- Email: [email protected]
Some comments on ‘On solutions to the diffusion equation’
NO’YES
1428
show that the expression used by \VILSON(~’ for the impurity concentration in cylindrical and spherical junctions was not a solution of the diffusion equation, and hence the results reported were invalid for calculating the sidewall cffcct in diffused ptz junctions. Gajda is correct in stating that
S
is not a solution of the diffusion equation for constant surface concentration in cylindrical or spherical co-ordinates. The correct solutions of the diffusion equation for diffusion from a point source of constant concentration into an infinite plane or an infinite volume, (i.e. in cylindrical or spherical co-ordinates), have the forms
FIG. 1. Cylindrical geometq
But from equation
(4b),
And observe that only at I’ = 0 is the concentration equal to a constant, N,, as the physical situation requires. Based on these considerations, we conclude that \%‘ilson’s results concerning cylindrical and spherical diffused functions are invalid. A brief word on earlier work in this area is also appropriate. LEE and SZE(~) considered abrupt and linearly graded impurity distributions in cylindrical and spherical geometries. But, in their case, these concentrations were clearly intended as models of a more complex situation. Electrical
Engineering
IVALTER
J. GAJDA, JR.
Department University Notre
of Notre
Dame,
Dame
Indiana
46556
References 1. P. R. WILSON, Solid-St. Electron. 12, 1 (1969). 2. P. R. WILSON, Solid-St. Electron. 12, 277 (1969). 3. T. P. LEE and S. hf. SZE, Solid-St. Electron. 10, 1105 (1967).
2
C = - =1Ei
1
-iiT
infinite
i C = ferfc
Y
pp. 1428-1429.
Pergamon Press 1970. Vol. 13, Printed in Great Britain
(1)
volume
(2)
1
__ infinite C2 \!‘I& i
whew ;1 and H are constants. It should be noted in passing that the impurit! distributions given by ~~ILSON'3' for gaussian diffused junctions arc solutions of the diffusion equation in cylindrical and spherical co-ordinates. KENNEDY and ~‘BRIE~*;~~) have calculated the impurity distribution near an infinitely long mask edge for a constant surface concentration diffusion diffusion, their and an instantaneous source results b&g confirmed espcrimentally by 11 I(‘DOXALLI and I':vERII.~RT'~~and TVII.S~N.‘~’ The impuritv concentration for a constant surfxc concentration diffusion is given by equation (8), Ref. (4). C(r, 8, t) = C,(1-2
2 77
Solid-Stare Electronics
plane
sin[(tL+&)@]
l7l=O
[Ys2v’Dt]‘n+
t’l’(tz + $):2
2rytt -t j?o Some comments on ‘ On solutions to the diffusion equation (Received
IT WAS
Gajda’s
30 January
intention
-,F,
1970)
in the above
paper’r)
I
N +
’
to
$
_-i---;
(I1 +
;);4<; . (3) 3)
Equation (3) is shown in Fig. (la) for i.e. along the surface of the semiconductor
0 = n under
1429
NOTES
the mask. Well away from the mask edge the impurity distribution normal to the surface is given by (4)
C= which
is shown
in Fig. (lb).
FIG. .l. (a) The impurity distribution along the surface under a mask, equation (3), the mask occupying the region. (b) The impurity distribution normal to the surface away from the mask edge, equation (4). (c) The impurity distribution from a constant concentration point source in a plane, equation (1).
The impurity distribution near the mask edge for a planar constant surface concentration diffusion then varies between the limits shown in Figs. la and b. Also shown, as Fig. (lc) is equation (l), the exact cylindrical distribution of impurities for a
constant C, process. This is very close to the exact distribution along the semiconductor surface under a diffusion mask, i.e. equations (1) and (4) give approximate limits on the impurity distribution near an infinitely long mask edge. It was stated in Ref. 2 that it was a reasonable approximation for a planar junction to assume that the curved region under the mask edge has a constant radius of curvature which was equal to the depth of the flat portion of the junction. With this assumption it is also reasonable to make the impurity distribution throughout this curved region the same as the impurity distribution in the plane part of the junction. Hence the results given in Ref. 2 are valid under the assumption that the curved portion of a planar junction has a constant radius of curvature. It is, of course, possible to calculate the depietion layer width at the surface, using equations (1) and (2) for cylindrical and spherical junctions, so as to obtain upper and lower limits to the depletion layer movement with voltage. This will be done at some future time, but the author does not feel that this is an urgent problem. Group Research Centre Joseph Lucas Ltd. Shirley, Solihull Warzuickshire, England
P. R. WILSON
References 1. 2. 3. 4.
W. J. GAJDA Jr., Solid-St. Electron. 13, 1427 (1970). P. R. WILSON, Solid-St. Electron. 12, 277 (1969). P. R. WILSON, Solid-St. Electron. 12, 1 (1969). D. P. KENNEDY and R. R. O’BRIEN, IBM JL Res. Dev. 9, 178 (1965). 5 . K. C. MACDONALD and T. E. EVERHART, J. appl. Plzys. 38, 3685 (1967). 6. P. R. WILSON, &lid-St. Electron. 12, 539 (196Y).
1428
show that the expression used by \VILSON(~’ for the impurity concentration in cylindrical and spherical junctions was not a solution of the diffusion equation, and hence the results reported were invalid for calculating the sidewall cffcct in diffused ptz junctions. Gajda is correct in stating that
S
is not a solution of the diffusion equation for constant surface concentration in cylindrical or spherical co-ordinates. The correct solutions of the diffusion equation for diffusion from a point source of constant concentration into an infinite plane or an infinite volume, (i.e. in cylindrical or spherical co-ordinates), have the forms
FIG. 1. Cylindrical geometq
But from equation
(4b),
And observe that only at I’ = 0 is the concentration equal to a constant, N,, as the physical situation requires. Based on these considerations, we conclude that \%‘ilson’s results concerning cylindrical and spherical diffused functions are invalid. A brief word on earlier work in this area is also appropriate. LEE and SZE(~) considered abrupt and linearly graded impurity distributions in cylindrical and spherical geometries. But, in their case, these concentrations were clearly intended as models of a more complex situation. Electrical
Engineering
IVALTER
J. GAJDA, JR.
Department University Notre
of Notre
Dame,
Dame
Indiana
46556
References 1. P. R. WILSON, Solid-St. Electron. 12, 1 (1969). 2. P. R. WILSON, Solid-St. Electron. 12, 277 (1969). 3. T. P. LEE and S. hf. SZE, Solid-St. Electron. 10, 1105 (1967).
2
C = - =1Ei
1
-iiT
infinite
i C = ferfc
Y
pp. 1428-1429.
Pergamon Press 1970. Vol. 13, Printed in Great Britain
(1)
volume
(2)
1
__ infinite C2 \!‘I& i
whew ;1 and H are constants. It should be noted in passing that the impurit! distributions given by ~~ILSON'3' for gaussian diffused junctions arc solutions of the diffusion equation in cylindrical and spherical co-ordinates. KENNEDY and ~‘BRIE~*;~~) have calculated the impurity distribution near an infinitely long mask edge for a constant surface concentration diffusion diffusion, their and an instantaneous source results b&g confirmed espcrimentally by 11 I(‘DOXALLI and I':vERII.~RT'~~and TVII.S~N.‘~’ The impuritv concentration for a constant surfxc concentration diffusion is given by equation (8), Ref. (4). C(r, 8, t) = C,(1-2
2 77
Solid-Stare Electronics
plane
sin[(tL+&)@]
l7l=O
[Ys2v’Dt]‘n+
t’l’(tz + $):2
2rytt -t j?o Some comments on ‘ On solutions to the diffusion equation (Received
IT WAS
Gajda’s
30 January
intention
-,F,
1970)
in the above
paper’r)
I
N +
’
to
$
_-i---;
(I1 +
;);4<; . (3) 3)
Equation (3) is shown in Fig. (la) for i.e. along the surface of the semiconductor
0 = n under
1429
NOTES
the mask. Well away from the mask edge the impurity distribution normal to the surface is given by (4)
C= which
is shown
in Fig. (lb).
FIG. .l. (a) The impurity distribution along the surface under a mask, equation (3), the mask occupying the region. (b) The impurity distribution normal to the surface away from the mask edge, equation (4). (c) The impurity distribution from a constant concentration point source in a plane, equation (1).
The impurity distribution near the mask edge for a planar constant surface concentration diffusion then varies between the limits shown in Figs. la and b. Also shown, as Fig. (lc) is equation (l), the exact cylindrical distribution of impurities for a
constant C, process. This is very close to the exact distribution along the semiconductor surface under a diffusion mask, i.e. equations (1) and (4) give approximate limits on the impurity distribution near an infinitely long mask edge. It was stated in Ref. 2 that it was a reasonable approximation for a planar junction to assume that the curved region under the mask edge has a constant radius of curvature which was equal to the depth of the flat portion of the junction. With this assumption it is also reasonable to make the impurity distribution throughout this curved region the same as the impurity distribution in the plane part of the junction. Hence the results given in Ref. 2 are valid under the assumption that the curved portion of a planar junction has a constant radius of curvature. It is, of course, possible to calculate the depietion layer width at the surface, using equations (1) and (2) for cylindrical and spherical junctions, so as to obtain upper and lower limits to the depletion layer movement with voltage. This will be done at some future time, but the author does not feel that this is an urgent problem. Group Research Centre Joseph Lucas Ltd. Shirley, Solihull Warzuickshire, England
P. R. WILSON
References 1. 2. 3. 4.
W. J. GAJDA Jr., Solid-St. Electron. 13, 1427 (1970). P. R. WILSON, Solid-St. Electron. 12, 277 (1969). P. R. WILSON, Solid-St. Electron. 12, 1 (1969). D. P. KENNEDY and R. R. O’BRIEN, IBM JL Res. Dev. 9, 178 (1965). 5 . K. C. MACDONALD and T. E. EVERHART, J. appl. Plzys. 38, 3685 (1967). 6. P. R. WILSON, &lid-St. Electron. 12, 539 (196Y).