Efficient Nonlinear Generation of THz Plasmons in Graphene and Topological Insulators

Surface plasmons in graphene may provide an attractive alternative to noble-metal plasmons due to their tighter confinement, peculiar dispersion, and longer propagation distance. We present theoretical studies of the nonlinear difference frequency generation (DFG) of terahertz surface plasmon modes supported by two-dimensional layers of massless Dirac electrons, which includes graphene and surface states in topological insulators. Our results demonstrate strong enhancement of the DFG efficiency near the plasmon resonance and the feasibility of phase-matched nonlinear generation of plasmons over a broad range of frequencies.

Figure 1

(a) Dispersion curve of a surface plasmon mode in monolayer graphene for $E_F=50\mathrm{meV}$. The dielectric constant ${\epsilon }_2=4$ corresponds to ${\mathrm{SiO}}_2$ in the THz region. (b) Dispersion curves for symmetric (${\omega }_{+}$) and antisymmetric (${\omega }_{-}$) plasmon modes in a thin TI film forming a double layer with a spacer thickness $d=6\mathrm{nm}$; $E_F=100\mathrm{meV}$. Electric field of the symmetric plasmon mode is shown in the inset. The value of ${\epsilon }_2=10$ typical for semiconductors was assumed. Shaded area is the Landau damping region.

Figure 2

(a) Geometry of the DFG process. Two pump fields at frequencies ${\omega }_a$ and ${\omega }_b$ incident at angles ${\theta }_a$ and ${\theta }_b$ on graphene/TI placed on a substrate generate a highly confined surface plasmon field $E_{\text{pl}}$ at their difference frequency and in-plane wave vector $q=q_b-q_a$. (b) Elementary three-wave-mixing processes involving two photons and a plasmon coupled to interband and intraband transitions, respectively. Gray shading indicates filled electron states.

Figure 3

(a) Magnitude of the 2D ${\chi }_{xxx}^{(2)}$ as a function of Fermi energy for a fixed incidence angle ${\theta }_a={\theta }_b=\theta =60°$ and fixed sum of the incident pump frequencies ${\omega }_a+{\omega }_b=400\mathrm{meV}$. The red line is the generated plasmon frequency which satisfies the phase-matching condition and energy conservation at each $E_F$. (b) The DFG efficiency $\eta =(I_{\text{pl}}/I_a{I}_b)$ and incidence angle $\theta$ as a function of the plasmon frequency under the conditions of frequency and phase matching. The sum of the incident pump frequencies is fixed to ${\omega }_a+{\omega }_b=400\mathrm{meV}$.

Figure 4

(a) Integrated waveguide geometry of DFG of surface plasmons by counterpropagating TM modes. Profiles of $B_y$ and $E_x$ field components are indicated in blue dashed and red lines, respectively. (b) Phase-matched plasmon frequency vs wavelength of the fundamental TM mode for one of the pump fields in a $1\text{−}\mu \mathrm{m}$ thick Si waveguide, for two values of Fermi energy. (c) DFG efficiency of surface plasmons for $E_F=50\mathrm{meV}$ and two values of the relaxation rate.