Systems and methods of actuated magnetic tunnel junctions

Abstract

The system and methods of the present disclosure include a system. The system can include a plurality of magnetic tunnel junction devices, where at least two of the plurality of magnetic tunnel junction devices are electrically coupled. The system can include a pulse generator electrically coupled to the plurality of magnetic tunnel junction devices to transmit an electrical pulse to actuate the plurality of magnetic tunnel junction devices. The system can include one or more processors to analyze a statistical property of each of the plurality of magnetic tunnel junction devices.

Description

SYSTEMS AND METHODS OF ACTUATED MAGNETICTUNNEL JUNCTIONSCROSS-REFERENCE TO RELATED APPLICATION

[0001] The present application claims priority to United States Provisional Patent Application No. 63 / 653,604, filed on May 30, 2024, the disclosure of which is incorporated herein by reference in its entirety and for all purposes.STATEMENT OF GOVERNMENT INTEREST

[0002] This invention was made with government support under N00014-23-1-2771 awarded by the Office of Naval Research. The government has certain rights in the invention.TECHNICAL FIELD

[0003] The present disclosure relates generally to actuated magnetic tunnel junctions (MTJs), specifically solving complex combinatorial optimization problems using arrays of stochastic actuated MTJ devices.BACKGROUND

[0004] Optimization problems can include making decisions on allocation of resources with multiple goals and constraints. A classic example is the traveling salesman problem in which N cities must each be visited exactly once in the minimum amount of time or minimum distance traveled. This problem is just one of a multitude of other optimization problems that are in a complexity class nondeterministic polynomial time (NP)-complete in computational theory. A time to find the optimal solution increases exponentially with a problem size N. Conventional deterministic complementary metal-oxide-semiconductor (CMOS) computers are limited in their efficiency of solving such problems, which include solving large-scale resource optimization problems (e.g., logistics management, supply chain optimization, energy management, predictive maintenance, automated planning, automated driving, etc.). Furthermore, conventional techniques to solve optimization problems that use simple continuous relaxation of an objective function followed by gradient descent minimization are inherently unable to avoid local optima, thus producing poor quality solutions.SUMMARY

[0005] As described further herein, the systems and methods of the present disclosure can be used to solve a decision (e.g., optimization) problem using arrays of interconnected magnetic tunnel junctions (MTJs). In some implementations, the present disclosure can be used to sample a solution to the optimization problem according to a given distribution (e.g., the uniform distribution over all solutions) using the arrays of interconnected magnetic tunnel junctions. In some implementations, the present disclosure can be used to estimate a total number of solutions to the optimization problem using the arrays of interconnected magnetic tunnel junctions.

[0006] At least one aspect of the present disclosure is directed to a system. The system can include a plurality of magnetic tunnel junction devices, wherein at least two of the plurality of magnetic tunnel junction devices are electrically coupled. The system can include a pulse generator electrically coupled to the plurality of magnetic tunnel junction devices to transmit an electrical pulse to actuate the plurality of magnetic tunnel junction devices. The system can include one or more processors to analyze a statistical property of each of the plurality of magnetic tunnel junction devices.

[0007] In some implementations, the plurality of magnetic tunnel junction devices is electrically connected in parallel or in series, where coupling of the plurality of magnetic tunnel junctions is determined by the electrical pulse, the coupling comprising at least one of a ferromagnetic or antiferromagnetic interaction. A strength of the at least one of the ferromagnetic interaction or the antiferromagnetic interaction can be variable. A characteristic of the electrical pulse can determine a strength of the coupling of the plurality of magnetic tunnel junctions, the characteristic comprising amplitude and duration. The electrical pulse can include a plurality of electrical pulses including a first electrical pulse and a second electrical pulse applied simultaneously or sequentially to at least one of the plurality of magnetic tunnel junction devices. The plurality of electrical pulses can include a third electrical pulse configured to determine a magnetic configuration of the plurality of magnetic tunnel junction devices following transmission of the electrical pulse.

[0008] In some implementations, the plurality of magnetic tunnel junctions is electrically coupled by complementary metal oxide semiconductor circuits. The pulse generator can generate the electrical pulse according to an optimization problem, the electrical pulse toencode the plurality of magnetic tunnel junction devices with the optimization problem. The statistical property can include a magnetization state of the plurality of magnetic tunnel junction devices. The one or more processors can determine the statistical property, following transmission of the electrical pulse, from combinations of magnetization states of at least two of the plurality of magnetic tunnel junction devices. The statistical property can include mappings to binary values 00, 01, 10, and 11. The electrical pulse generated by the pulse generator can have a duration between 0.1 nanoseconds and 1 microsecond, inclusive. A magnetization of the plurality of magnetic tunnel junctions can have a stability factor between 10 and 60, inclusive. A magnetization of the plurality of magnetic tunnel junctions can align perpendicular to a plane on which the plurality of magnetic tunnel junctions is disposed in equilibrium. The plurality of magnetic tunnel junction devices can be configured such that an effective z-z coupling is established between free layers of the plurality of magnetic tunnel junction devices following application of the electrical pulse.

[0009] Another aspect of the present disclosure is directed towards a method. The method can include encoding, by one or more processors, an optimization problem, into a plurality of magnetic tunnel junctions arranged in an array. The method can include applying, by a pulse generator, a first pulse to establish an initial state of the plurality of magnetic tunnel junctions. The method can include applying, by the pulse generator, a second pulse to magnetize the plurality of magnetic tunnel junctions. The method can include determining, by the one or more processors, a magnetic state of each of the plurality of magnetic tunnel junctions. The method can include determining, by the one or more processors, based on the magnetic state, a solution to the optimization problem.

[0010] In some implementations, applying at least one of the first pulse or the second pulse causes alignment of at least one layer in the plurality of magnetic tunnel junctions, where the magnetic state includes a resistance state of the plurality of magnetic tunnel junctions resulting from the alignment of the at least one layer.

[0011] Another aspect of the present disclosure is directed towards a method. The method can include applying at least one pulse to at least two magnetic tunnel junction devices, wherein the at least two magnetic tunnel junction devices are electrically coupled. The method can include tuning an amplitude and a duration of the at least one pulse to facilitate ferromagnetic or antiferromagnetic alignment between magnetizations of free layers of the atleast two magnetic tunnel junction devices. The method can include determining a resistance state following alignment of the magnetizations of the free layers.

[0012] In various implementations, the at least one pulse is a voltage pulse and includes a first pulse and a second pulse, the second pulse applied to the at least two magnetic tunnel junction devices after the first pulse or in parallel with the first pulse. Determining the resistance state can include applying a third pulse to the at least two magnetic tunnel junction devices.BRIEF DESCRIPTION OF THE FIGURES

[0013] The foregoing and other features of the present disclosure will become more fully apparent from the following description and appended claims, taken in conjunction with the accompanying drawings. Understanding that these drawings depict only several implementations in accordance with the disclosure and are therefore not to be considered limiting of its scope, the disclosure will be described with additional specificity and detail through use of the accompanying drawings.

[0014] FIG. 1 is a diagram of a pulse transformation of a Boltzmann distribution of magnetic states, in accordance with some implementations of the present disclosure.

[0015] FIG. 2 is a diagram of a probability of a bit flip depending on a pulse amplitude for a 1 nanosecond duration pulse, in accordance with some implementations of the present disclosure.

[0016] FIG. 3 is a schematic of a perpendicular magnetic tunnel junction integrated with complementary metal-oxide semiconductors (CMOSs), in accordance with some implementations of the present disclosure.

[0017] FIG. 4 is a schematic of a pulse activated magnetic tunnel junction (MTJ), in accordance with some implementations of the present disclosure.

[0018] FIG. 5A is a graph and a schematic of an electrically coupled MTJ system to implement an Ising model, in accordance with some implementations of the present disclosure.

[0019] FIG. 5B is a schematic of an example system including the MTJ, in accordance with some implementations of the present disclosure.

[0020] FIG. 6 is a schematic of encoding an array of MTJs with a combinatorial optimization problem, in accordance with some implementations of the present disclosure.

[0021] FIG. 7 is a graph of an Ising spin and macrospin, in accordance with some implementations of the present disclosure.

[0022] FIG. 8 is a table with physical values of example parameters for the array of MTJs, in accordance with some implementations of the present disclosure.

[0023] FIG. 9A is a histogram of final energies computed using Glauber and Landau- Lifshitz-Gilbert (LLG) dynamics applied to macrospins with Heisenberg coupling, in accordance with some implementations of the present disclosure.

[0024] FIG. 9B is a histogram of final energies computed using Glauber and LLG dynamics applied to macrospins with z-z coupling, in accordance with some implementations of the present disclosure.

[0025] FIG. 10A is a graph of average ground state energy per spin ((E) / n) as a function of n, computed using Glauber and LLG dynamics applied to macrospins with Heisenberg coupling, in accordance with some implementations of the present disclosure.

[0026] FIG. 10B is a graph of average ground state energy per spin ((E) / n) as a function of n, computed using Glauber and LLG dynamics applied to macrospins with z-z coupling, in accordance with some implementations of the present disclosure.

[0027] FIG. 11 A is a graph of average ground state energy per spin as a function of n'2 / 3, computed using Glauber and LLG dynamics applied to macrospins with Heisenberg coupling, in accordance with some implementations of the present disclosure.

[0028] FIG. 1 IB is a graph of average ground state energy per spin as a function of n'2 / 3, computed using Glauber and LLG dynamics applied to macrospins with z-z coupling, in accordance with some implementations of the present disclosure.

[0029] FIG. 12 is a schematic of an electrical circuit containing two magnetic tunnel junctions (MTJs) connected in parallel to a voltage source and a 2 k resistor and a histogramillustrating net resistance of the MTJ pair exhibiting four distinct levels corresponding to the four combinations of MTJ magnetic states: parallel / parallel (P / P), parallel / antiparallel (P / AP), antiparallel / parallel (AP / P), and antiparallel / antiparallel (AP / AP), which map to the binary configurations 00, 01, 10, and 11, respectively, in accordance with some implementations of the present disclosure.

[0030] FIG. 13 is a graph showing transition probabilities between four different initial and final states as a function of the applied voltage and a voltage-dependent transition matrix T(V), in accordance with some implementations of the present disclosure.

[0031] FIG. 14 shows an example pulse waveform used to set the transition probabilities between states, a histogram illustrating a case in which the majority of pulse events result in the states 01 and 10, indicating antiferromagnetic coupling, and a histogram illustrating a case in which the most probable resulting states are 00 and 11, consistent with ferromagnetic coupling, in accordance with some implementations of the present disclosure.

[0032] FIG. 15 is a flow diagram of an example method for solving optimization problems using MTJs, in accordance with some implementations of the present disclosure.

[0033] FIG. 16 is a flow diagram of an example method of controlling coupling between MTJs to solve optimization problems, in accordance with some implementations of the present disclosure.

[0034] Reference is made to the accompanying drawings throughout the following detailed description. In the drawings, similar symbols typically identify similar components, unless context dictates otherwise. The illustrative implementations described in the detailed description, drawings, and claims are not meant to be limiting. Other implementations may be utilized, and other changes may be made, without departing from the spirit or scope of the subject matter presented here. It will be readily understood that the aspects of the present disclosure, as generally described herein, and illustrated in the figures, can be arranged, substituted, combined, and designed in a wide variety of different configurations, all of which are explicitly contemplated and made part of this disclosure.DETAILED DESCRIPTION

[0035] The present disclosure can include semiconductor devices, specifically magnetic tunnel junctions (MTJs) that operate at room temperature, are nanometer-scale in size (e.g., can be arranged with high area density use low energy, and can be integrated with complementary metal-oxide semiconductors (CMOS) technology in a back end of a manufacturing line (e.g., after semiconductor components have been fabricated), thereby mitigating complex manufacturing and higher operating costs. An MTJ cell’s minimum area can be given by 4 , where / is the critical dimension of the semiconductor technology used. For example, for =20nm, a minimum MTJ cell area can be 1600nm2, corresponding to an areal density of 6.25xlO10MTJ / cm2, meaning that chips with greater than IO10MTJs (10G MTJs) are practical. Energy used per operation can be in the a of 0.1 to 10 picojoules (pJ).

[0036] MTJs are a semiconductor device that include at least two magnetic layers and an insulating barrier between the two magnetic layers. The two magnetic layers can be made of conducting ferromagnetic materials that can have different relative magnetic orientations (e.g., parallel or antiparallel). The insulating barrier can be thin and can be formed of materials such as aluminum oxide and magnesium oxide. The insulating barrier can allow electrons to tunnel through in a process known as quantum tunneling that provides electrical resistance. The change in resistance between parallel and antiparallel magnetic orientations is called magnetoresistance (TMR). TMR occurs when the probability of electrons quantum mechanically tunneling from one magnetic layer, through the insulating barrier, and to another magnetic layer depends on their relative magnetic orientations. A magnitude of TMR can depend on the electron spin polarization of the two magnetic layers and the characteristics of the insulating barrier. Applications of MTJs can include magnetic sensors, magnetic memory, spintronics devices, MTJ-based logic circuits, etc.

[0037] Combinatorial optimization problems can involve how to best allocate limited resources or arrange elements to optimize a desired objective function. Variables involved in combinatorial optimization problems are often discrete and binary (e.g., 0 and 1).

[0038] The Ising model is a mathematical model that investigates the behavior of magnetic materials. The Ising model includes discrete variables representing magnetic moments (e.g., spins) which can be either -1 or +1. The spins can be arranged on a lattice toallow spins to interact with its neighbors. The Ising model allows for the identification of phase transitions and thermodynamic quantities.

[0039] Physical systems can solve mathematical optimization problems and can rely on the correspondence between variables of mathematical problems and the degrees of freedom of an analog physical system, where the role of the objective function to be optimized is played by the energy function of the interacting physical system. Once this correspondence has been established, the system can be cooled down to zero temperature and settled to a ground state of the system (e.g., the state with minimal energy, which, in turn, represents the solution to the optimization problem). In many cases of practical interest, during the cooling process, the physical system can get stuck in long-lived metastable states, possibly without ever reaching the ground state on human-life timescale (e.g., no solution to the optimization problem is realized). Physical systems, such as analog physical systems, which are less prone to getting trapped in the local minima of the energy function can solve the optimization problem better than those which get easily stuck. Physical systems, such as analog physical systems with continuous degrees of freedom including XY or Heisenberg spin systems, which are less prone to getting trapped in the local minima of the energy function, can solve the optimization problem better than those which get easily stuck, including physical systems with bistable states such as Ising spin systems.

[0040] Many interesting optimization problems are discrete in nature, meaning that the variables involved are represented by integers, typically taking up only two values (e.g., 0 and 1). Consequently, using Ising-type physical systems to effectively map integer variables onto binary spins and then using Monte Carlo methods or alike to simulate the physical dynamics of these analog Ising machines can find a ground-state of the system and thereby the solution.

[0041] The systems and methods in accordance to the present disclosure include an energy-minimization framework to solve combinatorial optimization problems based on physical dynamics of macrospins described by Landau-Lifshitz-Gilbert (LLG) equations. Using an analog physical system can allow for a continuous dynamical system made up of 3- dimensional rotors, rather than up-down Ising spins. The rationale behind using the 3- dimensional rotors can include that the physical gambit that trajectories away from the z-axis, in particular those occurring halfway between North and South poles, might help the systems and methods in accordance to the present disclosure mitigate getting trapped in a localminima, thus facilitating the discovery of the true, or close to true, ground state. Systems and methods in accordance to the present disclosure have been benchmarked by a complex problem in combinatorial optimization, namely the Sherrington-Kirkpatrick (SK) spin glass model, where the value of the ground-state energy is analytically known in the infinite spin limit, and shows that the systems and methods of the present disclosure extrapolates to the exact value while performing systematically better than discrete methods based on Glauber dynamics of Ising spins.

[0042] The systems and methods of the present disclosure demonstrates that arrays of electrically connected MTJs can facilitate the discovery of nearly-optimal high-quality solutions to combinatorial optimization problems, thus the present disclosure can be more efficient at solving a broad class of optimization problems. The MTJs can solve the optimization problems by switching between a quick decrease along the gradient and a slow decrease close to the level curves of the objective function, which considerably improves a performance of the optimization resulting from sole gradient dynamics, as supported by extensive numerical simulations on a prototypical complex combinatorial optimization problem. The quick decrease along the gradient and slow decrease close to level curves can be due to a form of magnetization dynamics of the MTJs that can be described by equations as discussed herein. As discussed further herein, the present disclosure has benchmarked arrays of electrically connected MTJs on a complex problem with a theoretically known optimal solution using a state-of-the-art computer algorithm and found that arrays of electrically connected MTJs are as good or better than the best-known algorithms for this problem. Furthermore, arrays of electrically connected MTJs exhibit a favorable scaling of the time to solution with the problem size N. Arrays of electrically connected MTJs can be implemented in hardware, which can significantly reduce the time to solution compared to presently used software approaches.

[0043] Sub-nanosecond to nanosecond electrical pulses can cause the MTJ to fluctuate. Conventional stochastic MTJs can be designed to fluctuate spontaneously in response to thermal noise in the environment, a process known as superparamagnetism, which can be a slow process, typically requiring microseconds or more time.

[0044] As discussed further herein, connecting arrays of MTJs electrically can create a network of locally interacting processing elements called an Ising machine. The MTJ connections (e.g., connected in parallel or in series) or weights (e.g., placed on each of theMTJs) can determine the problem to be solved which can be considered the means of programming the devices. Appropriate electrical signals can be used to cause magnetic states of the array of MTJs to fluctuate and evolve to a state that represents the solution to the desired optimization problem.

[0045] MTJs can be electrically coupled (e.g., electrically connected) to provide a specific form of a mutual coupling of the MTJs. For example, two MTJs electrically connected in series that are both initially in a parallel magnetization (e.g., low resistance) state can predominantly be antiferromagnetically (e.g., antiparallel alignment) coupled. A voltage pulse applied to the two MTJs in series that leads to switching of one of the MTJs can lead to a decrease in a voltage drop across a second MTJ it less probable that the second MTJ will switch states (e.g., from parallel to antiparallel). Thus, a final state of the two MTJs can be more likely be antiferromagnetically aligned.

[0046] In contrast, two MTJs that are electrically connected in parallel that are both initially in a parallel magnetization state can be predominantly ferromagnetically coupled. A current pulse applied to the two MTJs in parallel that leads to switching of one of the MTJs can lead to an increase in a current through the second MTJ making it more probable that the second MTJ will switch. Thus, the final state of the two MTJs can more likely be ferromagnetically (e.g., parallel) aligned. The connection of the MTJs in series or parallel and actuated by voltage or current pulses can be accomplished with CMOS circuitry.

[0047] For MTJs connected in either series or parallel, the nature of the MTJ coupling — whether ferromagnetic or antiferromagnetic — can be controlled by characteristics of a sequence of voltage pulses applied simultaneously to both MTJs. Specifically, adjusting the amplitude and duration of the voltage pulse enables favoring of either parallel or antiparallel alignment of the magnetization of free layers of the MTJs. Applying such pulse sequences can enable dynamic control of the coupling behavior without change to the electrical configuration (e.g., series or parallel connection) of the MTJs.Magnetic Tunnel Junctions (MTJs)

[0048] FIGS. 1-3 depict various attributes of a stochastic actuated MTJ. Stochastic actuation can involve using random or probabilistic methods to initiate actions in a system. For example, FIG. 1 is a diagram of a pulse transformation of a Boltzmann distribution of magnetic states. As a result, a pulse can be a stochastic actuation of an MTJ and can transforman initial Boltzmann distribution of magnetic states into a random bit (e.g., 0 or 1). As described by FIG. 2, a probability of a bit flip (e.g., 0 to 1) depends on an amplitude of the pulse at a fixed pulse duration (Ins pulse duration in Fig. 2). The probability of a bit flip also depends on the pulse duration. Both pulse amplitude and duration together can determine bit flip probability. For example, if the bit is flipped from 0 to 1, the pulse can actuate the MTJ.

[0049] An application of stochastic actuated MTJ 302 in a system 300 can be seen in FIG. 3 where a perpendicular magnetic tunnel can be integrated with a CMOS circuit which can provide the pulse in a one transistor-one perpendicularly magnetized MTJ 302 IT / lpMTJ configuration. For example, in equilibrium the MTJ 302 layer magnetizations can align perpendicular to a plane on which the MTJ 302 is disposed. The system 300 can include at least a bit line 304, a bit line complement 306, and a word line 308. The bit line 304 and bit line complement 306 can be electrical connections to the MTJ 302. The bit line 304 can connect a top layer of the MTJ 302 with one terminal of a CMOS transistor in the CMOS circuit. The CMOS transistor can be connected to bottom layers of the MTJ 302 via another terminal. The word line 308 can be connected to the CMOS transistor and can turn the CMOS transistor on and off to select a particular MTJ 302 in an array of MTJs 302.

[0050] FIG. 4 is a schematic of the pulse actuated MTJ 302. The pulse actuated MTJ 302 can include at least a composite free layer 402, a layer of magnesium oxide (MgO) 404, a reference layer 406, a first synthetic antiferromagnetic layer 408, and a second synthetic antiferromagnetic layer 410, separated by at least one non-magnetic layer 412 (e.g., materials Ru or Ir). The non-magnetic layer 412 can be between the first synthetic antiferromagnetic layer 408 and the second synthetic antiferromagnetic layer 410 as well as between the first synthetic antiferromagnetic layer 408 and the reference layer 406. The MTJ 302 can include a synthetic antiferromagnetic portion 414 which can include the at least one non-magnetic layer 412, the first synthetic antiferromagnetic layer 408, and the second synthetic antiferromagnetic layer 410.

[0051] A duration of the pulse to activate the MTJ 302 can be 1 nanosecond. The duration can be greater than or less than 1 nanosecond. Bit probability p, which can be 50% ± 1.77%, can be single-exponentially sensitive to changes in an energy barrier (Eb) and temperature (T) throughBoltzmann's constant, and can be between approximately 10 and 60. v is the pulse amplitudeV divided by the pulse amplitude that leads to a 50% switching probability in a long pulse duration limit T > ns. T is the pulse duration and TDdepends on MTJ materials and geometry where a typical value is 0.1 nanosecond, nanoseconds.

[0052] FIGS. 5A-5B depict a graph and schematics of a system 500. The system 500 can include electrically coupled MTJs 302 (e.g., MTJ Ising devices) to implement an Ising model. As shown in FIG. 5A, electrically coupled MTJs 302 can couple Ising spins to solve combinatorial optimization problems. The coupled MTJs 302 can be identically initialized (e.g., with a pulse that aligns the magnetization of free layers of the MTJ in a same direction). A desired voltage (VT) bias can be applied to fix inputs or outputs. An actuation pulse 502 can anneal (e.g., iteratively adjusting elements of an array to achieve an optimal configuration) an array of electrically coupled MTJs 302. In situations where the MTJs 302 are not electrically coupled, the actuation pulse 502 can cause an infinite spin temperature with a 50% chance of each spin being +1 (e.g., up) or -1 (e.g., down). Following the actuation pulse 502, a device (e.g., processor 508) can determine if each MTJ 302 in the array is in a parallel or antiparallel magnetization state. Statistical properties of the MTJ 302 states (e.g., +1 or -1) can then be analyzed and the computation can be rerun.

[0053] The system 500 can include the pulse activated MTJ 302. In various implementations, the pulse activated MTJ 302 can be a first pulse activated MTJ 302A and the system 500 can further include a second pulse activated MTJ 302B, a third pulse activated MTJ 302C, and a fourth pulse activated MTJ 302D as shown in FIG. 5A. The first pulse activated MTJ 302A, the second pulse activated MTJ 302B, the third pulse activated MTJ 302C, and the fourth pulse activated MTJ 302D can be electrically coupled together in a circuit in an array of MTJs. The electrical coupling can include ferromagnetic (FM) coupling and antiferromagnetic (AFM) coupling. The system 500 can include a number of transistors, resistors, and amplifiers.

[0054] The system 500 can include the actuation pulse 502 (e.g., first pulse) and a reset pulse 504 (e.g., second pulse). The actuation pulse 502 can actuate the first MTJ 302A, the second MTJ 302B, the third MTJ 302C, and the fourth MTJ 302D. The spin of the first MTJ 302A, the second MTJ 302B, the third MTJ 302C, and the fourth MTJ 302D can then be determined following the actuation pulse 502. The reset pulse 504 can reset the first MTJ 302A, the second MTJ 302B, the third MTJ 302C, and the fourth MTJ 302D to initialize and redetermine the spin of the first MTJ 302A, the second MTJ 302B, the third MTJ 302C, andthe fourth MTJ 302D. The system 500 can repeat the initialization of the first MTJ 302A, the second MTJ 302B, the third MTJ 302C, and the fourth MTJ 302D with the reset pulse 504, apply a desired voltage, apply the actuation pulse 502 to anneal the first MTJ 302A, the second MTJ 302B, the third MTJ 302C, and the fourth MTJ 302D, and determining the spin (e.g., parallel or antiparallel) of the first MTJ 302A, the second MTJ 302B, the third MTJ 302C, and the fourth MTJ 302D. The statistical properties of the spin of the first MTJ 302A, the second MTJ 302B, the third MTJ 302C, and the fourth MTJ 302D can be determined and whether or not to repeat the initialization process (e.g., the reset pulse 504).

[0055] An optimization problem (e.g., combinatorial optimization problem) can be encoded into the system 500. The system 500 can then initialize the array of MTJs (e.g., MTJs 302A-D, herein referred to as MTJs 302) by the actuation pulse 502 to anneal the array of MTJs 302. The final states of each MTJ in the array of MTJs 302 can then be determined and digitally converted into a binary sequence which can encode a solution of the encoded optimization problem.

[0056] As shown in FIG. 5B, the system 500 can include at least one pulse generator 506. The pulse generator 506 can generate and apply the actuation pulse 502 and the reset pulse 504 to the MTJs 302. The pulse generator 506 can be electrically coupled to at least one of the MTJs 302. The pulse generator 506 can be a function generator, an arbitrary waveform generator, a dedicated pulse generator, or any other device capable of generating an electrical pulse, such as a voltage or current pulse. The pulse generator 506 can be programmed or set to define the pulse characteristics, such as the pulse shape, amplitude and duration. The pulse generator 506 can apply the actuation pulse 502 and the reset pulse 504 simultaneously or sequentially to the MTJs 302. The setting of the pulse generator 506 can determine and tune characteristics of the pulse. The characteristics of the pulse can include at least an amplitude and duration of the pulse, and the characteristics can be adjusted and determined by the pulse generator 506. The duration of the pulse can have a range between, for example, 0.1 to 10 nanoseconds, inclusive. The pulse generator 506 can apply a third pulse to determine a magnetic state (e.g., resistance state) of the MTJs 302. In some implementations, the characteristics of the pulse can determine at least a strength of the coupling of the MTJs 302, such as a strength of the at least one of the ferromagnetic or the antiferromagnetic interaction.

[0057] The system 500 can include at least one processor 508. The processor 508 can be electrically coupled to at least one of the pulse generator 506 or the MTJs 302. The processor508 can encode the optimization problem into the system 500, and can determine a magnetic state (e.g., parallel or antiparallel) of the MTJs 302 following application of pulses by the pulse generator 506. Based on at least the magnetic state, the pulse generator 506 can determine a solution to the encoded optimization problem. The processor 508 can determine final states of each MT J in the array of MTJs 302 and digitally convert the finals states into a binary sequence which can encode a solution of the encoded optimization problem.

[0058] In various implementations, the processor 508 can transmit instructions to the pulse generator 506 to generate pulses to apply to the MTJs 302. For example, the processor 508 can instruct the pulse generator 506 to repeatedly apply a sequence of pulses, until the solution determined by the pulse generator 506 converges. For example, the final states of the array of MTJs 302 can differ based on a same optimization problem, and the processor 508 can determine solutions until the solution converges. The sequence of pulses can include the actuation pulse 502, the reset pulse 504, and the third pulse to determine the magnetic state. In various implementations, based on the solution, the processor 508 can tune the characteristics of the pulses generated by the pulse generator 506. The processor 508 can determine the statistical properties of each for the MTJs 302 (e.g., states of the MTJs 302) to determine the solution.

[0059] The variables of a combinatorial optimization problem can be Boolean variables Xi that can be False (x£= 0) or True (x£= 1). An optimization problem can be a Boolean system of Boolean equations that are satisfied together. For example, a Boolean equation can be xx+ x2= a12, where a12is a Boolean variable and + is a binary exclusive OR operation (XOR). The optimization problem can be encoded into a spin model by associating each Boolean variable x£to a spin s£= (— 1)X[and. to Boolean variable a12, a coupling / 12= (— l)ai2. In the spin system (e.g., the system 500), the Boolean equation xx+ x2= a12can be reformulated as / I2S1S2= 1. For a system of m equations and in n variables, the Boolean equation can be generalized to H = — Z' ijJijsisj = ~m, where H is the energy of the interacting spins. A ground state of H is equivalent to finding a maximum number of Boolean equations that can be satisfied simultaneously which solves the optimization problem.

[0060] In a physical device (e.g., the system 500), each spin s can be replaced by an MTJ 302, and each coupling J is replaced by an electrical coupling. The physical device can be then actuated by electrical pulses (e.g., actuation pulse 502, reset pulse 504) to evolve to a ground state of the physical device, corresponding to effectively cooling of the physicaldevice to zero temperature. The solution to the optimization problem can then be obtained by reading (e.g., determining) a state of the MTJs 302 in the ground state. The processor 508 can determine the state of the MTJs 302 in the ground state to determine the solution.

[0061] As shown in FIG. 6, the solution to an optimization problem can be encoded into the ground state of the spin system’s (e.g., the array of MTJs 302) Hamiltonian. An example of the spin system can be the system 500. The spin system can be represented by the array (e.g., network) of electrically connected (e.g., coupled) MTJs 302. The electrical coupling of the MTJs 302 can be represented schematically by links of the array of MTJs 302 which can be denoted by / £j- which can be FM or AFM. An FM coupling of the MTJs 302 can favor alignment of the magnetization directions of free layers (e.g., layers with changeable magnetic orientation) of two MTJs 302 in a same direction (e.g., parallel). An AFM coupling of the MTJs 302 can favor alignment of the magnetization directions of the free layers in opposite directions (e.g., antiparallel). Macrospin, domain, and magnetic-domain can all refer to the spin of the free layer of the MTJ 302.

[0062] The parameters / £j- shown in FIG. 6 model the interactions between pairs of macrospins (e.g., spins of the MTJs 302 in the system 500). Couplings between neighboring spins can be seen in FIG. 6. In general, the couplings may not be restricted to the nearest neighbors (e.g., the first MTJ 302A can be coupled to the third MTJ 302C). Anisotropy is set along the z axis of each spin via a magnetic anisotropy field HAifavoring the North-South direction. After initializing (e.g., by the actuation pulse 502), the macrospins in random directions on the unit sphere and the spin system (e.g., MTJs 302) is cooled from a high temperature regime (e.g., range of temperatures including 400 Kelvin (K), etc.) to a low temperature regime (e.g., range of temperatures including 4 K, etc.). At high temperature, each macrospin experiences large thermal fluctuations with trajectories covering densely a unit sphere (e.g., shown in FIG. 1) of the MTJ 302. For example, the spin of the MTJ 302 can point in any direction in space which can be represented by a point on the unit sphere with a radius of 1. As the system (e.g., the system 500) cools to lower temperature, the fluctuations are reduced, and the spins trajectories tend to confine into small regions of the sphere. The dynamical trajectories of a single macrospin during the annealing process is depicted in FIG. 7. Eventually, the system can reach the end of the annealing schedule and spins stabilize around the north and south poles. The spins’ final states can be binarized based on up or down spin projection direction of the spins to receive a solution to the original problem.

[0063] The system (e.g., the system 500) to implement LLG dynamics can include n single-domain ferromagnets, and apply the macrospin approximation to each domain. The macrospin approximation can be a pool of several quantum spins which can be described by classical equations (e.g., LLG equations). The magnetization of the spins in the ith domain can be averaged to a single magnetic moment with magnitude mi = rrtim^ with magnitude mj > 0 and directionas illustrated in FIG. 6.

[0064] The micro-magnetic energy E of the n macrospins system can have three terms: the energy from the magnetic anisotropy determined by the crystallographic structure of the domain, Eanis, the energy from external magnetic field applied to the system, Eex, and the energy from the exchange interaction between each moments Eex,. Summarizing:E Eext d” Eanis+ Eex. (1)

[0065] The expression for each term is given by:where0is the vacuum permeability, HAicharacterizes the magnetic anisotropy of moment i, and eht is the unit vector characterizing the preferred crystallographic direction of moment i (which later can be taken along the z-axis, seen in FIG. 7). Vector Hext iis an external local magnetic field acting on the ith magnetic moment. Couplings Jtj characterize the exchange interaction between magnetic moments.

[0066] Each magnetic moment mtcan evolve according to the Landau-Lifshitz-Gilbert (LLG) equation (5):where y- = .0Yi, with Yt the gyromagnetic ratio of the ith moment, and aLis the damping constant. Vector leff t is the effective field experienced by the ith magnetic moment, given by equation (6):

[0067] To incorporate thermal fluctuations into the dynamics a thermal field Hth ican be added to

[0068] The thermal fields can be zero mean random variables delta-correlated in time, with correlation given by equations (8-9).where g, v = x, y, z, kBis the Boltzmann constant, and T is temperature. By integrating equation (9) over a small-time interval At, the standard deviation of Hth iis obtained as shown in equation (10):

[0069] In practice, for each component of Hth i, a value from a Gaussian distribution can be drawn with mean = 0 and standard deviation given by (Ji as seen in equation (10). Finally, to integrate the stochastic differential equation, Heun’s integration scheme can be applied.Simulation and Results of Macrospin Systems

[0070] In the systems and methods of the present disclosure, each macrospin i can be assumed to have the same crystallographic structure. All parameters related to the material properties can be the same across all n domains, therefore the index i of the following quantities are omitted: HAi== a, yt= y, y = y', and the crystallographic direction for each spin is set along the z-axis, ehi= z. Each magnetic moment can be assumed to have the same magnitude, mi = m for all n macrospins. This simplification models an array of identical macrospins. This simulation can be further simplified to only consider the evolution of the unit magnetic moment ini. Given the assumptions above, the system can be composed of n identical dynamical bits. At the end of the simulation, a projection on the z-axis, given by Si = sign (mi z), maps the macrospin onto a binary Ising-like spin s£=+l, as illustrated in FIG. 7. The values of the parameters used in the simulation can be seen in FIG. 8. With theparameters seen in FIG. 8, an annealing schedule can be implemented to mimic the physical cooling of the system from ambient temperature down to the low temperature regime. For example, the annealing schedule for running the actuation pulse 502 and the reset pulse 504 in the system 500 can depend on the parameters seen in FIG. 8.

[0071] Ideally, the annealing schedule should be performed slowly from Tamb= 300 / f to T = OK. In practice, an annealing schedule with nT= 30 intermediate temperatures 7 > T2> ••• > TnTdrawn from a distribution P(T) oc 1 / T5, with 6 = 1.5 can be chosen in order to have more values in the low temperature regime.

[0072] Simulation results for the Sherrington-Kirkpatrick (SK) model using the LLG dynamics as described above are described further herein. Glauber dynamics were also run for comparison. The Hamiltonian of the SK model for an n-spin system is given by:where Si=+1 are Ising spins,is the exchange interaction between the ith and jth spins. Couplings Jtj are independent and identically distributed (i.i.d.) random variables drawn from a Gaussian distribution with zero mean and variance <J2= 1. The summation may only count a pair i,j) once. The ground state energy Emin=minsHSK(s) has been calculated analytically in the limit of infinite system size n -> oo [9] and is given by equation (12): lim 0.76321 ... . (12) n— >oo

[0073] The Hamiltonian Hskcan be mapped onto the LLG model by mapping theterms in equation (11) to theterms in equations (4). The Hext iterm in equation (3) is set to zero since there is no external field applied in the SK model. With the anisotropy term favoring each spin to align along the z-axis, the ground state of the SK model can be mapped onto the ground state of the LLG system once the macrospins are binarized using their z- proj ections as illustrated in FIG. 7.

[0074] Thus, the analog LLG Hamiltonian reads:

[0075] The SK model can be solved for several system sizes ranging from n = 40 to n = 2000 using both Glauber and LLG dynamics and the value is recorded of the final energy E obtained with both methods. As shown in FIG. 9A, the method utilizing macrospins performs generally better than algorithmic methods such as Glauber dynamics (e.g., lower energies are consistently found for any value of ri).

[0076] In FIG. 10A the average ground state energy per spin, {E) / n, obtained from Glauber and LLG as a function of system size n is plotted. The curve corresponding to Glauber dynamic is observed to saturate at a higher asymptotic value than the LLG one as n increases. Furthermore, the LLG dynamics results reasonably extrapolate to the exact optimal value of equation (12) as n -> oo, as shown in FIG. 11 A, while the Glauber dynamics intercepts the y axis at a higher sub-optimal asymptotic value.

[0077] As shown in FIG. 9B, the SK model can be solved for several system sizes ranging from n=40 to n=1200 using both Glauber and LLG dynamics using a different form of coupling between spins (e.g., z-z coupling). FIG. 9B shows a variant in which the spin interactions are governed by z-z coupling rather than Heisenberg coupling, as defined in equation (13). In this case, the analog LLG Hamiltonian can be given by:

[0078] In FIG. 10B, an average ground state energy per spin, E) / n obtained from Glauber and LLG dynamics as a function of system size is plotted for the case of z-z coupling. The curve corresponding to Glauber dynamics saturates at a higher asymptotic value than the curve obtained from LLG dynamics. While the LLG results in this case do not extrapolate as closely to the exacted optimal value given in equation (12) as the LLG results do for Heisenberg coupling, the LLG results still outperform Glauber dynamics across all system sizes. As shown in FIG. 1 IB, the LLG data trends toward lower energy values with increasing n, approaching — but not reaching — a theoretical optimum value.

[0079] Beyond computer simulations, the systems and methods of the present disclosure offer the possibility of being realized experimentally in a lab by means of arrays of MTJ devices, where physical dynamics of the MTJ devices are governed precisely by the LLG equations.

[0080] FIG. 12 illustrates a configuration for connecting two MTJs 302 in a circuit such that free layers of the MTJs 302 can be coupled either ferromagnetically or antiferromagnetically. In the configuration shown in FIG. 12, the two MTJs 302 are connected in parallel to a voltage source and a resistor, such that the magnetization state (e.g., parallel or antiparallel) of each MTJ 302 influences the voltage drop and current flow through the other MTJ 302. The interaction of the MTJs 302 can result in an effective electrical coupling between the MTJs 302 that depends primarily on a component of the free-layer magnetizations aligned with a magnetization axis of the reference layers. Assuming the reference layers are aligned along the z-direction, this configuration can give rise to a z-z coupling, as described above. The combined resistance of the two MTJs 302 in parallel can assume four distinct values, corresponding to a relative magnetization directions of the free layers of the MTJs 302. Specifically, the states parallel / parallel (P / P), parallel / antiparallel (P / AP), antiparallel / parallel (AP / P), and antiparallel / antiparallel (AP / AP) yield unique resistance levels that correspond to the four distinct values, which can be mapped to binary configurations 00, 01, 10, and 11, respectively. The mappings can be shown in the histogram of FIG. 12. The difference observed between the 01 and 10 resistance levels can arise from minor variations in the resistances of the individual MTJs 302.

[0081] FIG. 13 illustrates characterization of the transition probabilities between all pairs of the four binary states (e.g., 00, 01, 10, and 11) as a function of the applied voltage pulse amplitude. These measurements can be used to construct a voltage-dependent transition matrix T(V) shown in FIG. 13 where each matrix element represents the probability of transitioning from one configuration to another under a given voltage. The steady-state distribution can be obtained by solving for the eigenvector of T(V) associated with eigenvalue 1. By varying the applied voltage, the steady-state probabilities of the four configurations can be selectively tuned, thereby controlling the effective coupling between the MTJs 302. In particular, the system (e.g., the system 500) can be driven to favor antiferromagnetic or ferromagnetic alignment depending on the applied voltages. For example, by varying the applied voltage, the steady-state distribution can be selectively tuned, enabling the effective coupling between MTJs 302 to vary between antiferromagnetic and ferromagnetic.

[0082] FIG. 14 shows a pulse sequence that can be used to control the effective coupling between MTJs 302. A set of three voltage pulses can be applied simultaneously to both MTJs 302, with pulses labeled 1, 2, and 3. Pulses 1 (e.g., first pulse) and 2 (e.g., second pulse) canbe used to set the transition probabilities through the amplitude and duration of the pulses 1 and 2. The pulse 3 (e.g., third pulse) can serve as a readout (e.g., determining signal, signal) to measure a resulting configuration of the MTJs 302. The histograms in FIG. 14 display the statistical distribution of final states after repeated application of the pulse sequence. The histogram on the lower left can correspond to pulse conditions that favor the states 01 and 10, consistent with antiferromagnetic coupling. In contrast, the histogram on the lower right can show a predominance of the 00 and 11 states, indicating ferromagnetic coupling.

[0083] FIG. 15 is a flow diagram of an example method 1500 for solving optimization problems using MTJs (e.g., MTJs 302). The method 1500, at block 1502, can include encoding an optimization problem into MTJs. The optimization problem can be encoded by one or more processors (e.g., processor 508). At least two of the MTJs can be electrically coupled. The MTJs can be electrically connected in parallel or in series. The MTJs can be electrically coupled by complementary metal oxide semiconductor circuits.

[0084] The method 1500, at block 1504, can include applying, by a pulse generator (e.g., pulse generator 506), a first pulse (e.g., actuation pulse 502) to establish an initial state of the MTJs. A characteristic of the first pulse can determine a strength of the coupling of the plurality of magnetic tunnel junctions, the characteristic including amplitude and duration. The initial state can be a magnetic state of the MTJs.

[0085] The method 1500, at block 1506, can include applying, by the pulse generator, a second pulse (e.g., reset pulse 504) to magnetize the MTJs. Coupling of the MTJs can be determined by an electrical pulse, such as the first pulse or the second pulse. The coupling can include at least one of a ferromagnetic or antiferromagnetic interaction. A strength of the at least one of the ferromagnetic interaction or the antiferromagnetic interaction can be variable. A characteristic of the second pulse can determine a strength of the coupling of the plurality of magnetic tunnel junctions, the characteristics including amplitude and duration. The first pulse and the second pulse can be applied to the MTJs simultaneously or sequentially by the pulse generator.

[0086] In some implementations, the pulse generator can generate at least one of the first pulse or the second pulse according to the optimization problem. The second pulse can encode the MTJs with the optimization problem. At least one of the first pulse or the second pulse can be generated by the pulse generator having a duration between 0.1 nanoseconds and 1microsecond, inclusive, such as 10 nanoseconds. Applying at least one of the first pulse or the second pulse can cause alignment of at least one layer in the plurality of magnetic tunnel junctions. The magnetic state can include a resistance state of the plurality of magnetic tunnel junctions resulting from the alignment of the at least one layer.

[0087] The method 1500, at block 1508, can include determining a magnetic state of each of the MTJs. The magnetic state can be determined by the processor. To determine the magnetic state, the method 1500 can include the pulse generator applying a third pulse to the MTJs. The third pulse can be configured to determine a magnetic configuration of the plurality of magnetic tunnel junction devices following application of the first and second pulses. A magnetization of the MTJs can have a stability factor between 10 and 60, inclusive. The magnetization of the MTJs in equilibrium can align perpendicularly to a plane on which the MTJs are disposed. The MTJs can be configured such that an effective z-z coupling is established between free layers of the MTJs following application of the electrical pulse.

[0088] The method 1500, at block 1510, can include determining whether the magnetic state of the MTJs has converged. Following a first application of the second pulse, the method 1500 at block 1510 can determine to return to block 1504. Following a second application of the second pulse, the processor can determine whether the magnetic state is converged. In response to determining that the magnetic state is not converged, the method 1500 can return to block 1504 to apply the first pulse. In response to determining that the magnetic state is converged, the method 1500 can proceed to block 1512.

[0089] The method 1500, at block 1512, can include determining, based on the magnetic state, a solution to the optimization problem. The processor can determine the solution by determining the statistical property of the MTJs. Statistical properties are can be determined by executing an instruction set as in FIG. 15 multiple times (e.g., repeating some steps of method 1500) and determining the magnetization state of the plurality of magnetic tunnel junction devices each time the instruction set is executed. The statistical property can include the distribution of magnetization states of the plurality of magnetic tunnel junction devices or the distribution of system energy which can be a function of the magnetization states of the plurality of magnetic tunnel junctions. Examples of the such functions can be Eqs. 13 and 14. The one or more processors can determine the statistical property, following application of the second pulse, from combinations of magnetization states of at least two of the MTJs. Thestatistical property can include mappings to binary values 00, 01, 10, and 11 with the generalization to N MTJs in an N bit binary sequence.

[0090] FIG. 16 is a flow diagram of an example method 1600 of controlling coupling between MTJs (e.g., MTJs 302) to solve optimization problems. The method 1600, at block 1602, can include applying at least one pulse to at least two magnetic tunnel junction devices, where the at least two magnetic tunnel junction devices are electrically coupled. The at least one pulse can be a voltage pulse and can include a first pulse and a second pulse. The second pulse can be applied to the at least two magnetic tunnel junction devices after the first pulse or in parallel with the first pulse. The pulses can be applied according to an optimization problem.

[0091] The method 1600, at block 1604, can include tuning an amplitude and a duration of the at least one pulse to facilitate ferromagnetic or antiferromagnetic alignment between magnetizations of free layers of the at least two magnetic tunnel junction devices. Tuning the amplitude and the duration can change the alignment of the magnetizations of the free layers. The method 1600, at block 1606, can include determining a resistance state following alignment of the magnetizations of the free layers. Determining the resistance state can include applying a third pulse to the at least two magnetic tunnel junction devices. The resistance state can be a magnetic state of the magnetic tunnel junction devices, and the resistance state can indicate a solution to the optimization problem.Definitions.

[0092] As used herein, the singular forms “a,” “an,” and “the” include plural referents unless the context clearly dictates otherwise. Thus, for example, the term “a member” is intended to mean a single member or a combination of members, “a material” is intended to mean one or more materials, or a combination thereof.

[0093] As used herein, the terms “about” and “approximately” generally mean plus or minus 10% of the stated value. For example, about 0.5 would include 0.45 and 0.55, about 10 would include 9 to 11, about 1000 would include 900 to 1100.

[0094] It should be noted that the term “exemplary” as used herein to describe various implementations is intended to indicate that such implementations are possible examples, representations, and / or illustrations of possible implementations (and such term is notintended to connote that such implementations are necessarily extraordinary or superlative examples).

[0095] As used herein, the terms “coupled,” “connected,” and the like mean the joining of two additional intermediate members being integrally formed as a single unitary body with one another or with the two members or the two members and any additional intermediate members being attached to one another.

[0096] It is important to note that the construction and arrangement of the various exemplary implementations are illustrative only. Although only a few implementations have been described in detail in this disclosure, those skilled in the art who review this disclosure will readily appreciate that many modifications are possible (e.g., variations in sizes, dimensions, structures, shapes and proportions of the various elements, values of parameters, mounting arrangements, use of materials, colors, orientations, etc.) without materially departing from the novel teachings and advantages of the subject matter described herein. Other substitutions, modifications, changes and omissions may also be made in the design, operating conditions and arrangement of the various exemplary implementations without departing from the scope of the present invention.

[0097] While this specification contains many specific implementation details, these should not be construed as limitations on the scope of any inventions or of what may be claimed, but rather as descriptions of features specific to particular implementations of particular inventions. Certain features described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a subcombination or variation of a subcombination.

Claims

WHAT IS CLAIMED IS:

1. A system, comprising: a plurality of magnetic tunnel junctions, wherein at least two of the plurality of magnetic tunnel junctions are electrically coupled; a pulse generator electrically coupled to the plurality of magnetic tunnel junctions to transmit an electrical pulse to actuate the plurality of magnetic tunnel junctions; and one or more processors configured to analyze a statistical property of each of the plurality of magnetic tunnel junctions.

2. The system of claim 1, wherein the plurality of magnetic tunnel junctions is electrically connected in parallel or in series, wherein coupling of the plurality of magnetic tunnel junctions is determined by the electrical pulse, the coupling comprising at least one of a ferromagnetic or antiferromagnetic interaction.

3. The system of claim 2, wherein a strength of the at least one of the ferromagnetic interaction or the antiferromagnetic interaction is variable.

4. The system of claim 3, wherein a characteristic of the electrical pulse determines the strength of the coupling of the at least two of the plurality of magnetic tunnel junctions, the characteristic comprising amplitude and duration.

5. The system of claim 1, wherein the electrical pulse comprises a plurality of electrical pulses comprising a first electrical pulse and a second electrical pulse applied simultaneously or sequentially to at least one of the plurality of magnetic tunnel junctions.

6. The system of claim 5, wherein the plurality of electrical pulses comprises a third electrical pulse configured to determine a magnetic configuration of the plurality of magnetic tunnel junctions following application of the electrical pulse.

7. The system of claim 1, wherein the plurality of magnetic tunnel junctions is electrically coupled by complementary metal oxide semiconductor circuits.

8. The system of claim 1, wherein the pulse generator generates the electrical pulse according to an optimization problem, the electrical pulse to encode the plurality of magnetic tunnel junctions with the optimization problem.

9. The system of claim 1, wherein the statistical property comprises a magnetization state of the plurality of magnetic tunnel junctions.

10. The system of claim 9, wherein the one or more processors determine the statistical property, following application of the electrical pulse, from combinations of magnetization states of at least two of the plurality of magnetic tunnel junctions.

11. The system of claim 10, wherein the statistical property comprise mappings to binary values 00, 01, 10, and 11.

12. The system of claim 1, wherein the electrical pulse generated by the pulse generator has a duration between 0.1 nanoseconds and 1 microsecond, inclusive.

13. The system of claim 1, wherein a magnetization of the plurality of magnetic tunnel junctions has a stability factor between 10 and 60, inclusive.

14. The system of claim 1, wherein a magnetization of the plurality of magnetic tunnel junctions align perpendicular to a plane on which the plurality of magnetic tunnel junctions are disposed in equilibrium.

15. The system of claim 1, wherein the plurality of magnetic tunnel junctions is configured such that an effective z-z coupling is established between free layers of the plurality of magnetic tunnel junctions following application of the electrical pulse.

16. A method, comprising: encoding, by one or more processors, an optimization problem, into a plurality of magnetic tunnel junctions arranged in an array; applying, by a pulse generator, a first pulse to establish an initial state of the plurality of magnetic tunnel junctions; applying, by the pulse generator, a second pulse to magnetize the plurality of magnetic tunnel junctions; determining, by the one or more processors, a magnetic state of each of the plurality of magnetic tunnel junctions; and determining, by the one or more processors, based on the magnetic state, a solution to the optimization problem.

17. The method of claim 16, wherein applying at least one of the first pulse or the second pulse causes alignment of at least one layer in the plurality of magnetic tunnel junctions, wherein the magnetic state comprises a resistance state of the plurality of magnetic tunnel junctions resulting from the alignment of the at least one layer.

18. A method, comprising: applying at least one pulse to at least two magnetic tunnel junction devices, wherein the at least two magnetic tunnel junction devices are electrically coupled; tuning an amplitude and a duration of the at least one pulse to facilitate ferromagnetic or antiferromagnetic alignment between magnetizations of free layers of the at least two magnetic tunnel junction devices; and determining a resistance state following alignment of the magnetizations of the free layers.

19. The method of claim 18, wherein the at least one pulse is a voltage pulse and comprises a first pulse and a second pulse, the second pulse applied to the at least two magnetic tunnel junction devices after the first pulse or in parallel with the first pulse.

20. The method of claim 19, wherein determining the resistance state comprises applying a third pulse to the at least two magnetic tunnel junction devices.

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