Design considerations for 1.3pm emission of GalnNAs/GaAs strained quantum-well lasers

D. Alexandropoulos and M.J. A d a m

Abstract: The authors explore theoretically different indium and nitrogen compositions and well widths for 1.3 pm emission of GalnNAs strained quantum well lasers. The nitrogen-induced conduction band non-parabolicity is accounted for through the band anti-crossing model, and valence-band mixing effects and strain are treated exactly. Basic design rules are outlined not only on the basis of the emission wavelength but also in terms of optimal device operation.

1 Introduction

The GaInNAs/GaAs material system was first proposed and grown by Kondow et al. [ I ] and has since been demonstrated by a number of research groups [2-61. This material is promising as an alternative material to InGaAsP at the important telecommunications window of 1.3 pm for the following reasons. The use of CalnNAs as the active material provides an efficient way to build vely deep quantum wells in the conduction band, leading to better carrier confinement and hence reduced carrier leakage at high temperatures. In addition, it can facilitate the fabrica- tion technology of the vertical-cavity surface-emitting laser (VCSEL) since it allows the use of the well established high refractive index contrast GaAs/AlAs distributed Bragg reflectors (DBRs) as opposed to InP/lnGaAsP DBRs (lattice matched to the InGaAsP active layer), which have low refractive index contrast.

In recent years, there has been active research activity in the growth and experimental characterisation of the mate- rial [6-111 and in theoretical investigations [ I 1-15]. As far as the theoretical investigations are concerncd, of particular importance is the band anti-crossing (BAC) model [ I I ] since, despite its simplicity, it managcs to explain the basic properties of the material and provide analytical expres- sions, such as conduction band edge dispersion relations and electron effective mass.

All of the above research activity, both experimental and theoretical, has provided a sufficient amount of knowledge about the physics and properties of CalnNAs to juslify reliable and relatively accurate composition mapping for 1.3 pm emission. It is our aim in this paper to produce this map and derive some basic design rules, not only from the viewpoint of the transition wavelength but also in terms of parameters related to laser performance, such as the differ- ential gain, transparency concentration and the transition matrix element.

9 IEE. 2003 IEE ~ ~ ~ ~ ~ ~ ~ ~ d i ~ ~ ~ . ~ ~ n i i n ~ no. 20030385 DOI: IO. I n49iip-0pt:znn3n385 Paper first received 22nd March and in rwised form 12th October 2002 The authors are with the Electronics Systcms Engineering Depanmenr. University of Essen, Wivenhoe Park, Colchester, Essen CO4 3SQ. UK

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2 Bandstructure of GalnNAs

The effect of the nitrogen on the conduction hand is accounted for through the BAC model. According this model, the incorporation of nitrogen into GaInAs (or GaAs) alloys leads to a strong interaction between the conduction band and a narrow resonant band formed by the nitrogen states (because of the highly localised nature of the perturbation induced by the nitrogen atoms). The overall effect is a splitting of the conduction band with a consequent reduction of the fundamental bandgap due to the lowering of the conduction band edge.

Mathematically, the interaction of the nitrogen states and thc extended like states of the matrix semiconductor is expressed through the following eigenvalue problem [ I I]:

where EILl are the conduction states of the matrix semi- conductor, EN the localised states of the nitrogen and V,,,N is the matrix element describing the interaction between the EIM and EA,. It is the VMN that leads to the mixing and anticrossing of these two states. In this context, the states of thc conduction sub-bands are represented by functions that are a mixture of extended conduction band and localised nitrogen states.

The matrix element V,MN is dependent upon the nitrogen concentration and the matrix semiconductor through the expression [13]: VM,=C,w,v2/.. where x is the nitrogen concentration and C,,,, is a parameter that depends on the matrix semiconductor and hence the indium composition.

In the present approach, the conduction bands are taken to be decoupled from the valence bands, which is a valid approximation given that the emission wavelength of interest is 1.3 pm, and thus the inclusion of the coupling between the conduction hands and valence bands will induce a minor correction [16].

For the calculation of the conduction-band structure, we use the BAC model and the envelope-function approxima- tion. For the calculation of the valence-band structure we use a 6 x 6 Hamiltonian accounting for the spin-orbit effects. This is further block diagonalised under the axial approximation, thus reducing the dimensionality of the matrix to 3 x 3 . The strain-dependent coupling of the heavy hole (HH), light hole (LH) and spin-orbit split-off

I05

(SO) valence sub-hands is accounted for in the context of the deformation potential theory, as described in [17].

Parameters for the quaternary compounds such as Luttinger parameters, lattice constants and strain-related parameters are obtained by linear interpolation between thc values of the binary compounds found in [IX]. For the nitrogen-related binaries the values correspond to the cubic form of these.

The eigenvalue problems are solved numerically using a finite-difference approach. The boundary conditions and the Hermitian properties of the Hamiltonian are obtained using the finite-difference formulas found in [19].

3 Theoretical results

3.7 C,, parameter and EN level Before proceeding with bandstructure calculations we deal first with the C,,,” parameter and the energetic position of the nitrogen level. Both parameters are involved in the eigenvalue problem of (1). Thus an accurate knowledge of these is essential for the calculation of the bandstructure of the GaInNAs and therefore of the transition energies. The motivation is that there is a plethora of C,,“ values in the literature for the same indium and nitrogen compositions [20]. The discrepancy in these values stems from structural differences, which are induced by the growth method and the different post-growth treatment techniques. Only recently have there been systematic studies, both theore- tical [21, 221 and experimental [21, 231, of the signatures of different environments that emerge under different annealing treatments. It was found that a set of discrete handgaps exists in GaInNAs alloys, and the experimentally observable bandgap is the outcome of the overlapping contributions of these five environments. This intrinsic property of the GalnNAs alloys originates from the five possible nearest neighbour (NN) configurations of the nitrogen sites, namely 4Ga, 3Ga, ZGa, IGa and 41n. Furthermore, it is argued that the effect of annealing is manifested in the alteration of the oscillator strength of the five environments. Hence the blue shift observed after annealing of GaInNAs alloys is attributed to the rearrange- ment of the oscillator strengths such that the 41n dominates the transitions.

According to [21], the nitrogen energy level for nitrogen compositions up to 3% varics little with the nitrogen content but depends on the indium content. The functional form of this dependence that is consistent with the results presented in [2 I], is as follows:

E,(,,,) =2.31 x IO- ’ .,? -0.0062 . y + 1.8918

= 2.39 x IO- ’ . yz - 0.0064 . y + 1.8384

EN,zoul = 2.82 x -0.0067 . y + 1.7808 (2)

E,v,3,voGa, = 3.66 x IO-’ .y2 - 0.0071 . y + 1.7258

EN(,,,) = 3.96 x IO-’ .y2 - 0.0074 . y + 1.6700

where y is the indium composition. To obtain (2) we have digitised Fig. 3 of [21] and subsequently interpolated a second-order polynomial. The corresponding change in the Ch,.vparameter according to [21] is about 2 for the 4Ga NN configuration and about 1.35 for the 41n NN configuration. The set of equations (2) and the corresponding CMN values provide the framework for the pair of ErenlN values to he used in our calculations. However, it is important to note that, apart from qualitative trends, more work is necessary in order to establish the conditions under which different NN configurations are favoured.

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3.2 Effect of In and N composition on the transition wavelength, transparency concentration and differential gain For the calculation of the indium and nitrogen composi- tions required to achieve a I .3 pm emission wavelength for various well widths, we employ the handstructure model described in Section 2, and for thc determination of the E,,., CMW values we choose a mean approach. That is, the EN is taken to vary with thc indium content, assuming that the NN environment is 2Ga, and C,,,,v is taken as the mean value of the two endpoint values, namely 1.675. This is justified if we consider that it is common practice to apply postgrowth treatments such as annealing, the magnitude of which varies. It is emphasised at this point that the choice of the dominant NN configuration is strongly dependent upon the growth technique and postgrowth treatment.

Following this approach, we calculate the indium and nitro- gen compositions optimised for 1.3 pm emission. Our results are shown in Fig. la. Any combination of indium and nitrogen composition that falls on the curve for the same well width will emit at 1.3 btm. High indium values favour the compressive- strain regime, whereas high nitrogen values favour the

N composition

a

N composition

b

Fig. 1 Composirion reqtrire,nenr.s ,for 1.3 pm emission of GaInNAslGaAs QWs a For the well widths shown b For 7 nm GalnNAsIGaAr for thrcc different NN environments: 4Ca. 2Ga and 41”

tensile-strain regime, hence the critical thickness parameter h, for a given well width increases for indium and nitrogen compositions that approach the lattice-matched condition. This is indeed the case for increased values of nitrogen and therefore the growth is facilitated. On the other hand high indium compositions approach the limit at which the lattice mismatch is no longer accommodated by elastic uniform elastic strains hut hy misfit dislocations, which leads to degradation of crystalline quality. Given the strong dependence of the func- tional form ofthe EN with the indium composition on the NN configuration, we plot in Fig. Ib the indium and nitrogen compositions for a 7 nm GaInNAs/GaAs QW that emits at I .3 pm for three cases, the two extremes 4Ga and 4111, and the mean approach, that is 2Ga.

Since both the incorporation of indium and the incor- poration of nitrogen lead to a redshift of the bandgap, the obvious route for 1.3 pm emission is either to favour large indium and low nitrogen compositions or the other way around. In order to distinguish these cases and explore the optimal route, that will ensure not only the desired emission wavelength but also enhanced laser performance, we calculate the peak differential gain, the transparency

I.6Ol

1 . 1 5 y s 0 2 4 6 8 10 12 14 16 I8

0.20 0'25L

-0.251 0 2 4 6 6 10 12 14 16 18

kli. m-I (x108)

b

Fig. 2 Band structure for 7nm GulnNAslGuAs QWenritting at 1.3 pm, .fur three drffeerent composition combinarions a Conduction band stmcmre b Valence band srmcmrc

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0.54 0.01 0.02 0.03 0.04 0.05

N composition

Fig. 3 Transparency concentration against N content for various well widths and In content such thot the emission wavelength is 1.3 Aim

concentration and the transition matrix element for differ- ent combinations of indium and nitrogen compositions. Ideally, for a laser structure one requires the maximum possible differential gain and the lowest possible trans- parency concentration, so that the carrier concentration required to reach the maximum gain and the carrier concentration required to reach positive material gain are minimised, respectively. In addition, a high differential gain enhances the modulation bandwidth of laser structures and also influences the saturation power of semiconductor optical amplifiers. The transition matrix element quantifies the overlap between the electron wavefunction and the hole wavefunction. Hence, maximum possible values are desired since the wavefunction overlap is then more efficient, which leads to higher values of gain.

The transparency Concentration, the peak differential gain and the transition matrix element are magnitudes that depend on the bandstructure of the active material, which in turn is modified for different indium and nitrogen compositions. Therefore it is instructive to calculate the handstructure of GaInNAs/GaAs for three different indium and nitrogen combinations that fall on the same curve of Fig. I . Fig. 2 shows the bandstructure of a 7 nm

0.01 0.02 0.03 0.04 0.05

N composition

Fig. 4 Peak di/fee,mrial gain at the transparency concentration shown in Fig. 3 agoinst N cunfenf fbr various well widths

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O.M,

.... 9 nm

1 7 nm brim \m

L 3 nm

0.01 0.02 0.03 0.04 0.05

N Composition

Fig. 5 (CI --f H H I ) against N conrenf

TE rransirion matrix element /or the lower fronsiiion

GalnNAs/GaAs single quantum-well structure for three different combinations of indium and nitrogen composi- tions that emit at I .3 pni,

Fig. 3 shows the transparency concentration for different nitrogen compositions and well widths. The indium compo- sition is such that for the nitrogen composition the emission wavelength is 1.3 pm. The corresponding peak differential gain at these transparency concentrations is shown in Fig. 4. The calculation or the peak differential gain is performed in the context of the free carrier theory allowing for Lorent- zian lineshape broadening of 0.1 ps. Details of the calcula- tion can be found elsewhere [24]. Finally we have calculated for the same conditions as above the transition matrix element, and this is shown in Fig. 5.

4 Discussion

Figure l h provides an explanation of the plethora of values of the combinations of compositions found in the litera- ture, for a given wavelength [20]. I t is easily seen that, according to the approach of [21], for thc C,,,E,,, pairs, the compositions of indium and nitrogen required for I .3 pm emission vary substantially, depending on the microscopic configuration. In the light of these results, the differences arise from the different post-growth treat- ments that induce different microstructural changes and hence allow different NN configurations to dominate the transitions.

The quantum confinement effect i n the lasing wave- length is quite straightfonvard and shows the usual trend. For increasing well width the transition energy is shifted downwards and therefore less indium and nitrogen are required to achieve an output wavelength of 1.3 pm (Fig. I ) . For the understanding of the relationship between the differential gain and the well width, thc bandstructure details and therefore the effect of strain have to be taken into account. Given that the emission wavelength is fixed at I .3 pm, for any given nitrogen concentration the decrease of the peak differential gain with increasing well width (Fig. 4) can be interpreted in terms of the amount of compressive strain. Indeed for increasing well width the indium fraction necessary for I .3 pm emission decreases and hence the compressive strain decreases, which trans- lates to an increase of the density of the states. This, however, is in competition with the quantum-confinement effect on the density of the states, but the differential gain

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behaviour is primarily determined by the strain effect. On the other hand in the case of the transparency concentra- tion (Fig. 3) thc quantum-confinement effect dominates, and the decreasing trend with increasing well width is attributed to the quantum-size-effect-induced modification of the density of states. Similar arguments hold for the transition matrix element’s dependence on the well width (Fig. 5). For decreasing well width the electron envclope function spreads over the well, leading to a decrease in the electron and hole overlap.

Since the primary use of GaInNAs is in laser configura- tions the differential gain, transparency concentration and transition matrix element must have suitable values. In other words, the differential gain and transition matrix element are the maximum possible and the transparency concentration the minimum possible, so as to assure optimal laser performance. In design terms, the optimal well width is determined by the interplay of various parameters. In particular, it is established from the discussion above that the differential gain reduces for increasing values of well width. On the other hand the optical confinement is more efficient for wider wells. Hence the choice is dependent upon the margins of tolerance for these magnitudes allowed by the design.

Turning our attention to the effect of nitrogen coniposi- tion for a fixed well width on these parameters, we see that with increasing nitrogen all deteriorate. Thesc effects can be understood in view of the handstructure of GalnNAs alloys (Fig. 2). As discussed in Section 2, the bandstructure of GalnNAs alloys results from the hybridisation of the extended like states of the matrix semiconductor, i.e. GalnAs, and the nitrogen localised states, and the resulting electron bands havc pronounced nonparabolicity. Owing to the nitrogen-induced nonparabolicity, the electron effectivc mass exhibits unusually large values. This property has been experimentally verified [25]. Large values of cffective mass translate to large values of the conduction band density of states.

Based on this framework, the transparency concentra- tion is expected to increase with increasing nitrogen content, as shown in Fig. 3. The effect on the transpa- rency concentration is clearer considering the dependence of the quasi-Fermi levels on the carrier concentration. Since the density of states is higher with increasing nitrogen content, the electron Fermi level moves slowly with injected carriers.

The same arguments can be used to explain the depen- dence of the peak differential gain (Fig. 4). However, the effect of nitrogen content is manifested not only through the increased density of states but also through the transi- tion matrix element. Increasing the nitrogen content, the localised part of the electron wavefunction is enhanced at the expense of thc cxtended part, which has the maximum overlap with the hole wavefunction. Hence, the transition matrix element is reduced with a consequent reduction of the peak differential gain.

5 Conclusion

The above discussion provides some insight into basic design rules. It is obvious that, when Considering the necessary indium and nitrogen compositions and well widths to achieve 1.3 pm, that it is not sufficient to do this only from the viewpoint of the emission wavelength but also one has to allow for the modifications that the indium and nitrogen contents will induce in laser-related parameters. Bearing this in mind when GalnNAs is

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intended for a laser configuration, then the route of highest possible indium and lowest possible nitrogen should be followed. In this way the effect of nitrogen is used only in the contribution to the reduction of the bandgap and nitrogen-induced nonparabolicity is suppressed. Conse- quently, the transparency concentration is minimised and the differential gain is maximised. Especially for the transition matrix element. the well has to be wide enough to allow thc maximum overlap between the electron and hole wavefunctions.

6 Acknowledgments

The authors would like to acknowledge Prof. S.L. Chuang (Univcrsity of Illinois at Urbana-Champaign) and Dr. S. Tomic (University of Surrey) for helpful discus- stons. The first author would like to acknowledge Agilent Technologies (UK) and EPSRC (UK) for financial support. This work is supported under an EPSRC research project.

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