Micromagnetic study of minimum-energy dissipation during Landauer erasure of either isolated or coupled nanomagnetic switches

The question of the minimum energy required to change the information content of physical switches is attracting a growing interest because current computer technology is facing fundamental challenges, such as increased power dissipation, and rapidly approaching the limits of scaling. In this context, bistable nanomagnetic switches can be used to store bits of information, associating each logic state to a different equilibrium orientation of the magnetization. Here we present the results of virtual experiments implemented by micromagnetic simulations on elongated dots of Permalloy, performing a reset to 1 operation, also known as Landauer erasure. We show that for isolated, realistic, elliptical dots with sub-100 nm lateral dimension, the minimum-energy consumption turns out to be consistent with the value expected for an ideal bistable switch, i.e., $E=k_{B}T\times ln\left(2\right)$, known as the Landauer limit. The only condition to achieve the correct limit is that the adopted erasure protocol includes a proper time interval where the probability distribution of the magnetization has the same amplitude in the two possible states. However, it is shown that significant deviations from the theoretical limit can be found either increasing the lateral dot dimension above about 100 nm or considering a rectangular (rather than elliptical) dot shape, where the presence of straight edges promotes a nonuniform behavior of the dot magnetization. Finally, we show that if the magnetic switch is not isolated, as it happens in potential devices where a regular array of dots is present, the dipolar interaction among adjacent switches can lead to either a reduction or an increase of the minimum dissipated energy, depending on the specific geometric arrangement of the magnetic dots.

Figure 1

[top, (a)–(e) Evolution of the energy landscape during the erasure process for a dot of 100 × 60 × 5 ${\mathrm{nm}}^3$. (center) Sequence of the applied external field for the Landauer erasure process. (f) values of the external $x$ and $z$ field components as a function of time. (g) Values of the average $x$ and $z$ magnetization components as a function of time for $T$ = 300 K and dot starting from the 1 state.

Figure 2

Graphical representation of the two integrals of Eq. (1) for the elliptical 100 × 60 × 5 ${\mathrm{nm}}^3$ element at $T$ = 300 K.

Figure 3

(points) Dissipated energy during a Landauer erasure process as a function of the total simulation time for two different values of the damping constant, α = 0.1 (circles) and α = 1.0 (triangles). The data are calculated for a 100 × 60 × 5 ${\mathrm{nm}}^3$ element (full red symbols) and for a 300 × 60 × 5 ${\mathrm{nm}}^3$ (open green symbols) at $T$ = 0 K. For the sake of comparison, theoretical values of the Landauer limit at $T$ = 300 K (black dotted line) and $T$ = 30 K (blue dotted line) are shown.

Figure 4

(Left) (a): (top) Modulus of the average magnetization as a function of time for elliptical (green) and rectangular (red) switches with size 100 × 60 × 5 ${\mathrm{nm}}^3$ at $T$ = 300 K, according to micromagnetic simulations; (bottom) dissipated energy as a function of the temperature for elliptical (green) and rectangular (red) switches from micromagnetic (points) and single-spin (solid lines) simulations, compared to the adjusted values of the Landauer limit (dashed lines): $k_{B}T\times ln\left(2\right)$ + energy dissipated at $T$ = 0 K. (Right) (b): the same as in (a), but for large dots of dimensions 200 × 120 × 10 ${\mathrm{nm}}^3$.

Figure 5

(points) Dissipated energy as a function of the temperature for elliptical switches with the same volume but different lateral dimensions and ellipticity: 100 × 60 × 5 ${\mathrm{nm}}^3,e$ = 1.7 and 175 × 35 × 5 ${\mathrm{nm}}^3,e$ = 5.0; (dashed line) adjusted value of the Landauer limit: $k_{B}T\times ln\left(2\right)$ + energy dissipated at $T$ = 0 K.

Figure 6

Statistical distribution of the dissipated power during the Landauer erasure process as a function of temperature, calculated by micromagnetic simulations (black curves) and by the macrospin approach (red dashed curves). The vertical green solid lines represent the adjusted value of the Landauer limit.

Figure 7

Left: comparison between the calculated (points) values of the dissipated energy and the adjusted Landauer limit (dashed lines). Right: width of the distribution of the values of the dissipated energy, calculated by micromagnetic simulations (points) and single-spin approach (solid curves).

Figure 8

Dissipated energy per switch as a function of the separation distance for the head to tail (blue) and side-by-side (red) configurations calculated with micromagnetic (points) and single-spin (solid lines) simulations. The energy corresponding to the Landauer limit $k_{B}T\times ln\left(2\right)$ and $\left[k_{B}T\times ln\left(2\right)\right]/2$ are plotted as reference (dashed lines).