Abstract
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. Twoband approximation of Kane’s dispersion law and parabolic dispersion law of charge carriers are considered. It is shown that electronpositron 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.
Keywords:
Quantum dot; Positron; Positronium; Narrowgap semiconductors; KleinGordon equation; Kane’s dispersion law; Size quantization; Binding energyBackground
Investigation of new physical properties of zerodimensional 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 [17]. In lowdimensional 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 BornOppenheimer 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 narrowgap semiconductors for which the CC standard (parabolic) dispersion law is violated, the effective masses of the electron and light hole are equal [1114]. 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 electronpositron 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 particleantiparticle pairs may form exotic atomic states, the most wellknown 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 [1719]. Ps lifetime is long enough that it has a welldefined atomic structure. Thus, in other studies [2023], 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 (Ps_{2}) [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 Ps_{2} 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 gammaray 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 CaF_{2} crystals. Greenberger, Mills, Thompson and Berko [30] have reproduced the results obtained in [29] for quartz and have observed Pslike quenching effects under the influence of the magnetic field. Authors interpreted their results as evidence for Pslike 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 selftrapped 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 narrowgap semiconductors are given in the paper [35]. It should be noted that in the narrowgap 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 [1114,3639] in the framework of Kane’s theory. For example, for the narrowgap QDs of InSb, the dispersion law of CCs is nonparabolic, and it is well described by Kane’s twoband mirror model [14,40]. Within the framework of the twoband 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 twodimensional (2D) case, the groundstate binding energy of the impurity increases four times compared to that of the threedimensional (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 electronpositron 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.
Methods
Theory
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:
where R_{0} is the radius of a QD. The radius of a QD and effective Bohr radius of a Ps a_{p} 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.
Figure 1. The electronpositron pair in the (a) spherical QD and (b) circular QD.
Strong size quantization regime In the regime of strong SQ, when the condition R_{0} ≪ a_{p} 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 narrowgap semiconductors is nonparabolic and is given in the following form [11,36]:
where S ~ 10^{8} cm/s is the parameter related to the semiconductor bandgap
where P_{e(p)} is the momentum operator of the particle (electron, positron),
where
where
Here,
where
where α_{n,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:
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:
Here n, l(n^{′}, l^{′}) are the principal and orbital quantum numbers, correspondingly. In more convenient
units, ϵ_{g} and
where
Weak size quantization regime In this regime, when the condition R_{0} ≫ a_{p} takes place, the system’s energy is caused mainly by the electronpositron 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 KleinGordon equation can be written as [41]:
where e is the elementary charge. After simple transformations, as in the case of a strong SQ regime, the KleinGordon equation reduces to the Schrödinger equation with a certain effective energy, and then the wave function of the system can be represented as:
where
where
or
where ϵ_{R} 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 electronpositron pair center of gravity motion, one can obtain, respectively, the following expressions:
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 electronpositron pair. The wave
function of the problem is sought in the form
where the following notations are introduced:
where the parameter
The solution satisfying the finiteness condition of the wave function is given as
Substituting the function (23) in Equation 21, one can get the Kummer equation [44]:
which solutions are given by the first kind degenerate hypergeometric functions:
The expression
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:
The expression of a Ps energy in a spherical QD with a parabolic dispersion law obtained in the work [28] is given for comparison:
where N^{′} is the principal quantum number of electronpositron 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:
For clarity, it makes sense to compare this expression to a similar result obtained in the case of a parabolic dispersion law [28]:
Here, it is necessary to make important remarks. First, in contrast to the case of
the problem of hydrogenlike impurities in a semiconductor with Kane’s dispersion
law, considered in [46,47], in the case of 3D positron, the instability of the groundstate energy is absent.
Thus, in the case of hydrogenlike impurity, the electron energy becomes unstable
when
A ‘free’ positronium regime (positronium in a bulk semiconductor) KleinGordon 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
Repeating the calculations described above, one can derive the expression for the wave functions:
where
As expected, the expression (33) follows from (27) in the limit case r_{0} → ∞. 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:
It follows from (34) that in the limiting case r_{0} → ∞, 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 twodimensional 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 electronpositron pair in an impermeable 2D circular QD with a radius R_{0} (see Figure 1b). The potential energy is written as:
The radius of QD and effective Bohr radius of the Ps a_{p} 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:
and solutions are given by the Bessel functions of the first kind J_{m}(η), where
where
Here n_{r}, m(n^{′}_{r}, 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:
Weak size quantization regime In this case, again, the system’s energy is caused mainly by the electronpositron
Coulomb interaction, and we consider the motion of a Ps as a whole in a QD. Solving
the KleinGordon 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
Next, consider the relative motion of the electronpositron pair. Seeking the wave
functions of the problem in the form
At ξ → 0, the solution of (42) sought in the form χ(ξ → 0) = χ_{0} ~ ξ^{λ}[45,46]. Here, in contrast to Equation 21, the quadratic equation is obtained with the following solutions:
In the 2D case, the solution satisfying the condition of finiteness of the wave function
is given as
A similar result for the case of a parabolic dispersion law is written as:
Here N^{′} = n_{r} + 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:
In the case of free 2D Ps with Kane’s dispersion law, the energy is:
Here again, the expression (47) follows from (44) at the limit r_{0} → ∞. 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:
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 groundstate 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 m^{2}, 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 aboveobtained
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 electronpositron 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 electronpositron pair on QD sizes is proportional
to
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 narrowgap 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 electronpositron pair by varying the material, dimension, and SQ.
Figure 2 shows the dependences of the ground and first excitedstate energies of the electronpositron
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:
Figure 2. Dependences of ground and first excitedstate energies of electronpositron pair. They are in a spherical QD on a QD radius in strong SQ regime.
The dependence of the energy of electronpositron 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 twoband 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.
Figure 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
R_{0} → ∞, the binding energy of nonparabolic case remains significantly greater than in
parabolic case. Thus, at
Figure 4. The dependence of the Ps binding energy in a spherical QD on a QD radius.
Finally, Figure 5 represents the comparative dependences of groundstate 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 electronpositron 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.
Figure 5. Dependences of groundstate 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.
Conclusions
In the present paper, sizequantized 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 twoband 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 electronpositron 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 groundstate 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.
Acknowledgments
This work is supported by the NSF (HRD0833184) and NASA (NNX09AV07A).
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