Chiral modulations and reorientation effects in MnSi thin films

We present an experimental and theoretical investigation of the influence of a uniaxial magnetocrystalline anisotropy on the magnetic textures that are formed in a chiral magnetic system. We show that the epitaxially induced tensile stress in MnSi thin films grown on Si(111) creates an easy-plane uniaxial anisotropy. The magnetoelastic shear stress coefficient is derived from SQUID magnetometry measurements in combination with transmission electron microscopy and x-ray diffraction data. Density functional calculations of the magnetoelastic coefficient support the conclusion that the uniaxial anisotropy originates from the magnetoelastic coupling. Theoretical calculations based on a Dzyaloshinskii model that includes an easy-plane anisotropy predict a variety of modulations to the magnetic order that are not observed in bulk MnSi crystals. Evidence for these states is found in the magnetic hysteresis and polarized neutron reflectometry measurements.

Figure 1

Modulated states in a thin layer of a cubic helimagnet with in-plane magnetization: (a) a conical phase that arises in the applied field perpendicular to the film surface, (b) a helicoid phase, which is a helix distorted by the applied field perpendicular to the propagation direction, (c) elliptically distorted conical phases formed in a strong in-plane magnetic field, and (d) a hexagonal skyrmion lattice with elliptical distortions imposed by a uniaxial anisotropy. The unit vector $\widehat{\mathbf{n}}$ indicates the direction of the uniaxial anisotropy.

Figure 2

Low anisotropy range of the magnetic phase diagram in reduced variables for uniaxial anisotropy $k=K_u/K_0$ and applied magnetic field $h=H/H_D$ for model (1) with an in-plane magnetic field [$H_D=2K_0/M_s$ is the saturation field for a bulk cubic helimagnet with $K_u=0$ and $K_0$ is an “effective stiffness” (Eq. (3))]. The regions of global stability for helical and skyrmionic states are indicated by different colors. Solid lines designate the first-order transitions: $H_1$ represents the transition between helicoid and distorted cone, and $H_{\mathrm{S}1}$ and $H_{\mathrm{S}2}$ are transitions from a distorted skyrmion lattice to helicoid and distorted cones, respectively. The dashed line indicates the critical field $H_{C2}^{\left|\right|}$ [Eq. (4)] for the distorted cone phase. $H_h={\pi }^2{K}_0/\left(16{\mu }_0{M}_s\right)=0.617H_D$ is the transition between the helicoid and the saturated states. Triangles and squares show experimental values of critical fields $H_{C2}^{\left|\right|}$ and $H_{\alpha }$ for MnSi films of different thickness $d$.

Figure 3

SQUID $M$-$H$ curves of MnSi thin films with an in-plane field applied along $\left[1\overline{1}0\right]$ (filled points) and out-of-plane field along [111] (open points) measured at $T=5$ K. The film thicknesses are $d=11.6$ nm (a), 17.6 nm (b), and 26.7 nm (c).

Figure 4

The in-plane saturation field (filled squares) and the out-of-plane saturation field (open circles) determined from $M$-$H$ curves measured at $T=5$ K, like those shown in Fig. 3

Figure 5

The uniaxial magnetocrystalline anisotropy (a) and the exchange stiffness (b), calculated from the $H_{C2}^{\perp }$ and $H_{C2}^{\parallel }$ in Fig. 4, as a function of MnSi film thickness

Figure 6

(a) In-plane elastic strain measured from plane-view TEM selected area diffraction patterns. (b) Out-of-plane strain determined from x-ray diffraction measurements of the MnSi(111) peak. In both sets of measurements, the Si substrate was used as an internal calibration standard. (c) Ratio of the out-of-plane strain to twice the in-plane strain. The three samples indicated by open red squares contain MnSi${}_{1.7}$ precipitates in excess of 20$\%$. All error bars are $\pm 1\sigma$.

Figure 7

(a) The volume strain as a function of three times the shear strain. The red line shows the polynomial fit to the data over the range cover by the $B_{2,\text{eff}}$ data. (b) The effective magnetoelastic shear stress, where the red line is a fit to the data. The fit in (a) and (b) gives $B_2=1.6\pm 1.0$ MJ/m${}^3$, $D_1=-78\pm 20$ MJ/m${}^3$, and $D_2=-460\pm 120$ MJ/m${}^3$. All error bars represent $\pm 1\sigma$.

Figure 8

The in-plane fields $H_{\alpha }$ and $H_{\beta }$, as shown in Fig. 3, determined from the peak in $dM/dH$ of the SQUID measurements with the magnetic field along $\left[1\overline{1}0\right]$. The filled (open) circles show $H_{\alpha }$ calculated from the decreasing-$H$ (increasing-$H$) branch of the $M$-$H$ curve measured at $T=$ 5 K. The red squares show $H_{\beta }$ for decreasing $H$ (filled squares) and increasing $H$ (open squares).

Figure 9

PNR with $\pm 1\sigma$ error bars of 20 nm Si/26.7 nm MnSi/Si(111) measured at $T=6$ K. The magnetic field applied along the in-plane $\left[1\overline{1}0\right]$ direction is (a) 1 mT, (b) 0.3 T, (c) 0.5 T, and (d) 0.8 T. The fits to the PNR data, shown by the solid lines, yield the magnetic moment profiles given in each of the insets.

Figure 10

XRR measurements of the 20 nm Si/26.7 nm MnSi/Si(111) sample shown in Fig. 9. The solid line is a fit to the data, which gives the x-ray SLD shown in the inset.

Figure 11

Results from density functional theory calculations for strained MnSi: (a) the dependence of the energy increase $\Delta E$ on the rhombohedral angle $\Xi$ with fixed lattice constant $a=0.4556$ nm and the calculated anisotropy for the magnetization along the rhombohedral axis, parallel to [111], and perpendicular to the axis along [1$\overline{1}$0]. (b) Dependence of strain components on $\Xi$ for the deformation with constant lattice parameter. Note the factor 10 for the volume strain ${\epsilon }_{\perp }+2{\epsilon }_{\parallel }$. (c) Corresponding evolution of spin moments for different sites in the rhombohedral cell. (d) Comparison of orbital magnetic moments on the different Mn sites for the magnetization parallel and perpendicular to the rhombohedral axis. In the latter case, the equivalence of the $3b$ sites is lost and the orbital magnetic moments split.

Figure 12

Diagram of a skewed conical phase in an in-plane magnetic field, where $\widehat{\mathbf{n}}$ is the film normal and $\widehat{\mathbf{x}}$ is the direction of the applied magnetic field $H$. The wavefronts of this phase remain parallel to the surface, while the axis of the cone $\widehat{\mathbf{c}}$ cants in the direction of the field by an angle $\phi$.

Figure 13

Experimental magnetization profile as normalized layer-averaged magnetization in field direction $M_x$ vs position in the layer (normalized by wavelength of helix modulation) for the $26.7$-nm-thick film in a 0.5-T field compared to numerical solutions for different modulated states. Data and solutions correspond to reduced anisotropy and field ($k=0.04$, $h=0.50$).