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

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

3. Motivation 3 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 4 Example – Vertically-integrated CMOS image sensor fabricated using flip-chip assembly

5. Motivation 5 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 6 Simplified 1D photodetector structure Semiconductor layer thickness – l

7. Problem Statement 7 What is the optimal semiconductor thickness, l opt, 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 implies low photocurrent If the semiconductor is made too thick then most photo- generated charge carriers recombine on their way to the contact, which also implies low photocurrent

8. Photodetector Model 8 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 9 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 10 Charge carrier equations  Poisson’s equation:  Continuity equations:  Drift-diffusion equations:

11. Photodetector Model 11 Boundary conditions Kirchoff’s voltage law – forces the sum of the voltage drops over the three resistors to equal the applied potential, V ab. 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 12 Boundary conditions  Kirchoff’s voltage law:  Kirchoff’s current law:  Charge neutrality:  Generation-recombination balance:

13. Photodetector Model 13 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 14 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 mean value local perturbation

15. Mathematical Method 15 Analytical solution Is derived by assuming  p (z) = 0 and  n (z) = 0 only. A systematic way was derived to solve all the remaining variables and, thereby, to find J = f (  0 ) for any  0. 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 16 AnalyticalNumerical Poisson’s equation  Continuity equations  Drift-diff. equations  K.’s voltage law (KVL)  K.’s current law (KCL)  Charge neutrality  Gen.-rec. balance 

17. Optimization of Thickness 17 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 18 In the absence of contact resistances, the contrast is maximal at l equals zero, which is nonsensical.

19. Optimization of Thickness 19 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 20 Total EHP generation rate increases with semiconductor thickness. However, mean excess carriers, and hence conductivity and contrast decrease as l grows large.

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

22. Conclusions 22 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 23 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