Abstract:
A simulation of a multiplication process includes tracing histories of a plurality of carriers, increasing a weight factor of a carrier to simulate a multiplication of the carrier, and summing the number of the plurality of carriers. Each of the plurality of carriers is multiplied by its respective weight factor.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority of U.S. Provisional Application No. 61/028,475, filed on Feb. 13, 2008, the disclosure of which is hereby incorporated by reference in its entirety. 
    
    
     BACKGROUND OF INVENTION 
     Many systems involve multiplication processes. For example, population grows through reproduction; monetary investment grows through gains such as stock market gain. In another example, avalanche photodiodes (APDs) operate by multiplying carriers (e.g., electrons and holes) being accelerated by electric fields. 
     APDs are particularly useful for photon counting, which finds applications in remote sensing, optical communication encryption, astronomy, ballistic missile defense, and ladar applications. 
     APDs can be operated in the Geiger mode for photon counting. A Geiger-mode APD is biased above its breakdown voltage such that a majority of the carriers (electrons and holes) continue to impact ionize in a runaway fashion, until an external circuit quenches the otherwise infinitely increasing gain. The Geiger mode APDs have may high dark currents (counts), and thus can be more susceptible to space radiations. 
     It would be useful to simulate Geiger-mode APDs to predict their behaviors such as breakdown (runaway) probabilities as functions of biases. Monte Carlo simulations can potentially provide more insights into the breakdown behaviors of APDs than analytical models, and may be used in designing optimal APD structures. However, conventional Monte Carlo simulations trace every carrier throughout their transport and impact ionization processes. In the Geiger mode, the high gains of the carriers make conventional Monte Carlo simulations computationally prohibitive. 
     SUMMARY OF INVENTION 
     In one aspect, embodiments disclosed herein relate to methods, computer readable medium, computer software, and computer systems for simulating a multiplication process, including tracing histories of a plurality of carriers, increasing a weight factor of a carrier to simulate a multiplication of the carrier; and summing the number of the plurality of carriers, wherein each of the plurality of carriers is multiplied by its respective weight factor. 
     In one embodiment, the multiplication process includes an impact ionization process in an avalanche photodiode (APD), and wherein the plurality of carriers include electrons and holes. 
     Following an impact ionization event caused by an electron, a new electron and a new hole can be simulated, wherein the new electron can be given a weight factor twice that of the electron. 
     Following an impact ionization event caused by a hole, a new hole and a new electron can be simulated, wherein the new hole can be given a weight factor twice that of the hole. 
     A breakdown voltage can be defined for the APD, and a breakdown probability of the APD can be calculated based on the defined breakdown voltage. The breakdown voltage can be defined based on one of a predetermined bias for the APD or a predetermined gain for an individual carrier. 
     An electrical current from the APD can be simulated, and a filter can be applied to the electrical current to determine a single photo detection probability. The filter can be determined from, for example a resolution of a measurement system, and the resolution can include at least one of an amplitude resolution or a frequency resolution of the measurement system. 
     Other aspects and advantages of the invention will become apparent from the following description and the attached claims. 
    
    
     
       BRIEF SUMMARY OF THE DRAWINGS 
         FIG. 1  illustrates simulated current pulses resulting from 10 photons received by an APD biased to an average gain of about 50; 
         FIG. 2  illustrates a method of reducing the number of carriers being simulated; 
         FIG. 3  shows a simulated single photon detection probability; 
         FIG. 4  illustrates gain curves for three APDs simulated using a method in accordance with an embodiment of the invention; and 
         FIG. 5  illustrates simulated breakdown probabilities of the three APDs. 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments of the present invention are described in detail below with respect to the drawings. Like reference numbers are used to denote like parts throughout for consistency. 
     Although the methods and systems are described below using a simulation of an APD as an example, the methods and systems can be applied to other simulations such as those of a population study, a finance analysis, an economics model, etc. In these simulations, individual “carriers,” e.g., an entity, a stock share, an electron, an individual, etc, are traced throughout their “histories,” such as birth, growth, multiplication, etc. 
       FIG. 1  shows simulated current pulses  10  resulting from an APD biased to an average gain of about 50, i.e., the APD is still in its linear mode. The APD has a thin (˜100 nm) InAlAs-based impact ionization region. A total of 10 photons are simulated to be absorbed a thick (˜1 μm) InGaAs absorption region. The photon arrival time follows a Poisson distribution. As shown, the pulses resulting from the individual photos have a large range of amplitudes and durations, an expected behavior for APDs operating in linear mode. 
     The electrical current in the APD is calculated using the well known Ramo&#39;s theorem, which states that the total current I(t)=Σ i q i v i /d, where q i  is the electrical charges of electrons or holes, v i  the carriers&#39; instantaneous velocity, and d is the distance between the p and n layers. The sum is over all carriers in the depletion region, including impact-ionization-generated carriers. 
     Using the simulated current pulses, in conjunction with known amplitude and frequency resolutions of measurement systems, single photo detection probability can be accurately predicted for a given APD in a given measurement system. Various quenching mechanisms can also be included in the Monte Carlo model. For example, when the current amplitude reaches a predetermined threshold value as determined from laboratory settings, the bias over the APD can be tuned below the breakdown voltage, while electrons and holes are continuously being traced in the simulations. 
     To more practically simulate APDs operating in the Geiger mode, the total number of carriers can be reduced accepting some sacrifice of statistical accuracy. It is noted that because of the high gains (&gt;10 4 ) of individual carriers, the total number of output carriers is enormous, making carrier number statistics less of an issue. 
     Some known techniques can be applied to simulations to reduce the total number of carriers. For example, in a technique known as the “Russian roulette,” particles moving away from the region of interest are “killed” at a certain probability. If a particle “survives,” its weight is increased by a factor inversely proportional to the kill probability. When applied appropriately, Russian roulette leaves Monte Carlo simulation unbiased while reducing the computing time. 
     In one embodiment as illustrated in  FIG. 2 , in a depletion region  20 , of an APD, when an electron (open circle)  22  causes an impact ionization (multiplication), instead of generating two electrons and one hole (solid circle) as in conventional Monte Carlo simulations, only one electron  24  and one hole  26  are generated. The electron  24  is given a “weight” twice its earlier value (double-sized circle). The weight is carried along in later impact ionizations. The weights of the carriers are applied to the output carrier count and the current, and when summing all the output carriers, each carrier being summed includes its weight accumulated throughout its transport and multiplication history. Thus, the total number of carriers being simulated is drastically reduced, particularly at high gains. 
     The methods in accordance with embodiments described herein make Monte Carlo simulations of Geiger-mode APDs more practical.  FIG. 3  illustrates one example of a simulated single-photon detection probability. In the simulation, a total of 100 photons are absorbed in the absorption region of the APD, and the photons follow a Poisson distribution in time. 
     The APD is biased above its breakdown voltage, V br , by a factor of (V−V br )/V br . The V br  is set to be, for example, the bias for the APD to reach an average gain of 100. In the following example, a cap for the gains of individual carriers is set to be g th =50,000, i.e., when a photoelectron reaches a gain of g th , the simulation stops, and the photon is considered “detected” by the single-photon detection system. 
     The scatters in the simulated data points reflect the stochastic nature of the APD and the simulation itself. The general behavior of the detection probability curve is consistent with measured data. This simulation only takes about 10 minutes on a personal computer. It is conceivable that conventional Monte Carlo simulations would take much longer. For example, for each of the data points  30  in  FIG. 3 , a total number of 5 million electrons and a similar number of holes would need to be traced when the detection probability is close to 1. 
     In other embodiments, g th  can be set even higher, e.g., 1 million. Current pulses similar to those illustrated in  FIG. 1  can be used in conjunction with a given experimental setup including a quenching circuit to simulate the detection probability. Based on the experimental setup, external current quenching can be applied to the simulations, and the photon detection probability can be simulated using a “filter” with known amplitude and frequency resolutions. Such a filter can be applied to the simulated current pulses to calculate the APD breakdown probability, or single-photon detection probability, specific to the APD and specific to the measurement set up. 
       FIG. 4  illustrates gain curves  40 ,  42 ,  44  for three APDs simulated using a method in accordance with an embodiment of the invention. The three APDs have InAlAs depletion regions with thicknesses of 140 nm, 500 nm, and 1100 nm, respectively.  FIG. 5  illustrates the breakdown probabilities  50 ,  52 ,  55  for the three APDs. 
     The simulations each are based on absorption of 1000 photons. In the embodiment used to generate  FIG. 3 , V br  is defined by all average gain. In the embodiment in connection with  FIGS. 4 and 5 , a cap of max gain for an individual carrier is set to be 500,000. In this cases, V br  is defined by the voltage at which at least one single carrier reaches the max gain of 500,000. The average gain for each thereby-defined V br  is about 800. 
     In the embodiments described above, the total number of carriers being traced in the simulation can be reduced significantly, thus reducing simulation time without biasing simulation results. With an experimental setup having known quenching circuits and current measurement accuracies, Monte Carlo simulations can help understanding of breakdown behaviors of APDs. Such an intuitive understanding can help optimize designs of APD structures for Geiger-mode operations. 
     While the invention has been described with respect to a limited number of embodiments, those skilled in the art, having benefit of this disclosure, will appreciate that other embodiments can be advised which do not depart from the scope of the invention as disclosed herein. 
     For example, although examples are described with respect to simulations of APDs, other simulations can employ the methods in accordance with embodiments of the invention. For example, in the simulation of population growth, instead of tracing each individual, the total number of simulated individuals can be reduced by simulating an individual with an increased weight factor instead of the individual reproducing one or more individuals. 
     In simulating a market, monetary gains can be simulated by an increased weight instead of increased investment entities or seeds, or shares of stocks. 
     It is noted that although in the examples described with respect to APDs, the multiplication factor for an individual carrier is in integer, in simulating other systems, non-integers can be used.