Novel coexistence of superconductivity with two distinct magnetic orders.

Abstract

The heavy fermion system exhibits properties that range from an incommensurate antiferromagnet for small to an exotic superconductor on the Ir-rich end of the phase diagram. At intermediate where antiferromagnetism coexists with superconductivity, two types of magnetic order are observed: the incommensurate one of and a new, commensurate antiferromagnetism that orders separately. The coexistence of -electron superconductivity with two distinct -electron magnetic orders is unique among unconventional superconductors, adding a new variety to the usual coexistence found in magnetic superconductors.

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arXiv:cond-mat/0509780v1 [cond-mat.supr-con] 29 Sep 2005
Novel Coexistence of Superconductivity with Two Distinct Magnetic Orders
A.D. Christianson,1,2, ∗A. Llobet,1Wei Bao,1J.S. Gardner,3,4, †I.P. Swainson,3J.W. Lynn,4J.-M.
Mignot,5K. Prokes,6P.G. Pagliuso,1, ‡N.O. Moreno,1J.L. Sarrao,1J.D. Thompson,1and A.H. Lacerda1
1Los Alamos National Laboratory, Los Alamos NM 87545
2Colorado State University, Fort Collins CO 80523
3NRC Canada, NPMR, Chalk River Laboratories, Chalk River, Ontario KOJ IJO, Canada
4NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899
5Laboratoire L´ eon Brillouin, CEA-CNRS, CEA/Saclay, 91191 Gif-sur-Yvette, France
6BENSC, Hahn-Meitner-Institut, Glienickerstrass 100, D-14109, Berlin, Germany
(Dated: February 2, 2008)
The heavy fermion CeRh1−xIrxIn5system exhibits properties that range from an incommensurate
antiferromagnet for small x to an exotic superconductor on the Ir-rich end of the phase diagram.
At intermediate x where antiferromagnetism coexists with superconductivity, two types of magnetic
order are observed: the incommensurate one of CeRhIn5 and a new, commensurate antiferromag-
netism that orders separately. The coexistence of f-electron superconductivity with two distinct
f-electron magnetic orders is unique among unconventional superconductors, adding a new variety
to the usual coexistence found in magnetic superconductors.
Magnetism and superconductivity are two major co-
operative phenomena in condensed matter, and the re-
lationship between them has been studied extensively.
Conventional superconductivity in phonon mediated s-
wave materials is susceptible to the Cooper-pair break-
ing by magnetic scattering[1, 2]. In cases where super-
conductivity and magnetic order coexist, such as rare-
earth based molybdenum chalcogenides, rhodium borides
and borocarbides, the superconducting d electrons and
localized f electrons are weakly coupled, and the com-
petition between superconductivity and magnetic order
is understood[2, 3] in terms of the theory of Abrikosov
and Gorkov[1].In contrast, strong magnetic fluctua-
tions have been observed in heavy fermion, cuprate and
ruthenate superconductors[4, 5, 6, 7], and magnetic ex-
citations have been proposed to mediate the Cooper
pairing in these unconventional superconductors[8, 9,
10, 11, 12, 13].Some of the U-based heavy fermion
compounds form an interesting subset of these super-
conductors where superconductivity develops out of a
magnetically ordered state.
magnetic state is a commensurate antiferromagnet; for
UNi2Al3[16, 17] an incommensurate antiferromagnet; for
UGe2[18] and URhGe[19] a ferromagnet. For UPt3[20],
three distinct superconducting phases[21, 22, 23] coex-
ist with a commensurate short-range antiferromagnetic
order. In analog to superfluid3He, intriguing coupling
between the magnetic and superconducting order param-
eters has been proposed[8].
For UPd2Al3[14, 15] the
Recently a new family of Ce-based heavy fermion ma-
terials has been discovered, which sets a record supercon-
ducting transition temperature of TC=2.3 K for heavy
fermion materials[24, 25, 26]. Coexistence of magnetic
order and superconductivity is observed in a wide com-
position range in CeRh1−xIrxIn5[27] [see Fig. 1(a)]; previ-
ously superconductivity was found only in heavy fermion
materials of the highest purity[18]. At one end of the
CeRh1−xIrxIn5series, CeIrIn5is a superconductor below
TC=0.4 K[25]. At the other end, CeRhIn5 orders mag-
netically below TNi=3.8 K in an antiferromagnetic spi-
ral structure with an ordering wave vector qi=(1
δ=0.297[28], that is incommensurate with the tetragonal
crystal lattice[29]. In the overlapping region of supercon-
ducting and antiferromagnetic phases, 0.25<
find unexpectedly an additional commensurate antiferro-
magnetic order that develops separately below TNc=2.7
K. This makes Ce(Rh,Ir)In5 unique among unconven-
tional superconductors in that the ground state exhibits
the coexistence of the superconducting order parameter
with two distinct magnetic order parameters.
2,1
2,±δ),
∼x<
∼0.6, we
Single crystals of CeRh1−xIrxIn5were grown from an
In flux with appropriate ratio of Rh and Ir starting
materials[27].Lattice parameters follow Vegard’s law
and the sample composition x is estimated to be within
±0.05 of the nominal composition[27]. Samples used in
this work were cut to a thickness ∼1.3 mm to minimize
problems due to the high neutron absorption of Rh, Ir
and In. The search for and collection of magnetic Bragg
neutron diffraction peaks in CeRh1−xIrxIn5(x =0.1, 0.2,
0.25, 0.3, 0.35, 0.4, and 0.5) were performed at the ther-
mal triple-axis spectrometers C5 and N5 of Chalk River
Laboratories, and BT2 and BT7 of NIST. Neutrons with
incident energy Ei=35 meV were selected using the (113)
reflection of a Ge crystal, the (002) of a Be crystal, or the
(002) of a pyrolytic graphite monochromator. At this en-
ergy, the neutron penetration length is longer than the
sample thickness. Polarized neutron diffraction exper-
iments were performed at BT2 of NIST with Ei=14.7
meV on a x=0.3 sample to verify magnetic signals and
to determine magnetic moment orientation. Pyrolytic
graphite filters were employed when appropriate to re-
move higher order neutrons. The sample temperature
was controlled by a top-loading pumped4He cryostat at
both Chalk River and NIST. Additional measurements of

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FIG. 1:
CeRh1−xIrxIn5. There are three distinct long-range orders:
superconducting (SC), incommensurate antiferromagnetic (I),
and commensurate antiferromagnetic (C). The diamonds rep-
resent the N´ eel temperature for the I phase, squares for the C
phase, and circles for TC of the SC phase. The open symbols
are from previous work[27], and the closed symbols from this
work. (b) Diamonds and squares represent squared magnetic
moments at 1.9 K for the I and C phases, respectively. The
circles represent the square of the total magnetic moment per
Ce in Eq. (4). The solid lines are guides to the eye. The
dashed lines delimit the I, I+C+SC and SC phases. Mag-
netic structures of the I and C phases are shown as insets to
(a) and (b), respectively.
(a) Temperature-composition phase diagram for
magnetic order parameters were performed at the triple-
axis spectrometer 4F2 of LLB-Saclay using a pumped
4He cryostat, and at the E4 diffractometer of BENSC
using a dilution refrigerator.
In a previous heat capacity study[27], a phase tran-
sition was observed in the range of 3.8 to 2.7 K for
0 ≤ x<
∼0.6, and was attributed to an antiferromagnetic
transition as in CeRhIn5[24] [open diamonds in Fig. 1(a)].
This is confirmed by the results of this neutron diffrac-
tion work which show that all samples (0.1 ≤ x ≤ 0.5)
have magnetic Bragg peaks characterized by the same in-
commensurate wave vector as in CeRhIn5[28]. The N´ eel
temperature, TNi, determined from the temperature vari-
ation of magnetic Bragg peaks is shown as filled diamonds
in Fig. 1(a).
Thermodynamic and transport measurements uncover
a superconducting state below 1 K in a wide composi-
tion range, 0.25<
∼x ≤ 1[27], [open circles in Fig. 1(a)].
In 0.25<
∼x<
∼0.6 where the superconducting and in-
commensurate antiferromagnetic phases coexist, we find
a second magnetic order below 2.7 K with a commensu-
rate antiferromagnetic ordering wave vector qc=(1
2,1
2,1
2)
TABLE I: Magnetic Bragg intensity, σobs, defined in Eq. (1),
observed at 1.9 K in units of 10−3barns per CeRh0.7Ir0.3In5.
The theoretical intensity, σcal, is calculated using Eq. (2) and
(3) for the incommensurate moment Mi = 0.73µB/Ce and
the commensurate moment Mc = 0.27µB/Ce, respectively.
q
σobs
σcal
q
σobs
σcal
(1
2
1
2δ )
21+δ )
1
22+δ )
1
23+δ )
1
24+δ )
1
25+δ )
(3
2
(3
2
(3
2
(3
2
(1
22
(1
22
(3
22
9.0(5)
9.6(3)
9.0(3)
5.7(3)
4.3(3)
2.9(3)
4.3(3)
5.4(15)
3.4(3)
2.5(4)
2.3(4)
0.8(2)
1.2(2)
9.3
12.1
10.8
7.7
4.9
2.7
4.2
4.2
3.9
3.1
3.2
1.1
1.1
(1
(1
(1
(1
(1
(1
(3
(3
(3
2
1
21-δ )
1
22-δ )
1
23-δ )
1
24-δ )
1
25-δ )
1
26-δ )
3
21-δ )
3
22-δ )
3
23-δ )
1
2
1
2
3
2
3
2
10.1(3)
9.9(3)
8.2(4)
5.7(3)
3.9(3)
2.7(5)
3.8(4)
5.0(4)
2.9(3)
2.3(2)
1.9(7)
1.3(2)
0.7(2)
10.8
12.0
9.6
6.5
3.9
2.1
4.2
4.1
3.6
2.6
2.7
1.1
1.0
(1
(1
(1
(1
(1
2
1
2
22
22
22
22
3
2δ )
21+δ )
3
22+δ )
3
23+δ )
1
2
3
2
2
(1
(1
(3
(3
2
1
2)
5
2)
1
2)
5
2)
3
2)
9
2)
3
2)
2
1
2
3
2
[filled squares in Fig. 1].
Table I lists integrated intensities of magnetic Bragg
peaks at 1.9 K for x = 0.3 of both commensurate and
incommensurate types, collected by rocking scans in the
two-axis mode and normalized to structural Bragg peaks
(002), (003), (004), (006), (111), (112), (113), (220),
(221) and (222) to yield magnetic neutron diffraction
cross-sections in absolute units, σ(q) = I(q)sin(2θ),
where 2θ is the scattering angle. In such units, the mag-
netic cross section is[30]
σ(q) =
?γr0
2
?2
M2
i,c|f(q)|2?
µ,ν
(δµν−? qµ? qν)F∗
µ(q)Fν(q),
(1)
where (γr0/2)2= 0.07265 barns/µ2
gered moment of the Ce ion in either the incommensu-
rate or commensurate antiferromagnetic structure, f(q)
the magnetic form factor which could be different for the
two type antiferromagnetic orders, ? q the unit vector of q,
and Fµ(q) the µth Cartesian component of the magnetic
structure factor per molecular formula.
The incommensurate magnetic structure for CeRhIn5
has been determined [inset in Fig. 1(a)] and the magnetic
cross section, Eq. (1), is reduced to[28]
B, Mi,c is the stag-
σi(q) =1
4
?γr0
2
?2
M2
i|f(q)|2?1 + |? q ·? c|2?,(2)
where ? c is the unit vector of the c-axis.
For the commensurate magnetic component, the sum-
mation in Eq. (1) is reduced to (1 − |? q ·?
where?
and |F(q)|2= 1. Polarized neutron diffraction measure-
ments at (1
2,δ), with a horizontal and
Mc|2)|F(q)|2,
Mc is the unit vector of the magnetic moment,
2,1
2,1
2) and (1
2,1

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3
FIG. 2:
surate (diamonds) magnetic Bragg peaks follow the magnetic
form factor of Ce3+(4f1) ion.
Both the commensurate (squares) and incommen-
vertical magnetic guide field respectively, indicate that
the magnetic moment in the commensurate antiferromag-
netic order, like in the incommensurate order[28], lies in
the tetragonal basal plane. The commensurate magnetic
structure is depicted in the inset in Fig. 1(b). There are
in general 8 symmetry-related?
and the easy axis within the basal plane cannot be de-
termined in a multidomain sample due to the tetragonal
symmetry. Assuming equal populations among these do-
mains during our unpolarized neutron diffraction exper-
iment, the magnetic polarization factor averages to[31]
?1 − |? q ·?
netic cross section for the commensurate antiferromag-
netic order is
?γr0
2
Mcdomains in the sample
Mc|2? =
?1 + |? q ·? c|2?/2. Therefore, the mag-
σc(q) =1
2
?2
M2
c|f(q)|2?1 + |? q · ? c|2?.(3)
Applying Eq. (2) and (3) to the data in Table I, a least-
square fit yields staggered magnetic moments Mi=0.73±
0.25µB and Mc=0.27 ± 0.10µB per CeRh0.7Ir0.3In5 at
1.9 K.
Measured magnetic intensities for CeRh0.7Ir0.3In5 in
Table I are plotted in Fig. 2 as a function of |q|
in scaled quantities 4σi(q)/?(γr0/2)2M2
and 2σc(q)/?(γr0/2)2M2
squared magnetic form factor, |f(q)|2, [Eq. (2) and (3)].
The solid line is |f(q)|2for Ce3+ion[32]. The fact that
the measured data for both commensurate (squares) and
incommensurate (diamonds) orders are well described by
the solid curve indicates not only the correctness of our
magnetic models in Eq. (2) and (3) for CeRh0.7Ir0.3In5,
but also that the Ce3+(4f1) ions are responsible for
both commensurate and incommensurate antiferromag-
netic orders. This is consistent with the thermodynamic
study[27] where entropy reaches the same value at ∼6
K for all samples from x=0 to x=1, suggesting that the
same f electrons are responsible for heavy fermion for-
mation, antiferromagnetic orders, and superconductivity
in CeRh1−xIrxIn5. Division of the f spectral weight was
i(1 + |? q · ? c|2)?
c(1 + |? q ·? c|2)?, which equal the
FIG. 3: Squared magnetic order parameters of the incom-
mensurate (diamonds) and commensurate (squares) antifer-
romagnetic transitions. The dotted line indicates the critical
temperature TC ≈ 0.5 K of the superconducting transition.
The solid lines are guides to the eye.
probed using neutron scattering in CeRhIn5[33] and was
further investigated in terms of a two-fluid model for this
family of heavy fermion materials[34, 35, 36].
Magnetic moments for other compositions were mea-
sured in a similar fashion, and are shown as squares and
diamonds for commensurate and incommensurate struc-
tures, respectively, in Fig. 1(b).
squared total magnetic moments per Ce,
The average of the
M2≡ ?(Mi+ Mc)2? = M2
i+ M2
c,(4)
is also shown in Fig. 1(b) as circles. A large staggered
magnetic moment of 0.85µB/U[15] coexisting with super-
conductivity in UPd2Al3has been considered a puzzling
anomaly[8], while UPt3[20] has a tiny moment of 0.02
µB/U, and the moment in UNi2Al3, 0.24 µB/U[17], is
also quite small. Here in the coexistence composition re-
gion of Ce(Rh,Ir)In5, M ranges from 0.27 to 0.78 µB/Ce,
bridging the gap between UNi2Al3and UPd2Al3.
In Fig. 3, the temperature dependences of the Bragg
peak intensities of (1
sented as the squared order parameters M2
of the incommensurate and commensurate antiferromag-
netic transitions, respectively. Data below 1.2 K were
taken at BENSC using a dilution refrigerator. There is no
thermal hysteresis for either magnetic phase transition.
When the commensurate phase transition occurs at 2.7
K, there is no abrupt change in the incommensurate order
parameter. Similarly, at the superconducting transition
TC ≈ 0.5 K (dotted line), no apparent anomaly occurs
in either magnetic order parameter. This is analogous
to the magnetic order parameter for the heavy fermion
superconductors UPd2Al3[37] and UPt3in the A and C
phases, but different from that for UPt3in the supercon-
ducting B phase[8, 20].
Note in Fig. 1(a) that the N´ eel temperatures TNiand
TNc both are nearly constant when they are non-zero.
2,1
2,1−δ) and (1
2,1
2,1
2) are pre-
i and M2
c

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4
This might suggest a phase separation scenario where
only a volume fraction of M2(x)/M2(0) is magnetically
ordered and the remaining is superconducting. However,
at x = 0.5, this would imply M2(x)/M2(0) = 0.08, which
is not consistent with a magnetic volume fraction of at
least 0.85 determined from a µSR study[38].
At the quantum critical point (QCP) of the supercon-
ducting phase near x = 0.25, the commensurate anti-
ferromagnetic order appears with Mcjumping from zero
[Fig. 1(b)]. This suggests a change in electronic struc-
ture at x ≈ 0.25 which produces a strong enough peak
at q = (1
2) in the RKKY interaction to induce the
new magnetic order. Several sheets of the Fermi sur-
face of differently enhanced masses have been observed
in the de Haas-van Alphen measurements of CeRhIn5
and CeIrIn5[39, 40] and the Fermi surface topology is
determined mostly by non-f electrons[41, 42]. The con-
currence of the superconductivity and commensurate an-
tiferromagnetic order observed here, therefore, suggests
that the responsible Fermi surface sheet for the supercon-
ductivity may be the one close to nesting at q = (1
Both Mi and Mc approach zero at a second QCP of
the antiferromagnetism near x ≈ 0.6, while the N´ eel
temperatures are insensitive to x (Fig. 1).
havior is similar to UPt3 under pressure[43], suggest-
ing that both belong to a distinct type of QCP. Recall
that TN ∝ JRKKYM2, where both the RKKY interac-
tion JRKKY and the saturated magnetic moment M are
controlled by the Kondo interaction. In both of these
systems the divergence of JRKKY and the reduction of
M appear to obey JRKKY ∝ M−2as this QCP is ap-
proached. Theoretical investigation of the physical pro-
cess maintaining the delicate proportional relation near
this type of QCP is warranted.
Insummary,theantiferromagnetic
Ce(Rh,Ir)In5 below ∼3.8 K is characterized by the
incommensurate antiferromagnetic spiral of wave vector
(1
2,±δ). In the coexistence composition region of su-
perconductivity and antiferromagnetism, an additional
phase transition to a commensurate antiferromagnet
characterized by (1
2) is discovered below 2.7 K.
The same f electron at each site, hybridizing with
other conduction electrons, is responsible for both the
superconductivity and the commensurate and incom-
mensurate antiferromagnetic orders.
the energy band(s) with Fermi surface nesting near the
(1
2) is responsible for the heavy fermion super-
conductivity.The novel coexistence of three different
types of cooperative ordered states adds a new variety
to the rich phenomena relating to the interplay between
magnetism and superconductivity.
We would like to thank G.D. Morris, R.H. Heffner,
C.M. Varma, Q. Si, S.A. Trugman, D. Pines, Ar. Abanov,
Y. Bang, P. Dai and S. Kern for useful discussions; A.
Cull for help at CRL, and P. Smeibidl and S. Gerischer
2,1
2,1
2,1
2,1
2).
This be-
phaseof
2,1
2,1
2,1
It is likely that
2,1
2,1
at HMI. Work at Los Alamos was performed under the
auspices of the US Department of Energy.
∗This work was part of ADC’s Ph.D. research project,
supervised by WB and AHL at LANL.
†Current address: Physics Department, Brookhaven Na-
tional Laboratory, Upton, New York 11973
‡Current address: Instituto de F´isica “Gleb Wathagin”,
UNICAMP, 13083-970, Campinas, Brasil
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