Critical Phenomena and the Quantum Critical Point of Ferromagnetic ${\mathrm{Zr}}_{1\ensuremath{-}x}{\mathrm{Nb}}_{x}{\mathrm{Zn}}_{2}$

Critical Phenomena and the Quantum Critical Point of Ferromagnetic ${Zr}_{1 - x} {Nb}_{x} {Zn}_{2}$

D. A. Sokolov, M. C. Aronson, W. Gannon, and Z. Fisk

Phys. Rev. Lett. 96, 116404 – Published 24 March 2006

Abstract

We present a study of the magnetic properties of ${Zr}_{1 - x} {Nb}_{x} {Zn}_{2}$ , using an Arrott plot analysis of the magnetization. The Curie temperature $T_{C}$ is suppressed to zero temperature for Nb concentration $x_{C} = 0.083 \pm 0.002$ , while the spontaneous moment vanishes linearly with $T_{C}$ as predicted by the Stoner theory. The initial susceptibility $χ$ displays critical behavior for $x \leq x_{C}$ , with a critical exponent which smoothly crosses over from the mean field to the quantum critical value. For high temperatures and $x \leq x_{C}$ , and for low temperatures and $x \geq x_{C}$ we find that $χ^{- 1} = χ_{0}^{- 1} + a T^{4 / 3}$ , where $χ_{0}^{- 1}$ vanishes as $x \to x_{C}$ . The resulting magnetic phase diagram shows that the quantum critical behavior extends over the widest range of temperatures for $x = x_{C}$ , and demonstrates how a finite transition temperature ferromagnet is transformed into a paramagnet, via a quantum critical point.

Received 6 December 2005

DOI:https://doi.org/10.1103/PhysRevLett.96.116404

Authors & Affiliations

D. A. Sokolov¹, M. C. Aronson¹, W. Gannon¹, and Z. Fisk^2,*

¹University of Michigan, Ann Arbor, Michigan 48109-1120, USA
²Florida State University, Tallahassee, Florida 32310-3706, USA

^*Present address: Department of Physics, University of California Davis, CA 95616-8677, USA.

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Vol. 96, Iss. 11 — 24 March 2006

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Figure 1
(a) Temperature dependence of the spontaneous magnetization $m_{0}$ in $μ_{B}$ per formula unit (f.u.) for ${Zr}_{1 - x} {Nb}_{x} {Zn}_{2}$ [ $x = 0$ (●), 0.03 (○), 0.05 (■), 0.06 (□)]. The solid lines are fits to $m_{0} (T) = m_{0} (0) [(T_{C} - T) / T_{C}]^{0.5}$ . Inset: Example of an Arrott plot for $x = 0.03$ , for temperatures from 1.8 K (●) to 20 K (▲), the solid lines are fits to $m^{2} \propto H / m$ , arrows mark extrapolated $1 / χ (T)$ (horizontal axis) and $m^{2} (T)$ (vertical axis). (b) Variation of Curie temperature $T_{C}$ (○) and $T_{C}^{4 / 3}$ (●) with Nb concentration $x$ . Solid lines are fits to $T_{C} \propto (x_{C} - x)^{3 / 4}$ and $T_{C}^{4 / 3} \propto (x_{C} - x)$ , with $x_{C} = 0.083 \pm 0.002$ . Inset: The zero temperature spontaneous moment $m_{0} (0) \propto T_{C}$ . Y, Ti, Hf doping (⊕) [37]; pressure, single crystal (○) [15]; pressure, ${ZrZn}_{1.9}$ polycrystal (□) [18]; Nb doping, polycrystals, this work (▲). Solid line is Stoner parameter $α$ [38].Reuse & Permissions
Figure 2
The temperature dependence of the inverse of the initial susceptibility $χ^{- 1}$ for Nb concentrations both larger and smaller than the critical concentration $x_{C} = 0.083 \pm 0.002$ .Reuse & Permissions
Figure 3
(a) $χ^{- 1} = χ_{00}^{- 1} τ^{γ}$ at different Nb concentrations: $x = 0$ (●), $x = 0.055$ (■), $x = 0.08$ (▲). Solid lines are the best fits to $χ^{- 1} = χ_{00}^{- 1} τ^{γ}$ . Inset: the critical exponent $γ$ (○) increases continuously as $x \to x_{C}$ . The solid line is a guide for the eye. (b) $1 / a χ^{*}$ (defined in the text) for $x \geq x_{c}$ is proportional to $T^{4 / 3}$ for $T \leq T^{*}$ . For $x \geq x_{c}$ , $1 / a χ^{*}$ deviates from $T^{4 / 3}$ (solid line) above a cutoff temperature $T^{*}$ marked by arrows, $x = 0.08$ (□), $x = 0.09$ (■), $x = 0.12$ (○), $x = 0.14$ (●). Inset: For $x \leq x_{c}$ $1 / χ^{*}$ deviates from $T^{4 / 3}$ behavior below a crossover temperature $T_{G}$ marked by arrows: $x = 0.08$ (□), $x = 0.06$ (△), $x = 0.03$ (▲). The solid line is a best fit of $1 / χ^{*} = a T^{4 / 3}$ for $x = 0.08$ .Reuse & Permissions
Figure 4
Top: The temperature-Nb concentration phase diagram for ${Zr}_{1 - x} {Nb}_{x} {Zn}_{2}$ . $T_{C} (x)$ (○), with the phase line $T_{C} \propto (x - x_{C})^{3 / 4}$ , the cutoff temperature $T_{G}$ (□) for $x \leq x_{C}$ with the crossover line $T_{G} \propto (x - x_{C})^{1.2 \pm 0.3}$ , the cutoff temperature $T^{*}$ (△) for $x \geq x_{C}$ ; the dashed line is a guide for the eye. The diamonds (◇) indicate the highest temperature investigated for $x \leq x_{C}$ . I, II, and III correspond, respectively, to the region of stable ferromagnetism, the crossover region, and the quantum critical region as defined in the text. Bottom: Variation with Nb concentration of the $T = 0$ inverse susceptibility $1 / χ_{0}$ for $x \geq x_{C}$ (■) and for $x \leq x_{C}$ (□). Solid lines are fits with $1 / χ_{0} \propto | x - x_{C} |$ . For $x = 0.14$ , $1 / χ_{0}$ is divided by 15 to appear on the plot.Reuse & Permissions

Physical Review Letters

Critical Phenomena and the Quantum Critical Point of Ferromagnetic Zr1−xNbxZn2

D. A. Sokolov, M. C. Aronson, W. Gannon, and Z. Fisk

Phys. Rev. Lett. 96, 116404 – Published 24 March 2006

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Critical Phenomena and the Quantum Critical Point of Ferromagnetic ${Zr}_{1 - x} {Nb}_{x} {Zn}_{2}$