ARTICLE pubs.acs.org/JPCA
First-Principle Electronic, Elastic, and Optical Study of Cubic Gallium Nitride Zahid Usman, Chuanbao Cao,* Ghulam Nabi, Dou Yan Kun, Waheed S. Khan, Tariq Mehmood, and Sajad Hussain Research Centre of Materials Science, School of Material Science and Engineering, Beijing Institute of Technology, Beijing 100081, People’s Republic of China ABSTRACT: The ab initio pseudopotential (PP) method within the generalized gradient approximation (GGA) has been used to investigate the electronic, elastic constants, and optical properties of zincblende GaN. An underestimated band gap along with higher DOS and squeezed energy bands around the fermi level is obtained. The d-band effect is briefly discussed for electronic band structure calculations. With the help of elastic constants, acoustic wave speeds are calculated in [100], [110], and [111] planes. The dielectric constant, refractive index, and its pressure coefficient are well illustrated. The effect of hydrostatic pressure is explicated for all these properties. The results of the present study are evaluated with the existing experimental and first-principle calculations.
1. INTRODUCTION Group-III nitride semiconductors GaN, AlN, and InN have been extensively studied because of their wide band gap, high thermal conductivity, low dielectric constant, and large bulk moduli.1 These characteristics lead to the commercial development of light-emitting diodes, modulated field effect transistors,2,3 and short-wavelength coherent lasers.4,5 Therefore, it is predicted that these coherent laser source nitrides will be key in the manufacturing of high-density data devices and undersea optical communication systems in the future. Research on wide band gap materials started in the early 1970s.6,7 Metal organic chemical vapor deposition (MOCVD) and molar beam epitaxy (MBE) methods made it possible to grow high-quality epitaxial layers and heterostructures, further enhancing the pace of advancements.8,9 However, large lattice mismatches and different thermal expansion coefficients between the epitaxial layer and the substrate obstructed the growth of GaN layers on crystalline substrates like sapphire, SiC, ZnO, Si, or GaAs. Mostly these materials are wurtzite in nature at room temperature; therefore, cleavage planes for epitaxial layers and substrates were unavailable. Now this problem is minimized by the introduction of highly symmetric zinc-blende materials grown on cubic substrates. This is one of the most rapidly flourishing techniques. Moreover, Yang et al.10 demonstrated the successful growth of cubic GaN on the GaAs (001) surface exhibiting better electrical and optical properties as compared to the wurtzite structures. Zinc-blende structures are different than its wurtzite counterpart in many aspects such as higher symmetry, low enthalpy, and appropriate n-type, p-type doping. Thus, it is advantageous to use cubic materials for device applications. However, the experimental techniques are usually costly and provide insufficient theoretical insight to build better materials r 2011 American Chemical Society
for novel device applications. Therefore, first-principle studies are highly encouraged which provide better experimental parameters leading to inexpensive and rapid development of novel material for useful practical application. Extensive theoretical efforts have been accomplished regarding proper description of electronic, structural, elastic, and optical properties. Most of these studies are based on the local density approximation within the density functional theory, either in the bulk electronic formalism or using the plane wave pseudopotential approximation (PW-PP).11 HartreeFock calculations1214 are computationally more expensive and always overestimate the bang gap energies significantly as compared to experimental values for such materials. The GW approach12 is also practiced, which provided experimentally comparable band gap values, but the main weakness it inherits is the implicit d-electron’s treatment. That is the reason this method is inaccurate for pd and sd hybridization. PP methods circumscribe the local density approximation (LDA)1519 and generalized gradient approximation (GGA). LDA is found to be underestimating the band gap values for semiconductors;20,21 therefore, GGA has been getting more attention recently. Our aim is to describe electronic, elastic constants, acoustic wave speeds, and optical studies of cubic GaN using ultrasoft pseudopotential (USP).22 A short view of the d-electron effect on the band structure is presented. The paper is organized in the following way. Section 2 presents brief computational details along with the method of calculation. Section 3 includes the band structure, density of Received: February 15, 2011 Revised: March 28, 2011 Published: May 23, 2011 6622
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Figure 2. (a) GGA band structure calculation at normal pressure, and (b) direct band gap variation with hydrostatic pressure. Figure 1. Total energy and band gap energy versus energy cutoff.
states (DOS), elastic constant, acoustic speeds, and optical properties calculations and discussions. In this study, pressure effects are also considered. Finally, a general conclusion is drawn on the basis of these results.
Table 1. Some Direct and Indirect Band Gaps (in eV) at Zero Pressure
2. COMPUTATIONAL DETAILS The first-principle calculations were performed with CASTEP operated within the Material Studio Software environment (Accelrys Software Inc.). This software is based on density functional theory (DFT) and uses the plane wave basis set for expansion of the electronic wave functions. Electron ion interactions are taken into account by ultrasoft PP, whereas PerdewWang (PW31) proposed the generalized gradient approximation23 is employed for electronelectron interaction. The pseudo-atomic configurations for Ga and N, respectively, are 3d104s24p1 and 2s22p3. Since d electrons have a profound effect on band structure calculations,24 d electrons are treated as valence electrons in Ga. The plane wave cutoff energy has been varied up to Ecut = 400 eV using periodic boundary conditions and Bloch’s theorem. However, the main part of the following results is obtained at 395 eV. In Figure 1 the total energy and energy band gap at the Γ point are plotted against Ecut for cubic GaN. It is apparent that the high-energy part of both physical quantities is flat as compared to the initial part. It manifests a good convergence of the total energy curve resulting in a better energy gap with respect to the cutoff energy. These curves are in good agreement with that of Palummo et al.25 performed with LDA-NCP. It is further noted that varying the k points had a very weak effect on total energy.25 Thus, a Monkhorst pack26,27 grid of 3 3 3 is considered to ensure better total energy convergence of less than 1 meV/atom. A denser K-point grid of 5 5 5 is used for Brillion zone integration for optical properties. First, the zinc-blende structure is geometrically optimized using the BroydenFletcher GoldfarbShenno (BFGS) technique with the residual force less than 2 meV/Å; then this equilibrium structure was further investigated for band structure, density of states, elastic constant, shear wave speeds, and optical properties calculations. In order to explore the effect of pressures on various physical properties, external pressures up to 40 GPa are applied through CASTEP, the structures are reoptimized, and the stressed physical quantities are compared with those at zero pressure. Whole calculations are subjected to the fact that the system is spin polarized.
3. RESULTS AND DISCUSSION
band gaps
ΓΓ
ΓX
ΓL
ΓM
XX
LL
MM
Eg (0)
1.691
4.64
4.59
6.37
6.07
5.519
7.30
3.1. Electronic Properties. The theoretical lattice constants and bulk modulus were calculated through fitting the total energy values with the Murnaghan equation of state.28 # " 0 B0 V ðV0 =V ÞB0 B0 V EðV Þ EðV0 Þ ¼ þ1 B00 B00 B0
Here, E(V) represents the ground state energy with the cell volume V, E(V0) is the DFT ground state energy with V0 (volume at zero pressure), B0 is the bulk modulus, and B0 0 is its pressure derivative at P = 0, whereas the changes of volume can be obtained with the help of pressure through the following equation h i B0 B00 =V Þ 1 ðV P ¼ 0 B00 Using the Murnaghan equation of state, equilibrium lattice parameters are found to be 4.554 Å, in good agreement with the experimental value a = b = c = 4.510 Å. These equilibrium lattice constants are 0.98% higher as compared to the provided experimental lattice constant because of the fact that GGA always overestimates the lattice constants. The bulk modulus and its pressure derivative are summarized in Tables 2 and 3, respectively, and will be discussed with elastic constants. The GGA band structure presented below is calculated at this theoretical value. This shows a larger band gap of 1.691 eV, significantly greater than previous calculations performed with USP.29 However, as compared to the experimental band gap, this value is nearly 50% smaller. Due to the cubic symmetry of zincblende GaN, the topmost valence band is found to be triply degenerate.30 The Fermi level (E = 0 eV) is set at the highest value of the valence band. The valence band maximum and conduction band minimum lie at the Γ point, making zinc-blende GaN a direct band gap material (ΓΓ). Some important features of this band gap depicted in Figure 2 are presented in Table 1. 6623
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Table 2. Elastic Constant Moduli (GPa) of Cubic GaN method present study
C11
C44
C12
B0
Cs
(GPa)
(GPa)
(GPa)
(GPa)
(GPa)
133
173
61
255
177
LDA-NCP25
176
experimental41
190
GGA-NCP40
204
67
LDA, FP-LMTO42
293
155
159
204.6
71
LDA, FP-APWþlo43
296
206
154
202
54.05
first-principle MD44
274.2
199
166.1
202
62
Table 3. Pressure Differentials of Elastic Moduli pressure gradients present work ∂C11/∂p ∂C12/∂p ∂C44/∂p ∂Cs/∂p B0 0
Figure 3. Total DOS for cubic GaN and PDOS for N and Ga at normal and hydrostatic pressures.
An interesting comparison exists between our calculated values of the direct band gap and that of Palummo et al.25 The minimum band gap calculated by Palummo et al. is 2.70 eV, which is closer to the experimental energy gap as compared to ours, 1.691 eV. However, the direct band gaps (ΓΓ), (MM), (XX), and (LL) are in close agreement with the (Γ15Γ1) = 2.65 eV, (K2K1) = 7.02 eV, (L3L1) = 6.00 eV, and (X3X1) = 5.48 eV band gaps proposed by Palumo et al., respectively. Our calculated direct and indirect band gaps agree well with DFT calculations made by Christensen and et al.31 using the selfconsistent linear muffin-tin-orbital method except a smaller value of (ΓΓ). In order to obtain more physical insight into the electronic band structure, a plot of DOS is presented in Figure 3. Total density of states is extended from about 16 to 11.9 eV, a narrow region where DOS is higher as compared to former calculations performed with same potential but a bit higher cutoff energy and different k points.32 A comparison of these two calculations reveals that the d band here has DOS greater than 13 eV, as compared to 10 eV.31 The width of the valence band is found to be 6.9 eV, which is in close agreement with NCP and GW calculations.33,34 The d band is located about 13.2 eV below the Fermi level, farther from the experimental value of 17.7 eV,35 which resonates fully with the N 2s valence band. This sd hybridization splits the N 2s band into two parts, one above and other below the Ga d band with a large dispersion away from Brillion zone center because of the fact that sd mixing is not allowed at the Γ point.36 Moreover, the d band splits into an inert doublet Ld3 as apparent from the electronic band structure.37 If we define the band gap from the total DOS plot as “the difference between conduction band edge
3.9 ( 0.673 4.4 ( 0.705 3.1 ( 0.540 0.25
ref 47
ref 48
ref 49
3.7 ( 0.3 4.9 ( 0.04
5.0 ( 0.2 5.0 ( 0.2
4.77 ( 0.1 4.79 ( 0.15
0.6 ( 0.3
1.1 ( 0.1
0.92 ( 0.03
ref 25
0.6 ( 0.4 0.04 ( 0.02 0.03 ( 0.007
4.3 ( 0.521
4.5 ( 0.4
5.0
4.79
2.66
and valence band edge”, we get a band gap of 3.21 eV nearly equal to the experimental value of 3.3 eV.38 PDOS reveals that the conduction band is mainly composed of s and p states of both Ga and N, whereas the top valence band contains N p states in abundance as compared to a minor contribution from Ga p states. The nonuniformity of the DOS in this region arises from the hybridization of atomic orbitals, and the difference in shape is because of the Ga cationic states effect. This consistency arises because of hybridization of the N p orbital with the Ga 3d orbital, finally resulting in a reduction of the band gap. N 2s is dominant in the lower parts of the valence band with some s character from Ga instead. Figure 2b shows the sublinear pressure dependence of the direct band gap (ΓΓ). This confirms that the direct band gap increases sublinearly with ambient pressure, consistent with the FP-LAPW method.39 Furthermore, the effect of pressure appears in the form of a broadening of the energy gap due to a shifting of the low-energy level to lower positions and high-energy levels to higher ones. Hence, the band gap widens. When pressure is applied on total DOS, the DOS decreases with the increase of pressure, but energy levels are shifted to their energy extremes, respectively, with a similar peak shape. It means that the number of states available to a specified level decreases with an increase of pressure.39 3.2. Elastic Constant Calculations. Unlike the electronic and optical properties, inadequate investigations have been made regarding pressure effects on elastic constants. The elastic constants of cubic GaN describe its response to an applied stress, and hence, the deformation produced within the system can be predicted. Both stress and strain have their tensile and shear components. These components are three in number each, giving six components as a whole. The linear elastic constants form a 6 6 symmetric matrix, having 27 different components. These components will reduce to three elements C11, C12, and C44 according to the symmetry of the cubic structure. The elastic constants C11 and C12 were extracted from the bulk modulus B0 and shear modulus (Cs) calculations by B0 = (C11þ 2C12)/3 and 6624
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∂2 v ∂2 v ∂2 v ∂2 v F 2 ¼ C11 2 þ C44 þ ∂t ∂y ∂x2 ∂z2
!
! ∂2 u ∂2 w þ þ ðC12 þ C44 Þ ∂x∂y ∂y∂z
ð2.bÞ ! ! ∂2 w ∂2 w ∂2 w ∂2 w ∂2 u ∂2 v þ ðC12 þ C44 Þ þ F 2 ¼ C11 2 þ C44 þ ∂t ∂z ∂x2 ∂y2 ∂x∂z ∂y∂z
ð2.cÞ
Figure 4. Pressure dependence of elastic constants Cij, bulk modulus, and shear modulus of cubic GaN.
Cs = (C11 C12)/2 expressions, respectively. Like bulk modulus, the shear modulus (Cs) is obtained from the energy fit of the Murnaghan equation of state versus tetragonal strain.40 Both B and Cs are presented in Table 2, while the pressure gradient of the bulk modulus is given in Table 3. In Table 2, we summarized our calculated elastic constants using GGA-USP at ambient pressure with other existing results. From this data, it is evident that our results for the bulk modulus are in good agreement with the experimental and LDADFT-based Palummo et al.25 results. However, the other theoretical methods seem to overestimate the bulk modulus as compared to the experimental value.41 The pressure derivative of the bulk modulus is estimated to be 4.3 (Table 3), which is quite consistent with the other DFT-based theoretical results,47,49,49 but its value is quite large as compared to those calculated by Palummo et al.25 We also determined the value of Poisson ratios (σo = C12/(C11 þ C12)) and Young modulus (Y0 = (C11 þ 2C12) 3 (C11 C12)/(C11 þ C12)) in the [100] plane as 0.314 and 164.3GPa, respectively, which are again very close to the calculated 0.352 and 181 GPa values.40 Figure 4 illustrates the variation of the elastic constants (C11, C12, and C44) bulk modulus and shear modulus with external pressure. This plot clearly shows that the elastic constants and bulk modulus increase linearly with applied external pressure, while the shear modulus decreases linearly with pressure. Moreover, Table 3 sketches a comparison of pressure derivatives ∂C11/∂p, ∂C12/∂p, ∂C44/∂p, and ∂Cs/∂p of these physical quantities by following the elastic stability criteria45,46 for cubic crystals under the desired pressure range. These criteria are given in eq 1 1=3ðC11 þ 2C12 þ PÞ > 0; 1=2ðC11 C12 þ 2PÞ > 0; C44 P > 0
ð1Þ
where F is the density, u, w, and z represent displacements in x, y, and z directions, and C’s represent elastic constants. In order to find the wave speed in the [100] direction, let us consider the solution of eq 2.a in the form of u ¼ u0 exp½iðKx ωtÞ
where the wave vector (K = 2π/λ) and particle motions are along the x direction and ω = 2πν. Then substituting eq 3 into eq 2.a provides the velocity of this longitudinal wave in the x direction, i.e. vl ¼ ðC11 =FÞ1=2
! ! ∂2 u ∂2 u ∂2 u ∂2 u ∂2 v ∂2 w F 2 ¼ C11 2 þ C44 þ þ þ ðC12 þ C44 Þ ∂t ∂x ∂y2 ∂z2 ∂x∂y ∂x∂z
ð2.aÞ
ð3.aÞ
Similarly, if the wave vector is along the x direction and particle displacement is along the y direction, the velocity of the transverse waves in the [100] direction is vt ¼ ðC44 =FÞ1=2
ð3.2Þ
Similarly, for shear waves in the [110] direction propagating in the xy plane with particle displacement (w) in the z direction and in the xy plane are respectively vt ¼ ðC44 =FÞ1=2
ð3.3Þ
vl ¼
1 ðC11 þ C12 þ 2C44 Þ 2F
vt^ ¼
1 ðC11 C12 Þ 2F
1=2 ð3.4Þ
1=2 ð3.5Þ
where vt^ represents the transverse component, perpendicular to vt and vl. In the [111] direction the longitudinal and transverse components of elastic waves are found to be 1=2 1 ðC11 þ 2C12 þ 4C44 Þ ð3.6Þ vl ¼ 3F
3.3. Acoustic Wave Speeds in Cubic GaN. In order to find acoustic wave speeds, we assume that pressure effects are small and uniform, so that the application of Hook’s law and Newton second law of motion could be valid. Within elastic limits, the general equations of motion in the x, y, and z directions, respectively, are
ð3Þ
vt ¼
1 ðC11 C12 þ C44 Þ 3F
1=2 ð3.7Þ
The detailed description and derivation of these shear speeds can be found in the books of Truell et al. and Kittle, respectively.50,51 As we know that cubic GaN has 2 atoms per primitive cell with lattice constant of 4.554 Å, we calculated the density (F) to be 5.99 g/cm3. All calculated values of acoustic speeds using eqs 3.23.7 are described in the form of Table 4. It is evident from Table 4 that the calculated acoustic speeds display a good comparison with that measured by Truell et al.50 6625
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Table 4. Summary of Acoustic Speed’s Comparison with Existing Literature physical quantity
present study (in 105 m/s)
other results50
Vl [100]
6.542
6.9
vt [100] vt [110]
5.435 5.435
5.02 5.02
vl [110]
6.42
7.87
vt^ [110]
3.19
3.3
vl [111]
8.26
8.17
vt [111]
4.07
3.96
Figure 6. (a) Refractive index (n) at normal and hydrostatic pressure. (b) Energy loss function. (c) Absorption coefficient. (d) Reflectivity.
Table 5. Calculated Values of the Dielectric Constant, Refractive Index, and Its Pressure Coefficient in Units of 102 GPa1 method
dielectric constant
refractive index
(ε1)
(n)
present study
5.58
Figure 5. (a) Calculated real and imaginary parts of the dielectric constants at normal pressure, and (b) pressure dependence of the imaginary part of the dielectric function (ε2).
GGA
5.7152
LDA-FPLAPW
5.49
3.4. Optical Properties. Optical properties are an important source of information about the electronic band structure and collective excitations in cubic GaN. These properties including dielectric constant, refractive index, reflectivity, absorption, and conductivity are not only frequency dependent but also are extractable from one another. The dielectric function describes the system’s linear response to electromagnetic radiations. It contains real and imaginary parts. The imaginary part of the dielectric function represents the absorption within the system, when light of a specific frequency is used. The imaginary part of the dielectric function can be calculated directly from electronic band structure calculations and the real part using the KramerKroning dispersion relation as shown in Figure 5a. The imaginary part of the dielectric function starts at about 2 eV with the first peak at 2.9 eV because of excitation from VBM to CBM. The second peak (5.36 eV), third peak (6.88 eV), and fourth peak (9.16 eV) occur because of L to M, Γ to A, and R5 to R3 transitions, respectively. The influence of hydrostatic pressure appears in the form of shifting of (ε2) peaks toward higher values as indicated in Figure 5 b. We computed the refractive index as a function of the real part of the dielectric constant n = (ε1)1/2 at low frequency. The refractive index is 2.36 and increases with energy, reaching a peak value of 2.95 at an energy of 4.92 eV in the ultraviolet region. Its minimum value lies at a photonic energy of 17 eV as shown in Figure 6a. The results are summarized in Table 5 for the dielectric constant, refractive index, and pressure coefficient of refractive index 1/n(dn/dp) 102 GPa1 in comparison with experimental as well as other theoretical results.
LDA-LMTO
4.7856
LDA-non relativistic
5.00557
2.24
LAPW experimental
5.758
2.3459
5.4753 55
2.36
1/n(dn/dp) 0.18
2.3154 2.34
0.28 0.20 0.19
Our calculated dielectric constant is slightly underestimated than some of the theoretical calculations52,56 but agree well with experiment.58 However, the refractive index is slightly overestimated than some of the first-principle studies54,57 still in good agreement with the experimentally calculated refractive index.59 The pressure derivative of the refractive index is negative and predicts the distortion produced in the material under stress. Since no experimental data is available for the pressure coefficient of the refractive index, in order to compare its validity, we rely only on other theoretical results. The calculated results for the pressure coefficients of the refractive index lie within the vicinity of other first-principle calculations as tabulated in Table 5. The complex dielectric function is sufficient to derive other physical properties like the energy loss function (L(ω)), absorption coefficient, and reflectivity as depicted in Figure 6b, 6c, and 6d, respectively. L(ω) is an important factor to estimate the dissipation of fast moving electron’s energy within the material medium. The prominent peaks are attributed to the interaction between plasma resonance and the corresponding frequency (plasma frequency ωp). The peaks of L(ω) are related to the trailing edges in the reflection spectra as shown in Figure 6d. The energy loss function attains its maximum value at ε1 = 0, whereas ε2 is observed to be nearly smooth in this region. In the (100) direction, zinc-blende GaN is transmitting electromagnetic radiation in the energy range only from 0 to 1.94 eV, while the 6626
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The Journal of Physical Chemistry A absorption coefficient at this stage is zero. It begins at 1.942 eV and increases with the electromagnetic radiation energy until 12 eV as shown in Figure 6c. This maximum absorption corresponds to the ε1 minima. The refractive index also follows a drop-down trend in this energy limit. Then the real part starts increasing and approaches zero at about 16.6 eV; here, the refractive index obtains it minimum value. At this energy, reflectivity (R) and L(ω) approach their maxima. However, for energy values higher than 20 eV, the material depicts transparency and the reflection coefficient approaches zero. This confirms the previous theoretical studies32 predicting nearly zero absorption occurring in the near-ultraviolet to visible light region.
’ CONCLUSION The conclusions of this study are summarized as follows. (1) The GGA proposed by PerdewWang is used along with the Ultra soft PP method to calculate the electronic band structure, DOS, elastic constants, acoustic waves, and optical properties. The band gap is approximated to be 1.691 eV, which is significantly higher than its previous GGA-USP study but underestimated as compared to the experimental value of 3.3 eV, LDA, and HartreeFock approximations. The DOS is observed to be higher, and energy levels are distributed in a smaller region around the Fermi level. The d-band effect appears in a profound reduction of the band gap through pd hybridization. N 2s has been seen to be splitting into two components, one above and other below the Ga d band. However, if we define the band gap from the total density of states as “the difference between conduction band edge and valence band edge”, we get a band gap of 3.21 eV nearly equal to the experimental value of 3.3 eV. The direct band gap widens with the hydrostatic pressure with the high-energy levels shifting to higher positions while the lower energy levels to the lower positions. Total DOS is found to be decreasing with external pressure. (2) The calculated elastic constants are in good agreement with the experimental and other first-principle studies. The bulk modulus and other elastic constants except shear modulus displayed a linearly decreasing response on hydrostatic pressures. The Young modulus and Poisson ratio are also comparable with the other results. (3) Elastic wave speeds showed an excellent consistency with that of Truell et al. calculations. (4) The dielectric constant and refractive index are identified with respect to the electronic band structure having values of 5.58 and 2.36 against experimental values of 5.7 and 2.34, respectively. The pressure coefficient of the refractive index also lies within the previously calculated domain. The pressure effects appear in a shifting of the imaginary part of the dielectric function (ε2) peaks toward higher values, while the shape of the peaks remains unchanged. The complex refractive index decreases with an increase of pressure. Cubic GaN seems to be transparent from the near-ultraviolet to the visible light regime. ’ AUTHOR INFORMATION Corresponding Author
*Phone: þ86 10 6891 3792. Fax: þ86 10 6891 2001. E-mail:
[email protected].
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’ ACKNOWLEDGMENT This work was supported by National Natural Science Foundation of China (20471007, 50972017) and the Research Fund for the Doctoral Program of Higher Education of China (20101101110026) ’ REFERENCES (1) Christensen, N. E.; Gorczyca, I. Phys. Rev. B 1993, 47, 4307. (2) Khan, M. A.; Shur, M. S.; Chen, Q. C.; Kuznia, J. N. Electron. Lett. 1994, 30, 2175. (3) Aktas, O.; Kim, W.; Fan, Z.; Botchkarev, A. E.; Salvador, A.; Mohammad, S. N.; Sverdlov, B.; Morkov, H. Electron. Lett. 1995, 31, 1389. (4) Nakamura, S.; Fasol, G. The Blue Laser Diode; Springer Verlag: Berlin and Heidelberg, 1997. (5) Fasol, G. Science 1997, 278, 1902. (6) Pankove, J. I. J. Lumin. 1973, 7, 114. (7) Kesamanly, F. P. Sov. Phys. Semicond. 1974, 8, 147. (8) Nakamura, S.; Isawa, N.; Senoh, M.; Mukai, T. Jpn. J. Appl. Phys. 1992, 31, 1258. (9) Kim, W.; Aktas, O.; Botchkarev, A. E.; Salvador, A.; Mohammad, S. N.; Morkoc, H. J. Appl. Phys. 1996, 79, 7657. (10) Yang, H.; Brandt, O.; Wassermeier, M.; Behrend, J.; Schonherr, H. P.; Ploog, K. H. Appl. Phys. Lett. 1996, 68, 244. (11) Payne, M. C.; Teter, M. P.; Allan, D. C.; Arias, T. A.; Joannopoulos, J. D. Rev. Mod. Phys. 1992, 64, 1045–1097. (12) Wright, A. F.; Nelson, J. S. Phys. Rev. B 1994, 50, 2159. (13) Palummo, L.; Reining, R. W.; Godby, C. M.; Bertoni; B€ornsen, N. Europhys. Lett. 1994, 26, 607. (14) Pandey, R.; Jaffe, J. E.; Harrison, N. M. J. Phys. Chem. Solids 1994, 55, 1357. (15) Fiorentini, V.; Methfessel, M.; Scheffler, M. Phys. Rev. B 1993, 471, 3353. (16) Fiorentini, M.; Methfessel; Scheffler, M. Phys. Rev. B 1993, 4713, 353. (17) Christensen, N. E.; Gorczyca, I. Phys. Rev. B 1994, 50, 4397. (18) Lambrecht, W. R. L.; Segall, B.; Strite, S.; Martin, G.; Agarwal, A.; Morkoc, H.; Rockett, A. Phys. Rev. B 1994, 50, 14155. (19) Xu, Y.-N.; Ching, W. Y. Phys. Rev. B 1993, 48, 4335. (20) Perdew, J. P.; Levy, M. Phys. Rev. Lett. 1983, 51, 1884. (21) Sham, L. J.; Schluter, M. Phys. Rev. Lett. 1983, 51, 1888. (22) Vanderbilt, D Soft self-consistent pseudopotentials in generalized eigenvalue formalism. Phys. Rev. B 1990, 41, 7892–7895. (23) Kleinman, L.; Bylander, D. M. Phys. Rev. Lett. 1982, 48, 1425–1428. (24) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Phys. Rev.B 1992, B 46. (25) Palummo, M.; Bertoni, C. M.; Reining, L.; Finocchi, F. Physica B 1993, 185, 404–409. (26) Monkhorst, H. J.; Pack, J. D. Phys. Rev. B 1976, 13, 5188. (27) Monkhorst, H. J.; Pack, J. D. Phys. Rev. B 1997, 16, 1748. (28) Murnaghan, F. D. Proc. Natl. Acad. Sci. U.S.A. 1944, 30, 244. (29) Stamp, C.; Van de Walle, C. G. Phys. Rev. B 1999, 59, 5521. (30) Angel, R.; Corkill, J. L.; Cohen, M. L.; Shirley, E. L.; Stephen, G. L. Phys. Rev. B 1993, 48, 16. (31) Christensen, N. E.; Gorczyca, I. Phys. Rev. B 1994, 50, 7. (32) Shuti, L.; Chuying, O. Physics Lett. A 2005, 336, 145–151. (33) Surh, M. P.; Louie, S. G.; Cohen, M. L. Phys. Rev. B 1991, 43, 9126. (34) Rubio, J. L.; Corkill, M. L.; Cohen, E. L.; Shirley, E. L.; Louie, S. G. Phys. Rev. B 1993, 48 (11), 810. (35) Vogel, D.; Kruger, P.; Pollman, J. Phys. Rev. B 1997-I, 55, 19. (36) Fiorentini, V.; Methfessel, M.; Scheffler, M. Phys. Rev. B 1993, 47, 20. (37) Ding, S. A.; Neuhold, G.; Weaver, J. H.; H€aberle, P.; Horn, K.; Brandt, O; Yang, H.; Ploog, K. J. Vac. Sci. Technol. A 1996, 14, 819. 6627
dx.doi.org/10.1021/jp201495e |J. Phys. Chem. A 2011, 115, 6622–6628
The Journal of Physical Chemistry A
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(38) Paisley, M. J.; Sitar, Z.; Posthill., J. B.; Davis, R. F. J. Vac. Sci. Technol. A 1987, 7, 701. (39) Arbouche, O.; Belgoumene, B.; Soudini, B.; Driz, M. Mater. Sci. 2009, 47, 432–438. (40) Right, A. F. J. Appl. Phys. 1997, 82, 6. (41) Sherwin, M. E.; Drummond, T. J. J. Appl. Phys. 1991, 69, 8423. (42) Kim, K.; Lambrecht, R. L.; Segall, B. Phys. Rev. B 1994, 50, 1502. (43) Kanoun, M. B.; Merad, A. E.; Merad, G.; Cibert, J.; Aourag, H. Solid State Electron. 2004, 48, 1601–1606. (44) Shmida, K; Sota, T; Suzuki, K. J. Appl. Phys. 1998, 84, 4951. (45) Yip, S.; Tang, J. Li; Wang, J. Mater. Sci. Eng., A 2001, 317. (46) Sin’ko, G. V.; Smimov, A. J. Phys., Condens. Matter 2002, 14, 6989. (47) Polian, A.; Grimsditch, M. Phys. Rev. B 1993, 60, 3. (48) Rimai, D. S.; Sladek, R. J. Solid State Commun. 1979, 30, 591. (49) Yogurtc,u, Y. K.; Miller, J.; Saunders, G. A. J. Phys. Chem. Solids 1981A, 42, 49. (50) Truell, R.; Elbaum, C.; Chick, B. B. Ultrasonic Methods in Solid State Physics; Academic Press: New York, 1969. (51) Kittle, C. Introduction to Solid State Physics; John & Wiley Sons: New York, 1996, pp 8090. (52) Gavrilenko, V. I.; Wu, R. Q. Phys. Rev. B 2000, 61, 2632. (53) Chin, V. W.L.; Tansley, T. L.; Osotchan, T. J. Appl. Phys. 1994, 75, 7365–7372. (54) Moss, J. S. Proc. Phys. Soc. B 1950, 63, 167. (55) Berrah, S.; Abid, H.; Boukortt, A. Semicond. Phys., Quantum Electron. Optoelectron. 2006, 9 (02), 12–16. (56) Christensen, N. E.; Gorczyca, I. Phys. Rev. B 1994, 50, 4403. (57) Riane, A. Z.; Matar, S. F.; Abdiche, A. Physica B 2010, 405, 985–989. (58) Perlin, P.; Gorczyca, I.; Christen, N. E.; Grzegory, I.; Teisseyre 67, H.; Suski, T. Phys. Rev. B 1992, 45 (13), 307. (59) Kohler, U.; As, D. J.; Schotttker, B.; Frey, T.; Lischka, K.; Scheiner, J.; Shokhovets, S.; Goldhahn., R. J. Appl. Phys. 1999, 85, 404.
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