Shockley approximation

Recombination is evidently controlled by trapping into defect states, consistent with the other recombination measurements. The recombination transitions through defects with two gap states are illustrated in Fig. 8.24, with electrons and holes captured into either of the two states. This type of recombination is analyzed by the Shockley-Read-Hall approach which distinguishes between shallow traps, for which the carrier is usually thermally excited back to the band edge, and deep traps, at which the carriers recombine. A demarcation energy, which is usually close to the quasi-Fermi energy, separates the two types of states. The occupancy of the shallow states is determined by the quasi-equilibrium and that of the deep states by the recombination processes. No attempt is made here at a comprehensive analysis of the photoconductivity, which rapidly becomes complicated. Instead some approximate solutions are derived which illustrate the processes. 

According to the IEEE standard test methods for the characterisation of OFETs [26], the Shockley equations for insulated gate field effect transistors (IGFETs) are used to approximate the field-indueed drain eurrent in the organic material between the drain and source contaets. It should be reeom-mended that the following assumptions form the basis of the applieability of the equations ... 

A stress applied to a crystal results in a strain. A phenomenological description of the electron energy levels under elastic strain was developed by Bardeen and Shockley [12]. It is referred to as the deformation potential approximation (DPA), in which the one-electron Hamiltonian is developed in a Taylor s series of the strain components The perturbation is written in cartesian coordinates, for a linear order in strain, as ... 

Figure A2.5.21. The heat eapacity of an order-disorder alloy like p-brass ealeulated from various anal5dic treatments. Bragg-Williams (mean-field or zeroth approximation) Bethe-1 (first approximation also Guggenheim) Bethe-2 (second approximation) Kirkwood. Each approximation makes the heat capacity sharper and higher, but still finite. Reproduced from [6] Nix F C and Shockley W 1938 Rev. Mod. Phys. 10 14, figure 13. Copyright (1938) by the American Physical Society.

Surface recombination is analyzed using Hall-Shockley-Read152,153 recombination analysis. With the usual approximation, the recombination flux, Jr, is given by... 

Furthermore, let gi () denote the piecewise linear approximation of the nonlinear flow versus effort relation of the t-tti diode according to Shockley s equation implemented by the switch model Sw Z),. 

Problem 2.2. The electronic surface states are shown in Fig. 2.26 by dashed lines. The parabolic shape of the upper surface band implies that it is well described by the nearly-free electron approximation and hence this state is classified as a Shockley state. The lower surface state has the character of a surface resonance in the region where it intersects the electronic bulk band. An electronic transition between the surface states is possible if the upper state is unoccupied, i.e., it is located above the Fermi level. The minimum energy of such a transition is about 1.8 eV. 

The contribution of the Shockley surface state to the physisorption bond on Pt(lll) can even be quantified [45] on adsorption of 0.33 ML of Xe, the occupied Pt(lll) surface-resonance band at F is shifted from —0.40 to —0.25 eV because of hybridi2ation with the occupied Xe 5p level. This translates into an energetic cost of r 17 meV/Pt-surface-atom or 50 meV/Xe-atom. On Pd where the Shockley surface state is a priori unoccupied, an upshift of the surface state does not affect the total energy balance and no surface-state-related destabilization occurs. Hence, approximately 60% of the Xe adsorption enthalpy difference of 80meV/atom between Pt and Pd stem from the surface state ... 

The evaluation of the decay rate involves a double integral of the self-energy bracketed with the initial-state wave functions (see Eq. 6.2). For an efficient evaluation, a free-electron approximation parallel to the surface simpHfies the calculations considerably. Different effective masses may be used for bulk and surface states. The results of such calculations for the occupied Shockley surface state on noble metal (111) surfaces is shown in Table 6.2. 

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