‘Squeezing’ near-field thermal emission for ultra-efficient high-power thermophotovoltaic conversion

We numerically demonstrate near-field planar ThermoPhotoVoltaic systems with very high efficiency and output power, at large vacuum gaps. Example performances include: at 1200 °K emitter temperature, output power density 2 W/cm2 with ~47% efficiency at 300 nm vacuum gap; at 2100 °K, 24 W/cm2 with ~57% efficiency at 200 nm gap; and, at 3000 °K, 115 W/cm2 with ~61% efficiency at 140 nm gap. Key to this striking performance is a novel photonic design forcing the emitter and cell single modes to cros resonantly couple and impedance-match just above the semiconductor bandgap, creating there a ‘squeezed’ narrowband near-field emission spectrum. Specifically, we employ surface-plasmon-polariton thermal emitters and silver-backed semiconductor-thin-film photovoltaic cells. The emitter planar plasmonic nature allows for high-power and stable high-temperature operation. Our simulations include modeling of free-carrier absorption in both cell electrodes and temperature dependence of the emitter properties. At high temperatures, the efficiency enhancement via resonant mode cross-coupling and matching can be extended to even higher power, by appropriately patterning the silver back electrode to enforce also an absorber effective surface-plasmon-polariton mode. Our proposed designs can therefore lead the way for mass-producible and low-cost ThermoPhotoVoltaic micro-generators and solar cells.


Fluctuation-Dissipation Theorem (FDT) 1 and Poynting's Theorem (PT)
Consider an isotropic object of relative dielectric permittivity = ′ + i ′′, that can absorb photons, namely ′′ > 0 [for exp(− ) convention]. If it is brought at an absolute temperature and optionally has a voltage across it, FDT states that its thermally excited, randomly fluctuating atoms/molecules can be modeled as current sources ( , ), with spatial correlation function is the mean number of generated photons of frequency in thermo-chemical quasi-equilibrium 3 at voltage and temperature , is the permittivity of free space, ℏ is the Planck constant divided by 2 , is the electron charge, is the Boltzmann constant, ( ′ − ′′) is the Dirac delta function and is the Kronecker delta. In Θ, the additional vacuumfluctuations term 1/2 is omitted, as it does not affect the energy exchange between objects.
PT states energy conservation: the electromagnetic power absorbed inside a volume is equal to that generated due to sources in minus that exiting the enclosing surface area :

Definition of thermal transmissivity for planar systems
The power absorbed by an object , at = 0, due to the thermal current-sources inside another object , at ≠ 0 and with optionally ≠ 0 across it, is , where the time-averaged power per unit frequency ( ) is identified in PT [Eq.(S2)] as A factor of 4 has been added in ( ), since only positive frequencies are considered in the Fourier decomposition of the time-dependent fields into frequency-dependent quantities 4 . The fields ( ) and ( ) due to the sources � � can be calculated via the Green's functions , � ; , � where is the magnetic permeability of free space. Using Eq.(S4), Eq.(S3) can be written Using the FDT from Eq.(S1), where = � = / is the wavevector of propagation and the speed of light, in free space.
Consider now a planar system (uniform in ) of layers (stacked in ) of dielectric permittivities and thicknesses . Then, , � ; , � can be written via their Fourier transforms Then in Eq.(S6), if , and ′ , ′ are the integration variables for the two instances, the integral ∫ gives ( − ′ ) � − ′ � and the integral ∫ gives the (large) transverse system area . Using also ∬ , the power per unit area becomes We define the thermal transmissivity of photons from layer to layer , � , �, via the net power per unit area absorbed by due to thermal sources in , when = 0: Then, from Eq.(S8) and (S9) we get The thermal transmissivity is dimensionless ( has units of ) and it depends on the geometry and materials of the entire photonic system (via ), but not on temperatures or voltages. Furthermore, when the system is reciprocal, � , � = � , �, so, from Eq.(S10), � , � = � , �.

Calculation of thermal transmissivity for planar systems
To calculate � , � via Eq.(S10) requires the evaluation of a difficult double integral. Instead, from PT [Eq.(S2)] and with the Levi-Civita symbol for vector outer products, ( ) in Eq.(S3) is written For thermal power transmission between different objects ( ≠ ), the first term on the right-hand side of Eq.(S11) is zero. Using , from Eq.(S4), the second term becomes Using the FDT from Eq.(S1), For a planar system, using Eq.(S7) and the definition Eq.(S9), can be calculated by the single integral where , now run only through , in Cartesian coordinates or , in cylindrical coordinates.
Eq.(S14) includes two evaluations at the boundaries of layer . Layers 1 and might be semi-infinite (unbounded) or bounded by a Perfect Electric Conductor (PEC) or a Perfect Magnetic Conductor (PMC). When < , the boundary term at , exists, only if layer is finite and there is another layer > with absorption or there is radiation at → +∞ for mode � , �. Similarly when > , for the term at , .
To calculate � , � via Eq.(S14), one can construct a semi-analytical expression for the Green's functions , using a scattering matrix formalism, and then perform the -integration analytically. We use the procedure outlined by Ref. 4, however, with the canonical scattering-matrix formulation 5 : For a two-port with ports and , waves incoming to the ports have amplitudes ,  Now, let � be the scattering matrix between the layers , , and assume that < . Then, the amplitude coefficients at layer relative to those at layer are: [Note: If layer = 1 and semi-infinite, then 1 = 1 = 0.

Definition and Calculation of thermal emissivity for planar systems
We also define the thermal emissivity of photons from layer , � , �, via the net power per unit area emitted outwards by due to its thermal sources, when all other ≠ = 0. Clearly, this power must be the sum of the powers transmitted to all other absorbing layers and the power potentially radiated into infinity, namely transmitted to the two semi-infinite boundary layers, so � , This thermal emissivity is a generalization of the regular ( -independent) emissivity 6 . Using Eq.(S14): With the same procedure, and � = 2 ′ exp�− , if layer is finite, or � = 0, ̃= 2 ′′ , � = 1, if it is semi-infinite, we can also calculate: � , � = ′ 2 � + � � Re� + � + Im�̃� + ��� + � � Re� + � + Im�̃� + ��� − Eq. (S18)| → (S20) As explained in the main text, the transmissivity and emissivity both have maximum value 1 for each of the two decoupled (for isotropic media TE/TM) polarizations.

Calculation of thermal self-transmissivity for planar systems
In some cases, we may want to calculate the power absorbed inside an object due to the thermal sources inside the same object, namely = . This may be useful to model thermal energy exchange between two different absorption mechanisms inside the same object, such as inter-band and free-carrier absorption in a semiconductor and the radiative recombination process, wherein photons absorbed by the former are re-emitted and then absorbed by the latter, e.g. increasing the saturation current of a pn-photodiode. For these cases, the first term on the right-hand side of Eq.(S11) is non-zero. Using from Eq.(S4) and using the FDT [Eq.(S1)] for the expectation value, we get Then, with a similar definition to Eq.(S9) for = , we arrive at the 'thermal form' of Poynting's Theorem [Eq.(S2)], for a layer in a planar system: for TM waves, the integral term in Eq.(S22), Note that, for the term , we are only interested in the (practical) case where layer is finite ( < ∞), otherwise , diverges. If a value is needed for a semi-infinite layer (for example, in Ref. 6), one can be provided by removing the divergent parts in Eq.(S23) and keeping only the structure-related terms: If layer = 1 and semi-infinite, ,1 = 4Re � ,1 If layer = and semi-infinite, , = 4Re � ,

Superposition principle
In a system of multiple objects, each one at temperature and with voltage across it, the thermallyexcited sources inside each object generate photons with mean number Θ ( ). To find the net rate of photons (or power equivalently) emitted by object , one must successively set to zero the temperatures of all objects but one each time, and then apply superposition. Using ( ) = ∑ ( ) If the system is reciprocal, then ( ) = ( ), so Eq.(S24) can also be written as 7,8 Consider the structure in Figure S1a, which is the same as in Figure 1e, but with an additional 'base' region in the semiconductor, which has a very large (bulk) width and is more lightly doped than the thin front (pn-junction 'emitter') region of permittivity from Eq. (7). We use ,base = 5 and doping level , ,base = , /2 = 0.2 / � ∞, , which would correspond, for example, to ≈ 2 × 10 17 −3 electrons at = 0.4 ( = 1200°). Therefore, the permittivity in the 'base' region is taken as

Surface-Plasmon-Polariton emitter and bulk semiconductor absorber
We keep the silver back electrode, as it helps prevent radiation of photons within the semiconductor lightline 7 . We optimize again the efficiency vs load power, at = 1200° with fixed = 0.07 and at = 3000° with fixed = 0.061 , to compare with the thin-film optimized structures at high power. The optimization parameters are , / , / , / .
The results are shown in Figure 3 with thin solid lines, cyan and orange for = 1200° and 3000° respectively. In Figure S1b-e are the TM emitter emissivities and emitter/load power densities for the two power levels 1 and 3 (shown with dots in Figure 3a) at 1200°.
At low power, clearly, the efficiency is significantly lower than in the thin-film absorber case ( Figure  3a). Most of the losses come from free-carrier absorption within the semiconductor bulk (Figure 3b). This is because this single-mode impedance matching requires a much smaller vacuum gap at the same power level (Figure 3d) and thus the emitter SPP couples strongly with the absorber free carriers. At the same time, the bulk absorber geometry means that the semiconductor light-cone is filled with modes, also below bandgap, which also attribute to loss by free-carrier absorption (Figures S1b, S1d). This is more pronounced for smaller load power, where lowmodes play a greater role. At the same power level