160 PROCEEDINGS O F THE IEEE February

Tunneling-Assisted Radiative Re- combination in GaAs-Diffused Junctions

Pankove [ l ] first proposed that the Aigrain Effect [2] (tunneling-assisted ra- diative recombination) could explain the observed shift of the electroluminescent emission peak with biasing current in GaAsdiffused p-n junctions. Archer et al. [ 3 ] showed that there is no need to consider virtual states since the transition can be described by a single-matrix element. Ac- cording to this model, recombination takes place wherever the overlap of the electron wave function with the hole wave function is not zero, and mainly inside the depletion layer where the overlap is maximum (Fig. 1). Engelmann [4] suggested an alternative model according to which the larger con- tribution to current is furnished by tunnel- ing of electrons and subsequent recombina- tion in the p-side (Fig. 1). This was sup- ported by the considerably larger effective mass of holes in comparison with that of electrons in GaAs. We have thus performed a computer calculation of both contribu- tions.

CHARGE I I

-a 0 x- I I

I I I I

Fig. 1. Charge and electric field y d energy distri- bution for a linearly graded lunct~on.

Calculations were made under the fol- lowing assumptions: 1) the impurity profile is taken to be of the linear gradient type (voltage vs. capacitance measurements for the diodes tested for this work supports this assumption). 2 ) The potential is determined by a carrier-free “space charge region, ” with the rest of the diode neutral. 3 ) The energies of the tunneling holes and electrons are taken to be approximately that of the p- side quasi-Fermi level for holes and that of the n-side quasi-Fermi level for electrons. (This is quite reasonable a t low tempera-

Manuscript received October 26. 1964.

tures since carriers near their respective quasi-Fermi levels face a lower and nar- rower potential barrier. 4) In order to sepa- rate the tunneling current from other spu- rious currents we have taken as a measure of the tunneling flux the integrated light intensity J. The normal recombination law for direct gap materials R =Bpn is assumed to be valid. B is calculated from absorption measurements.

The effective number of electrons N- (holes N + ) is considered to be equal to the net number of impurities N D - N A ( N A - N O ) when the electron energy is greater than that of the conduction band (hole energy lower than that of the valence band). Ordi- nary tunneling theory gives for N , inside the depletion layer the expressions

N&) = ,ax for x 5 , b,

for x < + a

where a=(3eV/2ac)”*, a is the measured impurity gradient, and V is the difference between the built-in potential and the ap- plied voltage (Fig. 1). Where b+ and b- are defined by 2a3 - 3 a z b ~ + b q -6c VT/ae = 0. For the purpose of analysis we have divided the current density into three components

J1 = eEJ->-(z)N+(x)dx

JZ = eE ~ * N - ( x ) N + ( r ) d x

J3 = e B J > - ( x ) N + ( d d x

J1 being the current density due to hole tunneling into the n region, J2 the current density due to electron tunneling into the p region, and J3 that generated by tunneling both of electrons and holes into the deple- tion layer and there recombining.

Fig. 2 shows the calculated current densities J2 and Ja as a function of V for a diode with the following characteristics:

N A - N D ( ~ side) = 5 X 1018 cm-3

a=6.0X10*1 cm-‘ (measured from ca- pacitance vs. voltage measurements)

B = 5.5 X lo-* cm+’/sec

ND-NA(n side)=1.2X101* C r r 3

m+ = 1Om- = 0.7 ma Ve = 94 mV

V A ~ (calculated from N D and N A )

An analysis of the results shows: 1) The large contribution to the total current den- sity comes from h , Le., from recombination inside the depletion layer. Only a t very large bias, i.e., when the depletion layer is greatly narrowed, does the contribution due to electron tunneling become equal to that of electron-hole tunneling (at those biasing voltages other injection mechanisms domi-

D

Fig. 2. Calculated current density J vs. the difference between built-in potential and applied voltage

cm-4. Vi-V.=V. for a charge gradient a=6.OX1O2’

nate [ 5 ] ) . 2 ) The calculated values of current density are of the same order of magnitude as the experimentally observed values (w.1 to -1 A/cmz). 3) The logarithmic slope of the calculated current density with applied voltage is constant, as observed for the logarithmic slope of the integrated light intensity with applied voltage. These slopes nevertheless disagree in magnitude (cal- culated sIope=8.28V-l whereas the ob- served sIope=40 V-l). The discrepancy is to be expected since in the voltage region of interest a large number of carriers penetrate the depletion layer by means of tunneling; this disturbs the charge distribution, making the approximation of a carrier-free “space charge region” too crude. A more precise calculation should take this effect into ac- count in a self-consistent formulation. At high-biasing currents the charge distribu- tion can be better approximated by taking an “effective charge gradient” a much lower than the nominal impurity gradient. In order to estimate this effect, we have re- peated the above calculations with an “ef- fective charge gradient” of 7.5~1019 cm-4 and the logarithmic slope increased to 4OV-1. Calculations were also performed for different doping levels and different gradients, and compared with experimental data. The above conclusions remain the same.

J. E. RIPPER^ R. C. C. LEITE~

Bell Telephone Labs., Inc. Murray Hill, N. J .

REFERENCES

. , - I - _ _ . P., private communication to A. G.

electron-impurity band tunnel i i in GaAs luminescent junctions, Phys. Rm. Leff. . vol10, Jun

141 Engelmann, R. W. H.. presented at the Meeting 1963, pp 483-485.

of the Am. Phys. Sax.. Washiiton. D. C.. Apr 1963.

[SI k i t e , R. C. C., J. C. Sarace, A. Yariv. .D. .H. Olson. B. G. Cohen. and J. M. Whelan, Injection mechanisms in GaAs diffused electroluminescent junctions (to be published).

1 On ,leave ol abeence from the In+tuto de P e s

* Present address: Instituto Tecnol6gico de quisas da Marinha. Rio de Janeiro. Brazd. Aerodutica, SHO Jus6 dos Campos, SP Brazil.