Evidence for a conical spin spiral state in the Mn triple layer on W(001): Spin-polarized scanning tunneling microscopy and first-principles calculations

The spin structure of a Mn triple layer grown pseudomorphically on a W(001) surface is studied using spin-polarized scanning tunneling microscopy (SP-STM) and density functional theory (DFT). In SP-STM images a $\mathrm{c}(4\times 2)$ superstructure is found. The magnetic origin of this contrast is verified by contrast reversal and using the $\mathrm{c}(2\times 2)$ antiferromagnetic state of the Mn double layer as a reference. SP-STM simulations show that this contrast can be explained by a spin spiral propagating along the [110] direction with an angle close to ${90}^{\circ }$ between magnetic moments of adjacent Mn rows. To understand the origin of this spin structure, DFT calculations have been performed for a large number of competing collinear and noncollinear magnetic states including the effect of spin-orbit coupling (SOC). Surprisingly, a collinear state in which the magnetic moments of the top Mn layer and the central Mn layer are aligned antiparallel and those of the bottom Mn layer are aligned parallel to those of the central layer is the energetically lowest state. We show that in this so-called “up-down-down” ($\uparrow \downarrow \downarrow$) state the magnetic moments in the Mn bottom layer are only induced by those of the central Mn layer. Flat spin spirals propagating in either one, two, or all Mn layers are shown to be energetically unfavorable to the collinear $\uparrow \downarrow \downarrow$ state even upon including the Dzyaloshinskii-Moriya interaction (DMI). However, conical spin spirals with a small opening angle of about ${10}^{\circ }$ are only slightly energetically unfavorable within DFT and could explain the experimental observations. Surprisingly, the DFT energy dispersion of conical spin spirals including SOC cannot be explained if only the DMI is taken into account. Therefore higher-order interactions such as chiral biquadratic terms need to be considered, which could explain the stabilization of a conical spin spiral state.

Figure 1

(a) Overview STM topographic scan of a Mn film on W(001) with an average coverage ${\theta }_{\mathrm{Mn}}^{\mathrm{av}}=(2.3\pm 0.2)\mathrm{p}$-ALs measured with a magnetic Mn/W tip. (b) Line profile measured between the points indicated by the double-headed black arrow in (a). (c) and (d) Spin-averaged atomic-resolution STM data taken on a DL Mn terrace (c) and a 3L Mn island (d). In both cases a square-shaped $1\times 1$ unit cell is observed, indicating pseudomorphic growth. (e) and (f) Atomically spin-resolved SP-STM scans of DL and 3L Mn, respectively. A $\surd 2\times \surd 2$ magnetic unit cell is reproduced for DL in (e) (compare with Ref. [4]), whereas a $2\surd 2\times \surd 2$ structure is observed for 3L Mn in (f). (g) Tunneling spectra measured on DL Mn terraces (green) and 3L Mn islands (blue). Scan parameters are as follows: In (a), $U=1\mathrm{V}, I=300\mathrm{pA}$; in (c) and (f), $U=10\mathrm{mV}, I=4\mathrm{nA}$; and in (d) and (e), $U=10\mathrm{mV}, I=5\mathrm{nA}$. Stabilization parameters in (g) are $U=2\mathrm{V}, I=300\mathrm{pA}$.

Figure 2

(a) SP-STM image of a surface area covered by a double-layer (DL) and triple-layer (3L) Mn/W(001). (b) Zoomed image of the DL showing the well-known AFM $\surd 2\times \surd 2$ magnetic unit cell [4]. (c) Zoomed image of 3L Mn/W(001) with the $2\surd 2\times \surd 2$ magnetic unit cell. (d) SP-STM scan of the same area as shown in (a) but now measured with a reversed out-of-plane magnetic Mn/W tip. (e) and (f) Magnified views showing the magnetic structure of the Mn DL and 3L, respectively. (g) Sum and (h) difference of the images (b) and (e) taken on the Mn DL before and after tip magnetization reversal, respectively. Note that while (g) shows the structural unit cell and emphasizes the topographic signature of defects, (h) is dominated by the magnetic unit cell, and defects are almost invisible. (i) Sum and (j) difference of the images (c) and (f) taken on 3L Mn before and after tip magnetization reversal, respectively. Scan parameters in (a) and (d) are $U=10\mathrm{mV}, I=4\mathrm{nA}$.

Figure 3

Simulation of SP-STM images of a flat spin spiral [(a) and (c)] and a conical spin spiral [(b) and (d)], propagating into the direction q indicated and exhibiting an angle of $87.5^{\circ }$ between the magnetic moments of Mn atoms in adjacent rows of the top layer. (a) and (b) show atomic magnetic moments in the top Mn layer. (c) and (d) show simulated magnetic contrast of SP-STM images according to the Tersoff-Hamann model for a magnetic tip with a magnetization direction perpendicular to the surface, i.e., in the $z$ direction with normalized contrast. In both simulated images a zigzag domain and a stripe domain can be found, which are also observed in the SP-STM images displayed in (e) and (f) as well as in their difference in (g). Note that (e)–(g) are the same as Figs. 2, 2, and 2.

Figure 4

Sketches of selected magnetic configurations investigated in the DFT study. The magnetic moments are indicated for Mn atoms in the top layer (white spheres), in the central layer (light gray spheres), and in the bottom layer (dark gray spheres). (a)–(e) Coplanar magnetic states with parallel magnetic moments in each layer. Note that the spin states in (a)–(d) do not lead to the observation of a magnetic superstructure in SP-STM images. The spin structures differ in the magnetic alignment of the bottom Mn layer. (f) Stereographic projection of the magnetic moments from the flat and conical spin spiral onto a sphere. Spin spirals lie at the intersection with a cone. The arrowheads show the sense of rotation of the moments when moving along q, as it is preferred by the Dzyaloshinskii-Moriya interaction. (g) $uudd$ state that leads to a $\mathrm{c}(4\times 2)$ magnetic superstructure and is created by a superposition of two ${90}^{\circ }$ spin spirals with opposite rotational sense. (h) Flat cycloidal spin spiral with an angle between adjacent moments within a Mn layer close to ${90}^{\circ }$. (i) Conical spin spiral with an opening angle of ${40}^{\circ }$ which is a superposition of the states in (c) and (h).

Figure 5

(a) Energy dispersion $E\left(\mathbf{q}\right)$ of spin spirals propagating in the Mn triple layer on W(001). Data points represent DFT total energies obtained without SOC, while connecting curves are a fit to the atomistic spin model. Calculations were carried out for the relaxed geometries (cf. Table 1) of four different magnetic states: the $\uparrow \downarrow \downarrow$ state (blue squares), the ${90}^{\circ }$ spin spiral state along $\overline{{\mathrm{\Gamma }}^{\prime }M}$ (green diamonds), the $\mathrm{c}(2\times 2)$ AFM state (yellow diamonds), and the $\uparrow \downarrow \uparrow$ [layered antiferromagnet (LAFM)] state (orange circles). The spin spiral vectors q are chosen along the high-symmetry directions of the first (white square) and second (gray diamond) Brillouin zones (see inset). (b) Magnetic moments for each Mn layer of a spin spiral with the ground state geometry of the ${90}^{\circ }$ spin spiral along $\overline{{\mathrm{\Gamma }}^{\prime }M}$ (shades of green) and the $\mathrm{c}(2\times 2)$ antiferromagnet (shades of yellow).

Figure 6

Energy dispersion $E\left(\mathbf{q}\right)$ of flat cycloidal spin spirals in the 3L Mn/W(001) for the ground state geometry of the $\mathrm{c}(2\times 2)$ AFM state. (a) Spin spiral energy without spin-orbit coupling (SOC). DFT total energies (yellow diamonds) are fitted with a model of bilinear exchange (yellow curve). Contributions of effective intralayer and interlayer exchange in the model are displayed in pink and blue, respectively. (b) SOC contribution to the energy dispersion of a cycloidal, counterclockwise rotating spin spiral, in first-order perturbation theory. The DFT data are fitted with the DMI of the effective spin model, which can also be separated into intralayer and interlayer contributions. Positive energy values imply the preference of a clockwise spiral by the same amount. Note that the interlayer interactions between the top Mn layer and the bottom Mn layer are mapped onto effective intralayer interaction in (a) and (b).

Figure 7

(a) Total DFT energies of conical ${90}^{\circ }$ spin spirals in the 3L Mn/W(001) along the $\overline{{\mathrm{\Gamma }}^{\prime }M}$ direction as a function of the opening angle $\theta$ neglecting SOC. The dashed curves represent fits with only Heisenberg exchange, while the solid curves include also the biquadratic exchange [see Eq. (4)]. All calculations were performed in three different ground state geometries as indicated. (b) SOC contributions to the energy of conical spin spirals obtained via DFT in first-order perturbation theory. The dashed curves represent fits with only DMI, i.e., the first term of Eq. (6), while the solid curves include also the chiral biquadratic pair interaction, i.e., both terms of Eq. (6). CW, clockwise; CCW, counterclockwise. (c) Sketch of conical spin spirals in selected Mn layers. Ferromagnetic layers point towards the viewer, while spiral parts rotate in a perpendicular direction. (d) Total DFT energies without SOC of conical spin spirals propagating only in selected Mn layers, with an opening angle of $\theta ={10}^{\circ }$. The first three columns of the table indicate in which of the layers the spirals propagate (top, middle, and bottom layers).