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Numerical Modeling of the Electrostatic Field inMetal-Insulator-Metal Structures

Emilia-Simona Malureanu and Daniel IoanElectrical Engineering FacultyNumerical Methods Lab.

Politehnica University of BuchareastEmail: [email protected]

[email protected]

Mihail-Iulian AndreiSpecial Electric Machines Dept.Icpe, Bucharest

E-mail:[email protected]

Abstract The accurate data about electric eld on the metal-insulator interface is essential for a series of phenomena, such aselectric breakdown, corona discharges, or electron emission. Asthe size of the structures used in electronics industry goes towardsnanaoscopic scale, the sharp edge electrostatic effect in thesestructures becomes more predominant and a correct estimation of

the electric eld intensity value implies using combined numericaland analytical methods.

I. INTRODUCTION

The accurate data about electric eld on the metal-insulator interface is essential for a series of phenomena,such as electric breakdown, corona discharges, or quantumtunneling [1], [2].

This paper proposes to analyse the electric eld inten-sity in structures with nanoscopic dimensions, considering anAl SiO 2 Al test structure, specially designed and realisedby silicon planar technology, for experimental study of eldelectron emission.

By modeling the test structure, the electrostatic eld wascomputed with the nite element method, using Comsol pro-gram . In the paper is presented the detailed geometrical model,as well as the solved equations and their boundary conditions.

The inuence of exact shape and surface smoothness areanalysed, considering the sharp corner effect in estimating thevalue of the electric eld in Al SiO2 Al test structuresand the results are validated, by comparison with those analyt-ically obtained. The electrostatic sharp corner effect imply theappearance of high intensity electric elds in the vecinity of charged electrodes with sharp corners and edges. The electricels intensity in the appex of the emiting electrode is in inverseratio to its bending radius. Modeling this effect is necessaryin studying the Corona phenomenon, in integrated circuits, asdetermining the precision of the electrostatic forces in MEMSdevices [3] or studying the electro-migration in integratedcircuits [4].

The precise modeling of this effect with the nite elementmethod implies using an extremely ne mesh around the edgesand corners with considerable consumption of computationalresources as memory and time. As the bending radius de-creases, the necessary of computational resources increases.If a geometrical model with sharp edges is adopted, witha zero bending radius, then a coarse mesh can be used in

order to determine the electric eld intensity. This type of model is imprecise in the vecinity of the edge, but accurateenough in the rest of the computational domain, where theelectric eld does not depend on the edges bending radius.This is a global correct model but locally imprecise. In orderto overcome this disadvantage and reduce the computing effortwithout dimishing the numerical result accuracy, Kraehenbuhlet al. [5] propose using the sharp edge model for determiningthe global electric eld and then, in the vecinity of therounded edges, the eld is determined by solving a localeld problem. Hagedorn and Hall [6] propose a method basedon the conformal mapping for determining the electric eldintensity in a sharp corners and edges geometry.

II. C ONCEPTUAL MODEL

The considered Al SiO 2 Al test structures can be seenin Figures (1). The geometrical parameters of the test structureare in Table (I).

Parameter Test structure1 Electrode length 4.5m2 Electrode width 3m3 Thickness ( gAl ) 0 .4m4 Electrodes appex r = 1 .5m5 SiO 2 layer thickness ( gSiO 2 ) 1.5m6 SiO 2 layer width 93.7nm7 Electrodes voltage 20V

TABLE I: Geometric parameters

(a) Optical microscope view. (b) Electronic microscope(SEM) view.

Fig. 1: Al SiO 2 Al test structures


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where S (r, ) = r sin ( ), with = 2/ 3 [5].

An approach based on conformal transformation will allowdetermining an increasing factor for the electric eld intensityon the electrode surface [6], [4]

E maxE 0

= 1.04

(r rac /d )1/ 3, (7)

where E max is the maximum value of the electric eld inten-sity in the considered test structure, E 0 = U/d is the electriceld, considered as an uniform one, d is the distance betweenelectrodes, rrac is the bending radius of the electrodes edge(transition radius) according to Fig. 6.

V. N UMERIC APPROACH

A. 2D top view

The geometry of electrodes depicted in Fig. 3 and 4 isconsidered. The approximate Al electrode is eliminated formthe computational domain since inside this the electrostaticeld is zero. Fig. 3 indicates the computational domain, itsshape and its dimensions, with values according to Tabel II,and the boundary conditions.

Fig. 3: 2D mathematical model.

Fig. 4: 2D parallel-plane model.

1) Preprocessing: Material properties

From COMSOL material library SiO 2 was chosen as theentire computational domain, where the electrostatic eld isdetermined.

2) Solving: The computational domain has a triangularmesh that generates 227.941 degrees of freedom by discretiz-ing Laplace equation with nite elements method. The problemwas solved in 10 seconds.

Parameter Value1 d 47 nm2 b 4.5m3 c 1.5m4 r 1.5m5 a 3m6 V 0 10V

TABLE II: Electrode dimensions

Fig. 5: Electric eld distribution.

3) Postprocessing: The main purpose of the 2D horizontalmodelling was to determine the value of the electric eldintensity when neglecting the the sharp edge electrostaticeffect.

4) Conclusions: As we expected, the eld is maximum inthe electrode appex. Its value is 214 MV/m as can be seen inFig. 5.

B. 2D side view

The geometry form Fig.6 is considered. The dimensions of the computational domain are accoording to Table IV.

Fig. 6: 2D geometric model. Lateral view.

The boundary conditions are according to Fig. 7:

with blue: V = 10V ; with red: V = 0 ; in rest: dV/dn = 0 .


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Fig. 7: Boundary conditions and materials in 2D verticalgeometry.


From COMSOL material library air, SiO 2 and Al werechosen according to Fig. 7.

2) Solving: For the rrac = 0 geometry, the computationaldomain has a triangular mesh that generates 110.350 degreesof freedom while for r rac = 0.05, the computational domainhas a triangular mesh that generates 132.002 degrees of freedom, by discretizing the electrostatic eld equation withnite elements method.

The problem was solved in 3s using 1GB.

3) Postprocessing: The main purpose of the 2D verticalmodelling was the study of the inuence of the sharp edgeelectrostatic effect on the value of the electric eld intensityand comparing the numerical results to the analytical onesobtained by conformal mapping method [6], [4].

Table III contains the analytical (determined according to7) and numerical values of the maximum electric eld intensity(E ) for different values of the transition radius ( r rac ).

The maximum value that the electric eld intensity canhave in the considered test structures is analytically determinedconsidering the transition radius equal to the Al atomic radius

r rac = rAl = 125 10 12 m. (8)

Also, for the numeric computation, the transition radius isr rac = 125 10 12 m 0

In Fig. 8, one can see the variation of the electric eld

depending on the transition radius, considering the values fromTable III.

C. 3D model

Considering the geometry from Fig. 9 and 10, the dimen-sions of the computational domain are according to Table IV.Fig. 10 describes the emission electrode geometry for twodifferent values of the transition radius ( r rac ).

The boundary conditions are dened as follows (Fig. 11):

on the boundary of the electrode: V = 10 V

Transition radius [m ] E [V /m ](analytic) E [V /m ](numeric)125 10 6 1.5971 10 9 1.03461 10 9

0.001 7 .986 10 8 8.593 10 8

0.00358 5 .22 10 8 5.305 10 8

0.00616 4 .356 10 8 4.467 10 8

0.00874 3 .877 10 8 4.009 10 8

0.01132 3 .55 10 8 3.716 10 8

0.01389 3 .322 10 8 3.505 10 8

0.01647 3 .139 10 8 3.345 10 8

0.01905 2 .99 10 8 3.220 10 8

0.02163 2 .866 10 8 3.118 10 8

0.02421 2 .76 10 8 3.034 10 8

0.02679 2 .66 10 8 2.963 10 8

0.02937 2 .588 10 8 2.903 10 8

0.03195 2 .517 10 8 2.851 10 8

0.03453 2 .452 10 8 2.805 10 8

0.03711 2 .394 10 8 2.765 10 8

0.03968 2 .341 10 8 2.729 10 8

0.04226 2 .2929 10 8 2.698 10 8

0.04484 2 .2481 10 8 2.669 10 8

0.04742 2 .206 10 8 2.643 10 8

0.05 2 .167 10 8 2.620 10 8

TABLE III: Analytical and numerical values of the electriceld intensity ( E ) for various values of the transition radius(r rac ).

Fig. 8: Electric eld intensity ( E ) variation depending thetransition radius ( r rac ).

gAl 0.4mgSiO 2 1.5m

r 1.5md 93nm

r f 0.63m

TABLE IV: Geometric parameters for 3D model.

on the Silicon wafer: V = 0 ; on the simmetry plane between the electrodes: V = 0 ; in rest: zero Neumann boundary conditions: dV/dn =

0.

1) Numeric model: The presented model was analysed inCOMSOL Multiphysics 4.3b, which uses the nite elementsmethod [7], [8].


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(a) Computational domain.

(b) Emission electrode.

Fig. 9: 3D geometric model.


From the COMSOL material library air, SiO2 and Al werechosen according to Fig.12.

3) Solving: For the rrac = 0 geometry, the computationaldomain has a a tetrahedral mesh that generates 26.849.967degrees of freedom by discretizing Laplace equation with niteelements method. The problem was solved in 2457 using36.93GB.

For the r rac = 10nm geometry, the computational domainhas a tetrahedral mesh that generates 23.055.956 degrees of freedom by discretizing Laplace equation with nite elementsmethod. The problem was solved in 2057 using 32.62GB.

4) Postprocessing: The main purpose of the 3D modellingwas the study of the inuence of the sharp edge electrostaticeffect on the value of the electric eld intensity.

5) Conclusions: The maximum value of the electric eldintensity was computed depending on the transition radius.This value appears punctual in the transition zone. For r rac 0, the maximum value of the electric eld intensity is E max =9.3414e8V/m 1 . For r rac = 10nm , the maximum value of theelectric eld intensity is E max = 5.6553e8V/m 2 . The electriceld distribution for the two values of the transition radiuscan be seen in Fig. 13. One can conclude that as the transition

1 This value is found using a COMSOL 4.3b function that allows ndingthe point of maximum electric eld intensity in the considered volume.

2 This value is found using a COMSOL 4.3b function that allows ndingthe point of maximum electric eld intensity in the considered volume.

(a) 3D geometric model with r rac 0 .

(b) 3D geometric model with r rac =10 nm .

Fig. 10: Transition radius.

Fig. 11: Boundary conditions for 3D model.

radius is decreasing, the value of the electric eld intensity isincreasing.


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