# Charge-stripe fluctuations in Nd4Ni3O8 as evidenced by optical spectroscopy

Jiahao Hao  Xinwei Fan  Qing Li  Xiaoxiang Zhou  Chengping He  Yaomin Dai
<EMAIL_ADDRESS>National Laboratory of Solid State Microstructures and
Department of Physics, Collaborative Innovation Center of Advanced
Microstructures, Nanjing University, Nanjing 210093, China  Bing Xu  Beijing
National Laboratory for Condensed Matter Physics, Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China  Xiyu Zhu  Hai-Hu Wen
National Laboratory of Solid State Microstructures and Department of Physics,
Collaborative Innovation Center of Advanced Microstructures, Nanjing
University, Nanjing 210093, China

(August 27, 2024)

###### Abstract

We present an investigation into the optical properties of Nd4Ni3O8 at
different temperatures from 300 down to 5 K over a broad frequency range. The
optical conductivity at 5 K is decomposed into IR-active phonons, a far-
infrared band $\alpha$, a mid-infrared band $\beta$, and a high-energy
absorption edge. By comparing the measured optical conductivity to first-
principles calculations and the optical response of other nickelates, we find
that Nd4Ni3O8 features evident charge-stripe fluctuations. The $\beta$ band is
attributed to electronic transitions between the gapped Ni-$d_{x^{2}-y^{2}}$
bands due to fluctuating charge stripes, while the high-frequency absorption
edge corresponds to the onset of transitions involving other high-energy
bands. Furthermore, an analysis of the temperature-dependent optical spectral
weight reveals a $T^{2}$ law, which is likely to originate from strong
correlation effects.

Since the discovery of unconventional superconductivity in copper oxide
compounds Bednorz and Müller (1986), tremendous efforts have been devoted to
the exploration of cuprate-like materials based on different transition-metal
ions in the hope of finding new superconducting systems and gaining insights
into the pairing mechanism in cuprates. Due to the proximity of Ni to Cu in
the periodic table, nickelates have attracted great attention Anisimov et al.
(1999); Lee and Pickett (2004); Poltavets et al. (2010); Zhang et al. (2016,
2017); Cheng et al. (2012); Pardo and Pickett (2010, 2012); Liu et al. (2012);
ApRoberts-Warren et al. (2011); Botana et al. (2016, 2017). Anisimov et al.
have pointed out that nickelate analogs to the superconducting cuprates may be
realized only if the Ni+ ($3d^{9}$) ions are forced into a planar oxygen
coordination, which mimics the parent compounds of the cuprates, and then
doped with low-spin Ni2+ ($3d^{8}$) holes Anisimov et al. (1999). On the other
hand, Lee and Pickett Lee and Pickett (2004) have compared LaNiO2 (Ni+,
3$d^{9}$) to CaCuO2 (Cu2+, 3$d^{9}$), and found very different behavior, such
as significantly reduced 2$p$-3$d$ hybridization and the existence of La 5$d$
at the Fermi level in LaNiO2, arguing against the analogy between nickelates
and cuprates. Recently, superconductivity with a critical temperature
($T_{c}$) up to 15 K has been reported in hole-doped infinite-layer nickelates
Nd1-xSrxNiO2 Li et al. (2019, 2020a); Zeng et al. (2020), reigniting the
debate on whether nickelates represent analogs of cuprates Botana and Norman
(2020); Karp et al. (2020a); Zhang et al. (2020); Jiang et al. (2020a).

Hole-doped $3d^{9}$ Ni ions in a planar oxygen coordination also exist in the
trilayer $R_{4}$Ni3O8 ($R=$ La, Pr or Nd). These compounds are 1/3 self hole
doped with a nominal 3$d^{8.67}$ filling, which should fall into the overdoped
Fermi-liquid regime in the phase diagram of hole-doped cuprates Botana et al.
(2017). From an experimental perspective, while Pr4Ni3O8 seems to be a metal
without any phase transition in the temperature range 2–300 K Zhang et al.
(2017), La4Ni3O8 exhibits a semiconductor-insulator transition at about 105 K,
resulting in a highly insulating ground state Poltavets et al. (2010); Cheng
et al. (2012); Zhang et al. (2016, 2017). The formation of long-range
antiferromagnetic order has been detected below the transition at 105 K by
La139 nuclear magnetic resonance (NMR) ApRoberts-Warren et al. (2011);
Poltavets et al. (2010) and neutron diffraction Zhang et al. (2019); a recent
synchrotron X-ray diffraction study has revealed that the ground state of
La4Ni3O8 is a quasi-two-dimensional charge-stripe-ordered insulator Zhang et
al. (2016). Nd4Ni3O8 also shows insulating behavior with no signature of a
phase transition from room temperature down to 2 K Li et al. (2021).
Theoretically, the ground state of $R_{4}$Ni3O8 can be metallic with a low-
spin (LS) configuration or insulating with a high-spin (HS) configuration
Pardo and Pickett (2010, 2012); Liu et al. (2012), depending on the relative
magnitude of Hund’s rule coupling ($J_{H}$) and the crystal field splitting
($\Delta_{\text{cf}}$) between the $d_{x^{2}-y^{2}}$ and $d_{3z^{2}-r^{2}}$
bands induced by the absence of apical oxygen ions. Moreover, Botana et al.
have demonstrated that a low-spin charge stripe (LS-CS) insulating ground
state can also be obtained from a combination of charge-order-related
structural distortions and magnetic order Botana et al. (2016).

To date, questions regarding the ground state of the trilayer $R_{4}$Ni3O8
compounds, and to what extent they are similar to the cuprates are still open
to debate Poltavets et al. (2010); ApRoberts-Warren et al. (2011); Cheng et
al. (2012); Zhang et al. (2016, 2017, 2019); Lin et al. (2020). We notice that
the theoretically proposed LS metallic, HS insulating, and LS-CS insulating
states for $R_{4}$Ni3O8 are characterized by entirely distinct band
structures, which consequently give rise to completely different optical
responses. Therefore, investigating the optical properties of $R_{4}$Ni3O8 and
comparing them with first-principles calculations and the optical response of
other nickelates may provide pivotal information on the nature of its ground
state.

In this paper, we present an optical study of Nd4Ni3O8 at 15 different
temperatures between 5 and 300 K in the frequency range 30–50 000
$\textrm{cm}^{-1}$ (3.75 meV–6.25 eV). The optical conductivity of Nd4Ni3O8 at
5 K consists of several different components: IR-active phonons, a far-
infrared band $\alpha$, a mid-infrared band $\beta$, and a high-energy
absorption edge. A comparison of our experimental results to theoretical
calculations and the optical properties of other nickelates manifests that
charge-stripe fluctuations exist in Nd4Ni3O8. The $\beta$ band and the high-
energy absorption edge can be associated with electronic transitions between
gapped Ni-$d_{x^{2}-y^{2}}$ states due to fluctuating charge stripes and the
onset of electronic transitions between bands lying further away from $E_{F}$,
respectively. In addition, the temperature-dependent optical spectral weight
exhibits a $T^{2}$ law even for a cutoff frequency as high as 12 000
$\textrm{cm}^{-1}$ ($\sim$1.5 eV), which is likely to arise from strong
correlation effects.

Figure 1: Reflectivity of Nd4Ni3O8 up to 50 000 $\textrm{cm}^{-1}$ at several
selected temperatures from 300 down to 5 K. Inset: Resistivity of Nd4Ni3O8 as
a function of temperature.

High-purity polycrystalline samples of Nd4Ni3O8 were obtained by reacting
Nd4Ni3O10 with CaH2 Li et al. (2021, 2020b). The resistivity $\rho$ of our
sample, as shown in the inset of Fig. 1, rises with decreasing temperature,
which is a clear signature of insulating behavior Li et al. (2021). The near-
normal incidence reflectivity $R(\omega)$ of Nd4Ni3O8 was measured at 15
different temperatures between 5 and 300 K in the frequency range from 30 to
12 000 $\textrm{cm}^{-1}$ using a Bruker Vertex 80v Fourier transform
spectrometer. An _in situ_ gold overfilling technique Homes et al. (1993a) was
utilized to obtain the absolute reflectivity. We then used an
AvaSpec-2048$\times$14 fiber optical spectrometer to extend $R(\omega)$ to 50
000 $\textrm{cm}^{-1}$ at room temperature.

First of all, we would like to point out that although single crystalline
samples of La4Ni3O8 and Pr4Ni3O8 have been synthesized using the high-pressure
floating zone technique Zhang et al. (2016, 2017); Zhang and Tao (2021), the
growth of single crystalline Nd4Ni3O8 requires much higher pressure that can
not be reached thus far. As a consequence, experimental studies of Nd4Ni3O8
are limited to polycrystalline samples Li et al. (2021). Optical data
collected on a polycrystal reflects the mixture of the optical responses from
all three crystal axes, which may not be reliable for a quantitative analysis.
Nevertheless, previous optical studies on polycrystals have shown that a
qualitative analysis of optical data from polycrystalline samples, in
combination with transport studies, can also provide useful information about
the charge dynamics of the materials Bonn et al. (1987a, b); Chen et al.
(2008); Dubroka et al. (2008); Inaba et al. (1995); Ueda et al. (2012, 2016);
Fujioka et al. (2021).

Figure 1 shows the measured $R(\omega)$ of Nd4Ni3O8 up to 50 000
$\textrm{cm}^{-1}$ at several representative temperatures from 300 down to 5
K. The far-infrared $R(\omega)$ below about 600 $\textrm{cm}^{-1}$ is
dominated by IR-active phonons (sharp features). Below the phonon bands,
$R(\omega)$ approaches a constant that is much lower than unity, which is the
prototypical optical response of an insulator Jiang et al. (2020b); Reijnders
et al. (2014); Lobo et al. (2007). This is consistent with the transport
measurement, as shown in the inset of Fig. 1, which also reveals insulating
behavior in the temperature range of 2–300 K Li et al. (2021). Above the IR-
active phonon bands, $R(\omega)$ exhibits a broad hump-like feature in the
frequency range 600–5000 $\textrm{cm}^{-1}$; a similar feature has been
reported in the charge-stripe-ordered La2-xSrxNiO4 Ido et al. (1991); Bi et
al. (1993) and La2NiO4+δ Homes et al. (2003).

Figure 2: The real part of the optical conductivity $\sigma_{1}(\omega)$ of
Nd4Ni3O8 at different temperatures from 300 down to 5 K. The inset shows
$\sigma_{1}(\omega)$ in the far-infrared range at $T=5$ K. The blue solid
circle at zero frequency denotes the dc conductivity from transport
measurements.

In order to obtain further information about the charge dynamics of Nd4Ni3O8,
the real part of the optical conductivity $\sigma_{1}(\omega)$ is determined
through a Kramers-Kronig analysis of $R(\omega)$. Since $R(\omega)$ exhibits a
tendency towards saturation in the far-infrared range, a constant was used for
the low-frequency extrapolation. On the high-frequency side, we adopted a
constant $R(\omega)$ up to 12.5 eV followed by a free-electron ($\omega^{-4}$)
response. Figure 2 displays $\sigma_{1}(\omega)$ of Nd4Ni3O8 at different
temperatures from 300 down to 5 K. The inset shows $\sigma_{1}(\omega)$ at 5 K
in the far-infrared range, and the blue solid circle at zero frequency denotes
the dc conductivity $\sigma_{\text{dc}}$ at 5 K from transport measurements.
The zero-frequency extrapolation of the optical conductivity
$\sigma_{1}(\omega\rightarrow 0)$ agrees quite well with $\sigma_{\text{dc}}$,
testifying to the self-consistency of our experimental results. The far-
infrared $\sigma_{1}(\omega)$ is dominated by IR-active phonon modes which are
characterized as sharp peaks below 600 $\textrm{cm}^{-1}$. The mid-infrared
$\sigma_{1}(\omega)$ features a broad absorption band lying in the frequency
range of 600–5000 $\textrm{cm}^{-1}$, which grows in amplitude as the
temperature is lowered. Above 10 000 $\textrm{cm}^{-1}$, $\sigma_{1}(\omega)$
increases sharply, leading to an absorption edge at about 20 000
$\textrm{cm}^{-1}$.

The measured $\sigma_{1}(\omega)$ of Nd4Ni3O8 can be fit to a series of
Lorentzian oscillators,

$\sigma_{1}(\omega)=\frac{2\pi}{Z_{0}}\sum_{j}\frac{\gamma_{j}\omega^{2}\Omega_{j}^{2}}{(\omega_{j}^{2}-\omega^{2})^{2}+\gamma_{j}^{2}\omega^{2}},$
(1)

where $Z_{0}\simeq 377\leavevmode\nobreak\ \Omega$ is the vacuum impedance;
$\omega_{j}$, $\gamma_{j}$ and $\Omega_{j}$ correspond to the resonance
frequency, linewidth and strength of the $j$th oscillator, respectively. The
red dashed line in Fig. 3 represents the fitting result at 5 K, which
reproduces the measured $\sigma_{1}(\omega)$ (blue solid line) quite well. As
illustrated in Fig. 3, the fit allows us to decompose $\sigma_{1}(\omega)$ of
Nd4Ni3O8 at 5 K into several different components: (i) IR-active phonon modes
which are not shown in the figure; (ii) a far-infrared (FIR) absorption band
$\alpha$ under the phonon modes (orange dashed line); (iii) a mid-infrared
(MIR) absorption band $\beta$ centered at about 2500 $\textrm{cm}^{-1}$ (green
dashed line); (iv) an absorption edge at about 20 000 $\textrm{cm}^{-1}$ (cyan
dashed line). In the following, we compare our experimental results to first-
principles calculations and the optical response of other nickelates, aiming
at understanding the origins of these components (except for the phonon modes)
in $\sigma_{1}(\omega)$ and the ground state of Nd4Ni3O8.

Figure 3: The blue solid curve denotes $\sigma_{1}(\omega)$ of Nd4Ni3O8
measured at 5 K. The red dashed line through the data is the fitting result,
which is decomposed into an FIR band $\alpha$ (orange dashed line), an MIR
component $\beta$ (green dashed line), and a high-energy absorption edge (cyan
dashed line).

First of all, it should be noted that a Drude component is absent in
$\sigma_{1}(\omega)$ of Nd4Ni3O8, agreeing well with the insulating behavior
revealed by transport measurements Li et al. (2021). Nevertheless, we would
like to mention that in systems with strong electronic anisotropy, such as
cuprates and iron pnictides, the optical response of a polycrystalline sample
is dominated by the weakly conducting $c$ axis Bonn et al. (1987b, a); Homes
et al. (1993b); Chen et al. (2008); Dubroka et al. (2008); Xu et al. (2020),
while the Drude response associated with the metallic $ab$ plane is
significantly suppressed. Hence, the absence of a Drude component in
$\sigma_{1}(\omega)$ of the polycrystalline Nd4Ni3O8 points towards two
possibilities: either this material is insulating in both the $ab$ plane and
the $c$ direction, or it is highly anisotropic with a metallic $ab$ plane and
an insulating $c$ direction. The latter possibility may be ruled out by the
insulating transport properties of the polycrystalline Nd4Ni3O8 Li et al.
(2021), because for compounds that are metallic in the $ab$ plane but
insulating along the $c$ axis, for example cuprates and iron pnictides,
$\rho(T)$ of a polycrystalline sample has been found to exhibit prominent
metallic behavior Dubroka et al. (2008); Takagi et al. (1992); Wu et al.
(2017).

Figure 4: (a), (c) and (e) display the calculated band structures of Nd4Ni3O8
for the LS metallic, HS insulating and LS-CS insulating states, respectively.
(b), (d) and (f) depict the calculated optical conductivity spectra based on
the band structures in (a), (c) and (e), respectively.

To gain more insight into the origins of the different components in
$\sigma_{1}(\omega)$, we calculated the band structure of Nd4Ni3O8 and
corresponding $\sigma_{1}(\omega)$ using density functional theory (DFT)
Schwarz et al. (2002); Schwarz and Blaha (2003); Abt et al. (1994); Ambrosch-
Draxl and Sofo (2006). The details about the calculations are given in the
appendix. Since $\sigma_{1}(\omega)$ of a polycrystal consists of
contributions from both the $ab$ plane and the $c$ direction, we calculated
both the in-plane $\sigma_{xx}(\omega)=\sigma_{yy}(\omega)$ and out-of-plane
$\sigma_{zz}(\omega)$ optical conductivities, and then compare the average
$\sigma_{\text{ave}}=(\sigma_{xx}+\sigma_{yy}+\sigma_{zz})/3=(2\sigma_{xx}+\sigma_{zz})/3$
to the measured $\sigma_{1}(\omega)$ of Nd4Ni3O8.

Figure 4(a) displays the calculated band structure of the LS state of Nd4Ni3O8
which reveals a metallic phase with the partially filled Ni-$d_{x^{2}-y^{2}}$
bands crossing the Fermi level, in good agreement with previous calculations
for La4Ni3O8 Poltavets et al. (2010); Sarkar et al. (2011); Pardo and Pickett
(2012) and Pr4Ni3O8 Botana et al. (2017); Karp et al. (2020b). The calculated
$\sigma_{xx}$ and $\sigma_{zz}$ for the LS state are shown as red and blue
dashed lines respectively in Fig. 4(b). A pronounced Drude component, the
optical signature of metallic behavior, can be seen in $\sigma_{xx}$. Although
the absence of Drude response in $\sigma_{zz}$ implies insulating behavior
along the $c$ direction, the metallic $\sigma_{xx}$ would inevitably produce
coherent electronic transport in the temperature-dependent $\rho(T)$ of the
polycrystalline Nd4Ni3O8. The robust insulating behavior in $\rho(T)$, as
shown in the inset of Fig. 1 and Ref. Li et al. (2021), suggests that the LS
metallic state fails in describing the experimental results of Nd4Ni3O8. Thus,
models yielding an insulating ground state should be invoked.

We recall that La4Ni3O8 is also highly insulating at low temperatures. In
order to account for the insulating behavior in La4Ni3O8, Pardo and Pickett
consider the Ni trilayer as a molecular trimer with in-plane AFM order, from
which on-site repulsion $U$ gives rise to a Mott insulating state, i.e. the HS
insulating state Pardo and Pickett (2010). Alternatively, Botana et al. have
pointed out that an insulating ground state, the LS-CS insulating state, can
also be obtained from a combination of charge-order-related structural
distortions and magnetic order Botana et al. (2016). We apply these two
approaches to Nd4Ni3O8 and derive $\sigma_{xx}$, $\sigma_{zz}$, as well as
$\sigma_{\text{ave}}$ from the corresponding electronic structure in each
case. The details about the theoretical calculations can be found in the
appendix.

The calculated band structure of Nd4Ni3O8 for the HS insulating state, which
is similar to the previous calculation for La4Ni3O8 Pardo and Pickett (2010),
is traced out in Fig. 4(c). A large (direct) gap of $\sim$1–1.5 eV opens
between the $d_{3z^{2}-r^{2}}$ states that are depicted as blue solid circles.
Figure 4(d) shows the optical conductivity spectra calculated from the band
structure in Fig. 4(c). A sharp peak arising from direct interband transitions
between the $d_{3z^{2}-r^{2}}$ bands can be identified at about 13 000
$\textrm{cm}^{-1}$ in $\sigma_{xx}$. On the low-frequency side, $\sigma_{xx}$
and $\sigma_{zz}$ vanish below 10 000 and 15 000 $\textrm{cm}^{-1}$,
respectively. It is immediately obvious that the measured $\sigma_{1}(\omega)$
of Nd4Ni3O8 contrasts with the calculated $\sigma_{\text{ave}}$ [purple solid
line in Fig. 4(d)]: (i) absorption bands, i.e. $\alpha$ and $\beta$, exist in
the low-frequency (below $\sim$10 000 $\textrm{cm}^{-1}$) region of the
measured $\sigma_{1}(\omega)$, while $\sigma_{\text{ave}}$ is vanishingly
small in the same spectral range; (ii) the sharp peak at 13 000
$\textrm{cm}^{-1}$ in the calculated $\sigma_{\text{ave}}$ is not observed in
the measured $\sigma_{1}(\omega)$. Such considerable discrepancies between the
calculated $\sigma_{\text{ave}}$ and the measured $\sigma_{1}(\omega)$ do not
favor the HS insulating state in Nd4Ni3O8.

Next, we consider the LS-CS insulating state. Figure 4(e) shows the calculated
band structure of Nd4Ni3O8 for the LS-CS insulating state, which also
resembles the previous result in La4Ni3O8 Botana et al. (2016). A much smaller
gap is formed between the Ni $d_{x^{2}-y^{2}}$ bands (red solid circles),
which is completely different from the HS insulating state. The calculated
$\sigma_{xx}$ and $\sigma_{zz}$ are shown in Fig. 4(f) as red and blue dashed
lines, respectively. While $\sigma_{xx}$ diminishes with decreasing frequency
and disappears at $\sim$3000 $\textrm{cm}^{-1}$, $\sigma_{zz}$ exhibits a
broad absorption peak at $\sim$6000 $\textrm{cm}^{-1}$ with a tail extending
down to $\sim$1000 $\textrm{cm}^{-1}$. The peak and the low-energy tail in
$\sigma_{zz}$ are predominantly contributed by interband electronic
transitions across the gap between the Ni $d_{x^{2}-y^{2}}$ bands.
Intriguingly, the calculated $\sigma_{\text{ave}}$ qualitatively reproduces
the measured $\sigma_{1}(\omega)$ in the MIR and high-frequency range, hinting
that the ground state of Nd4Ni3O8 might be (or is in close proximity to) an
LS-CS insulator as previously proposed for La4Ni3O8 Botana et al. (2016).
Within this framework, the $\beta$ band may be ascribed to electronic
transitions between the gapped $d_{x^{2}-y^{2}}$ bands, and the high-frequency
absorption edge is associated with the onset of electronic transitions
involving bands lying further away from $E_{F}$.

It is noteworthy that the optical response of Nd4Ni3O8 bears a remarkable
resemblance to that of charge-stripe-ordered La2-xSrxNiO4 Ido et al. (1991);
Bi et al. (1993); Katsufuji et al. (1996); Calvani et al. (1996) and La2NiO4+δ
Homes et al. (2003). An MIR peak akin to the $\beta$ band in Nd4Ni3O8 has also
been observed in $\sigma_{1}(\omega)$ of these materials Homes et al. (2003);
Ido et al. (1991); Bi et al. (1993); Katsufuji et al. (1996); Calvani et al.
(1996). While early studies have assigned the MIR peak to the formation of
small polarons Bi et al. (1993); Katsufuji et al. (1996); Calvani et al.
(1996); Jung et al. (2001), Homes et al. Homes et al. (2003) have taken into
account the fact that the holes doped into the NiO2 planes form a stripe order
which has been established by neutron and X-ray scattering studies Tranquada
et al. (1994); Yoshizawa et al. (2000), and consequently ascribed the MIR peak
to transitions from filled valence states to empty mid-gap states associated
with the charge stripes. Furthermore, as the MIR peak persists to temperatures
far above the charge-stripe-ordering transition temperature $T_{\text{CO}}$
Katsufuji et al. (1996); Homes et al. (2003), recent observations of short-
range-ordered or dynamic charge stripes on the same temperature scale have
naturally related it to the presence of charge-stripe fluctuations Abeykoon et
al. (2013); Coslovich et al. (2013); Anissimova et al. (2014). Given the
striking resemblance between the optical response of Nd4Ni3O8 and that of
La2NiO4+δ and La2-xSrxNiO4, it is plausible to associate the $\beta$ band in
Nd4Ni3O8 with the emergence of charge stripes or charge-stripe fluctuations.
Although the formation of charge stripes below the transition at 105 K in
La4Ni3O8 has been confirmed by the synchrotron X-ray diffraction Zhang et al.
(2016), no phase transition has been detected in Nd4Ni3O8 from room
temperature down to 2 K Li et al. (2021). In this case, the $\beta$ band in
Nd4Ni3O8 is most likely to signify charge-stripe fluctuations, i.e. short-
range-ordered or dynamic charge stripes.

The presence of charge-stripe fluctuations or even a static charge-stripe
order in Nd4Ni3O8 is not surprising, because it is structurally and
electronically identical to La4Ni3O8 which has been found to order in charge
stripes below the transition at 105 K Zhang et al. (2016). Although the
relatively smaller atomic radius of Nd may suppress the static long-range
order in Nd4Ni3O8, charge-stripe correlations should not be significantly
affected by such a small difference. In addition, similar to the case of
La4Ni3O8 Botana et al. (2016), the calculated LS-CS state has a lower free
energy than the HS molecular insulating state, representing a favorable ground
state for Nd4Ni3O8.

We further notice that neither the HS nor the LS-CS insulating state can
account for the $\alpha$ band in the measured $\sigma_{1}(\omega)$ of
Nd4Ni3O8. A similar FIR absorption band has been observed in doped
semiconductors Si:P and assigned to optical transitions between impurity
states and the conduction band Gaymann et al. (1995); Some bismuth-based
topological insulators, e.g. Bi2Te2Se and Bi2-xCaxSe3, also exhibit such an
impurity-related absorption band centered at about 200 $\textrm{cm}^{-1}$
Akrap et al. (2012); Di Pietro et al. (2012). By analogy with Si:P and
bismuth-based topological insulators, the $\alpha$ band in Nd4Ni3O8 may be
attributed to transitions involving in-gap impurity states. Alternatively, the
$\alpha$ band may be an artifact arising from the polycrystalline nature of
our sample. In La2NiO4+δ and La2-xSrxNiO4, the $ab$-plane $\sigma_{1}(\omega)$
exhibits a residual Drude response in the charge-stripe-fluctuation regime
Homes et al. (2003); Coslovich et al. (2013). Such a residual Drude response
may also exist in the $ab$-plane $\sigma_{1}(\omega)$ of Nd4Ni3O8 due to the
fluctuations of charge stipes. In this case, the mixture of a weak metallic
spectrum from the $ab$ plane and an insulating one from the $c$ direction may
create an artifact structure, i.e. the $\alpha$ band in $\sigma_{1}(\omega)$.
Another possible origin of the $\alpha$ band is the electronic excitations
within the stripes. If the stripe order is neither static nor of long range,
then low-energy excitations within the stripes may be present, giving rise to
the $\alpha$ band in the measured $\sigma_{1}(\omega)$ of Nd4Ni3O8.

Figure 5: Spectral weight up to different cutoff frequencies as a function of
$T^{2}$. The dashed lines are linear fits.

Finally, we examine the evolution of $\sigma_{1}(\omega)$ with temperature. It
is noticeable in Fig. 2 that $\sigma_{1}(\omega)$ in the FIR and MIR region
grows as the temperature is lowered. This is distinct from materials with
charge stripes or charge-stripe fluctuations, such as La2-xSrxNiO4 and
La2NiO4+δ, in which a suppression of the low-frequency $\sigma_{1}(\omega)$
alongside an enhancement of $\sigma_{1}(\omega)$ in the MIR peak region has
been observed Homes et al. (2003); Coslovich et al. (2013). Hence, the unusual
temperature dependence of the optical response in Nd4Ni3O8 is likely to be
dominated by a different effect.

The detailed temperature dependence of $\sigma_{1}(\omega)$ can be tracked by
inspecting the optical spectral weight $W$ defined as

$W=\int_{0}^{\omega_{c}}\sigma_{1}(\omega)d\omega,$ (2)

where $\omega_{c}$ represents a cutoff frequency. The symbols in Fig. 5 denote
$W$ as a function of $T^{2}$ for different cutoff frequencies, and the dashed
lines through the symbols are linear fits. $W$ increases upon cooling even for
$\omega_{c}$ = 12 000 $\textrm{cm}^{-1}$ ($\sim$1.5 eV), indicating a spectral
weight transfer from an energy range higher than $\sim$1.5 eV to the FIR-MIR
region. A more intriguing observation here is that $W$ varies linearly with
$T^{2}$ for all $\omega_{c}$’s we have chosen. Such behavior has been widely
observed and extensively studied in cuprates Molegraaf et al. (2002); Ortolani
et al. (2005); Toschi et al. (2005); Carbone et al. (2006a, b). In a normal
metal, e.g. gold Ortolani et al. (2005), $W$ is linear in $T^{2}$ for
$\omega_{c}<\omega_{p}$, but is temperature independent for
$\omega_{c}\geq\omega_{p}$. Here, $\omega_{p}$ is the plasma frequency
($\omega_{p}^{2}$ is proportional to the carrier density). However, in
cuprates Molegraaf et al. (2002); Ortolani et al. (2005); Toschi et al.
(2005); Carbone et al. (2006a, b), $W$ exhibits conspicuous linear dependence
on $T^{2}$ even for $\omega_{c}\geq\omega_{p}$, which is in sharp contrast to
a conventional metal. Dynamical mean-field theory calculations based on a
strongly correlated Hubbard model have demonstrated that the $T^{2}$ law of
$W$ in cuprates arises from strong correlation effects Toschi et al. (2005).
The phenomenon that $W$ varies linearly with $T^{2}$ in Nd4Ni3O8 is peculiar,
since it is an insulator without any observable free-carrier response.
Although detailed temperature dependence of the optical spectral weight in
insulating cuprates has not yet been reported, we notice that in the slightly
doped but still insulating Bi2Sr2-xLaxCuO6 (see Figs. 1(c) and 1(d) in Ref.
Lupi et al. (2009)), as the temperature is lowered, while the spectral weight
below about 80 $\textrm{cm}^{-1}$ is suppressed, a much more pronounced
increase of the spectral weight occurs between 80 and 2000 $\textrm{cm}^{-1}$,
resulting in a net spectral weight gain in the FIR-MIR range. Such an FIR-MIR
spectral weight increase at low temperatures can also be identified in the
optical conductivity of the slightly doped insulating YBa2Cu3Oy (Fig. 3 for
$y=6.28$ in Ref. Lee et al. (2005)). Considering these facts, the linear
evolution of $W$ with $T^{2}$ in Nd4Ni3O8 may be governed by the same
mechanism as that in cuprates, i.e. strong correlation effects. Therefore,
strong electronic correlation may also play an important role in Nd4Ni3O8.

To summarize, we experimentally obtained the optical conductivity of Nd4Ni3O8
at 15 different temperatures between 5 and 300 K in the frequency range 30–50
000 $\textrm{cm}^{-1}$ (3.75 meV–6.25 eV). Data fitting based on the Lorentz
model allows us to decompose the optical conductivity at 5 K into IR-active
phonons, an FIR $\alpha$ band, an MIR $\beta$ band, as well as a high-energy
absorption edge. A comparison of the measured optical conductivity to
theoretical calculations and the optical response of other nickelates suggests
that Nd4Ni3O8 features prominent charge-stripe fluctuations. The $\alpha$ band
may possibly be associated with impurities, the mixture of an in-plane
residual Drude response and our-of-plane insulating behavior, or electronic
excitations within the stripes; the $\beta$ band and the high-frequency
absorption edge can be ascribed to electronic transitions across the charge-
stripe-fluctuation-induced gap between the Ni-$d_{x^{2}-y^{2}}$ states and the
onset of transitions between other high-energy bands, respectively. The
optical spectral weight varies linearly with $T^{2}$ even for a cutoff
frequency as high as 12 000 $\textrm{cm}^{-1}$ ($\sim$1.5 eV). Such a $T^{2}$
law, which is also widely observed in cuprates, may be related to strong
correlation effects in Nd4Ni3O8.

###### Acknowledgements.

We thank Christopher C. Homes, Ricardo Lobo, Jiawei Mei, Yilin Wang, and
Shunli Yu for illuminating discussions. We gratefully acknowledge financial
support from the National Key R&D Program of China (Grant No. 2016YFA0300401),
the National Natural Science Foundation of China (Grants No. A2008/11874206,
E0209/52072170, 12061131001), the Strategic Priority Research Program of
Chinese Academy of Sciences (Grant No. XDB25000000), the Fundamental Research
Funds for the Central Universities with Grant No. 020414380095, and Jiangsu
shuangchuang program. Jiahao Hao, Xinwei Fan and Qing Li contributed equally
to this work.

## Appendix: theoretical calculations

The band structure of Nd4Ni3O8 and corresponding $\sigma_{1}(\omega)$ were
calculated using density functional theory (DFT) implemented in the full-
potential linearized augmented plane wave code WIEN2k Schwarz et al. (2002);
Schwarz and Blaha (2003); Abt et al. (1994); Ambrosch-Draxl and Sofo (2006).
The structural information of Nd4Ni3O8 was taken from Ref Li et al. (2021).
The Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA)
Perdew et al. (1996) was chosen as the exchange-correlation potential. To
avoid the ambiguity of the 4$f$ electrons of Nd around the Fermi energy, we
employed the so-called GGA$+U$ scheme with an effective Hubbard
$U_{\text{eff}}$ = 9 eV on the 4$f$ electrons of Nd to repel them away from
the Fermi level. This method has been used in previous calculations for
Nd1-xSrxNiO2 Choi et al. (2020); Ryee et al. (2020) and Nd4Ni3O8 Fan and Wen
(2020). We also applied $U_{\text{eff}}$ = 5 eV on the 3$d$ electrons of Ni,
since the correlation effect between 3$d$ electrons of transition-metal atoms
should be profound. Here, the value of $U_{\text{eff}}$ we chose for the 3$d$
electrons of Ni is close to those used in previous work on similar materials
such as La4Ni3O8 Poltavets et al. (2010); Pardo and Pickett (2010); Botana et
al. (2016), Nd4Ni3O8 Fan and Wen (2020), and Nd1-xSrxNiO2 Choi et al. (2020);
Ryee et al. (2020).

Based on the crystal structure of Nd4Ni3O8 shown in Fig. 6(a), our theoretical
calculations produce an LS metallic phase represented by the band structure in
Fig. 4(a). For this LS metallic state, no matter whether $U_{\text{eff}}$ is
applied, Nd4Ni3O8 always remains metallic. The same behavior has been found in
La4Ni3O8 Poltavets et al. (2010).

For the HS insulating state, we adopt the approach that was initially proposed
by Pardo and Pickett to account for the insulating behavior in La4Ni3O8 Pardo
and Pickett (2010). Specifically, we consider a
$\sqrt{2}a\times\sqrt{2}a\times c$ supercell with an in-plane AFM order, as
depicted in Fig. 6(b) where the orange and blue spheres denote Ni with
different spin directions. With this supercell, the calculated band structure
of Nd4Ni3O8, as shown in Fig. 4(c), exhibits a large gap between the Ni
$d_{3z^{2}-r^{2}}$ bands, closely resembling the previous calculation for
La4Ni3O8 Pardo and Pickett (2010). A larger $U_{\text{eff}}$ for the Ni 3$d$
electrons increases the gap, but makes no essential difference in the
electronic structure. However, a minimum $U_{\text{eff}}\simeq 3$ eV is
required to open a gap for the HS insulating state.

Figure 6: (a) The crystal structure of Nd4Ni3O8. The dashed frame highlights
the NiO2 trilayer. (b) The $\sqrt{2}a\times\sqrt{2}a\times c$ supercell with
an in-plane AFM order, which is used for the calculations of the HS insulating
state. (c) The charge ordering pattern we adopt for the calculations of the
LS-CS insulating state.

The LS-CS insulating state, which was obtained by Botana et al. Botana et al.
(2016) as the ground state of La4Ni3O8, requires a
$3\sqrt{2}a\times\sqrt{2}a\times c$ supercell with the charge/spin pattern
illustrated in Fig. 6(c). The green spheres correspond to non-magnetic Ni2+
ions; the orange and blue spheres represent Ni1+ ions with different spin
directions. Using such a supercell, we calculated the band structure of
Nd4Ni3O8 for the LS-CS state [Fig. 4(e)], which reveals a small gap between
the Ni $d_{x^{2}-y^{2}}$ states, similar to the previous calculation for
La4Ni3O8 Botana et al. (2016). While increasing $U_{\text{eff}}$ for the Ni
3$d$ electrons leads to an expansion of the gap, no substantial change in the
band structure is observed. In addition, a gap of 0.12 eV still remains even
without introducing a Coulomb $U$ on the Ni 3$d$ electrons for the LS-CS
state. These effects have also been noticed in previous work on La4Ni3O8
Botana et al. (2016).

Finally, we would like to remark that the LS metallic state is much more
unstable than the HS insulating state, in agreement with the calculations for
La4Ni3O8 Liu et al. (2012), while the LS-CS insulating state is about 0.4
eV/Ni more stable than the HS insulating state, which is also consistent with
previous calculations for La4Ni3O8 Botana et al. (2016).

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