A theoretical study on field emission diodes

applied surface science ELSEVIER

Applied Surface Science 76/77 (1994) 58-60

A theoretical study on field emission diodes LlU

Weldong

Northwestern Polytechmcal Uml erstty, Box 374, Xl'an 710072, China

Luo Enze * Xtdtan Unu,erslty, Box 271, Xt'an 710071, China

(Received 2 August 1993, accepted for pubhcatmn 10 November 1993)

Abstract The relationship between the structure and performance of field emission diodes is analyzed By using the difference of the geometric factor of the emitter and that of the collector, A/3, as the measurement of single-side conductivity of field emission diodes, the quantitative formula of single-side conductivity is derived Furthermore, the following results are obtained from the formula (1) when the tip-to-collector distance (d) is much larger than the tip radius, the geometric factor of the emitter will be only approximately dependent on the tip radius, and (2) when the tip-to-collector distance (d) is much smaller than the tip radius, the positive and reverse performance of the field emission diode will be consistent, and the geometric factor [3 = 1 / d Finally, some applications on the structure design of micro field emission diode devices are discussed

1. Introductaon E x t e n s i v e p r o g r e s s in v a c u u m m l c r o e l e c t r o n l c s has b e e n m a d e in r e c e n t years V a r i o u s v a c u u m m t c r o e l e c t r o n l c d evi c e s have b e e n f a b r t c a t e d a n d measured But theoretical research between their s t r u c t u r e s an d p e r f o r m a n c e s has b e e n l lm t te d In this p a p e r , w e a t t e m p t to e s t a b h s h a m o d e l of field emission d i o d e s t h r o u g h t h e " t i p e f f e c t " A h e l d emission v a c u u m m l c r o e l e c t r o n l c d i o d e can be s h o wn as Fig 1, w h e r e a is t h e a n o d e or collector, an d c is the c a t h o d e or tlp T h e tip-toc o l l e c t o r distance d is usually several m l c r o m e -

* Corresponding author Fax +86295262281

ters or less, and t h e tip radius R 0 ts usually tens o f n a n o m e t e r s Obviously, this is an ideal device, and does not take into a c c o u n t the i n f l u e n c e of

d

£°

7"2 Fig 1 A schematic diagram of a field emission vacuum electronic diode, a anode, c cathode, d tip-to-collector d~stance, R 0 tip radms

0169-4332/94/$07 00 © 1994 Elsevier Science B V All rights reserved SSDI 0169-4332(93)E0315-D

59

L Wetdong, L Enze/Apphed Surface Science 76 / 77 (1994) 58-60

+

the insulating regions b e t w e e n the c a t h o d e and the a n o d e W e usually use the formula E =/3AV

ka:o

Q)mHH J,umNm i

IS

I

t "W ~x

) I

I i

(1)

to calculate the field mtensity E of a p o m t on the surface of a cathode, where /3 is the so-called geometric factor of the c a t h o d e with dimensions of inverse length, and AV is the extracting voltage b e t w e e n the collector and the c a t h o d e It is k n o w n [1,2] that the geometric factor of a point on the surface of the c a t h o d e has the following relattonshtp with respect to surface curvature k

c~

¢b)

k,' k(v) I

j

\

I

I.

I

//

/

0

°"'d

I /3 = e x p ( - Z - k A n ) - 1 '

where An ts the force line path from a point on any equlpotenttal surface or a p o m t on the surface of the collector along a force line AV is the potential difference b e t w e e n two points, /¢ is called " t h e m e a n of k for the k - V curve", 1 ~aVk V /~ = A-V J0 ( ) dV

(3)

and ~: is called " t h e m e a n of k for the k - n curve"

do

2kc (5)

where k c is the m e a n curvature of the c a t h o d e

2. The single-side conductivity of field emission vacuum microelectronic diodes W e assume tic 1s the geometric factor of the cathode, and /3a IS that of the a n o d e As shown in Fig 2, /3c is the geometric factor of the point A (forward bias), /3a IS that of the point B (reverse bias), and A and B are two ends of the same force hne

(~)

Fig 2 Structure analysis of three typical field emission vacuum microelectromc diodes, AV extracting voltage, An force line path, k c curvature of cathode, k a curvature of collector (a) kc>0, k a > 0 , (b) k c > 0 , k a = 0 , (c) k c > 0 , k a < 0 , (d) k - V curve of three typical structures along the line from point A to point B

F r o m Eqs (2) t h r o u g h (4), we obtain 2k tic=

e x p ( - 2 7 ¢ A n ) -- 1 '

(4>

Here, k is the m e a n curvature of equlpotentlal surface or the " G r e e n ' s c u r v a t u r e " T h e approximate f o r m of Eq (2) is /3 = e x p ( - 2 k c A n ) - 1 '

Cc )

(2)

i.~

\

(6)

-2/7: /3a = exp(21cAn) -- 1

(7)

Therefore, the difference between /3c and /3a ( i e , the difference of the geometric factor of emitter with forward bias and with reverse bias) is a/3 =/3 c - / 3 a = - 2k

(8)

T h e difference of field intensity of forward bias and that of reverse bias IS AE=A/3

/IV=-2/~AV

(9)

T h e above equation represents the smgle-stde conductivity of field emission diodes Apparently, the larger the value of AE, the better the singleside conductivity T h r e e typical structures of field emission diodes and their k - V curves are shown in Fig 2 T h e value of ~:AV of the structure shown m Fig 2a is the largest a m o n g three structures, so its slngle-stde conductivity ts the best

L Wetdong,L Enze /Apphed Surface Sctence 76/77 (1994) 58-60

60

H o w e v e r , the A E o f the s t r u c t u r e s " a " a n d " b " 1s very a p p r o x i m a t e while k c >> k a ( w h e r e ka is the m e a n c u r v a t u r e of the collector), b u t the f a b r i c a t i o n o f t h e s t r u c t u r e " b " is easier, so structure " b " is also a g o o d s t r u c t u r e In s t r u c t u r e " c " , w h e n k~ = - k ~ , A E = 0 the p e r f o r m a n c e of f o r w a r d bias is the s a m e as that of reverse bias, f u r t h e r m o r e , the single-side c o n d u c t w l t y will be inverse while k c < - k a

a/3 =

3. The influence of structure p a r a m e n t s on performance

/3c = /3~ =

W e know [3] t h a t w h e n /c > 1~An, /3 1s only d e p e n d e n t on /~, a n d a l m o s t i n d e p e n d e n t of An, a n d w h e n 7¢ < 1~An, /3 1S only d e p e n d e n t on An, a n d a l m o s t i n d e p e n d e n t of k c In this section, we will analyze t h e influence of tip r a d i u s ( R 0 = ( 1 / k c ) ) a n d t i p - t o - c o l l e c t o r dist a n c e ( d ) on t h e g e o m e t r i c factor 13 with t h e a p p r o x i m a t e f o r m u l a (5) (1) d > > R 0, k~An >> 1, e x p ( - 2 k ~ A n ) = 0, t h e n /3 = - 2 k c -

(10)

l e , w h e n d >> R0, /3 IS a p p r o x i m a t e l y inversely p r o p o r t i o n a l to the tlp r a d i u s (2) d << R~, k~An << 1, c o n s i d e r i n g a field emission d i o d e , the diff e r e n c e o f the g e o m e t r i c f a c t o r of the tip a n d that of t h e c o l l e c t o r can be o b t a i n e d from Eqs (5) a n d (8),

(

A/3=2k c exp(-2k~An)-i

- exp(2k~-Sn) - 1

)

(11)

E x p a n d i n g the e x p o n e n t i a l t e r m s to the p o w e r series m t h e above f o r m u l a a n d r e t a i n i n g the first t i m e t e r m s smce kcAn << 1, we o b t a i n

A/3 = 0

series to s e c o n d time terms, t h e n 1

2 (13)

An kcAn - 1

B e c a u s e kcAn << 1, A/3 -~

2 An

(14) 1

1

An

d

(15)

Thus, the a p p r o x i m a t e r e l a t i o n s h i p b e t w e e n t i p - t o - c o l l e c t o r d i s t a n c e a n d g e o m e t r i c factor 13 ~s o b t a i n e d in which, w h e n kcAn << 1, 13 IS mversely p r o p o m o n a l to the t i p - t o - c o l l e c t o r d i s t a n c e a n d I n d e p e n d e n t of the tip r a d i u s T h e results o b t a i n e d are in a g r e e m e n t with t h e results of n u m e r i c a l analysis [5], b u t this m e t h o d ~s s i m p l e r a n d m o r e effective

4 Apphcatlons

2 R0

_1

In o r d e r to get the f o r m u l a of A3, while kcAn << 1, we r e t a i n the e x p a n s i o n s o f the p o w e r

(12)

T h a t is, w h e n t h e t i p - t o - c o l l e c t o r d i s t a n c e ( d ) is m u c h s m a l l e r t h a n t h e tip radius, the p o s m v e a n d r e v e r s e p e r f o r m a n c e of the field e m i s s i o n d i o d e wdl b e c o n s i s t e n t T h e result has b e e n c o n f i r m e d by S T M (scanning t u n n e h n g microscopy) e x p e r i m e n t s [4]

A s i n d i c a t e d above, a variety o f v a c u u m microe l e c t r o n i c d i o d e devices can be d e s i g n e d with d i f f e r e n c e p e r f o r m a n c e s by a d j u s t i n g two p a r a m eters - the tip r a d i u s R 0 a n d the t i p - t o - c o l l e c t o r d i s t a n c e d F o r e x a m p l e , if it is to be used as a high-voltage recttfier or switch device, it s h o u l d have a large t i p - t o - c o l l e c t o r d i s t a n c e a n d the s t r u c t u r e shown m Fig 2a or F i g 2b H o w e v e r , if the device is to be used in p r e s s u r e sensors, its t i p - t o - c o l l e c t o r d i s t a n c e should be small so as to o b t a m g o o d sensitivity Finally, t h e s t r u c t u r e shown in Fig 2c can be used in stabllovolt devices with reverse o p e r a t i o n

5. References [1] E Luo, J Phys D Appl Phys 19 (1986) 1 [2] E Luo, J Phys D Appl Phys 20 (1987)1609 [3] E Luo and C Win, Tech Digest, 4th Int Vacuum Microelectronics Conf Nagahama, Japan, 1991, p 122 [4] H Q Hguyen et al, IEEE Trans Flectron Devices 36 (1989) 2671 [5] H-C Lee and R-S Huang, IEEE Trans Electron Devices 39 (1992) 313