a typical level 2 is
to experimentally
Example
ones
can be derived
obtained
(EAM) [41] and the tight binding
(cohesive,
measured
At level 3, the ‘potentials’
itself.
by a
is also in this class [27].
directly by fitting properties
equation
on-
of atoms,
for which ‘repulsive
but are in fact (partially)
by fitting.
No
of such approaches
method
in the second
[28].
(d) 4th level: here the interatomic for example,
for example
Total energies,
are obtained
is made to solve the Schrodinger
approximation
(hopping
can treat thousands
TB ‘bond order’ method
structure,
atom method
band structure,
lattice parameters)
where
the eigenvalues
from a level-one formulation, directly
means are not easy to obtain.
Pettifor’s
by level-one
from electronic
are the embedded
separately
self-
which must
(TB) method
is diagonalized,
is very convenient, energies.
energies, elastic constants,
or to those calculated
is the Tight Binding
are calculated
electronic
in terms of parameters
which gives the TB parameters
The TB formulation
often gives valuable
attempt
An example
or by fitting to the electronic
theory.
which do not feature
is solved (the Hamiltonian
and the TB parameters
either from LMTO-ASA
calculations
must be expressed
‘from the outside’.
indeed the Schriidinger summed),
structure
Then the Hamiltonian
be introduced
271
MODELLING OF SOLUTIONS AND ALLOYS
is obtained
potentials
by summing
are strictly empirical
up pair energies
and the total energy,
(Lennard-Jones
potentials,
for
example). Molecular the exception
dynamics
ular dynamics, concerned
(MD) simulations
combining
we shall look for analytical perature/composition
cists must consider.
techniques
methods
obtained
for deriving
ab initio calculations
dependence,
ab-initio
with molec-
In what follows, we shall not be by brute force simulation.
thermodynamic
functions,
are rarely concerned
the two most important
variables
about
Instead, hence tem-
first principles
The introduction
electronic
of the Cluster
temperature
which thermodynami-
Until about 10 years ago, outside of brute force simulation,
be done about combining namics of alloys.
[29] which performs
phase diagrams.
performing
and concentration
with 4th or 3rd level approaches,
method
MD with level 1 approach.
with thermodynamic
Physicists
operate
of the famous Car-Parrinello
structure
calculations
Expansion
Method
little could
and thermody[30] has provided
T. ANDERSSON et a/.
272 a way to combine principles
quantum
and statistical
Though predicting
the aim of the empirical properties
of materials,
rely on global measurements
available
theoretical
Empirical individual
procedures,
functions
can produce,
thermodynamic
quantities:
i.e.,
methods
to deconvolute
parts treated
the
no data
by the best
enthalpies,
of ab initio methods
methods
are based on electronic for electronic
configurational,
vibrational,
properties,
entropies,
require no fitting
temperature-dependent
is improving
of supplanting
materials
properties.
results but often unreliable
ones) which can
ones.
are in the process
with electronic
models
very accurate
rely on sound physical
but often produce results (especially
widely from experimental
meaningful
are identical, Empirical
require (in principle)
from their component
by fitting,
ab initio methods
The accuracy
methods
are opposite.
and use ad hoc approximate
methods
structure
calculations,
applications
For example,
continuously
older empirical
so that ‘first principle’
ones.
Also, since ab initio
such methods
are particularly
which, almost by definition,
band gaps, charge densities
are concerned
are obtained
as by-
of the calculations.
A significant hoc expressions
advantage
of starting
hence can be calculated
itself can also be expressed Method
of thermodynamic These tools:
with sound physical
to be fitted, is that the coefficients
physical meaning,
Variation
a new field: ‘first
formulations.
methods
etc. By contrast,
and first principles
the methodologies
to the free energy. Ab initio methods
base and build thermodynamic
entropy
creating
of ab initio methods
various contributions
products
thereby
thermodynamics’.
2. Advantages
deviate
mechanics,
from ‘first principles’.
by cluster methods.
(CVM) which naturally properties first-principle
are now well developed
rather
than ad
acquire clear
Then the configurational
Such is the spirit of the Cluster
includes short range order in the calculation
of solid solutions. electronic
structure
calculations/cluster
and some general packages are being created
‘user friendly’ than what has been available up to now. 3. Application
formalisms,
of the cluster expansion
to semiconductor
systems
expansion/CVM which will be more
THERMODYNAMIC By and large, physicists have appealed
directly
(CALPHAD).
However,
at absolute defects)
interested
most computations
zero (total energies,
Very little thermodynamic pansion
formation
properties
method.
Lambrechts calculation
systems
energies
surface reconstruction, by molecular
‘potentials’.
to the rule is the remarkable important
by the cluster and abundant
exwork
effort along these lines comes
of Ito and Mohri [32] pl us sporadic
by these groups vary depending shortly.
of semiconductor
work in this field has been performed
(331 and de Fontaine
273
than that of the empiricists
they are often obtained
using semi-empirical
The main exception
and co-workers
be explained
rather
energies of compounds,
by the group of Zunger [31]. Another
from the collaboration
properties
to date have focused on structural
are required,
or Monte Carlo simulation
produced
in predicting
to the ab initio community
and when thermal
dynamics
MODELLING OF SOLUTIONS AND ALLOYS
cluster
expansion
(341. The nature
on the computational
First, let us see what are the problems
work by
and reliability
tools available,
of
as will
to be solved.
4. Tasks to be performed Let us consider temperature distinguish
strictly
and concentration,
(b) Solid-phase
words,
i.e., those
phase diagrams
involving
specifically
in all generality.
We
In principle, solid phase
of liquid phases
quasi-chemical
can be treated
have been explored
has been performed
one.
Following
lattice
of bond strengths,
these
(CE), but only
no thorough
parameters.
angles and coordination
of an
the liquid as
The CVM approximation Pelton
and others
was
[36], it would
the CVM pair, which is equivalent
approximations
study
exception
[35] w h o simply considered
the work of Blander,
Of course,
Actually,
by CE to date, with the possible
to try a simpler approximation, model.
by cluster expansions
systematically.
and Pasture1
with slightly expanded
the tetrahedron
ordering
defects
of Al-Ni by Colinet
an fee crystal
be interesting
phase separation,
all three problems
reactions
investigation
determination
reactions:
(c) Off-stoichiometry
phases.
in other
problems,
the following problems:
(a) Liquidus-Solidus
distribution
thermodynamic
to the
do not take into account which are characteristic
the
of liquid
274
T. ANDERSSON et al.
Solid-phase
equilibria
for quasi-binary contributions
have been investigated
sections
in ternary
in detail for semiconductors,
and higher systems.
have come from the group of Zunger (311. The technical
group’s successes can be understood are generally
dominated
by elastic effects, due to the often large disparity distortions:
cell-external
[37]. The first type can be handled by minimization to lattice parameter,
but the second requires,
Hence, first-principle
required.
As mentioned
pseudopotential
electronic
that in the Mohri-Ito to be neglected Other
collaborative
dynamics
studies
deviations
methods
tend not to use
and CE remains
local atomic
relaxation
are unfamiliar
from stoichiometry
with CE (note
calculations,
as top so
effects tend
also be taken into account. chemical reactions
of point defects in electronic In oxides,
defects,
these equilibrium
with definite equilibrium
to choice of defect reactions
constants
ab initio
materials.
semiconductor
vacancies
Such effects are traditionally
associated
in addition
a recent
[38]), and tend to
lines [29].
is no problem. create antisite
however
unless one includes very elaborate
In
compounds,
and defect clusters.
these defects may be ionized in different ways. Electron-hole
The problem,
are
codes are not ‘linear-response-ready’,
along Car-Parrinello
alloys, off-stoichiometry
thermore,
in FLAPW
included
codes and ab initio
of the latter technique
by the group of Ceder and Joannopoulos
We do not know of any CE investigations metallic
is met by FLAPW
work, important
specialists
on oxide systems
stay away from thermodynamic molecular
forces on individual
[32].
pseudopotential
collaboration
relaxations
codes with ‘force theorem’
above, that requirement
Ito’s pseudopotential
systems
in atomic ‘sizes’.
and cell-internal
in general, calculating
‘linear response’ codes. Specialists
Apparently,
for that
of total energy (at zero K) with respect
structure
the CE, thus only Zunger’s group, proficient practitioners.
reasons
as follows: mixing energies in semiconductor
There result two types of structural
atoms.
particularly
In this field, the most important
equilibrium
Furmust
studied by writing down symbolic constants
(Kroger-Vink
to consider,
or the AG“ for the reactions.
approach).
is to obtain
values of
Up to now, empirical
methods
have been used. Impressive performed authors
-
‘brute force’ ab initio calculations
point out some of the difficulties
of calculations,
of defect formation
see the review by Scherz and Scheffler [39] is particularly
for example
gaps for which tractable
problems and reliable
which one encounters
relative to electronic approaches
energies
in performing
excitation
have been
interesting.
these kinds
energies
are not yet available.
These
and band
Furthermore,
THERMODYNAMIC as stated
by these
a reliable
description
MODELLING OF SOLUTIONS AND ALLOYS
authors
‘no general
of all defect
theoretical
candidates’.
approach
exists
(of more than two partners)
and highly distorted
been investigated
Little is known about corresponding
although lattice
in detail.”
Scherz and Scheffler do describe vibrations
through
be of considerable
to ab initio calculation
to investigate
linear combination
expresses
is suited
defect
vibrational
of dynamical
have not entropies,
matrix elements methods.
the CE technique Moreover,
for
complexes
metal impurities
valence force-field
whether
of defect thermodynamics.
at least one of its implementations,
‘Compound
transition
calculations
the use of approximate
interest
that
As a consequence,
275
and
It would
could be applied
since the CE method,
or
the energy of a given ‘configuration’
of the energies of simple structures,
this approach
is reminiscent
Energy Model‘ [40], but in a more general way, not requiring
as a of the
the mean field
simplification. 5. Caveat The Cluster expansion
Expansion
in cluster
in addition,
method
functions
many components
minimization
becomes
prohibitively
in particular
The convergence especially
rapidly.
are present,
in the past with short expansions, experiment,
is no panacea:
converges
large.
the numbers Practical
small clusters,
leading to transition
of cluster expansions
Its application
calculations
often resulting
temperatures
only if the
must be used.
of variables
and the incorporation
where elastic and Madelung contributions
is practical
If not, large clusters
in the free energy
have been performed in poor agreement
hundreds
with
of degrees too high.
of more distant
are important,
If,
interactions,
need to be investigated
further. One important scopic properties structure
remark:
it is generally
(thermodynamic
calculations
not good strategy
ones, for example)
to attempt
directly
since the length scales involved are just too different.
to solve practical
technological
the computation
receives input from lower levels and produces
It is generally techniques culations produce
problems by a hierarchy of approaches,
too much to ask for any scientist
of more than perhaps themselves,
a hierarchy
TB parameters,
MD simulations.
to derive macro
from ab initio electronic
two successive
output
to be an expert
for the next level.
in the computational
levels. Even for electronic
of levels may be used: for example,
a density-of-states
is obtained
It is preferable
where, at each level,
structure
cal-
LMTO calculations
and its second moment
is used in
276
T. ANDERSSON et al. For thermodynamic
the interface coefficients
It follows that
may be obtained
structure
parameters
‘scale’ of the end product
required,
with what
theories.
Or a
based on a level-two
derived from level-one calculation. in this field is dictated
by the numbers
of levels in the hierarchy
in turn, dictate
types of specialists, language.
of a linear system
by level-one averaging)
provides
For example,
must be organized
Such considerations,
find a common
methods.
by inversion
of which may be obtained
the way research
must be considered.
another,
mechanical
may be used (direct configurational
with electronic
organized,
to us here), the cluster expansion
and statistical
expansion
the parameters
more direct method
(of interest
the quantum
of the cluster
of equations,
approach
properties
between
how research
teams
and how they shall communicate
This is a non-trivial
organizational
by the which must be with one
problem,
one of
‘vertical’ integration.
Thermodynamic Assessment of a Compound Semiconductor System -
the Cd-Te System
Introduction In order to illustrate of a compound the Cd-Te
the kind of procedures
semiconductor
system
was chosen.
possible optoelectronic infrared
system,
involved in the thermodynamic
including
Crystals
optimisation
based on this system
devices with adjustable
assessment
of the defect parameters, have attracted
band gap and as a substrate
interest
as
material
for
equilibria
the
detectors.
In the standard solid semiconductor
Calphad approach phase is modelled
described
in the next section
approach
can be extended
solid phase.
for the calculation as a stoichiometric
for the Cd-Te
by considering
system.
compound.
This procedure
is
We then go on to show how this
the lattice and electronic
Lastly, we show how some of the important
can be optimised.
of liquid--solid-gas
defect structure
defect parameters
of the
of the model
THERMODYNAMIC
MODELLING
OF SOLUTIONS
AND ALLOYS
277
1400-c
A 200 II0
0.2
0.4
0.6
0.8
1.0
MOLE_FFWCTION TE Figure 1: The assessed phase diagram for the Cd-Te system (after [15]).
Standard CALPHAD
Assessment
The Cd-Te binary system has been reviewed recently (151. The liquid phase was modelled by the associated to describe
solution model having CdTe as an associate.
the Gibbs energy of formation
and with Te. The composition in a Redlich-Kister was modelled dependent
dependence
polynomial
than the CdTe-Cd
as a stoichiometric
formation
enthalpy.
of the associate of the CdTe-Te
Eight coefficients
were used
and its interaction
with Cd
interation
interaction.
The CdTe solid phase
phase with an experimentally
Fig. (1) s h ows the calculated
is one order higher
adjusted
temperature
binary phase diagram.
Inclusion of the Defect Model Defects are introduced important fortunately,
during the growth and processing
role to play in determining the identification
the functional
and the determination
defects is a difficult task so that theoretical There are two aspects (a) obtaining
of semiconductors
properties
of a semiconductor.
of the concentration
calculations
and have an Un-
of the different
take on a particular
importance.
to these calculations: the isolated defect free energies (energies
tum mechanical
calculations
of the type described
and entropies)
from quan-
in an earlier section
278
T. ANDERSSON et a/. (b) using the defect
free energies
to calculate
the concentrations
of the various
defects under given conditions.
A theoretical
group at SRI International
(Cd,%n,Hg)-Te model
systems
which
contains
substitutional
eight
sublattices
interstitial
sublattice).
consider
although
native
point
In addition
of formation
(a) vibrational
are equally as important
two contributions -
model for the Coulombic (b) degeneracy
-
distortion
of the principal
It can be seen that
The other ionisation
the Cd interstitial important
as the energies of formation.
due to the introduction
the elastic contributions
and using a point charge
a direction
degeneracy
due
has been considered.
neutral
defects
is calculated
to dominate
at the highest
It also appears
with the important
energies are consistent
calculated
in CdTe at 700°C as a function
defect at low and medium
defect is the Te antisite.
ionisation
of
contributions.
energies that the Cd vacancy is a double acceptor
The calculated
The SRI
by fitting the total energy results to a valence force
the principal
Teed and Cdi, are donors,
[46].
[45]:
of Cd
Cd pressures
from calculations
interstitial
is
Cd pressures. of the
with the first ionisation
close to the valence band edge and the second being close to mid-gap. defects,
to i.e.,
at high temperatures,
in Fig (2) the results from Ref. [46] which shows the theoretically
concentrations
Vao, whilst
of defects
as well as the usual spin degeneracy,
to Jahn-Teller
to be isolated,
as being of importance
the change in the phonon spectrum
field model for calculating
on the two
and an anionic
there are ionised defects
are considered
to this entropy
for the
a quite complex
and vacancies
defects
The defects
the concentrations
the defect can be calculated
pressure.
(antisites
a bound defect pair has been suggested
group have considered
We reproduce
out this type of analysis
defects on both a cationic
to these neutral
and holes.
In order to be able to calculate the entropies
defects
plus two interstitial
as well as electrons
unbound,
have carried
in recent years [42-471. They have considered
level
The other
two
being a double donor.
with there being a p 3
n transition
in
CdTe at high Cd pressures. The model adopted
here is a slight modification
of that used by the SRI group.
have used only one interstitial
sublattice
degeneracy.
in order to be able to illustrate
This is sufficient
We
and have not allowed for any spin or Jahn-Teller the principles
involved
in
THERMODYNAMIC
Figure
2:
Calculated
MODELLING OF SOLUTIONS AND ALLOYS
neutral defect concentrations
279
for CdTe at 700% as a
function of Cd pressure (after [46]). this type of calculation.
The three-sublattice
model may be written:
(Cd,TeX,Te~,VaX,Va~,VaZ~)(Te,CdY,VaX)(Va,CdX,Cdf,CdZ+,TeX) where we have used the standard (X refers to a neutral The software permit
defect)
packages
the inclusion
Kriiger notation
[e,h]
for the effective charges on the defects
(191.
ChemSage
of electronic
and ThermwCalc
have been extended
defects as well as charged
lattice
defects
in order to [22,49].
The
input files require the following information:
1. the standard
chemical potentials
with those for electrons 2. the site exchange been considered
of Cd and Te in the zincblende
energies for the lattice species. in the present
potential
The only interactions
model are those between
the majority
which have components
sublattices.
the models in these packages are restricted
holes only, i.e., the chemical
together
and holes and the lattice defects.
Cd and Te on the two substitutional
At present,
structure,
of the electrons
to non-degenerate is given by:
electrons
and
280
T. ANDERSSON et al.
800 I
500 ! I 4998 E-4 4998
A
I
I 5002
5000
5004
MOLE_FFtACTION TE
( [48]) solidus in the
Figure 3: Comparison of the calculated and experimental
Cd-Te system. The calculated solidus results from the optimisation of the energies of formation of three defect energies. where y, is the concentration per sublattice
site so that the electron
lattice defect concentrations. the number
of electrons
and is conveniently
concentrations
This concentration
expressed
may be directly
as the fraction
compared
may, of course, be directly
per unit volume, which is the concentration
with the
converted
to
of choice by the semiconductor
industry.
Optimisation of the Defect Model Parameters As a starting
point for the defect free energies, those given by the SRI group for HgTe were
used. The most important solidus by Greenberg measurements
(48). Optimisation
is possible
Calc). In order to illustrate the energies, The calculated
of these were then optimised
of defect parameters
by using ChemOpt the possibilities
but not the entropies, solidus is compared
to the recent measurements to the experimental
(for ChemSage)
of extending
the standard
of the three most important with the experimental
or Parrot
of the solidus
(for Therm*
Calphad
approach,
defects were optimised.
data in Fig. (3).
THERMODYNAMIC F’ut
MODELLING OF SOLUTIONS AND ALLOYS
281
ure Development s/Improvements/Recommendations
It should
be emphasised
that
been used as the starting such calculated future
results
calculations
Calphad
theoretically
point
will be the best
on compound
community
their optimisation
source of thermodynamic An important
results,
particularly
have
It is likely that
data
for defects
contribution
in
from the
values through
for the solidus but also for
properties.
the Group have identified
species,
parameters
in CdTe.
can be to assist in the refining of these calculated by using experimental
and have shown how a system charged
thermodynamic
calculations
semiconductors.
the single phase thermodynamic Although
calculated
for the defect
an important
of interest
area for CALPHAD
can be modelled
development
to give the concentrations
of
there are several areas in which there is room for improvement:
1. There is a need for better values for defect entropies recent ab intio calculations the entropy
in semiconductors.
for defect energy formation
values are of equal importance
Although
the
are a welcome development,
in the calculation
of defect equilibria
at
high temperatures. 2. There are two ways in which the currently (a) by the incorporation
of degeneracy
sion of spin degeneracy distortions
in the electron/hole
the charged
3. Semiconductor
Cd-Zn-Te
defects
approximation modelling
The defect interaction
can be quite large (AE/LT
>>
energies between
0) so that
should be incorporated
into all the free energy minimisa-
community.
carrying out more detailed
calculations
These will consider not only the temperature-composition
grams of the kind shown in this report but will also include consideration and a calculation
of the functional
results will be published
the Bragg-
used in the present model is insufficient.
of the Group are currently
system.
The inclu-
is not a major hurdle [23] but the effects of Jahn-Teller
tion packages used by the CALPHAD
Some members
equilibria.
will be a more difficult problem.
(b) by the inclusion of defect interactions.
Williams
used defect models could be improved:
(semiconducting)
in due course.
properties
on the
phase diaof gasequilibria
of the materials.
These
282
T. ANDERSSON
Although combining
outside the terms of reference of the Working Group, there is also a need for
the thermochemical
tor industry
et a/.
modelling
can relate their processing
the final crystal. is in operation
with reactor modelling
variables to the charged species concentrations
The software interface, on both ThermoCalc
so that the semiconduc-
developed
by Sundman
and ChemSage,
of
and Eriksson and which
is an ideal tool for this purpose.
References [l] Papers presented and “Materials
at the symposia Science
“Materials
of Contacts
Annual Meeting of TMS, Anaheim, [2] G. Humpston,
D.M. Jacobson,
tional, Materials [3] A. Bolcavage,
and Metallization California,
Principles
and Interconnects”
February
of Soldering
125th
4-8 , , (1996) and Brazing
ASM Interna-
Park, OH, , (1993)
and Processing
of Materials
of Thermodynamics
(Eds.: P. Nash and B. Sundman),
TMS,
PA, ~171 (1995)
[4] B. Hallemans, [5] A. Boudene,
P. Wollants, J.R. Roos, Journal of Phase Equilibria, K. Hack, A. Mohammad,
S. Fries, H.-L. Lukas, R.A. Konetzki, C. Bernard,
C. Colinet,
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