Engineering Relaxation Pathways in Building Blocks of Artificial Spin Ice for Computation

Nanomagnetic logic, which makes use of arrays of dipolar-coupled single-domain nanomagnets for computation, holds promise as a low-power alternative to traditional computation with CMOS. Beyond the use of nanomagnets for Boolean logic, nanomagnets can also be exploited for nondeterministic computational schemes such as edge detection in images and for solving the traveling salesman problem. Here, we demonstrate the potential of arrangements of thermally active nanomagnets based on artificial spin ice for both deterministic and probabilistic computation. This is achieved by engineering structures that follow particular thermal relaxation pathways consisting of a sequence of reorientations of magnet moments from an initial field-set state to a final low-energy output state. Additionally, we demonstrate that it is possible to tune the probability of attaining a particular final low-energy state, and therefore the likelihood of a given output, by modifying the intermagnet distance. Finally, we experimentally demonstrate a scheme to connect several computational building blocks for complex computation.

Figure 1

Thermal relaxation and state network diagrams for four-loop and four-vertex structures. (a)–(c) Thermal relaxation in a four-loop structure from a field-set state to one of the two degenerate flux-closure ground states. Switched magnets are highlighted in gray, head-to-tail arrangements are indicated by green arrows, and the net magnetization is indicated by black arrows. (d) State network diagram for the four-loop structure. The states are indicated with black points that are located according to their energies along the vertical axis and the net magnetization of the state along the horizontal axis. Transitions between states m and l via single-nanomagnet reorientations are indicated by tapered lines. The midpoint of the lines corresponds to the energy $E_m+E_T$, where the barrier energy $E_T=E_b+(1/2)(E_l-E_m)$. The grayscale of the lines corresponds to the transition rate at T = 400 K according to Eq. (2) and the grayscale bar. The blue line represents the transitions between the states shown in (a–c), resulting in a monotonic relaxation path with successive lowering of dipolar energy. (e)–(g) In a four-vertex structure, thermal relaxation from a field-set state to the ground state requires a transition to a state of higher dipolar energy, which is associated with the creation of a magnetic charge at the vertex highlighted in red in panel (f). (h) State network diagram for the four-vertex structure. The red intermittent path is for the state evolution shown in (e)–(g), where an intermediate state with higher dipolar energy is accessed. The metastable states are indicated with triangles, while the ground states are indicated with larger black points.

Figure 2

Thermal relaxation and state network diagram of the $P$ gate. (a) Scanning-electron microscopy (SEM) image of the $P$ gate. The scale bar measures 250 nm. (b) Field-set initial state S₀, degenerate ground states (c) G1 and (d) G2, and metastable state (e) M, imaged by XPEEM. For 390 gates, G1 is reached in 24.6% of the gates, whereas G2 is reached in 48.5% of the gates. In addition, 16.0% of the gates get stuck in the metastable state M. The relaxation path taken is determined by the height of the successive activation barriers that need to be overcome during relaxation. (f) State network diagram of the $P$ gate at T = 295 K. The fastest relaxation path from the field-set state S₀ to the ground states G1 and G2, represented by the larger black points, are intermittent paths highlighted in blue and red. (g) Many different relaxation paths can be explored and there can even be a reversal of direction with the system traveling back along a given path. This reversal is illustrated with successive XPEEM imaging of the relaxation from an initial field-set state (1) to a final low-energy state (6). Here, a change in direction of relaxation is observed when transitioning from (1) to (2) to (3) and, subsequently, the structure traverses the path to G2. The corresponding path in (f) is highlighted in green. In the XPEEM images, black and white arrows indicate the magnetic moment orientations.

Figure 3

Tuning of ground-state occupation with modification of the nanomagnetic design. XPEEM images of (a) the field-set initial state S₀ of the original D-gate design and the ground states (b) G1 and (c) G2. The scale bar in (a) measures 500 nm and moment orientations are indicated with black and white arrows. (d) Schematic drawing of the D gate. In order to tune the probability of structures reaching the ground state G1, the lower right four-loop structure is moved by the distance d. (e) The percentage of gates ending up in the two ground states G1 and G2 for a four-loop structure offset by a distance d. The final percentages of low-energy states in 51 gates for different distances (points) are compared with the kinetic Monte Carlo simulations (solid lines).

Figure 4

Connecting several D gates. (a) Three equivalent D gates (highlighted in light gray) are coupled by strongly interacting parallel nanomagnets (in dark gray). These nanomagnets link the output magnets (O) of the outer two gates 1 and 2 to the input magnets (I) of the middle gate 3. (b) Initial field-set state and (c) final state imaged with XPEEM. After thermal relaxation, the outer gates 1 and 2 relax to the ground state G1, and the central gate 3 relaxes to G2. The scale bar in (b) measures 1 µm. Moment orientations are indicated with black and white arrows.