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Excitonic properties
Universal scaling of excitonic binding
energy
Room-temperature exciton binding
by remote dielectric confinement
Optical anisotropy
Excitonic properties
 During
the last decade the optical properties of quasi one-dimensional structures,
the so-called quantum wires (QWRs), have become a topic of increasing interest.
The initial motivation was related to the one-dimensional (1D) singularity in
the single-particle density-of-states (DOS), that was expected to induce sharp
peaks in the optical spectra, thereby leading to structures with improved optical
efficiency as compared to their two-dimensional (2D) and three-dimensional (3D)
counterparts. Coulomb correlations strongly modify this simple picture. This
was first pointed out by Ogawa and Takagahara
[1], using a idealized 1D
model. We have implemented a fully 3D approach based on the density-matrix
formalism, which results in a set of
generalized (multisubband) semiconductor Bloch equations (SBE), which is able
to treat both low-density (excitonic)
and high-density (gain) regimes on the same footing, while retaining the
full complexity of state-of-the-art
samples. For example, it has been used
to describe structures obtained by epitaxial growth on non-planar substrates (V-shaped wires)
[2]
or by cleaved-edge quantum well overgrowth (T-shaped wires)
[3], where the lateral
extension of the ground single-particle states is still significant,
the excited states gradually approach a 2D-like behaviour,
and the subband separation is relatively
small, so that the coupling between different
subbands may be important.
The figure shows the calculated absorption for a V-grooved QWR.
Dashed line correspond to single-particle calculations,
while full lines show the results including electron-hole correlations.
Our calculations have shown that
- The correlated absorption spectra of realistic
wires do show a strong quenching of the 1D single-particle
singularity, in agreement with Ref. 1; this effect does
not depend on details of the wire cross section. The Sommerfeld
factor, which is greater than unity in the bulk and in quantum
wells (the so-called Coulomb enhancement), is instead smaller
than unity in QWRs (Coulomb suppression), thus reducing
the influence of dimensionality on the optical spectra.
- The Coulomb-induced suppression of the 1D
singularity is found to hold not only in the linear regime
but also at higher carrier densities, and to persist in
the gain regime. The above results have had implications
for perspective devices, since the initial motivations -based
on single-particle models - are now recognized to be far
too simplified.
Relevant publications
- F. Rossi and E. Molinari, Coulomb-Induced Suppression of
Band-Edge Singularities in the Optical Spectra of Realistic Quantum-Wire Structures
Phys.
Rev. Lett. 76, 3642 (1996).
- F. Rossi and E. Molinari, Linear and nonlinear optical properties of realistic
quantum-wire structures: The dominant role of Coulomb correlation
Phys.
Rev. B 53, 16462 (1996) .
Top of page
Universal scaling of
excitonic binding energy
Given the Coulomb-induced suppression of the band-edge singularity,
the most relevantfeatures arising from electron-hole correlation are bound (below
band-gap)excitonic states, which, forrelatively low carrier densities, are found
to dominate the opticalresponse of the system. Therefore one of the most important
goals has become the achievement of a large exciton binding energy Eb,
as comparedto the thermal energy kTroom: this is indeeda prerequisite
for exploitingexcitonic nonlinearities in optical devices that can operateefficiently
at room temperature. In principle, enhancing Eb in QWRs is
an easy task: in quasi-1D(q1D) structures, in fact, Eb can
be made arbitrarily large provided that one is able to squeeze the electronand
hole wavefunctions to a sufficient extent, thereby increasing theCoulomb energy.
Note that this is at difference with 2D systems forwhich, even in the ideal
case of perfect 2D confinement, the bindingenergy of the ground-state exciton
is limited to four times the3D effective Rydberg. In practice, for the materials
ofchoice for technological application - GaAs/AlAs - this strategyis of difficult
implementation. While in quantum wells (QWs)Ebhas been indeedobserved
to approach the theoretical limit when the well thickness isprogressively reduced,
in QWRs reportedvalues of Eb areonly slightly larger than
in QWs, and still well below kTroom. In fact, the strong confinement
regime which is necessary toenhance the relatively low value of Eb
in bulk GaAs (» 4 meV) is still hard to obtain with the relatively shallow confinementspermitted
by present GaAs/AlAs based structures. Despite the largeexperimental efforts,
this limitation makesorder-kTroom exciton binding energy a
difficult goal. It should be noted that for agiven confinement length, (i.e.,
electron-hole Coulomb interactionenergy) Eb is determined
by the Coulomb-to-kinetic energy ratio,whose value is fixed to -2 in purely
2D and 3D Coulombic systems;since this ratio is ill-defined in purely 1D Coulombic
systems(both kinetic and Coulomb energies diverge) and, moreover,the virial
theorem does not hold in the presence of a confiningpotential, it might be hoped
that a moreconvenient (i.e., smaller) ratio could be obtained in properlydesigned
structures.
 In
this spirit, we have performed detailed investigations of excitonic confinement
for awide class of state-of-the art
GaAs/AlGaAs QWRs, with the aimto
investigate whether, in addition to the squeezing of the wavefunction (which, as we have discussed, is
somewhat limited by thechoice of the
materials), geometrical tailoring of the structure couldbe used to enhance the binding energy. Our
results, summarized in the figure, show that, ingeneral, q1D systems (points labelled V1-V2-T1-T2 for several QWR
geometries) are indeed advantageous with respect to2D ones (labelled QW), since smaller Coulomb-to-kinetic energy
ratios are possible inthe former
system in the strong confinement limit.However, such deviations scale in a universal manner withrespect to the wire cross-section.
Consistently, for all GaAs-basedQWRs
structures considered Eb is found to be very similarand always smaller than kTroom.
Relevant publications
- F. Rossi, G. Goldoni, and E. Molinari, Shape-Independent
Scaling of Excitonic Confinement in Realistic Quantum Wires,
Phys.
Rev. Lett. 78, 3527 (1997) .
Top of page
Room-temperature exciton binding
by remote dielectric confinement
Motivated by these results, we have recently proposed an alternativeapproach
to enhance Eb, which combines theeffects of quantum confinement
discussed so far, with those of dielectric confinement, i.e., confinement
effects induced bydielectric mismatch. As first pointed out by Keldysh
[4], theelectron-hole Coulomb attraction can be greatly
enhanced in layeredstructures with strong dielectric mismatch, due to the polarizationcharge
induced at the interfaces. For conventional semiconductor nanostructures such
as GaAs/AlGaAs- or GaAs/InGaAs-based samplesthis is a minor effect due to the
small dielectric mismatchbetween the constituents. On the other hand, interfacesbetween
III-V semiconductors and materials with very differentdielectric constants,
such as oxides, are usually very far from theexcellent optical qua lity of
the conventional ones. The proposedapproach is based on the idea that quantum
and dielectricconfinement can be spatially separated, since they are effective
overdifferent length scales. As shown in the figure, this may be obtained, for
example, by adding to aconventional GaAs/AlAs QWR “remote” insulating layers
whichinduce strong dielectric confinement, through the strong polarization charge
at the dielectrically mismatched semiconductor/oxide interface (shown in color),
without degrading the goodoptical properties ensuing from quantum confinement,
hence the term remote dielectric confinement (RDC). The question arises,
of course, about the best distance of the oxide layers, such that the interface
effects are small, but the Coulomb enhancement is still sufficiently strong
to give rise to room-temperature exciton binding.
When
dielectrically modulated structures are investigated, theelectron-hole interaction is no longer the
bare Coulomb interaction;instead, it
must be explicitly calculated, for a given structure, asthe Green's function of the Laplace operator
with a spatially dependentdielectric
constant. Accordingly, we have generalized our multisubbandSBE approach in order to include such renormalizedelectron-hole interaction as well as
self-energy effects. We have shown that RDC applied to state-of-the-artGaAs/AlAs-based QWRs, such as the one shown
in figure, may allow room-temperature exciton binding, as shown in the bottom
panel, where Eb is seen to exceed kTroom at
distances as large as 6 nm, where interface effects are certainly negligible.
On a very
general ground, RDC can be seen as a practical method toenhance the electron-electron and
electron-hole interaction within thewire,
without disturbing the quantum confined single particle states(apart from the small self-energy effects).
One is therefore tailoring the Coulomb interaction,similar to what is done -although in the opposite direction- by
use of screening from free-carriers.
Relevant publications
- G. Goldoni, F. Rossi, and E. Molinari, Strong Exciton Binding
in Quantum Structures through Remote Dielectric Confinement
Phys.
Rev. Lett. 80, 4995 (1998)
- F. Rossi, G. Goldoni, O. Mauritz, E. Molinari, Theory
of excitonic confinement in semiconductor quantum wires,
J. Physics: Cond. Matt. 11, 5969--5988 (1999)
Top of page
Optical anisotropy
The optical
spectroscopy of QWRs is more complex than for QWs of similar lateral dimension,
since in QWRs linewidths can be comparable to intersubband splittings. On the
other hand, a remarkable peculiarity of QWRs with respect to QWs is that the
optical activity is strongly anisotropic when light is linearly polarized, with
the electric field directed parallel or perpendicular to the wire axis. This
has long been recognized to be a band structure effect due to the quasi-1D
character of electronic state, combined with heavy- and light-hole (HH and LH)
mixing[5].
The anisotropic absorption is therefore used as a simple tool to reveal the 1D
character of electronic states in nanostructured materials.
 In
principle, the optical anisotropy can also be exploited to single out detailed
information on the electronic states, since it is very sensitive to specific
details of the band structure. In practice, this approach has been limited by
the lack of realistic calculations for complex geometries, as the popular
V-shaped and T-shaped QWRs. Indeed, common theoretical methods, even within
semi-empirical schemes as the tight-binding or the envelope function approach,
require a large scale computational effort. In order to keep calculations
tractable, often the optical properties of QWRs have been investigated
theoretically only for rather idealized structures yielding results that cannot
be directly compared with experimental spectra.
The figure
shows (top: experiment; bottom: theoretical calculation) that a combined
theoretical-experimental study of V-QWRs using accurate band structure
calculations for realistic structures provide quantitative predictions of the
anisotropy of photoluminescence excitation (PLE) spectra (the thick continuous
line). In particular, not only the average amount of anisotropy is reproduced
in this sample, but also the strong dip which leads to zero anisotropy at a
specific energy, which corresponds to the position of the LH transition, where
the rather broad PLE spectra show no particular feature, instead. Therefore,
detailed information on the valence band states can be singled out of the PLE
anisotropy, despite the dominant role of the light conduction electrons in the
optical spectra. Such calculations were based on a recently devised method
which provides the band structure for QWRs of arbitrary geometry at a
relatively small computational cost. The accuracy and the short computer times
make such calculations a practical characterization tool in conjunction with
experimental results, as well as a predictive tool for new devices.
Relevant publications
- G. Goldoni, F. Rossi, E. Molinari, A. Fasolino, R. Rinaldi, R. Cingolani,
Valence bandspectroscopy in V-grooved quantum wires,
Appl.
Phys. Lett. 69, 2965-2967 (1996).
- G. Goldoni, F. Rossi, E. Molinari, A. Fasolino, Band structure and optical
anisotropy in V-shaped and T-shaped semiconductor quantum wires, Phys.
Rev. B 55, 7110--7123 (1997).
Top of page
[1] T. Ogawa and
T. Takagahara, Phys. Rev. B 43, 14325 (1991); ibid. 44,
8138 (1991).
[2]
For reviews see H. Sakaki et al., in Proceedings of the Fifth International
Meeting on Optics of Excitons in Confined Systems, phys. stat. sol. (a)
164, 241 (1997); G. Goldoni, F. Rossi, and E. Molinari, ibid.
164, 265 (1997).
[3] L. Pfeiffer et al.,
Appl. Phys. Lett. 56, 967 (1990); W. Wegscheider et al.,
Appl. Phys. Lett. 65, 2510 (1994).
[4] L. V. Keldysh, Pis'ma Zh.
Eksp. Teor. Fiz. 29, 716 (1979) [JETP Lett. 29,
658 (1979)].
[5] P.C. Sercel and K.J. Vahala,
App. Phys. Lett. 57, 545 (1990).
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