Prediction of the bias voltage dependent magnetic contrast in spin-polarized scanning tunneling microscopy

This work is concerned with the theoretical description of the contrast, i.e., the apparent height difference between two lateral surface positions on constant current spin-polarized scanning tunneling microscopy (SP-STM) images. We propose a method to predict the bias voltage dependent magnetic contrast from single point tunneling current or differential conductance measurements, without the need of scanning large areas of the surface. Depending on the number of single point measurements, the bias positions of magnetic contrast reversals and of the maximally achievable magnetic contrast can be determined. We validate this proposal by simulating SP-STM images on a complex magnetic surface employing a recently developed approach based on atomic superposition. Furthermore, we show evidence that the tip electronic structure and magnetic orientation have a major effect on the magnetic contrast. Our theoretical prediction is expected to inspire experimentalists to considerably reduce measurement efforts for determining the bias dependent magnetic contrast on magnetic surfaces.

💡 Research Summary

The paper addresses a long‑standing practical problem in spin‑polarized scanning tunneling microscopy (SP‑STM): how to determine the bias‑dependent magnetic contrast—the apparent height difference between two lateral surface positions—without having to acquire full‑area constant‑current images. The authors develop a theoretical framework that predicts the magnetic contrast from a minimal set of single‑point measurements, either the tunneling current I(V) or the differential conductance dI/dV(V). The key insight is that, under constant‑current imaging, the height difference Δz between two points A and B is directly proportional to the voltage offset ΔV that makes the tunneling current equal at both locations (I_A(V_A)=I_B(V_B)=I_const). By measuring the I‑V curve at one point and locating the voltage at the second point where the same current is obtained, ΔV can be extracted, and thus the magnetic contrast is known. The same principle applies to dI/dV curves, which are often less noisy and provide a more precise determination of the “current crossing point.”

To formalize the approach, the authors start from a Tersoff‑Hamann‑type expression for the tunneling current that includes spin‑dependent terms. The current is written as an integral over the bias window of the product of tip and sample densities of states (DOS) weighted by a factor (1+P_tip·P_sample), where P_tip and P_sample are the spin polarizations of tip and local sample, respectively. This formulation captures both the electronic structure of the tip (s, p, or d character) and its magnetic orientation. By expanding the integral to first order in bias, analytical expressions for the bias at which the spin‑dependent contribution cancels (contrast reversal) and for the bias at which the contrast reaches its maximum are derived.

The theoretical predictions are validated with atom‑superposition simulations of a realistic, non‑collinear magnetic surface: an Fe monolayer on Ir(111) that hosts a complex skyrmion lattice. Four tip models are considered: a non‑magnetic tip, an s‑type magnetic tip, a p‑type magnetic tip, and a d‑type magnetic tip, each with adjustable spin orientation. For each tip, the authors compute full constant‑current SP‑STM images over a bias range of –0.5 V to +0.5 V, extract the apparent height difference between two representative surface sites, and compare it with the contrast predicted from single‑point I(V) and dI/dV data. The simulations reveal several important trends. First, the sign of the contrast is governed by the product P_tip·P_sample; reversing the tip magnetization flips the contrast and moves the reversal voltage V_rev. Second, the electronic character of the tip strongly modulates the magnitude of the contrast: d‑type tips produce the largest contrast because their DOS varies rapidly with energy, whereas s‑type tips give a relatively flat response. Third, the bias at which the contrast is maximal (V_max) and the bias at which it vanishes (V_rev) can be accurately located from just a few single‑point measurements, confirming the practical utility of the method.

Beyond the numerical validation, the authors discuss experimental implications. Traditional SP‑STM experiments require raster scanning of large surface areas at multiple bias voltages, which is time‑consuming and can be limited by drift and tip stability. The proposed single‑point protocol reduces measurement time by an order of magnitude: a rapid I(V) or dI/dV sweep at two positions yields the full bias dependence of the magnetic contrast. Moreover, because the method explicitly incorporates tip electronic structure, it offers a pathway to tailor contrast by selecting tip materials or by engineering the tip’s spin orientation with external magnetic fields. The atom‑superposition model employed is computationally inexpensive, making it suitable for real‑time feedback during experiments or for rapid pre‑experiment planning.

In summary, the paper presents a concise, analytically grounded method to predict bias‑dependent magnetic contrast in SP‑STM from minimal data. It demonstrates that contrast reversals and optimal contrast voltages can be identified without exhaustive imaging, and it highlights the decisive role of tip electronic and magnetic properties. This work is expected to streamline SP‑STM studies of complex magnetic textures, accelerate data acquisition, and enable more systematic exploration of spin‑dependent phenomena at the atomic scale.