Effect of reabsorbed recombination radiation on the saturation current of direct gap p-n junctions

Solrd-Srare Elecrronrcs Printed I” the U.S.A.

Vol. 27. No. 10. pp. 913-915,

EFFECT

1984

003x-1101/x4 $3.00 + MI Pcrgamon Press Ltd.

OF REABSORBED RECOMBINATION ON THE SATURATION CURRENT DIRECT GAP p-n JUNCTIONS OLDWIG

Jet Propulsion

Laboratory,

California

VON

Institute

RADIATION OF

Roes

of Technology,

Pasadena,

CA 91109,

U.S.A.

and

Jet Propulsion

Laboratory, California Physics Department,

HARRY MAVROMATIS Institute of Technology, Pasadena, CA 91109, U.S.A. American University, Beirut, Lebanon 365

(Received 19 November 1983; in revisedform

and

27 February 1984)

Abstract-The radiative transfer theory for semiconductors recently developed is applied to p-n junctions under conditions of low level injection. By virtue of the interaction of the radiation field with free carriers across the depletion layer or space charge region, the saturation current density j, in Shockley’s expression j = j,,[exp (qV/kT) - 1) for the diode current is reduced at high doping levels from the customary value which neglects radiation effects altogether. While the effect is insignificant in p-type material, it is noticeable in n-type material owing to the small magnitude of the electron effective mass in direct gap III-V compounds. At an equilibrium electron concentration of 2 X IO’* cm-j in GaAs, a reduction of j, by 15% is predicted.

It is well known that the photon-induced short-circuit current (photo current) of a p-n junction and the current generated by an applied (forward) voltage move in opposite directions. This is because the photo current is driven by the electric field of the junction and the current caused by an applied forward bias is driven against the junction field. p-n junctions consisting of direct band-gap material, as for instance GaAs, acquire a high density of photons in non-equilibrium situations, ’ even without illumination, by virtue of rather short band to band radiative recombination lifetimes. The photons generated in this manner will of course be quickly reabsorbed, subsequently re-emitted then reabsorbed again and so forth. The continuous interplay between absorption and emission will lead to a characteristic photon distribution within the semiconductor which we may call reabsorbed recombination radiation (RRR) to conform with customary usage. This self induced radiation field can be expected to act similar to an externally applied photon field and lower the

to p-n homojunctions again subject to low level injection conditions. The theory of a p-n junction under low level injection conditions was first given by Shockley as outlined in his classical book[2]. A brief review is felt necessary at this point in order to clarify further development. Consider an abrupt, one dimensional, p-n junction. The left half space is occupied by homogeneously doped n-type material and the right half space by homogeneously doped p-type material. A space charge layer or transition region of width W develops between the two different doped sides of the semiconductor. The minority carriers satisfy certain diffusion equations. If the edge of the space charge layer adjacent to the n-type material is located at z = 0 and the edge on the other side, adjacent to the p-type material is located at z = W, the boundary conditions for excess minority carriers read[2]

net current

Here P, is the equilibrium concentration of holes in n-type material, N,, the equilibrium concentration of electrons in p-type material and V is the applied voltage. Furthermore, within the space charge region the difference between the electron imref (quasi Fermi level) and the hole imref is constant and given by

of a p - n junction

in forward

bias.t

A theory of the RRR and its interaction with free. carriers in direct band-gap crystalline semiconductors has recently been developed for low level injection conditions[l]. In this note we like to apply this theory

p(O) = P,(eq”‘kr- 1),

n(W) = No(eqVikr-- 1).

EFn - EFp = qV. tA p-n junction solar cell can be looked at in this way. The dark current I, due to a forward bias developing across a load resistor is decreased by the short circuit current I,, generated by an external photon flux. The only difference is, that here ISC B I, so that the reduction of I, is actually quite large, reversing the sign of the net current.

(1)

(2)

Actually of course, eqn (1) follows from eqn (2). Neglecting recombination in the rather thin space charge layer, Shockley then succeeded in obtaining his celebrated current-voltage characteristic for the 913

0. VONRoos and H. MAVROMATIS

914

(3)

where B(o) is the unit step function (see [lA] for details) and have expanded all terms arising from photon transport assuming a,L z 1 with L as diffusion length. To lowest order in ( a0 L))‘, the diffusion equation for holes in n-type material reads.

(4)

(Dp+DRp)$; z

diode [2]

j =jo(e9v/kr -

l),

with

A(

Here

e”“/kT-l)Ez(-aoZ)(l-e”L-r)=O,

+ 27Rn (10) are the reverse saturation current densities for minority carriers. D, signifies the diffusion constant for electrons in p-type material and L,, is the diffusion length of electrons. Similar definitions hold for DP and L,. When radiative band to band recombination becomes important it has been shown[l] that the diffusion equations for minority carriers must be augmented by appropriate radiative transfer equations and the ensuing system of equations must be solved simultaneously. If f(z, 1, w ) signifies the nonequilibrium photon distribution function with q = cos 0 where 0 is the angle between the z-direction and the direction of the photon flux, f satisfies eqn (27) of [lA] in the n-type region of the p-n junction, to wit

valid for I I 0. D,, = (3~~7,~)~ ’ is the radiative diffusion constant introduced in [lA], rR,, and To,, are the radiative life times for p-type or n-type material respectively, also introduced in [lA] eqn (21). To be specific, if 7RP is given by eqn (21) of WU ‘R,r can be obtained from eqn (21) of [lA] by merely replacing PO, the equilibrium hole density in n-type material, with N,, the equilibrium density of electrons in p-type material, 7P signifies the hole life time governed by a Shockley-Read-Hall (SRH) mechanism[l]. E2 finally is the exponential integral

G(x)

=

m eeXf t2 dt. 11

In complete analogy, the diffusion equation trons in the p-type material of the junction

(11) for elecis given

by 1 $i = a(w)(e-“‘“PO-ip(z)

-f)

(6)

where a is the absorption coefficient at frequency w. In analogy, the radiative transfer equation for the p-type region of the p-n junction reads

fl g

= a(~)(eeh”‘kTN,-‘n(z)

-f).

(7)

By virtue of eqn (2) it can easily be shown that the radiative transfer equation within the depletion layer is given by: df q dz = a(co)(e-“‘k’(e9v’kr - 1) -f).

(8)

The three equations for the photon distribution function f valid in the three regions of the junction must be solved subject to the boundary conditions, f = 0 at z = f co and continuity across the edges of the space charge layer at z = 0 and z = W. The solutions obtained in this manner are then linked to the minority carrier diffusion equations exactly as was done in [lA]. The resultant integrodifferential equations, the analogue of eqn (36) of [lA], are then manipulated in the same manner as eqn (36) of [lA]. That is to say, we have approximated a(w) by a(w) = a&w

- wc)

(9)

(D,,+D,,)~-~+~(eY’/“T-l) n

RP

x E2( ao( z - w))(l

- e(~~““-~)

= 0, (12)

valid for z 2 w. Equations (10) and (12) must be solved subject to the boundary conditions that p vanishes at z = - co, n vanishes at z = + co and of course that eqns (1) are satisfied. Once the solutions of eqns (10) and (12) are found, the currents, being proportional to the derivatives of p and n at their respective edges of the depletion layer, z = 0 and z = W, may also be determined. It follows that the I-V characteristic of a direct band-gap p-n junction under conditions of low level injection, taking due account of the effect of the reabsorbed recombination radiation, is given by

j = (& + j&)(e9v’kT- 1). The reverse saturation in turn given by

j&=4

.h,=q~

(13)

current densities j& and j& are

D+D, ‘----Pp L; DnfD, L:,

(144 O p’ NoY,,

tl4b)

915

Reabsorbed recombination radiation in p-n junctions Table 1. The reduction factors 7” and y, as a function of majority carrier concentration diffusion lengths[4] -I

n-type

-l

GaAs

p-type -3

Polcm I

L

with the “reduction

GaAs

L ,[wl

Y”

101'

2.0

0.996

1017

7.5

1.0

5 x 101'

2.0

0.965

5 x 1017

6.8

0.998

1018

1.9

0.906

lOl8

5.7

0.995

2 x 1018

1.1

0.849

2 x 1018

4.0

0.988

factors”

and Y, = 1 - (%/27~,#+&). The function F(x) and the diffusion lengths L; are defined in the following way

(15b) LL and

F(x) = 0.5 - x In {(x + I)x-~‘~(x + 2)-‘14}. (17) It is noticed that if tR+a, then D,+O and j& as well as jb become the Shockley expressions (5). Although Shockley’s considered only nondegenerate material in his original theory[2], there is nothing to prevent application to degenerate material, for, (a) the minority carrier distribution stays obviously nondegenerate and (b) the SRH recombination terms n/7, or p/7, remain unchanged whether degenerate or nondegenerate material is contemplated[3]t. Consequently we computed 7” and 7, for a variety of doping levels both for degenerate and nondegenerate GaAs material. Table 1 lists the results. They are based on measurements of L’ by Casey et al. [4]. Casey et al. based their determination of L, or L, on the diffusion eqns (10) and (12) with V = 0 (short circuit condition) omitting D,. But since the structure of the diffusion equations remains unchanged by a renormalization of the diffusion constant from D, to D, + D, for instance, their measurement of the short circuit current and their method of analysis determined L’ rather than L. Table 1 has been prepared with the following inputs. For a given majority carrier concentration the diffusion constant D,, or DP was determined from mobilities given by Sze[S]. The radiative life times 7Rn and TV,, were determined from eqn (54) of [lA] tAt very high doping levels effects other than the Shockley-Read-Hall mechanism as for instance Auger recombination become important. But the onset of degeneracy in n-type GaAs starts at a much lower electron concentration than for instance in Si. Consequently Auger recombination is unimportant even though the material is highly degenerate..

and measured

or its equivalent for p-type material. The radiative diffusion constants DRn or DRP were computed from eqn (43a) of [lA] using 0~~= 2 X 104cm-‘. 7P and 7n were obtained using eqn (16), given the experimental values for L; and LA as reported by Casey et al. [4]. Thus, the reduction factors 7, and y,,were ascertained. From Table 1 we notice that the reduction of the diode current at any forward bias due to self-induced photon generation can be noticable for n-type material at high doping levels. The reason for this is clear. Owing to the small electron effective mass, the Fermi level moves, for high doping, into the conduction band and promotes re-emission of the internally generated radiation (Moss-Burstein effect [6]) thus creating a larger photon field than ordinary. The effect should be even larger in n-type InSb than in GaAs since the effective mass of electrons in InSb is about twice as small as the effective mass of electrons in GaAs. In conclusion we like to remark, that realistic GaAs p-n junctions possess of course a finite thickness. But we feel that an elaboration of the theory to encompass these cases is unwarranted, particularly since the overall effect of the reabsorbed recombination radiation is fairly small, as can be seen from Table 1. Acknowledgemenrs-The research described in this paper was carried out by the Jet Propulsion Laboratory, California Institute of Technology, and was sponsored by The Department of Energy through an agreement with the National Aeronautics and Space Administration. One of us (H.M.) would like to thank the National Aeronautics and Space Administration for a summer faculty fellowship. REFERENCES

1. 0. von Roos, J.Appl. Phys. 54, 1390 (1983). hereafter referred to as A; 0. von Roos, J. Appl. Phys. 54, 2495 (1983).

2. W. Schockley, Electrons and Holes in Semiconductors, p. 309ff. van Nostrand, Princeton, New Jersey (1950). 3. 0. von Roos, Solid-St. Electron. 21, 633 (1978). 4. H. C. Casey Jr., B. I. Miller and E. Pinkas, J. Appl. Phys. 44, 1281 (1973). 5. S. M. Sze, Physics of Semiconductor Devices, p. 40. WileyInterscience, New York, (1969). 6. N. Holonyak and M. H. Lee, Lasers, junctions, transport, p. 22. In Semiconductors and Semimetals, Vol. 14, Academic Press, New York, (1979).