Challenges in Drift-Diffusion Semiconductor Simulations

Abstract

We study and compare different discretizations of the van Roosbroeck system for charge transport in bulk semiconductor devices that can handle nonlinear diffusion. Three common challenges corrupting the precision of numerical solutions will be discussed: boundary layers, discontinuities in the doping profile, and corner singularities in L-shaped domains. The most problematic of these challenges are boundary layers in the quasi-Fermi potentials near ohmic contacts, which can have a drastic impact on the convergence order.

1 Introduction

The present paper aims at comparing different discretization philosophies for semiconductor problems. We study three major challenges for recent finite element and finite volume schemes which are designed to deal with nonlinear diffusion in a thermodynamic consistent way and are based on quasi-Fermi potentials as primary variables. In particular, we study the error and convergence rate of the numerical solutions in the presence of: boundary layers, discontinuous doping profile and corner singularities.

2 Modelling Semiconductors with Ohmic Contacts

2.1 Stationary van Roosbroeck System

The van Roosbroeck system is a drift-diffusion model, which describes the recombination and transport of charge carriers driven by diffusion and by electric fields within a semiconductor device. It consists of three nonlinear, coupled partial differential equations for the electrostatic potential $\psi :\varOmega \rightarrow \mathbb {R}$ as well as the non-negative electron and hole densities $n:\varOmega \rightarrow \mathbb {R}^+$ and $p:\varOmega \rightarrow \mathbb {R}^+$, namely a Poisson equation and two continuity equations. We consider a homogeneous material and some domain $\varOmega \subseteq \mathbb {R}^d$ for $d\in \{1,2,3\}$ in an isothermal setting. Then the stationary van Roosbroeck system is given by the system of elliptic partial differential equations

$$\begin{aligned} -\nabla \cdot \left( \varepsilon _0\varepsilon _r \nabla \psi \right)&=q\left( C + p(\psi ,\varphi _p) - n(\psi ,\varphi _n) \right) , \end{aligned}$$

(1a)

$$\begin{aligned} \nabla \cdot \mathbf {j}_n&= +q R, \end{aligned}$$

(1b)

$$\begin{aligned} \nabla \cdot \mathbf {j}_p&= -q R, \end{aligned}$$

(1c)

where q denotes the elementary charge, $\varepsilon _0$ is the vacuum permittivity and $\varepsilon _r$ is the relative permittivity of the material. The recombination rate R and the charge-carrier currents $\mathbf {j}_n$, $\mathbf {j}_p$ depend on the solution $n,p,\psi$ and vanish in thermal equilibrium. The given doping concentration $C:\varOmega \rightarrow \mathbb {R}$ (intentionally introduced impurities) varies spatially and can have discontinuities. The equations of state are given by

$$\begin{aligned}&n(\psi ,\varphi _n) = N_c \mathscr {F}\left( \frac{q(\psi -\varphi _n)-E_c}{k_BT}\right) , \end{aligned}$$

(2a)

$$\begin{aligned}&p(\psi ,\varphi _p) = N_v \mathscr {F}\left( \frac{q(\varphi _p-\psi )+E_v}{k_BT}\right) , \end{aligned}$$

(2b)

where the statistical distribution function $\mathscr {F}$ relates the electron and hole densities n, p to the quasi-Fermi potentials $\varphi _n,\varphi _p$. Working with quasi-Fermi potentials has all the advantages mentioned in the introduction, in particular from a modeling and computational point of view. Furthermore, we set the recombination rate to zero as it plays a minor role for most of our considerations.

The effective density of states for electrons in the conduction band $N_c$ and holes in the valence band $N_v$ as well as the corresponding band-edge energies $E_c,E_v$ and the band gap $E_g=E_c-E_v$ are material parameters and assumed to be spatially constant in this paper. Temperature and the Boltzmann constant are denoted with T and $k_B$. The three most important reference cases for the statistical distribution functions are the Boltzmann, Blakemore and Fermi-Dirac function. For each distribution function, the corresponding current densities in (1b) and (1c) are

$$\begin{aligned} \mathbf {j}_n&= -q \mu _n n \nabla \varphi _n=-q \mu _n n \nabla \psi + q D_n \nabla n,\end{aligned}$$

(3a)

$$\begin{aligned} \mathbf {j}_p&= -q \mu _p p \nabla \varphi _p=-q \mu _p p \nabla \psi - q D_p \nabla p. \end{aligned}$$

(3b)

Using the thermal voltage $U_T=\frac{k_BT}{q}$, the diffusion coefficients $D_n,D_p$ are linked to the carrier mobilities $\mu _n,\mu _p$ via generalized Einstein relations

$$\begin{aligned} \frac{D_n}{\mu _n}=U_T\,g\bigl (\eta _n\bigr ), \qquad \frac{D_p}{\mu _p}=U_T\,g\bigl (\eta _p\bigr ), \qquad g(\eta ) = \frac{\mathscr {F}(\eta )}{\mathscr {F}'(\eta )}, \end{aligned}$$

(4)

where g is the diffusion enhancement as motivated in [8].

The system (1) is supplied with mixed Dirichlet-Neumann boundary conditions.

3 Discretization of the van Roosbroeck System Using Potentials

In the following we are going to explain standard discretization methods to solve the van Roosbroeck system.

3.1 Finite Element Method

Assume $\varOmega \subset \mathbb {R}^2$ is a polygonal domain and let $\mathscr {T}_h$ be an admissible decomposition of $\varOmega$ into $N_\text {tria}$ triangles and $N_\text {vert}$ vertices, such that $\varOmega = \bigcup _{t=1}^{N_\text {tria}}\tau _t$ for $\tau _t\in \mathscr {T}_h$. Similar as in [2], we solve the stationary van Roosbroeck system (1) using a standard $P_1$ finite element method. We seek the electrostatic potential and the quasi-Fermi potentials $\mathbf {u}^h=(\psi ^h,\varphi ^h_n,\varphi ^h_p)\in V^h$, such that the van Roosbroeck system can be written in the weak form as

$$\begin{aligned} 0&= \int _\varOmega \Big (\varepsilon _0\varepsilon _r\nabla \psi ^h\cdot \nabla v_i - q\big (C+p(\psi ^h,\varphi _p^h)-n(\psi ^h,\varphi _n^h)\big )v_i\Big ){\,\mathrm d}x,\end{aligned}$$

(5a)

$$\begin{aligned} 0&= \int _\varOmega \Big (q\mu _n n(\psi ^h,\varphi _n^h) \nabla \varphi ^h_n\cdot \nabla v_j - qR\left( n(\psi ^h,\varphi _n^h),p(\psi ^h,\varphi _p^h)\right) v_j\Big ){\,\mathrm d}x,\end{aligned}$$

(5b)

$$\begin{aligned} 0&= \int _\varOmega \Big (q\mu _p p(\psi ^h,\varphi _p^h) \nabla \varphi ^h_p\cdot \nabla v_k + qR\left( n(\psi ^h,\varphi _n^h),p(\psi ^h,\varphi _p^h)\right) v_k\Big ){\,\mathrm d}x, \end{aligned}$$

(5c)

for all suitable test functions $\mathbf {v}^h=(v_i,v_j,v_k)\in V^h$, where $V^h\cong \mathbb {R}^{N_\text {vert}\times 3}$ is the $3N_\text {vert}$ dimensional space of vectorial continuous functions which are piecewise linear on each triangle $\tau _t$.

3.2 Finite Volume Method

In this section, we present a Voronoï finite volume technique [4, 6, 7]. Similar as for finite elements, we start by partitioning the domain $\varOmega$ into non-intersecting, convex polyhedral control volumes $\omega _k$ such that $\varOmega =\bigcup _{k=1}^{{N_\text {vert}}} \omega _k$. We associate with each control volume $\omega _k$ a node $\mathbf {x}_k\in \omega _k$. For every boundary intersecting control volume, we demand that this node lies on the boundary $\mathbf {x}_k\in \partial \varOmega \cap \omega _k$. Assuming the partition is admissible [3], i.e. for two adjacent control volumes $\omega _k$ and $\omega _l$, the edge $\overline{\mathbf {x}_k\mathbf {x}_l}$ of length $h_{kl}$ is orthogonal to $\partial \omega _k \cap \partial \omega _l$, the normal vectors to $\partial \omega _k$ can be calculated by $\varvec{\nu }_{kl}=(\mathbf {x}_l-\mathbf {x}_k)/\Vert \mathbf {x}_l-\mathbf {x}_k\Vert$. We note that the variables $(\psi ,\varphi _n,\varphi _p)$ are of interest only at the nodes, not at the edges.

For each control volume $\omega _k$, the finite volume discretization is given by the three equations:

$$\begin{aligned} \sum _{\omega _l \in \mathscr {N}\left( \omega _k\right) } |\partial \omega _k \cap \partial \omega _l| j_{\psi ;k ,l}&= q |\omega _k| \left( C_k + p(\psi _k,\varphi _{p;k})-p(\psi _k,\varphi _{n;k})\right) ,\end{aligned}$$

(6a)

$$\begin{aligned} \sum _{\omega _l \in \mathscr {N}\left( \omega _k\right) } |\partial \omega _k \cap \partial \omega _l | j_{n;k,l}&= +q |\omega _k| R(\psi _k,\varphi _{n;k},\varphi _{p;k}), \end{aligned}$$

(6b)

$$\begin{aligned} \sum _{\omega _l \in \mathscr {N}\left( \omega _k\right) } |\partial \omega _k \cap \partial \omega _l | j_{p;k,l}&= -q |\omega _k| R(\psi _k,\varphi _{n;k},\varphi _{p;k}). \end{aligned}$$

(6c)

We denote with $\mathscr {N}\left( \omega _k\right)$ the set of all control volumes neighboring $\omega _k$. In 2D, the measure $|\partial \omega _k \cap \partial \omega _l|$ corresponds to the length of the boundary line segment and in 3D to the area of the intersection of the boundary surfaces.

The unknowns $\psi _k$, $\varphi _{n;k}$, $\varphi _{p;k}$ correspond to the electrostatic potential as well as the quasi-Fermi potentials for electrons and holes evaluated at node $\mathbf {x}_k$. To approximate the fluxes in (6) using general $\mathscr {F}$, ideas from [1] are useful to derive a finite volume scheme for convection-diffusion problems in a thermodynamically consistent way by averaging the nonlinear diffusion term appropriately.

4 Numerical Examples

In this section, we are going to present numerical solutions of the van Roosbroeck system via FE and the Scharfetter-Gummel FV discretization. We focus on two challenges, which have an impact on the convergence rate of solutions: the size of a boundary layer and the regularity of the doping. Since in this section we are mostly concerned with numerical solutions, we will drop the superindex h. If necessary, we replace it with the acronym of the corresponding discretization method. Also we remind the reader that we solve the van Roosbroeck system without recombination, i.e., $R\equiv 0$. Throughout this section, we use the Blakemore distribution function.

Fig. 1

1D electron and hole densities n, p and doping C at bias $V_\mathrm{ext}=3\,\text {V}$ shown (left) with $\kappa =500$ and (right) with $\kappa =5\cdot 10^5$, the former yielding a smooth doping profile and the latter practically a discontinuous one

Fig. 2

1D quasi-Fermi potentials of electrons and holes $\varphi _n,\varphi _p$ and electrostatic potential $\psi$ (left) with bias $V_\mathrm{ext}=3\,\text {V}$ as well as boundary layers in the electron quasi-Fermi potential $\varphi _n$ near $x=0.3\,\upmu \text {m}$ for different mesh resolutions h (middle) for finite element and (right) Scharfetter-Gummel type finite volume discretization for $\kappa =500$

4.1 Resolution of Boundary Layer

In Fig. 1 the densities n, p and the doping C are shown for the two cases $\kappa =5\cdot 10^2$ and $\kappa =5\cdot 10^5$ at $V_\mathrm{ext}=3\,\text {V}$. Note that in both cases, the hole density p has a boundary layer at $x=0$ and the electron density n has a boundary layer at $x=0.3\,\upmu \text {m}$. This boundary layer, however, is on the length scale of $\lambda _D$ and therefore nicely resolved by the mesh. On the level of the plot, the difference in solutions corresponding to the two alternative doping profiles is not visible. In the left panel of Fig. 2 we show the potentials $(\psi ,\varphi _n,\varphi _p)$ for $V_\mathrm{ext}=3\,\text {V}$. While the electrostatic potential in both cases is a rather smooth function (blue line), the quasi-Fermi potentials have a boundary layer of size $\ell _J$ (green and red line) that can not be resolved on any of the uniform meshes. This logarithmic boundary layer is predicted by our analysis in [5]. As one can see in Fig. 2 (middle and right panel), the solution effectively jumps within the last interval before the ohmic contact.

4.2 Regularity of the Doping

Next, we discuss the influence of the smoothness of the doping on the convergence order for the different discretization methods. Whenever we compare a coarse discrete solution (of size 2h) to a finer one (of size h), we restrict the finer solution to the coarser mesh. Then we can subtract $\mathbf {u}^h$ from $\mathbf {u}^{2h}$ and slightly abusing the notation write $\Vert \mathbf {u}^{h}-\mathbf {u}^{2h}\Vert$ for the corresponding norm. Provided that the doping is sufficiently smooth and the carrier densities converge sufficiently fast, then the FV discretization of the Poisson equation is second order accurate, see the convergence for $n,p,\psi$ in the top right panel of Fig. 3. When the doping is discontinuous ($\kappa =5\cdot 10^5$), the bottom row in Fig. 3 shows that also the convergence order of the FV electrostatic potential becomes linear, which is plausible by standard FE error estimates. From Fig. 3 it appears that while the error in the FE method is dominated by the quasi-Fermi potentials, the error in the FV method is dominated by the lack of regularity in the doping.

Fig. 3

$L_2$ convergence rates in 1D for solution (left) of the FE discretization and (right) of the FV discretization with $\kappa =500$ in the top row and for $\kappa =5\cdot 10^5$ in the bottom row at $V_\mathrm{ext}=3\,\text {V}$

4.3 Corner Singularities and Boundary Adapted Meshes

Semiconductor devices may often be angular-shaped. However, in particular L-shaped domains pose numerical difficulties which we would like to study for the FE and FV methods. We consider a two-dimensional L-shaped domain

$$\begin{aligned} \varOmega = [0,2L]^2 \setminus [0,L]^2\subset \mathbb {R}^2, \end{aligned}$$

(7)

and impose ohmic contacts at the boundaries (x, 0) and (0, y) for $L\le x, y \le 2L$. All other boundaries are supplied with homogeneous Neumann boundary conditions. The p-i-n doping concentration $C:\varOmega \rightarrow \mathbb {R}$ is given by

$$\begin{aligned} C(\mathbf {x})={\left\{ \begin{array}{ll} +C_0 &{} 0\le x \le L/2, \\ -C_0 &{} 0 \le y \le L/2, \\ +2C_0(L-x)/L &{} L/2<x\le L, \\ -2C_0(L-y)/L &{} L/2<y\le L, \\ 0 &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

(8)

with $\mathbf {x}=(x,y)$ and as before $L=10^{-7}m=0.1\upmu \text {m}$ and $C_0=C_{\text {m}}=10^{23}\,\text {m}^{-3}$. With this choice we ensure that the convergence order does not suffer from the regularity of the doping. However, constructing a non-convex domain with a corner angle $\vartheta =\theta \pi$ and $\theta =3/2$ imposes a corner singularity of the form $\psi (\mathbf {x}) \sim r^{1/\theta }$ as $r\rightarrow 0$ for $r=\sqrt{(x-x_0)^2+(y-y_0)^2}$ at $x_0=y_0=0.1\upmu \text {m}$.

Fig. 4

Convergence of solutions on different meshes as a function of relative triangle size $h=2^{-\ell }$ for a electron quasi-Fermi potential $\varphi _n$ at $V_\mathrm {ext}=3\,\text {V}$, b electrostatic potential $\psi$ at $V_\mathrm {ext}=3\,\text {V}$, c electron quasi-Fermi potential $\varphi _n$ at $V_\mathrm {ext}=0.2\,\text {V}$, d electrostatic potential $\psi$ at $V_\mathrm {ext}=0.2\,\text {V}$

The upper left panel of Fig. 4 shows the convergence of the electron quasi-Fermi potentials at $V_\mathrm{ext}=3\,\text {V}$, where the FE and FV are compared on a sequence of uniform and a sequence of boundary adapted meshes. As in 1D, the FV method converges quadratically. Furthermore, for the FV discretization the error seems not to be influenced very much by the boundary adapted meshes. In contrast, the FE method again has a lower convergence order and local adaptivity improves the $L_2$ error of the solution by about one order of magnitude.

The lower panels of Fig. 4 show the solutions at $V_\mathrm{ext}=0.2\,\text {V}$, where the boundary layer is moderate and solutions are closer to thermal equilibrium. Hence, the lower left panel shows the general tendency to have lower errors. However, the convergence is slower with an order between $\mathscr {O}(h)$ and $\mathscr {O}(h^{4/3})$, indicating a stronger influence of the corner singularity. This effect is even more pronounced in the lower right panel, in which for all the used methods the convergence of the electrostatic potential nicely follows the $\mathscr {O}(h^{4/3})$ order predicted by the error analysis of the corner singularity.

5 Conclusion

Summarizing, in 2D both FE and FV discretizations deliver reasonable results. While the finite volume scheme often shows better convergence rates, the finite element method can be drastically improved by using meshes which are finer near ohmic contacts. We clearly observe that depending on the potential and the selected bias, the error is dominated by the boundary layer or the corner singularity. While the FV method generally handles the boundary layer well, the FE method in 2D introduces extra oscillations in the boundary layer, see [5] for details.