[58]  Benjamin Stadlbauer, Gregor MitschaBaude, and Clemens Heitzinger. Modeling singlemolecule stochastic transport in nanopore sensors. pages 17. In preparation. [ bib ] 
[57]  Clemens Heitzinger, Markus Schmuck, and Christoph Schwab. Stochastic twoscale convergence and covariance equation for elliptic multiscale PDEs. pages 126. In preparation. [ bib ] 
[56]  Leila Taghizadeh, Elisabeth Presterl, and Clemens Heitzinger. A biofilm model including quorum sensing. pages 112. Submitted for publication. [ bib ] 
[55]  Boaz Blankrot and Clemens Heitzinger. Efficient computational design and optimization of dielectric metamaterial devices. pages 118. Submitted for publication. [ bib ] 
[54]  Clemens Heitzinger and Leila Taghizadeh. Existence and local uniqueness for the stochastic driftdiffusionPoisson system. pages 124. Submitted for publication. [ bib ] 
[53]  Clemens Heitzinger and Leila Taghizadeh. Existence and local uniqueness for the StokesNernstPlanckdriftdiffusionPoisson system modeling nanopore and nanowire sensors. pages 125. Submitted for publication. [ bib ] 
[52]  Clemens Heitzinger and Gudmund Pammer. Existence, uniqueness, and a comparison of two nonintrusive methods for the stochastic nonlinear PoissonBoltzmann equation. pages 127. Submitted for publication. [ bib ] 
[51]  Amirreza Khodadadian, Leila Taghizadeh, and Clemens Heitzinger. Threedimensional optimal multilevel MonteCarlo approximation of the stochastic driftdiffusionPoisson system in nanoscale devices. J. Comput. Electron., pages 118, 2018. At press. [ bib  DOI  at publisher  PDF ] 
[50] 
Clemens Heitzinger, Gudmund Pammer, and Stefan Rigger.
Cubature formulas for multisymmetric functions and applications to
stochastic partial differential equations.
SIAM/ASA Journal on Uncertainty Quantification, 6(1):213242,
2018.
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The numerical solution of stochastic partial differential equations and numerical Bayesian estimation is computationally demanding. If the coefficients in a stochastic partial differential equation exhibit symmetries, they can be exploited to reduce the computational effort. To do so, we show that permutationinvariant functions can be approximated by permutationinvariant polynomials in the space of continuous functions as well as in the space of pintegrable functions defined on [0, 1]^{s} for 1 <=p < . We proceed to develop a numerical strategy to compute cubature formulas that exploit permutationinvariance properties related to multisymmetry groups in order to reduce computational work. We show that in a certain sense there is no curse of dimensionality if we restrict ourselves to multisymmetric functions, and we provide error bounds for formulas of this type. Finally, we present numerical results, comparing the proposed formulas to other integration techniques that are frequently applied to highdimensional problems such as quasiMonte Carlo rules and sparse grids.

[49] 
Amirreza Khodadadian, Leila Taghizadeh, and Clemens Heitzinger.
Optimal multilevel randomized quasiMonteCarlo method for the
stochastic driftdiffusionPoisson system.
Computer Methods in Applied Mechanics and Engineering (CMAME),
329:480497, February 2018.
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In this paper, an optimal multilevel randomized quasiMonteCarlo method to solve the stationary stochastic drift–diffusionPoisson system is developed. We calculate the optimal values of the parameters of the numerical method such as the mesh sizes of the spatial discretization and the numbers of quasipoints in order to minimize the overall computational cost for solving this system of stochastic partial differential equations. This system has a number of applications in various fields, wherever charged particles move in a random environment. It is shown that the computational cost of the optimal multilevel randomized quasiMonteCarlo method, which uses randomly shifted lowdiscrepancy sequences, is one order of magnitude smaller than that of the optimal multilevel MonteCarlo method and five orders of magnitude smaller than that of the standard MonteCarlo method. The method developed here is applied to a realistic transport problem, namely the calculation of randomdopant effects in nanoscale fieldeffect transistors.

[48] 
Clemens Heitzinger and Leila Taghizadeh.
Analysis of the driftdiffusionPoissonBoltzmann system for
nanowire and nanopore sensors in the alternatingcurrent regime.
Commun. Math. Sci., 15(8):23032325, 2017.
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The basic analytical properties of the driftdiffusionPoissonBoltzmann system in the alternatingcurrent (AC) regime are shown. The analysis of the AC case differs from the directcurrent (DC) case and is based on extending the transport model to the frequency domain and writing the variables as periodic functions of the frequency in a smallsignal approximation. We first present the DC and AC model equations to describe the three types of material in nanowire fieldeffect sensors: The driftdiffusionPoisson system holds in the semiconductor, the PoissonBoltzmann equation holds in the electrolyte, and the Poisson equation provides selfconsistency. Then the AC model equations are derived. Finally, existence and local uniqueness of the solution of the AC model equations are shown. Realworld applications include nanowire fieldeffect bio and gas sensors operating in the AC regime, which were only demonstrated experimentally recently. Furthermore, nanopore sensors are governed by the system of model equations and the analysis as well.

[47] 
Amirreza Khodadadian, Kiarash Hosseini, Ali Manzour ol Ajdad, Marjan Hedayati,
Reza Kalantarinejad, and Clemens Heitzinger.
Optimal design of nanowire fieldeffect troponin sensors.
Computers in Biology and Medicine, 87:4656, August 2017.
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We propose a design strategy for affinitybased biosensors using nanowires for sensing and measuring biomarker concentration in biological samples. Such sensors have been shown to have superior properties compared to conventional biosensors in terms of LOD (limit of detection), response time, cost, and size. However, there are several parameters affecting the performance of such devices that must be determined. In order to solve the design problem, we have developed a comprehensive model based on stochastic transport equations that makes it possible to optimize the sensing behavior.

[46] 
Gregor MitschaBaude, Andreas ButtingerKreuzhuber, Gerhard Tulzer, and Clemens
Heitzinger.
Adaptive and iterative methods for simulations of nanopores with the
PNPStokes equations.
J. Comput. Phys., 338:452476, June 2017.
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We present a 3D finite element solver for the nonlinear PoissonNernstPlanck (PNP) equations for electrodiffusion, coupled to the Stokes system of fluid dynamics. The model serves as a building block for the simulation of macromolecule dynamics inside nanopore sensors. The source code is released online at github.com/mitschabaude/nanopores. We add to existing numerical approaches by deploying goaloriented adaptive mesh refinement. To reduce the computation overhead of mesh adaptivity, our error estimator uses the much cheaper PoissonBoltzmann equation as a simplified model, which is justified on heuristic grounds but shown to work well in practice. To address the nonlinearity in the full PNP–Stokes system, three different linearization schemes are proposed and investigated, with two segregated iterative approaches both outperforming a naive application of Newton’s method. Numerical experiments are reported on a realworld nanopore sensor geometry. We also investigate two different models for the interaction of target molecules with the nanopore sensor through the PNPStokes equations. In one model, the molecule is of finite size and is explicitly built into the geometry; while in the other, the molecule is located at a single point and only modeled implicitly  after solution of the system  which is computationally favorable. We compare the resulting force profiles of the electric and velocity fields acting on the molecule, and conclude that the pointsize model fails to capture important physical effects such as the dependence of charge selectivity of the sensor on the molecule radius.

[45] 
Leila Taghizadeh, Amirreza Khodadadian, and Clemens Heitzinger.
The optimal multilevel MonteCarlo approximation of the stochastic
driftdiffusionPoisson system.
Computer Methods in Applied Mechanics and Engineering (CMAME),
318:739761, 2017.
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Existence and localuniqueness theorems for weak solutions of a system consisting of the driftdiffusionPoisson equations and the PoissonBoltzmann equation, all with stochastic coefficients, are presented. For the numerical approximation of the expected value of the solution of the system, we develop a multilevel MonteCarlo (MLMC) finiteelement method (FEM) and we analyze its rate of convergence and its computational complexity. This allows to find the optimal choice of discretization parameters. Finally, numerical results show the efficiency of the method. Applications are, among others, noise and fluctuations in nanoscale transistors, in fieldeffect bio and gas sensors, and in nanopores.

[44] 
Amirreza Khodadadian and Clemens Heitzinger.
Basis adaptation for the stochastic nonlinear PoissonBoltzmann
equation.
J. Comput. Electron., 15(4):13931406, 2016.
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A basisadaptation method based on polynomial chaos expansion is used for the stochastic nonlinear Poisson–Boltzmann equation. The uncertainty in this numerical approach is motivated by the quantification of noise and fluctuations in nanoscale fieldeffect sensors. The method used here takes advantage of the properties of the nonlinear Poisson–Boltzmann equation and shows an exact and efficient approximation of the real solution. Numerical examples are motivated by the quantification of noise and fluctuations in nanowire fieldeffect sensors as a concrete example. Basis adaptation is validated by comparison with the full solution, and it is compared to optimized multilevel MonteCarlo method, and the model equations are validated by comparison with experiments. Finally, various design parameters of the fieldeffect sensors are investigated in order to maximize the signaltonoise ratio.

[43] 
Caroline Geiersbach, Clemens Heitzinger, and Gerhard Tulzer.
Optimal approximation of the firstorder corrector in multiscale
stochastic elliptic PDE.
SIAM/ASA J. Uncertainty Quantification, 4(1):12461262, 2016.
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This work addresses the development of an optimal computational scheme for the approximation of the firstorder corrector arising in the stochastic homogenization of linear elliptic PDEs in divergence form. Equations of this type describe, for example, diffusion phenomena in materials with a heterogeneous microstructure, but require enormous computational efforts in order to obtain reliable results. We derive an optimization problem for the needed computational work with a given error tolerance, then extract the governing parameters from numerical experiments, and finally solve the obtained optimization problem. The numerical approach investigated here is a stochastic sampling scheme for the probability space connected with a finiteelement method for the discretization of the physical space.

[42] 
Martin Hermann Bernardi, Daniel Schmidlin, Robin Ristl, Clemens Heitzinger,
Arno Schiferer, Thomas Neugebauer, Thomas Wrba, Michael Hiesmayr, Wilfred
Druml, and Andrea Lassnigg.
Serum creatinine backestimation in cardiac surgery patients:
misclassification of AKI using existing formulae and a datadriven model.
Clin. J. Am. Soc. Nephrol. (CJASN), 11(3):395404, 2016.
(CJASN 2014 impact factor: 4.613; this publication was awarded the
Science Price 2017 by ÖGARI (Austrian Society for Anesthesiology,
Reanimation, and Intensive Medicine).).
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Conclusions: bSCr values backestimated using currently available eGFR formulae are inaccurate and cannot correctly classify AKI stages. Our model eSCr improves the prediction of AKI but to a still inadequate extent.

[41] 
Gerhard Tulzer and Clemens Heitzinger.
Brownianmotion based simulation of stochastic reactiondiffusion
systems for affinity based sensors.
Nanotechnology, 27(16):165501/19, 2016.
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In this work, we develop a 2D algorithm for stochastic reactiondiffusion systems describing the binding and unbinding of target molecules at the surfaces of affinitybased sensors. In particular, we simulate the detection of DNA oligomers using siliconnanowire fieldeffect biosensors. Since these devices are uniform along the nanowire, two dimensions are sufficient to capture the kinetic effects features. The model combines a stochastic ordinary differential equation for the binding and unbinding of target molecules as well as a diffusion equation for their transport in the liquid. A Brownianmotion based algorithm simulates the diffusion process, which is linked to a stochasticsimulation algorithm for association at and dissociation from the surface. The simulation data show that the shape of the cross section of the sensor yields areas with significantly different targetmolecule coverage. Different initial conditions are investigated as well in order to aid rational sensor design. A comparison of the association/hybridization behavior for different receptor densities allows optimization of the functionalization setup depending on the targetmolecule density.

[40] 
Amirreza Khodadadian and Clemens Heitzinger.
A transport equation for confined structures applied to the OprP,
Gramicidin A, and KcsA channels.
J. Comput. Electron., 14(2):524532, 2015.
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A transport equation for confined structures is used to calculate the ionic currents through various transmembrane proteins. The transport equation is a diffusiontype equation where the concentration of the particles depends on the onedimensional position in the confined structure and on the local energy. The computational significance of this continuum model is that the (6+1)dimensional Boltzmann equation is reduced to a (2+1)dimensional diffusiontype equation that can be solved with small computational effort so that ionic currents through confined structures can be calculated quickly. The applications here are three channels, namely OprP, Gramicidin A, and KcsA. In each case, the confinement potential is estimated from the known molecular structure of the channel. Then the confinement potentials are used to calculate ionic currents and to study the effect of parameters such as the potential of mean force, the ionic bath concentration, and the applied voltage. The simulated currents are compared with measurements, and very good agreement is found in each case. Finally, virtual potassium channels with selectivity filters of varying length are simulated in order to discuss the optimality of the filter.

[39] 
Gerhard Tulzer and Clemens Heitzinger.
Fluctuations due to association and dissociation processes at
nanowirebiosensor surfaces and their optimal design.
Nanotechnology, 26(2):025502/19, 2015.
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In this work, we calculate the effect of the binding and unbinding of molecules at the surface of a nanowire biosensor on the signaltonoise ratio of the sensor. We model the fluctuations induced by association and dissociation of target molecules by a stochastic differential equation and extend this approach to a coupled diffusionreaction system. Where possible, analytic solutions for the signaltonoise ratio are given. Stochastic simulations are performed wherever closed forms of the solutions cannot be derived. Starting from parameters obtained from experimental data, we simulate DNA hybridization at the sensor surface for different target molecule concentrations in order to optimize the sensor design.

[38] 
Clemens Heitzinger and Christian Ringhofer.
Hierarchies of transport equations for nanopores  equations derived
from the Boltzmann equation and the modeling of confined structures.
J. Comput. Electron., 13(4):801817, 2014.
Invited review paper.
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We review transport equations and their usage for the modeling and simulation of nanopores. First, the significance of nanopores and the experimental progress in this area are summarized. Then the starting point of all classical and semiclassical considerations is the Boltzmann transport equation as the most general transport equation. The derivation of the driftdiffusion equations from the Boltzmann equation is reviewed as well as the derivation of the NavierStokes equations. Nanopores can also be viewed as a special case of a confined structure and hence as giving rise to a multiscale problem, and therefore we review the derivation of a transport equation from the Boltzmann equation for such confined structures. Finally, the state of the art in the simulation of nanopores is summarized.

[37] 
Clemens Heitzinger and Christian Ringhofer.
Multiscale modeling of fluctuations in stochastic elliptic PDE
models of nanosensors.
Commun. Math. Sci., 12(3):401421, 2014.
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In this work, the multiscale problem of modeling fluctuations in boundary layers in stochastic elliptic partial differential equations is solved by homogenization. A homogenized equation for the covariance of the solution of stochastic elliptic PDEs is derived. In addition to the homogenized equation, a rate for the covariance and variance as the cell size tends to zero is given. For the homogenized problem, an existence and uniqueness result and further properties are shown. The multiscale problem stems from the modeling of the electrostatics in nanoscale fieldeffect sensors, where the fluctuations arise from random charge concentrations in the cells of a boundary layer. Finally, numerical results and a numerical verification are presented.

[36] 
Daniel Brinkman, Clemens Heitzinger, and Peter Markowich.
A convergent 2D finitedifference scheme for the DiracPoisson
system with magnetic potential and the simulation of graphene.
J. Comput. Phys., 257A:318332, 2014.
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We present a convergent finitedifference scheme of second order in both space and time for the 2D electromagnetic Dirac equation. We apply this method in the selfconsistent DiracPoisson system to the simulation of graphene. The model is justified for low energies, where the particles have wave vectors sufficiently close to the Dirac points. In particular, we demonstrate that our method can be used to calculate solutions of the DiracPoisson system where potentials act as beamsplitters or Veselago lenses.

[35] 
Gerhard Tulzer, Stefan Baumgartner, Elise Brunet, Giorgio C. Mutinati, Stephan
Steinhauer, Anton Köck, Paolo E. Barbano, and Clemens Heitzinger.
Kinetic parameter estimation and fluctuation analysis of CO at
SnO_{2} single nanowires.
Nanotechnology, 24(31):315501/110, August 2013.
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In this work, we present calculated numerical values for the kinetic parameters governing adsorption/desorption processes of carbon monoxide at tin dioxide singlenanowire gas sensors. The response of such sensors to pulses of 50ppm carbon monoxide in nitrogen is investigated at different temperatures to extract the desired information. A rateequation approach is used to model the reaction kinetics, which results in the problem of determining coefficients in a coupled system of nonlinear ordinary differential equations. The numerical values are computed by inversemodeling techniques and are then used to simulate the sensor response. With our model, the dynamic response of the sensor due to the gas–surface interaction can be studied in order to find the optimal setup for detection, which is an important step towards selectivity of these devices. We additionally investigate the noise in the current through the nanowire and its changes due to the presence of carbon monoxide in the sensor environment. Here, we propose the use of a wavelet transform to decompose the signal and analyze the noise in the experimental data. This method indicates that some fluctuations are specific for the gas species investigated here.

[34] 
Stefan Baumgartner and Clemens Heitzinger.
A onelevel FETI method for the driftdiffusionPoisson system
with discontinuities at an interface.
J. Comput. Phys., 243:7486, June 2013.
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A 3d FETI method for the driftdiffusionPoisson system including discontinuities at a 2d interface is developed. The motivation for this work is to provide a parallel numerical algorithm for a system of PDEs that are the basic model equations for the simulation of semiconductor devices such as transistors and sensors. Moreover, discontinuities or jumps in the potential and its normal derivative at a 2d surface are included for the simulation of nanowire sensors based on a homogenized model. Using the FETI method, these jump conditions can be included with the usual numerical properties and the original FarhatRoux FETI method is extended to the driftdiffusionPoisson equations including discontinuities. We show two numerical examples. The first example verifies the correct implementation including the discontinuities on a 2d grid divided into eight subdomains. The second example is 3d and shows the application of the algorithm to the simulation of nanowire sensors with high aspect ratios. The PoissonBoltzmann equation and the driftdiffusionPoisson system with jump conditions are solved on a 3d grid with realworld boundary conditions.

[33] 
Stefan Baumgartner, Clemens Heitzinger, Aleksandar Vacic, and Mark A. Reed.
Predictive simulations and optimization of nanowire fieldeffect
PSA sensors including screening.
Nanotechnology, 24(22):225503/19, June 2013.
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We apply our selfconsistent PDE model for the electrical response of fieldeffect sensors to the 3D simulation of nanowire PSA (prostatespecific antigen) sensors. The charge concentration in the biofunctionalized boundary layer at the semiconductorelectrolyte interface is calculated using the PROPKA algorithm, and the screening of the biomolecules by the free ions in the liquid is modeled by a sensitivity factor. This comprehensive approach yields excellent agreement with experimental currentvoltage characteristics without any fitting parameters. Having verified the numerical model in this manner, we study the sensitivity of nanowire PSA sensors by changing device parameters, making it possible to optimize the devices and revealing the attributes of the optimal fieldeffect sensor.

[32] 
Gerhard Tulzer, Stefan Baumgartner, Elise Brunet, Giorgio C. Mutinati, Stephan
Steinhauer, Anton Köck, and Clemens Heitzinger.
Inverse modeling of CO reactions at SnO_{2} nanowire surfaces for
selective detection.
Procedia Engineering, 47:809812, 2012.
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Nanowire gas sensors show high sensitivity towards various gases and offer great potential to improve present gas sensing. In this work, we investigate experimental results achieved with an undoped single SnO_{2} nanowire sensor device for CO pulses in N_{2} atmosphere at different operating temperatures. We calculated the reaction parameters according to the mass action law including frequency factors, activation energies, and numbers of intrinsic as well as extrinsic surface sites. With the values obtained, we then calculated the surface charge of the nanowire sensor by solving the corresponding differential equations. The simulated results agree very well with the experimental values at an operating temperature of 200°C and hence provide good understanding of the chemical reaction. This can be used to simulate the current through the transducer and consequently the sensitivity of the device, and the parameters provided here are useful for computational procedures to provide selectivity.

[31] 
Stefan Baumgartner, Martin Vasicek, and Clemens Heitzinger.
Modeling and simulation of nanowire based fieldeffect biosensors.
In G. Korotcenkov, editor, Chemical Sensors: Simulation and
Modeling. Volume 2: ConductometricType Sensors, pages 447469. Momentum
Press, 2012.
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A book chapter. Contents:

[30] 
Manuel Punzet, Dieter Baurecht, Franz Varga, Heidrun Karlic, and Clemens
Heitzinger.
Determination of surface concentrations of individual moleculelayers
used in nanoscale biosensors by insitu ATRFTIR spectroscopy.
Nanoscale, 4(7):24312438, 2012.
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For the development of nanowire sensors for chemical and medical detection purposes, the optimal functionalization of the surface is a mandatory component. Quantitative ATRFTIR spectroscopy was used insitu to investigate the stepbystep layer formation of typical functionalization protocols and to determine the respective molecule surface concentrations. BSA, antiTNFα and antiPSA antibodies were bound via 3(trimethoxy)butylsilyl aldehyde linkers to siliconoxide surfaces in order to investigate surface functionalization of nanowires. Maximum determined surface concentrations were 7.17×10^{13} mol cm^{2} for BSA, 1.7×10^{13} mol cm^{2} for antiTNFα antibody, 6.1×10^{13} mol cm^{2} for antiPSA antibody, 3.88×10^{13} mol cm^{2} for TNFα and 7.0×10^{13} mol cm^{2} for PSA. Furthermore we performed antibodyantigen binding experiments and determined the specific binding ratios. The maximum possible ratios of 2 were obtained at bulk concentrations of the antigen in the μg ml^{1} range for TNFα and PSA.

[29] 
Stefan Baumgartner and Clemens Heitzinger.
Existence and local uniqueness for 3d selfconsistent multiscale
models for fieldeffect sensors.
Commun. Math. Sci., 10(2):693716, 2012.
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We present existence and local uniqueness theorems for a system of partial differential equations modeling fieldeffect nanosensors. The system consists of the Poisson(Boltzmann) equation and the driftdiffusion equations coupled with a homogenized boundary layer. The existence proof is based on the LeraySchauder fixedpoint theorem and a maximum principle is used to obtain apriori estimates for the electric potential, the electron density, and the hole density. Local uniqueness around the equilibrium state is obtained from the implicitfunction theorem. Due to the multiscale problem inherent in fieldeffect biosensors, a homogenized equation for the potential with interface conditions at a surface is used. These interface conditions depend on the surfacecharge density and the dipolemoment density in the boundary layer and still admit existence and local uniqueness of the solution when certain conditions are satisfied. Due to the geometry and the boundary conditions of the physical system, the threedimensional case must be considered in simulations. Therefore a finitevolume discretization of the 3d selfconsistent model was implemented to allow comparison of simulation and measurement. Special considerations regarding the implementation of the interface conditions are discussed so that there is no computational penalty when compared to the problem without interface conditions. Numerical simulation results are presented and very good quantitative agreement with currentvoltage characteristics from experimental data of biosensors is found.

[28] 
Stefan Baumgartner, Martin Vasicek, Alena Bulyha, and Clemens Heitzinger.
Optimization of nanowire DNA sensor sensitivity using
selfconsistent simulation.
Nanotechnology, 22(42):425503/18, October 2011.
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In order to facilitate the rational design and the characterization of nanowire fieldeffect sensors, we have developed a model based on selfconsistent chargetransport equations combined with interface conditions for the description of the biofunctionalized surface layer at the semiconductor/electrolyte interface. Crucial processes at the interface, such as the screening of the partial charges of the DNA strands and the influence of the angle of the DNA strands with respect to the nanowire, are computed by a Metropolis Monte Carlo algorithm for charged molecules at interfaces. In order to investigate the sensing mechanism of the device, we have computed the current–voltage characteristics, the electrostatic potential and the concentrations of electrons and holes. Very good agreement with measurements has been found and optimal device parameters have been identified. Our approach provides the capability to study the device sensitivity, which is of fundamental importance for reliable sensing.

[27] 
Stefan Baumgartner, Martin Vasicek, and Clemens Heitzinger.
Analysis of fieldeffect biosensors using selfconsistent 3D
driftdiffusion and MonteCarlo simulations.
Procedia Engineering, 25:407410, 2011.
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Fieldeffect biosensors based on nanowires enjoy considerable popularity due to their high sensitivity and direct electrical readout. However, crucial issues such as the influence of the biomolecules on the chargecarrier transport or the binding of molecules to the surface have not been described satisfactorily yet in a quantitative manner. In order to analyze these effects, we present simulation results based on a 3D macroscopic transport model coupled with MonteCarlo simulations for the biofunctionalized surface layer. Excellent agreement with measurement data has been found, while detailed study of the influence of the most prominent biomolecules, namely doublestranded DNA and singlestranded DNA, on the current through the semiconductor transducer has been carried out.

[26] 
Clemens Heitzinger and Christian Ringhofer.
A transport equation for confined structures derived from the
Boltzmann equation.
Commun. Math. Sci., 9(3):829857, 2011.
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A system of diffusiontype equations for transport in 3d confined structures is derived from the Boltzmann transport equation for charged particles. Transport takes places in confined structures and the scaling in the derivation of the diffusion equation is chosen so that transport and scattering occur in the longitudinal direction and the particles are confined in the two transversal directions. The result are two diffusiontype equations for the concentration and fluxes as functions of position in the longitudinal direction and energy. Entropy estimates are given. The transport coefficients depend on the geometry of the problem that is given by arbitrary harmonic confinement potentials. An important feature of this approach is that the coefficients in the resulting diffusiontype equations are calculated explicitly so that the six position and momentum dimensions of the original 3d Boltzmann equation are reduced to a 2d problem. Finally, numerical results are given and discussed. Applications of this work include the simulation of charge transport in nanowires, nanopores, ion channels, and similar structures.

[25] 
Alena Bulyha and Clemens Heitzinger.
An algorithm for threedimensional MonteCarlo simulation of charge
distribution at biofunctionalized surfaces.
Nanoscale, 3(4):16081617, 2011.
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In this work, a MonteCarlo algorithm in the constantvoltage ensemble for the calculation of 3d charge concentrations at charged surfaces functionalized with biomolecules is presented. The motivation for this work is the theoretical understanding of biofunctionalized surfaces in nanowire fieldeffect biosensors (BioFETs). This work provides the simulation capability for the boundary layer that is crucial in the detection mechanism of these sensors; slight changes in the charge concentration in the boundary layer upon binding of analyte molecules modulate the conductance of nanowire transducers. The simulation of biofunctionalized surfaces poses special requirements on the MonteCarlo simulations and these are addressed by the algorithm. The constantvoltage ensemble enables us to include the right boundary conditions; the DNA strands can be rotated with respect to the surface; and several molecules can be placed in a single simulation box to achieve good statistics in the case of low ionic concentrations relevant in experiments. Simulation results are presented for the leading example of surfaces functionalized with PNA and with single and doublestranded DNA in a sodiumchloride electrolyte. These quantitative results make it possible to quantify the screening of the biomolecule charge due to the counterions around the biomolecules and the electrical double layer. The resulting concentration profiles show a threelayer structure and nontrivial interactions between the electric double layer and the counterions. The numerical results are also important as a reference for the development of simpler screening models.

[24] 
Clemens Heitzinger, Yang Liu, Norbert Mauser, Christian Ringhofer, and
Robert W. Dutton.
Calculation of fluctuations in boundary layers of nanowire
fieldeffect biosensors.
J. Comput. Theor. Nanosci., 7(12):25742580, 2010.
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Fluctuations in the biofunctionalized boundary layers of nanowire fieldeffect biosensors are investigated by using the stochastic linearized PoissonBoltzmann equation. The noise and fluctuations considered here are due to the Brownian motion of the biomolecules in the boundary layer, i.e., the various orientations of the molecules with respect to the surface are associated with their probabilities. The probabilities of the orientations are calculated using their free energy. The fluctuations in the charge distribution give rise to fluctuations in the electrostatic potential and hence in the current through the semiconductor transducer of the sensor, both of which are calculated. A homogenization result for the variance and covariance of the electrostatic potential is presented. In the numerical simulations, a cross section of a silicon nanowire on a flat surface including electrode and backgate contacts is considered. The biofunctionalized boundary layer contains singlestranded or doublestranded DNA oligomers, and varying values of the surface charge, of the oligomer length, and of the electrolyte ionic strength are investigated.

[23] 
Clemens Heitzinger, Norbert Mauser, and Christian Ringhofer.
Multiscale modeling of planar and nanowire fieldeffect biosensors.
SIAM J. Appl. Math., 70(5):16341654, 2010.
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Fieldeffect nanobiosensors (or BioFETs, biologically sensitive fieldeffect transistors) have recently been demonstrated experimentally and have thus gained interest as a technology for direct, labelfree, realtime, and highly sensitive detection of biomolecules. The experiments have not been accompanied by a quantitative understanding of the underlying detection mechanism. The modeling of fieldeffect biosensors poses a multiscale problem due to the different length scales in the sensors: the charge distribution and the electric potential of the biofunctionalized surface layer changes on the Angstrom length scale, whereas the exposed sensor area is measured in micrometers squared. Here a multiscale model for the electrostatics of planar and nanowire fieldeffect sensors is developed by homogenization of the Poisson equation in the biofunctionalized boundary layer. The resulting interface conditions depend on the surface charge density and dipole moment density of the boundary layer. The multiscale model can be coupled to any charge transport model and hence makes the selfconsistent quantitative investigation of the physics of fieldeffect sensors possible. Numerical verifications of the multiscale model are given. Furthermore a silicon nanowire biosensor is simulated to elucidate the influence of the surface charge density and the dipole moment density on the conductance of the semiconductor transducer. The numerical evidence shows that the conductance varies exponentially as a function of both charge and dipole moment. Therefore the dipole moment of the surface layer must be included in biosensor models. The conductance variations ob served in experiments can be explained by the field effect, and they can be caused by a change in dipole moment alone.

[22] 
Christian Ringhofer and Clemens Heitzinger.
Multiscale modeling and simulation of fieldeffect biosensors.
ECS Transactions, 14(1):1119, 2008.
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BioFETs (biologically sensitive fieldeffect transistors) are fieldeffect biosensors with semiconducting transducers. Their device structure is similar to a MOSFET, except that the gate structure is replaced by an aqueous solution containing the analyte. The detection mechanism is the conductance modulation of the transducer due to binding of the analyte to surface receptors. The main advantage of BioFETs, compared to currently available technology, is labelfree operation. We present a quantitative analysis of BioFETs which is centered around multiscale models. The technique for solving the multiscale problem used here is the derivation of interface conditions for the Poisson equation that include the effects of the quasiperiodic biofunctionalized boundary layer. The multiscale model enables selfconsistent simulation and can be used with any charge transport model. Hence it provides the foundation for understanding the physics of the sensors by continuum models.

[21] 
Clemens Heitzinger, Rick Kennell, Gerhard Klimeck, Norbert Mauser, Michael
McLennan, and Christian Ringhofer.
Modeling and simulation of fieldeffect biosensors (BioFETs) and
their deployment on the nanoHUB.
J. Phys.: Conf. Ser., 107:012004/112, 2008.
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BioFETs (biologically active fieldeffect transistors) are biosensors with a semiconductor transducer. Due to recent experiments demonstrating detection by a field effect, they have gained attention as potentially fast, reliable, and lowcost biosensors for a wide range of applications. Their advantages compared to other technologies are direct, labelfree, ultrasensitive, and (near) realtime operation. We have developed 2D and 3D multiscale models for planar sensor structures and for nanowire sensors. The multiscale models are indispensable due to the large difference in the characteristic length scales of the biosensors: the charge distribution in the biofunctionalized surface layer varies on the Angstrom length scale, the diameters of the nanowires are several nanometers, and the sensor lengths measure several micrometers. The multiscale models for the electrostatic potential can be coupled to any charge transport model of the transducer. Conductance simulations of nanowire sensors with different diameters provide numerical evidence for the importance of the dipole moment of the biofunctionalized surface layer in addition to its surface charge. We have also developed a web interface to our simulators, so that other researchers can access them at the nanohub and perform their own investigations.

[20] 
Clemens Heitzinger, Christian Ringhofer, and Siegfried Selberherr.
Finite difference solutions of the nonlinear Schrödinger equation
and their conservation of physical quantities.
Commun. Math. Sci., 5(4):779788, December 2007.
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The solutions of the nonlinear Schrödinger equation are of great importance for ab initio calculations. It can be shown that such solutions conserve a countable number of quantities, the simplest being the local norm square conservation law. Numerical solutions of high quality, especially for long time intervals, must necessarily obey these conservation laws. In this work we first give the conservation laws that can be calculated by means of Lie theory and then critically compare the quality of different finite difference methods that have been proposed in geometric integration with respect to conservation laws. We find that finite difference schemes derived by writing the Schrödinger equation as an (artificial) Hamiltonian system do not necessarily conserve important physical quantities better than other methods.

[19] 
Clemens Heitzinger and Christian Ringhofer.
An effective quantum potential for particleparticle interactions in
threedimensional semiconductor device simulations.
J. Comput. Electron., 6(4):401408, 2007.
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The classical Coulomb potential and force can be calculated efficiently using fast multipole methods. Effective quantum potentials, however, describe the physics of electron transport in semiconductors more precisely. Such an effective quantum potential was derived previously for the interaction of an electron with a barrier for use in particlebased Monte Carlo semiconductor device simulators. The method is based on a perturbation theory around thermodynamic equilibrium and leads to an effective potential scheme in which the size of the electron depends upon its energy and which is parameterfree. Here we extend the method to electronelectron interactions and show how the effective quantum potential can be evaluated efficiently in the context of manybody problems. Finally several examples illustrate how the momentum of the electrons changes the classical potential.

[18] 
Clemens Heitzinger and Gerhard Klimeck.
Computational aspects of the threedimensional featurescale
simulation of siliconnanowire fieldeffect sensors for DNA detection.
J. Comput. Electron., 6(13):387390, 2007.
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In recent years DNAsensors, and generally biosensors, with semiconducting transducers were fabricated and characterized. Although the concept of socalled BioFETs was proposed already two decades ago, its realization has become feasible only recently due to advances in process technology. In this paper a comprehensive and rigorous approach to the simulation of siliconnanowire DNAFETs at the featurescale is presented. It allows to investigate the feasibility of singlemolecule detectors and is used to elucidate the performance that can be expected from sensors with nanowire diameters in the decananometer range. Finally the computational challenges for the simulation of siliconnanowire DNAsensors are discussed.

[17] 
Clemens Heitzinger, Christian Ringhofer, Shaikh Ahmed, and Dragica Vasileska.
3D MonteCarlo device simulations using an effective quantum
potential including electronelectron interactions.
J. Comput. Electron., 6(13):1518, 2007.
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Effective quantum potentials describe the physics of quantummechanical electron transport in semiconductors more than the classical Coulomb potential. An effective quantum potential was derived previously for the interaction of an electron with a barrier for use in particlebased Monte Carlo semiconductor device simulators. The method is based on a perturbation theory around thermodynamic equilibrium and leads to an effective potential scheme in which the size of the electron depends upon its energy and which is parameterfree. Here we extend the method to electronelectron interactions and show how the effective quantum potential can be evaluated efficiently in the context of manybody problems. The effective quantum potential was used in a threedimensional MonteCarlo device simulator for calculating the electronelectron and electronbarrier interactions. Simulation results for an SOI transistor are presented and illustrate how the effective quantum potential changes the characteristics compared to the classical potential.

[16] 
Wilfried Wessner, Johann Cervenka, Clemens Heitzinger, Andreas Hössinger, and
Siegfried Selberherr.
Anisotropic mesh refinement for the simulation of threedimensional
semiconductor manufacturing processes.
IEEE Trans. ComputerAided Design of Integrated Circuits and
Systems, 25(10):21292139, October 2006.
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This paper presents an anisotropic adaptation strategy for threedimensional unstructured tetrahedral meshes, which allows us to produce thin mostly anisotropic layers at the outside margin, i.e., the skin of an arbitrary meshed simulation domain. An essential task for any modern algorithm in the finiteelement solution of partial differential equations, especially in the field of semiconductor process and device simulation, the major application is to provide appropriate resolution of the partial discretization mesh. The startup conditions for semiconductor process and device simulations claim an initial mesh preparation that is performed by socalled Laplace refinement. The basic idea is to solve Laplace’s equation on an initial coarse mesh with Dirichlet boundary conditions. Afterward, the gradient field is used to form an anisotropic metric that allows to refine the initial mesh based on tetrahedral bisection.

[15] 
Clemens Heitzinger, Alireza Sheikholeslami, JongMun Park, and Siegfried
Selberherr.
A method for generating structurally aligned grids for semiconductor
device simulation.
IEEE Trans. ComputerAided Design of Integrated Circuits and
Systems, 24(10):14851491, October 2005.
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The quality of the numeric approximation of the partial differential equations governing carrier transport in semiconductor devices depends particularly on the grid. The method of choice is to use structurally aligned grids since the regions and directions therein that determine device behavior are usually straightforward to find as they depend on the distribution of doping. Here, the authors present an algorithm for generating structurally aligned grids including anisotropy with resolutions varying over several orders of magnitude. The algorithm is based on a level set approach and permits to define the refined resolutions in a flexible manner as a function of doping. Furthermore, criteria on grid quality can be enforced. In order to show the practicability of this method, the authors study the examples of a trench gate metaloxidesemiconductor fieldeffect transistor (TMOSFET) and a radio frequency silicononinsulator lateral double diffused metaloxidesemiconductor (RF SOI LDMOS) power device using the device simulator MINIMOS NT, where simulations are performed on a grid generated by the new algorithm. In order to resolve the interesting regions of the TMOSFET and the RF SOI LDMOS power device accurately, several regions of refinement were defined where the grid was grown with varying resolutions.

[14] 
Dragica Vasileska, Hasanur Khan, Shaikh Ahmed, Christian Ringhofer, and Clemens
Heitzinger.
Quantum and Coulomb effects in nanodevices.
International Journal of Nanoscience, 4(3):305361, June 2005.
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In stateoftheart devices, it is well known that quantum and Coulomb effects play significant role on the device operation. In this paper, we demonstrate that a novel effective potential approach in conjunction with a Monte Carlo device simulation scheme can accurately capture the quantummechanical size quantization effects. We also demonstrate, via proper treatment of the shortrange Coulomb interactions, that there will be significant variation in device design parameters for devices fabricated on the same chip due to the presence of unintentional dopant atoms at random locations within the channel.

[13] 
Shaikh Ahmed, Dragica Vasileska, Clemens Heitzinger, and Christian Ringhofer.
Quantum potential approach to modeling nanoscale MOSFETs.
J. Comput. Electron., 4(12):5761, 2005.
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We propose a novel parameterfree quantum potential scheme for use in conjunction with particlebased simulations. The method is based on a perturbation theory around thermodynamic equilibrium and leads to an effective potential scheme in which the size of the electron depends upon its energy. The approach has been tested on the example of a MOScapacitor by retrieving the correct sheet electron density. It has also been used in simulations of a 25 nm nchannel nanoscale MOSFET with high substrate doping density. We find that the use of the quantum potential approach gives rise to a threshold voltage shift of about 220 mV and drain current degradation of about 30%.

[12] 
Hasanur Khan, Dragica Vasileska, Shaikh Ahmed, Christian Ringhofer, and Clemens
Heitzinger.
Modeling of FinFET: 3D MC simulation using FMM and
unintentional doping effects on device operation.
J. Comput. Electron., 3(34):337340, 2004.
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Novel device concepts such as dual gate SOI, Ultra thin body SOI, FinFETs, etc., have emerged as a solution to the ultimate scaling limits of conventional bulk MOSFETs. These novel devices suppress some of the Short Channel Effects (SCE) efficiently, but at the same time more physics based modeling is required to investigate device operation. In this paper, we use semiclassical 3D Monte Carlo device simulator to investigate important issues in the operation of FinFETs. Fast Multipole Method (FMM) has been integrated with the EMC scheme to replace the time consuming Poisson equation solver. Effect of unintentional doping for different device dimensions has been investigated. Impurities at the source side of the channel have most significant impact on the device performance.

[11] 
Stefan Holzer, Rainer Minixhofer, Clemens Heitzinger, Johannes Fellner, Tibor
Grasser, and Siegfried Selberherr.
Extraction of material parameters based on inverse modeling of
threedimensional interconnect fusing structures.
Microelectronics Journal, 35(10):805810, 2004.
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An approach for determining higher order coefficients of the electrical and thermal conductivities for different materials is presented. The method is based on inverse modeling using threedimensional transient electrothermal finite element simulations for electrothermal investigations of complex layered structures, for instance polycrystalline silicon (polysilicon) fuses or other multilayered devices. The simulations are performed with a threedimensional interconnect simulator, which is automatically configured and controlled by an optimization framework. Our method is intended to be applied to optimize devices with different material compositions and geometries as well as for achieving an optimum of speed and reliability.

[10] 
Clemens Heitzinger, Alireza Sheikholeslami, Fuad Badrieh, Helmut Puchner, and
Siegfried Selberherr.
Featurescale process simulation and accurate capacitance extraction
for the backend of a 100nm aluminum/TEOS process.
IEEE Trans. Electron Devices, 51(7):11291134, July 2004.
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One of the challenges that technology computeraided design must meet currently is the analysis of the performance of groups of components, interconnects, and, generally speaking, large parts of the IC. This enables predictions that the simulation of single components cannot achieve. In this paper, we focus on the simulation of backend processes, interconnect capacitances, and time delays. The simulation flows start from the blank wafer surface and result in device information for the circuit designer usable from within SPICE. In order to join topography and backend simulations, deposition, etching, and chemical mechanical planarization processes in the various metal lines are used to build up the backend stack, starting from the flat wafer surface. Depending on metal combination, linetoline space, and line width, thousands of simulations are required whose results are stored in a database. Finally, we present simulation results for the backend of a 100nm process, where the influence of void formation between metal lines profoundly impacts the performance of the whole interconnect stack, consisting of aluminum metal lines, and titanium nitride local interconnects. Scanning electron microscope images of test structures are compared to topography simulations, and very good agreement is found. Moreover, chargebased capacitance measurements were carried out to validate the capacitance extraction, and it was found that the error is smaller than four percent. These simulations assist the consistent fabrication of voids, which is economically advantageous compared to lowκ materials, which suffer from integration problems.

[9] 
Clemens Heitzinger, Andreas Hössinger, and Siegfried Selberherr.
An algorithm for smoothing threedimensional Monte Carlo ion
implantation simulation results.
Mathematics and Computers in Simulation, 66(23):219230, June
2004.
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We present an algorithm for smoothing results of threedimensional Monte Carlo ion implantation simulations and translating them from the grid used for the Monte Carlo simulation to an arbitrary unstructured threedimensional grid. This algorithm is important for joining various simulations of semiconductor manufacturing process steps, where data have to be smoothed or transferred from one grid to another. Furthermore different grids must be used since using orthogrids is mandatory because of performance reasons for certain Monte Carlo simulation methods. The algorithm is based on approximations by generalized Bernstein polynomials. This approach was put on a mathematically sound basis by proving several properties of these polynomials. It does not suffer from the ill effects of least squares fits of polynomials of fixed degree as known from the popular response surface method. The smoothing algorithm which works very fast is described and in order to show its applicability, the results of smoothing a threedimensional real world implantation example are given and compared with those of a least squares fit of a multivariate polynomial of degree 2, which yielded unusable results.

[8] 
Thomas Binder, Clemens Heitzinger, and Siegfried Selberherr.
A study on global and local optimization techniques for TCAD
analysis tasks.
IEEE Trans. ComputerAided Design of Integrated Circuits and
Systems, 23(6):814822, June 2004.
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We evaluate optimization techniques to reduce the necessary user interaction for inverse modeling applications as they are used in the technology computeraided design field. Four optimization strategies are compared. Two wellknown global optimization methods, simulated annealing and genetic optimization, a local gradientbased optimization strategy, and a combination of a local and a global method. We rate the applicability of each method in terms of the minimal achievable target value for a given number of simulation runs and in terms of the fastest convergence. A brief overview over the three used optimization algorithms is given. The optimization framework that is used to distribute the workload over a cluster of workstations is described. The actual comparison is achieved by means of an inverse modeling application that is performed for various settings of the optimization algorithms. All presented optimization algorithms are capable of evaluating several targets in parallel. The best optimization strategy that is found is used in the calibration of a model for silicon selfinterstitial cluster formation and dissolution.

[7] 
Clemens Heitzinger and Christian Ringhofer.
A note on the symplectic integration of the nonlinear Schrödinger
equation.
J. Comput. Electron., 3(1):3344, 2004.
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Numerically solving the nonlinear Schrödinger equation and being able to treat arbitrary space dependent potentials permits many application in the realm of quantum mechanics. The longterm stability of a numerical method and its conservation properties is an important feature since it assures that the underlying physics of the solution are respected and it ensures that the numerical result is correct also for small time spans. In this paper we describe symplectic integrators for the nonlinear Schrödinger equation with arbitrary potentials and perform numerical experiments comparing different approaches and highlighting their respective advantages and disadvantages.

[6] 
Clemens Heitzinger and Siegfried Selberherr.
On the simulation of the formation and dissolution of silicon
selfinterstitial clusters and the corresponding inverse modeling problem.
Microelectronics Journal, 35(2):167171, February 2004.
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The formation and dissolution of silicon selfinterstitial clusters is linked to the phenomenon of transientenhanced diffusion (TED) which in turn has gained importance in the manufacturing of semiconductor devices. Based on theoretical considerations and measurements of the number of selfinterstitial clusters during a thermal step, a model for the formation and dissolution of selfinterstitial clusters is presented including the adjusted model parameters for two different technologies (i.e. material parameter sets). In order to automate the inverse modeling part, a general optimization framework was used. In addition to solving this problem, the same setup can solve a wide range of inverse modeling problems occurring in the domain of process simulation. Finally, the results are discussed and compared with a previous model.

[5] 
Clemens Heitzinger, Andreas Hössinger, and Siegfried Selberherr.
On smoothing threedimensional Monte Carlo ion implantation
simulation results.
IEEE Trans. ComputerAided Design of Integrated Circuits and
Systems, 22(7):879883, July 2003.
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An algorithm for smoothing results of threedimensional (3D) Monte Carlo ion implantation simulations and translating them from the grid used for the Monte Carlo simulation to an arbitrary unstructured 3D grid is presented. This algorithm is important for joining various process simulation steps, where data have to be smoothed or transferred from one grid to another. Furthermore, it is important for integrating the ion implantation simulator into a process flow. One reason for using different grids is that for certain Monte Carlo simulation methods, using orthogrids is mandatory because of performance reasons.

[4] 
Clemens Heitzinger, Wolfgang Pyka, Naoki Tamaoki, Toshiro Takase, Toshimitsu
Ohmine, and Siegfried Selberherr.
Simulation of arsenic insitu doping with polysilicon CVD and its
application to high aspect ratio trenches.
IEEE Trans. ComputerAided Design of Integrated Circuits and
Systems, 22(3):285292, March 2003.
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Filling high aspect ratio trenches is an essential manufacturing step for state of the art memory cells. Understanding and simulating the transport and surface processes enables to achieve voidless filling of deep trenches, to predict the resulting profiles, and thus to optimize the process parameters and the resulting memory cells.

[3] 
Tibor Grasser, Hans Kosina, Clemens Heitzinger, and Siegfried Selberherr.
Characterization of the hot electron distribution function using six
moments.
J. Appl. Phys., 91(6):38693879, 2002.
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The shape of the hot electron distribution function in semiconductor devices is insufficiently described using only the first four moments. We propose using six moments of the distribution function to obtain a more accurate description of hot carrier phenomena. An analytic expression for the symmetric part of the distribution function as a function of the even moments is given which shows good agreement with Monte Carlo data for both the bulk case and inside n^{+}nn^{+} test structures. The influence of the band structure on the parameters of the distribution function is studied and proven to be of importance for an accurate description.

[2] 
Clemens Heitzinger and Siegfried Selberherr.
An extensible TCAD optimization framework combining gradient based
and genetic optimizers.
Microelectronics Journal, 33(12):6168, 2002.
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The SIESTA framework is an extensible tool for optimization and inverse modeling of semiconductor devices including dynamic load balancing for taking advantage of several, loosely connected workstations. Two gradientbased and two evolutionary computation optimizers are currently available through a uniform interface and can be combined at will. At a real world inverse modeling example, we demonstrate that evolutionary computation optimizers provide several advantages over gradientbased optimizers, due to the specific properties of the objective functions in TCAD applications. Furthermore, we shortly discuss some issues arising in inverse modeling and conclude with a comparison of gradientbased and evolutionary computation optimizers from a TCAD point of view.

[1] 
Tibor Grasser, Hans Kosina, Clemens Heitzinger, and Siegfried Selberherr.
Accurate impact ionization model which accounts for hot and cold
carrier populations.
Appl. Phys. Lett., 80(4):613615, January 2002.
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Conventional macroscopic impact ionization models which use the average carrier energy as a main parameter can not accurately describe the phenomenon in modern miniaturized devices. Here, we present a model which is based on an analytic expression for the distribution function. In particular, the distribution function model accounts explicitly for a hot and a cold carrier population in the drain region of metaloxidesemiconductor transistors. The parameters are determined by threeeven moments obtained from a solution of a sixmoments transport model. Together with a nonparabolic description of the density of states, accurate closed form macroscopic impact ionization models can be derived based on familiar microscopic descriptions.

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