Landauer theory

Electron transport through nanoscale devices is often handled at the Landauer theory level, where the electrons are assumed to move coherently from one electrode to the other without inelastic collisions. The current can be determined by computing the transmission function T(E, V), which depends on energy for a given applied voltage. The functional form of T(E, V) arises from the quantum states of the molecule that is coupled to the two electrodes. The most general computational methods used to compute those states have employed Green s function techniques to obtain the current self-consistently." ... 

Iand67] Landauer, R., Wanted a physically possible theory of physics , IEEE Spectrum 4 (1967) 105-109. 

The effective medium theory consists in considering the real medium, which is quite complex, as a fictitious model medium (the effective medium) of identical properties. Bruggeman [29] had proposed a relation linking the dielectric permittivity of the medium to the volumetric proportions of each component of the medium, including the air through the porosity of the powder mixture. This formula has been rearranged under a symmetrical form by Landauer (see Eq. (8), where e, is the permittivity of powder / at a dense state, em is the permittivity of the mixture and Pi the volumetric proportion of powder / ) and cited by Guillot [30] as one of the most powerful model. 

The physical sense of this expressions is quite transparent, they describe the quantum amplitudes of the scattering processes. Three functions scattering state in the Landauer-Biittiker theory. Note, that G here is the full GF of the nanosystem including the lead self-energies. 

Keywords Density functional theory (DFT) Green s functions Keldysh non-equilibrium Green s functions (NEGF) linear combination of atomic orbitals (LCAO) tunnel junction metal-fullerene-metal junction density of states (DOS) transmission function Landauer formula renormalized molecular levels (RMLs) I-V curves. 

Two other approaches have been taken to modelling the conductivity of composites, effective medium theories (Landauer, 1978) and computer simulation. In the effective medium approach the properties of the composite are determined by a combination of the properties of the two components. Treating a composite containing spherical inclusions as a series combination of slabs of the component materials leads to the Maxwell-Wagner relations, see Section 3.6.1. Treating the composite as a mixture of spherical particles with a broad size distribution in order to minimise voids leads to the equation ... 

The Landauer s seminal suggestion, that the current is transmission, dominates in mesoscopic physics and has applications to a variety of systems, including the electron transport in solids, liquids, quantum wires and dots. First and foremost, this theory describes conductance by purely dissipationless electrons scattering. Pursuing the NEM phenomena, we shall follow a similar reasoning, omitting the non-elastic effects on the microscopic scales. 

Kramers paper spurred an enormous amount of research on the theory of activated rate processes, especially in the physics community, as evidenced in numerous textbooks see, for example, Refs. 13 and 14. However, as noted by Landauer (15) in his subjective description of the history of noise activated escape from metastable states, up till the end of the seventies, the physical chemistry community largely ignored the theory of rates introduced by Kramers. The first experimental measurements of viscosity effects on activated rate processes were performed on the isomerization of frans-stilbene to c/s-stil-bene by Fischer and co-workers in 1968 (16). These authors were not aware of Kramers work and interpreted their results in terms of the free volume necessary for isomerization to occur. Since then, experimental work has proliferated see, for example, the recent textbook (17). 

R. Landauer, Noise in Nonlinear Dynamical Systems. Vol. 1. Theory of Continuous Fokker-Planck Systems (F. Moss and P. V. E. McClintock, eds.), Cambridge University Press, Cambridge, 1989, p. 1. 

Landauer s principle The principle put forward by Rolf Landauer in the 1960s that energy has to be expended to erase information. This principle links thermodynamics and information theory. 

Equation (2b) is a scaling law depicting the conductive behavior in the vicinity of the percolation threshold, the value of the critical exponent y being 1.6 to within 0.2. Equation (2c) expresses the composite conductivity dependence upon conductor concentration beyond the percolation threshold. Equation (2c) is a simplified form, valid in the case of conductor-insulator mixtures, of a more general equation derived in different ways by Bruggeman (70), Bottcher (71) and Landauer (72) and known as the Effective Medium Theory, (E.M.T.), formula ... 

In an effective media theory of a composite, a spherical or ellipsoidal grain is considered to be surrounded by a mixture, which has the effective conductivity of the composite medium. It is mainly used for composite materials with well-separated subphases for the prediction and explanation of large volume average values of electrical properties. An excellent overview has been provided by Landauer (1977). 

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