# Colossal Thermoelectric Power Factor in K7/8RhO2

Y. Saeed, N. Singh, and U. Schwingenschlgl  PSE Division, KAUST, Thuwal
23955-6900, Kingdom of Saudi Arabia

###### Abstract

We discuss the thermoelectric and optical properties of layered KxRhO2 (_x_ =
1/2 and 7/8) in terms of the electronic structure determined by first
principles calculations as well as Boltzmann transport theory. Our optimized
lattice constants differ significantly from the experiment, but result in
optical and transport properties close to the experiment. The main
contribution to the optical spectra are due to intra and inter-band
transitions between the Rh 4 _d_ and O 2 _p_ states. We find a similar power
factor for pristine KxRhO2 at low and high cation concentartions. Our
transport results of hydrated KxRhO2 at room temperature show highest value of
the power factor among the hole-type materials. Specially at 100 K, we obtain
a value of 3$\times$10-3 K-1 for K7/8RhO2, which is larger than that of
Na0.88CoO2 [M. Lee _et al_., Nat. Mater. 5, 537 (2006)]. In general, the
electronic and optical properties of KxRhO2 are similar to NaxCoO2 with
enhanced transport properties in the hydrated phase.

PACS: 71.15.Mb, 71.20.Dg, 72.15.Jf, 78.20-e

Keywords: Layered oxides, Density functional theory, Transport and Optical
properties,

Layered cobalt oxides are of technological interest because of their transport
properties key-1 ; key-2 ; key-3 ; key-4 ; Nat.Mat.-2006 . By varying the Na
concentration in NaxCoO2 the system changes its behaviour from metallic to
insulating and becomes superconducting near _x_ = 0.3 key-3 ; key-4 .
Recently, a strong enhancement of the Seebeck coefficient has been reported
for Na0.88CoO2 at T = 80 K Nat.Mat.-2006 , which greatly improves the
prospects for thermoelectric applications. The peak value of Seebeck
coefficient and the power factor were found to be 200 $\mu$V/K and
1.8$\times$10-3 K-1, respectivily, which is among the highest values for hole-
type materials below 100 K. This fact has promoted huge interest in the
isostructural and isovalent families AxCoO2 (A = K, Rb, Cs). Angle-resolved
photoemission spectroscopy points to similar electronic and optical properties
of K1/2CoO2 and Na1/2CoO2, which is also confirmed by band structure
calculations key-14 ; key-15 ; key-16 ; key-20 . The electronic structures of
hydrated and unhydrated NaxCoO2 is studied by first principles calculations
singh-h2o .

Analogous compounds with Rh in place of Co are found to be good thermoelectric
materials with reduced correlation effects key-20 ; key-23 ; key-25 ; key-26 ;
key-27 . Shibasaki _et al_. Rh-2011  have shown that the substitution of Rh
ions in La0.8Sr0.2Co1-xRhxO${}_{{}_{3-\delta}}$ diminishes the magnetic moment
of Co, where the thermopower is enhanced by a factor of 10 at _x_ = 1/2 as
compared to _x_ = 0 and 1\. The electronic structure, optical and
thermoelectric properties of K0.49RhO2 is investigated by Okazaki _et al._
optic-2011 ; krho-40uv/k  and found a qualitative similarities in the optical
conductivity spectra as compared to NaxCoO2. The experimental Seebeck
coefficient of 40 $\mu$V/K (300 K) is reported krho-40uv/k . The temperature
dependence of transport properties is different from those of NaxCoO2, which
also suggests that the correlation in the Rh oxides is weaker than in NaxCoO2.
An enhancement of the transport properties for an increasing concentration of
alkali cations has been reported for other layered oxides as well Li-PRB-2011
. Though, various 4 _d_ systems have been investigated, the electronic
structure in general is poorly understood so far Axrho-1 ; Axrho-2 .

In order to throw light on the inter connection between enhancement of
thermoelectric properties and a high cation concentration, first principles
calculations of electronic, optical, and thermoelectric properties for KxRhO2
(_x_ = 1/2 and 7/8) are performed. The experimental data of optical and
transport properties is taken from Refs. krho-40uv/k ; optic-2011 . A
comparison to NaxCoO2 is given in terms of the chemical nature of the Co 3 _d_
and Rh 4 _d_ orbitals. The optical transitions are explained by the electronic
band structure (BS) and density of states (DOS). The influence of a high
cation concentration on power factor is also discussed, which is key for
thermoelectric devices.

Figure 1: Volume optimization of K1/2RhO2 for different exchange correlation
functionals.

Our calculations are based on density functional theory (DFT), using the full-
potential linearized augmented plane wave approach as implemented in the
WIEN2k code wien2k . This method has been successfully applied to describe the
electronic structure of oxides U1 ; U2  including the optical spectrum N.Singh
. The transport is calculated by the semiclassical Boltzmann theory within the
constant scattering approximation, as implemented in the BoltzTraP code
BoltzTraP . Various exchange and correlation functionals (local density
approximation (LDA), generalized gradient approximation (GGA), GGA-sol, and
GGA-PBE0) are used for optimizing the _c/a_ ratio. Since the differences are
small, we will discuss in following only GGA-sol results for the electronic,
optical and transport properties.

The unit cell is divided into non-overlapping atomic spheres centered at the
atomic sites and an interstitial region. The convergence parameter RmtKmax,
where Kmax is the plane-wave cut-off and Rmt is the smallest of all muffin-tin
radii controls the size of the basis set. This convergance parameter is set to
7 together with Gmax=24. We use 66 k-points in the irreducible wedge of the
Brillouin zone for calculating the electronic structure and a dense mesh of
480 k-points in the optical calculations. For the transport calculations, we
use 4592 k-points. Self-consistency is assumed to be achieved for a total
energy convergence of $10^{-5}$ Ryd.

Figure 2: Energy band structures of (a) K1/2RhO2 and (b) K7/8RhO2.

KxRhO2 crystallizes in the $\gamma$-NaxCoO2 structure (space group
P6${}_{\text{3}}$/mmc) with the experimental lattice constants _a_ = 3.0647
and _c_ = 13.6 yubuta-exp a/c . The CdI${}_{\text{2}}$-type RhO${}_{\text{2}}$
layer and the K layer are stacked alternately along the _c_ -axis. The
experimental lattice constants of K1/2RhO2 are used as a starting point of the
optimization, obtaining the optimized volume and _c/a_ ratio presented in Fig.
1. Our LDA calculation yields a $\sim$ 14% reduction of the _c/a_ ratio from
4.44 (experimental) to 3.84 (calculated), which is close to other layered
Co/Rh oxides (_c/a_ ratio $\sim$ 3.8). We obtain lattice constants of _a_ =
3.02 and _c_ = 11.63 . The bonding length between Rh and O is reduced from
2.1326 to 2.0395 . To confirm the large deviation of the _c/a_ ratio from the
experimental structure parameters, we have optimized the __ structure __ by
more involved exchange correlation functionals (GGA, GGA-sol and PBE0).
However, we obtain almost the same results for all functionals. In addition,
our optimized lattice parameters are confirmed by calculations of the optical
and transport properties, see the discussion below. The calculated Seebeck
coefficient is overestimated for the experimental lattice constants, while the
optimized lattice constants yield a Seebeck coefficient close to the
experiment. The calculated optical conductivity of K1/2RhO2 (with optimized
lattice constants) at zero photon energy is found to be $\sim$ 2500
$\Omega^{-1}$cm-1,which is agrees excellently with the experiment.

The observed difference of the _c_ length between KxRhO2 and NaxCoO2 could be
attributed to the different ionic radii of K and Na. However, our optimized
value of _c_ is similar to other isostructural layered oxides such as SrxRhO2
SrRhO2 ; sro2006 , NaxCoO2 key-15 , LixNbO2 LiNbO2 , and LaxCoO2 LaCoO2 . In
addition, LiRhO2, NaRhO2, and KRhO2 layered oxides can form a hydrate (water
intercalated) phase Axrho-1  with an increased _c_ length Park . Takada _et
al._ key-4  showed that NaxCoO2 can be readily hydrated to form NaxCoO2 _y_
H${}_{\text{2}}$O, maintaining the CoO${}_{\text{2}}$ layers, but with a
considerably expanded _c_ axis, to accomodate the intercalated water. In the
following, the optimized lattice constants are used, unless stated otherwise.

Figure 3: DOS obtained for K1/2RhO2 and K7/8RhO2 .

In Figs. 2 (a) and (b) the calculated electronic BSs of K1/2RhO2 and
K7/8RhO2is presented. The BS of K1/2RhO2 is similar to isostructural and
isovalent Na1/2CoO2, except for a slightly larger pseudogap. Moreover, the
strongly dispersive bands and high hole concentration in K7/8RhO2 give rise to
a high thermoelectricity. The calculated DOS (Fig. 3) shows a crystal field
splitting experienced by the Rh4+ ions (splitting into $e_{g}$ and $t_{2g}$
states) is similar to the Co3+ case, but with a larger bandwidth. The
bandwidth of the $t_{2g}$ state is 1.52 eV for Rh4+ and 1.46 eV for Co3+ in
the case of K1/2CoO2 (not shown here). A similar increment is also observed
for the $e_{g}$ states, in agreement with the experiment optic-2011 . In
addition, a weak hybridization is observed between the Rh 4 _d_ and O 2 _p_
states at/below the Fermi level. The O _p_ states lie deep in the valence band
(below $-$2 eV) as compared to Na0.50CoO2. The DOS of K7/8RhO2 shows increased
bandwidths of the $e_{g}$ and $t_{2g}$ states, which reflects a reduction of
the electronic correlation effects in K7/8RhO2 as compared to NaxCoO2.

Figure 4: Optical reflectivity and conductivity of K1/2RhO2 and K7/8RhO2 along
with experimental data optic-2011 . Blue and black dots respresent the
calculated and experimental values at zero photon energy.

The optical properties of KxRhO2 (_x_ = 1/2 and 7/8) are studied and presented
in Figs. 4(a) and 4(b). The experimental results are taken from Ref. [17]. The
obtained reflectivities of K1/2RhO2 and K7/8RhO2 (Fig. 4(a)) are similar to
each other with a maximum value of $\sim$ 90% near 0 eV. A Drude-like edge in
the optical reflectivity of K1/2RhO2 is found at $\sim$ 1 eV (experiment: 1.2
eV), while for K7/8RhO2 this edge appears at $\sim$ 0.5 eV. At zero photon
energy, the calculated optical conductivity of $\sigma\sim$ 2500
$\Omega^{-1}$cm-1 for K1/2RhO2 is obtained in excellent agreement with the
experiment (blue and black dots in fig. 4(b)). Three well defined peaks are
observed: (i) near 1 eV due to the intra-band transition of Rh
($t_{2g}$-$t_{2g}$), (ii) at $\sim$ 3 eV due to the inter-band transition of
Rh 4 _d_ ($t_{2g}$-$e_{g}$), and (iii) around 5.5 eV due to the inter-band
transition from the O 2 _p_ to the Rh 4 _d_ $e_{g}$ states. These peaks are
also present in the experiment optic-2011  as well as for Na1/2CoO2 (0.5 eV,
1.6 eV, and 3 eV, respectively) key-14 , which again reflects the similarity
between these isostructural and isovalent compounds.

In the following, we will address both the experimental crystal structure
(hydrated) and optimized structure of KxRhO2. We have calculated the Seebeck
coefficient (S), thermal conductivity ($\kappa$), and power factor (Z). The
results are plotted in Figs. 5(a), (b), and (c) as a function of the
temperature from 0 to 700 K, and compared with the experimental Z of
Na0.88CoO2 Nat.Mat.-2006 . Fig. 5(a) shows that the calculated S of pristine
KxRhO2 is $\sim$ 50 $\mu$V/K at 300 K, in agreement with the experiment (40
$\mu$V/K) krho-40uv/k . The calculated S values of hydrated K1/2RhO2 and
K7/8RhO2 are strongly enhanced, amounting to $\sim$ 100 $\mu$V/K and $\sim$
140 $\mu$V/K, respectively. The calculated S of pristine KxRhO2 hardly depens
on the K concentration upto 300 K, while for higher temperature for K7/8RhO2
increases stronger to reach a value of 80 $\mu$V/K at 700 K. In contrast to
this behavior, the calculated S of hydrated KxRhO2 remains almost constant
above 300 K. According to Fig. 5(b) the thermal conductivity is similar for
the hydrated compounds and for pristine K7/8RhO2, while its much enhanced for
pristine K1/2RhO2.

Figure 5: Calculated thermoelectric properties of prestine (solid line) and
hydrated (dashed line) KxRhO2. Experimental data from Ref. Nat.Mat.-2006  is
included.

In Fig. 5(c), the calculated power factor Z of pristine and hydrated KxRhO2
along with experimental data for Na0.88CoO2 Nat.Mat.-2006  is presented. Upto
25 K, power factor of hydrated K7/8RhO2 behaves similar to the experimental
curve, but approaches a value of 3$\times$10-3 K-1 at 100 K, which is much
higher than that of other hole-type materials (like Na0.88CoO2 Z =
1.8$\times$10-3 K-1 at 80 K) in this temperature range. Even at room
temperature (300 K), the calculated Z value for hydrated KxRhO2 is much higher
than Na0.88CoO2, while for pristine KxRhO2 is just slightly greater. The large
power factor in hydrated KxRhO2 results from a decrease of the thermal
conductivity, increase in the electrical conductivity as reported for the
hydrated phase of NaRhO${}_{\text{2}}$ (Fig. 2 of Ref. Park ), and larger S as
compared to that of pristine KxRhO2. Therefore, the transport properties of
hydrated KxRhO2 are highly promising for technological applications.

Figure 6: KRhO2-hydrated

In conclusion, the electronic, optical, and transport properties of layered
KxRhO2 (_x_ = 1/2 and 7/8) are calculated, and compared to the isostructural
and isovalent compound NaxCoO2. Our optimized structure of K1/2RhO2 shows a
huge deviation in the _c/a_ ratio from the experiment but gives a good
agreement for the optical and transport properties. The large deviations, also
in comparison the other related compounds, indicate that the experimental
structure has been determined for the hydrated phase of KxRhO2. The Rh4+4 _d_
$e_{g}$ and $t_{2g}$ states of K1/2RhO2 are broader than the respective Co3+
states in NaxCoO2, which confirms previous reports. The calculated Seebeck
coefficient of pristine KxRhO2 (_x_ = 1/2 and 7/8) is S $\sim$ 50 $\mu$V/K at
room temperature, which is close to the experimental value of S = 40 $\mu$V/K.
Our calculations also show large values of S and Z for hydrated KxRhO2 in
whole temperature range from 0 to 700 K. At around 100 K, the calculated Z of
hydrated K7/8RhO2 is 3$\times$10-3 K-1, which is the highest value in any
hole-type material at this temperature. At room temperature, the calculated Z
value of pristine KxRhO2 is $\sim$ 0.4$\times$10-3 K-1, which is also slightly
greter than that of Na0.88CoO2. Therefore, our results suggest that hydration
can be used to elongate the structure and inducing the doping by the formation
of hydronium ions, results strong enhancement of thermoelectric properties of
this class of layered oxides.

Figure 7: Band Na50CoO2 c15.6A

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From Corollary 5.10, we conclude that the terms appearing in the right hand
side of the estimate (5.21) are uniformly bounded in probability over the
bounded subsets of initial conditions and for all $\epsilon\in(0,1]$.
Therefore, by [29, Corollary 3.2], for all $v\in\mathscr{P}_{2}^{M}$ and
$\epsilon\in(0,1]$, the family

$\displaystyle\\{I_{1,\epsilon}:\|\xi_{0}\|_{\mathrm{C}({\mathbb{T}^{2}})}\leq
N\\}$

are tight in $\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$. Using [29, Corollary
3.6], we have

$\displaystyle\\{I_{3,\epsilon}:\|\xi_{0}\|_{\mathrm{C}({\mathbb{T}^{2}})}\leq
N,\ v\in\mathscr{P}_{2}^{M},\ \epsilon\in(0,1]\\}$

is tight in $\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$.

We consider the penultimate term $I_{2,\epsilon}$ and estimate it using [29,
Lemma 3.1(iv)] (cf. [45, Lemma 2]), with $\gamma=2,\ q>\frac{4}{3},\
p=\frac{q}{q-1},\ \kappa_{p}=1-\frac{1}{q}$ and any
$\beta\in\big{(}0,\beta_{0}-\frac{1}{2}\big{)}$, where
$\beta_{0}=\min\big{\\{}2-\frac{2}{q},1\big{\\}},$ for
$x,x+y\in{\mathbb{T}^{2}}$, $t\in[0,T]$,

$\displaystyle|I_{2,\epsilon}(t,x)-I_{2,\epsilon}(t,x+y)|$ $\displaystyle\leq
C|y|^{\beta}\bigg{(}\int_{0}^{t}\|\xi_{v}^{\epsilon}(s)-\xi_{v}^{0}(s)\|_{\mathrm{L}^{q}}^{2}|v(s)|^{2}\mathrm{d}s\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq
CM|y|^{\beta}\sup_{s\in[0,T]}\|\xi_{v}^{\epsilon}(s)-\xi_{v}^{0}(s)\|_{\mathrm{L}^{q}}.$

Similarly, the Hölder’s continuity in time $t$ follows form [29, Lemma
3.1(ii)]. For any $q>2$ and $\alpha\in(0,\frac{1}{2}-\frac{1}{q})$, we have

$\displaystyle\|I_{2,\epsilon}(t)-I_{2,\epsilon}(s)\|_{\mathrm{L}^{\infty}}$
$\displaystyle\leq
C|t-s|^{\alpha}\bigg{(}\int_{0}^{T}\|\xi_{v}^{\epsilon}(s)-\xi_{v}^{0}(s)\|_{\mathrm{L}^{q}}^{2}|v(s)|^{2}\mathrm{d}s\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq
CM|t-s|^{\alpha}\sup_{ts\in[0<T]}\|\xi_{v}^{\epsilon}(s)-\xi_{v}^{0}(s)\|_{\mathrm{L}^{q}}.$

For large enough $q$, we observe that $I_{3,\epsilon}$ is Hölder’s continuous
in both space and time variables and the Hölder norm is uniformly bounded in
probability over $\epsilon\in(0,1],\ v\in\mathscr{P}_{2}^{M}$ and
$\|\xi_{0}\|_{\mathrm{C}({\mathbb{T}^{2}})}\leq N$. Therefore, the family
$\\{I_{2,\epsilon}:\ \|\xi_{0}\|_{\mathrm{C}({\mathbb{T}^{2}})}\leq N,\
\epsilon\in(0,1],\ v\in\mathscr{P}_{2}^{M}\\}$ is also tight in the space
$\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$. ∎

######  Theorem 5.13 (Convergence in probability in supremum norm).

For any $T>0,\ \delta>0,\ N>0$ and $M>0$

$\displaystyle\lim_{\epsilon\to
0}\sup_{\|\xi_{0}\|_{\mathrm{C}({\mathbb{T}^{2}})}}\sup_{v\in\mathscr{P}_{2}^{M}}\mathbb{P}\bigg{(}\|\xi_{v}^{\epsilon}-\xi_{v}^{0}\|_{\mathrm{C}([0,T]\times{\mathbb{T}^{2}})}>\delta\bigg{)}=0.$
(5.22)

###### Proof.

In Lemma 5.12, we established that for any
$\|\xi_{0}\|_{\mathrm{C}({\mathbb{T}^{2}})}\leq N$, the family

$\displaystyle\\{\xi_{v}^{\epsilon}-\xi_{v}^{0}:v\in\mathscr{P}_{2}^{M},\epsilon\in(0,1]\\}$

is tight in $\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$. Again, by Prokhorov’s
Theorem (see [3, Section 5]), we obtain that for any sequence $\epsilon_{m}\to
0,\ v_{m}\in\mathscr{P}_{2}^{M}$ and
$\|\xi_{0,m}\|_{\mathrm{C}({\mathbb{T}^{2}})}\leq N$, there exists a
subsequence (denoted by $(\epsilon_{m},v_{m},\xi_{0,m})$) such that
$\xi_{v}^{\epsilon}-\xi_{v}^{0}$ converges in distribution. Let $\xi^{*}$ be
the limit.

Since we have the embedding
$\mathrm{C}([0,T]\times{\mathbb{T}^{2}})\hookrightarrow\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$,
the above convergence gives
$\xi_{v}^{\epsilon}-\xi_{v}^{0}\Rightarrow\xi^{*}$, in the space
$\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$ topology also. Therefore,
Corollary 5.11 yields

$\displaystyle\sup_{t\in[0,T]}\|\xi^{*}(t)\|_{\mathrm{L}^{p}}^{p}=0,$

with probability 1 which implies that
$\|\xi^{*}\|_{\mathrm{C}([0,T]\times{\mathbb{T}^{2}})}=0$ with probability 1.
Since
$\|\xi_{v}^{\epsilon}-\xi_{v}^{0}\|_{\mathrm{C}([0,T]\times{\mathbb{T}^{2}})}$
is converging to $0$, we have

$\displaystyle\lim_{m\to
0}\mathbb{P}\big{(}\|\xi_{v_{m}}^{\epsilon_{m}}-\xi_{v}^{0}\|_{\mathrm{C}([0,T];{\mathbb{T}^{2}})}>\delta\big{)}=0.$

Due to arbitrary choice of $\epsilon_{m}\downarrow 0,\
v_{m}\in\mathscr{P}_{2}^{M}$, and $\|\xi_{0,m}\|_{\mathrm{L}^{p}}\leq N$, we
obtain the required result (5.22). ∎

###### Proof of Theorem 5.8.

Combining Theorems 5.3 and 5.13, we obtain the required result. ∎

Acknowledgments: The first author would like to thank Ministry of Education,
Government of India - MHRD for financial assistance. M. T. Mohan would like to
thank the Department of Science and Technology (DST) Science $\&$ Engineering
Research Board (SERB), India for a MATRICS grant (MTR/2021/000066).

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