# Tuning of Graphene-Based Optical Devices Operating in the Near-Infrared
## 2 Graphene Optical Properties
### Graphene Conductivity
#### 2.1.1 Dependence of Graphene Complex Permittivity  
The analysis of the influence of the parameters on the complex permittivity was carried out on the basis of equation [5] at \(\lambda\) = 1.55 \(\upmu\)m. Figure 1 shows the real (_Re_) and the imaginary (_Im_) parts of the graphene complex permittivity when \(t_{G}\) = 0.34 nm, \(\mu_{c}\) = 0 \(\div\) 1 eV, \(\Gamma\) = 2e12 s\({}^{-1}\) and \(T\) = 300 K.  
works [5; 8; 9; 11; 13; 14; 18; 19; 20; 21; 24; 25; 26]. For example, values equal to 0.5 nm or 1 nm are used in numerical simulations to relax the mesh/grid (e.g., about 1 nm could correspond to a 3-layer graphene). Figure 2a,b show the proportional dependence of the imaginary part of the dielectric constant on the change in the graphene thickness and refractive index (at about \(\sim\)0.4 eV), respectively. (c) Relaxation rate/scattering time  
Another parameter of the Kubo model and graphene conductivity is the relaxation rate or scattering time (\(\Gamma=\tau^{-1}=\left(ev_{F}{}^{2}\right)/(\mu\mu_{c})\)[26], \(\mu\)-charge carrier mobilities and \(v_{F}\)-Fermi velocity (=106 m/s)), which significantly affects the imaginary part of the dielectric constant (as seen in Figure 3). It is also worth noting that the \(\hbar\Gamma\) is the relaxation energy that defines the quality of graphene (a shorter relaxation time corresponding to a lower quality of graphene, and \(\tau\)-\(\infty\) corresponds to an ideally defect-free graphene sheet) [5; 14; 21]. It should be noted that the Kubo model from [13] distinguishes between the relaxation rates associated with interband and intraband transitions. Figure 3: (a) Real (blue curves) and imaginary (red curves) parts of the permittivity, (b) refractive index and (c) extinction coefficient as a function of the chemical potential for different values of relaxation rate from various publications (\(\lambda=1.55\)\(\upmu\)m, \(t_{G}=0.34\) nm, \(\mu_{c}=0\div 1\) eV, \(\Gamma=5\)e11 \(\div\) 5e14 s\({}^{-1}\), \(T=300\) K). Figure 2: (a) Real (blue curves) and imaginary (red curves) parts of the permittivity, (b) refractive index and (c) extinction coefficient as a function of chemical potential for different GE thicknesses from various publications (\(\lambda=1.55\)\(\upmu\)m, \(t_{G}=0.34\div 1\) nm, \(\mu_{c}=0\div 1\) eV, \(\Gamma=8\)e13 s\({}^{-1}\), \(T=300\) K). Figure 3a shows that the imaginary part of the dielectric constant decreases significantly for the Fermi levels \(\mid\mu_{c}\mid<0.4\) eV from the change in the scattering rate. In this case, a strong dependence of absorption on \(\Gamma\) is observed, since in-band absorption becomes the dominant process here. In this regime, lower values for \(\Gamma\), corresponding to high carrier mobility, give lower absorption [8]. The transient process has already been described in the section on the chemical potential, but, in addition to it, it is worth noting that the final values of the relaxation time have a significant effect since there is significant absorption due to intraband scattering interactions [21]. Imperfections in the graphene layer induce density inhomogeneities, varying the optical conductivity across the graphene sheet [61]. The typical value of the \(\Gamma\) in fabricated graphene-based devices operating in the NIR range, varies from \(5\)e\(12\) s\({}^{-1}\) to \(1\)e\(14\) s\({}^{-1}\), with a minimum value of \(4\)e\(9\) s\({}^{-1}\)[18]. On the experimental side, a typical value is \(\Gamma=8\)e\(13\) s\({}^{-1}\)[9]. Figure 3b,c show the refractive index and the extinction coefficient following Equation (4). (d) Temperature  
One of the constituent parameters of the Kubo model and graphene conductivity is the temperature (in some publications the temperature is taken as the room temperature \(\sim\)300 K).