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36 CHAPTER 3. FROM ATOMS TO SOLIDS Figure 3.2: Tetrahedral bonding in the diamond structure. The zincblende structure is the same but with two different atoms per unit cell Ionic semiconductors. In GaAs and cubic ZnS the total electron number from the pair of atoms satisfies the ”octet” rule, and they have the identical tetrahedral arrangement of diamond, but with the atoms alternating. This is called the zinclende structure. The cohesion in these crystals is now part ionic and part covalent. There is another locally tetrahedral arrangement called wurtzite which has a hexagonal lattice, favoured for more ionic systems. With increasing ionic components to the bonding, the structures change to reflect the ionicity: group IV Ge (diamond), III-V GaAs (Zincblende), II-VI ZnS (zincblende or wurtzite), II-VI CdSe (wurtzite), and I-VII NaCl (rocksalt). Metals Metals are generally characterised by a high electrical conductivity, arising because the electrons are relatively free to propagate through the solid. Close packing. Simple metals (e.g. alkalis like Na, and s-p bonded metals such as Mg and Al) usually are highly coordinated (i.e. fcc or hcp - 12 nearest neighbours, sometimes bcc - 8 nearest neigbours), since the proximity of many neighbouring atoms facilitates hopping between neighbours. Remember that the fermi energy of a free electron gas (i.e. the average kinetic energy per particle) is proportional to k 2 F ∝ a−2 ∝ n 2/3 (here a is the lattice constant and n the density; the average coulomb interaction of an electron in a solid with all the other electrons and V a(r) E a V b(r) E b Figure 3.3: A simple model of a diatomic molecule. The atomic hamiltonian is H i = T + V i (r), with T the kinetic energy −h¯ 2∇ 2 /2m and V i the potential. We keep just one energy level on each atom.
3.2. COMPLEX MATTER 37 the other ions is proportional to a −1 ∝ n 1/3 . Thus the higher the density, the larger the kinetic energy relative to the potential energy, and the more itinerant the electrons. 3 By having a high coordination number, one can have relatively large distances between neigbours - minimising the kinetic energy cost - in comparison to a loose-packed structure of the same density. Screening. Early schooling teaches one that a metal is an equipotential (i.e. no electric fields). We shall see later that this physics in fact extends down to scales of the screening length λ ≈ 0.1nm, i.e. about the atomic spacing (though it depends on density) - so that the effective interaction energy between two atoms in a metal is not Z 2 /R (Z the charge, R the separation), but Z 2 e −R/λ /R and the cohesion is weak. Trends across the periodic table. As an s-p shell is filled (e.g. Na,Mg,Al,Si) the ion core potential seen by the electrons grows. This makes the density of the metal tend to increase. Eventually, the preference on the right-hand side of the periodic table is for covalent semiconductor (Si, S) or insulating molecular (P, Cl) structures because the energy is lowered by making tightly bound directed bonds. Transition metals. Transition metals and their compounds involve both the outer s-p electrons as well as inner d-electrons in the binding. The d-electrons are more localised and often are spin-polarised in the 3d shell when they have a strong atomic character (magnetism will be discussed later in the course). For 4d and 5d transition metals, the d-orbitals are more strongly overlapping from atom to atom and this produces the high binding energy of metals like W (melting point 3700 K) in comparison to alkali metals like Cs (melting point 300 K). 3.2 Complex matter Simple metals, semiconductors, and insulators formed of the elements or binary compounds like GaAs are only the beginning of the study of materials. Periodic solids include limitless possibilities of chemical arrangements of atoms in compounds. Materials per se, are not perhaps so interesting to the physicist, but the remarkable feature of condensed matter is the wealth of physical properties that can be explored through novel arrangements of atoms. Many new materials, often with special physical properties, are discovered each year. Even for the element carbon, surely a familiar one, the fullerenes (e.g. C 60 ) and nanotubes (rolled up graphitic sheets) are recent discoveries. Transition metal oxides have been another rich source of discoveries (e.g. high temperature superconductors based on La 2 CuO 4 , and ferromagnetic metals based on LaMnO 3 ). f−shell electron metals sometimes produce remarkable electronic properties, with the electrons within them behaving as if their mass is 1000 times larger than the free electron mass. Such quantum fluid ground states (metals, exotic superconductors, and superfluids) are now a rich source of research activity. The study of artificial meta-materials becins in one sense with doped semiconductors (and especially layered heterostructures grown by molecular beam epitaxy or MBE), but this subject is expanding rapidly due to an influx of new tools in nanomanipulation and biological materials. Many materials are of course not crystalline and therefore not periodic. The physical description of complex and soft matter requires a separate course. 3 Note the contrast to classical matter, where solids are stabilised at higher density, and gases/liquids at lower density.
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