# __3D Simulation of the Defect Generation by Hydrogen at Si-SiO2 Interface__ [TOC] # Introduction Ionizing radiation generates charge in materials, whose interfaces are regions of especially high defect densities, even for $$SiO_2$$-on-$$Si$$ systems, where $$SiO_2$$ is a native oxide and well-matched to Si. Bulk oxide and interface defects can act as charge trapping sites. The study of this buildup of charge in device oxide layers and the changes, often degradation, in device and circuit characteristics that result is known as total ionizing dose (TID) effects. <div align="center"> [![illusation](http://data.xyzgate.com/ff23b7513096a4f03ae5bb9884b335ed.png "illusation")](undefined "illusation") </div>
Since the launch of even the first satellites, TID radiation effects have been observed to cause real life on-board anomalies. <div align="center"> ![](http://data.xyzgate.com/1faed7d6b6a32ea5d518b8ab930c1c8a.png ) </div> >__ An example of data from the CRRES satellite, which measured total dose at different distances from the Earth. The Earth is the blue ball __ ![](http://data.xyzgate.com/c8a5e38314c5a53f50332c8095f0602b.png)
### Chemical Reactions Here, we briefly summarize the influences of radiation-induced electrons and holes on metal-oxide-semiconductor (MOS) structures. ![](http://data.xyzgate.com/13e92a84e706975a8a2e08df060221a7.png) Oxygen vacancies are the dominant defects in $$SiO\_2$$. The neutral oxygen vacancy includes one $$Si-Si$$ bond instead of two $$Si-O$$ bonds. Because of the distinct energies, There are two detect species, detects $$V\_{o\delta}^+$$ and $$V\_{o\gamma}^+$$, and two neutral precursors, $$V\_{o\delta}$$ and $$V\_{o\gamma}$$. It is shown by observing the associated energies that $$V\_{o\delta}^+$$ is a shallow hole trap, while $$V\_{o\gamma}^+$$ is much deeper. These two kinds of defects can be hydrogenated or doubly hydrogenated to form $$V\_{o \gamma}H$$, $$V\_{o \gamma}H^+$$, $$V\_{o \gamma}H\_{2}$$, $$V\_{o \gamma}H\_{2}^+$$, $$V\_{o \delta}H$$, $$V\_{o \delta}H^+$$ and $$V\_{o \delta}H\_{2}$$, $$V\_{o \delta}H\_{2}^+$$, respectively.
![](http://data.xyzgate.com/4b42beb221311b1afb05d5f02a9c301d.png) - Radiation emerges Electron-hole pairs (EHPs). - The electric field pushes holes escaping initial recombination towards the interface, while electrons are towards the metal gate. - Neutral oxygen vacancies become positively charged when capturing holes and then are neutralized by electrons. - Protons and hydrogenated defects are created when molecular hydrogen recombines the positively charged defects. - Protons can also be cracked by positively-charged hydrogenated defects directly. - Interface traps are formed by the react of protons and the $$Si- H$$ bonds on the $$SiO_2/Si$$ interface. ```katex H^{+}+Si-H \Leftrightarrow N_{it}+H_2 ```
### Poisson-Nernst-Planck Equations In this model, every kind of species participates in chemical reactions and therefore we use generation and recombination terms in Nernst-Planck equation to simulate the ion transport process. The electrostatic field is determined by applied voltage and charged species, including electron-hole pairs (EHPs) and positively charged defects, which are simulated by Poisson equation shown as follows: ```katex \epsilon \nabla^2 \phi =-Q ``` ```katex Q_{SiO_{2}}=q(p+H^{+}+V_{o\delta}^{+}+V_{o\delta}H^{+}+V_{o\delta}H_{2}^{+}++V_{o\gamma}^{+}+V_{o\gamma}H^{+}+V_{o\gamma}H_{2}^{+}-n) ``` Electrons, holes and $$H^+$$ are charged and mobile, so the diffusion-convection process is considered. The $$U\_{radiation}$$ is the EHP generation term. During the radiation, recombination and generation of spices also arise by the chemical reactions, and they must also be accounted in the continuity equations, which we will describe later. ```katex \frac{\partial n}{\partial t}=\nabla\cdot(e\mu_{n}nE+D_{n}\nabla n)+U_{radiation}+G_n-R_n ``` ```katex \frac{\partial p}{\partial t}=-\nabla\cdot(e\mu_{p}pE-D_{p}\nabla p)+U_{radiation}+G_p-R_p ``` ```katex \frac{\partial H^{+}}{\partial t}=-\nabla\cdot(e\mu_{H^{+}}H^{+}E-D_{H^{+}}\nabla H^{+})+G_{H^+}-R_{H^+} ``` Hydrogen is mobile, but not charged. ```katex \frac{\partial H_{2}}{\partial t}=D_{H_{2}}\nabla H_{2}+G_{H_{2}}-R_{H_{2}} ``` The defects including $$V\_{o\gamma},V\_{o\gamma}^+,V\_{o\delta},V\_{o\delta}^+, V\_{o\gamma}H, V\_{o\gamma}H^+, V\_{o\delta}H, V\_{o\delta}H^+, V\_{o\gamma}H\_{2}, V\_{o\delta}H\_{2}, V\_{o\gamma}H\_{2}^{+}, V\_{o\delta}H\_{2}^{+} $$ are not mobile with no drift nor diffusion, but they still have recombination and generation terms. ```katex \frac{d T_{i}}{d t}=G_{i}-R_{i} ```
As the chemical reactions are complicate, we use recombination and generation terms to stand for chemical reactions in PNP equations, and these terms are nonzero after irradiation. Consider the bulk reactions: ```katex A+B \Leftrightarrow C \qquad (Reaction \quad 0) ``` ```katex A+M \Leftrightarrow C+N \qquad (Reaction \quad 1) ``` For non-equilibrium conditions, each specie has recombination and generation terms, we take spice $$A$$ for example, whose rates have the following form: the generation term of $$A$$ can be described as: ```katex G_A=k_{r_0} \cdot [C] + k_{r_1} \cdot [C] [N] ``` ```katex R_A=k_{f_0} \cdot [A] \cdot [B]+k_{f_1} \cdot [A] \cdot [M] ``` where $$k\_f$$ and $$k\_r$$ are reaction rates; the bracket represents concentration of the species in $$/cm^{3}$$. The recombination and generation terms of other spices can be defined as the same. Then we can add the recombination and generation terms to each continuity equation.
### Boundary Conditions [![](http://data.xyzgate.com/4f91d7d9a00c2c45629597abfc64fb90.png)](undefined) $$H^+$$ has to be emphasized on $$Si/SiO\_2$$ interface because of the reaction: ```katex H^{+}+Si-H \Leftrightarrow N_{it}+H_2 ``` Since the $$[H^+]\_{int},[SiH]\_{int}$$ is 2D with the unit of$$/cm^{2}$$ on the interface. ```katex \frac{d N_{it}}{dt}=\sigma_{int} \vec{J}_{H^{+}}\cdot\vec{n}([SiH]-[N_{it}]) ``` $$\sigma\_{int}$$ is the transformation coefficient with the unit of cm^2
# Multi-scale Methods Take $$V\_{o\gamma}H\_{2}^{+}$$ for example: ```katex \begin{aligned} \frac{\partial V_{o\gamma}H_{2}^{+}}{\partial t}=&(1.03e-13)[V_{o\gamma}H_2][h^+]+(1.03e-19)[V_{o\gamma}][H^+]+(4.02e-21)[V_{o\gamma}^+][H_2]\\ &+(3.21e-138)[V_{o\gamma}H_2]\\ &-(4.16e+3)[V_{o\gamma}H_2^+]-(3.81e+5)[V_{o\gamma}H_2^+]-(1.90e+5)[V_{o\gamma}H_2^+]\\ &-(2.06e-07)[V_{o\gamma}H_2^+][e^-]\\ \end{aligned} ``` Some parameter can be regarded as 0. And we can simplify it as: ```katex \frac{\partial V_{o\gamma}H_{2}^{+}}{\partial t}=-O(10^5)V_{o\gamma}H_{2}^{+} + C ``` [![](http://data.xyzgate.com/845e51f5a0fe19b4512b19cb19b5e004.png)](undefined "bad")
Step 1. Given the current state of the macro-variable reaction parameters, get the Generating terms and Reaction terms of $$e^-, h^+$$ of one macro time step $$\delta t$$, and take their result as current step. ```katex G_i(c_i,t+\delta t)=G(c_i(t),\tilde{c_i}(t),c^{'}_{i}(t));R_i(c_i,t+\delta t)=R(c_i(t),\tilde{c_i}(t),c^{'}_{i}(t)) ``` Step 2. Evolve the macro-variable of $$H^+, H\_2$$ for one macro time step using the macro-solver ```katex \int_{\Omega_s} \frac{c_i^{n}-c_i^{n-1}}{\delta t} \psi = \int_{\Omega_s} D_i (\nabla c_i^n \cdot \nabla \psi + z_i c_i^n \nabla u^n \cdot \nabla \psi) + \int_{\Omega_s} G_i(c_{i},t+\delta t) \psi - R_i(c_{i},t+\delta t) \psi, ``` Step 3. Calculate the micro-variable for $M$ micro time steps $$\delta \tau$$ with iteration method using the current step of $$e^-, h^+$$: ```katex \tilde{c_i}((m+1)\delta \tau) = \tilde{c_i}(m \delta \tau) + \delta \tau (G_i(\tilde{c_i},t+ m \delta \tau) - R_i(\tilde{c_i},t+ m \delta \tau)), m=0,...,M-1 ``` Step 4. Get macro-variable Generating terms and Reaction terms of $$H^+, H_2$$ ```katex G_i(c_{i}',t+\delta t)=G(c_i(t+\delta t),\tilde{c_i}(t+\delta t),c_{i}'(t));R_i(c_{i}',t+\delta t)=R(c_i(t+\delta t),\tilde{c_i}(t+\delta t),c_{i}'(t)) ``` Step 5. Evolve the macro-variable of $$H^+, H_2$$ for one macro time step using the macro-solver : ```katex \int_{\Omega_s} \frac{c_{i}^{'n}-c_{i}^{'n-1}}{\delta t} \psi = \int_{\Omega_s} D_i (\nabla c_{i}^{'n} \cdot \nabla \psi + z_i c_{i}^{'n} \nabla u^n \cdot \nabla \psi) + \int_{\Omega_s} G_i(c_{i}',t+\delta t) \psi - R_i(c_{i}',t+\delta t) \psi, ``` Step 6. set the current state of the macro-variable and repeat. [![good](http://data.xyzgate.com/613c055c95c0a04b42c9deea34c1b53e.png "good")](undefined "good")
# Linear System The new RAS preconditioner can be simply described as follows: ```katex M_{RAS}^{-1}=\Sigma_{i=1}^{N}R_{i,0}^{t}A_{i}^{-1}R_{i,\delta} ``` $$\delta$$ is the number of overlaps. Since $$R_{i}^{0} x$$ does not involve any data exchange with the neighbouring processors, it is much quicker and proved to be correct. It seems that the size of thickness has great influence on the calculation. Here is the data we test in 32 CPUs to simulate high doping semiconductors. |thickness | AS(1 overlap) | AS(2 overlaps) |RAS(1 overlap)|RAS(2 overlaps)| | ------------- | ------------- | | 4 | 93s | 92s | 105s | 113s | | 1 | 179s | 206s | 98s | 113s |
# numerical results ## H2 ![](http://data.xyzgate.com/1a6ae8efb8be802e25f0ff34fed1cff2.png) >__The trend of radiation-induced interface trap buildup versus molecular hydrogen concentration in the oxide.__ ![](http://data.xyzgate.com/6cbdd9561c354fdac9755dfe75291b55.png) >__data obtained from the experiments___
## EDLRS ![](http://data.xyzgate.com/1c831bda06d454f788043869987652cb.png) >__caculations showing qualitative match to ELDRS trend.The total dose is 0.1krad and the H2 density is $$3e14cm^{-3}$$__ ![](http://data.xyzgate.com/c8a5e38314c5a53f50332c8095f0602b.png) >__$$N_\{it}$$ versus dose rate for irradiation to 30 krad(Si) __
## Parallel efficiency Our tests start with 32 processes, whose parallel efficiency is regarded as $$100\%$$, and the parallel efficiency for p processes is defined as ```math E_p=\frac{32T_{32}}{pT_{p}} ``` where $$T_p$$ denotes the time to irradiate 0.1krad with the rate of 10 $rad/s$ for solving the PNPEs using p processes with one step.The parallel efficiencies obtained are satisfactory. TO get these data, we use a new mesh containing 1418778 vertices and 8637254 tetrahedra. |Num of procs | Time | Efficiency| |-----------|--------|--------| |32 | 182000s | 100%| |64 | 75888s | 120%| |128 | 30444s | 149%| |256 | 16566s | 137%| |512 | 14570s | 78%|
![](http://data.xyzgate.com/51ab0f047c6c2f1146810bbe142814d9.png)
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