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^{1,a)}, K. E. Kürten

^{1,2}and F. V. Kusmartsev

^{1}

### Abstract

The magnetic properties of a system of three interacting magnetic elliptical disks are examined. For the various levels of uniaxial anisotropy investigated a complicated series of phase transitions exist. These are marked by the critical lines of stability that are demonstrated in an applied magnetic field plane diagram.

D.M.F. thanks the EPSRC for funding under Grant No. EP/F005482/1.

### Key Topics

- Phase transitions
- 7.0
- Magnetic fields
- 6.0
- Cell membranes
- 4.0
- Magnetic phase transitions
- 4.0
- Phase diagrams
- 4.0

## Figures

The normalized net moment projected along the direction of the field as a function of the field. The magnetization angles in the three MLs are shown for and . There exist six branches of magnetization, B1–B6. A gradient descent method of iterative analysis shows the hysteresis path that occurs when is the starting configuration (bottom right illustration in (a)], i.e., beginning on B3. Dark (light) lines and arrows show the forward (reverse) evolution of the magnetization. Metastable branch B3 may be realized by a rapid cooling technique. The repetition of the fast cooling from high temperatures at different AMF strengths may drive the system to settle in the different valleys of the energy landscape. The top right illustration in (a) shows the six CPs (C1–C6) in the positive quadrant of the M-H plot at which there is a change in phase. The darkest (green online) areas (e.g. ) show , exclusively. The white areas (e.g., ) in the plots are where no perfectly non-P configurations (PNPCs) can appear. The lightly shaded areas represent the range of in which there exists the possibility of the appearance of the PNPCs that correspond to . CPs C1–C4 and C6 are characterized by Barkhausen jumps, whereas C5 marks the point along B1 when . As the AMF increases in strength tends to the field angle. The CPs C3 and C4 are offset from one another by [bottom inset of the main M-H in (a)]. The top inset magnification shows that B3 does not intercept B2. (b) The transition from C4 to B1 at (at this point on B1, are not quite P. As H increases in strength will transition into a P configuration with a second order phase transition, marked by a CP such as C5). The top vector in each column represents the magnetization vector in the top ML, depicted as ML-1. Likewise, the middle and bottom layers are denoted by ML-2 and ML-3, respectively. The first column is the orientation of the magnetization vectors at C4. The second column is after the system has switched to B1. (c) The transition from C1 to B5 at . (d) C3 to B1 at . (e) C2 to B2 at . (f) C5 to C2 at to , respectively. (g) C6 to B1 at .

The normalized net moment projected along the direction of the field as a function of the field. The magnetization angles in the three MLs are shown for and . There exist six branches of magnetization, B1–B6. A gradient descent method of iterative analysis shows the hysteresis path that occurs when is the starting configuration (bottom right illustration in (a)], i.e., beginning on B3. Dark (light) lines and arrows show the forward (reverse) evolution of the magnetization. Metastable branch B3 may be realized by a rapid cooling technique. The repetition of the fast cooling from high temperatures at different AMF strengths may drive the system to settle in the different valleys of the energy landscape. The top right illustration in (a) shows the six CPs (C1–C6) in the positive quadrant of the M-H plot at which there is a change in phase. The darkest (green online) areas (e.g. ) show , exclusively. The white areas (e.g., ) in the plots are where no perfectly non-P configurations (PNPCs) can appear. The lightly shaded areas represent the range of in which there exists the possibility of the appearance of the PNPCs that correspond to . CPs C1–C4 and C6 are characterized by Barkhausen jumps, whereas C5 marks the point along B1 when . As the AMF increases in strength tends to the field angle. The CPs C3 and C4 are offset from one another by [bottom inset of the main M-H in (a)]. The top inset magnification shows that B3 does not intercept B2. (b) The transition from C4 to B1 at (at this point on B1, are not quite P. As H increases in strength will transition into a P configuration with a second order phase transition, marked by a CP such as C5). The top vector in each column represents the magnetization vector in the top ML, depicted as ML-1. Likewise, the middle and bottom layers are denoted by ML-2 and ML-3, respectively. The first column is the orientation of the magnetization vectors at C4. The second column is after the system has switched to B1. (c) The transition from C1 to B5 at . (d) C3 to B1 at . (e) C2 to B2 at . (f) C5 to C2 at to , respectively. (g) C6 to B1 at .

The energy as a function of the AMF for the different stable states described in Fig. 1. These stable states are associated with the branches B1–B6. The local energy minima correspond to MM configurations which are found by inserting (the same as in Fig. 1) into the energy equation, Eq. (2). The orientations of the MMs at the critical points here are the same as in Fig. 1 and can be seen in Figs. 1(b)–1(g) for a comparison. The CPs all lie on the end of these branches, except for C5, which lies on B1. CPs C1, C2, C3, C4, and C6 are the points where first order phase transitions occur and there is a discontinuity in the first derivative of the energy with respect to the AMF. C5 is where a second order transition occurs between a nonsaturated and saturated state, i.e., near C5 the distinction between phases becomes almost nonexistent. The inset shows the hysteresis path shown in Fig. 1 as it appears through the energy branches.

The energy as a function of the AMF for the different stable states described in Fig. 1. These stable states are associated with the branches B1–B6. The local energy minima correspond to MM configurations which are found by inserting (the same as in Fig. 1) into the energy equation, Eq. (2). The orientations of the MMs at the critical points here are the same as in Fig. 1 and can be seen in Figs. 1(b)–1(g) for a comparison. The CPs all lie on the end of these branches, except for C5, which lies on B1. CPs C1, C2, C3, C4, and C6 are the points where first order phase transitions occur and there is a discontinuity in the first derivative of the energy with respect to the AMF. C5 is where a second order transition occurs between a nonsaturated and saturated state, i.e., near C5 the distinction between phases becomes almost nonexistent. The inset shows the hysteresis path shown in Fig. 1 as it appears through the energy branches.

(a) The CLS (blue lines online) are associated with local energy minima where some minima vanish or bifurcate into other ones. Here the critical lines are illustrated for a system of three interacting magnetic disks with anisotropy constant . For the situation whereby the AMF is applied at 30° to the easy axes of the layers, the critical points are outlined by the circles denoted by C1–C6. The line going through these points at 30° correlates with the illustrative cases in Figs. 1 and 2 that show the response of the system to an AMF at this angle. Here we show the CLS in the plane for to 90°. Here, . The three other quadrants of the diagram are symmetric to this first quadrant depiction. Beyond the line that C6 lies on (darkest area, green online) the magnetization vectors in the disks align in P with each other. The white areas in the phase diagram are when there is the possibility that are all equal or (i.e., configurations giving rise to branches B2 or B5 in Figs. 1 and 2 which have an AP configuration of at ).The lightly shaded, (purple online) areas are where there exist the PNPCs that have magnetization angles that are all different from each other. When the AMF is at an angle larger than , in this first quadrant of the phase diagram, the possibility of the phase with disappears [this occurs at the peak of the topmost cusp of the lightly shaded region at ]. The branches that were discussed in Figs. 1 and 2 have merged together. This can be seen to be occurring in (b), which shows the M-H plot for . The naming convention for the branches of Figs. 1 and 2 are preserved so that it can be seen that B3 and B6 have now become very small (the lightly shaded region). As the magnetic field angle approaches , B1, B3, and B5 begin to merge (as do B2, B4, and B6). Thus, the lightly shaded domain in (a) and (b) vanishes, leaving only the white domain.

(a) The CLS (blue lines online) are associated with local energy minima where some minima vanish or bifurcate into other ones. Here the critical lines are illustrated for a system of three interacting magnetic disks with anisotropy constant . For the situation whereby the AMF is applied at 30° to the easy axes of the layers, the critical points are outlined by the circles denoted by C1–C6. The line going through these points at 30° correlates with the illustrative cases in Figs. 1 and 2 that show the response of the system to an AMF at this angle. Here we show the CLS in the plane for to 90°. Here, . The three other quadrants of the diagram are symmetric to this first quadrant depiction. Beyond the line that C6 lies on (darkest area, green online) the magnetization vectors in the disks align in P with each other. The white areas in the phase diagram are when there is the possibility that are all equal or (i.e., configurations giving rise to branches B2 or B5 in Figs. 1 and 2 which have an AP configuration of at ).The lightly shaded, (purple online) areas are where there exist the PNPCs that have magnetization angles that are all different from each other. When the AMF is at an angle larger than , in this first quadrant of the phase diagram, the possibility of the phase with disappears [this occurs at the peak of the topmost cusp of the lightly shaded region at ]. The branches that were discussed in Figs. 1 and 2 have merged together. This can be seen to be occurring in (b), which shows the M-H plot for . The naming convention for the branches of Figs. 1 and 2 are preserved so that it can be seen that B3 and B6 have now become very small (the lightly shaded region). As the magnetic field angle approaches , B1, B3, and B5 begin to merge (as do B2, B4, and B6). Thus, the lightly shaded domain in (a) and (b) vanishes, leaving only the white domain.

Diagrams representative of the magnetic response of three interacting MLs. (a) The CLS are shown in a generic phase diagram for K as a function of H applied at . The central area of the diagram (lightest shade, pink online), between the two main critical lines, is representative of a SF phase. The phases in the regions prior to and immediately after the SF phase (intermediate hue, green online) are when the MMs in the layers are P. The small humps in the phase diagram either side of (darkest shade, red online) are phases that have an AP origin. (b) The generic phase diagram for [colors are as indicated in the description of (a)]. The CLS themselves are symmetric around . The asymmetry in the shading is because we start from a negative saturating field, P state, and start to increase H to the positive saturating field value (see Ref. 12). To see the diagrams of going from a positive saturation field to the negative one, simply perform a mirror reflection around . In (c)–(e) the phase diagram on the left shows the CLS that separate the P, AP, and SF phases for all applied field angles at distinct values of K. On the right, the top figure is the M vs H evolution for and the bottom figure is for . M is the component of magnetization in the applied field direction. The thick dark line (blue online) shows the magnetization as H is increased from a negative strength that corresponds to a P phase.

Diagrams representative of the magnetic response of three interacting MLs. (a) The CLS are shown in a generic phase diagram for K as a function of H applied at . The central area of the diagram (lightest shade, pink online), between the two main critical lines, is representative of a SF phase. The phases in the regions prior to and immediately after the SF phase (intermediate hue, green online) are when the MMs in the layers are P. The small humps in the phase diagram either side of (darkest shade, red online) are phases that have an AP origin. (b) The generic phase diagram for [colors are as indicated in the description of (a)]. The CLS themselves are symmetric around . The asymmetry in the shading is because we start from a negative saturating field, P state, and start to increase H to the positive saturating field value (see Ref. 12). To see the diagrams of going from a positive saturation field to the negative one, simply perform a mirror reflection around . In (c)–(e) the phase diagram on the left shows the CLS that separate the P, AP, and SF phases for all applied field angles at distinct values of K. On the right, the top figure is the M vs H evolution for and the bottom figure is for . M is the component of magnetization in the applied field direction. The thick dark line (blue online) shows the magnetization as H is increased from a negative strength that corresponds to a P phase.

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