Thermal Magnetization Fluctuation in Geometrically Constrained Magnetic Domain Wall on the Ferromagnetic Nanowire

We numerically investigated the thermal magnetization fluctuation of a geometrically constrained magnetic domain wall (CDW) in the ferromagnetic nanowire at 300 K. The CDWs are positioned at constrictions like notches and considered based on thermal fluctuation of the 2-D magnetization (M) vector, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\vec {M}(r, t) =m_{x}(r, t)\vec {x}+m_{y}(r, t)\vec {y}$ </tex-math></inline-formula>. We find that thermal fluctuation of the CDW exists at four kinds of specific frequencies—1.52 (mode 1), 2.28 (mode 2), 3.58 (mode 3), and 4.52 GHz (mode 4)—and the frequencies of CDW are distinguishable from those of domains of nanowire. Especially, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${M}_{x}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${M}_{y}$ </tex-math></inline-formula> show stepwise reductions at modes 1 and 3 with increasing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${H}_{t}$ </tex-math></inline-formula> in the transverse direction (−y-direction). It is caused by the thermal fluctuation in transverse M of vortex-type CDWs at notches. Thus, it is clearly seen that as the specific modes of thermal M fluctuations in the CDWs exist, the strength of the thermal M fluctuations in the CDWs depends on each mode. It reflects that the thermal M fluctuation of CDWs pinned at the notch is related to its depinning mechanism at 300 K. Besides, the thermal fluctuations at the domains of nanowire show a nonlocal distribution in the nanowire.