Non-boolean Computing With Nanomagnetics for Computer Vision Applications

Supplementary Material forNon-boolean computing with nanomagnets for computer vision ap-plications

Sanjukta Bhanja, D. K. Karunaratne, Ravi Panchumarthy, Srinath Rajaram, Sudeep Sarkar

This PDF file includes:

Supplementary text

1. Simulation experiments for the design of magnetic system.

2. Derivation and development of Virtual Vortex Model.

3. Derivation and development of magnetic Hamiltonian.

4. Layout for Nanomagnetic Disks.

5. Mechanism to Deselect the Cells.

6. Speed Comparison with State-Of-The-Art.

7. Fabrication Process.

References.

1

Non-Boolean computing with nanomagnets for computer vision applications

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NNANO.2015.245

NATURE NANOTECHNOLOGY | www.nature.com/naturenanotechnology 1

2015 Macmillan Publishers Limited. All rights reserved

1 Simulation Experiments for the Design of Magnetic System

Researchers have studied several magnetization states in permalloy nano-disks. Depending on the disksize, many meta-stable magnetization states such as single domain state, C-state and vortex states havebeen observed. Cowburn et al. 1 have fabricated isolated nanomagnetic disks of diameters ranging from55 nm to 500 nm and thickness ranging from 6 nm to 15 nm, and have experimentally identified a phasediagram between vortex state and single domain state. In 26 the authors have reported a phase diagramto show the dependence of the different meta-stable states with respect to disk diameters and thicknesses.These phase diagrams indicate that the magnetization state of a nanomagnetic disk is dependent on the di-mension of the disks. We have chosen the dimensions for the nanomagnetic disk near this phase boundaryin the vortex state region to study the dependence of the dipolar coupling energy between the nanomagneticdisks and the magnetization states of the nanomagnetic disks. In our study, we found that a nanomagneticdisk can exist either in the single domain state or in the vortex state depending on the coupling energy be-tween the neighbouring magnets. When the magnets are closer to each other the coupling energy betweenthe nanomagnetic disks would hold the magnetization state of the nanomagnetic disks in single domainstate and if they are separated far apart, the exchange energy, anisotropy energy and demagnetizationenergy is dominant and the nanomagnetic disk would settle to vortex state.

We have used LLG v2.46 micromagnetic simulation suite 7 to observe the coupling energy depen-dence on the magnetization states of nanomagnetic disks with diameter of 110 nm and thickness of 11 nm,by varying the centre-to-centre distance between the nanomagnetic disks from 110 nm to 360 nm in stepsof 5 nm. Table 1 shows material properties and experimental parameters used in the simulation model.As initial condition, we applied an external magnetic field in the in-plane direction, such that the magneticspins in the nanomagnetic disk would align with the external magnetic field resulting in a single domainstate. Once the external magnetic field is removed the magnetic spins would relax to its energy minimum.This external magnetic field can be in the form of stray fields from the neighbouring nanomagnetic disks.If the magnetic stray fields are stronger, the nanomagnetic disk would remain in a single domain state. Ifthe magnetic stray fields are weak, the magnetization of the nanomagnetic disk would settle to vortex state.For each separation, the magnetization states, the magnetic coupling energies and the magnetization statevectors were extracted for every disk.

Supplementary Fig. 1 shows the plot of the measured magnetic coupling energy with respect to thecentre-to-centre distance between the two nanomagnetic disks. It is evident from the plot that the magneticcoupling energy is high when the centre-to-centre distance is small. We have also observed that, whenthe centre-to-centre distance was small the magnetization states of nanomagnetic disks were in singledomain state. As the centre-to-centre distance increases the magnetic coupling energy falls and beyond250 nm, where the magnetic coupling energy is close to zero, the nanomagnetic disk settles to a vortexstate. Analysing this data we can conclude that when both the nanomagnetic disks are in single domainstate the coupling energy has a high value and when both the nanomagnetic disks are in vortex state thecoupling energy has minimal value. Using this kind of LLG simulation, we built a 2 state (vortex, singledomain) coupling energy model that approximates the LLG results. See equation 17 for a look ahead.

2


0.00.20.40.60.81.01.21.41.61.82.0

1.0 1.5 2.0 2.5 3.0

Pairw

ise

coup

ling

ener

gies

(in

J)

Disk Seperation from centre-to-centre (in nm)

LLG Simulation Data

Supplementary Figure 1: Pairwise coupling energy between disks by varying the centre-to-centre spacing for diskswith diameter 110 nm and thickness 11 nm using LLG simulations.

Supplementary Table 1: Material properties and experimental parameters used in simulation model

Parameters Description Value

Shape Shape of MTJ circular

D Diameter 110nm

FM1 Pinned layer material Co/Pd

NM Barrier layer material MgO

FM2 Free layer material NiFe

tfl Thickness of free layer 100A

tbl Thickness of barrier layer 35A

tpl Thickness of pinned layer 40A

Nx, Ny, Nz Unit element size 3x3x3nm3

Damping coefficient of free layer 0.015

T Temperature 300K

Ms Saturation magnetization of free layer 800.0e3A/m

3


a b c

Supplementary Figure 2: Conceptual idea of the virtual vortex model. The vector field is a cross section of a X-Yplane. The vector D represents the magnetization state in a nanomagnetic disk. (a) Single domain state. (b) C-state.(c) Vortex state.

2 Virtual Vortex Model

We first need a way to mathematically characterize the distribution pattern of magnetic vectors on a mag-netic disk using a small number of parameters, ideally one. For this, we propose the virtual vortex model.In our experiments, we observed single domain and vortex states as two stable magnetization states as ournanomagnetic disks dimension lie in the phase boundary. However, larger circular nanomagnetic disks canexhibit various configurations. Virtual vortex model is an effort to develop a comprehensive magnetizationrepresentation spanning single domain, C-state and vortex states in the same framework. In the singledomain state the magnetic spins in the nanomagnetic disk are aligned in-plane direction. Whereas, in thevortex state, the magnetic spins have a curling configuration around the disk centre.

In the virtual vortex model, the magnetization of a nanomagnetic disk is represented with a virtualvector field. The field has a curling formation around its virtual vortex centre. As the vectors in the fieldapproaches closer to the virtual vortex centre they gradually align from in-plane direction to out-of-planedirection. The state of any given magnetic disk can be approximated by a circular piece of this virtual vortexmodel. See supplementary Fig. 2 for an illustration. A magnetic disk in vortex state will be represented bya disk that aligns with the virtual vortex and disks with single domain arrangements will be at the peripheryof the virtual vortex model. Intermediate states, if any, can be represented by other locations in the plane.The disk centre could be between the vortex centre and at a point that is at infinite distance. If the diskcentre was at an infinite distance the vector field in the nanomagnetic disk would have a unidirectional in-plane configuration, representing the single domain state. Whereas if the disk centre was on the vortexcore, the vector field in the nanomagnetic disk would have a curling configuration around the nanomagnetsdisk centre. If the disk centre is at a point in-between the disk centre and an infinite point, the vector fieldwould have a C-state configuration. supplementary Fig. 2 illustrates the concept of the virtual vortex modelincorporating possible magnetization states (single domain state, C-state, vortex state).

4


DP

m x

y

n

mi

vi

0,0,01

a

bMY

KZ

R

X

r

Supplementary Figure 3: Vector diagram of a magnetic element m1 and vortex centre v1 of the magnetic vector M.The black dot represents the magnetic element and the red point represents the vortex center.

In the virtual vortex model, the nanomagnetic disk is segmented into magnetic elements and themagnetization of each magnetic element is represented with a single vector (magnetic vector - M = mi i +mj j + mkk ). We have assumed that as long as the size and the material of the magnetic elements in ananomagnetic disk are similar, the magnitude of the magnetization will remain constant but the direction ofthe magnetic vector will vary along the normal plane to the line segment connecting the magnetic elementand the vortex centre. We have modelled a magnetic vector of a magnetic element that is at an infinitedistance away from its vortex centre to have an in-plane direction (the k -component of the magnetic vectorwill be zero). As the magnetic element gets closer to the vortex centre the k -component of the magneticvector exponentially increases. The magnetic element at vortex center will only have k -component in themagnetic vector (i -component and j -component are zero).

The vector diagram in supplementary Fig. 3 represents the vector notations used for and to derivethe virtual vortex model. The circle in grey in supplementary Fig. 3 signifies a nanomagnetic disk with itscentre at point (a, b, 0) and a radius of r. The black dot on the circle signifies a magnetic element, mi, atthe point (x, y, 0) and it is represented with the vector R. The vector M represents the magnetization of themagnetic element. The vector M has its vortex centre, vi, at the point (m,n, 0) and it is represented withthe vector P. The vector from the vortex centre vi to the magnetic element mi is represented by the vectorD. The vector K starts at point (0, 0, 0) and is a unit vector in the Z-axis direction.

The vectors in supplementary Fig. 3 are expressed as:

5


R = xi + yj + 0k (1)

P = mi + nj + 0k (2)

D = RP (3)

K = 0i + 0j + k (4)

The magnetic vector M is expressed as:

M =

[pK+ q

(KD|K||D|

)](5)

where is based on the size and magnetic material of the magnetic element mi and p2 + q2 = 1.

We have modelled the amplitude p of the magnetic vector M to be an inverse exponential to themagnitude of vector D. The amplitude p is then expressed as:

p = e|D| (6)

where is the distance along the vectorD and it is a constant. To keep the magnitude of the magneticvector M constant, we have reduced the amplitude of the vector

(KD|K||D|

)by factor of q. The amplitude q is

modelled as:

q =

1 e2|D| (7)

Substituting the values of p and q to equation 5, we have expressed the magnetic vector M as:

6


M =

(e|D| K+ (

(1 e2|D| )) KD| K||D|

)(8)

It is evident from equation 8 that the vector M is only dependent on the vector D. Therefore, we canpredict the magnetization state of a nanomagnetic disk with the magnitude and the direction of vector D. (Avector can be expressed as a combination of its magnitude and the angle it makes with the X-axis.) If themagnitude of the vector D is |D| and the angle it makes with the X-axis is , then the magnetic vector M isdependent on |D| and and could be expressed as:

M(|D| , ) = (e|D| K+ (

(1 e2|D| )) KD| K||D|

)(9)

We have used the concept of the virtual vortex model to build a magnetic Hamiltonian for the magneticsystem proposed in this paper.

3 Development of Magnetic Hamiltonian

The magnetic coprocessor is designed to be a 2-dimensional grid with NxN nanomagnetic disks. Thesenanomagnetic disks are fabricated in critical dimensions such that they tend to be in single domain statewhen strongly coupled with neighbouring nanomagnetic disks and in vortex state when weakly coupled withneighbouring nanomagnetic disks. Hence, we abstract the magnetization state variable S whose magnitudecan be 0 for vortex state and 1 for single domain state. Magnetic pattern in any given magnetic disk is firstquantified using the virtual vortex model (the parameter D), which is then mapped into vortex and singledomain states. Note the single domain state has direction, so we need a vector of magnitude one torepresent it instead of just a scalar state. The magnetic Hamiltonian is designed and developed to estimatethe energy of the magnetic system based on the magnetization state variable (S) and the centre-to-centredistance (rij) between the nanomagnetic disks. The development of the magnetic Hamiltonian, depictedin supplementary Fig. 4, has 2 components: (1) Magnetization state abstraction model, (2) Internal energyand coupling energy approximation.

3.1 Magnetization State Abstraction Model (S)

We represent the magnitude of the magnetization state variable, |S|, either as 0 or 1, where 0represents a vortex state; 1 represent a single domain state. Note the direction of the variable S capturesthe direction of the single domain state. This model is developed using LLG simulation data from section 1and virtual vortex model from section 2.

The simulation experiments provided 64 instances of nanomagnetic disks with different magnetizationstates. We extracted the individual internal magnetic energies and their magnetization vectors for all the

7


Coupling Energy Computation

Input: Magnetization spin vector configurations of 60 nano-magnetic disks extracted from LLG simulations

Process: Compute coupling energies between all pairs nano-magnetic disks at different center-to-center distances using Dipole coupling energy equation

Output: Coupling energies between all combinations of nano-magnetic disk (64C2 * 32 = 64,512 energies)

|D| Computation

Input: Magnetization spin vector configurations of 60 nano-magnetic disks extracted from LLG simulations

Process: Compute |D| using virtual vortex model

Output: Magnetization state representation of 64 nano-magnetic disks with |D| (Range: 0 to infinity)

LLG Simulation Experiments: Magnetization spin configuration vectors

LLG Simulation Experiments: Individual nano-magnetic disk energies

Magnetization State Representation (D to S)

Input: Individual nano-magnetic disk energies extracted from LLG simulations and |D| values computed earlier

Process: Convert the |D| values into either 0 or 1, by finding relationship between disk energies and |D| values

Output: Magnetization state representation of 64 nano-magnetic disks with |S| (Either 0 or 1)

Individual Disk Energy Approximation

Input: Individual nano-magnetic disk energies extracted from LLG simulations and |S| equation

Process: Estimate by numerical approximation the individual disk energy in terms of |S|

Output: Individual disk energy approximation

Coupling Energy Approximation

Input: Coupling energies computed earlier and |S| equation

Process: Estimate by numerical approximation the coupling energy in terms of |S| and center-to-center distance

Output: Coupling energy approximation

Supplementary Figure 4: Flow chart of the several steps involved in the development of the magnetic Hamiltonian forthis computer vision problem.

8


nanomagnetic disks. The LLG micromagnetic simulation segments the nanomagnetic disk into elementsand calculates the magnetic energy and the direction of the magnetic moment for each element. Themagnitude of the magnetization is a constant value for all elements with the same dimensions and material.The direction of the magnetization is a variable represented with a unit vector. supplementary Fig. 5 showstwo such magnetizations of nanomagnetic disks. We analyzed the individual internal magnetic energies andobserved that the nanomagnetic disks in single domain state had much higher energy values than in vortexstate. This observation agrees well with our design principles of the nanomagnetic disk in supplementarysection 1.

Based on the virtual vortex model in section 2, we could represent the magnetization of nanomagneticdisk with the vector pointing from the vortex core to the disk centre (vector D in equation 8). We havecalculated the magnitude (|D|) and the direction () of the vector D for all the nanomagnetic disks. Thegraph in supplementary Fig. 6 represents the relationship between internal magnetic energy and the |D|value of the nanomagnetic disks extracted from the LLG simulation experiments. The observations fromLLG simulation experiments show that the internal magnetic energies of individual nanomagnetic disks insingle domain state have much higher energy values compared to vortex state. Similarly it is evident fromthe graph in supplementary Fig. 6 that when the nanomagnetic disk has a single domain state its internalmagnetic energy is higher and in-turn has large |D| values whereas in the vortex state the internal magneticenergy is lower and has small |D| values. The red curve in supplementary Fig. 6 represents the numericalapproximation that best fits the internal magnetic energy values. The values of |D| were derived valuesusing the virtual vortex model. Based on the simulations presented in the supplementary material section1, each nanomagnetic disk vector representation at energy minimum are extracted and using the virtualvortex model the values of |D| are derived. The missing data points from about 5.5 < |D| < 4.8 arebecause there are no corresponding energy minimum nanomagnetic disk representations obtained duringour simulations. The magnitude of the magnetization state variable |S| is a step function based on the baseten logarithmic of the |D| value. This can be expressed as:

|Si| =0, log10|Di| < 1, log10|Di| (10)

where |Di| is the magnitude of the vector pointing from the vortex core to the disk centre of the ithnanomagnetic disk and = 5.1.

3.2 Internal Energy and Coupling Energy Approximation

To find the total energy of a magnetic system in terms of state vector (S) and the distance between thenanomagnetic disks (rij), we require the coupling energy of all pairs of nanomagnetic disks in the system.

We used the internal magnetic energies extracted from the LLG simulation experiments and by nu-

9


b!a!

Disk Dimensions in cm Disk Dimensions in cm

Dis

k D

imen

sion

s in

cm

Dis

k D

imen

sion

s in

cm

Supplementary Figure 5: Examples of magnetization states. (a) Vortex state. (b) Single domain state.

9 8 7 6 5 4 3 2 1 00

0.5

1

1.5

2

2.5

3

3.5

4

Distance (|D| in cm) in Logarithmic scale from the virtual vortex core to the disk centre

Indiv

idual

Disk

Ene

rgies

(in

J)

Individual Disk Energies (in J)Step Function Fit

1 1.5 2 2.5 3 3.5x 107

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8 x 107

Supplementary Figure 6: Internal magnetic energy of a nanomagnetic disk with respect to its |D| value.

10


merical approximation, we have proposed a model to predict the internal magnetic energy of ith nanomag-netic disk when the magnetization state (Si) is known. The magnitude of the magnetization state |Si| iseither 0 or 1. The model can be expressed as:

Ei = |Si|+ (11)

where = 2.5 107 Joules and = 6.4 108 Joules

The LLG simulation experiments in section 1 provided coupling energies only between two single do-main nanomagnetic disks or between two vortex state nanomagnetic disks. But for a magnetic system withtwo or more nanomagnetic disks, we used the dipole energy equation 8,9 to calculate the coupling energiesbetween all possible combinations of single domain state and vortex state configurations of nanomagneticdisks and approximated the coupling energy in terms of the state representation Si and Sj.

The simulation experiments provided with vector field representations of the magnetizations of 64nanomagnetic disks. We selected all possible combinations of two vector fields, placed their disk centrescollinearly and calculated the magnetic coupling energy (E12) between the 1st and 2nd nanomagnetic diskusing the following dipole energy equation:

E12 =

(1

N1N2

) N1i=1

N2j=1

(mi mj 3(mi nij)(mj nij))

r3ij(12)

where mi is a unit vector in the 1st nanomagnetic disk and mj is a unit vector in the 2nd nanomagneticdisk. rij is the distance in meters between mi and mj. nij is the unit vector along the direction that connectsmi and mj. N1 is total number of unit vectors in the 1st nanomagnetic disk and N2 is the total number of unitvectors in the 2nd nanomagnetic disk. is a constant with units of Joules per cubic meter. We calculated themagnetic coupling energy between two nanomagnetic disks for 32 different separations ranging from 110nm to 320 nm. At each of the 32 separations, 64C2 coupling energies were calculated. The total number ofmagnetic coupling energies calculated are 64, 512.

We have used these magnetic coupling energies as ground truth and proposed the magnetic couplingenergy between two nanomagnetic disks in terms of the centre-to-centre distance (rij), state representation(S) and the direction of the magnetization of the nanomagnetic disk. The numerical approximation is in theform of exy. We have used this numerical approximation and proposed a model that is in the quadraticform to predict the magnetic coupling energy (E12) and is expressed as:

E12 = er12 |S1||S2|cos(1 2) = er12S1 S2 (13)

11


where r12 is the centre-to-centre distance between the 1st and the 2nd nanomagnetic disks. S1and S2 are the state values of the corresponding |D| values for the 1st and the 2nd nanomagnetic disksrespectively. Similarly 1 and 2 are the directions of the vector D for the 1st and the 2nd nanomagneticdisks respectively. and takes the values of 3.7 106Joules and 2.4 105cm1 .

3.3 Total Magnetic Energy in the Magnetic System

The total magnetic energy in the magnetic system can be calculated from the summation of all themagnetic coupling energies between each other and summation of the internal magnetic energy of all thenanomagnetic disks. The total magnetic energy of the magnetic system with N nanomagnetic disks can beexpressed as:

Etotal =Ni=1

Nj=1+1

Eij +Ni=1

Ei (14)

where Eij is the magnetic coupling energy between the ith and jth nanomagnetic disk and Ei is theinternal magnetic energy of the ith nanomagnetic disk.

Eij = erijSi Sj (15)

Ei = |Si|+ (16)

Etotal = Ni=1

Nj=i+1

erijSi Sj + Ni=1

|Si|+N (17)

3.4 Validation

In order to verify the magnetic Hamiltonian expressed in equation 17, we used the data from LLGsimulation experiments. The magnetic Hamiltonian expressed in equation 17 is in the terms of the mag-netization state representation (S) and hence we need to verify the magnetization states produced by theequation 17 are the same as the magnetization states produced in the simulation experiments. The simu-lation experiments provided the magnetization field vectors at energy minimum at 32 different separationsbetween two nanomagnetic disks. We calculated the magnetization state representation (S) of each nano-magnet at each of the 32 separations using equation 3.1.

12


Next, we calculated the energy between two nanomagnetic disks with all the possible magnetizationconfigurations (64C2) at each of the 32 separations as in the LLG simulation experiments. At each separationwe picked the pair of nanomagnetic disks with minimum energy and calculated their state representations(Si). We compared these with state representations obtained from simulation experiments at each of the 32separations. Except at separations 105 nm and 110 nm, all the remaining states matched., i.e., for exampleif the simulation experiments indicate a single domain state at a particular separation, our calculations forstate representation (S) also resulted in a single domain state.

4 Layout for Nanomagnetic Disks

The objective is to find the 2D placement coordinates of nanomagnets where each of them representing anedge segment and such that the coupling energy between two magnets i and j, is proportional to pairwiseedge affinity aij . We used an approach based on multidimensional scaling (MDS) 10. Let a matrix r beconstructed out of given weight such that: rij = 1log(aij) , zero diagonal values. We desire to find the 2Dcoordinate of each magnet, represented by the vector xi. Let the matrix of these coordinates for eachmagnet be X = [x1, ,xn]. The distance between the i-th and j-th coordinates should be proportional torij . In other words,

(xi xj)T (xi xj) = c rij . (18)

or equivalently

xTi xi 2xTi xj + xTj xj = c rij . (19)

These expressions involving pairwise distances can be consolidated and can be mathematically ex-pressed, as shown in 10, in the form:

XTX = c12HrH, (20)

where H = (I 1N 11T ) is referred to as the centering operator, with I as the identity matrix and 1 asthe vector of ones.

These coordinates X can be arrived at by classical MDS scheme 10. The solution is based onthe singular value decomposition of the centred distance matrix 12HrH = VV

T , where V, are theeigenvectors and eigenvalues respectively. Assuming that centred distance matrix represents the innerproduct distances of a Euclidean distance matrix, the coordinates are given by

13


a b

c

d e

f

Edges/Edges 5 8 20 26 28 29 43 44 47 53

CALC

ULAT

ED A

FFIN

ITY

VALU

ES

BETW

EEN

EDGE

S (a

ij)

CALCULATED AFFINITY VALUES BETWEEN EDGES (aij)

5 0.00 1.56 0.39 5.70 0.57 6.91 0.00 0.02 0.41 0.138 0.00 0.00 0.00 0.02 0.02 0.00 0.00 0.00 0.01

20 0.00 4.50 1.67 4.24 0.10 0.02 1.66 1.2226 0.00 0.96 34.08 0.45 0.13 7.46 5.0528 0.00 0.77 0.01 0.01 0.33 0.0329 0.00 0.65 0.22 9.26 6.6443 0.00 3.53 0.41 0.3544 0.00 0.18 0.2747 0.00 18.2253 0.00

Edges/Edges 5 8 20 26 28 29 43 44 47 53

CALC

ULAT

ED D

ISTA

NCE

MAP

BE

TWEE

N DI

SKS(

r ij)

CALCULATED DISTANCE MAP BETWEEN DISKS (rij)

5 0.00 208.33 447.70 64.94 245.70 125.04 129.74 112.49 180.00 64.948 0.00 305.04 176.72 179.41 92.00 81.78 120.39 104.88 176.72

20 0.00 451.03 209.45 379.55 365.57 336.87 409.92 451.0326 0.00 263.11 84.85 95.40 120.00 122.55 0.0028 0.00 211.79 198.24 143.11 266.77 263.1129 0.00 14.16 84.85 69.54 84.8543 0.00 75.50 77.66 95.4044 0.00 153.14 120.0047 0.00 122.5553 0.00

g

Supplementary Figure 7: Schematic of the steps involved in generating the 2D layout of the nanomagnetic disks.a, Grey scale satellite image of an urban area. b, Edge image with extracted edge segments of (a). c, Zoomed-inview of a section of labelled edge segments. d, Calculated partial affinity matrix (aij) between edge segments in (c).e, Calculated partial distance map matrix (rij) used for the placement of nanomagnetic disks, based on a statisticalmethod multidimensional scaling (MDS). f, 2D layout of nanomagnetic disks obtained using MDS from the distancemap matrix (rij). g, Zoomed-in section of the placement of nanomagnetic disks corresponding to the edge segmentsin (c).

14


X = (V12 )T (21)

Note that we have dropped the constant of proportionality, c, since the energy minimizing solutionsare invariant to scaling of the original function. Our nanomagnet placement solution is given by the first tworows of XMDS ; each column of this matrix gives us the coordinates of the corresponding nanomagnet toconsider.

5 Mechanism to Deselect the Cells

Mechanism to deselect the cells from the array has been predicted through LLG simulations shown insupplementary Fig. 8. For analysis purpose, we have shown a 3X3 programmable array. The magnetsthat need to be deselected from the array are between the dotted lines in supplementary Fig. 8. Thesenanodisks are deselected by passing a deselection current that takes these magnets into an oscillatingstate. The rest of the magnets are then clocked (in z direction) from its current state and released tosettle in its energy minimum state. The coupling energy between the deselected precising magnets with itsneighbour is close to zero. As one can see from supplementary Fig. 8, the final magnetization states of allnanomagnets in column (2) as well the cell in (2,1) location are deselected and the rest selected magnetssettle in their energy minimum states depending on its neighbour interaction. The isolated magnets settlein vortex state and coupled magnets settle in single domain state.

Column of magnetsdeselected

Initial state of 3x3 array

before programming

Final state of 3x3 array pattern

Coupledselectedmagnets

Isolatedselectedmagnets XY

ZLegend

Supplementary Figure 8: Hardware Schematic. A uniform 2D array of spin-transfer torque based MRAM reconfig-urable array with underlying circuitry. Only the selected magnets (magnets in single domain state) corresponding toan objective function participate in the computation after clocking. The deselected magnets will stay in precessionalstate (non-computing state).

15


6 Speed Comparison with State-Of-The-Art

0

100

200

300

400

500

600

0 200 400 600 800 1000 1200

Tim

e Ta

ken

in s

econ

ds

# of edge segments

IBM ILOG CPLEX running time with 96% sparsityIBM ILOG CPLEX running time with 98% sparsityMagnetic Computing (projected)IBM ILOG CPLEX running time with 96% sparsity - best fitIBM ILOG CLPEX running time with 98% sparsity - best fit

Supplementary Figure 9: Running time using Magnetic computing (projected) and average running times with errorbars over 5 runs of experiments with IBM ILOG CPLEX optimizer version 12.6.1 with 96% sparse affinity matrix suchthat each node has an average of 8 neighbours and with 98% sparse affinity matrix such that each node has an averageof 4 neighbours.

7 Fabrication Process Figures and Tables

Substrate (Si)

PMMA positive tone resist

Electron Beam Exposure

MIBK:IPA 1:3 developer

Liftoff

Thin-film(s)

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Supplementary Figure 10: A flow diagram to fabricate single layer nanomagnetic devices.

16


Supplementary Figure 11: SEM image of a contamination spot grown by an optimized electron beam.

Supplementary Table 2: RCA cleaning procedure.

Step Process Description

1 Rinse wafer with Deionizer water. Rinses off any superficial particles.

2 Dip in NH4OH : H2O2 : H2O (1:1:5) (SC1)at 60oC for 10 minutes.

Removes insoluble organic contami-nants.

3 Rinse wafer with deionizer water. Rinses off any residue.

4 Dip in HF (50:1) for 20 seconds. Removes native oxide layers.

5 Rinse the wafer with deionizer water. Rinses off any residue.

6 Dip in HCl : H2O2 : H2O (1:1:6) (SC1) at60oC for 10 minutes.

Removes ionic and heavy metal contam-inants.

7 Rinse wafer with deionizer water. Rinses off any residue.

8 Dry with Nitrogen Gas. Removes moisture off the wafer.

Supplementary Table 3: Resist coating procedure.

Step Process & Description

1 Place wafer on the spinner and a drop of PMMA/Anisole on the wafer.

2 Pre-ramp up: 0 - 500 rpm in 5 seconds.

3 Ramp up: 500 - 6000 rpm in 10 seconds.

4 Spin: 6000 rpm for 45 seconds.

5 Ramp down: 6000 - 0 rpm in 15 seconds.

6 Soft bake: 170oC for 30 minutes to remove solvent.

17


Supplementary Table 4: The electron beam lithography procedure.

Step Process & Description

1 Mark a specific location on the sample and insert it into the SEM chamber

2 Operating voltage 30 kV , working distance

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9. Meja-Lopez, J. et al. Vortex state and effect of anisotropy in sub-100-nm magnetic nanodots. Journalof applied physics 100, 104319104319 (2006).

10. Cox, T. F. & Cox, M. A. Multidimensional scaling (CRC Press, 2010).

Additional Information Research updates will be accessible at http://www.eng.usf.edu/bhanja/Research.html and http://marathon.csee.usf.edu/EMT/Project_Page.html.Correspondence and requests for materials should be addressed to S.B.

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Documents

Non-boolean Computing With Nanomagnetics for Computer Vision Applications