Editing Electron mobility (section)

=== Variable Range Hopping ===
At low temperature, or in system with a large degree of structural disorder (such as fully amorphous systems), electrons cannot access delocalized states. In such a system, electrons can only travel by [[Quantum tunnelling|tunnelling]] for one site to another, in a process called ''variable range hopping''. In the original theory of variable range hopping, as developed by Mott and Davis,<ref name=":3">{{Cite book|url=https://global.oup.com/academic/product/electronic-processes-in-non-crystalline-materials-9780199645336?cc=us&lang=en&|title=Electronic Processes in Non-Crystalline Materials|date=2012-03-24|publisher=Oxford University Press|isbn=978-0-19-964533-6|series=Oxford Classic Texts in the Physical Sciences|location=Oxford, New York}}</ref> the probability <math>P_{ij}</math>, of an electron hopping from one site <math>i</math>, to another site <math>j</math>, depends on their separation in space <math>r_{ij}</math>, and their separation in energy <math>\Delta E_{ij}</math>.

<math display="block">P_{ij} = P_{0}\exp\left(-2\alpha r_{ij} - \frac{\Delta E_{ij}}{k_{B} T}\right)</math>

Here <math>P_{0}</math> is a prefactor associated with the phonon frequency in the material,<ref>{{Cite journal|last=Emin|first=David|date=1974-02-11|title=Phonon-Assisted Jump Rate in Noncrystalline Solids|url=https://link.aps.org/doi/10.1103/PhysRevLett.32.303|journal=Physical Review Letters|volume=32|issue=6|pages=303–307|doi=10.1103/PhysRevLett.32.303|bibcode=1974PhRvL..32..303E|url-access=subscription}}</ref> and <math>\alpha</math> is the wavefunction overlap parameter. The mobility in a system governed by variable range hopping can be shown<ref name=":3" /> to be:

<math display="block">\mu=\mu_{0} \exp \left(-\left [ \frac{T_{0}}{T} \right ]^{-1/(d+1)}\right)</math>

where <math>\mu_{0}</math> is a mobility prefactor, <math>T_{0}</math> is a parameter (with dimensions of temperature) that quantifies the width of localized states, and <math>d</math> is the dimensionality of the system.