# Quantum Transport

We have seen in Chapter 8 how we could define current in classical Drude theory in terms of electrons or charges obeying Newton’s law with frictional forces giving rise to resistance. In Chapter 3 we had introduced the methodology of quantum mechanics and argued that classical physics was not really the right way of looking at dynamics on a microscopic scale. In practice it turns out that the classical theory of transport is very useful indeed, and one can go a long way in understanding transport phenomena in solid state physics and engineering using the classical method. But there comes a point beyond which the classical description does not work well anymore and we have to consider the quantum mechanical aspects. This happens on many occasion most of which we cannot discuss here, but we can consider a very simple and common situation where quantum mechanics is needed. Consider a beam of electrons injected for example in the conduction band of a semiconductor via an electrode and traveling to the other electrode. Now we can ask what is the current? In classical physics the answer is obvious if we know the velocity of the carriers. Now we can insert a potential barrier on the way, for example a material with a higher band gap as in Fig. 13.1 and ask: what is the resistance produced by the potential barrier on the electrons impinging on it? A classical Drude approach would obviously give us a totally oversimplified and misleading answer to this question. It would require the definition of ‘frictional force’ but which acts only in the form of one obstacle, and would not give a satisfactory picture of this well defined and concrete transport problem. So the right starting point in this case is the *quantum mechanical definition of the current (quantum current)*.

## Keywords

Fermi Level Landau Level Quantum Cascade Laser Resonant Tunneling Quantum Hall Effect## Preview

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