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Single Band Effective Mass Equation and Envolvent Functions

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Photon Absorption Models in Nanostructured Semiconductor Solar Cells and Devices

Part of the book series: SpringerBriefs in Applied Sciences and Technology ((BRIEFSAPPLSCIENCES))

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Abstract

The single-band effective mass Schrödinger equation to calculate the envelope functions is described and its grounds are shown. These envelope functions are used to multiply periodic part of the Bloch functions to obtain approximate eigenfunctions of the Hamiltonian of a nanostructured semiconductor. The Bloch functions, which are the product of a periodic function and a plane wave, constitute the exact solution of a homogeneous semiconductor; they are taken as a basis to represent the nanostructured Hamiltonian. The conditions that make possible the use of this single band effective mass Schrödinger equation are explained. The method is applied to the calculation of the energy spectrum of quantum dots for wavefunctions belonging to the conduction band. A box-shaped model of the quantum dots is adopted for this task. The results show the existence of energy levels detached from this band as well as eigenfunctions bound totally or partially around the quantum dot. The absorption coefficients of photons in the nanostructured semiconductor, which is our ultimate goal, are calculated. The case of spherical quantum dots is also considered.

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Notes

  1. 1.

    The division by \( \varOmega_{cell} \) is necessary, among other reasons, to keep the dimensionality.

  2. 2.

    Note that, in the non-relativistic limit, the energies in the Schrödinger equations do not have a defined energy origin. However, the origin for E and U L have to be the same and U must be considered as a potential energy that is added to U L .

  3. 3.

    Some authors, and more frequently those working in the k·p method, call Bloch functions to their periodic part, \( u_{{v,\varvec{k}}} (\varvec{r}) \).

  4. 4.

    In this equation, the imaginary unit i is conserved attached to the gradient sign \( \nabla \). This is because \( i\nabla \) is Hermitical. The advantage of using Hermitical operators is that \( \left\langle \varphi \right|i\nabla \left| \psi \right\rangle = \int {\varphi^{*} i\nabla \psi d^{3} r} = \int {(i\nabla \varphi )^{*} \psi d^{3} r.} \)

  5. 5.

    Exciton is an electron-hole pair linked by Coulombian attraction. At room temperature excitons are dissociated in a semiconductor. Electron-hole pairs and excitons are often considered synonymous, and is the case in this chapter.

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Authors and Affiliations

  1. Solar Energy Institute, Polytechnic University of Madrid, 28040, Madrid, Spain

    Antonio Luque & Alexander Virgil Mellor

  2. Nanostructured Solar Cells Laboratory, Ioffe Physical Technical Institute, St Petersburg, 194021, Russia

    Antonio Luque

  3. Imperial College of Science and Technology, London, SW7 2AZ, UK

    Alexander Virgil Mellor

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  1. Antonio Luque
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Correspondence to Antonio Luque .

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Luque, A., Mellor, A.V. (2015). Single Band Effective Mass Equation and Envolvent Functions. In: Photon Absorption Models in Nanostructured Semiconductor Solar Cells and Devices. SpringerBriefs in Applied Sciences and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-14538-9_2

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  • DOI: https://doi.org/10.1007/978-3-319-14538-9_2

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  • Online ISBN: 978-3-319-14538-9

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