Quantum percolation phase transition and magnetoelectric dipole glass in hexagonal ferrites

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Quantum percolation phase transition and magnetoelectric dipole glass in hexagonal ferrites

S. E. Rowley, T. Vojta, A. T. Jones, W. Guo, J. Oliveira, F. D. Morrison, N. Lindfield, E. Baggio Saitovitch, B. E. Watts, and J. F. Scott

Phys. Rev. B 96, 020407(R) – Published 17 July 2017

Abstract

Hexagonal ferrites not only have enormous commercial impact (£2 billion/year in sales) due to applications that include ultrahigh-density memories, credit-card stripes, magnetic bar codes, small motors, and low-loss microwave devices, they also have fascinating magnetic and ferroelectric quantum properties at low temperatures. Here we report the results of tuning the magnetic ordering temperature in $PbF e_{12 - x} G a_{x} O_{19}$ to zero by chemical substitution $x$ . The phase transition boundary is found to vary as $T_{N} \sim {(1 - x / x_{c})}^{2 / 3}$ with $x_{c}$ very close to the calculated spin percolation threshold, which we determine by Monte Carlo simulations, indicating that the zero-temperature phase transition is geometrically driven. We find that this produces a form of compositionally tuned, insulating, ferrimagnetic quantum criticality. Close to the zero-temperature phase transition, we observe the emergence of an electric dipole glass induced by magnetoelectric coupling. The strong frequency behavior of the glass freezing temperature $T_{m}$ has a Vogel-Fulcher dependence with $T_{m}$ finite, or suppressed below zero in the zero-frequency limit, depending on composition $x$ . These quantum-mechanical properties, along with the multiplicity of low-lying modes near the zero-temperature phase transition, are likely to greatly extend applications of hexaferrites into the realm of quantum and cryogenic technologies.

Received 25 April 2017
Revised 15 June 2017

DOI:https://doi.org/10.1103/PhysRevB.96.020407

Physics Subject Headings (PhySH)

Critical phenomena Dielectric properties Ferrimagnetism Ferroelectricity Magnetic phase transitions Magnetoelectric effect Percolation Quantum phase transitions

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

S. E. Rowley^1,2,3, T. Vojta⁴, A. T. Jones^1,5, W. Guo¹, J. Oliveira², F. D. Morrison⁶, N. Lindfield⁶, E. Baggio Saitovitch², B. E. Watts⁷, and J. F. Scott⁸

¹Cavendish Laboratory, Cambridge University, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
²Centro Brasileiro de Pesquisas Físicas, Rua Dr Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil
³Quantum Materials Laboratory, Cambridge CB3 9NF, United Kingdom
⁴Physics Department, Missouri University of Science and Technology, Rolla, Missouri 65409, USA
⁵Physics Department, Lancaster University, Lancaster LA1 4YB, United Kingdom
⁶School of Chemistry, St. Andrews University, St. Andrews KY16 9ST, United Kingdom
⁷IMEM-CNR, Parco Area delle Scienze 37/A, 43124 Parma, Italy
⁸Schools of Chemistry and of Physics and Astronomy, St. Andrews University, St. Andrews KY16 9ST, United Kingdom

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Vol. 96, Iss. 2 — 1 July 2017

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Images

Figure 1
Magnetic phase diagram and crystal structure of $PbF e_{12 - x} G a_{x} O_{19}$ . The Néel temperature, $T_{N} \approx 718 K$ , in $PbF e_{12} O_{19}$ separating paramagnetic and ferrimagnetic phases is suppressed via nonmagnetic Ga substitution and tuned through a geometrically driven percolation phase transition located at $T = 0 K$ and $x = x_{c} \approx 8.6$ as shown in (a). $x = 8.6$ is close to the calculated percolation threshold $x = 8.85$ referred to in the main text for the hexaferrite structure. The right inset shows a double unit cell of $PbF e_{12} O_{19}$ as explained more fully in the main text with the crystallographic $c$ direction indicated by the arrow. Values of $T_{N}$ were determined by Mössbauer and magnetization measurements [6]. The value of $T_{N}$ as a function of Ga $x$ in the related materials $BaF e_{12 - x} G a_{x} O_{19}$ and $SrF e_{12 - x} G a_{x} O_{19}$ differ from those shown above for $PbF e_{12 - x} G a_{x} O_{19}$ by only a few percent. The main figure shows $T_{N}^{3 / 2}$ (blue dots) plotted against $x$ and the straight line is a best fit to the data with an equation of the form $T_{N} / (718 K) = {(1 - x / x_{c})}^{ϕ}$ with critical Ga concentration $x_{c} = 8.56$ , and the power-law exponent determined as $ϕ = 0.67 \pm 0.02$ , i.e., 2/3. The region labeled (A) is where uniaxial quantum critical ferroelectric fluctuations have recently been reported in $BaF e_{12} O_{19}$ and $SrF e_{12} O_{19}$ [3, 9]. The regions labeled (B) and (C) are where an electric dipole glass state (ferroelectric relaxor) is observed, induced by magnetoelectric coupling as explained in the main text and later figures. The dashed line separates the classical paramagnetic phase and the paramagnetic phase composed of disconnected clusters of ferrimagnetic order. The region labeled (C) is where one might expect to search for exotic spin and thermodynamic states [27, 28]. In (b) the left image shows a projection into the $a − b$ plane of the relevant magnetic ions (one spinel block shown) used in the percolation calculation explained in the main text for the undoped parent compound $PbF e_{12} O_{19}$ . The right image shows a percolating magnetic cluster under the conditions of magnetic dilution in $PbF e_{12 - x} G a_{x} O_{19}$ with $x$ close to $x_{c}$ .
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Figure 2
Thermal and magnetization measurements in $PbF e_{3} G a_{9} O_{19}$ $(x = 9)$ . The heat capacity as a function of temperature in (a) demonstrates along with Mössbauer experiments [6] the absence of a bulk phase transition and thus no long-range order in a sample with $x > x_{c}$ . The weak hysteresis measured at 2 K in (b) indicates the contribution to the uniform magnetization from “rare regions”—small disconnected ferrimagnetic clusters—in the paramagnetic phase at low temperatures. Close to the zero-temperature percolation phase transition, the magnetic clusters are randomly distributed in space but perfectly correlated in time, leading to the possibility of singular thermodynamic functions over a range of tuning parameter variables [20, 28].
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Figure 3
(a),(b) Real and imaginary parts of the dielectric constant and (c) Vogel-Fulcher plots for $PbF e_{12 - x} G a_{x} O_{19}$ showing dipole glass behavior. (a) and (b) show the real ε and imaginary parts $ɛ^{''}$ of the dielectric constant measured at different frequencies plotted against temperature $T$ for samples of $PbF e_{12 - x} G a_{x} O_{19}$ with $x = 8.4$ and $x = 9.6$ , respectively. (c) A Vogel-Fulcher fit to the data of the peak temperature $T_{m}$ vs measurement frequency $f$ for the same two samples. The Vogel-Fulcher equation is of the form $f = f_{0} \exp [- T_{a} / (T_{m} - T_{f})]$ , where the constant $T_{a}$ is the activation temperature scale and $f_{0}$ is a characteristic frequency. The frequency-dependent variable $T_{m} (f)$ is defined as the temperature at which $ɛ^{'}$ has a peak in $T$ as in the examples shown in (a). The constant $T_{f}$ is the freezing temperature in the zero-frequency limit. For $x = 9.6$ , the fitting parameters were $T_{a} = 730 K, T_{f} = - 11.7 K$ , and $f_{0} = 3.02 \times 10^{11} Hz$ , and for $x = 8.4$ , they were $T_{a} = 611 K, T_{f} = 18.1 K$ , and $f_{0} = 3.00 \times 10^{11} Hz$ .
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