Fermi hole operator

The precursor to Kohn-Sham density-functional theory is Slater theory [12], In the latter theory, the nonlocal exchange operator of Hartree-Fock theory [25] is replaced by the Slater local exchange potential Vf(r) defined in terms of the Fermi hole p,(r, r ) as... 

In words, for virtual orbitals the holes operators bi", bj act exactly in the same manner as the particle operators aj do, while their role is reversed for occupied orbitals. Operator creates an electron in the virtual space, while it annihilates an electron in the Fermi see. This is equivalent to saying that it creates a hole in HF>. Similarly, operator bj creates an electron in 1HF>, while it annihilates one in the virtual subspace. This particle-hole formalism is in analogy with that of quantum field theory where, for instance, the holes correspond to positrons while the particles are electrons. 

Fermi Contact (FC) operator, 212, 251 Fermi correlation, 99 Fermi hole, 99 Field gradient, 213, 236 First quantization, 411 First-order corrections, in perturbation methods, 124, 125... 

There is one added layer which deserves special mention, namely a thin copper phthalocyaninc layer, which has been placed [103] between an 1TO anode and the hole transport layer. It is not an injection layer in the sense just discussed, because its HOMO is not well aligned with the 1TO Fermi energy and it slightly raises the operating voltage of the structure. It does, however, dramatically improve the stability of the device and appears to act as an adhesion layer for the organic materials above it. The inechanism(s) for these improvements is not yet well understood. 

The band edges are flattened when the anode is illuminated, the Fermi level rises, and the electrode potential shifts in the negative direction. As a result, a potential difference which amounts to about 0.6 to 0.8 V develops between the semiconductor and metal electrode. When the external circuit is closed over some load R, the electrons produced by illumination in the conduction band of the semiconductor electrode will flow through the external circuit to the metal electrode, where they are consumed in the cathodic reaction. Holes from the valence band of the semiconductor electrode at the same time are directly absorbed by the anodic reaction. Therefore, a steady electrical current arises in the system, and the energy of this current can be utilized in the external circuit. In such devices, the solar-to-electrical energy conversion efficiency is as high as 5 to 10%. Unfortunately, their operating life is restricted by the low corrosion resistance of semiconductor electrodes. 

Fig. 10-28. Polarization curves for cell reactions of photoelectrolytic decomposition of water at a photoezcited n-type anode and at a metal cathode solid curve M = cathodic polarization curve of hydrogen evolution at metal cathode solid curve n-SC = anodic polarization curve of oxygen evolution at photoezcited n-type anode (Fermi level versus current curve) dashed curve p-SC = quasi-Fermi level of interfadal holes as a ftmction of anodic reaction current at photoezcited n-type anode (anodic polarization curve r re-sented by interfacial hole level) = electrode potential of two operating electrodes in a photoelectrolytic cell p. sc = inverse overvoltage of generation and transport ofphotoezcited holes in an n-type anode.

The physics of free carriers is dominated by the Fermi surface. Looking at the linearized spectrum of Fig. 2, one realizes that electron-hole or electron-electron excitations involving quasiparticles (electrons or holes) on each side of the Fermi surface are gapless. This greatly influences the response functions to external fields. Let an external field Fa(q) couple to the operator Oa(q), where... 

The basic elements of the diagrams are shown in Figure 1. Figure 1 (a) shows the diagrammatic representation of a one-electron operator matrix element. Figure 1 (b) shows the representation of a two-electron matrix which in the Brandow scheme includes permutation of the two electrons involved. Upward (downward) directed lines represent particles (holes) created above (below) the Fermi level when an electron is excited. 

Ppor an explanation of -creation and -annihilation operators, see the earlier discussion of the particle-hole formalism in the section on The Fermi Vacuum and Particle-Hole Formalism. 

Considerations similar to those presented above show that illumination of a semiconductor leads to a shift of both the Fermi level and the quasi-levels of holes and electrons, and both the forward and reverse reactions, proceeding according to Eq. (1), are accelerated. In other words, the result of illumination is, above all, the efficient increase of the exchange current in the redox couple but this is not the only result. If a semiconductor under illumination is an electrode in an electrochemical cell and is connected through a load resistor with an auxiliary electrode, the cathodic and anodic reactions become spatially separated, as in the case of water photoelectrolysis (Fig. 11) considered above. The reaction with the minority carriers involved proceeds on the semiconductor surface, and that with the majority carriers involved, on the auxiliary electrode. Thus, the illumination of a semiconductor electrode gives rise to an electric current in the external circuit, so that some power can be drawn from the load resistor. In other words, the energy of light is converted into electricity. This is the way a photoelectrochemical cell, called the liquid junction solar cell, operates. 

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