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Superconductivity in YIn3 under positive pressure a,⁎

Surinder Singh , Ranjan Kumar a b

T

a,b

Department of Physics, Panjab University, Chandigarh 160014, India Department of Physics, King Abdulaziz University, Jeddah, Saudi Arabia

ARTICLE INFO

ABSTRACT

Keywords: Positive hydrostatic pressure Eliashberg spectral function Electron-phonon coupling constant Phonon linewidths Migdal-Eliashberg theory Logarithmic phonon frequency

Positive hydrostatic pressure was applied on YIn3 and variation in superconducting transition temperature (Tc) and average coupling constant (λep) was observed. The variation in phonon dispersion relation, Eliashberg spectral function, phonon DOS and average logarithmic phonon frequency was studied with an increase in positive hydrostatic pressure. The dispersion relation for YIn3 under ambient and all applied positive pressure shows positive frequency for all modes of vibrations signifying the structural stability of YIn3 under all applied pressures.

1. Introduction YIn3 is an intermetallic low coupling conventional superconductor having AuCu3-type structure and Pm-3 m space group. Using Eliashberg–Migdal formalism, Bellington et al. have calculated the electron-phonon coupling constant and superconducting transition temperature of YIn3 as 0.42 and 0.77 K, respectively [1]. This calculated value of superconducting transition temperature at ambient pressure is quite close to the experimental value of 0.88 K. Much research work has already been done to study the properties of conventional superconductors under applied external pressures. Under pressure theoretical study of superconducting properties of lithium has been done by Bazhirov et al. to investigate the effect of pressure on electronphonon coupling and the Fermi surface [2]. In another study, using first-principle methods Chan et al. have investigated the variation of electron-phonon coupling and superconducting transition temperature of phosphorus in the pressure range 20–70 GPa [3]. Experimentally the effect of tensile strain on superconducting properties like superconducting transition temperature of MgB2 thin films has been studied by Pogrebnyakov et al. and Hur et al. [4,5]. Theoretical application of negative pressure and its effect on the superconducting properties of MgB2 (i.e. increase in superconducting transition temperature) has been reported by Zhang et al. [6]. Till now no previously done theoretical study on YIn3 has been reported under positive hydrostatic pressure. The motivation behind this work is to study the superconducting properties and the variations of the same under applied positive hydrostatic pressure. In this research work positive hydrostatic pressure on YIn3 was applied in the range 1.04 GPa–4.1 GPa and the effect on superconducting transition ⁎

temperature, average electron-phonon coupling constant and phonon frequency across all modes was studied. This paper constitutes of four main sections: the second section of this research paper deals with methodology and computational details, the third section deals with results and their analysis and the last section concludes the research work of this paper. 2. Methodology Electron-Phonon Coupling and theoretical part related to it is the same as discussed in previous research work on YSn3 [7]. The Eliashberg–Migdal formalism for conventional superconductors was first discussed in two research publications by Eliashberg [8] and Migdal [9]. Eliashberg spectral function [10] is given by 2F (

)=

2 N ( f)

(

qj

qj

)

qj

qj

(1)

where qj is phonon line widths and N ( f ) is electronic density of states at Fermi level. The phonon line width [10] is given by qj

MK , K + q 2 [f ( k )

=

f(

k

+

q )]

(

q

k+q

+

k)

(2)

where MK , K + q represents electron-phonon matrix element. The average electron-phonon coupling constant and average logarithmic phonon frequency are given by the following mathematical expressions ep

=2

Corresponding author. E-mail addresses: [email protected] (S. Singh), [email protected] (R. Kumar).

https://doi.org/10.1016/j.physc.2019.05.003 Received 25 February 2019; Received in revised form 8 May 2019; Accepted 15 May 2019 Available online 16 May 2019 0921-4534/ © 2019 Elsevier B.V. All rights reserved.

2F 0

( )

d

(3)

Physica C: Superconductivity and its applications 564 (2019) 6–10

S. Singh and R. Kumar

ln

= exp

2F

2 ep

( )

log( ) d

Table. 1 Superconducting transition temperature, coupling constant and average logarithmic phonon frequency at ambient pressure and applied positive hydrostatic pressure.

(4)

0

where F(ω) is the phonon density of states (DOS). The superconducting transition temperature which was derived by McMillan [11] using Eliashberg–Migdal formalism and modified by Allen–Dyne [12] is

Tc =

ln

1.2K

1.04(1 +

exp ep

ep )

µ* (1 + 0.62

ep)

(5)

where μ* is dimensionless Coulomb pseudopotential, which represents electron-electron repulsion. And average logarithmic phonon frequency (ωln) is linearly dependent on Debye temperature (ϴD) [12].

Lattice constant increment

Pressure (GPa)

Logarithmic Average Frequency (ωln) (eV)

λep

µ*

Tc(K)

0.0% 0.5% 1.0% 1.5% 2.0%

0.0001013 1.04 2.1 3.1 4.1

9.6 × 10−3 9.9 × 10−3 10.2 × 10−3 10.5 × 10−3 10.9 × 10−3

0.38 0.36 0.32 0.29 0.26

0.07 0.07 0.07 0.07 0.07

0.69 0.53 0.31 0.17 0.079

transition temperature (Tc) of YIn3 decreases with an increase in the positive hydrostatic pressure from 0.69 K to 0.08 K. The electronphonon coupling constant (λep) also shows a decrease with an increase in the positive hydrostatic pressure. With an increase in the positive hydrostatic pressure logarithmic average phonon frequency (ωln) shows an increase from 9.6 × 10−3 eV to 10.9 × 10−3 eV. The dimensionless Coulomb Pseudo-potential (µ*) was taken as 0.07 for YIn3 for all the five calculations. The calculated value of superconducting transition temperature at ambient pressure came out to be 0.69 K, which is very near the experimental value of 0.88 K. The decrease in the superconducting transition temperature of YIn3 under positive hydrostatic pressure can be due to the corresponding decrease in the electronphonon coupling constant. The decrease in superconducting transition temperature with an increase in positive pressure for YIn3 is in agreement with our earlier calculations where YSn3 shows an increase in superconducting transition temperature with increase in negative hydrostatic pressure [7]. It effectively means that with negative pressure the transition temperature increases and with positive pressure the transition temperature decreases. YIn3 contains four atoms in a unit cell and so has 12 modes of vibrations out of which three are acoustic and nine optical modes of vibrations. Fig. 3 shows four dispersion relations under all four applied positive hydrostatic pressures. It can be observed that there is a gradual shift of all 12 modes of vibrations towards higher frequency as the positive hydrostatic pressure increases from 1.04 GPa to 4.1 GPa (0.5% lattice constant decrease to 2% lattice constant decrease). The topmost high energy optical modes at Γ-point gradually shift from 160–165 cm−1 to 165–170 cm−1 when pressure increases from 1.04 GPa to 2.1 GPa.These high energy optical modes further shift from 170–175 cm−1 to 175–180 cm−1 as the pressure increases from 3.1 GPa to 4.1 GPa. The comparison of the phonon dispersion relation for YIn3 under ambient pressure and 4.1 GPa applied positive hydrostatic pressure is shown in Fig. 4. The phonon dispersion relation shows a shift in the upward direction along the Y-axis with the increase in positive hydrostatic pressure. The upward shift in phonon dispersion relation implies an increase in frequency with the applied positive pressure for all 12 modes of vibrations. This shift towards higher frequency with positive hydrostatic pressure is consistent with the shift towards lower frequency with negative hydrostatic pressure (in our earlier done calculations for YSn3) [7]. The dispersion relation plots for YIn3 for all the five pressures were found to have positive frequency, implying that the YIn3 shows structural stability at all the applied pressures. Fig. 5. shows Eliashberg spectral function and phonon DOS at ambient pressure. Both these curves are similar in shape. It can be observed that there is a decrease in peak height (or no change in peak height) for 0.01 eV to 0.015 eV frequency range in Eliashberg spectra as compared with the phonon DOS curve, implying that this low-energy optical mode couple less strongly. Similarly, there is an increase in peak height in Eliashberg spectra as compared with the phonon DOS in the peak lying in 0.015 eV–0.02 eV frequency range, implying that this highest energy optical mode couple more strongly. The low-frequency

3. Computational details The calculations for YIn3 under ambient and applied positive hydrostatic pressure were performed using all electron full-potential linear augmented plane-wave (FP-LAPW) method as implemented in the ELK code [13]. The experimental value of lattice constant for YIn3 was taken as a = 4.593 Å or 8.6795 Bohr [14]. The crystal structure was relaxed up to a maximum force tolerance of 5 × 10−3 Hatree/ Bohr. The local density approximation (LDA) was used as exchange and correlation functional [15]. Good convergence was obtained for Interstitial region plane-wave cut-off of |G + K|max = 8/Rmt (where Rmt is average muffin-tin radius). The muffin-tin radius for both elements Y and In was taken as 2.6 a.u. The calculations for YIn3 were done taking 4 × 4 × 4 Monkhorst-Pack q-point grid and Brillouin zone integrations were done on a 24 × 24 × 24 k-mesh. For each of the five calculations, 120 dynamical matrices were generated using the Supercell method. 4. Results and discussion Band structure of YIn3 (Fig. 2.) was computed at ambient pressure and was found to be quite similar to the earlier reported band structure of YIn3 [1]. The superconducting properties of YIn3 were studied under ambient pressure and applied positive hydrostatic pressure. The positive hydrostatic pressure was applied by decreasing all the lattice constants by an equal percentage. Ram et al. have calculated the bulk modulus of YIn3 as 70 GPa [16]. Hence, the bulk modulus for calculating the positive hydrostatic pressure for YIn3 was taken as 70 GPa [16]. YIn3 is a conventional superconductor with phonon mediated electron-electron interaction. Table. 1. shows that the superconducting

Fig. 2. Band structure of YIn3 at ambient pressure. 7

Physica C: Superconductivity and its applications 564 (2019) 6–10

S. Singh and R. Kumar

Fig. 3. Phonon dispersion relation for YIn3 at (a) 0.5% (1.04 GPa) (b) 1% (2.1 GPa) (c) 1.5% (3.1 GPa) (d) 2% (4.1 GPa) lattice constant decrease.

pressure. This shift can be attributed to the increase in frequency with the applied positive pressure for all modes of vibrations. This shift is in tune with our earlier reported results for YSn3 where there is a shift towards lower frequency with the applied negative hydrostatic pressure [7]. Besides the frequency shift in both phonon DOS and Eliashberg spectral function, there is a slight decrease in peak strength or height with an increase in positive hydrostatic pressure. This can be attributed to the decrease in electron-phonon coupling strength with the increase in positive hydrostatic pressure. The α2F(ω) plots show that the peaks for 4.1 GPa pressure (2% lattice constant decrease) have grown lower in height in comparison to the ambient pressure curve. Comparison of each curve in Eliashberg spectral function, with the corresponding curve in phonon DOS, shows that there is an enhancement in peak strength for acoustic mode near 0.01 eV for all applied pressures. Similarly, peaks for 0.01 eV–0.015 eV frequency range in the Eliashberg spectra as compared to phonon DOS, show a decrease in peak height for all applied positive hydrostatic pressures. This means less coupling for these low-energy optical modes. The Peaks in the frequency range 0.020 eV–0.023 eV show an increase in peak height for Eliashberg spectral function as compared to the phonon DOS, for all applied positive hydrostatic pressures. This signifies strong coupling in these high-energy optical modes. Fig. 7. is the plot between α2F(ω)/F(ω) and frequency which effectively is the plot showing coupling strength (α2) along all modes of vibrations for ambient and all applied positive hydrostatic pressures. The plot shows two major bumps at 0.005 eV frequency and 0.015–0.020 eV frequency range for ambient and all applied positive hydrostatic pressures. The major bump near 0.005 eV frequency means

Fig. 4. Phonon dispersion relation for YIn3 at ambient pressure (black) and at (2% lattice constant increment) 4.1 GPa applied positive pressure (gray line).

acoustic mode peak near 0.005 eV shows negligible change in peak height (though the peak becomes broader) in α2 F(ω), compared to the phonon DOS. The contribution of these acoustic modes towards coupling become more clear in α2F(ω)/F(ω) plot. Fig. 6. shows phonon DOS and Eliashberg spectral function at various applied positive hydrostatic pressures. Both the phonon DOS and the Eliashberg spectral function show a shift from lower frequency towards higher frequency with an increase in positive hydrostatic 8

Physica C: Superconductivity and its applications 564 (2019) 6–10

S. Singh and R. Kumar

Fig. 5. (a) Eliashberg spectral function plots and (b) phonon DOS at ambient pressure.

that low-energy acoustic modes mediate the electron-electron interaction. Another major bulge in frequency range 0.015–0.020 eV means that the highest energy optical modes also mediate electron-electron interaction. These two coupling strength bulges are common among all five curves. Fig. 8. shows the variation of superconducting transition

temperature (Tc) and average coupling constant with applied positive hydrostatic pressures. The graph shows that the transition temperature decreases almost linearly with the increase in applied positive hydrostatic pressures. The average electron-phonon coupling constant also decreases almost linearly with applied positive hydrostatic pressure. The reason for the decrease in the transition temperature could be due

Fig. 6. (a) Eliashberg spectral function plots and (b) the phonon DOS for ambient and all applied positive hydrostatic pressures. 9

Physica C: Superconductivity and its applications 564 (2019) 6–10

S. Singh and R. Kumar

Fig. 7. α2F(ω)/F(ω) plot for ambient and all applied positive hydrostatic pressures.

to the decrease in the average coupling constant with an increase in positive hydrostatic pressure. These results are consistent with our earlier calculations where the negative pressure application showed an increase in the average coupling constant and superconducting transition temperature [7]. 5. Conclusion It can be concluded that the structure of YIn3 remains stable under ambient and all applied positive hydrostatic pressures as the dispersion relation plots for all pressures show positive frequency for all modes of vibrations. With an increase in the positive hydrostatic pressure, there is a linear decrease in superconducting transition temperature and average electron-phonon coupling constant of YIn3.The phonon frequency across all modes of vibrations increases with an increase in positive hydrostatic pressure. All the variations in phonon and superconducting properties under pressure for YIn3 are consistent with our earlier calculations under negative hydrostatic pressures for YSn3 [7]. Acknowledgments Few test calculations for YIn3 at ambient pressure were done at high-performance computing facility (HPC) at IUAC, Delhi and all calculations for YIn3 at ambient and applied pressures were done at National Param Supercomputing Facility (NPSF) at CDAC, Pune. References [1] D. Billington, T.M. Llewellyn-Jones, G. Maroso, S.B Dugdale, Supercond. Sci. Technol. 26 (2013) 085007. [2] T. Bazhirov, J. Noffsinger, M.L. Cohen, Phys. Rev. B 82 (2010) (2010) 184509. [3] K.T. Chan, Brad D. Malone, M.L. Cohen, Phys. Rev. B 86 (2012) 094515. [4] A.V. Pogrebnyakov, J.M. Redwing, S. Raghavan, V. Vaithyanathan, D.G. Schlom, S.Y. Xu, Q. Li, D.A. Tenne, A Soukiassian, X.X. Xi, M.D. Johannes, D. Kasinathan, W.E. Pickett, J.S. Wu, J.C.H. Spence, Phys. Rev. Lett. 93 (2004) 147006. [5] N. Hur, P.A. Sharma, S. Guha, M.Z. Cieplak, D.J. Werder, Y. Horibe, C.H. Chen, S.W. Cheong, Appl. Phys. Lett. (2001) 794180. [6] C. Zang, X. Zhang, Comput. Mater. Sci. 50 (2011) 1097–1101. [7] S. Singh, R. Kumar, J. Supercond. Nov. Magn. 31 (4) (2018) 943–1278. [8] G.M. Eliashberg, Zh Eksp. Teor. Fiz. 38 (1960) 966. [9] A.B. Migdal, Zh Eksp. Teor. Fiz. 34 (1958) 1438. [10] P.B. Allen, Phys. Rev. B 6 (1972) 2577. [11] W.L. McMillan, Phys. Rev. 167 (1968) 331. [12] P.B. Allen, R.C Dynes, Phys. Rev. B 12 (1975) 905. [13] http://elk.sourceforge.net. [14] V.B. Pluzhnikov, A. Czopnik, I.V. Svechkarev, Physica B 212 (1995) 375. [15] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [16] S. Ram, V. Kanchana, G. Vaitheeswaran, A. Svane, S.B. Dugdale, N.E. Christensen, J. Phys. 25 (2013) 155501.

Fig. 8. (a) Variation of transition temperature and (b) average coupling constant with percent lattice constant increment.

10