Reduced basis method for the electromagnetic scattering problem: a case study for FinFETs

Abstract

Optical 3D simulations in many-query and real-time contexts require new solution strategies. We study an adaptive, error controlled reduced basis method for solving parameterized time-harmonic optical scattering problems. Application fields are, among others, design and optimization problems of nano-optical devices as well as inverse problems for parameter reconstructions occurring e. g. in optical metrology. The reduced basis method presented here relies on a finite element modeling of the scattering problem with parameterization of materials, geometries and sources.

1 Introduction

Electromagnetic field solvers must be efficient to be of use for optimization tasks of 3D structures where computation times are of importance as typically a large number of computations with varying parameters have to be performed until the optimal structure is found. The same holds true for inverse problems where measured data is given and structural details are sought after. Hence there is a demand for highly accurate, error controlled results at low computation times. Especially for real time applications, like on-line process control in semiconductor manufacturing, new solution strategies for a repeated solution of 3D problems are required.

The reduced basis method (RBM) is such a strategy. Here we consider application of the RBM to time-harmonic Maxwell’s equations based on a finite element (FEM) discretization. The setup discussed here is light scattering off a fin field-effect transistor (FinFET) (analyzed previously without the RBM (Burger et al. 2015)). FinFETs are realized on wafers and measured after manufacturing by optical methods to control the actual geometrical shape. The measurement consists of an illumination of an array of periodically placed FinFETs and a determination of the reflectance in dependence of the wavelength and polarization.

2 Reduced basis method

The reduced basis method (Prudhomme et al. 2002; Rozza et al. 2008) allows for the construction of error controlled approximations to the relationship of an input, a specific configuration of parameters $\mu \in {\mathbb {R}}^d$, to an output quantity $s(\mu )=s({\varvec{E}}(\mu ))$ derived from a solution ${\varvec{E}}$ of the parameter dependent Maxwell’s equations. In the following we describe the method only briefly and refer to Pomplun (2010) and Pomplun et al. (2011) for details.

In this work we consider electromagnetic scattering problems where a parameter $\mu \in {\mathfrak {D}}\subset {\mathbb {R}}^d$ may describe geometry and/or properties of the illumination or materials. The weak formulation for an ${\mathcal {N}}$ dimensional FEM space $V_h$ reads:

For $\mu \in {\mathfrak {D}}$ find ${\varvec{E}} \in V_h$ such that:

$$\begin{aligned} a(\varphi ,{\varvec{E}};\mu ) = f(\varphi ;\mu ) \quad \forall \,\varphi \in V_h. \end{aligned}$$

(1)

The natural space for solutions of the electromagnetic scattering problem 1 is $H({\mathbf{curl }},\varOmega ) = \{ {\varvec{u}} \in (L^2(\varOmega ))^3 \,\vert \,\nabla \times {\varvec{u}} \in (L^2(\varOmega ))^3 \}$ with the corresponding norm $\Vert u \Vert _{H({\mathbf{curl }},\varOmega )} = ( \Vert {\varvec{u}} \Vert _{(L^2(\varOmega ))^3}^2 + \Vert \nabla \times {\varvec{u}} \Vert _{(L^2(\varOmega ))^3}^2 ) ^{1/2}$.

Essential for the RBM is the splitting of the solution process into two phases: a computationally expensive offline phase executed once and an inexpensive, fast online phase.

During the offline construction of the reduced basis the underlying, high-dimensional FEM model (commonly referred to as the truth approximation) is solved multiple times for certain parameter configurations to compute so-called snapshots. The selection of snapshots spanning the reduced basis can be carried out self-adaptively through a Greedy strategy by selecting the worst resolved parameter configuration contained within a finite training set ${\mathfrak {D}}_{train}\subset {\mathfrak {D}}$ of the parameter domain. An efficient and cheap a posteriori error estimator for the reduced basis approximation error is crucial for this selection (Pomplun and Schmidt 2010).

Subsequently, in the online phase, solutions are sought in the linear space $X_N\subset V_h$ spanned by these snapshots via a Galerkin projection. This reduced problem reads:

For $\mu \in {\mathfrak {D}}$ find ${\varvec{E}}_N \in X_N$ such that:

$$\begin{aligned} a(\varphi ,{\varvec{E}}_N;\mu ) = f(\varphi ;\mu ) \quad \forall \,\varphi \in X_N. \end{aligned}$$

(2)

As the number of snapshots N is typically in the order of 10–100, the reduced problem stated on the reduced basis requires only a low dimensional linear system to be solved. This operation can be done in real time and for many parameters if the assembly of the reduced system in the online phase remains independent of the FEM dimension ${\mathcal {N}}$ and only depends on the reduced basis dimension N.

Essential in this offline–online splitting is the parameterization of the Maxwell operator $a(\cdot ,\cdot ;\mu )$. It must thus be parameterized in a way that permits an effective offline–online decomposition. This is readily achieved for example for changes in the (scalar) permittivities of isotropic materials where the operator can be decomposed affinely, i.e. $a(v,u;\mu ) =a_1(v,u;\mu )+\varepsilon a_2(v,u;\mu )$. However, for geometrical variations or parameterizations of the illumination the operator is generally not affine in $\mu$ and has to be approximated by the Empirical Interpolation Method (EIM) (Barrault et al. 2004) to guarantee an affine decomposition.

In addition to the Maxwell operator also the output functional $s(\mu )=s({\varvec{E}}(\mu ))$ must admit an offline–online decomposition for the online phase to be independent of ${\mathcal {N}}$. This is trivial for linear outputs like the Fourier Transform of the electromagnetic field. We call the reduced basis together with the reduced output functional the reduced model.

3 Results

3.1 Optical model

The investigated structure is a periodic array of FinFETs (at the 22 nm technology node). These complex 3D architectures have become a crucial driver for down-scaling of structures on integrated circuits (Bunday et al. 2013).

Fig. 1

Left FEM-mesh side view of the FinFET geometry. The used designation (gate and fin) along with the material are inscribed on each particular component of the geometry. The surrounding air-domain is not shown. Center detailed view of FEM-mesh regions with geometrical meaning of radii $r_ {gate}$ (top) and $r_ {fin}$ (bottom). $r_{fin}$ also affects the roundings of adjacent layers. Right detailed view of displaced FEM-mesh to the most extreme values of $r_ {gate} = \{15.0, 1.0\,\text {nm}\}$ and $r_ {fin} = \{4.0, 0.5\,\text {nm}\}$ respectively. (Color figure online)

The device geometry of the unit cell is depicted as a FEM mesh in Fig. 1 (left) with designations and materials of the components. The device dimensions and optical parameters are taken from Bunday et al. (2013) and Palik (1998). The silicon fin sits on a 200 nm thick silicon oxide substrate (only partially shown) and the gate (silicon) wraps around the fin orthogonal to its axis in the center of the unit cell. A SiN layer (dark red) of 5 nm, a high-k layer (green) of 2 nm and a TiN layer (gray) of 7 nm complete the structure. The pitches of fin and gate are 44 and 88 nm respectively. The fin is 12.7 nm wide and 40 nm high with an undercut of 2.1 nm and a side-wall angle of 89.5 deg. The gate is 40 nm wide and 95 nm high with an undercut of 2 nm. Additionally a rounding of the fin and gate top edges is used. The FinFET is discretized with a parameterized tetrahedral mesh which accurately resolves the undercuts and corner roundings. The transparent boundary conditions at the top and bottom are realized with Perfectly Matched layers (Zschiedrich 2009).

The structure is illuminated by a plane wave at 390 nm and tilted 30 deg along the gate direction. Using 4th order ansatz functions and 190,089 degrees of freedom the FEM problem is solved in $\approx$270 s of CPU time.

Fig. 2

a Error estimates with increasing reduced basis dimension in a semi-logarithmic plot. The residual error estimate (black solid line) is normalized with the estimate for N $=$ 1. The error estimate for the Fourier transform (black dashed line) estimates the relative error in the ${\mathrm {L}}_{\infty }$ norm. b Snapshot positions in parameter space are marked as red circles and the number indicates the sequence of snapshot selection. The locations contained in the initial training set ${\mathfrak {D}}_{train}$ are marked by crosses. (Color figure online)

3.2 Reduced basis construction

We build a reduced basis for the corner rounding radii $r_{gate}$ and $r_{fin}$ in the parameter domain ${\mathfrak {D}}=[1\,{\text {nm}}, 15\,{\text {nm}}] \times [.5\,{\text {nm}}, 4\,{\text {nm}}]$. The definition of radii is depicted in Fig. 1 (center). A detailed view of the FEM-mesh for the extreme values is depicted on the right in Fig. 1. The reduced model is constructed for the Fourier transform of the electromagnetic field which is used to compute the reflectance of the structure.

The training set ${\mathfrak {D}}_{train}$ used for the Greedy searches is adaptively refined from a fine initial mesh of 11 by 11 equidistant values depicted as crosses in Fig. 2b. The EIM approximation uses 13 snapshots for the system matrix with an estimated relative approximation error of $5.3\times 10^{-7}$ in the ${\mathrm {L}}_{\infty }$ norm over the training set. A single snapshot is sufficient to approximate the right-hand side.

In the offline phase 38 truth solutions are computed to construct the reduced basis. Figure 2b depicts the snapshot order and location in the parameter domain ${\mathfrak {D}}$ (red numbered circles). The first snapshot is chosen in the center and the subsequent snapshots cover the corner of ${\mathfrak {D}}$ first. Generally the snapshot locations follow the boundaries of ${\mathfrak {D}}$ with a slight clustering at the corners.

The Greedy algorithm choses in each iteration the worst resolved parameter in the training set as the new snapshot location. Thus the error estimate generally decreases with increasing reduced basis dimension N. In Fig. 2 the residual error estimate (solid black line) with increasing N is shown in a semi-logarithmic plot. The initial estimate is used for normalization. The normalized estimate converges exponentially with N up to an estimated error of less than $2.5\times 10^{-11}$. The adaptive refinement of the training set leads to the non-monotonic convergence as new values are included.

The estimated error in the output quantity (dashed black line, Fourier transform in the z direction) shows a similar decrease although with a different rate due to different norms employed (relative ${\mathrm {L}}_{\infty }$ norm and H(curl, $\varOmega$) norm). The error estimate convergences up to an estimated error of less than $1\times 10^{-6}$.

3.3 Online evaluation of the reduced basis

In the online phase we evaluated the reduced system on 21 by 21 equidistant points evenly spread over the whole parameter space ${\mathfrak {D}}$. Each evaluation takes $\approx$200 ms. The contour plots in Fig. 3 (center column) show the computed reflectance of P and S polarization respectively. The plots next to the contour plots show the horizontal and vertical (left and right) cross-sections through the center of the parameter space supplemented by reference solutions (blue dots) and in the outermost plots the linear trend of the reference data is removed to highlight deviations. We observe a much stronger dependence on the corner rounding radius of the gate than of the fin in both polarizations. In both polarizations the deviation compared to the reference is on the order of $1\times 10^{-5}$.

Fig. 3

Reflectance for the P (top row) and S polarization (bottom row) over radii as contour plots (central column), as cross-sections through the parameter domain (left and right from center columns) and with removed linear trend (outermost columns). (Color figure online)

3.4 Error analysis

In the following we investigate these different errors by comparing the reduced basis approximations to reference solutions for a randomly selected set $\varXi$ of 100 parameter values. For $\mu \in \varXi$ we solve the truth approximation including the EIM approximation and compare the results.

In Fig. 2 the relative errors in the Fourier transform are shown in addition to the error estimate on the left. The mean relative error over $\varXi$ is depicted by the blue dashed linear whereas the maximum is shown in red. Both lines follow the error estimate very closely. The errors are reduced from about $1\times 10^{-1}$ to $1\times 10^{-6}$ with increasing reduced basis dimension. We conclude that the error estimator is very efficient for the given problem.

In Fig. 4 the max and mean of of the approximation error over $\varXi$ in the H(curl, $\varOmega$) norm are shown as well as a scatter plot relating the estimated error and the true error in the H(curl, $\varOmega$) norm. The maximum as well as the mean error exhibit identical convergence trends to the estimated error. With increasing N the error decreases exponentially from $1\times 10^{-2}$ to $1.2\times 10^{-12}$ at most with the mean being an order of magnitude smaller. The scatter plot demonstrates the good performance of the error estimator. The error and the estimate are highly correlated over more than 9 orders of magnitude.

Fig. 4

Maximum and mean error of $\varXi \subset {\mathfrak {D}}^{N}_{train}$ in H(curl, $\varOmega$) norm over the reduced basis dimension (left) and estimated reduced basis error over the error in H(curl, $\varOmega$) norm (right). The maximal and mean error decrease with increased reduced basis dimension. The estimated and actual error are highly correlated. (Color figure online)

4 Conclusions

The reduced basis method is well suited for many-query and real-time simulation tasks. We have demonstrated its application to optical scattering simulations of a parameterized FinFET where it allows for the computation of online solutions with a three order of magnitude speed up. The errors in both the field solution and the output are controlled satisfactorily. As a result we have an efficient, error controlled, real-time capable procedure for situations where the application of the direct problem would not be realistic.