Abstract:
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.

Description:
CROSS-REFERENCE TO RELATED PATENT APPLICATION 
       [0001]    This application claims the benefit of U.S. Provisional Applications 61/268,803 and 61/243,246, which were respectively filed on Jun. 16, 2009, and on Sep. 17, 2009, and which are incorporated herein in their entirety by reference. 
     
    
     GOVERNMENT INTERESTS 
       [0002]    This invention was funded, at least in part, with a government grant from the National Institute of Standards and Technology (NIST) pursuant to contract number 70NANB7H6168. This invention was also funded, at least in part, with a government support under Grant No. CCF-0621990 entitled “Nano: Applications, Architectures, and Circuit Design for Nano-scale Magnetic Logic Devices” awarded by the National Science Foundation (NSF). Accordingly, the United States government may therefore have certain rights in this invention. 
     
    
     FIELD OF THE DISCLOSURE 
       [0003]    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 
       [0004]    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 10 10  magnets switch at 10 8  times/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. 
         [0005]    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. 1   a - 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 the  FIG. 1   a - 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 in  FIG. 2   a - c.    
         [0006]    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. 
         [0007]    For example, reordering of magnetization states in an antiferromagnetic coupled horizontal line would proceed as shown in FIG.  3 - i  to 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 - i  to 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. 
         [0008]    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. 
         [0009]    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. 
         [0010]    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. 
         [0011]    As depicted in  FIGS. 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. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0012]    The invention will be better understood and aspects other than those set forth above will become apparent when consideration is given to the following detailed description thereof. Such description makes reference to the annexed drawings wherein: 
           [0013]      FIGS. 1A-C  are a Scanning Electron Micrograph (SEM) Image, Magnetic Force Microscopy (MFM) Image and an arbitary magnetic field vector corresponding to a ferromagnetically coupled bit line; 
           [0014]      FIGS. 2A-C  are a SEM Image, MFM Image and an arbitary magnetic field vector corresponding to an antiferromagnetically coupled bit line; 
           [0015]      FIGS. 3   i - iii  is a stylistic portrayal of a magnetic state reordering along an antiferromagnetically coupled bit line; 
           [0016]      FIG. 4  depicts one majority gate based AND logic gate layout showing four operational magnetic vector combinations and their respective outcome; 
           [0017]      FIG. 5  depicts one majority gate based OR logic gate layout showing four operational magnetic vector combinations and their respective outcome and their respective outcome; 
           [0018]      FIG. 6  illustrates some of the geometric and angular misalignment configurations through to be possible for MAMI of the present invention; 
           [0019]      FIG. 7  is one non-majority based AND logic gate layout and sweeping scenario showing four operational magnetic vector combinations and their respective outcome of the present invention; 
           [0020]      FIG. 8  depicts four possible layouts and sweeping scenarios forming non-majority based AND logic gates of the present invention; 
           [0021]      FIG. 9  is one non-majority based OR logic gate layout of the present invention showing all four operational magnetic vector combinations and their respective outcome; 
           [0022]      FIG. 10  depicts four layouts and sweeping scenarios forming non-majority based OR logic gates of the present invention; 
           [0023]      FIG. 11  depicts four layouts and sweeping scenarios forming non-majority based NAND logic gates of the present invention; 
           [0024]      FIG. 12  depicts four layouts and sweeping scenarios forming shaped based NOR logic gates of the present invention; 
           [0025]      FIG. 13  is a perspective view of a magnetic logic gate array of the present invention; 
           [0026]      FIG. 14  is a perspective view of a non-majority magnetic logic gate device of the present invention; 
           [0027]      FIG. 15   a  is an XOR logic gate configuration using majority gates; 
           [0028]      FIG. 15   b  is an XOR logic gate configuration using non-majority logic gate device the present invention; 
           [0029]      FIG. 16  is another XOR logic gate configuration using non-majority logic gate device of the present invention; 
           [0030]      FIG. 17  is Magnetization (My) as a function of time for slant edged MAMIs where the initial magnetization was originally along the x-axis and at the saturation magnetization; 
           [0031]      FIG. 18  is a demagnetizing energy profile as a function of magnetization angle for two MAMI asymmetric magnets (S) and a symmetric rounded rectangular magnet (RR); 
           [0032]      FIG. 19  is a magnetization profile along a y-axis (My) of a MAMI magnet as a function of applied magnetic field along the x axis (Hx) that depicts asymmetric behavior; 
           [0033]      FIG. 20  is M-H curves for symmetrical and asymmetrical magnets; 
           [0034]      FIG. 21   a  is a non-majority OR gate of the present invention; 
           [0035]      FIG. 21   b  is a truth table for the non-majority OR gate of the present invention; 
           [0036]      FIG. 21   c  is a non-majority AND gate of the present invention; 
           [0037]      FIG. 21   d  is a truth table for the non-majority AND gate of the present invention; 
           [0038]      FIG. 22  is a magnetization profile along a y-axis (My) of a MAMI magnet as a function of applied magnetic field along the x axis (Hx) showing different threshold energetics for three different transitions in a non-majority gate; and 
           [0039]      FIG. 23  is a magnetization profile along a y-axis (My) of a MAMI magnet as a function of applied magnetic field along the x axis (Hx) showing different threshold energetics for the same transition a non-majority gate as a function of Hx with or without a helper cell. 
       
    
    
       [0040]    The same reference numerals refer to the same parts throughout the various figures. 
       DETAILED DESCRIPTION 
       [0041]    Referring now to the drawings, and in particular  FIGS. 1-3 ,  6 - 16 , and  21  thereof, one envisioned example of the non-majority magnetic logic gate device  10 , such as those shown in  FIGS. 13-14 , comprises a substrate  20 ; a plurality of symmetrically aligned magnetic islands (SAMIs  30 ), a misaligned magnetic island (MAMI  60 ), a plurality of magnetic field inputs (MFIs  70 ) and at least one magnetic field output (MFO  80 ). The SAMIs  30  are disposed on the substrate  20  so that most of the SAMIs  30  are disposed either substantially perpendicular or substantially in parallel relative to a sweeping direction of an applied magnetic clock field  40  and such that the applied magnetic clock field  40  is applied substantially parallel or perpendicular to a plane  50  of the substrate  20 . Most of the SAMIs  30  have lengths longer than their widths which respectively define easy and hard magnetic axes. Most of the SAMIs  30  are electrically isolated from each other. Of those SAMIs  30  that are symmetrically aligned lengthwise side by side next to each other tend towards exhibiting antiferromagnetic coupling with each other. Of those SAMIs  30  that are symmetrically aligned widthwise side by side next to each other tend towards exhibiting ferromagnetic coupling with each other. The MAMI  60  is also disposed on the substrate  20  and the MAMI  60  also has a length longer than a width that respectively define easy and hard magnetic axes. The MAMI  60  is electrically isolated from the SAMIs  30  but the MAMI  60  is magnetically coupled and sandwiched in between two SAMIs  30 . The MAMI  60  is configured to exhibit a magnetization ground state bias which is dependent upon the sweeping direction of the applied magnetic clock field  40 . The MFIs  70  are disposed on the substrate  20  and are magnetically coupled to some of the SAMIs  30 . The MFO  80  is disposed on the substrate  20  and magnetically coupled to the MAMI  60 . 
         [0042]    The clocking of the applied magnetic clock field  40  may be at any frequency. Some envisioned clocking frequencies of that the applied magnetic clock field  40  are between 1 Hz to about 1 GHz. 
         [0043]    The strength of the applied magnetic clock field  40  is designed to be sufficiently strong to rotate magnetization moments of the SAMIs  30  and the MAMI  60  from 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. 
         [0044]    The distances between adjacent SAMIs  30  and the MAMI  60  disposed on the substrate  20  should be designed so that magnetic flux lines are sufficiently strong enough to magnetically influence each other. 
         [0045]    Although all of the SAMIs  30  do not necessarily have to be along the plane  50  of the substrate  20 , one envisioned configuration is that all of the SAMIs  30  are disposed along the plane  50  of the substrate  20 . 
         [0046]    Although all of the SAMIs  30  do not necessarily have to be disposed perpendicular or parallel to the sweeping direction of the applied magnetic clock field  40 , one envisioned configuration is that all of the SAMIs  30  are disposed either substantially perpendicular or substantially in parallel relative to the sweeping direction of the applied magnetic clock field  40 . 
         [0047]    Although all of the SAMIs  30  do not necessarily have lengths longer than their respective widths, some SAMIs  30  may even have circular shapes. One envisioned configuration is that all of the SAMIs  30  have lengths longer than their respective widths. Another envisioned geometric configuration is that some of the SAMIs  30  have elongated rectangular shapes. 
         [0048]    The MAMI  60  may be any geometrically and/or angularly misaligned magnetic island (2 or 3 dimensional) configuration in which some of 2 dimensional configurations are depicted in  FIG. 6 . One envisioned geometric configuration of the MAMI  60  is 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 field  40 . Another envisioned configuration is that the MAMI  60  has 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 field  40 . 
         [0049]    One mode of the sweeping direction of the applied magnetic clock field  40  is that it is left to right with respect to the substrate  20 . Another sweeping direction of the applied magnetic clock field  40  is right to left with respect to the substrate  20 . 
         [0050]    An envisioned orientation of the applied magnetic clock field  40  is that it is applied substantially parallel to the plane  50  of the substrate  20 . Another envisioned orientation of the applied magnetic clock field  40  is that it is applied substantially perpendicular to the plane  50  of the substrate  20 . 
         [0051]    An envisioned distributive arrangement of the SAMIs  30  is that some of the SAMIs  30  are aligned to form antiferromagnetic coupled binary wires  130 . Still another envisioned distributive arrangement of the SAMIs  30  is that some of the SAMIs  30  are aligned to form ferromagnetic coupled binary wires  140 . 
         [0052]    An envisioned arrangement of the MAMI  60  sandwiched between two SAMIs  30  is that it comprises an OR gate as shown in  FIGS. 9 ,  10 , and  21   a - b . Yet another envisioned arrangement of the MAMI  60  sandwiched between two SAMIs  30  is that it comprises an AND gate as shown in  FIGS. 7 ,  8  and  21   c - d . Still another envisioned arrangement of the MAMI  60  sandwiched between two SAMIs  30  is that it comprises a NAND gate as shown in  FIG. 11 . Still yet another envisioned arrangement of the MAMI  60  sandwiched between two SAMIs  30  is that it comprises a NOR gate as shown in  FIG. 12 . Accordingly the device  10  may comprise any number of logic functions which can include AND, OR, NAND, NOR and even comprise a logical memory like a bit of MRAM. 
         [0053]    A envisioned arrangement of the MAMI  60  sandwiched between two SAMIs  30  is that they are all together substantially aligned perpendicular to the sweeping direction of the applied magnetic clock field  40 . Still another envisioned arrangement of the MAMI  60  sandwiched between two SAMIs  30  is that they are all together aligned substantially in parallel with respect to the sweeping direction of the applied magnetic clock field  40 . 
         [0054]    An envisioned example of the device  10  is that the MAMIs  60  are magnetically coupled and sandwiched between pairs of SAMIs  30  to form an XOR gate as shown in  FIG. 15 . 
         [0055]    The compositional makeup of the SAMIs  30  and MAMIs  60  may be made of any known material so long as they exhibit magnetic properties. For example, it is envisioned that the SAMIs  30  and the MAMI  60  can be composed of magnetic material selected from the group consisting of a rare earth metal, ferrites, hipernom, hipernik, HyMu-80, monimax, Mo-permalloy, nilomag, remalloy, sanbold, supermumetal, ultraperm, vicalloy, 78 Permalloy, permalloy, supermalloy, AlNiCo, AlSiFe (sendust), Co, CoFe, CoFeB, CoFeV (supermendur), CoFeCr (hiperco), CoPt, CoZrTa, CoFe (permendur), CuZn, FeN, FeO, FeAlN, FeTaN, NiFe (permalloy), NiFeMo supermalloy, NiFe supermalloy, NiFeCuCr (mumetal), NiFeCo, MnZn, and combinations thereof. 
         [0056]    The non-majority magnetic logic gate device  10  may also further comprise a plurality of MAMIs  60  in which the MAMIs  60  are magnetically coupled and sandwiched between corresponding respective pairs of SAMIs  30 . 
         [0057]    The non-majority magnetic logic gate device  10  may also further comprise a plurality of MFOs  80  magnetically coupled to the MAMIs  60 . It is important to note that the MFOs  80  do not necessarily have to be immediately adjacent to the MAMIs  60 . That is either antiferromagnetic or ferromagnetic coupled binary wire  140   s  (as shown in  FIGS. 1-3  and  13 - 14  can be used to magnetically couple any of the MAMIs  60  to their respectively coupled MFOs  80 . 
         [0058]    The non-majority magnetic logic gate device  10  may also further comprise an XOR gate which is composed of three adjacent SAMIs  30  aligned linearly together and one MAMI  60 . The three linearly aligned together adjacent SAMIs  30  are magnetically coupled to each other and the one MAMI  60  is magnetically coupled to a middle one of the three adjacent SAMIs  30 . 
         [0059]    The non-majority magnetic logic gate device  10  may also further comprise an applied magnetic clock field  40  circuit  120  configured to produce and to sweep the applied magnetic clock field  40 . 
         [0060]    The non-majority magnetic logic gate device  10  may also further comprise a dielectric layer  90 , a cladding layer  100 , a fill layer  110 , and an applied magnetic clock field  40  circuit  120 . It is preferable that the dielectric layer  90  is disposed on the substrate  20  and the cladding layer  100  is on the dielectric layer  90 . It is also preferable that the fill layer  110  on the cladding layer  100  and applied magnetic clock field  40  circuit  120  buried in the fill layer  110 . 
         [0061]    Another envisioned example of the present invention is that it comprises a non-majority magnetic logic gate device  10 . The non-majority magnetic logic gate device  10  of this example comprises a substrate  20 , a plurality of SAMIs  30 , a MAMI  60 , a plurality of MFIs  70  and a MFO  80 . The plurality of SAMIs  30  are disposed on the substrate  20  such that most of the SAMIs  30  are disposed either substantially perpendicular or substantially in parallel relative to a sweeping direction of an applied magnetic clock field  40  so that the applied magnetic clock field  40  is applied substantially parallel or perpendicular to a plane  50  of the substrate  20 . Most of the SAMIs  30  have lengths longer than widths that respectively define easy and hard magnetic axes and that most of the SAMIs  30  are electrically isolated from each other. Of those SAMIs  30  that are symmetrically aligned lengthwise side by side next to each other tend towards exhibiting antiferromagnetic coupling with each other. Of those SAMIs  30  that are symmetrically aligned widthwise side by side next to each other tend towards exhibiting ferromagnetic coupling with each other. Three adjacent SAMIs  30  aligned linearly together are magnetically coupled to each other. The MAMI  60  is also disposed on the substrate  20  in which the MAMI  60  is configured to have a length longer than a width that respectively define easy and hard magnetic axes. Accordingly, the MAMI  60  is configured to exhibit a magnetization ground state bias which is dependent upon the sweeping direction of the applied magnetic clock field  40  and that the MAMI  60  is electrically isolated from the SAMIs  30 . The MAMI  60  is also disposed next to the three adjacent SAMIs  30  so that the MAMI  60  is magnetically coupled to only a middle one of the three adjacent SAMIs  30  in which the three adjacent SAMIs  30  and the MAMI  60  comprise an XOR gate. The MFIs  70  are also disposed on the substrate  20  and magnetically coupled to some of the SAMIs  30 . The MFO  80  is also disposed on the substrate  20  and magnetically coupled to the MAMI  60 . 
         [0062]    Yet another envisioned example is that present invention can comprise a magnetic logic gate array  150 . The magnetic logic gate array  150  comprises a substrate  20 ; a plurality of non-majority magnetic logic gate devices  10 , MFIs  70  and MFOs  80 . The non-majority magnetic logic gate devices  10  are disposed on the substrate  20  in which the devices  10  comprises SAMIs  30  and MAMIs  60 . The SAMIs  30  are disposed on the substrate  20  in which most of the SAMIs  30  are disposed either substantially perpendicular or substantially in parallel relative to a sweeping direction of an applied magnetic clock field  40  such that the applied magnetic clock field  40  is applied substantially parallel or perpendicular to a plane  50  of the substrate  20 . Most of the SAMIs  30  having lengths longer than widths that respectively define easy and hard magnetic axes and that most of the SAMIs  30  are electrically isolated from each other. Of those SAMIs  30  that are symmetrically aligned lengthwise side by side next to each other tend towards exhibiting antiferromagnetic coupling with each other. Of those SAMIs  30  that are symmetrically aligned widthwise side by side next to each other tend towards exhibiting ferromagnetic coupling with each other. The MAMIs  60  are also disposed on the substrate  20  in which the MAMIs  60  have lengths longer than widths that respectively define easy and hard magnetic axes and that the MAMIs  60  are electrically isolated from the SAMIs  30 . Most MAMIs  60  are magnetically coupled and sandwiched in between corresponding pairs of adjacent SAMIs  30 . The MAMI  60  is 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 field  40 . The MFIs  70  are disposed on the substrate  20  and magnetically coupled to some of the SAMIs  30 . The MFOs  80  are disposed on the substrate  20  and magnetically coupled to the MAMIs  60 . 
         [0063]    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 8.0×10 5  A/m and an exchange stiffness constant of 1.05×10 −11  J/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 nm 3 . 
       Simulation Example 1 
     Asymmetry and Magnetization Behavior 
       [0064]    Simulations of magnetic properties of magnets having “slanted” or “cut” edges as illustrated in the inserts in  FIG. 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. 
         [0065]    However, as magnetic moments tend to align along a magnet&#39;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 in  FIG. 17  where we considered three magnets with slanted edges. Slant edge magnets v1 and v2 have a 50×75×25 nm 3  footprints and v3 has a 40×60×20 nm 3  footprint. 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 
       [0066]    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. In  FIG. 19  this 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. 
         [0067]    The My state transition can be explained by plotting a device&#39;s demagnetizing energy, i.e., the internal energy that opposes the direction of magnetization, as a function of angle of magnetization  FIG. 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 in  FIG. 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. 
         [0068]    In our second set of simulations, we again considered a magnet in isolation with a slanted edge in isolation (specifically slant edge magnet v1 of  FIG. 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&#39;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 in  FIG. 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&#39;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 
       [0069]    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&#39;s fringing fields (i.e., Hy). 
         [0070]    This effect is depicted quantitatively in  FIG. 20  where a rounded rectangular and a slanted edge magnet were considered. Each simulated magnet was set to have a 50×75 nm 2  footprint and a constant  46 , 000  A/m bias was applied along a mangnet&#39;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&#39;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 
       [0071]      FIG. 21   a  depicts a 3 magnet ensemble which implements a logic OR function. If i 1  and i 2  have the same My state, output magnet ‘o’ will see a net Hy bias that induces ferromagnetic ordering in i 1 , o, and i 2 . If the magnet is sufficiently nulled, fringing fields from the two inputs will also influence the output magnet&#39;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 (See  FIG. 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 of  FIG. 21   b . As shown in  FIG. 21   c - d , a 2-input AND can be realized if the slanted edge is on the bottom left. 
         [0072]    In an actual circuit, the Hy biases will not be externally applied but instead will come from neighboring devices. For example, referring back to  FIG. 21   a , the fringing fields from i 1  and i 2  might 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 i 2  could represent (See  FIG. 21   b ). 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 i 1  and i 2  are 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 in  FIG. 21   c , a logic AND function would result ( FIG. 21   d ). 
         [0073]    The average Hy bias produced by a 50×75×25 nm 3  supermalloy magnet even 15 nm away from a potential target is ˜25,000 μm. Thus, even if both input magnets in  FIG. 21   a  have the same magnetization state,  FIG. 20  suggests 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&#39;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. 
         [0074]    We consider a circuit structure like that illustrated in  FIG. 21   a . 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&#39;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 in  FIG. 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 ↓. 
         [0075]    This can be explained quantitatively by considering the structure illustrated in  FIG. 21  when 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 i 2  than between o and i 1  (See Table 1). In essence, the position of the slant on the top of the target leads to weaker coupling between i 1  and o than between i 2  and 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. 
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Effect of By on target magnet with different spacings between 
               
               
                 input and target with slanted edges 
               
             
          
           
               
                   
                   
                 Distance 
                 Avg By 
               
               
                   
                 Case 
                 (nm) 
                 (mT) 
               
               
                   
                   
               
               
                   
                 Target - Bottom 
                 15 
                 −416 
               
               
                   
                 Target - Top 
                 15 
                 −336 
               
               
                   
                 Target - Top 
                 10 
                 −417 
               
               
                   
                 Target - Bottom 
                 15 
                 −413 
               
               
                   
                   
               
             
          
         
       
     
       Simulation Example 5 
     Asymmetry and Logic 
       [0076]    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 i 1  and o as depicted in  FIG. 21   a . This effect is illustrated quantitatively in  FIG. 23  where the ↑↓↓ to ↑↑↓ in case is again considered. Like the simulation results summarized in  FIG. 22 , we simulated an applied local field along the target magnet&#39;s hard axis. However, in this simulation, the spacing between i 1  and o was 10 nm instead of 15 nm. (The spacing between i 2  and o remained at 15 nm.) As seen in  FIG. 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. 
         [0077]    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 i 1  and i 2  will become weaker drivers as the clocking field will increase each magnet&#39;s x-component of magnetization. As also seen in  FIG. 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 
       [0078]    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. 
         [0000]    
       
         
               
             
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE II 
               
             
             
               
                   
               
               
                 Hy bias produced by 50 × 75 × 25 nm 3   
               
               
                 supermalloy target magnet 15 nm away 
               
             
          
           
               
                   
                   
                   
                 Avg. Hy bias 
               
               
                   
                 Magnet 
                 State 
                 (A/m) 
               
               
                   
                   
               
             
          
           
               
                   
                 Slant left 
                 up 
                 −17866 
               
               
                   
                 Slant left 
                 down 
                 17866 
               
               
                   
                 Slant right 
                 up 
                 −15133 
               
               
                   
                 Slant right 
                 down 
                 15133 
               
               
                   
                   
               
             
          
         
       
     
       Simulation Example 7 
     Clocking 
       [0079]    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.) 
         [0080]    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 
       [0081]    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 in  FIG. 15 . (A majority gate design appears in  FIG. 15   a  and a shape-based design appears in  FIG. 15   b .). 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). 
         [0082]    As to the manner of usage and operation of the present invention, the same should be apparent from the above description. Accordingly, no further discussion relating to the manner of usage and operation will be provided. 
         [0083]    While an envisioned example of the non-majority magnetic logic gate device has been described in detail, it should be apparent that modifications and variations thereto are possible, all of which fall within the true spirit and scope of the invention. With respect to the above description then, it is to be realized that the optimum dimensional relationships for the parts of the invention, to include variations in size, materials, shape, form, function and manner of operation, assembly and use, are deemed readily apparent and obvious to one skilled in the art, and all equivalent relationships to those illustrated in the drawings and described in the specification are intended to be encompassed by the present disclosure. 
         [0084]    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. 
         [0085]    Therefore, the foregoing is considered as illustrative only of the principles of the invention. Further, since numerous modifications and changes will readily occur to those of ordinary skill in the art, it is not desired to limit the invention to the exact construction and operation shown and described, and accordingly, all suitable modifications and equivalents may be resorted to, falling within the scope of the invention.