On solutions to the diffusion equation

NOTES trations

Pergamon Press 1970. Vol. 13, Printed in Great Britain

Solid-State Electronics pp. 1427-1428.

N(l, t) = C, exp On solutions (Received

to the

19 Decmbev

diffusion

equation

1969; in revised form 26 Februaq 1970)

t)

a

nr(x, t) = N,

1

1 -erf

x

-

1

(2)

where the surface concentration N(0, t) is fixed at a constant value, N,, by the design of the diffusion apparatus. Another often used solution is the Gaussian distribution,

s

LV(X, t) = ___ z//(nDt)

r

1 =

(3)

where the initial surface distribution is S atoms/ cm2 and no impurity atom flux crosses the system boundary (e.g. platinum box method of phosphorous diffusion). In two recent papers, WILSON(~~~)has used the following functions to describe impurity concen-

;;.

(5)

Where n = 0 for Cartesian coordinates, Y = x, n = 1 for cylindrical coordinates, r = r, n = 2 for spherical coordinates, r = p. and

t)

~ 2VVt)

(4b)

Where Y is taken as, alternatively, the Cartesian coordinate X, the cylindrical radial coordinate Y, or the spherical radial coordinate p. It is the purpose of this correspondence to point out that Wilson’s concentrations are not solutions to the diffusion equation and, as such, are not applicable to the problem of side-wall effects in diffused p-n junction structures. Upon assuming that the concentration depends on only one variable in each coordinate system, equation (1) becomes

This equation rests on two assumptions which are valid in many cases of experimental interest. First, it is assumed that the diffusion flux J is proportional to the gradient of the impurity distribution,

and second, the diffusion coefficient D is taken to be independent of the spatial coordinates. Solutions of equation (1) are well known for one-dimensional situations germane to the planar process. For example, with an initially impurityfree wafer of effectively infinite extent, the impurity concentration is

(4a)

r

2dPt)

= DV2N(P, t).

_i = - DVN(f,

1

-&

-V(r, t) = C, erfc-.

THE THEORETICALfoundations of integrated circuit diffusion technology are well known. Given a material system, the time-space behavior of an impurity atom concentration N(f, t) is described by the equation

iN(i,

1

Of the six impurity concentrations presented by Wilson [equations (4a) and (4b) for each of the three coordinate systems], only one [equation (4b) in Cartesian coordinates] is a solution to the diffusion equation (5). To be sure, the Gaussian distribution (3) can be written in the form of equation (4a) but, in that event, C, is not a constant surface concentration. Rather, it is a quantity depending on the diffusion time, t. This also is true for the Gaussian form in cylindrical and spherical coordinates. The incorrectness of the erfc and the Gaussian solutions in cylindrical and spherical coordinates can be verified by direct substitution as well as by considering the boundary conditions implicit in these expressions. Referring to Fig. 1, the proper boundary conditions at the water surface for an open tube diffusion process are

1427

LV(r, t) = No for 0 < r < R 0 = O”, 8 = 180”.

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