{'fraction_non_alphanumeric': 0.033489641657334826, 'fraction_numerical': 0.011583146696528556, 'mean_word_length': 4.654333201424614, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
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.

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. Achieving ultrahigh information density in a controlled way on surfaces of materials [1,2] is one of the ultimate goals of magnetic research nowadays for the purpose of future data storage technological applications. This can be established by the reduction of the size of magnetic information storage units going down to the nanoscale or even to single atoms [3]. Reading and writing information routinely from and to such magnetic units is a great challenge. Spin-polarized scanning tunneling microscopy (SP-STM) [4] employing a magnetic tip proved to be extremely successful for studying magnetism on surfaces in high spatial resolution. Recent experimental advances using this technique allow the investigation of complex magnetic structures (frustrated antiferromagnets, spin spirals, skyrmion lattices, etc.) [3,[5][6][7].
In the most routinely used constant current mode of the SP-STM the apparent height difference between differently magnetized surface atoms allows the discrimination of the individual atomic magnetic properties and the mapping of the magnetic structure. This apparent height difference is called the magnetic contrast. Finding the maximal magnetic contrast for a given surface-tip combination is crucial for a more efficient magnetic mapping. This can be done by using magnetic tips with large spin polarization, or by choosing the appropriate bias voltage. A very few works focused on the investigation of the bias dependent magnetic contrast so far. Among those, a magnetic contrast reversal was reported in two different magnetic systems [8,9], and the effect was related to the surface electronic structure rather than to the effect of the tip. In another work such contrast reversals were observed during the scanning with the STM tip at fixed bias, and this was identified to be due to the magnetic switching of the tip [10]. Moreover, under certain circumstances, a giant magnetic contrast can be obtained, and this effect was explained by chemically modified STM tips [11].
Distinguishing atoms with different magnetic properties on a complex magnetic surface can successfully be performed by spin-polarized scanning tunneling spectroscopy (SP-STS) as well. For example this technique has recently been used to read out information from an atomic scale all-spin-based logic device [12]. Here, through scanning the surface with a magnetic tip, the measured differential conductance (dI/dV ) values vary depending on the magnetic properties of the underlying surface atom. For such a spectroscopic detection of atomic magnetism, the contrast, i.e., the dI/dV difference above individual atoms, should also be tunable by changing the bias voltage [13].
In this work we propose a method to predict the bias voltage dependent apparent height difference between two lateral surface positions on constant current SP-STM images without the need of scanning the full surface magnetic unit cell, but using single point tunneling current data or differential conductance spectra. Therefore, a reconsideration of the relation between constant current and constant height STM images [14] in the SP-STM scenario is necessary, and we introduce two magnetic contrast formulas that contain quantities above the two lateral sites only. Depending on the number of single point measurements with oppositely aligned tip magnetizations different information on the magnetic contrast can be obtained: Taking tunneling current/spectra at one single point, the bias position of the contrast reversal can be identified. Measurements above two inequivalent lateral surface positions A and B at two different tip-sample distances z 1 and z 2 in the combination of (B, z 1 ), (A, z 1 ), and (A, z 2 ) (three points) together with one of the contrast formulas enable the determination of the bias dependent magnetic contrast between the given surface positions at the equivalent tip-sample distance z 1 . From this curve the bias position of the maximally achievable magnetic contrast can be obtained. Employing the other contrast formula requires the recording of the tunneling current/spectra at an extra tip position (B, z 2 ) (altogether four points). In addition, measurements at the specified four tip positions enable the determination of the bias dependent magnetic contrast between A and B surface sites for arbitrary tip-sample separations assuming an exponential decay for the magnitude of the contrast with respect to the tip-sample distance. We demonstrate the predictive capabilities of this method above a complex magnetic surface by performing numerical simulations based on first principles electronic structure data employing a recently developed atom superposition approach [9,15]. Comparing the bias dependence of the predicted magnetic contrasts to that of extracted from constant current SP-STM images we find excellent agreement, and based on that we propose this approach to be applied to SP-STM/STS experimental data as well. Moreover, we analyze the tip-sample distance dependence of the contrast, and also show evidence that the tip electronic structure and magnetic orientation have a major effect on the magnetic contrast.
The paper is organized as follows: The reconsideration of the relation between constant current and constant height STM images in the SP-STM setup together with the two pro-posed magnetic contrast formulas are reported in section II A. The procedure for obtaining different levels of information on the bias dependent magnetic contrast from single point tunneling current data or dI/dV spectra is presented in section II B. We validate our proposal by means of numerical simulations taking the complex magnetic surface of one monolayer (ML) Cr on a Ag(111) substrate and two tip models with different spin polarization characters. Details of the employed atom superposition approach are given in section II C, and the results are presented in section III. The summary of our findings is found in section IV.
Finally, in the appendix we report the derivation of the formula for determining the bias dependent magnetic contrast for arbitrary tip-sample separations, Eq. (19).

A. Contrast in SP-STM
We define the contrast between two lateral surface positions as their apparent height difference on a constant current contour.
On a nonmagnetic constant current STM image the contrast between atoms A and B on the surface at the average tip-sample distance z 1 and bias voltage V is [14,16] where ∆I AB (z 1 , V ) is the current difference above atoms A and B at the tip-sample distance z 1 , and I av (z, V ) is a laterally averaged current over the surface chemical unit cell at a tipsample distance of z. Since it is a tunneling current, it decays exponentially as z increases [14].
We adopt the above relation between constant current and constant height STM images to the SP-STM scenario. In this case the total tunneling current I T OT can be written as the sum of a non-spin-polarized (topographic) part, I T OP O , and a spin-polarized (magnetic) part, I M AGN , [9,16,17] and the total contrast can also be decomposed as the sum of topographic and magnetic contributions, Here, ∆I AB T OP O and ∆I AB M AGN are the respective topographic and magnetic current differences above atoms A and B. I av T OT has to be calculated by laterally averaging the total current over the surface (chemical and magnetic) supercell at a constant tip-sample distance, where N x and N y denote the number of grid points in the lateral x and y directions, respectively. Again, I av T OT (z, V ) is expected to decay exponentially as z increases. In the following, let us focus on the magnetic contrast only. Therefore, we assume an atomically flat sample surface consisting of chemically equivalent but magnetically inequivalent atoms. In this case the topographic contrast between any two surface atoms disappears since ∆I AB T OP O = 0. Hence, the total contrast between surface atoms is the magnetic contrast, ∆z AB (z 1 , V ) = ∆z AB M AGN (z 1 , V ). Since the calculation of the contrast requires the z-derivative of the exponentially decaying laterally averaged total current in Eq.(4), a full scanning of the surface magnetic unit cell at two constant heights is necessary. The measurement time of this is comparable to record the constant current contour above the same scan area, thus there is no advantage of using Eq.(3) for the contrast estimation. We would like to avoid any scanning above the surface but still predict the magnetic contrast between two surface atoms on a constant current contour. Therefore, the denominator in Eq.(3) needs to be reconsidered, and it is allowed to contain current quantities above the two lateral sites A and B only.
A motivation for constructing the magnetic contrast formula is suggested by the following analogy at constant current condition: In a nonmagnetic STM image the modulation due to the surface atoms is superimposed on the average tip-sample distance, whereas in an SP-STM image of a complex magnetic surface the magnetic modulation is superimposed on the topographic image. Therefore, taking Eq.(1) and generalizing to the SP-STM case, the small lateral variation of the current due to the magnetic modulation plays the role of the numerator, i.e., ∆I AB → ∆I AB M AGN , and the topographic current takes the role of the average current in the denominator, i.e., I av → I T OP O . This is a fortunate choice since the topographic currents above all surface atoms are the same due to the assumed chemical . Following this, we can define a magnetic contrast between atoms A and B on the surface at the tip-sample distance z 1 and bias voltage V as where we took advantage of the exponentially decaying character of the topographic tunneling current, We investigate the validity of this equation in section III B.
An even more straightforward idea is to approximate the lateral average of the total tunneling current over the magnetic unit cell as the average of the currents measured above the A and B sites, which is still supposed to decay exponentially with respect to z (for validation, see section III B), Using this quantity, another magnetic contrast between atoms A and B on the surface at the tip-sample distance z 1 and bias voltage V can be defined as Eq.(5) and Eq.(8) are the two key results of the present work for the bias dependent magnetic contrast estimation. We validate them by simulating SP-STM images above the Cr/Ag(111) surface, and extracting apparent height differences from constant current contours in section III B. In the following we consider how the ingredients for Eq.(5) and Eq. (8) can be obtained in SP-STM/STS experiments, and we report a procedure, which gives different levels of information on the bias dependent magnetic contrast from single point tunneling current data or dI/dV spectra measured with oppositely magnetized tips.

B. Magnetic contrast information from single point quantities
Let us assume that one can measure the bias dependence of the tunneling currents I A P (z 1 , V ) and I A AP (z 1 , V ) at the fixed tip position (A, z 1 ) (single point), where A denotes a lateral surface site, z 1 is the tip-sample distance, and P and AP denote parallel and antiparallel tip magnetization orientations, respectively, compared to a predefined direction.
From these data the spin-polarized contribution to the current in Eq.(2) can be obtained as In the following we refer to this quantity as the magnetic current. If the measurement of differential conductances is for any reason easier than of tunneling currents, then the magnetic current can still be obtained from dI A P /dV (z 1 ,Ṽ ) and dI A AP /dV (z 1 ,Ṽ ), which have to be recorded at the same tip position (A, z 1 ) and P and AP tip magnetization directions for a series of bias voltagesṼ . From these two series of tunneling spectra the spin-polarized contribution to the differential conductance can be calculated as Here, we used the fact that the differential conductance can be decomposed as the sum of non-spin-polarized and spin-polarized contributions [13,15], similarly to Eq.(2). Thus, using dI A M AGN /dV (z 1 ,Ṽ ) the magnetic current can be determined at arbitrary V bias voltages via integration: Here, the integral limits correspond to zero temperature.
Assuming structural stability for the tip and the surface during the above described measurements we can determine the bias voltage(s), where the magnetic current is zero, directly from Eq.(9), or by varying the upper integral limit V in Eq. (11). As demonstrated in section III, the zero magnetic current corresponds to a magnetic contrast inversion on the constant current SP-STM image. Thus, from P and AP single point current data or dI/dV , can similarly be calculated as above surface site A, i.e., following the procedure reported at the beginning of this section, Eq.(9)-Eq. (11).
For the denominator of Eq.(5) the topographic current I A T OP O (z 1 , V ), and its vacuum decay constant κ A T OP O (V ) have to be determined from experiments. The topographic current can be expressed using the already available I A P (z 1 , V ) and I A AP (z 1 , V ) current data at tip position (A, z 1 ) as or from the available tunneling spectra dI A P /dV (z 1 ,Ṽ ) and dI A AP /dV (z 1 ,Ṽ ). From these two series of tunneling spectra the non-spin-polarized contribution to the differential conductance can be calculated as Using this quantity the topographic current can be determined at arbitrary V bias voltages via integration: Again, the integral limits correspond to zero temperature.
In order to be able to calculate the vacuum decay constant κ A T OP O (V ) in Eq. (6), where an exponential decay was assumed, the topographic current has to be obtained at a different tip-sample distance z 2 above site A: (A, z 2 ). Therefore, measurements of tunneling currents or dI/dV spectra with oppositely magnetized tips at this tip position are necessary. From , following the procedure reported in the previous paragraph, Eq.(12)-Eq. (14). From expressing Eq.(6) at tip-sample distances z 1 and z 2 , the vacuum decay constant of the topographic current above atom A can be given as The derivation of this formula is identical to that of Eq.(A4) in the appendix. Note that though the absolute tip-sample distances z 1 and z 2 are unknown in experiments, the tip displacement z 2 − z 1 can be experimentally obtained.
Thus, by measuring single point current data or dI/dV spectra with oppositely magnetized P and AP tips at three well-defined positions above the surface: (B, z 1 ), (A, z 1 ), and (A, z 2 ), the bias dependent magnetic contrast between atoms A and B at the tip-sample distance z 1 , ∆z AB I (z 1 , V ), can be predicted following Eq.(5). This is the second level of information on the bias dependent magnetic contrast at a fixed tip-sample distance. From this function the bias position can be identified, where the maximal magnetic contrast can be achieved. We investigate the sensitivity of this bias position depending on the tip-sample distance and the magnetization direction of the tip in section III B.
We proposed a second option for the prediction of the bias dependent magnetic contrast in Eq.  (7), and its vacuum decay constant κ av.AB T OT (V ) have to be determined. Therefore, the bias dependence of the tunneling currents has to be measured at all of the mentioned four tip positions with a fixed P tip magnetization orientation: I J P (z i , V ), where J ∈ {A, B} and i ∈ {1, 2}. These data can also be obtained from tunneling spectra as The averaged currents at the two different tip-sample distances z 1 and z 2 are respectively. Using these and the assumed exponential decay in Eq. (7), the corresponding vacuum decay constant can be obtained as Thus, by measuring single point current data or dI/dV spectra with oppositely magnetized Assuming an exponentially decaying magnitude of the contrast with increasing tip-sample distance, the bias dependent magnetic contrast between atoms A and B at an arbitrary tip-sample distance z in the tunneling regime can be estimated as This is the third level of information on the bias dependent magnetic contrast. The derivation of this formula is reported in the appendix. We check the reliability of Eq.(19) based on contrasts calculated by using Eq. (5) in section III B.
The summary of the presented procedure to obtain different levels of information on the bias dependent magnetic contrast from single point differential conductance or current measurements is given in Table I. In the remaining of the paper we simulate the measurements following a simple model based on atomic superposition and first principles electronic structure data. T IP = (x J , y J , z) above surface site J can be determined as These quantities at different tip positions are necessary to predict the various levels of information on the bias dependent magnetic contrast (see Table I), as well as to simulate SP-STM images. Here, the sum over α includes a sufficient number of surface atoms [13] with position vector R α , projected charge DOS n α S , and magnetization DOS vector m α S [9]. Similarly, n T and m T denote the charge DOS and the magnetization DOS vector projected onto the tip apex atom, respectively. E S F and E T F are the Fermi energies of the surface and the tip. Moreover, the exponential expression is the tunneling transmission assuming spherical exponential decay of the electron wave functions and an effective rectangular tunnel barrier, where m is the electron mass, is the reduced Planck constant, and φ S and φ T are the surface and tip electron work functions, respectively. The effect of the electron orbitals on the tunneling is neglected (independent orbital approximation [16]). Note that a better description of the electron tunneling can be achieved by incorporating such orbital effects, e.g., by prescribing the tip orbital symmetry [23][24][25], or by taking into account symmetrydecomposed electronic structures and an orbital-dependent transmission function [26]. Using an orbital-dependent tunneling model, contrast reversals of nonmagnetic origin are expected [23,26]. In the present work we do not consider the topographic part of the contrast in Eq.(3), and we focus on the bias dependent magnetic contrast and its reversal only.

III. RESULTS AND DISCUSSION
In order to demonstrate the applicability of the proposed procedure for predicting the bias dependent magnetic contrast from single point tunneling data, we perform simulations on a sample surface with noncollinear magnetic order. One monolayer Cr on Ag(111) is a prototype of frustrated hexagonal antiferromagnets [16]. Due to the geometrical frustration of the antiferromagnetic exchange interactions between the Cr spin moments, its magnetic ground state is a noncollinear 120 • Néel state [17]. Taking the spin-orbit coupling into account, two Néel states with opposite chiralities can form, and one of them is energetically favored [9].
We performed fully noncollinear electronic structure calculations on the Cr/Ag(111) system, based on the density functional theory (DFT) within the generalized gradient approximation (GGA) and using the projector augmented wave (PAW) method, implemented in the Vienna ab initio simulation package (VASP) [27][28][29][30][31]. The computational details are reported elsewhere [9]. The ground state magnetic structure is shown in Figure 1 Moreover, it is clearly demonstrated in Figure 2 that the different tip spin polarization characters affect the magnetic current considerably. While the magnetic current above Cr1 is positive and monotonically changing in the negative bias range for the ideal tip, it is negative and has a local minimum for the Ni tip in the same bias range. At positive bias voltages the magnetic current is even more complicated, and it has a local maximum for the ideal tip (see inset), whereas it has two local maxima and one local minimum (see inset) for the Ni tip. In the studied bias range the global maximum and minimum is at -2.5 V and 2.5 V, respectively, for the ideal tip, and at 2.32 V and -1.38 V, respectively, for the Ni tip.
The magnetic contrast is reversed at zero magnetic current. In order to find out the corresponding bias positions for the two considered tips, we have to zoom in the region [0 V, 1 V]. The indicated rectangular area is shown in the inset of Figure 2. It is clearly seen that the sign change of the magnetic current occurs at 0.94 V and 0.74 V for the ideal and the Ni tip, respectively. This means that although the reversal is, in principle, due to the surface electronic structure [8,9], the tip plays a crucial role as well, since its electronic structure modifies the bias position of the reversal. Note that the determination of the above bias positions does not depend on whether the tip is placed above Cr1, Cr2, or Cr3 atoms.
Moreover, we find that the bias voltage of the contrast reversals is stable within ±0.01 V placing the tip above other surface positions.

B. Magnetic contrast between Cr1 and Cr3
In order to quantify the magnetic contrast between two surface sites at a given tip-sample distance, Eq.(5) or Eq.(8) has to be employed. This is the second level of information on the bias dependent magnetic contrast. Following Table I, we calculate all current ingredients for the magnetic contrast formulas using Eq. (20) and Eq.(21) above Cr1 and Cr3 atoms.
and thus, where we omitted the (z, V ) or (V ) arguments of the quantities, and assumed an exponential  Figure 3.
For validating the bias dependent magnetic contrast predictions, we calculate the apparent height difference between Cr1 and Cr3 atoms from constant current SP-STM images, which are also simulated within the atom superposition approach. The current contours are chosen in such a way that the apparent height of the Cr1 atom is 3.5Å at all considered bias voltages. The obtained data for the two considered tip models are shown in Figure 3 using circle symbols. The qualitative agreement with the predicted magnetic contrasts using Eq.(5) and Eq. (8) is obvious at the first sight. Having a closer look we find that Eq. (5) quantitatively reproduces the apparent height difference for ∆z Cr1−Cr3 > 0, and Eq.(8) for ∆z Cr1−Cr3 < 0. This means that both formulas are needed for a quantitative determination of the bias dependent magnetic contrast. In case one is interested in the identification of the bias voltage for obtaining the maximum contrast, the formula requiring less measurements or calculations, Eq.(5), can be applied.
In order to better visualize the bias dependent magnetic contrast, we simulated constant current SP-STM images. Figure 4 shows i.e., at -1.40 V and 2.50 V for the ideal tip, and at -0.84 V and 1.31 V using the Ni tip. The surface geometry and the magnetic structure as well as the scanning area are also shown.
The tip magnetization direction is fixed parallel to the Cr1 magnetic moment. For the zero bias images a temperature of 4.2 K was considered following Ref. [9] since there is no current at 0 K.
We find a similar type of magnetic contrast at 0 V and at the negative bias voltages for both tips, respectively, i.e., for V ≤ 0, ∆z Cr1−Cr3 > 0 (Cr1 appears higher than Cr2 and Cr3) for the ideal tip, and ∆z Cr1−Cr3 < 0 (Cr1 appears lower than Cr2 and Cr3) for the Ni tip, similarly as reported in Figure 3. At the corresponding reversal voltages, 0.94 V (ideal tip) and 0.74 V (Ni tip), all Cr atoms appear to be of equal height on the SP-STM image.
Here, the magnetic contrast is lost as the magnetic current is zero. This corresponds to a qualitatively similar image of performing the STM measurement with a nonmagnetic tip [9].
Thus, the magnetic current calculated at a single point above the surface was indeed able to find the correct bias position of the magnetic contrast reversal for both tips. Above the reversal voltage, the magnetic contrast is inverted. We illustrate this by showing the images calculated at 2.50 V for the ideal tip, and at 1.31 V for the Ni tip in Figure 4.
Let us analyze the effect of the tip-sample distance on the obtained results. Figure 5 shows the bias dependent magnetic contrast between Cr1 and Cr3 atoms calculated using Eq.(5) with the ideal magnetic tip at different tip-sample separations: z = 3.5, 4.0, 4.5, and 5.0Å. The tip magnetization direction is fixed parallel to the Cr1 magnetic moment.
It is clearly seen that the bias positions of the contrast reversal and the local maxima are practically unaffected by the tip-sample distance. We tested this for other distances as well, and the bias positions of the local maxima were found within ±0.02 V deviation. On the other hand, we find that the absolute contrasts are decreasing with increasing tip-sample distance. We would like to test whether this decay is exponential. In fact, assuming an exponential decay, Eq. (19) is derived in the appendix, which enables the determination of the bias dependent magnetic contrast at arbitrary tip-sample distances from two contrast functions at fixed heights z 1 and z 2 . This is the third level of information on the bias dependent magnetic contrast from single point measurements, see section II B. Taking the functions at z 1 = 3.5 and z 2 = 4.5Å, Eq.(19) is used to interpolate the bias dependent magnetic contrast to z = 4.0Å, and to extrapolate to z = 5.0Å. As the results obtained by the interpolation and the extrapolation agree quantitatively well with those calculated by using Eq.(5), the exponential decay of the absolute magnetic contrast with respect to the tip-sample distance and, thus, the validity of Eq. (19) are confirmed within the presented atom superposition framework. Note that orbital-dependent tunneling effects can modify this finding.
Finally, let us analyze the effect of the magnetization orientation of the tip on the magnetic contrast. Figure 6 shows the bias dependent magnetic contrast between Cr1 and Cr3 atoms spectively. This means, e.g., that the same contrast curve is obtained for a combination of (+0.866,+0.500) tip magnetization direction and P T = 0.866 as for the combination of (+0.500,+0.866) tip magnetization direction and P T = 1, the curve denoted by (Cr1) in Figure 6.

IV. CONCLUSIONS
In this work we presented a theoretical description of the contrast, i.e., the apparent height difference between two lateral surface positions on constant current SP-STM images. We Moreover, we showed 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.