TheRg , Lg ,Rc , and Cc , which were determined in the previous section, are listed in Table I. The reflection coefficients for the equivalent circuit model were calculated using these parameters. Figure 3 shows the measured (solid lines) and calculated (dashed lines) reflection coefficients of the CPWs from 1 to 15 GHz on a smith chart. The red, blue, and green lines correspond to the data of the reflection coefficients for CPWs with the graphene lines of L =10, 30, and 50 μm. The points in Figure 3 are the measured and calculated reflection coefficients at 5, 10, and 15 GHz. As shown in Figure 3, the reflection coefficients depended on L . The calculated reflection coefficients for this model nearly agreed with the measured ones from 1 to 15 GHz. Therefore, this model can be applied for the analyzing the contact properties between metal and graphene up to 15 GHz. The phase angle of the reflection coefficients for the graphene lines of L =50 μm was smaller than that for graphene lines ofL = 10 μm. We previously reported that the phase constant\(\text{β\ }\left(\approx\frac{1}{2}\omega\mu_{0}\sigma\right)\ deg/mm\)of graphene is smaller than that of metal due to the lower conductivity\(\sigma\) of graphene [9]. Therefore, the \(\beta\) of the graphene line decreases with increasing L . Consequently, the contact impedance was correctly characterized by Rc andCc obtained from the TLM, Hall measurement, and estimation, as shown in Table I.
We presented the contact impedance at the interface between a metal and graphene from 1 to 15 GHz. As mention the above,Rc and Cc were 50 Ω and 2.2 nF, respectively. In Figure 4 (a), the impedance ofCc is plotted as a function of frequency. It should be that the the impedance of Cc is represented as \(|\frac{1}{\omega C_{c}}|\) and is more than two orders of magnitude lower than that of Rc from 1 to 15 GHz. It also should be noted that the contact impedance consists ofRc and Cc . Therefore, these results indicate that the current flows mainly not throughRc , but through Cc in the microwave band. In the DC band, houwever, most of the current flows through Rc because the impedance value ofCc is approximately infinite. Therefore, it is suggested that the current flow from a metal into graphene in the microwave band is more capacitive and efficient than that in the DC band. In Figure 4 (b), the ratio of power consumption and power storage in the microwave band Pmw , to the total power consumption in the DC band Pdc , which represents the degree of energy dissipation at the metal/graphene contact, is plotted as a function of Cc . ThePmw /Pdc ratio is expressed as\(\frac{P_{\text{mw}}}{P_{\text{dc}}}=\frac{|\frac{1}{\omega C_{c}}|}{R_{c}+|\frac{1}{\omega C_{c}}|}\times 100\ [\%]\). The black, blue, and red lines correspond to thePmw /Pdc ratio for 5, 10, and 15 GHz, respectively. ThePmw /Pdc ratio decreased with increasing frequency and incresing Cc . Therefore, in the microwave band, Cc is a critical factor, hence, a reduction in the impedance ofCc should be achieved for more efficient carrier injection from metal to graphene. These results indicate that a higherCc is preferred in designing the feeding structure of graphene-based microwave devices such as antennas and transmission lines. For example, a larger contact area leads to a higherCc because of increasing ST. Furthermore, increasing the n of graphene may provide a higherCc because it is represented byCq of graphene, which is proportional ton .
Conclusions
The contact properties at the interface between metal and graphene in the microwave band from 1 to 15 GHz were analyzed through experimental measurements and caluculation with an equivalent circuit model using lumped circuit components. The calculated reflection coefficients based on the lumped circuit components nearly agreed with the measured ones, which indicated that the equivalent circuit model can be applied to analyze the contact properties between metal and graphene up to 15 GHz. The impedance of Cc was more than two orders of magnitude lower than that of Rc . This indicates that most of the carrier injection from metal to graphene flows not through Rc but through Ccin the microwave band, which suggests that the current flow between metal and graphene in the microwave band is more capacitive and efficient than that in the DC band. Therefore, the interface between metal and graphene with high capacitance is preferred in designing the feeding structure of graphene-based microwave devices. This work provides a building block that is essential for designing optically transparent graphene-based antennas.