We have investigated elastic, vibrational and thermal properties of B2 type ErCu intermetallic compound using density functional theory (DFT). The positive phonon frequencies reflect the dynamical stability of ErCu intermetallic compound in the B2 type cubic structure phase. Furthermore, the density functional perturbation theory (DFPT) as implemented in quasi-harmonic approximation (QHA) was used for the calculation of thermal properties such as vibrational energy ΔF, entropy S, internal energy ΔE and
constant-volume specific heat Cv of the ErCu. The entropy of ErCu is ~ 18 J∙K – 1∙mol – 1 that concluded that, the ErCu compound is not harder compound. The computed Poisson’s ratio (σ), Young’s modulus (E), bulk modulus (B) and shear modulus (GH) are 0.30, 66.65 GPa, 62.99 GPa and 25.18 GPa respectively. The B/GH ratio is 2.50, which confirmed the good ductility of ErCu intermetallic compound

Babariya, Bindiya

Gajjar, P.N.

Gupta, Sanjeev K.

Raval, Dhara

Electronic Sumy State University Institutional Repository

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ Vol. 12 No 2, 02031(4pp) (2020) Том 12 № 2, 02031(4cc) (2020)   2077-6772/2020/12(2)02031(4) 02031-1  2020 Sumy State University Ground State Stability and Thermal Properties of ErCu Using First Principles Study  Dhara Raval1, Bindiya Babariya1, Sanjeev K. Gupta2, P. N. Gajjar1, *  1 Department of Physics, University School of Sciences, Gujarat University, 380009 Ahmedabad, India 2 Computational Materials and Nanoscience Group, Department of Physics, St. Xavier's College, 380009  Ahmedabad, India  (Received 18 February 2020; revised manuscript received 15 April 2020; published online 25 April 2020)  We have investigated elastic, vibrational and thermal properties of B2 type ErCu intermetallic com-pound using density functional theory (DFT). The positive phonon frequencies reflect the dynamical stabil-ity of ErCu intermetallic compound in the B2 type cubic structure phase. Furthermore, the density func-tional perturbation theory (DFPT) as implemented in quasi-harmonic approximation (QHA) was used for the calculation of thermal properties such as vibrational energy ΔF, entropy S, internal energy ΔE and constant-volume specific heat Cv of the ErCu. The entropy of ErCu is ~ 18 J∙K – 1∙mol – 1 that concluded that, the ErCu compound is not harder compound. The computed Poisson’s ratio (), Young’s modulus (E), bulk modulus (B) and shear modulus (GH) are 0.30, 66.65 GPa, 62.99 GPa and 25.18 GPa respectively. The B/GH ratio is 2.50, which confirmed the good ductility of ErCu intermetallic compound.   Keywords: ErCu, Density Functional Theory, Elastic properties, Thermal properties.  DOI: 10.21272/jnep.12(2).02031 PACS numbers: 46.15. – x, 0.270. – c, 61.50 – f, 62.30 + d, 63.20 – e, 65.40 – b                                                            * pngajjar@gujaratuniversity.ac.in 1. INTRODUCTION  The intermetallic compounds received much atten-tion due to their superior physical, electrical, magnetic, mechanical and chemical properties [1-7]. It is well know that at elevated temperature, the intermetallic compounds have capacity to improve their performance in engineering applications. Owing to their high ductil-ity, oxidation resistance, high tensile strength, stiffness and low density; they are used in aerospace industry [2, 8]. The partially filled ‘f’ electrons of Er show the ErCu holds extraordinarily high ductility and crystal-lizes in the CsCl type cubic structure [9, 10]. Few stoi-chiometry of ErCu intermetallic compounds have been studied experimentally and theoretically with CsCl type structure [2, 9-16]. To best of our knowledge, there were no experimental and theoretical results for the comparison of vibrational and thermal properties of ErCu. Therefore, we have investigated vibrational and thermal properties of ErCu and explored phonon dis-persive curve, phonon density of states (PDOS) and thermal properties such as vibrational energy ΔF, en-tropy S, internal energy ΔE and constant-volume spe-cific heat Cv. Moreover, the elastic properties viz; bulk modulus B, shear modulus GH, Young‘s modulus E, Poisson‘s ratio , Pugh‘s ratio (B/GH) and elastic con-stants (C11, C12 and C44) of ErCu compound have been also investigated.  2. COMPUTATIONAL METHOD  The present study was carried out using density functional perturbation theory (DFPT) formalism with Troullier Martin type pseudopotential [17-18] for Er and ultrasoft exchange and correlation pseudopotential for Cu [19-23]. The qausi harmonic approximation (QHA) code developed by the Baroni et al. is used to investigate the thermal properties [24]. All calculation is explored with two atom basis located, where Er atom at (0.5, 0.5, 0.5) and Cu atom at (0.0, 0.0, 0.0) with cutoff energy 370 Ry and 16×16×16 Monk horst-pack grid [25]. Phonon code is used to find the vibrational and thermal proper-ties of ErCu intermatallic compound using the Quan-tum Espresso code [26, 27]. The vibrational properties have been obtained from optimized lattice parameter of ErCu intermatallic compound with spin-polarized case [28]. The phonon frequency spectra were studies along the high symmetry points Г - X - M - Г- R - X [29].  3. RESULTS AND DISCUSSION   In our previous work [27], we already optimized the geometrically stable lattice constant a0  3.414 Å of ErCu B2 type structure and based on the stability of system. Moreover, we had also appraised the electronic properties such as band structure, density of States, partial density of states, charge density and Fermi surface of ErCu for majority and minority spin cases. Hence, the same lattice constant of ErCu was used for further study.  3.1 Vibrational Properties  The calculated phonon dispersion relation of ErCu having two atoms in the system, the phonon dispersion curve has total six branches as shown in Fig. 1. The span of one LA (longitudinal acoustic), one LO (longitu-dinal optical), two TA (transverse acoustic) and two TO (transverse optical) is within the frequency 140 cm – 1 range. The positive phonon frequencies reflect the dy-namical stability of the ErCu in B2 phase. At the fre-quency of 50 cm – 1, LA, TA and LO branches were over-lapping at the M-point and near the X-point and the LA and LO branches are overlapping at the 101 cm – 1 fre-quency. It is noticed that one optical and two acoustic branches are triply degenerate at the Г-points that reveal the metallic nature of ErCu.  DHARA RAVAL, BINDIYA BABARIYA ET AL. J. NANO- ELECTRON. PHYS. 12, 02031 (2020)   02031-2   Fig. 1 – Phonon dispersion curves along the high symmetry direction in B2 type ErCu    Fig. 2 – Total and partial phonon density of states of ErCu in cubic phase along the high symmetry directions Table 1 – Calculated bulk modulus B, Young’s modulus E, shear modulus GH, Poission Ratio , Pugh’s constant B/GH and elastic constants C11, C12,C44  and comparison of other available data  Sr. No Property Present calculation Other calculation 1 Bulk Modulus B (GPa) 62.99  63.14 [10], 76.62 [2] 2 Yong Modulus E (GPa) 66.65  79.97 [10], 96.03 [2] 3 Shear Modulus GH (GPa) 25.18  30.84 [10], 36.64 [2] 4 Poisson Ratio  0.30  0.29 [10], 0.31 [2] 5 Pugh’s ratio (B/GH) 2.50  2.12 [10], 2.30 [2] 6 Elastic Constant C11 (GPa) 85.07  157.23 [10], 129.71 [2] 7 Elastic Constant C12 (GPa) 51.95  19.60 [10], 61.77 [2] 8 Elastic Constant C44 (GPa) 38.06  16.8 [10], 38.54 [2]  The total and partial phonon density of states of ErCu is shown in Fig. 2. In partial phonon DOS of Er atom has two major peaks one at 58.61 cm – 1 and se-cond at 85.73 cm – 1 frequency. While, in partial phonon DOS of Cu atom contribution leads four successive peaks at the 56.07 cm – 1, 86.01 cm – 1, 112.57 cm – 1 and 128.39 cm – 1 successively. Among them first peak aris-es due to flat curve of the acoustics mode frequency at the 57.20 cm – 1, second peak appeared due to the over-lapping of the LO and TO modes at the R-point with frequency 85.73 cm – 1, third and fourth peaks are re-sulted from the highly overlapping of LA and TO modes at R-point to X point at 112.57 cm – 1 and 128.67 cm – 1 respectively.  3.2 Elastic Properties  The elastic constants are important parameters to study elastic properties of the system. Moreover, elastic constants are requisite parameters that provide the information about the structural stabilities, ductility, inter-atomic forces and stiffness of the materials. The bulk modulus B (GPa) represents the resistance to fracture while shear modulus GH (GPa) represents the resistance to plastic deformation. In cubic crystals, there are only three independent elastic constant C11, C12 and C44. The bulk modulus B, Shear modulus GH, Young‘s modulus E, Poisson‘s ratio , Pugh‘s ratio (B/GH) and elastic constants (C11, C12 and C44) are tabu-lated in Table 1.   The mechanical stability of the ErCu cubic crystal is satisfied by Born-Huang stability criteria: C11-C12  0, C11  0, C44  0 and (C11 + 2C12)  0 [30]. Our calculated elastic constants of ErCu follow cubic stabil-ity conditions C12  B  C11. Which clearly indicate the stability of the ErCu compound in B2  cubic phase. It is also found that calculated values are in reasonable agreement with available other results [2, 10]. Accord-ing to Pugh‘s criteria material is brittle when, B/GH  1.75 or ductile B/GH  1.75 [31]. The B/GH ratio of ErCu intermatallic compound is 2.50. This is clearly confirms that ErCu possess good ductility in B2 type cubic structure.     GROUND STATE STABILITY AND THERMAL PROPERTIES… J. NANO- ELECTRON. PHYS. 12, 02031 (2020)   02031-3 3.3 Thermal Properties  The temperature plays significance role to deter-mine the electronic and optical properties of material. We have used a quasi-harmonic approximation to ob-tain thermal properties. In energy ΔF, internal energy ΔE, entropy S and at constant volume specific heat Cv from 0 K to 1000 K of Er, Cu and ErCu intermatallic compound in Fig. 3(a-d). which we listed the contribu-tion to the vibrational Fig. 3(a-d) depict that, at lower temperature (K), entropy S , Specific heat Cv and in-ternal energy ΔE values increase with temperature and converged to constant values with higher temperature (K). As shown in Fig. 3(b), vibrational energy ΔF de-crease of as temperature increases up to 1000 K range. As seen, Er atom has higher vibrational energy com-pare to that the Cu atom and ErCu compound. Moreo-ver, At the 110 K temperature, the vibrational energy of Cu atom and ErCu compound is same and at 140 K temperature, again, the Er atom and ErCu meet at same energy. While At 250 K temperature, the vibra-tional energy of Er and Cu atom are same that predict that the ErCu has a most convenient vibrational ener-gy with respect to the Er and Cu vibrational energy. From Fig. 3(c), we can see that, the entropy of ErCu is ~ 18 J∙K – 1∙mol – 1 that concluded that, the ErCu com-pound is not harder compound, which concludes that ErCu compound has a very high ductility.     a  b        c  d  Fig. 3 – Internal energy variations with temperature for Er, Cu and ErCu (a); vibrational energy variations with  temperature for Er, Cu and ErCu (b); entropy variation with temperature for Er, Cu and ErCu (c); at a constant volume, Specific heat variation with temperature for Er, Cu and ErCu  4. CONCLUSIONS  In summary, the phonon dispersion curves show the dynamical stability of the B2 ErCu system. The phonon dispersive curves and phonon density of states are corresponding to each other. The overlapping of acous-tic and optical modes brings the indication of the me-tallic nature of the ErCu. The entropy of ErCu is  ~ 18 J∙K – 1∙mol – 1 that concluded that, ErCu compound is not quite considering as harder compound. The B/GH ratio is 2.50, which also point out the excellent ductility of the ErCu compound in B2 type cubic phase.   DHARA RAVAL, BINDIYA BABARIYA ET AL. J. NANO- ELECTRON. PHYS. 12, 02031 (2020)   02031-4 AKNOWLEDGEMENTS  The computer facility developed under DST-FIST level-I (No.SR/FST/PSI-097/2006 dated 20th December 2006 and No.SR/FST/PSI-198/2014 dated 21 November 2014) programmes of Department of Science and Tech-nology, Government of India, New Delhi and support under DRS-SAP-I (No. F-530/10/DRS/2010 (SAP-I) dated November 2010 and No.F.530/17/DRS-II/2018 (SAP-I), dated 17 April 2018) of University Grants Commission, New Delhi are highly acknowledged. DR is thankful for the project fellowship under DRS-II-SAP (No.F.530/17/DRS-II/2018(SAP-I),dated 17/04/2018) of University Grants Commission, New Delhi.    REFERENCES  1. X. Tao, H. Chen, X. Li, Y. Ouyang, S. Liao, Phys. Scr. 83 No 4, 045301 (2011). 2. R.P. Singh, V.K. Singh, R.K. Singh, M. Rajagopalan, Am. J. Condens. Matter. Phys. 3 No 5, 123 (2013).  3. X. Tao, Y. Ouyang, H. Liu, F. Zeng, Y. Feng, Z. Jin, Physi-ca B: Condens. Matter 399 No 1, 27 (2007).  4. M. Davis, J. Kuriplach, Physica B: Condens. Matter. 205, 3 (1995).  5.  J. Pierre, P. Morin, D. Schmitt, B. Hennion, Journal de Physique 39 No 7, 793 (1978). 6. Jian Liu, Th. George Woodcock, Nils Scheerbaum, Ol. Gutfleisch, Acta Mater. 57 No 16, 4911 (2009). 7. R.L. Fleischer, R.J. Zabala, Metall. Trans. A 21 No 10, 2709 (1990). 8. Y.J. Shi, Y.L. Du, G. Chen, G.L. Chen, Mater. Trans. 49 No 11, 2480 (2008).  9. S. Meenakshi, Bull. Mater. Sci. 37 No 6, 1343 (2014).  10. M. Shugani, S.S. Chouhan, M. Aynyas, S.P. Sanyal, Adv. Phys. Theor. Appl. 19, 83 (2013).  11. V. Srivastava, G. Pagare, S.P. Sanyal, M. Rajagopalan, Phys. Stat. Sol. 246 No 6, 1206 (2009).  12. E. Mamontov, H. O'Neill, Q. Zhang, W. Wang and D.J. Wesolowski, J. Phys. Condens. Matter. 24 No 7, 064104 (2012).  13.  R. Sun, D.D. Johnson, Phys. Rev. B 87 No 10, 104107 (2013).  14.  I. Lukačević, S.K. Gupta, J. Alloy. Compd. 597, 148 (2014). 15. V. Mankad, S.K. Gupta, I. Lukačević, P.K. Jha, J. Phys. Conf. Ser. 377, 012076 (2012).  16.  V.H. Mankad, S.K. Gupta, I. Lukačević, P.K. Jha, Comp. Mater. Sci. 65, 536 (2012). 17.  C. Davide, Erbium pseudo potential, https://sites.google.com/site/dceresoli/pse 18. J.P. Perdew, A. Zunger, Phys. Rev. B 23, 5048 (1981). 19. N.E. Zein, Sov. Phys. Solid State 26, 3028 (1984). 20. D.K. Blat, N.E. Zein, V.I. Zinenko, J. Physics: Condens. Matter 3 No 29, 5515 (1991). 21. N.E. Zein, Phys. Lett. A 161 No 6, 526 (1992). 22. S.S. Baroni, P. Giannozzi, A. Testa, Phys. Rev. Lett. 58, 1861 (1987).  23. P. Giannozzi, S. de Gironcoli, P. Pavone, S. Baroni, Phys. Rev. B 43, 7231 (1991).  24. S. Baroni, S. de Gironcoli, A. Dal Corso, P. Giannozzi, Rev. Mod. Phys. 73, 515 (2001).  25. H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 No 12, 5188 (1976). 26. P. Giannozzi, J. Phys. Condens. Matter. 29, 465901 (2017). 27.  http://qeforge.qe-   forge.org/gf/project/thermo_pw 28. D. Raval, B. Babariya, S.K. Gupta, P.N. Gajjar, Physica B: Condens. Matter. 570, 122 (2019). 29. W. Setyawan, S. Curtarolo, Comp. Mater. Sci. 49 No 2, 299-312 (2010). 30. M. Born, K. Huang, Am. J. Phys. 23, 474 (1995). 31. S.F. Pugh, J. Sci. 45 No 367, 823 (1954)  

Ground State Stability and Thermal Properties of ErCu Using First Principles Study

SocioEconomic Challenges Journal (Sumy State University)

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ 
Vol. 12 No 2, 02031(4pp) (2020) Том 12 № 2, 02031(4cc) (2020) 
 
 
2077-6772/2020/12(2)02031(4) 02031-1  2020 Sumy State University 
Ground State Stability and Thermal Properties of ErCu Using First Principles Study 
 
Dhara Raval1, Bindiya Babariya1, Sanjeev K. Gupta2, P. N. Gajjar1, * 
 
1 Department of Physics, University School of Sciences, Gujarat University, 380009 Ahmedabad, India 
2 Computational Materials and Nanoscience Group, Department of Physics, St. Xavier's College, 380009  
Ahmedabad, India 
 
(Received 18 February 2020; revised manuscript received 15 April 2020; published online 25 April 2020) 
 
We have investigated elastic, vibrational and thermal properties of B2 type ErCu intermetallic com-
pound using density functional theory (DFT). The positive phonon frequencies reflect the dynamical stabil-
ity of ErCu intermetallic compound in the B2 type cubic structure phase. Furthermore, the density func-
tional perturbation theory (DFPT) as implemented in quasi-harmonic approximation (QHA) was used for 
the calculation of thermal properties such as vibrational energy ΔF, entropy S, internal energy ΔE and 
constant-volume specific heat Cv of the ErCu. The entropy of ErCu is ~ 18 J∙K – 1∙mol – 1 that concluded 
that, the ErCu compound is not harder compound. The computed Poisson’s ratio (), Young’s modulus (E), 
bulk modulus (B) and shear modulus (GH) are 0.30, 66.65 GPa, 62.99 GPa and 25.18 GPa respectively. The 
B/GH ratio is 2.50, which confirmed the good ductility of ErCu intermetallic compound.  
 
Keywords: ErCu, Density Functional Theory, Elastic properties, Thermal properties. 
 
DOI: 10.21272/jnep.12(2).02031 PACS numbers: 46.15. – x, 0.270. – c, 61.50 – f, 
62.30 + d, 63.20 – e, 65.40 – b 
 
                                                          
* pngajjar@gujaratuniversity.ac.in 
1. INTRODUCTION 
 
The intermetallic compounds received much atten-
tion due to their superior physical, electrical, magnetic, 
mechanical and chemical properties [1-7]. It is well 
know that at elevated temperature, the intermetallic 
compounds have capacity to improve their performance 
in engineering applications. Owing to their high ductil-
ity, oxidation resistance, high tensile strength, stiffness 
and low density; they are used in aerospace industry 
[2, 8]. The partially filled ‘f’ electrons of Er show the 
ErCu holds extraordinarily high ductility and crystal-
lizes in the CsCl type cubic structure [9, 10]. Few stoi-
chiometry of ErCu intermetallic compounds have been 
studied experimentally and theoretically with CsCl 
type structure [2, 9-16]. To best of our knowledge, there 
were no experimental and theoretical results for the 
comparison of vibrational and thermal properties of 
ErCu. Therefore, we have investigated vibrational and 
thermal properties of ErCu and explored phonon dis-
persive curve, phonon density of states (PDOS) and 
thermal properties such as vibrational energy ΔF, en-
tropy S, internal energy ΔE and constant-volume spe-
cific heat Cv. Moreover, the elastic properties viz; bulk 
modulus B, shear modulus GH, Young‘s modulus E, 
Poisson‘s ratio , Pugh‘s ratio (B/GH) and elastic con-
stants (C11, C12 and C44) of ErCu compound have been 
also investigated. 
 
2. COMPUTATIONAL METHOD 
 
The present study was carried out using density 
functional perturbation theory (DFPT) formalism with 
Troullier Martin type pseudopotential [17-18] for Er and 
ultrasoft exchange and correlation pseudopotential for 
Cu [19-23]. The qausi harmonic approximation (QHA) 
code developed by the Baroni et al. is used to investigate 
the thermal properties [24]. All calculation is explored 
with two atom basis located, where Er atom at (0.5, 0.5, 
0.5) and Cu atom at (0.0, 0.0, 0.0) with cutoff energy 370 
Ry and 16×16×16 Monk horst-pack grid [25]. Phonon 
code is used to find the vibrational and thermal proper-
ties of ErCu intermatallic compound using the Quan-
tum Espresso code [26, 27]. The vibrational properties 
have been obtained from optimized lattice parameter of 
ErCu intermatallic compound with spin-polarized case 
[28]. The phonon frequency spectra were studies along 
the high symmetry points Г - X - M - Г- R - X [29]. 
 
3. RESULTS AND DISCUSSION  
 
In our previous work [27], we already optimized the 
geometrically stable lattice constant a0  3.414 Å of 
ErCu B2 type structure and based on the stability of 
system. Moreover, we had also appraised the electronic 
properties such as band structure, density of States, 
partial density of states, charge density and Fermi 
surface of ErCu for majority and minority spin cases. 
Hence, the same lattice constant of ErCu was used for 
further study. 
 
3.1 Vibrational Properties 
 
The calculated phonon dispersion relation of ErCu 
having two atoms in the system, the phonon dispersion 
curve has total six branches as shown in Fig. 1. The 
span of one LA (longitudinal acoustic), one LO (longitu-
dinal optical), two TA (transverse acoustic) and two TO 
(transverse optical) is within the frequency 140 cm – 1 
range. The positive phonon frequencies reflect the dy-
namical stability of the ErCu in B2 phase. At the fre-
quency of 50 cm – 1, LA, TA and LO branches were over-
lapping at the M-point and near the X-point and the LA 
and LO branches are overlapping at the 101 cm – 1 fre-
quency. It is noticed that one optical and two acoustic 
branches are triply degenerate at the Г-points that 
reveal the metallic nature of ErCu. 
 DHARA RAVAL, BINDIYA BABARIYA ET AL. J. NANO- ELECTRON. PHYS. 12, 02031 (2020) 
 
 
02031-2 
 
 
Fig. 1 – Phonon dispersion curves along the high symmetry 
direction in B2 type ErCu 
 
 
 
Fig. 2 – Total and partial phonon density of states of ErCu in 
cubic phase along the high symmetry directions 
Table 1 – Calculated bulk modulus B, Young’s modulus E, shear modulus GH, Poission Ratio , Pugh’s constant B/GH and elastic 
constants C11, C12,C44  and comparison of other available data 
 
Sr. 
No 
Property Present calculation Other calculation 
1 Bulk Modulus B (GPa) 62.99 
 
63.14 [10], 
76.62 [2] 
2 Yong Modulus E (GPa) 66.65 
 
79.97 [10], 
96.03 [2] 
3 Shear Modulus GH (GPa) 25.18 
 
30.84 [10], 
36.64 [2] 
4 Poisson Ratio  0.30 
 
0.29 [10], 
0.31 [2] 
5 Pugh’s ratio (B/GH) 2.50 
 
2.12 [10], 
2.30 [2] 
6 Elastic Constant C11 (GPa) 85.07 
 
157.23 [10], 
129.71 [2] 
7 Elastic Constant C12 (GPa) 51.95 
 
19.60 [10], 
61.77 [2] 
8 Elastic Constant C44 (GPa) 38.06 
 
16.8 [10], 
38.54 [2] 
 
The total and partial phonon density of states of 
ErCu is shown in Fig. 2. In partial phonon DOS of Er 
atom has two major peaks one at 58.61 cm – 1 and se-
cond at 85.73 cm – 1 frequency. While, in partial phonon 
DOS of Cu atom contribution leads four successive 
peaks at the 56.07 cm – 1, 86.01 cm – 1, 112.57 cm – 1 and 
128.39 cm – 1 successively. Among them first peak aris-
es due to flat curve of the acoustics mode frequency at 
the 57.20 cm – 1, second peak appeared due to the over-
lapping of the LO and TO modes at the R-point with 
frequency 85.73 cm – 1, third and fourth peaks are re-
sulted from the highly overlapping of LA and TO modes 
at R-point to X point at 112.57 cm – 1 and 128.67 cm – 1 
respectively. 
 
3.2 Elastic Properties 
 
The elastic constants are important parameters to 
study elastic properties of the system. Moreover, elastic 
constants are requisite parameters that provide the 
information about the structural stabilities, ductility, 
inter-atomic forces and stiffness of the materials. The 
bulk modulus B (GPa) represents the resistance to 
fracture while shear modulus GH (GPa) represents the 
resistance to plastic deformation. In cubic crystals, 
there are only three independent elastic constant C11, 
C12 and C44. The bulk modulus B, Shear modulus GH, 
Young‘s modulus E, Poisson‘s ratio , Pugh‘s ratio 
(B/GH) and elastic constants (C11, C12 and C44) are tabu-
lated in Table 1.   
The mechanical stability of the ErCu cubic crystal 
is satisfied by Born-Huang stability criteria: C11-
C12  0, C11  0, C44  0 and (C11 + 2C12)  0 [30]. Our 
calculated elastic constants of ErCu follow cubic stabil-
ity conditions C12  B  C11. Which clearly indicate the 
stability of the ErCu compound in B2  cubic phase. It is 
also found that calculated values are in reasonable 
agreement with available other results [2, 10]. Accord-
ing to Pugh‘s criteria material is brittle when, 
B/GH  1.75 or ductile B/GH  1.75 [31]. The B/GH ratio 
of ErCu intermatallic compound is 2.50. This is clearly 
confirms that ErCu possess good ductility in B2 type 
cubic structure.  
 
 
 GROUND STATE STABILITY AND THERMAL PROPERTIES… J. NANO- ELECTRON. PHYS. 12, 02031 (2020) 
 
 
02031-3 
3.3 Thermal Properties 
 
The temperature plays significance role to deter-
mine the electronic and optical properties of material. 
We have used a quasi-harmonic approximation to ob-
tain thermal properties. In energy ΔF, internal energy 
ΔE, entropy S and at constant volume specific heat Cv 
from 0 K to 1000 K of Er, Cu and ErCu intermatallic 
compound in Fig. 3(a-d). which we listed the contribu-
tion to the vibrational Fig. 3(a-d) depict that, at lower 
temperature (K), entropy S , Specific heat Cv and in-
ternal energy ΔE values increase with temperature and 
converged to constant values with higher temperature 
(K). As shown in Fig. 3(b), vibrational energy ΔF de-
crease of as temperature increases up to 1000 K range. 
As seen, Er atom has higher vibrational energy com-
pare to that the Cu atom and ErCu compound. Moreo-
ver, At the 110 K temperature, the vibrational energy 
of Cu atom and ErCu compound is same and at 140 K 
temperature, again, the Er atom and ErCu meet at 
same energy. While At 250 K temperature, the vibra-
tional energy of Er and Cu atom are same that predict 
that the ErCu has a most convenient vibrational ener-
gy with respect to the Er and Cu vibrational energy. 
From Fig. 3(c), we can see that, the entropy of ErCu is 
~ 18 J∙K – 1∙mol – 1 that concluded that, the ErCu com-
pound is not harder compound, which concludes that 
ErCu compound has a very high ductility. 
 
 
 
 a  b 
 
     
 c  d 
 
Fig. 3 – Internal energy variations with temperature for Er, Cu and ErCu (a); vibrational energy variations with  temperature for 
Er, Cu and ErCu (b); entropy variation with temperature for Er, Cu and ErCu (c); at a constant volume, Specific heat variation 
with temperature for Er, Cu and ErCu 
 
4. CONCLUSIONS 
 
In summary, the phonon dispersion curves show the 
dynamical stability of the B2 ErCu system. The phonon 
dispersive curves and phonon density of states are 
corresponding to each other. The overlapping of acous-
tic and optical modes brings the indication of the me-
tallic nature of the ErCu. The entropy of ErCu is  
~ 18 J∙K – 1∙mol – 1 that concluded that, ErCu compound 
is not quite considering as harder compound. The B/GH 
ratio is 2.50, which also point out the excellent ductility 
of the ErCu compound in B2 type cubic phase.  
 DHARA RAVAL, BINDIYA BABARIYA ET AL. J. NANO- ELECTRON. PHYS. 12, 02031 (2020) 
 
 
02031-4 
AKNOWLEDGEMENTS 
 
The computer facility developed under DST-FIST 
level-I (No.SR/FST/PSI-097/2006 dated 20th December 
2006 and No.SR/FST/PSI-198/2014 dated 21 November 
2014) programmes of Department of Science and Tech-
nology, Government of India, New Delhi and support 
under DRS-SAP-I (No. F-530/10/DRS/2010 (SAP-I) 
dated November 2010 and No.F.530/17/DRS-II/2018 
(SAP-I), dated 17 April 2018) of University Grants 
Commission, New Delhi are highly acknowledged. DR 
is thankful for the project fellowship under DRS-II-SAP 
(No.F.530/17/DRS-II/2018(SAP-I),dated 17/04/2018) of 
University Grants Commission, New Delhi.  
 
 
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Ground State Stability and Thermal Properties of ErCu Using First Principles Study

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