APS/123-QED
Dimensional crossover of quantum critical behavior in CeCoIn5
J. G. Donath,1 P. Gegenwart,2 F. Steglich,1 E. D. Bauer,3 and J. L. Sarrao3
Max-Planck-Institute for Chemical Physics of Solids, D-01187 Dresden, Germany
I. Physik. Institut, Georg-August-Universität Göttingen, D-37077 Göttingen
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
(Dated: November 11, 2018)
The nature of quantum criticality in CeCoIn5 is studied by low-temperature thermal expansion
α(T ). At the field-induced quantum critical point at H = 5 T a crossover scale T ⋆ ≈ 0.3 K is
observed, separating α(T )/T ∝ T−1 from a weaker T−1/2 divergence. We ascribe this change to a
crossover in the dimensionality of the critical fluctuations which may be coupled to a change from
unconventional to conventional quantum criticality. Disorder, whose effect on quantum criticality
is studied in CeCoIn5−xSnx (0 ≤ x ≤ 0.18), shifts T ⋆ towards higher temperatures.
PACS numbers: 71.10.Hf,71.27.+a,74.70.Tx
Quantum criticality in heavy fermion (HF) systems
continues to attract interest due to the occurrence of
highly anomalous metallic states with severe deviations
from Landau Fermi liquid (LFL) behavior [1, 2] and
the emergence of unconventional superconductivity in
close vicinity to antiferromagnetic (AF) quantum criti-
cal points (QCPs) [3]. Neither the nature of the non-
Fermi liquid (NFL) normal state related to quantum crit-
icality, nor the superconducting (SC) pairing mechanism
has been clarified up to now. It is thus of great inter-
est to investigate whether quantum criticality in these
systems can be described by conventional theory within
the framework of a spin-density-wave (SDW) instability
[4, 5], or whether unconventional scenarios in which the
f-electrons localize at the magnetic QCP due to a de-
struction of the Kondo resonance [6, 7, 8] may be more
appropriate. For the formation of the latter, magnetic
frustration leading to a reduced dimensionality of the
critical fluctuations may be crucial.
The CeMIn5 (M=Rh, Ir, Co) systems are prototyp-
ical as they display a generic phase diagram with un-
conventional HF superconductivity in close vicinity to
an AF QCP [9]. They crystallize in a tetragonal struc-
ture which can be viewed as an alternating series of
CeIn3 and MIn2 layers. As a result of the layered
crystal structure, the Fermi surface displays a strongly
two-dimensional (2D) character with cylindrical sheets
along the crystallographic c-axis [10]. Compared to cu-
bic CeIn3, a HF superconductor with Tc = 0.2 K [3]
in a very narrow pressure range close to the magnetic
quantum phase transition at pc ≈ 2.6GPa, SC transition
temperatures of about 2 K are observed over wide pres-
sure ranges for the tetragonal CeRhIn5 (at p ≥ 1.6 GPa)
and CeCoIn5 (at ambient pressure) [11, 12]. This Tc
enhancement has been attributed to the layered crystal
structure and, relatedly, strongly anisotropic magnetic
fluctuations [12]. Indeed the nuclear magnetic relaxation
rate 1/T1 of CeCoIn5 displays a weak T
1/4 dependence
in the normal state between 2 and 40 K which signals
strongly anisotropic quantum critical fluctuations [13].
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Figure 1: (Color online) Phase diagram of CeCoIn5 for H ‖
c as determined from thermal expansion. Superconducting
phase in gray with first-order boundary below 0.7 K indicated
by thick black line. Regions where thermal expansion follows
2D- and 3D quantum critical behavior are marked in blue and
yellow, respectively. The inset displays the evolution of the
crossover with Sn-doping in CeCoIn5−xSnx at the respective
Hc2(x).
The aim of this Letter is a detailed investigation of the
nature of quantum criticality in CeCoIn5. We focus in
particular to the region very close to the upper critical
field Hc2 for superconductivity (5 T for H ‖ c, cf. Figure
1) which previously has been studied by heat and charge
transport [14, 15, 16] and specific heat measurements
[17]. Diverging coefficients of the T 2 contributions to the
electrical and the thermal resistivity prove the existence
of a magnetic-field induced QCP at 5 T. NFL behavior
in the temperature dependence of the electronic specific
heat coefficient at 5 T has been described in the frame of
the SDW theory [17]. However, below 0.3 K a large nu-
clear contribution arising from the Zeeman splitting of In-
nuclear moments needs to be subtracted. Therefore, the
data do not allow to distinguish between a saturation or
http://arxiv.org/abs/0704.0506v2
logarithmic divergence at lowest temperatures and thus
further thermodynamic measurements are needed to de-
termine the nature of quantum criticality in the system
(transport data will be discussed later).
Thermal expansion is ideally suited for this purpose.
It probes the pressure dependence of the entropy which
close to QCPs is accumulated at finite temperatures.
Scaling arguments have revealed that thermal expansion
α(T ) is far more singular than specific heat C(T ) in the
approach of any pressure-sensitive QCP [18]. Within
the SDW theory the leading contribution to α(T )/T di-
verges like T−1/2 and T−1 for 3D and 2D AF QCPs,
respectively [18]. Both can easily be distinguished from
α(T )/T = const. expected for a LFL. Especially impor-
tant in this context, thermal expansion, in contrast to
specific heat, is not affected by nuclear hyperfine contri-
butions.
For our study, we have used high-quality single crys-
tals of CeCoIn5−xSnx grown from In flux, whose low-
temperature specific heat and electrical resistivity are
discussed in [17, 19, 20]. For details on the sample char-
acterization see [19, 20, 21]. Thermal expansion has been
determined with the aid of high-resolution dilatometers
at temperatures down to 0.04 K and in magnetic fields up
to 10 T. We have measured the length change ∆Lc along
the c-axis and determined the linear (c-axis) expansion
coefficient α = ∂ lnLc/∂T .
Figure 2 displays our thermal expansion data on un-
doped CeCoIn5. At the upper critical field of 5 T, the
thermal expansion coefficient α(T )/T (cf. inset a) grows
much stronger upon cooling than the respective specific
heat coefficient which diverges only logarithmically [17].
Over more than one decade in temperature, i.e. for
0.3 K ≤ T ≤ 6 K, the data follow a 1/T divergence
which hints at 2D AF quantum critical fluctuations [18].
The latter may result from the layered crystal structure
[12, 13]. At T ⋆ ≈ 0.3 K, the temperature dependence
changes to α ∝ T 1/2 (see main part and inset b, which
also displays data obtained on a second sample down
to 40 mK). Note, that α/T does not show a saturation
excluding the formation of a LFL above the lowest mea-
sured temperature. The square-root behavior for α(T ) is
compatible with a 3D AF QCP of itinerant nature [18]
as observed for CeNi2Ge2 [2] and CeIn3−xSnx [22].
As H is increased above 5 T, T ⋆ increases and α(T )
becomes less singular, i.e. the coefficient of the square-
root contribution decreases (cf. Fig.2, inset b). This
suggests that the system is tuned away from the QCP,
compatible with previous studies [14, 17], although LFL
behavior is not yet fully established in thermal expansion.
Our data on CeCoIn5 are summarized in the main part
of Fig. 1. We have observed a crossover scale separating
2D from 3D quantum critical behavior. To provide fur-
ther evidence for this crossover and to investigate how
it is influenced by weak disorder, we now focus on the
series CeCoIn5−xSnx where the Sn-atoms preferentially
0 1 2 3 4 5 6
0.1 1 7
0.0 0.5 1.0 1.5
 T (K)
 1/2 CeCoIn
5T // c
  T (K)
α/T (10
   H (T)
 5 (S2)
 8  (S2)
 10 (S2)
 T (K)
Figure 2: (Color online) Temperature dependence of the lin-
ear thermal expansion coefficient of CeCoIn5 at H = 5 T
(‖ c). Dotted line and arrow indicate α ∝
T and crossover
temperature T ⋆, defined as upper limit for this T -dependence,
respectively. Inset (a) displays data from main part as α/T
vs T (on logarithmic scale). Solid line indicates T−1 depen-
dence. Inset (b) compares data from main part in the low-
temperature regime with 5, 8 and 10T data obtained from a
second sample (S2). Lines display square-root behavior.
x Tc (K) Hc2 (T)
0.00 (2.25 ± 0.05) (4.9± 0.1)
0.03 (1.80 ± 0.05) (4.5± 0.1)
0.06 (1.50 ± 0.05) (3.9± 0.1)
0.09 (1.15 ± 0.05) (3.4± 0.1)
0.12 (0.75 ± 0.05) (2.5± 0.1)
0.18 0 0
Table I: Values for the SC transition temperature Tc and up-
per critical magnetic field Hc2 for CeCoIn5−xSnx [19].
occupy the In-1 position within in the tetragonal plane
[21]. Sn doping weakens superconductivity, leading to a
linear suppression of Tc towards zero for x = 0.18 [19].
The temperature-magnetic field phase diagram of var-
ious CeCoIn5−xSnx single crystals has previously been
studied by low-temperature electrical resistivity and spe-
cific heat measurements [19, 20]. As Tc is reduced, a
corresponding reduction of Hc2 is observed (for the x-
dependence of Tc and Hc2, see Table I). For all differ-
ent Sn concentrations the temperature dependence of the
specific heat displays NFL behavior at the respective up-
per critical field and the formation of a LFL state at
fields exceeding Hc2(x) [19]. This suggests that field-
induced quantum criticality is always pinned at the upper
critical field Hc2 when the latter is reduced by Sn dop-
ing. Furthermore, the low-T specific heat coefficient at
0.1 1 6
            x
 0.00
 0.03
 0.06
 0.09
 0.12
 0.18
T (K)
CeCoIn
Figure 3: Linear thermal expansion coefficient as α vs. T on
a logarithmic scale for CeCoIn5−xInx at H ≃ Hc2(x) (given
in Table I). Note that data sets are shifted by 5× 10−6 K−2,
subsequently. Arrows indicate lower limit of α/T ∝ T−1 be-
havior.
H = Hc2(x) remains unchanged within the scatter of the
data for 0 ≤ x ≤ 0.12 [19]. On the other hand, the resid-
ual resistivity ρ0 shows a tenfold increase for x ranging
from 0 to 0.18, indicating the effect of disorder scatter-
ing due to the random distribution of Sn-atoms on the
in-plane In site [20]. The study of CeCoIn5−xSnx thus
allows to systematically investigate the disorder depen-
dence of NFL behavior without tuning the system away
from the QCP.
Figure 3 shows c-axis thermal expansion data for the
various studied CeCoIn5−xSnx single crystals at their re-
spective upper critical magnetic fields (for the zero-field
data see [23]). In all these samples, 2D-like quantum crit-
ical behavior α(T )/T ∝ T−1 is found from 6 K down to a
lower bound which increases from about 0.3 K for x = 0
to about 1.4 K for x = 0.18.
Like for x = 0, the low-T thermal expansion of all
samples studied is well described by α(T ) ∝
T . For
x = 0.18, this temperature dependence holds up to
T ⋆ ≈ 1.4 K, i.e. over more than one decade (see Fig. 4),
providing clear evidence for 3D AF quantum critical fluc-
tuations in the latter system. Clearly, the temperature
0 1 2 3 4
T (K)
CeCoIn
         x
 0.18
Figure 4: Linear thermal expansion coefficient α(T ) of
CeCoIn5 at H = 5 T (‖ c, open squares), as well as
CeCoIn4.82Sn0.18 at H = 0 (open circles). Dotted lines and
arrows indicate α ∝
T and crossover temperatures T ⋆, re-
spectively.
at which the dimensional crossover occurs is shifted with
Sn doping in CeCoIn5−xSnx to values above 1 K, cf. the
inset of Fig. 1. As stated above, the partial substitution
of the In-(1) site by Sn-atoms enhances impurity scat-
tering without tuning the system away from the QCP.
Our observation of a shift of the crossover scale T ⋆ with
x is then naturally attributed to the effect of isotropic
impurity scattering, which ”smears out” the magnetic
anisotropy. Crossovers have also been observed in the
electrical and heat transport [15, 16] as well as in the
Hall coefficient [24] for the current direction j ⊥ c. How-
ever, transport experiments are influenced by electronic
relaxational properties, which can give rise to compli-
cated behavior for anisotropic and multiband systems
like CeCoIn5. Indeed, for j ‖ c no crossover is visible
in ρ(T ), and the Wiedemann-Franz law, which is obeyed
for j ⊥ c, seems to be violated [16].
In order to clearly show that, at the QCP in
CeCoIn5−xSnx, a finite energy scale kBT
⋆ exists which
marks the crossover from 2D to 3D quantum critical
behavior, measurements either of the fluctuation spec-
trum in equilibrium, for example by inelastic neutron
scattering (INS), or of thermodynamic properties are re-
quired. q-scans of the INS over wide regions in recip-
rocal space, required to decide on the dimensionality of
the quantum critical fluctuations, are not possible at high
fields. Therefore, our thermodynamic measurements pro-
vide the only way to investigate this question and indeed
prove such a crossover.
We now address the nature of quantum criticality
(SDW-type or unconventional) in the regime where 2D-
like behavior is observed. Theory suggests that 2D
fluctuations are necessary for the occurrence of locally-
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Figure 5: (Color online) Critical Grüneisen ratio Γcr =
Vm/κT × αcr/Ccr, where αcr and Ccr denote thermal expan-
sion and specific heat after subtraction of non-critical back-
ground contributions [18], of YbRh2(Si0.95Ge0.05)2 (left axis,
[2]) and CeCoIn5−xInx (x = 0 at H = 5 T and x = 0.18
at 0 T, right axis). For the latter, the molar volume and
isothermal compressibility equal Vm = 9.57 · 10−5 m3/mol
and κT = (3.43 ± 0.16) × 10−3 GPa−1 [25], respectively.
A small background term [0.35 × 10−6 K−2] has been sub-
tracted from α/T for x = 0.12. No specific-heat background
contributions have been subtracted since C(T )/T ∝ log T at
T > T ⋆ (T ⋆ indicated by dotted arrows). The so-derived crit-
ical Grüneisen ratio is invalid for T < T ⋆, where the specific
heat is dominated by a non-critical contribution [26]. Lines
indicate power-law behavior at T > T ⋆.
critical quantum criticality [7]. The latter is well estab-
lished for the magnetic-field tuned AF QCP in the heavy
fermion system YbRh2Si2 and its slightly Ge-doped vari-
ant YbRh2(Si0.95Ge0.05)2, for which the critical field is
almost zero [2]. It is therefore very interesting to com-
pare the low-T thermodynamics of CeCoIn5−xSnx with
the latter system. Of particular importance is the tem-
perature dependence of the critical Grüneisen ratio Γcr,
i.e. the ratio of the critical components of thermal ex-
pansion to specific heat. It has previously been shown,
that Γcr(T ) ∝ T−ǫ with ǫ = 1 and 2/3 for conventional
and unconventional quantum criticality, respectively [2].
Figure 5 shows striking similarities in the temper-
ature dependence of the critical Grüneisen ratio of
YbRh2(Si0.95Ge0.05)2 and CeCoIn5−xSnx at tempera-
tures above T ⋆(x), i.e. in the 2D regime: A rather simi-
lar fractional Grüneisen exponent is found which is close
to the prediction for the locally-critical QCP scenario in
the presence of xy anisotropy [7]. Theoretically, it has
been shown that such behavior requires quasi-2D quan-
tum critical fluctuations [7] supporting further the lat-
ter at T > T ⋆ in CeCoIn5. In view of the lack of su-
perconductivity in YbRh2Si2 (at least for T > 10 mK),
it is highly desirable to check, whether or not a similar
crossover towards conventional behavior at temperatures
below the lower limit of previous studies (20 mK) takes
place in the latter material.
To summarize, we have found thermodynamic evidence
for a finite crossover scale T ⋆ at the magnetic-field tuned
QCP in CeCoIn5. We associate T
⋆ with a dimensional
crossover from 2D (T > T ⋆) to 3D (T < T ⋆) quantum
critical behavior. The introduction of disorder shifts the
crossover scale towards higher temperatures.
Stimulating discussions with M. Nicklas, Q. Si and S.
Wirth are gratefully acknowledged. Work at Dresden and
Göttingen was partially financed by the DFG Research
unit 960 (Quantum phase transitions), while work at Los
Alamos was carried out under the auspices of the U.S.
[1] G.R. Stewart, Rev. Mod. Phys. 73, 797-855 (2001), 78,
743-753 (2006).
[2] P. Gegenwart, Q. Si, F. Steglich, arXiv:0712.2045v2 and
refs. therein.
[3] N.D. Mathur et al., Nature 394, 39 (1998).
[4] A.J. Millis, Phys. Rev. B 48, 7183 (1993).
[5] T. Moriya and T. Takimoto, J. Phys. Soc. Jpn. 64, 90
(1995), and refs. therein.
[6] P. Coleman et al., J. Phys. Cond. Matt. 13 R723 (2001).
[7] Q. Si et al., Nature 413, 804 (2001).
[8] T. Senthil, M. Vojta, S. Sachdev, Phys. Rev. B 69,
035111 (2004).
[9] J.D. Thompson, et al., Physica B 329, 446 (2003) and
refs. therein.
[10] R. Settai et al., J. Phys. Condens. Matter 13 L627 (2001).
[11] H. Hegger et al., Phys. Rev. Lett. 84, 4986 (2000).
[12] C. Petrovic et al., J. Phys. Condens. Matter 13, L337
(2001).
[13] Y. Kawasaki et al., J. Phys. Soc. Jpn. 72, 2308 (2003).
[14] J. Paglione et al., Phys. Rev. Lett. 91, 246405 (2003).
[15] J. Paglione et al., Phys. Rev. Lett. 97, 106606 (2006).
[16] M.A. Tanatar, J. Paglione, C. Petrovich, L. Taillefer, Sci-
ence 316, 1320 (2007).
[17] A. Bianchi, R. Movshovich, I. Vekhter, P.G. Pagliuso,
J.L. Sarrao, Phys. Rev. Lett. 91, 257001 (2003).
[18] L. Zhu, M. Garst, A. Rosch, Q. Si, Phys. Rev. Lett. 91,
066404 (2003).
[19] E.D. Bauer et al., Phys. Rev. Lett. 94, 047001 (2005).
[20] E.D. Bauer et al., Phys. Rev. B 73, 245109 (2006).
[21] M. Daniel et al., Phys. Rev. Lett. 95, 016406 (2005).
[22] R. Küchler et al., Phys. Rev. Lett. 96, 256403 (2006).
[23] J.G. Donath, et al., Physica B 378-380, 98 (2006).
[24] S. Singh et al., Phys. Rev. Lett. 98, 057001 (2007).
[25] R.S. Kumar, A.L. Cornelius, J.L. Sarrao, Phys. Rev. B
70, 214526 (2004).
[26] For T < T ⋆, specific heat follows the predictions of the
3D SDW scenario [17], i.e. C(T )/T = γ′ − β′
T . Since
αcr ∝
T , Γcr ∝ T−1 (as expected [18]).
http://arxiv.org/abs/0712.2045

  The nature of quantum criticality in CeCoIn$_5$ is studied by low-temperature
thermal expansion $\alpha(T)$. At the field-induced quantum critical point at
H=5 T a crossover scale $T^\star\approx 0.3$ K is observed, separating
$\alpha(T)/T\propto T^{-1}$ from a weaker $T^{-1/2}$ divergence. We ascribe
this change to a crossover in the dimensionality of the critical fluctuations
which may be coupled to a change from unconventional to conventional quantum
criticality. Disorder, whose effect on quantum criticality is studied in
CeCoIn$_{5-x}$Sn$_x$ ($0\leq x\leq 0.18$), shifts $T^\star$ towards higher
temperatures.

APS/123-QED
Dimensional crossover of quantum critical behavior in CeCoIn5
J. G. Donath,1 P. Gegenwart,2 F. Steglich,1 E. D. Bauer,3 and J. L. Sarrao3
Max-Planck-Institute for Chemical Physics of Solids, D-01187 Dresden, Germany
I. Physik. Institut, Georg-August-Universität Göttingen, D-37077 Göttingen
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
(Dated: November 11, 2018)
The nature of quantum criticality in CeCoIn5 is studied by low-temperature thermal expansion
α(T ). At the field-induced quantum critical point at H = 5 T a crossover scale T ⋆ ≈ 0.3 K is
observed, separating α(T )/T ∝ T−1 from a weaker T−1/2 divergence. We ascribe this change to a
crossover in the dimensionality of the critical fluctuations which may be coupled to a change from
unconventional to conventional quantum criticality. Disorder, whose effect on quantum criticality
is studied in CeCoIn5−xSnx (0 ≤ x ≤ 0.18), shifts T ⋆ towards higher temperatures.
PACS numbers: 71.10.Hf,71.27.+a,74.70.Tx
Quantum criticality in heavy fermion (HF) systems
continues to attract interest due to the occurrence of
highly anomalous metallic states with severe deviations
from Landau Fermi liquid (LFL) behavior [1, 2] and
the emergence of unconventional superconductivity in
close vicinity to antiferromagnetic (AF) quantum criti-
cal points (QCPs) [3]. Neither the nature of the non-
Fermi liquid (NFL) normal state related to quantum crit-
icality, nor the superconducting (SC) pairing mechanism
has been clarified up to now. It is thus of great inter-
est to investigate whether quantum criticality in these
systems can be described by conventional theory within
the framework of a spin-density-wave (SDW) instability
[4, 5], or whether unconventional scenarios in which the
f-electrons localize at the magnetic QCP due to a de-
struction of the Kondo resonance [6, 7, 8] may be more
appropriate. For the formation of the latter, magnetic
frustration leading to a reduced dimensionality of the
critical fluctuations may be crucial.
The CeMIn5 (M=Rh, Ir, Co) systems are prototyp-
ical as they display a generic phase diagram with un-
conventional HF superconductivity in close vicinity to
an AF QCP [9]. They crystallize in a tetragonal struc-
ture which can be viewed as an alternating series of
CeIn3 and MIn2 layers. As a result of the layered
crystal structure, the Fermi surface displays a strongly
two-dimensional (2D) character with cylindrical sheets
along the crystallographic c-axis [10]. Compared to cu-
bic CeIn3, a HF superconductor with Tc = 0.2 K [3]
in a very narrow pressure range close to the magnetic
quantum phase transition at pc ≈ 2.6GPa, SC transition
temperatures of about 2 K are observed over wide pres-
sure ranges for the tetragonal CeRhIn5 (at p ≥ 1.6 GPa)
and CeCoIn5 (at ambient pressure) [11, 12]. This Tc
enhancement has been attributed to the layered crystal
structure and, relatedly, strongly anisotropic magnetic
fluctuations [12]. Indeed the nuclear magnetic relaxation
rate 1/T1 of CeCoIn5 displays a weak T
1/4 dependence
in the normal state between 2 and 40 K which signals
strongly anisotropic quantum critical fluctuations [13].
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Figure 1: (Color online) Phase diagram of CeCoIn5 for H ‖
c as determined from thermal expansion. Superconducting
phase in gray with first-order boundary below 0.7 K indicated
by thick black line. Regions where thermal expansion follows
2D- and 3D quantum critical behavior are marked in blue and
yellow, respectively. The inset displays the evolution of the
crossover with Sn-doping in CeCoIn5−xSnx at the respective
Hc2(x).
The aim of this Letter is a detailed investigation of the
nature of quantum criticality in CeCoIn5. We focus in
particular to the region very close to the upper critical
field Hc2 for superconductivity (5 T for H ‖ c, cf. Figure
1) which previously has been studied by heat and charge
transport [14, 15, 16] and specific heat measurements
[17]. Diverging coefficients of the T 2 contributions to the
electrical and the thermal resistivity prove the existence
of a magnetic-field induced QCP at 5 T. NFL behavior
in the temperature dependence of the electronic specific
heat coefficient at 5 T has been described in the frame of
the SDW theory [17]. However, below 0.3 K a large nu-
clear contribution arising from the Zeeman splitting of In-
nuclear moments needs to be subtracted. Therefore, the
data do not allow to distinguish between a saturation or
http://arxiv.org/abs/0704.0506v2
logarithmic divergence at lowest temperatures and thus
further thermodynamic measurements are needed to de-
termine the nature of quantum criticality in the system
(transport data will be discussed later).
Thermal expansion is ideally suited for this purpose.
It probes the pressure dependence of the entropy which
close to QCPs is accumulated at finite temperatures.
Scaling arguments have revealed that thermal expansion
α(T ) is far more singular than specific heat C(T ) in the
approach of any pressure-sensitive QCP [18]. Within
the SDW theory the leading contribution to α(T )/T di-
verges like T−1/2 and T−1 for 3D and 2D AF QCPs,
respectively [18]. Both can easily be distinguished from
α(T )/T = const. expected for a LFL. Especially impor-
tant in this context, thermal expansion, in contrast to
specific heat, is not affected by nuclear hyperfine contri-
butions.
For our study, we have used high-quality single crys-
tals of CeCoIn5−xSnx grown from In flux, whose low-
temperature specific heat and electrical resistivity are
discussed in [17, 19, 20]. For details on the sample char-
acterization see [19, 20, 21]. Thermal expansion has been
determined with the aid of high-resolution dilatometers
at temperatures down to 0.04 K and in magnetic fields up
to 10 T. We have measured the length change ∆Lc along
the c-axis and determined the linear (c-axis) expansion
coefficient α = ∂ lnLc/∂T .
Figure 2 displays our thermal expansion data on un-
doped CeCoIn5. At the upper critical field of 5 T, the
thermal expansion coefficient α(T )/T (cf. inset a) grows
much stronger upon cooling than the respective specific
heat coefficient which diverges only logarithmically [17].
Over more than one decade in temperature, i.e. for
0.3 K ≤ T ≤ 6 K, the data follow a 1/T divergence
which hints at 2D AF quantum critical fluctuations [18].
The latter may result from the layered crystal structure
[12, 13]. At T ⋆ ≈ 0.3 K, the temperature dependence
changes to α ∝ T 1/2 (see main part and inset b, which
also displays data obtained on a second sample down
to 40 mK). Note, that α/T does not show a saturation
excluding the formation of a LFL above the lowest mea-
sured temperature. The square-root behavior for α(T ) is
compatible with a 3D AF QCP of itinerant nature [18]
as observed for CeNi2Ge2 [2] and CeIn3−xSnx [22].
As H is increased above 5 T, T ⋆ increases and α(T )
becomes less singular, i.e. the coefficient of the square-
root contribution decreases (cf. Fig.2, inset b). This
suggests that the system is tuned away from the QCP,
compatible with previous studies [14, 17], although LFL
behavior is not yet fully established in thermal expansion.
Our data on CeCoIn5 are summarized in the main part
of Fig. 1. We have observed a crossover scale separating
2D from 3D quantum critical behavior. To provide fur-
ther evidence for this crossover and to investigate how
it is influenced by weak disorder, we now focus on the
series CeCoIn5−xSnx where the Sn-atoms preferentially
0 1 2 3 4 5 6
0.1 1 7
0.0 0.5 1.0 1.5
 T (K)
 1/2 CeCoIn
5T // c
  T (K)
α/T (10
   H (T)
 5 (S2)
 8  (S2)
 10 (S2)
 T (K)
Figure 2: (Color online) Temperature dependence of the lin-
ear thermal expansion coefficient of CeCoIn5 at H = 5 T
(‖ c). Dotted line and arrow indicate α ∝
T and crossover
temperature T ⋆, defined as upper limit for this T -dependence,
respectively. Inset (a) displays data from main part as α/T
vs T (on logarithmic scale). Solid line indicates T−1 depen-
dence. Inset (b) compares data from main part in the low-
temperature regime with 5, 8 and 10T data obtained from a
second sample (S2). Lines display square-root behavior.
x Tc (K) Hc2 (T)
0.00 (2.25 ± 0.05) (4.9± 0.1)
0.03 (1.80 ± 0.05) (4.5± 0.1)
0.06 (1.50 ± 0.05) (3.9± 0.1)
0.09 (1.15 ± 0.05) (3.4± 0.1)
0.12 (0.75 ± 0.05) (2.5± 0.1)
0.18 0 0
Table I: Values for the SC transition temperature Tc and up-
per critical magnetic field Hc2 for CeCoIn5−xSnx [19].
occupy the In-1 position within in the tetragonal plane
[21]. Sn doping weakens superconductivity, leading to a
linear suppression of Tc towards zero for x = 0.18 [19].
The temperature-magnetic field phase diagram of var-
ious CeCoIn5−xSnx single crystals has previously been
studied by low-temperature electrical resistivity and spe-
cific heat measurements [19, 20]. As Tc is reduced, a
corresponding reduction of Hc2 is observed (for the x-
dependence of Tc and Hc2, see Table I). For all differ-
ent Sn concentrations the temperature dependence of the
specific heat displays NFL behavior at the respective up-
per critical field and the formation of a LFL state at
fields exceeding Hc2(x) [19]. This suggests that field-
induced quantum criticality is always pinned at the upper
critical field Hc2 when the latter is reduced by Sn dop-
ing. Furthermore, the low-T specific heat coefficient at
0.1 1 6
            x
 0.00
 0.03
 0.06
 0.09
 0.12
 0.18
T (K)
CeCoIn
Figure 3: Linear thermal expansion coefficient as α vs. T on
a logarithmic scale for CeCoIn5−xInx at H ≃ Hc2(x) (given
in Table I). Note that data sets are shifted by 5× 10−6 K−2,
subsequently. Arrows indicate lower limit of α/T ∝ T−1 be-
havior.
H = Hc2(x) remains unchanged within the scatter of the
data for 0 ≤ x ≤ 0.12 [19]. On the other hand, the resid-
ual resistivity ρ0 shows a tenfold increase for x ranging
from 0 to 0.18, indicating the effect of disorder scatter-
ing due to the random distribution of Sn-atoms on the
in-plane In site [20]. The study of CeCoIn5−xSnx thus
allows to systematically investigate the disorder depen-
dence of NFL behavior without tuning the system away
from the QCP.
Figure 3 shows c-axis thermal expansion data for the
various studied CeCoIn5−xSnx single crystals at their re-
spective upper critical magnetic fields (for the zero-field
data see [23]). In all these samples, 2D-like quantum crit-
ical behavior α(T )/T ∝ T−1 is found from 6 K down to a
lower bound which increases from about 0.3 K for x = 0
to about 1.4 K for x = 0.18.
Like for x = 0, the low-T thermal expansion of all
samples studied is well described by α(T ) ∝
T . For
x = 0.18, this temperature dependence holds up to
T ⋆ ≈ 1.4 K, i.e. over more than one decade (see Fig. 4),
providing clear evidence for 3D AF quantum critical fluc-
tuations in the latter system. Clearly, the temperature
0 1 2 3 4
T (K)
CeCoIn
         x
 0.18
Figure 4: Linear thermal expansion coefficient α(T ) of
CeCoIn5 at H = 5 T (‖ c, open squares), as well as
CeCoIn4.82Sn0.18 at H = 0 (open circles). Dotted lines and
arrows indicate α ∝
T and crossover temperatures T ⋆, re-
spectively.
at which the dimensional crossover occurs is shifted with
Sn doping in CeCoIn5−xSnx to values above 1 K, cf. the
inset of Fig. 1. As stated above, the partial substitution
of the In-(1) site by Sn-atoms enhances impurity scat-
tering without tuning the system away from the QCP.
Our observation of a shift of the crossover scale T ⋆ with
x is then naturally attributed to the effect of isotropic
impurity scattering, which ”smears out” the magnetic
anisotropy. Crossovers have also been observed in the
electrical and heat transport [15, 16] as well as in the
Hall coefficient [24] for the current direction j ⊥ c. How-
ever, transport experiments are influenced by electronic
relaxational properties, which can give rise to compli-
cated behavior for anisotropic and multiband systems
like CeCoIn5. Indeed, for j ‖ c no crossover is visible
in ρ(T ), and the Wiedemann-Franz law, which is obeyed
for j ⊥ c, seems to be violated [16].
In order to clearly show that, at the QCP in
CeCoIn5−xSnx, a finite energy scale kBT
⋆ exists which
marks the crossover from 2D to 3D quantum critical
behavior, measurements either of the fluctuation spec-
trum in equilibrium, for example by inelastic neutron
scattering (INS), or of thermodynamic properties are re-
quired. q-scans of the INS over wide regions in recip-
rocal space, required to decide on the dimensionality of
the quantum critical fluctuations, are not possible at high
fields. Therefore, our thermodynamic measurements pro-
vide the only way to investigate this question and indeed
prove such a crossover.
We now address the nature of quantum criticality
(SDW-type or unconventional) in the regime where 2D-
like behavior is observed. Theory suggests that 2D
fluctuations are necessary for the occurrence of locally-
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Figure 5: (Color online) Critical Grüneisen ratio Γcr =
Vm/κT × αcr/Ccr, where αcr and Ccr denote thermal expan-
sion and specific heat after subtraction of non-critical back-
ground contributions [18], of YbRh2(Si0.95Ge0.05)2 (left axis,
[2]) and CeCoIn5−xInx (x = 0 at H = 5 T and x = 0.18
at 0 T, right axis). For the latter, the molar volume and
isothermal compressibility equal Vm = 9.57 · 10−5 m3/mol
and κT = (3.43 ± 0.16) × 10−3 GPa−1 [25], respectively.
A small background term [0.35 × 10−6 K−2] has been sub-
tracted from α/T for x = 0.12. No specific-heat background
contributions have been subtracted since C(T )/T ∝ log T at
T > T ⋆ (T ⋆ indicated by dotted arrows). The so-derived crit-
ical Grüneisen ratio is invalid for T < T ⋆, where the specific
heat is dominated by a non-critical contribution [26]. Lines
indicate power-law behavior at T > T ⋆.
critical quantum criticality [7]. The latter is well estab-
lished for the magnetic-field tuned AF QCP in the heavy
fermion system YbRh2Si2 and its slightly Ge-doped vari-
ant YbRh2(Si0.95Ge0.05)2, for which the critical field is
almost zero [2]. It is therefore very interesting to com-
pare the low-T thermodynamics of CeCoIn5−xSnx with
the latter system. Of particular importance is the tem-
perature dependence of the critical Grüneisen ratio Γcr,
i.e. the ratio of the critical components of thermal ex-
pansion to specific heat. It has previously been shown,
that Γcr(T ) ∝ T−ǫ with ǫ = 1 and 2/3 for conventional
and unconventional quantum criticality, respectively [2].
Figure 5 shows striking similarities in the temper-
ature dependence of the critical Grüneisen ratio of
YbRh2(Si0.95Ge0.05)2 and CeCoIn5−xSnx at tempera-
tures above T ⋆(x), i.e. in the 2D regime: A rather simi-
lar fractional Grüneisen exponent is found which is close
to the prediction for the locally-critical QCP scenario in
the presence of xy anisotropy [7]. Theoretically, it has
been shown that such behavior requires quasi-2D quan-
tum critical fluctuations [7] supporting further the lat-
ter at T > T ⋆ in CeCoIn5. In view of the lack of su-
perconductivity in YbRh2Si2 (at least for T > 10 mK),
it is highly desirable to check, whether or not a similar
crossover towards conventional behavior at temperatures
below the lower limit of previous studies (20 mK) takes
place in the latter material.
To summarize, we have found thermodynamic evidence
for a finite crossover scale T ⋆ at the magnetic-field tuned
QCP in CeCoIn5. We associate T
⋆ with a dimensional
crossover from 2D (T > T ⋆) to 3D (T < T ⋆) quantum
critical behavior. The introduction of disorder shifts the
crossover scale towards higher temperatures.
Stimulating discussions with M. Nicklas, Q. Si and S.
Wirth are gratefully acknowledged. Work at Dresden and
Göttingen was partially financed by the DFG Research
unit 960 (Quantum phase transitions), while work at Los
Alamos was carried out under the auspices of the U.S.
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http://arxiv.org/abs/0712.2045
