Landauer formula

One way to think about the Landauer formula is to say conductance is scattering [69]. In fact, conductance is elastic scattering, because in the original Landauer approach, all scattering is considered to be elastic - particles leave the electrode and are scattered elastically until they make it into the other electrode (or not). Inelastic events are not included, at least conceptually. 

The Landauer formula assumes elastic processes. If the electrons move coherently (that is without any loss of energy or of phase) they will tunnel if the energy gap through which they must tunnel becomes relatively small, they can tunnel a long way. Generally, the conduction in the tunneling regime is written as... 

Landauer proposed in 1957 the first mesoscopic theoretical approach to charge transport [176]. Transport is treated as a scattering problem, ignoring initially all inelastic interactions. Phase coherence is assumed to be preserved within the entire conductor. Transport properties, such as the electrical conductance, are intimately related to the transmission probability for an electron to cross the system. Landauer considered the current as a consequence of the injection of electrons at one end of a sample, and the probability of the electrons reaching the other end. The total conductance is determined by the sum of all current-carrying eigenmodes and their transmission probability, which leads to the Landauer formula of a ID system ... 

When internal interactions are neglected, the current is purely coherent (elastic), and the formalism yields expressions that are consistent with the Landauer formula. The current can be rewritten as... 

To calculate the total current we should substitute the expressions for the wave functions (37)-(39), and summarize all contributions [291]. As a result the Landauer formula is obtained. We present the calculation for the transmission function. First, after substitution of the wave functions we have for the partial current going through the system... 

The same current is given by the Landauer formula through the transmission function T(E)... 

This expression is equivalent to the Landauer formula (47), but the transmission function is related now to the tunneling matrix element. 

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. 

Once a pre-specified convergence criterion is reached, one proceeds to the last box labeled Analysis in Fig. 3 where physical quantities are determined. For transport problems, the basic interests are the current-voltage (I-V) curves and conductances. For coherent quantum conductors, one applies the Landauer formula [47,53],... 

Here G0 = 2e2/h is the quantum of conductance and the factor 2 is due to spin degree of freedom. The current determined by Landauer formula is the elastic current. Indeed, we did not include any interactions leading to inelastic transport channels into the Hamiltonian. 

Y. Meir, N.S. Wingreen, Landauer Formula for the Current through an Interacting Electron Region, Phys. Rev. Lett. 68 (16) (1992) 2512-2515. 

ELECTRICAL RESISTANCE, HALL EFFECT, DRUDE MODEL, TUNNELING, AND THE LANDAUER FORMULA... 

Equation 12.13 is the thermal Landauer formula derived previously in different... 

According to Samanta et ol. [62], who used a GF-based method to calculate the transmission function of electrons across the dithiol ligand, the resistance R per molecule was determined by the Landauer formula, R = hl2e )IT(Eg), where T( g) is the transmission function, T exp[2(m g) /H]. For the dithiol molecule, the barrier height was estimated to be g 2.8eV, and the resulting resistance calculated as R 30GQ. 

The conductance of a metal-molecule-metal junction is also given by the Landauer formula which relates the linear conductance to the electron transmission coefficient Tf... 

At the nanoscale, tunneling current can be described by Landauer formula for the linear conductance, G, of the electrode-molecule-electrode junction [24] ... 

Landauer Formula and Electron Transport through Molecular Junction... 

Landauer Formula and Electron Transport through Molecular Junction As discussed earlier, in a molecular junction, the two major parts are the electrodes (metals) and the molecule. In an applied bias voltage, when an electron is transformed from one electrode (say left electrode) to the other electrode (say right), the resultant process creates conductance through the molecule. 

Under the condition of a thermal equilibrium existing at the metal junction, the Landauer formula for the coherent conductance through the junction can be represented by Eq. 2. 

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