PDE models for nanotechnology

The topic of this project is mathematical modeling and simulation using deterministic partial differential equations (PDEs) and stochastic partial differential equations in multiscale settings with concrete applications in nanotechnology and metamaterials. Multiscale problems arise naturally in the applications due to the different length scales inherent in real-world structures, i.e., the nanometer length scale of single features and the macroscopic length scale of whole structures and devices. The main mathematical method for dealing with multiscale problems is homogenization. This project combines mathematical analysis, homogenization, stochastic aspects, and the development of improved numerical methods. There is a host of recent and concrete applications such as nanowire sensors, nanostructured metamaterials, random media, graphene, and silicene. Using stochastic PDEs allows us to go beyond calculating averages and hence to quantify uncertainties, fluctuations, and noise. Mathematical modeling means that we prove existence and uniqueness of the solutions of the model equations and that we solve new homogenization problems. The main model equations are nonlinear Poisson equations, elliptic eigenproblems, the Boltzmann transport equation, the Dirac equation, and systems thereof. The equations considered here contain stochastic coefficients and terms. In random media, for example, coefficients such as permittivity and permeability responsible for material properties are stochastic and give rise to nonlinear effects after homogenization. In nanowire sensors, the Brownian motion of biomolecules, surface gas reactions, and the trapping of charges are stochastic processes that give rise to stochastic terms with various probability distributions in boundary layers and hence lead to new homogenization problems. An important feature is that we use realistic stochastic processes and go beyond using normal distributions as the default assumption. Additionally, there is a strong numerical component in this project, as the development of numerical algorithms for multiscale stochastic PDE models is still in its infancy. The development of numerical algorithms for these models faces the challenge that the discretization of the stochastic terms adds dozens or hundreds of dimensions to 2D or 3D equations; this is often called the curse of dimensionality. Therefore, there is a definite need for improved methods in uncertainty quantification and we develop numerical algorithms based on results from the theory of stochastic homogenization. The model equations stem from applications in science and technology such as fluctuations and noise in nanowire field-effect sensors, the selectivity of nanowire gas sensors, the design of random media and of metamaterials, and charge transport in graphene and silicene. For all these applications, new simulation capabilities are developed to provide quantitative understanding and to enable rational design. The beauty of mathematics is that our results are very abstract and hence are relevant for many different applications. The numerical methods developed here make it possible to calculate what could not be calculated before.

Technological progress is often characterized by miniaturization, where nanotechnology is a prominent example. In the START project "PDE Models for Nanotechnology", new mathematical algorithms and computer programs were developed for the rational design of nanotechnological devices such as sensors and transistors and to extract as much information as possible from measurements. There are two main areas where the results of this project are the first in the world to achieve fundamental goals in this field. Firstly, noise, fluctuations, and process variations are increasingly important at nanometer scales, but preclude traditional simulation approaches. Therefore new algorithms were developed to address these challenges and to enable computer simulations that were not possible before. Secondly, in order to develop better sensors and to improve data analysis, we showed that computer simulations in conjunction with Bayesian statistical methods can serve as a mathematical magnifying glass and make it possible to extract information that was inaccessible before. A leading example of a real-world application are field-effect sensors, which can detect minute concentrations of biomolecules such as tumor markers and DNA strands. This combination of semiconductor technology and the biological world poses a number of challenges regarding modeling and simulation. Charge transport through nanoscale structures must be calculated; variations during manufacturing affect the devices, i.e., every transistor and every sensor is different due to inherent randomness in the production processes; and noise and fluctuations occur due to Brownian motion and in the semiconductor. The smaller the devices are, the more important these effects become. These issues also result in extremely computationally expensive simulations. In order to overcome these problems, we developed new mathematical algorithms that reduce the computational expense by orders of magnitude. Having developed these computer simulations, we asked how much can be learned from sensor measurements and simulations. How precisely can the number of biomolecules present in a sensor be measured? How much information can be extracted and how reliably? As it turns out, these questions can be answered by comparing measurements and simulations, adding to the usefulness of the computer simulations. In simple terms, if the simulation for a certain molecule concentration agrees well with the measurement, this concentration is likely; otherwise, it is unlikely. In this way, in view of all the noise and fluctuations, we calculate probability distributions of molecule concentrations and hence confidence intervals. Small confidence intervals mean that a sensor works very well. By this combination of mathematical modeling, computer simulations, and Bayesian statistics, we have made it possible for the first time to extract as much information as possible from sensor measurements. It also underlines the importance and usefulness of theory in interpreting measurements. Of course, this approach can also be applied to other sensor technologies.