# Crossover from itinerant to localized states in the thermoelectric oxide
[Ca2CoO3]0.62[CoO2]

H. Sakabayashi  R. Okazaki  Department of Physics, Faculty of Science and
Technology, Tokyo University of Science, Noda 278-8510, Japan

###### Abstract

The layered cobaltite [Ca2CoO3]0.62[CoO2], often expressed as the approximate
formula Ca3Co4O9, is a promising candidate for efficient oxide thermoelectrics
but an origin of its unusual thermoelectric transport is still in debate. Here
we investigate in-plane anisotropy of the transport properties in a broad
temperature range to examine the detailed conduction mechanism. The in-plane
anisotropy between $a$ and $b$ axes is clearly observed both in the
resistivity and the thermopower, which is qualitatively understood with a
simple band structure of the triangular lattice of Co ions derived from the
angle-resolved photoemission spectroscopy experiments. On the other hand, at
high temperatures, the anisotropy becomes smaller and the resistivity shows a
temperature-independent behavior, both of which indicate a hopping conduction
of localized carriers. Thus the present observations reveal a crossover from
low-temperature itinerant to high-temperature localized states, signifying
both characters for the enhanced thermopower.

##  I Introduction

The layered cobaltites provide fascinating platform for exploring functional
properties of oxides both in fundamental and applicational viewpoints [1].
Since the discovery of large thermopower in the metallic NaxCoO2 [2], this
class of materials has been recognized as potential oxide thermoelectrics that
possesses a high-temperature stability in air [3, 4, 5]. The crystal structure
of this family is composed of two subsystems of an insulating layer and
CdI2-type CoO2 conduction layer alternately stacked along the $c$ axis. As for
the origin of the large thermopower coexisting with the metallic conductivity,
Koshibae et al. have suggested a localized model in which a hopping conduction
of correlated $d$ electrons with spin and orbital degeneracies involves large
entropy flow [6]. This picture is supported by magnetic field dependence of
the thermopower [7] and is also discussed in semiconducting cobalt oxides [8].
On the other hand, Kuroki et al. have proposed an itinerant model based on
“pudding-mold” band structure, in which the difference in velocities of
electrons and holes is crucial [9]. Indeed, such a peculiar band shape is
observed by angle-resolved photoemission spectroscopy (ARPES) experiments [10,
11, 12, 13, 14] and a large value of thermopower is calculated accordingly
[15, 16], remaining the detailed conduction mechanism in this system
controversial.

The misfit oxide [Ca2CoO3]0.62[CoO2] [17, 18, 19] is a suitable compound to
shed light on above fundamental issue, because it straddles a border between
localized and itinerant states of Co $3d$ electrons, from which interesting
emergent phenomena appear in correlated electron systems [20]. As shown in
Figs. 1(a) and 1(b), this material has a rocksalt-type Ca2CoO3 block as an
insulating layer and its $b$-axis lattice parameter $b_{2}$ is different from
that of CoO2 layer $b_{1}$, leading to a misfit structure with incommensurate
ratio of $b_{1}/b_{2}\simeq 0.62$ while this system is often referred as the
approximate formula Ca3Co4O9. The metallic transport properties, along with a
spin-density-wave (SDW) formation at $T_{\rm SDW}\simeq 30$ K [22, 23, 24],
indicate an itinerant nature, although the magnetic structure revealed by
recent neutron experiments is quite unconventional [25]. On the other hand,
compared with that of NaxCoO2, this compound has moderately high resistivity
[26], which is close to the Ioffe-Regel limit [18]. Negative
magnetothermopower is also found in [Ca2CoO3]0.62[CoO2] [27]. In addition,
large thermopower of $Q\simeq 130$ $\mu$V/K near room temperature is well
explained in the extended Heikes formula based on the localized hopping
picture of correlated electrons [28], which prevails not only in the
conduction layer but also the rocksalt one as suggested by recent
spectroscopic studies [29]. These experimental facts imply a complicated
coexistence of itinerant and localized nature in this compound.

Figure 1:  Schematic view of the crystal structure of layered
[Ca2CoO3]0.62[CoO2] projected from (a) $a$ axis and (b) $c$ axis drawn by
VESTA [21]. While the $a$-axis lattice parameter $a$ is common, the $b$-axis
lattice parameter of the rocksalt layer $b_{2}$ is different from that of CoO2
layer $b_{1}$, resulting in a misfit structure with incommensurate ratio of
$b_{1}/b_{2}\simeq 0.62$. The Co ions in CoO2 (green) and rocksalt (blue)
layers are shown in different colors for clarity.

In this study, we argue the itinerancy and localization of conduction
electrons in [Ca2CoO3]0.62[CoO2] by means of in-plane transport anisotropy
measurements between $a$\- and $b$-axis directions. This is less investigated
so far compared to the strong anisotropy between in-plane and out-of-plane
directions [18, 30] but is essential for thorough understanding of the
underlying conduction mechanism. We find a considerable temperature dependence
of the in-plane anisotropy both in resistivity and thermopower. Below room
temperature, the in-plane anisotropy is relatively large and qualitatively
explained by the anisotropy of the electronic velocities near the Fermi energy
estimated from the results of ARPES experiments [31]. This is also consistent
with the results of recent band calculation [32], indicating the itinerant
nature in this system. Near $T_{\rm SDW}\simeq 30$ K, the in-plane anisotropy
drastically varies possibly due to a reconstruction of the Fermi surface.
Above room temperature, in contrast, we find that the in-plane anisotropy is
close to unity as temperature increases, which is captured as a localized
picture. The present results unveil the temperature-induced crossover from
itinerant to localized states in [Ca2CoO3]0.62[CoO2].

##  II Experiments

Figure 2:  (a) Photograph of a single crystal of [Ca2CoO3]0.62[CoO2] and (b)
Laue pattern. (c) Photographs of samples for the resistivity measurements. Two
samples with the rectangular shape of $\sim 1\times 0.3$ mm2 were obtained by
cutting one single crystal. (d) SEM image of [Ca2CoO3]0.62[CoO2] for the
sample thickness measurement.

Single crystals of [Ca2CoO3]0.62[CoO2] were grown by a flux method [33].
Powders of CaCO3 (99.9%) and Co3O4 (99.9%) were mixed in a stoichiometric
ratio and calcined two times in air at 1173 K for 20 h with intermediate
grindings. Then KCl (99.999%) and K2CO3 (99.999%) powders mixed with a molar
ratio of $1:4$ was added with the calcined powder as a flux. The concentration
of [Ca2CoO3]0.62[CoO2] was set to be 1.5% in molar ratio. The mixture was put
in an alumina crucible and heated up to 1123 K in air with a heating rate of
200 K/h. After keeping 1123 K for 1 h, it was slowly cooled down with a rate
of 1 K/h, and at 1023 K, the power of the furnace was switched off. As-grown
samples were rinsed in distilled water to remove the flux and then annealed in
air at 573 K for 3 h.

The typical dimension of obtained single crystals is $\simeq 4\times 4\times
0.01$ mm3 as shown in Fig. 2(a). The crystal orientation was determined by the
Laue method. Although the spot intensity is weak in such thin samples, three-
fold symmetry from CoO2 layer and four-fold symmetry from the rocksalt layer
are resolved in the Laue pattern shown in Fig. 2(b). To discuss the anisotropy
precisely, we cut one single crystal into two samples with the same
rectangular shape of $\simeq 1\times 0.3$ mm2 for the transport measurements
along the $a$ and $b$ axes as shown in Fig. 2(c). This method enables us to
compare the transport properties of the samples with the same oxygen contents.
Note that the resistivity anisotropy was also checked by utilizing the
Montgomery method near room temperature [34], although it may produce a fairly
large systematic error bar [35]. The sample thickness of $\simeq 5$ $\mu$m was
determined by the scanning electron microscopy (SEM) as shown in Fig. 2(d).
The resistivity and the thermopower were simultaneously measured by using a
conventional four-probe method and a steady-state method, respectively. The
thermoelectric voltage from the wire leads was carefully subtracted. We used a
Gifford-McMahon refrigerator below room temperature and an electrical furnace
for high-temperature measurement.

##  III Results and discussion

Figure 3:  Temperature variations of (a) resistivity $\rho_{i}$ ($i=a,b$), (b)
conductivity $\sigma_{i}=\rho_{i}^{-1}$, and (c) thermopower $Q_{i}$ measured
along the $a$\- (red) and $b$-axis (blue) directions in [Ca2CoO3]0.62[CoO2].

###  III.1 Temperature variations of resistivity and thermopower

Figures 3 summarize the temperature variations of the in-plane transport
properties in [Ca2CoO3]0.62[CoO2]. Hereafter we use an abbreviated form like
$\rho_{a}(=\rho_{aa})$ for the resistivity measured along the $a$-axis
direction. Overall behaviors are well reproduced compared with the earlier
reports in which the transport coefficients are measured with no distinction
among the in-plane directions [18]. In addition, the thermopower measured
along the $b$ axis $Q_{b}$ is larger than that along the $a$ axis $Q_{a}$,
consistent with recent theoretical calculation [32]. At low temperature below
$\sim 100$ K, the resistivity shows an insulating behavior while the
thermopower seems to be metallic, which are discussed in terms of carrier
localization [36], pseudogap opening [37], or quantum criticality [38]. Near
room temperature, the thermopower shows a temperature-independent behavior
with a relatively large value of $Q\simeq 130$ $\mu$V/K, quantitatively
explained by the extended Heikes formula of

$\displaystyle
Q=-\frac{k_{B}}{e}\ln\left(\frac{g_{3}}{g_{4}}\frac{y}{1-y}\right),$ (1)

where $k_{B}$ is the Boltzmann coefficient, $e$ the elementary charge, $g_{3}$
and $g_{4}$ the spin and orbital degeneracies of Co3+ and Co4+ ions,
respectively, and $y$ the Co4+ (hole) concentration [6]. Enhancement of the
thermopower above room temperature may be possibly attributed to a small spin-
state change near $T\simeq 380$ K [39], above which the degeneracy ratio
$g_{3}/g_{4}$ may vary with temperature. At this temperature, a small anomaly
has been observed in several quantities such as the resistivity, magnetic
susceptibility, heat capacity, and lattice constants [18, 22, 39]. On the
other hand, the magnitude of the resistive anomaly may be sample-dependent
[40] and is not resolved in the present samples. Although the present
measurements are limited below 600 K, the increase of thermopower may continue
up to 1000 K according to high-temperature transport experiments in this
compound [41].

Figure 4:  Temperature variations of the transport anisotropy in (a)
thermopower $Q_{b}/Q_{a}$ and (b) electrical conductivity
$\sigma_{b}/\sigma_{a}$ below room temperature. The SDW transition temperature
$T_{\rm SDW}\simeq 30$ K and its onset $T_{\rm SDW}^{\rm ON}\simeq 100$ K are
indicated by the dashed lines. The insets depict these anisotropies in the
whole temperature range measured in the present study.

###  III.2 Localized state at high temperatures

We first discuss high-temperature transport. The inset of Fig. 4(a) shows the
temperature dependence of the anisotropy of thermopower $Q_{b}/Q_{a}$. The
anisotropy $Q_{b}/Q_{a}$ decreases with increasing temperature and becomes
close to unity near 600 K. This behavior is well consistent with the localized
model, because the thermopower at high temperatures can be expressed by using
chemical potential $\mu$ as

$\displaystyle Q=-\frac{\mu}{eT},$ (2)

which leads to the Heikes formula of Eq. (1) [6], and the chemical potential
in the numerator is a thermodynamic quantity, which does not give the
anisotropic property. Note that other parameters to produce the anisotropy
such as velocity and relaxation time are cancelled out in Eq. (2). Therefore,
the thermopower anisotropy, which should be unity within the Heikes formula,
is an indicator for localized electronic state.

Moreover, as shown in Fig. 3(a), the resistivity becomes less temperature-
dependent in both directions at high temperatures. Such a behavior has also
been observed in high-temperature transport study while the in-plane
orientation is undetermined [41], and is described as the hopping conduction
at the Ioffe-Regel limit [42, 43, 44], at which the Fermi wavelength
$\lambda_{F}(=2\pi/k_{F})$ ($k_{F}$ being Fermi wavenumber) is comparable to
the mean free path $l$ of conduction electrons. Indeed, the dimensionless
value of $k_{F}l$ estimated from

$\displaystyle k_{F}l=\frac{2\pi\hbar c_{0}}{e^{2}\rho},$ (3)

where $\hbar$ is the reduced Planck constant and $c_{0}$ is the $c$-axis
length [37], is calculated as $k_{F}l\sim 0.54$ for the $a$\- and $0.50$ for
the $b$-axis directions. Although there is an unavoidable error bar mainly due
to the sample thickness, these values are close to unity, indicating the
hopping conduction. Thus, these transport coefficients indicate the localized
nature at high temperatures.

###  III.3 Itinerant state at low temperatures

We then focus on low-temperature transport anisotropy below room temperature.
The main panels of Figs. 4(a) and 4(b) show the temperature variations of the
anisotropies of thermopower $Q_{b}/Q_{a}$ and electrical conductivity
$\sigma_{b}/\sigma_{a}$ below room temperature, respectively. In temperature
range of 40 K $\lesssim T\lesssim$ 300 K, both $Q_{b}/Q_{a}$ and
$\sigma_{b}/\sigma_{a}$ increase with lowering temperature. Note that similar
behaviors in thermopower and conductivity are also observed in the related
layered cobalitite (Bi,Pb)2Sr2Co2Oy [45], implying a universal anisotropic
property in the CoO2-based materials as is discussed below.

Figure 5:  (a) Constant-energy surfaces near the Fermi energy calculated by
using the tight-binding model [Eq. (4)]. (b) Dispersion relations along
$\Gamma\to{\rm K}$ ($a^{*}$ direction, red) and $\Gamma\to{\rm M}$ ($b^{*}$
direction, blue).

To clarify the origin of the anisotropic transport at low temperatures, we
discuss the electronic band structure measured by ARPES at $T=40$ K [31]. Note
that the band structure is also proposed theoretically but there is a
difficulty originating from the misfit structure of this system, in which an
approximate formula is required for calculations [32, 46, 47, 48, 49]. In
fact, the calculated thermopower strongly depends on the calculation methods
[32, 49]. Thus, the present results may offer an experimental clue for the
challenging issue to theoretically obtain the transport properties precisely
in correlated electron systems with an incommensurate structure like this
material. Now the band structure experimentally obtained in
[Ca2CoO3]0.62[CoO2] is well fitted by a tight-binding dispersion relation for
the hexagonal lattice with primitive vectors $\vec{a_{h}}$ and $\vec{b_{h}}$
as [50]

$\displaystyle\varepsilon({\bm{k}})$
$\displaystyle=\varepsilon_{0}-2t\left\\{\cos(k_{a}a_{h})+2\cos\left(\frac{\sqrt{3}}{2}k_{b}a_{h}\right)\cos\left(\frac{k_{a}a_{h}}{2}\right)\right\\}$
$\displaystyle-2t^{\prime}\left\\{\cos(2k_{a}a_{h})+2\cos\left(\sqrt{3}k_{b}a_{h}\right)\cos\left(k_{a}a_{h}\right)\right\\},$
(4)

where $\varepsilon_{0}=-72.6$ meV is a constant and $a_{h}=2.82$ Å the lattice
constant. In this model, the conducting CoO2 layer is considered only, and
modeled as a hexagonal lattice. $t=-25.4$ meV and $t^{\prime}=7.2$ meV denote
the transfer integrals between atomic orbitals connected with $\vec{a_{h}}$
and those with $2\vec{a_{h}}$. As is discussed in Ref. 31, tight-binding fit
with $t^{\prime\prime}$, which is transfer integral along
$\vec{a_{h}}+\vec{b_{h}}$ direction, leads to wrong result probably due to a
one-dimensional $\sigma$-bond formation along $\vec{a_{h}}$ direction. Figure
5(a) shows the calculated contour plot for the constant-energy surfaces, in
which a Fermi surface with a hexagonal shape is confirmed as seen in ARPES
experiments [31]. Note that $k_{a}$ and $k_{b}$ directions correspond to $\sim
a$ and $b$ directions in real space, respectively [51].

Figure 6:  (a) Electronic velocity $v_{i}=1/\hbar(\partial\varepsilon/\partial
k_{i})$ as a function of energy measured from the Fermi energy for $a$ (red)
and $b$ (blue) directions. The velocity $v_{a}$ ($v_{b}$) is calculated for
$k_{b}=0$ ($k_{a}=0$). (b) A velocity ratio
$x_{i}\equiv(v^{2}_{i,h}-v^{2}_{i,e})/(v^{2}_{i,h}+v^{2}_{i,e})$ for $a$ (red)
and $b$ (blue) directions as a function of the energy measured from
$\varepsilon_{F}$ in magnitude.

Figure 5(b) shows the dispersion relations along the $\Gamma\to{\rm K}$ and
$\Gamma\to{\rm M}$ directions. The flat region in the top of these dispersions
indicates a pudding mold energy band, in which the thermopower is
approximately determined by the difference in the velocities of electrons and
holes as [9]

$\displaystyle
Q_{i}=\frac{k_{B}}{e}\frac{\sum(v^{2}_{i,h}-v^{2}_{i,e})}{\sum(v^{2}_{i,h}+v^{2}_{i,e})},$
(5)

where $v_{i,e}$ and $v_{i,h}$ are the velocities of electrons and hole for the
$i$ direction ($i=a,b$), respectively. Figure 6(a) presents the carrier
velocity calculated as $v_{a}=1/\hbar(\partial\varepsilon/\partial
k_{a})|_{k_{b}=0}$ and $v_{b}=1/\hbar(\partial\varepsilon/\partial
k_{b})|_{k_{a}=0}$ as a function of energy near the Fermi energy. Although the
summation in Eq. (5) is taken over all the ${\bm{k}}$ states within the
thermal energy $k_{B}T$, for simplicity we only consider these $v_{a}$ and
$v_{b}$, which largely contribute to the transport coefficients for each
direction. In this approximation, one obtains
$v_{i,e}=v_{i}(\varepsilon>\varepsilon_{F})$ and
$v_{i,h}=v_{i}(\varepsilon<\varepsilon_{F})$. Figure 6(b) depicts a velocity
ratio $x_{i}\equiv(v^{2}_{i,h}-v^{2}_{i,e})/(v^{2}_{i,h}+v^{2}_{i,e})$ for
each direction as a function of the energy from the Fermi energy in magnitude.
In the present energy range, which is comparable to the thermal energy
$k_{B}T$, $x_{b}>x_{a}$ holds, leading to $Q_{b}>Q_{a}$. This is consistent
with the present result shown in Fig. 3(c), indicating a validity of itinerant
picture based on the observed Fermi surface.

We mention the anisotropy in the SDW phase below 30 K. This material shows the
SDW transition at $T_{\rm SDW}\simeq 30$ K and its onset temperature is
$T_{\rm SDW}^{\rm ON}\simeq 100$ K. As shown in Fig. 4(a), although no
prominent feature is found at $T_{\rm SDW}^{\rm ON}$, we observed a cusp near
$T_{\rm SDW}$ in the temperature dependence of the thermopower anisotropy. The
cusp structure in the thermopower anisotropy indicates that the electronic
structure is drastically varied at $T_{\rm SDW}$, as is seen in the
significant change in thermopower anisotropy at the SDW transition in Fe-based
superconductors [52]. On the other hand, the conductivity anisotropy
$\sigma_{b}/\sigma_{a}$ is monotonically increased as temperature decreases
and shows no anomaly at $T_{\rm SDW}$ as seen in Fig. 4(b). This result
implies that, although the anisotropy of the velocities is crucial since
$v_{b}>v_{a}$ leads to $\sigma_{b}>\sigma_{a}$ from
$\sigma_{i}\sim\sum\left(v^{2}_{i,h}+v^{2}_{i,e}\right)\tau$, where $\tau$ is
the relaxation time, the conductivity anisotropy may be governed by the
relaxation time, which is cancelled out in the thermopower formula of Eq. (5).

##  IV Summary

To summarize, we have measured the anisotropies of the resistivity and the
thermopower in the layered [Ca2CoO3]0.62[CoO2] and found the considerable
temperature variations. In high-temperature range, the anisotropy in the
thermopower becomes close to unity, indicating a localized picture. On the
other hand, low-temperature anisotropies are qualitatively explained in the
itinerant band picture based on the results from ARPES, and the change in the
electronic structure associated with the SDW transition is probed as a cusp
behavior of the thermopower anisotropy. These results show a crossover from
low-temperature itinerant to high-temperature localized electronic states in
this material.

###### Acknowledgements.

The authors would like to thank S. Makino for an early stage of the present
study. We are grateful to S. Yoshioka for allowing us to use the scanning
electron microscope (SEM). We thank H. Yaguchi for discussion and H. Utagawa,
H. Hatada for experimental supports. This work was supported by JSPS KAKENHI
Grants No. 17H06136, No. JP18K03503, and No. JP18K13504.

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