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### Abstract

A single domain model is used to analyze the magnetic switching of two coupled magnetic layers, relevant for toggle magnetic random access memory, for the nonideal case where the layers are not identical. In general, we show that discontinuous transitions occur across the critical switching curve and continuous transitions occur around cusps in the critical switching curve. We illustrate this with examples of thickness imbalance, anisotropy mismatch, and Néel coupling and derive the corresponding switching behavior. Features in the direct-write switching can be used to experimentally distinguish between these various sources of asymmetry.

The author would like to thank the IBM MRAM team for experiments which stimulated this research and, in particular, Bill Gallagher, David Abraham, and Philip Trouilloud for helpful comments on the manuscript.

### Key Topics

- Anisotropy
- 6.0
- Critical point phenomena
- 4.0
- Liquid vapor phase transitions
- 2.0
- Magnetic anisotropy
- 2.0
- Magnetization reversals
- 2.0

## Figures

(Color online) (a) Energy landscape for a bit with , , , , , , and at a field of and . are the angles of the moments, with the easy axis. Blue (red) contours are low (high) energy; contours are apart with the lowest one at . The four minima are labeled L, D, R, and U. The dashed line is the valley floor. (b) Critical switching curve (CSC) for the same parameters as in (a), with cusps and dual cusps labeled with stars. The magnetic states at three fields are shown; the thicker arrow represents the moment of layer 1, and the arrows are offset for clarity. The easy axis is . The inset plots the energy along the valley floor shown as the dashed line in (a), with the minima labeled. (c) Zoom in of the lower portion of the CSC. The energy along the valley floor is plotted at the nine fields marked by dots (two dots are on top of stars) using the same scale as in (b).

(Color online) (a) Energy landscape for a bit with , , , , , , and at a field of and . are the angles of the moments, with the easy axis. Blue (red) contours are low (high) energy; contours are apart with the lowest one at . The four minima are labeled L, D, R, and U. The dashed line is the valley floor. (b) Critical switching curve (CSC) for the same parameters as in (a), with cusps and dual cusps labeled with stars. The magnetic states at three fields are shown; the thicker arrow represents the moment of layer 1, and the arrows are offset for clarity. The easy axis is . The inset plots the energy along the valley floor shown as the dashed line in (a), with the minima labeled. (c) Zoom in of the lower portion of the CSC. The energy along the valley floor is plotted at the nine fields marked by dots (two dots are on top of stars) using the same scale as in (b).

(Color online) (a) CSC for thickness imbalance, with , , , , , , and . Discontinuous transitions occur on dotted arrows across each branch of the CSC; the initial and final states are labeled for each arrow. Continuous transitions, on solid arrows, occur around each cusp, marked by a black star. Dual cusps, marked by colored stars, correspond to a point of indeterminate switching. The magnetic states are labeled by the letters L, D, R, and U in a circle. Each branch of the CSC is color coded to show which state disappears when crossing it. (b) Toggle map for word-line-first rectangular field excursions showing no change in white, direct write to R in red, and toggle in beige. Two examples of field excursions, both starting from the L state, are shown along with their transitions. (c) Toggle map for larger thickness imbalance, with . The dotted region is the direct-write area where only the R state exists. In (b) and (c), the scale for the axis is the same as for .

(Color online) (a) CSC for thickness imbalance, with , , , , , , and . Discontinuous transitions occur on dotted arrows across each branch of the CSC; the initial and final states are labeled for each arrow. Continuous transitions, on solid arrows, occur around each cusp, marked by a black star. Dual cusps, marked by colored stars, correspond to a point of indeterminate switching. The magnetic states are labeled by the letters L, D, R, and U in a circle. Each branch of the CSC is color coded to show which state disappears when crossing it. (b) Toggle map for word-line-first rectangular field excursions showing no change in white, direct write to R in red, and toggle in beige. Two examples of field excursions, both starting from the L state, are shown along with their transitions. (c) Toggle map for larger thickness imbalance, with . The dotted region is the direct-write area where only the R state exists. In (b) and (c), the scale for the axis is the same as for .

(Color online) (a) Critical switching curve for anisotropy mismatch, with , , , , , , and . See caption for Fig. 2. (b) Toggle map for word-line-first rectangular field excursions showing no change in white, direct write to R in red, direct write to L in green, and toggle in beige. Two examples of excursions, both starting from the R state, are shown along with their transitions. (c) Toggle map for bit-line-first rectangular excursions, which is simply a reflection of the word-line-first map about the easy axis. In (b) and (c), the scale for the axis is the same as for .

(Color online) (a) Critical switching curve for anisotropy mismatch, with , , , , , , and . See caption for Fig. 2. (b) Toggle map for word-line-first rectangular field excursions showing no change in white, direct write to R in red, direct write to L in green, and toggle in beige. Two examples of excursions, both starting from the R state, are shown along with their transitions. (c) Toggle map for bit-line-first rectangular excursions, which is simply a reflection of the word-line-first map about the easy axis. In (b) and (c), the scale for the axis is the same as for .

(Color online) Examples of the CSC for [(a) and (d)] thickness imbalance, [(b) and (e)] anisotropy mismatch, and [(c) and (f)] Néel coupling. Parts (d)–(f) also show the toggle maps for word-line-first rectangular field excursions. Except as noted, the parameters are , , , , , and .

(Color online) Examples of the CSC for [(a) and (d)] thickness imbalance, [(b) and (e)] anisotropy mismatch, and [(c) and (f)] Néel coupling. Parts (d)–(f) also show the toggle maps for word-line-first rectangular field excursions. Except as noted, the parameters are , , , , , and .

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