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Superconductivity origin of PdTe and pressure effect: Insights from ﬁrst-principles investigation Jianyong Chen a, *, Xing Wang b a b

Faculty of Science, Guilin University of Aerospace Technology, Guilin 541004, People's Republic of China Bowen College of Management, Guilin University of Technology, Guilin 541006, People's Republic of China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 July 2015 Received in revised form 23 November 2015 Accepted 1 December 2015 Available online 12 December 2015

The phonon spectra and electron-phonon interaction properties of hexagonal superconductor PdTe are studied systematically for the ﬁrst time by density functional perturbation theory (DFPT). We present phonon dispersion with non-negative frequency in the whole Brillouin zone and reveal its threedimensional character and strong vibrational coupling from both electronic and lattice dynamic viewpoint. First-principles calculation of logarithmically averaged frequency, Debye temperature, electronphonon coupling constant and transition temperature Tc agree well with experimental values. It is deﬁnitely believed superconductivity of PdTe originates from isotropical nonlocal electron-phonon interaction. Moreover, Unlike FeTe, the transition temperature of PdTe decreases with increasing of pressure. © 2015 Elsevier Masson SAS. All rights reserved.

Keywords: PdTe superconductor Phonon dispersion Eliashberg theory Pressure First-principles

1. Introduction The layered transition metal chalcogenides have attracted physicists for many decades. Various MX2-type transition metal dichalcogenides, for instance WTe2, IrTe2, and MoS2, have aroused great attention recently either for the novel phenomena or exotic physical nature, or for their potential applications [1,2]. However, systems consisting the VIII metal such as PdTe2, NiTe2 has been paid less effort. PdTe2 and NiAs-type PdTe shows superconductivity under 1.7 K [3,4] and 2.3 K - 4.5 K [5,6]. Compared with hightemperature superconductor Fe chalcogenides [7e9] the hexagonal PdTe can be regarded as the transformed structure of FeTe by shifting the Fe and Te layers. Although structure is similar, the local ligand ﬁeld is changed and brings about distinct physical properties. High pressure can enhance the transition temperature Tc of some Fe chalcogenides superconductors for a large scale [10]. A.B.Karki et al. declare that PdTe is a strong coupling superconductor [11]. Magnetic and transport measurements conﬁrm that PdTe is a type-II bulk superconductor below 4.5 K and suggest possible weak coupling simultaneously [6]. The structural,

* Corresponding author. E-mail address: [email protected] (J. Chen). http://dx.doi.org/10.1016/j.solidstatesciences.2015.12.003 1293-2558/© 2015 Elsevier Masson SAS. All rights reserved.

electronic, elastic and thermodynamic properties has been discussed both with the Local Density Approximation (LDA) and Generalized Gradient Approximation (GGA) approaches, which states the compound is unstable beyond 18.9 GPa [12]. The absence of essential features (quasi-two-dimensionality, half-ﬁlling electronic states, weaker covalency) for high-temperature superconductivity is studied from electronic properties and correlation effects with Wannier function method by comparing with Fe chalcogenides [13]. To the best of our knowledge, there is no literature focusing on superconductivity of PdTe from lattice dynamics and electronphonon coupling view within theoretical calculations. It is important to study the phonon involved properties to explore new phenomena and illuminating the superconductivity mechanism. In this work, we investigate the superconducting properties of PdTe based on the strong coupling mechanism of superconductivity established by Eliashberg [14,15], which extends original BardeenCooper-Schrieffer (BCS) theory for more general materials. The article is organized as follows. In Section 2 we present crystal structure and technical details. Section 3 is devoted to the electronic, lattice vibration, thermal dynamic and electronephonon interaction properties and analysis. Conclusions are drawn in Section 4.

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J. Chen, X. Wang / Solid State Sciences 52 (2016) 23e28

2. Crystal structure and technical details The calculations are performed with the Quantum Espresso [16] package employing density functional theory (DFT). The PerdewBurke-Ernzerhof parameterized generalized-gradient approximation (PBE-GGA) [17] without spin orbit coupling is used to describe the exchange correlation effect. An ultrasoft pseudopotential is utilized to represent the interaction between ionic cores and valence electrons. Plane-wave basis set for the electronic wave functions and the charge density with energy cutoffs are 50 R y and 500 R y respectively. The dynamical matrices and the electronphonon coupling constants are calculated using density functional perturbation theory (DFPT) [18] in the linear response regime. The electronic Brillouin zone integration in the phonon calculation is sampled with a 8 8 8 uniform k point mesh. The electron-phonon coupling converges with a ﬁner grid of 32 32 x 32 k points and a Gaussian smearing of 0.008 R y. The dynamical matrix is computed on a 4 4 4 mesh of phonon wave vectors q, phonon density of state (PHDOS) on 30 30 x 30 uniform mesh. The phonon dispersion is then obtained on a ﬁner 141 q mesh by Fourier interpolation of the real space interatomic force constants. The structures under different hydrostatic pressures are fully relaxed using variable-cell approach for each target external pressure. The hexagonal PdTe (prototype NiAs) crystallizes with space group P63/mmc (Patterson symbol) as shown in Fig. 1. Pd and Te atoms occupy 2a and 2c Wyckoff positions respectively. Pd(1): (0,0,1/2), Pd(2): (0,0,0); Te(1): (1/3,2/3,3/4) and Te(2): (2/3,1/3,1/4). Structure with different initial magnetic conﬁguration are calculated, it is found that nonmagnetic ground state converges with the lowest total energy and is stable at ambient condition which is in accordance with experiment [6]. Because nearly full-occupied Pdd orbitals favor neither superexchange mechanism nor ferromagnetic correlations. We utilize the experimental lattice parameters: a ¼ b ¼ 4.152 Å and c ¼ 5.672 Å as initial state to optimize the structure. The ﬁnal optimized lattice parameters are listed in

Table 1 Optimized structural parameters a (Å), c/a, zero-pressure bulk modulus B0 (GPa) and its pressure derivative B00 of PdTe at ambient condition, with other works for comparison.

This work GGAa Exp.b Exp.c a b c

a

c/a

B0

B00

4.220 4.221 4.1533 4.152

1.3671 1.355 1.3659 1.3658

98.3 102.9

4.87 5.25

Ref. [12]. Ref. [6]. Ref. [11].

Table 1. Deviation from experiment is less than 1.7%, indicating the reliability of our following calculations. 3. Results and discussion 3.1. Electronic structure Electronic band structure of PdTe shown in Fig. 2 is in good agreement with earlier calculations [12,13]. Both the conduction bands and valence bands cross the Fermi level. The crystal ﬁeld at the Pd site (D6h point symmetry) gives rise to the splitting of Pd d orbitals into a singlet a1g (d2z ) and two doublets e1g (dyz and dxz) and e2g (dxy and d2x2y). The crystal ﬁeld at the Te site (D3h point symmetry) causes the splitting of Te p orbitals into a singlet a4 (pz) and a doublet e2 (px and py). px, py, pz dominate almost equally the states at Fermi energy, indicating its three dimensionality character. While for superconductor MgB2 (Tc z 39 K), states at Fermi level are mainly composed of in-plane px and py states. The calculated band structure exhibits strong three-dimensionality as manifests from the different band structures in the G M K and A L H planes and the three-dimensional Fermi surface topology also directly support this picture. It is interesting to note that the Fermi energy EF is located near the valley of the overall density of states,

Fig. 1. (a) schematic view of PdTe crystal structure; (b) the Pd-centered edge and face sharing environment with octahedra; (c) The corresponding hexagonal prism Brillouin zone and high-symmetry points.

J. Chen, X. Wang / Solid State Sciences 52 (2016) 23e28

implying a low carrier density and it is relatively not difﬁcult to tune Tc by doping [19]. The electronephonon interaction mechanism is sensitive to the topology of Fermi surface [20,21]. Fermi surfaces of PdTe consist of one electron pocket located close to the zone center with purple and four hole pockets: two of them center at A and two at the K point. The three-dimensionality of PdTe electronic structure is obvious from the strong kz variation of the Fermi surface topology in Fig. 3. Such a strong three dimensionality indicates a much weaker Fermi surface nesting which is conﬁrmed by experiment [11].

3.2. Phonons and thermodynamic properties Fig. 4 presents phonon dispersion along high-symmetry directions and phonon density of states (PHDOS). There is no ﬁnite DOS below 0 cm1 and it thus reﬂects that there is not a single phonon branch which has negative frequency and the compound is fully dynamically stable at ambient condition in harmonic approximation. According to group theory, the ideal hexagonal structure (space group P63/mmc) with 4 atoms per unit cell displays 12 phonon modes, three of which are acoustic and nine are optical. The nine optical phonon modes at zone center belong to six irreducible representations, Namely E2u, E2g, A2u, B2g, E1u and B1u. E2g is Raman active and A2u, E1u is infrared active. E2u (E2g) and B1u

25

(B2g) are associated with PdePd (TeeTe) vibrations in counterphase only, with Te (Pd) atoms at rest. Representations E2u (E2g) are doubly degenerate and polarized in the xy plane, representations B1u (B2g) involves vibrations along z direction. Worth mention is that B1u branch shows high frequency, indicating strong Pd-layer interactions [22]. Other two mode are associated with displacements of all atoms. E1u are twofold degenerate and polarized in xy plane, with PdePd and TeeTe oscillating in-phase and different atoms counterphase. In A2u, PdePd and TeeTe oscillate in-phase and different atoms counterphase along z direction. Branches at zone center from low to high frequency in unit of cm1 are E2u (82.8926), E2g (111.9377), A2u (119.9919), B2g (122.4086), E1u (130.0336), B1u (166.5370) respectively. All the optical modes are strongly dispersive along most directions, reﬂecting the three dimensional character of PdTe. Resulting from the similar mass between Pd (106.42 a.u.) and Te (127.60 a.u.), there is no gap in phonon dispersion curves, suggesting relatively strong coupling of Pd and Te vibrations, which is also conﬁrmed from Fig. 4 (a), vibrations of two atoms contribute almost equally to the PHDOS except peak around 170 cm1 (Pd vibration only: mode B1u). However, there is a dispersionless optical branch and a gap around 4.5 THz in PdTe2, indicating weak interlayer interaction [3]. One of the main building blocks for hightemperature superconductivity is electronic and phononic degeneracy dispersion [23]. Compared with non-magnetic phonon

Fig. 2. (a) Calculated band structure of PdTe; (b,c,d) total and projected electron density of states (DOS), Fermi level is set at zero.

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J. Chen, X. Wang / Solid State Sciences 52 (2016) 23e28

Fig. 3. Calculated band resolved three-dimensional Fermi surface of PdTe.

dispersion of higher Tc (13 K with zero pressure) superconductor FeTe [7e9], phonon spectra of PdTe has less degeneracy in Brillouin zone especially in G M K plane. It is important to stress the ab initio calculations are carried out at ambient conditions (0 K and 0 GPa pressure), which may differ slightly from future neutron scattering experiment. Using the quasiharmonic approximation [24], We further obtain some thermo dynamic properties based on phonon density of states. The lattice contribution to speciﬁc heat at constant volume(Cv)can be evaluated by following expression [25]: u Zmax

Cv ¼ 3nNkB 0

Zu 2kB T

2

csc h2

Zu gðuÞdu 2kB T

(1)

Where n, kB, N, umax, g(u) are the number of atoms per unit cell, Boltzman constant, the number of unit cells, largest phonon frequency, phonon density of states respectively. According to Debye theory the speciﬁc heat can also be obtained by taking the derivative of the energy:

3 QZD =T T x4 e x Cv ¼ 9Nk dx QD ðex 1Þ2

(2)

0

Where QD is the only adjustable parameter Debye temperature. There are currently no experimentally measured phonon DOS data of PdTe. In Fig. 4 (b), we check our phonon calculation by ﬁtting Debye model with lattice speciﬁc calculated by Eq. (1). Debye temperature derived in this work is 197 K, which is in excellent accordance with experimental value 203 K, conﬁrming the validity of our work. In Fig. 4 (c,d), We carried out phonon dispersion under various hydrostatic pressuress. All branches are lifted upwards under hydrostatic pressure, with larger amplitude at lager pressure. It can be

accounted by the stifﬁng of bonds. This situation is consist with conventional pressure-induced phonon hardening [26,27].

3.3. Electron-phonon interaction and superconductivity We now turn our discussion to electron-phonon interaction. The Eliashberg function a2F(u) describes the averaged coupling strength between the electrons of Fermi energy and the phonons of energy u [20,28,29]:

a2 FðuÞ ¼

X gqy 1 d u uqy 2pNðεF Þ qy uqy

(3)

Where N(εF) is the electronic DOS at the Fermi level, uqy the phonon frequency, gqy is from Fermi “golden rule” as 2 P qy gqy ¼ 2puqy gkþqj0 ;kj dðεkj εF Þdðεkþqj0 εF Þ, kjj0

εkj is the electron eigenvalue with k and band index j, electron qy phonon matrix element gkþqj is deﬁned as: 0 ;kj 0

qy;j j

gkþqj0 ;kj ¼

X hqy ðR; vÞ D 0 dVeff pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ k þ q; j jk; j〉 dRy 2MR uqy R;y

(4)

Which describes the scattering probability of an electron with momentum k to a state with momentum kþq. hqy(R,v) is the eigenvectors of qy mode. The expression of electron-phonon coupling (EPC) constant l is also given in terms of the Eliashberg function:l¼2!dua2F(u)/u¼!dul(u). Based on the isotropic approximation in the Eliashberg theory, which describes the conventional electronephonon coupling superconductors appropriately, Transition temperature can be estimated from the Allen-Dynes formula [30,31]:

J. Chen, X. Wang / Solid State Sciences 52 (2016) 23e28

27

Fig. 4. (a) Total and atom projected phonon density of states (PHDOS) at ambient condition; (b) Lattice speciﬁc heat at constant volume Cv (in unit of R) of PdTe as a function of temperature from Eq. (1) and Debye model ﬁtting; Phonon dispersion of PdTe at ambient condition (black) and different pressure (red) with (c) 5 GPa (d) 12 GPa (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.).

2

lnðuÞ 5 du u

(5)

〈u〉log 1:04ð1 þ lÞ exp * 1:2 l m ð1 þ 0:62lÞ

(6)

〈u〉log ¼ exp4

2 l

a2 FðuÞ 0

Tc ¼

3

u Zmax

whereumax is the maximum phonon frequency, m* is the retarded Coulomb pseudopotential. It can be seen from mode resolved Eliashberg function Fig. 5 (a) and Fig. 4 (a) that the Eliashberg function are non-zero with all frequencies range and the variation trend is determined by the curves of total PHDOS. There demonstrates no diverge of main peaks positions of PHDOS and electron-phonon coupling function. Eliashberg has three peaks: 50 cm1, 95 cm1, 109 cm1 and 135 cm1 (the corresponding peaks in the PHDOS are situated at the 54 cm1, 95 cm1, 109 cm1 and 135 cm1 frequencies). Indicating that all modes contribute equally to electron-phonon coupling. The calculated logarithmically averaged frequency

calculated l is 0.66 which falls in the range of moderate electron phonon coupling. u* ¼ 0.10 is used in most theoretical calculations to determine the superconducting transition temperature of simple metals. Local-ﬁeld effect decreases the Coulomb repulsion parameter, while the exchangeecorrelation effect on the electron dielectric screening signiﬁcantly increases u* [32]. What's more, m* 0

varies slightly with pressure asm* ðPÞ ¼ m* ð0Þ½1 þ B0 P=B0 x=B0 , where x is volume dependent [33]. Exact calculation of u* under different conditions is devastating. For simplicity but without losing reliability, We treat m* as a adjustable parameter, estimated transition temperature is summarized in Table 2. It is expected that the discrepancy could be diminished by considering the spineorbit interaction, which would further soften phonons and a careful investigation of m* [34,35]. The Eliashberg function moves to higher frequency with increasing of pressure due to the hardening of overall phonon branches. Worse still, the amplitude of main peaks decrease with increasing of pressure, resulting a monotonously decreasing of Tc according to Eq. (5) and (6). Tc under different pressure are summarized in Table 2. Pressure can not enhance superconductivity of PdTe unlike on FeTe [36,37], which is a combining result of lacking of spin ﬂuctuations, nonlocality electron phonon coupling and overall hardening of phonons. 0

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J. Chen, X. Wang / Solid State Sciences 52 (2016) 23e28

(a)

(b)

Fig. 5. (a) Total and mode resolved Eliashberg function at ambient condition; (b) Eliashberg function under different pressures.

Table 2 Summary of calculated logarithmically averaged frequency

ambient 5 GPa 12 GPa GGAa Exp.b Exp.c Exp.d a b c d

Ref Ref Ref Ref

〈u〉log

QD(K)

l

Tc (K)

77.8 91.1 93.8

197

0.664 0.546 0.529 0.710

4.2e2.0 2.9e1.3 2.5e1.0

196 203 82

4.5 4.5 2.3

[12]. [6]. [11]. [5].

4. Conclusions Our calculated electronic properties of PdTe is in consistent with other works, phonon dispersion with non-negative frequency in the whole Brillouin zone and the agreement of Debye temperature with experiment result proves the validity of our calculation. Three dimension character and strong vibrational coupling from both electronic and lattice dynamic calculation is conﬁrmed. The electron-phonon spectral function has the same variation shape with PHDOS, indicating that superconductivity of PdTe originates from isotropically non-local electron-phonon coupling. It is also concluded that Tc decreases with increasing of pressure. Acknowledgments Support from the Scientiﬁc Research Fund of Guilin University of Aerospace Technology (NO. YJ1410) is highly appreciated for this work. References [1] G. Gruner, Rev. Mod. Phys. 60 (1988) 1129, http://dx.doi.org/10.1103/ RevModPhys.60.1129. [2] E. Morosan, H.W. Zandbergen, B.S. Dennis, R.J. Cava, et al., Nat. Phys. 2 (2006) 544, http://dx.doi.org/10.1038/nphys360. [3] T.R. Finlayson, Phy. Rev. B 33 (1986) 2473, http://dx.doi.org/10.1103/ PhysRevB.33.2473.

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