Generation-Recombination Models

1. Theory

• Definition: Generation-recombination models supply the $G-R$ term of the continuity equations: Shockley-Read-Hall recombination through traps, Auger recombination at high carrier density, impact ionization at high field and band-to-band tunnelling at very high field (Ayalew; Allpix; Ansys).
• SRH: The rate depends on the trap level and on electron and hole lifetimes, which fall with doping and are fitted by Fossum-Lee or Scharfetter forms (Ayalew; Allpix).
• Auger: Three-carrier collisions dominate the lifetime in heavily doped emitters and contacts (Allpix).
• Impact ionization: Hot carriers create electron-hole pairs at a rate $\alpha$ per unit length that rises steeply with field, setting avalanche breakdown (Allpix; Ansys).
• Tunnelling: At fields near $1\times 10^6$ $\mathrm{V}\mathrm{cm}$, electrons tunnel directly from valence to conduction band, causing GIDL and Zener breakdown (Wong et al.; Ansys).

2. Working Principle

• Trap-mediated capture: Traps near midgap capture electrons and holes alternately, so SRH is fastest for midgap levels and limited by the slower capture (Ayalew).
• Lifetime combination: Recombination rates add, so inverse lifetimes add, and the shortest mechanism dominates (Allpix).
• Field heating: Carriers gaining more than the ionization energy between optical-phonon collisions create new pairs; higher temperature means more phonon collisions and less ionization (Allpix; background knowledge for the mechanism).
• Local versus non-local: Local models evaluate rates from the field at each point, while non-local models integrate along tunnelling or acceleration paths, which is more accurate but slower (Wong et al.).

3. Applications

• Bipolar and solar devices: SRH and Auger lifetimes for base transport, emitter saturation current and cell efficiency (Allpix; Ayalew).
• Breakdown design: Avalanche voltage of power devices, ESD structures and avalanche photodiodes (Allpix; Ansys).
• Leakage: GIDL and junction tunnelling leakage in scaled MOSFETs and FinFETs (Wong et al.).
• Detector simulation: Charge trapping and gain in silicon particle detectors (Allpix).

Recombination

Name	Formula	Related Formulae	Variables/Constants
Shockley-Read-Hall recombination	$R_{SRH}=\frac{np-n_{i,e}^2}{{\tau }_p\left(n+n_1\right)+{\tau }_n\left(p+p_1\right)}$	$n_1=N_c\exp \left(\frac{E_T-E_C}{k_B{T}_L}\right),p_1=N_v\exp \left(\frac{E_V-E_T}{k_B{T}_L}\right),{\tau }_{300}=\frac{1}{{\sigma }_T{N}_T{v}_{th}}$	$R_{SRH}:netrecombinationrate$ $n_{i,e}:effectiveintrinsicdensity$ ${\tau }_n,{\tau }_p:electronandholelifetimes$ $n_1,p_1:trap-leveldensities$ $E_T:trapenergy$ $T_L:latticetemperature$ ${\sigma }_T:capturecross-section$ $\begin{array}{r}{N}_T:trapdensity,v_{th}=\sqrt{3k_B{T}_L/m}:thermal\\ velocity;Ayalew:theseequations,with{n}_1\\ =p_1=n_{i,e}andmaximumrateforamidgap\\ trap,andasurfacetermS/yinthe\\ lifetime;Bortolossi:constantlifetimes\\ of1\times 10^{-5}and3\times 10^{-6}sinaTCADtestcase\end{array}$ $(nocomputedvalue)$
Doping-dependent SRH lifetime	$\tau \left(N\right)=\frac{{\tau }_0}{1+N/N_{d0}}$	$\tau \left(N\right)={\tau }_{min}+\frac{{\tau }_{max}-{\tau }_{min}}{1+{\left(N/N_{ref}\right)}^{\gamma }},\tau \left(N,T\right)=\tau \left(N\right){\left(\frac{300\mathrm{K}}{T}\right)}^{3/2}$	${\tau }_0:low-dopinglifetime$ $N_{d0},N_{ref}:referenceconcentrations$ ${\tau }_{min},{\tau }_{max}:limits$ $\begin{array}{r}\gamma :exponent;Allpix:theFossum-Leeform\\ withelectrons{\tau }_0=1\times 10^{-5}s,N_{d0}=1\times 10^{16}\mathrm{c}{\mathrm{m}}^{-3}\\ andholes{\tau }_0=4.0\times 10^{-4}s,N_{d0}=7.1\times 10^{15}\mathrm{c}{\mathrm{m}}^{-3},\\ citingFossumandLeeandtheSentaurus\\ manual,andalow-temperature(300K/T{)}^{3/2}\\ factorafterSchenk;Ayalew:the\\ generalizedScharfetterformwith{\tau }_{min},N_{ref}\\ and\gamma asanempiricalfittosilicondata\\ (Fossum1982and1983),with{\tau }_{max}scalingas\\ (T_L/300{)}^{\alpha }(seetheQ/A)\end{array}$ $\begin{array}{r}(computed:electronlifetimesof9.1\mu \mathrm{s},\\ 0.91\mu \mathrm{s},99\mathrm{ns}and10\mathrm{ns}at1\times 10^{15},1\times 10^{17},\\ 1\times 10^{18}and1\times 10^{19}\mathrm{c}{\mathrm{m}}^{-3},andholelifetimesof\\ 351\mu \mathrm{s},26.5\mu \mathrm{s},2.8\mu \mathrm{s}and0.28\mu \mathrm{s})\end{array}$
Auger recombination	$R_{Auger}=C_n{n}^{2}p+C_{p}n{p}^2$	${\tau }_A=\frac{1}{C_a{N}^2},\frac{1}{\tau }=\frac{1}{{\tau }_{SRH}}+\frac{1}{{\tau }_A}(\text{minority carriers})$	$C_n,C_p:Augercoefficients,C_a=C_n+C_p$ $N:doping$ $\begin{array}{r}{\tau }_A:Augerlifetime;Allpix:R_{Auger}afterKerr,\\ C_a=3.8\times 10^{-31}\mathrm{c}{\mathrm{m}}^6/safterDziewior,\\ importanceabove5\times 10^{18}\mathrm{c}{\mathrm{m}}^{-3}(Fossumand\\ Lee),majoritycarrierswithinfinite\\ Augerlifetime,andthecombinedminority\\ lifetime;Bortolossi:C_n=2.9\times 10^{-31}and{C}_p=\\ 1.028\times 10^{-31}\mathrm{c}{\mathrm{m}}^6/s(seetheQ/A)\end{array}$ $\begin{array}{r}(computed:{\tau }_A=2.6\mu \mathrm{s}at1\times 10^{18},105\mathrm{ns}at\\ 5\times 10^{18}and26\mathrm{ns}at1\times 10^{19}\mathrm{c}{\mathrm{m}}^{-3};combinedwith\\ theelectronSRHlifetime,95,17and7.2\\ \mathrm{ns})\end{array}$

Impact Ionization and Tunnelling

Name	Formula	Related Formulae	Variables/Constants
Impact ionization: Chynoweth-type laws	$\alpha \left(E,T\right)=\gamma \left(T\right)a_{\infty }\exp \left(-\frac{\gamma \left(T\right)b}{E}\right)(\text{Van Overstraeten-De Man})$	$\gamma \left(T\right)=\frac{\tanh \left(\hslash {\omega }_{op}/2k_B{T}_0\right)}{\tanh \left(\hslash {\omega }_{op}/2k_{B}T\right)},\alpha =A{e}^{-B\left(T\right)/E},B=C+DT(\text{Massey})$	$\alpha :ionizationcoefficient(percm)$ $E:fieldalongthecurrent,a_{\infty }$ $\begin{array}{r}b:constants,\hslash {\omega }_{op}=0.063\mathrm{eV}:\\ optical-phononenergy,T_0=300\mathrm{K};Allpix:\\ VanOverstraeten-DeManelectrons{a}_{\infty }=\\ 7.03\times 10^5\mathrm{c}{\mathrm{m}}^{-1},b=1.231\times 10^6\mathrm{V}\mathrm{cm},holes\\ 1.582\times 10^6\mathrm{c}{\mathrm{m}}^{-1}and2.036\times 10^6\mathrm{V}\mathrm{cm}below{E}_0=\\ 4.0\times 10^5\mathrm{V}\mathrm{cm}and6.71\times 10^5\mathrm{c}{\mathrm{m}}^{-1}and1.693\times 10^6\\ \mathrm{V}\mathrm{cm}above;MasseyelectronsA=4.43\times 10^5\\ \mathrm{c}{\mathrm{m}}^{-1},C=9.66\times 10^5\mathrm{V}\mathrm{cm},D=499V/(cmK),\\ holes1.13\times 10^6,1.71\times 10^{6}and1.09\times 10^3;\\ RD50-optimizedvariants;Ansys:Selberherr\\ defaultsof7.03\times 10^5\mathrm{c}{\mathrm{m}}^{-1}and1.23\times 10^6\mathrm{V}\mathrm{cm}\\ (electrons)and1.58\times 10^6\mathrm{c}{\mathrm{m}}^{-1}and2.03\times 10^6\\ \mathrm{V}\mathrm{cm}(holes),matchingthelow-fieldVan\\ Overstraetenvalues\end{array}$ $\begin{array}{r}(computedat300\mathrm{K}:VanOverstraeten\\ electrons1.49\times 10^3,1.16\times 10^{4}and5.99\times 10^4\mathrm{c}{\mathrm{m}}^{-1}\\ andholes60,1.79\times 10^{3}and2.27\times 10^4\mathrm{c}{\mathrm{m}}^{-1}at\\ 2\times 10^5,3\times 10^{5}and5\times 10^5\mathrm{V}\mathrm{cm};Massey\\ electrons1.67\times 10^3,1.07\times 10^{4}and4.76\times 10^4\\ \mathrm{c}{\mathrm{m}}^{-1})\end{array}$
Impact ionization: Okuto-Crowell and Bologna	$\alpha \left(E,T\right)=a\left(T\right)E\exp \left[-{\left(\frac{b\left(T\right)}{E}\right)}^2\right](\text{Okuto-Crowell})$	$\alpha =\frac{E}{a\left(T\right)+b\left(T\right)e^{d\left(T\right)/\left(E+c\left(T\right)\right)}}(\text{Bologna})$	$\begin{array}{r}a(T)=a_{300}[1+c(T-T_0)],b(T)=b_{300}[1+d(T\\ -T_0)];Allpix:Okuto-Crowellelectrons{a}_{300}\\ =0.426/V,c=3.05\times 10^{-4},b_{300}=4.81\times 10^5\\ \mathrm{V}\mathrm{cm},d=6.86\times 10^{-4},holes0.243,5.35\times 10^{-4},\\ 6.53\times 10^{5}and5.67\times 10^{-4}(b(T)printedwith\\ a_{300},readas{b}_{300});theBolognamodelwith\\ polynomialandexponentialtemperature\\ terms;multiplicationgain⟨n⟩=exp(\int \alpha \\ dx)\end{array}$ $\begin{array}{r}(computed:Okuto-Crowellelectrons262,\\ 9.77\times 10^{3}and8.44\times 10^4\mathrm{c}{\mathrm{m}}^{-1}at2\times 10^5,3\times 10^{5}and\\ 5\times 10^5\mathrm{V}\mathrm{cm}(seetheQ/A))\end{array}$
Band-to-band tunnelling (Hurkx form)	$G_{BTBT}=AD{\left(\frac{F}{1V/cm}\right)}^P\exp \left(-\frac{B}{F}\right)$	$\text{local model: }F=\left|\vec{E}\right|\text{at each point};\text{non-local: integration along tunnelling paths}$	$G_{BTBT}:tunnellinggenerationrate,A,B$ $P:parameters$ $\begin{array}{r}D:afactoroforderonedependingonthe\\ quasi-Fermilevels(form,background\\ knowledge)\end{array}$ $\begin{array}{r}F:fieldmagnitude;Ansys:Hurkx\\ band-to-bandtunnellingwithA=4\times 10^{14}\\ \mathrm{c}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1},B=1.9\times 10^7\mathrm{V}\mathrm{cm}andexponent2.5,\\ Hurkxtrap-assistedtunnellingwith\\ tunnellingmass0.25,SRH,Augerand\\ radiativerecombination,Selberherrimpact\\ ionization,ap-type1\times 10^{18}\mathrm{c}{\mathrm{m}}^{-3}epitaxial\\ layerwitha1\times 10^{20}\mathrm{c}{\mathrm{m}}^{-3}surfacen-diffusion,\\ divergencenear2\mathrm{V}reversebiaswitha\\ deliberatelyloweredcriticalfield,and\\ junctionedgesof4\mathrm{nm}orless;Wonget\\ al.:thelocalHurkxmodelischeapbut\\ underestimatesleakageatlargebias,and\\ theirmodifiedmodelmatchesnon-local\\ BTBTfordiodeleakageandGIDLwithmore\\ thansix-foldspeed-upina3-DFinFET\end{array}$ $\begin{array}{r}(computed,D=1:2.2\times 10^{12},2.2\times 10^{21}and\\ 1.7\times 10^{26}\mathrm{c}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}at5\times 10^5,1\times 10^{6}and2\times 10^6\\ \mathrm{V}\mathrm{cm})\end{array}$

Important Questions

Ques: How fast does SRH lifetime fall with doping?

Roughly inversely with doping above the reference concentration. Allpix uses the Fossum-Lee form ${\tau }_0/(1+N/N_{d0})$ with electron values of $1\times 10^{-5}$ s and $1\times 10^{16}$ $\mathrm{c}{\mathrm{m}}^{-3}$ and hole values of $4.0\times 10^{-4}$ s and $7.1\times 10^{15}$ $\mathrm{c}{\mathrm{m}}^{-3}$, giving electron lifetimes of 9.1 $\mu \mathrm{s}$ at $1\times 10^{15}$, 0.91 $\mu \mathrm{s}$ at $1\times 10^{17}$, 99 $\mathrm{ns}$ at $1\times 10^{18}$ and 10 $\mathrm{ns}$ at $1\times 10^{19}$ $\mathrm{c}{\mathrm{m}}^{-3}$, and hole lifetimes of 351, 26.5, 2.8 and 0.28 $\mu \mathrm{s}$ (computed). Ayalew describes the generalized Scharfetter form with a minimum lifetime and an exponent, fitted to the same silicon data of Fossum. The fall comes from more defects and dopant-related traps in heavily doped material (background knowledge). Allpix adds a $(300\mathrm{K}/T{)}^{3/2}$ low-temperature factor after Schenk. Reference: Allpix Squared collaboration, ‘Charge Carrier Lifetime & Recombination’, Allpix Squared documentation, CERN (after Fossum and Lee 1982 and 1983, Schenk 1992, Kerr 2002 and Dziewior and Schmid 1977) (fetch-tool summary); T. Ayalew, ‘SiC Semiconductor Devices Technology, Modeling, and Simulation’, doctoral dissertation, Institute for Microelectronics, TU Wien, Section 3.5.1 (year not stated; Eqs. 3.107 and 3.108 truncated in the extract) (fetch-tool summary).

Ques: Is the hole SRH lifetime $4\times 10^{-4}$ s or $3\times 10^{-6}$ s?

Both appear, for different purposes. Allpix gives a hole ${\tau }_0$ of $4.0\times 10^{-4}$ s with $N_{d0}=7.1\times 10^{15}$ $\mathrm{c}{\mathrm{m}}^{-3}$, the Fossum-Lee fit to measured lifetimes in high-quality silicon, while Bortolossi’s TCAD test case uses constant lifetimes of $1\times 10^{-5}$ s for electrons and $3\times 10^{-6}$ s for holes, which are typical simulator defaults (background knowledge). The Fossum-Lee value describes the intrinsic material limit, about 130 times the default (computed); the default represents processed device silicon, where contamination and process damage shorten lifetimes. Allpix’s electron value, $1\times 10^{-5}$ s, matches the default, so only the hole value differs. For a real process, lifetimes should be calibrated from measured diode leakage or minority-carrier lifetime, since they vary by orders of magnitude between wafers and processes (background knowledge). Reference: Allpix Squared collaboration, ‘Charge Carrier Lifetime & Recombination’, Allpix Squared documentation, CERN (after Fossum and Lee 1982 and 1983, Schenk 1992, Kerr 2002 and Dziewior and Schmid 1977) (fetch-tool summary); A. Bortolossi, ‘3D Finite Element Drift-Diffusion Simulation of Semiconductor Devices’, Politecnico di Milano thesis (2014) (fetch-tool summary).

Ques: Are the Auger coefficients $3.8\times 10^{-31}$ or $3.93\times 10^{-31}$ cm^6/s in total?

They are close, from different fits. Allpix uses $C_a=C_n+C_p=3.8\times 10^{-31}$ $\mathrm{c}{\mathrm{m}}^6/s$ after Dziewior and Schmid, whose individual values are $2.8\times 10^{-31}$ and $0.99\times 10^{-31}$ $\mathrm{c}{\mathrm{m}}^6/s$, summing to $3.79\times 10^{-31}$ (background knowledge for the split). Bortolossi uses $2.9\times 10^{-31}$ and $1.028\times 10^{-31}$, summing to $3.93\times 10^{-31}$ $\mathrm{c}{\mathrm{m}}^6/s$ (computed), 3.6 per cent higher. Since the Auger lifetime $1/(C_a{N}^2)$ is 26 $\mathrm{ns}$ at $1\times 10^{19}$ $\mathrm{c}{\mathrm{m}}^{-3}$ with the Allpix value (computed), the difference shifts it by about 1 $\mathrm{ns}$, negligible against other uncertainties. Dziewior and Schmid’s values are the more commonly cited set (background knowledge), so $3.8\times 10^{-31}$ is the safer default; more recent work by Kerr, which Allpix cites for the rate form, includes Coulomb-enhanced Auger at lower doping (background knowledge). Reference: Allpix Squared collaboration, ‘Charge Carrier Lifetime & Recombination’, Allpix Squared documentation, CERN (after Fossum and Lee 1982 and 1983, Schenk 1992, Kerr 2002 and Dziewior and Schmid 1977) (fetch-tool summary); A. Bortolossi, ‘3D Finite Element Drift-Diffusion Simulation of Semiconductor Devices’, Politecnico di Milano thesis (2014) (fetch-tool summary).

Ques: When does Auger recombination dominate over SRH?

Only in very heavily doped regions with good material. Allpix states that Auger matters above about $5\times 10^{18}$ $\mathrm{c}{\mathrm{m}}^{-3}$, citing Fossum and Lee. With $C_a=3.8\times 10^{-31}$ $\mathrm{c}{\mathrm{m}}^6/s$, the Auger lifetime is 2.6 $\mu \mathrm{s}$ at $1\times 10^{18}$, 105 $\mathrm{ns}$ at $5\times 10^{18}$ and 26 $\mathrm{ns}$ at $1\times 10^{19}$ $\mathrm{c}{\mathrm{m}}^{-3}$ (computed). With the electron SRH lifetime of the Fossum-Lee form, the combined lifetimes are 95, 17 and 7.2 $\mathrm{ns}$ (computed): SRH still dominates for electrons at $5\times 10^{18}$, while for holes, whose SRH lifetime is 0.57 $\mu \mathrm{s}$ at that doping (computed), Auger already controls the lifetime. Auger therefore governs emitter recombination in solar cells and bipolar transistors and the lifetime under high injection, where the excess carrier density, not the doping, enters $n^{2}p$ (background knowledge). Reference: Allpix Squared collaboration, ‘Charge Carrier Lifetime & Recombination’, Allpix Squared documentation, CERN (after Fossum and Lee 1982 and 1983, Schenk 1992, Kerr 2002 and Dziewior and Schmid 1977) (fetch-tool summary).

Ques: Do the impact-ionization models agree?

Near $3\times 10^5$ $\mathrm{V}\mathrm{cm}$ they agree, but at lower fields they differ several-fold. With Allpix’s silicon parameters at 300 $\mathrm{K}$, the electron coefficient at $3\times 10^5$ $\mathrm{V}\mathrm{cm}$ is $1.16\times 10^4$ $\mathrm{c}{\mathrm{m}}^{-1}$ from Van Overstraeten-De Man, $1.07\times 10^4$ from Massey and $9.77\times 10^3$ from Okuto-Crowell; at $2\times 10^5$ $\mathrm{V}\mathrm{cm}$ they are $1.49\times 10^3$, $1.67\times 10^3$ and only 262, and at $5\times 10^5$ $\mathrm{V}\mathrm{cm}$ $5.99\times 10^4$, $4.76\times 10^4$ and $8.44\times 10^4$ (computed). The Okuto-Crowell form, $aE\exp [-(b/E{)}^2]$, falls faster at low field than the Chynoweth-type $\exp (-b/E)$. Since breakdown depends on the integral of $\alpha$ through the junction, models fitted to different field ranges give different breakdown voltages; the RD50-optimized variants in Allpix were recalibrated for detectors. Ansys’s Selberherr defaults equal the low-field Van Overstraeten values. Reference: Allpix Squared collaboration, ‘Impact Ionization’, Allpix Squared documentation, CERN (after Massey et al. 2006, Van Overstraeten and De Man 1970, Okuto and Crowell 1975, the Bologna model 1999 and the RD50 optimization, arXiv:2211.16543) (fetch-tool summary); Ansys Optics, ‘Silicon diode’ application example, Lumerical CHARGE knowledge base (fetch-tool summary).

Ques: Why does avalanche breakdown voltage rise with temperature?

Because phonon scattering reduces impact ionization. In the Van Overstraeten-De Man model, $\gamma (T)=\tanh (\hslash {\omega }_{op}/2k_B{T}_0)/\tanh (\hslash {\omega }_{op}/2k_{B}T)$ with $\hslash {\omega }_{op}=0.063$ $\mathrm{eV}$ multiplies both the prefactor and the exponent, and the Massey model raises B with temperature by 499 V/cm per kelvin for electrons (Allpix). At 400 $\mathrm{K}$, $\gamma =1.16$, which lowers the electron coefficient at $3\times 10^5$ $\mathrm{V}\mathrm{cm}$ from $1.16\times 10^4$ to $6.97\times 10^3$ $\mathrm{c}{\mathrm{m}}^{-1}$, 40 per cent (computed). A higher field, and hence a higher voltage, is needed for the same multiplication integral, so the breakdown voltage has a positive temperature coefficient (background knowledge for the circuit consequence). This distinguishes avalanche from Zener tunnelling breakdown, whose voltage falls with temperature as the band gap narrows. Reference: Allpix Squared collaboration, ‘Impact Ionization’, Allpix Squared documentation, CERN (after Massey et al. 2006, Van Overstraeten and De Man 1970, Okuto and Crowell 1975, the Bologna model 1999 and the RD50 optimization, arXiv:2211.16543) (fetch-tool summary).

Ques: How steep is band-to-band tunnelling, and why are local models inaccurate?

Extremely steep. With the Hurkx parameters in the Ansys silicon-diode example, A = $4\times 10^{14}$ $\mathrm{c}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$, B = $1.9\times 10^7$ $\mathrm{V}\mathrm{cm}$ and exponent 2.5, and an assumed prefactor D of 1, the generation rate is $2.2\times 10^{12}$ $\mathrm{c}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$ at $5\times 10^5$ $\mathrm{V}\mathrm{cm}$, $2.2\times 10^{21}$ at $1\times 10^6$ and $1.7\times 10^{26}$ at $2\times 10^6$ $\mathrm{V}\mathrm{cm}$ (computed), nine orders of magnitude for a doubling of field. Tunnelling is non-local: an electron leaves the valence band at one point and arrives in the conduction band some distance away, so evaluating the rate from the local field misrepresents the path. Wong et al. note that the local Hurkx model is cheap but underestimates leakage at large bias, and their modified local model matches non-local results for diode leakage and GIDL while running six times faster in a 3-D FinFET. Meshes of 4 $\mathrm{nm}$ or less are needed at such junctions (Ansys). Reference: Ansys Optics, ‘Silicon diode’ application example, Lumerical CHARGE knowledge base (fetch-tool summary); H. Y. Wong, D. Dolgos, L. Smith and R. V. Mickevicius, ‘Modified Hurkx band-to-band-tunneling model for accurate and robust TCAD simulations’, Microelectronics Reliability 104, 113552 (2020), doi:10.1016/j.microrel.2019.113552 (abstract only) (fetch-tool summary).

References

1. T. Ayalew, “SiC Semiconductor Devices Technology, Modeling, and Simulation,” doctoral dissertation, Institute for Microelectronics, TU Wien, Section 3.5.1 “Shockley-Read-Hall Recombination.” Available: https://www.iue.tuwien.ac.at/phd/ayalew/node72.html
2. Allpix Squared collaboration, “Charge Carrier Lifetime & Recombination,” Allpix Squared documentation, CERN. Available: https://allpix-squared.docs.cern.ch/docs/06_models/03_lifetime_recombination/
3. A. Bortolossi, “3D Finite Element Drift-Diffusion Simulation of Semiconductor Devices,” Laurea Magistrale thesis, Politecnico di Milano, 2014. Available: https://www.politesi.polimi.it/retrieve/a81cb05a-eff6-616b-e053-1605fe0a889a/2014_7_Bortolossi.pdf
4. Allpix Squared collaboration, “Impact Ionization,” Allpix Squared documentation, CERN. Available: https://allpix-squared.docs.cern.ch/docs/06_models/05_impact_ionization/
5. Ansys Optics, “Silicon diode,” Lumerical CHARGE application example. Available: https://optics.ansys.com/hc/en-us/articles/360042272934-Silicon-diode
6. H. Y. Wong, D. Dolgos, L. Smith and R. V. Mickevicius, “Modified Hurkx band-to-band-tunneling model for accurate and robust TCAD simulations,” Microelectronics Reliability, vol. 104, 113552, 2020, doi:10.1016/j.microrel.2019.113552. Available: https://scholarworks.sjsu.edu/faculty_rsca/1605

Related pages in this vault: TCAD Mobility Models, Drift-Diffusion Device Equations, Hydrodynamic and Energy-Balance Transport, HgCdTe Photodiode, Band-Gap Narrowing and Quantum Corrections, Thermal and Electro-Thermal Simulation, Lifetime Testers, PN Junction, Deep-Level Transient Spectroscopy (DLTS).