Indirect-to-direct band gap transition of the ZrS₂ monolayer by strain: first-principles calculations

Abstract

The elastic, electronic and optical properties of the ZrS₂ monolayer and the effects of different kinds of in-plane strains on its electronic structure are investigated using first-principles method. By analyzing the strain–energy relationship we show that the ZrS₂ monolayer has a Poisson's ratio of 0.22 and an in-plane stiffness of 75.74 N m⁻¹. The band structures of the ZrS₂ monolayer calculated using both generalized gradient approximation and hybrid functional present an indirect band gap feature. The optical properties of the ZrS₂ monolayer exhibit strong anisotropy. In the low-energy region, the perpendicular dielectric function dominates while in the high-energy range both perpendicular and parallel polarizations contribute. Moreover, strain has significant influence on the band structure. It is found that the band gap of the ZrS₂ monolayer can be continuously modified from zero to 2.47 eV always with an indirect band gap under symmetrical strain in the elastic regime. Remarkably, an indirect-to-direct band gap transition has been observed when the uniaxial strain is applied to the monolayer along either the zigzag or armchair directions.

---

1 Introduction

The discovery of graphene in 2004 (ref. 1) has set off a new wave of research on two-dimensional (2D) materials, such as graphyne,^2,3 silicene,^4,5 boron nitride (BN),^6,7 and transition metal dichalcogenide (TMD) monolayers.^8–10,13 These 2D materials not only open up new physics, but also provide many appealing opportunities for industrial applications including a new generation of transistors, photoemitting devices, hydrogen storage, and spintronics.^10–15 Graphene, due to the 2D honeycomb network and sp² hybridized orbitals, is one of the strongest materials ever tested.¹⁶ The unique electronic properties, for instance, the half-integer quantum Hall effect, mass-less carriers, high migration rate and saturation velocity,^1,17 suggest that graphene has great potential for ultrahigh-speed electronics.^18–21 However, graphene has zero band gap, which means it is not feasible for use in a field-effect transistor (FET). Subsequently, various kinds of methods haven been proposed in order to open the band gap of graphene, such as by applying in-plane strain^22,23 or external electric field combined with chemical modification.^24–27 On the other hand, 2D semiconductor materials formed by layered TMDs have a natural band gap, and recent progress on a TMD-monolayer-based FET¹² have stimulated great research interests. Each TMD layer consists of three atomic layers: a transition metal (M) layer sandwiched between two chalcogen (X) layers, forming the hexagon with M and X atoms alternately situated at the corners. In bulk MX₂, these molecular layers are held together by weak van der Waals (vdW) interactions with the Bernal stacking similar to graphite. Interesting, MoX₂ and WX₂ undergo an indirect-to-direct band gap transition when the dimensionality decreases from 3D to 2D, with the band gaps ranging from 1.5 to 2.0 eV,^28,29 which are suitable for optoelectronic applications and in devices for energy harvesting.^12,14,30,31 Recently, a new 2D TMD, the ZrS₂ monolayer, has been synthesized experimentally.³² ZrS₂-based FET devices with ultrafast response times and ultrahigh responsivity have been fabricated to study its electrical transport properties.³³ Subsequently, the same authors have also demonstrated that ZrS₂ nanostructures can be promising candidates for large-area solar cell applications with high short-circuit currents.³⁴ However, further developments and applications of 2D ZrS₂ are hampered by the absence of a systematic research work on 2D ZrS₂. Many properties of ZrS₂, such as in-plane stiffness and Poisson's ratio, optical reflection and adsorption coefficients, and strain effects on the band structure, which are important for material design, remains unclear. Therefore, a theoretical study on these properties is urgently needed. Moreover, the complicated and versatile electronic structures of 2D TMD monolayers inspire us to believe that the ZrS₂ monolayer deserves specific attention.

In this work, we systematically investigate the elastic, electronic and optical properties of the ZrS₂ monolayer and the intercoupling between strain and its electronic properties by means of density functional theory calculations. The Poisson's ratio and in-plane stiffness of the pristine ZrS₂ monolayer are calculated. The band structure of the ZrS₂ monolayer is studied using both GGA and HSE06. The frequency-dependent dielectric function ε, reflectivity R and adsorption coefficient I are computed to study the optical properties of the ZrS₂ monolayer. Furthermore, interesting variations of the band structures under different kinds of in-plane strains are observed. The mechanisms for these changes are discussed in detail.

2 Computational method

The calculations were performed using the projector augmented wave (PAW) method³⁵ with the generalized gradient approximation of Perdew–Burke–Ernzerhof (GGA-PBE)³⁶ exchange-correlation functional and the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional,³⁷ as implemented in the Vienna ab initio simulation package (VASP).³⁸ The energy cutoff for plane-wave expansion was set to 400 eV. A large vacuum layer of 12 Å was adopted to prevent the interaction between adjacent images. Brillouin zone (BZ) sampling was performed with Monkhorst Pack (MP) special k points meshes.³⁹ A k-points grid of 12 × 12 × 1 was chosen for the calculations. All the structures were fully relaxed using the conjugated gradient method until the Hellmann–Feynman force on each atom was less than 0.01 eV Å⁻¹.

3 Results and discussion

3.1 Structural and elastic properties

ZrS₂ crystallizes in the simplest 1T-CdI₂ type structure in which octahedrally-coordinated sandwich layers are stacked with a periodicity of one layer and a globally D_3d point-group symmetry. We calculate the properties of the bulk ZrS₂ first to test the reliability and accuracy of our method. Table 1 lists our test results along with other calculations and experimental values. Our results for bulk ZrS₂ are in agreement with earlier calculations^40–42 in the matter of band structure and band gap. The slight difference of the band gap between ref. 40 and ours may be due to different computational accuracy and the calculation packages adopted in the calculations. The ZrS₂ monolayer is a hexagonal lattice with two S atoms and one Zr atom per unit cell and a lattice basis defined by the vectors (a⃑₁, a⃑₂), as shown in Fig. 1(a). Two inequivalent crystallographic directions can be defined: the so-called armchair and zigzag directions, corresponding to x and y axes, respectively, also shown in Fig. 1(a).

Table 1 The test results of bulk ZrS₂ (the hexagonal lattice constants a and c (Å), the Zr–S bond length d (Å), the locations of VBM and CBM, and the indirect band gap E_g (eV)) as compared to previous results

	a	c	d	VBM	CBM	Eg
a Ref. 41.b Ref. 42.
Test	3.690	6.494	2.58	Γ	L	1.04
Ref. 40	3.687	6.659	2.58	Γ	L	0.79
Experimental	3.661^a	5.815^a	—	—	—	1.68^a, 1.7^b

---

Fig. 1 (a) Schematic representation of the ZrS₂ monolayer. The rectangular and rhombic shadows present the unit cells used to calculate the elastic and electronic properties, respectively. The green and yellow balls denote Zr and S, respectively. (b) The grid of data points (ε_a, ε_z) and the contours of total energy. (c) The three-dimensional plot of (ε_a, ε_z) and corresponding total energy.

The elastic property of ZrS₂ monolayer is described by two parameters: in-plane stiffness C and Poisson's ratio ν, which can be calculated by fitting the strain–energy relationship of the TMD monolayer, as described in ref. 43. The in-plane stiffness C can be defined as the second derivative of the total energy with respect to the strain at the equilibrium area, as specifically for two-dimensional systems, that is C = (1/S₀)(∂²E/∂ε²), where S₀ is the equilibrium area, E is the total energy, and ε is uniaxial strain (ε = Δa/a). The other parameter, Poisson's ratio ν, is the ratio of transverse strain and axial strain, namely, ν = −ε_trans/ε_axial. In order to calculate the elastic constants, we construct a rectangular unit cell of ZrS₂, which contains two ZrS₂ molecules, as seen in Fig. 1(a). a_z and a_a are the lattice constants along the zigzag and armchair directions, respectively. The two lattice constants are changed under the strain values between −0.02 and +0.02 with an increment of 0.01, as shown in Fig. 1(b). For each grid point, the structure is fully optimized and the total energy is calculated, as presented in Fig. 1(c). Following ref. 43, the data is fitted to the formula E = a₁ε_a² + a₂ε_z² + a₃ε_aε_z + E₀, where ε_a and ε_z are the strain along the armchair and zigzag directions, respectively, and E₀ is the total energy of the equilibrium configuration. Due to the isotropy in the honeycomb symmetry, a₁ is equal to a₂. By fitting the data, we get the specific formula E = 54.78ε_a² + 54.78ε_z² + 21.35ε_aε_z − 42.70. Assuming that the monolayer is under an uniaxial strain along the zigzag direction, the plane will spontaneously present a strain along the armchair direction due to the intrinsic properties of the material. The strain ε_a satisfies the expression ∂E/∂ε_a = 0, from which one obtains the Poisson's ratio ν = −ε_a/ε_z = a₃/2a₁ and the in-plane stiffness C = (1/S₀)(2a₁ − a₃²/2a₁). Hence the Poisson's ratio of the ZrS₂ monolayer is 0.22 and its in-plane stiffness is 75.74 N m⁻¹.

3.2 Electronic and optical properties

The band structure and partial density of states (PDOS) of the ZrS₂ monolayer are shown in Fig. 2. The general features of the band structures calculated by PBE and HSE06 are similar, except that the band gap calculated by HSE06 is much larger. The valence band maximum (VBM) and the conduction band minimum (CBM) are located at Γ and M, respectively, resulting in an indirect band gap of 1.12 eV for PBE and 1.93 eV for HSE06. The direct band gap of Γ–Γ is 1.63 eV for PBE (2.53 eV for HSE06). As the band gap of bulk ZrS₂ is around 1.7 eV,^41,42 the indirect band gap of ZrS₂ monolayer would be between 1.7 and 1.93 eV considering the quantum confinement effect for 2D systems. The band structure of the ZrS₂ monolayer can be divided into three main groups according to the angular momentum character identified from the PDOS of ZrS₂ (Fig. 2(b)). The first group between −6 and −2 eV mainly stems from S p states, Zr p and d states. A strong hybridization between Zr d and S p states is found here. The second group between −2 eV and the Fermi level (E_f) is composed of S p states and a small part of Zr d states. The third group from 1 eV and above has contributions mostly from Zr d states and a little from S p states.

Fig. 2 (a) The band structure of the ZrS₂ monolayer. The black lines and red dots present the band distributions calculated by PBE and HSE06, respectively. E_f is set at the VBM. (b) The PDOS of the ZrS₂ monolayer calculated by PBE only.

Due to the hexagonal symmetry, the dielectric tensor only has two independent components, namely, ε^⊥ and ε^∥, corresponding to the electric vectors E⃑ parallel and perpendicular to the c-axis, respectively. The real and imaginary parts of the frequency-dependent dielectric functions are calculated by HSE06 following the method in ref. 44, then the frequency-dependent reflectivity R and adsorption coefficient I are computed following the expressions in ref. 45. Strong anisotropy is observed between the imaginary parts of dielectric functions ε^⊥₂ and ε^∥₂ (Fig. 3(a)). While ε^⊥₂ dominates in the low-energy region (0 to 6 eV), both polarizations contribute in the higher energy region (above 6 eV). The variations of reflectivity and adsorption coefficients as the function of frequency also exhibit similar trends, as illustrated in Fig. 3(b) and (c). The spectra from 2 to 8 eV in Fig. 3(a) originate from the interband transitions between S p states (the second energy group) and Zr d states (the third energy group), and the spectra in the higher energy region (above 8 eV) are from the interband transitions from S p and Zr d states (the first energy group) to Zr d states (the third energy group).

Fig. 3 Calculated optical properties of the ZrS₂ monolayer: (a) Imaginary part of the dielectric function ε₂. (b) Reflectivity R. (c) Adsorption coefficient I.

3.3 Electronic properties of the ZrS₂ monolayer under strain

Direct band gap materials are usually preferred as photovoltaic materials due to the high efficiency of direct transition. However, the pristine ZrS₂ monolayer is an indirect band gap material. Nevertheless, it is well established that strain can efficiently modify the band structures of low dimensional materials. Hence, we investigate the influence of in-plane strain on the band structure modulation of ZrS₂. Three kinds of in-plane strains are applied to the ZrS₂ monolayer: (i) uniform strain, ε_u; (ii) uniaxial strain along the zigzag direction, ε_z; (iii) uniaxial strain along the armchair direction, ε_a. As the lattice vectors are changed upon straining a ZrS₂ monolayer, the associated reciprocal vectors (g⃑₁, g⃑₂) are affected accordingly. Fig. 4 presents the Brillouin zone (BZ) of the ZrS₂ monolayer under strain. The irreducible BZ of ZrS₂ under uniform strain remains unchanged, as the hexagonal symmetry is saved. For the ZrS₂ monolayer under uniaxial strain along either the zigzag or armchair directions, the structure of the BZ is deformed as the hexagonal symmetry is degenerated to rhombic symmetry, and the irreducible BZ evolves into a right angle trapezoid from the original triangular shape. Moreover, the high-symmetry k-points increase from three (Γ, K, M) to five (Γ, K, M, R, S).

Fig. 4 BZ of the ZrS₂ monolayer under (a) uniform strain (b) uniaxial strain along the zigzag direction (c) uniaxial strain along the armchair direction. The shaded areas are the corresponding irreducible BZ.

The electronic structure of the ZrS₂ monolayer is computed for each deformed configuration by means of PBE and HSE06. For uniform strain, the strain ranges from −10% to 15% with an increment of 1%, namely, both compressive strain and tensile strain are applied. Fig. 5(a) presents the variations of band gap (E_g) and the energy difference (ΔE) between deformed and undeformed ZrS₂ monolayers as functions of uniform strain. ΔE increases monotonously for both tensile and compressive strains, suggesting that the system is in the range of elastic deformation. E_g decreases almost linearly with the increasing compressive strain for both PBE and HSE06. When the strain reaches −10%, the band gap closes up. The VBM and CBM of the ZrS₂ monolayer overlap and it becomes semi-metal. For compressive strain from −1% to −10%, the VBM and the CBM are located at Γ and M, respectively, without changing. In the case of tensile strain, E_g increases almost linearly when the applied strain ranges from 1% to 7%, then decreases monotonously from 8% to 15%. While the VBM gradually moves from Γ to the middle between Γ and K, denoted as X in Fig. 5(b), the CBM moves from M to Γ. When the strain gradually changes from the maximum compressive strain (ε_u = −10%) to the maximum tensile strain (ε_u = 15%), the band structures of the ZrS₂ monolayer exhibit some notable features. It can be noted that the eigenstates of the conduction bands decrease with decreasing compressive strain or increasing tensile strain, i.e., all conduction bands move downwards, while the valence bands present the opposite variation trend. In addition, the width of each band decreases due to the decreasing orbital overlap resulting from the increasing Zr–S bond length, which means that the electrons become more localized during the tensile process. The variations of energy eigenvalues of the highest valence band (HVB) and the lowest conduction band (LCB) at some specific k-points under different uniform strains are investigated, as illustrated in Fig. 5(c). We note that the two bands at different high symmetry points shift with different rates of change. More specifically, the descent rate of the HVB at Γ point is much faster than that at X point with decreasing compressive strain, and when ε_u = 5%, the X point becomes higher than the Γ point, which accounts for the shift of the VBM. While the eigenvalue of the LCB at Γ point decreases with decreasing compressive strain and increasing tensile strain, that of the LCB at M point increases, and when ε_u = 6%, the Γ point becomes lower than the M point, which accounts for the shift of the CBM. In the pristine ZrS₂ monolayer, while the VBM (at Γ point) is composed of S p_x and p_y orbitals, the HVB at the X point is composed of S p_z and p_y orbitals. While the CBM (at M point) is composed mostly of Zr d_z², d_xy orbitals and S p_z orbitals, as well as a small part of Zr d_xy and d_xz orbitals, the LCB at Γ point is composed mainly of Zr d_xy, d_yz, d_xz and d_x²−y² orbitals. Therefore, the HVB at Γ point has much more x and y orbitals of p electrons than at the X point, and the HVB at X point has much more z orbitals of p electrons than at the Γ point. The LCB at Γ point has much more x and y orbitals of d electrons than at the M point, and the LCB at M point has much more z orbitals of p and d electrons than at the Γ point. As the strain is applied along the xy-plane, it is reasonable to deduce that the x and y orbitals are more sensitive to the in-plane strain than the z orbitals. Therefore, the states composed of more x and y orbitals will experience a faster change rate than the k-points composed of more z orbitals under in-plane strain. In other words, the different rates of change of the bands at different high-symmetry k-points can be explained by the different strain response rates corresponding to diverse orbital components.

Fig. 5 (a) Variations of E_g and ΔE between the deformed and undeformed ZrS₂ monolayer as functions of uniform strain. (b) Band structures of the ZrS₂ monolayer under uniform strain. (c) Variations of energy eigenvalues of the HVB and LCB at some specific k-points as functions of uniform strain. The VBM of a pristine ZrS₂ monolayer is set to zero as the reference energy point after the vacuum level has been aligned for all calculations.

In the case of uniaxial strain, the band structure changes remarkably. The strain ranging from 1% to 15% with an increment of 1% is taken into consideration for both zigzag and armchair directions. The variations of E_g and ΔE between deformed and undeformed ZrS₂ monolayers as functions of zigzag strain are shown in Fig. 6(a), and the band alignments are presented in Fig. 6(b). When ε_z < 8%, E_g increases with the applied strain, and the VBM and CBM are located at Γ and M points, respectively. When ε_z ≥ 8%, the CBM moves to Γ point, and the monolayer becomes a direct band gap material. But E_g decreases with the increasing strain in this case. When armchair strain is applied, the band structure also undergoes an indirect-to-direct band gap transition at ε_a = 8% and E_g also increases with the strain when ε_a < 8%, then decreases when ε_a ≥ 8%. Since zigzag and armchair strains have similar effects, in the following we'll mainly focus on the former. Fig. 6(c) illustrates the variations of energy eigenvalues of the HVB and LCB at different k-points as functions of zigzag strain. The states at the X point have lower energy than that of Γ for strain ε_z < 15%, thus the VBM remains unchanged. The LCB at the Γ point demonstrates a fast descent trend, while the LCB at M point exhibits a rising trend, which results in the change of the CBM. The lost hexagonal symmetry induced by the uniaxial strain causes the change of orbital components at the high-symmetry k-points, thus bringing about the different strain response rates, which accounts for the indirect-to-direct band gap transitions. The feature that the band gap of monolayer ZrS₂ can be modified within a wide range (0 to 2.47 eV for HSE06) by in-plane strain makes it a promising material for the solar cells because the band gaps match well with the range of visible light. In addition the monolayer can have a direct band gap under uniaxial strain.

Fig. 6 (a) Variations of E_g and ΔE between deformed and undeformed ZrS₂ monolayers as functions of zigzag strain. (b) Band structures of the ZrS₂ monolayer under zigzag tensile strain. (c) Variations of the energy eigenvalues of the HVB and LCB at some specific k-points as functions of zigzag strain. The VBM of a pristine ZrS₂ monolayer is set to zero as the reference energy point after the vacuum level has been aligned for all calculations.

4 Conclusion

In summary, we have studied the elastic, electronic and optical properties of the ZrS₂ monolayer and the influence of different kinds of in-plane strains on the band structure by first-principles calculations. The Poisson's ratio and the in-plane stiffness of the ZrS₂ monolayer computed by fitting the strain–energy relationship of the ZrS₂ monolayer are 0.22 and 75.74 N m⁻¹, respectively. The optical properties of the ZrS₂ monolayer are found to be strongly anisotropic. The perpendicular dielectric function has much larger values than the parallel one in the low-energy region, while they are similar in the high-energy region. The pristine ZrS₂ monolayer has an indirect band gap with the CBM and VBM located at M and Γ, respectively. When a symmetrical uniform strain is applied to the monolayer, the band structure of the ZrS₂ monolayer is remarkably modified and the band gap can be tuned from 0 to 2.47 eV for −10% < ε_u < 15%. When the uniaxial strain is applied along the zigzag or armchair direction, an indirect-to-direct band gap transition is observed for ε_z(a) > 8%. The direct band gaps range from 1.8 to 2.2 eV, which offers great opportunities for industrial applications.

Acknowledgements

J. Li gratefully acknowledges financial support from the Natural Science Foundation for Distinguished Young Scholar (Grant no. 60925016 and 91233120). This work was supported by the National Basic Research Program of China (Grant no. 2011CB921901) and the External Cooperation Program of Chinese Academy of Sciences.

References

1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov, Science, 2004, 306, 666.
2. D. Malko, C. Neiss, F. Viñes and A. Görling, Phys. Rev. Lett., 2012, 108, 086804.
3. B. G. Kim and H. Joon, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 86, 115435.
4. C. C. Liu, W. X. Feng and Y. Yao, Phys. Rev. Lett., 2011, 107, 076802.
5. P. Vogt, P. D. Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet and G. L. Lay, Phys. Rev. Lett., 2012, 108, 155501.
6. K. Watanabe, T. Taniguchi and H. Kanda, Nat. Mater., 2004, 3, 404.
7. G. Giovannetti, P. A. Khomyakov, G. Brocks, P. J. Kelly and J. van den Brink, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 76, 073103.
8. K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov and A. K. Geim, Proc. Natl. Acad. Sci. U. S. A., 2005, 102, 10451.
9. A. R. Botello-Méndez, F. López-Urías, M. Terrones and H. Terrones, Nanotechnology, 2009, 20, 325703.
10. Q. H. Wang, K. K. Zadeh, A. Kis, J. N. Coleman and M. S. Strano, Nat. Nanotechnol., 2012, 7, 699.
11. Q. Peng, C. Liang, W. Ji and S. De, Phys.Chem. Chem. Phys., 2013, 15, 2003.
12. B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti and A. Kis, Nat. Nanotechnol., 2011, 6, 147.
13. S. Tongay, J. Zhou, C. Ataca, J. Liu, J. S. Kang, T. S. Matthews, L. You, J. Li, J. C. Grossman and J. Wu, Nano Lett., 2013, 13, 2831.
14. O. Ochedowski, K. Marinov, G. Wilbs, G. Keller, N. Scheuschner, D. Severin, M. Bender, J. Maultzsch, F. J. Tegude and M. Schleberger, J. Appl. Phys., 2013, 113, 214306.
15. Y. Zhou, Z. Wang, P. Yang, X. Zu, L. Yang, X. Sun and F. Gao, ACS Nano, 2012, 6, 9727.
16. C. Lee, X. Wei, J. W. Kysar and J. Hone, Science, 2008, 321, 385.
17. A. K. Geim, Science, 2009, 324, 1530.
18. Y. M. Lin, K. A. Jenkins, A. Valdes-Garcia, J. P. Small, D. B. Farmer and P. Avouris, Nano Lett., 2009, 9, 422.
19. Y. M. Lin, C. Dimitrakopoulos, K. A. Jenkins, D. B. Farmer, H. Y. Chiu, A. Grill and P. Avouris, Science, 2010, 327, 662.
20. L. Liao, Y. C. Lin, M. Bao, R. Cheng, J. Bai, Y. Liu, Y. Qu, K. L. Wang, Y. Huang and X. Duan, Nature, 2010, 467, 305.
21. K. S. Novoselov, V. I. Fal'ko, L. Colombo, P. R. Gellert, M. G. Schwab and K. Kim, Nature, 2012, 490, 192.
22. G. Cocco, E. Cadelano and L. Colombo, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 241412.
23. V. Pereira, A. C. Neto and N. Peres, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 045401.
24. Y. Zhang, T. T. Tang, C. Girit, Z. Hao, M. C. Martin, A. Zettl, M. F. Crommie, Y. R. Shen and F. Wang, Nature, 2009, 459, 820.
25. E. Yu, D. Stewart and S. Tiwari, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 77, 195406.
26. B. M. Wong, S. H. Ye and G. O'Bryan, Nanoscale, 2012, 4, 1321.
27. M. Kim, N. S. Safron, C. Huang, M. S. Arnold and P. Gopalan, Nano Lett., 2012, 12, 182.
28. A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C. Y. Chim, G. Galli and F. Wang, Nano Lett., 2010, 10, 1271.
29. H. S. Matte, A. Gomathi, A. K. Manna, D. J. Late, R. Datta, S. K. Pati and C. N. Rao, Angew. Chem., Int. Ed., 2010, 49, 4059.
30. J. Kang, S. Tongay, J. Zhou, J. Li and J. Wu, Appl. Phys. Lett., 2013, 102, 012111.
31. J. Kang, S. Tongay, J. Li and J. Wu, J. Appl. Phys., 2013, 113, 143703.
32. Z. Zeng, Z. Yin, X. Huang, H. Li, Q. He, G. Lu, F. Boey and H. Zhang, Angew. Chem., Int. Ed., 2011, 50, 11093.
33. L. Li, X. S. Fang, T. Y. Zhai, M. Y. Liao, U. K. Gautam, X. C. Wu, Y. Koide, Y. Bando and D. Golberg, Adv. Mater., 2010, 22, 4151.
34. L. Li, H. Wang, X. S. Fang, T. Y. Zhai, Y. Bandoa and D. Golberga, Energy Environ. Sci., 2011, 4, 2586.
35. P. E. Blöchl, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 50, 17953.
36. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865.
37. J. Heyd, G. E. Scuseria and M. Ernzerhof, J. Chem. Phys., 2003, 118, 8207.
38. G. Kresse and J. Furthmuller, Phys. Rev. B: Condens. Matter Mater. Phys., 1993, 54, 11169.
39. H. J. Monkhorst and J. D. Pack, Phys. Rev. B: Condens. Matter Mater. Phys., 1976, 13, 5188.
40. H. Jiang, J. Chem. Phys., 2011, 134, 204705.
41. D. L. Greenaway and R. Nitsche, J. Phys. Chem. Solids, 1965, 26, 1445.
42. M. Moustafa, T. Zandt, C. Janowitz and R. Manzke, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 035206.
43. M. Topsakal, S. Cahangirov and S. Ciraci, Appl. Phys. Lett., 2010, 96, 091912.
44. M. Gajdos̆, K. Hummer, G. Kresse, J. Furthmüller and F. Bechstedt, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 73, 045112.
45. H. Shi, M. Chu and P. Zhang, J. Nucl. Mater., 2010, 400, 151.

---