⇦Chinese Physics Letters, 2022, Vol. 39, No. 6, Article code 067402 First-Principles Study of Hole-Doped Superconductors $R$NiO$_2$ ($R$ = Nd, La, and Pr) Juan-Juan Hao (蒿娟娟)1†, Pei-Han Sun (孙培函)2†, Ming Zhang (张铭)1, Xian-Xin Wu (吴贤新)3,4*, Kai Liu (刘凯)2*, and Fan Yang (杨帆)1* Affiliations 1School of Physics, Beijing Institute of Technology, Beijing 100081, China 2Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials & Micro-nano Devices, Renmin University of China, Beijing 100872, China 3CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China 4Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany Received 18 March 2022; accepted 2 May 2022; published online 29 May 2022 †Juan-Juan Hao and Pei-Han Sun contributed equally to this work.
*Corresponding author. Email: xianxinwu@gmail.com; kliu@ruc.edu.cn; yangfan_blg@bit.edu.cn Citation Text: Hao J J, Sun P H, Zhang M et al. 2022 Chin. Phys. Lett. 39 067402    Abstract Recent experiments have found that in contrast to the nonsuperconducting bulk $R$NiO$_2$ ($R$ = Nd, La, and Pr), the strontium-doped $R_{1-x}$Sr$_x$NiO$_2$ thin films show superconductivity with the critical temperature $T_{\rm c}$ of 9–15 K at $x=0.2$, whose origin of superconductivity deserves further investigation. Based on first-principles calculations, we study the electronic structure, lattice dynamics, and electron–phonon coupling (EPC) of the undoped and doped $R$NiO$_2$ ($R$ = Nd, La, and Pr) at the experimental doping level. Our results show that the EPC-derived $T_{\rm c}$'s are all about 0 K in the undoped and doped $R$NiO$_2$. The electron–phonon coupling strength is too small to account for the observed superconductivity. We hence propose that the electron–phonon interaction can not be the exclusive origin of the superconductivity in $R$NiO$_2$ ($R$ = Nd, La, and Pr).
cpl-39-6-067402-fig1.png
cpl-39-6-067402-fig3.png
 DOI:10.1088/0256-307X/39/6/067402 © 2022 Chinese Physics Society Article Text Superconducting materials with high transition temperature ($T_{\rm c}$) have been explored for many years. Traditionally, the pairing mechanism of superconductivity is described by the Bardeen–Cooper–Schrieffer (BCS) theory,[1] in which the phonons mediate the effective attraction between a Cooper pair through the electron–phonon coupling (EPC). Superconductors that can be well described by the BCS theory are usually referred to as conventional superconductors, such as Hg,[2] MgB$_{2}$,[3] and LaH$_{10}$,[4] and the remaining ones are called unconventional superconductors.[5] Their differences lie in the formation mechanism of Cooper pairs[6] and the symmetry of the superconducting gap function. For example, in copper oxide superconductors, the superconducting gap is anisotropic and d-wave,[7] and the formation of Cooper pairs is not caused by electron–phonon interactions, but may be related to magnetic fluctuations.[8] Recently, enormous attention has been paid to the research of undoped and hole-doped[9–39] infinite-layer nickelates $R$NiO$_2$ ($R$ = Nd, La, and Pr) from soft-chemistry topotactic reduction. Experiments show that, for $R$NiO$_2$ ($R$ = Nd, La, and Pr) materials, the hole doped $R$NiO$_2$ thin films exhibit superconductivity,[18–20] while bulk $R$NiO$_2$ does not.[39] For example, the infinite layer of strontium-doped NdNiO$_2$ film (Nd$_{0.8}$Sr$_{0.2}$NiO$_2$) is found to be superconducting and its critical temperature $T_{\rm c}$ reaches up to 15 K.[18] For La$_{1-x}$Sr$_x$NiO$_2$ ($x = 0.14$–0.20), a superconducting transition follows, and the maximum $T_{\rm c}$ (9 K) occurs at $x=0.20$.[19] In the PrNiO$_2$ film synthesized on an SrTiO$_3$ substrate, it was observed that the phase transition temperature of Pr$_{0.8}$Sr$_{0.2}$NiO$_2$ superconductor is 7–12 K.[20] Previous calculations[11] suggest that the EPC strength in undoped NdNiO$_2$ is not strong enough to regulate superconductivity around $T_{\rm c} \sim 10$ K and the EPC is excluded as the only source of superconductivity. However, experimentally undoped NdNiO$_2$ itself is not superconducting. Hence, in order to explain the origin of superconductivity in doped $R$NiO$_2$ ($R$ = Nd, La, and Pr), it is actually more reasonable to study it with the hole doping. According to the above experiments, the optimal hole doping in $R$NiO$_2$ is 0.2 with the maximum transition temperature $T_{\rm c}$ about 15 K.[18–20] Therefore, we choose the 0.2 hole-doping case to verify whether the superconductivities of NdNiO$_2$ and its sister compounds LaNiO$_2$ and PrNiO$_2$ originate from EPC. Based on the first-principles calculations and the EPC theory, we have made a comparative study on the electronic structure, phonon properties, EPC strength, and superconducting transition temperature of $R$NiO$_2$ ($R$ = Nb, La, and Pr) under the undoped and hole-doped conditions. The electronic structure, phonon spectrum, and EPC of undoped and doped $R$NiO$_2$ ($R$ = Nd, La, and Pr) were calculated by using Quantum ESPRESSO (QE) package[40] based on density functional theory[41,42] and density functional perturbation theory.[43] The generalized gradient approximation of Perdew–Burke–Ernzerhof type[44] was adopted for the exchange-correlation functional. The interactions between electrons and nuclei were described by the projected augmented wave method as implemented in PSlibrary.[45] The kinetic energy cutoff of the plane-wave function is 75 Ry and that of the charge density is 600 Ry. Both lattice constants and internal atomic positions were fully optimized. The hole doping effect was simulated by directly removing electrons from the system together with a compensating uniform charge background to maintain the charge neutrality. A $12\times12\times12$ $k$-grid was used for the self-consistent calculations. In the calculations of the dynamical matrix and the EPC, the Brillouin zone was sampled with a $6\times6\times6$ $q$-grid and a dense $48\times48\times48$ $k$-point mesh. Based on the BCS theory,[1] the Eliashberg spectral function $\alpha^{2} F(\omega)$ is defined as[46] $$ \alpha ^{2}F(\omega)=\frac{1}{2\pi N(\varepsilon _{\scriptscriptstyle{\rm F}})}\sum_{q\nu }^{}\delta (\omega -\omega _{q\nu})\frac{\gamma _{q\nu}}{\hbar\omega _{q\nu} },~~ \tag {1} $$ where $N(\varepsilon _{\scriptscriptstyle{\rm F}})$ is the density of states (DOS) at Fermi level $\varepsilon _{\scriptscriptstyle{\rm F}}$, $\omega _{q\nu}$ is the frequency of the $\nu$th phonon mode at wave vector $q$, and $\gamma _{q\nu}$ is the phonon linewidth. The total EPC constant $\lambda$ can be obtained by[46] $$ \lambda=\sum_{q\nu}^{}\lambda _{q\nu} = 2\int \frac{\alpha ^{2}F(\omega) }{\omega } d\omega.~~ \tag {2} $$ The superconducting transition temperature $T_{\rm c}$ can be determined by substituting the EPC constant $\lambda$ into the McMillan–Allen–Dynes formula,[47] $$ T_{\rm c}=\frac{\omega _{\log} }{1.2}{\exp}\Big[\frac{-1.04(1+\lambda)}{\lambda(1-0.62\mu ^{\ast })- \mu ^{\ast }}\Big],~~ \tag {3} $$ where the logarithmic average of Eliashberg spectral function $\omega _{\log}$ is defined as $$ \omega _{\log}={\exp} \Big[\frac{2}{\lambda}\int \frac{d\omega }{\omega } \alpha ^{2}F(\omega){\ln}(\omega)\Big],~~ \tag {4} $$ and $\mu ^{\ast}$ is the effective screened Coulomb repulsion constant with the empirical values between 0.08 and 0.15.[48,49] In our calculation, $\mu ^{\ast}$ was set to 0.1. The tetragonal crystal structure of infinite-layer nickelates $R$NiO$_2$ ($R$ = Nd, La, and Pr) with the space group $P4/mmm$ (No. 123) is shown in Fig. 1(a). Taking the NdNiO$_2$ as an example, the optimized lattice constants of undoped case are $a = 3.893$ Å and $c = 3.267$ Å, in good accordance with the reported experimental data.[50] With a doping concentration of 0.2 holes per formula unit (f.u.), the lattice constants decrease slightly to $a = 3.809$ Å and $c = 3.186$ Å, which is also consistent with the measured data.[11] Figure 1(b) displays the Brillouin region (BZ) of the primitive cell along with the high-symmetry $k$ points.
cpl-39-6-067402-fig1.png
Fig. 1. (a) Electronic band structure of NdNiO$_{2}$ without doping (black lines) and with a doping concentration of 0.2 holes per formula unit (red lines). The two bands across the Fermi level $E_{\rm F}$ are projected onto the Nd $5d$ and Ni $3d$ orbitals and denoted with green and blue dots, respectively. The size of the dots is proportional to the weights of the corresponding orbitals. (b) Total and partial density of states (PDOS) without doping (solid lines) and after doping (dashed lines). Inset shows the PDOS of Ni $3d$ orbitals in undoped NdNiO$_{2}$. [(c), (d)] The top and three-dimensional side views of the Fermi surfaces without and with doping, respectively.
We next explore the phonon properties and the EPC of undoped and doped $R$NiO$_2$ ($R$ = Nd, La, and Pr). We also take NdNiO$_2$ as an example to illustrate our results. Figures 3(a) and 3(d) show the phonon spectra before and after hole doping, respectively. We can see that there is no imaginary phonon mode in the whole BZ, indicating that they are both dynamically stable. In addition, we find that the phonon spectrum is hardened significantly after the hole doping, for which the maximum phonon frequency increases from 566 cm$^{-1}$ to 641 cm$^{-1}$. This is also reflected in the change in $\omega _{\log}$ (Table 1), which goes from 230 cm$^{-1}$ to 283 cm$^{-1}$ with the hole doping. The phonon density of states of undoped and doped NdNiO$_2$ are displayed in Figs. 3(b) and 3(e), respectively. We learn that the low-frequency branches are mainly contributed by the vibrations of Nd and Ni atoms, while the high-frequency branches are dominated by the O atomic vibrations. The calculated Eliashberg spectral function $\alpha^{2} F(\omega)$ and integrated EPC constant $\lambda$ of undoped and doped NdNiO$_2$ are shown in Figs. 3(c) and 3(f), respectively. It can be seen that the integrated $\lambda(\omega)$ (the red curve) decreases significantly after doping. Meanwhile, the peak at around 100 cm$^{-1}$ in $\alpha^{2} F(\omega)$ also reduces remarkably. The calculated total EPC constant $\lambda$ of undoped NdNiO$_2$ is only 0.172, which is further reduced to 0.128 with the hole doping of $x=0.2$. Such reduction of $\lambda$ can be attributed to the hardening of phonon spectra and the decrease of $\alpha^{2} F(\omega)$ [Fig. 3 and Eq. (2)]. Based on the McMillan–Allen–Dynes formula [Eq. (3)], we obtain the superconducting $T_{\rm c}$'s of both undoped and doped NdNiO$_2$ as nearly 0 K, see Table 1. Meanwhile, the calculated $T_{\rm c}$'s of LaNiO$_2$ and PrNiO$_2$ also approach 0 K in the undoped and doped cases. In order to display the influence of hole doping on $T_{\rm c}$, the values are re-represented by scientific notation, with their actual values and orders of magnitude. Take NdNiO$_2$ as an example, the $T_{\rm c}$ is $6.452 \times 10^{-7}$ K before doping and $1.373 \times 10^{-23}$ K after doping. Obviously, the latter is much less than the former, which indicates that the EPC is not suitable for this system. The same results were also observed in LaNiO$_2$ and PrNiO$_2$. The fact that the hole doping suppresses EPC contradicts with the fact that hole doping induces superconductivity in experiments.[18–20] These suggests that the superconducting origin in hole-doped $R$NiO$_2$ is unlikely to be EPC-dominated.
cpl-39-6-067402-fig3.png
Fig. 3. Phonon and EPC properties of NdNiO$_2$ without doping and with a doping concentration of 0.2 holes per formula unit. [(a), (d)] Undoped and doped phonon band structures. [(b), (e)] Undoped and doped phonon DOSs with atomic projection. [(c), (f)] Eliashberg function $\alpha^{2} F (\omega)$ for the undoped and doped cases, respectively. The solid red lines represent the cumulative values of $\lambda(\omega)$.
Note that in experiments, superconductivity is only found in thin films[18–20] instead of bulk materials,[39] which implies the special role of the interface in promoting superconductivity. Here we only focus on the bulk materials as the first-principle calculations for the thin films with substrates are difficult to perform. We leave such a topic for future study. In summary, we have systematically studied the electronic, phonon, and superconducting properties of undoped and doped $R$NiO$_2$ ($R$ = Nd, La, and Pr) using first-principles calculations. The effect of hole doping induces an overall upward shift of the electronic bands, the contraction of the electron-type Fermi pockets, and the significant reduction of DOS at $E_{\rm F}$. At the same time, the hole doping can harden the phonon spectrum and reduce the EPC constant effectively. All calculated superconducting $T_{\rm c}$'s remain 0 K for $R$NiO$_2$ with and without hole doping, which means that this kind of materials does not comply with the EPC mechanism of the BCS theory. Our theoretical calculations thus fully prove that $R$NiO$_2$ ($R$ = Nd, La, and Pr) superconductors after hole doping belong to the unconventional superconductors. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12074031, 11674025, and 12174443), the National Key R&D Program of China (Grant No. 2017YFA0302903), and the Beijing Natural Science Foundation (Grant No. Z200005). References Theory of SuperconductivitySuperconductivity at 250 K in lanthanum hydride under high pressuresSuperconductivity at 39 K in magnesium diboridePhenomenological theory of unconventional superconductivityBound Electron Pairs in a Degenerate Fermi GasPhase-sensitive tests of the symmetry of the pairing state in the high-temperature superconductors—Evidence for d x 2 − y 2 symmetryThe Resonating Valence Bond State in La 2 CuO 4 and SuperconductivityRobust d x 2 − y 2 -wave superconductivity of infinite-layer nickelatesSuperconductivity in infinite-layer nickelates: Role of f orbitalsFormation of a two-dimensional single-component correlated electron system and band engineering in the nickelate superconductor NdNiO 2 Similarities and Differences between LaNiO 2 and CaCuO 2 and Implications for SuperconductivityTheory of unconventional superconductivity in nickelate-based materials*Structural, magnetic, electronic, and phonon properties of NdNiO2 under pressure from first-principlesElectronic structure of rare-earth infinite-layer R Ni O 2 ( R = La , Nd ) Universal spin-glass behaviour in bulk LaNiO2 , PrNiO2 and NdNiO2A substantial hybridization between correlated Ni-d orbital and itinerant electrons in infinite-layer nickelatesSuperconductivity in an infinite-layer nickelateNickelate Superconductivity without Rare‐Earth Magnetism: (La,Sr)NiO2A Superconducting Praseodymium Nickelate with Infinite Layer StructureInduced magnetic two-dimensionality by hole doping in the superconducting infinite-layer nickelate Nd 1 − x Sr x NiO 2 Superconducting Dome in Nd 1 − x Sr x NiO 2 Infinite Layer FilmsSingle particle tunneling spectrum of superconducting Nd1-xSrxNiO2 thin filmsDesign of a multifunctional polar metal via first-principles high-throughput structure screeningPhase diagram of infinite layer praseodymium nickelate Pr 1 − x Sr x NiO 2 thin filmsPhase Diagram and Superconducting Dome of Infinite-Layer Nd 1 − x Sr x NiO 2 Thin FilmsEffective Hamiltonian for nickelate oxides Nd 1 − x Sr x NiO 2 Hund's metal physics: From SrNiO 2 to LaNiO 2 Optical Properties of the Infinite-Layer La 1 − x Sr x NiO 2 and Hidden Hund’s PhysicsMultiorbital Processes Rule the Nd 1 − x Sr x NiO 2 Normal StateSuperconductivity in nickel-based 112 systemsClassifying superconductivity in an infinite-layer nickelate Nd0.8Sr0.2NiO2Physical Properties Revealed by Transport Measurements for Superconducting Nd0.8 Sr0.2 NiO2 Thin FilmsDoping-dependent character and possible magnetic ordering of NdNiO 2 Negligible oxygen vacancies, low critical current density, electric-field modulation, in-plane anisotropic and high-field transport of a superconducting Nd0.8Sr0.2NiO2/SrTiO3 heterostructureEffect of lattice strain on the electronic structure and magnetic correlations in infinite-layer (Nd,Sr)NiO2Nickelate Superconductors: An Ongoing Dialog between Theory and ExperimentsCorrelated electronic structure of a quintuple-layer nickelateSynthesis and characterization of bulk Nd 1 − x Sr x Ni O 2 and Nd 1 − x Sr x Ni O 3 Advanced capabilities for materials modelling with Quantum ESPRESSOInhomogeneous Electron GasSelf-Consistent Equations Including Exchange and Correlation EffectsPhonons and related crystal properties from density-functional perturbation theoryGeneralized Gradient Approximation Made SimplePseudopotentials periodic table: From H to PuNeutron Spectroscopy of SuperconductorsFirst-principles calculations of the Coulomb pseudopotential μ * : Application to AlHigh Temperature Superconductivity in Metallic Hydrogen: Electron-Electron EnhancementsSynthesis of the infinite layer Ni(I) phase NdNiO2+x by low temperature reduction of NdNiO3 with sodium hydride
[1] Bardeen J, Cooper L N, and Schrieffer J R 1957 Phys. Rev. 108 1175
[2]Kamerlingh Onnes H 1911 Commun. Phys. Lab. Univ. Leiden 122 122
[3] Drozdov A P, Kong P P, Minkov V S, Besedin S P, Kuzovnikov M A, Mozaffari S, Balicas L, Balakirev F F, Graf D E, Prakapenka V B, Greenberg E, Knyazev D A, Tkacz M, and Eremets M I 2019 Nature 569 528
[4] Nagamatsu J, Nakagawa N, Muranaka T, Zenitani Y, and Akimitsu J 2001 Nature 410 63
[5] Sigrist M and Ueda K 1991 Rev. Mod. Phys. 63 239
[6] Cooper L N 1956 Phys. Rev. 104 1189
[7] Harlingen V D J 1995 Rev. Mod. Phys. 67 515
[8] Anderson P W 1987 Science 235 1196
[9] Wu X X, Sante D D, Schwemmer T, Hanke W, Hwang H Y, Raghu S, and Thomale R 2020 Phys. Rev. B 101 060504
[10] Bandyopadhyay S, Adhikary P, Das T, Dasgupta I, and Saha-Dasgupta T 2020 Phys. Rev. B 102 220502
[11] Nomura Y, Hirayama M, Tadano T, Yoshimoto Y, Nakamura K, and Arita R 2019 Phys. Rev. B 100 205138
[12] Botana A S and Norman M R 2020 Phys. Rev. X 10 011024
[13] Zhang M, Zhang Y, Guo H, and Yang F 2021 Chin. Phys. B 30 108204
[14] Zhang X M and Li B 2022 J. Solid State Chem. 306 122806
[15] Jiang P, Si L, Liao Z, and Zhong Z 2019 Phys. Rev. B 100 201106
[16] Lin H, Gawryluk D J, Klein Y M, Huangfu S, Pomjakushina E, von Rohr F, and Schilling A 2022 New J. Phys. 24 013022
[17] Gu Y, Zhu S, Wang X H J, and Chen H 2020 Commun. Phys. 3 84
[18] Li D, Lee K, Wang B Y, Osada M, Crossley S, Lee H R, Cui Y, Hikita Y, and Hwang H Y 2019 Nature 572 624
[19] Osada M, Wang B Y, Goodge B H, Harvey S P, Lee K, Li D, and Hwang H Y 2021 Adv. Mater. 33 2104083
[20] Osada M, Wang B Y, Goodge B H, Lee K, Yoon H, Sakuma K, Li D, Miura M, Kourkoutis L F, and Hwang H Y 2020 Nano Lett. 20 5735
[21] Ryee S, Yoon H, Kim T J, Jeong M Y, and Han M J 2020 Phys. Rev. B 101 064513
[22] Li D, Wang B Y, Lee K, Harvey S P, Osada M, Goodge B H, and Hwang H Y 2020 Phys. Rev. Lett. 125 027001
[23] Gu Q, Li Y, Wan S, Li H, Guo W, Yang H, and Wen H H 2020 Nat. Commun. 11 6027
[24] Li Q, He C, Si J, Zhu X, Zhang Y, and Wen H H 2020 Commun. Mater. 1 16
[25] Osada M, Wang B Y, Lee K, Li D, and Hwang H Y 2020 Phys. Rev. Mater. 4 121801
[26] Zeng S, Tang C S, Yin X, Li C, Li M, Huang Z, and Ariando A 2020 Phys. Rev. Lett. 125 147003
[27] Zhang H, Jin L, Wang S, Xi B, Shi X, Ye F, and Mei J W 2020 Phys. Rev. Res. 2 013214
[28] Wang Y, Kang C J, Miao H, and Kotliar G 2020 Phys. Rev. B 102 161118
[29] Kang C J and Kotliar G 2021 Phys. Rev. Lett. 126 127401
[30] Lechermann F 2020 Phys. Rev. X 10 041002
[31] Gu Q and Wen H 2021 Innovation 3 100202
[32] Talantsev E F 2020 Results Phys. 17 103118
[33] Xiang Y et al. 2021 Chin. Phys. Lett. 38 047401
[34] Lechermann F 2021 Phys. Rev. Mater. 5 044803
[35] Zhou X R, Feng Z X, Qin P X, Yan H, Wang X N, Nie P, and Liu Z Q 2021 Rare Met. 40 2847
[36] Leonov I 2021 J. Alloys Compd. 883 160888
[37] Botana A S, Bernardini F, and Cano A 2021 J. Exp. Theor. Phys. 132 618
[38] LaBollita H and Botana A S 2022 Phys. Rev. B 105 085118
[39] Wang B X, Zheng H, Krivyakina E, Chmaissem O, Lopes P P, Lynn J W, and Phelan D 2020 Phys. Rev. Mater. 4 084409
[40] Giannozzi P, Andreussi O, Brumme T, Bunau O, Nardelli M B, Calandra M, Car R, Cavazzoni C, Ceresoli D, Cococcioni M, and Baroni S 2017 J. Phys.: Condens. Matter 29 465901
[41] Hohenberg P and Kohn W 1964 Phys. Rev. 136 B864
[42] Kohn W and Sham L J 1965 Phys. Rev. 140 A1133
[43] Baroni S, de Gironcoli S, Dal C A, and Giannozzi P 2001 Rev. Mod. Phys. 73 515
[44] Perdew J P, Burke K, and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
[45] Corso A D 2014 Comput. Mater. Sci. 95 337
[46]Eliashberg G M 1960 Sov. Phys.-JETP 11 696
[47] Allen P B 1972 Phys. Rev. B 6 2577
[48] Lee K H, Chang K J, and Cohen M L 1995 Phys. Rev. B 52 1425
[49] Richardson C F and Ashcroft N W 1997 Phys. Rev. Lett. 78 118
[50] Hayward M A and Rosseinsky M J 2003 Solid State Sci. 5 839