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Open Access Nano Express

Theory of confined states of positronium in spherical and circular quantum dots with Kane’s dispersion law

Karen G Dvoyan*, Sergey G Matinyan and Branislav Vlahovic

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Department of Mathematics and Physics, North Carolina Central University, 1801 Fayetteville St., Durham, NC 27707, USA

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Citation and License

Nanoscale Research Letters 2013, 8:311  doi:10.1186/1556-276X-8-311

The electronic version of this article is the complete one and can be found online at: http://www.nanoscalereslett.com/content/8/1/311

Received:9 May 2013
Accepted:16 June 2013
Published:4 July 2013

© 2013 Dvoyan et al.; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Confined states of a positronium (Ps) in the spherical and circular quantum dots (QDs) are theoretically investigated in two size quantization regimes: strong and weak. Two-band approximation of Kane’s dispersion law and parabolic dispersion law of charge carriers are considered. It is shown that electron-positron pair instability is a consequence of dimensionality reduction, not of the size quantization. The binding energies for the Ps in circular and spherical QDs are calculated. The Ps formation dependence on the QD radius is studied.

Quantum dot; Positron; Positronium; Narrow-gap semiconductors; Klein-Gordon equation; Kane’s dispersion law; Size quantization; Binding energy


Investigation of new physical properties of zero-dimensional objects, particularly semiconductor quantum dots, is a fundamental part of modern physics. Extraordinary properties of nanostructures are mainly a consequence of quantum confinement effects. A lot of theoretical and experimental works are devoted to the study of the electronic, impurity, excitonic, and optical properties of semiconductor QDs. Potential applications of various nanostructures in optoelectronic and photonic devices are predicted and are under intensive study of many research groups [1-7]. In low-dimensional structures along with size quantization (SQ) effects, one often deals with the Coulomb interaction between charge carriers (CC). SQ can successfully compete with Coulomb quantization and even prevails over it in certain cases. In Coulomb problems in the SQ systems, one has to use different quantum mechanical approaches along with numerical methods. Thus, the significant difference between the effective masses of the impurity (holes) and electron allows us to use the Born-Oppenheimer approximation [8,9]. When the energy conditioned by the SQ is much more than the Coulomb energy, the problem is solved in the framework of perturbation theory, where the role of a small correction plays the term of the Coulomb interaction in the problem Hamiltonian [10].

The situation is radically changed when the effective mass of the impurity center (hole) is comparable to the mass of the electron. For example, in the narrow-gap semiconductors for which the CC standard (parabolic) dispersion law is violated, the effective masses of the electron and light hole are equal [11-14]. It is obvious that in the case of equal effective masses, adiabatic approximation is not applicable. A similar situation arises in considering the Coulomb interaction of the electron-positron pair. Antiparticle doping in semiconductor systems with reduced dimensionality greatly increases the possibilities of external manipulation of the physical properties of these nanostructures and widens the area of potential applications of devices based on them. On the other hand, such an approach makes real the study of the changes of the properties of antiparticles’ complexes formed in semiconductor media under the influence of SQ. Combinations of particle-antiparticle pairs may form exotic atomic states, the most well-known example being positronium (Ps), the bound state between an electron and positron [15,16]. There are two types of Ps: orthopositronium (parallel orientation of the spins) and parapositronium (antiparallel orientation). Orthopositronium has a lifetime τ ~ 1.4 × 10−7 s and annihilates with the emission of three gamma quanta, which by three orders exceed the lifetime of parapositronium [17-19]. Ps lifetime is long enough that it has a well-defined atomic structure. Thus, in other studies [20-23], the authors experimentally detected the occurrence of a positronium and its molecules in the structure of porous silicon and also detected positron lines of light absorption. Wheeler supposed that two positronium atoms might combine to form the dipositronium molecule (Ps2) [24]. Schrader theoretically studied this molecule [25]. Because Ps has a short lifetime and it is difficult to obtain low energy positrons in large numbers, dipositronium has not been observed unambiguously. Mills and Cassidy’s group showed that dipositronium was created on internal pore surfaces when intense positron bursts are implanted into a thin film of porous silica. Moreover, in another study [26], the authors report observations of transitions between the ground state of Ps2 and the excited state. These results experimentally confirm the existence of the dipositronium molecule. As a purely leptonic, macroscopic quantum matter–antimatter system, this would be of interest in its own right, but it would also represent a milestone on the path to produce an annihilation gamma-ray laser [27]. Further, in another work [21], porous silica film contains interconnected pores with a diameter d < 4 nm. From abovementioned follows that it is logically necessary to discuss size quantization effects related with this topic. In [28], additional quantization effects on the Ps states conditioned by QD confinement have been revealed along with quantization conditioned by Coulomb interaction in the framework of the standard (parabolic) dispersion law of CCs.

In the paper [29], the authors reported the first experimental observation of the Ps Bloch states in quartz and fcc CaF2 crystals. Greenberger, Mills, Thompson and Berko [30] have reproduced the results obtained in [29] for quartz and have observed Ps-like quenching effects under the influence of the magnetic field. Authors interpreted their results as evidence for Ps-like Bloch states. Later, Bloch states of Ps were observed in alkali halides and Ps effective mass was measured in NaBr and RbCl crystals [31]. In particular, the temperature dependence of the transition from a self-trapped Ps to the Bloch state was investigated. It is natural to assume that by creating a jump of the potentials on the boundaries of the media with the selection of specific materials with different widths of the bandgaps, it will be possible to localize the Bloch state of the Ps in a variety of nanostructures. There are many works devoted to the study of the Ps states in various solids or on their surfaces. The work functions of the positron and Ps for metals and semiconductors are calculated in [32]. It is remarkable that the Ps and metal surface interaction is mainly conditioned by the attractive van der Waals polarization interaction at large distances [33]. The interaction becomes repulsively close to the surface due to the Ps and surface electrons’ wave functions overlapping. The calculated energy of the formed bound state of Ps on the metal surface is in perfect agreement with the experimentally measured value [34].

Calculations of positron energy levels and work functions of the positron and Ps in the case of narrow-gap semiconductors are given in the paper [35]. It should be noted that in the narrow-gap semiconductors, in addition to reduction of the bandgap, the dispersion law of CCs is complicated as well. However, there are quite a number of papers in which more complicated dependence of the CC effective mass on the energy is considered [11-14,36-39] in the framework of Kane’s theory. For example, for the narrow-gap QDs of InSb, the dispersion law of CCs is nonparabolic, and it is well described by Kane’s two-band mirror model [14,40]. Within the framework of the two-band approximation, the electron (light hole) dispersion law formally coincides with the relativistic law. It is known that in the case of Kane’s dispersion law, the binding energy of the impurity center turns out more than that in the case of the parabolic law [40,41]. It is also known that reduction of the system dimensionality leads to the increase in Coulomb quantization. Hence, in the two-dimensional (2D) case, the ground-state binding energy of the impurity increases four times compared to that of the three-dimensional (3D) case [42].

As the foregoing theoretical analysis of Ps shows, the investigation of quantum states in the SQ semiconductor systems with Kane’s dispersion law is a prospective problem of modern nanoscience. In the present paper, the quantum states of the electron-positron pair in the spherical and circular QDs consisting of InSb and GaAs with impermeable walls are considered. The quantized states of both Ps and individually quantized electron and positron are discussed in the two SQ regimes - weak and strong, respectively.



Positronium in a spherical QD with Kane’s dispersion law

Let us consider an impermeable spherical QD. The potential energy of a particle in the spherical coordinates has the following form:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M1">View MathML</a>


where R0 is the radius of a QD. The radius of a QD and effective Bohr radius of a Ps ap play the role of the problem parameters, which radically affect the behavior of the particle inside a QD. In our model, the criterion of a Ps formation possibility is the ratio of the Ps effective Bohr radius and QD radius (see Figure 1a). In what follows, we analyze the problem in two SQ regimes: strong and weak.

thumbnailFigure 1. The electron-positron pair in the (a) spherical QD and (b) circular QD.

Strong size quantization regime In the regime of strong SQ, when the condition R0ap takes place, the energy of the Coulomb interaction between an electron and positron is much less than the energy caused by the SQ contribution. In this approximation, the Coulomb interaction between the electron and positron can be neglected. The problem then reduces to the determination of an electron and positron energy states separately. As noted above, the dispersion law for narrow-gap semiconductors is nonparabolic and is given in the following form [11,36]:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M2">View MathML</a>


where S ~ 108 cm/s is the parameter related to the semiconductor bandgap <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M3">View MathML</a>. Let us write the Klein-Gordon equation [43] for a spherical QD consisting of InSb with electron and positron when their Coulomb interaction is neglected:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M4">View MathML</a>


where Pe(p) is the momentum operator of the particle (electron, positron), <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M5">View MathML</a> is the effective mass of the particle, and E is the total energy of the system. After simple transformations, Equation 3 can be written as the reduced Schrödinger equation:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M6">View MathML</a>


where <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M7">View MathML</a>, <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M8">View MathML</a> is the effective Rydberg energy of a Ps, κ is the dielectric constant of the semiconductor, and <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M9">View MathML</a> is a Ps effective Bohr radius. The wave function of the problem is sought in the form <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M10">View MathML</a>. After separation of variables, one can obtain the following equation for the electron:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M11">View MathML</a>


where <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M12">View MathML</a> is a dimensionless energy. Seeking the wave function in the form <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M13">View MathML</a>, the following equation for the radial part of (5) could be obtained:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M14">View MathML</a>


Here, <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M15">View MathML</a>, l is the orbital quantum number, m is magnetic quantum number, <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M16">View MathML</a> is the reduced mass of a Ps, <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M17">View MathML</a> is dimensionless bandgap width, <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M18">View MathML</a> is the analogue of fine structure constant, and <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M19">View MathML</a> is the analogue of Compton wavelength in a narrow bandgap semiconductor with Kane’s dispersion law. Solving Equation 6, taking into account the boundary conditions, one can obtain the wave functions:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M20">View MathML</a>


where <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M21">View MathML</a>, Jl + 1/2(z) are Bessel functions of half-integer arguments, and Ylm(θ, φ) are spherical functions [44]. The following result could be revealed for the electron eigenvalues:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M22">View MathML</a>


where αn,l are the roots of the Bessel functions. The electron energy (8) is a constant of the separation of variables in the positron reduced Schrödinger equation:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M23">View MathML</a>


Solving the equation in a similar way, finally, in the strong SQ regime, one can derive the following expression for the total energy of the particles’ system:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M24">View MathML</a>


Here n, l(n, l) are the principal and orbital quantum numbers, correspondingly. In more convenient units, ϵg and <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M25">View MathML</a>, the expression of energy (10) can be written in a simpler form suitable for graphical representations:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M26">View MathML</a>


where <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M27">View MathML</a>. For comparison (see (10)), in the case of a parabolic dispersion law (e.g., for QD consisting of GaAs), the total energy in the strong SQ is given as [28]:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M28">View MathML</a>


Weak size quantization regime In this regime, when the condition R0ap takes place, the system’s energy is caused mainly by the electron-positron Coulomb interaction. In other words, we consider the motion of a Ps as a whole in a QD. In the case of the presence of Coulomb interaction between an electron and positron, the Klein-Gordon equation can be written as [41]:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M29">View MathML</a>


where e is the elementary charge. After simple transformations, as in the case of a strong SQ regime, the Klein-Gordon equation reduces to the Schrödinger equation with a certain effective energy, and then the wave function of the system can be represented as:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M30">View MathML</a>


where <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M31">View MathML</a>. Here, <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M32">View MathML</a> describes the relative motion of the electron and positron, while <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M33">View MathML</a> describes the motion of the Ps center of gravity. After switching to the new coordinates, the Schrödinger equation takes the following form:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M34">View MathML</a>


where <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M35">View MathML</a> is the mass of a Ps. One can derive the equation for a Ps center of gravity, after separation of variables, in the <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M36">View MathML</a> and ap units:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M37">View MathML</a>



<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M38">View MathML</a>


where ϵR is the energy of a Ps center of gravity quantized motion and L is the orbital quantum number of a Ps motion as a whole. For energy and wave functions of the electron-positron pair center of gravity motion, one can obtain, respectively, the following expressions:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M39">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M40">View MathML</a>


where N and M are, respectively, the principal and magnetic quantum numbers of a Ps motion as a whole.

Further, let us consider the relative motion of the electron-positron pair. The wave function of the problem is sought in the form <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M41">View MathML</a>. After simple transformations, the radial part of the reduced Schrödinger equation can be written as:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M42">View MathML</a>


where the following notations are introduced: <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M43">View MathML</a>. The change of variable <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M44">View MathML</a> transforms Equation 20 to:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M45">View MathML</a>


where the parameter <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M46">View MathML</a> is introduced. When ξ → 0, the desired solution of (21) is sought in the form χ(ξ → 0) = χ0 ~ ξλ[45,46]. Substituting this in Equation 21, one gets a quadratic equation with two solutions:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M47">View MathML</a>


The solution satisfying the finiteness condition of the wave function is given as <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M48">View MathML</a>. When ξ → ∞, Equation 21 takes the form <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M49">View MathML</a>. The solution satisfying the standard conditions can be written as <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M50">View MathML</a>[45]. Thus, the solution is sought in the form:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M51">View MathML</a>


Substituting the function (23) in Equation 21, one can get the Kummer equation [44]:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M52">View MathML</a>


which solutions are given by the first kind degenerate hypergeometric functions:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M53">View MathML</a>


The expression <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M54">View MathML</a> needs to be a nonnegative integer nr (radial quantum number) providing the finiteness of the wave functions everywhere:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M55">View MathML</a>


From the condition (26) for the positron energy in a spherical QD with a nonparabolic dispersion law (e.g. InSb) one can derive the following expression in dimensionless units:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M56">View MathML</a>


The expression of a Ps energy in a spherical QD with a parabolic dispersion law obtained in the work [28] is given for comparison:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M57">View MathML</a>


where N is the principal quantum number of electron-positron pair relative motion under the influence of Coulomb interaction only.

Determining the binding energy as the energy difference between the cases of the presence and absence of positron in a QD, one obtains finally the following expression:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M58">View MathML</a>


For clarity, it makes sense to compare this expression to a similar result obtained in the case of a parabolic dispersion law [28]:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M59">View MathML</a>


Here, it is necessary to make important remarks. First, in contrast to the case of the problem of hydrogen-like impurities in a semiconductor with Kane’s dispersion law, considered in [46,47], in the case of 3D positron, the instability of the ground-state energy is absent. Thus, in the case of hydrogen-like impurity, the electron energy becomes unstable when <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M60">View MathML</a> (Z is a charge number), and the phenomenon of the particle falling into the center takes place. However, in our case, the expression <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M61">View MathML</a> under the square root (see (27)) does not become negative even for the ground state with l = 0. In other words, in the case of a 3D Ps with Kane’s dispersion law, it would be necessary to have a fulfillment of condition for the analogue of fine structure constant <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M62">View MathML</a> to obtain instability in the ground state. However, obviously, it is impossible for the QD consisting of InSb, for which the analogue of fine structure constant is α0 = 0.123. It should be noted also that instability is absent even at a temperature T = 300 K, when the bandgap width is lesser and equals Eg = 0.17 eV instead of 0.23 eV, which is realized at lower temperatures.Second, for the InSb QD, the energy of SQ motion of a Ps center of gravity enters the expression of the energy (binding energy) under the square root, whereas in the parabolic dispersion law case, this energy appears as a simple sum (see (27) and (28) or (29) and (30)).Third, the Ps energy depends only on the principal quantum number of the Coulomb motion in the case of the parabolic dispersion, whereas in the case of Kane’s dispersion law, it reveals a rather complicated dependence on the radial and orbital quantum numbers. In other words, the nonparabolicity account of the dispersion leads to the removal of ‘accidental’ Coulomb degeneracy in the orbital quantum number [48]; however, the energy degeneracy remains in the magnetic quantum number in both cases as a consequence of the spherical symmetry.For a more detailed analysis of the influence of QD walls on the Ps motion, also consider the case of the ‘free’ Ps in the bulk semiconductor with Kane’s dispersion law.

A ‘free’ positronium regime (positronium in a bulk semiconductor) Klein-Gordon equation for a free atom of Ps can be written as (13). After separating the variables, the wave function is sought in the form of (14), where <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M63">View MathML</a>. Here, again, <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M64">View MathML</a> describes the relative motion of the electron and positron, while <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M65">View MathML</a> describes the free motion of a Ps center of gravity. Similar to (20), after simple transformations, one can obtain:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M66">View MathML</a>


Repeating the calculations described above, one can derive the expression for the wave functions:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M67">View MathML</a>


where <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M68">View MathML</a>. The energy of a free Ps atom in a narrow bandgap semiconductor with Kane’s dispersion law can be obtained from standard conditions:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M69">View MathML</a>


As expected, the expression (33) follows from (27) in the limit case r0 → ∞. For a clearer identification of the contribution of the SQ in a Ps energy, let us define the confinement energy as a difference between absolute values of energies of a Ps in a spherical QD and a free Ps:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M70">View MathML</a>


It follows from (34) that in the limiting case r0 → ∞, the confinement energy becomes zero, as expected. However, it becomes significant in the case of a small radius of QD. Note also that the confinement energy defined here should not be confused with the binding energy of a Ps since the latter, unlike the first, in the limiting case does not become zero.

Positronium in two-dimensional QD

As noted above, dimensionality reduction dramatically changes the energy of charged particles. Thus, the Coulomb interaction between the impurity center and the electron increases significantly (up to four times in the ground state) [42]. Therefore, it is interesting to consider the influence of the SQ in the case of 2D interaction of the electron and positron with the nonparabolic dispersion law.

Consider an electron-positron pair in an impermeable 2D circular QD with a radius R0 (see Figure 1b). The potential energy is written as:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M71">View MathML</a>


The radius of QD and effective Bohr radius of the Ps ap again play the role of the problem parameters, which radically affect the behavior of the particle inside a 2D QD.

Strong size quantization regime As it mentioned, the Coulomb interaction between the electron and positron can be neglected in this approximation. The situation is similar to the 3D case, with the only difference being that the Bessel equation is obtained for radial part of the reduced Schrödinger equation:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M72">View MathML</a>


and solutions are given by the Bessel functions of the first kind Jm(η), where <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M73">View MathML</a>. For the electron energy, the following expression is obtained:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M74">View MathML</a>


where <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M75">View MathML</a> are zeroes of the Bessel functions of the integer argument. The following result can be derived for the system total energy:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M76">View MathML</a>


Here nr, m(nr, m) are the radial and magnetic quantum numbers, respectively. For comparison, in the case of parabolic dispersion law for the 2D pair in a circular QD in the strong SQ regime, one can get:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M77">View MathML</a>


Weak size quantization regime In this case, again, the system’s energy is caused mainly by the electron-positron Coulomb interaction, and we consider the motion of a Ps as a whole in a QD. Solving the Klein-Gordon equation for this case, the wave functions of the system again can be represented in the form of (14); however, it must be taken into account that as <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M78">View MathML</a> and <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M79">View MathML</a>, also <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M80">View MathML</a> and <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M81">View MathML</a> are 2D vectors now. The wave functions and the Ps energy of the center of gravity motion, respectively, in the 2D case can then be obtained:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M82">View MathML</a>


<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M83">View MathML</a>


Next, consider the relative motion of the electron-positron pair. Seeking the wave functions of the problem in the form <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M84">View MathML</a>, after some transformations, the radial part of the reduced Schrodinger equation can be written as:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M85">View MathML</a>


At ξ → 0, the solution of (42) sought in the form χ(ξ → 0) = χ0 ~ ξλ[45,46]. Here, in contrast to Equation 21, the quadratic equation is obtained with the following solutions:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M86">View MathML</a>


In the 2D case, the solution satisfying the condition of finiteness of the wave function is given as <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M87">View MathML</a>. At ξ → ∞, proceeding analogously to the solution of Equation 21, one should again arrive at the equation of Kummer (24) but with different parameter λ. Finally, for the energy of the 2D Ps with Kane’s dispersion law one can get:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M88">View MathML</a>


A similar result for the case of a parabolic dispersion law is written as:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M89">View MathML</a>


Here N = nr + |m| is Coulomb principal quantum number for Ps. Again, determining the binding energy as the energy difference between cases of presence and absence of positron in a QD, one finally obtains the expression:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M90">View MathML</a>


In the case of free 2D Ps with Kane’s dispersion law, the energy is:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M91">View MathML</a>


Here again, the expression (47) follows from (44) at the limit r0 → ∞. Define again the confinement energy in the 2D case as the difference between the absolute values of the Ps energy in a circular QD and a free Ps energy:

<a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M92">View MathML</a>


Here, it is also necessary to note two remarks. First, in contrast to the 3D Ps case, all states with m = 0 are unstable in a semiconductor with Kane’s dispersion law. It is also important that instability is the consequence not only of the dimension reduction of the sample but also of the change of the dispersion law. In other words, ‘the particle falling into center’ [45] or, more correctly, the annihilation of the pair in the states with m = 0 is the consequence of interaction of energy bands. Thus, the dimension reduction leads to the fourfold increase in the Ps ground-state energy in the case of parabolic dispersion law, but in the case of Kane’s dispersion law, annihilation is also possible. Note also that the presence of SQ does not affect the occurrence of instability as it exists both in the presence and in the absence of SQ (see (44) and (47)).

Second, the account of the bands’ interaction removes the degeneracy of the magnetic quantum number. However, the twofold degeneracy of m of energy remains. Thus, in the case of Kane’s dispersion law, the Ps energy depends on m2, whereas in the parabolic case, it depends on |m|. Due to the circular symmetry of the problem, the twofold degeneracy of energy remains in both cases of dispersion law.

Results and discussion

Let us proceed to the discussion of results. As it is seen from the above-obtained energy expressions, accounting nonparabolicity of the dispersion law, in both 2D and 3D QDs in both SQ regimes, leads to a significant change in the energy spectrum of the electron-positron pair in comparison with the parabolic case. Thus, in the case of a semiconductor with a parabolic dispersion (for GaAs QD), the dependence of the energy of electron-positron pair on QD sizes is proportional to <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M93">View MathML</a> (r0 is QD radius), whereas this dependence is violated in the case of Kane’s dispersion law (for InSb QD).

Moreover, in a spherical QD, accounting of nonparabolicity of dispersion removes the degeneracy of the energy in the orbital quantum number; in a circular QD, in the magnetic quantum number. As it is known, the degeneracy in the orbital quantum number is a result of the hidden symmetry of the Coulomb problem [48]. From this point of view, the lifting of degeneracy is a consequence of lowering symmetry of the problem, which in turn is a consequence of the reduction of the symmetry of the dispersion law of the CC but not a reduction of the geometric symmetry. This results from the narrow-gap semiconductor InSb bands interaction. In other words, with the selection of specific materials, for example, GaAs or InSb, it is possible to decrease the degree of ‘internal’ symmetry of the sample without changing the external shape, which fundamentally changes the physical properties of the structure. Note also that maintaining twofold degeneracy in the magnetic quantum number in cases of both dispersion laws is a consequence of retaining geometric symmetry. On the other hand, accounting of nonparabolicity combined with a decrease in the dimensionality of the sample leads to a stronger expression of the sample internal symmetry reduction. Thus, in the 2D case, the energy of Ps atom with Kane’s dispersion law becomes imaginary. In other words, 2D Ps atom in InSb is unstable.

The opposite picture is observed in the case of a parabolic dispersion law. In this case, the Ps binding energy increases up to four times, which in turn should inevitably lead to an increase in a Ps lifetime. It means that it is possible to control the duration of the existence of an electron-positron pair by varying the material, dimension, and SQ.

Figure 2 shows the dependences of the ground- and first excited-state energies of the electron-positron pair in a spherical QD on the QD radius in the strong SQ regime. Numerical calculations are made for the QD consisting of InSb with the following parameters: <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M94">View MathML</a>, <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M95">View MathML</a>, Eg ≃ 0.23 eV, κ = 17.8, ap ≃ 103Å, <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M96">View MathML</a>, and α0 ≃ 0.123. As it is seen from the figure, at the small values of QD radius, the behavior of curves corresponding to the cases of parabolic and Kane’s dispersion laws significantly differ from each other. The energies of both cases decrease with the increase in QD radius and practically merge as a result of decreasing the SQ influence. The discrepancy of curves appears sharper in smaller values of QD radius because the dependence on QD sizes is proportional <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M97">View MathML</a> in the parabolic case, and in Kane’s dispersion case, the analogous dependence appears under the square root (see (10)). The slow growth of the particles’ energy with the decrease in QD radius in the case of Kane’s dispersion law is caused exactly by this fact. The situation is similar for excited states of both cases; however, the energy difference is considerably strong. Thus, at <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M98">View MathML</a>, the energy difference of ground states of parabolic and Kane’s dispersion cases is ΔEground ≃ 2.6Eg, whereas for excited states it is ΔEexcited ≃ 15.24Eg.

thumbnailFigure 2. Dependences of ground- and first excited-state energies of electron-positron pair. They are in a spherical QD on a QD radius in strong SQ regime.

The dependence of the energy of electron-positron coupled pair - a positronium - on a QD radius in a spherical QD in the weak SQ regime is illustrated in the Figure 3. As it is seen from the figure, in the weak SQ regime, when the Coulomb interaction energy of particles significantly prevails over the SQ energy of QD walls, the Ps energy curve behaviors in parabolic and Kane’s dispersion cases differ radically. With the decrease in radius, the energy of the Ps changes the sign and becomes positive in the parabolic case (see (28)). This is a consequence of SQ and Coulomb quantization competition. The situation is opposite in the case of the two-band Kane’s model approximation. In this case, the decrease in the radius changes the Coulomb quantization due to band interaction. In other words, in the case of nonparabolic dispersion law, the Coulomb interaction is stronger (see e.g., [42]). With the increase in radius, both curves tend to the limit of free Ps atoms of the corresponding cases (these values are given in dashed lines). The sharp increase in Coulomb interaction in the case of nonparabolicity accounting in the particles’ dispersion law becomes more apparent from the comparison of dashed lines.

thumbnailFigure 3. Dependence of Ps energy on a QD radius in a spherical QD in weak SQ regime.

Figure 4 illustrates the dependence of Ps binding energy in a spherical QD on the QD radius for both dispersion laws. As it is seen in the figure, with the increase in QD radius, the binding energy decreases in both cases of dispersion law. However, in the case of Kane’s dispersion law implementation, energy decrease is slower, and at the limit R0 → ∞, the binding energy of nonparabolic case remains significantly greater than in parabolic case. Thus, at <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M99">View MathML</a> in Kane’s dispersion case, the binding energy is <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M100">View MathML</a>, in the parabolic case, it is <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M101">View MathML</a>, and at value <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M102">View MathML</a>, they are <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M103">View MathML</a> and <a onClick="popup('http://www.nanoscalereslett.com/content/8/1/311/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/8/1/311/mathml/M104">View MathML</a>, respectively. Note the similar behavior as for the curves of the particle energies and the binding energies in the case of a 2D circular QD.

thumbnailFigure 4. The dependence of the Ps binding energy in a spherical QD on a QD radius.

Finally, Figure 5 represents the comparative dependences of ground-state energies on the QD radius in two QS regimes simultaneously: for Ps in weak SQ regime and for separately quantized electron and positron in strong SQ regime. As shown in the figure, the obtained energy of the coupled electron-positron pair - a positronium - is much smaller than the energy of separately quantized particles. Note that the jump between the energy curves corresponding to strong and weak SQ regimes is precisely conditioned by the formation of Ps atom. This is the criterion of the formation of a Ps as a whole at the particular value of the QD radius. It is seen from the figure that in the case of Kane’s dispersion law, the jump of the energy is significantly greater than that in the parabolic case. In other words, more energy is emitted at the formation of a Ps in a QD. Consequently, the binding energy of the Ps is much higher than in the case of parabolic dispersion law. As it was noted above, this is a consequence of the Coulomb quantization enhancement due to the interaction of bands.

thumbnailFigure 5. Dependences of ground-state energies on a QD radius. They are for the Ps in weak SQ regime and for separately quantized electron and positron in strong SQ regime.


In the present paper, size-quantized states of the pair of particles - electron and positron - in the strong SQ regime and the atom of Ps in the weak SQ regime were theoretically investigated in spherical and circular QDs with two-band approximation of Kane’s dispersion law as well as with parabolic dispersion law of CC. An additional influence of SQ on Coulomb quantization of a Ps was considered both in 3D and 2D QDs for both dispersion laws. The analytical expressions for the wave functions and energies of the electron-positron pair in the strong SQ regime and for the Ps as in the weak SQ regime and in the absence of SQ were obtained in the cases of the two dispersion laws and two types of QDs. The fundamental differences between the physical properties of a Ps as well as separately quantized electron and positron in the case of Kane’s dispersion law, in contrast to the parabolic case, were revealed. For the atom of Ps, the stability was obtained in a spherical QD and instability in all states with m = 0 in a circular QD in the case of Kane’s dispersion law. It was shown that the instability (annihilation) is a consequence of dimensionality reduction and does not depend on the presence of SQ. More than a fourfold increase in the binding energy for the Ps in a circular QD with parabolic dispersion law was revealed compared to the binding energy in a spherical QD. The convergence of the ground-state energies and binding energies to the free Ps energies for both cases of dispersion laws were shown. The jump between the energy curves corresponding to the cases of strong and weak SQ regimes (which is significantly greater in the case of Kane’s dispersion law), which is the criterion of the electron and positron coupled state formation - a positronium - at a particular radius of a QD, was also revealed. The removal of an accidental Coulomb degeneracy of energy in the orbital quantum number for a spherical InSb QD and in a magnetic quantum number for the circular QD, as a result of a charge carrier dispersion law symmetry degree reduction, was noticed.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

KD gave the main idea of the manuscript, did the calculations, and drafted the manuscript. SM provided theoretical guidance, did the calculations, and drafted the manuscript. BV performed the theoretical analysis of the results and drafted the manuscript. All authors read and approved the final manuscript.


This work is supported by the NSF (HRD-0833184) and NASA (NNX09AV07A).


  1. Harrison P: Quantum Wells, Wires and Dots, Theoretical and Computational Physics. New York: Wiley; 2005. OpenURL

  2. Bastard D: Wave Mechanics Applied to Semiconductor Heterostructures. Paris: Les editions de physique; 1989. OpenURL

  3. Bimberg D, Grundmann M, Ledentsov N: Quantum Dot Heterostructures. Chichester: John Wiley & Sons; 1999. OpenURL

  4. Herman D, Ong TT, Usaj G, Mathur H, Baranger HU: Level spacings in random matrix theory and Coulomb blockade peaks in quantum dots.

    Phys Rev B 2007, 76:195448.



  5. Dvoyan KG, Hayrapetyan DB, Kazaryan EM, Tshantshapanyan AA: Direct interband light absorption in strongly prolated ellipsoidal quantum dots’ ensemble.

    Nanoscale Res Lett 2009, 4(2):130-137. Publisher Full Text OpenURL

  6. Bayer M, Stern O, Hawrylak P, Fafard S, Forchel A: Hidden symmetries in the energy levels of excitonic ‘artificial atoms’.

    Nature 2000, 405:923. PubMed Abstract | Publisher Full Text OpenURL

  7. Ivchenko EL, Kavokin AV, Kochereshko VP, Posina GR, Uraltsev IN: Exciton oscillator strength in magnetic-field-induced spin superlattices CdTe/(Cd, Mn)Te.

    Phys Rev B 1992, 46:7713-7722. Publisher Full Text OpenURL

  8. Elliott RJ, Loudon RJ: Theory of the absorption edge in semiconductors in a high magnetic field.

    Phys Chem Solids 1960, 15:196-207. Publisher Full Text OpenURL

  9. Dvoyan KG, Kazaryan EM: Impurity states in a weakly prolate (oblate) ellipsoidal microcrystal placed in a magnetic field.

    Phys Status Solidi b 2001, 228:695-703. Publisher Full Text OpenURL

  10. Efros LA, Efros AL: Interband absorption of light in a semiconductor sphere.

    Sov Phys Semicond 1982, 16:772. OpenURL

  11. Kane EO: Band structure of indium antimonide.

    J Phys Chem Solids 1957, 1:249-261. Publisher Full Text OpenURL

  12. Atoyan MS, Kazaryan EM, Poghosyan BZ, Sarkisyan HA: Interband absorption and excitonic states in narrow band InSb spherical quantum dots.

    Physica E 2011, 43:1592. Publisher Full Text OpenURL

  13. Poghosyan BZ, Demirjian GH: Binding energy of hydrogenic impurities in quantum well wires of InSb/GaAs.

    Physica B 2003, 338:357-360. Publisher Full Text OpenURL

  14. Poghosyan BZ: Binding energy of hydrogen-like impurities in quantum well wires of InSb/GaAs in a magnetic field.

    Nanoscale Res Lett 2007, 2:515-518. Publisher Full Text OpenURL

  15. Rich A: Recent experimental advances in positronium research.

    Rev Mod Phys 1981, 53:127-165. Publisher Full Text OpenURL

  16. Berko S, Pendleton HN: Positronium.

    Ann Rev Nuclear Particle Sci 1980, 30:543. Publisher Full Text OpenURL

  17. Gidley DW, Frieze WE, Dull TL, Yee AF, Ryan ET, Ho H-M: Positronium annihilation in mesoporous thin films.

    Phys Rev B 1999, 60:R5157-R5160. Publisher Full Text OpenURL

  18. Charlton M, Humberston JW: Positron Physics. Cambridge: Cambridge University Press; 2001. OpenURL

  19. Barbiellini B, Platzman PM: The positronium state in quartz.

    Phys Status Solidi C 2009, 6:2523-2525. Publisher Full Text OpenURL

  20. Cassidy DB, Deng SHM, Tanaka HKM, Mills AP Jr: Single shot positron annihilation lifetime.

    Appl Phys Lett 2006, 88:194105. Publisher Full Text OpenURL

  21. Cassidy DB, Mills AP Jr: The production of molecular positronium.

    Nature 2007, 449:195-197. PubMed Abstract | Publisher Full Text OpenURL

  22. Cassidy DB, Mills AP Jr: Interactions between positronium atoms in porous Silica.

    Phys Rev Lett 2008, 100:013401. PubMed Abstract | Publisher Full Text OpenURL

  23. Cassidy DB, Hisakado TH, Tom HWK, Mills AP Jr: Photoemission of positronium from Si.

    Phys Rev Lett 2011, 107:033401. PubMed Abstract | Publisher Full Text OpenURL

  24. Wheeler JA: Polyelectrons.

    Ann NY Acad Sci 1946, 48:219. Publisher Full Text OpenURL

  25. Schrader DM: Symmetry of dipositronium Ps2.

    Phys Rev Lett 2004, 92:43401. OpenURL

  26. Cassidy DB, Hisakado TH, Tom HWK, Mills AP Jr: Optical spectroscopy of molecular positronium.

    Phys Rev Lett 2012, 108:133402. PubMed Abstract | Publisher Full Text OpenURL

  27. Mills AP Jr, Cassidy DB, Greaves RG: Prospects for making a Bose-Einstein-condensed positronium annihilation gamma ray laser.

    Mater Sci Forum 2004, 445:424. OpenURL

  28. Dvoyan KG: Confined states of a positronium in a spherical quantum dot.

    Physica B 2012, 407:131-135. Publisher Full Text OpenURL

  29. Brandt W, Coussot G, Paulin R: Positron annihilation and electronic lattice structure in insulator crystals.

    Phys Rev Lett 1969, 23:522. Publisher Full Text OpenURL

  30. Greenberger A, Mills AP, Thompson A, Berko S: Evidence for positronium-like Bloch states in quartz single crystals.

    Phys Lett 1970, 32A:72. OpenURL

  31. Kasai J, Hyodo T, Fujiwara K: Positronium in alkali halides.

    J Phys Soc Japan 1988, 57:329-341. Publisher Full Text OpenURL

  32. Boev OV, Puska MJ, Nieminen RM: Electron and positron energy levels in solids.

    Phys Rev B 1987, 36:7786-7794. Publisher Full Text OpenURL

  33. Cuthbert A: Positronium binding to metal surfaces.

    J Phys C 1985, 18:4561. Publisher Full Text OpenURL

  34. Saniz R, Barbiellini B, Platzman PM, Freeman AJ: Physisorption of positronium on quartz surfaces.

    Phys Rev Lett 2007, 99:096101. PubMed Abstract | Publisher Full Text OpenURL

  35. Bouarissa N, Aourag H: Positron energy levels in narrow gap semiconductors.

    Mat Sci Eng B 1995, 34:58-66. Publisher Full Text OpenURL

  36. Askerov B: Electronic and Transport Phenomena in Semiconductors. Moscow: Nauka; 1985. OpenURL

  37. Filikhin I, Suslov VM, Vlahovic B: Electron spectral properties of the InAs/GaAs quantum ring.

    Physica E 2006, 33:349-354. Publisher Full Text OpenURL

  38. Filikhin I, Deyneka E, Vlahovic B: Single-electron levels of InAs/GaAs quantum dot: comparison with capacitance spectroscopy.

    Physica E 2006, 31:99-102. Publisher Full Text OpenURL

  39. Filikhin I, Matinyan S, Nimmo J, Vlahovic B: Electron transfer between weakly coupled concentric quantum rings.

    Physica E 2011, 43:1669-1676. Publisher Full Text OpenURL

  40. Avetisyan AA, Djotyan AP, Kazaryan EM, Poghosyan BG: Binding energy of hydrogen-like impurities in a thin semiconductor wire with complicated dispersion law in a magnetic field.

    Phys Status Solidi b 2001, 225(2):423-431. Publisher Full Text OpenURL

  41. Avetisyan AA, Djotyan AP, Kazaryan EM, Poghosyan BG: Binding energy of hydrogen-like impurities in a thin semiconductor wire with complicated dispersion law.

    Phys Status Solidi b 2000, 218:441-447. Publisher Full Text OpenURL

  42. Branis SV, Gang L, Bajaj KK: Hydrogenic impurities in quantum wires in the presence of a magnetic field.

    Phys Rev B 1993, 47:1316-1323. Publisher Full Text OpenURL

  43. Galitsky VM, Karnakov BM, Kogan VI: Practical Quantum Mechanics. Moscow: Izd. Nauka; 1981. OpenURL

  44. Abramovitz M, Stegun I: Handbook on Special Functions. Moscow: Izd. Nauka; 1979. OpenURL

  45. Landau LD, Lifshitz EM: Quantum Mechanics. Moscow: Izd. Nauka; 1989. OpenURL

  46. Bethe H: Intermediate Quantum Mechanics. New York: Basic Books Inc; 1971. OpenURL

  47. Berestetski VB, Lifshitz EM, Pitaevski LP: Relativistic Quantum Theory. Moscow: Izd. Nauka; 1971. OpenURL

  48. Fock VA: Zur Theorie des Wasserstoffatoms.

    Z Phys 1935, 98:145-154. Publisher Full Text OpenURL