The Structural, Electronic, Optical and Elastic

Journal of Nanoelectronics and Optoelectronics

Copyright © 2016 by American Scientific Publishers All rights reserved. Printed in the United States of America

Vol. 11, pp. 561–567, 2016 www.aspbs.com/jno

ARTICLE

The Structural, Electronic, Optical and Elastic Properties of -Type Gallium Selenide: A First Principle Study Geoffrey Tse1, ∗ and Dapeng Yu1, 2

In this work, the structural, bandstructure, partial density of states (PDOS), elastic, charge density and optical calculations of a bulk -type Gallium Selenide (GaSe) were investigated, using ab initio density functional theory (DFT) calculations. To conclude, a direct-bandgap of up to 0.568 eV was opened. In addition, such quantity was reported to be smaller with the inclusion of the d electron orbital states from Ga atom. Furthermore, the lattice was found to be brittle, according to our elastic calculations, with the bulk to shear modulus ratio (B/G) ratio of 1.58. Finally, our optical data have shown a strong absorption peak with a value of 161.6 nm, identifying this material was suitable in extreme ultra violet (XUV) applications. Keywords: Electronic Structure, PDOS, Optical, Elasticity.

1. INTRODUCTION GaSe,1–11 a monochalcogenide III–VI semiconductor,12–22 consisted of different polytypes , , and .23, 24 Applications for this material included,25–34 the heterojunction devices35 and photoelectronics within visible spectrum,36 laser pulse,37 X-ray beam detection,38 detecting radiating waves at room temperature,39 generating Terahertz (THz) waves,40, 41 photoconductors42–45 and photovoltaics.46–48 It was also employed for spintronic applications, a potential use for memory devices.49 Ghalouci et al. have investigated the variation of structural and elastic material properties, depending on different polytypes.50 Rybkovskiy et al. have demonstrated, the dependence of band transitions, with different number of interlayers in the unit cell.51, 52 Zhou et al. have carried out an investigation in the second harmonic generation (SHG) intensity, in observing the structural and non-linear effect in GaSe atomic layers.53 Bashenov et al. have established his theoretical work in evaluating the elastic quantities on layered GaSe.54 Thornton et al. have explored 1 State Key Laboratory for Mesoscopic, and Electron Microscopy, School of Physics, Peking University, Beijing, 100871, P. R. China 2 Collaborative Innovation Center of Quantum Matter, Beijing, 100871, P. R. China ∗ Author to whom correspondence should be addressed. Email: [email protected] Received: 31 October 2015 Accepted: 27 April 2016

J. Nanoelectron. Optoelectron. 2016, Vol. 11, No. 5

the bandstructure and electroreflectance properties on this material.55 Similar computational modelling works were investigated for energy harvest applications.56–63 In this work, the structure -GaSe was investigated.64, 65 The bandstructure, PDOS, elastic material properties investigated here were compared in explicit, with the results performed by Ghalouci et al.,50 using Full Potential Linearized-Argumented-Plane Wave (FP-LAPW) methods66 of the Generalized-Gradient-Approximation (GGA).67

2. COMPUTATIONAL METHODS To make the electron system sufficient to converge, a total energy cutoff of 600 eV and the k-point grid of 8 × 8 × 4, under Monkhorst-Pack scheme,68 with 20 brillouin-zone sampling points were applied. For the bandstructure calculation, the self-consistent field (SCF) tolerance was set with a value of 2 × 106 eV/atom, with maximum force of 0.05 eV/Å, displacement of 0.002 Å and the maximum stress of 0.1 GPa. The interaction of the valence core was provided, with ultrasoft scheme,69 among the electron orbital 4s 2 4p1 3d 10 for Ga atom and 4s 2 4p4 for Se atom are deployed, rather than the norm-conserving pseudopotential.70–73 The lattice orientation of the C in Z plane, B along XZ plane was crucial when evaluating the optical properties. The degree of accuracy for the atomic description on material was specified, by comparing Cambridge-Sequential-Total-Energy-Package (CASTEP)74

1555-130X/2016/11/561/007

doi:10.1166/jno.2016.1984

561

The Structural, Electronic, Optical and Elastic Properties of -Type Gallium Selenide: A First Principle Study

with SIESTA,75 with the given factsheet.76 In this work, Ceperly, Alder, Perdew and Zunger, (CA-PZ)77, 78 of Local Density Approximation (LDA) was applied as exchange correlation potential. The spin-orbit effect on III–VIs were so small79 that spin-orbit interaction (SOI) was not included in this work. For material properties, the optical quantities were investigated, such as electron losses function L, the reflectivity R, refractive index n and absorption , using the real 1 and the imaginary part 2 of the dielectric tensor , calculated with density functional perturbation theory (DFPT). Those were shown in Eqs. (1)–(6), where term represented the frequency. 2 2 1 = 1 + p d (1) 0 2 − 2 2e2 kc uˆ × rkv Ekc − Ekv − E

0 1/2 2 2 1 + 2 − 1 =

2 =

1 + j2 − 1 2 R = 1 + j2 + 1 1/2 1 21 + 22 + 21 n = √ 2 2 L = 1 + 2

(2) (3) (4) (5) (6)

1 2

= (7) 4 Here, Eq. (1) utilized the formalism of Ehrenreich and Cohen.80 Equation (2) was simply derived from Eq. (1) with Kramers-Kronig relation. Equations (3)–(6) were found according to the previous study.81 Equation (7) was referred to the recent literature of Ambrosch-Draxl et al.82 Then, to testify the structural stability, the elastic properties were evaluated. Next, we used the following equations to calculate the bulk-modulus, shear-modulus and the Young’s Modulus of the material, Bv + BR 2 Gv + GR G= 2 9BG E= 3B + G 3B − 2G V= 23B + G B=

9Bv = 2c11 + c12 + c13 + c33 Gv =

562

1 c + c12 + 2c33 − c13 + c44 + c66 30 11 2 c11 + c12 c33 − 2c13 BR = c11 + c12 + 2c33 − 4c13

(8) (9) (10) (11) (12) (13) (14)

GR =

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2 c11 + c12 c33 − 2c13 c44 c66 2 c11 + c12 c33 − 2c13 c44 + c66 + 3Bv c44 c66

A1 =

2c66 c11 − c12

4c44 c11 + c33 − 2c13 B − BR Ac = v × 100 Bv + B R

A2 =

As =

Gv − G R × 100 Gv + GR

(15) (16) (17) (18) (19)

Here, providing the crystal symmetry, the number of elastic constants have been reduced down to 8 (c11 , c12 , c13 , c14 , c33 , c44 , c65 , c66 .83–85 The bulk B and shear G elastic quantity in Eqs. (8), (9) were represented with Ruess-Voigt-Hill approximations,86 while both Reuss (BR and GR 87 in Eqs. (14), (15) and Voigt (Bv and Gv 88 methods in Eqs. (12), (13) were being used, resulting in maximum and minimum elastic parameters. The E and V in Eqs. (10), (11) Indicated the Young’s Modulus and Poisson ratio respectively. A1 and A2 in Eqs. (16), (17) represented the shear isotropy factors,89 in active infrared mode. The anisotropy in shear (As and compressibility (Ac in Eqs. (18), (19) were also defined 90 in terms of percentage.

3. RESULTS AND DISCUSSION In this work, the monochalcogenide GaSe semiconductor in bulk was studied. The investigation consisted of the structural relaxation, bandstructure, PDOS, optical, and elastic properties. The Wyckoff atomic basis positions of Ga atom (0.6667, 0.3333, 0.15), and Se (0.3333, 0.6667, 0.425) were given. Then, structural optimization was then performed. The system required a total of 8 steps to make the electron energy sufficient to converge. To carry out a simulation run, a conventional unit cell was used, with a total of 10 Ga atoms and 10 Se atoms, as indicated in Figure 1. The lattice constants a = b and c in bulk after optimizing the cell were found to be 3.706 Å and 15.778 Å respectively, the bond angle of = = 90 and = 120 . The total energy was reported to be −9262.55 eV. The total energy plot against the lattice volume was shown in Figure 2. Such plot was useful in particular, to testify and see if the DFT calculations performed was at the equilibrium for the atomic configurations. This was in the case of over binding issues, in what the self-interaction between charges would be underestimated, in causing the theoretical to be smaller than experimental quantities.91 The point (0, 0, 0), among the high symmetry points of A(0, 0, 0,5), H(−0.333, 0.667, 0.5), K(−0.333, 0.667, 0), M(0, 0.5, 0) and L(0, 0.5, 0,5), were chosen within the zone boundary, in order to generate points in the desired k-paths. The direction was provided as → A → H → K → → M → L → H. The schematic diagram was found in Figure 3. J. Nanoelectron. Optoelectron., 11, 561–567, 2016

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–9261.8

Final Energy (eV)

Final Energy –9262.0

–9262.2

–9262.4

–9262.6 160

Fig. 2.

170

180 190 200 Lattice Volume (Å)

210

The total energy plot against the lattice volume of GaSe at bulk.

1.34 eV.50 The sub-bands I between 0.568 eV and 2.83 eV were the lower part of the conduction band, due to the mixed ps orbital from Ga and sp orbital from Se atom with high (low) density. Sub-bands II, III and IV formed the valance bands. The sub-bands II between −5 eV and 0 eV originated from the p states for Ga and the p state from Se atom. The sub-bands III between −3.73 eV and −6.89 eV were resulting from the small contribution from Ga s states and Se p states. The states near to the sub-band top contributed to the Se Pz states. The one near to the subband bottom and the middle corresponded to the Se Px and Py states respectively. The sub-bands IV between −7.9 eV and −6.87 eV turned out to be the large contribution from Ga s states and Se p states. The induction of an incident light was caused by the electron band-to-band transition from valence band to the conduction band. The power dissipation of electromagnetic wave, which was the total amount of photon energy absorbed in the material, corresponded to the imaginary

Fig. 1. The atomic structure of GaSe, indicating the atomic alignment of Gallium (Ga) and Selenium (Se), with the total of 20 atoms at bulk.

The bandstructure and PDOS were shown in Figures 4 and 5 respectively. The valance band top (VBT) was at the fermi level and the conduction band bottom (CBB) was at 0.568 eV. Brudnyi et al. reported with a value of 0.67 eV (0.89 eV), with(out) the inclusion of 3d orbital states from Ga atom.92 Ghalouci et al. obtained a value of J. Nanoelectron. Optoelectron., 11, 561–567, 2016

Fig. 3. The high symmetry points A, H , K, M, and L chosen on GaSe within the brillouin zone (BZ).

563

Eg, direct

0

Ef

–2 II

–4 –6

III IV

–8 G A

H K

G

M

L

part of the dielectric tensor 2 , as shown in Figure 6. Such plot represented the real 1 and the imaginary 2 parts of the dielectric function, which depended on the frequency, for the GaSe. The ionic and the electronic part of a non-polar material at high frequency, indicated in the static dielectric permittivity tensor (0). The dielectric constant at high frequency was reported to be 10.43. In Figure 6, the imaginary part of the dielectric tensor 2 indicated its first peak at 296 nm with the first edge is at 192.8 nm. The dielectric constant 2 which was

4s2 4p1 3d10

Ga

40

Partial Density of States (States/eV)

20 0 6

Se

4s2 4p4

4 2 0 60

GaSe

Total

40 20 0 –15

–10

–5

0

5

Energy (eV) Fig. 5. The partial density of states (PDOS) of GaSe plot at bulk.

564

α(ω)

300000 250000 200000 150000 100000 50000 0

R (ω)

1.0

10 5 0 –5 8

Re σ(ω) Im σ(ω)

0.8

4

0.6

0

0.4

–4

0.2 0.0

H

Fig. 4. The bandstructure of GaSe showing variation of electronic bands (I, II, III and IV) at bulk.

60

15

ε1(ω) ε2(ω)

Absorption (nm-1)

I

Refractive Index n(ω)

Energy (eV)

2

20

Reflectivity

Bandstructure

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4

n k

L (ω)

16 12

3 2

8

1

4

0

0 200 400 600 800 Wavelength (nm)

Electron Loss

4

Conductivity (fs-1) Dielectric function ε(ω)


200 400 600 800 Wavelength (nm)

Fig. 6. A diagram illustrating the optical properties of GaSe at bulk, with scissors operator = 0 eV, instrumental smearing of 0.5 eV, polarized, the polarization in direction of (10 00 00).

associated to the fundamental bandgap Eg , was identified to be 122 nm. This peak was related to the transition of the interband between VBT and the CBB. Figure 6 indicated the refractive index n and the extinction coefficient k. The static refractive index n0 was found to be 3.88. The refractivity spectrum R was displayed in Figure 6. The first edge value of this quantity was at 60.6 nm and the magnitude started to rise to attain the maximum at 92.3 nm. The absorption coefficient was found in Figure 6. The first absorption value was 63.2 nm, and the strong absorption coefficient was at 161.6 nm. In other words, there were low electron losses and strong absorption of the crystal at XUV range. Rybkovskiy et al. reported, with the absorption peak values of 415 and 560 nm respectively,51 using experimental data. This resulted in the fact that, the material was partially transparent in visible region. The conductivity was shown in Figure 6. The first edge was reported at the value of 6.17 (fs)−1 at wavelength of 189.6 nm. The value of 0.696 (fs)−1 at 122 nm was related to the fundamental bandgap. The maximum conductivity at 284.4 nm was found to be 9.59 (fs)−1 . In description of the energy loss on a fast electron, the prominent peaks found in the electron energy loss function (EELS) spectra L were responsible to a collective oscillation of the electrons in valance band, which was known as the plasma resonance. This was shown in Figure 6. Such peak was located at the plasma frequency p of 83.5 nm. At this frequency, the real part of dielectric tensor went through zero. J. Nanoelectron. Optoelectron., 11, 561–567, 2016

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C Ga

Ga

Layer 2 (lower)

Se2

Se1

Se1 Ga

Layer 1 Ga Se1

Se1

Se2

Layer 2

Ga

Ga

O

A

(upper)

Fig. 7. The contour plot of charge density in showing the lattice direction of A and C of GaSe.

Figure 7 showed the valence charge density contour plot of GaSe material. At this point, there were interactions between Ga–Ga and Ga–Se atoms. These were the covalent bonding. The Ga–Ga bonding was stronger than Ga–Se. On the other hand, the weak bonds between Se1–Se1, Se2–Se2 and Se1–Se2 were resulting from van de Waal force. The weak bonding on Se1–Se2 turned out Table I. The elastic constants cij (in GPa), young modulus E (in GPa), bulk modulus B, shear modulus G (in GPa), Poisson ratio V and the anisotropy percentage A for bulk material GaSe.

c11 c12 c13 c14 c33 c44 c65 c66 Bv BR B Gv GR G E V B/G A1 A2 Ac As

Our work

Ref. [50]

1009 2674 1994 0 4029 1707 0 3708 4170 3384 3777 2594 2187 2391 5923 0239 158 1 067 104 851

9783 4025 2567 – 3580 911 – 2879 4607 3388 3998 1873 1369 1621 4284 0321 247 1 044 1525 1556

J. Nanoelectron. Optoelectron., 11, 561–567, 2016

to be the interaction between the layer 1 and layer 2. Se1–Se2 was weaker than both Se1–Se1 and Se2–Se2. The formation of a unit cell structure was mainly caused by the covalent bond and Van de Waal interacting forces. Table I showed the isotropy quantities, which have been expressed in terms of percentage. The material would be identified as elastic, providing the isotropy value was reported to be 0%. On the other hand, the value of 100% would indicate that the material was related to the largest anisotropy. For two other values, material was shown to be isotropic if, A1 = A2 . As a matter of fact, the material was found to be anisotropic. The difference between the two clearly indicated the degree of elasticity on the material. The reference work according to Koroglu et al.,93 have highlighted that since the B/G critical value was found to be 1.75. The B/G work demonstrated here for bulk was below the critical value. Thus, material in bulk was thus determined to be brittle. Ghalouci et al.50 have reported, it was identified to be elastic. In general, the -GaSe structure was found to be stable. In Table I, it was observed, apart from As , all parameters in our work were in close agreement with the quantities reported by Ghalouci et al.50

4. CONCLUSION So far, there are no other data for valence charge density available. The → band transition has determined -GaSe to be a direct semiconductor, with an energy gap of 568 meV. The elastic parameters calculated in this work have determined the structural stability, and the material was found to be brittle. The bandstructure and the energy gap found in this work could be made, in close to the experimental by performing the Green’s function (G) and Coulombic interaction (W ) calculation, using GW approximations (GWA). This work has provided a comprehensive set of data, which is adequate for exploring the optical properties. The main applications for XUV would consist of photoelectron spectroscopy, solar imaging, and lithography. Our data will be useful to both theoretical and experimental researchers in the future, to unveil the new era of material properties. Acknowledgment: This work was funded by the National Natural Science Foundation of China (NSFC) project grants nos. 11234001 and 91433102, under the recipient of Dapeng Yu. Finally, the author would also like to thank Peking University (PKU) for the Postdoctoral Research Fellowship. This work made use of the computing facilities at International Centre of Quantum Materials (ICQM), supported by PKU.

References and Notes 1. E. D. Palik, Handbook of Optical Constants of Solids, Academic Press, Elsevier Inc., Netherland (1998), Vol. 3, ISBN: 978-0-12-544415-6.

565


2. O. Madelung, Semiconductors: Data Handbook, Springer Science and Business Media, Springer-Verlag, Berlin, Heidelberg (2012), ISBN: 978-3-642-18865-7. 3. N. C. Fernelius, Progress in Crystal Growth and Characterization of Materials 28, 275 (1994). 4. P. Liska, K. R. Thampi, M. Grätzel, D. Bremaud, D. Rudmann, H. M. Upadhyaya, and A. N. Tiwari, Appl. Phys. Lett. 88, 203103 (2006). 5. E. Mooser and M. Schlüter, Il Nuovo Cimento 18, 164 (1973). 6. X. Bu, N. Zheng, X. Wang, B. Wang, and P. Feng, Angewandte Chemie 116, 1528 (2004). 7. W. Shi, Y. J. Ding, X. Mu, and N. Fernelius, Appl. Phys. Lett. 80, 3889 (2002). 8. M. Gauthier, A. Polian, J. M. Besson, and A. Chevy, Physical Review B 40, 3837 (1989). 9. S. Chandra, T. H. Allik, G. Catella, R. Utano, and J. A. Hutchinson, Appl. Phys. Lett. 71, 584 (1997). 10. G. J. Hughes, A. McKinley, R. H. Williams, and I. T. McGovern, Journal of Physics C: Solid State Physics 15, L159 (1982). 11. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, Springer-Verlag, Berlin, Heidelberg (2013), Vol. 64, ISBN: 978-3-540-46793-9. 12. P. Schmid and J. P. Voitchovsky, Physica Status Solidi (B) 65, 249 (1974). 13. H. Peng, S. Meister, C. K. Chan, X. F. Zhang, and Y. Cui, Nano Lett. 7, 199 (2007). 14. D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey, Springer Science and Business Media, New York (2006), ISBN: 978-387-0-22022-2. 15. R. H. Bube and E. L. Lind, Physical Review 115, 1159 (1959). 16. J. L. Oudar, P. J. Kupecek, and D. S. Chemla, Optics Communications 29, 119 (1979). 17. J. H. Park, M. Afzaal, M. Helliwell, M. A. Malik, P. O’Brien, and J. Raftery, Chem. Mater. 15, 4205 (2003). 18. S. L. Stoll, E. G. Gillan, and A. R. Barron, Chem. Vap. Deposition 2, 182 (1996). 19. M. Di. Giulio, G. Micocci, P. Siciliano, and A. Tepore, J. Appl. Phys. 62, 4231 (1987). 20. V. G. Voevodin, O. V. Voevodina, S. A. Bereznaya, Z. V. Korotchenko, A. N. Morozov, S. Y. Sarkisov, and J. T. Goldstein, Opt. Mater. 26, 495 (2004). 21. N. B. Singh, T. Henningsen, V. Balakrishna, D. R. Suhre, N. Fernelius, F. K. Hopkins, and D. E. Zelmon, J. Cryst. Growth 163, 398 (1996). 22. F. Meyer, E. E. De Kluizenaar, and D. Den Engelsen, JOSA 63, 529 (1973). 23. K. Allakhverdiev, N. Ismailov, Z. Salaeva, F. Mikailov, A. Gulubayov, T. Mamedov, and S. Babaev, Appl. Opt. 41, 148 (2002). 24. A. Kuhn, A. Chevy, and R. Chevalier, Physica Status Solidi (A) 31, 469 (1975). 25. J. L. Brebner and J. A. Déverin, Helvetica Physica Acta 38, 650 (1965). 26. C. L. Marquardt, D. G. Cooper, P. A. Budni, M. G. Knights, K. L. Schepler, R. DeDomenico, and G. C. Catella, Appl. Opt. 33, 3192 (1994). 27. V. Capozzi and M. Montagna, Physical Review B 40, 3182 (1989). 28. J. F. Sánchez-Royo, D. Errandonea, A. Segura, L. Roa, and A. Chevy, J. Appl. Phys. 83, 4750 (1998). 29. A. Balzarotti, M. Piacentrini, E. Burattirii, and P. Piacentini, Journal of Physics C: Solid State Physics 4, L273 (1971). 30. R. H. Tredgold and A. Clark, Solid State Commun. 7, 1519 (1969). 31. B. B. Harbison, J. S. Sanghera, J. A. Moon, and I. D. Aggarwal, U.S. Patent No. 5,846,889, U.S. Patent and Trademark Office, Washington DC (1998).

566

Tse and Yu

32. V. K. Kapur, M. Fisher, and R. Roe, Nanoparticle oxides precursor inks for thin film copper indium gallium selenide (CIGS) solar cells, MRS Proceedings, Cambridge University Press, International Solar Electric Technology (ISET), Inglewood, CA, January (2001), Vol. 668, pp. H2-6. 33. K. R. Allakhverdiev, T. Baykara, S. Joosten, E. Günay, A. A. Kaya, A. Kulibekov, A. Seilmeier, and E. Y. Salaev, Optics Communications 261, 60 (2006). 34. D. V. Rybkovskiy, N. R. Arutyunyan, A. S. Orekhov, I. A. Gromchenko, I. V. Vorobiev, A. V. Osadchy, E. Y. Salaev, T. K. Baykara, K. R. Allakhverdiev, and E. D. Obraztsova, Physical Review B 84, 085314 (2011). 35. M. Di Giulio, G. Micocci, P. Siciliano, and A. Tepore, J. Appl. Phys. 62, 4231 (1987). 36. B. M. Ba¸sol, Thin Solid Films 361, 514 (2000). 37. A. G. Kyazym-Zade, A. A. Agaeva, V. M. Salmanov, and A. G. Mokhtari, Technical Physics 52, 1611 (2007). 38. C. Manfredotti, R. Murri, A. Quirini, and L. Vasanelli, Nuclear Instruments and Methods 131, 457 (1975). 39. A. Castellano, Appl. Phys. Lett. 48, 298 (1986). 40. A. Kenmochi, T. Tanabe, Y. Oyama, K. Suto, and J. T. Nishizawa, J. Phys. Chem. Solids 69, 605 (2008). 41. C. W. Chen, T. T. Tang, S. H. Lin, J. Y. Huang, C. S. Chang, P. K. Chung, S. T. Yen, and C. L. Pan, JOSA B 26, A58 (2009). 42. K. R. Allakhverdiev, M. Ö. Yetis, S. Özbek, T. K. Baykara, and E. Y. Salaev, Laser Physics 19, 1092 (2009). 43. W. Chen, J. Burie, and D. Boucher, Laser Physics-Lawrence 10, 521 (2000). 44. G. V. Rao, G. H. Chandra, P. S. Reddy, O. M. Hussain, K. R. Reddy, and S. Uthanna, Journal of Optoelectronics and Advanced Materials 4, 387 (2002). 45. J. Amzallag, H. Benisty, S. Debrus, M. May, M. Eddrief, A. Bourdon, A. Chevy, and N. Piccioli, Appl. Phys. Lett. 66, 982 (1995). 46. A. Segura, A. Chevy, J. P. Guesdon, and J. M. Besson, Solar Energy Materials 2, 159 (1980). 47. V. N. Katerinchuk, Z. D. Kovalyuk, and V. M. Kaminskii, Technical Physics Letters 26, 54 (2000). 48. S. Shigetomi and T. Ikari, J. Appl. Phys. 88, 1520 (2000). 49. N. Romeo, La Rivista del Nuovo Cimento (1971–1977) 3, 103 (1973). 50. L. Ghalouci, B. Benbahi, S. Hiadsi, B. Abidri, G. Vergoten, and F. Ghalouci, Computational Materials Science 67, 73 (2013). 51. D. V. Rybkovskiy, N. R. Arutyunyan, A. S. Orekhov, I. A. Gromchenko, I. V. Vorobiev, A. V. Osadchy, and E. D. Obraztsova, Physical Review B 84, 085314 (2011). 52. D. V. Rybkovskiy, I. V. Vorobyev, A. V. Osadchy, and E. D. Obraztsova, J. Nanoelectron. Optoelectron. 7, 65 (2012). 53. X. Zhou, J. Cheng, Y. Zhou, T. Cao, H. Hong, Z. Liao, and D. Yu, J. Am. Chem. Soc. 137, 7994 (2015). 54. V. K. Bashenov, I. Baumann, and D. I. Marvakov, Physica Status Solidi (B) 91, K125 (1979). 55. D. E. Thornton and S. Sampanthar, Journal of Physics C: Solid State Physics 4, L271 (1971). 56. G. Tse, J. Pal, U. Monteverde, R. Garg, V. Haxha, M. A. Migliorato, and S. Tomi´c, J. Appl. Phys. 114, 073515 (2013). 57. J. Pal, G. Tse, V. Haxha, M. A. Migliorato, and S. Tomi´c, Optical and Quantum Electronics 44, 195 (2012). 58. H. Y. S. Al-Zahrani, J. Pal, and M. A. Migliorato, Nano Energy 2, 1214 (2013). 59. H. Y. Al-Zahrani, J. Pal, M. A. Migliorato, G. Tse, and D. Yu, Nano Energy 14, 382 (2014). 60. G. T. D. Yu, Asian Journal of Current Engineering and Maths 4, 49 (2015). 61. G. T. D. Yu, Asian Journal of Current Engineering and Maths 4, 56 (2015). J. Nanoelectron. Optoelectron., 11, 561–567, 2016

Tse and Yu


62. G. Tse and D. Yu, Computational Condensed Matter 4, 59 (2015). 63. G. Tse and D. Yu, Modern Physics Letters B 30, 1650048 (2015). 64. L. Ghalouci, B. Benbahi, S. Hiadsi, B. Abidri, G. Vergoten, and F. Ghalouci, Computational Materials Science 67, 73 (2013). 65. Z. Rak, S. D. Mahanti, K. C. Mandal, and N. C. Fernelius, J. Phys. Chem. Solids 70, 344 (2009). 66. S. Cottenier, Density Functional Theory and the Family of (L) APW-methods: A Step-by-Step Introduction, Instituut Voor Kern-En Stralingsfysica, KU Leuven, Belgium (2002), Vol. 4, p. 41. 67. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 68. H. J. Monkhorst and J. D. Pack, Physical Review B 13, 5188 (1976). 69. D. Vanderbilt, Physical Review B 41, 7892 (1990). 70. R. Fu, M. H. Lee, and M. C. Payne, J. Phys.: Condens. Matter. 8, 2539 (1996). 71. J. S. Lin, A. Qteish, M. C. Payne, and V. Heine, Physical Review B 47, 4174 (1993). 72. A. M. Rappe, K. M. Rabe, E. Kaxiras, and J. D. Joannopoulos, Physical Review B 41, 1227 (1990). 73. V. Heine and D. Weaire, Solid State Physics, edited by H. Ehrenreich, F. Seitz, and D. Turnbull, Academic Press, New York (1970), Vol. 24, p. 1. 74. S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. I. Probert, K. Refson, and M. C. Payne, Zeitschrift für Kristallographie 220, 567 (2005). 75. D. Sanchez-Portal, I. Souza, and R. M. Martin, AIP Conf. Proc. 535, 111 (2000). 76. Supplementary Material (ESI) for Chemical Communications, The Royal Society of Chemistry (2002), http://www .rsc.org/suppdata/cc/b3/b309000k/b309000k.pdf. 77. J. P. Perdew and A. Zunger, Physical Review B 23, 5048 (1981).

J. Nanoelectron. Optoelectron., 11, 561–567, 2016

78. D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980). 79. M. O. D. Camara, A. Mauger, and I. Devos, Physical Review B 65, 125206 (2002). 80. H. Ehrenreich and M. H. Cohen, Physical Review 115, 786 (1959). 81. M. Arbi, N. Benramdane, Z. Kebbab, R. Miloua, F. Chiker, and R. Khenata, Mater. Sci. Semicond. Process. 15, 301 (2012). 82. C. Ambrosch-Draxl and J. O. Sofo, Comput. Phys. Commun. 175, 1 (2006). 83. B. Lee, R. E. Rudd, J. E. Klepeis, and R. Becker, Physical Review B 77, 134105 (2008). 84. J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, USA (1985). 85. J. J. Wang, F. Y. Meng, X. Q. Ma, M. X. Xu, and L. Q. Chen, J. Appl. Phys. 108, 034107 (2010). 86. R. Hill, The elastic behaviour of a crystalline aggregates, Proceedings of the Physical Society. Section A (1952), Vol. 65, p. 349. 87. W. Voight, Lehrbook der Kristallphysik, Teubner, Leipzig (1928), p. 962. 88. A. Reuss, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 9, 49 (1929). 89. P. Ravindran, L. Fast, P. A. Korzhavyi, B. Johansson, J. Wills, and O. Eriksson, J. Appl. Phys. 84, 4891 (1998). 90. D. H. Chung, W. R. Buessem, F. W. Vahldiek, and S. A. Mersol, Anisotropy in Single Crystal Refractory Compounds, Plenum, New York (1968), pp. 357–381. 91. G. Tse, Study of piezoelectricity on III/V semiconductors from atomistic simulations to computer modelling Doctoral Dissertation, The University of Manchester (2012). 92. V. N. Brudnyi, A. V. Kosobutsky, and S. Y. Sarkisov, Semiconductors 44, 1158 (2010). 93. U. Koroglu, S. Cabuk, and E. Deligoz, J. Alloys Compd. 574, 520 (2013).

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