Evidence of a Kondo Destroying Quantum Critical Point in YbRh₂Si₂

Rare-earth-based intermetallic compounds with heavy-fermion (HF) phenomena (“HF metals”) are well described within the framework of the Kondo lattice.¹⁾ In contrast to other families of correlated electron materials, HF metals exhibit a clear hierarchy of relevant energy scales, i.e., spin–orbit coupling, crystal-field (CF) splitting of the localized \(4f\) shell, single-ion Kondo energy and Kondo coherence.²⁾ Following the discovery of HF phenomena in thermodynamic and transport properties of CeAl₃,³⁾ the surprising observation of HF superconductivity in CeCu₂Si₂⁴⁾ and subsequently in a few U-based intermetallics, like UBe₁₃⁵⁾ and UPt₃,⁶⁾ initiated intense studies in this field. In addition to superconductivity, a second “revolution” sparked by HF research led to worldwide research activities on quantum criticality.^7–9) Antiferromagnetic (AF) quantum critical points (QCPs) in HF metals have served as prototypical settings to study non-Fermi-liquid (NFL) phenomena as well as quantum-criticality-driven novel phases, notably unconventional superconductivity.¹⁰⁾

In itinerant (d-electron-based) metals a spin-density-wave (SDW) QCP commonly occurs.^11–14) A SDW QCP may also be realized in HF metals, namely if the heavy (composite) charge carriers behave like d electrons in that they keep their integrity at the QCP. For example, the HF superconductor CeCu₂Si₂ exhibits a three-dimensional SDW QCP,¹⁵⁾ whose critical fluctuations were found to drive the formation of the Cooper pairs.^16–18)

For CeCu₂(Si_0.9Ge_0.1)₂ two separate superconducting domes exist at different pressure ranges:¹⁹⁾ One of them is centered at the SDW QCP at ambient pressure (\(p\)), resembling the dome of superconductivity which forms around the p-induced QCP in CePd₂Si₂.¹⁰⁾ The second dome occurs at about \(p = 5\) GPa and is believed to be located close to a weak valence transition, where Cooper pairing is ascribed to nearly quantum critical valence fluctuations,^20,21) as expected within a “valence-crossover” QCP scenario.^22,23) The latter has been assumed to also apply²²⁾ to the p-induced AF QCP in the HF superconductor CeRhIn₅.²⁴⁾ For this material, de Haas–van Alphen measurements²⁵⁾ performed in the field range \(10\leq B\leq 17\) T, i.e., not far above \(B_{\text{c2}}\approx 9\) T, revealed an abrupt reconstruction of the Fermi surface (FS) to occur at \(p_{\text{c}}\approx 2.3\) GPa, where the AF order is suppressed.²⁶⁾ However, the fact that this suppression of the AF order occurs smoothly^26,27) rules out the valence-crossover QCP scenario,²²⁾ as this description would require the AF quantum phase transition in CeRhIn₅ to be of first order. Therefore, the nature of the p-induced quantum criticality in CeRhIn₅ suggests an unconventional QCP.^28–30) In case of a “local” QCP^28,29) strong low-dimensional spin fluctuations are assumed to cause a critical destruction of the HFs, i.e., the propagating Kondo singlets. This has led to a new low-energy scale \(E^{*}(\delta)\), where δ is the control parameter. Distinct anomalies in physical properties observed at \(E^{*}(\delta)\), as discussed below, hint at an abrupt FS reconstruction at the QCP [see Fig. 1(a)]: The heavy quasiparticles exist only well below the Kondo temperature \(T_{\text{K}}\) on the paramagnetic side where they form a heavy Landau Fermi liquid below \(T_{\text{FL}}\). As here the \(4f\) states are delocalized, they contribute to a “large” FS. At an SDW QCP, \(E^{*}(\delta)\) is finite and the FS remains large [see Fig. 1(b)].

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Figure 1. (Color online) Schematic view of the temperature (T)–control parameter (δ) phase diagram near antiferromagnetic (AF) quantum critical points (QCPs) of Kondo break-down (a) and spin-density-wave (b) types. \(T_{0}\) indicates the onset of local Kondo screening, once all Kondo ions have adapted the lowest lying, crystal-field derived Kramers doublet. \(E^{*}(\delta)\) is a quantum critical energy scale indicating a Fermi surface crossover from “small” to “large”. From Ref. 9.

In the following, we discuss salient NFL phenomena as well as several pieces of evidence for a sudden FS reconstruction at the AF QCP in the tetragonal HF metal YbRh₂Si₂.^31–33) This compound has been shown to behave as a prototypical Kondo lattice system,³⁴⁾ with a characteristic Kondo temperature \(T_{\text{K}}\approx 30\) K, referring to the lowest-lying CF-derived Kramers doublet (labeled \(T_{0}\) in Fig. 1). At \(T_{\text{K}}\), where most of the Yb³⁺ ions are identical, spatial coherence among the charge carriers sets in.³⁴⁾

YbRh₂Si₂ is a weak antiferromagnet below \(T_{\text{N}}\approx 0.07\) K,³¹⁾ with a tiny ordered moment \(\mu_{\text{ord}}\approx 2\times 10^{-3}\) μ\(_{\text{B}}\)/Yb³⁺.³⁵⁾ A small magnetic field \(B_{\text{c}}\approx 0.06\) T (\(B\perp c\)) respectively 0.66 T (\(B\parallel c\)) is sufficient to suppress the AF order. This suppression occurs continuously when increasing the field through \(B_{\text{c}}\): Measurements of the magnetostriction reveal the AF phase transition to remain of second order to the lowest accessible transition temperature of 20 mK.³⁶⁾ From Fig. 2(a) one infers that upon cooling, the Sommerfeld coefficient of the electronic specific heat, \(\gamma(T) = C_{\text{el}}(T)/T\), is diverging at the critical field \(B = B_{\text{c}}\). In addition, below \(T\approx 0.15\) K, the electrical resistivity measured at \(B = B_{\text{c}}\) depends linearly on temperature, \(\rho(T) =\rho_{0} + aT\) [Fig. 2(b)]; \(\rho_{0}\) being the residual resistivity. Extrapolation of these T dependences of \(\gamma(T)\) and \(\rho(T)\) to zero temperature yields a NFL ground state with diverging quasiparticle mass \(m^{*}\) exactly at the field-induced QCP. At \(B\neq B_{\text{c}}\), on the other hand, YbRh₂Si₂ behaves as a heavy Fermi liquid, cf. Figs. 2(a) and 2(b).

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Figure 2. (Color online) Low-temperature thermodynamic and transport properties near the QCP in YbRh₂Si₂. (a) Sommerfeld coefficient of the electronic specific heat γ vs T at three magnetic fields applied within the basal, tetragonal plane (\({\perp}c\)). At the critical field (\(B_{\text{c}}\approx 0.06\) T), one observes \(\gamma\sim T^{-\varepsilon}\) (\(\varepsilon\approx 0.3{\text{--}}0.4\)) at the lowest temperatures and \(\gamma\sim\ln\) (\(T_{0}/T\)) at elevated temperatures (\(T_{0}\approx 24\) K, i.e., close to \(T_{\text{K}}\approx 30\) K). Inset: γ vs T at zero field over an extended T range. (b) Electrical resistivity ρ vs T at the same fields as in (a). From Ref. 9.

In this context, several remarkable observations are worth mentioning:

Direct FS studies at the QCP in YbRh₂Si₂ are not possible. Angle-resolved photoemission spectroscopy (ARPES) below \(T = 0.1\) K with a correspondingly high energy resolution is not available at present. On the other hand, measurements of magnetic quantum oscillations require the application of magnetic fields of several T, which would fully suppress the quantum critical fluctuations in YbRh₂Si₂.⁹⁾ Therefore, in order to investigate the evolution of the FS in the quantum critical regime of this material one has to rely on thermodynamic and transport measurements.

Isothermal studies of the magnetic field dependence of the initial Hall coefficient \(R_{\text{H}}\) (being almost identical with the normal Hall coefficient at \(T < 1\) K) revealed a large drop in \(R_{\text{H}}(B)\) at a crossover field \(B^{*}(T)\) and a substantial narrowing of this crossover upon cooling.⁴⁰⁾ Figure 3(a) displays the crossover line \(T^{*}(B)\) [\(=T(B^{*})\)], which is found to merge with both \(T_{\text{N}}(B)\) and \(T_{\text{FL}}(B)\) at \(B = B_{\text{c}}\) in the zero-temperature limit⁴¹⁾ and has been shown³⁶⁾ to represent a new thermodynamic energy scale.^28,29) Careful investigations of single crystals of widely differing quality showed, via various types of magnetotransport probes, that the crossover width is proportional to temperature⁴¹⁾ [cf. Fig. 3(b)]. This implies, upon extrapolation to \(T = 0\), an abrupt finite jump of \(R_{\text{H}}(B)\) at the field-induced QCP and verifies the nature of the Kondo breakdown QCP as predicted by Si et al.²⁸⁾ and Coleman et al.²⁹⁾ In the following, we shall address the dynamical processes associated with this unique type of instability.

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Figure 3. (Color online) (a) Position of the Fermi surface crossover in various magneto-transport experiments on samples of different quality in the temperature-field phase diagram of YbRh₂Si₂. Red horizontal bars are crossover widths, cf. (b). Dotted/dashed line marks the magnetic phase boundary \(T_{\text{N}}(B)\)/crossover to paramagnetic Landau-Fermi liquid phase \(T_{\text{FL}}(B)\). (b) Crossover width (full width at half maximum, FWHM) obtained from the same measurements exploited in (a). From Ref. 41.

The Wiedemann Franz (WF) law describes the combined heat and charge transport in a metal at absolute zero temperature, where all scatterings are elastic. Defining the thermal resistivity by \(w = L_{0}T/\kappa\), where κ is the thermal conductivity and \(L_{0} = (\pi k_{\text{B}})^{2}/3e^{2}\) Sommerfeld's constant, the WF law states that (as \(T\rightarrow 0\)) the residual thermal and electrical resistivities are identical: \(\rho_{0}/w_{0} = L/L_{0} = 1\). Here, \(L =\rho\kappa/T\) denotes the Lorenz number, while \(L/L_{0}\) is called the Lorenz ratio. The WF law, one of the fundamental laws in metal physics, can in principle be violated in two very different ways: If, in addition to the electronic quasiparticles, charge-neutral fermionic excitations (“spinons”) contribute to the heat current but not to the charge current, the Lorenz ratio \(L/L_{0} > 1\), even in the zero-temperature limit. On the other hand, a breakdown of Landau's quasiparticle concept may lead to \(L/L_{0} < 1\) at \(T = 0\).

In Fig. 4 the low-T behavior of \(\rho(T)\) and \(w(T)\) is displayed for a single-crystalline YbRh₂Si₂ sample of medium quality (\(\mathrm{RRR}\approx 40\)) at several different values of the control parameter, magnetic field B.⁴²⁾ For the highest applied fields of 0.6 T [Fig. 4(i)] and 1 T [Fig. 4(j)], both \(\rho(T)\) and \(w(T)\) show a very strong \(T^{2}\) dependence at sufficiently low temperature, characteristic of a heavy Fermi liquid phase. Extrapolating these results to \(T = 0\), one recognizes \(w_{0} =\rho_{0}\) within the experimental uncertainty; i.o.w., the WF law holds even in case of such an extremely heavy Landau Fermi liquid phase.

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Figure 4. (Color online) Thermal resistivity \(w(T) = L_{0}T/\kappa(T)\) (red) and electrical resistivity \(\rho(T)\) (blue) below \(T = 0.5\) K for \(B = 0\) (a), 0.02 (b), 0.06 (c), 0.08 (d), 0.1 (e), 0.2 (f), 0.3 (g), 0.4 (h), 0.6 (i), and 1 T (j), \(B\perp c\). Arrows indicate crossover to Fermi liquid (\(\rho -\rho_{0} = AT^{2}\)) behavior. Representative error bars are shown for a few selected temperatures. Dashed lines in (a)–(c) indicate linear regimes used for extrapolation to \(T = 0\). From Ref. 42.

In contrast, at \(B = 0\) both the electrical and thermal resistivities exhibit a linear temperature dependence above, respectively, \(T_{\text{N}} = 0.07\) K and \(T\approx 0.12\) K, where \(w(T)\) starts to drop [see Fig. 4(a)]. Below the Néel temperature, one finds \(\rho =\rho_{0} + AT^{2}\) with a huge value of the coefficient A. At the lowest accessible temperature of ≈25 mK the thermal resistivity is clearly smaller than its electrical counterpart: \(w <\rho\). This proves that, in addition to the electronic ones, another species of heat carriers is present at finite temperatures. These are the acoustic AF magnons, as identified with the aid of low-T specific heat results, see Fig. 5:³³⁾ The magnon specific heat obeys \(C_{\text{m}}\sim T^{3}\) below \(T\approx 0.05\) K, which implies a magnon thermal conductivity \(\kappa\sim T^{3}\) at sufficiently low temperatures, too. At \(T = 0\), the heat current is carried exclusively by the electronic quasiparticles. Because of the Fermi liquid phase, the WF law must hold in the antiferromagnetically ordered state (\(w_{0} =\rho_{0}\)). Unfortunately, this cannot be observed directly as the electronic heat transport is fully masked by the bosonic one, due to magnons below \(T_{\text{N}} = 0.07\) K and due to short-lived magnon excitations (“paramagnons”) in the temperature window \(T_{\text{N}} < T < 0.12\) K at \(B = 0\). At finite \(B < B_{\text{c}}\) an analogous behavior is observed [Fig. 4(b), \(B = 0.02\) T]. This paramagnon contribution is also visible below \(T\approx 0.07\) K at \(B\approx B_{\text{c}}\) [Fig. 4(c)], but becomes suppressed at sufficiently high magnetic fields, \(B\geq 0.2\) T [Figs. 4(f)–4(j)].

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Figure 5. (Color online) Specific heat of YbRh₂Si₂ as \(\Delta C/T\) vs \(T^{2}\). \(\Delta C(T) = C(T) - C_{\text{ph}}(T) - C_{\text{Q}}(T)\), where \(C_{\text{ph}}\) (\(C_{\text{Q}}\)) denotes the phonon (nuclear-quadrupole) contribution. Red line indicates a \(T^{3}\) contribution to \(\Delta C(T)\) below \(T\approx 0.05\) K. From Ref. 33.

The values of the residual electrical and electronic thermal resistivities at \(B = B_{\text{c}}\), \(\rho_{0}\) and \(w_{0}\), are obtained by extrapolating the NFL-type linear-in-T dependences of \(\rho(T)\) and \(w(T)\) to zero temperature, cf. Supplementary Information of Ref. 42. Such an extrapolation is necessary because of the influence of the above-mentioned paramagnons. It yields \(w_{0} >\rho_{0}\) or \(L/L_{0} < 1\), i.e., a violation of the WF law, exactly at the QCP.

Very similar experimental data have recently been reported by three groups,^43–45) who studied YbRh₂Si₂ single crystals of considerably higher quality (\(\mathrm{RRR}\gtrsim 100\)) compared to our sample. As is evident from Refs. 43–45, the paramagnon heat transport in cleaner samples appears to persist at higher fields.⁴⁶⁾ This, along with their limitation in applying large enough magnetic fields (compared to \(B_{\text{c}}\) of the respective sample orientation), prevented the authors of Refs. 43 and 44 to observe both the field-induced suppression of the bosonic contribution to the heat transport and the validity of the WF law in the paramagnetic phase. Therefore, they attribute the downturn in \(w(T)\) from the linear-in-T behavior seen at \(B = B_{\text{c}}\) to the electronic heat carriers and claim the WF law to hold at the field-induced QCP in YbRh₂Si₂. In contrast, more recent measurements to lower temperatures and higher fields⁴⁵⁾ confirm the existence of paramagnon excitations and their contribution to the heat transport.

In order to support our conclusion that the WF law is indeed violated as \(T\rightarrow 0\) at \(B = B_{\text{c}}\), we show in Fig. 6 the isothermal field dependence \(L(B)/L_{0}\) for \(0.1\leq T\leq 0.4\) K. Data at lower temperatures have been ignored because of the interfering bosonic contribution in the low-field range as discussed above. This means that all \(L(B)/L_{0}\) data in Fig. 6 are representative of purely electronic transport. Except for the results taken at \(T = 0.1\) K and the highest fields of 0.6 and 1 T which, within the error bars, are close to the “WF value” \(L/L_{0} = 1\), all Lorenz ratios are clearly smaller than 1. This indicates that the electronic heat carriers are subject to dominating inelastic scatterings, i.e., from both AF spin fluctuations and electronic charge carriers. At zero temperature these scatterings are frozen out, which is in accordance with the validity of the WF law in a Fermi liquid with sharp FS, e.g., at both \(B < B_{\text{c}}\) and \(B > B_{\text{c}}\) for YbRh₂Si₂.

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Figure 6. (Color online) Isothermal field scans of the Lorenz ratio \(L(B)/L_{0}\) at varying temperatures \(0.1\leq T < 0.5\) K. From Refs. 42 and 47. Inset: Data from isothermal field scans of the thermal conductivity and electrical resistivity at \(T = 0.49\) K and \(B\leq 12\) T. From Ref. 47.

In the quantum critical regime, where the FS is fluctuating, the quasiparticle weights taken at the respective small and large values of the Fermi wave vector satisfy dynamical, \(\omega/T\), scaling and smoothly vanish exactly at the QCP,⁴²⁾ cf. Fig. 7. This warrants the critical FS fluctuations to exist in the whole temperature range of quantum critical behavior, \(0\leq T\lesssim 1\) K. They represent a novel type of quantum critical fluctuations, i.e., they are fermionic in origin and operate as additional inelastic scatterers for the electronic quasiparticles. These FS fluctuations give rise to the distinct minimum in the \(L(B)/L_{0}\) isotherms close to \(T^{*}(B)\) (Fig. 6), which highlights an intimate relationship between those additional inelastic scatterings and the Fermi surface crossover.

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Figure 7. (Color online) Collapse of the quasiparticle weights across the local QCP. \(Z_{\text{L}}\) and \(Z_{\text{S}}\) are the quasiparticle weights for the “small” (left inset) and “large” (right inset) Fermi surfaces, respectively. At the QCP, the quasiparticles are critical on both the small and the large Fermi surfaces. From Ref. 42.

Recent results of isothermal measurements of \(\kappa(B)\) and \(\rho(B)\), performed on a high-quality YbRh₂Si₂ single crystal (\(\mathrm{RRR}\approx 150\)) at \(T = 0.49\) K⁴⁷⁾ fit well to those obtained from the T-scans on our medium-quality sample with \(\mathrm{RRR}\approx 40\), cf. Fig. 6. This proves that the minimum in the \(L(B)/L_{0}\) isotherms is a robust feature which, like the Hall crossover, becomes considerably narrower upon cooling (Fig. 6). As a consequence of the \(\omega/T\) scaling, the additional electronic inelastic scatterings extend to \(\omega = 0\) exactly at the QCP. This provides a very natural explanation for our main result that \(L(T\rightarrow 0)/L_{0} < 1\) at \(B = B_{\text{c}}\) and \(L(T\rightarrow 0)/L_{0} = 1\) at \(B\neq B_{\text{c}}\).

The fact that we find a reduction of the Lorenz ratio by “only” 10% is in full accord with the generalized quasiparticle–quasiparticle nature of the underlying scatterings. Though many-body in origin they include a finite moderate fraction of small-angle-scattering processes — in analogy to the electron–electron scatterings in simple metals.⁴⁸⁾

The data presented in Fig. 8 lend further support to our conclusion of the WF law being violated at the QCP in YbRh₂Si₂: The electrical resistivity \(\rho(T)\), measured by S. Lausberg on a medium-quality single crystal similar to the one used for our study of the heat conductivity, clearly displays that the sample is heated up only in the vicinity of \(B_{\text{c}} = 0.059\) T, namely, if an appropriately large current is applied at low temperatures. Obviously, an extra Joule's heat is generated here which, despite of the additional bosonic heat carriers, cannot be properly lead away. Very likely, this extra heat results from the presence of particularly strong inelastic scatterings, i.e., those discussed before. Some heating effect is still visible at \(B = 0.06\) T even when a rather low current is injected into the sample (see black trace). Upon proper extrapolation of these low-current data from the regime where \(\Delta\rho\sim T\) to \(T = 0\), the residual resistivity at \(B = 0.06\) T is found to be almost identical to the \(\rho_{0}\) values extrapolated for both \(B = 0.1\) and 0.2 T. As a result, \(\rho_{0}(B)\) is found to jump abruptly from a larger to smaller value upon raising the field through \(B_{\text{c}}\), manifesting an abrupt increase in the charge carrier concentration as already inferred from the Hall measurements discussed in Sect. 3.^40,41,49)

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Figure 8. (Color online) Temperature dependence of the electrical resistivity below \(T = 0.1\) K with two different excitation currents (black and red traces, respectively) at several magnetic fields. Where black and red traces overlap, the red ones are extrapolated (in green) to \(T = 0\) assuming a \(\Delta\rho\sim T^{2}\) dependence. The splitting between red and black traces close to the critical field, \(B_{\text{c}} = 0.059\) T, below \(T = 0.07\) K (marked by arrows) illustrates a heating of the sample under the applied larger current. This illustrates the violation of the Wiedemann Franz law at the QCP in YbRh₂Si₂. Very recent resistivity measurements on an YbRh₂Si₂ single crystal of similar quality by S. Hamann with \(j\perp B\parallel c\) reveal a similar heating effect at \(B = 0.6\) T, which is absent at \(B = 0\), 2, and 4 T, respectively. For \(B\perp c\), a sample heating below \(T = 0.06\) K is still visible at \(B = 0.06\) T under the smaller current, cf. extrapolated dashed straight line. Horizontal blue hatching displays an almost field-independent residual (\(T\rightarrow 0\)) resistivity in the paramagnetic regime. The weak field dependence of \(\rho_{0}\) inside the antiferromagnetic phase⁴¹⁾ is masked by the width of the grey hatching. The difference between the hatched horizontal regions indicates an abrupt decrease in \(\rho_{0}(B)\) upon increasing the field through \(B_{\text{c}}\). This illustrates a corresponding abrupt increase in the charge-carrier concentration at \(T = 0\).

Recently, \(L(B)/L_{0}\) was also determined for the HF metal YbAgGe.⁵⁰⁾ When the results taken at \(B = 4.5\) T are extrapolated to \(T = 0\), where a bi-quantum critical point is anticipated,⁵¹⁾ \(L_{\text{el}}/L_{0}\approx 0.92\) is obtained. Thus, for YbAgGe this fundamental type of violation of the WF law seems to be clearly demonstrated, too.⁵²⁾

In this paper, we have presented and discussed three pieces of evidence for a field-induced Kondo destroying AF QCP in YbRh₂Si₂. Upon reliably extrapolating our data taken at finite temperature to \(T = 0\), we recognize exactly at \(B = B_{\text{c}}\): (i) a merging of the quantum-critical energy scale \(T^{*}(B)\) with the magnetic phase boundary \(T_{\text{N}}(B)\) and the crossover line \(T_{\text{FL}}(B)\), (ii) an abrupt jump of the initial Hall coefficient \(R_{\text{H}}(B)\) and (iii) a violation of the WF law. While the first result locates the Kondo breakdown in the phase diagram and the second one quantifies the strength of this phenomenon, the latter result illustrates the dynamical processes, causing the apparent breakup of the Kondo singlets. Future research will focus on studies of this Mott-type instability in the absence of any interfering magnetism as expected for, e.g., Yb(Rh\(_{1-x}\)Ir_x)₂Si₂ with \(x\gtrsim 0.1\).⁵⁷⁾

Another promising area of future investigations concerns the occurrence of a ferromagnetic QCP in HF metals, like YbNi₄(P\(_{1-x}\)As_x)₂, \(x\approx 0.1\).⁵⁸⁾ This type of instability does apparently not exist in itinerant (d-electron-based) metallic materials.⁵⁹⁾ On the other hand, it was shown⁶⁰⁾ that for Kondo lattice systems the Kondo effect can be fully suppressed inside a ferromagnetically ordered phase. It has yet to be explored, whether in YbNi₄(P_0.9As_0.1)₂ the breakdown of the Kondo effect coincides with the ferromagnetic QCP, in analogy to the observation in the weak antiferromagnet YbRh₂Si₂.⁶¹⁾