Download Optical properties of semiconductor quantum dots

Optical properties of semiconductor quantum dots: Few-particle states and
coherent-carrier control
Ulrich Hohenester†
Istituto Nazionale per la Fisica della Materia (INFM) and Dipartimento di Fisica, Università di Modena e Reggio Emilia,
Via Campi 213A, I-41100 Modena, Italy
In quantum dots, often referred to as artificial atoms, electrons and holes are confined in all three space
directions. This strong quantum confinement results in a discrete, atomic-like carrier density of states.
In turn, the coupling to the solid-state environment (e.g., phonons) is strongly suppressed and Coulomb
correlations among charge carriers are strongly enhanced. Indeed, in the optical spectra of single dots
spectrally narrow emission peaks have been observed (indicating a small environment coupling), which
undergo discrete energy shifts when more carriers are added to the dot (indicating energy renormalizations
due to additional Coulomb interactions). In this paper we first give a short introduction into this rapidly
growing field of research and review some of the recent experimental and theoretical work. We then provide
details of a full configuration-interaction approach for the calculation of few-particle electron-hole states and
optical properties of semiconductor quantum dots. Finally, a short survey over the recent work of the author
is presented.
PACS numbers: 85.30.Vw,71.35.-y,73.20.Dx,78.66.Fd
CONTENTS
I. Introduction
II. Carrier states in semiconductor quantum dots
A. Physical system
B. Single-particle states
C. Coulomb interactions
D. Few-particle states
III. Optical properties of quantum dots
A. Light-semiconductor coupling
B. Linear absorption
C. Nonlinear absorption
D. Luminescence
E. Coherent-carrier control
IV. Results
A. Near-field spectra
B. Multi-excitons
C. Multi-charged excitons
D. Coherent-carrier control
V. Summary
Acknowledgments
A. Single-particle states and matrix elements
1. Single-particle states
2. Optical matrix elements
3. Coulomb matrix elements
B. Few-particle states
1. Direct diagonalization
2. Optical matrix elements
3. Biexcitons
References
I.
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INTRODUCTION
In recent years much attention is being devoted to the
properties of semiconductor quantum dots. In these sys-
† Electronic
address: [email protected]
tems, carriers are subject to a confining potential in all
spatial directions, giving rise to a discrete energy spectrum (“artificial atoms”) and to novel phenomena of interest for fundamental physics as well as for applications to electronic and optoelectronic devices (Bimberg
et al. 1998, Jacak et al. 1998). The extension and the
shape of the dot confining potential varies, depending
on the nanostructure fabrication technique: The dots
that are studied most extensively by optical methods
are induced by quantum well (QW) thickness fluctuations (Brunner et al. 1994, Flack et al. 1996, Gammon
et al. 1996, Hess et al. 1994, Zrenner et al. 1994), or
obtained by spontaneous island formation in strained
layer epitaxy (Grundmann et al. 1995, Leon et al. 1995,
Marzin et al. 1994), self-organized growth on patterned
substrates (Hartmann et al. 2000a), stressor-induced QW
potential modulation (Obermüller et al. 1999, Rinaldi
et al. 2000) cleaved-edge overgrowth (Schedelbeck et al.
1997, Wegscheider et al. 1997), as well as chemical selfaggregation techniques (Alivisatos 1996, Murray et al.
1995). The resulting confinement lengths fall in a wide
range between 1µm and 10 nm.
In spite of the continuing progress, all the available
fabrication approaches still suffer from the effects of inhomogeneity and dispersion in dot size, which leads to
large linewidths when optical experiments are performed
on large dot ensembles. A major advancement in the
field has come from different types of local optical experiments, that allow the investigation of individual quantum
dots thus avoiding inhomogeneous broadening (see Zrenner 2000, for a review). With these advances it became
possible to study in great detail a variety of novel physical phenomena, such as fine-structure splitting (Gammon et al. 1996), atomic-like excitation spectra (Brunner
et al. 1994, Gammon et al. 1995, Hessman et al. 1996),
2
Coulomb renormalizations of few-particle states (Bayer
et al. 2000a, Dekel et al. 1998, Hartmann et al. 2000a,
Landin et al. 1998, Motohisa et al. 1998, Warburton et al.
2000), interdot couplings (Schedelbeck et al. 1997), Zeeman splittings and diamagnetic shifts (Bockelmann et al.
1997, Cingolani et al. 1999, Findeis et al. 2000b, Heller
and Bockelmann 1997, Rinaldi et al. 1996, 1998).
The theoretical analysis of such experiments requires a
detailed understanding of carrier states in quantum dots
(accounting for effects such as band offsets, strain, or
piezoelectric fields) and of Coulomb-type renormalizations of few-particle carrier states. Single-particle electron and hole states have been computed for small quantum dots by use of ab-initio approaches (Rohlfing and
Louie 1998), and for larger dots by approximate techniques such as pseudopotential (Wang et al. 1999), eightband k.p (Jiang and Singh 1997, Pryor 1998, Stier et al.
1999), or effective-mass studies (Braskén et al. 1997,
Jacak et al. 1998). Coulomb renormalizations of fewparticle electron-hole states have been treated within restricted (Hawrylak 1999, Narvaez and Hawrylak 2000)
or full (Braskén et al. 2000, Hartmann et al. 2000a, Hohenester and Molinari 2000a) configuration-interaction
models, or by use of density-functional (Nair and Masumoto 2000) or density-matrix descriptions (Hohenester
and Molinari 2000a, Hohenester et al. 1999b).
In this paper we first review some of the recent work
about carrier states and optical properties of semiconductor quantum dots. Details of our computational approach will be presented. We have broken down this paper into the following parts: In Sec. II we discuss different
approaches for calculating single-particle and Coulombrenormalized few-particle states; Sec. III is devoted to a
discussion of optical spectroscopy techniques and of the
theoretical framework accounting for light-semiconductor
coupling; Sec. IV briefly discusses recent papers of the author and presents introductory material; finally, in Sec. V
we draw some conclusions.
II. CARRIER STATES IN SEMICONDUCTOR
QUANTUM DOTS
In this section we set out to review the basic ideas underlying the theoretical description of few-particle states
in semiconductor quantum dots.
A. Physical system
We shall consider the problem of carriers confined in a
single quantum dot under the action of external electromagnetic fields. The carriers will experience their mutual Coulomb interactions as well as the interaction with
phonons or other environmental degrees of freedom. It
turns out to be convenient to describe the problem by
the Hamiltonian:
H = (Hc + Hcc ) + Hop + Henv ,
(1)
where Hc accounts for non-interacting carriers (electrons
and holes) in presence of the dot confinement potential and of possible time-independent external electric
and magnetic fields; Hcc describes Coulomb interactions
among charge carriers; Hop accounts for the coupling to
light; finally, Henv describes the coupling to environmental degrees of freedom (e.g., phonons or carriers in the
wetting layer). 1
B. Single-particle states
The three-dimensional confinement of carrier states in
semiconductor quantum dots is due to the use of materials with different conduction and valence band offsets and due to strain fields and related effects (e.g.,
piezoelectric fields; Bimberg et al. 1998). In the following we briefly discuss the most widely used theoretical
approaches accounting for such effects, and develop the
theoretical framework used in this work.
1. Ab-initio approaches
True ab-initio calculations of carrier states in semiconductor quantum dots are inherently restricted to dots
containing only a few tens to hundreds of atoms; in this
respect, hydrogen-terminated silicon clusters represent
the most thoroughly studied class of systems (see, e.g.,
Rohlfing and Louie 1998). The theoretical framework for
the calculation of the electronic and optical properties is
usually based on the density-functional theory within the
local-density approximation (Dreizler and Gross 1990);
in addition the GW method (Arayasetiawan and Gunnarson 1998) is employed to account for electron-hole
Coulomb correlations (Albrecht et al. 1998, Rohlfing and
Louie 2000).
However, for the quantum dots of our present concern,
which typically consist of several thousands of atoms,
such first-principles techniques are not directly applicable. To overcome these limitations a number of different approaches have been proposed. Zunger and collaborators have developed an atomistic pseudopotential approach (see, e.g., Franceschetti et al. 1998, Wang et al.
1999, and references therein) with a suitably chosen parameterization of pseudopotentials; for details the reader
is referred to the literature.
1 Note that our choice of accounting for static electric and magnetic fields in Hc and for light coupling in Hop is somewhat arbitrary, and is primarily thought to highlighten the important role of
carrier-light coupling in our present study of the optical properties
of semiconductor quantum dots.
3
2. Envelope-function approximation
Another solution scheme, which we shall pursue in this
work, is provided by the envelope-function approximation
for the calculation of carrier states in quantum dots (see,
e.g., Haug and Koch 1993, Yu and Cardona 1996). Here,
the main assumption is that the dot confinement potential varies smoothly on a length scale of the lattice constant.2 The low-energy carrier wavefunctions then approximately consist of bulk-like atomic parts un,k∼
=0 (r)
(with n the band index and k the wavevector) which
are smoothly modulated by the envelope function φn (r)
(for details see Yu and Cardona 1996). Within the spirit
of the envelope-function approximation all details of the
atomic part un,k∼
=0 (r) of the carrier wavefunctions are
lumped into a few band-structure parameters (e.g., effective masses or optical bulk matrix elements, whose values
are either fitted to experiment or obtained from firstprinciples calculations), and only the envelope function
φn (r) is kept to describe the modulations of the atomic
bulk wavefunctions.
a. k.p approach
Conveniently the envelope-function approach comes
together with the k.p approximation (Kane 1966), which
provides an efficient approach for calculating the material
dispersion and wavefunctions. Details of this approach
depend on the number of bands considered in the calculation of the total wavefunction
ψ(r) =
X
un,k∼
=0 (r)φn (r).
(2)
n
For III-V semiconductors an exhaustive description of the
bandstructure around the Γ point is provided by an eightband model (Kane 1966, Trebin et al. 1979) containing: the s-like conduction-band states |s, ± 21 i (with spinup and spin-down orientation); the p-like valence-band
states | 32 , ± 32 i, | 32 , ± 12 i (heavy-hole and light-hole bands
in bulk semiconductors) and | 12 , ± 12 i (spin-orbit split-off
band). Within this scheme carrier states in quantum dots
are described by the effective one-particle Hamiltonian
(Jiang and Singh 1997, Pryor 1998, Stier et al. 1999):
h = t + hstr + v + δh,
(3)
with: t the kinetic term (in presence of possible external electric and magnetic fields), whose eigenvalues give
the bulk bandstructure; hstr the strain contributions; v
2 Note that the envelope-function approximation even provides
an excellent description for quantum-well-like confinement where
this assumption is no longer valid.
the confinement potential due to the conduction- and
valence-band offsets; finally, δh accounts for higher-order
contributions within the k.p approach which lead to a
mixing of electron (hole) states with different spin orientations (Khaetskii and Nazarov 2000, Maialle et al. 1993).
Eight-band k.p studies of carrier states in pyramidal
InAs/GaAs self-assembled quantum dots have been carried out by Jiang and Singh (1997), Pryor (1998), and
Stier et al. (1999). In the calculations for these dots the
authors have demonstrated the importance of treating
bandstructure effects at such high level of accuracy to account for the strong valence-band mixing due to the substantially different band offsets (gap energy Eg = 1.519
eV in GaAs vs. Eg = 0.418 eV in InAs) and due to the
pronounced effects of strain fields (lattice mismatch of
approximately seven percent).
We note here in passing that, despite the accuracy of
such 8×8 k.p studies, the details of carrier wavefunctions
in InAs dots are still a matter of intense debate. This controversy is due to the open issues of the detailed shape of
overgrown InAs/GaAs dots and of possible In diffusion
into the surrounding GaAs (Liu et al. 2000). Using Starkeffect spectroscopy together with theoretical modeling,
Fry et al. (2000) have shown that in capped InAs/GaAs
dots holes can become localized above electrons—in contrast to the predictions of calculations assuming pyramidal dot structures (Stier et al. 1999).
b. Effective-mass approximation
In this work we shall make further simplifications as
compared to the above envelope-function and k.p approach. First, we shall not treat strain-induced effects explicitly, but shall rather assume that an effective confinement potential —accounting for band offsets and strain
effects— can be provided. Such a procedure is well justified for epitaxially grown quantum dots (see, e.g., Hartmann et al. 1997, 1998) where strain can be safely neglected; in addition, because of the uncertainties regarding the detailed shape and material composition of many
dots, an appropriately parameterized confinement potential, with parameters possibly fitted to experiment, is
known to provide a sound approach for semi-quantitative
comparison with experiment (Bayer et al. 2000a, Hawrylak et al. 2000, Rinaldi et al. 2000). Second, we make
use of the effective-mass approximation; for electrons
we assume an isotropic mass me , and for heavy holes
↔
a diagonal effective-mass tensor mh = diag(mt , mt , ml )
with mt = mo /(γ1 + γ2 ) the transverse mass in (x, y)directions and ml = mo /(γ1 − 2γ2 ) the longitudinal mass
in z-direction (Haug and Koch 1993, Liu and Bryant
1999), where mo is the free-electron mass and γ1 , γ2 are
Luttinger’s parameters; contributions from the light-hole
band are neglected because of the usually narrow confinement in z-direction and the resulting large inter-subband
splitting. Inclusion of valence-band mixing is expected
to introduce minor quantitative changes in the electronic
4
energies and envelope functions of the quantum dot system (see Braskén et al. 1997, for a quantitative estimate
of valence-band mixing).
The single-particle properties for electrons and holes
in presence of an external magnetic field B = (0, 0, B) in
z-direction are then described by (Haug and Koch 1993,
Rossi 1998):
he = te (πe ) + ve (r) − eΦ(r) + µB ge Sz B + δhe
hh = th (πh ) + vh (r) + eΦ(r) − 31 µB gh Jz B + δhh ,
(4)
with the kinetic terms te (p) = p2 /(2me ) and th (p) =
↔ −1
1
e
p; the momenta πe,h = −i∇± 2c
(B×r) within
2 p(mh )
symmetric gauge; the confinement potentials ve (r) and
vh (r) for electrons and holes, respectively; the scalar potential Φ(r) accounting for an external electric field; the
Zeeman interaction energies µB ge Sz B and 31 µB gh Jz B for
electrons and holes, respectively (Blackwood et al. 1994,
van Kesteren et al. 1990), with µB Bohr’s magneton, ge
the electron g-factor, gh the hole g-factor, Sz = ± 12 the
electron spin orientation, and Jz = ± 23 the hole angular
momentum in z-direction; finally δhe and δhh are contributions which mix electron (hole) states with different
spin orientations. The eigenenergies eµ and hν for electrons and holes, respectively, and the corresponding envelope wavefunctions φeµ (r) and φhν (r) are obtained from
the solutions of Eq. (4), as outlined in the following section and in appendix A.1.
been demonstrated to be a good approximation for selfassembled quantum dots formed by strained-layer epitaxy (Jacak et al. 1998). Here:
v(r) =
1
d
mt ωo2 (x2 + y 2 ) + Vo θ(|z| − ),
2
2
(6)
where ωo is the level splitting due to the in-plane harmonic potential, Vo is the band offset, and d is the width
of the quantum well. With this choice of the confinement potential the envelope-function equation factorizes
into the in-plane and longitudinal parts, which both can
be solved analytically. The solutions of the in-plane part
provide the well-known Fock-Darwin states, as outlined
in some detail by Jacak et al. (1998) or Chakraborty and
Pietiläinen (1995).
C. Coulomb interactions
When the dot is populated by more than one electron
or hole the properties of few-particle carrier states are not
only governed by the single-particle energies and states
but, in addition, by mutual Coulomb interactions among
charge carriers. In nanoscale semiconductor quantum
dots such Coulomb renormalizations are enhanced because of the strong three-dimensional quantum confinement. Depending on the approximations employed in the
calculation of single-particle states Coulomb interactions
have to be treated on different levels of sophistication.
3. Solution of the single-particle envelope-function equation
1. Ab-initio approaches
In our computational approach we start from the
single-particle envelope-function Hamiltonian (with magnetic field in z-direction)
Ab-initio-type calculations, such as the densityfunctional–local-density-approximation or pseudopotential approach, provide wavefunctions ψ which show
rapid oscillations on the length scale of the lattice constant (atomic part) with smooth modulations on the
length scale of the semiconductor nanostructures (envelope part). When computing Coulomb matrix elements
within this ψ-basis dielectric screening has to be treated
with special care. Quite generally, dielectric screening in semiconductors is a dynamic process where one
charge carrier (electron or hole) polarizes its surrounding medium; in turn, a second carrier experiences not
only the bare Coulomb potential exerted by the first carrier but, in addition, also by the induced polarization
cloud. From bulk semiconductors it is well known that
such polarization-induced screening can reduce the bare
Coulomb interaction by a factor of the order of ten.3
The dielectric screening function κ(r, r 0 ) is a nonlocal quantity. Within ab-initio-type approaches κ is
h=
p2x + p2y
p2
B 2 (x2 + y 2 ) B lz
+ z + v(r) +
−
, (5)
2mt
2ml
2mt
2mt
where mt and ml are the transversal and lateral mass,
respectively, l = r×p is the angular-momentum operator
and B = ±eB/c. Here, on the right-hand side: the first
two terms account for the kinetic energy in the in-plane
and longitudinal directions; the third term accounts for
the diamagnetic shift in presence of an external magnetic
field; finally, the last term describes the Zeeman splitting
of states with non-zero angular momentum.
For arbitrary confinement potentials v(r), Eq. (5) is
solved by lattice discretization and direct diagonalization of the Hamiltonian matrix; details of our numerical
approach are presented in appendix A.1.
In this paper we shall alternatively assume a prototypical dot confinement which is parabolic in the (x, y)-plane
and box-like along z; such confinement potentials have
3 We shall neglect the frequency-dependence of the dielectric
screening function throughout and assume static screening.
5
treated at different levels of sophistication (Franceschetti
et al. 1998, Ögüt et al. 1997, Rohlfing and Louie 1998,
2000), where the details of dielectric screening are still
matter of intense debate (see, e.g., Delerue et al. 2000,
Franceschetti et al. 1999, Ögüt et al. 1999, Reboredo
et al. 1999).
2. Envelope-function approximation
Dielectric screening within the envelope function approximation is usually accounted for by reducing the bare
Coulomb matrix elements by the static dielectric constant κ. In this work we only keep monopole-monopole
Coulomb terms which conserve the number of electrons
and holes and concentrate on the two most important
interaction channels: direct Coulomb interactions, which
are responsible for the strong few-particle renormalizations; short-ranged electron-hole exchange interactions,
which give rise to the bright-dark splitting of excitons.
Other types of Coulomb interactions, such as Auger-type
processes or terms which lead to the splitting of longitudinal and transversal excitons, are discussed in some detail
by Axt and Mukamel (1998) and Maialle et al. (1993).
where the bright-dark splitting az of excitons in quantum dots is enhanced with respect to its bulk value because of motional narrowing (Maialle et al. 1993), i.e., the
confinement-induced increased overlap of electron and
hole wavefunctions; for InGaAs/GaAs dots Bayer et al.
(1999, 2000b) have reported a bright-dark exciton splitting of 200 µeV as compared to the 20 µeV bulk GaAs
value, i.e., an enhancement by a factor of ten. If not
noted differently, in the following we shall not consider
the fine splitting due to the electron-hole exchange interaction.
D. Few-particle states
We next show how to compute few-particle states of
quantum dots. In our calculations we account explicitly
for the dot confinement and external fields (Hc ) and for
mutual Coulomb interactions among electrons and holes
(Hcc ). It turns out to be convenient to introduce the
m †
Fermionic field operators (ψµσ
) which create an electron
(hole) in single-particle state µ with spin orientation σ =
±.4 With this notation the single-particle Hamiltonian
Hc of Eq. (1) is of the form Hoe + Hoh , with:
Hom =
a. Direct Coulomb interaction
X
m † m
m
µσ (ψµσ ) ψµσ ,
(8)
µσ
With the envelope functions φm
µ (r) (m = e, h) the direct Coulomb matrix elements are of the form (Haug and
Koch 1993, Rossi 1998):
and the Coulomb term Hcc reads:
Hcc =
∗
n∗ 0 n 0 m
φm
µ0 (r)φν 0 (r )φν (r )φµ (r)
,
κ|r − r 0 |
(7)
where qm = ∓e for electrons and holes, respectively. Details of our computational approach for calculation of V
are presented in appendix A.3. We note that in case of
cylinder symmetry of the dot confinement the angular
momentum in z-direction becomes a constant of motion
and the Coulomb-matrix elements conserve total angular
momentum (Jacak et al. 1998).
Vµmn
0 µ,ν 0 ν = qm qn
Z
drdr 0
b. Electron-hole exchange interaction
The electron-hole exchange interaction accounts for the
fact that an electron which is promoted from the valence
to the conduction band no longer experiences the exchange interactions with all valence-band electrons. In
the electron-hole picture this process gives rise to a repulsive interaction between electrons and holes with opposite spin orientations. Following the work of van Kesteren
et al. (1990) and Blackwood et al. (1994) the short-range
electron-hole exchange interaction is described in lowest
order by an effective Hamiltonian of the form az Jz Sz ,
1
2
X
X
m †
n
† n
m
Vµmn
0 µ,ν 0 ν (ψµ0 σ ) (ψν 0 σ 0 ) ψνσ 0 ψµσ .
mn,σ,σ 0 µµ0 ,νν 0
(9)
(Note that in Eq. (9) we have not included the electronhole exchange interaction.)
1. Direct diagonalization
In our computational approach the few-particle
electron-hole states are computed within a full
configuration-interaction model (McWeeny 1992). We
first construct a set of basis functions |`i within the
Hilbert space of electron-hole excitations; appropriate
truncation of the Hilbert space to approximately ten
single-particle states and 100–1000 few-particle states of
lowest energy for electrons and holes, respectively, is necessary to keep the numerics tractable (for details see appendix B.1). We then obtain for a given number of electrons and holes the few-particle energies Eλ and states
Ψλ` = h`|λi by direct diagonalization of the eigenvalue
problem:
4 For notational simplicity in the following we shall write for
holes σ = ± instead of Jz = ± 32 .
6
X
(Hc + Hcc )``0 Ψλ`0 = Eλ Ψλ` ,
(10)
`0
with H``0 = h`|H|`0 i. Such full configuration-interaction
(CI) approaches appear to be the method of choice for the
quantum dots of our present concern, since they allow to
treat Coulomb renormalizations with as few approximations as possible (i.e., a suitable truncation of the Hilbert
space) and since, within the strong-confinement regime,
only a limited number of basis functions has to be retained (thus keeping the numerics tractable for a moderate number of electrons and holes). Note that the rather
straightforward CI implementation of appendix B.1 without attempting major optimization of performance is due
to the strong-confinement regime and the resulting perturbative character of Coulomb interactions; a more sophisticated CI approach for carrier states in dots within
the weak-confinement regime is presented by Braskén
et al. (2000), where, however, the larger basis set |`i prohibited the calculation of optical spectra. We finally note
that other approaches for the calculation of few-particle
carrier states, such as truncated CI techniques (Hawrylak
and A. Wojs 1996, Jacak et al. 1998, Wojs and Hawrylak 1997), density-matrix descriptions (Hohenester et al.
1999b), or approaches based on density-functional theory have in general the drawback that they treat certain
types of Coulomb interactions within a mean-field approximation, which can lead to spurious effects in the
optical spectra.
III.
OPTICAL PROPERTIES OF QUANTUM DOTS
The coupling of light with optical (i.e., electron-hole)
excitations in quantum dots allows the study of fewparticle states by means of optical spectroscopy. This
section is devoted to a short survey over different spectroscopy techniques and the theoretical framework accounting for light-matter coupling.
A. Light-semiconductor coupling
In lowest order the light-semiconductor coupling is described by a process where a photon is absorbed and an
electron is promoted from the valence to the conduction
band (i.e., creation of electron-hole pair), and by the reversed process where an electron-hole pair is destroyed
and a photon is created; for simplicity, in this work we
shall neglect all higher-order processes (e.g., two-photon
absorption; Fedorov et al. 1996). The interaction of dot
carriers with light is then accounted for by a Hamiltonian
of the form −E.P (Haug and Koch 1993, Rossi 1998),
where E is the electric field of the light and P is the material polarization. We next employ a number of approximations (Axt and Mukamel 1998, Haug and Koch 1993,
Rossi 1998, Scully and Zubairy 1997): First, we neglect
optical intraband transitions; second, we make use of the
rotating-wave approximation; third, within our envelopefunction–one-band effective-mass approach we make use
of the dipole approximation and neglect the directional
dependence of the bulk dipole matrix element µo ; we furthermore assume that circularly polarized light creates
electron-hole pairs with well defined spin orientations.
Then (Haug and Koch 1993, Rossi 1998):
Hcl = − 12
XZ
dr [Eσ(+) (r, t)Pσ (r) + Eσ(−) (r, t)Pσ† (r)],
σ
(11)
where Eσ(±) is the electric field profile of the light with
polarization σ and with the superscript (±) accounting
for clockwise or counterclockwise rotating components,
e
h
P± (r) = µo ψ∓
(r)ψ±
(r) is the interband-dipole operator
with the field operators ψσm † (r), which create an electron
or hole with spin orientation σ at position r.5 Note that
in Eq. (11) we have explicitly assumed a real-space dependence of the electric field profile. Such a choice will
turn out to be convenient for the discussion of near-field
spectra; in case of far-field excitation the r-dependence of
E can be safely neglected. In the following we discuss the
cases where the electric field is treated either classically
(see absorption) or quantum-mechanically (see luminescence).
B. Linear absorption
Let us assume that the dot structure is weakly perturbed by a light field, which we shall treat on a classical level. Within the linear-response regime we assume
(without loss of generality) a monofrequent laser excitation at frequency ω with an (inhomogeneous) electric
field profile Eσ (r, t) = Eσ (r) exp(iωt). This external
light field induces an interband polarization Pσ (r, t) =
hPσ (r)it (Rossi 1998), where the brackets denote a suitable ensemble average. The total absorbed power α(ω)
at a given frequency ω is then proportional to the imaginary part of the Fourier-transformed interband polarization (Haug and Koch 1993, Mauritz et al. 1999, 2000):
Z
iωt
α(ω) ∝ =
drdt Eσ (r)e Pσ (r, t) .
(12)
The calculation of α(ω) thus requires the knowledge of
Pσ (r, t) in presence of the driving laser field. Within
5 These
m through:
field operators are related to ψµσ
X
m
m
ψσ
(r) =
φm
µσ (r)ψµσ .
µ
7
linear response it is possible to obtain under very general conditions Kubo-like expressions for the induced
interband polarization (Kubo et al. 1985); alternative
approaches for calculating Pσ (r, t) for nonequilibrium
semiconductor nanostructures are usually based on the
density-matrix (Rossi 1998) or non-equilibrium Green’s
function approaches (Haug and Jauho 1996).
2. Near-field vs. far-field absorption
Using Eqs. (12) and (13) we then get within timedependent perturbation theory7 for the absorption spectra (Mauritz et al. 1999, 2000):
α(ω) ∝
X Z
dr ξ(r − R)Ψx (r, r)2 Dγ (ω − Ex ), (15)
x
1. Excitons
In this work we pursue a somewhat simplified approach
based on a wavefunction description, and environment
coupling is introduced by adding a small imaginary part
iγ to the eigenenergies. Let us assume that before arrival of the driving laser the dot system is in its vacuum
state |0i, i.e., no electrons and holes present. The laser
then creates through the interband-polarization operator Pσ† (r) [Eq. (11)] an electron-hole pair, which propagates in presence of the dot confinement and of mutual
Coulomb interactions. Thus, the linear optical properties
are governed by the electron-hole (exciton) eigenergies Ex
and wavefunctions Ψx ,6 which we obtain from Eq. (10):
(eµ + hν )Ψxµν +
X
eh
x
x
Vµµ̄,ν
ν̄ Ψµ̄ν̄ = Ex Ψµν ,
(13)
µ̄ν̄
with Ψxµν = h0|ψνh ψµe |xi (note that for notational simplicity we have dropped all spin indices). Within the field
of semiconductor optics Eq. (13) is usually referred to
as the excitonic eigenvalue problem (Axt and Mukamel
1998, Haug and Koch 1993, Rossi and Molinari 1996a,b).
A particularly intriguing form of this equation is obtained
for the real-space representation of the exciton wavefunction:
Ψx (re , rh ) =
X
φeµ (re )Ψxµν φhν (rh ),
(14)
µν
where the left-hand side of Eq. (13) gets the form
[he (re ) + hh (rh ) − e2 /(κ|re − rh |)]Ψx (re , rh ), with the
single-particle Hamiltonians he and hh of Eq. (4), which
clearly emphasizes the Coulomb-type renormalization of
exciton states.
6 Few-particle states with more than one electron or hole only
contribute to the higher-order optical response (Axt and Mukamel
1998, Axt and Stahl 1995, Victor et al. 1995) and are thus neglected.
where Dγ (ω) = 2γ/(ω 2 + γ 2 ) is a broadened deltafunction and, for convenience, we have chosen an electricfield distribution of the form E(r) = Eo ξ(r − R), with
ξ(r) a normalized distribution. Two limiting cases can
be identified. For a spatially homogeneous electromagnetic field (ξ ∝ 1, i.e., far-field), the oscillator strength is
given
R by the spatial average of the excitonic wavefunction
∝ | dr Ψx (r, r)|2 . For near-field excitation ξ(r − R)
is centered around the tip position R of the near-field
probe; in the hypothetical limit of an infinitely narrow
probe, ξ(r − R) = δ(r − R), one is probing the local
value of the exciton wavefunction; within an intermediate regime of a narrow but finite probe, Ψλ (r, r) is averaged over a region which is determined by the spatial
extension of the light beam; therefore, excitonic transitions which are optically forbidden in the far-field may
become visible in the near-field (Bryant 1998, Liu and
Bryant 1999, Mauritz et al. 1999, 2000, Simserides et al.
2000).
C. Nonlinear absorption
We next turn to the discussion of nonlinear optical spectroscopy (Bonadeo et al. 1998b, 2000, Ikezawa
et al. 1997). Quite generally, this technique requires
two lasers at different frequencies: a strong pump
laser with electric field Eσpp exp(iωp t) and a weak probe
pulse with Eσt t exp(iωt t). Nonlinear absorption measures the absorption changes of the weak probe pulse
∝ χ(3) |Eσpp |2 Eσt t exp(iωt t), with the third-order susceptibility χ(3) , and thus provides information about the
optical properties of the dot system in presence of the
strong pump pulse. As compared to linear absorption,
this technique has the advantage of giving a much better noise-signal ratio, which is of crucial importance for
the measurement of single quantum dots (Bonadeo et al.
7 For
the time-dependent state vector we find:
Z
X [Ψx (r, r)]∗
|Ψit ∝ |0i − 12 µo dr E(r)e−iωt
|xi,
ω − Ex + iγ
x
where we have assumed that before arrival of the laser the system
is in |0i and we have performed the adiabatic approximation (note
that the damping constant γ, accounting for environment coupling
to, e.g., phonons, is essential to perform this limiting procedure).
8
1998b, 2000);8 In addition, nonlinear spectroscopy provides information about biexcitonic features in the optical spectra (Axt and Mukamel 1998).
1. Biexcitons
To gain some insight into the latter point, in the following we briefly discuss the nature of biexcitonic states.
Since the external light fields create electron-hole pairs
with given spin orientations, it turns out to be convenient
to define the exciton operators:
D. Luminescence
Luminescence (spontaneous emission) involves a process where one electron-hole pair is destroyed and
one photon is created. Its description thus requires
a quantum-mechanical treatment of light (Scully and
Zubairy 1997). Introducing a suitable quantization of
the light field according to Haug and Koch (1993) we
then obtain from Eq. (11) the carrier-photon interaction:
Hcl = −
X
1
(2πωkσ )− 2 [ia†kσ Pσ (r) − iakσ Pσ† (r)], (19)
kσ
b†xσ =
X
e
e
Ψxµν (ψµσ
)† (ψνσ
e
h
µν
)† σe =σ,σh =−σ
.
(16)
Apparently, b†xσ |0i creates an optically active exciton
state |xi. With these operators we then construct the
biexciton states |λi according to:
|λi = ησσ0
X
Ψ̄λxx0 b†xσ b†x0 σ0 |0i,
(17)
where the Bosonic field operators a†kσ create a photon
with polarization σ and wavevector k. Their free motion
P
is governed by kσ ωkσ a†kσ akσ with
√ ωkσ = c|k|/n, where
c is the speed of light and n ∼
= κ is the semiconductor
refraction index.
Assume that the dot is initially in the few-particle state
|λi. The luminescence spectrum is then obtained from
Fermi’s golden rule:
xx0
0
1
4
with ησσ0 = for parallel exciton spins (σ = σ ) and one
otherwise, and Ψ̄λxx0 the biexciton wavefunctions within
the exciton basis (16).9 As outlined in appendix B.3,
these wavefunctions are obtained, in analogy to the excitonic eigenvalue problem (13), from the four-particle
Schrödinger equation:
(Ex + Ex0 )Ψ̄λxx0 +
X
V̄xx0 ,x̄x̄0 Ψ̄λx̄x̄0 = Ēλ Ψ̄λxx0 ,
Iσ (ω) ∝
X
|(Pσ )λλ0 |2 Dγ (ω − Eλ0 + Eλ ),
(20)
λ0
Rwith the total interband-polarization operator Pσ =
dr Pσ (r). As Pσ removes one electron-hole pair, it connects states λ and λ0 with different numbers of electrons
and holes; in appendix B.2 we discuss how to compute
such matrix elements within our direct-diagonalization
approach.
(18)
x̄x̄0
E. Coherent-carrier control
with V̄ the exciton-exciton Coulomb matrix elements
[Eqs. (B7,B8)] and Ēλ the biexciton energies. As can be
inferred from Eq. (B8), the magnitude of these Coulomb
elements, which are responsible for biexciton renormalizations, depends on the asymmetry of electron and
hole wavefunctions of the exciton states contributing to
Ψ̄λxx0 (Hohenester and Molinari 2000a, Hohenester et al.
1999a).
8 Another
experimental technique circumventing this problem is
photo-luminescence excitation spectroscopy (PLE); for details see,
e.g., Bonadeo et al. (1998a, 2000), Hawrylak et al. (2000), Heller
and Bockelmann (1997) or Toda et al. (2000).
9 Here we have exploited that the exciton wavefunctions Ψx
µν
of Eq. (13) provide a complete set of basis functions
within the
P
x ∗ x
subspace of electron-hole excitations. Thus,
x (Ψµν ) Ψµ0 ν 0 =
P
0
x
∗
x
δµµ0 δνν 0 and µν (Ψµν ) Ψµν = δxx0 . Note that for parallel spins
the states b†xσ b†x0 σ |0i provide an overcomplete set of basis functions
(Axt and Mukamel 1998).
Up to now we have discussed situations where the optical properties of semiconductor nanostructures are investigated under steady-state conditions. Much of the
recent excitement about optical spectroscopy in semiconductors, on the other hand, is due to the possibility of manipulating electron-hole states by means of ultrafast coherent-carrier control (see, e.g., Heberle et al.
1995, Rossi 1998, Shah 1996). In its simplest form,
coherent-carrier control requires two temporally delayed
sub-picosecond laser pulses (Heberle et al. 1995): the first
one creates electron-hole pairs which propagate in phase
with the exciting laser pulse; when the second pulse arrives at later time, its action depends on the phase relation between the two pulses and on how much polarization induced by the first pulse is left in the system (i.e.,
dephasing). In absence of dephasing the second pulse
can, depending on the phase relation between the two
pulses, either remove all exciton population, or increase
it by at most a factor of four. However, such ideal performance is usually spoiled by dephasing processes which
are due to scatterings of excitons with environmental degrees of freedom (e.g., phonons) or due to exciton-exciton
9
IV.
RESULTS
In this section we turn to a brief discussion of the results obtained within our full configuration-interaction
approach. We start by discussing in Sec. IV.A model
calculations of optical near-field spectroscopy; Secs. IV.B
and IV.C are devoted to the optical properties of multiexciton and multi-charged excitons, respectively; finally,
we discuss in Sec. IV.D selected topics of coherent-carrier
control in quantum dots.
A. Near-field spectra
The major advancement in the field of quantum dot
spectroscopy is due to the improvement of different types
of local spectroscopy, which allow the study of individual dots and thus overcome the limitations related to
inhomogeneous broadening. Among local spectroscopies,
the approaches based on scanning near-field optical microscopy (SNOM; Hecht et al. 2000, Paesler and Moyer
1996) are especially interesting as they bring the spatial
resolution well below the diffraction limit of light: With
the development of small-aperture optical fiber probes,
sub-wavelength resolutions were achieved (λ/8 − λ/5 or
λ/40; Betzig et al. 1991, Saiki and Matsuda 1999) and
the first applications to nanostructures became possible
(Zrenner 2000). As the resolution increases, local optical techniques in principle allow direct access to the
space and energy distribution of quantum states within
the dot. This opens, however, a number of questions
regarding the interpretation of these experiments, that
were often neglected in the past.
First of all, for spatially inhomogeneous electromagnetic (EM) fields it is no longer possible to define and
measure an absorption coefficient that locally relates the
absorbed power density with the light intensity (since
the susceptibility χ(r, r 0 ) cannot be approximated by a
local tensor). In the linear regime, a local absorption
coefficient can still be defined, which is however a complicated function that depends on the specific EM field
distribution (Mauritz et al. 1999, 2000). The interpretation of near-field spectra therefore requires calculations
based on a reasonable assumption for the profile of the
EM field.
20
0.0 0.2 0.4 0.6 0.8 1.0
10
X (nm)
scatterings. Within the last couple of years the study of
such dephasing has attracted enormous interest (see, e.g.,
Rossi 1998, Shah 1996, and references therein) because
of the most interesting fundamental questions related to
the interplay of coherent and incoherent carrier dynamics, and of promising technological applications.
Implementation of coherent-carrier control in single
semiconductor quantum dots is an extremely demanding
task, and only recently first experimental results have become available (Bonadeo et al. 1998a, Toda et al. 2000).
There, the authors have reported that, because of severe
phase-space restrictions and the resulting strong suppressions of scatterings, dephasing rates are strongly reduced
as compared to semiconductors of higher dimensionality.
0
-10
-20
0
20
40
60
Photon Energy (meV)
80
100
FIG. 1. Contour plot of near-field spectrum (infinite resolution) of a single quantum dot (Hohenester and Molinari
2000a, Simserides et al. 2000). The solid line shows the optical far-field spectrum; because of symmetry only a small
portion of the excitons couples in the far field to the light.
Secondly, the quantum states that are actually probed
are few-particle states of the interacting electrons and
holes photoexcited in the dot. Even in the linear regime,
excitonic effects are known to dominate the optical spectra of dots since Coulomb interactions are strongly enhanced by the three-dimensional confinement. Nearfield spectra probe exciton wavefunctions (see Fig. 1),
and their spatial coherence and overlap with the EMfield profile will determine the local absorption (Mauritz
et al. 1999, 2000) The theoretical framework presented in
Secs. II and III is especially suited for a realistic description of the quantum states of the interacting electron and
holes photoexcited in the linear regime. In this respect,
our theoretical analysis presented in Simserides et al.
(2000) improves drastically over previous approaches,
which generally focused on a more detailed treatment of
the EM field distributions (Bryant 1998, Chavez-Pirson
and Chu 1999, Girard et al. 1998, Hanewinkel et al. 1997).
B. Multi-excitons
We next turn to the case of a dot populated by a larger
number of electron-hole pairs. In a typical single-dot experiment (Zrenner 2000), a pump pulse creates electronhole pairs in continuum states (e.g., wetting layer) in the
vicinity of the quantum dot, and some of the carriers become captured in the dot; from experiment it is known
10
that there is a fast subsequent carrier relaxation due to
environment coupling to the few-particle states of lowest energy (Bayer et al. 2000a, Dekel et al. 1998, Landin
et al. 1998, Zrenner 2000); finally, electrons and holes
in the dot recombine by emitting a photon. By varying
in a steady-state experiment the pump intensity and by
monitoring luminescence from the dot states, one thus
obtains information about the few-particle carrier states.
Fig. 2 shows luminescence spectra computed within
our direct-diagonalization approach [Eqs. (10,20)] for different numbers of electron-hole pairs (Hohenester and
Molinari 2000b, Rinaldi et al. 2000). Indeed, whenever one additional electron-hole pair is added to the dot
the optical spectrum changes because of the additional
Coulomb interactions. A comparison of such calculated
luminescence spectra with experiment is presented for
multi-excitons in Rinaldi et al. (2000) (see Bayer et al.
(2000a) for related work).
and Pötz 1999, Rossi 1998). Such a controlled “wavefunction engineering” is an essential prerequisite for a
successful implementation of ultrafast optical switching
and quantum-information processing in a pure solid state
system. In this respect, semiconductor quantum dots
appear to be among the most promising candidates because of their discrete atomic-like density of states and
the resulting suppressed coupling to the solid-state environment. It thus has been envisioned that optical excitations in quantum dots could be successfully exploited for
quantum-information processing (Imamoglu et al. 1999,
L. Quiroga 1999, Zanardi and Rossi 1998, 1999). In the
following we briefly discuss selected topics of coherentcarrier control in semiconductor quantum dots.
Coulomb interactions not only lead to renormaliztions
of multi-exciton states but also of multi-charged excitons,
i.e., complexes consisting of multiple electrons and a single hole. Experimental realization of such carrier complexes can be found, e.g., in Hartmann et al. (2000a),
where GaAs/AlGaAs quantum dots are remote-doped
with electrons from donors located in the vicinity of the
dot. Employing the mechanism of photo-depletion of the
quantum dot together with the slow hopping transport
of impurity-bound electrons back to the dot (Hartmann
et al. 2000a,b), it is possible to efficiently control the
number of surplus electrons in the dot from one to approximately six. Other possibilities of remote doping are
to place dots within a n-i field-effect structure (Warburton et al. 1997) and to apply an external gate voltage
to control the number of surplus electrons (Findeis et al.
2000a, Warburton et al. 2000). Quite generally, with increasing doping the spectral changes are similar to those
presented for multi-exciton states. More specifically, the
main peaks red-shift because of exchange and correlation
effects, and each few-particle state leads to a specific fingerprint in the optical response. This unique assignment
of peaks or peak multiplets to given few-particle configurations allows to unambiguously determine the detailed
few-particle configuration of carriers in quantum dots in
optical experiments; this fact is used in Hartmann et al.
(2000a,b) to study the impurity-dot transport.
D. Coherent-carrier control
Coherent-carrier control in semiconductor nanostructures has recently attracted an enormous interest because
it allows the coherent manipulation of carrier wavefunctions on a timescale shorter than typical dephasing times
(see, e.g., Bonadeo et al. 1998a, Heberle et al. 1995, Hu
Luminescence (arb. units)
C. Multi-charged excitons
-20
6X
(6e+6h)
5X
(5e+5h)
4X
(4e+4h)
3X
(3e+3h)
2X
(2e+2h)
X
(1e+1h)
-10
0
10
20
Photon Energy (meV)
30
FIG. 2. Luminescence spectra for a quantum dot (for details see Hohenester and Molinari 2000a) and for different
multi-exciton states (number of electron-hole pairs according
to expression in parentheses). The insets sketch the respective
electron-hole configurations before photon emission (spin orientation according to orientation of triangles). Photon energy
zero is given by the groundstate exciton X0 .
1. Quantum-information processing
Quantum-information processing (i.e., quantum computation, quantum cryptography, or quantum teleportation) represent exciting new arenas which exploit intrinsic quantum mechanical properties (see, e.g., Bennet and
DiVincenzo 2000, DiVincenzo 2000, Steane 1997, and references therein). Basic elements to process quantum information are quantum bits (qubits), which, in analogy to
classical bits, are defined as suitably chosen two-level sys-
11
Absorption (arb. units)
tems. Much of the present excitement about quantuminformation processing originates from the seminal discoveries of Shor and others, who showed that —provided
quantum-information processing can be successfully implemented for ∼100–1000 qubits— quantum algorithms
can perform some hard computations much faster than
classical algorithms, and can allow the reduction of exponentially complex problems to polynomial complexity
(Bennet and DiVincenzo 2000).
-10
0
10
20
30
Photon Energy (meV)
40
FIG. 3. Absorption spectra quantum dot (see Troiani et al.
2000a,b, for details), which is initially prepared in: (a) vacuum state; (b) exciton |X0 i state (exciton groundstate); (c)
exciton |X1 i state; (d) biexciton |Bi state.
Quite generally, only a few basic requirements are
needed for a successful implementation of quantuminformation processing, which, according to DiVincenzo
(2000), can be summarized in the following five points:
(i) A scalable physical system with well characterized
qubits; (ii) the ability to initialize the state of the qubits;
(iii) long relevant decoherence times, much longer than
typical qubit-manipulation times; (iv) a “universal” set
of quantum gates; and (v) a qubit-specific measurement
capability. Apparently, the main difficulty for a successful implementation of quantum-information processing in
a “real physical system” concerns the unavoidable coupling of qubits to their environment, which leads to the
process of decoherence where some qubit or qubits of the
computation become entangled with the environment,
thus in effect “collapsing” the state of the quantum computer.
Recently, we have demonstrated (Troiani et al.
2000a,b) that exciton states in quantum dots can serve
as qubits, where the requested quantum gates can be implemented by use of ultrafast spectroscopy and coherentcarrier control. The essence of our proposal is summarized in Fig. 3. Fig. 3(a) shows the absorption spectrum
of an empty dot (i.e., no electrons and holes present):
Two pronounced absorption peaks X0 and X1 can be
identified whose energy splitting is of the order of the dot
confinement (here ∼25 meV). As the central step within
our proposal we ascribe the different exciton states to the
computational degrees of freedom (qubits). Thus, for the
specific case of Fig. 1 we have two qubits (X0 and X1 ),
which have value one if exciton x is populated and zero
otherwise (x = X0 , X1 ).
Fig. 3 (b) reports the absorption spectrum for the case
that the first qubit (exciton X0 ) is set equal to one (is
populated): Due to state filling, the character of transition X0 changes from absorption to gain (i.e., negative absorption); in addition, the higher-energetic transition is shifted to lower energy, which is attributed to
the formation of a biexcitonic state B whose energy is
reduced by an amount of ∆ because of exchange interactions between the two electrons and holes, respectively (Hawrylak 1999, Hohenester and Molinari 2000a).
These findings can be formalized as briefly discussed in
the following. By use of the exciton operators (16) our
computational space is given by: The vacuum state |0i;
the single-exciton states (where only one qubit is equal
to one) |xi = b†x |0i (we only consider one spin orientation); and |11i = b†X0 b†X1 |0i, the state where both
qubits are equal to one (note that, quite generally, it
is not obvious that the biexciton state B corresponds to
the state |11i since Coulomb interactions can mix different single-exciton states x; however, as discussed in
Troiani et al. (2000a,b) at the example of a prototypical
dot confinement, in the strong confinement regime |Bi
is almost parallel to |11i, thus providing the producttype Hilbert space required for quantum-information processing). Then, the Hamiltonian describing the exciton dynamics in a single dot can be written in analogy
to nuclear-magnetic-resonance (NMR) based quantuminformation processing implementations (Bennet and DiVincenzo 2000, Gershenfeld and Chuang 1997):
H=
X
Ex n̂x − ∆ n̂X0 n̂X1 ,
(21)
x=X0 ,X1
with Ex the single-exciton energies and n̂x = b†x bx the
exciton occupation operator. From related NMR work it
is well known that a Hamiltonian of this form can account
for the conditional and unconditional qubit operations
required for QIP (see Troiani et al. 2000a,b, for details).
12
2. Adiabatic passage
Quantum coherence among charge states is known to
have strong impact on the optical properties of confined
systems. A striking example of such coherence is provided by trapped states, where two intense laser fields
drive an effective three-level system into a state which is
completely stable against absorption and emission from
the radiation fields (see, e.g., Scully and Zubairy 1997).
This highly nonlinear coherent-population trapping is exploited in the process of Stimulated Raman Adiabatic
Passage (STIRAP) (Bergmann et al. 1998) to produce
complete population transfer between quantum states:
Coherent control of optical excitations of atoms and
molecules, as well as of chemical-reaction dynamics has
been demonstrated in the last years (Bergmann et al.
1998).
Successful implementations of such schemes in semiconductor nanostructures would have a strong impetus on the development of novel opto-electronic devices. However, in contrast to atomic systems carrier
lifetimes in the solid state are much shorter because of
the continuous density-of-states of charge excitations and
stronger environment coupling. Although the possibility of trapped states and coherent population transfer
in semiconductor quantum wells has been demonstrated
theoretically (Binder and Lindberg 1998, Lindberg and
Binder 1995, Pötz 1997), its efficiency is expected to
be extremely poor as compared to atomic systems. It
has, on the other hand, been noted that the situation
could be much more favorable in semiconductor quantum dots (Pötz 1997) because of phase-space restrictions.
In Hohenester et al. (2000) we propose a solid-state implementation of STIRAP in two coupled semiconductor
quantum dots, which are remote-doped with a single surplus hole. The right-dot and left-dot hole states, respectively, are optically connected through the chargedexciton state. Proper combination of pulsed Stokes and
pump laser fields then allows a coherent population transfer between the two dot states without suffering significant losses due to environment coupling of the charged
exciton state (see Hohenester et al. 2000, for details).
2, that the fluorescence of the laser-driven system exhibits long dark periods, associated to the excitation of
the extremely weak 0 ↔ 2 transition, which are followed
by bright periods, where many photons are emitted from
the decay of the short-lived state 1.
In Hohenester (2000) we propose to use continuous
laser excitation and continuous monitoring of luminescence for the observation of single spin-flip processes in
semiconductor quantum dots. Contrary to the proposal
of Dalibard et al., within this scheme the rare scattering events originate from the coupling of the dot carriers
to the elementary excitations of their solid-state environment. Thus, the dot serves as a sensor of its environment, which makes possible the observation of the
otherwise almost inaccessible rare scattering events. Besides the most challenging prospect of measuring single
scatterings in solid state, the long-lived spin excitations
in quantum dots have recently attracted strong interest
(Gupta et al. 1999, Kamada et al. 1999, Khaetskii and
Nazarov 2000) because of their potential utilization for
quantum-information processing (Imamoglu et al. 1999,
Loss and DiVincenzo 1998).
V.
SUMMARY
In conclusion, we have discussed the theoretical framework for calculating carrier states in semiconductor quantum dots. We have developed a numerical approach
for computing single-particle states within the envelopefunction and effective-mass approximations and fewparticle electron-hole states by use of a full configurationinteraction model. The results have been used to predict
the optical spectra of quantum dots. More specifically, we
have shown that because of the strong quantum confinement Coulomb interactions between electrons and holes
are strongly enhanced, and that each electron-hole configuration leads to its characteristic signature in the optical spectra (absorption and luminescence). Our results
have been found to be in good agreement with experiment. Finally, we have shown how multi-excitonic states
can be exploited for a favourable implementation of novel
coherent-control techiques and quantum information processing in semiconductor quantum dots.
3. Quantum measurement
A final example which emphasizes the important role
of the interplay between coherent and incoherent carrier
dynamics in laser-excited single quantum dots is provided
by continuously monitored luminescence. The question
of how to theoretically describe environment coupling
and scatterings in a single system arose almost a decade
ago, when it became possible to store single ions in a Paul
trap and to continuously monitor their resonance fluorescence (see Plenio and Knight 1998, for a review). In the
seminal work of Dalibard et al. (1992) the authors showed
for a V-scheme, where the groundstate 0 of an atom is
coupled to a short-lived state 1 and to a metastable state
ACKNOWLEDGMENTS
I am indebted to Elisa Molinari for the opportunity
to work in her group and for most generous support,
both scientifically and personally, within the last couple of years. I gratefully acknowledge helpful discussions
and support from my collegous Costas Simserides, Filippo Troiani, and Rosa Di Felice. I thank the members
of the networks “Ultrafast” and “SQID” for lively and
stimulating discussions. In addition, I acknowledge successful and pleasant collaborations with Arno Hartmann,
13
Yann Ducommun, Ross Rinaldi, Roberto Cingolani, Artur Zrenner, Giovanna Panzarini, and Fausto Rossi. Special thanks are due to Claudia Ambrosch-Draxl and to
the members of the institute of theoretical physics of
Graz university for their kind hospitality. Finally, this
work would have been hardly possible without the love
and support of Moritz, Zoe, and Olga.
APPENDIX A: SINGLE-PARTICLE STATES AND
MATRIX ELEMENTS
Z
dr φhν (r)φeµ (r),
(A3)
with µo the dipole matrix element
of the bulk semiconP
ductor. Thus, Mνµ = µo i φhν,i φeµ,i for the latticediscretized wavefunctions.
3. Coulomb matrix elements
1. Single-particle states
In this appendix we show how to numerically solve
the single-particle Schrödinger equation (5) in absence of
magnetic fields. The confinement potential
P v(r) is assumed to be given at the positions ri = k λki ∆k êk of
a sufficiently dense real-space grid, with integers λki ∈
[ 21 Nk , 12 Nk − 1], and meshsize ∆k and unit vector êk in
the k-th Cartesian directions. Because of our use of the
envelope-function approximation, the low-energy singleparticle envelope functions φµ (r) have an only small rdependence. Introducing the abbreviations φµ,i = φµ (ri )
and vi = v(ri ), Eq. (5) can be approximately solved by
direct diagonalization of the lattice-discretized eigenvalue
problem:
X 2
∂i,k
−
+ vi φµ,i = µ φµ,i ,
2mk
Mνµ = µo
(A1)
k
In the calculation of the Coulomb matrix elements of
Eq. (7) the limit r − r 0 → 0 has to be treated with
some
care. First, we note Rthat Eq. (7) is of the form
R
dr ρ(r)Φ(r), with Φ(r) = dr 0 ρ(r 0 )/κ|r−r 0 |. P
The integrals canPbe approximately expressed through i ρi Φi
and Φi = j V̄ij ρj , where:
V̄ij =
3 Z
Y
k=1
1
2 ∆k
− 12 ∆k
dξk
1
κ|ri − rj − ξ|
(A4)
is the Coulomb potential averaged over a cube around
position ri −rj (note that this integral remains finite even
for ri − rj = 0), and we have used that, because of the
weak r-dependence of the envelope functions φ(r), within
each cube ρ can be approximated by ρi = ρ(ri ). In our
computational approach
we solve the convolution-type
P
summation Φi =
V̄
ρ
ij
j by Fourier transformation;
j
only those Coulomb matrix elements which are larger
than one percent of the largest element are kept.
with
APPENDIX B: FEW-PARTICLE STATES
φµ (ri − ∆k êk ) − 2φµ (ri ) + φµ (ri + ∆k êk )
.
=
∆2k
(A2)
Since the real-space representation (A1) of the Hamiltonian is a sparse matrix (i.e., the kinetic part only involves a few off-diagonal elements and the contributions
proportional to vi are diagonal) and since we are only
interested in the low-energy eigenstates, Eq. (A1) can be
easily solved for typical grid dimensions of 64 × 64 × 64
(we assume periodic boundary conditions).
2
∂i,k
φµ,i
2. Optical matrix elements
In the calculation of the electron-hole states and of
their optical properties we need the optical and Coulomb
matrix elements within the single-particle bases φeµ and
φhν for electrons and holes, respectively. By use of the
envelope-function and dipole approximations (Haug and
Koch 1993, Rossi 1998) the optical matrix element for
the transition between electron state µ and hole state ν
is of the form:
1. Direct diagonalization
In this Appendix we describe our direct-diagonalization approach for the calculation of the manyparticle states |λi for a given number of electrons and
holes. Since the Coulomb elements of Eq. (7) conserve
the spin orientations of electrons and holes, we construct
our basis functions |`i within the spin-restricted Hilbert
spaces:
|`i =
Y
m,σ
(ψµmN σ )† . . . (ψµm2 σ )† (ψµm1 σ )†
N =Nσm
|0i, (B1)
µ
~ =~
µm
`,σ
where |0i denotes the vacuum state (i.e., no electrons
and holes) and ` labels the set of quantum numbers µ
~
for electrons and holes with spin orientations σ = ±; for
given σ and m, we require (without loss of generality)
µN > · · · > µ2 > µ1 .
In order to keep the numerics tractable a suitable truncation of the Hilbert space is required. This truncation is
performed along two different lines: First, we only keep a
14
small number of single-particle states of lowest energy m
µ
for electrons and holes, respectively (typically ten states);
second, we only keep those µ
~ ’s for which the sum of single
P PNσm m
particle energies σ i=1
(µi σ )µ~ =~µm
< Eom (Eom cho`,σ
sen such that typically 100–1000 different configurations
for electrons and holes are kept). This separate truncation for electron and hole states turns out to be convenient because of the different magnitudes of level splittings e and h due to the substantially different electron
and hole masses. A further reduction of the Hilbert space
can be achieved in case of cylinder symmetry, where the
angular momentum in z-direction becomes a constant of
motion, and only subspaces with given total angular momentum Mz have to be considered (Jacak et al. (1998);
m
in case of cylinder symmetry these sub-Hilbert spaces
remain in general small enough that no additional Eo truncation is required).
Next, the many-particle eigenstates are obtained by
direct diagonalization of the Hamiltonian matrix (Hc +
For Hc , these contributions are diagonal
Hcc )``0 .
P
PNσm m
δ``0 m,σ i=1 (µi σ )µ~ =~µm
. For the case of Coulomb in`,σ
teractions we note that V can connect at most two of the
four subspaces (e, ±) and (h, ±). If V connects states
µ
~ =µ
~m
~0 = µ
~m
`,σ and µ
`0 ,σ within the same subspace (m, σ)
n
and if all µ
~ `,σ0 = µ
~ n`0 ,σ0 within the other subspaces, we
obtain the contribution:
m
Nσ k1 −1 Nσ k3 −1
X
X X X
(−1)k1 +k2 +k3 +k4 (Vµmm
0
k µ
1
k1 =2 k2 =1 k3 =2 k4 =1
0
k3 ,µk2 µk4
0
− Vµmm
0
k µ
2
0
k3 ,µk1 µk4
;k3 ,k4
)δµ~µ~ ;k
,
1 ,k2
(B2)
0
Q
where µ
~ ; k1 , k2 gives a vector of length Nσm − 2 with the entries at positions k1 and k2 removed, and δµ~µ~ = k δµk µ0k .
Correspondingly, if V connects states µ
~ =µ
~m
~m
~ n`0 ,σ0 within two different subspaces
ν =µ
~ n`,σ0 and µ
~0 = µ
ν0 = µ
`,σ , ~
`0 ,σ , ~
m̄
0
m̄
~ `0 ,σ̄ within the remaining subspaces, we obtain the contribution:
(m, σ), (n, σ ) and if µ
~ `,σ̄ = µ
m
m
n
n
Nσ Nσ Nσ0 Nσ0
X
X X X
0
(−1)k1 +k2 +k3 +k4 Vµmn
0
k µ
1
k1 =1 k2 =1 k3 =1 k4 =1
where µ
~ ; k1 gives a vector of length Nσm −1 with the entry
at position k1 removed.
In our computational approach it turns out be convenient to first compute and store the Coulomb matrix elements for all different combinations of sub-Hilbert spaces
(m, σ); these tabulated values are then used to construct
the Hamiltonian matrix (Hc + Hcc )``0 . Because of the
0
k2 ,νk3 νk4
0
;k2 ν ;k4
,
δ
δµµ ;k
1 ν ;k3
(B3)
Kronecker δ’s, only a small fraction of all matrix elements is non-zero (with the above mentioned truncation
of Coulomb elements only ∼1%), and the 100–1000 eigenstates of lowest energy Eλ are obtained, for a Hamiltonian matrix of typical dimension 104 , by direct diagonalization.
2. Optical matrix elements
We finally show how to compute the optical matrix elements (see also appendix A.2):
M±
``0 = h`|
X
h
e
Mνµ ψν∓
ψµ±
|`0 i.
(B4)
µν
Here we have assumed that photons with well-defined circular polarization σ = ± can create electrons and holes
with given (opposite) spin orientations; the dipole operator in Eq. (B4) connects few-particle states |`i and |`0 i with
e
0 e
h
0 h
e
0 e
different numbers of electrons and holes, N and N 0 (more specifically, N∓
= N∓
, N±
= N±
, and N±
= N±
− 1,
h
h
0
N∓ = N∓ − 1 for σ = ±). Thus:
0 e
0 h
N±
N∓
M±
``0 =
X X
(−1)ke +kh M(~µh0
ke =1 kh =1
where (~
µ)k denotes the k-th component of vector µ
~.
`
µ
~ e` ,±
µ
~h
µ
~ e` ,∓ µ
~h
` ,∓
` ,±
δ
δ
δ
δ
,
e
e
e
h
)
(~
µ
)
~ 0 ;ke µ
~ 0
~ 0 ;kh µ
µ
~ h0
,∓ kh
`0 ,± ke µ
` ,±
` ,∓
` ,∓
(B5)
` ,±
3. Biexcitons
In this appendix we show how to derive Eq. (18); our
analysis closely follows the work of Axt and Mukamel
15
(1998). First, we obtain from Eqs. (10) and (B2,B3) the
two-electron–two-hole Schrödinger equation:
Ēλ Ψλµν,µ0 ν 0 = (eµ + hν + eµ0 + hν0 )Ψλµν,µ0 ν 0
ee
λ
hh
λ
+Vµµ̄,µ
0 µ̄0 Ψµ̄ν,µ̄0 ν 0 + V
ν ν̄,ν 0 ν̄ 0 Ψµν̄,µ0 ν̄ 0
eh
λ
eh
λ
+Vµµ̄,ν
ν̄ Ψµ̄ν̄,µ0 ν 0 + Vµ0 µ̄0 ,ν 0 ν̄ 0 Ψµν,µ̄0 ν̄ 0
eh
λ
eh
λ
+Vµµ̄,ν
0 ν̄ 0 Ψµ̄ν,µ0 ν̄ 0 + V 0 0
µ µ̄ ,ν ν̄ Ψµν̄,µ̄0 ν 0 ,
(B6)
h
e
with Ψλµν,µ0 ν 0 = h0|ψνh0 σ0 ψµe 0 σe0 ψνσ
ψµσ
|λi, and we ase
h
h
sume an implicit summation over bared indices. Inserting ansatz (17) into Eq. (B6) we arrive after some lengthy
calculations at our final form (18), where in case of parallel exciton spins the antisymmetrization of the biexciton
wavefunctions has to be treated with some care (Axt and
Mukamel 1998). The explicit form of the exciton-exciton
Coulomb elements V̄ is:
0
0
Vxx0 ,x̄x̄0 =(Ψxµ0 ν̄1 )∗ (Ψxµ0 ν̄2 )∗ Vµee0 µ1 ,µ0 µ2 Ψx̄µ1 ν̄1 Ψx̄µ2 ν̄2
2
0
1
1
2
0
x̄
x̄
+ (Ψxµ̄1 ν 0 )∗ (Ψxµ̄2 ν 0 )∗ Vνhh
0 ν ,ν 0 ν Ψµ̄ ν Ψµ̄ ν
1 1
2 2
1
2
1
2
1
2
0
0
x̄
x̄
+(Ψxµ0 ν̄1 )∗ (Ψxµ̄2 ν 0 )∗ Vµeh
0 µ ,ν 0 ν Ψµ ν̄ Ψµ̄ ν
1 1
2 2
1
2
1
2
1
2
0
0
x̄
x̄
+(Ψxµ̄2 ν 0 )∗ (Ψxµ0 ν̄1 )∗ Vµeh
0 µ ,ν 0 ν Ψµ̄ ν Ψµ ν̄ ,
2 2
1 1
1
2
2
1
1
2
(B7)
and we assume summation over all µ’s and ν’s.
A particularly intriguing form of V̄ , which closely resembles its dipole-dipole character, is obtained by transforming Eq. (B7) to the real-space representation of the
exciton wavefunctions (14); we get:
V̄xx0 ,x̄x̄0 =
Z
drdr 0
µxx0 (r)µx̄x̄0 (r 0 )
,
κ|r − r 0 |
(B8)
with
Z
dre drh Ψx ∗ (re , rh )
0
×[δ(r − re ) − δ(r − rh ) Ψx (re , rh ).
µxx0 (r) = −e
(B9)
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