Energy Bands in General

The following constitutes a skeletal review of the basic ideas which are common to the entire field of band structure, i.e., independent of the details of the techniques used or the substances studied (with the exception of aperiodic systems). The material will be quite familiar to many readers and is therefore covered in a cursory fashion. The standard textbooks of elementary solid-state theory contain fuller descriptions of many of the matters which are reviewed. 

Any function, such as a potential energy function, which has the periodicity defined by (aj, 33,33) can only have Fourier components associated with the following values of q, the variable in the reciprocal space of the Fourier transform  

This defines a reciprocal lattice with primitive vectors gi, g2, ga- The Wigner-Seitz construction, in reciprocal space, defines the Brillouin zone. 

Bloch s theorem states that wave functions i/ (r) can always be chosen to have the periodicity of the Bravais lattice apart from a single multiplicative factor of exp(ik r), i.e., 

The principal properties of the energy as a function of k are as follows. Within the Brillouin zone E(k) is a continuous function. It is, of course, a multiple-valued function in the reduced zone scheme. At a Brillouin zone plane the gradient of (k) must be in the plane, except in certain exceptional cases. Finally, the band structure must be symmetric under inversion, k — k. This is usually referred to as time-reversal symmetry, but for simple Hamiltonians without spin-orbit coupling it follows simply from complex conjugation of Schrodinger s equation. 

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