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Effect of Temperature on − / Quantum Effect of temperature on 퐈퐧퐱퐆퐚ퟏ−퐱퐀퐬/푮풂푨풔 quantum dot lasing Mahdi Ahmadi Borji1 and Esfandiar Rajaei2 Department of Physics, The University of Guilan, Namjoo Street, Rasht, Iran ABSTRACT In this paper, the strain, band-edge, and energy levels of pyramidal 퐼푛푥퐺푎1−푥퐴푠/GaAs quantum dot lasers (QDLs) are investigated by 1-band effective mass approach. It is shown that while temperature has no remarkable effect on the strain tensor, the band gap lowers and the radiation wavelength elongates by rising temperature. Also, band-gap and laser energy do not linearly decrease by temperature rise. Our results appear to coincide with former researches. Keywords: quantum dot laser, strain tensor, band edge, nano-electronics, temperature effect I. INTRODUCTION Peter, 2012, Rossetti et al., 2009), size of the quantum dots (Baskoutas and Terzis, 2006, Semiconductor lasers are the most important Pryor, 1998), stoichiometric percentage of lasers that are used in cable television signals, constituent elements of the laser active region telephone and image communications, (Shi et al., 2011), substrate index computer networks and interconnections, (Povolotskyi et al., 2004), strain effect (Pryor CD-ROM drivers, reading barcodes, laser and Pistol, 2005, Shahraki and Esmaili, printers, optical integrated circuits, 2012), wetting layer (WL), and distribution telecommunications, signal processing, and a and density of quantum dots are shown to be large number of medical and military important in the energy levels and applications. Quantum dot semiconductor performance of QDLs. Therefore, finding the lasers due to the discrete density of states functionality of the impact of these factors have low threshold current and temperature can help in optimization of the performance dependence, high optical gain and quantum of quantum dot lasers. Temperature effects efficiency and high modulation speed are should be paid attention, since any adverse superior to other lasers. Effects of various effects that happen by change in the laser factors such as temperature (Chen and Xiao, usage conditions should be forecasted. 2007, Kumar et al., 2015, Narayanan and 1 Email: [email protected] 2 Corresponding author, E-mail: [email protected] 1 QD nanostructures have been the focus of In semiconductor (SC) hetero-structures many investigations due to their optical containing two or more semiconductors with properties arising from the quantum different lattice constants, band edge confinement of electrons and holes (Markéta diagrams show more complexity than usual ZÍKOVÁ, 2012, Ma et al., 2013, Danesh bulk semiconductors because of the Kaftroudi and Rajaei, 2010, Nedzinskas et important role of strain. Strain tensor depends al., 2012). By now, QD materials have found on the elastic properties of neighbor very promising applications in optical materials, lattice mismatch, and geometry of amplifiers and semiconductor lasers the quantum dot (Trellakis et al., 2006). (Bimberg et al., 2000, Gioannini, 2006, Danesh Kaftroudi and Rajaei, 2011, Asryan This research will study the band structure and Luryi, 2001). They are very important in and strain tensor of InxGa1−xAs quantum new laser devices and solar cells. Therefore, dots grown on GaAs substrate by quantum having a ubiquitous view of energy states, numerical methods to look for more efficient strain, and other physical features, and their QDs by changing the working temperature of change by varying some factors such as QD. temperature which affects the lasing process The rest of this paper is organized as follows: of a QD is instructive. Based on this fact, section II explains the model and method of many research groups attempt to develop and the numerical simulation. Our results and optimize QDs to fabricate optoelectronic discussions on the temperature effects are devices with better performance. presented in section III. Finally, we make a Finding a way to enhance the efficiency of a conclusion in section IV. QD with fixed size can be helpful. Among many materials, InGaAs/GaAs devices are faced by many scientists due to their II. MODELING AND NUMERICAL interesting and applicable features (Woolley SIMULATION et al., 1968, Nedzinskas et al., 2012, Hazdra et al., 2008, Fali et al., 2014, Yekta Kiya et Band structure of a zinc-blende crystal can be al., 2012, Azam Shafieenezhad, 2014). obtained through: However, lasers may be used in very low (Le- ′ Van et al., 2015, Rossetti et al., 2009, Tong et (퐻0 + 퐻푘 + 퐻푘.푝 + 퐻푠.표. + 퐻푠.표.)푢푛풌(풓) = al., 2007) or high (Ohse, 1988, Rouillard et 퐸푛(퐤)푢푛풌(풓) (1) al., 2000) temperature conditions which will affect their work. Alongside, it is proved that In which 푢푛풌(풓) is a periodic Bloch spinor 푖풌.풓 temperature affects the lasing process (i.e., 휓푛풌(풓) = 푒 푢푛풌(풓)) and through both change of the output 풑2 photoluminescence and the laser 퐻0 = + 푉0(풓, 휀푖푗) (2) 2푚0 characteristics (Rossetti et al., 2009) which are due to the dependence of carriers behavior ℏ2풌2 퐻푘 = (3) on temperature. 2푚0 2 ℏ 퐻푘.푝 = 풌. 풑 (4) 푚0 ℏ 퐻푠.표. = 2 2 (훔 × 훁푉0(풓, 휀푖푗)) . 풑 (5) 4푚0푐 ′ ℏ 퐻푠.표. = 2 2 (훔 × 훁푉0(풓, 휀푖푗)) . ℏ풌 (6) 4푚0푐 Here, 푉0(풓, 휀푖푗) is the periodic potential of the strained crystal, 흈 = (휎 , 휎 , 휎 ) is the 푥 푦 푧 Pauli spin matrix, 푐 is the light velocity, and 푚0 is the mass of electron. This equation can Fig. 1: Cross-section of a pyramidal InGaAs be solved by expansion of 푉0(풓, 휀푖푗) to first QD with square base of the area 17푛푚 × 17푛푚 and the height of 2/3 times the base order in strain tensor 휀푖푗 (Bahder, 1990). The resulting equation for a strained structure is width on 15푛푚 thick GaAs substrate and then 0.5푛푚 wetting layer. ℏ2풌2 ℏ (퐻0 + + 풌. 풑′) 푢푛풌(풓) = 2푚0 푚0 퐸푛(퐤)푢푛풌(풓) (7) In self-assembled InxGa1−xAs QDs, firstly WL with a few molecular layers is grown on where the substrate, and then, millions of QD islands grow on the wetting layer, each of ′ ℏ 풑 = 풑 + 2 (훔 × 훁푉0(풓, 휀푖푗)). (8) which having a random shape and size. The 4푚0푐 resulting system is finally covered by GaAs. This equation can be solved by the second Many shapes can be approximated for QDs order non-degenerate perturbation scheme in namely cylindrical, cubic, lens shape, which the latter terms are considered as the pyramidal (Qiu and Zhang, 2011), etc. QDs perturbation. Therefore, in the Cartesian are supposed to be far enough not to be space the solution is: influenced by other QDs. The one-band effective mass approach is used in solving the 2 ℏ 푘푖푘푗 1 Schrödinger equation. 퐸푛(퐤) = 퐸푛(ퟎ) + ( ∗ ) (9) 2 푚푛 푖,푗 Fig. 1 shows the profile of a pyramidal In which the tensor of the effective mas is In0.4Ga0.6As QD surrounded by a substrate defined as: and cap layer of GaAs (Pryor, 1998). This 1 1 ratio of indium is used in laser devices ( ∗ ) = ( ) 훿푖,푗 + (Kamath et al., 1997). The structure of both 푚푛 푖,푗 푚0 ′ ′ GaAs and InAs is zinc-blende. The pyramid 2 〈푛,0|푝푖 |푚,0〉〈푚,0|푝푗|푛,0〉 2 ∑푚≠푛 (10) has a square base of area 17푛푚 × 17푛푚 and 푚0 퐸푛(0)−퐸푚(0) the height of 2/3 times the base width on and 푖, 푗 ≡ 푥, 푦, 푧 (Galeriu and B. S., 2005). 15푛푚 thick GaAs substrate and 0.5푛푚 3 wetting layer. This structure is grown on Also, for 퐼푛푥퐺푎1−푥퐴푠 the parameters are (001) substrate index. As it can be observed calculated as follows: from the picture, WL is much thinner than QD height. The growth direction of the Lattice constant at T=300K (Adachi, 1983): structure is z. 푎 = (6.0583 − 0.405(1 − 푥)) Å (11) The parameters related to the bulk materials used in this paper are given in Table 1, in Effective electron mass at 300K (T.P.Pearsall, 1982): which it is benefitted from references (Jang et al., 2003, Singh, 1993, Yu, 2010). 푚푒 = (0.023 + 0.037(1 − 푥) + 0.003(1 − 푥)2)푚 (12) 표 Parameters used GaAs InAs Effective hole mass at 300K (N.M., 1999): Band gap (0K) 1.424eV 0.417eV 푚ℎ = (0.41 + 0.1(1 − 푥))푚표 (13) lattice constant 0.565325 0.60583 nm nm Effective light-hole masses at 300K: Expansion 0.0000388 0.0000274 푚푙푝 = (0.026 + 0.056(1 − 푥))푚표 (14) coefficient of lattice constant Effective split-off band hole-masses at 300K Effective electron 0.067mo 0.026mo is ~ 0.15 푚표 mass (Γ) Effective heavy 0.5mo 0.41mo hole mass III. RESULTS AND DISCUSSION Effective light hole 0.068mo 0.026m0 mass Strain is generally defined as the value of length increase relative to initial length (휀퐿 = Effective split-off 0.172mo 0.014 mass Δ퐿/퐿) which in a better definition it can be the summation of all infinitesimal length Nearest neighbor 0.2448 nm 0.262 nm increases relative to the instantaneous lengths distance (300K) (휀퐿 = ∑Δ퐿푡/퐿푡). Therefore, taking into Elastic constants C11 = 122.1 C11 = 83.29 account length change in all directions, one achieves the tensor as: C12 = 56.6 C12 = 45.26 1 푑푢푖 푑푢푗 C44 = 60 C44 = 39.59 휀푖푗 = ( + ) , 푖, 푗 ≡ 푥, 푦, 푧 (15) 2 푑푟푗 푑푟푖 Table 1. Parameters used in the model. where 푑푢푖 is the length change in i-th direction, and 푟푗 is the length in direction 푗 (Povolotskyi et al., 2004). The diagonal 4 elements are related to expansions along an working temperature. This shows that change axis (stretch), but the off-diagonal elements of temperature has no effect on the strain represent rotations. This tensor is symmetric tensor. (i.e., 휀푖푗 = 휀푗푖) although the distortion matrix 풖 may be non-semmetric.
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