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.

Figure 1

The surface geometry of 1 ML Cr on Ag(111) and its ground-state magnetic structure. The Cr and Ag atoms are denoted by spheres colored by green (medium gray) and purple (dark gray), respectively, and the magnetic moments of individual Cr atoms are indicated by (red) arrows. The ($\sqrt{3}\times \sqrt{3}$) magnetic unit cell is drawn by yellow (light gray) color, and the scanning area for the SP-STM simulations is shown by the black-framed rectangle. Moreover, the surface Cr positions are denoted by “x,” and Cr atoms in the magnetic unit cell are explicitly labeled by “1,” “2,” and “3.”

Figure 2

The simulated magnetic currents $I_{\mathrm{MAGN}}^J(z=3.5$ Å$,V)$ according to Eq. (21) $z=3.5$ Å above each Cr atom ($J\in \{$Cr1,Cr2,Cr3$\}$) in the ($\sqrt{3}\times \sqrt{3}$) magnetic unit cell (see Fig. 1), measured with an ideal electronically flat maximally spin-polarized tip (gray), or a model Ni tip (black). The tip magnetization direction is fixed parallel to the Cr1 magnetic moment. The inset shows the bias region between 0 and 1 V zoomed in. The sign change of the magnetic current occurs at 0.94 and 0.74 V for the ideal and the Ni tips, respectively. These correspond to magnetic contrast reversals.

Figure 3

The bias-dependent magnetic contrast between Cr1 and Cr3 atoms $\Delta z^{\mathrm{Cr}1-\mathrm{Cr}3}(z=3.5$ Å, $V)$ at $z=3.5$ Å tip-sample distance calculated using Eq. (5) (solid lines) and Eq. (8) (dashed lines) measured with the ideal magnetic (gray) and the Ni (black) tips. The tip magnetization direction is fixed parallel to the Cr1 magnetic moment. The local absolute maxima of the magnetic contrasts and their bias values are explicitly shown in both the negative and the positive bias ranges. Vertical dashed lines denote the contrast reversals, see also Fig. 2. For comparison, circle symbols show the apparent height difference between Cr1 and Cr3 atoms obtained from constant current SP-STM images, see text for details.

Figure 4

Simulated SP-STM images depending on the bias voltage and the considered tip: ideal magnetic (top row) and Ni (bottom row). The tip magnetization direction is fixed parallel to the Cr1 magnetic moment and is indicated by a vector (${\underline{M}}_{\mathrm{TIP}}$). The bias values have been chosen corresponding to the local absolute maxima of the magnetic contrasts and the contrast reversals in Fig. 3. The surface geometry and the magnetic structure of Cr/Ag(111) as well as the scanning area are also shown.

Figure 5

The bias-dependent magnetic contrast between Cr1 and Cr3 atoms $\Delta z^{\mathrm{Cr}1-\mathrm{Cr}3}(z,V)$ calculated using Eq. (5) with the ideal magnetic tip at different tip-sample separations: $z=3.5$ Å (gray solid line), $z=4.0$ Å (black circles), $z=4.5$ Å (gray dash-dotted line), and $z=5.0$ Å (black squares). The tip magnetization direction is fixed parallel to the Cr1 magnetic moment. Taking the functions colored by gray at $z=3.5$ and $4.5$ Å, Eq. (19) is used to interpolate the bias-dependent magnetic contrast to $z=4.0$ Å (black solid line) and to extrapolate to $z=5.0$ Å (black dashed line). The absolute contrast maximum of each curve is found at $-$1.40 V and is explicitly indicated.

Figure 6

The effect of the tip magnetization orientation on the bias-dependent magnetic contrast between Cr1 and Cr3 atoms $\Delta z^{\mathrm{Cr}1-\mathrm{Cr}3}(z=3.5$ Å$,V)$ at $z=3.5$ Å tip-sample distance calculated using Eq. (5) with the ideal magnetic tip. The unit vectors of the in-plane tip magnetization orientations are rotated in steps of ${30}^{\circ }$ and are explicitly shown. The parallel or antiparallel orientations with respect to the corresponding Cr magnetic moments are given in parentheses. The absolute contrast maximum of each curve is found at $-$1.40 V and is explicitly indicated.