Spin-Hall-Assisted Magnetic Random Access Memory

Spin-Hall-Assisted Magnetic Random Access Memory

Supplementary Information

A. van den Brink, S. Cosemans, S. Cornelissen, M. Manfrini, A. Vaysset, W. Van Roy, T. Min, H.J.M. Swagten, and B. Koopmans

1  Simulation details

Magnetization dynamics are simulated by solving the Landau–Lifshitz–Gilbert (LLG) equation[1]:

∂M∂t=-γµ0M×Heff+αMsM× ∂M∂t+cSHEMs2M×σSHE×M+cMTJMs2M×mref×M+βMTJMsM×mref, / (S1)

with M the free layer magnetization, γ the electron gyromagnetic ratio, m0 the vacuum permeability, Heff the effective magnetic field, α the Gilbert damping coefficient, and Ms≡M the saturation magnetization. The spin-Hall torque coefficient is given by cSHE=JSHEθSHEħγ/(2ed), JSHE the spin-Hall effect current density running underneath the free layer, θSHE the spin-Hall angle, ħ the reduced Planck constant, e the elementary charge, and d the free magnetic layer thickness. The spin-transfer torque coefficient is given by cSTT=JSTTPħγ/(2ed) with JSTT the spin-transfer torque current density running through the tunnel junction, and P the spin polarization which is assumed constant for simplicity. A field-like torque term is included with βMTJ=0.25 cMTJ as observed experimentally in MTJs[2].

The effective field Heff comprises four contributions: the applied magnetic field Happl, the effective anisotropy field Hani=2KU/(μ0Ms)z, with KU the uniaxial anisotropy energy density, the demagnetizing field HD which is approximated for a rectangular prism[3], and a Langevin thermal field HT. This thermal field is an isotropic Gaussian white-noise vector with variance σ2=2αkBT(μ0MsVτ) with kB the Boltzmann constant, T the absolute temperature, V the free layer volume, and τ the simulation time step. This particular stochastic contribution can be shown to yield appropriate thermal fluctuations[4]. Equation (S1) is solved numerically using an implicit midpoint rule scheme[5].

A thermal stability of Δ≡KeffV/(kBT)=40 at room temperature is imposed by setting KU=3.151×105 Jm-3, with Keff the effective anisotropy after correcting for the demagnetization field. Further notable parameters include α=0.1, as typical for PMA materials[6],[7], Ms=1.0×106 Am-1 for Co[8], θSHE=0.15 for Ta[9], and P = 0.5. All simulations are carried out at T = 300 K, with the initial magnetization drawn from an appropriate Maxwell-Boltzmann distribution around M=Msz. The Oersted field generated by JSHE is approximated by that of an infinite surface current, whereas Joule heating and current shunting effects are neglected.

2  Spin-Hall effect current pulse power consumption estimate

In the main text, the power consumed by a 0.5 ns pulse of JSHE = 28 MA/cm2 is mentioned to be very small, at 9 fJ. This is based on a calculation in a simplified system: JSHE runs through a 4 nm thick
β-Ta layer underneath the MTJ, which is 100 nm wide and 200 nm long, so that ISHE 0.112 mA. Given the typical resistivity value[10] of 200 µΩ-cm for sputtered thin films of β-Ta, the resistance of the thin wire segment is 1 kΩ. Assuming the connecting wires and transistor contribute another 0.5 kΩ of resistance to the current path, the total resistance is estimated at 1.5 kΩ. The required driving voltage for the SHE pulse is thus 0.168 V, yielding an energy consumption of 9.4 fJ for a pulse of 0.5 ns.

One could argue that additional capacitive losses in charging a separate line might affect the power consumption. This is difficult to quantify without an in-depth discussion of device integration, which is beyond the scope of this paper. However, a quick estimate shows that these losses are negligible compared to the static power consumption. Assuming each cell acts as a 1.8 fF capacitance (based on a 200 x 100 nm parallel plate capacitor with a dielectric constant of 10 and a separation of 1 nm) and 128 cells are addressed per line, the energy associated with charging a line (E=0.5CV2) is of the order of 10 fJ (STT pulse, unassisted), 1 fJ (STT pulse, assisted), or 0.1 fJ (SHE pulse), which in each case is negligible compared to the static write energy.

Finally, it should be noted that there is a practical lower limit to the driving voltage, imposed by transistor operation requirements. Furthermore, in a practical implementation of SHE-accelerated STT-MRAM, the device would likely be operated using a single voltage supply for both current pulses. Assuming a driving voltage of 0.9 V for both pulses increases the total power consumption to 0.32 pJ, which is still a factor 17 lower than the unassisted power consumption.

3  Parameter space exploration results

As mentioned in the main text, an extensive study was performed regarding the general validity of the obtained results by systematically varying all relevant system parameters. These include ambient conditions (temperature, applied magnetic field), system properties (dimensions, magnetic anisotropy, saturation magnetization, damping, spin-Hall angle), and current pulse properties (current densities, pulse durations, and delay time). While varying each parameter, all other parameters are set to their default value as listed above and in the main text, unless stated otherwise.

For each value of the parameter of interest, we generate a phase diagram of the switching probability Pswitch as a function of JSHE and JSTT. The number of averages per point is reduced to 64 for practical purposes. Typical examples of these phase diagrams are shown below where relevant. We extract two characteristic current densities from each phase diagram: the value of JSTT required to achieve Pswitch = 0.99 without SHE assistance, referred to as J0,STT, and the value of JSHE required to reduce JSTT to J0,STT/2 while maintaining Pswitch = 0.99, referred to as J½,SHE. This current density J½,SHE serves as a figure of merit describing the viability of the SHE-assisted write scheme for the given set of parameters. A lower value of J½,SHE indicates a more significant reduction in tunnel current density (and power consumption) using the SHE-assisted write scheme.

3.1  Temperature

The effect of system temperature is illustrated by means of two phase diagrams, at T = 100 K (Figure 1a) and T = 1000 K (Figure 1b). Increasing system temperature is seen to reduce the value of J0,STT (see also Figure 2), which is explained by an increase in thermal fluctuations which reduces incubation delay. At higher temperatures, a higher value of JSHE is therefore required for the SHE pulse to offer a benefit over thermal fluctuations, as mentioned in the main text and observed in Figure 1b. The reduction in J0,STT also results in an increase of J½,SHE with temperature, as seen in Figure 2, indicating that the SHE-assisted scheme becomes less effective with increasing temperature, as expected.

Figure 1: Switching probability Pswitch out of 64 attempts as a function of the pulse current densities JSTT and JSHE, respectively, for a system temperature of (a) 100 K and (b) 1000 K.

Figure 2: Values of J0,STT (purple squares) and J½,SHE (orange circles) as a function of system temperature. Lines are a guide to the eye.

3.2  In-plane magnetic field

As mentioned in the main text, application of a small magnetic field Bx (along the flow direction of JSHE) has a dramatic effect on the magnetization dynamics. Phase diagrams were created for a field range of 0 to 18 mT. The effective anisotropy field of the system is 28 mT; in-plane fields approaching this magnitude pull the magnetization significantly in-plane and cause precessions during the switching process. More importantly, for small values of Bx, the symmetry of the system is broken sufficiently to allow for directional switching without any STT current. This is clearly visible in the typical phase diagrams shown in Figure 3. The in-plane field also reduces the effective thermal stability of the system, however, reflected in an increase of J0,STT and a decrease of J½,SHE, shown in Figure 4.

Figure 3: Switching probability Pswitch out of 64 attempts as a function of the pulse current densities JSTT and JSHE, for an in-plane magnetic field Bx of (a) 6 mT and (b) 12 mT.

Figure 4: Values of J0,STT (purple squares) and J½,SHE (orange circles) as a function of applied in-plane magnetic field. Lines are a guide to the eye.

3.3  Lateral dimensions

To investigate the viability of the SHE-assisted scheme for different lateral dimensions, we simultaneously increase the bit length l and width w to maintain the same aspect ratio. This corresponds to a quadratic increase in the free layer volume V = d w l. The unassisted STT switching current density J0,STT is found to be constant under this variation (Figure 5a), corresponding to a quadratic increase in the critical current I0,STT = w l J0,STT (Figure 5b). This is in agreement with symmetry considerations. The spin-Hall current required to halve the required STT current is found to follow a quite different scaling behavior: it is independent of the lateral dimensions (Figure 5b). As I½,SHE = w de J½,SHE, with de the electrode thickness, this implies a 1/w dependence for J½,SHE, which is indeed observed in Figure 5a. This observed scaling behavior suggests that downscaling of SHE-assisted MRAM will pose a challenge.

Figure 5: Values of (a) J0,STT (purple squares) and J½,SHE (orange circles) as a function of bit length, while maintaining a constant aspect ratio by proportionally scaling the bit width, and (b) corresponding currents. The spin-Hall current is scaled by a factor 10 for clarity. Lines are a guide to the eye.

3.4  Lateral dimensions with thermal stability constraint

We explicitly study the lateral scaling behavior under constant thermal stability Δ=Keff VkBT=40, as such stable bits are interesting for memory applications. Reducing the free layer volume in this case requires an equivalent increase in the effective magnetic anisotropy Keff. Compared to the unconstrained scaling case discussed in the previous section, the STT switching current density is therefore expected to display an additional 1/(l x w) dependence, which is indeed observed (Figure 6). The spin-Hall current density is similarly affected, with J½,SHE approaching 100 MA/cm2 for a bit size of 80 x 40 nm, again demonstrating the challenge in downscaling SHE-based devices.

Figure 6: Values of (a) J0,STT (purple squares) and J½,SHE (orange circles) as a function of bit length, while maintaining a constant aspect ratio by proportionally scaling the bit width and a constant thermal stability by proportionally scaling the magnetic anisotropy constant, and (b) corresponding currents. Lines are a guide to the eye.

3.5  Aspect Ratio

As mentioned in the main text, a ‘tail’ is observed in the phase diagram at high JSHE when using the default system parameters. This is mentioned to result from precessional motion around the in-plane demagnetization field during the SHE pulse, implying that it should not occur in structures with an aspect ratio of 1. The phase diagrams shown in Figure 7 confirm these statements, showing no tail for an aspect ratio of 1 and an enhanced one for an aspect ratio of 50. The lowest value of J½,SHE is observed for an aspect ratio of 1 (Figure 8a), but a higher aspect ratio can be beneficial to reduce the SHE current (Figure 8b), and thus the Joule heating and power consumption, while maintaining thermal stability.

Figure 7: Switching probability Pswitch out of 64 attempts as a function of the pulse current densities JSTT and JSHE, for an aspect ratio (l/w) of (a) 1 and (b) 50. The dimensions are chosen such that in each case the junction area is identical to that of the 200 x 100 nm junction, i.e. 141 x 141 nm and 1000 x 20 nm, respectively.

Figure 8: Values of (a) J0,STT (purple squares) and J½,SHE (orange circles) as a function of bit aspect ratio (w/l), and (b) corresponding currents. The area is constrained to 0.02 µm2 for each aspect ratio. Lines are a guide to the eye.

3.6  Free layer thickness

Altering the magnetic free layer thickness, while constraining the thermal stability to Δ=Keff VkBT=40, is found linearly affect both J0,STT and J½,SHE (Figure 9).

Figure 9: Values of J0,STT (purple squares) and J½,SHE (orange circles) as a function of free layer thickness. The thermal stability is constrained to 40 for each thickness by adjusting the magnetic anisotropy. Lines are a guide to the eye.

3.7  Spin-Hall angle

The spin-Hall angle θSH is defined as the ratio between the spin current Is and the electric current Ie in a material: θSH≡Is/Ie. It is therefore expected that J½,SHE is inversely proportional to θSH, which is exactly what is observed (Figure 10). Trivially, the value of J0,STT is not affected by θSH.

Figure 10: Value of J½,SHE as a function of the bottom electrode spin-Hall angle. The line is a guide to the eye.

3.8  Thermal Stability

Increasing the thermal stability Δ=Keff VkBT by increasing the effective magnetic anisotropy Keff is found to have a linear effect on J0,STT, as expected. The value of J½,SHE is similarly affected, but for low thermal stability it saturates due to the diminishing role of JSHE compared to thermal fluctuations.

Figure 11: Values of J0,STT (purple squares) and J½,SHE (orange circles) as a function of thermal stability. Lines are a guide to the eye.

3.9  Damping

Changing the Gilbert damping parameter has a weak, linear effect on both J0,STT and J½,SHE (Figure 12). Both slightly increase with increasing damping.

Figure 12: Values of J0,STT (purple squares) and J½,SHE (orange circles) as a function of Gilbert damping parameter. Lines are a guide to the eye.

3.10  Saturation Magnetization

Changing the saturation magnetization of the free layer has proportional effect on J0,STT and J½,SHE (Figure 13). For very low values of the saturation magnetization, the value of J½,SHE is seen to saturate, which is analogous to the effect of increasing system temperature to very high values.

Figure 13: Values of J0,STT (purple squares) and J½,SHE (orange circles) as a function of free layer saturation magnetization. Lines are a guide to the eye.

3.11  Spin-Transfer Torque pulse duration

The value of J0,STT decays rapidly as a function of the STT pulse length tSTT, as expected (Figure 14). It is interesting to observe, however, that J½,SHE depends only weakly on tSTT, indicating that the SHE-assisted scheme is still a viable alternative to unassisted switching in applications where longer switching times are allowed. Note the upturn in J½,SHE for small tSTT, which is due to tSTT ≤ tSHE.