Quantum Effects in Confined Systems
In this chapter we give a short introduction to the basic concept of a particle in a box for the discussion of quantum effects in one dimension. This concept will then be expanded to three dimensions in cylindrical coordinates, which are the most adequate
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Quantum Effects in Confined Systems
In this chapter we give a short introduction to the basic concept of a particle in a box for the discussion of quantum effects in one dimension. This concept will then be expanded to three dimensions in cylindrical coordinates, which are the most adequate to describe rod shaped nanostructures. A system in which the motion of electrons or other particles (holes, excitons, etc.) is restricted in one or more dimensions, due to some potential profile, is usually referred to as a ‘‘low-dimensional’’ system and shows quantum confinement effects. Due to their dual wave-particle nature, electrons in a solid are treated as particles having an effective mass m* (accounting for the periodicity of the crystal potential) and a linear momentum arising from their wave-like nature ~ p ¼ h~ k. Here, h is the Planck’s constant divided by 2p, and ~ k represents the wavenumber of the associated wave of wavelength k ¼ 2p=k. The behavior of the electrons is strongly sensitive to the dimensions of the solid in which they move. In the bulk, the infinite extension of the solid is imposed by assigning the so-called periodic boundary conditions, such that the electrons are not affected by the borders of solid in terms of wave function and energy. If however the dimensions of the solid are reduced, the electrons start to ‘‘feel’’ the borders and the assumption of infinite extension of the solid in all the three spatial coordinates does not hold any more. In such a case the system is considered as ‘‘quantized’’.
1.1 One-Dimensional Quantum Box The simplest way to understand what happens to electrons in the case of a quantized system (in terms of wavefunctions and energies) is to consider the classical textbook case of an electron in a box, which is approached by solving the Schrödinger equation in one dimension (Fig. 1.1) [1]: ( ) 0; a=2 \ x\ a=2 d2 w 2mE þ 2 ¼ V with V ¼ ð1:1Þ dx2 b 1; x a=2; x a=2 R. Krahne et al., Physical Properties of Nanorods, NanoScience and Technology, DOI: 10.1007/978-3-642-36430-3_1, Ó Springer-Verlag Berlin Heidelberg 2013
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1 Quantum Effects in Confined Systems
Fig. 1.1 Electrons in a one dimensional quantum box: energy levels and wave functions
We seek for a solution of the form: wðxÞ ¼ Aeikx þ Beikx
ð1:2Þ
Since the potential is infinitely high at the border regions, we need to impose the conditions that the wave functions have to vanish at the borders (i.e. wða=2Þ ¼ 0 and wða=2Þ ¼ 0, see Fig. 1.1). Upon imposing wða=2Þ ¼ 0 we obtain: wðxÞ ¼ A eikx eikx ¼ 2iA sinðkxÞ ð1:3Þ and, after substitution in the Schrödinger equation we obtain the following expression for the energy: E¼
2 k 2 h 2m
ð1:4Þ
Since wða=2Þ ¼ 0, the following identity must hold: wða=2Þ ¼ 2iA sinðkaÞ ¼ 0
ð1:5Þ
The identity is verified only when ka ¼ np, with n = 1, 2, 3. Therefore the parameter k is quantized, and the separation between two consecutive values of k is Dk ¼ Dnp=a. The quantization energy is expressed as: En ¼
2 p2 h n 2ma2
ð1:6Þ
In the expression above, the
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