## 3 Results and Discussion
### Extraction of Schottky Parameters
#### 3.3.2 Method 2: d\(V\)/d(\(ln\)) vs. \(J\) and H(\(J\)) vs. \(J\)
The IV characteristic of the Schottky barrier diode is greatly influenced by metal-semiconductor interface inhomogeneity, which consequently affects the device performance. From Figure 7c and Table 1, we can see that with higher temperature, \(\phi_{B}\) increases for both GNW and TiSi\({}_{2}\) TMBS diodes. This characteristic is also reported in several papers with different types of metal and semiconductor materials used in the experiments [39; 40; 41; 42; 43; 32; 35; 33; 34]. The change in \(\phi_{B}\) value can be explained by [PERSON]'s model of barrier inhomogeneity [44]. The barrier inhomogeneity could be due to defects, uneven interface, inhomogeneity in thickness, or presence of insulating patches at the metal-semiconductor interface [45; 46; 47; 35; 41; 48]. At low temperature, the electrons do not have sufficient energy to surmount the high barrier, however, due to barrier inhomogeneity, current will flow through an area with lower barrier height. With increasing temperature, more and more electrons gain sufficient energy to surmount the higher barrier. As a result, the effective barrier height will increase as the temperature increases. The ideality factor, \(n\), indicates the conformity of a diode to pure thermionic emission. The \(n\) of the GNW diode in this study is higher than unity, but it shows Schottky diode rectifier behavior. It is well-known that graphene on silicon Schottky diodes has a high ideality factor. The values of \(\phi_{B}\) and \(n\) reported by other researchers are in the range of 0.41-0.72 and 1.01-7.69 eV, respectively [49; 50; 51; 52; 53]. The high \(n\) at low temperature is potentially attributed to other transport mechanisms such as generation-recombination of charges in the depletion layer, tunneling of current through the barrier layer, image force lowering and thermionic field emission [31; 32; 33; 50], and lateral distribution of barrier height inhomogeneity besides thermionic charge transport. As temperature increases, the value of \(n\) decreases and reaches unity. This behavior indicates that thermionic emission becomes the dominant carrier transport mechanism at a higher temperature [44; 50]. Figure 7: GNW-TMBS diode plots to extract Schottky parameters. (**a**) \(\mathrm{d}V/\mathrm{d}ln(\mathrm{\textit{I}})\) vs. \(\mathrm{\textit{J}}\) and (**b**) \(\mathrm{H}(\mathrm{\textit{J}})\) vs. \(\mathrm{\textit{J}}\). (**c**) Temperature dependence of ideality factor and barrier height for GNW and TiSi\({}_{2}\) based on [PERSON]’s method. The barrier height and ideality factor for GNW showed stable values in the range of 298-323 K, and started to change at 373 K. The change of these values indicates that the thermionic emission mechanism started to dominate the current transport mechanism at high temperature. From the I-V plot in Figure 5a, we can see that the current at the forward region is linear at voltage \(\leq\)0.5 V and starts to deviate from linearity due to series resistance, \(R_{s}\). The double bumps that were normally due to other transport mechanisms such as generation-recombination, thermionic field emission, and field emission, were not observed in this case [26; 54; 55]. This is also an indication that thermionic emission starts to dominate at the temperature range tested. At low temperature, two possible mechanisms are responsible for the current flowing through the Schottky junction. The first one is quantum tunneling, which can occur in both forward and reverse bias direction. The mechanism is known as field emission (FE) and if the temperature causes the probability of tunneling to increase, it is known as thermionic field emission (TFE). Quantum tunneling, however, occurs in heavily doped semiconductors with an extremely thin potential barrier, which is not the case for this study. Another mechanism is the generation-recombination of electron-hole pairs in the depletion region that can contribute to the main current component.