Spin Logic via Controlled Correlation in Nanomagnet–Dirac-Fermion Heterostructures

A hybrid structure combining the advantages of a topological insulator (TI), dielectric ferromagnet (FM), and graphene is investigated to realize the electrically controlled correlation between electronic and magnetic subsystems for low-power, high-functional applications. Two-dimensional Dirac-fermion states provide an ideal environment to facilitate strong coupling through the surface interactions with proximate materials. The unique properties of FM-TI and FM-graphene interfaces make it possible for active “manipulation” and “propagation,” respectively, of the information state variable based solely on the spin logic platform through electrical gate biases. Our theoretical analysis verifies the feasibility of the concept for logic application with both current-driven and currentless interconnect approaches. The device and circuit characteristics are also examined in realistic conditions, suggesting the desired low-power performance with the estimated energy consumption for copy or not operations as low as the attojoule level.

Figure 1

(a) Schematic illustration of the basic component that consists of a two-layer structure of TI and FM plus the control gates. With the gate bias, an effective out-of-plane anisotropy can be induced in the magnetic layer that rotates the magnetization by 90° from the in-plane orientation [33]. The graphene (Gr) channel interconnects the elemental cells. While not shown explicitly, note that the top and bottom gate electrodes are separated from the active region by a thin dielectric, respectively. (b) In the Bennett clocking scheme, the bias-induced energy minimum constitutes the neutral state, where the system resides at the end of the biasing stage (Null). A signal pulse applied subsequently provides an additional effective magnetic field to tilt the free-energy landscape (Active). The arrows indicate evolution of the magnetization in the ideal conditions. After the relaxation, the magnetization is locked to a stable state along the easy axis. (c) Snapshot of the magnetization evolution for 180° switches with Bennett clocking in the time domain (represented by $m_x$). Bias on the TI gate and signal pulses through the channel are indicated by the dashed lines.

Figure 2

(a) copy/not connection of two unit cells. The surface exchange interaction with the magnet induces a spin-dependent barrier in graphene. By controlling electron transmission through the spin-split bands (via the back-gate bias at ${\mathbf{\text{M}}}_1$), spin polarization of electrons arriving at ${\mathbf{\text{M}}}_2$ can be selected. (b) Calculated conductance polarization in the graphene interconnect as a function of gate voltage. The high-polarization windows are marked by the filled rectangles.

Figure 3

Separate input and output channels are defined for the information flow. The red curved arrow indicates the electron path between two neighboring cells. The fan-out is realized by placing multiple target cells along the same input channel.

Figure 4

(a) copy and (b) not operations in the currentless approach that are achieved via electrostatically controlled coupling between magnets when the downstream cell is in the active state. An energy well can be introduced by simultaneously applying a bias to the graphene back gates. The gap between the gates can be ignored so long as it is smaller than the screening length which is typically several tens of nanometers in graphene. (c) Effective signal field exerted on the downstream cell as a function of chemical potential in the graphene channel. The shaded region indicates the condition for error rate below $10^{-6}$.

Figure 5

(a) Circuit layout of a one-bit full adder. The insert shows the clocks and control signals. The graphene back-gate biases and the channel inputs share the same clock but may differ in values. A doubled strength of the outgoing signal from $C_{\text{out}}$ (specifically, $\overline{c_{\text{out}}}$) can be achieved by adjusting the cell size or the channel driving voltage. (b) Results of ten add operations performed with the input states set dynamically in the simulation. The top panel shows the magnetization of each cell, and the bottom two provide the spin parallel ($\hat{\mathbf{x}}$, black) and spin antiparallel ($-\hat{\mathbf{x}}$, red) currents through the input channels of $C_{\text{out}}$ and $S$, respectively. The magnetization $m_x$ varies between 1 and $-1$ in all five cases (e.g., $m_x=\pm 1$ for logic “1” and “0,” respectively). The heights are adjusted artificially to distinguish the curves from each other for easier viewing (top panel). Similarly, the spin antiparallel current is artificially shifted to the left by 0.5 ns to separate it from the spin parallel current (bottom panels).

Figure 6

Tentative circuit layout for the one-bit adder in the currentless approach. It uses the same control clocks and follows the two-stage operation as in the current-driven design.

Figure 7

Switching time (solid lines) and corresponding power consumption (dashed lines) of the current-driven relaxation process from $m_z=1$ to $m_x=1$ ($z\to x$). The relaxation is marked completed when $|m_x|>0.9$. The power consumption per ohm is calculated as $I^{2}t$, where $I$ is the total current assuming a 60-nm channel width and $t$ is the duration of relaxation.

Figure 8

Switching characteristics related to the operation error rates. (a) Intrinsic energy landscape of the magnet that drives the relaxation after the signal pulse. The green line shows a path along which the magnetization relaxes to the same side of the initial state, while the white one indicates that a low-energy valley can lead the magnetization to the opposite side. (b) Topology of switching success or failure with a signal current pulse of 0.2 ns at $0.2\mu \mathrm{A}/\mathrm{nm}$. The null-state magnetization that relaxes to $m_x=1$ is shown in blue (success), while that leads to $m_x=-1$ is marked in green (failure). The white and purple curves provide two sample paths (with the arrows pointing to the starting locations). The dot in the middle indicates the magnetization distribution at the end of a 0.5-ns biasing stage. The ellipses indicate the lowest-energy contours with an increment of $2k_{B}T$ from the neutral state.

Figure 9

Topology of switching success/failure with a signal current pulse of 0.4 ns in duration at different values of strength $J$ and anisotropy $K_y$. The null-state magnetization that relaxes to $m_x=1$ is shown in blue (success), while that which leads to $m_x=-1$ is marked in green (failure). The magnet has a dimension of $60\times 60\times 2{\mathrm{nm}}^3$. The ellipses indicate the energy contours of 2, 4, and $10k_{B}T$ from the neutral state.

Figure 10

Variation of the switching error rate over magnet size, signal current duration, signal current density, and hard-axis anisotropy. Identical line colors and symbols are used in (a) and (b) to denote different values of $K_y$, whereas (c) and (d) share the same notation on the pulse duration. In (c), the curves for duration over 0.4 ns essentially overlap each other. The orange-colored data points in (a) represent the results from the random field approach with $K_y=60\mathrm{fJ}/\mu {\mathrm{m}}^3$. All other curves are obtained by considering the thermal distribution of the null-state magnetization.