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InP quantum dots
Journal of Crystal Growth 223 (2001) 321–331
Controlling the self-assembly of InAs/InP quantum dots R.L. Williams, G.C. Aers*, P.J. Poole, J. Lefebvre, D. Chithrani, B. Lamontagne Institute for Microstructural Sciences, National Research Council, Montreal Road, Ottawa, Ont., Canada K1A 0R6 Received 10 October 2000; accepted 16 November 2000 Communicated by C.R. Abernathy
Abstract We examine, both experimentally and theoretically, the feasibility of positioning and sizing self-assembled InAs quantum dots on a variety of patterned InP templates. Such templates are fabricated using either chemically assisted ion beam etching or selective oxide patterning, and subsequent chemical beam epitaxial overgrowth. For templates of trapezoidal cross-section, we demonstrate experimentally how quantum dot formation can be localised on the top (0 0 1) surface of the structure. For templates of triangular cross-section, in which the (0 0 1) top surface has been eliminated and strained quantum wells have been embedded, we use finite element modeling of the stress distribution and elastic strain energy to examine the positioning and sizing of self-assembled InAs quantum dots on the sidewalls. We show that dots can be located near the intersection of the quantum well stressors with the template sidewalls, whilst the lateral dot dimension is influenced by the extent of the local strain distribution. Crown Copyright # 2001 Published by Elsevier Science B.V. All rights reserved. PACS: 85.30.Vw; 62.20.x; 62.20.Dc Keywords: A3. Quantum dots; A3. Selective epitaxy; B3. Semiconducting indium phosphide
1. Introduction Self-assembled quantum dot nanostructures have now been realized in a large number of strained semiconductor materials systems using a variety of crystal growth techniques [1–3]. For systems that display the Stranski–Krastanow growth mode [4], quantum dot formation is typically achieved following the deposition of a thin (5–10 A˚) wetting layer. This wetting layer is commensurate with the underlying semiconductor *Corresponding author. G.C. Aers, Tel.: +1-613-993-9403; fax: +1-613-990-0202. E-mail address: [email protected] (G.C. Aers).
substrate and has an elastic strain energy that increases approximately linearly with wetting layer thickness. Quantum dot formation has been shown to be a strain driven process in which the increasing strain energy associated with the 2D wetting layer is partially offset by the formation of 3D islands and the consequent redistribution of strain between island and substrate. In addition to this strain redistribution, which acts to lower the total energy, island formation is necessarily accompanied by the formation of surface facets which act to increase the total energy of the system. A number of recent publications [5,6] have examined these two competing effects in an attempt to understand the morphology of
0022-0248/01/$ - see front matter. Crown Copyright # 2001 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 0 ) 0 1 0 1 8 - 6
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self-assembled quantum dots, whilst others [7] have also invoked the dynamics of dot formation to explain the experimentally observed densities and faceting behavior. For Stranski–Krastanow growth on planar substrates, quantum dot nucleation is found to be random across the plane. Following nucleation, dots will grow either by consuming the 2D wetting layer or subsequently by the coalescence of multiple dots (Ostwald ripening [8]) until an equilibrium is reached or the dots are capped and further growth is prohibited. For many device applications, it would be advantageous to be able to control the nucleation sites for quantum dots and ultimately to control the size and electronic structure of individual dots. For semiconductor laser applications for example, dot size control would result in a larger joint density of states at the lasing transition energy and a corresponding reduction in the threshold current density for lasing. Control of dot location would enable the production of integrated optoelectronic devices with varying functionality across the wafer and ultimately may be of interest in the production of photonic band gap structures based on 3D arrays of band gap tailored quantum dots. The issues of quantum dot positioning and sizing have been addressed in a number of recent publications, where control of dot nucleation sites has been effected by such factors as dot stacking [9,10], spontaneous lateral ordering effects [11] and substrate patterning prior to crystal growth [12]. In the work presented here, we address the issue of controlling InAs quantum dot size and lateral position on patterned InP substrates. We demonstrate experimentally, for the first time in the InAs/ InP materials system, how InAs self-assembled quantum dots can be located on the top (0 0 1) surface of trapezoidal InP templates. Further growth on such templates is shown to eliminate the (0 0 1) top surface, producing a template of triangular cross-section bounded by either {1 1 1} or {0 1 1} sidewall facets. Using finite element modeling, we illustrate the use of elastic strain fields in encouraging the nucleation of quantum dots on sidewall facets and demonstrate how dot location and lateral size control can be achieved using fully in situ techniques, on length scales
determined by the growth of embedded quantum wells.
2. Template preparation Since self-assembled quantum dot nucleation is a process driven by the energetics of strain relaxation, lateral control of the nucleation process is likely to involve techniques for lateral control of the semiconductor composition and consequently the elastic strain. Lateral control of the semiconductor composition has been successfully demonstrated for InGaAs material through the growth of quantum well material on patterned substrates [13,14]. These substrates are typically wet etched prior to crystal growth to introduce low index crystal facets on which the indium adatom migration length is large compared to that on adjacent orientations. Surface diffusion away from these low index facets and onto the adjacent areas of the substrate can then be used to alter the semiconductor composition in a manner controlled through the geometry (facet angle, facet length and mesa width) of the etched structures. Although these techniques work well, they suffer from a number of disadvantages if one wishes to generate local changes in the semiconductor composition reproducibly. Since the surface diffusion process is sensitive to the geometry of the etched structure, this must be accurately reproduced using the wet etch processing procedure. This can prove difficult if angles, depths and widths must all be controlled simultaneously. In addition, if the template for quantum dot nucleation is prepared outside the crystal growth system then the lithography inherently limits the length scale over which composition modifications can be made. Inclusion of strain fields in a controlled manner is also difficult in these ex situ prepared structures. To overcome such problems, we have developed techniques in which tight constraints are placed upon the patterned substrate geometry by forming the template in situ using the crystal growth technique itself [15]. In addition to the advantage of reproducibility, this technique also allows one to include a number of strained quantum wells during growth of the patterned
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substrate template. The strain fields arising from these quantum well stressors can be manipulated on a length scale determined by the quantum well width. In this paper, we will be concerned with the growth of self-assembled InAs quantum dots grown by chemical beam epitaxy (CBE) on (0 0 1) InP substrates patterned prior to growth either by chemically assisted ion beam etching (CAIBE), or by selective oxide patterning. The growth of both InP and InAs is performed at 5008C with a growth rate of 0.5 mm/h, using trimethyl-indium and cracked AsH3 and PH3 as the sources. For the structures where exposed InAs dots were to be imaged by SEM or AFM the samples were cooled down immediately following the deposition of the InAs layer. A typical CAIBE patterned substrate prior to CBE growth is shown in Fig. 1. CAIBE patterning is performed using an Oxford Instruments, chlorine based system. Using this dry etching technique we pattern undercut mesa stripes for subsequent CBE growth. Alternatively, selective oxide patterning is performed using plasma enhanced chemical vapor deposition (PECVD) and subsequent e-beam lithography to open windows to the InP substrate. In both cases, subsequent growth of InP produces high quality {1 1 1}B or {0 1 1} facets, depending upon the mesa orientation, which eventually eliminate the (0 0 1) top surface of the mesa completely, leaving fully
Fig. 1. CAIBE etched (0 0 1) InP substrate prior to CBE growth. Ridge is oriented along the [1 1 0] direction.
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developed triangular templates such as that shown in Fig. 2(a). Because such facets are determined crystallographically, the angle and resulting surface diffusion properties are highly reproducible, whilst the facet length can be accurately and reproducibly determined through the growth time. All that is required to tightly control the template geometry is an accurate knowledge of the initial patterned width prior to growth. This is generally determined from calibration of the lithography process, but if necessary, it can be measured precisely by scanning electron microscopy (SEM). In the case of CAIBE patterning, Fig. 2(a)
Fig. 2. (a) Cross-sectional view of patterned (0 0 1) InP substrate after CBE growth. The lines with lighter contrast correspond to embedded InGaAs quantum wells. (b) Sidewall view of the same structure showing a defect free {1 1 1}B facet. The undercut profile obtained with the CAIBE isolates growth on top of the ridge from that on surrounding areas.
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shows the benefit of an undercut etched mesa. As crystal growth proceeds, facets develop both on and off the mesa, resulting in a geometry that effectively isolates the mesa growth from any effects occurring on the remainder of the substrate. Under optimized conditions, the CBE growth of these patterned InP templates can be defect free for many hundreds of microns, as illustrated in Fig. 2(b) which shows the sidewall of a typical structure after growth. However, these templates are highly sensitive to pre-growth cleaning procedures and defects will occur on the sidewalls if cleaning prior to growth is not optimized.
3. Self-assembled dots on the (0 0 1) surface of trapezoidal templates Prior to full completion of the template facets, the comparatively large value of the indium adatom migration length on the sidewall surface ensures that very little growth occurs on this facet. Deposited material migrates instead to the (0 0 1) top surface. In Fig. 3, we show an SEM micrograph of such a trapezoidal template prepared using selective oxide patterning. In this example the template is oriented to produce {0 1 1} sidewall facets and the growth has been terminated to leave
Fig. 3. SEM micrograph showing quantum dot positioning on top (0 0 1) surface of an InP template prepared by selective oxide patterning. Template orientation is along the [1 0 0] direction and results in {0 1 1} sidewall facets.
a (0 0 1) top surface with a width of approximately 20 nm. Following growth of the InP template, sufficient InAs has been deposited to exceed the critical thickness on the (0 0 1) after incorporating the material migrating from the sidewalls. A string of InAs quantum dots is observed along the top of the template. No dots are observed either on the template sidewalls or on the oxide surrounding the template. Such structures are observed routinely using the template preparation techniques discussed above. In contrast to dots grown on unpatterned substrates, where the lateral dot dimension is approximately 30–40 nm [16], the dots shown in Fig. 3 are found to have a lateral dimension limited by the width of the (0 0 1) surface. At the lower left end of the template, the dots have approximately equal dimensions in directions parallel and perpendicular to the template axis and show a clear tendency towards uniform separation. At the top right end of the template, the (0 0 1) top surface is slightly wider and the dots appear to form dimeric structures or ‘artificial molecules’. In Fig. 4 we show an image, taken using atomic force microscopy (AFM), of a section from a second trapezoidal template. In this second structure, dots are observed over hundreds of microns, but the uniformity of size and spacing is not maintained for the full length. From the AFM image, the dot height is seen to be approximately 20 nm. This is typical of the dot
Fig. 4. AFM micrograph showing 20 nm high quantum dots on top (0 0 1) surface of an InP template prepared by selective oxide patterning. Template orientation is along the [1 0 0] direction and results in {0 1 1} sidewall facets.
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height obtained from AFM for uncapped dots on unpatterned substrates, but it is not representative of the height measured using transmission electron microscopy (TEM) for capped dots in this materials system, which is typically 2–5 nm [16].
4. Strain field modeling of self-assembled dots on sidewall facets In Fig. 5, we show an SEM micrograph of the (1 1 1)B sidewall of a CAIBE-etched, CBE grown structure prior to full completion. InGaAs quantum wells, which are intended to be lattice matched on planar regions of the substrate, have been included so that the evolution of the sidewall can be monitored. The InGaAs quantum wells are seen to end abruptly at the (1 1 1)B facet, even though material has certainly been deposited on the exposed surface during crystal growth. Using AFM, the (1 1 1)B facet is found to be smooth on the monolayer scale with an rms roughness of approximately 0.5 nm. Once the (1 1 1)B facet is fully developed and the top mesa surface removed, as shown in Fig. 2, growth is forced to proceed on the (1 1 1)B facet. This facet now presents an ideal template for the growth of InAs self-assembled quantum dots, since the in situ preparation and long adatom
Fig. 5. SEM image showing lack of growth on {1 1 1}B facets prior to facet completion.
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migration length have resulted in a particularly flat surface into which one can incorporate a desired strain configuration using the embedded quantum wells. To investigate the effects of strain templating on self-assembled InAs quantum dots, we have developed a finite element model that describes the elastic deformation of quantum well and quantum dot materials within a lattice mismatched matrix. The calculation is performed using a plane strain approximation, so that we assume translational invariance along the length of a patterned mesa. For calculation of elastic energies, we then assume that the energy per unit length is the same for the quantum dot as for the actual quantum wire structure we calculate. In practice, we expect the elastic energy for the dot to be slightly smaller due to strain relaxation at the additional dot edges introduced on segmenting the quantum wire, although we do not expect the difference to be important for dot dimensions greater than a few nanometres. In the following we will refer to the InAs structure on the (1 1 1)B facet as a quantum dot, keeping in mind the considerations mentioned above. In these calculations, the main effect that determines the equilibrium elastic configuration will be the lattice mismatch between the quantum well or dot and the InP matrix. Differences in the elastic coefficients, Cij , between the semiconductor binaries InAs, GaAs and InP are small enough to be ignored. In Fig. 6(a), we show the equilibrium finite element mesh for a typical InP template in which we have incorporated a lattice mismatched InGaAs quantum well. For convenience we ignore the top triangular section of the template, as being unimportant for the elastic calculations. Of course, in reality the top segment is vital to prevent (0 0 1) growth of the type discussed above. For these calculations, we have used eight-node isoparametric quadrilaterals and six-node isoparametric triangles [17]. To help in visualizing the effects of strain, all displacements in these diagrams have been enhanced by a factor of 10. The mesh also includes an InAs quantum dot structure placed on the (1 1 1)B sidewall, just below the quantum well. For this example, we have chosen a 6 nm In0.7Ga0.3As quantum well, so that the well is
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Fig. 6. Finite element calculations showing (a) the equilibrium mesh, (b) x-stress, (c) y-stress and (d) shear-stress distribution for a 2 nm thick, 20 nm wide InAs quantum dot on the (1 1 1)B sidewall of an InP patterned substrate template incorporating a 6 nm compressively strained In0.7Ga0.3As quantum well. The maximum compressive, tensile and shear stresses are given in dynes/cm2. The color representation is non-linear, so that low stress concentrations are accentuated.
compressively strained in the bulk of the material. At the (1 1 1)B edge of the InP template, the free surface imposes a zero normal stress boundary condition, which allows the compressive quantum well to bow outwards from the surface. Such effects have recently been seen in cleaved edge InGaAs/InP material and the resulting surface displacements measured using STM techniques [18]. The InAs quantum dot structure placed on this surface is constrained by the InP lattice
parallel to the surface but is free to expand in the perpendicular direction. To quantify the details of elastic displacement, we extract the elastic stress distribution across the patterned InGaAs/InP structures. Calculated x-, y- and shear-stress maps are shown in Figs. 6(b), (c) and (d), respectively, for the structure discussed above. These stress maps are color coded with blue representing compressive stress and red representing tensile stress or clockwise and counter
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clockwise, respectively in the case of the shear stress. The color representation is non-linear, so that low stress concentrations are accentuated. In the bulk of the template, the InGaAs quantum well shows a strong compressive x-stress as expected from the difference in lattice constant between the InGaAs well material and the InP cladding. In the region around the (1 1 1)B facet, the quantum well x-stress is considerably relaxed and is even slightly tensile. In this same region, the quantum well y-stress becomes highly tensile, in contrast to a value near zero in the bulk of the template. For wide enough InP mesa structures, the quantum well y-stress vanishes in the bulk in agreement with the situation in a fully planar structure. The x- and y-stress in the InAs quantum dot are seen from Fig. 6 to be highly compressive as expected, although these diminish considerably at the dot edges. Conversely, the InAs dot induces a slight tensile stress in the underlying region of the template, except near the dot edges where the template stress is again compressive. Finally, the shear-stress distribution for this structure is shown in Fig. 6(d). For the quantum well, shear stress concentrations are developed at the (1 1 1)B surface in comparison with much lower values at the mesa center. Shear stresses in the quantum dot appear mainly as a result of the dot placement at an angle to the normal x- and y-direction. The anti-symmetry of the shear-stress map reflects the symmetry of the twisting motion involved in the elastic displacement field. Using stress maps such as that shown in Fig. 6 and the corresponding strain fields, we calculate the elastic energy of the system for various positions of the InAs quantum dot. The energy as a function of the vertical position of the dot center up the (1 1 1)B sidewall is shown in Fig. 7 for two different quantum dot thicknesses and for three different quantum dot widths. The energy displayed in these figures is DE=Ed ¼ ðEw Et Þ=Ed , where Ew is the energy of the complete structure including well and dot, Et is the energy of the template with the well but without the dot and Ed is the energy of the dot on the (1 1 1)B facet in the absence of the quantum well. This represents the additional energy due to
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Fig. 7. Elastic strain energy as a function of vertical position of the dot center up the (1 1 1)B facet for (a) a 20 nm wide InAs quantum dot and (b) a 2 nm thick InAs quantum dot. The zero of position corresponds to the lower edge of the template shown in Fig. 6, whilst the dotted lines represent the location of the quantum well.
the quantum dot and emphasizes the benefits of placing the dot near the quantum well. In Fig. 7(a) we see that placing the quantum dot in the vicinity of the compressive quantum well reduces the strain energy cost of creating the dot by approximately 30%. For these 20 nm wide quantum dots, the energy minimum is seen to occur somewhat above the position of the quantum well, i.e. nearer to the top of the InP template. Considering the stress distributions shown in Fig. 6, we believe this to be a result of the tensile y-stress induced in the structure above the quantum well, resulting in an expanded lattice constant more favorable to the InAs quantum dot. From Fig. 7(a), the position of the energy minimum is seen to be relatively insensitive to the quantum dot thickness. In
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Fig. 7(b), the position of the strain energy minimum is examined as a function of the quantum dot width for a fixed dot thickness of 2 nm. For a 10 nm wide dot, the energy minimum lies within the quantum well, whilst as the quantum dot width increases, the minimum position moves to positions slightly above the quantum well. The shift in the position of the energy minimum is also accompanied by a substantial broadening. These observations illustrate how the InAs quantum dot shifts position to avoid the more compressive stresses below the quantum well. The broadening of the energy minimum occurs when the dot width exceeds the extent of the underlying tensile stress induced by the quantum well, so that a convolution between the width of the quantum dot and the extent of the underlying tensile strain is observed.
so that we ignore any influence of the growth dynamics in determining the size. Under these approximations and assuming that the dots are positioned at the energy minimum discussed previously, the features determining the quantum dot lateral dimensions are shown schematically in Fig. 8. In addition to the template energy Et defined previously, an ensemble of dots will contribute both elastic and surface energy terms to the total energy of the system. For an area A of InAs deposited per unit length of sidewall, the additional energy per unit length, EðW; AÞ Et , for an ensemble of dots of lateral dimension W and thickness t is given by
5. Quantum dot size determination
The elastic contribution from the dot covered regions of the sidewall is AðEw Et Þ=W, where A=W reflects the fraction of sidewall covered by dots. The surface energy contribution, 4tEs A=W, represents the surface energy per unit length for the complete dot ensemble and reflects the reduction in ensemble surface area as the lateral extent of individual dots increases at fixed A. Es is the surface energy per unit area of InAs. In the calculations that follow, we use a value of
Based on the results discussed to this point, one should be able to position self-assembled InAs quantum dots on these strain templates within distances of approximately 10 nm. We have presented results for quantum dots placed on a {1 1 1}B surface in the absence of an InAs wetting layer. Using the finite element model we have verified that wetting layers of a few monolayers thickness have a negligible effect on the equilibrium location of InAs quantum dots positioned using the techniques described above. Given that one can locate InAs quantum dots using strained quantum well stressors, the question arises as to whether one can also influence the lateral dimension of the dots in a manner similar to that observed for dot growth on the constrained (0 0 1) surface shown in Fig. 3. To investigate this question using our present elastic model, we make an assumption regarding the equilibrium shape of the quantum dots. For CBE InAs/InP selfassembled quantum dots grown on (0 0 1) material, we have shown recently that the dots are approximately square [16] and we make the assumption that the dots remain square on the {1 1 1}B sidewalls. We also assume that the size of the quantum dot can be obtained as a balance between the surface energy and the elastic energy,
EðW; AÞ Et ¼
AðEw Et Þ 4tEs A þ : W W
ð1Þ
Fig. 8. Schematic diagram showing the contributions of elastic strain energy and surface energy to the total energy for an ensemble of dots located over a compressively strained quantum well. Each dot is assumed to be a square plate of side W.
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42 meV A˚2 for Es in agreement with recent data for InAs [6]. At small values of W, the deposited InAs is strained to fit the quantum well lattice constant, whilst at large values of W, material being added at the wire edges is strained to fit the InP matrix. In both limits this results in a strain energy, Ew , that varies linearly with W. The elastic contribution to the total energy of the dot ensemble is therefore expected to become independent of W at both high and low W as suggested in Fig. 8. The transition between these limiting values is expected at a width, W0 , determined by the lateral extent of the quantum well strain field on the (1 1 1)B facet. In Fig. 9 we show the additional energy per unit area of deposited InAs, with surface and elastic contributions, calculated for the 6 nm compressive In0.7Ga0.3As quantum well structure discussed previously. All energies are normalized to the template energy, Et . The surface and elastic strain contributions to the total energy have the general form discussed above, although the transition from the low to high W regimes is quite broad and not particularly well defined. In this case, the total energy decreases monotonically with increasing W, indicating that the lateral dimension of the dots is not constrained by any strain energy penalty arising from increased lateral dimension. To obtain the desired size constraint, it is
Fig. 9. Total energy per unit area of deposited InAs including elastic strain and surface energy contributions for an ensemble of dots as a function of lateral dot dimension W. The strain template corresponds to the single compressive quantum well structure of Fig. 6.
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necessary to increase this energy penalty. As mentioned above, at large values of W, the increasing strain energy arises from InAs accommodating the in-plane lattice constant of the InP matrix. At large W, we must therefore strain the InAs dot material to a lattice constant even less favorable than InP. This can be achieved if we incorporate tensile strained InGaAs quantum wells on either side of the original compressively strained quantum well. When the dot lateral dimensions are large enough to encompass the tensile quantum wells, we would expect a significant strain energy penalty. In Fig. 10, we show the equilibrium finite element mesh and the associated stress distributions for the type of structure suggested above. In this case, we have chosen a 10 nm In0.7Ga0.3As quantum well bounded on one side by a 10 nm In0.45Ga0.55As quantum well and on the other by a 5 nm In0.36Ga0.64As quantum well. The structure is chosen to be strain neutral overall in the bulk, so that the compressive strain of the center well is compensated by the tensile strain of the wells on either side. The total energy as a function of lateral dot dimension is shown for this structure in Fig. 11(a), again scaled to the energy for the template alone. Due to the presence of the tensile quantum wells, the total energy now has a minimum at a lateral dot dimension of approximately 30 nm, suggesting that such a structure should influence the quantum dot dimensions and hence the excited state spectrum of such dots. With this particular structure, which has not been optimized, the magnitude of the energy minimum is not particularly large, representing a reduction of approximately 15% in the asymptotic value at large dimensions. To enhance the dot size selectivity, a number of strategies are possible, including increasing the compressive stress of the center well and increasing the tensile stress of the bounding wells. In Fig. 11(b) we show the total energy variation with W for such a structure in which we have increased the composition of the top well to 82% gallium. The position of the energy minimum remains unchanged in this case, since it is determined primarily from the separation between the two tensile wells, but the depth of the energy minimum has increased substantially, representing
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Fig. 10. Finite element calculations showing (a) the equilibrium mesh, (b) x-stress, (c) y-stress and (d) shear-stress distribution for a 2 nm thick, 20 nm wide InAs quantum dot on the (1 1 1)B sidewall of an InP patterned substrate template. The template incorporates a 10 nm In0.7Ga0.3As quantum well in the center, a 10 nm In0.45Ga0.55As well below and a 5 nm In0.36Ga0.64As well above. The structure is designed to be strain compensated overall.
approximately 25% of the large W asymptotic value.
6. Conclusions To fully realise the potential of self-assembled quantum dot nanostructures, it will be important in the future to control both the location and dimensions of individual dots. In this paper, we have investigated the effects of a variety of
templates upon the growth of self-assembled InAs quantum dots. For templates of trapezoidal crosssection, we show experimentally that quantum dots can be placed selectively on the (0 0 1) top surface. For templates of triangular cross-section, in which the (0 0 1) top surface has been eliminated, elastic modeling suggests that control of both the dot location and lateral dimension should be possible using combinations of embedded quantum well stressors. In addition, we expect the quantum dot growth dynamics to be affected
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References
Fig. 11. Total energy per unit area of deposited InAs including elastic strain and surface energy contributions for an ensemble of dots as a function of lateral dot dimension W. Panels (a) and (b) correspond to the multi-well template structures discussed in the text.
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Acknowledgements The authors would like to thank J.W. Fraser and S. Moisa for SEM and AFM micrographs.
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Controlling the self-assembly of InAs/InP quantum dots R.L. Williams, G.C. Aers*, P.J. Poole, J. Lefebvre, D. Chithrani, B. Lamontagne Institute for Microstructural Sciences, National Research Council, Montreal Road, Ottawa, Ont., Canada K1A 0R6 Received 10 October 2000; accepted 16 November 2000 Communicated by C.R. Abernathy
Abstract We examine, both experimentally and theoretically, the feasibility of positioning and sizing self-assembled InAs quantum dots on a variety of patterned InP templates. Such templates are fabricated using either chemically assisted ion beam etching or selective oxide patterning, and subsequent chemical beam epitaxial overgrowth. For templates of trapezoidal cross-section, we demonstrate experimentally how quantum dot formation can be localised on the top (0 0 1) surface of the structure. For templates of triangular cross-section, in which the (0 0 1) top surface has been eliminated and strained quantum wells have been embedded, we use finite element modeling of the stress distribution and elastic strain energy to examine the positioning and sizing of self-assembled InAs quantum dots on the sidewalls. We show that dots can be located near the intersection of the quantum well stressors with the template sidewalls, whilst the lateral dot dimension is influenced by the extent of the local strain distribution. Crown Copyright # 2001 Published by Elsevier Science B.V. All rights reserved. PACS: 85.30.Vw; 62.20.x; 62.20.Dc Keywords: A3. Quantum dots; A3. Selective epitaxy; B3. Semiconducting indium phosphide
1. Introduction Self-assembled quantum dot nanostructures have now been realized in a large number of strained semiconductor materials systems using a variety of crystal growth techniques [1–3]. For systems that display the Stranski–Krastanow growth mode [4], quantum dot formation is typically achieved following the deposition of a thin (5–10 A˚) wetting layer. This wetting layer is commensurate with the underlying semiconductor *Corresponding author. G.C. Aers, Tel.: +1-613-993-9403; fax: +1-613-990-0202. E-mail address: [email protected] (G.C. Aers).
substrate and has an elastic strain energy that increases approximately linearly with wetting layer thickness. Quantum dot formation has been shown to be a strain driven process in which the increasing strain energy associated with the 2D wetting layer is partially offset by the formation of 3D islands and the consequent redistribution of strain between island and substrate. In addition to this strain redistribution, which acts to lower the total energy, island formation is necessarily accompanied by the formation of surface facets which act to increase the total energy of the system. A number of recent publications [5,6] have examined these two competing effects in an attempt to understand the morphology of
0022-0248/01/$ - see front matter. Crown Copyright # 2001 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 0 ) 0 1 0 1 8 - 6
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self-assembled quantum dots, whilst others [7] have also invoked the dynamics of dot formation to explain the experimentally observed densities and faceting behavior. For Stranski–Krastanow growth on planar substrates, quantum dot nucleation is found to be random across the plane. Following nucleation, dots will grow either by consuming the 2D wetting layer or subsequently by the coalescence of multiple dots (Ostwald ripening [8]) until an equilibrium is reached or the dots are capped and further growth is prohibited. For many device applications, it would be advantageous to be able to control the nucleation sites for quantum dots and ultimately to control the size and electronic structure of individual dots. For semiconductor laser applications for example, dot size control would result in a larger joint density of states at the lasing transition energy and a corresponding reduction in the threshold current density for lasing. Control of dot location would enable the production of integrated optoelectronic devices with varying functionality across the wafer and ultimately may be of interest in the production of photonic band gap structures based on 3D arrays of band gap tailored quantum dots. The issues of quantum dot positioning and sizing have been addressed in a number of recent publications, where control of dot nucleation sites has been effected by such factors as dot stacking [9,10], spontaneous lateral ordering effects [11] and substrate patterning prior to crystal growth [12]. In the work presented here, we address the issue of controlling InAs quantum dot size and lateral position on patterned InP substrates. We demonstrate experimentally, for the first time in the InAs/ InP materials system, how InAs self-assembled quantum dots can be located on the top (0 0 1) surface of trapezoidal InP templates. Further growth on such templates is shown to eliminate the (0 0 1) top surface, producing a template of triangular cross-section bounded by either {1 1 1} or {0 1 1} sidewall facets. Using finite element modeling, we illustrate the use of elastic strain fields in encouraging the nucleation of quantum dots on sidewall facets and demonstrate how dot location and lateral size control can be achieved using fully in situ techniques, on length scales
determined by the growth of embedded quantum wells.
2. Template preparation Since self-assembled quantum dot nucleation is a process driven by the energetics of strain relaxation, lateral control of the nucleation process is likely to involve techniques for lateral control of the semiconductor composition and consequently the elastic strain. Lateral control of the semiconductor composition has been successfully demonstrated for InGaAs material through the growth of quantum well material on patterned substrates [13,14]. These substrates are typically wet etched prior to crystal growth to introduce low index crystal facets on which the indium adatom migration length is large compared to that on adjacent orientations. Surface diffusion away from these low index facets and onto the adjacent areas of the substrate can then be used to alter the semiconductor composition in a manner controlled through the geometry (facet angle, facet length and mesa width) of the etched structures. Although these techniques work well, they suffer from a number of disadvantages if one wishes to generate local changes in the semiconductor composition reproducibly. Since the surface diffusion process is sensitive to the geometry of the etched structure, this must be accurately reproduced using the wet etch processing procedure. This can prove difficult if angles, depths and widths must all be controlled simultaneously. In addition, if the template for quantum dot nucleation is prepared outside the crystal growth system then the lithography inherently limits the length scale over which composition modifications can be made. Inclusion of strain fields in a controlled manner is also difficult in these ex situ prepared structures. To overcome such problems, we have developed techniques in which tight constraints are placed upon the patterned substrate geometry by forming the template in situ using the crystal growth technique itself [15]. In addition to the advantage of reproducibility, this technique also allows one to include a number of strained quantum wells during growth of the patterned
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substrate template. The strain fields arising from these quantum well stressors can be manipulated on a length scale determined by the quantum well width. In this paper, we will be concerned with the growth of self-assembled InAs quantum dots grown by chemical beam epitaxy (CBE) on (0 0 1) InP substrates patterned prior to growth either by chemically assisted ion beam etching (CAIBE), or by selective oxide patterning. The growth of both InP and InAs is performed at 5008C with a growth rate of 0.5 mm/h, using trimethyl-indium and cracked AsH3 and PH3 as the sources. For the structures where exposed InAs dots were to be imaged by SEM or AFM the samples were cooled down immediately following the deposition of the InAs layer. A typical CAIBE patterned substrate prior to CBE growth is shown in Fig. 1. CAIBE patterning is performed using an Oxford Instruments, chlorine based system. Using this dry etching technique we pattern undercut mesa stripes for subsequent CBE growth. Alternatively, selective oxide patterning is performed using plasma enhanced chemical vapor deposition (PECVD) and subsequent e-beam lithography to open windows to the InP substrate. In both cases, subsequent growth of InP produces high quality {1 1 1}B or {0 1 1} facets, depending upon the mesa orientation, which eventually eliminate the (0 0 1) top surface of the mesa completely, leaving fully
Fig. 1. CAIBE etched (0 0 1) InP substrate prior to CBE growth. Ridge is oriented along the [1 1 0] direction.
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developed triangular templates such as that shown in Fig. 2(a). Because such facets are determined crystallographically, the angle and resulting surface diffusion properties are highly reproducible, whilst the facet length can be accurately and reproducibly determined through the growth time. All that is required to tightly control the template geometry is an accurate knowledge of the initial patterned width prior to growth. This is generally determined from calibration of the lithography process, but if necessary, it can be measured precisely by scanning electron microscopy (SEM). In the case of CAIBE patterning, Fig. 2(a)
Fig. 2. (a) Cross-sectional view of patterned (0 0 1) InP substrate after CBE growth. The lines with lighter contrast correspond to embedded InGaAs quantum wells. (b) Sidewall view of the same structure showing a defect free {1 1 1}B facet. The undercut profile obtained with the CAIBE isolates growth on top of the ridge from that on surrounding areas.
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shows the benefit of an undercut etched mesa. As crystal growth proceeds, facets develop both on and off the mesa, resulting in a geometry that effectively isolates the mesa growth from any effects occurring on the remainder of the substrate. Under optimized conditions, the CBE growth of these patterned InP templates can be defect free for many hundreds of microns, as illustrated in Fig. 2(b) which shows the sidewall of a typical structure after growth. However, these templates are highly sensitive to pre-growth cleaning procedures and defects will occur on the sidewalls if cleaning prior to growth is not optimized.
3. Self-assembled dots on the (0 0 1) surface of trapezoidal templates Prior to full completion of the template facets, the comparatively large value of the indium adatom migration length on the sidewall surface ensures that very little growth occurs on this facet. Deposited material migrates instead to the (0 0 1) top surface. In Fig. 3, we show an SEM micrograph of such a trapezoidal template prepared using selective oxide patterning. In this example the template is oriented to produce {0 1 1} sidewall facets and the growth has been terminated to leave
Fig. 3. SEM micrograph showing quantum dot positioning on top (0 0 1) surface of an InP template prepared by selective oxide patterning. Template orientation is along the [1 0 0] direction and results in {0 1 1} sidewall facets.
a (0 0 1) top surface with a width of approximately 20 nm. Following growth of the InP template, sufficient InAs has been deposited to exceed the critical thickness on the (0 0 1) after incorporating the material migrating from the sidewalls. A string of InAs quantum dots is observed along the top of the template. No dots are observed either on the template sidewalls or on the oxide surrounding the template. Such structures are observed routinely using the template preparation techniques discussed above. In contrast to dots grown on unpatterned substrates, where the lateral dot dimension is approximately 30–40 nm [16], the dots shown in Fig. 3 are found to have a lateral dimension limited by the width of the (0 0 1) surface. At the lower left end of the template, the dots have approximately equal dimensions in directions parallel and perpendicular to the template axis and show a clear tendency towards uniform separation. At the top right end of the template, the (0 0 1) top surface is slightly wider and the dots appear to form dimeric structures or ‘artificial molecules’. In Fig. 4 we show an image, taken using atomic force microscopy (AFM), of a section from a second trapezoidal template. In this second structure, dots are observed over hundreds of microns, but the uniformity of size and spacing is not maintained for the full length. From the AFM image, the dot height is seen to be approximately 20 nm. This is typical of the dot
Fig. 4. AFM micrograph showing 20 nm high quantum dots on top (0 0 1) surface of an InP template prepared by selective oxide patterning. Template orientation is along the [1 0 0] direction and results in {0 1 1} sidewall facets.
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height obtained from AFM for uncapped dots on unpatterned substrates, but it is not representative of the height measured using transmission electron microscopy (TEM) for capped dots in this materials system, which is typically 2–5 nm [16].
4. Strain field modeling of self-assembled dots on sidewall facets In Fig. 5, we show an SEM micrograph of the (1 1 1)B sidewall of a CAIBE-etched, CBE grown structure prior to full completion. InGaAs quantum wells, which are intended to be lattice matched on planar regions of the substrate, have been included so that the evolution of the sidewall can be monitored. The InGaAs quantum wells are seen to end abruptly at the (1 1 1)B facet, even though material has certainly been deposited on the exposed surface during crystal growth. Using AFM, the (1 1 1)B facet is found to be smooth on the monolayer scale with an rms roughness of approximately 0.5 nm. Once the (1 1 1)B facet is fully developed and the top mesa surface removed, as shown in Fig. 2, growth is forced to proceed on the (1 1 1)B facet. This facet now presents an ideal template for the growth of InAs self-assembled quantum dots, since the in situ preparation and long adatom
Fig. 5. SEM image showing lack of growth on {1 1 1}B facets prior to facet completion.
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migration length have resulted in a particularly flat surface into which one can incorporate a desired strain configuration using the embedded quantum wells. To investigate the effects of strain templating on self-assembled InAs quantum dots, we have developed a finite element model that describes the elastic deformation of quantum well and quantum dot materials within a lattice mismatched matrix. The calculation is performed using a plane strain approximation, so that we assume translational invariance along the length of a patterned mesa. For calculation of elastic energies, we then assume that the energy per unit length is the same for the quantum dot as for the actual quantum wire structure we calculate. In practice, we expect the elastic energy for the dot to be slightly smaller due to strain relaxation at the additional dot edges introduced on segmenting the quantum wire, although we do not expect the difference to be important for dot dimensions greater than a few nanometres. In the following we will refer to the InAs structure on the (1 1 1)B facet as a quantum dot, keeping in mind the considerations mentioned above. In these calculations, the main effect that determines the equilibrium elastic configuration will be the lattice mismatch between the quantum well or dot and the InP matrix. Differences in the elastic coefficients, Cij , between the semiconductor binaries InAs, GaAs and InP are small enough to be ignored. In Fig. 6(a), we show the equilibrium finite element mesh for a typical InP template in which we have incorporated a lattice mismatched InGaAs quantum well. For convenience we ignore the top triangular section of the template, as being unimportant for the elastic calculations. Of course, in reality the top segment is vital to prevent (0 0 1) growth of the type discussed above. For these calculations, we have used eight-node isoparametric quadrilaterals and six-node isoparametric triangles [17]. To help in visualizing the effects of strain, all displacements in these diagrams have been enhanced by a factor of 10. The mesh also includes an InAs quantum dot structure placed on the (1 1 1)B sidewall, just below the quantum well. For this example, we have chosen a 6 nm In0.7Ga0.3As quantum well, so that the well is
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Fig. 6. Finite element calculations showing (a) the equilibrium mesh, (b) x-stress, (c) y-stress and (d) shear-stress distribution for a 2 nm thick, 20 nm wide InAs quantum dot on the (1 1 1)B sidewall of an InP patterned substrate template incorporating a 6 nm compressively strained In0.7Ga0.3As quantum well. The maximum compressive, tensile and shear stresses are given in dynes/cm2. The color representation is non-linear, so that low stress concentrations are accentuated.
compressively strained in the bulk of the material. At the (1 1 1)B edge of the InP template, the free surface imposes a zero normal stress boundary condition, which allows the compressive quantum well to bow outwards from the surface. Such effects have recently been seen in cleaved edge InGaAs/InP material and the resulting surface displacements measured using STM techniques [18]. The InAs quantum dot structure placed on this surface is constrained by the InP lattice
parallel to the surface but is free to expand in the perpendicular direction. To quantify the details of elastic displacement, we extract the elastic stress distribution across the patterned InGaAs/InP structures. Calculated x-, y- and shear-stress maps are shown in Figs. 6(b), (c) and (d), respectively, for the structure discussed above. These stress maps are color coded with blue representing compressive stress and red representing tensile stress or clockwise and counter
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clockwise, respectively in the case of the shear stress. The color representation is non-linear, so that low stress concentrations are accentuated. In the bulk of the template, the InGaAs quantum well shows a strong compressive x-stress as expected from the difference in lattice constant between the InGaAs well material and the InP cladding. In the region around the (1 1 1)B facet, the quantum well x-stress is considerably relaxed and is even slightly tensile. In this same region, the quantum well y-stress becomes highly tensile, in contrast to a value near zero in the bulk of the template. For wide enough InP mesa structures, the quantum well y-stress vanishes in the bulk in agreement with the situation in a fully planar structure. The x- and y-stress in the InAs quantum dot are seen from Fig. 6 to be highly compressive as expected, although these diminish considerably at the dot edges. Conversely, the InAs dot induces a slight tensile stress in the underlying region of the template, except near the dot edges where the template stress is again compressive. Finally, the shear-stress distribution for this structure is shown in Fig. 6(d). For the quantum well, shear stress concentrations are developed at the (1 1 1)B surface in comparison with much lower values at the mesa center. Shear stresses in the quantum dot appear mainly as a result of the dot placement at an angle to the normal x- and y-direction. The anti-symmetry of the shear-stress map reflects the symmetry of the twisting motion involved in the elastic displacement field. Using stress maps such as that shown in Fig. 6 and the corresponding strain fields, we calculate the elastic energy of the system for various positions of the InAs quantum dot. The energy as a function of the vertical position of the dot center up the (1 1 1)B sidewall is shown in Fig. 7 for two different quantum dot thicknesses and for three different quantum dot widths. The energy displayed in these figures is DE=Ed ¼ ðEw Et Þ=Ed , where Ew is the energy of the complete structure including well and dot, Et is the energy of the template with the well but without the dot and Ed is the energy of the dot on the (1 1 1)B facet in the absence of the quantum well. This represents the additional energy due to
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Fig. 7. Elastic strain energy as a function of vertical position of the dot center up the (1 1 1)B facet for (a) a 20 nm wide InAs quantum dot and (b) a 2 nm thick InAs quantum dot. The zero of position corresponds to the lower edge of the template shown in Fig. 6, whilst the dotted lines represent the location of the quantum well.
the quantum dot and emphasizes the benefits of placing the dot near the quantum well. In Fig. 7(a) we see that placing the quantum dot in the vicinity of the compressive quantum well reduces the strain energy cost of creating the dot by approximately 30%. For these 20 nm wide quantum dots, the energy minimum is seen to occur somewhat above the position of the quantum well, i.e. nearer to the top of the InP template. Considering the stress distributions shown in Fig. 6, we believe this to be a result of the tensile y-stress induced in the structure above the quantum well, resulting in an expanded lattice constant more favorable to the InAs quantum dot. From Fig. 7(a), the position of the energy minimum is seen to be relatively insensitive to the quantum dot thickness. In
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Fig. 7(b), the position of the strain energy minimum is examined as a function of the quantum dot width for a fixed dot thickness of 2 nm. For a 10 nm wide dot, the energy minimum lies within the quantum well, whilst as the quantum dot width increases, the minimum position moves to positions slightly above the quantum well. The shift in the position of the energy minimum is also accompanied by a substantial broadening. These observations illustrate how the InAs quantum dot shifts position to avoid the more compressive stresses below the quantum well. The broadening of the energy minimum occurs when the dot width exceeds the extent of the underlying tensile stress induced by the quantum well, so that a convolution between the width of the quantum dot and the extent of the underlying tensile strain is observed.
so that we ignore any influence of the growth dynamics in determining the size. Under these approximations and assuming that the dots are positioned at the energy minimum discussed previously, the features determining the quantum dot lateral dimensions are shown schematically in Fig. 8. In addition to the template energy Et defined previously, an ensemble of dots will contribute both elastic and surface energy terms to the total energy of the system. For an area A of InAs deposited per unit length of sidewall, the additional energy per unit length, EðW; AÞ Et , for an ensemble of dots of lateral dimension W and thickness t is given by
5. Quantum dot size determination
The elastic contribution from the dot covered regions of the sidewall is AðEw Et Þ=W, where A=W reflects the fraction of sidewall covered by dots. The surface energy contribution, 4tEs A=W, represents the surface energy per unit length for the complete dot ensemble and reflects the reduction in ensemble surface area as the lateral extent of individual dots increases at fixed A. Es is the surface energy per unit area of InAs. In the calculations that follow, we use a value of
Based on the results discussed to this point, one should be able to position self-assembled InAs quantum dots on these strain templates within distances of approximately 10 nm. We have presented results for quantum dots placed on a {1 1 1}B surface in the absence of an InAs wetting layer. Using the finite element model we have verified that wetting layers of a few monolayers thickness have a negligible effect on the equilibrium location of InAs quantum dots positioned using the techniques described above. Given that one can locate InAs quantum dots using strained quantum well stressors, the question arises as to whether one can also influence the lateral dimension of the dots in a manner similar to that observed for dot growth on the constrained (0 0 1) surface shown in Fig. 3. To investigate this question using our present elastic model, we make an assumption regarding the equilibrium shape of the quantum dots. For CBE InAs/InP selfassembled quantum dots grown on (0 0 1) material, we have shown recently that the dots are approximately square [16] and we make the assumption that the dots remain square on the {1 1 1}B sidewalls. We also assume that the size of the quantum dot can be obtained as a balance between the surface energy and the elastic energy,
EðW; AÞ Et ¼
AðEw Et Þ 4tEs A þ : W W
ð1Þ
Fig. 8. Schematic diagram showing the contributions of elastic strain energy and surface energy to the total energy for an ensemble of dots located over a compressively strained quantum well. Each dot is assumed to be a square plate of side W.
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42 meV A˚2 for Es in agreement with recent data for InAs [6]. At small values of W, the deposited InAs is strained to fit the quantum well lattice constant, whilst at large values of W, material being added at the wire edges is strained to fit the InP matrix. In both limits this results in a strain energy, Ew , that varies linearly with W. The elastic contribution to the total energy of the dot ensemble is therefore expected to become independent of W at both high and low W as suggested in Fig. 8. The transition between these limiting values is expected at a width, W0 , determined by the lateral extent of the quantum well strain field on the (1 1 1)B facet. In Fig. 9 we show the additional energy per unit area of deposited InAs, with surface and elastic contributions, calculated for the 6 nm compressive In0.7Ga0.3As quantum well structure discussed previously. All energies are normalized to the template energy, Et . The surface and elastic strain contributions to the total energy have the general form discussed above, although the transition from the low to high W regimes is quite broad and not particularly well defined. In this case, the total energy decreases monotonically with increasing W, indicating that the lateral dimension of the dots is not constrained by any strain energy penalty arising from increased lateral dimension. To obtain the desired size constraint, it is
Fig. 9. Total energy per unit area of deposited InAs including elastic strain and surface energy contributions for an ensemble of dots as a function of lateral dot dimension W. The strain template corresponds to the single compressive quantum well structure of Fig. 6.
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necessary to increase this energy penalty. As mentioned above, at large values of W, the increasing strain energy arises from InAs accommodating the in-plane lattice constant of the InP matrix. At large W, we must therefore strain the InAs dot material to a lattice constant even less favorable than InP. This can be achieved if we incorporate tensile strained InGaAs quantum wells on either side of the original compressively strained quantum well. When the dot lateral dimensions are large enough to encompass the tensile quantum wells, we would expect a significant strain energy penalty. In Fig. 10, we show the equilibrium finite element mesh and the associated stress distributions for the type of structure suggested above. In this case, we have chosen a 10 nm In0.7Ga0.3As quantum well bounded on one side by a 10 nm In0.45Ga0.55As quantum well and on the other by a 5 nm In0.36Ga0.64As quantum well. The structure is chosen to be strain neutral overall in the bulk, so that the compressive strain of the center well is compensated by the tensile strain of the wells on either side. The total energy as a function of lateral dot dimension is shown for this structure in Fig. 11(a), again scaled to the energy for the template alone. Due to the presence of the tensile quantum wells, the total energy now has a minimum at a lateral dot dimension of approximately 30 nm, suggesting that such a structure should influence the quantum dot dimensions and hence the excited state spectrum of such dots. With this particular structure, which has not been optimized, the magnitude of the energy minimum is not particularly large, representing a reduction of approximately 15% in the asymptotic value at large dimensions. To enhance the dot size selectivity, a number of strategies are possible, including increasing the compressive stress of the center well and increasing the tensile stress of the bounding wells. In Fig. 11(b) we show the total energy variation with W for such a structure in which we have increased the composition of the top well to 82% gallium. The position of the energy minimum remains unchanged in this case, since it is determined primarily from the separation between the two tensile wells, but the depth of the energy minimum has increased substantially, representing
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Fig. 10. Finite element calculations showing (a) the equilibrium mesh, (b) x-stress, (c) y-stress and (d) shear-stress distribution for a 2 nm thick, 20 nm wide InAs quantum dot on the (1 1 1)B sidewall of an InP patterned substrate template. The template incorporates a 10 nm In0.7Ga0.3As quantum well in the center, a 10 nm In0.45Ga0.55As well below and a 5 nm In0.36Ga0.64As well above. The structure is designed to be strain compensated overall.
approximately 25% of the large W asymptotic value.
6. Conclusions To fully realise the potential of self-assembled quantum dot nanostructures, it will be important in the future to control both the location and dimensions of individual dots. In this paper, we have investigated the effects of a variety of
templates upon the growth of self-assembled InAs quantum dots. For templates of trapezoidal crosssection, we show experimentally that quantum dots can be placed selectively on the (0 0 1) top surface. For templates of triangular cross-section, in which the (0 0 1) top surface has been eliminated, elastic modeling suggests that control of both the dot location and lateral dimension should be possible using combinations of embedded quantum well stressors. In addition, we expect the quantum dot growth dynamics to be affected
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References
Fig. 11. Total energy per unit area of deposited InAs including elastic strain and surface energy contributions for an ensemble of dots as a function of lateral dot dimension W. Panels (a) and (b) correspond to the multi-well template structures discussed in the text.
by local stress distributions, although the importance of these effects is not yet known.
Acknowledgements The authors would like to thank J.W. Fraser and S. Moisa for SEM and AFM micrographs.
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