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]

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

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 − 2 푥) )푚표 (12)

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. In our case, the (a) 휺풙풙 strain tensor is a diagonal matrix as follows:

휀푥푥 0 0 휀 = [ 0 휀푦푦 0 ] (16) 0 0 휀푧푧 in which 휀푥푥 = 휀푦푦 due to the symmetry. This tensor shows a biaxial in-plain strain defined as:

푎||−푎푠푢푏푠푡푟푎푡푒 휀푥푥 = 휀푦푦 = 휀|| = (17) 푎푠푢푏푠푡푟푎푡푒 and a perpendicular uniaxial strain defined as (b) 휺 (Peressi et al., 1998): 풛풛

2퐶푥푦 푎⊥−푎푠푢푏푠푡푟푎푡푒 휀푧푧 = 휀⊥ = − 휀푥푥 = (18) 퐶푥푥 푎푠푢푏푠푡푟푎푡푒 where 퐶푖푗 are components of the matrix which interconnects stress 휎 to strain (i.e., 휎 = 퐶휀) (Chuang and Chang, 1997). In Figs. 2(a) and 2(b) variation of strain is illustrated in different points of the middle cross-section of the structure. As it is seen, strain tensor is subjected to change in the both directions. However, obviously, the most variation is viewed at interfaces.

In Fig. 3 the non-zero elements of strain Fig. 2. Strain tensor in x and tensor are drawn by going up through z- z directions in different direction and at the middle of the structure. It points of the device at can be noticed that the existence of indium in T=300K one side of interfaces has caused a jump in the strain tensor for 휀푥푥 (휀푦푦) and 휀푧푧, meaning a tension in GaAs and compression in InGaAs lattice constants. This figure was found to be the same for different values of

This result seems logical, since as it was are different which can be explained as a proved, strain is only dependent on the ratio result of changes in the effective masses, of lattice constants which linearly change by since heavy hole band has the largest, and temperature (i.e., 푎 = 푏(푇 − 300) + 푎300퐾). split-off band has the smallest effective mass. So their ratio remains fixed and strain does All the figures show symmetry through x- not change by temperature. This means that axis. Moreover, the figures represent that our result can be acceptable from analytical band energy of the corners of the QD have point of view. matched their energy more than other points.

In Fig. 5 snapshots for conduction and valence Γ band-edges are shown in z- direction, and three first allowed energy states for electrons and holes are pointed out. All Heavy-Hole (HH), Light-Hole (LH), and Split-Off (SO) band-edges can also be seen in the valence band.

It is seen that in the mentioned physical conditions the first electronic eigenvalue separates from higher continuous states and lays down into the QD. This energy state is the ground state (GS) atom-like electron state with its special energy level which can be used in the recombination energy of electrons Fig. 3. Nonzero elements of the strain tensor and holes. Other states are among continuous at 푇 = 300퐾. states of the GaAs.

However, the interesting phenomenon is the Fig. 4(a,b,c,d) show the band-edge of each dependence of the banedges and energies to point of the of the pyramidal QDs at 300K for temperature. Obviously, by rise of Γ, Heavy-Hole (HH), Light-Hole (LH), and temperature from 300K to 700K, the GS Split-Off (SO) respectively. Both GaAs and levels of electrons have displaced. The InAs have direct band-gaps. As it is viewed, electronic states appear to decrease from the band-edges change in different points. 0.75eV to 0.55eV, showing that the e-h Fig. 4(a) indicates that deepest points into the energy difference has been under change by QD have the most energy difference with the temperature. This leads to different cap layer. Also, the cap layer band-edges wavelengths of the QD laser. In the same have been also subjected to change in the way, it can be observed that all the band- points close to the QD. Also, Figs 4(b,c and edges and the band-gap have been d) show the hole band-edges which have a temperature sensitive. less values; nevertheless, their band-edges

(a) Γ (b) HH

Fig. 4. Valence and conduction band-edges in x-y plane in the central point of the QD at T=300K. Note the numbers related to colors beside each figure.

Moreover, Fig. 6 illustrates a comparison of 푇2 퐸 = 퐸 − 2.76 × 10−4 band edge for different values of temperature. 𝑔 𝑔(푇=0) 푇 + 83 Obviously, effect of temperature on the band-edges is & 푇 ∈ [0,300] (20) remarkable in a wide range of temperature. for InAs, in which 퐸 is in eV and T is in As it is seen, rise of temperature has lowered 𝑔 Kelvin (Fang et al., 1990). Also, for different the band-edges of both substrate and quantum ratios of indium in 퐺푎 퐼푛 퐴푠, the behavior dot. 푥 1−푥 of energy gap is (Goetz et al., 1983): Functionality of energy gap by change of ( ) 2 temperature is shown to be 퐸𝑔 푥 = 0.36 + 0.63푥 + 0.43푥

2 & 푇 = 300퐾 (21) −4 푇 퐸𝑔 = 퐸𝑔(푇=0) − 5.405 × 10 푇 + 204 2 퐸𝑔(푥) = 0.4105 + 0.6337푥 + 0.475푥 & 푇 ∈ [0,1000] (19) & 푇 = 2퐾 (22) for GaAs, and

The melting point for GaAs and InAs is in hot conditions, the behavior can represent respectively 1511K and 1215K. In Fig. 7 the whole effect of temperature. energy gap and recombination energy of the Interestingly, impact of temperature on first eigenvalue are plotted as a function of energies is not linear, which is the important temperature for a wide temperature range. conclusion of this paper. Decrease of This range is previously used for bulk recombination energy results in the elongated materials in reference (Fang et al., 1990). laser wavelength which must be paid Although all the temperatures are not attention in the lasers. meaningful, since the laser structure corrupts

(a) T=300K (b) T=400K

Fig. 5. Conduction and valence 훤 band-edges of QDs in z-direction together with three first allowed energy states for electrons at different values of temperature. All Heavy-Hole (HH), Light-Hole (LH), and Split-Off (SO) band-edges can also be seen in the valence band.

IV. CONCLUSION

We investigated the band structure and strain

tensor of InxGa1−xAs QDs grown on GaAs substrate by quantum numerical solution. It was shown that in usual temperatures, rise of temperature has no remarkable effect on strain tensor, but slightly decreases the conduction and valence band-edges as well as electron- and hole-energy states. Moreover, the main result was that temperature does not have a linear impact on the bandgap and Fig. 6. Comparison of 훤 band edges at recombination energies; both decrease by different values of temperature. temperature rise.

Comparison of these results which are related ACKNOWLEDGEMENT to bulk semiconductors show that our results The authors give the sincere appreciation to can be logical. As it can be inferred, the Dr. S. Birner for providing the advanced 3D behavior of our results is almost similar to the Nextnano++ simulation program (Birner et formula found in bulk samples. al., 2007) and his instructive guides. We would like to thank numerous colleagues, namely Prof. S. Farjami Shayesteh, Dr. S. Salari, K. Kayhani, and Y. Yekta Kia for sharing their points of view on the manuscript.

REFERENCES:

ADACHI, S. 1983. J.Appl. Phys., vol. 54, 1844- 1848. ASRYAN, L. V. & LURYI, S. 2001. Tunneling- injection quantum-dot laser: ultrahigh temperature stability. Quantum Electronics, IEEE Journal of, 37, 905-910. Fig. 7: Energy gap and recombination AZAM SHAFIEENEZHAD, E. R., , SAEED YAZDANI energy of the first electron- and hole- 2014. The Effect of Inhomogeneous eigenvalues at different temperatures Broadening on Characteristics of Three- State Lasing Ingaas/Gaas Quantum Dot Lasers. International Journal of Scientific

Engineering and Technology, 3, 297- FALI, A., RAJAEI, E. & KAFTROUDI, Z. 2014. Effects 301. of the carrier relaxation lifetime and BAHDER, T. B. 1990. Eight-band k.p model of inhomogeneous broadening on the strained zinc-blende crystals. Physical modulation response of InGaAs/GaAs Review B, 41, 11992-12001. self-assembled quantum-dot lasers. BASKOUTAS, S. & TERZIS, A. F. 2006. Size- Journal of the Korean Physical Society, dependent band gap of colloidal 64, 16-22. quantum dots. Journal of Applied FANG, Z. M., MA, K. Y., JAW, D. H., COHEN, R. M. Physics, 99, 013708. & STRINGFELLOW, G. B. 1990. BIMBERG, D., GRUNDMANN, M., Photoluminescence of InSb, InAs, and HEINRICHSDORFF, F., LEDENTSOV, N. N., InAsSb grown by organometallic vapor USTINOV, V. M., ZHUKOV, A. E., KOVSH, phase epitaxy. Journal of Applied A. R., MAXIMOV, M. V., SHERNYAKOV, Y. Physics, 67, 7034-7039. M., VOLOVIK, B. V., TSATSUL’NIKOV, A. GALERIU, C. & B. S., M. S., M. A. 2005. k. p theory F., KOP’EV, P. S. & ALFEROV, Z. I. 2000. of semiconductor nanostructures. a Quantum dot lasers: breakthrough in dissertation submitted to the faculty of optoelectronics. Thin Solid Films, 367, the Worcester polytechnic institute for 235-249. degree of doctor of philosophy in physics, BIRNER, S., ZIBOLD, T., ANDLAUER, T., KUBIS, T., 11-13. SABATHIL, M., TRELLAKIS, A. & VOGL, P. GIOANNINI, M. 2006. Analysis of the Optical Gain 2007. nextnano: General Purpose 3-D Characteristics of Semiconductor Simulations. Electron Devices, IEEE Quantum-Dash Materials Including the Transactions on, 54, 2137-2142. Band Structure Modifications Due to the CHEN, S.-H. & XIAO, J.-L. 2007. Temperature Wetting Layer. IEEE Journal of Quantum Effect On Impurity-Bound Polaronic Electronics, 42, 331-340. Energy Levels In A Parabolic Quantum GOETZ, K. H., BIMBERG, D., JÜRGENSEN, H., Dot In Magnetic Fields. International SELDERS, J., SOLOMONOV, A. V., Journal of Modern Physics B, 21, 5331- GLINSKII, G. F. & RAZEGHI, M. 1983. 5337. Optical and crystallographic properties CHUANG, S. L. & CHANG, C. S. 1997. A band- and impurity incorporation of structure model of strained quantum- GaxIn1−xAs (0.44

KAMATH, K., PHILLIPS, J., JIANG, H., SINGH, J. & PERESSI, M., BINGGELI, N. & BALDERESCHI, A. BHATTACHARYA, P. 1997. Small-signal 1998. Band engineering at interfaces: modulation and differential gain of theory and numerical experiments. single-mode self-organized Journal of Physics D: Applied Physics, 31, In0.4Ga0.6As/GaAs quantum dot lasers. 1273. Applied Physics Letters, 70, 2952-2953. POVOLOTSKYI, M., DI CARLO, A., LUGLI, P., KUMAR, D., NEGI, C. M. S. & KUMAR, J. 2015. BIRNER, S. & VOGL, P. 2004. Tuning the Temperature Effect on Optical Gain of piezoelectric fields in quantum dots: CdSe/ZnSe Quantum Dots. In: microscopic description of dots grown LAKSHMINARAYANAN, V. & on (N11) surfaces. Nanotechnology, IEEE BHATTACHARYA, I. (eds.) Advances in Transactions on, 3, 124-128. Optical Science and Engineering. PRYOR, C. 1998. Eight-band calculations of Springer India. strained InAs/GaAs quantum dots LE-VAN, Q., LE ROUX, X., TEPERIK, T. V., HABERT, compared with one-, four-, and six-band B., MARQUIER, F., GREFFET, J. J. & approximations. Physical Review B, 57, DEGIRON, A. 2015. Temperature 7190-7195. dependence of quantum dot PRYOR, C. E. & PISTOL, M. E. 2005. Band-edge fluorescence assisted by plasmonic diagrams for strained III-V nanoantennas. Physical Review B, 91, semiconductor quantum wells, wires, 085412. and dots. Physical Review B, 72, 205311. MA, Y. J., ZHONG, Z., YANG, X. J., FAN, Y. L. & QIU, D. & ZHANG, M. X. 2011. The preferred JIANG, Z. M. 2013. Factors influencing facet orientation of GaAs pyramids for epitaxial growth of three-dimensional high-quality InAs and InxGa1−xAs Ge quantum dot crystals on pit- quantum dot growth. Scripta Materialia, patterned Si substrate. Nanotechnology, 64, 681-684. 24, 015304. ROSSETTI, M., FIORE, A., SĘK, G., ZINONI, C. & LI, MARKÉTA ZÍKOVÁ, A. H. 2012. Simulation of L. 2009. Modeling the temperature Quantum States in InAs/GaAs Quantum characteristics of InAs/GaAs quantum Dots. NANOCON 23, 10. dot lasers. Journal of Applied Physics, N.M., G. Y. A. A. 1999. Schmidt Handbook Series 106, 023105. on Semiconductor Parameters. World ROUILLARD, Y., BARANOV, A. N., YAREKHA, D. A., Scientific, London, vol. 2, 62-88. PERONA, A., VICET, A., JOULLIE, A. & NARAYANAN, M. & PETER, A. J. 2012. Pressure ALIBERT, C. Semiconductor mid-infrared and Temperature Induced Non-Linear lasers working at high temperature Optical Properties in a Narrow Band Gap application to gas analysis. Laser and Quantum Dot. Quantum Matter, 1, 53- Fiber-Optical Networks Modeling, 2000. 58. Proceedings of LFNM 2000. 2nd NEDZINSKAS, R., ČECHAVIČIUS, B., International Workshop on, 2000 2000. KAVALIAUSKAS, J., KARPUS, V., VALUŠIS, 6-8. G., LI, L., KHANNA, S. P. & LINFIELD, E. H. SHAHRAKI, M. & ESMAILI, E. 2012. Computer 2012. Polarized photoreflectance and simulation of quantum dot formation photoluminescence spectroscopy of during heteroepitaxial growth of thin InGaAs/GaAs quantum rods grown with films. Journal of Theoretical and Applied As(2) and As(4) sources. Nanoscale Physics, 6, 1-5. Research Letters, 7, 609-609. SHI, Z., HUANG, L., YU, Y., TIAN, P. & WANG, H. OHSE, R. W. 1988. Laser application in high 2011. Influence of V/III ratio on QD size temperature materials. Pure and Applied distribution. Frontiers of Optoelectronics Chemistry. in China, 4, 364-368.

SINGH, J. 1993. Physics of Semiconductors and Their Heterostructures, McGraw-Hill. T.P.PEARSALL 1982. GaInAsP Alloy Semiconductors. John Wiley and Sons. TONG, C. Z., YOON, S. F. & LIU, C. Y. 2007. Rate equation model of the negative characteristic temperature of InAs∕GaAs quantum dot lasers. Journal of Applied Physics, 101, 104506. TRELLAKIS, A., ZIBOLD, T., ANDLAUER, T., BIRNER, S., SMITH, R. K., MORSCHL, R. & VOGL, P. 2006. The 3D nanometer device project nextnano: Concepts, methods, results. Journal of Computational Electronics, 5, 285-289. WOOLLEY, J. C., THOMAS, M. B. & THOMPSON, A. G. 1968. Optical energy gap variation in GaxIn1−x As alloys. Canadian Journal of Physics, 46, 157-159. YEKTA KIYA, Y., RAJAEI, E. & FALI, A. 2012. Study of response function of excited and ground state lasing in InGaAs/GaAs quantum dot laser. J. Theor. Phys. , 1, 246-256. YU, C. 2010. Fundamentals of Semiconductors. Springer.