Connecting physics to systems with modular spin-circuits

Abstract

An emerging paradigm in modern electronics is that of CMOS + $\sf X$ requiring the integration of standard CMOS technology with novel materials and technologies denoted by $\sf X$. In this context, a crucial challenge is to develop accurate circuit models for $\sf X$ that are compatible with standard models for CMOS-based circuits and systems. In this perspective, we present physics-based, experimentally benchmarked modular circuit models that can be used to evaluate a class of CMOS + $\sf X$ systems, where $\sf X$ denotes magnetic and spintronic materials and phenomena. This class of materials is particularly challenging because they go beyond conventional charge-based phenomena and involve the spin degree of freedom which involves non-trivial quantum effects. Starting from density matrices $-$ the central quantity in quantum transport $-$ using well-defined approximations, it is possible to obtain spin-circuits that generalize ordinary circuit theory to 4-component currents and voltages (1 for charge and 3 for spin). With step-by-step examples that progressively become more complex, we illustrate how the spin-circuit approach can be used to start from the physics of magnetism and spintronics to enable accurate system-level evaluations. We believe the core approach can be extended to include other quantum degrees of freedom like valley and pseudospins starting from corresponding density matrices.

Figures (9)

• Figure 1: Physics to systems perspective with modular spin-circuits (a) Physics: Spin-circuits solve transport and magnetization dynamics self-consistently. (b) Devices: example stochastic MTJs (with spin-orbit and spin-transfer torque) using low-energy barrier magnets. (c) Circuits: stochastic neurons (p-bits) built out of stochastic MTJs (d) Architectures: Probabilistic architectures with interacting stochastic neurons (e) Networks: networks of p-bits mapped to computationally hard optimization problems (f) Algorithms: powerful algorithms that use replicas of probabilistic networks to help solve these optimization problems.
• Figure 2: 2-port formulation of spin conductances (a) Any 2-terminal spin conductor can be formulated in terms 2-port conductance matrices between its terminals. (b) The currents and voltages are related to each other by 4$\times$4 conductances $G_{ij}$ and currents and voltages are 4-component vectors. (c) Unlike charge currents, spin conductors may exhibit non-conservation of currents ($I_1 + I_2 \neq 0$) and non-reciprocity ($G_{12}\neq G_{21}$). Here, we show an example of a reciprocal spin conductor ($G_{12}=G_{21}=G_0$). Even with non-the conservative nature of spin-currents, it is possible to obtain a circuit description by introducing shunt conductances from the terminal to the ground to account for losses through spin-relaxation or coherent rotation mechanisms.
• Figure 3: Transport and magnetism (a) An example spin-valve built out of two interfaces is shown. Numerical results obtained from spin-circuits are compared with theory BRATAAStransport where the charge conductance shows magnetoresistance as a function of the relative angle between the ferromagnets. (b) Spin-circuit model illustrating the interaction between the magnetization dynamics (modeled by sLLG) and transport modules. The transport model receives two magnetization vectors from the stochastic LLG and produces 4-component spin currents carrying charge and spin information. sLLG receives spin currents and magnetic fields and produces a magnetization vector. (c) sLLG results are benchmarked with the Fokker Planck equation (FPE). 1000 low-barrier nanomagnets (with a very small perpendicular magnetic anisotropy) are prepared in the $-$1 direction and left to relax. The average magnetization $\left\langle m_z \right\rangle$ is measured over time and compared to FPE and the analytical solution (see text).
• Figure 4: Antiferromagnetic resonance (AFMR) with spin-circuits (a) Sketch of an antiferromagnet where two sublattices with opposing magnetizations. (b) Spin circuit model of the AFM with two antiferromagnetic layers analyzed by two LLGs coupled with exchange interactions. (c) Experimental results for AFMR in $\sf{MnF_2}$afmrexperiment. (d) Numerical results obtained from spin-circuits for AFMR, compared to known theory. (e) Easy-axis (z) component of the magnetization vector analysis over an external magnetic field applied in the +z direction. At a critical field, the sublattice spins enter the "spin-flop" region where they both develop a small $m_z$ component in the direction of the magnetic field. In all cases, spin-circuits provide excellent agreement with known theory.
• Figure 5: Non-local spin valve (NLSV) with low-barrier nanomagnets (LBM) (a) Physical structure consisting of networks of LBMs. A charge current is injected from LBMs going to a nearby local ground. Spin currents polarized in the direction of fluctuating LBMs are routed to one another. Inset shows an example of how magnetization dynamics $\hat{m}$ evolve over time for an LBM with low perpendicular anisotropy. The bottom panel shows the spin-circuit corresponding to the physical structure. (b) The average of the relative angle between LBM 1 and LBM 2 is measured as a function of injected charge currents, showing ferromagnetic (at positive $I_c$) and antiferromagnetic (at negative $I_c$) coupling. The numerical results are compared with those obtained from the Boltzmann law obtained from the Heisenberg Hamiltonian. This correspondence between the unitless Heisenberg Hamiltonian and spin-circuit requires a mapping factor $I_M$ with units of currents (see text and Ref. bunaiyan2023heisenberg). (c) A histogram of three LBMs at large negative currents where for better illustration the magnetizations $\hat{m}$ are binarized by thresholding at $\hat{m}_z =0$. The system shows frustration in the antiferromagnetic configuration.