Quantum yield in induced-junction silicon photodiodes at wavelengths around 400 nm

Photodiodes are the workhorses in radiometry and photometry and are extensively used as transfer standard detectors. Silicon photodiodes are especially interesting as they are sensitive over the visible spectral range, are highly uniform, fast, fairly stable, cheap and predictable. It is beneficial to have predictable standards as this enables independent realisations of spectral response with silicon photodiodes and fewer calibration points when used as transfer standard detectors only.

The quantum yield $y\left( \lambda \right)$ is defined as the ratio of the number of generated electron–hole pairs and the number of absorbed photons in a semiconductor photodiode. It is one of the quantities determining the spectral responsivity $R\left( \lambda \right)$ of a photodiode which is defined as the ratio of the photocurrent and the incident radiant power generating this photocurrent. The 3D simulation model used for the prediction of the spectral responsivity does not include quantum yield. Thus, the knowledge of the quantum yield is necessary to accurately model the spectral responsivity of a predictable quantum efficient detector (PQED) independently.

The quantum yield equals unity if the energy of the absorbed photon is between one and two times the band gap energy E_g. For silicon photodiodes with a band gap energy of 1.12 eV this is the wavelength range from about 550 nm to 1100 nm.

The quantum yield can exceed unity if one of the generated charge carriers, the electron or the hole, has enough energy to create a second electron–hole pair via impact ionisation. To achieve impact ionisation the photon energy must exceed 2E_g. However, because of the requirement of simultaneous energy and momentum conservation in combination with the band structure of silicon, a quantum yield above unity occurs only for photon energies larger than 2E_g. The longest wavelength at which a quantum yield above unity has been derived for silicon from experimental data varies between 365 nm and 415 nm [1–3] which correspond to photon energies of 3E_g and 2.6E_g, respectively.

At high photon energies above 12.4 eV, which corresponds to wavelengths shorter than 100 nm, the quantum yield in silicon seems to be proportional to the energy of the absorbed photon [4].

The quantum yield can be calculated from the measured spectral responsivity. This, however, requires a detailed knowledge of the reflectance losses and the internal losses of a photodiode. Whereas reflectance losses can be measured or calculated with low uncertainty, the estimation of the internal losses is difficult as it requires either the knowledge of quantum yield itself or a sophisticated physical model of the internal losses.

Models of the quantum yield are so far semiempirical and suffer from free parameters [3, 5] and inaccurate experimental data and are up to date far away from being applicable at the 100 ppm level. This is crucial for the attempt to establish a new primary detector standard for radiant power measurements, called PQED [6, 7]. On the other hand, the concept of the PQED equipped with specially designed induced-junction silicon photodiodes allows the prediction of the internal losses of the photodiode and thus, for the first time makes it possible to precisely estimate the quantum yield.

In the following, the determination of the quantum yield in induced-junction silicon photodiodes is described and a model for the quantum yield in the wavelength range from 360 nm to 500 nm is developed.

The external quantum efficiency $Q\left( \lambda \right)$ of a semiconductor photodiode is given by

$$\begin{align}Q\left( \lambda \right) = \left( {1 - \rho \left( \lambda \right)} \right)y\left( \lambda \right)\left( {1 - \delta \left( \lambda \right)} \right).\end{align} \tag{ 1 }$$

The parameters $\rho \left( \lambda \right)$ and $\delta \left( \lambda \right)$ describe the external losses due to the reflectance of the photodiode detector and the internal losses due to recombination of charge carriers generated by the incident photons, respectively. Equation (1) can be rearranged for the determination of the quantum yield:

$$\begin{align}y\left( \lambda \right) = Q\left( \lambda \right)\left( {1 + \rho \left( \lambda \right) + \delta \left( \lambda \right)} \right).\end{align} \tag{ 2 }$$

Here, it is assumed $\rho \left( \lambda \right) \ll 1$ and $\delta \left( \lambda \right) \ll 1$.

$Q\left( \lambda \right)$ can be calculated from the measured spectral responsivity $R\left( \lambda \right)$ via

$$\begin{align}Q\left( \lambda \right) = R\left( \lambda \right)/\left( {\frac{{e\lambda }}{{hc}}} \right)\end{align} \tag{ 3 }$$

where e is the elementary charge, h is the Planck constant, c is the speed of light in vacuum and λ is the vacuum wavelength.

The external losses $\rho \left( \lambda \right)$ can be measured and/or calculated. An experimental determination of the internal losses $\delta \left( \lambda \right)$ at wavelengths where the quantum yield $y\left( \lambda \right)$ is unknown is difficult as an experimental separation between both is not possible. Thus, $\delta \left( \lambda \right)$ must either be calculated or extrapolated from a wavelength region where the quantum yield is known to be equal unity. For both options, a model is required to obtain an estimation of the internal losses with sufficiently low uncertainty.

In the following, the quantum yield of induced-junction silicon photodiodes of a PQED will be determined according to equations (2) and (3).

The PQED is based on specially designed silicon photodiodes, the induced-junction photodiodes, and a sophisticated physical model of the photodiode's internal losses allowing for the prediction of the photodiode's spectral responsivity with low uncertainty. The induced-junction photodiodes are made of high purity silicon wafer with a very low doping level of about 2·10¹² cm⁻³. Usually, the p–n junction is achieved by doping the silicon crystal with impurities which creates defects in the semiconductor crystal. These defects increase the recombination of generated charge carriers and thus increase the internal losses of a photodiode. The responsivity is also highly sensitive to the doping profile given by the doping level and depth. The large number of defects in a p–n junction and sensitivity to the unknown doping profile makes it difficult to accurately predict the internal losses by a physical model. To avoid these disadvantages, the p–n junction of an induced-junction photodiode is generated by a surface charge trapped in the passivation layer of the silicon [8]. Thus, the purity of the silicon is conserved resulting in low internal losses and the possibility to predict these losses with high accuracy [9]. The concept of the PQED requires the knowledge of the quantum yield, and because of that, the PQED enables the investigation of the quantum yield because of the predictability of the low internal losses.

Two PQEDs [6, 7] studied here are each constructed from two induced-junction silicon photodiodes arranged in a wedge trap configuration. The windowless photodiodes (figure 1) were manufactured in the second processing round of the iMera Plus JRP 'qu-Candela'. They differ in the thickness of the silicon dioxide layer, being 220 nm and 301.4 nm thick in case of PQED_thin and PQED_thick, respectively. The angle between the photodiodes is 11.25° yielding nine reflections of an incoming laser beam if the PQED is aligned so that the reflected beam and the incident one are collinear.

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The spectral responsivity of the two PQEDs was measured in the wavelength range from 360 nm to 531 nm for p-polarised laser radiation. The measurements were performed at the laser-based cryogenic radiometer facility of the Physikalisch-Technische Bundesanstalt (PTB). The facility is equipped with a common Brewster window which means that cryogenic radiometer and PQED under test are irradiated through the same window. Thus, the correction for and the uncertainty contribution from the Brewster window transmittance can be avoided. The detector cavity of the cryogenic radiometer and the PQED are equipped with input apertures with the same diameter of 7 mm and were irradiated at the same position with respect to the laser beam. Thus, the uncertainty contribution arising from the scattered radiation around the laser beam is reduced. A relative standard uncertainty of spectral responsivity between 30 ppm and 50 ppm has been achieved. Figure 2 shows the external quantum efficiency $Q(\lambda)$ calculated from the measured spectral responsivity according to equation (3) and we observe that the external quantum efficiency differs by up to 600 ppm between the two PQEDs.

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In order to isolate the influence of quantum yield we need to decompose the influence of each of the loss parameters as will be discussed in the following chapters.

The estimation of the external losses due to reflectance of the PQED is based on the transfer-matrix method [10] taking into account the photodiode layered structure and material parameters combined with the multiple reflections between the photodiodes. Material parameters and layer thicknesses are taken from the literature [11, 12] or measured directly, e.g. using ellipsometry for oxide thickness [13]. Each parameter's uncertainty is taken into account to calculate the combined reflectance uncertainty. In addition, a photocurrent ratio between the two photodiodes in the trap structure was used to confirm that the used parameters are valid and the PQED is properly aligned relative to linearly polarised light [14]. The measured and modelled ratios agreed within the expanded (k = 2) uncertainties, which were 3.4% to 4.6% for modelled values, depending on wavelength. The uncertainties of the measured ratios were limited with electrical measurement capabilities and were in the order few tens of ppm. Difference between measured and modelled ratios were below 2% at all used wavelengths. In this way we estimated the reflectance losses (equations (1) and (2)) of the two different PQEDs.

The internal losses ($\delta \left( \lambda \right)$) are predicted using Genius simulation software from Cogenda [9]. The calculated $\delta \left( \lambda \right)$ of qu-Candela PQED photodiodes depends on parameters like oxide fixed charge (${Q_{\text{f}}}$), surface recombination parameter (${S_0}$), bulk doping concentration (${d_{{\text{bulk}}}}$) and bulk lifetime (${t_{{\text{bulk}}}}$). When sufficiently biased $\delta \left( \lambda \right)$, at short wavelengths below 800 nm, is mainly limited by the ${S_0}$ parameter which is directly proportional to the number of interface defect states. In the prediction of $\delta \left( \lambda \right)$ we have kept typical model parameters known from the manufacturing of the photodiodes and fitted the ${S_0}$ parameter to match the measured $\delta \left( \lambda \right)$ value at 531 nm, because it is assumed that the $y\left( \lambda \right)$ is equal to 1 at this wavelength. Uncertainty estimates for $\delta \left( \lambda \right)$ at short wavelengths were found by recalculating a new $\delta \left( \lambda \right)$ with a 30 ppm offset in the $\delta \left( \lambda \right)$ value at 531 nm. That means the ${S_0}$ parameter was fitted to match the 30 ppm offset in the $\delta \left( \lambda \right)$ value at 531 nm yielding a second value for the ${S_0}$ parameter with associated $\delta \left( \lambda \right)$ at short wavelengths. The difference between the original fit and second $\delta \left( \lambda \right)$ is used as the uncertainty in $\delta \left( \lambda \right)$. Table 1 shows the parameters used in the estimation of the internal quantum deficiency in each of the two PQEDs. From the calibration point at 531 nm we were now able to isolate the spectrally dependent $\delta \left( \lambda \right)$ for each of the two PQEDs.

Table 1. Model parameters used for the calculation of the internal losses due to recombination.

	PQED_thin	PQED_thick
${Q_{\text{f}}}$/cm−2	6 × 1011	6 × 1011
${S_0}$/(cm s−1)	25.1 × 103	46 × 103
${d_{{\text{bulk}}}}$/cm−3	2 × 1012	2 × 1012
${t_{{\text{bulk}}}}$/s	3 × 10−3	3 × 10−3

Table 2 shows the results on the measured external quantum efficiency $Q\left( \lambda \right)$ from measurements against the cryogenic radiometer, external reflectance losses $\rho \left( \lambda \right)$ and internal fitted and predicted losses $\delta \left( \lambda \right)$ as well as the quantum yield $y\left( \lambda \right)$ calculated from the first three quantities according to equation (2). In addition, the standard uncertainties of these quantities are stated. For completeness of the table, the calculated quantum yield values from the model are also included and the derivation of these values are explained in the following section. The uncertainty in the estimation of the quantum yield is limited by the propagated uncertainties from the fitted internal losses at 531 nm.

The measured quantum yield $y\left( \lambda \right)$ was modelled by using the interpolation function taking into account the effects of impact ionisation and electron–phonon scattering on probability of producing secondary electrons as described, e.g. in [3, 15]. The interpolation function used to model quantum yield ${y_{\text{m}}}\left( \lambda \right)$ is proposed as

$$\begin{align}{y_{\text{m}}}\left( \lambda \right) = 1 + B{\left[ {1 + \frac{{105\,A}}{{2\pi }}\frac{{{{\left( {h\nu - E_{\text{g}}^{{\text{in}}} - \hbar {\omega _0}} \right)}^{1/2}}}}{{{{\left( {h\nu - 2E_{\text{g}}^{{\text{in}}}} \right)}^{7/2}}}}} \right]^{ - 1}}\left( {h\nu - 2E_{\text{g}}^{{\text{in}}}} \right).\end{align} \tag{ 4 }$$

The following symbols and values are used in the interpolation function:

$B$ = 0.551 eV⁻¹ is the only fitting parameter used in equation (4). The parameter B includes effective density of states per unit energy and depends on band structure model used in the energy range studied and may depend on temperature [16]. In the present study, the band structure of silicon was not modelled and, therefore, the parameter B is adjustable in our interpolation function.

$A$ = 5.2 eV³ factor describing relative amounts of scattering by phonon emission and by impact ionisation [15]

$h\nu$ - photon energy of the applied radiation converted from wavelength

$\hbar {\omega _0}$ = 0.062 eV, phonon energy [15]

$E_g^{{\text{in}}}$ = 1.123 eV, indirect bandgap of silicon at 298 K [16].

The interpolation function equation (4) includes one free parameter B. At a given temperature, the parameter B is considered as a scaling factor which can include components arising from structure of silicon, e.g. effective density of states.

We have used the factor 2E_g ⁱⁿ as a threshold for impact ionisation in silicon, where E_g ⁱⁿ is indirect bandgap (E_g ⁱⁿ = 1.123 eV at room temperature). This is well in the range of the threshold values from 1.1 eV to 3.1 eV theoretically predicted by several authors, e.g. [17–19].

The estimated quantum yield of the two PQEDs is shown in figure 3. It can be noticed that measured external quantum efficiency and, consequently, reflectance and internal quantum deficiency depend on the PQEDs (table 2). We observe that when correcting for the individual loss parameters of the PQEDs the estimated quantum yield of the two PQEDs differ by 153 ppm or less (see table 2) and agree within their expanded (k = 2) uncertainties in the wavelength range from 531 nm to 395 nm. This is reassuring, as we expect the quantum yield to be a material specific parameter and less sensitive to individual devices. By using our proposed interpolation function equation (4), the quantum yield over the spectral range from 360 nm to 540 nm is calculated and shown with black solid line. The model agrees with measured values for both PQEDs within their expanded (k = 2) uncertainties.

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We see that the quantum yield increases with decreasing wavelength and is larger than unity even at wavelengths around 450 nm for the detectors used in this study. The agreement between modelled and measured quantum yield ensures that predictability of the PQED responsivity to 160 ppm is maintained down to 360 nm.

The quantum yield in induced-junction Si photodiodes has been derived from measured spectral responsivity, calculated external losses due to reflectance and predicted internal losses due to recombination. The results show that quantum yield increases with decreasing wavelength and is larger than unity even at wavelengths around 450 nm for the detectors used in this study. This is an important limitation when high accuracy modelling predicts the internal quantum deficiency by primary methods.

The measured quantum yield could be modelled by using an interpolation function. This opens the way for the first time to predict the spectral responsivity of induced-junction Si photodiodes even for wavelengths where the quantum yield is larger than one to uncertainties of 160 ppm down to 360 nm.

The projects 18SIB10 'chipS·CALe' and 22IEM06 'S-CALe Up' leading to this publication have received funding from the EMPIR program and from the European Partnership on Metrology, respectively, co-financed by the participating states and the European Union's Horizon 2020 and Horizon Europe research and innovation programmes.