Temperature dependence of carrier lifetime and Auger recombination in 1.3 μmInGaAsPB. Sermage, J. P. Heritage, and N. K. Dutta Citation: Journal of Applied Physics 57, 5443 (1985); doi: 10.1063/1.334820 View online: http://dx.doi.org/10.1063/1.334820 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/57/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On the hotcarrier effects in 1.3 μm InGaAsP diodes J. Appl. Phys. 73, 7978 (1993); 10.1063/1.353909 Measurement of radiative, Auger, and nonradiative currents in 1.3μm InGaAsP buried heterostructure lasers Appl. Phys. Lett. 50, 310 (1987); 10.1063/1.98234 Threshold temperature dependence of subnanosecond optically excited 1.3μm InGaAsP lasers Appl. Phys. Lett. 44, 578 (1984); 10.1063/1.94846 Photoexcited carrier lifetime and Auger recombination in 1.3μm InGaAsP Appl. Phys. Lett. 42, 259 (1983); 10.1063/1.93907 Influence of hot carriers on the temperature dependence of threshold in 1.3μm InGaAsP lasers Appl. Phys. Lett. 41, 1018 (1982); 10.1063/1.93395

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Temperature dependence of carrier Ufetime and Auger recombination in 1.3 f-lm InGaAsP

B. Sermage CN.E. T.. 196 rue de Paris. 92220 Bagneux. France

J. P. Heritagea)

Bell Communications Research. Inc .. Holmdel. New Jersey 07733

N. K. Dutta AT & T Bell Laboratories. Murray Hill, New Jersey 07974

(Received 15 October 1984; accepted for publication 15 January 1985)

Carrier lifetime has been measured by the luminescence decay of a 1. 3-,um-InGaAsP layer excited by a mode locked Y AG laser at 1.06/-lm. The measurements have been done as a function of excitation intensity for nearly three orders of magnitude of carrier concentration (4 X 1016

_

2x 1019 cm- 3) and for different temperatures (between 32 and 346 K). At low and moderate

carrier density, the lifetime l' follows the variation with excitation of the theoretical radiative lifetime. At high carrier density (above 1018 cm- 3

) the carrier decay rate increases more rapidly than the radiative one and around room temperature this can be accounted for by an additional recombination mechanism whose variation with excitation is typical of an Auger process. The Auger coefficient (Ca = 2.6X 10- 29 cm6 S-I) does not vary with temperature within experimental uncertainty. This suggests that though Auger recombination is for a large part responsible for the low To value of 1.3-,um InGaAsP lasers, the temperature dependence of the Auger coefficient does not contribute to it.

I. INTRODUCTION

InGaAsP lasers and light emitting diodes are important light sources for fiber-optic communications in the 1.2-1.6 ,urn spectral range. The utility of these lasers is, however, disturbed by the unusually strong temperature dependence of the threshold current. I Auger recombination is one of sev­eral proposed explanations for this effect.2-5 (Recent thresh­old measurements6 on short pulse optically pumped laser have established the importance of Auger recombination in these devices.) Lifetime measurements have been performed on the laser itself to determine the value of the Auger coeffi­cient. It is, however, very difficult to reach carrier concentra­tion larger than 3 X lOIS cm- 3 by this method because it is limited by stimulated recombination. For these levels of car­rier density, the Auger recombination rate is smaller than the radiative rate, and the large laser geometry favors stimu­lated recombination which modifies the carrier lifetime so that the so determined Auger coefficient is imprecise.

One method to eliminate this problem would be to use doped samples with different doping levels between 10 17 and 1019 em -3 and to study the luminescence decay following a small excitation.7 The advantage of this method is that the density of carrier is wen known. The disadvantage comes from the impurity concentration which induces band tailing and eventually modifies the recombination probability.

Another method which is the one we have used here is to excite locally a pure quartenary layer with a focused laser beam. 8 In this case, it is possible to achieve carrier concentra­tions as high as 10 19 cm - 3 with negligible stimulated recom-

.1 Work performed al AT & T Bell Laboratories.

bination. In a previous paper, 8 we have shown that at room temperature and at high carrier concentration, the variation of the lifetime as a function of carrier density was larger than that of the radiative lifetime showing the existence of Auger recombination. However, we did not determine the variation with temperature of the Auger coefficient and yet this vari­ation has to be known for the calculation of the variation with temperature of the threshold current in quarternary lasers.

Indeed the threshold current density J th is given by the sum of the radiative and the nonradiative contribution:

(1 )

where d is the thickness of the active layer. The threshold density nth is a function of temperature and can be calculat­ed from known parameters of InGaAsP band structure. The radiative lifetime 1', is a function of temperature and carrier density and can also be calculated. The nonradiative lifetime Tn, is supposed to be due to Auger recombination and in the first approximation varies with carrier density as follows:

(2)

The only unknown data are the value of Ca and its variation with temperature. Many authors have tried to calculate it theoretically in InGaAsp. 2

,9-14 However this calculation re­quires knowledge of the band structure far from the extrema in r and there are some uncertainties in the evaluation of the matrix element of the interaction. For these reasons there are large differences between the values calculated in these pa­pers (Ca - 33 X 10-29 cm6 s -I in Ref. 2 and Ca -4 X 10-29

cm6 s - 1 in Ref. 9). Furthermore the sign of the variation of

5443 J. Appl. Phys. 57 (12). 15 June 1985 0021-8979/85/125443-07$02.40 ® 1985 American Institute of Physics 5443

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C with temperature (at high carrier density) is not agreed u.;on by different authors. I 1.12 It is the purpose of this paper to study experimentally the variation of the Auger coeffi­cient with temperature.

II. EXPERIMENTAL PROCEDURE

The experiment consists in observing with a very fast photodiode the decay of the luminescence of an InGaAsP sample excited by a mode-locked laser. We have used differ­ent samples with quartenary layer thicknesses varying between 0.3 and 1.6,um, but most of the work was done on the 1.6-,um thick layer. The InGaAsP layers were grown by liquid phase epitaxy on an InP substrate and had a room­temperature emission wavelength of 1.3 ,urn at low excita­tion. They were not intentionally doped and the residual doping was n type between 2 X 1015 and 1016 cm -3. The lay­ers were covered by a l-,um thick InP cap layer to reduce surface recombination.

The experimental setup is the same as in Ref. 8 and is shown on Fig. 1. A mode-locked Y AG laser working at 1.06 ,urn gives pulses of 100 ps separated by 10 ns with an average power of 5 W. Since the delay between two pulses is too small to permit measurement ofluminescence decay times of 50 ns (which occur at low excitation), the beam passes through an accousto-optic modulator synchronized on the Y AG laser which extract one pulse every 250 ns.

The beam passes through a Glan-Thompson polarizer which has an extinction ratio of 5 X 10-6 to be certain of the degree of polarization. The beam is then focused with a 2 X or a 5 X microscope objective on the sample which is glued with silica oil on a sapphire window, which in turn is glued on a thermoelectric coolerlheater or a cryostat for the low temperature measurements. The luminescence coming out from the sample is collimated by a 20 X microscope objec­tive. The beam passes through another polarizer which sup­presses the 1.06-,um laser beam. Since the luminescence is unpolarized, one half of it passes through the polarizer. (This system is used in place of a 1.06-,um filter which cut the high energy part of the luminescence spectrum having the fastest

FIG. 1. Diagram of the luminescence decay measurement setup.

5444 J. Appl. Phys., Vol. 57, No. 12, 15 June 1985

decay and thus increases the observed luminescence decay time.) The beam is then focused with a 10 X microscope ob­jective on a fast photodiode (either a germanium APD with a response time of 500 ps for the low excitation cases or an InGaAs pin photodiode with a response of 100 ps, for the high excitation cases). The signal from the diode is sent to a sampling oscilloscope and then stored in a multichannel averager and recorded on a chart recorder.

III. DETERMINA rio IN OF THE DENSITY OF THE CARRIIER

The diameter of the beam waist has been measured by two methods: the first one consists in measuring the trans­mission of the beam through different pinholes. The maxi­mum of transmission is obtained when the pinhole is situated at the center of the beam waist. There, the value of the trans­mission gives the diameter of the beam waist if the beam is Gaussian which can be checked by using other pinholes. The other method takes advantage of the dynamic Burstein ef­fect: when we translate the InGaAsP sample along the beam, the transmission of the 1.06-,um beam is maximum when the sample is at the focus. The beam waist w is measured by a translation of the sample on which we have glued a l-,um­thick aluminum foil, perpendicular to the beam. The two measurements give the same value, inside a 5% uncertainty which corresponds to the precision of these measurements. The results for the two-microscope objective that we have used are the following: with the 2 X, W = 15.5,um, with the 5 X, w = 9 ,urn [the beam is Gaussian, i.e., its intensity is given by: 1= Ioexp( - 2r Iw2

)].

To calculate the density of carrier, we assume that each absorbed photon creates an electron-hole pair, so the elec­tron or hole density created at the center of the Gaussian is given by

n = E _2_TMo (I-R )(1- T) max hv 1Tdw2

X I: 1 - exp( ~ )] ~, (3)

where E is the pulses energy measured in front of the focus­ing microscope objective, hv is the energy of the 1.06-,um photon, R = 0.25 is the reflectivity of the sampl.e surface, T MO is the transmission of the microscope objective, Tis the sample transmission which depends on the excitation inten­sity, d is the thickness of the layer, tp is the pulse length (100 ps) and r is the lifetime of the carrier. ! The factor [1 - exp( - tJrlJ rlt, is a correcting factor not far from l.j

One major problem arises from the fact that the distri­bution of electrons and holes is Gaussian and we observe the luminescence coming from the entire excited spot. We thus consider an equivalent density of carrier n which is an aver­age over all the Gaussian spot. It can be shown (see Appen­dix) that this average density n is given by the foHowing equation:

n = 0.65nmu· (4)

Furthermore, we have checked that the diameter of the lu­minescence spot was the same as the diameter of the laser

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beam waist so that the lateral diffusion of carrier was negligi­ble.

IV. DETERMiNATION OF THE CARRIER LIFETIME

Figure 2 gives some examples of luminescence decay curves obtained at room temperature for different excitation intensities. As can be seen on the figure, the luminescence decay depends very strongly on the excitation intensity. The decay curves are then not exponential. We determine the luminescence decay time 7L by the tangent at t = O. 7L is defined as follows:

1 aIL (5)

where IL is the luminescence intensity. In fact the carrier lifetime 7 is generall1y not equal to 7L

since the luminescence I L is generally not proportional to the carrier density n. In all the foUowing, we suppose that the density of carrier brought by the excitation is much larger than the density of carrier at equilibrium: n = p>no,po. The luminescence intensity I L is proportional to the radiative recombination rate:

I an) d' . 1 L -- ra latlVe = n -, at 7,

(6)

where'T, is the radiative lifetime. Then the carrier lifetime 7 is given by

I an aIog (n) I _= __ = __ -= __ , (7) 7 n at aIog (h) 7 L 1 + Y 7 L

where y = a log(1/7r )/a log n. Thus the determination of the carrier lifetime from the

luminescence decay requires the knowledge of the variation of the radiative lifetime with the carrier density. We have

O,L--.-:::.-----:.---~-~ _ _;b.:___,:;,,-J o 5 10 15 20 25 DELAY (ns)

FIG. 2. Examples ofluminescence decay curves obtained for three different excitation intensities. 10 corresponds to a density of excitation of2.9X 10" W cm- 2 and a density of carrier of 8.6X 1016 cm- 3•

5445 J. Appl. Phys., Vol. 57, No. 12, 15 June 1985

thus calculated the radiative lifetime using Fermi golden rule as in Ref. 15. We assume that k conservation is valid (which is a good hypothesis since the doping is small), and we consider transitions between the lower conduction band and the three upper valence bands. The nonparabolicity of the bands is taken into account following Kane model, 16 the pa­rameters of which being determined from the values of the energies and the masses in r:EG = 0.953 eV, ..1 so = 0.25 eV, me = 0.0566, mhh = 0.465, mh 1 = 0.074.12 The matrix element of the dipolar transition between conduction and valence bands in the center of the Brillouin zone has been determined by the measurement of the absorption coeffi­cient at 1.06 flm: a = 1.6 X 104 cm - I. (We take into account the variation with k of the matrix element.)

The curves of the inverse of the radiative lifetime 1/7, at different temperatures are given on Fig. 3. As can be seen on the figure, I/r, is proportional to the carrier density at low concentration. It is the usual bimolecular recombina­tion. At high carrier density, the Fermi level for the holes penetrates into the valence band and the probability for an electron of the conduction band to find a free site in the valence band becomes equal to one, and then the radiative lifetime tends towards a constant value. At low and interme­diate density (S 1018 cm- 3

), one can approximate these curves by the following expansion obtained as the best fit of Fig. 3 curves:

I 2 -=B,n+C,n,

Br = 1.92 X 1O-1O(300/T)1.5 cm3 S-I,

Cr = - 4.7X 1O- 29(300IT)3 cm6 S-I.

(8)

Figure 4 gives the variation of y = a log( 1/7,)/ a log n as a function of the carrier density for different tempera­tures.

V. RESULTS AND DISCUSSION

Using Eq. (5) and thevaluesofytaken in Fig. 4, we have calculated carrier lifetimes 7 from the experimentallurnines­cence decay times 7 L •

Figure 5 gives the variation of 1/7 as a function of carn-

7' 109F~;:-,"""""""'r--.---r-..,........,...==:;;;~~~ $ W

~ UJ

~ ~108 ~ o < a:

~~7~~~~~~~~~~~1 ~ 1016 1017 1018 1019

~ CARRIERS DENSITY (cm-3)

FIG. 3. Variation of the inverse radiative lifetime as a function of carrier density calculated for \.3 /.lm InGaAsP at different temperatures.

Sermage, Heritage, and Dutta 5445

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1~~~~~~~~~~~

0.9 c 08 0\'

S 0.7 '0 -: 0.6 1: 05 ~ . ....... g 0.4

=c: 0.3 II ;> O.

0.1 O·~~~~~~~~~~~~~~

1016 1017 1018 1019 1020

CARRIERS DENSITY (cm- 3 )

FIG. 4. Variation ofr = alog(I/1',)/a log n with the carrier density calcu­lated for 1.3 p,m InGaAsP.

rier density at five temperatures: 32, 82,200,281, and 346 K. Let us note that the curves at 32, 82, and 200 K are not as precise (5-10 points) as the curves at 281 and 346 K which correspond to about 50 points each (Figs. 6 and 7).

At low carrier density (:S 1017 cm- 3), we reach long

lifetimes (lOOns) which shows that nonradiative recombina­tions via impurity centers or InP-InGaAsP interface recom­bination is small as in Refs. 7 and 8. At intermediate carrier density (1018 cm- 3

), the inverse lifetime 1/1- follows the ra­diative one and we do not observe an increase of the lifetime due to reabsorption of the emitted photons (recycling effect) as it has been observed by other workers. 7.17 This is due to the fact that the density of excitation (even for the lowest carrier density) is one or two orders of magnitude larger than those used in these references and also because in our case, the excited area is small and we detect only the luminescence coming from this area. Due to the filling of the bands, the absorption is quenched in the excited region and thus the emitted photons are reabsorbed outside this region where

lJJ :t i= lJJ IJ... :::; 108 lJJ III a: lJJ > ~

FIG. 5. Experimental curves of the inverse lifetime (full line) and calculated curves of the inverse radiative lifetime (dashed line) as a function of carrier density. The high carrier density, low temperature data are probably dis­torted by the presence of stimulated recombination which is not present for the 281 and 346 K data.

5446 J. Appl. Phys., Vol. 57, No. 12, 15 June 1985

InGaAsP 281K

FIG. 6. Inverse carrier lifetime at 281 Kin 1.3 p,m InGaAsP. The crosses (w = 15.5p,m) and the squares (w = 9 Jim) are the experimental inverse life­time. The dashed line is the calculated inverse radiative lifetime. The open circles are obtained by subtracting the radiative inverse lifetime from the experimental data and represent then the nonradiative part of the recombin­ation. The dashed dotted line is a Co n2 curve with an Auger coefficient of 2.6X 10- 29 cm6 s·'. The full line which passes through the experimental points is the sum of the dashed and the dashed dotted lines.

the luminescence of these created pairs does not reach our detector. The more intense the excitation is, the smaller the recycling will be which means that in any case, recycling cannot modify our results on Auger recombination. How­ever, since we have not measured the quantum efficiency, there could be at low carrier density some recycling and some nonradiative recombination whose opposite effect on

+

InGaAsP 346K

+

10,a CARRIERS DENSITY(cm- 3 )

10'9

FIG. 7. Inverse carriers lifetime at 346 K in l.3p,m InGaAsP. The significa­tion of curves and points are the same as in Fig. 6.

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lifetime would compensate. Different authors take different hypotheses on quantum efficiency: C. H. Henry et al. 7 sup­pose that at low doping level (10 17 cm-3

) and low excitation, quantum efficiency is 1 at room temperature. M. Asada and Y. Suematsul8 suppose that the quantum efficiency is 0.3 for the same conditions. This probably depends on the quality of the samples. Since in our case the residual doping is small (2 X 1016 em - 3), we think that nonradiative recombination is small. For all above reasons, we expect that recycling is small ( < 50%).

At low temperature (32 and 82 K), the decay rate is smaller than the theoretical radiative rate and we attribute that to a heating of the carrier that is larger at low than at room temperature. 19 This comes from the fact that when carrier temperature is not small compared to the LO phonon energy, carrier can emit LO phonons easily and then thenna­lize with the lattice more efficiently. The experimental curves at 32 and 82 K are close to the 60 and 100 K inverse radiative lifetime curves.

At high carrier density (;::; 1018 cm- 3), 1/T increases

more rapidly than 1/ T r • We do not interpret this effect at low temperature (32, 82, and 200 K) since this fast decay at high excitation could be due to stimulated recombination. At 281 and 346 K, we have checked that stimulated recombination was negligible by comparing the results obtained with two­microscope objective (2 X and 5 X ) i.e., with two diameters of the excited area (15.5 and 9,um). The data obtained in both cases fall on the same curve. If there were stimulated recom­bination, for the same carrier density it would be more im­portant, for a larger excited area. We have also checked that the lifetime was not dependent on the thickness (0.3-1.6 ,urn) of the active layer for the same carrier density (stimulated recombination would be enhanced by a thicker active layer). This gives us some confidence on the fact that the strong increase of 1/T at high carrier density is due to Auger recom­bination. Figures 6 and 7 show the experimental 1/T points at 281 and 346 K and the inverse of Auger recombination lifetime lira obtained by subtracting 1/T, (theoretical) to the experimental 1/T:

1 1 1 -=-+-. (9) T '7", '7"a

As can be seen on these figures, 1/'7"a points are rather wen fitted by the usual square law characteristic of an Auger process [Eq. (2)). The Auger coefficient Ca is found to be equal to 2.6X 10-29 cm6 S-I at both temperatures 281 and 346 K.

Let us note that though the dispersion of the experimen­tal results is important, we have recorded so many decay curves, especiaUy at high excitation that the value we obtain for the Auger coefficient is relatively precise ( ± 10%). How­ever there could be systematic errors due to uncertainty in the carrier density calculation. One error could come from the fact that conduction band offset between InP and In­GaAsP is not large enough to confine the electrons inside the quarternary layer. This offset is not well known. In the case of the InGaAs-InP interface, it has been measured that LJEc = 0.33 LJEG.20 In the case of Ino.87Gao.13As029P021-InP, LJEc was found to be equal to 0.66 LJEG. 21 This differ-

5447 J. Appl. Phys., Vol. 57, No. 12, 15 June 1985

ence could be due to the different compositions of the alloy or to the imprecision of these measurements. If the barrier height for the electrons is smaller than 200 meV, at high excitation, a portion of the electrons diffuse into InP so that the carrier density in InGaAsP is slightly smaller than the one we estimate. The charge density in InP curves the bands and it can be calculated using Poisson's equation that the number of carriers which penetrate in InP would be small compared to those which stay in the quarternary layer for the thicknesses we have used (;;00.3 ,urn).

Let us note that this value for the Auger coefficient is two orders of magnitUde larger than the one observed in GaAs: Ca _10- 31 cm6 S-I (Ref. 22) and four orders of mag­nitude smaller than in GaSb: Ca _10- 25 cm6 S-I.22 This is probably due to the relative difference between the band gap EG and the spin-orbit splitting .1 so. (When they are equal, Auger recombination with k conservation can be allowed in r.) This is nearly the case in GaSb(EG = 0.812 eV and LJ so = 0.749 eV). These energies are more different in In­GaAsP(EG = 0.935 eV and LJ so = 0.250 eV) and still more different in GaAs (EG = 1.42 eV and.1 so = 0.34 eV).22

The absolute value of the Auger coefficient agrees with the last experimental measurements: 2.3 X 10-29 cm6 S-I (Ref. 8) and 1.5 X 10- 29 cmo S-I.23 Let us note that the Ca value we determine here at room temperature is a little dif­ferent from the one obtained in Ref. 8 (2.3 X 10-29 cm6 S-I) with exactly the same experimental data. This is due to the fact that in Ref. 8 we used a simple parabolic band model to calculate the radiative lifetime. Here nonparabolic bands are included. Since the Auger lifetime is obtained by subtracting 1/T, from 1/'7", a difference in T, will give a slight difference in Ta and then in the Auger coefficient Ca. The theoretical determinations of the Auger coefficient are generally higher than our value since they vary between 4X 10-29 cmo S-I and 3.3 X 10-28 cm6 S-I.2.9-14 However, recently M. Take­shima24 has shown by taking into account the anisotropy of the bands, that the HHSC process (involving the splitted valence band) was predominant and calculated an Auger co­efficient of 2.9X 10-29 cmo S-I at room temperature. He also predicts a slight decrease in the Auger coefficient with temperature (2.95 X 10-29 cm6 S-I at 281 K and 2.8X 10- 29

cmo S-I at 346 K). The results of the calculations of Take­shima are in very good agreement with our measurements and fall inside our uncertainty limits. This constancy of Ca

(or its slight decrease with temperature) is in contradiction to what was generally thought previously.

VI. CONCLUSION

Finally, using the result of our measurement for the Auger coefficient (Ca -2.6X 10- 29 cm6 S-I) and the fact that it does not vary with temperature, we can calculate the value of To for 1.3-,um InGaAsP lasers. [The threshold cur­rent is supposed to vary with temperature as exp(T ITo).] We suppose that threshold is obtained for a gain of 100 em - I and we use the Kane model for the description of the con­duction and the three valence bands. We calculate the vari­ation of the threshold current between 300 and 350 K. We find at 300 K a carrier density at threshold of 1.42 X 1018

cm- 3 a current density of 5.3X 1~ A cm-2 ,urn-I, and a

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value for To of 132 K. If we take into account a recycling of the photon of 1.5 as in Ref. 7 due to the fact that the photons are guided inside the active layer, the current density is 4.0X 103 A cm- 2 pm- I and To = 121 K. If there was no Auger recombination, To would be equal to 209 K which corresponds approximately to the only variation of the threshold carrier density. This shows that Auger recombina­tion is an important mechanism which lowers the value of To. However it is not sufficient to explain the values of To observed on quaternary lasers which are usually between 50 and 9OK.

Let us note that the To value we have calculated as­sumes that the threshold carrier density and its variation with temperature can be obtained from a k'p type band structure calculation and for a threshold gain of 100 em - I.

In fact recent measurements6.25.26 seem to show that the car­rier density at threshold is about two times larger (3 X 10 18

cm- 3) and that its variation with temperature is larger than

what we have calculated, i.e., it varies like exp(T IT~) with a T~ of 120 K 6

•25 (other people have found 220 K.)27 This

larger carrier density makes Auger recombination more im­portant and in this case, the laser threshold current To value would by 50 and 107 K using 120 and 220 K for the thresh­old carrier density T~, 3 X lOIS cm- 3 for the carrier density at room temperature, 2.6 X 10-29 cm6 s -I for the Auger co­efficient and a recycling factor of 1.5. If the threshold carrier density T ~ is really 120 K, then the calculated laser thresh­old current To value is in very good agreement with the ex­perimental one. In this case, the question which would re­main is why the threshold carrier density is so high and why its temperature variation is larger than expected.

APPENDIX

The carrier density at the distance r from the center of the laser spot in the plane of the layer is

n(r,t) = nma>. (t )exp( - 2r Iw2),

where nmax is a function of time and w is independent on time, i.e., we assume that there is no diffusion.

For carrier density around nmax ' the radiative lifetime 1'r can be expressed in the following way

~=BlnY. 1'r

Then the luminescence intensity at the distance r from the laser is

IL(r,t)-~)rad= n(r,t) =Bln(r,t)Y+l, at Tr

IL (r,t )-B.nmax (t V+ lexp[ - 2r(r + 1)/w2).

The total luminescence intensity is then

IL =iooh(r)21Trdr=1TBln~:x. w2

o 2(r + 1) and

dIL 100 an -- = B\(r + 1)nY-21Trdr. dt 0 at

We suppose that the total carrier lifetime (radiative + nonradiative) vary with carrier density like

5448 J. Appl. Phys .• Vol. 57. No. 12. 15 June 1985

= 2(a + r+ 1)

and the luminescence d.ecay time 1'L is

_1_ =..!.. dIL = Bzn~ (r + 1)2 . 1'L IL dt a + r + 1

We have seen previously that the relation between the luminescence decay and. the carrier lifetime is

_1 =(r+1r!-. TL l'

The average density of carrier nay is defined by

~ = B2n~v' l'

then,

1 1 - = (r + 1)- = Bz(r + l)n~v' 1'L T

and finally

Let us consider two extreme cases: at low carrier den­sity a = r = 1, then nay = 0.67nmax and at high carriers density (n = 2x 10\9 cm- 3

), lifetime is commanded by Au­ger recombination (a = 2) and r = 0.34 from Fig. 4, then nay = 0.63nmax .

In conclusion, we can use with a good approximation the following relation:

nay = 0.65nmax ·

Iv. Horikoshi in In GaAsP Alloy Semiconductors, edited by T. P. Pearsall (Wiley, New York, 1982), p. 379.

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