Double-Free-Layer Magnetic Tunnel Junctions for Probabilistic Bits

Naturally random devices that exploit ambient thermal noise have recently attracted attention as hardware primitives for accelerating probabilistic computing applications. One such approach is to use a low barrier nanomagnet as the free layer of a magnetic tunnel junction (MTJ), the magnetic fluctuations of which are converted to resistance fluctuations in the presence of a stable fixed layer. Here, we propose and theoretically analyze a MTJ with no fixed layers but two free layers that are circularly shaped disk magnets. We use an experimentally benchmarked model that accounts for finite-temperature magnetization dynamics, bias-dependent charge, and spin-polarized currents as well as the dipolar coupling between the free layers. We obtain analytical results for statistical averages of fluctuations that are in good agreement with the numerical model. We find that the free layers with low diameters fluctuate to randomize the resistance of the MTJ in an approximately bias-independent manner. We show how such MTJs can be used to build a binary stochastic neuron (or a $p$-bit) in hardware. Unlike earlier stochastic MTJs that need to operate at a specific bias point to produce random fluctuations, the proposed design can be random for a wide range of bias values, independent of spin-transfer-torque pinning. Moreover, in the absence of a carefully optimized stabled fixed layer, the symmetric double-free-layer stack can be manufactured using present-day magnetoresistive random-access memory (MRAM) technology by minimal changes to the fabrication process. Such devices can be used as hardware accelerators in energy-efficient computing schemes that require a large throughput of tunably random bits.

Figure 1

The proposed device. (a) The standard MTJ of the MRAM technology using perpendicular easy-axis MTJs (PMTJs) with fixed and free layers. (b) We consider an MTJ with two free layers and no fixed layer. The magnetization of the free layers fluctuates in the plane in the presence of thermal noise that is turned into resistance fluctuations through tunneling magnetoresistance (TMR). One way to build the proposed device is to start from the standard PMA-MTJs and make both the free and the fixed layers thicker so that their magnetizations fall into the plane. We note that the $\mathrm{P}$MTJ stack sketch shown here is for illustrative purposes and that industrial $\mathrm{P}$MTJ stacks have many more additional layers that account for canceling dipolar fields, ensuring fixed-layer stability and other effects.

Figure 2

The dipolar model. (a) The geometry and parameters in the calculation of dipolar fields between the two free layers. (b) An illustrative vector plot of the dipolar field due to the bottom layer (dashed box) at the $y=0$ plane. (c) The dipolar field $H_X$ due to a source magnet (bottom, $+x$ polarized) with varying diameters ($2R=10,20,60,100\mathrm{nm}$) with a film thickness of $t=1$ nm at a distance $d=1$ nm (typical $\mathrm{Mg}\mathrm{O}$ thickness in MTJs) measured from the top of the source magnet with $M_s=800$ emu/cc $\approx 1$ T. At zero offset (circled points), the fields can be analytically calculated from Eq. (5). (d) The $z$-directed fields along the offset direction. In our model, these fields average out to zero when summed over the target magnetic volume.

Figure 3

The self-consistent model for the magnetization dynamics and transport. The coupled LLG equations provide instantaneous magnetizations to the MTJ model, which in turn produces a bias-dependent spin-polarized current, ${\overrightarrow{I}}_S(V)$ in the channel. $+{\overrightarrow{I}}_S(V)$ is incident to one ferromagnetic interface and $-{\overrightarrow{I}}_S(V)$ is incident to the other ferromagnetic interface. The dipolar model couples the two LLG solvers through fields that depend on instantaneous ${\hat{m}}_i$.

Figure 4

The zero-bias behavior. (a) The average $\cos {\theta }_{1,2}$ (${\theta }_{1,2}$ is the angle between magnetizations vectors) at zero bias. Equation (9) is compared with a finite-temperature LLG simulation at different diameters (measured in nanometers). The average is taken over $5\mu \mathrm{s}$ for all examples with a time step of $\mathrm{\Delta }t=1$ ps. The inset shows the average dipolar tensor component $D_{xx}=D_{yy}=D_0$ as a function of the diameter, calculated from Eq. (4) assuming the same geometric and magnetic parameters that are used in Fig. 2 and damping coefficient $\alpha =0.01$. (b) The normalized autocorrelation of $m^x$ for both layers and $\cos ({\theta }_{1,2})$ are shown for $2R=10$ nm as an example, where the total simulation period is $5\mu \mathrm{s}$. The inset shows the time dependence of the $m^x$ components for a short period.

Figure 5

The $R$ versus $V$ characteristics of a double-free-layer MTJ. (a) Representative fluctuations at different time instances for $2R=20$ nm free layers at $t=0$ ns, $t=0.5$ ns, and $t=20$ ns. (b) The $R$ versus $V$ characteristics for a $(2R)=10$ nm MTJ, where we assume a low-bias TMR of $115\mathrm{\%}$ [$P_0=0.65$ and $V=0$ in Eq. (12)] and a roll-off constant $V_0=50$ V, ensuring a symmetric bias dependence for the spin-polarized currents. We assume an $\mathrm{RA}$ product of $9\mathrm{\Omega }$-$\mu {\mathrm{m}}^2$ [56] to obtain $G_0$ for all MTJs in this paper. The plot is obtained by sweeping the voltage from ($-2\mathrm{V}$, 2 V) in 2 $\mu \mathrm{s}$ with 1-ps time steps; the dots correspond to 1-$\mu \mathrm{s}$ averages taken at that bias. (c) The average $\cos ({\theta }_{1,2})$ taken over 1 $\mu \mathrm{s}$ at different bias points for different diameters. All diameters roughly show bias-independent average angles or resistances.

Figure 6

A BSN with a double-free-layer MTJ. (a) The double-free-layer MTJ ($2R=20$ nm with all the same parameters that are used in previous figures) in a 1T-1MTJ circuit, where a $R_S=5k\mathrm{\Omega }$ resistance is used to shift the overall characteristics. (b) The drain voltage is measured while the input is swept from 0 V to 0.8 V over 0.5 $\mu \mathrm{s}$. The circled points are averages over 250 ns at each bias point. (c) The output of the inverter, which shows the BSN characteristics with the same measurement times as reported in (b).