Graphical method for the determination of junction parameters and of multiplication parameters

Solid-Stale Electronics Pergamon Press 1970. Vol. 13, pp. 25-36. Printed in Great Britain

GRAPHICAL METHOD FOR THE DETERMINATION OF

JUNCTION PARAMETERS AND OF MULTIPLICATION

PARAMETERS

F. VAN DE WIELE, R. VAN OVERSTRAETEN and H. DE MAN

Electronic Research Laboratories, Section Solid State Electronics, Catholic University of Louvain, 94 Kardinaal Mercierlaan, Heverlee, Belgium

(Received 14 Notember 1968; in recisedfovm 26 May 1969)

Abstract-A graphical method is presented, which allows the determination, for a p-n junction, of the breakdown voltage and also the electric field distribution, the width of the depletion layer and the multiplication factor, all at a given reverse voltage. This method is based on the approximation of the exact impurity profile by an exponential function. The use of reduced parameters makes the method independent of the material. Numerical examples are presented.

RBsumB-On prbente une mCthode graphique qui permet la d&ermination pour une jonction p-n de la tension de rupture et de la distribution de champ Clectrique, 1’Cpaisseur de la couche d’bpuisement et le facteur de multiplication & une tension inverse donnCe. Cette m&hode est basee sur l’approximation du profil d’impuretC exacte par une fonction exponentielle. L’emploi des parametres reduits rend la mCthode indkpendante du matCriau. Des examples numCriques sont prCsentCs.

Zusammenfassung-Eine graphische Methode wird dargestellt, die fti eine vorgegebene Sperrspannung bei einem p-n-ubergang die Bestimmung der Durchbruchspannung, der elektri- schen Feldverteilung, der Breite der Verarmungszone und des Multiplikationsfaktors erlaubt. Die Methode beruht auf der NSiherung des genauen Dotierungsprofils durch eine Exponentialfunktion. Da reduzierte Parameter benutzt werden, ist die Methode vom Halbleitermaterial unabhlngig. Sumerische Beispiele werden angegeben.

1. INTRODUCTION

FOR DIFFUSED junctions it is often useful to determine the breakdown voltage and for a given voltage, the electric field distribution, the partial widths of the depletion layer and the multiplication factor due to ionization. It will be shown that most diffused profiles can be approximated by an exponential function in the depletion layer. Introducing reduced variables it is possible to solve Poisson’s equation numerically for the electric field distribution and the partial widths of the depletion layer using Schottky’s model.

These computed results are plotted in function of the reduced parameters so that the graphs are independent of the material. The part concerning the determination of the breakdown voltage and of the multiplication factor for a given voltage is

restricted to materials for which Chynoweth’s law is valid, but may be extended to other materials also.

In Section 2, the approximation of the impurity profile is discussed. Sections 3 and 4 give respectively the graphical determination of the electric field distribution, of the multiplication factor and of the breakdown voltage.

2. IMPURITY PROFILE APPROXIMATION

Consider the impurity profile shown on Fig. 1. N is the net impurity concentration (donor concentration minus acceptor concentration), NO, the surface concentration of the diffused impurity and iV, the substrate concentration. The origin of the x-axis is taken at the metallurgical junction, at ;II distance x, from the surface.

25

26 F. VAN DE \VIELE, R. V=1S Ol-ERSTR_%ETEN and H. DE hIAS

/ N

n-type p-type

FIG. 1. Donor diffusion profile in a homogeneous p-type substrate.

No : surface concentration.

N,: substrate concentration.

For a complementary error function, one obtains:

(1)

and for a Gauss function:

i&r =_N, e-(~--J?i4Dt_1~s, (2)

with D the diffusion constant and t the diffusion

time. Introducing the normalized coordinate

E = x/2(dDt), (3)

and using the fact that x = 0 for N = 0, one

obtains from (1)

XJN, = erfc( - Ej) (Ej < 0). (4)

In most cases the depletion region extends only slightly around gj. A Taylor expansion around Ej of equation (l), written in normalized coordinates, gives after some manipulations :

WC) = - No exp( - tj’)

(d4Er [exp(Xjt) - 11. (5)

Another expansion of the error function”’ gil-es:

ezp( - ti”) erfc( -5,) = - -__-

(z’n)ti I l- ++;j;

I UC - :

135

23t16 f.. . 1. (‘,?

Taking this into account, equation (5) becomes for if;1 3 2:

-X-(f) m X,[exp(2tJ) - 11. G?

\iTith the original x coordinates, (7) becomes:

with

X = 2Dt/‘jxjj = (x/Dr)arg[erfc(:Vs,/lY,,)]. (9)

The condition 1 Fj[ > 2 imposes a maximum ratio given by :

:?;,;‘S, < erfc(2) w 4.7 x 10m3.

Since an error function profile normally occurs with a high surface concentration No this condition is practically always satisfied. For a Gaussian profile the same procedure leads to the same profile approximation given by equation (8) but now h is given by:

X = 2Dt/‘/sjj = [v’Dt,‘ln(So/,\:,)]. (10,

Equations (9) and (10) permit to calculate thr parameter X from the diffusion parameters. The parameters X and No can also be obtained from capacitance measurements.(l)

It should be remarked that the exponential profile (S) has the same absolute value of the gradient n = (n’,/h) at the metallurgical junction as profile (2) and approximately as profile (1) if \.~~lj2(\iD/) > 2. This gradient may also be obtained csperimentally from capacitance measure-

ments.

3. GRAPHICAL REPRESENTATION OF THE ELECTRIC FIELD DISTRIBUTION AND PARTIAL

JUNCTION WIDTHS

(a) Solution of Poisson’s equaiio?l

Using a reduced distance

Z = x/x, (11)

GRAPHICAL METHOD FOR THE DETERMINATION OF JUNCTION PARAMETERS 27

the exponential impurity profile (8) becomes

N = Ns(e-” - 1). (12)

For Schottky’s model, valid for reverse voltages, the electric field is determined by Poisson’s equation :

dE 1 dE qN,(e-‘-1) -=--= dx h dz

, E

(13)

with 4 the electronic charge and E the permittivity of the semiconductor material.

The reduced partial widths of the depletion layer will be denoted by z, = x,/h and zp = xp/X. The values of the electric field at these boundaries are, as usual, set equal to zero:

E(s,) = 0, (14)

E&z,) = 0. (15)

Integration of (13) and using boundary condition (14) yields:

E(z) = E,,(e-*~-e-Z-z++a,) (16)

O’sX with E,, = p.

E (17)

Put F(z) = eWr+z-1.

The electric field is then given by:

E(z) = E,[F(z,) - %41-

(18)

(19)

The second boundary condition (15) yields the neutrality condition:

or es2 “+,a, = e-%+z,. (20)

F(z) as a function of z is plotted in Fig. 2. The maximal electric field occurs at z = 0 and is given by:

E, = E(0) = EoF(zn). (21)

Given a value of z, (or given the applied voltage, as will be seen next), E,,,/Eo = F(z,) and also zp are read directly from Fig. 2. For this applied voltage, the reduced electric field E/E,, at an arbitrary distance z in the depletion layer is:

E/EC, = F(a,) -F(z), (22) FIG. 3. Plot of F(z) vs. z, and z,.

t F IL)

FIG. 2. Plot of F(z) = emz+z-1 vs. z.

and may be read directly from Fig. 2. For practical purposes F(z) is plotted on a double logarithmic scale in Fig. 3.

F. VAIi DE WIELE, R. VAN OVERSTRAETEN and H. DE MAN

FIG. 4a. Reduced potential o as a function of z,,, for 10-a =s li < 1.

The total voltage across the junction is obtained by a second integration:

V, + V, = TE dx = XE,, Ip(zJ --F(z)] dz,

2” 2”

with V, the applied and V, the built-in voltage. The sign convention for V, used here is that V, is positive for reverse bias. After introducing the reduced potential

va+ Vd

v=xE,’ (23)

one obtains

V = em2 n(Zp-.Zn-l)+e-“~-- @P ;d2_ (24)

From this equation and the neutrality condition (20), z, may be calculated as a function of zn. The

10:

V

10

I

Z”

FIG. 4b. Reduced potential u as a function of zn, for 1 < D < 103.

reduced potential v (zn), numerically computed, is plotted in Figs. 4a and b for 10e3 < v < 10”.

To get a better physical insight of the meaning of the reduced potential, we consider two limiting cases.

(a) Large v. Due to the asymmetry of the

impurity profile, zP becomes much larger than zn for large ~1. For example, for ZJ = 103, ztL equals 3.82 and from Fig. 3 we get zp = 42. Neglecting e-+ in the neutrality condition (20) we get

e-“n = 2 P

-2,.

Since xp -z, is much larger than unity (see Fig. 3), the reduced potential (24) may be approximated b>

Xp--Z n = (42v). (25)

Using (23) and (17), the width of the depletion laver becomes

GRXPHICAL METHOD FOR THE DETERMINATION OF JUNCTION PARAMETERS 29

which is nothing else than the well known formula for an abrupt junction. In this limiting case, the reduced partial widths are

.a, = - ln(d2v) (27)

zp = (1/2v)-ln(1/27_J), (28)

with an accuracy better than 0.5 per cent for z’ > 103. In this case F(z) is approximately equal to Z.

(b) SmaZl z‘. Figures 4 and 3 indicate that for small values of B, Z, and zp are small numbers and are approximately equal. Put

-an = xp = xi. (29)

A series expansion up to terms of the third order of the exponentials in (24) yields:

3 z* M -7l .

0

113

2 (30)

In this case, the junction width is

( 12h( v, + V,) 1 ‘3 w % 2X.Z, = -

qN, 1

126(Va’,+ V,) 1’3 =

( > (31)

4a

which is nothing else than the formula for a linearly graded junction. The reduced partial widths are given by (30)

3

0

l/3

-z,=zp= -v , 2

with an accuracy better than O-8 per cent for v < 10-a.

(B) Graphical representation and procedure

In order to determine graphically the electric field distribution in the depletion layer, and its width w, three parameters must be known. They are :

(a) The substrate concentration N,, determined by capacitance or resistivity measurements.

(b) The parameter A, defined by (9) or (10) in function of the diffusion parameters iV,,, N,, D, t. It may also be calculated from the gradient a = NJA, determined by capacitance measurements.(l)

(c) The built-in potential V,, determined experimentally out of capacitance measurements or calculated theoretically.

The problem to be solved is to determine W, E,,, and E(x), for a voltage V, applied to the junction, characterized by the preceding parameters. E, is calculated from (17) and v from (23). Figure 4 gives a, corresponding to v. For v >, lo3 and v 6 10d3, the analytical expression (27) or (32) has to be used. Figure 3 gives F(z,) and zp corresponding to 2,.

The width is then:

w = h(.Z,-a,). (33)

The maximal electric field is (21):

E, = E,,F(x,). (34)

The field at an arbitrary x or z is given by (19) :

-W = %[%4 - FWI, (35)

with F(z) read from Fig. 3.

(C) Numerical example

Given the experimental values of the parameters for a silicon diode:

N, = 3.25 x 1Ol5 cmp3, a = 160x 1020 cme4,

v, = 0.55 V,

one may calculate h = N,/a = 2.03 x 10-s cm and E, = 1.015 x lo4 V-cm-r. For a given bias of 80 V, the reduced voltage is v = 391. From Fig. 4, one obtains a, = -3.37 and from Fig. 3, zp = 24.5 and F(z,J = 24. The width of the depletion layer is (33): w = 5.65 x 10e4 cm and the maximum electric field (34) E,,, = 244x lo5 V-cm-l. The electric field at z = -2 is (19) 2.01 x lo5 V-cm-l, since F(z) = 4.2 from Fig. 3.

4. GRAPHICAL DETERMINATION OF THE MULTIPLICATION FACTOR AND OF THE

BREAKDOWN VOLTAGE

(A) Theory

Two assumptions are used:

(1) The ratio of the ionization rates for holes and electrons y = cc&, is constant, independent of the electric field.

(2) Chynoweth’s lawc3) M = a;oe-b’lEI is valid.

(36)

30 F. VAN DE WIELE, R. VAN OVERSTRAETEN and H. DE MAN

FIG. 5. a,, vs. I,, with y as parameter.

FIG. 6. QP vs. I,, with y as parameter.

GR-1PHICAL METHOD FOR THE DETERMINATION OF JUNCTION PARAMETERS 31

However, it should be remarked that these

assumptions are not completely restrictive for practical calculations, as will be discussed in the procedure.

\I:e have”)

2, z

with 111, the multiplication factor for electrons.

Putting

5P

In = u, dx s (37)

** n-e have

@‘n = &~~-exp[-U--~)l,D. (38)

(DT1 is plotted in function of I,, withy as a parameter in Fig. 5. One also can show that:(l)

mp=l+ y{l -exp[(l --~)I,11 (39) P Y-l

with flfp the multiplication factor for holes. Qr, is plotted in function of I,, with y as a parameter in Fig. 6. In Figs. 5 and 6, the range for the parameter y is from 0.05 to 1. This is sufficient for Si.(l) However, in Ge, ap is larger than an.(4) In this case, the same graphs may be used by inverting the roles of the electrons and the holes.

Making use of Chynoweth’s law, 1, may be written as

with

Y@$) =J” exp{ --~/IEo[W -clip dz. .? 73

I-(%, b/E,) is calculated using an IBM 360

L”

FIG. 7

computer and is plotted in Fig. 7 (I-IS), in function of z,, with b/E, as parameter. For materials for which Chynoweth’s law is not valid, the function Y must be calculated for the appro- priate law. This may be the case for GaP, GaAs, . . .

Formulae (38) and (39) are derived respectivel! for pure electron- and pure hole-injection. For mixed injection, with injection ratio(l) h (K = 0 for hole injection and K = co for electron injection)

Z,

FIG. 7-I


IO-'

,o-’

y 10-s

10-q

10~5 -2.0 -2.5 -3.0 -3.5 -40

Z”

FIG. 7-11

Z”

FIG. 7-111

4‘1

FIG. 7-IV

L”

Frc. 7-V


‘OG

10-l -2.0

L”

FIG. 7-W

Ln

FIG. 7-VII

F. VAN DE WIELE, R. VAN OVERSTRAETEN and H. DE MAN

zn Z”

FIG. ‘i-VIII FIG. 7-I);

FIG. 7 (I-IS). Y vs. zn, with b,‘E, as parameter, for different ranges of z,, and 1:

the expression for the multiplication factor is:(l)

ill =

esp - ( r(xn -s) dr)

and the definition of On, this expression bccomez

M = MJM, +k

(l+k)(l-On)’

with with a,,, for mixed injection. With

esp [ %(a,-~.,) dx] = 2, 1 - (1) P P

=#a

lk?,jMP = __ 1 - ‘I),


one finds for (I?,,,

(B) Procedure

In order to determine graphically the multiplication factor M at a given voltage V, and the breakdown voltage V,, the three parameters N,, h and V, must be known, as before, even as the ionization rates an and aP in function of the electric field E. The ionization rates are characterized by the parameters b and a, for the carriers with the largest ionization rate and a graph or an analytical expression of y in function of E.

(1) The first problem we are going to solve is the determination of the multiplication M at a given voltage V, and for a given injection ratio k. This problem may be of interest for example for a photo diode in avalanche, where the injection ratio may be calculated. The procedure to be followed is:

(u) Determine graphically the maximal electric field Em and the partial reduced width Z, at the given voltage V, by means of the procedure 3 B.

(b) Calculate b/E, and read Y (zn, b/E,) from Fig. 7.

(c) Read y according to E,,, from the graph of y vs. E or calculate it from the analytical expression.

(d) Calculate In = A a, Y (zn, b/E,) and read Q,, (In, r) and QD, (.sn, r) respectively from Fig. 5 and 6.

(e) Calculate Om = 1 - l/M using relation (42).

2. The second problem is the determination of the breakdown voltage V,. The procedure is :

(a) Calculate E, from (17). (b) R/lake a first estimate E,(l) of the maximal

electric field and read the corresponding y from the graph y vs. E or calculate it from the analytical expression.

(c) Since at breakdown @‘n = Qr, = 1, Fig. 5 or 6 give Inb. Figure 8 gives a graph of Inb vs. y.

(d) Calculate b/E, and from (40) Y = &/ha, and read Z, from Fig. 7.

(e) Read F(zJ from Fig. 3. (f) From (21) one obtains a second estimate

E,,,(2) of the maximal electric field. The procedure is now repeated from b on, until a sufficiently accurate EmCi) is obtained.

(g) Using the accurate value of z,, read ZJ from Fig. 4 and calculate (23) V, = hE,v- V,.

I”b

Y

FIG. 8. Ina vs. y.

Strictly speaking, this procedure is only valid for materials where the ionization rates may be expressed by Chynoweth’s law and where y equals a constant. However, since the overwhelming part of the ionization integral is due to the field range within 10 per cent of the maximal field at z = 0, this procedure may also be used in the case where Chynoweth’s law and the assumption of a constant y are not exactly valid, if one takes for CL %, b and y the values at E,.(l)

(C) Numerical examples

Consider the same experimental values of N,, a and V, as in Section 3 C and use the ionization data of Van Overstraeten and DE JIA#) for Si, namely

aal = 7.03 x lo5 cm-l,

b = l-231 x lo6 Vcm-I, (43)

and

y = Ti exp( -,&/lEj) i = 1, 2, (44)

with

rI = 0.955 PI = 4.62 x lo5 Vcm-’ for

E > 4 x lo5 Vcm-l

l?a = 2.25 pa = 8.05 x IO5 Vcm-’ for

E < 4 x lo5 Vcm-I.


(1) Determination of the multiplication factor at a given voltage V, and for a given injection ratio k. Take V, = SO V and k = 5. In section 3 C, it was found that E,,, = 2*44x lo5 Vcm-l and zn = - 3.37. _\ccording to (43), b/E, = 1.21 x 10’. From Fig. 7 III we read Y(z~, b/E,) = 5.6 x 10M2. Using (44), one finds y = O-083. With I, = 0.80 calculated from (40) and (43) one obtains from Fig. 5 and 6 CD, = O-567 and ar, = 0.095. Using (42) one finally finds @,,, = 1 - l/M = 0.526.

(2) Determination of the breakdown voltage. With a first estimate E,,,(l) = 3 x lo5 Vcm-I, one obtains from (44): Y = 0.154 and from Fig. 8, I,, = 2.22. From (40) and (43) we find Y = 0.155 and b/E, = 1.21 x lo2 and from Fig. 7 we read

-3.55. Figure 3 yields F(z,) = 28, and ?itZ (21) we obtain the second estimate

E,,,c2) = 2.84 x lo5 Vcm-I. Proceeding as before we successively obtain y = 0.133, Inb = 2.36, Y = O-165, z, = -3.57. This value of z, is sufficiently close to the value obtained with the first estimate. With .a, = -3.56, one finds from Fig. 4, v = 590, and from (23): V, = 121.6 VT. The experimental value is 119 V.

Acknowledgement-The authors are indebted to the Computer Center of the Catholic University of Louvain.

REFERENCES

1. R. VAN OVERSTRAETEN, H. DE MAN, submitted to Solid-St. Electron.

2. M. ABRAMOWITZ and A. STEGUN. Handbook of Mathematical Functions, p. 289: Dover Pubi. New York (1965).

3. A. G. CHYNOWETH, Phys. Rev. 109, 1537 (1958). 4. S. L. MILLER, Phys. Rev. 99, 1234 (1955).

Documents

Graphical method for the determination of junction parameters and of multiplication parameters