Strain Effect on the Absorption Threshold Energy of Silicon Circular Nanowires

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1 Commun. Theor. Phys. 65 (2016) Vol. 65, No. 1, January 1, 2016 Strain Effect on the Absorption Threshold Energy of Silicon Circular Nanowires R. Khordad and H. Bahramiyan Department of Physics, College of Sciences, Yasouj University, Yasouj , Iran (Received March 23, 2015; revised manuscript received October 14, 2015) Abstract In this work, the influence of strain on threshold energy of absorption in Silicon circular nanowires is investigated. For this purpose, we first have used the density functional theory (DFT) to calculate the electron and hole effective masses. Then, we have obtained absorption threshold energy with two different procedures, DFT and effective mass approximation (EMA). We have also obtained the band structures of Si nanowires both DFT and EMA. The results show that: i) the expansive strain increases the hole effective mass while compressive strain increases the electron effective mass, ii) the electron and hole effective masses reduce with decreasing the wire size, iii) the absorption threshold energy decreases by increasing strain for compressive and tensile strain and its behavior as a function of strain is approximately parabolic, iv) the absorption threshold energy (for all sizes) obtained using EMA is greater than the DFT results. PACS numbers: Gy, uf, Hj Key words: quantum wire, ab initio calculations, optical properties, mechanical properties 1 Introduction It is fully known that the investigation on nanostructures is important because it opens a new field in fundamental sciences. As the size of a device is reduced to the nanometer scale, its electronic, thermodynamic, optical, and magnetic properties begin to differ drastically from its bulk counterparts because effects like tunneling and quantum confinement play a dominant role. This means that the physical properties in nanometer regime are strongly depended on quantum size effects arising from the reduced dimensions. Among the nanostructures, nanowires have attracted extensive attention over the past several decades. [1 6] A nanowire is a nanostructure, with the diameter of the order of a nanometer. Alternatively, nanowires can be defined as structures that have a thickness or diameter constrained to tens of nanometers or less and an unconstrained length. Many different types of nanowires exist, including metallic (e.g., Ni, Pt, Au) and semiconducting (e.g., Si, InP, GaN, etc.). In particular, silicon (Si) nanowires are very interesting for fundamental research because they play a vital role in future mesoscopic electronic and optical devices, such as light-emitting diodes, field-effect transistors inverters, and nanoscale sensors. [7 10] In past few years, many researchers have studied Si nanowires, theoretically and experimentally. For example, Cui et al. [7] reported that Si nanowire fieldeffect transistors show high performance with increase in the average conductance. Holmes et al. [11] observed the visible band-edge photoluminescence in Si nanowires. To khordad@mail.yu.ac.ir; rezakh2025@yahoo.com c 2016 Chinese Physical Society and IOP Publishing Ltd obtain more information about physical properties of Si nanowires, the reader can refer to Refs. [12 18]. The study and survey of nanowires has always been of great attention both theoretically and experimentally. Nanowires have many interesting properties that are not seen in bulk or 3D materials. This is because electrons in nanowires are quantum confined laterally and thus occupy energy levels that are different from the traditional continuum of energy levels or bands found in bulk materials. Hitherto, nanowires with various cross sections have been prepared using several methods like lithography, vapor-liquid-solid synthesis, and solution-phase synthesis. Examples of the nanowires with various cross sectional shapes are T-shape, V-shape, triangular, square, and circular quantum wires. [19 25] It is fully known that the study of mechanical processes of nanostructures like strain is of great importance in many fields of science and engineering. It should be noted that the strain and stress fields in quantum nanostructures are important because they affect the electronic structure and optical properties. The previous works have shown that strain can shift the valence and conduction bands, change band gap, and cause trapping of carriers and excitons. [26 28] There are several methods for studying strains and stresses appearing in nanostructured materials such as experimental measurements, theoretical modeling, and computer simulations. [29 32] In the past few years, many works have been done on the light interband absorption coefficient and the absorption threshold frequency of nanostructures. [33 36] One of most interesting problems about absorption threshold fre-

2 88 Communications in Theoretical Physics Vol. 65 quency is the effect of various factors on this property. According to our knowledge, the influence of strain on absorption threshold frequency of quantum wires has not been studied so far. For this goal, we have intended to study this problem for a Si nanowire with circular cross section. We have considered the variation of effective mass and energy band gap with respect to confinement. 2 Energy Spectrum and Wave Functions With considering an electron (hole) in a quantum wire, we can write the Hamiltonian of the system in the effective mass approximation by H = 2 2m 2 + V (x, y, z), (1) where m is effective mass of electron (hole) and V (x, y) is confining potential as { 0, Inside, V (x, y) = (2), Outside. With solving the Schrödinger equation, one can obtain the wave functions and energy levels of a quantum wire with circular cross section as ψ mnnz (ρ, ϕ, z) = AJ n (χ nm ρ a ) e imϕ sin ( nz πz L ), (3) E mnnz = 2 χ 2 nm 2m a π 2 n 2 z 2m L 2, (4) where J n is n-th Bessel function and χ nm is m-th zero of n-th Bessel function. Also, A, a, and L are normalization constant, circle radius, and wire length. 3 Threshold Frequency of Absorption In this section, we briefly present direct interband light absorption coefficient and the absorption threshold frequency in a system. The light absorption coefficient can be written as [37 39] α = N Ψ e ηψ h η dr 2 δ( E e η Eη h ), (5) η η where N is a quantity proportional to the square of dipole moment matrix element modulus. It is to be noted that the quantity can be taken from Bloch functions. Also, ϖ and ε g in the relation = ϖ ε g are the frequency of incident light and the width of forbidden gap, respectively. The quantities Ψ e(h) and E e(h) show the wave function of electron (hole) and the electron (hole) energy. Here, the parameters η(n, m, k z ) and η (n, m, k z) are the set of quantum numbers of electron and the hole. With respect to Eq. (5), one can define the threshold value of absorption as ω 00 = ε g + E e + E h. (6) To obtain the absorption threshold frequency (ω 00 ) or absorption threshold energy ( ω 00 ), we require the energy of electron and hole. In this regards, we can apply Eq. (4) for obtaining absorption threshold frequency or absorption threshold energy in a Si circular quantum wire. But, we need the effective mass of electron and hole. In this work, we have calculated the effective mass of electron and hole as a function of strain. Then, we have used these effective masses for obtaining the absorption threshold energy. 4 Density Functional Theory In this section, we study the strain dependence of electron (hole) effective mass in a Si circular nanowire. For this purpose, we have used the density-functional theory to obtain electronic properties. We have employed VASP code with plane-wave pseudo potential total energy calculation. [40] This first principle calculation is based on density functional theory (DFT) within generalized gradient approximation (GGA) without of spin polarized. The electrostatic interaction between valence electron and ionic core is represented by ultra-soft pseudo potentials and the electronic exchange-correlation energy is treated under LDA. The kinetic energy cut-off for the plane wave basis set is 350 ev. These parameters are sufficient in leading to well-converged of total energy and geometrical configurations. In our calculations, we have found that the increase of the plane-wave cut-off energy will change the total energy less than 0.03 ev/atom. After calculating the electronic properties of the wire, one can obtain the effective mass by using m = 2 (d 2 ε/dk 2 ) 1. 5 Results and Discussion In this section, we have presented the numerical results for a Si circular quantum wire with three different sizes as a = 1.8 nm, 2.4 nm, and 2.9 nm. At first, we have calculated the electron and hole effective masses with DFT theory. Then, we have obtained threshold absorption energy ( ω 00 ) using both the DFT and effective mass approximation (EMA). The used parameter in this work E g-bulk = 0.65 ev. Fig. 1 The electron effective mass as a function of strain for a Si circular quantum wire with three different sizes. In Fig. 1, we have shown the electron effective mass as a function of strain for three different sizes. The electron

3 No. 1 Communications in Theoretical Physics 89 effective mass increases when the compressive strain (negative strain) increases and this is approximately constant by increasing the tensile strain (positive strain). Also, the bigger quantum wire (a = 2.9 nm) has a greater electron effective mass. has been plotted in Fig. 2. The hole effective mass enlarges by increasing the tensile strain until a maximum and then decreases. Also, the hole effective mass decreases when the compressive strain increases. The enlargement of hole effective mass is bigger for greater quantum wire (a = 2.9 nm). It is seen from the figures that the electron effective mass for all three sizes approaches constant value at high tensile strains whereas this behavior occurs at high compressive strains for the hole effective mass. Fig. 2 The variation of hole effective mass with strain for a Si circular quantum wire for three different sizes. Fig. 5 The threshold absorption energy with considering Coulomb interaction versus strain for a Si circular nanowire. The results obtained using both density functional theory (DFT) and effective mass approximation (EMA). Fig. 3 The threshold absorption energy as a function of strain for a Si circular nanowire with three different sizes. The results obtained using both density functional theory (DFT) and effective mass approximation (EMA). Fig. 6 The band structure of Silicon nanowire calculated by DFT and EMA for a = 2.9 nm. Fig. 4 The band structure of Silicon nanowire calculated by DFT for a = 2.9 nm. The variation of hole effective mass with strain for three different sizes as a = 1.8 nm, 2.4 nm, and 2.9 nm In Fig. 3, we have presented the threshold absorption energy ( ω 00 ) of a Si circular quantum wire for three different sizes as a function of strain. We have applied both DFT and EMA and compared these results. One can see from the figure that, the threshold absorption energy decreases by increasing strain for compressive and tensile strain and the behavior of the threshold absorption energy as a function of strain is approximately parabolic. The absorption threshold energy has biggest value at zero strain. Also, at a given strain, the absorption threshold energy decreases with enhancing the wire size. It is clear from the figure that the threshold absorption energy (for

4 90 Communications in Theoretical Physics Vol. 65 all sizes) obtained using EMA is greater than the DFT results. The results calculated by both EMA and DFT have similar behaviors. These results can be changed if we consider the electron-phonon interaction due to this fact that the polaronic energy correction on the ground state and so on the threshold absorption energy is mines. As we know, the electron-phonon interaction is greater for smaller quantum wires, so the EMA results are not suitable for small quantum wires. Figure 4 shows the band structures of circular Si nanowires obtained by DFT for three different strains with a = 2.9 nm. The figure clearly shows that the band gap depends on the strain. For the strain range used in this work, the Si nanowires show direct band structures. In Fig. 5, we have calculated the band structures of circular Si nanowires both DFT and EMA. The figure has been plotted for three different strains with a = 2.9 nm. It is seen from the figure that the band structures calculated by EMA are approximately close to DFT results. We know that the Coulomb interaction plays an important role in quasi-one-dimension nanowires. [41] Therefore, we have tried to study the Coulomb interaction effect on threshold absorption energy of Si circular nanowires. Figure 6 displays the threshold absorption energy of Si circular nanowires as a function of strain for three different sizes. In this figure, we have calculated threshold absorption energy with considering the Coulomb interaction and we have used both DFT and EMA and compared these results. It is observed from the figure that the threshold absorption energy calculated using EMA decreases with considering the Coulomb interaction. 6 Conclusion In the present work, we have investigated the strain effect on the absorption threshold energy of a Si circular nanowire. For this goal, effective masses of the electron and the hole, in Si 110 nanowires, have calculated using first principles density functional theory. Then, using the effective masses, we have computed the absorption threshold energy as a function of strain. We have also studied the band structures of Si nanowires both DFT and EMA. We have found that strain effect on effective masses of the electron and the hole is different. The compressive strain increases the effective mass of the electron, while tensile strain increases the effective mass of the hole. The absorption threshold energy obtained by both DFT and EMA has similar behaviors. In summary, for using Si nanowires in mesoscopic electronic and optical devices, these results can be used. References [1] H.Y. Zhou and S.W. Gu, Solid State Commun. 86 (1993) 403. [2] Q.H. Zhong, Phys. Lett. A 372 (2008) [3] Y.B. Yu, S.N. Zhu, and K.X. Guo, Solid State Commun. 132 (2004) 689. [4] C.Y. Chen, Y. You, X.H. Wang, and S.H. Dong, Phys. Lett. A 377 (2013) [5] Y.B. Yu, S.N. Zhu, and K.X. Guo, Solid State Commun. 139 (2006) 76. [6] R. Khordad, Solid State Sci. 19 (2013) 63. [7] Y. Cui, Z. Zhong, D. Wang, W. Wang, and C. Lieber, Nano Lett. 3 (2003) 149. [8] Y. Cui and C. Lieber, Science 291 (2001) 851. [9] J. Hahm and C. Lieber, Nano Lett. 4 (2004) 51. [10] Y. Cui, Q. Wei, H. Park, and C. Lieber, Science 293 (2001) [11] J.D. Holmes, K.P. Johnston, R.C. Doty, and R.A. Korgel, Science 287 (2000) [12] X.H. Peng, A. Alizadeh, S.K. Kumar, and S.K. Nayak, Int. J. Appl. Mech. 1 (2009) 483. [13] E. Durgun, D. Cakir, N. Akman, and S. Ciraci, Phys. Rev. Lett. 99 (2007) [14] E. Durgun, N. Akman, C. Ataca, and S. Ciraci, Phys. Rev. B. 76 (2007) [15] T. Vo, A.J. Williamson, and G. Galli, Phys. Rev. B 74 (2006) [16] A. Wang, A. Rahman, G. Klimeck, and M. Lundstorm, in: Proceedings of the IEDM, IEEE, New York (2005) pp [17] K. Nehari, N. Cavassilas, J.L. Autran, M. Bescond, D. Munteanu, and M. Lannoo, in: Proceedings of ESSDERC, Grenoble, France (2005) p [18] R. Khordad and H. Bahramiyan, Opt. Commun. 334 (2015) 85. [19] M. Shin, IEEE Trans. Nanotechnol. 6 (2007) 230. [20] Y. Cui, Z. Zhong, D. Wang, W.U. Wang, and C. M. Lieber, Nano Lett. 3 (2003) 149. [21] R. Khordad, Opt. Quant. Electron. 46 (2014) 283. [22] R. Khordad and H. Bahramiyan, Int. J. Mod. Phys. C 24 (2013) [23] P. Sen, O. Gulseren, T. Yildirim, I.P. Batra, and S. Ciraci, Phys. Rev. B 65 (2002) [24] V. Amar, M. Pauri, and A. Scotti, J. Math. Phys. 32 (1991) [25] W.K. Li and S.M. Blinder, J. Math. Phys. 26 (1985) [26] P. Raiteri, F. Valentinotti, and L. Miglio, Appl. Surf. Sci. 188 (2002) 4. [27] W. Yu and A. Madhukar, Appl. Phys. Lett. 79 (1997) 905. [28] Y. Kikuchi, H. Sugai, and K. Shintani, J. Appl. Phys. 89 (2001) [29] D.A. Faux, J.R. Downes, and E.P. O Reilly, J. Appl. Phys. 82 (1997) 3754.

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