Induced magnetic two-dimensionality by hole doping in superconducting infinite-layer nickelate
To understand the superconductivity recently discovered in NdSrNiO, we carried out LDA+DMFT (local density approximation plus dynamical mean-field theory) and magnetic force response calculations. The on-site correlation in Ni- orbitals causes notable changes in the electronic structure. The calculated temperature-dependent susceptibility exhibits the Curie-Weiss behavior, indicating the localized character of its moment. From the low-frequency behavior of self-energy, we conclude that the undoped phase of this nickelate is Fermi-liquid-like contrary to cuprates. Interestingly, the estimated correlation strength by means of the inverse of quasi-particle weight is found to increase and then decrease as a function of hole concentration, forming a dome-like shape. Another remarkable new finding is that magnetic interactions in this material become two-dimensional by hole doping. While the undoped NdNiO has the sizable out-of-plane interaction, hole dopings strongly suppress it. This two-dimensionality is maximized at the hole concentration . Further analysis as well as the implications of our new findings are presented.
Ever since its first discovery of high-temperature superconductivity in cuprates Bednorz and Müller (1986), tremendous theoretical and experimental efforts have been devoted to the intriguing physics of this family of materials Keimer et al. (2015); Lee et al. (2006); Scalapino (2012); Tsuei and Kirtley (2000); Damascelli et al. (2003). One important research direction is to find or design the cuprate-like superconductivity in alternative transition-metal systems in the hope that scrutinizing the similarities and differences enables us to understand and utilize the exotic superconductivity Norman (2016). Although the unambiguous understanding of its nature has not been reached yet, some ‘essential’ features for superconductivity have been highlighted. Obvious common features shared also with Fe-based superconductor include their two-dimensional electronic structure and magnetism Scalapino (2012). Spin moment interacting antiferromagnetically with each other, substantial – hybridization, and the large orbital polarization (one-band physics) are of key importance in cuprates but not much relevant to Fe-based materials.
Nickelates have been studied along this line as a promising candidate of non-Cu-based but cuprate-like superconductor Chaloupka and Khaliullin (2008); Uchida et al. (2011); Hansmann et al. (2009); Han et al. (2011); Boris et al. (2011); Zhang et al. (2017). This long quest ends in a success by the discovery of hole-doped infinite-layer nickelate, NdSrNiO grown on SrTiO substrate Li et al. (2019). While its mother compound, NdNiO, was originally synthesized more than a decade ago Hayward and Rosseinsky (2003), early theoretical studies excluded this form of nickelate from cuprate analogs due to the weak – hybridization and non-cuprate-like Fermi surface Lee and Pickett (2004). In this regard, the recent new discovery poses important challenges and possibilities for superconductivity research. Naturally, theoretical investigations intensively discuss its similarity with and the difference from cuprates in terms of electronic structure, gap symmetry and other details Botana and Norman (2019); Sakakibara et al. (2019); Wu et al. (2019); Hepting et al. (2019); Jiang et al. (2019); Nomura et al. (2019); Gao et al. (2019). While some important key features are identified to be similar with cuprates including the superconducting gap symmetry Sakakibara et al. (2019); Wu et al. (2019), it is far from clear if the obvious differences are then become irrelevant or unimportant in this nickelate superconductor Lee and Pickett (2004); Jiang et al. (2019).
In this paper, we present several key new findings obtained from first-principles calculations. (i) Our LDA+DMFT (local density approximation plus dynamical mean-field theory) calculation shows that the electronic correlation makes noticeable changes in the band structure and spectral weight. (ii) The calculated temperature-dependent local spin susceptibility shows that this material, albeit metallic, carries the local moment exhibiting Curie-Weiss behavior rather than Pauli-like. It is another notable similarity with cuprates. (iii) The undoped NdNiO exhibits Fermi-liquid-like self-energy which is in contrast to lightly-doped as well as undoped cuprates. (iv) The correlation strength as estimated by the inverse of quasi-particle weight is enhanced and then reduced by hole doping, thereby forming a dome-like shape. (v) Magnetic force response calculations demonstrate that hole doping makes this system magnetically cuprate-like. While the out-of-plane magnetic interactions are quite significant in NdNiO contrary to the cuprate situation, they are largely suppressed by hole doping. This magnetic two-dimensionality becomes maximized at around the doping concentration of the observed superconductivity.
Ii Computation Methods
We first carried out DFT (density functional theory) calculations within Perdew-Burke-Ernzherof parameterization of generalized gradient approximation (GGA) Perdew et al. (1996) as implemented in VASP Kresse and Furthmüller (1996). Following the literature Botana and Norman (2019); Sakakibara et al. (2019), we focused on LaNiO instead of NdNiO to avoid the numerical instabilities arising from Nd- electrons. To check the DFT magnetic ground state, four different magnetic orderings, namely ferromagnetic, A-type antiferromagnetic (AFM), C-type AFM, and G-type AFM orders, were compared using unitcell. The -grid of was used with energy cutoff of 600 eV. To further check the possible structural effects on the magnetic ground state, we fully optimized the crystal structure with the force criterion of meV/Å. It is found that the G-type AFM order is the lowest energy state. The calculated total energy of FM, A- and C-type AFM is greater than G-type by 38, 38 and 5 meV/Ni, respectively, and the moment size is . The results obtained from experimental lattice parameters of NdNiO Li et al. (2019) will be presented as our main data. We also double checked our results with OpenMX software package ope (); Ozaki (2003) and other exchange-correlation (XC) functionals including LDA Perdew and Zunger (1981) and LDA/GGA Anisimov et al. (1997), confirming that our conclusion is valid. For LDA/GGA, we adopted the charge density formulations to avoid the ambiguities stemming from the spin density XC energy Ryee and Han (2018). The on-site Coulomb interaction values of , 5, and 7 eV were tested, and we consistently obtained the G-type AFM order. It is found that, employing LDA gives the smaller magnetic moment ( /Ni) Ryee and Han (2017).
We performed MFT (magnetic force theory) calculations Oguchi et al. (1983); Liechtenstein et al. (1987) to obtain magnetic exchange couplings on top of DFT electronic structures ope (); Ozaki (2003); com (). The calculations were carried out using our recently-developed MFT code, named Jx Yoon et al. (2018, 2020). We adopted the most stable G-type AFM order obtained from GGA as the input. We also found that within the reasonable parameter range, LDA/GGA+ gives the same conclusion.
The charge self-consistent LDA+DMFT calculations were performed by using EDMFTF package whose DFT part is based on Wien2K Haule et al. (2010). The muffin-tin radius = 2.5, 1.98 and 1.7 a.u. for La, Ni and O, respectively. The cut-off parameter to expand the basis set of =7.0 was used. Paramagnetic phase has been considered with the k-grid of 111114. We took the energy window of [10, 10] eV to construct the local projector. The Coulomb interaction and Hund’s coupling of and eV for Ni- states were adopted. Since this is a reasonable choice considering recent studies Sakakibara et al. (2019); Nomura et al. (2019), the results of eV will be presented as our main data unless specified otherwise. We, however, have also performed calculations with larger values up to eV. Those results will also be presented below when necessary. The DMFT impurity problem was solved via hybridization expansion CTQMC algorithm Gull et al. (2011); Haule (2007). The so-called nominal double-counting scheme was used Pourovskii et al. (2007). To simulate the doping, we used virtual crystal approximation within DFT-LDA. For the double-counting potential in LDA+DMFT under doping, the nominal charge of Ni- state was adjusted accordingly assuming that the doped charge resides on the Ni-site.
iii.1 LDA+DMFT electronic structure
Having the realistic electronic structure and its analysis are the important first step toward further understanding of a given material. Due to the localized Ni- orbitals, the correlation effect can be significant in this LaNiO. Previous band structure calculations, however, are limited either to LDA/GGA (which largely misses the effect of on-site correlations) or to LDA/GGA (which is also quite limited, as a Hartree-Fock-like static approximation, in describing this metallic system). Here we adopt the LDA+DMFT method whose result is presented in Fig. 1(b). In the left panel, the calculated band dispersion by LDA is presented in dashed green lines and the LDA+DMFT results are presented by color plot. Dark and bright red regions represent the larger and smaller spectral weights, respectively. It is clearly noted that the effect of correlation changes the electronic structure noticeably. In particular, bandwidth is markedly reduced near the Fermi level (). See, for example, the states along –– line and –– line within the range of to 0 eV. In spite of this correlation effect, however, the system remains metallic being consistent with experiment. We found that, up to the largest value of eV we tried, the metallic solution is obtained. The electron pockets centered at and have been highlighted as a distinctive feature of this material as it self-dopes the holes into Ni- bands. These pockets are also maintained within LDA+DMFT. Upon hole doping, on the other hand, the electron pocket at gradually moves upward and eventually disappears from , thereby forming the cuprate-like one-band Fermi surface.
iii.2 The nature of magnetic moment
Another fundamentally important question is about the nature of spin moments based on which further theoretical investigation and modeling can be developed in different ways. Figure 2(a) presents our calculation results of temperature-dependent local spin susceptibility (where is an imaginary time, is the spin gyromagnetic ratio, is the temperature in eV, and is the local spin operator). Together with , it clearly indicates the Curie-Weiss behavior rather than Pauli-like. We note that at the larger value () shows the stronger local moment behavior and that the Curie-Weiss behavior is maintained even at (not shown). Thus it validates the various model studies that assume the local spin moments in this system.
iii.3 The nature of metallic phase
One obvious and physically important difference in between LaNiO and cuprate is the undoped phase which is metallic only in the former. This fact immediately raises an important question regarding the nature of this chemically undoped metallic phase and its evolution as a function of hole concentration toward the superconductivity. In hole-doped cuprates, the undoped parent compound has a well-developed AFM order and insulating gap both of which are destroyed by doping. Importantly, before superconductivity arises, the intriguing pseudo-gap phase emerges which has long been a central topic of cuprate research Norman et al. (2005). Further hole doping leads the system to be an exotic metallic phase, often called as ‘strange metal’. Fermi-liquid phase is eventually realized only in the heavy doping regime. Therefore, the question here is whether or not the undoped paramagnetic phase of LaNiO is Fermi-liquid-like. And if not, whether its nature has any similarity with strange metal or pseudogap phase of cuprates.
In order to address this issue, we try to fit our calculated local self-energies at the lowest Matsubara frequency () onto the polynomials. The ‘first-Matsubara-frequency rule’ states that the imaginary part of local self-energy at the lowest Matsubara frequency follows the -linear scaling behavior in Fermi-liquid regime Chubukov and Maslov (2012); Hausoel et al. (2017). We found that the calculated self-energies indeed exhibit the linear -dependence for both and 9 eV (see Fig. 2(c)). To further check the validity of this conclusion, we estimated the electron scattering rate (where is the quasi-particle weight and is the local self-energy on Matsubara frequency axis) by extrapolating the imaginary part of the Matsubara self-energy (Fig. 2(d)). It is clearly seen that the estimated of Ni- and exhibis the linear -dependence below K, also indicating the Fermi liquid. Hereby, we conclude that the undoped LaNiO is Fermi-liquid-like, which is in a sharp contrast to cuprates. This intriguing undoped phase (carrying the local moment feature and simultaneously the Fermi-liquid nature) deserves further study.
iii.4 The magnetic interactions
Magnetic interactions have been considered as the key to understand the unconventional superconductivity including not just cuprate family but also Fe-based and heavy fermion materials. The estimation of these couplings is therefore a crucial step. Here we performed the MFT calculations to obtain the magnetic coupling constant ’s in between Ni sites. The results are summarized in Fig. 3(a). The first thing to be noted is that LaNiO has a sizable out-of-plane coupling. The first neighbor out-of-plane coupling, , is indeed the nearest neighbor interaction among Ni sites (see Fig. 1(a)). Notably, this interaction is about 29% of the strongest interaction which refers to the in-plane nearest-neighbor coupling (the second nearest neighbor overall). This is another striking difference from cuprate. In cuprates, it is well known that the magnetic interactions are basically two-dimensional and the out-of-plane coupling is negligible. In fact, our calculation result of iso-structural infinite-layer cuprate compound confirms it: As shown in Fig. 3(a), for CaCuO is very small. It is therefore tempting to conclude that, if the magnetic interaction is responsible for superconductivity in any sense, the newly discovered nickelate is the different case from cuprate superconductivity.
Remarkably, however, this three-dimensional nature of magnetic interaction profile is changed to be two-dimensional by hole doping. As shown in Fig. 3(b), hole dopings reduce the size of significantly. At the hole doping of 0.2 – 0.25/f.u., it becomes basically zero. Note that the experimental value of hole doping for realizing superconductivity () is in good agreement with it. Although the nominal value of charge counting both in calculation and in experiment should not be considered as being identical, this excellent agreement is impressive and informative. Considering that the recent theoretical studies coincidentally report that the superconductivity in this nickelate is cuprate-like in terms of gap symmetry for example Sakakibara et al. (2019); Wu et al. (2019), our finding elucidates the effect of hole doping for inducing superconductivity: In this new nickelate superconductor, the main role of doping is neither to destroy the AFM order nor the insulating phase. But it is to make magnetic interaction be two-dimensional, and in this sense, cuprate-like.
Figure 4 is a schematic summary of our results for magnetism and correlation. It is reminiscent of typical phase diagram of cuprate and many other high-T materials. Here we also present the calculated total energy differences in between non-magnetic and the lowest energy AFM order (); see the orange squares. Note that this represents the stability of AFM order (within DFT) and corresponds to the AFM ordering temperature. We also considered the electron doping region which is represented by negative . Given that this system is ‘self-hole-doped’, the behavior of AFM region as a function of is quite similar with other well-established phase diagram in both electron-doped and hole-doped side. It hopefully stimulates the further experimental and theoretical investigations of electron-doped side of this material.
In order to represent the two-dimensionality of magnetic interactions, we present the calculated value of ; see the brown circles in Fig. 4. Its behavior as a function of forms also a dome-like shape; first increases and then decreases. As noted above, it reaches its maximum at 0.2 – 0.25 which is well compared with the experimental value. It suggests a possibility that the magnetic two-dimensionality is essential for superconductivity.
Another interesting feature is the behavior of correlation strength as a function of ; see the black squares in Fig. 4. As mentioned above, exhibits a dome-like feature although it varies at most in our calculation of eV (with eV, the variation becomes slightly larger ); see Fig. 2(b). According to our LDA+DMFT calculation, it reaches its maximum at around . Here the care needs to be paid when comparing in LDA+DMFT with that in DFT. In our LDA+DMFT calculations, the hole doping is simulated by virtual crystal approximation in DFT-LDA, and by adjusting the nominal value of double-counting in DMFT. Thus, the doped holes are basically constrained to reside in Ni- bands in our LDA+DMFT scheme (although the self-consistency can redistribute them) whereas, in DFT, holes can be distributed over the other bands. Considering this point, it seems feasible that the maximum point of can even be closer to . This feature can possibly have notable implications for, e.g., strong coupling theory.
We report several new findings and discuss their implications for recently discovered superconductivity in NdSrNiO. First, we present the realistic electronic structure by including the effect of on-site correlation within LDA+DMFT. The calculation of temperature-dependent spin susceptibility clearly identifies the localized nature of spin moments. The estimated correlation strength by means of the inverse quasi-particle weight is found to be enhanced and then reduced by hole doping. Interestingly, its maximum point is fairly close to the experimental value of nominal hole concentration to induce superconductivity. Finally, our MFT calculations demonstrate that the key role of hole dopings is to make the system magnetically cuprate-like. The out-of-plane magnetic coupling is largely suppressed by hole doping, and therefore the magnetic interactions become two-dimensional. This two-dimensionality also reaches its maximum at around .
S.R. is grateful to H. Sakakibara for helpful comments. This work was supported by BK21plus program, Basic Science Research Program (2018R1A2B2005204) and Creative Materials Discovery Program through NRF (2018M3D1A1058754).
- thanks: These three authors contributed equally to this work.
- thanks: These three authors contributed equally to this work.
- thanks: These three authors contributed equally to this work.
- J. G. Bednorz and K. A. Müller, Zeitschrift für Physik B Condensed Matter 64, 189 (1986).
- B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, Nature 518, 179 (2015).
- P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006).
- D. J. Scalapino, Rev. Mod. Phys. 84, 1383 (2012).
- C. C. Tsuei and J. R. Kirtley, Rev. Mod. Phys. 72, 969 (2000).
- A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys. 75, 473 (2003).
- M. R. Norman, Reports on Progress in Physics 79, 074502 (2016).
- J. c. v. Chaloupka and G. Khaliullin, Phys. Rev. Lett. 100, 016404 (2008).
- M. Uchida, K. Ishizaka, P. Hansmann, Y. Kaneko, Y. Ishida, X. Yang, R. Kumai, A. Toschi, Y. Onose, R. Arita, K. Held, O. K. Andersen, S. Shin, and Y. Tokura, Phys. Rev. Lett. 106, 027001 (2011).
- P. Hansmann, X. Yang, A. Toschi, G. Khaliullin, O. K. Andersen, and K. Held, Phys. Rev. Lett. 103, 016401 (2009).
- M. J. Han, X. Wang, C. A. Marianetti, and A. J. Millis, Phys. Rev. Lett. 107, 206804 (2011).
- A. V. Boris, Y. Matiks, E. Benckiser, A. Frano, P. Popovich, V. Hinkov, P. Wochner, M. Castro-Colin, E. Detemple, V. K. Malik, C. Bernhard, T. Prokscha, A. Suter, Z. Salman, E. Morenzoni, G. Cristiani, H.-U. Habermeier, and B. Keimer, Science 332, 937 (2011), https://science.sciencemag.org/content/332/6032/937.full.pdf .
- J. Zhang, A. S. Botana, J. W. Freeland, D. Phelan, H. Zheng, V. Pardo, M. R. Norman, and J. F. Mitchell, Nature Physics 13, 864 (2017).
- D. Li, K. Lee, B. Y. Wang, M. Osada, S. Crossley, H. R. Lee, Y. Cui, Y. Hikita, and H. Y. Hwang, Nature 572, 624 (2019).
- M. Hayward and M. Rosseinsky, Solid State Sciences 5, 839 (2003), international Conference on Inorganic Materials 2002.
- K.-W. Lee and W. E. Pickett, Phys. Rev. B 70, 165109 (2004).
- A. S. Botana and M. R. Norman, arXiv preprint arXiv:1908.10946 (2019).
- H. Sakakibara, H. Usui, K. Suzuki, T. Kotani, H. Aoki, and K. Kuroki, arXiv preprint arXiv:1909.00060 (2019).
- X. Wu, D. Di Sante, T. Schwemmer, W. Hanke, H. Y. Hwang, S. Raghu, and R. Thomale, arXiv preprint arXiv:1909.03015 (2019).
- M. Hepting, D. Li, C. J. Jia, H. Lu, E. Paris, Y. Tseng, X. Feng, M. Osada, E. Been, Y. Hikita, Y.-D. Chuang, Z. Hussain, K. J. Zhou, A. Nag, M. Garcia-Fernandez, M. Rossi, H. Y. Huang, D. J. Huang, Z. X. Shen, T. Schmitt, H. Y. Hwang, B. Moritz, J. Zaanen, T. P. Devereaux, and W. S. Lee, arXiv preprint arXiv:1909.02678 (2019).
- M. Jiang, M. Berciu, and G. A. Sawatzky, arXiv preprint arXiv:1909.02557 (2019).
- Y. Nomura, M. Hirayama, T. Tadano, Y. Yoshimoto, K. Nakamura, and R. Arita, arXiv preprint arXiv:1909.03942 (2019).
- J. Gao, Z. Wang, C. Fang, and H. Weng, arXiv preprint arXiv:1909.04657 (2019).
- J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
- G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
- T. Ozaki, Phys. Rev. B 67, 155108 (2003).
- J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).
- V. I. Anisimov, F. Aryasetiawan, and A. Lichtenstein, J. Phy.: Condens. Matter 9, 767 (1997).
- S. Ryee and M. J. Han, Sci. Rep. 8, 9559 (2018).
- S. Ryee and M. J. Han, Sci. Rep. 7, 4635 (2017).
- T. Oguchi, K. Terakura, and N. Hamada, Journal of Physics F: Metal Physics 13, 145 (1983).
- A. Liechtenstein, M. Katsnelson, V. Antropov, and V. Gubanov, Journal of Magnetism and Magnetic Materials 67, 65 (1987).
- “The current version of our Jx Yoon et al. (2020) is compatible only with openmx for the option of full band calculations. therefore we used the electronic structure obtained by openmx in the calculations of magnetic couplings. we carefully checked that the used of different code does not make any noticeable differences.” .
- H. Yoon, T. J. Kim, J.-H. Sim, S. W. Jang, T. Ozaki, and M. J. Han, Phys. Rev. B 97, 125132 (2018).
- H. Yoon, T. J. Kim, J.-H. Sim, and M. J. Han, Computer Physics Communications 247, 106927 (2020).
- K. Haule, C.-H. Yee, and K. Kim, Phys. Rev. B 81, 195107 (2010).
- E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Rev. Mod. Phys. 83, 349 (2011).
- K. Haule, Phys. Rev. B 75, 155113 (2007).
- L. V. Pourovskii, B. Amadon, S. Biermann, and A. Georges, Phys. Rev. B 76, 235101 (2007).
- M. R. Norman, D. Pines, and C. Kallin, Advances in Physics 54, 715 (2005), https://doi.org/10.1080/00018730500459906 .
- A. V. Chubukov and D. L. Maslov, Phys. Rev. B 86, 155136 (2012).
- A. Hausoel, M. Karolak, E. Şaşoğlu, A. Lichtenstein, K. Held, A. Katanin, A. Toschi, and G. Sangiovanni, Nature communications 8, 16062 (2017).