Figure 8.4(a) presents the magnetoresistivityρx,x of URu2Si2 sample #2 versus the magnetic fieldH atT = 1.5 K for different anglesθ1between the magnetic field and the c-axis. The magnetic field is turning from the transverse (H∥c;H⊥I,U; θ1
= 0◦) to the longitudinal (H∥ a; H∥ I,U; θ1 = 90◦) configurations. Contrary to sample #1 (presented in Sect. 8.2, RRR = 90), sample #2 (RRR = 225) does not show an anomaly at H∗ for H ∥ c, as seen in Section 7.3. When θ1 increases, the general form of the magnetoresistivity remains unchanged, but the anomalies from the metamagnetic transitions at H1,H2, andH3 and from the crossover at Hρ,maxLT are shifted to higher field values. For θ1 >50◦, the anomalies are shifted out of the field range (60 T) and the resistivity increases monotonically. The value of ρx,x at the maximum at Hρ,maxLT is also slightly increasing withθ1, which may be due to a small misalignment of the sample. The θ1-dependencies of the transition fieldsH1, H2, and H3 and the crossover fieldHρ,maxLT are presented in Figure 8.6(a).
Figure 8.4(b) presents the magnetoresistivityρx,xof sample #2 versus the magnetic fieldHatT = 1.6 K for different anglesθ2between the magnetic field and thec-axis.
θ2is the angle betweencandH, which lies in the (a,c)-plane and is perpendicular to the electric current. Hence the magnetic field is turning from the transverse (H∥c;
H ⊥ I,U) to the transverse (H ∥ a; H ⊥ I,U) configurations. At small angles, we observe that the same field-induced anomalies as that observed for sample #1 in Chapter 5. With increasing angle θ2, the transition fields H1, H2, and H3 and the crossover fieldHρ,maxLT shift to higher field values and are shifted out of the field range (upper limit: 60 T) for θ2 ≥65◦. The angle-dependence of H1, H2, H3, and
8.3 Crossover and transition fields for Hin the plane (a,c)
Figure 8.4:(a) Magnetoresistivity ρx,x of sample #2 versus the magnetic field H at T = 1.5 K for different anglesθ1 betweenH andc. The field is turning in the (a,c)-plane from the transverse (θ1= 0◦) to the longitudinal (θ1= 90◦) config-urations. (b) Magnetoresistivityρx,x of sample #2 versusH atT = 1.6 K for different angles θ2 between the magnetic field and the c-axis. The magnetic field is turning in the transverse (a,c)-plane.
Hρ,maxLT is presented in Figure 8.6(b). In this second configuration, the magnetic field remains transverse for all angles θ2 and the general form of the magnetoresistivity evolves differently than in the first configuration. For θ2 = 90◦, i.e., when H ∥ a, the transverse magnetoresistivity increases continuously up to the maximal field and no field-induced anomalies are observed. The oscillatory modulation ofρx,x(H) for θ2 = 90◦ is due to the Shubnikov-de Haas effect and will be analyzed in Chapter 9. Remarkably, the heights of the plateaus of phase V between H1 and H2 and phase III between H2 and H3 are independent of the orientation of the magnetic field relatively to the c-axis or to the current, as seen in Figures 8.4(a) and 8.4(b).
AboveH1, the magnetoresistivity is also sample-independent, as shown in Chapter 7, and has thus no observable orbital contribution. We conclude that the resistivity in phases III and V is dominated by the scattering of the charge carriers on the magnetic moments of the localized 5f-electrons. Within this picture, the scattering off of the f-electrons is sample-independent, since it corresponds to a scattering of the conduction electrons by the static or fluctuating magnetic moments from each 5f U-ion site, the distance - between two ions - involved in this process being smaller than the distance between two impurities.
A pronounced inflection point inρx,x(H) of sample #2 develops at a field well below Hρ,maxLT for θ2 ≥35◦. This is the same anomaly as that observed in the resistivity of sample #1 for H∥c atH∗, which is defined at the kink just after an inflection point [see Chapter 7], and corresponds to a Fermi surface-related crossover. Figures 8.5(a) and (b) present for both samples the evolution, with an increasing angle θ2,
8 Angle-Dependent Study of the High-Field Magnetoresistivity
sample #1 sample #2
2
sample #1 sample #2
2
Figure 8.5: (a) Definition of the anomaly at H∗ in the magnetoresistivity of samples #1 and #2 for different orientations of the magnetic field in the transverse (a,c)-plane. (b) Equivalent definition ofH∗ in the field-derivative of the resistivity.
of the kink-like anomaly in the resistivity and in the associated step-like anomaly in the field-derivative of the resistivity, respectively. Even if this anomaly at H∗ is not observed for sample #2 for H ∥ c, the figures 8.5(a) and (b) clearly shows that both samples exhibits a similar anomaly at H∗ forθ2 ≥35◦. Figure 8.6 shows the angle-dependence of the transition fields H1, H2, and H3, and the crossover fields Hρ,maxLT and H∗ in angles up to 60◦. All of these transition and crossover fields can be fitted with 1/cosθ-functions (θ = θ1,θ2). The angle-dependence of the low-temperature critical fields H1, H2, and H3 established by the resistivity measurements of this work is consistent with results from previous magnetization experiments for θ up to 15◦ [Sugiyama 1990] and resistivity experiments for θ up to 30◦ [Jo 2007]. The angle dependence of the crossover field Hρ,maxLT defined at the maximum of the resistivity is a new feature of the present work. The 1/cos θ-dependence indicates that the physics of these transitions and crossovers only depend on the projection of the magnetic field on thec-axis, at least up toθ= 60◦ (θ=θ1, θ2). The 1/cosθ-dependence of H∗ observed here is consistent with results from previous resistivity experiments for θ up to 40◦ [Shishido 2009, Aoki 2012]. The resistivity of our sample #2 does not show an anomaly at H∗ for θ = 0 but an extrapolation of the 1/cosθ-law leads toµ0H∗→20 T, when θ2 →0, which is much lower thanµ0H∗ ≃25 T in sample #1 or the values of 22.5 and 24 T reported in the literature [Shishido 2009, Altarawneh 2011, Aoki 2012]. Sample #2 has the highest quality and the anomaly at H∗ in its magnetoresistivity may be hidden, forH∥c, by an additional orbital contribution, whose intensity decreases at high angles θ2, while the intensity of the anomaly at H∗ increases at high θ2.