passive quenching electrical model of silicon photomultipliers (sspms) by kristen ann wangerin a thesis submitted to the graduate faculty of rensselaer polytechnic institute in partial fulfillment of the requirements for the degree of master of science major subject: nuclear engineering approved: _________________________________________ dr. yaron danon, thesis adviser rensselaer polytechnic institute troy, new york july 2008 (for graduation august 2008) ii contents passive quenching electrical model of silicon photomultipliers (sspms) ..................... i list of tables ........................................................................................................iv list of figures........................................................................................................v acknowledgment ...............................................................................................ix abstract...................................................................................................................x 1. introduction..............................................................................................................1 2. theory......................................................................................................................6 2.1 principle of operation .....................................................................................6 2.2 equivalent electrical circuit............................................................................9 2.3 avalanche .....................................................................................................12 2.4 passive quenching model .............................................................................15 2.5 recharging ....................................................................................................16 2.6 extraction of physical parameters..................................................................16 3. method of procedure ..............................................................................................18 4. materials and apparatus .........................................................................................20 4.1 sspms ..........................................................................................................20 4.2 measurement circuit .....................................................................................21 4.3 experiments ..................................................................................................21 4.4 modeling.......................................................................................................23 5. results....................................................................................................................24 5.1 1x1 mm2 devices ..........................................................................................24 5.1.1 experimental results .........................................................................24 5.1.2 preamplifier characterization.............................................................28 5.1.3 parameter extraction..........................................................................31 5.1.4 electrical model validation ...............................................................34 5.2 3x3 mm2 device............................................................................................42 iii 5.2.1 experimental results .........................................................................42 5.2.2 parameter extraction..........................................................................47 5.2.3 electrical model comparison.............................................................49 6. conclusions............................................................................................................57 7. appendix – matlab code – analyze data files .................................................58 8. appendix – matlab code – analyze results......................................................63 9. references..............................................................................................................71 iv list of tables table i. calculation of number of cells firing with increasing bias voltage. .............55 v list of figures figure 1. sspm model with a current pulse simulating the avalanche7 ...........................3 figure 2. sspm model with a current pulse simulating the avalanche and an additional parasitic grid capacitance. .......................................................................................3 figure 3. comparison between measured and simulated pulses with an itc-irst sspm and a (a) transimpedence amplifier and (b) voltage amplifier9 . .............................4 figure 4. electrical sspm model in this work................................................................5 figure 5. operating regions of an apd: photodiode (linear, no gain), avalanche (linear, gain of 10-200), geiger mode (photon = pulse out) ................................................7 figure 6. example of a 10 x 10 pixel array of apd cells 11 . ...........................................8 figure 7. the amplitude of the output detector signal is proportional to the number of apd cells hit with optical photons..........................................................................8 figure 8. sspm array of pixels connected in parallel11 . .................................................9 figure 9. sspm equivalent electrical circuit. .................................................................9 figure 10. photon detection efficiency as a function of overvoltage.............................12 figure 11. geometry and doping profile of an sspm diode12 .......................................14 figure 12. the electric field profile through the thickness of a sspm illustrating the differences in photon absorption location versus wavelength17 ..............................14 figure 13. voltage and current across the diode as a function of time15 ........................16 figure 14. sspm 25 um devices: (a) 1 mm (b) 1 mm cell array (c) 3 mm. ................20 figure 15. sspm 25 mm cells with quenching resistor for (a) 1 mm and (b) 3 mm devices..................................................................................................................20 figure 16. experimental circuit. the quenching resistor is included as part of the sspm. ..................................................................................................................21 figure 17. experimental measurement setup (a) without preamplifier (b) with preamplifier. .........................................................................................................22 figure 18. pulse waveforms captured with the digital oscilloscope at an applied voltage of (a) 71.1 v and (b) 71.9 v...............................................................................25 figure 19. histogram of cell pulse amplitudes and division into number of firing cells at a bias voltage of (a) 71.1 v and (b) 71.9 v.........................................................25 vi figure 20. division of all pulses into groups based on the number of cells firing at a bias voltage of 71.9 v. .................................................................................................26 figure 21. averaging of pulses after binning based on number of cells firing in the event at a bias voltage of 71.9 v.....................................................................................26 figure 22. normalization of all pulses at a bias voltage of 71.9 v, showing that pulse shape does not change with the number of firing cells...........................................27 figure 23. normalization of pulses for 2 cells firing to compare the pulse shape as a function of applied voltage or overvoltage. ...........................................................27 figure 24. increasing signal-to-noise ratio with increasing bias voltage. ......................28 figure 25. preamplifier pulse waveforms for (a) 1 ghz bandwidth and (b) 150 mhz bandwidth.............................................................................................................29 figure 26. preamplifier histograms for (a) 1 ghz bandwidth and (b) 150 mhz bandwidth.............................................................................................................30 figure 27. comparison of averaged pulse waveforms for (a) without and (b) with the preamplifier. .........................................................................................................30 figure 28. comparison of 1 ghz and 150 mhz bandwidths for signals (a) without and (b) with the preamplifier for 0, 2, and 4 cells firing. ..............................................31 figure 29. comparison of pulse shapes with and without the preamplifier for 0, 2, and 4 cells firing at (a) 1 ghz bandwidth and (b) 150 mhz bandwidth.........................31 figure 30. charge integration windows to obtain collected charge...............................32 figure 31. extracting the breakdown voltage by plotting bias voltage versus the gain..33 figure 32. extracting the breakdown voltage by integrating the first half of the pulse..33 figure 33. extraction of diode equivalent resistance by forward biasing the diode. ......34 figure 34. electrical model for 1 mm2 sspm with calculated parameters for one firing cell and the remaining cells represented as an equivalent circuit............................35 figure 35. comparison of modeled and experimental pulse shapes for different number of cells firing at a bias voltage of 71.1 v...............................................................36 figure 36. voltage (vm) across the diode as a function of time. ...................................37 figure 37. current (idiode) through diode as a function of time......................................37 vii figure 38. for increasing bias voltage and constant switch timing, comparison of experimental and modeled (a) pulses for increasing bias voltage and (b) voltage across (vm) and current through the diode.............................................................39 figure 39. increasing switch time with increasing bias voltage. ...................................39 figure 40. for increasing bias voltage and modified constant switch timing, comparison of experimental and modeled (a) pulses for increasing bias voltage and (b) voltage across (vm) and current through the diode.............................................................40 figure 41. comparison of sspm rise time for different hv decoupling capacitors. .....41 figure 42. comparison of output signals read with and without a 50 ohm coaxial cable. .............................................................................................................................41 figure 43. raw acquired single pulse waveforms with the preamplifier for a bias voltage of 70.5 v. .............................................................................................................43 figure 44. histogram of pulse amplitudes with the preamplifier for a bias voltage of 70.5 v...................................................................................................................43 figure 45. averaged pulses with the preamplifier for 0 to 6 cells firing for a bias voltage of 70.5 v. .................................................................................................44 figure 46. averaged pulses with the preamplifier for different bias voltages for three cells firing.............................................................................................................44 figure 47. normalization of pulses for 3 cells firing to compare the pulse shape as a function of applied voltage....................................................................................45 figure 48. averaged pulses with the preamplifier for different bias voltages for three cells firing that are corrected for the gain of the preamplifier. ...............................45 figure 49. averaged pulses (raw data) acquired without the preamplifier and with a strong laser pulse. .................................................................................................46 figure 50. averaged pulses acquired without the preamplifier normalized to the true sspm output magnitude. ......................................................................................46 figure 51. increasing signal-to-noise ratio with increasing bias voltage from preamplifier data...................................................................................................47 figure 52. total charge integrated for a signal pulse over a range of bias voltages. extrapolating the data to zero estimates the diode breakdown voltage...................48 figure 53. extraction of diode equivalent resistance by forward biasing the diode. ......48 viii figure 54. electrical model for 3x3 mm2 sspm with calculated parameters for one firing cell and the remaining cells represented as an equivalent circuit. .................49 figure 55. comparison of 3x3 mm2 sspm experimental and modeled pulses..............50 figure 56. voltage across the diode and current through the diode...............................51 figure 57. comparison of simulated 14399 cells firing to experimental data. simulated data is divided by 14399 to compare to experimental data of one cell firing. .........53 figure 58. comparison of modeled number of cells firing to experiment. ....................54 figure 59. comparison of normalized experimental pulse shapes with increasing bias voltage..................................................................................................................54 figure 60. effect of changing model parameters on pulse shape; increase cq to 10 ff, cd to 50 ff, decrease vbr to 67.1 v........................................................................55 figure 61. comparison of different over voltages from changing breakdown and bias voltages. ...............................................................................................................56 ix acknowledgment i would like to acknowledge my adviser, dr. danon, for his guidance of my thesis work. i am very grateful to my coworkers for all of their technical help and expertise, espe- cially jimmy wang, chang kim, floris jansen, yanfeng du, wen li, and kent burr. i am thankful for friends who were there to lend an ear when my research was on my mind and also their understanding when work had to prevail over play. finally, i would like to acknowledge my family for all of their support. x abstract sspm detectors can be studied and improved through electrical modeling of the di- ode and readout circuit to simulate, characterize, and predict their response for different geometries and configurations. an electrical model was developed to simulate and investigate the effect of increasing diode area on the response of sspms. passive components in the model are extracted from measurements and then used in the model to understand and predict device performance. the avalanche is represented with a switch in series with a voltage source and diode resistor, instead of a current source, which allows the change in potential, current through the diode, and timing of the avalanche to be simulated. pulse shapes are compared for two different size devices, 1x1 and 3x3 mm2 , to first validate the model and then demonstrate predictive capability. it is con- cluded that this electrical model can be used to better understand the design and development of sspms, particularly the effects of increasing parasitic capacitance on the timing and magnitude of the readout signal. 1 1. introduction solid-state photomultipliers (sspms) are a rapidly developing detector technology, with the potential to advance a range of applications, ranging from high-energy physics, biological sensors, nuclear medicine, dna sequencing, and homeland security1 . cur- rently, detectors are composed of a scintillator coupled to a photomultiplier tube (pmt). the scintillator converts gamma radiation into optical photons, and the pmt converts the optical photons to an electronic signal and amplifies it. the detector, however, is limited by the quantum efficiency of the photocathode of the pmt, and the bulky, fragile vacuum tube requires high voltage. standard pmts are also sensitive to magnetic fields, which distort electron trajectories to the first and second dynodes, and can be damaged by excess ambient light2,3 . sspms overcome many of the limitations of pmts, combining advantages of pmts and silicon detectors. they are compact, robust, stable, and low power devices. unlike pmts, sspms are unaffected by magnetic fields, making them an attractive option for applications such as pet-mri. sspms have high detection efficiency, high gain, and good energy resolution. energy resolution of 3% for 662 kev has been achieved with a labr3:ce scintillator4 . disadvantages are that they are currently small and expensive. a sspm is a photosensor consisting of an array of photodiodes, or cells, that are connected in parallel and operated above their breakdown voltage in geiger mode5,6 . when an optical photon strikes one of these cells, the cell undergoes an electrical breakdown, and its charge is collected onto a common electrode. the diode is in series with a quenching resistor, and the voltage across the diode drops during the avalanche. the decreasing potential slows the avalanche until the current is quenched and the cell begins to recharge. when used to detect pulses of light, the output signal of the sspm is proportional to the number of cells that are struck by optical photons. an sspm can be combined with a scintillator for detection of ionizing radiation, such as x-rays, gamma rays, or neutrons. in this case, a scintillator converts the ionizing radiation to a burst of optical photons, and the optical photons are detected by the sspm. the energy of the ionizing particle can be estimated based on the measured pulse height. 2 an electrical model of an sspm can enhance understanding of the design and be- havior of sspms. electrical models have been developed to simulate the behavior of solid-state photomultipliers7,8 . figure 2 shows the model developed by pavlov et al7 . the cells of the sspm are divided into one cell that undergoes breakdown and the remaining cells represented as an equivalent circuit. the time constants of the response of a cell undergoing breakdown and recharging are modeled with passive components. the avalanche of a geiger cell is simulated using a current pulse in the diode. using circuit parameters, the charge released from the diode is calculated, and the timing is estimated based on time constants of the circuit. there is a small inductor with a value of 10 nh/cm, presumably to account for stray inductances of the device. the motivation of the work was to simulate the pixel current over time in a 1.1 x 1.1 mm2 sspm to characterize the timing of the devices and define dead time and maximum count rate. the work simulated two readout resistors of 1000 and 50 w and concludes that the larger resistor slows down both the release of charge and recovery time of the pixel7 . another model was developed by corsi et al.8 to optimize the front-end readout with an sspm coupled to an ideal and finite bandwidth preamplifier. the model is shown in figure 2. a current source is again used to simulate the diode avalanche, and the charge is calculated based on model parameters of overvoltage and capacitance. there is an additional parasitic grid capacitance to correctly account for all time constants that characterize the shape of the signal waveform. a method for extraction of model parameters was described, and parameters are extracted for two unknown size sspms, one from itc-irst and one from photonique. the number of cells is 625 and 516. assuming a cell size near 50 mm, it is hypothesized they sspms are around 1 mm2 in size. the governing equations, which define the influence of parameters on the circuit time constants, were also developed. the time constants of the circuit were explored for readout resistor values of 20, 50, and 75 w as voltage versus time. finally, the responses of the experimental data and model for one of the sspms were compared using two different readout amplifiers, and the comparison plots are shown in figure 3. 3 figure 1. sspm model with a current pulse simulating the avalanche7 . figure 2. sspm model with a current pulse simulating the avalanche and an additional parasitic grid capacitance 9 . 4 (a) (b) figure 3. comparison between measured and simulated pulses with an itc-irst sspm and a (a) transimpedence amplifier and (b) voltage amplifier9 . in order to better relate the model to the physics, previous models are further devel- oped in this work to simulate the avalanche with a switch, diode resistor, and voltage source, shown in figure 4. the model is described in more detail in sections 2 and 4.4. the switch allows the potential across and current through the diode to be simulated based on the circuit and not the definition of the current source. experimental and modeled pulses are compared for 1x1 mm2 and 3x3 mm2 25 mm devices over a range of bias voltages. parameters in the model are extracted from experimental data for two types of device, and results are in good agreement without modification or addition of other parameters from device to device. the model will be able to predict the response of sspms for different device geometries and measurement circuit configurations. 5 figure 4. electrical sspm model in this work. 6 2. theory 2.1 principle of operation sspms are composed of an array of avalanche photodiode (apd) cells. apds convert light directly into charge. when an optical photon is absorbed in the semiconductor material, an electron-hole pair is created. a voltage across the detector creates a large electric field that accelerates the charge carriers toward the cathode and anode. as these carriers gain kinetic energy, they interact, liberating additional carriers. a cascading avalanche amplifies what would otherwise be a small signal from a single detected photon. the final output signal is large enough to be digitized by electronics with a high signal-to-noise ratio. apds can be operated in linear or geiger mode. figure 5 shows the photodiode and avalanche photodiode linear regions and the geiger mode region. in linear mode, the apds operate at just below the breakdown voltage, resulting in a low gain of only around 100 and poor signal-to-noise ratio. the signal-to-noise ratio is defined as the ratio of photon detection events to dark noise or spontaneous breakdown events. in geiger mode, the apds are operated above breakdown threshold. the number of collisions greatly increases, and the entire cell becomes ionized. at voltages signifi- cantly above breakdown voltage, more than 3 to 5 v, spontaneous breakdown begins to dominate. the avalanche is stopped by a quenching resistor; the large resistance limits the current available to flow through the diode to sustain the avalanche current. the output signal is thus limited by the discharge of the potential built up across the capaci- tance of the silicon. the size of this capacitance and potential determines the amount of charge that is collected for one photon10 . the potential is determined by the overvoltage, which is the difference between the applied and breakdown voltages. 7 figure 5. operating regions of an apd: photodiode (linear, no gain), avalanche (linear, gain of 10-200), geiger mode (photon = pulse out) 11 . the full discharge of one apd does not provide any information on the incoming energy of the photon, so the apd is subdivided into an apd cell array, similar to the pixilated anodes of solid-state czt and cdte detectors. cell size ranges from 25 to 100 microns12 . thousands of cells are connected in parallel to form an array13 from 1 x 1 mm2 to 3 x 3 mm2 in size. sspms can be tiled to make a detector as large as 1 cm2 in size. figure 6 shows a 10 x 10 pixel sspm array. the output detector signal is propor- tional to the number of apd cells that fire as a result of absorption of an optical photon, illustrated in figure 7. figure 8 shows an equivalent circuit of the diodes and quenching resistors connected in parallel. the diodes are reversed biased, and the output signal is read as a voltage across a resistor to ground. 8 figure 6. example of a 10 x 10 pixel array of apd cells 11 . figure 7. the amplitude of the output detector signal is proportional to the num- ber of apd cells hit with optical photons. 9 figure 8. sspm array of pixels connected in parallel11 . 2.2 equivalent electrical circuit the sspm can be described as an electrical circuit7 , shown in figure 9. the main electrical components are the quenching resistor (rq) and silicon resistor (rd) and capacitor (cd) of the diode. there is also parasitic capacitance associated with the quenching resistor (cq). other stray inductance and capacitance of the diode are not explicitly considered, as these values are small14 . figure 9. sspm equivalent electrical circuit. 10 the firing cell is independent, and the remaining cells are described as an equivalent circuit15 , where qeq q eq c1)(ncand 1n r r -= - = . (1) the equivalent circuit for the cells not firing can be derived using expressions for impedance and admittance with the properties that parallel admittance and series imped- ance add. the admittance for a resistor and capacitor are qc q r cjωy, r 1 y == . (2) the impedance of each is the reciprocal. in the sspm electrical model, the admittance of the top quenching resistance and capacitance is described as cjω r 1 +=qy , (3) where r = rq and c = cq. this admittance is in parallel with the diode capacitor. the diode switch remains open so the diode resistance is not included. the impedance of one cell is then described as ( ) ( )d q q d d q c cjωcjω r 1 cjω r 1 cjω cjω 1 cjω r 1 1 z ÷ ø ö ç è æ + ÷ ø ö ç è æ ++ =÷÷ ø ö çç è æ + ÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ + = . (4) the admittances of many cells in parallel are added together as ( ) ( ) ( ) ( ) ( ) ( ) , q d d q q d d q q d d q c cjω r 1 cjω cjωcjω r 1 1)(n ... cjω r 1 cjω cjωcjω r 1 cjω r 1 cjω cjωcjω r 1 y ÷ ø ö ç è æ ++ ÷ ø ö ç è æ + -= + ÷ ø ö ç è æ ++ ÷ ø ö ç è æ + + ÷ ø ö ç è æ ++ ÷ ø ö ç è æ + = (5) where (n-1) is the number of standby cells when one cell avalanches. 11 alternatively, the standby cells can be written in the form of an equivalent resis- tance and capacitance as cjω1)(nc, 1n r r eqeq -= - = . (6) the equivalent admittance is cjω1)(ny, r 1n y c_eqr_eq -= - = . (7) the admittance of the quenching circuit is written as c.jω1)(n r 1n -+ - =qy (8) the impedance of the equivalent cell circuit is then ( ) ( )d q q d d q c cjω1)(ncjω1)(n r 1n cjω1)(n r 1n cjω1)(n cjω1)(n 1 cjω1)(n r 1n 1 z -÷ ø ö ç è æ -+ - ÷ ø ö ç è æ -+ - +- =÷÷ ø ö çç è æ - + ÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ -+ - = . (9) pulling out the common factor, n-1, the admittance is ( ) ( ) q d d q c cjω r 1 cjω cjωcjω r 1 1)(ny ÷ ø ö ç è æ ++ ÷ ø ö ç è æ + -= , (10) which is equivalent to that derived in equation 5 for individual cells connected in parallel and proves the equivalent cells can be written as stated in equation 1. the equivalent electrical model of the sspm is valid for small numbers of cells fir- ing, and the modeled pulse shape is representative of experimental results. as the number of cells firing increases to hundreds or thousands, the differences in pulse shape increase. although this is a limitation of the model, the differences are small. alterna- tively, the firing cells could be pulled out individually and the equivalent cell circuit would account for fewer standby cells. this method would not be practical in a program such as pspice, but it could be more easily simulated in an analytical model. 12 2.3 avalanche for an avalanche to be initiated, a photon must be absorbed in the depletion region of the silicon and generate an electron-hole pair. the probability of a photon being absorbed in a cell increases with increasing overvoltage, as shown in figure 10. for a fully re- charged cell, a potential has been built-up across the capacitance of silicon, creating an electric field. the electric field accelerates the generated charge carriers, causing them to liberate more carriers, creating a cascading avalanche. the carriers move to the terminals of the diode, inducing a current. figure 10. photon detection efficiency as a function of overvoltage15 . the generation and transport of the charge carriers from the depletion region to the terminals of the device define the rise time of the current pulse. the variation in rise time is mostly determined by the avalanche buildup time for carriers that interact within the active depletion region. this is the drift time, and it is improved by increasing the electric field within this region. the tail in the rise time distribution is from minority carriers diffusing from the neutral region beneath the junction to the depletion region16 . this is the diffusion time. 13 the response time of the avalanche current, or the transit time for a carrier across the depletion region, is calculated as ps10 v w t sat depletion drift == , (11) where wdepletion is the thickness of the depletion region, assumed to be 1 mm17 ; and vsat is the saturation velocity of electrons, 107 cm/sec18 . at lower electric fields when the velocity is not saturated, the velocity is calculated as vdrift = m e. the mobility of elec- trons at 300 k is 1400 cm2 /v s18 . the applied voltage, v, can be assumed to drop across the depletion region, so the electric field, e, is v/wdepletion. for a thickness of 1 mm, the velocity saturates around 72 v. the diffusion time of electrons across the neutral region is defined as [sec] dπ w t 2 2 iftneutral_dr diffusion = , (12) where wdrift is the neutron drift region thickness, assumed to be between 5 and 10 mm17 , d is the diffusion coefficient for electrons, the minority carrier, and is equal to 20 cm2 /s19 . for drift region thicknesses of 5 and 10 mm, the diffusion time is equal to 0.8 and 3.4 ns, respectively, which contributes significantly more to the avalanche time than the drift time. these calculations show that the avalanche is largely characterized by location of the photon interaction. figure 12 shows the doping profile of an sspm. when a photon interacts in the n+ region, electrons will be collected, and holes will travel across the high-field avalanche region, or depletion region, where they liberate additional electron- hole pairs to generate the avalanche. electrons drift toward the cathode, and holes drift toward the anode, shown in figure 12. holes have a smaller probability than electrons of generating an avalanche, which reduces the quantum efficiency. therefore, the n+ region is referred to as the dead layer. interactions within the p regions have a higher probability of generating an avalanche; when a photon is absorbed in the p+ region, the holes will be collected, and the electrons will generate the avalanche. the photodetec- tion efficiency is, therefore, higher for shorter wavelengths, as these photons interact deeper within the silicon in the depletion region. 14 figure 11. geometry and doping profile of an sspm diode12 . figure 12. the electric field profile through the thickness of a sspm illustrating the differences in photon absorption location versus wavelength17 . - + cathode p-side anode n-side 15 the final contribution to the rise time is the discharge of the space charge in the sili- con, which dominates after the avalanche build-up. the time constant, tdischarge, represents the discharge of the silicon space charge and is calculated as dddischarge crτ = . (13) the pre-breakdown state of the diode can be described as a charged capacitor, the junction capacitance, cd, in series with the quenching resistance. when an avalanche is initiated, the system can then be modeled as the diode capacitance in series with the diode resistance, on the order of 1 kw. the diode resistance represents both the neutral regions inside the silicon and the space charge resistance, originally charged to the overvoltage, vbias – vbr. a voltage source, vbr, is added to represent the voltage to which the diode drops, described by the discharge time constant20 . for a diode capaci- tance of 25 ff, tdischarge is 25 ps. 2.4 passive quenching model as the carriers travel to the terminals, more carriers are liberated that sustain the ava- lanche current. the quenching resistor limits the amount of current available to the diode. without this current source, the avalanche is limited by the potential built up across the diode before the avalanche. as the charge built up across the diode dis- charges, this potential decreases, and the probability of electron-hole pair generation decreases. the avalanche current stops increasing when the space charge in the deple- tion region from generated electron-hole pairs collapses the voltage such that the internal field is below the avalanche field. the potential drops to the breakdown voltage, vbr, shown in figure 13, which is the voltage when an avalanche of zero current through the diode can be sustained, or the multiplication factor of electron-hole pair generation equals unity. the breakdown voltage is better described as a band or region, rather than a rigid voltage point. the diode current drops, as shown in figure 13, and when the current is on the order of 10-20 ma, the fluctuation of carriers will drop to zero and the avalanche will be quenched20 . 16 figure 13. voltage and current across the diode as a function of time15 . 2.5 recharging the resistances and capacitances of the diode determine the recovery time. after the avalanche, the quenching capacitance recharges almost instantaneously; the potential across the diode is very close to zero, this capacitance is small, and it is in parallel with the very large quenching resistance. thus, the recharging of the diode is first dominated by that of the quenching rc. the diode capacitance then begins to recharge, defining the pulse tail. as the voltage builds up across the diode, current flows through the quenching resistor, and the quenching capacitor will slowly discharge as the voltage across it changes. 2.6 extraction of physical parameters the current pulse from the diode can be considered a dirac delta pulse in time with charge, q, that can be expressed as )c)(cv(vq qdbfrbias +-= . (14) this relationship holds true because the time constants of the circuit, dominated by the capacitance, are longer than those of the avalanche. the value of the capacitances and breakdown voltage can be extracted using equa- tion 14. the charge associated with the discharge of a single cell is proportional to the gain as a function of bias voltage. the charge generated in an avalanche is the time integral of the current. current is the voltage measured over a resistor. plotting bias voltage versus the charge, the slope of the curve provides cd + cq. extrapolating to the 17 y-intercept provides the breakdown voltage, the point where the gain would equal zero21 . these capacitances act in parallel. the quenching capacitance is immediately recharged due to the large rq. once cq is fully recharged, cd begins to recharge. as the voltage across cq changes, cq will discharge in parallel with cd. the quenching resistor value can be extracted by forward biasing the diode8 . the slope of the forward iv curve is equal to the transconductance. taking the reciprocal gives the total resistance, rq_tot, which is equal to n r r q q_tot = , (15) where rc is the quenching resistance of one cell, and n is the number of cells. 18 3. method of procedure sspms events are triggered using a pulsed laser with an intensity such that only a few cells fire for each event. pulse waveforms are acquired with a digital oscilloscope. the sspm is mounted in a low noise aluminum box with a 1-mm diameter hole for the laser. all data is taken at room temperature. pulse waveforms are first acquired with and without a preamplifier to determine the effect of the preamplifier on pulse shape. the preamplifier increases the sspm signal above the noise; intrinsic noise of the oscilloscope noise is 1 ma. pulse shapes are compared, and it is found that the preamplifier has some shaping effect. data is acquired with 1 ghz or 150 mhz oscilloscope bandwidth. with 1 ghz bandwidth, high fre- quency noise is integrated into the signal. although 150 mhz bandwidth filters the noise, the pulse shape is slightly changed, so all data is acquired at 1 ghz the 1x1 mm2 devices are characterized without a preamplifier. the 3x3 mm2 de- vices are characterized with a preamplifier; because of their larger area and higher capacitance, these devices have more dark noise and smaller cell pulse amplitudes in the range of 50 to 200 mv. the true signal amplitude for the bigger devices is obtained by dividing the data taken with the preamplifier by the preamplifier gain. the pulse shape is obtained by increasing the laser intensity so that the signal is above the baseline noise without using the preamplifier. the offset of the pulse is estimated using the first 100 data points of every pulse, and it is used to set the baseline to zero. the maximum amplitude of each pulse is histogrammed. the histogram has peaks and valleys according to the number of cells that fired for each event. the spacing of the peaks defines the gain of one cell. data is taken over a range of voltages from near the breakdown voltage to a voltage when spontaneously breakdown begins to dominate. the optimum bias voltage is defined as that with the maximum signal-to-noise ratio and before excessive spontaneously break- down. an electrical model is developed in spice to first simulate the 1 mm2 sspm re- sponse in both pulse shape and amplitude. parameters are extracted based on extracted values from the experimental data as described in section 2.6. the breakdown voltage is extracted by plotting the gain as a function of voltage. the gain, in coulombs, is calcu- 19 lated by integrating the voltage pulse over time and dividing by the equivalent resis- tance. the cell capacitance is extracted using q=cv. the quenching resistance is extracted by forward-biasing the diode; the inverse slope of the current versus the voltage is the equivalent resistance of the diode. the avalanche is simulated using a switch, and the timing of the switch is determined by the current through the diode. the model is then compared to 3x3 mm2 sspm measurements to both validate the model and understand the effects of increasing sspm size on pulse shape. 20 4. materials and apparatus 4.1 sspms sspms are 1x1 mm2 and 3x3 mm2 multi-pixel photon counter devices from hamamatsu with 25 mm cell size12 , shown in figure 14. the geometry of the devices is shown in figure 15. (a) (b) (c) figure 14. sspm 25 um devices: (a) 1 mm (b) 1 mm cell array (c) 3 mm. (a) (b) figure 15. sspm 25 mm cells with quenching resistor for (a) 1 mm and (b) 3 mm devices. 21 4.2 measurement circuit the general experimental circuit used to test the both sspms is shown in figure 16. an rc filter between the voltage and diode decouples fluctuations in the bias voltage and cable capacitance and inductance from the rest of the measurement circuit. the oscillo- scope is terminated to 50 ohms. an additional 1 kw resistor is added in parallel for all experiments to convert current to voltage although it is only need when using voltage sensitive preamplifier. this additional resistor decreases the signal amplitude slightly. figure 16. experimental circuit. the quenching resistor is included as part of the sspm. 4.3 experiments a nanoled laser, model 05a, is used to generate optical photons that interact with the sspm. the laser has a maximum intensity of 30 mw, peak wavelength of 455 nm, and pulse duration of 0.8 to 1.4 ns fwhm22 . neutral density filters are use to attenuate the beam so that an average of one or two photons interact in each event. a neutral density filter of 5 attenuates the signal by a factor of 10-0.5 or 0.32. the amplitude from one cell is a few millivolts for a 1 mm2 sspm and hundreds of microvolts for a 3x3 mm2 sspm. the amplitude is less for the 3x3mm2 sspm due to a larger quenching resistor as is discussed in section 5.2.2. a tektronix tds5104b digital phosphor oscilloscope is used to acquire pulse waveforms. the bandwidth is 1 ghz with a sampling rate of 1 gs/s. the data is acquired with a division of 200 picoseconds per point for a total of 4000 data points per 22 pulse. for each trial, 40,000 waveforms are captured. the pulse is triggered using a sync/trigger from the laser with a frequency of 2 khz. it is assumed that the laser pulse is significantly shorter than the time response of the sspm and that the frequency is low enough that the sspm recovers fully between events. all measurements are performed with the sspm in a light-tight aluminum box to reduce noise. a 1-mm diameter hole in the box allows the beam to pass. a mini- circuits zx-60 4016e preamplifier is used to amplify the pulses. this amplifier has a wide bandwidth from 20 mhz to 4 ghz23 . the wide bandwidth is good for timing. the 1-kw resistor, which converts the current to a voltage, filters some of high frequency signal. when possible, measurements are taken without the amplifier to limit distortion of the pulse shape and in order to measure and analyze the raw signal response of the device. figure 17 shows the setup of the experimental measurements with and without the preamplifier. voltage supply test box with sspm laser signal out oscilloscope terminated to 50 w 1 kw resistor to ground voltage supply test box with sspm laser signal out preamppreamp oscilloscope terminated to 50 w 1 kw resistor to ground (a) (b) figure 17. experimental measurement setup (a) without preamplifier (b) with preamplifier. 23 4.4 modeling pspice, student version 9.1, is used to electrically simulate the sspm circuit to repro- duce both pulse shape and amplitude. the model diagram is shown in figure 34. each cell is composed of the following components. the quenching resistor is in parallel with the quenching capacitance. a capacitor simulates the charge buildup across the silicon diode and is in parallel with the diode resistor. the effect of each of the resistors and inductors is described in section 2.5. the avalanche is modeled using switches; one switch closes to initiate the avalanche and another opens to end it and allow the cell to recharge. when the switch closes, the diode capacitor acts in series with the diode resistance and a voltage source that represents the breakdown potential. the time when the second switch closes is determined by the current through the diode. before the electrical model of the microcells, there is an rc filter, which is required experimentally to decouple the cable and high voltage from the measurement circuit. after the microcells, there is an inductor and capacitor to ground for parasitic rc components of the circuit. the resistors to ground represent the termination of the oscilloscope and the additional 1 kw resistor required when using the preamplifier. 24 5. results data is taken with the 1 mm2 sspm, and results are compared to those from the electri- cal model. model parameters are extracted or estimated from data sheets. experimental results for the 3x3 mm2 device are then compared to the model. the effects of bias voltage and overvoltage on pulse shape are investigated. the effects of increasing diode area are also explored. 5.1 1x1 mm2 devices 5.1.1 experimental results experimental data is acquired over a range of applied voltages. for each bias voltage, digital waveforms are captured, shown in figure 18 for 71.1 and 71.7 v. the maximum amplitude of each pulse is histogrammed, shown in figure 19. as the bias voltage is increased, the pulse amplitude increases because more cells are firing. the histogram tells the number of cells firing for each event as well as the cell gain. as the voltage is increased, the distribution of cells firing per event broadens and shifts to higher values. cell gain is constant for a given applied voltage. from the histogram, events with the same number of firing cells are grouped and averaged to form a single pulse. figure 20 shows the division of all pulses into groups based on the number of cells firing, and figure 21 shows the averaged pulses. the averaged pulses are normalized in figure 22 to show that the shape of the pulse does not change with the number of cells that fire for low light detection levels. the tail of the pulses appears to increase for increasing bias voltage, shown in figure 23, which is caused from afterpulsing in the waveforms, or a release of traps in the multiplication region that can retrigger avalanche breakdown24 . 25 10 20 30 40 50 60 0 2 4 6 time (ns) amplitude(mv) 10 20 30 40 50 60 0 2 4 6 time (ns) amplitude(mv) (a) (b) figure 18. pulse waveforms captured with the digital oscilloscope at an applied voltage of (a) 71.1 v and (b) 71.9 v. 0 2 4 6 8 0 200 400 600 800 1000 1200 amplitude (mv) count peak amplitude noise 0 2 4 6 8 0 200 400 600 800 amplitude (mv) count peak amplitude noise (a) (b) figure 19. histogram of cell pulse amplitudes and division into number of firing cells at a bias voltage of (a) 71.1 v and (b) 71.9 v. 26 20 30 40 50 60 70 -1 0 1 2 3 4 5 time (ns) amplitude(mv) figure 20. division of all pulses into groups based on the number of cells firing at a bias voltage of 71.9 v. 20 30 40 50 60 70 0 1 2 3 4 5 time (ns) amplitude(mv) 0 cells 1 cell 2 cells 3 cells 4 cells 5 cells 6 cells 7 cells figure 21. averaging of pulses after binning based on number of cells firing in the event at a bias voltage of 71.9 v. 27 20 30 40 50 60 70 0 0.2 0.4 0.6 0.8 1 time (ns) amplitude(mv) 1 cell 2 cells 3 cells 4 cells 5 cells 6 cells 7 cells figure 22. normalization of all pulses at a bias voltage of 71.9 v, showing that pulse shape does not change with the number of firing cells. 20 30 40 50 60 70 0 0.2 0.4 0.6 0.8 1 time (ns) normalizedamplitude(mv) 71.1 v 71.3 v 71.5 v 71.7 v 71.9 v 72.1 v 72.3 v figure 23. normalization of pulses for 2 cells firing to compare the pulse shape as a function of applied voltage or overvoltage. 28 1 2 3 4 5 6 7 0.5 1 1.5 2 2.5 3 3.5 number of cells firing snr 71.1 v 71.3 v 71.5 v 71.7 v 71.9 v 72.1 v 72.3 v figure 24. increasing signal-to-noise ratio with increasing bias voltage. the signal, or gain, increases with increasing bias voltage because the number of firing cells increases. the signal-to-noise ratio also increases with increasing voltage, shown in figure 24. the signal is the peak in the histogram of pulse amplitudes, and the noise is the amplitude of the valley before the peak. the ratio is also a function of the average intensity and is expected to improve when the number of pixels is large. the ratio of signal-to-noise must be plotted because the signal and noise levels are dependent on the distribution of pulses acquired, and the signal-to-noise ratio is independent of this. it is shown that the signal increases faster than the noise for increasing bias voltage. therefore, the optimum applied voltage is not limited by the dark current but by the spontaneous breakdown of the device. 5.1.2 preamplifier characterization the pulse waveform is compared for different bandwidths and also with and without a preamplifier at a bias voltage of 71.5 v. the preamplifier amplifies the signal above the noise, improving signal to noise ratio and resolution of pulse amplitudes. the raw signal waveforms for full and 150 mhz bandwidth are shown in figure 25; events with differ- ent number of cells firing can clearly be distinguished. histograms are shown in figure 29 26. the gain of the preamplifier is calculated by averaging the pulse amplitude for four firing cells with and without the preamplifier. average waveforms are shown in figure 27. the gain is 11.3 for 1 ghz bandwidth and 10.7 for 150 mhz bandwidth data. the benefit of bandwidth limiting the data is that the high frequency noise is fil- tered, giving more differentiation between pulse amplitudes as a function of the number of cells that fired. the disadvantage is that there is distortion of the raw signal from the diode, shown in figure 28. the preamplifier also has an effect on the pulse shape, shown in figure 29 in the shortening of the tail of the pulse. pulse magnitudes have been normalized. the preamplifier signal is also corrected for a 1.25 ns delay in signal response. 20 30 40 50 60 0 10 20 30 40 time (ns) amplitude(mv) 20 30 40 50 60 0 10 20 30 40 time (ns) amplitude(mv) (a) (b) figure 25. preamplifier pulse waveforms for (a) 1 ghz bandwidth and (b) 150 mhz bandwidth. 30 0 10 20 30 40 0 500 1000 1500 amplitude (mv) count peak amplitude noise 0 10 20 30 40 0 500 1000 1500 amplitude (mv) count peak amplitude noise (a) (b) figure 26. preamplifier histograms for (a) 1 ghz bandwidth and (b) 150 mhz bandwidth. 20 30 40 50 60 0 1 2 3 time (ns) amplitude(mv) 20 30 40 50 60 0 10 20 30 time (ns) amplitude(mv) (a) (b) figure 27. comparison of averaged pulse waveforms for (a) without and (b) with the preamplifier. 31 20 30 40 50 60 0 0.5 1 time (ns) amplitude(mv) full bandwidth 150 mhz bandwidth 20 30 40 50 60 0 5 10 15 20 25 time (ns) amplitude(mv) full bandwidth 150 mhz bandwidth (a) (b) figure 28. comparison of 1 ghz and 150 mhz bandwidths for signals (a) without and (b) with the preamplifier for 0, 2, and 4 cells firing. 20 30 40 50 60 0 0.5 1 1.5 2 time (ns) amplitude(mv) without preamp with preamp 20 30 40 50 60 0 0.5 1 1.5 2 time (ns) amplitude(mv) without preamp with preamp (a) (b) figure 29. comparison of pulse shapes with and without the preamplifier for 0, 2, and 4 cells firing at (a) 1 ghz bandwidth and (b) 150 mhz bandwidth. 5.1.3 parameter extraction data is acquired over a range of bias voltages from near the breakdown voltage to when spontaneously breakdown in the diode begins to dominate the response. the breakdown voltage is estimated two ways. the first is by simply observing the voltage at which cell avalanches are detected on the oscilloscope, and it is found to be approximately 68.5 v. comparing to an extracted breakdown voltage, it is found that the actual breakdown voltage is approximately one volt less than the observed voltage. it is expected that 32 pulses would not be visible on the oscilloscope until the diode is biased slightly above breakdown. the second method to extract the breakdown voltage is to plot the bias voltage versus the gain of the diode. the y-intercept of a linear fit to the data is the breakdown voltage, when the gain of the diode would theoretically be zero. the gain or charge is calculated by integrating the signal over time and dividing by the equivalent resistance to convert to charge in a unit of coulombs, 50 1 1000 1 [c]chargeorgain 1- ÷ ÷ ø ö ç ç è æ w + w = ò dtv start_t end_t . to find the uncertainty in breakdown voltage, the uncertainty in charge is calculated by integrating over many pulses, taking the standard deviation, and propagating the error. the charge is first integrated over the window from 26 to 60 ns shown in figure 30. the plot of bias voltage versus gain is shown in figure 31. extrapolating linearly to a charge of zero results in a breakdown voltage of 69.9 v. this value is overestimated due to some after pulses after the main pulse in the integration window. integrating over the first half of the pulse, from 26 to 30 ns, results in a lower breakdown voltage of 68.3 v, shown in figure 32. 20 30 40 50 60 0 1 2 3 time (ns) amplitude(mv) figure 30. charge integration windows to obtain collected charge. 33 69.9 70.3 70.7 71.1 71.5 71.9 72.3 0 50 100 150 voltage (v) charge(pc) y = 34.96*x - 2443, vbr = 69.9 v figure 31. extracting the breakdown voltage by plotting bias voltage versus the gain. 0 5 10 15 20 25 68 69 70 71 72 charge (pc) voltage(v) v = 0.1607*c + 68.31, vbr = 68.3 v figure 32. extracting the breakdown voltage by integrating the first half of the pulse. the quenching and diode capacitances can then be extracted using pf.419 368971 769 vv q c breakdownbias = - = - = .. . , c = 34.96 * v - 2443, vbr = 69.9 v 34 where q is 69.7 ± 33.1 c for a bias voltage of 71.9 v, and vbreakdown is taken to be 68.3 v. the large uncertainty is due to the large pulse-to-pulse variation within the grouping of pulses based on number of cells firing. the capacitance of cquenching + cdiode is calcu- lated to be 19.4 pf. because these parameters can not be extracted independently, there is a degree of freedom on what their values are in the model to obtain the best fit pulse. the sspm is forward biased is extract the quenching resistance. the current versus the voltage is plotted in figure 33. the quenching resistance is equal to the slope of the linear region. for the 1 mm2 diode and 25 mm cell size, there are 1600 cells. using w=== 6113 1600 00880 1 n r r q q_total .. , where rq_total = 1/0.0088 = 113.6 w and n = 1600, the quenching resistance, rq, of one cell is 181.8 kw. y = 0.0088x - 0.0067 r2 = 0.9991 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.0 0.4 0.8 1.2 1.6 2.0 voltage (v) current(a) figure 33. extraction of diode equivalent resistance by forward biasing the diode. 5.1.4 electrical model validation extracted parameters for quenching resistance and capacitance and diode capacitance are compared to those input into the model, shown in figure 34. the breakdown voltage is taken to be 67.7 v. the model is validated by comparing the modeled pulse shapes to experimental data for three cells undergoing avalanche breakdown. the pulse shapes are compared through rise time, peak amplitude, and fall time, which are determined by the i = 0.0088 * v - 0.0067 35 time constants of the circuit. the change of voltage or potential across the diode is plotted, along with the current through the diode. figure 34. electrical model for 1 mm2 sspm with calculated parameters for one firing cell and the remaining cells represented as an equivalent circuit. 5.1.4.1 pulse shape – number of cells firing the pulse shape is compared in figure 35 between the modeled solid lines and experi- mental solid lines at a bias voltage of 71.7 v. the same pulse shape for multiple cell triggers shows that every cell has the same response. 36 25 30 35 40 45 50 0 1 2 3 4 time (ns) amplitude(mv) 7 cells 6 cells 5 cells 4 cells 3 cells 2 cells 1 cell 0 cells figure 35. comparison of modeled and experimental pulse shapes for different number of cells firing at a bias voltage of 71.1 v. 5.1.4.2 avalanche and quenching during an avalanche, the voltage across the diode drops to the breakdown voltage, shown in figure 36. as the potential decreases to the breakdown voltage, the number of charge carriers generated in the avalanche decreases, so the current also decreases, shown in figure 37. when the probability to generate new carriers becomes small, the avalanche is no longer self-sustaining, and the current drops to zero. the switch then closes and the cell begins to recharge. it is possible that the voltage across the diode may drop below the breakdown voltage, due to the space charge in the depletion region. this effect is not included in the modeling. 7 cells 6 cells 5 cells 4 cells 3 cells 2 cells 1 cell 0 cells model 6 cells model 3 cells 37 0 10 20 30 40 67 68 69 70 71 72 time (ns) voltage(v) 72.3 v-voltage 71.7 v-voltage 71.1 v-voltage figure 36. voltage (vm) across the diode as a function of time. 2 2.2 2.4 2.6 2.8 3 0 1000 2000 3000 4000 current(ua) time (ns) figure 37. current (idiode) through diode as a function of time. 5.1.4.3 pulse shape – overvoltage the effect of overvoltage on pulse shape and magnitude is investigated. the gain of the devices increases with overvoltage, shown in figure 38(a). the voltage and current are shown in figure 38(b). the modeled and experimental pulses are in good agreement at 72.3 v-voltage 71.7 v-voltage 71.1 v-voltage 38 lower bias voltages, as shown for 71.1 v. as the voltage is increased, the tail of the experimental pulse is much higher; this is the result of after pulses in the data and is not the real response from a single cell. when the collected waveforms are averaged to form a single pulse, the individual after pulses are collectively seen as an higher tail. with constant switch timing, it is found that the amplitude of the modeled pulse does not increase as much as the experimental pulses do as the bias voltage increases. as the bias voltage increases, the potential across the diode is initially greater, so it takes longer for the current to drop to where it is no longer self-sustaining. because the avalanche duration increases with increasing overvoltage, the switch should remain open longer at higher bias voltages in the model. modifying the switch timing so that the switch closes when the current reaches a constant value for all bias voltages, the total charge collected from the avalanche is greater, thus increasing the amplitude of the output pulse of the model so that it more matches that of the experimental pulses. the increase in switching time is found to be exponential, shown in figure 39, due to the exponential tailing of the current. the pulse amplitudes correlate well, shown in figure 40 (a). the cutoff current is assumed to be approximately 25 ua, near previously reported quenching current values20 . the voltage across and current through the diode as a function of time are shown in figure 40(b). the tail of the pulse is slightly longer for increasing voltage, as it requires more time to recharge to the higher pre-breakdown state. 39 25 30 35 40 45 50 55 0 0.5 1 1.5 2 time (ns) amplitude(mv) 71.1 v-exp 71.7 v-exp 72.3 v-exp 71.1 v-model 71.7 v-model 72.3 v-model 2 2.1 2.2 2.3 2.4 67 68 69 70 71 72 time (ns) voltage(v) 72.3 v-voltage 71.7 v-voltage 71.1 v-voltage 2 2.1 2.2 2.3 2.4 -100 -50 0 50 100 150 200 current(ua) 72.3 v-current 71.7 v-current 71.1 v-current (a) (b) figure 38. for increasing bias voltage and constant switch timing, comparison of experimental and modeled (a) pulses for increasing bias voltage and (b) voltage across (vm) and current through the diode. 2.15 2.19 2.23 2.27 2.31 2.35 71 71.2 71.4 71.6 71.8 72 72.2 72.4 bias voltage (v) switchclosingtime(s) figure 39. increasing switch time with increasing bias voltage. 40 20 30 40 50 60 0 0.5 1 1.5 2 2.5 time (ns) amplitude(mv) 72.3 v-exp 71.7 v-exp 71.1 v-exp 72.3 v-model 71.7 v-model 71.1 v-model 2 2.1 2.2 2.3 2.4 67 68 69 70 71 72 time (ns) voltage(v) 2 2.1 2.2 2.3 2.4 -100 -50 0 50 100 150 200 current(ua) (a) (b) figure 40. for increasing bias voltage and modified constant switch timing, com- parison of experimental and modeled (a) pulses for increasing bias voltage and (b) voltage across (vm) and current through the diode. 5.1.4.4 time constants the time constants of the sspm govern the diode recovery. the quenching resistance mostly affects the magnitude of the pulse. the quenching capacitance also affects the magnitude of the pulse and also the speed of response and the first part of the tail. the diode capacitance of the cell firing affects the middle and end parts of the tail. the equivalent cell diode capacitance mostly affects the end part of the tail. the equivalent capacitance, cquenching and cdiode, in the model matches well with that extracted. the diode and quenching capacitances are 30 pf, and the quoted terminal capacitance is 35 ff25 . some variation is expected between devices due to variations in silicon wafers. 5.1.4.5 inductance the inductor smoothes and slows the initial response of the pulse. this is hypothesized to be the result of the rc time constant of between the bias voltage cable and sspm. a 1 nf capacitor is added in parallel with the 220 nf capacitor, as the large capacitor has a slow response in comparison to the fast discharge of the sspm cell. the response of the sspm with this smaller capacitor is approximately one nanosecond faster. an even 41 smaller capacitor would have an even faster response, shown in figure 41. although the cable has inductance and capacitance, these do not affect the timing of the pulses, which are much faster than the time constant of the cable, as shown in figure 42. 10 20 30 40 50 60 0 5 10 15 20 25 30 time (ns) voltage(mv) 220 nf + 1 nf capacitor 220 nf capacitor figure 41. comparison of sspm rise time for different hv decoupling capacitors. 0 20 40 60 80 time (ns) 50 ohm 1m coaxial cable no coaxial cable figure 42. comparison of output signals read with and without a 50 ohm coaxial cable. 42 5.2 3x3 mm2 device the 3x3 mm2 device is characterized using the same procedure as for the 1 mm2 device. the components of the model are unchanged; parameters are only changed to reflect those extracted from the measurement data. measured and modeled data are compared and used to validate the model and consider the implications of increasing diode area on device response. 5.2.1 experimental results the 3x3 mm2 sspms are tested at bias voltages from 69 to 71 v. data is first taken with the preamplifier. the laser pulse is attenuated so that only one or a few number of cells fire for each event, shown in figure 43. the peak amplitudes of the pulses are histo- grammed in figure 44, and pulses are grouped according to the number of cells firing for the event. averaged pulses for each grouping are shown in figure 45. figure 46 shows pulses with three cells firing for a range of bias voltages. the pulses are normalized to the same amplitude in figure 47, showing that the shape does not change for small numbers of cells firing. the pulses are corrected to account for the gain of the preampli- fier to obtain the actual amplitude from the sspm and are shown in figure 48. the amplitude of the response from a single cell is less for the larger device, on the order of hundreds of millivolts, due to the parasitic capacitance of the other cells and also from a larger quenching resistor. the preamplifier distorts the pulse shape, so data is acquired without the preampli- fier and with a stronger laser pulse. the raw pulse waveforms are shown in figure 49. the number of cells firing for each event cannot be distinguished. the waveforms for each bias voltage are averaged to obtain one pulse. the pulses for one to five cells firing are scaled to the correct pulse amplitude using the amplitude of the gain-corrected preamplifier data and are shown in figure 50. 43 20 40 60 80 -2 0 2 4 6 8 time (ns) amplitude(mv) figure 43. raw acquired single pulse waveforms with the preamplifier for a bias voltage of 70.5 v. 0 2 4 6 8 0 200 400 600 amplitude (mv) count amplitude noise figure 44. histogram of pulse amplitudes with the preamplifier for a bias voltage of 70.5 v. 44 20 30 40 50 60 0 2 4 6 8 time (ns) amplitude(mv) figure 45. averaged pulses with the preamplifier for 0 to 6 cells firing for a bias voltage of 70.5 v. 20 30 40 50 60 0 1 2 3 4 5 time (ns) voltage(mv) 69.0 v 69.25 v 69.5 v 69.75 v 70.0 v 70.25 v 70.5 v 70.75 v 71.0 v figure 46. averaged pulses with the preamplifier for different bias voltages for three cells firing. 6 cells 5 cells 4 cells 3 cells 2 cells 1 cell 0 cells 71.0 v 70.75 v 70.5 v 70.25 v 70.0 v 69.75 v 69.5 v 69.25 v 69.0 v 45 20 30 40 50 60 0 0.5 1 time (ns) amplitude(mv) figure 47. normalization of pulses for 3 cells firing to compare the pulse shape as a function of applied voltage. 20 30 40 50 60 0 0.1 0.2 0.3 0.4 time (ns) voltage(mv) 69.0 v 69.25 v 69.5 v 69.75 v 70.0 v 70.25 v 70.5 v 70.75 v 71.0 v figure 48. averaged pulses with the preamplifier for different bias voltages for three cells firing that are corrected for the gain of the preamplifier. 71.0 v 70.75 v 70.5 v 70.25 v 70.0 v 69.75 v 69.5 v 69.25 v 69.0 v 46 0 20 40 60 80 100 0 5 10 15 time (ns) voltage(mv) 69.0 v 69.25 v 69.5 v 69.75 v 70.0 v 70.25 v 70.5 v 70.75 v 71.0 v figure 49. averaged pulses (raw data) acquired without the preamplifier and with a strong laser pulse. 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 time (ns) amplitude(mv) 69.0 v 69.25 v 69.5 v 69.75 v 70.0 v 70.25 v 70.5 v 70.75 v 71.0 v figure 50. averaged pulses acquired without the preamplifier normalized to the true sspm output magnitude. 71.0 v 70.75 v 70.5 v 70.25 v 70.0 v 69.75 v 69.5 v 69.25 v 69.0 v 71.0 v 70.75 v 70.5 v 70.25 v 70.0 v 69.75 v 69.5 v 69.25 v 69.0 v 47 1 2 3 4 5 0 2 4 6 8 10 number of cells firing snr 69.0 v 69.25 v 69.5 v 69.75 v 70.0 v 70.25 v 70.5 v 70.75 v 71.0 v figure 51. increasing signal-to-noise ratio with increasing bias voltage from preamplifier data. the signal-to-noise ratio is calculated from the preamplifier data and is shown in figure 51. the signals are the peak values in the histogram, and the noise is the ampli- tude of the valley before the peak. the ratio is seen to increase for increasing bias voltage. considering the trade-off between increasing snr and increasing dark current with overvoltage, the optimum bias voltage is taken to be 70.5 v. 5.2.2 parameter extraction integration of the current gives estimates of the breakdown voltage as well as the total cell capacitance. the corrected data that represents the true amplitude of the sspm pulse is integrated as voltage over time to get the total charge of the pulse. the variation in charge is smaller than for the 1 mm device because the variation in pulse amplitudes is small and the original pulse scaling is well above the noise. the afterpulsing is also to small to be seen. for any event, there may be a few cells that spontaneously break down, but these fall within the noise when the amplitude is on the order of millivolts. the charge versus the bias voltage is plotted in figure 52, and the breakdown voltage is calculated to be 67.9 v. observing the breakdown voltage on the oscilloscope, pulses are visible around 68.1 v. 48 the quenching and diode capacitances are extracted. the charge, q, is 52.3 ± 2.2 c for a bias voltage of 70.5 v. the breakdown voltage is taken to be 67.6 v, as this is the value later shown to match in the model. the capacitance of cq + cd is calculated to be 18.0 pf. the capacitances are kept equal to 5 pf for cq and 25 pf for cd, from the 1x1 mm2 model, as the device geometry is similar, and it is not expected that they would be dramatically different. 68 68.5 69 69.5 70 70.5 71 0 10 20 30 40 50 60 70 voltage (v) charge(pc) c = 20.23*v - 1373, vbr = 67.9 v figure 52. total charge integrated for a signal pulse over a range of bias voltages. extrapolating the data to zero estimates the diode breakdown voltage. y = 0.0577x - 0.0440 r2 = 0.9994 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.0 0.4 0.8 1.2 1.6 2.0 voltage (v) current(a) figure 53. extraction of diode equivalent resistance by forward biasing the diode. 49 the diode is forward biased to extract the quenching resistance. figure 53 shows the collected iv data and the curve fit to the linear portion. multiplying the inverse slope by the number of cells gives the quenching resistance, w==÷÷ ø ö çç è æ ÷ ø ö ç è æ =÷ ø ö ç è æ ÷÷ ø ö çç è æ = k6249 0.0577 14400 m25 mm3 0.0577 1 sizecell sizediode slope 1 r 22 q . m . 5.2.3 electrical model comparison the electrical model that was validated with the 1 mm2 device is tested against the 3x3 mm2 device. appropriate parameters are modified to represent the values extracted from 3x3 mm2 data, shown in figure 54. the quenching resistor is modified to 250 kw, and the breakdown voltage is modified to 67.6 v. the diode and quenching capacitances are kept constant. the pulse shapes and voltage and current of the diode are compared. differences in pulse shape are considered and explored. figure 54. electrical model for 3x3 mm2 sspm with calculated parameters for one firing cell and the remaining cells represented as an equivalent circuit. 50 5.2.3.1 pulse shape – overvoltage the sspm is simulated with one cell firing, and the remaining 14399 cells are grouped as an equivalent circuit. the experimental data to which the modeled results are com- pared is the data acquired without the preamplifier that has been scaled to the actual amplitude of the response from the device. the model and experimental data are com- pared in figure 55. the model follows the shape of the experimental pulses except for some small differences in pulse shape that shift with bias voltage. for a voltage of 69 v, the pulses follow closely over the rise time and first part of the tail. the second part of the tail is slower for the model. as the bias voltage increases, the differences between the pulse shapes shift to the rise time and first and middle parts of the tail. for a voltage of 71 v, the model has a slower rise time and a faster decay over the first part of the tail in comparison to experimental data. the final tail of the pulse matches well for all voltages. unlike for the 1mm2 data, the 3x3 mm2 does not show afterpulsing; there are many cells firing in each event so the after pulses from a few cells are not visible in the pulse. 20 40 60 80 100 0 50 100 150 time (ns) amplitude(uv) 71 v - exp 70 v - exp 69 v - exp 71 v - model 70 v - model 69 v - model figure 55. comparison of 3x3 mm2 sspm experimental and modeled pulses. 51 5.2.3.2 avalanche and quenching the switch timing is changed so that it always closes when the current reaches the quenching current of 25 ma. the voltage across the diode and the current through the diode are shown in figure 56. the current reaches the quenching value before the asymptotical part of the curve, so the switch timing does not change much over the range of bias voltages; the switch opens at 2.12 ns at 69 v and 2.16 ns at 71 v. 2 2.1 2.2 2.3 67 68 69 70 71 72 time (ns) voltage(v) 71 v-voltage 70 v-voltage 69 v-voltage 2 2.1 2.2 2.3 -100 -50 0 50 100 150 200 current(ua) 71 v-current 70 v-current 69 v-current figure 56. voltage across the diode and current through the diode. 5.2.3.3 time constants the differences in pulse shape with bias voltage are explored. it is hypothesized that the bigger diode behaves differently due to extra capacitance and inductance. the experi- mental pulse shape may also be different from the model because the model is simulating one cell firing, where as the experimental data is actually a pulse with many cells firing that is scaled to the amplitude of one cell. the spice model is limited by the number of cells that can be connected in parallel to simulate many cells firing. one way to simulate this is to modify the model so that all of the cells avalanche but one. this is implemented so that the equivalent cell circuit fires. the output pulse is divided by 14399 in order to compare it to experimental data 52 in figure 57. the amplitude of the modeled pulses is larger. it is expected that the pulse amplitude would be the same for the model with one cell firing or the model with 14399 cells firing and the amplitude divided by 14399. it is suspected this is because there is less parasitic capacitance when more cells are firing. when more cells are firing, the diode capacitance is active in the response instead of being a parasitic capacitance. with less parasitic capacitance, the signal amplitude increases. therefore, the gain changes with the number of cells that fire. this means the gain would change with bias voltage if the number of cells firing is not help constant, since the probability of avalanche in- creases with bias voltage. the gain is not seen to change in the preamplifier data because the number of cells firing is maintained at only a few cells as the voltage is increased. the shape of the model with 14399 cells firing also changes slightly, as shown in figure 58. the trend of the flattening of the tail in the model with increased number of cells firing is consistent with that observed experimentally. the experimental pulses are normalized to the same height to compare differences in pulse shape as the bias voltage is increased, as shown in figure 59. the differences in the time constants of the pulse tail may be due to an increased number of cells firing at higher bias voltage. table i shows the calculation of number of cells firing; the number of cells firing increases with increasing bias voltage. to calculate the number of cells firing, the raw signal amplitude obtained with the strong laser and without the preamplifier is divided by the actual sspm signal output for one cell firing. the model with one cell firing most closely matches the experimental response for 69 v, which is the voltage with the fewest number of cells firing. as the bias voltage is increased, more cells fire, and the experi- mental pulse shape changes, particularly in the middle of the tail. the model is modified so that 100 cells fire and 14299 cells are standing by to see if it matches more closely to the experimental data with 50 to 100 cells firing as is calculated. the modeled pulse shape closely matches that of 14399 cells firing, showing a limitation of the model. the shape of the middle of the tail of the pulse is dominated by the diode capaci- tance of firing cells; this time constant increases with increasing diode capacitance. if the diode capacitance is increased for firing and standby cells, the time constant is increased for both the middle and final parts of the tail. the effect of increasing the 53 diode capacitance is shown in figure 60. the peak of the pulse is also controlled by the diode capacitance, and it is also controlled by the overvoltage and quenching resistor and capacitance. the effects of changing overvoltage and quenching resistor are also shown in figure 60. the final portion of the tail is dominated by the parasitic capacitance. the difference in overvoltage is further explored in the model. it is found that the pulse response is the same regardless of how the overvoltage is changed; increasing the bias voltage has the same effect as decreasing the breakdown voltage, shown in figure 61. 20 40 60 80 100 0 50 100 150 time (ns) amplitude(uv) 71 v - exp 70 v - exp 69 v - exp 71 v - model 70 v - model 69 v - model figure 57. comparison of simulated 14399 cells firing to experimental data. simulated data is divided by 14399 to compare to experimental data of one cell firing. 54 20 40 60 80 0 20 40 60 80 100 time (ns) amplitude(uv) experimental model 1 cell model 14399 cells model normalize 14399 cells figure 58. comparison of modeled number of cells firing to experiment. 20 40 60 80 0 0.2 0.4 0.6 0.8 1 time (ns) 71 v - exp 70 v - exp 69 v - exp figure 59. comparison of normalized experimental pulse shapes with increasing bias voltage. 55 table i. calculation of number of cells firing with increasing bias voltage. bias voltage actual sspm amplitude (mv) (gain-corrected preamplifier data) raw pulse amplitude (mv) (strong laser data) number of cells firing 69 v 0.0591 3.2 54 70 v 0.0992 8.2 83 71 v 0.1359 14.4 106 0 20 40 60 80 0 0.02 0.04 0.06 0.08 0.1 time (ns) amplitude(mv) ideal values increase cq increase cd - firing cell increase cd - all cells decrease vbr figure 60. effect of changing model parameters on pulse shape; increase cq to 10 ff, cd to 50 ff, decrease vbr to 67.1 v. 56 0 20 40 60 0 0.05 0.1 time (ns) amplitude(mv) vb = 70, vbr = 67.3 vb = 70, vbr = 67.6 vb = 70, vbr = 67.9 vb = 69.7, vbr = 67.6 vb = 70.0, vbr = 67.6 vb = 70.3, vbr = 67.6 figure 61. comparison of different over voltages from changing breakdown and bias voltages. 57 6. conclusions an electrical model is developed in spice to simulate, validate, and predict the response of sspms and enable further understanding and development of these detectors. passive components are used to simulate the time constants of the response. the avalanche is modeled using a voltage source and switch, which allows for more accurate and intuitive modeling of the device behavior. model parameters are related to the physics of the device, and values are extracted based on experimental measurements. an rc circuit shields the effects of the voltage cable, and an lc circuit at the readout accounts for effects of the measurement setup. the model is developed and validated using a 1x1 mm2 device. experimental and modeled pulses are compared. the switch timing in the model is adjusted with increas- ing bias voltage so that the switch always closes when the diode current reaches the latching current. the pulse amplitudes are in agreement for pulses at different bias voltages and with different numbers of cells firing. the pulse rise time and decay times are also in agreement. the decay times of the experimental pulses appear slower for higher bias voltages, but this is a result of afterpulsing in the diode and not from the response of a single cell. a 3x3 mm2 device is used to demonstrate the predictive capabilities of the model. the model parameters are updated to reflect those extracted from the 3x3 mm2 device. all other parameters are kept constant. the experimental and modeled pulses are mostly in agreement. small differences in pulse rise time, amplitude, and fall time are hypothe- sized to be the effect of larger diode area as well as the result of the variation in number of cells firing on time constants. with the validated model, the properties of the readout signal of an sspm can be investigated as a function of bias voltage, readout circuit, and device design and geometry. 58 7. appendix – matlab code – analyze data files function [m_data, t2, start, finish, spacing, snr, cell_amp, charge] = ana- lyze3(filename,filenum) %% load in data fprintf('\nload in file: %s\n',datestr(clock)) load(filename); fprintf('%s\n',filename); data3d = reshape(y, 500, length(y)/500); many = size(data3d,2); fprintf('file loaded: %s\n',datestr(clock)) %% define pulse start and end % 1mm preamp comparison start = 120; finish = 225; % 3 mm preamp final start = 105; finish = 250; % 3 mm nopreamp weak laser start = 95; finish = 400; %% plot raw pulse waveforms offset = mean(mean(data3d(1:start-10,:))); % average offset for all waveforms factor = -1000; % flip polarity, convert to mv if offset>0 data3d = (data3d-abs(offset) ) * factor; else data3d = (data3d+abs(offset) ) * factor; end clear data3d figure(1); hold on; set(gca,'fontsize',20); xlabel('time (ns)'); ylabel('amplitude (mv)'); ylim([-1 max(max(data3d))]); xlim([0 100]) for i = 200:400 plot(t2(:)*1e9, data3d(:,i),'b') % plot pulse waveforms end plot(t2(start)*1e9, min(min(data3d)) : .02 : max(max(data3d))) plot(t2(finish)*1e9, min(min(data3d)) : .02: max(max(data3d))) %% play with data - pulse histograms fprintf('histogram pulses: %s\n',datestr(clock)) sta = start; fin = finish; temp1(size(data3d,2),finish-start+1) = 0; temp2(size(data3d,2),start) = 0; spec(1:2, size(data3d,2)) = 0; base(1:3, size(data3d,2)) = 0; data3o = data3d; % initialize for waveform offset % histogram using pulse max amplitude for k = 1 : many base(1,k) = sum(data3d(1:75,k))/75; % pulse & noise baseline data3o(:,k) = data3d(:,k) - repmat(base(1,k),size(data3d,1),1); % correct waveform temp1(k,:) = data3o(start:finish,k); % peak signal temp2(k,:) = data3o(1:start,k); % noise [amp tmp_m] = max(temp1(k,:)); % subtract baseline from pulse if tmp_m > 5 && tmp_m < finish-start-5 spec(1,k) = mean( temp1(k,tmp_m-1:tmp_m+1) ) - base(1,k); % peak hist value 59 else spec(1,k) = temp1(k,tmp_m) - base(1,k); % peak value when near hist edges end spec(2,k) = mean(temp2(k,:)) - base(1,k); % subtract baseline from noise end data3d = data3o; if filenum > 2 && preamp==1 % non-preamp data x = -20:.2:50; % preamp data end h = hist(spec(1,1:many), x); hh = hist(spec(2,1:many), x); figure(2); set(gca,'fontsize',18); xlabel('amplitude (mv)'); ylabel('count'); hold on plot(x,h); % plot pulses plot(x,hh,'r'); % plot noise xlim([-.2 x(find(h>0,1,'last'))+1]); ylim([-2 max(h(20:end))*1.1]) legend 'peak amplitude' 'noise' finish = fin; start = sta; %% plot pulses for number cell hits, average, compare to theory % find "finger valleys" fprintf('find valleys: %s\n',datestr(clock)) pos=1; neg=0; ind1 = find(x>0,1,'first')+0; ind2 = 0; ind3 = 0; jj=0; i=0; inc = 3; for m=1:5 % # valleys to find while pos==1 if jj < i pos = 0; neg = 1; end ind1 = ind1 + inc; ind2 = ind1 + inc; ind3 = ind2 + inc; i = mean(h(ind1:ind2)); jj = mean(h(ind2:ind3)); end while neg==1 if jj > i pos = 1; neg = 0; val(m) = x(ind2); % valley location end ind1 = ind1 + inc; ind2 = ind1 + inc; ind3 = ind2 + inc; i = mean(h(ind1:ind2)); jj = mean(h(ind2:ind3)); end end spacing = mean([val(2)-val(1), val(3)-val(2), val(4)-val(3)]); if filenum < 3 valley(3) = val(1); valley(2) = valley(3) - spacing; valley(1) = valley(2) - spacing; figure(5); close; figure(5); set(gca,'fontsize',18); xlabel('amplitude (mv)'); ylabel('count'); hold on plot(x,h); plot(x,hh,'r'); xlim([-1 x(find(h>0,1,'last'))+1]); ylim([0 max(h(5:end))*1.1]) legend 'peak amplitude' 'noise' plot(valley(1), 0:2:max(h), 'k') plot(valley(2), 0:2:max(h), 'k') plot(valley(3), 0:2:max(h), 'k') for i = 4:10 valley(i) = valley(i-1) + spacing; plot(valley(i), 0:1:max(h), 'k') end else valley(3) = val(2); valley(2) = valley(3) - spacing; valley(1) = valley(2) - spacing; 60 figure(5); close; figure(5); set(gca,'fontsize',18); xlabel('amplitude (mv)'); ylabel('count'); hold on plot(x,h); plot(x,hh,'r'); xlim([-1 x(find(h>0,1,'last'))+1]); ylim([0 max(h(5:end))*1.1]) legend 'peak amplitude' 'noise' plot(valley(1), 0:2:max(h), 'k') plot(valley(2), 0:2:max(h), 'k') plot(valley(3), 0:2:max(h), 'k') for i = 4:10 valley(i) = valley(i-1) + spacing; plot(valley(i), 0:1:max(h), 'k') end end %% find peak/valley ratio to get signal/noise ratio for i = 1:6 tmp_in = find(x<valley(i+1),1,'last'); % valley = units of mv noise(i) = min(h(tmp_in-2:tmp_in+2)); % median freq of noise tmp_nx = find(x<valley(i+2),1,'last'); [max_h(i), max_in(i)] = max(h(tmp_in:tmp_nx)); % limit to next valley max_in(i) = max_in(i) + tmp_in-1; % add tmp_in offset back signal(i) = max(h(max_in(i)-2:max_in(i)+2)); % median freq of signal end snr(1,:) = signal; snr(2,:) = noise; snr(3,:) = signal./noise; fprintf('signal-to-noise: %2.2f %2.2f %2.2f %2.2f %2.2f\n', snr(2:6)) %% divide pulses into "finger peaks" based on above valleys % preallocate variables fprintf('sort pulses: %s\n',datestr(clock)) pulse1 = []; pulse2 = []; pulse3 = []; pulse4 = []; pulse5 = []; pulse6 = []; pulse7 = []; pulse8 = []; pulse9 = []; t = 1:size(data3d,1); u = many; data1(t,u/4)=0; data2(t,u/2)=0; data3(t,u/2)=0; data4(t,u/4)=0; data5(t,u/4)=0; data6(t,u/4)=0; data7(t,u/4)=0; data8(t,u/4)=0; % decide which pulse goes where for i = 1 : many [disc,in] = max(data3d(start:finish,i)); if valley(1) <= disc && disc < valley(2); pulse1 = [pulse1 i]; data1(:,length(pulse1)) = data3d(:,i); elseif valley(2) <= disc && disc < valley(3); pulse2 = [pulse2 i]; data2(:,length(pulse2)) = data3d(:,i); elseif valley(3) <= disc && disc < valley(4); pulse3 = [pulse3 i]; data3(:,length(pulse3)) = data3d(:,i); elseif valley(4) <= disc && disc < valley(5); pulse4 = [pulse4 i]; data4(:,length(pulse4)) = data3d(:,i); elseif valley(5) <= disc && disc < valley(6); pulse5 = [pulse5 i]; data5(:,length(pulse5)) = data3d(:,i); elseif valley(6) <= disc && disc < valley(7); pulse6 = [pulse6 i]; data6(:,length(pulse6)) = data3d(:,i); elseif valley(7) <= disc && disc < valley(8); pulse7 = [pulse7 i]; data7(:,length(pulse7)) = data3d(:,i); elseif valley(8) <= disc && disc < valley(9); pulse8 = [pulse8 i]; data8(:,length(pulse8)) = data3d(:,i); elseif valley(9) <= disc; pulse9 = [pulse9 i]; end end % trim variables to the correct length data1 = data1(:,1:length(pulse1)); data2 = data2(:,1:length(pulse2)); data3 = data3(:,1:length(pulse3)); data4 = data4(:,1:length(pulse4)); data5 = data5(:,1:length(pulse5)); 61 data6 = data6(:,1:length(pulse6)); data7 = data7(:,1:length(pulse7)); data8 = data8(:,1:length(pulse8)); % plot figure(6); close; figure(6); hold on; set(gca,'fontsize',20); xlabel('time (ns)'); ylabel('amplitude (mv)'); ylim([min(min(data6)) max(max(data6))*1.05]); xlim([0 t2(finish)*1e9]); plot(1e9* t2(:), data3d(:,pulse1(1:10)),'r'); plot(1e9* t2(:), data3d(:,pulse2(1:10)),'y'); plot(1e9* t2(:), data3d(:,pulse3(1:10)),'g'); plot(1e9* t2(:), data3d(:,pulse4(1:10)),'b'); plot(1e9* t2(:), data3d(:,pulse5(1:10)),'m'); plot(1e9* t2(:), data3d(:,pulse6(1:10)),'k'); plot(1e9* t2(:), data3d(:,pulse6(1:10)),'k'); plot(1e9* t2(:), data3d(:,pulse7(1:10)),'c'); plot(1e9* t2(:), data3d(:,pulse8(1:10)),'r'); %% take average of pulses in each finger peak for j = 1 : 500 m_data(1,j) = mean(data1(j,:)); m_data(2,j) = mean(data2(j,:)); m_data(3,j) = mean(data3(j,:)); m_data(4,j) = mean(data4(j,:)); m_data(5,j) = mean(data5(j,:)); m_data(6,j) = mean(data6(j,:)); m_data(7,j) = mean(data7(j,:)); m_data(8,j) = mean(data8(j,:)); end % plot averaged pulses start2 = start-15; stop = min( max(start+(finish-start),200) , 250); stop2 = 400; figure(7); close; figure(7); hold on; set(gca,'fontsize',20); xlabel('time (ns)'); ylabel('amplitude (mv)'); axis([t2(start)*1e9 t2(stop)*1e9 -.5 ceil(max(max(m_data)))]); plot(t2(start2:stop2)*1e9, m_data(1,start2:stop2),'r') plot(t2(start2:stop2)*1e9, m_data(2,start2:stop2),'y') plot(t2(start2:stop2)*1e9, m_data(3,start2:stop2),'g') plot(t2(start2:stop2)*1e9, m_data(4,start2:stop2),'b') plot(t2(start2:stop2)*1e9, m_data(5,start2:stop2),'m') plot(t2(start2:stop2)*1e9, m_data(6,start2:stop2),'k') plot(t2(start2:stop2)*1e9, m_data(7,start2:stop2),'c') plot(t2(start2:stop2)*1e9, m_data(8,start2:stop2),'r') cell_amp = max(m_data(1:6,:)'); fprintf('cell amplitudes = %2.2f %2.2f %2.2f %2.2f %2.2f %2.2f\n',cell_amp(1:6)) %% integrate charge of individual pulses (correct amplitude of nopreampdata gain = 1; % no preamp cin = 1; % 1 is 1 cell firing! equivr = 1 / ((1/1000) + (1/50)); for pn = 1:many/5 %length(data3) % for 2 cells firing int_v = 0; for j = start:finish % m_data is voltage int_v = int_v + data(cin,j,pn)*(t2(2)-t2(1)); % integrate v wrt time end chargel(pn) = int_v / equivr / gain / cin; % convert to charge end charge(1) = mean(chargel); charge(2) = std(chargel); fprintf('charge = %s +/- %s\n\n', charge) % *1e12 to get pc, v [mv], t [s] 62 %% pulse normalization % normalize pulses to same height to compare pulse shapes for i = 1:6 m_data_n(i,:) = 1/max(m_data(i,:)) * m_data(i,:); end % plot normalized pulses figure(8);close; figure(8); hold on; set(gca,'fontsize',20); xlabel('time (ns)'); ylabel('amplitude (mv)'); axis([t2(start)*1e9 t2(stop)*1e9 -0.2 ceil(max(max(m_data_n)))*1.1]); %plot(t2(start:stop)*1e9,m_data_n(1,start:stop),'r') % don't normalize noise plot(t2(start:stop)*1e9,m_data_n(2,start:stop),'y') plot(t2(start:stop)*1e9,m_data_n(3,start:stop),'g') plot(t2(start:stop)*1e9,m_data_n(4,start:stop),'b') plot(t2(start:stop)*1e9,m_data_n(5,start:stop),'m') plot(t2(start:stop)*1e9,m_data_n(6,start:stop),'k') plot(t2(start:stop)*1e9,m_data_n(7,start:stop),'c') plot(t2(start:stop)*1e9,m_data_n(8,start:stop),'r') fprintf('analysis done: %s\n',datestr(clock)) 63 8. appendix – matlab code – analyze results %% filenames for 1-mm data % load('h:\sspm_matlab\hpk_1mm_1ucap\u1capdata.mat'); t2 = t2*1e9; filename(1,:) ='h:\hpk_1mm_1ucap\397_69.9_1000r_50ohm_fullbw_2.mat'; filename(2,:) ='h:\hpk_1mm_1ucap\397_71.1_1000r_50ohm_fullbw_2.mat'; filename(3,:) ='h:\hpk_1mm_1ucap\397_71.3_1000r_50ohm_fullbw_2.mat'; filename(4,:) ='h:\hpk_1mm_1ucap\397_71.5_1000r_50ohm_fullbw_2.mat'; filename(5,:) ='h:\hpk_1mm_1ucap\397_71.7_1000r_50ohm_fullbw_2.mat'; filename(6,:) ='h:\hpk_1mm_1ucap\397_71.9_1000r_50ohm_fullbw_2.mat'; filename(7,:) ='h:\hpk_1mm_1ucap\397_72.1_1000r_50ohm_fullbw_2.mat'; filename(8,:) ='h:\hpk_1mm_1ucap\397_72.3_1000r_50ohm_fullbw_2.mat'; filename(9,:) ='h:\hpk_1mm_1ucap\397_72.5_1000r_50ohm_fullbw_2.mat'; %% read in filenames for i = 1:size(filename,1) fname = filename(i, 1:length(find(isspace(filename(i,:))==0)) ); [m_data(i,:,:), t2(i,:,:), start(i), stop(i), gain(i), snr(i,:,:), cell_amp(i,:), charge(i,:,:)] = analyze5(fname, i); end %% params voltages = (71.1 : 0.2 : 72.3); % 1mm voltage steps data acquired v_opt = 71.9; % 1mm actual bias voltage voltages = (69.0 : 0.25 : 71.0); % 3mm voltages v_opt = 70.5; % 3mm bin = 3; % bias w/ highest s/n ratio (first m_data index) v = 3; % voltage cin = 2; % analyze pulse where x number of cells fired (second m_data index) %% plot averaged pulses for given bias voltage figure(20); close; figure(20); hold on; set(gca,'fontsize',20); xlabel('time (ns)'); ylabel('amplitude (mv)'); v=2; % voltage to plot pulses at plot(squeeze(t2(4,1,:)), squeeze(m_data(v,8,:)),'or','markersize',3) plot(squeeze(t2(4,1,:)), squeeze(m_data(v,7,:)),'oc','markersize',3) plot(squeeze(t2(4,1,:)), squeeze(m_data(v,6,:)),'ok','markersize',3) plot(squeeze(t2(4,1,:)), squeeze(m_data(v,5,:)),'om','markersize',3) plot(squeeze(t2(4,1,:)), squeeze(m_data(v,4,:)),'ob','markersize',3) plot(squeeze(t2(4,1,:)), squeeze(m_data(v,3,:)),'og','markersize',3) plot(squeeze(t2(4,1,:)), squeeze(m_data(v,2,:)),'oy','markersize',3) plot(squeeze(t2(4,1,:)), squeeze(m_data(v,1,:)),'or','markersize',3) legend '7 cells' '6 cells' '5 cells' '4 cells' '3 cells' '2 cells' '1 cell' '0 cells’ %% plot #cell pulses for bias voltages (can determine breakdown and noise/1 cell) figure(21); close; figure(21); hold on; set(gca,'fontsize',18); xlabel('time (ns)'); ylabel('amplitude (mv)'); cin = 4; % #cells firing to plot plot(squeeze(t2(1,:,:)),squeeze(m_data(1,cin,:)),'or','markersize',3) plot(squeeze(t2(2,:,:)),squeeze(m_data(2,cin,:)),'og','markersize',3) plot(squeeze(t2(3,:,:)),squeeze(m_data(3,cin,:)),'oy','markersize',3) plot(squeeze(t2(4,:,:)),squeeze(m_data(4,cin,:)),'ob','markersize',3) plot(squeeze(t2(5,:,:)),squeeze(m_data(5,cin,:)),'om','markersize',3) plot(squeeze(t2(6,:,:)),squeeze(m_data(6,cin,:)),'ok','markersize',3) plot(squeeze(t2(7,:,:)),squeeze(m_data(7,cin,:)),'oc','markersize',3) plot(squeeze(t2(8,:,:)),squeeze(m_data(8,cin,:)),'or','markersize',3) plot(squeeze(t2(9,:,:)),squeeze(m_data(9,cin,:)),'og','markersize',3) legend '71.1 v' '71.3 v' '71.5 v' '71.7 v' '71.9 v' '72.1 v' '72.3 v' 64 %% normalize to compare voltage bias voltages - pulse shape differences figure(22); close; figure(22); hold on; set(gca,'fontsize',18); axis([18 70 -.3 1.1]) cin = 3; % #cells firing to plot for i = 2:size(filename,1) m(i,:) = 1/max(m_data(i,cin,:)) * m_data(i,cin,:); t2b(i,:) = t2(i,1,:); end plot(t2b(9,start(9):stop(9) ), m(9,start(9):stop(9) ),'y') plot(t2b(8,start(8):stop(8) ), m(8,start(8):stop(8) ),'r') plot(t2b(7,start(7):stop(7) ), m(7,start(7):stop(7) ),'c') plot(t2b(6,start(6):stop(6) ), m(6,start(6):stop(6) ),'k') plot(t2b(5,start(5):stop(5) ), m(5,start(5):stop(5) ),'m') plot(t2b(4,start(4):stop(4) ), m(4,start(4):stop(4) ),'b') plot(t2b(3,start(3):stop(3) ), m(3,start(3):stop(3) ),'g') plot(t2b(2,start(2):stop(2) ), m(2,start(2):stop(2) ),'y') plot(t2b(1,start(1):stop(1) ), m(1,start(1):stop(1) ),'r') legend '71.1 v' '71.3 v' '71.5 v' '71.7 v' '71.9 v' '72.1 v' '72.3 v' '72.5 v' xlabel('time (ns)'); ylabel('normalized amplitude (mv)'); ylim([-.1 1.1]) %% integrate averaged!!! voltage/resistance=current pulses to find charge cin = 6; equivr = 1 / ((1/1000) + (1/50)); int_v(1:length(voltages)) = 0; charge(1:length(voltages)) = 0; % 3 mm nstart = start(4)+40; nstop = stop(4)-60; % 1 mm nstart = start(4)+30; nstop = stop(4)-200; sn = 2; % start and end voltage #’s en = 6; % length(voltages) for j=sn:en if j==3 pulse = m_data(j,cin-1,:); % pulse (# cells) to int. else pulse = m_data(j,cin,:); % pulse (# cells) to int. end for i=nstart:nstop % m_data is voltage int_v(j) = int_v(j) + pulse(i)*(t2(j,2)-t2(j,1)); % integrate v wrt time end gain = 1; % preamp gain, full bandwidth charge(j) = int_v(j) / equivr / gain / cin; % convert to charge fprintf('\ncharge for 1 cell = %2.2f pc',charge(j)*1e3) % time is in ns!! end ts(1:5/.01+1) = squeeze(t2(5,1,nstart)); tf(1:5/.01+1) = squeeze(t2(5,1,nstop)); plot(ts, -1 : .01 : 4,'r'); plot(tf, -1 : .01 : 4,'r') fprintf('\n\n') figure(24); close; figure(24); plot(charge(sn:en)*1e3,voltages(sn:en),'o','markersize',10); b = robustfit(charge(sn:en)*1e3,voltages(sn:en)); hold on; plot(charge(sn:en)*1e3, b(1)+b(2)*(charge(sn:en)*1e3),'r-') equation = sprintf('v = %2.4f pc + %2.2f',b(2),b(1)) text(150*((cin-2)/1.5),71.4,equation,'fontsize',16,'color','m') text(2.4,71.4,equation,'fontsize',16,'color','m') set(gca,'fontsize',18); xlabel('charge (pc)'); ylabel('voltage (mv)') %% plot gain versus voltage to extract breakdown % find max voltage of first few fingers & the "spacing" between them is the gain v_br = 68.3; % 1mm ~69 with fitting v_opt = 5; % 1mm v_br = 68.1; % 3mm 65 v_opt = 7; % 70.5 v %% plot charge versus voltage with uncertainty % load('h:\sspm matlab\sspm_code\charge1mm.mat') voltages = (69.0 : 0.25 : 71.0); % 3mm voltages aa = 1; bb = 9; figure(26); close; figure(26); hold on; set(gca,'fontsize',18); xlabel('voltage (v)'); ylabel('charge (pc)'); plot(voltages(aa:bb), charge(aa:bb,:,1)*1e12,'ro') % 1mm % set(gca,'xtick',[71.1 71.3 71.5 71.7 71.9 72.1 72.3]) % plot(voltages(2:8), charge(2:8,:,1)*1e12,'o') % errorbar(voltages(2:8), charge(2:8,:,1)*1e12, charge(2:8,:,2)*1e12,'o') % 3mm plot(voltages(1:9), charge(1:9,:,1)*1e12,'o') set(gca,'xtick',[69 69.5 70.0 70.5 71.0]) errorbar(voltages(1:9), charge(1:9,:,1)*1e12, charge(1:9,:,2)*1e12,'o') xlim([68.9 71.1]) %% use q = cv to extra capacitance v_opt = 6; % 1mm, 72.1 v v_br = 68.3; % 1mm v_opt = 7; % 3mm, 70.5 v v_br = 67.9; % 3mm cin = 1; cap = charge(v_opt,:,1) / (voltages(v_opt) - v_br) / cin; % c_quenching + c_diode fprintf('\nc_quenching + c_diode = %2.2f pf\n',cap*1e12) % time was not in ns!! % cap = 19.4 pf % 1mm % cap = 20.5 pf % 3mm cap_all = (charge(:,:,1) ./ (voltages(:) - v_br) / cin )*1e12 %% plot model and experimental data to compare % load('pulseshape.mat'; plot(t+12,one*6) figure(27); close; figure(27); hold on; %axis([15 70 -.3 4]); ax1 = gca; set(ax1,'xcolor','k','ycolor','k'); set(gca,'fontsize',18) plot(data(:,1),(data (:,3)),'color','r','marker','o','markersize',2,'parent',ax1); plot(data(:,1),(data (:,6)),'color','g','marker','o','markersize',2,'parent',ax1); plot(data(:,1),(data (:,9)),'color','b','marker','o','markersize',2,'parent',ax1); xlabel('time (ns)'); ylabel('voltage (v)'); axis([1.98 2.4 67 72]) ax1 = axes('position',get(ax1,'position'),... 'xaxislocation','bottom',... 'yaxislocation','right',... 'color','none',... 'xcolor','k','ycolor','k'); plot(data (:,1),(data (:,2).*1e6),'color','r','parent',ax1); plot(data (:,1),(data (:,5).*1e6),'color','g','parent',ax1); plot(data (:,1),(data (:,8).*1e6),'color','b','parent',ax1); ylabel('current (ua)','fontsize',18); axis([1.98 2.4 -100 500]); % legend '72.3 v-voltage' '71.7 v-voltage' '71.1 v-voltage' '72.3 v-current' '71.7 v- current' '71.1 v-current' set(gca,'fontsize',18) % 1mm, 1 cell fire in mode q = 2; plot(data(:,1)*1e9+23.6, (data(:,2)*1e3-.24) *q ,'y','linewidth',2) % 1 mm, 71.1 plot(data(:,1)*1e9+23.6, (data(:,2)*1e3-.257) *q ,'g','linewidth',2) % 1 mm, 71.3 plot(data(:,1)*1e9+23.6, (data(:,2)*1e3-.276) *q ,'b','linewidth',2) % 1 mm, 71.5 plot(data(:,1)*1e9+23.6, (data(:,2)*1e3-.293) *q ,'m','linewidth',2) % 1 mm, 71.7 plot(data(:,1)*1e9+23.6, (data(:,2)*1e3-.31) *q ,'k','linewidth',2) % 1 mm, 71.9 plot(data(:,1)*1e9+23.6, (data(:,2)*1e3-.327) *q ,'c','linewidth',2) % 1 mm, 72.1 plot(data(:,1)*1e9+23.6, (data(:,2)*1e3-.345) *q ,'r','linewidth',2) % 1 mm, 72.3 66 % 3 mm, preamp data figure(28); hold on; one = one*1e3; plot(squeeze(t2(1,:,:)),squeeze(one(1,:)),'or','markersize',3) plot(squeeze(t2(1,:,:)),squeeze(one(2,:)),'og','markersize',3) plot(squeeze(t2(1,:,:)),squeeze(one(3,:)),'oy','markersize',3) plot(squeeze(t2(1,:,:)),squeeze(one(4,:)),'ob','markersize',3) plot(squeeze(t2(1,:,:)),squeeze(one(5,:)),'om','markersize',3) plot(squeeze(t2(1,:,:)),squeeze(one(6,:)),'ok','markersize',3) plot(squeeze(t2(1,:,:)),squeeze(one(7,:)),'oc','markersize',3) plot(squeeze(t2(1,:,:)),squeeze(one(8,:)),'or','markersize',3) plot(squeeze(t2(1,:,:)),squeeze(one(9,:)),'og','markersize',3) legend '69.0 v' '69.25 v' '69.5 v' '69.75 v' '70.0 v' '70.25 v' '70.5 v' '70.75 v' '71.0 v' % 3 mm, normalize non-preamp data using preamp data load('h:\sspm_matlab\hpk_3mm_al\weaklaser_nopreamp\mm3nopreamp.mat'); load('h:\sspm_matlab\hpk_3mm_al\mediumnoise_final_preamp\final3mmrerun.mat'); t2 = t2*1e9; cin = 2; figure(29); hold on; % cin = 2 == 1 cell; set(gca,'fontsize',18); xlabel('time (ns)'); ylabel('amplitude (uv)') axis([10 100 -10 150]) t2b(1:6,1,:) = t2(1:6,1,:); t2b(7:9,1,:) = t2(7:9,1,:)+0.2; oneb(1:6,:) = one(1:6,:); oneb(7:9,:) = one(7:9,:) - 3e-5; for i=1:9 amp(i) = max(m_data(i,cin,:)) / 11.3; % [mv] max(oneb(i,:)) new3mm(i,:) = ( oneb(i,:) ./ (max(oneb(i,:))/amp(i)) )*1e3; % [uv] end plot(squeeze(t2b(9,1,:)),new3mm(9,:)-new3mm(9,1),'og','markersize',3) plot(squeeze(t2b(8,1,:)),new3mm(8,:)-new3mm(8,1),'or','markersize',3) plot(squeeze(t2b(7,1,:)),new3mm(7,:)-new3mm(7,1),'oc','markersize',3) plot(squeeze(t2b(6,1,:)),new3mm(6,:)-new3mm(6,1),'ok','markersize',3) plot(squeeze(t2b(5,1,:)),new3mm(5,:)-new3mm(5,1),'om','markersize',3) plot(squeeze(t2b(4,1,:)),new3mm(4,:)-new3mm(4,1),'ob','markersize',3) plot(squeeze(t2b(3,1,:)),new3mm(3,:)-new3mm(3,1),'oy','markersize',3) plot(squeeze(t2b(2,1,:)),new3mm(2,:)-new3mm(2,1),'og','markersize',3) plot(squeeze(t2b(1,1,:)),new3mm(1,:)-new3mm(1,1),'or','markersize',3) %% plot current versus forward bias to extract r_quenching % hpk 1 mm, 25 um, sample 397 f_bias = [0 0.2]; current = [35e-9 40e-6]; % hpk 3 mm, 25 um, sample 1 bias = [0 : .1 : 2]; current = [.3e-9 .274e-9 -.022e-9 -5.3e-9 -.143e-6 .0e-3 ... -25.35e-3 -30.52e-3 -35.94e-3 -41.18e-3 -47.1e-3 -52.7e-3 ... -58.89e-3 -64.5e-3 -70.9e-3]; figure(31); close; figure(31); set(gca,'fontsize',14); hold on; xlabel('voltage (v)'); ylabel('current (a)') plot(bias,-current) plot(bias(15:21),-current(15:21),'or') rq = 1/0.05833 * (3e-3/25e-6)^2 rq = 247 % kohm %% plot snr figure(33); hold off; mm=4 plot([1:mm],squeeze(snr(1,3,1:mm)),'-or'); hold on; plot([1:mm],squeeze(snr(2,3,1:mm)),'-og') plot([1:mm],squeeze(snr(3,3,1:mm)),'-ob') plot([1:mm],squeeze(snr(4,3,1:mm)),'-oc') plot([1:mm],squeeze(snr(5,3,1:mm)),'-om') 67 plot([1:mm],squeeze(snr(6,3,1:mm)),'-oy') plot([1:mm],squeeze(snr(7,3,1:mm)),'-ok') plot([1:mm],squeeze(snr(8,3,1:mm)),'-or') plot([1:mm],squeeze(snr(9,3,1:mm)),'-og') set(gca,'fontsize',18); xlabel('number of cells firing'); ylabel('snr') %legend '71.1 v' '71.3 v' '71.5 v' '71.7 v' '71.9 v' '72.1 v' '72.3 v' legend '69.0 v' '69.25 v' '69.5 v' '69.75 v' '70.0 v' '70.25 v' '70.5 v' '70.75 v' '71.0 v' set(gca,'xtick',[1 2 3 4 5 6 7]); xlim([1 6]) %% find uncertainity of charge / breakdown voltage cin = 4; gain = 1; % 11.3 == preamp gain, full bandwidth equivr = 1 / ((1/1000) + (1/50)); int_v(1:length(voltages)) = 0; charge(1:length(voltages)) = 0; sn = 4; % start/end voltage #’s en = 9; % length(voltages) for j = sn:en for pn = 100:200 pulse = m_data(j,cin,:); % pulse (# cells) to int for i = start(cin):stop(cin); % m_data is voltage int_v(j) = int_v(j) + pulse(i)*(t2(j,2)-t2(j,1)); % integrate v wrt time end charge(j,pn) = int_v(j) / equivr / gain / cin; % convert to charge end charge_std(j) = stdev(charge(j,:)); fprintf('\ncharge for 1 cell = %2.2f pc +/- %2.2f',charge(j)*1e3, charge_std) % [ns] end figure(34); close; figure(34); plot(charge(sn:en)*1e3,voltages(sn:en),'o','markersize',10); b = robustfit(charge(sn:en)*1e3,voltages(sn:en)); hold on; plot(charge(sn:en)*1e3, b(1)+b(2)*(charge(sn:en)*1e3),'r-') equation = sprintf('v = %2.4f pc + %2.2f',b(2),b(1)) text(150*((cin-2)/1.5),71.4,equation,'fontsize',16,'color','m') text(450,71.4,equation,'fontsize',16,'color','m') set(gca,'fontsize',14); xlabel('charge (pc)'); ylabel('voltage (mv)') %% 3mm data without preamp -- plot averaged pulses filename(1,:) = 'h:\hpk_3mm_al\weaklaser_nopreamp\shape_69.0_50ohmfbw_1kpa.mat '; filename(2,:) = 'h:\hpk_3mm_al\weaklaser_nopreamp\shape_69.25_50ohmfbw_1kpa.mat'; filename(3,:) = 'h:\hpk_3mm_al\weaklaser_nopreamp\shape_69.5_50ohmfbw_1kpa.mat '; filename(4,:) = 'h:\hpk_3mm_al\weaklaser_nopreamp\shape_69.75_50ohmfbw_1kpa.mat'; filename(5,:) = 'h:\hpk_3mm_al\weaklaser_nopreamp\shape_70.0_50ohmfbw_1kpa.mat '; filename(6,:) = 'h:\hpk_3mm_al\weaklaser_nopreamp\shape_70.25_50ohmfbw_1kpa.mat'; filename(7,:) = 'h:\hpk_3mm_al\weaklaser_nopreamp\shape_70.5_50ohmfbw_1kpa.mat '; filename(8,:) = 'h:\hpk_3mm_al\weaklaser_nopreamp\shape_70.75_50ohmfbw_1kpa.mat'; filename(9,:) = 'h:\hpk_3mm_al\weaklaser_nopreamp\shape_71.0_50ohmfbw_1kpa.mat '; for i = 1 : size(filename,1) load(filename(i,1:length(find(isspace(filename(i,:))==0)))); data3d = reshape(y, 500, length(y)/500) *-1; % plot raw pulses figure(35); hold on; for k = 200:300 plot(1e9*t2(:), data3d(:,k)*1e3,'b') % plot pulse waveforms end; set(gca,'fontsize',18); xlabel('time (ns)'); ylabel('voltage (mv)'); xlim([0 100]) % take average of all pulses for j = 1 : 500 one(i,j) = mean(data3d(j,:)) * 1000; % convert to mv end end figure(36); hold on; set(gca,'fontsize',18); xlabel('time (ns)'); ylabel('voltage (mv)'); xlim([0 100]) 68 normalize = 1; gain=11.3; plot(1e9*squeeze(t2(1,1,:)), squeeze(squeeze(pa_data(1,cin,:)))./gain,'r') plot(1e9*squeeze(t2(1,1,:)), squeeze(squeeze(pa_data(2,cin,:)))./gain,'g') plot(1e9*squeeze(t2(1,1,:)), squeeze(squeeze(pa_data(3,cin,:)))./gain,'y') plot(1e9*squeeze(t2(1,1,:)), squeeze(squeeze(pa_data(4,cin,:)))./gain,'b') plot(1e9*squeeze(t2(1,1,:)), squeeze(squeeze(pa_data(5,cin,:)))./gain,'m') plot(1e9*squeeze(t2(1,1,:)), squeeze(squeeze(pa_data(6,cin,:)))./gain,'k') plot(1e9*squeeze(t2(1,1,:)), squeeze(squeeze(pa_data(7,cin,:)))./gain,'c') plot(1e9*squeeze(t2(1,1,:)), squeeze(squeeze(pa_data(8,cin,:)))./gain,'r') plot(1e9*squeeze(t2(1,1,:)), squeeze(squeeze(pa_data(9,cin,:)))./gain,'g') legend '69.0 v' '69.25 v' '69.5 v' '69.75 v' '70.0 v' '70.25 v' '70.5 v' '70.75 v' '71.0 v' xlim([20 60]); ylim([-.03 .42]) %% 3 mm pulse manipulation % normalize preamp data for gain to get true amplitude load('d:\sspm\mediumnoise_final\final3mmrerun.mat'); t2 = t2*1e9; pa_data = m_data; % 9 voltages, 6 cells gain = 11.3; cin=4; figure(101); plot(squeeze(t2(1,1,:)),squeeze(pa_data(:,cin,:))); % 2 cell firing cpa_data = pa_data ./ gain; % corrected preamp figure(102); plot(squeeze(t2(1,1,:)),squeeze(cpa_data(:,cin,:))); % 2 cell firing for i=1:9 for j=1:6 f_cpa_data(i,j) = max(max(cpa_data(i,j,:))); % max pulse amp. end end; f_cpa_data % normalize non-preamp data to true amplitude load('d:\sspm\weaklaser_nopreamp\mm3nopreamp.mat'); t2 = t2*1e9; n_data = one * 1000; % normalize to correct amplitude figure(103); plot(t2(1,:),squeeze(one(:,:)*1000)); % no disting. cells!! just 1 pulse for i=1:9 for j=1:6 f_diff(i,j) = f_cpa_data(i,j)/max(n_data(i,:)); % one, n_data [mv] cn_data(i,1,:) = f_cpa_data(i,j)/max(n_data(i,:)) .* n_data(i,:); end end figure(104); plot(t2(1,:),squeeze(cn_data(:,:))); f_diff %% plot switch timing data - vout, v data = a_pastespecial; st = 1; et = 16000; figure(38); close; figure(38); set(gca,'fontsize',18) ax1 = gca; set(ax1,'xcolor','k','ycolor','k') hl2 = line(v71(st:et,1)*1e9,v71(st:et,3),'color','r','marker','o','markersize',2,'parent',ax1); hl2 = line(v70(st:et,1)*1e9,v70(st:et,3),'color','g','marker','o','markersize',2,'parent',ax1); hl2 = line(v69(st:et,1)*1e9,v69(st:et,3),'color','b','marker','o','markersize',2,'parent',ax1); xlabel('time (ns)'); ylabel('voltage (v)'); axis([1.98 2.3 67 72]) legend '71 v-voltage' '70 v-voltage' '69 v-voltage' ax2 = axes('position',get(ax1,'position'),... 'xaxislocation','bottom',... 'yaxislocation','right',... 'color','none',... 'xcolor','k','ycolor','k'); hl2 = line(v71(st:et,1)*1e9,v71(st:et,2)*1e6,'color','r','parent',ax2); hl2 = line(v70(st:et,1)*1e9,v70(st:et,2)*1e6,'color','g','parent',ax2); hl2 = line(v69(st:et,1)*1e9,v69(st:et,2)*1e6,'color','b','parent',ax2); ylabel('current (ua)','fontsize',18); axis([1.98 2.3 -100 200]); %legend '72.3 v-current' '71.7 v-current' '71.1 v-current' legend '71 v-current' '70 v-current' '69 v-current' set(gca,'fontsize',18) 69 figure(39); hold on; set(gca,'fontsize',18); xlabel('time (ns)'); ylabel('amplitude (mv)'); xlim([20 60]); ylim([-.2 2.5]); cin = 4; plot(squeeze(t2(8,:,:)),squeeze(m_data(8,cin,:)),'or','markersize',3) plot(squeeze(t2(5,:,:)),squeeze(m_data(5,cin,:)),'om','markersize',3) plot(squeeze(t2(2,:,:)),squeeze(m_data(2,cin,:)),'ob','markersize',3) plot(v723(:,1)*1e9+23.6, (v723(:,4)*1e3-.76),'r','linewidth',2) % 1 mm, 72.3 plot(v717b(:,1)*1e9+23.6, (v717b(:,4)*1e3-.65),'m','linewidth',2) % 1 mm, 71.7 plot(v711b(:,1)*1e9+23.6, (v711b(:,4)*1e3-.56),'b','linewidth',2) % 1 mm, 71.1 legend '72.3 v-exp' '71.7 v-exp' '71.1 v-exp' '72.3 v-model' '71.7 v-model' '71.1 v- model' legend '71.1 v-exp' '71.7 v-exp' '72.3 v-exp' '71.1 v-model' '71.7 v-model' '72.3 v- model' figure(112); hold on; set(gca,'fontsize',18); xlabel('time (ns)'); ylabel('amplitude (mv)') plot(normal(:,1)*1e9,(normal(:,2)-normal(1,2))*1e3,'k'); plot(cq10ff(:,1)*1e9,(cq10ff(:,2)-cq10ff(1,2))*1e3,'r'); plot(cd50ff(:,1)*1e9,(cd50ff(:,2)-cd50ff(1,2))*1e3,'m'); plot(cd50ff_720p(:,1)*1e9,(cd50ff_720p(:,2)-cd50ff_720p(1,2))*1e3,'b'); plot(vbr(:,1)*1e9,(vbr(:,2)-vbr(1,2))*1e3,'g'); legend 'ideal values' 'increase c_q' 'increase c_d - firing cell' 'increase c_d - all cells' 'decrease v_b_r' axis([0 80 -.005 0.1]) %% 1mm - compare preamp/no preamp shapes, bandwidths load('h:\hpk_1mm_pa\newgoodpreampdata.mat'); t2=t2*1e9; figure(40); close; figure(20); hold on; set(gca,'fontsize',18); xlabel('time (ns)'); ylabel('amplitude (mv)'); ylim([-6 30]); xlim([20 60]) cin = 4; % #cells firing to plot voltage pulses at: 4=3 cells firing plot(squeeze(t2(3,:,:)),squeeze(m_data(3,1,:)),'r') plot(squeeze(t2(4,:,:)),squeeze(m_data(4,1,:)),'b') plot(squeeze(t2(3,:,:)),squeeze(m_data(3,3,:)),'r') plot(squeeze(t2(3,:,:)),squeeze(m_data(3,5,:)),'r') plot(squeeze(t2(4,:,:)),squeeze(m_data(4,3,:)),'b') plot(squeeze(t2(4,:,:)),squeeze(m_data(4,5,:)),'b') legend 'full bandwidth' '150 mhz bandwidth' % without preamp full bandwidth / 150 mhz bandwidth figure(41); close; figure(41); hold on; set(gca,'fontsize',18); xlabel('time (ns)'); ylabel('amplitude (mv)'); ylim([-.1 1.5]); xlim([20 60]); cin = 3; plot(squeeze(t2(1,:,:)),squeeze(m_data(1,1,:)),'r') plot(squeeze(t2(2,:,:)),squeeze(m_data(2,1,:)),'b') plot(squeeze(t2(1,:,:)),squeeze(m_data(1,2,:)),'r') plot(squeeze(t2(1,:,:)),squeeze(m_data(1,3,:)),'r') plot(squeeze(t2(2,:,:)),squeeze(m_data(2,2,:)),'b') plot(squeeze(t2(2,:,:)),squeeze(m_data(2,3,:)),'b') legend 'full bandwidth' '150 mhz bandwidth' % no preamp versus preamp shape - full bandwidth figure(42); close; figure(42); hold on; set(gca,'fontsize',18); xlabel('time (ns)'); ylabel('amplitude (mv)'); ylim([-.6 3.4]); xlim([20 60]); cin = 3; plot(squeeze(t2(1,:,:)),squeeze(m_data(1,1,:)),'r') plot(squeeze(t2(3,:,:)),squeeze( m_data(3,1,:) * max(m_data(1,1,:))./max(m_data(3,1,:)) ),'b') plot(squeeze(t2(1,:,:)),squeeze(m_data(1,3,:)),'r') plot(squeeze(t2(1,:,:)),squeeze(m_data(1,5,:)),'r') plot(squeeze(t2(3,:,:)),squeeze( m_data(3,3,:) * max(m_data(1,3,:))./max(m_data(3,3,:)) ),'b') plot(squeeze(t2(3,:,:)),squeeze( m_data(3,5,:) * max(m_data(1,5,:))./max(m_data(3,5,:)) ),'b') legend 'without preamp' 'with preamp' 70 % no preamp versus preamp shape - 150 mhz bandwidth figure(43); close; figure(43); hold on; set(gca,'fontsize',18); xlabel('time (ns)'); ylabel('amplitude (mv)'); ylim([-.4 3.4]); xlim([20 60]); cin = 3; plot(squeeze(t2(2,:,:)),squeeze(m_data(2,1,:)),'r') plot(squeeze(t2(4,:,:)),squeeze( m_data(4,1,:) * max(m_data(2,1,:))./max(m_data(4,1,:)) ),'b') plot(squeeze(t2(2,:,:)),squeeze(m_data(2,3,:)),'r') plot(squeeze(t2(2,:,:)),squeeze(m_data(2,5,:)),'r') plot(squeeze(t2(4,:,:)),squeeze( m_data(4,3,:) * max(m_data(2,3,:))./max(m_data(4,3,:)) ),'b') plot(squeeze(t2(4,:,:)),squeeze( m_data(4,5,:) * max(m_data(2,5,:))./max(m_data(4,5,:)) ),'b') legend 'without preamp' 'with preamp' for i=1:6 factora(i) = 1/ (max(m_data(1,i,:))./max(m_data(3,i,:))); factorf(i) = 1/ (max(m_data(2,i,:))./max(m_data(4,i,:))); fa = mean(factora(2:5)) ff = mean(factorf(2:5)) end 71 9. references 1 sensl. spmmini low cost high gain apd. datasheet. v 1.0 rev 3.1. available at: http://www.sensl.com/pdfs/datasheets/spmmicro_datasheet.pdf. accessed june 23, 2008. 2 c. j. stapels, j. f. christian, g. entine, m. r. squillante, f. l. augustine, and w. g. lawrence. the solid-state photomultiplier for an improved gamma-ray detector. ieee conference on technolo- gies for homeland security 1104 (2005). 3 k. s. shah, j. glodo, m. klugerman, w. m. higgins, t. gupta, and p. wong. high energy resolu- tion scintillation spectrometers. ieee trans. nucl. sci., 51 (5) (2004) 2395-2399. 4 f. quarati. x-ray and gammy-ray response of a 2” x 2” labr3:ce scintillation detector. nuclear instruments and methods in physics research a 574 (2007) 115-120. 5 p. buzhan, et al. an advanced study of silicon photomultiplier. icfa instrumentation bulletin. 2001. available at: http://www.slac.stanford.edu/pubs/icfa/fall01/paper3/paper3.pdf. accessed june 23, 2008. 6 g. bondarenko, et al. limited geiger-mode silicon photodiode with very high gain. nuclear physics b (proc. suppl.) 61b (1998) 347-352. 7 n. pavlov, g. mæhlum, and d. meier. gamma spectroscopy using a silicon photomultiplier and a scintillator. ieee nuclear science symposium conference record (2005) 173-180. 8 f. corsi, et al. modeling a silicon photomultiplier (sipm) as a signal source for optimum front-end design. nuclear instruments and methods in physics research a 572 (2007) 416-418. 9 f. corsi, et al. electrical characterization of silicon photo-multiplier detectors for optimal front- end design. ieee nuclear science symposium conference record (2006) 1276-1280. 10 i. britvitch and d. renker. measurements of the recovery time of geiger-mode avalanche photodiodes. nuclear instruments and methods in physics research a 567 ( 2006) 260-263. 11 a. g. stewart, e. greene-o’sullivan, d. j. herbert, v. saveliev, f. quinlan, l. wall, p. j. hughes, a. mathewson, and j. c. jackson. study of the properties of new spm detectors. proceedings of spie 6119 (2006) 84-93. 12 hamamatsu, solid state division. mppc (multi-pixel photon counter). s10362-11 series. form ksp-0002 a. 13 j. barral. study of silicon photomultipliers. max planck institut fuer physik. july 2, 2004. 14 henry w. ott. noise reduction techniques in electronic systems. second edition. john wiley & sons, new york: 1988. 15 s. cova, et al. avalanche photodiodes and quenching circuits for single-photon detection. applied optics 35 (12) (1996) 1956-1976. 16 z. y. sadygov, et al. avalanche semiconductor radiation detectors. ieee transactions on nuclear science 43 (1996) 1009-1013. 72 17 j. ninkovic. recent developments in silicon photomultipliers. nuclear instruments and methods in physics research a 580 (2007) 1020-1022. 18 m. razeghi and a. rogalski. semiconductor ultraviolet detectors. applied physics reviews. journal of applied physics 79 (10) (1996) 7433-7473. 19 a. lacaita, m ghioni, and s cova.. double epitaxy improves single-photon avalanche diode performance. electronics letters 25 (13) (1989) 841-843. 20 c. piemonte. a new silicon photomultiplier structure for blue light detection. nuclear instru- ments and methods in physics research a 568 (2006) 224–232. 21 r. haitz. model for the electrical behavior of a microplasma. journal of applied physics 35 (5) (1964) 1370-1376. 22 nanoled blue led 05a. horiba jobin yvon. product specifications. available at: http://www.jyhoriba.it/itdivisions/fluorescence/ibh/blue_led_05a.htm. accessed june 23, 2008. 23 minicircuit preamplifier. datasheet. available at: http://www.minicircuits.com/pdfs/zx60- 4016e+.pdf. accessed june 23, 2008. 24 y. kang, et al. afterpulsing of single-photon avalanche photodetectors. lasers and electro-optics society. the 16th annual meeting of the ieee (2003) 775-776. 25 hamamatsu. mppc (multi-pixel photon counter). catalog no. kapd0002e02. july 2007.