Charge Carrier Lifetime & Recombination

Allpix Squared provides the possibility to simulate finite lifetimes of charge carriers as a function of the local doping concentration via non-radiative recombination processes. While most of these models require the total doping concentration $N_D + N_A$ as parameter, the doping profile used throughout Allpix Squared provides the effective doping concentration $N_D - N_A$ since this also encodes the majority charge carriers via its sign - an information relevant to some of the models. However, in the parts of a silicon detector relevant for this simulation, i.e. the sensing volume, the difference between effective and total concentration is expected to be negligible. Therefore the two values are treated as equivalent throughout the lifetime models and the doping concentration is taken as the absolute value $N = \left|N_D - N_A\right|$.

Whether a charge carrier has recombined with the lattice is calculated for every step of the simulation using the relation

$$p < 1 - e^{- dt / \tau(N)}$$

where $p$ is a recombination probability, drawn from a uniform distribution with $[0, 1]$, $dt$ is the last time step of the charge carrier motion and $\tau$ the lifetime for the local doping concentration calculated by the models described in the following. If the above equation evaluates to false, the charge carrier still exists, if it evaluates to true it has been recombined with the lattice.

Finite charge carrier lifetime can be simulated by all propagation modules and comprise the following models:

Shockley-Read-Hall Recombination

This model describes the finite lifetime based on Shockley-Read-Hall or trap-assisted recombination of charge carriers with the lattice [@shockley-read, @hall]. The lifetime is calculated using the Shockley-Read-Hall relation as given by [@fossum-lee]:

$$\tau(N) = \frac{\tau_0}{1 + \frac{N}{N_{d0}}}$$

where $\tau_0$ and $N_{d0}$ are reference lifetime and doping concentration, for electrons and holes respectively. The parameter values implemented in Allpix Squared are taken from [@fossum-lee] and the Synopsys Sentaurus TCAD software manual as

$$\begin{aligned} \tau_{0,e} &= 1\times 10^{-5} \,\text{s} \\ N_{d0,e} &= 1\times 10^{16} \,\text{cm}^{-3} \\ \\ \tau_{0,h} &= 4.0\times 10^{-4} \,\text{s} \\ N_{d0,h} &= 7.1\times 10^{15} \,\text{cm}^{-3} \end{aligned}$$

for electrons and holes, respectively.

The temperature dependence of the Shockley-Read-Hall lifetime is scaled following the low-temperature approximation model presented [@schenk] as:

$$\tau(N, T) = \tau(N) \cdot \left( \frac{300 \,\text{K}}{T} \right)^{3/2}$$

This model can be selected in the configuration file via the parameter recombination_model = "srh".

Auger Recombination

At high doping levels exceeding $5\times 10^{18} \,\text{cm}^{-3}$ [@fossum-lee], the Auger recombination model becomes increasingly important. It assumes that the excess energy created by electron-hole recombinations is transferred to another electron (e-e-h process) or another hole (e-h-h process). The total recombination rate is then given by [@kerr]:

$$R_{Auger} = C_n n^2p + C_p n p^2$$

where $C_n$ and $C_p$ are the Auger coefficients. The first term corresponds to the e-e-h process and the second term to the e-h-h process. In highly-doped silicon, the Auger lifetime for minority charge carriers can be written as:

$$\tau(N) = \frac{1}{C_{a} \cdot N^2}$$

where $C_{a} = C_{n} + C_{p}$ is the ambipolar Auger coefficient, taken as $C_{a} = 3.8\times 10^{-31} \,\text{cm}^6\,\text{s}^{-1}$ from [@dziewior].

This recombination mode applies to minority charge carriers only, majority charge carriers have an infinite life time under this model and the recombination equation will always evaluate to true.

This model can be selected in the configuration file via the parameter recombination_model = "auger".

Combined SRH/Auger Recombination

This model combines the charge carrier recombination from the Shockley-Read-Hall and the Auger model by inversely summing the individual lifetimes calculated by the models via

$$\begin{aligned} \tau^{-1}(N) &= \tau_{srh}^{-1}(N) + \tau_{a}^{-1}(N) &\quad \text{for minority charge carriers} \\ &= \tau_{srh}^{-1}(N) &\quad \text{for majority charge carriers} \end{aligned}$$

where $\tau_{srh}(N)$ is the Shockley-Read-Hall and $\tau_{a}(N)$ the Auger lifetime. The latter is only taken into account for minority charge carriers.

This model can be selected in the configuration file via the parameter recombination_model = "srh_auger".

Recombination with Constant Lifetimes

Some materials require constant lifetimes for charge carriers $\tau(N) = \tau$. This model requires the additional configuration keys lifetime_electron and lifetime_hole to be present in the module configuration section, for example:

# Constant lifetimes for electrons and holes in GaAs with Cr compensation:
recombination_model = "constant"
lifetime_electron = 30ns
lifetime_hole = 4.5ns

This model can be selected in the configuration file via the parameter recombination_model = "constant".

Custom Recombination Models

Allpix Squared provides the possibility to use fully custom recombination models. In order to use a custom model, the parameter recombination_model = "custom" needs to be set in the configuration file. Additionally, the following configuration keys have to be provided:

lifetime_function_electrons: The formula describing the electron lifetime.
lifetime_function_holes: The formula describing the hole lifetime.

The functions defined via these parameters can depend on the local doping concentration. In order to use the doping concentration in the formula, an x has to be placed at the respective position.

Parameters of the functions can either be placed directly in the formulas in framework-internal units, or provided separately as arrays via the lifetime_parameters_electrons and lifetime_parameters_electrons. Placeholders for parameters in the formula are denoted with squared brackets and a parameter number, for example [0] for the first parameter provided. Parameters specified separately from the formula can contain units which will be interpreted automatically.

Warning

Parameters directly placed in the recombination formula have to be supplied in framework-internal units since the function will be evaluated with the doping concentration in internal units. It is recommended to use the possibility of separately configuring the parameters and to make use of units to avoid conversion mistakes.

The following set of parameters re-implements the Shockley-Read-Hall recombination model using a custom recombination model. The lifetimes are calculated at a fixed temperature of 293 Kelvin.

# Replicating the Shockley-Read-Hall model at T = 293K
recombination_model = "custom"

lifetime_function_electrons = "[0]/(1 + x / [1])"
lifetime_parameters_electrons = 1.036e-5s, 1e16/cm/cm/cm

lifetime_function_holes = "[0]/(1 + x / [1])"
lifetime_parameters_holes = 4.144e-4s, 7.1e15/cm/cm/cm

Warning

It should be noted that the temperature passed via the module configuration is not evaluated for the custom recombination model, but the model parameters need to be manually adjusted to the required temperature.

The interpretation of the custom recombination functions is based on the ROOT::TFormula class [[@rootformula]] and supports all corresponding features, mathematical expressions and constants.