Non-majority MQCA magnetic logic gates and arrays based on misaligned magnetic islands

A non-majority magnetic logic gate device for use in constructing compact and power efficient logical magnetic arrays is presented. The non-majority magnetic logic gate device includes a substrate, symmetrically aligned magnetic islands (SAMIs), at least one misaligned magnetic island (MAMI), magnetic field inputs (MFIs), and at least one magnetic field output (MFO). The SAMIs and MAMI are electrically isolated from each other but are magnetically coupled to one another through their respective magnetic fringe fields. The MAMI is geometrically and/or angularly configured to exhibit a magnetization ground state bias which is dependent upon which direction the applied magnetic clock field is swept. Non-majority logic gates can be made from layouts containing the SAMIs and the MAMI which contain a smaller number of components as comparable majority logic gate layouts.

FIELD OF THE DISCLOSURE

The present disclosure relates to magnetic quantum-dot cellular automata, more particular to non-majority magnetic logic gates and arrays based on misaligned magnetic islands.

BACKGROUND

Nanomagnetic logic (NML), also known as, Magnetic Quantum-Dot Cellular Automata (MQCA) consists of using nanomagnetic islands arranged in such a way that allows logic functions to be performed by using NML circuits. Wires, gates, and inverters have already been demonstrated to function at room temperature. It is estimated that if 1010magnets switch at 108times/second, then the magnets would only dissipate about 0.1 W of power. These nanomagnet based devices can remain non-volatile provided that their size/shape remains above the superparamagnetic limit which means that these nanomagnets devices can be used to realize both logic and memory devices. If non-volatility can be sacrificed, research suggests that binary state in nanomagnets with feature sizes below the superparamagnetic limit should also be stable for around 1 millisecond. This retention time is sufficient to perform logic operations. Device switching times could also be reduced.

The fundamental building blocks for NML circuits can (i) be made with standard lithographic techniques and (ii) have all been experimentally demonstrated at room temperature. Wires that exhibit ferromagnetically ordering (FIG. 1a-c) can be formed by orienting rectangular magnets next to each other so that their magnetic poles are within a commonly shared axis as shown in theFIG. 1a-c. Likewise anti-ferromagnetically coupled bit wires can be formed by orienting rectangular magnets next to each other so that their magnetic poles are parallel to each other and not within a commonly shared axis as shown inFIG. 2a-c.

Because the energy difference between magnetization (binary) states in an NML device can be hundreds of kT at room temperature, an applied magnetic clock is needed to facilitate the re-evaluation of an NML ensemble subsequent to when input states are changed. The applied magnetic clock provides the necessary energy that modulates the barrier between magnetization states so that fringing fields from individual magnets can quickly bias neighboring magnets into their respective thermodynamically favorable magnetization state that corresponds to the logically correct output states associated with the input(s). This reordering of magnetization states is guided by either antiferromagnetic or ferromagnetic coupling which is dependent upon the relative positions of how the particularly adjacent magnets are geometrically arranged.

For example, reordering of magnetization states in an antiferromagnetic coupled horizontal line would proceed as shown in FIG.3-ito FIG.3-iii(where just 3 devices are shown for simplicity). After the field of the left most magnet is externally driven by an Input (not shown) to flip its magnetic state, the applied magnetic clock field (H) is then subsequently imposed on all of the magnets (e.g., in an antiferromagnetically ordered line) which drives the internal magnetic fields (sideways arrows) of the center and right most magnets to be biased along their hard (shorter) axes (as shown in the transition from FIG.3-ito FIG.3-ii). Note that the internal magnetic field (up arrow) of the left most magnet is unaffected by the applied magnetic clock field (H) because it remains biased along its easy (long) axis driven by the continued imposition of the external magnetic field at the Input (not shown). As a result of imposing the applied magnetic clock field which drives the internal magnetic fields of the center and right most magnets pointed towards their hard axes (i.e., nullify), the energetic barriers of the center and right most magnets needed to reach their new energetically favorable magnetization state are considerably lowered. Flux from neighboring magnets can then efficiently bias these magnets into a new magnetically stable state (FIG.3-iii) when the applied magnetic clock field (H) is removed.

It is known that fringing field-based interactions between single domain magnets with nanometer feature sizes can be used as a driving force to perform Boolean logic operations. With NML, logic functionality results from a complex interplay of shape anisotropy and magnet-to-magnet coupling. Magnet shape anisotropy, i.e., an elongated easy axis, creates a bi-stable system, and binary values (1/0) can be arbitrarily assigned to different magnetization directions. For many magnet shapes, the easy axis states are energetically equivalent for a magnet in isolation. When considering magnet ensembles, in clocked systems, fringing fields from individual devices can set the state of a neighboring device when that device is in a metastable logic state. It is known that these fringing field interactions can be used to implement majority voting gates and, in principle, implement any Boolean function.

To date, all known proposals for Boolean logic designs using NML architecture have either been majority gate based or assumed magnets with a uniform shape. Majority gates can be transformed into AND/OR or NAND/NOR gates and can be used to implement any Boolean function (as for AND/OR gates, inversion is possible with an antiferromagnetically ordered wire with an odd number of devices). One way of transforming a majority gate into either a AND/OR or NAND/NOR gate configuration is to permanently fix one of the inputs to a logic 0 or logic 1. Thus, Boolean logic can be realized using majority voting gates by arbitrarily setting one input of a majority gate to a logic ‘0’ or ‘1’, to transform the gate to a two input AND/OR gate. However, reducing a clocked majority gate to a 2-input AND/OR gate is non-trivial. The fixed/held input must be designed such that it does not impede the switching of the compute magnet (e.g., by providing too strong of a bias). If this does happen, a stuck-at fault will ensue as the two other inputs will not be able to drive the gate to a logically correct state.

Some advantages of NML designs include high scalability with ultra-low active power and essentially zero leakage power. NML are also thought to be inherently radiation resistant. To date, known NML designs have utilized elementary symmetrical shapes, i.e., rectangular and ellipsoid devices, have been used for majority gate logic designs.

As depicted inFIGS. 4-5, majority logic gates (MLG) have been used as a basis to demonstrate that magnetic quantum-dot cellular automata (MQCA) can be used to successfully implement various Boolean logic functions. Magnetic logic manipulates spin-polarized electrons in the magnetic material where information can be arbitrarily correlated with either “spin up” or “spin down” electrons. However, not all Boolean functions map well to majority voting gates (i.e. XOR). More specifically, an XOR gate constructed from majority gate-based AND/OR logic will likely require a relatively large footprint.

DETAILED DESCRIPTION

Referring now to the drawings, and in particularFIGS. 1-3,6-16, and21thereof, one envisioned example of the non-majority magnetic logic gate device10, such as those shown inFIGS. 13-14, comprises a substrate20; a plurality of symmetrically aligned magnetic islands (SAMIs30), a misaligned magnetic island (MAMI60), a plurality of magnetic field inputs (MFIs70) and at least one magnetic field output (MFO80). The SAMIs30are disposed on the substrate20so that most of the SAMIs30are disposed either substantially perpendicular or substantially in parallel relative to a sweeping direction of an applied magnetic clock field40and such that the applied magnetic clock field40is applied substantially parallel or perpendicular to a plane50of the substrate20. Most of the SAMIs30have lengths longer than their widths which respectively define easy and hard magnetic axes. Most of the SAMIs30are electrically isolated from each other. Of those SAMIs30that are symmetrically aligned lengthwise side by side next to each other tend towards exhibiting antiferromagnetic coupling with each other. Of those SAMIs30that are symmetrically aligned widthwise side by side next to each other tend towards exhibiting ferromagnetic coupling with each other. The MAMI60is also disposed on the substrate20and the MAMI60also has a length longer than a width that respectively define easy and hard magnetic axes. The MAMI60is electrically isolated from the SAMIs30but the MAMI60is magnetically coupled and sandwiched in between two SAMIs30. The MAMI60is configured to exhibit a magnetization ground state bias which is dependent upon the sweeping direction of the applied magnetic clock field40. The MFIs70are disposed on the substrate20and are magnetically coupled to some of the SAMIs30. The MFO80is disposed on the substrate20and magnetically coupled to the MAMI60.

The clocking of the applied magnetic clock field40may be at any frequency. Some envisioned clocking frequencies of that the applied magnetic clock field40are between 1 Hz to about 1 GHz.

The strength of the applied magnetic clock field40is designed to be sufficiently strong to rotate magnetization moments of the SAMIs30and the MAMI60from the easy axes to the hard axes into a neutral logic state such that fringing fields from the neighbor SAMI/MAMI can set a given device into a logically correct state.

The distances between adjacent SAMIs30and the MAMI60disposed on the substrate20should be designed so that magnetic flux lines are sufficiently strong enough to magnetically influence each other.

Although all of the SAMIs30do not necessarily have to be along the plane50of the substrate20, one envisioned configuration is that all of the SAMIs30are disposed along the plane50of the substrate20.

Although all of the SAMIs30do not necessarily have to be disposed perpendicular or parallel to the sweeping direction of the applied magnetic clock field40, one envisioned configuration is that all of the SAMIs30are disposed either substantially perpendicular or substantially in parallel relative to the sweeping direction of the applied magnetic clock field40.

Although all of the SAMIs30do not necessarily have lengths longer than their respective widths, some SAMIs30may even have circular shapes. One envisioned configuration is that all of the SAMIs30have lengths longer than their respective widths. Another envisioned geometric configuration is that some of the SAMIs30have elongated rectangular shapes.

The MAMI60may be any geometrically and/or angularly misaligned magnetic island (2 or 3 dimensional) configuration in which some of 2 dimensional configurations are depicted inFIG. 6. One envisioned geometric configuration of the MAMI60is that it has a slant edged rectangular shape which is substantially lined up parallel and/or perpendicular to the sweeping direction of the applied magnetic clock field40. Another envisioned configuration is that the MAMI60has an elongated rectangular shape (i.e., symmetrical in geometry) that is aligned so that it is not parallel and not perpendicular (i.e., angularly misaligned) to the sweeping direction of the applied magnetic clock field40.

One mode of the sweeping direction of the applied magnetic clock field40is that it is left to right with respect to the substrate20. Another sweeping direction of the applied magnetic clock field40is right to left with respect to the substrate20.

An envisioned orientation of the applied magnetic clock field40is that it is applied substantially parallel to the plane50of the substrate20. Another envisioned orientation of the applied magnetic clock field40is that it is applied substantially perpendicular to the plane50of the substrate20.

An envisioned distributive arrangement of the SAMIs30is that some of the SAMIs30are aligned to form antiferromagnetic coupled binary wires130. Still another envisioned distributive arrangement of the SAMIs30is that some of the SAMIs30are aligned to form ferromagnetic coupled binary wires140.

An envisioned arrangement of the MAMI60sandwiched between two SAMIs30is that it comprises an OR gate as shown inFIGS. 9,10, and21a-b. Yet another envisioned arrangement of the MAMI60sandwiched between two SAMIs30is that it comprises an AND gate as shown inFIGS. 7,8and21c-d. Still another envisioned arrangement of the MAMI60sandwiched between two SAMIs30is that it comprises a NAND gate as shown inFIG. 11. Still yet another envisioned arrangement of the MAMI60sandwiched between two SAMIs30is that it comprises a NOR gate as shown inFIG. 12. Accordingly the device10may comprise any number of logic functions which can include AND, OR, NAND, NOR and even comprise a logical memory like a bit of MRAM.

A envisioned arrangement of the MAMI60sandwiched between two SAMIs30is that they are all together substantially aligned perpendicular to the sweeping direction of the applied magnetic clock field40. Still another envisioned arrangement of the MAMI60sandwiched between two SAMIs30is that they are all together aligned substantially in parallel with respect to the sweeping direction of the applied magnetic clock field40.

An envisioned example of the device10is that the MAMIs60are magnetically coupled and sandwiched between pairs of SAMIs30to form an XOR gate as shown inFIG. 15.

The non-majority magnetic logic gate device10may also further comprise a plurality of MAMIs60in which the MAMIs60are magnetically coupled and sandwiched between corresponding respective pairs of SAMIs30.

The non-majority magnetic logic gate device10may also further comprise a plurality of MFOs80magnetically coupled to the MAMIs60. It is important to note that the MFOs80do not necessarily have to be immediately adjacent to the MAMIs60. That is either antiferromagnetic or ferromagnetic coupled binary wire140s(as shown inFIGS. 1-3and13-14can be used to magnetically couple any of the MAMIs60to their respectively coupled MFOs80.

The non-majority magnetic logic gate device10may also further comprise an XOR gate which is composed of three adjacent SAMIs30aligned linearly together and one MAMI60. The three linearly aligned together adjacent SAMIs30are magnetically coupled to each other and the one MAMI60is magnetically coupled to a middle one of the three adjacent SAMIs30.

The non-majority magnetic logic gate device10may also further comprise an applied magnetic clock field40circuit120configured to produce and to sweep the applied magnetic clock field40.

The non-majority magnetic logic gate device10may also further comprise a dielectric layer90, a cladding layer100, a fill layer110, and an applied magnetic clock field40circuit120. It is preferable that the dielectric layer90is disposed on the substrate20and the cladding layer100is on the dielectric layer90. It is also preferable that the fill layer110on the cladding layer100and applied magnetic clock field40circuit120buried in the fill layer110.

Another envisioned example of the present invention is that it comprises a non-majority magnetic logic gate device10. The non-majority magnetic logic gate device10of this example comprises a substrate20, a plurality of SAMIs30, a MAMI60, a plurality of MFIs70and a MFO80. The plurality of SAMIs30are disposed on the substrate20such that most of the SAMIs30are disposed either substantially perpendicular or substantially in parallel relative to a sweeping direction of an applied magnetic clock field40so that the applied magnetic clock field40is applied substantially parallel or perpendicular to a plane50of the substrate20. Most of the SAMIs30have lengths longer than widths that respectively define easy and hard magnetic axes and that most of the SAMIs30are electrically isolated from each other. Of those SAMIs30that are symmetrically aligned lengthwise side by side next to each other tend towards exhibiting antiferromagnetic coupling with each other. Of those SAMIs30that are symmetrically aligned widthwise side by side next to each other tend towards exhibiting ferromagnetic coupling with each other. Three adjacent SAMIs30aligned linearly together are magnetically coupled to each other. The MAMI60is also disposed on the substrate20in which the MAMI60is configured to have a length longer than a width that respectively define easy and hard magnetic axes. Accordingly, the MAMI60is configured to exhibit a magnetization ground state bias which is dependent upon the sweeping direction of the applied magnetic clock field40and that the MAMI60is electrically isolated from the SAMIs30. The MAMI60is also disposed next to the three adjacent SAMIs30so that the MAMI60is magnetically coupled to only a middle one of the three adjacent SAMIs30in which the three adjacent SAMIs30and the MAMI60comprise an XOR gate. The MFIs70are also disposed on the substrate20and magnetically coupled to some of the SAMIs30. The MFO80is also disposed on the substrate20and magnetically coupled to the MAMI60.

Yet another envisioned example is that present invention can comprise a magnetic logic gate array150. The magnetic logic gate array150comprises a substrate20; a plurality of non-majority magnetic logic gate devices10, MFIs70and MFOs80. The non-majority magnetic logic gate devices10are disposed on the substrate20in which the devices10comprises SAMIs30and MAMIs60. The SAMIs30are disposed on the substrate20in which most of the SAMIs30are disposed either substantially perpendicular or substantially in parallel relative to a sweeping direction of an applied magnetic clock field40such that the applied magnetic clock field40is applied substantially parallel or perpendicular to a plane50of the substrate20. Most of the SAMIs30having lengths longer than widths that respectively define easy and hard magnetic axes and that most of the SAMIs30are electrically isolated from each other. Of those SAMIs30that are symmetrically aligned lengthwise side by side next to each other tend towards exhibiting antiferromagnetic coupling with each other. Of those SAMIs30that are symmetrically aligned widthwise side by side next to each other tend towards exhibiting ferromagnetic coupling with each other. The MAMIs60are also disposed on the substrate20in which the MAMIs60have lengths longer than widths that respectively define easy and hard magnetic axes and that the MAMIs60are electrically isolated from the SAMIs30. Most MAMIs60are magnetically coupled and sandwiched in between corresponding pairs of adjacent SAMIs30. The MAMI60is configured, i.e., geometrically and/or angularly misaligned, to exhibit a magnetization ground state bias which is dependent upon the sweeping direction of the applied magnetic clock field40. The MFIs70are disposed on the substrate20and magnetically coupled to some of the SAMIs30. The MFOs80are disposed on the substrate20and magnetically coupled to the MAMIs60.

The following simulated examples were generated using the Object Oriented Micro-Magnetic Framework (OOMMF) developed by the National Institute of Standards and Technology (NIST). The OOMMF uses a Landau-Lifshitz ODE solver to relax 3D spins on a 2D mesh of square cells. Unless otherwise note, all of the following simulations assume magnets made from supermalloy (79% Ni, 15% Fe, and 5% Mo) which is a soft magnetic material with a uniaxial anisotropy constant of essentially zero. Thus there is no easy/hard axis associated with the magnetic material itself and any easy/hard axis is only defined by the magnet shape. Unless otherwise noted, a saturation magnetization of 8.0×105A/m and an exchange stiffness constant of 1.05×10−11J/m are used to model supermalloy. It is also assumed that the damping coefficient is 0.1, instead of 0.5, which corresponds more to experimental results. A stopping criteria, i.e., dm/dt, was chosen instead of a time cutoff. Each OOMMF simulation time step step was only considered to be complete when the maximum change in magnetization per unit time (across all spins associated with a given circuit element) fell below 1 degree/ns. A stopping criteria, i.e., dm/dt, was chosen instead of a time cutoff. Finally, unless otherwise noted, all simulated magnets were set to have a footprint of 50×75×25 nm3.

Simulation Example 1

Asymmetry and Magnetization Behavior

Simulations of magnetic properties of magnets having “slanted” or “cut” edges as illustrated in the inserts inFIG. 17. In the simulations three slanted magnets initially have a strong x-component of magnetization (e.g., they are biased along their hard/short axes). If no field is applied to keep a given device in this metastable state, and the magnetic material is polycrystalline (e.g., permalloy or supermalloy), then the devices should relax into a magnetization state determined by its easy/long axis. In clocked lines of magnets, the fringing fields from a neighboring device will ideally determine the sign of the final magnetization state. In the presence of no applied bias, what state a magnet with a rounded rectangular shape might eventually relax into would essentially be random.

However, as magnetic moments tend to align along a magnet's edge, a slanted edge can give a device a envisioned y-component of magnetization. Both the position of the slant and the initial x-component (direction) of magnetization both dictate what state a device will ultimately relax to. This effect is captured quantitatively inFIG. 17where we considered three magnets with slanted edges. Slant edge magnets v1 and v2 have a 50×75×25 nm3footprints and v3 has a 40×60×20 nm3footprint. Each slant edge magnet device was initialized with both a positive and negative x-component of magnetization and was allowed to relax with no external clocking field or Hy bias applied. The placement of the slant and the direction of the initial x-component of the magnetization consistently lead to a envisioned y-component of magnetization of each of the slant edge magnet devices (e.g., ↑ or ↓ states).

Simulation Example 2

Asymmetry and Magnetization Behavior

If a magnet with a slanted edge on its upper left initially has a strong positive y-component of magnetization and an external field is applied from left-to-right along its hard axis and then removed, there is no My state change. If the initial y-component of magnetization is negative and the same field is applied even with no Hy bias, there is an My state change. InFIG. 19this phenomenon is seen via OOMMF simulation of supermalloy magnets with various sizes and shapes. Slant position and direction of the Hx field/initial Mx state determine the final My state.

The My state transition can be explained by plotting a device's demagnetizing energy, i.e., the internal energy that opposes the direction of magnetization, as a function of angle of magnetizationFIG. 18. Each peak of the asymmetrical (i.e., slant) magnets is not centered at zero degrees as compared to the case for a symmetric rounded rectangular magnet. Rather each peak of the asymmetrical magnets is shown shifted to the left. This explains the My sign change inFIG. 18. The maximum magnetization energy is at an angle below horizontal. If the applied Hx field causes a device to move past this angle, even if some initial y-component of magnetization is retained, when Hx is removed, the devices relax such that My is positive.

In our second set of simulations, we again considered a magnet in isolation with a slanted edge in isolation (specifically slant edge magnet v1 ofFIG. 17). As in the first simulation example the slant edge magnet was initially magnetized such that its y-component of magnetization was equal to the saturation magnetization (Ms) of supermalloy (↑). We then applied an external field along the magnet's hard axis (in the positive x-direction) that increased in magnitude from 0 A/m to 120,000 A/m in 800 A/m increments. The field was then removed in a similar fashion. These simulations results are illustrated inFIG. 19. Note that this device always retains some of its initial y-component of magnetization state (↑). Next, the slant edge magnet was in an initial state such that My was negative (↓). Again, we applied a field along the device's hard axis, from left-to right, then gradually increased in magnitude. Given these initial conditions, when the external field (Hx) reaches 70,000 A/m, the position of the slant and the direction of the applied field include a transition from a down state (↓) to an up state (↑) even with no Hy bias is applied to the device.

Simulation Example 3

Asymmetry and Magnetization Behavior

Magnetic state changes (e.g., from ↑ or ↓) can also be facilitated with combinations of hard axis and fringing fields as well. The clocking fields (Hx) place each magnet in an ensemble into a state such that it can be switched into a new, logically correct state by its neighbor's fringing fields (i.e., Hy).

This effect is depicted quantitatively inFIG. 20where a rounded rectangular and a slanted edge magnet were considered. Each simulated magnet was set to have a 50×75 nm2footprint and a constant 46,000 A/m bias was applied along a mangnet's hard axis in each case. For each shape, a magnet can transition from a ↑ or ↓ state with a significantly lower Hy bias. However, while the M-H curve for the rounded rectangle device is symmetric (the same bias is required to facilitate an ↑ to ↓ and ↓ to ↑ transition), the M-H curve for the magnet with a slanted edge is asymmetric. Given the direction of the applied field (Hx) and the device's initial state, a stronger field is required to make the magnet transition to a state against that suggested by the position of its slant, e.g., 12,334 A/m field is required for a ↑ to ↓ transition while a field of 15,915 A/m is required for a ↑ to ↓ transition. Still, while a 30% larger Hy bias is needed, a state transition “against” the slant and direction is still possible.

Simulation Example 4

Asymmetry and Logic

FIG. 21adepicts a 3 magnet ensemble which implements a logic OR function. If i1and i2have the same My state, output magnet ‘o’ will see a net Hy bias that induces ferromagnetic ordering in i1, o, and i2. If the magnet is sufficiently nulled, fringing fields from the two inputs will also influence the output magnet's My state. The bias needed to facilitate a state transition for ferromagnetic ordering (in the presence of a given Hx field) can be determined via an M-H curve generated via OOMMF simulation (SeeFIG. 20). While the M-H curve for a magnet with a slanted edge is asymmetric, if the two input magnets could provide the higher Hy bias, they can facilitate the state transitions required for the top and bottom input combinations in the truth table ofFIG. 21b. As shown inFIG. 21c-d, a 2-input AND can be realized if the slanted edge is on the bottom left.

In an actual circuit, the Hy biases will not be externally applied but instead will come from neighboring devices. For example, referring back toFIG. 21a, the fringing fields from i1and i2might generate the Hy bias that ultimately sets the state of the slanted output magnet. With this configuration, there are four combinations of magnetization states that and i2could represent (SeeFIG. 21b). Based on the simulation results, if the “target” magnet with the slanted edge were initially biased such that Mx=Ms, and the input magnets were in opposite magnetization states, the target magnet would be expected to settle into a ↑ state—as the fringing fields from the two inputs effectively cancel. Similarly, if the target magnet is sufficiently biased along its hard axis, and the combined Hy biasing fields from i1and i2are sufficiently strong, then they are expected to be able to set the magnetization state My of the slant-edge target. If we equate a state to a binary 0 and a ↑ state to a binary 1, a magnet with a slanted edge could implicitly implement the logic OR function. Note that if the slanted edge were on the bottom left as shown inFIG. 21c, a logic AND function would result (FIG. 21d).

The average Hy bias produced by a 50×75×25 nm3supermalloy magnet even 15 nm away from a potential target is ˜25,000 μm. Thus, even if both input magnets inFIG. 21ahave the same magnetization state,FIG. 20suggests that the fringing fields from the magnets alone will be insufficient to switch the state of the target (as the MH curves there also include a 46,000 A/m Hx bias). This suggests that we will also need to leverage an external/clock field applied along a target magnet's hard axis in order to facilitate any potential state change given the biases that 2 inputs might actually provide at some reasonable distance away. We are particularly interested in facilitating a state transition with the smallest external field possible as larger currents would be needed to generate larger fields per the mechanism in FIG.14—which will only increase system energy demands.

We consider a circuit structure like that illustrated inFIG. 21a. The target magnet was initialized to a state opposite of that suggested by each of the four possible input combinations. (There are four possible state transitions.) In each instance, a field was applied parallel to the target magnet's hard axis. Our objective was to measure the magnitude of the external field required to facilitate a state transition. Three of the four simulation results are reported inFIG. 22. (The ↑↓↑ to ↑↑↑ case is the “easiest” and is now shown to improve graph readability.) As one can see, higher external fields are required for the case where inputs are in opposite magnetization states. (This makes sense as there is essentially no Hy helper bias.) However, the simulations where inputs with opposite magnetization states are not symmetric—e.g., a greater external field is required if the top input is ↑ and the bottom input is ↓ than if the bottom input is ↑ and the top input is ↓.

This can be explained quantitatively by considering the structure illustrated inFIG. 21when the line is in a ferromagnetically ordered state (e.g., all devices are ↓). If we measure the flux density 2 nm away from the top and bottom of the magnet with the slanted edge, the average flux density is approximately 13% higher between o and i2than between o and i1(See Table 1). In essence, the position of the slant on the top of the target leads to weaker coupling between i1and o than between i2and o. Thus, when the bottom magnet is in an ↑ state, and the top magnet is in a ↓ state, not only does the bottom magnet exert more control over the target, but it is also pushing the target to a logically correct state. For the opposite input combination, a greater external field is required as we essentially need to overcome the effects of a small “anti-bias” from the bottom magnet in a ↓ state.

Simulation Example 5

Asymmetry and Logic

As magnetic field strength is a function of distance, if allowable by a fabrication process, one way to ensure that both inputs have equal control over a slanted target is to change the distance between i1and o as depicted inFIG. 21a. This effect is illustrated quantitatively inFIG. 23where the ↑↓↓ to ↑↑↓ in case is again considered. Like the simulation results summarized inFIG. 22, we simulated an applied local field along the target magnet's hard axis. However, in this simulation, the spacing between i1and o was 10 nm instead of 15 nm. (The spacing between i2and o remained at 15 nm.) As seen inFIG. 23, the target magnet now changes state with a lower magnitude field (Hx). Similarly, from Table 1, in a ferromagnetically ordered line, the flux density between the top and bottom inputs and the target is essentially equal. Again, if enabled by a given fabrication process, these results suggest that asymmetric placement of the two inputs could allow for lower field and hence lower energy operations as the previous “worst case” is mitigated.

Finally, it is noted that realistically, an external field will not simply be applied to just a slanted target but rather the external field will be applied to all magnets in the ensemble. As such i1and i2will become weaker drivers as the clocking field will increase each magnet's x-component of magnetization. As also seen inFIG. 23, the net effect of this is that a higher external field is required to facilitate a state transition in symmetric ↑↓↓ to ↑↑↓ case.

Simulation Example 6

Fringing Field Magnitude

One important design parameter is magnet “drive strength”, e.g., how much of a bias will it produce on the neighboring device that it is supposed to drive. To determine which shape configuration produces a stronger Hy bias, we consider each configuration in both states via simulation and measure the field produced on an equivalently sized footprint 15 nm away (assuming a magnet will drive a neighbor to its right). As seen in Table II the “slant on the left” configuration is an approximately 18% stronger driver.

Simulation Example 7

Another important design parameter is the magnitude of the external field required to facilitate a state transition. As such, we also studied which slant placement makes it easier to put a device into a metastable state as suggested by the input combinations that require state transitions. Again, four different configurations are considered: (i) a target that is initially in a state dictated by the slant and the direction of the applied switching field and where the inputs would suggest a state transition against the envisioned direction of the slant, and (ii-iv) a target that is initially in a state against that dictated by the slant and the direction of the applied switching field and where we cycle through all input combinations to put it into a correct state. (Here, the “hard cases” occur when the fringing fields from each input cancel and the slant determines the state transition.)

For each of these configurations we considered a 3 magnet line terminated by the block. The first magnet in the line was slanted. Local fields were applied over the target/slanted magnet to mimic new input drivers and the external field applied to this system was increased from 0 until all of the magnets in the line switched into the logically correct state suggested by the applied local fields. We studied the state of the last magnet in the line as a function of the applied clock—as this captures proper switching behavior of the line. Again, we found that the “slant left” placement allows for each simulation to transition into a logically correct state with the lowest overall external field. (With the “slant right” configuration, there is less coupling between the input magnet (a MAMI) and its right neighbor (a SAMI) which makes it more difficult for the other magnets in the line to transition through a metastable state. It is worth noting that if inputs are asymmetric, larger external fields are required. This maximum external field would then become a system design parameter as it would ensure that all the lines will transition correctly for all input combinations. Given these results, a gate with a slant on the left should require lower fields to transition to a neutral state, and can be a stronger driver.

Simulation Example 8

Shape Gate v Majority Gate

A two input XOR gate is used as a vehicle to discuss how shape-based logic can impact system level performance. The implicit majority voting function associated with magnetic logic will not enable a more efficient two input XOR gate. Majority gates must be reduced to AND/OR gates to implement the Boolean function A′B+B′A. A schematic of what a gate might look like if it were to be constructed with nanomagnets appears inFIG. 15. (A majority gate design appears inFIG. 15aand a shape-based design appears inFIG. 15b.). The shape based design reduces extraneous interconnection which can in turn reduce gate delay by approximately 25% and reduce the gate footprint by almost 60%. As automata-like local interconnect and slower magnet switching times can only degrade performance, a shape based logic approach can only improve system level performance. Additionally, shape based logic gates appear to be controllable with the same fields necessary to control antiferromagnetic order bit lines of similarly sized magnets (good from the standpoint of system-level energy).

Those of ordinary skill in the art will appreciate that the apparatus and methods described herein are susceptible to variations and modifications other than those specifically described. It is to be understood that the invention includes all such variations and modification which fall within its spirit and scope.