Design rules for scalability in spin-orbit electronics

Spin-orbit electronics (spin-orbitronics) has been widely discussed for enabling nonvolatile devices that store and process information with low power consumption. The potential of spin-orbitronics for memory and logic applications has been demonstrated by perpendicular anisotropy magnetic devices comprised of heavy-metal/ferromagnet or topological-insulator/ferromagnet bilayers, where the heavy metal or topological insulator provides an efficient source of spin current for manipulating information encoded in the bistable magnetization state of the ferromagnet. However, to reliably switch at room temperature, spin-orbit devices should be large to reduce thermal fluctuations, thereby compromising scalability, which in turn drastically increases power dissipation and degrades performance. Here, we show that the scalability is not a fundamental limitation in spin-orbitronics, and by investigating the interactions between the geometry of the ferromagnetic layer and components of the spin-orbit torque, we derive design rules that lead to deeply scalable spin-orbit devices. Furthermore, employing experimentally verified models, we propose deeply scaled spin-orbit devices exhibiting high-speed deterministic switching at room temperature. The proposed design principles are essential for design and implementation of very-large-scale-integration (VLSI) systems that provide high performance operation with low power consumption.

operation. In contrast to the IMA originating from the device shape anisotropy, the PMA can originate from the interface between the magnetic and non-magnetic films within the device 4,5,20 . Thus, the PMA is not limited by the planar geometry of the device and may extend to relatively large values required for nonvolatility at smaller dimensions.
While rapid advances in fabrication of nonvolatile spin-orbit devices with scaled dimensions can be expected, it is of significant technological relevance to derive design rules that allow such devices to exhibit reliability, i.e., deterministic switching at room temperature. In this paper, we present design rules for achieving deeply scalable spin-orbit magnetic devices that exhibit deterministic switching operation at room temperature (≥300 K). We further show that the design rules enable high speed switching operation with low power consumption, suggesting the spin-orbitronics as a promising candidate for post-CMOS systems delivering high performance operation with low power consumption.

Geometry as a Degree of freedom for Scalability in Spin-orbitronics
In this section, we introduce and study the planar geometry of the ferromagnetic layer as a degree of freedom for achieving deeply scalable spin-orbit devices. The schematic of the studied spin-orbit magnetic device is illustrated in Fig. 1. This heterostructure represents the fundamental building block of the magnetic devices that employ spin-orbit torque to manipulate information encoded in the bistable magnetization state of a ferromagnet. The device comprises a ferromagnetic layer, referred to as the free layer, which is sandwiched between an oxide layer and a thin film with strong spin-orbit interaction, referred to as the channel. The magnetization of the free layer can be switched between two stable states along the z axis (+z and −z) via the spin-orbit torque produced by injecting a current pulse into the channel. To study the effect of geometry on device operation at scaled dimensions, the ferromagnetic layer is considered to be elliptical with a width and length of W and L, respectively. The magnetic dynamics of the device in response to the current-induced spin-orbit torque is investigated by numerical integration of the stochastic Landau-Lifschitz-Gilbert-Slonczewski (LLGS) equation 21 and via experimentally verified models of spin-orbit heterostructures 22 .
The magnetization of the free layer is determined by the effect of the torques and fields at room temperature. The spin-orbit torque (T SOT ) comprises two primary components, a damping-like torque (T DL ) and a field-like torque (T FL ). Denoting the unit vector along the magnetization by m and along the spin polarization by σ, T SOT is represented as (a) Layer architecture of the fundamental building block of magnetic devices that employ spin-orbit torque to manipulate a bit encoded in the bistable magnetization state of a ferromagnetic layer. The channel is a material with strong spin-orbit interaction, and the free layer is made of a ferromagnetic material. (b) Top view of the device. To study the effect of geometry on the device behavior, an elliptical free layer, instead of a conventional circular layer, is considered. The nominal device has the following parameters: channel width 40 nm, channel thickness 5 nm, free layer area equivalent to a circle with a radius of 12 nm, free layer thickness 1.
where ζ DL and ζ FL denote the efficiency of the channel current in producing the damping-like and field-like torque, respectively. Here, , e, J e , M s , and t FL denote the reduced Planck's constant, electron charge, channel current density, saturation magnetization, and free layer thickness, respectively. We choose the thickness of the free layer to be 1.2 nm and the area of the free layer to be equivalent to a circle with a radius of 12 nm. Accordingly, to ensure nonvolatility (Energy barrier = 40 k B T), the saturation magnetization and the PMA field are set to, respectively, . Furthermore, to enable the polarity of the channel current (+y or −y) to favor a unique stable state of the magnetization (+z or −z), an external magnetic field (H ext ) is applied along the channel to break the symmetry 4,5 . H ext is usually produced by integrating a nanomagnet with the device, and is taken to be small (≤0.1 H PA ) to avoid compromising device nonvolatility via reduction of the energy barrier between the stable magnetization states.
Sufficiently large PMA is essential for nonvolatile retention of a stable magnetization state at room temperature. However, since the switching process is influenced by the entire magnetic energy landscape, even a large PMA may not yield reliable switching in deeply scaled devices. We investigate shaping the magnetic energy landscape for achieving reliable switching at scaled dimensions. To analyze the effect of the magnetic energy landscape on switching reliability, we characterize the energy saddle points and investigate how the position of the saddle points with respect to the spin polarization axis (x axis) influences the switching reliability.  www.nature.com/scientificreports www.nature.com/scientificreports/ be zero (i.e., ζ = 0 FL eff ). The analysis is then continued by considering the effect of + H H FL Oe on device operation. In the analysis, we assume that ζ > 0 DL . Due to symmetry, all results are also valid for devices with ζ < 0 DL . Before going through the analysis, we define three distinguishable states which the magnetization takes during a switching event: the initial stable state, the final stable state, and the equilibrium state. The initial stable state refers the stable magnetization state before injecting a current pulse into the channel, the final stable state refers the stable magnetization state after current pulse injection, and the equilibrium state refers to the state where the resultant of all fields experienced by the magnetization is zero.
The magnetic energy landscape for three devices with the same volume but different aspect ratios of the free layer are illustrated in Fig. 3. For the device with free layer aspect ratio of η = 1 (circular cross section), the saddle point lies along the current flow (y axis), as illustrated in Fig. 3a. Since the spin-orbit torque tends to align the magnetization with the spin polarization direction (x axis), the magnetization reaches the equilibrium at an energetically excited state which is highly susceptible to thermal fluctuations, leading to stochastic switching. As shown in Fig. 4a,c, the switching probability for the device does not exceed 0.8 over a wide range of the current pulse duration (τ pulse ) and amplitude (J e ; , which is too low to be considered reliable. Although a PMA field of = H Oe 4, 000 PA is sufficient to retain a stable magnetization state for a long period of time at room temperature in this device, a prohibitively larger PMA field may be required to overcome thermal fluctuations and stabilize the magnetization from the equilibrium to the final stable state. Switching power dissipation is quadratically proportional to the PMA field 23 . Hence, relying on a prohibitively large PMA for reliable switching is not a viable solution as a large PMA drastically increases switching power dissipation, thus compromising scalability. We propose employing the planar geometry of the free layer as a degree of freedom for shaping the magnetic energy landscape to achieve reliable switching, without the need for increasing the PMA field beyond the level required for nonvolatility. By introducing in-plane magnetic shape anisotropy into the free layer via increasing the in-plane aspect ratio of the free layer, the saddle points of the magnetic energy landscape move toward the spin polarization axis (x axis). The position of the saddle points in the polar coordinate system is given by where H IA denotes the in-plane anisotropy field produced by the shape anisotropy. , and (c) = L n m 36 and = W nm 16 . By increasing the shape anisotropy (free layer aspect ratio), the energy saddle points move toward the spin polarization axis (x axis), and the equilibrium is reached at a significantly lower energy state. ϑ and ϕ denote the angle which the magnetization makes, respectively, with the z axis and with the x axis at each time instant.
Hence, in contrast to the circular geometry, the switching trajectory will pass through the saddle region of the magnetic energy landscape for a wide range of current pulse parameters, reaching equilibrium in a low-energy state within the duration of the current pulse, as shown in Fig. 3b,c. Hence, after current pulse injection, even a small PMA field is sufficient to absorb the magnetization effectively from the equilibrium state toward the final stable state, leading to robust switching operation in the presence of thermal fluctuations, as illustrated in Fig. 4b. For the device with the same magnetic parameters and volume of the free layer as described above, a free layer aspect ratio of η = 32 18 is sufficient to achieve a switching probability larger than 0.9995 over a wide rage of the amplitude and duration of the channel current (Switching Error Rate (SER) < 5 × 10 −4 ). Accordingly, nonvolatility and reliability can be achieved by tuning two independent degrees of freedom, the perpendicular magnetic anisotropy and the in-plane magnetic anisotropy, respectively. Consequently, the presence of in-plane magnetic anisotropy is essential for reliable operation at scaled dimensions.
In striking contrast to the two terminal magnetic devices operating via spin-transfer torque (STT) 24 , larger current densities may not necessarily lead to higher switching speeds in scaled devices operating on the basis of spin-orbit torque. The fields with z-component operating on magnetization are H DL and H PA ; at equilibrium, . Hence, as shown in Fig. 2, while driving the magnetization out of the initial stable state, the damping-like torque keeps the magnetization away from the final stable state once the magnetization reaches equilibrium (i.e., works opposite to the stabilizing torque exerted by H PA on the magnetization). By increasing the channel current density, the magnetization arrives more quickly at the equilibrium state. Nevertheless, the net destabilizing effect of the spin-orbit torque on the equilibrium state increases as well, leading to an equilibrium state that is energetically farther away from the final stable state, as shown in Fig. 4d. Thus, the magnetization takes longer to stabilize to the final stable state after current pulse injection and the switching process becomes prone to thermal fluctuations, which dramatically increases the switching error rate (more than three orders of magnitude).

Deeply Scalable Spin-orbit Devices with High Speed Switching at Room temperature
Having characterized the geometry as a degree of freedom for achieving deeply scaled spin-orbit devices, we introduce a design principle for achieving sub-nanosecond deterministic switching at deeply scaled dimensions. Spin-orbit heterostructures exhibit a wide range of damping-like and field-like torques. Furthermore, since the free layer is fabricated immediately on the channel, the effect of H Oe on the dynamics of the free layer magnetization is significant. So far, assuming that , we have not considered the effect www.nature.com/scientificreports www.nature.com/scientificreports/ of the field-like torque and H Oe on device operation. In the following, we show that the field-like torque and H Oe may dramatically affect both the switching speed and reliability at scaled dimensions.
H DL lies in the y-z plane, thus the damping-like torque induces a rotational motion of the magnetization between the stable magnetization states (+z and −z). H FL lies along the x axis, so the field-like torque produces a precessional motion of the magnetization around the x axis. Depending on whether the effective torque ratio, characterized by ζ ζ FL eff DL , is negative or positive, the z-component of the effective field-like torque (  =  ⋅ẑ  T  T   FL  z  FL   ) may, respectively, add to or subtract from the z-component of the damping-like torque ( = ⋅ẑ T T

DL z DL
) at equilibrium. Accordingly, for a negative ζ ζ FL eff DL , the stabilizing torque produced by H PA should overcome a larger destabilizing spin-orbit torque. The equilibrium state then lies at a relatively high-energy state close to the x-y plane, which is prone to thermal fluctuations and hence leads to stochastic switching. Alternatively, for a positive ζ ζ FL eff DL , T FL z largely cancels out T DL z at equilibrium, allowing the magnetization to relax to a low-energy equilibrium state close to the final stable state. The magnetization then quickly stabilizes toward the final stable state after current pulse injection, even in heterostructures without a large damping factor (α < . 0 03). As illustrated in Fig. 5, the switching can therefore be accomplished rapidly (<1 ns), and the process is highly robust to thermal fluctuations, leading to deterministic switching (SER < 10 −4 ), which is essential for memory and logic applications.
Accordingly, an effective approach to substantially accelerate the switching operation while maintaining the reliability is to cancel the effect of the damping-like torque at the equilibrium state to a greater degree. This can be achieved by increasing the positive torque ratio ζ ζ FL eff DL . The magnetization then not only moves more quickly toward the equilibrium state, but also reaches an equilibrium state closer to the final stable state, which shortens the stabilization time after current pulse injection. As illustrated in Fig. 5 with pulses as short as 500 ps and an amplitude in the range of the device critical current, without the need for a PMA field larger than that required to ensure nonvolatility. Surprisingly, subnanosecond switching time is enabled even at PMA fields smaller than those needed for nonvolatility, which is critical for applications requiring short retention time, such as high level cache memories and a wide range of logic systems. Hence, since power dissipation is reduced significantly (quadratically) by decreasing the PMA field, nonvolatility can be traded off for power dissipation, without compromising a device reliability. The derived design rules, therefore, may enable spin-orbitronics to serve as a universal memory technology, with high speed operation, low power dissipation, and high integration density, which has been a long-standing quest 25 . sufficient for achieving high-speed reliable switching at deeply scaled spin-orbit devices. As temperature increases, thermal fluctuations become stronger. Hence, the role of in-plane magnetic shape anisotropy in achieving deterministic switching increases at higher temperatures. Figure 6 illustrates the switching probability diagrams for two devices at three different temperatures, namely, the cryogenic temperature = T K 50 , the room temperature = T K 300 , and the on-chip temperature = T K 398 . All parameters but the aspect ratio of the free layer are the same for both devices. Particularly, for both devices = H Oe 4, 000

Discussion
, and the area of the free layer is equivalent to the area of a circle with a radius of 12 nm. As illustrated in Fig. 6, at cryogenic temperature both devices achieve deterministic switching for a wide range of current pulse parameters. At room temperature and on-chip temperature, however, the device with the larger aspect ratio η = 36 16 operates reliably over a relatively wider range of current pulse parameters. Although the presence of an in-plane magnetic shape anisotropy is essential for deterministic switching in scaled devices, the precise control of the free layer aspect ratio is not necessary. Particularly, deterministic switching can be achieved for a wide range of free layer aspect ratios, thus relaxing the fabrication process of scaled spin-orbit devices via eliminating the need for precise control of the aspect ratio. As illustrated in Fig. 6, aspect ratio of the free layer may significantly vary, yet the device exhibits reliable switching at scaled dimensions over a wide range of temperatures, verifying that for scalability the device aspect ratio is not required to be precisely controlled. Also, as illustrated in Fig. 6, by increasing the free layer aspect ratio largely from η = 32 18 to η = 36 16 , the switching current increases by about 10%. Accordingly, adding a suitable margin (about 10%) to the channel current with respect to the minimum switching current ensures deterministic switching in the worst case scenario (corresponding to η = 36 16 ).

conclusions
In conclusion, we show that scalability is not a fundamental limitation in spin-orbitronics and provide design rules for achieving deeply scalable spin-orbit devices with high speed switching at room temperature. We introduce and comprehensively study the geometry as a degree of freedom for scalability in spin-orbitronics. We show that the interaction of the spin-orbit torque components with the magnetic energy landscape, governed by the ferromagnetic layer geometry, is crucial to switching reliability, speed, and power dissipation in spin-orbit (2019) 9:13732 | https://doi.org/10.1038/s41598-019-49831-5 www.nature.com/scientificreports www.nature.com/scientificreports/ www.nature.com/scientificreports www.nature.com/scientificreports/ devices with deeply scaled dimensions. The design principles introduced and investigated in this paper pave the way toward the application of spin-orbitronics in very-large-scale-integration (VLSI) systems, such as memories, processors, and neural networks, with the capability to deliver high performance operation under the limitations imposed by low power consumption. , . Each component of H L follows a zero-mean Gaussian random process whose standard deviation is a function of temperature 26,27 ,

Methods
Here α, k B , T, M s , and v F denote, respectively, the damping factor, Boltzmann constant, temperature, saturation magnetization, and volume of the free layer. Δt is the duration of the constant effective thermal fluctuation field. The standard deviation of the thermal fluctuations is inversely proportional to the volume of the free layer. Hence, by scaling the free layer dimensions, H L becomes stronger, rendering the magnetization dynamics more susceptible to thermal fluctuations. calculation of the magnetic energy landscape. The magnetic energy landscape is governed by three components, (1) the uniaxial magnetic anisotropy, (2) the demagnetizing field, and (3) the external magnetic field. Accordingly, where E PA , E DM , and E H denote the energy components due to, respectively, the perpendicular magnetic anisotropy, demagnetizing field, and the external magnetic field. Here, K u is the perpendicular magnetic anisotropy, = N N N N ( , , )  . SPDs at each row correspond to switching events at, from left to right, = T K 50 , = T K 300 , and = T K 398 . All parameters but the free layer aspect ratio are the same for both devices. The device with a larger aspect ratio achieves deterministic switching over a relatively wider pulse parameters at higher temperatures. However, while having significantly different aspect ratios, both devices exhibit deterministic switching over a wide range of current pulse parameters, implying that the precise control of the aspect ratio is not necessary.