Finite-size scaling effect on N\'eel temperature of antiferromagnetic ${\mathrm{Cr}}_{2}{O}_{3}$ (0001) films in exchange-coupled heterostructures

Finite-size scaling effect on Néel temperature of antiferromagnetic ${Cr}_{2} O_{3}$ (0001) films in exchange-coupled heterostructures

Satya Prakash Pati, Muftah Al-Mahdawi, Shujun Ye, Yohei Shiokawa, Tomohiro Nozaki, and Masashi Sahashi

Phys. Rev. B 94, 224417 – Published 19 December 2016

Abstract

The scaling of antiferromagnetic ordering temperature of corundum-type chromia films has been investigated. Néel temperature $T_{N}$ was determined from the effect of perpendicular exchange bias on the magnetization of a weakly-coupled adjacent ferromagnet. For a thick-film case, the validity of detection is confirmed by a susceptibility measurement. Detection of $T_{N}$ was possible down to 1-nm-thin chromia films. The scaling of ordering temperature with thickness was studied using different buffering materials and compared with Monte-Carlo simulations. The spin-correlation length and the corresponding critical exponent were estimated, and they were consistent between experimental and simulation results. The spin-correlation length is an order of magnitude less than cubic antiferromagnets. We propose that the difference is from the change of number of exchange-coupling links in the two crystal systems.

Received 12 May 2016
Revised 8 November 2016

DOI:https://doi.org/10.1103/PhysRevB.94.224417

Physics Subject Headings (PhySH)

Exchange interaction

Antiferromagnets Heterostructures Thin films

Finite-size scaling Monte Carlo methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Satya Prakash Pati^*, Muftah Al-Mahdawi^†, Shujun Ye, Yohei Shiokawa, Tomohiro Nozaki, and Masashi Sahashi

Department of Electronic Engineering, Tohoku University, Sendai 980-8579, Japan

^*sppati@ecei.tohoku.ac.jp
^†mahdawi@ecei.tohoku.ac.jp

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Vol. 94, Iss. 22 — 1 December 2016

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Figure 1
The height distribution of a 1.5 nm ${Cr}_{2} O_{3}$ surface grown over an $Ir - {Fe}_{2} O_{3}$ buffer. The fitting (red line) to a log-normal distribution gives a shape parameter $s$ of 0.036. The inset shows the corresponding topography scan.
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Figure 2
(a) A schematic of $T_{N}$ detection method. The anisotropy of the Co layer is optimized such that the ordering of ${Cr}_{2} O_{3}$ spins changes the tilting of Co magnetization from an in-plane above $T_{N}$ to an out-of-plane direction below $T_{N}$ . A Ru metallic spacer layer is used to tune the exchange coupling energy $J_{K}$ and interfacial perpendicular anisotropy. The definitions of magnetic field $H$ and magnetization equilibration angle $θ_{0}$ are indicated. (b) Dependence of the total exchange coupling energy $J_{K}$ on Ru spacer thickness $t_{Ru}$ found from magnetization hysteresis loops at 100 K. $J_{K}$ monotonically decreases against the thickness of Ru. (c) Dependence of low-field magnetization normalized to saturation magnetization $M_{r} / M_{s}$ on $t_{Ru}$ . At $1.25 \leq t_{Ru} \leq 1.5$ nm, a large change in $M_{r} / M_{s}$ is found above and below $T_{N}$ . (d) The change of $M - H$ loops with changing $t_{Ru}$ is shown. For thin Ru ( $t_{Ru} \leq 1.0$ ), a dominant perpendicular anisotropy remains above and below $T_{N}$ . At an intermediate thickness of 1.25–1.50 nm, there is a large change of magnetization's easy direction upon crossing $T_{N}$ . In (b),(c), and (d) a 25 nm Pt buffer was used and the solid lines are eye guides.
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Figure 3
Comparison between low-field and high-field $M - T$ curves of Pt (25)/ ${Cr}_{2} O_{3}$ (1000)/Ru (1.25)/Co (1)/Pt (5). To remove the contribution of the substrate's diamagnetic response, the change in the total magnetization with respect to $T_{N}$ is plotted against temperature. The low-field $M - T$ curves correspond to the change in Co magnetization direction and to the ${Cr}_{2} O_{3}$ susceptibility response $χ H - T$ at high fields. Both measurements agree on a $T_{N}$ of $\approx 300$ K.
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Figure 4
(a) $M - T$ curves under $H = 50 Oe$ for ${Fe}_{2} O_{3}$ -buffered ${Cr}_{2} O_{3}$ ( $t_{Cr 2 O 3}$ )/Ru (1.25)/Co (1)/Pt (5), where $t_{Cr 2 O 3} = 20$ , 10, 5, 3, and 1 nm. (b) The shift of $T_{N}$ with $t_{Cr 2 O 3}$ is shown for different buffers, in addition to Monte-Carlo simulations. (c) A log-log plot of normalized $T_{N}$ versus $t_{Cr 2 O 3}$ . Solid lines are fittings to Eq. (5). The characteristic spin-correlation length $ξ_{0}$ is indicated by an extrapolation to the vanishing point of $T_{N}$ . Shift exponent $λ$ is determined from the slope. Both $ξ_{0}$ and $λ$ are in a reasonable agreement among experiments and simulation.
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Figure 5
Monte-Carlo (MC) simulation results of spin-correlation length using (a),(b) simulation method 1, and (c) simulation method 2. (a) A direct calculation of correlation function $Γ$ between center spin $S (0)$ and all other spins $S (r)$ in a $5 \times 5 \times 5 {nm}^{3}$ geometry. The temperature-dependent correlation length $ξ$ is found from a fitting to $Γ = r^{- 1.036} e^{- r / ξ}$ . (b) Temperature dependence of $ξ$ in the cases of including or disabling periodic boundary conditions. Both cases produce the same results. Inset is a log-log plot with a linear fitting to the power law in Eq. (6). (c) Thickness dependence of $T_{N}$ determined from the sublattice magnetization $〈 M_{sub} 〉$ . The finite thickness results in a decrease in $T_{N}$ when it approaches spin-correlation length. $T_{N}$ values are summarized in Figs. 4 and 4. The estimations of $ξ_{0}$ and $λ$ from both simulation methods are in a reasonable agreement [Figs. 4 and 5].
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Figure 6
Plane projections of (a) corundum and (b) rock-salt crystal structures on ( $11 \bar{2} 0$ ) and (100) planes, respectively. Differences in the coordination number of exchange interactions result in a different correlation length for each crystal system. It can be seen by counting the number of indirect connections between far neighbors, e.g., spins marked by numbers 1 and 2. The rock-salt structure has more connections in comparison to the corundum-type structure.
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Physical Review B

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Finite-size scaling effect on Néel temperature of antiferromagnetic ${Cr}_{2} O_{3}$ (0001) films in exchange-coupled heterostructures

Satya Prakash Pati, Muftah Al-Mahdawi, Shujun Ye, Yohei Shiokawa, Tomohiro Nozaki, and Masashi Sahashi

Phys. Rev. B 94, 224417 – Published 19 December 2016

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Finite-size scaling effect on Néel temperature of antiferromagnetic Cr2O3 (0001) films in exchange-coupled heterostructures

Satya Prakash Pati, Muftah Al-Mahdawi, Shujun Ye, Yohei Shiokawa, Tomohiro Nozaki, and Masashi Sahashi

Phys. Rev. B 94, 224417 – Published 19 December 2016

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Finite-size scaling effect on Néel temperature of antiferromagnetic ${Cr}_{2} O_{3}$ (0001) films in exchange-coupled heterostructures