{'fraction_non_alphanumeric': 0.027101078086842575, 'fraction_numerical': 0.006492346179045804, 'mean_word_length': 4.685743311886218, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Electronic structure of and Quantum size effect in III-V and II-VI semiconducting nanocrystals using a realistic tight binding approach

We analyze the electronic structure of group III-V semiconductors obtained within full potential linearized augmented plane wave (FP-LAPW) method and arrive at a realistic and minimal tight-binding model, parameterized to provide an accurate description of both valence and conduction bands. It is shown that cation sp3 - anion sp3d5 basis along with the next nearest neighbor model for hopping interactions is sufficient to describe the electronic structure of these systems over a wide energy range, obviating the use of any fictitious s* orbital, employed previously. Similar analyses were also performed for the II-VI semiconductors, using the more accurate FP-LAPW method compared to previous approaches, in order to enhance reliability of the parameter values. Using these parameters, we calculate the electronic structure of III-V and II-VI nanocrystals in real space with sizes ranging upto about 7 nm in diameter, establishing a quantitatively accurate description of the band-gap variation with sizes for the various nanocrystals by comparing with available experimental results from the literature.

of site and angular-momentum specific X-ray emission and absorption techniques, it is also possible to study the partial density of states (PDOS). Hence, the need for a physically sound, minimal model without any fictitious orbital and supported by accurate parameterization is required to be able to provide realistic descriptions of both valence and conduction bands in contrast to simulating only the bandgap of these semiconductors. Our attempts 9,12 in this direction on the II-VI semiconductors suggest that much of the difficulties arise from inaccurate parameterization of the bulk band structures. In order to overcome these difficulties, we carried out a detailed analysis of the bulk band structure obtained within the highly accurate FP-LAPW method, supported by the recently developed new generation Muffin-Tin Orbital (NMTO) method 15 to obtain physical, realistic and minimal model with accurate parameterizations. Such a method not only obviates the need for the fictitious s * orbital, but has also been found to explain quantum confinement effects quite well in the II-VI semiconductors. 10,11,12 Though the synthesis of high quality nanometric sized II-VI semiconductors is already very well established, the synthesis and studies of high quality III-V semiconducting nanocrystals are being increasingly reported in the recent literature. 16,17,18,19,20 III-V semiconductors provide a material basis for a number of already existing commercial products, as well as new cutting edge electronic and optoelectronic devices, like heterostructure bipolar transistors, diode lasers, light emitting diodes, electro-optic modulators 21 and in biology, as fluorescent labels. 22 Hence, it becomes necessary to have an electronic structure model with accurate predictive abilities to describe the quantum confinement effects in these nanocrystals.
In order to achieve this, we study the band dispersions as well as the DOS and PDOS obtained from the ab-initio Full Potential Linearized Augmented Plane-Wave (FP-LAPW) method 23 to establish the relative importance of various orbital degrees of freedom involved in describing the valence as well as some of the low-lying conduction bands in various III-V semiconductors. In order to identify the dominant hopping interactions, we employ the muffin-tin orbital (MTO)-based NMTO technique 15 which provides an unique scheme to derive first-principles TB Hamiltonian starting from full Local Density Approximation (LDA) calculation. The usefulness of this method has been demonstrated in a number of cases. 24 With inputs from these ab-initio methods, we construct a minimal TB model and carry out a least-squared-error minimization procedure to fit the TB dispersions to the ab-initio ones, thereby defining the values of the TB parameters. Using the TB parameters thus obtained, we carry out a real space calculation using Lanczos algorithm for different sizes of the nanocrystals to obtain the dependence of the electronic structure on the size of the nanocrystals. We compare these calculated results with the experimentally determined bandgaps of different nanocrystals as a function of size; the excellent agreement in each case establishes the validity of the present approach over the entire range of nanocrystal sizes. We have also carried out a similar analysis on all the II-VI semiconductors starting with the more accurate FP-LAPW results as inputs compared to the previous approach. 9 As these results are found to be slightly different and possibly more accurate in comparison to the earlier results, the new parameter values for the II-VI series are also reported here.

II. METHODOLOGY:
Ab initio band structures of all the III-V as well as II-VI compound semiconductors with the zinc blende structure were obtained using the FP-LAPW method. Self consistency was achieved using 30 k-points in the irreducible Brillouin zone. In order to obtain a realistic TB model, we first calculated the band dispersions and density of states (DOS). The atomic orbital contributions to the valence and conduction bands in the band dispersions and the DOS were determined in terms of orbital-projected band-structure, the so-called fatbands and the partial densities of states (PDOS), respectively. Analysis of these results establishes the minimal orbital basis for the TB model. To obtain a guideline for the range of relevant hopping interactions necessary for reliable descriptions of the valence and conduction bands, we carried out NMTO calculations which provide TB Hamiltonians derived in a first-principles way by constructing atom-centred, short-ranged Wannier orbitals, namely the NMTOs.
The TB Hamiltonian is given by where a † il and a il are respectively the creation and annihilation operators for electrons at the atomic site, i in the l th orbital. The onsite energy for the orbital l at the site i is given by ǫ il . The hopping interaction strengths t ll ′ ij depend on the type of orbitals and geometry of the lattice and are parameterized using the Slater-Koster parametrization scheme. 13 We start with the estimates of the values of onsite energies and the hopping integrals obtained from NMTO derived TB Hamiltonian and then carry out a least-squared error minimization fitting procedure at a number of high symmetry points in the band dispersion curves to fit the band dispersion obtained from FP-LAPW method. The parameters thus obtained are used for calculations of the electronic structure of nanocrystals.
We generated the clusters consisting of a central anion surrounded by the four nearest neighbor cations, followed progressively by alternate shells of anion and cations, similar to our previous studies on the different II-VI semiconductors, 11,12 and on Mn doped GaAs. 25 The effective diameter, d, of the nanocrystal is calculated assuming that the particles are spherical in shape using the formula where a is the bulk lattice parameters and N at is the number of atoms in the nanocrystal.
The largest nanocrystal for which the DOS was calculated has N at ∼ 10,000 atoms and d ∼ 7.5 nm containing approximately 65,000 orbitals. As it is virtually impossible to perform a complete diagonalization of such a large matrix, we obtain the eigen-value spectrum using the Lanczos algorithm. 26 We passivate the clusters with hydrogen atoms at the outermost layer to remove the dangling bonds and obtain the eigen-value spectrum for clusters with different sizes. From the eigen-spectrum, the top of the valence states (TVS) and the bottom of the conduction states (BCS) are obtained and the bare bandgap is calculated as the difference between them. However, the bare bandgap cannot be directly obtained from experimental data, usually based on optical absorption spectra, due to the presence of the excitonic peak close to the absorption edge. 6 Hence, we compare the theoretically obtained bandgap of a nanocrystal compared to that of the bulk, we do not attempt to correct the bandgap artificially, implicitly assuming that the errors in estimating the absolute bandgaps cancel out to a large extent. This assumption turns out to be a reasonable one, as will be shown later in the text, for the present series of III-V compounds and also for the II-VI compounds. 11 In order to understand the orbital contributions to the bulk band structure, we obtain contribution of each of the orbitals to the band wave-functions at each of the energy and momentum points, shown in terms of fatbands, in Fig. 1(b)-(f). In these panels, though the band dispersions are same as in Fig. 1(a), only the fatness associated with each band varies, with the size of the circles indicating the amount of the particular orbital character for that band at that k-point. For example, it can be seen that the lowest conduction band between 0 and 3 eV has contributions mainly from Ga-s ( Fig. 1(b)), with substantial mixing from As-s ( Fig. 1(d)) and As-p ( Fig. 1(e)) states. However, main part of As-s appears as nearly flat band at an energy of −11 eV ( Fig. 1(d)). From Figs. 1(b) and (e), it can be seen that the three strongly dispersive valence bands between −7 eV and 0 eV are made up of a mixture of mostly As-p and Ga-s orbitals. Though mostly prominent in the highly dispersive bands above the lowest conduction band, Ga-p contributions to some parts of the lowest conduction band (e.g. along Γ-X and near the X point) as well as to the top two valence bands are not negligible ( Fig. 1(c)). Hence, it is necessary to include at least the sp 3 orbitals of both Ga and As in the basis. Also the contribution of As-d ( Fig. 1(f)) to the bands of interest, though not very prominent, cannot be neglected, especially in the conduction band region.
Hence, we also included As-d in the basis. The contribution of Ga-d (not shown in the figure) however is virtually absent in the bands of interest, suggesting a Ga sp 3 -As sp 3 d 5 as the suitable basis, in contrast to the previously supported 7 sp 3 d 5 s * − sp 3 d 5 s * basis.
While the atomic orbital contributions to the band wave-functions along the high symmetry directions in terms of the fatbands provide a clear suggestion for the suitable basis for the system, this can be further supported by the analysis of the atomic orbital contributions to the overall momentum averaged electronic structure in terms of various PDOS shown in and p and the As-s, p and d, though the As-d states lie high in energy. This inclusion of As-d in the first step ensures that we use a realistic value of As-d for the fitting. In the next step, we fix the As-d on-site energy and re-optimize the parameters to fit the lowest 8 bands, in order to provide the most accurate description of the relevant valence and conduction states, primarily arising from the s and p orbitals of Ga and As. The best fit obtained this way is shown in Fig. 3(a). From the figure, it can be seen that though the basic features of the valence and the low-lying conduction bands are captured in this approach, there are too many important and gross discrepancies, such as, the curvature of the lowest conduction band at the Γ point, in the TB results compared to the ab-initio results. We have highlighted these discrepancies by boxes marked around such discrepancies in the figure.
These significant discrepancies suggest that the model adopted here misses out on some important interactions, thereby lacking the desired level of accuracy. Such problems have often prompted other groups to increase the basis, for example by the ad-hoc inclusion of an s * orbital on the cationic site. 7 However, this does not remedy the limitations of the model, as confirmed by us by obtaining the best description TB dispersions with Ga sp 3 s * -As sp 3 d 5 basis in comparison to the ab-initio approach; this comparison is shown in Fig. 3(b).
In order to obtain an insight into the possible origin of these discrepancies, we carried out analysis based on NMTO calculations that provide us a systematic and ab-initio way As we have a new set of parameter values (Table II) with a slightly different TB model for the II-VI series compared to the earlier report, 9 we have carried the electronic structure calculations for the nanocrystals of all these II-VI compounds also, for the sake of completion.
We find that the new results are in good agreement with the experimental data reported in Ref. [11] earlier. The variation of bandgap in the II-VI semiconductors are also fitted using the same expression ∆E g = 1/(ad 2 + bd + c) and the values of the fitting parameters are shown in Table IV.
In view of the recent experiments on II-VI semiconductors using high energy spectro- states are related inversely to the corresponding effective masses; in other words, the shifts in the valence and conduction states are controlled by 1/m * h and 1/m * e , respectively. Since the m * e is significantly larger than m * h for the III-V compounds, the conduction band is affected more pronouncedly compared to the valence band with a change in the size. This argument is valid only for the larger size limit where effective mass, determined for the bulk material remains to be a relevant quantity even for the nanocrystals. However, in the smaller size regime the trend appears to be reversed in several cases, such as in GaAs (Fig. 8 (f)), with the shift in the BCS being less than that in the TVS. This change in the behavior with size can be understood in the following way. At large sizes, the BCS is defined by the states belonging to the Γ point of the bulk band structure (see Fig. 6 (f) for GaAs) which has a low effective electron mass, being dominantly contributed by the Ga-s states. However, a rapid upward movement of these states with decreasing size inevitably makes BCS to be contributed primarily by the states belonging to X point of the bulk band structure (see Fig. 6(f)) which evidently corresponds to a larger effective mass, being primarily contributed by the Ga-p states and having a relatively flat dispersion. Therefore, this leads to a relatively less pronounced change in the BCS in the smaller size regime.

IV. CONCLUSIONS:
In this work, we present a systematic development of parameterized tight-binding model for an accurate description of the electronic structure of group III-V semiconductors. We analyze the nature and origin of bonding as well as the atomic orbital contributions to each band eigen-states to arrive at the necessary minimal model involving sp 3 orbitals at the cationic sites and sp 3 d 5 orbitals at the anionic sites, obviating the use of any fictitious s * orbital in the basis. We find that though the nearest-neighbor-only model provides an approximate description of the ab initio band dispersions over a wide energy range, it is necessary to include both the cation-cation and anion-anion second nearest neighbor interactions to obtain a satisfactorily accurate description of ab-initio band dispersions. We have also performed similar analysis for the II-VI semiconductors using the more accurate FP-LAPW ab-initio band structure in contrast to the previously used LMTO method. Using these optimized parameters, we perform real space calculations with the same tight binding model to obtain the variation of the bandgap as a function of the nanocrystal size. A comparison with the available experimental data of the bandgap variation with the size of these nanocrystals exhibits good agreement over the entire range of sizes; in sharp contrast to the results obtained with the EMA. We have also compared the present results with other calcu-lations and we find that the present results give a better description, wherever these differ.
A similar analysis was also carried out on the II-VI semiconductors using the newly obtained parameters and the calculated bandgap variation is are found to match well with existing experimental values. Ideally one would like to extend similar parameterized tight-binding Hamiltonian approach, not only for an accurate description of the electronic structure of such systems, but also to describe the cohesive energy and geometry optimization; this will, however, require an accurate description of the ionic contributions of the total energy along with the electronic contributions that has been modeled here.       Fig. 2(a), the thin solid line, dotted line and the dashed line represents PDOS of Ga-s, Ga-p and Ga-d orbitals respectively. In Fig. 2(b), the thin solid line, dotted line and dashed line represents the PDOS of As-s, As-p and As-d respectively.   Table I.