Theory of confined states of positronium in spherical and circular quantum dots with Kane’s dispersion law
© Dvoyan et al.; licensee Springer. 2013
Received: 9 May 2013
Accepted: 16 June 2013
Published: 4 July 2013
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.
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 .
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) . Schrader theoretically studied this molecule . 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 , 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 . Further, in another work , 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 , 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 , the authors reported the first experimental observation of the Ps Bloch states in quartz and fcc CaF2 crystals. Greenberger, Mills, Thompson and Berko  have reproduced the results obtained in  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 . 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 . It is remarkable that the Ps and metal surface interaction is mainly conditioned by the attractive van der Waals polarization interaction at large distances . 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 .
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 . 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 .
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
where N and M are, respectively, the principal and magnetic quantum numbers of a Ps motion as a whole.
where N ′ is the principal quantum number of electron-positron pair relative motion under the influence of Coulomb interaction only.
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 (Z is a charge number), and the phenomenon of the particle falling into the center takes place. However, in our case, the expression 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 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 E g = 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 ; 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.
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) . 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.
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.
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’  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 (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 . 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.
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.
This work is supported by the NSF (HRD-0833184) and NASA (NNX09AV07A).
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