Theoretical study of the torque curves in some magnetic multilayer systems

Abstract

Analytical expressions involving features from torque curves are derived for an exchange coupled ͓͑AF͒/͑F͔͒ bilayer, a coupled trilayer ͓͑F͒/͑N-M͒/͑F͔͒, and a magnetic tunnel junction ͑MTJ͒-like ͓͑AF͒/͑F͒/͑N-M͒/͑F͔͒ system. Included in the model are the exchange anisotropy field H E , the off-alignment angle ␤, the bilinear J 1 , and the biquadratic J 2 coupling strengths and the in-plane magnetocrystalline anisotropy fields of the two layers whose easy axes make an angle ␦. It will be shown how these parameters can be analytically derived from the torque curve. For the strong coupling case the MTJ-like system behaves as ͑AF͒/͑F͒ bilayer; for arbitrary ␦ value, the magnetic anisotropy field of the resulting system has a nonlinear relationship with the individual layer anisotropy fields.

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“…For the applied field angle in the 0 2π interval, the torque T vanishes at 0, 2, , 3 2 and 2 π π π π (i.e at 2 nπ , where n is an integer.) [79,81,82]. On the other hand, for (F)/(AF) bilayer with exchange anisotropy and in the aligned case ( ) 0 β = , the curve given by Eq.…”

“…Then the torque exerted on the magnetisation M by the magnetic field ∧ H, T = M H , is recorded. The torque method has been used to detect and measure the parameters pertaining to the exchange anisotropy such as the exchange anisotropy field HE and the off-alignment angle β (for the off-aligned case) [79,82].…”

“…Another interesting feature that can be used is the slope of the curve at a particular H value. The slope method has been applied to determine the field H E and the off-alignment angle β [79][80][81][82] as is discussed below. It was also used to measure the magnetic coupling in a trilayer system [83].…”

“…; the torque curve (inferred from Eq. ( 9)) is shifted compared to the aligned case ( ) [81,82]. The amount of shift (which will noted 0 α in the subsequent analysis) is related to the off-aligned angle β and can be used to experimentally derive the β angle.…”

“…The torque curve method has been used to measure the anisotropy constants of a ferromagnetic thin film, in multilayer and in (F)/(AF) bilayer [79][80][81][82][83]. It will be shown that the torque method can be used to detect and measure the exchange anisotropy field H E and also the off-alignment angle β (for the off-aligned case).…”