\section{Introduction}

Silicon photomultiplier (SiPM) has recently become popular photodetector for scintillator readout. 
Thanks to its compact size, solid state, low operating voltage and insensitivity to magnetic fields, SiPM has been a practical alternative to traditional photomultiplier tube (PMT), especially in space-borne applications. 
However, a known issue of SiPM is significant radiation damage and thus performance degradation caused by harsh space radiation environment. 

Several space missions utilizing SiPM have been launched so far as summarized in Table \ref{tab:missions}. 
Lazio-Sirad \cite{Bencardino2005449} studies the radiation environment inside International Space Station (ISS). 
The Hard X-ray Modulation Telescope (HXMT) \cite{LI201663} using SiPM in its automatic gain control system. 
SIRI-1 and SIRI-2 \cite{SIRI-1, mitchell2019strontium, SIRI-2} aims to qualify new ${\rm SrI_{2}:Eu}$ scintillator and SiPM technology. 
Mini-EUSO \cite{CAPEL20182954} observes the Earth from ISS in UV range. 
LabOSat \cite{BARELLA2020164490} is designed to technically validate SiPM with LEDs for excitation. 
Gravitational wave high-energy Electromagnetic Counterpart All-sky Monitor (GECAM) \cite{ZHANG20198} is a mission composed of two small satellites to localize Gamma-Ray Bursts (GRBs) and its main detector, Gamma-Ray Detectors (GRDs), are based on ${\rm LaBr_{3}:Ce}$ and SiPM array. 
GRBAlpha \cite{GRBAlpha}, a 1U CubeSat mission, acts as the demostration of CAMELOT \cite{CAMELOT}, a fleet of CubeSats to observe GRBs. 
The GAGG Radiation Instruments (GARI) \cite{GARI} are two identical instruments designed to space-qualify new gamma-ray detector technology for space-based astrophysical and defense applications. 
Gamma-Ray Integrated Detectors (GRID) \cite{Wen2019} is a space mission concept with the purpose of monitor GRBs by a constellation of CubeSats equipped with GAGG:Ce scintillator and SiPM. 
The first GRID detector GRID-01 was launched on Oct. 29, 2018 and the subsequent one GRID-02 launched on Nov. 6, 2020. 
Other similar missions under development including: BurstCube \cite{racusin2017burstcube}, HERMES \cite{FUSCHINO2019199}, EIRSAT-1 (GMOD) \cite{Murphy2021}, Glowbug \cite{Grove2019109}, CUBES \cite{Kushwah_2021}, GALI \cite{GALI}. 

\begin{table}
    \begin{center}
        \caption{Launched space missions utilizing SiPMs. }
        \begin{tabular}{cccc}
            \hline
            Project name        & Launch date   & Orbit                                     & Ref.                                  \\
            \hline
            Lazio-Sirad         & 2005.04.15    & 417 $\times$ 423 km 51.6$^{\circ}$ ISS    & \cite{Bencardino2005449}              \\
            HXMT                & 2017.06.15    & 533 $\times$ 543 km 43.0$^{\circ}$ LEO    & \cite{LI201663}                       \\
            GRID-01             & 2018.10.29    & 502 $\times$ 519 km 97.4$^{\circ}$ LEO    & \cite{Wen2019}                        \\
            SIRI-1              & 2018.12.03    & 567 $\times$ 589 km 97.7$^{\circ}$ SSO    & \cite{SIRI-1, mitchell2019strontium}  \\
            Mini-EUSO           & 2019.08.22    & 417 $\times$ 423 km 51.6$^{\circ}$ ISS    & \cite{CAPEL20182954}                  \\
            LabOSat             & 2020.01.15    & Unknown LEO                               & \cite{BARELLA2020164490}              \\
            GRID-02             & 2020.11.06    & 457 $\times$ 467 km 97.2$^{\circ}$ LEO    & \cite{Wen2019}                        \\
            GECAM               & 2020.12.09    & 586 $\times$ 604 km 29.0$^{\circ}$ LEO    & \cite{ZHANG20198}                     \\
            GRBAlpha            & 2021.03.22    & 534 $\times$ 563 km 97.5$^{\circ}$ LEO    & \cite{GRBAlpha}                       \\
            SIRI-2              & 2021.12.07    & 35784 $\times$ 35788 km 0.0$^{\circ}$ GEO & \cite{SIRI-2}                         \\
            GARI                & 2021.12.21    & 417 $\times$ 423 km 51.6$^{\circ}$ ISS    & \cite{GARI}                           \\
            GRID-03B \& GRID-04 & 2022.02.27    & 523 $\times$ 550 km 97.5$^{\circ}$ LEO    & \cite{Wen2019}                        \\
            \hline
        \end{tabular}
        \label{tab:missions}
    \end{center}
\end{table}

In-orbit radiation damage effect studies of SiPM were hardly reported so far, although several experiments have been conducted at accelerator facilities \cite{LI201663, BARTLETT2020163957, MITCHELL2021164798, HIRADE2021164673, ULYANOV2020164203}. 
SIRI-1 reports the only in-orbit result, which shows a significant increase in SiPM dark current \cite{MITCHELL2021164798}. 
In this article we present the in-orbit SiPM characterization results obtained by GRID-02 during the first few months after launch. 
Based on the basic measurement setups in the detector, we analyze the change of breakdown voltage and dark current due to radiation damage. 
An estimate of dark current increasing rate of SiPM operating in LEO is given. 
The energy resolution and lower trigger threshold deterioration of GRID-02 as a result of dark count noise increase is also studied. 

\section{In-orbit characterization setup and methods}

Besides the optimization for gamma-ray detection in GRID, several functions are specifically designed for performance study of SiPMs. 
The operating conditions, including current, bias voltage and temperature, of the SiPMs are monitored, and a charge injection module is used to analyze the noise contributions. 
Details about GRID instrument design could be found in \cite{Wen2021}. 
Here we briefly review the biasing and readout circuits, and focus on the characterization setup. 

The exploded view and a simplified block diagram of GRID detector are shown in Fig. \ref{fig:detector_exploded} and Fig. \ref{fig:block_diagram}. 
The detector comprises four independent channels designated as CH0 - CH3. 
Each channel consists of the scintillator crystal, the 4 $\times$ 4 array of MicroFJ-60035-TSV\footnote{\href{https://www.onsemi.com/pdf/datasheet/microj-series-d.pdf}{MicroFJ-60035-TSV datasheet}} SiPMs, the bias voltage power supply, the readout electronics, and the SiPM characterization circuits. 
A microcontroller (MCU) acts as control electronics, which monitors and controls the four channels. 

\begin{figure}
    \centering
    \includegraphics[width=0.8\textwidth]{detector_exploded.pdf}
    \caption{Exploded view of GRID detector. Details about GRID instrument design could be found in \cite{Wen2021}. }
    \label{fig:detector_exploded}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=1.0\textwidth]{block_function.pdf}
    \caption{A simplified block diagram of GRID detector. }
    \label{fig:block_diagram}
\end{figure}

Due to the limitation of power consumption and payload volume, there is no temperature control system in GRID detector. 
The temperature of GRID mainly depends on the overall temperature of the satellite. 
The temperature of each SiPM array is measured by a digital temperature sensor, TMP112\footnote{\href{https://www.ti.com/lit/ds/symlink/tmp112.pdf}{TMP112 datasheet}}, which offers 0.5$^{\circ}$C accuracy. 

Moreover, a charge injection module controlled by MCU is connected to the input of preamplifier so that a constant charge pulse could be injected at a specified frequency. 
Through charge injection with and without SiPM bias voltage, we are able to analyze the noise contributions, either from SiPM dark count noise or from electronics noise or other noise sources. 

Figure \ref{fig:schematic_layout} shows the schematic layout of ${I}\sim{V}$ measurement setup. 
The 16 SiPMs in the same array are simply connected in parallel, which means we can only characterize the whole array but not each single SiPM chips. 
The adjustable positive bias voltage on cathode is supplied by a switching regulator, LT3482\footnote{\href{https://www.analog.com/media/en/technical-documentation/data-sheets/3482fa.pdf}{LT3482 datasheet}}, and controlled by a 12-bit digital-to-analog converter (DAC), AD5672R\footnote{\href{https://www.analog.com/media/en/technical-documentation/data-sheets/ad5672r_5676r.pdf}{AD5672R datasheet}}. 
After a resistive divider and a voltage follower, it is monitored by a MCU internal 12-bit analog-to-digital converter (ADC). 
The current through SiPMs is converted to voltage by a 1 $\rm k\Omega$ grounding resistor on anode and measured by the same MCU internal ADC after a voltage follower. 
The difference between these two voltages is the so-called bias voltage ($V_{\rm bias}$) on SiPM. 
A proportional-integral-differential (PID) control method is applied to stabilize $V_{\rm bias}$. 

\begin{figure}
    \centering
    \includegraphics[width=0.8\textwidth]{schematic_layout.pdf}
    \caption{Schematic layout of ${I}\sim{V}$ measurement setup. }
    \label{fig:schematic_layout}
\end{figure}

The current, bias voltage and temperature values are recorded as housekeeping data once per second during scientific observation, which is used to analyze SiPM dark current. 
Besides, the breakdown voltage ($V_{\rm bd}$) of SiPM is determined by ${I}\sim{V}$ measurement at different bias voltages. 
This measurement method is encapsulated as a special mode triggered by an instruction. 
Once switched to this mode, the bias voltage will be successively changed to 40 preset values around and above $V_{\rm bd}$. 
The measurement lasts for about 40 seconds, during which the temperature change is negligible. 
However, when compare between different measurements, temperature correction is performed. 

Based on the characterization setup described above, we carefully scheduled characterization experiments and scientific observations. 
The payload debug phase began on Nov. 8, 2020 after launch and finished on Nov. 19, 2020. 
Then we moved to daily observation phase and arranged characterization experiments once per day since Dec. 4, 2020. 
Due to the existence of other payloads, the scientific observation time of GRID-02 is limited to about 20 ks (5 - 6 hours) per day. 

\section{Results}

\subsection{Breakdown voltage}
\label{sec:Vbd}

Several methods could be used to extract $V_{\rm bd}$ from ${I}\sim{V}$ measurement, as well discussed in \cite{NAGY201755} and \cite{KLANNER201936}. 
For convenience of comparison, the method mentioned in datasheet is applied \cite{J60035}. 
The $\sqrt{I}$ versus $V_{\rm bias}$ is linear fitted above $V_{\rm bd}$ and the $V_{\rm bd}$ is defined as the voltage intercept. 
Temperature correction is adopted through 
\begin{equation}
    \label{eq:Vbd}
    V_{\rm bd}(T_{21})=V_{\rm bd}(T)-k_{\rm bd}\cdot(T-T_{21}),
\end{equation}
where $T_{21}=21^{\circ}{\rm C}=294.15\ {\rm K}$ is the standard measurement condition specified in datasheet, $T$ is the absolute temperature of each ${I}\sim{V}$ measurement and $k_{\rm bd}=21.5\ {\rm mV/K}$ is the temperature coefficient of $V_{\rm bd}$ specified in datasheet \cite{J60035}. 
This $k_{\rm bd}$ value is also used in gain calibration procedure of GRID-02 \cite{Gao2021}. 
In Fig. \ref{fig:Vbr_SensL} we show $V_{\rm bd}$ as a function of time. 
$V_{\rm bd}$ has no obvious change in the error range. 
Other methods, such as polynomial fit of inverse logarithmic derivative ($ILD=({\rm d}\ {\rm ln}I/{\rm d}V_{\rm bias})^{-1}$), are going to be done in the future. 

\begin{figure}
    \centering
    \includegraphics[width=1.0\textwidth]{Vbr_SensL.pdf}
    \caption{SiPM breakdown voltage ($V_{\rm bd}$) as a function of time. The values are obtained by linear fit of $\sqrt{I}$ versus bias voltage ($V_{\rm bias}$) and corrected to 21$^{\circ}$C. }
    \label{fig:Vbr_SensL}
\end{figure}

\subsection{Dark current}

The SiPM dark current ($I_{\rm dark}$) could be expressed by 
\begin{equation}
    \begin{aligned}
        \label{eq:Idark}
        I_{\rm dark}&=DCR\cdot(Gain\cdot{e})\cdot{ECF}\\
        &=DCR\cdot{C}_{\rm pix}\cdot(V_{\rm bias}-V_{\rm bd})\cdot{ECF},
    \end{aligned}
\end{equation}
where $DCR$ is the dark count rate, $Gain$ the SiPM overall gain, $e$ the elementary charge, $ECF$ the excess charge factor and $C_{\rm pix}$ the pixel capacitance including quenching capacitance \cite{KLANNER201936}. 
The term $(V_{\rm bias}-V_{\rm bd})$ is also known as overvoltage ($V_{\rm ov}$). 

The $C_{\rm pix}$ is found independent of temperature or radiation damage \cite{GARUTTI201969} and the $ECF$ which depends on $V_{\rm ov}$ is assumed approximately unchanged. 
The $V_{\rm bias}$ is set to 28.5 V without any temperature compensation so the temperature dependence of $V_{\rm bd}$ needs to be taken into account as discussed in section \ref{sec:Vbd}. 
Field-Enhanced Shockley-Read-Hall (FE-SRH) model \cite{GARUTTI201969, 155882} indicates the relationship between $DCR$ and temperature: 
\begin{equation}
    \label{eq:FESRH}
    DCR(T)\propto\left(1+2\sqrt{3\pi}\frac{F_{\rm eff}}{(kT)^{3/2}}{\rm e}^{\left(\frac{F_{\rm eff}}{(kT)^{3/2}}\right)^2}\right)T^2{\rm e}^{-\frac{E_{\rm a}}{kT}},
\end{equation}
where $F_{\rm eff}$ is the effective electric field strength, $k$ the Boltzmann constant and $E_{\rm a}=0.605\ {\rm eV}$ the activation energy. 
Since $V_{\rm bias}$ is set to 28.5 V, $F_{\rm eff}$ is thought to remain unchanged. 

Finally the $I_{\rm dark}$ expression becomes 
\begin{equation}
    \label{eq:Idark_fit}
    I_{\rm dark}(T)\propto\left(1+2\sqrt{3\pi}\frac{F_{\rm eff}}{(kT)^{3/2}}{\rm e}^{\left(\frac{F_{\rm eff}}{(kT)^{3/2}}\right)^2}\right)T^2{\rm e}^{-\frac{E_{\rm a}}{kT}}\cdot(V_{\rm bias}-V_{\rm bd}(T)).
\end{equation}
We split data by date and assume that the radiation damage difference in the same day is negligible. 
Then we fit the plot of $I_{\rm dark}$ versus temperature with Eq. (\ref{eq:Idark_fit}) and take $F_{\rm eff}$ as the free parameter. 
The $I_{\rm dark}$ values are unified to 5$^{\circ}$C and shown in Fig. \ref{fig:IT_p5} as a function of time. 

\begin{figure}
    \centering
    \includegraphics[width=1.0\textwidth]{IT_p5.pdf}
    \caption{SiPM dark current ($I_{\rm dark}$) at 28.5 V bias voltage as a function of time. The values are the sum of 16 SiPMs in the same channel and are unified to 5$^{\circ}$C. }
    \label{fig:IT_p5}
\end{figure}

A linear fit shows that $I_{\rm dark}$ increases $\sim$93/96/98/110 $\mu$A/year per SiPM chip for CH0 - CH3 respectively, which is reasonably consistent with the result $\sim$132 $\mu$A/year per SiPM chip from SIRI-1 \cite{MITCHELL2021164798}. 
The difference among four channels might be caused by possible mechanical damage received before or during launch, while the difference between GRID-02 and SIRI-1 could be related to different orbit and different operating temperature. 
A discussion about dose difference is given in section \ref{sec:Dose}. 

We also extrapolate the fit of $I_{\rm dark}$ versus temperature to -20$^{\circ}$C to evaluate those SiPMs working with cooling system. 
The increasing rate of $I_{\rm dark}$ at -20$^{\circ}$C is $\sim$50 $\mu$A/year per SiPM chip, almost half of the value at 5$^{\circ}$C. 

\subsection{Dark count noise}
\label{sec:noise}

The energy resolution and noise contributions of the GRID detector is tested through charge injection. 
The charge injection peaks with and without bias voltage are fitted with Gauss functions to derive the standard deviation ($\sigma$) as shown in Fig. \ref{fig:sigma_onoff}. 
Negative date values represent results obtained before launch. 
The error bars are within data points and therefore not shown for visualization purpose. 
The results in ADC channels are converted to equivalent energy with a scale factor $\sim$0.027 keV/channel extracted from on-ground calibration data of GRID-02 \cite{Gao2021}. 
Note that this value is heavily dependent on detector design, such as the electronics gain and the light yield of the scintillator crystal, so it is only applicable to GRID-02. 

The $\sigma_{\rm elec}$ in Fig. \ref{subfig:sigma_off} shows no significant change since it is dominated by electronics noise and fluctuation of injected charge itself. 
But, when bias power on and dark count noise becomes significant, $\sigma_{\rm total}$ gradually grows up with $I_{\rm dark}$, as shown in Fig. \ref{subfig:sigma_on}. 
The total noise $\sigma_{\rm total}$ could be expressed by 
\begin{equation}
    \label{eq:sigma_total}
    \sigma_{\rm total}^2=\sigma_{\rm DC}^2+\sigma_{\rm elec}^2,
\end{equation}
where $\sigma_{\rm DC}$ is the dark current noise and Campbell's theorem shows its relationship with dark count rate ($DCR$) and $I_{\rm dark}$: 
\begin{equation}
    \begin{aligned}
        \label{eq:Campbell}
        \sigma_{\rm DC}^2&=DCR\cdot(Gain\cdot{e})^2\cdot\int{h}^2(t){\rm d}t\\
        &\propto{I}_{\rm dark}\cdot(V_{\rm bias}-V_{\rm bd}),
    \end{aligned}
\end{equation}
where $h(t)$ is the unit impulse response of the readout electronics. 
Note that the $I_{\rm dark}$ here is the actual measurement result and should not be temperature corrected. 
In Fig. \ref{fig:sigmatotal_I} we show the plot of $\sigma_{\rm total}$ versus $I_{\rm dark}$ and fit them with 
\begin{equation}
    \label{eq:sigma_Idark_fit}
    \sigma_{\rm total}=\sqrt{a_{\rm n}\cdot{}I_{\rm dark}\cdot(V_{\rm bias}-V_{\rm bd})+b_{\rm n}},
\end{equation}
where $a_{\rm n}$ and $b_{\rm n}$ are free parameters. 
The results indicate a noise increase of $\sim$7.5 keV/year for each channel of GRID-02. 
The energy resolution and lower trigger threshold would grow up as well. 

\begin{figure}
    \centering
    \subfigure[28.5 V bias voltage]{
        \includegraphics[width=0.47\textwidth]{sigma_off.pdf}
        \label{subfig:sigma_off}
    }
    \subfigure[bias voltage off]{
        \includegraphics[width=0.47\textwidth]{sigma_on.pdf}
        \label{subfig:sigma_on}
    }
    \caption{Standard deviation ($\sigma$) of charge injection peak as a function of time. }
    \label{fig:sigma_onoff}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=1.0\textwidth]{sigmatotal_I.pdf}
    \caption{Detector total noise ($\sigma_{\rm total}$) versus dark current ($I_{\rm dark}$). The current are origin values without any correction. }
    \label{fig:sigmatotal_I}
\end{figure}

\section{Discussions}

\subsection{Dark current estimation}

Several previous studies found a linear relationship between $I_{\rm dark}$ and radiation damage measured by dose or particle fluences \cite{LI201663, MITCHELL2021164798, GARUTTI201969}. 
By simply assuming that radiation damage of SiPMs in GRID is linearly correlated with time, and the $F_{\rm eff}$ in Eq. (\ref{eq:Idark_fit}) is independent of radiation damage, Eq. (\ref{eq:Idark_fit}) is modified as 
\begin{equation}
    \begin{aligned}
        \label{eq:Idark_fit_time}
        I_{\rm dark}(Time,T)&=(a_{\rm dc}\cdot{Time}+b_{\rm dc})\\
        &\mathrel{\phantom{=}}\cdot\left(1+2\sqrt{3\pi}\frac{F_{\rm eff}}{(kT)^{3/2}}{\rm e}^{\left(\frac{F_{\rm eff}}{(kT)^{3/2}}\right)^2}\right)T^2{\rm e}^{-\frac{E_{\rm a}}{kT}}\\
        &\mathrel{\phantom{=}}\cdot\left(V_{\rm bias}-V_{\rm bd}(T)\right),
    \end{aligned}
\end{equation}
where $a_{\rm dc}$, $b_{\rm dc}$ and $F_{\rm eff}$ are free parameters and $Time$ is the days after launch. 
A joint fit of $I_{\rm dark}$ versus time and temperature gives the estimate of $I_{\rm dark}$ at specified conditions. 
The fitting results are listed in Table \ref{tab:Idark_time_temp}. 

\begin{table}
    \begin{center}
        \caption{Fitting results of Eq. (\ref{eq:Idark_fit_time}). }
        \begin{tabular}{cccc}
            \hline
                & $a_{\rm dc}$  & $b_{\rm dc}$  & $F_{\rm eff}/k^{3/2}$ \\
            \hline
            CH0 & 737.694       & 4236.96       & 10125.2   \\
            CH1 & 643.053       & 5100.42       & 10285.4   \\
            CH2 & 733.691       & 5298.48       & 10178.4   \\
            CH3 & 870.865       & 8744.55       & 10122.8   \\
            \hline
        \end{tabular}
        \label{tab:Idark_time_temp}
    \end{center}
\end{table}

By substituting these parameters into Eq. (\ref{eq:Idark_fit_time}), we obtain an approximate empirical equation around room temperature which describes the $I_{\rm dark}$ of GRID-02 at 28.5 V bias voltage, at specified time, and at specified temperature: 
\begin{equation}
    \label{eq:Idark_Time_T_pred}
    I_{\rm dark}(Time,T)=16\cdot(0.26\cdot{}Time+1.96)\cdot{}{\rm e}^{0.03428\cdot(T-T_{5})},
\end{equation}
where $T_{5}=5^{\circ}{\rm C}=278.15\ {\rm K}$ is the reference temperature of GRID-02. 
The unit of $I_{\rm dark}$ is $\mu$A and the first term 16 means the result is the sum of 16 SiPMs as in GRID detector. 
The exponential term shows that $I_{\rm dark}$ is doubled for $\sim$20$^{\circ}$C temperature increase. 

\subsection{Preliminary dose estimation}
\label{sec:Dose}

The radiation damage effect on SiPMs depends on irradiation particle type, energy and fluence and is traditionally evaluated by dose or 1 MeV neutron equivalent fluence \cite{GARUTTI201969}. 
In this paper we give an estimate of dose refer to the method used by SIRI-1 \cite{MITCHELL2021164798}. 
The SPENVIS online program \cite{SPENVIS} could generate particle flux with orbit information as listed in Table \ref{tab:missions}. 
The AP-8 and AE-8 (version solar minimum) trapped particle radiation models are selected for proton and electron, respectively. 
Then, with SHIELDOSE-2 model \cite{SHIELDOSE2}, SPENVIS calculates cumulative dose for a given period as shown in Fig. \ref{fig:dose}. 
The asymptotic value, $\sim$50 rad(Si), is used as an approximate estimate of dose. 
And the empirical equation of $I_{\rm dark}$ becomes 
\begin{equation}
    \label{eq:Idark_Dose_T_pred}
    I_{\rm dark}(Dose,T)=16\cdot(1.9\cdot{}Dose+1.96)\cdot{}{\rm e}^{0.03428\cdot(T-T_{5})},
\end{equation}
where $Dose$ is the cumulative dose in rad(Si). 
We caution that this equation is applicable only when bias voltage is 28.5 V. 
Furthermore, detailed Monte Carlo dose simulation could be done with the mass model of the satellite and the in-orbit radiation environment models. 

\begin{figure}
    \centering
    \includegraphics[width=1.0\textwidth]{Dose.pdf}
    \caption{Annual dose in silicon as a function of aluminium shielding thickness for GRID-02 given by SHIELDOSE-2 from SPENVIS \cite{SPENVIS, SHIELDOSE2}. Trapped radiation models for proton and electron are AP-8 and AE-8 (version solar minimum), respectively. }
    \label{fig:dose}
\end{figure}

To test this empirical equation, the dark current increasing rate calculated by Eq. (\ref{eq:Idark_Dose_T_pred}) is compared with the measurement results of SIRI-1 and GRID-04, and they are in good accordance as listed in Table \ref{tab:Idark_pred}. 
This estimate of dark current can then be used with Eq. (\ref{eq:sigma_Idark_fit}) to calculate noise increase and evaluate life span of the instrument. 
Note that the free parameters, $a_{\rm n}$ and $b_{\rm n}$, in Eq. (\ref{eq:sigma_Idark_fit}) is very dependent on readout electronics design, such as the readout time constant and the electronics gain. 
A ground calibration should be carried out to obtain these parameters. 

\begin{table}
    \begin{center}
        \caption{Measured and estimated dark current increasing rate of SIRI-1 and GRID-04. }
        \begin{tabular}{ccccccc}
            \hline
                        & Operating         & Operating     &                                       & Dose in   & \multicolumn{2}{c}{Dark current increasing rate}  \\
            Mission     & temperature       & voltage       & Orbit                                 & silicon   & \multicolumn{2}{c}{per SiPM chip ($\mu$A/year)}   \\
            \cline{6-7}
                        & ($^{\circ}$C)     & (V)           &                                       & (rad)     & Measured              & Estimated                 \\
            \hline
            SIRI-1      & 7.75              & 28.5          & 567 $\times$ 589 km 97.7$^{\circ}$    & 90        & 132                   & 188                       \\
            GRID-04     & 5                 & 28.5          & 523 $\times$ 550 km 97.5$^{\circ}$    & 90        & 182                   & 171                       \\
            \hline
        \end{tabular}
        \label{tab:Idark_pred}
    \end{center}
\end{table}

\subsection{Methods of improve signal-to-noise ratio}

As discussed in section \ref{sec:noise}, the noise of radiation damaged SiPM is dominated by dark count noise described by Eq. (\ref{eq:Campbell}). 
Here we give a discussion on qualitative methods of improve signal-to-noise ratio (SNR). 

The first term is $DCR$ which depends on temperature as shown in Eq. (\ref{eq:FESRH}). 
This leads to a straight forward method which is lower the operating temperature. 
This might be difficult for CubeSats but is practicable for larger satellites. 
Fitting results in Table \ref{tab:Idark_time_temp} gives a practical approximation that $DCR$ reduced by half for 16$^{\circ}$C temperature decrease around room temperature. 

The second term is $Gain$ proportional to $V_{\rm ov}=V_{\rm bias}-V_{\rm bd}$. 
As a critical operating parameter, the domino effect is enormous if $V_{\rm ov}$ is decreased. 
On one hand, noise reduces with decrease of $Gain$ and $DCR$. 
On the other hand, signal reduces as well due to decrease of $Gain$ and photon detection efficiency ($PDE$). 
Care must be taken to find the optimum value of $V_{\rm ov}$. 

The third term is the response of readout electronics. 
In general, shorter readout time constant leads to lower dark count noise. 
However, a too short readout time constant might cause problems such as larger ballistic deficit and more electronics noise. 
The lower limit is usually due to the scintillation decay time. 

\section{Conclusions}

With the successful observation of several gamma ray bursts by GRID-02, the feasibility of SiPM in space application is once again proved. 
On the other hand, the significant radiation damage effect indicates the fact that life span evaluation requires more attention since the instrument design stage. 
As a CubeSat detector equipped with scintillator crystal and SiPM optimized for gamma-ray detection, GRID is specifically designed with SiPM characterization setups and conduct in-orbit characterization experiments. 
The characterization results of GRID-02 for the first few months show an increase in dark current of $\sim$100 $\mu$A/year per SiPM chip (model MicroFJ-60035-TSV) at 28.5 V and 5$^{\circ}$C, which leads to the increase in overall noise level (sigma) of $\sim$7.5 keV/year. 
An approximate empirical equation around room temperature is given for dark current estimation at specified dose and temperature. 
The measured dark current of SIRI-1 and GRID-04 are used to test this equation and it shows a good agreement. 
This equation also implies that reduction of operating temperature could greatly mitigate the increase in dark current and noise which means a cooling system is highly recommended. 
Carefully designed readout electronics and optimum operating voltage help improve signal-to-noise ratio as well. 

\section*{Acknowledgement}

This work is supported by the Tsinghua University Initiative Scientific Research Program and the National Natural Science Foundation of China (Grant No. 11961141015). 
The authors would like to thank Dr. Shaolin Xiong and his colleagues of the GECAM satellite group from the IHEP of CAS, for open and fruitful discussions on topics related to this work.

