This program, loto, analyzes dipole-dipole interactions in the
systems with periodic boundary conditions. 3-dimensional bulks,
2-dimensional films and 1-dimensional wires can be treated.
loto can plot dispersion diagrams of optical phonons for the
dipolar systems, as shown in examples below.
loto is named after a mysterious physics phenomenon; the
LO-TO splitting (longitudinal and transverse optical phonon
splitting).
Figure 1: Schematic illustration of a dipole crystal.
Thin solid lines indicate unit cell boundary.
Thick dashed lines are super cell boundary (BvonK boundary condition).
Arrows Are dipoles.
I is index for dipoles in a unit cell,
u_I is dipole moment or strength,
r_I is vector to each dipole in a unit cell, and
R is vector to each unit cell in a super cell.
References
K. De'Bell and A. B. Maclsaac and J. P. Whitehead:
Rev. Mod. Phys. Vol.72, pp.225-257 (2000), equation (14).
G. T. Gao and X. C. Zeng and W. C. Wang: J. Chem. Phys. Vol.106, pp.3311-3317 (1997), Appendix A.
Xavier Gonze et al.: Phys. Rev. B Vol.50, pp.13035-13038 (1994).
Xavier Gonze and Changyol Lee: Phys. Rev. B Vol.55, pp.10355-10368 (1997).
R. Brout: Phys. Rev Vol.113, pp.43-44 (1959).
M. H. Cohen and F. Keffer: Phys. Rev. Vol.99, pp.1128-1134 (1955).
Figure 2: First Brillouin zone for simple cubic (sc) crystal.
Symmetric k-points and axes are indicated.
Figure 3: Dipole-dipole interaction energy of simple cubic dipole lattice.
The LO-TO splitting is not yet corrected at the Gamma point.
Figure 4: Dipole-dipole interaction energy of simple cubic dipole lattice.
You have to introduce the LO-TO splitting at the Gamma point by your hands.
Minimum is at the M point.
Figure 5: Most stable antiferro structure of sc dipole infinite crystal.
The structure is of the M point in the first Brillouin zone.
-up-up-up-up-up- rods and -down-down-down-down-down- rods
are forming a checkerboard configuration.
Figure 6: Frequency of optical phonon.
Crystal which has one dipole per primitive cell does not has acoustic phonon.
There are only 3 optical phonon branches.
Figure 7: Schematic illustration of longitudinal oscillation of a
dielectric thin film. Polarization P_z
perpendicular to the film causes surface charges P_z.
The surface charges result in
depolarization field E_d=-4 pi P_z in the film.
Figure 13: First Brillouin zone for 2-dimensional square crystal.
Symmetric k-points (Gamma, X, and M) and axes (Delta, Z, and Sigma) are indicated.
Figure 14: Dipole-dipole interaction energy of 2-dimensional square dipole lattice.
Minimum is at the X point.
Red solid lines are in-plane modes.
Green dashed line is out-of-plane mode.
Figure 15: It is known that the ground-state of
the infinite periodic dipole square lattice
forms a continuously degenerate manifold of antiferromagnetic states.
Figure 16: 2-dimensional triangular crystal (left) and its first Brillouin zone (right).
In the figure of triangular crystal, real-space axes a and b are indicated.
a* and b* are primitive axes in reciprocal space.
In the first Brillouin zone, symmetric k-points (Gamma, M, and K) and
axes (Sigma, T, and T') are indicated.
Figure 17: Minimum is at the Gamma point.
Therefore, ferro-state is the most stable structure for the 2-dimensional triangular crystal.
loto is distributed in the hope that
it will be useful, but WITHOUT ANY WARRANTY.
You can copy, modify and redistribute loto,
but only under the conditions described in
the GNU General Public License (the "GPL").
For more detail, see COPYING.
