Figure 1

(Color online) (a) The apparatus diagram is shown at top. A differential measurement scheme using the reflected beam from a Glan-Thompson prism as a reference scaled and subtracted from the transmitted signal beam. A small air core coil is used to apply a low-frequency magnetic field in the plane of the sample and introduce an ac component in the measured MOKE signal. The transmitted signal is fed into a lock-in amplifier to detect the ac response while the dc signal is recorded simultaneously using a standard digital-to-analog converter board. Inset near the coil is a schematic of the typical asymmetric rectangular ac drive signal used. The bias field is provided by a permanent magnet mounted on a rail driven by a stepper motor. The $\widehat{x}$ and $\widehat{z}$ directions are indicated. (b) An AFM image of the disk is shown along with (c) an optical micrograph of the sample. The disks are $\left(1.98\pm 0.01\right)\mu \text{m}$ in diameter, $\left(32\pm 3\right)\text{nm}$ tall, and are separated by a center-to-center spacing of $4\mu \text{m}$. The scale bar on the optical image is $40\mu \text{m}$ long. The height is exaggerated in the AFM data to show the approximately linear edge-profile with a measured slope angle of $\sim 8°$ over an edge width of $\sim 230\text{nm}$. This is a side effect of the shadow-masking deposition process. The small defects are dust which settled on the sample over the course of measurements; AFM images of similar, fresh, samples show no significant defects.Reuse & Permissions

Figure 2

(Color online) (a) The dc-MOKE signal exhibits a typical single-sided hysteresis loop indicating the presence of a vortex state. The measured Kerr signal is linearly proportional to the magnetization of the sample. The slight offset from zero is arbitrary and is introduced by the preamplifiers in the differential measurement process. This loop is the average of 100 measurements. (b) The ac-MOKE signal is proportional to the susceptibility of the sample. These data were acquired with a 240 A/m peak-to-peak sinusoidal ac field with a frequency of 525 Hz in a single measurement.Reuse & Permissions

Figure 3

(Color online) (a) The numerical derivative of the dc MOKE signal (dashed red line) is plotted with the measured susceptibility signal (black solid line) over the annihilation range. The irreversible processes are immediately highlighted by the disagreement where a vortex annihilation appears as an increased signal in the dc derivative and a drop in the ac. (b) The analytical rigid vortex model allows extrapolation of the response of the disks into the annihilation region. The experimental data (solid black line) were acquired with an asymmetric 320 A/m square ac field while the fit (dashed red line) uses a saturation magnetization of 715 kA/m, a demagnetization factor of 0.0168, and an exchange length of 5.85 nm. Inset below is the extracted experimental distribution (solid black line) with a Gaussian fit (dashed red line).Reuse & Permissions

Figure 4

(Color online) (a) The fit results for the saturation magnetization as a function of temperature. The black squares indicate results from rapid temperature sweeps performed within one day while the open red circles and open blue triangles are results from longer term delay measurements performed at fixed temperatures of $23°\text{C}$ and $30°\text{C}$, respectively. The fit line uses a coefficient of $g=2.1\times {10}^{-5}{\text{K}}^{-3/2}$. (b) The fitted mean annihilation field as a function of temperature shows a distinct trend. Averaged data binned in $0.7°\text{C}$ bins is shown with black squares while the raw data are shown as light gray points. Five fit curves are shown. In order from bottom to top, the green (double dotted dashed), black (solid), and blue (single dotted dashed) lines use energy barriers of $E_o\left(0\right)=7.5\times {10}^{-18}\text{J}$, $8.5\times {10}^{-18}\text{J}$, and $9.5\times {10}^{-18}\text{J}$, respectively, in conjunction with a critical field of 14300 A/m while the red (dots) and magenta (dashed) lines use critical fields of 14500 and 14570 A/m with an energy barrier of $8.5\times {10}^{-18}\text{J}$.Reuse & Permissions

Figure 5

(Color online) Changing the sweep rate by adding delays at each data point results in a decrease in mean annihilation field. The trend is logarithmic, in agreement with the applied model. We additionally find that the temperature and sweep rate results agree with the same parameters within a 50 A/m offset. Averaged data are shown with black data points while the raw data are shown as light gray data points. The five fit curves shown are color coded to match the fit curves on the temperature plot. In order from bottom to top, the green (double dotted dashed), black (solid), and blue (single dotted dashed) lines use energy barriers of $E_o\left(0\right)=7.5\times {10}^{-18}\text{J}$, $8.5\times {10}^{-18}\text{J}$, and $9.5\times {10}^{-18}\text{J}$, respectively, in conjunction with a critical field of 14300 A/m while the red (dots) and magenta (dashed) lines use critical fields of 14500 and 14570 A/m with an energy barrier of $8.5\times {10}^{-18}\text{J}$.Reuse & Permissions