Persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors (Review Article)

The notion of persistent current comes back to orbital currents in normal metals, semiconductors and even insulators displaying diamagnetic behavior in weak magnetic fields, but came to focus at the discovery of current persistence and magnetic flux quantization at large fields in atomically big but...

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Datum:	2010
1. Verfasser:	Kulik, I.O.
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Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2010
Schriftenreihe:	Физика низких температур
Schlagworte:	Quantum coherent effects in superconductors and normal metals
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spelling	nasplib_isofts_kiev_ua-123456789-1175322025-02-09T20:22:32Z Persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors (Review Article) Kulik, I.O. Quantum coherent effects in superconductors and normal metals The notion of persistent current comes back to orbital currents in normal metals, semiconductors and even insulators displaying diamagnetic behavior in weak magnetic fields, but came to focus at the discovery of current persistence and magnetic flux quantization at large fields in atomically big but macroscopically small (mesoscopic) objects. The phenomenon bears much similarity with supercurrents in superconductive metals. We will review progress in developing of our understanding of the physical and technological aspects of this phenomenon. The exact solution for currents, magnetic moments and magnetomotive forces (torques) in crossed magnetic fields are presented. Time-dependent phenomena in crossed magnetic and electric fields, and in possibility of spontaneous persistent currents and of work extraction from static and dynamic quantum states are discussed. 2010 Article Persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors (Review Article) / I.O. Kulik // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1057–1065. — Бібліогр.: 141 назв. — англ. 0132-6414 PACS: 73.23.–b, 03.65.Ta, 71.10.pm https://nasplib.isofts.kiev.ua/handle/123456789/117532 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Quantum coherent effects in superconductors and normal metals Quantum coherent effects in superconductors and normal metals
spellingShingle	Quantum coherent effects in superconductors and normal metals Quantum coherent effects in superconductors and normal metals Kulik, I.O. Persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors (Review Article) Физика низких температур
description	The notion of persistent current comes back to orbital currents in normal metals, semiconductors and even insulators displaying diamagnetic behavior in weak magnetic fields, but came to focus at the discovery of current persistence and magnetic flux quantization at large fields in atomically big but macroscopically small (mesoscopic) objects. The phenomenon bears much similarity with supercurrents in superconductive metals. We will review progress in developing of our understanding of the physical and technological aspects of this phenomenon. The exact solution for currents, magnetic moments and magnetomotive forces (torques) in crossed magnetic fields are presented. Time-dependent phenomena in crossed magnetic and electric fields, and in possibility of spontaneous persistent currents and of work extraction from static and dynamic quantum states are discussed.
format	Article
author	Kulik, I.O.
author_facet	Kulik, I.O.
author_sort	Kulik, I.O.
title	Persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors (Review Article)
title_short	Persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors (Review Article)
title_full	Persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors (Review Article)
title_fullStr	Persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors (Review Article)
title_full_unstemmed	Persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors (Review Article)
title_sort	persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors (review article)
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2010
topic_facet	Quantum coherent effects in superconductors and normal metals
url	https://nasplib.isofts.kiev.ua/handle/123456789/117532
citation_txt	Persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors (Review Article) / I.O. Kulik // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1057–1065. — Бібліогр.: 141 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT kulikio persistentcurrentsfluxquantizationandmagnetomotiveforcesinnormalmetalsandsuperconductorsreviewarticle
first_indexed	2025-11-30T11:04:53Z
last_indexed	2025-11-30T11:04:53Z
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fulltext	© Igor O. Kulik, 2010 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11, p. 1057–1065 Persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors (Review Article) Igor O. Kulik Department of Physics and Astronomy, State University of New York at Stony Brook SUNY Stony Brook, NY 11794–3800, USA E-mail: i.o.kulik@gmail.com Received May 12, 2010 The notion of persistent current comes back to orbital currents in normal metals, semiconductors and even in- sulators displaying diamagnetic behavior in weak magnetic fields, but came to focus at the discovery of current persistence and magnetic flux quantization at large fields in atomically big but macroscopically small (mesos- copic) objects. The phenomenon bears much similarity with supercurrents in superconductive metals. We will review progress in developing of our understanding of the physical and technological aspects of this phenome- non. The exact solution for currents, magnetic moments and magnetomotive forces (torques) in crossed magnetic fields are presented. Time-dependent phenomena in crossed magnetic and electric fields, and in possibility of spontaneous persistent currents and of work extraction from static and dynamic quantum states are discussed. PACS: 73.23.–b Electronic transport in mesoscopic systems; 03.65.Ta Foundations of quantum mechanics; measurement theory; 71.10.pm Fermions in reduced dimensions (anyons, composite fermions, Luttinger liquid, etc.). Keywords: Aharonov–Bohm effect, Berry phase, qubit, Coulomb blocade, Luttinger liquid. Contents 1. Orbital and spin magnetism in solids: a historical perspective ........................................................... 1057 2. Persistent currents in normal metals, superconductors and dielectrics: a short survey ....................... 1059 3. One-dimensional normal metal ballistic ring in crossed magnetic fields ........................................... 1060 4. Conclusion and future perspectives .................................................................................................... 1062 References .............................................................................................................................................. 1063 1. Orbital and spin magnetism in solids: a historical perspective Magnetism of solids — metals, insulators and semicon- ductors — is a pure quantum mechanical property [1]. Or- bital magnetism means the existence of nonzero electron current in a state of thermodynamic equilibrium driven by the appliance of external magnetic field. Such non- dissipative, non-decaying currents can flow in the super- conductive metals at low temperature in presence of static magnetic field [2]. The current may also exist in the non- conducting solids with electrons confined to atoms, mole- cules or atomic clusters at the lattice sites. Magnetic mo- ment at the site is proportional to the applied magnetic field. The magnetic moment of current I equals (in CGS units) 1= ,M I S c (1) where 2=S Rπ is the effective surface of electron locali- zation perpendicular to the field. Quantum mechanical computation results in an expres- sion for M 2 2 2 04 ZeM B m c ⊥= < >r , (2) where 0m if the free electron mass. Magnetic field in a circular loop of radius R equals to /A R , where A is the vector potential at the loop whereas an expression in brackets of Eq. (2) is a square of the effective radius, R, of the (effective) loop. Z is the number of electrons in the loop. Assuming dielectric medium of N loops per cm3, Igor O. Kulik 1058 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 with the loop size and distance between the loops of order of the Bohr radius, 2 2 0 0= / ( e )a m , we receive magnetic susceptibility of the medium = NM B χ (3) of order 22 5 diel 10 2 e c −⎛ ⎞ χ ⎜ ⎟⎜ ⎟ ⎝ ⎠ ∼ ∼ (4) at temperature = 0.T Similar estimate of diamagnetic sus- ceptibility, in case of the bulk normal metal, is 22 50 metal 10 m e m c −⎛ ⎞ χ ⎜ ⎟⎜ ⎟ ⎝ ⎠ ∼ ∼ (5) assuming the ratio of the free electron mass to the effective mass of order 110− . The expression for magnetic suscepti- bility of bulk metal 2 2 2 2 1= ( ) = , 3 4 F B F e k N mc χ μ μ π (6) was received by Landau [3] and confirmed by Teller [4] and enters standard courses on quantum mechanics and condensed matter physics [5–8]. Here Fk = 2 /aπ is the Fermi wave number, Bμ = 0/ 2e m c is the Bohr magne- ton, and ( )N ε is the density of electron states at the ener- gy ε (the estimate of Eq. (5) assumes 0a a∼ ). Approximation made in the derivation of the Eq. (6) rested on the assumption that, in the bulk metal of size L much larger than the Larmor radius = /L Fr mcv eB , the contribution to the magnetic moment from the edge di- amagnetic currents (Fig. 1) is negligible in comparison with the contribution from bulk circular currents. However, as is known from the Van Leeuwen theorem [9], in classic- al theory both contributions, being of opposite direction of rotation, cancel each other. Quantum calculation in simple geometrical forms (2d disks, squares, stripes) showed larg- er amplitude oscillations [10–14] of magnetic moment compared to the Landau moment, in function of distance to the edge of a sample, in weak magnetic field. In clean met- als and strong magnetic fields, thermodynamic and kinetic oscillating phenomena — de Haas-van Alphen effect and Shubnikov–de Haas effect, respectively [6–8], have been discovered and studied revealing a large amount of infor- mation on the property of dynamics of mobile electrons considered as elementary excitations (quasiparticles) in metals. Theoretical prediction by Aharonov and Bohm [15] of an effect, called by their names, of the nonlocal interaction between charged particles and the electromagnetic field such that in certain topological geometries in which mag- netic field equals to zero in all space occupied by electrons, but the vector potential is not, the effect of interaction shows up and produces the physically observable effects. This stimulated investigations by several authors [16–23], of possible quantum effects in normal metals — the effects similar to the Josephson effect in superconductors. This ended by a prediction in 1970 by the author, of a non- decaying (later called «persistent») currents and flux quan- tization in a hollow thin-walled normal metallic cylinder and ring [24] threaded by a bunch of flux-lines of magnetic field confined within the inner cylinder (a magnetic coil) of a radius smaller than the radius of outer cylinder (Fig. 2) in which no electric and magnetic field is present. The magni- tude of persistent current in normal metal cylinder was estimated as [24] /2 0 1 1 01 e sin sin 2 4 TF F F F ev k L I k L L k L − ε ⎛ ⎞π φ⎛ ⎞≅ − π⎜ ⎟⎜ ⎟ φ⎝ ⎠ ⎝ ⎠ (7) where 0 1= /Fv Lε ; 1L is the circumference and 2L is the length of the cylinder. The current oscillates between di- amagnetic and paramagnetic in sign at non-integer value of Fig. 1. Electron orbits in a magnetic field for bulk (solid line) and edge state (dashed line) electrons. Fig. 2. Schematic of Aharonov–Bohm effect observation in a hollow normal-metal cylinder of length L and radius R threaded by a bunch of lines of force of magnetic field with total magnetic flux φ confined within an infinitely long cylinder of radius <r R such that the magnetic field outside the cylinder equals to zero but the current in the ring is not. The current varies with the flux periodically with a period /hc e . � r O RL J Persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1059 1Fk L in units of 2π in case of fixed chemical potential rather than the fixed particle number. At temperature 0>T ε , only the lower flux-oscillating harmonic remains. At 2 1=L L , Eq. (7) applies to a ring. In that case, the Aharonov–Bohm current estimates as 0 sin 2 .F AB ev I R ⎛ ⎞φ π⎜ ⎟φ⎝ ⎠ ∼ (8) Minimal size of the ring is of order of the Bohr radius 0a , and in weak field its magnetic moment equals 0 =AB B φ μ μ φ (9) will correspond to magnetic susceptibility of 3d insulating medium filled with isolated tightly packed rings 22 5e 10c −⎛ ⎞ χ ⎜ ⎟⎜ ⎟ ⎝ ⎠ ∼ ∼ (10) in accordance with an estimate of Eq. (4). The effect of Aharonov–Bohm persistent current in ma- croscopic loop was first regarded as doubtful, and at least hardly ready for the experimental realization at the time. Advances in nanotechnology in the next decade revived interest in subtle quantum mechanical phenomena and ap- pliances. The persistent current was next time considered in 1983 by Buttiker, Imry and Landauer [25] in the normal metallic ring, thus supporting the conclusion of previous papers and effectively stimulating advance in physics of mesoscopic systems and in technological progress of microelectronics. 2. Persistent currents in normal metals, superconductors and dielectrics: a short survey In the time interval between 1970 and 1983, the number of papers discussed possibility of experimental observation of persistent current through its oscillation with the magne- tic flux (the flux quantization phenomenon), and extension to different geometric configurations. In 1981, Altshuler, Aronov and Spivak [26] discovered new oscillating effect in the dirty normal metal with the period / 2hc e in which a pair of time-reversing electron trajectories at a scattering event interfere and show them- selves as 2e-charged pair. The effect of / 2hc e periodic oscillation in the nonsuperconducting melal is a kinetic phenomenon [27] whereas the /hc e periodicity is a ther- modynamic Aharonov–Bohm property [24]. The kinetic ef- fect was first time found in an experiment in normal metal by D.Yu. Sharvin and Yu.V. Sharvin [28], and in supercon- ductor above cT by Shablo et al. [29], and theoretically de- scribed in a ring geometry by Ambegaokar and Eckern [30]. Bogachek and Gogadze [31] considered flux quantiza- tion in two-dimensional disk due to edge electron orbits («whispering gallery» trajectories [32]) inside the disk what was shortly confirmed in an experiment in thin bulk Nb filaments [33]. This was the first experimental demon- stration of oscillations with single-electron flux quantum /hc e compared with two-electron flux quantum / 2hc e characteristic of superconductors. Similar effects in hollow cylinders, quantum dots and antidots, as well as in specific magnetic materials have been discussed in papers [34–37]. Landauer and Buttiker [38] calculated resistance of the ring at a condition when the current exceeds its maximal value or an ac component is added to the external magnetic field. Imry and Shiren [39] thoroughly discussed condition of persistent current observation regarding the effect of elastic and inelastic scattering to Aharonov–Bohm effect in solids. Their conclusion was that two-dimensional flat disk (quan- tum dot) whose lateral dimension is larger than the perime- ter of electron orbit in perpendicular magnetic field, and the mean free path of electron exceeds the dot diameter, de Haas–van Alphen oscillation will show up superimposed on Aharonov–Bohm persistent current oscillation, at very strong magnetic field such the electron Fermi energy Fε is larger than the Landau levels spacing / .zeB mc Zagoskin et al. [40,41] considered quantum /hc e — periodic oscillations of conductance in wide ballistic point contacts (e.g., see Ref. 42) superimposed on the conduc- tance jumps of 2 /e π and 22e /π height. Phase transitions induced by the Aharonov–Bohm field have been discussed by Azbel [43], Krive and Naftulin [44]. Bogachek, Krive, Kulik and Rozhavsky [45–48] consi- dered realization of the Aharonov–Bohm effect in dielec- tric crystals in the state of charge density wave, in which the conductance by charged solitons dominates [49]. This was later discussed in a number of papers [50–53] and ob- served in an experiment [54]. Strong electron-electron and electron-ion correlations in metals are represented as the interplay of mutually inter- connected phenomena such as Wigner crystallization [55], Coulomb blocade [56,57], and Luttinger liquid formation [58] which replaces the Fermi liquid of the non-interacting electrons. These phenomena have being discussed in the con- text of the Aharonov–Bohm effect. Glazman, Rudin and Shklowskii [59] considered quantum transport in a one-di- mensional Wigner crystal; Maslov, Stone, Goldbart and Loss [60] studied Josephson currents in Luttinger liquid; Sundstrom and Krive [61] discussed the effect of Coulomb blockade on persistent current in the Luttinger-liquid ring; Moskalets [62,63] discussed the effect of spin paramagnet- ism and of time-dependent magnetic field on Coulomb blockade in the ring; Krive, Sandstrom, Shekhter, Girvin and Jonson [64] thoroughly studied the persistent current and Aharonov–Bohm oscillations in the one-dimensional ring in a state of Wigner cryslal; Pletyukhov and Gritsev [65] investigated the persistent currents in Luttinger-liquid semiconducting rings; Krive, Palevsky, Shekhter and Jon- Igor O. Kulik 1060 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 son discussed resonant tunneling and Coulomb blockade in quantum wires [66]. Persistent quantized currents in mesoscopic loops, cy- linders, quantum dots and antidots as well as in regular and random ensembles of the dots have been discussed theoret- ically [67–69] and observed experimentally in many papers [70–78] etc. We refer especially to latest and most detailed contribution [79]. Much attention have been devoted to investigation of distinctions and similarities between the canonical and grand canonical averages in mesoscopic ensembles, and to the proportion in the orbital diamagnetic and paramagnetic magnitudes in corresponding susceptibil- ities [80–84]. Averin and Friedman [85,86] suggested using Aharo- nov–Bohm effect to study tunneling of quantum flux lines with the superconducting circuit incorporating Bloch tran- sistor (the single-electron device [87]). At the same period between 1970 and 1983, fundamen- tal discovery of Quantum Hall Effect took place [88], and Laughlin [89] conjectured that the clean hollow cylinder in orthogonal to each other and to the axis of symmetry of cylinder magnetic and electric fields would be a starting point for the explanation of the physical origin of the sharp plateaus in the Hall voltages observed by Klitzing et al. [88]. Zanchi and Montambaux [90] discussed similarity or imitation of Aharonov–Bohm effect in CDW system to the integer QHE. Sivan and Imry [91] discussed simultaneous appearance of Aharonov–Bohm and de Haas-van Alphen oscillations in a same quantum dot. Aharonov–Bohm effect in crossed magnetic fields [92–96], and in the electric field perpendicular to magnetic field [97–102] have been discussed in a number of papers. Particular interest is in the electromotive and magnetomo- tive forces accompanying persistent currents, and in the fast controllable transitions between the quantum states required for bit manipulations in quantum computers. The fundamental problem of mesoscopic physics is in the work extraction from the quantum states [103]. This problem can be formulated as transitions between the qua- sistationary persistent-current states [100], using of the hy- pothized quantum force in mesoscopic superconducting rings in magnetic field [104], and as the charge transfer or flux pumping in the adiabatic modulation of persistent cur- rent by the ac signals [105–108], and as the theory of sto- chastic pumps and reversible ratchets [109] and molecular motors in the stochastic environment [110]. Time-dependent behavior is an important problem in physics and control of mesoscopic Aharonov–Bohm de- vices [101,111–116]. Three-site discrete quantum structure [99,117] proved to be an interesting configuration which may allow performing basic operations required for reali- zation of qubit in hypothetical quantum computer. That is the only one known electronic instrument allowing to per- form full coherent quantum transformation (1,0) to (0,1) (a bit flip)in a finite time interval. Spontaneous quantum flux state in a system of odd number of electrons with the Kramers degeneracy in the ground state have been analyzed in papers [100–102,118] and [140,141]. It was proved [100] that the state survives inelastic scattering within the isolated Aharonov–Bohm loop but is suspected of weak lifting of the degeneracy due to electromagnetic radiation in the environment [119]. It is interesting to note that another type of spontaneous current have also been discussed [120–122] in supercon- ductive mesoscopic rings with Josephson contacts between the unconventional superconductors in which the angle between the orientation of d-vector in adjacent sides of contact (as well as a Josephson phase itself) is considered as a quantum variable. Interplay between the normal-metal and the supercon- ducting manifestations of the Aharonov–Bohm effect be- comes an intriguing option of the quantum theory of solids development. As early as in 1975, Bogachek, Gogadze and Kulik [123], and later Wei and Goldbart [124] noticed change of critical temperature versus magnetic flux oscillations from /hc e to / 2hc e — periodic, i.e. the doubling of the period of flux quantization in superconducting cylinders due to quantum effects in the normal state, and Zhang and Price [125] detected /hc e — periodic oscillation of cur- rent to flux susceptibility in superconducting Al in narrow temperature interval above the zero-field critical tempera- ture, in an exact agreement with the prediction of [123]. The interplay of attractive and repulsive Coulomb inte- ractions results in / 2hc e periodic oscillations of the ener- gy versus flux in strongly coupled electronic systems [126,127] regardless on whether the system is supercon- ducting or not. In a system of small number of electrons, superconducting behavior depends on whether the number of electrons is even or odd [128]. In the grand canonic en- semble, in which the number of electrons is not fixed, the two-fluid model shows different periodicities in the beha- vior of superfluid and normal-fluid (that of single-charged excitations) [129,130]. And in the last (but not the least important) case of unconventional superconductors [131– 134], both the normal-metal Aharonov–Bohm oscillations are displayed together with the / 2hc e superconductive oscillations. 3. One-dimensional normal metal ballistic ring in crossed magnetic fields One-dimensional flat ring allows for exact solution for the quantum energy states, persistent currents and electro- motive or magnetomotive forces (torques) acting on the ring in presence of the Aharonov–Bohm magnetic flux and arbitrary distribution of the external magnetic field around the ring. In this case, orbital magnetism is mixed with the paramagnetic magnetism of mobile electrons inside the ring. The simplest configuration shown in Fig. 3 includes thin solenoid piercing the ring and vertical magnetic field Persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1061 zB parallel to direction of solenoid. Hamiltonian of the system is 2 2 0 32 ˆ= ( ) 2 B zH n B mR − ν σ + μ σ (11) where R is radius of the ring and iσ the Pauli matrices: 1 2 3 0 0 1 0 1 0 1 0 = , = , = , = 1 0 0 0 1 0 1 i i −⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ σ σ σ σ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (12) and n̂ the operator 1ˆ = dn i dϑ , and 0= /ν φ φ . φ is the total magnetic flux threading the ring and 0 = /hc eφ is the flux quantum of the normal metal. Solution of Eq. (11) for the energy eigenvalues gives 2 2 2= ( ) . 2 n B zn B mR ε − ν ± μ (13) The diamagnetic current in the ring is received as a deriva- tive = /I cdE d− φ . Total magnetic moment of the ring including contribution from the current, 1 = / ,M IS c and paramagnetic contribution from spin in a magnetic field, 2 = / zM dE dB , is 3 0 e= .z BM n mc ⎛ ⎞φ − +μ σ⎜ ⎟φ⎝ ⎠ (14) Mention that φ is a total magnetic flux including contri- butions from the Aharonov–Bohm solenoid inside the ring and from the outside sources of magnetic field. m is the effective mass, not necessary equal to the mass entering to the expression for the Bohr magneton Bμ . Another configuration is that when the outside magnetic field is perpendicular to Aharonov–Bohm field localized inside the ring (Fig. 4) what corresponds to Hamiltonian 2 2 0 12 ˆ= ( ) 2 B xH n B mR − ν σ + μ σ (15) and to the equation for eigenenergies 2 2 2= ( ) 2 n B xn B mR ε − ν ± μ (16) and for the magnetization 0 =z eM n mc ⎛ ⎞φ −⎜ ⎟φ⎝ ⎠ , =x BM ±μ . (17) Leaving for a while a discussion of physical implication of diamagnetic and paramagnetic contributions to the mag- netization of the Aharonov–Bohm loop, we consider the cases of crossed fields with azimuthal (Fig. 5) and radial (Fig. 6) components of magnetic field orthogonal to the axes of symmetry of the ring. These configurations consi- dered in the papers [92–95] within the Berry-phase tech- nique [135,136] valid in the adiabatic approximation, and as an exact solution [96] in static fields. In case of azimuthal magnetic field Bϑ at any point at the ring, Hamiltonian of the ring is 2 2 02 ˆ= ( ) 2 BH n B mR ϑ ϑ− ν σ + μ σ (18) Fig. 3. Persistent current in presence of the Aharonov–Bohm flux φ and an external magnetic field .zB � J Bz Fig. 4. Persistent current in configuration in which magnetic field is perpendicular to the direction of Aharonov–Bohm flux. � J Bx Fig. 5. Persistent current in a ring produced by the Aharonov– Bohm flux φ and exsternal radial magnetic field Bϑ emerging from the line current extI in a wire along the symmetry axis of the ring .OZ � J B� Jext Igor O. Kulik 1062 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 where ϑσ is a Pauli matrix: 0 e = . e 0 i i i i − ϑ ϑ ϑ ⎛ ⎞− ⎜ ⎟σ ⎜ ⎟ ⎝ ⎠ (19) Psi-function Ψ is a two-component vector = = exp ( ),n nn in ∞ −∞ ψ⎛ ⎞ Ψ ϑ⎜ ⎟ϕ⎝ ⎠ ∑ (20) where for /ψ ϕ bound components we have 2 1( ) = 0,n nn ihϑ +⎡ ⎤− ν − ε ψ − ϕ⎣ ⎦ (21) ( )2 1[ 1 ] = 0,n nih nϑ ++ + −ν − ε ϕ which gives the nth energy level 2 2 21 1 1= ( ) ( ) 2 4 2n n n hϑε + − ν + ± + − ν + (22) where energy is given in units of characteristic energy 2 2/ mR , and hϑ is the value of magnetic field Bϑ in these units. The configuration of Fig. 6 is considered similarly to Eq. (18) in which last term replaces to B r rBμ σ and Pauli matrix rσ is 0 e = . e 0 i r i − ϑ ϑ ⎛ ⎞ ⎜ ⎟σ ⎜ ⎟ ⎝ ⎠ (23) Respectively, Eq. (21) for energy will be 2 2 21 1 1= . 2 4 2n rn n h⎛ ⎞ ⎛ ⎞ε + − ν + ± + − ν +⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (24) Combining all paramagnetic interactions with external magnetic fields (Fig. 7), we receive the Hamiltonian 22 02 0 ˆ= 2 B z z B B r rH n B B B mR ϑ ϑ ⎛ ⎞φ − σ + μ σ +μ σ +μ σ⎜ ⎟φ⎝ ⎠ (25) where φ is total magnetic flux from a thin solenoid inside the ring and from the z-component of external magnetic field penetrating through the full surface 24 Rπ enclosed by a ring; ext= 2 /B I cRϑ is the azimuthal component of the field at the ring, and rB the radial field component produced by two opposite-side solenoids inside the ring shown schematically in Fig. 7. Energy levels of electron with the wave function = e en in i t n ϑ − εψ⎛ ⎞ Ψ ⎜ ⎟ϕ⎝ ⎠ (26) are 2 2 21 1 1= 2 4 2n zn n h h⊥ ⎛ ⎞ ⎛ ⎞ε + −ν + ± + −ν − +⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (27) where 2 2= rh h h⊥ ϑ + is a perpendicular field at the ring in units of 2 2/ 2 .mR 4. Conclusion and future perspectives Origin of persistent current in a normal-metal ring is re- lied on two postulates: (a) Lifting of the time-reversal sym- metry in presence of vector potential A, and (b) Rigidity of the wave function under change of the magnetic field. The rigidity, as it was for the first time introduced by London [2] in an explanation of the phenomenon of superconduc- tivity of metals, means that Ψ remained constant under change of magnetic field. The same is valid in case of per- sistent currents in normal metals since the wave function of an electron in the ring = const·exp ( )inΨ ϑ (28) Fig. 6. Persistent current in a ring produced by the Aharonov– Bohm flux φ in the inner cylinder and by the radial magnetic field rB created by a pair of magnetic coils 1S and 2S with opposite direction magnetization. S1 J Br Ring S2 � S1 S2 J Br Br Bz Bz Jext � B� Fig. 7. Setup for observation of the Aharonov–Bohm effect in crossed magnetic fields zB , Bϑ , and rB together with the mag- netic flux φ in the inner solenoid. Persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1063 is rigid (i.e., doesn't change) within certain interval of mag- netic field 0 0 1 1< < 2 2 n n⎛ ⎞ ⎛ ⎞− φ φ + φ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (29) for a given n . In fact, we have here another type of super- conductivity with a very small critical parameters, even smaller than in typical Josephson junctions (termed some- times as a «weak superconductivity»). The critical current in a 1d ring is according to Eq. (14) and Eq. (1) is 2= / 2ABI e mRπ (30) compared to supercurrent in the same radius superconduct- ing ring [137] 3= 2 / e / ,SC SI e N A mR R∼ (31) assuming superelectron concentration sN ∼ N at = 0,T and wire cross section A of the order of the Bohr radius 0a squared. Then the ratio of expression (30) to (31) is of order of 0 / 2a Rπ , the inverse of the number of electrons in the ring. The Aharonov–Bohm «superconductivity» is therefore the one-electron effect whereas in the BCS su- perconductivity all electronic pairs contribute to current collectively. Aharonov–Bohm effect in crossed magnetic fields has an interesting property of anisotropy of magnetic suscepti- bility as discussed above in Ch. 3. The composition shown in Fig. 4 displays the magnetization component perpendi- cular to magnetic field. This means the presence of magne- tomotive force (a torque) = ×T M B (32) acting on a ring. Such effect is well known in bulk mono- crystalline normal metals where the weak rotation of the sample at change of static magnetic field serves as a most sensitive method [8] of studying the de Haas–van Alphen oscillations [6]. Similarly, levitating mesoscopic ring will display change of its orientation under control of the mag- netic field. 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Persistent currents, flux quantization and magnetomotive forces in normal metals and superconductors (Review Article)

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