Phys. Rev. B**52**, 8239 (1995).

J. Tulkki† and A. Heinämäki‡

†*Optoelectronics Laboratory, Helsinki University of Technology,
Otakaari 1
*

*FIN02150 Espoo, Finland,*

‡*VTT Electronics, Otakaari 7B, FIN02150 Espoo, Finland*

We have calculated the confinement effect in a InGaAs/GaAs quantum
well dot induced by a dislocation free InP stressor island. The energy
levels were calculated by including the strain interaction and the band
edge confinement in the Luttinger-Kohn Hamiltonian. The maximum level
spacing for dipole allowed interband
line spectrum was 20 meV.
Our calculation also gives excellent agreement with recent measurements
[H. Lipsanen, M. Sopanen, and J. Ahopelto, Phys. Rev. B**51**,
13868 (1995)]
and provides indirect evidence of screened Coulomb interaction,
tentatively addressed to slow carrier relaxation.

PACS numbers 71.50.+t, 78.55.-m

Progress in experimental and theoretical study of quantum dots (QD), restricting the motion of electrons and holes in three dimensions, depends critically on the possibility to study the dynamics of separate discrete energy levels by various spectroscopic probes. To make such experiments feasible, there has been extensive research [1-4] to achieve a proper combination of the following "quality factors": (1) large confinement effect or level spacing, (2) homogeneous size distribution of QDs, and (3) elimination of surface and interface states and defects, which readily destroy a simple state structure.

Stressor-induced two- and three-dimensional confinement in the underlying quantum well (QW) should provide a special advantage regarding the property (3) since even when the stressor itself may contain defects, the potential well induced by the stressor is located in a region of a nearly perfect crystal. Small confinement effect (level spacing meV [1,3]) and the inhomogeneous broadening have, however, prevented the spectroscopy of separate levels, whenever several QDs are probed simultaneously. A novel approach to achieve a large confinement effect by self-organized Stranski-Krastanow (S-K) growth [5], was recently suggested by Sopanen and coworkers [6]. In their approach an InP island produced a level spacing 20 meV in a quantum well dot.

In this work we report a complete computational analysis of the
confinement effect in these novel quantum structures, shown
schematically in Fig. 1. Our results, which include only material
parameters of the pertinent bulk III-V materials are in excellent
agreement with the
luminescence line spectrum reported by Sopanen *et al* [6].

The strain is calculated by determining a strain tensor
; *i*,
*j* = *x, y, z* for which the total strain energy
obtains the
minimum value under appropriate boundary conditions. The strain energy
density in a
zinc-blende structure can be expressed in terms of the elastic constants
,
and
[7]. Elastic
constants and all other material parameters used in the calculations
are given in Table I. The strain was included in the finite element
method (FEM) calculation by the
method. The quantum structure was first assumed unstrained, all
materials having the lattice constant of GaAs. Then the temperatures
of the QW and the
InP island were artificially increased so that these materials obtained
their real lattice constant. The total strain energy was then minimized
by assuming that at the bottom of the 100 nm thick structure as well as
at sidewalls 200 nm away from the dot axis (see Fig. 1) the structure
obtains the lattice constant of unstrained GaAs.

The strain interaction was calculated according to the theory of G. E. Pikus and G. L. Bir [8,9]. The strain interaction for the conduction band is given by the hydrostatic potential . Neglecting the spin-dependent terms, the strain-Hamiltonian for the valence band is given by

. (1)

where the strain potentials are given by
,
,
and
.
The deformation potential constants
,
, and
are given in Table I.
The non-diagonal terms* R* and *S* are not zero in our geometry
except far away from the dot axis, where our structure approaches a
strained QW, grown in the [001] direction. However, since the diagonal
shear term will
decouple the heavy and light hole bands in the QD by approximately 100 meV,
the correction of these non-diagonal terms in the strain-Hamiltonian in
Eq. (1) should be small near the band edge. Fig. 2 displays the sum of
the strain and band-edge confinement potentials seen by the electrons
and heavy holes for a QD having *R *= 40 nm , *W *= 8 nm ,
*D *= 6 nm and *X *= 30 nm. This particular combination of
dimensions of the quantum well dot will be used as a reference and we
will refer to it as QD0 in the following discussion.

The calculation of the confinement effect was carried out separately
for the QW and QD. We first calculated the conduction and valence band
levels corresponding to the QW well away (
150 nm ) from the dot axis
(see Fig. 1). In the calculation of QW energies we used the strain
potentials corresponding to the distance of 150 nm from dot axis. At
this distance the strain tensor is already very close to the asymptotic
value of a lateral
QW grown on a very thick GaAs substrate. The off-diagonal strain
potentials *S* and *R* in Eq. (1) are also very close to
zero at *r * = 150 nm.

The strain Hamiltonian Eq. (1) was added to the four-band Luttinger-Kohn
Hamiltonian [9] and the conduction and valence band confinement energies,
and
, respectively,
were determined by solving the eigenvalue problem by FEM. From the
conduction and valence band confinement energies we obtain the
luminescence energy
1316 meV for a QW having the width *W* = 8 nm and cladding layer *D*
= 6 nm. Our calculations neglect the indium segregation in the cladding
layer. This would lower the barrier potential in the cladding layer and
reduce the QW luminescence energy. Because of the compressive strain the
shear interaction potential
will raise the
heavy-hole band and lower the light-hole band increasing their energy
separation by about 100 meV in the QW. Since furthermore the first
excited conduction band state
is approximately
100 meV above the
level, we conclude that the near-band-gap absorption and emission are
dominated by
transitions.

The confinement energies in the conduction and valence bands of the quantum well dot were calculated by using the cylinder coordinate system to make maximum use of the symmetry of the problem. Accordingly the total envelope function for electrons or holes was written as a factorized product . Here fulfills the eigenvalue equation

. (2)

In Eq. (2) is
the band edge confinement potential as a function of the distance from
the surface. Here *D* and *W* are the dimension parameters and
*Y* the step function. The strain potential
is equal to
or
for electrons
and holes, respectively. For electrons
, whereas for
holes the lateral and vertical masses are given by
and
respectively.
In Eq. (2) we have neglected the coupling of *HH* and *LH*
bands included in the Luttinger-Kohn Hamiltonian to make the problem
numerically two dimensional. However, as pointed out above the * HH*
and *LH* bands are separated by the
strain potential,
which decreases the effect of band coupling. In the following we have
labeled the different angular momentum states by
for
etc.,
respectively. In analogy to the theory of a two-dimensional harmonic
oscillator there is always one
state between two
levels. If the potential
would be exactly harmonic, we would have
and
etc. We found
that for low *n* and *m *these equations are rather accurately
fulfilled, whereas for higher rotational levels we found a splitting of
1-2 meV. This splitting is caused by the large centrifugal barrier, which
drives the electrons and holes from the dot axis to the outer region where
the harmonic approximation is not valid.

The zeros of the potential energy scales in Fig. 2 were set to the band
edges of the unstrained
. For both electrons
and holes, there is an extremum, minimum for electrons and maximum for
holes, in the QW below the InP stressor indicating that electrons and
holes are diffused to these local extrema and relaxed to bound QD
states before finally recombining radiatively. In the vicinity of the
dot axis the effective potential as a function of* r* (for
fixed *z*) is approximately harmonic. In Fig. 2 the conduction band
minimum is 96 meV below the bottom of the QW band edge and valence band
maximum 19.7 meV above the top of the QW band edge. We found a large
variation in the ratio of confinement energies
. The maximum value
of 0.71 was obtained for a thick cladding layer (*R* = 40 nm,
*D* = 18 nm) and the minimum 0.22 for a large InP island and thin
cladding layer (*R* = 50 nm, *D* = 4 nm). This large variation
in the sharing of the confinement effect between electrons and holes
should be useful in the experimental study of carrier relaxation in
the QDs.

We define the redshift as the energy difference between the QD interband
luminescence energy
from the corresponding QW transition energy. This difference is given by
, where all energies
are measured from the band edges in the QW. We use the label
even for the set
of dipole allowed QD luminescence lines, since the electron and hole
wave functions (not shown) clearly exhibit the single vertical mode that
is characteristic to the corresponding QW states. Fig. 3 shows the
redshifts of interband
transitions as a function of *R *and *D.* We use QD0 as a
reference and change one of the parameters *R* or *D* at a time,
while keeping the other dimensions fixed to their values in QD0.

The calculated redshifts are in excellent agreement with measurements of
Sopanen *et al* [6]. In Figs 3a-b we have shown the experimental
data by open symbols. We have used *R* = 37.5 nm for experimental
data in order to account for a more accurate determination of the InP
island radius [6]. The experimental values were interpolated to
correspond to W = 8 nm by auxiliary calculation of the redshift as a
function of W. When we used *D* = 6 nm and *W *= 8 nm the
largest calculated level spacing was
20 meVand the largest
redshift 102 meV.

The dipole-allowed interband transition amplitudes are proportional to overlap integrals . In Fig. 4 we present the overlap integrals between ( odd indices) and ( even indices) states for the reference dot QD0. The transitions with are also marked in Fig. 4 although these integrals are exactly equal to zero. The electron and hole states are orthogonal to a very high accuracy (for the lowest states ). Only for the higher states there is some nonorthonormality. The state (number 7 in Fig. 4) is an exception, since this electron level is above the energy and can decay by tunneling to the QW.

Our calculation corresponds to a complete screening of Coulomb
interaction. For strong confinement, recent calculations by Bockelmann
and coworkers [11], including Coulomb interaction, predict a large
deviation from the even spacing of levels typical to the harmonic
potential approximation. By contrast the experimental level spacing
has been found to remain constant and regular also at high excitation
intensity [6]. The state filling resulting from the increasing excitation
intensity [6] should have had a large effect on exciton states predicted
by Bockelmann [11]. The screening of Coulomb interaction could be
explained by a fast diffusion of electrons and holes to QDs followed
by slow relaxation to bound levels. Since the dots cover only a small
fraction of the surface the density of the screening charge at QDs can
become rather high even at low excitation intensity. Thus the good
agreement between our calculations and the experiment [6], and the
absence of intensity dependence of energy levels in the experiment,
support indirectly the existence of *phonon bottleneck* [12] effect
i.e. the slow relaxation of electrons and holes to bound levels before
radiative recombination.

In conclusion we have shown that a model based on strain induced
confinement and the Luttinger - Kohn theory can explain the most
salient features in the luminescence spectrum of quantum well dots
fabricated by S-K growth method. A more complete calculation accounting
for Coulomb interaction, the *HH-LH* coupling and including a study
of relaxation mechanisms is in progress.

This work has been largely inspired by the experimental work of M. Sopanen, H. Lipsanen and J. Ahopelto. We gratefully acknowledge having results of their experimental work at our disposal prior publication and their numerous comments on this work. We also thank Timo Kahala and Sami Saarinen for assistance in the strain calculations. This work was funded by the Academy of Finland.

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Parameter | Unit | GaAs | |

eV | 1.520 | 1.163 | |

(from GaAs) | eV | 0.276 | |

(from GaAs) | eV | 0.081 | |

6.79 | 10.01 | ||

1.92 | 3.54 | ||

2.78 | 4.41 | ||

0.0665 | 0.0556 | ||

11.81 | 10.94 | ||

" | 5.38 | 5.17 | |

" | 5.94 | 5.45 | |

eV | -7.10 | -6.68 | |

eV | 1.16 | 1.12 | |

eV | -1.70 | -1.73 |