^{1,a)}and Sayeef Salahuddin

^{1}

### Abstract

We propose a generic formalism to estimate the anisotropies, exchange energies, and the surface antiferromagnet(AFM) moment of AFM-ferromagnet (FM) heterostructure systems that show spin glass like behavior. This scheme provides quantitative agreement with multiple experiments on epitaxial bismuth ferrite (BFO)-FM system that have been reported recently. We find that a single value of the interface coupling energy can reproduce both the exchange bias and the coercivity enhancement observed in experiments. We also find a surprisingly high surface AFM moment density that agrees well with measured values. This high moment on the BFO surface is indicative of a significant modulation of magnetic properties at the BFO-FM interface.

The authors would like to acknowledge many useful discussions with R. Ramesh, M. Gajek, and J. Heron. We also thank A. Mougin for clarification of their experimental data. This work supported in part by Nanoelectronic Research Initiative (NRI) and National Science Foundation (NSF).

I. INTRODUCTION

II. STRUCTURE AND THEORY

III. RESULTS AND DISCUSSION

A. Effect of antiferromagnet surface spin density on SEB and SG systems

B. Phase lag between AFM and FM during magnetization reversal

C. High coupling energy versus high AFM surface moment density

IV. COMPARISON TO EXPERIMENT

V. CONCLUSION

### Key Topics

- Atomic force microscopy
- 79.0
- Antiferromagnetism
- 50.0
- Exchange interactions
- 44.0
- Coercive force
- 36.0
- Magnetic hysteresis
- 35.0

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## Figures

(a) Schematic of the exchanged coupled BFO-FM system modeled. The FM is considered as a macrospin moment. The AFM has pinned and unpinned moments. The pinned moment creates an exchange bias and the unpinned moment rotates with the FM and causes coercivity enhancement. (b) Azimuthal exchange bias for SG (black lines) and simple exchange biased (red lines) system. The simple exchange bias shows usual cosine behavior with angular variation. The SG system shows a peak bias at a critical angle before reducing to zero at 90°.

(a) Schematic of the exchanged coupled BFO-FM system modeled. The FM is considered as a macrospin moment. The AFM has pinned and unpinned moments. The pinned moment creates an exchange bias and the unpinned moment rotates with the FM and causes coercivity enhancement. (b) Azimuthal exchange bias for SG (black lines) and simple exchange biased (red lines) system. The simple exchange bias shows usual cosine behavior with angular variation. The SG system shows a peak bias at a critical angle before reducing to zero at 90°.

Effect of AFM surface magnetic moment density on exchange coupled systems. Exchange bias versus the measuring angle for (a) the SEB system and (b) the SG system. In both cases, exchange bias decreases with increasing AFM moment density. A change in the shape of the overall behavior specific to the SG system is also observed. (c) Coercive field versus the measuring angle for SG system showing a change from convex to concave dependence with respect to the vertical symmetry axis with increasing AFM moment density.

Effect of AFM surface magnetic moment density on exchange coupled systems. Exchange bias versus the measuring angle for (a) the SEB system and (b) the SG system. In both cases, exchange bias decreases with increasing AFM moment density. A change in the shape of the overall behavior specific to the SG system is also observed. (c) Coercive field versus the measuring angle for SG system showing a change from convex to concave dependence with respect to the vertical symmetry axis with increasing AFM moment density.

(a) Magnetization versus applied field for the SG system at the critical angle where exchange bias is maximized. When the exchange coupling is an order of magnitude larger than the anisotropies (), no hysteresis is observed for the FM and the AFM. With comparable exchange coupling (), there is significant hysteresis for both FM and the AFM. (b) Normal components of the magnetization during field sweep at the critical angle. For low exchange coupling, both the FM and AFM show jump and hence cause hysteresis. With high exchange energy, both the FM and AFM rotate coherently. Hence, the hysteresis collapses. (c) The phase lag of the AFM with respect to the FM, in the forward branch of the hysteresis loop with varying AFM moment density. Here, . The pinning field is opposite to the forward branch.

(a) Magnetization versus applied field for the SG system at the critical angle where exchange bias is maximized. When the exchange coupling is an order of magnitude larger than the anisotropies (), no hysteresis is observed for the FM and the AFM. With comparable exchange coupling (), there is significant hysteresis for both FM and the AFM. (b) Normal components of the magnetization during field sweep at the critical angle. For low exchange coupling, both the FM and AFM show jump and hence cause hysteresis. With high exchange energy, both the FM and AFM rotate coherently. Hence, the hysteresis collapses. (c) The phase lag of the AFM with respect to the FM, in the forward branch of the hysteresis loop with varying AFM moment density. Here, . The pinning field is opposite to the forward branch.

(a) Azimuthal exchange bias and (b) Azimuthal coercive field with strong exchange coupling and high AFM moment. The exchange bias at the critical angle is considerably larger than the bias at the pinning direction when the coupling energy is high. In a strongly coupled system, for the same anisotropy, the coercive field along the pinning direction is considerably higher.

(a) Azimuthal exchange bias and (b) Azimuthal coercive field with strong exchange coupling and high AFM moment. The exchange bias at the critical angle is considerably larger than the bias at the pinning direction when the coupling energy is high. In a strongly coupled system, for the same anisotropy, the coercive field along the pinning direction is considerably higher.

a) Exchange bias vs. AFM thickness calculation for BFO-FM SEB system (black color online). It was assumed that the AFM anisotropy varies linearly with thickness. The anisotropy energy of the pinned moment is estimated by matching with the exchange bias at the critical thickness where the bias ensues. The experimental values are taken from Ref. 3. (b) Hysteresis loops as the ratio R = J_{ eb }/K_{AFM} is varied. The hysteresis becomes gradually slanted as R increases. The square like hysteresis in the experiment indicates R < 0.5.

a) Exchange bias vs. AFM thickness calculation for BFO-FM SEB system (black color online). It was assumed that the AFM anisotropy varies linearly with thickness. The anisotropy energy of the pinned moment is estimated by matching with the exchange bias at the critical thickness where the bias ensues. The experimental values are taken from Ref. 3. (b) Hysteresis loops as the ratio R = J_{ eb }/K_{AFM} is varied. The hysteresis becomes gradually slanted as R increases. The square like hysteresis in the experiment indicates R < 0.5.

Comparison to experiment. The angle dependence of (a) the exchange bias and (b) the coercive field for the device reported in Ref. 6. (c) The applied field versus the normalized longitudinal magnetization at 45° to the pinning direction. The collapsed hysteresis resembles the high AFM moment density. (d) Hysteresis parallel and perpendicular to the pinning direction for the device reported in Ref. 4 using a different FM material. The excellent agreement with experimental measurement indicates the robustness of the proposed scheme and calculated parameters.

Comparison to experiment. The angle dependence of (a) the exchange bias and (b) the coercive field for the device reported in Ref. 6. (c) The applied field versus the normalized longitudinal magnetization at 45° to the pinning direction. The collapsed hysteresis resembles the high AFM moment density. (d) Hysteresis parallel and perpendicular to the pinning direction for the device reported in Ref. 4 using a different FM material. The excellent agreement with experimental measurement indicates the robustness of the proposed scheme and calculated parameters.

(a) Direction of the magnetic moments and the fields with respect to the pinning direction. For a SEB system, . Since, we studied a soft FM in this work, . Since, both unpinned AFM moment and FM have negligible coercivity, they will remain exchange coupled during the magnetization rotation. The total surface moment density of the rotating moment is (). (b) Comparison of the numerical and the analytical exchange bias for two different densities of the surface AFM unpinned moments. The numerical calculation was performed by minimizing Eq.(1). On the other hand, the analytical values were calculated using Eq.(2). The parameters used in this comparison are , , was varied between 25 × 10^{−6} and 25 × 10^{−5} emu/cm^{2}, , , and .

(a) Direction of the magnetic moments and the fields with respect to the pinning direction. For a SEB system, . Since, we studied a soft FM in this work, . Since, both unpinned AFM moment and FM have negligible coercivity, they will remain exchange coupled during the magnetization rotation. The total surface moment density of the rotating moment is (). (b) Comparison of the numerical and the analytical exchange bias for two different densities of the surface AFM unpinned moments. The numerical calculation was performed by minimizing Eq.(1). On the other hand, the analytical values were calculated using Eq.(2). The parameters used in this comparison are , , was varied between 25 × 10^{−6} and 25 × 10^{−5} emu/cm^{2}, , , and .

## Tables

Calculated parameters of BFO-FM interface. The same parameters of BFO reproduce the hysteresis properties reported from two different groups using different FM materials.

Calculated parameters of BFO-FM interface. The same parameters of BFO reproduce the hysteresis properties reported from two different groups using different FM materials.

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