Electronic states Semiclassical approximation

Field emission is a tunneling phenomenon in solids and is quantitatively explained by quantum mechanics. Also, field emission is often used as an auxiliary technique in STM experiments (see Part II). Furthermore, field-emission spectroscopy, as a vacuum-tunneling spectroscopy method (Plummer et al., 1975a), provides information about the electronic states of the tunneling tip. Details will be discussed in Chapter 4. For an understanding of the field-emission phenomenon, the article of Good and Muller (1956) in Handhuch der Physik is still useful. The following is a simplified analysis of the field-emission phenomenon based on a semiclassical method, or the Wentzel-Kramers-Brillouin (WKB) approximation (see Landau and Lifshitz, 1977). 

A great deal of research has been done recently on approximate solutions of strongly coupled semiclassical equations. Some solutions have been extensively used for cross-section calculations, often without any estimation of possible errors. Final expressions for the nonadiabatic transition probability P between two electronic states will be written down for the most frequently used approximations. 

In the above semiclassical approximation, nuclear motion has been treated classically. We now consider the general case in which the motions of both electrons and nuclei obey the laws of quantum mechanics. Denoting by x s the nuclear coordinate normal to the intersection line L, and by z the set of electron coordinates, we write according to (3 1) the wave function for a given qiaantum state n of the total system... 

Method of calculating the transport coefficients of electrons in carbon nanotubes described in detail in Refs. [11-13], Evolntion of the electronic system was simulated in the semiclassical approximation of the relaxation time. Electron distribution function in the state with momentum p = (p, s) is of the x - approximation using Boltzmaim eqnation [10] ... 

The broadening Fj is proportional to the probability of the excited state k) decaying into any of the other states, and it is related to the lifetime of the excited state as r. = l/Fj . For Fjt = 0, the lifetime is infinite and O Eq. 5.14 is recovered from O Eq. 5.20. Unfortunately, it is not possible to account for the finite lifetime of each individual excited state in approximate theories based on the response equations (O Eq. 5.4). We would be forced to use a sum-over-states expression, which is computationally intractable. Moreover, the lifetimes caimot be adequately determined within a semiclassical radiation theory as employed here and a fully quantized description of the electromagnetic field is required. In addition, aU decay mechanisms would have to be taken into account, for example, radiative decay, thermal excitations, and collision-induced transitions. Damped response theory for approximate electronic wave functions is therefore based on two simplifying assumptions (1) all broadening parameters are assumed to be identical, Fi = F2 = = r, and (2) the value of F is treated as an empirical parameter. With a single empirical broadening parameter, the response equations take the same form as in O Eq. 5.4 with the substitution to to+iTjl, and the damped linear response function can be calculated from first-order wave function parameters, which are now inherently complex. For absorption spectra, this leads to a Lorentzian line-shape function which is identical for all transitions. 

An alternative approach to polaron transport in organic solids is in terms of electron transfer (ET). The process can be viewed as a special case of the non-radiative decay of an electronic state. The derivation of the theory is developed in various books or review papers [13-15]. The parameter of importance here is the transition probability per unit time (or transition rate) kif between an initial and a final state. The rate is estimated within the Franck-Condon approximation. In the high-temperature regime ( cOif < kT) the Franck-Condon-weighted density (FCWD) reduces to a standard Arrhenius equation, so the rate takes its semiclassical Marcus theory expression [16] ... 

A further simplification of the semiclassical mapping approach can be obtained by introducing electronic action-angle variables and performing the integration over the initial conditions of the electronic DoF within the stationary-phase approximation [120]. Thereby the number of trajectories required to obtain convergence is reduced significantly [120]. A related approach is discussed below within the spin-coherent state representation. 

Practical approximations for t are binary-encounter operators such as U3, which do not depend on the ion coordinates. The target and ion are therefore represented by the overlap (i+ 0), which is shown by experiments described in section 11.1.2 to be small unless i+) is the ground state 0+). These approximations were made by McCarthy and Shang (1993). They represented the final-state interaction in the long-range on-shell direct amplitude of (10.50) by the factor C(t])e "° (10.38). This factor was omitted from the resonant amplitude on the basis of the semiclassical picture of the two electrons emerging at different times. 

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