Na Diffusion in Hard Carbon Studied with Positive Muon Spin Rotation and Relaxation

The diffusive nature of Na⁺ in Na-inserted hard carbon (C Na), which is the most common anode material for a Na-ion battery, was studied with a positive muon spin rotation and relaxation (μ⁺SR) technique in transverse, zero, and longitudinal magnetic fields (TF, ZF, and LF) at temperatures between 50 and 375 K, where TF (LF) denotes the applied magnetic field perpendicular (parallel) to the initial muon spin polarization. At temperatures above 150 K, TF-μ⁺SR measurements showed a distinct motional narrowing behavior, implying that Na⁺ begins to diffuse above 150 K. The presence of two different muon sites in C Na was confirmed with ZF- and LF-μ⁺SR measurements; one is in the Na-inserted graphene layer, and the other is in the Na-vacant graphene layer adjacent to the Na-inserted graphene layer. A systematic increase in the field fluctuation rate (ν) with increasing temperature also evidenced a thermally activated Na diffusion, particularly above 150 K. Assuming the two-dimensional diffusion of Na⁺ in the graphene layers, the self-diffusion coefficient of Na⁺ (D _Na ^J) at 300 K was estimated to be 2.5 × 10^-11 cm²/s with a thermal activation energy of 39(7) meV.