PHYSICAL REVIEW LETTERS week ending PRL 96, 116404 (2006) 24 MARCH 2006
Critical Phenomena and the Quantum Critical Point of Ferromagnetic Zr1 xNbxZn2
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 Zr1 xNbxZn2, 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 1 1 4=3 1 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 C to the Fermi 2 features in the phase diagrams of different types of strongly liquid / T for > C [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 Zr1 xNbxZn2 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 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 Zr1 xNbxZn2 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 Ni1 xPdx 3Al [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 that the QCP, which occurs for c and 0, isotherms for Zr1 xNbxZn2 (x 0:03) presented as an dominates the magnetic phase diagram and generates a Arrott plot. Both the spontaneous magnetization m0 T d n =z phase line TC c , 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- 1 duce T limH!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 1 x 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 T m0 0 TC 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 x and The initial susceptibility for x TC with Nb doping is shown in Fig. 1(b), demonstrating -1 6 that the ferromagnetic phase line obeys the expected 10 d n =z /f.u.) 4=3 B TC T / x xC (d z 3, n 1) [22], ter- µ 0.00 ( 5 0.02 minating at a critical concentration xC 0:083 0:002, χ 10 0.03 1/ 0.05 analogous to the results of high pressure measurements 0.055 0.06 [19]. 4 0.08 10 0.09 The suppression of TC and m0 with Nb doping indicates 0.12 that the Stoner theory adequately describes the ferromag- 0.14 Nb concentration x 3 netism in ZrZn2, where the underlying control parameter 10 1 10 100 is the product of the Coulomb interaction and the density of T (K) states at the Fermi level. Stoner theory predicts that TC and m0 0 are proportional for 1. The inset of Fig. 1(b) FIG. 2. The temperature dependence of the inverse of the shows this proportionality is valid not only for our Nb initial susceptibility 1 for Nb concentrations both larger and doped samples, but also for those with Ti, Hf, and Y doping smaller than the critical concentration xC 0:083 0:002. 116404-2 PHYSICAL REVIEW LETTERS week ending PRL 96, 116404 (2006) 24 MARCH 2006 (a) (b) 100 -1 1.3 1200 x=0.03 6 /f.u.) 10 x=0.06 B 1.2 µ III γ II ( x=0.08 x 5 T(K) c χ∗ 6 1.1 10 10 1/ 10 4 -1 0 0.04 0.08 800 10 1 2 x 10 10 I 4/3 4/3 /f.u.) T (K ) χ∗ B 5 µ 10 a 1/ x=0.09 χ ( 1 1/ γ =1.08 5 400 4x10 4 x=0.12 10 γ =1.33 | 3 0 γ =1.23 χ x=0.14 3 |1/ 10 0 2 -2 0 2 10 10 10 0 400 800 1200 τ 4/3 4/3 =(T-Tc )/Tc T (K ) 1 1 1 0 FIG. 3. (a) 00 at different Nb concentrations: x 0 0 0.04 0.08 0.12 (᭹), x 0:055 (), x 0:08 (᭡). Solid lines are the best fits to 1 1 Nb concentration x 00 . Inset: the critical exponent ( ) increases con- tinuously as x ! xC. The solid line is a guide for the eye. FIG. 4. Top: The temperature-Nb concentration phase diagram (b) 1=a (defined in the text) for x xc is proportional to for Zr Nb Zn . T x ( ), with the phase line T / 4=3 4=3 1 x x 2 C C T for T T .Forx xc, 1=a deviates from T (solid x x 3=4, the cutoff temperature T (ᮀ) for x x with the C G C line) above a cutoff temperature T marked by arrows, x 0:08 crossover line T / x x 1:2 0:3, the cutoff temperature ᮀ ᭹ G C ( ), x 0:09 ( ), x 0:12 ( ), x 0:14 ( ). Inset: For x T (4) for x x ; the dashed line is a guide for the eye. The 4=3 C xc 1= deviates from T behavior below a crossover tempera- diamonds (᭛) indicate the highest temperature investigated for ᮀ ture TG marked by arrows: x 0:08 ( ), x 0:06 (4), x x x . I, II, and III correspond, respectively, to the region of ᭡ 4=3 C 0:03 ( ). The solid line is a best fit of 1= aT for x stable ferromagnetism, the crossover region, and the quantum 0:08. critical region as defined in the text. Bottom: Variation with Nb concentration of the T 0 inverse susceptibility 1= 0 for ᮀ x xC ( ) and for x xC ( ). Solid lines are fits with 1= 0 / Isolating the temperatures and Nb concentrations where jx xCj. For x 0:14, 1= 0 is divided by 15 to appear on the is dominated by the QCP is straightforward in the plot. paramagnetic regime (x xC). Near the QCP, the initial susceptibility for an itinerant, three dimensional ferromag- x net is given by 1 1 aT4=3, with 1 / x x is approximately independent of , as in the paramagnetic 0 0 C phase x>x . [22]. The variation of 1 with Nb concentration x is C 0 Our magnetization measurements establish that the plotted in the lower panel of Fig. 4. As predicted, 1 0 phase diagram of Zr Nb Zn has three different regimes, vanishes approximately linearly with (x x ), while a 1 x x 2 C depicted in Fig. 4. Ferromagnetic order is found in changes by less than 10%. The temperature dependent Region I, below a phase line TC x which is controlled part of the initial susceptibility is isolated by defining by the x x , T 0 QCP. The Stoner criterion which = 1 1 =a C C 1 0 . 1 is plotted in Fig. 3(b) as a describes the stability of ferromagnetism remains un- function of T4=3 for each of the three paramagnetic con- changed even as x ! xC, indicating that the reduction in centrations with x 0:09, 0.12, and 0.14, and for compari- the electronic density of states drives the QCP. The sponta- son x 0:08, which has T 1:2 0:1K. Figure 3(b) C neous moment obeys the mean field expression, m0 =a T4=3 T 0:5 demonstrates that 1 below a temperature m0 0 in Region I. which vanishes at x 0:15 (Fig. 4), the termination of Simple power-law divergences with reduced tempera- quantum criticality for x>xC. ture are found in Region II, 0 , and the Quantum criticality is also observed for x xC, enhancement of as x ! xC reveals that Region II is although the critical phenomena associated with the finite best considered a crossover region, controlled by the rela- temperature ferromagnetic transition ultimately dominate tive magnitudes of and (x xC). Specifically, for small 1 1 4=3 1 as T ! TC. Since 0 aT , with 0 / x xC , and large (x xC), we find the mean field behavior of a 1 the quantum critical susceptibility is largest for x xC, and finite transition temperature ferromagnet . In the extends over the broadest range of absolute temperatures, opposite limit (small (x xC) and large T) we find that 1 4=3 since TC ! 0. This is demonstrated in the inset of the QCP is dominant, yielding a bT . Accord- 1 1 Fig. 3(b), where we have plotted 1= 0 as a ingly, it is possible to identify the quantum critical behav- 4=3 4=3 1 1 function of T .Forx xC, the T behavior is only ior of a three dimensional ferromagnet 0 4=3 found above a temperature TG which grows rapidly with aT above a temperature TG which decreases rapidly as (x xC), as shown in Fig. 4. The lower panel of Fig. 4 x ! xC. shows that for x xC, 1= 0 decreases approximately lin- The quantum critical regime III is also extensive for 4=3 early with (x xC), while the magnitude a of the T term paramagnetic concentrations x>xC, occurring below a 116404-3 PHYSICAL REVIEW LETTERS week ending PRL 96, 116404 (2006) 24 MARCH 2006 temperature T which is almost 150 K for x xC, and [14] C. Pfleiderer and A. D. Huxley, Phys. Rev. Lett. 89, dropping rapidly at larger x. Region III extends to the 147005 (2002). [15] M. Uhlarz, C. Pfleiderer, and S. M. Hayden, Phys. Rev. highest temperatures investigated for x 116404-4