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) equation1:
𝜕𝐌
𝛼
𝜕𝐌
𝑐SHE
= −𝛾µ0 (𝐌 × 𝐇eff ) +
(𝐌 ×
) + 2 (𝐌 × 𝛔
̂SHE × 𝐌)
𝜕𝑡
𝑀𝑠
𝜕𝑡
𝑀𝑠
𝑐MTJ
𝛽MTJ
(𝐌 × 𝐦
̂ ref × 𝐌) +
̂ ref ),
+ 2 (𝐌 × 𝐦
𝑀𝑠
𝑀s
(S1)
with M the free layer magnetization, γ the electron gyromagnetic ratio, 0 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 = 𝐽SHE 𝜃SHE ħ𝛾/(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 = 𝐽STT 𝑃ħ𝛾/(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 MTJs2.
The effective field Heff comprises four contributions: the applied magnetic field Happl, the
effective anisotropy field 𝐇ani = 2𝐾U /(𝜇0 𝑀s )𝒛̂, with 𝐾U the uniaxial anisotropy energy density, the
demagnetizing field HD which is approximated for a rectangular prism3, and a Langevin thermal field
HT. This thermal field is an isotropic Gaussian white-noise vector with variance 𝜎 2 = 2𝛼𝑘B 𝑇⁄(𝜇0 𝑀s 𝑉𝜏)
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
fluctuations4. Equation (S1) is solved numerically using an implicit midpoint rule scheme 5.
A thermal stability of Δ ≡ 𝐾eff 𝑉/(𝑘B 𝑇) = 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 materials6,7, 𝑀s = 1.0 × 106 Am−1 for
1
Co8, 𝜃SHE = 0.15 for Ta9, 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 = 𝑀s 𝒛̂. 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 value10 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 = 0.5𝐶𝑉 2 ) 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 STTMRAM, 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.
2
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
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
20
0.8
0.8
15
0.6
0.6
10
0.4
5
0.2
sw itch
1
P
STT
10
0.4
J
J
STT
15
1
P
20
(MA/cm2)
(b) 25
(MA/cm2)
(a) 25
sw itch
that the SHE-assisted scheme becomes less effective with increasing temperature, as expected.
5
0
0
0
10
20
30
40
50
0.2
0
0
0
10
20
JSHE (MA/cm2)
30
40
50
JSHE (MA/cm2)
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.
Current density (MA/cm2)
25
20
J0,STT
J1/2,SHE
15
10
0
200
400
600
800
1000
Temperature
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.
4
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.
(b) 15
5
1
1
0.8
0.8
0.6
0.6
0.4
0
0
10
20
30
sw itch
5
0.2
0.2
0
P
0.4
sw itch
10
P
STT
(MA/cm2)
10
J
J
STT
(MA/cm2)
(a) 15
0
0
0
10
20
30
JSHE (MA/cm2)
JSHE (MA/cm2)
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.
Current density (MA/cm2)
22
J0,STT
20
J1/2,SHE
18
16
14
12
10
8
-2 0 2 4 6 8 10 12 14 16 18 20
Bx
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.
5
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.
(b) 6
80
J0,STT
70
J1/2,SHE
60
50
40
30
I0,STT
5
Current (mA)
Current density (MA/cm2)
(a) 90
10*I1/2,SHE
4
3
2
20
1
10
0
0
100
200
300
Bit length, w = 0.5*l (nm)
0
100
200
300
Bit length, w = 0.5*l (nm)
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.
6
3.4 Lateral dimensions with thermal stability constraint
We explicitly study the lateral scaling behavior under constant thermal stability Δ =
Keff 𝑉
𝑘B 𝑇
= 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.
(b) 10
J0,STT
I0,STT
J1/2,SHE
I1/2,SHE
Current (mA)
Current density (MA/cm2)
(a) 1000
100
10
1
0.1
0.01
0
100
200
300
Bit length, w = 0.5*l (nm)
0
100
200
300
Bit length, w = 0.5*l (nm)
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.
7
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 inplane 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.
0.4
5
0.2
0.4
sw itch
J
10
0.6
P
0.6
15
STT
10
0.8
sw itch
(MA/cm2)
15
1
0.8
20
J
STT
(MA/cm2)
20
1
25
P
(b)
(a) 25
5
0
0
0
0
10
20
30
40
50
0.2
0
0
10
20
JSHE (MA/cm2)
30
40
50
JSHE (MA/cm2)
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.
(b) 10
20
Current (mA)
Current density (MA/cm2)
(a) 22
18
16
J0,STT
14
J1/2,SHE
1
I0,STT
I1/2,SHE
0.1
12
10
0.1
1
10
Aspect Ratio (l/w)
100
0.01
0.1
1
10
100
Aspect Ratio (l/w)
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.
8
3.6 Free layer thickness
Altering the magnetic free layer thickness, while constraining the thermal stability to Δ =
Keff 𝑉
= 40, is found linearly affect both J0,STT and J½,SHE (Figure 9).
Current density (MA/cm2)
𝑘B 𝑇
40
J0,STT
J1/2,SHE
30
20
10
0.5
1.0
1.5
Free layer thickness (nm)
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.
Current density (MA/cm2)
60
J1/2,SHE
50
40
30
20
10
0
0.0
0.2
0.4
0.6
Spin-Hall Angle
Figure 10: Value of J½,SHE as a function of the bottom electrode spin-Hall angle. The line is a guide to the eye.
9
3.8 Thermal Stability
Increasing the thermal stability Δ =
Keff 𝑉
𝑘B 𝑇
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
Current density (MA/cm2)
thermal stability it saturates due to the diminishing role of JSHE compared to thermal fluctuations.
35
J0,STT
J1/2,SHE
30
25
20
15
10
0
20
40
60
80
100
Thermal Stability
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.
Current density (MA/cm2)
25
20
15
10
J0,STT
J1/2,SHE
5
0.00
0.05
0.10
0.15
0.20
Gilbert damping constant
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.
10
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.
Current density (MA/cm2)
30
J0,STT
J1/2,SHE
20
10
0
0.0
0.5
1.0
1.5
2.0
Saturation Magnetization (T)
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.
Current density (MA/cm2)
25
20
15
J0,STT
J1/2,SHE
10
5
0
0
2
4
6
8
10
STT pulse duration (ns)
Figure 14: Values of J0,STT (purple squares) and J½,SHE (orange circles) as a function of STT pulse duration. Lines
are a guide to the eye.
11
3.12 Spin-Hall effect pulse duration
Increasing the spin-Hall effect current pulse duration tSHE dramatically reduces the value of
J½,SHE for tSHE < 0.3 ns (Figure 15). Further increases in tSHE have little effect. Herein, the spin-Hall effect
pulse differs significantly from the STT pulse, due to the difference in spin orientation compared to the
initial magnetization. The observed scaling demonstrates the usefulness of short SHE pulses, but also
shows that reducing the current density by extending the SHE pulse duration is not an option.
Current density (MA/cm2)
40
J1/2,SHE
30
20
10
0.0
0.2
0.4
0.6
0.8
1.0
SHE pulse duration (ns)
Figure 15: Value of J½,SHE as a function of spin-Hall effect pulse duration. The line is a guide to the eye.
12
3.13 Delay between spin-Hall effect pulse and spin-transfer torque pulse
Simultaneous application of two current pulses along two different current paths may prove
impractical in actual devices. We therefore investigated the possibility of delaying the STT pulse. As
shown in Figure 16, J½,SHE increases moderately with the delay time, showing little increase below tdelay =
1.5 ns. In fact, J½,SHE is significantly reduced (almost halved) if the two pulses are subsequent rather than
simultaneous, as seen from the tdelay = 0.5 ns data point. In this case, the magnetization is maximally
destabilized prior to STT current application, resulting in efficient switching. It should be noted,
however, that the total switching time is then increased compared to the case of simultaneous pulses,
skewing the comparison.
Current density (MA/cm2)
50
J1/2,SHE
40
30
20
10
0
0
1
2
3
4
5
STT pulse delay time (ns)
Figure 16: Value of J½,SHE as a function of the delay between the spin-Hall effect pulse and the spin-transfer
torque pulse. A delay time of zero implies that both current pulses are started at t = 0. As the SHE pulse duration
is 0.5 ns, the lowest value of J½,SHE is observed if the STT pulse is started immediately after the SHE pulse is
ended. The line is a guide to the eye.
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