Nearly-free-electron model

The one-electron wave function in an extended solid can be represented with different basis sets. Discussed here are only two types, representing opposite extremes the plane-wave basis set (free-electron and nearly-free-electron models) and the Bloch sum of atomic orbitals basis set (LCAO method). A periodic solid may be considered constmcted by the coalescence of these isolated atoms into extended Bloch-wave functions. On the other hand, within the free-electron framework, in the limit of an infinitesimal periodic potential (V = 0), a Bloch-wave function becomes a simple... 

Fig. 4.2 A graphic display of the origin of band structure in the nearly free electron model.

Band theory provides a picture of electron distribution in crystalline solids. The theory is based on nearly-free-electron models, which distinguish between conductors, insulators and semi-conductors. These models have much in common with the description of electrons confined in compressed atoms. The distinction between different types of condensed matter could, in principle, therefore also be related to quantum potential. This conjecture has never been followed up by theoretical analysis, and further discussion, which follows, is purely speculative. 

Solid state physicists are familiar with the free- and nearly free-electron models of simple metals [9]. The essence of those models is the fact that the effective potential seen by the conduction electrons in metals like Na, K, etc., is nearly constant through the volume of the metal. This is so because (a) the ion cores occupy only a small fraction of the atomic volume, and (b) the effective ionic potential is weak. Under these circumstances, a constant potential in the interior of the metal is a good approximation—even better if the metal is liquid. However, electrons cannot escape from the metal spontaneously in fact, the energy needed to extract one electron through the surface is called the work function. This means that the potential rises abruptly at the surface of the metal. If the piece of metal has microscopic dimensions and we assume for simplicity its form to be spherical - like a classical liquid drop, then the effective potential confining the valence electrons will be spherically symmetric, with a form intermediate between an isotropic harmonic oscillator and a square well [10]. These simple model potentials can already give an idea of the reason for the magic numbers the formation of electronic shells. 

It has to be kept in mind, however, that the plasma frequency A>p as given in eq. (4) is modified due to limited validity of the nearly free electron model. Interference of plasmon excitation with interband and intraband transitions may shift the plasma frequency in a complicated way, as discussed by Raether (1965). 

The important physical properties of simple metals and, in particular, the alkali metals can be understood in terms of a free electron model in which the most weakly bound electrons of the constituent atoms move freely throughout the volume of the metal (231). This is analogous to the free electron model for conjugated systems (365) where the electrons are assumed to be free to move along the bonds throughout the system under a potential field which is, in a first approximation, constant (the particle-in-a-box model). The free electron approach can be improved by replaeing the constant potential with a periodic potential to represent discrete atoms in the chain (365). This corresponds to the nearly free electron model (231) for treating electrons in a metal. 

Figure 2.14 Free electron in a one-dimensional box of length / as function of wave number k (a) Completely free electron model and (b) nearly free electron model, reflection of the electrons on the ions.

Each kj. represents one orbital of the sohd. If the length of the chain reaches typical numbers of atoms in macroscopic dimensions, a very large number of crystal orbitals are formed. On the energy scale a single orbital is no longer distinguishable a band is formed. A similar result was obtained with the nearly free electron model. 

Nearly free-electron model of metals. A gas of free electrons into which a lattice of positive ions is immersed. 

Figure 5.6 Scheme of the near-free electron model of simple metals. The white circles represent the Wigner-Seitz cells, in which the point-positive charges are located. The lattice of the cells Is immersed in a free-electron gas. 

Figure 6.9a-f illustrates a variety of the accepted band structure representations for nearly-free electron model. The Figure introduces the repeated-zone, extended-zone and reduced-zone images. The original free-electron parabola E = fi k Klme) is shown in Figure 6.9a. To leading order in the weak one-dimension periodic potential this curve remains correct except the value of k near the reciprocal lattice vector g. One can imagine that in this point the Bragg plane reflects the electron wave since the Bragg condition holds. Another free-electron parabola is centered at fe = g, and two parabolas are crossed each other at the...

We have seen in previous chapters that the good examples of metallic crystals are the alkali metals, which can be correctly described by the near-free electron model. The valence electrons in these metals are completely separated from their ion cores and form a nearly uniform gas. 

The transition metals are not describable by the conventional near-free electron model of the metalhc bond since the valence d electrons remain relatively tightly bound to their parent atoms forming unsaturated bondings with their neighbors. These d bondings are responsible for the structural and cohesive properties of transition metals. 

Equations (2.9) and (2.10) are valid for both undoped and doped semiconductors. They are, however, not valid when the Fermi level is less than 3kT away from either one of the band edges. Under these conditions, the semiconductor is degenerate, and exhibits near-metallic behavior. The relationships for the effective densities of states were derived from the (nearly) free electron model, and may not be entirely accurate for transition metal oxides. Despite these limitations, (2.9) and (2.10) are exceedingly useful for describing the behavior of semiconducting photoelectrodes. 

In the physics literature models that assume delocalized valence electrons are more common than those based on localized electrons such as described here. Examples of such models for solids are band models based on molecular orbitals (the so-called tight-binding method) or models that describe the electrons in metals as being in an almost flat potential (the nearly-free-electron model). While physicists traditionally think in terms of delocalized electrons, chemists feel more at ease with localized electrons, and almost all the bonding models discussed here are based on localized valence electrons. The dilemma between stationary eigenstates and molecular structure is less problematic in these models than in those for symmetry-adapted delocalized electrons. 

Phillip s starting point is the similarity of the observed absorption band of a-Ge and the absorption band calculated on the basis of Penn s model (Penn (1962)). This is a nearly free electron model isotropic in three dimensions. It has only one energy gap Eq appearing at the Fermi surface. Its value in the crystal can be estimated from the formula suggested by Heine and Jones (1969)... 

Fig. 3.72. The Fermi surface of hep rare earths according to the nearly free electron model (Kasuya, 1966).

Fig. 3.73. The effect of magnetic ordering on the generalized susceptibility function of the nearly free electron -model (Elliott and Wedgwood, 1964).

Exchange models based on the nearly-free electron model have re-... 

The properties of semiconductor electrodes and their differences compared with metal electrodes can be understood by examining their electronic structure. Semiconductors are unique in their electronic properties due to their band structure. The origin of the energy bands is generally discussed using band or zone theory, where the motion of a single electron in the crystal lattice is considered. It is assumed that there is no interaction between individual electrons or between the electrons and lattice points (the potential energy is zero). Therefore, the model is sometimes referred to as the nearly free electron model. 

Y content, if one assumes the relation of 7 h =- INne, which can be derived from the nearly free-electron model, holds. Here, N is the atomic density and n is the number of electrons per atom. The strongly attractive interaction between Al and R atoms suggests that the s- and p-electrons in Al hybridize with s- and d-electrons in R, leading to a decrease of the free electrons which are attributed to electrical conductivity. As a result, the increase of the number of Al-R pairs in Al-R amorphous alloys with increasing... 

Figure 4.13. Electronic states in the nearly free electron model for a 1-D chain with unit ceU length, a. The energy states in (a) represent solutions to the Schrodinger equation (via the Bloch theorem) with an infinitesimal potential. In contrast, solution (b) represents a finite lattice potential, which results in the formation of gaps between the parabolas at the BZ boundary. That is, there is no longer a continuum of states from the lowest to infinity, but regions where no allowed states may exist. Reproduced with permission from Hofmann, P. Solid State Physics An Introduction, Wiley New York, 2008. Copyright 2008 Wiley-VCH Verlag GmbH Co.

A zero-order theory of the Hall effect is easy to construct, and its results are well known. In a nearly-free-electron model, the Hall coefficient, is given as... 

We will next employ a slightly more elaborate model to argue that surface states are localized near the surface and have energies close to the Fermi level. We consider again a ID model but this time with a weak periodic potential, that is, a nearly-free-electron model. The weak periodic potential in the sample will be taken to have oidy two non-vanishing components, the constant term Fq and a term which... 

Figure 11.5. Illustration of the features of the nearly-free-electron model for surface states. Left the potential Viz) (thick black line), and wavefunction f-kiz) of a surface state with energy k, as functions of the variable z which is normal to the surface. The black dots represent the positions of ions in the model. Right the energy Cjt as a function of q, where q = k - G/2. The energy gap is 2 Vg at = 0. Imaginary values of giverise to the surface states.

B. Comparison of the Nearly Free Electron Model with the SSH-HMO Hamiltonian... 

The variation of a is dependent on the electron concentration to the power for Ce3 xSe4, Pr3 xSe4, and Nd3 xSe4, as expected from the nearly free electron model [12]. 

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