Ab initio investigation of physical properties of LaT2B2C (T=ir, Rh) compounds: A density functional theory approach

Physica C: Superconductivity and its applications 568 (2020) 1353585

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Ab initio investigation of physical properties of LaT2B2C (T=ir, Rh) compounds: A density functional theory approach

T

⁎,a,b

H.Y. Uzunok a b

Sakarya Üniversitesi, Fen-Edebiyat Fakültesi, Fizik Bölümü, Sakarya, 54187, Turkey Sakarya Üniversitesi, BIMAYAM, Biyomedikal, Manyetik ve Yarıiletken Malzemeler Araştırma Merkezi, Sakarya, 54187, Turkey

ARTICLE INFO

ABSTRACT

Keywords: Intermetallics Density functional theory ab initiocalculations Electronic structure Superconductivity

The structural, electronic, lattice dynamical, and electron-phonon interaction properties of LaIr2B2C and LaRh2B2C crystallised in body-centred tetragonal structure are investigated by using the generalised gradient approximation of the density functional theory and the planewave pseudopotential method. The structural results are in good accordance with previous experimental studies. The electronic studies show that the density of states at the Fermi level of both borocarbide compounds are dominated by the d states of transition metal Ir(Rh). Also it is observed that the electron-phonon interaction in LaIr2B2C and LaRh2B2C are mainly carried out by the low-frequency phonon modes. By integrating Eliashberg spectral function, the superconducting transition temperature for LaIr2B2C and LaRh2B2C are obtained as 0.22 K and 0.20 K, respectively.

1. Introduction The superconductivity in quaternary borocarbides are taken interest for a long time. For this reason, numerous experimental and theoretical researches have been done to understand the mechanism of superconductivity in these compounds [1–49]. In particular, after the discovery of relatively high superconducting transition temperature (Tc) for Y-Pd-B-C borocarbide system at 23 K [2,11], these studies gather a pace. The experimental heat capacity measurements [7,8,12] and the effect of boron isotope experiments [36,37] suggest a BCS type phonon mediated superconductivity exist in these kind of compounds. Since the phonon-mediated superconductivity has found for these borocarbide compounds, several experimental and theoretical vibrational studies have done for LuNi2B2C-type compounds [38–41]. In 1994, new LuNi2B2C-type borocarbides [50,51] are synthesised LaIr2B2C and LaRh2B2C by using arc-melting technique. They could not find the superconductivity for these compounds down to 1.4 K. They have explained the reason of this occurrence by the position difference of the Fermi level. Ye and coworkers [52] synthesised different RRh2B2C (R= La to Er lanthanides except Eu) compounds to study the difference in structural properties. They have found that, if R goes from La to Er the a-lattice is shrinking while the c-lattice is expanding. In 2003, Paiva and colleagues [53] are suggested a layering model for superconductivity in the borocarbides. In this work they have propounded that, superconductivity is possible in these borocarbides only above a critical

⁎

Corresponding author. E-mail address: [email protected].

https://doi.org/10.1016/j.physc.2019.1353585 Received 14 November 2019; Accepted 4 December 2019 Available online 05 December 2019 0921-4534/ © 2019 Elsevier B.V. All rights reserved.

density and critical band offset. Anand and coworkers [54] is synthesised LaRh2B2C for using as a reference compound while investigating the PrRh2B2C. In this work, they have discovered that he electrical resistivity of LaRh2B2C is showed a metallic behaviour. In the literature search, there is no more experimental or theoretical information about the LaIr2B2C and LaRh2B2C compounds. Especially, the vibrational properties of a BCS-type superconductor is an essential study to understand the underlying mechanism of superconductivity. With this literature absence in mind, in this study, the structural,electronic, vibrational, and electron-phonon interaction properties of LuNi2B2C-type LaIr2B2C and LaRh2B2C compounds are investigated and presented. A generalised gradient approximation (GGA) scheme of the density functional theory has been used in these calculations. Hence the main goal of this study is to understand the underlying mechanism of superconductivity in borocarbide compounds, the Eliashberg spectral function, the average electron-phonon coupling constant, the logarithmic average frequency, and the Tc value for LaIr2B2C and LaRh2B2C compounds are calculated and compared with available experimental reports. 2. Method Quantum-Espresso package [55,56] is used to perform the calculations. While the Perdew-Burke-Ernzorhof GGA scheme [58] is used for solving Kohn-Sham [57] method, the scalar-relativistic ultrasoft

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H.Y. Uzunok

occurred from the C-B-M4 atoms. the equilibrium lattice constants a, and c, the internal parameter z, the bulk modulus parameter(B) and its pressure derivative (B ) are all calculated from the total energy which is taken as a function of lattice constant [62]. The calculated parameters are shown in Table 1 with available previous studies. Our structural calculations and corresponding previous experimental results [51] are in good accordance that suggests our structural results are trustworthy. For a better understanding of the electronic structure of studied materials, their electronic band structure and corresponding total and partial density of states are calculated and displayed in Fig. 2 for LaIr2B2C and in Fig. 3 for LaRh2B2C. As can be seen from the Fig. 2(a) and Fig. 3(a), both compounds show metallic behaviour due to the electronic bands cross the Fermi level. The distinguished feature for borocarbides are the flat bands along the Z-Γ and Γ-X which are mentioned in our previous studies [48,49]. We could see the flat bands along the Z-Γ direction in Fig. 2(a) for LaIr2B2C just above the Fermi level and in Fig. 3(a) for LaRh2B2C at the Fermi level. On the other hand, the electronic bands near the Fermi level window along the Γ-X direction show dispersive characters and no flat band is observed along this direction. Hence we could expect a lower density of states at the Fermi level (N(EF) for these studied borocarbides than the previous studied ones [48,49]. This could be a sign for a low strength electronphonon interaction for LaIr2B2C and LaRh2B2C hence due to McMillanHopfield equation the electron-phonon coupling parameter can be deduced from;

Fig. 1. The Body-centred tetragonal LuNi2B2C-type crystal structure of LaT2B2C (T=Ir or Rh) superconductor.

=

pseudopotentials [59] is used to explain interactions between core and valance electrons. An energy cutoff of 60 Ry is used to calculate wave functions in plane waves. The Monkhorst-Pack scheme [60] is used to realise a special k-points for the irreducible Brillouin-zone. While a (8 × 8 × 8) zone-centred grid is taken to obtain the structural parameters, electronic properties are calculated with a (24 × 24 × 24) zone-centred grid. Lattice dynamical properties are calculated by using the linear-response method [55] that allows the calculation of the dynamical matrix at arbitrary q vectors. On a 4 × 4 × 4 grid in q space, there are thirteen dynamical matrices have been calculated. The method for the electron-phonon interaction has been well described in the previous work [49].The electron-phonon matrix elements are performed using 24 × 24 × 24 k-mesh with a Gaussian width 0.02 Ry.

N (EF ) < I 2> . M < 2>

(1)

where < I2 > shows the averaged square of the electron-phonon matrix element, < ω2 > is the averaged square of the phonon frequency, and M is the involved mass. Since the BCS theory suggest that the Cooper pairs are mainly contributed by the electrons near the Fermi energy level [63], we have analysed the electronic density of states (DOS) for LaIr2B2C and LaRh2B2C. DOS for LaIr2B2C are presented in Fig. 2(b) and (c). The energy levels in the range from −5.0 eV to Fermi level, which is fixed at 0 eV, is mainly contributed by Ir 5d orbital electrons with some contribution coming from C 2p, La 5d, and B 2p orbitals. The contribution ratios from these orbitals to the DOS at the Fermi level (N(EF)) are around 56%, 20%, 12%, and 8%, respectively. These calculations suggest that, the N(EF) is mainly constituted by the d orbital of Ir atom. The contribution ratios to the N(EF) from atoms are 58% for Ir, 20% for C, 13% for La, and 9% for B. Similar observation is made for LaRh2B2C compound which DOS figures are presented in Fig. 3(b) and (c). The contribution ratios from Rh 5d, C 2p, La 5d, and B 2p orbitals to the N (EF) are 51%, 18%, 13%, and 13%, respectively. There is a difference in contributions from transition metals between LaIr2B2C and LaRh2B2C. We could say that, for both compounds, the N(EF) is mainly constituted by the d orbital of transition metal Ir and Rh atoms for both compound. The calculated N(EF) values for the studied compounds are 2.33 states/ eV for LaIr2B2C and 2.21 states/eV for LaRh2B2C. This observation shows that transition metal Ir and Rh electrons effect the superconducting properties of the studied compounds since Cooper pairs in the BCS theory can be formed by electrons which have energies close to the Fermi level. Also, since all the atoms contribute the N(EF) value, it is communicable that the studied compounds have a 3d electronic

3. Results 3.1. Structural and electronic properties LuNi2B2C-type body-centred tetragonal (BCT) structure of LaT2B2C (T=Ir, Rh) is shown in Fig. 1. This structure belongs to the I4/mmm space group that the atoms are placed as La at 2a (0, 0, 0), T (Ir, Rh) at 4d(0, 1/2, 1/4), B at 4e(0, 0, z) and C at 2b(0, 0, 1/2). The before mentioned z is known as internal parameter. Hence, we can deduce this BCT structure with three parameters; a, c, and z. The LuNi2B2C-type structure is formed by atomic layers along the c-axis. The transition metal square-plane is stayed between the interval of B atomic layers like a sandwich. It could be seen from the Fig. 1, there is a tetrahedral

Table 1 Static properties of the superconducting BCT compounds LaIr2B2C, and LaRh2B2C, and their comparison with previous experimental results.

LaIr2B2C Exp. [51] LaRh2B2C Exp. [51]

a(Å)

c(Å)

V(Å3)

z

dT

3.892 3.897 3.902 3.902

10.505 10.454 10.318 10.246

79.58 79.38 78.57 78.00

0.356 0.350 0.354 0.3518

2.752 2.755 2.759 2.759

2

T

dB

C (Å)

1.516 1.567 1.502 1.519

dT

B (Å)

2.241 2.212 2.229 2.212

B(GPa)

B

214.9

4.50

186.1

5.12

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Fig. 2. (a) The electronic band structure of LaIr2B2C along specific paths of the body-centred tetragonal Brillouin zone. The Fermi energy corresponds to 0 eV. (b), (c) The total and atomic projected electronic local density of states for LaIr2B2C.

Fig. 3. (a) The electronic band structure of LaRh2B2C along specific paths of the body-centred tetragonal Brillouin zone. The Fermi energy corresponds to 0 eV. (b), (c) The total and atomic projected electronic local density of states for LaRh2B2C.

structure.

phonon modes including three acoustic and fifteen optical phonon modes. From the group theory, the symmetries of the zone-centre optical phonon modes are obtained as:

3.2. Phonon and electron-phonon interaction properties The zone-centre phonon modes of BCT LaIr2B2C and LaRh2B2C are belongs to the point group D4h(4 mm). There are totally eighteen 3

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highest λ valued phonon modes contains Ir atom’s vibrations. Calculated phonon dispersion curves along the high symmetry directions and the total and partial density of states for LaIr2B2C and LaRh2B2C are presented in Fig. 4, and in Fig. 5, respectively. From the phonon dispersions, it is obvious that both of the studied compounds are dynamically stable since there are no imaginary phonon frequencies observed. Both LaIr2B2C and LaRh2B2C compounds’ phonon dispersions could be divided into five different regions which suggest us a phononic characteristic of these compounds. First region is lengthen from 0 to 6.0 THz for LaIr2B2C and 0 to 7.4 THz for LaRh2B2C that contain three acoustic and six optical phonon modes. The low optical phonon modes are showed a dispersive behaviour and overlapped with the acoustic branches. The second region for studied compounds contains three, third region contains four, fourth and fifth regions contain one for each phonon branches. All phonon branches show a dispersive attitude along the high symmetry directions. Even though the first frequency region is containing all the atoms vibration, the heavier atoms La and T (Ir or Rh) are dominated this region (see Fig. 4 for LaIr2B2C and Fig. 5 for LaRh2B2C). The higher frequency regions are formed by the vibrations of lighter B and C atoms. Due to this high mass difference, the first region is separated from the others with a clear gap such as 4.5 THz for LaIr2B2C and 3.0 THz for LaRh2B2C. In particular, the La atom’s vibrations are nearly disappeared at the frequencies greater than 4.2 THz for LaIr2B2C as it can be seen from the Fig. 4. Also the vibrations of Ir atom is also nearly disappeared after the first frequency region. Second frequency region between 10.5–12 THz is consisted of the B and C atoms’ vibrations. A high hybridisation between these atoms could be realised from this region. The third frequency region lays between 12.7 and 15.0 THz. Even though the section between 12.7 and 13.8 THz of

Table 2 Calculated zone-centre phonon modes (in THz), their electron-phonon coupling parameters and eigen characters for BCT LaIr2B2C, and LaRh2B2C. The notations of I, and R denote infrared and Raman active modes, respectively. LaIr2B2C

LaRh2B2C

Mode

ν

λ

Motion

ν

λ

Motion

Eu(I) A2u(I) B1g(R) Eg(R) Eu(I) A2u(I) Eg(R) Eu(I) A1g(R) A2u(I)

2.98 3.58 3.98 5.05 10.51 11.60 13.07 13.78 24.01 35.71

0.018 0.021 0.117 0.118 0.012 0.006 0.018 0.009 0.060 0.003

La+Ir+C La+Ir+B+C Ir Ir+B B+C B+C B B+C B B+C

3.37 3.90 3.47 4.77 10.33 10.25 12.20 13.63 24.84 36.43

0.009 0.022 0.070 0.050 0.007 0.004 0.017 0.003 0.042 0.003

La+Rh+C La+Rh+B+C Rh Rh+B B+C B+C B B+C B B+C

(D4h (4 mm)) = 3Eu + 2Eg + 3A2u + A1g + B1g

(2)

where E modes are double and others are single degenerate. Also subscripts “g” and “u” represent “gerade” and “ungerade” wave functions, respectively. While gerade functions are Raman active, the ungerade functions are infrared active. The eigendisplacements, frequencies and electron-phonon coupling parameters of these phonon modes for LaIr2B2C and LaRh2B2C are presented in Table 2. For both compounds the first Eg modes and the B1g mode have the relatively highest electronphonon interaction coupling (λ) among the zone-centre phonon modes. These modes are mainly consist of Ir atoms which have the highest contribution to the N(EF) value. Hence, it is not surprising that the

Fig. 4. (a) The phonon spectrum of LaIr2B2C along specific paths of the body-centre tetragonal Brillouin zone (right side) and the total and partial phonon density of states (on the left side). 4

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Fig. 5. (a) The phonon spectrum of LaRh2B2C along specific paths of the body-centred tetragoanl Brillouin zone (right side) and the total and partial phonon density of states (on the left side).

this region is only comprise of B vibrations, it is quite clear there is a strong hybridisation between B and C atoms at the rest of the region. These hybridisations are the evidence of strong B-C bond within the BCT structure of LaIr2B2C. The fourth frequency region is only formed by B atoms and the fifth region has the same hybridisation of B and C atoms while the C atom dominates this region. Similar observation could be made for LaRh2B2C compounds from the Fig. 5. Since the transition metal Ir(Rh) are mainly contribute the low frequency region, it is expected that the electron-phonon interaction mostly originate from this region due to these atoms domination over the N(EF).

The aim of this study is to analyse the electron-phonon interaction strength in BCT LaIr2B2C and LaRh2B2C compounds to understand the origin of superconductivity in these compounds. For this reason, the figure of the Eliashberg spectral function α2F(ω) and the electronphonon coupling parameter λ subject to frequency is illustrated in Fig. 6(a) for LaIr2B2C and in Fig. 6(b) LaRh2B2C. From these figures, we could say that the lowest frequency region contributes to the λ value around 69% for LaIr2B2C and 61% for LaRh2B2C. These results show that, more than the half of λ is originated from low frequency modes for both studied compound. Since the lowest frequency region is largely 5

Physica C: Superconductivity and its applications 568 (2020) 1353585

H.Y. Uzunok

[61];

TC =

ln

1.2

exp

1.04(1 + ) . µ* (1 + 0.62 )

(4)

The values of Tc are found to be 0.22 K for LaIr2B2C and 0.20 K for LaRh2B2C. As the previous experimental results suggest that both of the studied compounds have Tc values lesser than 1.4 K [2]. It is not possible to say a perfect match but there is an agreement between the presented calculations and previous experimental results. It is obvious that, both LaIr2B2C and LaRh2B2C do not have a high Tc values compared with the previously studied quetarnary borocarbide compounds such as YPd2B2C or LuNi2B2C [48,49]. 4. Summary In this study, the structural, electronic, lattice dynamical, and electron-phonon interaction properties of LaIr2B2C and LaRh2B2C crystallised in body-centred tetragonal structure are investigated by using the generalised gradient approximation of the density functional theory and the planewave pseudopotential method. The calculated structural parameters for these compounds are in accordance with available experimental results. The existence of a flat bands along the ΓZ directions near the Fermi energy level is the distinguishing characteristic of LuNi2B2C-type borocarbides. The density of states at the Fermi level is dominated by the d states of Ir and Rh transition metal atoms. The phonon dispersion, and phonon total and partial density of states are calculated by linear response method and presented in detail. Vibrational calculations confirm that both compounds are dynamically stable in their body-centred tetragonal phase. From the calculated Eliashberg spectral function and electron-phonon interaction calculations, it could be suggested that the low-frequency phonon modes are mainly involved in the process of scattering of electrons than the higher frequency phonon modes. Using the Allen-Dynes modified McMillian equation with the screened Coulomb pseudopotential parameter 0.10, the value of superconducting transition temperatures are found to be 0.22 K and 0.20 K for LaIr2B2C and LaRh2B2C, respectively (Table 3).

Fig. 6. The calculated Eliashberg spectral function α2F(ω) (black lines) and the electron-phonon coupling constant λ (red lines) for the (a) LaIr2B2C, and (b) LaRh2B2C. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Declaration of Competing Interest

Table 3 The calculated values of physical quantities connected to superconductivity in the BCT LaIr2B2C and LaRh2B2C. Method

N(EF)(States/eV)

ωln (K)

λ

Tc (K)

LaIr2B2C Exp. [50] LaRh2B2C Exp. [50]

2.33

272.4

0.32

2.20

254.11

0.31

0.22 < 1.4 0.20 < 1.4

This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue. I have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript and I declare that there is no conflict of interest. Acknowledgement I want to thank the Prof. Dr. H. M. Tütüncü for his invaluable advices. Some of the numerical calculations were performed using the Intel Nehalem (i7) cluster (ceres) at the University of Exeter.

contributed by the Ir atom for LaIr2B2C and 61% and Rh atom for LaRh2B2C, which are also dominated their compounds N(EF) values, it is an expected result because of the BCS theory. With the help of the α2F (ω), the average λ values are found to be 0.32 for LaIr2B2C and 0.31 for LaRh2B2C. These values show that both studied compounds have a weak electron-phonon interaction. From these weak electron-phonon interaction, we could estimate a low Tc for the studied compounds. With using calculated λ values, the logarithmically averaged phonon frequency ( ln ) is defined as; ln

= exp 2

d

1 0

2F

( ) ln

.

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(3)

The ln values are calculated as 272.4 K for LaIr2B2C and 254.1 K for LaRh2B2C. By using λ and ln values with a reasonable effective screened Coulomb repulsion constant μ* = 0.10, the theoretical TC values are obtained by the Allen-Dynes modificated McMillian formula 6

Physica C: Superconductivity and its applications 568 (2020) 1353585

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