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GaAs quantum dots coupled to wetting layer: Strain and size matter
Accepted Manuscript Investigation of Intersubband Transitions in Truncated Pyramid-Shaped InAs /GaAs Quantum Dots Coupled to Wetting layer: Strain and Size Matter
Roghaieh Parvizi PII:
S0749-6036(18)31234-5
DOI:
10.1016/j.spmi.2018.08.010
Reference:
YSPMI 5855
To appear in:
Superlattices and Microstructures
Received Date:
09 June 2018
Accepted Date:
05 August 2018
Please cite this article as: Roghaieh Parvizi, Investigation of Intersubband Transitions in Truncated Pyramid-Shaped InAs/GaAs Quantum Dots Coupled to Wetting layer: Strain and Size Matter, Superlattices and Microstructures (2018), doi: 10.1016/j.spmi.2018.08.010
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ACCEPTED MANUSCRIPT
Investigation of Intersubband Transitions in Truncated Pyramid-Shaped InAs/GaAs Quantum Dots Coupled to Wetting layer: Strain and Size Matter Roghaieh Parvizi Department of Physics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran [email protected]
Abstract: In this paper, we theoretically report competition behaviour of size and strain effects on the InAs/GaAs quantum dots (QDs) eigenenergies, the envelop functions and also transition energies of the z-polarized WL-to-S states with varying the QD height and WL thickness. The obtained result for the bounded ground state of S implies that although the size-dependence behaviour with the inclusive of the strain effects generally resembles that of unstrained case, the energy values have changed appreciably for the short dots with value of 200 meV and for tall dots with nearly 300 meV. The competition behaviour of strain and size effects on the electronic properties of the structure revealed that the WL state is more sensitive to the size and WL thickness in presence of strain. For the weakly bounded WL state, the envelop function was localized in the wetting layer or bulk regions as a continuum state for the thick WL within the strain-modified structure. The calculated optical wavelength from z-polarized transition of WL-to-S in the strain-modified structure agrees well with previous experimental photoluminescence data from other studies. Keywords: InGaAs/GaAs quantum dot, wetting layer, strain, continuum elastic theory.
1. Introduction Three dimensional (3D) arrays of quantum dots (QDs) have attracted intense research efforts in aspects of the promising potential for applications in the optoelectronic devices such as lasers, photodetectors and light emitting diode [1-7] as well as new physics on a nanoscale. An underlying understanding of such structures features is necessary for properly exploiting QD application potentials. The efficiency of QD-based optoelectronic devices is strongly related to the distribution density and their size, shape of dots in a QD array. Self-assembled QDs formed by strained epitaxy in the Stranski–Krastanow growth mode is the most effective and prominent growth technique for the fabrication of coherent, dislocation-free arrays of dots [8]. For example, self-assembled InAs (quantum dot) on the InxAs1-xGa (surrounding matrix) have shown the most prominent realization of these type of QD structures where firstly involves the growth of a wetting layer (WL) of
ACCEPTED MANUSCRIPT monolayer thick and then followed by a spontaneous coherent island formation, as dot, [9]. The engineering of this method relies on the strain induced by a mismatch of lattice constants of the dot and surface materials which giving rise to a growth of self-assembling dots (islands) on a WL as a substrate. This lattice mismatch-induced strain strongly modifies the potential profile along the growth axis, removes degeneracy (in the valance band). Accordingly, the introduction of strain may provide a facile way to fabricate tuneable wavelength-based optoelectronic devices. Meanwhile, the size and shape exert a significant influence on the responding wavelength for QD-based structures [10-14]. Therefore, it is clear that the much sensitivity of QD’s energy levels to the shape, size, and strain provides the optoelectronic devices with greater potential to obtain the ideal responding wavelength. Several studies have been reported the optical properties of QDs with respect to size by using different methods and approximations [15-18] without considering strain effects. In some works, the lattice mismatch was modelled by using the thermal strain in the areas of QD with temperature raised by 1K and the thermal strain coefficient was assumed to be the absolute values of the lattice mismatch [19]. A more detailed eight-band model has been employed for a single InAs/GaAs dot [20, 21], but it is too complicated for QDs’ structures with WL. This work mainly focused on the energy of electron in the conduction band of InAs/GaAs QDs with inclusion of WL under competition behaviour of the strain distribution and the size of QD structure. As the strain has an inevitable effect on the potential, this model is studied the influences of strained interfaces of semiconductor materials, such as InAs/GaAs, as deformation potentials of the band offset. In the author earlier publications and as reported in Ref. [22], the band structure including the strain effect has been calculated precisely in terms of the altering effective mass and the potentials deformation for a QD with a single size/shape structure. In these calculations, the strain was assumed to be a constant value between the InAs QDs and the surrounding InGaAs capping layer [23-25]. It should be noted that in the calculation of band structure the piezoelectric effect can be ignored for quantum dots with in-plane symmetry, with (001) growth direction, as reported by Bester and Zunger [26-27]. Consequently, the study of QDs electronic properties can be carried out without being concerned the strain-induced piezoelectricity. We also neglected the Coulomb interaction energy, due to the presence of the stronger confinement regime of the deep confining potential created by the QD heterostructure structure [28]. In this paper, we paid especial attention to the energies and transition properties of the two states of the ground, atomic orbital of s, and the second exited state, which is well known as WL-state as the atomic orbital d, by employing a simple one-band effective-mass model with taking the advantage of the finite element method. This transition is of a great importance for the QDIR photodetectors devices, because carrier (electron) can move to the continuum energy region, which is an essential requirement for such devices [29-31]. According to the theoretical and experimental reports, the ground state (S) to WL state transition in InAs/GaAs is strongly z-direction polarized [32, 33]. The strain obtained for
ACCEPTED MANUSCRIPT different QD height and WL thickness are used next as input parameters to calculate electron energy dependence to the QD size. The competition behaviour of size and strain effects on the QD eigenenergies, the envelop functions and also transition energies of the z-polarized WL-to-S states were investigated in detail with varying the QD height and WL thickness.
2. Model and theory The model system was considered based on the most reported practical samples to find realistic structure parameters. InAs quantum dot was grown on GaAs substrate in (001) direction, preferable in a self-assembled growth method, by using solid-state molecular beam epitaxy (MBE) [34]. As mentioned, our dot geometry is of a truncated pyramid-shaped in square base and 2 nm top length. The self-assembled QD situated on top of the WL, which is placed between GaAs as substrate and capping layers, is shown in Fig. 1. The wetting layer is a thin and flat layer with a material similar to that of applied in QD with reported thicknesses ranging from 0.5 to 2 nm [35, 36]. In this work, to cover all practical size/shape of QD structure, two directions of vertical- and lateral-expansion of truncated-pyramid shaped QD with a constant WL thickness of 1 nm and base length of 22 nm will be studied. Additionally, WL thickness effects will be considered for a constant size of QD with a varying thickness. To solve the strain and Schrödinger equations, we have taken the advantage of the finite element method (FEM) throughout our complex geometry. In this method, at first the continuous domain is segmented into a finite number of elements (meshes) which are connected via “nodes”. Regarding that we have to define geometry of the system with finite appropriate size. Therefore, QD is settled in a sufficiently large size as a computation domain which is segmented into a sufficient number of meshes. Tetrahedral meshes were applied for subdomain and triangular meshes were used for boundaries. In this work, the model is meshed with the intention to fully utilize the memory capacity within three dimensionally geometry, while not exceeding the limit and taking longer computational time. The z axis is considered to be perpendicular to the plane of the wetting layer and the x axis lies in the wetting layer plane, as shown in Fig.1. To simulate the isolated QD in numerical analysis, we consider both the substrate and the cap layer are thick enough as to be 50 nm. Once the geometry is properly built and meshed, it is ready for strain analysis.
z direction GaAs InAs quantum dot
WL GaAs
x direction
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Fig. 1 The pyramid-shaped quantum dot/wetting layer schematic structure In this work, we calculate the strain distribution within the continuum elasticity approximation. Taking into account length change (i.e. elastic displacement ui) in all directions, the total elastic strain εij is generally defined as: εij=(ui.j+uj.i)/2.
(1)
The total strain is given by
ij ije 0,ij
(2)
Where ije is the elastic strain and 0,ij is the initial lattice mismatch strain of QD system. The MBE rown self-assembled heterostructures QDs, such as InAs/GaAs QDs, consist of two different materials with two different lattice constants. The mismatch of lattice constants gives rise to strain fields in these QD structures and affects the electronic and optical properties of QDs [37]. For the InAs/GaAs structure, in-plane lattice mismatch parameter is defined as ε0,xx = ε0,yy=(ɑGaAs − ɑInAs) / ɑInAs
(3)
which is equal to -0.067. In this computation, the lattice constants of dot (InAs) and substrate (GaAs) materials are considered as 0.605 and 0.565 nm which indicates that lattice constant of dot exceeds the substrate. This fact leads to compressive strain field inside dot region and tensile strain field for outer matrix. This in-plane lattice mismatch is treated as the initial strain field in latter calculation. With assuming that the initial strain is elastically, the strain in perpendicular direction is given by Poisson’s effect, as:
0, zz 2
C12 0, xx C 11
(4)
where C11 and C12 are components of the elastic stiffness matrix which interconnects stress σ to strain. In this paper, the parameters 0,xx , 0, yy and 0,zz are regarded as the initial normal strains which will induce further strain fields in the whole QD and matrix structure. Therefore, they will be treated as inputs in the latter finite element analysis. We also assume that the initial strain is uniform inside the quantum dot and is zero outside. For convenience, the stress and strain tensors can be described via two columns vectors (as reported in the Appendix). According to the theory of linear elasticity, the strain components appear in the constitutive relation, Hooke’s law [38],
ACCEPTED MANUSCRIPT ij Cijkl kle . (5) By employing equation (2), it can be rewritten as
ij Cijkl ( kl 0,kl )
(6)
where Cijkl is the component of the fourth-order tensor of elastic moduli, i.e. stiffness matrix, and ε0,kl is the initial strain tensors expressed in equations (3) and (4) with ε0,kl =0, for k≠l. In Eq. (6), 1 for field point within quantum dot region and 0 in the matrix. The InAs and GaAs are both cubic symmetric materials. Thus, only three of the 81 components of stiffness matrix are independent. So the components can be simplified such as C11 = C1111 = C2222 = C3333; C12 = C1122 = C2233, etc., and C44 = C1212 = C2323, etc. Accordingly, the strain and stress can be determined by three independent elastic moduli, namely, C11, C12 and C44. The numerical values of elastic moduli used in this paper were reported in the Appendix. In this paper, the hydrostatic and the biaxial strains are considered respectively, as
hyb xx yy zz
(7)
and
b zz
( xx yy ) 2
(8)
The above mentioned physical properties of each domain are set; i.e. initial strain fields, the governing strain equation and the anisotropic elastic stiffness matrix. For strain field calculations, a stationary solver would be applied because strain field problem is a stationary problem. Strain field problem arising from lattice mismatch also is a kind of a linear elastic boundary value problem which is analysed via FEM technique. Thus, appropriate boundary conditions are necessary for each interface. In this work, we have followed the comprehensive boundary conditions reported in Ref. [39] as: all the nodes of the in-plane-outer surfaces are fixed against displacement in the z-direction (normal direction to the faces) due to periodic symmetric argument. The bottom of the substrate is constrained in the z direction, to avoid rigid body shift while the upper surface is kept traction free [39].The other internal interfaces of dot/wetting layer/matrix are not determined and should be left as unknown in the FEM solving process. The solutions from strain field calculations are stored as the inputs for the next step of band structure analysis. In this paper, the electronic properties of self-assembled QDs structure have been studied based on the single-band effective mass approximation theory (EMA). We know that this approximation is accurate
ACCEPTED MANUSCRIPT in case that QD dimensions are much larger than the lattice constants. Energy levels and envelop functions are calculated with the starting point of the quantum theory that deals with the behaviour of carries confined to a truncated pyramid-shaped InAs/GaAs QD as the solutions of the stationary Schrödinger equation with including strain effect as follow:
-
2 1 .( (r )) V (r ) (r ) E (r ) 2 me ( x )
(9)
where V (r )
(10)
V (r ) Eg ac h
including the strain effect on the band and ħ, me (r ) ; Eg ; E; and (r ) respectively designate the reduced Planck's constant, the material dependent electron effective mass, the unstrained band-gap energy, the energy eigenvalue, and the envelop function of the electron with regard to an electron at
position r . The parameter ac in Eq. (10) denotes the deformation potential for the conduction band. These main parameters were taken from Ref. [15].
3. Strain distribution Based on and the Stranski-Krastanov self-assembly technique for such materials of InAs and GaAs, the QD islands of InAs can be created with different shape and symmetry which thus has effect on the strain distribution in dot structure and the corresponding band energies. In the following, we have calculated the strain distributions for a quantum dot with 22 nm base length and height of 6 nm attached to the wetting layer with 1 nm thickness. The εxx, εyy, εzz and the shear strain εxz components of the strain tensor were plotted in Figs. 2 (a), (b), (c) and (d), respectively in the x–z cross section view. Fig. 2 clearly shows that the compressive strains (negative εxx and εyy) within the dot in the x-y plane, while interfaces of GaAs with WL and dot experience tension from InAs due to misfits of lattice constants of two materials. On the other hand, εzz is positive within the dot region. In this case, the matrix tries to induce the dot that its lattice constant to be that of GaAs equals to 5.65 A. The shear strain components, εxy, εyz and εxz, are negligible in the dot and surrounding matrix, as εxz shows in Fig. 2 (d), they could, however, be appreciable at the interfaces for the piezoelectric calculations [39]. Fig. 2 (e) shows the strain components along the growth direction which implies the symmetry of the square-based pyramid quantum dot against z axis. In this figure, the black solid-line, εxx, was coated by the red dash-line, εyy, thus, based on the mentioned symmetry the component εxx equals to the εyy. In the wetting layer and the bottom of the dot experiences a tensile (positive εzz) strain and a compressive (negative εzz) strain in the growth direction. The tensile εzz strain component inside the dot was moderately changed, while the εxx and εyy are different with negative trends. The in-plane strain
ACCEPTED MANUSCRIPT symmetry with respect to the in-plane direction is now broken. This effect is the most pronounced for the dots which may cause the splitting off the dot degeneracy in the p-states [40].
(a) xx
(c) zz
(b) yy
(d) xz
(e)
(f)
Fig. 2 (a), (b), (c) and (d) are the strain components of εxx, εyy, εzz and εxz in the x-z plane crosssectional view. The position-dependent strain profiles present in the quantum dot structure along both the growth (e) and the base of quantum dot directions (f) with the insets that show εxx cross section view with its profile along z (e) and x directions (f).
Fig. 3 demonstrates the distribution of the hydrostatic εhyb strain in the x-z plane cross-sectional view. The hydrostatic strain is compressive within the quantum dot while the interfaces experience a small hydrostatic tensile strain. Both εxx and εyy are nearly homogeneous and negative within the dot and resulting the presence of a region with hydrostatic compression. In the substrate near the InAs/GaAs interface, GaAs experiences tension from InAs and the same effect is shared by the dot and capping layer interfaces at top of QD. In order to find out the effect of the size and shape on the strain distribution, we have calculated and illustrated the hydrostatic and bi-axial strain distributions for a number of structures whit various heights of the quantum dot and different thickness’ of the wetting layer. These strain components were plotted as functions of position along the in-plane basis direction and z-axis. Fig. 3 (b and c) illustrate the strain relaxation profile for InAs dots of the same base width
ACCEPTED MANUSCRIPT (B = 22 nm) but different heights of 3, 6, 9 and 12 nm. In all of these figures, it can be seen that in the inner part of the QD, the hydrostatic strain value is almost constant and compressive, while the hydrostatic strain value in the interfaces is very small and tensile somehow is near zero for the barrier region. These results also reveal that when InAs self-assembled pyramidal dots are buried under a GaAs over layer, significant tensile stress is induced at the top of the dot; as we proposed a truncated pyramid to a more reliable sample. On the other hand, the stress at the edges of the dot is weak and compressive. With making a comparison as the obtained results show, we note that for dots with small heights (short dots), the strain components are more than that of dots with large heights (tall dots), while in the interior part of tall dots, the strain is more uniform. This difference in the hydrostatic strain is about 20% between the shortest dot and the dot with 12 nm height (suggested here as the tallest dot, as functions of both in-plane and vertical directions, as shown in Figs. 3 (b and c). The hydrostatic strain distributions with different WL thicknesses, 1, 2, 3 nm are visualized in Fig. 3 (d) and Fig. 4 (e) against both the in-plane and normal direction, respectively. The results show that the dependence of hydrostatic strain component on the WL variation is moderate and about less than 10% in both directions.
(a) Hydrostatic strain
(c)
(b)
(d)
Fig. 3 shows the hydrostatic strain in the x–z plane, (a) its comparison plotted along the z– and x –axis for different height with a fix WL thickness of 1 nm (b) and for different WL thickness with constant height of 6 nm.
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The details of the bi-axial strain distributions for the bi-axial strain can be found in Figs 4 (a, b, c, d and e), where the strain distributions scanning through the x-axis and growth direction are shown and cross section is in the x-z plane view (Fig. 4 (a)). The bi-axial strain suggests a significant strain transfer takes place between the different interfaces of substrate, dot and capping layer regions and outside the boundary region the bi-axial strain is quickly decreased to zero. In contrast to the hydrostatic strain, the bi-axial strain tends to be positive in the dot region and negative in the substrate and capping layer interfaces with the dot. It is not only homogenous within the dot region, but also is completely vanished near the centre of the dot and so, the strain would be entirely hydrostatic in this segment. Along the z direction, taller dots experiences a more relaxed strain rather than shorter ones (heights <6 nm) in the nearly centre region in the dot which is the main reason for the splitting of the hole envelop functions in the valance band structure [38]. In the z direction, a significant transfer of bi-axial strain to the barrier takes place in the interfaces, while in the in-plane direction this strain focused more in the centre region of quantum dot with a uniform trend. From Fig.4 (e), this comparison suggests that the tuning of the wetting layer thickness does not alter the confinement potential behaviour, but it only might affect the height of conduction band.
(a) Bi-axial strain
(b)
(c)
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(d)
(e)
Fig. 4 shows the shear strain in the x–z plane (a), its comparison plotted along the z– and x –axis for different height with a fix WL thickness of 1 nm (b and c), for different WL thickness with constant height of 6 nm along z–axis, (d) and hydrostatic and shear strains for different WL thickness with constant height of 6 nm along the pyramid basis direction.
Within the strained structure, we have calculated its potential profile. The potential profile is achieved by computing the band structure for the proposed geometry of quantum dot. Fig. 5 shows the calculated confinement potential distribution induced with the strain for different heights of dot and wetting layer thicknesses along the growth direction [001]. Also, this figure demonstrates potential profile of a structure without strain for a dot with height of 12 nm and WL with thickness of 1 nm. In the absence of strain, the confining potential for an electron is a square well formed by the difference in the absolute energy of the conduction band-edges in InAs and GaAs [30]. The hydrostatic strain, which is mainly affect the conduction band (CB), makes the height of CB lower such that it shifts up from 0.6 eV (unstrained InAs conduction band edge) to 1.0 eV (strained quantum dot structure). With strain-considered structure, from the left side of the confining potential well corresponding to the bottom side of dot (wetting layer region) to the centre of dot region doesn't change appreciably against variation of QD height, but the top of dot shows more confinement, nearly 50 meV and in the right up side of potential well tends to contribute into the semi-continuum region for taller dots. Additionally, the obtained results shown in Fig. 5 (b) revealed that within the strainmodified structure, the width of confining potential well is extended in z direction with increasing the thickness of WL, but the heights of this well are almost identical within only 10 meV variations.
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(b) (a)
Fig. 5 The calculated bandoffset with including strain effect (solid-line) (a) for different height and without strain effect within a QD Structure with 12 nm height, (b) for different WL thickness with a fixed QD height of 6 nm
4. Electronic structure and optical transitions The band structure of quantum dot structure is generally changed with the inclusion of the strain, both conduction and valance bands, which reduces the symmetry of structure. Strain modifies energy gaps and removes degeneracy [28]. In this section, we study the strain-included electronic structure of QD in the envelop function approximation using effective-mass approximation framework in one-band Hamiltonian, Eq. (9). The depth of the confining potential is shifted to a lower value which this value is treated as initial treatment in the following electronic calculations. It is worth noting that this value varies as a function of position, due to the variation of strain against position, as discussed in the earlier section. The strain-modified structure is defined via formation potential theory, Eq. 10, for the next step of solving Schrödinger equation [37]. For the conduction band of a pyramid-shaped quantum dot, there are usually only three states, which two of them are bounded (S and P states) and one of them is a weakly bounded, denoted as WL-state within the dot region. The obtained result for the first bounded state of S implies that although the size-dependence behaviour inclusive of the strain effects generally resembles that of unstrained case, the energy values have changed appreciably due to the compressive strain, see Fig. 6. It is attributed to this fact that this state is bounded in the dot region even for the case that strain is included, as shown in Fig. 9. For the short dots (height < 6 nm), the electron energy with neglecting the strain is about 200 meV less than of dots with same size with considering the strain effects. This value reaches to 300 meV for taller dots, as shown in Fig. 6. With a more carefully comparison, it can also be concluded that the strain-modified electronic energy
ACCEPTED MANUSCRIPT structure is likely not to be as sensitive to variations in the dot size and WL thickness as it was to changes in the unstrained structure, because of the lower confining potential.
Fig. 6 The calculated eigen-energies corresponding to S-state (bound state) in presence of strain effects and with neglecting strain for a range of WL thickness which is varying from 0.5 to 2 nm with step of 0.5 nm In this work, we focused on the WL state to clarify the strain inclusion effects on a weak bounded state. In the energy domain of such quantum dot structures, the electron can localized in the wetting layer and bulk which are forming continuum or semi-continuum states [41]. For every structure depending on the applied wetting layer thickness, this value is different. For example, for a dot with WL thickness of 1 nm in the strain-modified structure, this domain starts from 1.38 eV to 1.5 eV, with energy of GaAs domain as shown in Fig. 6 as a constant dashed-line in a structure with WL thickness of 1 nm. For the WL-state, inclusive of the strain effects for the thin WL of 0.5 nm thickness, the energy trend against height with including strain is so similar to that of without strain. However, for the thicker wetting layer, a plateau occurs in the energy trend, correspond to the used WL and then energy reduces as a common process. Indeed, this plateau is the continuum region where the electron’s envelop function is localized into the wetting layer or bulk region. These results suggest that thicker WL in presence of strain effect, this state is located within wetting layer, see Fig. (10, (a)to-(g)), while without considering strain effects, Fig 10 (h and i), it shows a bounded state localized within both dot and wetting layer regions. The obtained energy for a structure, neglecting strain, with WL of 2 nm resembles that of structure with 1 nm WL thickness with considering strain effects. The
ACCEPTED MANUSCRIPT obtained results revealed that for a structure with WL more than 1 nm, in presence of strain, the WL is not localized and its energy is in the continuum energy domain. Hence, as a realistic case with considering strain effects, for the structures with thick WL, the WL state is not considered as a bound state such that its wave function is towards outside of the structure.
Fig. 7 Eigen-energies corresponding to WL-state in presence of strain effects (solid-lines) and with neglecting strain (dotted-lines). The constant dashed line is related to the semi-continuum states in a structure with WL thickness of 1 nm.
Some insight can be gleaned by comparing the S to WL state transition energy as a function of dot height for different WL thicknesses with and without including strain effects. Fig. 8 displays that with considering strain effects, the structure displays an extended emission wavelength rather than a QD structure with neglecting strain. Moreover, another interesting result is found for the behaviour of transition energy for various WL thicknesses which is a consequence of mentioned result in Fig. 7. It concerns the particular WL thickness from which the WL state is not bounded anymore. Indeed, one can notice that only for thin WL the result of unstrained structure resembles that of strain-modified structure, while for thicker case the results become different since band-offset that corresponds to the strained structure is reduced. In Fig. 10, it is shown 3D plot the electron envelop function of WL-state for different dot height and WL thickness, for both unstrained and strained structures to explain the
ACCEPTED MANUSCRIPT reasons behind the obtained results. For WL-to-S transition, different transition energies have been reported based on the experimental measurements. This difference in values can be attributed to the fabrication conditions and stoichiometry, as well as the size and geometry of QDs [31]. The obtained results here with inclusive of strain effects is well matched with the reported value 160 meV which is found by Sauvage et al. for the z-polarized WL-to-S transition by applying photocurrent spectroscopy [42]. Another example for this transition reported the energy of 180 meV using Fourier transform infrared spectrometer [43].
Fig. 8 The transition energy of WL-to-S as a function of QD height for different WL from 0.5 to 2 nm with 0.5 nm step in presence of strain effects (dotted-lines) and with neglecting strain (solid-lines).
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Fig. 9 Cross section view and 3D plot of envelop function corresponding to S-state which is bounded in the dot region with considering strain effect.
(a) WL=1 nm, h= 3 nm
(d) WL=1.5 nm, h= 3 nm
(g) WL=2 nm, h= 11 nm
(b) WL=1 nm, h= 9 nm
(e) WL=1 .5 nm, h= 7 nm
(h) WL=2 nm, h= 3 nm, Without strain
(c) WL=1 nm, h= 13 nm
(f) WL=1.5 nm, h= 10 nm
(i) WL=2 nm, h=12 nm, Without strain
ACCEPTED MANUSCRIPT Fig. 10 shows envelop function corresponding to WL-state for different QD heights and WL thickness for both situations of with (a, b, c, d, e, f and g) and without strain (h and i)
Acknowledgment The authors express their appreciation to the Research Council of the University of Yasouj for financial support of this work.
5. Conclusion In summary, we have studied the strain distribution of self-assembled QD by the continuum elastic approach to describe the strain-driven self-assembled process of QD based on lattice mismatch. The envelop functions and energy levels of QD have calculated in a one band effective-mass approximation framework for different height and WL thickness but at the same base pyramid of 22 nm. The results indicate that the strain-modified bound energy of S-state is mainly controlled by the size, while the strain inclusion only shifts the energy value by at least 200 meV. However, the strain and the size both play key role in determining the weak bounded WL-state such that only for thin WL the result of unstrained structure shows similar trend with strain-modified structure, while for thicker case the results become different. For structures with WL thicker than 1 nm, the WL state is not considered as a bound state such that its envelop-function is towards outside of the structure. The results show that the difference of bound-to-continuum energy levels of the suggested truncated pyramid-shaped QD matches well with the experimental data with including the strain effects which is useful in designing the ideal QD-based photodetectors.
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ACCEPTED MANUSCRIPT 6. A. Luque, A. Martí, E. Antolín, and C. Tablero, “Intermediate bands versus levels in nonradiative recombination,” Physica B: Condensed Matter 382, 320 (2006). 7. J. H. Marsh, D. Bhattacharyya, A. Saher Helmy, E. A. Avrutin, and A. C. Bryce. “Engineering quantum-dot lasers,” Physica E: Low-dimensional Systems and Nanostructures 8, 154 (2000). 8. A. Baskaran and P. Smereka, “Mechanisms of Stranski-Krastanov growth,” Journal of Applied Physics 111, 044321 (2012). 9. Bimberg D, Grundmann M and Ledentsov N N 1999 Quantum Dot Heterostructures (West Sussex: Wiley) 10. Shahzadeh, Mohammadreza, and Mohammad Sabaeian. "Numerical analysis of optical properties of oblate semi-spheroid-shaped quantum dots coupled to wetting layer." JOSA B 32, 1097 (2015) 11. Yu-Min, Liu, Yu Zhong-Yuan, and Ren Xiao-Min. "Influence of strain-reducing layer on strain distribution of self-organized InAs/GaAs quantum dot and redshift of photoluminescence wavelength." Chinese Physics Letters 25.5 (2008): 1850 12. El Ghazi, Haddou, et al. "Wetting layer effect on impurity-related electronic properties of different (In, Ga) N QD-shapes." Physica B: Condensed Matter 537 (2018): 207-211 13. El-Yadri, M., et al. "Temperature and hydrostatic pressure effects on single dopant states in hollow cylindrical core-shell quantum dot." Applied Surface Science 441 (2018): 204-209. 14. Aghoutane, N., El-Yadri, M., Feddi, E., Dujardin, F., Sadoqi, M., & Long, G. (2018). Pressure effect on an exciton in a wurtzite AlN/GaN/AlN spherical core/shell quantum dot. MRS Communications, 1-6 15. H. Jiang, and J. Singh, “Self-assembled semiconductor structures: Electronic and optoelectronic properties,” Quantum Electronics, IEEE Journal 34, 1188 (1998). 16. M. Sabaeian, and M. Shahzadeh. “Self-assembled strained pyramid-shaped InAs/GaAs quantum dots: The effects of wetting layer thickness on discrete and quasi-continuum levels,” Physica E: Low-dimensional Systems and Nanostructures 61, 62 (2014). 17. R. V. N. Melnik, and M. Willatzen. “Bandstructures of conical quantum dots with wetting layers,” Nanotechnology 1, 1 (2004). 18. Niculescu, E. C., and D. Bejan. "Nonlinear optical properties of GaAs pyramidal quantum dots: Effects of elliptically polarized radiation, impurity, and magnetic applied fields." Physica E: Low-dimensional Systems and Nanostructures 74 (2015): 51-58 19. Yu-Min, Liu, Yu Zhong-Yuan, and Ren Xiao-Min. "Influence of strain-reducing layer on strain distribution of self-organized InAs/GaAs quantum dot and redshift of photoluminescence wavelength." Chinese Physics Letters 25.5 (2008): 1850. 20. O. Stier, M. Grundmann, and D. Bimberg. “Electronic and optical properties of strained quantum dots modeled by 8-band k⋅p theory,” Physical Review B 8, 5688 (1999). 21. Sun, C., Lu, P., Yu, Z., Cao, H., & Zhang, L. “Wetting layers effect on InAs/GaAs quantum dots,” Physica B: Condensed Matter, 407, 4440 (2012). 22. Parvizi, Roghaieh. "Interband optical transitions of a strained InxGa1− xAs/GaAs quantum dot/wetting layer with various In mole fractions." Physica B: Condensed Matter 456 (2015): 87-92. 23. Parvizi, R., and G. Rezaei. "Investigation and optimization of intraband electromagnetically induced transparency in strained InAs quantum dot/wetting layer structures." Optics Communications 358 (2016): 65-72 24. Parvizi, R., and G. Rezaei. "Bound to continuum intersubband transition optical properties in the strain reducing layer-assisted InAs quantum dot structure." Physica E: Low-dimensional Systems and Nanostructures 75 (2016): 336-344. 25. Sabaeian, Mohammad, and Maryam Riyahi. "Truncated pyramidal-shaped InAs/GaAs quantum dots in the presence of a vertical magnetic field: An investigation of THz wave emission and absorption." Physica E: Low-dimensional Systems and Nanostructures 89 (2017): 105-114. 26. Bester, Gabriel, et al. "Effects of linear and nonlinear piezoelectricity on the electronic properties of In As∕ Ga As quantum dots." Physical Review B 74.8 (2006): 081305.
ACCEPTED MANUSCRIPT 27. Bester, Gabriel, et al. "Importance of second-order piezoelectric effects in zinc-blende semiconductors." Physical review letters 96.18 (2006): 187602. 28. Stoleru, Valeria-Gabriela, Debdas Pal, and Elias Towe. "Self-assembled (In, Ga) As/GaAs quantum-dot nanostructures: strain distribution and electronic structure." Physica E: Lowdimensional systems and nanostructures 15.3 (2002): 131-152. 29. Cortez, S., Krebs, O., Voisin, P., & Gérard, J. M. (2002). Polarization of the interband optical dipole in InAs/GaAs self-organized quantum dots. Physica E: Low-dimensional Systems and Nanostructures, 13(2), 220-223. 30. S. Sauvage, P. Boucaud, J.-M. Gérard, and V. Thierry-Mieg, In-plane polarized intraband absorption in InAs/GaAs self-assembled quantum dots, Physical Review B, 58 (1998) 10562. 31. Y.-Y. Lin and J. Singh, Theory of polarization dependent intersubband transitions in p-type SiGe/Si self- assembled quantum dots, Journal of applied physics, 96 (2004) 1059-1063. 32. C. Kapteyn, F. Heinrichsdorff, O. Stier, R. Heitz, M. Grundmann, N. Zakharov, et al., Electron escape from InAs quantum dots, Physical Review B, 60 (1999)14265. 33. M. Sabaeian and M. Shahzadeh, Investigation of in-plane- and z-polarized intersubband transitions in pyramid-shaped InAs/GaAs quantum dots coupled to wetting layer: size and shape matter Journal of Applied Physics 116 (2014), 043102. 34. A. Baskaran and P. Smereka, “Mechanisms of Stranski-Krastanov growth,” J. Appl. Phys. 111, 044321 (2012). 35. Y. Yu, X.-J. Shang, M.-F. Li, G.-W. Zha, J.-X. Xu, L.-J. Wang et al., “Single InAs quantum dot coupled to different “environments” in one wafer for quantum photonics,” Appl. Phys. Lett. 102, 201103 (2013) 36. H. Zhang, Y. Chen, G. Zhou, C. Tang, and Z. Wang, “Wetting layer evolution and its temperature dependence during self-assembly of InAs/GaAs quantum dots,” Nanoscale Res. Lett. 7, 1–6 (2012). 37. Freund, L. B. "The mechanics of electronic materials." International Journal of Solids and Structures 37.1-2 (2000): 185-196. 38. Pan, E., F. Han, and J. D. Albrecht. "Strain fields in In As∕ Ga As quantum wire structures: Inclusion versus inhomogeneity." Journal of applied physics 98.1 (2005): 013534. 39. Kuo, M. K., et al. "Two-step strain analysis of self-assembled InAs/GaAs quantum dots." Semiconductor science and technology 21.5 (2006): 626. 40. Jiang, Hongtao, and Jasprit Singh. "Strain distribution and electronic spectra of InAs/GaAs self-assembled dots: An eight-band study." Physical Review B 56.8 (1997): 4696. 41. Saïdi, I., et al. "Electron and hole energy levels in InAs/GaAs quantum dots: Size and magnetic field effects." Journal of Applied Physics 109.3 (2011): 033703. 42. S. Sauvage, P. Boucaud, T. Brunhes, V. Immer, E. Finkman, and J.-M. Gerard, “Midinfrared absorption and photocurrent spectroscopy of InAs/GaAs self-assembled quantum dots,” Appl. Phys. Lett. 78, 2327–2329 (2001). 43. A. Adawi, E. Zibik, L. Wilson, A. Lema^ ıtre, J. Cockburn, M. Skolnick et al., “Comparison of intraband absorption and photocurrent in InAs/GaAs quantum dots,” Appl. Phys. Lett. 83, 602–604 (2003).
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Appendix: Elastic coefficients of In AS and GaAs in (001) direction The strain and stress were demonstrated as two column vectors [11 and 34]: u x xx xx v y yy yy zz w z zz = and xy (u y v x) 2 xy yz (v z w y ) 2 yz xz (u z w x) 2 xz
(A1)
where u, v, w are the displacements along the coordinates in the x, y and z directions, respectively and then the dependence of the strain components on stress can be displayed by applying a stiffness matrix D,
D( 0 ) 0
(A2)
where the initial stress in all the regions of structure was assumed to be zero. For a cubic crystal the D matrix of InAs and GaAs in the (001) direction was considered as follow:
83.29 45.26 45.26 DInAs 0 0 0 118.8 45.26 53.8 DGaAs 0 0 0
0 0 45.26 83.29 0 0 0 9 2 (10 N m ) and 0 0 39.59 0 0 0 0 0 39.59 0 0 0 0 0 39.59
45.26 45.26 83.29 45.26
0 0
0 0
0 0 53.8 118.8 0 0 0 9 2 (10 N m ) 0 0 59.4 0 0 0 0 0 59.4 0 0 0 0 0 59.4
53.8 53.8 118.8 53.8
0 0
0 0
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Highlights
Truncated pyramidal-shaped QDs have been investigated with considering strain and size effects.
The strain distribution was calculated using the continuum elastic theory. The energy of the bounded ground state of S shifted due to the strain effect. The WL-state is more sensitive to the size and WL thickness variation in presence of strain. The envelop function act as a continuum state for the thick WL within the strain-modified structure. The z-polarized transition of WL-to-S in the strain-modified structure extended to a longer wavelength.
Roghaieh Parvizi PII:
S0749-6036(18)31234-5
DOI:
10.1016/j.spmi.2018.08.010
Reference:
YSPMI 5855
To appear in:
Superlattices and Microstructures
Received Date:
09 June 2018
Accepted Date:
05 August 2018
Please cite this article as: Roghaieh Parvizi, Investigation of Intersubband Transitions in Truncated Pyramid-Shaped InAs/GaAs Quantum Dots Coupled to Wetting layer: Strain and Size Matter, Superlattices and Microstructures (2018), doi: 10.1016/j.spmi.2018.08.010
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ACCEPTED MANUSCRIPT
Investigation of Intersubband Transitions in Truncated Pyramid-Shaped InAs/GaAs Quantum Dots Coupled to Wetting layer: Strain and Size Matter Roghaieh Parvizi Department of Physics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran [email protected]
Abstract: In this paper, we theoretically report competition behaviour of size and strain effects on the InAs/GaAs quantum dots (QDs) eigenenergies, the envelop functions and also transition energies of the z-polarized WL-to-S states with varying the QD height and WL thickness. The obtained result for the bounded ground state of S implies that although the size-dependence behaviour with the inclusive of the strain effects generally resembles that of unstrained case, the energy values have changed appreciably for the short dots with value of 200 meV and for tall dots with nearly 300 meV. The competition behaviour of strain and size effects on the electronic properties of the structure revealed that the WL state is more sensitive to the size and WL thickness in presence of strain. For the weakly bounded WL state, the envelop function was localized in the wetting layer or bulk regions as a continuum state for the thick WL within the strain-modified structure. The calculated optical wavelength from z-polarized transition of WL-to-S in the strain-modified structure agrees well with previous experimental photoluminescence data from other studies. Keywords: InGaAs/GaAs quantum dot, wetting layer, strain, continuum elastic theory.
1. Introduction Three dimensional (3D) arrays of quantum dots (QDs) have attracted intense research efforts in aspects of the promising potential for applications in the optoelectronic devices such as lasers, photodetectors and light emitting diode [1-7] as well as new physics on a nanoscale. An underlying understanding of such structures features is necessary for properly exploiting QD application potentials. The efficiency of QD-based optoelectronic devices is strongly related to the distribution density and their size, shape of dots in a QD array. Self-assembled QDs formed by strained epitaxy in the Stranski–Krastanow growth mode is the most effective and prominent growth technique for the fabrication of coherent, dislocation-free arrays of dots [8]. For example, self-assembled InAs (quantum dot) on the InxAs1-xGa (surrounding matrix) have shown the most prominent realization of these type of QD structures where firstly involves the growth of a wetting layer (WL) of
ACCEPTED MANUSCRIPT monolayer thick and then followed by a spontaneous coherent island formation, as dot, [9]. The engineering of this method relies on the strain induced by a mismatch of lattice constants of the dot and surface materials which giving rise to a growth of self-assembling dots (islands) on a WL as a substrate. This lattice mismatch-induced strain strongly modifies the potential profile along the growth axis, removes degeneracy (in the valance band). Accordingly, the introduction of strain may provide a facile way to fabricate tuneable wavelength-based optoelectronic devices. Meanwhile, the size and shape exert a significant influence on the responding wavelength for QD-based structures [10-14]. Therefore, it is clear that the much sensitivity of QD’s energy levels to the shape, size, and strain provides the optoelectronic devices with greater potential to obtain the ideal responding wavelength. Several studies have been reported the optical properties of QDs with respect to size by using different methods and approximations [15-18] without considering strain effects. In some works, the lattice mismatch was modelled by using the thermal strain in the areas of QD with temperature raised by 1K and the thermal strain coefficient was assumed to be the absolute values of the lattice mismatch [19]. A more detailed eight-band model has been employed for a single InAs/GaAs dot [20, 21], but it is too complicated for QDs’ structures with WL. This work mainly focused on the energy of electron in the conduction band of InAs/GaAs QDs with inclusion of WL under competition behaviour of the strain distribution and the size of QD structure. As the strain has an inevitable effect on the potential, this model is studied the influences of strained interfaces of semiconductor materials, such as InAs/GaAs, as deformation potentials of the band offset. In the author earlier publications and as reported in Ref. [22], the band structure including the strain effect has been calculated precisely in terms of the altering effective mass and the potentials deformation for a QD with a single size/shape structure. In these calculations, the strain was assumed to be a constant value between the InAs QDs and the surrounding InGaAs capping layer [23-25]. It should be noted that in the calculation of band structure the piezoelectric effect can be ignored for quantum dots with in-plane symmetry, with (001) growth direction, as reported by Bester and Zunger [26-27]. Consequently, the study of QDs electronic properties can be carried out without being concerned the strain-induced piezoelectricity. We also neglected the Coulomb interaction energy, due to the presence of the stronger confinement regime of the deep confining potential created by the QD heterostructure structure [28]. In this paper, we paid especial attention to the energies and transition properties of the two states of the ground, atomic orbital of s, and the second exited state, which is well known as WL-state as the atomic orbital d, by employing a simple one-band effective-mass model with taking the advantage of the finite element method. This transition is of a great importance for the QDIR photodetectors devices, because carrier (electron) can move to the continuum energy region, which is an essential requirement for such devices [29-31]. According to the theoretical and experimental reports, the ground state (S) to WL state transition in InAs/GaAs is strongly z-direction polarized [32, 33]. The strain obtained for
ACCEPTED MANUSCRIPT different QD height and WL thickness are used next as input parameters to calculate electron energy dependence to the QD size. The competition behaviour of size and strain effects on the QD eigenenergies, the envelop functions and also transition energies of the z-polarized WL-to-S states were investigated in detail with varying the QD height and WL thickness.
2. Model and theory The model system was considered based on the most reported practical samples to find realistic structure parameters. InAs quantum dot was grown on GaAs substrate in (001) direction, preferable in a self-assembled growth method, by using solid-state molecular beam epitaxy (MBE) [34]. As mentioned, our dot geometry is of a truncated pyramid-shaped in square base and 2 nm top length. The self-assembled QD situated on top of the WL, which is placed between GaAs as substrate and capping layers, is shown in Fig. 1. The wetting layer is a thin and flat layer with a material similar to that of applied in QD with reported thicknesses ranging from 0.5 to 2 nm [35, 36]. In this work, to cover all practical size/shape of QD structure, two directions of vertical- and lateral-expansion of truncated-pyramid shaped QD with a constant WL thickness of 1 nm and base length of 22 nm will be studied. Additionally, WL thickness effects will be considered for a constant size of QD with a varying thickness. To solve the strain and Schrödinger equations, we have taken the advantage of the finite element method (FEM) throughout our complex geometry. In this method, at first the continuous domain is segmented into a finite number of elements (meshes) which are connected via “nodes”. Regarding that we have to define geometry of the system with finite appropriate size. Therefore, QD is settled in a sufficiently large size as a computation domain which is segmented into a sufficient number of meshes. Tetrahedral meshes were applied for subdomain and triangular meshes were used for boundaries. In this work, the model is meshed with the intention to fully utilize the memory capacity within three dimensionally geometry, while not exceeding the limit and taking longer computational time. The z axis is considered to be perpendicular to the plane of the wetting layer and the x axis lies in the wetting layer plane, as shown in Fig.1. To simulate the isolated QD in numerical analysis, we consider both the substrate and the cap layer are thick enough as to be 50 nm. Once the geometry is properly built and meshed, it is ready for strain analysis.
z direction GaAs InAs quantum dot
WL GaAs
x direction
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Fig. 1 The pyramid-shaped quantum dot/wetting layer schematic structure In this work, we calculate the strain distribution within the continuum elasticity approximation. Taking into account length change (i.e. elastic displacement ui) in all directions, the total elastic strain εij is generally defined as: εij=(ui.j+uj.i)/2.
(1)
The total strain is given by
ij ije 0,ij
(2)
Where ije is the elastic strain and 0,ij is the initial lattice mismatch strain of QD system. The MBE rown self-assembled heterostructures QDs, such as InAs/GaAs QDs, consist of two different materials with two different lattice constants. The mismatch of lattice constants gives rise to strain fields in these QD structures and affects the electronic and optical properties of QDs [37]. For the InAs/GaAs structure, in-plane lattice mismatch parameter is defined as ε0,xx = ε0,yy=(ɑGaAs − ɑInAs) / ɑInAs
(3)
which is equal to -0.067. In this computation, the lattice constants of dot (InAs) and substrate (GaAs) materials are considered as 0.605 and 0.565 nm which indicates that lattice constant of dot exceeds the substrate. This fact leads to compressive strain field inside dot region and tensile strain field for outer matrix. This in-plane lattice mismatch is treated as the initial strain field in latter calculation. With assuming that the initial strain is elastically, the strain in perpendicular direction is given by Poisson’s effect, as:
0, zz 2
C12 0, xx C 11
(4)
where C11 and C12 are components of the elastic stiffness matrix which interconnects stress σ to strain. In this paper, the parameters 0,xx , 0, yy and 0,zz are regarded as the initial normal strains which will induce further strain fields in the whole QD and matrix structure. Therefore, they will be treated as inputs in the latter finite element analysis. We also assume that the initial strain is uniform inside the quantum dot and is zero outside. For convenience, the stress and strain tensors can be described via two columns vectors (as reported in the Appendix). According to the theory of linear elasticity, the strain components appear in the constitutive relation, Hooke’s law [38],
ACCEPTED MANUSCRIPT ij Cijkl kle . (5) By employing equation (2), it can be rewritten as
ij Cijkl ( kl 0,kl )
(6)
where Cijkl is the component of the fourth-order tensor of elastic moduli, i.e. stiffness matrix, and ε0,kl is the initial strain tensors expressed in equations (3) and (4) with ε0,kl =0, for k≠l. In Eq. (6), 1 for field point within quantum dot region and 0 in the matrix. The InAs and GaAs are both cubic symmetric materials. Thus, only three of the 81 components of stiffness matrix are independent. So the components can be simplified such as C11 = C1111 = C2222 = C3333; C12 = C1122 = C2233, etc., and C44 = C1212 = C2323, etc. Accordingly, the strain and stress can be determined by three independent elastic moduli, namely, C11, C12 and C44. The numerical values of elastic moduli used in this paper were reported in the Appendix. In this paper, the hydrostatic and the biaxial strains are considered respectively, as
hyb xx yy zz
(7)
and
b zz
( xx yy ) 2
(8)
The above mentioned physical properties of each domain are set; i.e. initial strain fields, the governing strain equation and the anisotropic elastic stiffness matrix. For strain field calculations, a stationary solver would be applied because strain field problem is a stationary problem. Strain field problem arising from lattice mismatch also is a kind of a linear elastic boundary value problem which is analysed via FEM technique. Thus, appropriate boundary conditions are necessary for each interface. In this work, we have followed the comprehensive boundary conditions reported in Ref. [39] as: all the nodes of the in-plane-outer surfaces are fixed against displacement in the z-direction (normal direction to the faces) due to periodic symmetric argument. The bottom of the substrate is constrained in the z direction, to avoid rigid body shift while the upper surface is kept traction free [39].The other internal interfaces of dot/wetting layer/matrix are not determined and should be left as unknown in the FEM solving process. The solutions from strain field calculations are stored as the inputs for the next step of band structure analysis. In this paper, the electronic properties of self-assembled QDs structure have been studied based on the single-band effective mass approximation theory (EMA). We know that this approximation is accurate
ACCEPTED MANUSCRIPT in case that QD dimensions are much larger than the lattice constants. Energy levels and envelop functions are calculated with the starting point of the quantum theory that deals with the behaviour of carries confined to a truncated pyramid-shaped InAs/GaAs QD as the solutions of the stationary Schrödinger equation with including strain effect as follow:
-
2 1 .( (r )) V (r ) (r ) E (r ) 2 me ( x )
(9)
where V (r )
(10)
V (r ) Eg ac h
including the strain effect on the band and ħ, me (r ) ; Eg ; E; and (r ) respectively designate the reduced Planck's constant, the material dependent electron effective mass, the unstrained band-gap energy, the energy eigenvalue, and the envelop function of the electron with regard to an electron at
position r . The parameter ac in Eq. (10) denotes the deformation potential for the conduction band. These main parameters were taken from Ref. [15].
3. Strain distribution Based on and the Stranski-Krastanov self-assembly technique for such materials of InAs and GaAs, the QD islands of InAs can be created with different shape and symmetry which thus has effect on the strain distribution in dot structure and the corresponding band energies. In the following, we have calculated the strain distributions for a quantum dot with 22 nm base length and height of 6 nm attached to the wetting layer with 1 nm thickness. The εxx, εyy, εzz and the shear strain εxz components of the strain tensor were plotted in Figs. 2 (a), (b), (c) and (d), respectively in the x–z cross section view. Fig. 2 clearly shows that the compressive strains (negative εxx and εyy) within the dot in the x-y plane, while interfaces of GaAs with WL and dot experience tension from InAs due to misfits of lattice constants of two materials. On the other hand, εzz is positive within the dot region. In this case, the matrix tries to induce the dot that its lattice constant to be that of GaAs equals to 5.65 A. The shear strain components, εxy, εyz and εxz, are negligible in the dot and surrounding matrix, as εxz shows in Fig. 2 (d), they could, however, be appreciable at the interfaces for the piezoelectric calculations [39]. Fig. 2 (e) shows the strain components along the growth direction which implies the symmetry of the square-based pyramid quantum dot against z axis. In this figure, the black solid-line, εxx, was coated by the red dash-line, εyy, thus, based on the mentioned symmetry the component εxx equals to the εyy. In the wetting layer and the bottom of the dot experiences a tensile (positive εzz) strain and a compressive (negative εzz) strain in the growth direction. The tensile εzz strain component inside the dot was moderately changed, while the εxx and εyy are different with negative trends. The in-plane strain
ACCEPTED MANUSCRIPT symmetry with respect to the in-plane direction is now broken. This effect is the most pronounced for the dots which may cause the splitting off the dot degeneracy in the p-states [40].
(a) xx
(c) zz
(b) yy
(d) xz
(e)
(f)
Fig. 2 (a), (b), (c) and (d) are the strain components of εxx, εyy, εzz and εxz in the x-z plane crosssectional view. The position-dependent strain profiles present in the quantum dot structure along both the growth (e) and the base of quantum dot directions (f) with the insets that show εxx cross section view with its profile along z (e) and x directions (f).
Fig. 3 demonstrates the distribution of the hydrostatic εhyb strain in the x-z plane cross-sectional view. The hydrostatic strain is compressive within the quantum dot while the interfaces experience a small hydrostatic tensile strain. Both εxx and εyy are nearly homogeneous and negative within the dot and resulting the presence of a region with hydrostatic compression. In the substrate near the InAs/GaAs interface, GaAs experiences tension from InAs and the same effect is shared by the dot and capping layer interfaces at top of QD. In order to find out the effect of the size and shape on the strain distribution, we have calculated and illustrated the hydrostatic and bi-axial strain distributions for a number of structures whit various heights of the quantum dot and different thickness’ of the wetting layer. These strain components were plotted as functions of position along the in-plane basis direction and z-axis. Fig. 3 (b and c) illustrate the strain relaxation profile for InAs dots of the same base width
ACCEPTED MANUSCRIPT (B = 22 nm) but different heights of 3, 6, 9 and 12 nm. In all of these figures, it can be seen that in the inner part of the QD, the hydrostatic strain value is almost constant and compressive, while the hydrostatic strain value in the interfaces is very small and tensile somehow is near zero for the barrier region. These results also reveal that when InAs self-assembled pyramidal dots are buried under a GaAs over layer, significant tensile stress is induced at the top of the dot; as we proposed a truncated pyramid to a more reliable sample. On the other hand, the stress at the edges of the dot is weak and compressive. With making a comparison as the obtained results show, we note that for dots with small heights (short dots), the strain components are more than that of dots with large heights (tall dots), while in the interior part of tall dots, the strain is more uniform. This difference in the hydrostatic strain is about 20% between the shortest dot and the dot with 12 nm height (suggested here as the tallest dot, as functions of both in-plane and vertical directions, as shown in Figs. 3 (b and c). The hydrostatic strain distributions with different WL thicknesses, 1, 2, 3 nm are visualized in Fig. 3 (d) and Fig. 4 (e) against both the in-plane and normal direction, respectively. The results show that the dependence of hydrostatic strain component on the WL variation is moderate and about less than 10% in both directions.
(a) Hydrostatic strain
(c)
(b)
(d)
Fig. 3 shows the hydrostatic strain in the x–z plane, (a) its comparison plotted along the z– and x –axis for different height with a fix WL thickness of 1 nm (b) and for different WL thickness with constant height of 6 nm.
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The details of the bi-axial strain distributions for the bi-axial strain can be found in Figs 4 (a, b, c, d and e), where the strain distributions scanning through the x-axis and growth direction are shown and cross section is in the x-z plane view (Fig. 4 (a)). The bi-axial strain suggests a significant strain transfer takes place between the different interfaces of substrate, dot and capping layer regions and outside the boundary region the bi-axial strain is quickly decreased to zero. In contrast to the hydrostatic strain, the bi-axial strain tends to be positive in the dot region and negative in the substrate and capping layer interfaces with the dot. It is not only homogenous within the dot region, but also is completely vanished near the centre of the dot and so, the strain would be entirely hydrostatic in this segment. Along the z direction, taller dots experiences a more relaxed strain rather than shorter ones (heights <6 nm) in the nearly centre region in the dot which is the main reason for the splitting of the hole envelop functions in the valance band structure [38]. In the z direction, a significant transfer of bi-axial strain to the barrier takes place in the interfaces, while in the in-plane direction this strain focused more in the centre region of quantum dot with a uniform trend. From Fig.4 (e), this comparison suggests that the tuning of the wetting layer thickness does not alter the confinement potential behaviour, but it only might affect the height of conduction band.
(a) Bi-axial strain
(b)
(c)
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(d)
(e)
Fig. 4 shows the shear strain in the x–z plane (a), its comparison plotted along the z– and x –axis for different height with a fix WL thickness of 1 nm (b and c), for different WL thickness with constant height of 6 nm along z–axis, (d) and hydrostatic and shear strains for different WL thickness with constant height of 6 nm along the pyramid basis direction.
Within the strained structure, we have calculated its potential profile. The potential profile is achieved by computing the band structure for the proposed geometry of quantum dot. Fig. 5 shows the calculated confinement potential distribution induced with the strain for different heights of dot and wetting layer thicknesses along the growth direction [001]. Also, this figure demonstrates potential profile of a structure without strain for a dot with height of 12 nm and WL with thickness of 1 nm. In the absence of strain, the confining potential for an electron is a square well formed by the difference in the absolute energy of the conduction band-edges in InAs and GaAs [30]. The hydrostatic strain, which is mainly affect the conduction band (CB), makes the height of CB lower such that it shifts up from 0.6 eV (unstrained InAs conduction band edge) to 1.0 eV (strained quantum dot structure). With strain-considered structure, from the left side of the confining potential well corresponding to the bottom side of dot (wetting layer region) to the centre of dot region doesn't change appreciably against variation of QD height, but the top of dot shows more confinement, nearly 50 meV and in the right up side of potential well tends to contribute into the semi-continuum region for taller dots. Additionally, the obtained results shown in Fig. 5 (b) revealed that within the strainmodified structure, the width of confining potential well is extended in z direction with increasing the thickness of WL, but the heights of this well are almost identical within only 10 meV variations.
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(b) (a)
Fig. 5 The calculated bandoffset with including strain effect (solid-line) (a) for different height and without strain effect within a QD Structure with 12 nm height, (b) for different WL thickness with a fixed QD height of 6 nm
4. Electronic structure and optical transitions The band structure of quantum dot structure is generally changed with the inclusion of the strain, both conduction and valance bands, which reduces the symmetry of structure. Strain modifies energy gaps and removes degeneracy [28]. In this section, we study the strain-included electronic structure of QD in the envelop function approximation using effective-mass approximation framework in one-band Hamiltonian, Eq. (9). The depth of the confining potential is shifted to a lower value which this value is treated as initial treatment in the following electronic calculations. It is worth noting that this value varies as a function of position, due to the variation of strain against position, as discussed in the earlier section. The strain-modified structure is defined via formation potential theory, Eq. 10, for the next step of solving Schrödinger equation [37]. For the conduction band of a pyramid-shaped quantum dot, there are usually only three states, which two of them are bounded (S and P states) and one of them is a weakly bounded, denoted as WL-state within the dot region. The obtained result for the first bounded state of S implies that although the size-dependence behaviour inclusive of the strain effects generally resembles that of unstrained case, the energy values have changed appreciably due to the compressive strain, see Fig. 6. It is attributed to this fact that this state is bounded in the dot region even for the case that strain is included, as shown in Fig. 9. For the short dots (height < 6 nm), the electron energy with neglecting the strain is about 200 meV less than of dots with same size with considering the strain effects. This value reaches to 300 meV for taller dots, as shown in Fig. 6. With a more carefully comparison, it can also be concluded that the strain-modified electronic energy
ACCEPTED MANUSCRIPT structure is likely not to be as sensitive to variations in the dot size and WL thickness as it was to changes in the unstrained structure, because of the lower confining potential.
Fig. 6 The calculated eigen-energies corresponding to S-state (bound state) in presence of strain effects and with neglecting strain for a range of WL thickness which is varying from 0.5 to 2 nm with step of 0.5 nm In this work, we focused on the WL state to clarify the strain inclusion effects on a weak bounded state. In the energy domain of such quantum dot structures, the electron can localized in the wetting layer and bulk which are forming continuum or semi-continuum states [41]. For every structure depending on the applied wetting layer thickness, this value is different. For example, for a dot with WL thickness of 1 nm in the strain-modified structure, this domain starts from 1.38 eV to 1.5 eV, with energy of GaAs domain as shown in Fig. 6 as a constant dashed-line in a structure with WL thickness of 1 nm. For the WL-state, inclusive of the strain effects for the thin WL of 0.5 nm thickness, the energy trend against height with including strain is so similar to that of without strain. However, for the thicker wetting layer, a plateau occurs in the energy trend, correspond to the used WL and then energy reduces as a common process. Indeed, this plateau is the continuum region where the electron’s envelop function is localized into the wetting layer or bulk region. These results suggest that thicker WL in presence of strain effect, this state is located within wetting layer, see Fig. (10, (a)to-(g)), while without considering strain effects, Fig 10 (h and i), it shows a bounded state localized within both dot and wetting layer regions. The obtained energy for a structure, neglecting strain, with WL of 2 nm resembles that of structure with 1 nm WL thickness with considering strain effects. The
ACCEPTED MANUSCRIPT obtained results revealed that for a structure with WL more than 1 nm, in presence of strain, the WL is not localized and its energy is in the continuum energy domain. Hence, as a realistic case with considering strain effects, for the structures with thick WL, the WL state is not considered as a bound state such that its wave function is towards outside of the structure.
Fig. 7 Eigen-energies corresponding to WL-state in presence of strain effects (solid-lines) and with neglecting strain (dotted-lines). The constant dashed line is related to the semi-continuum states in a structure with WL thickness of 1 nm.
Some insight can be gleaned by comparing the S to WL state transition energy as a function of dot height for different WL thicknesses with and without including strain effects. Fig. 8 displays that with considering strain effects, the structure displays an extended emission wavelength rather than a QD structure with neglecting strain. Moreover, another interesting result is found for the behaviour of transition energy for various WL thicknesses which is a consequence of mentioned result in Fig. 7. It concerns the particular WL thickness from which the WL state is not bounded anymore. Indeed, one can notice that only for thin WL the result of unstrained structure resembles that of strain-modified structure, while for thicker case the results become different since band-offset that corresponds to the strained structure is reduced. In Fig. 10, it is shown 3D plot the electron envelop function of WL-state for different dot height and WL thickness, for both unstrained and strained structures to explain the
ACCEPTED MANUSCRIPT reasons behind the obtained results. For WL-to-S transition, different transition energies have been reported based on the experimental measurements. This difference in values can be attributed to the fabrication conditions and stoichiometry, as well as the size and geometry of QDs [31]. The obtained results here with inclusive of strain effects is well matched with the reported value 160 meV which is found by Sauvage et al. for the z-polarized WL-to-S transition by applying photocurrent spectroscopy [42]. Another example for this transition reported the energy of 180 meV using Fourier transform infrared spectrometer [43].
Fig. 8 The transition energy of WL-to-S as a function of QD height for different WL from 0.5 to 2 nm with 0.5 nm step in presence of strain effects (dotted-lines) and with neglecting strain (solid-lines).
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Fig. 9 Cross section view and 3D plot of envelop function corresponding to S-state which is bounded in the dot region with considering strain effect.
(a) WL=1 nm, h= 3 nm
(d) WL=1.5 nm, h= 3 nm
(g) WL=2 nm, h= 11 nm
(b) WL=1 nm, h= 9 nm
(e) WL=1 .5 nm, h= 7 nm
(h) WL=2 nm, h= 3 nm, Without strain
(c) WL=1 nm, h= 13 nm
(f) WL=1.5 nm, h= 10 nm
(i) WL=2 nm, h=12 nm, Without strain
ACCEPTED MANUSCRIPT Fig. 10 shows envelop function corresponding to WL-state for different QD heights and WL thickness for both situations of with (a, b, c, d, e, f and g) and without strain (h and i)
Acknowledgment The authors express their appreciation to the Research Council of the University of Yasouj for financial support of this work.
5. Conclusion In summary, we have studied the strain distribution of self-assembled QD by the continuum elastic approach to describe the strain-driven self-assembled process of QD based on lattice mismatch. The envelop functions and energy levels of QD have calculated in a one band effective-mass approximation framework for different height and WL thickness but at the same base pyramid of 22 nm. The results indicate that the strain-modified bound energy of S-state is mainly controlled by the size, while the strain inclusion only shifts the energy value by at least 200 meV. However, the strain and the size both play key role in determining the weak bounded WL-state such that only for thin WL the result of unstrained structure shows similar trend with strain-modified structure, while for thicker case the results become different. For structures with WL thicker than 1 nm, the WL state is not considered as a bound state such that its envelop-function is towards outside of the structure. The results show that the difference of bound-to-continuum energy levels of the suggested truncated pyramid-shaped QD matches well with the experimental data with including the strain effects which is useful in designing the ideal QD-based photodetectors.
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Appendix: Elastic coefficients of In AS and GaAs in (001) direction The strain and stress were demonstrated as two column vectors [11 and 34]: u x xx xx v y yy yy zz w z zz = and xy (u y v x) 2 xy yz (v z w y ) 2 yz xz (u z w x) 2 xz
(A1)
where u, v, w are the displacements along the coordinates in the x, y and z directions, respectively and then the dependence of the strain components on stress can be displayed by applying a stiffness matrix D,
D( 0 ) 0
(A2)
where the initial stress in all the regions of structure was assumed to be zero. For a cubic crystal the D matrix of InAs and GaAs in the (001) direction was considered as follow:
83.29 45.26 45.26 DInAs 0 0 0 118.8 45.26 53.8 DGaAs 0 0 0
0 0 45.26 83.29 0 0 0 9 2 (10 N m ) and 0 0 39.59 0 0 0 0 0 39.59 0 0 0 0 0 39.59
45.26 45.26 83.29 45.26
0 0
0 0
0 0 53.8 118.8 0 0 0 9 2 (10 N m ) 0 0 59.4 0 0 0 0 0 59.4 0 0 0 0 0 59.4
53.8 53.8 118.8 53.8
0 0
0 0
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Highlights
Truncated pyramidal-shaped QDs have been investigated with considering strain and size effects.
The strain distribution was calculated using the continuum elastic theory. The energy of the bounded ground state of S shifted due to the strain effect. The WL-state is more sensitive to the size and WL thickness variation in presence of strain. The envelop function act as a continuum state for the thick WL within the strain-modified structure. The z-polarized transition of WL-to-S in the strain-modified structure extended to a longer wavelength.