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Journal of Electronic Materials, Vol. 23, No. 12, 1994 Regular Issue Paper Three-Dimensional Simulation of Impurity Diffusion in Thin-Film Diffusion Barriers XIANG GUI,* STEVEN K. DEW, and MICHAEL J. BRETT Department of Electrical Engineering, University of Alberta, Edmonton, Alberta T6G 2G7, Canada Three-dimensional (3D) simulation of combined lattice and grain-boundary diffusion of impurities in thin-film diffusion barriers for semiconductor device metallizations is performed. Calculated results of impurity concentration pro- files demonstrate quantitatively an obvious underestimation of the frequently used two-dimensional (2D) analysis with respect to the influence of film ge- ometry and grain-boundary diffusion coefficient. As for the average concentra- tion at the backside of diffusion barriers, approximately a factor of two difference between the 2D and 3D simulation results is found over an interesting range of times and grain size structures. Graphs for predicting the effectiveness of diffusion barriers are presented with several normalized parameters associated with position and time. Particular application examples of aluminum diffusion in titanium nitride films justify the use of this material as an effective diffusion barrier in silicon microelectronic devices. Key words: Grain boundary diffusion, metallization, three-dimensional simulation, thin-film diffusion barriers, titanium nitride INTRODUCTION With the continuing trend of miniaturization of very large scale integration (VLSI) devices, the technologi- cal problems associated with metallization have be- come a central concern. 1,2 As metal-semiconductor and metal-metal interfaces are widespread in a solid- state device, undesired interdiffusion and reactions between these interfaces are significant sources of device failures. At increased power densities or el- evated temperatures, the degradation happens more rapidly. In order to avoid or retard the interdiffusion at metal-semiconductor interfaces and thus enhance *Also associated with the Reliability Physics Laboratory, Department of Electronic Engineering, Beijing Polytech- nic University, Beijing 100022, China. (Received April 14, 1994; revised August 28, 1994) device reliability, a diffusion-barrier layer is often interposed. ~,4 In device processing, metallic film bar- riers are normally polycrystalline due to the practical difficulties of realizing single crystalline films. As a result, atomic diffusion along grain boundaries, which normally provide much faster diffusion paths than the lattice diffusion, is an important limiting factor for determining the effectiveness of the thin-film diffusion barriers. Owing to complexity, the combined lattice and grain-boundary diffusion problems for thin-film sys- tems have generally been solved in two spatial dimen- sions. 5~ In addition, the film is often treated as one- grain thick, with a vertical grain boundary or an array of vertical grain boundaries separating different grains laterally. Lavine and Losee 3.10assumed alternative two-dimensional (2D) brick-wall and more irregular patterns of grain boundaries with multiple vertical 1309

Three-dimensional simulation of impurity diffusion in thin-film diffusion barriers

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Journal of Electronic Materials, Vol. 23, No. 12, 1994 Regular Issue Paper

Three-Dimensional Simulation of Impurity Diffusion in Thin-Film Diffusion Barriers

XIANG GUI,* STEVEN K. DEW, and MICHAEL J. BRETT

Depar tment of Electrical Engineering, Universi ty of Alberta, Edmonton, Alberta T6G 2G7, Canada

Three-dimensional (3D) simulation of combined lattice and grain-boundary diffusion of impurit ies in thin-film diffusion barriers for semiconductor device metallizations is performed. Calculated results of impuri ty concentration pro- files demonstra te quant i ta t ively an obvious underest imat ion of the frequently used two-dimensional (2D) analysis with respect to the influence of film ge- ometry and grain-boundary diffusion coefficient. As for the average concentra- tion at the backside of diffusion barriers, approximately a factor of two difference between the 2D and 3D simulation results is found over an interest ing range of t imes and grain size structures. Graphs for predicting the effectiveness of diffusion barriers are presented with several normalized parameters associated with position and time. Part icular application examples of a luminum diffusion in t i tanium nitride films just i fy the use of this mater ial as an effective diffusion barrier in silicon microelectronic devices.

Key words: Grain boundary diffusion, metallization, three-dimensional simulation, thin-film diffusion barriers, t i tanium nitride

INTRODUCTION

With the continuing trend of miniaturization of very large scale integration (VLSI) devices, the technologi- cal problems associated with metallization have be- come a central concern. 1,2 As metal-semiconductor and metal-metal interfaces are widespread in a solid- s tate device, undesired interdiffusion and reactions between these interfaces are significant sources of device failures. At increased power densities or el- evated temperatures , the degradation happens more rapidly. In order to avoid or retard the interdiffusion at metal-semiconductor interfaces and thus enhance

*Also associated with the Reliability Physics Laboratory, Department of Electronic Engineering, Beijing Polytech- nic University, Beijing 100022, China. (Received April 14, 1994; revised August 28, 1994)

device reliability, a diffusion-barrier layer is often interposed. ~,4 In device processing, metallic film bar- riers are normally polycrystalline due to the practical difficulties of realizing single crystalline films. As a result, atomic diffusion along grain boundaries, which normally provide much faster diffusion paths than the lattice diffusion, is an important limiting factor for determining the effectiveness of the thin-film diffusion barriers.

Owing to complexity, the combined lattice and grain-boundary diffusion problems for thin-film sys- tems have generally been solved in two spatial dimen- sions. 5~ In addition, the film is often t rea ted as one- grain thick, with a vertical grain boundary or an a r ray of vertical grain boundaries separating different grains laterally. Lavine and Losee 3.10 assumed al ternative two-dimensional (2D) brick-wall and more irregular pat terns of grain boundaries with multiple vertical

1309

1310 Gui, Dew, and Brett

a b

Fig. 1. An intermediate diffusion-barrier layer U, imposed between two materials A and B. The addition of this barrier layer makes Structure (b) much more chemically stable and thus reliable than Structure (a).

II III '' IlI Z

a

~, X

~. X

Y

b Fig. 2. Geometric models for the grain-boundary and lattice diffusion in the thin-film diffusion barrier illustrated by layer U in Fig. 1. (a) 2D model, and (b) 3D model.

grains to simulate the diffusion phenomena, using a Monte Carlo method. This is certainly useful for gaining some more relevant information. But typi- cally, deposited metal films are three-dimensional (3D) in nature and are composed of a matr ix of columnar microstructures tha t extend through the films, n Therefore, the first simplification aforesaid appears unjustified for the analysis of thin-film diffu- sion barriers, but the second might be applicable. This consideration of using a relatively simple 3D geo- metrical s t ructure also makes it possible to model the diffusion process with reasonable computational re- sources. A numerical approach is then necessitated,

which should be versat i le in dealing with the 3D geometry and boundary conditions.

The next section of this paper describes the 2D and 3D physical models used to s imulate impuri ty diffu- sion in thin-film diffusion barriers. This is followed by a number of calculated results, i l lustrat ing the influ- ence of film geometry and grain-boundary diffusion coefficient with a comparison between the 2D and 3D analyses. Possible implications involved in the diffu- sion effects are discussed accordingly. Specific ex- amples of a luminum penetrat ion in t i tanium nitride (TIN), a refractory metallic compound present ly em- ployed as a diffusion-barrier mater ia l of great inter- est in microcircuits, are given subsequently. The final section contains some concluding comments.

P H Y S I C A L M O D E L S

Figure 1 shows schematically a diffusion-barrier layer, U, inser ted be tween materials A and B to prevent their direct contact and reduce the ra te of interactions be tween A and B. Material A may repre- sent the commonly used A1 metallization, and B the Si substrate. A perfect diffusion barr ier is intrinsically impossible due to the abundance of diffusion pa ths through the barr ier and the existence of large concen- trat ion gradients in such a sandwiched thin-film structure. A polycrystall ine diffusion barr ier of a thickness 1 can be modeled in two and three dimen- sions, as shown in Figs. 2a and 2b, respectively. The grain boundaries are characterized by thin slabs of a uniform width 5. In the 2D case, there is an a r ray of parallel grain boundaries, lying orthogonal to the surface at an average spacing of 2 L . I n the 3D case, parallelopiped grains of the average spacings in the x and y directions, 2L x and 2 L , are separa ted by an

Y array of square columnar boundaries. Although the s t ructures described are idealized, the major effects can be s imulated and the results used as a basis for metallization design and reliability prediction in prac- tical VLSI devices.

Because of the symmetry, only a geometrically peri- odic s t ructure needs to be considered. This involves an elemental volume with reflective boundaries on its sides where concentration flux is zero. The top surface of the diffusion-barrier layer is a t tached to a reservoir of the diffusing species. Some of these species (atoms of mater ia l A) can diffuse all the way through the barr ier layer U, and the subsequent chemical interac- tions make their back diffusion unlikely, Therefore, the bot tom surface of the diffusion barr ier is a s sumed to be absorbing for the diffusing species. Such a boundary condition is different from a particle s ink with a zero or other constant concentration and varies as a function of position and elapsed diffusion time. This t r ea tment brings about a slightly smaller value at the same position and t ime when compared wi th solving the diffusion equations with the bot tom of the problem space extended to infinity. The difference in the results is due to the elimination of particle back- diffusion from region B. The average concentrat ion over the backside surface of the diffusion barr ier can

Three-Dimensional Simulation of Impurity Diffusion in Thin-Film Diffusion Barriers 1311

then be used as a measure for its efficiency. The transmission-line matrix (TLM) method for

solving diffusion problems has been well estab- lished.~2-14 In a recent paper,~5 we have demonstrated the novel use of the TLM modeling method for grain- boundary diffusion in thin films. This method is also adopted here owing to the convenience of incorporat- ing all the conditions specified and the efficiency of the computation. The backside boundary condition of material absorption described above is readily real- ized by matched-load terminations in the TLM method, where the reflection coefficient is set to be zero so t ha t the arriving species at the boundary will not diffuse back. For a detailed discussion of the TLM numerical techniques being used, the reader may refer to Refs. 15-18.

R E S U L T S AND DISCUSSION

For the solutions to be generally valid, it is more convenient to use the following dimensionless vari- ables. The positions along the x, y, and z directions are, respectively, normalized as

x X = ~-~, (1)

y Y = ~---, (2) Y

and Z = l '

and the reduced diffusion time is defined as

(3)

T - Dgt (4) 12 ,

where, D is the diffusion coefficient of species A in the grains o f the diffusion-barrier layer (U), and t is the time. The grain-boundary diffusion coefficient of A in U is denoted as D b. A normalized value of uni ty is employed for the constant concentration on the top surface of the diffusion barrier. Unless otherwise indicated for discussing variability of geometrical and/or physical effects, all of the calculated results presented in this paper are obtained by assuming the plausible representative parameter relationships 5,1~ as follows:

�9 the ratio between the boundary width and the barrier thickness is 5/1_ = 5 x10-3;

�9 the aspect ratio regarding the grain dimensions is l/L= 4, where L = L x = L ; and

�9 the diffusion coefficient a~ grain boundaries is four orders of magnitude higher than tha t within the grains, namely, Db/D ~ = 104.

Figure 3 displays concentration profiles at the back- side of the barrier layer (Z = 1) for several reduced diffusion times using the 2D simulations. For most of the diffusion region, an exponential decay of concen- trat ion against the distance away from the grain

boundary is clearly seen. However, the 2D calculation is commensurate with an infinite grain length along the y direction. In reality, grains should have approxi- mately equal size while looking from different angles in the x-y plane. This means tha t the 2D simulations could underes t imate the lateral diffusion effects to a large degree. In order to obtain an adequate estimate, a 3D calculation is performed and a representative result is shown in Fig. 4 as an isometric projection. Acting as fast diffusion paths, the grain boundaries effectively spread out the concentration with an iden- tical maximum value (-0.8), for a normalized diffu- sion time of 1 x 10 -2. This concentration is found to be

10 0

I--

X

10 -1

1 0 .2

N o r m a l i z e d

D i f f u s i o n T i m e

�9 .......... �9 l x 1 0 - 3

�9 ....... * 5 x l 0 "3

. . . . . . . . . . . l x 1 0 - 2

~ O : l I k ~ ' R

'~ ' i '-..

" , -+

.+ " - - . - . +_

'. .+

',+ -,

++ ' ,

~ +,~

1 0 -3 , s "', , i , i ,

0.0 0.2 0.4 0.6 0.8 1.0 X

Fig. 3. Two-dimensional simulations of the concentration C(X,1 ,T) vs the normalized distance, X, from the grain boundary into the grain at the backside of the thin-film diffusion barrier (Z = 1) and for several values of the normalized diffusion time (T).

Fig. 4. Three-dimensional simulation of the concentration C(X,Y, 1 ,T) vs the normalized distance, X and Y, at the backside of the thin-film diffusion barrier (Z = 1) and forthe normalized diffusion time T = 1 • 10 ~. The center of the plot (X=Y=O) corresponds to the intersection of per- pendicular grain boundaries.

1312 Gui, Dew, and Brett

v IO

1.0

0.8

0.6

0.4

0.2

0.0 0.0

' i i i i

X ~ ....... o 2D, T = l x l 0 -3 �9 ......... * 2D, T = l x l 0 "2 ............ 3D, T = l x l 0 -3

"',. �9 ........... 3D, T = l x l 0 -2

'-4~.~ ~'�9 1L L I ~ i I "U ' I I"IL I I "In* i I I L I I'RK1L l I1"11

'~

~ " ' ' . . . ".m

0.2 0.4 0.6 0.8

Z .0

Fig. 5. Compari_son of the 2D and 3D simulat ions of the average concentrat ion C (Z,T) vs the normal ized vert ical d istance Z for two va lues of the reduced dif fusion t ime (T).

25% greater than the corresponding 2D result. The comparison of the 2D and 3D simulations can be

fur ther i l lustrated in terms of an average impuri ty concentration C (Z,T) vs the normalized diffusion distance along the vertical direction Z. For the 2D case, this average concentration for a section parallel to the surface can be defined as

1

C2D (Z, T)= I C(X, Z, W)dX, (5) 0

and for the 3D case, it becomes 1 1

C3D(Z,T) = S S C(X,Y,Z, T)dYdX. (6) 0o

Shown in Fig. 5 are the calculated results. First, the discrepancy between the 2D and 3D predictions is significant. Second, one may notice tha t the barr ier layer manifests itself like a semi-infinite sample dur- ing the initial period of diffusion time, and that the absorbing effects for particles at the backside of the barrier layer come into play gradually as the time elapses. All this can be identified based on the shapes of the C ( Z, T ) vs Z curves, as similarly discussed in the previous investigations.~.~s Additional simulations indicate tha t the 3D curves shown in Fig. 5 may also be duplicated by the 2D analyses with an appropri- ately enlarged grain-boundary diffusion coefficient-- more evidence of underest imation by the 2D ap- proach. However, it is important to emphasize that this has to be restricted to averaged concentrations

3.0

2 .5

~- 2.0

- - 1.5

I ~ 1.0

0.5

...... O ......... 0 ......... 0 ...... ....... @ .... 0 ........ O ........ 0 ......... 0 ....... O .........

........ ~ ...... ~ ........ -~ ......... ~ ...... .~ ......... -~ ....... ~ ......... -~ ....... ~ ...... :

0 ....... 0 I = 4 L

...... -~ I = 8 L

I I I I 0.O 0 2 4 6 8 10

T ( 1 0 "3)

0.5

0.4

0.3

iC 0.2

0.1

..5 .._~ ......

/ / .... . ~ ......... Q ........

.."'"" ..O ......... O ......

"'""~ ;3 ..........

@.-. E ~

'" ..- 0 ....... 0 l = 4 L / / / . 0 .... �9 ........ ~ 1 = 8 L / " ..Y

.," 0"" , , / - : 7

I I I I 0.0 0 2 4 6 8 10

T ( 1 0 "3)

Fig. 6. The average concentrat ion at the backs ide of the di f fus ion barr ier C (1 .T) as a funct ion of the normal ized di f fusion t ime, T, for two va lues of the re lat ionship between I and L. (a) 3D/2D scal ing factor, and (b) 3D simulat ion results.

with some "equivalent" quanti t ies for fitting in the real ones. Other useful results such as tha t shown in Fig. 4 can only be achieved using the 3D calculation.

Since 2D process simulations remain the major approach nowadays, a direct 3D/2D comparison is shown in Fig. 6a to quantify the underes t imat ion by 2D solutions and generate a scaling factor for the average concentration as defined above. In order to facilitate the evaluation of the effectiveness of thin- film diffusion barriers, the resul ts are presented by plotting the average concentrations at the backside of the barriers (Z = 1 ) a s a function of the normalized diffusion time, i.e. C 3D(1, T)/C 2D(1, T) vs T, for several different geometrical s tructures. The 2D resul ts con- sis tently underes t imate the average concentrat ion by factors of approximately 2.2 and 1.9 for the ratio 1/L of 4 and 8, respectively. Shown in Fig. 6b are the 3D simulation results over the same range of t imes and grain size structures.

As an example of an application of our model, we

Three-Dimensional Simulation of Impurity Diffusion in Thin-Film Diffusion Barriers 1313

use the 3D curves shown in Fig. 6b to obtain a quali tat ive idea about the efficiency of a part icular barr ier mater ial of general interest , TiN, for pre- venting A1 penetrat ion in metallization systems of Si devices. Without this barr ier layer, intermixing of A1 and Si would spike shallow junctions and degrade circuit performance. It should be noted tha t TiN layers fabricated by different processing technology may have distinct physical s t ructures and phase content and exhibit a certain varied permeabil i ty to Si or A1 atoms. Recent measurements ~9,2~ indicated that the grain-boundary diffusion coefficients of Si and A1 in reactively evaporated TiN films with columnar microstructures are almost identical. The average grain size of these films is approximately 20 nm, corresponding to the magnitude of 2L in our model. Grain-boundary diffusion of A1 in the TiN films was characterized as 2~

10-18exp/. 30kJm~ 2 Db(AlinTiN)=3 • .jm s - , (7) \

where, R is the universal gas constant (8.31 J K -~ mol-~), and T is the tempera ture in kelvins. If grain- boundary diffusion is regarded as the prime limiting factor of the diffusion barrier and Eq. (7) can be extrapolated to a normal device operating tempera- ture of 350K, then according to Fig. 6a for thin TiN layers with a thickness of 40 nm (1 = 4L), the normal- ized diffusion t imes at which the average A1 concen- trat ion at the backside of the TiN layer reaches 5 and 10% of the concentration at the top surface are, respectively, 1 x 10 -3 and 2 x 10 -3. The corresponding real-life t imes are then determined by Eq. (4) and the ratio Db/Dg (= 104) to be roughly five and ten years. If the thickness of TiN layers were increased to 80 nm and the grain-boundary width remained unchanging so tha t l = 8L and 5/1 = 2.5 • 10 -3, then the curves shown in Fig. 6b would yield doubled values for the time. These results indicate that TiN films can serve as effective diffusion barriers and that a thickness on the order of tens of nanometers is sufficient for the pur- pose.

Figure 7 i l lustrates the average concentration at the backside of the diffusion barrier C(1, T) as a function of the ratio of boundary to matrix diffusivities Db/Dg. In general, the value ofD b is very much greater than tha t of Dg. However, one must consider the pronounced effects of"stuffed" barriers. 4 In this case, some species of ei ther host excess or foreign impuri ty atoms decorating the grain boundaries may result in a much reduced boundary diffusivity. It can be seen from Fig. 7 tha t there is a decrease of more than two orders of magni tude in C (1, T) when D b is varied to change the ratio Db/Dg from 10 4 to 10 2.

It is worthwhile to note that besides the concentra- tion gradient, an additional diffusion driving force associated with the current passag e perpendicular to the deposited metal film may enhance the impuri ty diffusion in thin-film diffusion barriers. This "vertical electromigration" phenomenon will occur over a met- allization topographical feature if the two factors

responsible for the impuri ty diffusion reinforce each other, especially under the condition of a large cur- rent densi ty and a high operating temperature . This effect will be addressed in future work.

In this section, we have compared calculated re- sults based on the 2D and 3D simulations. The 3D geometric s tructure employed is alogical and straight- forward extension from the idealized 2D model in common use. Nevertheless, the full 3D computat ion performed here is not complicated too much by vir tue of the use of TLM numerical approach. 15 As final remarks, we give fur ther examination of the val idi ty of the simplified 3D modeling assumption. Fair ly general and very computer-intensive Monte Carlo solutions for grain-boundary diffusion in semi-infi- nite samples have been reported by Hodge. 21 He compared three different geometries of grain bound- ary with the same characterist ic dimension, i.e., par- allel planar, square columnar, and randomly occur- ring boundaries. After a relatively long period of diffusion time, the 3D columnar and random bound- aries were found to exhibit essentially very similar concentration profiles which deviated from the profile for the 2D slabs. Of course, the complexity of the microstructure increases for the random-boundary situation, bu t it brings about primarily an increase in rapid diffusion paths (grain boundaries) not normal to the surface source as a resul t of the same average grain diameter used. In other words, diffusing species cannot find downward paths much more readily, and thus this change in the microstructure does not influ- ence substant ia l ly the overall vertical concentrat ion profiles. Consequent ly , simple square co lumnar boundaries may be considered as an acceptable ap- proximation especially for polycrystalline thin films tha t are characterized by the columnar microstruc- ture.

C O N C L U S I O N S

In summary, the 3D phenomenological solutions

o 4 • /u . . . . . . . . , . . . . . . . . , . . . . . . . . ,

. !

4 , . . . . . . . . �9 2D ....... 101 �9 --- �9 3D ............

. . f . " I1"" . . , , -

.., . . .

I - - .. ..,O:' ,._- 10 .2 =-" ../"

IO " ' / , :

Y

,,, ,/" 10-3 Ii" /"

."" T = 10 -2

104101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 10 3 10 4 t 0 s

Db/Dg

Fig. 7. The average concentrat ion at the backside of the diffusion barrier C(1 ,T) vs the ratio between grain-boundary and lattice diffu- sion coefficients, DJDg. This demonstrates the effectiveness of "stuff- ing" the grain boundary with excess or other atoms to lower D b.

1314 Gui, Dew, and Brett

have been presented for impurity penetration in thin- film diffusion barrier layers with coupled grain-bound- ary and lattice diffusion. A parallel comparison with the 2D analysis demonstrates clearly the 3D nature of the problem, hence 3D calculations are necessary for a more realistic modeling. To simplify the matter, a square columnar microstructure is suggested and used in the present investigation for thin-film diffu- sion barriers to s imulate the major effects. Impurity concentrations are il lustrated as a function of posi- tion and time, and the influence of the ratio of bound- ary to lattice diffusion coefficients is also qualita- tively studied. From the point of view of grain-bound- ary diffusion, TiN films with a thickness in the range of tens ofnanometers are shown to be effective barri- ers to prevent Al-Si intermixing in Si microelectronic devices for several years of operation.

A C K N O W L E D G M E N T S

This work was supported by the Alberta Microelec- tronic Centre, and the Natural Sciences and Engi- neering Research Council of Canada. The authors would like to thank Dr. Donard de Cogan and Mr. Marcos Lam for their useful discussions.

R E F E R E N C E S

1. P.B. Ghate, Physics Today 39 (10), 58 (1986). 2. See, for example, papers contained in VLSI Metallizations:

Physics & Technologies, ed. K. Shenai (Boston: Artech House, 1991).

3. M. Wittmer, J. Vac. Sci. Technol. A 2, 338 (1984).

4. H.P. Kattelus and M.-A. Nicolet, Diffusion Phenomena in Thin Films and Microelectronic Materials, ed. D. Gupta and P.S. Ho (Park Ridge, NJ: Noyes, 1988), p. 432.

5. G.H. Gilmer and H.H. Farrell, J. Appl. Phys. 47, 3792 (1976); ibid. 47, 4373 (1976).

6. C.Y. Chen and H.L. Huang, J. Vac. Sci. Technol. 18, 398 (1981).

7. D.L. Losee, J.P. Lavine, E.A. Trabka, S.-T. Lee and C.M. Jarman, J. Appl. Phys. 55, 1218 (1984).

8. J. Kucera, B. Million, J. Zidu and V. Hermansky, Thin Solid Films 230, 183 (1993).

9. J.P. LavineandD.L. Losee, J.Appl. Phys. 56,924(1984);ibid. 58, 4483 (E) (1985).

10. J.P. Lavine, J. Appl. Phys. 59, 1986 (1986). 11. R.N. Singh, VLSI Eleetronics, vol. 15: VLSI MetaUization, ed.

N.G. Einspruch, S.S. Cohen and G. Sh. Gildenblat (Orlando: Academic Press, 1987), p. 41.

12. P.B. Johns, Int. J. Numer. Method Eng. 11, 1307 (1977). 13. P.B. Johns and G. Butler, Int. J. Numer. Method Eng. 19,

1549 (1983). 14. P. Enders and D. de Cogan, Int. J. Numer. Model.: Electron.

Netw. Devices Fields, 6, 109 (1993). 15. X. Gui, S.K. Dew, M.J. Brett and D. de Cogan, J. Appl. Phys.

74, 7173 (1993). 16. X. Gui, P.W. Webb and G.B. Gao, IEEE Trans. Electron Dev.

ED-39, 1295 (1992). 17. X. Gui, P.W. Webb and D. de Cogan, Int. J. Numer. Model.:

Electron. Netw. Devices Fields 5, 129 (1992). 18. P.W. Webb andX. Gui, Int. J. Numer. Model.: Electron. Netw.

Dev. Fields 5, 251 (1992). 19. K.G. Grigorov, G.I. Grigorov, M. Stoyanova, J.L. Vignes, J.P.

Langeron, P. Denjean and J. Perriere, Appl. Phys. A 55, 502 (1992).

20. G.I. Grigorov, K.G. Grigorov, M. Stoyanova, J.L. Vignes, J.P. Langeron and P. Denjean, Appl. Phys. A 57, 195 (1993).

21. J.D. Hodge, J. Am. Ceram. Soc. 74, 823 (1991).