## 4 Modulation of Optical Bistability  
The graphene conductivity is a function of the momentum relaxation time \(\tau\), so \(\tau\) affects not only the transmittance, but also the bistable properties. Figure 5a provides the transmittance of light for different momentum relaxation time. We can see that there is a peak in each spectrum. As the momentum relaxation time is longer, the maximum of transmittance is larger. That is because the loss of light in graphene is inversely proportional to the momentum relaxation time. As the input light intensity is powerful enough, the nonlinear effect on the refractive index needs to be considered. Figure 5b gives the dependence of transmittance on the incident light intensity. For some larger \(\tau\), the slope of the transmittance curve can be negative with the increase of input intensity, which indicates that the bistable phenomena will be induced. At the same time, we can see that the maximum value of nonlinear transmittance is larger than that of linear transmittance, this could be attributed to the loss of graphene is reduced when the third-order nonlinear correction for the index of graphene is counted.  
Figure 5c shows the relationship between the output strength and the input strength, as the third-order nonlinear correction for the equivalent index of graphene is included, for different momentum relaxation times. For a smaller \(\tau\), the input-output relationship is linear within the given interval of input intensity. For a larger \(\tau\), the input-output curve is bistable. The most striking feature of bistability is that, as the input intensity of light enlarges, the slope of the curve changes from positive to negative and then to positive. Figure 5d demonstrates that the lower threshold and upper threshold change as the momentum relaxation time varies. It shows that the bistability occurs as the momentum relaxation time \(\tau>0.52\) ps. Afterwards, the lower threshold and upper threshold decrease with the increase of \(\tau\), but the width between the lower and upper thresholds increases. This shows that it can improve the three-order nonlinearity of graphene and switching discrimination through increasing the time constant.  
The input-output relationship of light intensity is dependent on the chemical potential in graphene. Figure 6a gives three input-output profiles of intensity for the chemical potential \(\mu=0.03\), \(0.06\), and \(0.09\) eV. The chemical potential of graphene can be modulated by chemical doping and gate voltages. One can see that the curve is non-bistable, as \(\mu=0.03\), while the curves are bistable as \(\mu=0.06\) and \(0.09\) eV. The bistable thresholds and the interval of thresholds are larger in bistabilities, as the chemical  
Figure 5: (**a**) Transmittance versus angular frequency for different momentum relaxation time \(\tau\)around P\({}_{7}\) in the transmission spectrum. (**b,c**) Transmittance and transmitted intensity varying with the incident intensity of light for different \(\tau\), respectively. (**d**) Upper and lower bistability threshold versus \(\tau\).  
potential is higher. For different given chemical potentials, Figure 6b shows the lower and upper thresholds of OB. One can see that, as the chemical potential \(\mu>\) 0.043 eV, the bistable profile could be exported. The thresholds and the interval of thresholds increases with the chemical potential. These phenomenon can be attributed to the fact that the nonlinear surface conductivity coefficient of graphene \(\sigma^{(3)}\) is inversely proportional to the chemical potential. The switch threshold can be adjusted flexibly though the chemical potential in graphene, as the optical bistability is applied to all-optical switches.