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Optoelectronic properties of GaN, AlN, and GaAlN alloys
Optik 126 (2015) 3357–3361
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Optik journal homepage: www.elsevier.de/ijleo
Optoelectronic properties of GaN, AlN, and GaAlN alloys Mingzhu Yang a,∗ , Benkang Chang a , Guanghui Hao a , Honggang Wang a , Meishan Wang b a b
Institute of Electronic Engineering and Optoelectronic Technology, Nanjing University of Science and Technology, Nanjing 210094, China School of Physics, Ludong University, Yantai 264025, China
a r t i c l e
i n f o
Article history: Received 9 June 2014 Accepted 14 July 2015 Keywords: GaN, AlN, and GaAlN alloys First principle calculation Band structure Mulliken charge distribution Optical properties
a b s t r a c t As advanced semiconductor materials, GaN, AlN, and GaAlN alloys are widely used in optoelectronic devices. In order to research the optoelectronic properties of GaN, AlN, and Ga1−x Alx N with different Al component, models of GaN, Ga0.875 Al0.125 N, Ga0.750 Al0.250 N, Ga0.625 Al0.375 N, Ga0.500 Al0.500 N, and AlN were built, then the atomic structure, band structure, density of states, Mulliken charge distribution, and optical properties of the six crystals were calculated based on first principle calculations. Results show that the lattice parameters decrease while the band gap increases with the increase of Al component. The GaN, AlN, and GaAlN alloys are all semiconductors with direct band gap. The global transfer index increases with the increase of Al component. When the Al component is bigger and bigger, the absorption peak shifts to higher energy, and the threshold wavelength of Ga1−x Alx N decreases. © 2015 Published by Elsevier GmbH.
1. Introduction
2. Method of calculation
The Ш-nitrides are wide-band-gap semiconductors with low compressibility, chemical and radiation inertness, and good thermal stability [1]. This makes the nitrides suitable for hightemperature optoelectronic devices, which can operate in harsh environments [2]. As a direct band gap material, GaAlN is well suited for detecting light at energies higher than its band gap energy and providing a large rejection at lower energies [3]. The band gap can be adjusted in order to correspond to a wavelength of 360 nm (for GaN), to 200 nm (for AlN). With an aluminum content of 45%, the cutoff wavelength is about 280 nm, the detectors become insensitive to solar radiations on earth, and is said to be solar blind. With an aluminum content of 30%, the band gap energy corresponds to a band gap of 305 nm adapted to control of burning furnace [4]. In order to clarify the optoelectronic properties of GaN, AlN, and GaAlN with different Al content. Models of GaN, Ga0.875 Al0.125 N, Ga0.750 Al0.250 N, Ga0.625 Al0.375 N, Ga0.500 Al0.500 N, and AlN were built, and the band structures, density of states(DOS), Mulliken charge distribution, absorption coefficient, dielectric function, reflectivity were calculated based on first principle calculations. This work lays theoretical foundation for the preparation and application of the GaN, AlN, and GaAlN based optoelectronic devices.
Wurtzite GaN and AlN both belong to P63mc space group, and 4 [5]. GaN has a hexagonal structure which is the symmetry is C6V formatted by reverse sleeve of hexagonal close packed Ga and N, while AlN is composed of Al and N. In the models of GaN (2 × 2 × 2) supercell shown in Fig. 1(a), there are 16 Ga atoms and 16 N atoms. When two of the Ga atoms are substituted by Al atoms, the model of Ga0.875 Al0.125 N was obtained. The Ga0.750 Al0.250 N, Ga0.625 Al0.375 N, and Ga0.500 Al0.500 N models are obtained by subsisting four, six, and eight of the Ga atoms with Al atoms. The six models are shown in Fig. 1. All calculations in this paper were performed with the quantum mechanics program of Cambridge Serial Total Energy Package (CASTEP) [6]. The Broyden Flecher Goldfarb Shanno (BFGS) algorithm was used to optimize the crystal models. The calculations were taken with an energy cutoff of 450 eV. All calculations were performed with a plane-wave pseudo potential method based on density function theory (DFT) combined with the generalized gradient approximation (GGA) [7,8]. The integral in the Brillouin zone was sampled with the Monkhorst–Pack [9] scheme and the number of k high symmetry points is 9 × 9 ×9. All calculations were carried in reciprocal space with Ga:3d10 4s2 4p1 , Al:3s2 3p1 and N:2s2 2p3 as the valence electrons. The scissors operator correction was used for the optical properties calculation. 3. Results and discussion 3.1. Structural properties
∗ Corresponding author. Tel.: +86 18066086052; fax: +86 025 84315177. E-mail address: [email protected] (M. Yang). http://dx.doi.org/10.1016/j.ijleo.2015.07.096 0030-4026/© 2015 Published by Elsevier GmbH.
The lattice parameters and average bond lengths of GaN, AlN, and GaAlN with different Al component are shown in Table 1.
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4.5 Band gap Vegard's Law
Band gap(eV)
4.0 3.5 3.0 2.5 2.0 1.5 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Al component Fig. 3. Band gaps and the band gap curves obtained based on Vegard’s law. Fig. 1. Models: (a) GaN, (b) Ga0.875 Al0.125 N, (c) Ga0.750 Al0.250 N, (d) Ga0.625 Al0.375 N, (e) Ga0.500 Al0.500 N, (f) AlN.
Due to the average bond length of Al–N is smaller than that of Ga–N, the lattice parameters decrease monotonically with the increase of Al component. According to Vegard’s law [10], the lattice parameters of Ga1−x Alx N can be calculated as follows: a(x) = x · aAlN + (1 − x)aGaN
(1)
a(x) = x · aAlN + (1 − x)aGaN
(2)
where a(x) and c(x) represent the lattice parameters of Ga1−x Alx N alloys, aGaN , cGaN , aAlN , and cAlN represent lattice parameters of GaN and AlN, respectively. x is the Al component. Meanwhile, the calculated values and the experimental values are both not in good agreement with the Vegard’s law, so deviation coefficients were used to correct the results: a(x) = x · aAlN + (1 − x)aGaN − ıa · x · (1 − x)
(3)
c(x) = x · cAlN + (1 − x)cGaN − ıc · x · (1 − x)
(4)
where ıa and ıc stand for bowing parameters. The calculated lattice parameters and the curves obtained based on Vegard’s law are shown in Fig. 2. By fitting the calculated results, the deviation coefficients of a and c are −0.006 ± 0.010 and −0.014 ± 0.022, respectively, which show that our calculated results are in good agreement with Vegard’s law.
values by 30−50% due to the well-known DFT band gap underestimation. According to Vegard’s law [10], the band gaps of Ga1−x Alx N can be calculated as follows: Eg (x) = x · Eg,AlN + (1 − x)Eg,GaN
(5)
where Eg (x), Eg,AlN , Eg,GaN represent the band gaps of Ga1−x Alx N, AlN, and GaN respectively, x is the Al component. Meanwhile there are some difference between the values calculated by first principles and the values based on Vegard’s law. So a bowing parameter b was introduced. Then the band gap can be obtained according to: Eg (x) = x · Eg,AlN + (1 − x)Eg,GaN − b · x · (1 − x)
(6)
The calculated band gaps and the band gap curves obtained based on Vegard’s law are shown in Fig. 3. By fitting, the bowing parameter b is 0.685 ± 0.531. The band structures of the six models are shown in Fig. 4, in which the dotted lines represent Fermi level. From Fig. 4, it can be found that GaN, AlN and GaAlN alloys are all semiconductors with direct band gap. The valence band maximum and the conduction band minimum are all located G point, which are useful for electrons traveling from valence band to conduction band. 3.3. Density of states The total DOS and partial DOS of the six models are shown in Fig. 5.
3.2. Band structure Based on first principle calculation, the obtained band gaps of six models are 1.63 eV, 1.879 eV, 2.261 eV, 2.272 eV, 2.646 eV, and 4.27 eV, respectively, which are smaller than the experimental
Fig. 2. Lattice parameters calculated by first principles and the curves based on Vegard’s law (a) a = b, (b) c.
Fig. 4. Band structures: (a) GaN, (b) Ga0.875 Al0.125 N, (c) Ga0.750 Al0.250 N, (d) Ga0.625 Al0.375 N, (e) Ga0.500 Al0.500 N, (f) AlN.
M. Yang et al. / Optik 126 (2015) 3357–3361
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Table 1 Average bond lengths and lattice parameters of GaN, AlN, and GaAlN models. Models
GaN Ga0.875 Al0.125 N Ga0.750 Al0.250 N Ga0.625 Al0.375 N Ga0.500 Al0.500 N AlN
Average bond length
Lattice parameters
(Ga–N)[0001]
(Ga–N)⊥[0001]
1.9857 1.9842 1.982 1.9854 1.9895
1.9744 1.9736 1.971 1.9727 1.9735
(Al–N) [0001] 1.9117 1.902 1.9072 1.9054 1.91217
(Al–N)⊥[0001]
a=b
c
1.9115 1.904 1.9061 1.9032 1.89981
3.226 3.213 3.199 3.186 3.180 3.126
5.282 5.250 5.221 5.189 5.145 5.012
From Fig. 5(a), it can be found that there is a peak located at −13.18 eV. The peak is mainly attributed to Ga 3d state electrons, so it decreases with the increase of the Al component. For GaN and AlN, the range of conduction band is 0–20 eV, while the conduction band of GaAlN alloys are narrower, which is in good agreement with the band structures shown in Fig. 4. For the six crystals, the valence conduction bands all consist of three parts. The third part located at −8–0 eV is mainly attributed by Ga 4s, Ga 4p, Al 3s, and Al 3p state electrons.
that stabilizes the linkage and the smaller the covalent contribution [12]. The Mulliken charge distribution and average bond population of GaN, AlN, and GaAlN are shown in Table 2. The ratio between the topological charge (˝) and the nominal oxidation state OS(˝), provides a measurement of the separation from the ideal ionic model for a given basin ˝. The global charge transfer index c of the crystal can then be obtained as the average of those ratios for all atoms that form the unit cell of the crystal [13]:
3.4. Mulliken charge distribution
1 (˝) = c= N OS(˝)
The electron density difference maps of the six crystals are shown in Fig. 6. According to relative electronegativity table of elements put forward by Pauling [11], the relative electronegativity values of Ga, Al, and N are 1.81, 1.61, and 3.04 respectively. The larger the difference in electronegativity, the larger is the electrovalent component
N
˝
(˝)
OS ˝
(7)
The global transfer of GaN, Ga0.875 Al0.125 N, Ga0.875 Al0.125 N, Ga0.750 Al0.250 N, Ga0.625 Al0.375 N, Ga0.500 Al0.500 N, and AlN are 0.3333, 0.3482, 0.3638, 0.3778, 0.3933, and 0.4533 respectively. Mori-Sánchez et al. [13] thinks that the global charge transfer index of Ш-nitrides are from 0.3 to 0.6. The global charge transfer index increases monotonously with the increase of Al component, showing that the polarity of crystals increases when the Al atoms are more and more. In the four GaAlN models, the average bond populations of Al–N bond are bigger than that of Ga–N bond, which also certifies that Al–N bond is more stable.
3.5. Optical properties According to the definitions of direct transition probabilities and Kramers–Krönig dispersion relations, the imaginary and the real parts of the dielectric function can be described [14,15] as follows:
Fig. 5. Total and partial DOS of GaN, AlN, and GaAlN with different Al component: (a) total DOS, (b) partial DOS of Ga atoms, (c) partial DOS of Al atoms, (d) partial DOS of N atoms.
Fig. 6. Electron density difference maps: (a) GaN, (b) Ga0.875 Al0.125 N, (c) Ga0.750 Al0.250 N, (d) Ga0.625 Al0.375 N, (e) Ga0.500 Al0.500 N, (f) AlN.
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Table 2 Mulliken charge distribution and average bond population. Models
Charge distribution
Average bond population
Ga GaN Ga0.875 Al0.125 N Ga0.750 Al0.250 N Ga0.625 Al0.375 N Ga0.500 Al0.500 N AlN
+1.000 +1.013 +1.020 +1.038 +1.0600
Al
N
Ga–N
+1.280 +1.300 +1.290 +1.300 +1.360
−1.000 −1.043 −1.093 −1.134 −1.180 −1.360
1.050 0.580 0.555 0.580 0.579
Al–N 0.620 0.615 0.619 0.624 1.000
Fig. 7. Complex refractive index of GaN, AlN, and GaAlN alloys: (a) refractive index, (b) extinction coefficient.
2e ε1 (ω) = 1 + · 2 ε0 m
BZ (2)
V,C
·
ε2 (ω) =
a · MV,C (K)2
2dK 2
Fig. 8. Absorption coefficient curves of GaN, AlN, and GaAlN alloys.
[EC (K) − EV (K)]/
1
(8)
[EC (K) − EV (K)]2 /2 − ω2
ε0
e 2 mω
·
V,C
2dK
BZ (2)
2
˛=
a · MV,C 2 ı[EC (K) − EV (K) − ω]
(9)
where ε0 is the vacuum dielectric constant, m and e represent the electron mass and electron charge respectively. ω is the angular frequency, C and V represent the conduction band and valence band respectively, and EC (K), EV (K) are the intrinsic level of the conduction band and valence band. BZ is the first Brillouin zone, K represents the electron wave vector, a is the unit vector of the vector potential A, MV,C is the unit of transition matrix. The complex refractive index can be expressed as: N(ω) = n(ω) + ik(ω)
(10)
which can be obtained as follows: 1 n(ω) = √ 2 1 k(ω) = √ 2
ε21 + ε22 ε21 + ε22
1/2 1/2
+ ε1 − ε1
range with positive correlation property is wider and wider as the Al component increases. Absorption coefficient indicates the percentage of light intensity attenuation during spreading through unit distance. Absorption coefficient can be calculated as [14]:
1/2
,
1/2 (11)
The complex refractive index of GaN, AlN, and GaAlN alloys are shown in Fig. 7. From Fig. 7, it can be found that the refractive index values of Ga1−x Alx N at 0 eV decrease as the Al component increases, which is in agreement with the experimental phenomenon [16]. When the refractive index increases as the energy increases, the crystal shows properties of normal dispersion, while the crystal shows abnormal dispersion property when the refractive index and the energy are inversely related. At the low energy range, there is positive correlation between the refractive index and the energy. The low energy
2ωk c
(12)
The absorption coefficient curves of GaN, AlN, and GaAlN alloys are shown in Fig. 8. The absorption coefficients of Ga1−x Alx N with different Al component are all in 105 /cm level, showing that Ga1−x Alx N own good absorption property. The absorption peak of Ga1−x Alx N shifts to higher energy when the Al component increases, and the threshold energy increases. 4. Conclusion Using first principles based on DFT, structural properties, band structures, density of states, Mulliken charge distribution, and optical properties of GaN, AlN and GaAlN alloys with different Al component were obtained. Results show that the lattice parameters decrease with the increase of Al component while the band gap increase monotonously. The deviation coefficients of lattice parameters and band gaps are −0.006 ± 0.010, −0.014 ± 0.022, and 0.685 ± 0.531, respectively. The global transfer index increase with the increase of Al component and the polarity of Al–N bond is bigger than that of Ga–N bond. The refractive index at the lower energy range decreases when the Al component increases, and the absorption peak shifts to higher energy. The threshold wavelength of Ga1−x Alx N is lower and lower when the Al component increases. This work gives theoretical guidance for the design and preparation of the GaAlN based optoelectronic devices. References [1] S. Strite, H. Morkoc, GaN, AlN, and InN. A review, J. Vac. Sci. Technol. B 10 (1992) 1237.
M. Yang et al. / Optik 126 (2015) 3357–3361 [2] C. Persson, R. Ahuja, A. Ferreira da Silva, B. Johansson, First-principle calculations of optical properties of wurtzite AlN and GaN, J. Cryst. Growth 231 (2001) 407–414. [3] G. Chris, C. Van de Walle, J. Stampfl, M.D. Neugebauer, McCluskey, N.M. Johnson, Doping of AlGaN alloys, MRS Internet J. Nitride Semicond. Res. 4S1 (G10.4) (1999). [4] J.L. Reverchon, G. Mazzeo, A. Dussaigne, J.Y. Duboz, Status of AlGaN based focal plane arrays for UV solar blind detection, Proc. SPIE 5964 (2005) 596402. [5] E.A. Albanesi, W.R.L. Lambrecht, B. Segall, Electronic structure and equilibrium properties of Gax Al1−x N alloys, Phys. Rev. B 48 (1993) 1784. [6] M.D. Segall, P.J.D. Lindan, M.J. Probert, C.J. Pickard, P.J. Hasnip, S.J. Clark, M.C. Payne, First-principles simulation: ideas, illustrations and the CASTEP code, J. Phys.: Condens. Matter 14 (2002) 2717–2744. [7] J.P. Perdew, A. Zunger, Self-interaction correction to density-functional approximations for many-electron systems, Phys. Rev. B 23 (1981) 5048–5079. [8] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77 (1996) 3865–3868.
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[9] H.J. Monkhorst, J.D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13 (1976) 5188–5192. [10] R.E. Nahory, M.A. Pollack, W.D. Johnston Jr., R.L. Barns, Band gap versus composition and demonstration of Vegard’s law for In1−x Gax Asy P1−y lattice matched to InP, Appl. Phys. Lett. 33 (1978) 659. [11] L. Pauling, The nature of the chemical bond. IV. The energy of single bonds and the relative electronegativity of atoms, J. Am. Chem. Soc. 54 (1932) 3570–3582. [12] L. Pauling, Nature of the Chemical Bond, 3rd ed., Cornell University Press, Ithaca, 1960, pp. 88. ˜ A classification of covalent, ionic, and [13] P. Mori-Sánchez, A.M. Pendás, V. Luana, metallic solids based on the electron density, J. Am. Chem. Soc. 124 (2002) 14721–14723. [14] K.R. Waters, M.S. Hughes, J. Mobley, J.G. Miller, Differential forms of the Kramers–Krönig dispersion relations, IEEE. T. Ultrason. Ferr. 50 (2003) 68–76. [15] J.P. Long, L.J. Yang, D.M. Li, H.C. Chen, First-principles calculations of structural, electronic, optical and elastic properties of LiEu2 Si3 , Solid State Sci. 20 (2013) 36–39. [16] J.F. Muth, J.D. Brown, M.A.L. Johnson, MRS Proc. 537 (1999) G5.2.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Optoelectronic properties of GaN, AlN, and GaAlN alloys Mingzhu Yang a,∗ , Benkang Chang a , Guanghui Hao a , Honggang Wang a , Meishan Wang b a b
Institute of Electronic Engineering and Optoelectronic Technology, Nanjing University of Science and Technology, Nanjing 210094, China School of Physics, Ludong University, Yantai 264025, China
a r t i c l e
i n f o
Article history: Received 9 June 2014 Accepted 14 July 2015 Keywords: GaN, AlN, and GaAlN alloys First principle calculation Band structure Mulliken charge distribution Optical properties
a b s t r a c t As advanced semiconductor materials, GaN, AlN, and GaAlN alloys are widely used in optoelectronic devices. In order to research the optoelectronic properties of GaN, AlN, and Ga1−x Alx N with different Al component, models of GaN, Ga0.875 Al0.125 N, Ga0.750 Al0.250 N, Ga0.625 Al0.375 N, Ga0.500 Al0.500 N, and AlN were built, then the atomic structure, band structure, density of states, Mulliken charge distribution, and optical properties of the six crystals were calculated based on first principle calculations. Results show that the lattice parameters decrease while the band gap increases with the increase of Al component. The GaN, AlN, and GaAlN alloys are all semiconductors with direct band gap. The global transfer index increases with the increase of Al component. When the Al component is bigger and bigger, the absorption peak shifts to higher energy, and the threshold wavelength of Ga1−x Alx N decreases. © 2015 Published by Elsevier GmbH.
1. Introduction
2. Method of calculation
The Ш-nitrides are wide-band-gap semiconductors with low compressibility, chemical and radiation inertness, and good thermal stability [1]. This makes the nitrides suitable for hightemperature optoelectronic devices, which can operate in harsh environments [2]. As a direct band gap material, GaAlN is well suited for detecting light at energies higher than its band gap energy and providing a large rejection at lower energies [3]. The band gap can be adjusted in order to correspond to a wavelength of 360 nm (for GaN), to 200 nm (for AlN). With an aluminum content of 45%, the cutoff wavelength is about 280 nm, the detectors become insensitive to solar radiations on earth, and is said to be solar blind. With an aluminum content of 30%, the band gap energy corresponds to a band gap of 305 nm adapted to control of burning furnace [4]. In order to clarify the optoelectronic properties of GaN, AlN, and GaAlN with different Al content. Models of GaN, Ga0.875 Al0.125 N, Ga0.750 Al0.250 N, Ga0.625 Al0.375 N, Ga0.500 Al0.500 N, and AlN were built, and the band structures, density of states(DOS), Mulliken charge distribution, absorption coefficient, dielectric function, reflectivity were calculated based on first principle calculations. This work lays theoretical foundation for the preparation and application of the GaN, AlN, and GaAlN based optoelectronic devices.
Wurtzite GaN and AlN both belong to P63mc space group, and 4 [5]. GaN has a hexagonal structure which is the symmetry is C6V formatted by reverse sleeve of hexagonal close packed Ga and N, while AlN is composed of Al and N. In the models of GaN (2 × 2 × 2) supercell shown in Fig. 1(a), there are 16 Ga atoms and 16 N atoms. When two of the Ga atoms are substituted by Al atoms, the model of Ga0.875 Al0.125 N was obtained. The Ga0.750 Al0.250 N, Ga0.625 Al0.375 N, and Ga0.500 Al0.500 N models are obtained by subsisting four, six, and eight of the Ga atoms with Al atoms. The six models are shown in Fig. 1. All calculations in this paper were performed with the quantum mechanics program of Cambridge Serial Total Energy Package (CASTEP) [6]. The Broyden Flecher Goldfarb Shanno (BFGS) algorithm was used to optimize the crystal models. The calculations were taken with an energy cutoff of 450 eV. All calculations were performed with a plane-wave pseudo potential method based on density function theory (DFT) combined with the generalized gradient approximation (GGA) [7,8]. The integral in the Brillouin zone was sampled with the Monkhorst–Pack [9] scheme and the number of k high symmetry points is 9 × 9 ×9. All calculations were carried in reciprocal space with Ga:3d10 4s2 4p1 , Al:3s2 3p1 and N:2s2 2p3 as the valence electrons. The scissors operator correction was used for the optical properties calculation. 3. Results and discussion 3.1. Structural properties
∗ Corresponding author. Tel.: +86 18066086052; fax: +86 025 84315177. E-mail address: [email protected] (M. Yang). http://dx.doi.org/10.1016/j.ijleo.2015.07.096 0030-4026/© 2015 Published by Elsevier GmbH.
The lattice parameters and average bond lengths of GaN, AlN, and GaAlN with different Al component are shown in Table 1.
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M. Yang et al. / Optik 126 (2015) 3357–3361
4.5 Band gap Vegard's Law
Band gap(eV)
4.0 3.5 3.0 2.5 2.0 1.5 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Al component Fig. 3. Band gaps and the band gap curves obtained based on Vegard’s law. Fig. 1. Models: (a) GaN, (b) Ga0.875 Al0.125 N, (c) Ga0.750 Al0.250 N, (d) Ga0.625 Al0.375 N, (e) Ga0.500 Al0.500 N, (f) AlN.
Due to the average bond length of Al–N is smaller than that of Ga–N, the lattice parameters decrease monotonically with the increase of Al component. According to Vegard’s law [10], the lattice parameters of Ga1−x Alx N can be calculated as follows: a(x) = x · aAlN + (1 − x)aGaN
(1)
a(x) = x · aAlN + (1 − x)aGaN
(2)
where a(x) and c(x) represent the lattice parameters of Ga1−x Alx N alloys, aGaN , cGaN , aAlN , and cAlN represent lattice parameters of GaN and AlN, respectively. x is the Al component. Meanwhile, the calculated values and the experimental values are both not in good agreement with the Vegard’s law, so deviation coefficients were used to correct the results: a(x) = x · aAlN + (1 − x)aGaN − ıa · x · (1 − x)
(3)
c(x) = x · cAlN + (1 − x)cGaN − ıc · x · (1 − x)
(4)
where ıa and ıc stand for bowing parameters. The calculated lattice parameters and the curves obtained based on Vegard’s law are shown in Fig. 2. By fitting the calculated results, the deviation coefficients of a and c are −0.006 ± 0.010 and −0.014 ± 0.022, respectively, which show that our calculated results are in good agreement with Vegard’s law.
values by 30−50% due to the well-known DFT band gap underestimation. According to Vegard’s law [10], the band gaps of Ga1−x Alx N can be calculated as follows: Eg (x) = x · Eg,AlN + (1 − x)Eg,GaN
(5)
where Eg (x), Eg,AlN , Eg,GaN represent the band gaps of Ga1−x Alx N, AlN, and GaN respectively, x is the Al component. Meanwhile there are some difference between the values calculated by first principles and the values based on Vegard’s law. So a bowing parameter b was introduced. Then the band gap can be obtained according to: Eg (x) = x · Eg,AlN + (1 − x)Eg,GaN − b · x · (1 − x)
(6)
The calculated band gaps and the band gap curves obtained based on Vegard’s law are shown in Fig. 3. By fitting, the bowing parameter b is 0.685 ± 0.531. The band structures of the six models are shown in Fig. 4, in which the dotted lines represent Fermi level. From Fig. 4, it can be found that GaN, AlN and GaAlN alloys are all semiconductors with direct band gap. The valence band maximum and the conduction band minimum are all located G point, which are useful for electrons traveling from valence band to conduction band. 3.3. Density of states The total DOS and partial DOS of the six models are shown in Fig. 5.
3.2. Band structure Based on first principle calculation, the obtained band gaps of six models are 1.63 eV, 1.879 eV, 2.261 eV, 2.272 eV, 2.646 eV, and 4.27 eV, respectively, which are smaller than the experimental
Fig. 2. Lattice parameters calculated by first principles and the curves based on Vegard’s law (a) a = b, (b) c.
Fig. 4. Band structures: (a) GaN, (b) Ga0.875 Al0.125 N, (c) Ga0.750 Al0.250 N, (d) Ga0.625 Al0.375 N, (e) Ga0.500 Al0.500 N, (f) AlN.
M. Yang et al. / Optik 126 (2015) 3357–3361
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Table 1 Average bond lengths and lattice parameters of GaN, AlN, and GaAlN models. Models
GaN Ga0.875 Al0.125 N Ga0.750 Al0.250 N Ga0.625 Al0.375 N Ga0.500 Al0.500 N AlN
Average bond length
Lattice parameters
(Ga–N)[0001]
(Ga–N)⊥[0001]
1.9857 1.9842 1.982 1.9854 1.9895
1.9744 1.9736 1.971 1.9727 1.9735
(Al–N) [0001] 1.9117 1.902 1.9072 1.9054 1.91217
(Al–N)⊥[0001]
a=b
c
1.9115 1.904 1.9061 1.9032 1.89981
3.226 3.213 3.199 3.186 3.180 3.126
5.282 5.250 5.221 5.189 5.145 5.012
From Fig. 5(a), it can be found that there is a peak located at −13.18 eV. The peak is mainly attributed to Ga 3d state electrons, so it decreases with the increase of the Al component. For GaN and AlN, the range of conduction band is 0–20 eV, while the conduction band of GaAlN alloys are narrower, which is in good agreement with the band structures shown in Fig. 4. For the six crystals, the valence conduction bands all consist of three parts. The third part located at −8–0 eV is mainly attributed by Ga 4s, Ga 4p, Al 3s, and Al 3p state electrons.
that stabilizes the linkage and the smaller the covalent contribution [12]. The Mulliken charge distribution and average bond population of GaN, AlN, and GaAlN are shown in Table 2. The ratio between the topological charge (˝) and the nominal oxidation state OS(˝), provides a measurement of the separation from the ideal ionic model for a given basin ˝. The global charge transfer index c of the crystal can then be obtained as the average of those ratios for all atoms that form the unit cell of the crystal [13]:
3.4. Mulliken charge distribution
1 (˝) = c= N OS(˝)
The electron density difference maps of the six crystals are shown in Fig. 6. According to relative electronegativity table of elements put forward by Pauling [11], the relative electronegativity values of Ga, Al, and N are 1.81, 1.61, and 3.04 respectively. The larger the difference in electronegativity, the larger is the electrovalent component
N
˝
(˝)
OS ˝
(7)
The global transfer of GaN, Ga0.875 Al0.125 N, Ga0.875 Al0.125 N, Ga0.750 Al0.250 N, Ga0.625 Al0.375 N, Ga0.500 Al0.500 N, and AlN are 0.3333, 0.3482, 0.3638, 0.3778, 0.3933, and 0.4533 respectively. Mori-Sánchez et al. [13] thinks that the global charge transfer index of Ш-nitrides are from 0.3 to 0.6. The global charge transfer index increases monotonously with the increase of Al component, showing that the polarity of crystals increases when the Al atoms are more and more. In the four GaAlN models, the average bond populations of Al–N bond are bigger than that of Ga–N bond, which also certifies that Al–N bond is more stable.
3.5. Optical properties According to the definitions of direct transition probabilities and Kramers–Krönig dispersion relations, the imaginary and the real parts of the dielectric function can be described [14,15] as follows:
Fig. 5. Total and partial DOS of GaN, AlN, and GaAlN with different Al component: (a) total DOS, (b) partial DOS of Ga atoms, (c) partial DOS of Al atoms, (d) partial DOS of N atoms.
Fig. 6. Electron density difference maps: (a) GaN, (b) Ga0.875 Al0.125 N, (c) Ga0.750 Al0.250 N, (d) Ga0.625 Al0.375 N, (e) Ga0.500 Al0.500 N, (f) AlN.
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Table 2 Mulliken charge distribution and average bond population. Models
Charge distribution
Average bond population
Ga GaN Ga0.875 Al0.125 N Ga0.750 Al0.250 N Ga0.625 Al0.375 N Ga0.500 Al0.500 N AlN
+1.000 +1.013 +1.020 +1.038 +1.0600
Al
N
Ga–N
+1.280 +1.300 +1.290 +1.300 +1.360
−1.000 −1.043 −1.093 −1.134 −1.180 −1.360
1.050 0.580 0.555 0.580 0.579
Al–N 0.620 0.615 0.619 0.624 1.000
Fig. 7. Complex refractive index of GaN, AlN, and GaAlN alloys: (a) refractive index, (b) extinction coefficient.
2e ε1 (ω) = 1 + · 2 ε0 m
BZ (2)
V,C
·
ε2 (ω) =
a · MV,C (K)2
2dK 2
Fig. 8. Absorption coefficient curves of GaN, AlN, and GaAlN alloys.
[EC (K) − EV (K)]/
1
(8)
[EC (K) − EV (K)]2 /2 − ω2
ε0
e 2 mω
·
V,C
2dK
BZ (2)
2
˛=
a · MV,C 2 ı[EC (K) − EV (K) − ω]
(9)
where ε0 is the vacuum dielectric constant, m and e represent the electron mass and electron charge respectively. ω is the angular frequency, C and V represent the conduction band and valence band respectively, and EC (K), EV (K) are the intrinsic level of the conduction band and valence band. BZ is the first Brillouin zone, K represents the electron wave vector, a is the unit vector of the vector potential A, MV,C is the unit of transition matrix. The complex refractive index can be expressed as: N(ω) = n(ω) + ik(ω)
(10)
which can be obtained as follows: 1 n(ω) = √ 2 1 k(ω) = √ 2
ε21 + ε22 ε21 + ε22
1/2 1/2
+ ε1 − ε1
range with positive correlation property is wider and wider as the Al component increases. Absorption coefficient indicates the percentage of light intensity attenuation during spreading through unit distance. Absorption coefficient can be calculated as [14]:
1/2
,
1/2 (11)
The complex refractive index of GaN, AlN, and GaAlN alloys are shown in Fig. 7. From Fig. 7, it can be found that the refractive index values of Ga1−x Alx N at 0 eV decrease as the Al component increases, which is in agreement with the experimental phenomenon [16]. When the refractive index increases as the energy increases, the crystal shows properties of normal dispersion, while the crystal shows abnormal dispersion property when the refractive index and the energy are inversely related. At the low energy range, there is positive correlation between the refractive index and the energy. The low energy
2ωk c
(12)
The absorption coefficient curves of GaN, AlN, and GaAlN alloys are shown in Fig. 8. The absorption coefficients of Ga1−x Alx N with different Al component are all in 105 /cm level, showing that Ga1−x Alx N own good absorption property. The absorption peak of Ga1−x Alx N shifts to higher energy when the Al component increases, and the threshold energy increases. 4. Conclusion Using first principles based on DFT, structural properties, band structures, density of states, Mulliken charge distribution, and optical properties of GaN, AlN and GaAlN alloys with different Al component were obtained. Results show that the lattice parameters decrease with the increase of Al component while the band gap increase monotonously. The deviation coefficients of lattice parameters and band gaps are −0.006 ± 0.010, −0.014 ± 0.022, and 0.685 ± 0.531, respectively. The global transfer index increase with the increase of Al component and the polarity of Al–N bond is bigger than that of Ga–N bond. The refractive index at the lower energy range decreases when the Al component increases, and the absorption peak shifts to higher energy. The threshold wavelength of Ga1−x Alx N is lower and lower when the Al component increases. This work gives theoretical guidance for the design and preparation of the GaAlN based optoelectronic devices. References [1] S. Strite, H. Morkoc, GaN, AlN, and InN. A review, J. Vac. Sci. Technol. B 10 (1992) 1237.
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