Avalanche breakdown voltage of diffused junctions in silicon

Solid-St&e

Elecrronics,

1973, Vol. 16, pp. 99 l-998.

AVALANCHE

Pergamon Press.

Printed in Great Britain

BREAKDOWN JUNCTIONS

VOLTAGE IN SILICON

OF DIFFUSED

P. R. WILSON Group

Research

Centre,

(Received

Joseph

Lucas

26 December

Limited,

Shirley,

Solihull,

1972; in revisedform

Warwickshire,

29 January

England

1973)

Abstract-A semi-theoretical relationship has been developed between the avalanche breakdown voltage and the product of the junction depth and background impurity concentration for plane, cylindrical and spherical diffused junctions. This has been confirmed experimentally for plane and cylindrical junctions, over the voltage range 10 < V’ < 9000 V.

INTRODUCTION

shown in Fig. 1. The curves labelled (a), (b) and (c) are for different diffusion processing as discussed later. While investigating the reasons for the failure of Warner’s breakdown criteria to apply to curved junctions a useful relationship was obtained between the breakdown voltage of diffused junctions, both plane and plaqar, and the product of the junction depth and background impurity concentration.

A PAPER has just been published by Warner-[11 in which he relates the breakdown voltage of a plane diffused junction to the thickness of the depletion layer at avalanche breakdown. The empirical relationship which he develops between the breakdown voltage V, and the depletion layer thickness XT is given by VB= (5.8 x 104)XT~~84.

(1) THEORY

The electric field across a diffused junction can be expressed in a normalised form [4]. For Gaussian diffused junctions these are

Equation (1) is valid for both abrupt and linearly graded junctions and hence also holds for diffused junctions. In 1967 1 published some experimental results for avalanche breakdown in planar (as opposed to plane) diodes[2] and showed that the breakdown voltage of circular planar diodes was adequately described by the theory of Sze and Gibbons[3] for linearly graded junctions, i.e. U5

Plane junction:

-=-

--

[erfc (kR,) - erfc (I&)]

,

-(R--R,)}.

(3)

Cylindrical junction: (2)

E(R) -=-&{&[e-

(RRtP _

e-(kR)z]

-+(R2

__R,Z)

.

xjCb

where A, is the gradient at the junction. Warner showed that the Sze and Gibbons breakdown criteria equation (2) could be expressed as equation (I), but he also stressed that equation (1) only applied to plane junctions and not to curved (planar) junctions. This was surprising as equation (2) adequately describes the breakdown voltage for planar junctions. In order to check this point the experimental data from Ref. [2] were replotted in the form suggested by Warner with the result

(4) Spherical junction:

-z -B(R~991

erfc (kR) + RI e-(kR1’Z- R e-(kH)2 I R,J)}

I

(5)

P. R. WILSON

992

where g = 7 for silicon and K is a constant. From equations (6), (7) and (9) the condition for avalanche breakdown can be written as

i.e. xjcb=

1

10

I

1111/1/

I

1

I

IO

XT*

I11111

IO2

tLm

Fig. 1. Breakdown voltage as a function of the depletion layer thickness for planar Gaussian diffused junctions. (---) Equation (1). (0), (+), (X) Experimental results from Ref. [2] for circular planar diodes fabricated in three different procedures, as outlined near the beginning of the Experimental section.

where E(R) is the electric field, R = X/Xj is the normalised distance from the surface, R, is the position of the edge of the depletion region on the highly doped side of the junction, E, is the permittivity of silicon and k = d(ln C,/C,) where Co and C,, are, respectively, the surface and background concentrations. In general, then, E(R)

Following breakdown

Fulop[S] is

= +f(R).

the

condition

(6) for

avalanche

(7) where (Y,~~ is an effective ionization coefficient and X, is the thickness of the depletion layer. Usually (Y,~~ is considered to have the form [S] aeff = a exp (- b/E)

(8)

but Fulop gives an approximate empirical relationship between the field and the effective ionization coefficient as

(es/q)

H2

117

/I J Lf(R)l’dR I K

R,

(10)

where R, and R, are the normalized positions of the edges of the depletion layer. For a given junction, R, and the normalized junction voltage can be calculated for values of R, [4]. Hence it is possible to plot graphs of the product of the junction depth and background impurity concentration as a function of the normalized breakdown voltage (VB/Xj”C,). Figures 2-5 show the product of the junction depth and the background impurity concentration plotted as a function of the normalized avalanche voltage for plane, cylindrical and spherical junctions. The value of K, given below in the Discussion section, was chosen to make the experimental and theoretical results agree, as described in the next section. EXPERIMENTAL

used in my original investigation[2] were planar p+n Gaussian diffused diodes with a circular geometry, and hence in cross section the planar junctions had approximately cylindrical sidewalls. (Diodes with a square or triangular geometry would have had sidewalls of both cylindrical and spherical shapes.) Details of diode fabrication are given in Ref. [2], but it should be noted here that the junction gradients were in the range 1Ol710*3cm--4 and hence the junctions covered the range of linear graded-diffused-abrupt types. The diodes fall into three batches according to the fabrication procedure. (a) Constant diffusion conditions; 2 hr diffusion time. Background concentration varied. (b) Constant diffusion; 64 hr diffusion time. Background concentration varied. (c) Diffusion time varied between 2 and 64 hr. Background concentration constant. As stated above, when the breakdown voltage is plotted as a function of the depletion layer thickness at breakdown, Fig. 1, the results fall on three lines, The devices

AVALANCHE

BREAKDOWN IN SILICON

993

IO”

IO’=

d5

u” x

lOI

lOI

lO-'3

0

16"

I

o-‘O

-B

IO

(a)

d2

IO"

v= ._ x do

IO9

Id9

lo-6

16"

Id4

16"

Fig. 2. Breakdown voltage vs rjCb for plane diffused junctions. The figures on the curves are values of CJC*. corresponding to fabrication methods (a), (b) and (c). In plotting Fig. 1 the depletion layer thicknesses were calculated for the plane part of the junction[4]. Replotting the results using the depletion layer thickness for cylindrical junctions also gave a graph with three distinct portions. However, Fig. 6 shows the product of the junction depth and background concentration plot-

SSE Vol. 16. No. 9 -C

ted as a function of the normalized breakdown voltage. The solid line is the theoretical relationship for cylindrical junctions. The value of K in equation (10) was chosen to make the two sets of results agree. Vincent et al.[6] have also published some experimental results for gaussian diffused p+n planar junctions. They used two different diffusion sched-

994

P. R. WILSON

v,

/x;c, (a)

(b) Fig. 3. Breakdown voltage vs xjC,, for cylindrical diffused junctions. The figures on the curves are values of C,IC,. ules to produce deep and shallow junctions, termed processes .4 and B respectively in their work, and varied the background concentration to obtain a range of breakdown voltages in the range I2 < V, < I50 V. Figure 7 shows their results replotted in the form suggested here. The solid lines are theoretical values for cylindrical junctions cor-

responding to the A and B processes. Warner[ l] quoted experimental results obtained by F. R. Carlson for plane n+~ erfc diffused diodes. A constant diffusion schedule was used and the background concentration was varied from 3.72 x lOI to 1.18 x 10’” crnm3 to produce a voltage range of IO < V, < I50 V. This data is shown in Fig. 8

AVALANCHE

IO

BREAKDOWN

IO -I0

IO

995

IN SILICON

-6 IO

IO-¶

v,/x;c, (4

d2

d

u”._ x

IO9 I

o-9

16"

16'

16'

I

o-3

v&, (b) Fig. 4. Breakdown voltage vs xIC, for spherical diffused junctions. The figures on the curves are values of C,/C,.

and the solid line is the theoretical result for plane The value of K is the same as that used for cylindrical junctions. Kokosa and Davies [7] have presented avalanche breakdown results for p+n erfc diffused plane junctions, with 100 < VB < 9000 V. These results are shown in Fig. 9.

junctions.

DISCUSSION

The previous sections have shown that there is a convenient semi-theoretical relationship between the junction parameters and the breakdown voltage for both plane and cylindrical diffused junctions. In the calculations the value of K, in equation (lo), was taken as 8.45 x 1O-36 in order to match the

P. R. WILSON

996

16'

U6

v,/x:c, (b) Fig. 5. Breakdown voltage vs .qCb for plane, cylindrical and spherical diffused junctions with Co/C0 = 105.

Fig. 6. Experimental and theoretical breakdown voltage for cylindrical diffused junctions. Experimental results taken from Wilson [2]. p+n Gaussian junction, 14 < V, < 900V.

theoretical and experimental curves, and this leads to the relationship between the electric field and the effective ionization coefficient shown in Fig. 10. Also shown in this figure is the relationship obtained by Maserjian[8] and the more recent relationship derived by van Overstraeten and de Man[9]. At very high fields the approximation to ~y,rrused here becomes increasingly less valid, but as zener breakdown will then be becoming increasingly important the breakdown criteria used here also becomes less valid.

Figure 11 shows the calculated maximum electric field within the depletion layer at avalanche breakdown for plane, cylindrical and spherical diffused junctions, with CO/CO= 105. The total range of the maximum field is approximately 1.5 to 4.5 X lo5 V/cm, and it is for E < 4.5 X lo5 V/cm that the empirical expression for the effective ionization coefficient has a good fit with the values given in Refs. [8] and [9]. The theoretical calculations were only done for gaussian diffused junctions, but with a slight modi-

AVALANCHE

BREAKDOWN

Fig. 7. Experimental and theoretical breakdown voltage for cylindrical diffused junctions. (0) Process A, (A) pro-

IN SILICON

997

Fig. 9. Experimental and theoretical breakdown voltage for plane diffused junctions. Experimental results taken from Kokosa and Davies171 n+p erfc iunction, 100 < v, < 9oooV. -

cess B; taken from Vincent et al. [6] p+n Gaussian junction, 12 < V, < 15OV.

v,/x;c, Fig. 8. Experimental and theoretical breakdown voltage for plane diffused junctions. Experimental results taken from Warner[l]. n+jr e&junction, IO < V, < I50 V.

fication they will also apply to erfc diffused junctions. The avalanche breakdown voltage is controlled by the high field region in the depletion region which in turn is a function of the impurity gradient near the junction. The following equation is the condition for a Gaussian and an erfc junction to have identical impurity gradients at the junction

0

I

2

E,

3

V/cm

4

5

6

x IO”

Fig. 10. The effective ionization coefficient as a function of electric field. (a) From Maserjian[8]; (b) from van Overstraeten and de Man[9]; (c) approximation used in this work, based on Fulop [5].

(11) where the suffixes e and g refer to erfc and Gaussian diffusions, respectively. For lo* < c&,/c0 < 10’

P. R. WILSON

998

This paper has presented a convenient relationship between thejunction parameters and the breakdown voltage for plane, cylindrical and spherical junctions. This has been confirmed by independent experimental results for plane and cylindrical, p+n and n+p Gaussian and erfc junctions over the range 10 < V, < 9000 v. REFERENCES 1. R. M. Warner, Jr., Solid-St. Electron. 15, 1303 (1972). 2. P. R. Wilson, hoc. IEEE 55, 1483 (1967). 3. S. M. Sze and G. Gibbons, So/id-St. Electron. 9, 83 1 (1966). 4. P. R. Wilson, So/id-St. Electron. 12, 1 (1969). 5. W. Fulop, Solid-St. Electron. lo,39 (1967). 6. D. A. Vincent, H. Rombeck, R. E. Thomas, R. M. Sirsi and A. R. Boothroyd, Solid-St. Electron. 14, 1193 (1971).

7. R. A. Kokosa and R. L. Davies, IEEE Trans. Electron Devices ED-12,874 (1966). 8. J. Maserjian, J. appl. Phys. 30, 1613 (1959). 9. R. Van Overstraeten and H. de Man, Solid-St. Electron. 13,583

(I 970).

APPENDIX E.

V/cmx105

Fig. 11. The maximum electric field in the depletion layer at avalanche breakdown for plane, cylindrical and spherical diffused junctions, with Co/C* = 105.

the solution of (11) lies in the range 2 < C,,/C, < 3. Hence, for practical purposes, the Gaussian curves can be used for erfc junctions provided the labelled ratios of surface to background concentrations are increased 2.5times. This change will have little practical effect on calculated breakdown voltages because of the small spread of the curves with concentration ratio. This is borne out by the comparison between the experimental plane erfc junction results and the theoretical plane Gaussian results shown in Figs. 8 and 9.

It was pointed out by Vincent et al.[6] that the maximum electric field in a curved diffused junction does not occur at the junction. If R, is the normalized position of the maximum electric field, then for cylindrical and spherical Gaussian junctions R, can be determined from; cylindrical junction:

&

e-‘kR1)2 +fR12 =2&

[1 + ~(/cR,)~] e&liRm’*

1 2 A?,”

(Al)

spherical junction: $$-

h1

$erfc

- kFi [gerfc -

h jRmY.

(/CR,) +R,

e-(kK1)2 ++R,3 1

(kR,)+R,[l+

(IcR,,)~] e-(kRrnlz) (AZ)