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, UkraineAbstract 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 anomalousdecrease in X-ray diffraction linewidth and an increase in the lattice thermal conductivity and heat capacity in this concentrationrange were observed. The experimental results are analyzed on the basis of percolation theory and ﬂuctuation theory of thesecond order phase transitions. From the experimental data, the critical exponents for the lattice thermal conductivity and latticeheat capacity are determined. It is suggested that self-organization processes (a short-range or long-range ordering of impurityatoms) accompany the percolation phenomena. The results obtained in this work represent another evidence to the propositionabout the universal character of critical phenomena accompanying the transition from an impurity discontinuum to an impuritycontinuum.q 2003 Elsevier Ltd. All rights reserved.Keywords: A. Semiconductors; A. Alloys; C. X-ray diffraction; D. Critical phenomena; D. Thermal conductivity1. 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 atommost widespread and having a great potential for practical ‘action sphere’, within which the crystal properties differapplications. In the framework of generally accepted notions considerably from those of the matrix, as R0 : In accordanceof 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 concentrationappearance of concentration anomalies of properties (percolation threshold xc ) at which the channels penetratingindicates the intersection of phase region boundaries. the whole system appear and an inﬁnite cluster consisting ofHowever, for a number of semiconductor solid solutions overlapping spheres of radius R0 is formed. The effectivewe observed [1 – 8] concentration anomalies of different value of xc depends on the range of interactions in theproperties (microhardness H; charge carrier mobility m; system, i.e. on R0 : The formation of the inﬁnite cluster islattice thermal conductivity lp ; etc.) in the range of small accompanied by critical phenomena, which must manifestimpurity concentrations (, 1.0 at.%), which indicated the themselves in the case of the solid solutions throughpresence of concentration phase transitions (PTs). We anomalies in the concentration dependences of differentsuggested  that these PTs have the universal character, properties. When the percolation threshold is reached, therecorresponding to the transition from an impurity disconti- appear channels, along which internal elastic stresses causednuum to an impurity continuum, and take place according to by the impurity atoms are partially compensated due to thea 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: firstname.lastname@example.org (E.I. Rogacheva). cross-section under the formation of percolation channels0022-3697/03/$ - 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) 1579–1583lead to a drop in H [1,7,8], a growth in lp  and m [1,3– 5]which we observed in the critical region. To prove the suggestion about the universal character ofthe concentration anomalies of physical properties in solidsolutions, it is necessary to expand the scope of objects andproperties to be studied, to perform a more detailed analysisof experimental data, and to further develop theoreticalgrounds. In this work, the new experimental results on theconcentration dependences of the X-ray diffraction (XRD)linewidth B; lattice thermal conductivity lp ; and speciﬁcheat C in the range of small impurity concentrations internary solid solutions based on IV– VI (SnTe, PbTe, andGeTe) semiconductor compounds , are presented. Thenew results are considered jointly with the previouslyobtained data and discussed within the framework of theabove mentioned ideas we have been developing in ourstudies.2. Results and discussion The experimental details of the sample preparation,XRD study, measurements of the thermal conductivity andthe heat capacity were described earlier in Refs. [2 – 8]. In Fig. 1, the room-temperature dependences of B on theimpurity content in PbTe– CdTe and PbTe –Bi2Te3 solidsolutions based on PbTe are presented. For comparison, wealso show previously obtained data for the PbTe –GeTe ,SnTe– Te  and CuInSe2 – CdS  systems. In deﬁcitsolid solutions (SnTe – Te ), the increase in the Te Fig. 1. The dependence of a relative change in the XRD linewidthcontent within the SnTe homogeneity region (50.15 – DB=B on the dopant concentration in solid solutions PbTe–CdTe50.8 at.% Te) corresponds to the increase in the concen- (a),(b), CuInSe2 –CdS  (c), SnTe–Te  (d), PbTe–GeTe 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 solidsolutions in a relatively narrow range of concentrations of asecond component (, 0.5 – 2.0 mol%), an anomalous homogeneity region relative to the stoichiometric compo-decrease in B is observed. sition . It is known that among the main factors that cause a A subsequent sharp decrease in B shows that internalbroadening of XRD lines, apart from instrumental factors stresses in the crystal decrease. This is in good agreementrelated 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  and wasregions . In homogeneous disordered solid solutions attributed to the decrease in internal stresses level with thewith a sufﬁciently large grain size, the main reason of a reaching of the percolation threshold and the formation ofbroadening 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 thestudied solid solutions were prepared and investigated under crystal lattice, sources of internal stresses and strainsthe same conditions, the effect of all variables except decreasing in an inverse proportion to the cube of themicrostresses could be ruled out. That is why the broadening distance . Since noticeable displacements of atoms areof XRD lines we observed after the introduction of the ﬁrst created within one or two interatomic distances from anportions 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 betweenincrease in B because alloys with concentrations of cation impurity atoms are much larger than R0 ; elastic ﬁelds createdvacancies smaller than , 0.5% do not exist in the by separate atoms practically do not overlap. As the impurityequilibrium 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 1581begin to overlap, which leads to a partial compensation ofelastic stresses of opposite signs. After percolation channelsvia deformational ﬁelds of separate atoms are formed, theinteraction of impurities becomes cooperative. As theimpurity concentration increases, the overlapping of defor-mational spheres and the compensation of microstressesgradually spread over the entire crystal (the density of‘inﬁnite cluster’ grows [9,10]), leading to a decrease in theoverall level of elastic strains in the crystal lattice, which, inturn, results in a decrease in B: Further introduction ofimpurity atoms into this new medium (‘impurity liquid’)causes new distortions of the crystal lattice and, consequentlya 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) soliddeformational spheres upon reaching the percolation solutions. a: T ¼ 300 K; b: T ¼ 170 K:threshold must stimulate redistribution of impurity atomsin the crystal lattice leading to the realization of theirconﬁguration with a minimum thermodynamic potential. As is seen, in the PbTe– Bi2Te3 system (Fig. 2(f)), also atElastic interactions between impurity atoms, similarly to least two anomalous regions are observed—in the vicinity ofCoulomb interactions, can lead to the formation of 1 mol% Bi2Te3 and near 3 mol% Bi2Te3. Under the dopingconﬁgurations of impurity atoms, which correspond to of PbTe with Bi2Te3, Bi atoms and vacancies are introducedminima of the elastic energy. Possible self-organization into the cation sublattice simultaneously. The Coulombprocesses may include a long-range ordering of impurity attraction between charged defects of opposite signs (Bi3þatoms (‘crystallization of impurity liquid’), a formation of and V22) stimulates processes of chemical interaction Pbcomplexes (short-range ordering), a change in the localiz- leading to the formation of neutral molecular complexesation 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 ofmore likely. Under heterovalent substitution when the percolation channels through these elements becomesstructure of a matrix differs from the structure of a dopant, possible. On the basis of the above considerations one canthe probability of a short-range ordering in solid solutions suggest that the ﬁrst anomaly in the B dependence on theincreases. The introduction of a dopant in the form of a impurity content is connected with the formation ofstable chemical compound stimulates the formation of percolation channels linking Bi atoms, while the secondneutral chemical complexes corresponding to the compo- anomaly is related to the formation of percolation channelssition of this compound. When a certain concentration of through Bi2Te3 complexes. This suggestion is supported bychemical complexes is reached, the formation of percolation the fact that it is after 3 mol% Bi2Te3 that the charge carrierchannels linking complexes and accompanied by a decrease concentration in the PbTe– Bi2Te3 system does not changein internal stresses becomes possible. any more . The very convincing argument in favor of self- In Fig. 2, the concentration dependences of the latticeorganization and ordering, which take place upon reaching thermal conductivity lp in SnTe –InTe and PbTe– GeTethe critical concentration of an impurity, is a dramatic solid solutions are presented. As is seen, in these systems andecrease in B; in some cases down to the value observed in anomalous increase in lp takes place in the range of smallan 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, aredifferent variants of ordering can be realized with observed. We have registered an anomalous increase in lpincreasing impurity concentration. To all appearances, in the range of small impurity concentration earlier in thetwo extrema observed in the concentration dependence of PbTe– MnTe system  and attributed it to a decrease in theB in the isovalent PbTe– GeTe system (Fig. 1(e)), can be effective phonon cross-section as a result of the formation ofattributed to the realization of different types of ordering percolation channels near the percolation threshold and a. A simple calculation shows that a composition of decrease in an overall level of elastic stresses in the crystal1 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 systemwith a ¼ 3 a0 ; whereas at , 1.6 mol% GeTe, a super-  is another argument in favor of the suggestion about thestructure 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 anof 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) 1579–1583PTs. In both cases, in the vicinity of a transition, the a dynamic scaling hypothesis for a second order PT, isproperties of a system are determined by strongly developed, equal to w ¼ 0:33; which is in perfect agreement with theinteracting ﬂuctuations, peculiarities of thermodynamic experimental data . This value is rather close to thequantities obey a power law, and their exponents are called critical exponents of lp we obtained for the PbTe– MnTecritical exponents. Both percolation and second-order PTs and SnTe– InTe systems, which represents another evidencemanifest themselves through critical phenomena, and are for a close analogy between second order PTs andcharacterized by the universality of critical exponents and percolation phenomena.scaling laws. The heat capacity C is a universal property showinganomalous behavior ðC , lT 2 TC l2a Þ in the vicinity of any 3. Conclusiontemperature second-order PT . It can be suggested that apercolation transition in solid solutions will also be New experimental data, which conﬁrm our earlieraccompanied by an anomaly in C ðC , lx 2 xc l2a Þ: In suggestion about the universal character of critical phenom-Ref. , for the ﬁrst time anomalous growth in C in PbTe– ena accompanying the transition from ‘an impurity vapor’ toMnTe solid solutions in the range of small impurity ‘an impurity condensate’, were obtained for IV –VI-basedconcentration (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 ofin the PbTe– MnTe system , plotting the Cðlx 2 xc lÞ XRD lines in the critical region occurs in all studied soliddependences in a double-logarithmic scale, we determined solutions including the case when the crystal is ‘doped’ withthe critical exponent for the speciﬁc heat as a ¼ 0:12 ^ vacancies (the SnTe– Te system ). The existence of the0:02: 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 . 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 percolationthe speciﬁc heat similar to the one observed in the PbTe– threshold stimulates self-organization processes (long- andMnTe system  (Fig. 3(b), curve 2). The pronounced short-range ordering) in the impurity subsystem. Thepeak in the isotherms of C proves the existence of critical narrowing of the XRD line width and the increase inphenomena. As far as the PbTe– GeTe system is concerned, the lattice thermal conductivity up to the values observed inthe accurate determination of the critical exponent is the impurity-free host compound, which were registered in acomplicated, since at least two anomalies are observed in number of systems, represent a convincing evidence forthe 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 ofthermal conductivity using the experimental data for the the speciﬁc heat and the lattice thermal conductivity criticalPbTe- MnTe system  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  that thetheoretical 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. References  E.I. Rogacheva, Critical phenomena in heavily-doped semi- conducting compounds, Jpn. J. Appl. Phys. 32 (Suppl. 32-3) (1993) 775 –777.  E.I. Rogacheva, V.I. Pinegin, T.V. Tavrina, Percolation Effects in Pb12xGexTe, Proc. SPIE 3182 (1998) 364–368.  E.I. Rogacheva, N. Sinelnik, O.N. Nashchekina, Concen- tration anomalies of properties in Pb12xGexTe, Acta Phys. Pol.Fig. 3. The dependence of the lattice thermal conductivity lp (a) and A 84 (1993) 729–732.speciﬁc heat C (b) of PbTe-based solid solutions on MnTe (a),(b)  E.I. Rogacheva, I.M. Krivulkin, V.P. Popov, A. Lobkovskaya,and GeTe (b) concentration. a: PbTe–MnTe (data from ); b: 1— Concentration dependences of properties in Pb12xMnxTe solidPbTe–MnTe (data from Ref. ), 2—PbTe–GeTe. solutions, Phys. Stat. Solidi A 148 (1995) K65–K67.
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