4.13: Comparing the quantum and Semi-Classical Drude models of conduction

The mean free path

The Drude model gives a physically incorrect picture of charge conduction. Nevertheless it works quite well. The quantum model shows that rather than all the electrons moving at the drift velocity, as in the Drude model, only the uncompensated electrons carrying current, each moving at approximately the Fermi velocity:\(^{3}\) Thus, the Drude model can be rearranged as

\[ J = qn^{‘}v_{F} \]

where the uncompensated charge density is

\[ n^{‘} = n\frac{v_{d}}{v_{F}} \]

We can also define the mean free path, \(L_{m}\), as the average distance an electron travels between scattering events. The mean free path is related to the Fermi velocity by:

\[ L_{m} = v_{F}\ \tau_{m} \]

Interestingly, the mean free path is approximately equal to the characteristic length \(L_{0}\) in the derivation of Ohm's law.

Equilibrium and Non-equilibrium current flow

We can demonstrate the differences between the classical and ballistic limits using the analogy of water flow from one reservoir to another. The application of bias across a wire is equivalent to depressing the height of the drain reservoir relative to the source reservoir. In the ballistic model water flowing from the source travels across the wire as a jet before relaxing to equilibrium in the drain. In the classical model the water minimizes its potential in channel.

Figure \(\PageIndex{1}\): A water flow analogy for ballistic and classical current flow. In the classical limit, the water is always in local equilibrium with the channel.

One way to think about classical transport is as the limit of a series of nanoscale ballistic wires interspersed by contacts. By definition, electrons in the contacts are in equilibrium. Thus contacts are different to the elastic scatterers we considered above, because electrons change energy in contacts. The limiting case of many closely spaced contacts is a continuously varying conduction band edge; see Figure 4.14.2.

Figure \(\PageIndex{2}\): In a classical wire, the conduction band edge varies continuously with position. The classical model can be imagined as the limiting case of many ballistic devices in series.

The length scale of ballistic conduction

To determine whether we should use the ballistic or semi-classical models of charge transport we need to know the likelihood of electron scattering in the channel. This depends on the channel length, and the quality of the semiconductor.

The number of scattering events in the channel is given by \(\tau /\tau_{m}\) where \(\tau\) is the transit time of the electron, and \(\tau_{m}\) is its average scattering time. Relating the transit time to the carrier velocity, and \(\tau_{m}\) to the definition of mobility in Equation (4.12.2) gives:

\[ \frac{\tau}{\tau_{m}} = \frac{l/v}{m_{eff}\ \mu/q}=\frac{l^{2}/\mu V_{SD}}{m_{eff}\ \mu/q} = \frac{ql^{2}}{m_{eff}V_{SD}\ \mu^{2}} \]

This expression is plotted in Figure 4.14.3 assuming a Si conductor with \(V_{DS} = 1V\), \(\mu = 300 cm^{2}/Vs\) and \(m_{eff} = 0.5 \times m_{0}\), where \(m_{0}\) is the mass of the electron. It shows that silicon is expected to cross into the ballistic regime for lengths of approximately l < 50nm.

Figure \(\PageIndex{3}\): The expected number of electron scattering events in Si as a function of the channel length. The threshold of ballistic operation occurs for channel lengths of approximately 50nm.