Quantum Wells, Wires and Dots
Quantum Wells, Wires and Dots provides all the essential information, both theoretical and computational, to develop an understanding of the electronic, optical and transport properties of these semiconductor nanostructures. The book will lead the reader through comprehensive explanations and mathematical derivations to the point where they can design semiconductor nanostructures with the required electronic and optical properties for exploitation in these technologies.
This fully revised and updated 4th edition features new sections that incorporate modern techniques and extensive new material including:
- Properties of non-parabolic energy bands
- Matrix solutions of the Poisson and Schrödinger equations
- Critical thickness of strained materials
- Carrier scattering by interface roughness, alloy disorder and impurities
- Density matrix transport modelling
Written by well-known authors in the field of semiconductor nanostructures and quantum optoelectronics, this user-friendly guide is presented in a lucid style with easy to follow steps, illustrative examples and questions and computational problems in each chapter to help the reader build solid foundations of understanding to a level where they can initiate their own theoretical investigations. Suitable for postgraduate students of semiconductor and condensed matter physics, the book is essential to all those researching in academic and industrial laboratories worldwide.
Quantum Wells, Wires and Dots
Semiconductors and heterostructures
1.1 The mechanics of waves
De Broglie (see reference ) stated that a particle of momentum p has an associated wave of wavelength _ given by:1.1
Thus, an electron in a vacuum at a position r and away from the influence of any electromagnetic potentials could be described by a state function, which is of the form of a wave, i.e.1.2
where t is the time, _ the angular frequency and the modulus of the wave vector is given by:1.3
The quantum mechanical momentum has been deduced to be a linear operator  acting upon the wave function _, with the momentum p arising as an eigenvalue, i.e.1.4
which, when operating on the electron vacuum wave function in equation (1.2), would give the following:1.6
and therefore1.7 1.8
Thus the eigenvalue1.9
which, not surprisingly, can be simply manipulated (p = _k = (h/2pi)(2pi/_)) to reproduce de Broglie's relationship in equation (1.1).
Following on from this, classical mechanics gives the kinetic energy of a particle of mass m as:1.10
Therefore it may be expected that the quantum mechanical analogy can also be represented by an eigenvalue equation with an operator:1.11
where T is the kinetic energy eigenvalue, and, given the form of _ in equation (1.5), then:1.13
When acting upon the electron vacuum wave function, i.e.1.14
Thus the kinetic energy eigenvalue is given by:1.16
For an electron in a vacuum away from the influence of electromagnetic fields, the total energy E is just the kinetic energy T. Thus the dispersion or energy versus momentum (which is proportional to the wave vector k) curves are parabolic, just as for classical free particles, as illustrated in Fig. 1.1.
Figure 1.1 The energy versus wave vector (proportional to momentum) curve for an electron in a vacuum
The equation describing the total energy of a particle in this wave description is called the time-independent Schrödinger equation and, for this case with only a kinetic energy contribution, can be summarised as follows:1.17
A corresponding equation also exists that includes the time dependency explicitly; this is obtained by operating on the wave function by the linear operator i__/_t, i.e.1.18
Clearly, this eigenvalue __ is also the total energy but in a form usually associated with waves, e.g. a photon. These two operations on the wave function represent the two complementary descriptions associated with wave-particle duality. Thus the second, i.e. time-dependent, Schrödinger equation is given by:1.20 1.2 Crystal structure
The vast majority of the mainstream semiconductors have a face-centred cubic Bravais lattice, as illustrated in Fig 1.2. The lattice points are defined in terms of linear combinations of a set of primitive lattice vectors, one choice for which is:1.21
Figure 1.2 The face-