Ferromagnetic levitation and harmonic trapping of
a milligram-scale Yttrium Iron Garnet sphere

Visible light shown onto the YIG sphere for illumination, can lead to local temperature increase of the levitated YIG sphere. Here a lens of focus length $f=750$ mm is used to focus the optical beam to a diameter of $d_{\mathrm{opt}}\sim 0.9$ mm at the YIG sphere. Out of this $P_{\mathrm{opt}}=5$ mW input beam,

$$\frac{P_{\mathrm{opt}}}{\sqrt{2\pi}d_{\mathrm{opt}}/2}\int_{-d/2}^{d/2}\exp\left(-\frac{x^{2}}{d_{\mathrm{opt}}}\right)dx\sim 2.8\,\mathrm{mW}$$

(A1)

hits the $d=0.5$ mm diameter YIG sphere. For a 532 nm laser, with reflectivity of $R_{\mathrm{532\,nm}}\sim 0.17$ YIGrefl for a polished bulk YIG, and absorption coefficient $\alpha=10^{3}\,\mathrm{cm}^{-1}$ YIGabsorb , $\dot{q}_{\mathrm{opt}}=2.35$ mW of the illuminated light is absorbed. During the measurement time of 10 s, the YIG sphere is optically heated by $\Delta E_{\mathrm{opt}}=23.5$ mJ.

Since the cooling power of our cryostat of 0.5 W for the superconductor bulk and YIG sphere is much larger than $\dot{q}_{\mathrm{opt}}$, we assume only the temperature of the levitated YIG sphere increases to $T_{2}$, and everything nearby remains at $T_{1}=4$ K. Bulk YIG has a thermal conductivity of $\kappa_{\mathrm{YIG}}(\mathrm{4\,K})=10.5$ W/m K and specific heat of $C_{\mathrm{v}}(\mathrm{4\,K})=0.02$ J/kg K, both of which increase with temperature YIGCv . The thermal conductivity throughout the YIG sphere is approximately,

$$\kappa_{\mathrm{YIG}}\times\pi\left(\frac{d}{2}\right)^{2}\times\frac{1}{d}\times(T_{2}-T_{1}).$$

(A2)

This gives 8.27 mW for $T_{2}=5$ K, which is larger than $\dot{q}_{\mathrm{opt}}$. Thus we can assume the temperature is uniform over the sphere.

The heat capacity of the YIG sphere can be modeled using the Debye model as

$$C_{\mathrm{v}}(T)\propto\left(\frac{T}{T_{\mathrm{D}}}\right)^{3}\int_{0}^{\frac{T_{\mathrm{D}}}{T}}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}\,dx$$

(A3)

with Debye temperature $T_{\mathrm{D}}=531$ K, and proportional coefficient 2085 J/kg K YIGCv . The optical heating can increase the YIG sphere temperature up to $T_{2}\sim 246$ K, which satisfies

$$\Delta E_{\mathrm{opt}}\sim m\int_{T_{1}}^{T_{2}}C_{\mathrm{v}}(\tau)\,d\tau=23.7\,\mathrm{mJ},$$

(A4)

with $C_{\mathrm{v}}(246\,\mathrm{K})=556.6$ J/kg K. This is due to the large absorption of visible light by YIG, as well as the small mass $m=0.339$ mg of the YIG sphere.

In this case, the radiative heat transfer from the levitated sphere to the glass plates on top and bottom of the superconductor bulk hole is negligible

$$\dot{q}_{\mathrm{rad}}=\sigma_{\mathrm{SB}}EA_{\mathrm{YIG}}(T_{2}^{4}-T_{1}^{4})\sim 8.6\,\mu\mathrm{W}.$$

(A5)

Here $\sigma_{\mathrm{SB}}=5.67\times 10^{-8}\,\mathrm{W/m^{2}\,K^{4}}$ is the Stefan-Boltzman constant,

$$E=\frac{\varepsilon_{1}\varepsilon_{2}}{\varepsilon_{2}+A_{\mathrm{YIG}}/A_{\mathrm{env}}(\varepsilon_{1}-\varepsilon_{1}\varepsilon_{2})}$$

with emmitivity $\varepsilon_{1}=\varepsilon_{2}=0.9$ for non-metalic surfaces, $A_{\mathrm{YIG}}=7.85\times 10^{-7}\,\mathrm{m}^{2}$ the surface area of the YIG sphere, and $A_{\mathrm{env}}=1.3\times 10^{-4}\,\mathrm{m}^{2}$ the surface area of the two glass plates EkinBook . Since the emmitivity scales as the square root of a material’s resistivity for metals, there is no heat transfer to the superconductor bulk, which reflects all radiative heat.

The heat conduction through gas in the free molecular regime is a negligible

$$\dot{q}_{\mathrm{gas}}=1.2\times\frac{0.5}{1+0.5A_{\mathrm{env}}/A_{\mathrm{YIG}}}PA_{\mathrm{YIG}}(T_{2}-T_{1})\sim 0.45\,\mu\mathrm{W},$$

(A6)

for $P=2\times 10^{-3}$ Pa EkinBook .

Thus, it is possible that the temperature of the levitated YIG increased to as high as $T_{2}\sim 246$ K. During the experiment, the YIG sphere was repetitively cooled back down to $T_{2}=T_{1}=4$ K in between measurements, when the YIG sphere is lowered back to its initial position at the bottom of the superconductor hole on top of a glass plate. For quantum technology applications, optical heating can be circumvented by attaching a metasurface mirror to the YIG sphere, or using an infrared laser near 1.55 $\mu$m, where YIG is transparent.

The superconductor critical temperature $T_{\mathrm{c}}$ can be determined by measuring the magnetic moment of the superconductor as a function of temperature, while the material is subjected to a low external magnetic field of 0.5 mT (5 Oe). Below $T_{\mathrm{c}}$, the superconductor will expel the magnetic field due to diamagnetic shielding effect, leading to a drop in the magnetic moment. The magnetic moment was measured using a superconducting quantum interference device (SQUID) magnetometer (MPMS-XL5, Quantum Design, Inc.). The critical temperature is determined as the temperature at which the magnetic moment begins to decrease rapidly.

We measured the critical temperature for the custom made high RRR Nb sample, which had a transition temperature of $T_{\mathrm{c}}\sim 9.3$ K (Fig. 6). This agrees with the bulk value of $T_{\mathrm{c}}=9.26$ K for Nb SupercondRev . The sharp drop in the magnetic moment of Nb reflects its purity and good superconducting properties. Note that since the YBCO ring is made from a commercial bulk, and its transition temperature of $T_{\mathrm{c}}=93$ K firstYBCO is well above our levitation conditions at 4 K, indicating that it can be assumed to be in a superconducting state albeit temperature fluctuations, the transition temperature of a machined sample has not been measured.

The critical current density $J_{\mathrm{c}}$ of a superconductor is the maximum electrical current density that can flow through it without causing energy dissipation due to vortex motion. The temperature and external magnetic field dependent critical current density can be calculated from the magnetic moment using the Bean model Bean-model ; Bean-Rev . This is a phenomenological model that assumes a spatially uniform critical current density independent of the magnetic field applied. Here the current density $j=J_{\mathrm{c}}$ inside the superconductor follows the Ampére’s law,

$$\displaystyle\frac{dB_{z}}{dx}=\mu_{0}j,$$

(B1)

where $z$ is the direction of the magnetic field $B$. The image field created inside the superconductor by this current is

$$\displaystyle B_{z}(x)=\mu_{0}J_{\mathrm{c}}x+const,$$

(B2)

which decreases linearly with the distance from the superconductor surface.

The M-H curve of the Nb sample used for $J_{\mathrm{c}}$ measurement is given in Fig. 7 (a). The fluctuations in the magnetic moment at 2 K and 3 K is due to flux jumps caused by magnetic flux avalanches, the rapid invasion of a large number of quantized magnetic flux due to thermo-magnetic instability. The negative value of the magnetic moment in the field-decreasing branch indicates that the irreversible magnetic moment is smaller than the reversible one due to weak flux pinning. This Nb sample is a cylinder, which creates a cone-shaped magnetic field profile and the magnetic moment is given by

$$\displaystyle m_{\mathrm{sc}}=\frac{1}{2}\int r\times J_{\mathrm{c}}\,dr=J_{\mathrm{c}}\int dr\int dz\,\pi r^{2}=\pi J_{\mathrm{c}}\frac{a^{3}}{3}h_{\mathrm{sc}},$$

(B3)

where $a$ and $h_{\mathrm{sc}}$ are the radius and height. The superconductor magnetization is

$$\displaystyle M_{\mathrm{sc}}=\frac{m_{\mathrm{sc}}}{V}=\pi J_{\mathrm{c}}\frac{a^{3}h_{\mathrm{sc}}}{3}\,\frac{1}{\pi a^{2}h_{\mathrm{sc}}}=\frac{J_{\mathrm{c}}a}{3},$$

(B4)

where $V$ the volume of the cylinder. Solving this for $J_{\mathrm{c}}$ gives

$$\displaystyle J_{\mathrm{c}}=\frac{3M_{\mathrm{sc}}}{a}.$$

(B5)

This critical current density is called the “magnetic critical current density”, which gives the maximum current density that a superconductor can carry before it starts to dissipate energy. From Fig. 7 (a), $m_{\mathrm{sc}}(4\,\mathrm{K},\,0\,\mathrm{mT})\sim 0.16\,\mathrm{emu}=1.6\times 10^{-4}\,\mathrm{A/m^{2}}$ for a Nb sample of $a=0.835\,\mathrm{mm},\,V=2.28\times 10^{-9}\,\mathrm{m^{3}}$ gives $M_{\mathrm{sc}}(4\,\mathrm{K},\,0\,\mathrm{mT})\sim 7.0\times 10^{4}\,\mathrm{A/m^{2}}$ and $J_{\mathrm{c}}(4\,\mathrm{K},\,0\,\mathrm{mT})\sim 2.5\times 10^{8}\,\mathrm{A/m^{2}}$. This is much lower than the value measured for Nb thin films at $2.0\times 10^{10}\,\mathrm{A/m^{2}}$ at 4.2 K Nb-Jc . The temperature and external magnetic field dependence of $J_{\mathrm{c}}$ is given in Fig. 8 (a).

The M-H curve for the YBCO sample used for $J_{\mathrm{c}}$ measurement is given in Fig. 7 (b). Due to the strong flux pinning force of YBCO, it retains a large amount of trapped magnetic flux after the external field is removed, and the M-H curve exhibits a large magnetic hysteresis loop. This YBCO sample is a cylindrical structure with a hole and two slits along the radial direction. Thus we model our superconductor as two rectangles with width $a$, length $b\,(a<b)$ and height $c$ of which $J_{\mathrm{c}}$ is given by

	$\displaystyle m_{\mathrm{sc}}^{\prime}$	$\displaystyle=2\int\int S(x)J_{\mathrm{c}}\,dxdz$
		$\displaystyle=2cJ_{\mathrm{c}}\int_{0}^{\frac{a}{2}}[4x^{2}+2x(b-a)]\,dx$
		$\displaystyle=\frac{1}{2}J_{\mathrm{c}}a^{2}bc\left(1-\frac{a}{3b}\right)$		(B6)

	$\displaystyle M_{\mathrm{sc}}^{\prime}$	$\displaystyle=\frac{m_{\mathrm{sc}}^{\prime}}{V^{\prime}}=\frac{1}{2}J_{\mathrm{c}}a^{2}bc\left(1-\frac{a}{3b}\right)\frac{1}{2abc}$
		$\displaystyle=\frac{1}{4}J_{\mathrm{c}}a\left(1-\frac{a}{3b}\right)$		(B7)

$\displaystyle J_{\mathrm{c}}=\frac{4M_{\mathrm{sc}}^{\prime}}{a\left(1-\frac{a}{3b}\right)}$

(B8)

with $S(x)$ being the cross sectional area normal to the external magnetic field, $a=(d_{\mathrm{sc}\,\mathrm{out}}-d_{\mathrm{sc}})/2,\,b=\pi(d_{\mathrm{sc}\,\mathrm{out}}+d_{\mathrm{sc}})/4$. From Fig. 7 (b), $m_{\mathrm{sc}}(4\,\mathrm{K},\,0\,\mathrm{mT})\sim 0.89\,\mathrm{emu}=8.9\times 10^{-4}\,\mathrm{A/m^{2}}$ for a YBCO sample of $a=0.596\,\mathrm{mm},\,b=2.124\,\mathrm{mm},\,V^{\prime}=2.63\times 10^{-9}\,\mathrm{m^{3}}$ gives $M_{\mathrm{sc}}(4\,\mathrm{K},\,0\,\mathrm{mT})\sim 3.4\times 10^{5}\,\mathrm{A/m^{2}}$ and $J_{\mathrm{c}}(4\,\mathrm{K},\,0\,\mathrm{mT})\sim 2.5\times 10^{9}\,\mathrm{A/m^{2}}$. This is higher than the critical current density of Nb, and much lower than the critical current density measured for YBCO wires at $2.0\times 10^{11}\,\mathrm{A/m^{2}}$ at 4.2 K YBCO-Jc . The temperature and external magnetic field dependence is given in Fig. 8 (b). To measure the full phase diagram of $J_{\mathrm{c}}$, it is necessary to measure the full magnetic moment hysteresis curve using high enough external magnetic fields. Since the external magnetic field of 5 T was not high enough, artifacts are introduced in the form of abrupt decrease in $J_{\mathrm{c}}$ above 4 T for temperatures below 10 K.

When a magnetic field is applied to a type II superconductor, vortices, which each carry a quantized magnetic flux, can penetrate the material. For low magnetic fields, the superconductor is in the vortex solid state, where vortices in a superconductor form a lattice. In the absence of any external forces, these vortices are pinned by defects and impurities in the material, resulting in a static arrangement. The superconductor is in a mixed state where both superconducting and normal conducting regions coexist within the material. When the temperature or magnetic field exceeds the critical limit, the superconductor transitions into a vortex liquid state. Vortices move freely throughout the superconductor in a disordered manner, due to thermal fluctuations, external magnetic fields, or interactions with each other, exhibiting fluid-like behavior. The motion of vortices causes substantial energy dissipation, and this phase is therefore not suitable for levitation.

This critical magnetic field can be estimated by the flux pinning force density $F_{p}=J_{\mathrm{c}}B$, which is a measure of the maximum pinning force per unit volume that can be exerted on vortices in a superconductor before they start to move (Fig. 9). This flux pinning force increases with applied magnetic field, because the magnetic field increases the number of vortices, enhances the interaction between the vortices and the pinning centers, making it more difficult for them to move or escape their pinned positions. However, beyond a certain magnetic field strength $H_{\mathrm{fp}}$, the stronger pinning centers become fully occupied or saturated, and additional vortices cannot be effectively pinned, reducing the ability to trap and immobilize vortices. Fig. 9 (a) shows that magnetic fields above $H_{\mathrm{fp}}\sim 100$ mT may cause vortex movement for temperatures below 4 K. This critical magnetic field decreases with increasing temperature. For $B_{z}\leq 75$ mT used for YIG levitation, vortex movement induced dissipation will arise for $T>6$ K. Fig. 9 (b) together with Fig. 7 (b) and Fig. 8 (b) shows that the peak in pinning force density is due to the artifacts in $J_{\mathrm{c}}$ measurement introduced by insufficient maximum external magnetic field strength. Thus $H_{\mathrm{fp}}>5$ T, and there is no vortex movement for magnetic fields below 5 T.

We estimate $H_{\mathrm{pen}}$, the maximum magnetic field our superconductor sample could expel by perfect cancellation of their magnetic moments, which depends not only on the type of superconductor, purity, and structural properties but also shape. This can be estimated from Fig. 7 as the external magnetic field when the modulus of the magnetic moment of a superconductor is maximum (Fig. 10).

For pure Nb, $H_{\mathrm{pen}}$ follows the same temperature dependence curve as the critical magnetic field $H_{\mathrm{c}}$, and the experimental data can be fitted to

$$\displaystyle H_{\mathrm{pen}}(T)=H_{\mathrm{pen}}(0)\left(1-\left(\frac{T}{T_{\mathrm{c}}}\right)^{\gamma}\right)$$

(B9)

with $T_{\mathrm{c}}=9.26$ K, $\gamma=2.13$ predicted from the BCS (Bardeen–Cooper–Schrieffer) theory. This shows that the Nb is in a diamagnetic state for magnetic fields below $H_{\mathrm{pen}}$. The maximum magnetic field our sample can tolerate at 0 K is $H_{\mathrm{pen}}(0)=122\pm 21$ mT. Assuming the temperature of the Nb bulk is equal to the base plate, for YIG levitation conditions of $T\leq 4.6$ K, $H_{\mathrm{pen}}(4.6\,\mathrm{K})\sim 94$ mT is larger than the magnetic fields applied. However, since the external magnetic field $B_{z}$ is close to $H_{\mathrm{pen}}$, local damage in our Nb sample can lead to flux penetrations and increase effective Nb hole inner diameter $d_{\mathrm{sc}}$ as expected from trap frequency measurements.

The magnetic field tolerance $H_{\mathrm{pen}}(4.6\,\mathrm{K})\sim 900$ mT proves the YBCO sample expelled magnetic flux completely during the entire experiment (Fig. 10 (b)). The sharp increase of $H_{\mathrm{pen}}$ at low temperatures is due to strong flux pinning in YBCO. Further research on how strongly pinned vortices affect the eddy current damping at higher external magnetic fields will be conducted elsewhere.

First the three-dimensional magnetic field distribution is calculated by a stationary study, using the Ampére’s law. Since the external magnetic field applied is lower than the saturation magnetization of pure YIG at low temperatures $M_{s}$ = 196 kA/m ($\mu_{0}M_{s}\sim 246$ mT), we use the relative permeability $\mu_{\gamma}=32$ YIGmu-init to calculate the constitutive relation $B=\mu_{0}\mu_{\gamma}H$ used during this calculation. This allows the YIG magnetization to both spatially and temporary respond to the changing magnetic field, which is consistent with the fact that it cannot be treated as a single domain magnetic dipole. The resulting force on the YIG sphere was calculated by numerically integrating the Maxwell stress tensor

$$\displaystyle\mathbf{F}_{em}=\int\int\frac{\mathbf{B}_{n}\cdot\mathbf{B}_{n}}{2\mu_{0}}\,dS$$

(C1)

where $\mathbf{B}_{n}$ is the normal component of the magnetic flux density.

In order to accurately calculate the force on the YIG sphere, constructing a fine enough mesh is crucial. We calculated the forces on the YIG sphere with varying maximum and minimum mesh sizes to resolve the small regions between the sphere and superconducting wall. A mesh conversion analysis showed that a minimum mesh size of $d/100=5\,\mu$m and largest mesh size of $d/20=25\,\mu$m was required to calculate the force on a YIG sphere of $d=0.5$ mm with an accuracy of over two scientific digits. This was used to calculate the force on the YIG sphere as a function of its position. Subsequently, this was numerically integrated to obtain the trapping potential (Fig. 11). This shows that the YIG sphere is indeed trapped in a harmonic potential with its center shifted around 20 $\mu$m from the center of the hole towards the slit.

The dynamics of the YIG sphere is simulated by a time domain study that calculates first-order ordinary differential equations

	$$\displaystyle\dot{v}=F_{em}/m$$		(C2)
	$$\displaystyle\dot{u}=u$$		(C3)

where $m$ is the mass, $u$ the position, $v$ the velocity of the YIG sphere respectively. The resulting oscillations are fitted to a sine-curve to determine the trap frequency.

Eddy current damping occurs when a magnetic object, in this case the YIG sphere, moves relative to a conductor, in this case the bobbin and surrounding oxygen free copper jigs. A damping force that opposes the motion of the YIG sphere arises from eddy currents, which create their own magnetic field that opposes the original magnetic field. The magnitude of this damping force is proportional to the conductivity of the conductor, strength of the magnetic field, and velocity of the magnet. It has been the dominant loss in many magneto-mechanical systems.

A time-dependent solver in COMSOL is used to calculate the electric energy dissipated per oscillation in the YIG sphere, superconductor coil bobbin, oxygen free copper lid, and base plate. The magnetic vector potential is calculated from

$$\displaystyle(j\omega\sigma-\omega^{2}\varepsilon)\mathbf{A}+\nabla\left(\frac{1}{\mu}\nabla\times\mathbf{A}\right)=0$$

(C4)

where $\sigma$ is the conductivity, $\varepsilon$ the permittivity, $\mu$ the permeability, $\omega$ the trap angular frequency, and $\delta=\sqrt{2/\omega\mu\sigma}$ the skin depth. For an oxygen free copper plate at low temperatures, the skin depth is nearly equal to the YIG diameter $\delta\sim d$. For the YIG sphere which is an insulator, the skin depth $\delta>1$ km is larger enough than the objects. In these cases, the eddy current can be calculated by volume integration of the Poynting vector as

$$\displaystyle P_{\mathrm{eddy}}=\frac{1}{2}(\mathbf{J}_{S}\cdot\mathbf{E}*)$$

(C5)

where $\mathbf{J}_{S}$ is the induced current and $\mathbf{E}$ is the electric field inside the object.

A time-dependent solver in COMSOL is used to calculate the electrical losses in both the YIG sphere, bobbin, lid and base plate for a single oscillation (Fig. 12). The YIG oscillation causes eddy currents in both the YIG sphere and its surroundings, which causes a displacement dependent dissipation. Since this dissipation in the bobbin and surrounding jigs is a first order induction effect caused by the YIG motion, while the eddy current in the YIG sphere is a second order induction effect caused by the magnetic field change due to eddy current in the bobbin and surrounding jigs, the former is larger than the latter. In the time dependent study, an oscillation is divided into time slots of $\delta t=1/f/100$ s, and the energy dissipation during each temporal duration is calculated. The energy dissipation per oscillation $\Delta E_{\mathrm{eddy}}$ is given by the average energy loss within the oscillation. Finally the $Q$-factor $Q_{\mathrm{eddy}}=2\pi E_{t}/(\Delta E_{\mathrm{eddy}})$ is estimated by comparing this to the kinetic energy $E_{t}=mA_{z}^{2}\omega_{z}^{2}/2$ where $m=\rho_{y}\,4/3\pi a^{3}$ is the mass of the YIG sphere and $A_{z}$ is the amplitude of oscillation taken from the position of the YIG.