Chapter 6 Conclusion and Future work
6.2 Future work
In fact, when observing the strong coupling effect cautiously, one can see that the intensities of two peaks are the same at about 37.875 K, the emission energies are closest at 37.75 K, and the FWHM are cross at 37.5 K as shown in Fig. 6.1. This result seems to be strange because the three points occur at different temperature, so that one cannot know that the strongest coupling occurs at which temperature. We suspect that this temperature-variation may be attributed to the difference in polarized directions of QD and cavity mode. In this system, there are two orthogonal-linear polarized QD
79
emissions and a linear polarized cavity mode. If the polarized direction of cavity mode is not completely parallel or perpendicular to that of QD emission, the two QD emission would both interact with the cavity mode, that maybe lead to the temperature-difference shown in Fig. 6.1. However, to verify whether this supposition is correct, a micro-PL measurement of all polarized directions is necessary to be
Figure 6.1: The difference in emission energy, FWHM, and intensity of two signals as a function of temperature.
80 gauge. According to quantum mechanism, the total Hamiltonian can be written as,
H = 1 In cylindrical coordinates, Eq. A.2 is transformed into
H = − ℏ2 of the Hamiltonian are well known Fock-Darwin states. According to Eq. A.3, one can see that the Schrodinger equation H𝜓(𝜌, 𝜙) = 𝐸𝜓(𝜌, 𝜙) is a separable partial differential equation, thus the eigenfunction 𝜓(𝜌, 𝜙) can be separated into radial part and angular part,
𝜓(𝜌, 𝜙) = 𝜐(𝜌)𝑒−𝑖𝑙𝜙 (𝐴. 4)
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𝑙 is the quantum number of angular momentum, 𝑙 = 0, ±1, ±2, ⋯. Bring Eq. A.4 into the Schrodinger equation, we get a radial ordinary differential equation as follows,
[𝜕2𝑣(𝜌) Bringing Eq. A.6 into Eq. A.5, we obtained Eq. A.7,
𝑑2𝐹(𝜌)
A.8, whose solution at 𝜒 = 0 is the Confluent Hypergeometric Series
𝜒𝑑2𝐹(𝜒)
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satisfy this, the eigenfunction exists only if
𝑎 = −𝑛 (𝐴. 11) We expand Eq. A.13 by Taylor series
𝐸𝑛𝑙 = (2𝑛 + |𝑙| + 1)ℏ𝜔0(1 +1 Further, Eq. A.16 can be expanded as
𝐸𝑛𝑙(𝐵𝑧) = (2𝑛 + |𝑙| + 1)ℏ𝜔0+ (𝑒2√〈𝜌02〉𝑛𝑙2
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B Hartree approximation
To consider the influence of Coulomb interaction on the magnetic responses of excitonic complexes, the Hartree approximation is a suitable method for solving such many-body problem. Beyond this approximation, the wave functions and energy of all particles in this system are approximated to stable status when considering Coulomb interaction. The Hamiltonian of total N particles system can be written as follows,
H = ∑ ℎ𝑠𝑝(𝑟⃗⃗⃗)𝑖
where ℎ𝑠𝑝 is the Hamiltonian of the 𝑖𝑡ℎ −particle without Coulomb interactions and 𝑉𝑐 is the Coulomb interaction term. We take the Slater function Ψ𝐻𝐹 to retain the state. The fermionic nature satisfies the anti-symmetrized condition of the Hartree wave function as follows,
Ψ𝐻𝐹(⋯ ⋯ 𝑟⃗⃗⃗⃗ ⋯ 𝑟𝑛 ⃗⃗⃗⃗⃗ ⋯ ) = −Ψ𝑚 𝐻𝐹(⋯ ⋯ 𝑟⃗⃗⃗⃗⃗ ⋯ 𝑟𝑚 ⃗⃗⃗⃗ ⋯ ) (𝐵. 3) 𝑛
Thus, the total energy of the N particle system is given by 𝐸𝐻𝐹 = ⟨Ψ𝐻𝐹|𝐻|Ψ𝐻𝐹
84
The second and third terms in Eq. B.4 are defined as direct and exchange interaction.
Taking the variation method, we can get Eq. B.5
δ[𝐸𝐻𝐹− 𝜀𝑖(⟨𝜙𝑖|𝜙𝑖 − 1)] = 0 ignore the exchange term and then get the Schrodinger equation of one particle from Eq. B.5, interactions among particles, which represents the electrostatic potential induced by other N-1 particles. ρ(𝑟⃗⃗⃗⃗) is the volume particle density equal to ∑ |𝜙′ 𝑁𝑗≠𝑖 𝑗(𝑟⃗⃗⃗⃗)|′ 2. It is clear that the wave function of 𝑖th particle 𝜙𝑖(𝑟⃗) is determined by the Hartree potential associated with other particles and the confined potential. Such problem is suitable to be solved by using the self-consist method, i.e., an iteratively calculated process until convergence.
85
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Vita
Name : Ying-Jhe Fu
(傅英哲)
Date of birth : December 26, 1983
Place of birth : Tainan, Taiwan, ROC
E-mail : [email protected]
Sex : Male
Education :
National Chiao Tung University Ph. D. September, 2007－July, 2014 Department of Electronics Engineering & Institute of Electronics
National Chiao Tung University M. S. September, 2006－June, 2007 Department of Electronics Engineering & Institute of Electronics
National Chiao Tung University B. S. September, 2002－June, 2006 Department of Materials Science and Engineering
Title of Ph. D. Dissertation :
Magneto-Optical Properties of Single InAs Quantum Dot and Their Coupling to Photonic Crystal Cavity
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Publication List
[1] Y. J. Fu, S. D. Lin, M. F. Tsai, H. Lin, C. H. Lin, H. Y. Chou, S. J. Cheng and W. H. Chang, "Anomalous diamagnetic shift for negative trions in single semiconductor quantum dots,", Physical Review B, vol. 81, p. 113307, 2010.
[2] Y. J. Fu, Yi-Shan Lee and Sheng-Di Lin, "Design and demonstration of high quality-factor H1-cavity in 2-D photonic crystal," Optics Letters, vol. 38, pp.
4915-4918, 2013.
[3] S. D. Lin, Y. J. Fu and C. Cheng, "Imbalanced initial populations between dark and bright states in semiconductor quantum dots," Optics Express, vol. 20, pp.
19850-19858, 2012.
[4] W. H. Chang, C. H. Lin, Y. J. Fu, T. C. Lin, H. Lin, S. J. Cheng, S. D. Lin, C.
P. Lee, "Impacts of coulomb interactions on the magnetic responses of excitonic complexes in single semiconductor nanostructures," Nanoscale Research Letters, vol. 5, pp. 680-685, 2010.
[5] Yu. B. Ovchinnikov, J. Hayes, D. J. Richardson, Y. J. Fu, S. D. Lin, P. See, A.G. Sinclair, "Wide spectral range confocal microscope based on endlessly single-mode fiber," Optics Express, vol. 18, pp. 18811-18819, 2010.
[6] Y. J. Fu, S. D. Lin, M. F. Tsai, H. Lin, C. H. Lin, S. Y. Wang, S. J. Cheng and W.
H. Chang, “Diamagnetic shift of exciton complexes in InAs quantum dots” 18th international conference on Electronic Properties of Two-Dimensional Systems (EP2DS18), Kobe, Japan (Jul. 19-24, 2009)