Sonia Francoual

The effects of rhenium doping on the temperature-versus-high magnetic field phase diagram of URu₂Si₂

Heavy fermions are actinide or rare-earth intermetallic alloys in which the hybridization between the localized f electrons and the conduction electrons yields strong correlations that cause important renormalization of the Fermi surface at low temperature. The primary electron–electron interactions at play are the local Kondo interaction associated with the screening of a virtual f-magnetic impurity by the conduction electrons and the indirect exchange long-range RKKY interaction between f magnetic moments. (RKKY stands for Ruderman-Kittel-Kasuya-Yosida and refers in a metal to the interaction of magnetic spins mediated by conduction electrons.)

These two interactions compete, yielding heavy fermion behavior and magnetic ordering, respectively. The fragile balance between the two interactions can be tuned by an external parameter such as pressure, magnetic field, or chemical doping, all of which may induce a quantum phase transition at absolute zero into a magnetically ordered state. On the way to and away from the transition, a variety of unusual electronic and magnetic behaviors is observed, from non-Fermi liquid behavior to unconventional superconductivity.

The uranium-ruthenium-silicon compound URu₂Si₂ is a moderately heavy electron compound crystallizing in the I4/mmm body-centered tetragonal structure of the thorium-chromium-silicon compound ThCr₂Si₂. At temperatures above the coherence temperature of 80 kelvin (K), the conduction electrons are scattered by isolated spins. At temperatures below 80 K, the Kondo lattice regime in which the resistivity becomes Fermi-liquid-like prevails and the mass enhancement due to electron correlations is approximately 25 m_e (m_e being the mass of the free electron). At lower temperatures, URu₂Si₂ orders with a transition at 17.5 K into a hidden-order (HO) phase presenting a small antiferromagnetic moment of 0.03 μ_B (μ_B stands for Bohr magneton, the unit used to express the electron magnetic dipole moment) and finally a transition at 1.2 K into a superconductive phase.

The transition at 17.5 K is abrupt and is observed in several measurable physical quantities such as resistivity, magnetic susceptibility, specific heat, thermal expansion, thermoelectric power, and thermal conductivity. However, the order parameter in the hidden-order phase remains unidentified. The recent neutron scattering observations of the gapping of itinerant-like incommensurate spin excitations and of the absence of crystalline electric field excitations below 10 millielectronvolts (Wiebe, Janik, MacDougall, and others in Nature Physics 3, 2007) and the recent field- and temperature-dependent measurements of the Hall Effect (Oh, Kim, Sharma, and others in Physical Review Letters 98, 2007) unveil important changes of the Fermi surface topology at the transition pointing toward an itinerant density-wave type order parameter.

In heavy-electron systems, the application of a magnetic field leads to the progressive breaking of the hybridization between the f electrons and the conduction electrons upon the alignment along the magnetic field of the spin degrees of freedom of the f quasiparticles. Their complete polarization suggests recovering of the full moment μ_B per f atom and is accompanied by a dramatic increase of the magnetization. The phenomenon—called itinerant electron metamagnetism—has been observed in URu₂Si₂ as has the non-Fermi liquid behavior associated with the putative existence of a quantum critical point at absolute zero between the low-field paramagnetic hidden-order-phase and the high-field polarized Fermi liquid.

A remarkable situation in URu₂Si₂ is that the divergence of the fluctuations expected at the quantum critical point is avoided, and multiple-field-induced ordered phases form instead, as shown in the figure at left. The ordering at the quantum critical point has been “ascribed to reentrant phenomena arising from the interplay of itinerant electron metamagnetism and the hidden order parameter as the hidden-order phase is suppressed at 36 T [tesla],” (Harrison, Jaime, and Mydosh in Physical Review Letters 90, 2003). An extensive body of research in strong magnetic fields has been carried out in URu₂Si₂ over the past five years to determine on one hand the properties of the Fermi liquid upon which the ordering manifests and on the other hand the nature of the ordering at high field in relation to the hidden order at low field.

A 5-percent-or-less transition-metal substitution of ruthenium atoms in URu₂Si₂ suppresses the hidden order at 17.5 K, while the overall heavy-electron behavior is preserved. Studies of the combined effects of rhodium doping and applied magnetic field reveal that the ordering in URu₂Si₂ at high field is significantly modified with the development of a single-field-induced robust phase (namely phase II in the figure on the previous page) around the putative quantum critical point located at approximately 34 T in URu_1.92Rh_0.08Si₂.

At low field, a heavy Fermi liquid appears where the hidden order is suppressed. Those properties follow the addition and the removal of conduction electrons associated with the transition-metal substitution and the modifications of the degree of hybridization between the f electrons and the conduction electrons of the ligand transition-metal atoms.

More-substantial transition-metal substitutions yield a complex magnetism that can be either antiferromagnetic ordering (rhodium, palladium, iridium, and platinum) or itinerant ferromagnetism (rhenium, technetium, and manganese), depending on the nature of the transition-metal element. Of particular recent interest is the doping of URu₂Si₂ with rhenium as ferromagnetic order develops for substitutions above 15 percent at which order coexists with non-Fermi liquid behavior at temperatures below 14 K.

In this research project, the high magnetic field phase diagram of URu_2-xRe_xSi₂ single crystals is mapped by magnetoresistance and magnetization measurements at low temperature in pulsed magnetic fields at the National High Magnetic Field Laboratory in Los Alamos. An accurate determination of the position in field and temperature of the phase transitions as a function of the rhenium doping was made on each sample through resistivity and magnetization measurements. Brian Maple and his group at the University of California, San Diego grew the samples. Maple’s group combined high-purity uranium, ruthenium, rhenium, and silicon via arc melting in an argon atmosphere to grow polycrystals from which single crystals were produced using the Czochralski method in a tri-arc furnace under flowing argon. The crystal quality and lattice parameters were confirmed by X-ray powder diffraction.

The crystals have a plane-parallel shape of approximately 0.7 x 1.7millimeters (mm), the longer edge being along the easy axis of magnetization c. The thickness is limited to 0.2 mm to avoid magnetocaloric effects and Joule heating of the sample during the pulse. Because the measured zero-field resistance of some of the samples is on the order of 5 milliohms (mΩ) or less at 4 K, the standard method of four-wire resistivity measurement in pulsed field becomes challenging. The alternative contactless tunnel diode oscillator technique consists of measuring the change in frequency and amplitude of the circuit oscillations of a self-resonant radio-frequency tank circuit composed of a coil (inductor) in the center of which the sample is placed.

Magnetoresistance curves measured at several different temperatures in URu1.96Re0.04Si2 using the tunnel diode oscillator (TDO) technique in the 50-tesla (T) mid-pulse magnet at the Los Alamos National High Magnetic Field Laboratory are shown in the figure above left. The sample is oriented with the c axis aligned with the magnetic field B. “X” and “O” symbols mark the field position of the resistivity maximum and the exit from the hidden-order phase, respectively. TDO magnetoresistance curves at low temperatures shown in the 33—41 T narrow magnetic field range are shown above left. Magnetoresistance curves measured at several different temperatures in URu1.92Re0.08Si2 using the four-wire standard technique in the 50-T mid-pulse magnet are shown in the figure above right. The sample is oriented such as the c axis is aligned with the magnetic field B and the current j is along the a axis. Curves have been shifted vertically in the aim of visual comparison with the TDO magnetoresistance curves in the figure above left.

Whereas the four-wire method gives direct access to the resistance through an AC excitation current being passed through the sample and an AC potential difference measured across the sample, changes in the tunnel diode oscillator resonant frequency measure changes in the depth of the radio-frequency field penetrating the sample, which is inversely proportional to the sample conductivity.

The tunnel diode oscillator experiments were carried out in a capacitor-driven 15-mm-bore 50-T nondestructive mid-pulse magnet. In this magnet, the 10-turns 1.9 mm-diameter compensated copper coil sits orthogonal to the magnetic field at the center of the field on the bottom part of a 1.5-m stick probe inserted in a helium-3 refrigerator. The tunnel diode, which is built into the probe, lies approximately 50 centimeters (cm) above the field center; the rest of the radio-frequency tank circuit sits outside of the helium-3, helium-4, and magnet spaces at a distance of approximately 1 meter (m).

The radio-frequency circuit, set at an initial frequency of 200 kilohertz (kHz), allows absolute value frequency variations up to 200 kHz. The 2.5-megahertz upper digitizing frequency of the computer-controlled data acquisition card samples the time-dependent oscillatory behavior of the output signal.

The figures above the magnetoresistance curves measured on the falling field of the pulse at several different temperatures in URu_1.96Re_0.04Si₂ using the tunnel diode oscillator technique and in URu_1.92Re_0.08Si₂ using the four-wire technique. (In the latter case, the sample has a bar-like shape elongated along the a tetragonal axis and a 10-mΩ resistance at 5.5 K).

The magnetization measurements were carried out using a wire-wound sample-extraction magnetometer. Two measurements were performed consecutively, one with the sample inserted in (coupled to) the detection coil and a second with the sample removed from the coil, enabling a fully compensated signal. The magnetization curves measured on the falling field in URu_1.98Re_0.02Si₂ and URu_1.92Re_0.08Si₂ are displayed in the figures on the next page. For all measurements, the sample was mounted and oriented so that the c axis was aligned with the field and the sample lay at the center of the field. The temperature was reduced to 0.5 K by pumping on the helium-3 and-4 baths and heated to 20 K by a heater placed close to the sample.

Quantum criticality is evidenced by the presence of a broad maximum in the field dependence of the magnetoresistance, the position of which (when followed in temperature) enables one to extrapolate the position of the quantum critical point at zero temperature. In the URu_2-x Re_xSi₂ samples studied (x < 10 %), this quantum critical (disordered) region is clearly identified in the magnetoresistance curves, a broad maximum emerging at fields around 34 T at 15 K, the width of which narrows and the position of which slightly shifts as the temperature is decreased.

Field dependence of the magnetization in URu1.98Re0.02Si2 measured at several different temperatures in the 50-tesla (T) mid-pulse magnet is shown in the figure at far left. The sample is oriented such that the c axis is aligned with the magnetic field B. Curves have been shifted vertically for clarity. The metamagnetic transition shows up at high temperatures as a broad magnetization increase for fields above 34 T. Below 4.0 kelvin (K), step-like features associated with phase transitions can be seen. Higher temperatures at which phase transitions are observed in the metamagnetic crossover region in the magnetization curves as a function of the rhenium doping in URu2-xRexSi2 are shown in the far left insert. Field dependence of the magnetization in URu1.92Re0.08Si2 measured at several different temperatures in the 65-T short-pulse magnet is shown in the figure at near left. The sample is oriented such that the c axis is aligned with the magnetic field B. Curves have been shifted vertically for clarity. The metamagnetic field Bm at 7.5 K is centered at 38.50 (0.5) T in URu1.92Re0.08Si2, whereas in the figure at far left it is centered at 37.3 (0.5) T in URu1.98Re0.02Si2 . Step-like features in the magnetization curves down to 0.5 K are not observed for this composition.

This maximum can be followed down to 2.1 K in URu_1.96Re_0.04Si₂ and down to 0.5 K in URu_1.92Re_0.08Si₂. In URu_1.96Re_0.04Si₂, the sharpening and discontinuous shift and increase in magnitude of this maximum signal the entrance into an ordered phase at 1.7 K. The hidden-order phase associated at zero field with an enhancement of the resistivity below 15 K and 12 K for x = 0.04 and 0.08 in URu_2-xRe_xSi₂, respectively, clearly appears in the magneto-resistance curve at fields up to approximately 35 T and approximately 31 T as shown in the figures on the previous page.

Although the increase in resistivity in the hidden order relative to the increase in the quantum critical region is considerably reduced from 0.04 to 0.08, the field required to suppress the hidden order, although reduced, remains high. Itinerant electron metamagnetism manifests in URu₂Si₂ as a large increase in the magnetization at a field B_m of 38 T above a cut-off critical temperature (T_c) of 6 K. At lower temperatures, successive magnetization plateaus (spin-flip-like features) appear inside the metamagnetic crossover region as the consequence of successive phase transitions between different ordered phases (II, III, IV, V in the figure on page 6.

The metamagnetic transition is shown to survive for substitutions of the ruthenium atoms by rhenium up to 4% (x = 0.08). However, the magnetization plateaus observed at x = 0.02 below 5.0 K and at x = 0.04 below 2.0 K are not observed at x = 0.08 down to 0.5 K in agreement with the magnetoresistance measurements. B_m as obtained from a fit of the centered position of a single broad maximum in the differential susceptibility, moves to higher fields as the rhenium content increases, with B_m at 7.5 K approximately 37.3 (0.5) T at x = 0.02 and approximately 38.5 (0.5) T at x = 0.08. This rise is consistent with a proposed qualitative picture in which the low rhenium doping contributes additional electrons to the system, making it more itinerant, and higher fields are required to break a strengthened hybridization between the f quasiparticles and the conduction band.

Marked differences are observed in the high-field phase diagram of URu₂Si₂ under rhenium doping as compared with the reported rhodium case. A progressive reduction in the ordering regions around the quantum critical point occurs as the rhenium content is increased, implying a complete suppression of the ordering at a rhenium doping of x ~ 0.05, suppression that becomes complete at x = 0.08. The underlying heavy Fermi liquid is one in which for x < 0.10 the f electrons gain in itinerancy, the opposite trend being observed in the rhodium case.

The present study sheds light on the effect of low transition-metal substitutions (rhenium doping versus rhodium doping on the ruthenium sites, x < 0.10) on the formation of novel-ordered phases near the quantum critical point in the heavy-fermion compound URu₂Si₂. We are now investigating how higher rhenium substitutions (between x = 0.10 and x = 0.30) that lead to the unusual coexistence of non-Fermi liquid behavior and emerging ferromagnetic order at zero field modify the itinerant electron metamagnetism properties and the spectrum of the quantum critical fluctuations at high magnetic field.

The experimental setup for the pulse-field measurements. (Schematic adapted from drawings by Ruminer and Swenson of the Los Alamos National High Magnetic Field Laboratory.) The oriented sample sits on the bottom of a 1.5-meter-long stick probe that is loaded into a helium-3 refrigerator and then inserted into a helium-4 cryostat. The temperature is tuned down to 0.5 kelvin (K) pumping on the helium-3 and -4 baths and heated to 20 K by applying a small amount of power on a heater situated close to the sample. A calibrated thermometer placed close to the sample measures the temperature before the pulse. The sample is placed at the center of the magnetic field at the beginning of the experiment. The magnetic field is measured during the pulse by means of a calibrated sensor coil placed close to the sample.

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