Calculation of the Structural, Electrical, and Optical Properties of κ-al 2 O 3 by Density Functional Theory

Transcription

1 CHINESE JOURNAL OF PHYSICS VOL. 46, NO. 2 APRIL 2008 Calculation of the Structural, Electrical, and Optical Properties of κ-al 2 O 3 by Density Functional Theory S. J. Mousavi, 1, S. M. Hosseini, 2 M. R. Abolhassani, 1, 3 and S. A. Sebt 1 1 Department of Physics, Science and Research Campus, Islamic Azad University, Tehran, Iran. 2 Department of Physics, Ferdowsi University of Mashhad (Material and Electroceramics Laboratory), Mashhad, Iran. 3 Department of Physics, Tarbiat Modares University, Tehran, Iran. (Received May 9, 2007) We have performed a first-principle study of the structural, electronic, and optical properties of κ-al 2 O 3 by means of density functional theory, using the full potential linearized augmented plane wave (FP-LAPW) method along with the generalized gradient approximation (GGA). The relaxed structural parameters are found to be in a very good agreement with the experimental data. The calculated electronic structure and charge density yield a direct band gap of 4.3 ev at the Γ point in the Brillouin zone. It is shown that orthorhombic κ-al 2 O 3 exhibits a biaxial birefringence; the calculated optical properties yield a refraction index of n a (0)=1.815, n b (0)=1.813, and n c (0)=1.808, which is close to the experimental values. PACS numbers: Mb I. INTRODUCTION Alumina (aluminum oxide) is a ceramic material of great interest, both for fundamental studies and for applications. It is a material of considerable technological and industrial significance, because of its hardness, abrasion resistance, mechanical strength, corrosion resistance, good electrical insulation, useful optical properties, fine particle size, high surface area, and catalytic surface activity. Its melting point and electrical conductivity are 2327 K and s/m at 20 C, respectively. Alumina or Al 2 O 3 exhibits a number of different phases, such as α, β, γ, κ, η, θ, and χ alumina. The electronic structure of alumina (Al 2 O 3 ) is increasingly of interest for its variety of applications in optical, electronic, and structural devices. For instance, α-al 2 O 3 is used in electronics, the γ-phase in catalysts, and the κ-phase in wear-resistance coating on cemented-carbide cutting tools [1, 2, 3]. Unlike the α-al 2 O 3 phase, whose structure has been known for a long time, only a few experimental and theoretical studies have been performed on κ-al 2 O 3. The difficulty in obtaining significant amounts of a pure sample of κ-al 2 O 3 and the poor degree of crystallinity have hampered the experimental determination of its electrical and optical properties [4]. Corresponding author. Tel./fax: c 2008 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA

2 VOL. 46 S. J. MOUSAVI, S. M. HOSSEINI, et al. 171 The aim of this paper is to determine some of the basic structural, electronic, and optical properties of κ-al 2 O 3 by first principles methods based on density-functional theory (DFT). II. STRUCTURE Transmission electron microscopy (TEM) and X-ray diffraction studies have shown that κ-al 2 O 3 belongs to the space group pna2 1 [5, 6]. κ-al 2 O 3 has an orthorhombic crystal structure containing eight Al 2 O 3 formula units per unit cell. The unit cell consists of four oxygen layers in a close-packed ABAC stacking sequence and four aluminum layers along the c-axes, with each oxygen layer comprising six oxygen atoms. The unit cell contains 40 atoms in total with 24 oxygen atoms and 16 aluminum atoms [7, 8, 9]. The crystal structure of κ-al 2 O 3 is shown in Fig. 1. A study by Olliver et al. based on XRD, TEM, and SEM concluded that the aluminum atoms inserted between the oxygen layer in both octahedral and tetrahedral positions are in a 3:1 ratio [10]. The number of octahedral and tetrahedral Al atoms in a unit cell is 12 and 4, respectively. The crystal four-fold symmetry (mm2) results in 10 independent atomic positions. The pna2 1 symmetry is as follows: (x,y,z), (-x,-y,z+1/2), (x+1/2,-y+1/2,z), (-x+1/2,y+1/2,z+1/2). FIG. 1: Crystal structure and atomic position of κ-al 2 O 3 [7]. III. METHOD OF CALCULATION The calculation of the structural, electric, and optical properties of κ-al 2 O 3 was carried out with a self-consistent scheme by solving the Kohn-Sham equations, using a FP-LAPW method in the framework of the density functional theory (DFT), along with the generalized gradient approximation (GGA) method [11, 12, 13, 14] using the Wien2k codes [15]. The calculation was performed with 3000 k-points and Rk max =7 (R is the

3 172 CALCULATION OF THE STRUCTURAL, ELECTRIC... VOL. 46 smallest muffin-tin radius and k max is the cut-off wave vector of the plane-wave basis set) for the convergence parameter, for which the calculation stabilizes and convergence in terms of the energy is achieved. The values of the other parameters are G max =14 a 1 0 (G max is the magnitude of the largest vector in the charge density Fourier expansion or the plane wave cutoff, and a 0 is the Bohr radius), R MT (Al)=1.6 au, and R MT (O)=1.7 au (muffin-tin radius). The iteration was halted when the difference charge density was less than e a 3 0 between steps, taken as a convergence criterion. The core cut off energy, which defines the separation of core and valence states, was chosen as 8 Ry. The imaginary part of the dielectric tensor can be computed from the knowledge of the electronic band structure of a solid. In the limit of linear optics, in the non-spinpolarized case, and within the frame of the random phase approximation, we can use the following well-known relations [16]: Iε αβ (ω) = 4πe2 m 2 ω 2 c,v Rε αβ (ω) = δ αβ + 2 π P And the optical conductivity is given by 0 0 dk c k P α v k v k P β c k δ(ε ck ε vk ω), (1) ω Iε αβ (ω ) ϖ 2 ω 2 dω. (2) R σ αβ (ω) = ω 4π Iε αβ(ω), (3) where c k and v k are the electron states in the conduction and valence bands, respectively, with the wave vector k, and P is the Kooshy integral section. Here, ε αβ (ω) = Rε αβ +iiε αβ = ε 1 + iε 2 is the complex dielectric tensor. Knowing the complex dielectric tensor one can calculate various optical constants from the following relations, which describe the system response in respect to the propagation of the electromagnetic wave through the material. The refractive index, n(ω), and the extinction coefficient, k(ω), are given by and ε(ω) + Rε(ω) n(ω) =, 2 (4) ε(ω) Rε(ω) k(ω) =. 2 (5) IV. RESULTS AND DISCUSSION IV-1. Electronic structure and bulk modulus It has long been accepted that the hardest materials possess strongly bonded crystal structures of high symmetry. Hardness is a function of both the strength of the interatomic

4 VOL. 46 S. J. MOUSAVI, S. M. HOSSEINI, et al. 173 bonding and of the rigidity of the lattice framework. Diamond is the hardest known bulk material (with a bulk modulus of approximately 443 GPa), due to strong covalent sp3 bonding in a tetrahedral lattice configuration. κ-al 2 O 3 is classified as a medium hard material; it can be used as a wear-resistance coating on cemented-carbide cutting tools. Thus the hardness, bulk modulus, and shear modulus are important parameters for estimating the hardness of a material in order to assess its use as cutting tool. From the literature, we know that the bulk elastic properties of a material determine how much it will compress under a given amount of external pressure. The ratio of the change in pressure to the fractional volume of compression is called the bulk modulus (B) of the material, which can be written as follows: B = V P V. (6) In term of energy, the bulk modules is also defined by the equation of state (EOS) and evaluated at the minimum: B = V 2 E V 2. (7) The position of the minimum of the EOS defines the equilibrium lattice parameter and unit cell volume at zero pressure. In this calculation, from a series of strained lattices the static lattice potential corresponding to total energy was calculated. From such results the equilibrium volume, bulk modulus, and its pressure derivative was derived. A series of total energy calculations as a function of volume can be fitted to an equation of state according to Murnaghan [17]: E(V ) = B 0V B 0 ( ) B V0 0 / V B C. (8) Here B 0 is an equilibrium bulk modulus that efficiently measures the curvature of the energy versus the volume curve about the relaxed volume V 0, and B 0 is the derivative of the bulk modulus. The calculations were first carried out applying the experimental data for lattice constants, then by minimizing the ratio of the total energy of the crystal to its volume (volume optimization) the theoretical lattice constants were obtained. Table I and II summarizes the results obtained after structural relaxation as well as results of experimental work for comparison [10]. In TABLE II, the residual forces on each atom after relaxation were less than 1 mry/bohr. The calculated lattice parameters a, b, and c as well as the atomic positions are found to be in excellent agreement with their corresponding experimental values. By a comparison of the experimental and calculated data of this table an accuracy of usually < 1% can be seen in the calculations. This agreement confirms the reliability of the calculations and is good encouragement for further study. The calculated bulk modulus and total energy are summarized in Table III.

5 174 CALCULATION OF THE STRUCTURAL, ELECTRIC... VOL. 46 TABLE I: Crystal parameter κ-al 2 O 3 Lattice constant In (Å) Calculated in this work FP-LAPW (GGA96) a b c Experiment [10] Calculated using pseudopotentials (ABINIT-codes) [18] TABLE II: Atomic position of κ-al 2 O 3 Atoms Al(1) Al(2) Al(3) Al(4) O(1) O(2) O(3) O(4) O(5) O(6) Calculated (this work) FP-LAPW (GGA96) Experimental [10] x y z x y z x y z Calculated using pseudopotentials (ABINIT-codes) [18] IV-2. Electronic properties The calculated electronic band structure of κ-al 2 O 3 is shown in Fig. 2. The zero of the energy was set at the top of the valance band. The energy scale is in ev, and the origin of the energy was arbitrarily set to be at the top of valance band. There is a direct band gap of 4.3 ev at the Γ point. The top of the valance band is very flat, which suggests a very large effective hole mass, and therefore we may conclude that the dominating charges in κ-al 2 O 3 carriers are electrons. TABLE III: The bulk modulus and total energy This work Others Bulk modulus (GPa) [7] E tot (ev/atom) [7]

6 VOL. 46 S. J. MOUSAVI, S. M. HOSSEINI, et al. 175 FIG. 2: Electronic band structure of κ-al 2 O 3 The total density of state (DOS) that describes the electron distribution in the energy spectrum is shown in Fig. 3. The electronic state is separated into three regions: lower valence band (LVB), upper valence band (UVB), and conduction band (CB). The LVB density of the state peak is about -16 ev and has a 3.5 ev width that is separated by a large gap of about 9.5 ev from the UVB states which have a width of about 7 ev. FIG. 3: Total density of state of κ-al 2 O 3. Fig. 4 shows the calculated partial density of state for only one oxygen and one aluminum (i.e., O(1) and Al(1) in Table II) atom. There is no sharp peak near the Fermi level for this O-2p, while from Fig. 3 it can be seen there is a narrow sharp peak in the UVB near 0.0 ev. The unit cell of κ-al 2 O 3 has 6 oxygen atoms with different positions. This means that the partial DOSs for each atom are not identical. The calculation of the total DOS of some oxygen atoms (i.e., O(4) in Table II) indicated a sharp peak close to

7 176 CALCULATION OF THE STRUCTURAL, ELECTRIC... VOL. 46 the Fermi level, furthermore the energy bands for each of the atoms are not identical. In this paper we include the total DOS of O for number 1, and also Al number 1 in which there is no peak near the Fermi level. But, when one calculates the total DOS of all the O atoms, all peaks from all the O atoms appear. The electronic states below the Fermi level are dominated by the O-2s and O-2p states for the LVB and UVB, respectively. FIG. 4: Partial density of state of O and Al atoms. IV-3. Optical properties The dielectric tensor for orthorhombic κ-al 2 O 3 is diagonal and has the following form: Iε xx 0 0 Iε = 0 Iε yy 0. (9) 0 0 Iε zz Figs. 5a and 5b show the real, ε 1 (ω), and imaginary, ε 2 (ω), parts of the dielectric functions calculated for κ-al 2 O 3 as a function of the incident photon energy. The value of the static refractive index along the a-axis obtained is n a (0) = Rε a (0) = 3.33 = which is close to the experimental value 1.79 [7]. Our calculated results for ε 2 (ω) show two main peaks at around 10 and 14 ev and two shoulders around 12 and 16 ev. These points are related to the interband transition from valence to conduction band states. Unfortunately, there is not any experimental data for comparison. The origin of different peaks in the dielectric functions is from the interband transition. Because of the selection rules, only transitions that imply a change ι = ±1 in angular momentum are allowed. The peak at 10 ev for ε 2 (ω) is related to the transition

8 VOL. 46 S. J. MOUSAVI, S. M. HOSSEINI, et al. 177 FIG. 5: The (a) real and (b) imaginary parts of the dielectric function of κ-al 2 O 3 from the Al-3s to the Al-3p state in the aluminum atom. The second peak at energy of 14 ev is related to transitions for higher empty states (i.e., Al-3p to Al-3d). These transition energies are much higher than visible light (2 3 ev) and are in the range of the ultraviolet ( ev) spectrum. From Figure (5a), the real part of the dielectric function and the static dielectric function ε 1 (0) are deduced; its value is 3.3. The optical conductivity of κ-al 2 O 3 in terms of energy was calculated and is shown in Fig. 6. The main peak around 15 ev was calculated. As we know from the literature, an exciton is a bound electron-hole pair, usually free to move together through the crystal. These excitons are produced when a photon of energy greater than the energy gap (4.8 ev for κ-al 2 O 3 ) is absorbed in a crystal. These excitons may move through the crystal transporting energy but not charge. Because of its charge neutrality it does not contribute directly to the electrical conductivity. If an insulator contains bound electron-hole pairs it is called an excitonic insulator. However, in the case of κ-al 2 O 3, from the optical conductivity we can see such excitonic features at 10, 15, and 17.5 ev. The EELS function can be deduced from the following relation [16]: I(ε) 1 = ε 2 ε (10) ε2 2 EELS is a valuable tool for investigating various aspects of materials. It has the advantage of covering the complete energy range, including non-scattered and elastically scattered electrons (zero loss), which excite the electrons of the atom s outer shell (valence loss) or valence interband transitions. In Figure 7 the energy loss function is plotted for κ-al 2 O 3. The energy of the main maximum is assigned to be the energy of the volume plasmon, hω p, and is equal to 24.5 ev. The results for the optical constants indicated that κ-al 2 O 3 exhibits biaxial birefringence, also known as trirefringence, which describes an anisotropic material that has more

9 178 CALCULATION OF THE STRUCTURAL, ELECTRIC... VOL. 46 FIG. 6: Optical conductivity of κ-al 2 O 3 FIG. 7: Electron energy loss spectrum of κ-al 2 O 3. than one axis of anisotropy. The refractive index tensor n, will, in general, have three distinct eigenvalues that can be labeled as n a, n b, and n c. The variation of the refraction index with energy is plotted in Fig. 8. The static refractive index is calculated to be 1.8. The calculated of optical properties of κ-al 2 O 3 are summarized in Table IV.

10 VOL. 46 S. J. MOUSAVI, S. M. HOSSEINI, et al. 179 FIG. 8: Refractive index of κ-al 2 O 3 TABLE IV: Optical constants of κ-al 2 O 3 Optical constants This work Static ε 1 (0) ε(xx)=3.30 ε(yy)=3.29 ε(zz)=3.27 Static refractive index n(0) n a (0)=1.815 n b (0)=1.813 n c (0)=1.808 Plasmon energy 24.5eV Excitonic feature Interband Transition 10, 15, and 17.5eV 10eV from Al-3s Al-3p 14eV from Al-3p Al-3d Reference [7] ε 1 = 3.2 n = , 14.5 and eV 15eV V. CONCLUSION We have calculated the structural, electronic, and optical properties of κ-al 2 O 3 using the full potential linearized augmented plane wave (FP-LAPW) method with the generalized gradient approximation (GGA) in the framework of density functional theory. The crystal structure has been optimized; the lattice constant and its equilibrium atomic position was calculated. The total density of state calculation shows that below the Fermi level the O-2p and O-2s states are dominate. The calculations show a static dielectric function along the a-axis 3.3, an electron energy loss spectrum of 24.5 ev, and a refractive index of

11 180 CALCULATION OF THE STRUCTURAL, ELECTRIC... VOL along the a-axis. It was found that the main interband transition in κ-al 2 O 3 is from the Al-3s to the Al-3p state, which is at an energy of 10 ev. References Electronic address: sjmmousavi@yahoo.com [1] R. H. French, J. Am. Ceram. Soc. 73, 477 (1990). [2] M. V. Finnis, J. Phys.: Condens. Matter 8, 5811 (1996). [3] D. G. Cahilla, S. M. Lee, and T. I. Selinder, J. Appl. Phys. 83, 5783 (1998). [4] H. L. Gross and W. Mader, Chem. Commun., (1997). [5] C. Verdozzi et al., Phys. Rev. Lett. 80, 5615 (1998). [6] G. Paglia, A. L. Rohl, C. E. Buckley, and J. D. Gale, J. Mater. Chem. 11, 3310 (2001). [7] B. Holm, R. Ahuja, Y. Yourdshahyan, B. Johansson, and B. I. Lundqvist, Phys. Rev. B 59, (1999) [8] R. W. G. Wyckoff, Crystal Structures, 2 nd ed. (Wiley, New York, 1964). [9] P. Liu and J. Skogsmo, Acta Crystallogr. B 47, 425 (1991). [10] B. Ollivier, R. Retoux, P. Lacorre, D. Massiot, and G. Ferey, Mater. Chem. 7, 1049 (1997). [11] K. Schwarz, J. Sol. St. Chem. 176, 319 (2003). [12] K. Schwarz, P. Blaha, Comput. Mat. Sci. 28, 259 (2003). [13] K. Schwarz, P. Blaha, G. K. H. Madsen, Comput. Phys. Commun. 147, 71 (2002). [14] M. Peterson et al., Comput. Phys. Commun., 126, 294 (2000). [15] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz, Inst. F. Mat. Chem., TU Vienna, [16] F. Wooten, Optical properties of solids (Academic Press, New York, 1972). [17] F. D. Murnaghan, The Compressibility of Media under Extreme Pressures, Proc. Nat. Acad. Sci. 30, 244 (1944). [18] R. Vali and S. M. Hosseini, Comput. Mat. Sci. 29, 138 (2004).