Speciﬁc heat of CeRhIn 5 : pressure-driven transition from antiferromagnetism to heavy-fermion superconductivity

CeRhIn 5 is known to show an unusual transition at a critical pressure of (cid:1) 15 kbar. Speciﬁc-heat data show a gradual change in the zero-ﬁeld ‘‘magnetic’’ speciﬁc-heat anomaly from one typical of antiferromagnetic (AF) ordering at ambient pressure to one more characteristic of a Kondo singlet ground state at 21 kbar. However, at 15 kbar there is a discontinuous change from an AF ground state to a superconducting ground state, and evidence of a weak ther-modynamic ﬁrst-order transition. Above the critical pressure, the low-energy excitations are characteristic of superconductivity with line nodes in the energy gap, and, at intermediate pressures, of extended gaplessness. (cid:1) 2003 Elsevier Science B.V. All rights reserved.

At ambient pressure heavy-fermion (HF) CeRhIn 5 is an antiferromagnetic (AF) that becomes superconducting (SC) abruptly at a critical pressure ðP c Þ $ 15 kbar [1]. Its phase diagram is different from that of any other Ce HF compound, and the abrupt change at P c suggests a ''firstorder-like transition'' [1]. The ground states, AF below P c and HF/SC above, both evolve continuously with increasing P , but at P c there is a discontinuous change.
Specific heats ðCÞ are shown in Fig. 1. The electronic component ðC e Þ is derived by subtracting C for LaRhIn 5 [1]. As P increases the ''magnetic'' specific-heat anomaly becomes broadened and reduced in amplitude. A second anomaly, associated with the SC state, first appears at 16 kbar. The characteristic T Õs derived from C e and the resistivity (q) [1] are compared in Fig. 2. The T of the magnetic-anomaly maximum (T max ) tracks the N e eel temperature ðT N Þ deduced from q for P 6 10 kbar, but then shifts to lower T . Values of the SC transition temperature ðT c Þ are in agreement with values determined from q.
The T dependence of C e at low T is discontinuous at P c . For all P , the lowest-order term in C e is cðH ÞT . For P < P c , the second term is B AFSW ðH ÞT 3 which corresponds to a spin-wave contribution expected for an AF; for P > P c , it is B 2 ðH ÞT 2 characteristic of certain HF SCÕs.
At 21 kbar and H ¼ 0, 50, and 70 kOe, C e is shown in Fig. 3. T c ðHÞ, assuming a parabolic T dependence, extrapolates to H c2 ð0Þ ¼ 159 kOe. The B 2 ð0ÞT 2 term indicates line nodes in the energy gap and is commonly attributed to a d-wave order parameter in HF compounds [2]. The value of cð0Þ ¼ 0 shows that the Fermi surface (except for the line nodes) is fully gapped, and the SC is both complete and bulk at 21 kbar.
cðHÞ is proportional to H as shown in Fig. 4(b), and extrapolation to H c2 ð0Þ gives c ¼ 382 mJ K À2 mol À1 for the normal-state. In Fig. 3 interpolation of the normalstate C e ðC en Þ between 1.7 and 0 K must give the same entropy ðS e Þ at T c as that for H ¼ 0, 50, and 70 kOe. The shape is similar to that of Kondo singlet-ground-state ordering, and very different from AF ordering. For the SC transition DC e ðT c Þ is relatively small as a consequence of the T dependence of both C en and C es , and the requirement of the equality of S e for the SC and normal states at T c .
The P dependence of cð0Þ is displayed in Fig. 4(a) with the AF values interpolated to the 21-kbar normal-state value and the SC values extrapolated to the AF curve at P c . The curves represent a normal-state c and a measure of the density of quasiparticle excitations. For H ¼ 0 and P P P c , the SC transition leaves a cð0Þ that varies between the normal-state value at P c and zero at 21 kbar. On the SC side of the phase boundary at P c , cð0Þ is the same in the SC and normal states and DC e ðT c Þ ¼ 0. With increasing P , cð0Þ ! 0 and DC e ðT c Þ increases, but with essentially no increase in T c . The extended gapless regions on the Fermi surface of SCÕs with d-wave pairing [3] suggest a basis for this behavior. Below a critical value of the pairing potential the gap vanishes and there is a density of lowenergy quasiparticle states. With increases in the pairing potential a gap appears and grows in amplitude while the quasiparticle density of states decline and go to zero for sufficiently high amplitudes.
Isotherms, S e ðP Þ vs. P , obtained by integration of C e ðT Þ=T , are shown in Fig. 5. Discontinuities near 12 kbar in ðoS e =oP Þ T and at 15 kbar in S e correspond to second-and first-order transitions. The second-order transition could be a change in the volume thermal expansion. The discontinuity in S e is a transition from the AF state to the SC state that includes a small first-order component, which terminates at a critical point in the vicinity of the magnetic ordering T .