Introduction

Understanding the interactions between superconductivity and magnetism in heavy fermion systems is one of the greatest challenges of condensed matter physics, often presented as a key to unveil the mechanism of unconventional superconductivity. In the generic phase diagram of these systems (see typically Fig. 1), superconductivity arises in proximity to a second order magnetic phase transition, possibly leading to a quantum critical point (QCP)1. The identification and the understanding of QCPs play a major role in the search of the pairing mechanism. A softening of an excitation mode (phonon, magnon, …) occurs at a QCP. Shall this mode be related to the pairing interaction, a maximum in the superconducting (SC) coupling would be observed2. In any case, it appears rather probable that the same mechanism causes the suppression of the magnetic state and the rise of superconductivity.

Figure 1
figure 1

Phase diagram of CeCo(In1−x1Cdx)5

with the equivalent “negative” chemical pressures adapted from Ref. 8 (empty blue and orange triangles) and positive hydrostatic pressures from Ref. 24 (empty orange circles). The red star represents the position of the putative QCP4. The black dotted lines indicate extrapolations of the AFM state as expected in absence of superconductivity. In one case the QCP is assumed to be driven by the AFM transition, while in the second case AFM is only found upon Cd doping. The full symbols indicate Tc and TN determined from specific heat measurements, for the samples presented in this study. Equivalent negative pressures were obtained by comparison of these transitions with literature values.

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The tetragonal system CeCoIn5 has the highest SC transition temperature (Tc ≃ 2.3 K) among the cerium based heavy fermion family3 and a pressure induced QCP at pc ≃ 0.4 GPa (red star in Fig. 1) as determined by SC parameters4 (maximum SC coupling and pair breaking). Extrapolation of pc from the normal state is more difficult and has led to a value in the range: −0.5 GPa to 1.3 GPa5,6,7. The antiferromagnetism, often believed to be at the origin of the QCP, can be revealed by different doping: (i) Cd or Zn at the In site8,9, (ii) Rh at the Co site10 or (iii) Nd at the Ce site11. Antiferromagnetism is also found by applying a large magnetic field in the (a,b)-plane12,13 (≃10–11.5 T). The antiferromagnetic (AFM) structure was fully resolved in the case of CeRhIn5 as incommensurate with in-plane moments mainly localized on the cerium site forming a spiral structure in the c-axis crystallographic direction14,15. In CeCoIn5 neutron diffraction experiments report a commensurate order when the AFM phase is reached upon Cd or Rh doping16,17 and an incommensurate one in an applied magnetic field12 or upon Nd doping11. The cadmium doped samples have the same phase diagram compared to the pure compound, beside a shift in the pressure scale. They are therefore presented as “equivalent” to negative hydrostatic pressure8. The “equivalence” can be understood in terms of a larger ionic or covalent radius for cadmium compared to indium. Cadmium atoms also contain an electron less than indium atoms and are therefore expected to increase the carrier concentration in the hole character bands of the Fermi surface of CeCoIn5. Similarly, hydrostatic pressure strongly modifies the electronic structure in CeCoIn5 as shown by the increase of the number density in the superconducting phase18 and the decrease of the electronic specific-heat coefficient γ with pressure19. These variations can be understood as a transfer of charge carriers between the different electronic bands under hydrostatic pressure. The two band hole sheets are the ones of strong 4f angular momentum character believed to play an important role in CeCoIn5, both for superconductivity and antiferromagnetism20.

The suppression of the AFM phase potentially leading to the QCP and superconductivity in CeCoIn5 is usually understood in term of “delocalization” of the Ce 4f electrons. In heavy fermions systems, at least three different mechanisms can “delocalize” the Ce 4f electrons: (i) the Kondo effect21, (ii) an increase in cerium valence22 and (iii) the formation/broadening of an electronic band of partial Ce 4f character (Mott delocalization)23. Each of these effects can independently introduce the Ce 4f electrons into the Fermi surface. Note that Ce 4f electrons belonging to the Fermi surface (referred to as itinerant) via one of the previous effects can still be “delocalized” by the two other mechanisms. A variation of the Ruderman-Kittel-Kasuya-Yoshida (RKKY) coupling is unlikely to be the cause of the suppression of the AFM phase as no ferromagnetism was reported under higher pressures24.

An example of a local—itinerant transition takes place between the parent AFM system CeRhIn5 in which most of the Ce 4f electrons do not participate to the Fermi surface and CeCoIn5 in which most of the Ce 4f electrons participate to the Fermi surface. Such transition can be observed experimentally via a modification of de-Haas-van-Alphen frequencies between CeCoIn525 and CeRhIn526. A comparison to the equivalent lanthanum system, which has no 4f electrons, leads to the same conclusion: The volume of the Fermi surface is modified upon lanthanum doping in Ce1−zLazCoIn5 and stays constant in Ce1−zLazRhIn527. The AFM transition in this family is however not directly related to the local—itinerant transition. In the intermediate system CeRh1−yCoyIn5 a reconstruction of the Fermi surface occurs at y = 0.4, associated with partial inclusion of the Ce 4f electrons into the Fermi surface for y > 0.4, while antiferromagnetism persists up to y ≃ 0.810. Similarly, in CeCo(In1−xCdx)5 the variation of the Fermi volume with x is weak and comparable to the case of LaCo(In1−xCdx)5 at least up to x = 0.075 which is already an AFM system28. The suppression of antiferromagnetism must therefore be ascribed to a “delocalization” of itinerant Ce 4f electrons.

The isostructural α—γ phase transition of pure cerium shows strong analogies to the AFM—paramagnetic transition of heavy fermion systems29. The large volume change of the unit cell at the α—γ phase transition is also attributed to one of the “delocalization” mechanisms discussed above30. The various proposed models are: (i) valence variation of ≈0.7 (promotional model)31, (ii) Kondo volume collapse model32 or (iii) formation of a band of partial Ce 4f angular momentum character (Mott transition)33.

The question addressed through the experiments of this study is the origin of the suppression of the magnetic order in CeCo(In1−xCdx)5, that is, identifying the “delocalization” mechanism and its relation to superconductivity. In CeCo(In1−xCdx)5, the AFM phase was probed by muon spin rotation (μSR) and neutron diffraction. The μSR technique also established the coexistence of antiferromagnetism with superconductivity, while the valence of cerium was determined by x-ray absorption spectroscopy (XAS). A comparison to metallic cerium allows resolving the nature of the transition. The combination of these three different experimental probes allows a deep understanding of the ground state of this system that cannot be achieved with a single technique.

Results

Crystals with negative hydrostatic pressure

Following the work of Pham et al. (Ref. 8) single crystals of composition CeCo(In1−xCdx)5 can be viewed as negative pressure of CeCoIn5. The actual doping is x < 0.03. See Supplementary Note S.1, Supplementary Fig S.1 and Supplementary Table S.1 for a detailed characterization. The SC and AFM transition temperatures for samples with various cadmium concentrations were used to determine the position of the samples into the phase diagram and their corresponding negative pressure (Fig. 1). Samples of a corresponding negative pressure of p = −0.7 GPa and p = −1.45 GPa were used for the μSR experiment while for the XAS experiment the corresponding pressure was p = 0 GPa (pure system) and p = −1.45 GPa. The neutron diffraction scans were recorded for the sample with p = −0.7 GPa. The samples p = −0.7 GPa and p = −1.45 GPa exhibit a Néel temperature of TN = 2.06(2) K and TN = 3.04(2) K, respectively, while superconductivity is found below Tc = 1.95(2) K and Tc = 0.52(2) K, respectively.

Magnetic order coexisting with superconductivity

The search for magnetic Bragg peaks was conducted using the neutron diffraction method. The neutron diffraction intensity was recorded in the direction corresponding to the lines (H, H, 0.5) for 0.4 ≤ H ≤ 1 and (0.5, 0.5, L) for 0.4 ≤ L ≤ 1 of the reciprocal space, below (1.45 K and 2.01 K) and above (4.98 K) TN. All the known propagation vectors in magnetically ordered compounds related to CeCoIn5 are located on these lines. A single additional Bragg peak at Q = (1/2, 1/2, 1/2) was identified below TN, corresponding to a commensurate propagation vector k = (1/2, 1/2, 1/2) (Fig. 2a,b). Since the position of the propagation vector k in the reciprocal space does not change with temperature, the temperature dependence of the neutron diffraction intensity at the center of the Bragg peak Q was recorded (Fig. 2c). Its variation is proportional to the square of the ordered magnetic moment. The intensity of the Bragg peak at Q = (1/2, 1/2, 1/2) vanishes above the Néel temperature of TN ≈ 2.1 K. This observation allows to relate the additional Bragg peak to the AFM order. The magnetic peaks are resolution limited at low temperatures (T = 1.45 K), the broadening of the peaks at T = 2.01 K allows to estimate in-plane and out-of-plane correlation lengths to ξa (2.01 K) ≈ ξc (2.01 K) ≈ 140 Å. Given the proximity to TN, these are large values for ξa and ξc. The amplitude of the ordered magnetic moment can be estimated to be m(p = −0.7 GPa) ≈ 0.4 μB, which is consistent with the estimated moments of m(p = −1.1 GPa) ≈ m(p = −1.7 GPa) ≈ 0.7 μB previously reported for higher cadmium concentrations16,34.

Figure 2
figure 2

Scans in the reciprocal space of the neutron diffraction intensity of CeCo(In1−xCdx)5

, performed along the lines Q = (H, H, 1/2) in (a) and Q = (1/2, 1/2, L) in (b). Reciprocal lattice units (r.l.u.) are used as coordinates of the reciprocal space. The CeCo(In1−xCdx)5 sample has a doping x corresponding to a negative pressure of p = −0.7 GPa. (c) Temperature dependence of the neutron diffraction intensity measured at Q = (1/2, 1/2, 1/2).

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Temperature dependent μSR experiments were performed in zero applied external magnetic field (μ0H = 0) and in an applied transverse field of μ0H = 50–54 mT. At 1.6 K, in zero applied magnetic field, two different configurations of the initial muon spin relative to the sample crystallographic c-axis, sketched in Fig. 3a,b, were investigated. From the zero field μSR asymmetry signal, the amplitude and the orientation of the magnetic field at the muon stopping site as well as the magnetic volume fraction were determined. The parameters magnetic fraction and moment orientation can be disentangled by the use of two different initial muon spin orientations, as indicated in eq. (3). At low temperatures, all the muons probe a local magnetic field at the muon stopping site. Therefore, the magnetic volume fraction reaches ≃100% at low temperatures in both samples, in agreement with the nuclear magnetic resonance (NMR) results34. For temperatures close to TN in sample p = −0.7 GPa, probably due to the intrinsic dopant distribution, the magnetic fraction is reduced and phase separation between paramagnetism and antiferromagnetism is observed (see Supplementary Note S.3 and Supplementary Fig. S.2 for details). The internal magnetic field at the muon stopping site is reduced by pressure, from Bm(−1.45 GPa) = 11.2(1) mT to Bm(−0.7 GPa) = 6.5(2) mT. It also rotates toward the axis under chemical pressure. Similarly, both a reduction and rotation of the cerium magnetic moment were observed in the parent system CeRhIn5 by nuclear quadrupole resonance under hydrostatic pressure35.

Figure 3
figure 3

μSR zero field asymmetry spectra of CeCo(In1−xCdx)5.

Zero applied magnetic field μSR asymmetry spectra have been recorded with the initial muon spin direction () oriented parallel (a) and perpendicular (b,c) to the crystallographic c-axis. In panel (a) the signals above and below TN are presented for CeCo(In1−xCdx)5 with a doping inducing an equivalent chemical pressure of p = −0.7 GPa (black) and p = −1.45 GPa (red). The oscillation in the asymmetry (A(t)) for T = 1.6 K indicates the presence of a long range magnetic order. In (b) and (c) the asymmetry corresponding to the two sets of detectors (crosses for up u and down d detectors relative to the initial muon momentum, full symbols for backward b and forward f) is presented for the two samples of CeCo(In1−xCdx)5 composition and equivalent pressure p = −0.7 GPa (b) and p = −1.45 GPa (c).

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The Fourier transform of the μSR asymmetry (Fig. 4) yields the magnetic field distribution at the muon stopping site p(Bμ). For commensurate magnetism, we would expect to observe a single magnetic field, or a discrete set of internal magnetic fields if there are several muon stopping sites. Here in contrary, independently of the chemical pressure, we observed a continuous distribution of internal magnetic field. The zero field μSR spectra are best described by a Bessel type depolarization function (blue lines in Fig. 3), which corresponds for p(Bμ) to a broadened distribution of the form (blue lines in Fig. 4):

Figure 4
figure 4

Internal magnetic field distribution p(Bμ) at the muon stopping site in CeCo(In1−xCdx)5

for a doping inducing an equivalent chemical pressure of p = −0.7 GPa (black) and p = −1.45 GPa (red). p(Bμ) is obtained by Fourier transformation of the magnetic contribution of the zero field μSR asymmetry spectra (A(t)) at 1.6 K of Fig. 3(a).

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In CeCo(In1−xCdx)5 a magnetic phase consisting of AFM droplets was proposed from NMR experiments34,36,37. Droplets of fixed magnetic moment would form around cadmium impurity atoms, the magnetic fraction increasing with doping. Such model appears highly unlikely regarding the internal magnetic field measured by μSR (Fig. 4). In a droplet model we would expect a magnetic volume fraction of less than 100%. Eventually, diluted microscopic droplets might induce a magnetic field in the surrounding non-magnetic regions, however in that case, we would expect the shape of the magnetic field distribution to change with doping. Furthermore, the value of Bm should be independent of doping in a droplet model in contrary to the experimental result.

At first sight the muon and neutron studies appears to be incompatible. Indeed, the simplest magnetic structure leading to the observed neutron magnetic Bragg peak consists of moments localized on the cerium atom ordering with the propagation vector k = (1/2, 1/2, 1/2) and forming a G-type antiferromagnetic order. In CeCoIn5 two muon stopping sites were identified. The first one (≃70% of the muons) is located at position (0.5,0.5,0.5), i.e. in the center of a parallelepiped formed by eight cerium atoms38. The same position was found for CeRhIn515. At this position, no magnetic field will be generated by such simple magnetic structure, in contradiction with the μSR results. Indeed, the first muon stopping site is a center of symmetry of the cerium crystallographic substructure of CeCo(In1−xCdx)5 and regardless of their orientation magnetic moments located at the cerium site ordering with a propagation vector k cannot produce a magnetic field at such location. The second muon stopping site was identified to the position (0.5,0,0) or (0,0.5,0)38 again in a localized G-type antiferromagnetic structure no magnetic field is produced at such positions irrespectively of the moment direction at the cerium atom for symmetry reasons. The position of the muon stopping site is the main difference between μSR and NMR that probe the two structurally different indium atoms. G-type antiferromagnetism can generate a magnetic field at both positions of the indium atoms, while it will not produce a magnetic field at the positions of the muon stopping sites. The values Bm = 6.5 mT and 11.2 mT obtained in the μSR measurements are not small magnetic field for this system, that could originate from defects in the magnetic structure. Indeed, the average magnetic field generated by a moment of m = 0.4 μB (0.7 μB value of Refs. 16 and 34 for samples of equivalent negative pressures p = −1.1 GPa and p = −1.7 GPa) on a sphere of radius R = 5 Å (distance cerium—1st muon stopping site) is Bav = 2μ0m/(4πR3) = 5.9 mT (10.3 mT), a value comparable to Bm. Fluctuations of localized static magnetic moments could only generate a Gaussian distribution of field at each muons stopping site centered at B = 0, with widths of a fraction of Bav.

However, both experiments appear to probe the same magnetic order. Indeed, both the amplitude and the pressure dependence of the internal field obtained in the μSR experiment and the values of the average field generated by the magnetic moments of the intensity given by the neutron experiment are similar. Therefore, a different magnetic structure must exist that complies with the results of both experiments.

The vortex state of CeCo(In1−xCdx)5 is characterized by a strong vortex pinning18, which was used to determine precisely the SC fraction of the sample p = −0.7 GPa. The procedure consists in cooling the sample in an external field of μ0H = 49.8 mT down to 20 mK. The field distribution at the muon stopping site is presented in Fig. 5a. The external magnetic field is then increased to μ0H = 54 mT. The green circles in Fig. 5b represent the obtained field distribution. Three different contributions can be identified. In the SC region, the magnetic field distribution remains centered at μ0H ≃ 50 mT due to vortex pinning. The narrow distribution of fields centered at the applied field (μ0H ≃ 49.8 mT or 54 mT) corresponds to muons stopping in the silver sample holder. In panel (b), a third contribution is obtained with a field distribution centered at μ0H ≃ 56 mT. A magnetic field larger than the applied one can be attributed to the contribution of superconducting stray field, originating either from muons stopping in the sample holder or in non-superconducting regions of the sample. In both cases, the muons would stop next to a superconducting region in the plane perpendicular to the applied magnetic field. An unpinned superconducting fraction is unlikely as the full superconducting fraction is pinned in pure CeCoIn518 and cadmium dopants are more likely to increase the number of pinning center than to reduce the pinning potential. By comparison with other spectra, as the mean magnetic field of this contribution is large and the fraction of muons substantial we believe that this third contribution is due to a sample partially non-superconducting. In Fig. 5b, the red line represents the fitted probability field distribution in the sample, while the blue line is the total fitted field distribution (sample + sample holder), assuming three independent Gaussian contributions. We found that 81(5)% of the muons stopping in the sample experience a magnetic field of ≃50 mT indicating pinning of the SC vortex lattice. This implies that at least 81(5)% of the sample is SC. At this temperature the sample is also ≃100% magnetic, demonstrating phase coexistence between magnetism and superconductivity. The remaining 19(5)% of the sample volume is probably non-superconducting. It might indicate the formation of non-SC droplets around the Cd impurity atoms, in analogy to the conclusion of the NMR work34 or be due to a small doping inhomogeneity (see Supplementary Note S.3). Non-SC droplets could also certainly explain the increase of the relaxation in the superconducting state with cadmium doping found in the NMR work34.

Figure 5
figure 5

Vortex pinning in the SC phase of CeCo(In1−xCdx)5, p = −0.7 GPa (field shift experiment).

The sample is cooled down to 20 mK in an external magnetic field μ0H ≃ 50 mT (field cooled) and the field distribution p(Bμ) is recorded [black squares in (a) and dashed gray line in (b)]. The external magnetic field is then increased to μ0H ≃ 54 mT and the evolution of p(Bμ) is analyzed (b). Due to vortices pinning the magnetic field distribution in the SC fraction of the sample remains centered at a value B ≃ 50 mT. In contrary muons stopping in the silver sample holder probe a narrow distribution of field centered at the value of the applied magnetic field. In (b) a third contribution is observed with muons probing a field distribution centered at μ0H ≃ 56 mT. It is attributed to muons stopping in a non-superconducting fraction of the sample.

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Absence of cerium valence transition and lowering of the electronic Ce 4f energy level

X-ray Absorption Spectra (XAS) of the M4,5 edges (electron excitations from 3d to 4f angular momentum level) were recorded for several cerium based systems. In Fig. 6a, typical XAS spectra for the two valence configurations Ce3+ and Ce4+ are shown, using powders of the Mott insulator system CeF3 and of the band insulator system CeO2, respectively (see Supplementary Note S.4 and Supplementary Fig. S.3 for an estimate of the exact valence). Spin orbit coupling of the 3d hole splits the spectrum in two edges: M4 ≈ 900 eV and M5 ≈ 882 eV. The substructures are mainly given by the atomic multiplet structure which is clearly different between the valences Ce3+ and Ce4+. Peaks A to C and A’ to C’ come from the Ce3+ multiplet while peaks D, E, D’ and E’ come from the Ce4+ multiplet. Using these features, comparison can be done with pure metallic cerium in the γ-phase (room temperature) and the α-phase (≃4 K) (Fig. 6b) as well as pure and cadmium doped CeCoIn5 (at ≃4 K Fig. 6c). The direct comparison is possible due to the absence of linear dichroism effects. No dichroism is present in the powders and in the amorphous cerium metal as all directions are averaged. The absence of dichroism in CeCoIn5 was reported in Ref. 39 and confirmed during the present experiment.

Figure 6
figure 6

Valence of different cerium based compounds.

M-edges XAS spectra were recorded for: (a) CeF3 (orange), CeO2 (green), (b) α-Ce (magenta), γ-Ce (black) and (c) CeCo(In1−xCdx)5 of corresponding hydrostatic pressure p = 0 and p = −1.45 GPa respectively. The two systems in (a) have a valence of 3+ and 4+, respectively, allowing to determine the intermediate valence state of the metallic cerium systems and CeCo(In1−xCdx)5 systems to be ≃3.38 and ≃3.15, respectively [dotted lines in (b) and (c)]. Some curves are shifted vertically for clarity.

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We obtained that both the α and γ metallic phases of cerium are composed of vf ≃ 62% of Ce3+, or a valence of ≃3.38. The variation of valence between the two phases is less than 0.03. We cannot exclude the formation of an oxide layer on the metallic cerium during the few minutes needed to transfer the sample into the chamber, but the value of vf is in good agreement with the 57% of Ce3+ found previously in the α phase on an in-situ evaporated cerium film40. The absence of valence variation is in agreement with Compton scattering experiments31. A positron annihilation experiment has revealed a valence in the range 3–3.5 for both α− and γ− Ce phases41.

Similarly, a value of vf = 85% Ce3+ or a valence of ≃3.15 is obtained for the two CeCoIn5 systems. The variation of valence with chemical pressure is less than 0.02. On the pure system a magnetic field up to μH = 6.5 T in the orientation was applied without modifying the valence. See Supplementary Note S.5, Supplementary Table S.3 and Supplementary Fig. S.4 for details. For CeCoIn5 both the valence value and the small influence of doping are in agreement with previous experiments39,42,43, even though the doping was different in those studies (Co → Rh,Ir or Ce → Yb).

As often discussed in the literature (see e.g. Ref. 41) the absence of strong valence variations invalidate the promotional model for the transition between the α and γ phases of metallic cerium. Similarly, no strong variation of valence occurs with doping in CeCo(In1−xCdx)5. The absence of valence modification across the field induced QCP (μ0HQCP = 4.8 T, Ref. 44) in CeCoIn5 is particularly interesting as it allows to dismiss any direct influence of the valence (promotional like model or valence fluctuations model) on the Ce 4f “delocalization” leading to the suppression of the AFM phase. Indeed, CeCoIn5 (p = −1.45, GPa H = 0) is AFM at low temperature while CeCoIn5 (p = 0 GPa, μ0H = 6.5 T) with the same valence is paramagnetic. The magnetic ground state of CeCoIn5 (p = 0 GPa, H = 0) is unclear due to the occurrence of superconductivity.

Figure 7 offers a closer view of the two absorption edges for the different systems. The four metallic systems display a systematic energy lowering of the absorption peaks compared to the insulator references, as indicated by the dashed vertical lines. Such an energy shift can either be due to an increase in the energy of the initial 3d state or a decrease of the energy of the final 4f state. Modification of coordination chemistry or in other words of the dispersion of the electron cloud surrounding each atom will modify the charge screening of the cerium ion. The variation in Coulomb interaction affects the energy of the surrounding electrons and particularly the ones on core orbitals such as the initial 3d state. In insulator systems such flow of charge is directly related to the ion valence. A linear relation between the absorption edge energy shift and the valence was found at the vanadium K-edge45, the rhenium L-edge46 and the cerium L-edge47, using for each element different alloys to obtain different valences. In the two first systems, the energy shift of the pure metallic element follows the same linear relation, assuming an ionic screening or equivalent valence of 0, corresponding to the formation of an electronic band.

Figure 7
figure 7

Energy lowering of the M-edge in metallic cerium and CeCo(In1−xCdx)5.

Details of the cerium XAS M5 (a) and M4 (b) edges (Fig. 6) are presented. Compared to the insulator references CeF3 and CeO2 the energy of the edges of the metallic samples is reduced by ≃150 meV, as indicated by the vertical dashed lines. The cerium α–γ transition and the evolution of CeCoIn5 with cadmium doping are characterized by a small variation in the cerium valence (peak height) and in the energy of the M4,5 edges (peak position) in opposite directions. The pattern enlarged in inserts is observed consistently on each peak. The error bars on the data in insets are of the size of the lines thickness, see Supplementary Note S.6 and Supplementary Fig. S.5.

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Taking the derivative of the XAS spectra, the energy position of the maximum of each absorption peak was determined. The shift of energy of each peak compared to the average value of the metallic cerium systems is then calculated. As this shift is found to be independent from the absorption peak energy (A–E, A’–E’) the average value and standard deviation are plotted in Fig. 8 versus the valence. The energy shift expected due to the variation of ionic screening for different valences is extracted from the insulators systems (red line in Fig. 8). The two metallic cerium phases and CeCo(In1−xCdx)5, are clearly located below this line and the shift is not directly related to the valence. In analogy to the case of vanadium and rhenium, this can be attributed to a stronger variation of the ionic screening due to the formation of a metallic band of partial Ce 4f angular momentum character.

Figure 8
figure 8

Energy shift in the XAS M4,5-edge for different cerium based systems.

The average difference in energy of the M4,5 absorption edges of each system compared to the average value of metallic cerium is plotted versus the valence for CeF3, CeO2 (black), CeCoIn5, CeCo(In1−xCdx)5 p = −1.45 GPa (blue), metallic cerium in the α and γ phase (green). The red line represent the shift expected due to the variation in ionic screening with the modification of the valence. The two metallic cerium phases and CeCo(In1−xCdx)5 are clearly located below this line, suggesting an at least partial band formation of 4f angular momentum character.

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Similarly, a weak hybridization of the Ce 4f final state, either due to the Kondo effect or to the formation of 4f metal ligands (Mott—like delocalization) would result in a shift of the M4,5-edge to lower energy. Such interpretation is corroborated by the observation of a comparable decrease ≃0.2 eV of the 4f state energy level in CeCoIn5 compared to CeRhIn5 in a x-ray photoemission experiment48. CeRhIn5 is usually considered as fully localized. We note that in pure cerium the Kondo energy must be substantially different between the γ and α phases, as the effective quasiparticles mass strongly varies between m* ≃ 6m0 in the γ-phase and m* ≃ 20m0 in the α-phase49. If the energy shift was due to the Kondo effect we would expect a substantially different shift between the α and the γ phase, in contrary to the experimental observations (Fig. 8). As the value of the shift is very similar in the CeCo(In1−xCdx)5 systems it is likely that the same effect is responsible for the shift in all four systems.

The two mechanisms that might decrease the energy of the absorption edge imply the formation of a narrow electronic band of partial Ce 4f angular momentum character. We therefore conclude that the presence of Ce 4f metal ligands is the main reason leading to the energy shift in the M4,5-edges XAS of α and γ-phase cerium and in CeCo(In1−xCdx)5. Similarly, scanning tunneling spectroscopy experiments50,51 have found indications for a narrow band of strong 4f electronic character in CeCoIn5 which is absent in CeRhIn5.

A closer look at the different XAS spectra of pure cerium (insets Fig. 7) reveals that the α-phase is slightly more localized than the γ-phase both in terms of valence (v(α) < v(γ)) and hybridization (dE(α) < dE(γ)). In contrary, the valence is higher in the cadmium doped than in the pure CeCoIn5 systems, while the hybridization is increased. The behaviour enlarged in insets of Fig. 7 is observed consistently on each peak. A quantitative estimate of these variations is given in Table S.3 using Eq. (4), assuming a fixed continuum contribution. Such opposite behaviours, indicate that the nature of the magnetic transition in CeCo(In1−xCdx)5 under chemical pressure and between the cerium α− and γ− phases is different.

Discussion

Due to theoretical considerations it is often assumed that Ce 4f orbitals do not directly participate to metallic bonding. The reason is that the radius of maximum electronic density for Ce 4f orbitals is a factor ≈4 smaller than the one of Ce 5d6s2 orbitals. In a compound, the average distance between the cerium and other nuclei is dominated by the Ce 5d6s2 character bands formation and therefore bands of partial Ce 4f orbital momentum character are assumed to have a higher energy than localized Ce 4f orbitals (see e.g. Ref. 30).

The large coherence length observed in the neutron diffraction experiment suggests homogeneous magnetic moments, which indicate that each site has the same mixed valence. This appears incompatible with a fully localized picture of Ce 4f electrons in which the valence of individual atoms can only take integer values. The μSR and neutron scattering experiments probe the magnetic order on different time and length scales. Indeed long range magnetic order is observed starting from a coherence of a few unit cells in the muon case, while the magnetic order revealed by the neutron Bragg peaks has a coherence length that exceeds 100 Å. One could therefore assume the existence of two separate magnetic orders. The first one, G-type commensurate localized antiferromagnetism would give no muon signal and the neutron Bragg peak, while the second incommensurate order, with short coherence length would be responsible for the muon signal. This scenario gives however no explanations why the two orders would have the same amplitude or why the short range magnetic order was not detected by neutron diffraction. A short range order would give a broader diffraction peak but should still be observed. A more convincing explanation is found by looking at the time scales of the two probes: τ ≈ μs for the muons, while τ ≈ ps for the neutrons. A magnetic order occurring in a band of partial Ce 4f character, such as the one revealed by the XAS measurements, is expected to have both localized and itinerant structures. On short time scales, the 4f character dominates, with electrons found preferentially at the cerium position and domains with magnetic moment localized on the cerium site will be probed. Due to the electronic motion of the Ce 4f electrons, the size of these commensurate magnetism domains will shrink at larger time scales and the incommensurate structure formed by the itinerant charge carriers will be probed. Such incommensurate magnetism is expected to generate a Bessel type depolarization function for the μSR asymmetry signal, as observed experimentally52. The time scale of such crossover depends on the electronic motion and can vary between different systems. In CeCo(In1−xCdx)5 we expect this time scale to be within the six orders of magnitude of difference in sensitivity between the neutrons and muons probes.

In this study we report three different indications for the presence of a band of partial Ce 4f character in CeCo(In1−xCdx)5. (i) The mixed valence of cerium is incompatible with Ce3+ 4f electrons of full localized orbital character. (ii) The energy shift observed in XAS measurement and (iii) the observed μSR and neutron signals of the magnetic order are expected in case of a narrow electronic band of partial Ce 4f character. In CeCo(In1−xCdx)5, Ce 4f electrons are therefore simultaneously “delocalized” by ligand formation, as indicated by the presence of a band of partial Ce 4f character, by the Kondo effect, as indicated by the large quasiparticles effective masses and through the mixed valence of the cerium atom. Under chemical or hydrostatic pressure one of these “delocalization” mechanisms is modified and leads to the disappearance of the AFM order.

For a valence fluctuations induced AFM phase transition we would expect a variation of valence of 0.1–0.722,31 at the transition, incompatible with the experimental results both in CeCo(In1−xCdx)5 and metallic cerium. As the effects on the XAS spectra of the cerium α–γ transition and upon cadmium doping in CeCo(In1−xCdx)5 are different, we may ascribe them to the two other “delocalization” mechanisms: Kondo effect and electronic band broadening (Mott “delocalization”). The strong variation of effective mass in metallic cerium between the α and γ phases49 suggests a modification of the Kondo hybridization for this transition53. By elimination we find that the suppression of magnetism in CeCo(In1−xCdx)5 with applied chemical/hydrostatic pressure, is certainly due to the broadening of a band of partial Ce 4f character.

In our previous μSR work on CeCoIn5 under pressure18 we had found a doubling of the SC carrier density, in a band of strong 4f orbital momentum character, between 0 GPa and 1 GPa. This is compatible with the band broadening expected for an increased cerium—cerium hybridization under pressure. As in the ferromagnetic case, band antiferromagnetism is driven by a “Stoner like” criterion for antiferromagnetism54. The magnetic order can be suppressed due to band broadening under pressure, while cadmium doping can both reduce the electronic bandwidth by “negative” chemical pressure and increase the hole density of state. Both effects are favourable for the realization of the “Stoner like” criterion for antiferromagnetism and would at some point lead to a phase transition. The equivalence between cadmium doping and hydrostatic pressure is in such a case naturally understood.

When the “delocalization” occurs via the Kondo effect, localized electrons are introduced into the main Fermi surfaces of low 4f angular momentum character. In contrary, Ce 4f ligand formation or band broadening, constitutes a band of substantial 4f angular momentum character. For fully itinerant systems such as CeCo(In1−xCdx)5, the interaction between the two “delocalization” mechanisms will produce an electron transfer between different Fermi pockets, reminiscent of the proposed Kondo breakdown scenario55, but also indicating that carrier concentration is an important parameter for the SC phase diagram in CeCoIn5 possibly as important as in the case of high- Tc cuprates.

In conclusion, we have investigated the antiferromagnetic state of CeCo(In1−xCdx)5 combining three different experimental probes: neutron scattering, muon spin rotation and X-ray absorption spectroscopy. Using neutron diffraction we confirmed the commensurate nature of the long range antiferromagnetic order at short time scales and the absence of secondary magnetic Bragg peak. We established the value of the magnetic moment for a different doping as the one reported in the literature (≈0.4 μB for a sample with Tc ≈ TN). We used muon spin rotation to probe the magnetic field at two centers of symmetry of the cerium sublattice. We measured a magnetic field that matches the expected average magnetic field generated by the magnetic moments of amplitude obtained in the neutron diffraction experiment. We observed that the position of the absorption edge in X-ray absorption spectroscopy is reduced toward lower energies in the case of CeCo(In1−xCdx)5 and metallic cerium compared to reference insulating materials. The presence of a narrow electronic band of partial 4f character was identified as a possible reason for this energy shift. Depending of the timescale such a narrow electronic band can display either localized or itinerant properties corresponding to the neutron and muon signals.

The magnetic and superconducting volume fractions (≃100% and ≃81% respectively) were extracted from the muon spin rotation experiment, directly assessing the microscopic coexistence of the two orders. The large magnetic fraction together with the evolution of the internal magnetic field strongly questions the proposal of magnetic island for this system and suggests that the concept of non-superconducting island is more appropriate.

A detailed analysis of the mixed valence in CeCo(In1−xCdx)5 demonstrated the absence of valence variation with doping and under magnetic field, notably across the quantum critical point. The evolution of fine structures in the X-ray absorption spectra indicate that the α–γ phase transition in metallic cerium has a different nature than the magnetic phase transition in CeCo(In1−xCdx)5. Assuming that the Kondo effect dominates the physics of metallic cerium we conclude that a broadening of the Ce 4f band is the main reason for the suppression of the magnetic order in CeCo(In1−xCdx)5.

These results provide a new approach on the superconducting coupling mechanism in this system by revealing the nature of an instability coexisting with superconductivity.

Methods

Neutron scattering

Measurements were carried out on the cold neutron three axis spectrometer IN12 located at ILL, Grenoble. The initial neutron beam is provided by a double focusing pyrolithic graphite (PG) monochromator. Higher order contamination is removed before the monochromator by a velocity selector. Diffraction measurements were carried out using a PG analyzer operated in a flat (non-focusing) geometry in order to reduce the background. The spectrometer was setup in W configuration with open-open-open collimations. The initial and final neutrons wave-vectors were ki = kf = 2.6 Å−1, chosen in order to minimize the neutron absorption of Co, In and Cd. The single crystal sample, of equivalent negative pressure p = −0.7 GPa and of dimensions: ≈5 × 7 × 0.2 mm3 is the largest of the pieces used for the μSR experiments. It was mounted in a helium-4 cryostat with the [1, −1, 0] axis vertical, the scattering plane being thus defined by [1,1,0] and [0,0,1]. The notations used are as follow : The scattering vector Q is decomposed into Q = τ + k, where τ is a reciprocal lattice vector corresponding to a Brillouin zone center position and k is the propagation vector for a given magnetic structure. The Cartesian coordinates, H and L, of the scattering vector Q  are expressed in reciprocal lattice unit (r.l.u.) (Q  = (H, H, L)). The Bragg peaks were fitted by a resolution limited Gaussian lineshape at 1.45 K and by a convolution of a Lorentzian with a resolution limited Gaussian at 2.01 K. The temperature variation of the magnetic intensity was fitted by the phenomenological function, I ∝ 1 − (T/TN)α with TN = 2.09(2) K and α = 6.4(3). To estimate the magnetic moment, the (3/2, 3/2, 1/2) Bragg peak was considered and its intensity was normalized to the structural Bragg peak reflection (1,1,0). The similar scattering geometry of these two peaks leads to comparable correction factors (Lorentz factor, absorption). For the magnetic peak, the Ce3+ magnetic form factor was considered. The magnetic reflection (3/2,3/2,1/2) is relevant to address the two cases of magnetic moment direction given by group theory in a purely localized scenario (c-axis or in plane orientation). For these two possible spin orientations, the moment is found to be ≃0.3 μB (c-axis orientation) and ≃0.5 μB (in-plane orientation), whose average is given in the main text.

Muon Spin Rotation (μSR)

Experiments were performed at the Swiss Muon Source (SμS), Paul Scherrer Institut (PSI), Switzerland, using the GPS and the LTF spectrometers. In a μSR experiment spin polarized positively charged muons are implanted into a sample. Due to their positive charge, the muons only stop at well defined locations in the unit cell, called the muon stopping sites. The initial muon spin orientation can be rotated by up to ≃60° before implantation in the sample of dimensions ≈8 × 12 × 0.2 mm3. Within the present study, a mosaic of 8 (p = −0.7 GPa) and 11 (p = −1.45 GPa) single crystals were used. In the presence of a local magnetic field at the muon stopping site Bμ the muon spin precesses at its Larmor frequency ωμ = γμBμ, where γμ = 135.54·106 HzT−1 is the gyromagnetic ratio of the muon. The muon decays with a life time of τμ = 2.2 μs into a positron and two neutrinos. Due to parity violation the decay positron is preferentially emitted along the muon spin direction. Forward (+) and backward (−) positron detectors with respect to the initial muon spin polarization are used to monitor the μSR asymmetry spectrum A(t):

where Ni(t) is the positron count histogram on detector i. N0i and Ci are constants.

In zero applied external magnetic field, the relaxation of A(t) in the paramagnetic state is due to the nuclear moments and follows the so-called Kubo-Toyabe (KT) relaxation function52. In the presence of a static or quasi–static long range magnetic order at the muon stopping sites, A(t) oscillates with time52. For CeCo(In1−xCdx)5, the evolution of A(t) is presented in Fig. 3a, with spectra taken above and below TN. While the frequency of the oscillations depends on the absolute value of the local magnetic field, the amplitude is related to the fraction and the relative orientation of . We may consider A(t) as the sum of four different contributions: The muons stopping in the silver sample holder give rise to a weak relaxation, visible at large time when contributions from the muons stopping in the sample have vanished (First term of eq. (3). A potential non-magnetic fraction would lead to a KT relaxation as observed above TN (Second term of eq. (3). A magnetic fraction gives rise to two additional terms in A(t) whose amplitude depends on the angle α between and the initial muon spin orientation : an oscillating term for (Third term of eq. (3)) and an exponential damping for (Fourth term of eq. (3)).

Here Ai (i = Bg & S) are initial asymmetries. fs = As/(As + ABg) represents the fraction of muons stopping in the sample and fM is the magnetic fraction. ΔKT and ΔBg are the field distributions due to nuclear moments in the sample and sample holder. The two other damping rates ΔL and ΔP are free parameters. We have used a standard value of ΔBg = 0.05 mT for the damping of the signal in the background, while ΔKT ≃ 0.45 mT was found above TN setting fM = 0 (empty symbols in Fig. 3a). Two different experimental configurations (inset Fig. 3) allow to implant muons with S either parallel () or perpendicular () to the sample -axis. Naming θ the angle between and , we obtain α = θ for the configuration and and for the configuration . Therefore, the asymmetries of the three different muon detectors configurations, schematically drawn in Fig. 3, can be fitted simultaneously using Eq. (3). The depolarization function PZ(t, Bm) can be either a sum of cosines or a Bessel function depending on the magnetic structure, with Bm the mean or respectively maximum field present at the muon stopping site. The field distribution p(Bμ) obtained by Fourier transformation of the sample contribution to the asymmetry (Fig. 4) has clearly more the form of Eq. (1) corresponding to the Bessel depolarization function than a sum of Dirac distributions centered at a finite magnetic field that would correspond to cosine depolarization functions. Detailed results of the fits performed in the time domain using the free software package MUSRFIT56 are presented in Supplementary Notes S.2 and S.3 and Supplementary Table S.2.

X-ray Absorption Spectroscopy (XAS)

Experiments were performed at the Swiss Light Source (SLS), Paul Scherrer Institut (PSI), Switzerland, using the X-Treme beamline, recording the total-electron-yield (TEY) current57. The metallic samples of a typical size of 2 × 2 × 0.1 mm3 were glued with silver epoxy on a copper sample holder, cleaved or scratched in a nitrogen atmosphere glove box before installation. The metallic cerium sample was annealed in vacuum at 750 K before the experiment to insure a full γ phase composition. The insulating powders were embedded in an indium foil to insure good electrical and thermal contact. In a XAS experiment, core level electrons are excited to the valence band by a monochromatic photon beam. When the core vacancies are refilled, Auger electrons are emitted, which scatter with other electrons in the material generating a TEY current between sample and ground. This current is normalized by the incoming beam intensity at each energy, monitored on a focusing mirror. The integral of the TEY spectra were normalized between 865 meV and 920 meV, after subtraction of the edge continuum (a Fermi step function centered at the M4-edge maximum was used). Each spectrum is the average of 15 energy scans of 3 minutes each. The experiment was repeated at different sample positions without any significant modifications. None of the samples displays a linear dichroism as reported in Ref. 39. The resolution is discussed in the Supplementary Note S.6.

In order to extract the valence of the metallic systems, the edges continuum was first subtracted on each XAS spectra. The spectra of CeF3 (XASCeF3(E)) and CeO2 (XASCeO3(E)) were then parametrized and used to fit the other spectra as:

with 4 − Vf the cerium valence of the analyzed system and dE an energy shift. The fitted results are presented by dotted lines in Fig. 6 and the parameters reported in table S.3.

Time and length scales of the μSR and neutron scattering experiments

The muon interaction time scale can be estimated using the precession frequency of the muon spin: γμBmτμ ≈ 1, →τμ ≈ μs. For a μSR experiment a static or quasi—static long range magnetic order means that clusters ranging from a few unit cells to the whole sample, order statically on a time scale of a few μs. For the setup used in the neutron scattering experiment, the magnetic ordering is long range on the scale of at least 300 Å and static on the a scale of ≈ps, the neutron scattering interaction time scale.

Additional Information

How to cite this article: Howald, L. et al. Evidence for Coexistence of Bulk Superconductivity and Itinerant Antiferromagnetism in the Heavy Fermion System CeCo(In1-xCdx)5. Sci. Rep. 5, 12528; doi: 10.1038/srep12528 (2015).