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While the physics of low-dimensional structures mainly involves their electronic properties, an understanding of the interaction between the electrons and holes and the phonons present within and around the confined layers is frequently needed if this physics is to be understood in detail. This introductory chapter gives examples of this and outlines the experimental methods that have been used to study the electron-phonon interactions. These include both optical and transport techniques and also techniques that involve the use of phonons as probes: phonon techniques. Phonons interact much more strongly with electrons than photons of the same frequency since their momenta are usually much closer to those of the confined electrons, and phonon techniques have been used to obtain information ranging from the confinement widths to the dispersion curve of quasiparticles in the fractional quantum Hall state.

During the 1970s, III-V compounds and epitaxial crystal growth provided the basis for low dimensional structures (or nanostructures). The best known example is a GaAs quantum well within AlGaAs barriers, electrons, and holes being confined in well defined energy levels that determine the optical properties. Quantum wires and dots are also described. The quantum well laser and the vertical cavity laser (VCSEL) show considerable advantages over their heterostructure predecessor. Another exciting development was that of the two-dimensional electron gas (2DEG) at an interface between semiconductors with different band gaps. By doping only the wide gap material so as to separate the doping atoms from the resulting free electrons, ionised impurity scattering can be minimised and extremely high electron mobilities achieved. Such samples led to the discovery of the fractional quantum Hall effect and to high mobility FETs (HEMTs) for microwave applications. Mesoscopic systems and heterojunction bipolar transistors (HBTs) are also described.

This chapter provides an overview of unsupervised learning algorithms that can be viewed as spectral methods for linear and nonlinear dimensionality reduction. Spectral methods have recently emerged as a powerful tool for nonlinear dimensionality reduction and manifold learning. These methods are able to reveal low-dimensional structure in high-dimensional data from the top or bottom eigenvectors of specially constructed matrices. To analyze data that lie on a low-dimensional submanifold, the matrices are constructed from sparse weighted graphs whose vertices represent input patterns and whose edges indicate neighborhood relations. The main computations for manifold learning are based on tractable, polynomial-time optimizations, such as shortest-path problems, least-squares fits, semi-definite programming, and matrix diagonalization.

Low-dimensional structures consisting of semiconductor heterostructures of thickness down to a very few atomic monolayers were grown ideally by MBE and resulted in a surge of interest in MBE itself. Short period superlattices, doping superlattices, quantum wells, wires and dots and two-dimensional electron gases (2DEGs) all proved of significance from scientific and commercial viewpoints. Superlattices and quantum wells showed negative electrical resistance, doping superlattices provided tuneable energy gaps, and quantum wells, quantum wires and quantum dots were characterised by confined energy states which offered photon energies tuneable by varying their dimensions. They also demonstrated increasingly sharp density of state functions, of interest for semiconductor lasers. 2DEGs showed very high electron mobilities by minimising ionised impurity scattering. The 2D localisation in 2DEGs resulted in discovery of the quantum Hall effect and fractional quantum Hall effect.