The Quantum Theory of Conduction
CRITIQUE OF THE BOLTZMANN EQUATION, REGIMES OF CONDUCTION
It was made clear in the previous chapter that the Boltzmann equation is valid only for particles. Electrons can only be thought as such if they have a wavefunction of the form of a wavepacket made out of Bloch functions, well localized in both the r and the к space. Then the maximum of the wavepacket moves according to Newtonian laws and exhibits particle behaviour with the real mass m replaced by the effective mass m*. How well the wavepacket should be localized? Obviously the extent of the wavepacket must be much smaller than the size of the system L (in one dimension). The former is usually of the wavelength X. Based on the above, one would think that we only need to distinguish two regimes L » X and L^X. When L ^ X any notion of a particle behaviour is meaningless and we are definitely within the regime of quantum transport, but when L » A. the situation is not so simple and we need an extra characteristic length to describe it. This extra characteristic length is the dephasing length or distance Тф.
This is the distance an electron has to travel before it suffers an inelastic collision. Note that since elastic collisions are more frequent then inelastic, the electron must have undergone through many elastic collisions before it suffers an inelastic one. The truly macroscopic regime is when L » Lr In this case an electron, on its way from one electrode to the other one, suffers many inelastic collisions and for every distance Тф it travels it loses the memory of its phase, the characteristic quantity which makes it a wave. In this case we have particle behaviour. What happens if L < Тф but we still have L > X? To better understand the various regimes we are discussing, we note that Тф is in the range of a few microns and the range of the wavelength of electrons in semiconductors is nanometers or tens of nanometers. In this case of [A, < L < Тф], we have what we call mesoscopic regime and the carriers of electricity, the electrons, preserve their wave nature while experiencing only elastic scattering in their path from one electrode to the other. In both the truly quantum regime and the mesoscopic regime, a wave treatment of conductance is necessary.
Present day electronic devices have channels in the nanometre range. At the time of writing this book, device manufacturers have accomplished the 14nm channel length and are working on the 7nm one. These lengths fall definitely below the values of the mean free path in both Si and GaAs. Then we have what we call ballistic transport. The electron traverses the distance between contacts without any scattering. This phenomenon belongs to both the truly quantum and mesoscopic regimes, but it was observed originally in mesoscopic systems. A serious conceptual problem arises: if there are no collisions, where is the energy supplied by the external voltage dissipated? We will tackle this problem later in Part III where we discuss devices.
Apart from the main problem of dealing with electrons exhibiting wave behavior, another problem occurring with the Boltzmann equation, if it were to be applied to nanosystems, is the definition of a Fermi level which is position dependent. This is rarely discussed. When we derived the Boltzmann equation, we took the derivative of the equilibrium distribution df0ldE assuming a functional dependence EF (r) in /0. However the Fermi level is unique and in principle defined only at equilibrium. How then can this assumption be justified? The essence of such an approximation, as explained in the previous chapter, is that every volume element dV, around point r, can be considered on the one hand as mathematically infinitesimal and on the other hand as physically containing many unit cells in equilibrium with its surroundings, so that a local Fermi level can be defined. Obviously such an approach requires an infinite or very large system. The Landauer formalism that we will describe in the next sections treats electrons strictly as waves and eliminates this latter problem by requiring the existence of only the Fermi levels of the electrodes which can always be considered as thermodynamic reservoirs with a fixed Fermi level.