Notes 345

S&j-Store Electmica, 1976, Vol. 19, pp. ?d5-346. Pcrgamon Press. Printed in hat kitain

PHOTOCURRENT IN A DIFFUSED p-n JUNCTION

(Receioed 26 My 1975; in revisedform 17 September 1975)

INTRODUCTION

An analytical study of diffused p-n junction photodiode has been made, with special reference to the effect of surface states on the generation of photocurrent at shorter wavelengths. The impurity distribution in the diiused base region has been assumed to be Gaussian, while that of collector has been assumed to be uniform. The results obtained have been compared with the spectral response of photodiodes with assumed exponential impurity distribution in the base. The fall in the photocurrent with increasing values of surface recombination velocity is more pronounced for a Gaussian distribution, especially for strong

absorption. This has been explained by considering the difference in the nature of built-m fields in these two cases.

Silicon solar cells have been the topic for intensive investigations [ 141 during the last two decades on account of their technological advantages and various other applications. In the present article, the authors wish to discuss certain aspects of spectral response in the short wavelength region of diffused silicon p-n junctions with Gaussian impurity protile in the base region. It has been noticed from the published literature that, although, the impurity distribution obtained from thermal diffusion[5] is, in general, described by either a complementary error function or a Gaussian function, it is customary to assume this distribution to be replaced by a suitable exponential function, presumably for mathematical simplicity. Admittedly though computer calculation is possible, we describe below a simple but sutliciently accurate analytical method for obtaining an expression for photocurrent resulting from a diffused junction as mentioned above. It is shown that this photocurrent depends signiticantly on the nature of impurity atom distribution in the base and also on recombination at the front surface. The magnitude of this photocurrent obtained for a Gaussian impurity profile ditIers significantly from that obtained on the basis of an exponential impurity distribution particularly in the shorter wavelength (higher absorption coefficient) region.

We assume that a diffused p-n junction with Gaussian impurity distribution in the p-type base region is exposed to monochroma- tic radiation and its front surface is parallel to the plane of the junction. The flow of photogenerated carriers in the base region will be governed by their concentration gradient and the built-in-field due to non-uniform doping. Under steady state condition the one-dimensional continuity equation for excess minority carrier electrons An in the ptype base region may be written in the usual manner [l]

where 4, p,, and T. are the average values of the diffusion constant, mobility and lifetime of electrons. F,= photon tlux incident at the front surface having zero

reflection coefficient a = absorption coefficient

E =_kTx 4 2L*=

(2)

is the built-in field due to Gaussian distribution of acceptor impruity atoms in the base region, which is expressed by the following equation

QO., is the fixed amount of impurity per unit area at the surface.

Normally, for silicon diffused junctions tbe impurity ditlusion length L,, may be assumed to be quite small as compared with the minority carrier diifusion length L. = ~(D.T.), allow@ us to make the following simplitication of the continuity equation

2

~_,~-An=_~2L~ze-“‘nL*’ (4) ”

where

Solution to eqn (4) may be obtained in terms of power series following a method suggested by Yang[6] as given below

An = a&r) + o&(r) + (5)

where

r2 r4 r6 f,(r)=lt-t-t-t+..

2 4x2 6x4~2

r3 r’ h(r)= I t-t

r’ 3x1

-+-+. . .

5X3X1 7X5X3X1

where a, and a2 are constants, which may be. determined from boundary conditions.

The mhtority carrier electrons are lost due to recombination at the front surface (x = 0), which is governed by the following equation

sAnl.-0 =D_$ i I-

,tp,,EAnIX-a.

Where s is the front surface recombination velocity. At the boundary of the depletion layer x = o, the concentration of excess carriers An is related to the bias u, across the junction according to the well-known diode equation

AnIx--=rb.[exp(-$)-I].

Where A is a numerical constant depending on the characteristics of the junction.

From eqns (5)-(7) the values of the constants a, and a2 are obtained as

0. n, exp -& -1

[ ()I --h(r)-+fi(o) I a’=st/2L,

[ &yflw+M~)]

t3

n, exp s -1 [( )I

- Ff.(+,(@) a2=

[ D”.fi(O) + Mu)] Sd2LA

where fl(@), f*(o) and h(o) are the values of f~(~)* f&) and fk) at x = 0.

346 Notes

If, at this stage, we assume that the doping concentration of the p-type base region is large compared with that of the collector region, and that the photogeneration of electron-hole pairs is

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limited to the pregion ione,-then, the photocurrent due td excess electron minority carriers crossing the depletion layer, may be assumed to be approximately equal to the minority carrier electron flow acrossthe p-n j&cti&. Neglecting recoibination- generation current in the depletion layer, the electron current component is given by

. . dAn

Jn = q&d, = +qp.tEAn)l.-w x Y

Using eqns (2) and (5) the expression for J. may be written as

, , . . 04 1.0 10

where Fig. 1. Normalized phoiocurrent vs (so/D.) with dimensionless absorption ao as parameter. Here s = surface recombination

J‘” = q&r,,

V2Ln [~flb)+f2w] (10)

velocity. The numerical values used are, for Gaussian distribution, o/q2L, =4, and for exponential case, fro = 5;

OIL. = 0.2.

has been assumed to be

(11) E(Gaussian)

E (exponential) =

The first term in eqn (9) is the dark current and the second represents the photocurrent. The above eqns (9)-(11) form the Now for higher values of (110 (shorter wavelengths) most of the basis of any investigation on photovoltaic properties of diffused carriers are generated in a very thin layer just below the surface. p-n junctions. In the present report, however, we restrict The rate of surface recombination of these carriers will be ourselves only to the study of .I RphD,o as a function of front surface obviously more pronounced for a Gaussian distribution, as it is recombination for various values of absorption coefficients (1. characterised by a negligible built-in electric field near the surface. Figure 1 gives a plot of normalised value of photocurrent as given Thus, the assumption of an exponential impurity distribution may by eqn (11) in terms of dimensionless functions of surface lead to erranous result for photocurrent, particularly, in the short recombination velocities (so/D.) and absorption coefficient (aw). wavelength region. The same graph also indicates photocurrent obtained from a similar p-n junction with an exponential doping protile as given in Acknowledgement-Thanks are due to the authorities, Kuruk- Ref. [4] for comparison. shetra University, for financial assistance to one of the authors

It is clear that decrease in the magnitude of photo-current with (AS.). increasing values of surface recombination is more sharp for the Gaussian distribution. This effect is more pronounced for higher Physics Department AMTABHA SINHA values of absorption coefficient, (YO, which, in effect, means Kurukshetra Unioersity S. K. CHATKIPADHYAYA shorter wavelengths. An explanation for this behaviour is easily Kurukshetra 132119 obtained from the nature of the built-in electric field in the base of India a photodiode, which is uniform for exponential distribution, while, for Gaussian distribution this field increases linearly with distance REFERENCES

from the front surface as indicated in eqn (2). According to this 1. A. G. Jordan and A. G. Milnes, Trans. IRE, ED-7,242 (1960). equation, when the distribution of impurity atoms is given by a 2. M. B. Prince, J. Appl. Phys. 26, 534 (1955). Gaussian function, the built-in field which aids the flow of 3. M. Wolf and E. L. Ralph, Trans. IEEE ED-12, 470 (1965). minority carrier electrons towards the depletion layer, is 4. R. B. Gangadhar and A. B. Bhattacharyya, ht. J. Electron. 25, negligibly small near the front surface (x = 0), even if the average 17 (1968). magnitude of the field is assumed to be large. This has exactly 5. H. F. Wolf, Silicon Semiconduckv Data, Chap. 5. Pergamon been done while plotting the graphs of Fig. 1, in which the ratio of Press, Oxford (1%9). electric fields at the edge of the depletion layer for the two cases 6. E. S. Yang, Solid St. Electron. 12, 399 (1%9).