Near-field radiative heat transfer in many-body systems

Many-body physics aims to understand emergent properties of systems made of many interacting objects. This review examines recent progress on the topic of radiative heat transfer in many-body systems consisting of thermal emitters interacting in the near-field regime. Near-field radiative heat transfer is a rapidly emerging field of research in which the cooperative behavior of emitters gives rise to peculiar effects that can be exploited to control heat flow at the nanoscale. Using an extension of the standard Polder and van Hove stochastic formalism to deal with thermally generated fields in $N$-body systems, along with their mutual interactions through multiple scattering, a generalized Landauer-like theory is derived to describe heat exchange mediated by thermal photons in arbitrary reciprocal and nonreciprocal multiterminal systems. In this review, this formalism is used to address both transport and dynamics in these systems from a unified perspective. The discussion covers (i) the description of nonadditivity of heat flux and its related effects, including fundamental limits as well as the role of nanostructuring and material choice; (ii) the study of equilibrium states and multistable states; (iii) the relaxation dynamics (thermalization) toward local and global equilibria; (iv) the analysis of heat transport regimes in ordered and disordered systems composed of a large number of objects, density, and range of interactions; and (v) the description of thermomagnetic effects in magneto-optical systems and heat transport mechanisms in non-Hermitian many-body systems. The review concludes with a listing of outstanding challenges and promising future research directions.

Figure 1

(a) Far-field radiative heat transfer between two infinite parallel plates (media 1 and 2) separated by a vacuum gap. In this scenario, the gap size $d$ is much larger than the thermal wavelength ${\lambda }_{\mathrm{Th}}$, and the two plates exchange heat only via propagating waves. The evanescent waves generated in the vacuum gap by total internal reflection are not able to reach the second plate and do not contribute to the heat transfer. (b) When $d<{\lambda }_{\mathrm{Th}}$ the tunneling of evanescent waves can give a significant contribution to the radiative heat transfer and in this way the Planckian (or blackbody) limit can be greatly overcome in this near-field regime.

Figure 2

Fluctuational electrodynamics. Schematic of radiative heat transfer in a two-body system. The two bodies of volumes $V_1$ and $V_2$ have temperature profiles $T_1(\mathbf{r})$ and $T_2(\mathbf{r})$ and frequency-dependent dielectric functions ${\epsilon }_1(\mathbf{r},\omega )$ and ${\epsilon }_2(\mathbf{r},\omega )$. Electromagnetic fields $\mathbf{E}$ and $\mathbf{H}$ are generated by the random currents $\mathbf{J}$ in the bodies due to their nonvanishing correlations given by the fluctuation-dissipation theorem. The net power exchanged by the two bodies is determined by the total transmission $\mathcal{T}$ that can be expressed as a sum of individual transmission coefficients ${\tau }_n$.

Figure 3

(a) Heat-transfer coefficient at room temperature (300 K) as a function of the gap size for two infinite thick parallel plates made of Au. The different lines correspond to the total contribution (black solid line) and to the contributions of propagating and evanescent waves for TE and TM polarizations. The horizontal line shows the result for two black bodies: $6.124\mathrm{W}/({\mathrm{m}}^2\mathrm{K})$. (b) The spectral heat flux (or conductance per unit area and frequency) as a function of the radiation frequency corresponding to the case in (a). The solid lines correspond to three different values of the gap size in the near-field regime, while the blue dashed line is the result for two black bodies. (c),(d) The same as in (a),(b) for ${\mathrm{SiO}}_2$.

Figure 4

(a) Schematic illustration of NFRHT measurement configuration used by [107]. The emitter microdevice is composed of a square mesa and Pt heater or thermometer suspended on a thermally isolated island. The receiver is a macroscopically large ($1\times 1{\mathrm{cm}}^2$) plate. (b) The corresponding heat flux vs gap size in the case of an emitter and an receiver made of ${\mathrm{SiO}}_2$. Measured data (red squares) are compared to the theoretical result (solid black line) obtained within FE. From [107].

Figure 5

(a) Heat-transfer coefficient for two parallel plates made of $n$-doped InSb at room temperature (300 K) as a function of the gap size for different values of a magnetic field applied parallel to the surfaces of the plates ($x$ direction). Inset: ratio between the zero-field coefficient and the coefficient for different values of the field in the near-field region. (b) The corresponding spectral heat flux as a function of the frequency (and wavelength) for a gap of $d=10\mathrm{nm}$ and different values of the parallel field. From [236].

Figure 6

Computed heat-transfer coefficient as a function of gap size for the multilayer system shown in the inset at room temperature (300 K). This structure comprises a thick, semi-infinite silica surface separated by a vacuum gap of size $d$ from a silica thin film coating on a semi-infinite Au surface. Different curves correspond to different thicknesses of the silica coating. Adapted from [317].

Figure 7

(a) Schematic of two doped-Si metasurfaces made of 2D periodic arrays of square holes placed on semi-infinite planar substrates and held at temperatures $T_1$ and $T_2$. (b) Room-temperature heat-transfer coefficient as a function of the gap size for the doped-Si metasurfaces of (a) with $a=50\mathrm{nm}$ and a filling factor of 0.9 (black line). For comparison, the plot also includes the results for the Si metasurfaces computed with an effective medium theory (orange dashed line), ${\mathrm{SiO}}_2$ parallel plates (blue line), and doped-Si parallel plates (red line). The horizontal dashed line shows the blackbody limit. From [105].

Figure 8

Comparison of the NFRHT rate between $\mathrm{graphene}\hspace{0.17em}\hspace{0.17em}(\mathrm{Gr})/\mathrm{Si}{\mathrm{O}}_2$ pair (red solid line) and ${\mathrm{SiO}}_2$ pair (blue solid line) with various gap sizes. The temperatures of the emitter and the receiver are 323.2 and 301.5 K, respectively. Lines show the calculated values and spheres are the average values of four repeated measurements at each point. Inset: schematic illustration of the $\mathrm{Gr}/\mathrm{Si}{\mathrm{O}}_2$ heterostructure. The blackbody limit has been plotted for comparison (black dashed line). From [309].

Figure 9

Transferred power $H_s$ by NFRHT between a ${\mathrm{SiO}}_2$ sphere with radius $R=5\mu \mathrm{m}$ at 300 K and a ${\mathrm{SiO}}_2$ plane at 0 K as a function of distance $d$. The transferred power is normalized to the power emitted by a blackbody with a surface area given by the cross section of the sphere. From [175].

Figure 10

Collage of selected computational methods. Schematics of basis functions, along with selected results, for spectral ([216]; [213]; [50]; [226]), finite-difference ([290]; [355]; [149]), volume-integral ([276]), and surface-integral ([287]; [291]; [292]) methods.

Figure 11

(a) Absorption cross section of spherical silver nanoparticles with respect to the wavelength. (b) Orthogonal and parallel configurational resonance frequencies for a dimer of silver nanoparticles ($R=10\mathrm{nm}$) in vacuum with respect to their separation distance $d$. The red horizontal line represents the plasmon resonance of an isolated particle. (c),(d) Absorption cross sections for a dimer of silver nanoparticles ($R=10\mathrm{nm}$) normalized by the absorption of a single particle. From [282].

Figure 12

Power flow exchanged between two SiC nanoparticles at $T_1=300\mathrm{K}$ (red line) and at $T_2=0\mathrm{K}$ (blue line) in the presence of a third SiC nanoparticle at temperature $T_3=0\mathrm{K}$ (gray line) and normalized by the power exchanged between two isolated particles, i.e., ${\phi }_{12}^{*}={\mathcal{P}}_{1\to 2}(T_1,T_2,T_3)/{\mathcal{P}}_{1\to 2}(T_1,T_2)$. From [23].

Figure 13

Transmission coefficient ${\mathcal{T}}_{21}$ (a) between two SiC nanoparticles and (b) between two SiC nanoparticles in the presence of a third SiC nanoparticles as in Fig. 12 for $d=0$. The dashed line marks the region where the particles would touch. The unphysical region beyond this line is shown to illustrate the hybridization mechanism of dipolar resonances, which can be seen in that region. From [23].

Figure 14

(a) Spectral conductance as a function of the energy for InSb cubes with a cube side of $1\hspace{0.17em}\hspace{0.17em}\mu \text{m}$ separated by a 500 nm gap at $T=300\mathrm{K}$, and for various values of the magnetic field $H$ applied along the $z$ direction. Inset: discretization geometry. The number of dipoles per cube is 4913 (each one has an edge length of 59 nm). From [103]. (b) Spectral heat flux between a silica probe and a silica surface. From [101].

Figure 15

Definition of reflection and transmission operators associated with an individual body. From [216].

Figure 16

Two planar slabs (bodies 1 and 2) are placed at distance $d$ and separated by vacuum. A particle of polarizability $\alpha$ is placed at distance $d_1$ from slab 1. From [240].

Figure 17

Nonadditive correction to the two-body heat flux $\mathrm{\Delta }{H}^{1\to 2}=\mathrm{\Delta }\mathrm{\Phi }$ [see Eq. (81)] in the presence of an atomic, i.e., pointlike, particle of polarizability $\alpha$. The upper curve corresponds to the near-field configuration $d=10\mathrm{nm}$, while the lower curve corresponds to the far field ($d=10\mu \mathrm{m}$). From [240].

Figure 18

Heat-flux amplification ${\mathrm{\Phi }}_{3\mathrm{s}}(d,\delta )/{\mathrm{\Phi }}_{2\mathrm{s}}(d)$ in a three-body configuration compared to a two-body configuration shown in the inset as a function of distance $d$ and thickness of the intermediate slab $\delta$. The black dashed line corresponds to the constant value ${\mathrm{\Phi }}_{3\mathrm{s}}(d,\delta )/{\mathrm{\Phi }}_{2\mathrm{s}}(d)=1$. From [218].

Figure 19

Heat flux exchanged between two slabs immersed in a thermal bath with respect to their separation distance $d$. Slab 1 has a fixed temperature of $T_1=400\mathrm{K}$, while the thermal bath is at $T_3=300\mathrm{K}$. The second slab, of thickness $\delta$, thermalizes to the equilibrium temperature at which the net flux it receives vanishes. From [188].

Figure 20

(a) Phase portrait (i.e., trajectories of temperatures) in a bistable system consisting of two membranes of ${\mathrm{SiO}}_2$ and ${\mathrm{VO}}_2$ in interaction with two thermal baths for different initial conditions. The green (red) points denote the stable (unstable) global steady-state temperatures. From [178]. (b) Self-oscillation of the temperature of a ${\mathrm{VO}}_2$ membrane in the vicinity of a ${\mathrm{SiO}}_2$ substrate when adding a specific external constant power $F_{\mathrm{ext}}$. From [92].

Figure 21

Left panels: and gate made with two ${\mathrm{SiO}}_2$ membranes (gates) suspended between a thermal ${\mathrm{SiO}}_2$ source and a ${\mathrm{VO}}_2$ drain. The color map represents the output temperature $T_D$ of the drain with respect to the two input temperatures $T_{G1}$ and $T_{G2}$ of the two gates. At the bottom is the truth table for the and gate. From [20]. Right panels: nor gate designed using a coupling of SiC and $\mathrm{V}{0}_2$ nanoparticles. From [158].

Figure 22

Top panels: dispersion curves (real part) of collective plasmonic modes along a chain of copper nanoparticles (10 nm radius) dispersed in vacuum in the case of dipolar moments ($\ell =1$), and for the multipolar moments of order $\ell =5$. Bottom panel: thermal conductance $G$ of linear chains of copper particles calculated from the kinetic theory for different multipole orders $\ell$ vs the separation distance $d$ normalized to the particle diameter $2a$. Inset: enlargement of the near-contact region. From [28].

Figure 23

(a) Thermal conductance of colloidal crystals made up of spheroidal SiC nanoparticles as a function of their horizontal radius. (b) Thermal conductivity of coplanar disk arrays for different diameters and separations at temperature $T=300\mathrm{K}$. From [257], and [283].

Figure 24

(a) Thermal conductance of Vicsek fractal structures as a function of the normalized gyration radius. From [252]. (b) Thermal conductance between two Ag nanoparticle clusters at various fractal dimensions. From [209].

Figure 25

(a) Illustration of the classical Coulomb drag effect. A drag electric current $I_d$ in a passive conducting wire is induced by a primary current $I$ flowing in a driving conductor placed close by. (b) Radiative drag effect in a many-body system: a drag heat flux $J_d$ carried by thermal photons between two particles is induced by a heat flux $J$ exchanged between two thermostatic objects in a many-body system. From [15].

Figure 26

(a) Schematic of a multitip SThM platform with three tips. Nanospheres (thermal emitters) are grafted on single scanning probe tips and held close to a substrate. Their temperatures and positions are individually controlled. (b) Normal component $⟨S_z⟩$ of the Poynting vector radiated through the substrate surface at $z=0$ by a three-tip SThM setup with glass nanoemitters at $T=300\mathrm{K}$. (c) Similar to (b) but with $T_2=T_3=350\mathrm{K}$ (red curve) and $T_1=300\mathrm{K}$ (blue curve). Inset: flux at $z=0$ for a single particle at $T=300\mathrm{K}$. (d),(e) Magnitude of Poynting vector field in the $(x,z)$ plane radiated by a multitip setup in (d) for an angular opening of $\theta =20°$ and in (e) for an angular opening $\theta =80°$. From [16].

Figure 27

(a) Heat flux between two nanoparticles at interparticle distance $d$ using coupling via the surface modes of an interface. From [87]. (b) Conductance ratio $G/G^{(0,0)}$ ($G$ conductance with interface and $G^{(0,0)}$ without interface) as a function of $d$ between two Au nanoparticles placed at distance $z=150\mathrm{nm}$ from a SiC substrate. The four lines correspond to the absence of graphene (black solid line), and to configurations with graphene having $\mu =0.1\mathrm{eV}$ (red dashed line), 0.3 eV (blue dot-dashed line), and 0.5 eV (orange dotted line). Inset: spectral conductance associated with the four same configurations. From [223].

Figure 28

(a) Sketch of heat flux between two nanoparticles through a hyperbolic multilayer metamaterial. (b) Exchanged power $\mathrm{\Phi }$ as a function of interparticle distance $L$ normalized to the exchanged power ${\mathrm{\Phi }}_0$, where the hyperbolic multilayer metamaterial has been replaced by vacuum. The numbers in the brackets give the distance $L$ (in $\mu \mathrm{m}$) and the amplification factor $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ at the maximum. From [364].

Figure 29

Time evolution of thermal state in a single-body (red curve), a two-body (black curve), and a three-body system (blue curve) of SiC nanoparticles with radii of 50 nm in a bath at temperature $T=300\mathrm{K}$. The distance between particles 1 and 2 is 400 nm, while the distances (solid line for dipole 1, dashed line for dipole 2, and dot-dashed line for dipole 3). From [227].

Figure 30

Distance dependence of the relaxation time $\tau ={\mathrm{\Gamma }}^{-1}$ of a nanoparticle above a substrate with temperature $T_b=300\mathrm{K}$ (top panel) for a gold nanoparticle above a gold surface, and (bottom panel) for a SiC nanoparticle above a SiC surface. We use ${\rho }^{\mathrm{Au}}{C}_{\mathrm{p}}^{\mathrm{Au}}=2.404\times {10}^6\mathrm{J}\hspace{0.17em}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$ and ${\rho }^{\mathrm{SiC}}{C}_{\mathrm{p}}^{\mathrm{SiC}}=2.212\times {10}^6\mathrm{J}\hspace{0.17em}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$. From [336].

Figure 31

(a) Normalized heat flux between two spheroidal nanoparticles with respect to the orientation of a third particle placed in between. From [250]. (b) Graphene-based heat-flux splitter. Three graphene disks with different Fermi levels controlled by external gating exchange thermal energy in the near-field through many-body interactions. The magnitude of heat flow from 1 to 2 and 1 to 3 can be controlled by an appropriate tuning of the Fermi level of the graphene disks 2 and 3. The thermal power exchanged in the near-field between graphene disks of 100 nm radius vs the separation distance in a three-body system. From [17].

Figure 32

(a) Radiative thermal transistor made of a three-terminal system composed of a ${\mathrm{SiO}}_2$ source, a ${\mathrm{VO}}_2$ gate, and a ${\mathrm{SiO}}_2$ drain. The gate is a layer based on a phase-change material and its temperature can be actively controlled around its local equilibrium value $T_G^{\mathrm{eq}}$ by an external thermostat, while the temperatures $T_S=360\mathrm{K}$ and $T_D=300\hspace{0.17em}\hspace{0.17em}\mathrm{K}$ of source and drain are fixed so that $T_S>T_D$. (b) Radiative fluxes ${\mathrm{\Phi }}_S$, ${\mathrm{\Phi }}_D$, and ${\mathrm{\Phi }}_G$ exchanged between the different parts inside the transistor. (c) Amplification factor with respect to the gate temperature. From [18].

Figure 33

Radiative heat pumping by modulation of control parameters in a three-particle system that is sketched in the inset. The three particles are made of SiC. In this case, particles 1 and 2 are thermostated at temperature $T_1=T_2$, while the temperature $T_3$ and the $x_3$ coordinate of particle 3 can be modulated with respect to time. Powers ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$ absorbed by particles 1 (solid red line) and 2 (dashed black line) as a function of time for the a periodic variation of the coordinate and temperature of particle 3 of frequency $\omega =2\pi \hspace{0.17em}\hspace{0.17em}{\mathrm{s}}^{-1}$ and amplitudes $\mathrm{\Delta }x=100\mathrm{nm}$ and $\mathrm{\Delta }T=5\mathrm{K}$ around $x_3=0$ and $T_3=300\mathrm{K}$. We have $d=600\mathrm{nm}$ and $y_3=300\mathrm{nm}$, and the radius of the particle is $R=50\mathrm{nm}$. Inset: geometrical configuration of a three-particle system where the position of particle 3 is periodically modulated. From [220].

Figure 34

Types of heat transport regimes in an $N$-body system. When an element (red) is heated up its heat spreads through the system by either (a) a classical Gaussian diffusion process or (b) an anomalous process. The trajectories correspond to random walks with a Gaussian and a non-Gaussian probability distribution function, respectively. Here the non-Gaussian process is a Levy flight with an algebraic PDF.

Figure 35

Top panel: thermal conductance $G$ in log-log scale along a chain of SiC spherical particles of 100 nm radius with different interparticle distances $h$ and different particle numbers $N$ as a function of the separation distance $\mathrm{\Delta }x=|\mathbf{r}-{\mathbf{r}}^{\prime }|$ at temperature $T=300\mathrm{K}$. From [30]. Bottom panel: heat-transfer coefficients $h_{\ell ,j}$ with respect to the normalized separation $z_{\ell ,j}/z_N$ in a dilute multilayer system made with 200 nm thick SiC layers separated by a distance $d=40\hspace{0.17em}\hspace{0.17em}\text{nm}$ at $T=300\mathrm{K}$. From [186].

Figure 36

Heat-transfer coefficient $h_{\ell ,j}$ with respect to the normalized separation $z_{l,j}/z_N$ in a dense multilayer system made with 200 nm thick SiC layers separated by a distance $d=5\mathrm{nm}$ at $T=300\mathrm{K}$. Inset: $h_{\ell ,j}$ decomposed into TE and TM polarization contributions. From [186].

Figure 37

Temperature profile as a function of the normalized position $z_j/z_N$ along a multilayer made with 200 nm thick $N=60$ SiC layers for different separation distances $d$ and at a fixed distance $D=500\mathrm{nm}$ from the thermostats. From [186].

Figure 38

(a) Giant thermal magnetoresistance along linear chains of InSb and InSb/Ag nanoparticles at $T=300\mathrm{K}$ as a function of the strength of an external magnetic field $B$ applied in the direction orthogonal to the chain axis. From [192]. (b) Anisotropic magnetoresistance between two InSb nanoparticles with respect to the orientation of the magnetic field. From [102].

Figure 39

Normalized mean Poynting vector $⟨\mathbf{S}⟩$ of thermal radiation emitted by an InSb nanoparticle at the origin of the coordinate system with radius of 300 nm at a temperature of 300 K into a cold environment (vacuum) at $T_b=0\mathrm{K}$ when a magnetic field is applied in $z$ direction. This circular heat flux persists in global thermal equilibrium. From [261].

Figure 40

Normalized mean Poynting vector $⟨\mathbf{S}⟩$ and its magnitude ($\mathrm{W}\hspace{0.17em}{\mathrm{m}}^{-2}$ in color scale) of thermal radiation emitted by three InSb nanoparticles with a radius of 300 nm having the same temperatures $T_1=T_2=T_3=300\mathrm{K}$ when a magnetic field is applied in the $z$ direction. From [264].

Figure 41

Heat-transfer spectra in a many-body system consisting of six InSb nanospheres placed at the vertices of a regular hexagon on the $x\text{−}y$ plane without (reciprocal) and with (nonreciprocal) an externally applied magnetic field in the $z$ direction. From [370], and [371].

Figure 42

Photon thermal Hall effect. A four-terminal junction with magneto-optical particles forming a square with $C_4$ symmetry is submitted to an external magnetic field $\mathbf{B}$ in the direction orthogonal particle plane. When a temperature gradient $\mathrm{\Delta }T=T_L-T_R$ is applied between the particles $L$ and $R$, a Hall flux transfers heat transversally between particles $B$ and $T$, thus bending the overall flux (red arrow) toward the top or the bottom. In this case the heat ${\mathcal{P}}_{i\to j}$ and ${\mathcal{P}}_{j\to i}$ exchanged between two particles $i$ and $j$ is not symmetric.

Figure 43

Magnetic-field strength dependence of the Righi-Leduc-like coefficient defined in Eq. (135) for four spherical InSb nanoparticles with a radius of 100 nm in a $C_4$-symmetric configuration as depicted in Fig. 42 choosing an interparticle distance of opposite particles of $d_a=500$ and $700\mathrm{nm}$. From [264].

Figure 44

(a) Sketch of the diode in the forward direction. Two InSb nanoparticles above an InSb substrate. The left particle is heated with respect to the other particle and the environment. (b) Sketch of the diode in the backward direction. (c) Normalized mean spectral in-plane Poynting vector and its amplitude ($\mathrm{J}\hspace{0.17em}{\mathrm{m}}^{-2}$; colorbar) for the $m=+1$ particle resonance for the diode in the forward direction. (d) Similar to (c) but for the backward case and the $m=-1$ particle resonance. See also [262].

Figure 45

Rectification coefficient from Eq. (137) for two InSb nanoparticles with a 100 nm radius 500 nm above an InSb substrate, as sketched in Fig. 44, as a function of the interparticle distance $d$ for different magnetic-field amplitudes. See also [262].