ЖЭТФ, 2021, том 159, вып. 4, стр. 711-718

NICKELATE SUPERCONDUCTORS: AN ONGOING DIALOG

BETWEEN THEORY AND EXPERIMENTS

A. S. Botana^a, F. Bernardini^b, A. Canoc*

^a Department of Physics, Arizona State University

Tempe, AZ 85287, USA

^b Dipartimento di Fisica, Università di Cagliari

IT-09042, Monserrato, Italy

^c Université Grenoble Alpes, CNRS, Grenoble INP, Institut Néel

38042, Grenoble, France

Received December 4, 2020,

revised version December 4, 2020

Accepted for publication December 5, 2020

Contribution for the JETP special issue in honor of I. E. Dzyaloshinskii’s 90th birthday

DOI: 10.31857/S0044451021040131

Unconventional superconductivity, understood

as superconductivity beyond the electron-phonon

paradigm, remains a defining problem in condensed

matter [1]. The challenge is exemplified by the high-T_c

cuprates, and nickelates joined the club last year after

Hwang and collaborators reported superconductivity

Fig. 1. Ball-and-stick model of the unit cell of the infinite-

in Sr-doped NdNiO₂ thin films [2]. Here, we provide

layer nickelates RNiO2 and sketch of the formal 3d⁹ electronic

configuration of a Ni¹⁺ atom in square-planar coordination.

an overview on nickelate superconductors.

The set of d orbitals is frequently divided into the subsets

Experimental facts. Rare-earth infinite-layer

eg-like = {dx2-y2 , dz2 } and t2g-like = {dxy, dxz, dyz} as in

octahedral coordination

nickelates RNiO₂ have been known for decades [3-6].

This special type of nickelates can be seen as the n = ∞

members of the series Rn+1Ni_nO2n+2, with each mem-

ber containing n-NiO₂ planes (see Fig. 1, R — ra-

Figure 2 shows the temperature-composition phase

re-earth). The synthesis of these materials is typically

diagram of Sr-doped NdNiO₂ and PrNiO₂ [13,14]. The

achieved by first growing the nickelate in its RNiO₃ per-

measured resistivity as a function of temperature shows

ovskite version, and then removing the apical oxygens

metallic behavior, with a low-temperature upturn sys-

from the NiO₆ octahedra with reducing agents such as

tematically observed (Fig.

2). This upturn can be

hydrogen [7]. This is the so-called topotactic reduction

attributed to weak localization, or it could be remi-

which, in practice, may have unwanted consequences

niscent of Kondo physics [15]. These systems can be

such as hydrogen intercalation into the sample [8]. Su-

seen as weak insulators or bad metals all along the

perconductivity in RNiO₂ (R = Pr, Nd) emerges via

phase diagram, in marked contrast to cuprates. Be-

charge carrier doping as in the cuprates. This has only

sides, no signature of long-range magnetic order has

been achieved by means of thin-film growth techniques

been reported so far for the parent infinite-layer nick-

[9], while superconductivity in doped bulk samples is

elates [4, 5, 16]. However, a recent NMR study has

yet to be reported [10-12].

pointed out the presence of antiferromagnetic fluctu-

ations and quasi-static antiferromagnetic order below

^* E-mail: andres.cano@neel.cnrs.fr

40 K in Nd0.85Sr0.15NiO₂ [17].

711

A. S. Botana, F. Bernardini, A. Cano

ЖЭТФ, том 159, вып. 4, 2021

Fig. 2. Temperature vs composition phase diagram of the superconducting infinite-layer nickelates reported so far — i. e. Sr-do-

ped NdNiO2 and PrNiO2 thin films on SrTiO3 substrates — and corresponding resistivity data (from [13] and [14]). Sr-doping

is equivalent to hole doping in these systems. The superconducting Tc reaches 15 K in the best superconducting samples. The

resistivity in the normal state shows metallic behavior as a function of temperature, with a Kondo-like upturn systematically

observed for both underdoped and overdoped samples. The metallicity is however rather poor, and hence these systems can

be seen as bad metals with weakly insulating features. At the same time, enhanced metallicity has been repeatedly reported in

superconducting samples when compared to the non-superconducting ones

The transport data reported so far is still sample de-

found to be inhomogenous, with different features at

pendent, with nominally equivalent samples displaying

different locations of the sample. One of the predomi-

finite resistivity or a complete drop [13, 18, 19]. Also,

nant features correspond to a V-shape spectrum. How-

whether the original LaNiO₂ reference compound hosts

ever, features corresponding to a full s-wave gap are

superconductivity or not remains an important open

equally observed and, in some cases, even a mixture of

question. This difference may be “simply” due to sam-

the two.

ple quality and/or the presence of topotactic hydrogen

in LaNiO₂ — i. e. the formation of LaNiO₂H instead

Theoretical considerations.

of LaNiO₂ [8]. Otherwise, it may be more intrinsically

related to the rare-earth elements themselves — i. e.

Single-particle picture. At the single-particle DFT

closed vs open 4f shells and the corresponding mag-

level, the calculations support the multiband picture

netic moments.

of the infinite-layer nickelates in agreement with exper-

imental data [20, 24-34]. These calculations reveal a

Figure 3 illustrates the measured Hall coefficient

large Fermi-surface sheet due to Ni-3d_x2_-y2 holes akin

for RNiO₂ that has been observed to change sign

to that in cuprates (see Fig. 4). However, the Fermi

both as a function of temperature and as a function

surface displays additional electron pockets due to 5d

of doping. This evidences an underlying multiband

states associated to the rare-earth 5d states (5d_z2 at

character in infinite-layer nickelates, which is another

Γ and 5d_xy at A). This can be seen as a self-doping

important fundamental difference when compared to

effect promoted by the hybridization of these formally

cuprates. Such a multiband picture has been con-

empty states with the Ni-3d bands. This effect is to-

firmed by X-ray spectroscopic techniques [20-22]. How-

tally absent in the cuprates. Moreover, the Ni-3d_z2

ever, in contrast to the Hall data, the changes in the

states turn out to be partially occupied and addition-

X-ray spectra observed as a function of Sr-doping have

ally hybridized with the R-5d ones. Thus, the full e_g-li-

been interpreted in terms of doped holes residing in the

ke = {d_x2_-y2 , d_z2 } sector of the Ni-3d states becomes

Ni-3d_x2_-y2 orbitals without necessarily invoking multi-

“active” in infinite-layer nickelates. In part, this results

band effects [21].

from the comparatively large difference in on-site ener-

The single particle tunneling spectrum has been

gies promoting charge from O-2p to Ni-3d orbitals, i. e.

measured on (Nd,Sr)NiO₂ [23]. The spectrum was

the so-called charge-transfer energy ∼ 4 eV.

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ЖЭТФ, том 159, вып. 4, 2021

Nickelate superconductors: an ongoing dialog. ..

argued to drive the Cooper pairing in these systems

[26, 27, 36, 37]. Specifically, d-wave superconductivity

was concluded from complementary random phase ap-

proximation (RPA) and fluctuation exchange (FLEX)

calculations for many-body multi-orbital Hamiltonians

in which the non-interacting part maps the relevant

DFT bands. More recently, these results have been con-

firmed using advanced techniques in which the starting

vertex is non-perturbative so that the local correlations

are fully included [38].

The same conclusion about the d-wave symmetry

of the superconducting gap was reached in [27] from a

standard t-J model constructed in a similar way for the

Ni-3d_x2_-y2 states. Further, the specific self-doping fea-

tures of the nickelates have inspired an extended t-J

model that generically addresses the strong-coupling

limit of similar multiband systems [39]. Alternatively,

if the Hund’s coupling J between d_x2_-y2 and d_z2 Ni

orbitals plays a dominant role, it has been pointed out

that the Cooper pairing can be interpreted within a

spin-freezing scenario as due to local-moment fluctu-

ations, rather than to pure antiferromagnetic fluctua-

tions [40].

Many-body correlation effects: Electronic structure.

The DFT picture has been revised using many-body

perturbation theory at the GW level, thus treating cor-

relations ab initio [41,42]. The low-energy physics re-

mains essentially unaffected, with only small changes

obtained in the interacting Fermi surface and in the

quasiparticle spectral weights near the Fermi level.

The Ni-3d_x2_-y2 bandwidth reduces slightly while the

Ni-3d_z2 one increases as the O-2p states are further

shifted to lower energies (1.5 eV further down from

the Fermi level) [41]. The latter, however, represents

a rather substantial change, and hence suggests that

the canonical charge-transfer-insulator picture is even

Fig. 3. Hall coefficient RH measured in NdNiO2 and PrNiO2

more unlikely for the infinite-layer nickelates in the GW

as a function of temperature and Sr doping [13,14]. In both

cases, the Hall coefficient below 100 K changes sign as a func-

framework. These changes are also tied to an impor-

tion of Sr content (x in R1-xSrxNiO2). The main charge car-

tant shift of the empty 4f states [41], which should be

riers have electron character in the parent compounds (x = 0),

taken as a warning regarding their role in the overall

which eventually changes to holes upon increasing doping.

physics of these materials.

This was the first experimental hint of an underlying multi-

band picture

DFT+DMFT (dynamical mean-field theory) calcu-

lations according to Hubbard-Hund interaction Hamil-

tonians have been performed to further scrutinize the

correlated nature of the different orbitals and clarify

Many-body correlation effects: Cooper pairing. The

the multiband nature in RNiO₂ [15, 35, 38, 40, 43-51].

standard electron-phonon mechanism has been ruled

In addition, DMFT has also been applied in combi-

out as the main reason for superconductivity in infini-

nation with the quasiparticle self-consistent GW ap-

te-layer nickelates [28].

Instead, repulsive interac-

proximation in a parameter-free fashion [42, 52]. The

tions mediated by spin-fluctuations were right away

overall multiband picture remains robust and the re-

713

A. S. Botana, F. Bernardini, A. Cano

ЖЭТФ, том 159, вып. 4, 2021

Fig. 4. Electronic structure of the reference material LaNiO2 obtained at the DFT level (fat-band plot, density of states (DOS)

and Fermi surface from [25] and [35]). In addition to the main Ni-3d_x2_-y2 cuprate-like band crossing the Fermi level, the Ni-3d_z2

states are not completely occupied in these systems. Further, the system has a multiband character due to La-5d states that

give rise to additional electron pockets and “self-dope” the Ni-3d_x2_-y2 hole-like Fermi surface. The O-2p states, in turn, are

comparatively far below the Fermi level

sults confirm the above trends. However, the effective

ble to spin-density disproportionation, breathing and

mass renormalization or inverse quasiparticle weight

half-breathing lattice distortions, and charge-density

m^∗/m = 1/Z undergoes substantial orbital-selective

disproportionation. A different point of view suggests

changes [35, 44, 46]. The Ni-d_x2_-y2 band is found to

that 5d conduction electrons could screen the Ni spins,

have a tendency towards localization such that a Mott

suppressing magnetism and giving rise to a Kondo ef-

gap can eventually open if the Hubbard interaction is

fect like that seen in heavy fermion materials [15]. Be-

large enough [44, 45]. The R-5d self-doping bands, in

yond that, an intrinsic difference that could be be-

contrast, remain much more weakly correlated [44,45].

hind the differing properties across the RNiO₂ series,

Beyond that, the DMFT results define an apparent

notably the emergence of superconductivity itself, is

Hubbard vs Hund dichotomy in which the multiband

the presence/absence of magnetic rare-earth elements

aspects of infinite-layer nickelates are emphasized dif-

[33, 59].

ferently [53].

Interfacial and surface effects. The local elec-

Magnetism. A magnetic ground state is consis-

tronic properties of the infinite-layer nickelate thin

tently obtained in theoretical studies [20, 24, 25, 48, 49,

films were first addressed in [60] and shortly after in

53-57]. In DFT+DMFT a near degeneracy of spin or-

[61]. The epitaxial growth of the RNiO₂/SrTiO₃ het-

ders is obtained that implies magnetic frustration es-

erostructures can in principle yield different atomic

pecially upon doping [48, 49]. This frustration arises

boundaries between the sample and the substrate [60].

due to the involvement of both 3d_x2_-y2 and 3d_z2 Ni

The most obvious configuration corresponds to a fully

orbitals and may be responsible for the experimental

reduced nickelate having its (RO →) R layer directly on

suppression of long-range magnetic order. A comple-

top of the TiO₂-terminated substrate. This configura-

mentary point of view is obtained via DFT calcula-

tion, however, was found to be energetically unstable,

tions, as the antiferromagnetic ground state portrays a

which is a local-scale manifestation of the thermody-

one-dimensional-like van Hove singularity of d_z2 char-

namic fragility of these phases. Thus, the infinite-layer

acter pinned at the Fermi level [58]. This singularity

nickelate prefers to face a RO layer to the substrate,

makes the antiferromagnetic phase eventually unsta-

and the same conclusion holds even for a direct-growth

714

ЖЭТФ, том 159, вып. 4, 2021

Nickelate superconductors: an ongoing dialog. ..

Fig. 5. Electronic band structure of the infinite-layer nickelates at the RNiO2/SrTiO3 interface (left) and at the surface (right)

for different atomic configurations (the colors highlight the main contributions of the interfacial/surface Ni-3d states near the

Fermi level; adapted from [60] and [62]). In both cases the nature of the self-doping effect — obtained from the R-5d states in

the bulk — changes or even disappear depending on the local atomic configuration. At the same time, other Ni-3d states are

pushed closer to the Fermi level and locally supplement the system with flat band features

process. This “chemical” reconstruction was further

considered at the DFT+U level [63]. This enhances

shown to produce drastic changes in the electronic

the itineracy of the Ni-3d_z2 orbitals at the interface

structure at the interface [60]. The interfacial chemical

with the substrate, while the magnetism is essentially

reconstruction according to the R → Sr → RO → SrO

suppressed at the surface to vacuum. Further, the ef-

sequence is to some extent equivalent to localized hole

fect of the nickelate termination anticipated in [60] is

doping. This local doping was found to deplete the

specifically considered in relation to superconductivity

self-doping R-5d states at the interface. Specifically,

in [62]. Thus, it is argued that the d-wave supercon-

the R-5d states are first replaced by Ti-3d ones, which

ducting gap expected for the bulk may transform into

are then pushed above the Fermi level for the SrO con-

a s_±-wave one at the NiO₂-terminated surface, and it

figuration (see Fig. 5). At the same time, the intefacial

is also suggested that a surface s + id-wave state may

Ni-3d_z2 states are driven closer to the Fermi energy

be realized under the appropriate conditions.

so that they manifestly participate in the low-energy

physics. Thus, the Kondo-lattice features are expected

Conclusions and perspectives. The recent dis-

to be fundamentally different at the interfaces where,

covery of superconductivity in infinite-layer nickelates

in addition, the Ni-e_g sector will likely be fully ac-

has created intense excitement. These systems have

tive. Besides, this sector is supplemented by a markedly

been rapidly scrutinized from many different angles,

flat band character of the interfacial Ni-3d_z2 states, so

using a battery of experimental and theoretical tools.

that interface-specific correlation effects may be pro-

The initial motivation of drawing analogies with the

moted [60].

high-T_c cuprates has thus been surpassed. Instead, the

accumulated results have now consolidated these sys-

This picture was subsequently confirmed for thin

tems as a new class of unconventional superconducting

films with asymmetric boundaries [61, 63, 64]. In that

materials.

case, the different polar discontinuities yield an effec-

tive built-in electric field across the film and polar lay-

Specifically, the rare-earth infinite-layer nickelates

ers can be formed at the surface and at the interface

have been confirmed to host a distinct multiband in-

[63]. These layers show antiparallel NiO₂ displace-

terplay, on top of which electronic correlations build

ments, but otherwise are decoupled. At the interface

and determine the main properties of these systems.

with the substrate, a two-dimensional electron gas ex-

This interplay is present already within the Ni sector

tending over several layers together with the aforemen-

itself, as not only the Ni-3d_x2_-y2 states but also the

tioned depletion of the self-doping R-5d states is ob-

Ni-3d_z2 ones are found to be active. This further intro-

tained [61, 63]. In addition, the combined effect of

duces specific correlation effects and the bad metallic,

magnetism (G-AFM order) and correlations has been

or weakly insulating behavior is now understood as a

715

A. S. Botana, F. Bernardini, A. Cano

ЖЭТФ, том 159, вып. 4, 2021

direct manifestation of these correlations. However, a

Funding. A. B. acknowledges the support from

Hubbard vs Hund dichotomy has emerged that is yet to

NSF DMR 2045826.

be clarified. In addition, the rare-earth states introduce

extra specific ingredients such as the self-doping effect

The full text of this paper is published in the English

and a 4f-ness that may qualitatively be even more im-

version of JETP.

portant. When it comes to the central question, that

is, the emergence of superconductivity in these mate-

rials, it has been ascribed to spin fluctuations (in a

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