2003 self-organization processes in impurity subsystem of solid solutions

Journal of Physics and Chemistry of Solids 64 (2003) 1579–1583
www.elsevier.com/locate/jpcs

Self-organization processes in impurity subsystem
of solid solutions
E.I. Rogacheva*
Department of Physics, Kharkov Polytechnic Institute, National Technical University,
21 Frunze St, Kharkov 61002, Ukraine

Abstract
New experimental results proving the existence of critical phenomena in the range of small impurity concentrations
(,1.0 at.%) in a number of ternary solid solutions based on IV– VI semiconducting compounds are presented. An anomalous
decrease in X-ray diffraction linewidth and an increase in the lattice thermal conductivity and heat capacity in this concentration
range were observed. The experimental results are analyzed on the basis of percolation theory and fluctuation theory of the
second order phase transitions. From the experimental data, the critical exponents for the lattice thermal conductivity and lattice
heat capacity are determined. It is suggested that self-organization processes (a short-range or long-range ordering of impurity
atoms) accompany the percolation phenomena. The results obtained in this work represent another evidence to the proposition
about the universal character of critical phenomena accompanying the transition from an impurity discontinuum to an impurity
continuum.
q 2003 Elsevier Ltd. All rights reserved.
Keywords: A. Semiconductors; A. Alloys; C. X-ray diffraction; D. Critical phenomena; D. Thermal conductivity

1. Introduction nature of impurity potential, for any type of the interaction
between dopants (deformational, electrostatic, dipole –
Solid solutions represent a broad class of substances, the dipole, etc) one can designate the radius of impurity atom
most widespread and having a great potential for practical ‘action sphere’, within which the crystal properties differ
applications. In the framework of generally accepted notions considerably from those of the matrix, as R0 : In accordance
of the physico-chemical analysis, the physical properties in with one of the problems of percolation theory, viz.
the solid solution region change in a monotonic way, and the ‘problem of spheres’ [9,10], there is a critical concentration
appearance of concentration anomalies of properties (percolation threshold xc ) at which the channels penetrating
indicates the intersection of phase region boundaries. the whole system appear and an infinite cluster consisting of
However, for a number of semiconductor solid solutions overlapping spheres of radius R0 is formed. The effective
we observed [1 – 8] concentration anomalies of different value of xc depends on the range of interactions in the
properties (microhardness H; charge carrier mobility m; system, i.e. on R0 : The formation of the infinite cluster is
lattice thermal conductivity lp ; etc.) in the range of small accompanied by critical phenomena, which must manifest
impurity concentrations (, 1.0 at.%), which indicated the themselves in the case of the solid solutions through
presence of concentration phase transitions (PTs). We anomalies in the concentration dependences of different
suggested [1] that these PTs have the universal character, properties. When the percolation threshold is reached, there
corresponding to the transition from an impurity disconti- appear channels, along which internal elastic stresses caused
nuum to an impurity continuum, and take place according to by the impurity atoms are partially compensated due to the
a percolation scenario [9,10]. Assuming that the properties overlapping of impurity deformational spheres. As a result,
are isotropic and taking into consideration a short-range the movement of dislocations and propagation of elemen-
tary excitations are facilitated. An increase in the dislocation
* Tel.: þ380-572-400-092; fax: þ380-572-400-601. mobility, a decrease in the effective phonon and electron
E-mail address: rogacheva@kpi.kharkov.ua (E.I. Rogacheva). cross-section under the formation of percolation channels
0022-3697/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/S0022-3697(03)00245-2

1580 E.I. Rogacheva / Journal of Physics and Chemistry of Solids 64 (2003) 1579–1583

lead to a drop in H [1,7,8], a growth in lp [6] and m [1,3– 5]
which we observed in the critical region.
To prove the suggestion about the universal character of
the concentration anomalies of physical properties in solid
solutions, it is necessary to expand the scope of objects and
properties to be studied, to perform a more detailed analysis
of experimental data, and to further develop theoretical
grounds. In this work, the new experimental results on the
concentration dependences of the X-ray diffraction (XRD)
linewidth B; lattice thermal conductivity lp ; and specific
heat C in the range of small impurity concentrations in
ternary solid solutions based on IV– VI (SnTe, PbTe, and
GeTe) semiconductor compounds [11], are presented. The
new results are considered jointly with the previously
obtained data and discussed within the framework of the
above mentioned ideas we have been developing in our
studies.

2. Results and discussion

The experimental details of the sample preparation,
XRD study, measurements of the thermal conductivity and
the heat capacity were described earlier in Refs. [2 – 8].
In Fig. 1, the room-temperature dependences of B on the
impurity content in PbTe– CdTe and PbTe –Bi2Te3 solid
solutions based on PbTe are presented. For comparison, we
also show previously obtained data for the PbTe –GeTe [2],
SnTe– Te [12] and CuInSe2 – CdS [13] systems. In deficit
solid solutions (SnTe – Te [12]), the increase in the Te Fig. 1. The dependence of a relative change in the XRD linewidth
content within the SnTe homogeneity region (50.15 – DB=B on the dopant concentration in solid solutions PbTe–CdTe
50.8 at.% Te) corresponds to the increase in the concen- (a),(b), CuInSe2 –CdS [13] (c), SnTe–Te [12] (d), PbTe–GeTe [2]
tration of cation vacancies (, 0.5– 3.2%) caused by the (e), and PbTe–Bi2Te3 (f). a: (400) diffraction line; b: (800)
deviation from stoichiometry and playing the role similar to diffraction line.
the role of impurity atoms. It is seen that in all studied solid
solutions in a relatively narrow range of concentrations of a
second component (, 0.5 – 2.0 mol%), an anomalous homogeneity region relative to the stoichiometric compo-
decrease in B is observed. sition [12].
It is known that among the main factors that cause a A subsequent sharp decrease in B shows that internal
broadening of XRD lines, apart from instrumental factors stresses in the crystal decrease. This is in good agreement
related to the experimental conditions, are (1) the micros- with a drop in H in the vicinity of the critical composition,
tresses in crystal and (2) a small size of coherent scattering which was observed in all studied systems [15] and was
regions [14]. In homogeneous disordered solid solutions attributed to the decrease in internal stresses level with the
with a sufficiently large grain size, the main reason of a reaching of the percolation threshold and the formation of
broadening of XRD lines is microstresses caused by a percolation channels.
difference in sizes of impurity and host atoms. Since all Impurity atoms are centers of local distortions in the
studied solid solutions were prepared and investigated under crystal lattice, sources of internal stresses and strains
the same conditions, the effect of all variables except decreasing in an inverse proportion to the cube of the
microstresses could be ruled out. That is why the broadening distance [16]. Since noticeable displacements of atoms are
of XRD lines we observed after the introduction of the first created within one or two interatomic distances from an
portions of the impurity (Fig. 1) is easy to explain. In the impurity atom, one can consider elastic fields as short-range.
SnTe– Te system (Fig. 1(d)), we do not observe the initial At small impurity concentrations, when distances between
increase in B because alloys with concentrations of cation impurity atoms are much larger than R0 ; elastic fields created
vacancies smaller than , 0.5% do not exist in the by separate atoms practically do not overlap. As the impurity
equilibrium state due to a significant shift of the SnTe concentration increases, elastic fields of neighboring atoms

E.I. Rogacheva / Journal of Physics and Chemistry of Solids 64 (2003) 1579–1583 1581

begin to overlap, which leads to a partial compensation of
elastic stresses of opposite signs. After percolation channels
via deformational fields of separate atoms are formed, the
interaction of impurities becomes cooperative. As the
impurity concentration increases, the overlapping of defor-
mational spheres and the compensation of microstresses
gradually spread over the entire crystal (the density of
‘infinite cluster’ grows [9,10]), leading to a decrease in the
overall level of elastic strains in the crystal lattice, which, in
turn, results in a decrease in B: Further introduction of
impurity atoms into this new medium (‘impurity liquid’)
causes new distortions of the crystal lattice and, consequently
a broadening of XRD lines.
Fig. 2. The lattice thermal conductivity lp vs. the dopant
The formation of continuous chains of impurity concentration in the SnTe– InTe (a) and PbTe–GeTe (b) solid
deformational spheres upon reaching the percolation solutions. a: T ¼ 300 K; b: T ¼ 170 K:
threshold must stimulate redistribution of impurity atoms
in the crystal lattice leading to the realization of their
configuration with a minimum thermodynamic potential. As is seen, in the PbTe– Bi2Te3 system (Fig. 2(f)), also at
Elastic interactions between impurity atoms, similarly to least two anomalous regions are observed—in the vicinity of
Coulomb interactions, can lead to the formation of 1 mol% Bi2Te3 and near 3 mol% Bi2Te3. Under the doping
configurations of impurity atoms, which correspond to of PbTe with Bi2Te3, Bi atoms and vacancies are introduced
minima of the elastic energy. Possible self-organization into the cation sublattice simultaneously. The Coulomb
processes may include a long-range ordering of impurity attraction between charged defects of opposite signs (Bi3þ
atoms (‘crystallization of impurity liquid’), a formation of and V22) stimulates processes of chemical interaction
Pb
complexes (short-range ordering), a change in the localiz- leading to the formation of neutral molecular complexes
ation of impurity atoms in the crystal lattice, etc. Under such as Bi2Te3. Thus, in addition to separate impurity atoms,
isovalent isomorphic substitution a long-range ordering is new structural elements appear, and the formation of
more likely. Under heterovalent substitution when the percolation channels through these elements becomes
structure of a matrix differs from the structure of a dopant, possible. On the basis of the above considerations one can
the probability of a short-range ordering in solid solutions suggest that the first anomaly in the B dependence on the
increases. The introduction of a dopant in the form of a impurity content is connected with the formation of
stable chemical compound stimulates the formation of percolation channels linking Bi atoms, while the second
neutral chemical complexes corresponding to the compo- anomaly is related to the formation of percolation channels
sition of this compound. When a certain concentration of through Bi2Te3 complexes. This suggestion is supported by
chemical complexes is reached, the formation of percolation the fact that it is after 3 mol% Bi2Te3 that the charge carrier
channels linking complexes and accompanied by a decrease concentration in the PbTe– Bi2Te3 system does not change
in internal stresses becomes possible. any more [17].
The very convincing argument in favor of self- In Fig. 2, the concentration dependences of the lattice
organization and ordering, which take place upon reaching thermal conductivity lp in SnTe –InTe and PbTe– GeTe
the critical concentration of an impurity, is a dramatic solid solutions are presented. As is seen, in these systems an
decrease in B; in some cases down to the value observed in anomalous increase in lp takes place in the range of small
an impurity-free host-material (the PbTe – Bi2Te3 and impurity contents. In the PbTe – GeTe system, two anom-
PbTe– CdTe systems). alous regions, whose locations correspond to those of
When the solid solution region is sufficiently wide, anomalies in the B dependences on the impurity content, are
different variants of ordering can be realized with observed. We have registered an anomalous increase in lp
increasing impurity concentration. To all appearances, in the range of small impurity concentration earlier in the
two extrema observed in the concentration dependence of PbTe– MnTe system [6] and attributed it to a decrease in the
B in the isovalent PbTe– GeTe system (Fig. 1(e)), can be effective phonon cross-section as a result of the formation of
attributed to the realization of different types of ordering percolation channels near the percolation threshold and a
[2]. A simple calculation shows that a composition of decrease in an overall level of elastic stresses in the crystal
1 mol% GeTe is optimal for an ordered distribution of lattice. The increase in lp up to the values of a host-
impurity atoms over the sites of a simple cubic lattice compound, which was observed in the PbTe – MnTe system
with a ¼ 3 a0 ; whereas at , 1.6 mol% GeTe, a super- [6] is another argument in favor of the suggestion about the
structure of impurity atoms with a fcc lattice and unit cell self-organization processes in impurity subsystem of crystal.
parameter of a ¼ 4 a0 (where a0 is the unit cell parameter In accordance with the modern views [9,10,18], there is an
of the studied alloy) can be formed. analogy between percolation phenomena and the second-order

1582 E.I. Rogacheva / Journal of Physics and Chemistry of Solids 64 (2003) 1579–1583

PTs. In both cases, in the vicinity of a transition, the a dynamic scaling hypothesis for a second order PT, is
properties of a system are determined by strongly developed, equal to w ¼ 0:33; which is in perfect agreement with the
interacting fluctuations, peculiarities of thermodynamic experimental data [18]. This value is rather close to the
quantities obey a power law, and their exponents are called critical exponents of lp we obtained for the PbTe– MnTe
critical exponents. Both percolation and second-order PTs and SnTe– InTe systems, which represents another evidence
manifest themselves through critical phenomena, and are for a close analogy between second order PTs and
characterized by the universality of critical exponents and percolation phenomena.
scaling laws.
The heat capacity C is a universal property showing
anomalous behavior ðC , lT 2 TC l2a Þ in the vicinity of any 3. Conclusion
temperature second-order PT [18]. It can be suggested that a
percolation transition in solid solutions will also be New experimental data, which confirm our earlier
accompanied by an anomaly in C ðC , lx 2 xc l2a Þ: In suggestion about the universal character of critical phenom-
Ref. [19], for the first time anomalous growth in C in PbTe– ena accompanying the transition from ‘an impurity vapor’ to
MnTe solid solutions in the range of small impurity ‘an impurity condensate’, were obtained for IV –VI-based
concentration (1 – 1.25 mol% MnTe) was detected. solid solutions. The analysis of these data and the data
From the experimental concentration dependences of C obtained in our earlier works shows that a narrowing of
in the PbTe– MnTe system [19], plotting the Cðlx 2 xc lÞ XRD lines in the critical region occurs in all studied solid
dependences in a double-logarithmic scale, we determined solutions including the case when the crystal is ‘doped’ with
the critical exponent for the specific heat as a ¼ 0:12 ^ vacancies (the SnTe– Te system [12]). The existence of the
0:02: This value is rather close to the value of a known from range of anomalous growth in the lattice thermal conduc-
the theory of the second-order PTs and confirmed tivity and heat capacity in this critical region is confirmed.
experimentally [18]. It is suggested that the formation of percolation channels
In the PbTe– GeTe system, we detected an anomaly in through impurity centers upon reaching the percolation
the specific heat similar to the one observed in the PbTe– threshold stimulates self-organization processes (long- and
MnTe system [19] (Fig. 3(b), curve 2). The pronounced short-range ordering) in the impurity subsystem. The
peak in the isotherms of C proves the existence of critical narrowing of the XRD line width and the increase in
phenomena. As far as the PbTe– GeTe system is concerned, the lattice thermal conductivity up to the values observed in
the accurate determination of the critical exponent is the impurity-free host compound, which were registered in a
complicated, since at least two anomalies are observed in number of systems, represent a convincing evidence for
the concentration dependences of B and lp : ordering.
An estimate of the critical exponent for the lattice On the basis of our experimental data, the estimates of
thermal conductivity using the experimental data for the the specific heat and the lattice thermal conductivity critical
PbTe- MnTe system [6] and for the SnTe– InTe system exponents are made in the approximation of percolation
(present work), which was made assuming lp , lx 2 xc l2w ; theory and fluctuation theory of the second order PTs.
yielded w ¼ 0:25 ^ 0:05: It is known [18] that the
theoretical value of the lp critical exponent w ðl ,
lT 2 TC l2w Þ; calculated for superfluid Helium 4 using
Acknowledgements

The author thanks Pinegin V.I. and Tavrina T.V. for their
assistance in carrying out the X-ray studies.

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