<br/> <font size='3'> **全国集成微系统科学技术及其建模与仿真学术交流会** <br/> <center> <br/> <font size='6.5'>**多组分漂移-扩散-反应非线性模型的高性能并行算法在电离损伤效应中的研究**</font> <br/> <font size='5'>**马召灿,卢本卓** **中国科学院** **数学与系统科学研究院** **计算数学与科学工程计算研究所** </font> <font size='5.5'> **zhaocanma@lsec.cc.ac.cn** **成都,2018.07.07** </font> </center> <br/>
# Introduction ![](http://data.xyzgate.com/5aabce9f6c56a0795a818762e126f90a.png) ![](http://data.xyzgate.com/13e92a84e706975a8a2e08df060221a7.png)
##EDLRS : Enhanced low dose rate sensitivity 以不同的剂量率辐射至相同的总剂量,剂量率较小时,总剂量效应更大; ![](http://data.xyzgate.com/c8a5e38314c5a53f50332c8095f0602b.png) >__$$N_\{it}$$ versus dose rate for irradiation to 30 krad(Si) from R. L. Pease, et al., The effects of hydrogen on the enhanced low dose rate sensitivity (ELDRS) of bipolar linear circuits, IEEE Trans. Nucl. Sci., vol. 55, no. 6, pp. 3170, Figure 3, Dec. 2008.__
##对H2环境的依赖 ![](http://data.xyzgate.com/afe4718d29fac5d3a582ef1ed648b90f.png) >Extracted $$N_{it}$$ and $$N_{ot}$$ measurements from Chen, et al. taken over a wide range of ambient H 2 concentrations at a high dose rate from X. J. Chen, et al., Mechanisms of enhanced radiation-induced degradation due to excess molecular hydrogen in bipolar oxides, IEEE Trans. Nucl. Sci., vol. 54, no. 6, pp. 1915, Figure 6, Dec.2007.
## 模型描述 ![](http://data.xyzgate.com/35f06a2896d0f9ed3707a01ae0ec8641.png) N. L. Rowsey 基于第一性原理计算构建的TID模型 软件:FLOODS( the FLorida Object Oriented Device Simulator) 模拟结果:1D MOS simulations ```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) ``` ![](http://data.xyzgate.com/4b42beb221311b1afb05d5f02a9c301d.png) ```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} ``` ```katex H^{+}+Si-H \Leftrightarrow N_{it}+H_2 ``` ```katex \frac{d N_{it}}{dt}=\sigma_{int} \vec{J}_{H^{+}}\cdot\vec{n}([SiH]-[N_{it}]) ```
#数值方法 ##全隐式时间格式 离散格式:Scharfetter-Gummel,有限元,一阶后向欧拉或者其他二阶格式。 ```katex \int_{\Omega_s} \frac{C_i^{n+1}-C_i^{n}}{\delta t} \psi = \int_{\Omega_s} D_i (\nabla C_i^{n+1} \cdot \nabla \psi + z_i C_i^{n+1} \nabla u^{n+1} \cdot \nabla \psi) + \int_{\Omega_s} k_f [A]^n [B]^n \psi - k_r[C_{i}^{n+1}] \psi, ``` ```katex \frac{C^{n+1}-C^{n}}{\delta t}=k_f[A]^n[B]^n -k_r[C]^{n+1} ``` ##稳定化方法(SUPG) ```math \dfrac{\partial p}{\partial t}=\nabla\cdot(D\nabla p+\mu q p\nabla\phi)+f ``` Let $$q=ze\_c$$, $$\beta=1/kT$$, $$u=e\_c\beta\phi$$, and the equation is rewritten as ```math \dfrac{\partial p}{\partial t}=\nabla\cdot D(\nabla p+zp\nabla u)+f ``` The weak form is ```math B(p,v)=\int_{\Omega}\dfrac{\partial p}{\partial t}vdx-\int_{\Omega}D(\nabla p\cdot\nabla v+zp\nabla u\cdot\nabla v)dx-\int_{\Omega}fvdx=0 ``` The SUPG term is ```math S(p,v_{supg})=\sum\limits_{K}\int_{K}\Big(\dfrac{\partial p}{\partial t}-\nabla\cdot D(\nabla p+zp\nabla u)-f\Big)\cdot v_{supg}dx ``` We define the Peclet number $$Pe_K=\dfrac{||a||_2h_K}{2D}$$, ```math v_{supg}=\sigma_K a\cdot\nabla v ``` Here, ```math a=-Dz\nabla u ``` ```math \sigma_K=\dfrac{h_K}{2||a||_2}\xi(Pe_K) ``` ```math \xi(Pe_K)=\begin{cases} \frac{Pe_K}{3},\quad 0\leq Pe_K\leq 3\\ 1,\quad Pe_K\geq 3 \end{cases} ``` <div class="row"><div class="col-lg-6">![](http://data.xyzgate.com/2136dc1ee91df45067368044ab52e13f.png) </div><div class="col-lg-6">![](http://data.xyzgate.com/34f01384fef5cceaa07926da071aea8d.png) </div><div class="col-lg-6 text-center">(a) standard FEM (h+) </div><div class="col-lg-6 text-center">(b) standard FEM (e-) </div></div> <div class="row"><div class="col-lg-6">![](http://data.xyzgate.com/b716e30d70d470fdb0b6ee6411ea9e56.png) </div><div class="col-lg-6">![](http://data.xyzgate.com/155d68b3613741f819db841d8116c128.png) </div><div class="col-lg-6 text-center">(a) SUPG method (h+) </div><div class="col-lg-6 text-center">(b) SUPG method (e-) </div></div> ###时间多尺度方法 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. ![](http://data.xyzgate.com/4e5594b4ffc9b5fc351a3cbf8ec905f8.png)
###PHG平台 Parallel-Hierarchical-Grid (PHG) 是我们自主设计的一个基于网格二分细 化适合大规模分布式存储并行计算机的三维并行自适应有限元软件平台。 ![](http://data.xyzgate.com/62d8b8f04dc45b10269cc31f03684467.png) ![](http://lsec.cc.ac.cn/phg/pic/heat2.gif) ###Online computing on xyzgate.com <iframe src="http://www.xyzgate.com/mycontainer" width="100%" height="500px" frameborder="0" scrolling="yes"></iframe>
#Results ##Interface trap denisty ![](http://data.xyzgate.com/51fa7c849b1210f4e5af8952380ccba3.png) ![](http://data.xyzgate.com/c69ea2a38b44b432ab1b5e0953d3006a.png) >Total Dose=10krad
##Trap denisty with H_2 ![](http://data.xyzgate.com/b660c2bc23d3ca11dcd79f8d72ba4590.png) >$$N_\{it}$$ versus dose rate for irradiation to 30 krad(Si) from N. L. Rowsey, et al., Radiation-Induced Oxide Charge in Low- and High-H 2 Environments, IEEE Trans. Nucl. Sci., Figure 1, 2012 ![](http://data.xyzgate.com/f39a80a87b611d95dce2b729ff808eb8.jpeg)
#Concentration ### defect concentrations ![](http://data.xyzgate.com/468f6f64be6a616c7775eb7a7194f0fc.png) floods结果 ![](http://data.xyzgate.com/1543af56fa3eae254e153132d0a9ac1e.png) >FLOODS calculation of the electron, $$V_{o\gamma}$$, and $$V_{o\gamma}H_2 ^+$$ defect concentrations versus oxide depth
### proton with time ![](http://data.xyzgate.com/6b15cd95373608da673e395296707d31.png) >Calculated H + concentration vs. oxide depth at various times following irradiation, from N. L. Rowsey, et al.,Radiation-Induced Oxide Charge in Low- and High-H 2 Environments, IEEE Trans. Nucl. Sci., Figure 3, 2012 ![](http://data.xyzgate.com/391355c645ece6945bdd4cb1e8530631.jpeg)
### hydrogen concentration vs. time and distance ![](http://data.xyzgate.com/e9ac5a272725c4c8b7026ffaeafe64a6.png) >Plot of molecular hydrogen concentration vs. time and distance in the oxide.X. J. Chen,et al Mechanisms of enhanced radiation-induced degradation due to excess molecular hydrogen in bipolar oxides, IEEE Trans. Nucl. Sci., Dec. 2007. ![](http://data.xyzgate.com/e71d1c1f0b50b587281de5f6dda7f5be.png)
### Defects vs. time ![](http://data.xyzgate.com/189966a3cb933b9717b64911cc5a5d44.gif) ![](http://data.xyzgate.com/741607d1ca41f34568cb65009b7345a0.gif) ![](http://data.xyzgate.com/6f337327c5814e6fbb4584cd6ddfa32f.gif)
![](http://data.xyzgate.com/1b35647deaa86fbb62ee55a92bea8129.png) >NMOS $$I_d - V_g$$ 曲线 ![](http://data.xyzgate.com/eea445634dfe377a96bfac010d4e07de.png) >This suggests that substrate reverse bias can effectively reduce the impact of TID-induced charge
## 并行效率测试 Mesh points: 93930 Mesh tetrahedra: 556000 DOF num: 93930 并行效率公式: ```katex E=\frac{36 * T_{36}}{p*T_p} ``` |进程数|时间|并行效率| | ------------- | ------------- |--------| |36|5.2016e+02s| 100%| |72|1.5520e+02s| 167%| |144|6.3946e+01s|203%| |288|5.1570e+01s|126%| |576|2.1718e+01s|149%| |864|3.3775e+01s|64%| |1008|3.6390e+01s|51%| |1152|5.5045e+01s|29%| |1728|9.2861e+01s|12%| |2304|2.6607e+02s|3%|
![Thanks](http://data.xyzgate.com/08799e3d46112614d45cfb970888b59e.png "Thanks")
/
Go to slide: