Exploring the Thermodynamic Limits of Computation in Integrated Systems: Magnetic Memory, Nanomagnetic Logic, and the Landauer Limit

Nanomagnetic memory and logic circuits are attractive integrated platforms for studying the fundamental thermodynamic limits of computation. Using the stochastic Landau-Lifshitz-Gilbert equation, we show by direct calculation that the amount of energy dissipated during nanomagnet erasure approaches Landauer’s thermodynamic limit of $kT\ln (2)$ with high precision when the external magnetic fields are applied slowly. In addition, we find that nanomagnet systems behave according to generalized formulations of Landauer’s principle that hold for small systems and generic logic operations. In all cases, the results are independent of the anisotropy energy of the nanomagnet. Lastly, we apply our computational approach to a nanomagnet majority logic gate, where we find that dissipationless, reversible computation can be achieved when the magnetic fields are applied in the appropriate order.

Figure 1

Top: timing diagram for external magnetic fields applied during “restore to one.” $H_x$ is applied along the magnetic hard axis to remove the uniaxial anisotropy barrier, while $H_y$ is applied along the easy axis to force the magnetization to one. Bottom: illustration of the magnetization of the nanomagnet at the beginning and end of each stage, and the direction of the applied field in the $x\mathrm{\text{−}}y$ plane.

Figure 2

(a) X-axis and (b) y-axis hysteresis loops of a nanomagnet during the “restore to one” operation at 300 K, obtained by solving the stochastic LLG equation. Each point on the curve is the average of 2000 simulations. The total area of the loops is very close to $kT\ln (2)$. Stage numbers correspond to the raising and lowering of external applied fields as described in Fig. 1. (c) Histogram of the energy dissipated at 300 K. The envelope curve was obtained using the Boltzmann thermal distribution calculation described in the supplementary material [17]. (d) Energy dissipation versus temperature. The solid line is the Landauer limit, $kT\ln (2)$, and the discrete points are the average dissipation over 2000 simulations carried out at each temperature. The simulations deviate from theory at low temperatures because the nanomagnets take longer to reach thermal equilibrium, weakening the quasistatic approximation.

Figure 3

(a) Two possible computation cycles for a nanomagnetic logic circuit containing a single majority logic gate (MLG) and (b),(c) their corresponding hysteresis loops and energy dissipation. In (a), the circuit computes the majority vote of three inputs (leftmost nanomagnets) and passes the result to an output (rightmost nanomagnet). After logic execution, the circuit is reset to its initial state irreversibly (top branch) or reversibly (lower branch). In (b), the hysteresis loops for the magnetic field applied to the nanomagnets to the right of the inputs is plotted. The nearest neighbor coupling energy was set to $50kT$ in this simulation. In (c), the energy dissipation of both computation cycles is plotted as a function of the nearest neighbor coupling energy between nanomagnets.