The modified fletcher boundary conditions

107

NOTES Solid-Stale

Electronics,

1975, Vol.

18, pp. 107-109.

Pergamon Press.

Printed in Great Britain

THE MODIFIED FLETCHER BOUNDARY CONDITIONS (Received

6 April

1974; in revised form

Consider a PN junction with the approximate boundary between the P region and space-charge layer at -xP, the corresponding N region boundary at xN, and the space-charge layer width w = xN -(-x,). The boundary conditions at these two planes were shown by Fletcherfl] to be

PP=

mop + mONd

, II,,=

,_d

( Wl
+ m,,,d)d I -cl2

(1)

3 1 May

1974)

not strictly correct, for, as shown by the present author[4), the work of van Vliet and Gummel does not remove all the inconsistencies. It is also necessary to take into account majority carrier injection, which results in a rather complicated behavior of the electrochemical potentials in both the space-charge and bulk regions. However, the boundary conditions remain valid if d in (5) is replaced by a new quantity d’, defined by d’ = exp [(qV,,-

qV, ~ c)/kT]

(8)

and pw =

d(tnw + ,_d’

m,>.wd)

, flhi=

mrlN + mcjrd

I-d’

(2)

so that ,I&, = dp,,,

nN = n,/d

(3)

where II and p denote hole and electron concentrations, respectively, the subscripts N and P refer to values at the boundaries s,~ and -xP of the space-charge region, the quantities ,u,,~ and nor2 are equilibrium majority carrier concentrations, defined by mrrr = prlr - nnp, and d is given

nruN = nu, -pON

(4)

by d = exp L-q(V,

~ V,J/kT]

(3

where V, is the applied forward bias and V,, is the diffusion or barrier potential. It has been shown by van Vliet[2] and by Gummel[3] that the Fletcher boundary conditions lead to theoretical inconsistencies, the cause being the assumption that the electrochemical potentials (or quasi-Fermi levels) FL. and F,, for electrons and holes, respectively, are constant throughout the space-charge region and separated by an amount equal to the applied bias energy -qV,. Since this stipulation is one of the two hypotheses upon which the Fletcher conditions rest, it is clear that a correction is necessary. This is accomplished by first realizing that both & and &, must have a non-zero slope everywhere. since the generalized Ohm’s laws .f,, = (oii le) d@,, /dx = pep,,E - eD, rs’;p

(6)

and J,, =((~,,/e)d&,idx

= neb,,E +eD,,$

where V, is the actual junction potential difference (which is larger that the separation of the electrochemical potentials at the junction but smaller than the applied bias V,) and incorporates the effects of majority carrier injection. Equations (l)-(3) retain the form shown, with d’ substituted for d. It is obvious that d’ in (8) reduces to d in (5) for the limit of very low injection, but what is not immediately clear is the quantitative effect of using the original, rather than the modified, Fletcher boundary conditions. A numerical calculation by Sah[S] establishes that the electrochemical potentials are constant across the spacecharge region, but it involves the depletion approximation, which is known to lead to very misleading conclusions[7]. A simpler theoretical approach has been given by van der Ziel,[h] who bases his treatment on (6) and (7). Integrating (6), for example, from -xP to xN and using the fact that the change in V corresponds to a barrier height of -elf,, - (-eV,) gives ph. = p,, exp L(qV,, - y V, - J,R,,. )/kT] where

(W is the hole resistance for unit area of the space-charge layer. The appearance of .f, outside the integral in (IO) means that recombination in the space-charge region is assumed to be negligible. Equation (9) has the form PN = d’p,, when we identify J,R,,, with l in (8). Incorporating space-charge neutrality al the boundaries into (6) and (7) by the relation dnldx = dpldx, defining dp/dx), as -pN/L,,. dnldx), as n,/L,, and eliminating the field E, gives

(7)

-4 = qQpav [ imply that the currents vanish when the gradients of the electrochemical potentials vanish. We therefore assume that both p,. and fi,, have a uniform positive slope e/w in the space charge region. Taking the slopes as identical is

(9)

[I +(hlnN)l/L,

+[I +(n/pp)l(nNlnp)llL, 1-

pdb

lppnN

1 (11)

with a similar expression for J,,. Returning to (IO), van der Ziel points out that p(x) decreases mononically in going

108

from

NOTES

-xp

to xN, so that

(19)

a~

R,,<

dx p=p

w

I _“P W”PN

and

WPPN

and

J,,~,~~~J,~,.+~~~~~,,(I+~). J,R,,,

Using

J,

P

I’(

(12)

N

Expressing

t1 +bvln~)lwlL,

e

I-

+[I +(np/p,)J(nN/n,)wiL,, pNnp IpPnN

we then J

J,,, = J,,,,. (3) and (6) require

JPh = J,,,,

=(~,,/~~)(~)~/n,~J-qD,,(dpid~~))~lI+(~)~)~~~~I

I +(&/I1,,)(PN/tI”) (7-Z)

Using

(I) and (2) in modified

For a typical concentrations

(15)

= d’.l,p

dA/dx

)F

d&/dx),’

(16)

ahrupt will be

silicon

x

lO”/cm’(P

x

1O’“/cm’( N region)

P-REGION

the

Inserting becomes

same

1h

this into (12), the inequality

J,&,>, <

T-REGION

N-REGION

p/kT jii,

/kT

15.-

IO--

5 -. _____---------------

0 ..

22.4 -2.5

MICRONS

2.5 MICRONS

Fig.

impurity

region)

10 ‘J+yD,,,

J,,_4.Sr. _

(18) through

the

N(, = 1 .S

Pb,

Going

diodeIS]

N, = 1.5

(17)

at -xp. gives

ph./nY i\

giving 9V,, a magnitude of 27.6 in units of LT. If we assume a forward bias of I6 units and a value for t of about 20 per cent of the bias, we find that tl’ i\ approximately 4.6~ IO-’ and (24) give\ ,),,/ny =~ 4.5 X IO-‘. Letting I*,~ be approximately equal to IJ-,,, equation (22) will then reduce to

Without a priori knowledge of the slopes of @,, at the two boundaries, we have no justification for taking their ratio such that (16) will reduce to the second of (14). An alternate approach is to generalize van der Ziel’s procedure by writing (6) and (7) at sN as

with a similar set procedure as before

the term

that

= ppq+p d&r /dx)p

d&/dx),

form,

(14)

and

.LN = P&I+

(21)

obtain

I’h

which we recognize as a valid approximation because the ratios w/L,,, w/L,,, pNInN and np/pp are all small. Hence E (=J,,R,,) is also small and the assumption that F,, and F,, are constant and parallel appears to be verified. We notice, however, that (13) is based on assumptions which may be expressed as

.L

as

J = Jr,,, + J,,, = J,,, T Jph

(13)

whereas

the total current

/p, from (I 1) gives


JR

<$$.

(20)

I

MICRONS

.--I

4.5 x 10 ‘Jw

I

YL.pN

(25)

L

to be verified

kT

II

now

(X)

NOTES

The second

term on the right R,. <

is small,

leaving

region can be ignored, and this contradicts behavior of silicon diodes [7].

4.5 x 10 ‘Jw r ‘I!&&‘NJph.

Electronics,

1975. Vol.

lg. pp. 109-110.

the observed

ALLEN

Since the injected hole current which reaches the N side of the space-charge region should be a significant fraction of the total current, the right-hand side of (27) is like the right-hand side of (12), except for the numerical factor which now invalidates the inequality. On the other, a similar calculation applied to (23) shows that JRp/np is small and van der Ziel’s conclusion is correct. This apparently contradictory result is in fact confirmed by an exact numerical calculation performed by Stone,]91 who obtained the behavior shown in Fig. 1 for a short diode with the doping specified above. We note that about 20 per cent of the total applied potential appears in the form of a change in GLpin the transition region, and the rest goe5 into the bulk region potential difference. It is thus apparent that the original Fletcher boundary conditions are not universally valid and must be modified to suit the device under study. In particular, equation (14) implies that the recombination current in the space-charge

Solid-State

109

Pergamon Press.

NLJSSBAUM

Department of Etectrical Engineering, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A.

REFERENCES I.

2. 3. 4. 5. 6. 7. 8. 9.

N. H. Fletcher, J. Electron. 2, 609 (1957). K. M. van Vliet, So/id-St. Electron. 9, 185 (1966). H. K. Gummel, Solid-St. Electron. 10, 209 (1961). A. Nussbaum, Solid-St. Electron. 12, 122 (1969). C. T. Sah, IEEE Trans. Electron Devices ED-13, 839 (1966). A. van der Ziel, Solid-St. Electron. 16, 1509 (1973). A. Nussbaum, Phys. Status Solidi 19A, 441 (1973). A. S. Grove, Physics and Technology of Serniconductor Devices, Wiley, New York (1967). P. S. Stone, Ph.D. Thesis, University of Minnesota (1971).

Printed in Great Britain

COMMENTS ON THE GENERALIZED EINSTEIN RELATION FOR SEMICONDUCTORS (Received 8 January

1974; in revised form 24 May 1974)

In a recent paper[l], Marshak and Assaf raised the question of the validity of the generalized Einstein relation given by

D _ kT Fm(tl) e F--dr)) P

(1)

under non-equilibrium conditions. They also derived[l] a different form of the relation which, according to them, is valid for a wide range of carrier concentrations both under equilibrium and non-equilibrium conditions. On the basis of this relation they also concluded that the diffusion equation formulation of Gassaway[2] for nonhomogeneous materials is in error. Although the problem has been discussed in some detail by Stratton[3] in a recent publication, which Marshak and Assaf may not have been aware of at the time of writing their paper, it is necessary to add the following comments to their work as otherwise some of the misleading statements of their paper would only increase the existing confusion in the literature. Firstly, the generalized Einstein relation in the form of equation (1) was first derived by Landsberg[4] and not by the authors listed in ref. [l]. Secondly, the difference between the relation of Marshak and Assaf and equation (l), which led them to comment that their relation is the most generalized one. resulted only from the difference in the definitions of D used to derive the two relations. In order to make this point clear we give below the derivation in short. The expression for the diffusion current derived in [I] is

given

by

In order to define the diffusion coefficient, expression has been written in [l] as

the above

When equation (3) is used for expressing the diffusion current, the diffusion coefficient is given by

(4)

We note that equation (2) may be reduced to equation (3) only if r is independent of x. &, is, however, the even part of the distribution function. For low electric fields, electrons will be in thermal equilibrium with the lattice. Electron-electron collisions then establish a local equilibrium amongst themselves and a Fermi-Dirac distribution is obeyed at the lattice temperature in degenerate materials with a quasi-Fermi energy varying with x. I/,, may, hence, be written as (this has also been assumed in ref. [l] for the evaluation of 0,) 1

“‘=gexp

I [(E - E,(x))/kT]

+ I’

(3