# Toward the Super Temporal Resolution Image Sensor with a Germanium Photodiode for Visible Light

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## Abstract

**:**

## 1. Introduction

## 2. Definitions of Temporal Resolution and Tools for Analyses

#### 2.1. Criteria for Temporal Resolution

#### 2.2. Temporal Resolution Based on DDF Models

#### 2.3. Monte Carlo Simulation Codes

_{1−x}Ge

_{x}compound, where x denotes the ratio of Ge. However, it is not directly applicable for our analysis. For example, the standard deviation to estimate the temporal resolution is not included in the output list, and the source of electrons cannot be placed in the bulk. The program can be easily modified, but the source program is not open. Therefore, we developed a method to utilize the code for our purpose without modification of the code (Appendix D).

## 3. Numerical Analyses

#### 3.1. Model Parameters

#### 3.2. Analysis for Si PD

- there are three ranges: the mixing (drift) effect is dominant for $0.14\text{}\mathsf{\mu}\mathrm{m}W100\text{}\mathsf{\mu}\mathrm{m}$, which covers the whole practically meaningful range, and the diffusion effect is dominant for the very thin range $W<0.14\text{}\mathsf{\mu}\mathrm{m}$ and for the very thick range $W>120\text{}\mathsf{\mu}\mathrm{m}\text{}$(not drawn);
- for $W>0.14\text{}\mathsf{\mu}\mathrm{m}$, the approximate expressions almost perfectly agree with the exact solution from the DDF model; for $W<0.14\text{}\mathsf{\mu}\mathrm{m}$, the discrepancy increases and the DDF model asymptotically converges to a constant value, ${\sigma}_{ddfL}=\sqrt{5}{D}_{c}/{v}_{c}^{2}$;
- the range used in practical applications, $\delta <W<3\text{}\delta $, is included in the drift-dominant range as concluded in our previous paper [1];
- The theoretical temporal resolution limit for $\left({W}_{T},\text{}{t}_{T}\right)$ is $\left(\delta ,\text{}2\sigma \right)=\left(1.73\text{}\mathsf{\mu}\mathrm{m},\text{}11.1\text{}\mathrm{ps}\right)$, and the practical limit for $\left({W}_{P},\text{}{t}_{P}\right)$ is $\left(3\delta ,\text{}3.3\sigma \right)=\text{}\left(5.19\text{}\mathsf{\mu}\mathrm{m},\text{}45.2\text{}\mathrm{ps}\right)$.

- the Monte Carlo simulation shows that ${\sigma}_{MC}$ monotonically decreases when the thickness of the photodiode decreases, while the numerical solution ${\sigma}_{ddf}$ of the DDF model converges to a constant value for an infinitesimal thickness;
- for $0.1\text{}\mathsf{\mu}\mathrm{m}W0.5\text{}\mathsf{\mu}\mathrm{m}$, ${\sigma}_{MC}$ departs from ${\sigma}_{ddf}$, and goes along the approximate expression ${\sigma}_{app};$
- for $W<0.1\text{}\mathsf{\mu}\mathrm{m},$ ${\sigma}_{MC}\text{}\mathrm{resulting}$ from our inhouse MC simulation code departs from ${\sigma}_{app},$ and, for $W<0.03\text{}\mathsf{\mu}\mathrm{m}$, it becomes parallel to the mixing (drift) component ${\sigma}_{mixL}=1/\sqrt{12}W/{v}_{c}$ with the slope proportional to $W$, suggesting that the motion of the signal electrons converges to a ballistic motion (see Appendix C).

#### 3.3. Super Temporal Resolution Limit for a Ge PD

#### 3.4. Toward Super Temporal Resolution through SWIR Imaging

## 4. Dark Current

## 5. Concluding Remarks

#### 5.1. Conclusions

#### 5.2. Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## List of Symbols

Symbol | Description |

$\alpha $ | Absorption rate of incident light |

$\delta $ | Penetration depth of incident light |

${v}_{c}$ | Drift velocity at saturation |

${D}_{c}$ | Diffusion coefficient at the drift velocity saturated |

$\mu $ | Mean |

${t}_{T}$ | Theoretical temporal resolution limit |

${t}_{P}$ | Practical temporal resolution |

$\sigma $ | Standard deviation of the arrival time of signal electrons |

${\sigma}_{ddf}$ | Standard deviation of the drift-diffusion-flux model |

${\sigma}_{ddfL}$ | Limit of the standard deviation of the drift-diffusion-flux model |

${\sigma}_{app}$ | Standard deviation of the approximation model |

${\sigma}_{mix}$ | Standard deviation of the mixing component |

${\sigma}_{diff}$ | Standard deviation of the diffusion component |

${\sigma}_{appL}$ | ${\sigma}_{app}$ for the infinitesimal thickness of the photodiode |

${\sigma}_{mixL}$ | ${\sigma}_{mix}$ for the infinitesimal thickness of the photodiode |

${\sigma}_{diffL}$ | ${\sigma}_{ddf}$ for the infinitesimal thickness of the photodiode |

${\sigma}_{MC}$ | Standard deviation calculated with the inhouse MC simulation code |

${\sigma}_{MCS}$ | Standard deviation calculated with the Sentaurus MC simulation code |

$W$ | Thickness of the sensor |

${W}_{T}$ | $W$ for the theoretical temporal resolution limit |

${W}_{P}$ | $W$ for the practical temporal resolution limit |

## Appendix A. Exact Formulation of the Temporal Resolution

Strict Expression | $\mathbf{Asymptotic}\text{}\mathbf{Expressions}\text{}\mathbf{for}\text{}\mathit{W}\text{}\to \text{}0$ | |
---|---|---|

The Gaussian drift-diffusion equation for a single pulse | $g\left(z,\text{}t\right)\text{}=1/\sqrt{4\pi Dt}\times \mathrm{exp}\left(-{\left(z-vt\right)}^{2}/4Dt\right)$ | |

The flux passing a detection plane | $h\left(z,t\right)\text{}=vg\left(z,t\right)-D\partial g\left(z,t\right)/\partial z=1/2\left(v+z/t\right)g\left(z,t\right)$ | |

The penetration depth distribution | $k\left(s\right)\text{}=\text{}\left(1/\delta \right){e}^{-s/\delta}$ | |

Convolution of $k\left(s\right)$ and $h\left(z,t\right)$ | $f\left(t\right)\text{}={{\displaystyle \int}}_{0}^{W}k\left(s\right)h\left(W-s,\text{}t\right)ds$ | |

The 0th moment (Absorption rate) | $p={{\displaystyle \int}}_{0}^{\infty}f\left(t\right)dt$ | ${W}^{\prime}=W/\delta $ |

The 1st moment | $E\left(t\right)\text{}=1/p\times {{\displaystyle \int}}_{0}^{\infty}tf\left(t\right)dt$ | ${D}^{\prime}/2=D/{v}^{2}$ |

The 2nd moment | $E\left({t}^{2}\right)\text{}=1/p\times {{\displaystyle \int}}_{0}^{\infty}{t}^{2}f\left(t\right)dt$ | $3/2{{D}^{\prime}}^{2}=6{\left(D/{v}^{2}\right)}^{2}$ |

Variance | ${\sigma}_{ddf}^{2}=E\left({t}^{2}\right)-E{\left(t\right)}^{2}$ | ${\sigma}_{ddfL}^{2}=5/4{{D}^{\prime}}^{2}=5{\left(D/{v}^{2}\right)}^{2}$ |

Temporal resolution | $\Delta {t}_{ddf}=2{\sigma}_{ddf}$ | $\sqrt{5}{D}^{\prime}=2\sqrt{5}D/{v}^{2}$ |

Normalized parameters | ${W}^{\prime}=W/\delta ,{t}^{\prime}=\delta /v,{D}^{\prime}=2D/{v}^{2}$ |

## Appendix B. An Explicit Approximate Expression for the Temporal Resolution

Approximation Formula | Solutions | Asymptotic Expressions $\mathbf{for}\text{}\mathit{W}\text{}\to \text{}0$ | |
---|---|---|---|

The penetration depth distribution | $k\left(s\right)\text{}=\text{}\left(1/\delta \right){e}^{-s/\delta}$ | ||

Absorption rate | $p={{\displaystyle \int}}_{0}^{W}k\left(s\right)ds$ | $1-{e}^{-{W}^{\prime}}$ | ${W}^{\prime}=W/\delta $ |

Average arrival time | ${t}_{r}=\left(W-s\right)/v$ | ||

$E\left({t}_{r}\right)$ | $\frac{1}{p}{{\displaystyle \int}}_{0}^{\infty}{t}_{r}k\left(s\right)ds$ | ${t}^{\prime}\left({W}^{\prime}/p-1\right)$ | |

$E\left({t}_{r}^{2}\right)$ | $\frac{1}{p}{{\displaystyle \int}}_{0}^{\infty}{t}_{r}^{2}k\left(s\right)ds$ | ${{t}^{\prime}}^{2}\left\{\left({{W}^{\prime}}^{2}-2{W}^{\prime}\right)/p+2\right\}$ | |

Mixing component of variance | ${\sigma}_{mix}^{2}=E\left({t}_{r}^{2}\right)-E{\left({t}_{r}\right)}^{2}$ | ${{\mathit{t}}^{\prime}}^{2}\left\{1-{{\mathit{W}}^{\prime}}^{2}\times \left(1-\mathit{p}\right)/{\mathit{p}}^{2}\right\}$ | ${\left({W}^{\prime}{t}^{\prime}\right)}^{2}/12={W}^{2}/\left(12{v}^{2}\right)$ |

Diffusion component of variance | ${\sigma}_{diff}^{2}=2D/{v}^{2}E\left({t}_{r}\right)$ | $\left({\mathit{D}}^{\prime}{\mathit{t}}^{\prime}\right)\left({\mathit{W}}^{\prime}-\mathit{p}\right)/\mathit{p}$ | ${D}^{\prime}{t}^{\prime}{W}^{\prime}/2=DW/{v}^{3}$ |

Variance | ${\sigma}_{app}^{2}={\sigma}_{mix}^{2}+{\sigma}_{diff}^{2}$ | ${D}^{\prime}{t}^{\prime}{W}^{\prime}/2=DW/{v}^{3}$ | |

Approximation for a whole thickness | ${\sigma}_{appS}^{2}={\sigma}_{app}^{2}+{\sigma}_{ddfL}^{2}$ | ${\sigma}_{appS}^{2}$ perfectly fits ${\sigma}_{ddf}^{2}$ (${\sigma}_{ddf}^{2}$ and ${\sigma}_{ddfL}^{2}$ are in Appendix A) | |

Temporal resolution | $\Delta {t}_{app}=2{\sigma}_{app}$ | ||

Normalized parameters | ${W}^{\prime}=W/\delta ,{t}^{\prime}=\delta /v,{D}^{\prime}=2D/{v}^{2}$ |

## Appendix C. Derivations of the Asymptotic Expressions

## Appendix D. A Method to Estimate the Standard Deviation by Using the Sentaurus MC Simulation Code

**k**-th calculation step. Although ${N}_{k}$ is not output from the simulation code, it can be calculated from the value of the current. The current ${I}_{D}$ (A/$\mathsf{\mu}\mathrm{m}$) is calculated by the following formula:

^{−19}

**C**is the unit electron charge, ${N}_{kn}$ is the number of electrons that have propagated to the front side, and $w$ (${\mathsf{\mu}\mathrm{m}}^{2}/{\mathrm{cm}}^{3}$) is a weight calculated by integration of the electron density over the device. Therefore, the average propagation time ${t}_{k}$ for the calculation step

**k**is:

- (1)
- ${N}_{k}$ is the number of all generated electrons, including electrons taking paths which are not our target, especially, the number of electrons absorbed at the source electrode and injected from another electrode,
- (2)
- ${N}_{k}$ is a random number.

**k**-th group with ${N}_{k}$ samples is ${\sigma}_{0}/\sqrt{{N}_{k}}$. Therefore, the standardized value ${X}_{k}=\left({x}_{k}-\mu \right)/{\sigma}_{k}=\left({x}_{k}-\mu \right)\times \sqrt{{N}_{k}}/{\sigma}_{0}$. The mean and the variance of a standardized value ${X}_{k}$, respectively, distribute around 0 and 1. Therefore, it is expected for a sufficiently large $K$ that the sum of the squares of ${X}_{k}$ approaches $K$, then,

**Figure A1.**The model to estimate the original standard deviation ${\sigma}_{0}$ by using the Sentaurus MC simulation code (W = 100, 50, and 10 nm; the front-side doping for the Ge PD is 10

^{15}cm

^{−3}).

**s**, the number of the electrons taking the path A in the figure is calculated from the current by assuming the propagation times of the electrons taking E, F, and G are negligibly small, and the generation probabilities of electrons taking C and D are suppressed, respectively, by the strong backside p-doping, and the adverse field and the front-side p-doping. The electrons taking B can be an error source, but the generation probability may not be so large, especially when

**s**increases. Then, the average propagation time for

**s**is calculated by dividing the time for the calculation step by the number of the electrons taking A. The simulation is repeated, and the variance is estimated by using Equation (A4) for a value of

**s**.

**s**= 0%, 10%, 20%, …, 100% of

**W**, and averaged by weighting the probability of the penetration depth of light to estimate the variance of the propagation time considering the penetration depth distribution. The standard deviation is calculated as the square root of the variance to estimate the temporal resolution.

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**Figure 1.**Evolution of ultra-high-speed image sensors and imaging devices having achieved the super temporal resolution. The theoretical temporal resolution limit for a Ge PD is presented later in Section 3.3 of this paper. The slanted dashed line assumes that a chip with in-situ driver circuits is stacked to the sensor chip.

**Figure 2.**Definitions of the temporal resolution. (

**a**) The theoretical temporal resolution limit, ${t}_{T}=2\sigma $, (

**b**) a practical temporal resolution, ${t}_{P}={t}_{95}-{t}_{05}\cong 3.3\sigma $, (

**c**) for reference.

**Figure 4.**A model to analyze the standard deviation σ of the arrival time of electrons to the detection plane. W: thickness of the photodiode, s: the generation site of an electron distributing in the depth direction due to the exponential distribution of the penetration depth of light. The bulk is made of an intrinsic silicon. From Figure 5a, the electric fields between the backside and the detection plane are set at 2.5 $\mathrm{V}/\mathsf{\mu}\mathrm{m}$ for simulations in Si and 0.42 $\mathrm{V}/\mathsf{\mu}\mathrm{m}$ for simulations in Ge at which the drift velocities are 95% of the fully saturated values (see also Table 1).

**Figure 7.**The temporal resolution of the Si PD with incident light of 550 nm along the <111> direction. Solid line ${\sigma}_{ddf}$: exact formulation of the DDF model; dashed line ${\sigma}_{app}$: our approximate expression; ${\sigma}_{ddfL}$: asymptotic value of ${\sigma}_{ddf}$ for an infinitesimal $W$; ${\sigma}_{diff}$ and ${\sigma}_{mix}$: diffusion and mixing components of ${\sigma}_{app}$ where mixing is due to a combined effect of the drift velocity and the distribution of the penetration depth.

**Figure 8.**Standard deviation of various relevant parameters for a Si PD for $0.01<W<1\text{}\mathsf{\mu}\mathrm{m}$. Wavelength of light $\lambda =550\mathrm{nm}$; ${\sigma}_{MC}$ (blue line) and ${\sigma}_{MCS}$ (red line): standard deviations of the arrival times calculated by our inhouse MC simulation code and by the Sentaurus MC simulation code; ${\sigma}_{diffL}$ and ${\sigma}_{mixL}$; asymptotic expression of ${\sigma}_{diff}$ and ${\sigma}_{mix}$; for an infinitesimal W; ${\sigma}_{diff}$ is a straight line proportional to ${W}^{-1}$, extending ${\sigma}_{diffL}$ (black line).

**Figure 9.**Standard deviation of arrival time and temporal resolutions of the Ge PD for 550 nm illumination with the average penetration depth of 20 nm. ${\sigma}_{MCS}$ is calculated by a newly proposed method based on Sentaurus MC simulations. The other results in the plots are calculated with equations in Appendix A and Appendix B for the values in Table 1. The diffusion component ${\sigma}_{diff}$ is not drawn since it completely fits ${\sigma}_{app}$ for $W<0.1\mathsf{\mu}$m for Ge PD.

**Figure 10.**Standard deviations for Ge PDs: (

**a**) wavelength of incident light $\lambda =1\mathsf{\mu}\mathrm{m}$ and the average penetration depth $\delta =0.526\text{}\mathsf{\mu}\mathrm{m},$ (

**b**) $\lambda =1.5\mathsf{\mu}\mathrm{m},\delta =2.04\mathsf{\mu}\mathrm{m}$.

Si $\overrightarrow{\mathit{E}}\parallel <111>$ | Ge $\overrightarrow{\mathit{E}}\parallel <100>$ | ||
---|---|---|---|

Wavelength | 0.55 $\mathsf{\mu}$m | 0.55 $\mathsf{\mu}$m | 1 $\mathsf{\mu}$m |

Penetration depth | 1.73 $\mathsf{\mu}$m | 20.0 nm | 526 nm |

Critical E-field | 25 kV/cm | 4.2 kV/cm | |

Drift velocity | 9.19 $\times $ 10^{6} cm/s | 5.8 $\times $ 10^{6} cm/s | |

Diffusion coefficient | 10.8 cm^{2}/s | 42.5 cm^{2}/s |

^{2}/s for 77, 160, 200, and 300 K, respectively. They take close values for 200 and 300 K. For Ge, the diffusion coefficients at critical fields are 48.5, and 42.5 cm

^{2}/s at 77 and 190 K. Therefore, the diffusion coefficient for Ge at 300 K is assumed to be 42.5 cm

^{2}/s.

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Ngo, N.H.; Nguyen, A.Q.; Bufler, F.M.; Kamakura, Y.; Mutoh, H.; Shimura, T.; Hosoi, T.; Watanabe, H.; Matagne, P.; Shimonomura, K.;
et al. Toward the Super Temporal Resolution Image Sensor with a Germanium Photodiode for Visible Light. *Sensors* **2020**, *20*, 6895.
https://doi.org/10.3390/s20236895

**AMA Style**

Ngo NH, Nguyen AQ, Bufler FM, Kamakura Y, Mutoh H, Shimura T, Hosoi T, Watanabe H, Matagne P, Shimonomura K,
et al. Toward the Super Temporal Resolution Image Sensor with a Germanium Photodiode for Visible Light. *Sensors*. 2020; 20(23):6895.
https://doi.org/10.3390/s20236895

**Chicago/Turabian Style**

Ngo, Nguyen Hoai, Anh Quang Nguyen, Fabian M. Bufler, Yoshinari Kamakura, Hideki Mutoh, Takayoshi Shimura, Takuji Hosoi, Heiji Watanabe, Philippe Matagne, Kazuhiro Shimonomura,
and et al. 2020. "Toward the Super Temporal Resolution Image Sensor with a Germanium Photodiode for Visible Light" *Sensors* 20, no. 23: 6895.
https://doi.org/10.3390/s20236895