Magnetic structure of CeRhIn$_{5}$ under magnetic field
Magnetic structure of CeRhIn5 under magnetic field
S. Raymond, E. Ressouche, G. Knebel, D. Aoki and J.
Flouquet
CEA-Grenoble, DRFMC, SPSMS, 38054 Grenoble Cedex 9, France
Abstract. The magnetically ordered ground state of CeRhIn5 at ambient pressure
and zero magnetic field is an incomensurate helicoidal phase with the propagation
vector k=(1/2, 1/2, 0.298) and the magnetic moment in the basal plane of the
tetragonal structure. We determined by neutron diffraction the two different
magnetically ordered phases of CeRhIn5 evidenced by bulk measurements under
applied magnetic field in its basal plane. The low temperature high magnetic phase
corresponds to a sine-wave structure of the magnetization being commensurate with
k=(1/2, 1/2, 1/4). At high temperature, the phase is incommensurate with k=(1/2,
1/2, 0.298) and a possible small ellipticity. The propagation vector of this phase is the
same as the one of the zero-field structure.
http://arxiv.org/abs/0704.1187v1
Magnetic structure of CeRhIn5 under magnetic field 2
The interplay between magnetism and superconductivity is one of the most
important point of interest in the study of heavy fermion systems [1]. The widely open
question concerns the cooperative versus the antagonist nature of the two ground states.
The generic phase diagram obtained in heavy fermion systems, high-Tc superconductors
and organics is composed of a superconducting region appearing in the vicinity of a
vanishing magnetic phase. Because magnetic fluctuations are expected to be strong in
this region of the phase diagram, they are invoked to be responsible for the Cooper pair
formation [2]. However no definitive symbiosis between experiment and theory firmly
establishes this point in the same manner than the classical scenario of phonon mediated
conventional superconductivity does. In contrast, the SO(5) theory describes the two
phenomena, magnetism and superconductivity, in a unified picture via a superspin order
parameter that encompases both states [3]. This leads to a rich variety of possible phase
diagrams with yet limited experimental examples to test in detail this theory [4]. To
this respect, CeRhIn5 (and the related CeTIn5 compounds with T=Ir, Co [5]) provides a
unique experimental case where the Néel (TN ) and the superconducting transition (Tc)
temperatures are of the same order (of about 1 K) and can be tuned by pressure (p) or
magnetic field (H). The resulting intricated magnetic and superconducting (p, T , H)
phase diagram of CeRhIn5 points toward competition between antiferromagnetism and
superconductivity [6, 7, 8] : At zero magnetic field, there is a pressure range (1.6-1.9
GPa) where magnetism and superconductivity coexist with TN > Tc. From 2 GPa,
where TN ≈ Tc, a pure superconducting phase emerges and antiferromagnetism is
suppressed. The application of a magnetic field in this phase induces a spectacular
reentrance of the long range magnetic order. Beyond NMR experiments [9], microscopic
probes are lacking to address the nature of the magnetically ordered phases in the (p, T ,
H) phase diagram. In this viewpoint, and as a starting point for further investigations
to be performed under pressure, we have investigated the magnetically ordered phases
of CeRhIn5 under magnetic field applied in its basal plane at zero pressure.
CeRhIn5 crystallises in the tetragonal space group P4/mmm [10]. The sample was
obtained by the In self flux method. A rectangular-shaped platelet of width 1 mm normal
to the c-axis was cut from this batch, the other dimensions being 4.3 mm along [1, -1, 0]
and 2.7 mm along [1, 1, 0]. This geometry is aiming to minimize the strong absorption
cross section from In and Rh for the study of the ([1, 1, 0], [0, 0, 1]) scattering plane. The
measurements were performed on the two-axis D23-CEA-CRG (Collaborating Research
Group) thermal-neutron diffractometer equipped with a lifting detector at the Institut
Laue Langevin (ILL), Grenoble. A copper monochromator provides an unpolarized
beam with a wavelength of λ=1.276 Å. The sample was mounted in a vertical field 4He
flow cryomagnet with the [1, -1, 0] axis along the magnetic field.
The (T , H) phase diagram obtained by calorimetry measurements for the field
applied perpendicular to the tetragonal axis is shown in Fig.1. It is composed of three
magnetically ordered phases (two being induced by the magnetic field) consistently
with the data obtained by other goup using calorimetry [11], thermal expansion and
Magnetic structure of CeRhIn5 under magnetic field 3
Figure 1. (T , H) phase diagram of CeRhIn5 determined by specific heat for field
applied in the basal plane at ambient pressure. It shows three different ordered phases.
Open (respect. full) symbols correspond to first (respect. second) order transition.
magnetostriction [12]. In the diffraction experiment, we apply the field along [1, -1, 0]
and refer to this phase diagram by neglecting the in-plane anisotropy. The magnetic
structure at zero field is known to be incommensurate with slightly different propagation
vectors reported in the literature, k=(1/2, 1/2, 0.297) [13] or k=(1/2, 1/2, 0.298) [14].
The helicoidal nature of the order, as opposed to a sine-wave modulated structure, is
known from the distribution of hyperfine field observed in NQR measurements [15].
In the present experiment, the lattice parameters were obtained from the centering
of 18 independent reflections of the crystal and a refinement of the nuclear structure was
performed at 1.9 K with 181 Bragg peaks yielding the structural parameters shown in
Table 1 and the scale factor for calculation the magnetic structure. These parameters are
consistent with the one of the literature [10] as concern the lattice parameters and the
fractional coordinate z. The principal mean square atomic deplacements u have typical
values of such intermetallic compounds. All refinements were corrected from extinction
and absorption with the linear absorption coefficient µ=0.49 mm−1. As far as magnetic
scattering is concerned, the measured neutron Bragg intensity after correction for scale
factor, extinction, absorption and Lorentz factor, is the square of the component of the
magnetic structure factor perpendicular to Q : |FM⊥(Q)|2. In the present case with
only one magnetic Ce atom/unit cell at the origin, the magnetic structure factor is :
FM(Q) = pf(Q).mk.e
−WCe (1)
where p ≈ 0.27×10−12 cm is the scattering amplitude at Q=0 for a single magnetic
moment of 1 µB, f(Q) is the Ce magnetic form factor, WCe is the Debye-Waller factor
of Ce. mk is the Fourier component of the magnetic moment distribution. The magnetic
Magnetic structure of CeRhIn5 under magnetic field 4
Table 1. Structural parameters at T = 1.9 K.
a = 4.638 Å
c = 7.521 Å
z 0.30526 (14)
uCe 0.0014 (5) Å
uRh 0.0006 (4) Å
uIn1 0.0018 (5) Å
uIn2 0.0015 (4) Å
R = 0.0532
structures of interest for the present paper are (i) the collinear sine-wave structure, for
which :
iΦk (2)
and (ii) the non-collinear elliptical structure :
(muuk + im
iΦk (3)
where Ak is the amplitude of the sine-wave, uk and vk are unit vectors, Φk is a phase
factor and mu, mvare the component of the magnetic moment along the unit vectors uk
and vk. The helicoidal order corresponds to the particular case m
u=mv.
The obtained propagation vector for the zero field magnetic structure is found to be
k=(1/2, 1/2, 0.298) in agreement with the literature. The structure was determined by
measuring 16 magnetic peaks and by performing a least square fitting of the helicoidal
model. The comparison between the observed intensities and the calculated ones is
shown in Table 2 with the given weighted least square factor R. A magnetic moment
mI=0.59 (1) µB is found at 1.9 K, a value a little lower than the one found in the
literature 0.75 (2) µB at 1.4 K [13]. Given the rather flat temperature evolution
of the order parameter between 1.4 and 1.9 K [13], the difference in the magnetic
moment determination is not due to the difference in the measurement temperature. We
believe that this difference is related to the data treatment, the present work including
absorption and exctinction corrections.
Figure 2 show Q-scans performed along the c-axis for H = 3 and 5 T (Phase III)
with the same scan performed at H = 0 T as a reference (Phase I). The propagation
vector is now commensurate being (1/2, 1/2, 1/4). For this phase, 7 magnetic reflections
were collected at H = 3 T and T = 1.9 K. The best refinement is obtained for a colinear
sine-wave structure (See Table 3) with the moment perpendicular to the field i.e. along
[1, 1, 0]. Refinement with an helical structure does not work. For completeness, an
elliptic structure was refined and yields, within the error bars, zero component of the
magnetic moment along the field and thus confirms the sine-wave refinement. The
propagation vector Q=(1/2, 1/2, 1/4) corresponds to a particular case of the sine-wave.
For a phase Φk=-π/4 in eq.(2), all the magnetic moments have the same length and the
magnetic structure corresponds to the so-called ++ - - structure consisting in up, up,
Magnetic structure of CeRhIn5 under magnetic field 5
Table 2. Magnetic refinement with an helicoidal structure at zero field in phase I at T
= 1.9 K. The Q vector is the Brillouin zone center +/- the propagation vector k=(1/2,
1/2, 0.298).
Q |FM⊥(Q)|2calc |FM⊥(Q)|2obs
(1, 1, 0) - 1.10 1.17
(0, 0, 0) + 1.10 1.03
(0, 0 ,1) + 1.52 1.50
(-1, -1, 1) + 1.52 1.39
(0, -1, 1) + 1.52 1.43
(1, 1, 1) - 1.30 1.36
(1, 1, 1) + 0.76 0.46
(0, 0, 2) + 1.54 1.50
(1, 1, 2) - 1.57 1.63
(0, 0, 2) - 1.57 1.53
(1, 0, 2) - 1.57 1.47
(2, 2 ,2) - 0.78 0.57
(1, 1, 3) - 1.47 1.62
(0, 0, 4) + 1.08 1.34
(1, 1, 4) - 1.24 1.14
(1, 1, 5) - 0.97 1.94
R = 0.0696
Table 3. Magnetic refinement with a sine-wave structure in phase III for H = 3 T
and T = 1.9 K.
Q |FM⊥(Q)|2calc |FM⊥(Q)|2obs
(0, 0, 1) + 1.08 1.02
(1, 1, 1) - 0.62 0.65
(0, 0, 2) + 1.40 1.42
(1, 1, 2) - 1.32 1.31
(1, 1, 3) - 1.38 1.30
(0, 0, 4) + 1.08 1.00
(1, 1, 4) - 1.19 1.82
R = 0.0934
down, down sequence of magnetic moment when moving along the c-axis. This structure
is favorized at low temperature because it reduces the magnetic entropy. The obtained
magnetic amplitude of the sine-wave at 1.9 K is AIII=0.84 (2) µB. For the peculiar
++ - - structure, the magnetic moment mIII is related to the sine wave amplitude by
mIII=AIII/
2. We thus obtain mIII=0.59 µB, the same value than mI . Note that the
maximum in plane magnetic moment sustended by the doublet ground state is 0.92 µB as
deduced from crystal field spectroscopy [16]. The difference between the paramagnetic
moment of the doublet ground state and the saturated ordered moment is often ascribed
to the Kondo effect in cerium compounds.
Phase II was investigated by performing Q-scans at 3.7 K and 4 T. An example
of such a scan along the c-axis is shown on Fig.3a) for Q=(0.5, 0.5, L) with the same
Magnetic structure of CeRhIn5 under magnetic field 6
Figure 2. Q-scans performed along the c-axis for H= 0, 3 and 5 T at 1.9 K.
Table 4. Magnetic refinement with a sine-wave structure in phase II for H = 4 T and
T = 3.6 K.
Q |FM⊥(Q)|2calc |FM⊥(Q)|2obs
(1, 1, 0)- 0.04 0.07
(0, 0, 1)+ 0.31 0.21
(1, 1, 1)- 0.16 0.19
(1, 1, 2)- 0.36 0.37
R = 0.1902
scan performed at 3.1 K in phase III as a reference. The propagation vector is found
to be the same than the helicoidal phase, i.e. k=(1/2, 1/2, 0.298). Figure 3b) shows
the temperature variation of the magnetic Bragg peak Q=(0.5, 0.5, 1.298) at 4 T. The
difficulty to study this phase is that it exits in a reduced temperature range in the
vicinity of the Néel temperature, where magnetic moment is barely developped. As a
consequence the magnetic signal is weak. Figure 4 shows the field dependence of the
magnetic Bragg peak intensity measured at Q=(1/2, 1/2, 1.298) at 3.6 K. Since the
intensity is constant in both phases, this suggests that the propagation vector does not
change as a function of field. Because of the weak signal, only 4 magnetic reflections were
collected in phase II at 3.6 K and 4 T and the result of a refinement with a sine-wave
structure is given in Table 4. For H=4 T and T=3.6 K, the magnetic amplitude is found
to be AII=0.44 (2) µB. Refinement with an elliptical phase is slightly better (R=0.1449
instead of R=0.1902) and gives a non zero component along the field m[1,−1,0]=0.12 (5)
µB, the component perpendicular to the field being then m[1,1,0]=0.4 µB. We cannot
definitivelly conclude on the elliptical nature of this phase given the weak number of
collected reflections.
In the previous paragraphs, we neglect the possible ferromagnetic component
Magnetic structure of CeRhIn5 under magnetic field 7
Figure 3. a) Q-scans performed along the c-axis for H= 4 T at 3.1 and 3.7 K. b)
Temperature dependence of the Bragg peak intensity at Q=(0.5, 0.5, 1.295) for H =
4 T. Solid lines are guides for the eyes. Dashed line represents the background.
Figure 4. Magnetic field dependence of the Bragg peak intensity at Q=(0.5, 0.5,
1.298) at 3.6 K. The solid line is a guide for the eyes. The dashed line corresponds to
the background.
along the applied field. The corresponding signal was not observed in the present
experiment due to its location on the top of the nuclear peaks. The resulting structure
obtained by combining the sine-wave and the ferromagnetic component is a so-called
fan structure. The fact that mIII and mI are equal within the error bars indicate
that this ferromagnetic component is anyway very weak at least at low temperature.
Magnetization measurements performed at 1.3 K in the basal plane give an induced
ferromagnetic moment of about 0.08 µB at 5 T [17]. The helicoidal nature of the
ordering at zero field is certainly due to the RKKY interactions that allow the conditions
Magnetic structure of CeRhIn5 under magnetic field 8
for stabilizing such a state due to their oscillating nature. We invoke RKKY interactions
rather than Fermi surface nesting because dHvA experiments suggest the localized
nature of the magnetism of CeRhIn5 at ambient pressure [18]. The effect of a magnetic
field applied in the plane of an helix is known from a long time and was worked out
shortly after the discovery of the helix structure [19]. The resulting sinusoidal oscillating
structure or elliptical arrangement depends of the anisotropy and the magnetic field and
the details of the complete (T , H) phase diagram depend on the precise Hamiltonian.
On general ground and at the mean field level, the possible transition from helix to
commensurate structure under field was also predicted in the earlier works for peculiar
values of the propagation vectors [20]. A field induced transition to the antiferromagnetic
state is expected for k ≈ 1/2 and to the ++ - - structure for k ≈ 1/4, the situation
encoutered in the present work.
Despite the proximity of the zero field propagation vector to the one of the ++
- - structure, another commensurate structure is reported at zero field for CeRhIn5
based systems with this time k=(1/2, 1/2, 1/2). This antiferromagnetic order occurs in
CeRh1−xIrxIn5 (x) [21] and in CeRh0.6Co0.4In5 [22]. Interestingly it is reported to coexist
with the incommensurate order and also with the superconducting ground state. On
cooling the incommensurate order appears first followed by the commensurate order
and the superconducting state. On another hand, it is worthwhile to note that the
commensurate order with k=(1/2, 1/2, 1/2) alone is reported for the related CeCoIn5
compound doped with 10 % Cd both in the antiferromagnetic and antiferromagnetic
plus superconducting phases [23]. Contrastingly, the occurence of commensurate order
is not reported in the diffraction studies performed on CeRhIn5 under pressure. However
different groups obtain different results. Either the incommensurate order is reported
to change weakly with pressure up to 1.63 GPa [24] or at opposite, the propagation
vector changes to k=(1/2, 1/2, 0.396) at 0.1 GPa [14]. This confusing situation asks
for new experiments under pressure. The occurence of different commensurate and
incommensurate phases in the (T , H , p, x) phase diagram of CeRhIn5 deserves further
investigation especially for the interplay between magnetic order and superconductivity.
We have determined the two different magnetic ordering states in CeRhIn5 at
ambient pressure under magnetic field applied in its basal plane. The low temperature
phase is characterized by the commensurate propagation vector k=(1/2, 1/2, 1/4) and a
colinear structure with the magnetic moment perpendicular to the field. The saturated
magnetic moment of 0.6 µB is the same as the one found in the zero field phase. The
high temperature phase is incommensurate with the same propagation vector as the
zero field incommensurate helix, k=(1/2, 1/2, 0.298). The structure is colinear at first
approximation with an eventual ellipticity of about 1/3.
Magnetic structure of CeRhIn5 under magnetic field 9
Acknowledgements
We acknowledge M. Zhitomirsky for illuminating discussion concerning helicoidal
structures under applied magnetic field.
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http://arxiv.org/abs/cond-mat/0411359
http://arxiv.org/abs/cond-mat/0703703
