(Color online) Sensor cross-geometry with definitions of variables. w is the width of the central area, βw is the length of each arm. θ c, θx , and θy are the angles between the average magnetization and the x-axis in the central area, the x-axis arms and the y-axis arms, respectively. In PHE measurements, a current Ix is injected in the x-direction and the voltage Vy is measured in the y-direction in response to a magnetic field Hy applied in the y-direction. In AMR measurements, the current is injected in the y-direction, and the voltage Vy is measured in the y-direction.
(Color online) (a) PHE response calculated vs applied field for the indicated values of μ0 H ms using the presented analytical model. The arrows indicate the field sweep direction. (b) Values of μ0 ΔH = H 2 – H 1 as function of μ0 H ms, where H 2 and H 1 are the switching fields in hysteresis loops calculated using the presented analytical model. All calculations were carried out with μ0 H ex = 2.4 mT and μ0 H K = 0.66 mT.
(Color online) PHE measurements on crosses with (a) w (μm) = 40 and 20, and (b) w (μm) = 10, 7, 5, 4, and 3. The curves show the low-field region of sweeps carried out between ±50 mT.
(Color online) (a) AMR measurements of the resistance of the y-axis arm vs μ0 Hy for the indicated values of w. The values of Ry, ⊥ and were obtained from measurements in a saturating field (50 mT) applied along the x- and y-directions, respectively. (b) Schematic of interpretation of the magnetic microstructure for the studied sensors with (I) small values of w (w ≤ 5 μm), (II) intermediate values of w (7 μm ≤ w ≤ 10 μm), and (III) large values of w (w ≥ 20 μm).
(Color online) Parts (a) and (b) show the remanent micromagnetic configuration calculated in LLG for a cross with w = 4 μm and β = 1.5 after application of μ0 Hy = +50 mT and μ0 Hy = −50 mT, respectively. Parts (c) and (d) show Hz calculated in llg at a height of 50 nm above the film from the configurations in (a) and (b). Parts (e) and (f) show the MFM images obtained on a sensor cross with w = 4 μm in the remanent state after application of μ0 Hy = +50 mT and μ0 Hy = −50 mT, respectively. The contrast line A is in both the images tangential to the corner that presents a bright contrast, the line B to the dark contrast.
Results of analytical calculations and micromagnetic simulations on the vertical arm of the cross considered as a separate domain. Columns two and three list the equivalent ellipsoid demagnetizing factors and calculated along the width and length, respectively, for a uniformly magnetized prism with length × width × thickness = L × w × t FM using the method described in Ref. 18 with the values of L and t FM given in Sec. II. Column four gives the field value to be inserted in Eq. (5). The last two columns are the average angles of the remanent magnetization calculated analytically from Eq. (6) and using the micromagnetic software LLG where periodic boundary conditions were implemented to represent a prism of infinite length.
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