*Corresponding author. Tel.: #86 25 3363222; fax: #86 253363222; e-mail: lw@seu.edu.cn.

Nuclear Instruments and Methods in Physics Research A 423 (1999) 213—222

Simulation of field emitter array in diode model

Lei Wei*, Wang Baoping, Yin Hanchun

Department of Electronic Engineering, Southeast University, Nanjing 210096, People+s Republic of China

Received 22 January 1998; received in revised form 27 August 1998

Abstract

In the application of field emitter arrays, it is useful to understand its emission performance. This paper calculates theelectric field of a diode emitter array in three-dimensional space. From this calculation, the emission uniformity of a diodearray is studied. The dependance of the tip-to-tip spacing and of the cathode—anode distance on the uniformity of thearray is also analyzed. The pressure sensor based on the diode model is simulated. From the dependence of the emissioncurrent for different tip densities, the influence of the emission uniformity can be obtained. ( 1999 Published byElsevier Science B.V. All rights reserved.

PACS: 85.45.Db; 41.20.Cv; 79.70.#q

Keywords: Field emitter array; Diode model; Three-dimensional space; Uniformity; Emission current; Pressure sensor

1. Introduction

The performance of single field emitters has beenintensively studied over the past few years [1—3].However, it is desirable to employ field emitterarrays in the application of field emission displays,microsensors and other devices. It is important tounderstand the emission performance of field emit-ter array. Few papers have been devoted to specifythe emission characteristic of a periodic structuredarray [4,5].

Because of the interaction of the field betweenthe tips and the influence of fringe field, the emis-sion currents from the tips are usually not uniformover the field emitter array [6,7]. This paper studies

the emission uniformity of a diode array. An im-proved finite difference method is used to calculatethe potential distribution of the whole emitter arrayin Cartesian coordinates. The Fowler—Nordheimequation is applied to compute the emission cur-rent density [8], and the emission performances ofdifferent tips are obtained. The geometrical para-meters of the array, such as spacing between tips,distance between cathode and anode, etc., arechanged and the variation of the emission uniform-ity is calculated. It has been found that the emis-sions of the tips in the center part of the array aresensitive to the tip-to-tip spacing. The uniformity ofthe emission of the array can be improved if thedistance between the cathode and the anode isdecreased.

The pressure sensor based on the diode array isalso analyzed. The distributions of the currents

0168-9002/99/$ — see front matter ( 1999 Published by Elsevier Science B.V. All rights reserved.PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 1 2 1 5 - 7

Fig. 1. Grids in irregular mesh.

from the tips are obtained for different pressurevalues. The influence of the emission uniformity isdiscussed.

2. Simulation method

The finite difference method is employed to cal-culate the electrostatic potential of the whole arrayin three dimensions. The method is improved to fitarbitrary boundaries in order to calculate the po-tential distribution of the emitter array accurately.

Neglecting the space charge effect in vacuumbetween the tip and the anode, the electrostaticpotential can be calculated from Laplace equation

L2uLx2

#

L2uLy2

#

L2uLz2

"0. (1)

To fit arbitrary boundaries, the mesh used is irregu-lar. From Fig. 1, Eqs. (2) and (3) can be obtained

u(i,j,k)

!u(i~1,j,k)

"[x(i,j,k)

!x(i~1,j,k)

]Lu

Lxl(i,j,k)

#[y(i,j,k)

!y(i~1,j,k)

]Lu

Lyl(i,j,k)

#[z(i,j,k)

!z(i~1,j,k)

]Lu

Lzl(i,j,k)

,

u(i,j,k)

!u(i,j~1,k)

" [x(i,j,k)

!x(i,j~1,k)

]Lu

Lxl(i,j,k)

#[y(i,j,k)

!y(i,j~1,k)

]Lu

Lyl(i,j,k)

#[z(i,j,k)

!z(i,j~1,k)

]Lu

Lzl(i,j,k)

,

u(i,j,k)

!u(i,j,k~1)

" [x(i,j,k)

!x(i,j,k~1)

]Lu

Lxl(i,j,k)

#[y(i,j,k)

!y(i,j,k~1)

]Lu

Lyl(i,j,k)

#[z(i,j,k)

!z(i,j,k~1)

]Lu

Lzl(i,j,k)

.

(2)

LuLx

l(i`1,j,k)

!

LuLx

l(i,j,k)

"

L2uLx2

l(i,j,k)

[x(i#1, j, k)

!x(i, j,k)]#L2u

LxLyl(i,j,k)

[y(i#1, j, k)

!y(i, j, k)]#L2uLxLz

l(i,j,k)

][z(i#1, j, k)!z(i, j, k)],

Lu

Lxl(i,j`1,k)

!

Lu

Lxl(i,j,k)

"

L2uLx2

l(i,j,k)

[x(i, j#1, k)

!x(i, j,k)]#L2u

LxLyl(i,j,k)

[y(i, j#1,k)

!y(i, j, k)]#L2uLxLz

l(i,j,k)

][z(i, j#1, k)!z(i, j, k)],

Lu

Lxl(i,j,k`1)

!

Lu

Lxl(i,j,k)

"

L2uLx2

l(i,j,k)

[x(i, j, k#1)

!x(i, j,k)]#L2u

LxLyl(i,j,k)

[y(i, j, k#1)

!y(i, j, k)]#L2uLxLz

l(i,j,k)

[z(i, j, k#1)

!z(i, j, k)]. (3)

214 L. Wei et al. /Nucl. Instr. and Meth. in Phys. Res. A 423 (1999) 213—222

Fig. 2. Simulated diode structure (not to scale).

Fig. 3. Six tips are selected to describe the performance of thearray.

The potential of grid (i, j, k) reduces to the simpleform

u(i,j,k)

"

!1

¹0

[¹1u(i`1,j,k)

#¹2u

(i`1,j~1,k)

#¹3u

(i`1,j,k~1)#¹

4u

(i,j`1,k)

#¹5u(i~1,j`1,k)

#¹6u

(i,j`1,k~1)

#¹7u(i,j,k`1)

#¹8u(i~1,j,k`1)

#¹9u(i,j~1,k`1)

#¹10

u(i~1,j,k)

#¹11

u(i,j~1,k)

#¹12

u(i,j,k~1)

], (4)

where coefficients ¹0—¹

12are determined by the

coordinates of grids. Because we want to study theemission performance of the whole array, all thetips in the array are covered with a single mesh.A coarse electric field is calculated on this mesh.Due to the limitation of computer memory andCPU time, the number of grids in this mesh is lessthan 4 000 000. At least three submeshes are de-fined around each tip in the array. With these smallmeshes, the precise electric field around the tips canbe obtained.

From the electric field, the current density isobtained by Fowler—Nordheim equation [1]

J"1.54]10~6E2

/t2(y)

]expA!6.87]107/3@2l(y)

E B A/cm2, (5)

where E is the electric field at the tip in V/cm, l(y)and t2(y) are electric field dependent elliptical func-tions, / is the work function of the emitter materialin eV, and y is the image charge lowering contribu-tion to the work function, is given by y"3.79]10~4 E1@2//. For simplicity, it is generallyassumed that t2(y)"1.1 and l(y)"0.95!y2.

3. Emission uniformity of the tips in the array

The emitter array in the diode mode is analyzedin this paper. Fig. 2 shows an array which has 36tips. Because of the symmetry, six tips (shown inFig. 3) are selected to represent the performance ofthe array.

In the simulation, it is assumed that the radius ofthe tip is 10 nm, the tip-to-collector distance is3 lm, the height of the emitter is 4 lm and theapplied voltage is 50.0 V. In the field emitter array,the tip-to-tip spacing affects the electric field.Figs. 4 and 5 give the equi-potential curves fordifferent tip-to-tip spacings. Figs. 6 and 7 show theeffect of the tip-to-tip spacing on the electric fieldand the emission current density. The electric fieldat tip 6 is largest in the array. Due to the fieldinteraction, the electric field is smaller at the centertips than that at corner tips. Therefore, the densityof the emission current at tip1 is much smallerthan the current density at tip 6. The ratios ofE5*11, 2, 5*15

to E5*16

and J5*11,2, 5*15

to J5*16

are usedto represent the uniformity of the array.

When the tip-to-tip spacing decreases, the fieldinteraction between the tips becomes stronger.Hence, the differences of electric field and emission

L. Wei et al. /Nucl. Instr. and Meth. in Phys. Res. A 423 (1999) 213—222 215

Fig. 4. Equi-potential curves with tip-to-tip spacing of 10.0 lm.

Fig. 5. Equi-potential curves with tip-to-tip spacing of 5.0 lm.

Fig. 6. Effect of the tip-to-tip spacing on the electric field at the tips.

current density between tip 1 and tip 6 increases.The uniformity of the electric field and the emissioncurrent density becomes worse if tip-to-tip spacingdecreases.

The variation of the performance of the array asthe density of the tips increases is also studied here.It is assumed that the number of tips in the fieldemitter array is increased for a constant length andwidth of the array (60.0 lm). Figs. 8 and 9 show theelectric field and the current density as function ofthe number of tips. Fig. 10 gives the variation oftotal emission current.

In Figs. 8 and 9, E.*/

and E.!9

represent theminimum and the maximum electric field at tips inthe array, J

.*/and J

.!9represent the minimum and

maximum current density of the tips in the array.As shown in these figures, the uniformity of theelectric field and of the current density becomesworse as the density of tips increases. When thenumber of tips is 144, the ratio of emission currentdensity between center tip and corner tips is only0.44. Therefore, more emitters do not necessarilylead to higher emission current. In Fig. 10, theemission current is maximum when the number oftips is 64.

216 L. Wei et al. /Nucl. Instr. and Meth. in Phys. Res. A 423 (1999) 213—222

Fig. 7. Effect of the tip-to-tip spacing on the emissions of the tips.

Fig. 8. The variation of uniformity of electric field as the number of tips is increased, E$"E

.*//E

.!9.

L. Wei et al. /Nucl. Instr. and Meth. in Phys. Res. A 423 (1999) 213—222 217

Fig. 9. The variation of the uniformity of the current density as the number of tips is increased, J$"J

.*//J

.!9.

Fig. 10. The variation of total emission current as the number of tips is increased.

218 L. Wei et al. /Nucl. Instr. and Meth. in Phys. Res. A 423 (1999) 213—222

Fig. 11. Effect of cathode—anode distance d on electric field at tips.

Fig. 12. Effect of cathode—anode distance d on emission of tips.

L. Wei et al. /Nucl. Instr. and Meth. in Phys. Res. A 423 (1999) 213—222 219

Fig. 14. Distribution of the emission current from the tips, the tip-to-tip distance is 15.0 lm.

Fig. 13. Pressure sensor based on field-emission diode array.

Another parameter which affects the uniformityof emission of the array is the distance betweencathode and anode. We have studied the variationof the emission performance of tips while the para-meter d is decreased. In this calculation, the tip-to-tip spacing is kept at 7.0 lm.

Fig. 11 shows that the electric field at thetips increases with decreasing distance betweencathode and anode. The density of the emission

current also increases while d decreases. Theuniformity of the electric field and the uniformity ofthe emission current density of the array are im-proved as the cathode—anode distance decreases(Fig. 12).

4. Analysis of the pressure sensor

The pressure sensor based on the diode array isshown in Fig. 13.

The shape of the anode diaphragm changes un-der pressure. For the application of the pressuresensor, it is important to know the distribution ofthe emission currents from the tips. This distribu-tion depends on the pressure applied on anodediaphragm. However, the emission uniformity ofthe diode array also influences this distribution.The pressure sensor shown in Fig. 13 is simulatedhere. In the calculation, the radius of the tips isassumed to be 10 nm, the tip-to-collector distancewithout pressure is 3.1 lm.

220 L. Wei et al. /Nucl. Instr. and Meth. in Phys. Res. A 423 (1999) 213—222

Fig. 15. Distribution of the emission current from the tips, the tip-to-tip distance is 20.0 lm.

Fig. 16. Distribution of the emission current from the tips, the tip-to-tip distance is 25.0 lm.

L. Wei et al. /Nucl. Instr. and Meth. in Phys. Res. A 423 (1999) 213—222 221

The variation of a thin silicon diaphragm can beexpressed as

DL4wLx4

#2HL4w

Lx2Ly2#D

L4wLy4

"p,

where h is the diaphragm thickness. D" 1.42]106 h3 N cm, H"1.12]106 h3 N cm for silicondiaphragm and w is the deflection of diaphragm.w can be expressed as the sum of three components.These components represent the deflection ofa simply supported diaphragm and the deflectioncaused by the bending moments distributed alongtwo pairs of edges [4].

The emission currents from different tips are cal-culated as function of the deflection of diaphragm.Figs. 14—16 give the dependence of the tip currentfor different tip densities.

In the calculation of curves 1—3, the pressure is200, 150 and 100 lN/cm2, respectively. The max-imum currents I

.!9emitted from tips are different

in these figures. In Fig. 16, the value of I.!9

islargest. The curves in Figs. 14 and 15 are nor-malized by this value.

Because the distance between the tip and theanode is large in the simulation model, the emissioncurrent from tip is not very sensitive to the deflec-tion of the diaphragm. In these figures, the tipdensities are different, so the curves of emissioncurrent are also different for the same pressure. Dueto the influence of the emission uniformity, thecurve of the emission current in Fig. 16 is mostsensitive to the pressure.

5. Conclusion

The field emitter array has already been em-ployed in field emission display, microsensor andother devices. It is important to understand theemission performance of array. In this paper, thefield emitter array in diode model is simulated withthe finite difference methods in three dimensions.

The simulation shows that the emissions of the tipsare not the same in the array. The emission currentdensity of the tips near the edge of the array islarger than that of tips in the center. As the tip-to-tip spacing decreases, the uniformity of the emis-sion from tips became worse. From the result of thesimulation, it can be seen that the total emissioncurrent decreases if the density of tips is too high.

The distance between cathode and anode is an-other parameter which affects the uniformity of theemission. The electric field and current density in-crease with decreasing distance between cathodeand anode. The uniformity of emission is improvedin this case. A pressure sensor based on diode arraymodel is discussed. Due to the different emissionuniformity, the sensitivity of sensor to the pressureis different for different tip density.

Acknowledgements

The authors wish to thank Jean Coolen, Gerardvan Poppel and Dany Benoy for their encourage-ments and support.

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