Emerging spin transfer torque magnetoresistive random access memories (STT MRAM) are nonvolatile and offer high speed and endurance. They are promising for stand-alone and embedded applications in the automotive industry, microcontrollers, Internet of Things, frame buffer memory, and slow SRAM. The MRAM cell usually includes a CoFeB fixed reference layer and a free magnetic layer (FL), separated by a tunnel barrier. To design ultra-scaled MRAM cells it is necessary to accurately model the torques acting on the magnetization in composite magnetic layers with one or several nonmagnetic inclusions between the ferromagnetic parts. The magnetization dynamics is governed by the Landau-Lifshitz- Gilbert (LLG) equation supplemented with the corresponding torques. The torques depend on nonequilibrium spin accumulation generated by an electric current. The electric current and the spin accumulation also depend on the magnetization. Therefore, the LLG and the spin-charge transport equations are coupled and must be solved simultaneously.
We apply the finite element method (FEM) to numerically solve this coupled system of partial differential equations. To develop an open source solver, we use well-developed C++ FEM libraries. The computationally most expensive part is the demagnetizing field calculation. It is performed by a hybrid finite element-boundary element method. This confines the simulation domain for the field evaluation to the ferromagnets only. Advanced compression algorithms for large, dense matrices are used to optimize the performance of the demagnetizing field calculation in complex structures. To evaluate the torques acting on the magnetization, a coupled spin and charge transport approach is implemented. For the computation of the torques acting in a magnetic tunnel junction (MTJ), a magnetization-dependent resistivity of the tunnel barrier is introduced. A fully three-dimensional solution of the equations is performed to accurately model the torques acting on the magnetization. The use of a unique set of equations for the whole memory cell including the FL, fixed layer, contacts, and nonmagnetic spacers is one of the advantages of our approach. To incorporate the temperature increase due to the electric current, we solve the heat transport equation coupled to the electron, spin, and magnetization dynamics, and we demonstrate that the FL temperature is highly inhomogeneous due to a non-uniform magnetization of the FL during switching.
Spin-orbit torque (SOT) MRAM is fast-switching and thus well suitable for caches. By means of micromagnetic simulations, we demonstrate the purely electrical switching of a perpendicular FL by the SOTs due to two orthogonal short current pulses. To further optimize the pulse sequence, a machine learning approach based on reinforcement learning is employed. Importantly, a neural network trained on a fixed material parameter set achieves switching for a wide range of material parameter variations.
We employ a reinforcement learning strategy for finding switching schemes for deterministic switching of a spin-orbit torque magnetoresistive random access memory cell. The free layer of the memory cell is perpendicularly magnetized, and the spin-orbit torques are generated by currents through two orthogonal heavy metal wires. A rewarding scheme for the reinforcement learning approach is defined such that the objective of the algorithm is to find a pulse sequence that leads to fast deterministic field-free switching of the memory cell. The reliability of the found switching scheme is tested by performing micromagnetic simulations. The results show that a neural network model trained on fixed material parameters is able to reverse the memory cell magnetization for a wide range of material parameters and can be used to derive a writing pulse sequence for fast and deterministic spin-orbit torque switching of a perpendicular free layer.
We employ a finite element discretization scheme to solve numerically the coupled spin and charge transport equations in spin-transfer torque magnetoresistive random access memory cells. To adapt the drift-diffusion formalism to the case of a magnetic tunnel junction, we model the tunnel barrier as a material with a low magnetization-dependent conductivity and a large spin diffusion constant. This generalized spin and charge drift-diffusion approach is applied to determine the torques entering the Landau-Lifshitz-Gilbert equation to describe the magnetization dynamics. In particular, the switching times under a fixed voltage, a fixed current, and a fixed current density are compared.
We demonstrate that a magnetic field-free two-pulse scheme previously proposed to switch an in-plane magnetized free layer is also suitable for switching a perpendicularly magnetized layer. In the case of a symmetric square free layer, deterministic switching is achieved by running the second pulse over a part of the free layer. Applying the same approach to a rectangular free layer results in switching times as short as 0.25 ns. The optimal overlap of the second heavy metal wire with the free layer is found to be between 30-60%. It is shown that the switching scheme yields a large window for the time delay/overlap between the two pulses, still maintaining the switching times as short as 0.25 ns. Consequently, the scheme is extremely robust against pulse duration fluctuations and pulse synchronization failures.
Spin correlations at hopping are responsible for large magnetoresistance at trap-assisted resonance tunneling between normal metallic and ferromagnetic electrodes. The reason for the spin correlations at hopping is the spin-selective escape rate, which results in non-zero average spin at a trap. This causes a dependence of the trap occupation and, therefore, the current on the average spin. Surprisingly, strong spin dephasing enhances the amplitude of the magnetoresistance at trapassisted tunneling from a normal metal to a ferromagnet. Spin dephasing can also boost the tunneling magnetoresistance in magnetic tunnel junctions. Spin relaxation, however, reduces the spin correlations and associated effects, as expected.
Since the spin on the trap is a vector quantity, it produces unusual correlations in multi-terminal devices. Our analysis of a three-terminal device with normal metallic and ferromagnetic electrodes and trap-assisted hopping implies that the spin correlations result in current-voltage dependences characteristic to a single-electron transistor. Importantly, the transfer characteristics are determined by the spin correlations and the spin blockade alone as, because of the finite transition rate between the trap and the normal metallic electrodes, the current is not Coulomb blocked and it always flows through the trap-source, trap-drain, and trap-gate junctions. However, when both the gate and the source electrodes are ferromagnetic with high interface spin polarizations and anti-parallel, the current through all junctions is either suppressed or it flows only between source and drain depending on the voltages applied, in complete analogy to a single electron transistor.
There are two major obstacles impeding computing systems from further advancements: The power dissipation due to leakage and the energy spent for the information transfer between memory and processor(s). The first issue is commonly handled by shutting down unused circuit parts, however, when the dormant circuits are turned on again, their previous state must be recovered. This is commonly realized by retrieving the required information from the memory, which exacerbates the limited bandwidth between memory and processor(s). In order to circumvent these limitations, we have proposed a non-volatile buffered magnetic logic grid with instant-on capability. Non-volatile magnetic flip flops and spin-transfer torque majority gates are combined to a compact regular structure, which enables a small layout foot print as well as it guarantees the reduction of the information transfer due to a shared buffer. In the proposed structure the information is passed from one magnetic layer to another by first running a current through the magnetic layer to be read, which subsequently generates a magnetization orientation encoded spin-transfer torque, when the polarized electron spins enter the next layer. Since the current passing through the junction also exerts a spin-transfer torque on the read layer, its magnetization orientation could be destabilized which might cause a read disturbance. However, during our simulations it was also found out that the stray fields of neighboring layers have a non-negligible influence on the proposed copy operation. In this work we investigate these potential read disturbances in detail for a 2-bit shift register for varying stray field strength by changing the thickness of the interconnection layer. We found that for closer proximity the acting stray fields not only stabilize but also speed up the copy procedure, while for increasing interconnection layer thickness oscillating domain walls are formed and the copy operation becomes unreliable.
We investigate the switching statistics dependence on cell geometry by means of systematic micromagnetic simulations. We find that MTJs with a free layer composed of two ellipses with the axes a/2 and b inscribed into a rectangle a × b are characterized by the same switching speed and thermal stability as MTJs with a composite free layer (C-MTJs). As has been shown, the C-MTJs demonstrate a substantial decrease of the switching time and the switching current as compared to conventional MTJs with a monolithic free layer. Thus, while preserving all the advantages of the C-MTJs, the newly proposed structure does not require a narrow gap between the two parts of the composite layer and therefore can be easily fabricated.
An analytical two-band k•p model for the conduction band of silicon is compared with the numerical nonlocal empirical
pseudo-potential method and the sp3d5s* nearest-neighbor tight-binding model. The two-band k•p model gives results
consistent with the empirical pseudo-potential method and describes the conduction band structure accurately. The tight-binding
model overestimates the gap between the two lowest conduction bands at the valley minima, which results in an
underestimation of the non-parabolicity effects. When shear strain is introduced, the two-band k•p model predicts an
analytical expression for the strain-dependence of the band structure, which is in good agreement with results of pseudo-potential
simulations.
Access to the requested content is limited to institutions that have purchased or subscribe to SPIE eBooks.
You are receiving this notice because your organization may not have SPIE eBooks access.*
*Shibboleth/Open Athens users─please
sign in
to access your institution's subscriptions.
To obtain this item, you may purchase the complete book in print or electronic format on
SPIE.org.
INSTITUTIONAL Select your institution to access the SPIE Digital Library.
PERSONAL Sign in with your SPIE account to access your personal subscriptions or to use specific features such as save to my library, sign up for alerts, save searches, etc.