Dipolar Field-Assisted Domain Wall Depinning in Trilayer Structures

The depinning of dipolar-coupled domain walls (DWs) in 3-D permalloy/non-magnetic/permalloy trilayers from notches was investigated using micromagnetic simulation. Due to the coupling, DWs in the top and bottom layers exhibited opposing magnetizations, resulting in either clockwise (CW) or anti-CW (ACW) circulation of magnetization. When propagating the DWs toward a notch positioned on one side of the wire, three distinct phases were observed: no depinning (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${P} _{0}$ </tex-math></inline-formula>), only one wall depinning (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${P} _{1}$ </tex-math></inline-formula>), or both walls depinning (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${P} _{2}$ </tex-math></inline-formula>). Depinning thresholds in the trilayer system were lower than in isolated layers due to the transverse component of the dipolar-coupled field between coupled DWs. While the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${P} _{0}$ </tex-math></inline-formula>–<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${P} _{1}$ </tex-math></inline-formula> phase boundary was unaffected by stacking order, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${P} _{1}$ </tex-math></inline-formula>–<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${P} _{2}$ </tex-math></inline-formula> transition required up to 170% higher fields for CW walls than ACW walls, due to the flux closure causing slight DW offsets and selecting the magnetization of the leading wall. This study demonstrates how dipolar-coupled stray field shape the depinning field, revealing unique 3-D properties beyond those of individual layers.