Anisotropic assessment of ultrasonic wave velocity and thermal conductivity in ErX (X:N, As)

Transcription

1 Indian Journal of Pure & Applied Physics Vol. 54, January 216, pp Anisotropic assessment of ultrasonic wave velocity and thermal conductivity in ErX (X:N, As) Vyoma Bhalla 1, 2* & Devraj Singh 2 1 Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida , India 2 Department of Applied Physics, Amity School of Engineering and Technology, Bijwasan, New Delhi 11 61, India * vyoma.bhalla@student.amity.edu Received 31 July 215; revised 23 September 215; accepted 26 November 215 The elastic and ultrasonic properties of erbium nitride (ErN) and erbium arsenide (ErAs) have been investigated in the temperature range -3K. The second order elastic constants (SOECs) have been obtained using Coulomb and Born-Mayer potentials up to second nearest neighbours. SOECs have been used to evaluate various mechanical and thermal parameters which provide knowledge about the future performance of ErX. Mechanical properties such as Young's modulus, bulk modulus, shear modulus, Zener anisotropy factor, Poisson's ratio and toughness to fracture ratio (G/B) have also been calculated. The chosen materials are found to be brittle and anisotropic in nature. The ultrasonic wave velocities for longitudinal and shear modes of propagation along <1>, <11> and <111> crystallographic directions have also been computed at room temperature using evaluated values of SOECs and density of the chosen materials. These properties have been visualized using MTEX software for ErAs. The ultrasonic wave velocity computed along different orientations is an important physical quantity in studying the thermal properties such as Debye temperature and anisotropic thermal conductivity. These properties play a significant role in quality control in material producing industries. The obtained results are discussed in correlation with mechanical and thermophysical properties of the materials. Keywords: Elastic constants, Ultrasonic velocity, Anisotropic thermal conductivity, MTEX 1 Introduction Material characterization using ultrasonic nondestructive testing is a versatile tool for evaluating the microstructural properties and classifying the different types of materials as metallic, semi-metallic, insulator or intermetallic 1. Erbium monopnictides are the semimetallic, rare-earth pnictides used in photo-optic applications. A lot of attention has been given to the growth of single crystals 2, 3 of ErX (X: N, As). The rareearth monopnictides i.e. group V-compounds have strong magnetic and electrical properties due to the presence of partially filled f-electron orbitals of rare earth atom 4. These properties are useful for the application in electronic industry, spintronic devices and for scientific research and experimentation. There are a number of physical studies on ErX materials using different approaches. The first-principle calculations for structural, magnetic and electronic properties of ErN were performed by Dergal et al 5. The analogous computation to understand the electronic structure of R-V (R-Gd, Er; V- N, P, As) compounds were also carried out by Petukhov et al 6. and Komesu et al 7. The electronic, magnetic and transport properties have been studied for rare-earth monopnictides by Duan et al 8. Pandit et al 9. studied the half metallic character of ErN using LDA approximation. The erbium monopnictides i.e. ErX (X: N, As) are chosen for the ultrasonic study of elastic and thermal properties. In the present study, the second order elastic constants (SOECs) have been evaluated using Coulomb and Born-Mayer potential model in the temperature range -3 K. These evaluated SOECs plays a significant role in determining various material properties. Using the calculated values of SOECs the wave velocity for longitudinal and shear modes of propagation along <1>, <11> and <111> crystallographic directions have also been computed. The ultrasonic wave propagation relates the microstructure and physical properties of materials and thus, the inspection of physical properties and ultrasonic velocities is performed using the open source software 1, 11 MTEX To the best of our knowledge, there are no theoretical studies regarding the thermal behaviour of these compounds in the literature. 2 Theoretical Approach 2.1 Elastic Stiffness Constants The Coulomb and Born-Mayer potentials has been used to evaluate SOECs of single crystalline ErX which is given below 13 :

2 BHALLA & SINGH: ULTRASONIC WAVE VELOCITY AND THERMAL CONDUCTIVITY IN ErX 41 R C B (1) where is the long-range electrostatic/coulomb potential and is the short-range repulsive/born- Mayer potential: C B 2 e r C and B Aexp r b (2) where e is the electronic charge, ± sign apply to like and unlike ions, respectively, r is the nearestneighbour distance, b is the hardness parameter and A is the strength parameter 14. Following Brugger s 15 definition of elastic constants at absolute zero, the SOECs at particular temperature are obtained by adding the vibrational energy to static elastic constants i.e. C ( T) C C (3) vib IJ IJ IJ where superscript denotes static elastic constant at K and superscript vib denotes vibrational part of elastic constant at required temperature. The expressions for and the corresponding vibrational part are given in literature 16. The elastic constant plays an important role for the characterization of the materials in the field of external forces. C IJ 2.2 Mechanical properties Elastic constants are efficient tools to evaluate the mechanical properties of solids. For a cubic material to be mechanically stable the Born criterion must be satisfied. The condition for Born stability is given as: B ( C 2 C ) / 3 ; ( C11 C12) / 3 ; C (4) where B is the bulk modulus related to the elastic constants. Zener anisotropy factor (A) is a dimensionless ratio which provides a convenient criterion for identifying materials that are elastically anisotropic 17 : 2C A 44 C C (5) The range of the bounds is highly dependent upon this parameter. As A approaches unity the crystal becomes isotropic and the bounds vanish. The bounds for anisotropic materials are generally given by Voigt s and Reuss s approximations 18. The average elastic property of the sample can be determined by averaging the elastic constant in the limits of these bounds. This is given as: G V ( C11 C12 3 C44 ) ; 5 G R 5 ( C C ) C C 3( C C ) (6) where subscripts V and R refer to the upper bound follows from Voigt s assumption that the total strain is uniform within the polycrystal while the lower bound follows from Reuss original assumption that the stress is uniform, respectively. The average of two as proposed by Voigt-Reuss-Hill is thus defined by shear modulus 19 : ( GV GR) G (7) 2 It helps further in estimating the Young s modulus (Y) and Poisson s ratio (ν) of the specimen and is quite helpful in the characterization of the chosen materials. Y 9GB 3B 2G ; G 3 B 6B 2G (8) 2.3 Directional dependence of Tensorial Properties The wave velocity propagating along three symmetry directions <1>, <11> and <111> provides information about the crystallographic texture by taking advantage of the effects of directionality (anisotropy) that exists in the material. The longitudinal and shear wave velocities along <1>, <11> and <111> crystallographic directions are calculated using the Christoffel equation 2 : 2 ( Cijkld jdl V ik ) pk (9) where C ijkl is SOEC; d is the propagation unit vector; V is the ultrasonic wave velocity; ρ is the density; δik is the Kronecker delta and p is the polarization unit vector. On solving Eq. (9), we get the eigen values which are related to ultrasonic velocities V 1, V 2 and V 3 denoted by V P, V S1 and V S2, respectively. The expressions to obtain ultrasonic velocities along <1>, <11> and <111> crystallographic directions

3 42 INDIAN J PURE & APPL PHYS, VOL 54, JANUARY 216 are given in our previous paper 21. Here, P and S correspond to the longitudinal and shear modes of propagation, respectively. 2.4 Directional Anisotropic Thermal Conductivity The Debye average velocity, V D is computed using the values of V P, V S1 and V S2 with a simple formula given by 22 : 1/ VD 3 3 ; 3 VP VS1 along <1> and <111> directions 1/ ; 3 VP VS1 VS 2 along <11> direction...(1) V D is used to find out the Debye temperature 23 (θ D). θ D is used to characterize the excitation of phonons and to describe the various lattice thermal phenomena. The thermal conductivity is closely related to ultrasonic wave velocity of the materials associated with the lattice vibration. Consequently, the minimum thermal conductivity of the lattice along different crystal orientations 12 is given by: kb 2/3 min q ( VP VS1 VS 2) (11) 2.48 where is the Boltzmann constant and q is the atomic number per unit volume. k B 3 Results and Discussion The present calculations are done using expressions for the overall average elastic constants of a polycrystalline in terms of its single crystal components. 3.1 Crystal Elastic and Mechanical Properties The SOECs are computed in the temperature range -3 K using two basic parameters viz. lattice parameter 9, a= 4.83Å and 5.73 Å for ErN and ErAs, respectively and hardness parameter 16, b=.313å for both the materials. The SOECs are computed in the temperature range -3 K using Eqs. (1)-(3) and are shown in Table 1. It is clear from Table 1 that the values of C 11 and C 44 increase linearly with increase of temperature whereas C 12 decreases with increase of temperature. The variations of elastic constants with temperature follow a systematic trend identical as observed in ytterbium monopnictides 24. The computed elastic constants provide the information about the stability and stiffness of materials, especially the bulk and shear moduli. The bulk modulus provides a comparison of interatomic bonding strength among the materials and is calculated using Eq. (4). The mechanical properties computed at room temperature using the SOECs are listed in Table 1. It is clear from Table 1 that the bulk modulus of ErN is higher than that of ErAs. Thus, we may say that ErN is harder than ErAs. Further, we compute shear modulus, G using Voigt 25, Reuss 26 and Hill 19 models as in Eqs. (6)-(7). It is correlated to the material hardness in a consistent manner and is found largest for ErN. So it is concluded that ErN possesses more resistance for the deformation than ErAs. Table 1 depicts that the value of bulk modulus for ErN obtained using above model is comparable to the value obtained by Natali et al 27. using first principle calculations. Further, the Born stability criterion of cubic crystals given by Eq. (4) is satisfied by these materials. According to Pugh s criterion 28, when the toughness/fracture ratio, (G/B) is less than.5, the materials exhibit toughness and for G/B greater than.5, the materials exhibit brittleness. It is observed Table 1 Elastic constants, Cij in GPa; bulk Modulus (B), shear modulus (G), Young s modulus (Y) in GPa, toughness/fracture ratio (G/B), anisotropic ratio (A) and Poisson s ratio (ν), density (ρ) in g/cc of ErX in the temperature range -3K. Materials Temp (K) C11 C12 C44 B G Y G/B ν A ρ ErN a 51.2 b.293 b ErAs a [Ref. 27], b MTEX values

4 BHALLA & SINGH: ULTRASONIC WAVE VELOCITY AND THERMAL CONDUCTIVITY IN ErX 43 from Table 1 that the results indicate that both the materials are brittle in nature. The Young s modulus (Y) and Poisson ratio, (ν) is calculated using Eq. (8) for a polycrystalline system. The Young s modulus, Y provides a measure of stiffness, i.e., for large value of Y, the material is stiffer. So ErN is stiffer than ErAs in the present investigation. The value of Poisson ratio, ν is typically.33 for ductile metallic materials and is smaller for brittle covalent materials. Table 1 presents the value of υ is approximately.24 which indicates that chosen monopnictides are brittle covalent materials 29, 3. So it is sure that the chosen materials are of brittle nature. The value of A provides the information about anisotropic material behaviour which is calculated by Eq. (5). For isotropic crystals A=1 and the values greater or smaller than unity measure the degree of elastic anisotropy. From Table 1, it is found that the materials are to be highly anisotropic, which motivated us to further evaluate the acoustic and thermal properties along <1>, <11> and <111> crystallographic directions. The anisotropy of elastic properties is the condition which is responsible for the formation of microcracks and lattice distortion in the materials. 3.2 Crystal Structure The crystal symmetry (cs) is defined as cubic and specimen symmetry (ss) as triclinic. The variable of type symmetry stores all the information regarding crystal coordinate system. Using the elastic constants from Table 1, the 6x6 elastic stiffness matrix for cubic ErN is represented 31 as: M = [ ; ; ; ; ; ]; C 11= 51.9 GPa relates the compression stress and strain along X, Y, Z-axes while C 44= 14.2 GPa relates shear stress and strain in the same direction. C 12= 12.7 GPa relates the compression stress in one direction to the strain in another say X-and Y-directions. Similarly, the stiffness matrix for ErAs is constructed. 3.3 Ultrasonic Velocity and Thermal Conductivity The Christoffel tensor given in Eq. (9) serves as the basis for computation of elastic wave velocity along three crystallographic directions as well as the polarization directions. The propagation velocities for the longitudinal wave and shear acoustic waves of ErX (X: N, As) along the three crystallographic directions <1>, <11> and <111> are calculated using the expressions for the computation of wave velocities 32. The computed values of ultrasonic velocities along different crystallographic directions are given in Table 2. From Table 2, it is observed that the highest velocity is of longitudinal waves along <111> direction for ErN and along <1> direction for ErAs. The lowest shear velocity is along <11> polarized along <1 1 > for ErN and polarized along <1> for ErAs.The Debye average velocity, V D is computed using Eq. (1) which is further used to calculate the Debye temperature. From Table 3, V D is found the highest along <1> for ErN in contrast to ErAs having highest V D along <111> direction. Therefore, the ultrasonic wave propagation will be more appropriate along <1> in case of ErN which is similar in nature to intermetallics 33. The Debye temperature, θ D is found to be the highest along <1> for ErN and along <111> for ErAs. θ D is,generally, used to distinguish the high- and lowtemperature regions of the solid. Thus, we have carried out our calculations of minimum thermal conductivity using Eq. (11) under high temperature conditions i.e. T>> θ D (at 3 K). In this region, all Table 2 Ultrasonic velocities of erbium monopnictides; VP for longitudinal waves and VS for shear waves at room temperature (all velocities are in 1 3 m/s) Materials Ultrasonic velocities along different directions (at room temperature) <1> <11> <111> VP VS1=VS2 *c VP VS1 *d VS2 *e VP VS1=VS2 *f ErN ErAs *c shear wave polarized along <1> direction *d shear wave polarized along <1> direction *e shear wave polarized along <1 1 >direction *f shear wave polarized along < 1 1> direction

5 44 INDIAN J PURE & APPL PHYS, VOL 54, JANUARY 216 Table 3 Debye average velocity, VD (in 1 3 m/s), Debye temperature, θd (in K), the minimum thermal conductivity under high temperature (at 3 K), min (in W/(m.K)) Materials Directions VD θ D ErN ErAs <1> <11> <111> <1> <11> <111> min the models will have energy of k BT where k B is the Boltzmann constant. One of the important present studies is to analyze the effect of crystal orientations on the lattice thermal conductivity using a simple model based on the velocity of ultrasonic waves associated with the lattice vibration 12. As shown in Table 3, the minimum thermal conductivity of the polycrystalline samples is anisotropic. The thermal conductivity is the highest for ErN and lowest for ErAs along <1> direction. Thus, ErAs is the ideal (a) (b) Fig. 1 (a) Young s Modulus (in GPa) and (b) Poisson s ratio of ErAs at room temperature (a) (b) Fig. 2 Wave velocities plot for ErAs along any direction on the hemispehere (a) direction dependent P-wave velocity, VP and (b) the speed difference between the S1 and S2 waves together with the fast shear-wave polarization (all in 1 3 m/s)

6 BHALLA & SINGH: ULTRASONIC WAVE VELOCITY AND THERMAL CONDUCTIVITY IN ErX 45 thermal barrier material as compared to ErN, and it may have great potential application in hightemperature conditions. 4 Graphic Visualization using MTEX In order to visualize the above computed quantities, we have used MTEX software 1, 34. The computed parameters are plotted on a projection of the unit hemisphere for ErAs (Figs 1 and 2). The graphical visualizations are based on single crystal elastic constants (here, SOECs) of ErN and ErAs. Figure 1 shows the mechanical properties of ErAs. Figure 2 shows the plot for wave speed, P and S with polarization normal to Y-axis. It helps in understanding the propagation of wave in anisotropic media. 5 Conclusions On the basis of the obtained results of present investigation, it is concluded that 1 The SOECs and elastic moduli are the highest for ErN which make them suitable for mechanical purpose. 2 The Born stability criteria are satisfied for these materials. So these materials are mechanically stable. 3 The analysis on the elastic and mechanical properties shows that ErN is harder than ErAs. The chosen materials are brittle in nature as toughness/fracture ratio is greater than.5. 4 The chosen materials have anisotropic nature. This results in lattice distortion of the material. 5 The ultrasonic velocity in ErN is greater than ErAs. Hence, there is more possibility for propagation of ultrasonic wave in ErN in comparison to ErAs. 6 The assessment of anisotropic behaviour of thermal conductivity is also being done. The minimum thermal conductivity, min for ErN is higher than that of ErAs. Thus, the conductive heat losses will be reduced in ErAs along <1> direction which makes it suitable for potential application in the field of semiconductors. The behaviour of ultrasonic velocity and thermal conductivity shows important characteristic feature of these materials for electronic industry and research and development. References 1 Singh D, Bhalla V, Kumar R & Tripathi S, Indian J Pure & Appl Phys, 53 (215) Dargis R, Smith R, Clark A, Arkun E & Lebby M, US Patent No.19 13/939, 721 (215). 3 Klenov D O, Zide J M, Zimmerman J D, Gossard A C & Stemmer S, App Phys Lett, 86 (25) Petit L, Tyer R, Szotek Z, Temmerman W M & Svane A, New J Phys, 12 (21) 2 pp. 5 Dergal S, Merad A E & Brahmi B N, Am J Mat Sci Tech, 2 (213) 4. 6 Petukhov A G, Lambrecht W R L & Segall B, Phys Rev B, 53 (1996) Komesu T, Jeong H K, Choi J, Borca C N, Dowben P A, Petukhov A G et al., Phys Rev B, 67 (23) 12pp. 8 Duan C G, Sabirianov R F, Mei W N, Dowben P A et al., J Phys: Cond Matt, 19 (27) 3pp. 9 Pandit P, Srivastava V, Rajagopalan M & Sanyal S P, Phys Rev B, 45 (21) Bachmann F, Hielscher R and Schaeben H, Solid State Phenom, 16 (21) Hielscher R & Schaeben H A, J App Cryst, 41 (28) Cahill D G, Watson S K & Pohl R Lower, Phys Rev B, 46 (1992) Born M & Mayer J E, Zeitschrift für Physik, 75 (1931) Leibfried G & Ludwig W. Theory of anharmonic effect in crystal, In: Seitz F & Turnbull D, Editors, Solid State Phys, (Academic Press, New York), 1966, Vol 12, pp Brugger K, Phys Rev, 13 (1964) A Mori S & Hiki Y, J Phys Soc Jpn, 45 (1978) Zener C, Elasticity and Anelasticity of Metals (University of Chicago, Chicago), Ledbetter H M & Reed R P, J Phys Chem, 2 (1973) Hill R, J Mech Phys Solids, 11 (1963) Christoffel E B, Annali di Matematica Pura ed Applicata. Serie. II, 8 (1877) Kaushik S, Bhalla V & Singh D, J Pure Appl Ultrason, 36 (214) Bhalla V, Kumar R, Tripathy C & Singh D, Int J Mod Phys B, 27 (213) 28 pp. 23 Singh D, Kaushik S, Bhalla V, Tripathi S & Gupta A K, Arab J Sc Eng, 39 (214) Pandey D K, Singh D, Bhalla V, Tripathi S, Yadav R R, Indian J Pure & Appl Phys, 52 (214) Voigt W, Lehrbuch der Kristallphysik Teubner, Leipzig 191; reprinted (1928) with an additional appendix, Leipzig, Teubner (New York, Johnson Reprint), Reuss A, Z Angew Math Mech, 9 (1929) Natali F, Ruck B J, Plank N OV, Trodahl H J et al., Prog Mater Sci, 58 (213) Pugh S F, Philos Mag, 45 (1954) Bhajanker S, Srivastava V, Pagare G & Sanyal S P, J Phys: Conf Ser, 377 (212) Ciftci Y, Mogulkoc Y & Evecen M, Int Conf App Phys Simul Comp, Nye J F, Physical Properties of Crystals: their Representation by Tensors and Matrices (Oxford University Press, London) Nye J F, Physical Properties of Crystals (Oxford university press, London) Singh D & Pandey D K, Pramana, 72 (29) Mainprice D, Bachmann F, Hielscher R, Schaeben H &Lloyd G E, Calculating Anisotropic Piezoelectric Properties from Texture Data using the MTEX Open Source Package (Geological Society, London)