The Solid as a Many-Particle Problem
As mentioned in Sect. 1.1, we understand the solid as being composed of ions (nuclei and closed electron shells) and valence electrons. A more rigorous approach would start from nuclei and electrons, but a simple consideration of the spatial extension of
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As mentioned in Sect . 1.1, we understand the solid as being composed of ions (nucl ei and closed elect ron shells) and valence elect rons. A more rigorous approach would start from nuclei and elect rons, but a simple consideration of the spat ial extension of elect rons in different shells of t he isolated at oms shows immediate ly that t his is not necessary. The wave fun ctions of elect rons in inn er shells (the core elect rons) with binding energies of hundreds or thousa nds of eV exte nd over a distance mu ch smaller t han the lattice spacing in a solid, as visualized in Fig. 2.1. In fact , when the atoms are assembled int o the configuration of a cryst allattice (or likewise of a molecule, cluster , liquid...) it will be the outermost , weakly bound valence elect rons which first experience the pr esence of near est neighbors. They will rearran ge from their states in the isolated atoms into those which est ablish the chemical binding . Together with the elect rostat ic energy of the ion configurat ion, this defines the st abl e structure. Some t extbooks on Solid St ate Theory st art with a det ailed description of this st ruc t ure of cryst allin e solids (e.g. [4,7,9,11]) which is only bri efly rep eated here. Inst ead, we follow the approach of [5, 14, 21] with a pr esentation of the basic Hamiltonian , which defines the solid as a quantum-rnechanical many-body problem .
V(r), 'I'(r)
1- -- -- -
• r
- core electrons
Fig. 2.1. Schematic view of asolid: periodic pot enti al (dashed lin e) and wave functions of core and valence elect rons (solid lin es)
U. Rössler, Solid State Theory © Springer-Verlag Berlin Heidelberg 2004
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2. The Solid as Many-Particle Problem
The effect iveness of chemical binding dep ends on the overl ap of the elect ronic wave functions at neighboring lattice sit es and on t heir coordination number , Thus, met als prefer a close-packed structure, nam ely the bodycentered cubic (bcc) and face-centered cubic (fcc) lattic es, with delocalized elect rons acting as glue between the positively cha rged ions (m etallic binding) , while in (bin ar y) ionic crys t als, elect rons are tran sferred from the cat ion t o the anion to complet e their outer shells (ionic or heteropolar bin ding) and form lattices dom inated by elect rostatic int eraction (like the ro cksalt struct ure ). Rar e gases with closed shell configurations and lar ger molecules form crys talline solids due to t he weak van der Waal s1 forces and are stable only at low te mperature s. Elements of the fourth group of t he periodic t able shar e each of their four valence elect rons with four near est neighbors in dir ect ed covalent bonds (covalent or homopolar binding) , which results in t he diamond st ructure . A mixtur e of covalent and ionic binding, whereby the cont ribut ion of the latter increases with t he pol arity of the material , is typic al for the zinc blende structure realized in A3B5 , A 2B 6 , and A 1B7 compounds. A dom inant covalent binding is ty pical for semiconductors. In most cases, the dist
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