condensed matter - What is meant by phase coherence of electrons?

I came across a term "phase coherent electron devices" in the book Condensed Matter Field Theory by Atland and Simons. What is meant my phase coherent electrons (coherence is a term generally talked about in the context of light) in a system and when does a bunch of electrons behave in such fashion?

Answer

Strictly speaking, a phase coherent electron device is an electronic device whose dimensions is smaller than the phase coherence length of the electrons. This definition is the one adopted in mesoscopic physics.

So, what is a phase coherence length? To each electron, one associates a wave-function $\Psi=\Psi_{0}e^{i\varphi}$, with $\varphi$ the phase of the wave function of amplitude $\Psi_{0}$. The length associated to the phase coherence is the length after which the phase has changed significantly, say by $2\pi$ to quantify the concept.

The notion of phase coherence is important for modern electronics of small size devices (especially at low temperatures), since it means that for devices smaller than the phase coherence length, quantum effects associated to the phase of the wave functions are no more negligible. Among those effects, interference effects are certainly the immediate ones we can think of. The interference effects are one of the many signatures of the quantum regime, hence the importance of the concept of phase coherence.

The branch of physics studying the interference effects of electrons in metallic, superconducting and semi-conducting structures is called the mesoscopic physics. This is also the topic where one studies the effects destroying quantum phenomena (i.e. decoherence, though this word is sometimes use in the more restrictive meaning of destructing the entanglement). For instance, the phase coherence length associated to the electrons becomes smaller at high temperature, since the phonon bath allows more incoherent scatterings between electrons, finally changing the phase of the electrons wave-function by a significant amount, say $2\pi$. When the phase coherence length becomes smaller than the size of the electronic device, the quantum effects wash out and only classical effect survive.

An important example of phase coherence of electron is the Landauer quantisation formula of the conductance: when one sees an electron as a pure wave, the conductance of a resistive element should follow the concept of tunneling in quantum structures. This clearly disagrees with the Ohm's law of classical resistors. The reason why is now understood as the effects of temperature and size of the device: taking a reservoir of Landauer resistors finally restaures the Ohm's law.

One may at first think that the electron phase coherence is easier to maintain in ballistic structures, when the electrons mean-free-path (basically, the distance between two scattering events of an electron between two impurities) is larger than the size of the device. Nevertheless, phase coherence plays important role in diffusive systems, both electronic or photonic ones. In such systems, an important effect if the universal conductance flutuations. About diffusive systems, please open the book by Akkermans and Montambaux, Mesoscopic Physics of Electrons and Photons, Cambridge University Press (translated from the french original edition).