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Ab initio study of the structural and optical properties of orthorhombic ternary nitride crystals
View the table of contents for this issue, or go to the journal homepage for more 2001 J. Phys.: Condens. Matter 13 10417
(http://iopscience.iop.org/0953-8984/13/46/312)
J. Phys.: Condens. Matter 13 (2001) 10417–10431 PII: S0953-8984(01)24829-2
Ab initio study of the structural and optical properties
of orthorhombic ternary nitride crystals
Jung Y Huang1,3, L-C Tang1and M H Lee21Institute of Electro-Optical Engineering, Chiao Tung University, Hsinchu 305, Taiwan, Republic of China
2Department of Physics, Tamkang University, Taipei 251, Taiwan, Republic of China
E-mail: [email protected]
Received 15 May 2001, in final form 27 September 2001 Published 2 November 2001
Online atstacks.iop.org/JPhysCM/13/10417
Abstract
The structural, optical and mechanical properties of orthorhombic ternary nitride crystals have been analysed theoretically with first-principles calculation. Our results indicate that these nitrides possess fairly large optical bandgap (4–6 eV), second-order nonlinear optical susceptibility (13– 18 pm V−1), and bulk modulus 170–370 GPa. Therefore these materials could be useful for optical and protective coating applications. Our analysis with the band-by-band and atomic species projection techniques not only yields useful information about material properties, but also provides deep insight into the fundamental understanding of the mechanical and optical properties of orthorhombic ternary nitrides.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Nitrogen, like carbon, has an inclination to form multiple bonds, which distinguishes it from the other group VB elements. The properties of the numerous covalent nitrides vary greatly depending on the element with which nitrogen is combined. For example, the optical bandgap of compound semiconductors formed from group III elements and nitrogen varies from 6.5 eV for AlN to 2 eV for InN. Therefore group III–N alloys can be used to fabricate optical active devices with output wavelengths covering from the red to the ultraviolet [1]. Nitrides of a number of metals can also form hard and highly stable refractory materials with very high melting points. These materials recently have attracted significant interest for their potentials as protective or wear resistant coating layers.
The chemistry of nitrogen in many aspects of the structure of alloys is still not well understood [2]. Orthorhombic ternary nitrides are relatively new ceramic materials with 3 Addressee for correspondence.
Figure 1. The crystal structure of orthorhombic ternary nitride MgGeN2withP na21symmetry.
potentially interesting thermal, mechanical and optical properties [3]. These crystals are therefore of technological and scientific importance. However to fully realize their potential, sufficient understanding in these ternary nitrides is essential. Electronic structure calculations can provide much insight into materials with existing and possibly new structures.
In this paper, theoretical studies on the electronic structures, mechanical and optical properties of orthorhombic ternary nitrides with first-principles calculation are presented. We do not intend to produce most accurate values for comparing with experimental results since no such data are currently available. Instead we aim at improving the fundamental understanding of the mechanical and optical properties of orthorhombic ternary nitrides.
2. Method
The crystal structure of an orthorhombic ternary nitride ABN2is shown in figure 1. The unit
cell hasP na21symmetry [3]. The metal atoms A, B and the two crystallographically different
N atoms occupy the general positions (x, y, z; ¯x, ¯y, z + 1/2; x + 1/2, ¯y + 1/2, z; ¯x + 1/2, y + 1/2, z + 1/2). Both metal atoms are tetrahedrally coordinated by N and vice versa both N atoms are tetrahedrally coordinated by the two metal atoms. This crystal structure with lattice constantsa, b and c can also be deduced from a hexagonal structure (a = b,c) in which a ≈√3a,b ≈ 2aandc = c≈√8/3a.
The calculations were performed using the ab initio plane-wave-pseudopotential approach within the framework of density-functional theory (DFT) implemented in CASTEP software [4]. The summation over the Brillouin zone (BZ) was carried out with a k-point sampling using a Monkhorst–Pack grid [5]. A kinetic-energy cutoff of 550 eV and 18k special points and 96 bands were used to ensure the convergence in the calculations of optical properties. In order to save computation time, the size of the special point set was reduced to eightk points (3 × 3 × 3 mesh) for the calculations of the equilibrium lattice constants and mechanical properties. We found that the smaller set ofk points causes less than 2% difference in the equilibrium lattice constants.
2.1. Calculations of structural properties
The equilibrium lattice constants and fractional atomic coordinates are determined from the total-energy minimization procedure. We carried out the relaxation of the lattice parameters and atomic positions within constraints imposed by the space-group symmetry elements. Energy– volume relations were obtained by varying unit-cell volumes and the results were fitted to the Murnaghan equation of state [6]
Etot(V ) = B0V B 0 V 0 V B 0 1 B 0− 1 + 1 +E0. (1)
From the fit, an estimate of the static bulk modulus at zero pressureB0, and the first-order
pressure derivative of the bulk modulusB0 can be obtained. This procedure has proved to be quite reliable in the computational design of advanced materials.
In this work, we derive the phonon frequencies of ABN2 with the Hellmann–Feynman
(HF) forces and the direct method [7]. The calculations start from building a 1×1×1 supercell with an optimized geometry. For this configuration, the maximal magnitude of the HF forces does not exceed 0.001 eV Å−1. The HF forces are then calculated for displaced atoms, one at a time. We have displaced atoms A, B, N(1) and N(2) alongx, y, and z directions by an amplitude of 0.03 Å. Each displaced configuration generated 3× 16 = 48 components of HF forces and a total of 576 components have been used to construct the dynamical matrix and deduce the phonon frequencies [8].
2.2. Calculations of electronic properties
To understand the nature of optical transitions and other relevant effects on the calculated optical properties, we study the local densities of states and other electronic properties of these nitride crystals.
Letnk(r) be the self-consistent wave function of the crystal at the nth band and kth point in the BZ; we can decomposenk(r) into a summation over the atomic orbitals {(i)lm(r)} of each atomic speciesi by
nk(r) = i∈{atoms} l m=+l m=−l Cnk,lm(i) (i)lm(r) (2) whereCnk,lm(i) =V 0nk(r) (i)∗
lm (r) dV . The lth orbital of the α species contributes to nk(r)by a fraction ofh(α)nk,l[9] h(α)nk,l = i∈{α} m=+l m=−l C(i)nk,lmCnk,lm(i)∗ i∈{α,β,γ,...} l m=+l m=−l Cnk,lm(i) Cnk,lm(i)∗ −1 . (3)
An atom projection concept was then employed for resolving interesting components from the total density of states (TDOS)
LDOS(α , E) = n k l h(α)nk,lδ(E − Enk) PDOS(α l, E) = n k h(α)nk,lδ(E − Enk). (4)
The partial density states (PDOS) and local density of states (LDOS) can be used to provide valuable insight into the formation of the energy bandgap and the nature of transitions from which the linear and nonlinear optical properties originate.
2.3. Calculations of linear and nonlinear optical properties
In a crystalline solid, the most important optical transitions do not change the momentum or the spin of the electrons involved in the transition. In terms of an energy band structure this means that one has to consider optical excitations from an occupied to unoccupied state of the same spin for eachk vector in the BZ.
The linear optical properties of a dielectric crystalline material can therefore be described with a dielectric function ofεij(q, ω) at q = 0. When the incident photon energy is higher than the bandgapEg, the material can attenuate the photon flux. The absorption coefficient αij(ω) is related to the imaginary part of the dielectric function by [10]
Imεij(ω) = λn(ω) 2π αij(ω) = 8π2¯h2e2 m2V k cv (fc− fv)p i cv(k)pjvc(k) E2 vc δ[Ecv(k) − ¯hω] =5.2324 × 103 V (Å3) k cv pi cv(k, Å−1)pjvc(k, Å−1) E2 vc δ[Ecv(k, eV) − ¯hω]. (5) Herefcrepresents the Fermi distribution of the conduction bandc; picv(k, Å−1) denotes the momentum matrix element (MME, in unit of Å−1) from the conduction bandc to the valence bandv at the k point of the BZ. The real part of the dielectric function was obtained from the imaginary part with Kramer–Kronig transformation.
For the second-order nonlinear optical response, the theoretical description is much more complex [11]. However, it is fairly easy to derive the NLO susceptibility at the zero frequency limits χijk(2)(0) = 1 V e¯h m 3 k vc c 1 EccE2cvEc2v
(Dijkvcc+Dcvcijk+Dcijkcv)
− v
1 EvvE2cvE2cv
Dvijkcv+Dijkvvc+Dcvvijk
(6) whereDnmlijk = Im [pinm(pmlj plnk +pmlk pjln]/2. We are interested in decomposing χijk(2)(0) into various contributions from atomic species or orbitals. This can be properly done by calculating the contribution from theαth species as [12]
χijk(2)(α, E) = 1 V e¯h m 3 l k vc [h(α)vk,lδ(E − Evk) + h(α)ck,lδ(E − Eck)] × c
(Dvcijkc+Dcvcijk+Dijkccv)
EccEcv2Ec2v
−
v
(Dvijkcv+Dvvijkc+Dcvvijk)
EvvE2cvEcv2
. (7)
Equation (7) is very similar to equation (4) except that the second-order nonlinear optical strengths serve as the weighting factor. We shall use this equation to yield insight into the underlying mechanism of NLO susceptibility.
3. Results
3.1. Crystal structure and lattice dynamical properties
The calculated atomic positions of six orthorhombic ternary nitride crystals are summarized in table 1. The tetrahedral coordination of the N atoms is distorted with the presence of two
Table 1. Calculated fractional atomic coordinates of the orthorhombic ternary nitrides. Nitride crystal Fractional coordinates Fractional coordinates Fractional coordinates ABN2(P na21) of metal A of metal B of species N
BeCN2 Be (0.088, 0.677, 0.000) C (0.057, 0.130,−0.007) N1 (0.058, 0.100, 0.345) N 2(0.124, 0.659, 0.390) MgCN2 Mg (0.088, 0.677, 0.000) C (0.046, 0.125, 0.002) N 1 (0.035, 0.061, 0.330) N2 (0.117, 0.670, 0.444) BeSiN2 Be (0.086, 0.656, 0.003) Si (0.078, 0.129,−0.007) N1 (0.083, 0.126, 0.372) N2 (0.096, 0.633, 0.360) MgSiN2 Mg (0.076, 0.625, 0.000) Si (0.068, 0.125, 0.010) N1 (0.051, 0.093, 0.355) N2 (0.112, 0.657, 0.421) BeGeN2 Be (0.115, 0.660, 0.000) Ge (0.082, 0.129,−0.012) N1 (0.070, 0.126, 0.368) N2 (0.091, 0.635, 0.358) MgGeN2 Mg (0.076, 0.625, 0.000) Ge (0.069, 0.125, 0.010) N1 (0.053, 0.096, 0.357) N2 (0.110, 0.654, 0.418)
types of metal–N bond. The average bond length is calculated to be 1.75 Å for Si–N and 2.09 Å for Mg–N. These results agree well with the recent measured values of MgSiN2with
the neutron diffraction method [13]. The contours of charge difference in the MgSiN2crystal
on a plane perpendicular to thez axis are shown in figure 2. The electron density is much lower around the group IV atom (Si) relative to the N, indicating strong directionality of Si–N bonding. Comparing to the group-IV homo-nucleus semiconductors, the stronger bonding between group IV and N atoms leads to a shorter bond length and therefore a larger energy bandgap. This tendency can be better appreciated in figure 3, where comparisons of the optical bandgap of orthorhombic ternary nitride crystals, column IV and IV nitrides as a function of bond length are shown.
In figure 4, we present the calculated energies as a function of unit cell volume for the two orthorhombic ternary nitride crystals BeCN2 and MgCN2. From the calculated results,
the equation-of-state parameters such as the bulk modulus and its pressure derivative were deduced. The results are summarized in table 2. The bulk modulus for MgCN2 was found
to be 234 GPa, which decreases to 175 GPa for MgSiN2 and 171 GPa for MgGeN2. The
tendency is expected from the decreasing bonding strength (and therefore larger unit cell volume) from C–N, Si–N to Ge–N. By replacing Mg with Be, the bulk modulus becomes even larger with a value reaching 374 GPa for BeCN2. These calculated values support the
notion of the orthorhombic ternary nitrides as potential protective coating materials. Note that our calculated bulk modulus of MgSiN2 also agrees with the linear thermal compressibility
K = 6.84 × 10−12Pa−1 (K ≡ −1/V (∂V /∂P )T = 1/B) reported in the literature [13]. For
further comparison, note that Karch et al estimated the bulk modulus ofβ-Ge3N4to be about
214 GPa [15] and Gavrilenko reported 2H-GaN to be 215 GPa [16]. Both crystals possess a similar structure as the ternary nitrides studied here.
The phonons of the orthorhombic ternary nitrides at the. point can be classified by the irreducible representation of the point group C2v. The group theory predicts the following
symmetry of the modes: 12A1+ 12A2+ 12B1+ 12B2, where A1, B1and B2modes are both
Raman and infrared active while A2are infrared active only.
There exists a gap from 20 to 250 cm−1 in the zone-centre phonon frequencies. We are interested in the high frequency phonon modes since these vibration modes could potentially influence the infrared absorption edge of the ternary nitride crystals. Calculated frequencies
Figure 2. Charge difference contours on a plane perpendicular to thez axis of MgSiN2at a height of 2.05 Å.
of the high-frequency zone-centre photon modes of MgSiN2and MgGeN2are summarized in
table 3. By replacing Si with Ge, a large frequency downshift appears in the phonon modes whose lattice distortion involves a significant Si motion. These vibrations include the B1
-symmetry modes at 465 cm−1 and 549 cm−1and A1-symmetry modes at 346, 354, 614, 824
and 836 cm−1. The highest frequency mode at 836 cm−1 is Raman and infrared active and therefore could play the major role in determining the infrared absorption edge of MgSiN2.
By allowing the possibility of biphonon absorption, MgSiN2could therefore be transparent up
to 6µm in the infrared.
3.2. Electronic structures and density of states
The electronic band structures of BeSiN2and MgSiN2near the optical bandgap are presented
in figure 5. It can be seen that these ternary nitrides are not direct band crystals. The valence band maximum (VBM) of MgSiN2occurs at the U(1/2, 0, 1/2) point while the conduction
band minimum (CBM) lies at.. For comparison, the VBM of BeSiN2appears at. and the
CBM occurs at X(1/2, 0, 0). From further comparison with the direct band result of GaN, the major cause for the indirect bandgap in these ternary nitrides could be attributed to the lower crystal symmetry of the orthorhombic structure.
1.50 1.75 2.00 2.25 2.50
Bond length (Angstrom)
0 2 4 6
Optical gap (eV)
Si3N4 MgSiN2 Si (b) 1.50 1.75 2.00 2.25 2.50 0.0 2.5 5.0
Optical gap (eV)
Ge3N4 MgGeN2
Ge
(a)
Figure 3. Comparison of the optical gap of orthorhombic ternary nitride crystal MgXN2(P na21, X= Si, and Ge), with column IV and IV nitrides as a function of bond length.
0.7 0.9 1.1 1.3
Normalized Unit Cell Volume
-1 1 3 ∆ E (eV/unit cell) MgCN2 0.7 0.9 1.1 1.3 -1 1 3 ∆ E (eV/unit cell) BeCN2 (b) (a)
Figure 4. Calculated energy difference (0E = E(V ) − E(V0)) as a function of normalized unit cell volume (V /V0) for the ternary nitrides (a) BeCN2and (b) MgCN2. The solid curves are the fits to equation (1).
Table 2. The calculated lattice constants (in Å) and mechanical properties of the orthorhombic ternary nitrides. Experimental data are given in parentheses whenever are available.
Nitride crystal Unit cell dimensions (Å)
(P na21) (α = β = γ = 90◦) B (GPa) B= (∂B/∂P )0 BeCN2 a = 4.430 374 6.2 b = 5.301 c = 4.265 MgCN2 a = 4.598 234 3.7 b = 6.219 c = 4.481 BeSiN2 a = 4.942(4.98)a 270 12.7 b = 5.748(5.75)a c = 4.651(4.67)a MgSiN2 a = 5.267(5.28)a 175 1.0 b = 6.483(6.46)a c = 5.011(4.98)a BeGeN2 a = 4.986 263 10.4 b = 5.802 c = 4.694 MgGeN2 a = 5.301(5.49)a 171 10.8 b = 6.524(6.61)a c = 5.064(5.17)a Ge3N4(P 63/m)b a = 7.40 214 4.4 c = 3.20
a Data taken from Inorganic Crystal Structure Database (ICSD), Fachinformationszentrum, Eggenstein-Leopoldshafen 2, D-7514 Karlsruhe, Germany.
b[14].
Table 3. Computed frequencies of zone-centre phonons (.), classified by symmetry label. Nitride crystal (P na21) K (Group) Irrep. Frequencies (cm−1) MgSiN2 .(C2v) A1(z, x2+y2, z2) 346, 354, 421, 544 614, 824, 836 B1(x, xz) 465, 549, 706, 804 MgGeN2 .(C2v) A1(z, x2+y2, z2) 223, 324, 412, 524, 541, 738, 780 B1(x, xz) 323, 487, 681, 768
To gain more insight into the calculated optical properties we studied PDOS and LDOS spectra. The calculated PDOSs for MgSiN2 are given in figure 6. For the ternary nitride
crystals studies here the valence bands are dominated by the bonding states of nitrogen. More specific features in the PDOS are presented as follows. The valence bands with bonding energy from −14.3 to −10.8 eV are mainly contributed from the N 2s states. The N 2p states dominate the electronic bands that are near the top of the valence bands ranging from −5.7 to 2.4 eV. The N bonding states are much higher than those of the electronic states of metal atoms (Mg and Si) in the absence of the metal d states. In contrast, all the antibonding states from nitrogen, and metal atoms, play significant roles in the conduction bands. Similar results were also found on group III nitride crystals XN (X: B, Al, Ga, In) [16].
(b) (a)
Figure 5. Electronic band structures of (a) BeSiN2and (b) MgSiN2near the optical gap.
3.3. Linear and second-order nonlinear optical susceptibilities
The calculated energy gap, dielectric constants, and second-order nonlinear susceptibilities are summarized in table 4. The anisotropy of static dielectric constants of these ternary nitrides is not significant. Although a small dissimilarity appears between thexx and yy components, the major birefringence lies between thezz and the xx components with a magnitude of about 0.1. Replacing Si with Ge increases the averaged dielectric constant from 4.61 to 4.79 accompanied with a decrease in the energy bandgap.
The frequency-dependent dielectric functions of MgSiN2are presented in figure 7. The
first two peaks appearing at 4.5 and 5.0 eV in the imaginary part arise from the optical transitions from the VBM to CBM, which determine the absorption edge of MgSiN2. From the PDOS
analysis, we can find these transitions to have the characters of N 2p (at VBM) and the s orbitals of Mg, Si and N (at the CBM). Therefore the decreasing optical bandgap shown in table 4 can originate from a difference in the antibonding properties of metal atoms at the CBM.
Equation (6) has been used for analysing the static second-harmonic susceptibility of the ternary nitride crystals and the results are presented in table 4. For these materials,χzzz(2) is negative with a fairly constant magnitude lying between−13.8 and −17.3 pm V−1. In contrast all other susceptibility components such as χxzx(2) and χyzy(2) vary significantly with material. Note that bond additivity model predicts a simple relation ofχzzz(2) = −2χxxz(2) for wurtzite structure [17]. For ternary nitrides, we found that only MgGeN2follows this simple
rule, while all other nitrides deviate considerably. For example, the ratio ofχxzx(2) andχzzz(2)for MgSiN2is near to unity, but for BeSiN2 the ratio can be as small as 0.1. Similar behaviour
was also observed in group III nitrides [17], indicating that the simple bond additivity model is inappropriate for elucidating the nonlinear optical response in nitride crystals.
(b)
(c) (a)
Figure 6. Partial densities of states of MgSiN2projected onto (a) Mg, (b) Si and (c) N species at various atomic orbitals.
Figure 7. Real (solid curve) and imaginary (dashed) part of the optical dielectric function of MgSiN2as a function of incident photon energy.
Table 4. Calculated optical properties of the orthorhombic ternary nitrides. Nitride crystal Bandgap Dielectric Second-order
(P na21) Eg(eV) constants nonlinearity (pm V−1) BeCN2 5.75 εxx= 4.74 χzzz(2)= −12.90 εyy= 4.71 χxzx(2) = χzxx(2) = 4.73 εzz= 4.89 χyzy(2) = χzyy(2)= 1.16 MgCN2 4.76 εxx= 5.10 χzzz(2)= −13.81 εyy= 5.00 χxzx(2) = χzxx(2) = 2.61 εzz= 4.96 χyzy(2) = χzyy(2)= 13.38 BeSiN2 5.08 εxx= 4.72 χzzz(2)= −17.09 εyy= 4.77 χxzx(2) = χzxx(2) = 1.66 εzz= 4.90 χyzy(2) = χzyy(2)= 1.12 MgSiN2 4.15 εxx= 4.59 χzzz(2)= −17.30 εyy= 4.57 χxzx(2) = χzxx(2) = 15.96 εzz= 4.66 χyzy(2) = χzyy(2)= 14.82 BeGeN2 5.24 εxx= 4.75 χzzz(2)= −18.14 εyy= 4.72 χxzx(2) = χzxx(2) = 6.22 εzz= 4.90 χyzy(2) = χzyy(2)= 3.47 MgGeN2 3.97 εxx= 4.80 χzzz(2)= −14.49 εyy= 4.71 χxzx(2) = χzxx(2) = 8.80 εzz= 4.86 χyzy(2) = χzyy(2)= 7.58
In such complex compounds as ternary nitrides, it is useful to identify the role played by each atomic species and orbital. For this purpose, we use equation (7) for resolvingχ(2)(0) into contributions from each atomic species at each energy interval in either the initial valence bands or intermediate conduction bands. The results are presented in figure 8 for BeSiN2and
figure 9 for MgSiN2. The main contributions to the SH susceptibilities of the orthorhombic
ternary nitrides were found to originate from the electronic states near the optical bandgap. By comparing figure 8(b) with 9(b), we can also find that the variousχxzx(2) contributions from electronic bands of BeSiN2have more or less the same magnitude but with opposite signs. In
contrast, for MgSiN2 the positive terms dominate, which results in a largerχxzx(2). Transition
moments, which reflect the degree of inhomogeneity in the charge distribution of electronic bands, determine the sign of each term in equation (7). From the species and band-by-band decomposition, we found that all the atomic species contribute to the SH susceptibilities and there are no dominant species. Unlike borate crystals [18], the SH response in these ternary nitride crystals cannot be clearly separated into various terms from isolated atomic groups or clusters. This is expected from the ternary nitride structure whose metal atoms are tetrahedrally coordinated by N and vice versa both N atoms are tetrahedrally coordinated by the metal atoms 2× A and 2 × B. The full band method is therefore needed for correctly calculating the NLO response of these crystals.
Finally to reveal the reliability of our modelling, we applied the calculation method to wurtzite GaN. The choice of GaN is based upon its structural similarity to the ternary nitrides and the availability of numerous experimental and simulation results. The results are summarized in table 5. Our calculated unit cell dimensions are among the closest results to the experimental values. In addition, our bulk modulus agrees reasonably well with the measured values and also with other calculations. The calculated energy bandgap is about 30% smaller
-2 1 4 7 10 -3 -1 1 dχ /dE (pm/V.eV) Si N Be (a) (b) (c) χ(zzz)= -17.09 pm/V -2 1 4 7 10 -1 0 1 dχ /dE (pm/V.eV) Si N Be χ(xzx)=1.66 pm/V Eband ( eV ) -1 0 1 dχ /dE (pm/V.eV) Si N Be χ(yzy)=1.12 pm/V -2 1 4 7 10
Figure 8. Second-order nonlinear optical susceptibility components (a)zzz, (b) xzx and (c) yzy of BeSiN2projected onto various atomic species and energy bands by using equation (7).
than the experimental observation; however, this is to be expected at the level of the LDA methodology. The smaller bandgap also overestimates optical dielectric constants. Although the smaller bandgap obtained with LDA can be corrected with a simple scissors approximation or more sophisticated GW correction, we did not intend to do so in this study. Our calculated second-order susceptibilities, which are about one half of the experimental values [8, 9], agree reasonably well with other calculated results with a similar theoretical sophistication [17]. Based on these comparisons, we can confidently conclude that the simulation results reported in this study reliably reflect the real physical properties of the ternary nitrides.
4. Conclusions
In summary, the structural, mechanical and optical properties of orthorhombic ternary nitride crystals have been analysed theoretically with a first-principles calculation scheme. These nitrides possess fairly large optical bandgap, second-order nonlinear optical susceptibility and bulk modulus. Therefore they could be useful for both optical and protective coating applications. Our calculations with band-by-band and atomic species decomposition not only yield values of material properties, but also provide deep insight into the fundamental understanding of the mechanical and optical properties of orthorhombic ternary nitrides.
ternary
nitrides
10429
Table 5. Lattice constants, elastic, and optical properties of wurtzite GaN.
a = b (Å) c s B0(Gpa) B0 Eg(eV) ε⊥ ε| εav χxzx(2) (pm V−1) χzzz(2)(pm V−1) Present 3.178 5.167 209 5.0 2.31 5.91 5.92 5.92 −9.0 15.9 calculation Exp. [19] 3.190 5.200 Exp. [20] 3.160 5.120 Exp. [21] 188 3.2 Exp. [22] 237 4.3 Exp. [23] 245 4.0 Exp. [24] 3.52 Exp. [25] 3.30 Exp. [26] 5.35 Exp. [27] 5.80 Exp. [28] 14.4 29.7 Exp. [29] 16.0 33.0 Other calc. [16] 3.240 5.240 2.40 5.87 5.89 5.88 −11.7 20.8 Other calc. [30] 3.170 5.130 Other calc. [31] 3.162 5.142 Other calc. [32] 3.143 5.111 215 5.9 5.21 5.41 5.28 Other calc. [33] 3.220 5.260 Other calc. [15] 3.143 5.111 Other calc. [34] 3.146 5.125 Other calc. [35] 3.170 5.135 Other calc. [36] 3.162 5.141 Other calc. [37] 3.126 5.120 190 2.9 Other calc. [38] 3.210 5.237 197 —-Other calc. [39] 200 3.8 4.71 4.62 4.68 Other calc. [40] 3.80 Other calc. [17] 5.54 5.60 5.74 −6.9 11.5 Other calc. [41] −4.2 7.0 Other calc. [42] −8.5 12.1
-4 0 4 8 12 -2 0 2 dχ /dE (pm/V.eV) -4 0 4 8 12 -1 1 3 dχ /dE (pm/V.eV) χ(xzx)=15.96 pm/V χ(zzz)= -17.30 pm/V -0.5 0.5 1.5 dχ /dE (pm/V.eV) χ(yyz)=14.82 pm/V -4 0 4 8 12 Eband ( eV ) (a) (b) (c) Si N Be Si N Be Si N Be
Figure 9. Second-order nonlinear optical susceptibility component (a)zzz, (b) xzx and (c) yzy of MgSiN2projected onto various atomic species and energy bands by using equation (7).
Acknowledgment
The authors are indebted for the financial support from the National Science Council of the Republic of China under grant NSC 89-2112-M-009-069.
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