Proceedings of the 13th WSEAS International Conference on COMPUTERS
Numerical Modeling of Neural Switch Programming Field Using Finite Element Analysis MAHMOUD Z. ISKANDARANI Faculty of Science and Information Technology Al-Zaytoonah Private University of Jordan P.O.BOX 911597, Post Code: 11191 JORDAN
[email protected] Abstract: - The design and numerical modeling using Finite Element Analysis (FEA) of electric field strength in programmable neural switch is carried out. The obtained model provided good approximation to the derived complex analytical solution. Effect of electrode separation and field spread in both x and y directions are studied and explained. Boundary effect on field strength representation is discussed and numerically reduced through increasing the number of nodes for each element in the finite grid. Edge effect on field strength is also eliminated using semi-infinite coplanar electrode approximation. Key-Words: - Neural, Numerical Modeling, Finite Element, Synapse, Memory, Information Processing excellent chemical and thermal stability and has ability to switch between two states, namely conducting (oxidized) and insulating (reduced), similar to the real neuron. Combining these polymers produced an ionicelectronic interacting mixed conductor. The resulting device was tested and proved to be able to store the injected charge, affecting its conduction and switches states due to device resistance change. Such a change would be stable occupying a specified level until another charge is injected or removed [10, 11, 12, 13].
1 Introduction The architecture of the brain can be viewed with greater or less granularity. Although understanding of lower-level processing is relevant to higher levels, it is possible to begin understanding the higher levels without a complete picture of the lower ones. At the level of single neurons, the several ion species present are assumed to obey a set of partial differential equations. Such equations can be numerically modeled using Finite Element Analysis, which proved to be a good approximation to the complex analytical solution and a fast tool for validation of the analytical model and future variations [1, 2, 3, 4]. In this paper, the programming field of a polymer based neural switch device is numerically modeled using various types of elements. Effect of electrode separation and distance from the origin are analyzed. Mathematical representation of FEA is illustrated and discussed.
Ag Gate
PEO10LiCF3SO3 Au Source
SiO2 Substrate
Fig.1: Neural Switch Structure
2 The Modeled Neural Switch Figure 1 illustrates the designed and modeled neural switch, which is made as realistic in construction as possible using polyethylene oxide - lithium trifluoromethane sulphonate (PEO10LiCF3SO3), which is known as material for high energy, long life, flexible, and re-chargeable just like a neural switch and has good mechanical and stable electrochemical properties [5, 6, 7, 8, 9]. The material is used together with a highly conducting polymer, called polypyrrole (ppy) that possess
ISSN: 1790-5109
Au Drain PPY Film
3 Numerical Modeling The used numerical model approximates an analytical solution of the neural switch described in equation 1 using semi-infinite co-planar electrode approximation with SCHWARZ-CHRISTOFFEL transformation as illustrated in figure 2. A check is carried out that covers model continuity, distortion index, element connectivity, and orientation, in conjunction with the properties of the
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Proceedings of the 13th WSEAS International Conference on COMPUTERS
polymeric material used and analytical solution boundary conditions to ensure that the model is physically and numerically correct [18, 19, 20].
{
Z-Plane
⎡ N1 , x, N 2 , x N3 , x N 4 , x ⎤ ⎢ ,y ,y ,y ,y⎥ ⎣ N1 N 2 N3 N 4 ⎦
v π (z2 - d 2 )1 / 2
(5.1)
(5.1) is the displacement matrix where, x and y are given by: 4
4
i =1
i =1
x = ∑ N i X i , y = ∑ N i Yi
Y (Im)
I
(5)
Where [B] is given by:
dw d ⎛ Vo -1 ⎛ z ⎞ ⎞ ⎜⎜ |E|= E = = cosh ⎜ ⎟ ⎟⎟ dz dz ⎝ π ⎝ d ⎠⎠
=
β,x } = [B] {β e} β, y
K
N
2d J
L
(5.2)
Now, the element characteristic matrix [K], with material property q and element thickness t is given by:
X (Re)
M
-Π
[K] = ∫ ∫ [B ]T q[B] tdxdy
Fig. 2: Geometrical Construction.
(6) 3.1
Four-Node Plane Isoparametric Element
1 1
= ∫ ∫ [B ]T q[B] tJdudv
Let the field quantity be β = β(x,y) = β(u,v), where u and v are dimensionless cartesian co-ordinates used to generate elements in the (x,y) plane. If an element has side lengths of 2a and 2b then u and v are given by:
u=
x y ,v= a b
-1 -1
Where; J is the Jacobian of the transformation from uv plane to x-y plane. In equation (6) second form, [B] is a function of u and v and β is expressed in terms of u and v so equation (5) is re-written as:
(2)
{
For a four-node element, β is interpolated from nodal values {βe} = [β1,β2,β3,β4]T, (3)
i =1
⎡ N1 u, N 2 , u N3 , u N4 , u ⎤ [D N ] = ⎢ ⎥ ⎣ N1 , v N 2 , v N3 , v N 4 , v⎦
Where Ni is the interpolation or shape function given by: 1 1 N1 = (1 - u) (1 - v), N2 = (1 + u) (1 - v) 4 4 1 1 N3 = (1 + u) (1 + v), N 4 = (1 - u) (1 + v) 4 4
(7.1)
From equation (3): (4.1)
1 1 N1 , u = - (1 - v), N1 , v = - (1 - u) 4 4
(4.2)
(7.2)
And so on.
Where u = + 1, v = + 1
Now, the relationship between (β,u), (β,v) and (β,x), (β,y) is given by:
For an isoparametric element:
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(7)
Where,
4
β = [N] β e or β = ∑ Ni β i
β,u } = [D N ] {β e} β,v
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Proceedings of the 13th WSEAS International Conference on COMPUTERS
{
β,x β,u β,u } = [J ]-1 { } = [Γ ] { } β, y β,v β,v
3.2
(8)
By placing an extra node on each side of the fournode quadrilateral element described previously a quadratic quadrilateral element is produced. As with the four-node bilinear element, the sides of the quadratic element are at u = + 1 and v = + 1. Axes u and v may be curved in the quadratic element like the ones used in the analysis. The complete set of shape functions is as shown in table 1:
From equations (5), (7) and (8) we obtain:
[B] [ β e] = [Γ] [D N ] [ β e]
(9)
From (9):
[B] = [Γ] [D N ]
(9.1)
Where;
N1 =
Γ11 = u, x, Γ12 = v, x, Γ21 = u, y, Γ 22 = v, y
⎡ 4 ⎢ ∑ Ni , ux i ⎡ x, u y, u ⎤ ⎢ i =1 [J] = ⎢ = ⎥ 4 ⎣ x, v y, v ⎦ ⎢ ⎢∑ Ni , Vx i ⎣ i =1
1 1 (1 + u) (1 - v) - ( N5 + N6) 4 2 1 1 N3 = (1 + u) (1 + v) - ( N6 + N7) 4 2 N2 =
⎤
4
∑ N , uy ⎥ i
i
⎥ ⎥ Ni , Vyi ⎥ ∑ i =1 ⎦ i =1
(10)
4
⎡ x1 y1 ⎤ ⎢ ⎥ x 2 y2 ⎥ ⎢ = [D N] ⎢ x 3 y3 ⎥ ⎢ ⎥ ⎢⎣ x 4 y4 ⎥⎦
1 1 (1 - u) (1 + v) - ( N7 + N8) 4 2 1 2 N5 = (1 - u ) (1 - v) 2
N4 =
(10.1)
1 (1 + u) (1 - v2) 2 1 2 N7 = (1 - u ) (1 + v) 2 N6 =
Where; DN is given by: 1 ⎡- (1 - v) (1 - v) (1 + v) - (1 + v)⎤ 4 ⎢⎣- (1 - u) - (1 + u) (1 + u) (1 - u)⎥⎦
(10.2)
N8 =
From equations (9.2) and (10), Γ can be written as:
[Γ] = [J ]-1 =
1 ⎡J 22 - J12 ⎤ ∝ ⎢⎣ - J 21 J11 ⎥⎦
1 (1 - u) (1 - v2) 2
Table 1: Eight Nodes Shape Function
(11)
To determine the characteristic matrix, the same procedures are followed as before. It is important to realize that isoparametric elements are geometrically isotropic, so, the numerical values of coefficients in [K] do not depend on the order of numbering of nodes. However, the order must be maintained and must run counter clockwise if J is not to become negative over part or the entire element.
Where; ∝ = det [J] = (J11 J22 - J21 J12) The JACOBIAN is a function of u and v and it is regarded as a scale factor that transforms the area dxdy in the first form of (6) from dudv in the second form of (6). Now all the necessary expressions are developed and by substituting them into equation (6) the characteristic matrix of a four-node quadrilateral element is evaluated using GAUSS QUADRATURE formula.
ISSN: 1790-5109
1 1 (1 - u) (1 - v) - ( N8 + N5) 4 2
(9.2)
So, the required JACOBIAN matrix [J] is given by:
[DN] =
Eight-node plane isoparametric element
4 Analysis, Discussion and Conclusion When programmed, the bias applied between the gate and the shorted source/drain terminal will set up a field that will oxidize or reduce the polymer, and hence when the programming voltage is
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Proceedings of the 13th WSEAS International Conference on COMPUTERS
removed and a constant current is applied between the source and drain, the measured conductance will be related to the previously applied bias and will be fixed until another latching potential is introduced [14, 15, 16]. Such actions are related to the established electric field within the switching device where a slow and gradual (power) decrease in the programming field value is realized (figure 3) as we travel along the x-axis. This is consistent with all field theories and with the geometrical co-planar electrode construction shown in figure 2. Also, a peak point occurred at x=5µm, which corresponds to the internal edge of the electrode structure as it equals half the separation between the electrodes (2d = 10µm) according to figure 2. Such a peak point is valid as the field strength at the inner edge is at its peak, while the one at the far edge is at its lowest. This is also consistent with the semi-infinite coplanar electrode assumption. A decrease in the field value as we travel up the y-axis (figure 4) is evident. This is also consistent with field theory and the increment of distance away from the field lines linking the device structure. Further analysis lead to the following conclusions regarding the modeled field strength: (1) As the separation between the electrodes increases, the field value decreases as shown in figure 5. (2) The decrement in the field value as a function of distance along the y-axis is gradual and has a slow, power like characteristics, where no peak point is present as the plot considers the effect of travel along the y-axis as shown in figure 6. (3) Increasing the nodes per modelling element in the finite mesh resulted in more uniform field strength and better representation of the derived analytical solution, as field approximation and numerical simulation is affected by the total number of nodes used in constructing the grid needed for analysis. Figure 7 illustrates the reduction of error as a function of used nodes.
Numerically Simulated Electric Field Strength (En) for d=7.5 um
Electric Field Strength (mV/um)
0.6 0.5 0.4
En(Y=2.5)
0.3
En(Y=4.5) En(Y=6.5)
0.2
En(Y=8.5) En(Y=10.5)
0.1
En(Y=12.5)
0
En(Y=14.5)
0
10
20
Travelled Distance in um ( X-axis)
Fig. 4: Field Characteristics for Neural Switch
Electric Field Strength (mV/um)
Numerically Simulated Electric Field Strength (En) for X=0 um 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
d=5 um d=7.5um
0
5
10
15
Travelled Distance along Y-axis (um)
Fig. 5: Effect of Electrode Separation on the Programming Field of Neural Switch
Electric Field Strength (mV/um)
Numerically Simulated Electric Field Strength (En) for X=3 um 0.7 0.6 0.5 0.4
d=5 um
0.3
d=7.5um
0.2 0.1 0 0
5
10
15
Travelled Distance along Y-axis (um)
Fig. 6: Effect of Electrode Separation on the Programming Field of Neural Switch Numerical Simulation
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.5
Average Error
Electric Field Strength (mV/um)
Numerically Simulated Electric Field Strength (En) for d=5um
En(Y=2.5) En(Y=4.5) En(Y=6.5) En(Y=8.5) En(Y=10.5)
0.4 0.3 0.2 0.1
En(Y=12.5)
0
En(Y=14.5)
0
0
5
10
15
Travelled Distance in um (X-axis)
40
60
80
100
120
Fig. 7: Effect of Number of Nodes on modeling Accuracy
Fig. 3: Field Characteristics for Neural Switch
ISSN: 1790-5109
20
Percentage nodes used in the simulation
20
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ISBN: 978-960-474-099-4
Proceedings of the 13th WSEAS International Conference on COMPUTERS
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