Magnetism and superconductivity driven by identical 4f states in a heavy-fermion metal

Abstract

The apparently inimical relationship between magnetism and superconductivity has come under increasing scrutiny in a wide range of material classes, where the free energy landscape conspires to bring them in close proximity to each other. Particularly enigmatic is the case when these phases microscopically interpenetrate, though the manner in which this can be accomplished remains to be fully comprehended. Here, we present combined measurements of elastic neutron scattering, magnetotransport, and heat capacity on a prototypical heavy fermion system, in which antiferromagnetism and superconductivity are observed. Monitoring the response of these states to the presence of the other, as well as to external thermal and magnetic perturbations, points to the possibility that they emerge from different parts of the Fermi surface. Therefore, a single 4f state could be both localized and itinerant, thus accounting for the coexistence of magnetism and superconductivity.

The ground state properties of a system are of fundamental importance and the starting point for considering the excitations that enliven real systems. The prevalent electronic ground states of metals, magnetism, and superconductivity, are typically mutually exclusive quantum many body phenomena. This antagonism can be evaded by spatial separation (e.g., in some Chevrel phases (1)) or by subdividing the 5f states in some actinide compounds into more localized and more itinerant parts giving rise to magnetism or participating in superconductivity, respectively (see, e.g., (2)). The quest for microscopic coexistence of these phenomena involving identical electrons is fueled by the expectation for insight into the complex behavior of new materials with intertwined ground states as, e.g., the cuprate superconductors. Experimentally, this endeavor not only requires finding an appropriate material, but also calls for a concerted investigation of both the charge and the spin channel and hence, judiciously chosen measurement methods.

The heavy-fermion metals offer an interesting playground where magnetism and superconductivity can both compete and coexist. In these systems, the hybridized f electrons are not only responsible for long-range magnetic order, but are also involved in superconductivity. In this context the CeMIn₅ (where M = Co, Ir, or Rh) family of heavy-fermion metals has been in vogue due to their rich electronic phase diagrams in which an intricate interplay between superconductivity and magnetism is observed (3). For instance, in CeCoIn₅, a superconducting ground state is found below a transition temperature T_c ≈ 2.3 K whereas CeRhIn₅ orders antiferromagnetically below T_N ≈ 3.7 K (3). On the other hand, superconductivity is observed in the latter compound by application of pressure whereas the proximity to magnetism in CeCoIn₅ is demonstrated by the likely existence of a zero temperature magnetic instability (3). Moreover, neutron scattering experiments indicate strong antiferromagnetic (AF) quasielastic excitations at wavevectors and equivalent positions in the paramagnetic regime (4). However, the excitations become fully inelastic when entering the superconducting state, resulting in the appearance of a spin resonance. These findings underline the analogy to the cuprate high-temperature superconductors (5 and 6).

We conducted a comprehensive investigation of the magnetic order and superconductivity in CeCo(In_0.9925Cd_0.0075)₅. Neutron scattering, magnetotransport, and heat capacity, i.e., microscopic and macroscopic, spin and charge sensitive studies have been combined on flux-grown single crystals of the same batch. These combined efforts not only allow to unambiguously pin down the associated effects but also to cross fertilize the methods. We find a local duality of the electronic 4f degrees of freedom implying electronic phase separation on the Fermi surface.

The specific composition with x = 0.0075 was chosen since T_c ≈ 1.7 K and T_N ≈ 2.4 K are closest within the CeCo(In_1-xCd_x)₅ series (7), Fig. 1A. Consequently, the interplay between superconductivity and antiferromagnetism is expected to be most pronounced (8). Earlier studies (9 and 10) were conducted on samples with x≥0.01 where T_c is small compared to T_N. Thus, the condensation of conduction electrons into Cooper pairs effectively took place in a state where fluctuations of the AF order parameter were not appreciably large, i.e., where the balance of the two phenomena is already shifted toward AF order. As a result, no significant anomaly in the magnetic intensity as determined by neutron scattering was observed on entering the superconducting regime (10).

Fig. 1.

(A) Doping x dependence of AF and superconducting (SC) transition temperatures in CeCo(In_1-xCd_x)₅ for Cd-content x ≤ 0.02 (after ref.  7). The crystals investigated here (arrow) exhibit both transitions. (B) B–T phase diagram of CeCo(In_0.9925Cd_0.0075)₅ obtained from magnetotransport (circle, square), neutron scattering (triangle, upside-down triangle) and heat capacity (+, ×) measurements with B⊥c. (C) B–T diagram for B∥c from magnetotransport and heat capacity. Indications of a transition within the AF phase are found (diamond).

Results

The resulting magnetic field—temperature (B–T) phase diagrams are presented in Fig. 1 B and C for B⊥c and B∥c, respectively. The excellent agreement of results obtained by three very different methods evidences that bulk properties are probed. The strikingly equivalent behavior of the superconducting and AF phase boundary, in particular for B⊥c, is indicative of a mutual influence of the two phenomena. The steep initial slope of T_c(B) of approximately -13(-4) T/K for B⊥(∥)c indicates a large effective quasiparticle mass, i.e., heavy-fermion superconductivity.

Initial elastic scans at the lowest temperature (T = 0.5 K) across the nuclear peaks confirmed the tetragonal crystal structure, with lattice parameters a = 4.595 Å and c = 7.533 Å. To search for magnetic intensity, scans along high-symmetry directions were performed. Well below T_N ≈ 2.4 K and T_c ≈ 1.7 K, scans along revealed additional magnetic intensity at and symmetry equivalent positions, see data at T = 0.5 K in Fig. 2A. Above T_N this magnetic superstructure peak completely vanished, cf. Fig. 2A for data at T = 3 K. The scans did not indicate additional intensity at other, e.g., incommensurate, positions. In particular, no magnetic superstructure peaks were detected around or , which have been observed in the closely related system CeRhIn₅ (11). The commensurate magnetic structure is therefore in close agreement with that reported earlier on the 1% cadmium (Cd) doped sample (10). Elastic scans across at T = 0.5 K and for several magnetic fields are displayed in Fig. 2B. Obviously, a magnetic field of B = 12 T suffices to fully suppress antiferromagnetism at this temperature. More importantly, the observation of a magnetic superstructure peak in zero magnetic field well inside the superconducting state clearly demonstrates the coexistence of AF order and superconductivity on a microscopic scale. Based on our heat capacity measurements we emphasize that both, the transition into the AF ordered and the superconducting state, are bulk transitions.

Fig. 2.

Elastic neutron scattering scans in CeCo(In_0.9925Cd_0.0075)₅ along [001] and across : (A) at different temperatures in zero magnetic field and (B) for several magnetic fields at T = 0.5 K. Solid lines indicate fits with Gaussian lineshape to the data. (C) Temperature dependence of the magnetic intensity at in B = 0 along with fits from mean-field expectations to the data (solid and dashed lines, see text). (D) Same as (C) in addition with data for several magnetic fields. The paramagnetic background contribution was subtracted from data in (C) and (D).

In order to scrutinize the possible influence of superconductivity on the AF order, the magnetic intensity at was recorded as a function of temperature for different magnetic fields, Fig. 2 Cand D. In zero magnetic field the magnetic intensity increases below T_N and displays a kink at T_c (marked by arrows) with no further change in intensity at lower temperatures. For increasing magnetic field, T_N and the overall magnetic intensity are reduced. No magnetic intensity was detected for B = 12 T. The assignment of this kink to T_c is corroborated by the magnetotransport and heat capacity measurements. An attempt to fit the zero-field magnetic intensity by a mean-field model for the sublattice magnetization (using a Brillouin function for an effective spin- system) fails to describe the whole temperature dependence, as indicated by the dashed line in Fig. 2C. On the other hand, a fit restricted to the temperature range T_c < T < T_N reproduces these data reasonably well (solid line in Fig. 2C) and results in an expected magnetic intensity for T → 0 of about 40% larger than the experimentally observed saturation value. Obviously, the onset of superconductivity prevents a further rise of magnetic intensity below T_c without suppressing the AF order itself.

The magnetic intensities measured as a function of applied field for different temperatures and different protocols are directly compared to magnetotransport ρ_xx(B) in Fig. 3 facilitating again a clear assignment of the observed features. The disappearance of magnetic intensity, signaling the transition from the antiferromagnetically ordered phase to a paramagnetic one, nicely concurs with the strong change in slope in ρ_xx(B). On the other hand, the kink in the field-dependent neutron intensity can be identified as the superconducting upper critical field B_c2 coinciding with the approach to zero resistivity. The latter is also supported by the similarity of the field-dependent neutron intensity (Fig. 3B) and its temperature dependence, Fig. 2D.

Fig. 3.

Comparison of neutron scattering and resistivity measurements. (A) Field-dependent magnetic intensity at and T = 0.5 K after zfc and field cooling (fc, upside-down triangle). Arrows indicate the direction of magnetic field variation. The different field conditions give rise to different domain population as shown in the inset: For fc and B > B_c2 (B_c2 is marked by the dashed line) only the depicted spin configuration in the basal plane is found, below B_c2 also the second displayed domain (left) is increasingly occupied. (B) Magnetic intensity after zfc at T = 0.5 K and 1.5 K. (C) Field dependence of resistivity ρ_xx for B⊥c. A protocol analogous to (A) has been followed, yet no significant hysteresis was found. (D) Resistivity ρ_xx as function of B(⊥c) displayed for selected temperatures.

Discussion

Interestingly, a pronounced hysteresis is seen for the neutron scattering intensities taken at increasing zero-field cooling (zfc) and decreasing magnetic field, Fig. 3A. Whereas the aforementioned kink is observed for increasing magnetic field, in decreasing field the magnetic intensity grows steadily and only reaches for B → 0 the values of the zfc measurements. In the pristine CeCoIn₅, a multicomponent ground state (also discussed as a possible Fulde-Ferrel-Larkin-Ovchinnikov phase) with characteristics of a first order phase transition has been observed at low temperatures (T < 0.3 K) in fields B > 10 T applied along the direction (12 and 13). However, in accord with the sensitivity of such a state to disorder, its existence in the Cd substituted system has been dismissed (14). It is to be noted that enhanced disorder arising from Cd substitution increases the typical resistivity values in this system by an order of magnitude in comparison to undoped CeCoIn₅. Moreover, the range of magnetic fields within which this hysteresis is observed in neutron scattering implies that the hysteretic behavior is seen mainly above B_c2 within the AF phase, ruling out shielding effects. An alternative scenario would involve that the field-driven transition from an AF phase into a paramagnetic one is first order in nature. To investigate this possibility, we have performed resistivity measurements in slowly increasing and decreasing fields at T = 0.5 K. As shown in Fig. 3C, no significant hysteresis is observed indicating that the field-driven transition is continuous in character (at least for T≥0.06 K).

With the first order scenario likely ruled out, the observed hysteresis in our neutron scattering data (and the lack of it in ρ_xx) can only be explained by invoking the possibility of two different domain populations in the field cooled and zero-field cooled measurements. Though relatively little explored in comparison to ferromagnets, the existence of magnetic domains is well established for anisotropic antiferromagnets. A particularly well investigated example is elemental chromium for which the influence of different domain populations as a function of measuring protocols has been observed (15 and 16). Our neutron data indicate an unequal domain population upon entering the magnetically ordered state at low temperatures and high magnetic fields. Decreasing the magnetic field at low T and crossing the phase boundary into the AF state, one domain configuration (with magnetic moments ⊥B, cf. Fig. 3A) is strongly favored over the other (with magnetic moments ∥B) resulting in a substantially reduced magnetic intensity in neutron scattering measurements (see Materials and Methods section). Further reducing the magnetic field and inside the superconducting state the second domain successively becomes populated balancing the domain population when reaching B = 0, identical to the zfc case. The associated domain walls strongly influence the magnetotransport only if the electronic mean free path ℓ is comparable to or larger than the domain wall thickness δ (15). The lack of hysteresis in our transport measurements suggests that this criterion is not met in the B⊥c direction, i.e., δ≫ℓ. Note that even in undoped CeCoIn₅, ℓ is reduced to a few 10 Nanometer (nm) already in moderate fields (17). However, we were able to resolve a tiny hysteresis (≲0.2 T at 0.2 K) in the magnetoresistance for B∥c within the AF regime (18). This observation is consistent with an enhanced (factor of 1.5) dynamic spin correlation length within the ab plane compared to the c direction (4 and 19) which indicates a reduced δ in c direction.

In order to trace the anisotropic nature of the superconductivity and magnetism in this system, we also measured ρ_xx with field applied along the crystallographic c axis. Selected results of these measurements are shown in Fig. 4, with the zero-resistance superconducting state and the field-induced destabilization of AF order being clearly marked out. At lowest temperature and for decreasing field, the sharp increase in resistivity at B ∼ 7 T indicates carrier localization due to the onset of AF order. On further reduction of the magnetic field, a drop in ρ_xx(B) for B ≲ 6 T is observed within the AF state. This could possibly originate from a spin rearrangement as found in CeCu₅Au (20), a scenario that would also account for the observed anisotropy in ρ_xx(B). Alternatively, one might speculate that the drop in ρ_xx(B) may be caused by a change in ordering vector as, e.g., observed CeCu₂Si₂ (21). The signatures of these two transitions merge as they become broadened at increasing temperatures. The magnetoresistance is negative down to T = 0.06 K for 7 T ≲ B ≤ 15 T manifesting that there is no Fermi liquid regime in the investigated field range. This effectively rules out the presence of a quantum critical point in the B∥c direction. Interestingly, for B⊥c the destruction of long-range AF order is succeeded by a field range of positive magnetoresistance which indicates that the system enters into a regime with coherent Kondo scattering (22). Analyzing ρ_xx(T) for constant fields did not reveal any signature of a T² dependence eliminating the possibility of Fermi liquid behavior also for B⊥c (a Kohler’s scaling analysis is hampered by large uncertainties in ρ_xx(0)).

Figure 4.

(A) Magnetic field dependence of the resistivity ρ_xx as measured in CeCo(In_0.9925Cd_0.0075)₅ at different temperatures with B∥c. The phase boundaries associated with superconductivity and antiferromagnetism are marked.

The almost constant neutron intensity below T_c is intriguing. Its analysis above and the electronic transport measurements indicate a second order phase transition at T_c without spatial phase separation. Then, the deviation of the neutron intensity from its expected value below T_c implies coexistence and, more importantly, mutual influence of AF and superconducting order. These conclusions go well beyond those drawn from earlier Nuclear Magnetic Resonance measurements (9): Although the microscopic coexistence of AF and superconducting order was inferred, the interplay between the two different types of order was not observed. Based on our new measurements we speculate that the low-energy magnetic fluctuations are gapped by superconductivity and likely shifted to higher energies (possibly to the resonance at 0.6 meV observed in undoped CeCoIn₅ (4)), a similar mechanism as discussed for the cuprates (23). The delicate, unprecedented balance of the two states may result from the proximity of T_c and T_N in the chosen compound. Since the commensurability of the AF order with and NMR studies (9) suggest mainly local magnetism, the single 4f state spans both local and itinerant character in momentum space. We note that the local coexistence of AF and superconducting order is corroborated by the spin-spin correlation length ξ_m clearly exceeding the superconducting one, ξ_GL. The former can be estimated from the broadened (beyond resolution) Gaussians of the neutron scattering intensities (cf., e.g., Fig. 2A), ξ_m ≈ 32 nm, whereas an upper bound ξ_GL ≲ 10 nm can be inferred from the estimated critical field B_c2(T → 0).

In conclusion, magnetotransport, heat capacity, and elastic neutron scattering measurements were combined to unambiguously identify the respective features of antiferromagnetic and superconducting order in the heavy-fermion alloy CeCo(In_1-xCd_x)₅ with x = 0.0075 resulting in a highly consistent B–T phase diagram. Below T_c, superconductivity and magnetism correlate via identical 4f states resulting in a delicate balance of local coexistence. The phase transitions appear to be continuous in nature, and the pronounced hysteresis observed in neutron scattering measurements likely arises from different domain populations dictated by the sample history. The destruction of antiferromagnetic order at lowest temperature is not followed by Fermi liquid behavior for a substantial field range leaving the ground state unresolved.

Materials and Methods

A 12 mg platelet-like single crystal was used for neutron scattering as well as for heat capacity measurements in a commercial Physical Property Measurement System with ³He insert. Magnetoresistance measurements were conducted with B ≤ 15 T applied both ∥c and ⊥c for 0.06 K ≤ T ≤ 6 K.

Elastic neutron scattering measurements were conducted in the temperature range 0.5 K ≤ T ≤ 3 K within the [110]–[001] horizontal scattering plane, both in field cooled (fc) and zfc conditions. Magnetic fields B ≤ 12 T were applied within the basal ab plane. These measurements were performed on the thermal triple-axis spectrometer E1 at the BER-II reactor of the Helmholtz-Zentrum Berlin für Materialien und Energie in Berlin/Germany.

With respect to our discussion of magnetic domains it should be noted that the neutrons can only couple to moments perpendicular to the momentum transfer Q. In our case, this allows to unambiguously assign a largely reduced magnetic intensity at and high magnetic fields B to such domains with magnetic moments perpendicular to B since only for these domains there are components of the magnetic moment parallel to Q. In contrast, for the domains with magnetic moments parallel to B all individual moments are aligned perpendicular to Q which results in a much higher neutron intensity.