Planar forces in polar crystals

Once the electric fields caused by displacement of atomic planes in polar crystals are understood. It is easy to calculate the phonon dispersion ab initio, using the method explained on Ge in Section 5. Fig. 7.0.1 shows the displacement pattern employed it is essentially the same one as in Ge, except that twice as many calculations are now needed the Ga and As atoms have to be interchanged and the calculation repeated with the As-planes displaced. The equivalent linear chain is shown in Fig. 7.0.2 together with the definition of force constants to make the definition of k ) clear, the displaced plane (Ga or As ) is shown in Fig. 7.0.2 aSways at origin. The orientation of the +[100] direction is the same as used in Ge (Section 5.3). With two types of atoms in each unit cell the force constants need an additional label cation-cation or anion-anion but the labels cation-anion or anion-cation at the odd-neighbor k are not needed since k is the same as k, to any order in u - a 

The orientation used in the vibrational calculations on the linear chain is the one showed in Fig. 7.0.2a the cation plane is placed at the origin, and anion planes are at the sites 1 plane K =0 contains the atom at (000), the plane =-M is the one containing the site a/4(lll). The dynamical matrix is given by eq. (5.7.1) and treatment of enharmonic effects - which are small in the [100] direction - is as in Ge, Chapter 5.4. Even the conclusions about the spatial extent of forces turn out to be similar to Ge (Chapter 5.5), including the even/odd ii zig-zag convergence. The essential difference is in presence of a macroscopic electric field. 

When we attempt, starting from the pattern of Fig. 7.0.1, to determine the longitudinal force constants, a constant electric field will appear along the supercell - as explained in Section 6.2 (see Figs, 6.2.2a and 6.2.4). The pattern of Fig. 7.0.1 does not 

So far, repeating the displacement periodically has not caused any major inconvenience - except in the case where the supercell chosen was shorter than the range of forces this was, however, merely a quantitative problem, adding slightly, perhaps. 

The longitudinal dispersion plotted in Fig. 7.1.1 was calculated with forces extending to 3rd neighbors only. The corresponding force constants are displayed graphically in Fig. 

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