Journal of Crystal Growth 189/190 (1998) 580—584

Excitonic enhanced optical gain of GaN/AlGaN quantum wellswith localized states

Takeshi Uenoyama*Central Research Laboratories, Matsushita Electric Industrial Co., Ltd., 3-4 Hikaridai, Seika-cho, Souraku-gun, Kyoto 619-02, Japan

Abstract

We have evaluated the optical gain of GaN/AlGaN quantum well structures with localized states, taking into accountthe Coulomb interaction. The localized states are introduced in the well as quantum dot-like subband states. We haveused the temperature Green’s function formalism to treat the many-body effects and have found a new excitonicenhancement of the optical gain involved the localized states. This enhancement is stronger than the conventionalCoulomb enhancement. It might play an important role to reduce the threshold carrier density. ( 1998 Published byElsevier Science B.V. All rights reserved.

1. Introduction

A short wavelength is one of the essential condi-tions of laser diodes (LDs) for high-density storagedevices. Recently, continuous-wave operation ofthe InGaN multi-quantum-well LDs with a lifetimeof 35 h has been achieved [1,2]. However, there isa key issue, such as high threshold carrier density,to lead them to commercial production. So far, wederived the electron and hole effective mass para-meters of the group-III nitrides, using the first-principles calculation [3] and evaluated the opticalgain of the wurtzite GaN/AlGaN quantum welllasers within a free carrier model [4,5]. As a result,

*Corresponding author. Tel.: #81 774 98 2519; fax: #81774 98 2576; e-mail: takeshi@crl.mei.co.jp.

the threshold current density of the GaN/AlGaNLDs was estimated very high compared to conven-tional GaAs/AlGaAs quantum well LDs. The rea-son is the large density of states due to the strongelectronegativity and the weak spin—orbit couplingof the N atom in group-III nitrides. To overcomethe problem, we proposed the strain effects [6] andthe incorporation [7] of the GaAs or GaP in thewell layer to reduced the density of states. On theother hand, the strong Coulomb interaction forwide-gap semiconductors is also expected to reducethe threshold carrier density. In this paper, weinvestigate the role of the electron—hole Coulombinteraction in the optical gain spectra and try tofind a possibility of the reduction of the thresholdcarrier density. The LD structure we study here isa quantum well with localized states. The localizedstates are introduced as the quantum dot-like sub-band states. Since the radiative recombination

0022-0248/98/$19.00 ( 1998 Published by Elsevier Science B.V. All rights reserved.PII S 0 0 2 2 - 0 2 4 8 ( 9 8 ) 0 0 2 0 4 - 8

attributed to the localized excitons by the alloyfluctuation were measured recently [8,9], it be-comes important to study the relation between thelocalized states and the optical gain. Generally,since the wide-gap semiconductors have the strongCoulomb interaction, the optical gain is enhancedby the Coulomb interaction, compared to the onewithin a free carrier model. This is so-called ‘ex-citonic enhancement’ or ‘Coulomb enhancement’.We have found another excitonic enhancement ofthe optical gain involved the localized states. Thisenhancement is stronger than the conventional oneand it might play an important role to reduce thethreshold carrier density.

2. Optical gain with localized states

The configuration of the quantum dot-like struc-tures in the well is shown in Fig. 1a. The carriersare confined along the growth direction in thequantum well and in the disk with the radiusR along the well layer. The disks are assumed to bedistributed randomly in the well and not to becorrelated among them. Furthermore, only theholes are assumed to be trapped at the lowestsubbands in the disk, since the hole mass is muchheavier than the electron mass. The wave functionof the localized states might be given by

tj(r, z)"S

ap

exp (!a2(r!R

j)2)f

0(z), (1)

Fig. 1. (a) Configuration of the quantum dot-like structures.(b) Energy diagrams of the three-band model.

where Rjis the position of the disk in the well and

f0(z) is the wave function of the lowest subband in

the quantum well. a is the parameter which can bedetermined by the variational principle for thegiven radius R and potential depth of the disk. Theenergy diagram of the three-band model can bedepicted schematically in Fig. 1b. The Hamiltonianof the model including the Coulomb interaction isgiven by

H"+k

e%kaskak#+

k

e)kbskbk#

N0

+i/1

edidsidi

#»%%#»))#»%), (2)

with

»%%"1

2+q

+k1, k2

»(q)ask1`q

ask2~q

ak2ak1,

»))"1

2+q

+k1, k2

»(q)bsk1`q

bsk2~q

bk2bk1

#

1

2+q

+j1, j2

»(q)e*q(Rj1~Rj2)e~q2@2adsj1dsj2dj2dj1

#

1

2+q

+k1, j2

»(q)e~*,Rj2e~q2@4absk1`q

dsj2dj2dk1

#

1

2+q

+j1, k2

»(q)e~*,Rj1e~q2@4adsjibsk2~q

bk2dji,

»%)"+q

+k1, k2

»(q)ask1`q

bsk2~q

bk2ak1

#+q

+k1, k2, j

»(q)e*(k2~q)Rje~(k2~q)2@2a

]2

¸Sp

aask1`q

dsjbk2ak1

#+q

+k1, k2, j

»(q)e~*k2Rje~k22@2a

]2

¸Sp

aask1`q

bsk2~q

djak1

#+q

+k1, j

»(q)e~*2Rje~q2@4a2

¸Sp

aask1`q

dsjdjak1,

where e$i, (d

i, ds

i), e%

k, (a

k, as

k) and e)

k, (b

k, bs

k) are the

energy levels (annihilation and creation operators)

T. Uenoyama / Journal of Crystal Growth 189/190 (1998) 580–584 581

of the localized states, the conduction and the val-ence subband states, respectively. N

0is the number

of the localized states. We assumed that the localiz-ed energy levels are equal to e$. The energy disper-sions of the conduction and valence subbandsin-plane of the hetero-junction are assumed to beparabolic. »%%, »)) and »%) are the Coulomb inter-actions of electron—electron, hole—hole and elec-tron—hole, respectively and their diagrams areshown in Figs. 2a—2c. »(q) is the two-dimensionalFourier components of the bare Coulomb poten-tial. Here, we ignore the higher-order Coulombinteractions with respect to the N

0/¸2, correspond-

ing to the Fig. 2d. There are two kinds oftransitions; one is between the localized and con-duction subband states (channel-1 transition) andthe other, between the conduction and valence sub-band states (channel-2 transition). The recombina-tion current operator is expressed by

J"+k, j

w1,k,j

akdj#+

k

w2,k

akbk#h.c., (3)

where w1,k,j

("SkDeA/mc ) pDvtjT) and w

2, k("SckDeA/mc ) pDvkT) are the optical transitionmatrix elements, respectively. The current—currentcorrelation function is defined as

P(iu)"Pb

0

dq e*uqS¹qJ(q)J(0)T

"N0+k, j

w*1,k,j

p1,j

(k, iu)#+k, j

w*2,k

p2(k, ix).

(4)

The detail derivation of P(iu) is discussed in Ref.[10]. According to the recipe of the temperatureGreen’s function, the optical gain a(u) is given bythe imaginary part of the current—current correla-tion function P(u # id). The two-particle Green’sfunctions, n

1,j(k, iu) and n

2(k, iu) are obtained from

the coupled Bethe—Salpeter equations [11]:

n1,j

(k, iu)"w1,k,j

n01,j

(k, iu)#n01,j

(k, iu)

]+q

»s(q)Ae~*2Rje!q2/4ap

1,j(k#q, iu)

#e*,Rje~q2@2a2

¸Sp

asp2(k#q, iu)B ,

Fig. 2. Diagrams of the Coulomb interaction. (a) V%%, (b) V)), (c)V%), and (d) a higher-order diagram.

p2(k, iu)"w

2,kn02(k, iu)#p0

2(k, iu)+

q

»s(q)

]A+j

e~*(k`q)Rje~(k`q)2@2a2

¸Sp

ap1, j

](k#q, iu)#p2(k#q, iu)B ,

where »4(q) is the static screened Coulomb poten-

tial »(q)/e(q) and the single plasmon-pole approxi-mation [11] was adopted in the dielectric functione(q). n0

1,j(k, iu) and n0

2(k, iu) are the noninteracting

two particle Green’s functions, expressed as

n01,j

(k, iu)"1!f

e(e%k)!f

h(edj)

iu!e%k!ed

j

,

n02(k, iu)"

1!f%(e%k)!f

h(ehk)

iu!e%k!e)

k

,

where f%and f

)are the distribution functions for the

electron and the hole, respectively.

3. Results and discussion

At first, we show the optical gain spectra of theGaN/Al

0.2Ga

0.8N quantum well in the two-band

model in Fig. 3 where the localized states are ig-nored, to clarify the effects by the Coulomb interac-tion. The carrier density is changed from zero to

582 T. Uenoyama / Journal of Crystal Growth 189/190 (1998) 580–584

Fig. 3. Optical gain spectra of the two-band model for variouscarrier densities from zero to 6]1012 cm~2.

6]1012 cm2. The parameters used in the calcu-lation were m

%"0.18m

0, m

)"1.65m

0and

¸"40 A_ . The transparent carrier density is about5]1012 cm2, which is the almost same as the onewithin free carrier model. The band edge absorp-tion and the sharp exciton absorption can beobserved clearly at the carrier density n"0. Theexciton binding energy can be estimated as 40 meVfrom them. Then the binding energy and the ab-sorption strength become reduced as increasing thecarrier density. At n"1]1012 cm2, the band edgeabsorption is not observed clearly, but the excitonicabsorption can still occur. However, when theoptical gain appears, the excitonic absorption isgone. The reason is that the phase space fillingfactor 1!f

%(e%

k)!f

)(e)

k) kills the bound states at

the population inversion. After the populationinversion, the electron—hole Coulomb interactionstill enhances the oscillator strength, so-calledCoulomb enhancement. But the enhancement fac-tor is roughly 1.3—1.5 [12] from the free carriermodel and the factor is less than that before thepopulation inversion. The red shift of the spectra isshown as increasing the carrier density, due to theband-gap renormalization. After all, the elec-tron—hole Coulomb interaction or excitonic effectjust enhance the oscillator strength of the transitionand the exciton-like absorption is not obtainedwith the optical gain simultaneously.

Fig. 4. Optical gain spectra of the three-band model, whichincludes the effects of the localized states and the Coulombinteraction.

Next, we plot the optical gain spectra in Fig. 4,including the effects of the localized states and theCoulomb interaction. Both the carrier densityn and the localized state density n

0were

1]1012 cm~2. The localized state energy was37 meV from the top of the valence subband stateby adjusting the radius R and the energy depth ofthe disk. The dash line shows the gain spectrumwithout the coupling between the channel-1 andthe channel-2 transitions. The many-body effectsother than the coupling are considered in eachtransition. At this carrier density, the quantum wellis almost transparent or has the gain at the chan-nel-1 transition and the excitonic absorption occursat the channel-2 transition, simultaneously. Then,switching on the coupling between the twotransition, the tiny gain is enhanced more than2500 cm~1 at the channel-1 transition and thestrength of the excitonic absorption is enhanced aswell. This enhancement occurs near the zero pointof the gain spectrum. In that sense, it correspondsto the conventional Coulomb enhancement but thestrength is much bigger than the conventional one.Note that the enlargement of the gain and theobservation of the excitonic absorption are causedby the localized states. When the population inver-sion at the band edge transition starts with increas-ing the carrier density, this enhancement becomessoftened.

T. Uenoyama / Journal of Crystal Growth 189/190 (1998) 580–584 583

4. Conclusions

In summary, we have studied on the optical gainof the wurtzite GaN/AlGaN quantum wells withlocalized states, using the many-body approach. Inthe two-band model, the electron—hole Coulombinteraction enhances the oscillator strength at theany carrier density but the optical gain and theexcitonic absorption cannot observed simulta-neously. In the three-band model which includes theeffects of the Coulomb interaction and the localizedstates, we have found a new excitonic enhancementof the optical gain involved the localized states.This enhancement is stronger than the conven-tional Coulomb enhancement. This results mightlead a possibility of low threshold carrier density.

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