Transition processes in semiconductor lasers

LETTERS Transition processes in semiconductor (Received

22 March

TO THE lasers

1963)

THERE has

been considerable interest in the nature of the electron transitions which occur in gallium arsenide lasers.(l, 293) The three possible types of optical transitions are band-to-band, band-to-acceptor, and donor-to-acceptor. These transitions are illustrated schematically in Fig. 1. NATHAN and BURNS@) have shown that the recombination process involves an acceptor.

Band to band

transitions

\

Localized

EDITOR

slightly displaced toward the band, so that q/KT > 2. Hence the valence band states are occupied with high probability whereas the acceptor states are nearly empty. In this note we report a calculation of the transition probabilities and absorption constants for the band-to-acceptor and donor-to-acceptor transitions. It is possible, fortunately, to use experimental information concerning the absorption associated with band-to-band transitions@* 7) in order to determine an important matrix element involved in these calculations. Band-acceptor transitions have also been studied by DuMKE.(*) The calculation follows the procedures of standard first order time dependent perturbation theory. The wave functions for donor and acceptor states are determined in principle from the theory of LUTTINGER and KoHN.@) The formal theory of transitions involving impurity states has been given by BOWLDEN. Since no calculations of the acceptor wave functions in GaAs have been reported, it is necessary to assume that these are hydrogenic with an effective Bohr-radius, a,, determined from the observed binding energy Ea = 0.04 eV. We obtain the following absorption coefficient for acceptor-band transitions:

k-c

=

[1+2m*/h2a~(E,+Aw-E,)]4 X

Z/(hw+Ea--Eg) (1)

hw

in FIG. 1. Energy band structure of GaAs showing important optical transitions (not to scale)

Although the original theoretical studies which suggested the feasibility of semiconductor lasers considered only band-to-band transitions,@) these are not likely to be important. Under normal operating conditions (77”K, NA z 3 x lOls cm-s, No fi: 1017 cm-s), the Fermi level which at absolute zero would be located midway between the valence band and the acceptor levels is only

which Eg is the band gap and m* is the effective mass of electrons in the conduction band. The quantity K is given by K _ 2+

l

Pc?l12

rr2m2Ync~ where

(2)

E is the polarization vector of the electromatrix magnetic field, PC, is the momentum element for band-to-band transitions and n is the index of refraction. K can be determined from optical absorption in the direct transition, since

1063

LETTERS

1064

this has an absorption

TO

constant.

acv = 23’2K(&‘2 + /L;‘~)

2/(fim-&) ~iw

(3)

in which ~1 and pa are reduced effective masses of the conduction band and the heavy and light hole bands. Broadening of the acceptor levels has not been included in (1). In the case of donor-acceptor transitions, the finite width of the donor and acceptor levels must be explicitly included. This is caused by differences in the local environment of each donor and acceptor. We will assume that this distribution is Gaussian, characterized by a width 7. There is an additional parameter in this case: the average overlap integral between donor and acceptor wave functions, which we denote by Sad. In the case Nd < Na, we find ~~3lati3~~ au-d = K ?)w

(4) The overlap integral has been computed for hydrogenic wave functions and its square averaged over a Poisson distribution of donor-acceptor distances. For NA between 1 x 1018 cm-3 and 5 x 101* cme3, Sid Z 0.2. We use the experimental value for the band to band transitions(a)

with ~(0 = 3 x lo4 cm -eV-li2.

(5)

The level width 7 can be inferred from the experiment of NATHAN and BURNS@) to be r] M 0.017 eV. We then find for the band-acceptor transitions

(2 1

d/(fiw+-%-47)

au-c = PO

(6)

where PO = 7 x 103 cm-l/eVlJa and for the donor transitions CC.&.d = 110 cm-l (?&) The

absorption

EDITOR

seem to be adequately large to permit the operation of a laser.* Direct test of the calculated absorptions would be possible in part through the study of the optical properties of heavily compensated n-type GaAs (although in that case Na would appear in equation 7 instead of Nd). The acceptorband transitions discussed here would produce an absorption edge lying approximately 0.04 eV below the principal edge at 1 a51 eV at low temperatures. The acceptor-donor transitions would appear as a moderately broad background. Aeronutronic Division of Ford Motor Co., Newport Beach, Calif.

calculated

here

JOSEPH CALLAWAY~

* No account has been taken of the degeneracies of the donor and acceptor states. This should be correct for transitions in which the initial state is a localized acceptor state. When the initial state is in the conduction band, or associated with a donor, the relevant formulae should be multiplied by factors of four and two, respectively. t Permanent address, Dept. of Physics, University of California, Riverside, California.

References 1. HALL R. N., FENNER G. E., KINGSLEY J. D., SOLTYS T. J. and CARLSON R. O., Phys. Rev. Letters 9, 366 (1962). 2. NATHANM. I., DUMKE W. P., BURNS G., DILL F. H. and LATHER G., Appl. Phys. Letters 1, 62 (1962). 3. QUIST T. M., RED&~ R. k., KEYES G. J.,. KRA& W. E.. LAX B.. MCWHORTER A. L. and ZEIGLER H. J.,‘AppZ. Phys. Letters 1, 91 (1962). 4. NATHAN M. I. and BURNS G., Appl. Phys. Letters 1, 89 (1962). 5. DUMKE W. P., Phys. Rev. 127, 1559 (1962). 6. Moss T. S., J. a&d. Phys. 32; 2136 (1961). 7. STURGE M. D.. Phvs. Rev. 127. 768 (1962). 8. DUMKE W. P.,‘Buli. Amer. phys. Soc.B, 261 (1963). 9. LUTTINGER J. M. and KOHN W., Phys. Rev. 97, 869 (1955). 10. BOWLDEN H. J., Phys. Rev. 106, 427 (1957).

The distribution of zinc between solid GaSb and Ga-Sb melts (Received

exp [ -E”+T~wEd)2]

coefficients

THE

17 April

1963)

IN A RECENT paper,(l) the distribution coefficient for zinc in the indium-antimony system was shown to depend rather sharply on zinc concentration. The present work was carried out to