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Optical properties of InAs/InP ultrathin quantum wells

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Page 1: Optical properties of InAs/InP ultrathin quantum wells

*Corresponding author. V. Albe, ENFA, B.P. 87, 31326 Cas-tanet Tolosan, France. Tel.: #33-561-753224; fax: #33-561-750309.E-mail address: [email protected] (V. Albe).

Physica B 301 (2001) 233}238

Optical properties of InAs/InP ultrathin quantum wells

Virginie Albe*, Laurent J. Lewis

De&partement de Physique et Groupe de Recherche en Physique et Technologie des Couches Minces (GCM), Universite& de Montre&al,Case Postale 6128, Succursale Centre-Ville, Montre&al, Que&bec, Canada H3C 3J7

Received 6 June 2000; received in revised form 6 December 2000

Abstract

The optical properties of ultrathin InAs impurity layers embedded in bulk InP are investigated. Our calculations arebased on a tight-binding description of the electronic structure, with spin-orbit interactions and strain e!ects included ina consistent manner. It is shown that the energy gap increases with decreasing number of InAs monolayers. In the limit ofa single InAs monolayer, the energy gap is found to be 120meV less than that of bulk InP. Our results are in goodagreement with experimental data as far as the heavy-hole}electron transition is concerned. The energy di!erencebetween optical transitions is, however, in disagreement with both experiment and e!ective-mass calculations. � 2001Elsevier Science B.V. All rights reserved.

PACS: 78.66.!w; 78.66.Fd; 78.20.Bh

Keywords: Tight-binding model; Electronic structure; Energy transitions; InAs/InP; InAs/GaAs

1. Introduction

In recent years, considerable attention has beendevoted to the experimental study of ultrathinlayers of impurity atoms embedded in bulk hostmaterial, i.e., quantum wells (QWs) in the case ofsemiconductor materials [1}7].

These structures show large radiative e$cienciesand are therefore promising candidates for high-speed and optoelectronic-device applications[8}11]. Their physical properties, however, remainlargely unresolved. From a theoretical viewpoint,

QWs are usually described by macroscopic modelssuch as the envelope-function approximation(EFA) and it is assumed that such models are valideven in the case of ultrathin layers. There has beenmuch less e!ort in describing QWs at the micro-scopic level.

Recent tight-binding calculations [12] indicatethat the EFA is not adequate to describe the elec-tronic properties of InAs impurity layers in GaAs[13]; because of the large lattice mismatch betweenInAs and GaAs (about 6.7%), QW thickness islimited to the range 0}1.6 monolayers (MLs) in thiscase [3,14,15]. In view of these results, the InAs/InPsystem, which has a much smaller mismatch (about3.1%), can provide an interesting test of the validityof electronic structure models. Indeed, InAslayers can be grown pseudomorphically on InPwith thickness up to a few MLs, thus allowing a

0921-4526/01/$ - see front matter � 2001 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 2 6 9 - 1

Page 2: Optical properties of InAs/InP ultrathin quantum wells

comparison between experimental data and theor-etical values. The photoluminescence spectra ofInAs/InP ultrathin QWs generally show multiplelines which are associated with InAs layers of inte-ger thickness [5,25,26]. The positions of the opticaltransitions can be deduced from the photolumines-cence spectra and compared to theoretical predic-tions.

Up to now, atomic-scale studies of ultrathinInAs/InP QWs have only been used to determinethe energy gap and valence band o!set at theInAs/InP interface [16]. In this paper, we examinethis system for InAs thickness varying from 1 to6 MLs using an empirical tight-binding model todescribe the electronic structure and opticaltransitions. We "nd that the energy gap increaseswith decreasing number of InAs monolayers; in thelimit of a 1-ML thick InAs QW, the energy gap is120meV less than that of bulk InP. Our results arein good agreement with experimental data as far asthe heavy-hole}electron transition is concerned.The energy di!erence between optical transitions,however, is in disagreement with e!ective-mass cal-culations. The heavy hole is localized in the InAsQW, while the lowest electron state is located in thehost InP material, which is con"rmed by recentphotoluminescence experiments.

2. Model

The calculations reported here are based on thesp�s* tight-binding [17], with both spin}orbit in-teractions and strain e!ects taken into account inthe electron Hamiltonian. We considered ultrathinInAs/InP QWs constructed as supercells contain-ing between 164 and 184 atoms; periodic boundaryconditions were used to eliminate surface e!ects.

In-plane atoms occupy the positions of a perfectzinc-blende lattice, thus corresponding topseudomorphic growth conditions. The distancesbetween In and As atomic planes are, however, notthose of the zinc-blende lattice; they have beencalculated within the macroscopic theory of elastic-ity for a lattice mismatch of 3.13% between InAsand InP. Indeed, recent ab-initio calculations [18]and high-resolution X-ray di!raction measure-ments [19] indicate that even in the single mono-

layer limit, the macroscopic theory of elasticitydoes not break down. At the InAs/InP interface, wehave scaled the on-site energies of the commoncation (In) atom by a factor of �

�in order to smooth

out a little bit the interface boundary; this methodhas been used within empirical pseudopotentialcalculations [20] for various AlAs/GaAs systems,as well as tight-binding calculations [21] forInAs/GaAs structures, and found to be very re-liable.

In order to account for the e!ect of strains on theelectronic structure arising from the lattice mis-match, the interatomic matrix elements were scaledaccording to Harrison's law [21]

H��"H����d�d �

���,

where � and � are atomic orbitals, d�

and d are theequilibrium and distorted bond lengths, respective-ly, and n�� are parameters adjusted so as to repro-duce the deformation potentials. In order toperform this "t, we have set all n��"2 exceptn��

"3.54, following Ref. [12]; we thus obtain theband-gap and uniaxial deformation potentials,a"!6.2 eV and b"!1.8 eV, respectively, closeto the experimental values a"!6.0 eV andb"!1.8 eV [22]. For InAs, we used the para-meters of Ref. [12]. To account for strain e!ects, theon-site energies of p-symmetry orbitals have beenmodi"ed as follows:

E���

"E�#b

�(���

!���

)

and

E��

"E�!2b

�(���

!���

),

where ���

and ���

are the in-plane and interplanestrain components, respectively. b

�is a constant

"tted to reproduce the deformation potentialb (b

�"0.7).

If nearest-neighbour interactions only are con-sidered, 23 parameters are needed to describe InPand pseudomorphically strained InAs; they aregiven in Table 1. In the tight-binding model, thevalence band o!set �E

�between the two materials

is assumed to be a constant, and is added to thediagonal elements of the Hamiltonian matrix. Inour case, it is the InAs on-site energies that are

234 V. Albe, L.J. Lewis / Physica B 301 (2001) 233}238

Page 3: Optical properties of InAs/InP ultrathin quantum wells

Table 2Calculated transition energies (in eV) between the hh and CS1 for InAs/InP quantum wells as a function of thickness (in MLs); alsoshown for comparison are the results of calculations within the envelope function approximation (EFA), as well as experimental data

ML This work EFA [16] Expt. [5] Expt. [26] Expt. [25]

1 1.292 1.32 1.34 1.282 1.198 1.24 1.28 1.32 1.253 1.166 1.16 1.18 1.23 1.164 1.153 1.07 1.11 1.09

Table 1Tight-binding parameters (in eV) for InAs [12] and InP [31] inthe sp�s* basis including spin}orbit interactions. a and c standfor anion and cation, respectively. �

�and �

�are anion and cation

spin}orbit parameters, respectively

InAs InP

E(s, a) !9.5381 !8.5274E(s, c) !2.7223 !1.4826E(p, a)

���0.6757 0.8285

E(p, a)�

0.9685 0.8285E(p, c)

���3.4858 4.0851

E(p, c)�

3.7786 4.0851E(s*, a) 7.2724 8.2129E(s*, c) 6.6090 7.0726<(s, s) !5.9828 !5.3615<

��� ��1.4677 1.8801

<��

2.7912 1.8801<

��4.1215 4.2084

<��� ��

4.7372 4.2084<(sa, pc)

���2.9784 2.2227

<(sa, pc)�

3.4234 2.2227<(sc, pa)

���5.3143 5.5642

<(sc, pa)�

6.1083 5.5642<(s*a, pc)

���3.1742 3.4081

<(s*a, pc)�

3.6485 3.4081<(s*c, pa)

�� �3.6715 4.4187

<(s*c, pa)�

4.2200 4.4187��

0.1385 0.0244��

0.1290 0.1426

shifted by �E�

(relative to bulk InP), sinceits valence-band edge is higher in energy whenit forms an interface with InP. The value�E

�"0.42 eV, taken from the "rst-principles

calculations of Ref. [23], has been used. The elec-tronic states*energies and wavefunctions*are then obtained by direct diagonalization; herewe consider only the � point.

3. Results and discussion

We have calculated the energy levels of ultrathinInAs/InP QWs with InAs thickness varying from1 to 4 MLs. Excitonic e!ects have been neglected asthey are expected to be small. Indeed, the free-exciton binding energy in bulk InAs is estimated tobe about 1meV [22]. For the GaAs/Al

�Ga

���As

system, the exciton binding energy for thin layers ispossibly higher than in bulk material [24], but inInAs/GaAs, it is estimated to be about 15 meV.A tight-binding analysis including excitonic e!ects[13] indicates a binding of 12.9meV for 1 MLand 15.7meV for 2 MLs. These results arecorroborated by the observation of high gain lasingat room temperature [11] for 1.5 ML of InAs inbulk GaAs. For the InAs/InP system, the excitonbinding energy is expected to be roughly 5meV[16].

Our results for the `"rst transition energya, i.e.,the transition from the heavy-hole (hh) valencestate to the "rst conduction state (CS1), are re-ported in Table 2. We also give, for comparison, theEFA results of Ref. [16], the photoluminescencedata of Refs. [5] and [25], as well as the excitedphotoluminescence data of Ref. [26]. When thenumber of InAs MLs decreases, quantum con"ne-ment causes the "rst transition energy to increase.For the 1-ML thick QW, the energy gap is found tobe 120meV less than that of bulk InP (1.29 vs1.41 eV). Good overall agreement with experimentis found. The EFA results are also in good agree-ment with photoluminescence measurements.However, it should be noted that the thickness ofthe QW's, in EFA calculations, is "tted to experi-ment [16], in contrast to tight-binding calculations,where the only adjustable parameter*once the

V. Albe, L.J. Lewis / Physica B 301 (2001) 233}238 235

Page 4: Optical properties of InAs/InP ultrathin quantum wells

Fig. 1. Anion and cation electron densities (in arbitrary units) atthe zone centre for the 1-ML thick InAs QW in InP for (a) theheavy-hole (hh) state; (b) the light-hole (lh) state; and (c) thelowest electron state (CS1). Triangles and full lines are forcations, while squares and dotted lines are for anions. The InAsML corresponds to layer number 41 (or, equivalently, 0, sincethe system is periodically replicated).

Table 3Calculated and measured (Ref. [26]) transition energies (in eV)between lh and CS1 for InAs/InP quantum wells as a function ofthickness (in MLs); also shown are the energy di!erences be-tween lh-CS1 and hh-CS1 (cf. Table 2) transitions

ML lh-CS1 lh-CS1}hh-CS1

This work Expt. This work Expt.

1 1.318 0.0262 1.1223 1.37 0.025 0.053 1.191 1.32 0.025 0.094 1.178 0.0255 1.170 0.025

model has been optimized to correctly describebulk properties*is the valence band o!set �E

�.

The tight-binding approach is an atomisticmodel and no assumptions are made on the`shapea of the wavefunctions or interface matchingconditions. Fig. 1(a) shows the electron density ofthe hh valence state for a 1-ML thick InAs QWembedded in bulk InP (40-ML thick). The hh stateis strongly localized on the InAs layer and itswavefunction consists mostly of the p

�and

p�

states of the As anion. In contrast, the light-hole(lh)valence state is localized on the InP layers, asshown in Fig. 1(b), and its wavefunction arisesmainly from the p

�states of the P anion. These

results are a consequence of the presence of

a uniaxial strain in InAs, which causes the degener-ate valence-band manifold to split into the hh stateof p

�, p

�symmetry and the lh state of p

�symmetry.

Fig. 1(c) shows the electron density of the "rstconduction state for the same 1-ML thick QW; it isalso found to be localized on the InP layers and ismainly composed of the s-symmetry states of the Incation.

The 2-, 3-, and 4-ML thick QWs exhibit a similarbehaviour of the wavefunctions. Recent photo-luminescence experiments of InAs self-organizedquantum dots on InP substrates [27] suggest thatthe recombination process involves the electronslocalized in the InP substrate and the holes localiz-ed in the InAs dots. This is consistent with ourcalculations of the lowest transition energy, whichinvolves the recombination of CS1 electrons in InPwith hh localized in InAs.

We have also calculated the optical transitions ofhigher energies for InAs/InP QWs with InAs thick-ness ranging between 1 and 6 MLs. The results forthe transition from the lh valence state to the "rstconduction state are reported in Table 3. Experi-mental data (only the 2- and 3-ML thick QWs areavailable) obtained from excited photolumines-cence spectra by Leonelli et al. [26] are also givenfor comparison. Again, here, the transition energiesdecrease with increasing QW thickness, character-istic of quantum con"nement. The calculatedvalues are in reasonable agreement with experi-ment. We also compare in Table 3 the energy di!er-ence between the hh-CS1 and the lh-CS1transitions. Our tight-binding calculations predict

236 V. Albe, L.J. Lewis / Physica B 301 (2001) 233}238

Page 5: Optical properties of InAs/InP ultrathin quantum wells

Table 4Calculated and measured (Ref. [9]) transition energies (in eV)between hh and CS1 and between lh and CS1 for InAs QWs inGaAs (rather than InP); also shown are the energy di!erencesbetween the two transitions

ML hh-CS1 lh-CS1 lh-CS1}hh-CS1

This work Expt. This work Expt. This work Expt.

1 1.428 1.463 1.506 1.477 0.078 0.0142 1.318 1.385 1.471 1.461 0.153 0.0763 1.276 1.371 1.448 1.444 0.172 0.073

this di!erence to be essentially independent of thethickness of the QW (between 1 and 6MLs) atabout 25meV. In contrast, excited photolumines-cence experiments [26] indicate di!erences of 50and 90meV for the 2- and 3-ML thick InAs QWs,respectively. Using the EFA, Bastard and Marzin[28,29] "nd 89 and 136meV.

In order to ensure that the above discrepancy isnot the result of some limitation of our tight-bind-ing approach, we have calculated the hh-CS1 andlh-CS1 transition energies in the case of ultrathinInAs QWs in GaAs (rather than InP) using exactlythe same procedure as above (with �E

�"0.04 eV

following Ref. [30]) The results are listed in Table 4.The calculated values are in good agreement

with experiment. The energy di!erence between thehh-CS1 and the lh-CS1 transitions, further, is foundto follow*at least qualitatively*the behaviourobserved in experiment, thus reassuring us of thevalidity of our calculations in the case of InAs/InP.

4. Conclusions

We have studied the electronic structure andoptical properties of ultrathin InAs/InP QWs with-in a tight-binding model. We "nd that (i) the cal-culated hh-CS1 transition is in good agreementwith the EFA results and experimental data. (ii)The CS1 state is located in InP while the hh valencestate is localized in InAs; this is con"rmed by recentphotoluminescence experiments [27]. (iii) The en-ergy di!erence between the hh-CS1 and the lh-CS1

states compares well with experimental data for theInAs/GaAs system, while for InAs/InP, our resultsdi!er from both experimental data and EFA calcu-lations.

Acknowledgements

We are grateful to R. Leonelli and P. Paki foruseful discussions, and for providing a copy of Ref.[26] prior to publication. This work was supportedby grants from the Natural Sciences and Engineer-ing Research Council (NSERC) of Canada and the`Fonds pour la formation de chercheurs et l'aidea la recherchea (FCAR) of the Province of QueH bec.We are grateful to the `Services informatiques del'UniversiteH de MontreH ala for generous allocationsof computer resources.

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