Solid-State Electronics, Vol. 17, pp. 1163-1169. Pergamon Press. Printed in Great Britain

A S I M P L E M O D E L OF A B U R I E D C H A N N E L C H A R G E C O U P L E D D E V I C E

A. W. LEES* Allen Clark Research Centre, The Plessey Company Limited, Caswell, Towcester,

Northampton, England

and

W. D. RYAN

Queens University, Belfast, N. Ireland

(Received 17 January 1974)

Abstraet--A one-dimensional model of a buried channel CCD is presented. The potential equations, using the depletion region approximation are solved for a range of doping levels and stored charge. In this way values of device capacitance are calculated.

It is shown that for a device with uniform layer and substrate dopings, the equations have an analytic solution. These solutions take a rather complicated form, but have a simple geometrical representation. A geometrical construction, based on field plots, is given. This construction may be used in the study of more general doping profiles.

EO

Eox

P~ p2

N, Nz

e B,C,D, E,F,G

L Q

NOTATION

T oxide thickness ~-' 'effective' oxide

~ l potentials

E relative permittivity of silicon = 11-7 permittivity of free space relative permittivity of oxide layer charge density substrate charge density layer doping substrate doping electronic charge

are integration constants layer depth stored charge per unit area

1. INTRODUCTION

The calculation of the potential in a buried channel charge coupled device presents a nonlinear two dimensional problem which is very demanding in terms of computing time. In the later stages of the design of such a device, a full two dimensional calculation must be carried out, but initially a

*Address as from 13th "May: Scientific Services Department, Central Electricity Generating Board, Ratcliffe-on-Soar, Nr. Nottingham, England.

simpler model is needed to give some insight into device behaviour.

Amelio[1] analysed the potential in a surface CCD using finite difference techniques. The machine requirements for such a calculation are severe. McKenna and Schryer [2] have presented a technique based on Fourier transformation of the potential in both surface and buried channel devices. Their analysis, however, assumes a com- pletely depleted layer. It is of great interest to consider the effect of the presence of mobile charge in the layer. This problem has been considered recently by Kent[3], who solved the nonlinear Poisson equation.

In the present paper a simpler approach, based on the depletion layer approximation, is presented. The analysis is given for a device with constant layer doping, though it is easily extended to linear doping profiles.

A one dimensional model is considered. Figure 1 shows the geometry of a typical device; it may be seen that the variation of potential under the centre of one of the plates is much greater in the direction perpendicular to the plate than in the direction of charge transfer. This observation suggests that we may gain useful insight by considering the variation of potential along a line, from the centre of one of the electrodes to the substrate. Consider the

1163

1164 A. W. LEES and W. D. RYAN

Transfer Input ga te ga te Output

Input\,, ~\\ "/ / g a t e / O u t p u t

, / /' / / /'

' / / \ j

p type layer

~ _ _ ~ Alurnmium

~ , L ~ - ~ p+ "isolat ion" wail

n +

-- Depie- ion layer boundaries

Fig. 1. The bur ied channe l s t ructure.

overall charge neutrali ty, a constant potential results. Beneath the channel, the remainder of the layer is depleted and so has a parabolic potential. In the substrate there is deplet ion up to a depth L. Thus the potential in this region has a parabolic form with posit ive curvature (for a device with a p - type substrate). At each region interface con- tinuity of potential and field are required. The overall potential distribution is of the form shown in Fig. 2. In section 2 the equat ions are solved for a device of constant doping. Sect ion 3 gives a simple geometr ical interpretat ion to the results.

2. EQUATIONS OF POTENTIAL

A device with constant layer doping N~ and substrate doping N2 is considered. Figure 2 is used to define some of the symbols used.

7' is the equivalent thickness of the oxide layer defined as

e T T p - -

potential distribution of such a model. In the oxide, since there is zero space charge, a linear potential plot results in this region. In the layer, which in this instance is taken as being constant ly doped, the signal charge (majori ty carriers) reside in a small region a < x < b about the maximum potential . To a first approximat ion, we may assume the signal charge distributes itself in such a way as to neutralise the space charge of the dopant ions. This results f rom the neglect of diffusion. Thus in the first region of the layer (i.e. o < x < a) the constant doping gives a parabolic potential curve. In the charged channel region (a < x < b ) , as there is

where T is the actual thickness of the oxide layer, e and eo~ being the dielectric constants of the oxide and semiconductor , respect ively.

In the oxide, i.e.

and we know that

- T < x < o

& , ( T ' ) = V (1)

being the potential applied to the gate. By

~ J

/

X=o X=a X=D X , = d

S u b s t r a t e Layer

L X = L

Fig. 2. Potential profile,

Model of a buried channel CCD 1165

considering the oxide as having it effective rather than actual thickness, continuity of both field and potential is imposed at the oxide-semiconductor interface, i.e. at x =0. This assumes that the surface charge (Qss) in the oxide is zero. The treatment of a general value of Q, presents no fundamental difficulties but does complicate the algebra.

In 0 < x < a, the potential is &fix)

of the device. This stored charge is an injected signal of majority carrier charge which will there- fore neutralise the charge on the dopant ions.

In the region below the channel, i.e. b < x < d

493(x)- P~ x2 + Dx + E (4) - - - - 2 8 8 0

for L > x > d

where

V2~2 -- pl 880

pt = N~e

e is being the electronic charge. p,(x) is the charge density of the dopant ions, N,

is their number density. Therefore

= -P~X--+B &fix) 288o x + C. (2)

At the surface the field is

and hence

dx x=0 = - B

hence

V:~4 - P: EEO

o,r '- &4(x)= - ~ + F x + G

2880

and for x I> L

O ( x ) = o .

(5)

The problem of fitting potential curves to the known quantities pj, p2, ~" d, Q, V is that of determining a, b, B, C, D, E, F, G, L.

To solve for these nine unknowns we have the set of equations:-

at x = 0

V + B r ' =02(0)

i.e.

V + 8BT= &frO). (3) 8ox

The region a < x < b represents the charge and carrying channel of the device. The mobile charge will distribute itself in such a way that the electric field just counteracts the diffusive force. In the doping regime of CCD's (i.e. about 10 ~5 per cent 3) the Debye length in silicon is small on the scale of the device. It follows that to a good approximation the effect of diffusive forces on the one dimensional potential distribution may be neglected. In this case some region a < x < b where the electric field must and be zero and hence the density of mobile carriers is equal to the doping since

V2dO = 0 = 1 [ _ p , _ Q / ( a _ b ) ] 88o

at b > x > a

also we have

8 T B V + = C (6)

E'ox

&ffa) = ~b3(b) (7)

b - a = Q / p ~ (8)

d 6 3 ( x ) ~-x ~=. = 0 (9)

d4,3(x) = 0 dx x=h

At the layer-substrate junction,

(10)

where Q is the stored charge per unit cross-section &3(d) = &g(d) (11)

1166

and

d(b3(x) = d4~,(x) dx ..... dx ....

A. W. LEES and W. D. RYAN

The final cond i t ions are gove rned by the dep le t ion

edge, L.

¢G(L ) = 0

d d~4(x ) ~XX ~=l. = 0 .

Equa t ions (13) and (14), giving the dep th of deple t ion in the subs t ra te , L, make the p rob lem

nonl inear . The solut ions of the set of equa t ions (6)-1141

involves s t r a igh t fo rward bu t r a the r ted ious a lgebra ; only the resul t is r eco rded here.

We define th ree new quant i t i es

QL 2p2ee,,

a = p,2/8eeop2

= Q + p,Q + p,L(p2-p,) /3 2 . . . . 4esop----~2 2eeop~

( p , - p 2 ) L 2 + Q2 + (p2-p, )ZL 2

3/ 2ee, ~ 2eeop,_

It may easily be shown tha t the sur face potent ia l , C, is g iven by

X -+ N/(X 2 - 4 Z )

The po ten t ia l at any poin t in the dev ice may be ob ta ined by back subs t i tu t ion . A l though these

(12) expres s ions are r a the r compl ica ted in appea rance , they may be eva lua ted ex t r eme ly rapidly.

The p a r a m e t e r s of par t icu lar in te res t are the po ten t ia l and pos i t ion of the channe l , the sur face field, and the deple t ion dep th in the subs t ra te . Table 1 shows some spec imen results . V repres-

113) ents the vol tage on the plate, V~,~ is the potent ia l of the charged channe l , and E,, is the e lectr ic field at

(14) the sur face of the semiconduc to r . The p a r a m e t e r s specified in the ca lcu la t ion are plate vol tage, oxide th ickness , layer doping and th ickness , and sub- s t ra te doping.

where

C =

X = 2V 4 eT p , ~ 4_ Q T

T2p, 2

+ 4aeo% ~,,

T2pl'-/3 2 e2T~Q 2 Z = V 2+ 16a2e02e~,~ 4- 4e2el2e~

VTp,~ VTQ 2aeoeox eoeo,

o~ J4eo2e ~,"

15)

An exp re s s ion may be g iven for the dep th of the channe l (centre)

Table 1. Layer thickness = 5 gm: oxide thickness = 0-1 ~xm

N, Q Eo (xl0,4cm 2) N: (10 ~Ccm ~) V V .... (V//xm)

30 3-0 11.0 1t 38.42 17.53 311 -3.0 5.0 0 19.05 11-96 30 -3.0 0.0 5.0 42-28 17.25 30 3.0 5.0 5.0 23.35 11.72 30 15.0 0-0 0.0 26.14 14.23 30 15-0 5.(! 0 14.09 8-36 30 -15.0 0.0 5.0 29.35 13-69 311 -15 '0 5-0 5.0 14.12 7-44 41) -4-0 0.0 0 51.23 23 "39 40 4.0 2-0 0 42"38 21.09 41) 4"0 4.0 0 34.45 18.86 41) -4.11 0.11 5 55.09 23-09 411 - 4"0 2.0 5 46.34 2/I-82 40 4.0 4-0 5 38-59 18.6 411 -4.0 11.11 10 58.94 22-82 411 --4.0 2.(1 10 50"38 20.56 40 4-0 4-0 10 42-77 18-35

Figure 3 shows the channe l potent ia l p lo t ted as a func t ion of f ree channe l charge. The slope of these cu rves at zero charge appear s to be insens i t ive to doping levels, ove r the range shown. It is, howeve r , s t rongly d e p e n d e n t on the layer th ickness . The slope of the cu rves in Fig. 3 r ep re sen t ing the dif ferent ia l s u s c e p t a n c e s of the dev ice is an i m p o r t a n t des ign pa rame te r . In all cases , the po ten t ia l s s h o w n are those for zero vol t s appl ied to the e lec t rode .

The des igner also needs some measu re of the efficiency with which the channe l potent ia l is con t ro l led by the e lec t rode voltage. The 'modu la - t ion efficiency' may be defined as

d V ...... change in channe l V d V change in applied V "

y _ a + b _ - /3 ± X / [ / 3 2 + 4 a ( y + V)] 116) 2 4 a U n d e r opera t iona l condi t ions , one requi res tha t

Model of a buried channel CCD

5 0 ~ ~ (5~m La'fer}

40 0.1-~- X+X/~ X {xl(~ 5eni ) (~IO jcl~ )

\ ! ! i i

,.E 2 0

\

0 5 ~0 15

Stored charge Q, x IO sC cm -~

Fig . 3. D i f f e ren t i a l s u s c e p t a n c e .

I 2O

1167

80

.o

_o ~ 6 o :

50

x• n:=5x I0 ~5 cm 3

x

\x ~ x

~ x

I J I I

-0.2 -0 4 -0 6 -0 8

%,

Fig. 4. Modulation efficiency.

-x

_[I 0

1168 A. W. LEES and W. D. RYAN

the change in the channel voltage with the introduction of mobile charge must be less than the change in channel voltage due to the change in potential of neighbouring electrodes. This condition is necessary to prevent spillage of signal charge into neighbouring wells. Typically, this limits the voltage change to about four volts. From Fig. 3 it is seen that this implies that the charge is limited to small (on the range considered) values. Figure 4 shows the modulation efficiency (taken at zero charge) for devices with a 5 p~ m layer. This case is presented as an example, since it will be seen from equations (15) and (16) that there must be some dependence on the layer thickness through the parameters a and /3.

The modulation efficiency is, to a good approxi- mation, dependent only on the ratio of the two doping levels p2/p~, and not on their absolute values. In all cases, the modulation efficiency increases with stored charge.

3. P I C T O R I A L R E P R E S E N T A T I O N

The analysis of the previous section leads to expressions which may be easily evaluated. Even in the simplest case of constant doping, the algebra does obscure a simple physical picture of what is happening. As is often the case with nonlinear algebraic or differential equations, a simpler formu- lation (if a less directly applicable one) may be obtained by interchanging dependent and indepen- dent variables. In the present case, let the position of the channel be assumed. The system is now

described by the relations

2

V ...... _ - p , a t - B a + V (17) 2 8 8 o

B - p,a (]8)

from potential continuity at X = d

- p , d 2 + p~ bd+ D = -p:d_~" + Ed + F 2ee,, ee,, 2ee,,

19)

and field continuity gives

p,d+ C = - p z d + E. (20) 8~t, ,EFo

D is determined from condition (10) and F is related to E through (13) and (14). These equations have a very simple geometrical interpretation, if field rather than potential is plotted as the basic variable. The simplicity of this approach is illus- trated in Fig. 5. In this instance a device is considered with a totally depleted layer, again with uniform doping in both layer and substrate. In both regions of the device the gradient of the field, being proportional to the charge density, is constant. In the oxide layer the field is constant. The unknown in this diagram is the location of the zero field axis. The choice of the position of the channel located point 1, is the point having zero field. If the zero field axis is now constructed through point 1 a

(=

D

L a y e r

i I

S u b s t r a t e [

[E'

Fig. 5. Field plot with depleted layer.

0~ /,

Model of a buried channel CCD

. . . . . i . . . .

1169

Fig. 6. Field plot with channel charge.

second intercept is found at 2. This point corres- ponds to the limit of depletion in the substrate. The potential difference between the gate and the channel is given by the combined area of rect- angle a b c d and triangle c d l, while the channel-substrate voltage is represented by the area of the triangle le2. Since the voltage at 2 in the substrate is known to be zero, the potentials throughout the device may be calculated.

This representation of the device is useful in gaining insight into the behaviour of the potentials in the device. For example, it may be seen that increasing the magnitude of the surface field Eo drives the channel away from the gate, increases the gate-channel voltage, and decreases the channel-substrate voltage and the depletion depth. It is physically reasonable that the surface field is an important parameter of the device since it is the field which drives carriers away from the gate to form a depleted layer. In this sense the model may be said to be field controlled.

The effect of a given channel charge Q may also be determined from the field profile, as illustrated in Fig. 6. In the region a, b, the field becomes zero. The total field change produced by the channel charge is represented by E j - E: = Q/eeo.

The analysis may be extended to non-uniform doping in a straightforward way.

Acknowledgements--We wish to thank B. L. H. Wilson, K. D. Perkins and V. A. Browne for several discussions on CCD; also Drs. R. A. Abram and G. J. Rees for helpful comments on form of presentation. Thanks are also due to the Directors of The Plessey Company Limited for permission to punish and to the Ministry of Defence (Procurement Executive) for support.

REFERENCES

1. G. F. Amelio, Bell Syst. Tech. J. 51,705 (1972). 2. J. McKenna and N. L. Schryer, Bell Syst. Tech. J. 52,

667 (1973). 3. W. D. Kent, Bell Syst. Tech. J. 52, 1009 (1973).