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The electronic band structures of InNxAs1−x , InNxSb1−x and InAsxSb1−x alloys
Journal of Alloys and Compounds 469 (2009) 504–511
The electronic band structures of InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys Rezek Mohammad, S¸enay Katırcıo˘glu ∗ Middle East Technical University, Physics Department, Inonu Bulvari, 06530 Ankara, Turkey Received 19 October 2007; received in revised form 1 February 2008; accepted 2 February 2008 Available online 17 March 2008
Abstract The band gap bowings of InNx As1−x , InNx Sb1−x , and InAsx Sb1−x alloys defined by the optimized lattice constants are investigated using empirical tight binding (ETB) method. The present ETB energy parameters which take the nearest neighbor interactions into account with sp3 d2 basis are determined to be sufficient to provide a typical feature for the band gap bowings of the alloys. The band gap bowing parameter is found to be relatively large in both InNx As1−x and InNx Sb1−x compared to InAsx Sb1−x alloys. Moreover, the variation of the fundamental band gaps of InNx Sb1−x alloys is sharper than that of InNx As1−x alloys for small concentrations of N. Besides, a small amount of nitrogen is determined to be more effective in InNx Sb1−x than in InNx As1−x alloys to decrease the corresponding effective masses of the electrons around Γ points. © 2008 Elsevier B.V. All rights reserved. Keywords: Empirical tight binding method (ETB); InNx As1−x ; InNx Sb1−x ; InAsx Sb1−x ; Band gap bowing parameter; Ternary alloys
1. Introduction There is currently considerable interest in InN, InAs and InSb, because of their optical and high temperature device applications. InN is a potential material for the fabrication of high-speed heterojunction transistors [1] and low-cost solar cells with high efficiency [2]. InAs and InSb are interesting narrow gap semiconductors from the point view of optical spectroscopy and optoelectronic applications [3]. Since InAs has a high electron mobility, it may provide an important material for use in high-speed electronics [4,5]. InSb calls the attention to have the smallest energy gap of any of the binary III–V materials. This property often allows the fabrication of infrared imaging systems, free space communications and gas phase detection systems [6,7]. A new class of semiconductor alloys in which one of the constituent elements in compound is replaced by an element with highly dissimilar properties has been discovered recently. These ternary semiconductor alloys exhibit a range of unexpected characteristics. Particularly, the addition of a small amount of nitrogen in InAs and InSb compounds leads to spectacular
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changes of the electronic properties [8–17]. On the other hand, the substitution of a small amount of As in InSb indicates the similar drastic changes on the electronic properties of InSb compound [18–28]. The most important effect of the substitutions in these InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys is large reduction of the fundamental band gap (EgΓ ) of InAs and InSb. In the literature, the aim dependent reduction of the fundamental energy gap has been tuned by the substituent compositions in InNAs, InNSb and InAsSb alloys [8,11–18,20–27] grown by molecular beam epitaxy (MBE) [12,14–17,21,22,24], metal organic chemical vapor deposition (MOCVD) [8,11,13,23,25], Bridgman [26] and hotwall epitaxy (HWE) [28] methods. Therefore, these ternary alloys are potential materials for room temperature infrared detectors, gas sensors and lasers operating in near-infrared (0.9–1.3 m), mid-infrared (2–5 m) and far infrared (8–12 m) regions [13,17,21–26]. In the literature, the most of the works on InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys are experimental. To the knowledge of authors, there are a few works [9,10,14–17,19,27] in which the electronic band structures of these ternary alloys have been studied theoretically. In most of these theoretical works, the ternary alloys were defined by only specific concentrations of N and As [9,14–17,27]. In the present work, the electronic band structures of InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys are calculated by empirical tight binding (ETB) method. The calculations
R. Mohammad, S¸. Katırcıo˘glu / Journal of Alloys and Compounds 469 (2009) 504–511
are performed on the alloys to introduce mainly the band gap bowing corresponding to the total range of nitrogen and arsenic concentrations. The paper is organized as follows: the steps of calculations are explained in Section 2, the variation of the lattice constant and the fundamental band gap (EgΓ ) of the alloys as a function of N and As concentrations varying in the range of 0.0 < x < 1.0 are reported in Section 3. The variation of the conduction band edge of the alloys around Γ point for small concentrations of N and As is also given in Section 3, finally, a conclusion is introduced in Section 4. 2. Calculational Method In the present work, the cubic unit cells of three alloys, InNx As1−x , InNx Sb1−x , and InAsx Sb1−x , are defined by the corresponding equilibrium lattice constants. The unit cells of InNx As1−x and InNx Sb1−x ordered alloys are obtained by the replacement of As and Sb atoms by N in InAs and InSb compounds with a fraction of x, respectively. Similarly, the unit cells of InAsx Sb1−x ordered alloys are defined by the replacement of Sb atoms by As atoms in InSb compound with a fraction of x, respectively. The total energies of the alloys with specific concentrations of constituents (x = 0.0, 0.25, 0.50, 0.75,and 1.0) are calculated by full potential-linearized-augmented plane waves (FP-LAPW) method based on density functional theory (DFT). The exchange-correlation energy of electrons is defined by gradient generalized density approximation based on Perdew–Burke–Ernzerhof functional [29]. In FP-LAPW calculational steps implemented in WIEN2k code [30], the unit cell is partitioned into nonoverlapping muffin-tin spheres around the atomic sites. Basis functions are expanded in combinations of spherical harmonic functions inside the nonoverlapping spheres. In the interstitial region, a plane wave basis is used and expansion is limited with a cutoff parameter, RMT KMAX = 7. Here, RMT is the minimum radius of the sphere in the unit cell, KMAX is the magnitude of the largest K vector used in the plane wave expansion. The muffin-tin radius is adopted to be 2.0, 1.5, 1.8, and 2.0 a.u. for In, N, As, and Sb atoms, respectively. In the calculations, the electrons of In, N, As, and Sb atoms in 4s2 4p6 4d10 5s2 5p1 , 2s2 2p3 , 3d10 4s2 4p3 , and 4p6 4d10 5s2 5p3 shells, respectively, are treated as valence electrons by choosing a cutoff energy of −8 Ry. The expansion of spherical harmonic functions inside the muffin-tin spheres is truncated at l = 10. The cutoff for fourier expansion of the charge density and potential in √ the interstitial region is taken to be GMAX = 14 Ry. The self consistent calculations are carried on until the total energy difference is less than 0.1 mRy. The electronic band calculations of the alloys are performed using 3 × 3 × 3 grids correspond to 27 k points in the irreducible wedge of the Brillouin zone. In the present work, the total energies of the alloys calculated for certain values of x are minimized with respect to different volumes of the unit cells. The equilibrium lattice constants of InNx As1−x , InNx Sb1−x , and InAsx Sb1−x alloys correspond to the minimum total energies are obtained by fitting of the variation of the total energy to the Murnaghan’s equation of state [31]. In the next step, the variation of the lattice constant of the alloys as a function of x is fitted to the polynomial function aABx C1−x = xaAB + (1 − x)aAC − bx(1 − x)
(1)
to extract the equilibrium lattice constants of the alloys having concentrations of constituents different than x = 0.0, 0.25, 0.50, 0.75, and1.0. Here, aAB , aAB , and aABx C1−x are the equilibrium lattice constants of the compounds and the alloys considered in this work, respectively. The bowing parameter, b, shows the deviation of the equilibrium lattice constants from Vegard’s linear rule [32]. Since the application of the first principles calculational methods on nonsymmetric systems consumes too much calculational time, the semiempirical and empirical methods have been required to calculate the electronic band structures of the alloys for full range of the composition(s). In the present work, electronic band structures of InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys are calculated by ETB for total range of x(0.0 < x < 1.0). ETB method has received considerable attention because of its intuitive simplicity and its realistic description of structural and dielectric properties in terms of chemical bonds. Slater–Koster model [33] of ETB was extensively used with minimal sp3 basis
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and interaction only between nearest neighbor atoms to find out the valence band energy dispersion of elements and compounds. But this model fails to calculate the indirect gap of semiconductors satisfactorily, especially at X point. This deficiency was tried to be overcome by including an excited s state, s∗ , to the interactions in Vogl’s ETB model [34]. On the other hand, the successfully reproduced indirect energy gaps of III–V compounds by pseudopotential (PP) method [35] showed that, the influence of d-orbitals to the lowest conduction state is large at Γ , X and L symmetry points. In the present nearest neighbor ETB calculations this fact is considered and the excited first and second d-orbitals, d1 (dz2 ) and d2 (dx2 −y2 ), are taken into account as well as sp3 . In our previous work [36], the nearest neighbor interactions between anion and cation with sp3 d2 basis was defined by 13 different types of ETB energy parameters. They are Esa , Epa , Eda , Esc , Epc , Edc , Vss , Vxx , Vxy , Vpasc , Vpcsa , Vdapc , Vdcpa . In ETB calculations of InN, InAs and InSb, they were formulated by the solution of secular equation at Γ and X symmetry points. In the same work, ETB energy parameters of InN, InAs and InSb were derived from the first principles band structure calculations of the corresponding compounds by the fitting process. Since ETB interaction energy parameters reproduced the energy band gaps of the compounds correctly at Γ and X [36], we have surely used them in the band gap engineering calculations of InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys. Since only the nearest neighbor interactions are taken into account, the self energy parameters (Esa , Epa , Eda , Esc , Epc and Edc ) of In, N, As, and Sb are included in the present calculations directly. The other ETB energy parameters (Vss , Vxx , Vxy , Vpasc , Vpcsa , Vdapc , Vdcpa ) needed for the electronic band structure calculations of the alloys are defined by Vegard’s law [32] and scaled by r −2 [37]; particulary, Vss is formulated as VssABx C1−x =
2 ×V 2 x(dAB ssAB ) + (1 − x)(dAC × VssAC ) 2 dAB x C1−x
(2)
2 , d 2 and d 2 Here, dAB AC ABx C1−x are the square of the bond length of the compounds and the alloys, respectively. VssAB and VssAC define s-orbital interactions between an anion and cation in AB and AC compounds, respectively. The rest of the interaction parameters of InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys are calculated using similar formulations. In the next step of ETB calculations, electronic band structures of InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys are obtained by the solution of secular equation in the unit cells of the alloys for different concentrations of constituents along different momentum directions. Electronic band structures of the alloys are analyzed for the variation of the fundamental energy gap and the conduction band edge around Γ point with respect to the concentration of constituents. The bowing parameter, b, which shows the deviation of the fundamental energy gap from linearity is calculated by the best fit of our results to the following expression:
EgΓAB
x C1−x
(x) = xEgΓAB + (1 − x)EgΓAC − bx(1 − x)
Here, EgΓAB , EgΓAC and EgΓAB
x C1−x
(3)
are the fundamental band gap energies of the
compounds and alloys considered in this work, respectively.
3. Results and discussion 3.1. InNx As1−x In III–V semiconductors, the replacement of a few percent of the group V element by small, highly electronegative and isoelectronic nitrogen atoms results in a drastic reduction of the fundamental band gap of approximately 100 meV per atomic percent of nitrogen [38,39]. This effect of N has been confirmed for InNx As1−x alloys experimentally [8,12,13] with the concentration x < 18%. InNx As1−x alloys with N concentration range of 0.0 < x < 0.061 have been grown first by MOCVD work of Naoi et al. [8]. They indicated that, all their samples had direct transition band structures and the increase in nitrogen content made the band energy smaller at each step.
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R. Mohammad, S¸. Katırcıo˘glu / Journal of Alloys and Compounds 469 (2009) 504–511 Table 1 Empirical matrix elements of the sp3 d2 Hamiltonian in eV
Esa Esc Epa Epc Eda Edc Vss Vxx Vxy Vpcsa Vpasc Vdapc Vdcpa
Fig. 1. Lattice constant vs. composition (x) for InNx As1−x alloys.
In their work, the minimum band gap energy was reported to be 0.12 eV for InAs0.939 N0.061 alloy. Therefore, the band gap of InAs was able to be decreased by approximately 260 meV by inclusion of only 6.1% of nitrogen at room temperature. In another work [12], the photoluminescence (PL) peak energy was observed to be decreased with N composition in the InAsN layer of InAsN/InGaAsP multiple quantum well; 1% nitrogen increase in alloy composition caused a shrinkage of 31 meV on the transition energy at 10 K. The smallest transition energy was reported as 0.190 eV when the nitrogen content in InAsN wells was determined to be 18% for InNAs/GaAs multiple quantum wells [13]. In theoretical side, the variation of the band gap of InNx As1−x ordered alloys as a function of x has been studied by Yang et al. [9] using tight binding method for x = 0.25, 0.5 and 0.75. The band gap of InAs calculated in this work [9] was decreased by ∼ 0.48 eV by the inclusion of 50% of nitrogen. In a recent work [10], the tight-binding calculations have been carried on both maximally N-rich and As-rich cluster configurations in InNx As1−x alloys defined in a supercell. This study has indicated that the band gap of InNx As1−x decreased to zero when x = 0.17 and x = 0.50 for maximally N-rich and As-rich cluster configurations, respectively. In the same work, the variation of the band gap of these configurations in InNx As1−x alloys was also studied for x > 0.75 but not for the metallic region. In the present work, the cubic cell (working cell) of each InNx As1−x alloy is defined by the corresponding equilibrium lattice constant obtained by volume optimization. The variation of the equilibrium lattice constant of InNx As1−x alloys as a function of nitrogen concentration is plotted in Fig. 1. The equilibrium lattice constants are found to be deviated from linearity ˚ The large bowing with an upward bowing parameter of −0.64 A. parameter is originated from the large mismatching between the ˚ and InN (4.97 A) ˚ compounds. lattice constants of InAs (6.18 A) Therefore, Vegard’s linear rule [32] which was assumed to be
InN
InAs
InSb
−11.6558 −2.6424 4.6235 4.8241 15.7403 12.6942 −5.8787 5.0545 7.3168 0.8742 4.8020 4.1776 0.0000
−7.9244 −2.8927 0.0000 3.7660 9.3469 7.7964 −5.1076 0.7172 3.6286 4.1592 3.4948 4.7535 0.1638
−5.5942 −3.2074 0.0000 2.8711 7.7705 5.9425 −4.4532 0.2715 2.9182 4.0969 2.8601 4.1220 0.2738
acceptable in the previous two theoretical works [9,10] is not valid for the determination of the lattice constants of InNx As1−x alloys. In the present ETB study, the electronic band structures of InNx As1−x alloys defined by optimized lattice constants are calculated for the full range of nitrogen concentration (0.0 < x < 1.0). ETB energy parameters of InNx As1−x alloys based on sp3 d2 basis and nearest neighbor interactions are given in Table 1. According to electronic band structures, InNx As1−x alloys are all direct gap materials as it was reported in Refs. [8–10]. Fig. 2 displays the variation of EgΓ as a function of x for InNx As1−x alloys. EgΓ starts to decrease from the value of bulk InAs (0.417 eV at 0 K) and reaches to its minimum value at x = 0.51. In Fig. 2, EgΓ passes to negative values at x = 0.1 (cross point). Therefore, InNx As1−x alloys express a metallic behavior in the range of 0.1 < x < 0.89. The variation of EgΓ as a function of nitrogen concentration is described by a large overall bowing parameter of 5.21 eV. The bowing parameter of EgΓ was recommended as 4.22 eV in Ref. [40]. The variation of EgΓ of InNx As1−x alloys defined by Vegard’s linear lattice constants is
Fig. 2. Band-gap energy vs. composition (x) for InNx As1−x alloys defined by optimized (solid line) and Vegard’s linear rule (dotted line) lattice constants.
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also plotted as a function of x in Fig. 2, for comparison. Since the experimental results at room temperature belong to the epitaxially grown InNAs instead of bulk samples, we could not compare the present EgΓ values of 0 K with the PL peak energies directly, for x < 0.061 [8]. In Ref. [8], the PL peak energies correspond to the fundamental gap energies of InNx As1−x were measured to be 0.34, 0.28 and 0.21 eV for x = 0.014, 0.019 and 0.034, respectively. Therefore, the band gap of InNx As1−x is decreased by an amount of 0.04, 0.1, and 0.17 eV, respectively, compared to the band gap value of InAs (0.38 eV) at room temperature. In the present work, the same amount of band gap decrease energies are calculated approximately for the same values of x; they are 0.046, 0.09 and 0.14 eV, respectively compared to the band gap energy of InAs (0.417 eV at 0 K) considered in this work. On the other hand, the present cross point of the bowing (x = 0.10) is more agreement with that (x = 0.15) of the tight-binding calculations for maximally N-rich clusters in InNx As1−x alloys [10] than that (x = 0.5) of ETB calculations for InNAs ordered alloys [9]. The negative energy gaps of InNx As1−x alloys obtained in the present work were not calculated in these works [9,10], but they were predicted. The conduction band profiles of InNx As1−x alloys for x = 0.0, 0.02, 0.06 and 0.12 are shown in Fig. 3. It is observed that the conduction band edge is nonparabolic for intrinsic InAs as it was reported in the literature [37]. In view of Fig. 3, we note that the increase of N concentration decreases the energy of the conduction band edge and causes the interaction between the nitrogen level and the conduction band edge in InNx As1−x alloys. Furthermore, the increase of N concentration in InNx As1−x alloys increases the curvature of the conduction band edge around Γ point. Therefore, a small amount of N concentration decreases the corresponding electron effective masses in InNx As1−x alloys around Γ points.
Fig. 3. The conduction band dispersion calculated for InNx As1−x alloys with x = 0% (solid line), 2% (dotted line), 6% (dashed line) and 12% (star line) around Γ point at 0 K.
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3.2. InNx Sb1−x InSb has the smallest band gap of any of the binary III–V semiconductors with an atmospheric transition window in the rage of mid-infrared (3–5 m). It was experimentally indicated that [14–17], this atmospheric transition can access the range of 8–15 m when a small fraction of antimony is replaced by nitrogen in InSb. Time resolved optical measurements [15] have been used to observe an absorption edge of 15 m at 290 K in an alloy with a nominal composition of InN0.006 As0.994 . In a recent work [17], the variation of the band gap of InSb through incorporation of nitrogen was assessed optically by observation of the emission spectra from InNSb LEDs and the transmission of light through the material. In the same work [17], the experimental results − →→ at room temperature were modeled by k .− p method and the sharp band gap decrease of InNx Sb1−x was reported for nitrogen concentration in the range of 0.0 < x < 0.01. The smallest band gap has been calculated to be 0.065 eV with respect to nitrogen concentration of 0.01 [17]. On the other hand, the highresolution electron energy loss spectroscopy (HREELS) of the nitrided layer in InNx Sb1−x alloys indicated that a negative band gap alloy has been formed by the nitrogen-induced shift of the conduction band [14]. In the same work, the room temperature − →→ HREELS results of InNSb alloys were modeled by k .− p calculations and the negative bowing of the band gap was obtained − →→ for a wider range of 0.0 < x < 0.1. Besides, the k .− p calculational results [14] have given negative band gap in the nitrogen range of 0.02 < x < 0.1; the minimum of the conduction band was below the maximum of the valance band indicated that the alloys were semimetallic. In the present work, the cubic unit cells of InNx Sb1−x alloys are defined by the corresponding optimized lattice constants. The variation of the equilibrium lattice constant of InNx Sb1−x alloys as a function of nitrogen concentration in the range of 0.0 < x < 1.0 is plotted in Fig. 4. It is observed that the equi-
Fig. 4. Lattice constant vs. composition (x) for InNx Sb1−x alloys.
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Fig. 5. Band-gap energy vs. composition (x) for InNx Sb1−x alloys defined by optimized (solid line) and Vegard’s linear rule (dotted line) lattice constants.
librium lattice constants of InNx Sb1−x alloys are deviated from ˚ The linearity with an upward bowing parameter of −1.13 A. large bowing parameter due to the large mismatching between ˚ and InN (4.97 A) ˚ comthe lattice constants of InSb (6.63 A) pounds indicates that Vegard’s linear rule is not valid for the unit cell definition of InNx Sb1−x alloys. The corresponding ETB energy parameters used in the electronic band structure calculations of InNx Sb1−x alloys are tabulated in Table 1. According to the electronic band structure calculations, InNx Sb1−x alloys are direct gap materials for each nitrogen concentration in the range of 0.0 < x < 1.0. Fig. 5 displays the variation of EgΓ due to the concentration of nitrogen for InNx Sb1−x alloys. The variation has a large bowing with the smallest band gap energy of 0.9 eV at x = 0.54. InNx Sb1−x alloys express a metallic behavior in the range of 0.06 < x < 0.91. The overall bowing parameter is calculated to be 4.78 eV. The variation of EgΓ of InNx Sb1−x alloys defined by Vegard’s linear lattice constants is also plotted as a function of x in Fig. 5, for comparison. According to the recent resolved optical measurements [15], HREELS [14] and response wavelength measurements of LEDs [17], the band gap of the epitaxially grown InNx Sb1−x alloys had a very sharp variation for small concentrations of nitrogen; the EgΓ decreased by ∼ 100 meV with respect to room temperature band gap value of InSb (0.18 eV) in nitrogen concentration range of 0.0 < x < 0.006. In the same range, the present reduction is calculated to be only 15 meV with respect to the band gap − →→ value of InSb (0.235 eV) at 0 K. Therefore, the k .− p model calculations [14,17] of the experimental band gap values give a cross point (x = 0.022, 0.011) smaller than ours (x = 0.06) normally. In the present work, the smallest band gap energy belongs to InN0.54 Sb0.46 is −0.9 eV, but in Ref. [14] it belongs to InN0.1 Sb0.9 alloy with the value of −0.5 eV. Fig. 6 displays the variation of the conduction band edge of InNx Sb1−x alloys around Γ point for x = 0.0, 0.02, 0.06 and 0.14. In Fig. 6, the conduction band edge is decreased in
Fig. 6. The conduction band dispersion calculated for InNx Sb1−x alloys with x = 0% (solid line), 2% (dotted line), 6% (dashed line) and 14% (star line) around Γ point at 0 K.
InNx Sb1−x alloys with increase of nitrogen concentration. Furthermore the curvature of the conduction band edge is increased by increase of nitrogen concentration. Therefore, a small amount of nitrogen concentration in InNx Sb1−x alloys is sufficient to increase the speed of the conduction electrons. This result was − →→ also obtained by k .− p model calculations of InNx Sb1−x alloys [15,16]. 3.3. InAsx Sb1−x InAsx Sb1−x alloys with small amount of As contents have been attracted considerable interest because of their potentials for the fabrication of infrared sources and detectors. Although the energy gap of InAs (0.417 eV at 0 K) is larger than that of InSb (0.235 eV at 0 K) by 0.18 eV, a small amount of As in InSb is sufficient to decrease the band gap of InSb to the values correspond to the long wavelengths in 8–12.5 m spectral range. The variation of the optical properties of InAsx Sb1−x was first investigated on bulk-grown polycrystalline samples by Woolley and Warner [18]. The lowest energy gap was measured to be 0.1 eV in this work [18] for x = 0.35. InAsx Sb1−x materials over the complete composition range (0.0 < x < 1.0) have been grown on InAs and characterized by X-ray diffraction and optical absorption measurements at room temperature and 10 K [21,22]. The band gap energies of these materials correspond to the cutoff wavelengths of 12.5 and 8.7 m were measured to be 0.099 and 0.14 eV for x = 0.32 at room temperature and 10 K, respectively [21,22]. In Ref. [41], the lowest band gap of InAsx Sb1−x has been reported to be 0.1 eV for x = 0.4. Huang et al. [25] investigated the band gap variation of InAsx Sb1−x alloys as a function of complete composition range by PL spectra and measured the lowest band gap value of 0.14 eV for x = 0.4 at 10 K. In a recent work [26], the infrared transmission spec-
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Fig. 7. Lattice constant vs. composition (x) for InAsx Sb1−x alloys.
tra of InAsx Sb1−x alloys in single bulk crystal structure show a continuous decrease in optical energy gap with the increase of As content in InSb. In this work, the absorption edges correspond to the fundamental energy gaps were measured to be 0.16 and 0.15 eV at room temperature for x = 0.02 and 0.05, respectively. To the knowledge of authors, there are only two theoretical works [19,27], in which the band gap bowing of the of InAsx Sb1−x alloys has been predicted by empirical pseudopotential calculations. In the present work, the cubic unit cells of InAsx Sb1−x ordered alloys are defined by the optimized lattice constants. The variation of the equilibrium lattice constant of InAsx Sb1−x alloys as a function of x is plotted in Fig. 7. It is found that the deviation of the optimized lattice constants from linearity is relatively small. Therefore, Vegard’s linear rule can be a good approximation to obtain the lattice constants of InAsx Sb1−x alloys. The small bowing parameter due to the small mismatching ˚ and InAs (6.18 A) ˚ between the lattice constants of InSb (6.63 A) ˚ The present lattice compounds is calculated to be −0.063 A. ˚ belong to InAs0.05 Sb0.95 constant values of 6.61 and 6.47 A and InAs0.35 Sb0.65 alloys, respectively are found to be close ˚ [21] measured at room to the values of 6.4606 [26] and 6.3 A temperature for the same alloys. In the present work, the electronic band structures of InAsx Sb1−x are calculated by ETB using the energy parameters given in Table 1. It is found that InAsx Sb1−x alloys are direct gap materials for each As concentration in the range of 0.0 < x < 1.0. Fig. 8 shows the variation of EgΓ as a function of x. In the same figure the fundamental gap energies of InAsx Sb1−x alloys determined by the lattice constants of Vegard’s linear rule are also plotted with respect to x, for comparison. In view of Fig. 8, we note that the overall bowing for Γ point transition is large and downward. The energy gap starts to decrease from the band gap value of InSb (0.235 eV at 0 K) and reaches to the lowest value of −0.095 eV at x = 0.41. Furthermore, InAsx Sb1−x alloys are metallic in the range of 0.19 < x < 0.65. The pre-
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Fig. 8. Band-gap energy vs. composition (x) for InAsx Sb1−x alloys defined by optimized (solid line) and Vegard’s linear rule (dotted line) lattice constants.
dicted bowing parameter by the best fit of our results to the expression given in Eq. (3) is 1.72 eV. In the literature, the bowing parameter was predicted to be 0.596 eV at room temperature by optical absorption [20]. The bowing parameter of InAsSb alloys estimated from low temperature PL measurements was reported to be higher than the value measured at room temperature; it was 0.6853 [22] and 0.672 eV [23] at 0 K. The bowing parameters predicted by two EPP calculations were 0.7 [19] and 0.72 eV [27] for InAs0.5 Sb0.5 alloy. Since the present ETB energy parameters are adjusted to have only the band gap values of bulk InSb and InAs at 0 K, the present overall bowing parameter is found to be quiet high with respect to the reported values in the literature [19,20,22,23,27]. The present band gap values of InAsx Sb1−x alloys have been compared with the PL spectra energies measured at T = 10 and 79 K for certain concentrations of As. The present band gap value of 0.158 eV is 21% smaller than the PL value [22] of 0.20 eV for x = 0.05. The discrepancy increases to 40% when the concentration of As is 0.08 [22]. On the other hand, the present band gap value of 0.231 eV is 25% smaller than the PL value of 0.31 eV for x = 0.87 (As-rich InAsSb alloy) [22]. In this experimental study [22], the end point energy (at x = 1) corresponds to the band gap value of InAs was measured to be ∼ 0.5 eV instead of ∼ 0.4 eV. However, the present end point energy at x = 1 is in good agreement with the value of literature. The present band gap value of 0.03 for x = 0.15 is much smaller than the corresponding PL value [25] of 0.135 eV measured at 10 K. On the other hand, the calculated band gap value of 0.18 eV at x = 0.83 (As-rich InAsSb alloy) is greater than the corresponding low temperature PL value of 0.13 eV [25]. The present band gap value of 0.298 eV for x = 0.917 is found to be 12% smaller than the measured value of 0.34 eV for InAs0.917 Sb0.083 alloy grown on a lattice matched substrate (GaSb) at 79 K [24]. The PL peak energies of InAs0.71 Sb0.29 and InAs0.85 Sb0.15 alloys at 10 K were reported to be 0.2 and 0.28 eV, respectively [23]. The corresponding values are found to be 0.050 and 0.210 eV by the
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InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys express a metallic behavior in the range of 0.1 < x < 0.89, 0.06 < x < 0.91, and 0.19 < x < 0.65, respectively. The variation of the fundamental band gaps of InNx Sb1−x alloys is sharper than that of InNx As1−x alloys for small concentrations of N. The band gap bowing parameter is relatively large in both InNx As1−x and InNx Sb1−x compared to InAsx Sb1−x alloys. Since the band structure of the materials depend closely on temperature, there is not a good agreement between the present band gaps of the alloys (calculated at 0 K) and the ones measured at low and room temperatures. However, the present ETB results are able to provide a typical feature for the band gap bowings of InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys. As a final conclusion, a small amount of N and As in InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys, respectively are sufficient to decrease the corresponding effective masses of the electrons around Γ points. This decrease is more pronounced in InNx Sb1−x than in InNx As1−x alloys. References Fig. 9. The conduction band dispersion calculated for InAsx Sb1−x alloys with x = 0% (solid line), 2% (dotted line), 6% (dashed line) and 15% (star line) around Γ point at 0 K.
present ETB calculations. According to the present results, the concentration of As (x = 0.41) corresponds to the lowest band gap of InAsx Sb1−x is very close to the value of x = 0.38 given by PL measurements [22]. Fig. 9 displays the variation of the conduction band edge of InAsx Sb1−x alloys around Γ point for x = 0, 0.02, 0.06 and 0.15. It is found that the increase of the concentration of As in InAsx Sb1−x increases the curvature of the conduction band edge around Γ point relatively. Therefore, a small amount of As concentration in InAsx Sb1−x alloys decreases the corresponding effective masses of the conduction electrons around Γ points. 4. Conclusion In the present work, the electronic band structures of ternary alloys, InNx As1−x , InNx Sb1−x , InAsx Sb1−x , are studied as a function of nitrogen and arsenic concentrations in the range of 0.0 < x < 1.0 by ETB method using sp3 d2 basis and nearest neighbor interactions. The cubic cells of the alloys are defined by the optimized lattice constants. Because of the large mismatching between the lattice constants of InN and InAs, InN and InSb adjacent compounds, a large deviation of the equilibrium lattice constants of InNx As1−x and InNx Sb1−x alloys from Vegard’s linear lattice constants is built. However, this deviation is found to be relatively small for InAsx Sb1−x alloys due to the small mismatching between the lattice constants of InSb and InAs adjacent compounds. Therefore, Vegard’s linear rule can be a good approximation for the definition of the lattice constants of InAsx Sb1−x alloys, but it can be considered to be weak for InNx As1−x and InNx Sb1−x alloys. The present band structure calculations showed that all the alloys considered in this work are direct gap materials. Besides,
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The electronic band structures of InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys Rezek Mohammad, S¸enay Katırcıo˘glu ∗ Middle East Technical University, Physics Department, Inonu Bulvari, 06530 Ankara, Turkey Received 19 October 2007; received in revised form 1 February 2008; accepted 2 February 2008 Available online 17 March 2008
Abstract The band gap bowings of InNx As1−x , InNx Sb1−x , and InAsx Sb1−x alloys defined by the optimized lattice constants are investigated using empirical tight binding (ETB) method. The present ETB energy parameters which take the nearest neighbor interactions into account with sp3 d2 basis are determined to be sufficient to provide a typical feature for the band gap bowings of the alloys. The band gap bowing parameter is found to be relatively large in both InNx As1−x and InNx Sb1−x compared to InAsx Sb1−x alloys. Moreover, the variation of the fundamental band gaps of InNx Sb1−x alloys is sharper than that of InNx As1−x alloys for small concentrations of N. Besides, a small amount of nitrogen is determined to be more effective in InNx Sb1−x than in InNx As1−x alloys to decrease the corresponding effective masses of the electrons around Γ points. © 2008 Elsevier B.V. All rights reserved. Keywords: Empirical tight binding method (ETB); InNx As1−x ; InNx Sb1−x ; InAsx Sb1−x ; Band gap bowing parameter; Ternary alloys
1. Introduction There is currently considerable interest in InN, InAs and InSb, because of their optical and high temperature device applications. InN is a potential material for the fabrication of high-speed heterojunction transistors [1] and low-cost solar cells with high efficiency [2]. InAs and InSb are interesting narrow gap semiconductors from the point view of optical spectroscopy and optoelectronic applications [3]. Since InAs has a high electron mobility, it may provide an important material for use in high-speed electronics [4,5]. InSb calls the attention to have the smallest energy gap of any of the binary III–V materials. This property often allows the fabrication of infrared imaging systems, free space communications and gas phase detection systems [6,7]. A new class of semiconductor alloys in which one of the constituent elements in compound is replaced by an element with highly dissimilar properties has been discovered recently. These ternary semiconductor alloys exhibit a range of unexpected characteristics. Particularly, the addition of a small amount of nitrogen in InAs and InSb compounds leads to spectacular
∗
Corresponding author. Tel.: +90 3122105058; fax: +90 3122105099. E-mail address: [email protected] (S¸. Katırcıo˘glu).
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changes of the electronic properties [8–17]. On the other hand, the substitution of a small amount of As in InSb indicates the similar drastic changes on the electronic properties of InSb compound [18–28]. The most important effect of the substitutions in these InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys is large reduction of the fundamental band gap (EgΓ ) of InAs and InSb. In the literature, the aim dependent reduction of the fundamental energy gap has been tuned by the substituent compositions in InNAs, InNSb and InAsSb alloys [8,11–18,20–27] grown by molecular beam epitaxy (MBE) [12,14–17,21,22,24], metal organic chemical vapor deposition (MOCVD) [8,11,13,23,25], Bridgman [26] and hotwall epitaxy (HWE) [28] methods. Therefore, these ternary alloys are potential materials for room temperature infrared detectors, gas sensors and lasers operating in near-infrared (0.9–1.3 m), mid-infrared (2–5 m) and far infrared (8–12 m) regions [13,17,21–26]. In the literature, the most of the works on InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys are experimental. To the knowledge of authors, there are a few works [9,10,14–17,19,27] in which the electronic band structures of these ternary alloys have been studied theoretically. In most of these theoretical works, the ternary alloys were defined by only specific concentrations of N and As [9,14–17,27]. In the present work, the electronic band structures of InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys are calculated by empirical tight binding (ETB) method. The calculations
R. Mohammad, S¸. Katırcıo˘glu / Journal of Alloys and Compounds 469 (2009) 504–511
are performed on the alloys to introduce mainly the band gap bowing corresponding to the total range of nitrogen and arsenic concentrations. The paper is organized as follows: the steps of calculations are explained in Section 2, the variation of the lattice constant and the fundamental band gap (EgΓ ) of the alloys as a function of N and As concentrations varying in the range of 0.0 < x < 1.0 are reported in Section 3. The variation of the conduction band edge of the alloys around Γ point for small concentrations of N and As is also given in Section 3, finally, a conclusion is introduced in Section 4. 2. Calculational Method In the present work, the cubic unit cells of three alloys, InNx As1−x , InNx Sb1−x , and InAsx Sb1−x , are defined by the corresponding equilibrium lattice constants. The unit cells of InNx As1−x and InNx Sb1−x ordered alloys are obtained by the replacement of As and Sb atoms by N in InAs and InSb compounds with a fraction of x, respectively. Similarly, the unit cells of InAsx Sb1−x ordered alloys are defined by the replacement of Sb atoms by As atoms in InSb compound with a fraction of x, respectively. The total energies of the alloys with specific concentrations of constituents (x = 0.0, 0.25, 0.50, 0.75,and 1.0) are calculated by full potential-linearized-augmented plane waves (FP-LAPW) method based on density functional theory (DFT). The exchange-correlation energy of electrons is defined by gradient generalized density approximation based on Perdew–Burke–Ernzerhof functional [29]. In FP-LAPW calculational steps implemented in WIEN2k code [30], the unit cell is partitioned into nonoverlapping muffin-tin spheres around the atomic sites. Basis functions are expanded in combinations of spherical harmonic functions inside the nonoverlapping spheres. In the interstitial region, a plane wave basis is used and expansion is limited with a cutoff parameter, RMT KMAX = 7. Here, RMT is the minimum radius of the sphere in the unit cell, KMAX is the magnitude of the largest K vector used in the plane wave expansion. The muffin-tin radius is adopted to be 2.0, 1.5, 1.8, and 2.0 a.u. for In, N, As, and Sb atoms, respectively. In the calculations, the electrons of In, N, As, and Sb atoms in 4s2 4p6 4d10 5s2 5p1 , 2s2 2p3 , 3d10 4s2 4p3 , and 4p6 4d10 5s2 5p3 shells, respectively, are treated as valence electrons by choosing a cutoff energy of −8 Ry. The expansion of spherical harmonic functions inside the muffin-tin spheres is truncated at l = 10. The cutoff for fourier expansion of the charge density and potential in √ the interstitial region is taken to be GMAX = 14 Ry. The self consistent calculations are carried on until the total energy difference is less than 0.1 mRy. The electronic band calculations of the alloys are performed using 3 × 3 × 3 grids correspond to 27 k points in the irreducible wedge of the Brillouin zone. In the present work, the total energies of the alloys calculated for certain values of x are minimized with respect to different volumes of the unit cells. The equilibrium lattice constants of InNx As1−x , InNx Sb1−x , and InAsx Sb1−x alloys correspond to the minimum total energies are obtained by fitting of the variation of the total energy to the Murnaghan’s equation of state [31]. In the next step, the variation of the lattice constant of the alloys as a function of x is fitted to the polynomial function aABx C1−x = xaAB + (1 − x)aAC − bx(1 − x)
(1)
to extract the equilibrium lattice constants of the alloys having concentrations of constituents different than x = 0.0, 0.25, 0.50, 0.75, and1.0. Here, aAB , aAB , and aABx C1−x are the equilibrium lattice constants of the compounds and the alloys considered in this work, respectively. The bowing parameter, b, shows the deviation of the equilibrium lattice constants from Vegard’s linear rule [32]. Since the application of the first principles calculational methods on nonsymmetric systems consumes too much calculational time, the semiempirical and empirical methods have been required to calculate the electronic band structures of the alloys for full range of the composition(s). In the present work, electronic band structures of InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys are calculated by ETB for total range of x(0.0 < x < 1.0). ETB method has received considerable attention because of its intuitive simplicity and its realistic description of structural and dielectric properties in terms of chemical bonds. Slater–Koster model [33] of ETB was extensively used with minimal sp3 basis
505
and interaction only between nearest neighbor atoms to find out the valence band energy dispersion of elements and compounds. But this model fails to calculate the indirect gap of semiconductors satisfactorily, especially at X point. This deficiency was tried to be overcome by including an excited s state, s∗ , to the interactions in Vogl’s ETB model [34]. On the other hand, the successfully reproduced indirect energy gaps of III–V compounds by pseudopotential (PP) method [35] showed that, the influence of d-orbitals to the lowest conduction state is large at Γ , X and L symmetry points. In the present nearest neighbor ETB calculations this fact is considered and the excited first and second d-orbitals, d1 (dz2 ) and d2 (dx2 −y2 ), are taken into account as well as sp3 . In our previous work [36], the nearest neighbor interactions between anion and cation with sp3 d2 basis was defined by 13 different types of ETB energy parameters. They are Esa , Epa , Eda , Esc , Epc , Edc , Vss , Vxx , Vxy , Vpasc , Vpcsa , Vdapc , Vdcpa . In ETB calculations of InN, InAs and InSb, they were formulated by the solution of secular equation at Γ and X symmetry points. In the same work, ETB energy parameters of InN, InAs and InSb were derived from the first principles band structure calculations of the corresponding compounds by the fitting process. Since ETB interaction energy parameters reproduced the energy band gaps of the compounds correctly at Γ and X [36], we have surely used them in the band gap engineering calculations of InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys. Since only the nearest neighbor interactions are taken into account, the self energy parameters (Esa , Epa , Eda , Esc , Epc and Edc ) of In, N, As, and Sb are included in the present calculations directly. The other ETB energy parameters (Vss , Vxx , Vxy , Vpasc , Vpcsa , Vdapc , Vdcpa ) needed for the electronic band structure calculations of the alloys are defined by Vegard’s law [32] and scaled by r −2 [37]; particulary, Vss is formulated as VssABx C1−x =
2 ×V 2 x(dAB ssAB ) + (1 − x)(dAC × VssAC ) 2 dAB x C1−x
(2)
2 , d 2 and d 2 Here, dAB AC ABx C1−x are the square of the bond length of the compounds and the alloys, respectively. VssAB and VssAC define s-orbital interactions between an anion and cation in AB and AC compounds, respectively. The rest of the interaction parameters of InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys are calculated using similar formulations. In the next step of ETB calculations, electronic band structures of InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys are obtained by the solution of secular equation in the unit cells of the alloys for different concentrations of constituents along different momentum directions. Electronic band structures of the alloys are analyzed for the variation of the fundamental energy gap and the conduction band edge around Γ point with respect to the concentration of constituents. The bowing parameter, b, which shows the deviation of the fundamental energy gap from linearity is calculated by the best fit of our results to the following expression:
EgΓAB
x C1−x
(x) = xEgΓAB + (1 − x)EgΓAC − bx(1 − x)
Here, EgΓAB , EgΓAC and EgΓAB
x C1−x
(3)
are the fundamental band gap energies of the
compounds and alloys considered in this work, respectively.
3. Results and discussion 3.1. InNx As1−x In III–V semiconductors, the replacement of a few percent of the group V element by small, highly electronegative and isoelectronic nitrogen atoms results in a drastic reduction of the fundamental band gap of approximately 100 meV per atomic percent of nitrogen [38,39]. This effect of N has been confirmed for InNx As1−x alloys experimentally [8,12,13] with the concentration x < 18%. InNx As1−x alloys with N concentration range of 0.0 < x < 0.061 have been grown first by MOCVD work of Naoi et al. [8]. They indicated that, all their samples had direct transition band structures and the increase in nitrogen content made the band energy smaller at each step.
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Esa Esc Epa Epc Eda Edc Vss Vxx Vxy Vpcsa Vpasc Vdapc Vdcpa
Fig. 1. Lattice constant vs. composition (x) for InNx As1−x alloys.
In their work, the minimum band gap energy was reported to be 0.12 eV for InAs0.939 N0.061 alloy. Therefore, the band gap of InAs was able to be decreased by approximately 260 meV by inclusion of only 6.1% of nitrogen at room temperature. In another work [12], the photoluminescence (PL) peak energy was observed to be decreased with N composition in the InAsN layer of InAsN/InGaAsP multiple quantum well; 1% nitrogen increase in alloy composition caused a shrinkage of 31 meV on the transition energy at 10 K. The smallest transition energy was reported as 0.190 eV when the nitrogen content in InAsN wells was determined to be 18% for InNAs/GaAs multiple quantum wells [13]. In theoretical side, the variation of the band gap of InNx As1−x ordered alloys as a function of x has been studied by Yang et al. [9] using tight binding method for x = 0.25, 0.5 and 0.75. The band gap of InAs calculated in this work [9] was decreased by ∼ 0.48 eV by the inclusion of 50% of nitrogen. In a recent work [10], the tight-binding calculations have been carried on both maximally N-rich and As-rich cluster configurations in InNx As1−x alloys defined in a supercell. This study has indicated that the band gap of InNx As1−x decreased to zero when x = 0.17 and x = 0.50 for maximally N-rich and As-rich cluster configurations, respectively. In the same work, the variation of the band gap of these configurations in InNx As1−x alloys was also studied for x > 0.75 but not for the metallic region. In the present work, the cubic cell (working cell) of each InNx As1−x alloy is defined by the corresponding equilibrium lattice constant obtained by volume optimization. The variation of the equilibrium lattice constant of InNx As1−x alloys as a function of nitrogen concentration is plotted in Fig. 1. The equilibrium lattice constants are found to be deviated from linearity ˚ The large bowing with an upward bowing parameter of −0.64 A. parameter is originated from the large mismatching between the ˚ and InN (4.97 A) ˚ compounds. lattice constants of InAs (6.18 A) Therefore, Vegard’s linear rule [32] which was assumed to be
InN
InAs
InSb
−11.6558 −2.6424 4.6235 4.8241 15.7403 12.6942 −5.8787 5.0545 7.3168 0.8742 4.8020 4.1776 0.0000
−7.9244 −2.8927 0.0000 3.7660 9.3469 7.7964 −5.1076 0.7172 3.6286 4.1592 3.4948 4.7535 0.1638
−5.5942 −3.2074 0.0000 2.8711 7.7705 5.9425 −4.4532 0.2715 2.9182 4.0969 2.8601 4.1220 0.2738
acceptable in the previous two theoretical works [9,10] is not valid for the determination of the lattice constants of InNx As1−x alloys. In the present ETB study, the electronic band structures of InNx As1−x alloys defined by optimized lattice constants are calculated for the full range of nitrogen concentration (0.0 < x < 1.0). ETB energy parameters of InNx As1−x alloys based on sp3 d2 basis and nearest neighbor interactions are given in Table 1. According to electronic band structures, InNx As1−x alloys are all direct gap materials as it was reported in Refs. [8–10]. Fig. 2 displays the variation of EgΓ as a function of x for InNx As1−x alloys. EgΓ starts to decrease from the value of bulk InAs (0.417 eV at 0 K) and reaches to its minimum value at x = 0.51. In Fig. 2, EgΓ passes to negative values at x = 0.1 (cross point). Therefore, InNx As1−x alloys express a metallic behavior in the range of 0.1 < x < 0.89. The variation of EgΓ as a function of nitrogen concentration is described by a large overall bowing parameter of 5.21 eV. The bowing parameter of EgΓ was recommended as 4.22 eV in Ref. [40]. The variation of EgΓ of InNx As1−x alloys defined by Vegard’s linear lattice constants is
Fig. 2. Band-gap energy vs. composition (x) for InNx As1−x alloys defined by optimized (solid line) and Vegard’s linear rule (dotted line) lattice constants.
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also plotted as a function of x in Fig. 2, for comparison. Since the experimental results at room temperature belong to the epitaxially grown InNAs instead of bulk samples, we could not compare the present EgΓ values of 0 K with the PL peak energies directly, for x < 0.061 [8]. In Ref. [8], the PL peak energies correspond to the fundamental gap energies of InNx As1−x were measured to be 0.34, 0.28 and 0.21 eV for x = 0.014, 0.019 and 0.034, respectively. Therefore, the band gap of InNx As1−x is decreased by an amount of 0.04, 0.1, and 0.17 eV, respectively, compared to the band gap value of InAs (0.38 eV) at room temperature. In the present work, the same amount of band gap decrease energies are calculated approximately for the same values of x; they are 0.046, 0.09 and 0.14 eV, respectively compared to the band gap energy of InAs (0.417 eV at 0 K) considered in this work. On the other hand, the present cross point of the bowing (x = 0.10) is more agreement with that (x = 0.15) of the tight-binding calculations for maximally N-rich clusters in InNx As1−x alloys [10] than that (x = 0.5) of ETB calculations for InNAs ordered alloys [9]. The negative energy gaps of InNx As1−x alloys obtained in the present work were not calculated in these works [9,10], but they were predicted. The conduction band profiles of InNx As1−x alloys for x = 0.0, 0.02, 0.06 and 0.12 are shown in Fig. 3. It is observed that the conduction band edge is nonparabolic for intrinsic InAs as it was reported in the literature [37]. In view of Fig. 3, we note that the increase of N concentration decreases the energy of the conduction band edge and causes the interaction between the nitrogen level and the conduction band edge in InNx As1−x alloys. Furthermore, the increase of N concentration in InNx As1−x alloys increases the curvature of the conduction band edge around Γ point. Therefore, a small amount of N concentration decreases the corresponding electron effective masses in InNx As1−x alloys around Γ points.
Fig. 3. The conduction band dispersion calculated for InNx As1−x alloys with x = 0% (solid line), 2% (dotted line), 6% (dashed line) and 12% (star line) around Γ point at 0 K.
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3.2. InNx Sb1−x InSb has the smallest band gap of any of the binary III–V semiconductors with an atmospheric transition window in the rage of mid-infrared (3–5 m). It was experimentally indicated that [14–17], this atmospheric transition can access the range of 8–15 m when a small fraction of antimony is replaced by nitrogen in InSb. Time resolved optical measurements [15] have been used to observe an absorption edge of 15 m at 290 K in an alloy with a nominal composition of InN0.006 As0.994 . In a recent work [17], the variation of the band gap of InSb through incorporation of nitrogen was assessed optically by observation of the emission spectra from InNSb LEDs and the transmission of light through the material. In the same work [17], the experimental results − →→ at room temperature were modeled by k .− p method and the sharp band gap decrease of InNx Sb1−x was reported for nitrogen concentration in the range of 0.0 < x < 0.01. The smallest band gap has been calculated to be 0.065 eV with respect to nitrogen concentration of 0.01 [17]. On the other hand, the highresolution electron energy loss spectroscopy (HREELS) of the nitrided layer in InNx Sb1−x alloys indicated that a negative band gap alloy has been formed by the nitrogen-induced shift of the conduction band [14]. In the same work, the room temperature − →→ HREELS results of InNSb alloys were modeled by k .− p calculations and the negative bowing of the band gap was obtained − →→ for a wider range of 0.0 < x < 0.1. Besides, the k .− p calculational results [14] have given negative band gap in the nitrogen range of 0.02 < x < 0.1; the minimum of the conduction band was below the maximum of the valance band indicated that the alloys were semimetallic. In the present work, the cubic unit cells of InNx Sb1−x alloys are defined by the corresponding optimized lattice constants. The variation of the equilibrium lattice constant of InNx Sb1−x alloys as a function of nitrogen concentration in the range of 0.0 < x < 1.0 is plotted in Fig. 4. It is observed that the equi-
Fig. 4. Lattice constant vs. composition (x) for InNx Sb1−x alloys.
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Fig. 5. Band-gap energy vs. composition (x) for InNx Sb1−x alloys defined by optimized (solid line) and Vegard’s linear rule (dotted line) lattice constants.
librium lattice constants of InNx Sb1−x alloys are deviated from ˚ The linearity with an upward bowing parameter of −1.13 A. large bowing parameter due to the large mismatching between ˚ and InN (4.97 A) ˚ comthe lattice constants of InSb (6.63 A) pounds indicates that Vegard’s linear rule is not valid for the unit cell definition of InNx Sb1−x alloys. The corresponding ETB energy parameters used in the electronic band structure calculations of InNx Sb1−x alloys are tabulated in Table 1. According to the electronic band structure calculations, InNx Sb1−x alloys are direct gap materials for each nitrogen concentration in the range of 0.0 < x < 1.0. Fig. 5 displays the variation of EgΓ due to the concentration of nitrogen for InNx Sb1−x alloys. The variation has a large bowing with the smallest band gap energy of 0.9 eV at x = 0.54. InNx Sb1−x alloys express a metallic behavior in the range of 0.06 < x < 0.91. The overall bowing parameter is calculated to be 4.78 eV. The variation of EgΓ of InNx Sb1−x alloys defined by Vegard’s linear lattice constants is also plotted as a function of x in Fig. 5, for comparison. According to the recent resolved optical measurements [15], HREELS [14] and response wavelength measurements of LEDs [17], the band gap of the epitaxially grown InNx Sb1−x alloys had a very sharp variation for small concentrations of nitrogen; the EgΓ decreased by ∼ 100 meV with respect to room temperature band gap value of InSb (0.18 eV) in nitrogen concentration range of 0.0 < x < 0.006. In the same range, the present reduction is calculated to be only 15 meV with respect to the band gap − →→ value of InSb (0.235 eV) at 0 K. Therefore, the k .− p model calculations [14,17] of the experimental band gap values give a cross point (x = 0.022, 0.011) smaller than ours (x = 0.06) normally. In the present work, the smallest band gap energy belongs to InN0.54 Sb0.46 is −0.9 eV, but in Ref. [14] it belongs to InN0.1 Sb0.9 alloy with the value of −0.5 eV. Fig. 6 displays the variation of the conduction band edge of InNx Sb1−x alloys around Γ point for x = 0.0, 0.02, 0.06 and 0.14. In Fig. 6, the conduction band edge is decreased in
Fig. 6. The conduction band dispersion calculated for InNx Sb1−x alloys with x = 0% (solid line), 2% (dotted line), 6% (dashed line) and 14% (star line) around Γ point at 0 K.
InNx Sb1−x alloys with increase of nitrogen concentration. Furthermore the curvature of the conduction band edge is increased by increase of nitrogen concentration. Therefore, a small amount of nitrogen concentration in InNx Sb1−x alloys is sufficient to increase the speed of the conduction electrons. This result was − →→ also obtained by k .− p model calculations of InNx Sb1−x alloys [15,16]. 3.3. InAsx Sb1−x InAsx Sb1−x alloys with small amount of As contents have been attracted considerable interest because of their potentials for the fabrication of infrared sources and detectors. Although the energy gap of InAs (0.417 eV at 0 K) is larger than that of InSb (0.235 eV at 0 K) by 0.18 eV, a small amount of As in InSb is sufficient to decrease the band gap of InSb to the values correspond to the long wavelengths in 8–12.5 m spectral range. The variation of the optical properties of InAsx Sb1−x was first investigated on bulk-grown polycrystalline samples by Woolley and Warner [18]. The lowest energy gap was measured to be 0.1 eV in this work [18] for x = 0.35. InAsx Sb1−x materials over the complete composition range (0.0 < x < 1.0) have been grown on InAs and characterized by X-ray diffraction and optical absorption measurements at room temperature and 10 K [21,22]. The band gap energies of these materials correspond to the cutoff wavelengths of 12.5 and 8.7 m were measured to be 0.099 and 0.14 eV for x = 0.32 at room temperature and 10 K, respectively [21,22]. In Ref. [41], the lowest band gap of InAsx Sb1−x has been reported to be 0.1 eV for x = 0.4. Huang et al. [25] investigated the band gap variation of InAsx Sb1−x alloys as a function of complete composition range by PL spectra and measured the lowest band gap value of 0.14 eV for x = 0.4 at 10 K. In a recent work [26], the infrared transmission spec-
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Fig. 7. Lattice constant vs. composition (x) for InAsx Sb1−x alloys.
tra of InAsx Sb1−x alloys in single bulk crystal structure show a continuous decrease in optical energy gap with the increase of As content in InSb. In this work, the absorption edges correspond to the fundamental energy gaps were measured to be 0.16 and 0.15 eV at room temperature for x = 0.02 and 0.05, respectively. To the knowledge of authors, there are only two theoretical works [19,27], in which the band gap bowing of the of InAsx Sb1−x alloys has been predicted by empirical pseudopotential calculations. In the present work, the cubic unit cells of InAsx Sb1−x ordered alloys are defined by the optimized lattice constants. The variation of the equilibrium lattice constant of InAsx Sb1−x alloys as a function of x is plotted in Fig. 7. It is found that the deviation of the optimized lattice constants from linearity is relatively small. Therefore, Vegard’s linear rule can be a good approximation to obtain the lattice constants of InAsx Sb1−x alloys. The small bowing parameter due to the small mismatching ˚ and InAs (6.18 A) ˚ between the lattice constants of InSb (6.63 A) ˚ The present lattice compounds is calculated to be −0.063 A. ˚ belong to InAs0.05 Sb0.95 constant values of 6.61 and 6.47 A and InAs0.35 Sb0.65 alloys, respectively are found to be close ˚ [21] measured at room to the values of 6.4606 [26] and 6.3 A temperature for the same alloys. In the present work, the electronic band structures of InAsx Sb1−x are calculated by ETB using the energy parameters given in Table 1. It is found that InAsx Sb1−x alloys are direct gap materials for each As concentration in the range of 0.0 < x < 1.0. Fig. 8 shows the variation of EgΓ as a function of x. In the same figure the fundamental gap energies of InAsx Sb1−x alloys determined by the lattice constants of Vegard’s linear rule are also plotted with respect to x, for comparison. In view of Fig. 8, we note that the overall bowing for Γ point transition is large and downward. The energy gap starts to decrease from the band gap value of InSb (0.235 eV at 0 K) and reaches to the lowest value of −0.095 eV at x = 0.41. Furthermore, InAsx Sb1−x alloys are metallic in the range of 0.19 < x < 0.65. The pre-
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Fig. 8. Band-gap energy vs. composition (x) for InAsx Sb1−x alloys defined by optimized (solid line) and Vegard’s linear rule (dotted line) lattice constants.
dicted bowing parameter by the best fit of our results to the expression given in Eq. (3) is 1.72 eV. In the literature, the bowing parameter was predicted to be 0.596 eV at room temperature by optical absorption [20]. The bowing parameter of InAsSb alloys estimated from low temperature PL measurements was reported to be higher than the value measured at room temperature; it was 0.6853 [22] and 0.672 eV [23] at 0 K. The bowing parameters predicted by two EPP calculations were 0.7 [19] and 0.72 eV [27] for InAs0.5 Sb0.5 alloy. Since the present ETB energy parameters are adjusted to have only the band gap values of bulk InSb and InAs at 0 K, the present overall bowing parameter is found to be quiet high with respect to the reported values in the literature [19,20,22,23,27]. The present band gap values of InAsx Sb1−x alloys have been compared with the PL spectra energies measured at T = 10 and 79 K for certain concentrations of As. The present band gap value of 0.158 eV is 21% smaller than the PL value [22] of 0.20 eV for x = 0.05. The discrepancy increases to 40% when the concentration of As is 0.08 [22]. On the other hand, the present band gap value of 0.231 eV is 25% smaller than the PL value of 0.31 eV for x = 0.87 (As-rich InAsSb alloy) [22]. In this experimental study [22], the end point energy (at x = 1) corresponds to the band gap value of InAs was measured to be ∼ 0.5 eV instead of ∼ 0.4 eV. However, the present end point energy at x = 1 is in good agreement with the value of literature. The present band gap value of 0.03 for x = 0.15 is much smaller than the corresponding PL value [25] of 0.135 eV measured at 10 K. On the other hand, the calculated band gap value of 0.18 eV at x = 0.83 (As-rich InAsSb alloy) is greater than the corresponding low temperature PL value of 0.13 eV [25]. The present band gap value of 0.298 eV for x = 0.917 is found to be 12% smaller than the measured value of 0.34 eV for InAs0.917 Sb0.083 alloy grown on a lattice matched substrate (GaSb) at 79 K [24]. The PL peak energies of InAs0.71 Sb0.29 and InAs0.85 Sb0.15 alloys at 10 K were reported to be 0.2 and 0.28 eV, respectively [23]. The corresponding values are found to be 0.050 and 0.210 eV by the
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InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys express a metallic behavior in the range of 0.1 < x < 0.89, 0.06 < x < 0.91, and 0.19 < x < 0.65, respectively. The variation of the fundamental band gaps of InNx Sb1−x alloys is sharper than that of InNx As1−x alloys for small concentrations of N. The band gap bowing parameter is relatively large in both InNx As1−x and InNx Sb1−x compared to InAsx Sb1−x alloys. Since the band structure of the materials depend closely on temperature, there is not a good agreement between the present band gaps of the alloys (calculated at 0 K) and the ones measured at low and room temperatures. However, the present ETB results are able to provide a typical feature for the band gap bowings of InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys. As a final conclusion, a small amount of N and As in InNx As1−x , InNx Sb1−x and InAsx Sb1−x alloys, respectively are sufficient to decrease the corresponding effective masses of the electrons around Γ points. This decrease is more pronounced in InNx Sb1−x than in InNx As1−x alloys. References Fig. 9. The conduction band dispersion calculated for InAsx Sb1−x alloys with x = 0% (solid line), 2% (dotted line), 6% (dashed line) and 15% (star line) around Γ point at 0 K.
present ETB calculations. According to the present results, the concentration of As (x = 0.41) corresponds to the lowest band gap of InAsx Sb1−x is very close to the value of x = 0.38 given by PL measurements [22]. Fig. 9 displays the variation of the conduction band edge of InAsx Sb1−x alloys around Γ point for x = 0, 0.02, 0.06 and 0.15. It is found that the increase of the concentration of As in InAsx Sb1−x increases the curvature of the conduction band edge around Γ point relatively. Therefore, a small amount of As concentration in InAsx Sb1−x alloys decreases the corresponding effective masses of the conduction electrons around Γ points. 4. Conclusion In the present work, the electronic band structures of ternary alloys, InNx As1−x , InNx Sb1−x , InAsx Sb1−x , are studied as a function of nitrogen and arsenic concentrations in the range of 0.0 < x < 1.0 by ETB method using sp3 d2 basis and nearest neighbor interactions. The cubic cells of the alloys are defined by the optimized lattice constants. Because of the large mismatching between the lattice constants of InN and InAs, InN and InSb adjacent compounds, a large deviation of the equilibrium lattice constants of InNx As1−x and InNx Sb1−x alloys from Vegard’s linear lattice constants is built. However, this deviation is found to be relatively small for InAsx Sb1−x alloys due to the small mismatching between the lattice constants of InSb and InAs adjacent compounds. Therefore, Vegard’s linear rule can be a good approximation for the definition of the lattice constants of InAsx Sb1−x alloys, but it can be considered to be weak for InNx As1−x and InNx Sb1−x alloys. The present band structure calculations showed that all the alloys considered in this work are direct gap materials. Besides,
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