Electronic structure and optical properties of (BeTe)_n/(ZnSe)_m superlattices

Electronic configuration of superlattices (BeTe)_n/(ZnSe)_m in the ground state is Be:[He] 2s², Te:[Kr] 4d¹⁰ 5s² 5p⁴, Zn:[Ar] 3d¹⁰ 4s² and Te:[Se] 4d¹⁰ 4p⁴ 4s². As a first step, we determined the structural properties of the binary compounds in the zinc-blende (ZB) structure. The lattice constant (a₀) was obtained by fitting the total energy as a function of volume to Birch’s [12] equation of state. We obtained the lattice constant, the bulk modulus and its pressure derivative from this numerical fitting procedure. The calculated structural parameters of binary compounds for LDA are presented in Table 2, along with the previous theoretical calculations and experimental data. Inspection of Table 2 shows that our LDA results are in reasonable agreement with experimental and available theoretical values. In the second step, we were interested in the quantum well superlattice consisting of binary compounds. In our case, (ZnSe) played the role of the barrier while (BeTe) acted as the well.

In the present work, the superlattices consist of a sequence of alternating n and m layers of ZnSe and BeTe along a specified growth direction. X-axis as the growth axis and supperlattice (m, n) with tetragonal symmetry have been chosen (n is the number of ZnSe monolayers, and m is the number of BeTe monolayers). The superlattice SL(1,1) is made up of alternate monolayers of BeTe and ZnSe, each monolayer containing two atoms; this alternation increases with m and n. The elementary cell volume is proportional to the number of monolayers. For the considered structures, we have performed structural optimization of the parameter of the host compound by minimizing the total energy with respect to the cell parameters. The total energies calculated as a function of the unit cell volume are fitted to the Birch-Murnaghan equation of state [12].

The deduced results are illustrated in Table 3. The data show that the SL11 lattice parameter interpolates the bulk material values, giving a good epitaxial interface and minimizing the creation of defects; for the other SL with m = n, the lattice parameter increases with the number of layers used. Finally, for n ≠ m, we observe a slight difference of the lattice parameter, especially when the number of BeTe layers increases. This difference is due to the atomic radius of the compound, which causes compression of the superlattice.

For the binary compounds and their super-lattices, the electronic band structures have been calculated at the equilibrium lattice. The most prominent features of the calculated band gaps are shown in Table 2 and Table 4. It is interesting to note that BeTe has an indirect band gap with the valence band maximum at G and conduction band minimum at X, whereas ZnSe has a direct band gap with the valence band maximum (VBM) and conduction band minimum (CBM) at G. The calculated band gaps for the binary compounds are in good agreement with the available theoretical results. We can see also that, due to the approximations used in our work, the band gap values are underestimated compared with the experimental data.

In Fig. 1 and Fig. 2, we show the calculations of band structures for superlattices at m = n and m ≠ n. Examination of these figures shows that they exhibit both types of band gaps: for m = n superlattices there is a direct band gap at Γ whereas for m > n, the superlattice band gap is indirect, with VBM located at G and CBM at R (except for m = 4 and n = 2, where the gap is direct but is close to G-R). The values of these gaps are 1.427 and 2.10 eV, respectively. No previous superlattice results are available for comparison; from the binary compound data, we conclude that the underestimation of band gaps by LDA also applies to superlattices. This underestimation is due to strongly correlated 3d electrons of Zn.

Fig. 1

Fig. 2

To describe the number of states that are available to be occupied by electrons per interval of energy at each energy level, we have also calculated the total and the partial densities of states (DOS) of these compounds, as displayed in Fig. 3 and Fig. 4.

Fig. 3

Fig. 4

Inspection of Fig. 3 shows that there are differences in the conduction band densities of states of the two binary compounds, with 3 regions present in the ZnSe DOS and two regions present in the BeTe DOS. This difference is due to the existence of the middle region dominated by the 3d-Zn states of the ZnSe compound; by contrast, the Be d sates are empty for the BeTe compound.

Examination of Fig. 4 shows that the topology of the superlattices densities of states is the same as that of the ZnSe binary compound. Two examples are shown in Fig. 4: m = n =1, 2, 3 and m ≠ n with m = 3 and n = 1. The lower occupied bands, located between ∼ 13 and − 10.8 eV, are primarily formed by the Se 4s and Be 2s states, with small contributions by the Zn 4s/4p/3d states. The states in the middle region between ∼ 7 and − 6 eV occur mainly because of the Zn 3d states, with small contributions due to the other Se and Zn states and the Te 5p/5d states. The states in the last region between ∼ 5.8 eV and the Fermi level are dominated by the Te 5p and Se 4p states. Hybridization between these states is apparent in this region. In the conduction band, the dominant contribution to the band structure from 2 to 10 eV varies depending on the relative contributions of the states of all the elements, with strong hybridizations between the Te 5p and Te 5d states, as well as the Se 4p and Se 4d states.

In this section, we discuss the optical properties of a material that must be investigated to determine its potential usefulness in optoelectronic applications. For this reason, we only chose the materials that showed a direct band gap character in our LDA study. When examining the optical response of the compounds under investigation, it is convenient to take into account the transitions of electrons from the occupied energy bands to the unoccupied energy bands, particularly at the high symmetry points in the Brillouin zone. The real part ε₁(ω) of the dielectric function can be determined from the imaginary part ε₂(ω) by the Kramers-Kronig relationship. A fully detailed description of the calculation of the optical properties was presented previously by Ambrosch-Draxl and Sofo [32].

To calculate the optical spectra of the dielectric function, ε(ω), a dense mesh of uniformly distributed k-points is required. Hence, the Brillouin zone integration was performed with 400 k-points in the irreducible part of the Brillouin zone for SL(1, 1), SL(2, 2), SL(3, 3), SL(1, 3), SL(2, 4), SL(4, 2), and SL(1, 5) without broadening. Using the optical relationships described above, we have calculated the real and imaginary components of the frequency dielectric function and then used these functions to determine the refractive index n(ω) and reflectivity R(ω). Due to the similarities of the topology of the optical properties of SLs, in Fig. 5 we only present the real and imaginary parts of ε(ω) for SL(1, 1), SL(3, 3), SL(4, 2) and SL(1, 5).

Fig. 5

To illustrate a minor difference in the description of these properties, the dielectric function ε₂(ω) exhibits a structure that varies depending on the composition of the superlattice; thus, we can see that the ε₂(ω) peaks depend on the ZnSe compound layers. The threshold for direct optical transitions for the Γ-Γ band gap is between the valence band maximum and the conduction band minimum. Beyond these thresholds, the ε₂(ω) curve rises rapidly because the number of the points contributing to ε₂(ω) increases abruptly. The main peaks in the spectra are located between 4.82 and 5.73 eV. The real part ε₁(ω) of the frequency-dependent dielectric function was obtained according to the Kramers-Kronig dispersion relation and is displayed in Fig. 5. We note that peak intensities in these spectra typically occur between 3.86 and 3.96 eV, i.e between the peaks of the binary compounds located at 3.334 and 4.38 eV for ZnSe and BeTe, respectively. Subsequently, ε₁(ω) becomes negative between 6.75 and 8.06 eV, depending on the superlattice configuration.

In Fig. 6 and Fig. 7, the refractive index n(ω) and the reflectivity spectrum R(ω) are plotted for several different superlattices (SL(1, 1), SL(3, 3), SL(4, 2) and SL(1, 5)). The optical spectra of the superlattices are similar. From our examination of the reflectivity spectra of the superlattices, we note that R(ω) increases by up to 70 % and then starts to decrease at approximately 26 eV. This result suggests that these superlattices behave like semiconductors. The static dielectric constant ε₁(0) is given by the low energy limit of ε₁(ω). Note that we do not include the phonon contributions to dielectric screening, and ε₁(0) corresponds to the static dielectric constant of Ȉ 2.01. To our knowledge, there are no available experimental or theoretical results for the optical properties of these superlattices, so the present work can be considered to be a predictive study.

Fig. 6

Fig. 7