Diagrammatic Monte Carlo Method

In an exciton, the electron and the hole are bound together by an electric attraction-known as the Coulomb force-in a fashion very similar to that of an electron and a positron in a hydrogen atom. The presence of the host lattice and its thermal and magnetic excitations that consist of phonons and magnons, respectively-known collectively as the ‘bosonic’ field-can affect the excitons considerably.

The researchers, including Andrei Mishchenko from the RIKEN Advanced Science Institute in Wako, aimed to develop a technique to study the excitons’ interaction with phonons in an exact way. In particular, they focused on taking into consideration the fact that phonons do not act instantaneously as occurs in the Coulomb attraction. “Previously, the only way to treat the exchange [between electrons and holes] by bosons was an instantaneous approximation, where the influence of particle–boson interaction was included into the model by renormalization of the instantaneous coupling,” explains Mishchenko.

Mishchenko and colleagues’ technique is known as a Diagrammatic Monte Carlo Method and is based on the diagrams that the Nobel laureate Richard Feynman introduced to quantum field theory. The method per se existed already and was normally used with all variables expressed as a function of spatial coordinates. This, however, limits the size of the area that can be examined in a calculation. The team therefore formulated the algorithm for momentum space. This provides the “possibility to overcome the limitation of the direct space method [for] finite systems and handle the problem [in] a macroscopic system,” says Mishchenko.

Like any new theoretical method, the team’s numerical technique must be compared with known scenarios to verify its validity, so Mishchenko and colleagues used it to study excitons with different values for the electron and hole masses. They found very good agreement with previous theories within the limit in which it is reasonable to neglect any retardation effect. Importantly however, the results show that in standard conditions it is incorrect to neglect the retardation.

As Mishchenko explains: “Our ‘free-from-approximations’ results show that the domain of validity of the instantaneous approximation is very limited.”

*Burovski, E., Fehske, H. & Mishchenko, A.S. Exact Treatment of Exciton-Polaron Formation by Diagrammatic Monte Carlo Simulations. Physical Review Letters 101, 116403 (2008).



The Wiedemann-Franz law: When you are static and cold, you have more of electric conductivity in you. That is because electrons transportation properties diminish by velocity because of higher collisions between electrons. The same is true for heat, which increase collisions of electrons hence decrease their conductivity. Electric conduction depends on free electrons moving in one direction. the Wiedemann and Franz proved that proportionality of thermal and electric conductivity is the same for all metals at the same temperature.

Thermal conductivity / electric conductivity = Temperature x Constant L
K/ σ = L T (1)

There is difficulty in nano-scale materials to conduct electrical measurements. But STM with a fixed tip (2) has been used to measure conductivity. Also electrostatic force microscopy EFM that applies voltage between the tip and the sample made image of local charge domains on a sample surface. (3) Resistance to electrical conductivity decrease with size because of improved order and lesser amount of defects in the lattice, p (resistance) = 1/ σ (4) But scattering of free electrons on the nanosurfaces may have adverse effect on electric conductivity.

(1) http://en.wikipedia.org/wiki/Wiedemann-Franz_law
(2) Kelsall et al, 2007, Nanoscale science and technology, Willey, p 126
(3) Kelsall et al, 2007, Nanoscale science and technology, Willey, p 92
(4) Owens et al, 2007, The physics and chemistry of Nanosolids, p 81

- Jones, William; March, Norman H. (1985). Theoretical Solid State Physics. Courier Dover Publications.