Ab Initio Calculation of Accurate Electronic and Transport Properties of Zinc Blende Gallium Antimonide (zb-GaSb)

This article reports the results of our investigations on electronic and transport properties of zinc blende gallium antimonide (zb-GaSb). Our ab-initio, self-consistent and non-relativistic calculations used a local density approximation potential (LDA) and the linear combination of atomic orbital formalism (LCAO). We have succeeded in performing a generalized minimization of the energy, using the Bagayoko, Zhao and Williams (BZW) method, to reach the ground state of the material while avoiding over-complete basis sets. Consequently, our results have the full physical content of density functional theory (DFT) and agree with available, corresponding experimental data. Using an experimental room temperature lattice constant of 6.09593 Ȧ , we obtained a direct band gap of 0.751 eV, in good agreement with room temperature measurements. Our results reproduced the experimental locations of the peaks in the total density of valence states as well as the measured electron and hole effective masses. Hence, this work points to the capability of ab-initio DFT calculations to inform and to guide the design and the fa-brication of semiconductor based devices—provided a generalized minimization of the energy is performed.


Introduction
Gallium antimonide (GaSb) is one of the semiconductor compounds of the III-V family, derived from gallium and antimony; its stable crystal structure is zinc blende. It is a direct gap semiconductor which has the possibility of being p-or n-type doped, with good mobility. It also has a significant electro-optical potential in the infrared domain [1]. GaSb has emerged in recent years as a very technologically attractive semiconductor, given its applications in high-efficiency thermo-photovoltaics, mid-infrared lasers, photodetectors, high-speed electronic devices, and non-linear optics. Such applications derive from several interesting properties of the material, such as the high hole mobility (850 -10,800 cm 2 /Vs), the low carrier effective masses and the small direct band gap value [2]. The importance of its technological applications makes it an extensively studied semiconductor, both theoretically and experimentally. Table 1  The calculations with ab-initio, generalized gradient approximation (GGA) potentials follow the LDA ones in the table. The resulting, calculated band gap values of zb-GaSb are not satisfactory. Indeed, of the first thirteen (13) ab-initio GGA results for the band gap, the four (4) are negative and the remaining 9 are zero. The following twenty-seven (27) results in the table, obtained with GGA ab-initio potentials, range from 0.06 eV to 1.015 eV. Five (5) of these results were obtained with the Engel and Vosko GGA which tends to overestimate band gaps. One of these results, 0.726 eV for an indirect band gap, disagrees qualitatively with experiment that reports a direct band gap.
The above ab-initio calculations are followed in the table by several calculations with ad hoc potentials. In particular, two (2) of these calculations, with an empirical pseudopotential approach, reported band gaps of 0.62 eV and 0.715 eV. Two (2) of the calculations used a version of the modified Becky and Johnson potential and eight (8)  calculations in this paragraph, which used ad hoc DFT potentials, have no predictive capabilities.
The theoretical results discussed above and shown in Table 1 disagree with each other and with the experimental values of the band gap. Seven (7) of these experimental values, in the last 14 rows of Table 1, range from 0.67 eV to 0.75 eV, for eight (8) measurements at room temperature; and seven (7) other values are from 0.78 eV to 0.822 eV for low temperature experiments. The apparent, experimental agreement on a band gap around 0.81 eV, for low temperatures, and around 0.73 eV, for room temperature, contrasts strongly with the case of theoretical studies with 24 negative values or zero for the band gap of zb-GaSb. The disagreement between the theoretical values of the ab-initio DFT calculations and between these values and the accepted experimental values is the main motivation for this work. The importance of a correct, calculated band gap resides in the need to produce accurate optical and dielectric properties, densities of states, and effective masses. These quantities, among others, cannot be correctly calculated using an incorrect, theoretical band gap.
Our motivation to resolve the discrepancies noted above is further supported by the fact that the previous works of our group accurately described or predicted properties of semiconductors, using ab-initio DFT potentials [57] [58]. This feat was made possible by our use of the Bagayoko, Zhao and Williams (BZW) method or its enhancement by Ekuma and Franklin (BZW-EF). Bagayoko explained that these methods seek out and reach the ground state of a material without using over-completes basis sets. To our knowledge, none of the calculations in Table 1 used successive, self-consistent calculations, with increasingly large, augmented basis sets, to reach verifiably the ground state of the material. The attainment of the ground state is required by the second DFT theorem and is essential for the results of a DFT calculation to have the full, physical content of DFT and to be consistent with experiment [57].
After this introduction, focused on the importance of the material and the previous theoretical and experimental studies, we describe in Section 2 our method of calculation and the details relating to the replication of our work. Section 3 is devoted to the presentation and discussion of our results for the band structure and the band gap, as well as the total and partial densities of states, and the effective masses of electrons and holes. A relatively succinct conclusion is set out in Section 4.

Computational Method
We used in this work the local density approximation (LDA) potential of Ceperley and Alder [59], as parameterized by Vosko, Wilk and Nusair [60]. We have applied the linear combination of atomic orbitals. Gaussian functions constitute the radial parts of these orbitals. We used a software package that was developed and refined over a decade at the US Department of Energy's Ames Laboratory, Ames, Iowa [61]. Our non-relativistic calculations utilized an experimental lattice constant, at room temperature. We applied the BZW method to perform the linear combination of atomic orbitals (LCAO). Our method allows the conco- The necessary and sufficient criterion for stopping the computation is to have three consecutive calculations that produce identical, occupied energies.
In accordance with the above, our calculations necessarily begin with a small basis set which accommodate all the electrons of the system under study. Calculation II follows, with a basis set comprising that of Calculation I plus an orbital representing an excited state. We compare the occupied energies of the two self-consistent calculations: invariably, some occupied energies from calculation II are lower than their corresponding values from calculation I. We augment the basis set of calculation II, with an orbital, to perform calculation III. We compare the occupied energies from calculations II and III. We continue this process until three consecutive calculations produce the same occupied energies, indicating that we reached the ground state. Among these three (3) consecutive calculations, only the first, which has the smallest basis set, provides the DFT description of the ground state of the material [57] [58] [63]- [69]. The basis set of this calculation is the optimal basis set. The optimal basis set, once self-consistent is achieved, leads to the ground state charge density of the material. Likewise, basis sets which are larger than the optima one, and which contain the optimal one, lead to the ground state charge density upon reaching self-consistency. As explained by our group in the publications referenced above and others, the use of these larger basis sets also lowers certain unoccupied energies: these lowered unoccupied energies are not due to a physical interaction in the Hamiltonian which has not changed from its value obtained with the optimal basis set. Incidentally, since the occupied energies do not change once, we reach the optimal basis set, the further lowering of some unoccupied energies because of using larger basis sets containing the optimum is one plausible explanation of the almost universal underestimation of band gaps and energy gaps by mainstream calculations [70]. Indeed, these calculations, to date, have used a single basis set which was deliberately chosen to be large in order to ensure completeness. Most often, these basis sets were over-complete for the description of the ground state [57].
Details relevant to the replication of our work follow. The crystal structure of GaSb is Zinc Blende [1]. We have used a measured value of the lattice constant at room temperature, 6.09593 Å. The self-consistent calculations for Ga 3+ and
Various characteristics of the electronic band structure can be explained in more detail using the table of calculated eigenvalues at high points symmetry in the Brillouin zone and figures for the total and partial densities states. Figure 2 and Figure 3 show the total state density (DOS) and the partial densities of state (pDOS), respectively, as derived from the bands in Figure 1. Many parts of our calculated density of state are very close to corresponding experimental values obtained by X-ray photoemission spectroscopy measurements [71]. Figure Table 3 which shows the calculated eigenvalues at high symmetry points in the Brillouin zone.
The total valence band width is 15.862 eV. The width of the lowest laying group of bands is 0.106 eV. The widths of the middle and upper most groups valence bands are respectively 2.651 eV and 6.847 eV. These widths, derived from Table 3, can also be estimated using the content of the figure below for TDOS. A purpose of this table is to be able to make comparisons with possible future experimental measurements of X-ray, ultraviolet (UV) or other spectroscopy.
According to our calculated pDOS in Figure 3, the lowest lying group of valence bands comes almost entirely from Ga d. The middle group of valence bands is largely from Sb s, with a small contribution from Ga s. The upper most group of valence bands is unmistakably dominated by Sb p, Ga p, and Ga s, with a very small contribution of Sb s.    We used the electron band structure from Calculation IV (in Figure 1) to calculate the effective masses of the electron, at the bottom of the conduction band, and of the holes, at the top of the valence band. We present our results, as well as other theoretical and experimental values, in Table 4. The effective masses of the electrons are indicated by me while those of the heavy and light holes are respectively noted as m hh and m lh . The first column shows the effective masses with the specific directions in which they are calculated. It is gratifying to note that our calculated effective masses are in general agreement with experience [55] [74] [75] [76], and with parts of some previously calculated ones [12] [26] [72] [73]. Out of the nine (9) effective masses in Table 4, the only one for which there is a difference between an experimental value and our result is m hh (Γ-L) for which we found 0.682 m 0 while the experiments report from 0.24 m 0 to 0.59 m 0 [74] [75] [76].
For Reference [a], atheo and aexp indicate results obtained with a theoretical and an experimental lattice constant.
The above agreement between our calculated effective masses and the corresponding, experimental values indicates the correct rendering, by our calculations, of the curvatures of the bands at the conduction band minimum (CBM) and the valence band maximum (VBM).
A discussion of our results, compared to previous ones, follows. We first recall that our main motivation for this work was the resolution of the glaring disagreement between measured band gaps of approximately 0.81 eV and 0.73 eV, Table 4. Calculated, effective masses for zb-GaSb (in units of the free electron-mass, m0): me indicates an electron effective mass at the bottom of the conduction band; mhh, and mlh represent the heavy and light hole effective masses, respectively. Theo.: theory, expt.: experiment.

Our work
Theo.
[ Our results, due to our strict adherence to the conditions of validity of the DFT, have the entire physical content of the DFT. They agree with the available, corresponding, experimental data, as was the case for the band gap at room temperature (i.e., 0.75 eV). Our work also made it possible to obtain the experimentally identified locations of the peaks in the total density of valence states.; This agreement points to the overall correct description of the ground state band structure by our calculations. In addition, this overall description of the bands is further indicated by our correct rendering of noted curvatures of the bands, in accordance with our results for the effective masses.
The results of Reference 12 for the electron effective masses confirmed our contention that as the bottom of the conduction band is spuriously lowered,  [12] found electron effective masses of 0.036, 0.035, and 0.035 m 0 ; for the much smaller band gap of zero, these electron effective masses are respectively 0.008, 0.004, and 0.006 m 0 . Both sets of values are lower than the corresponding ones from our calculation. Our truly ground state calculations [57] did not need to invoke self-interaction correction of the derivative discontinuity of the exchange correlation energy in order for their outcomes to agree with available, corresponding, experimental ones. The same was true for several previous works of our group [57] [58] [63]- [69].

Conclusion
We have performed ab initio, self-consistent calculations of electronic energy bands, total and partial densities of states and of effective masses for zb-GaSb. With the method of Bagayoko, Zhao and Williams (BZW), we performed a generalized minimization of the energy to reach the ground state, verifiably, without using over-complete basis sets. Our work reproduced not just the correct, experimental room temperature band gap, but also the locations of the peaks in the total density of states, and the electron and holes effective masses. With this overall accuracy, our calculated results can inform and guide the design and fabrication of semiconductor based devices, as envisioned by the Materials Genome Initiatives (MGI).