A low noise and low power cryogenic amplifier for single photoelectron sensitivity with large arrays of SiPMs
P. Carniti, A. Falcone, C. Gotti, A. Lucchini, G. Pessina, S. Riboldi, F. Terranova
PPrepared for submission to JINST
A low noise and low power cryogenic amplifier for singlephotoelectron sensitivity with large arrays of SiPMs
P. Carniti, a A. Falcone, a C. Gotti, a A. Lucchini, b G. Pessina, a S. Riboldi b and F. Terranova a a INFN Milano-Bicocca and University of Milano-Bicocca, Department of PhysicsPiazza della Scienza 3, Milano, 20126, Italy b INFN Milano and University of Milano, Department of PhysicsVia Celoria 16, Milano, 20133, Italy
E-mail: [email protected]
Abstract: This paper presents a low noise amplifier for large arrays of silicon photomultipliers(SiPMs) operated in cryogenic environments, especially liquid argon (87 K) and liquid nitrogen(77 K). The goal is for one amplifier to read out a total photosensitive surface of tens of cm whileretaining the capability to resolve single photoelectron signals. Due to the large capacitance ofSiPMs, typically a few nF per cm , the main contributor to noise is the series (voltage) component.A silicon-germanium heterojunction bipolar transistor (HBT) was selected as the input deviceof the cryogenic amplifier, followed by a fully differential operational amplifier, operated in anunconventional feedback configuration. The input referred voltage noise of the circuit at 77 K isjust below 0.4 n V /√ H z white (above 100 kHz) and 1 n V /√ H z at 10 kHz. The value of the basespreading resistance of the HBT at 77 K was determined from noise measurements at different biascurrents. Power consumption of the full circuit is about 2.5 mW. The design gives the flexibility tooptimally compensate the feedback loop for different values of the input capacitance, and obtain again-bandwidth product in the GHz range. The signal-to-noise ratio obtained in reading out SiPMsis discussed for the case of a 300 kHz low pass filter and compared with the upper limit that wouldderive from applying optimum filtering algorithms.Keywords: Front-end electronics for detector readout, analogue electronic circuits, Photon detec-tors for UV, visible and IR photons (solid-state), Noble liquid detectors (scintillation, ionization,double-phase). Corresponding author. a r X i v : . [ phy s i c s . i n s - d e t ] J a n ontents In recent years, silicon photomultipliers (SiPMs) emerged as a viable alternative to vacuum-basedphotomultipliers (PMTs) to sense scintillation and Cherenkov light signals in many kinds of particledetectors, since they offer similar or higher efficiency in a smaller and more robust package, andare not affected by magnetic fields. Their main drawback is the significantly higher dark countrate (DCR), which, however, can be effectively mitigated by lowering the operating temperature.For detectors that use liquid argon as a scintillator (such as the photon detection system of DUNE[1] and the DarkSide experiment [2]), it is natural to take advantage of the cryogenic environmentand operate the SiPMs inside the liquid (87 K). At such temperature, dark counts of thermal originbecome negligible, and those originated by tunneling dominate the DCR. Proper shaping of theelectric field and high silicon purity allow to reach DCR at the level of 0.01 Hz/mm at cryogenictemperatures [3, 4]. By connecting several SiPMs side by side, photosensitive surfaces of tens ofcm can then be realised, capable of resolving faint light signals down to single photons [5].From the electrical point of view, with an approximation that will suffice for the rest of thispaper, the source impedance of PMTs and SiPMs is capacitive. But while for large area PMTs itis just the parasitic capacitance of the readout electrode and its connections, typically of the orderof 10 pF or less, the source capacitance of SiPMs is given by the total capacitance of the cells it iscomposed of, and is then directly proportional to the photosensitive surface, with typical values ofabout 50 pF/mm . When several SiPMs are connected in parallel (ganged) to instrument an areaof tens of cm , a source capacitance of tens or hundreds of nF is to be expected. Despite havingsimilar gain and signal characteristics, large arrays of SiPMs have then a significantly lower sourceimpedance than large area PMTs. The importance of the parallel (current) noise of the amplifieris reduced, while its series (voltage) noise becomes the leading contributor. Hence the need for afront-end amplifier designed for the lowest possible series noise.The quest for low series noise is balanced by the need for low power consumption. If the SiPMsand the amplifier are submerged in a cryogenic liquid, boiling should be avoided, since it would– 1 – D I R P1 R P2 R P3 R P5 C P5 R P4 R F C F R O1 R O2 diff. outputSiPM R I U C D input V CM V CC R T DAQV EE V CC Room TCryo T C T C P6 R P6 C L R L R B C B Figure 1 . Schematic of the amplifier. Q is a silicon-germanium heterojunction bipolar transistor (InfineonBFP640). U is a fully differential operational amplifier (Texas Instruments THS4531). A typical choice ofcomponent values is summarized in table 1. interfere with the propagation of photons and their detection by the SiPMs. Another useful featureis the ability to drive the (differential) transmission lines that connect the output of the circuit to theoutside world, i.e. the data acquisition systems located at room temperature, up to several metersaway. Figure 1 shows the schematic of the amplifier. The SiPM is a 2-terminal device: the choice of anodeor cathode readout affects the polarity of the bias voltage and of the current signals. The input ofthe circuit is connected to the anode (cathode) of the SiPM, modeled by the source capacitance C D .The cathode (anode) of the SiPM, not shown, is connected to a bias voltage generator, bypassed toground with a large value capacitor close to the SiPM. The SiPM connected at the input is modeledby its capacitance C D , expected to range up to ∼
100 nF for a ∼
20 cm photosensitive area.The circuit is based on a general and well known topology: a discrete transistor Q followedby the operational amplifier U , although the choice of a fully differential opamp results in anunconventional feedback configuration, to be discussed in the following. Q is a silicon-germaniumheterojunction bipolar transistor (HBT), designed for radiofrequency applications. The presenceof germanium atoms in the base makes its band-gap smaller than that of the emitter, allows higherdoping levels, and results in a very small base spreading resistance, hence low series noise, and widebandwidth even at low bias currents (below 1 mA). These characteristics also make most HBTs ableto work effectively at cryogenic temperatures, although in some cases with higher low frequencynoise [6–9]. Several cryogenic amplifiers are reported in literature that take full advantage of SiGeHBTs [10–17]. The HBT we used in this work is the BFP640 from Infineon.Due to the large value of C D , abrupt changes in the bias voltage of the SiPMs could easilypropagate to the input of the amplifier. The Schottky diode D I protects Q by guarding againstreverse bias of its base-emitter junction. – 2 –he second stage U is a fully differential operational amplifier. The device we used as U is the THS4531 from Texas Instruments, with 27 MHz differential gain-bandwidth product,0.25 mA supply current and rail to rail outputs at room temperature. Being designed in a BiCMOStechnology, its ability to operate at 77 K is not to be taken for granted [18]. Band-gap narrowing,which can take place in the highly doped emitter of standard (homojunction) bipolar transistors,could degrade the current gain at low temperature and impair the performance. MOS transistors,on the contrary, are generally able to work in cryogenic environments. But even for a fully CMOSopamp, the ability to work at 77 K might also depend on other parameters related to circuit design(stability of voltage or current references, etc.), and needs in any case to be tested. Several samplesof the THS4531 were tested in liquid nitrogen, and were all observed to work, with supply currentincreased by about 50%, and bandwidth almost doubled, reaching a gain-bandwidth product of50 MHz.The outputs of the circuit are connected to a data acquisition system (DAQ) through a differ-ential transmission line of characteristic impedance R T or, equivalently, two lines of characteristicimpedance R T /
2. The termination is AC-coupled through C T to avoid DC current flowing in theoutput lines. When the outputs of the THS4531 are terminated with high impedance (1 M Ω ),the output dynamic range at 77 K is still almost rail to rail, although a small oscillation with afrequency of about 20 MHz was observed. The frequency of the oscillation does not appear todepend on the loop gain of the amplifier. It could be related with the compensation of the outputstages of the opamp, which operate at unity gain. A complete study of this behaviour cannot beperformed without a detailed description of the THS4531. The oscillation disappears when theoutputs are AC-terminated with 50 Ω , since in this case the loop gain of the rail-to-rail output stageis reduced; but in this case we observed the output dynamic range to be limited to about ± V .The best compromise was then to couple the 50 Ω termination above a few MHz, by choosing R T = Ω , C T =
330 pF. This proved to be effective in suppressing the aforementioned oscillationand preventing reflections in the output lines even when long cables (up to 12 m) were used, whileat the same time limiting the load at the output of the amplifier, allowing it to work with the widedynamic range it shows on a high impedance load. This remains true as long as the timescale of thesignals is larger than their propagation time on the output lines, of the order of tens of ns.The signal from the SiPM can be modeled as a current pulse I D ( t ) with a fast rise ( ≤
10 ns) anda slower recovery ( τ D ∼
100 ns or larger). Let us now assume U to be ideal, with infinite open-loopgain and bandwidth, and let us neglect R I . The gain block made of Q and U can be seen as asingle high-gain opamp with (negative) feedback provided by R F and C F . If C F is small enough( C F R F < τ D ), there is no significant integration of the SiPM signals, and the transimpedance closedloop gain is equal to R F . Similarly, R I can be considered “small enough” if there is no integrationat the input node, that is for C D R I < τ D . Under these assumptions, the differential signal acrossthe outputs is simply given by V O ( t ) = − R F I D ( t ) . In most of the measurements presented in thispaper, we worked without C F , and with R I = Ω , so the assumption to neglect them here is indeedjustified.The DC voltage at the non-inverting output of U is the V BE of Q , around 0.6 V at roomtemperature, 1 V at 77 K. The other output of U is at V CM − V BE , where V CM is the voltage appliedat the common-mode input of U . V CM can be chosen to maximize the output dynamic range, thenatural choice being halfway between the power supplies of U . The purpose of R L , C L is to filter– 3 –igh frequency noise and disturbances from V CC . Let us neglect the voltage drop across R L . AtDC, the inverting input of U is held at a fixed voltage by the divider made by R P and R P . Whenthe feedback loop is closed, this sets the voltage at the collector of Q . If R P = R P , and neglecting R L , the collector of Q is at V CC /
2. The current through R P is then V CC / R P , while the biascurrent of Q is I C = V CC (cid:18) R P − R P (cid:19) (2.1)where clearly R P needs to be larger than R P . Table 1 lists the typical values of the components usedin the measurements throughout the paper, unless specified otherwise. With V CC = R P = Ω , R P = R P = R P =
16 k Ω , the V CE of Q is 1.5 V and its bias current is approximately 400 µ A.Since R L = Ω gives a voltage drop of about 200 mV, I C is actually 370 µ A.Let us neglect R P and C P for now. The role of R P and R P , which is AC coupled through C P , is to make the inputs of U as symmetrical as possible. In particular, R P = R P , while R P is chosen so that R P (cid:107) R P = R P . C P is as large as possible, compatibly with the requirementto work at cryogenic temperature, which led us to choose C0G (NP0) ceramic capacitors. In caselarger values are needed, solid tantalum capacitors could also be used [19]. This arrangementimproves substantially the capability of the circuit to reject disturbances from the power supply V CC , which is then limited only by the resistor precision (typically to about 40 dB for 1% resistors).With C P =
100 nF, R P = .
69 k Ω (obtained as 3 . Ω in series with 390 Ω ), the rejection iseffective starting from a few hundred Hz. At high frequency, above about 1 MHz, the invertinginput of U goes to ground through R P = Ω and C P = . U from feeding a part of the signal from the collector of Q to the inverting input of U , which would result in an additional pole in the loop gain, leadingto a reduction of the available bandwidth. R P and C P break the symmetry between the inputsof U , but in this frequency range V CC is effectively bypassed to ground by R L and C L =
100 nF,eliminating disturbances altogether.
Table 1 . Choice of component values for the schematic of figure 1. With V CC = V EE = − I C = µ A. The “-” stands for “not present”. Unless otherwise noted, theseare the values used in the measurements presented in this paper. Q BFP640 U THS4531 D I SB01-15C R I Ω R F Ω C F - R B - C B - R O Ω R O Ω R L Ω C L
100 nF R P Ω R P
16 k Ω R P
16 k Ω R P
16 k Ω R P Ω + Ω C P
100 nF R P Ω C P o+ x1x1 V o-Vi+Vi- Z+Z-gd(Vi+-Vi-) gc ( -VCM ) Vo++Vo-2
VCM
Figure 2 . Model of the fully differential operational amplifier U . Transistor Q is operated at gain G = − g m R P , where g m is its transconductance and R P = R P (cid:107) R P is the load impedance at the collector. We neglect poles due to stray capacitances at the collector of Q , which would be in parallel with R P . We also neglect the capacitances of Q , which are all wellbelow 1 pF ( C BC , which is multiplied by the Miller factor, is 0.08 pF according to the datasheet).Since g m = I C / V T , where I C is the collector current and V T = kT / q , where k is the Boltzmannconstant and q is the elementary charge, the transconductance of Q at a given I C is larger at 77 Kby a factor ∼ G =
36 atroom temperature, which becomes G =
135 at 77 K.The stability of the feedback loop can be analyzed in the domain of the complex frequency s .Let us first consider the frequency response of U . A fully differential amplifier can be modeledwith the schematic shown in figure 2. The input stage is a voltage-controlled current generatorwith transconductance g d . It rejects common mode signals at the input and generates a currentproportional to the differential input voltage V i + − V i − . If we denote the load impedance of the twobranches with Z + and Z − , as noted in the schematic, the differential gain is given by V o + − V o − V i + − V i − = g d ( Z + + Z − ) . (3.1)while the common mode gain is given by V o + + V o − V i + − V i − = g d ( Z + − Z − ) . (3.2)In case of perfect matching ( Z + = Z − = Z ) the differential gain becomes 2 g d Z and the commonmode gain vanishes. The role of the common mode current generators with transconductance g c is then just to fix the output common mode voltage to be equal to V CM . Their bandwidth isunimportant, as long as V CM is constant. If matching is not perfect, these generators also help insuppressing the residual common mode gain.Let us assume to work in perfect matching. The load impedance Z gives the dominant pole ofthe opamp: we can define A and τ so that 2 g d Z = A /( + s τ ) . In reality there will be at leastanother pole at higher frequency, for instance due to the output buffers, with time constant η . The– 5 – igure 3 . Edges of the differential output signals for β (cid:48) = .
3, driving 12 m output cables. The horizontalscale is 0.2 µ s/div, the vertical scale is 20 mV/div. differential gain can then be modeled as V o + − V o − V i + − V i − = G ( s ) = A + s τ + s η (cid:39) A s τ + s η . (3.3)Assuming U to be unity-gain stable, and not overcompensated, the frequency of the second polecorresponds to the gain-bandwidth product of the amplifier, that is the frequency where | G ( s )| (cid:39) η ∼ . η ∼ . V o + and V i − and between V o − and V i + , in a perfectly symmetrical configuration. In the circuit describedhere, the feedback loop includes the input transistor Q , inherently single-ended, and thereforeinvolves only one of the outputs. This implies that the feedback factor is halved, and is given by β ( s ) = g m R P Z I Z F + Z I , (3.4)where Z I = R I + / sC D and Z F = R F (cid:107) / sC F . (Note that, since β ( s ) is halved, bandwidth will behalved as well, compared to the case where feedback is applied symmetrically on both branches.)The loop gain is given by T ( s ) = − G ( s ) β ( s ) = − A s τ + s η (cid:20) g m R P + s ( C F R F + C D R I ) + s C F R F C D R I s ( C D R F + C D R I + C F R F ) + s C F R F C D R I (cid:21) . (3.5)– 6 –t is clear that if both R I and C F are zero, the last term reduces to a pole at the origin, and theloop gain is unstable. Either R I or C F need to be present to compensate one of the poles. Atthe same time, as already discussed, their value should not be too large to avoid integration of theSiPM signals. The presence of both C F and R I gives the terms in s , that become dominant athigh frequency. In principle, the presence of R I is sufficient: indeed in the measurements presentedhere, C F was not used, see table 1. Setting C F = T ( s ) = − A s τ + s η (cid:20) g m R P + sC D R I sC D ( R F + R I ) (cid:21) . (3.6)The expression has three poles and one zero. If C D R I > η , which is true with the choice ofcomponents in table 1 and C D above 10 nF, the zero compensates the pole due to C D ( R F + R I ) before the frequency of the third pole is reached. For smaller values of C D , the value of R I needsto be increased, for instance to 5.1 Ω . The condition for stability is then for the magnitude of theloop to be below unity before reaching the frequency of the pole due to η , which is the bandwidthlimit of U . Since in eq. 3.3 we assumed | G ( s = / η )| (cid:39)
1, and using the fact that R F (cid:29) R I , thiscondition is satisfied if β (cid:48) = g m R P R I R F < . (3.7)In other words, the amplifier is stable if the attenuation of the signal fed from ouput of U back tothe input of Q is larger than the gain | G | = g m R P provided by the first stage. The bandwidth of thefull circuit is the frequency where | T ( s )| =
1, and coincides with the bandwidth of U multiplied by β (cid:48) . Or, to see it differently, the gain-bandwidth product of the gain block composed by Q and U is G times the gain-bandwidth product of U , and gives 6.7 GHz at 77 K. The closed loop bandwidthof the circuit is then obtained by dividing this by 2 R F / R I . With the values of table 1, eq. 3.7 is wellsatisfied, since β (cid:48) = /
18 at 77 K. The validity of the calculations above was checked by increasingthe value of R I to 5 . Ω . In this case β (cid:48) = .
3. Since the gain-bandwidth product of the THS4531at 77 K was measured to be close to 50 MHz, the bandwidth of the full circuit with R I = . Ω isexpected to be about 15 MHz. A 10% to 90% risetime of 23 ns was indeed observed with 2 m longoutput cables, in good agreement with the expected bandwidth. If the length of the output cables isincreased to 12 m, the edges of the two outputs are just slightly deteriorated to 26 −
27 ns, as shownin figure 3.There is another possible source of instability of the circuit, which is not related with the entirefeedback loop discussed above, but rather with self-oscillations of Q . Silicon-germanium HBTsare designed for radiofrequency applications, and they exhibit bandwidth in the tens or hundreds ofGHz when they are biased with typical currents of tens of mA. When biased at lower currents, as inour case, the bandwidth reduces but is still in the GHz range. Parasitic inductance and capacitancein the layout of the circuit board can introduce unwanted resonances, which can become criticalat cryogenic temperature, where the transconductance is largest. A parasitic oscillation of Q canappear at the output of the circuit as an oscillation at lower frequency, due to the interplay betweenthe high frequency oscillation of Q and the rest of the circuit. To avoid this, the following measureswere found to be effective: minimizing the stray inductance at the emitter, by connecting it to theground plane of the board with very short traces and several vias; adding a small resistor in serieswith the base, a role that is already filled by R I ; adding the series combination of a capacitor ( ∼ nF)– 7 – R P1 R P2 R P3 R P5 C P5 R P4 R F R O1 R O2 R I U V CM V CC SpectrumV EE V CC Room TCryo TC P6 R P6 C L R L Room T Cryo T
Function AnalyzerGenerator x20
Figure 4 . Setup for noise measurements. All component values are those in table 1, except where specified.The bias current of Q was changed by using R P = . Ω , 4 . Ω , 3 k Ω , 2 . Ω , giving I C = µ A . and resistor (a few Ω ) between the base and ground, close to the base, which are C B and R B in theschematic of figure 1. R I , R B and C B can also help in achieving loop stability in case the SiPM or itsconnecting wires have an inductive component, which would appear in series with C D . Althoughthey are foreseen in the schematic of figure 1, R B and C B were not always used, and do not appearin table 1. The noise of U is referred to the base of Q by dividing it by the gain of the first stage G . With thecomponent values shown in table 1, G =
36 at room temperature and G =
135 at 77 K, makingthe noise of U negligible. The series noise of the amplifier is then uniquely determined by Q and R I . The white component is given by e n = kT g m + kT R BB (cid:48) + kT R I , (4.1)where k is the Boltzmann constant, T is the temperature, g m and R BB (cid:48) are the transconductanceand the base spreading resistance of Q . The first and last term of eq. 4.1 can easily be calculated,while the second is harder to determine. The value of R BB (cid:48) for a given HBT at room temperature, ifnot explicitly listed in the datasheet, can be extracted from the quoted noise figure, and is typicallya few Ω at currents of 1 mA and above. Its value, however, may be different for currents in thehundred µ A range, and may depend on temperature. We accounted for such dependence with a firstorder expansion in powers of 1 / I C : R BB (cid:48) ( I C ) = R ∗ BB (cid:48) + α I C (4.2)where R ∗ BB (cid:48) is the value at high collector currents, and α accounts for a possible increase of R BB (cid:48) atlow values of I C . Both R ∗ BB (cid:48) and α can be expected to depend on temperature. By using 4.2, and– 8 –ince g m = qI C / kT , where q is the elementary charge, eq. 4.1 can be rearranged as e n = (cid:18) k T q + kT α (cid:19) I C + kT (cid:0) R ∗ BB (cid:48) + R I (cid:1) . (4.3)The sum is composed of four terms, grouped in pairs. Measurements at different values of I C allowto disentangle the first and second from the third and fourth. The first term, coming from the g m of Q , has the strongest temperature dependence. We will show that, in our measurements, thefirst term dominates over the second at room temperature. At cryogenic temperature, the relativeimportance of the second term becomes larger, and we used eq. 4.3 to determine the values of R ∗ BB (cid:48) and α from the measured noise spectra.The low frequency noise of transistors at cryogenic temperature depends strongly on thepresence of impurities in the band gap, and, being Q a heterojunction device, at the interfacebetween the different semiconductors. It is device-dependent, and possibly also batch-dependent.The relative weight of the low frequency contribution in affecting the signal-to-noise ratio dependson the bandwidth of interest. For a typical application that needs to detect scintillation light, thelowest frequency of interest can be associated with the scintillation time constant, a few µ s in thecase of liquid argon, corresponding to a few tens of kHz. The noise contributions below the lowestfrequency of interest can often be filtered out, and their weight reduced.The current (parallel) noise is due to the base current of Q and to the thermal noise of R F .The base current of Q is expected below a few µ A, and R F = . Ω . Any base current below100 µ A and feedback resistors above 100 Ω give a white current noise below 10 p A /√ H z . For largevalues of C D , and for signal frequencies above the kHz range, this contribution is negligible.The noise spectra were measured in the configuration shown in figure 4. The amplifier wasoperated at a differential gain close to 900 ( R F = .
35 k Ω , R I = . Ω ). The input node, before R I , was connected to ground through the series connection of a small 0.25 Ω resistor and a largecapacitance, composed of a 470 µ F tantalum capacitor at room temperature, connected to the inputwith a 20 cm cable, in parallel with 100 nF C0G and 10 µ F X7R ceramic capacitors at cold. Thedifferential output of the circuit was converted to single-ended on a second stage amplifier basedon a AD8055 with gain 20, closed-loop bandwidth 10 MHz, and measured with a Rohde&SchwarzFSV4 spectrum analyzer. To characterize the transfer function, a test signal from an Agilent 33250A(white noise, 100 mV peak to peak, bandwidth 80 MHz) was injected through a 50 + Ω sourceimpedance across the 0.25 Ω resistor, resulting in a 0.25 mV peak to peak signal at the input nodeof the circuit. The source was then disconnected to measure the output noise spectrum, which wasthen divided by the measured transfer function.Figure 5 shows the results at room temperature (300 K) and liquid nitrogen (77 K) for differentvalues of the collector current I C . (Each spectrum was obtained by concatenating several measure-ments taken on different time scales.) The white noise depends on I C as expected. At larger valuesof I C the improvement becomes less evident, due to the presence of the constant terms in eq. 4.3.The low frequency part of the spectra does not depend on I C , and is slightly more pronounced at77 K. The curves show lorentzian “bounces” below 10 kHz. The squared spectra were interpolatedwith the sum of five lorentzian functions and a constant (white) term: N ( ω ) = A ( ω ) + e n = (cid:213) i = A i + ω τ i + e n (4.4)– 9 –
00 1k 10k 100k 1M
Frequency [Hz] N o i s e [ n V / H z ] T = 300 K
120 A210 A370 A480 A
100 1k 10k 100k 1M
Frequency [Hz] N o i s e [ n V / H z ] T = 77 K
120 A210 A370 A480 A370 A fit
Figure 5 . Noise spectra measured for different values of the collector current I C at room temperature (top)and liquid nitrogen (bottom). At 77 K, the spectra at 370 µ A and 480 µ A are almost indistinguishable. Theblack dotted curve shows the interpolation of the data at 77 K, 370 µ A. Each lorentzian term is expected to be due to the presence of a specific class of generation-recombination centers (traps) with a definite time constant τ i [20, 21]. The most prominent arefound at τ i = µ s, corresponding to 1.5 kHz, and at about 100 Hz, accounting for the rise at thelower end of the spectrum. The resulting curve for the spectrum at 370 µ A is shown with a dottedblack line in figure 5.From the fitting curves, the value of the white component e n was extracted. Figure 6 gives thesquared white noise e n as a function of 1 / I C . From eq. 4.3, a linear fit of the data allows to extractthe unknown parameters R ∗ BB (cid:48) and α . The intercept corresponds to the condition 1 / I C →
0, wherethe first two terms in eq. 4.3 vanish. From the intercept values, which are 0.203 nV /Hz at 300 Kand 0.064 nV /Hz at 77 K, the values of R ∗ BB (cid:48) + R I can be determined. Knowing that R I = . Ω ,we obtain R ∗ BB (cid:48) ∼ Ω at 300 K and R ∗ BB (cid:48) ∼ Ω at 77 K. The value at 300 K is compatible withthe value extracted from the noise figure in the BFP640 datasheet.– 10 – Inverse current 1/I C [1/ A] S qua r ed w h i t e no i s e e n2 [ n V / H z ]
300 K data300 K fit77 K data77 K fit
Figure 6 . Squared white noise extracted from the spectra of figure 5, plotted as a function of the inversecollector current of Q . The slope of the fitting line at 300 K is 239 µ A(nV) /Hz. It is about 10% larger than2 k T / q = µ A(nV) /Hz. About half of this excess noise at room temperature can be explainedby the noise of the second stage U : the THS4531 has a white voltage noise of 10 n V /√ H z at roomtemperature, which divided by G =
36 gives about 0.3 n V /√ H z at the input. The contributionof the second term of eq. 4.3 at 300 K is then small, and does not allow a precise determinationof α . In other words, the value of R BB (cid:48) at room temperature is small enough not to give sizeablecontributions to the total noise of the circuit.At 77 K, the slope of the fitting curve gives 32.7 µ A(nV) /Hz at 77 K, which is twice as largeas what is expected from the first term of eq. 4.3 alone, 2 k T / q = µ A(nV) /Hz. The noiseof the THS4531 was observed to decrease to about 3.3 n V /√ H z at 77 K, while the gain of thefirst stage increases to G =
135 at 77 K, making it completely negligible. Since the circuit wassubmerged in liquid nitrogen, and the power consumption is low, we ruled out the possibility that Q was operating at a temperature significantly higher than 77 K. The difference gives α = µ A Ω .The estimated value of R BB (cid:48) is then 18, 22, 30, 46 Ω at I C = µ A respectively.Even though partially contributed by R BB (cid:48) , the total white noise of the amplifier is remarkablylow, considered the low power consumption. In all the measurements presented in the followingsections, we worked at 370 µ A, with a white noise of 0.4 n V /√ H z . As can be clearly seen in figure5, working with larger currents than those considered did not give any advantage at 77 K. Given the measured noise spectra, we address the determination of the signal-to-noise ratio in thereadout of actual SiPMs. Figure 7 shows an example of a SiPM read out at 77 K (liquid nitrogen)with the amplifier described in this paper. The SiPM has an area of 0.96 cm , a total capacitanceof 4.8 nF, and a recovery time τ D (cid:39)
800 ns. It was biased at 24 V (3 V overvoltage) by setting the– 11 – igure 7 . Light signals from a pulsed LED, detected by a SiPM, read out with the amplifier described in thispaper. The horizontal scale is 500 ns/div, the vertical scale is 4 mV/div. cathode voltage to 25 V, since the anode, connected to the input of the circuit, was at 1 V. The gainof the SiPM at 3 V overvoltage was about 2 . × . It was illuminated with a pulsed LED throughan optical fiber, and the light intensity was adjusted so that a few photoelectrons were detected oneach pulse. Figure 7 shows excellent separation between signals corresponding to different numbersof photoelectrons.To determine the signal-to-noise ratio in the case of larger photosensitive area, we addeddifferent values of capacitance at the input node: 10 nF, 32 nF and 79 nF. Considering as referencea SiPM capacitance of 50 pF/mm , these values correspond to roughly 2 cm , 6.4 cm and 15.8 cm respectively, to be added to the 0.96 cm of the device. For the first point at 10 nF, the value of R I was increased from 1 Ω to 5.1 Ω to guarantee stability of the loop gain. A warm second stage basedon a OP27 opamp, with bandwidth 300 kHz and gain 23.5, was used to convert the differentialsignals to single ended. The bandwidth was further filtered at 1 MHz on the oscilloscope. Thesignal-to-noise ( S / N ) for single photoelectron signals was determined by dividing the observedsignal amplitude by the RMS noise of the baseline. The resulting data points are shown in figure8 as “Measured SiPM 1”. The data are in reasonable agreement with the expected behaviour,which is described by the curve labelled “LP 300 kHz [ τ D =
800 ns]”. The curve was obtained bycalculating the expected signal amplitude for single photoelectron signals from a SiPM with gain2 . × , τ D =
800 ns, filtered with a single-pole low-pass at 300 kHz, and dividing it by theRMS noise obtained by integrating numerically from 100 Hz to 1 MHz the noise spectrum of theamplifier, low-pass filtered at 300 kHz.The same measurement was repeated for a different device, with an area of 1.7 mm , acapacitance of 35 pF, and a shorter recovery time τ D (cid:39)
100 ns. The gain of the device at 3 Vovervoltage is expected to be about 1 . × , hence it was operated at a slightly higher overvoltage,adjusted to obtain a gain close to 2 . × . As done in the previous case, capacitances of 10 nF, 32 nFand 79 nF were added in parallel with the SiPM to simulate a larger total area. The measured S / N – 12 –
7 10 20 30 50 70 100 200
Cd [nF] S / N Measured SiPM 1 [ D = 800 ns]Measured SiPM 2 [ D = 100 ns]LP 300 kHz [ D = 800 ns]LP 300 kHz [ D = 100 ns]Optimum [ D = 800 ns]Optimum [ D = 100 ns] Figure 8 . Signal-to-noise ratio for single photoelectron signals from two SiPMs operated at a gain of2 . × . The SiPMs differ in the values of the recovery time τ D . The measured data at different capacitancevalues for two devices, labelled “Measured SiPM 1” and “Measured SiPM 2”, are in reasonable agreementwith the expected curves “LP 300 kHz”, calculated from the gain and τ D of the devices and the measurednoise spectra in 5, using just a low pass filter at 300 kHz. The curves can also be compared with the expected“Optimum” S / N , obtained by integrating 5.5 numerically. is shown in figure 8 as “Measured SiPM 2”. Again, the data are in reasonable agreement with theexpected values, calculated for a 300 kHz low pass filter, labelled as “LP 300 kHz [ τ D =
100 ns]”.By comparing the two sets of measurements, it appears that a shorter recovery time τ D givesa better S / N by almost a factor two, although the difference becomes smaller at higher values of C D . This is due to the larger weight of the low frequency noise components for larger values of τ D .In many applications, however, a relatively long recovery time is beneficial for the suppression ofafterpulses. In any case, even at larger values of τ D , the measurements show a good S / N , alwaysabove 4 up to 100 nF capacitance, which would correspond to a photosensitive surface of 20 cm for a 50 pF/mm SiPMs.It could be argued that the 300 kHz low pass filter applied in these measurements is not theoptimal one, and the S / N could improve with better filtering algorithms, to be applied offline. Fromthe theory of optimal filtering of signals from particle detectors [22–24], the best signal-to-noiseratio that can be achieved for any amplitude measurement is given by (“OF” stands for optimumfilter): (cid:18) SN (cid:19) = π ∫ ∞−∞ (cid:12)(cid:12) ˜ V O ( ω ) (cid:12)(cid:12) N ( ω ) d ω (5.1)where ˜ V O ( ω ) is the Fourier transform of the output signal V O ( t ) , and N ( ω ) is the output noisespectral density. Any other filter or processing algorithm is bound to give a sub-optimal signal-to-noise ratio. The signal of a single SiPM cell, corresponding to a single photoelectron, can bedescribed by a current step with an exponential decay, carrying a total charge Q . The differential– 13 –ignal at the output of the amplifier can then be expressed as V O ( t ) = R F I D ( t ) = R F Q τ D e − t τ D . (5.2)To match the gain of the SiPMs we measured, we set Q = . × q , where q is the elementarycharge, and we consider two values for τ D , 100 ns and 800 ns. Here and in the following we neglectthe integration at the input node, by considering R I =
0, and neglect the finite bandwidth of theamplifier. (Even if we wanted to consider these effects in eq. 5.2, they would affect the followingexpression 5.4 for the output noise in the same way. They would therefore cancel out from theintegrand of eq. 5.5.) The magnitude of the Fourier transform of the output signal is (cid:12)(cid:12) ˜ V O ( ω ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) R F Q + i ωτ D (cid:12)(cid:12)(cid:12)(cid:12) = R F Q + ω τ D . (5.3)The output noise spectral density, neglecting parallel contributions, is given by N O ( ω ) = (cid:12)(cid:12)(cid:12)(cid:12) ( Z F + Z I ) Z I (cid:12)(cid:12)(cid:12)(cid:12) (cid:16) A ( ω ) + e n (cid:17) = | + i ω C D R F | (cid:16) A ( ω ) + e n (cid:17) = (cid:16) + ω C D R F (cid:17) (cid:16) A ( ω ) + e n (cid:17) (5.4)where we used Z F = R F and Z I = / sC D . The terms A ( ω ) and e n account for the low frequencyand white noise respectively, as expressed by eq. 4.4. The integral in eq. 5.1 becomes (cid:18) SN (cid:19) = π ∫ ∞−∞ R F Q (cid:0) + ω τ D (cid:1) (cid:0) + ω C D R F (cid:1) (cid:0) A ( ω ) + e n (cid:1) d ω (5.5)Neglecting the low frequency noise allows to calculate the integral analytically, but leads to anoverestimated result. We can instead take the actual noise spectrum, as shown in figure 5, andevaluate the integral numerically. The resulting curves are shown in figure 8, labelled as “Optimum”for the two values of τ D . The dependence of the curves on τ D is small at low values of C D , andcompletely negligible for larger values of C D . The curves show that, with respect to the measured S / N discussed previously, there is margin for improvement by up to factors of 3 − S / N above10 would be expected for SiPM capacitance up to 200 nF, corresponding to approximately 40 cm ,independently of the SiPM recovery time. We presented a front-end amplifier designed to readout large arrays of SiPMs in cryogenic environ-ments. Compared with similar amplifiers based only on operational amplifiers, the present designgives lower noise at significantly lower power dissipation. The resulting gain-bandwidth productis in the GHz range. The circuit topology offers high flexibility in compensating the loop gain,which allows to obtain a signal rise time down to ∼
20 ns with large values of input capacitance,up to ∼
100 nF. The conditions for close-loop stability were discussed in detail. The circuit shows– 14 –nput-referred white voltage noise of 0.4 n V /√ H z at 77 K, at a power consumption of 2.5 mW. Thebase spreading resistance of the input transistor was determined from white noise measurements,and was observed to depend on its bias current. At low frequency, noise is also contributed bylorentzian terms. The amplifier allowed to measure S / N > . Furtherimprovements would derive from applying an optimum filtering algorithm, with the expected ca-pability to readout a 100 nF photosensitive surface with S / N (cid:39)
20, independently of the SiPMrecovery time.
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