Abstract
A continuous phase transition driven to zero temperature by a nonthermal parameter, such as pressure, terminates in a quantum critical point (QCP). At present, two main theoretical approaches are available for antiferromagnetic QCPs in heavyfermion systems. The conventional one is the quantum generalization of finitetemperature phase transitions, which reproduces the physical properties in many cases^{1,2,3,4,5}. More recent unconventional models incorporate a breakdown of the Kondo effect, giving rise to a Fermisurface reconstruction^{6,7,8}— YbRh_{2}Si_{2} is a prototype of this category^{5,9,10,11}. In YbRh_{2}Si_{2}, the antiferromagnetic transition temperature merges with the Kondo breakdown at the QCP. Here, we study the evolution of the quantum criticality in YbRh_{2}Si_{2} under chemical pressure. Surprisingly, for positive pressure we find the signature of the Kondo breakdown within the magnetically ordered phase, whereas negative pressure induces their separation, leaving an intermediate spinliquidtype ground state over an extended range. This behaviour suggests a new quantum phase arising from the interplay of the Kondo breakdown and the antiferromagnetic QCP.
Main
In heavyfermion systems, the Kondo effect leads to the formation of composite quasiparticles of the f and conductionelectron states with largely renormalized masses forming a Landau Fermiliquid ground state in the paramagnetic regime well below the Kondo temperature T_{K}. These quasiparticles are assumed to stay intact at the quantum critical point (QCP) in the conventional models in which magnetic order arises through a spindensitywave (SDW) instability. However, the observation of magnetic correlations in CeCu_{5.9}Au_{0.1} being of local character^{11} prompted a series of theoretical descriptions that discard this basic assumption. Rather, they focus on the breakdown of the Kondo effect, which causes the f states to become localized and decoupled from the conductionband states at the QCP where one expects the Fermi surface to be reconstructed^{7}. Consequently, a new energy scale is predicted reflecting the finitetemperature T crossover of the Fermisurface volume. This picture has been scrutinized in tetragonal YbRh_{2}Si_{2} (T_{K}≈25 K; ref. 12), a stoichiometric and very clean heavyfermion metal that seems to be ideally suited for this kind of study^{9,12}: antiferromagnetic order sets in at a very low temperature T_{N}=0.07 K and can easily be suppressed by a small magnetic field of μ_{0}H_{N}=60 mT (, with c being the magnetically hard axis). Halleffect experiments^{13} have detected a rapid change of the Hall coefficient along a line that converges with H_{N}, the width of the Hall crossover extrapolating to zero for T→0. This change was considered evidence for an abrupt change of the Fermisurface volume indicating a correspondence between and the Kondobreakdown energy scale. Subsequent thermodynamic and transport investigations confirmed to be a new energy scale^{14}. The a.c. susceptibility χ′(T) turned out to exhibit a pronounced maximum at , which can be particularly well distinguished from the sharp signature at T_{N}. The magnetoresistance exhibits a steplike crossover similar to the Hall coefficient. In fact, recent calculations for a Kondo lattice predict such a feature for both transport properties^{15}. Furthermore, the magnetization shows a smeared kink at between two almost linear regimes with different slopes^{14}. The anomaly at in all isothermal measurements can be described either by the same crossover function proposed for the Hall effect or its integral version resembling a smeared kink. The maximum in χ′(T) is a natural consequence of the magnetization kink being smeared and shifted to higher fields as the temperature is raised.
The Fermisurface reconstruction may also occur away from the antiferromagnetic QCP as observed in CeIn_{3} and CeRh_{1−x}Co_{x}In_{5} (refs 1617). It is therefore very important to understand the interplay between the phenomena assigned to the Kondo breakdown and the magnetic order. We address this issue by investigating YbRh_{2}Si_{2} under positive and negative chemical pressure, which was realized by partial isoelectronic substitution of smaller Co or larger Ir for Rh, respectively (the analogy of chemical and external pressure is discussed in Supplementary Information A). In Yb systems, pressure yields a stabilization of magnetism, in particular an increase of T_{N} (ref. 18). On the other hand, negative pressure, corresponding to a lattice expansion, reduces T_{N} .
The T–H phase diagrams of Yb(Rh_{0.94}Ir_{0.06})_{2}Si_{2} and Yb(Rh_{0.93}Co_{0.07})_{2}Si_{2} (labelled 6% Ir and 7% Co in the following) are compared with that of YbRh_{2}Si_{2} in Fig. 1. This set emphasizes the evolution of the various energy scales. First, the magnetic one follows the expected pressure dependence: for 6% Ir, T_{N} is depressed below 0.02 K, whereas in the case of 7% Co, T_{N} is enhanced to 0.41 K. Second, the energy scale virtually does not change its position in the T–H phase diagram. Consequently, is separated from T_{N}(H) in 6% Ir, whereas they intersect in 7% Co. Finally, at fields above the respective critical fields (for 6% Ir) and H_{N} (for 7% Co) at which and T_{N}(H) vanish, the Fermiliquid phase forms below T_{FL}(H).
These findings were mainly deduced from the temperaturedependent a.c. susceptibility χ′(T) shown in Fig. 2. For 6% Ir, no signature of magnetic order is observed as the zerofield curve increases monotonically with decreasing T. An antiferromagnetically ordered state is anticipated at even lower temperatures (see Supplementary Information A). On the other hand, χ′(T) of 7% Co exhibits a sharp kink at T_{N}=0.41 K and a cusp at T_{L}=0.06 K, the critical temperature of a second, presumably also antiferromagnetic, transition. In external fields, these two transitions are shifted to lower temperatures, with the lower one bifurcating as detailed in Supplementary Information B. For 6% Ir and 7% Co, a maximum in χ′(T) assigned to is observed above 45 mT and 55 mT, respectively. In an increasing field, this maximum shifts to higher temperatures. Remarkably, for 7% Co the maximum appears both below T_{N} at small fields, and above T_{N} at fields of 150 mT and higher. This clearly illustrates that the energy scales, T_{N}(H) and , indeed intersect.
In Fig. 3, we compare the signatures of in magnetoresistance as well as in magnetization for stoichiometric YbRh_{2}Si_{2}, 7% Co and 6% Ir. At 0.5 K, the magnetoresistance curves of these three samples exhibit an almost identical, steplike crossover. In particular, the inflection point assigned to (ref. 14) is nearly unchanged by chemical pressure, resembling the results from susceptibility and magnetization (discussed below). In the case of 7% Co, this holds true above T_{N}, whereas at lower temperatures the inflection point is locked to the antiferromagnetic phase boundary (not shown).
We analysed the magnetization M as outlined in supporting online material for ref. 14 as . In Fig. 3b, we focus on the data at the lowest temperatures, which unambiguously prove the existence of the anomaly also within the antiferromagnetic phase of 7% Co. exhibits a broadened kink at between two linear regimes with different slopes^{14}. In addition to, but clearly distinct from, this kink at , a small peak is observed at 220 mT for 7% Co, which is related to the critical field of the ordered phase. Therefore, our magnetization and susceptibility results on 7% Co yield striking evidence for the crossover at to also occur inside the antiferromagnetic ordered phase. A reexamination of existing magnetization data^{19} confirms this finding for YbRh_{2}Si_{2} under external pressure, supporting the equivalence to chemical pressure. To check for possible disorder effects, a comprehensive study of YbRh_{2}Si_{2} under hydrostatic pressure is in preparation.
Exactly such an intersection of T_{N}(H) and , as observed for 7% Co, is expected in the threedimensional (3D) SDW theory. As shown in Supplementary Information D, the field dependence of the Néel temperature in the vicinity of H_{N} indeed follows the anticipation of the 3D SDW theory. However, ref. 6 predicts that critical fluctuations ought to be 2D once T_{N}(H) and converge at the antiferromagnetic QCP, as observed for pure YbRh_{2}Si_{2} (refs 13, 14).
We now turn to the interesting case of 6% Ir where, according to our results, the critical fields H_{N} and seem to become separated from each other (Fig. 1, top panel). The resistivity versus temperature curves measured in various fields are shown in Fig. 4. In zero field, ρ(T) is quasilinear below 1 K with a slight upward curvature at the lowest temperatures. In small external fields, this curvature is reduced, yielding the steepest curve at 50 mT. A T^{2} form indicative of a Fermiliquid ground state is observed only at fields exceeding 50 mT (see the lines in Fig. 4). The A coefficient in the Fermiliquid regime, which is proportional to the effective quasiparticle–quasiparticle scattering crosssection, follows a (H−H_{c}^{A})^{−1} divergence with a critical field of μ_{0}H_{c}^{A}=30(5) mT (Fig. 4, bottom inset) close to H_{FL}=35(5) mT, the field at which T_{FL} vanishes in the zero temperature limit. An important finding of this study is that H_{FL} is substantially larger than the critical field of the antiferromagnetically ordered phase μ_{0}H_{N}≈15 mT. In addition, the line extrapolates to a critical field in close vicinity to H_{FL} and H_{c}^{A} (see Fig. 4, top inset). Consequently, our results reveal a finite field range within which the resistivity exhibits nonFermiliquid behaviour. Obviously, neither the crossover at nor that at T_{FL}(H) is linked to the critical field of the antiferromagnetically ordered phase. We note that similar behaviour was observed for Yb_{0.95}La_{0.05}Rh_{2}Si_{2}, for which ρ(T) is linear in fields up to 40 mT, where neither a magnetically ordered nor a Fermiliquid ground state was found^{20}.
The divergence of the A coefficient backs the presence of a QCP connected with the vanishing energy scale . Further support for a QCP at stems from the analysis of the resistivity exponent n in ρ(T)−ρ_{0}∝T^{n}, shown in Fig. 4, top inset, as a coloured contour plot. The blue region (n=2) reflects Fermiliquid behaviour. Deviations are ascribed to the quantum criticality^{9}: in fact, the red region (n=1) is clearly linked to and well separated from the critical field of the antiferromagnetic order. To underline that the QCP at cannot be of magnetic origin, Supplementary Information E shows the line to be still at the same position in the phase diagram for 17% Ir substitution, although here no antiferromagnetic order is expected at all^{21}. This indicates a separation of the antiferromagnetic QCP from the Kondobreakdown QCP. In the intermediate field range, the local f moments are expected to be neither Kondo screened nor antiferromagnetically ordered. This highlights a new metallic ‘spinliquid’type ground state that has to be explored in more detail. The existence of a ‘spin liquid’ in a Kondo lattice has been speculated^{8,22}, but the conditions under which it might be realized remain uncertain. The quasilinear resistivity predicted in ref. 22 resembles our experimental observations. Also for YbAgGe, a finite field range was reported where the resistivity exhibits similar nonFermiliquid behaviour to the lowest temperatures^{23}. However, the specific heat of YbAgGe shows a saturation of C(T)/T in this field range, discarding a spinliquid ground state^{24}. In contrast, preliminary measurements on 6% Ir down to 0.06 K reveal a strong divergence of C(T)/T with decreasing temperature in the field range below 50 mT (not shown), supporting our claim of a spin liquid. In addition, the susceptibility continues to increase towards the lowest temperatures (see Fig. 2a). The experimental evidence of such a new, nonmagnetic ground state is fascinating and will certainly motivate future experimental and theoretical studies.
Figure 5 shows the evolution of the two different QCPs as a function of Ir/Co substitution. The following main results can be deduced from this figure. (1) The antiferromagnetic state is stabilized through the application of positive chemical pressure, as expected. (2) The position of the suggested breakdown of the Kondo effect depends only weakly on chemical pressure—although the Kondo effect itself is known to be strongly pressure dependent. (3) As a consequence, for positive pressure, the antiferromagnetic QCP at H_{N} is located in the regime with intact Kondo screening () where the SDW theory is expected to be applicable in accordance with our observations. (4) For negative chemical pressure, on the other hand, H_{N} is separated from towards lower fields with an intermediate spinliquidtype ground state emerging. Obviously, here, antiferromagnetic order and the Fermiliquid ground state are not connected by a single QCP, but are separated by a spin liquid, that is, a nonFermiliquid range as previously observed for MnSi (ref. 25) and, perhaps, in βYbAlB_{4} (refs 26, 27).
To conclude, the application of chemical pressure provides a wider view on the global phase diagram of YbRh_{2}Si_{2} by lifting the coincidence of the multiple energy scales in the stoichiometric compound. The results and their interpretation presented here pose a formidable challenge for those theories describing the breakdown of the Kondo effect near an antiferromagnetic QCP in Kondo lattice systems. It remains to be explored under which conditions antiferromagnetic ordering and the Fermisurface reconstruction may eventually become separated as observed for YbRh_{2}Si_{2} with Ir substitution. Equally important, it needs to be understood why in pure YbRh_{2}Si_{2}, the antiferromagnetic QCP coincides with the Kondo breakdown.
Methods
Single crystals were grown from In flux, analogous to the stoichiometric samples described earlier^{12}. The In flux was subsequently removed in hydrochloric acid. The presented results prove the absence of residual In. Xray diffraction confirms the single crystallinity. All lowtemperature measurements were carried out with the magnetic field aligned perpendicular to the crystallographic c axis, . The a.c.susceptibility measurements were carried out at low frequencies with a modulation field amplitude of 4 μT down to 0.02 K. As no imaginary signal was detected, the real part χ′ is a direct measure of the field derivative of the magnetization. The temperaturedependent susceptibility χ′(T) was measured in selected static magnetic fields. The isothermal susceptibility χ′(H) was measured as a function of a field applied in addition to the modulation field. The electrical resistivity ρ was monitored by a standard fourpoint lockin technique at low frequencies down to 0.02 K. An extremely small outofphase signal of less than 1% proves the high quality of the spotwelded contacts. With the help of lowtemperature transformers, a very high sensitivity of better than 0.1 nV was realized. In all samples, the resistivity was measured perpendicular to the crystallographic c axis, and the magnetic field was applied parallel to the current. The magnetic field dependence of the magnetization M(H) was isothermally measured in a highresolution Faraday magnetometer down to 0.05 K (ref. 28). Background contributions from the sample platform and the torque exerted on the sample have been subtracted. The magnetization was analysed in the form by fitting
to the data from which the crossover field H_{0} was obtained^{14}. is preferred for the analysis as it enables a more precise fitting compared with M itself, although the conclusions drawn from M(H) are identical (see ref. 14 and its supporting online material).
References
Hertz, J. A. Quantum critical phenomena. Phys. Rev. B 14, 1165–1184 (1976).
Millis, A. J. Effect of a nonzero temperature on quantum criticalpoints in itinerant fermion systems. Phys. Rev. B 48, 7183–7196 (1993).
Moriya, T. & Takimoto, T. Anomalous properties around magnetic instability in heavy electron systems. J. Phys. Soc. Jpn. 64, 960–969 (1995).
Löhneysen, H. v., Rosch, A., Vojta, M. & Wölfle, P. Fermiliquid instabilities at magnetic quantum phase transitions. Rev. Mod. Phys. 79, 1015–1075 (2007).
Gegenwart, P., Si, Q. & Steglich, F. Quantum criticality in heavyfermion metals. Nature Phys. 4, 186–197 (2008).
Si, Q., Rabello, M. S., Ingersent, K. & Smith, J. L. Locally critical quantum phase transitions in strongly correlated metals. Nature 413, 804–808 (2001).
Coleman, P., Pépin, C., Si, Q. & Ramazashvili, R. How do Fermi liquids get heavy and die? J. Phys. Condens. Matter 13, R723–R738 (2001).
Senthil, T., Vojta, M. & Sachdev, S. Weak magnetism and nonFermi liquids near heavyfermion critical points. Phys. Rev. B 69, 035111 (2004).
Custers, J. et al. The breakup of heavy electrons at a quantum critical point. Nature 424, 524–527 (2003).
Park, T. et al. Isotropic quantum scattering and unconventional superconductivity. Nature 456, 366–368 (2008).
Schröder, A. et al. Onset of antiferromagnetism in heavyfermion metals. Nature 407, 351–355 (2000).
Trovarelli, O. et al. YbRh2Si2: Pronounced nonFermiliquid effects above a lowlying magnetic phase transition. Phys. Rev. Lett. 85, 626–629 (2000).
Paschen, S. et al. Halleffect evolution across a heavyfermion quantum critical point. Nature 432, 881–885 (2004).
Gegenwart, P. et al. Multiple energy scales at a quantum critical point. Science 315, 969–971 (2007).
Coleman, P., Marston, J. B. & Schofield, A. J. Transport anomalies in a simplified model for a heavyelectron quantum critical point. Phys. Rev. B 72, 245111 (2005).
Harrison, N. et al. Fermi surface of CeIn3 above the Néel critical field. Phys. Rev. Lett. 99, 056401 (2007).
Goh, S. K. et al. Fermisurface reconstruction in CeRh1−xCoxIn5 . Phys. Rev. Lett. 101, 056402 (2008).
Goltsev, A. V. & AbdElmeguid, M. M. Origin of the pressure dependence of the Kondo temperature in Ce and Ybbased heavyfermion compounds. J. Phys. Condens. Matter 17, S813–S821 (2005).
Tokiwa, Y. et al. Fieldinduced suppression of the heavyfermion state in YbRh2Si2 . Phys. Rev. Lett. 94, 226402 (2005).
Weickert, F., Gegenwart, P., Ferstl, J., Geibel, C. & Steglich, F. Lowtemperature electrical resistivity of Yb1−xLaxRh2Si2 . Physica B 378–380, 72–73 (2006).
Westerkamp, T., Gegenwart, P., Krellner, C., Geibel, C. & Steglich, F. Lowtemperature magnetic susceptibility of Yb(Rh1−xMx)2Si2 (M=Ir, Co) single crystals. Physica B 403, 1236–1238 (2008).
Pépin, C. Selective Mott transition and heavy fermions. Phys. Rev. B 77, 245129 (2008).
Niklowitz, P. G., Knebel, G., Flouquet, J., Bud’ko, S. L. & Canfield, P. C. Fieldinduced nonFermiliquid resistivity of stoichiometric YbAgGe single crystals. Phys. Rev. B 73, 125101 (2006).
Tokiwa, Y. et al. Lowtemperature thermodynamic properties of the heavyfermion compound YbAgGe close to the fieldinduced quantum critical point. Phys. Rev. B 73, 094435 (2006).
DoironLeyraud, N. et al. Fermiliquid breakdown in the paramagnetic phase of a pure metal. Nature 425, 595–599 (2003).
Nakatsuji, S. et al. Superconductivity and quantum criticality in the heavyfermion system βYbAlB4 . Nature Phys. 4, 603–607 (2008).
Nevidomskyy, A. H. & Coleman, P. Layered Kondo lattice model for quantum critical βYbAlB4 . Phys. Rev. Lett. 102, 077202 (2009).
Sakakibara, T., Mitamura, H., Tayama, T. & Amitsuka, H. Faraday force magnetometer for highsensitivity magnetization measurements at very low temperatures and high fields. Jpn. J. Appl. Phys. 33, 5067–5072 (1994).
Acknowledgements
The authors would like to thank P. Coleman and Q. Si for motivating discussions. We acknowledge partial support by the DFG Research Group 960 ‘Quantum Phase Transitions’.
Author information
Authors and Affiliations
Contributions
S.F. set up, carried out and analysed the resistivity measurements. T.W. set up, carried out and analysed the a.c.susceptibility measurements. M.B. set up, carried out and analysed the magnetization measurements. C.K. and C.G. grew the single crystals for the study. F.S., P.G., S.W. and N.O. planned and headed the project. S.F. wrote the paper with assistance from F.S., N.O, M.B., P.G. and S.W.
Corresponding authors
Supplementary information
Supplementary Information
Supplementary Information (PDF 306 kb)
Rights and permissions
About this article
Cite this article
Friedemann, S., Westerkamp, T., Brando, M. et al. Detaching the antiferromagnetic quantum critical point from the Fermisurface reconstruction in YbRh_{2}Si_{2}. Nature Phys 5, 465–469 (2009). https://doi.org/10.1038/nphys1299
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nphys1299
Further reading

Transport evidence for decoupled nematic and magnetic criticality in iron chalcogenides
Communications Physics (2022)

Magnetic phase diagram of the solid solution LaMn2(Ge1−xSix)2 (0 ≤ x ≤ 1) unraveled by powder neutron diffraction
Scientific Reports (2022)

Quantum phases driven by strong correlations
Nature Reviews Physics (2021)

Quantumcritical phase from frustrated magnetism in a strongly correlated metal
Nature Physics (2019)

Frustration can be critical
Nature Physics (2019)