Accurate calibration of thermophotovoltaic efficiency

Abstract

The new record efficiency in Thermophotovoltaics relies upon a highly reflective rear mirror. The excellent rear mirror boosts voltage by enhancing the luminescence extraction, and separately also reflects low energy photons, which would otherwise be useless in thermophotovoltaics. The reflected low energy photons reheat the thermal emitter, and regenerate above-bandgap energy photons. The efficiency calibration for such regenerative thermophotovoltaics depends on several factors, yet predominantly on the accurate measurement of the rear mirror reflectivity. Here, we report on the technique for accurate measurement of mirror reflectivity, and of record thermophotovoltaic efficiency 29.1 ± 0.6%, at 1207 °C.

Introduction

Thermophotovoltaics utilizes a photovoltaic cell to generate electricity from the thermal radiation of a hot emitter [1]. We can achieve high electrical power from the thermophotovoltaic cells, requiring no moving parts. Thus, it is an attractive power source for hybrid and unmanned vehicles [2, 3], deep space probes [4,5,6], co-generation [7,8,9], concentrated solar energy [10], and compatible with thermal energy storage [11,12,13].

Recently we reported a record thermophotovoltaic efficiency of 29.1% at 1207 °C, by employing a highly reflective rear mirror [14]. Photons with energy smaller than the band-edge of the semiconductor do not get absorbed (Fig. 1a), and are wasted in typical thermophotovoltaic cells. The use of a highly-reflective mirror at the rear of the cell ensures that those photons can be returned back to the thermal source, thus re-heating the emitter. Through this concept of photon regeneration, the thermophotovoltaic power conversion efficiency is tremendously improved. Extremely low parasitic absorption in the thermophotovoltaic cell is the most critical requirement for high thermophotovoltaic power conversion efficiency. This idea of regenerative thermophotovoltaics has further been employed to demonstrate an even higher efficiency record recently, 32% at 1182 °C [15].

Fig. 1

Power conversion in regenerative thermophotovoltaics. Photons with energies below the band-edge of thermophotovoltaic cell are reflected by the mirror to reheat the emitter. This results in photon regeneration and a tremendous efficiency boost. Mirror inefficiency leads to below band-edge loss. Above the band-edge, losses are mostly due to defects in the cell material, Auger recombination as well as thermalization of high energy charge carriers

In this paper, we describe an accurate method to measure the heat to electricity conversion efficiency in regenerative thermophotovoltaics. This is based on an accurate measurement of the reflectivity of the thermophotovoltaic cell. We also use the short-circuit current of the cell to calibrate the incident heat radiation precisely. This provides a very simple and sensitive calibration procedure for regenerative thermophotovoltaics. This further improves the calibration accuracy we previously reported for a record 29.1% TPV efficiency, at an emitter temperature of 1207 °C.

Power conversion efficiency in thermophotovoltaics

In regenerative thermophotovoltaics, any unabsorbed portion of the radiation can be reflected by the thermophotovoltaic cell to re-generate heat at the emitter. As such, heat-to-electricity conversion efficiency at an emitter temperature T_s is defined as:

$$ {\eta}_{thermophotovoltaic}\left({T}_s\right)=\frac{P_{electrical}\left({T}_s\right)}{P_{absorbed}\left({T}_s\right)}=\frac{P_{electrical}\left({T}_s\right)}{\int_0^{\infty}\left\{1-R(E)\right\}{P}_{incident}\left(E, Ts\right) dE\ } $$

(1)

where, P_electrical is the electrical power generated by the thermophotovoltaic cell, and P_incident is the total optical power incident on the cell. P_absorbed is the power absorbed by the cell and E is the photon energy. R(E) is the spectrally-resolved reflectivity of the thermophotovoltaic cell. Calibrating the thermophotovoltaic efficiency, therefore, depends on the accurate calibration of (i) electrical power generated by the cell P_electrical (T_s), (ii) thermal power incident on the cell P_incident (E, T_s), and (iii) thermal radiation reflectivity R(E) of the thermophotovoltaic cell. In the next section, we describe the methods to measure these three factors.

Experimental methods

We use a graphite ribbon as the emitter in our experiment. We accomplish the Joule heating of the emitter by passing an alternating current through the graphite thermal emitter. Beneath the emitter, the thermophotovoltaic cell is placed under an Au-coated Cu baffle. The baffle minimizes any stray photons from reaching the cell. Gold coating on the baffle ensures a very high (~ 98%) reflectivity of the stray photons. These photons would otherwise heat the baffle and the chamber through parasitic absorption. The positive and negative electrodes of the cell are wire-bonded to electrical pads for measuring the current-voltage characteristics. The thermophotovoltaic cell is kept at a standard temperature of 25 °C by active water cooling. For our experiment, we used thermophotovoltaic cells with an In_0.55Ga_0.45As active layer. Details of the cell design and the growth can be found in ref. [14].

The entire setup is placed in a 10^− 5 Torr vacuum to minimize conductive and convective heat transfer, between the emitter and the photovoltaic cell. This also prevents oxidation of the graphite emitter at elevated temperatures.

Results and discussion

The thermophotovoltaic efficiency is the ratio of the electrical power extracted to the power absorbed by the cell. The measurement of the electrical power extracted from the thermophotovoltaic cell, P_electrical, is routine. We make wire-bonds to the cell electrodes, as shown in Fig. 2, and measure the current-voltage response.

Fig. 2

Experimental setup for the thermophotovoltaic efficiency calibration. The graphite emitter is heated by Joule heating. The cell sits under a baffle, to minimize any stray photons reaching the cell. The wire bonds are made separately, to the top and bottom electrodes, for electrons and holes respectively. The cell is cooled to a standard temperature, 25 °C. Current (I) and voltage are measured by wire-bonding the cell to electrical pads

Measurement of the absorbed power depends on the measurement of the cell reflectivity and the incident thermal radiation. Incident thermal radiation on the cell depends on three factors; (i) temperature of the emitter (T_s), (ii) emissivity ε of the graphite emitter, and (iii) the geometric view factor F, which is the solid angle subtended by the emitter as seen from the thermophotovoltaic cell, controlled by a geometric baffle. Together with the surface area A_cell of the thermophotovoltaic cell, the incident power is P_incident(E, T_s) = A_cellεF_effb_s(E, T_s) dE where, b_s(E, T_s)dE=$\frac{2\pi {E}^3}{c^2{\hslash}^3\left(\mathit{\exp}\left(\frac{E}{k_BT}\right)-1\right)} dE$ is the black body power density in units of W/m² radiated from the emitter at temperature T_s, and ε is the graphite emissivity. We are introducing an effective view factor F_eff to take into account the multiple photon bounces. The effective view factor is also dependent on the graphite emissivity owing to multiple photon reflections between the graphite emitter and the thermophotovoltaic cell. In our method of calibration detailed geometric analysis is unnecessary since εF_eff can be obtained directly from the observed short-circuit current.

Using the expression of P_incident, we can measure the power absorbed P_absorbed by the thermophotovoltaic cell as:

$$ {P}_{\mathrm{absorbed}}\left({T}_s\right)={A}_{\mathrm{cell}}\varepsilon {F}_{\mathrm{eff}}{\int}_0^{\infty}\left\{1-R(E)\right\}{b}_{\mathrm{s}}\left(E,{T}_{\mathrm{s}}\right) dE $$

(2)

Note that we are ignoring the photon energy dependence of the emissivity, and its temperature dependence, based on the experimental evidence that such dependency is very weak [16]. We now describe the procedure to calibrate the emitter temperature T_s, and the power absorbed P_absorbed (T_s) by the thermophotovoltaic cell. These can be calibrated directly from the short-circuit current of the thermophotovoltaic cell.

Under illumination from a thermal emitter at temperature T_s, we can express the short-circuit current as:

$$ {J}_{\mathrm{s}\mathrm{c}}\left({T}_{\mathrm{s}}\right)=q{A}_{\mathrm{cell}}\varepsilon {F}_{\mathrm{eff}}{\int}_0^{\infty } EQE(E)\cdot \frac{bs\left(E, Ts\right)}{E} dE $$

(3)

We can isolate εF_eff on the left as:

$$ \varepsilon {F}_{\mathrm{eff}}=\frac{J_{\mathrm{s}\mathrm{c}}\left({T}_{\mathrm{s}}\right)}{q{A}_{\mathrm{cell}}{\int}_0^{\infty } EQE(E)\cdot \frac{bs\left(E, Ts\right)}{E} dE\ } $$

(3')

where q is the electron charge and the external quantum efficiency, and EQE is the probability that a photon incident on the photovoltaic cell generates an electron-hole pair and is electrically extracted. We can write EQE as C×(1-R), where C is the internal quantum efficiency of the thermophotovoltaic cell. C is zero below the bandgap. We can replace EQE with C×(1-R) and change from 0 to E_g the lower limit of the denominator integral:

$$ \varepsilon {F}_{\mathrm{eff}}=\frac{J_{\mathrm{s}\mathrm{c}}\left({T}_{\mathrm{s}}\right)}{q{A}_{\mathrm{cell}}C{\int}_{E_{\mathrm{g}}}^{\infty}\left\{1-R(E)\right\}\cdot \frac{bs\left(E, Ts\right)}{E} dE\ } $$

(4)

where we have pulled C out of the integral. We assume spectrally-averaged C to be relatively constant.

We now discuss our procedure for measuring R(E) and C to measure emitter temperature T_s, using Eq. (4). Once we know the reflectivity R(E) and internal quantum efficiency C at emitter temperature T_s, the corresponding short-circuit current J_sc (T_s) calibrates the emissivity-view factor product εF_eff.

We measure the spectral reflectivity R(E) of the cell using a Fourier Transform Infrared (FTIR) spectrometer. The measured spectral reflectivity is shown in Fig. 3a, averaging about 94.5% sub-bandgap reflectivity. The above bandgap reflectivity of the thermophotovoltaic cell is 34.5%, the Fresnel reflectivity of the front surface (this is unchanged even beyond our above bandgap measurement range, 0.75 eV–1.0 eV). Thus the spectral absorptivity {1-R(E)} in Eq. (4) is calibrated.

Fig. 3

Reflectivity and external quantum efficiency of one of the thermophotovoltaic cells. For this cell, sub-bandgap reflectivity is 94.5%. The above-bandgap reflectivity is the 34.5% Fresnel reflectivity at the air-semiconductor interface. The cell converts 65% of the above-bandgap incident photons to electron-hole pairs, as given by the external quantum efficiency on the right

We calibrate the emitter to a reference temperature T_s = 1085 °C by placing a copper particle on top of the graphite ribbon emitter and monitor its melting point by a change in color. We then measure the reference short-circuit current and the open-circuit voltage at T_s = 1085 °C. We can infer the average internal quantum efficiency C from the measured voltage at the reference temperature T_s = 1085 °C.

Photon absorption primarily happens in the active layer. The generated electron-hole pairs are then efficiently transported to the electrical contacts. The internal quantum efficiency of the thermophotovoltaic cell is the product of the transport efficiency, and the optical absorption fraction inside the active layer. We can calculate the transport efficiency using the familiar base transport efficiency [17] expression T_r = [1-(L/L_D)²], where L is the diffusing distance, and L_D is the diffusion length √(Dτ). In the Appendix, we show that L_D ~ 19 μm. The active layer thickness is 2.5 μm. In our case, the diffusing distance L is half the active layer thickness, L = 1.25 μm for the following reason. The photon recycling events spread the minority carriers evenly throughout the active thickness, and so the average diffusing distance is halved. This produces a transport efficiency T_r = 99.6%, which must be multiplied by the optical absorption fraction [1-exp{−2αL}] = 99.3% for double-pass absorption with α = 10⁴/cm, near the band-edge. The product of T_r and the absorption fraction is C = 98.9%, and we note EQE≡C×(1-R).

Alternately, we measure EQE from spectrally resolved short-circuit, as shown in Fig. 3b. The internal quantum efficiency C = 99.2% estimated from spectrally resolved short circuit current is a close match to the C = 98.9% estimated from the diffusion length.

Eq. (4), combined with known graphite temperature, known short- circuit current, measured absorptivity {1-R}, and measured internal quantum efficiency calibrates the emissivity-view factor product εF_eff = 0.32. For the given emissivity-solid angle product, only 32% of the potential short circuit current was collected.

We can then use this measured εF_eff, for calibration of emitter temperatures other than T_s = 1085 °C, by the changed short circuit current at each temperature. At each temperature T_s, we use the measured C and the spectrally resolved absorptivity {1-R(E)} to obtain the emitter temperature. The calibrated temperatures are shown in Fig. 4a.

Fig. 4

Calibrating graphite emitter temperature by means of short circuit current measurement. In the first step, short-circuit current is at a reference temperature T = 1085 °C, the melting point of copper, on top of the graphite strip emitter, as shown with the inset in (a). The short circuit current in (b) calibrates temperatures at T ≠ 1085C from the corresponding black body spectra at each temperature

We monitor the electrical power extracted from the thermophotovoltaic cell at the corresponding temperatures. The current-voltage response of the thermophotovoltaic cell is shown in Fig. 4b. The emitter temperature T_s and the electrical power generated P_electrical (T_s), are steps toward the thermophotovoltaic cell efficiency η (T_s) ≡P_electrical (T_s)/P_absorbed (T_s). We now describe the procedure for calibrating the denominator P_absorbed (T_s).

We can explicitly determine P_absorbed at emitter temperature T_s, by plugging Eq. (3') into Eq. (2) as follows:

$$ {P}_{absorbed}\left({T}_s\right)=\frac{J_{sc}\left({T}_s\right){\int}_0^{\infty}\left\{1-R(E)\right\} bs\left(E, Ts\right) dE}{q\int_0^{\infty } EQE(E)\cdot \frac{bs\left(E, Ts\right)}{E} dE\ }. $$

(5)

We can substitute EQE with C×(1-R) and change the lower limit on the denominator integral from 0 to E_g, similar to Eq. (4):

$$ {\displaystyle \begin{array}{cc}{P}_{absorbed}\left({T}_s\right)=& \frac{J_{sc}\left({T}_s\right){\int}_0^{\infty}\left\{1-R(E)\right\} bs\left(E, Ts\right) dE}{q\int_{E_g}^{\infty }C\ \left\{1-R(E)\right\}\cdot \frac{bs\left(E, Ts\right)}{E} dE\ }\\ {}\kern5.5em =& \frac{J_{sc}\left({T}_s\right)}{q}\left[\left(\frac{1-{R}_{below}}{1-{R}_{above}}\right)\frac{1}{C}\frac{\int_0^{E_g}{b}_s\left(E,{T}_s\right) dE}{\int_{E_g}^{\infty}\frac{bs\left(E, Ts\right)}{E} dE}+\frac{1}{C}\frac{\int_{E_g}^{\infty }{b}_s\left(E,{T}_s\right) dE}{\int_{E_g}^{\infty}\frac{bs\left(E, Ts\right)}{E} dE}\right]\end{array}} $$

(6)

where, R_below and R_above are the spectral average reflectivities, below and above the band-edge photon energy, respectively. The reflectivities R_below and R_above, averaged over round-trip oscillations, are taken as constant and removed from under the integrals. We have already measured the internal quantum efficiency C = 98.9% in Eq. (6). There are three black-body integrals in Eq. (6) are also exactly known since the temperature T_s is accurately calibrated. The power conversion efficiency, Eq. (1), is a ratio of useful electrical power to the total thermal power absorbed by the thermophotovoltaic cell. Those quantities are shown in Fig. 5a. The corresponding efficiency is in Fig. 5b. We obtain a record 29.1 ± 0.6% efficiency at 1207 °C. We now describe the procedure of error measurement in our thermophotovoltaic experiment.

Fig. 5

Record thermophotovoltaic efficiency. Measured total incident thermal radiation absorbed, and the maximum electrical power generated by the thermophotovoltaic cell at those emitter temperatures, are shown in (a). Power conversion efficiency, the fraction of the absorbed heat that gets converted to the electrical power, is given in (b)

Error analysis

In this section, we measure the accuracy of the thermophotovoltaic efficiency calibration using the cell reflectivity, internal quantum efficiency, and short-circuit current. The accuracy of this method depends largely on temperature calibration using the melting point of copper and then convolving the corresponding black-body spectrum with the measured short circuit current. The reduction in short circuit current relative to the full black-body spectrum is accounted for by the emissivity-solid angle factor, εF_eff. We obtain this emissivity-solid angle factor by measuring the short-circuit current and the internal quantum efficiency.

The thermophotovoltaic efficiency is the ratio of electrical power generated by the thermophotovoltaic cell, to the thermal power absorbed by the cell. We measure the electrical power generated by the thermophotovoltaic cell very precisely with a Keithley 2400 source meter (with nanovolt precision). As such, the error in our efficiency measurement is entirely due to the error in the estimation of P_absorbed with δη/η = |δP_absorbed/P_absorbed|.

The absorbed power at emitter temperature T_s depends on the Planck spectrum b_s(E, T_S), the spectral absorptivity {1-R(E)}, and the internal quantum efficiency C, as shown in Eq. (6). We now show how each of these physical parameters affects the accuracy of thermophotovoltaic efficiency measurement.

The incident power is determined from black body radiation at the reference temperature T_s = 1085 °C, the melting point of copper. We use a very slow temperature ramp (20 °C/min) while increasing the emitter temperature from 25 °C to 1085 °C, during the calibration of εF_eff. This ensures an accurate determination of εF_eff = 0.32. A faster ramp can overshoot the emitter temperature beyond the melting point of copper, and reduce the accuracy of the method. We further confirm the solid-angle subtended F_eff by geometric modeling of the view factor. Similarly, we compare the measured emissivity ε = 0.90 against the reported value of graphite emissivity ε = 0.91 [16] and find a close match.

Once the temperature is well-defined using the short-circuit current, we can know the Planck spectrum b_s(E, T_s) accurately. The uncertainty in measuring P_absorbed then depends entirely on the accuracy of measuring R_below, R_above, and C. Taking partial derivatives with respect to each of these three variables in Eq. (6) and then normalizing by P_absorbed, we obtain (derivation in appendix):

$$ \frac{\mathit{\partial \eta }}{\eta }=\frac{\mathit{\partial}{P}_{\mathrm{abs}\mathrm{orbed}}}{P_{\mathrm{abs}\mathrm{orbed}}}=\sqrt{\left({\left|\frac{\partial R}{1-{R}_{\mathrm{below}}}\right|}^2+{\left|\frac{\partial R}{1-{R}_{\mathrm{above}}}\right|}^2\right){f_{\mathrm{abs},\mathrm{below}}}^2+{\left|\frac{\partial C}{C}\right|}^2} $$

(7)

where, f_{abs, below} represents the sub-bandgap fraction of the absorbed power. We measure the spectral reflectivity above and below the band-edge with the same spectrometer. As such, we can assume δR_above = δR_below = δR, the systematic error in the reflectivity measurement. The emitter temperature being accurately calibrated from the short-circuit current, we can measure f_{abs, below}. The uncertainty in thermophotovoltaic efficiency δη/η can be measured from the uncertainty of measured reflectivity δR and the internal quantum efficiency δC.

The rear reflectivity R of a regenerative thermophotovoltaic cell with an excellent rear mirror is ≥94%. This results in a very small absorptivity a ≡ (1-R) of the cell. For an accurate measurement of thermophotovoltaic efficiency, we need the relative uncertainty δa/a = δR/(1-R) to be small. Thus the δR of the reflectivity measurement needs to be very small. We monitored the reflectivity of a reference gold sample (NIST SRM#1928) to be 98 ± 0.2%, in our FTIR infrared spectrometer. By averaging 200 successive scans, the remaining ±0.2% error is thought to be a systematic error in the instrument rather than a statistical error.

An additional contribution to δR is due to scattering from the electrical grid lines, on top of the thermophotovoltaic cells, as shown in Fig. 2. The electrical grid lines on the top of the thermophotovoltaic cell are 5um wide and 200 μm apart. We measure the reflectivity with a 150 μm spot size, in between the grid lines. The reflectivity on the Au grid-lines is 98%, but at the air-semiconductor interface, the net reflectivity is 94.5%. From linear interpolation, the front surface reflectivity can be 94.6%, including the grids. We make this interpolation taking 2.5% front surface coverage by the gridlines into account. As such, our measured reflectivity has an additional uncertainty of 0.1%. This results in net reflectivity uncertainty 98 ± 0.3%.

We can measure the errors in the internal quantum efficiency δC. We obtain C = 98.9% from the measured open-circuit voltage, as discussed in the appendix. Given the 50% uncertainty in the diffusion length, the uncertainty on our internal quantum efficiency is 98.9 ± 0.9%.

Now using the measured δC and δR we can calculate the efficiency uncertainty δη given by Eq. (7). The relative uncertainty δη/η as a function of emitter temperature is shown in Fig. 6. As the emitter temperature increases, more photons are emitted above the band-edge than below. The error given by δR×f_{abs, below} decreases, while the signal 1-R_above increases with increasing emitter temperature. Therefore, the signal to noise ratio improves with temperature. This is shown in Fig. 6. At 1207 °C, the calibrated thermophotovoltaic efficiency is 29.1%, with an uncertainty 29.1 ± 0.6%. In our previous results [14], the uncertainty was 29.1 ± 0.4%. However, the previous calibration depended on the calibration of emitter temperature using short-circuit current and external quantum efficiency and then convolving the Planck spectrum with the thermophotovoltaic cell absorptivity. Using a different approach based on the carrier collection efficiency, we arrive at the same efficiency, without relying on a separate measurement of external quantum efficiency.

Fig. 6

Accurate calibration of record thermophotovoltaic efficiency. The uncertainty in efficiency is mainly determined by the relative uncertainty of the sub-bandgap reflectivity. As the emitter temperature increases, more photons are emitted above the band-edge than below. This results in a higher signal-to-noise ratio with increasing temperature

We can compare the accuracy of our method to the traditional calorimetric method [11]. Precise calorimetry requires complete minimization of parasitic heat absorption to achieve any reasonable accuracy, which can be challenging. In our approach, we rely on the electrical measurement of short-circuit current and open-circuit voltage, and optical measurement of reflectivity. All of these measurements are routine. Thus we can potentially achieve better accuracy compared to calorimetric measurement of thermophotovoltaic cell efficiency.

The thermophotovoltaic cells were scanned 200 times to reduce the corresponding random error in sub-bandgap reflectivity. The efficiency calibration using the copper melting point for temperature, and then convolving the corresponding black-body spectrum with the measured short-circuit current, provides a simple and precise absolute calibration method in thermophotovoltaics.

Conclusion

We conclude that thermophotovoltaic efficiency can be accurately calibrated through a simple procedure. The melting point of copper calibrates the absolute source temperature, and the short-circuit current of the thermophotovoltaic cell calibrates the relative source temperature. The geometric solid angle factors and the source emissivity thus become known. With a precise measurement of the sub-bandgap reflectivity of the cell, we can accurately determine the heat to the electricity conversion efficiency for the regenerative thermophotovoltaic case, in which sub-bandgap radiation is recycled to the heat source. In the regenerative case, the need for accuracy in the sub-bandgap reflectivity is quite critical. We improved that reflectivity measurement by repeated scans to diminish the statistical errors. Our method calibrates an efficiency of 29.1% at 1207 °C, with a corresponding error 29.1 ± 0.6%.

Appendix

Minority Carrier Diffusion Length

We have done a careful calibration of the internal photo-luminescence efficiency in the presence of photon recycling, taking mirror reflectivity into account. As in many modern photovoltaic cells, the internal photoluminescence efficiencies in our thermophotovoltaic cells are quite high, ~ 85% [14]. This is not as good as ~ 96% photo-luminescence efficiency of the record-breaking GaAs cells [18]. The luminescent photons are absorbed and re-emitted many times. Thus the actual radiative minority carrier lifetime is 1/(1–0.85) ~ 6.7× longer than in the case of the lifetime in optically absorbing surroundings.

To determine the internal quantum efficiency C, we need the diffusion length √(Dτ) which contains the carrier lifetime τ ≡ 1/(BN) where B is the bimolecular radiative recombination co-efficient, and N is the majority carrier doping. But then we need to multiply that lifetime by 6.7, for photon recycling. We used an active n-layer doping N = 10¹⁷ cm^− 3. The minority hole diffusion constant in the n-type InGaAs active layer is D = 7.5 cm²s^− 1_, from [19]. This value was measured in the lattice-scattering dominated regime of mobility. Ahrenkiel et al. reported a radiative recombination coefficient B = 1.43 × 10^− 10 cm³s^− 1 [20] for an air-In_0.53Ga_0.47As-InP structure. Therefore τ ~ 70 nsec. Accounting for photon recycling by multiplying by 6.7, the radiative minority lifetime becomes τ ~ 470 nsec. The non-radiative in these high-quality materials is even longer τ ~ 2 μsec.

This gives us a minority carrier diffusion length = $\sqrt{D\tau}$ ~ 19 μm.

Mathematics of Error Estimation

The electrical power was measured very accurately with a nano voltmeter. As such, the error in our efficiency measurement is solely due to the error in the estimation of P_absorbed, δη/η = |δP_absorbed/P_absorbed|.

We can substitute εF_eff in Eq. (2) from Eq. (3') as follows:

$$ {P}_{absorbed}=\frac{J_{sc}\left({T}_s\right)}{q}\times \frac{\int_0^{\infty}\left\{1-R(E)\right\}{b}_s\left(E,{T}_s\right) dE}{\int_0^{\infty}\left\{1-R(E)\right\}C(E)\ {b}_s\left(E,{T}_s\right)/E\bullet dE} $$

(8)

We can write EQE = C×{1-R(E)}, where internal quantum efficiency C expresses the probability of an absorbed photon to generate electron-hole pair. C is 0 below the bandgap E_g. Separating the super-bandgap and sub-bandgap parts of the integral in Eq. (8), we get,

$$ {P}_{absorbed}=\frac{J_{sc}\left({T}_s\right)}{q}\times \frac{\int_0^{E_g}\left\{1-R(E)\right\}{b}_s\left(E,{T}_s\right) dE}{\int_{E_g}^{\infty}\left\{1-R(E)\right\}C(E)\ {b}_s\left(E,{T}_s\right)/E\bullet dE}+\frac{J_{sc}\left({T}_s\right)}{q}\times \frac{\int_{E_g}^{\infty}\left\{1-R(E)\right\}{b}_s\left(E,{T}_s\right) EdE}{\int_{E_g}^{\infty}\left\{1-R(E)\right\}C(E)\ {b}_s\left(E,{T}_s\right)/E\bullet dE} $$

(9)

At a given temperature, b_s(E, T_s) and J_sc (T_s) are precisely known. For the ease of derivations, let’s define R_below, R_above, Ć to denote the sub-bandgap reflectivity, super-bandgap reflectivity, and the internal quantum efficiency respectively, all the terms being spectrally averaged within the limits of the respective integral. We can now re-write Eq. (9) as:

$$ {\displaystyle \begin{array}{cc}{P}_{absorbed}=& \frac{J_{sc}\left({T}_s\right)}{q}\times \frac{\left(1-{R}_{below}\right){\int}_0^{E_g}{b}_s\left(E,{T}_s\right) dE}{\left(1-{R}_{above}\right)\acute{\mathrm{C}}{\int}_{E_g}^{\infty }\ {b}_s\left(E,{T}_s\right)/E\bullet dE}+\frac{J_{sc}\left({T}_s\right)}{q}\times \frac{\left(1-{R}_{above}\right){\int}_{E_g}^{\infty }{b}_s\left(E,{T}_s\right) dE}{\left(1-{R}_{above}\right)\acute{\mathrm{C}}{\int}_{E_g}^{\infty }\ {b}_s\left(E,{T}_s\right)/E\bullet dE}\\ =& \frac{J_{sc}\left({T}_s\right)}{q}\times \frac{\left(1-{R}_{below}\right){\int}_0^{E_g}{b}_s\left(E,{T}_s\right) dE}{\left(1-{R}_{above}\right)\acute{\mathrm{C}}{\int}_{E_g}^{\infty }\ {b}_s\left(E,{T}_s\right)/E\bullet dE}+\frac{J_{sc}\left({T}_s\right)}{q}\times \frac{\int_{E_g}^{\infty }{b}_s\left(E,{T}_s\right) dE}{\acute{\mathrm{C}}{\int}_{E_g}^{\infty }\ {b}_s\left(E,{T}_s\right)/E\bullet dE}\end{array}} $$

(10)

Now, we can take partial derivatives with respect to R_below, R_above, Ć, and P_absorbed on both sides of Eq. (10). Before we do that, let us call the first term on the RHS of Eq. (10) as A and the second term as Ḃ. As such, P_absorbed = A+Ḃ. Now, taking the derivatives on both sides of Eq. (10), we get,

$$ {\displaystyle \begin{array}{cc}\partial {P}_{absorbed}=& -\frac{\partial {R}_{below}}{\left(1-{R}_{below}\right)}\left({P}_{absorbed}-\dot{\mathrm{B}}\right)+\frac{\partial {R}_{above}}{\left(1-{R}_{above}\right)}\left({P}_{absorbed}-\dot{\mathrm{B}}\right)-\frac{\partial \acute{\mathrm{C}}}{\acute{\mathrm{C}}}\left({P}_{absorbed}-\dot{\mathrm{B}}\right)-\frac{\partial \acute{\mathrm{C}}}{\acute{\mathrm{C}}}\left({P}_{absorbed}-A\right)\\ {}\kern5.25em =& -\frac{\partial {R}_{below}}{\left(1-{R}_{below}\right)}\left({P}_{absorbed}-\dot{\mathrm{B}}\right)+\frac{\partial {R}_{above}}{\left(1-{R}_{above}\right)}\left({P}_{absorbed}-\dot{\mathrm{B}}\right)-\frac{\partial \acute{\mathrm{C}}}{\acute{\mathrm{C}}}\left(2\times {P}_{absorbed}-\dot{\mathrm{B}}-A\right)\\ {}\kern5.25em =& -\frac{\partial {R}_{below}}{\left(1-{R}_{below}\right)}\left({P}_{absorbed}-\dot{\mathrm{B}}\right)+\frac{\partial {R}_{above}}{\left(1-{R}_{above}\right)}\left({P}_{absorbed}-\dot{\mathrm{B}}\right)-\frac{\partial \acute{\mathrm{C}}}{\acute{\mathrm{C}}}{P}_{absorbed}\end{array}} $$

(11)

As such, the root-mean-square error is,

$$ {\displaystyle \begin{array}{l}\;\partial {P}_{abs orbed}=\kern0.5em \sqrt{{\left|\frac{\partial {R}_{below}}{\left(1-{R}_{below}\right)}\;\left({P}_{abs orbed}-\dot{\mathrm{B}}\right)\right|}^2+{\left|\frac{\partial {R}_{above}}{\left(1-{R}_{above}\right)}\;\left({P}_{abs orbed}-\dot{\mathrm{B}}\right)\right|}^2+{\left|\frac{\partial \acute{\mathrm{C}}}{\acute{\mathrm{C}}}{P}_{abs orbed}\right|}^2}\\ {}\kern4.199998em =\kern0.5em \sqrt{{\left|\frac{\partial {R}_{below}}{\left(1-{R}_{below}\right)}\right|}^2\kern0.24em {\left({P}_{abs orbed}-\dot{\mathrm{B}}\right)}^2+{\left|\frac{\partial {R}_{above}}{\left(1-{R}_{above}\right)}\right|}^2{\left({P}_{abs orbed}-\dot{\mathrm{B}}\right)}^2+{\left|\frac{\partial \acute{\mathrm{C}}}{\acute{\mathrm{C}}}{P}_{abs orbed}\right|}^2}\\ {}\begin{array}{cc}\kern4.079998em =& \sqrt{{\left|\frac{\partial R}{\alpha_{below}}\right|}^2\;{\left({P}_{abs orbed}-\dot{\mathrm{B}}\right)}^2+{\left|\frac{\partial R}{\alpha_{above}}\right|}^2\;{\left({P}_{abs orbed}-\dot{\mathrm{B}}\right)}^2+\left|\frac{\partial C}{C}\;{P}_{abs orbed}\right|\;}\end{array}\\ {}\begin{array}{cc}\frac{\partial {P}_{abs orbed}}{P_{abs orbed}}=& \sqrt{{\left|\frac{\partial R}{\alpha_{below}}\right|}^2\;{\left(1-B\right)}^2+{\left|\frac{\partial R}{\alpha_{above}}\right|}^2{\left(1-B\right)}^2+{\left|\frac{\partial C}{C}\right|}^2}\end{array}\\ {}\begin{array}{cc}\kern2.999999em \frac{\partial \eta }{\eta }=& \sqrt{\left({\left|\frac{\partial R}{\alpha_{below}}\right|}^2+{\left|\frac{\partial R}{\alpha_{above}}\right|}^2\right)\;{f_{abs, below}}^2+{\left|\frac{\partial C}{C}\right|}^2}\end{array}\end{array}} $$

(12)

This gives us the expression of relative error of the absorbed power, and in turn the relative error in thermophotovoltaic efficiency. Here, we use δR_below = δR_above = δR, the systematic error in our FTIR measurement. Also, a denotes the absorptivity, with the subscript indicating the relevant part of the spectrum. From the previous section, we know C = Ć, as quantum efficiency does not vary with the emitter temperature for our experiment. Finally, we use B = Ḃ/P_absorbed = 1–f_{abs, below}. The temperature-dependent factor f_{abs, below} tells us the fraction of the thermal radiation absorbed by the cell that is below the bandgap.