Thermoelectric response near a quantum critical point: the case of
  CeCoIn5
Thermoelectric response near a quantum critical point: the case of CeCoIn5
K. Izawa1,2,3, K. Behnia4, Y. Matsuda3,5, H. Shishido5,6, R.Settai6, Y. Onuki6 and J. Flouquet2
1Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo, 152-8551 Japan
2DRFMC/SPSMS, Commissariat à l’Energie Atomique, F-38042 Grenoble, France
3Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
4Laboratoire Photons Et Matière(CNRS), ESPCI, 75231 Paris, France
5Department of Physics, Kyoto University, Kyoto 606-8502, Japan and
6Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
We present a study of thermoelectric coefficients in CeCoIn5 down to 0.1 K and up to 16 T in order
to probe the thermoelectric signatures of quantum criticality. In the vicinity of the field-induced
quantum critical point, the Nernst coefficient ν exhibits a dramatic enhancement without saturation
down to lowest measured temperature. The dimensionless ratio of Seebeck coefficient to electronic
specific heat shows a minimum at a temperature close to threshold of the quasiparticle formation.
Close to Tc(H), in the vortex-liquid state, the Nernst coefficient behaves anomalously in puzzling
contrast with other superconductors and standard vortex dynamics.
PACS numbers: 74.70.Tx, 72.15.Jf, 71.27.+a
CeCoIn5 is an unconventional superconductor with an
intriguing normal state[1]. Its behavior is peculiar near
the upper critical field, where the energy scale governing
various electronic properties is vanishingly small and in-
creases with increasing magnetic field[2, 3], a behavior ex-
pected in presence of a Quantum Critical Point(QCP)[4].
The proximity of this QCP to the upper critical field in
CeCoIn5 is puzzling[5, 6, 7]. The possible existence of
a FFLO state[8] and/or an elusive magnetic order are a
subject of recent intense research. On the other hand,
even in the absence of magnetic field, the normal state
presents strong deviation from the standard Fermi-liquid
behavior[1, 9]. The application of pressure leads to the
destruction of superconductivity and the restoration of
the Fermi liquid[6, 10, 11]. The link between the field-
induced and the pressure-induced routes to the Fermi
liquid is yet to be clarified.
During the last three years, the anomalous properties
of CeCoIn5 near the field-induced QCP have been re-
ported thanks to measurements of specific heat[3], elec-
tric resistivity[2], thermal transport[7] and Hall effect[13].
In this paper, new insight on the quantum criticality is
given via thermoelectric response down to 0.1K. As far
as we know, this is the first experimental investigation
of the thermoelectric tensor in the vicinity of a QCP, a
subject of several theoretical studies[16, 17, 18]. Single
crystals were grown by self-flux method. Thermoelec-
tric coefficients were measured with one heater and two
RuO2 thermometers in magnetic field along c-axis. The
heat current was applied along the basal plane. Previous
studies of thermoelectricity in CeCoIn5 detected a large
Nernst coefficient and a field-dependent Seebeck coeffi-
cient in the non-Fermi liquid regime above Tc[14] and an
additional field scale at 23 T[15]. Here, we find that the
most spectacular thermoelectric signature of quantum
criticality is a drastic enhancement of the Nernst coeffi-
cient, ν. The vanishingly small Fermi energy, which was
previously detected by a nearly diverging enhancement of
the A coefficient of resistivity (ρ = ρ0 +AT
2)[2] and the
Sommerfeld coefficient of specific heat (γ = Cel/T )[3],
leads also to an apparently diverging ν/T . These results
show two distinct anomalies close to Hc2(0) and Tc(0)
which are different in the origin. This conclusion cannot
be derived from other probes mentioned above. We also
find a milder enhancement of the Seebeck coefficient near
the QCP. Moreover, the ratio of thermopower to elec-
tronic specific heat, expressed in appropriate units[19],
remains close to unity even in the vicinity of the QCP.
The temperature dependence of this ratio presents a min-
imum at a temperature roughly marking the formation
of well-defined quasi-particles[7].
Figure 1 presents the data obtained by measuring the
Nernst and the Seebeck coefficients at various magnetic
fields. Since the thermoelectric response of Fermions is
expected to be T -linear well below their Fermi temper-
ature, what is plotted in the figure is the temperature
dependence of the two coefficients divided by temper-
ature. As seen in fig. 1(a), the Seebeck coefficient, S
vanishes in the superconducting state. In the normal
state, S/T increases with decreasing temperature for all
fields. For fields exceeding 5.4 T, the normal state ex-
tends down to zero temperature and a finite S/T in the
zero-temperature limit can be extracted. For a field of
16 T (which is well above the quantum critical region)
S/T saturates to a value of about 13 µVK−2. For fields
between 5.4 T and 16 T, S/T presents a non-monotonous
temperature dependence. An upturn below 0.15 K is vis-
ible for µ0H ≃ 5.5 T (i.e. in the vicinity of the QCP)
curves. Note that this upturn leads to a moderate en-
hancement of S/T . The overall change in the magnitude
of S/T is about 70 %. On the other hand, the tem-
perature dependence of the Nernst coefficient divided by
temperature ν/T reveals a more dramatic signature of
Quantum criticality. As seen in figure 1(b), for µ0H =
5.5 T and µ0H = 6 T, below 1 K, ν/T is steadily increas-
ing with decreasing temperature. No such enhancement
occurs for µ0H =16 T, far above QCP. At the lowest
measured temperature (∼ 0.1 K), ν/T is five-fold en-
http://arxiv.org/abs/0704.1970v2
sample
2 3 4 5 6 7
2 3 4
T   ( K )
5.45T
5.2T 5T 4.5T
2 3 4 5 6 7
2 3 4
T   ( K )
 0.5 T
 5.3 T
 5.2 T  5 T  4.5 T  4 T
 5.4 T
 16 T
FIG. 1: (a) The Seebeck coefficient divided by temperature
as a function of temperature for different magnetic fields in
a semi-log plot. Note the upturn near the QCP.(b) Temper-
ature dependence of the Nernst coefficient divided by tem-
perature ν/T . Close to the QCP, this quantity never satu-
rates. The inset defines the convention used for the sign of
the Nernst coefficient(see text).
hanced near the QCP (∼6 T) compared to its 16 T value.
Since the thermal Hall conductivity κxy in CeCoIn5 be-
comes large at low temperatures due to enhancement of
the mean-free-path of the electrons [12], the transverse
thermal gradient ∇yT could generate a finite transverse
electric field Ey. Therefore, the (measured) adiabatic
and the (theoretical) isothermal Nernst coefficients are
not identical in CeCoIn5. However, using the value of
|∇yT |/|∇xT | ∼ 0.1 at 5.2 T reported in Ref. [12], the dif-
ference between these two is estimated to be about 10 %,
indicating that the observed enhancement is not due to a
finite ∇yT . We will argue below that this enhancement
reflects a concomitant decrease in the magnitude of the
normalized Fermi energy as previously documented by
specific heat and resistivity measurements.
The thermoelectric response of CeCoIn5 in the vicin-
ity of QCP can be better understood by complementing
our data with the information extracted by other experi-
mental probes[2, 3], which originally detected a quantum
critical behavior near Hc2. In particular, an interesting
issue to address is the fate of the correlation observed
between thermopower and specific heat of many Fermi
liquids in the zero-temperature limit In a wide range of
systems, the dimensionless ratio linking these two is of
the order of unity (q = SNAe
≃ ±1, with γ = Cel/T , NA
the Avogadro number and e the charge of electron)[19].
What happens to such a correlation at a quantum criti-
cal point? Combining the specific heat data reported by
Bianchi et al.[3] with our thermopower results allows us
to address these questions. Fig. 2(a) presents q computed
in this way as a function of temperature. The first feature
to remark is that q remains of the order of unity even in
the quantum critical region. Note that, theoretically, this
correlation arises because S/T and γ are both inversely
proportional to the normalized Fermi energy and thus q is
expected to be of the order of (and not rigorously equal
to) unity[17]. According to our result (q ≃ 0.9 at 6 T
and 0.1 K), this correlation holds even when the normal-
15105
m0H  ( T )
1/2 ( µΩ
 n/T  (0.1 K)
 A1/2 
 g  (0.1 K) (a.u.)
2 3 4 5 6 7 8
T  ( K )
FIG. 2: (a) The temperature dependence of q, the dimen-
sionless ratio of thermopower to electronic specific heat at
three magnetic fields. The temperature marked by the arrow
designs the threshold of the quasi-particle formation accord-
ing to the temperature dependence of the Lorenz number as
reported by Paglione et al.[7] (b) A comparison of the field
dependence of ν/T , γ (as reported in ref. [3]) and A1/2 (taken
from ref. [2]).
ized Fermi Energy becomes vanishingly small. The sec-
ond feature of interest in figure 2(a) is the temperature
dependence of q, which presents a minimum. For both
fields, the temperature at which this minimum occurs
is close to the one where the Lorenz number(L = κ
linking thermal, κ, and electric,σ, conductivities present
also a minimum. Paglione and co-workers, who report
this latter feature, argue that this temperature marks
the formation of well-defined quasi-particles[7]. This is a
temperature below which both thermal and electric re-
sistivities display a T 3/2 temperature dependence. Re-
markably, Miyake and Kohno, who provided a theoretical
framework in a periodic Anderson model for the corre-
lation between thermopower and specific heat, predicted
that q should deviate downward from unity in presence
of an antiferromagnetic (AF) QCP leading to hot lines
on the Fermi surface[17].
We now turn to the Nernst coefficient. In a simple
picture, it is proportional to the energy derivative of the
relaxation time at the Fermi energy[20]. In a first ap-
proximation, it tracks a magnitude set by the cyclotron
frequency, the scattering time and the Fermi energy[21].
Since it scales inversely with the Fermi Energy, there
is no surprise that it becomes large in heavy-fermion
metals[14, 15] and in particular in heavy-Fermion semi-
metals[22, 23], where both the heavy mass of electrons
and the smallness of the Fermi wave-vector contribute to
its enhancement(ν/T ∝ 1/(kF ǫF )). Now, since the Fermi
energy (broadly defined as the characteristic energy scale
of the system) becomes very small near a QCP, one would
expect a large Nernst coefficient in agreement with the
experimental observation reported here.
With these phenomenological considerations in mind
let us compare the behavior of the Nernst coefficient with
specific heat and resistivity. Both γ and A, the T 2 term
of the resistivity (ρ = ρ0 +AT
2) inversely scale with the
Fermi Energy, ǫF . Therefore, both are enhanced when
the Fermi energy is small. Since these two quantities are
linked by the Kadowaki-Woods relation (γ2 ∝ A), the
enhancement is more pronounced in A than in γ. Fig-
ure 2(b) compares the field-dependence of A1/2, γ and
ν/T . In a naive picture, the enhancement of the three
quantities are comparable in magnitude. This quanti-
tative correlation suggests that the main reason for the
enhancement of ν/T near QCP is due to a small ǫF . It
is instructive to trace a contour plot of this quantity in
the temperature-field plane. This is done in Fig. 3 with a
logarithmic color scale in order to enhance the contrast.
Note that contrary to the other probes, there is no need
to subtract an offset from the Nernst data. In the case of
specific heat, one should subtract the Schottky contribu-
tion at low temperature[3] and high-field, and the phonon
contribution at high temperature. In the case of resistiv-
ity the T 2 behavior is interrupted at low temperature
and high-fields by an upturn due to the temperature-
dependent magnitude of ωcτ [2]. As seen in Fig. 3, ν/T
becomes very large near the QCP, which constitutes the
main hearth of the figure. However, there is a second one
at zero field just above Tc, which was identified by previ-
ous measurements[14]. This zero-field hot region corre-
sponds to a purely linear resistivity and anomalously en-
hanced Hall coefficient[9] due to strong anisotropic scat-
tering by AF fluctuations[11], which can also enhance the
Nernst coefficient[24]. On the other hand, close to the
QCP, the magnitude of Hall coefficient[13] is compara-
ble to its value at room-temperature or in LaCoIn5[11].
Therefore, there appears to be two distinct sources for
the enhancement of the Nernst coefficient. In the zero-
field regime just above Tc, it is enhanced mostly because
of strong inelastic scattering associated with AF fluctu-
ations, but in the zero-temperature regime just above
Hc2, it becomes large because of the smallness of the
Fermi energy. The occurrence of superconductivity im-
pedes to explore the route linking together these two hot
regions of the (B,T) plane. The inset of the figure com-
pares the evolution of energy scales detected by different
experimental probes near the QCP.
We now turn to the puzzling behavior of the Nernst
coefficient in the vicinity of the superconducting transi-
tion. Deep into the superconducting state, there is no
measurable Nernst signal, as illustrated by the existence
of the black area in Fig. 3. On the other hand, close
to Hc2(T) (or alternatively, near Tc(H)), vortices can
move and an additional contribution to the Nernst sig-
nal is expected. In the entire range of our study, the
Nernst coefficient keeps the same sign which is presented
in the inset of Fig. 1. Such a Nernst coefficient is neg-
ative according to a textbook convention on the sign of
the thermoelectric coefficients[25]. However, the liter-
ature on the vortex Nernst effect[26] usually takes for
positive the Nernst signal generated by vortices moving
from hot to cold, which leads to an opposite convention.
The sign of the Nernst effect in CeCoIn5 is negative ac-
cording to the textbook convention[25], but positive ac-
cording to the vortex one[26, 27]. Indeed, contrary to
quasi-particles, the Nernst signal produced by vortices
0 1 2 3 4
0.2 0.4 0.6
 T(K)
0.01000
0.02185
0.04775
0.1044
0.2280
0.4983
1.089
2.380
5.200
FIG. 3: Contour plot of ν/T in the (B,T) plane. The color
scale is logarithmic. Note the presence of two hot regions
close to Hc2 and Tc. The inset is a zoom on the region near
Hc2. The variation of three temperature scales, the onset
of T 2 resistivity(solid squares), the minimum in L/L0(open
squares) the and minimum in q(open circles) with magnetic
field is also shown.
should have a fixed sign. A thermal gradient ∇xT gen-
erates a force on a vortex because its core has an excess
of entropy. The direction of this force is thus thermo-
dynamically determined; vortices move along the ther-
mal gradient from hot to cold region. The orientation of
electric field is also unambiguously set by the direction
of the vortex movement and the vortex Nernst signal is
not expected to have an arbitrary sign. In order to sepa-
rate the vortex and the quasi-particle contributions to the
Nernst signal, we put under careful scrutiny the effect of
superconducting transition on three coefficients : ρ(T ),
S(T ) and N(T ) . As illustrated in fig. 4(a) and 4(b),
with the onset of superconductivity, the Nernst signal,
N , collapses faster than both resistivity and the Seebeck
coefficient. This robust feature was observed for all mag-
netic fields. On the other hand, the collapse in ρ(T ) and
S(T ) closely track each other. This latter feature, which
was also observed in cuprates[27], suggests that the See-
beck response is essentially generated by quasi-particles.
Therefore, the most natural assumption regarding their
contribution to the Nernst signal in the vortex liquid
regime is that Nqp(T ) also follows ρ(T ) and S(T ) and
the vortex contribution to the Nernst signal can be ob-
tained by subtracting the normalized Seebeck coefficient
off the normalized Nernst one. Fig. 4(c) and 4(d) show
that this procedure clearly resolves a signal of opposite
sign. Thus, the most straightforward interpretation of
the faster collapse of N(T ) implies an additional source
of Nernst signal in the vortex liquid regime with a sign
opposite to the predominant one and also to the one ex-
pected for vortices moving along the heat flow.
This result appears incompatible with the standard
picture of vortex dynamics driven by a thermal gradient.
However, one shall not forget that additional forces on
vortices besides thermal force may be present. CeCoIn5
1.51.00.5
T   ( K )
2.42.22.01.8
T   ( K )
 S/Sn
 N/Nn
 r/rn
FIG. 4: (a)(b) Normalized magnitudes r of r = N/Nn(open
circles), S/Sn(solid circles) and ρ/ρn(solid line) in the vicinity
of superconducting transition for 1 T and 5 T. For all fields,
the onset of superconductivity leads to a faster collapse of the
Nernst signal. Note also the small shoulder at 1 T. (c)(d) The
additional contribution to transverse thermoelectricity in the
vortex liquid regime obtained by subtracting the normalized
Seebeck coefficient off the normalized Nernst signal.
is distinguished from other superconductors by the possi-
ble occurrence of an anti-ferromagnetic state in the nor-
mal core of its vortices. This feature could decrease the
entropy excess of the vortices and reduce the intensity
of the thermal force, which can therefore be vanquished
by another source of vortex movement. As first noted
by Ginzburg[28], in a superconductor subject to a ther-
mal gradient, a quasi-particle current (which carry heat)
and a supercurrent (which does not) counterflow in or-
der to keep the charge current zero[27]. In ordinary con-
ditions, this counterflow generates a transverse Magnus
force on vortices[29]. Its role in the context of superclean
CeCoIn5[12] needs an adequate theoretical treatment.
Another remarkable feature of Fig. 4 is the presence of
a small shoulder in the temperature dependence of the
Nernst effect at the end of the transition. The shoulder is
present in an extended range of magnetic fields and only
disappears in the proximity of Hc2. There seems to be
a narrow temperature window, where a thermal gradient
can create a transverse electric field, but a current does
not produce any electric field. The simplest explanation
for such a discrepancy would imply a threshold force to
depin vortices, fdp attained by the applied temperature
gradient, but not by the applied current. However, this
feature was found to be robust and no change was de-
tected by modifying the magnitude of the applied ther-
mal gradient. Clearly, the sign and the fine structure of
the Nernst effect in the vortex liquid regime of CeCoIn5
need further investigation.
We thank J-P. Brison, H. Kontani, N. Kopnin, K. Maki
and K. Miyake for helpful discussions and specially N. P.
Ong for his illuminating input on the sign of the Nernst
effect. K.I. acknowledges a European Union Marie Curie
fellowship. This work was supported by the Agence Na-
tionale de la Recherche through the ICENET project.
[1] C. Petrovic et al., J. Phys. Condens. Matter 13, L337
(2001).
[2] J. Paglione et al.,Phys. Rev. Lett. 91, 246405 (2003).
[3] A. Bianchi et al., Phys. Rev. Lett. 91, 257001 (2003).
[4] G. R. Stewart, Rev. Mod. Phys. 73, 797 (2001).
[5] E. D. Bauer et al., Phys. Rev. Lett. 94, 047001 (2005).
[6] F. Ronning et al., Phys. Rev. B 73, 064519 (2006).
[7] J. Paglione et al.,Phys. Rev. Lett. 97, 106606 (2006).
[8] Y. Matsuda and H. Shimahara, J. Phys. Soc. Jpn 76,
051005 (2007).
[9] Y. Nakajima et al., J. Phys. Soc. Jpn. 73, 5 (2004).
[10] V. A. Sidorov et al., Phys. Rev. Lett. 89, 157004 (2002).
[11] Y. Nakajima et al., J. Phys. Soc. Jpn. 76, 024703 (2007).
[12] Y.Kasahara et al., Phys. Rev. B 72, 214515 (2005).
[13] S. Singh et al., Phys. Rev. Lett. 98, 057001 (2007).
[14] R. Bel et al., Phys. Rev. Lett. 92, 217002 (2004).
[15] I. Sheikin et al., Phys. Rev. Lett. 96, 077207 (2006).
[16] I. Paul and G. Kotliar, Phys. Rev. B 64, 184414 (2001).
[17] K. Miyake and Kohno, J. Phys. Soc. Jpn. 74, 254 (2005).
[18] D. Podolsky et al., Phys. Rev. B 75, 014520 (2007).
[19] K. Behnia et al., J. Phys.: Condens. Matter 16, 5187
(2004).
[20] E. H. Sondheimer, Proc. R. Soc. London, Ser. A 193,
484(1948).
[21] K. Behnia et al., Phys. Rev. Lett. 98, 076603 (2007).
[22] R. Bel et al., Phys. Rev. B 70, 220501(R) (2004).
[23] A. Pourret et al. Phys. Rev. Lett. 96, 176402 (2006).
[24] H. Kontani, Phys. Rev. Lett. 89, 237003 (2002).
[25] G. S. Nolas et al., Thermoelectrics, Springer (2001).
[26] Y. Wang et al., Phys. Rev. B 73, 024510 (2006).
[27] R. P. Huebener, Supercond. Sci. Technol. 8, 189 (1995).
[28] V. L. Ginzburg, Sov. Phys. Usp. 34, 101(1991).
[29] H. -C. Ri et al., Phys. Rev. B 47, 12312 (1993).
