Optimization of Photodetector Thickness in Vertically-Integrated Image Sensors

1. Optimization of Photodetector Thickness in Vertically-Integrated Image Sensors Orit Skorka, Dan Sirbu, and Dileepan Joseph University of Alberta, Canada

2. Outline • Motivation • Problem statement • Photodetector model • Mathematical method • Optimization of thickness • Conclusions

3. Motivation • Image sensors are required to have high SNR, high dynamic range, high resolution, and high frame rate • All these features may not be achievable with current planar technologies (CCD and CMOS) • There has been an increased interest in fabricating image sensors in which the photodetectors are vertically integrated with the CMOS circuitry

4. Motivation Example – Vertically-integrated CMOS image sensor fabricated using flip-chip assembly

5. Motivation • The photodetectors and the electronics can be optimized independently of each other in VI-CMOS image sensors. • There are more degrees of freedom in the photodetector design. • Material – no longer restricted to c-Si. • Photodetector thickness – the vertical dimension can now be controlled. • Advantages of Vertical-Integration

6. Problem Statement • Simplified 1D photodetector structure • Semiconductor layer thickness – l

7. Problem Statement • What is the optimal semiconductor thickness, lopt, for the photodetector to have a maximum contrast? • Contrast Initial Hypothesis • If the semiconductor is made too thin then very little light is absorbed, which implieslow photocurrent • If the semiconductor is made too thick then most photo-generated charge carriers recombine on their way to the contact, which also implieslow photocurrent

8. Photodetector Model • Three-resistor system • The illumination decays exponentially in the semiconductor • The absorbed photons generate extra electron-hole pairs (EHPs), or excess carriers, that improve conductivity

9. Photodetector Model Charge carrier equations • Poisson’s equation – relates electric potential or electric field to concentration of charge carriers • Continuity equations – ensures charge carriers are neither created nor destroyed at any point • Drift-diffusion equations – describe the current density as a sum of a drift current, which arises from existance of electric field, and diffusion current, which arises from concentration gradients

10. Photodetector Model Charge carrier equations • Poisson’s equation: • Continuity equations: • Drift-diffusion equations:

11. Photodetector Model Boundary conditions • Kirchoff’s voltage law – forces the sum of the voltage drops over the three resistors to equal the applied potential, Vab. • Kirchoff’s current law – forces the sum of the hole current and the electron current in the semiconductor to equal the current drawn from the power supply. • Charge neutrality – as electrons and holes are generated and recombined in pairs, the semiconductor must remain neutral, assuming initial charge neutrality. • Generation-recombination balance – in the steady state, every EHP generation must be offset by an EHP recombination (perhaps elsewhere) in the semiconductor.

12. Photodetector Model Boundary conditions • Kirchoff’s voltage law: • Kirchoff’s current law: • Charge neutrality: • Generation-recombination balance:

13. Photodetector Model • EHP boundary conditions are often defined through the “recombination velocity” of charge carriers at the boundaries. It relates the concentration of charge carriers and current density on the boundaries. • We have defined EHP boundary conditions by charge neutrality and generation-recombination balance, which are easier to interpret. These conditions, fewer than those used in the literature, prove sufficient to solve the problem without inconsistency.

14. Mathematical Method • The solution process is based on the mean value of the different variables in the semiconductor, and on the deviation of the local quantity from its mean value. The mean is computed over the length l. Example – Hole (p) and electron (n) concentration local concentration local perturbation mean value

15. Mathematical Method Analytical solution • Is derived by assuming gp(z) = 0 and gn(z) = 0 only. • A systematic way was derived to solve all the remaining variables and, thereby, to find J = f (F0) for any F0. • However, the solution does not satisfy all equations. Numerical solution • Is based on an iterative finite-differences method. • It shows the equations are consistent and complete.

16. Mathematical Method

17. Optimization of Thickness • Our initial hypothesis proved wrong • If the semiconductor is too thin, very little light is absorbed. However, the electric field becomes very strong for a constant applied voltage. • According to the model, the contrast is low at this end because the resistance of the semiconductor is low in comparison to the contact resistances. The device is dominated by the contacts.

18. Optimization of Thickness • In the absence of contact resistances, the contrast is maximal at lequals zero, which is nonsensical.

19. Optimization of Thickness • Our initial hypothesis proved wrong • If the semiconductor is too thick, there is more EHP recombination in the device, as predicted. However, the total generation rate of EHPs also increases with semiconductor thickness. • According to the model, the contrast is low here because the mean value of excess charge carriers per unit volume decreases with device length and, thus, the mean photoconductivity also decreases.

20. Optimization of Thickness • Total EHP generation rate increases with semiconductor thickness.However, mean excess carriers, and hence conductivity and contrast decrease asl grows large.

21. Optimization of Thickness • Optimal thickness was found analytically and numerically • See the paper for material (a-Si:H) and other parameters • Simulation results: lopt = 400 nm for maximum contrast

22. Conclusions • We presented a new approach to solve semiconductor charge carrier equations in 1D photodetectors. • Our boundary conditions used Kirchoff’s laws, charge neutrality, and generation-recombination balance. • Our approach was based on mean values of variables and on deviations of local values from the mean values. • The method was used for optimization of photodetector thickness in a vertically-integrated image sensor.

23. Acknowledgments The authors gratefully acknowledge the support of • Alberta Ingenuity • The Natural Sciences and Engineering Research Council (NSERC) of Canada • The Mary Louise Imrie Graduate Student Award, University of Alberta