Dispersion characteristics of thin films capacitances caused by surface trapping

Thin Solid Films - Elsevier Sequoia

S.A., Lausanne

- Printed

in Switzerland

R17

Short Communication

Dispersion characteristics A. VAN CALSTER

Laboratory

(Received

caused by surface trapping

AND H. J. PAUWELS

of Electronics,

February

of thin film capacitances

Ghent State University,

Ghent (Belgiumj

5, 1971)

The study of thin film transistors has revealed that the instability of the characteristics is, in part, caused by ion migration in the insulating layer. To explain this drift mechanism, Tickle’ proposed a surface trapping model, in which it is assumed that mobile ions are generated from surface traps, move quickly through the bulk and are retrapped at the other side. This model can be used to gain insight in the dispersion characteristics of thin film capacitors at very low frequencies. Results obtained by Argall and Jonscher’ and explained by them in terms of slow carrier transport through the bulk of the insulator and carrier accumulation or depletion at the contacts, can be explained by the model of Tickle. We consider a parallel plate capacitor in which ions are trapped at the surfaces of the insulator in potential wells of energy depth W (Fig. 1). A potential V is applied to the capacitor. Since a rapid transit through the insulating film L

Fig.

I. Energy diagram

of’ions

in thin film capacitor

Thin Solid Films, 7 (1971) R17--R20

i 18

SH()R I' ('OMMUNI('ATIONS

is a s s u m e d a priori, no space charge is present m the bulk. Le), J/~ and t72 be n u m b e r o f ions t r a p p e d at the left-hand-side and r i g h t - h a n d - s i d e surface unit area. respectively, a n d let />, be the p r o b a b i l i t y per unit time tbr an to escape a t r a p at the left a n d be r e c a p t u r e d by a t r a p at tire l'ighl: /~2 is p r o b a b i l i t y per unit time for the inverse process. We o b t a i n filch dn

tile pc~ ion the

I

.....

iq 11)1 +112132.

de

{11

dll 2

(2)

tl11) 1 --1121> 2 .

d!

In thermal e q u i l i b r i u m , d H j d l = tt and. t h c r e | o r e

l i i = P2

(3)

On the o t h e r hand, if the traps at left arid right are identical ira nature a n d in n u m b e r , we m u s t have a t h e r m a l e q u i l i b r i u m (F'ig. 1) tll

C qV.'k't'

{4)

H2

where q is the charge o f the ion. "fherefore P2

e

(5)

qV,,kT.

P1 We m a y t h e r e f o r e a s s u m e Pl = eql 2kTf{l )" I72

e d 2/(1['(1").

(6)

where f ( I ' ) will d e p e n d on a m o r e detailed m o d e l for thc traps. F r o m eqns. (1) and (2), it follows t h a t the total n u m b e r o f ions involved is a c o n s t a n t , say' N, and therefore

dt'l + dt

I l l ( P l - k t r ) 2 ) - - - ,Np2 '

(71

11 we a s s u m e thai 1," is small e n o u g h for l i n e a r i s a t i o n o f eqn. {6t in terms o f l . we o b t a i n

p2=f(0) l)l --1>2 =

[1

1 qV

f'(O) lq

--2 kT + f(0) j"

2f(O)[1 + I" f'(O)] " ?io)l

If we neglect the term p r o p o r t i o n a l to b" in eqn. (9). we o b t a i n Thin Solid t'7/,11v 7 (1971) RI7 R20

(,R) (9)

SHORT COMMUNICATIONS

R 19

dna nl N + dt z 2z

(10)

A V, z

where

1

2f(0)' A= f N q

r

kT

2f'(0)]

(11)

f(O) 3"

If we assume an exp(jmt) time dependence for V, we obtain from eqn. (10) for the a.c. component of n~, in complex representation: -A V. nl - 1 +j~o~

(12)

If ~ is the charge per unit area on the left-hand-side metal electrode, we have for the current in the external circuit dT~ j

= --.

(13)

dt

If e~ is the dielectric constant and L the thickness of the insulating layer, we have V e ~ -L = rc + q n l . (14) In complex notation, we obtain therefore goe

J = -~jwV-ja~qn

jmVIe~+

1

AqL

]

(15)

c

The insulating layer therefore behaves as if its dielectric constant were given by

B 1 +je)z

e = e~ + -

; B = AqL.

(16)

The results obtained by Jonscher and Argall 2 for their type B dispersion are given by

B = e ~ + 1 +(jcoz) t - "

(17)

but it is well known 3 that eqn. (17) is derived from eqn. (16) by assuming an appropriate distribution of the time constant z. The result, eqn. (16), is independent of a specific model for the trapping centers (apart from the value of B and z), provided the transit time through the insulating layer is very small with respect to z, and the voltage V is small enough to permit linearisation of eqns. (6). In order to gain insight we have Thin Solid Films, 7 (1971) RI7-R20

R20

SttOR I ('OMMUNI('ATIONS

analysed a specific model in which the ions from the traps are injected at a distance x o from the surface (x o ~ m e a n free path), travel by drift and diffusion to the surfaces, and are recaptured in the traps with zero time constant. We find for eqns. (6) and positive V, ] t91 =

ve

W'kT

e(

q|','kT}(x,l ll

1

e qv,kr

, P2 = P1 e

q ~ r '1~ "~ r

{18)

where I+" is the energy depth o f the traps and v is the oscillation frequency of the ions in the traps 4. The conditions for linearisation of eqns. (18) is q}~ << 1

(19)

kT

and the constants r and A are given by 1 = 2re

r

~,'ktXo

. A

L

1+

4 kT ~

t20)

L!

The value of 0.8 eV given by Argatl and Jonscher= for the activation energy 1+ of the time constant r corresponds with this o f the forward drift measured by Hofstein. F r o m the value of a in eqn. (17) one should in principle be able to gain information about the distribution o f the trap depth [4". Finally the transit time through the insulating layer is o f the order L2/2D where D is the diffusion constant: this time should be much smaller than r. REFERENCES

I 2 3 4 5

E.J. SW~Slt:M A'qD A. C. TI('~:IE, IElili lraH.~, lf/c~, l)cvicc~, l!D.14(I967) 760 F. ARGALL AND A. K. JONSCHER. Thin Solid Fi[m3, 2 (1968) 185, K . S . COLF A~D R. H. COLE, .L Chemical Physics, 9 (1941) 341. C. Kn"IH, Introduction to Solid State Physics, Wiley. New York. 1956, p. 485. S. R. HOFSTUN, I E E E 1)'arts. Elec. Devices, ED-13 (1966) 222.

Thill So/M Fi/,~rs, 7 (1971) RI7 R20