ABSTRACT
To deal with the spin and electronic properties of wurtzite IIInitride semiconductors and understand the specific features that differentiate them from zinc blende III-V materials, one has to know the energy spectrum. The energy spectrum gives us all necessary information about how electron spin is related to its momentum; and that is the key information we need in order to use the material in various spintronic applications. 1.1 Symmetry
Ga(Al,In)N crystallizes in two modifications: zinc blende and wurtzite. The crystal structure of the wurtzite GaN belongs to the space group P63mc (International notation) or C 6u4 (Schönflies notation). The unit cell is shown in Fig. 1.1.In the periodic lattice potential, the electron Hamiltonian is invariant to lattice translations, so the wave function should be an eigenfunction y(r) of the translation operator:( + ) = ( )exp(i ),y yr R r kR (1.1)where exp(ikR) is the eigenvalue of the translation operator, and R is the arbitrary lattice translation. This condition is the Wide Bandgap Semiconductor Spintronics Vladimir Litvinov Copyright © 2016 Pan Stanford Publishing Pte. Ltd. ISBN 978-981-4669-70-2 (Hardcover), 978-981-4669-71-9 (eBook) www.panstanford.com
consequence of symmetry only and it presents a definition of the wave vector. In an infinite crystal, the wave vector would be a continuous variable. Since we are dealing with a crystal of finite size, we have to impose boundary conditions on the wave function. This can be done in two ways. First, we may equate the wave function to zero outside the boundaries of the crystal. This would correspond to taking the surface effects into account. If we are not interested in finite-size (or surface) effects, there is a second option: We assume that the crystal comprises an infinite number of the periodically repeated parts of volume V (volume of a crystal) and then impose the Born-von Karman cyclic boundary conditions:
Figure 1.1 Unit cell of a GaN crystal. Large spheres represent Ga sites. ( + ) = ( ),iN r b ri(1.2) where bi are basis vectors of the Bravais lattice. From Eqs. (1.1) and (1.2)
exp( )= 12 ,= = 0, ±1, ...,ii i i i
i
m m L kL
k (1.3) where Li = biNi is the linear size of the crystal of volume V in the direction bi. Thus, the wave vector takes discrete values, so all integrals over the wave vectors that may appear in the theory should be replaced by summation over the discrete variable k.
