Coherent phenomena in semiconductors

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Coherent phenomena in semiconductors

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1998 Semicond. Sci. Technol. 13 147

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Semicond. Sci. Technol. 13 (1998) 147–168. Printed in the UK PII: S0268-1242(98)78411-1

TOPICAL REVIEW

Coherent phenomena insemiconductors

Fausto Rossi †

Istituto Nazionale per la Fisica della Materia (INFM) and Dipartimento di Fisica,Universita di Modena, via Campi 213/A, I-41100 Modena, Italy

Received 2 April 1997

Abstract. A review of coherent phenomena in photoexcited semiconductors ispresented. In particular, two classes of phenomena are considered. The first isconcerned with the role played by optically induced phase coherence in theultrafast spectroscopy of semiconductors; the other with the Coulomb-inducedeffects on the coherent optical response of low-dimensional structures.

All the phenomena discussed in this review are analysed in terms of atheoretical framework based on the density-matrix formalism. Due to its generality,this quantum-kinetic approach allows a realistic description of coherent as well asincoherent, i.e. phase-breaking, processes, thus providing quantitative informationon the coupled—coherent versus incoherent—carrier dynamics in photoexcitedsemiconductors.

The primary goal of this review is to discuss the concept ofquantum-mechanical phase coherence as well as its relevance to and implicationsfor semiconductor physics and technology. In particular, we will discuss thedominant role played by optically induced phase coherence on the process ofcarrier photogeneration and relaxation in bulk systems. We will then review typicalfield-induced coherent phenomena in semiconductor superlattices such as Blochoscillations and Wannier–Stark localization. Finally, we will discuss the dominantrole played by Coulomb correlation on the linear and nonlinear optical spectra ofrealistic quantum-wire structures.

1. Introduction

The resonant excitation by coherent optical radiationof an electronic transition in a semiconductor, e.g. avalence to conduction band excitation, creates a quantum-mechanical coherent superposition of the initial and finalstates of the transition, called optical polarization. Thenonlinear optical properties of this coherent superposition,together with its time evolution, can be used to provide asensitive measurement of many fundamental parameters ofa semiconductor material [1] including elastic and inelasticscattering, energy level splittings between nearly degeneratestates, energy relaxation, as well as associated informationsuch as Lande g-factors and band-mixing.

In general, nonlinear laser spectroscopy is a veryestablished and well understood means of probing thesephenomena [2–9]. However, the description of thenonlinear optical response of a semiconductor crystal canbe considerably more complex than for simple isolatedand noninteracting atoms [8]. This is particularly truefor the case of the ultrafast optical spectroscopy used forthe study of the sub-picosecond dynamics of photoexcitedcarriers in bulk systems as well as in semiconductor

† E-mail: [email protected]

heterostructures [1, 10, 11].The lifetime of the coherent quantum-mechanical

superposition generated by an ultrafast laser excitation,called ‘dephasing time’, determines the typical time-scale on which coherent phenomena can be observed.Such dephasing time reflects the role played by thevarious ‘incoherent’, i.e. phase-breaking, mechanisms indestroying the phase coherence induced by the laserphotoexcitation. Since semiconductors are characterizedby very short electron–hole dephasing times, of theorder of few hundreds of femtoseconds [8], coherentphenomena manifest themselves only through ultrafastoptical experiments with sub-picosecond time resolutions[1]. On this time-scale, the ultrafast evolution ofphotoexcited electron–hole pairs will reflect the strongcoupling between coherent and incoherent dynamics, thusproviding invaluable information on the non-equilibriumrelaxation processes occurring in the semiconductor, e.g.carrier–carrier and carrier–phonon scattering.

The aim of this article is to review the basic aspectsrelated to coherent phenomena in semiconductors. Inparticular, we will focus on the ultrafast—coherent versusincoherent—carrier dynamics as well as on Coulomb-correlation effects in photoexcited semiconductors.

0268-1242/98/020147+22$19.50 c© 1998 IOP Publishing Ltd 147

F Rossi

The review is organized as follows. In the remainderof this section, after a brief historical account of coherentexperiments in solids, we will try to gain more insightinto the concept of coherence by introducing a simplifieddescription of the light–matter interaction in terms ofa two-level model. In section 2 we will discussthe theoretical approach commonly used for a realisticdescription of both coherent and incoherent phenomenain various semiconductor structures, e.g. bulk systems,semiconductor superlattices, quantum wells and wires.Section 3 deals with ultrafast carrier photoexcitation andrelaxation in bulk semiconductors; in particular, we willdiscuss the dominant role played by coherence in the carrierphotogeneration process. In section 4 we will review anddiscuss typical field-induced phenomena in superlattices,namely Bloch oscillations and Wannier–Stark localization,as well as their dephasing dynamics. Section 5 is devoted tothe analysis of the coherent optical response of quasi-one-dimensional systems; more specifically, we will discuss thestrong modifications induced by Coulomb correlation on thelinear and nonlinear optical spectra of realistic quantum-wire structures. Finally, in section 6 we will summarizeand draw some conclusions.

1.1. Historical background

Coherent phenomena in atomic and molecular systems havebeen investigated for a long time [2]. The first spin echoexperiment [12] was performed in 1950 on protons in awater solution of Fe+++ ions. Pulses in the radiofrequencyrange were generated by means of a gated oscillator withpulse widths between 20µs and a few milliseconds. Withthese pulses, dephasing times of the order of 10 ms havebeen measured.

In the 1960s echo experiments were brought into thevisible range [13, 14]. A Q-switched ruby laser producedpulses of approximately 10 ns duration which were usedto observe photon echoes from ruby. In this case thedephasing times were of the order of 100 ns.

As already pointed out, for the observation of anycoherent dynamics the pulse width has to be shorter than thetypical dephasing time. Since in semiconductors electron–hole dephasing times are much shorter—they are in therange of a few picoseconds down to some femtoseconds—coherent experiments in semiconductors had to wait untilthe development of suitable lasers able to generate sub-picosecond pulses.

1.2. Ultrafast spectroscopy in semiconductors

The physical phenomena governing the ultrafast carrierdynamics in photoexcited semiconductors can be dividedinto two classes: coherent phenomena, i.e. phenomenarelated to the quantum-mechanical phase coherence inducedby the laser photoexcitation, andincoherent phenomena,i.e. phenomena induced by the various phase-breakingscattering mechanisms. The above classification in terms ofcoherent and incoherent phenomena is not purely academic;It corresponds to rather different experimental techniquesfor the investigation of these two different regimes.

From an historical point of view, optical spectroscopyin semiconductors started with the analysis of relativelyslow incoherent phenomena (compared with the electron–hole dephasing time-scale). The investigation ofnonequilibrium carriers started with the analysis of theincoherent energy-relaxation processes in the late 1960susing continuous wave (cw) lasers [15]. In the 1970spulse sources for the study of photoexcited carriers becameavailable [16] and this initiated time-resolved studies ofthe energy-relaxation process. Many experiments basedon different techniques have been performed [1]: band-to-band luminescence [17, 18] which monitors the product ofelectron and hole distribution functions, band-to-acceptorluminescence [19–21] which provides information on theelectron distribution functions only, and pump-and-probemeasurements [22–26] where the measured differentialtransmission is proportional to the sum of electron and holedistribution functions.

The theoretical analysis of these relaxation phenomenais commonly based on the semiclassical Boltzmann theory.The Boltzmann equations for both electron and holedistribution functions are commonly solved by means ofsemiclassical Monte Carlo simulations [27–29].

In addition to the analysis of incoherent energy-relaxation processes, ultrafast optical spectroscopy hasallowed the investigation of coherent phenomena. Asalready pointed out, a coherent laser field creates, inaddition to a non-equilibrium carrier distribution, a coherentpolarization. The investigation of the coherent dynamicsin semiconductors started in the 1980s. Different aspectshave been investigated [1]: the optical Stark effect [30–33], the dephasing of free carriers [34] and excitons[35–37], quantum beats related to various types of levelsplittings [38–41], charge oscillations in double-quantum-well systems [42, 43] and superlattices [44], many-particleeffects [45–48], and the emission of coherent terahertz(THz) radiation [49, 50].

These experiments cannot be analysed within theframework of the Boltzmann equation. The reasonis that this coherent polarization reflects a well-definedphase relation between electrons and holes, which isneglected within the semiclassical Boltzmann theory.Any proper description requires a quantum-mechanicaltreatment where, in addition to the distribution functionsof electrons and holes, the interband polarization is alsotaken into account as an independent variable. Severalapproaches have been used in the literature [8]: Bogoliubovtransformations [51], nonequilibrium Green’s functions[52–56], band-edge equations based on the real-spacedensity matrix [57], and the density matrix formalism inmomentum space [58–64].

During the last decade the time resolution has beenfurther improved down to a few tens of femtoseconds [65].On such an extremely short time-scale, coherent effects canno longer be neglected and are found to play a dominantrole also for the case of typical incoherent measurementssuch as time-resolved and time-integrated luminescence.In such conditions, the carrier dynamics is the resultof a strong interplay between coherent and incoherent

148


phenomena. Therefore, the traditional separation† betweencoherent and incoherent approaches for the theoreticalinvestigation of photoexcited semiconductors is no longervalid. What is needed is a comprehensive theoreticalframework able to describe on the same kinetic level bothclasses of phenomena as well as their mutual coupling. Tothis purpose, a generalized Monte Carlo method for theanalysis of both coherent and incoherent phenomena hasbeen recently proposed [66–70]. The spirit of the methodis to combine the advantages of the conventional MonteCarlo approach [27–29] in treating the incoherent, i.e.phase-breaking, dynamics with the strength of a quantum-kinetic approach in describing coherent phenomena [8].In particular, the coherent contributions are evaluated bymeans of a direct numerical integration while the incoherentones are ‘sampled’ by means of a conventional Monte Carlosimulation in three-dimensional (3D)k space.

This theoretical approach has ben applied successfullyto the analysis of various ultrafast optical experiments,e.g. four-wave-mixing studies of many-body effects in bulkGaAs [71, 72] and luminescence studies of the hot-carrierphotogeneration process [73–75].

1.3. The meaning of coherence

In order to clarify the concept of phase coherence, let usconsider the simplest physical model for the description oflight–matter interaction, i.e. an optically driven two-levelsystem [1, 2]. Within a two-level picture, a ground statea with energyεa and an excited stateb with energyεbare mutually coupled by a driving force, e.g. an externalfield, and/or by their mutual interaction, e.g. Coulombcorrelation. The two-level-system Hamiltonian

H =H0+H ′ (1)

is the sum of a free-level contribution

H0 = εaa†aaa + εba†bab (2)

and of a coupling term

H ′ = Ubaa†baa + Uaba†aab. (3)

Here, the usual second-quantization picture in terms ofcreation (a†) and destruction (a) operators has beenintroduced [8]. The two terms forming the couplingHamiltonianH ′ will induce transitions from statea to b andvice versa according to the coupling constantUba = U ∗ab.

As a starting point, let us consider a single-electronsystem. In the absence of interlevel coupling (Uab = 0),we have two stationary states

|a(t)〉 = exp

(− iεat

h

)a†a|0〉

|b(t)〉 = exp

(− iεbt

h

)a†b|0〉

(4)

† From an historical point of view, the theoretical approachescommonly used for the interpretation of coherent phenomena providea rather qualitative description of phase-breaking processes in terms ofphenomenological energy-relaxation and dephasing rates. On the otherhand, incoherent phenomena were traditionally investigated in terms ofsemiclassical Monte Carlo simulations, which provide a microscopicdescription of the non-equilibrium scattering dynamics.

(|0〉 being the vacuum state), corresponding to a singleelectron in levela or b respectively. In contrast, in thepresence of interlevel coupling, the state of the system is,in general, a linear superposition of the noninteracting statesin equation (4)

|ψ(t)〉 = ca(t)|a(t)〉 + cb(t)|b(t)〉 (5)

whose coefficients obey the following equations of motion:

d

dtca = Uab

ihexp

(i(εa − εb)t

h

)cb

d

dtcb = Uba

ihexp

(i(εb − εa)t

h

)ca.

(6)

Again, we see that in the absence of interlevel coupling(U = 0) there is no time variation of the coefficients, i.e.if the system is prepared in state|a〉 or |b〉 it will remainin such an eigenstate. In contrast, the interlevel couplinginduces a time variation of the coefficients.

The above two-level model provides the simplestdescription of light–matter interaction in atomic andmolecular systems [2] as well as in solids [8]. The couplingterm in equation (3) has the same structure of a field–dipoleinteraction Hamiltonian and the driving forceu can beregarded as a coherent light field. We will now consider twolimiting cases: the ultrashort- and continuous-excitationregimes.

An ultrashort optical excitation can be described interms of a delta-like light pulse:Uab(t) = ηδ(t). In thiscase, the equations of motion (6) can be solved analytically.Due to this excitation, the two-level system will undergo aninstantaneous transition from its ground state{ca = 1, cb =0} to the excited state{ca = cosα, cb = −i sinα}, withα = η/h. Therefore, after the pulse the system will remainin the excited state

|ψ(t)〉 = (cosα)|a(t)〉 + (−i sinα)|b(t)〉 (7)

which is a coherent quantum-mechanical superposition ofthe two noninteracting states. In addition to a finiteoccupation probability|cb|2 = sin2 α of the excited state,there exists a well-defined phase coherence between theground- and the excited-state contributions, i.e. apart fromtheir amplitudes, the coefficientsca andcb differ in phase byπ/2. This is what is generally meant by optically inducedphase coherence.

Let us now consider a continuous optical excitationresonant with our two-level system:Uab(t) = U0eiωLt withhωL = εb − εa. Also for this case, equations (6) can besolved analytically. In particular, for the initial conditionat t = 0 given by the ground state{ca = 1, cb = 0}, thesolution is given by

|ψ(t)〉 = cos( 12ωRt)|a(t)〉 − i sin( 1

2ωRt)|b(t)〉 (8)

where ωR = 2U0/h is the so-called Rabi frequency.Compared with the previous case, the continuous excitationgives rise to a periodic population and depopulation of theexcited state according to|cb|2 = sin2( 1

2ωRt). This purelycoherent phenomenon is known as the ‘Rabi-oscillation

149

F Rossi

regime’. As for the previous case, the excited state inequation (8) reflects a well-defined optically induced phasecoherence.

As mentioned above, the interlevel-coupling Hamilto-nian in equation (3) may also describe, in addition to anoptical excitation, some interlevel coupling due, for exam-ple, to the Coulomb interaction. Also in this case, in thepresence of this interlevel coupling the quantum state of thesystem will result in a coherent superposition of the twostationary states in (4). Again, this coupling will induce awell-defined phase coherence between the two levels.

From the above considerations we see that coherentphenomena can be divided into two basic classes:opticallyinduced and Coulomb-inducedphenomena. In order todescribe the optical response of an electron gas withina semiconductor crystal, the above analysis, based on asingle two-level system, has to be replaced by a statistical-ensemble approach, i.e. a description based on a collectionof independent two-level systems. Any statistical ensembleof quantum systems is properly described by its single-particle density matrix [64]

ρnn′ = 〈a†n′an〉 (9)

where n denotes a generic set of quantum numbers. Itsdiagonal elementsfn = ρnn provide the average occupationnumbers while the non-diagonal terms describe the degreeof phase coherence between statesn andn′. For the case ofour two-level system,ρnn′ reduces to a two-by-two matrix:the diagonal elementsfa = ρaa and fb = ρbb describe,respectively, the average occupation of levelsa andb, whilethe non-diagonal termsp = ρba andp∗ = ρab reflect theaverage degree of phase coherence between the ground andthe excited states. Starting from the Heisenberg equationsof motion for the creation and destruction operators, i.e.

ihd

dta†n = [a†n,H] ih

d

dtan = [an,H] (10)

we obtain the following equations of motion for the abovedensity-matrix elements

d

dtfb = − d

dtfa = 2

hRe(iUabp)

d

dtp = εb − εa

ihp + Uba

ih(fa − fb).

(11)

They are known as optical Bloch equations [1, 8] in analogywith the equations first derived by Bloch [76] for the spinsystems. This statistical-ensemble description reduces tothe previous one for the case of a so-called ‘pure state’,i.e. the case in which all the two-level systems formingthe ensemble are in the same quantum-mechanical state|ψ〉. In this case, the above set of kinetic equationsand the equations of motion (6) are totally equivalent.However, as we will discuss in section 2, the density-matrix approach introduced so far allows, in additionto the study of coherent phenomena, the analysis ofincoherent phenomena, which is not possible within asimple Schrodinger-equation formalism.

The optical Bloch equations (10) provide the simplestdescription of light–matter interaction. Let us consider

again the case of a continuous optical excitation resonantwith our two-level system: Uab(t) = U0eiωLt . If wechoose as the initial condition at timet = 0 the state{fa = f0, fb = 0, p = 0}, the solution of the above opticalBloch equations is given by:

fa(t) = f0 cos2( 12ωRt)

fb(t) = f0 sin2( 12ωRt)

p(t) = f0

4

[e−iω+t − e−iω−t

] (12)

whereω± = ωL ± ωR. As for the case of a single two-level system, the above solution describes a Rabi-oscillationregime. In particular, the interlevel density-matrix elementp originates from the superposition of the two frequencycomponentsω+ and ω−. They differ from ωL by theRabi frequencyωR. Such a modification of the two-levelfrequencyωL due to its coupling with the external field isknown as ‘Rabi splitting’ [6].

If we now rewrite the interlevel density-matrix elementp in (12) as

p(t) = − if0

2e−iωLt sin(ωRt) (13)

we see that, apart from the quantum-mechanical phasefactor corresponding to the energy separation ¯hωL = εb −εa, its amplitude exhibits Rabi oscillations according tosin(ωRt). More specifically, we obtain

|p|2 ∝ sin2(ωRt) ∝ cos2( 12ωRt) sin2( 1

2ωRt) ∝ fa(t)fb(t)(14)

i.e. the quantity|p|2 is proportional to the product ofthe two occupation numbers, thus reflecting the total(or macroscopic) dipole moment of the two-level systemat time t . This elucidates the link between opticallyinduced phase coherence and polarization: a coherentoptical excitation gives rise to a coherent quantum-mechanical superposition of the two states which results in amacroscopic polarization of the system. Such a polarizationfield is fully described by the non-diagonal matrix elementp in equation (12).

The above simplified description of light–matterinteraction neglects any kind of incoherent, i.e. phase-breaking, phenomena. As we will see in section 2, suchdephasing processes will lead to a decay of the interbandpolarizationp, thus destroying the optically induced phasecoherence in the carrier system.

As a final remark, let us discuss the concept of phasecoherence in connection with the choice of representation.On the basis of the density-matrix formalism consideredso far, the phase coherence is described by the non-diagonal density-matrix elementsρnn′ in (9). However, thisseparation intodiagonalandnon-diagonalterms is clearlyrepresentation dependent: what is diagonal in a given basisis in general not diagonal in a different basis and viceversa. If one considers, as a basis set, the eigenstates of thetotal HamiltonianH (which includes the external drivingforce), the density matrixρ in (9) is always diagonal, i.e.no phase coherence. Thus, in order to speak of phasecoherence, we need to regard the total HamiltonianH

150


as the sum of the system HamiltonianH0 plus a driving-force termH ′ (see equation (1)). Provided there is sucha separation between thesystem of interestand thedrivingforce, the non-diagonal density-matrix elements within therepresentation given by the eigenstates ofH0 will describethe degree of coherence induced in the system by thedriving forceH ′.

Two typical cases, discussed respectively in sections 4and 5, may help in clarifying the meaning of coherence.The first case is that of a semiconductor superlattice in thepresence of a constant and homogeneous electric field. Aswe will see, within the Wannier–Stark representation theBloch-oscillation dynamics is the result of phase coherencebetween the various Wannier–Stark states. In contrast,within the Bloch representation the same phenomenon ispurely described in terms of carrier populations, i.e. nooff-diagonal density-matrix elements. The second caseis that of Coulomb-induced phase coherence: within afree-particle representation, even at the simplest Hartree–Fock level, Coulomb interaction induces phase coherencebetween free-carrier states. This is clearly not the casewithin an exciton basis, where the density matrix isdiagonal.

2. Theoretical background

In this section we will review in a systematic way the basicideas used in the theoretical analysis of ultrafast carrierdynamics in semiconductors. The approach we are going topresent is based on the density-matrix formalism introducedin section 1.3.

2.1. Physical system

In order to study the optical and transport properties ofsemiconductor bulk and heterostructures, let us consider agas of carriers in a crystal under the action of an appliedelectromagnetic field. The carriers will experience theirmutual interaction as well as the interaction with the phononmodes of the crystal. Such a physical system can bedescribed by the following Hamiltonian

H =Hc +Hp +Hcc +Hcp +Hpp. (15)

The first term describes the noninteracting-carrier systemin the presence of the external electromagnetic field whilethe second one refers to the free-phonon system. Thelast three terms describe many-body contributions: theyrefer, respectively, to carrier–carrier, carrier–phonon andphonon–phonon interactions.

In order to discuss their explicit form, let us introducethe usual second-quantization field operatorsΨ†(r) andΨ(r). They describe, respectively, the creation and thedestruction of a carrier inr. In terms of the above fieldoperators the carrier HamiltonianHc can be written as

Hc =∫

drΨ†(r)[(−ih∇r − e

cA(r, t)

)2

2m0+ eϕ(r, t)

+V l(r)]Ψ(r). (16)

Here,V l(r) denotes the periodic potential due to the perfectcrystal whileA(r, t) andϕ(r, t) denote, respectively, thevector and scalar potentials corresponding to the externalelectromagnetic field. Since we are interested in the electro-optical properties as well as in the ultrafast dynamicsof photoexcited carriers, the electromagnetic field actingon the crystal—and the corresponding electromagneticpotentials— will be the sum of two different contributions:the high-frequency laser field responsible for the ultrafastoptical excitation and the additional electromagnetic fieldacting on the photoexcited carriers on a longer time-scale.More specifically, by denoting with the labels 1 and 2 thesetwo contributions, we can write

A(r, t) = A1(r, t)+A2(r, t)

ϕ(r, t) = ϕ1(r, t)+ ϕ2(r, t)(17)

and recalling that

E(r, t) = −1

c

∂

∂tA(r, t)−∇rϕ(r, t)

B(r, t) = ∇r ×A(r, t)(18)

we haveE(r, t) = E1(r, t)+E2(r, t)

B(r, t) = B1(r, t)+B2(r, t).(19)

Equations (18), which give the electromagnetic fields interms of the corresponding vector and scalar potentials,reflect the well-known gauge freedom: there is an infinitenumber of possible combinations ofA andϕ which giverise to the same electromagnetic field{E,B}. We willuse such freedom of choice for the laser field (term 1):we assume a homogeneous (space-independent) laser fieldE1(t) fully described by the scalar potential

ϕ1(r, t) = −E1(t) · r. (20)

This assumption, which corresponds to the well-knowndipole approximation, is justified as long as the space-scaleof interest is small compared with the light wavelength.The explicit form of the laser field considered here is

E1(t) = E+(t)+ E−(t) = E0(t)eiωLt + E∗0(t)e−iωLt (21)

where E0(t) is the amplitude of the light field andωLdenotes its central frequency.

With this particular choice of electromagnetic potentialsdescribing the laser field, the Hamiltonian in (16) can berewritten as

Hc =H0c +Hcl (22)

where

H0c =

∫drΨ†(r)

[(−ih∇r − ecA2(r, t)

)2

2m0+ eϕ2(r, t)

+V l(r)]Ψ(r) (23)

describes the carrier system in the crystal under the action ofthe electromagnetic field 2 only (see equation (19)), while

Hcl = e∫

drΨ†(r)ϕ1(r, t)Ψ(r) (24)

151

F Rossi

describes the carrier–light (cl) interaction due to the laserphotoexcitation.

In analogy with the carrier system, if we denote thecreation and destruction operators for a phonon of modeλ

and wavevectorq by b†q,λ and bq,λ respectively, the free-phonon Hamiltonian takes the form

Hp =∑qλ

hωqλb†qλbqλ (25)

whereωqλ is the dispersion relation for the phonon modeλ.

Let us now discuss the explicit form of the many-bodycontributions. The carrier–carrier interaction is describedby the two-body Hamiltonian

Hcc = 1

2

∫dr∫

dr′Ψ†(r)Ψ†(r′)Vcc(r − r′)Ψ(r′)Ψ(r)(26)

whereVcc denotes the Coulomb potential.Let us now introduce the carrier–phonon interaction

Hamiltonian

Hcp =∫

drΨ†(r)Vcp(r)Ψ(r) (27)

where

Vcp =∑qλ

[gqλbqλe

iq·r + g∗qλb†qλe−iq·r]

(28)

is the electrostatic phonon potential induced by the latticevibrations. Here, the explicit form of the couplingfunction gqλ depends on the particular phonon modeλ(acoustic, optical etc) as well as on the coupling mechanismconsidered (deformation potential, polar coupling etc).

Let us finally discuss the phonon–phonon contributionHpp. The free-phonon HamiltonianHp in (25), whichdescribes a system of noninteracting phonons, by definitionaccounts only for the harmonic part of the lattice potential.However, non-harmonic contributions of the interatomicpotential can play an important role in determining thelattice dynamics in highly excited systems [77], sincethey are responsible for the decay of optical phonons intophonons of lower frequency. In our second-quantizationpicture, these non-harmonic contributions can be describedin terms of a phonon–phonon interaction which induces, ingeneral, transitions between free-phonon states. Here, wewill not discuss the explicit form of the phonon–phononHamiltonianHpp responsible for such a decay. We willsimply assume that such a phonon–phonon interaction isefficient enough to maintain the phonon system in thermalequilibrium. This corresponds to neglecting hot-phononeffects [78].

It is well known that the coordinate representation usedso far is not the most convenient one in describing theelectron dynamics within a periodic crystal. In general, itis more convenient to employ the representation given bythe eigenstates of the noninteracting-carrier Hamiltonian—or a part of it—since it automatically accounts for someof the symmetries of the system. For the moment we willsimply consider an orthonormal basis set{φn(r)} without

specifying which part of the Hamiltonian is diagonal insuch a representation. This will allow us to write downequations valid in any quantum-mechanical representation.Since the noninteracting-carrier Hamiltonian is, in general,a function of time, the basis functionsφn may also be timedependent. Here, the labeln denotes, in general, a setof discrete and/or continuous quantum numbers. In theabsence of electromagnetic field, the above wavefunctionswill correspond to the well-known Bloch states of thecrystal and the indexn will reduce to the wavevectorkplus the band (or subband) indexν. In the presence ofa homogeneous magnetic field the eigenfunctionsφn mayinstead correspond to Landau states. Finally, for the case ofa constant and homogeneous electric field, there exist twoequivalent representations: the accelerated Bloch states andthe Wannier–Stark picture. Such equivalence turns out tobe of crucial importance in understanding the relationshipbetween Bloch oscillations and Wannier–Stark localizationand, for this reason, it will be discussed in more detail insection 4.1.

Let us now reconsider the system Hamiltonianintroduced so far in terms of such aφn representation. As astarting point, we may expand the second-quantization fieldoperators in terms of the new wavefunctions

Ψ(r) =∑n

φn(r)an Ψ†(r) =∑n

φ∗n(r)a†n. (29)

The above expansion defines the new set of second-quantization operatorsa†n and an; they describe, respec-tively, the creation and destruction of a carrier in staten.

For the case of a semiconductor structure (the only oneconsidered here), the energy spectrum of the noninteracting-carrier Hamiltonian (23)—or a part of it—is alwayscharacterized by two well-separated energy regions: thevalence and the conduction band. Also in the presenceof an applied electromagnetic field, the periodic latticepotentialV l gives rise to a large energy gap. Therefore, wedeal with two energetically well-separated regions, whichsuggests the introduction of the usual electron–hole picture.This corresponds to a separation of the set of states{φn}into conduction states{φei } and valence states{φhj }. Thus,

the creation (destruction) operatorsa†n (an) introduced inequation (29) will be divided into creation (destruction)electron and hole operators:c†i (ci) andd†j (dj ). In terms ofthe new electron–hole picture, the expansion (29) is givenby

Ψ(r) =∑i

φei (r)ci +∑j

φh∗j (r)d†j

Ψ†(r) =∑i

φe∗i (r)c†i +

∑j

φhj (r)dj .(30)

If we now insert the above expansion into equation (23),the noninteracting-carrier Hamiltonian takes the form

H0c =

∑ii ′εeii ′c

†i ci ′ +

∑jj ′εhjj ′d

†j dj ′ =H0

e +H0h (31)

where

εe/h

ll′ = ±∫

dr φe/h∗l (r)

[(−ih∇r − ecA2)2

2m0+ eϕ2+ V l

−ε0

]φe/h

l′ (r) (32)

152


are just the matrix elements of the Hamiltonian in theφrepresentation. The± sign refers, respectively, to electronsand holes whileε0 denotes the conduction-band edge. Here,we neglect any valence-to-conduction band coupling due tothe external electromagnetic field and vice versa. This iswell fulfilled for the systems and field regimes we are goingto discuss in this review. As already pointed out, the aboveHamiltonian may be time dependent. We will discuss thisaspect in the following section, where we will derive ourset of kinetic equations.

Let us now write in terms of our electron–holerepresentation the carrier–light interaction Hamiltonian (24)

Hcl = −∑i,j

[µehij E

−(t)c†i d†j + µeh∗ij E+(t)dj ci

]. (33)

The above expression has been obtained within the well-known rotating-wave approximation [8] by neglectingintraband transitions, absent for the case of opticalexcitations. Here,µehij denotes the optical dipole matrixelement between statesφei andφhj .

Similarly, the carrier–carrier Hamiltonian (26) can berewritten as

Hcc = 1

2

∑i1i2i3i4

V cci1i2i3i4c†i1c†i2ci3ci4

+1

2

∑j1j2j3j4

V ccj1j2j3j4d†j1d†j2dj3dj4

−∑i1i2j1j2

V cci1j1j2i2c†i1d†j1dj2ci2 (34)

where

V ccl1l2l3l4

=∫

dr∫

dr′ φ∗l1(r)φ∗l2(r′)V cc(r − r′)φl3(r′)φl4(r)

(35)are the Coulomb matrix elements within ourφ representa-tion. The first two terms describe the repulsive electron–electron and hole–hole interaction while the last one de-scribes the attractive electron–hole interaction. Here, weneglect terms that do not conserve the number of electron–hole pairs, i.e. impact-ionization and Auger-recombinationprocesses [79], as well as the interband exchange interac-tion. This monopole–monopole approximation is justifiedas long as the exciton binding energy is small comparedwith the energy gap.

Finally, let us rewrite the carrier–phonon interactionHamiltonian introduced in equation (27)

Hcp =∑ii ′,qλ

[geii ′,qλc

†i bqλci ′ + ge∗ii ′,qλc†i ′b†qλci

]−∑jj ′,qλ

[ghjj ′,qλd

†j bqλdj ′ + gh∗jj ′,qλd†j ′b†qλdj

](36)

with

ge/h

ll′,qλ = gqλ∫

dr φe/h∗l (r)eiq·rφe/h

l′ (r). (37)

In equation (36) we can clearly recognize four differentcontributions corresponding to electron and hole phononabsorption and emission.

2.2. Kinetic description

Our kinetic description of the ultrafast carrier dynamics insemiconductors is based on the density-matrix formalismintroduced in section 1.3. Since this approach has beenreviewed and discussed in several papers [64, 80] andtextbooks [1, 8], here we will simply recall in our notationthe kinetic equations relevant for the analysis of carrierdynamics in semiconductor heterostructures, generalizingthe approach presented in [64] to the case of a time-dependent quantum-mechanical representation.

The set of kinetic variables is the same as thatconsidered in [64]. Given our electron–hole representation{φei }, {φhj }, we will consider the intraband electron and holesingle-particle density matrices

f eii ′ =⟨c†i ci ′

⟩f hjj ′ =

⟨d†j dj ′

⟩(38)

as well as the corresponding interband density matrix

pji =⟨dj ci

⟩. (39)

Here, the diagonal elementsf eii andf hjj correspond to theelectron and hole distribution functions of the Boltzmanntheory while the non-diagonal terms describe intrabandpolarizations. In contrast, the interband density-matrixelementspji describe interband (or optical) polarizations.

In order to derive the set of kinetic equations, i.e.the equations of motion for the above kinetic variables,the standard procedure starts by deriving the equations ofmotion for the electron and hole operators introduced in(30)

ci =∫

dr φe∗i (r)Ψ(r) dj =∫

dr φh∗j (r)Ψ†(r).

(40)By applying the Heisenberg equation of motion for the fieldoperatorΨ, i.e.

d

dtΨ = 1

ih[Ψ,H] (41)

it is easy to obtain the following equations of motion

d

dtci =

1

ih

[ci,H

]+ 1

ih

∑i ′Zeii ′ci ′

d

dtdj =

1

ih

[dj ,H

]+ 1

ih

∑j ′Zhjj ′dj ′

(42)

with

Ze/h

ll′ = ih∫

dr

(d

dtφe/h∗l (r)

)φe/h

l′ (r). (43)

As for the case of equation (31), here we neglectagain valence-to-conduction band coupling and viceversa. Compared with the more conventional Heisenbergequations of motion, they contain an extra term, the lastone. This accounts for the possible time dependence ofour φ representation which will induce transitions betweendifferent states according to the matrix elementsZll′ .

By combining the above equations of motion withthe definitions of the kinetic variables in (38) and (39),

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F Rossi

the resulting set of kinetic equations can be schematicallywritten as

d

dtf ei1i2 =

d

dtf ei1i2

∣∣∣∣H

+ d

dtf ei1i2

∣∣∣∣φ

d

dtf hj1j2= d

dtf hj1j2

∣∣∣∣H

+ d

dtf hj1j2

∣∣∣∣φ

d

dtpj1i1= d

dtpj1i1

∣∣∣∣H

+ d

dtpj1i1

∣∣∣∣φ

.

(44)

They exhibit the same structure as the equations of motion(42) for the electron and hole creation and destructionoperators: a first term induced by the system HamiltonianH (which does not account for the time variation ofthe basis states) and a second one induced by the timedependence of the basis functionsφ.

Let us start discussing this second term, whose explicitform is

d

dtf ei1i2

∣∣∣∣φ

= 1

ih

∑i3i4

[Zei2i4δi1i3 − Zei3i1δi2i4

]f ei3i4

d

dtf hj1j2

∣∣∣∣φ

= 1

ih

∑j3j4

[Zhj2j4

δj1j3 − Zhj3j1δj2j4

]f hj3j4

d

dtpj1i1

∣∣∣∣φ

= 1

ih

∑i2j2

[Zhj1j2

δi1i2 + Zei1i2δj1j2

]pj2i2

.

(45)

As we will see in section 4.1, these contributions play acentral role in the description of Zener tunnelling withinthe vector-potential representation.

Let us now come to the first term. This, in turn, isthe sum of different contributions, corresponding to thevarious parts of the Hamiltonian. The total Hamiltoniancan be regarded as the sum of two terms, a single-particlecontribution plus a many-body one

H =Hsp +Hmb

= (H0c +Hcl +Hp

)+ (Hcc +Hcp +Hpp

). (46)

The explicit form of the time evolution due to the single-particle HamiltonianHsp (noninteracting carriers pluscarrier–light interaction plus free phonons) is given by

d

dtf ei1i2

∣∣∣∣sp

= 1

ih

{∑i3i4

[εei2i4δi1i3 − εei3i1δi2i4

]f ei3i4

+∑j1

[Ui2j1p

∗j1i1− U ∗i1j1

pj1i2

] }d

dtf hj1j2

∣∣∣∣sp

= 1

ih

{∑j3j4

[εhj2j4

δj1j3 − εhj3j1δj2j4

]f hj3j4

+∑i1

[Ui1j2p

∗j1i1− U ∗i1j1

pj2i1

] }d

dtpj1i1

∣∣∣∣sp

= 1

ih

{∑i2j2

[εhj1j2

δi1i2 + εei1i2δj1j2

]pj2i2

+∑i2j2

Ui2j2

[δi1i2δj1j2 − f ei2i1δj1j2 − f hj2j1

δi1i2] }

(47)

with Ui1j1 = −µehi1j1E−(t).

This is a closed set of equations, which is a consequenceof the single-particle nature ofHsp. In addition, we stressthat the structure of the two contributions entering equation(44) is very similar: one can include the contribution (45)in (47) by replacingε with ε + Z.

Let us finally discuss the contributions due to themany-body part of the Hamiltonian: carrier–carrier andcarrier–phonon interactions (the phonon–phonon one is notexplicitly considered here). As discussed in [64], forboth interaction mechanisms one can derive a hierarchyof equations involving higher-order density matrices. Inorder to close such equations with respect to our setof kinetic variables, approximations are needed. Thelowest-order contributions to our equations of motion aregiven by first-order terms in the many-body Hamiltonian:the Hartree–Fock level. Since we will neglect coherent-phonon states, the only Hartree–Fock contributions willcome from carrier–carrier interactions. They simply resultin a renormalization

1εe/h

l1l2= −

∑l3l4

V ccl1l3l2l4fe/h

l3l4(48)

of the single-particle energy matricesεe/h as well as in arenormalization

1Ui1j1 = −∑i2j2

V cci1j1j2i2pj2i2

(49)

of the external fieldU†. We stress that the Hartree–Fock approximation, which consists in factorizing averagevalues of four-point operators into products of twodensity matrices, is independent of the quantum-mechanicalpicture. It is then clear that the above kinetic equations arevalid in any quantum-mechanical representation.

All the contributions to the system dynamics discussedso far describe a fully coherent dynamics, i.e. no scatteringprocesses. In order to treat incoherent phenomena, e.g.energy relaxation and dephasing, one has to go one stepfurther in the perturbation expansion taking into accountsecond-order contributions (in the perturbation HamiltonianHmb) as well. The derivation of these higher-ordercontributions, discussed in [64], will not be repeated here.Again, in order to obtain a closed set of equations (withrespect to our set of kinetic variables (38) and (39))additional approximations are needed, namely the mean-field and Markov approximations. As for the Hartree–Fock case, the mean-field approximation allows us to writethe various higher-order density matrices as products ofsingle-particle ones. The Markov approximation allows usto eliminate the additional higher-order kinetic variables,e.g. phonon-assisted density matrices, providing a closedset of equations still local in time, i.e. no memory effects[81–85]. This last approximation is not performed in thequantum-kinetic theory discussed in [64] where, in additionto our single-particle variables, one considers two-particle[79] and phonon-assisted [84] density matrices.

† The explicit form of the renormalization terms considered in this reviewaccounts for the Fock contributions only, i.e. no Hartree terms. Thegeneral structure of Hartree–Fock contributions, relevant for the case of astrongly non-homogeneous system, is discussed in [64].

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While the mean-field approximation is representation-independent, this is unfortunately not the case for theMarkov limit. This clearly implies that the validity of theMarkov approximation is strictly related to the quantum-mechanical representation considered.

The above kinetic description, based on intra- andinterband density matrices, allows us to evaluate any single-particle quantity. In particular, for the analysis of theultrafast carrier dynamics in photoexcited semiconductorstwo physical quantities play a central role: the intra- andinterband total (or macroscopic) polarizations

P e/h(t) =∑ll′Me/h

ll′ fe/h

l′l (t) P eh =∑ij

µehij pji(t)

(50)whereMe/h and µeh denote, respectively, the intra- andinterband dipole matrix elements in ourφ-representa-tion. The time derivative of the intraband polarizationP e/h describes the radiation field induced by the Bloch-oscillation dynamics (which for a superlattice structureis in the THz range) while the Fourier transform of theinterband (or optical) polarizationP eh provides the linearand nonlinear optical response of the system.

3. Coherent ultrafast dynamics in bulksemiconductors

In this section, we will discuss the dominant role playedby optically induced phase coherence in the ultrafastgeneration and relaxation of photoexcited carriers in bulksemiconductors. In particular, we will compare thedescription based on the theoretical approach discussedabove with the more conventional picture based on theBoltzmann theory.

3.1. Bloch model

In order to study the ultrafast carrier dynamics in bulksystems, let us consider a two-band semiconductor model.For the case of a homogeneous system, the only relevantterms of the single-particle density matrix ink space arethe diagonal ones. This property, due to the translationalsymmetry of the Hamiltonian, reduces the set of kineticvariables (38) and (39) to the following electron and holedistribution functions (intraband density-matrix elements)

f ek =⟨c†kck

⟩f h−k =

⟨d†−kd−k

⟩(51)

together with the corresponding polarizations (interbanddensity-matrix elements)†

pk =⟨d−kck

⟩. (52)

† Here, the standard electron–hole picture introduced in section 2.1 hasbeen applied to our plane-wave states. In particular, due to the charge-conjugation symmetry, the hole states are still labelled in terms of thecorresponding valence-electron states, i.e.kh ≡ −ke.

The explicit form of the kinetic equations (44) within theabove two-band picture is given by

d

dtf ek = gk(t)−

∑k′

[Wek′kf

ek

(1− f ek′

)−We

kk′fek′(1− f ek

)]d

dtf hk = g−k(t)−

∑k′

[Whk′kf

hk

(1− f hk′

)−Wh

kk′fhk′(1− f hk

)]d

dtpk =

1

ih

[(εek + εh−k

)pk + Uk

(1− f ek − f h−k

)]−∑k′

[W

p

k′kpk −Wp

kk′pk′]

(53)

with the generation rate

gk = 1

ih

[Ukp

∗k − U ∗kpk

]. (54)

Here, the incoherent contributions are treated within theusual Markov limit as described in [64, 67, 70]. Within suchan approximation scheme, the incoherent contributions tothe polarization dynamics exhibit the same structure as forthe distribution functions in terms of the following in- andout-scattering rates

Wp

k′k = 12

∑ν=e,h

[Wνk′k(1− f νk′

)+Wνk′kf

νk′]

(55)

whereWe/h

k′k are the usual scattering rate of the semiclassicalBoltzmann theory. This Boltzmann-like structure of thescattering term in the polarization equation is the startingpoint of the generalized Monte Carlo method discussed insection 1 [66–70].

The above kinetic equations are known assemiconduc-tor Bloch equations(SBEs)‡. As we can see, here thecarrier photogeneration is a two-step process: the exter-nal field U induces a coherent polarizationp which, inturn, generates electron–hole pairs via its coupling with thefield according to equation (54). The generation rate isthus determined by the interband polarizationpk which, inturn, is influenced by the various density-dependent scatter-ing mechanisms, e.g. carrier–carrier processes. Therefore,in contrast to the semiclassical case discussed below, thegeneration rate within the SBE model is clearly densitydependent.

3.2. Boltzmann model

The conventional Boltzmann model is obtained from theabove SBEs by performing an adiabatic elimination ofthe interband polarizationpk, i.e. by inserting a formalsolution of the polarization equation into the generationrate (54) and then performing a Markov limit withrespect to the electron and hole distribution functions[67, 70]. This corresponds to assuming that the carrier

‡ In the absence of Coulomb correlation and scattering, the SBEs in (53)reduce to the optical Bloch equations introduced in section 1.3, i.e. theydescribe a collection of independent (noninteracting) two-level systems.

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F Rossi

distributions are slowly varying functions on the time-scaleof the laser photoexcitation, i.e. the carrier generation andrelaxation are treated as independent processes. Withinsuch an approximation scheme, except for phase-space-filling effects, the generation rate entering the semiclassicalBoltzmann equations (BEs) for electrons and holes is fullydetermined by the temporal and spectral characteristics ofthe laser pulse. Contrary to the SBE case, all effectsrelated to the optically induced phase coherence—and toits temporal decay—are neglected.

3.3. Coherent carrier photogeneration

In order to compare the different models of carrierdynamics in photoexcited semiconductors, let us reviewsome simulated experiments [74] based on the generalizedMonte Carlo solution of the SBEs [66–70] discussed insection 1. The corresponding BE has been solved, forcomparison, by a standard ensemble Monte Carlo (EMC)simulation [27, 29]. Both BEs and SBEs have been solvednumerically for the case of GaAs bulk excited by a 150 fslaser pulse.

In figure 1 the generation rate at different times asobtained from the SBE is plotted as a function of thecarrier wavevector for three different densities. At thelowest density the behaviour is essentially the same asin the case without carrier–carrier scattering: energy–timeuncertainty leads to an initially very broad generation rate;with increasing time the line narrows and, in the tails,exhibits negative parts due to a stimulated recombination ofcarriers initially generated off-resonance [67, 70]. After thepulse, the distribution function of the generated carriers isin good agreement with the BE result. Scattering processesdestroy the coherence between electrons and holes which isnecessary for the stimulated recombination processes. As aconsequence, with increasing density the negative tails arestrongly reduced and the generation remains broad for alltimes, resulting in a much broader carrier distribution thanin the BE case.

3.4. Comparison with experiments

Hot-carrier luminescence has proven to be a powerfultechnique in studying the ultrafast dynamics of photoexcitedcarriers in semiconductors [18]. In band-to-acceptor (BA)luminescence experiments [19, 21, 86, 87], due to their highsensitivity, carrier densities as low as several 1013 cm−3

have been reached. Thus, the transition between adynamics dominated by carrier–phonon scattering to adynamics dominated by carrier–carrier scattering (with itsconsequences for the generation process discussed above)is experimentally accessible.

In the BA luminescence experiment reviewed here [73–75], a 3µm thick GaAs layer doped with Be acceptors ofa concentration of 3× 1016 cm−3 is excited at a photonenergy of 1.73 eV by transform-limited 150 fs pulsesfrom a mode-locked Ti:sapphire laser. Time-integratedluminescence spectra are recorded [73]. The right columnof figure 2 shows measured BA luminescence spectra forthree different values of the carrier density. At low density

we observe an initial peak of the generated carriers (markedbold) and pronounced replicas due to the emission of aninteger number of optical phonons. With increasing densitythe peaks become broader and at the highest density onlya slight structure related to the phonons is still visible.

These measured spectra are compared with correspond-ing simulated experiments based on the BE and SBE mod-els [74] (left and middle column in figure 2). We findseveral pronounced differences between the two models.In the semiclassical model the unrelaxed peak is clearlyvisible up to the highest density in contrast to the coherentmodel, where, in agreement with the experiment, this peakis strongly broadened. At the higher densities the semiclas-sical spectra exhibit an increase in the broadening of sub-sequent replicas which is not present in the Bloch-modelcalculations or in the experimental results.

To illustrate this difference quantitatively, in figure 3we have plotted the full width at half-maximum (FWHM)of the three highest peaks in the spectra. From thesemiclassical model we obtain a strong increase in thedensity dependence of subsequent peaks. This behaviourcan be easily understood: due to the emission time ofan optical phonon of about 150 fs, the carriers populatesubsequent replicas at increasing times and, therefore,the efficiency of carrier–carrier scattering processes inbroadening the peaks increases. However, the measuredspectra exhibit a different behaviour which is quantitativelyreproduced by the SBE model: already the unrelaxed peakexhibits a strong increase with increasing density and thedensity dependence of all replicas is approximately thesame. The reason is that the broadening of the generationprocess as discussed above obviously dominates over thebroadening due to subsequent scattering processes. Asdiscussed in [75], this is due to the fact that the generationprocess is strongly influenced by the decay of the interbandpolarization which, in turn, is due to electron plus holescattering, while the broadening of the electron distributionduring the relaxation process is due to electron scatteringonly†.

The above theoretical and experimental analysisconstitutes a clear demonstration of the importance ofdephasing processes for the analysis of luminescencespectra. The density dependence of the spectra in theinteresting transition region between carrier–phonon andcarrier–carrier dominated dynamics can only be explainedby a coherent modelling of the carrier generation includingthe dynamics of the interband polarization. The mainmechanism determining the width of the peaks is thebroadening of the generation rate. It can be shownthat for band-to-acceptor spectra the broadening of therecombination process, neglected in the present model,plays a minor role [74]. However, for band-to-band spectrathis phenomenon should be taken into account. For thecase discussed above the background due to band-to-bandluminescence has been found to be of negligible importance[75].

† For the case of bulk GaAs considered here, due to the different electronand heavy-hole effective masses, the hole–hole scattering is about fivetimes larger than the electron–electron scattering.

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Figure 1. Generation rate of electrons as a function of k at different times and densities. After [74].

Figure 2. Calculated and measured BA spectra for different carrier densities. After [74].

4. Bloch oscillations and Wannier–Starklocalization in superlattices

Ever since the initial applications of quantum mechanicsto the dynamics of electrons in solids, the analysis ofBloch electrons moving in a homogeneous electric field hasbeen of central importance. By employing semiclassicalarguments, in 1928 Bloch [88] demonstrated that, awavepacket given by a superposition of single-band statespeaked about some quasi-momentum, ¯hk, moves with a

group velocity given by the gradient of the energy-bandfunction with respect to the quasi-momentum and that therate of change of the quasi-momentum is proportional to theapplied fieldF . This is often referred to as the ‘accelerationtheorem’

hk = eF . (56)

Thus, in the absence of interband tunnelling and scatteringprocesses, the quasi-momentum of a Bloch electron in ahomogeneous and static electric field will be uniformlyaccelerated into the next Brillouin zone in a repeated-zone

157

F Rossi

Figure 3. FWHM of the unrelaxed peak and the first and second phonon replica as a function of carrier density obtained fromthe calculated and measured spectra. The lines connecting the theoretical values are meant as guides to the eye. After [74].

Figure 4. A schematic illustration of the field-inducedcoherent motion of an electronic wavepacket initiallycreated at the bottom of a miniband. Here, the width of theminiband exceeds the LO-phonon energy ELO , so thatLO-phonon scattering is possible. After [117].

scheme (or equivalently undergoes an Umklapp processback to the first zone). The corresponding motion of theBloch electron through the periodic energy-band structure,shown in figure 4, is called ‘Bloch oscillation’; it ischaracterized by an oscillation periodτB = h/eFd, whered denotes the lattice periodicity in the field direction.

There are two mechanisms impeding a fully periodicmotion: interband tunnelling and scattering processes.Interband tunnelling is an intricate problem and is still atthe centre of a continuing debate. Early calculations of thetunnelling probability into other bands in which the electricfield is represented by a time-independent scalar potentialwere made by Zener [89] using a Wentzel–Kramers–Brillouin generalization of Bloch functions, by Houston[90] using accelerated Bloch states (Houston states), andsubsequently by Kane [91] and Argyres [92] who employedthe crystal-momentum representation. Their calculationsled to the conclusion that the tunnelling rate per Blochperiod is much less than unity for electric fields up to106 V cm−1 for typical band parameters corresponding toelemental or compound semiconductors.

Figure 5. A schematic representation of the transitionsfrom the valence to the conduction band of a superlattice inthe Wannier–Stark localization regime. After [44].

Despite the apparent agreement among these calcula-tions, the validity of employing the crystal-momentum rep-resentation or Houston functions to describe electrons mov-ing in a non-periodic (crystal plus external field) potentialhas been disputed. The starting point of the controversy wasthe original paper by Wannier [93]. He pointed out that,due to the translational symmetry of the crystal potential, ifφ(r) is an eigenfunction of the scalar-potential Hamiltonian(corresponding to the perfect crystal plus the external field)with eigenvalueε, then anyφ(r + nd) is also an eigen-function with eigenvalueε+n1ε, where1ε = eFd is theso-called Wannier–Stark splitting (d being the primitive lat-tice vector along the field direction). He concluded that thetranslational symmetry of the crystal gives rise to a discreteenergy spectrum, the so-called Wannier–Stark ladder. Thestates corresponding to these equidistantly spaced levels arelocalized states, as schematically shown in figure 5 for thecase of a semiconductor superlattice.

The existence of such energy quantization was disputedby Zak [94], who pointed out that for the case of an infinitecrystal the scalar potential−F · r is not bounded, whichimplies a continuous energy spectrum. Thus, the main pointof the controversy was related to the existence (or absence)

158


of Wannier–Stark ladders. More precisely, the point wasto decide whether or not interband tunnelling (neglected inthe original calculation by Wannier [93]) is strong enoughto destroy the Wannier–Stark energy quantization (and thecorresponding Bloch oscillations).

It is only during the last decade that this controversycame to an end. From a theoretical point of view, most ofthe formal problems related to the non-periodic nature ofthe scalar potential (superimposed on the periodic crystalpotential) were finally removed by using a vector-potentialrepresentation of the applied field [95, 96]. Within such avector-potential picture, upper boundaries for the interbandtunnelling probability have been established at a rigorouslevel, which show that an electron may execute a number ofBloch oscillations before tunnelling out of the band [96, 97],in qualitatively good agreement with the earlier predictionsof Zener and Kane [89, 91].

The second mechanism impeding a fully periodicmotion is scattering by phonons, impurities etc (seefigure 4). This results in lifetimes shorter than the Blochperiod τB for all reasonable values of the electric field,so that Bloch oscillations should not be observable inconventional solids.

In superlattices, however, the situation is much morefavourable because of the smaller Bloch periodτB resultingfrom the small width of the mini-Brillouin zone in the fielddirection [98].

Indeed, the existence of Wannier–Stark ladders aswell as Bloch oscillations in superlattices has beenconfirmed by a number of recent experiments [1].The photoluminescence and photocurrent measurementsof the biased GaAs/GaAlAs superlattices performed byMendezet al [99], together with the electroluminescenceexperiments by Voisinet al [100], provided the earliestevidence of the field-induced Wannier–Stark ladders insuperlattices. A few years later, Feldmannet al [101]were able to measure Bloch oscillations in the timedomain through a four-wave-mixing experiment originallysuggested by von Plessen and Thomas [102]. A detailedanalysis of the Bloch oscillations in the four-wave-mixingsignal (which reflects the interband dynamics) has been alsoperformed by Leo and coworkers [103, 104].

In addition to the above interband-polarization analysis,Bloch oscillations have also been detected by monitoringthe intraband polarization which, in turn, is reflectedby anisotropic changes in the refractive index [1].Measurements based on transmittive electro-optic sampling(TEOS) have been performed by Dekorsy and coworkers[105, 106]. Finally Bloch oscillations have recently beenmeasured through a direct detection of the THz radiationin semiconductor superlattices [44, 107].

4.1. Two equivalent pictures

Let us now apply the theoretical approach presented insection 2 to the case of a semiconductor superlattice inthe presence of a uniform (space-independent) electricfield. The noninteracting carriers within the superlatticecrystal will then be described by the HamiltonianH0

c inequation (23), where now the electrodynamic potentials

A2 and ϕ2 (in the following simply denoted byA andϕ) correspond to a homogeneous electric fieldE2(r, t) =F (t).

As pointed out in section 2.1, the natural quantum-mechanical representation is given by the eigenstates ofthis Hamiltonian[(−ih∇r − e

cA(r, t)

)2

2m0+ eϕ(r, t)+ V l(r)

]φn(r)

= εnφn(r). (57)

However, due to the gauge freedom discussed insection 2.1, there is an infinite number of possiblecombinations ofA and ϕ—and therefore of possibleHamiltonians—which describe the same homogeneouselectric field F (t). In particular, one can identify twoindependent choices: the vector-potential gauge

A(r, t) = −c∫ t

t0

F (t ′) dt ′ ϕ(r, t) = 0 (58)

and the scalar-potential gauge

A(r, t) = 0 ϕ(r, t) = −F (t) · r. (59)

As shown in [108], the two independent choicescorrespond, respectively, to the well-known Bloch-oscillation and Wannier–Stark pictures. They simply reflecttwo equivalent quantum-mechanical representations and,therefore, any physical phenomenon can be described inboth pictures.

More specifically, within the vector-potential picture(58), the eigenfunctionsφn in (57) are the so-calledaccelerated Bloch states (or Houston states) [90, 95, 96]. Asdiscussed in [108], such a time-dependent representationconstitutes the natural basis for the description ofBloch oscillations, i.e. it provides a rigorous quantum-mechanical derivation of the acceleration theorem (56),thus showing that this is not a simple semiclassicalresult†. Within such a representation, Bloch oscillationsare fully described by the diagonal terms of theintraband density matrix (38) (semiclassical distributionfunctions). Therefore, non-diagonal elements, describingphase-coherence between different Bloch states, do notcontribute to the intraminiband dynamics. However,they are of crucial importance for the description ofinterminiband dynamics, i.e. field-induced Zener tunnelling,which in this Bloch-state representation originates from thetime variation of our basis states (see equation (45)).

In contrast, within the scalar-potential picture (59), theeigenfunctionsφn in (57) are the well-known Wannier–Stark states [93]. Contrary to the previous Bloch picture,within such a representation the intraminiband Blochdynamics originates from a quantum interference betweendifferent Wannier–Stark states, thus involving non-diagonalelements of the intraband density matrix (38).

† The acceleration theorem (56) and the corresponding Bloch-oscillationdynamics are usually regarded as a semiclassical result compared withthe Wannier–Stark picture. In fact they correspond to two different fullyquantum-mechanical pictures.

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4.2. Some simulated experiments

In this section, we will review recent simulated experimentsof the ultrafast carrier dynamics in semiconductorsuperlattices [109–114]. They are based on a generalizedMonte Carlo solution [66–70] of the set of kinetic equations(so-called semiconductor Bloch equations) derived insection 2.2. In this case, the Bloch representation discussedin section 4.1 has been employed, limiting the set ofinterband density-matrix elements in (39) to the diagonalones, i.e. i = j . In addition, incoherent scatteringprocesses have been treated within the usual Markov limitas discussed in [64, 67, 70]. Within such an approximationscheme, the explicit form of the kinetic equations (44)coincides with the SBEs (17) (obtained for the bulk case)[108], provided we replace the wavevectork with kν, νbeing the superlattice miniband index.

In the simulated experiments reviewed here thefollowing superlattice model has been employed. Theenergy dispersion and the corresponding wavefunctionsalong the growth direction (k‖) are computed within thewell-known Kronig–Penney model [98], while for the in-plane direction (k⊥) an effective-mass model has been used.Starting from these 3D wavefunctionsφ0

kν , the variouscarrier–carrier as well as carrier–phonon matrix elementsare numerically computed (see equations (35) and (37)).They are, in general, functions of the various minibandindices and depend separately onk‖ and k⊥, thus fullyreflecting the anisotropic nature of the superlattice structure.

Only coupling to GaAs bulk phonons has beenconsidered. This, of course, is a simplifying approximationwhich neglects any superlattice effect on the phonondispersion, such as confinement of optical modes in thewells and in the barriers and the presence of interfacemodes [115, 116]. However, while these modificationshave important consequences for phonon spectroscopies(such as Raman scattering), they are far less decisive fortransport phenomena†.

We will start discussing the scattering-induced dampingof Bloch oscillations. In particular, we will show that inthe low-density limit this damping is mainly determined byoptical-phonon scattering [109, 110] while at high densitiesthe main mechanism responsible for the suppression ofBloch oscillations is found to be carrier–carrier scattering[114].

This Bloch-oscillation analysis in the time domain isalso confirmed by its counterpart in the frequency domain.As pointed out in section 4.1, the presence of Blochoscillations, due to a negligible scattering dynamics, shouldcorrespond to Wannier–Stark energy quantization. Thisis confirmed by the simulated optical-absorption spectra,which clearly show the presence of the field-inducedWannier–Stark ladders [113].

4.2.1. Bloch-oscillation analysis All the simulatedexperiments presented in this section refer to thesuperlattice structure considered in [111]: 111A GaAs

† Indeed, by now it is well known [116] that the total scattering ratesare sufficiently well reproduced if the phonon spectrum is assumed to bebulk-like.

Figure 6. Full Bloch-oscillation dynamics corresponding toa laser photoexcitation resonant with the first-minibandexciton. (a) Time evolution of the electron distribution as afunction of k‖. (b) Average kinetic energy, (c) current and(d) THz signal corresponding to the Bloch oscillations in (a).

wells and 17A Al 0.3Ga0.7As barriers. For such a structurethere has been experimental evidence for a THz emissionfrom Bloch oscillations [107].

In the first set of simulated experiments, an initialdistribution of photoexcited carriers (electron–hole pairs)is generated by a 100 fs Gaussian laser pulse in resonancewith the first-miniband exciton (¯hωL ≈ 1540 meV). Thestrength of the applied electric field is assumed to be4 kV cm−1, which corresponds to a Bloch periodτB =h/eFd of about 800 fs.

In the low-density limit (corresponding to a weaklaser excitation), incoherent scattering processes do notalter the Bloch-oscillation dynamics. This is due to thefollowing reasons. In agreement with recent experimental[107, 117] and theoretical [109, 110, 111] investigations, forsuperlattices characterized by a miniband width smallerthan the LO-phonon energy—as for the structure consideredhere—and for laser excitations close to the bandgap,at low temperatures carrier–phonon scattering is notpermitted. Moreover, in this low-density regime carrier–carrier scattering plays no role. Due to the quasi-elasticnature of Coulomb collisions, in the low-density limitthe majority of the scattering processes is characterizedby a very small momentum transfer. As a consequence,the momentum relaxation along the growth direction isnegligible. As a result, on this picosecond time-scalethe carrier system exhibits a coherent Bloch-oscillationdynamics, i.e. a negligible scattering-induced dephasing.This can be clearly seen from the time evolution of thecarrier distribution as a function ofk‖ (i.e. averaged overk⊥) shown in figure 6. During the laser photoexcitation(t = 0) the carriers are generated aroundk‖ = 0, where the

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Figure 7. (a) Total THz signals for eight different spectral positions of the exciting laser pulse: 1540, 1560, . . . ,1680 meV(from bottom to top). (b) Individual THz signal of the electrons and holes in the different bands for a central spectral positionof the laser pulse of 1640 meV. After [113].

transitions are close to resonance with the laser excitation.According to the acceleration theorem (56), the electronsare then shifted ink space. When the carriers reach theborder of the first Brillouin zone they are Bragg reflected.After about 800 fs, corresponding to the Bloch periodτB ,the carriers have completed one oscillation ink space. Asexpected, the carriers execute Bloch oscillations withoutlosing the synchronism of their motion by scattering. Thisis again shown in figure 6, where we have plotted: (b)the mean kinetic energy, (c) the current and (d) its timederivative which is proportional to the emitted far field, i.e.the THz radiation. (It can be shown that, by neglectingZener tunnelling, the time derivative of the intrabandpolarizationP e/h in equation (50) is proportional to thecurrent.) All these three quantities exhibit oscillationscharacterized by the same Bloch periodτB . Due to the finitewidth of the carrier distribution ink space (see figure 6(a)),the amplitude of the oscillations of the kinetic energy issomewhat smaller than the miniband width. Since forthis excitation condition the scattering-induced dephasingis negligible, the oscillations of the current are symmetricaround zero, which implies that the time average of thecurrent is equal to zero, i.e. no dissipation.

As already pointed out, this ideal Bloch-oscillationregime is typical of a laser excitation close to the gapin the low-density limit. Let us now discuss, still at lowdensities, the case of a laser photoexcitation high in theband. Figure 7(a) shows the THz signal as obtained froma set of simulated experiments corresponding to differentlaser excitations [111]. The different traces correspond tothe emitted THz signal for increasing excitation energies.We clearly notice the presence of Bloch oscillations inall cases. However, the oscillation amplitude and decay(effective damping) is excitation dependent.

For the case of a laser excitation resonant with the first-miniband exciton considered above (see figure 6), we havea strong THz signal. The amplitude of the signal decreaseswhen the excitation energy is increased. Additionally, there

are also some small changes in the phase of the oscillations,which are induced by the electron–LO-phonon scattering.

When the laser energy comes into resonance with thetransitions between the second electron and hole minibands(hωL ≈ 1625 meV), the amplitude of the THz signalincreases again. The corresponding THz transients show aninitial part, which is strongly damped and some oscillationsfor longer times that are much less damped. For a betterunderstanding of these results, we show in figure 7(b) theindividual THz signals, originating from the two electronand two heavy-hole minibands for the excitation withhω = 1640 meV. The Bloch oscillations performed by theelectrons within the second miniband are strongly dampeddue to intra- and interminiband LO-phonon scatteringprocesses [109, 111]. Since the width of this secondminiband (45 meV) is somewhat larger than the LO-phononenergy, intraminiband scattering is also possible, wheneverthe electrons are accelerated into the high-energy region ofthe miniband. The THz signal originating from electronswithin the first miniband shows an oscillatory behaviourwith a small amplitude and a phase which is determined bythe time the electrons need to relax down to the bottom ofthe band.

At the same time, the holes in both minibands exhibitundamped Bloch oscillations, since the minibands are soclose in energy that for these excitation conditions no LO-phonon emission can occur. The analysis shows that atearly times the THz signal is mainly determined by theelectrons within the second miniband. At later times theobserved signal is due to heavy holes and electrons withinthe first miniband.

The above theoretical analysis closely resemblesexperimental observations obtained for a superlatticestructure very similar to the one modelled here [107].In these experiments, evidence for THz emission fromBloch oscillations has been reported. For some excitationconditions these oscillations are associated with resonantexcitation of the second miniband. The general behaviour

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Figure 8. (a) Total THz radiation as a function of time;(b) incoherently-summed polarization as a function of time.After [114].

of the magnitude of the signals, the oscillations and thedamping are close to the results shown in figure 7.

Finally, in order to study the density dependence ofthe Bloch-oscillation damping, let us go back to the caseof laser excitations close to the gap. Figure 8(a) showsthe total (electrons plus holes) THz radiation as a functionof time for three different carrier densities. With increasingcarrier density, carrier–carrier scattering becomes more andmore important. Due to Coulomb screening, the momentumtransfer in a carrier–carrier scattering increases (its typicalvalue being comparable with the screening wavevector).This can be seen in figure 8(a), where for increasingcarrier densities we realize an increasing damping of theTHz signal. However, also for the highest carrier densityconsidered here we deal with a damping time of the orderof 700 fs, which is much larger than the typical dephasingtime, i.e. the decay time of the interband polarization,associated with carrier–carrier scattering. The dephasingtime is typically investigated by means of four-wave-mixing(FWM) measurements and such multi-pulse experimentscan be simulated as well [71, 72]. From a theoretical pointof view, a qualitative estimate of the dephasing time isgiven by the decay time of the ‘incoherently summed’

polarization (ISP) [67]. Figure 8(b) shows such ISP asa function of time for the same three carrier densities offigure 8(a). As expected, the decay times are always muchsmaller than the corresponding damping times of the THzsignals (note the different time-scale in figures 8(a) and(b)). This difference, discussed in more detail in [108, 114],can be understood as follows. The fast decay times offigure 8(b) reflect the interband dephasing, i.e. the sum ofthe electron and hole scattering rates. In particular, forthe Coulomb interaction this means the sum of electron–electron, electron–hole, and hole–hole scattering. As forthe case of bulk GaAs discussed in section 3, this lastcontribution is known to dominate and determines thedephasing time-scale. On the other hand, the total THzradiation in figure 8(a) is the sum of the electron and holecontributions. However, due to the small value of the holeminiband width compared with that of the electron, theelectron contribution will dominate. This means that theTHz damping in figure 8(a) mainly reflects the damping ofthe electron contribution. This decay, in turn, reflects theintraband dephasing of electrons which is due to electron–electron and electron–hole scattering only, i.e. no hole–holecontributions.

From the above analysis we can conclude that the decaytime of the THz radiation due to carrier–carrier scatteringdiffers considerably from the corresponding dephasingtimes obtained from a FWM experiment. The former isa measurement of the intraband dephasing while the latterreflects the interband dephasing†.

4.2.2. Optical-absorption analysis Let us now discussthe frequency-domain counterpart of the Bloch-oscillationpicture considered so far. Similar to what happens inthe time domain, for sufficiently high electric fields, i.e.when the Bloch periodτB = h/eFd becomes smaller thanthe dephasing time, the optical spectra of the superlatticeare expected to exhibit the frequency-domain counterpartof the Bloch oscillations, i.e. the Wannier–Stark energyquantization discussed in section 4.1. In the absence ofCoulomb interaction, the Wannier–Stark ladder absorptionincreases as a function of the photon energy in a step-like fashion. These steps are equidistantly spaced. Theirspacing, named Wannier–Stark splitting, is proportional tothe applied electric field.

The simulated linear-absorption spectra correspondingto a superlattice structure with 95A GaAs wells and15 A Al 0.3Ga0.7As barriers are shown in figure 9 [113].As we can see, the Coulomb interaction gives rise toexcitonic peaks in the absorption spectra and introducescouplings between these Wannier–Stark states. Suchexciton peaks, which are no longer equidistantly spaced,are often referred to as excitonic Wannier–Stark ladders[118] of the superlattice.

† We stress that this difference between intraband and interband dephasingin superlattices is the same discussed in section 3 for the caseof bulk semiconductors, where the broadening of the photoexcitedcarrier distribution is mainly determined by the decay of the interbandpolarization (interband dephasing) while the subsequent energy broadeningof the electron distribution is due to electron scattering only (intrabanddephasing) (see figure 2).

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Figure 9. Absorption spectra for various static appliedelectric fields for a GaAs/Al0.3Ga0.7As superlattice (well(barrier) width 95 (15) A). The vertical displacementsbetween any two spectra are proportional to the differencebetween the corresponding fields. The Wannier–Starktransitions are labelled by numbers, the lower (higher) edgeof the combined miniband by E0 (E1). After [113].

Since for the superlattice structure considered in thissimulated experiment [109, 113] the combined minibandwidth is larger than the typical two-dimensional (2D) and3D exciton binding energies, it is possible to investigate thequasi-3D absorption behaviour of the delocalized minibandstates as well as localization effects induced by the electricfield.

For the free-field case, the electron and hole states arecompletely delocalized in our 3Dk space. The perturbationinduced by the application of a low field (here≈5 kV cm−1)couples the states along the field direction, and the Franz–Keldysh effect, well known from bulk materials [8], appearsin the spectra: one clearly notices oscillations whichincrease in amplitude with the field and shift withF 2/3

from then = 0 andn = 1 levels towards the centre of thecombined miniband.

For increasing field the potential drop over the distanceof a few quantum wells eventually exceeds the minibandwidth and the electronic states become more and morelocalized. Despite the field-induced energy differenceneFd, the superlattice potential is equal for quantum wellsseparated bynd. Therefore, the spectra decouple into aseries of peaks corresponding to the excitonic ground statesof the individual electron–hole Wannier–Stark levels. EachWannier–Stark transition contributes to the absorption witha pronounced 1−s exciton peak, plus higher bound excitonand continuum states. The oscillator strength of a transitionn is proportional to the overlap between electron and holewavefunctions centred at quantum wellsn′ and n + n′respectively. The analysis shows that this oscillator strengthis almost exclusively determined by the amplitude of theelectron wavefunction in quantum welln′+n since for fieldsin the Wannier–Stark regime the hole wavefunctions arealmost completely localized over one quantum well due totheir high effective mass (see figure 5). Thus, the oscillatorstrengths of transitions to higher|n| become smaller withincreasing|n| and field.

At high fields (here'8 kV cm−1) the separationbetween the peaks is almost equal toneFd. For example,the peak of then = 0 transition which is shifted by theWannier–Stark exciton binding energy with respect to thecentre of the combined miniband, demonstrates that theincreasing localization also increases the exciton bindingenergy. This increased excitonic binding reflects thegradual transition from a 3D to a 2D behaviour, discussedin detail in [111].

For intermediate fields there is an interplay between theWannier–Stark and the Franz–Keldysh effect. Coming fromhigh fields, first the Wannier–Stark peaks are modulatedby the Franz–Keldysh oscillations. However, as soon asthe separationeFd between neighbouring peaks becomessmaller than their spectral widths, the peaks can no longerbe resolved individually so that only the Franz–Keldyshstructure remains.

5. Coulomb-correlation effects in semiconductorquantum wires

The importance of Coulomb-correlation effects in theoptical spectra of semiconductors and their dependenceon dimensionality has now been long recognized [119].More recently, increasing interest has been devoted to one-dimensional (1D) systems [120], prompted by promisingadvances in quantum-wire fabrication and application, e.g.quantum-wire lasers. The main goal is to achieve structureswith improved optical efficiency as compared to 2D and3D ones. A common argument in favour of this effortis based on the well-known van Hove divergence in the1D joint density-of-states (DOS), which is expected togive rise to very sharp peaks in the optical spectra of1D structures. Such a prediction is, however, based onfree-carrier properties of ideal 1D systems and it ignoresCoulomb-correlation effects.

Early theoretical investigations by Ogawa and Takaga-hara [121, 122], based on a 1D single-subband model ofthe wires, showed that the inverse-square-root singularityin the 1D DOS at the band edge is smoothed when exci-tonic effects are taken into account.

In this section, we will review recent theoretical results[123–125] based on a full 3D description of realisticquantum-wire structures made available by the presenttechnology, such as structures obtained by epitaxial growthon non-planar substrates (V-shaped wires) [126] or bycleaved-edge quantum well overgrowth (T-shaped wires)[127]. On the one hand, this approach allows an accuratedetermination of 1D-exciton binding energies; on the otherhand, it allows us to investigate the nonlinear (gain) regimeas well. In both cases, one finds a strong suppression ofthe 1D band-edge singularity, in agreement with previousresults based on simplified 1D models [121, 122].

This theoretical approach [123, 124] is based on thegeneral kinetic theory presented in section 2, which allowsa full 3D description of Coulomb interaction within amultiminiband scheme. In particular, we focus on thequasi-equilibrium regime where the solution of the coupledkinetic equations (44) simply reduces to the solution of theinterband-polarization equation. This is performed by direct

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Figure 10. Typical cross-section of V-grooved wiresderived from transmission electron micrographs and usedto define the confinement potential entering the 2Dsingle-particle Schrodinger equation. The wire width alongy at the apex of the V is about 10 nm. The dashed linesidentify the region (about 70 nm× 25 nm), wheresingle-particle and excitonic wavefunctions are localized.After [124].

numerical evaluation of the polarization eigenvalues andeigenvectors [124], which fully determine the absorptionspectrum as well as the exciton wavefunctions. Themain ingredients entering this calculation are the single-particle energies and wavefunctions, obtained numericallyfor the 2D confinement potential deduced, for example fromtransmission electron microscopy (TEM) as in [126].

The above theoretical scheme has been applied torealistic V- and T-shaped wire structures. In particular, herewe show results for the GaAs/AlGaAs V-wire structure of[126], whose cross-section is shown in figure 10.

5.1. Linear response: excitonic-absorption regime

Let us start by considering the optical response of thesystem in the low-density limit. In figure 11 we showthe linear-absorption spectra obtained when taking intoaccount the lowest wire transition only. Results of ourCoulomb-correlated (CC) approach are compared to thoseof the free-carrier (FC) model. As we can see, electron–hole correlation introduces two important effects. First, theexcitonic peak arises below the onset of the continuum, witha binding energy of about 12 meV, in excellent agreementwith recent experiments [126]. Second, one finds a strongsuppression of the 1D DOS singularity. A detailed analysisof the physical origin of such suppression [123, 124] hasshown that the quantity which is mainly modified by CCis the oscillator strength (OS). In figure 12(a) the ratiobetween the CC and FC OS is plotted as a function ofexcess energy (solid curve). This ratio is always less thanunity and goes to zero at the band edge, and reflects a sortof hole in the electron–hole correlation functiong(z) asshown in figure 12(b). Such vanishing behaviour is foundto dominate the 1D DOS singularity and, as a result, theabsorption spectrum exhibits regular behaviour at the bandedge (solid curve in figure 11).

The above analysis also shows that for realisticquantum-wire structures electron–hole correlation leads toa strong suppression of the 1D band-edge singularity inthe linear-absorption spectrum, contrary to the 2D and 3Dcases.

Figure 11. Linear-absorption spectra of realistic V-shapedquantum-wire structures obtained by including the firstelectron and hole subbands only. After [125].

5.2. Nonlinear response: gain regime

Most of the potential quantum-wire applications, i.e. 1Dlasers and modulators, operate in strongly nonlinear-response regimes [120]. In such conditions, the abovelinear-response analysis has to be generalized, taking intoaccount additional factors such as: (i) screening effects, (ii)band renormalization, (iii) space-phase filling.

Figure 13 reports quantitative results for nonlinearabsorption spectra of realistic V-shaped wire structuresat different carrier densities at room temperature. As areference we also show the results obtained by including thelowest subband only (figure 13(a)). In the low-density limit(case A:n = 104 cm−1) we clearly recognize the excitonpeak. With increasing carrier density, the strength of theexcitonic absorption decreases due to phase-space fillingand screening of the attractive electron–hole interaction,and moreover the band renormalization leads to a red-shift of the continuum. At a density of 4× 106 cm−1

(case D) the spectrum already exhibits a negative regioncorresponding to stimulated emission, i.e. gain regime.As desired, the well pronounced gain spectrum extendsover a limited energy region (smaller than the thermalenergy). However, its shape differs considerably fromthe ideal FC shape (curve marked with diamonds in thesame figure). In particular, the band-edge singularity in theideal FC gain spectrum is clearly smeared out by electron–hole correlation. The overall effect is a broader and lesspronounced gain region.

Finally, figure 13(b) shows the nonlinear spectracorresponding to the realistic case of a 12-subband V-shaped wire. In comparison with the single-subband case(figure 13(a)), the multisubband nature is found to play animportant role in modifying the typical shape of the gainspectra, which for both CC and FC models extends overa range much larger than that of the single-subband casefor the present wire geometry. In addition, the Coulomb-induced suppression of the single-subband singularities,here also due to intersubband-coupling effects, tends toreduce the residual structures in the gain profile. Therefore,even in the ideal case of a quantum wire with negligibledisorder and scattering-induced broadening, our analysis

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Figure 12. (a) OS ratio and DOS versus excess energy; (b) electron–hole correlation function. After [125].

Figure 13. Nonlinear absorption spectra:(a) single-subband case; (b) realistic 12-subband case.After [125].

indicates that, for the typical structure considered, theshape of the absorption spectra over the whole densityrange differs considerably from the sharp FC spectrum offigure 11.

This tells us that, in order to obtain sharp gain profiles,one of the basic steps in quantum-wire technology isto produce structures with increased subband splitting.However, the disorder-induced inhomogeneous broadening,not considered here, is known to increase significantly thespectral broadening and this effect is expected to increase

with increasing subband splitting. Therefore, extremelyhigh-quality structures (e.g. single-monolayer control) seemto be the only possible candidates for successful quantum-wire applications.

6. Summary and conclusions

A review of coherent phenomena in optically excited semi-conductors has been presented. Our analysis has allowedthe identification of two classes of phenomena:opticallyinduced and Coulomb-inducedcoherent phenomena. Wehave shown that both classes can be described in terms ofa unique theoretical framework based on the density-matrixformalism. Due to its generality, such a quantum-kineticapproach allows a realistic description of coherent as wellas incoherent, i.e. phase-breaking, processes, thus providingquantitative information on the coupled—coherent versusincoherent—carrier dynamics in photoexcited semiconduc-tors.

The primary goal of this review was to discuss theconcept of quantum-mechanical phase coherence as wellas its relevance to and implications for semiconductorphysics and technology. In particular, we have analysed thedominant role played by optically induced phase coherenceon the process of ultrafast carrier photogeneration. We havethen discussed typical field-induced coherent phenomena insemiconductor superlattices, e.g. Bloch oscillations and thecorresponding THz radiation, as well as their dephasingdynamics. Finally, we have analysed the dominant roleplayed by Coulomb correlation on the linear and nonlinearoptical spectra of realistic quantum-wire structures, namelythe Coulomb-induced suppression of ideal 1D-band-edgesingularities.

Our analysis shows that the conventional separationbetween coherent and incoherent regimes is no longervalid for most of the recent ultrafast optical experimentsin semiconductors. The reason is that on such extremelyshort time-scales the carrier dynamics is the result of astrong interplay between coherence and relaxation, thuspreventing a time-scale separation between photogenerationand relaxation dynamics. Similar considerations applyto the various Coulomb-induced phenomena. The strongmodifications induced by Coulomb correlation on the

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optical response of low-dimensional semiconductors play asignificant role in the phase-breaking relaxation dynamicsas well. It is thus evident that any theoretical analysisof the optical response of semiconductor heterostructuresmust provide a proper description of both coherent andincoherent phenomena as well as of their mutual couplingon the same kinetic level.

Acknowledgments

I am particularly grateful to Tilmann Kuhn for his essentialcontribution in understanding most of the ideas andconcepts discussed in this topical review. I wish to thankStephan W Koch, Torsten Meier and Peter Thomas, aswell as Elisa Molinari, for their relevant contributions tothe research activity reviewed in the paper. I am alsograteful to Roberto Cingolani, Thomas Elsaesser, AlfredLeitenstorfer and Peter E Selbmann for stimulating andfruitful discussions.

This work was supported in part by the EC Commissionthrough the Network ‘ULTRAFAST’.

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Coherent phenomena in semiconductors

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