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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 Solid solutions represent a broad class of substances, the most widespread and having a great potential for practical applications. In the framework of generally accepted notions of the physico-chemical analysis, the physical properties in the solid solution region change in a monotonic way, and the appearance of concentration anomalies of properties indicates the intersection of phase region boundaries. However, for a number of semiconductor solid solutions we observed [1 – 8] concentration anomalies of different properties (microhardness H; charge carrier mobility m; lattice thermal conductivity lp ; etc.) in the range of small impurity concentrations (, 1.0 at.%), which indicated the presence of concentration phase transitions (PTs). We suggested [1] that these PTs have the universal character, corresponding to the transition from an impurity discontinuum to an impurity continuum, and take place according to a percolation scenario [9,10]. Assuming that the properties are isotropic and taking into consideration a short-range * Tel.: þ380-572-400-092; fax: þ380-572-400-601. E-mail address: [email protected] (E.I. Rogacheva).
nature of impurity potential, for any type of the interaction between dopants (deformational, electrostatic, dipole – dipole, etc) one can designate the radius of impurity atom ‘action sphere’, within which the crystal properties differ considerably from those of the matrix, as R0 : In accordance with one of the problems of percolation theory, viz. ‘problem of spheres’ [9,10], there is a critical concentration (percolation threshold xc ) at which the channels penetrating the whole system appear and an infinite cluster consisting of overlapping spheres of radius R0 is formed. The effective value of xc depends on the range of interactions in the system, i.e. on R0 : The formation of the infinite cluster is accompanied by critical phenomena, which must manifest themselves in the case of the solid solutions through anomalies in the concentration dependences of different properties. When the percolation threshold is reached, there appear channels, along which internal elastic stresses caused by the impurity atoms are partially compensated due to the overlapping of impurity deformational spheres. As a result, the movement of dislocations and propagation of elementary excitations are facilitated. An increase in the dislocation mobility, a decrease in the effective phonon and electron 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
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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 content within the SnTe homogeneity region (50.15 – 50.8 at.% Te) corresponds to the increase in the concentration of cation vacancies (, 0.5– 3.2%) caused by the deviation from stoichiometry and playing the role similar to 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 decrease in B is observed. It is known that among the main factors that cause a broadening of XRD lines, apart from instrumental factors related to the experimental conditions, are (1) the microstresses in crystal and (2) a small size of coherent scattering regions [14]. In homogeneous disordered solid solutions with a sufficiently large grain size, the main reason of a broadening of XRD lines is microstresses caused by a difference in sizes of impurity and host atoms. Since all studied solid solutions were prepared and investigated under the same conditions, the effect of all variables except microstresses could be ruled out. That is why the broadening of XRD lines we observed after the introduction of the first portions of the impurity (Fig. 1) is easy to explain. In the SnTe– Te system (Fig. 1(d)), we do not observe the initial increase in B because alloys with concentrations of cation vacancies smaller than , 0.5% do not exist in the equilibrium state due to a significant shift of the SnTe
Fig. 1. The dependence of a relative change in the XRD linewidth DB=B on the dopant concentration in solid solutions PbTe–CdTe (a),(b), CuInSe2 –CdS [13] (c), SnTe–Te [12] (d), PbTe–GeTe [2] (e), and PbTe–Bi2Te3 (f). a: (400) diffraction line; b: (800) diffraction line.
homogeneity region relative to the stoichiometric composition [12]. A subsequent sharp decrease in B shows that internal stresses in the crystal decrease. This is in good agreement with a drop in H in the vicinity of the critical composition, which was observed in all studied systems [15] and was attributed to the decrease in internal stresses level with the reaching of the percolation threshold and the formation of percolation channels. Impurity atoms are centers of local distortions in the crystal lattice, sources of internal stresses and strains decreasing in an inverse proportion to the cube of the distance [16]. Since noticeable displacements of atoms are created within one or two interatomic distances from an impurity atom, one can consider elastic fields as short-range. At small impurity concentrations, when distances between impurity atoms are much larger than R0 ; elastic fields created by separate atoms practically do not overlap. As the impurity concentration increases, elastic fields of neighboring atoms
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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 deformational 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. The formation of continuous chains of impurity deformational spheres upon reaching the percolation threshold must stimulate redistribution of impurity atoms in the crystal lattice leading to the realization of their configuration with a minimum thermodynamic potential. Elastic interactions between impurity atoms, similarly to Coulomb interactions, can lead to the formation of configurations of impurity atoms, which correspond to minima of the elastic energy. Possible self-organization processes may include a long-range ordering of impurity atoms (‘crystallization of impurity liquid’), a formation of complexes (short-range ordering), a change in the localization of impurity atoms in the crystal lattice, etc. Under isovalent isomorphic substitution a long-range ordering is more likely. Under heterovalent substitution when the structure of a matrix differs from the structure of a dopant, the probability of a short-range ordering in solid solutions increases. The introduction of a dopant in the form of a stable chemical compound stimulates the formation of neutral chemical complexes corresponding to the composition of this compound. When a certain concentration of chemical complexes is reached, the formation of percolation channels linking complexes and accompanied by a decrease in internal stresses becomes possible. The very convincing argument in favor of selforganization and ordering, which take place upon reaching the critical concentration of an impurity, is a dramatic decrease in B; in some cases down to the value observed in an impurity-free host-material (the PbTe – Bi2Te3 and PbTe– CdTe systems). When the solid solution region is sufficiently wide, different variants of ordering can be realized with increasing impurity concentration. To all appearances, two extrema observed in the concentration dependence of B in the isovalent PbTe– GeTe system (Fig. 1(e)), can be attributed to the realization of different types of ordering [2]. A simple calculation shows that a composition of 1 mol% GeTe is optimal for an ordered distribution of impurity atoms over the sites of a simple cubic lattice with a ¼ 3 a0 ; whereas at , 1.6 mol% GeTe, a superstructure of impurity atoms with a fcc lattice and unit cell parameter of a ¼ 4 a0 (where a0 is the unit cell parameter of the studied alloy) can be formed.
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Fig. 2. The lattice thermal conductivity lp vs. the dopant concentration in the SnTe– InTe (a) and PbTe–GeTe (b) solid solutions. a: T ¼ 300 K; b: T ¼ 170 K:
As is seen, in the PbTe– Bi2Te3 system (Fig. 2(f)), also at least two anomalous regions are observed—in the vicinity of 1 mol% Bi2Te3 and near 3 mol% Bi2Te3. Under the doping of PbTe with Bi2Te3, Bi atoms and vacancies are introduced into the cation sublattice simultaneously. The Coulomb attraction between charged defects of opposite signs (Bi3þ and V22 Pb ) stimulates processes of chemical interaction leading to the formation of neutral molecular complexes such as Bi2Te3. Thus, in addition to separate impurity atoms, new structural elements appear, and the formation of percolation channels through these elements becomes possible. On the basis of the above considerations one can suggest that the first anomaly in the B dependence on the impurity content is connected with the formation of percolation channels linking Bi atoms, while the second anomaly is related to the formation of percolation channels through Bi2Te3 complexes. This suggestion is supported by the fact that it is after 3 mol% Bi2Te3 that the charge carrier concentration in the PbTe– Bi2Te3 system does not change any more [17]. In Fig. 2, the concentration dependences of the lattice thermal conductivity lp in SnTe –InTe and PbTe– GeTe solid solutions are presented. As is seen, in these systems an anomalous increase in lp takes place in the range of small impurity contents. In the PbTe – GeTe system, two anomalous regions, whose locations correspond to those of anomalies in the B dependences on the impurity content, are observed. We have registered an anomalous increase in lp in the range of small impurity concentration earlier in the PbTe– MnTe system [6] and attributed it to a decrease in the effective phonon cross-section as a result of the formation of percolation channels near the percolation threshold and a decrease in an overall level of elastic stresses in the crystal lattice. The increase in lp up to the values of a hostcompound, which was observed in the PbTe – MnTe system [6] is another argument in favor of the suggestion about the self-organization processes in impurity subsystem of crystal. In accordance with the modern views [9,10,18], there is an analogy between percolation phenomena and the second-order
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PTs. In both cases, in the vicinity of a transition, the properties of a system are determined by strongly developed, interacting fluctuations, peculiarities of thermodynamic quantities obey a power law, and their exponents are called critical exponents. Both percolation and second-order PTs manifest themselves through critical phenomena, and are characterized by the universality of critical exponents and scaling laws. The heat capacity C is a universal property showing anomalous behavior ðC , lT 2 TC l2a Þ in the vicinity of any temperature second-order PT [18]. It can be suggested that a percolation transition in solid solutions will also be accompanied by an anomaly in C ðC , lx 2 xc l2a Þ: In Ref. [19], for the first time anomalous growth in C in PbTe– MnTe solid solutions in the range of small impurity concentration (1 – 1.25 mol% MnTe) was detected. From the experimental concentration dependences of C in the PbTe– MnTe system [19], plotting the Cðlx 2 xc lÞ dependences in a double-logarithmic scale, we determined the critical exponent for the specific heat as a ¼ 0:12 ^ 0:02: This value is rather close to the value of a known from the theory of the second-order PTs and confirmed experimentally [18]. In the PbTe– GeTe system, we detected an anomaly in the specific heat similar to the one observed in the PbTe– MnTe system [19] (Fig. 3(b), curve 2). The pronounced peak in the isotherms of C proves the existence of critical phenomena. As far as the PbTe– GeTe system is concerned, the accurate determination of the critical exponent is complicated, since at least two anomalies are observed in the concentration dependences of B and lp : An estimate of the critical exponent for the lattice thermal conductivity using the experimental data for the PbTe- MnTe system [6] and for the SnTe– InTe system (present work), which was made assuming lp , lx 2 xc l2w ; 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
a dynamic scaling hypothesis for a second order PT, is equal to w ¼ 0:33; which is in perfect agreement with the experimental data [18]. This value is rather close to the critical exponents of lp we obtained for the PbTe– MnTe and SnTe– InTe systems, which represents another evidence for a close analogy between second order PTs and percolation phenomena.
3. Conclusion New experimental data, which confirm our earlier suggestion about the universal character of critical phenomena accompanying the transition from ‘an impurity vapor’ to ‘an impurity condensate’, were obtained for IV –VI-based solid solutions. The analysis of these data and the data obtained in our earlier works shows that a narrowing of XRD lines in the critical region occurs in all studied solid solutions including the case when the crystal is ‘doped’ with vacancies (the SnTe– Te system [12]). The existence of the range of anomalous growth in the lattice thermal conductivity and heat capacity in this critical region is confirmed. It is suggested that the formation of percolation channels through impurity centers upon reaching the percolation threshold stimulates self-organization processes (long- and short-range ordering) in the impurity subsystem. The narrowing of the XRD line width and the increase in the lattice thermal conductivity up to the values observed in the impurity-free host compound, which were registered in a number of systems, represent a convincing evidence for ordering. On the basis of our experimental data, the estimates of the specific heat and the lattice thermal conductivity critical exponents are made in the approximation of percolation theory and fluctuation theory of the second order PTs.
Acknowledgements The author thanks Pinegin V.I. and Tavrina T.V. for their assistance in carrying out the X-ray studies.
References
Fig. 3. The dependence of the lattice thermal conductivity lp (a) and specific heat C (b) of PbTe-based solid solutions on MnTe (a),(b) and GeTe (b) concentration. a: PbTe–MnTe (data from [6]); b: 1— PbTe–MnTe (data from Ref. [19]), 2—PbTe–GeTe.
[1] E.I. Rogacheva, Critical phenomena in heavily-doped semiconducting compounds, Jpn. J. Appl. Phys. 32 (Suppl. 32-3) (1993) 775 –777. [2] E.I. Rogacheva, V.I. Pinegin, T.V. Tavrina, Percolation Effects in Pb12xGexTe, Proc. SPIE 3182 (1998) 364–368. [3] E.I. Rogacheva, N. Sinelnik, O.N. Nashchekina, Concentration anomalies of properties in Pb12xGexTe, Acta Phys. Pol. A 84 (1993) 729–732. [4] E.I. Rogacheva, I.M. Krivulkin, V.P. Popov, A. Lobkovskaya, Concentration dependences of properties in Pb12xMnxTe solid solutions, Phys. Stat. Solidi A 148 (1995) K65–K67.
E.I. Rogacheva / Journal of Physics and Chemistry of Solids 64 (2003) 1579–1583 [5] E.I. Rogacheva, I.M. Krivulkin, The temperature and concentration dependences of the charge carrier mobility in PbTe – MnTe solid solutions, Semiconductors 36 (2002) 966–970. [6] E.I. Rogacheva, I.M. Krivulkin, Isotherms of thermal conductivity in PbTe–MnTe solid solutions, Fiz. Tverd. Tela 43 (2001) 1000–1003. [7] E.I. Rogacheva, T.V. Tavrina, I.M. Krivulkin, Anomalous composition dependence of microhardness in Pb12xGexTe semiconductors solid solutions, Inorg. Mater. 35 (1999) 236–239. [8] E.I. Rogacheva, A.S. Sologubenko, I.M. Krivulkin, Microhardness of Pb12xMnxTe semimagnetic solid solutions, Inorg. Mater. 34 (1998) 545– 549. [9] D. Stauffer, A. Aharony, Introduction to Percolation Theory, Taylor & Francis, London, 1992, pp. 15–88. [10] B.I. Shklovskii, A.L. Efros, Electronic Properties of Doped Semiconductors, Nauka, Moscow, 1979, Springer-Verlag, New York, 1984. [11] A.V. Lyubchenko, E.A. Sal’kov, F.F. Sizov, Physical Foundations of Semiconductor Quantum Photoelectronics, Naukova Dumka, Kiev, 1984.
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[12] E.I. Rogacheva, G.V. Gorne, S.A. Laptev, A.V. Arinkin, T.B. Vesene, Concentration dependences of properties in SnTe homogeneity region, Izv. AN SSSR, Neorgan. Mater. 22 (1986) 41– 44. [13] E.I. Rogacheva, T.V. Tavrina, Effect of CdS doping on structure and properties of CuInSe2, Funct. Mater. 8 (2001) 635 –641. [14] Ya.S. Umanskii, X-ray Study of Metals and Semiconductors, Metalurgiya, Moscow, 1969, p. 38. [15] E. Rogacheva, Concentration-dependent microhardness in semiconductor solid solutions, Izv. AN SSSR, Neorgan. Mater. 25 (1989) 754–757. [16] T. Suzuki, H. Yoshinaga, S. Takeuchi, Dislocation Dynamics and Plasticity, Syokabo, Tokyo, 1986, Mir, Moscow. [17] E.I. Rogacheva, S.A. Laptev, V.S. Ploskaya, B.A. Efimova, Solid solutions based on PbTe in the Pb –Bi–Te system, Izv. AN SSSR, Neorgan. Mater. 20 (1984) 1350–1353. [18] A.Z. Patashinskiy, V.L. Pokrovskiy, The Fluctuation Theory of Phase Transitions, Nauka, Moscow, 1982. [19] E.I. Rogacheva, I.M. Krivulkin, Concentration anomaly of heat capacity in the Pb12xMnxTe semimagnetic semiconductors, Inst. Phys. Conf. Ser. 152 (1998) 831– 834.
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 Solid solutions represent a broad class of substances, the most widespread and having a great potential for practical applications. In the framework of generally accepted notions of the physico-chemical analysis, the physical properties in the solid solution region change in a monotonic way, and the appearance of concentration anomalies of properties indicates the intersection of phase region boundaries. However, for a number of semiconductor solid solutions we observed [1 – 8] concentration anomalies of different properties (microhardness H; charge carrier mobility m; lattice thermal conductivity lp ; etc.) in the range of small impurity concentrations (, 1.0 at.%), which indicated the presence of concentration phase transitions (PTs). We suggested [1] that these PTs have the universal character, corresponding to the transition from an impurity discontinuum to an impurity continuum, and take place according to a percolation scenario [9,10]. Assuming that the properties are isotropic and taking into consideration a short-range * Tel.: þ380-572-400-092; fax: þ380-572-400-601. E-mail address: [email protected] (E.I. Rogacheva).
nature of impurity potential, for any type of the interaction between dopants (deformational, electrostatic, dipole – dipole, etc) one can designate the radius of impurity atom ‘action sphere’, within which the crystal properties differ considerably from those of the matrix, as R0 : In accordance with one of the problems of percolation theory, viz. ‘problem of spheres’ [9,10], there is a critical concentration (percolation threshold xc ) at which the channels penetrating the whole system appear and an infinite cluster consisting of overlapping spheres of radius R0 is formed. The effective value of xc depends on the range of interactions in the system, i.e. on R0 : The formation of the infinite cluster is accompanied by critical phenomena, which must manifest themselves in the case of the solid solutions through anomalies in the concentration dependences of different properties. When the percolation threshold is reached, there appear channels, along which internal elastic stresses caused by the impurity atoms are partially compensated due to the overlapping of impurity deformational spheres. As a result, the movement of dislocations and propagation of elementary excitations are facilitated. An increase in the dislocation mobility, a decrease in the effective phonon and electron 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
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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 content within the SnTe homogeneity region (50.15 – 50.8 at.% Te) corresponds to the increase in the concentration of cation vacancies (, 0.5– 3.2%) caused by the deviation from stoichiometry and playing the role similar to 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 decrease in B is observed. It is known that among the main factors that cause a broadening of XRD lines, apart from instrumental factors related to the experimental conditions, are (1) the microstresses in crystal and (2) a small size of coherent scattering regions [14]. In homogeneous disordered solid solutions with a sufficiently large grain size, the main reason of a broadening of XRD lines is microstresses caused by a difference in sizes of impurity and host atoms. Since all studied solid solutions were prepared and investigated under the same conditions, the effect of all variables except microstresses could be ruled out. That is why the broadening of XRD lines we observed after the introduction of the first portions of the impurity (Fig. 1) is easy to explain. In the SnTe– Te system (Fig. 1(d)), we do not observe the initial increase in B because alloys with concentrations of cation vacancies smaller than , 0.5% do not exist in the equilibrium state due to a significant shift of the SnTe
Fig. 1. The dependence of a relative change in the XRD linewidth DB=B on the dopant concentration in solid solutions PbTe–CdTe (a),(b), CuInSe2 –CdS [13] (c), SnTe–Te [12] (d), PbTe–GeTe [2] (e), and PbTe–Bi2Te3 (f). a: (400) diffraction line; b: (800) diffraction line.
homogeneity region relative to the stoichiometric composition [12]. A subsequent sharp decrease in B shows that internal stresses in the crystal decrease. This is in good agreement with a drop in H in the vicinity of the critical composition, which was observed in all studied systems [15] and was attributed to the decrease in internal stresses level with the reaching of the percolation threshold and the formation of percolation channels. Impurity atoms are centers of local distortions in the crystal lattice, sources of internal stresses and strains decreasing in an inverse proportion to the cube of the distance [16]. Since noticeable displacements of atoms are created within one or two interatomic distances from an impurity atom, one can consider elastic fields as short-range. At small impurity concentrations, when distances between impurity atoms are much larger than R0 ; elastic fields created by separate atoms practically do not overlap. As the impurity concentration increases, elastic fields of neighboring atoms
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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 deformational 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. The formation of continuous chains of impurity deformational spheres upon reaching the percolation threshold must stimulate redistribution of impurity atoms in the crystal lattice leading to the realization of their configuration with a minimum thermodynamic potential. Elastic interactions between impurity atoms, similarly to Coulomb interactions, can lead to the formation of configurations of impurity atoms, which correspond to minima of the elastic energy. Possible self-organization processes may include a long-range ordering of impurity atoms (‘crystallization of impurity liquid’), a formation of complexes (short-range ordering), a change in the localization of impurity atoms in the crystal lattice, etc. Under isovalent isomorphic substitution a long-range ordering is more likely. Under heterovalent substitution when the structure of a matrix differs from the structure of a dopant, the probability of a short-range ordering in solid solutions increases. The introduction of a dopant in the form of a stable chemical compound stimulates the formation of neutral chemical complexes corresponding to the composition of this compound. When a certain concentration of chemical complexes is reached, the formation of percolation channels linking complexes and accompanied by a decrease in internal stresses becomes possible. The very convincing argument in favor of selforganization and ordering, which take place upon reaching the critical concentration of an impurity, is a dramatic decrease in B; in some cases down to the value observed in an impurity-free host-material (the PbTe – Bi2Te3 and PbTe– CdTe systems). When the solid solution region is sufficiently wide, different variants of ordering can be realized with increasing impurity concentration. To all appearances, two extrema observed in the concentration dependence of B in the isovalent PbTe– GeTe system (Fig. 1(e)), can be attributed to the realization of different types of ordering [2]. A simple calculation shows that a composition of 1 mol% GeTe is optimal for an ordered distribution of impurity atoms over the sites of a simple cubic lattice with a ¼ 3 a0 ; whereas at , 1.6 mol% GeTe, a superstructure of impurity atoms with a fcc lattice and unit cell parameter of a ¼ 4 a0 (where a0 is the unit cell parameter of the studied alloy) can be formed.
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Fig. 2. The lattice thermal conductivity lp vs. the dopant concentration in the SnTe– InTe (a) and PbTe–GeTe (b) solid solutions. a: T ¼ 300 K; b: T ¼ 170 K:
As is seen, in the PbTe– Bi2Te3 system (Fig. 2(f)), also at least two anomalous regions are observed—in the vicinity of 1 mol% Bi2Te3 and near 3 mol% Bi2Te3. Under the doping of PbTe with Bi2Te3, Bi atoms and vacancies are introduced into the cation sublattice simultaneously. The Coulomb attraction between charged defects of opposite signs (Bi3þ and V22 Pb ) stimulates processes of chemical interaction leading to the formation of neutral molecular complexes such as Bi2Te3. Thus, in addition to separate impurity atoms, new structural elements appear, and the formation of percolation channels through these elements becomes possible. On the basis of the above considerations one can suggest that the first anomaly in the B dependence on the impurity content is connected with the formation of percolation channels linking Bi atoms, while the second anomaly is related to the formation of percolation channels through Bi2Te3 complexes. This suggestion is supported by the fact that it is after 3 mol% Bi2Te3 that the charge carrier concentration in the PbTe– Bi2Te3 system does not change any more [17]. In Fig. 2, the concentration dependences of the lattice thermal conductivity lp in SnTe –InTe and PbTe– GeTe solid solutions are presented. As is seen, in these systems an anomalous increase in lp takes place in the range of small impurity contents. In the PbTe – GeTe system, two anomalous regions, whose locations correspond to those of anomalies in the B dependences on the impurity content, are observed. We have registered an anomalous increase in lp in the range of small impurity concentration earlier in the PbTe– MnTe system [6] and attributed it to a decrease in the effective phonon cross-section as a result of the formation of percolation channels near the percolation threshold and a decrease in an overall level of elastic stresses in the crystal lattice. The increase in lp up to the values of a hostcompound, which was observed in the PbTe – MnTe system [6] is another argument in favor of the suggestion about the self-organization processes in impurity subsystem of crystal. In accordance with the modern views [9,10,18], there is an analogy between percolation phenomena and the second-order
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PTs. In both cases, in the vicinity of a transition, the properties of a system are determined by strongly developed, interacting fluctuations, peculiarities of thermodynamic quantities obey a power law, and their exponents are called critical exponents. Both percolation and second-order PTs manifest themselves through critical phenomena, and are characterized by the universality of critical exponents and scaling laws. The heat capacity C is a universal property showing anomalous behavior ðC , lT 2 TC l2a Þ in the vicinity of any temperature second-order PT [18]. It can be suggested that a percolation transition in solid solutions will also be accompanied by an anomaly in C ðC , lx 2 xc l2a Þ: In Ref. [19], for the first time anomalous growth in C in PbTe– MnTe solid solutions in the range of small impurity concentration (1 – 1.25 mol% MnTe) was detected. From the experimental concentration dependences of C in the PbTe– MnTe system [19], plotting the Cðlx 2 xc lÞ dependences in a double-logarithmic scale, we determined the critical exponent for the specific heat as a ¼ 0:12 ^ 0:02: This value is rather close to the value of a known from the theory of the second-order PTs and confirmed experimentally [18]. In the PbTe– GeTe system, we detected an anomaly in the specific heat similar to the one observed in the PbTe– MnTe system [19] (Fig. 3(b), curve 2). The pronounced peak in the isotherms of C proves the existence of critical phenomena. As far as the PbTe– GeTe system is concerned, the accurate determination of the critical exponent is complicated, since at least two anomalies are observed in the concentration dependences of B and lp : An estimate of the critical exponent for the lattice thermal conductivity using the experimental data for the PbTe- MnTe system [6] and for the SnTe– InTe system (present work), which was made assuming lp , lx 2 xc l2w ; 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
a dynamic scaling hypothesis for a second order PT, is equal to w ¼ 0:33; which is in perfect agreement with the experimental data [18]. This value is rather close to the critical exponents of lp we obtained for the PbTe– MnTe and SnTe– InTe systems, which represents another evidence for a close analogy between second order PTs and percolation phenomena.
3. Conclusion New experimental data, which confirm our earlier suggestion about the universal character of critical phenomena accompanying the transition from ‘an impurity vapor’ to ‘an impurity condensate’, were obtained for IV –VI-based solid solutions. The analysis of these data and the data obtained in our earlier works shows that a narrowing of XRD lines in the critical region occurs in all studied solid solutions including the case when the crystal is ‘doped’ with vacancies (the SnTe– Te system [12]). The existence of the range of anomalous growth in the lattice thermal conductivity and heat capacity in this critical region is confirmed. It is suggested that the formation of percolation channels through impurity centers upon reaching the percolation threshold stimulates self-organization processes (long- and short-range ordering) in the impurity subsystem. The narrowing of the XRD line width and the increase in the lattice thermal conductivity up to the values observed in the impurity-free host compound, which were registered in a number of systems, represent a convincing evidence for ordering. On the basis of our experimental data, the estimates of the specific heat and the lattice thermal conductivity critical exponents are made in the approximation of percolation theory and fluctuation theory of the second order PTs.
Acknowledgements The author thanks Pinegin V.I. and Tavrina T.V. for their assistance in carrying out the X-ray studies.
References
Fig. 3. The dependence of the lattice thermal conductivity lp (a) and specific heat C (b) of PbTe-based solid solutions on MnTe (a),(b) and GeTe (b) concentration. a: PbTe–MnTe (data from [6]); b: 1— PbTe–MnTe (data from Ref. [19]), 2—PbTe–GeTe.
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