ultrasonic wave propagation in rare-earth monochalcogenides

Cent. Eur. J. Phys. • 7(1) • 2009 • 198-205DOI: 10.2478/s11534-008-0130-1

Central European Journal of Physics

Ultrasonic wave propagation in rare-earthmonochalcogenides

Research Article

Devraj Singh∗, Dharmendra K. Pandey, Pramod K. Yadawa

Department of Applied Physics, AMITY School of Engineering and Technology,Bijwasan, New Delhi-110 061, India

Received 14 May 2008; accepted 16 September 2008

Abstract: The ultrasonic attenuation in thulium monochalcogenides TmX (X=S, Se and Te) has been studied the-oretically with a modified Mason’s approach in the temperature range 100 K to 300 K along 〈100〉, 〈110〉and 〈111〉 crystallographic directions. The thulium monochalcogenides have attracted a lot of interest dueto their complex physical and chemical characteristics. TmS, TmSe and TmTe are trivalent metal, mixedvalence state, and divalent semiconductor, respectively. Coulomb and Born-Mayer potential is applied toevaluate the second- and third-order elastic constants. These elastic constants are used to compute ul-trasonic parameters such as ultrasonic velocities, thermal relaxation time, and acoustic coupling constantsthat, in turn, are used to evaluate ultrasonic attenuation. A comparison of calculated ultrasonic parameterswith available theoretical/experimental physical parameters gives information about classification of thesematerials.

PACS (2008): 43.35.+d; 62.20.de; 63.20kg

Keywords: thulium monochalcogenides • elastic constants • acoustic coupling constants • ultrasonic attenuation© Versita Warsaw and Springer-Verlag Berlin Heidelberg.

1. Introduction

Monochalcogenides of the rare-earth elements (MeX, withMe=rare-earth element and X=S, Se and Te) comprise alarge class of materials that crystallize in a simple NaCl-type structure. MeX exhibits interesting electrical, opticaland magnetic properties [1]. Vaitheeswaran et al. inves-tigated high-pressure structural behaviors of lanthanummonohalcogenides (LaX: X=S, Se and Te) by theoret-ical and experimental study [2]. The theoretical studyof structure, elastic constants, and high-pressure prop-∗E-mail: [email protected]

erties of CeX (X=S, Se and Te) are presented by Bouhe-madou et al. [3].A lot of discussion on elastic, electronic, optical, and mag-netic properties of U, Np, Pu and Am telluride single crys-tals is available in the literature [4]. A simple theoreticalmodel is developed for determining the electrical proper-ties of rare-earth monochalcogenide (Eu, Sm, Yb) com-pounds under pressure using two parameters-lattice con-stant and activation energy by Ariponnammal and Natara-jan [5]. For neodymium monochalcogenides (NdX: X=S,Se and Te), it is shown in the literature that the criticalfield of 90 kOe induces an antiferromagnet transition at4.2 K and 1.6 K [6]. The temperature dependent resistiv-ity, thermo e.m.f., thermal conductivity, and ultrasonic pa-

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Devraj Singh, Dharmendra K. Pandey, Pramod K. Yadawa

rameters are presented for ReX(Re=La, Ce, Pr, Nd, Sm,Eu, Tm; X=S, Se and Te) by various investigators [7–15].Bucher et al. studied the magnetic and thermal prop-erties of all known sulfides, selenides and tellurides ofpraseodymium and thulium [16]. Batlogg interpreted va-lence of the Tm ions in mixed valent TmSe by alloyingwith TmTe and EuSe in order to study intermediate-valentrare-earth ions as a function of the degree of valence mix-ing [17]. The electronic structure, elastic, and spectro-scopic properties of TmX (X=S, Se and Te) are investi-gated by Lebegue et al. [18]. Nakamishi et al. inves-tigated the second-order elastic constants (C11, C12, andC44) and the bulk modulus of TmS first time by meansof ultrasonic measurement [19]. Ohashi et al. measuredresistivity of TmSe under pressure up to 8 GPa belowroom temperature [20]. Resistivity, magnetic susceptibil-ity, elastic constants, specific heat, and magnetization ofmagnetic semiconductor TmTe have been measured in de-tail by Matsumura et al. [21, 22].The thulium monochalcogenides TmX (X=S, Se and Te)have NaCl-type structure. Tm compounds exhibit VanVleck paramgnetism at low temperatures as a consequenceof the crystal-field singlet ground states [16]. TmS, TmSeand TmTe are golden metal, red brown colored interme-diate valance system, and silver blue semiconductor, re-spectively [1]. These materials are technologically impor-tant, having many applications that range from catalysisto microelectronics. In the present investigation, we havechosen thulium monochalcogenides for a theoretical studyof ultrasonic attenuation and other associated parameterssuch as ultrasonic velocities, acoustic coupling constants,etc., along 〈100〉, 〈110〉 and 〈111〉 directions in the tem-perature range 100-300 K. The calculated ultrasonic pa-rameters are discussed with related thermophysical prop-erties (e.g., specific heat and thermal conductivity) for thecharacterization of the chosen materials.2. Theory

2.1. Second- and third-order elastic con-stants by Coulomb and Born-Mayer potential

The potential used for evaluation of second- and third-order elastic constants (SOEC and TOEC) is taken as thesum of the Coulomb and Born-Mayer potentials.φ(r) = φ(C ) + φ(B), (1)

where φ(C ) is the electrostatic/Coulomb potential andφ(B) is the repulsive/Born-Mayer potential, given as

φ(C ) = ±(e2r

) and φ(C ) = A exp(−rb ), (2)where e is electronic charge, r is the nearest neighbor dis-tance, b is the hardness parameter, and A is the strengthparameter.According to lattice dynamics (developed by Leibfried andLudwig [23]), lattice energy changes with temperature.Hence adding vibrational energy contribution to the staticelastic constants, one gets second- and third-order elasticconstants (Cij and Cijk ) at the required temperature.

Cij = C 0ij + CV ib

ij and Cijk = C 0ijk + CV ib

ijk , (3)where superscript 0 has been used to denote SOEC andTOEC at 0 K (static elastic constants) and superscript V ibhas been used to denote the vibrational part of SOEC andTOEC at a particular temperature. The expressions for Cijand Cijk are given in our previous paper [11].2.2. Ultrasonic velocities along 〈100〉, 〈110〉and 〈111〉 crystallographic directionsThere are three types of ultrasonic velocities-one longitu-dinal (VL) and two shear (VS1 and VS2)-for each directionof propagation in cubic crystals [24, 25]. The expressionsfor velocities are as follows:

• Along the 〈100〉 crystallographic direction:VL =√C11

ρ , VS1 = VS2 =√C44ρ . (4)

• Along the 〈110〉 crystallographic direction:VL =√C11 + C12 + 2C442ρ ,

VS1 =√C44ρ , VS2 =√C11 − C12

ρ . (5)• Along the 〈111〉 crystallographic direction:

VL =√C11 + 2C12 + 4C443ρ ,

VS1 = VS2 =√C11 − C12 + C443ρ . (6)199

Ultrasonic wave propagation in rare-earth monochalcogenides

The ultrasonic velocities can be computed using second-order elastic constants and density (ρ). The Debye aver-age velocity (VD) is useful for knowledge of Debye tem-perature and the thermal relaxation time of the materi-als. The Debye temperature (TD) is an important physicalparameter of solids, defining a division line between thequantum and classical behaviors of phonons [26]. The fol-lowing expressions have been used for the evaluation ofDebye average velocity and Debye temperature [26, 27].( 1VD

)3 = 13 ∑i

14π∫ 1V 3i

dΩ, (7)

TD = ~VD(6π2na)1/3kB

, (8)where Vi is constituent velocity along a particular direc-tion of wave propagation, ~ is the quantum of action equalto Planck’s constant divided by 2π , kB is the BoltzmannConstant, and na is the atom concentration.2.3. Ultrasonic attenuation due to phonon-phonon interaction and due to thermoelastic re-laxation mechanismsThe main causes of ultrasonic attenuation for perfectcrystals are due to electron-phonon interaction, phonon-phonon interaction, and thermoelastic mechanism. Atroom temperature, the electron mean free path is not equalto the phonon mean free path, hence no coupling will takeplace. Thus, at high temperature, the attenuation dueto electron-phonon interaction will be absent. The twodominant processes that will give rise to an appreciableeffect on ultrasonic attenuation at higher temperature arephonon-phonon interaction (also known as Akhieser loss)and thermoelastic relaxation phenomenon.The expressions for ultrasonic attenuation produced froma phonon-phonon interaction effect for longitudinal andshear waves under condition ωτ 1 (ω - angular fre-quency, τ - thermal relaxation time) are [28]

AL = E0ω2 (DL3 ) τL2ρV 3L

, (9)

AS = E0ω2 (DS3 ) τS2ρV 3S

, (10)where E0 is the thermal energy density, and where VLand VS are the longitudinal and shear wave velocities,respectively.

The rate at which thermal energy is interchanged betweenvarious modes is called thermal relaxation time. The netattenuation rate is governed by the rate at which energyis removed by the transverse waves to the high-frequencythermal waves. Since half of the built-up low frequencyphonons are shear and half longitudinal, and since onlyshear phonons can convert their energy to high-frequencythermal waves, the relaxation time for longitudinal waves(τL) is twice the thermal relaxation time (τth), and relax-ation time for shear waves (τS) is equal to thermal relax-ation time [28]. The expression for τth is:τth = 12τL = τS = 3k

CVV 2D, (11)

where k is thermal conductivity and CV is the specificheat per unit volume. The acoustic coupling constant (DLor DS) can be evaluated with the help of the followingexpression [28]:D = 9〈(γji)2

〉 − 3〈γji 〉2ρCVTE0 , (12)

where γji are the Grüneisen parameters corresponding to aparticular direction of propagation and polarization. Theultrasonic attenuation caused by thermoelastic mechanismis calculated from the formula [28]ATh = ω2〈γji 〉2kT2ρV 5

L. (13)

3. Results and discussionThe higher order elastic constants of the chosen TmX ma-terials (X: S, Se, Te) have been calculated using only twobasic parameters: (a) nearest neighbor distance [16] and(b) hardness parameter [23]. The calculated elastic con-stants are presented in Table 1.The value of specific heat per unit volume Cv and energydensity E0 have been obtained by TD/T tables in AIPHandbook [29]. The thermal conductivity (k) of TmX hasbeen taken from literature [16]. The ultrasonic wave ve-locities have been calculated using Equations (4)-(7) andare presented in Table 2. The thermal relaxation time (τth)have been calculated using Eq. (11) and is also presentedin Table 2.The acoustic coupling constant (D) is a function of theGrüneisen parameter, specific heat per unit volume, en-ergy density, and absolute temperature. Grüneisen pa-rameters follow from second- and third-order elastic con-stants for rock salt type TmX [28]. Acoustic coupling con-stants for longitudinal (DL) and shear (DS) waves are cal-culated using Eq. (12). The ratio of DL and DS are shownin Figs. 1-4.

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Figure 1. DL/DS vs temp. along 〈100〉 direction.

Figure 2. DL/DS1 vs temp. along 〈110〉F direction.

Figure 3. DL/DS2 vs temp. along 〈110〉# direction.

Table 1. Second- and third-order elastic constants (SOEC andTOEC) in the unit of 1010 Nm−2 of thulium monochalco-genides in the temperature range 100-300 K.

Material Temp. [K] C11 C12 C44TmS 100 5.748 1.693 1.785200 5.914 1.609 1.792300 6.101 1.525 1.799TmSe 100 4.801 1.351 1.732200 4.964 1.277 1.438300 5.135 1.202 1.444TmTe 100 4.155 1.042 1.116200 4.308 0.973 1.121300 4.467 0.904 1.125Material Temp.[K] C111 C112 C123 C144 C166 C456TmS 100 -90.789 -6.949 2.449 2.945 -7.298 2.923200 -91.457 -6.644 1.975 2.967 -7.324 2.923300 -92.331 -6.335 1.501 2.988 -7.354 2.923TmSe 100 -76.680 -5.538 1.933 2.380 -5.844 2.362200 -77.434 -5.259 1.504 2.399 -5.869 2.362300 -78.278 -4.980 1.073 2.417 -5.895 2.362TmTe 100 -67.945 -4.243 1.455 1.886 -4.534 1.871200 -68.703 -3.974 1.038 1.901 -4.554 1.871300 -69.521 -3.706 0.621 1.916 -4.575 1.871

Figure 4. DL/DS vs temp. along 〈111〉 direction.

Finally ultrasonic attenuation due to phonon-phononinteraction and thermoelastic relaxation mechanismsare computed using Eqs. (9), (10), and (13) andare listed in Table 3. The total attenuation[(A/f2)L + (A/f2)S + (A/f22)Th] are also shown inFigs. 5-8. The validation of the present work has be madeby the comparison with available theoretical and experi-mental results. It is clear from the Table 1 that, out ofnine elastic constants, four (i.e., C12, C111, C123, and C166)are decreasing with temperature while others (i.e., C11,

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Table 2. Longitudinal and shear velocities (Vl and Vs) in the unitof 103 ms−1, Debye average velocity (VD) in the unit of103 ms−1 and thermal relaxation time (τth) in the unit of sec10−12 sec of thulium monochalcogenides in the temperaturerange 100-300 K.

Material TmS TmSeTemp. [K] 100 200 300 100 200 300〈100〉 Vl 2.611 2.650 2.693 2.328 2.368 2.409♣ Vs 1.455 1.459 1.463 1.271 1.274 1.277

VD 1.601 1.606 1.612 1.401 1.406 1.411τth 2.328 4.635 7.487 0.474 0.986 1.775

〈110〉 Vl 2.556 2.568 2.583 2.256 2.269 2.283F Vs1 1.455 1.459 1.463 1.271 1.274 1.278# Vs2 1.551 1.599 1.649 1.395 1.443 1.491

VD 1.643 1.666 1.689 1.454 1.475 1.495τth 2.211 4.310 6.822 0.440 0.895 1.580

〈111〉 Vl 2.537 2.540 2.545 2.231 2.235 2.240$ Vs 1.520 1.554 1.590 1.355 1.389 1.423VD 1.661 1.695 1.730 1.480 1.513 1.546τth 2.162 4.164 6.501 0.425 0.851 1.478Material TmTeTemp.[K] 100 200 300

〈100〉 Vl 2.161 2.202 2.244♣ Vs 1.120 1.122 1.126

VD 1.239 1.243 1.249τth 19.144 43.204 81.013

〈110〉 Vl 2.043 2.057 2.073F Vs1 1.120 1.123 1.126# Vs2 1.322 1.370 1.417

VD 1.319 1.337 1.355τth 16.886 37.345 68.767

〈111〉 Vl 2.002 2.007 2.013$ Vs 1.259 1.293 1.327VD 1.369 1.420 1.436τth 1.567 3.396 6.126

♣ - shear wave polarized along 〈100〉,F - shear wave polarized along 〈001〉,# - shear wave polarized along 〈110〉,$ - shear wave polarized along 〈110〉.

C44, C112, and C144) increase with temperature and whilethe value C456 remains unchanged. The increase/decreasein stiffness constants is due to the increase/decrease inatomic interaction with temperature. If inter-atomic dis-tance increases/decreases with temperature, then inter-action potential decreases/increases, a result that causesthe decrease/increase in stiffness constants. This type ofbehavior has been found already in other NaCl-type ma-terials such as lanthanum and praseodymium monochalco-genides [11, 14]. The calculated bulk moduli of TmS, TmSe

Figure 5. Total atten vs temp. along 〈100〉 for TmS and TmSe.


and TmTe are 31 GPa, 25 GPa, and 21 GPa, respectively.The experimental bulk moduli for the same materials are81 GPa, 25 GPa, and 30 GPa, respectively [30–32]. A com-parison of bulk moduli shows justification for the calcu-lated higher-order elastic constants. The negative valuesof C111, C112, and C166 be with the literature [11, 14].The listed data of ultrasonic velocities in Table 2 indicatethat, due to increase in second-order elastic constants,the velocities increase with temperature in TmX materials202



Figure 8. Total atten vs temp. for TmTe.

along each crystallographic direction.It can be seen from Table 2 that the order of τth forTmTe, TmS, and TmSe are 10−11 sec, 10−12 sec, and10−12-10−13 sec, respectively. The order of τth for di-electrics [33], semiconductors [12, 34], metals [35], andintermetallics [11, 14] are 10−10-10−11 sec, 10−11 sec,10−12 sec, and 10−12-10−13 sec, respectively. It is veryclear that TmS, TmSe, and TmTe have metallic, intermet-tallic, and semiconducting behavior.The variation of DL/DS with temperature (Figs. 1-4) indi-cates that the acoustic coupling constant decreases with

Table 3. Ultrasonic attenuation[ (A/f2)L, (A/f2)S and

(A/f2)Th ] on

the order of 10−15 Nps2m−1 of thulium monochalcogenidesin the temperature range 100-300 K.

Material TmS TmSeTemp.[K] 100 200 300 100 200 300〈100〉 (

A/f2)L 0.198 1.100 2.570 0.049 0.251 0.669♣

(A/f2)S 0.043 0.256 0.646 0.011 0.060 0.175(A/f2)Th 0.016 0.060 0.122 0.004 0.013 0.029

〈110〉 (A/f2)L 0.270 1.486 3.386 0.066 0.334 0.873

F(A/f2)S1 0.159 0.792 1.699 0.034 0.155 0.383# (A/f2)S2 0.695 3.587 7.923 0.164 0.781 1.967(A/f2)Th 0.037 0.133 0.266 0.008 0.030 0.065

〈111〉 (A/f2)L 0.176 0.972 2.231 0.043 0.219 0.574$ (A/f2)S 0.487 2.549 5.696 0.117 0.562 1.428(A/f2)Th 0.016 0.059 0.124 0.004 0.014 0.031Material TmTeTemp. [K] 100 200 300〈100〉 (A/f2)L 2.265 11.639 31.321♣

(A/f2)S 0.546 3.059 9.012(A/f2)Th 0.143 0.546 1.255

〈110〉 (A/f2)L 3.023 15.326 40.577F

(A/f2)S1 1.119 5.341 13.652# (A/f2)S2 6.851 32.961 84.126(A/f2)Th 0.345 1.290 2.921

〈111〉 (A/f2)L 1.960 9.932 26.240$ (A/f2)S 4.980 24.105 61.734(A/f2)Th 0.163 0.635 1.490

♣ - shear wave polarized along 〈100〉,F - shear wave polarized along 〈001〉,# - shear wave polarized along 〈110〉,$ - shear wave polarized along 〈110〉.

temperature for waves propagating along the 〈100〉, 〈110〉,and 〈111〉 directions and polarized along the 〈100〉, 〈110〉,and 〈110〉 directions, respectively, while, for waves prop-agating along the 〈110〉 direction with polarization along〈001〉, the DL/DS are found to increase with temperature.The D is measure of acoustic energy converted to thermalenergy under the relaxation process, thus the decrease inDL/DS with temperature shows that longitudinal loss de-creases with temperature or vice versa. A comparison forD has been made with the same structured material LiF(Table 4) [36]. The obtained results are in good agreementwith experimental results for LiF.It is obvious from Table 3 that the values of ultrasonicattenuation due to phonon-phonon interaction (

A/f2)Land (A/f2)S increases with temperature in the chosenmonochaclogenides, TmS, TmSe, and TmTe. The valuesof (A/f2)L are found greater than those of (A/f2)S alongthe 〈100〉 and 〈110〉 directions (polarized along the 〈001〉203


Table 4. Comparative results of acoustic coupling constants (D) atroom temperature.

Propagation Polarisation Dexp. [36] Dtheo. [36] Dtheo.direction LiF LiF [Present work]TmX〈100〉 Long. 38 38 16〈100〉 Shear 4.5 3 1.3〈110〉 Long. 17 15 20〈110〉 Shear 〈001〉 4.5 12 3.6〈110〉 Shear 〈110〉 35 18 24〈111〉 Long. 11 ∼=18 13〈111〉 Shear 〈110〉 21 16

direction), while the (A/f2)L values are found to be lessthan those of (A/f2)S along the 〈110〉 direction (polar-ized along the 〈110〉 direction) and the 〈111〉 direction(polarized along the 〈110〉 direction). Such properties ofattenuation due to phonon-phonon interaction were alsofound in metals [35], intermetallics [11, 14], semiconduc-tors [12, 37, 38], and dielectrics [36, 39]. Ultrasonic atten-uation due to thermoelastic loss (A/f2)Th is negligible incomparison to attenuation over frequency square due tophonon-phonon interaction in all the thulium monochalco-genides. From Equations 9 and 10, it seems that the ratioof AL and AS is AL/AS = 2DLV 3S

DSV 3L

. The ratio is 3.98, 3.82and 3.48 for TmS, TmSe, and TmTe, respectively, along the〈100〉 direction. The values of the ratio are in good agree-ment with the available experimental data 3.14 and 3.45for Ge and Si, respectively, along the 〈100〉 direction [28].At room temperature, (A/f2)L and (A/f2)S for pure ger-manium are found to be 3.44× 10−16 Nps2m−1 and 1.02×10−16 Nps2m−1 along the 〈100〉 direction [28], while thetotal attenuation [(A/f2)L + (A/f2)S + (A/f2)Th] for theother semiconducting materials, GaAs [40] and InSb [37],are 1.85× 10−16 Nps2m−1 (along the 〈110〉 direction) and1.598×10−16 Nps2m−1 (along the 〈111〉 direction), respec-tively. However, the experimental values for semiconduct-ing materials are mostly located lower than the obtainedresults for TmTe semiconducting material. Then again, thequantum of attenuation is found approximately.Figures 5-8 show that the total attenuation[A = (A/f2)total = (A/f2)L + (A/f2)S + (A/f22)Th]increases with temperature. The fit curve to totalattenuation implies that the total attenuation in thesematerials can be written as a function of temperature,A = ∑n=2

n=0AnT n. The value of An depends on specificheat per unit volume, energy density, thermal relaxationtime, thermal conductivity, elastic constants, and density.It is observed from Figures 5-8 that the values of totalattenuation are found lowest in TmSe and highest in

TmTe. Thus it may be predicted that intermetallic TmSebehaves in its purest form and is more ductile (as evincedby minimum attenuation) while semiconductor TmTe isleast ductile. Therefore, impurity will be least in TmSe.4. ConclusionsOn the basis of the above discussion, the following con-clusions can be drawn:

1. The theory for evaluation of higher-order elasticconstants is also valid for monochalcogenides.2. The order of τth for TmTe, TmS, and TmSe are10−11 sec, 10−12 sec, and 10−12-10−13 sec, respec-tively. This reconciles with the finding that TmS,TmSe, and TmTe have metallic, intermettallic, andsemiconducting behavior.3. Total attenuation in these materials is given by theexpression A = ∑n=2

n=0AnT n. The value of An de-pends on specific heat per unit volume, energy den-sity, thermal relaxation time, thermal conductivity,elastic constants, and density.4. The lowest attenuation is found in TmSe. It followsthat this material has excellent purity and ductilityin comparison to TmS and TmTe.5. The present study confirms the validity of theMason-Bateman approach for the evaluation of ul-trasonic attenuation in thulium monochalcogenidesalong the 〈100〉, 〈110〉, and 〈111〉 directions.6. On the basis of ultrasonic attenuation, the clas-sification of materials can be made, i.e., it is ei-ther metallic, intermediate valence, semiconductor,or dielectrics.

The results of the present investigation may be theoreti-cally related to many other physical properties of solids,such as specific heat, infrared lattice resonance frequency,X-ray scattering, and various transport properties. Theseresults can be used for further investigations and indus-trial purposes.AcknowledgementsWith great pleasure, we express our sincere gratitude toProf. B. P. Singh, Senior Director, and Prof. Prem Prakash,Director, Amity School of Engineering and Technology,New Delhi, for their inspiration and encouragement inbringing out the work.

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ultrasonic wave propagation in rare-earth monochalcogenides

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