1 Two-Gap Superconductivity and Decisive Role of Rare-Earth d Electrons in Infinite-Layer Nickelates

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Two-Gap Superconductivity and Decisive Role of Rare-Earth d Electrons in

Infinite-Layer Nickelates

Zhenglu Li1,2,3 and Steven G. Louie1,2,*

1Department of Physics, University of California at Berkeley, Berkeley, CA, USA.

2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA.

3Mork Family Department of Chemical Engineering and Materials Science, University of Southern

California, Los Angeles, CA, USA.

*Email: sglouie@berkeley.edu

Abstract

We present a theoretical prediction of a phonon-mediated two-gap superconductivity in infinite-

layer nickelates Nd1-xSrxNiO2 by performing ab initio GW and GW perturbation theory calculations.

Electron GW self-energy effects significantly alter the characters of the two-band Fermi surface

and enhance the electron-phonon coupling, compared with results based on density functional

theory. Solutions of the fully k-dependent anisotropic Eliashberg equations yield two dominant s-

wave superconducting gaps – a large gap on a band of rare-earth Nd d and interstitial orbital

characters and a small gap on a band of transition-metal Ni d character. Increasing hole doping

induces a non-rigid-band response in the electronic structure, leading to a rapid drop of the

superconducting Tc in the overdoped regime in agreement with experiments.

Still holding the record for the highest superconducting Tc under ambient conditions, cuprates

remain among the most mysterious materials in condensed matter physics, despite extensive

studies in the last four decades [1]. In the pursuit of new superconductors as cuprate analogs, the

recent success in discovering superconducting infinite-layer nickelates with rather high Tc raises

the hope of gaining more insights into unconventional superconductivity [2-5]. Infinite-layer

nickelates and cuprates share a similar transition-metal-oxygen planar structure motif, and Ni+ and

Cu2+ are in the same 3d9 atomic valence configuration in the undoped parent compounds. Upon

hole doping of x = 0.1 – 0.3 in Nd1-xSrxNiO2, superconductivity is observed [1-5] and reaches a

highest reported Tc of 23 K at x = 0.15 [5]. To date, several nickelate superconductors have been

discovered [2-5,6-9], including Nd1-xSrxNiO2, Pr1-xSrxNiO2, La1-xSrxNiO2, La1-xCaxNiO2, and

Nd6Ni5O12. The similarity in the crystal structure and the presumed electronic structure of these

materials to the cuprates [10,11] lead to a prevailing perspective of superconductivity in the

infinite-layer nickelates as being unconventional and mainly originated from the Ni 𝑑-band

[2,12-15].

While the nature of superconductivity in nickelates remains under debate, the conventional

mechanism of pairing due to electron-phonon (e-ph) coupling was not supported from early on

[12,13,15]. Previous DFT calculations showed that the e-ph coupling constant λ in NdNiO2 is only

~0.2, accounting for a phonon-mediated Tc < 1 K [16]. Existing theoretical model studies [17-22]

on exploring unconventional mechanisms (typically with assumed interaction forms, downfolded

subspaces, and parametrized coupling strengths) are mostly focused on the Ni d band. Meanwhile,

scanning tunneling spectroscopy (STS) experiment [23] on nickelate thin films interestingly

observed two types of superconducting gaps (as well as their mixture) depending on the tip position

in the measurements: one is a V-shape dI/dV profile which has been interpreted as an

unconventional d-wave gap, and the other is a U-shape profile which is a typical signature of an s-

wave gap. Moreover, recent superfluid density experiments [24,25] revealed both nodal and fully-

gapped behaviors in infinite-layer nickelates. These intriguing results do not provide a

straightforward and self-consistent picture for superconductivity in the infinite-layer nickelates.

It is important to note that Kohn-Sham orbitals of DFT are not constructed to, and often may

not, describe well the true quasiparticle excitation energies (e.g., the band structure) [26-29] as

well as the e-ph coupling strength [30-33]. Thus, conclusions based on DFT calculations may be

questionable, especially for materials with non-negligible electron correlations. On the other hand,

the ab initio GW approach [26,34,35] has achieved much success in describing, from first

principles, the quasiparticle properties of many materials [26,28,29] including the e-ph coupling

[30,31]. Moreover, the fully k-dependent anisotropic Eliashberg theory [36-39] has become a

standard computational method for solving for superconductivity in real materials [33], e.g., MgB2

[40] and hydrides [41]. However, with a few exceptions, virtually all existing ab initio anisotropic

Eliashberg theory calculations are performed in conjunction with DFT approaches [33,38-41]. The

GW band structures and e-ph coupling, while being more accurate, have not been fully exploited

in use with anisotropic Eliashberg theory.

Here, by combining the GW band structures and GW e-ph interactions with the anisotropic

Eliashberg theory, our ab initio calculations reveal a phonon-mediated s-wave two-gap

superconductivity in Nd0.8Sr0.2NiO2 with a theoretically predicted Tc of ~ 22 K (experimental Tc =

19.3 K at x = 0.2 [5]). We hence arrive at an opposite conclusion from that drawn from standard

DFT results on the role of phonons in infinite-layer nickelate superconductors. We find that many-

electron (self-energy) effects rearrange the bands near the Fermi energy (EF) and drastically

reshape the Fermi surface (FS) and its orbital characters compared with DFT – by introducing a

substantial amount of Nd 𝑑 and interstitial orbital (IO) components [16,21,42] at EF. The Nd-IO

states host strong e-ph coupling strength. Our computed results show that the two-band FS exhibits

two distinct s-wave superconducting gaps, with characteristic values of 4.7 meV on the Nd-IO

band and 2.1 meV on the Ni band. The predicted two-gap superconductivity provides a strong

explanation of the intriguing observations of multiple forms of gap behavior in the tunneling [23]

and superfluid density [24,25] experiments by considering inhomogeneity in sample quality.

Moreover, upon increasing hole doping, the Nd-IO band rapidly moves away from EF, explaining

the decrease of Tc towards the overdoped regime as observed experimentally [3-5].

Fig. 1(a) and 1(b) show the DFT (with generalized gradient approximation (GGA)) and GW

band structures of Nd0.8Sr0.2NiO2, respectively, interpolated and analyzed with maximally

localized Wannier functions (MLWFs) [43]. Two bands cross EF, where one band is of Ni 𝑑

character (Fig. 1(f)), and the other band is mostly of Nd 𝑑 character (Fig. 1(g)) and of IO

character (Fig. 1(h)) centering around a hollow site coplanar with the Nd atoms, with some

hybridizations with other characters (e.g., the Ni 𝑑 and Nd 𝑑 orbitals hybridize near the Γ

point). We denote these two bands by their dominant characters as the Ni band and the Nd-IO band.

Comparing with DFT-GGA results, the GW self-energy effects bring significantly more Nd-IO

characters to the FS, by shifting the Nd-IO band (Ni band) downward (upward) relative to EF. The

Ni 𝑑 orbital is more spatially localized than the Nd 𝑑 and the IO orbitals (Fig. 1(f-h)),

therefore the two bands experience different electron self-energy effects originating from the

Coulomb interaction. Refined DFT exchange-correlation functionals such as meta-GGA or hybrid

functional [44,45], as well as GW plus dynamical mean-field theory (DMFT) [46], also give band

renormalization effects following a similar trend as the GW results. Our results further show that

the self-energy effects (more Nd-IO characters for states at EF) are robust within typical GW

numerical uncertainties across the hole-doping regime (Supplemental Material [47]). We note that

the renormalized band structures vary with different rare-earth elements in infinite-layer nickelates

[46,94]. Fig. 1(i) shows the projected density of states (DOS) of Nd0.8Sr0.2NiO2 from DFT-GGA

and GW. Unlike the case of DFT-GGA where the Ni 𝑑 states dominate the DOS at EF, the

Nd-IO components are comparable with Ni d states in the GW results, which as discussed below

is critical for having two gaps and a significant high Tc in phonon-mediated superconductivity.

The microscopic theory of superconductivity based on e-ph coupling makes use of the

quasiparticle band energy 𝜀𝐤 and the e-ph matrix element 𝑔(𝐤,𝐪) – the scattering amplitude

between the states |𝑛𝐤⟩ and |𝑚𝐤 + 𝐪⟩ induced by the electron potential change from the phonon

mode |𝐪𝜈⟩ – among other ingredients. Here, m and n are electron band indices, k and q are

wavevectors, and ν is the phonon branch index. These ingredients can be computed from first

principles [33]. The prevalent ab initio approaches have been based on band structures and e-ph

matrix elements from DFT methods, which however, do not describe true quasiparticle properties

and can fail qualitatively in more correlated materials [28,29]. The GW method for quasiparticle

band energies [26] and the recently developed GW perturbation theory (GWPT) [30,31] for e-ph

matrix elements properly account for the self-energy effects in materials. They provide more

accurate ingredients for the first-principles computation of e-ph coupling and phonon-mediated

superconductivity. In this work, we denote the DFT results as those computed with 𝜀𝐤

 and

𝑔

(𝐤,𝐪), and the GW results as those computed with 𝜀𝐤

 and 𝑔

 (𝐤,𝐪), unless otherwise

stated.

Fig. 2(a) and 2(b) show the state-resolved (band- and wavevector-resolved) e-ph coupling

strength [37-39] 𝜆𝐤 =∑ |𝑔(𝐤,𝐪)|𝛿𝜀𝐤𝐪 − 𝐸 × 

ℏ𝐪

𝐪 for states near EF, where 𝜔𝐪 is

the phonon frequency. 𝜆𝐤 gives the e-ph coupling strength for the state |𝑛𝐤⟩ coupling with all

electronic states on the FS induced by all phonon modes. The Nd-IO states have much stronger e-

ph coupling strength than the Ni states, in both DFT and GW calculations. Electron correlation

(through the GW self-energy) notably gives rise to two important effects: 1) it brings more states

of Nd-IO characters (with stronger e-ph coupling) to the FS (Fig. 1); and 2) it significantly

enhances the magnitude of the e-ph matrix elements. Fig. 2(c) shows the Eliashberg function

𝛼𝐹(𝜔) which describes the phonon-frequency dependent e-ph coupling strength. The GW e-ph

coupling is remarkably higher than that of DFT basically at all frequencies. Consequently, the

Fermi-surface averaged e-ph coupling constant [33,37] 𝜆 = 

∑𝜆𝐤𝛿(𝜀𝐤 − 𝐸)

𝐤 (where NF is

the total DOS at EF) is 0.71 at the GW level but only 0.13 at the DFT level, showing an

enhancement by a factor of 5.5. Experimentally, a temperature-dependent Hall effect was observed

in Nd1-xSrxNiO2 [3,4], indicating a two-carrier (of very different characters) transport scenario,

which is consistent with the theoretical results of a hole-like and an electron-like band at EF. Due

to the stronger e-ph coupling in the Nd-IO band, combining with any residual scattering

mechanisms (such as defects), the mobility of the Nd-IO band is suppressed more significantly

than that of the Ni band at elevated temperatures with increased phonon population. Moreover, an

earlier DMFT study [95] revealed that the electron-electron self-energy for the Ni 𝑑 states is

strongly temperature dependent. These self-energy effects from both the e-ph coupling and

electron-electron interaction, together with defect scattering, provide a conceptual understanding

of the measured temperature-dependent Hall effect. Future studies are however needed for a more

detailed understanding of the transport behaviors of infinite-layer nickelates.

In this study, we construct the fully anisotropic Eliashberg equations [36-39] using the GW

quasiparticle band structure and the GWPT e-ph matrix elements to solve for the superconducting

properties of Nd1-xSrxNiO2. The k- and q-dependence is densely sampled via Wannier interpolation

techniques [43,96]. We have also considered possible short-range antiferromagnetic fluctuations

of local Ni moments on the electronic structure and find that they have little effects on the central

Nd-IO band [47].

Fig. 3(a) shows the distribution of the superconducting gap [33,36-40] Δ𝐤 on the FS of

Nd0.8Sr0.2NiO2 at temperature T = 5 K, obtained by solving the anisotropic Eliashberg theory at the

full GW level. The effective Coulomb potential [33,37-40] 𝜇∗ is set to 0.05, and the dependence in

𝜇∗ for superconducting properties such as Tc is very weak [47]. The gap values are highly band-

and k-dependent and vary dramatically on the FS, leading to a bimodal distribution of gaps (Fig.

3(b)) – i.e., Nd0.8Sr0.2NiO2 hosts a fundamentally two-gap superconductivity. Fig. 3(b) shows the

density distribution of Δ𝐤 with orbital projections. The Ni band has small gaps of 0.5 – 2.5 meV

with an average of 1.7 meV, whereas the Nd-IO band has large gaps of 3 – 5 meV with an average

of 4.1 meV.

To understand the origin of the two-gap nature of the superconductivity, we plot in Fig. 3(c)

the density distribution of the state-pair-resolved e-ph coupling strength [38,40] 𝜆𝐤,𝐤𝐪 =

𝑁∑

ℏ𝐪 |𝑔(𝐤,𝐪)|

, which represents the e-ph coupling strength between the pair of states

|𝑛𝐤⟩ and |𝑚𝐤 + 𝐪⟩ induced by all phonon branches. The intra-band e-ph coupling distributions

are quite distinct for the Ni band and the Nd-IO band. The Ni ↔ Ni coupling is narrowly distributed

at low coupling values (with an average 𝜆 ↔ 

. = 0.22) whereas the Nd-IO ↔ Nd-IO coupling

is broadly distributed up to a high coupling value (with 𝜆 ↔ 

. = 1.88 and some

individual values up to ~4.0). The Nd-IO ↔ Ni inter-band e-ph coupling shows an asymmetric

distribution within a relatively moderate coupling range (with 𝜆 ↔ 

. = 0.58 ). These

distinctive distributions of 𝜆𝐤,𝐤𝐪 result in a well separated two-peak structure in the state-

resolved 𝜆𝐤 density distribution (with 𝜆

. = 1.14 and 𝜆

. = 0.37) as shown in Fig. 3(d) by

taking the average of 𝜆𝐤,𝐤𝐪 over all the |𝑚𝐤 + 𝐪⟩ states on the FS. Consequently, two-gap

superconductivity arises in Nd0.8Sr0.2NiO2 (Fig. 3(b)).

Solving the anisotropic Eliashberg equations [36-40] (with 𝜇∗ = 0.05) at different temperatures

yields superconducting gaps as a function of T (Fig. 4(a)). We obtain a theoretical Tc = 22.3 K for

Nd0.8Sr0.2NiO2, agreeing well with the experimental Tc = 19.3 K at x = 0.2 [5], but is in strong

contrast with the DFT result of Tc = 0 K [47]. Fig. 4(b) shows the computed superconducting

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1Two-GapSuperconductivityandDecisiveRoleofRare-EarthdElectronsinInfinite-LayerNickelatesZhengluLi1,2,3andStevenG.Louie1,2,*1DepartmentofPhysics,UniversityofCaliforniaatBerkeley,Berkeley,CA,USA.2MaterialsSciencesDivision,LawrenceBerkeleyNationalLaboratory,Berkeley,CA,USA.3MorkFamilyDepartmentofChemic...

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