1.2 Where the polarization comes from

Above about 1140 °C (congruent crystal) the lithium ions sit in the oxygen sheets and the niobium ions sit at the centres of their octahedra; the arrangement looks the same upside down, so it carries no net electric dipole P. On cooling through the Curie temperature both cations shift the same way along c relative to the oxygen frame. Every unit cell now holds a dipole pointing the same way: a built-in polarization, about 0.7 C per square metre of crystal face (0.71 or 0.80 C/m² conflict), that no applied field created P-comp.

The two possible directions, up and down c, are mirror images. A strong enough field pushes lithium back through its oxygen triangle and niobium through the middle of its octahedron, and the crystal's polar axis flips. That reversibility is what makes LiNbO₃ a ferroelectric, and what lets engineers write patterns of up and down regions into it (1.8, 1.9).

z, Å |                                                  
 +5.0-                                                  
     |                                                  
     |       O--------O-------------O                   
     |       | \    /   \         / |                   
 +4.0-       |  \  /     \       /  |                   
     |       |   \/    Nb \\    /  |                    
     |       |   / \        \  /   |     NbO6           
     |       |  /   \        \/    |                    
 +3.0-       | /     \\     / \\   |                    
     |       |/        \   /    \  |                    
     |       /          \ /      \|                     
     |      O------------O--------O                     
 +2.0-      |\\        /  \\     /|                     
     |      |  \     //     \   /  |                    
     |      |   \  //        \//   |                    
     |     |     \\          /\\    |    LiO6           
 +1.0-     |    // \        /   \   |                   
     |     |   /    \  Li  /     \  |                   
     |     | //      \\   /       \\ |                  
     |     |/          \ /          \|                  
  0.0-    O-------------O------------O   O3 face        
     |     \           / \         //|                  
     |     |\\        /   \\     //  |                  
     |     |  \     //      \   /    |                  
 −1.0-      |  \   /         \//     |                  
     |      |   \\/     ·    /\\     |   empty          
     |      ||   /\        //   \    |                  
     |       |  /  \     //      \  |                   
 −2.0-       |//    \\  /         \\|                   
     |       O--------O-------------O                   
     |                                                  
     |                                                  
                                                        
      along c from s = 0, O frame                       
    −1.0      −0.5        0       +0.5      +1.0        
      |----+----|----+----|----+----|----+----|         
 Nb                        ====>                 +0.275 
 Li                        =============>        +0.713 
 O                        |                       0.000 

     |                                        : 
 2858-  .                ....                .  
     |  :              ..    ..              :  
     |  :            ..        ..            :  
     |   .          .            .          .   
     |   :         .              .         :   
     |    .       .                .       .    
     |     .    ..                  ..    .     
    0-      ....                      ..@.      
     |                                          
      +---------------------------------------- 
             −1            0           +1     s 
U, K per 10-atom cell                           

Drag s from +1 to −1: niobium slides 0.550 Å through the middle of its cage, lithium threads its O₃ face and drops into the empty octahedron below, and the polarization reverses. Switch the frame: the same relative motion is shared differently among Li, Nb and O. At the congruent coercive field, 21 kV/mm, the field term splits the two minima by 229 K against a 2858 K barrier and removes the barrier only near 807 kV/mm (template coupling template). Real crystals do not switch cell by cell: reversal starts at nuclei on the surface and spreads by sideways motion of domain walls P-comp (1.8, 1.9).

Displacements from the centrosymmetric reference, computed from the Hsu et al. 1997 coordinates; relative to Nb at z = 0 the reference has Li at z = 1/4 and O layers at z = 1/12 + n/6 (R̅3c) derived. Values along +c in three frames derived:

Only the relative displacements are physical; the three frames show why quoted numbers differ. Volk & Wöhlecke 2008 quotes Li 44 pm from the oxygen plane and Nb 26–27 pm from the octahedron centre P-comp: the Nb value matches the O-fixed frame and the Li value matches the Nb-fixed frame derived. Total travel on reversal in the O-fixed frame: Nb 0.550 Å, Li 1.426 Å, through the O₃ triangle into the adjacent, previously empty octahedron derived.

Evaluated with P_s = 0.71 C/m². Bare polar faces never show this field: it is about 136 times the coercive field of congruent material (21 kV/mm P-comp, 1.8; 154 times with 0.80 C/m²), so surface charge from the surroundings or from internal conduction screens P_s almost completely; only changes of P_s (pyroelectric, piezoelectric) appear as charge derived.

Displacive or order-disorder. Weis and Gaylord class LiNbO₃ with displacive ferroelectrics such as BaTiO₃ P. Inbar and Cohen review conflicting experimental claims for both pictures and find, from first principles, a double well 2858 K deep, far deeper than kT_c, which they read as order-disorder character, driven by oxygen–lithium coupling and strong Nb d – O p hybridization, the same family of mechanism as in the perovskites P. Neutron diffraction at high temperature sees lithium distributed over split positions P-comp. The debate is open.

Charge density. A synchrotron deformation-density analysis computes P_s = 0.43 C/m² from Hirshfeld charges against 0.71 C/m² measured (0.80 C/m² elsewhere conflict), and reports electron density transferred from Nb toward the large O₃ triangle near Li P. In the refined structure that triangle is the face Li crosses on reversal, the largest O₃ triangle of the column derived. Point-charge estimates of P_s depend on the charge partition.

Why the switching animation is a cartoon at the atomic scale. The coercive field (about 21 kV/mm) does not remove a 2858 K per-cell barrier: with the template coupling of the figure it lowers the barrier to 2744 K (96.0%), and a homogeneous reversal would need about 807 kV/mm, 38 times E_c template. Real reversal proceeds by nucleation at surfaces and sideways motion of domain walls P-comp (1.8, 1.9).

How the figure is computed. The down structure is the refined up structure inverted through the centroid of the Nb O₆ octahedron; each atom is paired with the nearest same-element atom of the inverted structure, and positions move linearly, r(s) = r₀ + s (r_up − r₀), with s = 0 the centrosymmetric midpoint derived. At s = 0 Li lies in its O₃ face and Nb at its O₆ centroid (offsets below 0.001 Å); O moves only sideways, by at most 0.043 Å derived. Frames differ by a rigid shift along c. The energy is a template curve through the computed depth: