PHYSICAL REVIEW LETTERS week ending PRL 96, 116404 (2006) 24 MARCH 2006 Critical Phenomena and the Quantum Critical Point of Ferromagnetic Zr1ÿxNbxZn2 D. A. Sokolov,1 M. C. Aronson,1 W. Gannon,1 and Z. Fisk2,* 1University of Michigan, Ann Arbor, Michigan 48109-1120, USA 2Florida State University, Tallahassee, Florida 32310-3706, USA (Received 6 December 2005; published 24 March 2006) We present a study of the magnetic properties of Zr1ÿxNbxZn2, using an Arrott plot analysis of the magnetization. The Curie temperature TC is suppressed to zero temperature for Nb concentration xC 0:083 0:002, while the spontaneous moment vanishes linearly with TC as predicted by the Stoner theory. The initial susceptibility displays critical behavior for x xC, with a critical exponent which smoothly crosses over from the mean field to the quantum critical value. For high temperatures and x xC, and for ÿ1 ÿ1 4=3 ÿ1 low temperatures and x xC we find that 0 aT , where 0 vanishes as x ! xC. The resulting magnetic phase diagram shows that the quantum critical behavior extends over the widest range of temperatures for x xC, and demonstrates how a finite transition temperature ferromagnet is trans- formed into a paramagnet, via a quantum critical point. DOI: 10.1103/PhysRevLett.96.116404 PACS numbers: 71.10.Hf, 75.10.Lp, 75.40.Cx Zero temperature phase transitions and their attendant heat capacity is maximized [24–28] while the electrical 1 quantum critical fluctuations have emerged as dominant resistivity evolves from / T for ÿ ÿC to the Fermi 2 features in the phase diagrams of different types of strongly liquid / T for ÿ > ÿC [19,29–31]. The low field mag- correlated electron systems, from oxide superconductors netization is anomalous near the QCP [13,19], but a de- [1] and heavy fermion compounds [2,3], to low dimen- tailed study spanning the ordered and paramagnetic phases sional materials [4]. These fluctuations qualitatively mod- is still lacking. We provide this study of the magnetization ify the electronic states near a quantum critical point of ZrZn2 here, discussing how the QCP is generated with (QCP), leading to unusual temperature divergences of the Nb doping, and the subsequent evolution of the critical susceptibility and heat capacity, to anomalous power-law phenomena as the QCP is approached. behavior in the electrical transport, and even to scale Zr1ÿxNbxZn2 is ideal for such a study. Neutron form invariance in the magnetic responses [5–9]. The funda- factor measurements [32] show that the magnetic moment mental excitations near QCPs are qualitatively unlike those is spatially delocalized, consistent with the small sponta- of conventional metals, representing in some cases entirely neous moment [33]. We establish a magnetic phase dia- new classes of collective states [10,11]. A central issue is gram, and show that it is dominated by a QCP at whether these unusual properties require the exceptionally xC 0:083 0:002 and TC 0K. Stoner theory de- rich physics of these host systems, derived from low di- scribes the ferromagnetism of ZrZn2 well, indicating that mensionality and strong correlations, or whether only variation in the density of states at the Fermi level controls proximity to a zero temperature phase transition is re- both the Curie temperature TC and the zero temperature quired. Thus, it is important to identify electronically spontaneous moment m0 0. Our measurements of the simple systems, and to study the evolution of their critical initial magnetic susceptibility T describe how the criti- phenomena as the ordered phases are suppressed to zero cal phenomena evolve with Nb doping, crossing over from temperature. mean field behavior when the reduced temperature is low Itinerant ferromagnets are particularly attractive hosts and when Nb concentrations are far from the critical value, for such a study, as they lack the complex interplay of to a regime at small x ÿ xC where the susceptibility is itinerant and localized character found near the QCPs of controlled by the QCP over an increasingly broad range heavy fermion systems [8,12]. Pressure and compositional of temperatures. variation have been used to suppress the finite temperature Polycrystalline Zr1ÿxNbxZn2 samples with Nb con- magnetic ordering transition, finding that the magnetically centrations 0 x 0:14 were prepared by solid state ordered phase can vanish discontinuously as in pressurized reaction [34]. X-ray diffraction confirmed the C-15 MnSi [13], UGe2 [14], and perhaps ZrZn2 [15–19], or ZrZn2 structure [33] at each composition, as well as re- continuously as in Ni1ÿxPdx3Al [20]. While disorder sidual amounts of unreacted Zr and Zn. The magneti- can affect the order of the quantum critical phase transition zation was measured using a Quantum Design magnetome- in itinerant ferromagnets [21], it is generally found in ter at temperatures from 1.8 to 200 K and in magnetic fields systems with continuous transitions tuned by a parameter up to 7 T. The inset of Fig. 1(a) shows the magnetic T ÿ that the QCP, which occurs for ÿ ÿc and 0, isotherms for Zr1ÿxNbxZn2 (x 0:03) presented as an dominates the magnetic phase diagram and generates a Arrott plot. Both the spontaneous magnetization m0 T dn=z phase line TC ÿ ÿ ÿc, where d 3, z n 2 limH!0m H; T and the initial susceptibility T 3 [13,19,22,23]. Near the QCP, the electronic part of the limH!0dm H; T=dH were determined by extrapolation 0031-9007=06=96(11)=116404(4)$23.00 116404-1 © 2006 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 96, 116404 (2006) 24 MARCH 2006 (a) (b) 50 [37], those with modified stoichiometry [16,38] and even α=1.04 -2 0.2 when high pressures are applied [15,18,19]. Values of 2x10 /f.u.) 2 B 4/3 40 are indicated in the inset of Fig. 1(b), suggesting that modi- µ 1 ) (K /f.u.) B 0.1 (0) ( fications to the underlying electronic structure and not the µ 0.12 0 ( C 2 4/3 m m T /f.u.) 0 B 5 disorder associated with doping are primarily responsible 0 1 3 5x10 µ 30 α=1.0 ( H/m(Oe*f.u./µ ) 0 B 0.00 for altering the stability of ferromagnetism in ZrZn2. In- m 0 20 40 (K), T (K) 0.08 C C T deed, only a 2% reduction in is necessary to drive the 20 ferromagnetism in undoped ZrZn2 to the brink of instabil- ity, whether by doping or by the application of pressure. 0.04 10 The initial magnetic susceptibility is considerably modified as the system is driven from a finite temperature 0 0 5 10 15 20 0.0 0.02 0.04 0.06 0.08 instability in undoped ZrZn2, through a QCP, and into the T (K) Nb concentration x paramagnetic phase. Arrott plot analyses are used to de- ÿ1 duce TlimH!0dH=dm H; T, shown in Fig. 2 for FIG. 1. (a) Temperature dependence of the spontaneous mag- a wide range of Nb concentrations x. diverges at T in netization m in per formula unit (f.u.) for Zr Nb Zn [x C 0 B 1ÿx x 2 the ferromagnetic samples (x x ), with little sign of 0 (᭹), 0.03 ( ), 0.05 (᭿), 0.06 (ᮀ)]. The solid lines are fits to C 0:5 critical rounding. The sample with x 0:08 displays a m0 Tm0 0 TC ÿ T=TC . Inset: Example of an Arrott plot for x 0:03, for temperatures from 1.8 K (᭹)to20K nearly power-law response in absolute temperature, as (᭡), the solid lines are fits to m2 / H=m, arrows mark extrapo- expected near a QCP [22,39]. Finally, for x xC, T 2 lated 1= T (horizontal axis) and m T (vertical axis). approaches a constant value 0 as T ! 0, signalling that 4=3 ᭹ (b) Variation of Curie temperature TC ( ) and TC ( ) with long range ferromagnetic order is no longer possible. 3=4 Nb concentration x. Solid lines are fits to TC /xC ÿ x and The initial susceptibility for x<xC is well described by 4=3 ÿ TC /xC ÿ x, with xC 0:083 0:002. Inset: The zero tem- a power law 00 over at least two decades of perature spontaneous moment m0 0/TC. Y, Ti, Hf doping () reduced temperature T ÿ TC=TC, and for absolute [37]; pressure, single crystal ( ) [15]; pressure, ZrZn1:9 poly- temperatures as large as 100 K [Fig. 3(a)]. The inset of crystal (ᮀ) [18]; Nb doping, polycrystals, this work (᭡). Solid Fig. 3(a) shows that increases smoothly from the near- line is Stoner parameter [38]. mean field value 1:08 0:05 previously observed in ZrZn2 [36], to 1:33 0:01 for x xC. Since the interactions which lead to magnetic order in itinerant ferromagnets of data from fields larger than 4.5 T. Previous de Haas– are long ranged, the intrinsic exponents related to the van Alphen experiments on a single crystal of ZrZn found 2 underlying symmetries are only found at reduced tempera- a field induced transition in the magnetization near 5 T tures which are much smaller than those accessed in our [35], and a complex pressure-temperature phase diagram experiments [40–42]. We conclude that the variation of was proposed for this sample [15]. However, our Arrott with Nb concentration is the result of a gradual crossover plots are linear in fields from 1–7 T, at least at low from the mean field behavior associated with a finite temperatures and for small x, indicating that this field- temperature ferromagnetic transition for x<x to quan- driven transition is absent, as it was in earlier work C tum criticality as x ! xC.