We present a combined theoretical and experimental study of the anisotropy in
the optical absorption of V-shaped quantum wires. By means of realistic band
structure calculations for these structures, we show that detailed information
on the heavy- and light-hole states can be singled out from the anisotropy
spectra {\em independently of the electron confinement}, thus allowing accurate
valence band spectroscopy.Comment: To be published in Appl. Phys. Lett. (8 pages in REVTeX, two
  postscipt figures

A. Fasolino

Ando

Bockelmann

E. Molinari

F. Rossi

G. Goldoni

Gailhanou

Goldoni

Kapon

Luttinger

R. Cingolani

R. Rinaldi

Rinaldi

Sercel

Tiwari

English

arXiv

We present a combined theoretical and experimental study of the anisotropy in the optical absorption of V‐shaped quantum wires. By means of realistic band structure calculations for these structures, we show that detailed information on the heavy‐ and light‐hole states can be singled out from the anisotropy spectra independently of the electron confinement, thus allowing accurate valence band spectroscopy

Goldoni, G.

Rossi, Fausto

Molinari, E.

Fasolino, A.

Rinaldi, R.

Cingolani, R.

PORTO Publications Open Repository TOrino

Politecnico di Torino
Porto Institutional Repository
[Article] Valence band spectroscopy in V-grooved quantum wires
Original Citation:
Goldoni G., Rossi F., Molinari E., Fasolino A., Rinaldi R., Cingolani R. (1996). Valence band
spectroscopy in V-grooved quantum wires. In: APPLIED PHYSICS LETTERS, vol. 69 n. 20,
pp. 2965-2967. - ISSN 0003-6951
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DOI:10.1063/1.117745
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Valence band spectroscopy in Vgrooved quantum wires
G. Goldoni, F. Rossi, E. Molinari, A. Fasolino, R. Rinaldi et al. 
 
Citation: Appl. Phys. Lett. 69, 2965 (1996); doi: 10.1063/1.117745 
View online: http://dx.doi.org/10.1063/1.117745 
View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v69/i20 
Published by the American Institute of Physics. 
 
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Valence band spectroscopy in V-grooved quantum wires
G. Goldoni,a) F. Rossi, and E. Molinari
Istituto Nazionale di Fisica della Materia (INFM) and Dipartimento di Fisica, Universita` di Modena,
Via Campi 213/A, I-41100 Modena, Italy
A. Fasolino
Institute of Theoretical Physics, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen,
The Netherlands, and Istituto Nazionale di Fisica della Materia (INFM) and Dipartimento di Fisica,
Universita` di Modena, Via Campi 213/A, I-41100 Modena, Italy
R. Rinaldi and R. Cingolani
Istituto Nazionale di Fisica della Materia (INFM) and Dipartimento di Scienza dei Materiali,
Universita` di Lecce, Via Arnesano, I-73100 Lecce, Italy
~Received 9 July 1996; accepted for publication 5 September 1996!
We present a combined theoretical and experimental study of the anisotropy in the optical
absorption of V-shaped quantum wires. By means of realistic band structure calculations for these
structures, we show that detailed information on the heavy- and light-hole states can be singled out
from the anisotropy spectra independently of the electron confinement, thus allowing accurate
valence band spectroscopy. © 1996 American Institute of Physics. @S0003-6951~96!04646-3#
In semiconductor quantum wires, the anisotropic absorp-
tion of light linearly polarized parallel or perpendicular to the
wire axis has long been recognized to be the effect of the
quasi-one-dimensional ~quasi-1D! character of electronic
states, combined with heavy- and light-hole ~HH and LH!
mixing.1,2 The observation of optical anisotropy has since
been considered as evidence of the 1D character of nano-
structured materials.3 However, even in samples showing the
best optical properties, the interpretation of optical spectra is
difficult owing to the large broadening of the peaks, which is
comparable to the typical inter-subband splittings, and to the
lack of realistic band structure calculations, so far limited to
wires with model geometries.2,4–6 Moreover, the
confinement-induced blueshift is dominated by the light ef-
fective mass of conduction electrons, and information con-
cerning hole states are even more difficult to extract from the
spectra.
The aim of this letter is to show, by a combination of
optical measurements and band structure calculations, that in
realistic structures the details of the optical anisotropy can be
traced to HH and LH states and, therefore, allow for a spec-
troscopy of hole states. We shall focus on quasi-1D struc-
tures obtained by epitaxial growth on non-planar substrates
~V-shaped wires!7–9, which rank among the best available
samples from the point of view of optical properties.
Our samples are arrays of GaAs quantum wires embed-
ded in ~GaAs!8~AlAs!4 superlattices ~SL!, grown by molecu-
lar beam epitaxy ~MBE! on V-grooved substrates obtained
by holographic lithography and wet etching. In our case, the
thickness of the crescent-shape wire which self-organizes at
the bottom of the V-groove —as obtained by transmission
electronic microscopy ~TEM! analysis— is approximately
80 Å. More details on sample parameters are reported in
Ref. 10.
In Fig. 1a we show the low-temperature ~10 K! photolu-
minescence excitation ~PLE! spectra ~thin lines! for the
sample just described. The spectra are obtained with light
polarized along the wire axis and perpendicular to it ~and to
the growth direction!. We shall focus on the excitonic con-
tinuum, i.e., the energy range above 1.575 eV. The broad
peaks in PLE spectra are very different from the singularities
which one would naively expect from the ideal joint density
of states ~DOS! of quasi-1D electrons and holes, because of
the relatively small intersubband splittings compared to the
inhomogeneous broadening11 and the effects of Coulomb
interaction12, so that the interpretation is not easy. As we
shall demonstrate, however, additional information can be
gained by considering the optical anisotropy between the two
a!Electronic mail: goldoni@unimo.it
FIG. 1. a! Low-temperature PLE spectra measured with incoming light
linearly polarized perpendicular to the wire axis ~thin solid line!, parallel to
the wire axis ~dotted line!, and relative anisotropy ~thick solid line!. b!
Calculated absorption spectra with light linearly polarized perpendicular to
the wire axis ~thin solid line!, parallel to the wire axis ~dotted line!, parallel
to the z direction ~dashed line!, and relative anisotropy ~thick solid line!.
2965Appl. Phys. Lett. 69 (20), 11 November 1996 0003-6951/96/69(20)/2965/3/$10.00 © 1996 American Institute of Physics
Downloaded 06 Jul 2012 to 192.167.204.100. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions
polarizations ~thick line in Fig. 1a!, defined as
100*(I i2I')/I i , where I i , I' are the PLE intensities for the
two polarizations. In this range of energy, the anisotropy lies
in the range 10–20%, but goes through a deep minimum
around 1.59–1.60 eV, where it is almost suppressed.
We have performed realistic band structure calculations
within the envelope function approximation. The HH–LH
mixing was modelled by the Luttinger Hamiltonian13, with
proper account of crystallographic directions14. As in Ref.
10, the wire potential profile was extracted from a TEM mi-
crograph of the sample. In the present complex geometry, the
common approach of first solving the Schro¨dinger equation
for uncoupled HHs and LHs and then using them as a basis
set to represent the Luttinger Hamiltonian, results in a very
large computation due to the poor convergence with respect
to the number of subbands included in the basis. To over-
come this difficulty, our approach is to set up a basis set from
the eigenstates of only one type of particle, with a fictitious
mass to be specified later, confined in the V-shaped wire
potential. These are obtained numerically in terms of a plane
wave expansion as in Ref. 10. In this scheme the additional
cost in the calculation of matrix elements of the Luttinger
Hamiltonian is more than compensated, because only one
type of particle needs to be calculated, and, more important,
the convergence can be improved by properly choosing the
fictitious mass. In practice, we find it efficient to choose a
value close to the HH mass15.
Band parameters are taken from GaAs bulk values. The
only unknown parameters are the barrier heights for conduc-
tion and valence electrons: in fact, the embedding short pe-
riod superlattice ~SL! is modeled by a homogeneous barrier
with an effective band offset. The conduction band offset
was taken to be 150 meV, a value that was independently
proved to reproduce PLE and magnetoluminescence
experiments10. The criterium for the choice of the valence
band offset is a by-product of the present work and will be
discussed below.
The calculated absorption spectra and the relative optical
anisotropy, obtained from the full band structure within the
dipole approximation, are shown in Fig. 1b. A Gaussian
broadening of 64.5 meV has been included. The anisotropy
~thick line! shows a minimum, and then reaches a value of
;20 %. The comparison with experiments is quite favour-
able for energies around the minimum, while the agreement
worsens in the high energy range, where the calculated an-
isotropy drops rapidly and finally changes sign. This discrep-
ancy might be due to our ‘‘effective’’ description of the bar-
riers which affects particularly the higher-lying hole states.
As for the calculated absorption spectra, apart from a
redshift of the continuum onset with respect to the experi-
mental curves ~which is within the uncertainty in the wire
geometry!, the main difference lies in the relative intensities
of the two broad features at 1.59 eV and 1.61 eV in the ex-
periment. This might be due to the Coulomb interaction,
which is neglected in the present calculation, and causes a
change in the excitonic continuum. Note, however, that
while each single polarization is significantly changed by
excitonic effects, the relative anisotropy seems to be much
less affected.
The suppression of the optical anisotropy at
;1.59 meV is due to states which are mainly of LH charac-
ter: As an indication of the role of LH states, we also show in
Fig. 1b the calculated absorption spectra for light linearly
polarized parallel to the growth direction (z direction!. Al-
though measurements with this polarization are actually dif-
ficult because of the small active volume, it is instructive to
calculate, since z-polarized light only couples with the LH
component of hole states.2 Figure 1b shows that the mini-
mum in anisotropy actually lies in the range where the
z-polarized absorption ~and therefore the LH character! has a
sudden increase.
Due to the large DOS, the electron and hole states at
k50 (k is the 1D wave vector along the wire axis! are
mainly responsible for the features in the spectra. Due to the
quasi-1D confinement provided by the wire barriers, and in
contrast to the case of quantum wells, these states are of
mixed HH and LH character.2 Figure 2 shows the calculated
HH character of k50 hole states ~labeled by the quantum
number nh) vs the confinement energy.16 Even in a strongly
confined system as the present V-shaped wire, the ground
state is almost a pure HH state ~92%); the transition involv-
ing this hole state (nh51) and the first electron state
(ne51) is mainly responsible for the first peak in the calcu-
lated spectrum.17 The excited hole states show an increas-
ingly mixed HH–LH character, so that the classification in
HH and LH states, familiar from quantum wells, becomes
inappropriate. Simultaneous to the decrease in the HH char-
acter, the wave functions of excited states become more and
more extended along the ‘‘wings’’ of the wire, in analogy to
the simpler case of conduction electrons10, and, conse-
quently, they couple to excited electron subbands, mostly
with ne5nh); starting from the nh52 level in Fig. 2, they
contribute to the broad features above ;1.59 in the calcu
FIG. 2. Percentage of HH character for the lowest hole subbands at k50 vs
subband energy. The arrow indicates the first state which is mainly
(56 %) of LH character. Insets: total charge density, and HH- and LH-
projected charge densities for the state indicated with the arrow (nh513).
Darker regions correspond to larger values of the charge density; thin lines
represent the potential profile used in the calculation. In-plane axes are in
nm ~note the different scale in the two directions!. For clarity, in the three
panels grey levels are set to different scales.
2966 Appl. Phys. Lett., Vol. 69, No. 20, 11 November 1996 Goldoni et al.
Downloaded 06 Jul 2012 to 192.167.204.100. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions
lated spectra. However, this progression is interrupted by the
state indicated with an arrow in Fig. 2. This state is of mainly
LH character ~56 %!, and for brevity it will be indicated
below as the LH state. Although this is a highly excited state
(nh513), its wave function, shown in the insets, is localized
in the center of the wire. Therefore, due to the large overlap,
this state couples primarily with the first conduction subband
(ne51), making a contribution to the low energy part of the
spectrum. The large LH component, moreover, makes that
the intensity for the two polarizations is reversed with re-
spect to the strongly HH ground state, causing the dip in the
anisotropy at 1.59 meV. We have checked, in a set of wires
with different thicknesses at the bottom of the V-groove and
different barrier compositions, that a more or less pro-
nounced dip in the anisotropy is always present in the low
energy range of the spectrum, extending over a range com-
parable to the linewidth of the broadened spectra, and super-
imposed to a background of an otherwise large and positive
anisotropy.
An immediate consequence of the above reasoning is
that, since both the ground HH state and the LH state couple
with the lowest electron subband, the difference in energy
between the onset of the continuum and the position of the
dip in the anisotropy is a direct measure of the energy split-
ting between the ground HH and the first LH states, indepen-
dent of the electron confinement. This splitting, which in the
experiment of Fig. 1a is ;16 meV, has been used in our
calculation to estimate the effective valence band offset; its
value was finally taken as 85 meV.18
In conclusion we have shown by realistic band structure
calculations that the optical anisotropy of V-shaped quantum
wires contains detailed information on the valence subbands,
which can be singled out from the conduction electron con-
tribution, allowing for a spectroscopy of the hole states.
1P. C. Sercel and K. J. Vahala, Appl. Phys. Lett. 57, 545 ~1990!.
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3For a review see: R. Cingolani and R. Rinaldi, Riv. Nuovo Cimento 16, 1
~1993!; see also: R. Rinaldi et al., Phys. Rev. B 50, 11 795 ~1994!.
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10R. Rinaldi, R. Cingolani, M. Lepore, M. Ferrara, I. M. Catalano, F. Rossi,
L. Rota, E. Molinari, P. Lugli, U. Marti, D. Martin, F. Morier-Gemoud, P.
Ruterana, and F. K. Reinhart, Phys. Rev. Lett. 73, 2899 ~1994!.
11E. Molinari, F. Rossi, L. Rota, P. Lugli, R. Rinaldi, M. Lepore, and R.
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13 J. M. Luttinger, Phys. Rev. 102, 1030 ~1956!.
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15The choice of such fictitious mass is purely dictated by computational
convenience: once convergence is achieved, it does not affect the final
results in any way.
16The angular momentum quantization axis is chosen along the direction of
strongest confinement, i.e. the @001# growth direction.
17To help with the interpretation of the theoretical spectra we record here
the electron confinement energy of the three lowest subbands ne51,2,3:
43.3 meV, 57.3 meV, and 65.3 meV. The band gap is 1.519 eV.
18To give an idea of the precision of this assessment, a variation of the
valence band offset of 615 meV shifts the HH–LH splitting of
61.3 meV.
2967Appl. Phys. Lett., Vol. 69, No. 20, 11 November 1996 Goldoni et al.
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Valence band spectroscopy in V-grooved quantum wires

We present a combined theoretical and experimental study of the anisotropy in the optical absorption of V-shaped quantum wires. By means of realistic band structure calculations for these structures, we show that detailed information on the heavy- and light-hole states can be singled out from the anisotropy spectra independently of the electron confinement, thus allowing accurate valence band spectroscopy

Valence band spectroscopy in V-grooved quantum wires

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