# AMaSiS 2021 - Abstract

**Onsager, Claire**

*Mapping conductivity with electrical impedance tomography*

Coauthors: Charles Costakis, Lauren Lang, Suzan van der Lee and Matthew A. Grayson

Northwestern University, USA

In semiconductor fabrication, the conductivity distribution provides essential information about inhomogeneities, such as variations in deposition layer thickness, dopant concentration gradients, and local defects that affect electronic device performance. Electrical impedance tomography (EIT) is a fast characterization method whereby applied current and voltage measurements on the periphery of a semiconductor map its internal conductivity [1]. In this so-called inverse problem, a model space is created to span all possible conductivity distributions while a data space spans all possible boundary measurements, and the mapping problem is equivalent to transforming a vector from data space to model space. Present day EIT methods are constrained by resolution limitations of their contact configurations, as well as by the construction of the inverse mapping problem. But by increasing the number of contacts one can expand the data space for significant improvements in resolution and accuracy. And by defining a reduced model space of orthonormal basis functions, computation speeds can be significantly enhanced.

In this work, we include more contacts than standard methods to expand the data space and fewer basis functions to reduce the model space. In standard EIT, the Sheffield measurement protocol is used to define the data space, whereby two adjacent current contacts are paired with two adjacent voltage contacts which are themselves cyclically permuted around the sample. However, the Sheffield protocol produces a data space whose measurements contain a significant amount of redundant information, and our method uses a Monte Carlo search of the expanded data space to identify optimal measurement configurations for signal-to-noise improvement. Simultaneously, we reduce the model space by defining a smaller set of continuous orthonormal basis functions over the volume to eliminate the underdetermined nature of the EIT problem. In standard EIT, a finite element mesh forms the model space thereby requiring regularization to converge to a solution. This reduced set of basis functions, on the other hand, results in a well-defined problem while minimizing the computational time.

With the above improvements to the data space and model space, the Jacobian relating the two can be deconstructed using singular value decomposition (SVD). The comparison of vector properties and singular values provides metrics to select measurement protocols that provide new information with high signal to noise. The method above is observed to enhance the signal to noise ratio by over 600% compared to the standard EIT method employing the Sheffield protocol for the same resolution and same number of measurements. Phantom models representing different inhomogeneity scenarios will be presented along with the improvements in mapping resolution and accuracy.

**Acknowledgments**: This work was supported by NSF ECCS‐1912694.

**References**

[1] K.B. Tushar, Applications of Electrical Impedance Tomography (EIT): A Short Review, *OP Conf. Ser.: Mater. Sci. Eng.*, ** 331** (2018), 012004.