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.

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*_{cell}*εF*_{eff}*b*_{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/m^{2} 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.

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{−2*αL*}] = 99.3% for double-pass absorption with *α* = 10^{4}/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.

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.

### 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 δ*R*×*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.

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.