Two dimensional (2D) peak finding is a common practice in data analysis for physics experiments, which is typically achieved by computing the local derivatives. However, this method is inherently unstable when the local landscape is complicated, or the signal-to-noise ratio of the data is low. In this work, we propose a new method in which the peak tracking task is formalized as an inverse problem, thus can be solved with a convolutional neural network (CNN). In addition, we show that the underlying physics principle of the experiments can be used to generate the training data. By generalizing the trained neural network on real experimental data, we show that the CNN method can achieve comparable or better results than traditional derivative based methods. This approach can be further generalized in different physics experiments when the physical process is known.
Recent advances on experimental techniques in condensed matter physics boost the generation of large volume high-quality data. In order to present these data, 2D (and even 3D) representations of the data becomes more and more popular, such as in angular resolved photo-emission spectroscopy (ARPES) 1 , scanning tunneling microscopy (STM) 2 , resonant inelastic x-ray scattering (RIXS) 3 , etc. In these experiments, as the data quality is often limited by the instrumental resolution and different intrinsic physical processes, the retrieving of physical quantities with high precision can therefore benefit from effective data analysis methods.
As an example, a typical 2D ARPES experiment data set is shown in Fig. 1. Ideally, the ARPES spectra should follow the theoretical energy-momentum dispersion shown in Fig. 1A. However, intrinsic broadening effects 1 such as electronic correlation, as well as extrinsic factors such as crystal defects can broaden the spectrum in both energy and momentum dimensions. Together with the resolution limitation, sometimes it is difficult to resolve the energy bands in the 2D measurements (e.g. in Fig. 1B). To enhance the features in the 2D band dispersions, several derivative based methods have been proposed, such as the Maximum Curvature (MC) method 4 and the Minimum Gradient (MG) method 5 . The MC method calculates the local curvature and assumes the pixels with large curvature are the positions of the energy bands; the MG method calculates the average gradient and assumes the positions with small average gradient represent the energy band location. However, these methods can only give reasonable results in high signal-to-noise ratio data and tend to fail in complicated situations when multiple bands are close to each other or when the data are too noisy.
On the other hand, recent development of machine learning and deep learning techniques such as convolutional neural network (CNN) provide great performance in improving 2D data qualities by solving a series of inverse problems, such as super-resolution, denoising and patching [6][7][8] . As highquality 2D data (i.e. images) are subjected to various degrading transformations in experiments, the objective of our data analysis becomes finding an appropriate inverse transformation that can best recovers the original images -which can be formulated as solving an inverse problem.
Motivated by this insight, we consider the energy band extraction problem in a 2D ARPES image (broadened by the intrinsic and extrinsic processes as discussed above) as a problem of inversing the physical processes that blurs the spectral peaks along the band dispersions, and thus it can be treated using CNN. In other words, we effectively look for a map between the “broadened experimental dispersions” and the “original dispersions”. Moreover, leveraging the existing knowledge of the intrinsic and extrinsic broadening processes, massive simulated data can be generated for the training purpose before we apply the trained model to the real experimental data.
In this work, we use the simulated ARPES data to train a modified Super-Resolution Convolutional Neural Network (SR-CNN) that fits the ARPES experiments’ spectral intensity map with the extrema feature map, thus visualizes the positions of the energy bands. SR-CNN was originally proposed for mapping low resolution image patches to corresponding high resolution patches, which is an inverse problem of down sampling in image processing. 9 Comparing to other recent neural network architectures in solving this problem that uses very deep net 7,8 , the SR-CNN has only three layers and the function of each layer can be well understood. We demonstrate that this method can resolve complicated features in ARPES data and outperforms previous traditional algorithms in noiseresilience, sharpness and accuracy.
For reference, the theoretically calculated energy band is presented at Fig. 1A to compare with the experiment data in Fig. 1B. The key feature showing in the calculation is that there are three energy bands crossing the Fermi energy (EF) and the band No.2 terminates around EF and merges with other faint bands. In addition, band No. 1 has a degenerate point at k = 0 about 60meV below the EF. Ideally, a successful data analysis algorithm should resolve all these features from the ARPES intensity map.
Due to the experiment restriction, the number of pixels in the current data along k-direction is only 29. The low resolution makes it even more difficult to directly track the features from the experiment data. The MC and MG methods show significant enhancement of the energy band features (as shown in Fig. 1 C andD) but are either noisy (MC) or still blurry (MG). The result of our method is shown in Fig. 1E, demonstrating all three EF-crossing bands and the crossing below FS. It also shows that the band in the middle (No.2) ends at about -15meV below the FS, which matches the theoretical calculation.
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