Anisotropic Confinement, Electronic Coupling and Strain Induced Effects Detected by Valence-Band Anisotropy in Self-Assembled Quantum Dots

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.


Introduction
Recent attention has been given to the study of coupled quantum dot (QD) arrays for their potential application in quantum information processing [1][2][3]. The selfassembling 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][5][6][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][9][10][11][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][14][15][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 selfassembled 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 selfassembling growth of 1D dot arrays (QD chains) [19] 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.

Theoretical Model
A multi-band k · p model based on the standard Kohn-Luttinger [20] and parabolic Hamiltonians to probe the electronic structure of holes and electrons, respectively, in dots grown along the [100] 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 [21]. 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.
The double quantum dots structures under investigation are schematically illustrated in Figure 1. According to realistic dimensions the dots are assumed to have semi-cylindrical shape with radius ρ, laterally separated by an inter-dot barrier of thickness d. Since the main focus is concentrated in the tunneling along the lateral direction [ ] 011 , the confining potential is defined as V(r, z) = V(r) + V(z), where the infinite barrier model have been used, as represented in Figure 1b (Figure 1c), at top (left) and bottom (right) interfaces, whereas the finite barrier model at the internal interfaces have been adopted, as represented in Figure 1c, in order to account for inter-dot coupling effects.
For the GaInAs alloys in consideration, at the center of the Brillouin zone, the split-off band is energetically well separated from the topmost valence subbands. In the limit of decoupled split-off band, the four-band Hamiltonian provides a good description of low-lying hole states by considering the coupling between the heavy-hole (hh) (J = 3/2, j z = ±3/2) and the light-hole (lh) (J = 3/2, j z = ±1/2). In the effective-mass approximation, when spanned in this basis, the kinetic energy of the hole is described by the 4 × 4 Kohn-Luttinger Hamiltonian with the Luttinger parameters γ i (i = 1, 2, 3), and the momentum operatorsˆˆˆ, k k ik i x y . The Hamiltonian of the hole in the quantum dot system is where V(r) is an infinite barrier outside of the semicylindrical 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 [21].
By exploring cylindrical symmetry in the Kohn-Luttinger model, the wave function of a hole state can be written in the form The indexes (j, n, m) label the quantization along the z-direction (j) and in-plane (n, m) quantum numbers, respectively, α denotes the spin-up (| ↑ >) and spindown (| ↓ >) periodic Bloch function character, namely: |hh ↑〉, |lh ↑〉, |hh ↓〉 or |lh ↓〉 and, finally, C j n m , ( )  are the weight coefficients in the basis set of envelope wavefunctions, F j (z)f n,m (r, ), at a position (r, z) inside the dot. The solutions for the in-plane motion, f n,m (r, ), are given by [24] f a J J a n n n m n nm for semi-cylindrical confinement (Figure 1c). In these expressions, μ n,m is the mth zero of the Bessel function of order n, J n (x), whereas the form of function F j (z) depends on the profile potential along z-direction between the dots. The depth of the quantum well is determined by the offset between the valence-band edges in the dot and the barrier materials. For the GaAs/In 0.4 Ga 0.6 As interface, the valence-band offset can be estimated as ΔE v = 214 meV. By analytically solving the Schrödinger equation for holes and regarding the mismatch between the Luttinger parameters in the GaAs/In 0.4 Ga 0.6 As interfaces, the transcendental equation is derived, which determines all subband energies (j) and the corresponding wavefunctions (see Appendix 1). The signal (±) in the Eq. 16 provides them, respectively, with symmetric or asymmetric character F z j ± ( ) . Taking advantage of this fact, the Hilbert space for the hole wavefunctions Ψ v (r) can be split into two orthogonal subspaces, labeled I and II, that are classified according to the parity of the quantum number j. As a result, the Hilbert subspace I(II) gathers spinor states with spin-up (spin-down) components having odd j-values (even j-values) that are coupled with states with spin-down (spin-up) and odd j-values (even j-values). Hence, the eigenvalue problem for the Hamiltonian in Eq. 3 can be solved independently for each class of states I and II. The hole state wavefunction (4) for a given subspace can then be written as 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 operatorsˆ,k k ± ± 2 andk z involved in the off-diagonal terms Eq. 2 of the Hamiltonian ℋ KL are given in Appendix 2.
As shown schematically in Figure 1, effects associated with isotropic and anisotropic spatial confinements are simulated in the calculation by changing the lateral sizes, D 011 and D [ ] 011 , in the (100) plane as well as the inter-dot distance (d). Two geometry cases will be studied: (i) Uncorrelated dots, which consider isotropic spatial confinement in the (100) plane, with D D ≠ ) and include inter-dot coupling ( Figure 1b) that leads to a chain-like 1D dot alignment. Here, the spin quantization axis (z-axis) must be set along the [ ] 011 direction [25,26]. 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.
The oscillator strength for optical electric fields linearly polarized along the [ ] 011 and parallel to [011] directions (see Figure 1) can be calculated as . For uncorrelated dot arrays, showing mostly random distribution (case (i)), they are given by 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, 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.
In correlated arrays showing preferential dot diffusion, the compressive strain can be relaxed by forming 1D arrangement, as occurring for strain distribution in freestanding superlattices. In this case, the in-plane strain is defined by ε || = ε xx = ε yy = (a ⊥ -a w )/a w , where the lateral lattice constant (a ⊥ ) can be calculated as [27] Here, S α = (S 11 + S 12 ) α is the sum of elastic compliance constants, L α (a α ) 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 effects associated with the anisotropic confinement, within the inter-dot coupled model and simulated by a semi-cylindrical dot shape (see Figure 1c), uses only the subset of the expansion functions in Eq. 5 that complies with null boundary conditions at the flat part of the semi-cylinder. The corresponding linear crossed polarized optical matrix elements, for this correlated dot array model (case (ii)), are given by Pol 011 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.
First of all, let us analyze the effect of the spatial confinement in the case of a single dot with the semicylindrical shape, namely: the limiting case d ∞ shown in Figure 1c. As the strength of the spatial confinement is relaxed along the direction [ ] 011 by the QD size increase, the topmost valence band becomes occupied by a state with a stronger lh-character and reduced hh-contribution [4,28,29]. This effect is caused by the strong hole mass anisotropy, namely: m m where 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 C hh  = [ ] 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 selfassembled QDs grown along the [100] 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.
Analogously to the exchange of ground-state character induced by anisotropic confinement and shear strain fields, this effect can be also produced by electronic coupling between nearest-neighboring QDs, an effect that leads to the enhancement of the effective value 〈 〉 D [ ] 011 . The interchange of ground-state character is highly favored in coupled dots by increasing the inter-dot tunneling, as can be seen in Figure 3, which leads to the envelope function spreading along the coupling direction, [ ] 011 . In order to show this effect, we have used the combination of dots with finite inter-dot separation, d. Note, in Figure 3, that coupled dots will show a left-shifted crossing point for equal oscillator strength, when compared to the uncorrelated dot case. As discussed before, this shift can be further modified by shear strain fields.
For the limiting cases (see Figure 4)  , respectively, and these results are attributed to the anisotropy of hole effective masses. The crossing point where the polarized emissions have equal intensities can be shifted by the shear strain contribution to hhand lh-energy level positions. In Figure 5, it can be observed that the crossing points are shifted to the right as the strain order factor and/or inter-dot distance are increased. Furthermore, two asymptotic limits D D

Experimental Confirmation of the Purposed Theory
Experiments that confirm this modeling were performed using In 0.4 Ga 0.6 As 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 [ ] 011 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 As 2 or As 4 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. [19].
Two sets of samples A were grown under As 4 background. In one set, the layer of dots was left uncapped for morphology analysis, and in the other, the QDs were buried with GaAs for low-temperature PL analysis. The other two sets samples B were grown under the same conditions as the sets A, except that under As 2 background. Surface morphologies of the two uncapped samples were performed by using atomic force microscopy (AFM), as shown in Figure 6, imaged by Nanoscope IV in the tapping mode and using a high-resolution Silicon tip. The (1 × 1) μm AFM images show the morphologies of the In 0.4 Ga 0. 6 As uncapped dot samples.
The mean dot size and the center-to-center distance along the [ ] 011 direction of both sets are displayed in Table 1. The AFM pictures show clearly the effect of different Arsenic background both on dot formation and distribution. The predominantly anisotropic dot shape and distribution obtained along [ ] 011 direction is for samples grown under As 4 environment. Finally, these sets of samples enable us to use sample (B) as the reference for uncoupled QD arrays with mostly isotropic distribution on the (100) plane.
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° [30].  [011] [011] [011] [011]  (022) reflection exhibits a much broader profile than the scan measured with the sample rotated by 90°, close to the GaAs( ) 022 reflection. Such a behavior indicates that a more effective strain relaxation for the islands may take place along the [022] direction, while a more strained lattice profile is found along the [ ] 022 direction. The intensity distribution in both profiles of Figure 7a is almost symmetric with respect to the GaAs peak position, denoting the existence of compressively strained InGaAs inside the QDs, as well as on the GaAs matrix surrounding the QDs [31]. Similar diffraction profiles are observed in the longitudinal scans performed on sample B (Figure 7b). For this sample, the difference of widths of diffuse intensity on (022) and ( ) 022 scans is not as pronounced as observed for sample A, indicating a less anisotropic relaxation.
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 [022] and [ ] 022 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 L L 022 , 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.
Some considerations must be drawn before extending the analysis of the data shown in Figure 7c and 7d. For uncapped QDs, the relaxation of lattice parameter is monotonic from their base to their apex [32]. Capped QDs, in contrast, exhibit a non-monotonic gradient, with lateral and vertical strain variations. This condition generally implies in the existence of similar in-plane strain status on the island base and apex, both in contact with the GaAs matrix. It is therefore impossible to resolve vertically the position of iso-strain areas for the capped QDs with our GID measurements. Nevertheless, the lateral sizes observed represent a good approximation of the in-plane area of iso-strain regions projected on the substrate surface plane. Such approach allows for a visualization of the anisotropic strain relaxation. Since the diffraction signal observed at (|H|, |K|) > 2 is related to the existence of compressively strained GaAs surrounding the QDs [31], maps with the projected view of iso-strain areas were extracted from the experimental data by taking into account the L values for (|H|, |K|) < 2, which are directly related to compressively strained In 0.4 Ga 0.6 As from the islands (Tensile strained GaAs at the bottom and at the apex of the island also contribute to the diffracted intensity at |H| = |K| < 2. However, the total volume of material with local lattice parameter larger than a GaAs outside the island is considerably smaller than the amount of material contained inside the islands. For a discussion on tensile strained substrate material see [33]). These projection maps for QDs from samples A and B are shown in Figure 8a and 8b, respectively. The iso-strain projection areas were drawn following the condition that they are contained on curves delimited by where x and y are the in-plane coordinates along the [011] and [ ] 011 directions, respectively, considering the plane origin at the central QD position. Figure 8 shows the iso-strain areas for an in-plane region of approximately 1,100 Å × 700 Å, which contains 9 QDs for sample A and 4 QDs for sample B (see Table 1). The color scale in these maps refers to the in-plane strain with respect to bulk GaAs.
From Figure 8a, one clearly observes that iso-strain contour lines from one QD of sample A almost reach the neighbor QDs along 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 In 0.4 Ga 0.6 As 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. [34]. 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 [35]. Finally, a 500 Å-thick cap layer was added to the simulation, as represented in Figure 9a. Two-dimensional cuts of the simulated data are shown in Figure 9b, d, and 9f for sample A and Figure 9c, e, and 9g for sample B. The selected cuts are schematically depicted at Figure 9a and were chosen to be at the island bottom (b) and (c), middle (d) and (e), and top (f) and (g). Since the representation used in Figure 8 cannot be directly correlated to the Cartesian in-plane strain components x and y, the maps of Figure 9b-g were drawn as a function of the axial (first) principal strain component. Such principal component analysis allows for the reduction of the dimensionality of the data set, providing a resulting representation with radial symmetry. The axial strain component is given by [36] where ε xy is the in-plane shear strain and ε xx , ε yy the normal in-plane strains. For all principal strain maps, the color scale represents the deviation of the local lattice parameter with respect to the bulk local lattice parameter. Therefore, higher principal strain values are found in positions where the In 0.4 Ga 0.6 As lattice of the islands is fully strained to the GaAs lattice constant. Finite values of the axial strain component are also observed in regions surrounding the islands, in which the GaAs local lattice is affected by the proximity to the island. Selected contour level edges were marked by dark lines in all maps as a guide to the eyes.
The maps generated by in-plane cuts in the simulation of QDs in sample A clearly exhibit elongated contours along the [ ] 011 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 [ ] 011 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.
The effects of different dot size distributions and interdot coupling have been analyzed by low-temperature linear polarized PL measurements carried out on the samples A and B buried with GaAs. The samples were placed in a closed-cycle Helium cryostat (Janis-CCS-150) and excited using a 532-nm continuous wave YAG laser (Coherent Verdi V10-10 W). The PL signal was carried out by a monochromator (SpectraPro 2500i-0.5 m focal length) and detected by a liquid-nitrogencooled InGaAs photodiode detector array (Princeton Instruments-model 7498-0001). Figure 10a and  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 interdot 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 [ ] 011 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 [ ] 011 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. − 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 [ ] 011 than along the [110] 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 [39].

Conclusions
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 Villegas-Lelovsky et al. Nanoscale Res Lett 2011, 6:56 http://www.nanoscalereslett.com/content/6/1/56 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.