Theory of the Optical Response of Singleand Coupled Semiconductor Quantum Dots
Due to their quasi-zero-dimensional structure, quantum dots show optical properties which are different from those of nanostructures with spatial confinement in less than three dimensions. In this chapter, the theory of both the linear optical properties
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Abstract. Due to their quasi-zero-dimensional structure, quantum dots show optical properties which are different from those of nanostructures with spatial confinement in less than three dimensions. In this chapter, the theory of both the linear optical properties and nonlinear dynamics of semiconductor quantum dots is discussed. The main focus is on the experimentally accessible quantities such as absorption/luminescence and pump-probe spectra. The results are calculated for single and coupled quantum dots (Förster coupling) as well as quantum dot ensembles. The focus is on obtaining a microscopic understanding of the interactions of optically excited quantum dot electrons with the surrounding crystal vibrations (electron–phonon coupling). The discussed interactions are important for applications in, e.g., quantum information processing and laser devices.
9.1 Introduction This chapter discusses the linear and nonlinear optical properties of semiconductor quantum dots (QDs). QD electrons experience different interactions due to their environment: coupling to phonons and photons. Due to the discrete energy states caused by the three-dimensional spatial confinement, the optical dephasing witnessed in the linear spectra as well as the nonlinear dynamics is quite different compared to that in bulk semiconductors [1]. We start by considering the optical properties of a single QD, both in the linear regime with absorption and resonance fluorescence spectra as well as in the nonlinear regime with Rabi oscillations and differential transmission spectra. The concept of THz acoustoluminescence in QDs is discussed. In a next step, we generalize our approach to two coupled QDs and discuss their dynamics with respect to Coulomb and radiative coupling. Finally, we present the nonlinear response of an ensemble of QDs as well as the optical properties of a set of coupled QDs.
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9.2 Theory This section focuses on the theoretical foundations. After a discussion of the QD model, we present the interaction Hamiltonians. Then we introduce the formalisms to describe the optical response. 9.2.1 Quantum Dot Model For the QDs treated in this chapter, we use the envelope function ansatz for the wave functions: a product ansatz of the Bloch part (to take into account the periodic potential) uk≈0 (x) at the band edge and an envelope function ξ(x) (to describe the additional confinement of the QD) [2]: ϕ(x) = uk≈0 (x) ξ(x),
(9.1)
where in effective mass approximation ξ(x) (x is the three-dimensional spatial coordinate) is a solution of the eigenvalue equation h¯ 2 − ∗ + U (x) ξ(x) = (E − Ek≈0 )ξ(x) (9.2) 2m with the effective mass m∗ and the mesoscopic confinement potential U (x) (for models going beyond the above ansatz; see, e.g., [3, 4]). In this work, we apply different models for ξ(x). Spherical Harmonic Oscillator Model [1]: The advantage of the spherically symmetric three-dimensional harmonic oscillator model, described by the potential U (x) = 1/2 m∗ ω2 |x|2 , is its simplicity. The ground state envelope
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