# How does temperature affect the performance of an SiPM?

Slawomir Piatek, PhD, Hamamatsu Corporation & New Jersey Institute of Technology
January 4, 2017

## Introduction

A silicon photomultiplier (SiPM) is a solid-state device capable of detecting single photons due to its very high internal gain (106 – 107). A SiPM is a rectangular matrix of square microcells (10 – 100 µm on the side) numbering from several hundreds to several tens of thousands depending on the model. Each microcell is a series combination of an avalanche photodiode (APD) operating in Geiger mode and a quenching resistor RQ. All of the microcells are identical and connected in parallel. In a typical operation, a SiPM is reversed biased with the voltage VBIAS that is a few volts above the breakdown voltage VBD of the APDs. The difference VBIAS - VBD is known as the overvoltage ΔV. Overvoltage is one of the key parameters controlling the operation and opto-electronic characteristics of the detector. In response to light or internally generated charge carriers, a SiPM acts like a current source — the output is a superposition of current pulses produced by the activated, or fired, microcells.

Temperature plays a significant role in the operation of all solid-state devices, including SiPMs. Its varied effects include influencing the concentration of charge carriers in doped and intrinsic semiconductors, thus affecting their resistivity; controlling the phonon spectrum, thus affecting electron-phonon and hole-phonon interactions; and setting the rate of thermal generation of charge carriers, thus affecting the amount of output dark current. This technical note examines and summarizes the present understanding of how changing temperature affects the key opto-electronic characteristics of a SiPM. It relies on published results in peer-reviewed articles and is organized into semi-autonomous sections, each discussing a particular characteristic.

## Quenching resistance

RQ passively terminates Geiger discharge in a microcell. Its value affects the amplitude of a single photoelectron current pulse (often referred to as 1 p.e. pulse or waveform) and, together with the junction capacitance CJ, the recovery time. If the value of RQ were to fall below a certain threshold, quenching would not have occurred, leaving the microcell in the state of a permanent Geiger discharge and insensitivity to light. The value of RQ can be deduced from the forward-bias I-V characteristic. As the bias increases, the current eventually becomes linearly proportional to the applied voltage because, as the resistance of the depletion region in the APDs approaches zero, only RQs limit the current. Consequently, the slope of the linear section equals N/RQ, where N is the total number of microcells. Fitting for the slope and knowing N gives RQ. I-V characteristics obtained at different temperatures yield information on how RQ varies with T.

Recently, Otte et al. (2016) has performed a comprehensive study of SiPMs from three different manufacturers. Among the many tests, temperature effects within the range –40 °C to +40 °C are prominent. The left panel of Figure 1 shows I-V characteristics for the Hamamatsu S13360-3050CS SiPM, while the right panel shows how RQ varies with T. The values of RQ come from linear fits depicted as red lines in the I-V characteristics (left panel).

A straight line with negative slope is a good fit to the RQ versus T data. The implied relative change of RQ with T is about 0.2% per degree Celsius. The negative temperature coefficient implies that RQ is made from a semiconductor material. Dinu et al. (2015) uses a different approach to measure RQ. The study finds that the junction capacitance of the investigated SiPMs does not vary with T, therefore, the recovery time of the output current pulses at different temperatures is only affected by RQ(T). Figure 2 depicts RQ versus T for two Hamamatsu SiPMs for the temperature range from -175 °C to +55 °C. Note that for the wider range of temperatures, the relationship is no longer linear. Instead, as expected from thermal properties of semiconductors, RQ is well fit by $a+b{T}^{1/2}{e}^{\left(c/T\right)}$, where a, b and c are adjustable coefficients. The solid lines in Figure 2 are the best fits.

## Breakdown voltage

Impact ionization of silicon atoms in the avalanche (or high-field) section of the depletion region is the gain mechanism in the constituent APDs of a SiPM. Here, a charge carrier (an electron or a hole) gains energy from the electric field, collides with a silicon atom ionizing it and, thus, creating an electron-hole pair. The probability of impact ionization depends on the strength of the electric field and on temperature. As temperature increases, scattering of charge carriers from the crystal lattice becomes more likely resulting in the net loss of carrier’s energy hindering impact ionization. This can be succinctly stated that the electron and hole ionization rates, α and β, are functions of temperature, both decreasing with T. To compensate the lower ionization rates, the electric field has to be strengthened by increasing the reverse bias. One expects, therefore, that the breakdown voltage VBD is a function of temperature.

Reverse-bias I-V characteristics at different temperatures can be used to determine VBD(T). The plot of $\frac{dln\left(I\right)}{dV}$ versus V shows a clearly delineated peak around VBD with the maximum corresponding to V = VBD. The left panel of Figure 3 shows a family of such plots for different temperatures, while the right panel depicts the corresponding plot of VBD versus T. Both figures are for Hamamatsu S13360-3050CS SiPM and are taken from Otte et al. (2016). The figures imply that within the temperature range of the experiments, VBD increases linearly with T at the rate of about 54 mV/°C — a significant change. Other studies, for example Dinu et al. (2016) and Ferri et al. (2014), also find a linear relationship between VBD and T for a similar temperature range.

In another study, Collazuol et al. (2011) uses I-V characteristics to determine VBD for a much wider temperature range (50 K < T < 320 K, or -223 °C < T < 47 °C). Figure 4 shows that for any given small range of temperatures the relationship is approximately linear but over a wider range, it deviates from linearity albeit consistent with theoretical calculation (see, for example, Crowell & Sze (1966)).

The gain µ of a SiPM equals the number of electrons that contribute to the 1 p.e. output current pulse i(t). Therefore,

Equation 1

$\mu ={\int }_{0}^{\infty }i\left(t\right)dt\approx \frac{\left({V}_{BIAS}-{V}_{BD}\right){C}_{J}}{e}=\frac{∆V{C}_{J}}{e}$

where CJ is the junction capacitance of a single APD and e is the elementary charge. Stability of the gain under changing ambient temperature is a common requirement in practical applications. This can be accomplished by adjusting VBIAS to compensate changes of VBD and, potentially, CJ with T.

## Gain and junction capacitance

As discussed above, the gain of a SiPM can be measured by integrating 1 p.e. waveforms. Doing so as a function of VBIAS and for different temperatures offers another method of determining VBD and a way of finding how CJ depends on T. The left panel of Figure 5 shows plots of µ versus VBIAS at several temperatures, while the right panel shows the corresponding plot of CJ versus T for Hamamatsu S13360-3050CS SiPM. Both figures are from Otte et al. (2016). For a given temperature, the plot of µ versus VBIAS is well fit by a straight line, implying no dependence of CJ on VBIAS. The value of VBD equals VBIAS when µ = 1, which is essentially the X intercept. The slope equals CJ/e, allowing an easy determination of CJ. The right panel of Figure 5 shows that CJ has a statistically weak dependence on T for this particular SiPM; however, other SiPMs in the study have a more significant dependence, as illustrated in Figure 6 for SensL J-series 30035 SiPM. The straight lines at different temperatures are parallel to each other implying that µ is independent of T provided that ∆V remains the same.

## Signal shape

The shape of the output current pulse depends on the values of RQ and CJ (and also on the bandwidth of the detection system and load resistance). Since RQ does and CJ may vary with T, the shape too will be a function of T, as illustrated in Figure 7 (adapted from Dinu et al. 2015). The figure shows that a higher T makes a SiPM "faster" (shorter recovery time), which is highly desirable in some applications. However, the tradeoff is an increase in dark counts whose consequence is a higher minimum detectable power, or lower sensitivity.

## Dark counts

A thermally generated (or due to quantum tunneling) charge carrier may enter (or be generated where) the high field region of an APD and trigger Geiger discharge. This possibility is known as a dark count. It results in an output that is indistinguishable from that resulting from absorption of a photon. Counting the number of the output waveforms with amplitude greater than one half of that for 1 p.e. waveform in a time interval τ gives the rate in Hertz (counted number divided by τ). The rate is a function of ΔV and T, as shown in the left panel of Figure 8. Note that the abscissa is actually the relative overvoltage:

Equation 2

$∆{V}_{r}\equiv \frac{\left({V}_{BIAS}-{V}_{BD}\right)}{{V}_{BD}}=\frac{∆V}{{V}_{BD}}$

The right panel shows how the rate changes with T relative to the rate at 40 °C for a fixed value of ΔVr. The rate doubles every 5.3 °C for this particular SiPM.

Noteworthy is a measurement of dark count rate as a function of T for a fixed overvoltage (1.5 V) by Collazuol et al. (2011), shown in Figure 9. The wider temperature range allowed the study to identify distinct sources of dark counts. Shockley-Read-Hall generation enhanced by trap-assisted tunneling is mostly responsible for dark counts in the region labeled a), whereas band-to-band tunneling dominates the region b). The study notes that region c) is still under investigation.

## Optical crosstalk

Optical crosstalk is a type of correlated noise ubiquitous in SiPMs. It comes in two types: direct and delayed. When a microcell undergoes Geiger discharge, it emits about 2.9 x 10-5 photons per electron that can be reabsorbed by silicon (Lacaita et al. 1993). The total number of emitted photons is $2.9×{10}^{-5}\mu =3×{10}^{-5\phantom{\rule[-0.1em]{0.4em}{0.5em}}}\frac{∆V{C}_{J}}{e}$, or about 29 for µ = 106. A direct crosstalk occurs when a photon from the primary discharge moves either directly, or after a reflection, to the avalanche region of a neighboring microcell and triggers there a secondary discharge. The two resulting current pulses are nearly simultaneous producing a 2 p.e. output waveform. However, if the emitted photon is absorbed outside of the avalanche region but within the depletion region, one of the charge carriers can drift into the avalanche region and triggers a secondary discharge. The two resulting current pulses are spaced chronologically by the drift time. This is delayed crosstalk, which can resemble afterpulsing (discussed in the next section). The following discussion applies only to direct crosstalk.

The probability of optical crosstalk PCT is expected to increase with ΔV because the number of the emitted photons is linearly proportional to ΔV and because the probability of Geiger discharge PG increases with ΔV. The dependence of PCT on T for a fixed ΔV will be mostly determined by the dependence CJ with T (since CJ affects the gain and, therefore, the number of emitted photons). Figure 10 is a plot of PCT as a function of ΔVr at several temperatures for Hamamatsu S13360-3050CS SiPM. For a given ΔVr, there is only a weak dependence of PCT on T, reflecting a weak trend seen in Figure 5. In contrast, Collazuol et al. (2011) finds no evidence that PCT depends on T, which is possible if CJ of the investigated SiPM is independent of T.

## Afterpulsing and delayed crosstalk

Impurities in silicon and crystal lattice defects can act as charge traps. A trap can capture a charge carrier created during an avalanche and then release it at a later time. If the release occurs during the recovery, the number of electrons produced in the afterpulse is less than µ and the resulting afterpulse waveform partly overlaps in time the primary; the output is their superposition. In contrast, if a trap releases charge after the complete recovery, the resulting afterpulse waveform does not overlap the primary and it is indistinguishable in shape from a 1 p.e. waveform.

Delayed crosstalk has a somewhat different signature. Here, the secondary waveform has the same amount of charge as the primary. If the drift time is less than the recovery time, the output consists of a superposition of two 1 p.e. waveforms, whereas if it is longer, two separate 1 p.e. waveforms result. The characteristic times of afterpulsing and delayed crosstalk are expected to be different because the physical processes governing them are different. Therefore, one can estimate their rates, or probabilities, from a histogram of time differences between consecutive pulses (for more details of this rather difficult measurement see Otte et al. (2016)).

The plot in the left panel of Figure 11 shows how the probability of delayed optical crosstalk depends on relative overvoltage at several temperatures. As expected, the probability increases with ΔVr, though for a nominal ΔVr (marked with an arrow), it amounts to only about 2%. For fixed ΔVr, the probability increases with decreasing T (surprisingly, other SiPMs in the study show the opposite effect). The plot of afterpulsing probability as a function of ΔVr, shown in the right panel of Figure 11, does not exhibit any significant dependence on T. This is surprising because the trapping time constant is known to decrease exponentially with T, therefore, afterpulsing probability should decline with T. The probabilities of afterpulsing for the other two SiPMs investigated in the study show the opposite dependence with T. These surprising and contradictory results may have to be clarified with additional future testing. Otte et al. (2016) does note that the afterpulsing data may have been contaminated with delayed crosstalk.

## Photon detection efficiency

Photon detection efficiency (PDE) is a probability that a SiPM produces an output signal in response to an incident photon. PDE is defined as a product of a geometrical fill factor, quantum efficiency (QE), and probability of Geiger discharge PG. PDE is expected to depend strongly on ΔV (through PG) and may show some dependence on T. The reasons for the latter are: 1. The band gap energy is a function of temperature, causing QE = QE(T) especially for the wavelengths close to the threshold wavelength; 2. Carrier mobility increases with T, thus PG may increase with T for fixed overvoltage; 3. Free carrier concentration depends on T, thus affecting all of the parameters that depend on the concentrations. There is very little reliable research investigating how T affects PDE. Near room temperatures, the effect is likely to be negligible but for a wider range, this may not be so, as indicated by Figure 12 from Collazuol et al. (2011). As noted by the study, more work is needed to fully understand PDE(T).

## Conclusions

Changing temperature can have a significant impact on the operation of a SiPM. Two strong dependencies on T are the value of the breakdown voltage and the dark count rate. For a stable gain, it is necessary that the ambient temperature is constant or that VBIAS is adjusted to keep the overvoltage fixed. The latter requires using a temperature-compensation circuitry. Higher dark count rate increases the minimum detectable power. In low-light applications, it may be necessary to cool the SiPM to increase its sensitivity.

## References

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