% LaTex input for the phonons in cubic bulk PbTiO$_{3}$ 
% at the border of the zone
% Version 0.0.1, 2010/02/17

\documentclass[aps,prb,superscriptaddress,onecolumn,floatfix,showpacs,amsmath]{revtex4-1}
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\marginparwidth 2.7in
\def\jjm#1{\marginpar{\small JJ: #1}}
\def\mhcm#1{\marginpar{\small MHC: #1}}
%\jjm{Are PACS numbers real, or just borrowed from a previous paper?}
%\usepackage{citesupernumber}

\begin{document}

\title{Phonons in cubic bulk PbTiO$_{3}$ with {\sc Siesta}}

\author{ Javier Junquera }
\affiliation{ Departamento de Ciencias de la Tierra y 
              F\'{\i}sica de la Materia Condensada, Universidad de Cantabria, 
              Avda. de los Castros s/n, 39005 Santander, Spain}
\date{\today}

\begin{abstract}
\end{abstract}

\pacs{73.30.+y,73.40.Rw,73.40.Qv,73.90.+f}
% 73.30.+y: Surface double layers, Schottky barriers, and work functions
% 73.40.Rw: Metal-insulator-metal structures
% 73.40.Qv: Metal-insulator-semiconductor structures
%           (including semiconductor-to-insulator)
% 73.90.+f: Other topics in electronic structure and electrical properties
%           of surfaces, interfaces, thin films, and low-dimensional structures
%           (Restricted to new topics in section 73)

 
\maketitle

\section{Phonons in cubic bulk PbTiO$_{3}$ at the border of the first
         Brillouin zone}
\label{sec:phonons-border}

 \begin{itemize}
    \item Monkhorst-Pack grid: $6 \times 6 \times 6$.
    \item Mesh cutoff: 1200 Ry 
    \item Force constant matrix computed in a $2 \times 2 \times 2$ 
          supercell in real space (40 atoms in the supercell).
    \item All the atoms in the supercell in real space 
          displaced along the three cartesian directions
          $\pm 0.03$ bohrs.
    \item Force constant matrix computed at the corresponding
          theoretical lattice constant:
          \begin{itemize}
             \item LDA lattice constant          = 7.36051 bohr = 3.895012 \AA.
             \item GGA-Wu Cohen lattice constant = 7.40679 bohr = 3.919505 \AA.
          \end{itemize}
 \end{itemize}

 \newpage

% \begin{table}
%    \caption[ ]{ Phonons at $\Gamma$. LO-TO splitting not considered. 
%                 Frequencies in cm$^{-1}$. 
%                 400 Ry of Mesh Cutoff. Displacement: 0.02 bohr
%               }
%    \begin{center}
%       \begin{tabular}{cccc}
%          \hline
%          \hline
%          Eigenvalue number                &
%          Degeneration                     &
%          LDA (Ceperley-Alder)             & 
%          GGA (Wu-Cohen)                   \\
%	  \hline
%          1                               &
%          triple                          &
%          -270                            &
%          -169                            \\
%          2                               &
%          triple                          &
%          0                               &
%          0                               \\
%          3                               &
%          triple                          &
%          85                              &
%          100                             \\
%          4                               &
%          triple                          &
%          222                             &
%          241                             \\
%          5                               &
%          triple                          &
%          529                             &
%          489                             \\
%          \hline
%          \hline
%       \end{tabular}
%    \end{center}
%    \label{table:phonongamma400}
% \end{table}
%
% \begin{table}
%    \caption[ ]{ Phonons at X. Frequencies in cm$^{-1}$.
%                 400 Ry of Mesh Cutoff. Displacement: 0.02 bohr
%               }
%    \begin{center}
%       \begin{tabular}{cccc}
%          \hline
%          \hline
%          Eigenvalue number                &
%          Degeneration                     &
%          LDA (Ceperley-Alder)             & 
%          GGA (Wu-Cohen)                   \\
%	  \hline
%          1                               &
%          doble                           &
%          -244                            &
%          -66                             \\
%          2                               &
%          doble                           &
%          64                              &
%          44                              \\
%          3                               &
%          single                          &
%          119                             &
%          111                             \\
%          4                               &
%          doble                           &
%          174                             &
%          171                             \\
%          5                               &
%          single                          &
%          215                             &
%          266                             \\
%          6                               &
%          single                          &
%          269                             &
%          278                             \\
%          7                               &
%          doble                           &
%          282                             &
%          288                             \\
%          8                               &
%          single                          &
%          465                             &
%          464                             \\
%          9                               &
%          doble                           &
%          508                             &
%          495                             \\
%          10                              &
%          single                          &
%          751                             &
%          713                             \\
%          \hline
%          \hline
%       \end{tabular}
%    \end{center}
%    \label{table:phononx400}
% \end{table}
%
% \begin{table}
%    \caption[ ]{ Phonons at M. Frequencies in cm$^{-1}$.
%                 Eigenfrequencies ordered according to
%                 the LDA results. 
%                 400 Ry of Mesh Cutoff. Displacement: 0.02 bohr
%               }
%    \begin{center}
%       \begin{tabular}{cccc}
%          \hline
%          \hline
%          Eigenvalue number                &
%          Degeneration                     &
%          LDA (Ceperley-Alder)             & 
%          GGA (Wu-Cohen)                   \\
%	  \hline
%          1                               &
%          single                          &
%          -242                            &
%          -47                             \\
%          2                               &
%          single                          &
%          -38                             &
%          41                              \\
%          3                               &
%          doble                           &
%          23                              &
%          34                              \\
%          4                               &
%          single                          &
%          72                              &
%          52                              \\
%          5                               &
%          doble                           &
%          206                             &
%          236                             \\
%          6                               &
%          doble                           &
%          304                             &
%          309                             \\
%          7                               &
%          doble                           &
%          358                             &
%          411                             \\
%          8                               &
%          single                          &
%          439                             &
%          416                             \\
%          9                               &
%          single                          &
%          474                             &
%          439                             \\
%          10                              &
%          single                          &
%          541                             &
%          488                             \\
%          11                              &
%          single                          &
%          810                             &
%          776                             \\
%          \hline
%          \hline
%       \end{tabular}
%    \end{center}
%    \label{table:phononm400}
% \end{table}
%
% \begin{table}
%    \caption[ ]{ Phonons at R. Frequencies in cm$^{-1}$.
%                 400 Ry of Mesh Cutoff. Displacement: 0.02 bohr
%               }
%    \begin{center}
%       \begin{tabular}{cccc}
%          \hline
%          \hline
%          Eigenvalue number                &
%          Degeneration                     &
%          LDA (Ceperley-Alder)             & 
%          GGA (Wu-Cohen)                   \\
%	  \hline
%          1                               &
%          triple                          &
%          -77                             &
%          -45                             \\
%          2                               &
%          triple                          &
%          76                              &
%          64                              \\
%          3                               &
%          triple                          &
%          317                             &
%          376                             \\
%          4                               &
%          triple                          &
%          374                             &
%          392                             \\
%          5                               &
%          doble                           &
%          511                             &
%          459                             \\
%          6                               &
%          single                          &
%          843                             &
%          810                             \\
%          \hline
%          \hline
%       \end{tabular}
%    \end{center}
%    \label{table:phononr400}
% \end{table}
%
% \begin{table}
%    \caption[ ]{ Phonons at $\Gamma$. LO-TO splitting not considered. 
%                 Frequencies in cm$^{-1}$.
%               }
%    \begin{center}
%       \begin{tabular}{cccccc}
%          \hline
%          \hline
%          Monkhorst-Pack mesh           &
%          Mesh Cutoff (Ry)              &
%          Atomic displacements (bohr)   &
%          Eigenvalue                    &
%          Degeneration                  &
%          LDA (Ceperley-Alder)          \\
%          \hline
%          $ 6 \times 6 \times 6 $       &
%          400 Ry                        &
%          0.02                          &
%          1                             &
%          triple                        &  
%          -270                          \\
%                                        &
%                                        &
%                                        &
%          2                             &
%          triple                        &  
%          0                             \\
%                                        &
%                                        &
%                                        &
%          3                             &
%          triple                        &  
%          85                            \\
%                                        &
%                                        &
%                                        &
%          4                             &
%          triple                        &  
%          222                           \\
%                                        &
%                                        &
%                                        &
%          5                             &
%          triple                        &  
%          529                           \\
%          \hline
%          $ 8 \times 8 \times 8 $       &
%          400 Ry                        &
%          0.02                          &
%          1                             &
%          triple                        &  
%          -273                          \\
%                                        &
%                                        &
%                                        &
%          2                             &
%          triple                        &  
%          0                             \\
%                                        &
%                                        &
%                                        &
%          3                             &
%          triple                        &  
%          86                            \\
%                                        &
%                                        &
%                                        &
%          4                             &
%          triple                        &  
%          221                           \\
%                                        &
%                                        &
%                                        &
%          5                             &
%          triple                        &  
%          529                           \\
%          \hline
%          $ 6 \times 6 \times 6 $       &
%          1200 Ry                       &
%          0.02                          &
%          1                             &
%          triple                        &  
%          -95                           \\
%                                        &
%                                        &
%                                        &
%          2                             &
%          triple                        &  
%          0                             \\
%                                        &
%                                        &
%                                        &
%          3                             &
%          triple                        &  
%          166                           \\
%                                        &
%                                        &
%                                        &
%          4                             &
%          triple                        &  
%          220                           \\
%                                        &
%                                        &
%                                        &
%          5                             &
%          triple                        &  
%          519                           \\
%          \hline
%          $ 6 \times 6 \times 6 $       &
%          400  Ry                       &
%          0.04                          &
%          1                             &
%          triple                        &  
%          -160                          \\
%                                        &
%                                        &
%                                        &
%          2                             &
%          triple                        &  
%          0                             \\
%                                        &
%                                        &
%                                        &
%          3                             &
%          triple                        &  
%          101                           \\
%                                        &
%                                        &
%                                        &
%          4                             &
%          triple                        &  
%          222                           \\
%                                        &
%                                        &
%                                        &
%          5                             &
%          triple                        &  
%          523                           \\
%          \hline
%          $ 6 \times 6 \times 6 $       &
%          1200  Ry                      &
%          0.04                          &
%          1                             &
%          triple                        &  
%          -131                          \\
%                                        &
%                                        &
%                                        &
%          2                             &
%          triple                        &  
%          0                             \\
%                                        &
%                                        &
%                                        &
%          3                             &
%          triple                        &  
%          122                           \\
%                                        &
%                                        &
%                                        &
%          4                             &
%          triple                        &  
%          220                           \\
%                                        &
%                                        &
%                                        &
%          5                             &
%          triple                        &  
%          518                           \\
%          \hline
%          \hline
%       \end{tabular}
%    \end{center}
%    \label{table:phononsstabilityresults}
% \end{table}

 \begin{table}
    \caption[ ]{ Phonons at $\Gamma$. LO-TO splitting not considered. 
                 Frequencies in cm$^{-1}$. 
                 1200 Ry of Mesh Cutoff. Displacement: 0.03 bohr
               }
    \begin{center}
       \begin{tabular}{cccc}
          \hline
          \hline
          Eigenvalue number                &
          Degeneration                     &
          LDA (Ceperley-Alder)             & 
          GGA (Wu-Cohen)                   \\
	  \hline
          1                               &
          triple                          &
          -112                            &
          -138                            \\
          2                               &
          triple                          &
          0                               &
          0                               \\
          3                               &
          triple                          &
          145                             &
          120                             \\
          4                               &
          triple                          &
          220                             &
          238                             \\
          5                               &
          triple                          &
          519                             &
          490                             \\
          \hline
          \hline
       \end{tabular}
    \end{center}
    \label{table:phonongamma1200}
 \end{table}

 \begin{table}
    \caption[ ]{ Phonons at X. Frequencies in cm$^{-1}$.
                 1200 Ry of Mesh Cutoff. Displacement: 0.03 bohr.
               }
    \begin{center}
       \begin{tabular}{cccc}
          \hline
          \hline
          Eigenvalue number                &
          Degeneration                     &
          LDA (Ceperley-Alder)             & 
          GGA (Wu-Cohen)                   \\
	  \hline
          1                               &
          doble                           &
          45                              &
          48                              \\
          2                               &
          single                          &
          110                             &
          112                             \\
          3                               &
          doble                           &
          140                             &
          90                              \\
          4                               &
          doble                           &
          170                             &
          171                             \\
          5                               &
          single                          &
          266                             &
          274                             \\
          6                               &
          doble                           &
          280                             &
          285                             \\
          7                               &
          single                          &
          289                             &
          276                             \\
          8                               &
          doble                           &
          495                             &
          465                             \\
          9                               &
          single                          &
          508                             &
          501                             \\
          10                              &
          single                          &
          740                             &
          714                             \\
          \hline
          \hline
       \end{tabular}
    \end{center}
    \label{table:phononx1200}
 \end{table}

 \begin{table}
    \caption[ ]{ Phonons at M. Frequencies in cm$^{-1}$.
                 Eigenfrequencies ordered according to
                 the LDA results. 
                 1200 Ry of Mesh Cutoff. Displacement: 0.03 bohr
               }
    \begin{center}
       \begin{tabular}{cccc}
          \hline
          \hline
          Eigenvalue number                &
          Degeneration                     &
          LDA (Ceperley-Alder)             & 
          GGA (Wu-Cohen)                   \\
	  \hline
          1                               &
          single                          &
          -50                             &
          9                               \\
          2                               &
          doble                           &
          42                              &
          43                              \\
          3                               &
          single                          &
          55                              &
          55                              \\
          4                               &
          single                          &
          150                             &
          105                             \\
          5                               &
          doble                           &
          233                             &
          237                             \\
          6                               &
          doble                           &
          302                             &
          306                             \\
          7                               &
          single                          &
          436                             &
          419                             \\
          8                               &
          doble                           &
          438                             &
          424                             \\
          9                               &
          single                          &
          458                             &
          437                             \\
          10                              &
          single                          &
          526                             &
          491                             \\
          11                              &
          single                          &
          800                             &
          778                             \\
          \hline
          \hline
       \end{tabular}
    \end{center}
    \label{table:phononm1200}
 \end{table}

 \begin{table}
    \caption[ ]{ Phonons at R. Frequencies in cm$^{-1}$.
                 1200 Ry of Mesh Cutoff. Displacement: 0.03 bohr
               }
    \begin{center}
       \begin{tabular}{cccc}
          \hline
          \hline
          Eigenvalue number                &
          Degeneration                     &
          LDA (Ceperley-Alder)             & 
          GGA (Wu-Cohen)                   \\
	  \hline
          1                               &
          triple                          &
          -84                             &
          -60                             \\
          2                               &
          triple                          &
          61                              &
          66                              \\
          3                               &
          triple                          &
          371                             &
          374                             \\
          4                               &
          triple                          &
          425                             &
          408                             \\
          5                               &
          doble                           &
          496                             &
          461                             \\
          6                               &
          single                          &
          833                             &
          811                             \\
          \hline
          \hline
       \end{tabular}
    \end{center}
    \label{table:phononr1200}
 \end{table}

\bibliographystyle{apsrev}
\bibliography{/home/javier/Biblio/misc,/home/javier/Biblio/ferro,/home/javier/Biblio/dft,/home/javier/Biblio/bandoffset}

\end{document}
