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Abstract Current induced spin-orbit torque (SOT) holds great promise for next generation magnetic-memory technology. Field-free SOT switching of perpendicular magnetization requires the breaking of in-plane symmetry, which can be artificially introduced by external magnetic field, exchange coupling or device asymmetry. Recently it has been shown that the exploitation of inherent crystal symmetry offers a simple and potentially efficient route towards field-free switching. However, applying this approach to the benchmark SOT materials such as ferromagnets and heavy metals is challenging. Here, we present a strategy to break the in-plane symmetry of Pt/Co heterostructures by designing the orientation of Burgers vectors of dislocations. We show that the lattice of Pt/Co is tilted by about 1.2° when the Burgers vector has an out-of-plane component. Consequently, a tilted magnetic easy axis is induced and can be tuned from nearly in-plane to out-of-plane, enabling the field-free SOT switching of perpendicular magnetization components at room temperature with a relatively low current density (~1011 A/m2) and excellent stability (> 104cycles). This strategy is expected to be applicable to engineer a wide range of symmetry-related functionalities for future electronic and magnetic devices.
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Three‐dimensional elastic beam frames: Rigid joint conditions in variational and differential formulation
Abstract We consider three‐dimensional elastic frames constructed out of Euler–Bernoulli beams and describe a simple process of generating joint conditions out of the geometric description of the frame. The corresponding differential operator is shown to be self‐adjoint. In the special case of planar frames, the operator decomposes into a direct sum of two operators, one coupling out‐of‐plane displacement to angular (torsional) displacement and the other coupling in‐plane displacement with axial displacement (compression). Detailed analysis of two examples is presented. We actively exploit the symmetry present in the examples and decompose the operator by restricting it onto reducing subspaces corresponding to irreducible representations of the symmetry group. These “quotient” operators are shown to capture particular oscillation modes of the frame.
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- Award ID(s):
- 1815075
- PAR ID:
- 10445793
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Studies in Applied Mathematics
- Volume:
- 148
- Issue:
- 4
- ISSN:
- 0022-2526
- Page Range / eLocation ID:
- p. 1586-1623
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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