**2. Spectral responsivity scale in the visible range based on single silicon photodiodes.**

A spectral responsivity scale means that the responsivity is known at every wavelength within the response range of interest and it would be desirable to know it for all the other parameters associated with a beam: angle of incidence, divergence or polarization.

Aspectral responsivity scale in the visible range can be created by calibrating a silicon trap detector at several laser wavelengths against ahigh accuracy primary standard such as an electrically calibrated cryogenic radiometer. This method provides a very certain value for the responsivity at specific wavelengths as those of lasers (for instance 406.7 nm, 441.3 nm, 488.0 nm, 514.5 nm, 568.2 nm, 647.1 nm and 676.4 nm). From there single elements detectors, most suitable for some applications, can be calibrated against that trap detector at those wavelengths to define the working scale.

The spectral responsivity of silicon photodiodes is given by the well-known equation

$$R(\lambda) = (1 - \rho(\lambda))\varepsilon(\lambda)\frac{\lambda}{k} \tag{2}$$

This chapter describes the results obtained for the responsivity of the photodiodes by using a model to calculate the diode's reflectance from experimental measurements and a model for the internal quantum efficiency, which is also fitted to experimental values. Based on the mod‐ els, the fitting errors and the uncertainty of reflectance and responsivity measurements, the uncertainty of the responsivity scale is calculated according to the ISO recommendations.

#### **3. Reflectance evaluation of silicon photodiodes**

From the reflectance point of view, a silicon photodiode can be considered as a system formed by a flat transparent film over an absorbing medium. The flat film is the silicon ox‐ ide and the absorbing medium is the silicon substrate. The reflectance of such a system is given by [5]

$$\rho = \frac{\left[r\_{12}^2 + \rho\_{23}^2 + 2r\_{12}\rho\_{23}\cos(\phi\_{23} + 2\beta)\right]}{1 + r\_{12}^2\rho\_{23}^2 + 2r\_{12}\rho\_{23}\cos(\phi\_{23} + 2\beta)}\tag{3}$$

trap detector, the photodiodes' internal quantum efficiency can be calculated according to (2). Using a model based in physical laws rather than experimental equations allows obtain‐ ing the physical quantity for different circumstances, such as different angles of incidence,

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**Figure 1.** Spectral reflectance values of photodiode CIRI and fitted values according to equation (3).

To spectrally know the internal quantum efficiency we have used the model developed by Gentile *et al* [2], based on that from Geist and Baltes [14] and improved by Werner *et al* [15].

where *Pf* is the collection efficiency at the front, *T* is the junction depth, *Pb* is the collection efficiency at the silicon bulk region, which starts at depth *D*, *h* is the photodiode's length, *R*back is the reflectance at the photodiode's back surface and *α* is the absorption coeffi‐ cient.According to Gentile *et al* [2] a simplified model can be used if the model is to be ap‐ plied to wavelengths shorter than 920 nm. This model is obtained from the previous equation by deleting the last two terms. Then, the quantum efficiency can be obtained from

(*<sup>D</sup>* <sup>−</sup> *<sup>T</sup>* )*α*(*λ*) {exp −*Tα*(*λ*) −exp *Dα*(*λ*) } −

(4)

**5. Internal quantum efficiency of silicon photodiodes**

*<sup>α</sup>*(*λ*)*<sup>T</sup>* {1−exp <sup>−</sup>*Tα*(*λ*) } <sup>−</sup> <sup>1</sup> <sup>−</sup> *Pb*

The internal quantum efficiency is given by

*Pb*expexp *hα*(*λ*) + *Rback*exp *hα*(*λ*) *Pb*

1 − *Pf*

*ε*(*λ*)=*Pf* +

for instance.

where *r*12 is the amplitude of the reflection coefficient from air to silicon oxide, *ρ*23 is the amplitude of the reflection coefficient from silicon oxide to silicon, *φ*23 is the phase change at the interface silicon oxide–silicon and *β* = 2*πn*2*h* cos*(θ*2*)/λ*0, with *h* the thickness of SiO2, *n*2 the refractive index of SiO2 and *θ*2 the refraction angle at the air–oxide interface. These variables change with the angle of incidence and the light polarization, so the reflectance value will be known if the silicon oxide thickness, the angle of incidence, the refractive index and the light polarization status are known. This reflectance model has been already tested for another type of silicon photodiode from the same manufacturer [6].

Spectral values of the refractive index are available in the literature. In this work values have been obtained from those given in [7]. The index of refraction of silicon oxide has been inter‐ polated by fitting a polynomial to data; the real part of the refractive index of silicon has been obtained by fitting a polynomial in 1*/λ* and the imaginary part by fitting an exponen‐ tial decay in *λ*.Reflectance was measured with an angle of incidence of 4 in our reference spectrophotometer, using p-polarized light, at the laser wavelengths for which the diodes were calibrated against the trap: 406.7 nm, 441.3 nm, 488.0 nm, 514.5 nm, 568.2 nm, 647.1 nm and 676.4 nm. By fitting equation 3 to measurement results, the silicon oxide thickness was obtained for every photodiode, as shown in table 1. The fitting error in this table is the pa‐ rameter given by the fitting software.


**Table 1.** Silicon oxide thickness fitted to reflectance measurements

The fitting is very good for wavelengths longer than 500 nm, getting worse for shorter wavelengths, as can be seen in figure 1 for one of the photodiodes studied. The same results are obtained for the three photodiodes studied in this work.

This agrees also with [2]. Probably it is due to the measurement bandwidth. For conven‐ ience, the reflectance was measured in our reference spectrophotometer with a bandwidth of 5 nm in order to have a good signal-to-noise ratio at the shortest wavelengths. But in this region the first and second derivatives of reflectance are higher than in the middle visible, so the increased bandwidth produces an effective reflectance value that differs significantly from the spectral value. For this reason, reflectance values below 500 nm were not used in the final fitting process to obtain the thickness.

Using thickness values given in table 1, the reflectance of the photodiodes at normal inci‐ dence can be calculated, and from them and the responsivity values measured against the trap detector, the photodiodes' internal quantum efficiency can be calculated according to (2). Using a model based in physical laws rather than experimental equations allows obtain‐ ing the physical quantity for different circumstances, such as different angles of incidence, for instance.

*<sup>ρ</sup>* <sup>=</sup> *<sup>r</sup>*<sup>12</sup> <sup>2</sup> <sup>+</sup> *<sup>ρ</sup>*<sup>23</sup>

1 + *r*<sup>12</sup> <sup>2</sup>*ρ*<sup>23</sup>

for another type of silicon photodiode from the same manufacturer [6].

rameter given by the fitting software.

176 Photodiodes - From Fundamentals to Applications

**Table 1.** Silicon oxide thickness fitted to reflectance measurements

the final fitting process to obtain the thickness.

are obtained for the three photodiodes studied in this work.

<sup>2</sup> <sup>+</sup> <sup>2</sup>*r*12*ρ*23cos(*ϕ*<sup>23</sup> <sup>+</sup> <sup>2</sup>*β*)

where *r*12 is the amplitude of the reflection coefficient from air to silicon oxide, *ρ*23 is the amplitude of the reflection coefficient from silicon oxide to silicon, *φ*23 is the phase change at the interface silicon oxide–silicon and *β* = 2*πn*2*h* cos*(θ*2*)/λ*0, with *h* the thickness of SiO2, *n*2 the refractive index of SiO2 and *θ*2 the refraction angle at the air–oxide interface. These variables change with the angle of incidence and the light polarization, so the reflectance value will be known if the silicon oxide thickness, the angle of incidence, the refractive index and the light polarization status are known. This reflectance model has been already tested

Spectral values of the refractive index are available in the literature. In this work values have been obtained from those given in [7]. The index of refraction of silicon oxide has been inter‐ polated by fitting a polynomial to data; the real part of the refractive index of silicon has been obtained by fitting a polynomial in 1*/λ* and the imaginary part by fitting an exponen‐ tial decay in *λ*.Reflectance was measured with an angle of incidence of 4 in our reference spectrophotometer, using p-polarized light, at the laser wavelengths for which the diodes were calibrated against the trap: 406.7 nm, 441.3 nm, 488.0 nm, 514.5 nm, 568.2 nm, 647.1 nm and 676.4 nm. By fitting equation 3 to measurement results, the silicon oxide thickness was obtained for every photodiode, as shown in table 1. The fitting error in this table is the pa‐

> **Photodiode SiO2 thickness/nm Fitting error/nm** CIRI 29.58 0.19 SiN 28.84 0.17 Si1 29.93 0.19

The fitting is very good for wavelengths longer than 500 nm, getting worse for shorter wavelengths, as can be seen in figure 1 for one of the photodiodes studied. The same results

This agrees also with [2]. Probably it is due to the measurement bandwidth. For conven‐ ience, the reflectance was measured in our reference spectrophotometer with a bandwidth of 5 nm in order to have a good signal-to-noise ratio at the shortest wavelengths. But in this region the first and second derivatives of reflectance are higher than in the middle visible, so the increased bandwidth produces an effective reflectance value that differs significantly from the spectral value. For this reason, reflectance values below 500 nm were not used in

Using thickness values given in table 1, the reflectance of the photodiodes at normal inci‐ dence can be calculated, and from them and the responsivity values measured against the

<sup>2</sup> <sup>+</sup> <sup>2</sup>*r*12*ρ*23cos(*ϕ*<sup>23</sup> <sup>+</sup> <sup>2</sup>*β*) (3)

**Figure 1.** Spectral reflectance values of photodiode CIRI and fitted values according to equation (3).

#### **5. Internal quantum efficiency of silicon photodiodes**

To spectrally know the internal quantum efficiency we have used the model developed by Gentile *et al* [2], based on that from Geist and Baltes [14] and improved by Werner *et al* [15]. The internal quantum efficiency is given by

$$\begin{aligned} \varepsilon(\lambda) &= P\_f + \frac{1 - P\_f}{\alpha(\lambda)\Gamma} [1 - \exp[-T\alpha(\lambda)]] - \frac{1 - P\_b}{(D - T)\alpha(\lambda)} [\exp[-T\alpha(\lambda)] - \exp[D\alpha(\lambda)]] - \\ &- P\_b \text{expexp}\{ha(\lambda)\} + R\_{\text{back}} \exp[ha(\lambda)] P\_b \end{aligned} \tag{4}$$

where *Pf* is the collection efficiency at the front, *T* is the junction depth, *Pb* is the collection efficiency at the silicon bulk region, which starts at depth *D*, *h* is the photodiode's length, *R*back is the reflectance at the photodiode's back surface and *α* is the absorption coeffi‐ cient.According to Gentile *et al* [2] a simplified model can be used if the model is to be ap‐ plied to wavelengths shorter than 920 nm. This model is obtained from the previous equation by deleting the last two terms. Then, the quantum efficiency can be obtained from

$$\mathcal{L}\left(\boldsymbol{\lambda}\right) = \text{Pf} + (\mathbf{1} - \mathbf{P}\boldsymbol{\lambda}) / \mathcal{a}\left(\boldsymbol{\lambda}\right) \text{T} \left\{ \mathbf{1} - \exp\left[ -\mathbf{T}\boldsymbol{a}\left(\boldsymbol{\lambda}\right) \right] \right\} - (\mathbf{1} - \mathbf{P}\boldsymbol{b}) / \left( (\mathbf{D} - \mathbf{T})\boldsymbol{a}\left(\boldsymbol{\lambda}\right) \right) \left\{ \exp\left[ -\mathbf{T}\boldsymbol{a}\left(\boldsymbol{\lambda}\right) \right] - \exp\left[ \mathbf{D}\boldsymbol{a}\left(\boldsymbol{\lambda}\right) \right] \right\} \tag{5}$$

the calculated values and those measured against the trap is excellent as can be seen in fig‐ ure 3 for one of the photodiodes studied. This result is just a check of the consistency of the method. Nevertheless, it can be seen that most calculated values are smaller than the meas‐ ured ones. This might be due to the independent fitting of reflectance and quantum efficien‐ cy values and their functional forms, but it may also be due to the presence of a systematic error in the measurements. Some research will have to be done in the future to clarify this.

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**Figure 3.** Spectral Difference between calculated and measured spectral responsivity values for photodiode CIRI as a

**7. Spectral responsivity scale in the near IR range based on single InP/**

As in the visible range, semiconductor photodiodes are the best choice for establishing spec‐ tral responsivity scales in the near IR range. The first attempt was to use germanium photo‐ diodes, since its gap allowed to obtain a device responding to wavelengths lower than 1.6 μm, approximately, depending on temperature. However germanium photodiodes have got a rather high dark current and lower shunt resistance than silicon, then they are not so use‐ ful for optical radiation detection. Since optical communications were demanding better de‐ tectors to enlarge their use, other photodiodes were developed in this spectral region of great interest. Since no other single element semiconductor was possible, semiconductor hetero-junctions were developed. A hetero-junction is a junction formed between two semi‐ conductors with different band-gaps. Of course building such devices is not straightforward since the lattice parameters have to be matched, but this is not the subject of this chapter and

function of wavelength.

**InGaAs photodiodes**

many good references may be found in literature [17].

This model has been fitted to the calculated internal quantum efficiency values by a non-lin‐ ear squared method.

The parameters' initial values were taken from Gentile *et al* [2]. The goodness of the fit can be seen in figure 2, where values for one of the studied photodiodes are shown. The same results are obtained for the three photodiodes studied in this work. The main difference be‐ tween the fitted values of the internal quantum efficiency and those calculated from the re‐ sponsivity and reflectance measurements is about 10-3, which agrees well with results given by other authors, e.g.[2,9].

**Figure 2.** Experimental internal quantum efficiency values ofphotodiode SiN and fitted values according to equation (5) againstthe absorption coefficient.

Another point that can be discussed is how far the internal quantum efficiency can be ex‐ trapolated. Using this simplified model and fitting with values corresponding to wave‐ lengths shorter than 700 nm, quantum efficiency values continue to increase very slightly to 900 nm at least. This is not what really happens in the photodiode, so there will be an upper limit for the extrapolation. This limit will depend on the uncertainty allowable to the re‐ sponsivity value and will be discussed in the following section.

#### **6. Spectral responsivity values of silicon photodiodes**

Responsivity of detectors has been calculated with the model described previously and the parameters obtained by the fitting process by using (2), (3) and (5). The agreement between the calculated values and those measured against the trap is excellent as can be seen in fig‐ ure 3 for one of the photodiodes studied. This result is just a check of the consistency of the method. Nevertheless, it can be seen that most calculated values are smaller than the meas‐ ured ones. This might be due to the independent fitting of reflectance and quantum efficien‐ cy values and their functional forms, but it may also be due to the presence of a systematic error in the measurements. Some research will have to be done in the future to clarify this.

e l

ear squared method.

by other authors, e.g.[2,9].

(5) againstthe absorption coefficient.

a l

178 Photodiodes - From Fundamentals to Applications

( ) = +- Pf 1 Pf / T 1 exp T 1 Pb / D T exp T exp D ( )

( ) } ( ) (( )

This model has been fitted to the calculated internal quantum efficiency values by a non-lin‐

The parameters' initial values were taken from Gentile *et al* [2]. The goodness of the fit can be seen in figure 2, where values for one of the studied photodiodes are shown. The same results are obtained for the three photodiodes studied in this work. The main difference be‐ tween the fitted values of the internal quantum efficiency and those calculated from the re‐ sponsivity and reflectance measurements is about 10-3, which agrees well with results given

**Figure 2.** Experimental internal quantum efficiency values ofphotodiode SiN and fitted values according to equation

Another point that can be discussed is how far the internal quantum efficiency can be ex‐ trapolated. Using this simplified model and fitting with values corresponding to wave‐ lengths shorter than 700 nm, quantum efficiency values continue to increase very slightly to 900 nm at least. This is not what really happens in the photodiode, so there will be an upper limit for the extrapolation. This limit will depend on the uncertainty allowable to the re‐

Responsivity of detectors has been calculated with the model described previously and the parameters obtained by the fitting process by using (2), (3) and (5). The agreement between

sponsivity value and will be discussed in the following section.

**6. Spectral responsivity values of silicon photodiodes**

al

 al

( ) ) { é ùé ù

 al

ë ûë û - - ( ) ( ) } (5)

( ) { - - -- - é ù ë û a l

**Figure 3.** Spectral Difference between calculated and measured spectral responsivity values for photodiode CIRI as a function of wavelength.
