Genesis and fading away of persistent currents in a Corbino disk geometry
GGenesis and fading away of persistent currents in a Corbino disk
Yuriy Yerin, V.P. Gusynin, S.G. Sharapov, and A.A. Varlamov Dipartimento di Fisica e Geologia, Universit´a degli Studi di Perugia, Via Pascoli, 06123 Perugia, Italy Bogolyubov Institute for Theoretical Physics, National Academy of Science of Ukraine,14-b Metrologicheskaya Street, Kiev, 03143, Ukraine CNR-SPIN, c/o Dipartimento DICII, Universit´a “Tor Vergata”, Viale del Politecnico 1, I-00133 Rome, Italy (Dated: February 23, 2021)The detailed analytical and numerical analysis of the electron spectrum and persistent currentsand their densities for a Corbino disk in a constant magnetic field is presented. We calculate thecurrent density profiles and study their dependence on the inner and outer radius of the Corbinodisk. We study evolution of the persistent currents and track their emergence and decay for differentlimiting cases of such a geometry, starting from a nanodot and ending by a macroscopic circle. Theconsistency of our results for the currents is confirmed by the agreement between the direct inte-gration of the corresponding current densities and the application of the Byers-Yang formula. Thequalitative comparison with the well-known results for quasi-one dimensional mesoscopic metallicrings is provided. Our study can be applicable for more accurate treatment and interpretation ofthe experimental data with measurements of the persistent currents in different doubly-connectedsystems.
PACS numbers:
I. INTRODUCTION
While the Landau diamagnetism of free electron gas [1]is often regarded as a standard textbook knowledge, thediscussion of the role of the edge states is only one yearyounger [2]. This analysis addressed the naturally risenquestion, why the Landau’s calculation of magnetizationand similar calculations of the non-dissipative transportcoefficients are correct for large enough system [3]. Yetthese studies revealed the presence of the macroscopicnon-dissipative persistent edge currents flowing along theboundaries of the sample.The persistent currents can also exist in doubly-connected systems due to the Aharonov-Bohm effect[4]. The best known illustration of them is the fluxquantization in the superconductor rings [5]. Further-more, it was predicted [6, 7] that in a hollow thin-walled normal metallic cylinder or ring with the smallenough radius R threaded by a magnetic flux Φ thepersistent current can flow. Its magnitude oscillates as I ∼ ( | e | v F /R ) sin(2 π Φ / Φ ), where v F is the Fermi ve-locity and Φ = hc/e is the magnetic flux quantum. Thediamagnetic currents in the restricted geometry, includ-ing rings, were studied in between 60-70’s e.g. in [8–10].It was also demonstrated that the account for such geo-metrical effects can lead to the magnetic response of themagnitude larger than the Landau diamagnetic moment(see the reviews in Refs. 11, 12.)Further insight into the nature of these currents wasobtained after the discovery of the quantum Hall effect.It was shown in [13] that the quantized Hall current maybe expressed as the difference between diamagnetic cur-rents flowing along the two edges (see also Ref. 14 fora more recent discussion of link between quantum Halleffect and diamagnetism).When in a state of thermodynamic equilibrium the chemical potential of these edges is the same, the edgecurrents cancel each other and the total one caused bythe applied external magnetic field is zero.Interestingly, a more simple rectangular geometry wasconsidered in [13] two years after the same problem wasstudied in the annular geometry [16]. This is not sur-prising because the annular geometry of the Corbino diskrepresents a practical realization of the cylinder geometrysuggested by Laughlin for the gedanken experiment. Thesizes of the disk in [16] are assumed to be macroscopic, i.e.its inner ( r ) and outer ( r ) radii along with the width ofthe ring ( r − r ) strongly exceeds the electron magneticlength l = (cid:112) (cid:126) c/ | eB | , viz. r , r , ( r − r ) (cid:29) l . Againif the chemical potential of the two edges is the same,the currents at the inner and outer edge flow in the op-posite directions and there is no net current around theannulus.It could seem that between the two above discussed ap-proaches the contradiction exists. The one-dimensionaltreatment of the thin ring demonstrates the existence ofthe persistent current, while in the microscopic deriva-tion of the edge currents in two-dimensional Corbino diskshows that the net current is zero. This imaginary con-troversy is related to the reconstruction of the electronspectrum in the Corbino disk as it becomes of the micro-scopic size. Indeed, when the latter approach the mag-netic length, the staircase of Landau energy levels un-dergoes non-negligible alteration, and the edge currentsstart to overlap.Subsequently the properties of persistent currents werestudied in [17–19] within different approaches for ballis-tic and diffusive regimes of conductivity. It was revealedthat the condition of their feasibility consists of the re-quirement that the size of a system should be mesoscopic,i.e. the radius of a ring has to be of the order of the elec-tron mean free path at zero temperature. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Persistent currents owing to edge effects of the dia-magnetic response in the bulk disks and due to theAharonov–Bohm effect in a mesoscopic samples were ini-tially predicted as the tiny effects, and they were hardlydetectable experimentally at those times. Nevertheless,enormous advances in nanotechnology of the last twodecades renewed interest to this elusive quantum me-chanical phenomenon. Recently the magnetic responseof individual gold rings with the typical radii of the or-der of 1 µm and the comparable width indeed has beenmeasured at very low temperatures. It has been foundthat the response of sufficiently small rings in appliedmagnetic field can be attributed to the predicted in Ref.[17–19] persistent currents. Their amplitudes were foundin a rather good agreement to the corresponding theoryfor quasi-one-dimensional rings [20, 21].It is important to note that in all Refs. [6, 17–19] theconsideration of persistent currents was carried out solelyfor a quasi-one-dimensional mesoscopic ring within theassumption of its infinitesimal small width. Contrary, inRef. [22] the current density distribution was studied forthe macroscopic Corbino disk placed in classically strongmagnetic field and under the assumption that r , r , ( r − r ) (cid:29) l .The authors of Ref. 23 investigated the persistent cur-rent in Corbino disk of the relatively large size r , r , ( r − r ) (cid:29) l in ballistic regime as a function of electron den-sity. They found the violent fluctuations of the latter(in sign and in absolute value) which is quite unusualfor systems without disorder. These fluctuations resultfrom the alternative contributions of the overlapping in-ner and outer edge states. The same authors in Ref. 24considered numerically the case of the large disk withthe infinitesimal inner radius in the strong magnetic field ( r (cid:29) l ).The goal of the present paper is to establish a bridgebetween the mentioned above different approaches forcalculation of the persistent current flowing in a ring.We perform our calculations for the Corbino disk start-ing from the microscopic solution of the eigenvalues andeigenfunctions problem for the electron in a magneticfield in the case of doubly connected geometry. Bas-ing on it we succeed in the unified way to track genesisof the current in microscopic rings [18, 19]), to repro-duce the strong current oscillations found in Ref. [23] formesoscopic disk, and to observe fading away of the cur-rent when the sizes of Corbino disk become substantial[13, 22]). II. MODEL AND GENERAL RELATIONSA. Model
We consider a Corbino disk with the inner and theouter radii r and r , correspondingly, subjected to aconstant magnetic field B applied perpendicularly to itsplane described by the Hamiltonian H = p m e = 12 m e (cid:16) − i (cid:126) ∇ + ec A ( r ) (cid:17) , (1)where m e is the effective electron mass. Due to the ax-ial symmetry of the problem it is convenient to studyit in the polar coordinates, where the vector potentialis written in the symmetric gauge A ( r ) = ( A r , A ϕ ) = (0 , Br ). Consequently, the Schr¨odinger equation ac-quires the form (cid:20) − (cid:126) m e (cid:18) ∂ ∂r + 1 r ∂∂r + 1 r ∂ ∂ϕ (cid:19) − i (cid:126) ω c ∂∂ϕ + 18 m e ω c r (cid:21) ψ ( r, ϕ ) = Eψ ( r, ϕ ) . (2)Here − e < ω c = eB/ ( m e c ) is the cyclotron frequency. The boundary con-ditions imposed on the wave function ψ ( r, ϕ ) correspondto the impenetrability of the disk edges, i.e. the manda-tory requirement ψ ( r , ϕ ) = ψ ( r , ϕ ) = 0 (3). Separation of the variables in Eq. (2) ψ ( r, ϕ ) = f ( r ) e − imϕ (4)reduces it to the differential equation for the radial com- ponent of the wave function (cid:20) − (cid:126) m e (cid:18) ∂ ∂r + 1 r ∂∂r − m r (cid:19) + 18 m e ω c r (cid:21) f ( r ) = ˜ Ef ( r ) , (5)where the energy ˜ E is shifted in respect to E as ˜ E = E + (cid:126) ω c m/
2. Introducing the dimensionless energies ε = E/ ( (cid:126) ω c ), ˜ ε = ˜ E/ ( (cid:126) ω c ) and the dimensionless variable ρ = r/l (where l = (cid:112) (cid:126) c/ ( eB ) is the magnetic length)one can simplify Eq. (5): (cid:20) ∂ ∂ρ + 1 ρ ∂∂ρ − ρ ε − m ρ (cid:21) f ( ρ ) = 0 . (6)Its general solution can be written in terms of two Whit-taker functions M κ,µ ( z ) and W κ,µ ( z ): f ( ρ ) = 1 ρ (cid:20) C M ˜ ε, | m | (cid:18) ρ (cid:19) + C W ˜ ε, | m | (cid:18) ρ (cid:19)(cid:21) . (7)Below we will analyze the spectral properties of Eq. (6)and the asymptotic behavior of radial functions (7). B. Dispersive Landau levels in the Corbinogeometry
Applying of the boundary conditions (3) at Eq. (7)yields a transcendental equation for the energy levels˜ ε n,m ≡ ˜ ε : W ˜ ε, | m | (cid:18) ρ (cid:19) M ˜ ε, | m | (cid:18) ρ (cid:19) − M ˜ ε, | m | (cid:18) ρ (cid:19) W ˜ ε, | m | (cid:18) ρ (cid:19) = 0 . (8)Here the two quantum numbers appear, viz. the first n corresponds to the level number in Landau problem,while the second one, m , characterizes the angular mo-mentum and the latter is an analogue of the wave-vectorcomponent k y in the Landau gauge [1]. Let us recall,that in the case of homogeneous Landau problem thequantum number k y determines the position of the po-tential minimum x = l k y in the coordinate space. Inthe case under consideration its role passes to the value r m = (cid:112) | m | l , which is nothing else as the position alongthe radial coordinate of the maximum in the probabil-ity of the electron state with quantum number m for thegiven n .The same impenetrability boundary conditions allowto simplify the form of radial part of the wave function(7) f nm ( ρ ) = Cρ (cid:20) W ˜ ε, | m | (cid:18) ρ (cid:19) M ˜ ε, | m | (cid:18) ρ (cid:19) − M ˜ ε, | m | (cid:18) ρ (cid:19) W ˜ ε, | m | (cid:18) ρ (cid:19)(cid:21) , (9)leaving in it the only constant C . The latter is deter-mined from the normalization condition:( Cl ) − = 2 π ρ (cid:90) ρ dρρ (cid:20) W ˜ ε, | m | (cid:18) ρ (cid:19) M ˜ ε, | m | (cid:18) ρ (cid:19) − M ˜ ε, | m | (cid:18) ρ (cid:19) W ˜ ε, | m | (cid:18) ρ (cid:19)(cid:21) . (10)Accordingly, the eigenfunction of the one particle prob-lem is ψ nm ( r ) = f nm ( r ) e − imϕ (11)(see Eqs. (4) and (9)).Let us note that the Whittaker functions in Eq. (8) inthe case when the parameter˜ ε n,m = n + 12 ( | m | + 1) , n = 0 , , . . . , (12) m n , m (a) m n , m (b) -5 0 5 10 15 20 25 30 35 40 45 50 55 60 m n , m (c) m n , m (d) FIG. 1: Energy levels as a function of angular quantum num-ber m for a Corbino disk with a fixed inner radius ρ = 3 anddifferent outer radii ρ = 6 (a), ρ = 9 (b), ρ = 10 (c) and ρ = 13 (d). turn out linearly dependent and they reduce to Laguerrepolynomials. It is why these trivial solutions, correspond-ing to the infinite system with the spectrum (this problemin the symmetric gauge was addressed in [25]) ε n,m = n + 12 ( | m | − m + 1) , n = 0 , , . . . , m ≥ − n, (13)we exclude from consideration.In Figure 1 one can see the series of the energy levels ε n,m as the function of the angular quantum number m for different inner and outer radius of the disk obtainedfrom the numerical solution of Eq. (8). One observesthat at the outer radius ρ increases the Landau levelsinside the ring flatten and approach the values (13) forthe infinite system. C. Persistent currents
1. General expressions
The current density operator in the representation offield operator ψ is ˆj = ie (cid:126) m e ( ψ ∗ ∇ ψ − ∇ ψ ∗ ψ ) − e m e c A ψ ∗ ψ. (14)For the Fermi gas with chemical potential µ the currentdensity expectation value reads [26] j ( r ) = − e m e ( πππ ( r ) + πππ ∗ ( r (cid:48) )) × ∞ (cid:88) m = −∞ n =0 ψ nm ( r ) ψ ∗ nm ( r (cid:48) ) | r (cid:48) = r n F ( E n,m ) , (15)where πππ = − i (cid:126) ∇ + ec A ( r ) is the gauge invariant momen-tum operator, ψ nm is given by Eq. (11), and n F ( E ) =[exp(( E − µ ) /T ) + 1] − is the Fermi-Dirac distribu-tion function. In the considered limit of low tempera-tures ( T (cid:28) (cid:126) ω c ) is reduced to the Heaviside function θ ( µ − E n,m ).Based on the expressions for radial and tangential com-ponents of a momentum π r = − i (cid:126) ∂∂r , π ϕ = − i (cid:126) r ∂∂ϕ + eBr c , (16)one can find the corresponding expressions for the currentdensity: j r = − ie (cid:126) m e (cid:18) ∂∂r − ∂∂r (cid:48) (cid:19) ∞ (cid:88) m = −∞ n =0 θ ( µ − E n,m ) × ψ nm ( r, ϕ ) ψ ∗ nm ( r (cid:48) , ϕ ) | r (cid:48) = r ≡ j ϕ ( r ) = e (cid:126) m e l ∞ (cid:88) m = −∞ n =0 (cid:18) r m r sgn( m ) − r (cid:19) ×| ψ nm ( r, ϕ ) | θ ( µ − E n,m ) . (18)Full current flowing in the disk is I = (cid:90) r r j ϕ ( r ) dr = ∞ (cid:88) m = −∞ n =0 I nm θ ( µ − E n,m ) (19)with the partial component I nm carried by the state withdefinite quantum numbers n, m : I nm = e (cid:126) m e r (cid:90) r (cid:18) mr − rl (cid:19) f nm ( r ) dr = e (cid:126) m e m r (cid:90) r drr f nm ( r ) − πl , (20)where in the second term we used the normalization con-dition (cid:90) d r | ψ nm ( r, ϕ ) | = 2 π (cid:90) r r rdrf nm ( r ) = 1 . (21)One can see that Eqs. (19) and (20) are in agreementto Eq. (7) in [16]. Note that the first paramagnetic partin Eqs. (18) and (20) originates from the gradient termin Eq. (14) and is related to the spacial inhomogeneityof the current flow in the disk. The second term of thecorresponding equations ∼ B is diamagnetic.
2. Byers-Yang formula
It is worth to mention that a more complicated prob-lem with the Corbino disk placed in the constant mag-netic field and threaded by the flux Φ can still be consid-ered basing on the solution (7) in terms of the Whittaker functions. This occurs because adding a vector potential A η ( r ) = (0 , Φ η/ (2 πr )) corresponding to the magneticfield B η ( r ) = ∇ × A η = e z Φ ηδ ( r ) does not alter thestructure of Eq. (2). One can easily check that the cor-responding solution for the problem that involves a su-perposition of the constant field and flux can be writtenby mere replacement m → m − η . The energy spectrumof the infinite system is still given by Eq. (8) with theshifted azimuthal quantum number.Under certain conditions (see Ref. [24] for a detaileddiscussion) the current I nm carried by the state withdefinite quantum numbers n, m can be found using theByers-Yang formula [27] I nm = − eh ∂E n,m ( η ) ∂η = eh ∂E n,m ( η ) ∂m . (22)Although initially the Byers-Yang formula was intro-duced for a system with a hole threaded by the flux,the second equality in Eq. (22) allows one to use it forthe description of the current in the system with hole ina constant magnetic field with η = 0. Yet, consideringthe spectrum for the infinite system, Eq. (8), one cansee that this formula is ill defined for m = 0. We notethat the Byers-Yang formula follows from Eq. (20) if oneutilizes the Hellman-Feynman theorem. We will use thisformula below to estimate the values of I nm .Finally we note that in what follows we use the dimen-sionless units for the brevity of notations. Yet we willrestore the units to underline the physics. III. THE CASES AMENABLE TOANALYTICAL SOLUTIONA. Asymptotic analysis of the eigenvalue problemfor wide Corbino disk
The solution of transcendental equation (8) admits therich variety of asymptotic representations. Below we an-alyze the approximation of a wide Corbino disk ρ (cid:29) ρ ,and, besides the first case with m = 0, in the followingthe large angular momentum limit ( m (cid:29)
1) will be inthe focus of our discussion. For the sake of conveniencethey are graphically classified in Figure 2.
1. The states with zero angular momentum in a disk with asmall hole
Let us start from consideration of the electron spec-trum in a wide disk ( ρ (cid:29)
1) with the small hole( ρ (cid:28) m = 0.Basing on the asymptotic expressions of the Whittakerfunctions with m = 0 for small arguments one can arriveto Eq. (A6) (see details of the derivation in AppendixA in parts 2 and 3). Corresponding energy levels for not (a) (b)(c) (d) FIG. 2: Graphical representation of applicability of the ana-lytical expressions for energy levels in a Corbino disk underthe conditions specified in figures (see details in the text). (a)Eq. (23) is valid in the entire disk. (b) Eq. (26) is for thebulk region, while inner and outer orange domains correspondEq. (24) and Eq. (25) respectively. (c) Eq. (33) correspondsfor a narrow Corbino disk (Corbino ring). (d) Eq. (45) isapplicable for the small Corbino diskwith the infinitesimal in-ner radius. The proportions are not to scale. The figures areillustrative in nature. very large quantum numbers n + 1 (cid:28) /ρ are given by: ε n, = n + 12 + 1ln (cid:16) ρ (cid:17) − ψ ( n + 1) − γ , (23)where ψ ( z ) is the digamma function and γ = 0 . ... isthe Euler-Mascheroni constant. One can see, that theLandau equidistant “ladder” distorts, and this distortionincreases with the growth of the level number.
2. The states with large angular momentum close to theedges
In a Corbino disk with the radius of the inner holelarger than magnetic length ( ρ (cid:38)
1) the edge states areformed in the both border areas. Near the edges whenthe center of the wave function ρ m = (cid:112) | m | satisfies thecondition | ρ m − ρ i | < i = 1 ,
2) the energy levels havethe following form (see Eq. (A20) in the Appendix A) ε n,m = 2 (cid:18) n + 34 (cid:19) − Γ( n + 1 / πn ! (2 n + 1) ρ m − ρ ρ m + 2 G (2 n + 2) (cid:18) Γ( n + 3 / πn ! (cid:19) ( ρ m − ρ ) ρ m , (24) ε n,m = 2 (cid:18) n + 34 (cid:19) − Γ( n + 1 / πn ! (2 n + 1) ρ − ρ m ρ m + 2 G (2 n + 2) (cid:18) Γ( n + 3 / πn ! (cid:19) ( ρ − ρ m ) ρ m , (25) respectively, with G (2 z ) = ψ ( z +1 / − ψ ( z ). They corre-spond to the energies of the skipping electrons subjectedto the parabolic potential with the minimum shifted withrespect to the inner/outer edge. In the first term of theseexpressions one can recognize the spectrum for the har-monic oscillator with reflecting wall at the minimum ofpotential (see the Problem 2.12 in Ref. [15]). These pecu-liarities of the electron energy levels near the edges of theribbon placed in magnetic field were anticipated in [16](also cf. Eqs. (7) and (9) in Refs. [3, 22], respectively).
3. The states with large angular momentum far from theedges
In the bulk of the disk not too close to the edges, when ρ < ρ m < ρ and | ρ m − ρ i | (cid:29)
1, the electron statesare localized. The corresponding spectrum tends to theLandau one (see details of the derivation in Appendix Ain part 5): ε ( i ) n,m = n + 12 + 1 √ πn ! x n +1 i e − x i / ,x i = ρ m ( ρ i − − ln ρ i ρ m ) , ρ m = r m l , (26)and corresponds to the energy spectrum of the ribbon inmagnetic field [3].
4. Currents carried by the states with large angularmomentum
To find the current I nm carried by the state with def-inite quantum numbers n, m in the different regions ofthe Corbino disk we apply Byers-Yang formula (22).First, it is easy to see from Eq. (26) that inside the disk, ρ < ρ m < ρ and | ρ m − ρ i | (cid:29) I nm = 0,because the Landau levels are flat up to the exponentialcorrection.Near the edges of the disk, | ρ m − ρ i | <
1, from Eqs. (24)and (25) one obtains the currents I n ( ρ i ) ≡ I nm ( ρ i ) = ( − i Γ( n + 1 / n + 1) π n ! eω c ρ i , (27)where we set ρ m = ρ i , so that the partial contributions I nm ( ρ i ) are independent on m . Let us note that thecurrents at the inner and outer edges of the disk do notcoincide due to the difference in curvatures.The partial contribution to the full persistent currentis I tot n = I n ( r ) + I n ( r )= − Γ( n + 1 / n + 1) π n ! e (cid:126) m e l r − r r r . (28)and in the simple rectangular geometry when r , r → ∞ it turns zero [3].One can rewrite the factor e (cid:126) /m e in Eq. (28) as e (cid:126) /m e = 2 π J a , where J = m e e π (cid:126) ≈ . × − A (29)is the current carried by an electron in hydrogen atom asestimated in the Bohr’s model, and a = (cid:126) / ( m e e ) ≈ .
053 nm is the Bohr radius. Then one can recast theEq. (28) in the following form I tot n = −J n + 1 / n + 1) πn ! a r r r − r l , (30)where the magnetic length l = 26 nm / (cid:112) B [T] and thefield is measured in Tesla.Let us consider the effect of discussed persistent cur-rent on the quantum Hall effect measurements in themacroscopic (2 r = 0 . r = 3 . I tot nm induced by the disk curvature.Since the filling factor in Ref. [28] was below 3, we re-strict our evaluation only by one channel, what gives | I tot n | = 6 . × − A for B = 10 T and n = 0. Suchsmall additional persistent current cannot affect the mea-surements of the quantum Hall effect.However, for the case of mesoscopic ring geometry evenin the case of a weak magnetic field the presence of thepersistent current becomes detectable and cannot be ne-glected. Indeed, the persistent current was discovered inrecent experiments with mesoscopic gold rings [21]. Inthese noninvasive studies (performed by means of scan-ning SQUID microscope) of the rings with the radii ofthe order of 1 µ m and the width 350 nm the persistentcurrent was generated by ac magnetic field with the am-plitude B = 45 G. The measured peak values were foundto be approximately of 1 nA.In order to estimate the magnitude of the total per-sistent current in such a ring basing on Eq. (30) it isnecessary to perform in it the summation over the quan-tum numbers n, m . The principal quantum number n changes over all filled states below the chemical poten-tial up to its maximal value µ/ ( (cid:126) ω c ) (cid:29)
1, while the az-imuthal quantum number m can be fixed by the valueof radius r = (cid:112) | m | l . Summation in Eq. (30) is easilyperformed using the Stirling’s formula what results in: I tot = − J π (cid:18) µ (cid:126) ω c (cid:19) / a l (cid:18) r − r (cid:19) . (31)One can see that the persistent current is inversely pro-portional to magnetic field and is determined by the dif-ference of the edge curvatures.Taking the density of the two-dimensional electrongas to be n ≈ . × cm − and using the relation µ/ ( (cid:126) ω c ) = πl n , we obtain from Eq. (31) the value | I tot | = 1 .
53 nA. This is in a good agreement with theabovementioned experimental results [21].
B. The spectral problem and persistent currents ina narrow Corbino disk
1. Energy spectrum
For a Corbino disk with the large inner radius as it be-comes more and more narrow ( ρ → ρ ) the energy levelstend to grow up (see Fig. 8 (right panel) in Appendix A).This fact is easy to understand, bearing in mind that inthe problem under consideration we have the interplaybetween the Landau and size quantization of the energylevels. When the disk becomes more and more narrowthe first level of size quantization rises up like in the nar-row quantum well with infinitely high walls. As shownin Appendix B, Eq. (8) for eigenenergies acquires a muchsimpler form that involves a combination of the Besselfunctions of the first and second kind: J | m | (cid:16) √ ερ (cid:17) Y | m | (cid:16) √ ερ (cid:17) − J | m | (cid:16) √ ερ (cid:17) Y | m | (cid:16) √ ερ (cid:17) = 0 . (32)When the magnetic field disappears, (cid:126) ω c →
0, then √ ερ , → r , √ m e E/ (cid:126) , so that Eq. (32) reduces tothe well-known equation that describes, for example, mo-tion of a free particle on the Corbino disk [29].The equation (32) can be solved in the high energyapproximation (see Eq. (B5) in Appendix B) and we findthe following spectrum E n,m = (cid:126) m e (cid:18) π n d + m − / r (cid:19) − (cid:126) ω c m (33)with n = 1 , , . . . , and m = −∞ , . . . , , . . . , ∞ and d = r − r (cid:28) r being the width of the Corbino disk.The first term of Eq. (33) is nothing else that the energyspectrum of the free electron gas confined in quantumwell of the width d . Writing Eq. (33) we also neglectedthe term ∼ d which is present in Eq. (B5) in AppendixB and replaced r by r ≈ r . Note that the term in theparenthesis of Eq. (33) agrees with Eqs. (3.5) and (3.6)of Ref. [29].
2. Wave function
The radial wave function corresponding to the eigenen-ergies given by Eq. (32) reads f nm ( ρ ) = C (cid:104) Y | m | (cid:16) √ ερ (cid:17) J | m | (cid:16) √ ερ (cid:17) − J | m | (cid:16) √ ερ (cid:17) Y | m | (cid:16) √ ερ (cid:17)(cid:105) . (34)The last expression can be further simplified in thelarge energy limit and it acquires the form (B6) in Ap-pendix B which is more convenient for calculation thecurrent I nm .
3. Persistent current in a large and narrow Corbino disk
Let us now consider the current I nm carried by thestate with definite quantum numbers n, m . Substitut-ing the integral (B9) in the second line of Eq. (20) andassuming r = r ≈ r (cid:29) l we obtain I m ≡ I nm = e (cid:126) πm e (cid:18) mr − l (cid:19) , (35)where within our approximation for the wave functionEq. (34) the partial contributions I nm are independenton the principal quantum number n , thereby I nm ≡ I m (see details in Appendix B)). One can easily check thatexactly the same expression for I nm follows directly fromthe Byers-Yang formula (22) with the derived above spec-trum (33).The found spectrum (33) and wave function (34) in theabove approximations will allow us to obtain the valueof full current flowing in the narrow Corbino disk vs itsradius and width.Substitution of Eq. (34) into Eq. (18) for the tangen-tial component of the current density and subsequentstraightforward integration leads to the cumbersome for-mula that can be expressed in terms of sine and cosineintegral functions respectively. However, within our as-sumption about the large scale Corbino disk ( r ≈ r = ρl , ρ (cid:29)
1) with the small width d (cid:46) l ( δ = d/l (cid:46)
1) wecan simplify the expression for the current to the form[see Eqs. (B7) - (B9) in Appendix B] I = ∞ (cid:88) m = −∞ n =1 I nm θ ( µ − E n,m )= e (cid:126) πm e l N max (cid:88) n =1 M max (cid:88) m = M min (cid:18) mρ − (cid:19) . (36)The summations in Eq. (36) can be performed exactly.Both limits of summation are determined by the theta-function and the explicit expression for spectrum (33).The condition that the argument of theta-function inEq. (36) remains positive for fixed value of the chemicalpotential yields the constraints for the azimuthal quan-tum number m : M min = − ρ (cid:115) µ (cid:126) ω c − π n δ + ρ ρ ,M max = ρ (cid:115) µ (cid:126) ω c − π n δ + ρ ρ . (37)with [ . . . ] denoting the integer part.What concerns N max it can be determined fromthe condition of the positiveness of the square root inEq. (37): N max = δπ (cid:115) µ (cid:126) ω c + ρ . (38) In the following we assume that N max (cid:29)
1, i.e. thewidth of the disk is not too small and is limited by theconditions (cid:2) µ/ ( (cid:126) ω c ) + ρ / (cid:3) − / (cid:28) δ (cid:46)
1. If this werenot so, and N max would become less than 1, then becauseof such a strong size quantization there would be no levelleft under the chemical potential.The summation over m in Eq. (36) is trivial and resultsin I = e (cid:126) πm e l N max (cid:88) n =1 (cid:20) πρδ (cid:112) N − n + 1 (cid:21) . (39)The remaining summation over n in view of N max (cid:29) I = e (cid:126) ρ m e δl (cid:20) πN − N arcsin (cid:18) N max (cid:19) − (cid:112) N − δπρ N max (cid:21) . (40)Keeping the leading term of Eq. (40) and substitutingexpression Eq. (38) we arrive at I = e (cid:126) rd πm e l (cid:18) µ (cid:126) ω c + r l (cid:19) . (41)Magnetic fields here are restricted from below by the con-dition ω c (cid:29) (cid:126) / ( m e r ), which follows from the require-ment l (cid:28) r .In the limit of weak enough fields, when (cid:126) / ( m e r ) (cid:28) ω c (cid:28) (cid:112) µ/ ( m e r ), the magnitude of the current increaseslinearly with growth of magnetic field: I = e drµ πc (cid:126) B. (42)When the field becomes strong enough, yet remainsin the limit of classical ones (cid:112) µ/ ( m e r ) (cid:28) ω c (cid:28) µ/ (cid:126) ,(supremacy of the second term under the square root inEq. (38)), Eq. (41) reduces to I = e dr πc (cid:126) m e B , (43)and the magnitude of the persistent current increasesmore rapidly ( I ∼ B instead of I ∼ B ).In turn, numerical analysis indicates on the sign chang-ing character of the effect. As one can see in Fig. 3 fora Corbino disk with the width d = 0 . l alteration of thesign in I is observed. Such a discrepancy can be explainedby the neglecting of the higher order terms in the expan-sion of the wave function Eq. (34) during the derivationof Eq. (36).
4. Persistent current in a single transmission channelCorbino ring
When the disk becomes very narrow, d →
0, it is suf-ficient to restrict ourselves by considering in Eq. (33)
11 11.25 11.5 11.75 12 12.25-1-0.500.51
I = 0.0584I = -0.0290I = -0.0047I = 0.0029
FIG. 3: Current density as a function of a dimensionless radialcoordinate for a narrow Corbino disk with a fixed width d =0 . ρ = 11, ρ = 11 . ρ = 11 . ρ = 11 .
75 (blue), ρ = 11 . ρ = 12(green) and ρ = 11 . ρ = 12 .
25 (red). The chemicalpotential is chosen to be µ = 20 (cid:126) ω c and T = 0. Inset showsthe values of the current in units of m e l πe (cid:126) in a Corbino disk,where the color lines correspond to an appropriate currentdensity profile of the system. only the lowest, n = 1 level. Yet the spectrum (33)seems do not agree with the spectrum of an electron inone-dimensional ring [6, 18]. The origin of the discrep-ancy is obvious, viz. the presented here results are ob-tained for the constant magnetic field, while the spectrain Refs. [6, 18] were obtained for the ring threaded by theflux. Accordingly, to recover the results of Refs. [6, 18]one should follow the prescription discussed in Sec. II C 2replacing m → m − η in the term ∼ m and omitting thelast term. C. The spectral problem and persistent current ina small Corbino disk with the infinitesimal innerradius
1. Energy spectrum and wave function
In the case when the inner radius of a Corbino disk issmaller than the magnetic length one can forget about itand approximate the disk by the solid one without holein the centre at all ( ρ = 0). The energy spectrum insuch a case is determined by the zeroes of the confluenthypergeometric function (of the first kind or Kummer’sconfluent hypergeometric function) (see e.g. Ref. [30]):Φ (cid:18) − ε nm + | m | − m , | m | + 1 , ρ (cid:19) = 0 . (44)Being interested in the properties of a “small” disk weassume that its the outer radius is smaller than the mag-netic length: r (cid:46) l (weak magnetic field approximation).Corresponding spectrum acquires the form (cf. Eq. (33)) (a) -1.5-1-0.50 (b) (c) FIG. 4: (a) The full current as a function of a dimensionlessradius ρ for a small Corbino disk with a infinitesimal ρ . Thechemical potential is equal to µ = 20 (cid:126) ω c and T = 0. Inset in(a) shows the zoom of the plot, where the current has negativevalues. (b) The negative contribution I to the full current,while (c) is the positive contribution I . E n,m = (cid:126) m e j nm r − (cid:126) ω c m, (45)where j nm is the n-th zero of J | m | ( z ).The radial component of the wave function can be alsoevaluated (see Eqs. (C1)-(C5) in Appendix C) f nm ( r ) = 1 √ πr J | m | +1 ( j nm ) J | m | (cid:18) j nm rr (cid:19) . (46)As in the case of the previous subsection these analyticalfindings will allow us to study the nontrivial full currentgenesis vs the size of such small Corbino disk.
2. Persistent current
In the case of a disk of the outer radius r (cid:46) l with thehole much smaller magnetic length ( r →
0) the totalcurrent can be determined by Eq. (18) for the tangen-tial component of the current density and correspondingexpression for radial part of the wave-function (46). Inte-gration can be explicitly performed in terms of the Besselfunctions (see Appendix C): I = I + I . (47)Here I = − e (cid:126) πm e l ∞ (cid:88) m = −∞ n =1 θ ( µ − E n,m ) (48)is the diamagnetic current and I = e (cid:126) πm e r ∞ (cid:88) m = −∞ n =1 A nm sgn( m ) J | m | +1 ( j nm ) θ ( µ − E n,m ) (49)is the paramagetic current with A nm given by Eq. (C7)and sgn(0) = 0.
3. Derivation using Byers-Yang formula
One can also verify that Eqs. (47) - (49) follow directlyfrom the Byers-Yang formula (22). Differentiating thespectrum (45) one obtains I nm = e (cid:126) m e (cid:18) sign( m ) πr j nm ∂j nν ∂ν (cid:12)(cid:12)(cid:12) ν = | m | − πl (cid:19) , (50)where it was taken into account that roots j nm of theequation J | m | ( z ) = 0 depend on | m | . Substituting thederivative of these roots ∂j nm /∂ | m | [see Eq. (C9) in Ap-pendix C] in Eq. (50) we arrive at the final result I nm = e (cid:126) m e (cid:32) A nm sign( m ) πr J | m | +1 ( j nm ) − πl (cid:33) . (51)As one can easily see, the second term of Eq. (51) corre-sponds to I and the first term to I , respectively.
4. Emergence of the current states in the small Corbinodisk
The numerical simulation of Eq. (47) is represented inFigure 4. With the chemical potential equal to 20 (cid:126) ω c and in the vicinity of the zero temperature two leaps ofthe current are clearly observed. The behaviour of theseleaps can be easily understood if we consider separatelytwo contributions of the currents I and I , where I given by Eq. (48) is the negative and independent of theradius ρ and I represented by Eq. (49) is inversely pro-portional to the value of outer radius. As long as all en-ergy levels Eq. (45) of the small Corbino disk are locatedabove the given value of the chemical potential there isno current in a Corbino disk (see inset in Fig. 4a). Thefirst leap is connected with the emergence of the quan-tum state with m = 0 below the chemical potential. Dueto this the full current dependence has only one contri-bution from I given by Eq. (48) with zero part from I and, therefore, is the constant until the certain valueof ρ (Fig. 4 b), where another energy levels with the nonzero azimuthal number m give rise a new leap and anew constant.Further increase of the radius allows to involve otherenergy levels with m (cid:54) = 0. This leads to the activation ofthe second contribution I given by Eq. (49). As a resulttogether with the second leap in I (Fig. 4b) similar effectoccurs for I (Fig. 4c). IV. CORBINO DISK OF THE ARBITRARYSIZES: NUMERICAL ANALYSIS
In principle, the expression for the current density (18)allows to investigate its profile for the disk of arbitraryradii (in practice the computation for a wide enough diskis very time-consuming). (a) (b)
FIG. 5: Current density as a function of a dimensionless radialcoordinate for a Corbino disk with a fixed inner radius ρ = 3and different outer radii ρ = 5 .
25 (black) ρ = 5 . ρ = 6 (blue), ρ = 6 . ρ = 7 (red) in (a) and ρ = 9 (black) ρ = 10 (blue), ρ = 11 (green), ρ = 12(yellow) and ρ = 13 (red) in (b). The chemical potential ischosen to be µ = 1 . (cid:126) ω c and T = 0. We studied the current density as a function of a radialcoordinate for a Corbino disk with a fixed inner radius ρ = 3 and the set of outer radii from ρ = 5 .
25 to ρ = 13 .
0. The corresponding results are presented inFigures 5 and 6.0Figure 5 shows the the current density as the functionof the dimensionless radial coordinate of a Corbino disk ρ for the fixed inner radius ρ = 3. Relatively narrowCorbino disk shows almost a sinusoidal type of the be-haviour with the maximal positive value of the currentdensity near the inner radius and the minimal negativeone near the outer edge of a system (Fig. 5a, black line).With the increasing of the width disk (the same withthe increasing of the outer radius) the deformation of thecurrent density profile is begun. One can see in Figure 5 a(red line) that in the middle part of the Corbino disk thecurrent density profile starts to be flattened with the zerovalue. This suppression is clearly seen in Fig. 5, wherethe evolution of the current density profile for the widedisk is shown. Such a behaviour can be easily understoodfrom the energy spectrum for a wide disk when electronsstrive to approach Landau levels as it can be seen in Fig.1 b and as a result make infinitesimal contributions to thecurrent density distribution inside the disk. Moreover,together with the flattened zero part of ϕ the doublechanging of the current density sign is observed near inthe vicinity of the inner and outer edge of the Corbinodisk.The occurrence of such complicated current densityprofiles should be taken into account in experiments withpersistent currents in a ring, where as we have shown al-ready in Fig. 5 the local magnetic response of a systemcan be changed significantly even for a ring with the rel-atively small width.The numerical calculations of the full current as thefunction of the inverse ρ are presented in Fig. 6. Theywere obtained by numerical integration of the previouslyobtained current densities for the fixed radius ρ . he de-pendence of total current on ρ − exhibits unambiguouslythe decay of the persistent current with the increase ofthe outer radius of the disk. This result is not surpris-ing and is in agreement with Eq. (31), obtained fromByers-Yang formula. V. CONCLUSIONS
We have presented a comprehensive study of the elec-tron spectra and occurrence of persistent current in the2DEG filling the Corbino disk of arbitrary dimensionssubjected to constant magnetic field. The results ob-tained in this article can be summarized as follows.i) When the outer radius of the disk Corbino is smallwith respect to magnetic length, while that one of theinner hole is infinitesimaly small r (cid:28) r (cid:46) l (the case ofnanodot, see Fig. 2d) no current flows in the system. Theonly available states here correspond to zero azimuthalnumber which do not carry paramagnetic current (seeEq. (49) and take into account that A n = 0). The dia-magnetic contribution is also absent, because E n, > µ .As the outer radius increases, we observe how the firstcurrent state appears simultaneously with the emergenceof the first m (cid:54) = 0 state (see Fig. 4). FIG. 6: Total current as a function of a dimensionless inverseradial coordinate 1 /ρ for a Corbino disk with a fixed innerradius ρ = 3. The chemical potential is equal to µ = 1 . (cid:126) ω c and T = 0. Insets show current density profiles for ρ = 7and ρ = 13. ii) The next case amendable for the analytic solutionis the narrow Corbino disk with the radius much largerthe magnetic length r − r (cid:46) l (cid:28) r , r (see Fig. 2c).In this case we have succeeded to find the expression forthe persistent current that is valid for the arbitrary rela-tion between the Landau and geometrical (longitudinal)quantizations of electron motion [see Eqs. (40) – (41)].iii) We have found the nontrivial magnetic field depen-dencies [see Eqs. (42) – (43)] of the persistent currentin the cases of supremacy of one or another effect. Thenumerical analysis presented in Fig. 3 allows to see thealteration of the current direction frequently observed inexperiment with nanorings (see Ref. [21]).iv) In the limit of extremely narrow ring ( r − r (cid:28) l ),when only single channel remains in it, our formula forthe spectrum (33) allows to reproduce the well-knownexpression of Cheung et al. for the persistent currentinduced in a nanoring threaded by the flux [18].v) We have succeeded to make a considerable progressin study of the general case of the Corbino disk of arbi-trary dimensions r (cid:29) r (cid:38) l . Detailed analysis of thespectral problem for the edge states resulted in Eqs. (24),(25) (Fig. 2b) and further application of the Byers-Yangformula allowed us to reveal that the total current in widediskis proportional to the strength of magnetic field andis determined by the difference in curvatures ( r − − r − )of the inner and outer edges [see Eq. (31)]. This clearlydemonstrates its fading away in the case of the standardrectangular geometry (see Refs. [3, 13]) and the annulargeometry but neglecting the curvature effects [16, 22].The obtained results allowed us to apply them for anal-ysis of some experimental findings (see Ref. [21]) andfind very reasonable coincidence [see the estimates afterEq. (31)]. Their validity can be restricted both by disor-der and by electron-electron interactions (see e.g. [31].)Our study points out clear evidence of the geometrysignificance for more precise and accurate interpretations1of experiments with persistent currents in mesoscopicrings and similar systems. Acknowledgments
V.P.G. and S.G.Sh. acknowledge a support bythe National Research Foundation of Ukraine grant (2020.02/0051) ”Topological phases of matter and ex-citations in Dirac materials, Josephson junctions andmagnets”. Y.Y. acknowledges support by the CarESSproject. A.A.V. is grateful to Yu. Galperin, A. Ka-vokin, and V.B. Shikin for valuable discussions. S.G.Shthanks V. Kagalovsky for useful discussion. The au-thors are grateful to C. Petrillo for critical reading ofthe manuscript and valuable comments.
Appendix A: Derivation of asymptotic expressions for the eigenvalue problem1. Alternative form of the equation for eigenenergies of the Corbino disk
Using the relation (in notations of [33]) between Whittaker functions and confluent hypergeometric functions of thefirst Φ( a, b, z ) and the second kind Ψ( a, b, z ), respectively, Eq. (8) can be written the following formΨ (cid:18) − (cid:15) + | m | − m , | k | + 1; ρ (cid:19) Φ (cid:18) − (cid:15) + | m | − m , | m | + 1; ρ (cid:19) − Φ (cid:18) − (cid:15) + | m | − m , | m | + 1; ρ (cid:19) Ψ (cid:18) − (cid:15) + | m | − m , | m | + 1; ρ (cid:19) = 0 . (A1)This form turns out to be useful for finding different analytical asymptotic solutions in some limits.
2. Eigenvalue equation for the disk with large outer radius
We start investigation of the energy levels distribution from the case of the disk with large outer radius ρ (cid:29) ρ < ρ . The general Eq. (8) in this case can be simplified using the asymptotic expressions forWhittaker functions for large arguments [36], M ˜ ε, | m | (cid:18) ρ (cid:19) ≈ e ρ / (cid:18) ρ (cid:19) − ˜ ε (cid:34) Γ (1 + | m | )Γ (cid:0) + | m | − ˜ ε (cid:1) + O (cid:18) ρ (cid:19)(cid:35) , (A2)and W ˜ ε, | m | (cid:18) ρ (cid:19) ≈ e − ρ / (cid:18) ρ (cid:19) ˜ ε (cid:20) O (cid:18) ρ (cid:19)(cid:21) . (A3)In result one arrive to the following equation for the energy levels W ˜ ε, | m | (cid:18) ρ (cid:19) = 0 . (A4)In other words, the energy dispersion relation for a Corbino disk with the very large external radius and fixed internalradius is determined by zeros of the Whittaker function. It is worth to mention that the roots of Eq. (A4) can beapproximated by those ones of Bessel or Airy functions (see, e.g. [32]). The numerical solutions of Eq. (A4) arepresented in Fig. 7.
3. Solutions with zero angular momentum a. Numerical analysis of the case of arbitrary hole
Now we analyze the energy levels distribution basing on Eq. (8) in the particular case of the quantum number m = 0 and consider their dependence on the external radius ρ with fixed internal radius ρ . Figure 8 illustrates their2 m FIG. 7: (Left) The energy levels (cid:15) n,m of the very large disk (see Eq. (A4)) for the inner radius ρ = 0 .
25 (black lines) and ρ = 5(red lines) as a function of continuous variable m . (Right) Three-dimensional representation of energy levels as a function ofthe quantum number m considered as a continuous variable and the inner radius ρ . FIG. 8: The dependence of of the energy levels with m = 0 on the external radius ρ for the two values of the internal radius: ρ = 0 .
25 (left panel) and ρ = 3 (right). characteristic deviations from the standard Landau spectrum. One can see that for ρ = 0 .
25 (left panel) the energylevels tend to constant values that differ from the half integer Landau spectrum even for large ρ . This obviously isthe consequence of the inner hole presence. It will be shown below that the dispersion relation returns to the Landauspectrum with small corrections when ρ →
0, i.e. the central hole disappears.For sufficiently large ρ > m = 0 (see Fig. 1d). In other words, the flattening of the spectrum and the approaching usualhalf-integer Landau levels takes place for sufficiently large m only as clearly see from Fig. 1 (d). b. The states with zero angular momentum in a disk with a small hole The mentioned above deviation of the spectrum from the standard Landau one can be found out analytically byconsidering the specific limit ρ (cid:28) ρ → ∞ (plane with a small hole). Using asymptotic expansions for theWhittaker function for the case m = 0 (i.e. ˜ ε = ε ) with a small radius ρ W ε, (cid:18) ρ (cid:19) = 12 √ (cid:16) − ψ (cid:0) − ε (cid:1) − γ + 2 ln (cid:16) ρ (cid:17) + ln 2 (cid:17) Γ (cid:0) − ε (cid:1) ρ + O ( ρ ) . (A5)3one arrives at the transcendental equation with the digamma function ψ ( z ): − ψ (cid:18) − ε (cid:19) − γ + 2 ln (cid:18) ρ (cid:19) + ln 2 = 0 , (A6)which for ρ (cid:28) ρ →
4. Asymptotic of solutions with m → ∞ Let us pass to the analysis of the energy levels with large angular momentum: m → ±∞ . We will do this basingon the same Eq. (A4), obtained in the assumption of ρ (cid:29)
1, but do not requiring any more ρ (cid:28)
1, i.e. we just fix ρ < ρ . a. General relations One can rewrite Eq. (A4) using the relation between the Whittaker function and the confluent hypergeometricfunction of the second kind Ψ( a, b, z ) [33]:Ψ (cid:18) − (cid:15) + | m | − m , | m | + 1; ρ (cid:19) = 0 . (A7)The last equation also follows directly from Eq. (A1) in the limit ρ → ∞ . It is valid for the states with arbitraryangular momentum, while for m ≥ (cid:18) − (cid:15), m + 1; mλ m (cid:19) = 0 . (A8)Here we introduced the parameter λ m = ρ / (2 m ).The function Ψ( a, b ; x ) for fixed a and fixed λ = x/b > a, b ; x ) ∼ b − a e ζ b (cid:34) λ (cid:18) λ − ζ (cid:19) a − U ( a − , ζ √ b ) − (cid:32) λ (cid:18) λ − ζ (cid:19) a − − (cid:18) ζλ − (cid:19) a (cid:33) U ( a − , ζ √ b ) ζ √ b (cid:35) + O (cid:18) b (cid:19) (A9)for b → ∞ uniformly in compact λ -intervals of (0 , ∞ ) and compact real a -intervals. Here ζ = (cid:112) λ − − ln λ ) withsign( ζ ) = sign( λ − U ( a, x ) is related to the parabolic cylinder function U ( a, x ) = D − a − / ( x ).The functions U ( a, x ) and U ( a, − x ) at large positive x behave as U ( a, x ) (cid:39) e − x x − a − (cid:20) − ( a + 1 / a + 3 / x + O (cid:18) x (cid:19)(cid:21) , x → ∞ ,U ( a, − x ) (cid:39) √ π Γ(1 / a ) e x x a − − sin( πa ) e − x x − a − , x → ∞ , (A10)therefore we can neglect the second term in the expansion (A9) and the equation (A8) reduces to (note that λ < ζ ) < U ( − (cid:15), − ζ √ m ) = 0 , ζ = (cid:112) λ − − ln λ ) > . (A11) b. The energy spectrum of skipping electrons Let us consider now the case when the center of wave function ( ρ m ) is located quite close to one of the edges: | m − ρ | < | m − ρ | = 0 one can rewrite Eq. (A7), using the asymptotic expression for fixed a and large b in confluenthypergeometric function (see Eq. (13.8.7) in [34]):Ψ( a, b ; b ) = √ π (2 b ) − a (cid:34) (cid:0) a +12 (cid:1) − ( a + 1) (cid:112) /b Γ (cid:0) a (cid:1) + O (cid:18) b (cid:19)(cid:35) , (A12)that gives 1Γ (cid:0) − (cid:15) (cid:1) − (3 / − (cid:15) ) (cid:112) /m Γ (cid:0) − (cid:15) (cid:1) = 0 . (A13)Since m → ∞ , the energy levels are given by the poles of Γ (cid:0) − (cid:15) (cid:1) . which are at (cid:15) = 2 n + 3 / , n = 0 , , , . . . . Solvingthe last equation we obtain the behavior of energy levels at edges of the disk: ε n,m (cid:39) n + 3 / / − n )Γ( n + 3 / πn ! (cid:112) /m, m → ∞ . (A14)ii) When ρ m is near the edge ρ but | m − ρ | < λ close to 1 ( λ (cid:46)
1) where ζ ( λ ) (cid:39) − λ and U ( − (cid:15), ( λ − √ m ) = 0 . (A15)Using the formulas (19.3.5) from [36], U ( a,
0) = √ π a + Γ (cid:0) + a (cid:1) , U (cid:48) ( a,
0) = − √ π a − Γ (cid:0) + a (cid:1) , U (cid:48)(cid:48) ( a,
0) = a √ π a + Γ (cid:0) + a (cid:1) , (A16)and keeping the terms up to x in the expansion, we get the equation1 + ax / (cid:0) − (cid:15) (cid:1) = √ x Γ (cid:0) − (cid:15) (cid:1) , x = ( λ − √ m. (A17)As x →
0, the energies are given by poles of gamma function Γ (cid:0) − (cid:15) (cid:1) which are at (cid:15) = 2 n + 3 / n = 0 , , . . . . Nearthe poles, writing (cid:15) = 2 n + 3 / δ we haveΓ ( − n − / − δ/ − n − δ/
2) = √ x + O (cid:0) x (cid:1) . (A18)Expanding in δ the equation reduces to δ (cid:20) − δ G (2 n + 2) (cid:21) = √ n + 3 / π Γ( n + 1) x, (A19)where G (2 z ) = ψ ( z + 1 / − ψ ( z ) is the known function and we used the relation for digamma function ψ ( − n − /
2) = ψ ( n + 3 / ε n,m = 2 n + 3 / n + 1 / √ πn ! (2 n + 1) ρ − ρ m √ m + G (2 n + 2) (cid:18) Γ( n + 3 / πn ! (cid:19) ( ρ − ρ m ) m (A20)[compare with Eq. (7) in Ref. 3]. We see the doubling of the frequency (with a shift) and the linear and quadraticcorrections due to the edge. c. The energy spectrum of the electrons rotating on the cyclotron orbits far from the edge Now let us pass to the energy spectrum of electrons rotating on the cyclotron orbits far from the edges: | m − ρ | > √ π Γ(1 / − (cid:15) ) + sin( π(cid:15) ) e − x x (cid:15) = 0 , x = ζ ( λ ) √ m, ζ ( λ ) = (cid:112) λ − − ln λ ) . (A21)This is the same equation as for the energy spectrum of the states in the bulk of a ribbon [3, 22] hence we can write ε n,m = n + 12 + 1 √ πn ! x n +1 e − x / , x = ζ ( λ ) √ m, λ = ρ ρ m < . (A22)and arrive at Eq. (26).5
5. Back to Corbino disk
Until now we analyzed the case of the infinite system with a hole. We checked that a similar consideration is validfor the Corbino disk with the two edges described by the general equation for eigenvalues (A1). Using the asymptoticexpression (A9) for Ψ-functions and the analogous one forΦ( a, b ; λb ) (cid:39) b − a e ζ b (cid:34) λ (cid:18) λ − ζ (cid:19) a − U ( a − , − ζ √ b ) + (cid:32) λ (cid:18) λ − ζ (cid:19) a − − (cid:18) ζλ − (cid:19) a (cid:33) U ( a − , − ζ √ b ) ζ √ b (cid:35) + O (cid:18) b (cid:19) , (A23)for the fixed λ = ρ / (2 m ) and λ = ρ / (2 m ) Eq. (A1) acquires the form U ( − (cid:15), ζ √ m ) U ( − (cid:15), − ζ √ m ) − U ( − (cid:15), − ζ √ m ) U ( − (cid:15), ζ √ m ) = 0 , ζ i = ζ ( λ i )sign( λ i − . (A24)When ρ < √ m < ρ we have λ < λ >
1, hence in this case we can write U ( − (cid:15), − ζ √ m ) U ( − (cid:15), − ζ √ m ) − U ( − (cid:15), ζ √ m ) U ( − (cid:15), ζ √ m ) = 0 , (A25)with positive ζ i = (cid:112) λ i − − ln λ i ). Thus, for m → ∞ , the equation splits in two: U ( − (cid:15), − ζ √ m ) = 0 and U ( − (cid:15), − ζ √ m ) = 0 . (A26)In the bulk, when ρ < √ m < ρ , where √ m − ρ (cid:29) ρ − √ m (cid:29) ρ (cid:29) ρ − ρ (cid:29) ε n,m = n + 12 + 1 √ πn ! x n +1 e − x / , x = ζ ( λ i ) √ m, λ i = ρ i ρ m . (A27)The energy spectra of the electrons skipping along the edges are determined by Eqs. (24) - (25). Appendix B: Large and narrow Corbino disk
As one can see from Fig. 8 (right panel) for a narrow Corbino disk with the large inner and outer radii energy tendsto be large. By means of the asymptotic expressions of Whittaker functions for large parameter (cid:15) (see Eqs. (13.21.15)and (13.21.15) in [34]) M ˜ ε, | m | (cid:18) ρ (cid:19) ≈ ρ √ | m | + 1) ˜ ε − | m | J | m | (cid:16) √ ερ (cid:17) , (B1)and W ˜ ε, | m | (cid:18) ρ (cid:19) ≈ ρ √ (cid:18) ˜ ε + 12 (cid:19) (cid:20) sin (cid:18) π ˜ ε − π | m | (cid:19) J | m | (cid:16) √ ερ (cid:17) − cos (cid:18) π ˜ ε − π | m | (cid:19) Y | m | (cid:16) √ ερ (cid:17)(cid:21) . (B2)we arrive at the new equation (32) for energy levels, which is much simpler than Eq. (8) or the equivalent Eq. (A1).Denoting in Eq. (32) x = √ (cid:15)ρ and λ = ρ /ρ one can rewrite it in the form coinciding with Eq. (10.21.45) in [34] J | m | ( x ) Y | m | ( λx ) − J | m | ( λx ) Y | m | ( x ) = 0 . (B3)Then for λ > λ − (cid:28) n th positive zeros ofthe Bessel functions cross-product (see Eq. (10.21.50) in [34]) x n = πnλ − m − πn λ − λ , n = 1 , , . . . . (B4)Accordingly, the solutions of Eq. (B3) are˜ (cid:15) = 12 (cid:20) πnδ + m − / πn δρ ρ (cid:21) , n = 1 , , . . . , (B5)6where δ = ρ − ρ . Restoring units and neglecting the term ∼ δ one arrives at the final expression (33) for thespectrum of the narrow disk.Using the large argument asymptotic of Bessel functions (see Eqs. (9.2.1) and (9.2.2) in [36]) one obtains fromEq. (34) the following expression for the wave function f nm ( ρ ) = sin( √ (cid:15) ( ρ − ρ )) l √ πδρ , (B6)where the normalization constant C is determined by the condition (21).The formula (20) for the current I n,m carried by the state with definite quantum numbers n, m contains the followingintegral (cid:82) ρ ρ dρf nm ( ρ ) /ρ . For large narrow Corbino disk we can use the found above asymptotic of the wave function(B6) and obtain ρ (cid:90) ρ dρρ f nm ( ρ ) = √ (cid:15)πl δ (cid:34) − √ (cid:15)δ )2 √ (cid:15)ρ + sin(2 √ (cid:15)ρ ) (cid:16) Ci(2 √ (cid:15)ρ ) − Ci(2 √ (cid:15)ρ (cid:17) − cos(2 √ (cid:15)ρ ) (cid:16) Si(2 √ (cid:15)ρ ) − Si(2 √ (cid:15)ρ ) (cid:17)(cid:105) , (B7)where Si( z ) and Ci( z ) are sine integral and cosine integral functions, respectively. Using their asymptotic at largeargument, Si( z ) ≈ π − cos zz , Ci( z ) ≈ sin zz , z (cid:29) , (B8)and that in the leading approximation from Eq. (B5) follows that √ (cid:15)δ = πn , we arrive at the following simple result ρ (cid:90) ρ dρρ f nm ( r ) = 12 πl ρ ρ . (B9) Appendix C: Small Corbino disk with infinitesimal inner radius
In this case one can approximate a Corbino disk as a solid disk without a hole in the centre with the conditions ρ < ρ = 0. The solution for energy spectrum and is given by zeroes of the confluent hypergeometric function(44) (see e.g. [30]). We rewrite this expression by means of the formula that represents the confluent hypergeometricfunction as the series of the Bessel functions of the first kind (see Eq. (13.3.7) in [36])Φ (cid:18) − ε + | m | − m , | m | + 1 , ρ (cid:19) = Γ( | m | + 1) e ρ ˜ ε − | m | ∞ (cid:88) p =0 A p (cid:18) ρ ε (cid:19) p J | m | + p (cid:16) √ ερ (cid:17) , (C1)where coefficients satisfy the recurrence relation( n + 1) A n +1 = ( n + | m | ) A n − − εA n − , (C2)and A = 1, A = 0, A = | m | + .If the expansion parameter ρ / (2 √ ε ) is small then we can keep the first term in Eq. (C1). This leads to theequation for eigenvalues J | m | (cid:16) √ ερ (cid:17) = 0 , (C3)which has the solutions √ ερ = j nm with j nm being the n-th root of the equation J | m | ( z ) = 0. The correspondingenergy spectrum E n,m is given by Eq. (45) in the main text. The expansion parameter ρ / (2 √ ε ) < ρ / (2 j nm ) < ρ / (2 j ). Thus, using the value of the lowest root j ≈ .
4, we estimate that r < . l which determines the range of validity of the considered approximation. Note that Eq. (C3) also follows directly fromEq. (32) by taking there ρ = 0.7In turn, the radial component of the wave function has the form f nm ( r ) = CJ | m | (cid:16) √ ερ (cid:17) = CJ | m | (cid:18) j nm rr (cid:19) . (C4)The constant C can be found from the normalization condition (21) (see Eq. (6.521.1) in [38]): C = 1 πr J | m | +1 ( j nm ) . (C5)Therefore, the radial component of the wave function is given by Eq. (46) in the main text. In this case the fullcurrent given by Eqs. (19) and (20) acquires the following form I = e (cid:126) πm e r ∞ (cid:88) m = −∞ n =1 θ ( µ − E n,m ) J | m | +1 ( j nm ) r (cid:90) dr (cid:18) mr − rl (cid:19) J | m | (cid:18) j nm rr (cid:19) . (C6)The integration of the first term in the bracket can be done using the following formula (Eq. (1.8.3.17) in [39]) A nm ≡ | m | (cid:90) drr J | m | ( j nm r ) = 1 + J ( j nm ) − | m |− (cid:88) k =0 J k ( j nm ) , | m | ≥ , (C7)while the second term is integrated using the normalization condition (21). Thus we arrive at Eq. (47) in the maintext.To verify that Eqs. (47) - (49) follow directly from the Byers-Yang formula (22) one needs to calculate explicitlythe derivative ∂j nν /∂ν of the roots j nν of the equation J ν ( j nν ) = 0 with ν = | m | . It can be expressed as follows [seeRef. 40 Section 15.6, Eq. (2)]: ∂j nν ∂ν = 2 νj nν J ν +1 ( j nν ) (cid:90) dxx J ν ( j nν x ) . (C8)Accordingly, one obtains ∂j nm ∂ | m | = 2 | m | j nm J | m | +1 ( j nm ) (cid:90) dzz J | m | ( j nν z ) = A nm j nm J | m | +1 ( j nm ) , (C9)where in the last identity is taken into account the definition (C7) for A nm . [1] L.D. Landau, Diamagnetismus der Metalle , Zeitschriftf¨ur Physik , 629 (1930).[2] E. Teller, Der diamagnetismus von freien elektronen ,Zeitschrift f¨ur Physik , 311 (1931).[3] M. Heuser and J. Hajdu, On Diamagnetism and Non-Dissipative Transport , Z. Physik , 289 (1974).[4] Y. Aharonov and D. Bohm,
Significance of Electromag-netic Potentials in the Quantum Theory , Phys. Rev. 115,485 (1959).[5] M. Tinkham,
Introduction to Superconductivity , DoverPublications, 2 edition (2004).[6] I.O. Kulik,
Flux quantization in a normal metal , JETPLett. , 407 (1970).[7] I.O. Kulik, Persistent currents, flux quantization, and magnetomotive forces in normal metals and superconduc-tors (Review Article) , Low Temperature Physics , 841(2010).[8] R.E. Prange, Three Geometrical Modifications of theSurface-Impedance Experiment in Low Magnetic Fields ,Physical Review , 737 (1968).[9] E.N. Bogachek and G.A. Gogadze, Oscillation effects ofthe flux quantization type in normal metals , Zh. Eksp.Teor. Fiz. , 1839 (1972).[10] S.S. Nedorezov, Size effects in the magnetic susceptibilityof metals , Zh. Eksp. Teor. Fiz. , 624 (1973) [Sov. Phys.JETP , 317 (1973)].[11] K. Richter, D. Ullmo, and R.A. Jalabert, Orbital mag-netism in the ballistic regime: geometrical effects , Physics Reports , 1 (1996).[12] E. Gurevich and B. Shapiro,
Orbital Magnetism in Two-dimensional Integrable Systems , J. Phys. I France , 807(1997).[13] A. H. MacDonald and P. Stˇreda, Quantized Hall effectand edge currents , Phys. Rev. B , 1616 (1984).[14] V.P. Mineev, de Haas–van Alphen effect versus integerquantum Hall effect , Phys. Rev. B , 193309 (2007).[15] V. Galitski, B. Karnakov, and V. Kogan, ExploringQuantum Mechanics: A Collection of 700+ Solved Prob-lems for Students, Lecturers, and Researchers , OxfordUniversity Press (2013).[16] B.I. Halperin,
Quantized Hall conductance, current-carrying edge states, and the existence of extended statesin a two-dimensional disordered potential , Phys. Rev. B , 2185 (1982).[17] M. Buttiker, Y. Imry, and R. Landauer, Josephson behav-ior in small normal one-dimensional rings , Phys. Lett. A96, 365 (1983).[18] Ho-Fai Cheung, Y. Gefen, E.K. Riedel, and Wei-HengShih,
Persistent currents in small one-dimensional metalrings , Phys. Rev. B , 6050 (1988).[19] Ho-Fai Cheung, E.K. Riedel, and Y. Gefen, PersistentCurrents in Mesoscopic Rings and Cylinders , Phys. Rev.Lett. , 587 (1989).[20] A. C. Bleszynski-Jayich, W. E. Shanks, B. Peaudecerf,E. Ginossar, F. von Oppen, L. Glazman, J. G. E. Harris, Persistent Currents in Normal Metal Rings , Science ,272 (2009).[21] H. Bluhm, N.C. Koshnick, J.A. Bert, M.E. Huber, andK.A. Moler,
Persistent Currents in Normal Metal Rings ,Phys. Rev. Lett. , 136802 (2009).[22] A.V. Kavokin, B.L. Altshuler, S.G. Sharapov, P.S. Grig-oryev, and A.A. Varlamov,
The Nernst effect in Corbinogeometry , PNAS , 2846 (2020).[23] Y. Avishai, Y. Hatsugai, and M. Kohmoto,
Persistentcurrents and edge states in a magnetic field , Phys. Rev.B , 9501 (1993).[24] Y. Avishai and M. Kohmoto, Quantization of persistentcurrents in quantum dot at strong magnetic fields , Phys-ica A , 504 (1993).[25] Ya.I. Frenkel and M.P. Bronstein,
Quantization of FreeElectrons in Magnetic field , J. Russian Phys. and Chem.Soc. (Physical section) , 485 (1930).[26] A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski, Meth- ods of Quantum Field Theory in Statistical Physics ,Dover, New York (1975).[27] N. Byers and C.N. Yang,
Theoretical ConsiderationsConcerning Quantized Magnetic Flux in SuperconductingCylinders , Phys. Rev. Lett. , 46 (1961).[28] V.T. Dolgopolov, A.A. Shashkin, N.B. Zhitenev,S.I. Dorozhkin, and K. von Klitzing, Quantum Hall effectin the absence of edge currents , Phys. Rev. B , 12560(1992).[29] A. Suzuki and S. Matsutani, Quantum mechanics of aparticle in a ballistic Corbino ring with finite-potentialbarriers , Nuovo Cimento B , 593 (1996).[30] M.E. Rensink,
Electron Eigenstates in Uniform MagneticFields , American Journal of Physics , 900 (1969).[31] D.B. Chklovskii, B.I. Shklovskii, and L.I. Glazman, Elec-trostatics of edge channels , Phys. Rev. B , 4026 (1992).[32] B. Gabutti and L. Gatteschi, New Asymptotics for theZeros of Whittaker’s Functions , Numerical Algorithms , 159 (2001).[33] H. Bateman and A. Erdelyi, Higher TranscendentalFunctions , Vol. I, McGraw-Hill book Co., New York(1953).[34]
NIST Handbook of Mathematical Functions , CambridgeUniversity Press, 2010.[35] N.M. Temme,
Uniform asymptotic expansions of conflu-ent hypergeometric functions , J. Inst. Math. Appl. ,pp. 215 (1978).[36] M. Abramowitz and I.A. Stegun, (Eds.). Handbookof Mathematical Functions with Formulas, Graphs,andMathematical Tables , 9th printing. New York: Dover,1972.[37] T. Champel, V. P. Mineev,
De Haas-van Alphen effect intwo- and quasi two-dimensional metals and superconduc-tors , Philosophical Magazine B , 55 (2001).[38] I.S. Gradstein and I.M. Ryzhik, Table of Integrals, Series,and Products , Fifth Edition (Academic Press, New York,1994).[39] A.P. Prudnikov, Yu.A. Brychkov, and O.I. Marichev,
In-tegrals and Series. Special functions.
V.II, M., Nauka,1983.[40] G. N. Watson,