Vacuum state Fermi

Let us adopt I 0 > as a new vacuum state. We shall call it the Fermi vacuum state. 

With respect to the Fermi vacuum state I 0 > we can now define new creation and annihilation operators which in contrast to the X+, X operators shall be designated... 

From this definition it is evident that application of Yj to the Fermi vacuum is equivalent to annihilation of a particle (or creation of a hole) in 14>0 >. The effect of YA on the Fermi vacuum state is the creation of a particle (or annihilation of a hole) in I 0>. The effect of YA" on the Fermi vacuum is the creation of a particle in the virtual spin-orbitals and finally, the effect of YA" is the annihilation of a particle in virtual spin-orbitals. Thus e.g., a singly excited Slater determinant I ) can be described as... 

The reference state of A-electron theory becomes a reference vacuum state 4>) in the field theory. A complete orthonormal set of spin-indexed orbital functions fip(x) is defined by eigenfunctions of a one-electron Hamiltonian Ti, with eigenvalues ep. The reference vacuum state corresponds to the ground state of a noninteracting A-electron system determined by this Hamiltonian. N occupied orbital functions (el < pi) are characterized by fermion creation operators a such that a] ) =0. Here pt is the chemical potential or Fermi level. A complementary orthogonal set of unoccupied orbital functions are characterized by destruction operators aa such that aa < >) = 0 for ea > p and a > N. A fermion quantum field is defined in this orbital basis by... 

The diagrams are interpreted in terms of the particle-hole formalism. The Fermi level is defined such that all single particle states lying below it are occupied and all above it are unoccupied. In the particle-hole picture, the reference state is taken to be a vacuum state, containing no holes below the Fermi level and no particles above it. Excitation leads to the creation of particle-hole pairs, with particles above the Fermi level and holes below it. 

In describing many-body systems in their ground or low lying excited states it is convenient to redefine the vacuum state to contain the single particle states occupied in the ground state, . This is usually termed the Fermi vacuum. A set of creation and annihilation operators can be defined with respect to the Fermi vacuum as follows... 

This classification of the one-electron orbitals is displayed in Figure 10.1. While, due to this definition, it is possible to have completely unoccupied core-valence orbitals or (completely) occupied valence orbitals in the model space, they could be eliminated always by a proper re-definition of the many-electron vacuum state 0). The distinction of the valence states into core-valence and valence orbitals has the great advantage that we need not to deal explicitly with the particular choice of the vacuum in the derivation of perturbation expansions. As we shall see below, namely, the projection upon the vacuum appears rather frequently in the derivations for open-shell structures and can be carried out formally, if all the core and core-valence orbitals (i.e. all the orbitals up to the Fermi level) are taken into account separately from the rest of the orbital functions. 

As seems from Eq. (12.10c), calculation of E involves the evaluation of the quantities Wqk, lo where O labels the ground state Fermi vacuum)... 

The chemical potential of electrons in a Fermi distribution is also called the Fermi level. The energy required to remove an electron from the Fermi level to infinity (the vacuum state) is the work function. Since the difference in chemical potential determines the flow of particles, when two materials with different Fermi levels are brought together as illustrated in Figure 15.2, electrons will flow from the material with the higher Fermi level (smallest work function) to the material with the lower Fermi level until equilibrium is reached. This transfer of charge results in the contact potential between the two materials. 

It is now possible to form heterostructures in which the transition from one system to the other takes place over one atomic layer. When two such semiconductors are brought into intimate contact, the electron affinities (energy required to remove an electron from the bottom of the conduction band to vacuum state) must line up, and at thermal equilibrium, the Fermi levels must also line up. The result is band bending with discontinuities in both the valence (AEy) and conduction band (AEc) at the interface as shovm in Figure 22.13. (Recall that in homojunctions between n- and p-type material, the vacuum levels already lined up because both sides of the junction were the same material so no such discontinuities appeared.)... 

Figure 3.2. Simple illustration of the particle-hole formalism. This figure shows the states given in the particle formalism in Figure 3.1 when depicted in the particle-hole picture, (a) shows the reference configuration or vacuum state with no particles above the Fermi level and no holes below it. (6) corresponds to a single excitation which creates a hole below the Fermi level and a particle above it. (c) is a doubly excited state with two holes and two particles and (d) is associated with a triply excited state with three holes and three particles. The particle-hole formalism focusses attention on the excitation process the particles and holes created during an excitation. The other electrons in the studied many-body system are merely spectators to the excitation process.

In using the particle-hole formalism, we adopt < o) as a new vacuum state. We call this reference state the Fermi vacuum state. With respect to this Fermi vacuum state l o), we can now define new creation and annihilation operators. To distinguish them from the operators X+ and X described above, the creation and annihilation operators in the particle-hole formalism shall be designated Y+ and Y, respectively. If we label the occupied spin-orbitals as A ) and the virtual spin-orbitals as A"), then the creation (Y+) and annihilation (F) operators in the particle-hole formalism are then defined in the following manner ... 

From the above mentioned relations it is easy to see that the vacuum expectation value of the electronic Hamiltonian (3.4) is zero. The particle-hole formalism implies a redefinition of the vacuum state. Since correlation energy is defined with respect to the Hartree-Fock energy, we redefine the vacuum state as being the occupation-number vector corresponding to the converged HF determinant, the Fermi vacuum. This leads to a redefinition of creation... 

Adsorption related charging of surface naturally affects the value of the thermoelectron work function of semiconductor [4, 92]. According to definition the thermoelectron work function is equal to the difference in energy of a free (on the vacuum level) electron and electron in the volume of the solid state having the Fermi energy (see Fig. 1.5). In this case the calculation of adsorption change in the work function Aiqp) in... 

For electrons in a metal the work function is defined as the minimum work required to take an electron from inside the metal to a place just outside (c.f. the preceding definition of the outer potential). In taking the electron across the metal surface, work is done against the surface dipole potential x So the work function contains a surface term, and it may hence be different for different surfaces of a single crystal. The work function is the negative of the Fermi level, provided the reference point for the latter is chosen just outside the metal surface. If the reference point for the Fermi level is taken to be the vacuum level instead, then Ep = —, since an extra work —eoV> is required to take the electron from the vacuum level to the surface of the metal. The relations of the electrochemical potential to the work function and the Fermi level are important because one may want to relate electrochemical and solid-state properties. 

Figure A.l 1 shows the change in density of states due to chemisorption of Cl and Li. Note that the zero of energy has been chosen at the vacuum level and that all levels below the Fermi level are filled. For lithium, we are looking at the broadened 2s level with an ionization potential in the free atom of 5.4 eV. The density functional calculation tells us that chemisorption has shifted this level above the Fermi level so that it is largely empty. Thus, lithium atoms on jellium are present as Li, with 8 almost equal to 1. Chemisorption of chlorine involves the initially unoccupied 3p level, which has the high electron affinity of 3.8 eV. This level has shifted down in energy upon adsorption and ended up below the Fermi level, where it has become occupied. Hence the charge on the chlorine atom is about-1.

First, we describe the various system parameters, primarily adapted from Newns (1969). From the energy dispersion relation (2.32), the bulk states are distributed through a band, centered at a, and with width Wb = 4 / . The Fermi level Ef is taken to be at the center of this band, and is chosen to be the energy zero (so that Ef = a = 0, for all systems). The position of /, relative to the vacuum level, is given by the work function (j>, whence the isolated H adatom level, relative to Ef is... 

Further we want to study the nonadiabatic corrections to the ground state. Therefore /o> will be the unperturbed ground state wave function (we shall use Hartree-Fock ground state Slater determinant -Fermi vacuum) and % ) will be boson ground state-boson vacuum 0). 

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