Percolation and Tunneling
When raindrops fall on a road, they - of course - fall randomly. Yet despite this randomness we can describe extremely well the typical size of a group of raindrops, and the probability that a network of connected raindrops forms a pathway across the road. The key to our understanding of this system, and very many others, is percolation theory 1, 2, 3. Furthermore, percolation allows an understanding of transport properties, such as flow through porous rocks, the propagation of forest fires or diseases, and the conductivity of random networks1, 2, 3. These are all classical percolation problems but increasingly our ability to fabricate interesting nanoscale systems means that quantum mechanics can dominate the system’s properties, presenting new challenges for percolation theory.
As expected from percolation theory, a random two dimensional assembly of nanoparticles has a critical surface coverage, the percolation threshold, pc, above which a connected network exists across the system and classical conduction occurs. We have previously shown that (for surface coverages p>pc) the size of connected structures 4 and the conductivity of the system1 are well described by percolation critical exponents1. For p<pc the conductivity of the system should be zero. However when there is quantum tunneling between particles, percolation theory is confronted with an enormous problem: tunneling probabilities decay exponentially, and there is (in principle) no limit to the range of tunneling. Why, in this case, should there be a percolation threshold?
A comprehensive description of 2D percolating-tunneling (PT) assemblies of nanoparticles is required to allow an understanding of a range of important systems. In particular we want to develop this understanding by exploring memristor behaviour and superconductivity which appear in the same 2D PT systems.
- D. Stauffer and A. Aharony, Introduction to Percolation Theory, (2nd Ed.), Taylor and Francis, Philadelphia 1991.
- A. Hunt and R. Ewing, Percolation Theory for Flow in Porous Media, Lect. Notes Phys. 771, Springer, Berlin, 2009.
- R. Meester and R. Roy, Continuum Percolation, Cambridge Tracts in Mathematics, Cambridge University Press (1996).
- J. Schmelzer jr, S. Brown, A. Wurl, M. Hyslop, and R. Blaikie, ‘Finite-Size Effects in the Conductivity of Cluster Assembled Nanostructures’, Phys. Rev. Lett. 88, 226802 (2002).
- A. Dunbar, J. Partridge, M. Schulze and S. Brown, ‘'Morphological differences between Bi, Ag and Sb nano-particles and how they affect the percolation of current through nano-particle networks’, Eur. Phys. J. D. 39, 415 (2006).