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Copyright © 2011 American Scientific Publishers All rights reserved Printed in the United States of America Journal of Computational and Theoretical Nanoscience Vol. 8, 1–9, 2011 The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots V. Lozovski and V. Piatnytsia∗ Department of Mathematics, Theoretical Physics and Computer Sciences, Institute of High Technologies, National Taras Shevchenko University of Kyiv, Kyiv, 03022, Ukraine The analytical expressions for electron wave function of pyramid-like and cone-like quantum dot located on the semiconductor surface are obtained. The energy spectrum of electron states in the quantum dots is calculated. The dependence of energy levels on the angle at the vertex of an edge-shaped nanodots is studied. The distribution of electron density for different electronic states is calculated for each case. The dependence of function of linear response to the external field on vertex angle is calculated in the frame of proposed approach. The local field distributions for both pyramid- and cone-like particles are calculated. Keywords: Analytical Study, Quantum Dot, Pyramid Shape, Cone Shape, Energy Levels, Probability Density Distribution, Effective Susceptibility, Lamb Shift, Local-Field. Due to heavy growth of nano-technologies the study of effects of spatial quantization uses in the practice now. There are numerous papers devoted to studies of quantumdimensional effects (see, for example, the monograph and reviews).1–5 Modeling of electron properties of quantumdimension structures customary as a rule is performed for the particles of simple geometric shape.6–9 The technologies obtaining quantum dots (especially, quantum dots situated at the surface), however, allow to obtain the quantum dots of different shapes.10–12 For example, QD obtained by MBE usually are pyramid-like.13 Nowadays the technique of obtaining the cone-shaped particles at the surface of Si by scanning of the surface with laser beam of high intensity were reported in Refs. [14–15]. As a result, the different types of cone-shaped particles were obtained—as needle-like (with small vertex angle), as nano-hills (with large vertex angle).16 Modeling of electronic properties of edge-like quantum dots is a nontrivial problem. One can point out only several papers in which the electron states of cone-like quantum dots were calculated.17–19 It should be noted that direct numerical calculations were used in these works, except, may be Ref. [19] where the analytical solution of Schrödinger equation for cone-shape restricted ∗ Author to whom correspondence should be addressed. J. Comput. Theor. Nanosci. 2011, Vol. 8, No. 11 potential as the infinite series of the complicated functions, which expressed via Bessel functions was found. In contrast pyramid-shaped QDs widely investigated problem but only by the numerical methods. Approach based on numerical solutions of Schrödinger equation sometimes is not convenient for studying some electronic properties of quantum dots. It is clear that for practical use one needs to have simple analytical expressions for wave functions of electronic states. 2. PYRAMID-LIKE QD. MODEL AND MAIN EQUATIONS Let one consider pyramid-like small particle (Fig. 1). Let one supposes that confinement potential has very high wells. This assumption will be correct for quantum dots embedded in wide band gap matrix or for quantum dot situated in the vacuum at the surface of solid. It is clear that in the case of high barrier the solution is not strongly differ from solution for infinite well. Especially when one needs to know the ground state energy level and few lowest excited levels. Strongly speaking the electron energy in quantum dot consists of several terms. Namely, U r = Uconf r + Ustrain r + UC r, where Uconf r is confinement potential, Ustrain r is strain-induced potential, UC r is Coulomb interactions between electrons. In the present 1546-1955/2011/8/001/009 doi:10.1166/jctn.2011.1965 1 RESEARCH ARTICLE 1. INTRODUCTION The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots y x φ h z P RESEARCH ARTICLE Fig. 1. l Geometry of pyramid-like QD problem. work one considers the particles with characteristic dimensions about 10 nm. Because of doped semiconductor characterized by electron concentration about 1016 –1018 sm−3 , one can suppose that there is not more than one electron per the particle. This means, that electron–electron interaction can be neglected in the present consideration. Then, one should to consider electron states for a single electron inside the particle. One can point to rather large number of the works in which the studies of influences of mechanical stresses on the electronic properties of quantum dots were considered.21–25 It was established that the stresses can distort the structure of energy levels of electron in quantum dot. These distortions are depended on the value and distribution of mechanical stresses, say on shape and dimension of a particle. One can think that developing of regular analytical approach to calculate the electron energy structure of cone-shaped quantum dot with taking into account the stress potential is impossible at this time. Then, one should to restrict oneself by consideration of a one side of the problem, however the strain induced distortions of electron potential energy usually can be accounted in the frame of perturbation theory. As a result, in this work one did not take into account the effects of mechanical stresses. On the other hand, one will be needed to take into account the strain-induced effects when the concrete experiment would like to describe. Then, one consider electron states inside potential box having pyramid-like shape and characterize by infinitely high potential barrier. This model, for example, can be used for semiconductor pyramid-like and cone-like quantum dots which is located at the dielectric flat surface. The restrictive surface of the quantum dot in this case is specified by expressions ⎧ z = ax x > 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪z = −ax x < 0 ⎪ ⎪ ⎪ ⎨ (1) z = ay y > 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z = −ay y < 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z ∈ 0 h −1 where a = tan /2, with vertex angle of pyramid. The solution of Schrödinger equation inside the 2 Lozovski and Piatnytsia pyramid-shaped domain [Eq. (1)] even for infinitely high potential wells is rather hard problem due to complicated boundary conditions. From the other hand, one can simplify the problem, to solve Schrödinger equation in the domain restricted by right parallelepiped.26 In this case the domain formed by coordinate lines of coordinate system and boundary conditions has a simple form. Thus, one can propose the approach which could transform the pyramidlike domain to the parallelepipedal one. To realize this idea one proposes the new variables u v w according to (Fig. 2) ⎧ ⎧ ⎪ ⎪ u = x/z ⎪ ⎪ ⎪ ⎪x = uw ⎪ ⎪ ⎨ ⎨ (2) v = y/z ⇒ y = vw ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩w = z ⎩z = w In the new variables u v w the domain P has a form of right parallelepiped of rib length 2/a and altitude h and describes with the next expressions ⎧ ⎪ u ∈ −1/a 1/a ⎪ ⎪ ⎪ ⎨ (3) v ∈ −1/a 1/a ⎪ ⎪ ⎪ ⎪ ⎩w ∈ 0 h Then, Laplace operator in these variables can be written in the form26 1 2 u uvw = 2 + w u2 u2 + v2 + 1 u 1 2 u + 2 + w v2 u2 + v2 + 1 v 1 2 + 2 2 w (4) w w u + v2 + 1 w As a result, Schrödinger equation inside the domain P will be written as 2 − = E (5) 2m uvw One can easily see that Eq. (5) has a form of Helmholtz equation uvw + k2 = 0 (6) w z P P' Transformation y x u v Fig. 2. Visualisation of mathematical transformation. J. Comput. Theor. Nanosci. 8, 1–9, 2011 Lozovski and Piatnytsia The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots √ with k = 2mE/. Let we suppose that the quantum dot has a pyramid shape with small vertex angle < 90 . This approximation means, that u v 1. Then, Eq. (6) can be reduced to equation with separate variables. Then, one can write the equations for each of components of wave function separately ⎧ 2 ⎪ ⎪ + ku2 u = 0 +u ⎪ ⎪ ⎪ u2 u u ⎪ ⎪ ⎪ ⎪ ⎨ 2 (7) +v + kv2 v = 0 ⎪ v2 v v ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 k2 + k2 ⎪ ⎪ w 2 w + k2 − u 2 v w = 0 ⎩ 2 w w w w Let one consider the first equation of the system above. The solution in this direction is not trivial. Albeit one can use the simplification. Let one suppose that 2 u / u2 u u / u. This supposition has a sense because u 1. It means that the first derivative of wave function on the variable u should to be not so large. This assumption one can verify when the solution of equation will be obtained. Using this approximation one can neglect the second term in the first line of Eq. (7) and obtain 2 + ku2 u = 0 (8) u2 u u = cCosku u + dSinku u (9) To find the solution (eigenstates) of the first equation of Eq. (7) in domain u ∈ −1/a 1/a, one have to fulfill boundary conditions ⎧ ⎨u −a−1 = 0 (10) ⎩ −1 u a = 0 C w w = √ J√k2 +k2 − 3 kw un vm 4 w As a result, the wave function of pyramid-like quantum dot in Cartesian coordinates looks like C x y z = √P J knmj z z x −1 × sin kun a cos kun z x + cos kun a−1 sin kun z y × sin kvm a−1 cos kvm z y + cos kvm a−1 sin kvm z Now one can verificate the supposition discussed above. Let one set solution Eq. (10) under first and second derivative orders and compare. One obtains the condition ku > u which is satisfied because of ku ≈ a, u ≈ a−1 and a 1. It is clear that solution for v direction is similar to function of u direction v v = C2 Sin kvm a−1 Cos kvm v + Cos kvm a−1 Sin kvm v kvm = ma/2 (12) J. Comput. Theor. Nanosci. 8, 1–9, 2011 (14) with constant CP being founded from wave function x y z dx dy dz = 1, where normalization Vpyramid integration is over volume of the pyramid, and = 2 + k2 − 3/4. To find the eigenvalues and eigenfunckun vm tions it is necessary to fulfill the boundary condition P = 0, for two lateral directions which were found above. For third direction one should to solve the system of equations ⎧ kun = na/2 n = 1 2 ⎪ ⎪ ⎪ ⎪ ⎨ m = 1 2 kvm = ma/2 (15) ⎪ ⎪ ⎪ ⎪ ⎩J√ knmj h = 0 k2 +k2 − 3 un After substitution of explicit form of wave function [Eq. (9)] in Eq. (10) one obtains the equation for eigenstates finding Sinknu 2a−1 = 0. Thus the u direction wave function can be written as u u = C1 Sin kun a−1 Cos kun u + Cos kun a−1 Sin kun u kun = na/2 (11) (13) vm 4 The conditions of solution existence give us the energy spectrum of electron in the pyramid-like quantum dot 2 2 knmj Enmj = (16) 2meff where knmj is the wave vector satisfying system of Eq. (15). It is interesting to examine the dependence of electron energy structure on the shape of QD, particularly the alternation of the spectrum when shape of the particle changes from the sharp to the blunt form. And it is easy to investigate this idea in our analytical solution. The calculations are performed for GaAs QDs located in the dielectric or wide-band semiconductor. Then, the dependence of electron eigenenergies in the pyramid-like quantum dot on the vertex angle is shown in Figure 3 for two different cases. Panel “a” corresponds to the case when the volume 3 RESEARCH ARTICLE The solution of Eq. (8) is well known As one can see the structure of third equation of the system [Eq. (7)] is similar to standard Bessel function differential equation. After usual polynomial substitution w = w s w s = −1/2 it reduces to the ordinary cylindrical equation for the function w. It is well known that the solution the equation has a simple form26 The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots 222 (a) 1.5 Energy, eV 113 231 1.2 131 112 121 111 0.3 35 40 45 50 55 60 65 Vertex angle, φ0 222 (b) 1.8 113 231 Energy, eV 1.5 131 1.2 221 112 0.9 121 0.6 111 0.3 35 40 45 50 55 60 65 Vertex angle, φ0 RESEARCH ARTICLE the wave functions obtained earlier with numerical calculations (see, for example Ref. [29]). This can serve as evidence of truth of proposed analytical approach, which one will use for calculation of electron energy structure of pyramidal-shaped quantum dot. 221 0.9 0.6 Fig. 3. Dependence of electron spectrum the single pyramid-like GaAs quantum dot on the vertex angle. Panel “a” corresponds to the case when the volume of quantum dot is the constant (r0 = 5 nm, a0 = 50 ). The panel “b” corresponds to the case when the radius of the base of quantum dot is the constant. The set of quantum numbers nmj > are given at the curves. of quantum dot is constant r0 = 5 nm, 0 = 50 ) with alternation of vertex angel. Panel “b” corresponds to the case when the base edge of the pyramid is constant and vertex angel changes. As one can see, the energy structure strongly depends on the vertex angle. One should note that there are some values of the angle when the energy levels degenerate. Obtained structure of spectrum with analytical method is similar to calculated with numerical method presented in Refs. [13, 27] earlier. The distribution of probability density ∗ = f x y z of electron location inside the particle is shown in Figure 4 for quantum dot with vertex angle = 60 for different states to which are manifested at the bottom of each of the figure. One should note that in three-dimension probability density distribution are belt with the surface of ∗ = const, with the constant chosen that inside the bodies the probability to find the electron is more than 0.9 (Fig. 4 panel “a”). The probability densities depicted as a function ∗ = f x 0 z in cross-section by plane XOZ are shown in the panel “b.” One should point once more that spatial structure of obtained wave functions are similar to 4 Lozovski and Piatnytsia 3. CONE-LIKE QD. MODEL AND MAIN EQUATIONS Cone-shaped quantum dot energy structure calculation is the aim of this section. To solve this problem one will use the approach of analytical calculation developed for pyramid-like QD. Then, one can describe the cone-like QD domain as ⎧ ⎨z = a x 2 + y 2 (17) ⎩ z ∈ 0 h with a = tan−1 /2, and the vertex angle of the cone. It is clear that coordinate transformations should be the similar to the transformations used in the previous consideration. But in contrast to the case of pyramid-shaped QD, the present problem has an axial symmetry, that is why the transformation domain is a cylinder. Then in the new variables the domain C’ has a form of right cylinder of radii and altitude h and is described with the next expressions ⎧ u = cos ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨v = sin (18) √ ⎪ ⎪ ⎪ = u2 + v2 ⎪ ⎪ ⎪ ⎪ ⎩ w ∈ 0 h where = 0 1/a. Then, Laplace operator in variables given by [Eq. (18)] can be written in the form 2 1 2 1 1 w = 2 + + + w 2 2 + 1 2 2 1 + w2 (19) w w 2 2 w +1 To solve Schrödinger equation in this case one should to simplify the problem. Namely, to obtain the analytical solution of the problem one needs to suppose that the quantum dot has a cone shape with vertex angle < 90 . In this case parameter will change in the range of 0 ≤ < 1. This fact allows one to neglect certain of the terms in wave equation, namely, the term proportional to 2 can be neglected in comparison with 1 and term proportional to can be neglected as compared with 1/. Then, one can write 1/2 + 1 ≈ 1 and 1/ + /2 + 1 ≈ 1/. As a result, the wave equation can be reduced to equation with J. Comput. Theor. Nanosci. 8, 1–9, 2011 Lozovski and Piatnytsia The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots |111> (a) |112> y x φ z h C |211> r |311> Fig. 5. Geometry of cone-like QD problem. C1 eim . The solutions of second and third equations of system [Eq. (21)] can be presented via√Bessel function, namely, = C J k and = C / w J kw, with 2 m w 3 |112> |111> 0 0,1719 0,3438 0,5156 0,6875 0,8594 1,031 1,100 z , nm 8 6 4 2 –4 –2 0 2 8 z , nm (b) 6 4 2 –4 4 –2 2 4 |311> |211> 8 6 6 z , nm 8 4 4 2 2 –4 –2 0 2 4 –4 x , nm –2 0 2 4 x , nm Fig. 4. The probability density of electron in quantum dot (a) for different eigenstates which are shown in the brackets. In the panel (b) the cross-sections by plane XOZ of the probability density of electron are shown. with constant CC being founded from wave function normalization. To find the wave functions satisfying the boundary condition C = 0, one should to solve the system of equations ⎧ ⎪ m = 0 ±1 ±2 ⎪ ⎪ ⎨ kmi Jm = 0 (24) a ⎪ ⎪ ⎪ ⎩J√ 2 3 k h = 0 mij k − i separate variables. It means that one can write the equations for the each component and w w of wave function w = w w separately ⎧ 2 ⎪ ⎪ + m 2 = 0 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎨ 2 1 m2 2 = 0 + − + k ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k2 1 w ⎪ ⎪ 2 2 ⎩ 2 w + k − 2 w = 0 w w w w (20) (21) It is clear that the solution of the first equation of the system [Eq. (21)] has an exponential form = J. Comput. Theor. Nanosci. 8, 1–9, 2011 4 One can easily find numerically the energy spectrum from the system [Eq. (24)] it expresses via zeros of different orders Bessel functions. The dependence of energy levels of electron in the cone-like quantum dot on the vertex angle is shown in Figure 6. The spectrum of electron energy inside the cone-like quantum dot in general is similar to pyramid-shaped (extremely degenerated along the shape alternation). The distribution of probability density ∗ = f x y z of electron location inside the cone-like particle is shown in Figure 7 for quantum dot with vertex angle = 60 . It can be easily seen that the probability density has cylindrical symmetry. One should note that in three-dimension pictures of probability density distribution are belt with the surface of ∗ = const, with the constant chosen that inside the bodies the probability to find the electron is more than 90%. 5 RESEARCH ARTICLE z , nm 0 x , nm x , nm = k2 − 3/4. Thus, the solution of wave equation for cone-like quantum dot has a form26 C w = √ J kwJm k eim = k2 − 3/4 w (22) This solution in Cartesian coordinate system can be rewritten as 2 CC x + y 2 im x y z = √ J kzJm k e z z y = arctan (23) x The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots 4. EFFECTIVE SUSCEPTIBILITY OF A SINGLE PYRAMID-LIKE AND CONE-LIKE QUANTUM DOT The optical properties of the system with quantum dot can be easily studied in the frame of effective susceptibility concept.28 This concept assumes calculation of optical response to the external field which can be performed with taking into account self-acting field.29 In other words, one shall take into account processes of action of the local field caused by electron transition currents on the electron inside the quantum dot. Because this local field is defined by induced inside the particle currents of electron transitions, this local field is the field of self action. As a result, there exist effects caused by local field action or local-field effects. For example, the Lamb shift of electron energy levels and broadening of the energy levels are caused by local field effect.29 General scheme of calculation of effective susceptibility consists in finding self-consistent local field, which acts at arbitrary point inside the quantum dot. This (a) 1.5 Energy, eV self-consistent field obeys Lippmann-Schwinger equation Ei R = Ei0 R − i0 dR Gij R R × V dR jk R R Ek R (25) V where Gij R R is electrodynamical Green function, which describes the electrodynamical properties of the medium in which the quantum dot is embedded. When one supposes that the quantum dot is situated at the surface of dielectric, one can use the Green function for two homogeneous isotropic semi-spaces with the flat interface.30 Supposing that the electron transitions between the ground state and few excited states mainly contribute in the paramagnetic part of the response and taking into account that (a) 013 112 |011> |111> |012> |021> 021 1.2 RESEARCH ARTICLE Lozovski and Piatnytsia 211 0.9 012 111 0.6 (b) 35 40 45 50 55 60 65 Vertex angle, φ0 z, nm 013 112 4 2 021 1.5 0 211 –4 –2 0 2 8 6 4 2 0 –4 4 –2 x , nm 1.2 012 011 0.3 35 40 45 50 55 60 65 Vertex angle, φ0 Fig. 6. Dependence of eigenenergies on the vertex angle of the single cone-like Si quantum dot located at the dielectric surface. Panel “a” corresponds to the case when the volume of quantum dot is the constant. The panel “b” corresponds to the case when the radius of the base of quantum dot is the constant. The set of quantum numbers mij > are given at the curves. 2 4 2 4 |021> 8 8 6 6 z, nm 0.6 0 x , nm |012> 111 0.9 z, nm Energy, eV |111> 1,000 0,9000 0,7500 0,6000 0,4500 0,3000 0,1500 0 6 (b) 1.8 6 |011> 8 z, nm 011 0.3 4 2 4 2 0 –4 –2 0 x , nm 2 4 0 –4 –2 0 x , nm Fig. 7. The probability density of electron in quantum dot (a) for different eigenstates which are shown in the brackets. In the panel (b) the cross-sections by plane XOZ of the probability density of electron are shown. J. Comput. Theor. Nanosci. 8, 1–9, 2011 Lozovski and Piatnytsia The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots only few (∼1) electrons can be localized inside the particle, one can write the linear response to the local field in the form.30 31 The diamagnetic term in linear response is proportional to electron number32 inside the object and can be neglected because of about one electron per the quantum dot in the case under consideration. ji0n Rjjn0 R i ij R R = n + i + E0 − En jin0 Rjj0n R (26) − + i − E0 + En where e 0∗ (27) i n − n i 0∗ 2im is the current when electron transmits from state 0 to state n characterizing by energy levels E0 and En , respectively. Damping constant describes the life time of electron in the energy level. The linear response to the local (total) field can be rewritten in the more convenient form i a ji0n Rjjn0 R ij R R = n n (28) 2En − E0 an = 2 + i2 + En − E0 2 ji0n = − Then, knowledge of effective susceptibility allows us to write the mesoscopical current inside the particle induced by external field. It means that one obtains the local field at arbitrary point of the system Ei R = Ei0 R − i0 × dR Gij R R Xjk R Ek0 R V (35) Now we introduce the calculations performed for the conelike quantum dot because of transitions intensities for lowest energy levels are comparable against pyramid-like nanoparticle where probability of transitions extremely differs. The dependence of effective susceptibility on the energy and vertex angle is given in Figure 8 panel ‘a’. The calculations were performed for damping constant (a) with n = dRjin0 REi R Jin0 = V RESEARCH ARTICLE Substitution of [Eq. (28)] into [Eq. (25)] gives us the integral equation with multiplicative kernel which can be easily solved. Following Refs. [29, 33], one obtains −1 n = mn + an Nmn Jin0 Ei0 (29) dRjin0 R (30) V and self-energy part Nmn = 0 dR dR jim0 RGij R R jj0n R V V (31) Because the dipole momentum induced at the particle can be defined as i dRji R (32) pi = (b) 6 ImXzz arb. units φ = 45º 5 φ = 55º 4 3 φ = 65º V and current density ji R = 2 dR ij R R V (33) Ej R = Xij R Ej0 R Using obtained solution of Lippmann-Schwinger equation [Eq. (29)], one can write the effective susceptibility of the quantum dot in the form.33 34 −1 1 Xij R = an Ji0n mn + an Nmn Jj0m 2 n (34) J. Comput. Theor. Nanosci. 8, 1–9, 2011 1 0 0.6 0.8 1.0 1.2 1.4 1.6 hω, eV Fig. 8. Panel (a) dependence of zz-component effective susceptibility on energy and vertex angle. The transitions (013) → (011) and (021) → (011) were taken into account. Panel (b) dependence of effective susceptibility zz-component on energy for cone particles with different angles. 7 The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots RESEARCH ARTICLE = 005n0 . As it can be easily seen, the spectral dependence of Xzz is nontrivial because strong dependence of energy levels on vertex angle (shape of quantum dot). Narrow < 52 and wide ≥ 60 cones are characterized by two peaks curve of Xzz , but cones which vertex angle lies in the range of 52 < < 60 is characterized by single peak curve of Xzz . Cross sections of the frequency dependence of imaginary part of Xzz at different are shown in Figure 8 panel ‘b’. Then, the change of vertex angle of cone quantum dot can carry out drastic change of optical properties of the system with quantum dot. The knowledge of effective susceptibility allows us to calculate the local field distributions within the quantum dots and outside of the particles for both pyramid shaped and cone shaped quantum dots, which are shown in Figures 9 and 10. The strange on the face of it behaviour of local field as function of coordinates is observed in Figures 9 and 10. Indeed, the field enhancement at the top of the cone could it might be observed. But one can see, that the maximum of local field intensity is inside the cone not at the vertex. This fact is very similar to near-field image of pyramid-like quantum dot,35 where the local field intensity is not located at the vertex. As it was mentioned above, the self-action processes lead to change of energy levels of the quantum dot, socalled Lamb shifts. The Lamb shift for electron transi(a) 1,000 12 0,8333 9 z, nm 6 0,6667 3 0,5000 0 –3 –9 –6 –3 0 3 6 9 x, nm (b) 1,000 12 0,8500 z, nm 9 6 0,7000 3 0,5500 0 –3 –9 –6 –3 0 3 6 9 x, nm Fig. 9. Cross sections (y = 0, panel “a” and y = r, panel “b”) of localfield distribution in the pyramid-like quantum dot. 8 Lozovski and Piatnytsia Δ /ωα 0.15 0.12 0.09 0.06 40 45 50 55 60 ϕº Fig. 11. Dependence of lamb shift on the vertex angle. tion between states 011 and 012 (energy of transition equals to ) dependence on the vertex angle is shown in Figure 11. One can see that relative values of Lamb shifts can achievement up to 14%. One should note that unlike to Refs. [33, 36] contribution of interaction between the particle and surface in the case of cone quantum dot is smaller than the self-action via vacuum contribution. Other electron transitions are characterized by smaller Lamb shifts. 5. SUMMARY AND DISCUSSIONS The semi-analytical method for calculation of electron properties of edge-shaped quantum dots is proposed in the work. The analytical form of wave functions of electron in pyramid-like and cone-like quantum dots was obtained. One should to note that the analytical method of solving Schrödinger equation proposed in this work was used for calculation of energy structure of electron states in the edge-like quantum dots. The dependence of the electron energy levels on the vertex angle of the particle was analysed. The energy levels and its dependences on the vertex angle have the similar form as reported in Refs. [13, 27] calculated by direct numerical methods. This fact allows us to hope that proposed here analytical method gives the true results for cone-shaped quantum dot too. The distributions of electron densities within the particles are obtained. The structure of wave functions is the same as the symmetry of coordinate systems (Cartesian for pyramid-like and cylindrical for cone-like QDs). The distributions of electronic transition currents which define paramagnetic part of electric response of quantum dots were calculated. The self-consistent approach to calculation of optical response of the quantum dots was used. The effective susceptibility of cone quantum dot was calculated and its dependence on quantum dot shape (vertex angle) was firstly demonstrated. The self-action effects in particle, lead to energy levels shifts—so called Lamb shifts.29 It was established that relative Lamb shift in the J. Comput. Theor. Nanosci. 8, 1–9, 2011 Lozovski and Piatnytsia The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots case under consideration can achieve of values more than 10% for transition between ground and first excited state. It was established that the main contribution to the shift give self-action via vacuum. The self-action effects by exchange of virtual photons with the substrate give smaller contribution. On the base of obtained effective susceptibility, the local field distributions were calculated. The both electronic and local-field properties are strongly dependent on the shape of the particle (vertex angle, in the case under consideration). Indeed, the currents of electron transitions J nm R and the self-energy part Nnm are defined by the shape of the quantum dot. Moreover, one should note that the self-energy part is defined by particle shape via transition currents as integration over the volume of the particle [Eq. (20)]. References Received: 19 January 2011. Accepted: 26 February 2011. J. Comput. Theor. Nanosci. 8, 1–9, 2011 9 RESEARCH ARTICLE 1. V. V. Mitin, V. A. Kochelap, and M. A. Stroscio, Quantum Heterostructures, Cambridge University Press, Cambridge (1999). 2. D. Bimberg, M. Grudmann, and N. N. Ledentsov, Quantum Dot Heterostructures, Wiley, Chichester (1999). 3. F. Rossi, Semiconductor Macroatoms, Basic Physics and QuantumDevice Applications, Imperial College Press, London (2005). 4. W. H. Suh, Y.-H. Suh, and G. D. Stucky, Nano Today 4, 27 (2009). 5. A. D. 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