## 3 Optical Bistability, Tristability and Multistability
One can see that the peaks in reflection spectra are just at the valleys in transmission spectra for a given chemical potential, that is, the maximum transmittance is the conjugate of the minimum reflectivity. For a shorter incident wavelength, interband transitions of electrons in graphene are dominant and graphene can be viewed as metals, so the loss of graphene is greater, and the transmittance is lower. As the incident wavelength increases, intraband transition dominates and the loss of light in graphene decreases. Meanwhile, most of the light has been transmitted. Interband transition only occurs as the incident photon satisfies the condition \(\hbar\omega_{1}>2|E_{F}|\). The electrons emitted by interband transition lead to  
Figure 4: (**a,b**) Transmission and reflection spectra for different chemical potentials of graphene, respectively. considerable loss of light. Therefore, with the increasing of the chemical potential, the lower limit of frequency of photons required for interband transition increases, which results in a shift in the \(R\) and \(T\) curves towards lower wavelengths, which could be utilized to modulate the reflectivity and transmittivity of light. The third-order nonlinearity, known as Kerr effect, needs to be considered as the incident intensity of light is strong enough. The transmittance, therefore, may change with the input intensity of light as a strong laser impinges. Figure 5a gives the nonlinear transmittance for different values of chemical potential. One can see that, for a given incident intensity, the nonlinear transmittance for \(\mu_{c}\) = 0.40 eV is larger than those for \(\mu_{c}\) = 0.45 and 0.50 eV. This characteristic results from the fact that the linear transmittance for the chemical potential, 0.40 eV, is significantly higher than the two others. With the same incident intensity of light, larger nonlinear transmittance results in stronger transmitted intensity. The negative value in the curve slope of the nonlinear transmittance indicates that optical bistability may be realized if given an appropriate incident intensity. Figure 5b provides the input-output relationship for different chemical potentials. It can be seen that, for \(\mu_{c}\) = 0.40 eV, two bistablities appear in the curve within a given input intensity interval. As the intensity of incident light increases, there are two upward jumps, namely I and III, while the incident light intensity decreases, there are two downward jumps, namely II and IV. This specialty can be applied to polymorphic switches. The upper and lower thresholds of bistability can be tuned by an external voltage on graphene. In order to explore the mechanism of formation of these jumps in the transmission spectra, we here investigate the relation between the incident wavelength and the linear surface conductivity of graphene. For three given different chemical potentials \(\mu_{c}\) = 0.40, 0.45 and 0.50 eV, Figure 6a gives the real part Re(\(\sigma_{g}\)) of the surface conductivity of graphene. For shorter incident wavelengths, Re(\(\sigma_{g}\)) approximates to 6 S and keeps constant as the wavelength increases. Otherwise, Re(\(\sigma_{g}\)) suddenly jumps downward as the wavelength is continuing to increase. The real value always is as low as zero by increasing the wavelength sequentially. We here have viewed graphene as a dielectric with an equivalent thickness, so the imaginary part in the refraction index of graphene is the counterpart of Re(\(\sigma_{g}\)) representing loss or gain of materials. Therefore, larger Re(\(\sigma_{g}\)) means greater loss and weaker transmitted intensity. Figure 5: **(a,b)** Transmittance of light and transmitted intensity versus incident intensity of light for different chemical potentials of graphene, respectively. Figure 6b provides the real part of surface conductivity \(\sigma_{g}\) in the parameter space. The chemical potential \(\mu_{c}\) in graphene and the incident wavelength compose of the parameter space.