Cryogenics 50 (2010) 476–479

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Cryogenics

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Technical Note

Effect of electrical resistivity on ultrasonic attenuation in NpTe

Devraj Singh *, Pramod K. Yadawa, Saurabh K. SahuDepartment of Applied Physics, AMITY School of Engineering and Technology1, Bijwasan, New Delhi 110 061, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 October 2009Received in revised form 12 November 2009Accepted 24 April 2010

Keywords:A. SemimetallicC. Ultrasonic properties

0011-2275/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.cryogenics.2010.04.005

* Corresponding author. Tel.: +91 11 2806 2106/14E-mail address: dsingh1@aset.amity.edu (D. Singh

1 Affiliated to G.G.S.I.P. University, Delhi.

Ultrasonic attenuation due to electron–phonon interaction (EPI) has been computed in semimetallic sin-gle crystal neptunium telluride (NpTe) in low temperatures 5–80 K. For the same evaluation, we havealso evaluated ultrasonic velocity, electronic viscosity and second order elastic constants (SOEC). TheSOEC of NpTe have been evaluated using the Born model of ionic solid. The behaviour of ultrasonic atten-uation is quite similar to its inverse resistivity. The ultrasonic attenuation due to EPI is most significant at40 K. Computed results of ultrasonic parameter have been compared and discussed.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Ultrasonic non-destructive testing is a resourceful techniquethat can be appropriate for study of different types of materials.This is useful for depiction of microstructures, appraisal of defectsand assessment properties of objects. By virtue of this, ultrasonicmeasurements during production and heat action allow ensuringthe lack of unacceptable discontinuities and the company of a par-ticular microstructure with preferred properties. The interaction ofultrasound with microstructure is significant for many materialproblems. Attenuation and backscattering reduce the detectabilityof flaws, especially in materials with coarse grains or complexmicrostructures such as semimetallics. Further, quantification ofthese wave propagation properties provide information about themicrostructure that can be used in materials explanation studies,e.g. non-destructive determination of grain size. Wave propagationvelocity is another key factor in ultrasonic characterization, whichin combination with attenuation can provide imperative tools inunderstanding, the scrutinizing ability of materials; for example,it can provide information about crystallographic texture. Ultra-sonic velocity is directly related to elastic constants, i.e. V =

p(C/

q), where C is the elastic constant and q, the density of that partic-ular material. The elastic constants, in particular, provide valuableinformation on the stability and stiffness of materials [1,2].

The semimetallic NpTe is quite interesting as observed abnor-mal physical properties, has attracted much attention in recentyears [3–8], because it is a typical low carrier, strongly correlatedsystem with simple rock salt type structure. Although a numberof ultrasonic studies have been made in metals at low temperature

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region [9–11]. To the knowledge of authors, such theoretical ultra-sonic studies in NpTe have not been reported in literature. Yet inorder to study the behaviour of ultrasonic attenuation due to EPIwith electrical resistivity, NpTe has been taken in presentinvestigation.

2. Theory

Theory of present investigation is divided into two parts:

2.1. Second order elastic constants

The temperature dependent SOEC have been evaluated follow-ing Brügger’s definition at 0 K [12] and using Born model [13].The SOEC at particular temperature are obtained using Mori andHiki approach [14].

2.1.1. Expressions of the SOEC are given below

C11 ¼32

e2

r40

Sð2Þ5 þ1

br0

1r0þ 1

b

� �/ðr0Þ þ

2br0

ffiffiffi2p

2r0þ 1

b

!/

ffiffiffi2p

r0

� �þ f ð1;1ÞG2

1 þ f ð2ÞG2 ð1Þ

C12 ¼32

e2

r40

Sð1;1Þ5 þ 2br0

ffiffiffi2p

2r0þ 1

b

!/

ffiffiffi2p

r0

� �þ f ð1;1ÞG2

1 þ f ð2ÞG1;1 ð2Þ

C44 ¼32

e2

r40

Sð1;1Þ5 þ 2br0

ffiffiffi2p

2r0þ 1

b

!/

ffiffiffi2p

r0

� �þ f ð2ÞG1;1 ð3Þ

where r0 is the short range parameter, b, the hardness parameter, /(r0), the Born–Mayer potential and /(r0) = A exp(�r0/b) and/ð

ffiffiffi2p

r0Þ ¼ A expð�ffiffiffi2p

r0=bÞ, A is the strength parameter is given as

D. Singh et al. / Cryogenics 50 (2010) 476–479 477

A ¼ 3be2

r20

Sð1Þ31

6 expð�q0Þ þ 12ffiffiffi2p

expð�ffiffiffi2p

q0Þð4Þ

with q0 = r0/b

2.1.2. Expressions of f(n) and Gn are

f ð2Þ ¼ 12r0

�hx0

4coth x and f ð1;1Þ ¼ 1

2r0

�hx0

48�hx0

2kTsinh2xþ coth x

� �

3

3.2

3.4

3.6

/f2 ) lo

ng [

in 1

0-19 N

ps2 /m

]

where x ¼ �hx0

kTand x2

0

¼ 1Mþþ 1

M�

� �1

br0

r0

b� 2

� �/ðr0Þ þ 2

r0

b�

ffiffiffi2p� �

/ðffiffiffi2p

r0Þh i

where M+ and M� are the masses of ions and T is the particulartemperature

G1 ¼ 2fð2þ 2q0 � q20Þ/ðr0Þ þ 2ð

ffiffiffi2pþ 2q0 �

ffiffiffi2p

q20Þ/ð

ffiffiffi2p

r0ÞgHG2 ¼ 2fð�6� 6q0 � q2

0 þ q30Þ/ðr0Þ þ ð�3

ffiffiffi2p� 6q0

�ffiffiffi2p

q20 þ 2q3

0Þ/ðffiffiffi2p

r0ÞgH and

G1;1 ¼ fð�3ffiffiffi2p� 6q0 �

ffiffiffi2p

q20 þ 2q3

0Þ/ðffiffiffi2p

r0ÞgH

where H ¼ 1ðq0�2Þ/ðr0Þþ2ðq0�

ffiffi2pÞ/ðffiffi2p

r0Þ

The values of lattice sum are

Sð2Þ5 ¼ �1:04622 and Sð1;1Þ5 ¼ 0:23185

2.8

0 10 20 30 40 50 60 70 80

Temperature [in K]

(

Fig. 1. Temperature versus calculated (A/f2)long of NpTe.

0 10 20 30 40 50 60 70 80

1.6

1.7

1.8

1.9

2

2.1

Temperature [in K]

shea

r [in

10

-18

Np

s2/m

](

/f2 )

Fig. 2. Temperature versus calculated (A/f2) of NpTe.

2.2. Ultrasonic attenuation due to EPI

A review [15] of the studies of ultrasonic attenuation in solids ina wide temperature range shows that the ultrasonic attenuationvaries from material to material in different orientations and alsoaccording to the temperature regions. At room temperature(ffi300 K) and above, the phonon–phonon interaction and thermo-elastic mechanisms are the playing their vital role to study ultra-sonic attenuation in almost all type of materials like metallic,dielectric, semimetallics, intermetallics, semiconductor, alloys,etc. In the low temperature region the electron mean free path in-creases and becomes same as the mean free path of acoustical pho-nons at high frequency. Hence probability of interaction [9]between conducting electron and phonons increases as explainedby Pippard [16]. When an ultrasonic wave is passed through a so-lid, a coupling between conduction electrons and acoustical pho-nons occurs below 100 K. The concept is that, in the normal statea lattice vibration can communicate energy to the electron gas.

Table 1Calculated SOEC (in the unit of 1010 N/m2), ultrasonic velocities VL and VS (in the orderof 103 m/s), density q (in the unit of 103 kg/m3) and electronic viscosity ge (in the unitof 10�6 kg/m s) of NpTe in the temperature range 5–80 K.

Temperature(in K)

C11 C12 C44 VL VS q ge

5 3.972 0.940 0.956 1.944 0.954 10.501 1.04810 3.972 0.939 0.956 1.970 0.966 10.231 1.00420 3.972 0.938 0.956 1.973 0.968 10.202 0.97730 3.976 0.932 0.956 1.977 0.969 10.106 0.90640 3.985 0.926 0.956 1.980 0.970 10.016 0.86650 3.995 0.919 0.957 1.997 0.977 10.015 0.87760 4.007 0.912 0.957 2.002 0.978 9.991 0.89670 4.020 0.905 0.957 2.006 0.979 9.985 0.91780 4.034 0.898 0.958 2.015 0.981 9.935 0.859

The attenuation caused by energy loss due to compressionaland shear viscosities of lattice [15] at particular temperature is gi-ven by:

A=f 2long ¼

2p2

qV3L

43ge þ v

� �ð5Þ

A=f 2shear ¼

2p2

qV3S

ge ð6Þ

where electron viscosity

ge ¼9� 109�h2ð3p2NÞ2=3

5e2Rð7Þ

the velocity of longitudinal wave Vlong ¼ffiffiffiffiffiffiC11q

q, the velocity of shear

wave Vshear ¼ffiffiffiffiffiffiC44q

q, f, the frequency of the wave, v, the compres-

sional viscosity, q, the density of material, R, the electrical resistiv-ity of the material, N, the electronic density and �h, the Planck’sconstants divided by 2p.

shear

0.13

0.14

0.15

0.16

0.17

0 10 20 30 40 50 60 70 80Temperature [in K]

(1/R

) [i

n 10

2m

ho/m

]

Fig. 3. Temperature versus (1/R)of NpTe.

Table 2Comparative value of second order elastic constants at 80 K (1010 N/m2).

SOEC NpTe GdP [19] GdAs [19] GdSb [19] GdBi [19] CeP [20] CeSb [20]

C11 4.034 5.180 4.876 4.102 4.179 4.515 3.800C12 0.898 1.337 1.184 0.799 0.856 1.175 0.809C44 0.958 1.409 1.252 0.863 0.918 1.238 0.876

Table 3Comparative value of ultrasonic attenuation due to EPI at 80 K (10�15 Np s2/m).

Ultrasonic attenuation NpTe GdP [19] GdAs [19] GdSb [19] GdBi [19] CeP [20] CeSb [20]

(A/f2)long 0.00031 0.00699 0.00942 0.02278 0.03080 0.015 0.0007(A/f2)shear 0.00184 0.03695 0.05431 0.17719 0.22416 0.005 0.007

478 D. Singh et al. / Cryogenics 50 (2010) 476–479

3. Results and discussion

The calculated SOEC and ultrasonic velocity for longitudinal andshear waves, density and electronic viscosity of NpTe are presentedin Table 1. The temperature variation of ultrasonic attenuation dueto EPI for longitudinal and shear waves and reciprocal of electricalresistivity (1/R) are illustrated in Figs. 1–3, respectively.

It is evident from the Table 1 that the calculated values of C11,C12 and C44 are in good agreement with experimental valuesC11 = 4.39 � 1010 N/m2, C12 = 0.80 � 1010 N/m2 and C44 = 1.00 �1010 N/m2 (all values are at room temperature) [5]. TheC11 > C44 > C12 is found in this survey, this provision is theoreticallyand experimentally established for NaCl-type crystals by Kukin(C11 = 4.94 � 1010 N/m2, C12 = 1.27 � 1010 N/m2 and C44 = 1.28 �1010 N/m2) [17]. The values C12 and C44 in this temperature regimeare very close to each other, which confirms the soundness of Cau-chy’s relation C0

12 ¼ C044 at 0 K. The values of SOEC of NpTe are in

good conformity with the same values of supplementary semimet-allics GdP, GdAs, GdSb, GdBi, CeP and CeSb in the same tempera-ture range [18–20] as given in Table 2. The temperature variationin NpTe is found to be bare minimum. The computed ultrasonicvelocities for longitudinal and shear waves boost with temperatureas observed in other metals [9,21] and semimetallics [18–20]. Theultrasonic velocities for longitudinal waves are found greater thanthat of shear waves because the value of C11 is greater than C44. It isdepicted in Table 1 that the difference D1 = C12 � C44 of NpTe in-creases non-linearly with increasing temperature. This type ofbehaviour is also found in NaCl [17].

The evaluated values of ultrasonic attenuation at different tem-peratures are visualized in Figs. 1 and 2. The comparison of atten-uation of NpTe due EPI is not available directly for the same crystalin this temperature regime. So we compare our results with metalsand other semimetallics, and compared results of semimetallicsare presented in Table 3. It can be seen from the Figs. 1 and 2 thatthe attenuation decreases up to 40 K temperature and then in-creases for further temperatures. The ultrasonic attenuation dueto EPI is most significant at 40 K in the present material. This nat-ure of ultrasonic absorption of NpTe is fairly different from othermetals [9,21] and semimetallics [18–20]. This type of behaviourof NpTe has been obtained due to its resistivity values at differenttemperature. The values of ultrasonic attenuation of shear wavesare approximately 10 times greater than that of attenuation of lon-gitudinal waves as C11 > C44. Hence ultrasonic attenuation for shearwaves (A/f2)shear is prime over total attenuation [(A/f2)Total =(A/f2)long + (A/f2)shear]. It is obvious from the Figs. 1–3 and Eqs.(5)–(7) that the ultrasonic attenuation is directly proportional toelectronic viscosity, and electronic viscosity is directly propor-tional to inverse electrical resistivity or in other words, the temper-ature dependent ultrasonic attenuation due to EPI for longitudinaland shear waves is analogous to reciprocal of resistivity (1/R) as

probable in other materials [9,18–20]. The order of attenuation inNpTe is established less than that of other materials like GdX(X = P,As, Sb and Bi) [18,19], CeP and CeSb [20] as shown in Table 3.So NpTe is more proper for engineering applications in comparisonto other materials [9,18–20]. We find a relation (Ashear/Along) = (3/4)(VL/VS)3 ffi 6.0 from Eqs. (5) and (6) to evaluate coefficients of ultra-sonic attenuation and velocities [9], which holds good in presentinvestigation. Hence present theoretical investigation is acceptablefor ultrasonic attenuation due to EPI in semimetallic NpTe in thistemperature range.

4. Conclusions

The tendency of our computations of SOEC bears a similitude tothe experimental values. The inconsistency between experimentand theoretical explanation of SOEC of former works in low tem-perature region is considerably reduced by the present approach.The ultrasonic velocities for longitudinal and shear waves enlargewith elastic constants. The electrical resistivity is the core domi-nating factor for ultrasonic attenuation of NpTe. The activities ofultrasonic attenuation are quite different from other metals andsemimetallics. It can be perceived from the Figs. 1 and 2 that EPImechanism is the most noteworthy at 40 K. Hence 40 K tempera-ture is the key temperature to characterize the NpTe. Hence ourtheoretical approach appears to be valid for ultrasonic character-ization of NpTe semimetallic material having low carrier concen-tration in this temperature range.

The ultrasonic parameters may be correlated with the opticaland thermoelectric parameters over low temperature region. Thesecomputed results with other well known physical properties ofNpTe, may expand future prospects for its applications.

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