Magnetic phase diagram of UCoAl

We report precision c-axis magnetic measurements on a high-quality single crystal of the heavy fermion metamagnet UCoAl. The metamagnetic transition at HM changes from 1st order at low temperature to a crossover at high temperature. HM is nearly linearly increasing with increasing temperature up to a critical temperature T0. The critical temperature T0 is determined from both the field and the temperature dependences of magnetization to be ∼ 11 K. The field dependence of the Sommerfeld coefficient γ is estimated from M(T) by using a Maxwell relation. γ(H) shows a step-like decrease at HM. This behavior is consistent with the previous reports of specific heat and resistivity measurements at low temperatures.


I. INTRODUCTION
Metamagnetism and quantum criticality in strongly correlated electron systems have attracted much attention because of the possibility of new quantum phases, and appear related to unconventional superconductivity and non-Fermi liquid behavior. In itinerant ferromagnets with tricritical points, the metamagnetic transition from the paramagnetic(PM) phase to the ferromagnetic(FM) phase possesses the wing structure of the first order transition in the temperature-pressure-magnetic field phase diagram [1][2][3][4]. The first-order metamagnetic transition at low temperature reaches a critical end point above which the transition changes to a crossover. UCoAl is one of the best materials in which to investigate such a * E-mail: tmatsuda@tmu.ac.jp metamagnetic transition from a PM to a FM phase.
The heavy fermion UCoAl is known to be a unique system with a metamagnetic transition from a paramagnetic to a ferromagnetic ground state in uranium-based compounds. Its crystal structure is the ZrNiAl-type hexagonal structure (space group: P62m, No. 189) without inversion symmetry. When a magnetic field is applied along the c-axis (magnetic easy axis), the paramagnetic ground state becomes a field-induced ferromagnetic state as the magnetic field increases through the metamagnetic transition at H M [5,6]. The magnetization curves are very anisotropic between H||c-axis and H ⊥ c-axis, indicating an Ising-like magnetic behavior. From investigations of chemical pressure and uniaxial pressure effects, UCoAl is recognized as a nearly ferromagnetically ordered system [7,8]. Although the ground state at ambient pressure is paramagnetic, a negative pressure (P c ∼ -0.2 GPa) will induce a ferromagnetic ground state [9]. In this paper, we report the results of magnetic measurements performed on a high quality single crystal, comparing our results with those of previous investigations. We also present the magnetic phase diagram for fields along the c-axis at ambient pressure.

II. EXPERIMENTAL
A single crystal ( Fig. 1) has been grown by the Czochralski pulling method using a tetra arc furnace. The starting materials of U (purity: 99.95%-3N5), Co (3N), and Al (6N) were melted on a water-cooled copper hearth under a high-purity Ar gas atmosphere. The ingot was turned over and was melted again. This process was repeated several times in order to obtain a homogeneous polycrystalline ingot. The ingot was subsequently pulled at a rate of 15 mm/h for the crystal growth. A barshaped sample was cut from the ingot for the magnetic measurements. The residual resistivity ratio was approximately 27 for a current along the c-axis. This indicated that the crystal was the highest quality reported up to now. The magnetic measurements were performed using a commercial SQUID magnetometer (Quantum Design MPMS) at temperature down to 1.8 K in magnetic fields up to 7 T. Figure 2 shows magnetization curves at selected temperatures. At 2 K, a sharp jump in the magnetization is observed at a metamagnetic transition field H M near 0.7 T. With increasing temperature, the metamagnetic transition field shifts to higher field and becomes broader. Finally, the transition smears at high temperatures. We determined the transition field from the peak position of the derivative of the magnetization with respect to the field and plot those fields in the temperature vs magnetic field phase diagram shown in Fig. 3. As mentioned in previous reports, this transition is first order and has magnetic hysteresis around H M at low temperatures. Figure 2(b) shows the hysteresis loop of the magnetization curve below 11 K. The hysteresis width decreases with increasing temperature as shown in Fig. 3. At temperature above 12 K, the hysteresis can not be observed within the experimental accuracy. This fact indicates that the first order transition terminates around T 0 ∼11 K. This result is in good agreement with the recent results of magnetoresistance and magnetostriction [4].

III. RESULTS AND DISCUSSION
From thermodynamics, we expect a slope of dH M /dT to be zero at T = 0 K. However, at temperature above 4 K, H M is roughly linearly increasing with increasing temperature up to T 0 . This positive slope of the firstorder boundary indicates that the entropy S decreases with increasing field across the H M . To estimate ∆S, we use the Clausius-Clapeyron relation ∆S at 4 K is estimated to be 55 mJ/K·mol where dH M /dT ∼ 0.033 T/K and ∆M ∼ 0.3µ B /f.u.. This value is in good agreement with the previous results for the magnetocaloric effect [10]. We also precisely measured the temperature dependence of magnetization M (T ) for fields along the c-axis at selected fields. Figure 4 displays these results for increasing temperature. At fields below 0.6 T, M (T ) shows a broad peak structure around 20 K. This behavior is typical for heavy-electron systems, such as CeRu 2 Si 2 , UPt 3 , and URu 2 Si 2 . This peak structure shifts to lower temperature with increasing field, and it seems to merge with the field-induced ferromagnetic state at low tem-  perature. The interesting point is that the peak position seems to be connected to the end point of the first-order transition, as shown in Fig. 3.
Using the thermodynamic Maxwell relation d 2 M/dT 2 = dγ/dH, we try to estimate the field dependent part of the Sommerfeld coefficient ∆γ(H) based on the data of M (T, H) as shown in Fig. 5, although there is a phase boundary of first order at H M . Here we took the γ 0 value of UCoAl as 73 mJ/K 2 · mol at zero field [10]. Although the step-like behavior of γ in Fig. 5 is qualitatively consistent with both the field dependence of C/T and variation of the A coefficient of the resistivity [4,10], the absolute value of ∆γ across H M is only half that of the reported data [4]. This underestimate of ∆γ(H) can be attributed to the discontinuous volume change at H M , which was neglected in this estimation for simplicity. Indeed, a large negative magnetostriction has been reported accompanying the metamagnetic transition for fields along the c-axis [4,11].

IV. CONCLUSIONS
We grew a high quality single crystal of UCoAl and performed precision magnetic measurements for field along the c-axis. The metamagnetic transition at H M changes from 1st order at low temperature to a crossover at high temperature. H M is nearly linearly increasing with increasing temperature up to a critical temperature determined to be approximately 11 K by both field and temperature dependence of magnetization. The field dependence of the Sommerfeld coefficient γ was estimated from M (T ) using a Maxwell relation. γ(H) shows a step like decrease at H M . This behavior is consistent with the previous reports on specific heat and resistivity at low temperatures.

ACKNOWLEDGMENTS
The present work was financially supported by Grantsin-Aid for Young Scientists (B)(22740241) and Scientific Research on Innovation Areas"Heavy Electrons" (20102002, 23102726), from the Ministry of Education, Culture, Sports, Science and Technology (MEXT). This work was also supported by the EC program gTransnational Accessh (EuromagNET II), ERC starting grant (NewHeavyFermion) and French ANR project (COR-MAT, SINUS).