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VLSI DESIGN2001, Vol. 13, Nos. 1-4, pp. 23-36Reprints available directly from the publisherPhotocopying permitted by license only

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Simulation of Optoelectronic Devices

PAOLO LUGLI*, FABIO COMPAGNONE, ALDO DI CARLO and ANDREA REALE

INFM-Dipartimento di Ingegneria Elettronica, Universita di Roma "Tor Vergata" 00133 Roma, Italy

In the spirit of reviewing various approaches to the modeling and simulation of opto-electronic devices, we discuss two specific examples, related respectively to Semicon-ductor Optical Amplifiers and to Quantum Cascade Lasers. In the former case, atight-binding analysis is performed aimed at the optimization of the polarizationindependence of the device. Further, a rate-equation model is set up to describe thedynamics of gain recovery after optical pumping. A Monte Carlo simulation of asuperlattice quantum cascade laser is then presented which provides an insight into themicroscopic processes controlling the performance of this device.

Keywords: Simulation; Modelling; Photonics; Semiconductor devices; Quantum cascade lasers(QCL); Semiconductor optical amplifiers (SOA)

1. INTRODUCTION

Driven mainly by the growing request of fast com-munication links, the field of optoelectronic device,especially solid-state based, has been witnessing inrecent years an increasing attention [1]. One of thecrucial issues for optical links is the possibility toamplify the optical power attenuated during thepropagation along the fiber without having to resortto an optical-electrical conversion. Therefore,optical amplification of the power transmitted in afiber link has become a critical point in recent years[2]. Traditionally, optical amplification has beenachieved using both semiconductor devices (semi-conductor optical amplifiers, SOA) and doped

fibers, such as in the erbium doped fiber amplifiers(EDFA). While the latter is by far the most widelyused technique at present, SOA’s present a series ofpotential advantages such as integrability, com-pactness, tunability of the operating wavelength,large gain bandwidth, and non-linear functions(e.g., for use a wavelength converters). Moreover,semiconductor amplifiers can be efficiently used toamplify short optical pulses and can replace othertraditional amplifiers used in femtosecond systems.The use ofmodern advanced growth techniques haslead to the optimization ofthe performances ofsuchdevices thanks to the quantization of the density ofstates and to band structure engineering. Moreover,the strict control of epitaxial growth permits the

* Corresponding author.

23

24 P. LUGLI et al.

realization of strained multilayers in an activeMQW region. The introduction of strain in MQW-SOAs relates to the need to reduce the polarizationdependence of the gain [3].

In the following, we will present a theoreticalstudy of the optical and dynamical properties ofSOAs that not only explains the operation of thedevices, but opens also the way to its optimization.

In addition to optical communication, severalother areas of great economical interest pose anincreasing demand for new solid state opticaldevices. Among them, environmental control anddata storage. The latter has pushed researchertowards the development of green-blue lasersbased on nitride heterostructures, the former hasmotivated the interest towards semiconductorinfrared and far-infrared (FIR) sources. A descrip-tion of a modeling approach for nitride-baseddevices can be found in these Proceedings [4] andwill not be treated here. Rather, the numericalsimulation of the best solid-state device for theFIR region, namely the Quantum Cascade Laser(QCL), will be discussed in detail. Actually, QCLsare the only valuable semiconductor light sourcefor far-infrared application in the range 4-11 lm[5, 6].

2. MODELLING SEMICONDUCTOROPTICAL AMPLIFIERS

Optical amplification in SOA’s exploits the radia-tive transitions from states near the top valenceband of a direct gap semiconductor to states nearthe bottom of the conduction band. While SOA’sbased on bulk materials are already in a veryadvanced stage, better performance can be achievedvia quantum well (QW) structures. Although ex-perimentally such effects are well known andreproducible, a clear theoretical explanation is stillmissing. The next section describes a tight-binding(TB) approach to the study of the electronic andoptical properties of SOAs. In addition to suchfundamental issues, the problem of the dynamicalnon linear response of the SOA to propagating

optical pulses is also crucial. The need for useful andprecise time domain models of device is twofold.First, one has to check and/or predict performancesof real optoelectronic devices in telecommunica-tions applications. On the other side, the investiga-tion of time dependent properties can lead to anunderstanding of the role played by the variousrecombination processes taking place inside MQW-SOAs [7, 8]. A crucial aspect of this goal is a properdescription of the light-carriers interaction. In thesecond section we will introduce a rate-equationmodel, and present some comparison with experi-mental results.

2.1. A Tight-binding Description of Electronicand Optical Properties

In QW amplifiers, the increased confinement ofcarriers and radiation field leads to a significantreduction of threshold currents and power dissi-pation. Unfortunately, the radiation process sup-porting optical amplification in QWs resultsnormally in a strong polarization dependence ofthe amplified signal. In fact, in a typical twodimensional structure the dominant transitionoccurs between the quantized heavy hole (HH)state and the correspondent conduction bandstate. Such transition has a strong TE component,which is therefore amplified much more than theTM component related to the light hole (LH)states. Moreover, the difference between the TEand TM confinement factor enhances the polari-zation sensitivity of the gain. In order to achievethe desired polarization insensitivity, several SOAstructures with strain between the different layershave been proposed [9-11]. The essential featureof the strain is to shift HH and LH bands in-dependently. Thus, alignment between the firstHH and the first LH level can be obtained, whichresults on balanced TM and TE contributions.

Traditionally, in fact, the k.p method within the"Effective Mass Approximation" (EFA) is used inthe study of two dimensional (2D) structures.However, SOAs have several peculiar character-istics which cannot be investigated in the context

OPTOELECTRONIC DEVICES 25

of EFA. First of all, EFA very poorly reproducesthe alignment condition to achieve polarizationindependence, since it completely fails in describ-ing the kll----0 mixing when valence band states areresonant in energy [12]. A second characteristic,which is typical of the device we study in thisarticle, consists in the presence of thin (___91t)GaAs -strained layer. It is well known that EFAcannot be applied to very thin layers [13]. A thirdpoint is given by the presence of many differentalloys in the same structure. Such make the EFAinapplicable since it is based on the assumptionthat the periodic part of the Bloch functions donot change between materials composing thestructure. While this holds for GaAs/A1GaAsthick quantum structures, it is not clear, up tonow, if it can be extended when ultra-thin layers ofdifferent materials are used in the same structure.Due to these limitations, we resort to a more

rigorous treatment based on the tight-binding (TB)description of the nanostructure. Details of themethod and its application to a variety of opto-electronic and microelectronic devices can befound in Ref. [14, 15], and references therein. Itsmain advantage is to relax all approximations ofEFA without inducing the unpracticable computa-tion cost typical of more ab-initio methods. Opticalproperties can be easily calculated within the tight-binding scheme without introducing new fittingparameters [16, 17].

Strain can be included in the tight-bindingmodel by scaling the hopping matrix elements.The scaling properties of the matrix element hasbeen extensively discussed by Harrison and others[18, 19], who showed that an inverse square de-pendence on the nearest-neighbor distance (-)between two atoms reasonably reproduces thechemical properties of several materials, andfurther improved by several authors [20-22].

In the present work we use the sp3s tight-binding model [23] with spin-orbit interaction. Thenumerical implementation of the TB approach isof crucial importance. By itself, the method iscomputationally quite heavy since the diagonaliza-tion of very large matrices is needed. In order to

speed up the calculations, we have introduced ahybrid method to diagonalize the tight-bindingHamiltonian which uses a standard (LAPACK[24]) routine to calculate eigenvalues and an inverseiteration scheme [25] to calculate eigenvectors. Theadvantage ofthis procedure over other, relies on thefact that only few eigenvectors are needed, namelythose closed to the energy band gap.The kll integration needed to calculate the

absorption/gain coefficient is performed in the2D Brillouin zone by using a uniform k-points gridin the irreducible wedge, which is obtained byconsidering the symmetry properties of theHamiltonian and is defined by the followingequations

-<k< 1; Ikyl<l-k.(1)

Since we are only interested in the absorption/gaincoefficient close to the energy gap, we do limit theintegration to the regionlkll < 0.1 27fla.

In each optical transition, two nearly spin-degenerate valence and conduction subbands areinvolved. Each squared optical matrix element issummed over the two final conduction states andaveraged over the two initial valence states. [12] Inour procedure we first calculate (and store) theenergy levels and the squared optical matrixelements for each kll, then we evaluate the absorp-tion/gain coefficient by performing the sums overthe carrier distribution functions. To reduce thenumerical fluctuations induced by the finite numberof kll points considered in this sum ( 1600), wesum over a much finer kll grid ( 160000 points).The energy levels and squared matrix elements atthese new kll points are obtained by using a bilinearinterpolation of the calculated quantities. This isallowed since the variation ofboth energy levels andsquared matrix elements in the irreducible wedgeare quite smooth.The reference structure for our study consists of

153]k wide (52 monolayers) In0.533Ga0.a67As

26 P. LUGLI et al.

quantum well surrounded by In0.74 Gao.:26Aso.56P0.44 barriers, lattice matched to an InP substrate.We investigate the differences in optical matrixelements and gain coefficient between the referencestructure and one where 3 monolayers (ML) ofInGaAs in the middle of the well are replacedby strained GaAs. The latter system has beenshown to guarantee a good degree of polarizationinsensitivity [11].The energy band profile of the two systems, i.e.,

without and with GaAs b-strain is shown inFigures a and lb, respectively. The GaAs 5-strainregion gives rise to a discontinuity (with respect toInGaAs) AEe=0.234eV in the conduction bandand AEHH 0.267 eV, AELH 0.183 eV in theheavy and the light hole bands respectively.

Figures 2a and 2b show the valence banddispersions for the structure without and withGaAs b-strain, respectively. In these figure thesplitting of the spin degenerate bands is not shownsince its value is less than 2meV. For theunstrained QW, the first two valence bands haveheavy hole character (close to kl1-0), while thethird one has light hole character. This can bededuced by looking at the squared optical matrixelements, shown in Figure 3a, which are related tothe transitions between valence states and the first

1.3

0.8

" 0.3HH, LHISO

a)

>" -0.3- 1.3LU C

0.8

0.3SO

-0.30

]k/

50 100Depth [ME]

FIGURE Band edge profile for the SOA (a) without and (b)with 6-strain.

0.35

0.34- 0.33

0.32

0.31

0.30

0.29-0.050 -0.025 0.000 0.025 0.050

[110] k [2/a] [oo]

0.35

0.34- 0.33

o) 0.32

LU 0.31

0.30

0.29-0.050 -0.025

[o]0.000 0.025 0.050

k [2/a] [aoo]

FIGURE 2 Valence band dispersion for the SOA (a) withoutand (b) with 6-strain.

conduction state. For the transition VIC1(between the first valence band and the firstconduction band) the TM contribution vanishesat kll 0, while the TE contribution dominates. Onthe other hand, for the V3 C1 transition theratio between TM and TE contributions (at kll 0)is close to 4, while the ratio of the TM contributionbetween this transition and the TE contributionfor the V1 C1 transition is close to 4/3. Suchratios are those typical [26] of a heavy hole (firstvalence band) and a light hole (third valence


band) transitions. The second level has a vanishingTE and TM contribution for kll 0. This marks aforbidden transition namely the transition betweenthe second heavy-hole level and the first conduc-tion band level. Moving away from kll=0, thecharacter of the bands mixes, and both TE andTM contribution are present.When 5-strain is present the first light-hole level

lifts up in energy, as discussed in the previoussection, while the first heavy-hole level shifts down,leading to a band degeneration at kll=0 (seeFig. 2b). The character of these states at zonecenter can be deduced, as for the case without5-strain, by looking at the squared optical matrixelements (Fig. 3b). We notice that the first valenceband has a light hole character (first LH level),while the second valence band has a heavy-holecharacter (first HH level). Very interesting is thethird valence band (second HH band ), where the

40.0

’-:" 20.0

0.0

40.0x

E 2O.O

o 0.0

40.0

20.0

V2->C1TETM

FIGURE 3 Squared optical matrix element as a function ofthe inplane k vector along [110] and [100] directions for theSOA (a) without and (b) with 5-strain. The contribution of eachindividual transition is distinguished.

40.0

"’:" 20.0

0.0E

40.0x

E 20.0

o

0 0.0

40.0

20.0

0.0-0.05

[1101

V2->C1 TE"TM

V3->C1

-0.03 0.00 0.03 0.05k [2g/a] [OOl

FIGURE 3 (Continued).

transition to C1 presents at kll 0 a TE contribu-tion. This transition, which cannot be accounted inthe k. p-EFA model, follows from band mixing at

kll =0. The same mixing effect is responsible forthe TM polarized V2 - C1 transition at zone

center.The total absorption coefficient for the two

systems is shown in Figures 4a and 4b. The sharpstructures in the absorption coefficient are due tothe fact that no broadening has been consideringin the calculation. The comparison with experi-ments would of course require some type ofphenomenological broadening of the radiativetransition to be considered, which will smoothenthe calculated features. The absorption edge forthe InGaAs well without 6-strain occurs at0.774eV. At this energy, only the TE mode isabsorbed, since absorption involves the transitionbetween the first valence state, with heavy holecharacter, and the first conduction band. How-ever, due to the band mixing the TM mode also

28 P. LUGLI et al.

80000

60000

40000

20000

0760

700 800 900 1000 11

780 800 820 840E [meV]

80000

60000

40000

20000

TETM

b)

760 780

700 800 900 1000 11C

800 820 840E [meV]

FIGURE 4 Absorption coefficient for the SOA (a) withoutand (b) with 6-strain. The inserts show an enlarged energy scale.The noise seen in the result is due to the use of a numerical kllintegration.

begins to be absorbed at slightly higher energies(0.783 eV). Band mixing is also responsible for thecontribution to the absorption coefficient from thetransition V2 C1, normally forbidden at kll 0.For energies higher than 0.790eV, the transitionV3-+ C1 takes place. Since the V3 band has alight-hole character, the TM mode is now ab-sorbed strongly, at least for kll 0.When b-strain is introduced, the first LH and

the first HH valence states move closer in energy,as can be seen from the absorption coefficient ofFigure 4b. Here the absorption edge, 20meVhigher than without b-strain is due to the firsttwo valence state (LH and HH character). Thepeak in the absorption coefficient around the

absorption edge deserve some mention. As alreadyobserved by Chang and Schulman [12], such asharp feature is due to negative effective mass ofthe valence states. Indeed, the first valence statehas a negative mass around kll=0. A moredetailed inspection reveals that this mass is similarto the conduction band mass, i.e., rn,,1 =0.07 m0.When the conduction and valence bands areparallel, vertical optical transitions between thesebands occur at energy even for kll 0. Thus severalk-points give contributions at the same energy.With respect of Ref. [12], however, the peculiarityin our system consists in the fact that the valenceband with negative effective mass corresponds tothe first valence band, i.e., the ground state. Thisimplies that such absorption enhancement couldbe used in a laser structure in order to reduce thecurrent threshold (in a similar way to what ispursued with quantum wire and quantum dotslasers).Due to the simultaneous transitions from both

LH and HH states, the final absorption coefficientwould tend to be equal for TM and TE modes.However, a difference between the TE and TMpolarizations appears in our results (Fig. 4b). Twoeffects are responsible for such polarizationdependence, namely band mixing for kll- 0, and,more important, the presence of the third valencebased close in energy to the first two. Such bandmixes even at kll =0 with HH1 and LH1, thuscontributing to the global transition to C1. Thiseffect will not be expected from the simple EFAtheory. Actually, the sum of the contributions tothe absorption coefficient of the first two valencebands is, indeed, very similar for the TM and theTE. However, the contribution of the third valenceband, which occur in the same spectral region,produces the enhancement of the TM absorption.The above observations lead us to conclude

that, in order to have polarization insensitive

absorption we need to separate in energy the firsttwo valence bandfrom the rest of the valence bands.Such requirement could indeed be very difficult tofulfill with the 6-strained system, since the pertur-bation tends to reduce the separation between the


first two HH bands. Possible alternatives havebeen suggested in [15].

In order to calculate the gain coefficient, a properchoice of the quasi-Fermi levels is required. Thoseare obtained by considering the charge injectionfrom the contacts. We follow a simplified approachwhere we assume the charge density in the well to beknown. The corresponding quasi-Fermi levels arethen used to calculate the gain. We assume a roomtemperature Fermi-Dirac distribution for electronand holes. Many-body effects are not included,however, a generalized Elliot formula can be usedto account for such effects [27].The calculated gain coefficient for the well with

and without GaAs b-strain is shown in Figure 5band Figure 5a, respectively. In both cases wechoose a quasi-Fermi level for electron of 150 meV

40000

30000

20000

10000

0760

TETM a)

800 840 880 920 960E [meV]

40000

30000

20000

10000

076O

TM b)

800 840 880 920 960E [meV]

FIGURE 5 Gain coefficient for the SOA (a) without and (b)with b-strain.

from the conduction band edge (see Fig. 1). Thequasi-Fermi level of holes is chosen in order to

give an equal density of hole with respect toelectrons. For the structure without -strain we

have a hole quasi-Fermi level of 7.88 meV abovethe top of the valence band, while for the-strained structure the hole quasi-Fermi level is3.28 meV below the top of valence band. Such bigdifference in the quasi-Fermi level between holesand electron is due to the hole density of the stateswhich is very high compared to the electron one.As for the absorption coefficient, there is a sensibledifference for the TE and TM polarization for thesystem without the b-strain. The overall differencebetween TM and TM reduces for the well with-strain. However, the presence of the TM spikeand the presence of the third valence band makethe TM greater than the TE gain close to the bandedges. We need to point out, however, that a

prevalence of TM contribution to the gain isnecessary is SOAs, since it balances the higherconfinement factor, of the TE mode with respect tothe TM one [28].

2.2. Rate Question Model for SOAs

In general, a "rate equation" model is obtained byintegrating the current continuity equation, as-

suming charge neutrality along any device station.Such model has been successfully applied to thedescription of laser performances [8-29]. For thecase of SOAs, the main difference with respect tothe laser case is the absence of resonating cavity.We consider here the MQW structure shown inFigure 6. The structure includes two separate con-finement hetero-layers (SCH) connecting thecontact regions to the wells. When coherentphenomena can be neglected (see e.g. [30-31]),the mean electron density Ni of each layer as [32]:

dNi Idt T]inj- R(Ni) Rst (2)

where I is the injected current, inj is the injectionefficiency, and L is the thickness of the layer. The

30 P. LUGLI et al.

ohmic contact cladding sch welt barrier barrier well sch substrate

n-lnP

SCH SCH

p-lnP

Wl WlO

FIGURE 6 Schematic structure of conduction band in a MQW-SOA with more important redistribution mechanisms sketched.

equations are coupled through the parameters thatmodel the carrier transport processes as explainedin [33, 34].For the two SCH (labeled and 2 for left and

right regions respectively) one can write explicitly:

dNschdt

linjI Nschl NschlqLsch % 7-nsch NschN1 Lwel/

FNsch2

Te Lsch "is(3)

dNsch2dt

Nsch2 Nsch2"l 7"nsch(Nsch2NM Lwell Nschl7"e Lsch

(4)

Apart from the injection term appearing in the firstequation, the remaining terms account for lossesdue to diffusive transport and subsequent capturein the adjacent QW (Nh/-s, where 7-s= -d/ff +-cap), losses due to non radiative or spontaneousrecombination (Nch/-ch, where the recombinationtime -cn depends on carrier density), carrieraccumulation due to thermionic emission from

the adjacent QW (the total number (Nqw: Lwell)/7-eis normalized with respect to SCH layer width Lsch)and exchange term between the two SCH due toleakage current ((Nch/rs), where is the couplingfactor, between 0 and 1).

In MQW systems, one must distinguish betweenthe lateral wells adjacent to the SCH, and innerones. For the lateral QWs, the following equationshold:

dtNschl Lsch N1 NI N

7"s Lwell 7"e TcN1 MFwVgg(N1)S,

and

dNmdt

Nsch2 Lsch NM7" Lwell Te

NM NM-1 NM7"c 7-n(NM) M’wVgg(NM)S

(6)

which include the exchange terms with the SCH(capture and thermionic emission) and with the


adjacent QW (inter-well transport). The last termon the right hand side of Eqs. (5) and (6) describesthe amplification of the photon density S whichexperiences a total confinement factor MPw.Pwand g(N) are, respectively, the confinement factorand gain of each QW.The density of the central QWs is finally

described by:

dNi Ni-1 Ni Ni Ni+dt % %

N n(Ui)

Mrwvgg(Ni)S. (7)

The following formulation, suitable especiallyfor large signal analysis [35], is used for the gaincoefficient of the i-th QW:

g(Ni) Go InANth + BN2th + CN3th

N >_ NthN < Nth

(8)

where a is the absorption coefficient [36], andAN+BN2/ CN3 accounts explicitly for the role ofeach recombination process through the recombi-nation coefficients A for capture at trap centers, Bfor spontaneous emission and C for Augerrecombination.The time constant for the recombination mecha-

nisms is given by:

rn(Ni) --A + BNi + CNZi(9)

The set of values for the coefficients A, B, and Care taken from the literature, and checked versusthe comparison with experimental results (seebelow). The escape of carriers from the QWs isrelated to thermionic emission over the barriers:

The transport time 7"s--7"diff/7"cap is modeledconsidering ambipolar diffusion times in the SCH,

while capture time is estimated considering theeffect of capture by means of LO phonon emissionand also carrier-carrier scattering, which is incre-asingly important at increasing bias current [37].The interwell time constant models two concur-

ring processes: tunneling through the barriers onone dies, and thermionic emission plus transportalong the barriers with subsequent capture in theadjacent QW on the other (see Fig. 6), i.e.:

/ (10)Tc (7"e / Td / q"c)

A list of parameters used in the model is given inTable I.We want to point out that the definition of the

recombination coefficients turns out to be veryuseful for a compact expression of recombinationrates appearings in Eqs. (3)-(7) that two extremesituations are considered for each case, whichcorrespond, respectively, to the minimum andmaximum value of the coefficients A, B and Cthat can be found in the literature for the materialsand structure consistent with our sample.

TABLE Parameter of rate equation model

inj. Eft. r/;nj 0.95width (SCH) Lsch 130 nmwidth (QW) Lw 4.8 nmwidth (BARR) Lb 6 nmbarrier heigth (QW-SCH) EB 110 meVcapture time rc pstunnel time rt psel mob. (SCH) SCH 0.8 104cm2/Vholes mob. (SCH) SCH

.P$cH 0.005" 104cm2/V.el mob. (QW) n 1.1. 104cm2/V.sholes mob. (QW) #paw 0.02.104cma/V.temperature T 300 Koptical conf. Pw 0.02group vel. Vg 8.5.107m/sThreshold dens. Nth 9. 1017cmmin traps coef. Amin 10 S-min spont, coef f Bmin 4.10- llcm3s-min auger coef f. Cmi 5.10-3cm6s-1max traps coef f. Amax 10 S-max spont, coef f Bmax 2.10- lcm3s-max auger coef f. Cma 2.10-29cm6sinitial dens. (SCH) Nscho 4.10lcm -3initial dens. (QW) Nwo 4.10cmleakage factor 0.2

32 P. LUGLI et al.

From Figure 8, we conclude that Auger re- spontaneous recombination processes for all thecombination is the dominant mechanism for the current (and density) values examined. Indeed, ifgain compression recovery, thereby being more one considers the case in which all the threeeffective than the Shockley-Read-Hall and processes are considered in the Rate Equation

Model, the calculated time constant takes on

measuredsimulated

" measuredsimulated

measuredsimulate6 generated carriers induce population inversion,

thereby pushing the SOA above the threshold for0.0 2000.0 4000.0 6000.0 gain. After that, the stationary regime is recovered

time [ps] by stimulated recombination, with a characteristic

FIGURE 7 Gain compression under high current injection time of tens of picoseconds. Even in this case,regime, theory and experiment agree quite well.

almost the same values as that for the case ofAuger recombination alone. Our results validatesimple models (see [8]) which consider only Augercontributions to the gain coefficient.

Different considerations apply to the gainenhancement process observed at low current bias.At currents lower than 40mA a sharp peak ofduration of less than lOOps appears in themeasured output power (see Fig. 9). In thisconditions the SOA is absorbing, and when thepump pulse arrives, it is absorbed. The photo-

0.900

0.700

0.500

SHR AUGER

-measured

Amin (out of scale: 10 ns)

0.30080 100 120 140

SPONT.

-measured

13na

%*

measured

0.900

0.700

0.500

0.300160 80 100 120 140 160 80 100 120 140 160

current [mA]

FIGURE 8 Relative importance of the more typical recombination mechanisms on the overall gain compression recovery processin the Rate Equation Model analysis. The time constants extrapolated by the experimental data are also shown.


45mA I/simulate.d

measured

m.easured

500.0 2500.0 4500.0 6500.0time [ps]

FIGURE 9 Gain dynamics at low injection current regime:from gain compression to gain enhancement.

3. MONTE CARLO SIMULATIONOF SUPERLATTICE QUANTUMCASCADE LASER

A very interesting realization of QCLs is based on

the use of periodic superlattices (SL) as activeregions [6, 39]. This new design concept exhibitsnew device physics and offers several performanceadvantages compared with conventional QCLbased on coupled quantum wells. So far, thetheoretical investigation on the electron dynamicsin SL minibands has been very scarce [40]. On theother hand, the modeling of intra-miniband andinter-miniband electronic transitions will even-tually lead to a complete understanding of inter-miniband lasing and further improvements ofsuperlattice QCL devices. In the following we

present a theoretical investigation of electrondynamics in InGaAs/InA1As superlattice QCLstructure [40, 41].For the SL active region of InGaAs/InA1As

structure W=60]k B= 18]k (well and barrierwidth, respectively), we have calculated the elec-tron energies and wave functions and the scatter-ing rates for electron-phonon interactions [42].Figure 10 shows the miniband profile and the

electron-LO phonon emission scattering rates,plotted as a function of the electron energy(measured from the bottom of the miniband) atdifferent values of the parallel kp and perpen-dicular kz k-vector. For instance, for the 2 2transition, the lowest curve corresponds to themaximum kz (at the miniband edge), as only stateson the kp dispersion are involved in the transition,while the largest scattering rate is found at kz O,that is at the top of the miniband. The reversesitutaion holds for the intraminiband transitionswithin the first miniband. A Monte Carlo simula-tion has been set up based on the scattering ratesdescribed above [43]. In addition, electron-elec-tron scattering is included, the interaction beingcalculated assuming a statically screened Coulombpotential. Pauli exclusion principle is accountedfor in the usual way. In order to reach a steadystate condition in the MC simulation, electrons are

extracted from the bottom of the first miniband ata fixed rate determined by a time constant -t,nn

and injected in the second miniband at energy Ei,until stationarity is reached.The stationary distribution functions for the

first and second minibands are plotted in the insetof Figure 11. Three different densities are con-

sidered. As the density increases, the enhancedintercarrier scattering broadens the distribution

r Kz=O "i

00.0 0.2 0.4 0.6 0.8 1.0Energy [eV]

1.5 - Kz=O t1.0

’’ ’’ ’I11.5

Energy [eV]

St"’’""’’’’"/2---2Kz=O’’J/ 400 f" tL

00.0 0.2 0.4 0.6 0.8 1.0 -0.04-0.02 0 0.02 0.04

Energy [eV] kz [A-1]

FIGURE 10 Scattering rates and miniband dispersion of theInGaAs/InA1As structure.

34 P. LUGLI et al.

1.0

0.5

0.00.10 0.40

=10., t=10ntot=lO cm

,’

o.

" First miniband

’, t, ’,,, 0.0 ,I,

0.0

0.15 0.20 0.25 0.30 0.35Energy [eV]

FIGURE 11 Electroluminescence spectra in stationary condi-tion. The inset shows the stationary electron distributionfunctions of the first and the second minibands.

functions, this reducing the overlap in k-spacebetween the population of the two miniband. Theeffect of such broadening on the lasing capabilitiesof the SL can be evaluated by looking at theinversion gain, gin, which we define as the ratio ofthe electron density at the bottom of the upperminiband with respect to the density at the top ofthe lower miniband. The higher the value ofgin, thehigher the probability that the structure will lase(and the lower the threshold current for lasing).An interesting result is found as a function of

doping density. As the density moves from1016cm -3 to 1018cm -3, the inversion gain of theInGaAs/InA1As structure drops from 26 to 1.2,indicating that low doping of the active region hasto be preferred. The physical reason can beunderstood from the rates for electron-phononand electron-electron scattering calculated duringthe simulation [40]. While the phonon rate doesnot change appreciably (indicating that Pauliexclusion principle does not effect the electrontransport around the miniband edge, at least up tothe densities considered here), the intercarrier rateincreases with the electron density, both as intraand as inter-miniband process. The former con-tributes to the thermalization and spreading ink-space of the electrons within each miniband, thelatter causes a redistribution between minibands,

in particular favoring the scattering out the upperinto the lower miniband. Indeed a significantreduction of the threshold current and room

temperature operation in QCL with intrinsic SLregions has been reported [44]. The electrolumi-nescence spectra are plotted in Figure 11. Thespectra get broader at higher densities, a directconsequence of the electron thermalization de-tected in the distribution functions. The influenceof various simulation parameters has also beentested. In short, we found that the inversion gain ishigher when the injection energy is lower. This iscaused by a strong reduction of the phononscattering within the second miniband whenelectrons are injected closer to the miniband edge,while the interminiband 2 and the intramini-band do not show any sizable dependencefrom Eij. This result indicates the importance ofthe injection condition in determining the perfor-mance of superlattice QCLs. For instance, it is

possible to compensate for the loss of inversiongain characteristic of high densities by properlytuning the injection energy. Figure 3 shows thecomparison between the electroluminescence spec-tra of the InGaAs/InA1As SL measured at 5Kand 300 K. The injected current is 75 mA in bothcases. A detailed description of the experimental

1.2Experimental data T=5KExperimental data T=300K

/’ [] MC simulation T=300K

/", o.o,. Sedminiband 1

0.2 - o.o 0.4Y o u, o.oi- nergy [eV]

0.00.5 0.20 0.25 0.30 0.35

Energy [eV]

FIGURE 12 Comparison between calculated and measuredelectroluminescence spectra. The inset shows the electrondistribution function at low and room temperature.


apparatus can be found in Ref. [45]. The mainpeak is associated with radiative transitions at theminigap between electronic states having wavevec-tors close to the edge of the SL mini Brillouin zone.The high energy tail corresponds to transitionsbetween states with lower kz. The calculated spectra(solid dots in Fig. 12) reproduce well the peak andthe high energy tail. The corresponding electrondistribution functions are plotted in the inset ofFigure 12. We can see that at low temperature thephonon replicas appear clearly in the distributions(but they are washed out in the spectra), and thatthe population ofthe edge state is much higher thanat room temperature. This is the main reason of thebroader spectrum measured at 300 K.

4. CONCLUSIONS

We have presented a series ofmodeling approachesfor semiconductor optoelectronic devices, focusingon semiconductor optical amplifiers and quantumcascade lasers. The influence ofa delta-strain on themodal absorption/gain characteristic of a semicon-ductor optical amplifier have been studied by meansof a tight-binding calculation. The relation betweenlevel alignment, valence band mixing, TE and TMoptical transition matrix element and opticalabsorption/gain has been evidenced via a tightbinding approach. The SOA studied under thepump-probe scheme at high injection currentregime has also been addressed. The role of therecombination processes have been clarified, fromthe point of view of gain compression mechanismsexploiting Cross Gain Modulation for WDMapplications. We have confirmed that Auger is themost important recombination mechanism for

1.551am SOA, even if we consider an Augermechanism reduced in intensity due to valenceband modifications induced by strain. Finally aMonte Carlo simulation ofthe electron dynamics insuperlattice QCLs has been presented, which showsthe role of phonon and intercarrier scatteringprocesses.

Acknowledgment

This work was partially supported by INFMunder PRA project "SUPERLAS", and by theEuropean TMR Network "Ultrafast QuantumElectronics". We greatfully acknowledge Prof. N.Scamarcio and Dr. D. Campi for the experimentalresults and for useful discussions.

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