Moment of inertia of superconductors J. E. Hirsch Department of Physics, University of California, San Diego La Jolla, CA 92093-0319 We find that the bulk moment of inertia per unit volume of a metal becoming superconducting 2 2 increases by the amount me/(πrc), with me the bare electron mass and rc = e /mec the classical 2 electron radius. This is because superfluid electrons acquire an intrinsic moment of inertia me(2λL) , with λL the London penetration depth. As a consequence, we predict that when a rotating long cylinder becomes superconducting its angular velocity does not change, contrary to the prediction of conventional BCS-London theory that it will rotate faster. We explain the dynamics of magnetic field generation when a rotating normal metal becomes superconducting. PACS numbers: I. INTRODUCTION and is called the “London moment”. In this paper we propose that this effect reveals fundamental physics of A superconducting body rotating with angular veloc- superconductors not predicted by conventional BCS the- ity ~ω develops a uniform magnetic field throughout its ory [3, 4]. For a preliminary treatment where we argued interior, given by that BCS theory is inconsistent with the London moment we refer the reader to our earlier work [5]. Other non- 2m c B~ = − e ~ω ≡ B(ω)ˆω (1) conventional explanations of the London moment have e also been proposed [6, 7]. with e (< 0) the electron charge, and me the bare electron Eq. (1) has been verified experimentally for a variety mass. Numerically, B =1.137 × 10−7ω, with B in Gauss of superconductors [8–15]. The same result is obtained and ω in rad/s. The magnetic field is parallel to the an- whether the sample is first cooled and then set into ro- gular velocity. The phenomenon was predicted in 1933, tation, or cooled while rotating, as predicted by London just before the Meissner effect was discovered, by Becker, [2]. Rotation speeds used in experiment are typically of Heller and Sauter [1] for a perfect conductor set into ro- order 20 to 100 revolutions/second. Equation (1) follows tation. Subsequently London predicted that the same from London’s equation [4] final state should result when a rotating normal metal is e cooled into the superconducting state [2]. The resulting ∇×~ ~vs = − B.~ (2) m c magnetic moment will depend on the shape of the body e with ~vs the superfluid velocity. In the interior of the body, the superfluid rotates together with the body, ω hence the superfluid velocity at distance r from the ro- tation axis is given by ~vs = ~ω × ~r (3) ωR ω(R-2λ L) and using ∇×~ (~ω × ~r)=2~ω in Eq. (2), Eq. (1) results. ωr ωr Consider the simplest geometry, a long cylinder of ra- λ dius R, where no demagnetization effects exist. A section L of such a cylinder is shown in Figure 1. For a supercur- rent flowing within the London penetration depth (λL) of the surface, the superfluid velocity that excludes an arXiv:1808.02857v3 [cond-mat.supr-con] 23 Mar 2019 ~ B applied magnetic field B is eλL ~vs = − B.~ (4) mec FIG. 1: Cylinder rotating with angular velocity ω. In the Therefore, to generate the interior London field Eq. (1) interior, at radial distance r from the rotation axis, ions and in a rotating superconductor in the absence of applied electrons move with the same speed ωr. Near the surface, ions external field, the superfluid velocity in a rim of thickness move with speed ωR and electrons move with slower speed λL at the surface has to be ω(R−2λL). A magnetic field B exists throughout the interior (dashed arrows), generated by the surface current resulting 2λL from electrons lagging behind the ions. ~vs = ~ω × ~r(1 − ) (5) R 2 π 4 so the superfluid velocity in the rim lags the rotation of Ie = nsmehR (9b) the body by the small amount 2 where h is the height of the cylinder. I is the moment ∆vs = −2λLω. (6) e of inertia of the superfluid electrons. As the system goes The full behavior of superfluid velocity versus radius is superconducting, electrons in the rim slow down accord- given by Laue [16] for a cylindrical geometry and by ing to Eq. (5), and the electronic angular momentum Becker et al [1] and London [2] for a spherical geome- decreases by try. The London penetration depth is given by [4] 2 ∆Le = (2πRλLhns) × (me∆vsR) (10) 1 4πnse 2 = 2 (7) λL mec with ∆vs given by Eq. (6). In Eq. (10), the first factor is with ns the superfluid density. the number of superfluid electrons in the surface rim, and For a perfect conductor that is set into rotation, Eq. the second factor is the change in angular momentum of (1) follows from Maxwell’s equations, as discussed by one electron. Using Eq. (7), Eq. (10) can be rewritten Becker et al [1], assuming the superfluid electrons are as completely detached from the lattice. As the body is set mec 2 2 ∆Le = −( ) R hω. (11) into rotation the moving ions generate an electric current e and hence a time-dependent magnetic field that generates a Faraday electric field that pushes the superfluid elec- The kinetic energy of the superfluid electrons in the trons to follow the motion of the ions, albeit with a small normal state due to the body rotation is lag near the surface that gives rise to the surface current 2 1 2 Le that generates the magnetic field Eq. (1). The derivation Ke = Ieω = (12) is reviewed in Appendix A. 2 2Ie In this paper we explain the behavior of rotating su- and when the system goes superconducting and the elec- perconductors using the theory of hole superconductivity trons in the rim slow down it decreases by the amount [17], and we argue that the conventional theory of super- conductivity is inconsistent with the physics of rotating Le∆Le ∆Ke = = ∆Leω (13) superconductors. Ie or, using Eqs. (11) and (1) II. THE CONVENTIONAL VIEW 2 B 2 ∆Ke = − (πR h). (14) We consider a normal metallic cylinder rotating around 4π its axis with angular velocity ~ω that is cooled into the su- For the ions, the kinetic energy of rotation is perconducting state. We assume the cylinder is floating 2 in a gravitation-free environment and there is no friction. Li Ki = (15) Therefore, total angular momentum is conserved. 2Ii In the normal state, electrons and ions rotate together at the same speed. When the cylinder becomes super- with Ii the moment of inertia for the ions. From Eq. (8) conducting, electrons near the surface have to sponta- and conservation of total angular momentum we deduce neously slow down to the velocity Eq. (5) upon entering the superconducting state. The conventional theory of ∆Li = −∆Le, (16) superconductivity does not explain the dynamics of this therefore, angular momentum has to be transferred from process, which London himself considered to be “quite the electrons to the body. This gives rise to a change in absurd” from the perfect conductor viewpoint [18] and the kinetic energy of the ions, did not provide an explanation for. Furthermore, the body has to spontaneously speed up to compensate for Li ∆Ki = ∆Li = ∆Liω = −∆Ke (17) the electronic angular momentum change. The conven- Ii tional theory of superconductivity does not explain what is the physical process that causes the body to increase hence its rotation velocity. The total mechanical angular momentum of electrons ∆Ke + ∆Ki =0. (18) plus ions is Therefore, conservation of angular momentum requires L~ = L~ e + L~ i. (8) that the change in ionic and electronic kinetic energies of rotation exactly compensate each other. The body’s The electronic angular momentum in the rotating normal rotation speed slightly increases to compensate for the state is slowing down of the rim electrons, and its kinetic energy Le = Ieω (9a) of rotation slightly increases. The conventional theory of 3 superconductivity does not explain how the kinetic en- when the system becomes superconducting, because the ergy saved by the lagging electrons is transferred to the energy saved in the slowing down of the rim electrons was body to make it speed up. The frequency of rotation entirely used up in speeding up the body to satisfy an- when the rotating normal metal becomes superconduct- gular momentum conservation. Therefore, we argue that ing will be larger than ω by the amount the conventional theory cannot account for the physics of 2 rotating superconductors without violating either conser- ∆Li me ns 8λL vation of energy or conservation of angular momentum. ∆ω = = 2 ω (19) Ii mpA n R The implausibility of the conventional picture also fol- lows from the following argument. From the Meissner where mp is the nucleon mass, A the atomic weight and effect we learned that the final state of a superconductor n the ionic number density. in a magnetic field B is unique, independent of history. For a type I superconductor it is expected [19] that Whether we apply a magnetic field B to a normal metal when the body is rotating at frequency ω the transition and then cool it to the superconducting state, or we apply to the superconducting state will occur when the system the same magnetic field B to a metal already supercon- is cooled below Tc to the temperature T for which ducting, the final state of the system is exactly the same. Similarly here, the rotation frequency ω plays the role 2mec Hc(T )= − ω = B. (20) that B plays in the Meissner effect. Whether we apply e the rotation ω to the normal metal and then cool it to with Hc(T ) the thermodynamic critical field.