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(CdSe)n superlattices
Chinese Journal of Physics 60 (2019) 462–472
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FP-LMTO study of structural, electronic and optical properties of wurtzite (CdS)n/(CdSe)n superlattices
T
M. Merabeta,b, S. Benaliaa,b, L. Djoudia,b, O. Cherefa, N. Bettahara, D. Racheda, , R. Belacela ⁎
a b
Laboratoire des Matériaux Magnétiques, Faculté des Sciences, Université Djillali Liabès de Sidi Bel-Abbès, Sidi Bel-Abbès 22000, Algerie University of Tissemsilt, Institute of Science and Technology, Tissemsilt 38000, Algeria
ARTICLE INFO
ABSTRACT
Keywords: C. FP- LMTO A. Superlattices D. Electronic properties D. Optical properties
In this paper, we have conducted a first-principles study of the structural, electronic and optical properties of (CdS)n/(CdSe)n superlattices (where n is numbers of monolayers) in the wurtzite phase (B4), using the Full-Potential Linear Muffin-Tin Orbital (FP-LMTO) method within the Local Density Approximation (LDA) technique, in order to describe the exchange correlation energy. The calculated electronic properties indicate that all (CdS)n/(CdSe)n superlattices configurations, possess a semiconductor behavior with same energy gaps. We have seen more carefully and accurately that the different superlattices configurations have no effect on the electronic properties; in particular, we did not observe any dependence between the band gap behavior and the used layers.
1. Introduction Wide-bandgap II–VI compounds are expected to be one of the most vital materials for high-performance optoelectronics devices such as light emitting diodes (LEDs) and laser diodes (LDs), which operate in the blue or ultraviolet spectral range. Additionally, the high ionicity of these compounds makes them good candidates for high electro-optical and electromechanical coupling [1]. The Cd-chalcogenides are the most important binary semiconductors used in the fabrication of optoelectronic devices [2,3] especially light-emitting devices in the short-wavelength region of visible light, because of their direct band gap. Unlike most III–V and II–VI semiconductors, CdX (X=S, Se and Te) compounds exist in both zinc blende (ZB) and wurtzite (WZ) structures or in mixed (ZB)/(WZ) phases [3,4]. Depending on the growth conditions, one of these two crystal structures can be stabilized either by epitaxial strain on proper substrates or on buffer layers, or by controlling the growth temperature. This additional structure freedom provides an opportunity for making more efficient and reliable devices by choosing the appropriate polytypism of the compounds. Due to the technological importance of these materials, the various experimental and theoretical efforts have been made to predict the fundamental properties of these materials in recent decades [5–11]. The current availability of advanced epitaxial technologies that can grow II–VI heterostructures at nanoscale sizes with distinctive layers has motivated researchers to examine their optical characteristics. The exploration of heterostructure semiconductors incorporating various layers tends to focus exclusively on the production of self-organized quantum wells (QWs) and superlattices (SLs) which they have a wide range of highly promising properties for optoelectronic applications. Semiconductor heterostructures are ultra-thin layered materials whose thicknesses are in the range of a few atomic layers. The
⁎
Corresponding author. E-mail address: [email protected] (D. Rached).
https://doi.org/10.1016/j.cjph.2019.05.026 Received 11 March 2019; Received in revised form 19 April 2019; Accepted 9 May 2019 Available online 24 May 2019 0577-9073/ © 2019 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
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Fig. 1. Full Potential Linear Muffin-Tin Orbital (FP-LMTO) method. Table 1 Parameters used in the calculations: number of plane wave (NPW), energy cutoff (in Rydberg) and the muffin-tin radius (RMT in atomic units). Configuration
(CdS)1/(CdSe)1 (CdS)2/(CdSe)2 (CdS)3/(CdSe)3
NPLW (Total)
13,128 26,262 39,374
Ecut(Ryd)
102.8657 102.7102 103.6485
MTS (ua) Cd
S
Se
2.494 2.497 2.485
2.431 2.424 2.414
2.459 2.472 2.458
presence of these ultrathin layers leads to the “quantum size effect” when the physical dimensions of the layers are comparable to the De Broglie wavelength of the charge carrier [12]. This new effect is of great interest in both fundamental physics and devices applications [13]. A superlattice is a periodic heterostructure formed by two types of semiconducting materials, one type that acts as a quantum well and the other acting as a quantum barrier. Similarities between the CdS and CdSe lattices and the suitability of their bandgaps for superlattice construction [14] have dictated the choice of these both semiconductors for this present investigation. Both CdS and CdSe semiconductors have a wide band-gap (2.42 and 1.76 eV at room temperature respectively). Also it has been observed [15] that luminescence from superlattices can be very strong, so that structures using semiconductors that emit light in the visible or near-visible region, are of special interest. Further, from a structural viewpoint, the system is interesting as the first reported hexagonal superlattice mixing system. In order to fully take advantage of (CdS)n/(CdSe)n superlattices properties for eventual technological applications, a theoretical investigation is necessary. For this purpose, we have conducted that investigation on the structural, electronic and the optical properties of the (CdS)n/(CdSe)n (n number of monolayers; n = 1, 2, 3) superlattices in the wurtzite (WZ) structure, using the fullpotential linear muffin-tin orbital (FP-LMTO) method within the density functional theory (DFT) as implemented in the new version Lmtart (Lmtart 8: Last updated September 2018) computer code. The organization of the paper is as follows. In Section 2, we have described the employed method and the details of calculations. Results and discussions of the structural, electronic and the optical properties of the (CdS)n/(CdSe)n (n number of layers; n = 1, 2, 3) superlattices in the wurtzite (WZ) structure are presented in Section 3. Finally, conclusions and remarks are given in Section 4. 2. Calculation methodology Based on the Density Functional Theory (DFT) [16,17], we have calculated the structural, electronic and optical properties of (CdS)n/(CdSe)n superlattices, (n is numbers of layers). We have employed the Full Potential Linear Muffin-Tin Orbital (FP-LMTO) method [18,19] to solve the Kohn–Sham equations [20,21] within the Local Density Approximation (LDA) technique [22,23] in order to describe the exchange correlation energy. In the FP-LMTO method, the crystalline space is divided into non-overlapping muffin-tin spheres (MTSs) centered at the atomic sites region, (also known as heart region), and the interstitial region (IR). The potential inside MTSs (r ≤ RMT) is assumed to be spherically symmetric, while in IR (r > RMT), it is assumed to be constant, where RMT is the radius of the MTSs, and r is a position (Fig. 1). The FP-LMTO method used in our study, [24] is based on the smooth Hankel functions [25] as the envelope function of the basis set. The smoothing radii and values (Hankel function decay parameters) were carefully adjusted to optimize an efficient basis set with one s, p, and d state for each Cd, S, and Se site. The smooth interstitial quantities are calculated using a fine Fourier transform mesh, and the Brillouin-zone integrations were carried out with a well-converged k mesh based on a uniform division of reciprocal space. In order to achieve energy eigenvalues convergence, the charge density and potential inside the MTSs are represented by spherical harmonics up lmax = 6. The self-consistent calculations are considered to be converged when the total energy of the system is stable at 10−6 Ry. The k integration over the Brillouin zone is performed using the tetrahedron method [26]. To avoid the overlap of atomic spheres, the MTSs radii for each atomic position are taken differently for each composition. We point out that the using of the fullpotential calculation ensures that the calculation is not completely independent of the choice of spheres radii. The values of the 463
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Fig. 2. Total energy in terms of cell volume of all (CdS)n/(CdSe)n superlattices.
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Fig. 3. Total energy in terms of c/a ratio of all (CdS)n/(CdSe)n superlattices.
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Table 2 Structural parameters and formation energies of all (CdS)n/(CdSe)n superlattices. Configuration
a0(a.u)
c/a
B0(GPa)
B0
EF (eV)
(CdS)1/(CdSe)1 (CdS)2/(CdSe)2 (CdS)3/(CdSe)3
8.0615 8.0676 8.031
1.6433 3.3010 4.9384
52.1921 57.04376 62.77754
4.54284 4.3107 4.20025
−0.75757 −1.38907 −1.91693
spheres radii (MTSs), number of plane waves (NPLW) and the plane waves cut-off (Ecut) values, used in our calculation, are summarized in Table 1. 3. Results and discussion 3.1. Structural properties Majorities of II–VI compounds and in particular CdS and CdSe, crystallize in the hexagonal structure, especially wurtzite phase. This structure is composed of two interpenetrating hexagonal compact hcp lattices, where, Cd atoms are occupying the position 1/3, 2/3, 0 and 2/3, 1/3, 1/2, and S or Se atoms are occupying the position 1/3, 2/3, u and 2/3, 1/3, u + 1/2, where u is the dimensionless internal parameter (Noted that literature search has failed to reveal a precisely determined value for u, and it is often taken to have the ideal value 3/8 in the case when the ratio c/a is equivalent to 8/3 [27]). Wurtzite (CdS)n/(CdSe)n superlattices are constituted by a sequence of alternating layers of nCdS and nCdSe respectively, in a well-defined growth direction. These layers form a periodic arrangement of quantum wells and potential barriers along the growth direction. In our study, we chose the axis [0001] as growth axis (z is taken to be perpendicular to the layers while x and y axes are parallel to them). The massive binaries have a Wurtzie phase, and the superlattices (CdS)n/(CdSe)n have a hexagonal symmetry. Quantum well superlattices (CdS)n/(CdSe)n have an interesting properties from a structural, electronic and optical viewpoint. The most interesting feature of these compounds is the presence of the quantum wells and potential barriers along of the structure. The possibility of penetration of the charge carriers in the potential barriers generates a coupling between quantum wells, which leads to a continuity of wave functions of electrons and holes according to conduction mini bands and valence bands along of the growth direction. The structural ground state properties of wurtzite (CdS)n/(CdSe)n superlattices in its ordered unit cell are determined by performing a self-consistent calculation of the total energy in terms of cell volume (Figs. 2 and 3); in order to calculate the lattice constants a, at ideal values of the ratio c/a (such that c/a ideal = 8/3 ). Then, we optimize the ratio c/a at aop. After that, and for accuracy, we use the optimized values, meaning, the lattice constants aop, and the ratio (c/a)op, for recalculating the ratio c/a and the lattice constant a of equilibrium cell. The equilibrium lattice constants a, bulk modules B and its pressure derivatives B’ are optimized by an adjustment of the total energy obtained in terms of cell volume (ETotal(V)) using the fitting Birch’s equation of state [28]. However, the c/a ratio is optimized by an adjustment of their total energies obtained in terms of c/a (E Total(c/a)) using a polynomial fitting. The results for the different structural parameters are grouped in the Table 2. According to the Table 2 analysis, we have noticed that the calculated equilibrium parameters of all superlattices are constant; and this is due to the lack of growth in the direction of this axis, also the number of atoms in this direction remains the same in all three configurations. For c/a, we have noticed that in all configurations, this ratio is proportional to the number of layers, so that when the number of layers are increased, the c/a ratio increased evenly. This increase is due to the fact that this direction is following the growth direction of our superlattices. We note that the c/a ratio increases according to the relationship (c/a)nn = n(c/a)11. This proves that when the number of layers increases, the mesh increases uniformly following the growth direction. 1. The superlattices (CdS)n/(CdSe)n crystallize in the wurtzite structure (CdS and CdSe are energetically favorable to the wurtzite structure) where meshes contain 4, 8 and 12 atoms for the (CdS)1/(CdSe)1, (CdS)2/(CdSe)2 and (CdS)3/(CdSe)3 configurations respectively. We can clearly see from the Table 2 that the calculated bulk modules of our superlattices increases with the increasing of the number of layers, which suggests the same increasing for the compressibility of each compound. These compounds became harder when the number of layers increase. It represents bond strengthening or weakening effects induced by changing the composition [29]. In order to test if (CdS)n/(CdSe)n superlattices can be synthesized and form a stable phase experimentally, the formation energy Ef were calculated. The formation energy Ef per formula unite for all superlattices is calculated as follows [30]:
Ef = E [(CdS )n (CdSe)n]
n {E [(CdS )1 (CdS )1]+E [(CdSe )1 (CdSe )1]} 2
The formation energy calculation is presented in Table 2. From the Table 2 analysis, it can be found that formation energy for these superlattices have negative values, which imply that these superlattices phases are energetically favorable. Thus, these values guarantee the physical stability of these compounds against their decomposition. Also the negative values reflecting the formations of these superlattices are exothermic and also can be synthesized experimentally in laboratory conditions. 466
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Fig. 4. Band structures along high-symmetry directions of all (CdS)n/(CdSe)n superlattices.
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Table 3 Bandgaps values of all (CdS)n/(CdSe)n superlattices. Configuration
Eg (eV )
Gap nature
(CdS)1/(CdSe)1 (CdS)2/(CdSe)2 (CdS)3/(CdSe)3
0.619529 0.614942 0.625481
Direct Direct Direct
3.2. Electronic properties 3.2.1. Band structure and electronic behavior The band structures along high-symmetry directions of the superlattices are calculated at equilibrium lattice constants, with LDA approach, and they are illustrated in Fig. 4. In our superlattices configurations, we have noted that the CdSe plays the role of a quantum well, while the CdS plays the role of a potential barrier. In addition, the alignment type of this quantum wells superlattices belong in the type I alignment [31]. Type I superlattices correspond to the case when a large gap semiconductor is brought into contact with another small gap semiconductor, where the extremes of the conduction and valence bands are located in the same layer (same semiconductor). The charge carriers (electrons and holes) are confined in the small gap material that forms the quantum well embedded in the larger gap material that forms the potential barriers, therefore, the electrons and holes are trapped in the same material, so, their recombination will be important and the distribution of the forbidden band differences between the valence band and the conduction band depends on the nature of the both different used materials [32–34]. According to Fig. 4 analysis, it shows a similar electronic variation for all superlattices. The band structures clarify a band gap in the vicinity of the Fermi level, which is a semiconductor nature. Consequently, these compounds have a semiconductor behavior with same energy gap values. It is also observed that the band structures of all superlattices configurations possess a direct band gaps, since the energies gaps (Eg) are situated at the Γ Brillouin zone point (the valence band maximum (VBM) and the conduction band minimum (CBM) are located at Γ Brillouin zone point). The band structure of SL33 is more numerous than SL22, which is more numerous than SL11, due to the increasing of the layers in each time. The calculated gap values of all our superlattices are shown in Table 3. We have concluded that the number of layers have no effect on the electronic properties. 4. It is clear that the calculated gap values obtained by LDA are small to compare to 2.42 and 1.76 eV of CdS and CdSe. This underestimation of the gap values is mainly due to the fact that the LDA does not take account the energy independence of the quasi-particles correctly [35]. Also it's due to the internal electric fields, which they have strongly modified the band structure [36]. This system (CdS/CdSe) is the first reported semiconductor superlattice with this crystal structure and it possesses new behavior because of the existence of extremely large piezoelectric fields within the strained layers; these fields drastically modify the energy band structure [37]. 3.2.2. Density of states In order to understand the electronic properties and the origin of the band structure of all superlattices, we have calculated their total and partial density of states (TDOS and PDOS) using LDA approach. The corresponding densities of states obtained are represented in Fig. 5. By analyzing this figure, we can say that the TDOS and PDOS have presented a similar behavior for all superlattices. The densities of states of the valence and conduction bands of our superlattices are characterized by two regions separated by gaps. Following always Fig. 5, it shows that the range between −6 and −4.6 eV have no DOS. The first part of the first region in the valence band in the range −4.6 to −3 eV usually comes from the p(S and Se) states, and the s(Cd) states with a small contributions of d(Cd) and p(Cd) states; The second part of the first region in the valence band in the range between −3 eV to −0.3 eV (close to the Fermi level), DOS originates from the p(S and Se) and hybridization of s, p and d states of the Cd atoms. At Fermi levels, the compounds have no DOS, which demonstrate the semiconductor behavior of these materials. The second region is the region in the conduction band, this region for all configurations has the same contribution in the total density of states. This contribution is formed by hybridization of s, p and d states of the three atoms (Cd, S and Se). 3.3. Optical properties The optical properties of the material can be described by the complex dielectric function ε(ω), which represents the response of a system to an external electromagnetic field. Indeed, during the interaction of a material with a wave, the response of the material is governed by its complex dielectric function (or permittivity) ε(ω). Knowing the real and imaginary parts of the dielectric function allows the calculation of different functions and also makes it possible to predict the behavior of the wave inside the material. The complex dielectric function can be expressed as: ε(ω) = ε1(ω) + iε2(ω) [38,39], where ε1(ω) and ε2(ω) are the real and imaginary parts of the dielectric function, respectively. Generally, the contributions of ε(ω) result from intra-band transitions and inter-band transitions. The contribution of intra-band transitions is important only in the case of metals [40]. The inter-band transitions can be direct or indirect. In our calculations, we will ignore the indirect transitions that involve diffusion by a phonon where the contribution of ε(ω) is small compared to direct transitions. We have discussed the optical properties of a material that must be studied in order to determine its potential utility in optoelectronic applications. For this reason, we have selected all materials of our study since they showed a direct band gap character. The 468
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Fig. 5. Total and partial density of states of all (CdS)n/(CdSe)n superlattices.
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Fig. 6. Calculated dielectric functions (real and imaginary parts) for (CdS)n/(CdSe)n superlattices at direct band gap.
real part ε1 of the dielectric function can be determined from the imaginary part ε2 by the Kramers–Kronig relation. Ambrosch-Draxl and Sofo [41] have already presented a detailed description of the calculation of optical properties. To calculate the optical spectra of the dielectric function, ε(ω), a dense mesh of uniformly distributed k-points is required. Therefore, the Brillouin zone integration was performed with 600 k- points in the irreducible part for all superlattices without broadening. The imaginary and real parts of the dielectric functions for all superlattices in the energy range [0–13.6] eV are illustrated in Fig. 6. As can be seen from this figure analysis, the optical spectra show a considerable anisotropy between the ordinary components ɛ⊥ and the extraordinary components ɛ∥. The analysis also showed, for the curves of the imaginary part of the dielectric functions ε2 for all configurations, that the first critical point of the dielectric function occurs at the energy of 0.68029 eV. This value corresponds to the electronic transition value (Гv→Гc), and it represents the fractionation Гv→Гc which gives the threshold of the direct optical transitions between the highest state of the valence band and the lowest state of the conduction band. This is also known as the fundamental absorption edge. Beyond 470
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these points, the curve increases rapidly. This is due to the fact that the number of points contributing to ε2 increases sharply. The main peaks of 2 for SL11, SL22 and SL33 are about 4.725984, 4.725984 and 6.394726 eV respectively, while, the main peaks of 2 for SL11, SL22 and SL33 are equal, and they are located at 6.304726 eV for all configurations. For the real part of the dielectric function of all configurations, their spectra begin by the static dielectric constants which they are located at 3.892359. The extraordinary component 1 vanishes three times at energies 4.710059, 5.372491 and 6.4422255 eV for all configurations, while the ordinary component 1 vanishes once at energy 6.326179 eV for all configurations. The minimum of ε1 spectra corresponds to the energy 7.931364 eV for the extraordinary components 1 and also for the ordinary components 1 . Then the spectra of ε1 slowly increase to zero at high energy for all configurations. We have concluded that the number of layers used has no effect on the real parts of the dielectric functions for all configurations, and they have an important effect on the extraordinary component of the imaginary part of the dielectric function 2 in the SL33. The Fig. 5 analysis shows clearly that the critical points values do not change with the number of layers for the real parts of the dielectric functions for the three configurations, while the imaginary parts of the dielectric functions have a different values for the critical points of the extraordinary components 2 of SL11 and SL22 compared to SL33. 4. Conclusion In this work, we carried out a detailed study of the structural, electronic and optical properties of (CdS)n/(CdSe)n superlattices using the FP-LMTO method.
• The calculated lattice constants of all superlattices are almost equal; where c/a ratios are proportional to the number of layers. The • • •
calculated bulk modules of our superlattices increases with the increasing of the number of layers, which suggests the same increasing for the compressibility of each compound. For the electronic proprieties, these compounds have a semiconductor behavior with same energy gaps values. It is also observed that the band structure of all superlattices possesses a direct band gap. The real part of the dielectric functions is the same for all configurations, while the extraordinary component 2 of the imaginary part of the dielectric function is the same for SL11 and SL22, and different for SL33 due to the number of layers used in this configuration. We did not observe any dependence, either between the band gap behavior and the number of layers used, or between the real part of the dielectric function and the number of layers used, where this last have an important effect on the extraordinary component 2 of the imaginary part of the dielectric function of SL11 and SL22 compared to SL33.
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[27] E.H. Kisi, M. Elcombe, u parameters for the wurtzite structure of ZnS and ZnO using powder neutron diffraction, Acta Cryst. C 45 (1989) 1867. [28] F. Birch, Finite strain isotherm and velocities for single‐crystal and polycrystalline NaCl at high pressures and 300°K, Geophys. Res. 83 (1978) 1257. [29] M. Labidi, S. Ghemid, H. Meradji, S. Labidi, F. ElHajHassan, Density functional calculations of Pb1−xCaxSySe1−y alloys lattice matched to different substrates, Phys. Chem. Solids 73 (2012) 608. [30] C. Jedrzejek, J. Konior, S. Kaprzyk, Formation Energy and Electronic Structure of II-VI/IV Semiconductor Superlattices, Acta Phys. Pol. A 87 (1995) 349. [31] R. Tsu, Superlattice to Nanoelectronics, Elsevier Ltd, 2005 2-3-34. [32] E.L. Ivchenko, G.E. Pikus, Superlattices and Other Heterostructures, Springer-Verlag, Berlin Heidelberg, 1997, pp. 3–4 1995. [33] G. Allan, G. Bastard, N. Boccara, M. Lannoo, M. Voos, Heterojunctions and Semiconductor Superlattices, Springer Verlag, Berlin Heidelberg, 1986, pp. 4–5. [34] E.E. Mendez, K. von Klitzing, Physics and Applications of Quantum Wells and Superlattices, Plenum Press, New York, 1987, pp. 4–6. [35] S.N. Rashkeev, W.R.L. Lambrecht, Second-harmonic generation of I-III-VI2 chalcopyrite semiconductors: Effects of chemical substitutions, Phys. Rev. B 63 (2001) 165212. [36] M.P. Halsall, J.E. Nicholls, CdS/CdSe intrinsic Stark superlattices, J. Appl. Phys. 71 (1992) 907. [37] M.P. Halsall, J.J. Davies, J.E. Nicholls, B. Cockayne, P.J. Wright, Photoluminescence of CdS/CdSe wurtzite superlattices and evidence for type II behaviour modified by piezoelectric fields, J. Lumin 48–49 (1991) 735–739. [38] W. Khan, M. Shoaib, U. Shah, S.H. Khan, G. Murtaza, Ab Initio Investigation of the Structural, Electronic and Optical Properties of the Li2In2XY6 (X = Si, Ge; Y = S, Se) Compounds, J. Electron. Mater. 47 (2018) 566–576. [39] K.M. Wong, S.M. Alay-e-Abbas, Y. Fang1, A. Shaukat, Y. Lei1, Spatial distribution of neutral oxygen vacancies on ZnO nanowire surfaces: An investigation combining confocal microscopy and first principles calculations, J. Appl. Phys. 114 (2013) 034901. [40] M. Merabet, D. Rached, R. Khenata, S. Benalia, B. Abidri, N. Bettahar, S. Bin Omran, Electronic structure of (BP) n/(BAs) n (0 0 1) superlattices, Physica B 406 (2011) 3247–3255. [41] C. Ambrosch-Draxl, J.O. Sofo, Linear optical properties of solids within the full-potential linearized augmented planewave method, Comput. Phys. Commun. 175 (2006) 1.
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FP-LMTO study of structural, electronic and optical properties of wurtzite (CdS)n/(CdSe)n superlattices
T
M. Merabeta,b, S. Benaliaa,b, L. Djoudia,b, O. Cherefa, N. Bettahara, D. Racheda, , R. Belacela ⁎
a b
Laboratoire des Matériaux Magnétiques, Faculté des Sciences, Université Djillali Liabès de Sidi Bel-Abbès, Sidi Bel-Abbès 22000, Algerie University of Tissemsilt, Institute of Science and Technology, Tissemsilt 38000, Algeria
ARTICLE INFO
ABSTRACT
Keywords: C. FP- LMTO A. Superlattices D. Electronic properties D. Optical properties
In this paper, we have conducted a first-principles study of the structural, electronic and optical properties of (CdS)n/(CdSe)n superlattices (where n is numbers of monolayers) in the wurtzite phase (B4), using the Full-Potential Linear Muffin-Tin Orbital (FP-LMTO) method within the Local Density Approximation (LDA) technique, in order to describe the exchange correlation energy. The calculated electronic properties indicate that all (CdS)n/(CdSe)n superlattices configurations, possess a semiconductor behavior with same energy gaps. We have seen more carefully and accurately that the different superlattices configurations have no effect on the electronic properties; in particular, we did not observe any dependence between the band gap behavior and the used layers.
1. Introduction Wide-bandgap II–VI compounds are expected to be one of the most vital materials for high-performance optoelectronics devices such as light emitting diodes (LEDs) and laser diodes (LDs), which operate in the blue or ultraviolet spectral range. Additionally, the high ionicity of these compounds makes them good candidates for high electro-optical and electromechanical coupling [1]. The Cd-chalcogenides are the most important binary semiconductors used in the fabrication of optoelectronic devices [2,3] especially light-emitting devices in the short-wavelength region of visible light, because of their direct band gap. Unlike most III–V and II–VI semiconductors, CdX (X=S, Se and Te) compounds exist in both zinc blende (ZB) and wurtzite (WZ) structures or in mixed (ZB)/(WZ) phases [3,4]. Depending on the growth conditions, one of these two crystal structures can be stabilized either by epitaxial strain on proper substrates or on buffer layers, or by controlling the growth temperature. This additional structure freedom provides an opportunity for making more efficient and reliable devices by choosing the appropriate polytypism of the compounds. Due to the technological importance of these materials, the various experimental and theoretical efforts have been made to predict the fundamental properties of these materials in recent decades [5–11]. The current availability of advanced epitaxial technologies that can grow II–VI heterostructures at nanoscale sizes with distinctive layers has motivated researchers to examine their optical characteristics. The exploration of heterostructure semiconductors incorporating various layers tends to focus exclusively on the production of self-organized quantum wells (QWs) and superlattices (SLs) which they have a wide range of highly promising properties for optoelectronic applications. Semiconductor heterostructures are ultra-thin layered materials whose thicknesses are in the range of a few atomic layers. The
⁎
Corresponding author. E-mail address: [email protected] (D. Rached).
https://doi.org/10.1016/j.cjph.2019.05.026 Received 11 March 2019; Received in revised form 19 April 2019; Accepted 9 May 2019 Available online 24 May 2019 0577-9073/ © 2019 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
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Fig. 1. Full Potential Linear Muffin-Tin Orbital (FP-LMTO) method. Table 1 Parameters used in the calculations: number of plane wave (NPW), energy cutoff (in Rydberg) and the muffin-tin radius (RMT in atomic units). Configuration
(CdS)1/(CdSe)1 (CdS)2/(CdSe)2 (CdS)3/(CdSe)3
NPLW (Total)
13,128 26,262 39,374
Ecut(Ryd)
102.8657 102.7102 103.6485
MTS (ua) Cd
S
Se
2.494 2.497 2.485
2.431 2.424 2.414
2.459 2.472 2.458
presence of these ultrathin layers leads to the “quantum size effect” when the physical dimensions of the layers are comparable to the De Broglie wavelength of the charge carrier [12]. This new effect is of great interest in both fundamental physics and devices applications [13]. A superlattice is a periodic heterostructure formed by two types of semiconducting materials, one type that acts as a quantum well and the other acting as a quantum barrier. Similarities between the CdS and CdSe lattices and the suitability of their bandgaps for superlattice construction [14] have dictated the choice of these both semiconductors for this present investigation. Both CdS and CdSe semiconductors have a wide band-gap (2.42 and 1.76 eV at room temperature respectively). Also it has been observed [15] that luminescence from superlattices can be very strong, so that structures using semiconductors that emit light in the visible or near-visible region, are of special interest. Further, from a structural viewpoint, the system is interesting as the first reported hexagonal superlattice mixing system. In order to fully take advantage of (CdS)n/(CdSe)n superlattices properties for eventual technological applications, a theoretical investigation is necessary. For this purpose, we have conducted that investigation on the structural, electronic and the optical properties of the (CdS)n/(CdSe)n (n number of monolayers; n = 1, 2, 3) superlattices in the wurtzite (WZ) structure, using the fullpotential linear muffin-tin orbital (FP-LMTO) method within the density functional theory (DFT) as implemented in the new version Lmtart (Lmtart 8: Last updated September 2018) computer code. The organization of the paper is as follows. In Section 2, we have described the employed method and the details of calculations. Results and discussions of the structural, electronic and the optical properties of the (CdS)n/(CdSe)n (n number of layers; n = 1, 2, 3) superlattices in the wurtzite (WZ) structure are presented in Section 3. Finally, conclusions and remarks are given in Section 4. 2. Calculation methodology Based on the Density Functional Theory (DFT) [16,17], we have calculated the structural, electronic and optical properties of (CdS)n/(CdSe)n superlattices, (n is numbers of layers). We have employed the Full Potential Linear Muffin-Tin Orbital (FP-LMTO) method [18,19] to solve the Kohn–Sham equations [20,21] within the Local Density Approximation (LDA) technique [22,23] in order to describe the exchange correlation energy. In the FP-LMTO method, the crystalline space is divided into non-overlapping muffin-tin spheres (MTSs) centered at the atomic sites region, (also known as heart region), and the interstitial region (IR). The potential inside MTSs (r ≤ RMT) is assumed to be spherically symmetric, while in IR (r > RMT), it is assumed to be constant, where RMT is the radius of the MTSs, and r is a position (Fig. 1). The FP-LMTO method used in our study, [24] is based on the smooth Hankel functions [25] as the envelope function of the basis set. The smoothing radii and values (Hankel function decay parameters) were carefully adjusted to optimize an efficient basis set with one s, p, and d state for each Cd, S, and Se site. The smooth interstitial quantities are calculated using a fine Fourier transform mesh, and the Brillouin-zone integrations were carried out with a well-converged k mesh based on a uniform division of reciprocal space. In order to achieve energy eigenvalues convergence, the charge density and potential inside the MTSs are represented by spherical harmonics up lmax = 6. The self-consistent calculations are considered to be converged when the total energy of the system is stable at 10−6 Ry. The k integration over the Brillouin zone is performed using the tetrahedron method [26]. To avoid the overlap of atomic spheres, the MTSs radii for each atomic position are taken differently for each composition. We point out that the using of the fullpotential calculation ensures that the calculation is not completely independent of the choice of spheres radii. The values of the 463
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Fig. 2. Total energy in terms of cell volume of all (CdS)n/(CdSe)n superlattices.
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Fig. 3. Total energy in terms of c/a ratio of all (CdS)n/(CdSe)n superlattices.
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Table 2 Structural parameters and formation energies of all (CdS)n/(CdSe)n superlattices. Configuration
a0(a.u)
c/a
B0(GPa)
B0
EF (eV)
(CdS)1/(CdSe)1 (CdS)2/(CdSe)2 (CdS)3/(CdSe)3
8.0615 8.0676 8.031
1.6433 3.3010 4.9384
52.1921 57.04376 62.77754
4.54284 4.3107 4.20025
−0.75757 −1.38907 −1.91693
spheres radii (MTSs), number of plane waves (NPLW) and the plane waves cut-off (Ecut) values, used in our calculation, are summarized in Table 1. 3. Results and discussion 3.1. Structural properties Majorities of II–VI compounds and in particular CdS and CdSe, crystallize in the hexagonal structure, especially wurtzite phase. This structure is composed of two interpenetrating hexagonal compact hcp lattices, where, Cd atoms are occupying the position 1/3, 2/3, 0 and 2/3, 1/3, 1/2, and S or Se atoms are occupying the position 1/3, 2/3, u and 2/3, 1/3, u + 1/2, where u is the dimensionless internal parameter (Noted that literature search has failed to reveal a precisely determined value for u, and it is often taken to have the ideal value 3/8 in the case when the ratio c/a is equivalent to 8/3 [27]). Wurtzite (CdS)n/(CdSe)n superlattices are constituted by a sequence of alternating layers of nCdS and nCdSe respectively, in a well-defined growth direction. These layers form a periodic arrangement of quantum wells and potential barriers along the growth direction. In our study, we chose the axis [0001] as growth axis (z is taken to be perpendicular to the layers while x and y axes are parallel to them). The massive binaries have a Wurtzie phase, and the superlattices (CdS)n/(CdSe)n have a hexagonal symmetry. Quantum well superlattices (CdS)n/(CdSe)n have an interesting properties from a structural, electronic and optical viewpoint. The most interesting feature of these compounds is the presence of the quantum wells and potential barriers along of the structure. The possibility of penetration of the charge carriers in the potential barriers generates a coupling between quantum wells, which leads to a continuity of wave functions of electrons and holes according to conduction mini bands and valence bands along of the growth direction. The structural ground state properties of wurtzite (CdS)n/(CdSe)n superlattices in its ordered unit cell are determined by performing a self-consistent calculation of the total energy in terms of cell volume (Figs. 2 and 3); in order to calculate the lattice constants a, at ideal values of the ratio c/a (such that c/a ideal = 8/3 ). Then, we optimize the ratio c/a at aop. After that, and for accuracy, we use the optimized values, meaning, the lattice constants aop, and the ratio (c/a)op, for recalculating the ratio c/a and the lattice constant a of equilibrium cell. The equilibrium lattice constants a, bulk modules B and its pressure derivatives B’ are optimized by an adjustment of the total energy obtained in terms of cell volume (ETotal(V)) using the fitting Birch’s equation of state [28]. However, the c/a ratio is optimized by an adjustment of their total energies obtained in terms of c/a (E Total(c/a)) using a polynomial fitting. The results for the different structural parameters are grouped in the Table 2. According to the Table 2 analysis, we have noticed that the calculated equilibrium parameters of all superlattices are constant; and this is due to the lack of growth in the direction of this axis, also the number of atoms in this direction remains the same in all three configurations. For c/a, we have noticed that in all configurations, this ratio is proportional to the number of layers, so that when the number of layers are increased, the c/a ratio increased evenly. This increase is due to the fact that this direction is following the growth direction of our superlattices. We note that the c/a ratio increases according to the relationship (c/a)nn = n(c/a)11. This proves that when the number of layers increases, the mesh increases uniformly following the growth direction. 1. The superlattices (CdS)n/(CdSe)n crystallize in the wurtzite structure (CdS and CdSe are energetically favorable to the wurtzite structure) where meshes contain 4, 8 and 12 atoms for the (CdS)1/(CdSe)1, (CdS)2/(CdSe)2 and (CdS)3/(CdSe)3 configurations respectively. We can clearly see from the Table 2 that the calculated bulk modules of our superlattices increases with the increasing of the number of layers, which suggests the same increasing for the compressibility of each compound. These compounds became harder when the number of layers increase. It represents bond strengthening or weakening effects induced by changing the composition [29]. In order to test if (CdS)n/(CdSe)n superlattices can be synthesized and form a stable phase experimentally, the formation energy Ef were calculated. The formation energy Ef per formula unite for all superlattices is calculated as follows [30]:
Ef = E [(CdS )n (CdSe)n]
n {E [(CdS )1 (CdS )1]+E [(CdSe )1 (CdSe )1]} 2
The formation energy calculation is presented in Table 2. From the Table 2 analysis, it can be found that formation energy for these superlattices have negative values, which imply that these superlattices phases are energetically favorable. Thus, these values guarantee the physical stability of these compounds against their decomposition. Also the negative values reflecting the formations of these superlattices are exothermic and also can be synthesized experimentally in laboratory conditions. 466
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Fig. 4. Band structures along high-symmetry directions of all (CdS)n/(CdSe)n superlattices.
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Table 3 Bandgaps values of all (CdS)n/(CdSe)n superlattices. Configuration
Eg (eV )
Gap nature
(CdS)1/(CdSe)1 (CdS)2/(CdSe)2 (CdS)3/(CdSe)3
0.619529 0.614942 0.625481
Direct Direct Direct
3.2. Electronic properties 3.2.1. Band structure and electronic behavior The band structures along high-symmetry directions of the superlattices are calculated at equilibrium lattice constants, with LDA approach, and they are illustrated in Fig. 4. In our superlattices configurations, we have noted that the CdSe plays the role of a quantum well, while the CdS plays the role of a potential barrier. In addition, the alignment type of this quantum wells superlattices belong in the type I alignment [31]. Type I superlattices correspond to the case when a large gap semiconductor is brought into contact with another small gap semiconductor, where the extremes of the conduction and valence bands are located in the same layer (same semiconductor). The charge carriers (electrons and holes) are confined in the small gap material that forms the quantum well embedded in the larger gap material that forms the potential barriers, therefore, the electrons and holes are trapped in the same material, so, their recombination will be important and the distribution of the forbidden band differences between the valence band and the conduction band depends on the nature of the both different used materials [32–34]. According to Fig. 4 analysis, it shows a similar electronic variation for all superlattices. The band structures clarify a band gap in the vicinity of the Fermi level, which is a semiconductor nature. Consequently, these compounds have a semiconductor behavior with same energy gap values. It is also observed that the band structures of all superlattices configurations possess a direct band gaps, since the energies gaps (Eg) are situated at the Γ Brillouin zone point (the valence band maximum (VBM) and the conduction band minimum (CBM) are located at Γ Brillouin zone point). The band structure of SL33 is more numerous than SL22, which is more numerous than SL11, due to the increasing of the layers in each time. The calculated gap values of all our superlattices are shown in Table 3. We have concluded that the number of layers have no effect on the electronic properties. 4. It is clear that the calculated gap values obtained by LDA are small to compare to 2.42 and 1.76 eV of CdS and CdSe. This underestimation of the gap values is mainly due to the fact that the LDA does not take account the energy independence of the quasi-particles correctly [35]. Also it's due to the internal electric fields, which they have strongly modified the band structure [36]. This system (CdS/CdSe) is the first reported semiconductor superlattice with this crystal structure and it possesses new behavior because of the existence of extremely large piezoelectric fields within the strained layers; these fields drastically modify the energy band structure [37]. 3.2.2. Density of states In order to understand the electronic properties and the origin of the band structure of all superlattices, we have calculated their total and partial density of states (TDOS and PDOS) using LDA approach. The corresponding densities of states obtained are represented in Fig. 5. By analyzing this figure, we can say that the TDOS and PDOS have presented a similar behavior for all superlattices. The densities of states of the valence and conduction bands of our superlattices are characterized by two regions separated by gaps. Following always Fig. 5, it shows that the range between −6 and −4.6 eV have no DOS. The first part of the first region in the valence band in the range −4.6 to −3 eV usually comes from the p(S and Se) states, and the s(Cd) states with a small contributions of d(Cd) and p(Cd) states; The second part of the first region in the valence band in the range between −3 eV to −0.3 eV (close to the Fermi level), DOS originates from the p(S and Se) and hybridization of s, p and d states of the Cd atoms. At Fermi levels, the compounds have no DOS, which demonstrate the semiconductor behavior of these materials. The second region is the region in the conduction band, this region for all configurations has the same contribution in the total density of states. This contribution is formed by hybridization of s, p and d states of the three atoms (Cd, S and Se). 3.3. Optical properties The optical properties of the material can be described by the complex dielectric function ε(ω), which represents the response of a system to an external electromagnetic field. Indeed, during the interaction of a material with a wave, the response of the material is governed by its complex dielectric function (or permittivity) ε(ω). Knowing the real and imaginary parts of the dielectric function allows the calculation of different functions and also makes it possible to predict the behavior of the wave inside the material. The complex dielectric function can be expressed as: ε(ω) = ε1(ω) + iε2(ω) [38,39], where ε1(ω) and ε2(ω) are the real and imaginary parts of the dielectric function, respectively. Generally, the contributions of ε(ω) result from intra-band transitions and inter-band transitions. The contribution of intra-band transitions is important only in the case of metals [40]. The inter-band transitions can be direct or indirect. In our calculations, we will ignore the indirect transitions that involve diffusion by a phonon where the contribution of ε(ω) is small compared to direct transitions. We have discussed the optical properties of a material that must be studied in order to determine its potential utility in optoelectronic applications. For this reason, we have selected all materials of our study since they showed a direct band gap character. The 468
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Fig. 5. Total and partial density of states of all (CdS)n/(CdSe)n superlattices.
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Fig. 6. Calculated dielectric functions (real and imaginary parts) for (CdS)n/(CdSe)n superlattices at direct band gap.
real part ε1 of the dielectric function can be determined from the imaginary part ε2 by the Kramers–Kronig relation. Ambrosch-Draxl and Sofo [41] have already presented a detailed description of the calculation of optical properties. To calculate the optical spectra of the dielectric function, ε(ω), a dense mesh of uniformly distributed k-points is required. Therefore, the Brillouin zone integration was performed with 600 k- points in the irreducible part for all superlattices without broadening. The imaginary and real parts of the dielectric functions for all superlattices in the energy range [0–13.6] eV are illustrated in Fig. 6. As can be seen from this figure analysis, the optical spectra show a considerable anisotropy between the ordinary components ɛ⊥ and the extraordinary components ɛ∥. The analysis also showed, for the curves of the imaginary part of the dielectric functions ε2 for all configurations, that the first critical point of the dielectric function occurs at the energy of 0.68029 eV. This value corresponds to the electronic transition value (Гv→Гc), and it represents the fractionation Гv→Гc which gives the threshold of the direct optical transitions between the highest state of the valence band and the lowest state of the conduction band. This is also known as the fundamental absorption edge. Beyond 470
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these points, the curve increases rapidly. This is due to the fact that the number of points contributing to ε2 increases sharply. The main peaks of 2 for SL11, SL22 and SL33 are about 4.725984, 4.725984 and 6.394726 eV respectively, while, the main peaks of 2 for SL11, SL22 and SL33 are equal, and they are located at 6.304726 eV for all configurations. For the real part of the dielectric function of all configurations, their spectra begin by the static dielectric constants which they are located at 3.892359. The extraordinary component 1 vanishes three times at energies 4.710059, 5.372491 and 6.4422255 eV for all configurations, while the ordinary component 1 vanishes once at energy 6.326179 eV for all configurations. The minimum of ε1 spectra corresponds to the energy 7.931364 eV for the extraordinary components 1 and also for the ordinary components 1 . Then the spectra of ε1 slowly increase to zero at high energy for all configurations. We have concluded that the number of layers used has no effect on the real parts of the dielectric functions for all configurations, and they have an important effect on the extraordinary component of the imaginary part of the dielectric function 2 in the SL33. The Fig. 5 analysis shows clearly that the critical points values do not change with the number of layers for the real parts of the dielectric functions for the three configurations, while the imaginary parts of the dielectric functions have a different values for the critical points of the extraordinary components 2 of SL11 and SL22 compared to SL33. 4. Conclusion In this work, we carried out a detailed study of the structural, electronic and optical properties of (CdS)n/(CdSe)n superlattices using the FP-LMTO method.
• The calculated lattice constants of all superlattices are almost equal; where c/a ratios are proportional to the number of layers. The • • •
calculated bulk modules of our superlattices increases with the increasing of the number of layers, which suggests the same increasing for the compressibility of each compound. For the electronic proprieties, these compounds have a semiconductor behavior with same energy gaps values. It is also observed that the band structure of all superlattices possesses a direct band gap. The real part of the dielectric functions is the same for all configurations, while the extraordinary component 2 of the imaginary part of the dielectric function is the same for SL11 and SL22, and different for SL33 due to the number of layers used in this configuration. We did not observe any dependence, either between the band gap behavior and the number of layers used, or between the real part of the dielectric function and the number of layers used, where this last have an important effect on the extraordinary component 2 of the imaginary part of the dielectric function of SL11 and SL22 compared to SL33.
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