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About this book

Designs in nanoelectronics often lead to challenging simulation problems and include strong feedback couplings. Industry demands provisions for variability in order to guarantee quality and yield. It also requires the incorporation of higher abstraction levels to allow for system simulation in order to shorten the design cycles, while at the same time preserving accuracy. The methods developed here promote a methodology for circuit-and-system-level modelling and simulation based on best practice rules, which are used to deal with coupled electromagnetic field-circuit-heat problems, as well as coupled electro-thermal-stress problems that emerge in nanoelectronic designs. This book covers:

(1) advanced monolithic/multirate/co-simulation techniques, which are combined with envelope/wavelet approaches to create efficient and robust simulation techniques for strongly coupled systems that exploit the different dynamics of sub-systems within multiphysics problems, and which allow designers to predict reliability and ageing;

(2) new generalized techniques in Uncertainty Quantification (UQ) for coupled problems to include a variability capability such that robust design and optimization, worst case analysis, and yield estimation with tiny failure probabilities are possible (including large deviations like 6-sigma);

(3) enhanced sparse, parametric Model Order Reduction techniques with a posteriori error estimation for coupled problems and for UQ to reduce the complexity of the sub-systems while ensuring that the operational and coupling parameters can still be varied and that the reduced models offer higher abstraction levels that can be efficiently simulated.

All the new algorithms produced were implemented, transferred and tested by the EDA vendor MAGWEL. Validation was conducted on industrial designs provided by end-users from the semiconductor industry, who shared their feedback, contributed to the measurements, and supplied both material data and process data. In closing, a thorough comparison to measurements on real devices was made in order to demonstrate the algorithms’ industrial applicability.

Table of Contents


Chapter 1. Nanoelectronic Coupled Problems Solutions – Highlights from the nanoCOPS Project

This introductory chapter summarizes the objectives and the highlights of the FP7-ICT project nanoCOPS from which the outcomes of this book emerged. It is based on the Brochure that was made at the end of the project to broadly advertise the achievements of the project. We identify the role of mathematics to solve the coupled problems that were posed by the partners from semiconductor industry. The size of the problems asked for developing reduction techniques that allowed for parameterization. Applications needed uncertainty quantification for reliability purposes and optimization. All methods developed have been tested and validated on industrial test examples.
Caren Tischendorf, E. Jan W. ter Maten, Wim Schoenmaker

Equations, Discretizations


Chapter 2. EM-Equations, Coupling to Heat and to Circuits

In this chapter we present the underlying physical equations of the systems that are addressed in the remainder of this book. Two kind of couplings are considered, in particular: (1) the coupling of electromagnetic fields with circuits and (2) the electro-thermal coupling in power transistors. These cases are not exhaustive but are used as typical for identifying generic guidelines for addressing coupled problems.
Wim Schoenmaker, Hans-Georg Brachtendorf, Kai Bittner, Caren Tischendorf, Christian Strohm

Chapter 3. Bond Wire Models

In this chapter, models for the heating in bond wires are presented. First, an analytic model is developed, which allows a fast characterisation of the bond wire properties, e.g., its time to failure. Secondly, a more accurate model using the finite integration technique for spatial discretisation is constructed. This model is also used to study the impact of manufacturing uncertainties, e.g., variable bond wire lengths, on the performance of bond wires.
David J. Duque Guerra, Thorben Casper, Sebastian Schöps, Herbert De Gersem, Ulrich Römer, Renaud Gillon, Aarnout Wieers, Tomas Kratochvil, Tomas Gotthans, Peter Meuris

Chapter 4. Discretizations

This chapter discusses the techniques for converting continous systems to discrete systems. Specific attention is paid to the mimetic discretization approach. Furthermore the set up of coupled problems is addressed. Finally, it is shown that discretization requires a careful handling of the various terms in the discretization procedure, otherwise numerical unstable equations will appear.
Wim Schoenmaker, Hans-Georg Brachtendorf, Kai Bittner, Caren Tischendorf, Christian Strohm

Chapter 5. Automated Generation of Netlists from Electrothermal Field Models

The equivalence between field and circuit models is shown and formalised. Accordingly, an algorithm for the automated generation of circuit netlists out of field models is presented. Such netlists can be used by standard circuit simulators. The field-to-netlist extraction is organised for coupled electrothermal field models. The resulting circuit model is an exact representation of the semi-discrete field model. A 3D electrothermal field model of a microelectronic chip package serves as an illustration of the overall procedure.
Thorben Casper, David J. Duque Guerra, Sebastian Schöps, Herbert De Gersem

Time Integration for Coupled Problems


Chapter 6. Holistic / Monolithic Time Integration

This chapter discusses a number of techniques to solve the discrete systems of equations that result from the coupling of the circuit and electromagnetic field equations. We briefly summarize the modified nodal analysis for the lumped modeling of circuits. Then we discuss the coupling relations for including EM models into the circuit simulation systems and combine them with the spatially discretized Maxwell equations. Finally, we apply an adaptive time stepping scheme to the resulting coupled differential-algebraic equation system and in particular focus on a number of technical steps to find the solutions of the discretized coupled equations.
Wim Schoenmaker, Hans-Georg Brachtendorf, Kai Bittner, Caren Tischendorf, Christian Strohm

Chapter 7. Non-Intrusive Methods for the Cosimulation of Coupled Problems

Many industrial applications require transient simulations of coupled problems. The coupling can involve multiples physical disciplines, multiple scales in space and time and different requirements on simulation accuracy and efficiency. In many cases, the coupling necessitates a decoupling of the time-stepping procedure, either because different software packages each have their own time stepper, or in order to improve the overall simulation efficiency, e.g. if multirate behaviour is present. In this chapter, two advanced cosimulation approaches are presented: dynamic iteration (waveform relaxation) and a fractional splitting equipped with error correction. These schemes provide an efficient numerical set-up despite large differences in time scales or the high nonlinearity of the coupling terms. The methods are illustrated for a field-circuit coupled magnetoquasistatic finite-element model of a power transformer and for an electrothermal finite-integration model of an electronic package.
Sebastian Schöps, David J. Duque Guerra, Herbert De Gersem, Andreas Bartel, Michael Günther, Roland Pulch

Chapter 8. Multirate Circuit - EM - Device Simulation

Radio frequency (RF) integrated circuits are at the core of modern mobile communication. They basically comprise the analog front-end, the analog-to-digital and vice versa the digital-to-analog conversion, and the digital signal processing. These days, both the analog and digital parts, are integrated on the same die. The analog front-end mainly performs amplification, filtering and mixing to or vice versa from the RF regime to baseband. The waveforms of voltages and currents of such an IC are described by a system of ordinary Differential-Algebraic Equations (DAEs) resulting from the well-known Modified Nodal Analysis (MNA). Standard solvers for initial value problems, also referred to as transient analysis, are however prohibitively slow, since the time step or vice versa its inverse the sampling rate, are limited by Shannon’s sampling theorem. The sampling theorem predicates that the sampling rate must be at least twice as high as the highest relevant frequency components of the spectra of all waveforms of the circuit. Since modern RF integrated circuits operate in the GHz range, solving the initial value problem of these DAEs is extremely slow. This chapter addresses the simulation problem of RF circuits by generalizing the method of the Equivalent Complex Baseband (ECB) for circuits and systems described by nonlinear DAEs.
Kai Bittner, Hans-Georg Brachtendorf

Uncertainty Quantification


Chapter 9. Uncertainty Quantification: Introduction and Implementations

This chapter provides a short introduction to Uncertainty Quantification based on Monte Carlo simulations, on Generalized Polynomial Chaos expansions and on Worst Case Corner Analysis. Furthermore, it also covers remarks how the chosen implementation should fit within existing simulation environments.
Roland Pulch, Piotr Putek, E. Jan W. ter Maten, Wim Schoenmaker

Chapter 10. Robust Shape Optimization under Uncertainties in Device Materials, Geometry and Boundary Conditions

We address the shape optimization problem of electronic and electric devices under geometrical and material uncertainties. Thereby, we aim at reducing undesirable phenomena of these devices such as hot-spots or torque fluctuations. The underlying minimization is based on the computation of a direct problem with random input data. To investigate the propagation of uncertainties through two- and three-dimensional, spatial models the stochastic collocation method (SCM) has been used in our work. In particular, uncertainties, which result from imperfections of an industrial production, are modelled by random variables with known probability distributions. Then, the polynomial chaos expansion (PCE) is used to construct a suitable response surface model, which can be effectively incorporated into the robust optimization framework. Correspondingly, the gradient directions of a cost functional, comprised of the expectation and the variance value, are calculated using the continuum design shape sensitivity and the PCE in conjunction with the SCM. Finally, the optimization results for the relevant electronics/electrical engineering problems demonstrate that the proposed method is robust and efficient. Overall, this work demonstrates, how recent techniques from shape and topology optimization can be combined with uncertainty quantification to solve complicated real-life problems.
Piotr Putek, E. Jan W. ter Maten, Michael Günther, Andreas Bartel, Roland Pulch, Peter Meuris, Wim Schoenmaker

Chapter 11. Going from Parameter Estimation to Density Estimation

A common approach to model variability in integrated circuits is to select a normal distribution for input variables and express variability as a non-linear function of the input variables. Even for simple non-linear functional forms as quadratic polynomials this causes the variability to no longer be normally distributed. It is thus important to be able to estimate the probability distribution of the output. In this chapter we give a brief introduction to the statistical theory of density estimation, which allows to estimate the density of a probability distribution without assuming a specific form of this density. In order to give a self-contained story accessible for non-statisticians, we first present the basic definitions and results of parameter estimation. Then we go beyond parametric estimation by discussing kernel density estimators in detail. We will indicate links with common notions from mathematical analysis such as convolution, Fourier analysis and approximation theory.
Alessandro Di Bucchianico

Chapter 12. Inverse Modeling: Glue-Package-Die Problem

In mathematical modeling, the physical or geometrical parameters are often affected by uncertainties. For example, imperfections of an industrial manufacturing generate undesired variations in the produced devices. We consider an uncertainty quantification, where parameters are replaced by random variables. Consequently, the probability distributions of the parameters have to be predetermined as an input to the stochastic model. However, the variability of input parameters is often not directly accessible by measurements, whereas the output quantities are available. We investigate a problem from nanoelectronics: a piece of glue connecting a die and a package. A randomness in both the formation and the quality of the piece of glue cause uncertain geometrical parameters and material parameters. The task consists in a fitting of input probability distributions for the random parameters to measurements of the output. This problem can be seen as a form of a stochastic inverse problem. The cumulative distribution function is approximated by a piecewise linear function. We apply a minimization, which yields a nonlinear least squares problem. Numerical results are illustrated for this test problem.
Roland Pulch, Piotr Putek, Herbert De Gersem, Renaud Gillon

Model Order Reduction


Chapter 13. Parametric Model Order Reduction for Electro-Thermal Coupled Problems

We consider automatic parametric model order reduction for electro-thermal (ET) coupled problems arising from (nano-)microelectronic simulations. We show that the PMOR method based on multi-moment matching proposed in [2] can be applied to the discretized ET models. The error bound in [6] is used to check the accuracy of the reduced models. Based on the error bound, an adaptive algorithm for computing the reduced ET models is proposed in [7]. As a result, the reduced ET models are constructed automatically and reliably. This chapter reviews the PMOR method, the error bound, the adaptive algorithm, and their applications to ET models.
Lihong Feng, Peter Benner

Chapter 14. Sparse (P)MOR for Electro-Thermal Coupled Problems with Many Inputs

This chapter reviews the recently proposed (parametric) model order reduction methods for electro-thermal (ET) coupled problems with many inputs, arising from (nano-)microelectronic simulation. For non-parametric problems, we discuss the block-diagonal structured model order reduction (MOR) methods (BDSM-ET) proposed in [6, 7]. For parametric problems, we discuss the parametric MOR method for ET coupled problems with many inputs (IpBDSM-ET) proposed in [3]. By construction, these methods lead to sparse reduced-order models (ROMs) and their efficiency is demonstrated using (non-)parametric ET coupled problems from industrial applications.
Nicodemus Banagaaya, Lihong Feng, Peter Benner

Chapter 15. Reduced Models and Uncertainty Quantification

Uncertainty quantification analyses the variability of system outputs with respect to process variations and thus represents a useful tool for robust design. Often statistics of a dynamical system’s outputs are quantities of interest. A sampling of the outputs requires many transient simulations. Due to the complexity of systems in nanoelectronics, methods of model order reduction (MOR) are applied for accelerating the uncertainty quantification. We consider coupled problems or multiphysics systems. We employ parametric MOR techniques to build parameter-dependent reduced-order models, which can be used for fast computations at all parameter samples. Samplingbased techniques like the Latin hypercube method, for example, or quadrature rules yield the parameter values. We apply this approach to an electrothermal coupled system. Furthermore, we illustrate a co-simulation technique with different quadrature grids for the subsystems. Now just some parts of a coupled problem are substituted by parametric MOR, if the others cannot be reduced efficiently. This method is applied to a circuit-electromagnetic coupled system.
Yao Yue, Lihong Feng, Peter Benner, Roland Pulch, Sebastian Schöps

Robustness, Reliability, Ageing


Chapter 16. Estimating Failure Probabilities

System failure describes an undesired configuration of an engineering device, possibly leading to the destruction of material or a significant loss of performance and a consequent loss of yield. For systems subject to uncertainties, failure probabilities express the probability of this undesired configuration to take place. The accurate computation of failure probabilities, however, can be very difficult in practice. It may also become very costly, because of the many Monte Carlo samples that have to be taken, which may involve time consuming evaluations. In this chapter we present an overview of techniques to realistically estimate the amount of Monte Carlo runs that are needed to guarantee sharp bounds for relative errors of failure probabilities. They are presented for Monte Carlo sampling and for Importance Sampling. These error estimates apply to both non-parametric and parametric sampling. In the case of parametric sampling we propose a hybrid algorithm that combines simulations of full models and approximating response surface models. We illustrate this hybrid algorithm with a computation of bond wire fusing probabilities.
E. Jan W. ter Maten, Theo G. J. Beelen, Alessandro Di Bucchianico, Roland Pulch, Ulrich Römer, Herbert De Gersem, Rick Janssen, Jos J. Dohmen, Bratislav Tasić, Renaud Gillon, Aarnout Wieers, Frederik Deleu

Chapter 17. Fast Fault Simulation for Detecting Erroneous Connections in ICs

Imperfections in manufacturing processes can be modelled as unwanted connections (defects, or faults) that are added to the nominal, "golden", fault-free design of an electronic circuit to study their impact. Testing in a structured way using fault simulation techniques to obtain information on the impact of faults and guaranteeing defect coverage and test quality is not a common practice during the design of analog or mixed signal ICs. Fault simulation involves defect extraction and injection of defects into the netlist of the analog or mixed signal circuit and performing analogue simulation (DC, AC, or Transient) of the tests. The major drawback is the long CPU time associated with the many analogue simulations. For example, if simulation of the test suite takes one hour, it may take several years to perform all simulations for more than 10,000 defects (when not exploiting parallelism).
In the transient simulation the solution due to an inserted fault is compared to a golden, fault-free, solution. A strategy is developed to efficiently simulate the faulty solutions until their moment of detection. We obtain a significant speed-up of over 100x over sequentional approaches, while a useful estimate of the detection status and the defect coverage can still be ensured. Our strategy can also be used when exploiting parallelism.
Jos J. Dohmen, Bratislav Tasic, Rick Janssen, E. Jan W. ter Maten, Theo G. J. Beelen, Roland Pulch, Michael Günther

Chapter 18. Calibration of Probability Density Function

The capability performance index (Cpk) is often used to measure the capability of the production process and to predict yield. However, this Cpk is only defined for the Gaussian distribution. At NXP Semiconductors an on-chip calibration technique is frequently used to reduce the effect of process variations. The resulting distribution has a much flatter peak than a Gaussian density and consequently the Cpk is significantly underestimated. In this chapter we propose two possible approaches to address accurate Cpk calculation for non-normal distributions. One approach is to use the so-called Generalized Gaussian distribution function and to estimate its defining parameters. We propose a numerical fast and reliable method for computing these parameters and a simple formula to calculate the Cpk value from these defining parameters. Another approach is to transform data as a way to deal with non-normal distributions. We show that both approaches significantly outperform the standard Cpk calculation for the non-normal distributions of interest.
Jos J. Dohmen, Theo G. J. Beelen, Oryna Dvortsova, E. Jan W. ter Maten, Bratislav Tasić, Rick Janssen

Chapter 19. Ageing Models and Reliability Prediction

We overview reliability related activities like ageing and life time prediction. We want to predict the number of thermal stress cycles an IC can handle before showing passivation cracks. For this a sufficient model for electro-migration was used that can be applied to an IC with multiple drivers and knowing a required thermal profile. At first state-of-the-art reliability concepts are reviewed. Next a new framework is introduced that aims at simplifying and speeding-up the process of assessing the reliability of complex application profiles.
Renaud Gillon, Aarnout Wieers, Frederik Deleu, Tomas Gotthans, Rick Janssen, Wim Schoenmaker, E. Jan W. ter Maten

Test Cases, Measurements, Validation and Best Practices


Chapter 20. Test Cases for Power-MOS Devices and RF-Circuitry

This chapter gives an overview of realistic test cases, provided by industrial partners, for validation of the simulation tools, that were developed in the last years. They allowed to compare the results of measurements and simulations for real-life size applications.
Rick Janssen, Renaud Gillon, Aarnout Wieers, Frederik Deleu, Hervé Guegnaud, Pascal Reynier, Wim Schoenmaker, E. Jan W. ter Maten

Chapter 21. Measurements for RF Amplifiers, Bond Wire Fusing and MOS Power Cells

This chapter focuses on several areas regarding measurements (including methodology, test-bed development and measurement of results itself) of experimental results in the framework of an established cooperation with several partners from industry. We give attention to measurements for RF amplifiers, bond wire fusing and MOS power cells. We also include on-chip measurements of RF passive components and pay attention to heat, stress and reliability measurements. Measurements set-ups have been made in close cooperation with ON Semiconductor Belgium (Oudenaarde, Belgium) and with ACCO Semiconductor (Louveciennes, France).
Tomas Kratochvil, Jiri Petrzela, Roman Sotner, Jiri Drinovsky, Tomas Gotthans, Aarnout Wieers, Renaud Gillon, Pascal Reynier, Yannick Poupin

Chapter 22. Validation of Simulation Results on Coupled Problems

The Test Set for the Power-MOS Devices and RF-Circuitry, as described in Chapter [11], and the Measurements for RF-Amplifiers, Bond Wire Fusing and MOS Power Cells (see Chapter [18]) were used to validate simulation results of coupled problems by comparing with measurements, as well as by comparing to outcomes with other simulation tools (when possible, mostly without full coupling).
Rick Janssen, Renaud Gillon, Aarnout Wieers, Frederik Deleu, Hervé Guegnaud, Pascal Reynier, Wim Schoenmaker, E. Jan W. ter Maten

Chapter 23. Methodology and Best-Practice Guidelines for Thermally Optimized Driver Design

An important outcome of the validation activities is the formulation of methodologies to address coupled problems in EDA. This chapter describes typical coupled problems arising in the field of activities of the industrial partners. It also highlights the challenges and makes the link to some test cases. In order to address these challenges and solve the design problems illustrated by the test cases, several design flows were put in place based on outcomes of the project activities. We describes these flows as well as the tools set-up in order to support them. Finally, the recommended usage of these flows is proposed.
Renaud Gillon, Aarnout Wieers, Frederik Deleu, Rick Janssen, Wim Schoenmaker, Bart De Smedt, Hervé Guegnaud, Pascal Reynier


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