Kadowaki-Woods Ratio of Strongly Coupled Fermi Liquids
Kadowaki-Woods Ratio of Strongly Coupled Fermi Liquids
Takuya Okabe
Faculty of Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu 432-8561,Japan
(Dated: November 28, 2018)
On the basis of the Fermi liquid theory, the Kadowaki-Woods ratio A/γ2 is evaluated by using a
first principle band calculation for typical itinerant d and f electron systems. It is found as observed
that the ratio for the d electron systems is significantly smaller than the normal f systems, even
without considering their relatively weak correlation. The difference in the ratio value comes from
different characters of the Fermi surfaces. By comparing Pd and USn3 as typical cases, we discuss
the importance of the Fermi surface dependence of the quasiparticle transport relaxation.
PACS numbers: 71.10.Ay, 71.18.+y, 71.20.Be, 71.27.+a, 72.15.-v
It is widely known as a universal feature of heavy
fermion systems that there holds the Kadowaki-Woods
(KW) relation A/γ2 ≃ 1 × 10−5µΩ cm(mol K/mJ)2 be-
tween the electronic specific heat coefficient γ of C = γT
and the coefficient A of the resistivity ρ = AT 2 in the
clean and low temperature limit.[1] According to the
Fermi liquid theory, this is interpreted as an indication
of the fact that A is squarely proportional to quasiparti-
cle mass enhancement due to strong electron correlation.
On the other hand, transition metal systems are reported
since before to obey a similar relation with a more than
an order of magnitude smaller value of A/γ2.[2, 3] In view
of the observation that there seems to exist several types
of systems in this regard, the recent finding by Tsujii et
al.[4] is quite impressive that many Yb-based compounds
show the KW ratio A/γ2 as small as the transition met-
als. Kontani derived the small ratio as a result of the
large orbital degeneracy of the the 4f13 state of trivalent
Yb by applying the dynamical mean field approximation
to a periodic Anderson model of an orbitally degenerate f
electron states coupled with a single conduction band.[5]
To discuss the KW ratio A/γ2 and the many-body
mass enhancement effect, a simple model is usually
adopted at the cost of neglecting material specific individ-
ual factors. In the present work, we are interested in such
an effect as caused by a system-dependent factor, that is,
the Fermi surface dependence of quasiparticle current re-
laxation. The system should have a large enough Fermi
surface relative to the Brillouin zone boundary in order
for the quasiparticle current to dissipate effectively into
an underlying lattice through mutual quasiparticle scat-
terings. In other words, the effectiveness of the trans-
port relaxation may depend on the size and shape of the
Fermi surface. To investigate this point definitely, we
discuss the quasiparticle transport by taking account of
the momentum dependence of quasiparticle scattering on
the basis of realistic band structures. This has been ham-
pered so far by a task required for not so simple Fermi
surfaces of many band systems as could be simply mod-
elled analytically. In terms of fairly realistic energy bands
obtained from a first principle calculation, we evaluate
those quantities which are not affected severely by the
electron correlation effect. The theory in use is essen-
tially within the phenomenological Fermi liquid theory
described by renormalized quantities, and unlike a model
calculation no bare microscopic quantities appear explic-
itly. Schematic results using simple abstract models have
been given before, in which a tight binding square lattice
model and a two-band model are investigated.[6, 7, 8]
For the ratio A/γ2 we make use of the expression,
= 21.3αFa [µΩ (mol K/mJ)
], (1)
which corresponds to Eq. (4.11) in Ref. 7 where we set
a = 4Å for the lattice constant. In what follows we sub-
stitute a calculated value for a. Below we follow how to
derive αF , where α is a coupling constant, and F is a
factor determined by the Fermi surface.
Following a microscopic analysis of the quasiparticle
transport with vertex corrections properly taken into
account,[9] we may derive a phenomenological linearized
Boltzmann equation.[7] Generalizing the theory to take
a many-band effect into account, in the low temperature
T → 0 we end up with the equation
∗Electronic address: ttokabe@ipc.shizuoka.ac.jp
vipµ = (πT )
pp′kρ
p′+k(l
pµ + l
p′µ − l
p′+kµ − l
p−kµ), (2)
where vipµ and ρ
p = δ(µ − ε
p) are the velocity compo- nent and the local density of state of the renormalized
http://arxiv.org/abs/0704.0843v2
mailto:ttokabe@ipc.shizuoka.ac.jp
(mass-enhanced) quasiparticle with the crystal momen-
tum p in the i-th band. The superscripts i and j are the
band indices, while the subscript µ = x, y, z are Carte-
sian coordinates. In the right hand side of Eq. (2), the
2nd to 4th terms in the parenthesis represent vertex cor-
rections in the microscopic formulation. In terms of the
solution lipµ, which physically represents stationary devi-
ation of the Fermi surface in an applied electric field Eµ,
the conductivity is given by
σ ≡ σµ = 2e
pµ, (3)
The above equations (2) and (3) correspond to
Eqs. (3.10) and (3.15) of Ref. 8 respectively. We may
suppress the index µ (= x) in Eq. (3) as we discuss the
cubic systems in what follows.
Instead of solving the simultaneous matrix equations
(2) exactly, we use trial functions for lipµ as commonly ap-
plied in a variational principle formulation of the trans-
port problems.[10] Assuming
lipµ ∝ e
|vipµ|
we obtain
αijci,j
ρ2|vx|
, (4)
where
ci,j =
k1,k2,k3,k4
k1+k2=k3+k4
ρik1ρ
ρik4(e
−eik4)
2/4ρiρj ,
ρ|vx| ≡
ρip|v
px|. (6)
We define coupling constants αij = ρiρj〈W
ij〉/π, where
p, is the density of states of the i-th band at
the Fermi level and 〈W ij〉 denotes the quasiparticle scat-
tering probability W
pp′k averaged over the momenta p, p
and k. As the double sum in (2), dominated by Umk-
lapp processes, covers a complicated shaped phase space
over the Fermi surface, it is generally a good approxi-
mation to take W
pp′k out of the momentum sum as an
averaged quantity. The total density of states ρ =
is substituted for γ = 2π2ρ/3.
In heavy fermion systems, the momentum dependence
pp′k could be generally neglected, for the quasiparti-
cle scattering W
pp′k is primarily caused by strong on-site
Coulomb repulsion U . Then we can make an order of
magnitude estimate of αii in terms of Landau parame-
ters F
and F
. For an anisotropic Fermi liquid, as
in an isotropic case, one can derive that the charge and
spin susceptibilities are given by χic = 2ρi/(1+F
) and
χis = 2ρi/(1 + F
), respectively. Thus, for the systems
in which charge fluctuations are suppressed, χic → 0, we
obtain F
≫ 1. On the other hand, in terms of A
/(1 +F
), one obtains a rough estimate of the cou-
pling αii = 1
)2 + 1
. There-
fore, under the normal condition that the spin enhance-
ment is moderate, (1+F
)−1 ∼ 1, αii should universally
stay around a constant of an order of unity.[7] This corre-
sponds to the condition to make the Wilson ratioRW = 2
in the impurity model.[11, 12] We discuss a normal state
that the system is well away from critical instabilities,
around which A/γ2 will be strongly enhanced at vari-
ance with experimental results under consideration.[13]
We evaluate F numerically for α = αij = 1 to obtain
A/γ2, and investigate the Fermi surface dependence.
It is noted that the factor F is determined by the shape
and extent of the Fermi surfaces relative to the Brillouin
zone boundary. Microscopically, the mass enhancement
due to the many-body effect is represented by the ω-
derivative of the electron self-energy Σ(q, ω), or by the
renormalization factor zip as ρ
p = ρ
0,p/z
p, where ρ
0,p is
a bare density of states. It is easily checked that the
factor z cancels in F when zip is independent of i. Oth-
erwise, in case that a dominant contribution to the re-
sistivity comes from an electron-correlated main band,
then the other bands may be neglected and A/γ2 be-
comes independent of z of the main band. As we see
below numerically, it is found indeed that F is domi-
nated by a few scattering channels within a main band
or two. Hence, we elaborate on a numerical estimate of
F on the basis of a realistic band calculation reproducing
reliable Fermi surfaces of relevant bands, even if it may
not take account of local many-body correlation effects
fully enough for the renormalized quantities like ρi and
vip to be separately compared with experiments. As a
matter of course, we must exclude the extreme case in
which strong correlation modifies electron states around
the Fermi level qualitatively from those of a band calcu-
lation. We apply our theory to those itinerant electron
systems in which correlation strength is not negligible
but not so strong.
To calculate F for some typical cubic d and f itiner-
ant electron systems in the fcc and Cu3Au structures, we
have performed ab initio band calculations within den-
sity functional theory using the plane wave pseudopoten-
tial code VASP with the Perdew-Wang 1991 generalized
gradient approximation to the exchange correlation func-
tional Exc.[14, 15, 16, 17] By minimizing the total energy
we obtain the lattice constant a, which is accurate enough
to be used in Eq. (1).
To evaluate F numerically, we have to broaden the
delta function ρip = δ(µ − ε
p) by ∆ to pick up electron
states around the Fermi level. The width ∆ of the order
of real temperature should be decreased as the number of
the k-points is increased until we confirm to have a con-
vergent result. For the number L of subdivisions along re-
TABLE I: Calculated results.
a (Å) ρ|vx|
a F N A/γ2 b
USn3 4.60 3.1 4.0 3 0.39
UIn3 4.61 4.9 1.6 3 0.16
UGa3 4.24 3.9 2.5 3 0.23
Pd 3.86 7.4 0.23 3 0.019
Pt 3.91 8.4 0.15 4 0.012
aIn unit of a = 1.
bIn unit of [10−5 µΩ cm (mol K/mJ)2].
ciprocal lattice vectors, band calculations are performed
with Lband ∼ 50, from which we obtain the band energies
εik on the finer k-mesh of L ∼ 200 by interpolation. As
the four-fold k-sum in the numerator of Eq. (4), especially
for the most important terms coming from the main d or
f correlated bands, constitutes the most time consuming
part of the calculation, we have to reduce the numerical
task by some symmetry considerations not only on the
cubic symmetry of the quasiparticle states, but on the
relative directions of the four momentum vectors of the
scattering quasiparticle states and the x-direction of the
current flow. The reduction is particularly effective for
the intra-band scatterings i = j.
The calculated results are shown in Table I, where F
and A/γ2 for α = αij = 1 are shown along with the
lattice constant a, the number N of metallic bands con-
tributing to the resistivity, and ρ|vx| defined in Eq. (6).
We find that our results explain well the experimental
tendency of an order of magnitude small values of the
ratio A/γ2 for the transition metal systems. As for the
absolute values of the ratio, our results are a few times
smaller than observed evenly, but the accuracy of this
order should not be taken seriously here. Among other
things, the results indicate that different characters of
the Fermi surfaces play an important role.
To show the relative contribution to the resistivity
from relevant bands, relative magnitudes of ci,j in the nu-
merator of Eq. (4) are shown for Pd and USn3 in Figs. 1
and 2, respectively. For Pd, the contribution to F comes
from the 4th to 6th bands, among which dominant is the
5th hole band of the 3d character. Similarly, the 5th band
contributes majorly not only to ρ, i.e., ρ5 ≃ 5.4ρ4 ≃ 12ρ6,
but to ρ|vx| in Eq. (6). On the other hand, for USn3,
while the 14th heavy electron band plays a central role,
the 12th and 13th hole bands also make non-negligible
contributions through the inter-band scatterings. Hence,
as the first point to note, numerical importance of the
inter-band contributions makes F large in the f electron
system. This is partly because ρi for i = 12, 13, 14 are
comparable with each other, namely, ρ14 ≃ 2ρ13 ≃ 3ρ12.
Moreover, it is remarked that the large and nearly spher-
ical shape of the Fermi surfaces are essential too. As the
second point to note, the importance of the Fermi surface
geometry can be understood within a single band model
by comparing contribution from the main band. We find
that c5,5/ρ
5 = 0.097 for Pd is an order of magnitude
 6  4
FIG. 1: cij (i, j = 4, 5, 6) for Pd. The contribution from the
5th band is dominant for the resistivity.
 14 12
FIG. 2: cij (i, j = 12, 13, 14) for USn3. The interband con-
tribution with the 14th band is important too.
smaller than c14,14/ρ
14 = 0.93 for USn3. The difference
comes from the different characters of the Fermi surfaces.
According to an elementary formula σ = e2ρv2τ =
e2ρvl, the conductivity σ depends on ρv as well as l. In
this context, the mean free path l is not a single particle
property determined by a lifetime of the particle state,
but it is the transport property which characterizes how
efficiently the total electric current decays into a lattice
system, e.g., in our case, through mutual Umklapp scat-
tering processes between the current carriers. In partic-
ular, regardless of interaction, electrons in free space will
not have resistivity.[9] Thus, to evaluate the transport
property l correctly, it is crucial to take account of the
momentum dependence of the scattering states and their
conservation modulo the reciprocal lattice vectors.
Note that ρ|vx| defined in Eq. (6) is related to the sur-
face area S of the Fermi surfaces, as ρdε = Sdk⊥/(2π)
Hence, ρ|vx| too is independent of the mass renormaliza-
tion z as F is, and for free electrons we obtain ρ|vx| ∝
∝ n2/3. One can see a correlation between F and
ρ|vx| in Table I. In fact, Pd and Pt have twice as large
ρ|vx| as the uranium compounds. The difference can-
not be simply explained by the difference in the Fermi
surface volume n. It is caused by the fact that the f -
electron systems have the nearly isotropic Fermi surfaces
while the d-electron systems have complicated ones with
FIG. 3: The intersection of the Fermi surfaces of Pd d-hole
states with the (111̄) plane.
FIG. 4: The intersection of the Fermi surfaces of USn3 with
the (100) plane
relatively large area compared to their total volume, as
indicated in Figs. 3 and 4. The different characters of
the surfaces affect not only the single particle quantity
ρ|vx| but also the transport property of the total cur-
rent relaxation. As the order of magnitude difference in
F is not explained merely by ρ|vx|, we have to have re-
sort to the other factor, that is, the transport property
depending on the Fermi surfaces. It originates from the
detailed k-dependence of the scattering states, as repre-
sented in ci,j , or by the phase space volume available for
all possible scattering channels under strict restrictions
of energy and momentum conservations. Thus our quan-
titative analysis concludes the important effect on the
quasiparticle transport due to the shape and complexity
of the Fermi surfaces.
In summary, we evaluated the Kadowaki-Woods ratio
A/γ2 of some itinerant d and f electron systems numeri-
cally on the basis of the Fermi liquid theory using quasi-
particle Fermi surfaces obtained by band calculations.
In a single framework, we find the d electron systems
have smaller ratio than the f systems, as observed, and
among others we pointed out an important effect to the
transport coefficient A originating from a commonly ne-
glected specific feature depending on the characters of
the Fermi surfaces. The effect is not understood fully
as a single-particle property of interacting systems, but
we stress the importance of the phase space restriction
due to momentum conservation in two-body scattering
processes to dissipate a total electric current. In short,
to realize effective dissipation, the system should have a
large and regular shaped Fermi surface. In future we will
examine that the Fermi-surface dependent efficiency of
mutual quasiparticle scatterings may depend on a type
of transport current to be relaxed.
Acknowledgment
The author is grateful to N. Fujima, S. Kokado and
T. Hoshino for providing assistance in the numerical cal-
culations. He also acknowledges computational resources
offered from YITP computer system in Kyoto University.
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