Controlling the motional quality factor of a diamagnetically levitated graphite plate

The levitation of matter has a variety of applications ranging from quantum science and technology through to industrial development of levitated actuators, motors, and robots. Typically to levitate or trap an object requires a source of power, e.g., optical trapping uses strong laser fields, while dynamic electric fields can hold charged objects in Paul traps. These so-called “active” systems suffer from various types of noise and thereby motional heating. In contrast, “passive” diamagnetic levitation, which requires no active driving or energy input, has the potential for very low noise, low heating, levitation, and motional control. Moreover, macroscopic objects can easily be supported and manipulated.¹ For this reason, diamagnetic levitation has recently attracted much attention and since its original exposition,^2–7 it is now firmly regarded as one of the main techniques in levitodynamics.^8–10 

Levitation can be used to explore many fundamental questions, including testing ideas in thermodynamics¹¹ and alternative theories of quantum mechanics,¹² and searching for new types of forces in nature.^13–15 Tabletop experiments have been proposed which explore the relationship between quantum theory and gravity.^16–21 Another major application of levitodynamics is the development of novel sensors, including inertial sensors,^22–27 detecting gravitational waves²⁸ and dark matter/energy,²⁹ magnetometry,³⁰ measuring mass,³¹ and light pressure sensing.³² It is only in levitated systems (using optical trapping) that a macroscopic quantum superposition has ever been achieved.^33–36 Within levitodynamics, plates of pyrolytic graphite are gaining increasing attention as a highly promising platform. These have already been used to experimentally test theories of dark matter²⁹ and may outperform other types of levitated systems.³⁷ 

In addition to quantum science, another important area of levitodynamics is the development of levitated actuators, motors, and robots. These typically use magnetic levitation, due to its ability to support large mass loads against gravity. An example includes the development of miniature robots which move on a planar surface and can be controlled via localized currents.^38–42 Researchers have recently developed photo-thermal methods to actuate the motion of a diamagnetically levitated graphite plate, which has led to a large interest in developing photo-activated 2D positioners.^43–50 We particularly note recent works using diamagnetic levitated graphite for optical energy harvesting,⁵¹ and a multiple-degree-of-freedom nanopositioner.⁵² 

For diamagnetic levitation of macroscopic masses, for example, those used in actuators, the ideal material is Highly Oriented Pyrolytic Graphite (HOPG) due to its extremely high magnetic susceptibility normal to the slab face.^1,53 However, HOPG is an excellent conductor; thus, eddy currents are induced as it moves through magnetic fields. These cause damping, reducing the quality factor of the motional mode, and producing heat leading to thermal noise. Higher quality factors may be achieved in materials which are electrical insulators, such as diamond, polymers, some glasses, and organic materials.^27,54,55 However, these tend to be only weakly diamagnetic,^6,56 thus unsuited to levitation of centimeter-sized objects (see Sec. I of the supplementary material for a brief numerical comparison).

For applications, such as robots and actuators, it may not be necessary to completely eliminate eddy damping, which may even be useful in some situations. For other applications, however, such as motional sensors, it is desirable to completely remove the eddy damping. In electrical transformers, engineers reduce eddy losses either by interrupting the currents with layers of laminated material or by making the core of the transformer out of a highly resistive ferrite magnetic material. The latter route was recently adapted to the diamagnetic levitation of graphite,³⁷ where micrometer-scale graphite particles were encapsulated in an electrically insulating resin. Eddy currents and the associated damping were significantly reduced, leading to quality a factor of $5×105$⁠. At the same time, however, this results in a lowering of the diamagnetic lift. The filling fraction of graphite within the resin cannot be too high without compromising structural integrity, and random orientation of the particles within the resin lowers the effective magnetic susceptibility.

In our work, we seek to control eddy damping using a method more similar to the lamination technique in electrical transformers. We consider the diamagnetic levitation of a solid slab of HOPG, which we then pattern with very narrow through-cut slots, the purpose of which is to interrupt the path of eddy currents. We machine several samples with increasing slot densities. We hypothesize that as we modify the density of slotted interruptions to the eddy currents, then these currents will be modified and so too will the associated eddy damping. The advantage of this method to control the eddy damping is that one retains virtually all the diamagnetic lift of the original HOPG slab, as the through-slots are very narrow.

In the following, we outline the experiment, analysis, and simulation of the levitodynamics of the slotted graphite plates. We find that the through-cut slots allow us to systematically control the motional quality factor in a highly predictable manner, with excellent agreement between theory and experiment.

A sketch of the experimental setup is presented in Fig. 1. A plate of pyrolytic graphite is levitated by four NdFeB magnets placed in an alternating polarity checkerboard pattern. The magnets are rigidly held within a holder which is fixed to a five axis vacuum-compatible motorized stage. The magnet/motor platform is mounted on a small optical breadboard which, in turn, sits on four vibration isolation supports. The position of the graphite sample is monitored by an interferometric displacement sensor. This displacement sensor is based on a compact Michelson interferometer and enables high precision measurements in real time with a resolution of picometers at a high bandwidth.

The whole structure is positioned in a vacuum chamber, which is evacuated by a system consisting of a turbopump, an ion pump, and associated roughing pump. During the measurement periods, the turbopump is switched off to avoid unwanted mechanical vibrations, while the ion-pump operates continuously to maintain high vacuum (⁠$10−7 hPa$⁠). The vacuum chamber and ion pump are supported by a damped and vibration isolated optical table, while the turbopump is supported by a separate vibration damped and isolated platform.

The experiment aims to increase the motional quality factor of a diamagnetically levitated slab of pyrolytic graphite. Four $10×10 mm2$ samples were machined from a single piece of graphite, to ensure that they all possessed similar electric and magnetic properties. The thickness of each sample was approximately $0.77 mm$⁠; however, this was not uniform since the graphite surface was very coarse. Each plate had a pattern of ring-like slits machined into it whose purpose was to interrupt the eddy currents and hence lower the resulting eddy damping forces. These slits were created by femtosecond laser machining. The slit designs and photographs of the machined samples are shown in Fig. 2. A small mirror was then glued onto the center of each piece, to allow an interferometer to measure vertical displacement. Each graphite sample was levitated for a period of 20 min, and its vertical position was recorded using the interferometer. The resulting power spectral density was then analyzed to find the motional quality factor. Further details of the setup are given in Sec. II of the supplementary material.

Due to its positioning, the interferometer is most sensitive to vertical motion of the plate. To understand the frequencies observed in the experiment, we simulated the motional modes for the $N=0$ plate. These modes are determined by the forces and torques experienced as it moves through the inhomogeneous magnetic field above the magnet array. As discussed in Sec. IV of the supplementary material and in agreement with Chen et al.,¹ there are six modes in total. Three of them, oscillations in the horizontal plane or rotation about the vertical axis, have frequencies bunched around 4 Hz. The other three involve vertical motion, namely, vertical oscillation and tilting about the horizontal axes and are predicted to have frequencies around 17 Hz. It is these vertical modes that our interferometric setup will be most sensitive to.

As the graphite plate moves, the magnetic field it experiences, and hence the magnitude and direction of the force on the plate, varies in a complex manner. This will cause a nontrivial coupling between all of the motional modes. For simplicity, we will approximate the three vertical modes as a single effective mode of a one-dimensional oscillator. This oscillator is primarily damped by eddy currents, induced by motion of the plate through the magnetic field. Damping due to air should be negligible at the pressures we consider, see Sec. IV of the supplementary material for detailed calculation.

The power spectral density for a harmonic oscillator is given in Sec. X of Ref. 57. As discussed in Sec. V of the supplementary material, the power spectral densities for the experimental data traces were fitted to the theoretical values, allowing us to extract the effective damping rates $γ$ and natural frequencies $f0$⁠, from which we could calculate the quality factor Q of each oscillator. These are shown in Fig. 3.

As expected, the oscillation frequencies are all approximately 17 Hz. Moreover, there is a slight upward shift in $f0$ as the number of slots increases. As discussed in Sec. IV of the supplementary material, simulations indicate that this is due to the inhomogeneous removal of material from the graphite plate. In total, the rings are able to increase the oscillator quality factor by a factor of ∼40.

To estimate the expected increase in the quality factor due to the slits, we simulated the eddy currents in each of the graphite plates in Fig. 2. The currents are induced by an effective electric potential which depends on the geometry of the plate and its motion through the external magnetic field.^1,58–60 This current then exerts a force on the plate due to the magnetic field, which can be integrated to find the induced eddy damping. We developed both a two-dimensional model in Mathematica and three-dimensional COMSOL simulation, the details of which are described in Sec. VI of the supplementary material.

The two-dimensional model used the finite element method in Mathematica 13.0 to simulate the currents. Moving from $N=0$ to $N=8$⁠, the total current was found to have decreased by an order of magnitude. Around the edges of a slit, very large currents could occur, however, as these occur in infinitesimal areas they do not contribute significantly to eddy damping.

At low pressures, eddy currents are solely responsible for motional damping $γeddy$ of the plate. The quality factor of a mode is inversely proportional to the damping rate: $qN∝mN/γeddy n$⁠, for sample N with mass $mN$⁠. Hence, we can predict the ratio of quality factors of the different plates, by calculating the ratios of their corresponding $γeddy$ values and masses. These are shown in Fig. 4. We can see that the simulated values agree well with experiment. The $N=8$ sample does appear to have a slightly larger quality factor than predicted, which as we will discuss later is most likely due to the machined slots being wider than the design.

We also built a three-dimensional model using the commercial FEM package COMSOL. The eddy currents are plotted in Fig. 5 (these do not significantly differ from those generated in Mathematica). Using this, we were able to predict the absolute values of the quality factors, rather than simply their ratios. The results of the COMSOL simulation are also plotted in Fig. 4, agreeing with the results from the experiment and Mathematica.

The motional quality factor of the graphite plates was measured to increase as more slots were cut into the surface. This increase was consistent with both the two-dimensional Mathematica and three-dimensional COMSOL simulations of the eddy current damping. This indicates that the increase in the quality factor is, indeed, due to suppression of eddy currents. Overall currents were suppressed by an order of magnitude, corresponding to an increase in a quality factor of 40.

In Fig. 4, we can see that for $N=8$⁠, the simulated quality factors are slightly lower than the experimental value. We attribute this primarily to a discrepancy between the designed slot patterns and what is created by the femtosecond laser cutting. In Fig. 2, the slot patterns of the laser-cut samples are clearly wider than the CAD designs. Moreover, the machining process carves V-shaped slits which remove more graphite than expected, an effect which is more pronounced at high slit density. Wider slots yield less eddy currents and, thus, reduced eddy damping, leading to higher quality factors in the actual samples than what is predicted by our models.

To motivate the design of future systems, we explore a simple analytic model of a conductive metal plate oscillating above a cylindrical magnet in Sec. VII of the supplementary material. The quality factor is found to be inversely proportional to conductivity, independent of plate thickness, and proportional to $1/a2$⁠, where a is the length of the plate.

Pyrolytic graphite is one of the most strongly diamagnetic materials known and has great potential for use in levitated technologies. However, due to its high electrical conductivity, it exhibited strong eddy damping. The ability to engineer this damping while retaining a strong diamagnetic susceptibility will permit researchers in a wide range of disciplines the ability to apply such conducting diamagnetic materials to situations where fast motional control is required. We showed that by patterning the graphite plate with through-slots, we can interrupt the eddy currents in a controlled manner and gain detailed control over the eddy damping while retaining the strong diamagnetic lift. In this study, we have not optimized the slotted pattern, and it is an interesting question whether one can produce designs which remove the least material, maintain the structural integrity of the plate, and control the eddy damping to the maximal extent.

See the supplementary material for a discussion about the comparison with diamagnetic electrical insulators, details of the experimental setup, the laser machining setup, analysis of the motional mode frequencies, power spectral density, and eddy damping both numerically and analytically.

This work was supported by the Okinawa Institute of Science and Technology Graduate University, Japan and Macquarie University, Sydney, Australia. The authors acknowledge the technical assistance from A. Elarabi and P. Kennedy from the OIST Engineering Section. The graphite slab machining work was performed in part at the OptoFab node of the Australian National Fabrication Facility (ANFF) utilizing Commonwealth and NSW State Government funding.