# Geiger Mode Simulation of Avalanche Photodiodes in ATLAS

Single-photon counting detectors are used in a wide range of applications, including astronomy, optical communications, biological sensors, and military uses. Photomultiplier tubes (PMTs) have fulfilled these needs in the past. Now, avalanche photodiodes (APDs) operating in Geiger mode as single-photon counters offer advantages over PMTs which include higher quantum efficiency, smaller size, and lower breakdown voltages. Geiger mode APDs have demonstrated single-photon sensitivites from the far infrared to the deep ultraviolet wavelengths.

In Geiger mode operation, an APD will be pulsed to a reverse bias beyond its avalanche breakdown voltage and held there for a while. If that voltage is maintained, eventually an avalanche breakdown will occur, possibly due to thermal generation of carriers, stray carriers from crystalline defects, or traps. If, before a spontaneous breakdown occurs, a photon creates an electron-hole pair in the device, then one or both of the created carriers may pass through the high-field region of the APD, gaining enough energy to cause a self-sustaining avalanche breakdown. The probability that an avalanche will occur will depend on the strength of the field and the initial location of the charged carrier.

Simulating an APD would ideally be carried out using a full-band Monte Carlo (MC) device model that incorporates realistic band structures for each APD geometry. Unfortunately, these MC simulations are very compute-intensive. MC models with simplified carrier transport can be more practical, but the calculation of avalanche initiation probablities would imply many reruns of the MC software, expecially for 2- or 3-dimensional APD designs.

While they cannot simulate an individual avalanche directly, ATLAS S-Pisces and Blaze can calculate the avalanche initiation probability as a post-processing step in an approach suggested by McIntyre[1]. The calculation method is similar to the method of ionization integrals used to determine breakdown voltage, in which ionization rates are integrated along electric field lines from anode to cathode. In this case, however, the integrals along the field lines involve the probabilities of avalanche initiation and are used to solve the following set of equations:

where s(x,y) is the distance along the field line, P_{e}(x,y) is the probability that an electron generated at (x,y) will initiate an avalanche, P_{h}(x,y) is the probability that a hole generated at (x,y) will cause an avalanche, and P_{p}(x,y) is the joint probability that an electron-hole pair generated at (x,y) will initiate an avalanche. α_{e} and α_{h} are the electron and hole ionization rates.

**Geiger Mode Simulation of a One-dimensional APD**

This one dimensional reach-through APD is an example of an APD with separate absorption and multiplication regions and is taken from the optoelectronics example 18. The absorption region is InGaAs and the multiplication region is silicon. In operation, photogenerated electrons from the InGaAs diode will enter the silicon multiplication region where they may initiate an avalanche.

Figure 1. Doping profiles in the one-dimensional APD. |

Since a Geiger-mode APD will be operated beyond the breakdown voltage, it is important to find that BV accurately. A full simulation of this APD with Poisson’s equation and the carrier continuity equations with impact ionization turned on shows a breakdown voltage of about 22 volts. At breakdown, the lightly doped region of the silicon diode has a high field (peak at 5.7e5 V/cm) and a high impact ionization rate (peak at 3e25 /(s*cm3)), as shown in Figure 2.

Figure 2. Field and impact generation rate in the 1D APD at breakdown. |

Overlaying the dark current and Illuminated current, Figure 3 shows a high signal-to-noise ratio, but the product of the gain and the internal quantum efficiency, as shown in Figure 4, is low when the voltage is below breakdown, but quite high (5e4) above breakdown.

Figure 3. Dark and illuminated IV curves. |

Figure 4. 1D APD IV and gain * IQE. |

The geiger mode simulation in ATLAS allows us to look at the probability that a carrier entering the multiplication region will cause an avalanche. With the Geiger mode turned on, a structure file saved at a bias will contain the electron, hole, and pair probabilities at every point. Figure 5 shows a vertical slice through the APD with the electron and hole probabilities.

Figure 5. Electron and hole avalanche initiation probabilities in the 1D APD. |

If a probe statement specifying the electron avalanche probability at a location in the APD is included, then the log file can display the avalanche probability versus the applied bias voltageas shown in Figure 6.

Figure 6. Electron avalanche initiation probability versus applied bias. |

The designer can use this log file to determine the bias necessary for a specified probability that a photon-generated electron will produce an avalanche event which can be counted.

**Geiger-mode Simulation of a Three-dimensional APD**

A three dimensional, cylindrically symmetric avalanche photodetector, can be represented in two dimensions (radius, depth) in ATLAS. In cylindrical coordinates a silicon APD with a diffused anode might look like Figure 7.

Figure 7. A silicon APD in cylindrical coordinates with a vertical cutline. |

From the simulation of breakdown with impact ionization we find that BV=36.1 V from the following IV curve shown in Figure 8.

Figure 8. Breakdown IV curve of the 2D APD. |

At breakdown, the electric field and impact generation are shown in Figure 9.

Figure 9. Electric field and impact generation rate at breakdown. |

Obviously, the spontaneous avalanche breakdown will occur where the field is highest – at the edge of the active region. The designer can use the geiger mode in Atlas determine the voltage at which a photon-generated carrier will cause an avalanche anywhere in the APD - not just at the edge of the anode. For instance, at 40 volts bias the three avalanche probabilities look like Figure 10.

Figure 10. Electron, hole, and joint electron-hole probabilities for avalanche initiation. |

Although the probability that an electron will cause an avalanche at 40 volts is limited to a small region at the circumference of the anode, any hole which drifts toward the high field region from outside the radius of the anode has about a 35% chance of causing an avalanche, as shown in Figure 11.

Figure 11. Joint electron-hole avalanche probability in 2D and a horizontal slice through the high-field region. |

For the geiger simulations we probed a point near the center of the device during the bias ramp. The log file showing the electron, hole, and joint probabilities for this 3D APD are shown in Figure 12.

Figure 12. Electron, hole, and joint probabilities vs. bias. |

At 70 volts, the joint avalanche probability at the center of the APD approaches 95%.

**Conclusion**

The ATLAS Geiger mode capability avoids the time-consuming, computationally intensive Monte Carlo simulation of individual avalanches of an APD, and instead calculates avalanche initiation probabilities from the electric field solutions as a post-processing step. This capability was developed at a user’s request and is expected to become an indispensible tool for designers of Geiger-mode avalanche photodectors.

**References**

- McIntyre, R., “On the Avalanche Initiation Probability of Avalanche Diodes Above the Breakdown Voltage”, IEEE Trans. Elec. Dev., V. ED-20, N. 7l (July 1973): 637-641.