Ultrafast carrier dynamics in semiconductor nanostructures: interplay between coherence and relaxation

Superlattices and Microstructures, Vol. 26, No. 2, 1999 Article No. spmi.1999.0766 Available online at http://www.idealibrary.com on

Ultrafast carrier dynamics in semiconductor nanostructures: interplay between coherence and relaxation FAUSTO ROSSI† Istituto Nazionale per la Fisica della Materia (INFM) and Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy (Received 7 May 1999)

The strong coupling between coherent and incoherent ultrafast phenomena in the electrooptical response of semiconductor nanostructures is discussed theoretically within a density matrix formalism. In particular, the problem of scattering-induced damping of Bloch oscillations in superlattices is reviewed. Moreover, a generalization to ‘open systems’ of the conventional semiconductor Bloch equations is discussed. The presence of spatial boundary conditions manifests itself through self-energy corrections and additional source terms in the kinetic equations. As an example, some simulated experiments of quantum transport phenomena through double-barrier structures are reviewed. c 1999 Academic Press

Key words: ultrafast processes, coherent phenomena, Bloch oscillations, dephasing.

1. Introduction Coherent and incoherent effects induced by electric and laser fields in semiconductors have received considerable attention for a long time [1–3]. More recently, the progress in the generation of ultrashort laser pulses, together with the development of spectroscopies on this timescale, has led to a series of experiments which give new insight into the microscopic carrier dynamics in semiconductors [4]. Also, recent progress in the fabrication and characterization of semiconductor heterostructures and superlattices has allowed a detailed study of a new class of phenomena induced by an applied electric field, such as Bloch oscillations and dynamic localization. Both classes of phenomena typically occur on a pico- or femtosecond timescale, where the coupling between coherent and incoherent carrier dynamics may play a dominant role [5]. Therefore, an adequate theoretical model of the ultrafast dynamics on this timescale must account for both coherent and incoherent effects on the same kinetic level. The aim of this paper is to review and discuss, from a theoretical point of view, the strong interplay between phase coherence and relaxation in the electro-optical response of semiconductor nanostructures. The paper is organized as follows: in Section 2, our theoretical approach, based on a density matrix formulation, is recalled and discussed. In Section 3, we review the problem of the scattering-induced damping of Bloch oscillations and of the corresponding THz emission [6–9]. Section 4 presents a generalization to open systems [10] of the † E-mail: [email protected]

0749–6036/99/080129 + 12 $30.00/0

c 1999 Academic Press

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density matrix approach introduced in Section 2; the method is then applied to the study of quantum transport phenomena in a double-barrier structure. Finally, in Section 5, we summarize and draw some conclusions.

2. Theoretical formulation 2.1. Physical system In order to properly describe ultrafast carrier dynamics in semiconductor nanostructures, let us consider a generic electron–phonon system, whose Hamiltonian can be schematically written as: ! X 0 0 H = H◦ + H = (Hc + Hp + HEM ) + Hs . (1) s

The single-particle term H◦ is the sum of the noninteracting-carrier and phonon Hamiltonians Hc and Hp plus the Hamiltonian HEM describing the interaction of the carrier system with an electromagnetic (EM) (static and/or optical) field. The many-body contribution H0 may include various interaction mechanisms, e.g. carrier–carrier, carrier–phonon, etc. More specifically, the single-particle Hamiltonian Hc describes the noninteracting carrier system within the potential profile of our mesoscopic structure. By denoting with φα (r) = hr|αi

(2)

the wavefunction of the single-particle state α and with α the corresponding energy, the noninteracting carrier Hamiltonian reads: X Hc = α aα† aα . (3) α

Here, the usual second-quantization picture in terms of creation (aα† ) and destruction (aα ) operators has been employed. The explicit form of the many-body contribution Hs0 in (1) depends on the particular interaction mechanism s considered. 2.2. Kinetic description The basic quantity in our theoretical approach is the single-particle density matrix [1, 5, 11]: ραβ = haβ† aα i.

(4)

The diagonal elements (%α ≡ ραα ) correspond to the usual distribution functions of the semiclassical Boltzmann theory [2] while the nondiagonal terms (α 6= β) describe the degree of quantum coherence (interlevel polarization) between levels α and β. In order to investigate transport as well as energy relaxation phenomena, our aim is to study the time evolution of single-particle quantities such as mean kinetic energy, charge current, etc. The average value of a generic single-particle operator A can be written in terms of its matrix elements Aαβ according to: X hAi = Aαβ ρβα = Tr(Aρ). (5) αβ

Therefore, in the Heisenberg picture the time evolution of A is fully described by the time evolution of the density matrix ρ.

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Starting from the Heisenberg equations of motion for the destruction operators aα , i.e. d 1 1 1 d d aα = [aα , H] = [aα , H◦ ] + [aα , H0 ] = aα + aα 0 , (6) H◦ dt H dt i~ i~ i~ dt it is possible to derive a set of equations of motion for the density matrix elements ραβ —often called semiconductor Bloch equations (SBE) [1]—whose general structure is given by: d d d ραβ = ραβ + ραβ 0 . (7) H◦ dt H dt dt The time evolution induced by the single-particle Hamiltonian H◦ can be evaluated exactly yielding: 1 1 X d ραβ = (α − β )ραβ + Tαβ,α 0 β 0 ρα 0 β 0 . (8) H◦ dt i~ i~ 0 0 αβ

Here, the first term (diagonal within the αβ-representation) is induced by Hc while the second one (describing a coherent interlevel coupling) is induced by the carrier-field Hamiltonian HEM . P On the contrary, the time evolution due to the many-body Hamiltonian H0 = s Hs0 involves, in general, phonon-assisted as well as higher-order density matrices. Thus, in order to ‘close’ our set of equations of motion (with respect to our kinetic variables in (4)) approximations are needed. A detailed discussion of the various approximation schemes—based on a dynamical expansion in powers of the interaction Hamiltonian H0 —is given in [11]. In particular, the ‘mean-field’ approximation together with the Markov limits allows us to derive a set of closed equations of motion still local in time. More specifically, for a number of interaction mechanisms, including carrier–phonon as well as carrier–carrier scattering, their contributions to the time evolution can be cast into the general form: X d in ∗ in∗ 1 0 0 0 ραβ 0 = 2 [(δαα − ραα )0α 0 β + (δβα − ρβα 0 )0α 0 α ] H dt α0 X out ∗ out∗ 1 0 (9) − 2 [ραα 0α 0 β + ρβα 0 0α 0 α ]. α0

Here, the matrices 0 can be regarded as generalized in- and out-scattering rates, in analogy with the Boltzmann collision term of the semiclassical theory [2]. They are, in general, complex quantities: their real parts describe energy relaxation and dephasing rates while their imaginary parts describe energy renormalization effects. Their explicit form for the case of carrier–phonon as well as carrier–carrier interactions are given in [11]. In the semiclassical limit the nondiagonal elements of the density matrix ρ are neglected, i.e. ραβ = %α δαβ , and the many-body contribution in eqn (9) reduces to: d in out %α 0 = (1 − %α )0αα − %α 0αα . (10) H dt The explicit form of the scattering matrices 0 given in [11] in the semiclassical limit reduces to X X in out 0αα = Wαα 0 %α 0 , 0αα = Wαα 0 (1 − %α0 ), (11) α0

α0

where Wαα 0 are transition probabilities as given by Fermi’s golden rule. By combining eqns (10) and (11) we then recover the usual Boltzmann transport equation: X d %α 0 = [(1 − %α )Wαα 0 %α0 − %α Wαα 0 (1 − %α 0 )]. (12) H dt 0 α

This clearly shows that in the semiclassical limit the above scattering model reduces to the conventional

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Boltzmann theory. As we can see from eqn (9), in the quantum case the total effect is also the result of a balance between in- and out-scattering contributions. In particular, for the case of quasi-elastic processes it is possible to show [11, 12] that, in analogy to the semiclassical case, the two in and out contributions strongly cancel, thus giving rise to long dephasing times. 2.3. Generalized Monte Carlo method From our previous analysis, it is easy to conclude that the SBE (7) can be regarded as the result of a coherent single-particle dynamics plus incoherent many-body contributions. Therefore, they can be solved by means of the same Monte Carlo (MC) simulation scheme described in [12]. The method is based on a time-step separation between coherent and incoherent dynamics. The former accounts in a rigorous way for all quantum phenomena induced by the potential profile of the device. The latter, described within the basis given by the eigenstates α of the potential profile, accounts for all the relevant scattering mechanisms by means of a generalized ‘ensemble’ MC simulation [12].

3. Application to ‘closed systems’: ultrafast photoexcitation regime In this section we will apply the general density matrix formulation discussed so far to the study of photoexcited electron–hole pairs in superlattices in the presence of a static electric field. To this end, it is convenient to move from the full electron description considered so far to the more familiar electron–hole picture. In this case the set of generic states α splits into electron (e) and hole (h) states, and the single-particle density matrix ρ introduced in (4) can be regarded as a 2 × 2 block matrix of the form:  e  f p† ρ= . (13) p fh More specifically, for an infinite superlattice structure the electron (hole) states are labeled by the quasimomentum k plus an electron (hole) miniband index i ( j). As kinetic variables, we consider here the electron e = hc† c i ( f h = hd † d i) as well as the (hole) distribution functions in minibands i ( j), i.e. f i,k i,k i,k j,k j,k j,k interband polarizations p ji,k = hd j,−k ci,k i. Within this electron–hole picture, the explicit form of the SBE in (7) in the presence of both a static field F and an optical field E is given by:   e ∂ f i,k ∂ eF 1 X e ∗ ∗ + ·∇k f i,k = (U i j 0 ,k p j 0 i,k − Uij0 ,k p j 0 i,k ) + ∂t ~ i~ ∂t inco

j0

 ∂ eF h + ·∇k f j,−k = ∂t ~   ∂ eF + ·∇k p ji,k = ∂t ~



Here,

h ∂ f i,−k 1 X (Ui 0 j,k p ∗ji 0 ,k − Ui∗0 j,k p ji 0 ,k ) + i~ 0 ∂t inco i

1 X e (E 0 δ j j 0 + E hjj 0 ,−k δii 0 ) p j 0 i 0 ,k i~ 0 0 ii ,k i j

∂ p ji,k 1 e h + Ui j,k (1 − f i,k − f j,−k ) + . i~ ∂t inco Ui j,k = µi j,k E(t) −

0

0

X

V (ik−kj 0−kkj i 0 ) p j 0 i 0 ,k0

X

e V (iikk00 ikk 0 i 00 ) f i 00 ,k00

i 0 j 0 ,k0

and e Eiie 0 ,k = i,k δii 0 −

i 00 ,k00

00

00

(14)

(15)

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E hjj 0 ,k =  hj,k δ j j 0 −

133

X

j 00 ,k00

00

00

kk h V (kk j 0 j 00 j j 00 ) f j 00 ,k00

(16)

are, respectively, the renormalized Rabi-frequencies and electron and hole energies [1, 6]. Here, V is the Coulomb matrix element while µ is the optical dipole matrix element between conduction- and valenceband states. The last terms on the right-hand sides of eqns (14) refer to incoherent contributions, i.e. various scattering processes. They are treated within the usual Markov approximation neglecting terms involving second or higher powers of the interband polarization, e.g. polarization scattering. As for the case of bulk semiconductors [12], within such an approximation scheme, the scattering terms have the structure of the usual Boltzmann collision term. We shall now review recent simulated experiments of the ultrafast carrier dynamics in semiconductor superlattices [6–9]. The superlattice model employed in our simulated experiments is described in [13]: the energy dispersion and the corresponding wavefunctions along the growth direction (kk ) are computed within the well-known Kronig–Penney model while for the in-plane direction (k⊥ ) an effective mass model has been used. Starting from these three-dimensional wavefunctions, the various carrier–carrier as well as carrier–phonon matrix elements are numerically computed. They are, in general, functions of the various miniband indices and depend separately on kk and k⊥ , thus fully reflecting the anisotropic nature of the superlattice structure. We will now start discussing the scattering-induced damping of Bloch oscillations. In particular, we will show that in the low-density limit this damping is mainly determined by optical–phonon scattering [6], while at high densities the main mechanism responsible for the suppression of Bloch oscillations is found to be carrier–carrier scattering [8]. All the simulated experiments presented in this section refer to the superlattice structure considered in Ref. [6]: 111 GaAs wells and 17 Al0.3 Ga0.7 As barriers. For such a structure, there has been experimental evidence for a THz emission from Bloch oscillations [14]. In the first set of simulated experiments an initial distribution of photoexcited carriers (electron–hole pairs) is generated by a 100 fs Gaussian laser pulse in resonance with the first miniband exciton (~ωL ≈ 1540 meV). The strength of the applied electric field is assumed to be 4 kV cm−1 , which corresponds to a Bloch period τB = h/eFd of about 800 fs. In the low-density limit (corresponding to a weak laser excitation), incoherent scattering processes do not alter the Bloch oscillation dynamics. This is due to the following reasons: in agreement with recent experimental [14, 15] and theoretical [6, 13] investigations, for superlattices characterized by a miniband width smaller than the LO phonon energy—as for the structure considered here—and for laser excitations close to the band gap, at low temperatures carrier–phonon scattering is not permitted. Moreover, in this lowdensity regime carrier–carrier scattering plays no role: due to the quasi-elastic nature of Coulomb collisions, in the low-density limit the majority of the scattering processes are characterized by a very small momentum transfer. As a consequence, the momentum relaxation along the growth direction is negligible. As a result, on this picosecond timescale, the carrier system exhibits a coherent Bloch oscillation dynamics, i.e. a negligible scattering-induced dephasing. This can be clearly seen from the time evolution of the carrier distribution as a function of kk (i.e. averaged over k⊥ ) shown in Fig. 1. During the laser photoexcitation (t = 0) the carriers are generated around kk = 0, where the transitions are close to resonance with the laser excitation. According to the acceleration theorem, the electrons are then shifted in k-space. When the carriers reach the border of the first Brillouin zone they are Bragg reflected. After about 800 fs, corresponding to the Bloch period τB , the carriers have completed one oscillation in k-space. As expected, the carriers execute Bloch oscillations without losing the synchronism of their motion by scattering. This is again shown in Fig. 1, where we have plotted: B, the mean kinetic energy; C, the current; and D, its time derivative which is proportional to the emitted far field, i.e. the THz radiation. (It can be shown that, by neglecting Zener tunneling, the time derivative of the intraband polarization P e/ h is proportional to the current.) All three quantities exhibit

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Kinetic energy (meV)

134

A

200 fs

B 10 5 0

Current (a.u.)

C 400 fs

0

600 fs

THz signal (a.u.)

Distribution function (a.u.)

0 fs

15

800 fs

D

0

1 ps

–1.0

–0.5

0.0 k|| (πd)

0.5

1.0

0

1

2

3

4

Time (ps)

Fig. 1. Full Bloch oscillation dynamics corresponding to a laser photoexcitation resonant with the first miniband exciton: A, time evolution of the electron distribution as a function of kk ; B, average kinetic energy; C, current; and D, THz signal corresponding to the Bloch oscillations in A (see text). From Ref. [8].

oscillations characterized by the same Bloch period τB . Due to the finite width of the carrier distribution in k-space (see Fig. 1A), the amplitude of the oscillations of the kinetic energy is somewhat smaller than the miniband width. Since for this excitation condition the scattering-induced dephasing is negligible, the oscillations of the current are symmetric around zero, which implies that the time average of the current is equal to zero, i.e. no dissipation. The above theoretical analysis closely resembles experimental observations obtained for a superlattice structure very similar to the one modelled here [14]. In these experiments, evidence for THz emission from Bloch oscillations has been reported. In order to study the density dependence of the Bloch oscillation damping, let us return to the case of laser excitations close to gap. Figure 2A shows the total (electrons plus holes) THz radiation as a function of time for three different carrier densities. With increasing carrier density, carrier–carrier scattering becomes more and more important: due to Coulomb screening, the momentum transfer in a carrier–carrier scattering increases (its typical value being comparable with the screening wavevector). This can be seen in Fig. 2A, where for increasing carrier densities we realize an increasing damping of the THz signal. However, for the highest carrier density also considered here, we deal with a damping time of the order of 700 fs, which is much larger than the typical dephasing time, i.e. the decay time of the interband polarization, associated with carrier–carrier scattering. The dephasing time is typically investigated by means of four-wavemixing (FWM) measurements and such multi-pulse experiments can be simulated as well [16]. From a theoretical

Superlattices and Microstructures, Vol. 26, No. 2, 1999 n = 1 × 1013 n = 1 × 1015 n = 1 × 1017

A

n = 1 × 1013 n = 1 × 1015 n = 1 × 1017

B

2.0 Polarization

Total THz radiation (a.u.)

4.0

135

0.0

–2.0

–4.0 0

1

2 Time (ps)

3

4

0.0

0.5

1.0

Time (ps)

Fig. 2. Total THz radiation A, and incoherently summed polarization B, as a function of time (see text). From Ref. [8].

point of view, a qualitative estimate of the dephasing time is given by the decay time of the ‘incoherently summed’ polarization (ISP) [12]. Figure 2B shows such ISP as a function of time for the same three carrier densities of Fig. 2A. As expected, the decay times are always much smaller than the corresponding damping times of the THz signals (note the different timescale in Figs 2A and B). This difference, discussed in more detail in [8], can be understood as follows: the fast decay times of Fig. 2B reflect the interband dephasing, i.e. the sum of the electron and hole scattering rates. In particular, for the Coulomb interaction this means the sum of electron–electron, electron–hole, and hole–hole scattering. As for the case of bulk GaAs, this last contribution is known to dominate and determines the dephasing timescale. On the other hand, the total THz radiation in Fig. 2A is the sum of the electron and hole contributions. However, due to the small value of the hole miniband width compared to the electron one, the electron contribution will dominate. This means that the THz damping in Fig. 2A mainly reflects the damping of the electron contribution. This decay, in turn, reflects the intraband dephasing of electrons which is due to electron–electron and electron–hole scattering only, i.e. no hole–hole contributions. From the above analysis we can conclude that the decay time of the THz radiation due to carrier–carrier scattering differs considerably from the corresponding dephasing times obtained from a FWM experiment: the first one is a measurement of the intraband dephasing while the second one reflects the interband dephasing.

4. Generalization to ‘open systems’: quantum transport regime The analysis presented so far is typical of a so-called ‘closed’ system, i.e. defined over the whole coordinate space. However, this is not the case of interest for the study of quantum transport phenomena in mesoscopic devices, where the properties of the carrier subsystem are strongly influenced by the spatial boundaries with the external environment. In this section we shall review a generalization to open systems [10] of the density matrix formulation discussed in Section 2. To this aim, let us start rewriting the SBE (7) as: X d ραβ = L αβ,α 0 β 0 ρα 0 β 0 . (17) dt 0 0 αβ

Here, L is an effective Liouville operator, whose explicit form, given in [10], involves the single-particle energies  as well as the in- and out-scattering operators 0 in (9).

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A

Density

t = 0 fs

t = 100 fs

t = 250 fs

0

200

400

600

Position (nm) B

Density

t = 0 fs

t = 100 fs

t = 250 fs

0.00

0.10

0.20

0.30

0.40

Energy (eV) Fig. 3. Carrier density at different times as a function of position A, and energy B, corresponding to an electron wavepacket injected into a RTD structure in the absence of scattering processes (the two barriers are schematically depicted as black vertical lines). From Ref. [10].

The study of open systems requires a real-space description, which can be obtained in terms of the phasespace formulation of quantum mechanics originally proposed by Wigner [17]. In our case, this corresponds to the following unitary transformation u connecting our αβ representation to the desired phase-space r, k: Z     0 − 32 dr0 φα r + 12 r0 e−ik·r φβ∗ r − 12 r0 . (18) u αβ (r, k) = (2π ) By applying the above Weyl transform to the single-particle density matrix ρ in (4), we obtain the so-called Wigner function [18]: X f W (r, k) = ραβ u αβ (r, k). (19) αβ

W

For a closed system, f is defined for any value of the real-space coordinate r and its time evolution is fully determined by its initial condition. In contrast, for an open system f W is defined only within a given region  of interest and its time evolution is determined by the initial condition inside such region as well as

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A

Density

t = 0 fs

t = 100 fs

t = 250 fs

0

200

400

600

Position (nm) B

Density

t = 0 fs

t = 100 fs

t = 250 fs

0.00

0.10

0.20

0.30

0.40

Energy (eV) Fig. 4. Same as in Fig. 3 but in the presence of scattering processes (see text). From Ref. [10].

by its values f bW on the boundary rb of the domain at any time t 0 > t◦ . More specifically, by applying the Green’s function theory to the equation of motion for f W —which is obtained by applying to eqn (17) the Weyl–Wigner transform (19)—we obtain: Z Z f W (r, k; t) = dr0 dk0 G(r, k; r0 , k0 ; t − t◦ ) f W (r0 , k0 ; t◦ )  Z Z Z t + drb dk0 dt 0 G(r, k; rb , k0 ; t − t 0 ) f bW (rb , k0 , t 0 )v(k0 ), (20) t◦

where G(r, k; r0 , k0 ; τ ) =

X

αβ,α 0 β 0

u αβ (r, k)[e Lτ ]αβ,α 0 β 0 u ∗α0 β 0 (r0 , k0 )

(21)

is the evolution operator, while v(k) is the component of the carrier group velocity normal to the boundary surface. We clearly see that the value of f W is obtained from the propagation of the initial condition f W (t◦ )

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inside the domain  plus the propagation of the boundary values f bW from the points of the surface at any time t 0 to the point r, k of interest. Given the above Wigner formulation for open systems, we now introduce a corresponding density matrix description via the following ‘inverse’ Weyl–Wigner transform: Z Z ρ αβ = dr dku ∗αβ (r, k) f W (r, k). (22) 

By applying the above transformation to eqn (20) and then performing its time derivative, we finally obtain: X d ρ αβ = L αβ,α 0 β 0 ρ α0 β 0 + S αβ , (23) dt 0 0 αβ

where L = U LU −1 is the Liouville tensor in (17) ‘dressed’ by the transformation Z Z Uαβ,α 0 β 0 = dr dku ∗αβ (r, k)u α 0 β 0 (r, k), 

(24)

while S αβ =

X α0 β 0

Uαβ,α 0 β 0

Z

drb

Z

dku ∗α 0 β 0 (rb , k)v(k) f bW (rb , k)

(25)

is a source term induced by our spatial boundary conditions. Equation (23) is the desired generalization to the case of open systems of the conventional SBE in eqn (17). In addition to the source term in eqn (25), the presence of spatial boundary conditions induces modifications on the Liouville operator L of the system via the transformation U in eqn (24). In order to illustrate the power and flexibility of the above generalized SBE, we have simulated quantum transport phenomena in rather different physical systems, namely double-barrier structures and superlattices [10]. Since we were mainly interested in low-temperature and low-carrier density conditions, only optical–phonon scattering has been considered. Here, we review and discuss simulations of an electron wavepacket entering the double-barrier structure‡ of a GaAs/AlGaAs resonant tunneling diode (RTD). Figure 3 shows the time evolution of the wavepacket in the absence of scattering as a function of position A, and energy B. It is easy to recognize the wellestablished resonance scenario typical of any purely coherent dynamics: as the wavepacket enters our RTD structure, a part of it is transmitted and a part is reflected (see Fig. 3A). Since in this simulation scattering is not included, the wavepacket central energy is conserved, i.e. no energy relaxation occurs (see Fig. 3B). On the contrary, in the presence of incoherent scattering processes the resonance dynamics of Fig. 3A are strongly modified by the scattering itself, as shown in Fig. 4A. In particular, the presence of phase-breaking scattering processes is found to reduce both the interference peaks and the transmitted wavepacket. This is confirmed by the corresponding energy distribution in Fig. 4B, where we clearly recognize the granular nature of the dissipation process through the formation of optical–phonon replica. This is the fingerprint of any full microscopic treatment of energy relaxation, thus confirming the microscopic nature of our quantum mechanical simulation, in contrast with all previous phenomenological approaches.

5. Conclusions In summary, we have reviewed the problem of coherent versus incoherent carrier dynamics in semiconductor nanostructures from a theoretical point of view. In particular, we have recalled and discussed the density matrix theory as applied to the ultrafast-photoexcitation regime. In this context we have reviewed the problem ‡ The barrier height is 0.5 eV while the barrier width and separation are, respectively, 20 and 60 .

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of scattering-induced damping of Bloch oscillations in superlattices as well as the problem of inter- versus intraband dephasing. Moreover, we have presented a generalization to open systems of the conventional SBE. As for the damping of Bloch oscillations in superlattices, this generalization to open systems allows for a proper description of the strong coupling between coherent and incoherent dynamics. Indeed, all the simulated experiments reviewed in this paper clearly show the failure of any purely coherent or incoherent approach in describing typical quantum transport phenomena in semiconductor nanostructures. Acknowledgements—I wish to thank Stephan W. Koch, Torsten Meier, and Peter Thomas, as well as Aldo Di Carlo and Paolo Lugli for their relevant contributions to the research activity reviewed in the paper. I am also grateful to Carlo Jacoboni, Tilmann Kuhn, and Elisa Molinari for stimulating and fruitful discussions. This work was supported in part by the EC Commission through the Network ‘ULTRAFAST QUANTUM OPTOELECTRONICS’.

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