Plane-Wave Expansion - The Free-Electron Models

1 Plane-Wave Expansion - The Free-Electron Models 

Soon afterwards, Hendrik Lorentz treated the free electrons as classical distinguishable particles that obey the MaxweU-Boltzmann distribution laws (Lorentz, 1904-1905). 

Further development of Sommerfeld s theory of metals would extend well outside the intended scope of this textbook. The interested reader may refer to any of several books for this (e.g. Seitz, 1940). Rather, this book will discuss the band approximation based upon the Bloch scheme. In the Bloch scheme, Sommerfeld s model corresponds to an empty lattice, in which the electronic Hamiltonian contains only the electron kinetic-energy term. The lattice potential is assumed constant, and taken to be zero, without any loss of generality. The solutions of the time-independent Schrodinger equation in this case can be written as simple plane waves, = exp[/A r]. As the wave function does not change if one adds an arbitrary reciprocal-lattice vector, G, to the wave vector, k, BZ symmetry may be superimposed on the plane waves to reduce the number of wave vectors that must be considered  

Note that the periodic Uk(r) function that appears in E q. 4.14, is absent since the periodic potential V r) is assumed to be infinitesimally small (i.e. in the limit V r) 0). In the free-electron model, the conduction electrons are regarded as free to move throughout 

From the form of Eq. 4.30, it is seen that 8 is a simple parabolic function of k [s(A ) = e k + G)]. The band structure describes this dependence of e k) on k, and it is an n + dimensional quantity, where n is the number dimensionality of the crystal. To visualize it, e k) is plotted along particular projectories between high-symmetry points. Hence, for a one-dimensional crystal, the band stmcture will consist of a single parabola in the free-electron approximation. The parabola shows all the degenerate (positive and negative) values for k. For three-dimensional crystals, a single paraboloid is obtained. 

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