The capacitor design is shown in the schematic in Fig. 1(a). In this work, we studied the effect of several parameters, including magnetic disk diameters, the magnetic strip width and length at the capacitor entrance, corner geometries, particle sizes, and the operating frequency, the results of which will be provided in this section. A magnetic disk or a half-disk exposed to an externally applied rotating magnetic field gets magnetized, and based on our simulation results, energy minima (i.e., energy wells) are formed at positions where the normal component of the magnetic thin film curvature is in parallel or antiparallel to the external magnetic field direction [see dark blue areas in Fig. 2(a)]. The magnetic particles tend to move to these low energy spots. By rotating the external magnetic field, the energy wells move around the disks [see Fig. 2(b)] and meet at the intersection of the two half-disks [see Fig. 2(c)]. Thus, the particles following the energy wells switch from one half-disk to the next one and ultimately move along the magnetic track. The black dashed lines in Figs. 2(b)–2(d) illustrate the magnetic particle trajectory. The insets in Fig. 2 are the corresponding experimental microscopy images that show the particle movement from one disk to another. In these experimental images, compare the position of the blue and red arrows, which stand for the initial particle position, and the particle position at the depicted magnetic field frequency, to see the distance it moves during each step. As a result, this design behaves as a conductor and transports the particles to the desired spots on the chip, while particle positions are synched to the external magnetic field. In each cycle, each particle moves exactly two half-disks.As shown in Fig. 1(a), to engineer a capacitor by placing two magnetic tracks in a T-shaped junction, an additional asymmetry is introduced in the design, followed by a closed-loop magnetic path. We first study the T-shaped junction section in Fig. 3. In the “loading” mode (i.e., when the external magnetic field rotates clockwise), the particle approaching the junction moves from left to the right of the horizontal track over the vertical strip (i.e., it moves in the +x direction). This successful particle transport over the vertical track is possible because, as shown in Fig. 3(c), an energy slope over this track toward the right is formed, and a deep energy well inside the capacitor (on the right side of the vertical magnetic strip) is created. Four timepoints of this process are illustrated in Figs. 3(a)–3(d). After overcoming this barrier, the particle is inside the capacitor and circulates inside it.In the counter-clockwise driving field, the particles on the left side of the vertical strip move away from the capacitors (toward the −x direction). However, the particles inside the capacitor and on the right side of the vertical magnetic strip, initially move along the −x direction toward the vertical magnetic strip [see Figs. 3(e) and 3(h)]. When the particle approaches the T-junction, as shown in Fig. 3(g), the energy well remains on the right of the vertical track, and then it moves upward, as illustrated in Fig. 3(h). Thus, the slight energy barrier on the vertical magnetic track prevents the particle from exiting the capacitor.

A. Strip geometries

We studied the effect of several parameters on the capacitor performance. The geometry of the vertical entrance strip, including its width and length, plays important role in the capacitor operation. Based on our simulation results, as shown in Fig. 4, when the vertical magnetic strip is narrow enough, the particle approaching it at point P1 in Fig. 4(a) experiences a negative energy slope toward the +x direction. However, when the vertical strip is wide, as shown in Fig. 4(b), two energy wells appear on both sides. Hence, the approaching particle at point P1 sees an energy barrier on the strip and cannot enter the capacitor. Figure 4(c), in which we plot the energy distribution along the line P1P2, better illustrates the negative slope of the energy (blue curve) and the energy barrier (red curve) in the cases shown in Figs. 4(a) and 4(b), respectively. We ran a stochastic numerical analysis of the storing efficiency of capacitors too to study the effect of the strip width. These results are shown in curves in Figs. 4(d) and 4(e) for particles with mean diameters of 2.8 and 5 μm, respectively. As shown in these figures, the narrower strip width results in better capacitor loading performances. This fact is more important at higher external magnetic field frequencies and for larger particles. The strip widths in the range of 2–4 μm are studied as our simulation results have suggested that a strip width of 2 or 3 μm is a good choice, but strips wider than 4 μm do not allow the particles to move into the capacitor. The strips narrower than 2 μm are typically hard to fabricate in laboratory setups; thus, we do not include their results here.The effect of the strip length is shown in Fig. 5. When the vertical magnetic strip is short, the particle approaching it at point P1 in Fig. 5(a) experiences an energy barrier, preventing it from entering the capacitor. However, when the vertical strip is long enough, as shown in Fig. 5(b), the approaching particle experiences a negative energy slope and moves inside the capacitor in the +x direction. Figure 5(c), in which we plot energy distribution along the line P1P2, better depicts the negative energy slope (blue curve) and the energy barrier (red curve) in the aforementioned cases. The energy barrier in the red curve (i.e., the 1 μm strip) prevents particle transport into the capacitor. We also ran the stochastic analysis to complete our study on the effect of the vertical strip length. The results are illustrated in Figs. 5(d) and 5(e) for particles with mean diameters of 2.8 and 5.7 μm, respectively. As seen in these figures, shorter strip lengths result in lower capacitor loading efficiencies. This effect is more important at higher frequencies and for larger particles. Based on our results, a strip length of 3 or 4 μm, a strip width of 2 or 3 μm, and a disk radius of 8 μm are good choices for storing tested particles. Again, because of typical challenges in fabricating strips shorter than 2 μm, we do not include their analysis. The strips taller than 4 μm also result in good capacitor loading; however, large strips result in difficulties in particle transport between the magnetic disks. A large strip between the two magnetic disks moves their energy wells far from each other, resulting in a double-well [see Fig. 8(e)]. Thus, the particle cannot appropriately switch between the two magnetic disks. Hence, we do not include the strips taller than 4 μm in Fig. 5.

The thickness of the magnetic thin film is 100 nm only. Also, the chip is covered with a 250 nm thick layer of SiO2, which smooths the surface further. Thus, particles in the size range of micrometers do not see a noticeable physical barrier when moving over magnetic thin films. Hence, we have not included this negligible effect in our analysis.

At low external magnetic field frequencies, the balance between the magnetic force and the drag force results in phase-locked motion at which the particles can follow the energy wells rotating around a disk with a phase lag (θ) [see Fig. 6(a)]. For a fixed frequency below a critical frequency (fc), the phase lag is fixed (i.e., phase-locked regime), but by increasing the applied frequency, the higher particle velocity causes a stronger viscous fluid drag, which increases the phase lag. At frequencies higher than the critical frequency, the particles cannot follow the energy wells and enter the phase-slipping regime. At this regime, the phase lag is no longer constant, and because of the increase in viscous force, the particles periodically slip out of their potential energy wells back into the nearest adjacent energy well. In magnetophoretic circuits, appropriate particle transport is achieved when the particles’ move synced with the external magnetic field at frequencies lower than the critical frequency (i.e., in the phase-locked mode).

B. Magnetic disk and particle sizes

In addition to the external field frequency, the phase lag is dependent on the disk and particle size. Since the particles are synced with the external magnetic field (i.e., in each cycle, they pass two half-disks), they need to move a larger distance circulating larger magnetic disks compared to them circulating smaller disks, in a certain time. Hence, they experience a larger drag force and follow the energy wells around larger disks with a larger phase lag. Particle size plays its role in two ways. First, since they occupy a larger volume, they experience a larger drag force compared to the smaller ones. Also, since we calculate the force at its center, and their center is farther away from magnetic disks, compared to small particles, they experience weaker magnetic forces. Both mentioned effects increase the phase lag for large particles.

Introducing the vertical strip at the T-shaped junction decreases the critical frequency in capacitors by either making the energy wells smaller in size [compare the energy well near the magnetic disk 1 with the one near the magnetic disk 2 in Fig. 6(b)] or splitting them into two smaller wells with an energy barrier in between them. Hence, for appropriate capacitor loading, lower operational frequencies are needed.However, the particle size is even a more sophisticated parameter. In addition to the effect explained above, if the particle diameter is too much larger than the vertical strip length, it experiences energy barriers on its surfaces on the ±y side [see Fig. 6(c)]. Thus, for appropriate capacitor loading and to prevent repelling forces on mentioned areas on particles, it is better to design the vertical strip length bigger than the particle diameter.We ran a series of experiments to study the effects of all the parameters mentioned above. In these experiments, we used three different bead sets with average diameters of 2.8, 5.7, and 8.4 μm. We also fabricated capacitors with disk radii of 6, 8, 10, 15, and 20 μm. The connecting strips in between these disks are 3 μm wide with lengths of 2, 4, 5, and 8 μm. Each recorded loading efficiency is at least based on studying the trajectory of ten particles. The results of these experiments are plotted in Fig. 7. One can use these curves to design the capacitor geometries based on bead sizes of interest in specific applications. One can say that designs based on the disk radius of ∼8 μm and strip length of ∼4 μm result in appropriate loading efficiencies for particles with diameters in the range of 2.8–5.7 μm. Also, loading the capacitors with larger particles can only happen at low frequencies.

C. Corner geometries

Inside the capacitor, the particle experiences four corners. Since our experiments showed that they might get stuck in inappropriately designed corners, carefully engineering them is crucial. As illustrated in Fig. 8(a), in an inappropriate design, the overlap of the energy wells of magnetic disks 1 and 3 may result in an unwanted closed loop particle trajectory, and the particle may get stuck in the corner. An example of an experimental particle trajectory is shown by blue dots in Fig. 8(b), in which the particle has got stuck in a corner. To avoid this problem, the corner must be designed such that these two magnetic disks are far enough and their energy wells do not overlap. An example of such a design, in which the strips in between the half-disks are enlarged, is illustrated in Fig. 8(c). However, a too large distance between the half-disks results in a bistable energy well in between them, as shown in Fig. 8(e).Also, particle size plays an important role in this phenomenon. The energy wells of the two mentioned magnetic disks do not overlap at planes close to the chip surface (i.e., the case for small particles). The smaller the particles, the farther the energy wells are produced by magnetic disks 1 and 3. The energy simulation for such a case is shown in Fig. 8(d), where capacitor geometry is the same as the one used in Fig. 8(a). Thus, the design of Fig. 4(a) cannot transport large particles, but it appropriately transports small particles.

Shifting the disks at the corners toward the center of the capacitor fills the gaps between the magnetic disks. This approach is an alternative method to prevent particles from being trapped at the corners of the capacitors.

D. Complete capacitor design and pilot biological studies

Microscopy images of the capacitors engineered based on our design criteria mentioned above are shown in Fig. 9. The chosen disk diameter, strip length, and strip widths here are 15, 3, and 3 μm, respectively. In this figure, the detected experimental particle trajectories are depicted by blue dotted lines. The capacitor introduced in Fig. 9(a) stores three 2.8 μm magnetic particles at an operating frequency of 0.1 Hz without a problem. However, as an example of a failed loading experiment, as illustrated in Fig. 9(b), this capacitor is not able to store the particle at 1 Hz. We also repeated the experiment with large beads (8.4 μm), which did not enter the capacitor with the mentioned geometries (results not shown here). These results agree well with what we expect based on our simulation outcomes. We also performed experiments with magnetically labeled CD4+ T cells. Figure 9(c) depicts the cell trajectories when the capacitor is loaded with them. The loaded particles circulate inside the capacitors. Movie 1 in the supplementary material shows capacitor loading with magnetized cells.Since the proposed magnetophoretic chips are introduced with the goal of being used in biological studies, here we show their capability in running biological studies. Toward this goal, we ran some pilot drug screening experiments, which show that the cells behave normally on our proposed chips. A typical experiment in analyzing cancer cells is to treat them in a Petri dish with candidate drugs. Two known drugs for MOLM-13 cells are quizartinib and ponatinib. Hence, after loading the capacitors with MOLM-13 cells, we treated them with the two mentioned drugs, on different chips. We studied their effects on cell viability, the results of which are compared with our test results in the culture dishes. [See Figs. 9(d) and 9(e). After adding the drug to the cells, we used a live/dead staining test to evaluate the cell viability. Each experiment is repeated three times. As shown in Figs. 9(d) and 9(e) at different drug concentrations, the cell viabilities in the two experiments are similar. The observed good agreements here confirm our initial pilot studies and proof that our proposed magnetophoretic chips can be used in real biological cell and drug studies.]