## Abstract

Broken inversion and time reversal symmetries affect the electromagnetic wave modes supported by continuous media, which in turn governs thermal radiation and enables control of radiative heat, linear momentum, and angular momentum transfer. We identify opportunities for exploring thermal radiation in inversion symmetry- and time reversal symmetry-breaking materials and compare and contrast radiative transport phenomena in these systems, especially nonreciprocity. Application of these phenomena can lead to novel methods of thermal management, tunability, and object manipulation at short length scales.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Thermal radiation and its associated momentum transfer from materials highlight the fundamental phenomenon of symmetry in condensed matter physics. Understanding and control of these effects is crucial to the development of next-generation optoelectronic and photonic devices. Maxwell’s equations with thermally fluctuating current sources [1,2], along with the appropriate constitutive equations (e.g., ${\boldsymbol D} = \overline{\overline \varepsilon } {\boldsymbol E},$ ${\boldsymbol B} = \overline{\overline \mu } {\boldsymbol H},$ and ${\boldsymbol J} = \overline{\overline \sigma } {\boldsymbol E}$ to a linear approximation) and boundary conditions, can provide a full description of thermal radiation. Spectral, angular, and polarization characteristics of the electromagnetic waves supported by a material and their thermal population manifest in the characteristics of the spectral directional heat flux. The modal spectrum of the medium is in turn described by its geometry as well as its conductivity, permittivity, and permeability tensors, which incorporate its symmetries (or lack thereof), electronic and phononic band structures, and their correlations.

Modern theories reveal that fundamental laws governing thermal radiation such as Planck’s law and Kirchhoff’s law of radiation are only conditionally valid. Planck’s law is valid if the peak wavelength of thermal radiation ${\lambda _{max}},$ given by Wien’s law [3], is much shorter than the sizes of and the distances between bodies. Rytov [1] and Polder and van Hove [2] expanded the theory to include near-field radiative heat transfer, where the coupling of surface and evanescent waves between materials strongly enhances the spectral heat flux (Fig. 1(b)). The dominance of the evanescent contribution in the near-field leads to the so-called breakdown of Planck’s law, where the total heat flux exceeds the blackbody limit of $\sigma {T^4},$ where $\sigma = 5.67 \times {10^{ - 8}}\; \textrm{W}{\textrm{m}^{ - 2}}{\textrm{K}^{ - 4}}$ is the Stefan-Boltzmann constant and *T* is the temperature of the body. Such super-Planckian heat flux has led to the development of devices such as thermal switches [4–8], thermal diodes [9,10], and near-field thermophotovoltaic (TPV) systems [11–13].

Kirchhoff’s law of radiation (the equality of spectral directional emissivity and absorptivity [14–19]) relies on the Lorentz reciprocity theorem, which is valid when the material’s permittivity, conductivity, and permeability tensors, $\; \overline{\overline \varepsilon } $, $\overline{\overline \sigma } $, and $\overline{\overline \mu } ,$ are (1) linear, (2) symmetric, and (3) time-independent [20]. Evading at least one of these conditions creates a nonreciprocal system, which not only can violate Kirchhoff’s law, but also opens up vast opportunities for controlling thermal radiation. In this Perspective, we review the characteristics of thermal radiation in systems either exhibiting or lacking configurational symmetry – the symmetry of the bodies exchanging heat, their surfaces, and the materials constituting them [21] – as well as material symmetry – the symmetry of crystal structure, such as inversion (${\cal I}$) and time reversal (${\cal T}$) symmetry. We discuss recent developments in this emerging field, and future opportunities.

One of the properties crucial to achieving nonreciprocity is the dielectric tensor $\overline{\overline \varepsilon } $, which encompasses the behavior of electric polarization within a material. In general, an electric polarization of a material resulting from external or internal electric and magnetic fields takes the form ${P_i} = {\varepsilon _0}({{\chi_{ij}}{E_j} + {\chi_{ijk}}{\nabla_k}{E_j} + {\chi_{ijk}}{E_j}{B_k} + {\chi_{ijkl}}{B_l}{\nabla_k}{E_j} + {\chi_{ijl}}{E_j}{E_l} + \cdots } ),$ where $\chi $ with indices are susceptibility tensors and $\cdots $ represents higher-order contributions [23]. In the frequency domain, the dielectric tensor corresponding to the polarization terms that are linear in the electric field can be expressed ${\varepsilon _{ij}}({{\boldsymbol q},\omega ,\; {\boldsymbol B}} )= \varepsilon _{ij}^0 + {\alpha _{ijk}}{q_k} + {\beta _{ijk}}{B_k} + {\gamma _{ijkl}}{q_k}{B_l},$ where ${q_i}$ is the photon momentum, $\varepsilon _{ij}^0$ and ${\gamma _{ijkl}}$ are symmetric tensors, and ${\alpha _{ijk}}$, ${\beta _{ijk}}$ are antisymmetric tensors [24]. Most work on radiative heat and momentum transfer has focused on the first term, $\varepsilon _{ij}^0$. However, breaking the ${\cal I}$ and/or ${\cal T}$ symmetry of materials gives rise to nonzero contributions from other terms, offering opportunities to discover new thermal radiation-driven phenomena in a design space that has not been fully explored.

Figure 1 illustrates the roles of ${\cal I}$ and ${\cal T}$ symmetries in governing the radiative heat and momentum transfer. In the presence of ${\cal I}$ and ${\cal T}$ symmetries, only the first term in ${\varepsilon _{ij}}({{\boldsymbol q},\omega ,\; {\boldsymbol B}} )$ is nonzero, resulting in reciprocal heat ($U$) and linear momentum (${\boldsymbol p}$) transfer via bulk and surface modes (Fig. 1(a)). That is, the transport from/to a material surface in any given opposing angular directions are identical (but not necessarily angularly or spectrally isotropic). The near-field radiative heat transfer between two materials is written as ${q_{1 \to 2}} = \mathop \smallint \nolimits_0^\infty \frac{{d\omega }}{{2\pi }}[{\mathrm{\Theta }({\omega ,{T_1}} )- \mathrm{\Theta }({\omega ,{T_2}} )} ]\mathop \smallint \nolimits_{ - \infty }^\infty \frac{{{d^2}{\boldsymbol q}}}{{{{({2\pi } )}^2}}}\tau ({\omega ,{\boldsymbol q}} ),$ where $\mathrm{\Theta }({\omega ,T} )= \hbar \omega /\left( {{e^{\frac{{\hbar \omega }}{{{k_B}T}}}} - 1} \right)$ is the mean energy of a Planck oscillator and $\tau ({\omega ,{\boldsymbol q}} )$ is the energy transmission coefficient [25,26]. Figure 1(b) plots $\tau ({\omega ,{\boldsymbol q}} )$ as a function of photon energy and momentum. It reveals the reciprocity of the contributions from the bulk modes (the bright area including low momenta within the light cone) and surface modes (bright lines with large momenta outside of the light cone at the frequencies corresponding to the excitation of surface modes), seen in $\tau ({\omega ,{\boldsymbol q}} )$ being symmetric in the in-plane wavevector ${\boldsymbol q}$, i.e., $\tau ({\omega ,{\boldsymbol q}} )= \tau ({\omega , - {\boldsymbol q}} ))\; $[26].

Breaking either ${\cal I}$ or ${\cal T}$ symmetry gives rise to non-zero contributions from the second and third terms in ${\varepsilon _{ij}}({{\boldsymbol q},\omega ,\; {\boldsymbol B}} ),$ respectively, and to the material possessing optical activity; that is, the ability to rotate the plane of polarization of radiation [20]. This phenomenon is characterized by the gyration vector ${\boldsymbol g}$ of these systems, which affects the polarization vector according to ${\boldsymbol P} \propto {\boldsymbol E} \times {\boldsymbol g}.$ As seen in ${\varepsilon _{ij}}({{\boldsymbol q},\omega ,\; {\boldsymbol B}} )$, in ${\cal I}$ symmetry-breaking systems, ${\boldsymbol g}$ is set by the wavevector ${\boldsymbol q\; }$, whereas in ${\cal T}$ symmetry-breaking systems, ${\boldsymbol g}$ is set by the magnetic field, to a linear approximation. As a result, bulk waves are right- and left-handed elliptically polarized at different wavenumbers, and carry angular momentum ${\boldsymbol L}$ (Fig. 1(c), (d)).

Radiative heat and momentum transfer are nonreciprocal in ${\cal T}$ symmetry-breaking systems, meaning the transport from/to a surface in any given opposing angular directions are different (and necessarily anisotropic). These systems include metals and semiconductors under external magnetic fields [18,27], as well as magnetic Weyl semimetals (where an internal magnetization ${\boldsymbol M}$ breaks ${\cal T}$ symmetry [19,28,29]). In these materials, the permittivity and conductivity tensors become asymmetric – $\overline{\overline \varepsilon } \ne {\overline{\overline \varepsilon } ^T}$ and $\overline{\overline \sigma } \ne {\overline{\overline \sigma } ^T}$ – thus breaking Lorentz reciprocity. For bulk and surface waves with momentum ${\boldsymbol q}$, there are two limiting geometries: the Voigt configuration (${\boldsymbol q}$ normal to ${\boldsymbol g}$) and the Faraday configuration (${\boldsymbol q}$ parallel to ${\boldsymbol g}$), and a mixture of the two cases in other directions. While bulk waves in ${\cal T}$ symmetry-breaking media have a reciprocal dispersion, surface waves can show a nonreciprocal dispersion, i.e., $\omega ({\boldsymbol q} )\ne \omega ({ - {\boldsymbol q}} ),$ in the presence of a nonzero component of ${\boldsymbol g}$ normal to ${\boldsymbol q}$ as a result of the lower symmetry of the surface (the surface intrinsically breaks mirror symmetry). In this case, surface modes become nonreciprocal (Fig. 1(d)), and different thermal population of these modes such that one dominates can lead to highly directional and nonreciprocal emission and absorption. In the Voigt configuration, surface waves possess the strongest nonreciprocity and bulk waves are linearly polarized. By contrast, surface waves in the Faraday configuration are totally reciprocal and bulk waves are elliptically polarized [29].

A prominent feature of radiative heat transfer between ${\cal T}$ symmetry-breaking nonreciprocal media is its controllability in the near-field. In the presence of a second medium, the surface waves can couple to each other, and their reciprocity or lack thereof depends on the configurational inversion symmetry of the two-body system (as opposed to the material symmetries of each body such as ${\cal I}$ and ${\cal T}$, based on atomic configuration) [30,31]. In Fig. 1(e), the system possesses configurational inversion symmetry and the coupled surface modes are reciprocal even though each slab supports nonreciprocal single-interface SPPs by itself. The single-interface SPPs on each slab cannot couple to the identical nonreciprocal mode on the opposing slab since their field rotation directions (or phases, which are linked to ${\boldsymbol q}$) do not match [32]. As a consequence, the energy transmission coefficient is reciprocal $\tau ({\omega ,{\boldsymbol q}} )= \tau ({\omega , - {\boldsymbol q}} )$ and exhibits weak contributions from the surface modes. When configurational inversion symmetry is broken by rotating one of the slabs by 180° in Fig. 1(f), the phases of the single-interface SPPs on each slab become well-matched, resulting in strong coupling and a nonreciprocal energy transmission coefficient $\tau ({\omega ,{\boldsymbol q}} )\ne \tau ({\omega , - {\boldsymbol q}} )$ with dominant contributions from the coupled surface modes. This opens up exciting opportunities for rotational control and modulation of near-field radiative heat transfer between identical or dissimilar materials [6,32]. In fact, it has been shown that the radiative heat transfer between a ${\cal T}$ symmetry-breaking material and a ${\cal T}$-symmetric material is nonreciprocal [26,33], consistent with the broken configurational inversion symmetry of the system. Figure 1(d) is a particular case where air is the ${\cal T}$-symmetric material facing the ${\cal T}\; $ symmetry-breaking material.

In addition to control of the magnitude of the near-field radiative heat transfer, its directionality can also be controlled in many-body ${\cal T}$ symmetry-breaking systems, i.e., directional flow of heat ${q_{i \to j}} \ne {q_{j \to i}}$ can be achieved. In such systems, the possibilities of persistent heat flow in thermal equilibrium [34], a photonic analog of the thermal Hall effect [35], and their applications for thermal routing [32] and rectification [36,37] are predicted. These phenomena can also be re-contextualized in terms of their configurational symmetries, as in [21] and as we did here for the two-body system in Fig. 1. In fact, configurational symmetry was recently used as a design tool for more complicated semitransparent structures that violate Kirchhoff’s law [38].

In turn, media without ${\cal I}$ symmetry (Fig. 1(c)) can possess non-zero ${\alpha _{ijk}}$ and exhibit spatial dispersion-induced phenomena such as natural optical activity and linear birefringence. These effects enable anisotropic emission and polarization control of thermal radiation in both the near- and far-field [39–42], as well as radiative transfer of angular momentum. ${\cal I}$-symmetry breaking can also result in spontaneous polarization ${\boldsymbol P},$ as well as second-order nonlinear optical and optoelectronic effects such as the bulk photovoltaic effect (BPVE), which is prohibited in centrosymmetric material systems [41,43,44]. If these nonlinear effects are weak, however, the surface modes and radiative transfer are still reciprocal.

Angular momentum-carrying thermal radiation from materials with broken symmetries (illustrated in Fig. 2 for the case of a microsphere with a broken ${\cal T}$ symmetry), paves the way toward sculpting near-field Casimir forces [45] and torques [46], generating thrust that leads to translational, rotational, and spinning particle motions. Similarly, the Casimir torque arises between bodies with broken ${\cal I}$ symmetry when their optical axes are not aligned [47].

To summarize, symmetry breaking is a powerful tool to engineer and manipulate thermal radiation, nanoscale forces, and directional charge carrier transport. To make full use of this tool, it is crucial to identify realistic materials and conditions to break symmetry in frequency ranges relevant to thermal radiation (i.e., in the infrared spectrum where modes can be thermally populated [48]). Weyl semimetals [19,28,29,41], oxide perovskites, alkali-metal chalcogenides [49], narrow-gap semiconductors under strain gradients [43] and external magnetic fields [27,50,51], and ferroelectrics under external electric fields [52] show high promise for these applications. Thermal radiation-driven nonlinear effects in ${\cal I}$ symmetry-breaking materials such as the BPVE [49,53–55] offer opportunities for infrared detection and energy harvesting, and recent observations of strong BPVE in gapless materials such as Weyl semimetals [56–58] open up a previously inaccessible energy regime. Additional spectral and polarization control of both linear and nonlinear effects can be achieved via electronic confinement in nanostructures [59–61], strain engineering [43,49,62], and metamaterial design [63–65].

## Funding

DEVCOM Soldier Center (0601102A); Army Research Office (W911NF-13-D-0001, W911NF-19-1-0279); Lincoln Laboratory, Massachusetts Institute of Technology (ACC-777).

## Acknowledgments

We acknowledge funding from the DEVCOM Soldier Center and the Assistant Secretary of the Army for Acquisition Logistics and Training, specifically 0601102A Defense Research Sciences Congressional Funding as well as funding from the Advanced Concepts Committee of the MIT Lincoln Laboratory. We also thank Prof. Gang Chen, Dr. Richard M. Osgood III, and Dr. Mark J. Polking for discussions.

## Disclosures

The authors declare no conflict of interest.

## Data availability

Data underlying the results presented in this paper are available from the corresponding author upon a reasonable request.

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