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Authors:A Srivani, V Rama Murthy, G Veeraraghavaiah Department of Physics & Nanotechnology Acharya Nagarjuna University 1. Introduction. Studies of the electronic and structural properties of binary semiconductors have received considerable attention, both experimentally and theoretically. Some experimental studies of the band structure parameters for InP and its alloys have been carried out. Rochon and Fortin [1] found the low-temperature direct band gap to be 1.423eV. Since exciton rather than inter band transitions are usually observed in such absorption measurements the binding energy of 5meV has been added to the spectral position of the resonance [2, 1]. InP is a direct-gap semiconductor of great technological significance [3, 4, 5], since it serves as the substrate for most optoelectronic devices operating at the communications wave length of 1.55µm. InAs has assumed increasing importance in recent years as the electron quantum well material for InAs/GaSb/AlSb based electronics [6] and long-wave length optoelectronic [7] devices. The vast majority of experimental low-temperature energy gaps fall in the 0.41–0.42eVrange [8, 9], although some what higher values have also been reported [10, 11]. The value 0.417eV that was obtained from recent measurements on a high-purity InAs sample, [12] is generally adopted. The InSb is the III–V binary semiconductor with the smallest band gap. For many years it has been a touchstone for band structure computational methods, [13] partly because of the strong band mixing and nonparabolicity that result from the small gap. The primary technological importance of InSb arises from mid-infrared optoelectronics applications [14]. Numerous studies of the fundamental energy gap and its temperature dependence have been conducted over the last three decades [15, 16, 10, 11, 17, 18, 9, 19]. While there is a broad consensus that Eg(T=0) = 0.235eV, several different sets of Varshni parameters have been proposed [15, 17, 18, 19, 20]. However, despite interesting properties related to their narrow energy gaps [21], the In-V semiconductors have been the object of fewer investigations [22, 23], in comparison to the extensive work performed on other III-V zinc-blende-structure materials like GaAs and AlAs. In this work, intended as a first step towards a detailed study of his superlattice compound, we study the structural and electronic properties of the InP, InAs, InSb semiconductors using the self-consistent FP-LAPW method. The comparison of our results with other results indicates the importance of the In-4d states, which play a role in the electronic structure of these materials similar to that of the Ga-3d states in GaAs and the d states in the zinc-blende-structure II-VI compounds [24, 25]. Furthermore, this study shows that, within the local-density approximation (LDA), InAs and InSb have metallic properties as a result of the well-know failure of the LDA in the description of the excitation properties of semiconductors. On the other hand, excellent agreement is found with experiment for the equilibrium properties and good agreement (generally within the experimental resolution) is found for occupied energy bands. We focus too on the bonding properties, related to the covalent character of the bonds. 2. details In the following calculations, we have distinguished the In(1s22s22p63s23p63d104s24p6), and P(1s22s22p63s23p6), As(1s22s22p63s23p63d104s2 4p6), and Sb(1s22s22p63s23p63d104s24p6) inner-shell electrons from the valence band electrons of the In(4d105s25p1), P(3s23p3), As(3d104s24p3), and Sb(4d105s25p3) shells. For these three compounds, zinc-blende structure (ZB) has the lowest minimum total energy. It is the most stable phase of this compound at ambient pressure. The In and P, As, Sb atoms are in fcc positions as follows: In(0, 0, 0); P, As, Sb(1/4, 1/4, 1/4). We have considered a set of lattice parameters taken from Ref [26] as aeq= 5.869, 6.058, 6.479 Å respectively for InP, InAs, and InSb. Then, using these parameters, we optimized the volume. The equilibrium lattices constants and bulks moduli are calculated by fitting the total energy versus volume according to Murnaghan’s equation of state [27]. The calculations were performed using non relativistic full-potential linearized augmented plane wave (FP-LAPW) [28] approach within the framework of the density-functional theory (DFT). The exchange-correlation energy of the electrons is described in the local-density approximation (LDA) using the Perdew and Wang [29] functional. Basis functions were expanded in combinations of spherical harmonic functions inside non-overlapping spheres surrounding the atomic sites (muffin-tin spheres) and in a Fourier series in the interstitial region. In the muffin-tin spheres, the l-expansion of the non-spherical potential and charge density was carried out up to lmax=10. In other to achieve energy eingenvalues convergence, the wave functions in the interstitial region were expanded in plane waves with a cut-off of kmax=10/RMT (where kmax is the maximum modulus for the reciprocal lattice vector, and RMT is the average radius of the MT spheres), equivalent to approximately 750 and 440 basis functions per atom. Furthermore, we have adopted the values of 2.35 Bohr for indium and 2.20, 2.35, 2.25 respectively for phosphor, arsenide, and antimonide as the MT radii value. The k integration over the Brillouin zone is performed up to Monkhorst and Pack [30] mesh (ten points in the irreductible wedge of the Brillouin zone (IBZ) are used). The iteration process was repeated until the calculated total energy of the crystal converged to less than 1 mRyd. A total of 7 iterations were necessary to achieve self-consistency. 3. Results and discussion The table.1 lists the equilibrium lattice constants, bulk moduli (B), and their derivatives (B’) for the three materials. For the lattice parameters, the agreement with experiment [31] is excellent; the error is about 0.2%, 0.01%, and 0.04% for InP, InAS, and InSb respectively. Good agreement is found for the bulk moduli (B), within 2% for the worst case InAs. Theoretical calculations within LDA give 5.729, 5.921, 6.346 Å respectively for InP, InAs, and InSb using Plane wave pseudopotential method [32] and 5.74, 5.94, 6.36Å respectively for InP, InAs, and InSb using norm-conserving pseudopotential method [33]. These last calculations underestimate the equilibrium lattice parameters by about 2%. This may be attributed to the contribution of the in-4d states, which is neglected in this approach [33], since similar differences between all-electron and Table 1. Calculated and experiment values of a, B, B’ of InP. References are given where appropriate. a (Å) B (Mb) B’ InP 5.867 0.72 - Expt[31] 5.729 0.736 4.79 PW-PP[32] 5.74 0.76 4.5 PP[33] 5.842 0.716 4.846 This Work. InAs 6.054 0.58 - Expt[31]. 5.921 0.617 4.545 PW-PP[32] 5.94 0.63 4.7 PP[33] 6.053 0.566 4.739 This Work. InSb 6.472 0.46 - Expt.[31] 6.346 0.476 4.688 PW-PP[32] 6.36 0.48 4.8 PP[33] 6.476 0.467 4.307 This Work. available pseudopotential results were found in GaAs and ZnSe as a result of the contribution of the Ga and Zn 3d states [34, 35]. Previous pseudopotential calculations of the lattice constants of these materials gave results with small differences with the experiment ones: Zhang and Cohen [36] obtained 5.87, 6.04, and 6.49Å for InP, InAs, and InSb respectively, and other calculations are smaller than experiments results: Singh and Varshni [37] obtained 6.24Å for InSb, and Boguslawski and Balderschi [38] obtained 5.75 and 5.95Å for InP, and InAs, respectively. The electronic energy eingenvalues at high-symmetry points, calculated at present work lattice constant, are listed in table 2, 3, 4 respectively for InP, InAs and InAs for the occupied and lowest-lying unoccupied states and are compared with the experimental results [39, 40], with empirical pseudopotential calculations (PP) [41], and with the results of norm-conserving pseudopotential method [33] using the Cerperley-Adler (CA)[42] form of the exchange-correlation potential Our predicted band gap values are 0.51, -0.47, and -0.43eV respectively for InP, InAs, and InSb. We find that for all the three compounds, the differences between the predicted and the experimental band gaps are about 1.0eV. It is interesting to notice that although LDA underestimates the band gaps. However, our results are in agreement with the FLAPW calculations using the relativistic (R) approach [33], and the Hedin-Lundqvist (HL) [43] form of the exchange-correlation potential. Table 2. Electronic energy levels for InP. Experimental values from Ref. [39] unless otherwise indicated. EPP FLAPW PP Expt. Present (R,HL) (SR,CA) (XPS) work -11.42 -11.60 -11.16 -11.0 -11.64 0.0 -0.12 0.0 0.0 0.0 1.50 0.38 0.98 1.42a 0.51 -8.91 -9.20 -9.14 -8.9 -9.58 -6.01 -5.97 -5.59 -5.9 -5.37 -2.09 -2.47 -2.12 -2.0 -2.25 2.44 1.60 1.68 - 1.63 -9.67 -9.87 -9.73 -10.0 -9.92 -5.84 -5.90 -5.54 -5.9 -5.79 -1.09 -0.98 -0.85 -1.0 -0.99 2.19 1.23 1.60 - 1.31 a Reference [40, 44] Table 3. Electronic energy levels for InAs. Experimental values from Ref. [39] unless otherwise indicated. EPP FLAPW PP Expt. Present (R,HL) (SR,CA) (XPS) work -12.69 -12.06 -11.54 -12.3 -11.92 0.0 0.0 0.0 0.0 0.0 0.37 -0.63 -0.03 0.42a -0.47 -10.20 -10.07 -9.89 -9.8 -10.20 -6.64 -6.08 -5.66 -6.3 -5.49 -2.37 -2.48 -2.13 -2.4 -2.19 2.28 1.33 1.60 - 1.48 -10.92 -9.87 - - -10.5 -6.23 -5.90 -5.54 -5.9 -5.74 -1.00 -1.00 -0.88 -0.9 -0.98 1.53 0.63 1.04 - 0.76 a Reference [40] Table 4. Electronic energy levels for InSb. Experimental values from Ref. [39] unless otherwise indicated. InSb EPP FLAPW PP Expt. Present (R,CA) (SR,CA) work -11.71 -11.11 -10.46 -11.2b -10.80 0.0 -0.76 0.0 0.0b 0.0 0.25 -0.74 -0.01 0.24a -0.43 -9.20 -9.15 -8.69 -9.5b -9.16 -6.43 -6.29 -5.76 -6.4b -5.51 -2.24 -2.48 -2.11 -2.4b -2.14 1.71 1.04 1.32 - 1.10 -9.95 -9.75 -9.23 -10.5b -9.48 -5.92 -5.98 -5.45 -6.4b -5.60 -0.96 -1.06 -0.91 -1.4b -0.99 1.03 0.13 0.63 - 0.39 aXPS, Reference [40] bUPS, Reference [39] Our total valence-band width calculations both for InAs and InSb, respectively -11.92eV and -10.80eV are slightly smaller than X-ray-photoemission-spectroscopy results (-12.3eV) for InAs, and than Ultraviolet-photoemission-spectroscopy (-11.2eV) for InSb, but larger for InP (-11.64eV) in comparison with (XPS) results (-11.0eV). This last result is somehow surprising, since the bottom of the valence band is formed by the tightly bound anion s-bonding states, whose binding energy is expected to be underestimated by the LDA [33]. The electronic band structure of InP, InAs, and InSb are shown in figure 1(a)1(b)1(c), respectively. We show clearly that the bad gap for the three materials is direct at point G. The valence band maxima are derived from p-like orbitals px, py, pz, which remain degenerate under the tetrahedral group of the zinc blende lattice as report in Ref.[45]. We observe that the width of the In-4d band increase in going from InSb (0.08eV) to InP (0.15eV), as a consequence of the lattice constant decrease. Figures 1. Self-consistent band structure of (a) InP, (b) InAs, and (c) InSb along the principal high-symmetry directions in the Brillouin zone. (The energy zero is taken at the valence band maximum). To visualize the nature of the bond character and to explain the charge transfer and the bonding properties of the three semiconductors, we have investigated in some details the effect of In-d states on the charge density of selected states and on the total charge density. The total valence charge densitys for InP, InAs, and InSb, are displayed along In-P, In-As, and In-Sb bonds respectively in Figure 2(a), 2(b), and 2(c). And the (110) plane containing In and P, In and As, In and Sb atoms in Figure 3(a), 3(b), 3(c) respectively. Figures 2. Line plots of the total valence charge density along the In-P (a), In-As (b) and In-Sb (c) direction respectively. The calculated electron charge distribution indicates that there is strong ionic character as can seen along the In-P, In-As and In-Sb bonds. For the three compounds, the essential of the charge is not centred in the medium of the bond, but that this charge exhibits a preference for the anion in particular. At the cation we show an almost absence of the charge density, there is just charge who represent the In-4d states. At the interstitial region and there is almost a few charge. The net charge transfer from the cation (In) to the anion (p, As, Sb), which indicates the degree of the ionicity of the bonding. Figure 3. Contour plots of the total valence charge density in the (110) plane for (a) InP, (b) InAs, and (c) InSb. The competition between the ionic and covalent character in boron compounds can be related to the charge transfer between cation and anion. We estimate the ionicity factor of InP from the asymmetry of the valence charge distribution. Two different approaches have been used to calculate the ionicity factor of InP compound: (i) the Garcia-Cohen approach based on the valence charge density calculation [46], and (ii) the Pauling definition based on the electronegativity values of the elements. The scaling law introduced by Garcia and Cohen was successful in predicting the fi behavior for a wide variety of semiconductors. The Garcia-Cohen ionicity factor is defined as (1) where and are the measures of the strength of the symmetric and antisymmetric components of the charge density, respectively, and are defined as [46]. (2) We also use the Pauling definition [47] of the ionicity of a single bond and the Phillips electronegativity values for In and P [48] for comparison; a rapid estimation of the ionicity factor is obtained by using the Pauling equation: (3) where and are the electronegativities of atoms A and B, respectively. Table 3. The calculated ionicity factor fi for InP, InAs and InSb. [48] [49] [50] InP 0.483a 0.26b 0.421 0.534 0.419 InAs 0.583a 0.26a 0.357 0.553 0.360 InSb 0.347a 0.25a 0.321 0.303 0.328 aCalculated using Gacia et al approach [46]. bEstimated using Pauling definition [47]. The calculated ionicity values of InP, InAs, and InSb compared with those of Phillips [48], Christensen [49], and Al-Douri [50] are summarized in the table3. The calculated value 0.483, 0.583, and 0.347 respectively for InP, InAs and InSb are close to those given by Christensen but very different to these estimated using Pauling definition. Since InP, InAs and InSb have a tetrahedral coordinated structure according to the Phillips scale of ionicity [48]. 4. Conclusions In this paper, we present our results on the electronic structure investigation of InP. We present the band structure results. We provide information about the charge density which allows us to evaluate the nature of the bond character. The main conclusions can be summarized as follows. 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