Limits to the Optical Response of Graphene and Two-Dimensional Materials

Abstract

2D materials provide a platform for strong light-matter interactions, creating wide-ranging design opportunities via new-material discoveries and new methods for geometrical structuring. We derive general upper bounds to the strength of such light-matter interactions, given only the optical conductivity of the material, including spatial nonlocality, and otherwise independent of shape and configuration. Our material figure of merit shows that highly doped graphene is an optimal material at infrared frequencies, whereas single-atomic-layer silver is optimal in the visible. For quantities ranging from absorption and scattering to near-field spontaneous-emission enhancements and radiative heat transfer, we consider canonical geometrical structures and show that in certain cases the bounds can be approached, while in others there may be significant opportunity for design improvement. The bounds can encourage systematic improvements in the design of ultrathin broadband absorbers, 2D antennas, and near-field energy harvesters.2D materials [1, 2] and emerging methods [3][4][5][6][7][8] for patterning 2D layers and their surroundings are opening an expansive design space, exhibiting significantly different optical [9][10][11] (and electronic) properties from their 3D counterparts. In this Letter, we identify energy constraints embedded within Maxwell's equations that impose theoretical bounds on the largest optical response that can be generated in any 2D material, in the near or far field. The bounds account for material loss as encoded in the real part of a material's conductivityin the case of a spatially local conductivity tensor σ, they are proportional to σ † (Re σ) −1 σ -and are otherwise independent of shape and configuration. We derive the bounds through convex constraints imposed by the optical theorem [12][13][14] and its near-field analogue, leveraging a recent approach we developed for spatially local 3D materials [15]. In addition to accommodating nonlocal models, this work demonstrates starkly different near-field dependencies of 2D and 3D materials. For graphene, the 2D material of foremost interest to date, the bounds bifurcate into distinctive low-and highenergy regimes: the low-energy bounds are proportional to the Fermi level, whereas the high-energy bounds are proportional to the fine-structure constant, α, for any geometrical configuration. We find that far-field bounds on the extinction cross-section can be approached by elliptical graphene disks, whereas the near-field bounds on the local density of states [16][17][18][19][20] and radiative heat transfer rate [21][22][23][24][25][26] cannot be approached in prototypical flatsheet configurations. The bounds presented here provide a simple material figure of merit to evaluate the emerging zoo of 2D materials, and offer the prospect of greater optical response via computational design. The material * Corresponding author: owen.miller@yale.edu figure of merit can guide ongoing efforts in 2D-material discovery, while the general bounds can shape and ...

Nonreciprocal radiative heat transfer between two planar bodies

“…Understanding radiative heat transfer [1][2][3] is essential in many applications ranging from radiative cooling [4][5][6] and thermal diodes [7] to thermal transistors [8][9][10][11] and thermophotovoltaic systems [12][13][14][15][16]. The majority of works investigating radiative heat transfer consider materials that satisfy Lorentz reciprocity [17][18][19][20][21][22][23][24][25][26][27][28][29][30]. On the other hand, it is known that breaking the constraint of reciprocity is necessary in order to reach the thermodynamic limit of thermal radiation harvesting [31][32][33].…”

Fundamental Limits to Near-Field Optical Response over Any Bandwidth

“…* hyungki.shim@yale.edu † owen.miller@yale.edu icant ongoing debate about whether a plasmonic or an all-dielectric approach is better, and in which scenarios 2D materials might be better than conventional bulk materials. Unlike all previous bounds and sum rules [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43], the material figure of merit we derive here enables general quantitative answers to these questions. In a frequencybandwidth phase space, we map out which materials are optimal and where the critical thresholds, from dielectric to plasmonic and bulk to 2D, occur.…”

“…At the other extreme, single-frequency limits to power extinction and other physical observables have been discovered in both the near field [24,36,37,43] and far field [38][39][40][41][42]60] based on energy-conservation principles, but they necessarily fail to account for the effects of nonzero bandwidth. (As an example, they predict infinite maximal response for any lossless material.…”

“…With suitable boundary conditions, a Hilbert transform (i.e., a Kramers-Kronig-like transform) then enables a sum rule, relating integrated response over all frequencies to that of a single frequency. Conversely, energy-conservation bounds-recognized primarily within the past decade [24,[36][37][38][39][40][41][42][43]60]-exploit the power-quantity-by-amplitude form in a different way. In writing a power quantity as the imaginary part of an amplitude, the amplitude itself is linear in the electromagnetic fields and/or currents (holding the incident field fixed).…”

Optimal Nanoparticle Forces, Torques, and Illumination Fields

“…For nonspherical scatterers, optical forces and torques generally require simulation of Maxwell's equations [37][38][39][40][41], providing numerical results but little insight. This contrasts strongly with the more detailed knowledge of power flow in such systems, ranging from bounds [6,7,10,11,[42][43][44] to sum rules [45][46][47] to spherical-particle design criteria [48]. The disparity between the broad understanding of power flow versus the relative paucity for momentum flow may reflect the complexity of the Maxwell stress tensor relative to the Poynting vector.…”