Simplified schematic diagram of an STT-RAM memory element. The nanomagnets (NMs) are on the y-z plane and are shaped like elliptical cylinders. NM-1 is magnetically hardened along the z-axis so that its magnetization direction is fixed. The magnetization direction of NM-2 can be rotated with an in-plane spin polarized current that delivers a spin transfer torque. The magnetization orientation of the free layer NM-2 with respect to the z-axis (0 ◦ and 180 ◦ ) encodes 

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... with ∼100 nm size 6 and it has been demonstrated in numerous exper- iments involving both spin-valves 7 and magnetic tunnel junctions (MTJs) 8 . MTJs, consisting of an insulating layer sandwiched between two ferromagnetic layers (one hard and the other soft ), are becoming the staple of non- volatile magnetic random access memory (RAM) 4,9 (see Fig. 1). Switching the soft layer of an MTJ with the STT- mechanism (STT-RAM) allows for high integration den- sities, but usually requires a high current density (> 10 7 A/cm 2 ) resulting in significant energy dissipation 10 ...

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... a free nanomagnet (see Fig. 1) in the shape of an elliptical cylinder whose elliptical cross section lies in the y-z plane with its major axis and minor axis aligned along the z-direction and the y-direction, respectively. The dimension of the major axis is a, that of the minor axis is b, and the thickness is l. The volume of the nano- magnet is Ω = (π/4) abl. Let ...

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... easy axis was obtained by averaging over 10,000 MC simulations for either temper- ature. The scattered points in the figure are the data for one representative MC simulation. Clearly, averaging over many simulations reduces the spread and random- ness in the data. The single simulation is representative of a single experimental measurement. In Fig. 10, we find that the mean switching delay τ decreases with increasing saturation magnetization M s . This can be explained as follows. At lower M s , the spin- transfer torque is weakened because it is proportional ...

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... M s since the mean value of the deflection from the easy axis θ init , at any temperature, is not sensitive to M s . This result shows that cooling a nanomagnet down from room temperature to 4.2 K will be counter-productive in at least one respect; it will increase the thermally averaged switching delay at any M s (remember that I s ∝ M 2 s ). In Fig. 10, the random scattering of the data points about the thermal mean is not just due to thermal fluc- tuations, since similar behavior was observed in Ref. [36] which did not consider any thermal fluctuation at all. The scatter is more prominent at smaller values of M s , which corresponds to lower I s (I s ∝ M 2 s ). At lower I s , the ...

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... mean is not just due to thermal fluc- tuations, since similar behavior was observed in Ref. [36] which did not consider any thermal fluctuation at all. The scatter is more prominent at smaller values of M s , which corresponds to lower I s (I s ∝ M 2 s ). At lower I s , the magnetization dynamics is more complex since there are more ripples (see Fig. 13 and Fig. 14 later). As a re- sult, there is more variability in the switching dynamics with changing I s when the latter is small. This variability contributes to the ...

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... just due to thermal fluc- tuations, since similar behavior was observed in Ref. [36] which did not consider any thermal fluctuation at all. The scatter is more prominent at smaller values of M s , which corresponds to lower I s (I s ∝ M 2 s ). At lower I s , the magnetization dynamics is more complex since there are more ripples (see Fig. 13 and Fig. 14 later). As a re- sult, there is more variability in the switching dynamics with changing I s when the latter is small. This variability contributes to the ...

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... Fig. 11, we plot the thermal means of the energy dissipation E total at 4.2 K and 300 K as a function of the saturation magnetization M s , while keeping the in-plane shape anisotropy barrier constant. E total is overwhelm- ingly dominated by the component I 2 s Rτ , and the inter- nal energy dissipation E d has a minor contribution (see Fig. ...

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... Fig. 11, we plot the thermal means of the energy dissipation E total at 4.2 K and 300 K as a function of the saturation magnetization M s , while keeping the in-plane shape anisotropy barrier constant. E total is overwhelm- ingly dominated by the component I 2 s Rτ , and the inter- nal energy dissipation E d has a minor contribution (see Fig. 12). The switching current I s varies as the square of M s , so that I 2 s varies as M 4 s . Furthermore, if we reduce M s , we have to increase the shape anisotropy (or the aspect ratio a/b) to keep the in-plane shape anisotropy energy barrier constant. If the switching current flows along the minor axis of the elliptical nanomagnet (al- ...

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... because the in-plane shape anisotropy barrier is kept constant). Consequently, the power dissipation I 2 s R increases with M s more rapidly than M 4 s . Unless the switching delay τ has a stronger dependence on M s than τ ∝ M −4 s , we will expect the energy dissipation to decrease with decreasing M s and that is precisely what we observe in Fig. ...

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... at 4.2 K than at 300 K, the mean energy dissipation is also larger at 4.2 K. Moreover, the difference between the thermal means of the energy dissipation at 4.2 K and 300 K is proportional to M 4 s R [∆τ ], where ∆τ is the difference between the mean delays at the two temperatures. Since ∆τ is ap- proximately independent of M s , as seen in Fig. 10, the difference between the mean energy dissipations at the two temperatures should increase as ∼M 4 s , which is what we ...

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... order to understand how we can maintain a con- stant switching delay while scaling M s , let us consider the relationship between switching current and switch- ing delay. In Fig. 13, we plot the magnetization dynam- ics without considering any thermal fluctuations during the switching when M s = 4.09 × 10 5 A/m. Compare this s . In Fig. 13, we have assumed the same initial orientation as in Fig. 7. The square-law scal- ing however results in an increased switching delay since the latter has obviously increased by ...

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... order to understand how we can maintain a con- stant switching delay while scaling M s , let us consider the relationship between switching current and switch- ing delay. In Fig. 13, we plot the magnetization dynam- ics without considering any thermal fluctuations during the switching when M s = 4.09 × 10 5 A/m. Compare this s . In Fig. 13, we have assumed the same initial orientation as in Fig. 7. The square-law scal- ing however results in an increased switching delay since the latter has obviously increased by a factor of 2 (from 1.05 ns to 2.1 ns). This has happened because of more ripples generating from more precessional motion of the magnetization vector seen in ...

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... . In Fig. 13, we have assumed the same initial orientation as in Fig. 7. The square-law scal- ing however results in an increased switching delay since the latter has obviously increased by a factor of 2 (from 1.05 ns to 2.1 ns). This has happened because of more ripples generating from more precessional motion of the magnetization vector seen in Fig. 13. In order to main- tain the same switching delay of 1.05 ns as before, we will have to deviate from the square-law scaling and in- crease the switching current by nearly two times to 1.05 mA. Thus, we need to pump an excess current of 1.05 mA -0.523 mA = 0.527 mA in order to maintain the same switching speed. The corresponding ...

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... to deviate from the square-law scaling and in- crease the switching current by nearly two times to 1.05 mA. Thus, we need to pump an excess current of 1.05 mA -0.523 mA = 0.527 mA in order to maintain the same switching speed. The corresponding magnetization dynamics without considering any thermal fluctuations during the switching is shown in Fig. 14, where we have clearly recovered the 1.05 ns delay. The energy dissipa- tion (dominated by I 2 s Rτ ) now goes up by a factor of two [I s increases by a factor of two while τ decreases by a fac- tor of two]. Thus, we find that if we wish to maintain a constant switching delay, then we need to inject some ex- cess current over that ...

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... illustrative purposes, we show in Fig. 15 the mag- netization dynamics in the presence of thermal fluctua- tions at 300 K for the same parameters as in Fig. 14. This is one representative run picked out from 10,000 Monte Carlo simulations. Note that there is only some quan-titative differences, but not much qualitative difference, between Figs. 14 and 15. The ripples are ...

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... illustrative purposes, we show in Fig. 15 the mag- netization dynamics in the presence of thermal fluctua- tions at 300 K for the same parameters as in Fig. 14. This is one representative run picked out from 10,000 Monte Carlo simulations. Note that there is only some quan-titative differences, but not much qualitative difference, between Figs. 14 and 15. The ripples are somewhat larger in amplitude and the precessional motion is slightly exac- erbated. The switching delay has increased by ...