#### 3.2. Comparison of the Approximaate Expression of Theoretical Highest Frame Rate with Numerical Calculation of the Strictly Formulated Expression

Temporal resolution is dependent on the distribution of the arrival time of signal electrons.

Figure 4 shows example trajectories of generated electrons and the relation between the travel time and the travel distance from the backside for a BSI MCG image sensor with the potential separation by the p-well. As shown in the figure, the major cause spreading the arrival time is the horizontal motion of signal electrons travelling over the p-well to the center of the pixel.

A silicon pipe with an infinitesimal diameter perfectly suppresses the horizontal motion. The remaining vertical motion has two governing factors: mixing of electrons due to the exponential distribution of the penetration depth of light, and pure diffusion due to the vertical random motion of generated signal charges. Based on the assumption, the temporal resolution of photoelectron conversion layers, including silicon layers, was theoretically analyzed. Fortunately, a simple approximate expression of the theoretical temporal resolution limit was derived. The accuracy was confirmed in comparison with the temporal resolution calculated by Monte Carlo simulations [

15,

16]. The expression of the travel time distribution can be derived with no approximation, while it cannot be expressed by elementary functions. In this paper, our approximate expression is compared with the numerical calculations of the strictly derived expression. Almost perfect agreement is confirmed by the comparison for a range used in practical applications.

Figure 1 shows the superposition of distribution functions of two signal packets generated by double instantaneous illuminations to the backside of the sensor and dispersed during the travel, where Δt is the interval of the double illuminations, and σ is the standard deviation of the arrival time of signal electrons at the collecting gate on the front side. If a Gaussian distribution is assumed as the arrival time distribution, for Δt > 2σ, a dip appears at the center of the superposed distribution. Therefore, the no-dip condition, Δt = 2σ, was employed for the separability criterion for the temporal resolution [

15]. This is a similar concept to the Rayleigh’s criterion for the spatial resolution applied to the superposed Airy’s diffraction patterns with a 16% dip at the center [

14].

The expression of the arrival time distribution can be derived through a strict theoretical analysis. However, the resultant expression cannot be expressed with elementary functions, requiring numerical calculations to observe the characteristics. A common method to obtain an approximate expression from a rigorous analytical expression is to expand the original expression to a series under a specific condition and employ the lower order terms. However, we employed a different approach. The arrival time distribution asymptotically approaches the Gaussian for a large

W or a large

D, where

W is the thickness of the photoelectron conversion layer, and

D is the vertical diffusion coefficient. By assuming the Gaussian distribution at the arrival section, we derived an explicit approximate solution of the temporal resolution limit, which is two times of the standard deviation of the arrival time, as follows [

15]:

where

where

${\sigma}_{sA}^{2},{\sigma}_{m}^{2}\mathrm{and}{\sigma}_{d}^{2}$ denote respectively the approximate expression of the variance of the arrival time and those caused by the mixing effect due to the penetration distribution

$\mathrm{k}\left(s\right)$ of photons and the pure diffusion effect due to the random motion of generated electrons;

${W}^{\prime}=W/\delta ,{t}^{\prime}=\delta /v,\mathrm{and}\text{}{D}^{\prime}=2D/{v}^{2};$ δ represents the average penetration depth and

$v$ representsthe drift velocity.

When the values of the four parameters, W, δ, $v$ and D are assigned, the temporal resolution limit Δt_{A} is calculated. The values of δ, $v$ and D are dependent on the wavelength of the incident photon, the material and the environmental conditions including the electric field, temperature and pressure.

Equation (1) provides not only the expression of the temporal resolution limit, but also examining the conditions of these parameters to decrease Δt_{A} leads to the sensor structure which minimizes the temporal resolution limit. For example, the value of the drift velocity $v$ saturates and the vertical diffusion coefficient D takes the minimum value at the critical field 25 kV/cm, which minimizes Δt_{A}. The crystal orientation <111> of the Silicon layer provides slightly smaller D than the <100> layer, while the availability of the <111> wafer is low in practice.

Factors governing the temporal resolution limit are apparently observed from the simple expression of Equation (1). The parameters, $W{}^{\prime}$, ${t}^{\prime}$ and $\text{}D{}^{\prime}t{}^{\prime}$ are standardized with the average penetration depth $\delta $, respectively representing the thickness, the drift duration, and the diffusion per the average penetration depth.

The unit drift duration ${t}^{\prime}$ is a dominant factor; for $\mathrm{t}{}^{\prime}\gg 1\text{}\mathrm{and}\text{}\mathrm{t}{}^{\prime}\ll 1$, the mixing and the diffusion respectively governs the temporal resolution. By taking the limits of the parameters, Equation (1) is reduced to simple expressions, which describe the basic characteristics of the temporal resolution limit.

To examine the accuracy of Equation (1), the expression of the temporal resolution limit is derived with the strictly theoretical formulation. The one dimensional spatial distribution at a time

t and a depth

z of an electron generated at

t =

z = 0 is expressed by a Gaussian distribution [

21]:

However, the temporal distribution passing through

z =

W skews from the Gaussian distribution with a slightly acute front and a longer tail, since electrons arriving at

W earlier and later are respectively affected less and more by diffusion. The temporal distribution, i.e., the flux distribution at

W, is derived by inserting the spatial distribution, Equation (3), to the drift diffusion equation as follows:

Electrons generated at the depth

s travels

$\left(W-s\right)$. The flux distribution of the electrons at

W is expressed by inserting the travel distance

$\left(W-s\right)$ to

z in Equation (4). The probability distribution of the flux weighted by the distribution of the penetration depth is the product of Equation (2) and Equation (4). The total flux distribution is the integration of the product between

$\left(0,\text{}W\right)$ with respect to

s.

The 0th, 1st and 2nd moments of the arrival time with respect to

t are as follows:

The temporal resolution limit is:

where

Equation (6) is formulated in the strict manner. However, it cannot be expressed with elementary functions. The value is numerically calculated.

The approximate expression Δt

_{A} is compared with the numerical value of the exact solution Δt

_{E}. For the comparison, an intrinsic silicon layer of the crystal orientation <111> is assumed, which provides a lower drift velocity

$v$ and a larger diffusion coefficient

D than those of a <100> silicon layer, resulting in a shorter temporal resolution limit. In this case, the values of

$v$ and

D are respectively 9.19 × 10

^{6} cm/s and 10.8 cm

^{2}/s at 300 K under the critical electric field of 25 kV/cm [

15,

22,

23]. The wavelength of the incident light and the energy of the incident X-ray are assumed 550 nm (green light, 2.25 eV) and 10 keV, for which the average penetration depths are respectively 1.733 μm [

24] and 126.6 μm [

25]. As the wavelength of visible light is between 400 nm and 700 nm, light with the wavelength of 550 nm was selected as a representative visible light.

The results are shown in

Figure 5. The approximate expression perfectly agree with the numerically calculated exact solution for

$W>0.4\text{}\mathsf{\mu}\mathrm{m}$ both for green light of 550 nm and an X-ray of 10 keV. The range covers the values used in practice. Furthermore, for

$0.4\text{}\mathsf{\mu}\mathrm{m}W3\text{}\mathsf{\mu}\mathrm{m}$ for the green light and

$0.4\text{}\mathsf{\mu}\mathrm{m}W300\text{}\mathsf{\mu}\mathrm{m}$ for the X-ray, the temporal resolution limit is approximated by the following relation within the 1.5% error:

where

$v=0.0919$ μm/ps for the critical field, and the values of

$\Delta \mathrm{t}$ and

$W$ are in ps and μm. The values of the constants in Equation (7) are slightly different from Equation (11) in the prior paper [

15]. The reason is that the latter one was derived by omitting the second term in the square root in Equation (10).

When ${W}^{\prime}=1$, i.e., the thickness W is equal to the average penetration depth δ, the temporal resolution limit for the representative visible light of 550 nm estimated from Equation (1) is compared with the exact solution calculated from the strict formulation Equation (6). The values are respectively 11.108 ps and 11.119 ps. The difference is only 0.1%. The temporal resolution limit, 11.1 ps, is reconfirmed by the exact solution. The theoretical highest frame rate is the inverse, 90.9 Gfps.

If a BSI silicon image sensor is designed by strictly following the conditions introduced in the theoretical analysis, the sensor will achieve the theoretical temporal resolution limit. However, some of the conditions conflict with other performance parameters of silicon image sensors, such as sensitivity and crosstalk. The temporal resolution $2\mathsf{\sigma}$ represents the limit for the non-dip condition. In practice, the frame interval of $3\mathsf{\sigma}\text{}\mathrm{to}\text{}4\mathsf{\sigma}$ sufficiently suppresses the temporal cross talk. When ${W}^{\prime}=W/\delta =3$, instead of 1, the absorption rate (sensitivity) $p=\text{}$95.0%, and the crosstalk due to photons remaining after the absorption is reduced to a practically negligible level.

The parameter values in Equation (1) are selected, depending on applications. For the high-speed X-ray image sensor developed by Claus et al. [

26],

${W}^{\prime}$ is around 1, since the circuit layer on the front side is much thinner than the penetration depth, causing less crosstalk due to electrons generated in the circuit layer, while the signal generation layer should be thick enough to keep a reasonable absorption ratio.

#### 3.3. Suppression of Horizontal Motion of Electrons with Convex Pyramid Charge Collector

A narrow square silicon pipe is assumed, where both the incident light and generated electrons are guided to the front side, and disperse at the bottom end of the pipe. This pipe architecture can be implemented by vertical etching of a silicon surface with the crystal orientation <100>, which is a well-known existing technology. The pipe was named a light-charge guide pipe (LCGP).

The efficiency for suppression of the horizontal motion was evaluated through simulations by changing the diameter and the length to adjust the tradeoffs between the frame rate, sensitivity and crosstalk. The critical field is 25 kV/cm. The result of the practical optimization is shown in the third column of

Table 2, where the temporal resolution of 49 ps is achieved.

Even though the LCPG can be made with an existing technology, it requires an effective light focusing component attached on the backside, in spite that the major advantage of the backside illumination is the 100% fill factor. Hence, we will propose a convex silicon pyramid as shown in

Figure 6. A <111> silicon surface appears by etching the <100> surface with an angle of 54.7 degrees under an appropriated condition. With the technique, concave and convex silicon pyramids can be formed [

27,

28,

29,

30,

31]. The field in the direction along the pyramid surface is 81.6% (sin 54.7 degrees) of the vertical one. Therefore, it is expected that a temporal resolution may be close to the resolution achieved by the LCPG.

An array of concave silicon pyramids (pyramid-shaped holes) have been applied to solar cells to reduce the reflection factor at silicon surfaces [

28,

29]. Yokogawa et al. applied the concave silicon pyramid array to their infrared image sensor to decrease dark current by enhancing diffraction of incident light with the pyramid array and making the silicon layer thinner [

31]. Before the application, they improved the quality of the concave pyramid array to sufficiently suppress dark current from the pyramid array. While, at this moment, a high quality convex pyramid array is not available, if a good application is presented, it will not take a long time to develop a technology to improve the quality.

Apart from the process technology, a crucial problem associated with the convex pyramid array is how to guide signal electrons to the outlet at the bottom, avoiding collision of the electrons to the pyramid surface. The crossing angle between the equi-potential contours and the pyramid surface must be more than 90 degrees. Then, electrons move inward in the pyramid. A simulation study is performed to confirm the technical feasibility of the structure.

The thickness of the total silicon layer of the simulation model is 13.1 μm, consisting of the backside hole accumulation layer of 0.1 μm, the generation layer of 12.0 μm (three times the penetration depth 4 μm of 650-nm red light), and the circuit layer of 1.0 μm. The pixel is a 12.73 μm square. The critical field of 25 kV/cm is applied to the generation layer.

A thin Boron layer is applied over the pyramid surface, and a small circular Phosphorous implant is introduced at the center of the outlet of the pyramid. Then, concentrations of the dopants was adjusted by simulations to increase the electron collection ratio (the number of electrons collected by the collecting gate/the number of generated electrons). The resultant potential field is shown in the right half of

Figure 6, which collects more than 98% of the generated electrons as shown in the fourth column of

Table 2. Furthermore, the sensor can achieve the ultimate high signal-to-noise ratio (S/N). The fill factor is 100%, and the photo-conversion rate can be more than 90% for

W = 3

δ.

Figure 7 shows a convex silicon pyramid array fabricated by Ando. This is a preliminary one with a large size due to the limited performance of the MEMS facility of Ritsumeikan university. Still, it proves technical feasibility of the technology not only based on simulations shown in

Figure 6 and

Table 2, but also on a physical experiment. Further research on the fabrication technology is necessary, especially, for stacking a monocrystalline silicon layer to the top of the pyramids. One possible method may be the Si–Si direct bonding with high-temperature annealing.

The pyramid funnel has a huge application potential for BSI global shatter image sensors, 100% fill factor ultra-fast image sensors with in-pixel memories, detectors for imaging TOF MS with direct ion or electron bombardment on the backside, a device to connect a bundle of optical fibres with a Silicon or Germanium detector array for ultra-high-speed communication, and so on.

The size is too large at this moment due to limitation of our MEMS facility; the top of each pyramid should be shrunk more; a silicon layer is stacked on the top and the circuit is fabricated in the layer.