Anisotropic Confinement, Electronic Coupling and Strain Induced Effects Detected by Valence-Band Anisotropy in Self-Assembled Quantum Dots
© Villegas-Lelovsky et al. 2010
Received: 6 July 2010
Accepted: 9 September 2010
Published: 1 October 2010
A method to determine the effects of the geometry and lateral ordering on the electronic properties of an array of one-dimensional self-assembled quantum dots is discussed. A model that takes into account the valence-band anisotropic effective masses and strain effects must be used to describe the behavior of the photoluminescence emission, proposed as a clean tool for the characterization of dot anisotropy and/or inter-dot coupling. Under special growth conditions, such as substrate temperature and Arsenic background, 1D chains of In0.4Ga0.6 As quantum dots were grown by molecular beam epitaxy. Grazing-incidence X-ray diffraction measurements directly evidence the strong strain anisotropy due to the formation of quantum dot chains, probed by polarization-resolved low-temperature photoluminescence. The results are in fair good agreement with the proposed model.
Recent attention has been given to the study of coupled quantum dot (QD) arrays for their potential application in quantum information processing [1–3]. The self-assembling process and its control become essential concerns in the search for new proposals of optoelectronic and quantum computing devices. Also, the spinor states in quasi-zero dimensional systems and their electronics have become features of renewed interest [4–7]. High uniformity of size, shape and distribution control of dot arrays are required in many application proposals like detectors, low-threshold lasers and photonic crystals. The lack of control over the self-assembly process of formation of these QDs leads to inhomogeneous broadening in size and/or shape that may degrade the quality of a device application. Therefore, the need for probing size, shape and effective inter-dot coupling has become an important area of research in recent years [8–12].
The anisotropy observed in linearly polarized PL-emissions from self-assembled QDs has been studied in recent years, and several works have detected some correlation with the anisotropic shape of the QD array [13–16]. There is also an agreement about the complexity of valence-band effects in QDs as a relevant issue when dealing with optical response from transitions between these completely localized states [7, 17, 18].
In the present work, we addressed mechanisms of testing simultaneously one-dimensional (1D) lateral ordering of dots, inter-dot coupling and 2D anisotropy of self-assembled QDs from studies of grazing-incidence X-ray diffraction (GID) and polarized photoluminescence (PL) emissions under different excitation power. This work has been motivated by the plausibility of controlled self-assembling growth of 1D dot arrays (QD chains)  and their potential use for testing important quantum effects such as correlation of information and optical coupling between dots where the relevant aspects of effects associated with inter-dot coupling and QD shape, size and distribution deserve special attention. It is also discussed the interplay between shape and strain fields with the inter-dot correlation that is revealed in the GID measurements and PL-emission spectra from QD arrays. Two sets of samples are investigated: one shows chain-like 1D correlation between neighboring dots and the other exhibits a mostly random island distribution. Two different QD shape models are used in order to calculate and test the polarized optical emission spectra dependence with spatial dot correlation and local geometry. The experimental confirmation included in this work highlights and supports the importance of probing correlated distribution in QD arrays for the characterization and improving of the growth-controlled processes.
A multi-band k · p model based on the standard Kohn–Luttinger  and parabolic Hamiltonians to probe the electronic structure of holes and electrons, respectively, in dots grown along the  direction was developed. Due to strong valence-band admixture, such a procedure provides straightforward information on the relaxation of the inter-band optical transition selection rules, using lower computational efforts than in tight-binding calculation model, for example [13, 14]. The built-in strain field distribution, which lead to the formation of self-assembled QD arrays, has been considered within the Bir–Pikus deformation potential model . Uniform strain tensors are assumed, a model that neglects effects caused by variations at the QD interfaces [22, 23]. This approximation works reasonably well for the study of ground-state properties of medium (~150 Å) and large (>250 Å) size dots.
with the Luttinger parameters γ i (i = 1, 2, 3), and the momentum operators .
where V(ρ) is an infinite barrier outside of the semi-cylindrical cross-section, and V(z) is a double quantum well potential with infinite high outside walls, whose finite barrier is due to the offset between the band edges in the well and barrier materials; ℋ BP is the Bir–Pikus Hamiltonian .
The hole states of the semi-cylindrical QDs system are calculated by exact diagonalization of the Hamiltonian ℋ, on a finite basis set expansion given by Eq. 6 using a standard numerical diagonalization technique. The matrix elements of the momentum operators and involved in the off-diagonal terms Eq. 2 of the Hamiltonian ℋ KL are given in Appendix 2.
These two models were tested and compared in order to search for the main qualitative differences between optical emission probabilities for light polarized along and perpendicular to the z-axis, respectively. This modeling tests the different behavior of optical emissions associated fundamentally with the difference between heavy-hole (hh) and light-hole (lh) longitudinal and transversal ellipsoidal effective masses as well as the effects originated from the strain fields on these hole energy levels.
where P = ⟨s|p x |x⟩ = ⟨s|p y |y⟩ = ⟨s|p z |z⟩ is the isotropic conduction-valence-band momentum matrix element between functions at the Γ-point, is the overlap between j th electron and hole envelope functions along z-axis, and the factor 2 is due to double spin degeneracy.
according to Eq. 4, and this identity is independent of QD size. Besides, neither hydrostatic nor axial strain contributions would induce changes to Eq. 8 in this symmetric case (unless anisotropic strains are applied). Therefore, a distribution of cylindrical uncorrelated dots over the (100) plane would lead to identical linear PL-emission intensities polarized along and perpendicular to the z-axis.
Here, Sα = (S11 + S12)α is the sum of elastic compliance constants, Lα (aα) is the width (bulk lattice constant) of the corresponding layers regions α = w (well) or b (barrier). In this way, a 3% strain can be relaxed to a value near 1%. Although shear strain contribution, which affects the separation between hh and lh subbands, becomes relaxed, the hydrostatic strain component leads to the effective reduction of the inter-dot potential barrier, which enhances the inter-dot coupling and tunneling. The envelope function spreading along the direction favors the confinement of a carrier with higher in-plane effective mass, which leads to the exchange of the ground-state character, since .
Here, the factor 2 occurs due to the summation over subbands j = 1,2 since these states are nearly degenerate for large inter-dot separation, d. It is clear that the identity in Eq. 8 has changed and no longer holds for all values of the inter-dot distance and QD sizes. We will be showing below that mass anisotropy of hole ground-state might be hold responsible for these anisotropic optical emission intensities once the dot confinement strength becomes relaxed in certain directions, whether by dot size anisotropy and/or by inter-dot coupling tuned by the strain fields.
where ⟨D⟩ and denote mean confining lengths. Consequently, by tuning the confinement anisotropically, the condition E lh < E hh can be attained due to the mass anisotropy of carriers. As a result, the corresponding envelope functions must be more extended in one direction than the other. Thus, the corresponding PL transitions allowed for certain light polarization can probe the anisotropic character of the Bloch functions that, in the multi-band calculations, are determined by the values of the expansion coefficients in Eq. 4. It is noted, from Eq. 10, that a state having small hh-character and, consequently, small values of coefficients and , produces smaller oscillator strength for optical transition polarized along the inter-dot coupling direction .
Figure 2 shows the oscillator strength values calculated for two coupled semi-cylindrical QDs with two values of the transverse diameter, D, as a function of the axial length, (see Figure 1c). Here, we have estimated the strain strength to hold with the uncorrelated dot array condition and confirm that the bigger the transverse size of dot array is (Figure 1b) the smaller must the strain order factor be. Furthermore, for compressive strain ε|| > ε⊥, the crossing point can be shifted toward the dotted line. For dilation strain, with ε|| < ε⊥, the crossing point is shifted away from the uncorrelated dot condition, and this condition can be attained in self-assembled QDs grown along the  direction. Certainly, shear strain field distribution is able to tune the equal oscillator strength condition for these mutually perpendicular polarized emissions in isolated anisotropic QDs.
Experimental Confirmation of the Purposed Theory
Experiments that confirm this modeling were performed using In0.4Ga0.6As QDs grown by molecular beam epitaxy on semi-insulating (100)GaAs. The QDs were obtained using the Stranski–Krastanov growth mode. Two set of samples were prepared for the experiments: (A) QDs with strong anisotropy in shape along direction and with partial ordering along that; (B) QDs with weak or no anisotropy on the (100) surface and large separation in both in-plane directions. The shape and the distribution of QDs were controlled by the Arsenic background. The use of As2 or As4 background during the growth allows the control of group III element diffusion on GaAs (100) surfaces, providing choices for different dot samples with the same composition but different shapes and distribution. Details of growth mechanisms and the processes involved in diffusion controlling by the background Arsenic environment are described in Ref. .
Average QD parameters with dispersion obtained from a Gaussian fit of the AFM data
3.9 × 1010
280 ± 12
220 ± 10
63 ± 12
160 ± 30
1.9 × 1010
350 ± 15
350 ± 15
90 ± 15
330 ± 75
Grazing-incidence X-ray diffraction (GID) measurements were performed in both samples at the XRD2 beamline of the Brazilian Synchrotron Light Laboratory (LNLS), using a 4 + 2 axis diffractometer. The X-ray photon energy was fixed to 10 keV. Since both samples were capped by a GaAs 50 nm layer, the incident angle was fixed at 0.28°, slightly above the GaAs critical angle, maximizing the signal from the buried quantum dots. The diffracted signal was measured by integrating the exit angle from 0 to 1.2° .
In order to quantify the strain relaxation inside QDs in both samples, transversal scans were performed at several positions along the longitudinal profiles shown in Figure 1a and 1b. These scans (not shown here) are measured by fixing the θ - 2θ condition and varying the sample rotation angle θ solely. In momentum transfer space, the angular momentum transfer q a = (4π/λ)sin(2θ/2)sing(Δθ) is varied, where Δθ = (θ/2θ)/2. Such a procedure allows to obtain the average lateral size L of regions inside the QDs with constant strain status by evaluating the width Δq a of transversal scans, L = 2π/Δq a [30, 32]. Values obtained for the local lateral size of iso-strain regions as a function of the in-plane strain status for samples A and B in the  and directions are shown in Figure 7c and 7d, respectively. For both samples, the lateral size of iso-strain regions along the QDs chain direction is larger than along the axis parallel to the chains. The ratio , which is a quantitative indicator of the anisotropic lattice relaxation inside QDs, is larger for sample A than for sample B, corroborating the qualitative information inferred from the widths of longitudinal scans.
From Figure 8a, one clearly observes that iso-strain contour lines from one QD of sample A almost reach the neighbor QDs along the chain direction. An asymmetric ratio of 1.7 is found for the broader iso-strain contour lines of QDs in this sample, pointing out again to a more pronounced strain relaxation along the  axis. The physical presence of very close QDs along the chains may therefore induce a modulation of the strain field that allows for a gentle strain relaxation in the direction. In sample B (Figure 8b), the asymmetric shape of iso-strain regions is still observed, but with a ratio of 1.35. Although an elongation is observed along the direction, the QDs are too apart from each other and do not strongly influence the strain field of the neighbor QDs in this direction.
Since the GID measurements do not reveal directly the height above the substrate of each iso-strain region finite element method, simulations were performed using a commercial software package to provide complementary information on the strain configuration of capped islands. In our simulations, a three-dimensional box containing a single GaAs capped In0.4Ga0.6As island was created for each sample, with periodic contour conditions at all lateral edges in order to take into account the symmetry of QD chains and the possible interaction with the strain field from neighbor QDs. A 15 Å thick wetting layer of nominal concentration was inserted between the islands and the substrate, following Ref. . The island profiles used in this simulation were extracted from the AFM measurements in uncapped islands (Figure 6) that resulted in the dimensions from Table 1. The nominal composition was kept, assuming thus a negligible deviation of island stoichiometry from the nominal values (Anomalous grazing-incidence diffraction measurements performed at the Ga - K edge do not point out to deviations (within an error bar of 7%) from the nominal In/Ga content inside QDs.). Such assumptions consider that islands do not undergo dramatic changes in morphology or composition under capping, which is a valid approximation for the growth temperatures used here and the reduced strain with respect to pure InAs islands . Finally, a 500 Å-thick cap layer was added to the simulation, as represented in Figure 9a.
The maps generated by in-plane cuts in the simulation of QDs in sample A clearly exhibit elongated contours along the direction, most notably for the cuts at the island basis and middle. This indicates that for lower in-plane strain conditions, the lattice surrounding the islands behave as semi-continuous wires along the direction. In the QDs of sample B, an elongation of axial strain contour levels is also observed along the chain directions for all maps. However, the anisotropic effect is much more reduced with respect to the results obtained for sample A.
In Figure 10a, one may see a polarization degree around 6%, as might be expected due to the elongation in the quantum dots profile revealed by the AFM images (Figure 6) and strain distribution (Figure 8). As highlighted in Figures 2 and 3, the oscillator strength grows for emissions linearly polarized along the larger dot size direction. This behavior is enhanced for inter-dot separation up to d ~ 160 Å. When d is further reduced, the inter-dot tunneling probability increases considerably, and this behavior is also enhanced. The PL intensity polarized along coupling direction is also enhanced in coupled QDs by the reduction of barrier heights due to hydrostatic strain of the order of 1%. Besides, the anisotropic PL-emissions from sample A, as shown in Figure 5a, can be qualitatively reproduced by the oscillator strengths, shown in Figure 3, calculated by using the nominal values for both samples. As seen in Figure 3, an effective increase in inter-dot tunneling (distance d ~ 160) Å would lead to the relaxation of the confinement along the direction. These effects would lead to a hole ground-state character exchange from predominant-hh to -lh, and to the intensity difference between these cross-polarized emissions, experimentally confirmed by Figure 10a.
For the isotropic case, PL-emissions occur when , a condition well-fulfilled for the cylindrical model of Figure 1b. By changing the dot shape and coupling along direction in (100) plane, the model shows that condition can be obtained for semi-cylindrical geometry only for a small combination of values that emulates uncoupled dot distribution in the (100) plane if strain effects are included into the Hamiltonian. According to the theoretical modeling, an isotropic dot distribution on the (100) plane (case (i)) accounts for isotropic crossed polarized PL-emissions, as shown in Figure 5b for sample B. However, according to Figure 10b, a small polarization degree is still present in symmetric QDs, associated with the elongation that remains, as revealed by the Figure 8b. Such feature might come from the anisotropic diffusion rate of Indium atoms during the growth, which presents a higher mobility than the Gallium atoms. Furthermore, the Indium diffusion coefficient is faster along the than along the  direction, and as a result, the quantum dots of sample B are not completely symmetric [37, 38].
To confirm the results from X-ray measurements, Figure 10c displays the shift in peak position of the spectra as a function of the excitation intensity. Note, for sample A, a shift toward higher energies as the excitation intensity grows. Such a blue-shift for the elongated dots has been associated with the screening of the built-in electric field due to the presence of strain. On the other hand, for sample B no remarkable energy shift is observed showing that the strain is not so pronounced as in the previous case .
The control and simulation of size anisotropy and effective inter-dot tunneling effects, as described in this work, is an important issue to be addressed during the characterization of ordered sets of coupled dots. The strain fields, present during the growth process of these QDs have led to the appearance of anisotropic geometric shapes, mostly elongated along the preferential direction. These effects can be probed by polarized optical responses from different samples. In summary, we have shown that the shape, spatial distribution and the inter-dot coupling of InGaAs self-assembled QDs can be probed and characterized by using linearly polarized PL-emissions. Valence-band effects due to admixing between hole states and strong anisotropic effective masses have led to different PL intensities in samples with lateral QD ordering forming "chain-like" structures. The envelope function model used here to describe the polarized optical responses showed fairly good agreement with structural AFM and X-ray data and may be used to predict or characterize the strength of inter-dot coupling and/or anisotropic dot shape and distribution.
Appendix 1: Double Quantum Well Potential
Appendix 2: Matrix Elements
In the particular case where t = t' Eq. 20 can be reduced to
The other matrix elements for the high-order operators are evaluated numerically using Eqs. 18–21 and the matrix identity .
where index j stands for the piecewise wavefunctions (17). The resulting integrals in z-direction are solved numerically.
Authors acknowledge financial support from the agencies FAPESP and CNPq (GEM, VL-R), CONACYT/Mexico and FAPEMIG (LV-L) and LNLS-MCT (AM) and ICTP/Trieste (CT-G) and the National Science Foundation of the U.S. trough Grant DMR-0520550 (BLL, YuIM). LV-L thanks E. Gomez for technical assistance.
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