# And forms a tetrahedral system of covalent bonds with

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and forms a tetrahedral system of covalent bonds with four neighboring atoms, as indicated in the figure. Note: For instance, in silicon the 1 1 1 ( , , ) 4 4 4 atom is shared by neighboring atoms.

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2.57 Fall 2004 – Lecture 8 3 In certain crystals, such as GaAs, both the covalent and ionic bonding are important. Fundamentally, the covalent bonding force is still due to charge interaction. Unlike the van der Waals force in molecular crystals or the electrostatic force in ionic crystals, however, it is more difficult to write down simple expressions for the covalent crystals. In covalent bonds, electrons are preferentially concentrated in regions connecting the nucleus, leaving some regions in the crystal with low charge concentration. Metals and their associated metallic bonds can be considered as an extreme case of the covalent bonds, when the covalent bonds begin to overlap and all regions of the crystal get filled up with charges. In the case of total filling of the empty space, it becomes impossible to tell which electron belongs to which atom. This can be shown by the distribution of wavefunction ( ) r Ψ G , in which the probability * ( ) ( ) r r Ψ Ψ G G is more uniform in a metal. 3.1.4 Reciprocal lattice If a time function ( ) f t is periodic with a period of T (i.e. ( ) ( ) f t T f t + = ), its can be expanded into a Fourier series as, ( ) + = π + π = −∞ = ω ω −∞ = n t in n t in n n n n e ' b e ' a t T n 2 cos b t T n 2 sin a ) t ( f Here the angular frequency ω =2 π /T is the Fourier conjugate of the time periodicity such that e i ω T =1, which ensures that f(t) is periodic. A spatial function, f(x), with a periodicity a, ( ) ( ) f x f x a = + , can be similarly expanded into a Fourier series, ( ) + = −∞ = n x ink n x ink n x x e ' b e ' a ) x ( f where the wavevector, k x =2 π /a, is the Fourier conjugate to spatial periodicity a. a L 0 5 10 15 -4 -3 -2 -1 0 1 2 3 4 Normalized Wavevector 2 2 8 ma h E o = E / E 0 , E 0 = h 2 /(8ma 2 ) k / ( π /a)
2.57 Fall 2004 – Lecture 8 4 In the above figure, the electron energy dispersion shows a period of 2 π /a because of the periodic potential field ( ) ( ) u x a u x + = (recall the Bloch theorem). Note: (1) The discussed function f(x) can represent not only the charge density but also other periodically distributed properties. (2) The Born-von Karman boundary condition requires that the wave functions at the two end points be equal to each other. This results in 2 2 k L Na π π = = , not x 2 k a π = . (3) For personal interests, you may want to compare the

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