Superconductivity and magnetic order in CeRhIn$_{5}$; spectra of
  coexistence
Superconductivity and magnetic order in CeRhIn5; spectra of coexistence.
J.V. Alvarez and Felix Yndurain
Departamento de F́ısica de la Materia Condensada,
Universidad Autónoma de Madrid, 28049 Madrid, Spain
(Dated: November 14, 2018)
We discuss the fixed-point Hamiltonian and the spectrum of excitations of a quasi-bidimensional
electronic system supporting simultaneously antiferromamagnetic ordering and superconductivity.
The coexistence of these two order parameters in a single phase is possible because the magnetic
order is linked to the formation of a metallic spin density wave, and its order parameter is not
associated to a spectral gap but to an energy shift of the paramagnetic bands. This peculiarity
entails several distinct features in the phase diagram and the spectral properties of the model, which
may have been observed in CeRhIn5. Apart from the coexistence, we find an abrupt suppression
of the spin density wave when the superconducting and magnetic ordering temperatures are equal.
The divergence of the cyclotron mass extracted from de Haas-van Alphen experiments is also
analyzed in the same framework.
PACS numbers: 71.10.Pm,74.50.+r,71.20.Tx
The interplay between magnetism and superconduc-
tivity is a recurrent area of research in condensed matter
physics. This interest has being activated in the last
years due to the experimental findings of their coexis-
tence in materials based on Ce, particularly in the 1-1-
5 CeMIn5 family [1] [2]. A prominent member of this
family is CeRhIn5, which grows in tetragonal form, al-
ternating CeIn3 and RhIn2 planes along the c crystallo-
graphic axis. The structural anisotropy, induces quasi-
bidimensionality in the electronic bonding and the Fermi
surface, as evidenced in a series of de Haas-van Alphen
measurements and band structure calculations [3][4][5].
At ambient pressure CeRhIn5 becomes an antiferromag-
net (AFM) below TN ∼ 3.8K [1], with a small staggered
magnetization aligned in the ab plane. Within the stan-
dard Doniach’s Kondo lattice paradigm, applying pres-
sure in a weak AFM heavy-fermion system opens a route
to very interesting effects. As the pressure increases this
theoretical scenario predicts: i) A reduction of TN due
to Kondo compensation. ii) The eventual suppression of
the AFM order in a quantum critical point (QCP), which
alike other heavy-fermion compounds, would be respon-
sible of the anomalies in the metallic phase. iii) The set-
ting of unconventional superconductivity. However, all
calorimetric [6], NQR [7], transport [8] and susceptibil-
ity [9] measurements provide a consistent picture for the
pressure-temperature phase diagram (presented schemat-
ically in Fig. 1) in conflict with the aforementioned the-
oretical scenario. Surprisingly, TN first increases with
pressure and it only starts to decrease for pressures higher
than 0.7GPa. Superconductivity shows up before TN has
gone down to zero, i.e. AFM and SC coexist. Finally,
AFM disappears abruptly at Pc = 1.9GPa exactly when
TN = TSC , in a first order transition and before a QCP
could have taken place. Despite the lack of a QCP in the
pressure-temperature phase diagram, the metallic phase
still might be understood in the framework of a quan-
0 0.5 1 1.5 2 2.5 3
P(GPa)
T_N_Park
T_SC_Park
T_SC_Chen
T_N_Chen
T_N_Flouquet
T_SC_Flouquet
AFM+SC
FIG. 1: Schematic experimental pressure-temperature phase
diagram of CeRhIn5 at zero-magnetic field adapted from ref-
erences [6], [9] and [11].
tum criticality if, changing another experimental knob,
one could find a QCP nearby in the phase diagram. The
natural choice is using a magnetic field to quench the
superconductivity, and in that way, continue the point
Pc(H = 0) into a line of first order transitions down to
zero temperature, ending with a QCP at H∗c2. A new
surprise appeared on this type of experiments [6, 10].
For pressures higher than Pc, the AFM reenters applying
an in-plane magnetic field Hm < Hc2. Besides, trans-
port measurements suggest [1, 8] that magnetic order in
CeRhIn5 may not be associated to a gap in the single-
particle spectrum. Actually, the resistivity as a function
of the temperature does not show the conventional mini-
mum characteristic of a metal-insulator transition at any
temperature. The small anomaly observed in the resistiv-
ity close to TN seems related to a change on the scattering
mechanism when the AFM sets in.
http://arxiv.org/abs/0704.1082v1
In vivid contrast with quantum critical and quasi-one-
dimensional systems, the understanding of the individual
AFM or SC states rely on simple but accurate mean-field
theories. In this Letter, we propose that a basic compre-
hension of the microscopic coexistence of AFM and SC
and the first-order transition between these two phases
can also be achieved within a mean-field scenario. We dis-
cuss this phenomenon at the microscopic level in terms
of a quasi-bidimensional model of interacting electrons
proposed by one of us [12, 13, 14]. We will enumerate
some implications of that model, discussing to which ex-
tent can be related to the phenomenology observed in
CeRhIn5.
The model Hamiltonian contains the terms that nat-
urally establish superconductivity and antiferromag-
netism.
H = Hk +HV +HU (1)
ε(k)c
kσckσ (2)
(k+q)↑
(−k−q)↓
c−k↓ck↑ (3)
k′−σckσck′−σ (4)
Where U is a Hubbard on-site repulsion, and Vkq the
effective interaction in the Cooper channel, which we will
take to be attractive. Only s-wave pairing will be con-
sidered throughout this work. Superconductivity with
an order parameter having a symmetry related to Vkq
is favored by HV but the presence of a Hubbard repul-
sion establish a competition, which in terms of the gap
is given by ∆ = ∆2 −∆1, such that:
∆2 = V
−k−σ〉 (5)
∆1 = U
−k−σ〉 (6)
where the prime restricts the summation to states with
an energy (measured from the FL) smaller than a cut-off
energy Ec. For those states we assume a very weak mo-
mentum dependence of Vkq . In the absence of magnetic
order and setting a constant density of states at the FL
(i.e. far from a logarithmic divergence) , the competition
between terms favoring and disfavoring the SC is clearly
shown in the McMillan-like formula for the critical tem-
perature.
TSC = 1.13Ec exp
(V − U∗)D(EF )
where U∗ = U
1+UD(EF ) ln(W/Ec)
FIG. 2: Calculated electronic structure of paramagnetic
CeRhIn5. (a) Density of states and (b) band structure. The
arrow indicates a saddle point singularity at the X point of
the Brillouin zone. The inset shows the variation of the sad-
dle point energy along the z R-X-R direction (it crosses the
Fermi Level at approximately the (π/a, 0, 0.6π/c) point).
An essential ingredient of the model is the quasipar-
ticle dispersion relation ε(k) in the paramagnetic phase
taken to set the Fermi level (FL) very close to the sad-
dle point of a 2D system of itinerant electrons. Based
on first principles calculations, Hall et al.[3] find at the
FL a sharp peak in the Density of States characteris-
tic of a 2D van Hove logarithmic singularity. We have
also performed a full first principles calculation of the
electronic structure of tetragonal paramagnetic CeRhIn5
using the VASP code [15]. The generalized gradient ap-
proximation of Perdew et al.[16] for the exchange and
correlation was adopted. The results for the experimen-
tal lattice parameters [17] are reported in Figure 2. In
Figure 2(a) we observe, like Hall et al.[3], a sharp peak
in the density of states near the Fermi level. In addi-
tion, in Figure 2(b) we display the band structure at the
vicinity of the Fermi level. We find a saddle point very
close to the Fermi level whose dispersion along the z-
direction (R-X-R in the standard notation) is about 0.3
eV indicating the two-dimensional character of the sad-
dle point singularity. Under these conditions we have,
besides the Van Hove singularity in the density of states
that favors superconductivity and magnetic order, an im-
portant kinematic restriction, namely ε(k) = ε(k + Q),
where Q = (π/a, π/a) is the vector connecting two equiv-
alent saddle points within the Brillouin zone. The above
restriction determines the gapless nature of the SDW like
in the CDW case proposed by Rice and Scott [18].
The SDW order parameter is given by,
γσ = U
kσck+Q−σ〉 (8)
with γσ = −γ−σ = γ. The resulting quadratic Hamil-
tonian is solved by a Bogoliubov transformation and the
SC and SDW order parameters obtained selfconsistently.
The Hamiltionian eigenvalues are obtained by solving:
εk − Ek −γ↑ −∆ 0
−γ↑ ε(k+Q) − Ek 0 −∆
−∆ 0 ε−k − Ek γ↓
0 −∆ γ↓ ε−(k+Q) − Ek
The four solutions are E1 = −E−, E2 = −E+, E3 =
E−, E4 = E+ where E±(k) =
ε(k)2 +∆2 ± γ. No-
tice that for a 1D or 2D nested system ε(k) = −ε(k+Q)
and therefore E±(k) = ±
ε(k)2 +∆2 + γ2.
The system is solved self-consistently for an effective
half bandwidth at ambient pressure of W0 and the phase
diagram for V = 4W0, Ec = 0.7W0 , n = 0.92 electrons
and U = 2.25W0 , U = 2.50W0 and U = 2.75W0, is
presented in Fig. 3. To simulate the effect of the pres-
sure we have considered a linear variation of the band-
width with the pressure and a bandwidth independent
electron-electron interaction U. These assumptions have
being found to be reasonable with a first principles calcu-
lation using the SIESTA code [19] taking Ni as a bench-
mark. Also, for simplicity, a constant V interaction and
hole concentration independent of pressure are assumed.
The results of our model agree qualitatively with the ex-
perimental findings depicted in Fig. 1and Fig. 3(d). We
indeed find a competition between SDW and SC but, as
seen experimentally, they can coexist in a non negligi-
ble region of the phase diagram. In addition, the SDW
disappears abruptly when the two critical temperatures
became equal, i.e., the SDW transition temperature can
not be lower than the superconducting one. This numeri-
cal result is similar to the analytical finding by Bilbro and
McMillan [20] concerning superconductivity and marten-
sitic transformation in A15 compounds.
The proximity of the FL to a saddle point is an ingredi-
ent of model (1) necessary for the formation of the metal-
lic SDW. We ask ourselves whether in CeRhIn5 there is
experimental evidence for such a proximity. A recent de
Haas-van Alphen study [4] shows a divergence in the cy-
clotron mass at a pressure Pc2 ∼ 2.4GPa, accompanied
with a change in the quasi-2D Fermi surface. Those ex-
periments were performed for values of magnetic fields
and pressures in which the system is AFM. Following
Park et al. Pc2 is very close to the pressure at which TN
would extrapolate to zero in absence of SC (see upper
inset in Fig. 4 ). The cyclotron mass in a 2D elec-
tronic system mc = h̄
2/2π(∂A(E)/∂E) where A is the
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
0 5 10 15 20 25
(a) (b)
(c) (d)
FIG. 3: Phase diagram (temperature versus pressure) ob-
tained using the model Hamiltonian (1) and a two dimensional
band structure with the FL close to a saddle point. The vari-
ables are in units of the half bandwidth W0. TN and TSC
stand for the SDW and SC critical temperatures respectively.
Panels (a), (b) and (c) are the calculated results for the pa-
rameters given in the text and U = 2.25W0 , U = 2.50W0 and
U = 2.75W0 respectively. The inset in panel (c) indicates the
metallic DOS in the three (SDW, SDW+SC and SC) different
regimes. The shaded area indicates occupied levels. Panel (d)
represents the experimental results of ref. [9]
area enclosed in the isoenergetic contour line E(k) = E.
Close to the 2D Van Hove singularity one expects the
cyclotron mass to diverge logarithmically. Actually, the
precise functional form has been computed in Ref. [21]
and found to be
mc = mc0
C +D ln
|EF − Evhs|
In a tetragonal crystal structure C and D are numbers
nearly independent of the pressure and mc0 is the cy-
clotron mass at the bottom of the band. The divergence
is driven by the denominator in the argument of the log-
arithm which in our model is: EF (γ) − Evhs ∼ γ(P ) ∼
TN(P ). In other words, in model (1), mc is enhanced
as the AFM vanishes because the FL in the SDW ap-
proaches the saddle point at X (see the inset in Fig. 4).
To elucidate if the experimental results are compatible
with this argument and with expression (9), we have ex-
tracted the pressure dependence of TN , fitting the ex-
perimental data in Ref ([6]) (see upper inset in Fig. 4
to a cubic polynomial law in (Pc2 − P ) in the range of
pressures between P = 0.65 and P = 2.4 GPa. We do
not attempt to justify physically this fit here, since our
only goal is extracting an analytic expression for TN (P )
for the range of pressures of interest, to insert it in (9).
The results are presented in the main panel of Fig. 4.
Our model reproduces reasonably well these experimen-
tal findings.
Within this model, we expect an anomaly in the spe-
cific heat at TN , which is not associated to a SDW spec-
0 0.5 1 1.5 2 2.5 3
P(Gpa)
0 0.5 1 1.5 2 2.5 3
P(Gpa)
FIG. 4: Cyclotron mass (mc) as a function of pressure. Tri-
angles are the experimental values from Ref. [4] showing the
mc divergence close to P ∼ 2.4 GPa, where the magnetic
order disappears. The solid line is a fit to the theoretical
model. Close to a Van Hove singularity the cyclotron mass
diverges logarithmically as the difference between EF and the
Evhs vanishes (see lower inset). According to the model this
energy difference is proportional to TN and its pressure de-
pendence can be extracted, for instance, from Ref. [6](see
upper inset).
tral gap but to the entropy released when the magnetic
order disappears. The electronic part of the specific heat
Ce−V = −2β
Ei(k)
∂Ei(k)
× (10)
Ei(k) +
ε(k)2 +∆2
where i=3,4. The second term inside the parenthesis
gives the SC anomaly at TC and the third term gives
the antiferromagnetic anomaly at TN . The SC anomaly
is much weaker in the coexisting phase, because the FL
lies in a depression of the density of states (see central
graph on the panel (c) of Fig. 3) created by the under-
lying SDW, while in the purely SC phase the FL is very
close to a divergence in the density of states (right graph).
Remarkably, this enhancement has been also observed in
calorimetric measurements on CeRhIn5 [6]
To summarize: beyond the detailed boundary shape
in the phase diagram of CeRhIn5 we have identified two
unequivocal, clear-cut features of the phenomenology of
CeRhIn5 in the discussed model, namely; coexistence and
abrupt disappearance of AFM when TN = TSC . The es-
sential ingredient in the model is the metallic SDW, fa-
vored for the proximity of Fermi level to a Van Hove
logarithmic singularity in the density of states. The
gapless nature of the SDW implies the lack of a metal-
insulator transition at, or close to, TN as shown by resis-
tivity measurements. The kinematic conditions needed
for the metallic SDW to be formed seem to be present in
CeRhIn5 as shown by the logarithmic divergence of the
cyclotron mass.
We appreciate very much discussions with G. Gomez-
Santos, H. Suderow and S. Vieira. Financial support of
the Spanish Ministry of Science ( Ramon y Cajal contract
and Grant BFM2003-03372) is acknowledged.
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http://arxiv.org/abs/cond-mat/0512078
