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First principle calculations of crystal field splittings and magnetic parameters for Nd3+ in NdBa2Cu3O7
Physica C: Superconductivity and its applications 565 (2019) 1353517
Contents lists available at ScienceDirect
Physica C: Superconductivity and its applications journal homepage: www.elsevier.com/locate/physc
First principle calculations of crystal field splittings and magnetic parameters for Nd3+ in NdBa2Cu3O7
T
Bang-Xing Lia,b, , Wen-lin Fenga,b, Shuang Xiana, De-Jiu Fenga, Chen Zhenga ⁎
a b
Department of Physics and Energy, Chongqing University of Technology, Chongqing, 400054, China Chongqing Key Laboratory of Green Energy Materials Technology and Systems, Chongqing 400054, China
ARTICLE INFO
ABSTRACT
Keywords: First-principles Crystal field parameters Magnetic susceptibility Energy levels Nd3+ ions
Nine crystal field (CF) parameters of Nd3+ in NdBa2Cu3O7 high-Tc superconductor with D2h symmetry were theoretically investigated on the basis of first-principle approach with the WIEN2k software package in the help of the Wannier projection. The calculated CF splitting energy levels, obtained through a reliable computer package with no other adjustable parameters by diagonalization of an effective Hamiltonian including spherically symmetric atomic parameters and ab initio calculated CF parameters, of Nd3+ ions in orthorhombic NdBa2Cu3O7 agree well with experimental values given by previous researchers. The magnetic parameters, including the magnetic susceptibility and g factors, were calculated through the perturbation method and the obtained results are in reasonable agreement with some observed values available in literature.
1. Introduction Since the YBa2Cu3O7, with a critical transition temperature above the boiling point of liquid nitrogen, has been discovered as the first super-conductor in the late 1980s, compounds of the type RBa2Cu3O7 (R denoted as rear earth ions) form an important family of superconducting materials with high current density when the temperature is below the critical temperature [1, 2]. When the Nd replaces Y to form a single phase, NdBa2Cu3O7 becomes an important high-Tc superconductors with maximum critical current density of 4.6 × 104 A/cm2 at 77 K, promising high field applications, higher critical temperature (near 92 K) and magnetic susceptibility [3–6]. Currently, the magnetic properties of this material are found to be closely related to the electronic and crystal-field properties of the Nd3+ site [7, 8]. Therefore, the crystal-field behaviors of NdBa2Cu3O7 have been studied with neutron spectroscopy [9], inelastic neutron scattering (INS) [10, 11] and inelastic magnetic neutron scattering [12]. The theoretical work on the CF splitting energy levels and magnetic properties of Nd3+ site in NdBa2Cu3O7 also becomes a valuable work. First principle calculations of the properties of molecules and solids have become a common tool of solid state physics and quantum chemistry. Comparing to traditional method, the first principle method can obtain the reliable values with less variable parameters, or even without any variable parameter. While the traditional method for
calculating the crystal field energy levels of rare-earth (RE) ions in crystals [13–15], all parameters, such as Slater parameters, two body configuration interaction parameters, three body parameters, magnetic parameters even including crystal field parameters (CFPs), are treated as the fitting parameters determined by matching the calculated and observed energy level schemes. Usually, the calculated energy level values can fitting very well with the experimental values with the condition of complicated fitting process and employment of a great number of fitting parameters (which can be as great as14 for D-ions and 27 for f-ions) [16]. For Nd3+ ions in orthorhombic structure with the symmetry of mmm (D2h), its ground multiplet will fully split into 5 Kramers doublets. The limited observations of energy levels (four levels for Nd) prevent the independent determination of all the crystal-field parameters (nine CF parameters). An over fit problem will emerge if we adopt the traditional fitting method to calculate the CF splitting energy levels. Soderholm et al. [10] calculated four energy levels of Nd3+ in NdBa2Cu3O7 with four adjustable parameters (B40, B44, B64 and an adjustable rescaling factor). It's a bad method for calculating 4 energy levels with 4 adjustable parameters, even the calculated energy levels are in good agreement with the experimental values. The big discrepancy between calculated energy levels and experimental values can be obtained from using three scaled crystal-field parameters for Nd3+ based on scaling ratios relative to Ho3+ of 0.886, 1.207 and 1.681 for
⁎ Corresponding author at: Department of Applied Physics, Chongqing University of Technology, No.69, Hongguang Road, Banan District, Chongqing, Chongqing 400054, China E-mail address: [email protected] (B.-X. Li).
https://doi.org/10.1016/j.physc.2019.1353517 Received 7 July 2019; Received in revised form 8 August 2019; Accepted 17 August 2019 Available online 19 August 2019 0921-4534/ © 2019 Elsevier B.V. All rights reserved.
Physica C: Superconductivity and its applications 565 (2019) 1353517
B.-X. Li, et al.
k = 2, 4 and 6, respectively, by Goodman [11]. In order to make more reliable analysis for limited observations of energy levels with CF theory, the CF splitting energy levels, magnetic susceptibility and the g factors of Nd3+ ions in NdBa2Cu3O7 superconductors were calculated uniformly based on the first principle approach with no other adjustable parameters except a fixed typical hybridization parameter, obtained by best fitting theory and experiment for Nd3+ in YAlO3 [17] and NdGaO3 [18], Δ=−8.2 eV for Nd3+ during the calculation process of CF parameters.
1 The experimental orthorhombic lattice parameters of NdBa2Cu3O7 [23] were used for calculation in this paper without any further optimization. The electronic structure, which feeds back a spherically symmetrical potential and density, of NdBa2Cu3O7: Nd3+ was obtained by standard self-consistent procedure with 4f orbitals treated as the core states and procedure stopped with the total energy smaller than 0.00001 Ryd. The eigenvalue problem was solved in four points of the irreducible Brillouin zone, the number of K points was 48, and the number of basis functions was ∼9200 (corresponding to parameter RKmax = 6.13). The muffin-tin radii for Nd, Ba, Cu, O atoms were 2.5, 2.5, 1.89, 1.63 respectively. 2 The non-self-consistent calculations were carried out with the 4f orbitals and 2 s and 2p states of oxygen treated as the valence states. Relative position of 4f and oxygen states was adjusted using the ‘hybridization’ parameter Δ (we adopt a fixed value Δ=−8.2ev for Nd3+ ions) 3 The 4f band states are transformed to Wannier basis and the local 4f Hamiltonian in the Wannier basis is extracted and expanded in series of spherical tensor operators. The coefficients of expansion are CF parameters.
2. Theory 2.1. Effective Hamiltonian The effective Hamiltonian operating on the 4f states without any external field can be expressed as [17]
^ =H ^ +H ^ H AT CF
(1)
^ is the spherically symmetric atomic Hamiltonian, which can Where H AT be written as [19] n
^ =E H AT AVE +
F kfk + k
4f si . li
+ HADD
2.3. Magnetic susceptibility and g factors
(2)
i=1
The ground multiplet of the Nd3+ ion in NdBa2Cu3O7 with the point symmetry of D2h will fully split into 5 Kramers doublets. Under an externally applied magnetic field, the Kramers energy levels for Nd3+ ions will be split due to the Zeeman Effect. The splitting energies for rare earth ions of odd number of 4f electrons depend on the magnetic field can be described as
Where
HADD = L (L + 1) + G (G2 ) + G (R7 ) + P k pk k
+
+
j
M jm
s
T sts (3)
j
k = 2, 4, 6; j = 0, 2, 4 and i is the sum over all electrons. The fk represent the angular operator part of the two-electron radial integrals, s and l are spin and orbital angular momentum operators, and the remaining operators related to the spin–spin and spin-other orbit interactions (mj) and the effect of additional configurations upon the spin–orbit interaction (pk). In Eq. (1). The atomic parameters of HAT are only weakly material dependent and we used the values determined by Yeung et al. [19] for ^ is the crystal-field term. In the Wybourne Nd3+ ions in LaCl3. H CF notation [20] it has the form
^ = H CF
k max
k
k Bqk C^q
±
n =
=
B
(5)
2
gJ2 NµB2 2
Ei(1) kB T
i (1)
2Ei(2) exp
Ei(0) kB T
(6)
(2)
where Ei , Ei and Ei are the zeroth, first- and second order perturbed Zeeman energies, respectively, and the other symbols have their usual meaning.
k
q
^B
B
(0)
where C^q is a spherical tensor operator of rank k acting on the 4f electrons of the rare earth ions, for which k = 2, 4 and 6 and q is summed from –k to k. The coefficients Bqk are the crystal-field para^ meters. Hermiticity of H requires that (B k )* = ( 1)qB k . For the CF
1 B 2
where ε(0) is the energy in zero field, g is the effective g-factor and χ is the Van Vleck susceptibility, which can be expressed as [23]
(4)
k=0 q= k
1
= (0) ± 2 µB B g ( n )
3. Results and discussion
q
NdBa2Cu3O7 crystals, which have orthorhombic Pmmm structure, the Nd cations are located on sites of D2h point symmetry, which leads to nine real independent crystal-field parameters.
3.1. CF parameters and CF splitting energy levels As shown in Table 1, the values of CF parameters, calculated from the novel method using the WIEN2k software package, of Nd3+ in NdBa2Cu3O7 have same order of magnitude with other methods, fitting or scaled. Combining the atomic parameters of HAT, five Kramers CF splitting energy levels can be obtained by diagonalization of an effective Hamiltonian. The calculated CF parameters of B22, B62 and B20, B60, B64 had different sign from [10,24], respectively. However, the
2.2. First principle calculation of CF parameters Values of the CFP are calculated from the novel method using the WIEN2k [14] software as proposed by Novák et al. [17, 18, 21, 22]. Which can be described as following main three steps.
Table 1 Values of Bkq (in meV) of Nd3+ ions at D2h site symmetry in NdBa2Cu3O7 calculated by first principle method.
Present [10]a [11]b [24]a a b
B20
B22
B40
B42
B44
B60
B62
B64
B66
87.1 51.4 50.4 −93.2
−52.9 22.2 10.1 58.4
−284 −327.4 −317.3 −369.7
−3.8 −2.5 −5.3 −14.4
166.9 210.5 148 221.7
83.3 74.9 104.4 −97.5
24.9 −31.8 −18 31.7
218.2 247 273.2 −317
−6.3 −0.5 −0.5 −17.6
Denote fitting method. Denote scale method. 2
Physica C: Superconductivity and its applications 565 (2019) 1353517
B.-X. Li, et al.
3.2. Magnetic susceptibility
Table 2 Comparison of experimental and Ab initio calculated CF energy levels (in meV) for Nd3+ ion in NdBa2Cu3O7. Cal.
[11]scale
Expt. [10]
Expt. [9]
Expt. [12]
0 10.68 20.52 36.98 103.07
0 3.85 25.23 38.36 117.4
0 12 20.8 36 117
0 12 20 36
0 12.6 21.5 40.5
The calculated and experimental (which are extracted from [24, 25]) temperature dependences of the susceptibilities of NdBa2Cu3O7 were presented in Fig. 1. According to this figure, the calculated susceptibilities have the same trend with the experimental values and the calculated average susceptibilities are same as the experimental ones [24, 25] when the temperature is higher than 100 K. The experimental ones are located between the calculated average susceptibilities and susceptibilities along a axial. This results may come from the difference from the axial chosen for experiment and for calculation. The g factors of five lowest energy levels of Nd3+ ions in NdBa2Cu3O7 have been calculated and compared with those in [23, 26] (see Table 3). According to the Table 3, the calculated values of average gx and gy based on the 1st and 5th Kramers doublets are in reasonable agreement with those in [26], and the calculated average value of gz is between the experimental values given by [23,26]. 4. Conclusion The crystal field parameters, energy levels, magnetic susceptibilities and g factors of Nd3+ in NdBa2Cu3O7 have been calculated by first principle method with no other adjustable parameters. According to our calculation, the method proposed by P. Novak, can be successfully used in the calculations of CFPs in cuprate superconductor materials. The crystal field splitting energy levels of Nd3+ in NdBa2Cu3O7 can be obtained from diagonalization of the effective Hamiltonian. The calculated energy values can agree very well with the experimental values given by previous papers. The calculated magnetic susceptibilities and g factors also have been compared with the experimental values given by relevant references. The good agreements between the calculated and experimental values are obtained.
Fig. 1. Comparison of calculated (denoted as χ_calc: average susceptibility, χ(x)_calc: susceptibility along a axial, χ(yz)_calc: susceptibility perpendicular to a axial) and experimental (χ_exp: extracted from [24], χ(c)_exp: susceptibility along the c axial extracted from [25], χ(ab)_exp: susceptibility parallel to ab plane extracted from [25]) temperature dependently magnetic susceptibility of NdBa2Cu3O7.
Acknowledgements This work was supported by the Project of Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJ1600913, KJQN201801133), and the National Natural Science Foundation of China (Grant No. 11547146), and University Innovation Team Building Program of Chongqing (No. CXTDX201601030).
Table 3 Comparison of experimental and calculated values of g factors of five Kramers doublets of Nd3+ ion in NdBa2Cu3O7 originating from 4I9/2 multiplet by the first principle. Doublet
gx
gy
gz
Cal. 1st Cal. 2nd Cal. 3rd Cal. 4th Cal. 5th Cal. Ave.* [23] [26]
3.00 −3.88 0.66 1.22 2.69 2.84 2.13 2.83
1.65 −1.37 −2.94 3.64 2.71 2.18 2.13 2.2
2.99 1.06 −1.40 −1.73 2.24 2.62 3.6 2.49
References [1] M.K. Wu, J.R. Ashburn, C.J. Torng, P.H. Hor, R.L. Meng, L. Gao, Z.J. Huang, Y.Q. Wang, C.W. Chu, Superconductivity at 93 K in a new mixed-phase Y-Ba-Cu-O compound system at ambient pressure, Phys. Rev. Lett. 58 (1987) 908–910. [2] X.X. Wei, R.S. Nagarajan, J.Xue E.P., J. Wang, J. Ding, Fabrication of YBa2Cu3O7-x (YBCO) superconductor bulk structures by extrusion freeforming, Ceram. Int 42 (2016) 15836–15842. [3] W.G. Suharta, H. Mugirahardjo, S. Pratapa, D. Darminto, S. Suasmoro, X-ray and high-resolution neutron diffraction studies on NdxY1−xBa2Cu3O7−δ superconductors, J. Supercond. Nov. Magn. 26 (2013) 3209–3214. [4] S.I. Yoo, N. Sakai, H. Takaichi, T. Higuchi, M. Murakami, Melt processing for obtaining NdBa2Cu3Oy superconductors with high Tc and large Jc, Appl. Phys. Lett. 65 (1994) 633–635. [5] P. Schätzle, W. Bieger, U. Wiesner, P. Verges, G. Krabbes, Melt processing of (Nd,Y) BaCuO and (Sm,Y)BaCuO composites, Supercond. Sci. Technol. 9 (1996) 869–874. [6] W. Bieger, U. Wiesner, G. Krabbes, P. Schätzle, A. Bauer, P. Verges, L. Zelenina, Melt texturing and properties control of Nd1+yBa2−yCu3Ox bulk materials, J. Low Temp. Phys. 105 (1996) 1445–1450. [7] N.H. Babu, M. Kambara, D.A. Cardwell, A.M. Campbell, Effect of oxygen content variation on flux pinning in Nd–Ba–Cu–O top-seeded melt grown superconductor, Supercond. Sci. Technol. 15 (2002) 702–707. [8] V. Likodimos, N. Guskos, H.G. Seale, A. Koufoudakis, M. Wabia, J. Typek, H. Fuks, Copper magnetic centers in oxygen deficient RBa2Cu3O6+x (R=Nd, Sm): an EPR and magnetic study, Phys. Rev. B. 54 (1996) 12342–12352. [9] P. Allenspach, A. Furrer, P. Bruesch, P. Unternaehrer, Crystal field in RBa2Cu3O7 (R = rare earth) determined by neutron spectroscopy, Phys. B 156-157 (1989) 864–866. [10] L. Soderholm, C.K. Loong, G.L. Goodman, B.D. Dabrowski, Crystal-field splittings and magnetic properties of Pr3+ and Nd3+ in RBa2Cu3O7, Phys. Rev. B 43 (1991)
Average g factors were calculated based on the g factors of 1st and 5th Kramers doublets. ⁎
calculated first four Kramers energy levels of Nd3+ ions in NdBa2Cu3O7 (shown in first column of Table 2) with these calculated parameters on the condition of no other adjustable parameters just have up to 1.32 meV discrepancy with experimental energy values given by L. Soderholm [10], P. Allenspach [9] and U. Walter [12]. The first four Kramers energy levels obtained with CF parameters (shown in fourth row of Table 1) scaled from Ho3+ with 3 adjustable parameters in [11] have as high as 8.15 meV discrepancy with experimental energy values given by [9], which shown in second column in Table 2. From the comparison of the calculated results, the novel method using the WIEN2k software package with no other adjustable parameters for calculating the CF splitting energy levels is much more reliable than scaled method from Ho3+. 3
Physica C: Superconductivity and its applications 565 (2019) 1353517
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[20] [21] [22] [23] [24] [25] [26]
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across the lanthanide series: the case of LaCl3:Ln3+, J. Phys. Chem. A 119 (2015) 6309–6316. B.G. Wybourne, Spectroscopic Properties of Rare Earth, Interscience, New York, 1965. P. Novák, V. Nekvasil, K. Knížek, Crystal field and magnetism with wannier functions: orthorhombic rare-earth manganites, J. Magn. Magn. Mater. 358-359 (2014) 228–232. P. Novák, J. Kuneš, K. Knížek, Crystal field of rare earth impurities in LaF3, Opt. Mater. 37 (2014) 414–418. N. Guskos, M. Calamiotou, S.M. Paraskevas, A. Koufoudakis, C. Mitros, H. GamariSeale, J. Kuriata, L. Sadlowski, M. Wabia, On the influence of oxygen deficiency on EPR spectra of NdBa2Cu3O7−δ, Phys. Status. Solidi. B 162 (1990) k101–k105. N. Ravindran, T. Sarkar, S. Uma, G. Rangarajan, V. Sankaranarayanan, Magnetic susceptibility, specific heat and crystal-field effects in neodymium cuprate superconductors, Phys. Rev. B 52 (1995) 7656–7662. N. Senthilkumaran, T. Sarkar, G. Rangarajan, C. Chen, J.W. Hodby, M. Spears, B.M. Wanklyn, Magnetic susceptibilities and crystalline electric field effects in single crystals of NdBa2Cu3O7-δ, Phys. B 223-224 (1996) 565–567. V. Likodimos, N. Guskos, H. Gamari-Seale, A. Koufoudakis, M. Wabia, J. Typek, H. Fuks, Copper magnetic centers in oxygen deficient RBa2Cu3O6+x (R=Nd, Sm): an EPR and magnetic study, Phys. Rev. B 54 (1996) 12342–12352.
Contents lists available at ScienceDirect
Physica C: Superconductivity and its applications journal homepage: www.elsevier.com/locate/physc
First principle calculations of crystal field splittings and magnetic parameters for Nd3+ in NdBa2Cu3O7
T
Bang-Xing Lia,b, , Wen-lin Fenga,b, Shuang Xiana, De-Jiu Fenga, Chen Zhenga ⁎
a b
Department of Physics and Energy, Chongqing University of Technology, Chongqing, 400054, China Chongqing Key Laboratory of Green Energy Materials Technology and Systems, Chongqing 400054, China
ARTICLE INFO
ABSTRACT
Keywords: First-principles Crystal field parameters Magnetic susceptibility Energy levels Nd3+ ions
Nine crystal field (CF) parameters of Nd3+ in NdBa2Cu3O7 high-Tc superconductor with D2h symmetry were theoretically investigated on the basis of first-principle approach with the WIEN2k software package in the help of the Wannier projection. The calculated CF splitting energy levels, obtained through a reliable computer package with no other adjustable parameters by diagonalization of an effective Hamiltonian including spherically symmetric atomic parameters and ab initio calculated CF parameters, of Nd3+ ions in orthorhombic NdBa2Cu3O7 agree well with experimental values given by previous researchers. The magnetic parameters, including the magnetic susceptibility and g factors, were calculated through the perturbation method and the obtained results are in reasonable agreement with some observed values available in literature.
1. Introduction Since the YBa2Cu3O7, with a critical transition temperature above the boiling point of liquid nitrogen, has been discovered as the first super-conductor in the late 1980s, compounds of the type RBa2Cu3O7 (R denoted as rear earth ions) form an important family of superconducting materials with high current density when the temperature is below the critical temperature [1, 2]. When the Nd replaces Y to form a single phase, NdBa2Cu3O7 becomes an important high-Tc superconductors with maximum critical current density of 4.6 × 104 A/cm2 at 77 K, promising high field applications, higher critical temperature (near 92 K) and magnetic susceptibility [3–6]. Currently, the magnetic properties of this material are found to be closely related to the electronic and crystal-field properties of the Nd3+ site [7, 8]. Therefore, the crystal-field behaviors of NdBa2Cu3O7 have been studied with neutron spectroscopy [9], inelastic neutron scattering (INS) [10, 11] and inelastic magnetic neutron scattering [12]. The theoretical work on the CF splitting energy levels and magnetic properties of Nd3+ site in NdBa2Cu3O7 also becomes a valuable work. First principle calculations of the properties of molecules and solids have become a common tool of solid state physics and quantum chemistry. Comparing to traditional method, the first principle method can obtain the reliable values with less variable parameters, or even without any variable parameter. While the traditional method for
calculating the crystal field energy levels of rare-earth (RE) ions in crystals [13–15], all parameters, such as Slater parameters, two body configuration interaction parameters, three body parameters, magnetic parameters even including crystal field parameters (CFPs), are treated as the fitting parameters determined by matching the calculated and observed energy level schemes. Usually, the calculated energy level values can fitting very well with the experimental values with the condition of complicated fitting process and employment of a great number of fitting parameters (which can be as great as14 for D-ions and 27 for f-ions) [16]. For Nd3+ ions in orthorhombic structure with the symmetry of mmm (D2h), its ground multiplet will fully split into 5 Kramers doublets. The limited observations of energy levels (four levels for Nd) prevent the independent determination of all the crystal-field parameters (nine CF parameters). An over fit problem will emerge if we adopt the traditional fitting method to calculate the CF splitting energy levels. Soderholm et al. [10] calculated four energy levels of Nd3+ in NdBa2Cu3O7 with four adjustable parameters (B40, B44, B64 and an adjustable rescaling factor). It's a bad method for calculating 4 energy levels with 4 adjustable parameters, even the calculated energy levels are in good agreement with the experimental values. The big discrepancy between calculated energy levels and experimental values can be obtained from using three scaled crystal-field parameters for Nd3+ based on scaling ratios relative to Ho3+ of 0.886, 1.207 and 1.681 for
⁎ Corresponding author at: Department of Applied Physics, Chongqing University of Technology, No.69, Hongguang Road, Banan District, Chongqing, Chongqing 400054, China E-mail address: [email protected] (B.-X. Li).
https://doi.org/10.1016/j.physc.2019.1353517 Received 7 July 2019; Received in revised form 8 August 2019; Accepted 17 August 2019 Available online 19 August 2019 0921-4534/ © 2019 Elsevier B.V. All rights reserved.
Physica C: Superconductivity and its applications 565 (2019) 1353517
B.-X. Li, et al.
k = 2, 4 and 6, respectively, by Goodman [11]. In order to make more reliable analysis for limited observations of energy levels with CF theory, the CF splitting energy levels, magnetic susceptibility and the g factors of Nd3+ ions in NdBa2Cu3O7 superconductors were calculated uniformly based on the first principle approach with no other adjustable parameters except a fixed typical hybridization parameter, obtained by best fitting theory and experiment for Nd3+ in YAlO3 [17] and NdGaO3 [18], Δ=−8.2 eV for Nd3+ during the calculation process of CF parameters.
1 The experimental orthorhombic lattice parameters of NdBa2Cu3O7 [23] were used for calculation in this paper without any further optimization. The electronic structure, which feeds back a spherically symmetrical potential and density, of NdBa2Cu3O7: Nd3+ was obtained by standard self-consistent procedure with 4f orbitals treated as the core states and procedure stopped with the total energy smaller than 0.00001 Ryd. The eigenvalue problem was solved in four points of the irreducible Brillouin zone, the number of K points was 48, and the number of basis functions was ∼9200 (corresponding to parameter RKmax = 6.13). The muffin-tin radii for Nd, Ba, Cu, O atoms were 2.5, 2.5, 1.89, 1.63 respectively. 2 The non-self-consistent calculations were carried out with the 4f orbitals and 2 s and 2p states of oxygen treated as the valence states. Relative position of 4f and oxygen states was adjusted using the ‘hybridization’ parameter Δ (we adopt a fixed value Δ=−8.2ev for Nd3+ ions) 3 The 4f band states are transformed to Wannier basis and the local 4f Hamiltonian in the Wannier basis is extracted and expanded in series of spherical tensor operators. The coefficients of expansion are CF parameters.
2. Theory 2.1. Effective Hamiltonian The effective Hamiltonian operating on the 4f states without any external field can be expressed as [17]
^ =H ^ +H ^ H AT CF
(1)
^ is the spherically symmetric atomic Hamiltonian, which can Where H AT be written as [19] n
^ =E H AT AVE +
F kfk + k
4f si . li
+ HADD
2.3. Magnetic susceptibility and g factors
(2)
i=1
The ground multiplet of the Nd3+ ion in NdBa2Cu3O7 with the point symmetry of D2h will fully split into 5 Kramers doublets. Under an externally applied magnetic field, the Kramers energy levels for Nd3+ ions will be split due to the Zeeman Effect. The splitting energies for rare earth ions of odd number of 4f electrons depend on the magnetic field can be described as
Where
HADD = L (L + 1) + G (G2 ) + G (R7 ) + P k pk k
+
+
j
M jm
s
T sts (3)
j
k = 2, 4, 6; j = 0, 2, 4 and i is the sum over all electrons. The fk represent the angular operator part of the two-electron radial integrals, s and l are spin and orbital angular momentum operators, and the remaining operators related to the spin–spin and spin-other orbit interactions (mj) and the effect of additional configurations upon the spin–orbit interaction (pk). In Eq. (1). The atomic parameters of HAT are only weakly material dependent and we used the values determined by Yeung et al. [19] for ^ is the crystal-field term. In the Wybourne Nd3+ ions in LaCl3. H CF notation [20] it has the form
^ = H CF
k max
k
k Bqk C^q
±
n =
=
B
(5)
2
gJ2 NµB2 2
Ei(1) kB T
i (1)
2Ei(2) exp
Ei(0) kB T
(6)
(2)
where Ei , Ei and Ei are the zeroth, first- and second order perturbed Zeeman energies, respectively, and the other symbols have their usual meaning.
k
q
^B
B
(0)
where C^q is a spherical tensor operator of rank k acting on the 4f electrons of the rare earth ions, for which k = 2, 4 and 6 and q is summed from –k to k. The coefficients Bqk are the crystal-field para^ meters. Hermiticity of H requires that (B k )* = ( 1)qB k . For the CF
1 B 2
where ε(0) is the energy in zero field, g is the effective g-factor and χ is the Van Vleck susceptibility, which can be expressed as [23]
(4)
k=0 q= k
1
= (0) ± 2 µB B g ( n )
3. Results and discussion
q
NdBa2Cu3O7 crystals, which have orthorhombic Pmmm structure, the Nd cations are located on sites of D2h point symmetry, which leads to nine real independent crystal-field parameters.
3.1. CF parameters and CF splitting energy levels As shown in Table 1, the values of CF parameters, calculated from the novel method using the WIEN2k software package, of Nd3+ in NdBa2Cu3O7 have same order of magnitude with other methods, fitting or scaled. Combining the atomic parameters of HAT, five Kramers CF splitting energy levels can be obtained by diagonalization of an effective Hamiltonian. The calculated CF parameters of B22, B62 and B20, B60, B64 had different sign from [10,24], respectively. However, the
2.2. First principle calculation of CF parameters Values of the CFP are calculated from the novel method using the WIEN2k [14] software as proposed by Novák et al. [17, 18, 21, 22]. Which can be described as following main three steps.
Table 1 Values of Bkq (in meV) of Nd3+ ions at D2h site symmetry in NdBa2Cu3O7 calculated by first principle method.
Present [10]a [11]b [24]a a b
B20
B22
B40
B42
B44
B60
B62
B64
B66
87.1 51.4 50.4 −93.2
−52.9 22.2 10.1 58.4
−284 −327.4 −317.3 −369.7
−3.8 −2.5 −5.3 −14.4
166.9 210.5 148 221.7
83.3 74.9 104.4 −97.5
24.9 −31.8 −18 31.7
218.2 247 273.2 −317
−6.3 −0.5 −0.5 −17.6
Denote fitting method. Denote scale method. 2
Physica C: Superconductivity and its applications 565 (2019) 1353517
B.-X. Li, et al.
3.2. Magnetic susceptibility
Table 2 Comparison of experimental and Ab initio calculated CF energy levels (in meV) for Nd3+ ion in NdBa2Cu3O7. Cal.
[11]scale
Expt. [10]
Expt. [9]
Expt. [12]
0 10.68 20.52 36.98 103.07
0 3.85 25.23 38.36 117.4
0 12 20.8 36 117
0 12 20 36
0 12.6 21.5 40.5
The calculated and experimental (which are extracted from [24, 25]) temperature dependences of the susceptibilities of NdBa2Cu3O7 were presented in Fig. 1. According to this figure, the calculated susceptibilities have the same trend with the experimental values and the calculated average susceptibilities are same as the experimental ones [24, 25] when the temperature is higher than 100 K. The experimental ones are located between the calculated average susceptibilities and susceptibilities along a axial. This results may come from the difference from the axial chosen for experiment and for calculation. The g factors of five lowest energy levels of Nd3+ ions in NdBa2Cu3O7 have been calculated and compared with those in [23, 26] (see Table 3). According to the Table 3, the calculated values of average gx and gy based on the 1st and 5th Kramers doublets are in reasonable agreement with those in [26], and the calculated average value of gz is between the experimental values given by [23,26]. 4. Conclusion The crystal field parameters, energy levels, magnetic susceptibilities and g factors of Nd3+ in NdBa2Cu3O7 have been calculated by first principle method with no other adjustable parameters. According to our calculation, the method proposed by P. Novak, can be successfully used in the calculations of CFPs in cuprate superconductor materials. The crystal field splitting energy levels of Nd3+ in NdBa2Cu3O7 can be obtained from diagonalization of the effective Hamiltonian. The calculated energy values can agree very well with the experimental values given by previous papers. The calculated magnetic susceptibilities and g factors also have been compared with the experimental values given by relevant references. The good agreements between the calculated and experimental values are obtained.
Fig. 1. Comparison of calculated (denoted as χ_calc: average susceptibility, χ(x)_calc: susceptibility along a axial, χ(yz)_calc: susceptibility perpendicular to a axial) and experimental (χ_exp: extracted from [24], χ(c)_exp: susceptibility along the c axial extracted from [25], χ(ab)_exp: susceptibility parallel to ab plane extracted from [25]) temperature dependently magnetic susceptibility of NdBa2Cu3O7.
Acknowledgements This work was supported by the Project of Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJ1600913, KJQN201801133), and the National Natural Science Foundation of China (Grant No. 11547146), and University Innovation Team Building Program of Chongqing (No. CXTDX201601030).
Table 3 Comparison of experimental and calculated values of g factors of five Kramers doublets of Nd3+ ion in NdBa2Cu3O7 originating from 4I9/2 multiplet by the first principle. Doublet
gx
gy
gz
Cal. 1st Cal. 2nd Cal. 3rd Cal. 4th Cal. 5th Cal. Ave.* [23] [26]
3.00 −3.88 0.66 1.22 2.69 2.84 2.13 2.83
1.65 −1.37 −2.94 3.64 2.71 2.18 2.13 2.2
2.99 1.06 −1.40 −1.73 2.24 2.62 3.6 2.49
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