Accounting for the resistivity contribution of grain boundaries in metals: critical analysis of reported experimental and theoretical data for Ni and Cu

Abstract

In the present paper, reported literature data on the grain-size dependence of resistivity of Ni and Cu are critically evaluated by two conceptually different methods. One is the phenomenological approach of Andrews (Phys. Lett. 19: 558, 1965) according to which in a polycrystalline metal there is a resistivity contribution inversely proportional to the average grain diameter, the proportionality constant defined as the Andrews parameter A. The other method is the customary Mayadas–Shatzkes (MS) model (Phys Rev B 1: 1382, 1970) yielding a grain-boundary reflection coefficient R. During the analysis, special care was taken to rely only on data for which the surface scattering resistivity contribution was definitely negligibly small and the grain size was determined by direct microscopy imaging. By sorting out with this analysis the most reliable grain-size-dependent resistivity data for polycrystalline Ni and Cu metals with random grain boundaries, we have then derived the current best room-temperature values of the Andrews parameter A, the specific grain-boundary resistivity and the reflection coefficient R. We have also found a fairly good relation between the parameters A and R and compared the experimental values with their theoretical estimates reported in the literature. Then, the conceptual differences between the two approaches are discussed and the deficiencies of the MS model, especially in connection with the validity of Matthiessen’s rule, are highlighted. A major conclusion is that by the Andrews method one can derive a model-independent reliable parameter characterizing the grain-boundary contribution to the resistivity of metals.

Understanding Grain Boundary Electrical Resistivity in Cu: The Effect of Boundary Structure

“…The measured resistivities of the GBs (1–28·10 –12 Ωcm 2 ) fit the resistivity values obtained by localized electrical measurements of random high angle GBs in Cu thin films (20–40 × 10 –12 Ωcm 2 ) reported by ref . However, the values are higher by 1 order of magnitude than the values reported for Cu by macroscopic measurements and predicted by simulations (0.1–4 × 10 –12 Ωcm 2 ). ,,,,, This might arise from the way GB resistivity values are simulated with DFT, where a relatively low amount of atoms in a defect-free periodic structure is considered, whereas real GB structures are never defect free. ,,, Consequently, the calculated values only give lower bounds for the GB resistivity. In addition, our findings also overestimate the resistivity compared to macro-scale experiments.…”

“…Therefore, we assume that E gb and ΔV represent the deviation of a GB from the background crystalline potential. An increase in excess properties leads to an increase in the fluctuating atomic potential of a GB relative to the bulk, and consequently, a higher scattering potential. , …”

“…An increase in excess properties leads to an increase in the fluctuating atomic potential of a GB relative to the bulk, and consequently, a higher scattering potential. 13,17 To explore such a correlation between GB resistivity and its excess properties, we searched for GB structures with MD annealing simulations using an embedded atom method (EAM) potential 44 that has been successful in reproducing experimental GB structures. 28,31 For each of Σ21a, Σ7, Σ19b, and Σ3, we used both possible symmetric GB planes.…”

“…This creates a fluctuation in the periodic atomic potential from the adjacent crystals across the boundary, leading to electron scattering at the boundary by a potential wall. The magnitude of the potential wall is associated with the GB structure and its chemical bonding. , Moreover, the distinct atomic arrangements at the GB also locally change the density of states and electron density compared to the grain interior as confirmed by density functional theory (DFT) simulations. ,, However, its experimental observation is challenging because of the difficulties in isolating a specific GB and characterizing solely its resistivity. ,, Hence, cumulative scattering events on the different GBs blur out all details of the influence of GB type and character on resistivity. To overcome this challenge, there is a need to probe the electrical resistivity of an individual GB segment.…”

“…12,14,16 However, its experimental observation is challenging because of the difficulties in isolating a specific GB and characterizing solely its resistivity. 13,17,18 Hence, cumulative scattering events on the different GBs blur out all details of the influence of GB type and character on resistivity. To overcome this challenge, there is a need to probe the electrical resistivity of an individual GB segment.…”