An HDG Method for the Time-dependent Drift–Diffusion Model of Semiconductor Devices

Abstract

We propose a hybridizable discontinuous Galerkin (HDG) finite element method to approximate the solution of the time dependent drift–diffusion problem. This system involves a nonlinear convection diffusion equation for the electron concentration u coupled to a linear Poisson problem for the electric potential $\phi$. The non-linearity in this system is the product of the $\nabla \phi$ with u. An improper choice of a numerical scheme can reduce the convergence rate. To obtain optimal HDG error estimates for $\phi$, u and their gradients, we utilize two different HDG schemes to discretize the nonlinear convection diffusion equation and the Poisson equation. We prove optimal order error estimates for the semidiscrete problem. We also present numerical experiments to support our theoretical results.

Appendix

In this section we give a proof for (17a) and (17b). The proof of (17c) is similar and we do not provide details.

1.1 Error Equations

We start be deriving equations satisfied by standard projections [see (12)] of the exact solution.

Lemma 6.1

Let $({\varvec{q}}, u)$ be components of the solution of (2), then we have

holds for all $({\varvec{r}}_1, w_1, \mu _1)\in {\varvec{Q}}_h\times V_h\times {\widehat{V}}_{h}(0)$.

Proof

By the definition of and in (14c) and (14e) respectively, the projections and integrating by parts, we get

where we have also used (2a). In addition,

Hence, again using the projections, we have

Since, using (2c), $\nabla \cdot {\varvec{q}} = \nabla \cdot ({\varvec{p}} u) - u_t$, then we have

$$\begin{aligned} (\nabla \cdot {\varvec{q}}, w_1)_{{\mathcal {T}}_h} = -(u_t, w_1)_{{\mathcal {T}}_h} + \langle {\varvec{p}}\cdot {\varvec{n}} u, w_1 \rangle _{\partial {\mathcal {T}}_h} - ({\varvec{p}}u, \nabla u)_{{\mathcal {T}}_h}. \end{aligned}$$

This implies that

and completes the proof of the lemma. $\square$

To simplify notation, we define

$$\begin{aligned} \eta _h^{{\varvec{q}}}:={\varvec{\varPi }}_k^o{\varvec{q}}-{\varvec{q}}_{Ih},\ \ \ \eta _h^{u}:=\varPi _{k+1}^ou-u_{Ih},\ \ \ \eta _h^{{\widehat{u}}}:=\varPi _k^{\partial }u-{\widehat{u}}_{Ih}. \end{aligned}$$

We then subtract the equation in Lemma 6.1 from (15) to get the following lemma.

Lemma 6.2

Under the conditions of Lemma 6.1, we have the error equation

(26)

holds for all $({\varvec{r}}_1, w_1, \mu _1)\in {\varvec{Q}}_h\times V_h\times {\widehat{V}}_{h}(0)$.

1.2 Main Error Estimate

We can now prove (17b).

Lemma 6.3

For h small enough, we have the error estimates

$$\begin{aligned} \Vert {{\varvec{q}}}-{{\varvec{q}}}_{Ih} \Vert _{\mathcal {T}_h}+\Vert h_K^{-1/2}(\varPi _k^{\partial }u_{Ih}-{\widehat{u}}_{Ih} )\Vert _{\partial \mathcal {T}_h}\le Ch^{k+1}|u|_{k+2}. \end{aligned}$$

Proof

We take $({\varvec{r}}_1, w_1, \mu _1) = (\eta _h^{{\varvec{q}}},\eta _h^u,\eta _h^{{\widehat{u}}})$ in (26). First

Next,

For h small enough, we obtain

On the other hand,

Next, we estimate $\{R_i\}_{i=1}^4$ term by term. For the first term $R_1$, Lemma 2.7 gives

$$\begin{aligned} R_1&\le Ch^{k+1}|{\varvec{q}}|_{k+1}\Vert h_K^{-1/2}(\eta _h^u-\eta _h^{{\widehat{u}}})\Vert _{\partial \mathcal {T}_h},\\&\le Ch^{k+1}|{\varvec{q}}|_{k+1}\left( \Vert \eta _h^{{\varvec{q}}}\Vert _{\mathcal {T}_h}+\Vert h_K^{-1/2}(\varPi _k^{\partial }\eta _h^u-\eta _h^{{\widehat{u}}})\Vert _{\partial \mathcal {T}_h}\right) . \end{aligned}$$

For the term $R_2$, by Lemmas 2.9 and 2.7 to get

$$\begin{aligned} R_2&\le Ch^{k+2}|u|_{k+2}\Vert \nabla \eta _h^u\Vert _{\mathcal {T}_h}\\&\le Ch^{k+2}|u|_{k+2}\left( \Vert \eta _h^{{\varvec{p}}}\Vert _{\mathcal {T}_h}+\Vert h_K^{-1/2}(\varPi _k^{\partial }\eta _h^u-\eta _h^{{\widehat{u}}})\Vert _{\partial \mathcal {T}_h}\right) . \end{aligned}$$

For the term $R_3$, we use Lemma 2.7 to get

$$\begin{aligned} R_3&= \langle {\varvec{p}}\cdot {\varvec{n}} (\varPi _k^{\partial } u-u), \eta _h^u-\eta _h^{{\widehat{u}}} \rangle _{\partial \mathcal {T}_h}\\&\le Ch^{k+1}|u|_{k+1}\Vert h_K^{-1/2}(\eta _h^u-\eta _h^{{\widehat{u}}})\Vert _{\partial \mathcal {T}_h}\\&\le Ch^{k+1}|u|_{k+1}\left( \Vert \eta _h^{{\varvec{p}}}\Vert _{\mathcal {T}_h}+\Vert h_K^{-1/2}(\varPi _k^{\partial }\eta _h^u-\eta _h^{{\widehat{u}}})\Vert _{\partial \mathcal {T}_h}\right) . \end{aligned}$$

Moreover, for the last term we have

$$\begin{aligned} R_4 \le Ch^{k+1}|u|_{k+1} \Vert h_K^{-1/2}(\varPi _k^{\partial }\eta _h^u-\eta _h^{{\widehat{u}}}) \Vert _{\partial \mathcal {T}_h}. \end{aligned}$$

Use the Cauchy-Schwarz inequality for the above estimates of $\{R_i\}_{i=1}^4$, we get

$$\begin{aligned} \Vert \eta _h^{{{\varvec{q}}}}\Vert _{\mathcal {T}_h}+\Vert h_K^{-1/2}(\varPi _k^{\partial }\eta _h^{u}-\eta _h^{{\widehat{u}}} )\Vert _{\partial \mathcal {T}_h} \le Ch^{k+1}|u|_{k+2}. \end{aligned}$$

Use of the triangle inequality and estimates (13a) and (13b) completes the estimate. $\square$

1.3 Duality Arguments

To obtain a $L^2$ norm estimate of $\Vert \eta _h^{u}\Vert _{{\mathcal {T}}_h}$, we use the dual problem (8) with corresponding a priori estimate (9). To perform the error analysis, the main difficulty is to deal with the nonlinearity. We define a new form which is related to the trilinear form :

(27)

Next, we give a property of the operators and . We omit the proof since it is very straightforward.

Lemma 6.4

For all $(u_h ,{\widehat{u}}_h, w_1,\mu _1)\in V_h\times {\widehat{V}}_{h}(0)\times V_h\times {\widehat{V}}_{h}(0)$, we have

Similarly to Lemma 6.1, we have the following lemma.

Lemma 6.5

Assuming M is chosen sufficiently large, let $({\varPhi },{\varvec{\varPsi }})$ solve (8) then we have the equation

holds for all $({\varvec{r}}_1, w_1, \mu _1)\in {\varvec{Q}}_h\times V_h\times {\widehat{V}}_{h}(0)$.

With the above preparation we can now derive estimate (17a).

Theorem 6.6

Let u and $u_{Ih}$ be the solutions of (2) and (15), respectively. If h is small enough, then we have the error estimate

$$\begin{aligned} \Vert u-u_{Ih}\Vert _{\mathcal {T}_h}\le Ch^{k+2}\Vert u\Vert _{k+2}. \end{aligned}$$

Proof

We take $({\varvec{r}}_1,w_1, \mu _1)=(\eta _h^{{\varvec{q}}},-\eta _h^u,-\eta _h^{{\widehat{u}}})$ and $\varTheta =-\eta _h^u$ in Lemma 6.5 to get

By Lemma 6.4 we have

This implies

On the other hand, we have

Comparing the above two equations, we get

$$\begin{aligned} \Vert \eta _h^u\Vert ^2_{\mathcal {T}_h}&= -\langle {\varvec{\varPi }}_k^o{\varvec{q}}\cdot {\varvec{n}}-{\varvec{q}}\cdot {\varvec{n}},\varPi _{k+1}^o\varPhi -\varPi _k^{\partial }\varPhi \rangle _{\partial \mathcal {T}_h} -\langle h_K^{-1}(\varPi _{k+1}^ou-u),\varPi _k^{\partial }\varPi _{k+1}^o\varPhi \\&\quad -\varPi _k^{\partial }\varPhi \rangle _{\partial \mathcal {T}_h} -\,({\varvec{p}} (\varPi _{k+1}^ou-u), \nabla \varPi _{k+1}^o\varPhi )_{\mathcal {T}_h}\\&\quad +\langle {\varvec{p}}\cdot {\varvec{n}} (\varPi _k^{\partial } u-u), \varPi _{k+1}^o\varPhi - \varPi _{k}^\partial \varPhi \rangle _{\partial \mathcal {T}_h}\\&\quad +\langle {\varvec{p}}\cdot {\varvec{n}} (\eta _h^u-\eta _h^{{\widehat{u}}}), \varPi _{k+1}^o\varPhi -\varPi _{k}^{\partial }\varPhi \rangle _{\partial \mathcal {T}_h}\\&\quad -\,\langle {\varvec{\varPi }}_k^o{\varvec{\varPsi }}\cdot {\varvec{n}}-{\varvec{\varPsi }}\cdot {\varvec{n}},\eta _h^{{\widehat{u}}}-\eta _h^u \rangle _{\partial \mathcal {T}_h}- \langle h_K^{-1}(\varPi _{k+1}^o\varPhi -\varPhi ),\varPi _k^{\partial }\eta _h^u-\eta _h^{{\widehat{u}}}\rangle _{\partial \mathcal {T}_h}\\&\quad +\,( {\varvec{p}} (\varPi _{k+1}^o\varPhi -\varPhi ), \nabla \eta _h^u)_{\mathcal {T}_h} -\langle {\varvec{p}}\cdot {\varvec{n}} (\varPi _k^{\partial } \varPhi -\varPhi ), \eta _h^u - \eta _h^{{\widehat{u}}} \rangle _{\partial \mathcal {T}_h}\\&\quad +\,(\nabla \cdot {\varvec{p}} (\varPi _{k+1}^o\varPhi -\varPhi ),\eta _h^u)_{\mathcal {T}_h}\nonumber \\&=\sum _{i=1}^{10}S_i. \end{aligned}$$

We estimate $\{S_i\}_{i=1}^{10}$ as follows (we omit some of the details):

$$\begin{aligned} S_1&=- \langle {\varvec{\varPi }}_k^o{\varvec{q}}\cdot {\varvec{n}}-{\varvec{q}}\cdot {\varvec{n}},\varPhi -\varPi _{k+1}^{o}\varPhi \rangle _{\partial \mathcal {T}_h} \le Ch^{k+2}|{\varvec{q}}|_{k+1}\Vert \varPhi \Vert _{2},\\ S_2&= - \langle h_K^{-1}(\varPi _{k+1}^ou-u),\varPi _{k+1}^{o}\varPhi -\varPhi \rangle _{\partial \mathcal {T}_h}\le Ch^{k+2} |u|_{k+2}\Vert \varPhi \Vert _{2},\\ S_3&=-({\varvec{p}} (\varPi _{k+1}^ou-u), \nabla \varPi _{k+1}^o\varPhi )_{\mathcal {T}_h} \le Ch^{k+2}|u|_{k+2}|\varPhi |_1,\\ S_4&=\langle {\varvec{p}}\cdot {\varvec{n}}(\varPi _k^{\partial }u-u),\varPi _{k+1}^o\varPhi -\varPhi \rangle _{\partial \mathcal {T}_h}\le Ch^{k+2}|u|_{k+1}\Vert \varPhi \Vert _{2},\\ S_5&\le C\Vert h_K^{-1/2}(\eta _h^u-\eta _h^{{\widehat{u}}})\Vert _{\partial \mathcal {T}_h}h|\varPhi |_1\le Ch^{k+2}|u|_{k+2}|\varPhi |_1,\\ S_6&\le Ch\Vert h_K^{-1/2}(\eta _h^u-\eta _h^{{\widehat{u}}})\Vert _{\partial \mathcal {T}_h}\Vert {\varvec{\varPsi }}\Vert _1\le Ch^{k+2 }\Vert {\varvec{\varPsi }}\Vert _{1},\\ S_7&\le Ch\Vert h_K^{-1/2}(\eta _h^u-\eta _h^{{\widehat{u}}})\Vert _{\partial \mathcal {T}_h}\Vert {\varvec{\varPhi }}\Vert _{2}\le Ch^{k+2}|u|_{k+2}\Vert {\varvec{\varPhi }}\Vert _{2},\\ S_8&\le Ch^{2}\Vert \varPhi \Vert _{2}\Vert \nabla \eta _h^u\Vert _{\mathcal {T}_h}\le Ch^{k+2}\Vert \varPhi \Vert _{2}|u|_{k+2},\\ S_9&= - \langle {\varvec{p}}\cdot {\varvec{n}}(\varPi _k^{\partial }\varPhi -\varPhi ),\eta _h^u-\eta _h^{{\widehat{u}}}\rangle _{\partial {\mathcal {T}}_h}\le Ch^{k+2}|\varPhi |_1|u|_{k+2},\\ S_{10}&\le Ch^{2}\Vert \varPhi \Vert _{2}\Vert \eta _h^u\Vert _{\mathcal {T}_h}. \end{aligned}$$

Summing the above estimates, we get

$$\begin{aligned} \Vert \eta _h^u\Vert ^2_{\mathcal {T}_h}\le Ch^{k+2}\Vert u\Vert _{k+2} \Vert \eta _h^u\Vert _{\mathcal {T}_h} +Ch^2 \Vert \eta _h^u\Vert ^2_{\mathcal {T}_h}. \end{aligned}$$

Let h be small enough, we have

$$\begin{aligned} \Vert \eta _h^u\Vert _{\mathcal {T}_h}\le Ch^{k+2}\Vert u\Vert _{k+2}. \end{aligned}$$

A simple application of the triangle inequality finishes the proof. $\square$