1 New solid state photomultiplier Dmitry Shushakov and Vitaly Shubin P. N. Lebedev Physical Institute, Department of Solid State Physics, Moscow, Russia. ABSTRACT The physical principles of a new high-sensitive photodetector developed on the base of Avalanche heterostructures with Negative Feedback (ANF) are considered. The main difference between such a structure and a conventional avalanche photodiode (APD) is nonstationarity of the electrical field strength in an avalanche region caused by the feedback. It is shown that when the non-stationarity is manifested at amplification of a single photocarrier, it changes radically the main characteristics of the avalanche process. A qualitative physical model of ANF process and the results of numerical simulation and experiments are presented. They exhibit an ability of ANF-based device to provide a unique for solid state photosensor a combination of low noise, high gain and low response time. It is also shown that at the same time the negative feedback enables an avalanche device to be manufactured in multi-element design without serious difficulties. The experimental distribution of gain resulting from single-electron-initiated avalanche events was obtained on a SiC-Si ANF heterostructure, and confirmed the main model predictions. The results, illustrating the process of a few-photon light pulse registration are also presented. The opportunities of ANF-based devices further development are discussed. 2. INTRODUCTION Currently there is no any solid state device that can replace a vacuum-tube photomultiplier (PMT) for solving the tasks, where both extra-high sensitivity and fast photoresponce are necessary. Even an avalanche photodiode (APD), with its higher quantum efficiency and high enough gain, can not provide the same abilities in detecting of a weak light pulse. It is shown  that the process of avalanche amplification in APD is too noisy at high gain due to the basic physical reasons. Our study of the avalanche process has shown that the negative feedback changes its dynamics, and improves the device s internal amplifier characteristics. An avalanche-with-negative-feedback (ANF) process provides high gain (up to 10 5 ) both with much lower excess noise and better response time, especially, at high gain. Moreover, ANF process enables one to solve the problem of multi-element photosensor manufacturing that is known to be very complicated for both APD and PMT.
2 The new properties of the ANF devices are due to new physics of a Single-Carrier-initiated avalanche process (SC ANF process). The model of this process presented in the paper allows one to estimate the main features of the ANF-based devices. 3. THE MODEL OF SC ANF PROCESS The charge negative feedback means the dependence of the avalanche process rate on the quantity of charge resulting from the process. It may be realized by different ways, but the simplest one is introducing of a potential barrier for minority carriers between the avalanche region and the electrode. So, the carriers, resulting from the impact ionization in the avalanche layer, do not reach the electrode immediately, but are accumulated at the barrier. Their charge screens the field and suppresses the avalanche process. If accumulated charge does not spread along the barrier (i.e., the feedback is local), than the coefficient of feedback is high enough, and accumulation of even a small charge leads to essential decreasing of field in the avalanche region. Thus, even a single-carrier-initiated ionization chain is developing in the non-stationary conditions, and each subsequent ionization occurs at the lower field than the previous one. The main peculiarities of the SC ANF process may be estimated from comparing it with the same single-carrier-initiated multiplication process in APD. In both cases the photocarrier injected in the avalanche layer, generates few electron-hole pairs by an impact ionization mechanism. They are divided by the field, and the carriers drift in opposite directions, generating new pairs. These pairs, in their turn, produce new ones, supporting the avalanche process even after the initial carrier leaves the avalanche region. One can say that the photocarrier initiates the generation of a current pinch, which then develops independently. The pinch may peter out at any moment due to statistical fluctuation if all the carriers leave it without further ionizations. As the resulting gain (G) is equal to the total number of carriers, generated during the pinch life time (τ av ), this value is also probabilistic, depending on the fluctuations of τ av and current density. So, the form of gain probability distribution is defined by statistical laws of the current pinch evolution. There are two possible types of the current pinch evolution depending on the value of the maximal electric field strength E m in the avalanche region. When it is lower than some critical value E cr, then average 1 number of carriers in the pinch decreases in time exponentially. For E m > E cr (overcritical regime) the current in the pinch increases exponentially, leading to thermal breakdown in APD case. So, E cr corresponds to breakdown field in APD. The negative feedback makes the ionization chain to stay finite even when the electric field strength in the avalanche region exceeds the breakdown for APD value. Moreover, an operating with overvoltage is the most efficient regime for the ANF structure. At the beginning of the first stage, the current in the pinch is relatively small. So, the rate of the field changing caused by the minority carriers accumulation at the barrier is also not high, and the influence of the negative feedback is negligible. Hence, the number of carriers in the pinch increases in time exponentially as well as in a p-n junction, operating in Geiger mode. SC ANF process continues to grow with very low probability to stop until the field is lowered by the feedback. 1 Averaged by the large number of realizations.
3 The second stage begins when the field reaches its critical value. The current is maximal at this moment, thus, the field quickly decays, and the multiplication process ceases. So, nearly each SC ANF process results in the charge package (i.e., gain), not lower, than it is necessary to screen the overvoltage. At the same time it can not be much higher than this value as the probability of ionizing quickly decays with the field lowering. Thus, the gain of SC ANF process depends mainly on the overvoltage value and the negative feedback coefficient 1, but not on the fluctuations of current in the pinch, or its life time. When several photons are falling on to the structure simultaneously (but not at the same place), they are amplified independently due to the negative feedback locality, and produce a summarized output signal. This enables ANF structure to operate in the regime of proportional amplification while being in the overcritical state. The system of non-stationary differential equations describing the avalanche process developed in a small-area 2 MOS structure was proposed in  to simulate the dynamics of the current pinch evolution under the influence of the negative feedback. It allows one to obtain the dependence of avalanche current in the pinch vs time for various negative feedback coefficients and electric field strength. Such important characteristics of SC ANF process, as the dependencies of the average gain and response time on the electric field strength may be also calculated from it. Unfortunately, this system of differential equations does not allow one to investigate probability distribution of gain as well as noise characteristics, following from it. And a theory of the branching processes describing the avalanche multiplication process, appears to lead to a very complicated multi-dimensional model in the non-stationary case. So, a Monte Carlo simulation was used to confirm the main predictions of the qualitative physical model. The structure assumed to be MOS-like, p-si-based, and with uniform doping. The charge accumulated at the interface was supposed to spread over the fixed area 3 S 0 10 μm 2 during the pinch life time. Current flowing through the insulator layer was postulated to be small as compared with the current in pinch, and not affecting the result. The mechanism of multiplication process was postulated to be the same as described by McIntyre, excluding the field stationarity and a fixed relationship between ionization coefficients for electrons and holes. 4. THE BASIC FEATURES OF SC ANF PROCESS. The main properties of the ANF structures are shown in Fig.1. There are three basic features of the SC ANF process following directly from its model: the gain self-calibration, smoothing of the gain vs voltage dependence, and decreasing of the avalanche response time. These features define the applied properties of the ANF-based device. 1 Feedback coefficient means the value of voltage on depleted layer lowering ΔU after generating of ΔG electrone-hole pairs. 2 The square accumulated charge is spreaded over during the τ av must be limited to provide sufficient coefficient of negative feedback. 3 This value of S 0 was estimated experimentally  on uniform avalanche MIS structures, where charge spreading area is limited by electrons drift velocity along interface.
4 Negative feedback Overvoltage SC ANF process gain selfcalibration gain vs voltage smoothing avalanche time decreasing low excessive noise high probability of a weak pulse detection high gain large active area (or multielement) fastness solid state photomultiplier Fig. 1. Schematic, illustrating the main features of ANF process Self-calibration of the gain Gain probability distribution for a conventional APD, where the electric field strength in the avalanche region does not depend on time, was derived by McIntyre in . He has shown, that for single-carrierinitiated avalanche events, the distribution has a long high-gain tail that makes the avalanche process in APD so noisy, increasing the dispersion of gain. In the SC ANF the behavior of gain probability distribution changes radically. Negative feedback cuts off high-gain tail of distribution and, at the same time, diminishes probability of low-gain realizations by moving them to one narrow peak located close to the average gain value (high-gain peak). The mechanics of the gain self-calibration follows directly from the physics of SC ANF process. The negative feedback does not allow the gain to be too high because the resulting charge screens the field and overvoltage do not allow the multiplication process to stop while the accumulated charge (i.e., gain) is too low.
5 One can say, that a wide APD s gain probability distribution is shrunk to a narrow peak in the SC ANF case, and this leads to a cardinal decrease of the noise-factor. Two gain probability distributions obtained by Monte Carlo simulation of SC ANF process are shown in fig. 2. Distribution 2 was calculated for a relatively low applied voltage, slightly exceeding the critical value. Its low-gain part is nearly the same as it would be in APD, but the high-gain part is shrunk to a peak (nearly 30 percents of all realizations). The peak enlarges with voltage and shifts to higher gain (distribution 2). The peak growth is due to adding of a part of low-gain realizations. P And the shift corresponds to higher charge, which must be accumulated to screen higher overvoltage. Nearly 50 percent of all realizations are located in the peak. 1 Fig. 2. Calculated probability P that a single Gain, in- G jected charge carrier will result in G carriers. Solid line corresponds to higher applied voltage and dashed line to lower one. So, the share of the peak increases with overvoltage, and this leads to an essential diminishing of the 2 G noise factor F =, which appears to be as low as 1.77 for the second distribution. This value is nearly 2 G two hundreds of time lower, than it would be for APD, operating with the same average gain 1. The calculated noise factor vs average gain dependencies for APD (1) and SC ANF (2) are F shown in Fig.3. At low gains corresponding to low electric field, the noise factor in both cases behaves similarly. And at high gain, when SC ANF enters the overcritical mode, its noise factor rapidly decays due to the mentioned effect of the gain self-calibration. The value of gain corresponding to the critical field depends on the feedback coefficient. For higher feedback it shifts to lower gains, and the same effect is exhibited at lower supplied voltage. Fig. 3. Calculated dependence of noaverage ise factor gain, F on G To confirm the validity of results provided by average gain G for APD (1) and SC ANF (2). the Monte Carlo simulation the same points for APD were calculated from analytical equations, derived in . They are marked by asterisks on line 1. One can see their correspondence with the results obtained from Monte Carlo simulation (filled triangles) is quite well In assumption of the same relationship between ionization coefficients for electrons and holes.
6 One more characteristic, important for a photosensor, is a probability of a weak light signal detection. As is known , the APD does not provide high detection probability of a few-photon light pulse despite to its high enough quantum efficiency and gain. This is due to the mentioned above fact that most of photocarriers in APD result in a very small (undetectable) charge package, and high gains are rare. SC ANF-based structure has the same quantum efficiency as APD, but its detection probability is much higher. The main reason is higher electric field intensity, which is not limited by the breakdown value. Thus, much higher ionization coefficients may be reached. P r Average gain, G Fig. 4. Calculated probability P r that a single electron will give a detectable pulse when the average gain is G, and discriminator is set at 1000 electrons. 1 SC ANF; 2 APD. The calculated dependencies of a singleelectron-detection probability P s vs average gain G when discriminator level is set at 1000 carriers, are shown in figure 4. One can see, this probability is saturated for APD (line 2) at high gain, but continues to increase with gain for SC ANF process (line 1), tending theoretically to 1. At higher discriminator level, the advantage of SC ANF is even more essential as its detection probability is not noticeably affected by the discrimination level setting. Obviously P r is the probability of any carrier detecting. Never APD nor SC ANF allows one to distinguish a dark carrier from that one initiated by light. But a signal pulse consisting of several photoelectrons may be distinguished in the ANF structure from a single-dark-carrier-initiated event due to its higher amplitude. Both low noise factor and high probability of each carrier detection makes the output signal to be nearly proportional to the signal amplitude. This feature is very important as it allows not only to distinguish a few-photons-initiated signal from the dark noise, but, to measure its amplitude in the analog regime, i.e., without photons counting Dependence of average gain on applied voltage The dependencies of average gain on voltage U d, applied to the depleted layer, are shown in fig. 5. Line 1 corresponds to APD, and lines 2,3, to SC ANF. Voltage is counted off from its critical value. It is seen that this dependence is very sharp in APD, especially at high voltage, tending to its critical value. This does not allow APD to operate with high gain because even a small fluctuation of voltage, caused by power supply instability or the device area non-uniformity, leads to a high variation of gain. If a part of active area appears to be in the overcritical regime due to breakdown voltage inhomogeneities, then microplasma may occur, leading to extra-high excessive noise. This is the sharp gain vs voltage dependence that makes the manufacturing of multi-element APD devices so difficult.
7 The same dependencies for the SC ANF were calculated from the system of non-stationary differential equations  based on the model of a small-area MOS structure. Line 2 corresponds to higher capacity of an insulator (lower feedback coefficient), and line 3, to lower capacity. As mentioned above, there is a very small probability for multiplication process to be terminated until accumulated charge screens the excessive voltage ΔU = Ud Ucr. This means the resulting charge is nearly proportional to initial overvoltage on the depleted layer when overvoltage is high enough. So, at high voltages this dependence may be approximated by a simple equation G = e 1 Ci Δ Ud, where e is the elementary charge value, and C i, the capacity of insulator layer. G Fig. 5. Calculated dependence of average gain UG d - U cr, V from voltage U d on depleted layer. 1 APD; 2,3 SC ANF with different coefficients of negative feedback. At low initial voltage the accumulating charge is also small and its influence on the avalanche process is negligible. So, low-voltage part of the dependence is similar to that in APD. Linearity of the dependence at high gains leads to a small (proportional) response of gain on the voltage fluctuations and structure inhomogeneities. This allows one to reach much higher gain in the ANF structure and enables designing of multielement devices Dependence of avalanche response time on the electric field strength. Figure 6 represents the influence of the negative feedback on the characteristic avalanche response time (τ av ). This time in APD (line 1) increases with field strength as fast as the average gain G due to the stationary rate of the avalanche process. τ av, ρs Fig. 6. Calculated dependence of avalanche time τ av on the maximal electric field strength E m E m in - E cr, V/cm the avalanche layer for APD (1) and SC ANF (2). In the SC ANF (line 2) overvoltage allows one to increase of the ionization coefficients for electrons and holes, thus the rate of the process also increases, and the same gain is reached for a shorter time. The dependence was calculated from the non-stationary system of differential equations. τ av was defined as the time of the current in pinch decaying e times by its maximal value. The results are in a good agreement with the results of Monte Carlo simulation, that allows one to obtain the pinch life time directly.
8 5. EXPERIMENTAL RESULTS. Investigation of the ANF process was carried out with different types of heterostructures. The simplest one is a uniform MOS structure. It was shown in  that such a structure can operate in self-stabilized avalanche regime, being depleted by a trapezoidal pulse of supplied voltage. The locality of the negative feedback in such a structure is due to the limitations of speed of accumulated charge spreading along interface. So, most of the minority carriers, resulting from the current pinch, are spread over a small area ( 10 μm 2 ) during the life time of the current pinch. Negative feedback is not full in such a structure as not all the carriers take part in the field screening, but even the partial feedback enables operation with overvoltage. Avalanche MOS structure appears to be the best object for SC ANF process modeling and experimental investigation 1. Large-area and multi-element avalanche MOS structures were manufactured and they exhibited both high gain and sensitivity. Though these structures operate only with a pulse supplied voltage, they are much less expensive than APD, and may be used in some applications. ANF structures operating with a constant supplied voltage, were obtained on the base of SiC-Si heterostructure with a special profiling of Si substrate. It may be imagined as consisting of small (a few square micrometers) cells operating independently, and producing summarized output signal. When ANF structure is illuminated by a light pulse, each photon initiates the self-calibrated SC ANF process in a separate cell resulting in the charge package at the Si-SiC heterobarrier. This charge, in turn, induces charging of a metallic electrode, which is common for all cells. As the area of each cell is small, it is hardly probable that several photons may hit in the same cell simultaneously. So, a momentary charge on an electrode is proportional to the number of photons falling on the structure during the typical time of a cell relaxation τ rel. This time depends on the properties of SiC layer, and means the time of charge flowing through it. If such a structure, operating in a few-photon pulse registration mode, is incidentally illuminated by an intensive light pulse, no damage occurs, as it would be in the case of PMT or ANF. As the cells do not have time to relax between the avalanche events, inversion charge is accumulated all over the structure area, screens the field thus, decreasing the average gain. One can say average gain at high illumination is tuned automatically in accordance with light intensity. Experimental measuring of gain noise factor on such a structure was carried out in . The results of that study confirm its lower, than in the APD, excessive noise. As shown above, distribution of the gains provides the most full information about the structure internal amplifier. This distribution was measured on SiC-Si heterostructure experimentally. During the experiment, the pulses resulting from dark carriers amplification, were measured by a charge-sensitive preamplifier, followed by a waveform recorder. The discrimination level of the scheme was close to 1000 electrons. All the pulses, exceeding this level were detected, and their gains were measured. Probability that a pulse is resulted from more than one initial carrier was estimated to be nearly Accumulated charge spreading along interface may be blocked, for example, by introducing of a potential relief over the structure area.
9 The obtained gain distribution is shown in Fig. 6 both with the same distribution calculated for APD for the same average gain. The distribution was not normalized due to unknown share of realizations with gains, that are lower than discrimination level. It is seen from the figure, that the peak on experimental distribution really presents, and this confirms the fact of gain self-calibration in ANF structures. Small peaks, other than the main one, are not essential as they are probably due to the structure s non-uniformity. This means some cells are operating at another voltage than most of the cells have. In the simple laboratory specimen used for the experiment, a non-uniformity of the SiC layer properties is quite possible. N Gain G Fig. 6. Experimental distribution of gain G (solid line 2). N is number of realizations with gain G. Dashed line (1) corresponds to calculated distribution which would take place in APD for the same average gain. The following parameters were estimated for this distribution: 35 percents of all realizations are in the main peak, gain Gav=7400, noise factor F=1.26. And the value of the noise factor calculated for APD operating with the same average gain, exceeds two hundred. So, even the simplest ANF structure has exhibited much higher gain and lower excess noise, than is theoretically possible for APD. The second experiment, carried out on the same structure illustrates this result. The output signal from the structure illuminated by a few-photon light pulse, was recorded after amplification by the charge-sensitive preamplifier. A horizontal axis corresponds to time, and a vertical one, to the number of electrons at the input of preamplifier. Two vertical dashed lines correspond to the moments of light turning on and off. N q 10 3 Light pulse Fig. 8. Experimentally recorded on SiC-Si structure light pulse, resulted in several photoelectrons. Vertical axis corresponds to number of charges Nq at the input of charge sensitive amplifier, following the structure. Time t, Each of the detected photocarriers result in some additional charge at the input of the preamplifier. As the amplification time is much less, than the waveform recorder resolution (50 ns), only vertical steps correspond to each photoelectron-initiated avalanche event. The amplitude of a step is proportional to the number of the carriers, resulting from this event (i.e., gain). A slow decay after a sharp rise is caused by the preamplifier feedback time, and is not related to the structure properties. As the signal amplification parameters correspond to the values calculated from the gain distribution, this experiment is only a visualization of the previous one.
10 The experiment exhibits the importance of a low noise-factor of the ANF structure that makes the resulting signal to be proportional to the light pulse amplitude. It may be also mentioned that variations of gain in APD would be hundred times larger for the same average gain. 6. DISCUSSION As shown above, the SC ANF process is realized in the avalanche structure with negative feedback at the electric field strength, exceeding the critical one for a conventional p-n junction. The main features of this process are so distinguished from those of p-n junction, that one can say about a new generation of avalanche devices. Let us overview some hot points of this trend. An ideal amplifier is the main idea, following directly from the physics of SC ANF process. It means that the every signal carrier gives rise after amplification nearly the same charge package with low fluctuations. The output packages from different charge carriers are summarized, and give the output signal that is proportional to the input one. A solid state analog of a vacuum-tube photomultiplier must have three main features: high gain, low noise and small response time. It seems that the SC ANF photodetector is able to satisfy these requirements. As a solid state device, the SC ANF detector has many advantages, and the first one is its higher quantum efficiency. A few-photon pulse analog registrator is a hypothetical device, which may be designed on the basis of the SC ANF process. The requirements for such a device are even more strict, as it have to provide also a high detection probability for each photocarrier. Designing of a high-sensitive multi-element solid state videosensor is a problem, that may be solved radically by using the internal amplification. A weak sensitivity of the SC ANF process to the voltage fluctuations and spatial inhomogeneities, as well as its high gain and low noise factor, makes it promising in this area too. And, finally, the avalanche devices based on different kinds of semiconductor materials may be designed due to a weak dependence of the SC ANF process features on the relationship of ionization coefficients for electrons and holes. This would allow one to manufacture avalanche photosensors for different urgent spectral ranges. 7. CONCLUSION A concept of a novel solid state photosensor with an internal amplification based on the avalanche process with the negative feedback was considered in the paper. It is shown that the advantages of such a device follow from the physics of a single-photocarrierinitiated avalanche process developed under the influence of feedback. Numeric simulation confirms the high gain, low noise factor, and small response time of the SC ANF-based device, following from its qualitative model. Experimental results obtained on the SC ANF-based structure exhibit its promising as a prototype of a new generation of avalanche photosensors and, in particular, solid state photomultipliers.
11 These devices may be used for a wide range of applications, such as systems of optical location and communication. The ability of multi-element construction makes them very promising as high-sensitive videosensors, which may be used in different areas from medicine to a space remote sensing. 8. ACKNOWLEDGMENTS This research is supported by the Russian Basic Research Foundation. Project REFERENCES 1. R. J. McIntyre. The distribution of gain in uniformly multiplying avalanche photodiode, IEEE Trans. Electron Devices., Vol. ED-19, pp S. V. Bogdanov, V. E. Shubin, D. A. Shushakov. The influence of electric field non-stationatity on dinamics of a single charge carrier multiplication, Kratkie soobshenija po physike, Vol. 5/6, pp. 3-10, (Bullettin of the Lebedev Physical Institute, Allerton Press Inc.) 3. S. V. Bogdanov, A. B. Kravchenko, A. F. Plotnikov, and V. E. Shubin. Model of avalanche multiplication in MIS structures, Phys. Stat. Sol. (a) Vol. 93, pp , T. M. Burbaev, V. V. Kravchenko, V. A. Kurbatov, V. E. Shubin. Noise of avalanche photodetector, basing on Si-SiO 2 -Ni structure, Kratkie soobshenija po physike, Vol. 4, pp , (Bullettin of the Lebedev Physical Institute, Allerton Press Inc.)