Strain patterning of flexomagnetism

Abstract

Flexomagnetism, the coupling of magnetic ordering to strain gradients, provides access to novel symmetry-broken magnetic phases that cannot be accessed via uniform strain. However, flexomagnetism is hard to understand because it is extremely difficult to control a spatially varying strain. Here, we develop a top-down strategy to pattern transverse strain gradients using helium ion implantation through a lithographically defined mask. Using epitaxial films of the antiferromagnetic nodal line semimetal GdAuGe, we demonstrate that transverse strain gradients $\partial \varepsilon_{zz}/\partial x$ induce near-room-temperature ferromagnetic response, compared to the retained para or antiferromagnetism for homogeneously strained GdAuGe. We spatially correlate the magnetic response with the regions of largest strain gradient, via magnetic force microscopy and nanobeam x-ray diffraction, respectively, to confirm the flexomagnetic response. Our approach opens new avenues for the precise control of magnetic phases in thin films of quantum materials via a patterned strain gradient.

Figures (5)

• Figure 1: Types of strain gradients. (a) Bending induced strain gradients. The $\pm$$\varepsilon_{xx}$ refers to the tensile and compressive strain present in the lattice in a bent geometry which can be considered as half of a single wrinkle. In such geometry both transverse strain gradient and longitudinal strain gradient occur. Break down of the dominant strain gradient components: (b) longitudinal, $\partial \varepsilon_{xx} /\partial x$ and (c) transverse, $\partial \varepsilon_{zz} /\partial x$. The intrinsic longitudinal strain gradient resembles an inhomogeneous uniaxial deformation, while the transverse component reflects in-plane spatial variations of the transverse normal strain. Shear component of strain is not shown.
• Figure 2: Homogeneous strain doping. (a) X-ray diffraction $\theta-2\theta$ scans about the (0002) GdAuGe reflection fo He implanted GdAuGe films on graphene/Ge (111) substrates. (b) Off-axis 10$\bar{1}$2 reflection. (c) Variation of the out-of-plane (c-axis) and in-plane (a-axis) lattice constants with helium dose, with the percentage values indicating the relative change compared to the undosed state. (d) The unit cell of GdAuGe showing the unit cell constants.
• Figure 3: Patterning of transverse strain gradients. (a)-(c) Representative 2D reciprocal space maps (RSM) using hard-xray-nanobeam of 40 nm effective beam width at uniformly strained, boundary and unstrained positions (denoted by markers). The vertical axis $Q_z$ follows the reciprocal lattice vector of (0004) reflection. (d) Spatial strain map constructed by analyzing the 2D RSMs recorded at each (x,y) sample position over an area of 20 $\mu$m $\times$10 $\mu$m with steps of 100 nm. The color-scale on the left denotes the out of plane strain percentage. Details of strain calculation is described in supplementary text. (e) Experimental strain profile extracted from the strain-map shown in (d). (f) Theoretical strain relaxation profile over a single strain-unstrained phase boundary, calculated using phase-field simulations. The strain components shown are out of plane ($\varepsilon_{zz}$), in plane ($\varepsilon_{xx}$) and shear ($\varepsilon_{zx}$) strains.
• Figure 4: Strain gradient-induced ferro/ferrimagnetism. (a) Temperature dependence of magnetization under 100 Oe magnetic field oriented inplane. The strain patterned film has alternative stripes with $\varepsilon_{zz}=0\%$ and $\varepsilon_{zz}=2\%$. The uniform sample is homogeneously doped to $\varepsilon_{zz}=2\%$. (b) Isothermal magnetization measured at 150 K with magnetic field variation upto $\pm$ 0.1 Tesla. The strain-patterned sample shows a non-linear ferromagnetic hysteresis loop with remanence of 0.06 $\mu_B$ and coercivity of $~$100 Oe, whereas the uniformly strained sample shows a linear response with negligible remanence and coercivity. Field orientation is inplane along x. (c) Field-direction dependence of isothermal magnetization measured at 150 K in the strain-patterned sample. A nonlinear hysteresis loop which saturates with field, appears only when applied field is along the direction of strain propagation (x). Inset shows the zoomed-in view of the hysteresis in all three direction.
• Figure 5: Magnetic force microscopy (MFM) of the strain-patterned GdAuGe. (a) Atomic force microscopy topographic image of the sample surface, higher steps (brighter) are helium-exposed or strained regions. The randomly distributed white spots and stripes are residual photoresist. (b)-(e) Representative MFM images taken at 20 K under 0, 1, 3, and 7 Tesla out of plane magnetic fields $H_z$. (f) Line profiles of the MFM signal ($\Delta f$) taken at the position of the dotted line in (c). The shaded background marks the strained and unstrained regions.(g) Micromagnetic simulations of MFM signal $\Delta f$ considering a stray field distributions shown as schematic in (j). (h) Experimental strain-gradient derived from nanobeam diffraction profile in Fig.\ref{['pattern']}(b). (i) Magnetic field dependence of the $\Delta f$ values at locations marked in (f). (j) Schematic describing the stray field in the strain patterned GdAuGe film.