Chiral Plasmons Without Magnetic Field

Chiral plasmons without magnetic field Justin C. W. Songa,b,c,1 and Mark S. Rudnerd aWalter Burke Institute of Theoretical Physics, California Institute of Technology, Pasadena, CA 91125; bInstitute of Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125; cDepartment of Physics, California Institute of Technology, Pasadena, CA 91125; and dCenter for Quantum Devices and Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark Edited by Allan H MacDonald, The University of Texas at Austin, Austin, TX, and approved February 19, 2016 (received for review September 24, 2015) Plasmons, the collective oscillations of interacting electrons, possess new platform for achieving a range of magneto-optical effect an- emergent properties that dramatically alter the optical response of alogs that are magnetic field free and “on demand.” Aprime metals. We predict the existence of a new class of plasmons— example is optical nonreciprocity (10), which is central for optical chiral Berry plasmons (CBPs)—for a wide range of 2D metallic device components, e.g., optical isolators and circulators. Above a systems including gapped Dirac materials. As we show, in these threshold frequency (ωth; shaded area in Fig. 1A), the single uni- materials the interplay between Berry curvature and electron– directionally propagating mode ωedge allows for chiral transport of electron interactions yields chiral plasmonic modes at zero mag- − light via coupling to CBPs. Such CBP-mediated waveguides pro- netic field. The CBP modes are confined to system boundaries, – even in the absence of topological edge states, with chirality man- vide a novel paradigm for deep subwavelength (11 13), linear, and ifested in split energy dispersions for oppositely directed plasmon magnetic field-free strong nonreciprocity, crucial for miniaturizing waves. We unveil a rich CBP phenomenology and propose setups optical components. In particular, we predict that CBPs can en- for realizing them, including in anomalous Hall metals and opti- able nonreciprocity over a large technologically important band- cally pumped 2D Dirac materials. Realization of CBPs will offer a width (terahertz to mid-IR). powerful paradigm for magnetic field-free, subwavelength optical The intrinsic chirality of CBPs starkly contrasts with that achieved nonreciprocity, in the mid-IR to terahertz range, with tunable split- via cyclotron motion of charged particles in a magnetic field, with tings as large as tens of THz, as well as sensitive all-optical diag- important quantitative and qualitative consequences. In the latter nostics of topological bands. case, chirality arises via the Lorentz force and gives rise to con- ventional magnetoplasmons (14–18). There, the cyclotron fre- topological materials | interactions | nonreciprocal response | Berry curvature quency, Zωc = ZeB=m (19–21), determines the constant splitting between magnetoplasmon modes of opposite chirality, which can be ’ n electronic systems, chirality expresses the system s ability to oftheorderofafewmillielectronvolts for accessible field strengths. discriminate between forward and backward propagation of I In contrast, chirality in CBPs arises from the combined action of electronic signals along certain directions. This technologically useful and hotly sought-after property can be achieved through plasmonic self-generated electric fields and the anomalous velocity the application of external magnetic fields. However, the need of Bloch electrons, the phase space dual to the Lorentz force. This for strong applied fields “on-chip” brings many challenges for combination makes the CBP mode splitting directly sensitive to applications. Recently, materials exhibiting chirality in the ab- plasmon wavelength, the Berry flux, and interaction strength, in sence of a magnetic field have started to gain prominence. contrast to magnetoplasmon splittings, which only depend on † These materials include metals exhibiting anomalous (1) and magnetic field (17, 22). As a result, for short wavelengths and quantum anomalous (2–6) Hall effects, as well as nonmagnetic unscreened interactions, large splittings ZΔω of several tens to a materials pushed out of equilibrium, where, for example, a hundred millielectron volts can be achieved (Eq. 3). zero-field charge Hall effect was recently demonstrated (7). In each case, zero-field chirality arises from Bloch band Berry Self-Induced Anomalous Velocity curvature, a fundamental property of Bloch eigenstates that The origin of CBPs in two dimensions can be understood from dramatically affects single-particle electronic motion and ma- the Euler equations for electron density, nðr, tÞ (23) terial responses (1, 8, 9). Here we show that Berry curvature can work in concert with Significance interactions, leading to new types of collective modes in 2D topological metals, with nonvanishing Berry flux (i.e., net Berry curvature), F. In particular, F gives rise to chiral plasmonic A class of collective excitations is introduced, arising from the excitations—propagating charge density waves with split disper- combined action of electron interactions and Berry curvature. sion for oppositely directed modes—in the absence of a magnetic These excitations manifest as chiral plasmonic modes at zero field (Fig. 1). We refer to these collective modes as chiral Berry magnetic field. These collective modes are predicted to arise in a plasmons (CBPs). Notably, these chiral modes are localized to range of anomalous Hall metals, yielding a fundamental char- the edge of the 2D metal, even in the absence of topological edge acteristic of interacting topological metals with non-zero Berry states, and exhibit a rich phenomenology (see below). flux. Featuring a rich set of properties, these modes will enable We expect CBPs to be manifested in a wide variety of mag- versatile new tools for molding the flow of light. netic and nonmagnetic materials. The former are materials that Author contributions: J.C.W.S. and M.S.R. designed the research, performed the research, exhibit anomalous Hall effects, wherein time reversal symmetry and wrote the paper. (TRS) breaking is encoded in the Bloch band Berry curvature. The authors declare no conflict of interest. The latter include a range of readily available gapped Dirac materials, e.g., transition metal dichalcogenides and graphene/ This article is a PNAS Direct Submission. 1 hBN, wherein TRS breaking is achieved by inducing a non- To whom correspondence should be addressed. Email: [email protected]. equilibrium valley polarization (7). In both cases CBPs are char- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. acterized by clear optical signatures such as split peaks in optical 1073/pnas.1519086113/-/DCSupplemental. † ’ absorption. We note that Kohn s theorem guarantees that splittings are unchanged for parabolic bands; weak renormalizations are expected for linearly dispersing systems such as gra- From a technological perspective, CBPs in nonmagnetic ma- phene, where it has been measured to be at most of order 10% (42). In contrast, for terials are particularly appealing because they provide an entirely CBPs, the splitting grows directly proportional with interaction strength (Eq. 3). 4658–4663 | PNAS | April 26, 2016 | vol. 113 | no. 17 www.pnas.org/cgi/doi/10.1073/pnas.1519086113 Downloaded by guest on September 24, 2021 Crucially, va depends directly on the self-generated electric ∂tnðr, tÞ + ∇ · vðr, tÞ = 0, [1a] field −∇ϕðrÞ. Consequently, the magnitude of the splitting is governed by the wave vector q and the strength of Coulomb ∂ p r t − n r t e∇ϕ r t = [1b] t ð , Þ ð , Þ ð , Þ 0, interactions. We emphasize, however, that CBPs are a linear Δω v r t p r t phenomenon, with the mode splitting independent of the where ð , Þ and ð , Þ are the velocity and momentum density magnitude of ϕ. As we will show, the q-dependent CBP splitting fields, ϕðrÞ is the scalar electric potential, and −e < 0 is the elec- can be large tron charge. We note that in principle, the force equation, Eq. 1b, also includes contributions arising from the stress density of the À Á e2 electronic fluid. However, at long-wavelengths these contribu- ZΔω = Z ωedge − ωedge ≈ q [3] + − A κ Fj j, tions yield only subleading corrections to the plasmon dispersion (24). As our aim here is to clearly and most simply demonstrate the where A is a numerical prefactor of order unity that depends existence and main features of CBPs, in this work we neglect small on geometry, and we used the 2D Coulomb potential WðqÞ = corrections due to the stress density. For an alternative formulation ‡ 2πe=ðκjqjÞ with backgroundffiffiffi dielectric constant κ.Foredge in terms of currents and conductivities, see SI Appendix. p CBPs, we find A = 8 2π=9 (see below), yielding large splittings To fully specify the dynamics, Eq. 1 must be supplemented by − ZΔω ≈ 6 − 60 meV for q = 1 − 10 μm 1 (here we used F = 1, κ = 1). a set of constitutive relations that relate velocity, momentum, e2=κ 3 1 The appearance of on the right side of Eq. signals the crucial density, and the electric potential. Plasmons emerge from Eq. role interactions play in Δω. as self-sustainedR collective oscillations of nðr, tÞ, with the potential ϕðr, tÞ = d2r′ Wðr, r′Þδnðr′, tÞ generated by the plasmon’s Chiral Edge Berry Plasmons δn = n r t − n n density fluctuations ð , Þ 0. Here 0 is the average car- We now analyze collective motion described by Eq. 1, treating W r r′ rier density, and ð , Þ is the Coulomb interaction. the electric potential ϕ self-consistently. For an infinite bulk, As we argue, in the presence of Berry curvature, the con- applying ∂t to the continuity relation in Eq. 1a, using Eq. 2, and stitutive relations take on an anomalous character. The form substituting in the force equation, Eq. 1b, yields of the constitutive relations can be found by starting with Z ∞ the quasiparticle semiclassical equations of motion (8), n0e ∂«p −∂2δn = ∇2ϕ ϕ r = dr′ W r − r′ δn r′ [4] vp = + 1 p_ × ΩðpÞ, p_ = e∇ϕðrÞ, where vp = r_ is the quasiparticle t , ð Þ ð Þ ð Þ.

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