# Dynamical structural instability and its implication on the physical
properties of infinite-layer nickelates

Chengliang Xia1,2, Jiaxuan Wu2, Yue Chen1 and Hanghui Chen2,3 1Department of
Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong
SAR, China
2NYU-ECNU Institute of Physics, NYU Shanghai, Shanghai 200122, China
3Department of Physics, New York University, New York, NY 10003, USA

###### Abstract

We use first-principles calculations to find that in infinite-layer nickelates
$R$NiO2, the widely studied tetragonal $P4/mmm$ structure is only dynamically
stable for early lanthanide elements $R$ = La-Sm. For late lanthanide elements
$R$ = Eu-Lu, an imaginary phonon frequency appears at $A=(\pi,\pi,\pi)$ point.
For those infinite-layer nickelates, condensation of this phonon mode into the
$P4/mmm$ structure leads to a more energetically favorable $I4/mcm$ structure
that is characterized by an out-of-phase rotation of “NiO4 square”. Special
attention is given to two borderline cases: PmNiO2 and SmNiO2, in which both
the $P4/mmm$ structure and the $I4/mcm$ structure are local minimums, and the
energy difference between the two structures can be fine-tuned by epitaxial
strain. Compared to the $P4/mmm$ structure, $R$NiO2 in the $I4/mcm$ structure
has a substantially reduced Ni $d_{x^{2}-y^{2}}$ bandwidth, a smaller Ni $d$
occupancy, a “cleaner” Fermi surface with a lanthanide-$d$-derived electron
pocket suppressed at $\Gamma$ point, and a decreased critical
$U_{\textrm{Ni}}$ to stabilize long-range antiferromagnetic ordering. All
these features imply enhanced correlation effects and favor Mott physics. Our
work reveals the importance of structure-property relation in infinite-layer
nickelates, in particular the spontaneous “NiO4 square” rotation provides a
tuning knob to render $R$NiO2 in the $I4/mcm$ structure a closer analogy to
superconducting infinite-layer cuprates.

##  I Introduction

The discovery of superconductivity in infinite-layer nickelates SrxNd1-xNiO2
Li et al. (2019) has drawn great attention Hepting et al. (2020); Zeng et al.
(2020); Li et al. (2020a); Gu et al. (2020a); Goodge et al. (2021); Wang et
al. (2021); Zhao et al. (2021); Lu et al. (2021) because the parent material
NdNiO2 has similar crystal and electronic structures to those of infinite-
layer cuprate CaCuO2 Sawatzky (2019), which exhibits high-temperature
unconventional superconductivity upon doping Smith et al. (1991); Azuma et al.
(1992). Both NdNiO2 and CaCuO2 crystallize in a simple tetragonal $P4/mmm$
structure in which Ni (Cu) and O atoms form a flat “NiO4 (CuO4) square”
Siegrist et al. (1988); Li et al. (2020b); Wang et al. (2020a). The $P4/mmm$
crystal structure has only two degrees of freedom: lattice constants $a$ and
$c$. As for the non-interacting electronic structure, first-principles
calculations show that CaCuO2 has only one Cu $d_{x^{2}-y^{2}}$-derived band
that crosses the Fermi level, while NdNiO2 has two bands crossing the Fermi
level Jiang et al. (2019); Botana and Norman (2020); Karp et al. (2020a);
Adhikary et al. (2020); Zhang et al. (2020a). One is Ni
$d_{x^{2}-y^{2}}$-derived band and the other band is derived from Nd $d$
orbitals and an interstitial $s$ orbital Nomura et al. (2019); Gu et al.
(2020b); Hirayama et al. (2020). So far, the minimum theoretical model that is
adequate to describe the low-energy physics of infinite-layer NdNiO2 has been
under intensive debate and several different mechanisms for superconductivity
in SrxNd1-xNiO2 have been proposed Hu and Wu (2019); Wang et al. (2020b);
Jiang et al. (2020); Si et al. (2020); Geisler and Pentcheva (2020);
Sakakibara et al. (2020); He et al. (2020); Wu et al. (2020); Werner and
Hoshino (2020); Zhang et al. (2020b); Wang et al. (2020c); Zhang and
Vishwanath (2020); Bernardini et al. (2020); Bernardini and Cano (2020); Liu
et al. (2021); Wan et al. (2021); Plienbumrung et al. (2021); Malyi et al.
(2021); Peng et al. (2021); Choubey and Eremin (2021); Kang and Kotliar
(2021). Albeit there are many important differences, one thing in common is
that all first-principles calculations use the $P4/mmm$ crystal structure of
NdNiO2 (either experimental one or theoretical optimized one), based on which
one-particle band structure calculations (using density functional theory and
its Hubbard $U$ extension) Krishna et al. (2020); Choi et al. (2020a); Liu et
al. (2020); Choi et al. (2020b); Zhang et al. (2020c, 2021) or more
sophisticated many-body electronic structure calculations (such as dynamical
mean field theory and GW) Katukuri et al. (2020); Olevano et al. (2020); Karp
et al. (2020b); Ryee et al. (2020); Lechermann (2020a, b); Leonov et al.
(2020); Petocchi et al. (2020); Lechermann (2021); Kitatani et al. (2020);
Karp et al. (2021); Leonov (2021); Kutepov (2021) are performed. On the
experimental side, in addition to SrxNd1-xNiO2, recently superconductivity is
also observed in SrxPr1-xNiO2 Osada et al. (2020a, b); Ren et al. (2021),
SrxLa1-xNiO2 Osada et al. (2021) and CaxLa1-xNiO2 Zeng et al. (2021); Puphal
et al. (2021). Thus it is anticipated that superconductivity should be
observed in the entire lanthanide series of infinite-layer nickelates $R$NiO2.
In particular, Refs. Kapeghian and Botana (2020); Been et al. (2021) perform a
systematic study on the electronic structure of $R$NiO2 in the $P4/mmm$
structure as $R$ traverses the lanthanide series and find promising trends
that favor superconductivity.

In this work, we use first-principles calculations to show that the widely
studied $P4/mmm$ structure of infinite-layer nickelates is only dynamically
stable for early lanthanide elements $R$ = La-Sm. For late lanthanide elements
$R$ = Eu-Lu, an imaginary phonon mode appears at $A=(\pi,\pi,\pi)$ point in
the $P4/mmm$ structure of $R$NiO2. The imaginary phonon mode corresponds to an
out-of-phase rotation of “NiO4 square” about the $z$ axis. Condensation of
this unstable phonon mode into the $P4/mmm$ structure leads to a more
energetically favorable crystal structure with lower symmetry (space group
$I4/mcm$). Attention is given to two borderline cases PmNiO2 and SmNiO2, which
have two local minimums: the $P4/mmm$ structure and the $I4/mcm$ structure.
Epitaxial strain can be used to fine-tune the energy difference between the
two crystal structures.

Compared to the $P4/mmm$ structure, infinite-layer $R$NiO2 in the new $I4/mcm$
structure has a distinct electronic structure: the Ni $d_{x^{2}-y^{2}}$
bandwidth is substantially reduced (by about 0.5 eV) and Ni $d$ occupancy
decreases; the Fermi surface becomes “cleaner” because one
lanthanide-$d$-derived electron pocket disappears at the $\Gamma$ point; the
critical $U_{\textrm{Ni}}$ to stabilize a long-range antiferromagnetic
ordering in $R$NiO2 is smaller. All these features imply that with the “NiO4
square” rotation, correlation effects will be enhanced and Mott physics will
play a more prominent role in the new $I4/mcm$ structure than in the $P4/mmm$
structure, when local interaction is added on Ni $d$ orbitals Lee et al.
(2006). In particular, our results suggest that among the lanthanide series of
infinite-layer nickelates, SmNiO2 is the most promising candidate to
crystallize in the $I4/mcm$ structure, which renders it a closer analogy to
superconducting infinite-layer cuprates.

##  II Computational Details

We perform density functional theory (DFT) Hohenberg and Kohn (1964); Kohn and
Sham (1965) calculations within the ab initio plane-wave approach, as
implemented in the Vienna Ab-initio Simulation Package (VASP) Payne et al.
(1992); Kresse and Furthmuller (1996). We use projected augmented wave (PAW)
pseudopotentials with the 4$f$ electrons placed in the core (except for La),
explicitly to avoid complication that arises from treating the localized 4$f$
electrons. We employ generalized gradient approximation (GGA) for the
exchange-correlation functional with Perdew-Burke-Ernzerhof (PBE)
parametrization Perdew et al. (1996). The theoretical lattice constants of
$R$NiO2 are in good agreement with the available experimental structure
information (see Supplementary Materials SI  Sec. I) Wang et al. (2020a);
Zhang et al. (2021); Lin et al. (2021). We use an energy cutoff of 600 eV.
Charge self-consistent calculations are converged to 10-7 eV. Both cell and
internal atomic positions are fully relaxed until each force component is
smaller than 1 meV/Å and pressure on the cell is smaller than 0.1 kB. We use
the finite-displacement method to calculate the full phonon dispersion with
the aid of Phonopy Togo and Tanaka (2015). A supercell that consists of
$3\times 3\times 3$ primitive cells is used to calculate the force constants
and dynamical matrices. The primitive cell of the $P4/mmm$ structure has one
formula of $R$NiO2 (i.e. 4 atoms), while the primitive cell of the $I4/mcm$
structure has two formulae of $R$NiO2 (i.e. 8 atoms) in order to accommodate
the rotation of “NiO4 square”. For the 4-atom $P4/mmm$ cell, we use a
Monkhorst-Pack k mesh of $14\times 14\times 14$ to sample the first Brillouin
zone. For the 8-atom $I4/mcm$ cell, we use a Monkhorst-Pack k mesh of
$10\times 10\times 10$ to sample the first Brillouin zone. For ease of
comparison, when calculating the electronic structure, Fermi surface and long-
range magnetic ordering, we use the 8-atom cell for both $P4/mmm$ and $I4/mcm$
structures. This cell-doubling is also necessary to accommodate the rocksalt
antiferromagnetic ordering in the $P4/mmm$ structure. To study the energy
evolution as a function of “NiO4 square” rotation, we use a linear
interpolation and generate an intermediate crystal structure between the
fully-relaxed $P4/mmm$ and $I4/mcm$ crystal structures. The “NiO4 square”
rotation angle continuously changes with an interpolation parameter $\lambda$
(more details are found in Supplementary Materials SI  Sec. II). When we
impose a bi-axial strain on infinite-layer nickelates, the out-of-plane $c$
axis is fully relaxed in order to minimize the total energy. When calculating
Sr${}_{x}R_{1-x}$NiO2, we use virtual crystal approximation (VCA) Bellaiche
and Vanderbilt (2000). To break spin symmetry and study magnetic order, we use
the charge-only DFT+$U$+$J$ method Park et al. (2015); Chen et al. (2015);
Chen and Millis (2016) (by setting LDAUTYPE = 4 in VASP). This method is such
that the exchange-correlation functional only depends on charge density but
not on spin density; thereby, spin symmetry is only broken by the $U/J$ terms
that are added to the Kohn-Sham potential, while the exchange splitting that
arises from the spin-dependent exchange-correlation functional is disabled. By
setting the parameter $U/J=0$ in the charge-only DFT+$U$+$J$ method, the non-
spin-polarized DFT results are recovered. The calculations of Ni $d$ projected
magnetic moment and Ni $d$ occupancy use the default VASP value for the radius
of sphere which is 1.11 Å. We also use the charge-only DFT+$U$+$J$ method to
test phonon spectrum for a few representative $R$NiO2 (see Supplementary
Materials SI  Sec. VIII) Lin et al. (2021); Hayward et al. (1999); Hayward and
Rosseinsky (2003).

##  III Results and Discussion

###  III.1 Phonon spectrum

Figure 1:  Phonon properties of infinite-layer $R$NiO2 in the $P4/mmm$
structure. a: The series of lanthanide elements. Different colors are used to
distinguish four categories of phonon dispersions, whose prototypes are shown
in b-e. b: Phonon dispersion of infinite-layer NdNiO2 in the $P4/mmm$
structure. c: Phonon dispersion of infinite-layer SmNiO2 in the $P4/mmm$
structure. d: Phonon dispersion of infinite-layer EuNiO2 in the $P4/mmm$
structure. e: Phonon dispersion of infinite-layer TmNiO2 in the $P4/mmm$
structure. f: Frequency of the lowest phonon mode at $A=(\pi,\pi,\pi)$ point
$\omega^{*}_{A}$ for the entire lanthanide series of infinite-layer $R$NiO2 in
the $P4/mmm$ structure. g: The lowest phonon mode at $A$ point of infinite-
layer $R$NiO2 in the $P4/mmm$ structure.

We calculate the phonon dispersion of the fully-relaxed $P4/mmm$ structure for
the entire lanthanide series of infinite-layer $R$NiO2. We find that the
complete set of phonon dispersions (see Supplementary Materials SI  Sec. II)
can be classified into four categories, which we use colors to distinguish in
panel a. The first category includes $R$=La-Nd (denoted by green). NdNiO2 is
the prototype, whose full phonon dispersion is shown in panel b. In this
category, the full phonon dispersion is free of imaginary modes and the
$P4/mmm$ crystal structure is dynamically stable. The second category includes
$R$=Pm, Sm (denoted by yellow). SmNiO2 is the prototype, whose full phonon
dispersion is shown in panel c. In this category, a soft phonon develops at
$A=(\pi,\pi,\pi)$ point, implying a potentially unstable mode. The third
category includes $R$=Eu-Dy (denoted by orange). EuNiO2 is the prototype,
whose full phonon dispersion is shown in panel d. In this category, the
frequency of the lowest phonon mode at $A$ point (marked as $\omega^{*}_{A}$)
becomes imaginary and the $P4/mmm$ crystal structure is dynamically unstable.
The last category includes $R$=Ho-Lu (denoted by red). TmNiO2 is the
prototype, whose full phonon dispersion is shown in panel e. In this category,
the lowest phonon modes at multiple q points become imaginary in the phonon
dispersion, indicating that the $P4/mmm$ crystal structure is far from stable.
In panel f, we compare the frequency of the lowest phonon mode at $A$ point
$\omega^{*}_{A}$ for the entire lanthanide series. We find that
$\omega^{*}_{A}$ monotonically decreases from La to Lu and becomes imaginary
when $R$=Eu and beyond. In panel g, we show the lowest phonon mode at $A$
point of the $P4/mmm$ structure, which is an out-of-phase rotation of “NiO4
square” about the $z$ axis. Infinite-layer nickelates with early lanthanide
elements such as NdNiO2 are stable against this “NiO4 square” rotation, and
their equilibrium structure is the widely-studied $P4/mmm$ structure. However,
infinite-layer nickelates with late lanthanide elements become dynamically
unstable when the “NiO4 square” rotates. Condensation of this unstable mode
into the $P4/mmm$ structure will lead to a more energetically favorable
crystal structure, which in turn results in a new electronic structure.

###  III.2 Rotation of “NiO4 square” and the new $I4/mcm$ structure

When the lowest phonon frequency at the $A$ point in the $P4/mmm$ structure of
$R$NiO2 becomes imaginary ($R$ = Eu-Lu), it means that condensation of this
unstable phonon mode into the $P4/mmm$ structure can decrease the total energy
and will result in a new crystal structure with lower symmetry. Such a crystal
structure is shown in panel a of Fig. 2, which has space group $I4/mcm$. The
primitive cell of the $I4/mcm$ structure has 8 atoms, which has three degrees
of freedom: in addition to the two lattice constants $a$ and $c$, there is an
angle $\theta$ that characterizes the out-of-phase rotation of “NiO4 square”
about the $z$ axis. When $\theta=0$, the $I4/mcm$ structure is reduced to the
$P4/mmm$ structure. For ease of comparison with the $I4/mcm$ structure, we
show the two-Ni unit cell of the $P4/mmm$ structure (doubling the 4-atom
primitive cell along the $[111]$ direction) in panel b. For subsequent
electronic structure, Fermi surface and long-range magnetic ordering
calculations of infinite-layer $R$NiO2, we use the 8-atom cell for both the
$I4/mcm$ structure and the $P4/mmm$ structure.

Figure 2:  a: The primitive cell of infinite-layer nickelate $R$NiO2 in the
$I4/mcm$ structure. $\theta$ is the rotation angle of “NiO4 square” along the
$z$ axis. b: A doubled unit cell of infinite-layer nickelate $R$NiO2 in the
$P4/mmm$ structure (the $P4/mmm$ primitive cell is doubled along [111]
direction). $\theta=0^{\circ}$ in the $P4/mmm$ structure. c: The energy
evolution of $R$NiO2 ($R$ = Nd, Pm, Sm and Eu) as a function of the “NiO4
square” rotation angle $\theta$. The $P4/mmm$ structure is used as the energy
reference. d: The zoom-in view of the energy evolution of PmNiO2 and SmNiO2 as
a function of the “NiO4 square” rotation angle $\theta$. e and f: The phonon
dispersions of infinite-layer PmNiO2 and SmNiO2 in the $I4/mcm$ structure,
respectively.

To get a better understanding of the new $I4/mcm$ structure, we calculate the
energy evolution of $R$NiO2 as a function of the rotation angle $\theta$ (see
panel c of Fig. 2, the calculation details can be found in Supplementary
Materials Sec. II SI ). We select $R$ = Nd, Pm, Sm and Eu, which are near the
phase boundary where the lowest phonon frequency at $A$ point becomes
imaginary. We find that the total energy of NdNiO2 monotonically increases
with the rotation angle $\theta$, indicating that the $P4/mmm$ structure (i.e.
$\theta=0$) is stable against the rotation of “NiO4 square”. By contrast, the
total energy of EuNiO2 first decreases with the rotation angle $\theta$ and
then increases. The energy minimum is at $\theta=7.7^{\circ}$. This clearly
shows that the $P4/mmm$ structure is not dynamically stable in infinite-layer
EuNiO2. PmNiO2 and SmNiO2 (the second category) exhibit more interesting
features in that they have two local minimums: one is at $\theta=0$ ($P4/mmm$
structure) and the other is at $\theta>0$ ($I4/mcm$ structure), as is shown in
panel d. For PmNiO2, the energy of the $P4/mmm$ structure is slightly lower
than that of the $I4/mcm$ structure by 0.5 meV/f.u. For SmNiO2, the energy
order is reversed and the $I4/mcm$ structure becomes more stable than the
$P4/mmm$ structure by 10.5 meV/f.u. Furthermore, we calculate the energy
barrier from the $P4/mmm$ structure to the $I4/mcm$ structure. We find that
the barrier decreases from 2.8 meV/f.u. for PmNiO2 to 0.5 meV/f.u. for SmNiO2.
Next we test that after condensation of the “NiO4 square” rotation mode, the
$I4/mcm$ structure becomes dynamically stable in some infinite-layer
nickelates. We perform the phonon calculation of the $I4/mcm$ structure for
PmNiO2 and SmNiO2. The phonon dispersions are shown in panels e and f. We find
that the phonon dispersion of the $I4/mcm$ structure is free from imaginary
frequencies for PmNiO2 and SmNiO2. We make two comments here. First, while
SmNiO2 has two local minimums, considering the facts that 1) its $I4/mcm$
structure is energetically more favorable than the $P4/mmm$ structure, 2) the
energy barrier for SmNiO2 to transition from the $P4/mmm$ structure to the
$I4/mcm$ structure is tiny (0.5 meV/f.u.), and 3) its $I4/mcm$ structure is
dynamically stable, we argue that in experiments SmNiO2 is most likely
stabilized in the $I4/mcm$ structure. Second, the “NiO4 square” rotation is
the first structural distortion that will appear in the $P4/mmm$ structure of
infinite-layer nickelates $R$NiO2 when the lanthanide element $R$ traverses
from La to Lu. For late lanthanide elements (such as Ho-Lu), more complicated
structural distortions are expected to emerge in infinite-layer $R$NiO2. The
purpose of the current study is to show that just by including one more degree
of freedom in the crystal structure of $R$NiO2 (i.e. “NiO4 square” rotation),
the resulting electronic structure trends can be qualitatively different from
those of the $P4/mmm$ structure (see discussion below).

###  III.3 Epitaxial strain

Figure 3:  a: The lowest phonon frequency at $A=(\pi,\pi,\pi)$ point of
infinite-layer $R$NiO2 in the $P4/mmm$ structure ($R$ = Pm, Sm, Eu and Gd) as
a function of biaxial strain. Negative (positive) strain means compressive
(tensile) biaxial strain. b and c : The phonon dispersions of infinite-layer
SmNiO2 in the $P4/mmm$ structure. The cyan curves are the phonon dispersions
of fully-relaxed SmNiO2 (i.e. without strain). The red curves in b and c are
the phonon dispersions of SmNiO2 under 1% and 2% compressive strains,
respectively. The black arrows highlight that with compressive strain, the
lowest phonon mode at $A=(\pi,\pi,\pi)$ point is “hardened”, while the lowest
phonon modes at $Z=(0,0,\pi)$ and $R=(\pi,0,\pi)$ are “softened”. d: The
energy evolution of infinite-layer SmNiO2 as a function of “NiO4 square”
rotation angle $\theta$. The red, cyan and green symbols correspond to 1%
compressive strain, no strain and 1% tensile strain, respectively. For each
case, the $P4/mmm$ structure is used as the energy reference.

Before we carefully compare the physical properties of $R$NiO2 between the
$I4/mcm$ structure and the $P4/mmm$ structure, we study epitaxial strain
effects first. That is because superconductivity in infinite-layer nickelates
is observed in thin films rather than in bulk Li et al. (2019, 2020b, 2020a);
Zeng et al. (2020); Osada et al. (2020a, b); Ren et al. (2021); Osada et al.
(2021); Zeng et al. (2021); He et al. (2021). We investigate how epitaxial
strain influences the phonon dispersion of infinite-layer $R$NiO2, in
particular, whether it may remove the imaginary phonon mode at $A$ point and
thus stabilize the $P4/mmm$ structure. Experimentally, oxide thin films are
grown along the $z$ axis with a biaxial strain imposed by substrates in the
$xy$ plane. The biaxial strain (either compressive or tensile) typically
ranges within 3% Schlom et al. (2008).

Panel a of Fig. 3 shows the lowest phonon frequency at $A$ point
$\omega_{A}^{*}$ of a few infinite-layer $R$NiO2 in the $P4/mmm$ structure as
a function of biaxial strain $\xi$. We select $R$ = Pm, Sm, Eu and Gd, which
are close to the phase boundary where $\omega_{A}^{*}$ becomes imaginary (the
complete phonon dispersions of those four nickelates under epitaxial strain
are found in Supplementary Materials SI  Sec. III). The biaxial strain is
defined as $\xi=(a_{\textrm{sub}}-a)/a\times 100\%$, where $a_{\textrm{sub}}$
is the theoretical substrate lattice constant and $a$ is the DFT optimized
lattice constant of infinite-layer $R$NiO2 in the $P4/mmm$ structure. For each
infinite-layer $R$NiO2, we vary the strain $\xi$ and find that
$\omega_{A}^{*}$ decreases with tensile strain and increases with compressive
strain. However, we note that for infinite-layer nickelate GdNiO2, a
compressive strain up to 3$\%$ can not remove the “NiO4 rotation” instability
in the $P4/mmm$ structure. This is also true for other infinite-layer
nickelates $R$NiO2 with late lanthanide elements ($R$ = Gd-Lu). More
importantly, we find that while compressive strain helps remove the phonon
instability at $A$ point in the $P4/mmm$ structure, it may induce other phonon
instabilities. Panel b compares the phonon dispersions of infinite-layer
nickelate SmNiO2 in the $P4/mmm$ structure under 1% compressive strain versus
without epitaxial strain. It shows that compressive strain “hardens” the
lowest phonon frequency at $A$ point but “softens” the lowest phonon
frequencies at $Z$ and $R$ points. Under a compressive strain of 2% or larger
(see panel c and Fig. S3 in the Supplementary Materials SI ), the lowest
phonon frequencies at $Z$ and $R$ points become imaginary in the $P4/mmm$
structure. To summarize, for infinite-layer nickelates, tensile strain
increases the phonon instability at $A$ point in the $P4/mmm$ structure; small
compressive strain helps remove the phonon instability at $A$ point but larger
compressive strain can cause other phonon instabilities at $Z$ and $R$ points.
Hence, epitaxial strain alone cannot substantially increase the stability of
the $P4/mmm$ structure in infinite-layer nickelates $R$NiO2.

On the other hand, we find that for SmNiO2, epitaxial strain can tune its
energetics and structural properties. Panel d of Fig. 3 shows the energy
evolution of SmNiO2 as a function of “NiO4 square” rotation angle $\theta$. We
compare three different epitaxial strains: 1% compressive (-1%), no strain
(0%) and 1% tensile (+1%). From 1% compressive strain to 1% tensile strain,
the energy difference between the $I4/mcm$ structure and the $P4/mcm$
structure monotonically increases from 4.8 to 17.5 meV/f.u. in its magnitude
(indicating that the $I4/mcm$ structure gradually becomes more stable than the
$P4/mmm$ structure). At the same time, the energy barrier from the $P4/mmm$
structure to the $I4/mcm$ structure decreases from 1.5 meV/f.u. (1%
compressive strain) to 0.5 meV/f.u. (no strain) and disappears (1% tensile
strain). The disappearance of the energy barrier indicates that under 1%
tensile strain, the $P4/mmm$ structure is no longer a local minimum in SmNiO2
and it spontaneously transitions into the $I4/mcm$ structure. Furthermore, the
equilibrium “NiO4 square” rotation angle $\theta$ in the $I4/mcm$ structure
also increases from $6.4^{\circ}$ to $7.1^{\circ}$ when the epitaxial strain
changes from 1% compressive to 1% tensile strain. This shows that epitaxial
strain can be used as a fine-tuning knob to delicately control the structural
stability of infinite-layer SmNiO2.

In the next three sections, we will study the entire lanthanide series of
infinite-layer nickelates in the $I4/mcm$ structure and in the $P4/mcm$
structure. We compare the trends in electronic properties and magnetic
properties between the two crystal structures. For demonstration, we use
SmNiO2 as a prototype.

###  III.4 P4/mmm versus I4/mcm structures: structural properties

Fig. 4 compares the structural properties of $R$NiO2 between the $I4/mcm$
structure and the $P4/mmm$ structure. For infinite-layer nickelates in the
first category ($R$ = La-Nd), we only study the $P4/mmm$ structure because the
$I4/mcm$ structure cannot be stabilized in those nickelates. Panels a shows
the lattice constants $a$ and $c$ of the $I4/mcm$ and $P4/mmm$ structures. For
ease of comparison to the $P4/mmm$ structure, we convert the lattice constants
of the $I4/mcm$ structure into the pseudo-tetragonal lattice constants $a$ and
$c$ (see Supplementary Materials SI  Sec. I). The general trend is similar in
the two structures that $a$ and $c$ get smaller when $R$ traverses the
lanthanide series Been et al. (2021). For a given $R$, the $a$ and $c$ lattice
constants are larger in the $I4/mcm$ structure than in the $P4/mmm$ structure.
Panel b shows the volume per Ni atom of the $I4/mcm$ and $P4/mmm$ structures.
Consistent with the trends of the lattice constants, the volume decreases as
$R$ traverses the lanthanide series. The key difference between the $I4/mcm$
structure and the $P4/mmm$ structure lies in the “NiO4 square” rotation. In
panel c, we show the “NiO4 square” rotation angle $\theta$. In the $P4/mmm$
structure, $\theta=0$ by definition. We find that $\theta$ increases in the
$I4/mcm$ structure, as we traverse from Pm to Lu. A finite $\theta$ means that
the in-plane Ni-O-Ni bond angle is reduced from the ideal 180∘. A direct
consequence of $\theta$ is the elongation of Ni-O bond. In the $P4/mmm$
structure, the Ni-O bond length is simply half of the lattice constant $a$,
which decreases as $R$ traverses the lanthanide series. By contrast, in the
$I4/mcm$ structure, Ni-O bond length is elongated compared to that in the
$P4/mmm$ structure and it slowly increases as $R$ traverses the lanthanide
series. We note that the “NiO4 square” rotation $\theta$ and the volume
reduction are two competing forces on the Ni-O bond length. The former, which
is absent in the $P4/mmm$ structure, is more dominating in the $I4/mcm$
structure. A similar picture of these two competing forces is also found in
LiNbO3 under hydrostatic pressure Xia et al. (2021). The different behaviors
of Ni-O-Ni bond angle and Ni-O bond length in the $I4/mcm$ structure versus in
the $P4/mmm$ structure will have important influences on the electronic
properties of $R$NiO2.

Figure 4:  Crystal information of infinite-layer nickelates $R$NiO2 in the
$I4/mcm$ structure (red symbols) and in the $P4/mmm$ structure (blue symbols).
a: The lattice constants $a$ (circle symbols) and $c$ (square symbols). For
ease of comparison, we convert the lattice constants of the $I4/mcm$ structure
into the pseudo-tetragonal lattice constants $a$ and $c$. b: Volume per
$R$NiO2 formula (f.u.). c: “NiO4 square” rotation angle $\theta$. In the
$P4/mmm$ structure, $\theta=0^{\circ}$. d: Ni-O bond length.

###  III.5 P4/mmm versus I4/mcm structures: electronic properties

Figure 5:  Electronic properties of infinite-layer nickelates $R$NiO2 in the
$I4/mcm$ structure and in the $P4/mmm$ structure. a and b: bandwidth of Ni
$d_{x^{2}-y^{2}}$ orbital and Ni $d$ occupancy. The red (blue) symbols refer
to the $I4/mcm$ structure (the $P4/mmm$ structure). c and d: Electronic
structure of SmNiO2 in the $P4/mmm$ structure and in the $I4/mcm$ structure,
respectively. The red (blue) symbols highlight the atomic projection onto the
Ni $d_{x^{2}-y^{2}}$ orbital (Sm $d$ orbitals). The purple arrows in c and d
highlight that a Sm-$d$-derived band is removed from the Fermi level and is
pushed up to higher energy via the “NiO4 square” rotation in the $I4/mcm$
structure. e and f: Fermi surface of SmNiO2 in the $P4/mmm$ structure and in
the $I4/mcm$ structure, respectively. The red (blue) arrows highlight the
Fermi surface sheets that are composed of Ni $d_{x^{2}-y^{2}}$ orbital (Sm $d$
orbitals).

Panel a of Fig. 5 compares the Ni $d_{x^{2}-y^{2}}$ bandwidth of the $I4/mcm$
structure and the $P4/mmm$ structure. Consistent with the previous studies
Kapeghian and Botana (2020); Been et al. (2021), the Ni $d_{x^{2}-y^{2}}$
bandwidth of the $P4/mmm$ structure monotonically increases when $R$ traverses
the lanthanide series. That is because the Ni-O bond length of the $P4/mmm$
structure monotonically decreases, which increases the Ni-O hopping and thus
the bandwidth. By contrast, we find that the Ni $d_{x^{2}-y^{2}}$ bandwidth of
the $I4/mcm$ structure monotonically decreases when $R$ traverses the
lanthanide series. That is consistent with the trend of a decreasing Ni-O-Ni
bond angle and an increasing Ni-O bond length, both of which reduce the
overlap between Ni-$d_{x^{2}-y^{2}}$ and O-$p$ orbitals and thus suppress the
Ni-O hopping Kumah et al. (2014). More importantly, for a given lanthanide
element $R$, the Ni $d_{x^{2}-y^{2}}$ bandwidth of the $I4/mcm$ structure is
substantially smaller than that of the $P4/mmm$ structure. For example, for
SmNiO2, its Ni $d_{x^{2}-y^{2}}$ bandwidth is 3.2 eV in the $P4/mmm$ structure
and is reduced to 2.7 eV in the $I4/mcm$ structure. This indicates that for
the same value of $U$ on Ni $d_{x^{2}-y^{2}}$ orbital, correlation strength is
increased in the $I4/mcm$ structure, compared to the $P4/mmm$ structure. In
addition, we also compare the Ni $d$ orbital occupancy $N_{d}$ between the
$I4/mcm$ structure and the $P4/mmm$ structure in panel b. Ref. Wang et al.
(2012) shows that the metal $d$ orbital occupancy $N_{d}$ is a good measure of
$p$-$d$ hybridization in complex oxides. We find that as $R$ traverses the
lanthanide series, $R$NiO2 in the $P4/mmm$ structure has a progressively
increased $N_{d}$. This is consistent with the previous study Been et al.
(2021) which shows that the O-$p$ content monotonically decreases across the
lanthanide series. However, for $R$NiO2 in the $I4/mcm$ structure, $N_{d}$
almost stays a constant as $R$ traverses the lanthanide series. By analyzing
the density of states of $R$NiO2 (see Supplementary Materials SI  Sec. VI), we
find that in the $P4/mmm$ structure, running across the lanthanide series, the
centroid of O-$p$ states monotonically decreases to lower energy, by about 1
eV from La to Lu Been et al. (2021). This change increases the charge-transfer
energy and decreases the $p$-$d$ hybridization. But in the $I4/mcm$ structure,
the “NiO4 square” rotation counteracts this effect and the centroid of O-$p$
states almost does not move across the lanthanide series. As a result, the Ni
$d$ occupancy $N_{d}$ and $p$-$d$ hybridization change marginally across the
lanthanide series of $R$NiO2. We note that for a given lanthanide element $R$,
$N_{d}$ is smaller in the $I4/mcm$ structure than in the $P4/mmm$ structure. A
smaller $N_{d}$ corresponds to a smaller critical $U$ value for the metal-
insulator transition Wang et al. (2012) , i.e. $R$NiO2 in the $I4/mcm$
structure is closer to the Mott insulating phase than that in the $P4/mmm$
structure.

Next we study the electronic band structure and Fermi surface of SmNiO2 as a
prototype (very similar results are also obtained in infinite-layer nickelates
close to the phase boundary $R$NiO2 with $R$ = Pm, Eu and Gd, see
Supplementary Materials SI  Sec. V). Panel c of Fig. 5 shows the band
structure of SmNiO2 in the $P4/mmm$ structure. Due to the cell-doubling, there
are four bands that cross the Fermi level: two are
Ni-$d_{x^{2}-y^{2}}$-derived bands and the other two are Sm-$d$-derived bands.
In the unfolded Brillouin zone (BZ), the Sm-$d$-derived band crosses the Fermi
level and results in two electron pockets: one is at $\Gamma$ point and the
other is at $A$ point. After band folding, in the body-centered-tetragonal
Brillouin zone (BCT-BZ) Been et al. (2021), the electron pocket that is
originally at $A$ point in the unfolded BZ is mapped to $\Gamma$ point,
leading to two electron pockets at $\Gamma$ point. This is clearly seen in
panel e, which shows the Fermi surface of SmNiO2 in the $P4/mmm$ structure.
These $\Gamma$-centered electron pockets are one of the main differences
between infinite-layer nickelates and superconducting cuprates and their role
is still under debate Jiang et al. (2019); Botana and Norman (2020); Karp et
al. (2020a); Adhikary et al. (2020); Zhang et al. (2020a); Nomura et al.
(2019); Gu et al. (2020b); Hirayama et al. (2020). Panel d of Fig. 5 shows the
band structure of SmNiO2 in the $I4/mcm$ structure. Compared to the $P4/mmm$
structure, the “NiO4 square” rotation removes one Sm-$d$-derived band
(highlighted by the purple arrows) away from the Fermi surface and pushes it
to higher energy. In the corresponding Fermi surface (panel f), one
$\Gamma$-centered electron pocket vanishes. This is an interesting result in
that 1) compared to the $P4/mmm$ structure, the Fermi surface of SmNiO2 in the
$I4/mcm$ structure more closely resembles that of CaCuO2; and 2) the “NiO4
square” rotation in the $I4/mcm$ structure effectively acts as hole doping in
SmNiO2. To demonstrate the second point more clearly, we calculate the band
structure and Fermi surface of Sr0.2Sm0.8NiO2 in the $P4/mmm$ structure (see
the Supplementary Materials SI  Sec. VII) and we find that they are similar to
pristine SmNiO2 in the $I4/mcm$ structure.

###  III.6 P4/mmm versus I4/mcm structures: magnetic properties

Next we study the magnetic properties of $R$NiO2 and compare the $I4/mcm$
structure and the $P4/mmm$ structure. We use the charge-only DFT+$U$+$J$
method Park et al. (2015); Chen et al. (2015); Chen and Millis (2016) in which
the spin polarization is broken by the $U/J$ extension rather than the spin-
dependent exchange-correlation functional. The advantage of using this method
is that when $U/J$ parameters approach zero, we will recover our non-spin-
polarized DFT results. We first fix the optimized crystal structure that is
obtained from the non-spin-polarized (nsp) DFT calculations, upon which the
electronic structure calculations are performed. This is the convention of
some previous DFT+$U$ and DFT+dynamical mean field theory (DFT+DMFT) studies
Been et al. (2021); Ryee et al. (2020); Choi et al. (2020a); Liu et al.
(2020); Choi et al. (2020b); Lechermann (2020a, b); Leonov et al. (2020);
Petocchi et al. (2020); Lechermann (2021). Then we relax the crystal structure
within the charge-only DF+$U$+$J$ method and discuss the relaxation effects on
magnetic properties.

Figure 6:  Magnetic properties of infinite-layer nickelates $R$NiO2 in the
$I4/mcm$ structure (red symbols) and in the $P4/mmm$ structure (blue symbols).
a: The magnetic moment on Ni atom of SmNiO2 in rocksalt antiferromagnetic
ordering (AFM) as a function of $U_{\textrm{Ni}}$. b: The critical
$U_{\mathrm{Ni}}$ for rocksalt antiferromagnetic ordering across the
lanthanide series of infinite-layer nickelates $R$NiO2. c: The energy
difference between rocksalt antiferromagnetic ordering and ferromagnetic
ordering of SmNiO2 as a function of $U_{\mathrm{Ni}}$. The dashed lines
highlight the critical $U_{\mathrm{Ni}}$ for rocksalt antiferromagnetic
ordering. d: The energy difference between rocksalt antiferromagnetic ordering
and ferromagnetic ordering at $U_{\mathrm{Ni}}$ = 3 eV across the lanthanide
series of infinite-layer nickelates $R$NiO2.

We first study rocksalt antiferromagnetic ordering (ordering wave vector
$\textbf{q}=(\pi,\pi,\pi)$) in SmNiO2 Liu et al. (2020). Panel a of Fig. 6
shows the magnetic moment on Ni atom as a function of $U_{\textrm{Ni}}$
(throughout the calculations, we set $J_{\textrm{Ni}}=0.15U_{\textrm{Ni}}$
Giovannetti and Capone (2014)). We compare the Ni magnetic moment between the
$I4/mcm$ structure and the $P4/mmm$ structure. We find that within the charge-
only DFT+$U$+$J$ method, the critical effective $U_{\textrm{Ni}}$ for rocksalt
antiferromagnetic ordering is reduced from 1.8 eV in the $P4/mmm$ structure to
1.5 eV in the $I4/mcm$ structure. This is consistent with the bandwidth
reduction effect that correlation strength is increased on Ni
$d_{x^{2}-y^{2}}$ orbital in the $I4/mcm$ structure, which favors the
formation of long-range magnetic ordering. Next in panel b, we study the
entire lanthanide series of infinite-layer nickelates $R$NiO2 and compare the
critical $U_{\textrm{Ni}}$ for the $I4/mcm$ structure and for the $P4/mmm$
structure. We find that the critical $U_{\textrm{Ni}}$ for the $I4/mcm$
structure is always smaller than that for the $P4/mmm$ structure, and from
PmNiO2 to LuNiO2, the reduction in the critical $U_{\textrm{Ni}}$ becomes more
substantial. This feature is consistent with the trend of Ni $d_{x^{2}-y^{2}}$
bandwidth reduction (see Fig. 5a). In addition, we study the energy difference
between rocksalt antiferromagnetic ordering and ferromagnetic ordering $\Delta
E=E_{\textrm{AFM}}-E_{\textrm{FM}}$ as a function of $U_{\textrm{Ni}}$. Panel
c shows that for SmNiO2, both in the $I4/mcm$ structure and in the $P4/mmm$
structure, when $U_{\textrm{Ni}}$ exceeds the critical value (highlighted by
the two dashed lines), rocksalt antiferromagnetic ordering has lower energy
than ferromagnetic ordering ($\Delta E$ is negative). However, the magnitude
of $|\Delta E|$ is larger in the $I4/mcm$ structure than in the $P4/mmm$
structure when the long-range magnetic order is stabilized in SmNiO2. The
results in panel c indicate that given the same value of $U_{\textrm{Ni}}$,
the “NiO4 square” rotation in the $I4/mcm$ structure further stabilizes the
rocksalt antiferromagnetic ordering over the ferromagnetic ordering. We repeat
the same calculations for the entire lanthanide series of infinite-layer
nickelates $R$NiO2 and show in panel d the energy difference $\Delta E$ at
$U_{\textrm{Ni}}=3$ eV. We find that $\Delta E$ is negative and its magnitude
is larger in the $I4/mcm$ structure than in the $P4/mmm$ structure for the
entire series of $R$NiO2 ($R$ = Pm-Lu).

Next within the charge-only DFT+$U$+$J$ method, we relax the crystal structure
of SmNiO2 for each given $U$ and $J$ (see Supplementary Materials SI  Sec.
VIII). We find that adding $U$ and $J$ terms does not considerably change the
optimized lattice constants and the “NiO4 square” rotation angle. However, it
is noted that a weak “cusp” feature emerges at the critical $U_{\textrm{Ni}}$
when the long-range magnetic ordering is stabilized. Using the optimized
crystal structure from the charge-only DFT+$U$+$J$ method, we still find that
1) the critical $U_{\textrm{Ni}}$ for the $I4/mcm$ structure is smaller than
that for the $P4/mmm$ structure, 2) rocksalt-antiferromagnetic ordering is
more stable than ferromagnetic ordering in both crystal structures, and 3) the
magnitude of the energy difference between rocksalt-antiferromagnetic ordering
and ferromagnetic ordering $|\Delta E|$ is larger in the $I4/mcm$ structure
than in the $P4/mmm$ structure. All these results are qualitatively consistent
with the previous ones that are obtained by using the nsp-DFT optimized
crystal structure.

##  IV Conclusion

In conclusion, we perform first-principles calculations to study structural,
electronic and magnetic properties of the entire lanthanide series of
infinite-layer nickelates $R$NiO2. We find that the widely-studied $P4/mmm$
structure is only dynamically stable when $R$ is an early lanthanide element
(La-Sm). For late lanthanide elements (Eu-Lu), an unstable phonon mode appears
at $A$ point in the $P4/mmm$ structure, which corresponds to an out-of-phase
“NiO4 square” rotation about the $z$ axis. For infinite-layer nickelates with
late lanthanide elements, condensation of this phonon mode in the $P4/mmm$
structure lowers the total energy and leads to a new $I4/mcm$ crystal
structure. Special attention is paid to two borderline cases PmNiO2 and
SmNiO2, in which both the $P4/mmm$ structure and the $I4/mcm$ structure are
local minimums. When epitaxial strain is imposed on infinite-layer nickelates,
tensile strain further increases the dynamical instability at $A$ point in the
$P4/mmm$ structure, while compressive strain “hardens” the phonon at $A$ point
but “softens” the phonons at $Z$ and $R$ points in the $P4/mmm$ structure.
Furthermore, epitaxial strain can fine-tune the energy difference between the
$I4/mcm$ structure and the $P4/mmm$ structure when they are both dynamically
stable in $R$NiO2 (such as SmNiO2).

We use the new $I4/mcm$ crystal structure to study the trends of electronic
and magnetic properties of $R$NiO2. We find that compared to the $P4/mmm$
structure, the Ni $d_{x^{2}-y^{2}}$ bandwidth of $R$NiO2 is substantially
reduced in the $I4/mcm$ structure, which implies an increased correlation
strength in the new $I4/mcm$ structure. In addition, the Ni $d$ occupancy of
$R$NiO2 gets smaller in the $I4/mcm$ structure than that in the $P4/mmm$
structure, which means a small critical $U_{\textrm{Ni}}$ for the metal-
insulator transition Wang et al. (2012). Furthermore, the electronic structure
and Fermi surface of $R$NiO2 become “cleaner” in the $I4/mcm$ structure than
in the $P4/mmm$ structure, because one lanthanide-$d$-derived band is removed
from the Fermi level and thus a lanthanide-$d$-derived electron pocket
disappears at $\Gamma$ point. Finally, the critical $U_{\textrm{Ni}}$ to
stabilize the rocksalt antiferromagnetic ordering in $R$NiO2 is reduced from
the $P4/mmm$ structure to the $I4/mcm$ structure. All these results imply that
correlation effects are enhanced and Mott physics plays a more important role
in the new $I4/mcm$ crystal structure of infinite-layer $R$NiO2. Hence, if
$R$NiO2 in the $I4/mcm$ crystal structure can be synthesized in experiment, it
will provide a closer analogy to infinite-layer cuprate CaCuO2. Our work
suggests that among the lanthanide series of infinite-layer nickelates, the
most promising candidate to crystallize in the $I4/mcm$ structure is SmNiO2.

We finally note that structure-property relations have been widely studied in
complex oxides, such as perovskite nickelates and manganites Medarde (1997);
Salamon and Jaime (2001); Catalan (2008); Middey et al. (2016); Catalano et
al. (2018). Spontaneous structural distortions, such as Jahn-Teller,
breathing, rotations and tilts of oxygen octahedra, turn out to have
substantial impacts on the physical properties of complex oxides Rondinelli
and Fennie (2012); Benedek and Fennie (2013); Zhai et al. (2014); Liao et al.
(2016); Chen et al. (2018). Our work reveals a similar coupling between
crystal structure and electronic structure in infinite-layer nickelates
$R$NiO2: by substituting the lanthanide element $R$, we can control the
rotation of “NiO4 square”, which tunes the underlying electronic structure and
may potentially favor superconductivity.

Note added: after the completion of our work, we became aware of Refs.
Bernardini et al. (2021); Álvarez et al. (2021); Zhang et al. (2022), which
also study structural distortions in infinite-layer nickelates $R$NiO2 and
YNiO2. The authors of Ref. Bernardini et al. (2021) find that YNiO2 is also
prone to the “NiO4 square” rotation. The authors of Ref. Álvarez et al. (2021)
show that due to the $R$-to-Ni cation mismatch, the “NiO4 square” is prone to
various types of rotations. For $R$ of a small ionic radius such as Y, the
ground state structure has the orthorhombic $Pbnm$ symmetry with the
$a^{-}a^{-}c^{+}$ rotation pattern. For $R$ of a moderate ionic radius such as
Gd, the ground state structure has the tetragonal $I4/mcm$ symmetry with the
$a^{0}a^{0}c^{-}$ rotation, which is consistent with our results. The authors
of Ref. Zhang et al. (2022) also find the dynamical instability of “NiO4
square” rotation in infinite-layer nickelates $R$NiO2. In addition, they study
the electronic and magnetic properties of $R$NiO2 at finite temperatures.

## Acknowledgement

We are grateful to Andrew Millis for useful discussions.

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