# Modelling proximity effects in Transition Edge Sensors to investigate the influence of lateral metal structures

## TL;DR

This paper develops a numerical model based on Usadel's equations to predict how lateral metal structures in Transition Edge Sensors affect their superconducting and electrothermal properties, aligning well with experimental observations.

## Contribution

It extends existing models to include normal-metal features in TESs, enabling prediction of their influence on key device parameters and magnetic field effects.

## Key findings

- Increasing metal structures decreases T_c, α, and β.
- Critical temperature scales as L^{-0.7} with device length.
- Model predicts magnetic flux dependence consistent with experiments.

## Abstract

The bilayers of Transition Edge Sensors (TESs) are often modified with additional normal-metal features such as bars or dots. Previous device measurements suggest that these features improve performance, reducing electrical noise and altering response times. However, there is currently no numerical model to predict and quantify these effects. Here we extend existing techniques based on Usadel's equations to describe TESs with normal-metal features. We show their influence on the principal TES characteristics, such as the small-signal electrothermal parameters $\alpha$ and $\beta$ and the superconducting transition temperature $T_{c}$. Additionally, we examine the effects of an applied magnetic field on the device performance. Our model predicts a decrease in $T_{c}$, $\alpha$ and $\beta$ as the number of lateral metal structures is increased. We also obtain a relationship between the length $L$ of a TES and its critical temperature, $T_{c} \propto L^{-0.7}$ for a bilayer with normal-metal bars. We predict a periodic magnetic flux dependence of $\alpha, \beta$ and $I_{c}$. Our results demonstrate good agreement with published experimental data, which also show the reduction of $\alpha$, $\beta$ and $T_{c}$ with increasing number of bars. The observed Fraunhofer dependence of critical current on magnetic flux is also anticipated by our model. The success of this model in predicting the effects of additional structures suggests that in the future numerical methods can be used to better inform the design of TESs, prior to device processing.