Non-random distributions in binary crystals : from
order to total disorder
J. Duran, J.P. Lemaistre
To cite this version:
J. Duran, J.P. Lemaistre.
Non-random distributions in binary crystals : from order to total disorder.
Journal de Physique Lettres, 1982, 43 (24), pp.845-851.
<10.1051/jphyslet:019820043024084500>. <jpa-00232134>
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J.
Physique
-
LETTRES 43
(1982) L-845 - L-8511
15 DÉCEMBRE
1982,J
L-845
Classification
Physics Abstracts
61.70W
.
Non-random distributions in binary
from order to total disorder
crystals :
J. Duran
Laboratoire d’Optique de la Matière Condensée (Luminescence)
Tour 13, 4, place Jussieu, 75230 Paris Cedex 05, France
(*),
Université P. et M. Curie,
and J. P. Lemaistre
Centre de Physique Moléculaire Optique et Hertzienne (**), Université de Bordeaux 1, 33405 Talence
et Centre de Mécanique Ondulatoire Appliquée, 23, rue du Maroc, 75019 Paris, France
(Reçu le 17 juin 1982, accepte le 28 octobre 1982)
Ce travail propose un modèle théorique pour la génération de distributions non aléatoires
Résumé.
d’ions ou de molécules dans les cristaux binaires. Il consiste à simuler par ordinateur la croissance
épitaxiale d’un cristal dans lequel on tient compte des interactions entre les différents constituants.
La définition d’un paramètre d’interaction et de paramètres d’ordre translationnels permet une
description continue allant du système totalement désordonné au système parfaitement ordonné.
2014
Abstract
2014
The basic idea for
generating non-statistical distributions
in
binary crystals lies
in the
computer simulation of an epitaxial crystal growth where the interactions between the components
are taken into account. The resulting non-statistical distributions are described in terms of three
pertinent parameters (interaction and translational order parameters). We show that, by varying the
interaction parameter, we are able to cover continuously the range between totally disordered and
purely ordered crystals.
The general problem concerning the distribution of molecules or ions in
1. Introduction.
binary crystals has been underlying most of the recent theoretical and experimental studies.
They are dealing with concentrated systems where inter-particle interaction plays a significant
role such as spin-glasses, mixed crystals above the delocalization transition.. ~ From a theoretical
point of view, all the works dealing with these problems consider that the site disorder problem
could be handled assuming a statistical distribution of the particles in the mixture reducing the
problem to the consideration of diagonal and off diagonal disorders alone [2].
A number of recent experiments, dealing with doped crystals, have shown various situations
where the dopants were not statistically distributed in the matrices [3-9]. Several mechanisms have
been put forward in order to explain the non-statistical character of the particle distribution.
-
(*) Equipe de Recherche Associee au CNRS.
(**) Laboratoire associé au CNRS 283.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019820043024084500
JOURNAL DE PHYSIQUE - LEITRFS
L-846
be summarized
the
following processes :
the interaction between A and B particles in the Am B,, crystal plays a significant role during
the crystal growth at the interface solution-crystal.
at high enough temperatures, the particles in the crystal may reorganize in order to lead
to the minimization of the thermodynamic energy of the system. This mechanism is connected
to a characteristic time and may or may not be efficient depending on the heat treatment which
follows the crystal growth [10].
While the latter process is difficult to handle, even numerically, because it leads to the solution
of a time-dependent many body problem, it turns out that the former mechanism can lead to a
tractable solution. It is the purpose of this letter to provide a technique for generating non random
distributions (NRD) and to show how, introducing adequate parameters, these NRD can be
described and used for future calculations dealing with non-statistical site disorder.
This work provides computer simulations of NRD for a two-dimensional model using an
interaction parameter Z related to the strength of the attractive or repulsive interactions between
the A and B particles during the crystal growth. In order to describe the various amounts of
ordering introduced in the crystal due to the inter-particle interactions, we introduce translational order parameters (TOP). The interaction parameter Z and the tragslational order parameters P are shown to be related and turn out to be pertinent adjustable parameters for experimental investigation.
They
can
invoking
-
-
Computer simulation of the crystal growth. Starting from a solution containing A and B
particles the crystal growth corresponds to the pseudo-chemical reaction [10] :
2.
-
The general solution of the problem should take into account the various attractive or repulsive
interactions between the A-A, B-B and A-B pairs of particles, and the solution cannot be obtained
unless we consider an epitaxial growth which greatly reduces the degree of freedom of the system.
Seen from this point of view, the problem can be solved either using a row after row or a site after
site growth. For sake of simplification we choose here the second procedure considering that the
other technique would lead to the same set of NRD. Moreover the site after site process is generally
looked upon as a realistic description of the real crystal growth.
If we take into account the inter-particle interaction in a site after site crystal growth, the
probability of introducing an A particle from the solution into the crystal depends on whether
or not A particles are present in the crystal in a nearest-neighbour (NN) or next nearest-neighbour
(NNN) position. According to this statement we may define two conditional probabilities
PA/A and P A/B. PA/A is the probability to crystallize an A particle when there is already, at least,
an A particle in a NN or NNN position in the crystal. P AIB is the probability to crystallize an A
particle when there is no A particle in the crystal in a NN or NNN position. In the absence of
any interaction between the A particles, these two probabilities are obviously identical and are
equal to the concentration C~ of the A particles in the,solution. Now, if we take into account the
inter-particle interactions we can modulate the two probabilities by setting :
with K1 ~ K2 i= 1 for non-random distributions.
A simple relationship between Kand K2 can be found if we consider the following procedure :
our model can be looked upon as describing a modulation of the local concentration of the
solution at the liquid crystal interface depending on the already crystallized neighbourhood.
For example, if there are repulsive interactions between the A particles, the K11 factor (which
NON-RANDOM DISTRIBUTIONS IN BINARY CRYSTALS
L-847
therefore turns out to be smaller than 1) will reduce the effective local concentration of the neighbouring solution if there is an A particle in the crystal in a NN or NNN position.
The conservation of the A particles implies that an increase (decrease) in the probability 7B/A
corresponds to an equivalent decrease (increase) of the probability PAIB so that :
This last condition postulates that the volume of the solution is significantly larger than that
of the crystal during the crystal growing process.
The purpose of the present work is to set up a technique starting from a relatively simple
situation. Nevertheless any other situation involving A-B and B-B interactions in addition to
the A-A interactions can be handled following the same procedure.
The modulation of the local probabilities 7B~ and P A/B may be described using a single parameter Z defined as follows :
limiting cases corresponding to the Z values - 1, 0,
describe respectively the case of maximum attraction, purely random and maximum repulsion between the A particles.
A computer simulation of the system has been made using the following procedure :
we consider a two-dimensional square lattice made up of 104 sites. The distribution of the A
particles in the first row may be generated either at random or by including the modulation
technique described above. In both cases the results were identical as soon as one considers a
sufficiently large part of the crystal. The distribution of the following rows is calculated using the
modulation function defined in the preceding equations. Three illustrative examples of the calculated distributions obtained in the three limiting cases are reported in figure 1. As can be seen
in this figure, the introduction of the Z parameter leads to enhancing or inhibiting the clusterization process depending on whether the interaction is attractive or repulsive respectively. As
will be realized below, the consideration of these interactions allows the full range between
ordered and disordered crystals to be covered.
It is instructive to consider here the three
+ 1 which will
3. Translational order parameters. - Besides providing a technique for generating NRD for
future calculation of observable quantities which are to be compared to experimental results,
this paper is aimed at defining pertinent parameters relevant to the description of these unusual
distributions. As can be seen in figure 1 the introduction of the interaction parameters leads to a
certain ordering of the crystal with respect to the classical random topology. In order to introduce
the translational order parameters we first investigate the simplest case of a linear chain.
Let us consider a system of np particles randomly distributed along a linear chain of N equidistant sites. For a random distribution the number q" of clusters of n consecutive particles is
given by :
--
where C
np/N is the particles concentration.
Now let us define the two first translational order parameters that we shall denote PI and P2
in the following way : P, (P 2) is the ratio of the number of superpositions of occupied sites obtained
after translating the chain by one (two) inter-site distance, to the total number of particles. According to this definition :
=
... r
,
..
JOURNAL DE PHYSIQUE - LETTRES
L-848
1. 2013 Computer simulation for a two-dimensional distribution of particles at concentration ranging
between 16 and 25 % for three cases of interest : a) Z
0, C 21 % : random distribution; b) Z= -0.75,
C = 20 % : strong attraction; c) Z
1, C 16 % : maximum repulsion, no total order; d) Z 0.95,
C = 25 % : nearly maximum repulsion, nearly pure order. (Representative sections of the 104 - sites lattice).
Fig.
=
=
=
=
=
Concerning the calculation of P2 the total number of superpositions of occupied sites is
y (n - 2) q" if the clusters are separated by at least two unoccupied sites, while it is equal to
N
ti=2
N
y
(n - 1) q,, if they
separated by
are
one
unoccupied
site.
Consequently, the
total number of
,,=1
coincidences may be written
NC~(1 - C)
where ql
=
chain,
get :
we
as :
is the number of isolated
.
unoccupied
sites. So, for
an
infinite linear
relationships between P1,P2 and C hold for any random distribution, whatever the
dimensionality, since, in this particular case, the distributions, in any direction, are uncorrelated.
These two
NON-RANDOM DISTRIBUTIONS IN BINARY CRYSTALS
L-849
Due to the fact that the introduction of the interaction parameter Z will lead to some correlation between the particles in the crystal, these parameters PI and P2 are expected to differ significantly from the normal C value, and therefore are likely to give information about the amount
of ordering introduced through inter-particle interaction.
We have computed these translational parameters for the chosen two-dimensional square
lattice keeping in mind that, for obvious reasons the P1 (P2) parameter should be more relevant
to the attractive (repulsive) cases.
It is still interesting to consider the extreme cases for the P values which are 0 and + 1. As can
1 corresponds to a total ordering of the crystal since, in this
be seen from the definition P
particular case, the system is translationally invariant. The P1 and P2 parameters are relevant to
the present problem which deals with NN and NNN interactions only. It should be kept in mind
that longer range interactions would require the definition of higher order translational parameters. So, under the present conditions, the smaller the value of P, the more disordered the
=
crystal.
4. Results of the calculations for various non random distributions. - The first consequence
connected with the consideration of interaction parameters lies with the fact that the concentration
of the crystal will differ from that of the solution. This very common feature reported by crys0153ls
makers is illustrated in figure 2. As can be expected, repulsive interactions will prevent the dopant
from being introduced in the matrix, whereas attractive interactions will increase the dopant
concentration. Independently from other considerations connected to the crystal growth technique (reciprocal solubility, melting temperatures, eutectics...), these curves provide the first
numerical data taking into account the affinities of the components of a binary crystal. Undoubtedly, it would be interesting to compare numerical data coming from crystal makers in order
to correlate the Z parameters to thermodynamical quantities.
Figure 3 reports the P1 (or P2) dependence on the Z value at different concentrations.
A striking evidence which can be seen from this figure is related to the fact that all the calculated
structures are more ordered than the random distribution which turns out to be the most disor-
Particle concentration in the crystal (Cc) versus particle concentration in the solution (Cs). These
have been calculated for the following values of the interaction parameter Z : + 1 (a); + 0.75 (b);
+ 0.5 (c) ; 0 (d) ; - 0.5 (e) ; - 0.75 (f) and - 1 (g).
Fig.
2.
curves
-
L-850
JOURNAL DE PHYSIQUE
-
Dependence of the translational order parameters on the particle concentration in the crystal Cc.
(a) (smoothed curve); Z 0.5 (b); Z 0.8 (c); Z 0.95 (d); Z 1 (e). Attractive interaction :
full
0 (a) (smoothed curve); Z = - 0.5 (b); Z
curves calculated for : Z
1 (c); for P, and P2. Pt
dotted curve.
curve; P 2
Fig.
Z
=
3.
0
-
=
=
=
=
=
=
=
-
=
dered one. In other words, we conclude that any kind of interaction between the components of a
binary crystal will introduce a certain amount of ordering in the mixture, therefore leading to
drastic consequences as far as random walk processes or energy diffusion are considered.
As predicted in the preceding section we observed that both P parameters are equal to C in the
random distribution. The satisfactory numerical fit of the analytical equations can be considered
as a good test for the random generator used in the computation and also as a confirmation
that the number of particles considered in the present calculation is sufficiently large to insure a
convenient reliability of the reported results.
Let us consider successively the curves obtained in the attractive and repulsive cases as reported
in figure 3.
When Z is negative (attractive interaction) we observe that the larger deviation, with respect
to the random distribution, is obtained at concentrations ranging between 10 and 20 ~, and
turns out to be larger in the small concentration region.
Although our calculation, being restricted to 104 particles and short-range interactions, is
obviously unable to lead to reliable estimates of the clusterization effect in the small concentration region (below 2 %), it turns out that the calculation can be quite easily extended in this
concentration range.
An interesting feature arises when considering high concentrations (above 60 %) where all the
curves coalesce with the random value. This feature shows that the distributions tend to be the
same whatever the attractive interaction which is considered. This is an important point; however
one might raise the question as to whether this particular behaviour is related to the definition
of the P parameters. As we shall show in a forthcoming paper this effect is quite real and is connected to experimentally undistinguishable distributions.
As quoted in figure 2 the crystal concentration is always smaller than that of the solution when Z
is positive (repulsive case). The obtained limiting value of 25 % obtained for Z
+ 1 can be
the
of
if
we
consider
interaction
NN
to
and NNN
(restricted
particular type
easily explained
distribution
B
of
A
and
an
be
which
of
alternate
can
seen
is
favour
as
in
in
particles
only)
figure 1.
For this particular value of Z the system displays a steep dependence of the order parameter on C,
corresponding to the order-disorder transition as will be discussed in a more detailed paper.
=
NON-RANDOM DISTRIBUTIONS IN BINARY CRYSTALS
L-851
5. Conclusion. - This work relies on a phenomenological definition of the interaction parawhich is considered as characterizing the modulation of the solution concentration at
the liquid crystal interface during the crystal growth. This introduction of an interaction parameter leaves open any interpretation of the physical nature of the involved process. Apart from
the natural view of Z describing the Coulomb interaction in an ionic binary crystal, this parameter may as well be considered as taking into account the steric problem in a mixture of two
components with different ionic or molecular sizes...
An interpretation of the Z value can be attempted a priori if we consider the pseudo-chemical
reaction quoted in section 2 and relate the concentrations by a mass action law. This procedure
leads to a straightforward connection between Z and the Gibbs free energy for the crystal growth.
However, to the best of our knowledge, we feel that the involved quantities cannot be estimated
numerically, since the crystal growing process involves a number of unknown processes. Another
technique which has been applied recently involves the calculation of the total energy of the binary
alloy as developed in [11]. Using results presented in that paper would allow us to correlate our
data to thermodynamical quantities. However since we are, in this paper, more interested in
calculating the detailed distribution for future connection to experiments, it seems more reliable to
try to relate the Z value, a posteriori, to experimentally available data such as inhomogeneous
bandshapes due to the distributions of monomers, pairs, triads, etc..., which are strongly dependent on the clusterization degree of the binary mixture. In order to do that, we have performed a
calculation of the expected bandshapes in the various distribution calculated in the present
paper. The results of these calculations will be reported in a forthcoming paper.
meter Z
References
[1] This work was undertaken while one of the authors (J. D.) was at the Laboratoire d’Optique Physique
de l’ESPCI (ER 5 du CNRS).
J. M., Models of disorder (Cambridge University Press) 1979.
ZIMAN,
[2]
[3] PETIT, R. H., EVESQUE, P., DURAN,J., J. Phys. C (Solid State Physics) 14 (1981) 5081.
[4] LIDIARD, A. B., Phys. Rev. 94 (1954) 29.
[5] TALLANT, D. R., MOORE, D. F., WRIGHT, J. C., J. Chem. Phys. 67 (1977) 2897.
[6] PELL, E. M., J. Appl. Phys. 31 (1960) 1675.
[7] HAYES, W., J. Appl. Phys. 33S (1962) 330.
[8] BLEANEY, B., J. Appl. Phys. 33S (1962) 338.
[9] VORON’KO, Y. K., KAMINSKII, A. A. and OSIKO, V. V., Sov. Phys.-JETP 22 (1966) 501.
[10] KROGER, F. A., The Chemistry of Imperfect Crystals (North-Holland Publishing Company) 1964.
[11] ROBBINS, M. O. and FALICOV, L. M., Phys. Rev. B 25 (1982) 2343.