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Acta Materialia 56 (2008) 5954–5963

The high-temperature elastic moduli of polycrystalline PbTemeasured by resonant ultrasound spectroscopy

Fei Ren a, Eldon D. Case a,*, Joseph R. Sootsman b, Mercouri G. Kanatzidis b, Huijun Kong c,Ctirad Uher c, Edgar Lara-Curzio d, Rosa M. Trejo d

a Department of Chemical Engineering and Materials Science, Michigan State University, 2527 Engineering Building, East Lansing, MI 48824-1226, USAb Department of Chemistry, Northwestern University, Evanston, IL, USA

c Physics Department, University of Michigan, Ann Arbor, MI, USAd High Temperature Materials Laboratory, Oak Ridge National Laboratory, Oak Ridge, TN, USA

Received 5 March 2008; received in revised form 30 July 2008; accepted 31 July 2008Available online 4 September 2008

Abstract

The thermoelectric material PbTe has been used in a wide variety of power generator applications. However, there is limited mechan-ical property data available for PbTe for temperatures above room temperature. This paper reports dynamic elastic moduli measured viaresonant ultrasound spectroscopy on undoped and PbI2-doped polycrystalline PbTe between room temperature and 773 K; in addition,the room temperature carrier concentration was measured by a Hall effect experiment. The Young’s modulus and shear modulus of PbTedecreased linearly with temperature above room temperature, while the Poisson’s ratio exhibited either monotonic increase or decreasewith temperature. The Young’s modulus and shear modulus values obtained during heating and cooling agreed to within 1%. Thedynamic elastic moduli data obtained in this study compared well in general with literature data. The difference observed betweenthe current study and other literature studies is explained in terms of the carrier concentration effect on elastic moduli.� 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Semiconductor compounds; Dynamic mechanical analysis; Elastic behavior; Temperature-dependent

1. Introduction

1.1. PbTe as a thermoelectric material

PbTe is a thermoelectric (TE) material widely used inTE power generation applications [1–3]. The well-knownelectronic and thermal properties of PbTe [4] are centralto its function as a TE material. In contrast, its mechan-ical properties, such as the Young’s modulus and Pois-son’s ratio, have not been reported for temperaturesabove 303.2 K [5]. Nevertheless, the elastic moduli areimportant to understanding the materials’ response tothermal and mechanical stresses imposed by the in-serviceenvironment.

1359-6454/$34.00 � 2008 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2008.07.055

* Corresponding author. Tel.: +1 517 353 6715.E-mail address: casee@egr.msu.edu (E.D. Case).

To date, the authors are aware of only two studies inthe open literature of the temperature dependence of theelastic constants of PbTe [5,6]. Houston et al. measuredthe elastic stiffness of single-crystal PbTe using ultrasonicpulse echo technique from nearly 0 to 303.2 K [5]. UsingHouston et al.’s single-crystal values for PbTe, the aggre-gate elastic moduli (Young’s modulus, shear modulusand Poisson’s ratio) for polycrystalline PbTe are calcu-lated based on the Reuss-Voigt-Hill averages of the sin-gle-crystal data [7]. Salamov et al. utilized the low-frequency torsional pendulum method to determine therelative change in shear modulus, G/GRT (where GRT isthe shear modulus at room temperature), of undopedPbTe and Cr-doped PbTe single crystals as a functionof temperature between 20 and 600 �C [6]. In additionto the temperature-dependent modulus measurementsby Houston et al. [5] and Salamov et al. [6], a third

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F. Ren et al. / Acta Materialia 56 (2008) 5954–5963 5955

study of the elastic moduli of PbTe was performed atroom temperature only [8].

In the current study, the Young’s modulus, shear mod-ulus and Poisson’s ratio of two polycrystalline PbTe spec-imens (an undoped PbTe specimen and a PbTe specimendoped with 0.031 at.% PbI2) were measured using resonantultrasound spectroscopy (RUS) between room temperatureand 773 K. The Young’s modulus, shear modulus andPoisson’s ratio results from this study are compared to datafrom the literature. The differences observed in both thesecond-order elastic constants and temperature derivativesof the second-order elastic constants between this studyand the literature results may be due to carrier concentra-tion effects which are reviewed in Section 3.4.

2. Background: temperature-dependent elastic moduli for

polycrystalline brittle materials

The temperature-dependence of the Young’s modulus ofbrittle polycrystalline materials can be described by theempirical Wachtman equation [9]:

EðT Þ ¼ E0 � bET exp�T 0

T

� �; ð1Þ

where E0 is the Young’s modulus as T ? 0 K. The con-stants bE and T0 are fitting parameters. Anderson showedbE is related to the Gruneisen parameters. T0 is roughlyhD/2, where hD is Debye temperature [10].

At temperatures higher than roughly 0.3hD � 0.5hD, theWachtman equation simplifies to a linear relationship, suchthat [9]:

EðT Þ ¼ ERT½1� bE�RTðT � RT Þ�; ð2Þwhere ERT is the room-temperature Young’s modulus, bE-

RT is a fitting parameter, and RT denotes room tempera-ture. Thus in Eq. (2), at T = RT, E(T) = ERT. Houstonet al. determined that hD for undoped PbTe is 175 K [5].In this study the elastic moduli were measured from roomtemperature to 773 K, and as expected the Young’s modu-lus of undoped PbTe was found to exhibit a linear depen-dence on temperature (Fig. 1a).

Relationships analogous to the empirical relationshipsfor Young’s modulus (Eqs. (1) and (2)) also apply to thetemperature-dependent shear modulus for polycrystallinematerials [9]. Thus, for shear modulus as a function of tem-perature, T:

GðT Þ ¼ G0 � bGT exp�T 0

T

� �; ð3Þ

and for T greater than about 0.3hD:

GðT Þ ¼ GRT½1� bG-RTðT � RT Þ�; ð4Þwhere G0 and GRT are the shear modulus as T ? 0 K andat room temperature, respectively. The parameters bG andbG-RT are obtained via a least-squares fit to the modulusdata. Thus for Eq. (4), G(T) = GRT for T = RT.

As we shall discuss further in Section 4.1.2, when wecompare our data with elastic moduli from the literature,

we shall distinguish between the static and dynamic mod-ulus measurements, where static measurements give iso-thermal elastic constants and dynamic measurementsyield adiabatic elastic constants. However, it is worth-while to note, for example, that for cubic NaCl, KCland aluminum and trigonal quartz, there is no differencebetween the isothermal and adiabatic values of the com-pliance s44 [11], which implies that the shear modulusobtained by static and dynamic techniques should beidentical for isotropic materials. (In terms of the compli-ance sij and the stiffness cij, for crystals of cubic symmetry,1/s44 = c44 [12]. For isotropic materials, c44 = G = shearmodulus [12].)

3. Materials and methods

In this study, for the temperature-dependent elasticmodulus measurements, two polycrystalline ingots, namelyan undoped PbTe ingot and a PbTe ingot doped with 0.031at.% PbI2, were prepared using the Bridgman method. Theingots were then cut into parallelepiped specimens thatwere approximately 10 � 7 � 2 mm for RUS measure-ments. Two additional specimens (one specimen each ofthe undoped PbTe and the PbI2-doped PbTe compositions)were prepared by Bridgman method for room-temperatureHall effect measurements.

The RUS experiments were performed on a commercialunit (RUSpec, Quasar International, Albuquerque, NM).The RUS specimen stage and transducers were placed inan environmental chamber. A continuous 100 ml min�1

Ar flow was supplied to the chamber during the measure-ments to minimize specimen oxidation. Preliminary RUSmeasurements above 773 K resulted in slight discolorationon the specimen surfaces. Thus the maximum measurementtemperature was 773 K in this study. RUS scans weretaken at 50 K intervals with a heating and cooling rate of5 K min�1. Two consecutive 5-min isothermal holds wereapplied at each measurement temperature such that thefirst hold allowed the specimen to thermally equilibratewhile the RUS scans were taken during the second hold.Each RUS measurement was typically completed within2 min. The RUS scan frequency ranged between 30 and350 kHz for the undoped PbTe specimen, and between 50and 500 kHz for the PbI2-doped PbTe specimen such thatthe first 40 resonant frequencies in the lower frequencyrange were obtained.

The Young’s modulus, shear modulus and Poisson’sratio were calculated from the RUS spectra using commer-cial software (RPModel, Quasar International) [13]. Foreach of the 36 RUS scans included in this study, the overallgoodness-of-fit, as characterized by the root mean square(rms) error, was less than 0.5%. At each measurement tem-perature, the errors in individual Young’s modulus, shearmodulus and Poisson’s ratio calculations were determinedby linear propagation analysis from the uncertainties dc11

and dc44 in the elastic stiffness c11 and c44 where dc11 anddc44 were reported at a 95% confidence level by the RUS

Fig. 1. Temperature-dependent (a) Young’s modulus, (b) shear modulus and (c) Poisson’s ratio for the undoped PbTe specimen in this study. For (a) and(b), the solid curves represent the least-squares best fit of the Young’s and shear modulus data to Eqs. (2) and (4), respectively, while for (c) the solid curverepresents the least-squares best fit of the Poisson’s ratio data to Eq. (5). The error bars were calculated from elastic constants c11 and c44 and theirderivatives dc11 and dc44 given by the RUS software package RPModel. Note that the error bars for Young’s modulus and shear modulus were smallerthan the plotting symbols.

5956 F. Ren et al. / Acta Materialia 56 (2008) 5954–5963

commercial software package RPModel (Quasar Interna-tional). The errors are graphically presented as error bars(Figs. 1–4), where for the Young’s modulus and shear mod-ulus values obtained in this study, the error bars were smal-ler than the plotting symbols.

The room-temperature carrier concentration and thecarrier type for the undoped PbTe and the PbI2-dopedPbTe specimens were determined from Hall effect measure-ments. The Hall effect was measured using a QuantumDesign MPMS (magnetic property measurement system)and a Linear Research AC bridge with 16 Hz excitation.Within the MPMS dewar, a cryogenic probe integrated a5 T superconducting magnet with a SQUID detection sys-tem, which provided measurements over a temperaturerange of 1.9–300 K. Four-wire AC Hall measurementswere performed on two parallelepiped samples with dimen-sions of approximately 1 � 3 � 5 mm. The current and

Hall voltage leads are fine copper wires and the contactsare made of indium.

4. Results and discussion

4.1. Elastic moduli of polycrystalline undoped PbTe and

PbI2-doped PbTe

4.1.1. Room-temperature moduli

Table 1 compares the room-temperature values ofYoung’s modulus, shear modulus and Poisson’s ratio forthis study’s undoped PbTe and the PbI2-doped PbTe spec-imens with room-temperature modulus values computedfrom the single-crystal data for undoped PbTe studied byHouston et al. [5] and Einsprunch and Manning [8]. Themean value of Young’s modulus is 57.5 GPa for the four

Fig. 2. Temperature-dependent (a) Young’s modulus, (b) shear modulus, and (c) Poisson’s ratio for the PbI2-doped PbTe specimen in this study. For (a)and (b), the solid curves represent the least-squares best fit of the Young’s and shear modulus data to Eqs. (2) and (4), respectively, while for (c) the solidcurve represents the least-squares best fit of the Poisson’s ratio data to Eq. (5). The error bars were calculated from elastic constants c11 and c44 and theirderivatives dc11 and dc44 given by the RUS software package RPModel. Note that the error bars for Young’s modulus and shear modulus were smallerthan the plotting symbols.

F. Ren et al. / Acta Materialia 56 (2008) 5954–5963 5957

specimens included in Table 1. Thus, the four Young’smodulus values listed in Table 1 differ from the mean valueby less than 1%. Similarly, the mean shear modulus for thespecimens in Table 1 is 22.64 GPa, so that the values of theindividual shear moduli differ from the mean by less than1.4%. Therefore the differences among the room tempera-ture Young’s or the shear modulus values are no more thanabout 2–3%, respectively (Table 1). The room-temperaturecarrier concentrations for the four specimens listed inTable 1 vary by a factor of up to 8.5 (Table 1). (The PbTe-and Cr-doped PbTe specimens of Salamov et al. [6] willalso be discussed further in Section 4.3.)

Except for the undoped PbTe specimen measured in thisstudy, the room temperature Poisson’s values range from0.260 for this study’s PbI2-doped PbTe to 0.268 for Einspr-unch and Manning’s undoped PbTe specimen [8]. Theroom-temperature value of the Poisson’s ratio for theundoped PbTe specimen included in this study is 0.293.Given that Poisson’s ratio, m, is a function of both theYoung’s and shear moduli (m = (E � 2G)/2G), the observeddifferences in the room-temperature Poisson’s ratio values

probably represent the propagation of error based onuncertainties in both the Young’s and the shear modulusvalues (Table 1).

4.1.2. Temperature-dependent elastic moduli

For the polycrystalline undoped PbTe and PbI2-dopedPbTe specimens included in this study, both the Young’smodulus and shear modulus decrease linearly with increas-ing temperature. The modulus values obtained upon heat-ing and cooling are in good agreement such that theaverage relative differences between heating data and cool-ing data were 0.17% and 0.35% for the Young’s modulusand shear modulus of the undoped PbTe, and 0.22% and0.17% for the Young’s modulus and shear modulus ofthe PbI2-doped PbTe, respectively (Table 2). In addition,at a given temperature, the maximum relative differencebetween the heating and cooling data were 0.57% for theYoung’s modulus and 1.0% for the shear modulus data,respectively (Table 2, Fig. 2).

The Young’s modulus and the shear modulus dataobtained during both the heating and the cooling cycles

Fig. 3. Comparison of the shear modulus normalized to room temper-ature shear modulus, G/GRT, as a function of temperature between thisstudy and literature [5–8]. The error bars for the shear modulus valuesobtained in this study, which were smaller than the data symbols, werecalculated from elastic constants c11 and c44 and their derivatives dc11 anddc44 given by the RUS software package RPModel. No error bars areincluded for the plots of the literature data since no uncertainties werereported in these studies [5,6,8].

Fig. 4. Comparison of temperature dependent elastic moduli between thisstudy and the literature [5,7,8] for (a) the Young’s modulus, and (b) thePoisson’s ratio for the combined data sets: PbI2-doped PbTe materials(this study) and undoped PbTe specimens from Houston et al. [5] andEinsprunch and Manning [8]. The error bars for the shear modulus(smaller than the data symbols) and the Poisson’s ratio obtained in thisstudy were calculated from elastic constants c11 and c44 and theirderivatives dc11 and dc44 given by the RUS software package, RPModel.No error bars are included for the plots of the literature data since nouncertainties were reported in these studies [5,6,8].

5958 F. Ren et al. / Acta Materialia 56 (2008) 5954–5963

were least-squares fit to Eqs. (2) and (4), respectively, yield-ing coefficients of determination, r2, between 0.991 and0.999 (Tables 3a and b). Also, a fit of Houston et al.’sYoung’s and shear modulus data [5] to Eqs. (2) and (4),respectively, over the temperature range 50–303 K (Table3a and b) shows that both the Young’s modulus and shearmodulus for Houston’s undoped PbTe specimen [5]decrease linearly with increasing temperature.

Although the room temperature Young’s and shear mod-uli differ by only a few per cent among the four specimenslisted in Table 1, the differences among the temperaturederivatives were much larger. From a linear least-squaresfit to Eq. (2), the slope of the Young’s modulus vs. temper-ature relationship was 4.50 ± 0.04 � 10�4 K�1 for the

Table 1Room-temperature Young’s modulus, shear modulus, Poisson’s ratio, and carrier concentration of undoped and doped PbTe materials measured in thisstudy and from literature

Source Material Young’s modulus (GPa) Shear modulus (GPa) Poisson’s ratio Carrier concentration (carrier cm�3)

This study Polycrystalline PbTe 57.5 ± 0.3 22.32 ± 0.04 0.293 ± 0.003 p = 4.8 � 1017

Polycrystalline PbI2-doped PbTe 57.5 ± 0.2 22.81 ± 0.02 0.260 ± 0.003 n = 7.3 � 1018

[5] Single-crystal PbTe 58.05a 22.95a 0.264a p = 4.1 � 1018

[8] Single-crystal PbTe 56.95a 22.48a 0.268a Not statedb

[6] PbTe Not reported 1.3 � 10�5c Not reported p = 1.2 � 1019

Cr-doped PbTe Not reported 3.14 � 10�5c Not reported n = 4.1 � 1018

The elasticity values listed for the single-crystal PbTe [5,8] are the Voigt–Reuss–Hill averages [7] computed from the single-crystal data.a Aggregate values [7].b Although Einsprunch and Manning reported an electrical resistivity of 0.002 X m for their undoped PbTe specimen [8], the carrier concentration

cannot be calculated since the carrier mobility is unknown for the Einsprunch and Manning study.c The absolute shear modulus values at room temperature were calculated by comparing the oscillation frequencies of PbTe specimens to those obtained

from V, Cu and Al specimens with identical geometry [6].

Table 2The differences in elastic moduli between the heating data and the coolingdata for the undoped and PbI2-doped PbTe specimens included in thisstudy

Average relativedifference (%)

Maximum relativedifference (%)

PbTe E 0.17 0.37G 0.35 1.0m 1.95 6.9

PbI2-doped PbTe E 0.22 0.57G 0.17 0.61m 0.80 2.0

The relative differences in each parameter, Prd, were calculated as|PHT � PCL|/PHT � 100%, where the subscript ‘‘HT” and ‘‘CL” denoteheating and cooling, respectively. The average and the maximum values ofPrd are given for P = E, G and m as below.

F. Ren et al. / Acta Materialia 56 (2008) 5954–5963 5959

undoped PbTe specimen, which was about 27% lower thanthat for the PbI2-doped PbTe specimen (Table 3a). Theslope of the shear modulus vs. temperature relationship asobtained from linear least-squares fit to Eq. (4) was3.7 ± 0.1 � 10�4 K�1 for the undoped PbTe specimen,which was about 54% of the value of 6.9 ± 0.1 � 10�4 K�1

for the PbI2-doped PbTe specimen (Table 3b). The slopesof the linear least-squares fit lines to Eq. (2) for the PbTeand the PbI2-doped PbTe in this study are approximately21% lower and about 9% higher, respectively, than thoseof Houston’s PbTe data (Table 3a and Fig. 4a).

Table 3aResults of the linear least-squares fit of Young’s modulus obtained during bothis the number of data points included in the fit and r2 is the coefficient of det

Specimen Temperature range (K) N

PbTe (this study) 298–773 2PbI2-doped PbTe (this study) 298–673 1PbTe [5] 50–303 6

The least-squares-determined parameters ERT and bE-RT are defined in Eq. (2

Table 3bResults of the linear least-squares fit of the shear modulus obtained during bothis the number of data points included in the fit and r2 is the coefficient of det

Specimen Temperature range (K)

PbTe (this study) 298–773PbI2-doped PbTe (this study) 298–673PbTe [5] 50–303

The least-squares-determined parameters GRT and bG-RT are defined in Eq. (4

Table 3cResults of the linear least-squares fit of the Poisson’s ratio obtained during boththe number of data points included in the fit and r2 is the coefficient of determ

Specimen Temperature range (K) N

PbTe (this study) 298–773 20PbI2-doped PbTe (this study) 298–673 16Combined data set, PbTe [5,8] and

PbI2-doped PbTe (This study)4–673 24

The least-squares-determined parameters mRT, e1 and e2 are defined in Eq. (4)

The temperature dependence of Poisson’s ratio for bothPbTe and PbI2-doped PbTe specimens in this study can bedescribed by the empirical equation:

tðT Þ ¼ tRT þ e1ðT � T RTÞ þ e2ðT � T RTÞ2; ð5Þwhere mRT is the Poisson’s ratio at room temperature, ande1 and e2 are fitting parameters. In this study, TRT (roomtemperature) was 298 K. For this study’s undoped PbTespecimen, the Poisson’s ratio decreased with increasingtemperature where the scatter in the data lead to coefficientof determination, r2 = 0.86 (Table 3c and Fig. 1c). Thetemperature-dependent Poisson’s ratio for the PbI2-dopedPbTe specimen is also described well by the parabolic tem-perature relationship (Eq. (5)) with an r2 value of 0.96 forthe least-squares fit. However, in contrast to the undopedPbTe, the Poisson’s ratio for the PbI2-doped PbTe speci-men increases monotonically with temperature (Table 3cand Fig. 2c).

A nonlinear least-squares fit to Eq. (5) on three datasets, i.e. the Poisson’s ratio of the PbI2-doped PbTebetween 298 and 673 K in the current study, the aggregatePoisson’s ratio values between 4 and 302 K from Houstonet al.’s PbTe study [5], and the 302 K Poisson’s ratio fromEinsprunch and Manning’s work [8] yielded mRT = 0.261 ±0.001, e1 = 6.0 ± 5.0 � 10�5 K�1 and e2 = 1.6 ± 0.2 � 10�7

K�2, with an r2 of 0.96 (Table 3c and Fig. 4b). Thusalthough the temperature intervals for the modulus mea-

heating and cooling cycles as a function of temperature to Eq. (2), where N

ermination

ERT (GPa) bE-RT (1/K) r2

0 57.52 ± 0.07 4.50 ± 0.04 � 10�4 0.9986 57.70 ± 0.09 6.20 ± 0.09 � 10�4 0.996

58.31 ± 0.06 5.60 ± 0.08 � 10�4 0.999

).

heating and cooling cycles as a function of temperature to Eq. (4) where N

ermination

N GRT (GPa) bG-RT (1/K) r2

20 21.97 ± 0.06 3.7 ± 0.1 � 10�4 0.99116 22.93 ± 0.06 6.9 ± 0.1 � 10�4 0.9966 23.06 ± 0.03 6.0 ± 0.1 � 10�4 0.999

).

heating and cooling cycles as a function temperature to Eq. (5), where N isination

mRT e1 (1/K) e2 (1/K2) r2

0.298 ± 0.005 6.0 ± 5.0 � 10�5 �4.3 ± 1.0 � 10�7 0.860.258 ± 0.002 8.0 ± 2.0 � 10�5 1.2 ± 0.6 � 10�7 0.960.261 ± 0.001 6.0 ± 0.5 � 10�5 1.6 ± 0.2 � 10�7 0.96

.

5960 F. Ren et al. / Acta Materialia 56 (2008) 5954–5963

surements in this study and the studies by Houston et al. [5]and Einsprunch and Manning [8] overlap only near 300 K,the Poisson’s ratio vs. temperature trends agree well in thetwo studies (Fig. 4b).

Unlike the temperature dependence of Young’s andshear modulus (where the Young’s and shear modulidecrease monotonically with increasing temperature),experimental observations of brittle materials show that mcan increase, decrease, or remain relatively constant as afunction of temperature. For example, a study by Sogaand coworkers [14] showed m increased linearly from roomtemperature to 1300 K for polycrystalline Al2O3, and forMgO m increased linearly between room temperature and1150 K. In contrast, the Poisson’s ratio of b’-Sialon firstdecreased from 0.273 at 300 K to approximately 0.267 at1350 K, then rapidly increased to 0.28 at 1750 K, followedby a drop to 0.27 at 1800 K [15]. In addition, a yttria-sta-bilized tetragonal zirconia polycrystalline specimen showedan increase in m from 0.316 at 300 K to 0.34 at 1000 Kbefore reaching a plateau at 0.34 between 1000 and1600 K. Finally, m increased to 0.35 at 1800 K [15]. Sincem = (E � 2G)/2G, the temperature dependence of Poisson’sratio is related to the relative change in the Young’s mod-ulus and the shear modulus as a function of temperature.

In the next section, we compare our dynamic elasticitydata (RUS) with the dynamic pulse-echo elasticity datafrom Houston et al. [5] and Einsprunch and Manning [8]and with torsional pendulum measurements by Salamovet al. [6]. Static techniques (such as indentation, tensionand bending tests) give isothermal elastic constants whiledynamic experiments provide adiabatic moduli [11,17].Under adiabatic conditions, a material tends to be stiffer(have a higher elastic modulus) than under isothermal con-ditions [16]. However, the differences between adiabaticand isothermal moduli are often relatively small[11,12,17,18]. According to Nye [12], the relative differencebetween the adiabatic and isothermal elastic constants isroughly on the order of 1 part in 103, although in somecases the differences can be substantial. For KCl at roomtemperature the difference between the isothermal and adi-abatic values of the elastic compliance s12 is approximately18% [11].

It is worthwhile to note that for cubic systems, includingNaCl, KCl, aluminum and trigonal quartz, there are nodifferences between the isothermal and adiabatic values ofs44 [11], which implies that the shear modulus obtainedby static and dynamic techniques should be identical forisotropic materials.

4.2. Comparison of the relative shear moduli, G/GRT, as a

function of temperature

Salamov et al. reported the relative shear moduli, G/

GRT, as a function of temperature and for undoped PbTeand Cr-doped PbTe as well as the room temperature shearmodulus values. Before comparing G/GRT as a function oftemperature with the other work discussed in this paper, a

comment needs to be made about the room temperatureshear modulus measurements.

Salamov et al. state that ‘‘It is determined that at roomtemperature the PbTe and PbTe:Cr samples are character-ized by absolute shear moduli that equal 1.3 � 104 and3.14 � 104 N/m2, respectively” [6]. The room-temperatureshear modulus values for Salamov et al.’s undoped PbTe(13 kPa) is more than six orders of magnitude lower thanthe room temperature shear moduli found in three inde-pendent studies (this study, Ref. [5] and Table 1 in Ref.[8]). As an additional comparison, the shear modulus ofsolid argon (a van der Waals bonded solid) is extremelylow (approximately 1.7 GPa at 4.2 K and 1.0 GPa at76.8 K [7]), but even the shear modulus of solid argon isnearly five orders of magnitude higher than the room-tem-perature modulus of undoped PbTe reported by Salamovet al. [6]. Thus, it appears that the room-temperature shearmoduli reported by Salamov et al. for both PbTe (13 kPa)and Cr-doped PbTe (31.4 kPa) are much too low to bephysically reasonable. In addition, the shear modulus val-ues reported by Salamov et al. for undoped PbTe andCr-doped PbTe differ by more than 140%, which is muchlarger than the several per cent difference observed in otherstudies for carrier concentration related shifts in second-order elastic constants, as will be discussed in Section 4.3.

In order to make comparisons among the data sets, therelative shear modulus, G/GRT from this study, Houstonet al.’s study [5], Einsprunch and Manning’s study [8],and Salamov et al.’s study [6] are plotted together inFig. 3. The temperature dependence of G/GRT for thePbI2-doped PbTe specimen agrees well with the undopedPbTe in Houston et al.’s study [5,7] (Fig. 3). However,the undoped PbTe in the current study has a smaller slopeas determined by linear least-squares fit to Eq. (4) (Table3b).

Salamov et al. [6] observed a monotonic decrease in G/GRT with increasing temperature for both the undopedPbTe and the Cr-doped PbTe data obtained from torsionalpendulum measurements [6] (Fig. 3). However, the rates ofchange of G/GRT with temperature for both the undopedPbTe and the Cr-doped PbTe specimens studied by Sala-mov et al. [6] are much smaller than those observed forthe undoped PbTe specimen and the PbI2-doped PbTespecimen included in this study (Fig. 3). In this study, theG/GRT values (and their temperature derivatives indicatedin parenthesis) at 673 K were 0.848 ± 0.002(3.7 ± 0.1 � 10�4 K�1) for the undoped PbTe specimenand 0.742 ± 0.001 (6.9 ± 0.1 � 10�4 K�1) for the PbI2-doped PbTe specimen. In contrast, the G/GRT values at673 K were approximately 0.95 (1.3 � 10�4 K�1) and 0.98(5.4 � 10�5 K�1) for the undoped and Cr-doped PbTespecimens in Salamov et al.’s study [6] (Fig. 3). The temper-ature derivatives of the normalized shear moduli (slope ofG/GRT vs. T) increase monotonically with increasingroom-temperature carrier concentration for the undopedPbTe (p-type) and the PbI2-doped PbTe (n-type) in thisstudy and the undoped PbTe (p-type) included in Houston

F. Ren et al. / Acta Materialia 56 (2008) 5954–5963 5961

et al.’s study [5] (Tables 1 and 3b), which (if we ignore thecarrier type) is consistent with the general trend discussedin Section 4.3 in which the electronic contribution to theelastic moduli increases as carrier concentration increasesfor n-type semiconductors [19,20]. However, the tempera-ture derivatives for Salamov et al.’s G/GRT showed theopposite trend with carrier concentration for the p-typeundoped PbTe and the n-type Cr-doped PbTe [6] (Tables1 and 3b).

The techniques used to measure the shear modulus inthis study and in previous studies [5,6,8] were different: inthis study, the elastic moduli were measured by RUS inthe frequency range between 30 and 500 kHz; Houstonet al. [5] and Einsprunch and Manning [8] utilized a MHzultrasonic pulse echo technique; while the frequency usedby Salamov et al. was between 4.8 and 5 Hz in their tor-sional pendulum study [6] (Table 4). Thus the elastic mod-uli determined in this study as well as those in the studiesby Houston et al. [5] and Einsprunch and Manning [8]are dynamic (adiabatic) moduli, while Salamov et al.’s tor-sional pendulum shear modulus values [6] are quasi-static(isothermal).

However, as discussed in Section 4.1, no significant dif-ference was observed between the isothermal and adiabaticelastic compliance s44 [11], which suggests the shear modu-lus measured via static and dynamic techniques should beidentical. Thus the different slopes observed in the temper-ature-dependent G/GRT among the current study and theliterature data [5,6] may be due at least in part to the differ-ences in carrier concentrations between the specimens(Table 1) as will be described in the next section.

4.3. Effects of carrier concentration on the elastic moduli of

semiconductors

Electronic carrier concentrations can affect the elasticproperties (especially c44) of semiconducting materials[21–24] such as Ge, Si and PbTe. The electronically inducedchanges in elastic modulus are somewhat complex since theeffects of carrier concentrations are functions of tempera-ture, doping level and/or the material’s deviation fromstoichiometry.

For example, for a p-doped Si single crystal at 78 K, thechange d(c11 � c12) in the elastic stiffness (c11 � c12) was8 GPa as the carrier concentration increased from1.0 � 1017 to 6.5 � 1019 cm�3 [23]. At 302 K d(c11 � c12)was 5 GPa for the same range of carrier concentrations[23]. (In terms of the physical interpretation of (c11 � c12),

Table 4Comparison of measurement techniques and frequency between this studyand literature

Researchers Technique Frequency

This study RUS 30–500 kHzHouston et al. [5] Ultrasonic pulse echo �MHzEinspruch and Manning [8] Ultrasonic pulse echo 10 MHzSalamov et al. [6] Torsional pendulum 4.8–5 Hz

for isotropic materials, such as randomly oriented poly-crystals, c44 = 1/2(c11 � c12) = G = shear modulus [16].However, in cubic crystals, 1/2(c11 � c12) is not necessarilyequal to G, the relationship of (c11 � c12) to the engineeringmoduli depends on crystallographic direction.) From theliterature [7], (c11 � c12) = 102.9 GPa at 77.2 K and101.8 GPa at 298 K, so the 8 and 5 GPa changes representrelative changes of 7.8% and 4.9% in (c11 � c12),respectively.

For Ge, carrier concentration induced changes in boththe second-order elastic constants and their temperaturederivatives have been discussed in the literature. For exam-ple, using a multivalley model, Keyes [21] predicted thatdc44/c44 (the change in the elastic constant c44), wasapproximately 8.6% as the carrier concentration increasedfrom 2 � 1018 to 5 � 1019 cm�3. In a subsequent experi-mental study, Bruner and Keyes [22] used the ultrasonicpulse echo technique to measure the elastic constants ofundoped Ge (carrier concentration <1014 electrons cm�3)and n-type As-doped Ge (carrier concentra-tion = 3.5 � 1019 electrons cm�3). At 4.2 K the c44 valueof As-doped Ge was 64.2 GPa, which was approximately5.6% lower than the c44 value of 68 GPa for the undopedGe [22]. In addition, between 100 and 240 K the slopes ofc44 vs. temperature were approximately 6.3 and3.7 MPa K�1 for the undoped and the As-doped Ge mate-rials, respectively [22]. If we normalize the c44 vs. T slopesnormalized with respect to the room temperature c44 valueof 67.1 GPa [25], the slopes were approximately 9.4 � 10�5

and 5.5 � 10�5 K�1 for the undoped and the As-doped Gematerials, respectively [22]. Thus, the temperature deriva-tive of c44 for Ge increased by a factor of 1.7 as the carrierconcentration increased from <1014 to 3.5 � 1019 electronscm�3 due to the As doping.

Sreedhar et al. [24] used two models to predict the elec-tronic contribution to the elastic constants of PbTe: (i)Cohen’s nonellipsoidal nonparabolic (NENP) band model;and (ii) an ellipsoidal parabolic (EP) band model. At 0 K,as the carrier concentration increased from 1018 to 1021

holes cm�3, the NENP model predicted that c44 at 0 Kwas lowered by approximately 33% in contrast to 5.5%as predicted by the EP model [24]. In addition, Sreedharet al. [24] predicted the change, Dc44, to be a function oftemperature at a given carrier concentration. For example,at a carrier concentration of 1 � 1021 holes cm�3, theNPNE model predicted that Dc44 at 300 K was approxi-mately the same as the Dc44 at 0 K; while the PE model pre-dicted that Dc44 at 300 K was approximately 150% of Dc44

at 0 K [24]. Thus, the temperature derivative of c44 is pre-dicted to be a function of carrier concentration, but themagnitude of the change in the temperature derivative ofc44 depends on the specific band model [24].

Following up the theoretical work by Keyes [21] andSreedhar et al. [24], over the past 15 years a series of theo-retical calculations have been performed on the electroniccontribution to elastic constants of semiconductors[19,20,26,27]. The calculations by Ghatak et al. [19,20] indi-

5962 F. Ren et al. / Acta Materialia 56 (2008) 5954–5963

cate that for bulk n-type semiconductors, the electroniccontribution to the elastic constants increases monotoni-cally with increasing carrier concentration, although Gha-tak et al. note that there are limited experimental resultsfor the electronic contribution to the elastic moduli. Fur-thermore, the details of the electronic contribution to theelastic constants depend on the details of the band struc-ture [19,20,26,27], thus the electronic contribution to elasticmodulus will depend on the particular semiconductor-dop-ant material pair, the dopant concentration, and the result-ing carrier concentration and the details of the electronicband structure.

The dependence of the elastic modulus on carrier con-centration discussed above for Ge, Si and PbTe is compa-rable in magnitude to the dependence of the shear modulusvalues observed for the various PbTe specimens discussedin this study. The room temperature carrier concentrationsmeasured in this study were 4.8 � 1017 holes cm�3 forundoped PbTe and 7.3 � 1018 electrons cm�3 for thePbI2-doped PbTe (Table 1). The room-temperature carrierconcentration of the PbTe specimen studied by Houstonet al. [5] was 4.1 � 1018 holes cm�3 (Table 1). For the PbTespecimen and the Cr-doped PbTe specimen included in Sal-amov et al.’s study [6], the room-temperature carrier con-centrations were 1.2 � 1019 hole cm�3 and 4.1 � 1018

electrons cm�3 (Table 1).For undoped binary semiconductors such as PbTe, devi-

ations from stoichiometry can induce changes in the major-ity charge carrier concentration and/or carrier type. ForPbTe in particular, excess Pb results in a p-type semicon-ductor while excess Te yields an n-type semiconductor.Thus, small shifts in stoichiometry in the undoped PbTecan account for the differing carrier concentration and car-rier types listed in Table 1.

Thus, the different room temperature elastic constantsbetween the two specimens in this study and the differencebetween measurements from this study and the literaturedata [5–7] may, as least in part, be due to the different car-rier concentration, which also affects the temperature deriv-atives of the elastic moduli. In the literature the changes inthe elastic moduli due to the electronic contribution are aslarge as 6–8% [21–23] while the changes in the numericalvalues of the temperature derivatives can be as large as170% [22,24]. Thus the electronic contribution to the sec-ond-order elastic constants (Young’s modulus and shearmodulus) is smaller than the contribution to the tempera-ture derivatives. In Table 1 (which includes data from thisstudy and literature data from Houston et al. [5]) the differ-ences in the room-temperature Young’s modulus and shearmodulus were less than 1% and 1.4%, respectively (Table1). Also, the changes in the temperature derivatives as afunction of carrier concentration were up to 27% forYoung’s modulus (Table 3a) and 46% for shear modulus(Table 3b). Thus, for this study the differences in the sec-ond-order elastic moduli and their temperature derivativesas a function of carrier concentration are consistent withthe literature [21–24].

Due to the difference in the temperature derivativesbetween the updoped PbTe and the PbI2-doped PbTe, thedifferences in the second-order elastic moduli increasedwith increasing temperature (Figs. 3 and 4a), but at673 K the differences in Young’s and shear moduli were3.6 GPa (7.6%) and 1.9 GPa (10.4%), respectively. Thus,even though the temperature derivatives of undoped PbTeand the PbI2-doped PbTe differed by roughly 30–50%, thedifferences in the Young’s and shear moduli were stillwithin approximately 10%.

5. Summary and conclusions

This study is the first to determine the dynamic elasticmoduli, including Young’s modulus, and Poisson’s ratio,m, above 303.2 K for polycrystalline undoped PbTe andPbI2-doped PbTe materials. For both the undoped andthe doped PbTe specimens, E and G exhibited linear tem-perature dependence between room temperature and 773and 673 K, respectively; while the Poisson’s ratio of theundoped PbTe and the doped PbTe either increased ordecreased monotonically with increasing temperature.

Near room temperature, E and G agreed to within 2–3%with the undoped PbTe data from Houston et al. [5] andEinspruch and Manning [8]. Both Houston et al. andEinspruch and Manning used the ultrasonic pulse echotechnique to measure elastic moduli, although Einspruchand Manning’s measurements were taken at 302 K only,while Houston et al. measured the moduli over the temper-ature interval from 4 to 303 K.

For the range of temperatures between 4 and 673 K, thetemperature-dependent Poisson’s ratio for three combineddata sets (PbTe data from Houston et al. from 4 to 303 K[5], the 302 K PbTe data from Einsprunch and Manning [8]and the 298–673 K PbI2-doped PbTe data from this study)were fit by a single parabolic curve with an r2 of 0.96(Fig. 4b and Table 3c). Thus, although the temperatureranges are disjoint except near 300 K, the three data setsappear to follow a single m–T trend.

However, the temperature-dependent Poisson’s ratio forthe undoped PbTe in this study exhibited a different trendthan those for the PbI2-doped PbTe and the undoped PbTein Houston et al.’s study, which decreased monotonicallywith increasing temperature (Fig. 1c). As discussed previ-ously, the Poisson’s ratio is very sensitive to the relativechange between the Young’s modulus and the shear mod-ulus, and thus the effect of carrier concentration on theshear modulus may play a very important role affectingthe Poisson’s ratio as a function of temperature.

Salamov et al.’s [6] torsional pendulum measurements ofthe relative shear modulus for undoped PbTe and Cr-dopedPbTe showed a different temperature dependence comparedto the current study and that observed by Houston et al. [5].These differences may be due in part to the different mea-surement techniques used in these three studies.

Moreover, the undoped PbTe and PbI2-doped PbTematerials in this study, the undoped PbTe in Houston

F. Ren et al. / Acta Materialia 56 (2008) 5954–5963 5963

et al.’s work [5], and the undoped and Cr-doped PbTe in Sal-amov et al.’s study [6] possessed different carrier types andcarrier concentrations, which may also affect the measureddynamic moduli as well as their temperature dependence.

In addition to being directly applicable to PbTe, theresults of this study may impact the study of the mechani-cal properties of LAST (Pb–Sb–Ag–Te) [28,29] andLASTT (Pb–Sb–Ag–Te–Sn) [29,30] TE materials, whereLAST and LASTT are derivatives of PbTe with antimonyand silver (LAST) or antimony, silver and tin (LASTT).Thus, the understanding of the mechanical behavior ofPbTe may give insight into the mechanical properties ofLAST and LASTT materials.

Acknowledgments

The authors acknowledge the financial assistance of theOffice of Naval Research MURI Grant No.N000140310789 and the US Department of Energy GrantDE-FC26-04NT42281. The research work at the HighTemperature Materials Laboratory was sponsored by theAssistant Secretary for Energy Efficiency and RenewableEnergy, Office of Freedom CAR and Vehicle Technologies,as part of the High Temperature Materials LaboratoryUser Program, Oak Ridge National Laboratory, managedby UT-Battelle, LLC, for the US Department of Energyunder Contract No. DE-AC05-00OR22725.

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