In this section, we introduce some recent applications of spin-STM to unravel non-collinear spin ordering and spin-dependent electronic structures in nanometer scale magnetic structures, based on the quantitative characterizations of tips as discussed in the last section.

### 4.1 Non-collinear spin order probed by field-tuned tips

#### 4.1.1 Experimental proof of noncollineartiy of a periodic spin-dependent d*I*/d*V* pattern

A spatially periodic magnetic state could originate from either a collinear [56, 57] or non-collinear [9] spin-density wave (SDW). A collinear-SDW is characterized by a periodic change of the magnitude of the spin moments, while the spin orientation is fixed. On the other hand, a non-collinear SDW is composed of spin moments of constant magnitude but with changing orientation. The d*I*/d*V* signal in spin-STS depends on \(\hat{\varvec{e}}_{\text{T}} \cdot \hat{\varvec{e}}_{\text{S}}\) (Eq. 7) [15]. If one uses a tip of in-plane oriented *M*
_{T} at zero field, as induced by an in-plane magnetic anisotropy, the tip is sensitive only to the in-plane component of *M*
_{S} (*M*
_{S,||}) at zero field. On the other hand, the tip will be sensitive only to the out-of-plane component of *M*
_{S} (\(M_{{{\text{S}},{ \bot }}}\)) at a field large enough to saturate *M*
_{T} along the field direction. Accordingly, magnetic-field-dependent d*I*/d*V* measurement for a collinear-SDW will result only in a change of magnitude, but not of the phase, with changing field. In contrast, measurement of a non-collinear SDW with the same tip will show a field-dependent phase shift of the stripe pattern of the d*I*/d*V* signal, since the tip gets more sensitive to \(M_{{{\text{S}},{ \bot }}}\) as the field increases [9].

#### 4.1.2 Revealing noncollinearity of stripe patterns in BLH Fe nanoisland

Figure 8a, b show CC-STM and d*I*/d*V* images of a Fe|Co island, measured at 0 T. Figure 8c is a hard sphere model representing the atomic stacking in Fe region 1 of b. The Fe regions at the three corners of the island show a stripe contrast along three different directions as indicated by the solid lines superposed along the stripes with the labels 1–3. To resolve the spin ordering in the stripe patterns, we perform in-field spin-STS with the SW tip as characterized by spin-STM on a BLH Co on Cu(111) (Fig. 7). In this tip the direction of *M*
_{T} is tuned by an external field. The magnetic easy axis of the tip is canted by 55 ± 1° from the sample normal. Therefore, it is sensitive to both \(M_{{{\text{S}},{ \bot }}}\)
*M*
_{T,||} and \(M_{{{\text{S}},{ \bot }}}\) at 0 T, while it is sensitive only to the \(M_{{{\text{S}},{ \bot }}}\) at a field large enough to saturate \(M_{{{\text{S}},{ \bot }}}\). Figure 8d shows the field dependence of the d*I*/d*V* profile along the line AA’ perpendicular to the stripe direction of region 1 in Fig. 8b. The wavelength of the stripe pattern, as denoted, is identical with that (1.28 nm) observed in the pure Fe island [7, 20]. Figure 8e shows a zoom-in of the field dependence between two maxima in the profiles, as shown within the grey dashed line in Fig. 8d, for clarity. With increasing magnetic field, the positions of maxima and minima move monotonically from right to left, while the distance between the extrema remains constant. A phase shift of ∆*P* ~ 0.18 nm is measured upon a change of the magnetic field from 0 to 1.5 T. This observation rules out a collinear SDW. Rather, in-field spin-STS identifies a non-collinear-SDW as the spin texture of the stripe contrast.

#### 4.1.3 Simulation of phase shift in the field dependence of the stripe pattern

Based on Wortmann et al.’s discussion [15], the d*I*/d*V*|_{mag} (Eq. 7) signal is proportional to the projection of *M*
_{S} to *M*
_{T} at the tip position, i.e., \(\hat{\varvec{e}}_{\text{T}} \cdot \hat{\varvec{e}}_{\text{S}}\). Here we calculate a normalized \(\varvec{M}_{\text{T}} \cdot \varvec{M}_{\text{S}}\) (\(\left. {\varvec{M}_{\text{T}} \cdot \varvec{M}_{\text{S}} } \right|_{\text{norm}}\)) with respect to the external field. We model *M*
_{T} as a SW magnet of a magnetic easy axis canted by *α* = 55° from the external field direction, as discussed in Fig. 7c and d. An ab initio study of the sample configuration [7] predicted a Néel-type non-collinearity, where the plane of the spin rotation is parallel with the wave vector of a periodicity *λ*
_{SH} = 1.28 nm. Thereby, we call this spin order a “spin-cycloid” in the rest of the paper. Then both the out-of-plane (*M*
_{S,z}) and in-plane (*M*
_{S,x}) components of the *M*
_{S} show a sinusoidal position dependence along the *x*-axis. Thus, \(\hat{\varvec{e}}_{{\text{S,z}}}\) and \(\hat{\varvec{e}}_{{\text{S,x}}}\) can be described by the equations

$$\hat{\varvec{e}}_{{\text{S,z}}} = \sin \left( {\frac{2\pi x}{{\lambda_{\text{SH}} }}} \right),\quad\hat{\varvec{e}}_{{\text{S,x}}} = \sin \left( {\frac{2\pi x}{{\lambda_{\text{SH}} }} \mp \frac{\pi }{2}} \right),$$

(10)

where the signs ‘−’ and ‘+’ in the formula of \(\hat{\varvec{e}}_{{\text{S,x}}}\) indicate a right-rotating (RR) and left- rotating (LR) cycloid, respectively.

We calculate the signal induced by the stripe pattern in region 1 in Fig. 8b. Figure 9a shows a description of *M*
_{T}, with the polar (*θ*
_{T}) and azimuthal (*φ*
_{
T
}) angles, in the Cartesian coordinate system. A sketch of the geometric relation between the in-plane component of the tip magnetization *M*
_{T,||} and the wave vector *k*
_{1} of the stripe pattern for region 1 in Fig. 8b is presented. Figure 9b shows the field dependence \(\left. {\varvec{M}_{\text{T}} \cdot \varvec{M}_{\text{S}} } \right|_{\text{norm}}\) as a function of position *x* (lower figure), calculated with *α* = 55° and *φ*
_{T} = 170° for a RR-cycloid, with the *x*-position dependent out-of-plane (black dotted) and in-plane (red dotted) components of *M*
_{S}(*x*) (upper figure). Figure 9c shows the results of the corresponding calculation but for a LR-cycloid. Note the left-to-right (right-to-left) shift of the maxima of \(\left. {\varvec{M}_{\text{T}} \cdot \varvec{M}_{\text{S}} } \right|_{\text{norm}}\) from 0 to 1.5 T for the RR- (LR-) cycloid for the given azimuthal angle of *M*
_{T}. For the stripe contrast in region 1 of the island in Fig. 8, we observe that the extrema of the d*I*/d*V* curves shift from right to left. Interestingly, the three cases show different amounts of phase shifts with increasing field (Supplementary Information of [7]). We perform a quantitative analysis of the phase shift and its dependence on stripe orientation and field. An excellent agreement between the experiments and simulations reveals that the field dependence of the phase shifts is determined by a given orientation of the tip magnetization.

### 4.2 Spin-STM of a non-collinear magnetic state with a superparamagnetic tip

Figure 10a is CC-STM image of a BLH Fe island on Cu(111) in bridge-site stacking of the topmost atoms, measured with a Fe-coated W tip. Figures 10b–d are d*I*/d*V* images of the same island as that of Fig. 10a measured at 0 T, +1.5 T, and −1.5 T, respectively. The stripe patterns in Fig. 10c, d indicate a cycloidal spin order as discussed in the text above describing Figs. 8 and 9. However, no stripe contrast is observed in the d*I*/d*V* map at zero field. We obtained d*I*/d*V* images under an external field from −3 to +3 T along the sample normal. We show in Fig. 10e the stripe contrast at 5 field values, measured along a direction perpendicular to the stripe patterns as indicated by the white lines in Fig. 10b–d. Figure 10f shows the field dependence of the stripe contrast (red), peak-to-peak amplitude of the contrast oscillation at each field followed by a normalization with the saturation value from the Langevin fitting (Eq. 8). The inspection of Fig. 10e reveals two aspects: (1) The contrast at a given position increases for increasing applied field. No stripe contrast is observed at zero field. The stripe contrast saturates at ~±1.5 T. (2) The contrast depends only on the magnitude of the field, but not on the sign of the field. The extrema positions of the stripe patterns as indicated by the yellow and blue dashed lines in Fig. 10c and d remain unchanged. To obtain a quantitative insight into the field dependence of the stripe contrast (red curve in Fig. 10f), we also obtained the field dependence of d*I*/d*V* signals measured at the center of the Co core of a Fe|Co island (inset) with the same tip. This measurement on the Co reference sample definitely reflects the response of the tip to the applied field (blue in Fig. 10f). This tip behaves as a superparamagnetic particle. The larger slope near zero field for measurements on Co (blue curve) as compared to measurements on Fe (red curve), clearly implies a sizable contribution of the response of the sample magnetic order to the measured quantity of the d*I*/d*V* signal (red). In addition, the results imply a ‘non-hysteretic’ and ‘monotonic’ response of the Fe magnetic order to the applied field.

The field dependence of the magnetization *M*
_{T} of a superparamagnetic tip can be written as *M*
_{T}(*H*) = −*M*
_{T}(−*H*) (blue curve in Fig. 10f). The field dependence of the d*I*/d*V* signal of the pure Fe island shown in Fig. 10 reveals a relation of d*I*/d*V* (−*H*) = d*I*/d*V* (*H*) (red). Then, the relation given by Wortmann et al., d*I*/d*V*|_{mag} = *m*
_{T} · *m*
_{Fe}, resolves the influence of the sign reversal in *H* on the magnetization density of state *m*
_{Fe} (see Sect. 1.2) through a simple calculation

$$\left. {\frac{dI}{dV}\left( { - H} \right)} \right|_{\text{mag}} = \varvec{m}_{\text{T}} \left( { - H} \right) \cdot \varvec{m}_{\text{Fe}} \left( { - H} \right) \Leftrightarrow \left. {\frac{dI}{dV}\left( H \right)} \right|_{\text{mag}} = \varvec{m}_{\text{T}} \left( H \right) \cdot \varvec{m}_{\text{Fe}} \left( H \right),$$

(11)

which leads to a relation *m*
_{S}(−*H*
_{ext}) = −*m*
_{S}(*H*
_{ext}). This, concurrent with the above-mentioned ‘non-hysteretic’ and ‘monotonic’ field dependence, provides an important conclusion on the magnetic property of the non-collinear spin order in the Fe island. The stripe order in Fe island responds to the external field as an effective magnetic moment, which is thermally fluctuating at a given measurement temperature. We show in the next section the significance of this aspect to obtain a spin-polarization map of the Fe stripe phase from the data set shown in Fig. 10a–c.

### 4.3 Spin-polarization of magnetically ordered nanostructures

#### 4.3.1 Differential conductance asymmetry

The magnetic configuration of a sample is determined by its spin-dependent electronic structure, giving rise to spin polarization of the electronic density of states. To investigate the spin polarization of a sample with spin-STM, the asymmetry of the differential conductance, *A*
_{dI/dV
}, is introduced. The asymmetry is defined as [5]

$$A_{{{\text{dI}}/{\text{dV}}}} = \frac{{\left. {{\text{d}}I/{\text{d}}V} \right|_{\text{AP}} - \left. {{\text{d}}I/{\text{d}}V} \right|_{\text{P}} }}{{\left. {{\text{d}}I/{\text{d}}V} \right|_{\text{AP}} + \left. {{\text{d}}I/{\text{d}}V} \right|_{\text{P}} }}$$

(12)

In case of a ferromagnetic sample of bistable magnetization, *A*
_{dI/dV} is calculated from the d*I*/d*V* signals recorded under *P* and *AP* to the unit vector of magnetization configurations, i.e. *ê*
_{T}·*ê*
_{S} = ±1, as schematically illustrated in Fig. 11a. As introduced in Sect. 1.2, Wortmann et al. derived the description of the d*I*/d*V* signal measured by spin-STM [15] (Eq. 7). Substitution of their result into the Eq. 12 leads to

$$A_{{{\text{dI}}/{\text{dV}}}} = - P_{\text{T}} P_{\text{S}} ,$$

(13)

which links the d*I*/d*V* asymmetry, *A*
_{dI/dV}, to the spin polarization of the sample at the tip apex position, *P*
_{S}(*R*
_{T}).

Equations 11 and 12 were applied to the Co island ‘A’ in Fig. 5a to extract the spatial distribution of its spin-polarization. Figure 11a, b are two d*I*/d*V* images on the island measured at μ_{0}
*H* = −1.1 T, with (b) *AP* and (c) *P* magnetization configurations, as indicated in Fig. 5c. Figure 11d is the *A*
_{dI/dV} map, at the given bias voltage *V*
_{b}, calculated from the d*I*/d*V* images in Fig. 11b and c. Oka et al. extracted a set of energy-resolved *A*
_{dI/dV} maps of this Co island, which led to the disclosure of the electronic nature of “spin-dependent quantum interference within a single nanostructure” [5].

#### 4.3.2 Differential conductance asymmetry of non-collinear magnetic order

In case of a helical (or cycloidal) spin order, the local magnetization rotates with a spatial period. Thus, the spatially averaged magnetization is zero. As discussed in Sect. 4.2, a sign reversal of the external field induces a corresponding reversal in that of the local magnetization, *M*
_{S}(*r*; −*H*) = −*M*
_{S}(*r*; *H*), indicative of two distinct antiparallel magnetic states at each position *r*. In addition, the power of spin-STM, which allows to resolve the local magnetic signal down to the atomic scale, makes a study on the local spin-polarization of the cycloidal order in the bilayer Fe island feasible. Although the definition of *AP* and *P* configurations is not applicable to this case due to the periodic change of the local magnetization, one can clearly distinguish two distinct magnetic states, as sketched in Fig. 11e. Combined with the tip magnetic state, we introduce two magnetic configurations *α* and *β*, hence *M*
_{S,}(*r*) = −*M*
_{S,β
}(*r*), analogous to the *AP* and *P* configurations in the case of Fig. 11a. The asymmetry *A*
_{dI/dV
} for the cycloidal spin order in the Fe island is defined as

$$A_{{{\text{dI}}/{\text{dV}}}} = \frac{{\left. {{\text{d}}I/{\text{d}}V} \right|_{\alpha } - \left. {{\text{d}}I/{\text{d}}V} \right|_{\beta } }}{{\left. {{\text{d}}I/{\text{d}}V} \right|_{\alpha } + \left. {{\text{d}}I/{\text{d}}V} \right|_{\beta } }}$$

(14)

With substitution of the Eqs. 7, 14 becomes

$$A_{{{\text{dI}}/{\text{dV}}}} = \frac{{n_{\text{T}} n_{\text{S}} + \varvec{m}_{\text{T}} \cdot \varvec{m}_{{{\text{S}},\alpha }} - n_{\text{T}} n_{\text{S}} - \varvec{m}_{\text{T}} \cdot \varvec{m}_{{{\text{S}},\beta }} }}{{n_{\text{T}} n_{\text{S}} + \varvec{m}_{\text{T}} \cdot \varvec{m}_{{{\text{S}},\alpha }} + n_{\text{T}} n_{\text{S}} + \varvec{m}_{\text{T}} \cdot \varvec{m}_{{{\text{S}},\beta }} }} = - \frac{{\varvec{m}_{\text{T}} \cdot \varvec{m}_{{{\text{S}},\alpha }} }}{{n_{\text{T}} n_{\text{S}} }}$$

(15)

This leads to a link between the symmetry *A*
_{dI/dV} and the spin polarization of the sample at the tip apex position, *P*
_{S}(*R*
_{T}) in the form

$$A_{{{\text{dI}}/{\text{dV}}}} = - P_{\text{T}} P_{\text{S}} \left( {\varvec{R}_{\text{T}} } \right)\cos \theta ,$$

(16)

where *θ* is the angle between *M*
_{T} and *M*
_{S} (see Fig. 1a). The asymmetry *A*
_{dI/dV} corresponds to the projection of *P*
_{S} onto the tip magnetization direction.

If a superparamagnetic tip (Fig. 6) is used in spin-STM/S of the cycloidal spin order in the bilayer Fe island, one is not able to have the above-mentioned two magnetization configurations (*α* and *β*) because the tip magnetization will also be reversed by the sign reversal of the applied field, as discussed in Fig. 10. This always results in the numerator of the Eq. (14) to be zero. We introduce a procedure to overcome this obstacle in the following discussion. A careful inspection of the right hand side of Eq. 15 indicates that the denominator and numerator are no other than non-magnetic and magnetic terms of the d*I*/d*V* signals for the configuration *α*, respectively. These two contributions are provided by the d*I*/d*V*|_{
H=0} and d*I*/d*V*|_{
α
} – d*I*/d*V*|_{
H=0} data, respectively. Figure 11f, g show the d*I*/d*V*|_{
H=0} and d*I*/d*V*|_{
α
} maps of a bilayer Fe nanoisland measured with a superparamagnetic tip, and Fig. 11h is its *A*
_{dI/dV
} map, derived from Fig. 11f and g as given by Eq. 15. Fischer et al. extracted a set of energy-resolved *A*
_{dI/dV} maps of this Fe island, which were successfully utilized to reveal the “spinor nature of electronic states in nanosize non-collinear magnets” [20].