Phys. Rev. B52, 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. B51, 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 , was recently suggested by Sopanen and coworkers . 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 .
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 . 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
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  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  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
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 . 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 . 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 , 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 . The state filling resulting from the increasing excitation intensity  should have had a large effect on exciton states predicted by Bockelmann . 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 , and the absence of intensity dependence of energy levels in the experiment, support indirectly the existence of phonon bottleneck  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.