Positron Characteristics in Cadmium and Zinc Chalcogenides

Transcription

1 Commun. Theor. Phys. 65 (2016) Vol. 65, No. 5, May 1, 2016 Positron Characteristics in Cadmium and Zinc Chalcogenides N. Bouarissa, 1, Z. Rouabah, 2 and C. Champion 3 1 Laboratory of Materials Physics and its Applications, University of M sila, M sila, Algeria 2 Laboratoire Matériaux et Systèmes Electroniques, Université de Bordj-Bou-Arreridj, El-Anasser, Bordj-Bou-Arreridj, Algeria 3 Université de Bordeaux 1, CNRS/IN2P3, Centre d Etudes Nucléaires de Bordeaux-Gradignan (CENBG), France (Received September 8, 2015; revised manuscript received November 2, 2015) Abstract Electron energy levels and positron states have been calculated for cadmium and zinc chalcogenide compounds within the pseudo-potential approach and the independent particle model. Furthermore, the present contribution deals with the electron and positron chemical potentials allowing the calculation of the positron affinity to different materials of interest and hetero-structures formed by these materials. Besides, we here determine the positron diffusion constant by means of the positron deformation potential. An attempt has been made to scale positron affinity and diffusion constant with the lattice constant and the band gap energy, respectively. Such scaling is found to be not possible. The information gathered by the present study is of prime importance for a better understanding of positron trapping at interfaces and precipitates and should be useful in slow positron beam experiments. PACS numbers: Nr, Z Key words: positron affinity, positron diffusion constant, chalcogenides, interfaces, hetero-structures 1 Introduction The zinc and cadmium chalcogenides, also known as II-VI compounds, are the wide-gap semiconductors that have potential applications in optical, electro-optical and acoustic-optical devices, working in visible, infrared and THz ranges of spectrum. [1 7] ZnTe is a semiconductor material with a direct band-gap of ev. [8] It is important for the development of various semiconductor devices, including light-emitting diodes, laser diodes, solar cells, and components of microwave generators. CdTe is also a direct band-gap semiconductor with a gap of ev. [9] It is used as an infrared optical windows and a solar cell material. It is usually sandwiched with cadmium sulfide to form a p-n junction photovoltaic solar cell. ZnSe is a semiconductor material with a direct band-gap of 2.70 ev. [10] It is widely used for infrared components, windows and lenses, and to form II-VI light emitting diodes and diode lasers. It emits blue light. Positron motion in solids is of interest in various kinds of experiments, such as positron trapping at lattice defects or momentum-density measurements. [11] Following the progress in the production of low energy positron beams, [12] positron annihilation in near surface layers and thin films is increasingly studied. [11 12] These experiments are in many cases directly involved with positron diffusion to the surface. Alongside with the evolution of the experimental techniques and applications it has, therefore, been highly important to develop the theoretical machinery, which is needed to give a satisfactory interpretation and analysis to the experimental annihilation characteristics. [13 15] The slow-positron beam technique is now a wellestablished tool for the investigation of defects in solids and for the studying solid surfaces. [16 21] In fact, many of the implanted positrons in the primary beam can diffuse back to the surface and be emitted into the vacuum either as free positrons or as Positronium atoms. Several scenarios are then possible. The versatility of scenarios implies that positrons can be used to extract different kinds of physical information of the surface and near-surface region. [22 23] The positron work function is of interest in the study of the behavior of positrons at surfaces [24] and the positron deformation potential is related to the positron diffusion in metals and semiconductors. [25] On the other hand, although the positron affinity is a bulk material property, its practical usage concerns materials surfaces and interfaces. [26] Besides, precise knowledge of the positron diffusion coefficient in solids is of great interest for fundamental understanding of positron interaction in solids. [27] In the present paper, we report a theoretical investigation of positron behavior in cadmium and zinc chalcogenides. The aim of this work is to highlight positron diffusion and surface emission. We calculate the positron band structure, in particular the bottom of the lowestenergy band, parallel with the electron structure. In perfect bulk crystal the positron and electron band structures directly give the positronium work function. If the surface dipole potential is known, one can deduce the positron work function. Furthermore, the knowledge of the volume dependence of the positron and electron energy levels Corresponding author, n bouarissa@yahoo.fr c 2016 Chinese Physical Society and IOP Publishing Ltd

2 618 Communications in Theoretical Physics Vol. 65 may be used so as to estimate the positron-phonon coupling strength which in turn allows the calculation of the positron diffusion constant. The basis of the present calculations is the pseudo-potential approach. In the case of positron states, the wave function is calculated in an identical manner by employing the point core approximation for the ionic potential. Computational details are given in Sec. 2 while Sec. 3 is devoted to the discussion of results. A summary and some concluding remarks are presented in Sec Computations The present calculations are mainly based on the empirical pseudo-potential method (EPM). [28] It is first used in determining the electron structure. Briefly, the fundamental concept involved in pseudo-potential calculation is that ion core can be emitted. Computationally this is crucial for it means that the deep ion core has been removed and a simple plane-wave basis will yield rapid convergence. The EPM involves adjusting pseudo-potential form factors to achieve good agreement with the experimental results. The experimental and theoretical (when experimental data are not available) band-gap energies for ZnTe, CdTe and ZnSe at Γ, X, and L high symmetry points in the Brillouin zone used in the fitting procedure are shown in Table 1. Table 1 Band-gap energies for ZnTe, CdTe and ZnSe fixed in the fits. Compound E Γ Γ /ev EX Γ /ev EL Γ /ev ZnTe a 3.05 b 2.38 b CdTe c 3.48 d 2.47 d ZnSe 2.70 e 3.18 f 3.31 f a Experimental value reported in Ref. [8]; b Experimental values quoted by Ref. [3]; c Experimental value reported in Ref. [9]; d Theoretical values reported in Ref. [29]; e Experimental value reported in Ref. [10]; f Theoretical values reported in Ref. [8]. The electron wave functions are expanded into plane waves with a cut-off energy of 14 Ry. Since the potential can be Fourier expanded in plane waves, an eigen-value equation for determining an energy-wave vector relationship can be established. The dimension of the eigen-value problem is ( ) matrix. The empirical pseudopotential parameters are optimized using the non-linear least-squares method of Kobayasi and Nara. [30 31] The final adjusted symmetric V S and anti-symmetric V A pseudopotential from factors and used lattice constants for ZnTe, CdTe and ZnSe compounds are given in Table 2. Table 2 Pseudopotential parameters for ZnTe, CdTe, and ZnSe. Compound Form Factors (Ry) V S (3) V S (8) V S (11) V A (3) V A (4) V A (11) Lattice constant/å ZnTe CdTe ZnSe The positron wave function has been evaluated assuming that there is only one positron for many electrons; there is no exchange part because there is no positronpositron interaction. The positron potential is purely Coulombic in nature. There is a repulsive ion core potential and an attractive Hartree potential. In addition to these two there is a third part, which comes from the electron-positron correlation. The electron-positron potential is a slow function of the electron density. It is generally flat in the interstitial region swamped by the ionic and Coulomb potentials in the ion core region. Hence, it is not considered here. More details about the approach can be found in Ref. [32]. 3 Results and Discussion Figure 1 displays the calculated electron band structure of zinc blende ZnTe along several symmetry points in the Brillouin zone. The zero energy reference is taken to be at top of the valence band. It is formed by the triply degenerate hybridized Zn (4p, 3d)-Te (5p) like orbitals in an anti-bonding manner. The minimum of conduction band is at the zone center Γ and is the anti-bonding Zn (4s)-Te (5s) singlet state. The valence band maximum is at Γ and the conduction band minimum occurs at Γ as well indicating thus a Γ Γ direct gap nature of ZnTe with a fundamental energy gap of ev. Figure 2 shows the calculated positron band structure along the principal symmetry lines of the Brillouin zone for the zinc blende ZnTe. In order to facilitate the comparison with electron band structure (Fig. 1), the zero-energy reference is also taken to be at the top of the fourth band. Note that the positron band structure resembles that of the electron with the exception that it does not show a band gap at the Γ point. The positron band is rather free-particle-like and resembles closely the lowest valence electron band. This is consistent with the fact that the positron bands are all conduction bands. Interestingly the lowest positron energy state is the Γ state at K + = 0, which corresponds to the positron thermalization energy. The internal electron and positron chemical potentials denoted by µ and µ +, respectively are purely bulk properties that can be obtained from a band structure calcu-

3 No. 5 Communications in Theoretical Physics 619 lation. The electron chemical potential, i.e., the position of the Fermi level, depends on the details of the electron band structure. The positron chemical potential is the lowest energy of the positron. The calculated electron and positron chemical potentials for the semiconductors of interest are listed in Table 3. By using the positron affinities given in Table 3, one can estimate the positron affinity to different heterostructures formed by different materials under load, for example, for ZnTe and CdTe in contact, the positron favors ZnTe with the affinity difference of 0.47 ev. Based on the results of Table 3, we may proceed in the same manner for other heterostructures. Thus, the positron favors ZnSe for ZnTe/ZnSe and for CdTe/ZnSe heterostructures with affinities difference of 1.3 ev and 1.77 ev, respectively. One should note that if the material M 2 is a precipitate, the relation A M1,M2 + > 0 represents a necessary condition for positron trapping in the precipitate. Fig. 1 Electron band structure for Zincblende ZnTe. Table 3 Calculated electron and positron chemical potentials, and positron affinity for ZnTe, CdTe, and ZnSe. Compound µ /ev µ + /ev A + /ev ZnTe CdTe ZnSe Usually, people used to correlate the positron lifetimes, affinities and deformation potentials for different semiconductors with their lattice constants. [14,34] Recently, the positron bulk lifetime in semiconductors has been scaled with a 3/2 0, [35] where a 0 is the lattice constant of the semiconductor. In this respect, we have tentatively scaled A + with a 3/2 0. Our results are shown in Fig. 3. As clearly seen from this figure, such a scaling is not possible. Fig. 2 Positron band structure for Zincblende ZnTe. The positron affinity A + is a basic bulk characteristic of materials. Nevertheless, A + is important for the understanding of positron trapping at interfaces and at precipitates. [26] It is simply defined by the relation, [33] A + = µ + µ +. (1) The calculated positron affinities of the materials in question are presented in Table 3. One can observe that CdTe has a larger negative value of A +. As a consequence, the Coulomb repulsion felt by positrons in the larger open interstitial regions results in strong positron affinity. For distinct materials M 1 and M 2 in contact, the requirement that the Fermi level is the same everywhere in the solid (hetero-structure) formed by the materials M 1 and M 2, makes the electron chemical potentials align mutually and, therefore, the difference between the positron affinities in these two materials can be simply the difference between positron levels, [13,33] A M1,M2 + = A M1 + AM2 +. (2) Fig. 3 Dependence of the positron affinity on the inverse of the square root of the cell volume in chalcogenides studied. The absolute value of the positron diffusion coefficient D + is a measure of the nature and strength of the various positron-solid interaction mechanisms. [27] The deformation potential theory gives the positron diffusion constant due to acoustic phonon scattering as, [ 8π ] 1/2 4 C ii D + = 9 (m p )5/2 (k B T) 1/2 E +2, (3) d where m p is the positron effective mass, T the absolute temperature, E + d the positron deformation potential,

4 620 Communications in Theoretical Physics Vol. 65 and C ii the elastic constant associated with longitudinal waves and averaged over the directions of propagation. In the present paper, m p is determined from the positron band structure (see, for example Fig. 2) by calculating the positron energy in few k points near the bottom of the lowest energy band and by taking the curvature, [13,36] m p = 2[ d 2 E ] 1. (4) dk 2 The positron effective masses obtained in the present calculations are found to be 1.17, 1.67 and 1.17 m 0 (m 0 is the positron free particle mass) in ZnTe, CdTe, and ZnSe, respectively. The absolute temperature T is taken to be 300 K. For evaluating the positron deformation potential ), we used the expression reported in Ref. [13], (E + d E + d = V da + dv, (5) which is the sum of the volume (V ) derivatives of positron and electron chemical potentials. The volume derivative of A + is determined by performing EPM and IPM electronic and positronic band structure calculations, for a few slightly different lattice constants. In Table 4, we list the calculated volume derivative of A + along with the positron deformation potential for all the materials of interest. The calculation of the average in C ii is a complicated numerical task. In the present paper we followed the approximation reported by Boev et al. [13] and calculated it as, C ii = 1 2 (C 11 + C C 44 ). (6) For evaluating the elastic constants C 11, C 12 and C 44 for the different materials under study, we followed the approach of Bouarissa [37] that was based essentially on the work of Baranowski. [38] Our results regarding C ii for ZnTe, CdTe and ZnSe are collected in Table 4. By substituting all these calculated parameters in Eq. (3), the positron diffusion constant D + has been obtained for all the materials in question. Our results are shown in Table 4. The main importance of the theoretical results for diffusion constant shown in Table 4 is in the trends between different hosts. The magnitudes are certainly affected by the dependence of the effective mass on the host, especially when the lattice and band structures differ considerably. We have also tentatively checked whether the positron diffusion constant in cadmium and zinc chalcogenides can be scaled with their energy bandgap. Our results are shown in Fig. 4. We observe that ZnTe does not fit into the line. We may then conclude that no such scaling is possible. Fig. 4 Dependence of the positron diffusion constant on the energy band-gap of chalcogenides studied. Table 4 Calculated volume derivatives of positron affinity, positron deformation potential, the average in elastic constants and the positron diffusion constant. Compound A + / V /(ev/at.unit 3 ) E + d /ev C ii /(10 11 dyn/cm 2 ) D + /(cm 2 /s) ZnTe CdTe ZnSe Conclusion We have performed band-structure calculations for electron and positron energetic in cadmium and zinc chalcogenides. The zinc-blende structure is considered for all compounds of interest. Electron structure and related properties are then obtained within the EPM. Besides, positron energy bands are calculated using IPM for ionic potential. The chemical potentials for positron and electrons have been also calculated for all the materials of interest allowing then the determination of the respective positron affinity. Furthermore, we report the volume dependency of the chemical potential which turn is used to determine the positron diffusion constant. A scaling of positron affinity with the lattice constant and the positron diffusion constant with the energy band-gap are found to be not possible. Although there is no experimental data available so far for all studied quantities, to our knowledge, the present results are able to predict quantities directly measurable by slow-positron beam techniques and might be seen as a support to experiments dealing with bulk and surface solids.

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