Abstract
1. Introduction
Many semiconductors such as CdTe, ZnO, AlN and their related II–VI and III–V compounds would not be very useful without being doped[
In consideration of the full-electronic potential, linearized augmented plane wave (LAPW)[
For systems with strong electron interaction, the calculated values are not accurate and we usually adopt LDA + U[
In this paper, we systematically calculate the lattice constants, model structures, electronic band structures, formation energies and transition energies of defects in CdTe by different calculation methods. The target is to find out a method which can reflect the physical properties of defects better in CdTe. For most of the cases, LDA + U gives better results than the standard LDA. Since this demands a low computing power relative to LAPW, the LDA + U approach is an attractive means to study the supercells with large number of atoms and strong electron interactions.
2. Method of calculation
In this paper, DFT was used to perform all the structural optimizations, energy band structures and total energy calculations. The exchange-correlation functional was handled by both standard LDA and LDA + U methods and the VASP code was used in all cases[
It is well-known that formation energies of the defects can reflect the degree of difficulty or ease commendably during the defect formation in bulk materials. The larger the formation energy is, the more difficultly the defect forms, and vice versa. To determine the defect formation energy and defect transition energy levels, first, it is necessary to identify the defects with low formation energies for every position of the Fermi level and every possible stoichiometry. Second, we need to calculate the total energy
where ni is the number of elements which constitute the supercell, which added to (ni> 0) or taken from (ni< 0) the bulk crystal to create the defect, andq is the number of electrons which transfer from the supercell to the reservoirs while forms the defect cell; μi is the atomic chemical potential with constituenti referenced to elemental solid or gas with Ei; Ev is the valance-band maximum (VBM); μe is the electron chemical potential which was measured relative to the VBM Ev. The last term (μe + Ev) represents the electron Fermi energy Ef.
In addition, the transition energy
The calculation results are sensitive to the k-points sampling, thus Γ-point-only approach[
For donor defect (q > 0), the ionization energy referenced to CBM is given by:
where
Under equilibrium growth conditions, there are some thermodynamic limits on the achievable value of the atomic chemical potentials μi in Eq. (1). By means of some certain conditions, we can obtain the formation energy simply without using the atomic chemical potentials μi, then after deducing from Eqs. (1) and (2), the formation energy of charge state is given by:
where
3. Results and discussion
There is a slight discrepancy on the lattice constant for undoped CdTe by different calculation methods, and the result is given as follows: it is 6.421 Å by standard LDA, and 6.408 Å by LDA + U, and 6.541 Å by LAPW[
It is widely-believed that band gap is always considered as the most important aspect to describe the electronic properties of semiconductors for optoelectronic applications. LAPW predicts a band gap of 1.48 eV (1.59 eV at the experimental lattice constant), which agrees well with the experimental value[
3.1. Formation energies of the neutral point defects
The calculated defect formation energy
In this paper, the results of defect formation energy calculated by LAPW method are obtained from Wei’s work[
For interstitial defects, two cases always should be considered in the zinc-blende structure, these are Aia and Aic, which indicate the interstitial A atom is surrounded by anion atoms or cation atoms[
Figure 1.(Color online) (a) The stable structure is that the Te atom is at the center of the tetrahedron, and (b) a more stable structure is that the anion atom Te and two cation atom Cds are almost in a line.
For anti-site defects, it is obvious that the result of LDA + U is inferior to that of standard LDA for group-IB elements substituting the Cd site, XCdIB (IB = Cu, Ag, Au with d electrons and without considering plus U). However, LDA + U is distinctly superior to standard LDA when the group-IIIA elements substitute Te site, YTeIIIA (IIIA represents Br, In with d electrons without considering plus U). For Te vacancy and Cd vacancy, the results of LDA + U are closer to LAPW results than standard LDA as well. In general, we can make a conclusion that the majority of calculated formation energies by standard LDA and LDA + U are lower than LAPW.
3.2. Transition energies of the charged state defects
In this section, we discuss and analyze the calculated optical transition energies of charged state defects in CdTe. Figs. 2 and 3 describe the transition energy positions of CdTe, which are calculated by different methods. Similarly, Table 2 presents the calculated acceptor defect (q< 0) transition energies andTable3 gives the donor defect (q> 0) transition energies of tetrahedron CdTe. These results are acquired by using Eqs. (3) and (4). One can also deduce the formation energies of charged state defects from Eqs. (1) and (2). Meanwhile,k-point, core level, and electrostatic correction were also taken into consideration during the transition energies calculation.
Figure 2.(Color online) The acceptor defect (
Figure 3.(Color online) The donor defect (
Table 2 shows that the intrinsic defect Teia (0/–2) has a deep acceptor level above the VBM[
For the intrinsic defect VCd, which has relative shallow transition energy levels with respect to Teia. It is the most common intrinsic acceptor in CdTe, whose (0/–1) transition energy level is high with 0.13 and 0.173 eV by LAPW and LDA +U, while standard LDA gives a value of 0.091 eV which is small enough to reach high hole density at room temperature. After comparing with the experimental results, LDA + U might be, in this case, the worst method.
For extrinsic acceptor, the results of XCdIB (IB represents Cu, Ag, Au, Na) and YTeIIIA (IIIA represents Sb, As, P, N) by standard LDA and LDA + U indicate these defects all have relative shallow (0/–1) levels compared with Teia, and there is no apparent difference between these methods. As shown in Table 2, a distinct phenomenon is that most of the calculated (0/–1) transition energy results obtained by LDA + U are improved obviously relative to that of standard LDA, but some become worse slightly, such as AuCd, NaCd and BiTe, whose results of standard LDA are closer to that of LAPW. No matter how disparate they are, the properties of these defects have not changed. They are all shallow acceptors, except AuCd and BiTe, the transition energies of which are far above the threshold of 0.15 eV. While for other defects, it shows that both standard LDA and LDA + U results can reflect their defects properties well. For PTe and NTe, the calculated transition energies of (0/–1) by standard LDA and LDA + U are –0.034/–0.058 eV and 0.06/0.013 eV, respectively. There is not much difference between them with respect to the absolute shallow transition energy level 0.01 eV by LAPW. It can easily be deduced that PTe and NTe are suited to p-type doping. For some extrinsic defects which contain atoms (Cu, Ag, Au, Sb, As) with d electrons, the results are also improved, although these atoms only occupy ~1.56% in this model.
Above all, there is not much difference between the results of standard LDA and LDA + U, that is, their properties do not change. We also find that the approach of LDA + U, in most instances, is better to reflect the physical properties of the material with d electrons by comparing with standard LDA.
In the following, we will adopt the same way to calculate transition energies of acceptor defect and give the ionization energies of donor defect.
From Table 3, we find the native defects CdTe(+2/0) is a shallow level of 0.10 eV by LAPW in the band gap. Standard LDA gives a value of 0.01 eV and LDA + U shows a value of 0.05 eV, which indicates that LDA + U gives a better result. However, for other native defects with deep levels in the band gap, such as TeCd(+1/0), the results by different methods are 0.34, 0.345, and 0.625 eV, respectively, indicating that the result of standard LDA can reflect the case better. This conclusion is also true for Cdic(+2/0). From the data of VTe(+2/0) in Table 3, we find that all the methods show deep donors. LDA + U gives a value of 0.662 eV, because the atomic positions are sensitive to their charge states in negative U systems, such as CdTe, Cdic, and VTe. It can be explained that charge states influence the transition/ionization energy levels powerfully, that is, the electrostatic correction value affects the result with charge square.
For extrinsic impurity donor, the calculated (+1/0) transition energy levels of ACd (where A= Al, In, Ga) by LDA + U are at –0.038 eV (above the CBM), 0.347 eV and 0.316 eV (below the CBM). After comparing the results, our calculation suggests that gallium could be a good n-type dopant for CdTe rather than aluminum and indium. For BTe (where B represents I, Br, Cl or F), the calculated transition energy levels at 0.293, 0.341, 0.371, and 0.739 eV are gained by LDA + U, which indicates these defects all have deeper levels. Comparatively speaking, the iodine has the superiority to be as a good n-type dopant in CdTe, while Fluorine is the worst candidate for its very deep level.
For interstitial defects Cdic, Cui and Nai, from previous work, we find that the stable structures of them for (+1/0) are Cuia and Naic. LAPW indicates that both of them can be good n-type dopant candidates in CdTe, and the results by LDA + U are better than standard LDA.
4. Conclusion
In this study, the effects of two different calculation methods were investigated. We have systematically employed the methods of standard LDA and LDA + U to calculate the lattice constants, model structures, electronic band structures, the formation energies and transition energies of defects in CdTe, these results were analyzed and compared to the data of LAPW, which derived from other work[
After comparing the calculation results of LAPW with standard LDA and LDA + U, we can safely draw a conclusion that the approach of LDA + U not only improves the accuracy from standard LDA method but also reduces the computationally demanding with respect to LAPW method in calculation of defects in CdTe. Therefore, it has a huge potential to model supercells with a large number of atoms and strong electron interactions.
Acknowledgements
The authors gratefully acknowledge the financial support of the National Nature Science Foundation of China (No. 51902005) and Young Talent Fund of University Association for Science and Technology in Shaanxi Province of China (No. 20180507).
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