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Quantum Well42. Quantum Wells, Superlattices,and Band-Gap Engineering

This chapter reviews the principles of band-gap engineering and quantum confinementin semiconductors, with a particular emphasison their optoelectronic properties. The chapterbegins with a review of the fundamentalprinciples of band-gap engineering and quantumconfinement. It then describes the optical andelectronic properties of semiconductor quantumwells and superlattices at a tutorial level, beforedescribing the principal optoelectronic devices.The topics covered include edge-emitting lasersand light-emitting diodes (LEDs), resonant cavityLEDs and vertical-cavity surface-emitting lasers(VCSELs), quantum cascade lasers, quantum-wellsolar cells, superlattice avalanche photodiodes,inter-sub-band detectors, and quantum-welllight modulators. The chapter concludes witha brief review of current research topics, includinga discussion of quantum-dot structures.

42.1 Principles of Band-Gap Engineeringand Quantum Confinement .................. 102242.1.1 Lattice Matching ....................... 102242.1.2 Quantum-Confined Structures .... 1023

42.2 Optoelectronic Propertiesof Quantum-Confined Structures........... 102442.2.1 Electronic States in Quantum

Wells and Superlattices .............. 102442.2.2 Interband Optical Transitions ...... 102642.2.3 The Quantum-Confined Stark

Effect ....................................... 102842.2.4 Inter-Sub-Band Transitions........ 102942.2.5 Vertical Transport ...................... 103042.2.6 Carrier Capture and Relaxation ... 1031

42.3 Emitters.............................................. 103242.3.1 Interband Light-Emitting Diodes

and Lasers ................................ 103242.3.2 Quantum Cascade Lasers ............ 1033

42.4 Detectors ............................................ 103442.4.1 Solar Cells................................. 103442.4.2 Avalanche Photodiodes.............. 103442.4.3 Inter-Sub-Band Detectors .......... 103542.4.4 Unipolar Avalanche Photodiodes . 1035

42.5 Modulators ......................................... 1036

42.6 Future Directions ................................. 1037

42.7 Conclusions ......................................... 1038

References .................................................. 1038

The need for efficient light-emitting diodes and lasersoperating over the whole of the visible spectrum and alsothe fibre-optic windows at 1.3 µm and 1.55 µm drivesresearch into new direct-gap semiconductors to act asthe active materials. Since the emission wavelength ofa semiconductor corresponds to its band-gap energy,research focuses on engineering new materials whichhave their band gaps at custom-designed energies. Thisscience is called band-gap engineering.

In the early years of semiconductor optoelectronics,the band gaps that could be achieved were largely deter-mined by the physical properties of key III–V materialssuch as GaAs and its alloys such as AlGaAs and InGaAs.Then in 1970 a major breakthrough occurred when Esakiand Tsu invented the semiconductor quantum well andsuperlattice [42.1]. They realised that developments inepitaxial crystal growth could open the door to new

structures that exploit the principles of quantum confine-ment to engineer electronic states with custom-designedproperties. They foresaw that these quantum-confinedstructures would be of interest both to research scien-tists, who would be able to explore uncharted areas offundamental physics, and also to engineers, who wouldlearn to use their unique properties for device applica-tions. Their insight paved the way for a whole new breedof devices that are now routinely found in a host of ev-eryday applications ranging from compact-disc playersto traffic lights.

The emphasis of the chapter is on the optoelectronicproperties of quantum-well and superlattice structures.We begin by outlining the basic principles of band-gap engineering and quantum confinement. We willthen discuss the electronic states in quantum-confinedstructures and the optical properties that follow from

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42

1022 Part D Materials for Optoelectronics and Photonics

them. In Sects. 42.3–42.5 we will explain the principlesof the main optoelectronic devices that employ quan-tum wells and superlattices, namely emitters, detectorsand modulators. Finally we will indicate a few interest-ing recent developments that offer exciting prospectsfor future devices before drawing the chapter to its

conclusion. A number of texts cover these topics inmore detail (e.g. [42.2–5]), and the interested readeris referred to these sources for a more comprehen-sive treatment. A description of the purely electronicproperties of low-dimensional structures may be foundin [42.6].

42.1 Principles of Band-Gap Engineering and Quantum Confinement

42.1.1 Lattice Matching

The art of band-gap engineering relies heavily on de-velopments in the science of crystal growth. Bulkcrystals grown from the melt usually contain a largenumber of impurities and defects, and optoelectronicdevices are therefore grown by epitaxial methods suchas liquid-phase epitaxy (LPE), molecular-beam epi-taxy (MBE) and metalorganic vapour-phase epitaxy(MOVPE), which is also called metalorganic chem-ical vapour deposition (MOCVD) (Chapt. 14). The basicprinciple of epitaxy is to grow thin layers of veryhigh purity on top of a bulk crystal called the sub-strate. The system is said to be lattice-matched whenthe lattice constants of the epitaxial layer and thesubstrate are identical. Lattice-matching reduces thenumber of dislocations in the epitaxial layer, but italso imposes tight restrictions on the band gaps thatcan be engineered easily, because there are only a rel-atively small number of convenient substrate materialsavailable.

Band gap (eV)

Lattice constant (nm)0.3

6

4

2

00.4 0.5 0.6

Direct band gap

Indirect band gap

AIN

GaN

InNInPGaAs

GaP

AIP

AIAs

InAs

Fig. 42.1 Room-temperature band gap of a number of im-portant optoelectronic III–V materials versus their latticeconstant

We can understand this point more clearly by ref-erence to Fig. 42.1. This diagram plots the band-gapenergy Eg of a number of important III–V semicon-ductors as a function of their lattice constant. Themajority of optoelectronic devices for the red/near-infrared spectral regions are either grown on GaAsor InP substrates. The simplest case to consider isan epitaxial layer of GaAs grown on a GaAs sub-strate, which gives an emission wavelength of 873 nm(1.42 eV). This wavelength is perfectly acceptable forapplications involving short-range transmission downoptical fibres. However, for long distances we requireemission at 1.3 µm or 1.55 µm, while for many otherapplications we require emission in the visible spectralregion.

Let us first consider the preferred fibre-optic wave-lengths of 1.3 µm and 1.55 µm. There are no binarysemiconductors with band gaps at these wavelengths,and so we have to use alloys to tune the band gap by vary-ing the composition (Chapt. 31). A typical example isthe ternary alloy GaxIn1−xAs, which is lattice-matchedto InP when x = 47%, giving a band gap of 0.75 eV(1.65 µm). Ga0.47In0.53As photodiodes grown on InPsubstrates make excellent detectors for 1.55-µm radi-ation, but to make an emitter at this wavelength, wehave to increase the band gap while maintaining thelattice-matching condition. This is achieved by incorpo-rating a fourth element into the alloy – typically Al orP, which gives an extra design parameter that permitsband-gap tuning while maintaining lattice-matching.Thus the quaternary alloys Ga0.27In0.73As0.58P0.42 andGa0.40In0.60As0.85P0.18 give emission at 1.3 µm and1.55 µm, respectively, and are both lattice-matched toInP substrates.

Turning now to the visible spectral region, it is a con-venient coincidence that the lattice constants of GaAsand AlAs are almost identical. This means that we cangrow relatively thick layers of AlxGa1−xAs on GaAssubstrates without introducing dislocations and other de-fects. The band gap of AlxGa1−xAs varies quadratically

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Quantum Wells, Superlattices, and Band-Gap Engineering 42.1 Principles of Band-Gap Engineering and Quantum Confinement 1023

with x according to:

Eg(x) = (1.42+1.087x +0.438x2) eV , (42.1)

but unfortunately the gap becomes indirect forx > 43%. We can therefore engineer direct band gaps of1.42–1.97 eV, giving emission from 873 nm in the nearinfrared to 630 nm in the red spectral range. Much workhas been done on quaternary alloys such as AlGaInP,(Chapt. 31) but it has not been possible to make blue-and green-emitting devices based on GaAs substrates todate, due to the tendency for arsenic and phosphorouscompounds to become indirect as the band gap increases.

The approach for the blue end of the spectrum pre-ferred at present is to use nitride-based compounds.(Chapt. 32) Early work on nitrides established thattheir large direct gaps made them highly promis-ing candidates for use as blue/green emitters [42.7].However, it was not until the 1990s that this poten-tial was fully realised. The rapid progress followedtwo key developments, namely the activation of p-type dopants and the successful growth of strainedInxGa1−xN quantum wells which did not satisfy thelattice-matching condition [42.8]. The second point goesagainst the conventional wisdom of band-gap engi-neering and highlights the extra degrees of freedomafforded by quantum-confined structures, as will nowbe discussed.

42.1.2 Quantum-Confined Structures

A quantum-confined structure is one in which the mo-tion of the electrons (and/or holes) are confined inone or more directions by potential barriers. The gen-eral scheme for classifying quantum-confined structuresis given in Table 42.1. In this chapter we will beconcerned primarily with quantum wells, although wewill briefly refer to quantum wires and quantum dotsin Sect. 42.6. Quantum size effects become importantwhen the thickness of the layer becomes comparablewith the de Broglie wavelength of the electrons or holes.

Table 42.1 Classification of quantum-confined structures.In the case of quantum wells, the confinement direction isusually taken as the z-axis

Structure Confineddirections

Free directions(dimensionality)

Quantum well 1 (z) 2 (x, y)

Quantum wire 2 1

Quantum dot (or box) 3 none

If we consider the free thermal motion of a particle ofmass m in the z-direction, the de Broglie wavelength ata temperature T is given by

λdeB = h√mkBT

. (42.2)

For an electron in GaAs with an effective mass of0.067m0, we find λdeB = 42 nm at 300 K. This im-plies that we need structures of thickness ≈ 10 nmin order to observe quantum-confinement effects atroom temperature. Layers of this thickness are routinelygrown by the MBE or MOVPE techniques describedin Chapt. 14.

Figure 42.2 shows a schematic diagram ofa GaAs/AlGaAs quantum well. The quantum confine-ment is provided by the discontinuity in the band gap atthe interfaces, which leads to a spatial variation of theconduction and valence bands, as shown in the lower halfof the figure. The Al concentration is typically chosento be around 30%, which gives a band-gap discontinuityof 0.36 eV according to (42.1). This splits roughly 2 : 1between the conduction and valence bands, so that elec-trons see a confining barrier of 0.24 eV and the holes see0.12 eV.

If the GaAs layers are thin enough, according to thecriterion given above, the motion of the electrons andholes will be quantised in the growth (z) direction, giving

Conduction band

Valence band

d

GaAssubstrate

AlGaAs GaAs AlGaAs

z

GaASEgAlGaASEg

Fig. 42.2 Schematic diagram of the growth layers and re-sulting band diagram for a GaAs/AlGaAs quantum well ofthickness d. The quantised levels in the quantum well areindicated by the dashed lines. Note that in real structuresa GaAs buffer layer is usually grown immediately abovethe GaAs substrate

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42.1


rise to a series of discrete energy levels, as indicated bythe dashed lines inside the quantum well in Fig. 42.2.The motion is in the other two directions (i. e. the x–yplane) is still free, and so we have quasi two-dimensional(2-D) behaviour.

The quantisation of the motion in the z-directionhas three main consequences. Firstly, the quantisationenergy shifts the effective band edge to higher energy,which provides an extra degree of freedom in the art ofband-gap engineering. Secondly, the confinement keepsthe electrons and holes closer together and hence in-creases the radiative recombination probability. Finally,the density of states becomes independent of energy,in contrast to three-dimensional (3-D) materials, wherethe density of states is proportional to E1/2. Many ofthe useful properties of the quantum wells follow fromthese three properties.

The arrangement of the bands shown in Fig. 42.2in which both the electrons and holes are confinedin the quantum well is called type I band alignment.Other types of band alignments are possible in whichonly one of the carrier types are confined (type IIband alignment). Furthermore, the flexibility of theMBE and MOVPE growth techniques easily allows thegrowth of superlattice (SL) structures containing manyrepeated quantum wells with thin barriers separatingthem, as shown in Fig. 42.3. Superlattices behave likeartificial one-dimensional periodic crystals, in whichthe periodicity is designed into the structure by therepetition of the quantum wells. The electronic statesof SLs form delocalised minibands as the wave func-tions in neighbouring wells couple together throughthe thin barrier that separates them. Structures con-taining a smaller number of repeated wells or withthick barriers that prevent coupling between adjacentwells are simply called multiple quantum well (MQW)structures.

Single QW Superlattice

Conductionband

Valenceband

db

Fig. 42.3 Schematic diagram of a superlattice, showing theformation of minbands from the energy levels of the corre-sponding single quantum well (QW). The structure formsan artificial one-dimensional crystal with period (d +b),where d and b represent the thickness of the QW and barrierregions respectively. The width of the minibands dependson the strength of the coupling through the barriers. It isfrequently the case that the lowest hole states do not couplestrongly, and hence remain localised within their respectivewells, as shown in the figure

In the next section we will describe in more detail theelectronic properties of quantum wells and superlattices.Before doing so, it is worth highlighting two practicalconsiderations that are important additional factors thatcontribute to their usefulness. The first is that band-gaptunability can be achieved without using alloys as theactive material, which is desirable because alloys in-evitably contain more defects than simple compoundssuch as GaAs. The second factor is that the quantumwells can be grown as strained layers on top of a lat-tice with a different cell constant. A typical example isthe InxGa1−xN/GaN quantum wells mentioned above.These layers do not satisfy the lattice-matching condi-tion, but as long as the total InxGa1−xN thickness is lessthan the critical value, there is an energy barrier to theformation of dislocations. In practice this allows consid-erable extra flexibility in the band-gap engineering thatcan be achieved.

42.2 Optoelectronic Properties of Quantum-Confined Structures

42.2.1 Electronic States in Quantum Wellsand Superlattices

Quantum WellsThe electronic states of quantum wells can be understoodby solving the Schrodinger equation for the electronsand holes in the potential wells created by the band dis-continuities. The simplest approach is the infinite-wellmodel shown in Fig. 42.4a. The Schrodinger equation in

the well is

− �2

2m∗w

d2ψ(z)

dz2= Eψ(z) , (42.3)

where m∗w is the effective mass in the well and z is the

growth direction. Since the potential barriers are infinite,there can be no penetration into the barriers, and we musttherefore have ψ(z) = 0 at the interfaces. If we chooseour origin such that the quantum well runs from z = 0

PartD

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Quantum Wells, Superlattices, and Band-Gap Engineering 42.2 Optoelectronic Properties of Quantum-Confined Structures 1025

E2

V0

E1

0

E

d d

n = 1

a) Infinite well b) Finite well

n = 2

Fig. 42.4a,b Confined states in a quantum well of width d.(a) A perfect quantum well with infinite barriers. (b) A finitewell with barriers of height V0. The wave functions for then = 1 and n = 2 levels are sketched for both types of well

to z = d, d being the width of the well, the normalisedwave functions take the form (see e.g. [42.9]):

ψn(z) = √2/d sin knz , (42.4)

where kn = (nπ/d) and the quantum number n is aninteger (≥ 1). The energy En is given by

En = �2k2

n

2m∗w

= �2

2m∗w

(nπ

d

)2. (42.5)

The wave functions of the n = 1 and n = 2 levels aresketched in Fig. 42.4a.

Although the infinite-well model is very simplified,it nonetheless provides a good starting point for under-standing the general effects of quantum confinement.Equation (42.5) shows us that the energy is inversely pro-portional to d2, implying that narrow wells have largerconfinement energies. Furthermore, the confinement en-ergy is inversely proportional to the effective mass,which means that lighter particles experience larger ef-fects. This also means that the heavy- and light-holestates have different energies, in contrast to bulk semi-conductors in which the two types of hole states aredegenerate at the top of the valence band.

Now let us consider the more realistic finite-wellmodel shown in Fig. 42.4b. The Schrodinger equation inthe well is unchanged, but in the barrier regions we nowhave:

− �2

2m∗b

d2ψ(z)

dz2+ V0ψ(z) = Eψ(z) , (42.6)

where V0 is the potential barrier and m∗b is the effective

mass in the barrier. The boundary conditions require that

the wave function and particle flux (1/m∗)dψ/dz mustbe continuous at the interface. This gives a series of evenand odd parity solutions which satisfy

tan(kd/2) = m∗wκ

m∗bk

, (42.7)

and

tan(kd/2) = − m∗bk

m∗wκ

, (42.8)

respectively. k is the wave vector in the well, given by

�2k2

2m∗w

= En , (42.9)

while κ is the exponential decay constant in the barrier,given by

�2κ2

2m∗b

= V0 − En . (42.10)

Solutions to (42.7) and (42.8) are easily found by simplenumerical techniques [42.9]. As with the infinite well,the eigenstates are labelled by the quantum number n andhave parities of (−1)n+1 with respect to the axis of sym-metry about the centre of the well. The wave functionsare approximately sinusoidal inside the well, but decayexponentially in the barriers, as illustrated in Fig. 42.4b.The eigen-energies are smaller than those of the infinitewell due to the penetration of the barriers, which meansthat the wave functions are less well confined. There isonly a limited number of solutions, but there is alwaysat least one, no matter how small V0 might be.

As an example we consider a typical GaAs/Al0.3Ga0.7As quantum well with d = 10 nm. The con-finement energy is 245 meV for the electrons and125 meV for the holes. The infinite well modelpredicts E1 = 56 meV and E2 = 224 meV for the elec-trons, whereas (42.7), (42.8) give E1 = 30 meV andE2 = 113 meV. For the heavy (light) holes the infinite-well model predicts 11 meV (40 meV) and 44 meV(160 meV) for the first two bound states, instead of themore accurate values of 7 meV (21 meV) and 29 meV(78 meV) calculated from the finite-well model. Notethat the separation of the electron levels is greater thankBT at 300 K, so that that the quantisation effects willbe readily observable at room temperature.

Strained Quantum WellsEven more degrees of freedom for tailoring the elec-tronic states can be achieved by epitaxially stackingsemiconductor layers with different lattice constantsto form strained quantum wells. Examples include

PartD

42.2


InxGa1−xAs on GaAs, and Si1−xGex on Si. Large bi-axial strain develops within the x–y plane of a quantumwell grown on a substrate with a different lattice con-stant. In order to avoid the buildup of misfit dislocationsat the interfaces, the strained layers need to be thin-ner than a certain critical dimension. For example,a defect-free strained InxGa1−xAs layer on GaAs re-quires a thickness less than around 10 nm when x = 0.2.Since the band gap is related to the lattice constant, thestrain induces a shift of the band edges which, in turn,affects many other properties. It is due to some of theseeffects that strained QW structures have become widelyexploited in optoelectronic devices. (Chapt. 37)

The most significant effect of the strain is to alterthe band gap and remove the valence-band degeneracynear the Γ valley. The splitting of the valence band isa consequence of the lattice distortion, which reducesthe crystal symmetry from cubic to tetragonal [42.10].There are essentially two types of strain. Compressivestrain occurs when the active layer has a larger latticeconstant than the substrate, for example in InxGa1−xAson GaAs. In this case, the band gap increases and theeffective mass of the highest hole band decreases, whilethat of next valence band increases. The opposite caseis that of tensile strain, which occurs when the activelayer has a smaller lattice constant than the substrate,such as Si1−xGex on Si. The ordering of the valencebands is opposite to the case of compressive strain, andthe overall band gap is reduced.

SuperlatticesThe analytical derivation of the allowed energy valuesin a superlattice (SL) is similar to that for a single QW,with the appropriate change of the boundary conditionsimposed by the SL periodicity. The mathematical de-scription of a superlattice is similar to a one-dimensionalcrystal lattice, which allows us to borrow the formalismof the band theory of solids, including the well-knownKronig–Penney model [42.9]. Within this model, theelectron envelope wave function ψ(z) can be expressedas a superposition of Bloch waves propagating alongthe z-axis. For a SL with a barrier height V0, the al-lowed energy is calculated numerically as a solution ofthe transcendental equation involving the Bloch wavevector:

cos (ka) = cos (kd) cos(κ′b

)

− k2 +κ′2

2kκ′ sin (kd) sin(κ′b

),

E > V0 , (42.11)

cos (ka) = cos (kd) cos (κb)

− k2 −κ2

2kκsin (kd) sin (κb) ,

E < V0 , (42.12)

where a ≡ (b+d) is the period, and k and κ are givenby (42.9) and (42.10), respectively. The decay constantκ′ is given by:

E − V0 = �2κ′2

2m∗b

. (42.13)

The electronic states in superlattices can be understoodin a more qualitative way by reference to Fig. 42.3 andmaking use of the analogy with the tight-binding modelof band formation in solids. Isolated atoms have dis-crete energy levels which are localised on the individualatom sites. When the atoms are brought close together,the energy levels broaden into bands, and the overlap-ping wave functions develop into extended states. In thesame way, repeated quantum-well structures with largevalues of the barrier thickness b (i. e. MQWs) have dis-crete levels with wave functions localised within thewells. As the barrier thickness is reduced, the wavefunctions of adjacent wells begin to overlap and thediscrete levels broaden into minibands. The wave func-tions in the minibands are delocalised throughout thewhole superlattice. The width of the miniband dependson the cross-well coupling, which is determined by thebarrier thickness and the decay constant κ (42.10). Ingeneral, the higher-lying states give rise to broader mini-bands because κ decreases with En . Also, the heavy-holeminibands are narrower than the electron minibands, be-cause the cross-well coupling decreases with increasingeffective mass.

42.2.2 Interband Optical Transitions

AbsorptionThe optical transitions in quantum wells take placebetween electronic states that are confined in the z-direction but free in the x–y plane. The transition ratecan be calculated from Fermi’s golden rule, which statesthat the probability for optical transitions from the ini-tial state |i〉 at energy Ei to the final state | f 〉 at energyE f is given by:

W(i → f ) = 2π

�|〈 f |er ·E |i〉|2 g(�ω) , (42.14)

where er is the electric dipole of the electron, E is theelectric field of the light wave, and g(�ω) is the jointdensity of states at photon energy �ω. Conservation of

PartD

42.2


energy requires that E f = (Ei +�ω) for absorption, andE f = (Ei −�ω) for emission.

Let us consider a transition from a confined holestate in the valence band with quantum number n toa confined electron state in the conduction band withquantum number n′. We apply Bloch’s theorem to writethe wave functions in the following form:

|i〉 = uv(r) exp(ikxy ·rxy)ψhn(z)

| f 〉 = uc(r) exp(ikxy ·rxy) ψen′ (z) , (42.15)

where uv(r) and uc(r) are the envelope function for thevalence and conduction bands, respectively, kxy is thein-plane wave vector for the free motion in the x–yplane, rxy being the xy component of the position vector,and ψhn(z) and ψen′ (z) are the wave functions for theconfined hole and electron states in the z-direction. Wehave applied conservation of momentum here so that thein-plane wave vectors of the electron and hole are thesame.

On inserting these wave functions into (42.14) wefind that the transition rate is proportional to both thesquare of the overlap of the wave functions and the jointdensity of states [42.11]:

W ∝ |〈ψen′ (z)|ψhn(z)〉|2 g(�ω) . (42.16)

The wave functions of infinite wells are orthogonal un-less n = n′, which gives a selection rule of ∆n = 0. Forfinite wells, the ∆n = 0 selection rule is only approx-imately obeyed, although transitions between states ofdifferent parity (i. e. ∆n odd) are strictly forbidden. Thejoint density of states is independent of energy due tothe quasi-2-D nature of the quantum well.

Figure 42.5a illustrates the first two strong transi-tions in a typical quantum well. These are the ∆n = 0transitions between the first and second hole and elec-tron levels. The threshold energy for these transitions isequal to

�ω = Eg + Ehn + E en . (42.17)

The lowest value is thus equal to (Eg + Eh1 + E e1),which shows that the optical band gap is shifted by thesum of the electron and hole confinement energies. Oncethe photon energy exceeds the threshold set by (42.17),a continuous band of absorption occurs with the absorp-tion coefficient independent of energy due to the constant2-D density of states of the quantum well. The differ-ence between the absorption of an ideal quantum wellwith infinite barriers and the equivalent bulk semicon-ductor is illustrated in Fig. 42.5b. In the quantum wellwe find a series of steps with constant absorption co-efficients, whereas in the bulk the absorption varies as

Absorption coefficient

(hω – Eg) in units of (h2/8d 2µ)0 84

Bulk

n = 1

n = 2

b)

Conduction band

Valence band

e2

e1

h1h2

n = 1 n = 2

a)

Fig. 42.5a,b Interband optical transitions between con-fined states in a quantum well. (a) Schematic diagramshowing the ∆n = 0 transitions between the n = 1 andn = 2 sub-bands. (b) Absorption spectrum for an infinitequantum well of thickness d with a reduced electron–holemass µ in the absence of excitonic effects. The absorptionspectrum of the equivalent bulk semiconductor in shown bythe dashed line for comparison

(�ω− Eg)1/2 for �ω > Eg. Thus the transition from 3-Dto 2-D alters the shape of the absorption curve, and alsocauses an effective shift in the band gap by (Eh1 + E e1).

Up to this point, we have neglected the Coulombinteraction between the electrons and holes which areinvolved in the transition. This attraction leads to theformation of bound electron–hole pairs called excitons.The exciton states of a quantum well can be modelled as2-D hydrogen atoms in a material with relative dielectricconstant εr. In this case, the binding energy EX is givenby [42.12]:

Ex(ν) = µ

m0

1

ε2r

1

(ν−1/2)2 RH , (42.18)

where ν is an integer ≥ 1, m0 is the electron mass, µ isthe reduced mass of the electron–hole pair, and RH isthe Rydberg constant for hydrogen (13.6 eV). This con-trasts with the standard formula for 3-D semiconductorsin which EX varies as 1/ν2 rather than 1/(ν−1/2)2, andimplies that the binding energy of the ground-state ex-citon is four times larger in 2-D than 3-D. This allowsexcitonic effects to be observed at room temperature inquantum wells, whereas they are only usually observedat low temperatures in bulk semiconductors.

Figure 42.6 compares the band-edge absorptionof a GaAs MQW sample with that of bulk GaAs atroom temperature [42.13]. The MQW sample contained77 GaAs quantum wells of thickness 10 nm with thickAl0.28Ga0.72As barriers separating them. The shift of

PartD

42.2


Absorption coefficient (105 m–1)

Energy (eV)1.40 1.60

10

5

01.45 1.50 1.55

4

3

2

1

0

Bulk GaAsHeavy hole

Light hole

n = 1

n = 2

300 K

GaAs MQW

Fig. 42.6 Absorption spectrum of a 77-period GaAs/Al0.28Ga0.72As MQW structure with 10-nm quantum wellsat room temperature. The absorption spectrum of bulk GaAsis included for comparison. (After [42.13], c© 1982 AIP)

the band edge of the MQW to higher energy is clearlyobserved, together with the series of steps due to each∆n = 0 transition. The sharp lines are due to excitons,which occur at energies given by

�ω = Eg + Ehn + E en − Ex . (42.19)

Equation (42.18) predicts that EX should be around17 meV for the ground-state exciton of an ideal GaAsquantum well, compared to 4.2 meV for the bulk. The ac-tual QW exciton binding energies are somewhat smallerdue to the tunnelling of the electrons and holes into thebarriers, with typical values of around 10 meV. How-ever, this is still substantially larger than the bulk valueand explains why the exciton lines are so much betterresolved for the QW than the bulk. The absorption spec-trum of the QW above the exciton lines is approximatelyflat due to the constant density of states in 2-D, whichcontrasts with the rising absorption of the bulk due tothe parabolic 3-D density of states. Separate transitionsare observed for the heavy and light holes. This followsfrom their different effective masses, and can also beviewed as a consequence of the lower symmetry of theQW compared to the bulk.

EmissionEmissive transitions occur when electrons excited in theconduction band drop down to the valence band andrecombine with holes. The optical intensity I(�ω) isproportional to the transition rate given by (42.14) mul-tiplied by the probability that the initial state is occupied

hω

F

Thermal emission

Tunneling

Fig. 42.7 Schematic band diagram and wave functions fora quantum well in a DC electric field F

and the final state is empty:

I(�ω) ∝ W(c → v) fc(1− fv) , (42.20)

where fc and fv are the Fermi–Dirac distribution func-tions in the conduction and valence bands, respectively.In thermal equilibrium, the occupancy of the states islargest at the bottom of the bands and decays expo-nentially at higher energies. Hence the luminescencespectrum of a typical GaAs QW at room tempera-ture usually consists of a peak of width ∼ kBT atthe effective band gap of (Eg + Ehh1 + E e1). At lowertemperatures the spectral width is affected by inhomoge-neous broadening due to unavoidable fluctuations in thewell thickness. Furthermore, in quantum wells employ-ing alloy semiconductors, the microscopic fluctuationsin the composition can lead to additional inhomogeneousbroadening. This is particularly true of InGaN/GaNquantum wells, where indium compositional fluctua-tions produce substantial inhomogeneous broadeningeven at room temperature.

The intensity of the luminescence peak in quantumwells is usually much larger than that of bulk materialsbecause the electron–hole overlap is increased by theconfinement. This leads to faster radiative recombina-tion, which then wins out over competing nonradiativedecay mechanisms and leads to stronger emission. Thisenhanced emission intensity is one of the main reasonswhy quantum wells are now so widely used in diodelasers and light-emitting diodes.

42.2.3 The Quantum-Confined Stark Effect

The quantum-confined Stark effect (QCSE) describesthe response of the confined electron and hole states inquantum wells to a strong direct-current (DC) electricfield applied in the growth (z) direction. The field is

PartD

42.2


usually applied by growing the quantum wells insidea p–n junction, and then applying reverse bias to thediode. The magnitude of the electric field F is given by:

F = V built-in − V bias

L i, (42.21)

where V built-in is the built-in voltage of the diode, V bias

is the bias voltage, and L i is the total thickness ofthe intrinsic region. V built-in is approximately equal tothe band-gap voltage of the doped regions (≈ 1.5 V fora GaAs diode).

Figure 42.7 gives a schematic band diagram ofa quantum well with a strong DC electric field applied.The field tilts the potential and distorts the wave func-tions as the electrons tend to move towards the anodeand the holes towards the cathode. This has two impor-tant consequences for the optical properties. Firstly, thelowest transition shifts to lower energies due to the elec-trostatic interaction between the electric dipole inducedby the field and the field itself. At low fields the dipole isproportional to F, and the red shift is thus proportionalto F2 (the quadratic Stark effect). At higher fields, thedipole saturates at a value limited by ed, where e is theelctron charge and d the well width, and the Stark shift islinear in F. Secondly, the parity selection rule no longerapplies due to the breaking of the inversion symmetryabout the centre of the well. This means that forbid-den transitions with ∆n equal to an odd number becomeallowed. At the same time, the ∆n = 0 transitions grad-

Photocurrent (arb. units)

Energy (eV)1.4 1.5 1.6 1.7

GaAs MQW, 300 K

0 V

– 10 V

Fig. 42.8 Room-temperature photocurrent spectra fora GaAs/Al0.3Ga0.7As MQW p–i–n diode with a 1-µm-thicki-region at zero bias and −10 V. The quantum well thick-ness was 9.0 nm. The arrows identify transitions that areforbidden at zero field. (After [42.14], c© 1991 IEEE)

ually weaken with increasing field as the distortion tothe wave functions reduces the electron–hole overlap.

Figure 42.8 shows the normalised room-temperaturephotocurrent spectra of a GaAs/Al0.3Ga0.7As MQW p–i–n diode containing 9.0-nm quantum wells at 0 V and−10 V applied bias. These two bias values correspondto field strengths of around 15 kV/cm and 115 kV/cmrespectively. The photocurrent spectrum closely resem-bles the absorption spectrum, due to the field-inducedescape of the photoexcited carriers in the QWs into theexternal circuit (Sect. 42.2.5). The figure clearly showsthe Stark shift of the absorption edge at the higher fieldstrength, with a red shift of around 20 meV (≈ 12 nm) at−10 V bias for the hh1 → e1 transition. The intensity ofthe line weakens somewhat due to the reduction in theelectron–hole overlap, and there is lifetime broadeningcaused by the field-assisted tunnelling. Several parity-forbidden transitions are clearly observed. The two mostobvious ones are identified with arrows, and correspondto the hh2 → e1 and hh1 → e2 transitions, respectively.

A striking feature in Fig. 42.8 is that the exciton linesare still resolved even at very high field strengths. In bulkGaAs the excitons ionise at around 5 kV/cm [42.11],but in QWs the barriers inhibit the field ionisation,and excitonic features can be preserved even up to≈ 300 kV/cm [42.14]. The ability to control the ab-sorption spectrum by the QCSE is the principle behinda number of important modulator devices, which will bediscussed in Sect. 42.5.

In the case of a superlattice, such as that illustratedin Fig. 42.3, a strong perpendicular electric field canbreak the minibands into discrete energy levels local toeach QW, due to the band-gap tilting effect representedin Fig. 42.7. The possibility of using an electric field tomodify the minibands of a superlattice is yet anotherremarkable ability of band-gap engineering to achievecontrol over the electronic properties by directly usingfundamental principles of quantum mechanics.

42.2.4 Inter-Sub-Band Transitions

The engineered band structure of quantum wells leadsto the possibility of inter-sub-band (ISB) transitions,which take place between confined states within theconduction or valence bands, as illustrated schemati-cally in Fig. 42.9. The transitions typically occur in theinfrared spectral region. For example, the e1 ↔ e2 ISBtransition in a 10-nm GaAs/AlGaAs quantum well oc-curs at around 15 µm. For ISB absorption transitionswe must first dope the conduction band so that there isa large population of electrons in the e1 level, as shown

PartD

42.2


hω hωn = 2

n = 1Electrons

n = 2

n = 1

e–a) b)

Fig. 42.9 (a) Inter-sub-band absorption in an n-type quan-tum well. (b) Inter-sub-band emission following electroninjection to a confined upper level in the conduction band

in Fig. 42.9a. This is typically achieved by n-type dopingof the barriers, which produces a large electron density asthe extrinsic electrons drop from the barriers to the con-fined states in the QW. Undoped wells are used for ISBemission transitions, and electrons must first be injectedinto excited QW states, as shown in Fig. 42.9b.

The basic properties of ISB transitions can beunderstood by extension of the principles outlinedin Sect. 42.2.2. The main difference is that the envelopefunctions for the initial and final states are the same,since both states lie in the same band. The transition ratefor a conduction-band ISB transition then turns out tobe given by:

W ISB ∝ |〈ψen′ (z) |z|ψen(z)〉|2 g(�ω) , (42.22)

where n and n′ are the quantum numbers for the initialand final confined levels. The z operator within the Diracbracket arises from the electric-dipole interaction andindicates that the electric field of the light wave must beparallel to the growth direction. Furthermore, the oddparity of the z operator implies that the wave functionsmust have different parities, and hence that ∆n must bean odd number. The use of ISB transitions in infraredemitters and detectors will be discussed in Sects. 42.3.2,42.4.3 and 42.4.4.

42.2.5 Vertical Transport

Quantum WellsVertical transport refers to the processes by whichelectrons and holes move in the growth direction. Is-sues relating to vertical transport are important for theefficiency and frequency response of most QW opto-electronic devices. The transport is generally classifiedas either bipolar, when both electrons and holes areinvolved, or unipolar, when only one type of car-rier (usually electrons) is involved. In this section wewill concentrate primarily on bipolar transport in QWdetectors and QCSE modulators. Bipolar transport inlight-emitting devices is discussed in Sects. 42.2.6

and 42.3.1, while unipolar transport in quantum cascadelasers is discussed in Sect. 42.3.2.

In QW detectors and QCSE modulators the diodesare operated in reverse bias. This produces a strong DCelectric field and tilts the bands as shown in Fig. 42.7.Electrons and holes generated in the quantum wells byabsorption of photons can escape into the external cir-cuit by tunnelling and/or thermal emission, as illustratedschematically in Fig. 42.7.

The physics of tunnelling in quantum wells is essen-tially the same as that of α-decay in nuclear physics. Theconfined particle oscillates within the well and attemptsto escape every time it hits the barrier. The escape rate isproportional to the attempt frequency ν0 and the trans-mission probability of the barrier. For the simplest caseof a rectangular barrier of thickness b, the escape timeτT is given by:

1

τT= ν0 exp(−2κb) , (42.23)

where κ is the tunnelling decay constant givenby (42.10). The factor of 2 in the exponential arisesdue to the dependence of the transmission probabilityon |ψ(z)|2. The situation in a biased quantum well ismore complicated due to the non-rectangular shape ofthe barriers. However, (42.23) allows the basic trendsto be understood. To obtain fast tunnelling we need thinbarriers and small κ. The second requirement is achievedby keeping m∗

b as small as possible and by working witha small confining potential V0. The tunnelling rate in-creases with increasing field, because the average barrierheight decreases.

The thermal emission of electrons over a confiningpotential is an old problem which was originally appliedto the heated cathodes in vacuum tubes. It has beenshown that the thermal current fits well to the classicalRichardson formula:

JE ∝ T 1/2 exp

(− eΦ

kBT

), (42.24)

with the work function Φ replaced by [V (F)− En],V (F) being the height of the barrier that must be over-come at the field strength F [42.15]. The emission rate isdominated by the Boltzmann factor, which represents theprobability that the carriers have enough thermal kineticenergy to escape over the top of the barrier. At low fieldsV (F) ≈ (V0 − En), but as the field increases, V (F) de-creases as the barriers tilt over. Hence the emission rate(like the tunnelling rate) increases with increasing field.The only material-dependent parameter that enters theBoltzmann factor is the barrier height. Since this is in-sensitive both to the effective masses and to the barrier

PartD

42.2


hωhωn p

–

+

– – – – –

–

– –

+ + + +

+

+ +

Fig. 42.10 Schematic representation of the drift of injectedcarriers and their subsequent capture by quantum wells.Light emission occurs when electrons and holes are cap-tured in the same quantum well and then recombine witheach other

thickness, the thermal emission rate can dominate overthe tunnelling rate in some conditions, especially at roomtemperature in samples with thick barriers. For exam-ple, the fastest escape mechanism in GaAs/Al0.3Ga0.7AsQWs at room temperature can be the thermal emissionof holes, which have a much smaller barrier to overcomethan the electrons [42.16].

SuperlatticesThe artificial periodicity of superlattice structures givesrise to additional vertical transport effects related to thephenomenon of Bloch oscillations. It is well known thatan electron in a periodic structure is expect to oscil-late when a DC electric field is applied. This effect hasnever been observed in natural crystals because the os-cillation period – equal to h/eFa, where a is the unitcell size – is much longer than the scattering times ofthe electrons. In a superlattice, by contrast, the unit cellsize is equal to (d +b) (See Fig. 42.3) and the oscil-lation period can be made much shorter. This allowsthe electron to perform several oscillations before be-ing scattered. The oscillatory motion of the electronsin a superlattice was first observed by two groups in1992 [42.17, 18]. The following year, another group di-rectly detected the radiation emitted by the oscillatingelectron wave packet [42.19]. The subject has since de-veloped greatly, and THz-frequency emission has nowbeen achieved from GaAs/AlGaAs superlattices even atroom temperature [42.20].

42.2.6 Carrier Capture and Relaxation

In a QW light-emitting device, the emission occurs af-ter carriers injected from the contacts are transported tothe active region and then captured by the QWs. The

capture and subsequent relaxation of the carriers is thusof crucial importance. Let us consider the band-edgeprofile of a typical QW diode-laser active region, as il-lustrated in Fig. 42.10. The active region is embeddedbetween larger band-gap cladding layers designed toprevent thermally assisted carrier leakage outside the ac-tive region. Electrons and holes are injected from the n-and p-doped cladding layers under forward bias and lightemission follows after four distinct process have takenplace: (1) relaxation of carriers from the cladding layersto the confinement barriers (CB); (2) carrier transportacross the CB layers, by diffusion and drift; (3) carriercapture into the quantum wells; and (4) carrier relaxationto the fundamental confined levels.

The carrier relaxation to the CB layers occurs mainlyby longitudinal optical (LO) phonon emission. The CBlayer transport is governed by a classical electron fluidmodel. The holes are heavier and less mobile than theelectrons, and hence the ambipolar transport is dom-inated by the holes. Carrier nonuniformities, such ascarrier pile up at the p-side CB region due to the lowermobility of the holes, are taken into account in the de-sign of the barrier layers. The carrier capture in theQWs is governed by the phonon-scattering-limited car-rier mean free path. It is observed experimentally thatthe capture time oscillates with the QW width. Detailedmodelling reveals that this is related to a resonancebetween the LO phonon energy and the energy differ-ence between the barrier states and the confined stateswithin the well [42.21]. As another design guideline,the QW widths must be larger or at least equal to thephonon-scattering-limited carrier mean free path at theoperating temperature in order to speed up the carriercapture. Finally, the relaxation of carriers to the low-est sub-band occurs on a sub-picosecond time scaleif the inter-sub-band energy separation is larger thanthe LO phonon energy. Carrier–carrier scattering canalso contribute to an ultrafast thermalisation of carri-ers, on a femtosecond time scale at the high carrierdensities present inside laser diodes. Many of theseprocesses have been studied in detail by ultrafast laserspectroscopy [42.22].

Carrier capture and escape are complementary ver-tical transport mechanisms in MQW structures. In thedesign of vertical transport-based MQW devices, oneprocess must often be sped up at the expense of makingthe other as slow as possible. For example, in order toenhance the performance of QW laser diodes, the carrierconfinement capability of the MQW active region mustbe optimised in terms of minimising the ratio betweenthe carrier capture and escape times [42.23].

PartD

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42.3 Emitters

42.3.1 Interband Light-Emitting Diodesand Lasers

Quantum wells have found widespread use in light-emitting diode (LED) and laser diode applications fora number of years now. As discussed in Sects. 42.1.2and 42.2.2, there are three main reasons for this. Firstly,the ability to control the quantum-confinement energyprovides an extra degree of freedom to engineer theemission wavelength. Secondly, the change of the den-sity of states and the enhancement of the electron–holeoverlap leads to superior performance. Finally, the abil-ity to grow strained layers of high optical quality greatlyincreases the variety of material combinations that canbe employed, thus providing much greater flexibility inthe design of the active regions.

Much of the early work concentrated on lattice-matched combinations such as GaAs/AlGaAs on GaAssubstrates. GaAs/AlGaAs QW lasers operating around800 nm has now become industry-standard for applica-tions in laser printers and compact discs. Furthermore,the development of high-power arrays has opened upnew applications for pumping solid-state lasers such asNd : YAG. Other types of lattice-matched combinationscan be used to shift the wavelength into the visible spec-tral region and also further into the infrared. QWs basedon the quaternary alloy (AlyGa1−y)xIn1−xP, are used forred-emitting laser pointers [42.24], while Ga0.47In0.53AsQWs and its variants incorporating Al are used for theimportant telecommunication wavelengths of 1300 nmand 1550 nm.

The development of strained-layer QW lasers hasgreatly expanded the range of material combinationsthat are available. The initial work tended to focus onInxGa1−xAs/GaAs QWs grown on GaAs substrates.The incorporation of indium into the quantum wellshifts the band edge to lower energy, thereby givingemission in the wavelength range 900–1100 nm. An im-portant technological driving force has been the need forpowerful sources at 980 nm to pump erbium-doped fi-bre amplifiers [42.25]. Furthermore, as mentioned inSect. 42.2.1, the strain alters the band structure andthis can have other beneficial effects on the deviceperformance. For example, the compressive strain inthe InxGa1−xAs/GaAs QW system has been exploitedin greatly reducing the threshold current density. Thisproperty is related to the reduced effective mass of theholes and hence the reduced density of states. An exten-sive account of the effects of strain on semiconductor

layers and the performance of diode lasers is givenin [42.26].

At the other end of the spectral range, a spectac-ular development has been the InxGa1−xN/GaN QWsgrown on sapphire substrates. These highly strainedQWs are now routinely used in ultrabright blue andgreen LEDs, and there is a growing interest in de-veloping high-power LED sources for applications insolid-state lighting [42.27]. Commercial laser diodes op-erating around 400 nm have been available for severalyears [42.8], and high-power lasers suited to appli-cations in large-capacity optical disk video recordingsystems have been reported [42.28]. Lasers operatingout to 460 nm have been demonstrated [42.29], and alsohigh-efficiency ultraviolet light-emitting diodes [42.30].At the same time, much progress has been made in theapplication of AlGaN/GaN quantum-well materials inhigh-power microwave devices [42.31, 32].

A major application of quantum wells is in vertical-cavity surface-emitting lasers (VCSELs). These lasersemit from the top of the semiconductor surface, andhave several advantages over the more-conventionaledge-emitters: arrays are readily fabricated, which facil-itates their manufacture; no facets are required, whichavoids complicated processing procedures; the beam iscircular, which enhances the coupling efficiency into

Contact Contact

Output

n-GaAs substrate

Oxidized orimplanted

regions

QWs

p-DBR

n-DBR

Fig. 42.11 Schematic diagram of a vertical-cavity surface-emitting laser (VCSEL). The quantum wells (QWs) thatcomprise the gain medium are placed at the centre ofthe cavity formed between two distributed Bragg reflec-tor (DBR) mirrors. Oxidised or proton-implanted regionsprovide the lateral confinement for both the current and theoptical mode

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Quantum Wells, Superlattices, and Band-Gap Engineering 42.3 Emitters 1033

optical fibres; and their small size leads to very lowthreshold currents. For these reasons the developmentof VCSELs has been very rapid, and many local-areafibre networks operating around 850 nm currently em-ploy VCSEL devices. This would not have been possiblewithout the high gain coefficients that are inherent to theQW structures.

Figure 42.11 gives a schematic diagram of a typicalGaAs-based VCSEL. The device contains an active QWregion inserted between two distributed Bragg reflec-tor (DBR) mirrors consisting of AlGaAs quarter-wavestacks made of alternating high- and low-refractive-index layers. The structure is grown on an n-type GaAssubstrate, and the mirrors are appropriately doped n-or p-type to form a p–n junction. Electrons and holesare injected into the active region under forward bias,where they are captured by the QWs and produce gainat the lasing wavelength λ. The quantum wells are con-tained within a transparent layer of thickness λ/2n0,where n0 is the average refractive index of the activeregion. The light at the design wavelength is reflectedback and forth through the gain medium and adds upconstructively, forming a laser resonator. Oxidised orproton-implanted regions provide lateral confinement ofboth the current and the optical mode. Reviews of the de-sign and properties of VCSELs may be found in [42.34]and [42.35].

The conventional VCSEL structures grown onGaAs substrates operate in the wavelength range700–1100 nm [42.36]. Some of these structures arelattice-matched, but others – notably the longer-wavelength devices which incorporate strained InGaAsquantum wells – are not. Much work is currentlyfocussed on extending the range of operation tothe telecommunication wavelengths of 1300 nm and1550 nm. Unfortunately, it is hard to grow DBR mir-rors with sufficient reflectivity on InP substrates dueto the low refractive-index contrast of the materials,and thus progress has been slow. Recent alternative ap-proaches based on GaAs substrates will be mentionedin Sect. 42.6.

Resonator structures such as the VCSEL shownin Fig. 42.11 can be operated below threshold asresonant-cavity LEDs (RCLEDs). The presence of thecavity reduces the emission line width and hence in-creases the intensity at the peak wavelength [42.37].Furthermore, the narrower line width leads to an in-crease in the bandwidth of the fibre communicationsystem due to the reduced chromatic dispersion [42.38].A review of the progress in RCLEDs is givenin [42.39].

42.3.2 Quantum Cascade Lasers

The principles of infrared emission by ISB transitionswere described in Sect. 42.2.4. Electrons must first beinjected into an upper confined electron level as shownin Fig. 42.9b. Radiative transitions to lower confinedstates with different parities can then occur. ISB emis-sion is usually very weak, as the radiative transitionshave to compete with very rapid nonradiative decayby phonon emission, (Sect. 42.2.6). However, when theelectron density in the upper level is large enough,population inversion can occur, giving rapid stimulatedemission and subsequent laser operation. This is theoperating concept of the quantum cascade (QC) laserfirst demonstrated in 1994 [42.40]. The laser oper-ated at 4.2 µm at temperatures up to 90 K. Althoughthe threshold current for the original device was high,progress in the field had been very rapid. A compre-hensive review of the present state of the art is givenin [42.33], while a more introductory overview may befound in [42.41].

Activeregion

Injector

Activeregion

3

21

3

21

Miniband

Fig. 42.12 Conduction band diagram for two active regionsof an InGaAs/AlInAs quantum cascade laser, together withthe intermediate miniband injector region. The levels ineach active region are labelled according to their quantumnumber n, and the corresponding wave function probabilitydensities are indicated. Laser transitions are indicated bythe wavy arrows, while electron tunnelling processes areindicated by the straight arrows. (After [42.33], c© 2001IOP)

PartD

42.3


The quantum-well structures used in QC lasersare very complicated, and often contain hundredsof different layers. Figure 42.12 illustrates a rel-atively simple design based on lattice-matchedIn0.47Ga0.53As/Al0.48In0.52As quantum wells grown onan InP substrate. The diagram shows two active regionsand the miniband injector region that separates them.A typical operational laser might contain 20–30 suchrepeat units. The population inversion is achieved byresonant tunnelling between the n = 1 ground state ofone active region and the n = 3 upper laser level ofthe next one. The basic principles of this process wereenunciated as early as 1971 [42.42], but it took morethan 20 years to demonstrate the ideas in the labora-tory. The active regions contain asymmetric coupledquantum wells, and the laser transition takes place be-tween the n = 3 and n = 2 states of the coupled system.The separation of the n = 2 and n = 1 levels is care-fully designed to coincide with the LO-phonon energy,so that very rapid relaxation to the ground state oc-

curs and the system behaves as a four-level laser. Thislatter point is crucial, since the lifetime of the upperlaser level is very short (typically ≈ 1 ps), and popu-lation inversion is only possible when the lifetime ofthe lower laser level is shorter than that of the upperone. The lasing wavelength can be varied by detaileddesign of the coupled QW active region. The transitionenergy for the design shown in Fig. 42.12 is 0.207 eV,giving emission at 6.0 µm. Further details may be foundin [42.33].

A very interesting recent development has been thedemonstration of a QC laser operating in the far-infraredspectral region at 67 µm [42.43]. Previous work in thisspectral region had been hampered by high losses dueto free-carrier absorption and the difficulties involved indesigning the optical waveguides. The device operatedup to 50 K and delivered 2 mW. These long-wavelengthdevices are required for applications in the THz fre-quency range that bridges between long-wavelengthoptics and high-frequency electronics.

42.4 Detectors

Photodetectors for the visible and near-infrared spectralregions are generally made from bulk silicon or III–Valloys such as GaInAs. Since these devices work verywell, the main application for QW photodetectors is inthe infrared spectral region and for especially demandingapplications such as avalanche photodiodes and solarcells. These three applications are discussed separatelybelow, starting with solar cells.

42.4.1 Solar Cells

The power generated by a solar cell is determined by theproduct of the photocurrent and the voltage across thediode. In conventional solar cells, both of these parame-ters are determined by the band gap of the semiconductorused. Large photocurrents are favoured by narrow-gapmaterials, because semiconductors only absorb photonswith energies greater than the band gap, and narrow-gapmaterials therefore absorb a larger fraction of the solarspectrum. However, the largest open-circuit voltage thatcan be generated in a p–n device is the built-in voltagewhich increases with the band gap of the semiconduc-tor. Quantum-well devices can give better performancethan their bulk counterparts because they permit sepa-rate optimisation of the current- and voltage-generatingfactors [42.44]. This is because the built-in voltage is

primarily determined by the band gap of the barrier re-gions, whereas the absorption edge is determined bythe band gap of the quantum wells. The drawback inusing quantum wells is that it is difficult to maintainhigh photocurrent quantum efficiency in the low-fieldforward-bias operating conditions in solar cells.

Recent work in this field has explored the addedbenefits of the versatility of the design of the QW ac-tive region [42.45] and also the possibility of usingstrained QWs. In the latter case, a tradeoff arises be-tween the increase in both the absorption and the numberof interface dislocations (which act as carrier traps)with the number of QWs. A way round this compro-mise is to use strain balance. An example is the caseof InxGa1−xAs/GaAs0.94P0.06 QW solar cells grownon GaAs substrates, in which the compressive strainof the InGaAs QWs is compensated with the tensile-strained GaAs0.94P0.06 barriers, such that the overallactive region could be successfully lattice-matched tothe substrate [42.46].

42.4.2 Avalanche Photodiodes

Avalanche photodiodes (APDs) are the detectors ofchoice for many applications in telecommunicationsand single-photon counting. The avalanche multiplica-

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Quantum Wells, Superlattices, and Band-Gap Engineering 42.4 Detectors 1035

n-electrode

p-electrode

Passive layer

p-InP field control

InGaAsP / InAlAssuperlattice

p-InP

p-InGaAs

n-InP buffer

n-InP substrate

Incident light AR coating

Fig. 42.13 Schematic representation of an InGaAs/InP/InGaAsP/InAlAs superlattice avalanche photodiode (SL-APD). Light is absorbed in the bulk InGaAs layer and theresulting photocurrent is multiplied by the avalanche pro-cess in the InGaAsP/InAlAs superlattice region. The spatialperiodicity of the superlattice reduces the dark current

tion mechanism plays a critical role in determining thephotodetection gain, the noise and the gain–bandwidthproduct. Commercially available III–V semiconductorAPDs are typically engineered with different band-gap materials in the absorption and multiplicationregions [42.47]. The absorption layer has a relativelynarrow band gap (such as In0.53Ga0.47As) to allow forlarger absorption, whereas the multiplication region hasa wider band gap (such as InP or In0.52Al0.48As) to re-duce the dark current at the high electric fields required.It has been demonstrated that the dark current can befurther reduced by incorporating adequately designedsuperlattices into the multiplication layer to form a su-perlattice avalanche photodiode (SL-APD) as shownin Fig. 42.13 [42.48]. Essentially, the miniband forma-tion (Fig. 42.3) in the SL corresponds to a larger effectiveband gap and thus reduces the probability of band-to-band tunnelling at the high electric fields requiredfor carrier impact ionisation and subsequent carriermultiplication.

It has been proposed that the excess noise factorcan be reduced further by designing multiplication lay-ers with SL [42.49] or staircase [42.50] structures toenhance the ionisation rate of one carrier type rel-ative to the other due to their different band-edgediscontinuities [42.51]. However, this proposal is stillunder active theoretical and experimental scrutiny, be-cause most of the electrons populate the higher energysatellite valleys (X and L) at the high electric fields re-quired for avalanche gain [42.52]. Recent Monte Carlo

simulations [42.53], backed up by experimental evi-dence [42.54], have shown that the ratio of the electronand hole ionisation rates, commonly used as a figure ofmerit in bulk multiplication layers [42.55], is not crit-ically affected by the band-gap discontinuities. In fact,these recent models emphasise that it is the spatial mod-ulation of the impact ionisation probability, associatedwith the periodic band-gap discontinuity, which leads toa reduction in the multiplication noise.

42.4.3 Inter-Sub-Band Detectors

The principles of infrared absorption by ISB transitionswere described in Sect. 42.2.4. Infrared detectors arerequired for applications in defence, night vision, as-tronomy, thermal mapping, gas-sensing, and pollutionmonitoring. Quantum-well inter-sub-band photodetec-tors (QWIPs) are designed so that the energy separationof the confined levels is matched to the chosenwavelength. A major advantage of QWIPs over theconventional approach employing narrow-gap semicon-ductors is the use of mature GaAs-based technologies.Furthermore, the detection efficiency should in principlebe high due to the large oscillator strength that followsfrom the parallel curvature of the in-plane dispersionsfor states within the same bands [42.56]. However, tech-nical challenges arise from the requirement that theelectric field of the light must be polarised along thegrowth (z) direction. This means that QWIPs do notwork at normal incidence, unless special steps are takento introduce a component of the light field along thez-direction. Various approaches have been taken for op-timum light coupling, such as using bevelled edges,gratings or random reflectors [42.57].

Despite their promising characteristics, QWIPs haveyet to be commercialised. The main issue is the highdark current at higher operating temperatures. The darkcurrent is governed by the thermionic emission ofground-state electrons directly out of the QW above45 K [42.58]. Overcoming such technical difficulties hasmade possible the demonstration of long-wavelengthlarge-format focal-plane array cameras based on ISBtransitions [42.59].

42.4.4 Unipolar Avalanche Photodiodes

The combination of resonant inter-sub-band (ISB) pho-todetection (Sect. 42.4.3) and avalanche multiplicationhas been studied for exploiting the possibility of design-ing a unipolar avalanche photodiode (UAPD) [42.60].UAPDs rely on impact ionisation involving only one

PartD

42.4


hω

Ea

F

(1)

(1)(2)

Fig. 42.14 Unipolar carrier multiplication in a multiple QWstructure at field strength F: (1) is the primary electronresulting from photodetection, while (2) is the secondaryelectron resulting from the impact ionisation of the QW bythe primary electron

type of carrier in order to achieve gain in photocon-ductive detectors for mid- and far-infrared light. Theunipolar impact ionisation occurs when the kinetic en-ergy of the primary carrier exceeds an activation energyEa defined as the transition energy between the QWground state and the QW top state (Fig. 42.14). A sin-gle QW then releases an extra electron each time it issubject to an impact with an incoming electron.

The impact ionisation probability for this processis given by the product between the carrier captureprobability and the carrier escape probability under

a mechanism of carrier–carrier scattering in the QW.The subsequent electron transport towards further mul-tiplication events occurs through a sequence of escape,drift and kinetic energy gain under the applied electricfield, relaxation, capture and QW ionisation. Ultimately,the multiplication gain in an UAPD is governed by theQW capture probability, the number of QWs and thefield uniformity over the QW sequence. The unipolarnature of the multiplication process must be preservedin order to avoid field nonuniformities stemming fromspatial-charge variation caused by bipolar carrier trans-port across the multiplication region.

Interest in a purely unipolar multiplication mech-anism was originally motivated by the possibilityof reduced noise in comparison to bipolar APDs(Sect. 42.4.2), where band-to-band transitions lead togain fluctuations manifested as excess noise [42.55].For this purpose, the QWs in an UAPD structure aretypically tailored such that the inter-sub-band activa-tion energy Ea is smaller than the inter-band impactionisation activation energy that would be responsiblefor bipolar avalanche multiplication. However, recentstudies have shown that unipolar avalanche multipli-cation is also accompanied by an excess noise factor,such that the noise gain exceeds the photoconductivegain [42.61], thus limiting the practical applicationsof UAPDs.

42.5 Modulators

In Sect. 42.2.3 we noted that the optical properties ofquantum-well diodes are strongly modified by the appli-cation of voltages through the quantum-confined Starkeffect (QCSE). Referring to Fig. 42.8, we see that atwavelengths below the heavy-hole exciton at 0 V, theabsorption increases as the voltage is applied, whichprovides a mechanism for the modulation of light. Forexample, the amount of light transmitted at wavelengthsclose to the band edge would change with the voltageapplied. Moreover, since changes of absorption are ac-companied by changes of the refractive index throughthe Kramers–Kronig relationship, it is possible to makeQCSE phase modulators as well [42.62]. In additionto the standard GaAs/AlGaAs devices operating around800 nm, QCSE modulators have been demonstrated inseveral other material systems, such as GaInAs-basedstructures for the important telecommunications wave-length at 1.5 µm [42.63].

The operation of GaAs-based QCSE transmis-sion modulators at normal incidence is hampered

by the fact that the substrates are opaque at theiroperating wavelength. One way round this problem

–+

Laser

Proton implantationBragg grating

Modulatedoutput

CommonMQWs EAM

n-type substrate

Fig. 42.15 Schematic diagram of an integrated quantum-well waveguide electroabsorption modulator (EAM) anddistributed feedback (DFB) laser. The laser is forward-biased while the EAM is reverse-biased. The p-contactsof the two electrically independent devices are separatedby proton implantation. The light emitted by the laser isguided through the EAM region, resulting in a modulatedoutput beam when data pulses are applied to the EAM

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Quantum Wells, Superlattices, and Band-Gap Engineering 42.6 Future Directions 1037

is to etch the substrate away, but this is a dif-ficult process, and a much better solution is toinclude a mirror underneath the quantum wells so thatthe modulated light does not have to pass throughthe substrate [42.64]. In many practical applications,however, mostly involving the integration of QCSEmodulators with MQW light emitters on a commonsubstrate, the waveguide geometry is the configura-tion of choice [42.65]. In this architecture, the lightbeam propagates along the waveguide from the emit-ter to the electroabsorption modulator (EAM) region,as shown in Fig. 42.15. The QCSE modulator trans-mits the incoming laser light when no voltage is appliedand absorbs the beam when the MQW stack is suitablybiased.

The most successful commercial impact of QCSEmodulators has been in the integration of EAMs with dis-tributed feedback (DFB) or distributed Bragg reflector(DBR) MQW diode lasers in waveguide configurations,as shown in Fig. 42.15. These devices have been usedfor optical coding in the C-band (1525–1565 nm), at10 Gb/s or higher data transmission speeds [42.66]. The

combination of a continuous laser and a high-speedmodulator offers better control of the phase chirp ofthe pulses than direct modulation of the laser output it-self [42.67]. In particular, the chirp factor is expectedto become negative if the photogenerated carriers canbe swept out fast enough in the EAM, which is desir-able for long-distance data transmission through opticalfibers [42.68].

A promising step toward the merger betweenband-gap-engineered semiconductors and mature very-large-scale integration (VLSI) silicon architectures hasbeen achieved when III–V semiconductor QCSE mod-ulator structures have been integrated with state-of-the-art silicon complementary metal–oxide–semiconductor(CMOS) circuitry [42.69]. Through this hybrid tech-nology, thousands of optical inputs and outputs couldbe provided to circuitry capable of very complex in-formation processing. The idea of using light beamsto replace wires in telecommunications and digi-tal computer systems has thus become an attractivetechnological avenue in spite of various challengesimplied [42.70].

42.6 Future Directions

The subject of quantum-confined semiconductor struc-tures moves very rapidly and it is difficult to see far intothe future. Some ideas have moved very quickly from theresearch labs into the commercial sector (e.g. VCSELs),while others (e.g. quantum cascade lasers) have takenmany years to come to fruition. We thus restrict our-selves here to a few comments based on active researchfields at the time of writing.

One idea that is being explored in detail is theeffects of lower dimensionality in quantum-wire andquantum-dot structures (Table 42.1). Laser operationfrom one-dimensional (1-D) GaAs quantum wires wasfirst demonstrated in 1989 [42.72], but subsequentprogress has been relatively slow due to the difficultyin making the structures. By contrast, there has been anexplosion of interest in zero-dimensional (0-D) struc-tures following the discovery that quantum dots canform spontaneously during MBE growth in the Stranski–Krastanow regime. A comprehensive review of thissubject is given in [42.73].

Figure 42.16 shows an electron microscope imageof an InAs quantum dot grown on a GaAs crystal bythe Stranski–Krastanow technique. The dots are formedbecause of the very large mismatch between the lat-tice constants of the InAs and the underlying GaAs.

The strain that would be produced in a uniform layeris so large that it is energetically favourable to formsmall clusters. This then leads to the formation of is-lands of InAs with nanoscale dimensions, which canthen be encapsulated within an optoelectronic structureby overgrowth of further GaAs layers.

The ability to grow quantum-dot structures directlyby MBE has led to very rapid progress in the deploy-ment of quantum dots in a variety of applications. Itremains unclear at present whether quantum dots reallylead to superior laser performance [42.74]. The intrin-

InAs quantum dot

10 nm GaAs

Fig. 42.16 Transmission electron microscope image of anuncapped InAs quantum dot grown on GaAs by theStranski–Krastanow technique. (After [42.71], c© 2000APS)

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42.6


sic gain of the dots is higher than that of a quantumwell [42.75], and the threshold current is less sensitiveto temperature [42.76]. However, the volume of the gainmedium is necessarily rather small, and the benefits ofthe lower dimensionality cannot be exploited to the full.At present, one of the most promising applications forquantum dots is in long-wavelength lasers [42.77]. Asmentioned in Sect. 42.3.1, the production of VCSELs at1300 nm and 1550 nm has proven to be difficult usingconventional InP-based QW structures due to the lowrefractive-index contrast of the materials that form theDBR mirrors. The use of InAs/GaAs quantum dots asthe active region circumvents this problem and allowsthe benefits of mature GaAs-based VCSEL technology.

Another very exciting potential application forquantum dots is in quantum information processing.High-efficiency single-photon sources are required forquantum cryptography and also quantum computationusing linear optics. Several groups have demonstrated

single photon emission after excitation of individualInAs quantum dots (see e.g. [42.78, 79]), and onegroup has demonstrated an electrically driven single-photon LED [42.80]. After these proofs of principle,the challenge now lies ahead to establish the quantum-dot sources in working quantum information-processingsystems.

At the same time as exploring the effects of lowerdimensionality, many other groups are working on newQW materials. One of the most promising recent devel-opments is the dilute nitride system for applications inlong-wavelength VCSELs and solar cells [42.81]. It hasbeen found that the inclusion of a small fraction of nitro-gen into GaAs leads to a sharp decrease in the band gapdue to very strong band-bowing effects. This then allowsthe growth of InGaAsN structures that emit at 1300 nmon GaAs substrates [42.77, 82]. The field is developingvery rapidly, with 1300-nm VCSELs and 1500-nm edgeemitters already demonstrated [42.83, 84].

42.7 Conclusions

Semiconductor quantum wells are excellent examples ofquantum mechanics in action. The reduced dimension-ality has led to major advances in both the understandingof 2-D physics and the applied science of optoelectron-ics. In some cases, QWs have enhanced the performanceof conventional devices (e.g. LEDs and edge-emittinglasers), and in others, they have led to radically newdevices (e.g. VCSELs, quantum cascade lasers, QCSE

modulators). At present, the main commercial use forQW optoelectronic devices is in LEDs, laser diodes andQCSE modulators. It remains to be seen whether someof the other devices described here (QW solar cells, SL-APDs, QWIPs) will come to commercial fruition, andwhether systems of lower dimensionality will eventuallyreplace QWs in the same way that QWs have replacedbulk devices.

References

42.1 L. Esaki, R. Tsu: IBM J. Res. Develop. 14, 61–5 (1970)42.2 G. Bastard: Wave Mechanics Applied to Semi-

conductor Heterostructures (Wiley, New York1988)

42.3 M. Jaros: Physics and Applications of Semiconduc-tor Microstructures (Clarendon, Oxford 1989)

42.4 C. Weisbuch, B. Vinter: Quantum SemiconductorStructures (Academic, San Diego 1991)

42.5 S. O. Kasap: Optoelectronics and Photonics: Prin-ciples and Practices (Prentice Hall, Upper SaddleRiver 2001)

42.6 M. J. Kelly: Low-Dimensional Semiconductors(Clarendon, Oxford 1995)

42.7 Paul J. Dean: III–V Compound Semiconductors. In:Electroluminescence, ed. by J. I. Pankove (Springer,Berlin, Heidelberg 1977) pp. 63–132

42.8 S. Nakamura, S. Pearton, G. Fasol: The Blue LaserDiode, 2nd edn. (Springer, Berlin, Heidelberg 2000)

42.9 S. Gasiorowicz: Quantum Physics, 2nd edn. (Wiley,New York 1996)

42.10 E. P. O’Reilly: Semicond. Sci. Technol. 4, 121–137(1989)

42.11 M. Fox: Optical Properties of Solids (Clarendon, Ox-ford 2001)

42.12 M. Shinada, S. Sugano: J. Phys. Soc. Jpn. 21, 1936–46 (1966)

42.13 D. A. B. Miller, D. S. Chemla, D. J. Eilenberger,P. W. Smith, A. C. Gossard, W. T. Tsang: Appl. Phys.Lett. 41, 679–81 (1982)

42.14 A. M. Fox, D. A. B. Miller, G. Livescu, J. E. Cun-ningham, W. Y. Jan: IEEE J. Quantum Electron. 27,2281–95 (1991)

PartD

42

Quantum Wells, Superlattices, and Band-Gap Engineering References 1039

42.15 H. Schneider, K. von Klitzing: Phys. Rev. B 38,6160–5 (1988)

42.16 A. M. Fox, R. G. Ispasoiu, C. T. Foxon, J. E. Cun-ningham, W. Y. Jan: Appl. Phys. Lett. 63, 2917–9(1993)

42.17 J. Feldmann, K. Leo, J. Shah, D. A. B. Miller,J. E. Cunningham, T. Meier, G. von Plessen,A. Schulze, P. Thomas, S. Schmitt-Rink: Phys. Rev.B 46, 7252–5 (1992)

42.18 K. Leo, P. Haring Bolivar, F. Brüuggemann,R. Schwedler, K. Köhler: Solid State Commun. 84,943–6 (1992)

42.19 C. Waschke, H. G. Roskos, R. Schwedler, K. Leo,H. Kurz, K. Köhler: Phys. Rev. Lett. 70, 3319–22(1993)

42.20 Y. Shimada, K. Hirakawa, S.-W. Lee: Appl. Phys.Lett. 81, 1642–4 (2002)

42.21 P. W. M. Blom, C. Smit, J. E. M. Haverkort,J. H. Wolter: Phys. Rev. B 47, 2072–2081 (1993)

42.22 J. Shah: Ultrafast Spectroscopy of Semiconduc-tors and Semiconductor Nanostructures, 2nd edn.(Springer, Berlin, Heidelberg 1999)

42.23 R. G. Ispasoiu, A. M. Fox, D. Botez: IEEE J. QuantumElectron. 36, 858–63 (2000)

42.24 P. Blood: Visible-emitting quantum well lasers.In: Semiconductor Quantum Optoelectronics, ed. byA. Miller, M. Ebrahimzadeh, D. M. Finlayson (Insti-tute of Physics, Bristol 1999) pp. 193–211

42.25 N. Chand, S. N. G. Chu, N. K. Dutta, J. Lopata,M. Geva, A. V. Syrbu, A. Z. Mereutza, V. P. Yakovlev:IEEE J. Quantum Electron. 30, 424–40 (1994)

42.26 E. P. O’Reilly, A. R. Adams: IEEE J. Quantum Electron.30, 366–79 (1994)

42.27 M. R. Krames, J. Bhat, D. Collins, N. F. Gargner,W. Götz, C. H. Lowery, M. Ludowise, P. S. Mar-tin, G. Mueller, R. Mueller-Mach, S. Rudaz,D. A. Steigerwald, S. A. Stockman, J. J. Wierer: Phys.Stat. Sol. A 192, 237–245 (2002)

42.28 M. Ikeda, S. Uchida: Phys. Stat. Sol. A 194, 407–13(2002)

42.29 S. Nagahama, T. Yanamoto, M. Sano, T. Mukai:Phys. Stat. Sol. A 194, 423–7 (2002)

42.30 S. Kamiyama, M. Iwaya, H. Amano, I. Akasaki: Phys.Stat. Sol. A 194, 393–8 (2002)

42.31 L. F. Eastman, V. Tilak, V. Kaper, J. Smart, R. Thomp-son, B. Green, J. R. Shealy, T. Prunty: Phys. Stat. Sol.A 194, 433–8 (2002)

42.32 W. S. Tan, P. A. Houston, P. J. Parbrook, G. Hill,R. J. Airey: J. Phys. D: Appl. Phys. 35, 595–8 (2002)

42.33 C. Gmachl, F. Capasso, D. L. Sivco, A. Y. Cho: Rep.Prog. Phys. 64, 1533–1601 (2001)

42.34 K. J. Eberling: Analysis of vertical cavity surfaceemitting laser diodes (VCSEL). In: Semiconduc-tor Quantum Optoelectronics, ed. by A. Miller,M. Ebrahimzadeh, D. M. Finlayson (Institute ofPhysics, Bristol 1999) pp. 295–338

42.35 M. SanMiguel: Polarisation properties of verticalcavity surface emitting lasers. In: Semiconduc-

tor Quantum Optoelectronics, ed. by A. Miller,M. Ebrahimzadeh, D. M. Finlayson (Institute ofPhysics, Bristol 1999) pp. 339–366

42.36 O. Blum Spahn: Materials issues for vertical cavitysurface emitting lasers (VCSEL) and edge emittinglasers (EEL). In: Semiconductor Quantum Opto-electronics, ed. by A. Miller, M. Ebrahimzadeh,D. M. Finlayson (Institute of Physics, Bristol 1999)pp. 265–94

42.37 E. F. Schubert, Y.-H. Wang, A. Y. Cho, L.-W. Tu,G. J. Zydzik: Appl. Phys. Lett. 60, 921–3 (1992)

42.38 N. E. Hunt, E. F. Schubert, R. F. Kopf, D. L. Sivco,A. Y. Cho, G. J. Zydzik: Appl. Phys. Lett. 63, 2600–2(1993)

42.39 R. Baets: Micro-cavity light emitting diodes. In:Semiconductor Quantum Optoelectronics, ed. byA. Miller, M. Ebrahimzadeh, D. M. Finlayson (In-stitute of Physics, Bristol 1999) pp. 213–64

42.40 J. Faist, F. Capasso, D. L. Sivco, C. Sirtori,A. L. Hutchinson, A. Y. Cho: Science 264, 553–6(1994)

42.41 F. Capasso, C. Gmachl, D. L. Sivco, A. Y. Cho: Phys.Today 55(5), 34–40 (2002)

42.42 R. A. Kazarinov: Sov. Phys. Semicond. 5, 707–9(1971)

42.43 R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere,E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti,F. Rossi: Nature 417, 156–9 (2002)

42.44 K. Barnham, I. Ballard, J. Barnes, J. Connolly,P. Griffin, B. Kluftinger, J. Nelson, E. Tsui,A. Zachariou: Appl. Surf. Sci. 113/114, 722–733(1997)

42.45 R. H. Morf: Physica E 14, 78–83 (2002)42.46 N. J. Ekins-Daukes, K. W. J. Barnham, J. P. Connolly,

J. S. Roberts, J. C. Clark, G. Hill, M. Mazzer: Appl.Phys. Lett. 75, 4195–5197 (1999)

42.47 J. Wei, J. C. Dries, H. Wang, M. L. Lange, G. H. Olsen,S. R. Forrest: IEEE Photon. Technol. Lett. 14, 977–9(2002)

42.48 A. Suzuki, A. Yamada, T. Yokotsuka, K. Idota,Y. Ohiki: Jpn. J. Appl. Phys. 41, 1182–5 (2002)

42.49 F. Capasso, W. T. Tsang, A. L. Hutchinson, G. F. Wil-liams: Appl. Phys. Lett. 40, 38–40 (1982)

42.50 G. Ripamonti, F. Capasso, A. L. Hutchinson,D. J. Muehlner, J. F. Walker, R. J. Malek: Nucl.Instrum. Meth. Phys. Res. A 288, 99–103(1990)

42.51 R. Chin, N. Holonyak, G. E. Stillman, J. Y. Tang,K. Hess: Electron. Lett. 16, 467–9 (1980)

42.52 C. K. Chia, J. P. R. David, G. J. Rees, P. N. Robson,S. A. Plimmer, R. Grey: Appl. Phys. Lett. 71, 3877–9(1997)

42.53 F. Ma, X. Li, S. Wang, K. A. Anselm, X. G. Zheng,A. L. Holmes, J. C. Campbell: J. Appl. Phys. 92, 4791–5 (2002)

42.54 P. Yuan, S. Wang, X. Sun, X. G. Zheng, A. L. Holmes,J. C. Campbell: IEEE Photon. Technol. Lett. 12, 1370–2(2000)

PartD

42


42.55 M. A. Saleh, M. M. Hayat, P. P. Sotirelis, A. L. Holmes,J. C. Campbell, B. E. A. Saleh, M. C. Teich: IEEE Trans.Electron. Dev. 48, 2722–31 (2001)

42.56 L. C. West, S. J. Eglash: Appl. Phys. Lett. 46, 1156–8(1985)

42.57 S. D. Gunapala, G. Sarusi, J. S. Park, T. Lin,B. F. Levine: Phys. World 7(10), 35–40 (1994)

42.58 S. D. Gunapala, S. V. Bandara: Quantum well in-frared photodetector (QWIP) focal plane arrays. In:Semiconductors and Semimetals, Vol. 62, ed. byM. C. Liu, F. Capasso (Academic, New York 1999)pp. 197–282

42.59 S. D. Gunapala, S. V. Bandara, J. K. Liu, E. M. Lu-ong, N. Stetson, C. A. Shott, J. J. Block, S. B. Rafol,J. M. Mumolo, M. J. McKelvey: IEEE Trans. Electron.Dev. 47, 326–332 (2000)

42.60 B. F. Levine, K. K. Choi, C. G. Bethea, J. Walker,R. J. Malik: Appl. Phys. Lett. 51, 934–6 (1987)

42.61 H. Schneider: Appl. Phys. Lett. 82, 4376–8 (2003)42.62 J. S. Weiner, D. A. B. Miller, D. S. Chemla: Appl. Phys.

Lett. 50, 842–4 (1987)42.63 R. W. Martin, S. L. Wong, R. J. Nicholas, K. Satzke,

M. Gibbons, E. J. Thrush: Semicond. Sci. Technol. 8,1173–8 (1993)

42.64 G. D. Boyd, D. A. B. Miller, D. S. Chemla, S. L. McCall,A. C. Gossard, J. H. English: Appl. Phys. Lett. 50,1119–21 (1987)

42.65 A. Ramdane, F. Devaux, N. Souli, D. Delprat,A. Ougazzaden: IEEE J. Quantum Electron. 2, 326–35(1996)

42.66 T. Ido, S. Tanaka, M. Suzuki, M. Koizumi, H. Sano,H. Inoue: J. Lightwave Technol. 14, 2026–33(1996)

42.67 Y. Miyazaki, H. Tada, T. Aoyagi, T. Nishimura,Y. Mitsui: IEEE J. Quantum Electron. 38, 1075–80(2002)

42.68 G. Agrawal: Fiber Optic Communication Systems(Wiley, New York 1993)

42.69 K. W. Goosen, J. A. Walker, L. A. D’Asaro, S. P. Hui,B. Tseng, R. Leibenguth, D. Kossives, D. D. Ba-con, D. Dahringer, L. M. F. Chirovsky, A. L. Lentine,

D. A. B. Miller: IEEE Photon. Technol. Lett. 7, 360–2(1995)

42.70 D. A. B. Miller: IEEE J. Sel. Top. Quantum Electron.6, 1312–7 (2000)

42.71 P. W. Fry, I. E. Itskevich, D. J. Mowbray, M. S. Skol-nick, J. J. Finley, J. A. Barker, E. P. O’Reilly,L. R. Wilson, I. A. Larkin, P. A. Maksym, M. Hop-kinson, M. Al-Khafaji, J. P. R. David, A. G. Cullis,G. Hill, J. C. Clark: Phys. Rev. Lett. 84, 733–6(2000)

42.72 E. Kapon, D. M. Hwang, R. Bhat: Phys. Rev. Lett.63, 430–3 (1989)

42.73 D. Bimberg, M. Grundmann, Nikolai N. Ledentsov:Quantum Dot Heterostructures (Wiley, Chichester1998)

42.74 M. Grundmann: Physica E 5, 167–84 (2000)

42.75 M. Asada, Y. Miyamoto, Y. Suematsu: IEEE J. Quan-tum Electron. 22, 1915–21 (1986)

42.76 Y. Arakawa, H. Sakaki: Appl. Phys. Lett. 40, 939–41(1982)

42.77 V. M. Ustinov, A. E. Zhukov: Semicond. Sci. Technol.15, R41–R54 (2000)

42.78 P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld,P. M. Petroff, L. Zhang, E. Hu, A. Imamoglu: Science290, 2282–5 (2000)

42.79 C. Santori, M. Pelton, G. Solomon, Y. Dale, Y. Ya-mamoto: Phys. Rev. Lett. 86, 1502–5 (2001)

42.80 Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. Shields,C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie,M. Pepper: Science 295, 102–5 (2002)

42.81 A. Mascarenhas, Y. Zhang: Current Opinion SolidState Mater. Sci. 5, 253–9 (2001)

42.82 A. Yu. Egorov, D. Bernklau, B. Borchert, S. Illek,D. Livshits, A. Rucki, M. Schuster, A. Kaschner,A. Hoffmann, Gh. Dumitras, M. C. Amann,H. Riechert: J. Cryst. Growth 227–8, 545–552(2001)

42.83 G. Steinle, H. Riechert, A. Yu. Egorov: Electron. Lett.37, 93–5 (2001)

42.84 D. Gollub, M. Fischer, A. Forchel: Electron. Lett. 38,1183–4 (2002)

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