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^{1,a)}, P. L. Trouilloud

^{1}and W. J. Gallagher

^{1}

### Abstract

The authors analyze the critical switching curve for two identical coupled magnetic free layers, as used in toggle magnetic random access memory. The continuous and discontinuous transitions between different magnetic states are described. A general criteria for toggling is derived by summing up the number of clockwise and counterclockwise transitions, leading to a larger toggle region than previously reported. It also leads to a significant chirality effect, wherein the toggle region shifts depending on the order in which the word and bit line fields are applied. Finally, the authors discuss a type of switching useful for experimentally measuring the critical switching curve.

The authors would like to thank the IBM MRAM team for experiments which stimulated this analysis. Work supported in part by the Defense Microelectronics Activity.

### Key Topics

- Demagnetization
- 5.0
- Ap stars
- 3.0
- Chiral symmetries
- 3.0
- Activation energies
- 2.0
- Stellar spectral lines
- 2.0

## Figures

(Color online) Energy landscapes at representative fields showing the available magnetic states (for a with , , and ). are the angles of the moments, with the easy axis. Blue (red) contours are low (high) energy, with between contours. Dots mark energy minima. (a) , , and the lowest contour is at . At low fields only the two antiparallel (AP) states exist. (b) , , and the lowest contour is at . At fields inside the critical switching curve (CSC) both the two AP states and the two scissor (S) states exist. (c) , , and the lowest contour is at . At large fields only the two S states exist. There is a continuous phase transition between AP and S states outside the CSC. (d) CSC showing the AP states (red arrows) and S states (blue arrows) for each of (a)–(c) (the easy axis is ).

(Color online) Energy landscapes at representative fields showing the available magnetic states (for a with , , and ). are the angles of the moments, with the easy axis. Blue (red) contours are low (high) energy, with between contours. Dots mark energy minima. (a) , , and the lowest contour is at . At low fields only the two antiparallel (AP) states exist. (b) , , and the lowest contour is at . At fields inside the critical switching curve (CSC) both the two AP states and the two scissor (S) states exist. (c) , , and the lowest contour is at . At large fields only the two S states exist. There is a continuous phase transition between AP and S states outside the CSC. (d) CSC showing the AP states (red arrows) and S states (blue arrows) for each of (a)–(c) (the easy axis is ).

(Color online) (a) CSC with transitions labeled (for a with , , and ). For fields smaller than the outer (red) curve, the AP states are stable, and for fields larger than the inner (blue) curve the scissor (S) states are stable. AP states switch, with either a clockwise (CW) or counterclockwise (CCW) rotation, on blue arrows into an S state. S states switch to AP states on red arrows. Dotted arrows are discontinuous transitions across the CSC, and solid arrows are continuous transitions around the cusps marked with black stars. Examples of the dependence of the CSC on (b) bit length and (c) ratio of dipole to demagnetization field are also shown.

(Color online) (a) CSC with transitions labeled (for a with , , and ). For fields smaller than the outer (red) curve, the AP states are stable, and for fields larger than the inner (blue) curve the scissor (S) states are stable. AP states switch, with either a clockwise (CW) or counterclockwise (CCW) rotation, on blue arrows into an S state. S states switch to AP states on red arrows. Dotted arrows are discontinuous transitions across the CSC, and solid arrows are continuous transitions around the cusps marked with black stars. Examples of the dependence of the CSC on (b) bit length and (c) ratio of dipole to demagnetization field are also shown.

(Color online) Shaded toggle regions for rectangular field excursions for a with , , and . Word line first (a) and bit line first (b) field excursions shift the toggle region up and to the left, and down and to the right, respectively (chirality effect). Three examples of field excursions that result in a toggle are shown. Each is labeled with two arrows, solid for a continuous transition or dotted for a discontinuous transition, showing transitions to first a scissor state (blue arrows) and then back to an antiparallel state (red arrows).

(Color online) Shaded toggle regions for rectangular field excursions for a with , , and . Word line first (a) and bit line first (b) field excursions shift the toggle region up and to the left, and down and to the right, respectively (chirality effect). Three examples of field excursions that result in a toggle are shown. Each is labeled with two arrows, solid for a continuous transition or dotted for a discontinuous transition, showing transitions to first a scissor state (blue arrows) and then back to an antiparallel state (red arrows).

(Color online) Shaded toggle regions for L-shaped and U-shaped field excursions for a with , , and . L-shaped word line first (a) and bit line first (b), and U-shaped word line first (c) and bit line first (d) field excursions each map out toggle regions along one of the four edges of the critical switching curve. Two examples of field excursions (one that results in a toggle and one that does not) are shown; both start and end at the origin. Each is labeled with arrows, solid for a continuous transition or dotted for a discontinuous transition, showing transitions to scissor states in blue and to antiparallel states in red. The order in which the transitions occur is given by the numbers 1–4.

(Color online) Shaded toggle regions for L-shaped and U-shaped field excursions for a with , , and . L-shaped word line first (a) and bit line first (b), and U-shaped word line first (c) and bit line first (d) field excursions each map out toggle regions along one of the four edges of the critical switching curve. Two examples of field excursions (one that results in a toggle and one that does not) are shown; both start and end at the origin. Each is labeled with arrows, solid for a continuous transition or dotted for a discontinuous transition, showing transitions to scissor states in blue and to antiparallel states in red. The order in which the transitions occur is given by the numbers 1–4.

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