Chris Spencer Boston University Electrical and Computer Engineering 770

Graphene has garnered a lot of interest for its unique electronic properties and its potential applications. Graphene is a two-dimensional form of carbon in which atoms are arranged in a honeycomb lattice. It is a zero-gap semiconductor, this allows the ability to dope graphene to have high concentrations by applying a voltage. This property of electric gating is also used to control conductivity and the dielectric constant of graphene. There is extensive research on the many properties of this monolayer of carbon but I will focus on conductivty, surface waves supported by graphene, and research into applications with optical modulators.
There are two absorption properties that are involved in light-graphene interaction, interband and intraband absorption for small signals. This is described by the complex conductivity, $\sigma_g=\sigma_{intra}(\omega,\mu_c,\Gamma,T)+\sigma_{inter}(\omega,\mu_c,\Gamma,T)$, where $$\omega$$ is the angular frequency of light, $$\mu_c$$ is the chemical potential, $$\Gamma$$ is scattering rate, and $$T$$ is the temperature [1].Whats interesting is that the chemical potential can be tuned with electrical gating, thus the conductivity can be controlled by the same process. Chemical potential also controls what absorption process is occurring.Interband corresponding to absorption from the valence band to conduction band and intraband absoprtion corresponding to absoprtion from a semiconductor like optical property to a metal like optical property [2]. For incoming light with energy $$\hbar \omega$$, interband absoprtion dominates when $$\mu_c < \frac{\hbar \omega}{2}$$ and intraband dominates when $$\mu_c > \frac{\hbar \omega}{2}$$ and it is theoretically predicted that intraband absoprtion would be dominate when $$\mu_c \approx \frac{\hbar \omega}{1.67}$$ [2]. Zhaolin Lu et al investigates graphene conductivty at $$T=300 K$$ with a scattering rate $$\hbar \Gamma=5 meV$$. Figure 1 plots real and imaginary parts of conductivity vs chemical potential and the dielectric constant as a funciton of chemical potential. Notice the sensitivity of conductivity with little change in chemical potential. Here $$\epsilon_{eff}=1-\frac{\delta_g}{i\omega\epsilon_0\Delta}$$ where $$\Delta=0.7 nm$$ is the effective thickness of graphene. Notice the dip in in the magnitude of the effective dielectric constant, it is near zero at $$\mu_t$$. Physically this is the marking of the transition from a dielectric type graphene to a metal type graphene. The transition chemical potential is $$\mu_t=0.515 eV$$ for this experiment. Theoretical value from [1] predicts $$\mu_t=0.479 eV$$ which has about 7 $$\%$$ error. Since conductivity can be tuned by electrical gating so too can the effective dielectric constant.

The real and imaginary parts of graphene conductivity as a function of chemical potential at λ0=1550nm and the real and imaginary parts of the effective dielectric constant as a chemical potential.

Consider the types of waves that graphene can support, in hopes to integrate this into waveguides. Plasmons are high-frequency, collective density oscillations of an electron liquid and occur in many metals and semiconductors [3]. Surface plasmons are the collective oscillations of charges at the surface of plasmonic materials and SPs coupled with photons form composite particles of surface plasmon polaritons [4]. It is found that two types of electromagnetic surface waves can exist in graphene, TE and TM modes [4]. The TE mode should come as a surprise. It is know that SPPs can exist at an interface between a dielectric and a metal for TM modes but TE modes don’t exist since they don’t satisfy boundary conditions at the interface [5]. Visually this is shown in figure 2 for a TM mode at a metal dielectric interface[7]. It is predicted that TE modes exist in graphene, a mode that does not exist in systems with parabolic electron dispersion [6]. Further it is shown that TE modes lie in the range $$1.667 <\frac{\hbar\omega}{\mu}<2$$ [6]. TM modes are supported when $$0<\frac{\hbar\omega}{\mu}<1.667$$ [4]. Dispersion relations for these modes are given by $$k_{TM}=k_0\sqrt{1-(\frac{2}{\sigma \eta_0})^2}$$ and $$k_{TE}=k_0\sqrt{1-(\frac{\sigma\eta_0}{2})^2}$$ for an isolated graphene and $$\eta_0$$ is the impedance of free space [4].