The quantum critical point in CeRhIn_5: a resistivity study
ABSTRACT The pressure--temperature phase diagram of CeRhIn_5 has been studied under high magnetic field by resistivity measurements. Clear signatures of a quantum critical point has been found at a critical pressure of p_c = 2.5 GPa. The field induced magnetic state in the superconducting state is stable up to the highest field. At p_c the antiferromagnetic ground-state under high magnetic field collapses very rapidly. Clear signatures of p_c are the strong enhancement of the resistivity in the normal state and of the inelastic scattering term. No clear T2 temperature dependence could be found for pressures above T_c. From the analysis of the upper critical field within a strong coupling model we present the pressure dependence of the coupling parameter lambda and the gyromagnetic ratio g. No signatures of a spatially modulated order parameter could be evidenced. A detailed comparison with the magnetic field--temperature phase diagram of CeCoIn_5 is given. The comparison between CeRhIn_5 and CeCoIn_5 points out the importance to take into account the field dependence of the effective mass in the calculation of the superconducting upper critical field H_c2. It suggests also that when the magnetic critical field H_(0) becomes lower than H_c2 (0)$, the persistence of a superconducting pseudo-gap may stick the antiferromagnetism to H_c2 (0). Comment: 15 pages, 20 figures, to be published in J. Phys. Soc. Jpn
Typeset with jpsj2.cls <ver.1.2.1>Full Paper
The quantum critical point in CeRhIn5: a resistivity study
Georg Knebel∗, Dai Aoki, Jean-Pascal Brison, and Jacques Flouquet
Commissariat ` a l’´Energie Atomic, INAC, SPSMS, 17 rue des Martyrs, 38054 Grenoble, France
The pressure–temperature phase diagram of CeRhIn5 has been studied under high magnetic
field by resistivity measurements. Clear signatures of a quantum critical point has been found
at a critical pressure of pc ≈ 2.5 GPa. The field induced magnetic state in the superconducting
state is stable up to the highest field. At pc the antiferromagnetic ground-state under high
magnetic field collapses very rapidly. Clear signatures of pc are the strong enhancement of the
resistivity in the normal state and of the inelastic scattering term. No clear T2temperature
dependence could be found for pressures above Tc. From the analysis of the upper critical field
within a strong coupling model we present the pressure dependence of the coupling parameter
λ and the gyromagnetic ratio g. No signatures of a spatially modulated order parameter could
be evidenced. A detailed comparison with the magnetic field–temperature phase diagram of
CeCoIn5 is given. The comparison between CeRhIn5 and CeCoIn5 points out the importance
to take into account the field dependence of the effective mass in the calculation of the su-
perconducting upper critical field Hc2. It suggests also that when the magnetic critical field
HM(0) becomes lower than Hc2(0), the persistence of a superconducting pseudo-gap may stick
the antiferromagnetism to Hc2(0).
KEYWORDS: CeRhIn5, heavy fermion superconductor, quantum critical point, upper critical field
The interplay of long range magnetic order and super-
conductivity is one of the central questions in the physics
of heavy fermion systems. Usually small amounts of mag-
netic impurities lead to suppress the superconducting
state in conventional superconductors, while in several
heavy fermion compounds it is found that superconduc-
tivity (SC) appears just close to a quantum phase tran-
sition or can even coexist with magnetic order.1,2)It is
generally believed that quantum fluctuations are respon-
sible for the attractive interaction to form Cooper pairs.
Both scenarios, magnetic fluctuations close to a quantum
critical point (QCP) where long range magnetic order
is suppressed,3)as well as density fluctuations due to a
valence transition can lead to an attractive interaction
to form Cooper pairs.4)Close to such a quantum phase
transition the normal state properties show strong devi-
ations from the usual Fermi liquid behavior of a metal at
low temperature, notably the resistivity deviates strongly
from the T2temperature dependence and the specific
heat divided by temperature γ = C/T increases to low
The heavy fermion family CeMIn5(M = Co, Rh, or
Ir) offers an ideal opportunity to study the competi-
tion between antiferromagnetism (AF) and SC.6)While
CeCoIn5 and CeIrIn5 are superconducting at ambient
pressure and antiferromagnetism can be induced either
by doping on the M-site or on the In site,7–9)CeRhIn5is
antiferromagnetically ordered below TN= 3.8 K at am-
bient pressure. It orders in an incommensurate magnetic
structure with an ordering vector ? q = (1/2,1/2,0.297).
In zero magnetic field AF is suppressed rapidly for pres-
sures p > p?
purely superconducting with most probably d-wave sym-
c= 1.95 GPa and the ground state is a
metry.10–15)At this pressure p?
transition temperatures and the superconducting tran-
sition temperature coincides, TN= Tc≈ 2.2 K. It shows
up that when Tc> TNno long range magnetic ordering
can appear as at least large parts of the Fermi surface
are gapped due to the onset of SC. Therefore, at zero
magnetic field the QCP in CeRhIn5is hidden by SC. Be-
SC is reported for p > 1 GPa and even at ambient pres-
sure.16–18)However, the nature of this superconducting
state below p?
For pressures above p?
field H ? ab plane as well as for H ⊥ ab leads to
a new phase inside the superconducting state13,14,19)
which has been detected by ac calorimetry. This new
phase is most probably a re-entrance of the magnetic
phase. It is very reminiscent to the high magnetic field
phase in CeCoIn5.20)However, in difference to CeCoIn5,
the re-entrance field seems to persist also for fields higher
than the upper critical field Hc2, as has been observed
first in resistivity measurements.21)The field induced
phase is suspected to collapse at the critical pressure
pc≈ 2.5 GPa. Interestingly the shape of the Fermi sur-
face, as detected in de Haas van Alphen experiments
changes abruptly close to pc and the effective mass of
the observed orbits increases strongly in the vicinity of
pc.22)A detailed study of the electrical transport prop-
erties under high pressure at rather high temperatures
has been published recently.23)
In this paper we will give a detailed study of the low
temperature electrical resistivity of CeRhIn5under high
pressure and high magnetic field H ? ab. The aim will be
to study the magnetic QCP by applying magnetic fields
H > Hc2to suppress SC. Furthermore a detailed study
of the pressure dependence of the upper critical field will
be given and a comparison to CeCoIn5is given.
c(TN> Tc) coexistence of antiferromagnetism and
cis still under debate.14)
cthe application of a magnetic
arXiv:0808.3687v1 [cond-mat.str-el] 27 Aug 2008
2 J. Phys. Soc. Jpn.Full PaperAuthor Name
field for different pressures. The arrows indicate the magnetic
transition for p = 0.3 and 1.7 GPa.
(Color online) Resistivity of CeRhIn5 in zero magnetic
2. Experimental details
The sample used in these experiments was cut from the
same single crystal used in our specific heat experiments
under high pressure.12,14)The dimension of the sample is
0.16 × 0.09 × 0.05 mm3. At ambient pressure the resid-
ual resistivity ratio ρ(300K)/ρ(0K) ≈ 200 indicates the
high quality of the sample. The electrical resistivity was
measured using a standard four point lock-in technique
at 17 Hz. Electrical contacts to the sample have been re-
alized by spot-welding 10 µm Au wires to the sample. A
current of maximal 100 µA was used to measure the re-
sistivity at low temperature. The temperature was mea-
sured with a calibrated Ge thermometer which is fixed
on the mixing chamber of the dilution refrigerator in a
field compensated region of the cryostat. The pressure
cell has been thermalized to the mixing chamber using a
Cu rod with 10 mm diameter. A magnetic field of max-
imal 16 T could be applied within the ab plane of the
crystal perpendicular to the current direction.
High pressure measurements have been performed in
a diamond anvil pressure cell with argon as pressure
medium. The pressure has been fixed at ambient tem-
perature and determined by measuring the fluorescence
of ruby before and after the experiment at liquid nitrogen
temperature. The difference of these pressure determina-
tion was less than 0.15 GPa in each case.
Resistivity in zero magnetic field
Figure 1 shows the resistivity of CeRhIn5in zero mag-
netic field for different pressures. The antiferromagnetic
transition for pressures below p?
visible. No SC is observed for p = 0.3 GPa in this sample.
For p = 1.7 GPa a superconducting transition appears
= 2.12 K below TN = 2.65 K. It is remark-
able that Tc determined from resistivity appears much
higher in temperature compared to the the previous spe-
cific heat experiment performed on a sample cut from the
same single crystal with Tc(C) = 1.27 K.14)Such a dis-
crepancy of the transition temperatures on an identical
sample has been already observed in previous NQR ex-
c= 1.95 GPa is clearly
for different magnetic fields H ⊥ c. (b) Derivative dρ/dT of the
resistivity versus temperature. The arrows indicate the temper-
ature of the magnetic transitions or the superconducting transi-
tion. Curves are shifted by 5 µΩcm/K respectively.
(Color online) (a) Resistivity of CeRhIn5at p = 1.7 GPa
periment at p = 1.72 GPa where the onset of Tcdetected
by the ac susceptibility at Tonset
field transition is at lower temperature TMF
determined from NQR relaxation rate.24)Thus the ob-
servation that at least the appearance of superconduc-
tivity in the pressure range below p?
seems to be a general feature. Above 2 GPa, close to
pc≈ 2.5 GPa, a very sharp superconducting transition is
observed with a width of ∆Tc≈ 30 mK. At high pressure
p > pc, the superconducting transition broadens remark-
ably and Tcdecreases. No superconductivity is observed
above 5 GPa.
= 2 K but the mean
= 0.9 K
3.2Resistivity under magnetic field
Next we will discuss the resistivity under magnetic
field for a fixed pressure. Fig. 2(a) shows the resistivity
at p = 1.7 GPa as function of temperature. To determine
the phase diagram, we plotted the derivative dρ/dT vs.
temperature in Fig. 2(b). At low fields H < 3 T one
magnetic transition appears at TNabove the supercon-
ducting transition at Tc. For higher fields, two distinct
magnetic anomalies can be seen in the derivative. From
this data the phase diagram can be drawn, as shown in
Fig. 3; it is reminiscent to the one obtained at ambient
pressure25,26)where three different magnetic phases can
be distinguished. At zero pressure it has been shown in
detailed neutron scattering experiments, that the incom-
mensurate magnetic structure of phase AF I with an or-
dering vector ? qic=(1/2, 1/2, 0.298) gets commensurable
(phase AF III) under magnetic field at low temperatures
with ? qc=(1/2, 1/2, 1/4).26)Phase AF II at ambient pres-
sure has the same structure than the incommensurate
phase AF I, but the ordered moment is reduced. The on-
J. Phys. Soc. Jpn. Full Paper Author Name3
derived from the present resistivity measurements. Three dif-
ferent magnetic phases can be distinguished, the labeling cor-
responds to the different magnetic phases obtained at ambient
pressure.25,26)The superconducting transition has been derived
from the midpoint of the transition. Stars corresponds to field
sweeps, circles to temperature sweeps.
(Color online) Phase diagram of CeRhIn5at p = 1.7 GPa
different magnetic fields. (Inset) Derivative dρ/dT of the resis-
tivity versus temperature for H = 15 T. Arrows indicate the
temperature of the magnetic transitions at TNand TN1.
(Color online) Resistivity of CeRhIn5at p = 2.4 GPa for
set of superconductivity does not allow to draw the phase
line between the antiferromagnetic phases AF I and AF
III to lower temperatures. Remarkably, no accident can
be observed in the T dependence of the upper critical
field Hc2close to the crossing point of the phase line of
the incommensurate to commensurate transition (phase
AF I to AF III) and the Hc2(T) line. It seems as if the
superconducting phase is superimposed to the magnetic
phase diagram without interplay; the same phenomenon
will appear above p?
Increasing the pressure above p?
ducting ground state and in zero magnetic field the anti-
ferromagnetism is suppressed. The main panel of Fig. 4
presents the resistivity for different magnetic fields at
cleads to a supercon-
2.4 GPa derived from the electrical resistivity (circles and di-
amonds) and from our previous ac calorimetry measurements
(squares).14)The pressure of resistivity and specific heat mea-
surement may be slightly different. Open and closed circles give
zero resistivity and the midpoint of the superconducting transi-
tion in the resistivity; filled and half-filled diamonds correspond
to TNand TN1determined from the derivative dρ/dT, respec-
tively. (See inset Fig.4.)
(Color online) (H,T) phase diagram of CeRhIn5 at p =
for different pressures. (Inset) Derivative dρ/dT of the resistivity
versus temperature for H = 9 T and 15 T. No magnetic anomaly
can be seen in the resistivity measurements.
(Color online) Resistivity of CeRhIn5 at p = 2.8 GPa
p = 2.4 GPa. The superconducting transition at low
field is very sharp, broadening appears for fields above
7 T. The midpoint of the superconducting transition for
10 T is at Tc= 0.51 K by a width of ∆Tc≈ 140 mK,
Hc2(0) can be extrapolated to 10.62 T. For magnetic
fields H > 9 T two further anomalies can be detected
above the superconducting transition. The maximum of
the derivative dρ/dT marks the transition temperature
TN1and the shoulder the N´ eel temperature TN(see inset
of Fig. 4). Even at the highest field, these two transitions
can be observed. From these data together with previous
4J. Phys. Soc. Jpn. Full Paper Author Name
CeRhIn5 for p = 2.6 GPa (triangles) and 2.8 GPa (circles) .
Closed symbols mark the onset of the transition, open symbols
correspond to the temperature of ρ = 0. With increasing pres-
sure the width of the superconducting transition increases sig-
nificantly to low temperatures.
specific heat results14)the magnetic phase diagram at
this pressure can be plotted as shown in Fig. 5 (small
differences in pressure between specific heat and resistiv-
ity data explain the small shift of TNon crossing Hc2).
The application of a magnetic field leads to a phase tran-
sition inside the superconducting state where AF and SC
coexist.13,14)However, the AF state is very stable, even
far above Hc2and can be followed in the resistivity up to
at least 15 T. Remarkably, again no anomaly in Hc2(T)
occurs close to the crossing of Hc2(T) and TN(H).
Above the critical pressure pc≈ 2.5 GPa the antifer-
romagnetism is completely suppressed. Fig. 6 presents
the resistivity and the inset dρ/dT as function of tem-
perature for p = 2.8 GPa. No magnetic transition can
be observed. The broad maximum in dρ/dT corresponds
to the change of curvature in the resistivity and is not
due to any magnetic anomaly. Even at zero field the su-
perconducting transition is slightly broader than at the
maximum of Tc, ∆Tc ≈ 60 mK; it is associated to the
change in the slope of dTc/dp. With increasing magnetic
field the transition broadens significantly. In Fig. 7 we in-
dicate the onset of the transition and zero resistivity as
function of magnetic field for p = 2.6 GPa and 2.8 GPa.
The pressure dependence of the upper critical field will
be discussed below in detail.
Pressure and field dependence in CeRhIn5
The (p,T,H) phase diagram of CeRhIn5is extremely
rich (see Fig. 8). At p?
TNand Tcmerge into one point. In a first approach the
crossing point looks like a bi-critical point: as function
of pressure a direct transition from AF to SC occurs. In
the phase diagram in Fig. 8 such direct transition corre-
sponds to the vertical hatched area, without the emer-
gence of a AF+SC regime. However, in a real experiment
such a transition is difficult to realize under pressure, in-
homogeneities (in the pressure as well as in the sample)
may always impede such a ‘clear’ phase diagram. Due
to inhomogeneities an AF+SC regime can appear; how-
c, the two critical temperatures
CeRhIn5in zero magnetic field from ac calorimetry (circles),14)
ac susceptibility (triangles)12)and resistivity (this work, dia-
monds for TNand stars for Tc). At low pressure the ground state
is antiferromagnetic. Below p?
cboth, antiferromagnetism (AF)
and superconductivity (SC) coexists. At p?
suddenly before the quantum critical point at pcis reached under
pressure. Above p?
ca purely superconducting the ground state
appears in zero magnetic field. The dashed line gives the ex-
pected pressure dependence of the N´ eel temperature in absence
(Color online) Pressure–temperature phase diagram of
cthe AF is suppressed
ever, it would not be homogeneous and phase separation
into AF and SC parts is expected. Another possibility is
this scenario comes from the homogeneous character of
the nuclear spin dynamics in the AF+SC domain at low
temperature (T < Tc).15)In recent nuclear-quadrupole-
resonance (NQR) experiments the observation of a tetra-
critical point in zero magnetic field has been reported
and it has been suggested that a uniformly homoge-
neous AF+SC phase exist below p?
existence of AF and SC in this pressure range is also
followed from the fact that the NQR relaxation (1/T1) is
mono-exponential, independent on the investigated In-
site.15)This led to the suggestion that both, the anti-
ferromagnetic and the superconducting order parameter
are strongly coupled as it is proposed in the SO(5) the-
ory.28,29)However, the superconducting phase transition
at Tc is at least inhomogeneous below p?
ferent experimental probes different transition temper-
atures are detected. The vertical hatched line describes
then only the trend that the AF+SC domain is highly
non-symmetrical by respect to p?
just above p?
difficult to draw precisely the AF+SC boundary.
For p < p?
magnetic component. The superconducting phase tran-
sition observed below p?
in nature. However zero resistivity has been observed
at p = 1.7 GPa and the upper critical field determined
by the resistivity is rather large. The magnetic ordered
state seems not to change dramatically under high pres-
sure. The magnetic (H–T) phase diagram observed at
cis a tetracritical point.27,28)Strong support for
c. The uniformly co-
c, as with dif-
c: AF needs to disappear
c. From experimental point of view it is very
c≈ 2 GPa the ground state has an antiferro-
cin the resistivity is not bulk
J. Phys. Soc. Jpn. Full Paper Author Name5
p = 1.7 GPa is qualitatively unchanged in comparison to
low pressure with the appearance of different magnetic
phases (see Fig. 3).
At zero pressure it has been shown in detailed neu-
tron scattering experiments, that the incommensurate
magnetic structure of phase AF I with an ordering vec-
tor ? qic =(1/2, 1/2, 0.298) gets commensurable (phase
AF III) under magnetic field at low temperatures with
? qc=(1/2, 1/2, 1/4).26)Phase II at ambient pressure has
the same structure than the incommensurate phase AF I,
but the ordered moment is reduced. To identify the mag-
netic structures under high pressure, neutron scattering
or NMR experiments are indispensable. However, up to
now no successful neutron scattering experiments have
been performed under application of magnetic field and
high pressure for CeRhIn5. No definite conclusion can be
given on the magnetic ordering vector under pressure in
the different phases. All neutron scattering experiments
performed up to now report an incommensurate ordering
vector up to 1.7 GPa in zero magnetic field.30–32)In the
most recent neutron scattering experiments at 1.7 GPa
? qic(1.7 GPa) = (1/2,1/2,0.4) has been observed in zero
field at T = 0.4 K inside the superconducting state.32)
Nevertheless, from our transport measurements here and
also from the ac calorimetry under pressure14)the differ-
ent magnetic phases seem almost unchanged under high
pressure up to p?
not allow to draw the phase line between the antiferro-
magnetic phases AF I and AF III to lower temperatures.
Up to now it is unclear, if the magnetic ordering
changes its commensurability under pressure. The ob-
servation of the magnetic signal by neutron scattering,
however, is not a direct prove of coexistence of AF
and SC on a microscopic scale as the detected mag-
netic intensity is an average of the magnetic moment
in the crystal volume. More detailed microscopic in-
formations can be obtained from NQR measurements.
As discussed above, these experiments point to an uni-
form coexistence of both AF and SC in this pressure
range.15)Furthermore, it is stated from measurements
of the NQR spectra at the In(2) site that the magnetic
structure in this coexistence regime should be commen-
surate.34)Regarding to the phase diagram in Fig. 3 it
is difficult to imagine that there is a profound change in
the magnetic structure. However, for the doped systems
CeRh1−xIrxIn535,36)and CeRh1−xCoxIn537)as well as
for Cd doped CeCo(In1−xCdx)538)remarkably AF and
SC coexists only when an AF ordering with commensu-
rate ? qAF = (1/2,1/2,1/2) is observed which is the or-
dering vector of the cubic CeIn3.39,40)This is also the
characteristic wavevector in CeCoIn5where a sharp spin
resonance develops in the superconducting state.41)In
doped systems, the commensurate ordering vector ? qcof
phase III of CeRhIn5has never been reported.
In the pressure range p?
field, the superconducting phase transition is well de-
fined; bulk superconductivity appears at p?
tiferromagnetic state is rapidly suppressed. A natural
explanation is that the opening of an superconducting
gap on large parts of the Fermi surface leads to prevent
the onset of long range antiferromagnetism. Spectacu-
c.33)The onset of superconductivity does
c< p < pc at zero magnetic
cand the an-
indicating the Fermi surface topology in the different states of the
phase diagram. The boundary between the localized Fermi sur-
face (localized description of the 4f electron) and of the itinerant
paramagnetic phase (itinerant description of the 4f electron) is
indicated. One yet unsolved question is the Fermi surface topol-
ogy in the AF+SC state with the strong interplay between anti-
ferromagnetism and superconductivity. One can speculate that
at H = 0 an itinerant FS persists down to p?
(Color online) (H,p) phase diagram of CeRhIn5at T → 0
larly, the antiferromagnetic order is recovered inside the
superconducting state under application of a magnetic
field.13,14)In difference to p < p?
the antiferromagnetic transition TNis lower than the su-
perconducting transition Tc. The present resistivity mea-
surements at p = 2.4 GPa indicate that the antiferro-
magnetic state is robust up to high magnetic fields (see
Fig. 5) and the field dependence of the antiferromag-
netic transition TN(H) and Tc(H) intersect in one point
(T?,H?). The re-entrant phase occurs for fields H < H?
as in the mixed state antiferromagnetism can be induced
in the vortex core. A description of a homogeneous mixed
superconducting and antiferromagnetic order parameter
can be found in the frame of SO(5) theory.28,29)The
interesting effect is that antiferromagnetism can extend
far from the vortex core. It is predicted that the antifer-
romagnetic signal will increase under magnetic field as
the vortex number will be proportional to H. This re-
sults seems to be in agreement with the data of ref. 13.
However, in our previous experiment the re-entrant sig-
nal disappears below at least 3 T.14)The ac calorimetry
experiments have evidenced clearly that the intersection
point (T?,H?) shifts to higher fields and lower temper-
atures as function of pressure13,14)and it was expected
that the antiferromagnetically ordered phase collapses at
the critical pressure pc≈ 2.5 GPa.
For p > pcindeed, no indication of re-entrance of an-
tiferromagnetism under field is observed. The collapse
of the antiferromagnetic state coincides with the strong
change of the Fermi surface. (Small differences in the ab-
solute value of the critical pressure pchave been reported
in various experiments, see e.g. refs. 13–15,22).
A schematic (H,p) phase diagram for T = 0 is shown
in Fig. 9 indicating the evolution of the Fermi surface
cin this pressure range