(Invited) The Landauer Approach to Electron and Phonon Transport
Authors: Maassen Jesse , Lundstrom Mark 
Published: February 2020

Semiclassical electron and phonon transport is typically simulated by solving the electron or phonon Boltzmann transport equation.  More recently, quantum transport of electrons and phonons has been simulated by the non-equilibrium Green’s function (NEGF) approach. The Landauer approach is a simple, physical approach to analyze electron and phonon transport.  It is widely used to treat ballistic electron transport at the nanoscale.  Less well known is that under the right conditions, it works all the way from the ballistic to diffusive limits in nanoscale to macroscale structures.  The Landauer approach can also be used for phonon transport – and also from the ballistic to diffusive limits and from the nano- to macro-scales.  Because the Landauer approach can be derived from the Boltzmann equation or from the NEGF equations, it provides a direct connection to more rigorous and fine-grained treatments.  We believe that it should be more widely-used for three reasons:  1) it provides a smooth bridge from the ballistic to diffusive limits and from small to large structures, 2) it provides a common “language” for electron and phonon transport, and 3) it provides a clear connection to more fundamental semiclassical and quantum approaches to transport.   Our goals in this talk are: 1) to review the Landauer approach to electron and phonon transport,  2) to illustrate how it compares to more rigorous treatments with some new benchmarking calculations, and 3) to illustrate how this approach helps identify good thermoelectric materials.

The electrical conductance can be written as the quantum of electrical conductance, times the number of thermally active channels for electrical conduction, times the transmission.  The quantum of conductance is a set of fundamental constants.  The number of thermally active channels depends only on electronic structure and can be rigorously computed from first principles. The transmission is one for a ballistic conductor and small for a diffusive conductor. For large structures, this approach reduces to the traditional (Boltzmann equation) expressions of near-equilibrium electron transport. For small structures, it describes quantized, ballistic conduction.  It is common to use the Landauer approach for nanostructures and the Boltzmann equation for large structures, but there is no need for two separate approaches – the Landauer approach covers both regimes of transport. 

The Landauer approach also applies to phonon transport using the same physical concepts described above. For example, the thermal conductance can be written as the quantum of thermal conductance, times the number of thermally active channels for phonon conduction, times the phonon transmission.  This talk will present a brief overview of this technique applied to electron and phonon transport, and will discuss some new examples that benchmark the Landauer approach and illustrate the benefits and limitations of the method.

It is sometimes felt that electrons and phonons are different because electrons are fermions and phonons are bosons.  It turns out that this makes little difference. It does affect the thermal window functions that determine the number of thermally active channels, but the window functions turn out to be quite similar in shape.  There are two significant differences – in practice, but not in principle.  The first is that the energy bandwidth of the electron dispersion is typically much, much larger than the width of the thermal window function for electrons, so that simplified dispersions often work very well (e.g. parabolic energy bands).  For phonons, the width of the thermal window function is comparable to the width of the phonon dispersion, so simplified phonon dispersions usually lead to large errors.  The second major difference has to do with the electron and phonon mean-free-paths.   Both mean-free-paths are energy-dependent, but since electrons typically occupy only a small range of energies near the bottom or top of a band, the use of a constant mean-free-path is often sufficient.  In contrast, there is a very strong energy dependence of the phonon mean-free-path, and phonons with a very wide range of mean-free-paths contribute to the thermal conductivity.  In spite of these technical differences, the formal equations and computational approaches are essentially identical for electrons and phonons.

For most electronic devices, both electron and phonon transport play a role, and for thermoelectric devices, the intimate coupling of the two flows is critical. For problems like thermoelectrics, the use of a common approach to analyze electron and phonon transport is beneficial and can provide new insights.  In this talk, we will illustrate what makes a good thermoelectric material from a Landauer perspective considering both traditional bulk and novel 2D materials as examples.

Figure 1