Finite size effect of proton-conductivity of amorphous silicate thin films based on mesoscopic fluctuation of glass network

The finite size effect of proton conductivity of amorphous silicate thin films, a-M_0.1Si_0.9O_x (M = Al, Ga, Hf, Ti, Ta, and La), was investigated. The proton conductivity across films, σ, was measured in dry air by changing the thickness in the range of 10-1000 nm. σ of the films with M = Al, Ga, and Ta was elevated in a power law by decreasing thickness into less than a few hundred nanometers, and the increment was saturated at a thickness of several 10's of nanometers. On the other hand, σ of the films with M = Hf, Ti, and La was not related to the decrease of the thickness in the range of >10 nm. Thickness-dependent conductivity of the former could be numerically simulated by a percolative resistor network model that involves the randomly distributed array of two kinds of resistors R ₁ and R₂ (R₁ > R₂) in the form of a simple cubic-type lattice. High-resolution TEM clarified that a-M _0.1Si_0.9O_x films involved heterogeneous microstructures made of the condensed domain and the surrounding uncondensed matrix due to the fluctuation of glass networks on the nanometer scale. The condensed domain had a wormlike shape with an average length of several 10's of nanometers and performed the role of the proton conduction pathway penetrating through the poorly conducting matrix. It was concluded that the thickness-dependent conductivity could be identical to finite-size scaling of the percolative network of the interconnected domains in the nanometer range.