Simultaneous Extraction of the Grain Size, Single-Crystalline Grain Sheet Resistance, and Grain Boundary Resistivity of Polycrystalline Monolayer Graphene

Abstract

The electrical properties of polycrystalline graphene grown by chemical vapor deposition (CVD) are determined by grain-related parameters-average grain size, single-crystalline grain sheet resistance, and grain boundary (GB) resistivity. However, extracting these parameters still remains challenging because of the difficulty in observing graphene GBs and decoupling the grain sheet resistance and GB resistivity. In this work, we developed an electrical characterization method that can extract the average grain size, single-crystalline grain sheet resistance, and GB resistivity simultaneously. We observed that the material property, graphene sheet resistance, could depend on the device dimension and developed an analytical resistance model based on the cumulative distribution function of the gamma distribution, explaining the effect of the GB density and distribution in the graphene channel. We applied this model to CVD-grown monolayer graphene by characterizing transmission-line model patterns and simultaneously extracted the average grain size (~5.95 μm), single-crystalline grain sheet resistance (~321 Ω/sq), and GB resistivity (~18.16 kΩ-μm) of the CVD-graphene layer. The extracted values agreed well with those obtained from scanning electron microscopy images of ultraviolet/ozone-treated GBs and the electrical characterization of graphene devices with sub-micrometer channel lengths.

Keywords: CVD graphene; GB distribution; grain boundary (GB); grain size; polycrystalline; sheet resistance; single-crystalline grain; transmission-line model measurement.

Figure 1

Parallel-resistance model for theoretical computation of the sheet resistance depending on the channel dimension. (a) A schematic of the electrical device with a poly-graphene channel simulated using the VT method. For poly-graphene resistance modeling, the channel width is divided into extremely narrow elements (∆W_ch = W_ch/10⁵). The resistance of the poly-graphene channel is calculated from the parallel connection of the divided elements. (b) The calculated channel resistance as a function of channel length for polycrystalline graphene (with an average grain size of 5 μm) simulated using the VT method. Inset: the calculated channel resistances in relatively short channels (denoted by the dashed box), which shows that the slope (dR_ch/dL_ch) varies with the channel length. (c) The calculated channel sheet resistance as a function of the channel length. Inset: the calculated channel sheet resistances in relatively short channels (denoted by the dashed box), which shows that the sheet resistance decreases significantly as the channel length approaches the average grain size.

Figure 2

Histogram distributions of the proportion of the GB number within the channel region when the channel length is (a) less than the average grain size, (b) equal to the average grain size, and (c) greater than the average grain size. Each distribution was evaluated by counting the GB number within narrow width elements divided into 10⁵. The black dashed lines in (a–c) indicate the locations where the number of GBs changes. For all channel lengths, the envelope of the proportional distribution of the number of GBs follows the continuous probability density function of the gamma distribution (gamma PDF), and the discrete proportion distribution of the GB number can be estimated from the cumulative distribution function of the gamma distribution (gamma CDF) with an accuracy greater than 98%.

Figure 3

GB distribution-based analytical resistance model for the simultaneous extraction of the grain parameters (the average grain size, grain sheet resistance, GB resistivity) of polycrystalline graphene. (a) Considering the proportion distribution of the GB number estimated from the gamma CDF, the divided width elements (ΔW_ch) can be grouped and rearranged by the number of GBs. Based on the rearranged width elements (W_ch,n), a sheet-resistance model composed of the three-grain parameters can be induced as a function of the channel length. (b) The developed analytical resistance model was used to fit the calculated channel sheet resistance as a function of the channel length. The sheet-resistance dependence on the channel length can be estimated with a high fitting accuracy greater than 99.98% by adjusting three fit (i.e., grain) parameters, from which the three-grain parameters can be extracted from the analytical resistance model fitted to the R_sh–L_ch curve.

Figure 4

Experimental verification of the GB distribution-based analytical resistance model and parameter extraction method. (a) OM images of fabricated TLM patterns comprising the CVD-graphene FETs with varying channel lengths (L_ch of 2–100 μm). (b) Measured channel sheet resistance as a function of channel length and fitting result using the analytical resistance model. The three-grain parameters extracted from the fitted model are ~5.95 μm (for the average grain size), ~321 Ω/sq (for the average grain sheet resistance), and ~18.16 kΩ·μm (for the average GB resistivity). (c) The representative SEM image of CVD graphene grown on a Cu foil, with UV/ozone-treated GBs highlighted in yellow. The average grain size estimated from 376 grains is 5.88 ± 1.5 μm. (d) OM and SEM image of a fabricated TLM pattern comprising the graphene FETs with sub-micrometer channel lengths (L_ch of 0.18–0.75 μm). (e) The measured width-normalized channel resistance as a function of the channel length, in which the linear slope indicates the sheet resistance of single-crystalline grains due to the extremely low probability of the presence of GBs within the short-channel regions. Inset: the histogram distribution of the number of GBs when the channel length is significantly smaller than the average grain size (L_ch = 0.18 μm and l_G = 5 μm). (f) Summary of experimental results for the GB resistivity reported in the literature [4,5,6,25,26,27,28], where all represented resistivity values were extracted at the charge neutrality point. This summary plot shows that the GB resistivity extracted in this study falls within the range of the reported resistivity values.