Band-structure and anomalous contributions to the Hall effect of YbRh 2 Si 2

We report on Hall effect measurements on YbRh 2 Si 2 single crystals with different residual resistivity and on LuRh 2 Si 2 single crystal. The temperature dependence of the linear-response Hall coefﬁcient of YbRh 2 Si 2 is described by the anomalous Hall effect and the normal contribution incorporating multi-band effects. Sample dependencies at low T are found and explained by slight changes in the charge-carrier concentrations.

YbRh 2 Si 2 is a stoichiometric heavy fermion (HF) compound showing non-Fermi-liquid (NFL) properties in a wide temperature range above the antiferromagnetic transition at T N ¼ 70 mK [1]. T N is found to be continuously suppressed with small magnetic fields to T ¼ 0 accessing the quantum critical point (QCP) [2]. The QCP in YbRh 2 Si 2 is best described by the local scenario [3]. This is mainly supported by earlier Hall effect studies [4] in which the Hall effect of YbRh 2 Si 2 was measured while tuning the material through the QCP. Here, the Hall coefficient R H as a mirror of the Fermi surface volume was found to show a crossover which sharpens to a discontinuous jump if extrapolating T ! 0. It was concluded that the Fermi surface volume abruptly changes at the QCP. This crossover was followed up to 500 mK introducing a new energy scale which is confirmed by other transport properties as well as thermodynamic quantities [5]. In order to understand differences found in R H ðTÞ of newer samples we extended the measurements to samples of varying quality including the sample used in Ref. [4] and to the non-magnetic analog LuRh 2 Si 2 .
Single crystals of YbRh 2 Si 2 and LuRh 2 Si 2 were synthesized with an In flux-growth technique as described earlier [1]. All samples were polished to thin plates and prescreened via rðT; BÞ to ensure In free samples. Samples of thickness t between 30 and 80 mm were contacted via spot welded gold wires. The current I was applied in the crystallographic ab-plane and the magnetic field B along the crystallographic c-axis of the samples inducing the Hall voltage V xy perpendicular to I and B (see inset of Fig. 1). The Hall resistivity was derived from the asymmetric component of the field-reversed Hall voltage, r H ðBÞ ¼ t½V xy ðþBÞÀ V xy ðÀBÞ=2I. The comparison of the symmetric component with the transversal magnetoresistivity measured simultaneously was used as a consistency check. R H was calculated in the low-field limit from the initial slope of the Hall resistivity, R H ¼ ðdr H ðBÞ=dBÞ B¼0 . Measurements down to 2 K were performed in a commercial PPMS, measurements between 17 mK and 6 K in a dilution refrigerator newly set up for these experiments. The characteristics of LuRh 2 Si 2 are typical of a nonmagnetic intermetallic system. The resistivity r is linear in T above % 100 K with rð300 KÞ ¼ 20 mO cm. The magnetoresistance is quadratic for small fields. All these properties are consistent with a simple metallic system.
In LuRh 2 Si 2 R H ðTÞ is positive at all T and constant below 20 K and above 100 K. Between these temperatures the Hall coefficient changes its value by a factor of 1.6. Band-structure calculations for LuRh 2 Si 2 [7] reveal two hole-like bands at the Fermi energy. Therefore, we approximate the observed T dependence by a simple twoband model in the low-field limit [8] where R H ðTÞ is the mobility weighted sum of two bands of positive charge carriers. The data are well described assuming one band dominating at high temperatures due to its high mobility arising from lower phonon scattering rates and the other band dominating the low temperature Hall coefficient by its enhanced mobility due to a lower scattering rate at residual scatterers. The crossover temperature between these limits is determined by the Debye temperature known from specific heat measurements. Surprisingly, very similar behavior of R H ðTÞ has been observed in the non-magnetic La analogs of the Ce-115's [9].
A negative sample-independent R H ðTÞ is found for all YbRh 2 Si 2 samples at high temperatures, followed by a minimum at T % 100 K and a sign change at T % 20 K. However, below 25 K sample dependencies are obvious.
R H ðTÞ exhibits a maximum at % 1 K for all samples but with different absolute values and at slightly different temperatures. At lowest temperatures R H ðTÞ saturates at different values R 0 H . Furthermore, for sample 2 of highest quality, a plateau between 7 and 13 K is found at a value quite close to R 0 H . For the other samples a shoulder is observed in this temperature range roughly pointing towards R 0 H for each sample. The Hall effect of HF compounds is usually the sum of a normal contribution, R n H ðTÞ, reflecting the band-structure and an anomalous contribution due to skew scattering of the conduction electrons on the orbital f moment, R a H ðTÞ ¼ CrðTÞwðTÞ, with w the susceptibility and C a constant [10]: Previously, the Hall effect of sample 1 was described by Eq. (1) Fig. 1). If we incorporate the multi-band effects observed in LuRh 2 Si 2 we can understand R H ðTÞ of these samples down to lowest T. The step observed for LuRh 2 Si 2 is also present in YbRh 2 Si 2 . Particularly, sample 2 shows the same slope of R H ðTÞ for 7 KpTp100 K, even the step height is identical (although shifted by À5 Â 10 À10 m 3 C À1 ). Thus, R H ðTÞ may be described by Eq. (1) if we take into account the T-dependent normal Hall coefficient seen in LuRh 2 Si 2 and the constant C being lowered below the coherence temperature as often found in HF [10]. Furthermore, we are now able to account for the sample dependencies within the two-band model used for LuRh 2 Si 2 . Here, the low-T value R 0 H strongly depends on the charge-carrier concentration n l of the low-T dominating band. Thus, the different R 0 H are reasonably explained by small deviations of n l suggesting the normal contribution to be dominant below 100 K.
The dominance of R n H ðTÞ at low T is further supported by the absence of any scaling between r 0 and R 0 H expected for the anomalous Hall effect in the T ¼ 0 limit where R a H ðT ¼ 0Þ ¼ Cr 0 wðT ¼ 0Þ with wðT ¼ 0Þ usually less sample dependent than r 0 . The slight variation of n 1 would also request for a scaling of r 0 and R 0 H but this might be hidden by a change of the impurity concentration for different samples.
According to this analysis R H ðTÞ is expected to be almost temperature independent below 10 K. By contrast the data show a pronounced maximum around 1 K superposed to the underlying constant contribution. We associate this maximum with the NFL properties of YbRh 2 Si 2 .
To conclude, the Hall effect of YbRh 2 Si 2 is described by contributions of the anomalous and the normal Hall effect reflecting the band structure. The band-structure  1) to the data of sample 2 (1) above 25 (7) K, where R n H was assumed to be constant. At lower temperatures they are obtained by extrapolation according to Eq. (1) with C and R n H obtained from the high temperature fit. The inset sketches the setup. effect is deduced by analyzing the Hall effect of LuRh 2 Si 2 in terms of a two-band model. Below 100 K, the normal Hall effect dominates and is superposed by a broad maximum at T % 1 K assigned to the NFL properties. Sample dependencies are explained in the two-band model by small deviations of the charge-carrier concentration.