IEEE SENSORS JOURNAL 1. Theoretical Approach to CMOS APS PSF and MTF Modeling Evaluation

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1 SENSORS JOURNAL 1 Theoretical Approach to CMOS APS PSF and MTF Modeling Evaluation Igor Shcherback, Dan Grois, Tatiana Danov, and Orly Yadid-Pecht Abstract In this work, a fully theoretical CMOS active pixel sensor (APS) modulation transfer function model is formulated, evaluated, and compared with practical results. The model is based on a two-dimensional diffusion equation solution and covers the symmetrical photocarriers diffusion effect together with the impact of the pixel active area geometrical shape. Thorough scanning results obtained by means of a unique submicron scanning system (the S-cube system) from various APS chips, implemented in a standard CMOS 0.35 m technology, are compared with our theoretical predictions. The agreement of the presented comparison results indicates that for any potential active area shape, an analytical reliable estimate of image performance is possible. Index Terms Active pixel sensor (APS), CMOS image sensor, diffusion process, modeling, modulation transfer function (MTF), point spread function (PSF). I. INTRODUCTION MODULATION transfer function (MTF) is an important figure of merit in focal plane array sensors. It determines the upper limits to the image quality, i.e., the image resolution or sharpness and describes the image quality in terms of contrast as a function of spatial frequency. Understanding the tradeoff between different figures of merit enables to achieve the best design for specific missions. Unfortunately, the characterization of MTF has typically been one of the more difficult and errorprone performance testing procedures. Commonly used MTF estimation methods have been described elsewhere [1] [10]. When designing a CMOS active pixel sensor (APS), the area of the photosensitive element has a certain geometrical shape, which does not cover the whole pixel area [5]. Since fill factor of CMOS APS is always below 100%, and imagers lose resolution as the result of the photocarrier diffusion [10], their performance prediction has a great value. To improve the design performance it is important to understand the effect it has on the device sensitivity and the resulting MTF. For insight into the pixel geometric MTF, especially for complex pixel topologies, maximum information is gained by spot scanning of the pixel. Spot scan and direct PSF measurements enabled by the unique submicron scanning system [11], [12] Manuscript received August 12, 2004; revised February 4, The associate editor coordinating the review of this paper and approving it for publication was Prof. Francisco Arregui. I. Shcherback, D. Grois, and T. Danov are with The VLSI Systems Center, Ben-Gurion University, Beer-Sheva 84105, Israel ( shcherba@ee.bgu.ac.il; grois@bgu.ac.il; tdanov@bgu.ac.il). O. Yadid-Pecht is with The VLSI Systems Center, Ben-Gurion University Beer-Sheva 84105, Israel. She is also with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB [AUTHOR: PLEASE PROVIDE POSTAL CODE] Canada ( oyp@ee.bgu.ac.il). Digital Object Identifier /JSEN Fig. 1. Schematic cross section of the considered pixel structure. Parameter d represents the depth (in Y direction). In the presented examples, it represents the depth where the photogeneration rate approaches zero. (S-cube system) permit experimental estimation of the photodiode aperture response behavior and the depletion/diffusion structure of APS devices, i.e., it is possible to gain extremely detailed insight into the pixel aperture response; moreover, diffusion effects are readily apparent and easy to interpret in the MTF data. Direct insight into the pixel spatial response is possible. In this work, we extend the recently presented empirical analysis [10], and formulate the theoretical approach for the overall point spread function (PSF) estimation for CMOS image sensors. Note that all the calculations are based on the parameters reported for the standard CMOS m technology process data, and compared with the results obtained by thorough scanning (via the S-cube system) of various 7 m pitch N_plus/P_well photodiode APS s fabricated in the same standard m CMOS technology. The system enables full PSF extraction for imagers via submicron spot light stimulation. This is unique to our system. Other systems provide MTF measurements, and cannot acquire the true PSF, therefore, limiting the evaluation of the sensor and its performance grading. The system enables a detailed, point by point, quantitative determination of the contributions to the total output signal from each particular region of the pixel itself and its surroundings. The system obtains an optical signal as input and returns an electrical signal as output. It is capable to hit onto a desirable well-defined point within the scan area. It is able to focus the incoming light signal into the spot of a desirable diameter size (e.g., below 0.35 m, according to the technological demands) after penetration through the certain transparent media depth without beam broadening, i.e., the desirable spot size is maintained during the scan. The use of different light sources enables coverage of visible spectrum X/$

2 2 SENSORS JOURNAL Fig. 2. Normalized excess minority-carrier concentration distribution over the semiconductor depth for =632nm, obtained from the 1-D diffusion equation solution [in (6), the equivalent value of d, based on the process data, was taken as 10 L ]. The spot scanning method enabled by the S-cube system is inherently two-dimensional (2-D), which is extremely important, keeping in mind the arbitrariness of the modern photodiode shapes and the fact that it is not easily separable to simple rectangular shapes. It is important to understand that spatial extension of the detector is only a fraction of the pixel pitch and that these fractions might be different for each of the two axes. The integrated signal level and charge transfer efficiency varies thereby with relative position of the sensor at which MTF is measured. In the absence of an electrical field, the minority carrier distribution ( ) is obtained from the solution of the three-dimensional (3-D) diffusion equation where is the thermal-equilibrium minority carriers density, is the diffusion coefficient, is the diffusion length, and is the optical generation rate, which is often expressed as where is the maximal optical generation rate at the surface and is the wavelength dependant light absorption depth. In the following two sections, we provide both: the one-dimensional (1-D) and the 2-D diffusion equation analysis. It will be shown that the 2-D approach can be used as an approximation tool for photocarrier diffusion consideration, keeping in mind that all the pixels are situated on a common substrate and, therefore, for a given pixel array the photocarrier diffusion behavior within the substrate is common too. (1) (2) II. ONE- AND TWO-DIMENSIONAL DIFFUSION EQUATION ANALYSIS In the absence of an electric field, the excess minority-carrier concentration within the semiconductor depth, is obtained from the solution of the 1-D diffusion equation By assuming a constant depletion depth (the schematically cross section of the considered structure is shown in Fig. 1) and the fact that the junction acts as a perfect sink, we can write the following boundary conditions: (3) (4) for (5) where is the junction depth and represents the depth where the photogeneration rate approaches zero. Equation (5) arises from the assumption that the minority carrier concentration eventually decreases to zero at the depth far beyond the optical absorption depth. The solution of (3) is where c1 and c2 that satisfy the boundary conditions of (4) and (5) are (6) (7)

3 SHCHERBACK et al.: THEORETICAL APPROACH TO CMOS APS PSF AND MTF MODELING 3 Fig. 3. The normalized averaged symmetrical excess minority-carrier concentration distribution for =632nm, obtained from the 2-D diffusion equation solution (16). Fig. 4. Comparison between (A) the design layout and (B) the actual electrical scan result, obtained by the S-cube system for a pixel fabricated in standard CMOS 0.35-m technology (after [3]). In (B), the lighter areas indicate a stronger response. [see (8), shown at the bottom of the page]. The normalized excess minority-carrier concentration as a function of the depth for nm is shown in Fig. 2. Assuming that carrier presented carrier distribution serves a source for their subsequent diffusion and recombination in direction, we can write the following boundary condition: for (9) where is the middle point of the imaging site (pixel). Since the minority carrier concentration gradient eventually decreases to zero at the distance beyond diffusion length, we can write the second boundary condition for (10) (8)

4 4 SENSORS JOURNAL Fig. 5. Plot of the actual measured PSF (after [2] [4]). Cross sections used for fitting are located along the arrows, normal to the layout surface. Here, and in the following images, the lighter the area, the stronger the pixel response. Fig. 6. Comparison of our analytical solution (16) with the scanning data (e.g., A 0 A cross section in Fig. 5) and empirical fitting result (17). The solution of 2-D equation (for - )is where c3 and c4 that satisfy the boundary conditions of (9) and (10) are (12) (13) Since the diffusion within the substrate is symmetrical, the axis and axis have the same distribution. Therefore, analogically substituting with, we get (14) III. TWO-DIMENSIONAL DIFFUSION APPROACH; PSF AND MTF MODELED AND EXPERIMENTAL COMPARISON Integration of (11) over the axis from to (and normalization to the integrating range) gives an average planar representation of the minority-carrier distribution; see (15), keeping in mind the - symmetry assumption (11) (15)

5 SHCHERBACK et al.: THEORETICAL APPROACH TO CMOS APS PSF AND MTF MODELING 5 Fig. 7. Comparison between the normalized measured (on the left) and the normalized modeled symmetrical (on the right) PSF representations. X and Y axes presents the relative scanner counted steps; each step 14 m/step. The lighter the area, the stronger the pixel response. Fig. 8. Difference between the corresponded modeled and measured PSF distributions (in correspondence to Fig. 7). The maximum relative difference is up to 15% of maximum pixel response. The lighter the area, the stronger the difference in pixel response. where, 1, 2 and 3 are the constants dependent on the values of, and. This result is represented graphically in Fig. 3. In order to verify these pure analytical result, s we have compared them with the carriers distribution obtained by thorough scanning of the various CMOS m APS via our unique scanning system (previously reported in [2] [4]). A comparison between the design layout and the actual electrical scan result, obtained by the S-cube system is presented in Fig. 4 (after [3]). In Fig. 5, the value at each point of a 2-D signal map represents the electrical outcome of the 3-D photocarrier integration process (i.e., the integration over the depth at each point). It is proportional to the excess photocarrier concentration and also to the conversion gain of the corresponding pixel. Note that this is the true measured result as obtained from the system. 1 A =(1=d 0 y ) c (y)dy. 2 B =(1=d 0 y ) c (y)dy. 3 C =(1=d 0 y ) (G 0 L ) =D ) 1 e dy. (b) Fig. 9.(a) MTF contour plot calculated via the symmetrical 2-D Fourier transform from the measured PSF. (b) Modeled MTF contour representation obtained from the analytically calculated PSF, via the 2-D Fourier transform. (a) Fig. 6 shows the comparison of our analytical solution (15) with the scanning data (e.g., cross section on Fig. 4)

6 6 SENSORS JOURNAL and empirical fitting result. The function used for the empirical fitting (analogously to [2]), is sensor performance and resolution abilities can be easily extracted based on the modeled PSF data for each wavelength of interest. The degradation of the imager performance caused by the diffusion effects is calculated from the MTF. The analytical PSF/MTF model based on a comprehensive 3-D diffusion equation solution is subject to further research. (16) where,, and are the constants dependant on the design, technology and the incoming radiation parameters. Note that this comparison is performed without actual array structure consideration, i.e., the distribution data (i.e., from the measurements) was taken at equal distances from the readout photodiode edge. Thereby we consider the symmetrical case neglecting the disturbances influenced by the actual array design, e.g., the opaque metal lines, logic gates, etc. Good functional agreement can be noticed for the presented comparison, such that the maximum difference between the analytical [diffusion equation s solution, (15)] and the scanning data is up to 3.7%, while the difference between the analytical and the fitting function (16) is up to 3%. Based on the presented analytical solution, we built a normalized 2-D symmetrical diffusion matrix (there is no diffusion direction priority within the uniform silicon substrate), considering its dimensions in a way analogously to the one reported in [2]. The convolution of this matrix with the matrix that represents the actual pixel photodiode geometry produces the PSF of the pixel [2]. Note that unlike the work [2], the PSF obtained here represents a pure theoretical result. The comparison between the modeled and the measured PSF is presented in Figs. 7 and 8. Fig. 7 shows a comparison between the normalized measured (on the left) and the normalized modeled symmetrical (on the right) PSF representations, while, in Fig. 8, the absolute difference between corresponding PSF distributions is shown, emphasizing the symmetrical character of modeled function such that the difference is negligible in the center (pixel region) and grows toward the periphery. The palpable difference (up to 15% of the maximum pixel response) resulting from our symmetrical approach can be significantly reduced by taking appropriate consideration of the actual array topography, i.e., by solving the complete 3-D diffusion equation with necessary boundary conditions, including the design asymmetry. Furthermore, the comprehensive 3-D approach must include the basic physical models such as generation/recombination, mobility, and carrier lifetime models in dependence on the dopants concentration, wavelengths, etc. The precise geometrical information can be extracted via the submicron scanning system and inserted to the theoretical model for better fitting. Fig. 9(a) and (b) shows the MTF contour plots obtained from the measured and modeled PSF (Fig. 7) distribution correspondingly via the 2-D symmetrical Fourier transform. The resemblances that can be observed from these two contour plots comparison shows that the proposed symmetrical approach can be used as a first approximation, enabling pure theoretical estimation of the pixel level MTF, including its geometry and interpixel diffusion effects. True 2-D pixel MTF characterizing IV. SUMMARY In this work, a fully analytical 2-D symmetrical approach is suggested for the minority carrier diffusion effect for modeling and estimation of CMOS APS PSF/MTF. The model includes the effects of the photocarrier diffusion within the substrate in addition to the effects of the pixel sampling aperture shape and size. The good agreement between the values obtained by the theoretical and true measurement approaches, verifies the viability of this pure theoretical methodology. These results can be further improved by utilizing a comprehensive 3-D solution. However, for full geometry effect consideration, real data, obtained via accurate submicron scanning, is necessary. The pure theoretical approach, as proven here for the first time, can give a reasonable approximation and serve as a first stage for performance evaluation. REFERENCES [1] D. H. Seib, Carrier diffusion degradation of modulation transfer function in charge coupled imagers, Trans. Electron Devices, vol. ED 21, no. 3, pp , Mar [2] J. P. Lavine, E. A. Trabka, B. C. Burkey, T. J. Tredwell, E. T. Nelson, and C. N. Anagnosyopoulos, Steady-state photocarrier collection in silicon imaging devices, Trans. Electron Devices, vol. ED-30, no. 9, pp , Sep [3] E. G. Stevens, A unified model of carrier diffusion and sampling aperture effects on MTF in solid-state image sensors, Trans. Electron Devices, vol. 39, no. 11, pp , Nov [4] E. G. Stevens and J. P. Lavine, An analytical, aperture and two-layer diffusion MTF and quantum efficiency model for solid-state image sensors, Trans. Electron Devices, vol. 41, no. 10, pp , Oct [5] O. Yadid-Pecht, The geometrical Modulation Transfer Function (MTF) for different pixel active area shapes, Opt. Eng., vol. 39, no. 4, pp , [6] J. Lee and R. Hornsey, Photoresponse of photodiode arrays for solidstate image sensors, J. Vac. Sci. Technol. A, vol. 18, no. 2, Mar./Apr [7] T. Dutton, T. Lomheim, and M. Nelson, Survey and comparison of focal plane MTF measurement techniques, Proc. SPIE, vol. 4486, pp , [8] D. Abbott, Modulation transfer function and quantum efficiency in thin semiconductor photodetectors, EDL, vol. 38, no. 16, Aug [9] C. Lin et al., Analytical charge collection and MTF model for photodiode-based CMOS imagers, Trans. Electron Devices, vol. 49, no. 5, pp , May [10] I. Shcherback and O. Yadid-Pecht, CMOS Imagers: From Phototransduction to Image Processing. Norwell, MA: Kluwer, [11], CMOS APS crosstalk characterization via a unique submicron scanning system, Trans. Electron Devices, vol. 50, no. 9, pp , Sep [12] I. Shcherback, B. Belotserkovsky, and O. Yadid-Pecht, A unique submicron scanning system use for CMOS APS crosstalk characterization, presented at the Proc. SPIE, Jan

7 SHCHERBACK et al.: THEORETICAL APPROACH TO CMOS APS PSF AND MTF MODELING 7 Igor Shcherback received the B.Sc. degree in physics and the M.Sc. degree in electro-optical engineering from the Ben-Gurion University (BGU), Beer-Sheva, Israel, in 1998 and 2003, respectively. He is currently pursuing the Ph.D. degree in electro-optics at BGU. In 1999, he joined The VLSI Systems Center, BGU, where he has been engaged in the research, development, and design of CMOS imager devices. His areas of interest are the physics and technology of semiconductor devices, CMOS imaging, active ixel sensors, smart sensors, image processing, and algorithm implementation. Tatiana Danov received the M.Sc. degree in electrical engineering from the University of Lobachevsky, Nizhniy Novgorod, Russia, in She is currently with The VLSI Systems Center, Ben-Gurion University, Beer-Sheva, Israel. Her current research is on the optimization and design of CMOS imager device structures in deep-submicron CMOS technology. Dan Grois was born in Kharkov, Ukraine, in He received the B.Sc. degree in electrical and computer engineering from the Ben-Gurion University (BGU), Beer-Sheva, Israel, in He is currently pursuing the M.Sc. degree in electro-optics engineering at BGU. His research interests include image and video processing, imaging systems, image and video compression standards, and buffer and bit-rate control. Orly Yadid-Pecht received the B.Sc. degree from the Electrical Engineering Department and the M.Sc. and D.Sc. degrees from the Technion Israel Institute of Technology, Haifa, in 1983, 1990, and 1995, respectively. She was a Research Associate of the National Research Council (USA) from 1995 to 1997 in the area of advanced image sensing at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena. Since 1997, she has been with the Electrical and Electro-Optical Engineering Departments at the Ben-Gurion University, Beer-Sheva, Israel, where she has established the The VLSI Systems Center, with accompanying lab facilities and courses. Since 2003, she has been with the Electrical and Computer Engineering Department, University of Calgary, Calgary, AB, Canada, promoting the area of biomedical sensors. Her areas of interest are CMOS active pixel sensors, smart sensors, image processing, neural nets, pattern recognition algorithms implementation, and mixed-mode VLSI design. Dr. Yadid-Pecht was an Associate Editor for the TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, and she is now a Deputy Editor-in-Chief for the TRANSACTIONS ON CIRCUITS AND SYSTEMS I, REGULAR PAPERS. She is also the Chair of the Sensory Systems Technical Committee of the Circuits and Systems Society. In addition, she is a member of the technical committee for the Biannual Workshop on CCDs and Advanced Image Sensors, and a member of the SPIE Solid State Sensor Arrays Program Committee.