WannierSTM: Efficient superconducting scanning-probe spectroscopy from maximally localized Wannier functions

To understand the work described here, one must first picture the microscopic world that modern science seeks to map. At the scale of atoms and molecules, electricity does not flow like water in a pipe; instead, it behaves as a wave of probability, jumping across gaps and interacting with the magnetic spins of individual atoms. Scientists use a tool called a scanning tunneling microscope to probe this realm. It works by lowering an incredibly sharp metal tip, often just a single atom wide, toward a surface. When the tip hovers just above the material, electrons can "tunnel" through the empty space between them, creating a tiny electric current. By measuring how this current changes as the tip moves or as the voltage is adjusted, researchers can create a detailed map of the material's electronic structure, revealing where electrons are likely to be found and how they move.

However, interpreting these maps is notoriously difficult, especially when the material involves superconductors. Superconductors are special materials that conduct electricity with zero resistance, but they also possess a unique quantum property where electrons pair up and move in a coordinated state. When a superconductor is combined with magnetic materials or complex molecules, the resulting behavior becomes a tangled web of competing forces. Theoretical models have long struggled to bridge the gap between the messy, complex reality of real-world materials and the simplified math needed to simulate them. Existing methods either oversimplify the material to the point of losing accuracy or require so much computing power that they cannot be used for the detailed, repeated scans needed to match real experiments.

This is where a new software tool called WannierSTM steps in. Developed by a team of researchers in Finland and Spain, this program offers a streamlined way to simulate how electrons behave in these complex, superconducting hybrid systems. The researchers built the software to act as a translator, taking the raw, heavy data generated by standard atomic simulations and converting it into a compact, efficient format that retains all the essential physical details. Instead of trying to calculate every single electron in a massive system, the software focuses on the most relevant energy levels, allowing it to run simulations that are both fast and accurate. This efficiency is crucial because it enables scientists to generate the kind of detailed spectroscopic maps that experimentalists see in the lab, but with the precision of a first-principles calculation.

The core innovation lies in how the software handles the quantum mechanics of superconductivity. In these systems, electrons and their "holes" (the absence of an electron) are deeply linked, a relationship that requires a specific mathematical framework to describe correctly. The new program incorporates this framework directly, allowing it to model how superconducting pairing interacts with magnetic spins and the spin-orbit coupling found in heavy atoms. By treating the connection between the scanning tip and the sample in a simplified but physically sound way, the software avoids the need for slow, repetitive numerical integrations. Instead, it uses a direct analytical approach to calculate the current and conductance, meaning it can sweep through a wide range of voltages and spatial positions in a fraction of the time previously required.

To prove the tool works, the team tested it on several different scenarios, ranging from simple theoretical chains to realistic molecular structures. In one test, they simulated a chain of atoms with a magnetic impurity, successfully reproducing the specific energy signatures known as Yu-Shiba-Rusinov states, which are a hallmark of magnetic atoms sitting on a superconductor. In another, they modeled a nanowire designed to host Majorana zero modes, a type of exotic particle that is of great interest for future quantum computers. The software correctly identified the conditions under which these states appear and how they change as the wire's properties are tweaked. These simulations served as a rigorous check, confirming that the program could handle the delicate balance between magnetism and superconductivity without losing physical accuracy.

The researchers then moved on to more complex, real-world materials. They applied the software to two-dimensional magnetic materials made of iron, gallium, and tellurium, which are promising candidates for spintronic devices. Using data from standard atomic simulations, they constructed a finite patch of the material and simulated how a scanning tip would interact with it. The results showed how the material's magnetic properties and the induced superconductivity reshaped the flow of electrons, creating distinct patterns in the conductance maps. The simulations revealed that even in these strongly magnetic systems, the superconducting influence could be clearly seen, redistributing the electron density in ways that would be visible to an experimentalist.

In a final demonstration, the team simulated a single molecule known as titanocene bis(cyclooctatetraenyl), which acts as a magnetic qubit. They calculated how the molecule's electronic signature would look when probed by a scanning tip at different positions and voltages. The software successfully predicted how the molecule's magnetic state would interact with a superconducting environment, showing how the energy levels shifted as the superconducting strength increased. This ability to model a single molecule with high precision demonstrates the tool's versatility, proving it can handle both large, periodic materials and isolated, complex structures.

The significance of this work is that it provides a unified path from the raw data of atomic simulations to the specific images and graphs produced by scanning tunneling microscopes. Before this tool, researchers often had to choose between using a simplified model that missed important details or a full-scale simulation that was too slow to be practical for mapping. WannierSTM removes this trade-off, offering a method that is computationally light enough to run on standard computers while remaining grounded in the rigorous physics of real materials. By making these complex calculations accessible and efficient, the software opens the door for scientists to design and understand new quantum materials with a level of detail that was previously out of reach, bringing theoretical predictions and experimental observations closer together.