Percolation effects in semimetallic Bi-Sb solid solutions

Phys. Status Solidi A 207, No. 2, 344–347 (2010) / DOI 10.1002/pssa.200925144 p s sa

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Percolation effects in semimetallicBi-Sb solid solutions

E. I. Rogacheva*, A. A. Drozdova, and O. N. Nashchekina

National Technical University ‘‘Kharkov Polytechnic Institute’’, Frunze St. 21, 61002 Kharkov, Ukraine

Received 30 March 2009, revised 28 September 2009, accepted 30 September 2009

Published online 17 November 2009

PACS 61.72.-y, 62.20.-x, 64.70.-p, 72.15.-v

* Corresponding author: e-mail [email protected], Phone: þ38 057 7076092, Fax: þ38 057 7076601

The dependences of the microhardness, electrical conductivity,

the Hall coefficient, charge carrier mobility, magnetoresistance,

and the Seebeck coefficient on Sb concentration (0–2 at.%) for

Bi-Sb solid solutions were obtained in the temperature range

77–300 K. It was established that the concentration depend-

ences of the properties exhibit a non-monotonic behavior. The

existence of the peculiarities in the concentration curves is

attributed to critical phenomena accompanying the transition of

percolation type from an impurity discontinuum to an impurity

continuum in Bi–Sb solid solutions. It was shown that

the behavior of the curves is similar for cast and pressed

samples and is independent of the annealing time. The results

obtained in this work represent another evidence for our

proposition about the universal character of critical phenomena

that occur in solid solutions under the transition to an impurity

continuum.

� 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Solid solutions are commonly dividedinto diluted and concentrated ones; however, there are noclear criteria for distinguishing between these two types ofsolid solutions. It is usually considered that properties withinthe solid solution region change continuously unless somephase transitions take place, for example the ordering ofatoms or precipitation of a second phase. At the same time,one can expect a qualitative change in properties of solidsolution under increasing concentration of a dopant, whenthe interaction between impurity atoms becomes coopera-tive, which can be described as the formation of ‘‘theimpurity condensate’’.

It is known, that in the semiconductor crystals, theoverlap of the electronic wave functions of impurity atoms attheir concentration Nc leads to the dielectric-metal transition(the Mott transition) [1]. The percolation theory substanti-ates the existence of such transition and gives a simplifieddescription of its basic regularities assuming that impurityatoms are distributed randomly [2]. According to percolationtheory, for the infinite crystal there exists a definiteconcentration Nc of the impurity atoms (percolationthreshold), at which the electronic wave functions ofimpurity atoms form a linked system, passing through thewhole crystal (so-called ‘‘infinite cluster’’), and the disturb-ances of the crystal lattice becomes delocalized. In the finitecrystal, instead of a precisely defined impurity concentration,

there is a certain concentration range, usually sufficientlynarrow.

The Mott transition is just one manifestation of the inter-particle interaction between impurity atoms. The nature ofinteraction between impurities and defects can be different:Coulomb, deformational, dipole-dipole, etc. The complexityof the processes occurring under the interaction betweenimpurity atoms determines the variety of manifestations ofthis interaction, and consequently a wide range of properties,in whose concentration dependences these manifestationscan be observed.

Earlier in a number of solid solutions based on IV–VIsemiconductor compounds, we observed anomalies in theconcentration dependences of properties at � 0.5–1 mol.%of the impurity, which indicated the presence of concen-tration phase transitions (see, for example, Refs. [3–9]). Wesuggested [10] that these phase transitions correspond to thetransition from an impurity discontinuum to an impuritycontinuum (‘‘impurity liquid’’), and can be described usingpercolation theory [2]. There is a threshold impurityconcentration xc, at which the qualitative change of solidsolution state associated with the formation of percolationchannels occurs. Such transition is expected to be accom-panied by critical phenomena, which, in turn, manifestthemselves through anomalies in the concentration depen-dences of different properties.


Phys. Status Solidi A 207, No. 2 (2010) 345

Original

Paper

In accordance with the modern views [11], there is ananalogy between percolation phenomena and second-orderphase transitions. In both cases, in the vicinity of thetransition, properties of the system are determined bystrongly developed interacting fluctuations, peculiarities ofthermodynamic quantities obey a power law, and theirexponents are called critical exponents.

For further confirmation of the suggestion regarding theuniversal character of phase transitions connected with theformation of the impurity continuum, it is desirable toexpand the range of objects to be studied as well as spectrumof properties to be measured. For example, it would be ofinterest to choose metals or semimetals, as objects of study,instead of semiconductors.

The semimetals Bi and Sb form a continuous series ofsolid solutions, which are known as promising low-temperature thermoelectric and magnetothermoelectricmaterials for refrigeration devices at temperatures below �200 K [12]. In Ref. [13] we reported our observing anomaliesin the isotherms of properties in the concentration range 0.5–2.0 at. % Sb for Bi-Sb solid solutions annealed at 520 K for100 h. We attributed the existence of these anomalies to themanifestation of critical phenomena accompanying thetransition from diluted to concentrated solid solutions.

The goal of the present work is a detailed study of thedependences of the mechanical, galvanomagnetic, andthermoelectric properties of Bi-Sb solid solutions on Sbconcentration in the range of 0–2 at.% Sb with a view towardrevealing the concentration-dependent anomalies in theproperties connected with a percolation transition. It was alsoof interest to find out whether the sample preparationtechnique and heat treatment determining the microstructureand degree of impurity distribution homogeneity influencethe manifestation of critical phenomena of percolation type.For this purpose, properties of samples of Bi-Sb solidsolutions prepared by different methods were studied.

It was shown that in the range of small Sb concentrations(0.5–2.0 at.%), there are anomalies in the property-composition dependences, whose presence does not dependon the annealing time nor on the technique of samplepreparation (cast or pressed samples).

2 Experimental Polycrystalline samples of Bi-Sbsolid solutions were prepared by melting high purityelements (99.999%) in quartz ampoules evacuated down to10�3 Pa at � 1020 K for �5 h with application of vibrationalstirring. After the synthesis the samples for measurementswere fabricated using four different techniques: (i) the meltwas cooled down in air and then annealed at 520� 5 K for200 h (T1); (ii) the melt was cooled down in air and thenannealed at 520� 5 K for 1200 h (T2); (iii) after T2 thesamples were ground up and pressed under a pressure ofP¼ 400 MPa at room temperature (T3); (iv) after TO3 cold-pressed samples were annealed at (520� 5) K for 250 h (T4).The chemical composition and homogeneity of the sampleswere controlled using X-ray fluorescent method, electronicmicroprobe analysis (JSM - 6390 LV, Jeol Ltd., Japan) and

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X-Ray Photoelectronic Spectroscopy (XPS-800 Kratos). Itwas shown that the chemical compositions of the obtainedsamples corresponded to what was intended with anaccuracy not less than 5% (relative), and the degree of thesample homogeneity was satisfactory. Microhardness H wasmeasured using a PMT-3 apparatus with a pyramidaldiamond indentor at a load of 0.49 N. The H values wereobtained as the mean value of the measurement of 30indentations, and the relative standard error of the meanvalue was � 0.5 – 1%. The measurements of the Seebeckcoefficient S, electrical conductivity s, Hall coefficient RH,and magnetoresistance Dr/r were carried out in thetemperature range 77–300 K with an accuracy not less than5%. RH and Dr/r were measured by the method of constantmagnetic field and direct current through the sample. TheHall mobility mH was calculated as mH¼RH s. The Seebeckcoefficient S was measured by a compensation methodrelative to copper. All samples exhibited n-typeconductivity.

3 Results In Fig. 1, the isotherms of the electricalconductivity, Hall charge carrier mobility, and magnetore-sistance (at 77 and 300 K) measured on cast samples after T2are presented.

It is seen that the introduction of the first portions of Sb(up to � 0.5 at.%) into Bi results in a decrease in s, mH, andDr/r. Under further increase in Sb concentration up to� 1.5at.% these parameters increase and then decrease again.

As far as microhardness and the Seebeck coefficient areconcerned, under the introduction of the first portions ofantimony impurity atoms (up to � 0.5 at.%), both theseparameters increase (Figs. 2,b,c, curves 2). However underincreasing Sb concentration to � 1.0 –1.5 at.%, the Seebeckcoefficient decreases, and the microhardness does notpractically change. At higher Sb concentrations, S and Hincrease again.

Thus, in the range of 0.5–1.5 at.% Sb, the isotherms of allproperties for samples subjected to T2 exhibit anomalousbehavior.

In Fig. 2, the room-temperature dependences ofelectrical conductivity, the Seebeck coefficient, and micro-hardness on Sb concentration for samples prepared bydifferent techniques are presented. One can see that thebehavior of the dependences does not practically depend onthe sample preparation method, and the presence ofanomalous sections is observed for all types of heat treatmentused in this work.

It is not difficult to explain the drop in s and mH at thesimultaneous increase in H occurring under increasingimpurity concentration up to � 0.5 at.% Sb on the basis ofgeneral properties of solid solutions. In disordered solidsolutions, impurity atoms are centers of local distortions ofthe crystal lattice, sources of internal stresses and strains.Therefore, on the one hand, impurity atoms block thedislocation movement, which leads to crystal hardening andincrease in microhardness, and on the other hand, serve asadditional centers for scattering electrons, which leads to a


346 E. I. Rogacheva et al.: Percolation effects in semimetallic Bi-Sb solid solutionsp

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Figure 1 The isotherms of charge carrier mobility mH, magneto-resistanceDr/r, electrical conductivity s for the cast Bi-Sb samplesannealed for 1200 h for T¼ 77 K (curve 1) and T¼ 300 K (curve 2).

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Figure 2 The dependences of electrical conductivity s (a), theSeebeck coefficient S (b) and microhardness H (c) on Sb concen-tration at 300 K for Bi-Sb polycrystals prepared using differentpreparation techniques and heat treatment: 1 – cast samplesannealedfor 200 h; 2 – cast samples annealed for 1200 h and 3 – pressedsamples annealed for 250 h.

drop in s and mH. A decrease in charge carrier mobility withincreasing impurity concentration results naturally in a dropin magnetoresistance, as Dr/r � mH [14]. The decrease incharge carrier concentration, taking place in Bi-Sb solidsolutions [12], leads to the increase in the Seebeck coefficient.

In the second concentration range �0.5–1.5 at.% Sb, thebehavior of the dependence reverses. The non-monotoniccharacter of the isotherms of the mechanical, galvanomag-netic, and thermoelectric properties indicates qualitativechanges in the electron and lattice subsystems of the crystalunder changing composition, supporting the idea that in thestudied concentration range of Bi-Sb solid solutions aconcentration phase transition takes place.

It is natural to suggest that the observed transition is ofpercolation type and is connected with the transformationfrom diluted to concentrated solid solutions. On the basis ofthis suggestion, we attribute the anomalous growth in s, RH,mH, and Dr/r, decrease in S and constancy in H in theconcentration range of � 0.5–1.5 at.% Sb to the manifes-tation of the above mentioned critical phenomena.


Since noticeable displacements of atoms are createdwithin one or two interatomic distances from an impurityatom, one can consider elastic fields as short-range ones andintroduce a radius of deformational interactions R0.

At small impurity concentrations, the elastic fieldscreated by separate atoms practically do not overlap. Asthe impurity concentration increases, the elastic fields ofneighboring atoms begin to overlap, which leads to a partialcompensation of the elastic stresses with opposite signs.After percolation channels via deformational fields ofseparate atoms are formed, the process of the microstresscompensation intensifies, spreading over the entire crystaland leading to a sharp decrease in the overall level of elasticstrains in the crystal lattice, which, in turn, results in aslowing down of the increase in microhardness and ofthe decrease in electrical conductivity, growth in themagnetoresistance and drop in the Seebeck coefficient.

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Phys. Status Solidi A 207, No. 2 (2010) 347

Original

Paper

Further introduction of impurity atoms into this new medium(‘‘impurity liquid’’) causes new distortions of the crystallattice and, consequently, a growth in microhardness and theSeebeck coefficient, and a simultaneous decrease inelectrical conductivity and magnetoresistance.

In the approximation of short-range interactions, assum-ing statistical distribution of impurity atoms and using theideas of percolation theory [2], one can estimate on the basisof R0 values the concentration xC (percolation threshold) atwhich an uninterrupted chain of overlapping deformationfields penetrating the entire crystal (infinite cluster) isformed. One can also solve an opposite problem, i.e.knowing the value of the critical concentration xC, calculatethe radius of the impurity atom ‘‘action sphere’’ R0 from thecondition [1, 2]

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4=3pNcð2R0Þ3 � 2:7; (1)

where Nc is the average number of sphere centers pervolume unit. Assuming that xC corresponds to �0.005–0.015, we obtain R0¼ (1.3–0.9) a0, where a0¼ 0.65 nm isthe pseudocubic unit cell parameter calculated as a0¼V1/3,where V is the volume of the Bi crystal unit cell. Thecalculated value R0 indicates a short-range character of theforce fields created by impurity atoms. Such character istypical for example for deformational fields.

The formation of continuous chains of impurity atomsupon reaching the percolation threshold can stimulatesuch redistribution of impurity atoms in the crystal lattice,which would lead to the realization of their configurationcorresponding to a minimum of the thermodynamicpotential. Elastic interactions between impurity atoms,similarly to Coulomb interactions, can lead to the formationof certain configurations of impurity atoms, which corre-spond to minima of the elastic energy. Possible self-organization processes may include a long-range orderingof impurity atoms (‘‘crystallization of impurity liquid’’), theformation of complexes, a change in the localization ofimpurity atoms in the crystal lattice, etc.

The fact that the annealing time and additionalhomogenization of alloys by pressing samples does notpractically affect not only the behavior of the dependencesshows that for the investigated concentration range of Bi-Sbsolid solutions even annealing for 200 h ensures sufficientlyhigh level of homogeneity, which was proved by the resultsof both electron microprobe analysis and electrophysical andmechanical studies.

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4 Conclusions In Bi-based Bi-Sb semimetallic solidsolutions, in the concentration range 0.5–1.5 at.% Sb,anomalies were revealed in the isotherms of microhard-ness, electrical conductivity, charge carrier mobility,magnetoresistance, the Hall coefficient and the Seebeckcoefficient, and these anomalies were attributed to atransition from localized impurities to an impuritycontinuum. The observed effect represents another proofof the suggestion about the universal character of criticalphenomena that occur in solid solutions under thetransition to an impurity continuum.

Acknowledgements This work was supported by theUkrainian Fundamental Research Foundation (grant # FU/408-2008).

References

[1] B. I. Shklovskii and A. L. Efros, Electronic Properties ofDoped Semiconductors (Nauka, Moscow, 1979 Springer-Verlag, New York, 1984).

[2] D. Stauffer and A. Aharony, Introduction to PercolationTheory (Taylor & Francis, London/Washington, DC,1992), pp. 15–88.

[3] E. I. Rogacheva, N. A. Sinelnik, and O. N. Nashchekina, ActaPhys. Pol., A 84, 729 (1993).

[4] E. I. Rogacheva, I. M. Krivulkin, V. P. Popov, and T. A.Lobkovskaya, Phys. Status Solidi A 148, K65 (1995).

[5] E. I. Rogacheva, V. I. Pinegin, and T. V. Tavrina, Proc. SPIE3182, 364 (1998).

[6] E. I. Rogacheva, T. V. Tavrina, and I. M. Krivulkin, Inorg.Mater. 35, 236 (1999).

[7] E. I. Rogacheva and I. M. Krivulkin, Semiconductors 36, 966(2002).

[8] E. I. Rogacheva and I. M. Krivulkin, Fiz. Tverd. Tela 43,1000 (2001).

[9] E. I. Rogacheva, J. Phys. Chem. Solids 64, 1579 (2003).[10] E. I. Rogacheva, Jpn. J. Appl. Phys. 32, Suppl. (32–3), 775

(1993).[11] A. Z. Patashinskii and V. L. Pokrovskii, Fluctuation Theory

of Phase Transitions (Nauka, Moscow, 1982), p. 382.[12] L. I. Anatychuk, Thermoelements and Thermoelectric

Devices, Reference book (Naukova Dumka, Kyiv, 1979),p. 376.

[13] E. I. Rogacheva, A. A. Yakovleva, V. I. Pinegin, and M. S.Dresselhaus, J. Phys. Chem. Solids 69, 580 (2008).

[14] E. V. Kuchis, Galvanomagnetic Effects and the Methods ofIts Studing (Radio and Communication, Moscow, 1990),p. 264.


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Percolation effects in semimetallic Bi-Sb solid solutions