Anomalously high Na⁺ and low Li⁺ mobility in intercalated Na₂Ti₆O₁₃

We report an anomalous diffusion behavior in intercalated Na₂Ti₆O₁₃. Using first-principles calculations, the direct migration of inserted Na⁺ along the tunnel direction is predicted to have a barrier of 0.24-0.44 eV, while the migration of inserted Li⁺ along the tunnel direction has a barrier of 0.86-1.15 eV. Although Li⁺ can also diffuse along a zig-zag path in the tunnel, the barrier of 0.86-0.99 eV is still much higher than that for Na⁺. Our results surprisingly lead to the conclusion that the diffusion of larger Na⁺ is 4-8 orders of magnitude faster than Li⁺ in the same host lattice, and explain the experimentally observed exceptional rate capability of Na₂Ti₆O₁₃ as the Na-ion battery anode. The anomalous diffusion behavior is attributed to the geometric features of Na₂Ti₆O₁₃. For migration of Li⁺ it is necessary to weaken Li-O bonds and to overcome the repulsion between Li and host Na ions simultaneously, while for Na⁺ diffusion the improved Na-O bonding at the transition state partially compensates for the energy penalty from the repulsion of host Na ions.