Initial Slope of the Hysteresis Curve

An analytical expression for the initial slope T of the hysteresis curve is derived for a stripe domain structure in a thin magnetic film, giving that T-' is proportional to t ' I2 ( t = film thickness). This is confirmed by measurements on RF sputtered CoCr films with 20 nm 5 t 5 950 nm. Theory and comparison with experimental a f ~ results Using the equilibrium condition = 0 we find ap 2tX One of the numerical quantities used to describe the BfT = 2 (ii (3) ( ) 2) p2 hysteresis curve is its initial slope T (or initial suscep aP n3 tibility). Wielinga et al. [l] used this parameter as a function of the film thickness t for RF sputtered CoCr. (4) In this article an analytical expression is derived that and when tends to the period becomes is very convenient for analysing experimental data be87r3Xt cause we found a linear relationship between T-' and t ' /2. (5) The energy of a stripe domain structure in a thin magnetic film was calculated by Kooy and Enz [2] . The total free energy per unit area FT of the film is the sum of the energy of the magnetization in the external field, the wall energy and the demagnetization energy and is given by: (1) period, u, the domain wall energy density, M the magnetization and M, its saturation value. In order to calculate the initial slope an approximation of the Kooy and Enz model for low magnetization is calculated, i.e. E << 1. Futhermore it is assumed that t / P > 0.5 in order to obtain a 4 % accuracy in the approximation of the series summation. By expanding the series summation into powers of c2 and dividing the energy by p o ~ f it can be shown that where ~T=F'/~~M?, h = H/Ms and X = a , / , u o ~ ~ (the characteristic material length). Applying the a f ~ equilibrium condition = 0 we obtain a& Substituting (5) into (3) we obt;&in the initial slope