Sequential vortex hopping in an array of artificial pinning centers

We use low-temperature magnetic force microscopy (MFM) to study the hopping motion of vortices in an array of artificial pinning centers (APCs). The array consists of nanoscale holes etched in a niobium thin film by Ar-ion sputtering through an anodic aluminum-oxide template. Variable-temperature magnetometry shows a transition temperature of 7.1 K and an enhancement of the magnetization up to the third matching field at 5 K. Using MFM with attractive and repulsive tip-vortex interaction, we measure the vortex-pinning strength and investigate the motion of individual vortices in the APC array. The depinning force for individual vortices at low field ranged from 0.7 to 1.2 pN. The motion of individual vortices was found to be reproducible and consistent with movement between adjacent holes in the film. The movements are repeatable but the sequence of hops depends on the scan direction. This asymmetry in the motion indicates nonuniform local pinning, a consequence of array disorder and hole-size variation.

Figure 1

Plan-view SEM image of the 75-nm-thick Nb nanohole array. Insets are SEM images of the MFM cantilever [MikroMasch (NSC-18/Al)] and probe (top) and MFM probe tip (bottom) used for this work. The probe was coated on one side (right) with about a 60 nm Fe layer extending down from the probe tip.

Figure 2

(Color online) (a) Magnetization half loops, $M\left(h\right)$, at different temperatures for the 75-nm-thick Nb nanohole array. The vertical dashed lines mark integer matching fields, $H_M\approx 2170\text{Oe}$. The arrows indicate the direction of the field change. The inset is a plan view SEM image of the nanohole array overlaid with unit cell $\left(a=105\text{nm}\right)$. The outer red circles indicate vortex positions for the first matching field and the inner green and blue filled circles indicate vortex position for the second and third matching fields. A quarter loop of a uniform 75-nm-thick Nb film at 6.4 K is shown for comparison (red lower line). (b) Estimate of the critical current calculated from the magnetization half loops using the Bean critical-state model for a square thin film of side 5 mm.

Figure 3

(Color online) (a) MFM image of the nanohole array cooled to 5 K in the ambient field with tip retracted. The image shows one newly depinned vortex (solid circle), six previously depinned vortices (dashed circle), and two stable vortices (no circle). The tip-sample separation of this scan is 160 nm. The color scale range is $7.7\text{pN}/\mu \text{m}$. (b) Number of vortices first depinned at each scan height (left axis) and fitted maximum force gradient (vortex peak height) as a function of tip-sample separation (right axis). (c) The fraction of vortices still pinned as a function of maximum lateral force (lower axis) and critical current density (top axis). (d) Probability of a vortex first depinning as a function of maximum lateral force (upper axis) and tip-sample separation (bottom axis). The critical current density is given by $J_c=\text{max}\left(F_{\text{lat}}\right)/\left(t{\Phi }_0\right)$. The maximum lateral force axis of (c) and (d) are the same.

Figure 4

(Color online) MFM forward and reverse images at 5 K for an attractive tip-vortex interaction with a tip-sample separation of 100 nm. The discontinuities in the images are due to the vortex hopping between five distinct vortex positions. The color scale range is $14.2\text{pN}/\mu \text{m}$. The relative positions of the images were corrected for piezohysteresis.

Figure 5

(Color online) (a) Forward scan image of an attractive vortex-tip interaction $\left(z=100\text{nm}\right)$ with metastable vortex-position boundaries highlighted: forward hop between 2 and 1, 4 (white solid), forward hop between 3 and 2 (white dashed), forward hop between 3 and 5 (white dash dot) and reverse hop between 1 and 3 (white dot). The two horizontal dashed lines indicate scan lines used to determine lateral shift of vortex from position 2 to 1. Color scale range is $12.1\text{pN}/\mu \text{m}$. The inset shows an enlarged view of the black framed region where the vortex hops from position 2 to position 1. Inset color scale range is $5.2\text{pN}/\mu \text{m}$. (b) Line cuts number 50 and 51 from panel (a). Solid circles are data from the unperturbed vortex in position 1. Solid squares are from the following forward scan line, clearly showing the shifted vortex minimum signal at position 2 and the abrupt decrease in $\partial F_z/\partial z$ as the vortex hops from position 2 back to position 1. The solid lines are Lorentzian fits to the data. The error in the fitted positions, $\pm 6\text{nm}$, is shown on the vertical bars which indicate the center positions of the fits. The image was acquired at 5 K.

Figure 6

(Color online) (a) Reverse scan image with $z=100\text{nm}$. The filled circles indicate the estimated vortex positions and their size the associated error. The solid white circles show the positions and nominal diameter of the APCs while the white dashed circles indicate the associated errors in the positions of the APCs. (b) Forward scan image with $z=100\text{nm}$ showing the enumerated vortex positions, hopping directions, and number of repeats of the hopping cycle. The white circles highlight the locations of the APCs. Color scale range is $9.0\text{pN}/\mu \text{m}$ for both images. (c) Table of depinning events showing the direction of the vortex movement and the number of repeats in the cycle. (d) Polar plot illustrating the distribution of the depinning events measured from the vortex to the MFM tip before (solid) and after (open) a depinning event. The gray ring indicates the region, $0.55\pm 0.06\mu \text{m}$, of maximum lateral force between the vortex and the MFM tip. Initial and final scan line numbers of an event group are displayed at top and bottom of a symbol group. The images were acquired at 5 K.