Generation and morphing of plasmons in graphene superlattices

Recent experimental studies on graphene on hexagonal boron nitride (hBN) have demonstrated that hBN is not only a passive substrate that ensures superb electronic properties of graphene's carriers, but that it actively modifies their massless Dirac fermion character through a periodic moiré potential. Here we present a theory of the plasmon excitation spectrum of massless Dirac fermions in a moiré superlattice. We demonstrate that graphene-hBN stacks offer a rich platform for plasmonics in which control of plasmon modes can occur not only via electrostatic gating but also by adjusting, e.g., the relative crystallographic alignment.

Figure 1

(a) shows the SBZ with its high-symmetry points and the first star of reciprocal lattice vectors $\pm {\mathit{g}}_1,\pm {\mathit{g}}_2$, and $\pm {\mathit{g}}_3$. Here, the $\tilde{\Gamma }$ point (i.e., the center of the SBZ) coincides with the $\nu =K,K^{\prime }$ points of the original graphene's Brillouin zone. (b) illustrates the concept of plasmon morphing. Two Dirac-band crossings are located at different points of the SBZ and are separated in energy. When the chemical potential $\mu$ lies at the position of the horizontal solid line, the system displays the plasmon [17, 18] of a 2D Dirac system with $n$-type doping (green shaded area). Upon increasing the chemical potential (dashed line), the $\tilde{\Gamma }$-point plasmon morphs into the $\tilde{K}/{\tilde{K}}^{\prime }$-point plasmon of a 2D Dirac system with $p$-type doping (orange shaded area).

Figure 2

Graphene-hBN superlattice minibands along the $\tilde{\Gamma }\text{−}\tilde{M}\text{−}\tilde{K}\text{−}\tilde{\Gamma }$ direction (twist angle $\varphi \simeq 0.03\mathrm{rad},\lambda \simeq 7\mathrm{nm}$). Thick lines denote the first two conduction minibands. All the numerical results in this figure have been obtained by including $\mathcal{N}=37$ RLVs $\left(\left|\mathit{G}\right|\le 3|{\mathit{g}}_1|\right)$ in Eq. (2). (a) refers to a moiré potential with $V_{\mathrm{s}}=30\mathrm{meV}$ and $V_{\Delta }=0$. (b) gives a 3D representation of the moiré superlattice miniband structure for the same parameters as in (a). (c) and (d) refer to a moiré potential with $V_{\mathrm{s}}=0$ and $V_{\Delta }=30\mathrm{meV}$. In (c) black solid (red dashed) lines label the miniband structure around the $K \left(K^{\prime }\right)$ principal valley.

Figure 3

A 2D density plot of the RPA loss function $L(\mathit{q},\omega )$ for the superlattice miniband structure in Fig. 2 and $\left|\mathit{q}\right|=0.007{\mathrm{nm}}^{-1}$. The vertical thin solid line denotes the bottom edge ${\varepsilon }_{\tilde{M},2,K}$ of the $n=2$ conduction miniband—see Fig. 2. The long-dashed line represents the chemical potential dependence of a DP with a Fermi velocity $v_{\mathrm{F}}$ equal to the bare value $v_{\mathrm{F}}\sim {10}^6\mathrm{m}/\mathrm{s}$. For chemical potentials above ${\varepsilon }_{\tilde{M},2,K}$, a satellite $\tilde{M}$-point plasmon is generated: Its chemical potential dependence (dotted line) follows that of a plasmon in a 2D parabolic-band electron gas. 1D cuts of this 2D density plot for different values of the chemical potential $\mu$ are reported in Ref. [28].

Figure 4

Same as in Fig. 3 but for the miniband structure shown in Fig. 2 and $\left|\mathit{q}\right|=0.021{\mathrm{nm}}^{-1}$. The long-dashed and dotted lines have the same physical meaning as in Fig. 3. The short-dashed line represents the chemical potential dependence of a $\tilde{K}$-point plasmon stemming from a satellite Dirac point. Note that ${\varepsilon }_{\tilde{K},1,K^{\prime }}$ coincides with ${\varepsilon }_{\tilde{M},2,K}$.

Figure 5

Wave vector and energy dependence of the loss function $L(\mathit{q},\omega )$ for $\mu =330\mathrm{meV}$. All the other parameters are as in Fig. 4. In this plot $L(\mathit{q},\omega )$ has been evaluated for nine values of $q$. The gray-shaded area in (b) illustrates the energy dependence of $L(\mathit{q},\omega )$ for $q=0.009{\mathrm{nm}}^{-1}$. Solid and dashed curves represent $\text{Re}\left[{\epsilon }_{\mathbf{0},\mathbf{0}}(\mathit{q},\omega )\right]$ and $\text{Im}\left[{\epsilon }_{\mathbf{0},\mathbf{0}}(\mathit{q},\omega )\right]$, respectively. The values of these functions can be inferred from the vertical axis on the right. While the sharp peak at $\hslash \omega \simeq 20\mathrm{meV}$ corresponds to a true zero of the macroscopic dielectric function ${\epsilon }_{\mathbf{0},\mathbf{0}}(\mathit{q},\omega )$, the broad peak at $\hslash \omega \simeq 100\mathrm{meV}$ does not.