#  Fermi Surface Effect on Lorenz Number of Correlated Metal

Takuya Okabe  Faculty of Engineering, Shizuoka University, 3-5-1 Johoku,
Hamamatsu 432-8561,Japan

###### Abstract

We investigate an effect that an ideal Lorenz number $L_{\rm i}$ of correlated
metal shows peculiar Fermi surface dependence, which is caused by the onset of
a particular channel of Umklapp scattering. We evaluate $L_{\rm i}$ for some
simple models and transition metals, and note that $L_{\rm i}$ for NaxCoO2
decreases sensitively as $x$ approaches an Umklapp threshold around
$x_{c}\simeq 0.6$.

###### pacs:

74.25.Fy, 71.10.Ay, 71.10.Fd, 71.20.Be

##  I Introduction

The discovery of large thermopower in NaxCoO2Terasaki et al. (1997) has
prompted quest for related good thermoelectric (TE) materials and encouraged
experimental and theoretical research on TE properties of strongly correlated
electron systems. In practical TE application, the important material
parameter is the dimensionless figure of merit $ZT=S^{2}/(\kappa/\sigma T)$,
where $S$ is the Seebeck coefficient, $\kappa$ the thermal conductivity,
$\sigma$ the electrical conductivity, and $T$ the absolute temperature. Hence
it is equally as important to enhance $S$ as it is to decrease the Lorenz
number $L$ defined by $L=\kappa/\sigma T$Mahan (1998). The latter however is
usually hampered by the Wiedemann-Franz (WF) law, according to which $L$
should be a universal constant, e.g., $L=\frac{\pi^{2}}{3}(k/e)^{2}$ for
impure metals. In fact, normal metals, which generally have low $S$, will not
be good TE devices with $ZT>1$, unless the WF law is overcome. It would thus
be interesting to investigate a possible material dependence of $L$ to see if
it is a controllable variable in principle.

We discuss the ideal value $L_{\rm i}$ (which is simply denoted as $L$ below)
of a correlated electron system on a rigid lattice without impurity in the low
temperature limit for the purpose of elucidating its material specific
dependence. Indeed, for pure transition metals, the Lorenz numbers have been
observed to vary from metal to metalSchriempf (1968); White and Tainsh (1967);
Anderson et al. (1968); Beitchman et al. (1970); Wagner et al. (1971).
Theoretically, Herring had derived a universal constantHerring (1967), while
the problem had also been addressed specifically by treating $s$-$d$
hybridization as perturbation, namely, by a two band model of conductive $s$
electrons scattering off localized $d$ statesRice (1968); Schriempf et al.
(1969). Herring’s argument based on the Fermi liquid theory should be the
proper approach to the problem at low temperatures. We derive a dimensionless
factor (Eq. (10)) so as to modify his constant result, with which we
investigate the Fermi surface (FS) dependence of $L$ for some typical cases
concretely. After showing nontrivial results obtained for simple models, we
give results of numerical evaluation of the Lorenz numbers of some transition
metals. Lastly, we find it interesting to investigate a model of the sodium
cobalt oxide NaxCoO2, because a nontrivial $x$ dependence of $L$ is expected
theoretically owing to its simple cylindrical FSSingh (2000).

##  II Fermi Liquid theory

We begin with the linearized transport equation of a Fermi liquid under the
temperature gradient $\nabla T$ and the electric field ${\bf E}$,

$\frac{\partial
n_{p}}{\partial\varepsilon_{p}}\left(-\frac{\varepsilon_{p}-\mu}{T}\nabla
T\cdot{\bf v}_{p}+e{\bf E}\cdot{\bf v}_{p}\right)=I[\varphi],$ (1)
$\displaystyle TI[\varphi]$ $\displaystyle=$
$\displaystyle\sum_{p^{\prime},k}W^{pp^{\prime}}_{k}n_{p}n_{p^{\prime}}(1-n_{p-k})(1-n_{p^{\prime}+k})(\varphi_{p}+\varphi_{p^{\prime}}$
$\displaystyle-\varphi_{p-k}-\varphi_{p^{\prime}+k})\delta(\varepsilon_{p}+\varepsilon_{p^{\prime}}-\varepsilon_{p-k}-\varepsilon_{p^{\prime}+k}),$

where $n_{p}=n(\varepsilon_{p})=1/({\rm e}^{(\varepsilon_{p}-\mu)/T}+1)$ is
the Fermi distribution function, and $W^{pp^{\prime}}_{k}$ represents the
transition probability of quasiparticle scattering; $p,p^{\prime}\rightarrow
p-k,p^{\prime}+k$. In terms of the solution $\varphi_{p}$ of Eq. (1), the
electric and heat currents carried by quasiparticles are respectively given by

${\bf J}=2e\sum_{p}{\bf v}_{p}\frac{\partial
n}{\partial\varepsilon_{p}}\varphi_{p},\quad{\bf Q}=2\sum_{p}{\bf
v}_{p}(\varepsilon_{p}-\mu)\frac{\partial
n}{\partial\varepsilon_{p}}\varphi_{p},$

where $\varepsilon_{p}$ and ${\bf v}_{p}$ are energy and velocity of
quasiparticles, respectively. In comparison with the phenomenological formulae
${\bf J}=\sigma{\bf E}-\sigma S\nabla T$ and ${\bf Q}=T\sigma S{\bf
E}-\kappa_{0}\nabla T,$ we obtain the transport coefficients $\sigma$, $S$,
and $\kappa_{0}$, the thermal conductivity at zero electric field. The thermal
conductivity at zero current, $\kappa$, is given by $\kappa=\kappa_{0}-T\sigma
S^{2}$Mahan (1998). Formally, the above results expressed in terms of the
renormalized quantities $\varepsilon_{p}$ and ${\bf v}_{p}$ bear resemblance
to those of a weakly interacting Fermi gas. Nevertheless, many body effects
are included not only in the renormalization for the individual quasiparticle,
but also in the field induced shift of the quasiparticle distribution
$\varphi_{p}$, which implicitly includes a collective Fermi liquid effect
depending on Landau parametersPines and Nozières (1966); Okabe (1998). We do
not write down the explicit expressions for them as they are irrelevant for
our purposes in what follows.

For definiteness, let us assume that the currents ${\bf J}$ and ${\bf Q}$ flow
in the $x$-direction. Then, for the two functions $l_{1}$ and $l_{2}$ defined
by $\varphi_{p}=-eE_{x}l_{1}+{\partial_{x}T}l_{2}$, we obtain the equations

$-\frac{\partial n_{p}}{\partial\varepsilon_{p}}{v}_{px}=I[l_{1}],$ (2)

and

$-\frac{\partial n_{p}}{\partial\varepsilon_{p}}t_{p}{v}_{px}=I[l_{2}],$ (3)

where $t_{p}=(\varepsilon_{p}-\mu)/T$. For an isotropic system, the collision
integral $I$ has been evaluated analytically by an elaborate
approximationAbrikosov and Khalatnikov (1959). As we cannot assume the
predominance of normal scattering processes in general cases of anisotropic
Fermi liquids of our concern, we have to make approximations in another way.
To simplify the multiple momentum sum in $I$, we replace
$(1-n(\varepsilon_{p^{\prime}+k}))\delta(\varepsilon_{p}+\varepsilon_{p^{\prime}}-\varepsilon_{p-k}-\varepsilon_{p^{\prime}+k})$
with
$(1-n(\varepsilon_{p}+\varepsilon_{p^{\prime}}-\varepsilon_{p-k}))\delta(\varepsilon_{p^{\prime}+k}-\mu),$
for significant contributions to the collision term should come from a thermal
neighborhood of the FS in any case. Moreover, as in the isotropic case, to
describe the momentum dependence of the solutions $l_{i}$, we decouple the
crystal momentum variable ${\bf p}$ into the radial (energy) direction
$t_{p}=(\varepsilon_{p}-\mu)/T$ and the perpendicular component
$\bar{\Omega}_{p}$, and set $l_{i,p}=M_{i}(\bar{\Omega}_{p})N_{i}(t_{p})$
($i=1,2$). Then the momentum sum is written as $\sum_{p}=T\int\rho{\rm
d}t\int{\rm d}\bar{\Omega}_{p},$ where $\rho$ represents the density of
quasiparticle states (DOS) under the normalization condition $\int{\rm
d}\bar{\Omega}_{p}=1$. By integrating over the energy variables, we obtain
equations for $M_{i}(\bar{\Omega}_{p})$.

First we note that we can reproduce the same conductivity formula as derived
and discussed previouslyYamada and Yosida (1986); Okabe (1998). By setting
$N_{1}(t_{p})$ as a constant, the energy integrals which appear in the four
terms in $I[l_{1}]$ of Eq. (2) can be calculated analytically. As a result, we
obtain

$\displaystyle v_{px}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{2}\rho^{2}T^{2}\int{{\rm
d}\bar{\Omega}_{p^{\prime}}}\int{{\rm
d}\bar{\Omega}_{p^{\prime\prime}}}W^{pp^{\prime}}_{p-p^{\prime\prime}}({M}_{1,p}+{M}_{1,p^{\prime}}-{M}_{1,p^{\prime\prime}}-{M}_{1,p+p^{\prime}-p^{\prime\prime}})\delta(\varepsilon_{p+p^{\prime}-p^{\prime\prime}}-\mu)$
(4) $\displaystyle=$
$\displaystyle\frac{\pi^{2}T^{2}}{2}\sum_{p^{\prime},p^{\prime\prime}}W^{pp^{\prime}}_{p-p^{\prime\prime}}({M}_{1,p}+{M}_{1,p^{\prime}}-{M}_{1,p^{\prime\prime}}-{M}_{1,p+p^{\prime}-p^{\prime\prime}})\rho_{p^{\prime}}\rho_{p^{\prime\prime}}\rho_{p+p^{\prime}-p^{\prime\prime}},$

and

$\sigma={2e^{2}}\rho\int{v}_{px}M_{1}(\bar{\Omega}_{p}){{\rm
d}\bar{\Omega}_{p}}.$ (5)

In Eq. (4), we used $\rho_{p}=\delta(\varepsilon_{p}-\mu)$. Assuming the
angular dependence ${M}_{1,p}\propto v_{px}$, we obtain the resistivity
coefficient

$A=\frac{\pi^{2}\sum_{p,p^{\prime},p^{\prime\prime}}W^{pp^{\prime}}_{p-p^{\prime\prime}}\rho_{p}\rho_{p^{\prime}}\rho_{p^{\prime\prime}}\rho_{p+p^{\prime}-p^{\prime\prime}}v_{p_{x}}({v}_{p{x}}+{v}_{p^{\prime}{x}}-{v}_{p^{\prime\prime}{x}}-{v}_{p+p^{\prime}-p^{\prime\prime}x})}{4e^{2}\left(\sum_{p}\rho_{p}{v}_{px}^{2}\right)^{2}}$
(6)

for the electrical resistivity $\sigma^{-1}=AT^{2}$.

On the other hand, we have to make a further approximation for $\kappa_{0}$.
By taking the $t$-derivative at $t_{p}=0$ of Eq. (3), while adopting
$N_{2}(t)\propto t/(t^{2}+\pi^{2})$Herring (1967), we finally obtain similar
equations, namely,

$\displaystyle v_{px}$ $\displaystyle=$ $\displaystyle\rho^{2}T^{2}\int{{\rm
d}\bar{\Omega}_{p^{\prime}}}\int{{\rm
d}\bar{\Omega}_{p^{\prime\prime}}}W^{pp^{\prime}}_{p-p^{\prime\prime}}({M}_{2,p}-c({M}_{2,p^{\prime}}$
(7)
$\displaystyle+{M}_{2,p^{\prime\prime}}+{M}_{2,p+p^{\prime}-p^{\prime\prime}}))\delta(\varepsilon_{p+p^{\prime}-p^{\prime\prime}}-\mu),$

where

$c=\int_{0}^{\infty}{\rm
d}t\left({t}\coth\frac{t}{2}-2\right)\frac{2t/(t^{2}+\pi^{2})}{\sinh{t}}\simeq
0.162,$

and $\kappa_{0}=2\pi^{2}\rho T\int_{-\infty}^{\infty}\frac{\partial
n}{\partial t}\frac{t^{2}{\rm
d}t}{t^{2}+\pi^{2}}\int{v}_{px}M_{2}(\bar{\Omega}_{p}){{\rm
d}\bar{\Omega}_{p}},$ from which we obtain the thermal resistivity
$\kappa_{0}^{-1}=BT$. It is remarked that the difference in the integrand
kernels of Eqs. (4) and (7) stems from the fact that $N_{2}(t)$ for
$\kappa_{0}$ is an odd function, while $N_{1}(t)$ for $\sigma$ is even. To
evaluate the Lorenz number $L={\kappa}/{\sigma T}$ concretely, assuming
$M_{2,p}\propto v_{px}$ as above, we obtain

$L<L_{0}\equiv\frac{\kappa_{0}}{\sigma T}=\frac{\pi^{2}}{12}(12-\pi^{2}){\cal
L}\left(\frac{k}{e}\right)^{2},$ (8)

where the Boltzmann constant $k$ is written explicitly111 In the same manner,
we get $\sigma S=\frac{2\pi^{2}e}{3}T\int\frac{{\rm d}(\rho v_{px})}{{\rm
d}\varepsilon}M_{1}{\rm d}\bar{\Omega}_{p}.$ For the parabolic band, we obtain
$S=\frac{2\pi^{2}}{3e}\frac{\rho^{\prime}}{\rho}T=\frac{\pi^{2}}{3e}\frac{T}{\varepsilon_{F}}$,
the same result as that for free electrons.. The constant prefactor in Eq. (8)
is separated as it comes from the energy integral, and corresponds to the
result first derived by HerringHerring (1967), though our result differs by
$\pi^{2}/12$. In addition, the dimensionless factor ${\cal L}$ in Eq. (8)
originates from the directional dependence of scattering, and is given
explicitly by

$\displaystyle{\cal L}=\frac{A}{B}$ $\displaystyle=$
$\displaystyle\frac{\displaystyle\int{{\rm d}\bar{\Omega}_{1}}\int{{\rm
d}\bar{\Omega}_{2}}\int{{\rm
d}\bar{\Omega}_{3}}W^{p_{1}p_{2}}_{p_{1}-p_{3}}v_{p_{1}x}({v}_{p_{1}x}+{v}_{p_{2}x}-{v}_{p_{3}x}-{v}_{p_{1}+p_{2}-p_{3},x})\delta(\varepsilon_{p_{1}+p_{2}-p_{3}}-\mu)}{\displaystyle\int{{\rm
d}\bar{\Omega}_{1}}\int{{\rm d}\bar{\Omega}_{2}}\int{{\rm
d}\bar{\Omega}_{3}}W^{p_{1}p_{2}}_{p_{1}-p_{3}}v_{p_{1x}}({v}_{p_{1}x}-c\left({v}_{p_{2}x}+{v}_{p_{3}x}+{v}_{p_{1}+p_{2}-p_{3},x}\right))\delta(\varepsilon_{p_{1}+p_{2}-p_{3}}-\mu)}$
(9) $\displaystyle=$
$\displaystyle\frac{\displaystyle\sum_{1,2,3}W^{p_{1}p_{2}}_{p_{1}-p_{3}}\rho_{p_{1}}\rho_{p_{2}}\rho_{p_{3}}\rho_{p_{1}+p_{2}-p_{3}}v_{p_{1}x}\left({v}_{p_{1}x}+{v}_{p_{2}x}-{v}_{p_{3}x}-{v}_{p_{1}+p_{2}-p_{3},x}\right)}{\displaystyle\sum_{1,2,3}W^{p_{1}p_{2}}_{p_{1}-p_{3}}\rho_{p_{1}}\rho_{p_{2}}\rho_{p_{3}}\rho_{p_{1}+p_{2}-p_{3}}v_{p_{1}x}({v}_{p_{1}x}-c\left({v}_{p_{2}x}+{v}_{p_{3}x}+{v}_{p_{1}+p_{2}-p_{3},x}\right))}.$
(10)

Note that one would find a trivial result ${\cal L}=1$, a constant Lorenz
ratio, if one neglects both of the three vertex correction terms following
${v}_{p_{1}x}$ in the parentheses of the denominator as well as the numerator
of Eq. (9). To have a nontrivial effect ${\cal L}\neq 1$, it is essential not
to disregard detailed momentum structure of the relevant quasiparticle
scatterings on the FS. In fact, ${\cal L}$ quantifies the effect of FS
geometry on the availability of phase space for the quasiparticle scatterings
to relax the transport currents. Essentially, the above expression represents
the fact that thermal and electrical resistivities are determined mainly by
different types of scattering processes, that is, normal processes are
important for the thermal resistivity $B$, while the electrical $A$ is caused
by Umklapp processes. In fact, for the numerator, or the resistivity
coefficient $A$, to take a finite value, there must exist at least a set of
four momenta ${\bf p}_{i}$ ($i=1,2,3,4$) on the FS satisfying the Umklapp
condition ${\bf p}_{1}+{\bf p}_{2}={\bf p}_{3}+{\bf p}_{4}+{\bf G}$, where
${\bf G}(\neq 0)$ is a reciprocal lattice vector. This is not met if the FS is
too small. Indeed, there is a limit Fermi radius estimated by the relation
$|{\bf p}_{i}|=|{\bf G}|/4$ which holds at the threshold where all the vectors
are parallel or antiparallel. In the vicinity of the threshold, one should
expect a strong FS dependence of ${\cal L}$. This is a matter of our concern
in the following.

##  III results

###  III.1 Simple Model

The factor ${\cal L}$ may be sensitive to the momentum dependence of the
scattering probability
$W^{pp^{\prime}}_{k}=\frac{2\pi}{\hbar}\left(|A_{\uparrow\downarrow,k}^{pp^{\prime}}|^{2}+\frac{1}{2}|A_{\uparrow\uparrow,k}^{pp^{\prime}}|^{2}\right),$
where $A_{\uparrow\uparrow,k}^{pp^{\prime}}$ and
$A_{\uparrow\downarrow,k}^{pp^{\prime}}$ are the scattering amplitudes for
quasiparticles with parallel and antiparallel spins. In particular, the
momentum dependence can give rise to a conspicuous effect in the vicinity of
quantum critical points where there are quantum fluctuations localized in $k$
space. To take this into account, we may assume

$A_{\uparrow\uparrow,k}^{pp^{\prime}}\propto
A_{\uparrow\downarrow,k}^{pp^{\prime}}\propto
A_{k}^{pp^{\prime}}=\frac{1}{{1+\xi^{2}\gamma({{\bf k}-{\bf Q}})}}$ (11)

in order to represent the effect of fluctuations peaked around ${\bf k}={\bf
Q}$. In addition to the correlation length $\xi$, one may adopt $\gamma({\bf
k})\propto\sum_{{\bf d}}\left(1-{\rm e}^{{\rm i}{\bf k}\cdot{\bf d}}\right)$,
where the sum is taken over the nearest neighboring lattice vectors ${\bf d}$,
as it gives a simple lattice periodic function to give $\gamma({\bf k})\simeq
k^{2}$ in the long wavelength limit $k\rightarrow 0$.

To illustrate how ${\cal L}$ varies, let us first investigate a simple two-
dimensional (2D) model. We should make a special remark, however. In a
strictly 2D system, one will find that normal forward scatterings make the
denominator $B$ of Eq. (9) logarithmically divergent, as so for the inverse
lifetime of quasiparticle which corresponds to the denominator of Eq. (9) with
$c=0$Hodges et al. (1971). On the other side, the numerator $A$ still remains
finite as noted by Fujimoto $et$ $al.$Fujimoto et al. (1991). In effect, the
divergence is suppressed by a small decay rate $\Gamma$ assigned to the
quasiparticle states, $\rho_{p}=\delta(\varepsilon_{p}-\mu)$. In a real
system, such a cutoff must be provided by an inevitable effect of three
dimensionality of the system or by a finite density of impurities. As the
dependence on $\Gamma$ is logarithmic and weak numerically, here we present a
typical behaviour assuming $\Gamma$ as a given constant for simplicity.

Figure 1:  For a parabolic band on a square lattice, the dimensionless Lorenz
factor ${\cal L}$ in Eq. (8) is shown as a function of the filling $n$ per
orbital. For $\xi=0$ (solid line), for $\xi=1$ and ${\bf Q}=(\pi,\pi)$ (dashed
line, AFM), and for $\xi=1$ and ${\bf Q}=(0,0)$ (dotted line, FM). In all
cases, ${\cal L}$ must vanish at the threshold $n_{c,1}\simeq 0.2$.

First we discuss the simplest case of a parabolic band on a square lattice. It
is easy to show that the result is independent of the quasiparticle mass, so
that we may simply use $\varepsilon_{p}=p^{2}$. We obtain Fig. 1 for
$\Gamma=0.02$, which representatively shows ${\cal L}$ as a function of the
electron number $n$ per orbital for three types of $W^{pp^{\prime}}_{k}$
according to Eq. (11). As expected, characteristic anomalies are clearly
observed. The solid line for $\xi=0$ ($W^{pp^{\prime}}_{k}=$const.) typically
shows the first onset of the Umklapp processes involving the smallest
reciprocal lattice vector ${\bf G}=(2\pi,0)$ at $n_{c,1}=\pi/16\simeq 0.20$,
as well as the second one at $n_{c,2}\simeq 0.39$ for ${\bf G}=(2\pi,2\pi)$.
It is noted that the threshold fillings can be easily evaluated as they are
geometrically determined by the given FS. The dashed line indicates that the
latter structure for $n>n_{c,2}$ is particularly emphasized by the
commensurate antiferromagnetic (AFM) fluctuations with ${\bf Q}=(\pi,\pi)$. On
the other hand, the ferromagnetic (FM) fluctuations with ${\bf Q}=(0,0)$
generally suppress ${\cal L}$, as they strengthen the relative weight of the
normal processes contributing to the thermal resistivity $B$.

Figure 2:  ${\cal L}(\equiv A/B)$ of a parabolic band on a sc lattice and a
fcc lattice. In the inset, the resistivity coefficients
$A\propto\sigma^{-1}/T^{2}$ and $B$ for the sc lattice are shown as a function
of $n$.

Similarly, one may obtain results for three dimensional systems, in which no
cutoff is required. In Fig. 2, we observe the lattice-structure dependence of
${\cal L}$ for the parabolic band in a simple cubic (sc) lattice and a face
centered cubic (fcc) lattice with $W_{k}^{pp^{\prime}}=$const. For the sc
lattice the first threshold lies at $n_{c,1}\simeq 0.065$ due to ${\bf
G}=(2\pi,0,0)$, while it is at $n_{c,1}\simeq 0.085$ with ${\bf
G}=(2\pi,2\pi,-2\pi)$ for the fcc lattice. For the latter, the secondary kinks
expected at $n_{c,2}\simeq 0.13$ and $n_{c,3}\simeq 0.37$ corresponding to
${\bf G}=(4\pi,4\pi,0)$ and $(4\pi,0,0)$, respectively, are not so conspicuous
as that found clearly at $n_{c,2}\simeq 0.19$ due to ${\bf G}=(2\pi,2\pi,0)$
for the sc lattice. In the inset of Fig. 2, the resistivity coefficients $A$
and $B$ are shown respectively for the sc lattice. We remark that the presence
of the threshold $n_{c,1}$ may be more easily anticipated from a relatively
gradual $n$-dependence of ${\cal L}$ than from the electrical resistivity
coefficient $A$, which drops abruptly at $n_{c,1}$. The results exemplify that
the ideal Lorenz ratio is not a constant number but shows the lattice
structure dependence interestingly.

###  III.2 Transition metals

Table 1:  Calculated values for the ideal Lorenz ratio are compared with experiments.  | ${\cal L}$ |  $L_{0}$ ($10^{-8}$V2/K2) |  $L_{\rm exp}$ ($10^{-8}$V2/K2)
---|---|---|---
Pd ($\xi=0$) | 1.0 | 1.3 |
Pd ($\xi=5$Å) | 0.60 | 0.78 | 1.1 Schriempf (1968)
Ni | 1.1 | 1.4 | 1.0 White and Tainsh (1967)
Pt | 1.0 | 1.3 | 0.1 Anderson et al. (1968)
Fe | 0.87 | 1.1 | 1.1 Beitchman et al. (1970)
W | 0.61 | 0.79 | 0.2-0.4 Wagner et al. (1971)

In principle, we can evaluate ${\cal L}$ for real materials though it would
generally require a hard task numerically. Let us evaluate them for transition
metals in the same manner as described elsewhereOkabe (2007), i.e., from the
result of a first principle band calculation, we pick up a main band with the
largest DOS, for which we apply Eq. (10). Table 1 shows calculated values
along with experimental resultsSchriempf (1968); White and Tainsh (1967);
Anderson et al. (1968); Beitchman et al. (1970); Wagner et al. (1971). For
these to be evaluated, we regarded $W_{k}^{pp^{\prime}}$ as a constant, except
Pd for which presented also is the result with a paramagnon ferromagnetic
correlation effect with the correlation length $\xi=5$ÅHicks et al. (1968),
which is taken into account as above in Eq. (11). Considering the
approximations made to derive Eq. (10), for the scattering amplitudes and so
on, we conclude that we could explain a small $L_{\rm exp}$ of tungsten, among
others. This must be primarily due to a peculiar FSLoucks (1965). For
platinum, however, the observed valueAnderson et al. (1968) is inexplicable by
our single band result.

Figure 3:  For a single band tight-binding model of NaxCoO2, ${\cal L}$ as a
function of $x$ as well as the Fermi surfaces at $x=0.6$ are shown to indicate
the strong effect caused by a slight deformation due to the hopping integrals
$t_{2}$ and $t_{3}$.

###  III.3 NaxCoO2

Lastly, we discuss a tight-binding model of NaxCoO2. Though this material has
attracted much interest by its large thermopower, it is of particular interest
for us because of its simple cylindrical FS. As a nearly 2D system on a
triangular lattice, it has an almost circular hole surface centered at the
$\Gamma$ point with a slight hexagonal distortion bulging into the sides of
the hexagonal Brillouin zoneSingh (2000); Qian et al. (2006); Geck et al.
(2007). Therefore, it provides us with an unique case to realize the
nontrivial effect as discussed above. Indeed the FS suggests that the Umklapp
threshold must exist around $x_{c}\sim 0.6$, and one should expect that even a
slight change in the shape of the FS around the threshold could have a
striking effect on the doping dependence of the genuine quasiparticle
transport properties and the ideal Lorenz factor. In fact, if we assume a
undistorted parabolic band, it is straightforward to show that we should
obtain $x_{c}=2n_{c,1}-1\simeq 0.55$. Nevertheless, experimentally, the
$T^{2}$ dependence of the electrical resistivity has been observed up to
$x\simeq 0.7$Li et al. (2004); Foo et al. (2004), around which, therefore, it
must be necessary to take into account a three dimensional lattice
distortionQian et al. (2006); Geck et al. (2007). For definiteness and
simplicity, we restrict ourselves to the doping regime where the three
dimensional effect is irrelevant. We do not consider incipient ferromagnetic
in-plain correlationsBoothroyd et al. (2004), as they would not modify the
result qualitatively. Below, we pay special attention to an expected
nontrivial behavior that the factor ${\cal L}$ should tend to vanish as we
approach a threshold $x\lesssim x_{c}$. We present results for a 2D tight-
binding model on a triangular lattice, in which up to the third neighbor
hopping integrals $t_{i}$ ($i=1,2,3$) are considered as in Ref. Kuroki and
Arita (2007), where Kuroki and Arita have discussed the thermopower and the
electrical conductivity by a relaxation time approximation with a single time
scale $\tau$. Here the approximation must be abandoned from the outset, since
otherwise one would only obtain a trivial result of a constant value
regardless of the FS, as mentioned above.

To see how ${\cal L}$ tends to vanish around $x\lesssim x_{c}$ qualitatively,
we show Fig. 3 in which the $x$-dependence of ${\cal L}$ around $x\lesssim
x_{c}$ is shown for $t_{1}>0$ and $\Gamma/t_{1}=0.02$, along with the Fermi
surface for $x=0.6$. In comparison with the dashed line for $t_{2}=t_{3}=0$,
the solid line shows the strong effect around $x\lesssim x_{c}$ due to a
slight modification of the FS caused by $t_{2}$ and $t_{3}$. In effect, as the
threshold value $x_{c}\sim 0.6$ itself depends sensitively on portions of the
FS closest to the Brillouin zone boundary, the result cannot be regarded as a
quantitative prediction. Furthermore, unfortunately, it is not easy to compare
this nontrivial prediction directly with experimental results, because the
intrinsic electron term of the thermal resistivity ($\propto BT$) for the
perovskite oxides has been completely outweighed by contributions due to
phonons and impuritiesLi et al. (2004); Foo et al. (2004). Those extrinsic
terms have to be separated out properly to verify the nontrivial filling
dependence.

Figure 4:  As in Fig. 3, shown are the filling $x$ dependences of the factor
$F$, proportional to the KW ratio $A/\gamma^{2}$.

Lastly, as the vanishment of the factor ${\cal L}$ essentially reflects that
of the electrical resistivity, it would be interesting also to calculate the
$x$ dependence of the resistivity coefficient $A$, or the Kadowaki Woods (KW)
ratio $A/\gamma^{2}$. To evaluate the resistivity coefficients themselves,
however, we need the absolute value of the scattering amplitude. Assuming the
strong coupling $\rho^{2}W^{pp^{\prime}}_{p-p^{\prime\prime}}\simeq\pi$Okabe
(2007), e.g., then we obtain $A/\gamma^{2}={9F}/{16\pi e^{2}}$ with

$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{\displaystyle\sum_{1,2,3}\rho_{p_{1}}\rho_{p_{2}}\rho_{p_{3}}\rho_{p_{3}-p_{1}-p_{2}}v_{p_{1x}}({v}_{p_{1x}}+{v}_{p_{2x}}-{v}_{p_{3x}}-{v}_{p_{1}+p_{2}-p_{3},x})}{\displaystyle\rho^{4}\left(\sum_{p}\rho_{p}{v}_{px}^{2}\right)^{2}}.$
(12)

As in Fig. 3 for ${\cal L}$, we show $F$ as a function of $x$ in Fig. 4. The
ratio $A/\gamma^{2}$ also vanishes as $x$ approaches $x_{c}$, as expected.
However, numerically, we find that it should remain of the order of a common
value, $A/\gamma^{2}\simeq 1\times 10^{-5}\mu\Omega$ cm(mol K/mJ)2. Li $et$
$al.$Li et al. (2004) have observed a strongly enhanced deviation from this
standard value for $x=0.7$, which should be well beyond the threshold $x_{c}$.
It is difficult to regard the numerical deviation simply as a FS effect. Apart
from taking into account a relevant three dimensional effect, one would have
to assume an enhanced scattering
$\rho^{2}W^{pp^{\prime}}_{p-p^{\prime\prime}}\gg 1,$ which may be due to
strong scatterings caused by a proximity to some sort of instabilityLi et al.
(2004).

##  IV Summary

In summary, on the basis of anisotropic Fermi liquid theory, we investigated
the ideal Lorenz ratio for correlated metals by taking due care of the
momentum dependence of transport relaxation processes due to mutual elastic
scatterings between quasiparticles. It was shown explicitly that the ideal
Lorenz ratio of a correlated electron system is not a constant, but may vary
drastically in circumstances and even be made vanishingly small, not only by
quantum fluctuations but by a filling control around thresholds of Umklapp
scattering channels. Although it might not be easy to extract the ideal Lorenz
value in practice, theoretically we pointed out and discussed that such a
nontrivial effect should be expected in such a simple single-band system as
NaxCoO2.

The numerical calculations were carried out on Altix4700 at Shizuoka
University Information Processing Center.

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