charge_collection_efficiency#
- optika.sensors.charge_collection_efficiency(absorption, thickness_implant=<Quantity 2317. Angstrom>, cce_backsurface=0.21, cos_incidence=1)[source]#
Compute the average charge collection efficiency using the differential charge collection efficiency profile described in Stern et al. [1994].
- Parameters:
absorption (Quantity | AbstractScalar) – The absorption coefficient of the light-sensitive material for the wavelength of interest.
thickness_implant (Quantity | AbstractScalar) – The thickness of the implant layer, the layer where recombination can occur. Default is the value given in Stern et al. [1994].
cce_backsurface (Quantity | AbstractScalar) – The differential charge collection efficiency on the back surface of the sensor. Default is the value given in Stern et al. [1994].
cos_incidence (float | AbstractScalar) – The cosine of the angle of the incident light’s propagation direction inside the substrate with the surface normal
- Return type:
Examples
Plot the charge collection efficiency as a function of wavelength.
import matplotlib.pyplot as plt import astropy.units as u import named_arrays as na import optika # Define a grid of wavelengths wavelength = na.geomspace(10, 10000, axis="wavelength", num=1001) * u.AA # Compute the absorption coefficient for silicon absorption = optika.chemicals.Chemical("Si").absorption(wavelength) # Compute the CCE vs wavelength cce = optika.sensors.charge_collection_efficiency( absorption=absorption, ) # Plot the effective and maximum quantum efficiency fig, ax = plt.subplots(constrained_layout=True) na.plt.plot( wavelength, cce, ax=ax, ); ax.set_xscale("log"); ax.set_xlabel(f"wavelength ({wavelength.unit:latex_inline})"); ax.set_ylabel("charge collection efficiency");
Notes
The charge collection efficiency is the fraction of photoelectrons that are measured by the sensor [Janesick, 2001], and is an important component of the quantum efficiency of the sensor
In Stern et al. [1994], the authors define a differential charge collection efficiency, \(\eta(z)\), which is the probability that a photoelectron resulting from a photon absorbed at a depth \(z\) will be measured by the sensor. In principle, \(\eta(z)\) depends on the exact implant profile on the backsurface of the sensor, however Stern et al. [1994] and Boerner et al. [2012] have shown that a piecewise-linear approximation of \(\eta(z)\),
(1)#\[\begin{split}\eta(z) = \begin{cases} \eta_0 + (1 - \eta_0) z / W, & 0 < z < W \\ 1, & W < z < D, \end{cases}\end{split}\]is sufficient, given the uncertainties in the optical constants involved.
The total charge collection efficiency is then the average value of \(\eta(z)\) weighted by the probability of absorbing a photon at a depth \(z\),
(2)#\[\text{CCE}(\lambda) = \frac{\int_0^\infty \eta(z) e^{-\alpha z} \, dz} {\int_0^\infty e^{-\alpha z} \, dz}.\]Plugging Equation (1) into Equation (2) and integrating yields
(3)#\[\text{CCE}(\lambda) = \eta_0 + \left( \frac{1 - \eta_0}{\alpha W} \right)(1 - e^{-\alpha W}).\]Equation (3) is equivalent to the term in curly braces of Equation 11 in Stern et al. [1994], with the addition of an \(e^{-\alpha W}\) term which represents photons absorbed inside the epitaxial layer but outside the implant layer.
Equation (3) is only valid for normally-incident light. We can generalize it to obliquely-incident light by making the substitution
(4)#\[x \rightarrow \frac{x}{\cos \theta}\]where \(\theta\) is the angle between the propagation direction inside the silicon substrate and the normal vector.
Substituting (4) into Equation (3) and solving yields
(5)#\[\text{CCE}(\lambda, \theta) = \eta_0 + \left( \frac{1 - \eta_0}{\alpha W \sec \theta} \right) (1 - e^{-\alpha W \sec \theta})\]