Numerical study of geometrical frustration in a planar soft-hard magnetic system

We computationally study the frustrated macroscopic magnetic configurations of a thin soft-magnetic layer on top of a hard magnet geometrically patterned as three equidistant disks. The interplay between competing exchange, magnetostatic, and Zeeman interactions, enhanced by the geometric constraints, results in rich magnetization configurations when an external magnetic field is rotated around the sample. We identify four successive ranges of the external field magnitude in each of which the system exhibits unique energetics and chooses different sets of energy minima during the rotation of the external magnetic field. The system can be used as a ternary logic storage device, with read and write operations achieved by utilizing the unique behavior of the system in each of the four regimes.

Figure 1

(Color online) (a) Two-dimensional soft-hard magnetic composite system. The magnetic structure in the soft-magnetic region is determined by the boundary condition fixed by the hard magnetic disks and external field $\mathbf{H}$. One of the many possible energy minima, with two magnetic influx and one outflux between the disks, is shown. The length scale of the system, $L$, defined as the distance between the two hard magnetic disks, is shown. (b) Side view of the device.

Figure 2

Variation in energy of the system at different regimes with respect to the angle of externally applied field, starting from the initial configuration shown in Fig. 1. Plot (a) shows the variation in energy in the first regime. No transitions are seen. The external field is not strong enough to dislodge the system from the energy minimum that it stays in. (b) shows the behavior of the system in the second regime. One transition occurs in an interval of $2\pi$ radian rotation of external field. A static domain-wall shield forms and vanishes during transition. (c) shows the energy variation in third regime. The energy never drops to the initial low-energy state the system started in. The system stays in metastable states and all the transitions are between three equivalent metastable states characterized by switching of the domain-wall shield from one pair of hard disks to another. (d) shows the variation in energy in the last regime. High external field aids the system to escape from one minimum to another easily. Three transitions happen in a $2\pi$ radian rotation of external field.

Figure 3

Magnetization configurations of the system at different regimes. (a) shows the domain-wall formation in the second regime at an external field of 500 Oe. This domain wall does not switch to another pair of hard magnetic disks as the external field is rotated. (b) and (c) show domain-wall shields in the third regime at an external field of 1000 Oe. In this regime, the domain-wall shield switches to the next pair of hard magnetic disks as the external field is rotated. The system stays in these metastable states and never drops down to the ground state. (d) shows the system in the fourth regime at an external field of 2500 Oe. The magnetization follows the external field faithfully. The arrows to the top left of the figures show the direction of the external field. The appearance of domain-wall shield is enhanced by a dotted line to aid visualization.

Figure 4

Domain formation in the three-disk system. The figure shows the three-disk system with its length scale double that of previous figures, with radius of hard magnets, distance between hard magnetic centers, radius of the soft-magnetic circle, and thickness of the sample taking the values 40, 150, 300, and 20 nm, respectively. The length scale $L$ has to be compared with that of Fig. 1. Domain-wall formation becomes energetically favorable and frustration in the system is neutralized by domains which shield the influence of one hard disk on soft magnet from that of another.

Figure 5

An illustration of the three successive operations needed to change the logic state of the system. In the first phase, denoted by section AB of the figure, the system is taken from first to third regime by fixing the field direction at $\frac{\pi }{3}$ radians and increasing the magnitude from 0 to 1000 Oe. The average magnetization angle in the first phase is denoted as ${\theta }_0$. The initial state of the system is shown as an inset in the leftmost part of the figure. In the second phase, denoted by section BD, the system is in the third regime, and the magnetic field is rotated through $2\pi +\frac{2\pi }{3}$. The logic state of the system changes as shown in the inset, with hardly noticeable change in the average magnetization near point C. The rightmost part shows the third phase, denoted by section DG, in which the magnetic-field magnitude is reduced from 1000 to 0 Oe, with its direction fixed. The abrupt transition from the third to first regime can be seen in the average magnetization plot, denoted by section EF. The average magnetization angle in the third phase is ${\theta }_0+2\pi +\frac{2\pi }{3}$. The system’s changed logic state is shown as an inset of the rightmost part.

Figure 6

(Color online) Schematic of a magnetic random access memory using the three-disk system. The dots are the magnetic tunnel junctions with the top soft-magnetic layer replaced by the three-disk system. Appropriately designed current pulses sent through horizontal and vertical wires create a rotating magnetic field at the position of dot A, changing its logic state. Judicious choice of geometrical parameters of the memory device and the absence of $x$, $y$, or both components of the write field at neighboring dots B, C, and D would lead to a field strength below the transition field to the third regime at these dots, thereby leaving their logic states unchanged. The side view shows the modified magnetic tunnel junction. The middle and the lower sections are the tunnel barrier and the fixed ferromagnetic layer, respectively. The magnetization direction of the fixed ferromagnetic layer is chosen such that the resistance of the tunnel junction is different for the three different logic states of the system. Leads for reading logic states are not shown.