Modeling Temperature-Dependent Avalanche Characteristics of InP

Avalanche photodiodes (APDs), and single photon avalanche diodes (SPADs), with InP avalanche regions and InGaAs absorption regions, are used for detecting weak infrared light at ∼1.55 μm wavelength. These devices are often cooled to below room temperature during operation yet both validated simulation models and impact ionization coefficients that accurately describe the avalanche characteristics of InP are lacking in the temperature range of interest (200 K to room temperature). In this article we present an accessible, validated temperature dependent simulation model for InP APDs/SPADs. The model is capable of simulating avalanche gain, excess noise, breakdown voltage, and impulse current at 150–300 K. Temperature dependent ionization coefficients in InP, which may be used with other APD/SPAD simulation models, are also presented. The data reported in this article is available from the ORDA digital repository (DOI: 10.15131/shef.data.c.4373006).


I. INTRODUCTION
L IGHT detection and ranging (LiDAR) systems [1] for longrange three-dimensional imaging are increasingly important in applications such as autonomous vehicles, surveillance, remote sensing, and gas detection. The majority of LiDAR systems use a silicon Avalanche Photodiode (APD) or a silicon Single Photon Avalanche Diode (SPAD) to detect the weak optical signals. The operation wavelength for these Si detectors is up to ∼1 µm, which has several drawbacks for long-range LiDAR since optical signals at these wavelengths are attenuated heavily by atmospheric obscurants (e.g., smoke, dust, and water vapor), and are not eye-safe. The level of signal attenuation may be significantly reduced by increasing the operating wavelength to 1.55 µm and optical signals at 1.55 µm can also have a ∼10 5 times greater energy c.f. 633 nm whilst still remaining eye-safe (assuming 1 ns -10 µs pulse duration) [2]. To detect 1.55 µm wavelength optical signals, the Si detector must be replaced Manuscript  by an appropriate alternative, such as In 0.53 Ga 0.47 As/InP APD or SPAD. In LiDAR applications, In 0.53 Ga 0.47 As/InP APDs or SPADs are often cooled to below room temperature (e.g., 200-240 K, [3], [4]) to reduce reverse leakage currents or dark counts respectively, to improve signal-to-noise performance.
In addition to LiDAR systems, In 0.53 Ga 0.47 As/InP SPADs are used in receiver modules of Quantum Key Distribution (QKD) systems [5] and they are similarly cooled during operation. In In 0.53 Ga 0.47 As/InP APDs and SPADs, many of the important characteristics are determined by the impact ionization process taking place in the devices' InP avalanche region. Since In 0.53 Ga 0.47 As/InP APDs and SPADs are cooled in operation, accurate simulation models for the impact ionization process at 200 K to room temperature capable of simulating carrier distribution, and scattering processes, will assist the optimization of In 0.53 Ga 0.47 As/InP APDs and SPADs.
Data from InP impact ionization at low temperatures is more limited. Reported work, [13]- [15], has relied on diffused junctions to create their InP diodes, so are likely to suffer from significant uncertainties in the electric field values. Crucially, ionization coefficients were not presented in [13]. Independent efforts to extract the ionization coefficients from their presented experimental results would be difficult due to incomplete knowledge of the doping profiles of the measured diodes. Temperature dependent ionization coefficients for InP were reported in [14]. However, their values for room temperature are significantly lower than other reports [6]- [8], probably due to further uncertainties arising from using only a single InP diode structure and incorrect experimental values of avalanche gain data (avalanche gain values from mixed carrier injection were assumed as values from pure hole injection in [14]).
Several simulation models have been reported for temperature dependence of breakdown voltage of In 0.53 Ga 0.47 As/InP APDs [12], [16]- [18], but most did not present rigorous validation. All, except [18], were validated against only a single In 0.53 Ga 0.47 As/InP APD design. A Simple Monte Carlo (SMC) model for the impact ionization process, first reported in 1999 [19], has recently been made available [20], with parameter files published for a range of avalanche materials including Si [21]. The SMC model is far less computationally intensive compared to Analytical and Full Band Monte Carlo models [22], [23], whilst incorporating sufficient impact ionization statistics (including dead space effects) to simulate a wide range of APD/SPAD designs, e.g., with thin avalanche regions and/or rapidly varying electric field profiles.
In this paper we present an SMC parameter set for InP at 150 to 300 K, for use with our SMC simulator [20]. We comprehensively validated against a range of experimental data, including material characteristics such as saturation velocities, impact ionization coefficients for electrons and holes (α and β), as well as device characteristics such as M(V) and F(M)f r o m multiple APD structures. In addition, we present temperature dependent impact ionization coefficients of InP that can be used with other models requiring ionization coefficients as inputs.

II. MODEL
The SMC model has been extensively described in previous publications [19], [21] so only a brief description will be included here with an emphasis on the temperature dependence [24]. In the SMC model, a free carrier drifts in the depletion region for a random distance before undergoing one of three mechanisms: intervalley phonon absorption; intervalley phonon emission; or impact ionization. After the free carrier has undergone one of these mechanisms, a new random scattering direction is chosen to update the momentum of the carrier. The intervalley phonon emission and intervalley phonon absorption rates, R em and R ab ,are where T is the temperature, N(T) is the temperature dependent phonon occupation factor, λ(T) is the temperature dependent mean free path between scattering events, E is the energy of the carrier, ℏω is the mean phonon energy, and m * is the effective mass of the carrier. N(T)isgivenby where k is Boltzmann's constant. Since λ ∼ (1 + 2N ) −1 the temperature dependent mean free path takes the form of where N(300 K) and λ(300 K) are the phonon occupation factor and mean free path respectively at 300 K. Within this model, temperature dependence is described by N(T) and λ(T). The other SMC parameters are either known to change very slowly with temperature or are assumed to be temperature independent. The rate of impact ionization, R ii , is calculated using the Keldysh equation [25], where γ is the softness factor of impact ionization, C ii is the prefactor of impact ionization rate, and E th is the threshold energy of impact ionization. Ramo's theorem [26] is used to calculate the instantaneous current from the movement of the carriers within the device, which make up the impulse current. Avalanche gain from the injection of a single carrier (electron or hole) is given by the area under the impulse current versus time characteristics. The impulse current versus time data can also be used to calculate theoretical response times and bandwidths.
Examples of average impulse current characteristics from the model are shown in Fig. 1. The impact ionization coefficient for a given electric field, is determined by the inverse of the mean distance travelled between 20000 consecutive impact ionization events. When simulating a given device design, the electric field profile at a given reverse bias was calculated using a 1D Poisson field solver.

III. MODEL VALIDATION
A value of 42 meV was used for ℏω, based on the longitudinal optical phonon frequency of 345 cm −1 in InP [27]. The effective masses of the carriers were adjusted so that the saturation velocities at 300 K produced by the SMC model matched the reported values of 6.8 × 10 6 and 7.0 × 10 6 cm.s −1 for electrons and holes respectively [28], [29], as shown in Fig. 2(a). Values of C ii and λ(300 K) were adjusted so that the model produces α(ζ) and β(ζ) in line with [6]- [8], as shown in Fig. 2(b).
Further adjustments to C ii and λ(300 K) values relied on comparisons between simulated and reported M(V) and F(M) characteristics for six InP APDs at 300 K [8]. These are devices A-F, whose structural details are summarized in Table I. The doping information was deduced from fitting room temperature capacitance-voltage data of the devices (multiple different-sized Fig. 2. Comparisons of (a) drift velocities and (b) impact ionization coefficients at room temperature of the InP SMC model (symbols) with reference data (lines), which are taken from [28,29] for (a) and [6,7,8] for (b).

TABLE I STRUCTURE DETAILS OF DEVICES USED IN MODEL VALIDATION AT ROOM TEMPERATURE (A-F) AND LOW TEMPERATURES (C-G)
diodes for each device). The unintentional doping in all structures, except device E, is n-type. The simulated M(V) and F(M) characteristics are in good agreement with data from [8], as shown in Fig. 3(a) and (b). The resultant InP SMC model parameter set is summarized in Table II.
Validation of the InP SMC model at lower temperatures consisted of simulations of M(V) characteristics of devices C-G at 290, 250, 200, 150, 100, and 77 K. These simulations are compared to data from [18] in Fig. 4. Simulated and experimental data for all devices are in agreement for temperatures  from 290 K down to 150 K. As temperature decreases below 150 K, there is growing discrepancy between the simulations and experimental data, particularly for the devices with wider avalanche regions. Hence the model is deemed to be valid from 290 K to 150 K. The discrepancy observed for comparisons at 100 and 77 K could be caused by insufficient knowledge of doping profiles for temperatures <150 K where there is a lower level of dopant activation.
where d e and d h are the electron and hole deadspaces respectively (where deadspace is defined as the minimum distance an electron or hole can travel before it can undergo impact ionization) and x is the carrier path length. Fitting the ionization path length PDFs to h e (x) and h h (x) allowed us to extract α * (E)andβ * (E) in the form for each temperature, with values of fitting parameters A f , B f , C f summarized in Table III.
Using the provided expressions for α * (E) and β * (E) with threshold energies of 2.8 eV for electrons and 3.0 eV for holes [8], allowed for the temperature dependent gain simulation of devices C-G using the recurrence equations [30], as shown in Figs. S1-4. A comparison of room temperature F(M)i sa l s o presented in Fig. S5.

V. C ONCLUSION
We have presented a simulation model for temperature dependent InP avalanche characteristics for 150-300 K. This model has been validated against experimental results of avalanche gain and excess noise factor from InP APDs at 300 K, with avalanche widths from 0.125-2.5 µm, along with experimental avalanche gain results for InP APDs at temperatures from 150-300 K and with avalanche widths of 0.125-1.63 µm. This validated model has subsequently been used to calculate the temperature dependent impact ionization coefficients of InP between 150-290 K.