Infrared Topological Plasmons in Graphene

TL;DR

This paper introduces a graphene-based plasmonic system with topologically protected one-way edge states at infrared frequencies, enabling robust, defect-immune plasmonic devices through engineered band topology and magnetic field effects.

Contribution

It presents a novel design of periodically patterned graphene supporting topological edge states at infrared frequencies, demonstrating a practical high-frequency topological plasmonic platform.

Findings

Supports topologically nontrivial bandgaps up to tens of terahertz

Hosts topologically protected one-way edge plasmons

Edge states are immune to backscattering from defects

Abstract

We propose a two-dimensional plasmonic platform - periodically patterned monolayer graphene - which hosts topological one-way edge states operable up to infrared frequencies. We classify the band topology of this plasmonic system under time-reversal-symmetry breaking induced by a static magnetic field. At finite doping, the system supports topologically nontrivial bandgaps with mid-gap frequencies up to tens of terahertz. By the bulk-edge correspondence, these bandgaps host topologically protected one-way edge plasmons, which are immune to backscattering from structural defects and subject only to intrinsic material and radiation loss. Our findings reveal a promising approach to engineer topologically robust chiral plasmonic devices and demonstrate a realistic example of high-frequency topological edge state.

Equations

$$\hat{\mathcal{H}}{\mathbf{U}}_{\nu }={\omega }_{\nu }{\mathbf{U}}_{\nu },$$

$$\text{with}{\mathbf{U}}_{\nu }\equiv \left(\begin{array}{c}{\omega }_{\text{f}}\Phi \\ \mathbf{J}\end{array}\right)\text{and}\hat{\mathcal{H}}\equiv \left(\begin{array}{cc}0& {\omega }_{\text{f}}\hat{V}{\hat{\mathbf{p}}}^{\text{T}}\\ \alpha \hat{\mathbf{p}}& {\omega }_{\text{c}}{\sigma }_2\end{array}\right).$$

$${\bar{C}}^{(n)}\equiv \sum _{n^{\prime }=-\infty }^{n-1}{C}^{(n^{\prime })}=-\mathrm{sgn}B+\sum _{n^{\prime }=1}^{n-1}{C}^{(n^{\prime })},$$

Full text

Time-reversal-symmetry $(\mathcal{T})$ breaking, a necessary condition for achieving quantum Hall phases Klitzing et al. (1980); Haldane (1988), has now been successfully implemented in several bosonic systems, as illustrated by the experimental observation of topologically protected one-way edge transport of photons Wang et al. (2009); Skirlo et al. (2015) and phonons Nash et al. (2015). More generally, two-dimensional (2D) $\mathcal{T}$-broken topological bosonic phases have been proposed in a range of bosonic phases, spanning photons Haldane and Raghu (2008), phonons Prodan and Prodan (2009); Wang et al. (2015), magnons Shindou et al. (2013), excitons Yuen-Zhou et al. (2014), and polaritons Karzig et al. (2015). The operating frequency of these systems is typically small, however – far below $\mathrm{T}\mathrm{H}\mathrm{z}$ – limited by the spectral range of the $\mathcal{T}$-breaking mechanism. For example, the gyromagnetic effect employed in topological photonic crystals is limited by the Larmor frequency of the underlying ferrimagnetic resonance, on the order of tens of $\mathrm{G}\mathrm{H}\mathrm{z}$ Wang et al. (2009). In phononic realizations, the attainable gyrational frequencies limit operation further still, to the range of $\mathrm{k}\mathrm{H}\mathrm{z}$ Fleury et al. (2014). Towards optical frequencies, proposals of dynamic index modulation Fang et al. (2012) and optomechanical coupling Hafezi and Rabl (2012) are promising but experimentally challenging to scale to multiple coupled elements Tzuang et al. (2014); Shen et al. (2016); Fang et al. (2016).

Recently, Jin et al. (2016) pointed out that the well-known magnetoplasmons of uniform 2D electron gases Mast et al. (1985); Volkov and Mikhailov (1988) constitute an example of a topologically nontrivial bosonic phase hosting unidirectional edge states. However, as the topological gap exists only below the cyclotron frequency $\omega_{\text{c}}$, the spectral operation remains limited to low frequencies. In this letter, we show that by suitably engineering the plasmonic band structure of a periodically nanostructured 2D monolayer graphene, see Fig. 1(a), the operation frequency of topological plasmons can be raised dramatically, to tens of $\mathrm{T}\mathrm{H}\mathrm{z}$, while maintaining large gap–midgap ratios even under modest $\mathbf{B}$-fields. Bridging ultrafast electronics and infrared topological photonics, the proposed platform can be seamlessly integrated with well-established CMOS technology, allowing dynamically gate-tunable topological states across a broad spectral range.

Graphene distinguishes itself as an ideal platform for topological plasmonics in three key aspects: first, it supports large, tunable carrier densities $n\!\sim\!${10}^{11}{10}^{14}\text{,}{\mathrm{cm}}^{-2} Efetov and Kim ([2010](#bib.bib21)); Liu *et al.* ([2011](#bib.bib22)); Fang *et al.* ([2014](#bib.bib23)), or equivalently, large, tunable Fermi energies $E_{\scriptscriptstyle\mathrm{F}}=\hbar v_{\scriptscriptstyle\mathrm{F}}\sqrt{\pi n}$ (Fermi velocity, $v_{\scriptscriptstyle\mathrm{F}}\approx$9.1\text{\times}{10}^{7}\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1} Note (1)); second, it exhibits an ultrasmall, tunable Drude mass $m^{*}\equiv E_{\scriptscriptstyle\mathrm{F}}/v_{\scriptscriptstyle\mathrm{F}}^{2}$ (e.g. at $E_{\scriptscriptstyle\mathrm{F}}=$0.2\text{,}\mathrm{eV}$$, $m^{*}/m_{\text{e}}\approx 4\%$), allowing ultrahigh cyclotron frequencies $\omega_{\text{c}}\equiv eB/cm^{*}=eBv_{\scriptscriptstyle\mathrm{F}}^{2}/cE_{\scriptscriptstyle\mathrm{F}}$ up to the THz range; and third, high-quality graphene can exhibit exceptionally long intrinsic relaxation times $1/\gamma$, extending into the picosecond range Du et al. (2008); Bolotin et al. (2008). These properties enable topological plasmons of unprecedentedly high frequency, short wavelength, long propagation, and large topological bandgaps.

The plasmonic properties of a general graphene domain $\mathbf{r}\in\Omega\subseteq\mathbb{R}^{2}$ under an external magnetic field $\mathbf{B}=B\hat{\mathbf{z}}$ is described by a linear eigenvalue problem with three field components: the scalar potential $\Phi$ and the surface electric current density $\mathbf{J}\equiv\mathrm{J}_{x}\hat{\mathbf{x}}+\mathrm{J}_{y}\hat{\mathbf{y}}$ Jin et al. (2016). For an eigenstate indexed by $\nu$ and frequency $\omega_{\nu}$, this eigenproblem is specified by Note (2)

[TABLE]

Here, $\hat{\mathbf{p}}\equiv-{\rm i}\bm{\nabla}$ is the in-plane momentum operator, $\hat{V}[f](\mathbf{r})\equiv\int_{\Omega}{\rm d}\mathbf{r}^{\prime}\,f(\mathbf{r}^{\prime})/|\mathbf{r}-\mathbf{r}^{\prime}|$ the Coulomb operator, \sigma_{2}\equiv\Big{(}\begin{smallmatrix}0&-{\rm i}\\ {\rm i}&0\end{smallmatrix}\Big{)} a Pauli matrix, $\omega_{\text{{f}}}\equiv E_{\scriptscriptstyle\mathrm{F}}/\hbar$ the Fermi “frequency”, and $\alpha\equiv e^{2}/\pi\hbar$ a prefactor of graphene’s intraband conductivity $\alpha\omega_{\text{{f}}}\omega^{-1}$. Conceptually, Eqs. (1) and comprise the Coulomb, continuity, and constitutive equations. The no-spill boundary condition $\mathbf{J}\cdot\hat{\mathbf{n}}=0$ applies along the perimeter of $\Omega$ (edge normal, $\hat{\mathbf{n}}$). Under a suitable inner product Eq. (1a) is Hermitian (see SM).

We explore the band topology of 2D plasmons in periodically structured graphene under magnetic-field induced $\mathcal{T}$-breaking. Figure 1(a) illustrates our design: a triangular antidot lattice of periodicity $a$ and antidot diameter $d$ is etched into a suspended sheet of graphene Note (3). The domain $\Omega$ in Eqs. (1) is then the torus defined by the rhombic unit cell of Fig. 2(a). Band folding splits the eigenindex $\nu$ into a band index $n=1,2,\ldots$ and a crystal wave vector $\mathbf{k}$ restricted to the hexagonal Brillouin zone (BZ) of Fig. 2(b). Accordingly, the eigenvectors assume the Bloch form $\mathbf{U}_{n\mathbf{k}}(\mathbf{r})=\mathbf{u}_{n\mathbf{k}}(\mathbf{r})\mathrm{e}^{{\rm i}\mathbf{k}\cdot\mathbf{r}}$, with periodic component $\mathbf{u}_{n\mathbf{k}}\equiv(\omega_{\text{{f}}}\phi,\mathbf{j})_{n\mathbf{k}}^{\text{T}}$.

First, we consider the simple but instructive $d=0$ scenario, i.e. the uniform sheet, see Fig. 1(b). This “empty lattice” captures the essential impact of band-folding: by folding the uniform sheet plasmon dispersion, $\omega(k)=\sqrt{2\pi\alpha\omega_{\text{{f}}}k+\omega_{\text{c}}^{2}}$ Note (4), over the hexagonal BZ, three-fold Dirac-like point degeneracies arise between the $n=$123 bands at the $\mathbf{K}$ (and $\mathbf{K}^{\prime}$) point. For $B=0$, the lattice’s $C_{6v}$ symmetry guarantees that two-fold degenerate Dirac points remain between the $n=$12 bands even when $d\neq 0$. The uniform-sheet Dirac point plasmon frequency, $\omega_{\mathbf{K}}\equiv\sqrt{(\omega_{\mathbf{K}}^{0})^{2}+\omega_{\text{c}}^{2}}$ with $\omega_{\mathbf{K}}^{0}\equiv\sqrt{2\pi\alpha\omega_{\text{{f}}}|\mathbf{K}|}$ and $|\mathbf{K}|=4\pi/3a$, along with the cyclotron frequency $\omega_{\text{c}}$, then define the characteristic frequencies of the problem and are indicated in Fig. 1(b). By applying a finite $B$-field to the $d\neq 0$ system, the Dirac point degeneracy is split, inducing a gap linearly proportional to $\omega_{\text{c}}$. As a result, topological plasmons with both high frequency and sufficient topological gap require simultaneously large $\omega_{\mathbf{K}}^{(0)}$ and $\omega_{\text{c}}$.

The parameter space involved in simultaneously maximizing $\omega_{\mathbf{K}}^{0}$ and $\omega_{\text{c}}$ is illustrated in Fig. 1(c). The monotonic $E_{\scriptscriptstyle\mathrm{F}}$-dependence of the two characteristic frequencies is opposite, highlighting an inherent trade-off between the operating frequency and the gap size. In addition, the accessible parameter space is restricted by several constraints, indicated by gray regions in Fig. 1(c): first, intrinsic Drude loss estimated at $\gamma/2\pi\sim$1\text{,}\mathrm{T}\mathrm{H}\mathrm{z} smears out the gap region, necessitating $\omega_{\text{c}}\gtrsim\gamma$; second, interband dispersion is non-negligible when $\omega_{\mathbf{K}}\gtrsim\omega_{\text{{f}}}$, eventually introducing significant loss through Landau damping; and third, Landau quantization of the charge carriers ultimately invalidates a semiclassical description Gusynin *et al.* ([2007](#bib.bib30)); Ferreira *et al.* ([2012](#bib.bib31)) when $E_{\scriptscriptstyle\mathrm{F}}\lesssim E_{\text{{l}}}\equiv v_{\scriptscriptstyle\mathrm{F}}\sqrt{2\hbar eB/c}$ (the first Landau level), or equivalently, when $\hbar\omega_{\text{c}}\lesssim\tfrac{1}{2}E_{\text{{l}}}$, see Supplemental Material (SM). Overall, we find that an experimentally favorable region exists for Fermi energies $E_{\scriptscriptstyle\mathrm{F}}\sim$0.20.3\text{\,}\mathrm{eV}, periodicities $a\sim$400600\text{,}\mathrm{nm}, and magnetic fields $B\sim$28\text{\,}\mathrm{T}.

Next, we turn to the nanostructured system, settling on a periodicity $a=$400\text{,}\mathrm{nm}, antidot diameter $d=$200\text{\,}\mathrm{nm}, see Fig. 2(a), and a Fermi level $E_{\scriptscriptstyle\mathrm{F}}=$0.2\text{,}\mathrm{eV} (equivalent, at $B=0$, to a carrier density $n\approx$3\text{\times}{10}^{12}\text{\,}{\mathrm{cm}}^{-2}). Antidot lattices like these are well-within experimental capabilities Bai et al. (2010); Zhu et al. (2014); Yeung et al. (2014); Liu et al. (2015). The eigenvalue problem, Eqs. (1), is solved numerically by discretizing in an unstructured triangular mesh, employing linear nodal functions, and with the lattice-specific Coulomb interaction evaluated by Ewald summation (see SM). Figure 2(c) depicts the calculated plasmon dispersion $\omega_{n}(\mathbf{k})$ along the boundary of the irreducible BZ for increasing magnetic field strength $B=$0148\text{,}\mathrm{T}$$.

In the nonmagnetic scenario, $B=0$, the lattice disperses like the uniform sheet under the substitution $k\rightarrow\zeta_{n}(\mathbf{k})/a$, i.e. as $\omega_{n}^{0}(\mathbf{k})=\sqrt{2\pi\alpha\omega_{\text{{f}}}\zeta(\mathbf{k})/a}$, with modal parameter $\zeta_{n}(\mathbf{k})$ solely dependent on $a/d$ and the relative location of $\mathbf{k}$ in the BZ Christensen (2017); e.g. at $a/d=2$ we find $\zeta_{1,2}(\mathbf{K})\approx 2.535$. Near the $\mathbf{\mathbf{\Gamma}}$ point \zeta_{1}(\mathbf{k})\mathrel{\vbox{ \offinterlineskip\halign{\hfil#\cr\propto\cr\kern 0.2pt\cr\sim\cr\kern-2.5pt\cr}}}|\mathbf{k}|, yielding the conventional long-wavelength 2D plasmon dispersion \omega\mathrel{\vbox{ \offinterlineskip\halign{\hfil#\cr\propto\cr\kern 0.2pt\cr\sim\cr\kern-2.5pt\cr}}}\!\sqrt{k}. Particle-hole symmetry ($\mathcal{C}$) of Eqs. (1) entails the existence of a corresponding set $\{n=-1,-2,\ldots\}$ of negative energy states, $\omega_{-n}(\mathbf{k})=-\omega_{n}(-\mathbf{k})$ (and a trivial zero-frequency band, $n=0$) Jin et al. (2016): accordingly, besides the Dirac point degeneracy at $\mathbf{K}$ between the $n=1$ and $2$ bands, an implicit degeneracy exists at $\bm{\Gamma}$ between the $n=\pm 1$ (and $n=0$) bands. By applying a magnetic field, the bands are linearly perturbed from $\omega_{n}^{0}(\mathbf{k})$ to $\omega_{n}(\mathbf{k})\simeq\omega_{n}^{0}(\mathbf{k})+\xi_{n}(\mathbf{k})\omega_{\text{c}}+\mathcal{O}(\omega_{\text{c}}^{2})$ (see SM); the modal perturbation parameter $\xi_{n}(\mathbf{k})$ is obtained numerically at the degeneracy points as $\xi_{1}(\bm{\Gamma})\approx 0.63$ and $\xi_{1,2}(\mathbf{K})\approx\mp 0.27$ at $a/d=2$ Note (5). This is illustrated in Fig. 2(d): the degeneracies at $\mathbf{\Gamma}$ and $\mathbf{K}$ are linearly and evenly gapped when $B\neq 0$. As we explain shortly, the low-frequency gap opened at $\bm{\Gamma}$ supports a topological edge state entirely analogous to its uniform sheet counterpart. The high-frequency (${\approx}$15\text{,}\mathrm{THz}$$) gap opening at $\mathbf{K}$, however, introduces a new, qualitatively distinct topological edge state.

Next, we describe the topological properties of the plasmonic lattice as quantified by the band Chern number, $C^{(n)}\equiv\frac{1}{2\pi{\rm i}}\oint_{\scriptscriptstyle\partial\text{BZ}}{\rm d}\mathbf{k}\ \langle\mathbf{u}_{n\mathbf{k}}|\nabla_{\mathbf{k}}|\mathbf{u}_{n\mathbf{k}}\rangle$ (evaluated numerically from the computed eigenvectors Fukui et al. (2005)). Figure 2(b) depicts the evolution of $C^{(n)}$ across $B=$0148\text{,}\mathrm{T}. At $B=$0\text{\,}\mathrm{T} the Berry flux is identically zero cf. time-reversal and parity symmetry; the band-structure is topologically trivial. For $B\neq 0$, $\mathcal{T}$ is broken, allowing nonzero Berry fluxes and nontrivial topology: the 1st and 2nd bands have $C^{(1)}=C^{(2)}=0$, independent of $B$ since finite gaps separate them from distinct bands throughout the considered $B$-range. Conversely, the higher order bands, $n=3,4,\ldots$, display Chern numbers covering a broader range, up to $\pm 2$. A few bands exhibit point-degeneracies within numerical accuracy and are assigned a composite Chern number $C^{(n\oplus n+1)}$. As the $B$-field is increased, there is an exchange of Chern numbers between the $n=$456$$ bands as gaps close and reopen, illustrating the mechanism of Berry flux monopole exchange. For even stronger $B$-fields (see SM), all six bands eventually separate completely, leaving $C^{(1)}=C^{(2)}=0$, $C^{(3)}=-2$, and $C^{(4)}=C^{(5)}=C^{(6)}=+1$.

By the bulk-edge correspondence, the existence of topologically protected edge states is intimately linked with the bulk topology, i.e. with $C^{(n)}$. As recently pointed out in Ref. 18, the presence of $\mathcal{C}$ symmetry, and the concomitant existence of a set of negative-frequency states $\{n=-1,-2,\ldots\}$, necessitates a global perspective of the band topology for the definition of associated gap Chern numbers. Specifically, the total Chern number of positive ($+$) and negative ($-$) frequency bands is $C_{\pm}\equiv\sum_{n=1}^{\infty}C^{(\pm n)}$. In uniform graphene $C_{\pm}=\pm\operatorname{sgn}B$ Jin et al. (2016). Since Chern numbers can be annihiliated or created (pairwise) only under band closings, this result holds in nanostructured graphene as well cf. the finite bandgap separating positive and negative bands. With this in mind, we define the $n$th gap Chern number $\bar{C}_{n}$ associated with the gap immediately below the $n$th band as

[TABLE]

specializing to positive-frequency gaps at the last equality. For lattice terminations adjacent to vacuum, bulk-edge correspondence then requires that the number of left minus right propagating topological edge states equal $\bar{C}^{(n)}$ Hatsugai (1993).

These considerations predict the existence of single-mode one-way edge states in the first and second gaps when $B\neq 0$ and multi-mode one-way edge states in the gap between the $n=$34 bands at $B=$48\text{\,}\mathrm{T}, cf. Fig. 2(c). We confirm these predictions in Fig. 3 by numerically calculating the edge states supported by a broad ribbon (20 unit cells wide) extended along $x$ with the particular edge termination of Fig. 3(a). The bulk states are folded into the projected 1D BZ, $k_{x}\in(-\pi/a,\pi/a)$, see Fig. 3(b), due to breaking of Bloch periodicity along $y$. Additionally, edge states emerge: they are identified and post-selected from the ribbon-spectrum by their edge confinement and bulk-gap habitation (in emulating single-boundary physics, edge states localized on the bottom ribbon edge are omitted). The resulting edge-dispersion is shown in Fig. 3(c) for $B=$048\text{,}\mathrm{T}. At $B=0$, all edge states are non-topological; states at $\pm\mathbf{k}$ travel in opposite directions and edge connections between bulk bands are trivial. For $B\neq 0$, topological one-way edge states appear in the bandgaps, consistent with the obtained gap Chern numbers. They connect upper and lower bulk bands, occasionally by circling the 1D BZ, separated by nontrivial $\bar{C}^{(n)}\neq 0$ gaps. The edge states propagate to the right, consistent with the sign (chirality) of $\bar{C}^{(n)}\neq 0$. They are topologically protected from backscattering only in the complete bandgap: above it, any defect may scatter them to either bulk or counterpropagating edge states. The low-frequency $\bar{C}^{(1)}=-1$ gap hosts edge states entirely analogous to the edge magnetoplasmons of the uniform sheet – an edge-state mirror of the bulk dispersion-agreement (${\mathrel{\vbox{ \offinterlineskip\halign{\hfil$#$\cr\propto\cr\kern 0.2pt\cr\sim\cr\kern-2.5pt\cr}}}}\sqrt{k}$) between the $n=1$ band and the uniform sheet. In contrast, the high-frequency (${\approx}$15\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}) edge state in the $\bar{C}^{(2)}=-1$ gap result directly from band-engineering, and is a qualitatively new type of edge magnetoplasmon. Finally, a multimode triple of edge states appear in the $\bar{C}^{(4)}=-3$ gap. Though the gap is comparatively small, it can be widened by tuning $a/d$. Figure 3(d) illustrates the sharp spatial Bloch mode confinement of the edge states, $|\phi_{nk_{x}}(\mathbf{r})|$, for a few select $n$ and $k_{x}$ at $B=$8\text{,}\mathrm{T}$$. The degree of confinement correlates positively with the size of the topological bandgap, i.e. implicitly with $B$, paralleling the uniform 2D electron gas Volkov and Mikhailov (1988).

The edge states can be efficiently excited by nearby point sources, as demonstrated in Fig. 4: a $y$-polarized dipole near the edge, emitting in the gap-center ($14.6\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$) of the $n=$12$$ bands, excites the edge plasmon at $k_{x}=0$ (for computational details, see SM). In the absence of intrinsic material loss, the edge state propagates unidirectionally to the right with constant amplitude as seen in Figs. 4(b-d). Topological protection ensures that even structural defects, such as the sharp trench in Fig. 4(d), are traversed without backscattering. The increased edge-confinement with mounting magnetic field is exemplified by Figs. 4(b-c).

The edge state’s topological nature does not shield from intrinsic material or radiation loss. While the latter is negligible, owing to the strongly localized and electrostatic nature of graphene plasmons [cf. the nearly vertical light cone in Fig. 2(c)], the former can be appreciable, as in all plasmonic systems. Finite relaxation $\gamma$ is readily incorporated in Eqs. (1) by the substitution $\omega_{\nu}\rightarrow\omega_{\nu}+{\rm i}\gamma$. This introduces an imaginary spectral component, $\operatorname{Im}\omega_{\nu}\simeq-\tfrac{1}{2}\gamma(1+\xi_{\nu}\omega_{\text{c}}/\!\operatorname{Re}\omega_{\nu}^{0})$ for $\gamma\ll\operatorname{Re}\omega_{\nu}^{0}$. This impacts the propagation of edge states in two aspects: first, it blurs the gap region, allowing small but finite loss-induced coupling between edge and bulk states (see SM); second, states exhibit a finite life-time, or, equivalently, finite propagation length {\mathrel{\vbox{ \offinterlineskip\halign{\hfil#\cr\propto\cr\kern 0.2pt\cr\sim\cr\kern-2.5pt\cr}}}}\,1/\gamma.

In conclusion, we have demonstrated the band topology of 2D plasmons in periodically patterned graphene under a $\mathcal{T}$-breaking magnetic field. Multiple sets of topologically protected one-way edge plasmons corresponding to nontrivial gap Chern numbers are discovered. Their operating frequencies can be as high as tens of THz, i.e. in the far-infrared regime. They can be experimentally verified by terahertz near-field imaging Chen et al. (2012); Fei et al. (2012) and Fourier transform infrared spectroscopy Skirlo et al. (2015). Our findings suggests a new direction in the synthesis of high-frequency $\mathcal{T}$-broken topological bosonic phases, and can be directly extended to non-magnetic schemes based on valley polarization Kumar et al. (2016); Song and Rudner (2016).