A Complementary Topographic Feature Detection Algorithm Based on Surface Curvature for Three-Dimensional Level-Set Functions

Abstract

The level-set method is widely used in expanding front simulations in numerous fields of computational research, such as computer graphics, physics, or microelectronics. In the latter, the level-set method is employed for topography simulations of semiconductor device fabrication processes, being driven by complicated physical and chemical models. These models tend to produce surfaces with critical points where accuracy is paramount. To efficiently increase the accuracy in regions neighboring these critical points, automatic hierarchical domain refinement is required, guided by robust feature detection. Feature detection has to be computationally efficient and sufficiently accurate to reliably detect the critical points. To that end, we present a fast parallel geometric feature detection algorithm for three-dimensional level-set functions. Our approach is based on two different, complementary curvature calculation methods of the zero level-set and an optimized feature detection parameter to detect features. For performance reasons, our algorithm can be in principal linked to different curvature calculation methods, however, as will be discussed, two particularly attractive options are available: (i) A novel extension of the standard curvature calculation method for level-set functions, and (ii) an often disregarded method for calculating the curvature due to its purported low numerical accuracy. We show, however, that the latter is still a viable option, and that our algorithm is able to reliably detect features on geometries stemming from complicated, practically relevant geometries. Our algorithm and findings are applicable to other fields of applications such as surface simplification.

Full text

Journal of Scientific Computing (2023) 94:71
https://doi.org/10.1007/s10915-023-02133-5
A Complementary Topographic Feature Detection Algorithm
Based on Surface Curvature for Three-Dimensional Level-Set
Functions
Christoph Lenz1·Luiz Felipe Aguinsky1·Andreas Hössinger2·Josef Weinbub1
Received: 25 July 2022 / Accepted: 12 January 2023 / Published online: 8 February 2023
© The Author(s) 2023
Abstract
The level-set method is widely used in expanding front simulations in numerous fields of
computational research, such as computer graphics, physics, or microelectronics. In the latter,
the level-set method is employed for topography simulations of semiconductor device fabri-
cation processes, being driven by complicated physical and chemical models. These models
tend to produce surfaces with critical points where accuracy is paramount. To efficiently
increase the accuracy in regions neighboring these critical points, automatic hierarchical
domain refinement is required, guided by robust feature detection. Feature detection has to
be computationally efficient and sufficiently accurate to reliably detect the critical points. To
that end, we present a fast parallel geometric feature detection algorithm for three-dimensional
level-set functions. Our approach is based on two different, complementary curvature calcu-
lation methods of the zero level-set and an optimized feature detection parameter to detect
features. For performance reasons, our algorithm can be in principal linked to different cur-
vature calculation methods, however, as will be discussed, two particularly attractive options
are available: (i) A novel extension of the standard curvature calculation method for level-
set functions, and (ii) an often disregarded method for calculating the curvature due to its
purported low numerical accuracy. We show, however, that the latter is still a viable option,
and that our algorithm is able to reliably detect features on geometries stemming from com-
plicated, practically relevant geometries. Our algorithm and findings are applicable to other
fields of applications such as surface simplification.
BChristoph Lenz
lenz@iue.tuwien.ac.at
Luiz Felipe Aguinsky
aguinsky@iue.tuwien.ac.at
Andreas Hössinger
andreas.hoessinger@silvaco.com
Josef Weinbub
weinbub@iue.tuwien.ac.at
1Christian Doppler Laboratory for High Performance TCAD, Institute for Microelectronics,
TU Wien, Gußhausstraße 27-29, 1040 Vienna, Austria
2Silvaco Europe Ltd., Compass Point, St Ives, Cambridge PE27 5JL, UK
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71 Page 2 of 21 Journal of Scientific Computing (2023) 94 :71
Keywords Level-set method ·Curvature ·Feature detection ·Parallel algorithm
1 Introduction
The level-set method was introduced by Osher and Sethian and has proven itself to be a
valuable tool in advancing front simulations [1–3]. The front is described by the zero level-
set of a level-set function φin a domain ⊂Rn
S={x∈:φ(x)=0},(1)
which is discretized on a grid with resolutions (x1...xn). If the grid resolution is equiv-
alent in all coordinate directions then it is called a regular grid with resolution x.This
implicit representation allows the level-set method to robustly handle the merging of fronts
without self-intersections.
In this work we focus in particular on topography simulations for modeling semiconductor
device fabrication processes, since the merging of fronts often occurs there [4]. However, the
here presented algorithm and findings are applicable to other fields in a similar manner. In
general, semiconductor topography simulations are a vital part of the set of process technology
computer-aided design (TCAD) simulation toolset. Dedicated simulators are available, such
as, commercial (e.g., Silvaco’s Victory Process [5]) and open source tools (e.g., ViennaTS
[6]). In semiconductor topography simulations, the surface of a wafer undergoing processing
is usually described by the zero level-set of a level-set function.
During an entire process TCAD simulation workflow of a practically relevant semicon-
ductor device, several different physical and chemical models are computed to represent, e.g.,
the formation of different materials (deposition) or the removal of material regions from the
wafer (etching). In a multi-layer structure representation, these materials are represented by
individual level-set functions and a stacking process using Boolean operations is utilized to
keep the description of thin material layers consistent (layer wrapping) [7]. The effects of the
different physical and chemical models on each material layer are simulated by propagating
the zero level-sets of these material layers. The temporal evolution of the zero level-set of
each layer is described by the level-set equation
∂φ(x,t)
∂t+V(x,t)∇φ(x,t)=0,(2)
where V(x,t)stands for a velocity field that drives the evolution of the zero level-set. The
modeling of the physical and chemical processes is achieved through the construction of
V(x,t)[8,9]. Certain velocity fields are so intricate that the properties of the Hamiltonian
H=V(x,t)∇φ(x,t)in (2) change, e.g., Honly has a weak solution [10]. Thus, different
methods of numerically solving (2) using finite differences are required for certain simulation
workflows [10–12]. The velocity fields are directly linked to the discrete representation of the
zero level-sets of the affected material layers, which can impose strict quality requirements
on the feature detection (e.g. noise in the zero level-set).
To calculate the finite differences required to solve the level-set Eq. (2) and to obtain the
curvatures of the zero level-set, the level-set values (φ-values) are required on all grid points.
This is achieved by a re-distancing step [13]. Depending on the used level-set framework,
the level-set function may fulfill the signed distance property
∀x∈∇φ(x)=1,(3)
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Journal of Scientific Computing (2023) 94 :71 Page 3 of 21 71
after the re-distancing step. This is achieved by using, for example, the fast marching method
(FMM) [14]. Level-set frameworks based on the sparse field approach drop the signed dis-
tance property for a faster re-distancing algorithm [15]. To further improve performance,
so-called narrow-band methods are used that only calculate the φ-values for grid points that
are required for the calculation of finite differences [16].
Topographies originating from process TCAD simulations are often characterized by
large, essentially flat or only slightly bent areas, and small areas with pronounced geometric
variation. We refer to these areas with pronounced geometric variation as features. These
properties of the topography motivate the use of hierarchical domain refinement such that only
areas around a feature are resolved in more detail, to minimize the impact on performance
[17]. The selective refinement of the simulation domain is a commonly used strategy for
handling practically relevant numerical simulations [18–23]. To that end and the focus of
this work, a new algorithm is needed to detect the dynamic changes in the wafer surface
(features) during topography simulation, enabling external fast and efficient hierarchical
domain refinement.
Geometric feature detection or extraction of three-dimensional (3D) data sets is a widely
studied field, where the surface curvature, which describes the local changes in the geometry,
is used to achieve feature detection [24–27]. The curvature of surfaces is of high interest in
other computational domains as well, such as fluid dynamics, where the relation between
surface curvature and surface tension is of interest [28]. However, in these applications the
numerical accuracy of the calculated curvature values is the most important factor, whereas
for feature detection the computation time of the curvature calculation is of equal or greater
concern. Recently, machine learning approaches have been proposed to approximate the
curvature of the zero level-set of two dimensional level-set functions [29]. Nonetheless,
these methods require the same number of grid points as those based on finite differences.
Moreover, the machine learning model has to be re-trained for each value of x,whichis
a user-supplied parameter in most process TCAD simulations and thus of limited practical
use.
In this work, we tackle the problem of efficient and automatic feature detection for hierar-
chical grid refinement. We present a parallel topographical feature detection algorithm based
on the curvatures of the zero level-sets of 3D level-set functions. Our algorithm uses comple-
mentary methods of calculating the curvatures depending on the applied process model (i.e.,
the velocity field) to provide robust and accurate feature detection. We show that in particular
two curvature calculation methods are of interest due to their complementary application
goals: accuracy or performance. We discuss the run-time of the feature detection and eval-
uate our feature detection algorithm on several geometries originating from process TCAD
simulations. Furthermore, we use these geometries to calibrate a numerical feature detection
parameter tailored to process TCAD simulations.
2 Feature Detection Algorithm
Before we describe the feature detection algorithm, we discuss the curvatures of a 3D surface.
Furthermore, to ensure the robustness of the feature detection algorithm we discuss the impact
of minimal surfaces. These considerations allow us to define feature points on the zero level-
set.
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2.1 Curvatures of a 3D Surface
A point on a 3D orientable surface (e.g., zero level-set) has two principal curvatures κ1and
κ2[30]. The principal curvatures define the mean curvature
H=κ1+κ2
2,(4)
and the Gaussian curvature
K=κ1κ2.(5)
The mean and Gaussian curvatures of the zero level-set can be calculated directly from the
φ-values using various methods which are presented in Sect. 3. The principal curvatures can
be calculated from the mean and Gaussian curvatures [30]
κ1=H+H2−K,(6)
κ2=H−H2−K.(7)
The maximal mean curvature a level-set function can describe is bound by ±1/x[3], since
the smallest circle radius which can be represented with a given resolution is x.
The mean curvature captures the local geometric variation of the zero level-set for most
geometries that typically occur during level-set based simulations. Thus, we will focus our
primary discussion on the mean curvature and only return to the Gaussian curvature to
improve the robustness of the feature detection.
2.2 Minimal Surfaces
A minimal surface is one that has a mean curvature Hof zero. However, κ1and κ2might
not necessarily be equal to zero on each point on the surface (see Eq. 4). In this case, every
point on a minimal surface is a saddle point. Examples of such a surface are the catenoid, the
helicoid, or the plane [31]. It is possible that segments of the zero level-set locally describe
a minimal surface that is not a plane (e.g., a local saddle point). Thus, it is essential to detect
these segments of the zero level-set as features even though their mean curvature is zero.
Simply considering the mean curvature of the zero level-set of a 3D level-set function is not
sufficient to determine if a surface point is a feature or not. Only if a point on the zero level
set fulfills H=K=0 its principal curvatures also fulfill κ1=κ2=0, thus, the point is
part of a plane. Therefore, if a point on the zero level-set has a mean curvature of H=0and
its Gaussian curvature is K= 0 it is part of a minimal surface and should be considered part
of a feature.
2.3 Feature Definition
Before describing the feature detection algorithm, we first need to specify what parts of
the zero level-set we consider to be a feature. When the zero level-set locally describes a
plane, then H=K=0. These regions have no geometric variation and are not features
for our purposes. Consequently, a point on the zero level-set is considered to be a feature
if it is not part of a plane, i.e., H= 0orK= 0. However, this definition is too strict
when considering the following: (i) The curvature values calculated for the zero level-set are
numerical approximations, thus small numerical errors can cause the calculated values to
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Journal of Scientific Computing (2023) 94 :71 Page 5 of 21 71
deviate from an analytical plane. (ii) If the zero level-set only bends slightly, e.g., has a small
numerical mean curvature value, it should not be considered a feature since the geometric
variation is smooth enough to not negatively impact the propagation of the zero level-set
when solving (2). We thus define a feature of a level-set function as a point on the zero-level
set that has an absolute mean curvature value higher than a threshold parameter C>0and
is not part of a plane.
This approach is similar to calculating the curvedness of a surface [32]. However, our
approach is more computationally efficient since it avoids several additional calculation
required for the curvedness. In our approach the principal curvatures do not need to be
explicitly calculated, additionally our approach avoids calculating the Gaussian curvature if
the mean curvature of a point is not 0.
2.4 Algorithm
The features of the zero level-set are detected by processing the grid points. For each grid
point on the zero level-set the absolute mean curvature |H|is calculated. If |H|is bigger
than the feature threshold parameter C, then the point on the zero level-set is considered to
be a feature. Otherwise, if |H|<Cthe point is checked if it is part of a minimal surface
by calculating the Gaussian curvature Kand comparing it against |K|>C. Therefore, the
algorithm detects a feature when either of the following conditions is met:
|H|>C,or
|H|<Cand |K|>C.(8)
The feature detection algorithm is parallelized in a straightforward manner since it only
depends on the curvature calculation which only needs to access the φ-values (provided by
the re-distancing step). In this work we use the domain decomposition approach presented
in [33].
3-Value Based Curvature Calculation Methods
In this section we discuss three established methods and introduce one novel method for
calculating the curvatures of the zero level-set with finite differences. To achieve a better
understanding of the performance of the different methods we discuss the required stencils
to calculate the curvatures. Our novel method of calculating the curvatures of the zero level-
set uses more grid points than the standard methods without increasing the size of the finite
difference stencil. This is achieved by incorporating under-utilized information from points
already present in the stencil. The primary computational effort of the feature detection is the
movement of the finite difference stencil over the grid points inside the narrow-band. Thus, the
size of the finite difference stencil is significant when considering computational performance.
Furthermore, this analysis allows us to choose complementary curvature calculation methods
dependent on the surface properties of the level-set functions in the feature detection algorithm
(see Sect. 2.4).
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71 Page 6 of 21 Journal of Scientific Computing (2023) 94 :71
Fig. 1 Plane stencil used to calculate the mean curvatures of the zero level-set
3.1 General Formula
The most common method to calculate the mean curvature of the zero level-set is the general
formula for implicit surfaces [34]:
H=
Dx(φ)2(Dyy (φ) +Dzz (φ))
+Dy(φ)2(Dxx(φ) +Dzz (φ))
+Dz(φ)2(Dxx(φ) +Dyy (φ))
−2Dx(φ) Dy(φ)Dxy(φ)
−2Dy(φ) Dz(φ)Dyz(φ)
−2Dz(φ) Dx(φ)Dxz(φ)
2∇φ3.(9)
The derivatives required in (9) are calculated by using the following finite difference approx-
imations
Dx(φi,j,k)≈φi+1,j,k−φi−1,j,k
2x,(10)
Dxx(φi,j,k)≈φi+1,j,k−2φi,j,k+φi−1,j,k
x2,(11)
Dxy(φi,j,k)≈φi+1,j+1,k−φi−1,j+1,k−φi+1,j−1,k+φi−1,j−1,k
4xy,(12)
where φi,j,kstands for the φ-value of the level-set function at the grid point (i,j,k). We will
refer to this method of calculating the mean curvature as the General Formula method.
These finite differences require the level-set values of 19 different grid points to calculate
the mean curvature of the zero level-set. These 19 points lie in the three intersecting planes
that are created by the three coordinate direction (plane stencil) as shown in Fig. 1.
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Journal of Scientific Computing (2023) 94 :71 Page 7 of 21 71
Fig. 2 Star stencil used to calculate curvatures of the zero level-set
The Gaussian curvature can be calculated in a similar way to the mean curvature [34]:
K=
Dx(φ)2(Dyy (φ) Dzz (φ) −Dyz (φ)2)
+Dy(φ)2(Dxx(φ) Dzz (φ) −Dxz(φ )2)
+Dz(φ)2(Dxx(φ) Dyy(φ) −Dxy(φ)2)
+2Dx(φ) Dy(φ)(Dxz(φ) Dyz (φ) −Dxy(φ) Dzz (φ))
+2Dx(φ) Dz(φ)(Dxy(φ) Dyz (φ) −Dxz(φ) Dyy(φ))
+2Dy(φ) Dz(φ)(Dxy(φ) Dxz(φ ) −Dyz(φ) Dxx(φ))
(Dx(φ)2+Dy(φ)2+Dz(φ)2)2.(13)
Calculating the Gaussian curvature requires the same derivatives as calculating the mean
curvature thus no changes in the stencil are required.
3.2 Shape Operator
When the level-set function φfulfills the signed distance property (see Eq. 3) the mean
curvature can be calculated as follows [3]
H=trace(Hφi,j,k)
2≈Dxx(φi,j,k)+Dyy(φi,j,k)+Dzz (φi,j,k)
2,(14)
where Hφi,j,kstands for the Hessian of φat the grid point (i,j,k). We will refer to this
method as the Shape Operator method.
Calculating the mean curvature of the zero level-set using the Shape Operator method
only requires the level-set values of 7 grid points, which lie on the grid lines that pass trough
the grid point (i,j,k)(star stencil) as shown in Fig. 2.
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71 Page 8 of 21 Journal of Scientific Computing (2023) 94 :71
The Shape Operator method is an often overlooked method of calculating the mean
curvature of a level-set function: In performance-oriented applications, usually the signed
distance property of the level-set function is approximated with a first-order scheme (e.g.
using FMM). Which, introduces numerical errors into the calculated mean curvature values.
However, in Sect. 4we demonstrate that this error is negligible when this method is used for
feature detection.
Alongside with requiring a smaller stencil, the Shape Operator method has an additional
performance advantage: It is possible to avoid explicitly calculating the Gaussian curvature to
determine if a point on the zero level-set is a feature. Both conditions of the feature detection
algorithm (8) are simultaneously checked if:
|Dxx(φi,j,k)|+|Dyy(φi,j,k)|+|Dzz (φi,j,k)|>2C.(15)
Let the grid point (i,j,k)be part of a minimal surface that is not a plane, so H=0and
without loss of generality:
−Dyy(φi,j,k)=Dxx(φi,j,k)+Dzz (φi,j,k)= 0.(16)
Thus (15) only holds if Dxx (φi,j,k)=Dyy(φi,j,k)=Dzz (φi,j,k)≈0, which is fulfilled if
the considered point on the zero level-set is a feature.
3.3 Variation of Normal
The third method calculates the mean curvature of the zero level-set by calculating the Euler-
Lagrange derivative of the normal vector [35,36]
H=∇· ∇φ
∇φ≈1
2xD+
x(φ)
g+
x−D−
x(φ)
g−
x+
D+
y(φ)
g+
y−D−
y(φ)
g−
y+D+
z(φ)
g+
z−D−
z(φ)
g−
z,(17)
where g±
xis defined as follows:
g±
x:= (D±
x(φi,j,k), 1
2(Dy(φij,k)+Dy(φi±1,j,k)),
1
2(Dz(φi±1,j,k)+Dz(φi,j,k))), (18)
g±
y:= (D±
y(φi,j,k), 1
2(Dx(φi,j±1,k)+Dx(φi,j,k)),
1
2(Dz(φi,j±1,k)+Dz(φi,j,k))), (19)
g±
z:= (D±
z(φi,j,k)), 1
2(Dx(φi,j,k±1)+Dx(φi,j,k)),
1
2(Dy(φi,j,k±1)+Dy(φi,j,k))). (20)
To calculate g±
xthe first-order finite forward and backwards differences have to be calculated
by using the finite difference formulas:
D+
x(φi,j,k)≈φi+1,j,k−φi,j,k
x,(21)
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Journal of Scientific Computing (2023) 94 :71 Page 9 of 21 71
D−
x(φi,j,k)≈φi,j,k−φi−1,j,k
x.(22)
This method again requires a 19-point plane stencil (see Fig. 1) to calculate the mean curvature
of the zero level-set. We are going to refer to this method as the Variation of Normal method.
To calculate the Gaussian curvature, the missing derivatives for (13) have to be calculated.
3.4 Big Stencil
Considering that a 19 point finite difference stencil is present to calculate the finite differences
to determine the mean curvature of the zero level-set (see Sect. 3.1 and Fig. 1), we propose
using an existing, more accurate finite difference approximation [37] to calculate the mean
curvature. The number of grid points used to calculate the finite differences Dxand Dxx can
be increased without increasing the stencil by using the finite difference formulas
˜
Dx(φi,j,k)≈φi+1,j+1,k−φi−1,j+1,k+φi+1,j−1,k−φi−1,j−1,k
4xand (23)
˜
Dxx(φi,j,k)≈
φi+1,j+1,k−2φi,j+1,k+φi−1,j+1,k+φi+1,j,k−2φi,j,k
+φi−1,j,k+φi+1,j−1,k−2φi,j−1,k+φi−1,j−1,k
3x2.(24)
When the mean curvature is calculated using (9) and the finite difference approximations
(12), (23), and (24) this novel method is henceforth referred to as the Big Stencil method.
This new method uses the same finite difference stencil as the General Formula method (9)
(see Fig. 1). However, it utilizes more information of the level-set function to calculate the
derivatives required in (9). Thus, the numerical accuracy of the calculated mean curvature
values is improved. The Gaussian curvature can simply be calculated by using the more
accurate finite difference approximations and (13).
4 Results
We investigate the applicability of the curvature calculation methods presented in Sect. 3
and the presented feature detection algorithm in Sect. 2for several representative geometries
(Sect. 4.1). We run a parameter search for the parameter Cto find a suitable parameter value
for process TCAD simulations (Sect. 4.2) and perform an empirical evaluation of the calcu-
lated curvature values and detected features (Sect. 4.3). Furthermore, the performance of the
curvature calculation methods is evaluated (Sect. 4.4). We implemented our feature detec-
tion algorithm in the open-source topography simulator ViennaTS [38] and used OpenMPfor
parallelization. The presented benchmarks have been compiled with GCC 9.3.0 and executed
on a single node of VSC-41, which is equipped with two 24-core Intel Xeon Platinum 8174
processors.
4.1 Geometries and Mean Curvature Values
The first geometry we investigate is a sphere, where the mean curvature value can be analyt-
ically calculated and subsequently compared to numerical approximations. Most geometries
1https://vsc.ac.at//systems/vsc-4/
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Fig. 3 Mean curvature values for a sphere with radius 9.7 and grid resolution 0.27 after 5 time units
that occur during level-set based simulations, Most geometries that occur during level-set
based simulations, however, do not have a readily available analytical solution.
however, do not have a readily available analytical solution. Therefore, to evaluate the
applicability of our methods for realistic scenarios, we additionally explore two representative
and technologically relevant examples from the field of process TCAD: A stacked nanosheet
field-effect transistor (FET) [39] and a selectively grown epitaxial crystal [10].
The analytical mean curvature of a sphere with radius ris known to be 1/r. As previously
discussed, every step in a level-set based simulation is a numerical approximation, often of
first-order, which introduces small but compounding numerical errors. Thus, we subject the
sphere to a velocity field to make the calculated mean curvature values comparable to values
that occur during a simulation workflow. We initialize the sphere with a radius of 15. After
initialization, the sphere is subjected to a constant velocity field with a velocity of −1for
5 time units so that, analytically, the final sphere should have a radius of 10 and a mean
curvature Hof 0.1. A first-order Osher scheme is used to solve the level-set equation [11].
The distribution of the calculated mean curvature values of all points on the zero level-set,
for all methods introduced in Sect. 3, is shown in Fig. 3. Due to numerical errors introduced
by the used scheme, the distance from all points on the zero level-set to the center of the
sphere is between 9.9and9.6, with an average distance of 9.7. The smaller radius compared
to the analytical sphere is explained by a loss in volume originating from the normalization
into the discrete grid. Thus, we expect the calculated mean curvature values to be distributed
around the curvature of a sphere with radius 9.7, which is indicate by the black line in Fig. 3.
The mean curvature values computed with the General Formula and the Variation of
Normal methods do not differ in a significant way from each other. The computed values
using the Shape Operator show the largest deviation from the value of the analytical solution.
This is expected from the reduced size of its stencil, as discussed in Sect. 3.2. The results
obtained from the Big Stencil method provide the best match to the expected values from
the analytical solution. All curvature calculation methods except for Big Stencil tend to
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Journal of Scientific Computing (2023) 94 :71 Page 11 of 21 71
Fig. 4 Calculated mean curvature values of two material layers of a stacked nanosheet FET. Calculated with
Shape Operator
Fig. 5 Distribution of the calculated absolute mean curvature values of the two material layers of a stacked
nanosheet FET shown in Fig. 4
overestimate the mean curvature of the shrunken sphere. Calculations done with the Shape
Operator method overestimate the mean curvature up to 75%. However, from our empirical
analysis, we postulate that the Shape Operator method is still accurate enough to be viable
for the purpose of feature detection, as will be further discussed in Sect. 4.3.
The next geometries under investigation originate from an exemplary process TCAD sim-
ulation of a stacked nanosheet FET [39]. A first-order Osher scheme is used to solve the
level-set equation. As previously discussed in Sect. 1, different level-set functions represent-
ing the individual material layers are combined in a layer wrapping to model the topography
of a semiconductor device. Therefore, it is sufficient to only investigate certain material lay-
ers, since the features of the underlying layers are already captured in the upper material layer.
Figure 4shows two representative material layers (layer 3 and layer 5) after the 24th process
step of a stacked nanosheet FET fabrication simulation (see Table II of [39]). To arrive at the
geometries shown in Fig. 4each material layer has undergone significant geometric changes
due to the employed velocity fields. These changes generate complicated geometries with
pronounced features. The distribution of the absolute mean curvature values in these layer
functions are shown in Fig. 5.
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71 Page 12 of 21 Journal of Scientific Computing (2023) 94 :71
(a) General Formula method
−1.5
0
+1.5
(b) Big Stencil method
Fig. 6 Calculated mean curvature values of the zero level-set of the material layer of a heteroepitaxially grown
SiGe crystal on a Si fin. The noise on the crystal facets introduced by the crystallographic orientation-dependent
velocity field is reduced when using the Big Stencil method
The calculated mean curvature values for the two material layers of the stacked nanosheet
FET shown in Fig. 5display a similar behavior as on the sphere in Fig. 3. Namely, the
Shape Operator method overestimates the mean curvature values compared to the other
calculation methods which behave in a similar way. A significant amount (about 96%) of
the absolute mean curvature values fall between 0.0and0.5, which indicates that most of
the geometry does not have sharp features and is primarily flat or slightly bent. The absolute
mean curvature values in the indicated band between the values of 0.0and0.5 agree to a large
extent, except for some runaway values when using the Shape Operator method. However, as
will be discussed later (see Sect. 4.3), the Shape Operator method is still able to distinguish
between the features and flat parts of the geometries.
Crystallographic orientation-dependent growth requires velocity fields that depend upon
special interpolation schemes to calculate the velocities for all points on the zero level-set
[40]. Simulating such processes results in level-set functions with sharp corners which have
to be maintained throughout a simulation to uphold accuracy. Furthermore, the simulation
of such growth processes result in non-convex Hamiltonians that require a special numerical
dissipation scheme to solve (2)[10]. Figure 6shows the SiGe material layer of a heteroepi-
taxially grown SiGe crystal on a Si fin. The distribution of the calculated absolute mean
curvature values for the SiGe material layer are shown in Fig. 7.
Figure 6shows that the calculated mean curvature values of the facets, i.e., the planes
formed by the crystallographic growth, are higher than expected when using the General
For m u la (see Fig. 6a). This is due to noise that is introduced by the numerical technique to
solve this physical process. However, when we consider the mean curvature values calculated
by the Big Stencil method (see Fig. 6b), the effects of the noise on the mean curvature
values is reduced significantly. The lesser impact of noise and the thus created difference in
calculated mean curvature values are explained by the additional grid points that are taken into
consideration when using (23) to approximate the second-order derivatives. Since the level-
set function is an approximation to a signed distance function, nearly all information about
the curvature is gained from the second-order derivatives in the same coordinate direction
(e.g., ˜
Dxx,...), see (14). Figure 7confirms this observation as there are more mean curvature
values close to zero and fewer mean curvature values larger than 2 for the SiGe material layer.
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Journal of Scientific Computing (2023) 94 :71 Page 13 of 21 71
Fig. 7 Distribution of the
calculated absolute mean
curvature values for the SiGe
material layer shown in Fig. 6
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
0
5000
10000
15000
20000
25000
30000
Feature Threshold Parameter C
Flagged Grid Points
General Formula
Big Stencil
Shape Operator
Variation of Normal
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
0
5000
10000
15000
20000
25000
30000
Feature Threshold Parameter C
Flagged Grid Points
General Formula
Big Stencil
Shape Operator
Variation of Normal
Fig. 8 Flagged points of the zero level-set compared to the chosen feature detection parameter Cfor the
stacked nanosheet FET shown in Fig. 4
4.2 Parameter Study
To determine an appropriate choice for the feature threshold parameter C(8) (which will be
used in Sect. 4.3), we conduct a parameter search which starts with a value of 0.01 up to
1/xwith a step length of 0.01. Figure 8shows the number of flagged grid points compared
to the chosen feature threshold parameter Cfor the two representative material layers of the
stacked nanosheet FET. Figure 9shows the results of the feature threshold parameter search
for the heteroepitaxially grown SiGe crystal material layer.
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71 Page 14 of 21 Journal of Scientific Computing (2023) 94 :71
Fig. 9 Flagged points of the zero
level-set compared to the chosen
flagging parameter Cfor the
heteroepitaxially grown SiGe
crystal shown in Fig. 6
As expected, the Shape Operator method identifies slightly more surface points as features
than the other three methods. Nevertheless, as will be discussed in Sect. 4.3, it still reliably
detects features. The other three methods identify approximately the same number of surface
points as features with only minor deviations. Figure 8suggests that the feature detection
parameter Cshould be between 0.1and1.0 since with smaller values the entire surface would
be flagged as a feature and, conversely, with a larger parameter some features may be missed.
Furthermore, Fig. 5shows that a feature detection parameter of C=0.5 identifies most
features and simultaneously avoids capturing only slightly bent parts of the zero level-set, as
discussed in Sect. 2.
The parameter search for the feature threshold parameter on the SiGe crystal shows a
different behavior (see Fig. 9) than for the stacked nanosheet FET. Using the Big Stencil
method for the curvature calculation initially identifies fewer points on the zero level-set as
features than other methods. As previously discussed (Sect. 4.1), the difference is explained
by the error in the mean curvature calculation introduced by the noise of the velocity field.
The amount of grid points flagged as features equalizes at a feature threshold parameter of
C=1.1 for all methods. However (see Fig. 11), feature detection using this feature threshold
parameter is not satisfactory: The noise in the level-set function is so large that the feature
detection is not able to detect the edges of the crystal as features. Yet, the numerical noise on
the crystal facet is still detected. Reconsidering the results shown in Fig. 9with these newly
gained insights, we infer that the Big Stencil method provides a substantially higher quality
in the feature detection. The amount of flagged grid points is constant between the feature
threshold parameters 0.4and1.1 when using the Big Stencil method. We thus pick 0.5(same
as in the previous case, the stacked nanosheet FET) as the feature threshold parameter Cfor
the following empirical evaluation.
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Journal of Scientific Computing (2023) 94 :71 Page 15 of 21 71
(a) Shape Operator material layer 3 (b) Shape Operator material layer 5
(c) Big Stencil material layer 3
Flat
Feature
(d) Big Stencil material layer 5
Fig. 10 Detected features of the stacked nanosheet FET using a feature detection parameter Cof 0.5
4.3 Empirical Evaluation
This section evaluates if our proposed feature detection algorithm combined with the feature
detection parameters Cidentified in Sect. 4.2 reliably detects the features of the investigated
geometries. Figure 10 shows the detected features for the stacked nanosheet FET with a
feature detection parameter of 0.5usingtheShape Operator and the Big Stencil methods,
the features detected by the other methods are similar to these two methods. Figures 11 and
12 show the detected features of the SiGe crystal with a feature detection parameter of 1.1and
0.5, respectively. We only show the results of the feature detection for the General Formula
and Big Stencil method in Fig. 12 as the results of the Shape Operator,andVariation of
Normal are similar to the results of the General Formula.
Figure 10 shows that a feature detection parameter of 0.5 is able to distinguish features
form flat parts of the geometry, thus, confirming that it is a good choice for reliably detecting
the feature of the stacked nanosheet FET.
The features detected in Fig. 11a–c show that the feature detection parameter of 1.1is
too large: This particular parameter choice does not allow for identifying the features at the
edges of the SiGe crystal, while still falsely detecting points on the facets as features. The
Big Stencil method does not detect any features on crystal facets but, due to the high feature
detection parameter, it also fails to capture several grid points on the edges that should be
detected as features, see Fig. 11d.
Figure 12 shows that the Big Stencil method is able to reliably detect the features of the SiGe
crystal with a feature detection parameter of 0.5, while the other methods wrongly detect the
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71 Page 16 of 21 Journal of Scientific Computing (2023) 94 :71
Fig. 11 Detected features of a SiGe crystal with a feature detection parameter Cof 1.1
Fig. 12 Detected features of a SiGe crystal with a feature parameter Cof 0.5
noise on the crystal facet as features. These observations suggest that it can be advantageous
to use complementary curvature calculation methods dependent on the requirements of the
application: The Shape Operator method reliably detects the features of the geometries
relying on simple velocity fields as used, for example, for the fabrication simulation of the
stacked nanosheet FET. In applications where the accuracy of the mean curvature is of utmost
importance, the Big Stencil method should be used in virtue of its higher accuracy. This is
the case, e.g., for non-convex Hamiltonians originating from crystallographic orientation-
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Journal of Scientific Computing (2023) 94 :71 Page 17 of 21 71
Fig. 13 Run-time and speedup of the feature detection algorithm on a sphere with radius 9.7 and grid resolution
0.27
Fig. 14 Run-time and speedup of the feature detection algorithm for all material layers of the stacked nanosheet
FET after 24 process steps
dependent velocity fields, where the Big Stencil method reduces the effects of noise on the
curvature calculation.
4.4 Performance
This section evaluates the averaged performance (run-time and parallel speedup) of our
feature detection algorithm based on the geometries introduced in Sect. 4.1. For this analysis
all curvature calculation methods presented in Sect. 3are considered.
As can be seen in Figs. 13,14,and 15,theShape Operator method is the fastest method
while the other three methods perform similarly to each other. The speedup in Fig. 13aisa
reflection of the best case scenario, since the convexity of the geometry enables the domain
decomposition algorithm to create sub-domains with approximately the same number of
surface points. This optimal parallelization scenario is of course not the case for practically
relevant cases.
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71 Page 18 of 21 Journal of Scientific Computing (2023) 94 :71
Fig. 15 Run-time and speedup of the feature detection algorithm for the SiGe crystal
The principal differences in run-time and speedup performance between Figs. 14 and
15 as compared to Fig. 13 are due to load-balancing issues in the domain decomposition:
Even though the domain decomposition splits the narrow-band into roughly equally sized
sub-domains (considering the entire grid), the number of surface points differs in each sub-
domain.
In addition, the primary computational effort of the mean curvature calculations of the
zero level-set is the application of the finite difference stencil. This fundamental process is
inherently memory-bound and thus detrimental for parallel speedup.
The performance analysis reinforces the observations made in Sect. 4.3: Due to its higher
accuracy and insignificant increase in run-time compared to the other methods the Big Stencil
method should be the default choice only for complicated velocity fields where accuracy is
paramount. However, for applications where the accuracy is a less significant factor and the
primary concern is performance, the Shape Operator method is the optimal choice.
5 Conclusion
We introduce a parallel automatic feature detection algorithm for 3D level-set functions with a
focus on computational performance, efficiency, and accuracy. The algorithm is based on two
different but complementary methods for calculating the curvatures of the zero level-set. Our
algorithm can utilize the optimal curvature calculation method based on the properties of the
velocity field that is used to propagate the zero level-set. The Shape Operator method uses the
signed distance property of the level-set function to calculate the mean curvature. This allows
the Shape Operator method to use the smallest possible finite difference stencil to calculate
the mean curvature of the zero level-set. It is, therefore, the fastest method to detect geometric
features and is used for structures with little to no noise in the level-set layer functions such
as found in the discussed simulation of an exemplary stacked nanosheet FET. The Big Stencil
method can better cope with the noise introduced by crystallographic orientation-dependent
velocity fields to allow for the exact detection of crystal facets. Thus, the Big Stencil method
should be favored for complicated process models involving, e.g., strong dependencies on
the surface normal. Furthermore, we determined that a good choice for the feature detection
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Journal of Scientific Computing (2023) 94 :71 Page 19 of 21 71
parameter Cis 0.5 in process TCAD simulations for both complementary surface curvature
methods, considering the discussed, representative examples.
We compare the methods in our algorithm with two other methods of calculating the mean
curvature of the zero level-set in our investigations: The General Formula method, and the
Variation of Normal method. In our experiments, these two methods perform comparably
to each other regarding accuracy and performance. However, the Shape Operator method
significantly outperforms both methods in terms of run-time while being sufficiently accurate
for feature detection. Simultaneously, the Big Stencil method has an equivalent run-time to
both methods yet it has an overall higher numerical accuracy and is less susceptible to models
which introduce numerical noise.
Acknowledgements The financial support by the Austrian Federal Ministry for Digital and Economic Affairs,
the National Foundation for Research, Technology and Development,andtheChristian Doppler Research
Association is gratefully acknowledged. The computational results presented have been achieved using the
Vienna Scientific Cluster (VSC).
Author Contributions All authors contributed to the study conception and design. The first draft of the
manuscript was written by CL and all authors commented on previous versions of the manuscript. All authors
read and approved the final manuscript.
Funding Open access funding provided by TU Wien (TUW).
Data Availibility Statement The data sets generated and analysed during the current study are available from
the corresponding author on reasonable request.
Declarations
Conflict of interest The authors have no relevant financial or non-financial interests to disclose.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence,
and indicate if changes were made. The images or other third party material in this article are included in the
article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is
not included in the article’s Creative Commons licence and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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