Transparent and ‘opaque’ conducting electrodes for ultra-thin highly-efficient near-field thermophotovoltaic cells

Transparent conducting electrodes play a fundamental role in far-field PhotoVoltaic systems, but have never been thoroughly investigated for near-field applications. Here we show, in the context of near-field planar ultra-thin ThermoPhotoVoltaic cells using surface-plasmon-polariton thermal emitters, that the resonant nature of the nanophotonic system significantly alters the design criteria for the necessary conducting front electrode. The traditional ratio of optical-to-DC conductivities is alone not an adequate figure of merit, instead the desired impedance matching between the emitter and absorber modes along with their coupling to the free-carrier resonance of the front electrode are key for optimal device design and performance. Moreover, we demonstrate that conducting electrodes ‘opaque’ to incoming far-field radiation can, in fact, be used in the near field with decent performance by taking advantage of evanescent photon tunneling from the emitter to the absorber. Finally, we identify and compare appropriate tunable-by-doping materials for front electrodes in near-field ThermoPhotoVoltaics, specifically molybdenum-doped indium oxide, dysprosium-doped cadmium oxide, graphene and diffused semiconductors, but also for ‘opaque’ electrodes, tin-doped indium oxide and silver nano-films. Predicted estimated performances include output power density ~10 W/cm 2 with >45% efficiency at 2100 °K emitter temperature and 60 Ω electrode square resistance, thus increasing the promise for high-performance practical devices.

; note that the optimized efficiency is fairly insensitive to both 'convergence layers' in most cases, so the convergence of Figs. (g,h) is not perfect. Figure S5. Optimization results vs load power per surface area for the structure of Fig.1 with some of the electrode materials of Fig.4a, with absorber designed to support a single mode (dotted curves, same as those in Figs.4b and S4f) and two modes (solid curves): (a) Efficiency ; single-mode design is more efficient in most cases, even after ignoring the radiative recombination inside the two-mode absorber. (b) Normalized semiconductorabsorber thickness ; two-mode absorber is considerably thicker (as expected), so the depletion region cannot extend throughout it.

Accuracy of simplified pn-junction electronic modeling
To estimate the order of magnitude of the error of our simplified pn-junction electronic modeling, we will use some standard pn-junction analytical formulas on the structure of Figure 2 at the highest-efficiency operating point ,1 . The bandgap is = 0.72 and the optimization results include / Carnot = 0.98 ⇒ = 0.605 (see Figure S1a), / = 0.033 ⇒ = 56.5 (see Figure S1b) and = 67 / 2 , where is the 'dark' recombination current per surface area of the PV cell.
We will use material parameters for GaxIn1-xAsySb1-y semiconductors from Ref.
In the calculation of we assumed that it is entirely due to radiative recombination throughout the absorber, so we can extract the radiative-recombination coefficient via and ℎ = � ℎ ℎ = 16.7 , which are much larger than the absorber thickness, the preferred scenario for a 'short' PV diode, so that bulk recombination in quasi-neutral regions is minimized 4 . Therefore, even if one were to calculate the precise solution (via the drift-diffusion equations) for the minority-carrier distributions and the recombination current, those should not depend on the diffusion lengths and ℎ , and the relevant dimensions shall be the absorber thickness and depletion-region width .
This will also hold for the recombination current due to non-radiative mechanisms. Considering that the n 'base' occupies almost entirely the absorber, we will conservatively (and for simplicity) assume that non-radiative recombination of minority holes occurs throughout the absorber (namely also in the depletion region ). Thus we can get a rough estimate for the current due to each non-radiative mechanism 5 : -Very low surface-recombination velocities down to = 10 / (and below) have been experimentally demonstrated both for GaAs 6,7 and for black Si 8,9,10 , so we can estimate surfacerecombination current = 2 / • = 23 / 2 . -Very long Shockley-Read-Hall (SRH) recombination lifetimes of around = 1 have been measured experimentally 3,11,12,6 and even larger are estimated 13 for record-efficiency GaAs solar cells 14 , so we can estimate SRH-recombination current = 2 / • / = 13 / 2 .
Since the above analysis is quite approximate, in Figure S6a, we plot the efficiency of the system of Figure 2, when the recombination current is doubled from the value calculated using only radiative recombination, and we find that efficiency drops at most by 2.4%. Moreover, in practice the PV cell will likely include window layers for passivation, so the minority carriers will be decoupled from the electrodes and, no matter what their precise distribution inside the absorber may be, there should be little modification to their radiative recombination rates inside the ultra-thin absorber film compared to our simple assumption of constant quasi-Fermi levels. Therefore, we do not expect the efficiency error due our simplified electronic modeling and negligence of non-radiative recombination to exceed 3% in most cases discussed in this article.
An exception may be the case of 'opaque' electrodes: We conservatively limited the thickness of the 'convergence layers' to 0.005 , so that they are only a small perturbation to the real system. Then the optimal front 'convergence layer' is clamped at maximum thickness in Figure S1c in the 'opaque' frequency regime and in Figure S4g for the 'opaque' electrodes ITO and Ag. Instead, if we allow their thicknesses to be unlimited, in Figure S6, we show that a substantial increase in the 'opaque' efficiency is observed. This leads us to believe that the performance evaluation of 'opaque' electrodes may be more sensitive to precise electronic modeling of the pn junction.  Fig.1 with model electrode ℏ = 0.0072 + 0.04ℏ , at = 2100°, = 300°, = 60Ω and with = 4 = 0.72 , with upper limit 0.005 for the 'convergence layers' (solid curves, same as those in Figs.2a and S1c) and without upper limit (dashed curves): (a) Efficiency ; allowing for thicker front 'convergence layer' results in significantly increased efficiency in the 'opaque' frequency regime ( > ), thus hinting that the 'opaque' results are quite sensitive to precise modeling of the electronic details of the pn-junction; also shown in dotted curve the efficiency when the recombination current is doubled compared to that in the solid curve, leading to a drop of at most 2.4%. (b) Normalized front and back 'convergence layer' thicknesses and .