Kubo-Greenwood formula

Here, we will shortly summarize these results again. After this, we will focus on our recent results for the electrical conductivity. This quantity we calculated using the Kubo-Greenwood formula [8]. 

For comparison, we applied also a simplified LCAO-DFT method to get the conductivity by means of the Kubo-Greenwood formula. This method is a hybrid between ab initio and empirical methods and is described in detail in Ref. [12]. It allows a faster computation of the electronic properties and the consideration of larger supercells than the Car-Parrinello method. Within this scheme it is also possible to split the total DOS into fractions referring to the sodium and tin atoms, respectively, i.e. to get the partial densities-of-states. 

We confine ourselves here to situations where the mean free path is due to elastic collisions with impurities or, in alloys, liquids and amorphous materials, with the non-periodic field in such materials. According to the Kubo-Greenwood formula, the current at temperature T is given by (31), where, instead of (32), o(E) is defined by the equation... 

Here the average is over all states separated by a fixed energy difference hcom and N0(Ef) is the density of states per spin. This formula has been derived by various methods (MacMilliah 1981, Kaveh and Mott 198 la, b, Imry et a/. 1982, Lee 1982). For non-zero q, we find, making o>->0, as in the derivation of the Kubo-Greenwood formula (34),... 

The next step in the theory is to calculate the conductivity above and below the mobility edge. In the Anderson model, locali2ed states are defined by a decreasing probability that the electron diffuses a larger distance from its starting point. Mott and Davis (1979 Chapter 1) prove that the dc conductivity in the localized states is zero at T = 0 K. They use the Kubo-Greenwood formula for the conductivity,... 

The proportionality on the right-hand side of Eq. (4.16) follows from the Kubo-Greenwood formula, Eq. (2.6), so long as the scattering matrix element is only weakly energy-dependent near Ep. Interpreted... 

In the absence of the electron-phonon interaction or when the important phonon energies are substantially less than k0T, we can write the Kubo-Greenwood formulas for the dc conductivity in the following form. 

The thermopower, Q, can be calculated conveniently by performing the energy derivative, eq. (41), within the Kubo-Greenwood formula, resulting in... 

When the mean free path I is on the order of the distance between atoms, the free-electron approximation (i.e., Drude model) breaks down. This occurs particularly in noncrystaUine materials and for impurity conduction. In this situation, we make use of the Kubo-Greenwood formula [1176], which has universal validity. This formula uses elementary quantum mechanics to calculate the conductivity of a metal at frequency transition probabilities from occupied states below Epio empty states above Ep and then letting w tend to 0 for dc conductivity [1125, 1156, 1157]. The main steps in the calculation are as follows we suppose that an electric field F cos (ot acts on a specimen of volume fl. Since the quantity o mean rate of loss of ener per unit volume, the conductivity o(ti>) is given by the product of the following factors ... 

The subscript av represents an average over all states with energies E. Equation (4.10) is called the Kubo-Greenwood formula. For a long mean free path approximation, Eq. (4.10) reduces to the Drude model. 

According to the general Kubo-Greenwood formula [26], compensation effects between various transport mechanisms appear in the thermoelectric power expression [18]... 

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