Levitation and Suspension Effects in Superconductors
Brian B. Schwartz
Department of Physics, Brooklyn College, and The American Physical Society
Introduction
The discovery of high-temperature superconductors has added a new image to our media-oriented visual society: that of a small magnet floating above a pellet of superconductor cooled to liquid nitrogen temperature, 77 K. A picture much like that in Figure 1 has appeared in scientific journals, newspapers, and popular as well as business-oriented magazines. There are two major hallmarks of superconductivity. First there exists a critical temperature Tc below which the resistance goes to zero; second, below Tc, the magnetic field is excluded from the interior of the superconductor. This field exclusion effect is called the Meissner-Ochsenfeld effect. The levitation of a magnet above a superconductor, or vice versa, was well known before the discovery of high- Tc superconductors; however, it was difficult to demonstrate. A special see-through glass-walled dewar operating at liquid helium temperature, 4 K, was required. The discovery of high- Tc superconductors changed the ease with which the demonstration of levitation is possible. All one needs now is a Styrofoam cup and relatively cheap and easy-to-handle liquid nitrogen. Even more importantly, the demonstrator and the observers are now able to get a hands-on feel for the levitation effect. This ability to "tinker" is especially important because one can feel directly some of the special levitation properties of the high- Tc materials. For example, the levitating superconductor exhibits lateral stability, and requires a force to move it sideways. This would not be the case if the superconductor simply excluded the magnetic field. Another easily observable effect is the fact that the levitation height can vary and thus is not unique. Even more surprising was the recent discovery of the suspension effect,1 in which a chip of a high-temperature superconductor with strong pinning can be raised from a liquid nitrogen container and held suspended below the magnet in a "stable" position. (See Fig. 2.)
In this essay, we present a discussion of the levitation and suspension effects associated with superconductivity. In all cases, we will try to present a physical picture in terms of a simple understanding of the magnetic behavior of superconductors. We shall use the force law between the gradient of the magnetic field of a magnet and a magnetizable material. As a background, we define and describe the magnetization of type I superconductors, and another kind of superconductor called type II, which allows for some penetration of magnetic field in special quantized flux units called vortices. The flux vortices, produced by small circulating currents, can be effectively pinned and thus become unable to move sideways or into or out of the superconductor easily. The pinning effects are instrumental to understanding the suspension effect and the lateral stability as well as the possibility of many "stable" levitation and suspension heights.
Perfect Diamagnetism and the Meissner Effect
If an electric field E in a metal acts on an electron of charge e and mass m, Newton's force law gives F = eE = m(dv/dt). If one assumes perfect conductivity in which no resistive term scatters the electron velocity, and uses Maxwell's equations, one obtains the magnetic behavior for perfect conductors in which the magnetic field lines deep inside the perfect conductor do not change with time. As soon as one attempts to change the internal magnetic field, lossless currents are set up which maintain the internal field, Bin, and thus within a perfect conductor (dB/dt) = 0. In 1933, however, careful magnetic measurements by Meissner and Ochsenfeld showed that not only does (dB/dt) = 0 hold within superconducting materials, but, in addition, the internal magnetic field itself is also zero, B = 0. In other words, superconductors are not simply perfect conductors but also exhibit perfect diamagnetism. The magnetic field inside the superconductor is related to the external magnetic field H and the magnetization M of the superconductor through the equation
B = m 0H + M
If, as Meissner observed for superconductors, B = 0, then M is
M = -(1/m 0)H = c H
where c = (-1/m 0) is the magnetic susceptibility and corresponds to perfect diamagnetism. Superconductors which exhibit a complete Meissner effect with perfect diamagnetism are called type I superconductors. In Figure 3, we plot B and M versus the applied magnetic field H. Since excluding magnetic field from the interior of a superconductor raises its free energy, there exists a critical field Hc, above which it is energetically favorable for the superconductor to revert to the normal state where 0 = m 0H and M = 0.
Incomplete Magnetic Flux Exclusion, Type II Superconductors
Almost all the elemental metals which exhibit superconductivity (except Nb) are type I superconductors. There is another class of magnetic behavior for some alloys (such as NbTi), compounds (such as Nb3Sn), and the new high-Tc superconductors, called type II superconductors. The magnetic field penetrates type II superconductors in quantized units of magnetic flux called Abrikosov vortices, corresponding to swirls of supercurrent producing a flux value of 2.07 x 10-15 Wb. One must take into account the balance between the increase in energy required to create a vortex, and the lowering of the magnetic exclusion energy if flux can penetrate into the superconductor. Above a minimum field, it becomes energetically favorable to allow some flux to penetrate into the type II superconductor. The separation between type I and type II behavior is determined essentially by the ratio k of the penetration depth of the decaying magnetic field at the surface of a superconductor l to the superconducting correlation length, x 0, which corresponds to the length over which the electrons of the Cooper pair which are responsible for superconductivity are "bound." Type I superconductors have k < 1 whereas type II superconductors have k > 1. Figure 4 illustrates the magnetic behavior of an ideal type II superconductor. At a lower critical field Hc1, it first becomes favorable for the flux vortices to penetrate. The material remains superconducting until the upper critical field value Hc2 is reached. In the region H c1 < H < H c2, the field inside the superconductor, B is smaller than the external field H corresponding to a negative magnetization. At and above H c2, the superconductor returns to the normal state with B = m 0H and M = 0.
The Magnetic Behavior of Pinned Type II Superconductors
For ideal type II superconductors, the flux vortices can move into and out of the superconductor without any hindrance. The superconductor can always reach its thermodynamically most stable state in a magnetic field. The magnetization curves are reversible and each external field value gives a unique value of internal field B. If however, the flux vortex or bundles of vortices which are responsible for the internal field are impeded in their movement (pinned), then a situation can arise in which the field inside the superconductor differs from that of an ideal type II superconductor.
A comparison can be made between the ideal type II behavior shown in Figure 4, and the pinned, irreversible behavior shown in Figure 5. Above H c1 for increasing magnetic fields, the flux for the pinned case does not penetrate freely into the superconductor. As a result, a screening current is set up such that the internal field is smaller than that in the thermodynamically ideal state. The change in the internal field and the magnetization at the lower critical field H c1 is not very sharp. The field is prevented from penetrating the superconductor until the pinning force is overcome by the Lorentz force produced by the screening current acting on the quantized flux vortices. This model for flux motion is called the Beam model and produces the negative magnetization curve for increasing external field shown in Figure 5. The stronger the pinning force, the more difficult it is for flux to penetrate. The upper critical field H c2 remains the same since it is the magnetic field value for which the superconducting state returns to the normal state. If we now lower the external field from H c2, we get the opposite situation. The flux which is inside the superconductor at H c2 cannot move out freely. As a result, the internal field remains higher than the external field, leading to a positive magnetization. This positive magnetization remains even if the external field is reduced to zero. Figures 4b and 5b are sketches of the internal and external field behavior for an ideal and pinned type II superconductor in increasing and decreasing magnetic fields.
The Force of a Magnetic Field on a Magnetizable Object
For simplicity, let us consider the case of the vertical force by a magnetic field H on a small (unit volume) specimen which is magnetized in the z direction parallel to B. The magnitude of the force is given by
where (dH/dz) is the gradient of the external magnetic field and M(H) is the magnetization of the material in the external field. Note that the force can be either positive or negative depending upon the signs of the gradient field and the magnetization. (This same force equation can be used to explain the attraction of a magnetizable material, like soft iron, to a permanent magnet, and the attraction and repulsion between two permanent magnets.)
For ideal type I or ideal type II superconductors, some external field is always excluded from the superconductor. The force between a magnet and a superconductor is always repulsive, and if the force is strong enough to balance the downward gravitational force, one obtains levitation. For example, if we consider a magnetic dipolar field along the z axis, then one has H a z-3 and (dH/dz) a -z-4. For type I superconductors M(H) = -(1/m 0)H. Therefore, the force between the dipolar field and a type I superconductor is proportional to z-7 and thus is in a positive direction for z > 0 (repulsive) and in the negative direction for z < 0 (again repulsive with respect to the dipole). As a result, the superconductor can levitate above a magnet or a magnet above the superconductor (see Fig. 1). The force on an ideal type II superconductor by a dipolar field is always negative as well since M(H) is always diamagnetic (negative) and again levitation can be obtained.
The Force of a Magnetic Field on a Pinned Type II Superconductor
The situation for a pinned type II superconductor is quite different. The vertical magnetic force can be either attractive or repulsive depending upon the sign of the magnetization. A superconducting chip can be suspended below a magnet if the field gradient times the positive magnetization of the superconductor gives rise to a magnetic force with sufficient attraction to counterbalance the downward force of gravity. In such a situation, the superconducting chip can hang suspended in the gradient of the magnetic field as shown in Figure 2.
The suspension effect can be observed in special samples of high-Tc yttrium-barium-copper oxide (123 material) doped with silver oxide, for which strong pinning occurs. These doped materials possess the magnetization behavior sketched in Figure 6. To observe the suspension effect, a special procedure must be used, and is shown schematically in Figure 7. The permanent magnet is first lowered toward the superconducting chip, which is immersed in liquid nitrogen (Fig. 7a). We assume axial symmetry so only forces in the z direction must be considered. The superconductor responds to the field of the magnet in such a way as to be repelled by the permanent magnet. This corresponds to the negative (diamagnetic) initial magnetization region in Figure 6 in which the diamagnetic superconductor and the permanent magnet repel one another. The chip is forced against the bottom of the cup by both gravity and the magnetic field. As the permanent magnet is lowered further, the repulsive force can be even stronger (Fig. 7b) because the sample is more diamagnetic. At this point, the permanent magnet is moved away from the superconductor. This corresponds to reducing the external field, which in turn results in flux trapping and a positive magnetization (upper side of the loop in Fig. 6). This positive magnetization leads to an attraction of the superconductor to the permanent magnet (Fig. 7c). If the force of attraction is strong enough to counterbalance the force of gravity, the superconductor can be lifted out of the liquid nitrogen and will hang suspended in space below the permanent magnet (Fig. 7d and the photograph in Fig. 2), as long as it remains superconducting.
The attraction and ultimate suspension of the superconductor below the permanent magnet is unlike the case of the attraction due to a soft iron magnet. The force of attraction gets stronger as an iron specimen gets closer to the permanent magnet. In the case of the pinned type II superconductor, the attractive force gets stronger the further away the magnet is pulled from the superconductor. To understand this phenomenon, one must take into account the negative slope of the upper part of magnetization curve in Figure 6. If the suspended superconductor is disturbed slightly such that it moves closer to the permanent magnet (stronger magnetic field) than the original suspension point, as in Figure 7e, the flux remains constant in the superconductor and the screening current is reduced. This response is similar to what one expects for a perfect conductor. The positive magnetization of the superconductor is reduced and the attraction weakens. The gravitational force causes the superconductor to "fall back" towards the original suspension point. If the superconductor is disturbed such that it moves farther from the permanent magnet (weaker magnetic field), as in Figure 7f, than the original suspension point, the positive magnetization increases along the magnetization curve. The attraction gets stronger and the superconductor "moves up" towards the original suspension point. In effect, the superconductor acts as if it has a built-in servomechanism that acts to increase or decrease its magnetization so as to return it toward the original suspension point.
The irreversibility and hysteresis of the magnetization does not lead to a unique suspension point. The only requirement for suspension (or levitation) is that the upward force counterbalances the downward force of gravity. If the magnet in Figure 7b is initially lowered even further (larger B in the hysteresis loop), the reversal of the sign in magnetization upon pulling away would occur at a higher field. One would obtain a smaller suspension height, for a dipole, where the larger field gradient is closer to the magnet. In addition, if the superconductor is disturbed while it hangs suspended, a different magnetization path will be traced out, leading to another possible suspension height. The vertical behavior near the suspension point is discussed more fully elsewhere.5 Similar arguments for lateral forces show that lateral stability is also due to flux pinning.
Conclusion
The experimental discovery of magnetic suspension was a direct result of working with the new high-Tc superconductors in a liquid nitrogen environment. Palmer Peters of NASA expected the usual repulsive response of superconductors to a permanent magnet. In some cases, however, he observed that some of the superconducting chips seemed to be sticking to his magnet. At first he thought it was water condensation or some direct contact surface-sticking effect. He noted, however, that the superconducting chip was truly hanging suspended in space below the magnet. Shortly after this observation, it was realized that pinning with flux tapping would give rise to a positive magnetization and could explain the suspension effect. Finally, one should note that the suspension of the superconductor at a fixed distance from a magnet does not depend upon a gravitational field. (For levitation, the magnetic force is always repulsive. In the absence of gravity, the superconductor and magnet move further and further away from each other.) In the procedure sketched in Figure 7, when the magnet direction is reversed, for zero gravity, the suspension would occur at the point where the magnetization M = 0. At this point, there is zero net force on the superconductor and it will be pulled out of the liquid nitrogen and remain at a distance corresponding to the field value of the M = 0 crossing.
References
Questions
Problems
Figure Legends
Figure 1
Photograph of a small permanent magnet levitated above a pellet of the Y1Ba2Cu3,O7-d
(123) superconductor cooled to liquid nitrogen temperature. (Courtesy of IBM Research)
Figure 2
A Y1Ba2Cu3,O7-d
(123) superconductor doped with AgO (the small black chip) is shown suspended below a magnet (the shiny cylinder handing from a string). The suspended superconductor has just been raised from the liquid nitrogen in the Styrofoam cup and is hence at 77 K.
Figure 3
The internal magnetic field and the magnetization vs. external field for a type I superconductor. The internal magnetic field profile sketched correcsponds to parts (a) and (b).
Figure 4
The internal magnetic field and the magnetization vs. external field for an ideal type II superconductor. The internal magnetic field profile sketched corresponds to parts (a), (b), and (c).
Figure 5
The internal magnetic field and the magnetization vs. increasing and decreasing external field for a pinned, irreversible type II superconductor. The internal magnetic field sketched corresponds to parts (a), (b), (c), (d), and (e).
Figure 6
A schematic representation of the magnetization curve corresponding to parts (a), (b), (c), (d), (e), and (f) in Figure 7. The suspension corresponds to point d. The curves de and df correspond to the situation in which the superconductor is disturbed such that it moves closer and further from the suspending magnet.
Figure 7
A schematic drawing of the steps necessary to observe and understand the suspension effect: (a) Lowering the permanent magnet, negative magnetiation. (b) Further lowering leads to stronger repulsion of the superconductor chip. (c) Pulling away of the magnet, with the reversal of the magnetization and an attraction to the magnet. (d) Suspension of the superconductor at a distance z0. At this point, the upward magnetic force counterbalances the downward force of gravity. (e) The superconductor is disturbed such that it moves closer to the magnet. This results in a reduction of the positive magnetization. Superconducting screening currents flow without flux motion so the slope is steep corresponding to diamagnetic screening much like a perfect conductor. (f) The superconductor is disturbed such that it moves further from the magnet. This results in an increase of the positive mangetization.