Scaling of nonlinear susceptibilities in an artificial permalloy honeycomb lattice

A. Dahal, Y. Chen, B. Summers, and D. K. Singh

Phys. Rev. B 97, 214420 – Published 18 June 2018

Abstract

Two-dimensional artificial magnetic honeycomb lattice is predicted to manifest several magnetic phase transitions as a function of reducing temperature. We have performed the analysis of nonlinear susceptibility to explore the equilibrium nature of phase transition in artificial honeycomb lattice of ultrasmall connected permalloy ( ${Ni}_{0.81} {Fe}_{0.19}$ ) elements, typical length of $≃ 12$ nm. The nonlinear susceptibility $χ_{n 1}$ is found to exhibit an unusual crossover character in both temperature and magnetic field. The higher order susceptibility $χ_{3}$ changes from positive to negative as the system traverses through the spin solid phase transition at $T_{s} = 29$ K. Additionally, the static critical exponents, used to test the scaling of $χ_{n 1}$ , do not follow the conventional scaling relation. We conclude that the magnetic phase transition, especially to the low temperature spin solid order, is not conventional in nature at this length scale.

Received 14 December 2017
Revised 24 March 2018

DOI:https://doi.org/10.1103/PhysRevB.97.214420

Physics Subject Headings (PhySH)

Frustrated magnetism Magnetic phase transitions Magnetic susceptibility

Nanostructures

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

A. Dahal, Y. Chen, B. Summers, and D. K. Singh^*

Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA

^*singhdk@missouri.edu

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Issue

Vol. 97, Iss. 21 — 1 June 2018

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Images

Figure 1
Magnetization as a function of field. Here magnetization is plotted as a function of field at different temperatures. Magnetization data exhibits a crossover behavior in field. While the higher temperature susceptibility is stronger at low field, the magnetization at low temperature is larger above the crossover field $H ≃ 0.04$ –0.06 T. Inset shows the scanning electron micrograph of a typical artificial honeycomb lattice of ultrasmall elements. Magnetic field was applied in-plane to the lattice.
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Figure 2
Nonlinear susceptibility $χ_{n 1}$ as functions of field and temperature. (a) $χ_{n 1}$ is estimated using Eqs. (2, 3, 4) where $χ_{1}$ is obtained from fitting $M$ vs $H$ plot at low field. Two features are immediately obvious in this figure: a change in the sign of overall nonlinear susceptibility across $T ≃ 30$ K and a crossover regime in field and temperature. As shown in the inset of the figure, the slope of the curve changes from negative to positive at some field value. We call it characteristic crossover field, which increases as temperature decreases. (b) and (c) Higher order susceptibility $χ_{3}$ as a function of temperature across the crossover field. $χ_{3}$ increases as a function of temperature in high field regime 2 (b) and becomes more negative in low field regime 1 (c).
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Figure 3
Estimation of static critical exponents $γ$ and $δ$ . (a) To estimate the critical exponent $γ$ , the coefficient $a_{3}$ (see text for detail) is plotted as a function of $τ = (T / T_{s}) - 1$ for different $T_{s}$ values, across the spin solid transition at $T = 30$ K. $γ$ is estimated by fitting the fixed number of points in the divergence regime of the curve using Eq. (5). Best fit to the experimental data is obtained for the critical exponent $| γ | = 1.9$ (inset shows the plot of fitting parameter $χ^{2}$ vs $γ$ ). (b) Similar analysis is performed for the low field regime 1, with estimated $| γ | = 1.4$ . In both regimes, best fit corresponds to spin solid transition at $T_{s} = 29$ K. Nonlinear susceptibility $χ_{n 1}$ is plotted as a function of field at temperature near $T_{s}$ . Experimental data are fitted using the asymptotic function in Eq. (6) to obtain critical exponent $| δ |$ (c) in high field regime 2, $≃ 2.4$ and (d) low field regime 1, $≃ 2.5$ .
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Figure 4
Scaling analysis of nonlinear susceptibilities in artificial honeycomb lattice. (a) Nonlinear susceptibilities exhibit scaling behavior for $| γ | = 1.5$ , $δ = 10$ , and $| β | = 0.1 .$ The critical scaling coefficients do not satisfy the scaling relation in Eq. (7). (b) Similar analysis was performed in the low field regime 1. Interestingly, the nonlinear susceptibilities exhibit scaling behavior for the same set of critical exponents, as in high field regime 2.
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