Mesoscopic persistent currents in a strong magnetic field
Eran Ginossar, Leonid I. Glazman, Teemu Ojanen, Felix von Oppen, William E. Shanks, Ania C. Bleszynski-Jayich, J. G. E. Harris
MMesoscopic persistent currents in a strong magnetic field
Eran Ginossar, Leonid I. Glazman,
1, 2
Teemu Ojanen, Felix von Oppen, William E. Shanks, Ania C. Bleszynski-Jayich, and J. G. E. Harris
1, 2 Department of Physics, Yale University, New Haven, Connecticut 06520, USA Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA Dahlem Center for Complex Quantum Systems and Fachbereich Physik,Freie Universit¨at Berlin, 14195 Berlin, Germany (Dated: November 21, 2018)Recent precision measurements of mesoscopic persistent currents in normal-metal rings rely onthe interaction between the magnetic moment generated by the current and a large applied mag-netic field. Motivated by this technique, we extend the theory of mesoscopic persistent currents toinclude the effect of the finite thickness of the ring and the resulting penetration of the large mag-netic field. We discuss both the sample-specific typical current and the ensemble-averaged currentwhich is dominated by the effects of electron-electron interactions. We find that the magnetic fieldstrongly suppresses the interaction-induced persistent current and so provides direct access to theindependent-electron contribution. Moreover, the technique allows for measurements of the entiredistribution function of the persistent current. We also discuss the consequences of the Zeeman split-ting and spin-orbit scattering, and include a detailed and quantitative comparison of our theoreticalresults to experimental data.
PACS numbers: 73.23.Ra, 73.23.-b, 05.30.Fk
I. INTRODUCTION
Persistent currents in normal-metal rings threadedby an Aharonov-Bohm flux constitute a paradigm ofquantum-coherence effects in the thermodynamic prop-erties of mesoscopic systems. While the history of per-sistent currents dates back to the early days of quantummechanics and of superconductivity, they were studiedintensively starting with the seminal paper by Buttiker,Imry, and Landauer. Most experiments to date detected persistent currentsusing SQUIDs (superconducting quantum interferencedevice) as magnetometers. A different technique was re-cently developed by Bleszynski-Jayich et al. which ismuch more sensitive and allows for precision measure-ments with lower back action and over a wider rangeof magnetic fields. The high-precision cantilever torquemagnetometer relies on the interaction of the magneticmoment associated with the persistent current and alarge applied magnetic field. This interaction shifts theresonance frequency of a microcantilever on which therings are located. Measurements of the frequency shiftallow one to extract the persistent current quantitatively.This paper extends the existing theory of mesoscopicpersistent currents to include a large applied magneticfield. We focus on metallic samples with diffusive elec-tron dynamics for which the applied magnetic fields are non -quantizing. Our results hold for both normal-metalrings as well as rings made of nominally superconductingmaterials (provided that the magnetic field significantlyexceeds the superconducting critical field H c ) and in-clude the effects of spin, namely Zeeman splitting andspin-orbit scattering.Within an independent-electron model, the flux-periodic persistent current is strongly sample specific with both magnitude and sign depending on the detailsof the disorder configuration and the geometry of thering. As a result, its ensemble average (cid:104) I (cid:105) is, even in thecanonical ensemble, small compared to its second mo-ment (cid:104) I (cid:105) so that the latter describes the typical mag-nitude of the persistent current of an individual ring. Thetypical persistent current is φ -periodic and, in a diffu-sive metallic ring, has an amplitude of the order of e/τ D ,where τ D denotes the diffusion time of an electron aroundthe ring.The disorder-averaged persistent current is dominatedby the contribution of electron-electron interactions and is φ / λ ( e/τ D ), where λ is an effective electron-electroncoupling constant. While λ is of order unity in lowest or-der perturbation theory, higher-order contributions areexpected to reduce its magnitude significantly. Inrings made of superconducting materials, a related mech-anism leads to a current due to superconducting fluctu-ations above the critical temperature T c . These theoretical expectations have been testedin several experiments, including metallic, semiconducting, as well as superconducting rings.While results of early experiments with metallic ringswere in apparent strong disagreement with theoreticalpredictions, a more recent SQUID-based experiment yielded data reasonably close to theory. Finally, themeasurement of the typical persistent current reportedin Ref. 4 agrees, without any adjustable parametersand over a wide range of experimental variables, withthe predictions of the model of noninteracting diffusiveelectrons described here. We also note in passing thatthere is a closely related set of works, both experimentaland theoretical, which explores the magnetic responseof singly-connected mesoscopic systems, see e.g., Refs. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b as well as magnetic impurities within the ring. Thisindicates that accurate measurements and understandingof persistent currents in various settings would address anumber of interesting questions in many-body condensedmatter physics.This paper is organized as follows. In Sec. II, we dis-cuss the flux dependence of the persistent current. Sec.III discusses the effects of the strong magnetic field onthe persistent current within the independent-electronmodel, including the effects of the Zeeman energy andspin-orbit scattering. Sec. IV focuses on the interactioncontribution to the persistent current. Sec. V contains adetailed comparison between our theoretical results andthe experimental data of Ref. 4. We conclude in Sec. VI.
II. FLUX PERIODICITY OF THE PERSISTENTCURRENT
Conventionally, persistent currents are discussed in thelimit of a pure Aharonov-Bohm flux threading the ring. In this case, gauge invariance implies flux periodicity, I ( φ ) = I ( φ + φ ) , (1)where the period is given by the flux quantum φ = h/e ,and time-reversal invariance gives the relation I ( φ ) = − I ( − φ ) . (2)As a result, the persistent current vanishes at integer andhalf-integer multiples of the flux quantum and can be ex-pressed as a Fourier series I ( φ ) = (cid:80) ∞ p =1 I p sin(2 πpφ/φ ).It is instructive to deduce the consequences of thisFourier decomposition for the current-current correlationfunction C ( φ, φ (cid:48) ) = (cid:104) I ( φ ) I ( φ (cid:48) ) (cid:105) . (3)Here, (cid:104) . . . (cid:105) denotes a disorder average. We anticipatethat the Fourier components I p are mutually uncorre-lated, i.e., (cid:104) I p I p (cid:48) (cid:105) = (cid:104) I p (cid:105) δ pp (cid:48) . Then, we obtain C ( φ, φ (cid:48) ) = ∞ (cid:88) p =1 (cid:104) I p (cid:105) sin(2 πpφ/φ ) sin(2 πpφ (cid:48) /φ ) = ∞ (cid:88) p =1 (cid:104) I p (cid:105) { cos[2 πp ( φ − φ (cid:48) ) /φ ] − cos[2 πp ( φ + φ (cid:48) ) /φ ] } . (4)Within the diagrammatic approach to diffusive electronicsystems, the two terms depending on ( φ − φ (cid:48) ) and ( φ + φ (cid:48) ) have immediate interpretations as the diffuson andcooperon contributions, respectively. Both contributionsare of the same magnitude but depend differently on themagnetic flux.In the presence of an additional large magnetic field B penetrating the metal ring, one expects that the cooperoncontribution is strongly suppressed. This leads to achange in the flux dependence of C ( φ, φ (cid:48) ) which canalso be obtained directly from symmetry considerations.While gauge invariance and hence the flux periodicitypersist, the additional magnetic field changes the time-reversal relation into I ( B, φ ) = − I ( − B, − φ ). As a result,the current is no longer odd in the Aharonov-Bohm flux φ alone, and the Fourier series takes the more generalform (at fixed B ) I ( φ ) = ∞ (cid:88) p =1 { I (+) p cos(2 πpφ/φ )+ I ( − ) p sin(2 πpφ/φ ) } . (5)If we again anticipate that the Fourier components aremutually uncorrelated, (cid:104) I ( ± ) p I ( ± ) p (cid:48) (cid:105) = (cid:104) [ I ( ± ) p ] (cid:105) δ pp (cid:48) (6) (cid:104) I (+) p I ( − ) p (cid:48) (cid:105) = 0 (7)and that, moreover, (cid:104) [ I (+) p ] (cid:105) = (cid:104) [ I ( − ) p ] (cid:105) , we find C ( φ, φ (cid:48) ) = ∞ (cid:88) p =1 (cid:104) [ I (+) p ] (cid:105) [sin(2 πpφ/φ ) sin(2 πpφ (cid:48) /φ ) + cos(2 πpφ/φ ) cos(2 πpφ (cid:48) /φ )]= ∞ (cid:88) p =1 (cid:104) [ I (+) p ] (cid:105) cos[2 πp ( φ − φ (cid:48) ) /φ ] . (8) r'r r r r r r r r r r r'r r r r r r r r FIG. 1: (a) Diffuson and (b) cooperon diagrams for the au-tocorrelation function of the density of states. Full lines rep-resent electronic Green functions, dashed lines correspond todisorder scattering.
In agreement with expectations, our analysis implies thatin the presence of a large magnetic field B , the current-current correlation function has the flux dependence ofa diffuson contribution. Note that the magnitude of thepersistent current, (cid:104) I ( φ ) (cid:105) , becomes independent of flux.As a special case, this also implies that the persistentcurrent can be nonzero at zero flux.It is interesting to note that in the presence of a largemagnetic field, the flux dependence of the persistent cur-rent can also be written as I ( φ ) = ∞ (cid:88) p =1 I p cos(2 πpφ/φ − α ) . (9)Comparing with Eq. (5) yields the identities I (+) p = I p cos α and I ( − ) p = I p sin α . Then, we automatically re-produce Eqs. (6) and (7) by assuming that the phaseoffset α has a uniform distribution over the disorder en-semble. This also yields the relation (cid:104) I p (cid:105) = 2 (cid:104) [ I ( ± ) p ] (cid:105) .In the next section, we verify these flux dependenciesexplicitly within the model of diffusive non-interactingelectrons. III. INDEPENDENT-ELECTRONCONTRIBUTIONA. Current-current correlation function
The persistent current is obtained as the flux-derivative of the thermodynamic potential I = − ∂ Ω ∂φ . (10) For non-interacting electrons, the (grand-canonical) ther-modynamic potential Ω can be expressed asΩ( µ, B ) = − T (cid:90) dE ν ( E, B ) ln[1 + e − β ( E − µ ) ] (11)in terms of the density of states ν ( E, B ). Here, β = 1 /T denotes the inverse temperature. (We use units k B = 1and ¯ h = 1.) Here, the magnetic field B includes boththe Aharonov-Bohm flux φ threading the ring and themagnetic field penetrating the ring. For definiteness, wewill from now on decompose the full magnetic field into apure Aharonov-Bohm contribution and an in-plane field B (cid:107) penetrating the ring. Accordingly, we will drop thevector nature of B in the following although it should bekept in mind that in principle, the persistent current isnot an isotropic function of magnetic field.The thermodynamic potential Ω( µ, B ) at finite tem-perature can be related to its zero-temperature limitΩ ( µ, B ) = (cid:90) µ −∞ dE ( E − µ ) ν ( E, B ) (12)as Ω( µ, B ) = (cid:90) ∞−∞ dE (cid:18) − ∂f µ ( E ) ∂E (cid:19) Ω ( E, B ) , (13)in terms of the Fermi-Dirac distribution f µ ( E ). Thus,the current-current correlation function C I ( B, B (cid:48) ) = (cid:104) I ( B ) I ( B (cid:48) ) (cid:105) takes the form C I ( B, B (cid:48) ) = (cid:90) dE dE (cid:48) (cid:18) − ∂f µ ( E ) ∂E (cid:19) (cid:18) − ∂f µ ( E (cid:48) ) ∂E (cid:48) (cid:19) ∂ ∂φ∂φ (cid:48) (cid:104) Ω ( E, B )Ω ( E (cid:48) , B (cid:48) ) (cid:105) = (cid:90) d(cid:15) ∂ (cid:15) (cid:18) (cid:15) − exp( − β(cid:15) ) (cid:19) ∂ ∂φ∂φ (cid:48) (cid:104) Ω ( E, B )Ω ( E (cid:48) , B (cid:48) ) (cid:105) = (cid:90) d(cid:15) ∂ (cid:15) (cid:18) (cid:15) − exp( − β(cid:15) ) (cid:19) C (0) I ( E, B ; E (cid:48) , B (cid:48) ) (14)Here, we used in the second identity that the correlator depends only on the energy difference (cid:15) = E − E (cid:48) so that wecan perform the integral over the sum σ = E + E (cid:48) . Thus, we are led to consider the zero-temperature autocorrelationfunction C (0) I ( E, B ; E (cid:48) , B (cid:48) ) = (cid:104) I ( E, B ) I ( E (cid:48) , B (cid:48) ) (cid:105) of currents at different chemical potentials E and E (cid:48) as well as fields B and B (cid:48) .Within a model of non-interacting, diffusive electrons, the calculation of C (0) I ( E, B ; E (cid:48) , B (cid:48) ) = (cid:90) E −∞ dE (cid:90) E (cid:48) −∞ dE ( E − E )( E − E (cid:48) ) ∂ ∂φ∂φ (cid:48) (cid:104) ν ( E , B ) ν ( E , B (cid:48) ) (cid:105) (15)starts from the familiar diagrams in Fig. 1 for the disorder-averaged autocorrelation function of the density of states. Note that both the diffuson and the cooperon diagram contribute to the persistent current. The diffuson diagramdepends on the difference A − = A − A (cid:48) of the magnetic vector potentials, the cooperon diagram on the sum A + = A + A (cid:48) . Performing the integrations over the fast Green-function arguments in the diagrams of Fig. 1, onearrives at the expression C (0) I ( E, B ; E (cid:48) , B (cid:48) ) = 12 π ∂ ∂φ∂φ (cid:48) Re (cid:88) ± (cid:90) ∞ dσ (cid:90) σ − σ d(cid:15) (cid:20) σ − (cid:15) (cid:21) Tr (cid:18) − D [ ∇ − ie A ± ] + i ( (cid:15) + E − E (cid:48) ) (cid:19) . (16)Rewriting the square of the diffusion pole as a derivative with respect to (cid:15) and integrating by parts yields C (0) I ( E, B ; E (cid:48) , B (cid:48) ) = − π ∂ ∂φ∂φ (cid:48) (cid:88) ± (cid:90) ∞ dσ Im (cid:90) σ − σ d(cid:15) (cid:15) Tr (cid:18) − D [ ∇ − ie A ± ] + i ( (cid:15) + E − E (cid:48) ) (cid:19) . (17)Here, D denotes the diffusion constant and we limit at-tention to spinless systems. (Effects of spin will be dis-cussed separately in Sec. III C.)In Eq. (17), the trace is over a space of wavefunctions ψ satisfying the condition ˆn · [ ∇ − ie A ± ] ψ | Σ = 0 . (18)at the surface Σ of the metallic ring. ( ˆn denotes denotesa unit vector normal to the surface.) In general, thisboundary condition makes the evaluation of Eq. (17) atedious problem.To simplify this problem, we use a model in whichthe in-plane magnetic field is taken to be of constantmagnitude and to point along the azimuthal directionaround the ring. While this toroidal-field model is clearlydifferent from experimental realizations, we expect thatit gives a qualitatively and, for certain quantities, evenquantitatively correct account of the consequences of alarge magnetic field penetrating the ring. Specifically,we expect that the predictions for the correlation field B c are parametrically correct while the numerical prefactorwould reflect the particular field configuration. At thesame time, predictions for the typical current amplitudewill be quantitatively correct because the large in-planefield drops out of the final expressions.Some considerations for more general field configura-tions are collected in an Appendix. B. Toroidal magnetic field
The simplification of the toroidal-field model derivesfrom the fact that in this case, the eigenvalue problem − D [ ∇ − ie A ± ] ψ = E ψ (19)together with the boundary condition in Eq. (18) can besolved by separation of variables. Let us consider a ringdefined as a cylinder of length L (along the z -direction)and radius R (in the x − y -plane) with periodic boundaryconditions in the z -direction. The total vector potential A is a sum of the Aharonov-Bohm contribution A ⊥ =( φ/L ) ˆz describing the flux threading the ring and the vec-tor potential A (cid:107) = ( B (cid:107) / ˆz × r of the in-plane magneticfield penetrating the ring. Then, the eigenvalue problemin Eq. (19) separates with ψ ( x, y, z ) = χ ( x, y ) exp( ikz )where E = E c ( n − ϕ ± ) + (cid:15) ⊥ (20)with n = 0 , ± , ± . . . and − D [( ∂ x − ieB y ) + ( ∂ y + ieB x ) ] χ = (cid:15) ⊥ χ. (21)Here, we defined the Thouless energy E c = 4 π DL (22)and the dimensionless flux variable ϕ ± = φ ± /φ . Notethat in order not to introduce unnecessary numericalprefactors into equations, this definition of the Thouless energy differs by a factor of four from the definitions em-ployed in Refs. 4 and 9.Inserting these eigenvalues into Eq. (17), we find C (0) I ( E, B ; E (cid:48) , B (cid:48) ) = − π ∂ ∂φ∂φ (cid:48) (cid:88) ± (cid:88) (cid:15) ⊥ (cid:88) n (cid:90) ∞ dσ Im (cid:90) σ − σ d(cid:15) (cid:15) E c ( n − ϕ ± ) + (cid:15) ⊥ + i ( (cid:15) + E − E (cid:48) ) . (23)Performing the sum over n by Poisson summation and measuring all energy variables in units of the Thouless energy,one obtains C (0) I ( E, B ; E (cid:48) , B (cid:48) ) = − E c π ∂ ∂φ∂φ (cid:48) (cid:88) ± (cid:88) (cid:15) ⊥ ∞ (cid:88) p =1 cos(2 πpϕ ± ) (cid:90) ∞ dσ Im (cid:90) σ − σ d(cid:15) (cid:15) exp( − πp (cid:113) (cid:15) ±⊥ + i ( (cid:15) + E − E (cid:48) )) (cid:113) (cid:15) ±⊥ + i ( (cid:15) + E − E (cid:48) ) . (24)The integrals over (cid:15) and σ can be readily done to yield C (0) I ( E, B ; E (cid:48) , B (cid:48) ) = − E c π ∂ ∂φ∂φ (cid:48) (cid:88) ± (cid:88) (cid:15) ⊥ ∞ (cid:88) p =1 cos(2 πpϕ ± ) F p ( z ± ) (25)where z ± = [ (cid:15) ±⊥ + i ( E − E (cid:48) )] /E c and where we defined the function F p ( z ) = Re (cid:20)(cid:18) πp ) + 3 √ z (2 πp ) + z (2 πp ) (cid:19) e − πp √ z (cid:21) (26)We note that this result is valid for spinless fermions.Effects of spin will be discussed below in Sec. III C.It is interesting to compare the result in Eq. (25) withthe corresponding correlation function for the conduc-tance fluctuations of a metallic ring. Indeed, the flux-sensitive contributions to the correlation function of theconductance at different magnetic fields differ from ourresult for the persistent current (apart from an overallprefactor) only by the preexponential factor in the func-tion F p ( z ).In the absence of the in-plane magnetic field, we needto retain only the lowest transverse eigenvalue (cid:15) ±⊥ = 0 toexponential accuracy in 2 L/R . Then, we find (cid:104) I ( φ ) I ( φ (cid:48) ) (cid:105) = 6 E c π φ ∞ (cid:88) p =1 p sin(2 πpϕ ) sin(2 πpϕ (cid:48) ) (27)for the current-current correlation, which reproduces theresult obtained in Ref. 9.In the limit of a large in-plane magnetic field, thecooperon contribution is strongly suppressed since timereversal symmetry is broken. This can be seen explicitlyby computing the lowest transverse eigenvalue (cid:15) ±⊥ pertur-batively in B , for both the cooperon and the diffuson con-tributions. This perturbative approach is valid as long as R (cid:28) (cid:96) B , where (cid:96) B has to be evaluated for the appropri-ate in-plane magnetic fields entering the cooperon (+)and diffuson ( − ) contributions. (Here, (cid:96) B = (1 /eB (cid:107) ) / denotes the magnetic length.) Due to the boundary con- dition of zero normal current, the ground state wavefunc-tion | gs (cid:105) of Eq. (21) at zero B (cid:107) is a constant with zerotransverse eigenvalue. Thus, the leading correction to theeigenvalue is given by (cid:15) ⊥ = (cid:28) gs (cid:12)(cid:12)(cid:12)(cid:12) De ( B (cid:107) ) x + y ) (cid:12)(cid:12)(cid:12)(cid:12) gs (cid:29) = D (cid:96) B (cid:18) R(cid:96) B (cid:19) , (28)and we find that (cid:15) ⊥ E c = 132 π (cid:18) LR(cid:96) B (cid:19) . (29)For the cooperon contribution, the magnetic field is of theorder of twice the applied magnetic field. Thus, by Eq.(25) this contribution is exponentially suppressed oncethe relevant in-plane field is larger than one flux quantumpenetrating the ring.We first focus on the typical persistent current at zerotemperature. In this case, the effective in-plane fieldvanishes for the diffuson contribution, while it stronglysuppresses the cooperon contribution. Thus, assum-ing from now on that B (cid:107) is sufficiently large to make (cid:15) ⊥ (2 B ) (cid:29) E c , we need to retain only the diffuson con-tribution and obtain (cid:104) I ( φ ) I ( φ (cid:48) ) (cid:105) = 3 E c π φ ∞ (cid:88) p =1 p cos(2 πp [ ϕ − ϕ (cid:48) ]) . (30) D B ° (cid:144) B c X I H Φ , B ° L I H Φ , B ° + D B ° L \ - - - FIG. 2: (Color online) Current-current correlation function (cid:104) I ( φ, B (cid:107) ) I ( φ, B (cid:107) + ∆ B (cid:107) ) (cid:105) (in units of ( E c /φ ) ) at zero tem-perature (solid line). The dashed and dotted lines correspondto the contributions from the first and the second harmonics,respectively. The inset shows the same curves but plottedlogarithmically along the vertical axis. Comparing with Eqs. (8) and (9), we find (cid:104) [ I (+) p ] (cid:105) = (cid:104) [ I ( − ) p ] (cid:105) = 12 (cid:104) I p (cid:105) = 3 E c π φ p (31)for the harmonics of the persistent current.Equation (25), (26), and (29) also imply that the cor-relation function of the persistent current at different val-ues of the in-plane magnetic fields falls off exponentiallywith the magnetic-field difference once the in-plane fieldchanges by more than a flux quantum through the crosssection of the ring, i.e., on the scale of the correlationfield B c = √ π φ LR . (32)Note that the functional dependence of the correlationfield on L and R remains the same for much more gen-eral field configurations but that the numerical prefactorin Eq. (32) is specific to the toroidal-field model. A plotof the correlation function (cid:104) I ( φ, B (cid:107) ) I ( φ, B (cid:107) + ∆ B (cid:107) ) (cid:105) isshown in Fig. 2. Its exponential fall-off has importantramifications in experiment. The decay of the correla-tion function implies that measurements of the persistentcurrent at in-plane fields which are significantly separatedfrom each other on the scale set by B c are statisticallyindependent. We are thus led to the ergodic hypothesisthat averaging over a sufficiently wide range of in-planefields is equivalent to averaging over the disorder ensem-ble. This observation is particularly pertinent in view ofthe novel technique of measuring persistent currents em-ployed in Ref. 4 which allows one to obtain the persistentcurrent over a wide range of in-plane magnetic fields.We close this section by discussing the temperature de-pendence of the persistent current at large in-plane mag-netic fields. At finite temperatures, the persistent currentcorrelation function depends on ∆ B and temperature T via the two dimensionless variables, ∆ B/B c and T /E c . D B ° (cid:144) B c X I H Φ , B ° L I H Φ , B ° + D B ° L \ FIG. 3: (Color online) Current-current correlation function (cid:104) I ( φ, B (cid:107) ) I ( φ, B (cid:107) + ∆ B (cid:107) ) (cid:105) (in units of ( E c /φ ) ) vs. ∆ B (cid:107) atfinite temperatures. The curves are normalized to their valueat ∆ B (cid:107) = 0 and correspond to T = 0 . , . , . , . , . × E c (from bottom to top). The inset shows the same curves butplotted logarithmically along the vertical axis. The correlation function can be readily evaluated by com-bining Eq. (14) with Eq. (25). Performing the remainingintegral numerically, we obtain the results shown in Fig.3 for the current-current correlation function and in Fig.4 for the temperature dependence of the typical current.We see from Fig. 4 that the temperature dependence canbe approximated as exponential with reasonable (thoughuncontrolled) accuracy. (Numerical values of the fit arequoted in the figure caption.) Moreover, we observe thatthe typical persistent current becomes rapidly dominatedby the first harmonic as temperature increases. T (cid:144) E c X I H Φ L \ - - - FIG. 4: (Color online) Temperature dependence of the typicalcurrent (cid:104) I ( φ ) (cid:105) (blue line). The dependence can be well fittedby an exponential (cid:104) I (cid:105) ≈ c ( E c /φ ) exp( − αT /E c ) with c =0 .
036 and α = 8 . C. Effects of spin
In weak magnetic field and in the absence of spin-orbit scattering, spin enters the persistent current simplythrough a degeneracy factor of two. Thus, Eq. (27) ismodified into (cid:104) I ( φ ) I ( φ (cid:48) ) (cid:105) = 24 E c π φ ∞ (cid:88) p =1 p sin(2 πpϕ ) sin(2 πpϕ (cid:48) ) . (33)This result includes both the diffuson and cooperon con-tributions.In a large applied magnetic field, but still without spin-orbit scattering, the cooperon contribution is suppressedand we have to take the Zeeman energy into account.The corresponding spinless result was given in Eq. (30).We can include the spin and Zeeman energies by writingthe persistent current as a sum of the contributions ofspin-up and spin-down electrons, I = I ↑ + I ↓ . Once theZeeman energy becomes large compared to the Thoulessenergy, there are no correlations between I ↑ and I ↓ andas a result, we find (cid:104) I ( φ ) I ( φ (cid:48) ) (cid:105) = 6 E c π φ ∞ (cid:88) p =1 p cos(2 πp [ ϕ − ϕ (cid:48) ]) . (34) The recent precision measurements of the persistentcurrent were performed on samples whose spin-orbit scat-tering length is smaller than or of order of the circum-ference of the rings, as deduced from weak-localizationmeasurements. For this reason, we now turn to a morethorough discussion of the consequences of the electronspin, which in addition accounts for the spin-orbit scat-tering. This can be done by a standard extension of thediagrammatic technique for diffusive systems. To bespecific, we focus on sufficiently large magnetic fields thatthe cooperon no longer contributes significantly. Exten-sions to include the cooperon contribution at weak fieldswould pose no additional complications.Including spin indices, we define the diffuson D s s s (cid:48) s (cid:48) ( r , r (cid:48) , (cid:15) ) as shown in Fig. 5 and view it as a 4 × D ( r , r (cid:48) , (cid:15) ) where ( s , s (cid:48) ) labels the rows and( s , s (cid:48) ) the columns. With the ordering ( s, s (cid:48) ) = ( ↑↑ , ↑↓ , ↓↑ , ↓↓ ), one obtains the equation (cid:104) − D ( ∇ − ie A − ) + i(cid:15) + H Z + H so (cid:105) D ( r , r (cid:48) , (cid:15) ) = 12 πN (0) τ δ ( r − r (cid:48) ) (35)by the standard procedure, starting with the diagrammatic representation shown in Fig. 5. ( N (0) denotes the densityof states at the Fermi energy and τ is the elastic scattering time.) Here, the contribution of the Zeeman energy E Z yields the term H Z = − iE Z iE Z
00 0 0 0 , (36)while spin-orbit scattering is included through H so = 23 τ so −
10 2 0 00 0 2 0 − (37)in terms of the spin-orbit scattering time τ so .By retracing the steps leading up to Eq. (17) in the presence of spin effects, we obtain for the correlation functionof the persistent current, C (0) I ( E, B ; E (cid:48) , B (cid:48) ) = − π ∂ ∂φ∂φ (cid:48) (cid:90) ∞ dσ Im (cid:90) σ − σ d(cid:15) (cid:15) Tr (cid:18) − D [ ∇ − ie A − ] + i ( (cid:15) + E − E (cid:48) ) + H Z + H so (cid:19) , (38)where Tr now denotes a trace over configuration space and the four-dimensional spin space.In the limit of large Zeeman splitting, E Z (cid:29) E c , the modes ↑↓ and ↓↑ are exponentially suppressed. For negligiblespin-orbit scattering, we then obtain two massless modes ↑↑ ± ↓↓ . As a result, the correlation function is twice largerthan the result for spinless electrons given in Eq. (25), in agreement with Eq. (34). As the spin-orbit scatteringincreases, only the density mode ↑↑ + ↓↓ remains massless and in the limit of strong spin-orbit scattering, we recoverthe result in Eq. (25) for spinless electrons.More generally, we can discuss the crossover between the limits of weak and strong spin-orbit scattering rate. Onefinds C (0) I ( E, B ; E (cid:48) , B (cid:48) ) = − π ∂ ∂φ∂φ (cid:48) (cid:90) ∞ dσ Im (cid:90) σ − σ d(cid:15) (cid:15) × Tr (cid:32) − D [ ∇ − ie A − ] + i ( (cid:15) + E − E (cid:48) ) + 1 − D [ ∇ − ie A − ] + i ( (cid:15) + E − E (cid:48) ) + τ so + 1 − D [ ∇ − ie A − ] + i ( (cid:15) + E − E (cid:48) + 2 E Z ) + τ so + 1 − D [ ∇ − ie A − ] + i ( (cid:15) + E − E (cid:48) − E Z ) + τ so (cid:33) , (39)where the trace is now over configuration space only. Specifying again to the toroidal-field model, we obtain C (0) I ( E, B ; E (cid:48) , B (cid:48) ) = − E c π ∂ ∂φ∂φ (cid:48) ∞ (cid:88) p =1 cos(2 πpϕ ± ) (cid:20) F p (cid:18) i E − E (cid:48) E c + (cid:15) −⊥ E c (cid:19) + F p (cid:32) i E − E (cid:48) E c + (cid:15) −⊥ + τ so E c (cid:33) + F p (cid:32) i E − E (cid:48) + 2 E Z E c + (cid:15) −⊥ + τ so E c (cid:33) + F p (cid:32) i E − E (cid:48) − E Z E c + (cid:15) −⊥ + τ so E c (cid:33)(cid:35) (40)where the function F p ( z ) had been defined in Eq. (26).Combining Eq. (40) with Eq. (14) and setting (cid:15) −⊥ = 0,we can obtain the crossover of the typical current be-tween the limits of weak and strong spin-orbit scatter-ing for arbitrary temperature. (Note that the results forthe typical current are not restricted to the toroidal-fieldmodel.) Corresponding numerical results in the limit oflarge Zeeman splitting (where the last two terms in thesquare bracket in Eq. (40) can be neglected) are plottedin Fig. 6, which show that the crossover becomes sloweras temperature increases. IV. INTERACTION CONTRIBUTION
We now turn to a discussion of the interaction-contribution to the persistent current in high magneticfields. Adapting the first-order correction in the interac-tion V derived in Ref. 10 to the case of a finite magneticfield, one finds for the disorder-averaged contribution tothe grand canonical potential∆Ω = N (0) ¯ Vπ (cid:90) ∞ dE coth( E T ) E = + s s s s s s s s' s' s' s' s' s' s' D D r r ' r r ' r FIG. 5: Diagrammatic representation of the equation of mo-tion for the diffuson D s s s (cid:48) s (cid:48) ( r , r (cid:48) , (cid:15) ). Full lines represent elec-tronic Green functions and dashed lines denote disorder andspin-orbit scattering. × ReTr 1 − D ( ∇ − ie A ) + iE (41)Here, ¯ V is the Fourier component of the screenedCoulomb interaction potential averaged in momentumspace. In a field much stronger than the upper criticalfield of the ring we may constrain considerations to thelowest-order correction, Eq. (41). To estimate the inter-action contribution to the average persistent current, weagain employ the toroidal-field model introduced in Sec.III. Then, the eigenvalue problem and boundary condi-tions for the cooperon here are identical to those in Eqs.(18) and (19), respectively.We denote the cooperon eigenvalues by (cid:15) ( l ) n,m,φ with l, n, m being the radial, longitudinal and azimuthal quan- (cid:144) E c Τ so X I H Φ L \ FIG. 6: (Color online) Crossover of the typical current (cid:104) I ( φ ) (cid:105) as function of the spin-orbit scattering rate. The curves, cor-responding to temperatures T = 0 . , . , . , . × E c (frombottom to top), are normalized to the value of (cid:104) I ( φ ) (cid:105) in thelimit of vanishing spin-orbit scattering rate. All curves areplotted in the limit of large in-plane field where the cooperoncontribution is suppressed and the Zeeman energy is largecompared to the Thouless energy. tum numbers, respectively. Due to cylindrical symmetrythe cooperon modes can be found by separation of vari-ables, with the replacement n → n − φ/φ added totake into account the Aharonov-Bohm flux. In distinc-tion from Sec. III, the vector potential A in Eq. (41)corresponds to the total field so that (cid:96) B (cid:28) R . In thislimit, the radial equation can be approximated to lowestorder in (cid:96) B /R as D(cid:96) B (cid:18) − ∂ ∂x + ( κ m − x ) + (cid:96) B k n,φ (cid:19) χ ( x ) = (cid:15) ( l ) n,m,φ χ ( x )(42)where x = r/(cid:96) B is a scaled distance from the center of thecross section, κ m = m(cid:96) B /R and k n,φ = 2 π ( n − φ/φ ) /L .Note that the ratio between the radial and the longitudi-nal terms in Eq. (42) is dominated by L/(cid:96) B . The eigen-values can be written as (cid:15) ( l ) n,m,φ = D (cid:18) πL (cid:19) (cid:34) ( n − φφ / + (cid:18) L π(cid:96) B (cid:19) λ l ( κ m ) (cid:35) (43)where the values of λ ( κ m ) for the lowest branch of eigen-states ( l = 0) can be estimated by using the variationalmethod with a Gaussian trial solution. The function λ ( κ ) has a shallow minimum λ ∗ = (1 − /π ) / at κ ∗ m =( π − π ) − / . Using the eigenvalues (cid:15) ( l ) n,m,φ to evaluatethe trace in Eq. (41), it is straightforward to show thatthe contribution to persistent current ∆ I = − ∂ ∆Ω /∂φ isperiodic in φ → φ + φ /
2, and for T = 0 can be writtenas ∆ I = N (0) ¯ Vπ (cid:18) πL (cid:19) hDφ ∞ (cid:88) p =1 pg p sin (cid:18) πp φφ / (cid:19) . (44)In the regime of experimental interest, L (cid:29) R (cid:29) (cid:96) B , thecoefficients g p , g p = 12 p π ∞ (cid:88) m =0 e − pL/(cid:96) B √ λ ( κ m ) (cid:18) p L(cid:96) B (cid:112) λ ( κ m ) (cid:19) (45)can be estimated by evaluating the sum in the saddle-point approximation, g p ≈ . p − . (cid:18) R √ (cid:96) B L (cid:19) (cid:20) p L(cid:96) B (cid:112) λ ∗ (cid:21) e − L(cid:96)B p √ λ ∗ . (46)All harmonics of the average persistent current are ex-ponentially suppressed; the higher the harmonic p , thestronger is the suppression. Note that this implies thatfor sufficiently strong magnetic field, measurements of theaverage current, e.g., by employing large arrays of rings,should be dominated by the canonical-ensemble contri-bution of the free-electron model. V. COMPARISON TO EXPERIMENT
The recent development of cantilever-based torsionalmagnetometers with integrated mesoscopic rings re- T (K) q h I p i ( n A ) Sample
Sample
Sample
Sample
Sample ,, FIG. 7: (Color online) Temperature dependence of the typ-ical current contribution from the p th harmonic (cid:112) (cid:104) I p ( T ) (cid:105) .The markers are the data first presented in Ref. 4. The solidcurves represent new fits to the data using Eqs. (14) and (40)while the dashed curves show the fits from Ref. 4. The sam-ple parameters and best-fit parameters are given in Table I.Closed and open markers denote measurements taken duringdifferent cooldowns, over different field ranges, and at differ-ent magnetic field orientations. In the case of the p = 1 datafrom Sample sulted in measurements of the rings’ persistent currentin the presence of large magnetic fields. Here we brieflyreview these measurements and compare them with thecalculations from the preceding sections. This compar-ison is most readily performed by fitting the measuredtemperature dependence of the current to the form pre-dicted in Eqs. (14) and (40).The parameters characterizing each sample are col-lected in Table I. The temperature dependence of the p th harmonic (cid:113) (cid:104) I p (cid:105) of each sample’s typical current wasdetermined as follows. At a single temperature T , themean square amplitude of the p th harmonic of the currentwas extracted from a measurement of I ( B ) taken over arange of B spanning many B c . This large span ensuredthat the mean was determined from a large number ofindependent measurements, as discussed at the end ofSec. III B. For each sample, the form of I ( B ) was foundto be independent of temperature except for an overallscaling. This scaling was determined by measuring I ( B )over a smaller field range (with bounds denoted by B min and B max ) at each subsequent temperature and compar-ing the magnitude of each harmonic with the value mea-sured over the same field range at T . This procedure, aswell as other details of the measurements, are describedin detail in Ref. 4. The resulting values of (cid:113) (cid:104) I p (cid:105) are0 TABLE I: Sample parameters. “Marker” refers to the markers used in Fig. 7, with closed and open markers representing twodifferent cooldowns of the same sample. For the closed markers the angle between the magnetic field and the plane of the ringswas 6 ◦ and T = 323 mK, while for the open markers the angle was 45 ◦ and T = 365 mK. N denotes the number of rings inthe sample. The ring circumference and linewidth are given by L and w . The thickness of each sample was 90 nm. The spinorbit scattering length L so = 1 . ± . µ m. B min and B max give the bounds for measurements of I ( B ) taken over smaller fieldranges. D L and D ZSO are extracted from fitting the persistent current data. D L is the best-fit value of the diffusion constantfound in Ref. 4, which assumed the limit of strong spin-orbit scattering and large Zeeman splitting. D ZSO is the best-fit valueof the diffusion constant found by taking into account the finite spin-orbit scattering rate and Zeeman splitting as described inSection V. The estimated uncertainty in all fit coefficients is 6%.Sample Marker p N L ( µ m) w (nm) B min (T) B max (T) D L (cm /s) D ZSO (cm /s) shown in Fig. 7.In Ref. 4, this data was analyzed by assuming the limitof strong spin orbit scattering: 1 /τ so (cid:29) { E c , T } , andlarge Zeeman splitting, E Z (cid:29) { E c , T } . As can be seenfrom the sample parameters listed in Table I, this as-sumption is fairly accurate though not exact. For thesesamples 0 . < /E c τ so < .
47 while 0 . < T /E c < .
7. From Fig. 6 it is clear that these parameters are notfully within the strong spin orbit scattering limit. Addi-tionally, for the smallest rings (Sample
T /E Z ≈ .
36 and deviationsfrom the large Zeeman splitting limit change (cid:113) (cid:104) I p (cid:105) by asmuch as 5%. For Samples T /E Z < . <
1% deviations of (cid:113) (cid:104) I p (cid:105) from the large Zee-man splitting limit. As a result the measurements of Ref.4 were not fully in the strong spin orbit scattering, largeZeeman splitting limit, so we reanalyze the data here,taking into account the full dependence of (cid:113) (cid:104) I p (cid:105) on spinorbit scattering and Zeeman splitting.We fit the data from Ref. 4 (Fig. 7) using the expres-sion for (cid:113) (cid:104) I p ( T, D, L so , E z ) (cid:105) derived from Eqs. (14) and(40). The only fitting parameter is the electron diffu-sion constant D . The spin orbit length L so ≡ √ Dτ so =1 . ± . µ m was determined independently from magne-totransport measurements of a wire codeposited with therings. Since each data point in Fig. 7 is extracted frommeasurements of I ( B ) made over a range of B , we can-not use a single value of the Zeeman splitting; instead,we average over the magnetic field range to obtain thefitting function (cid:113) (cid:104) I p ( T, D, L so , B min , B max ) (cid:105) = (cid:115) (cid:82) B max B min dB (cid:104) I p ( T, D, L so , E Z ( B )) (cid:105) B max − B min . (47) The best-fit values of the diffusion constant D ZSO aregiven in Table I. The corresponding fits are shown inFig. 7 as solid lines. For comparison the values of thediffusion constant found in Ref. 4, D L , are also given inTable I and the corresponding fits are shown as dashedlines in Fig. 7.Figure 7 and Table I show that the finiteness of thespin-orbit scattering rate and the Zeeman energy result insmall but noticeable changes to the fitted curves and theextracted values of D . We find that most of the differenceis due to the finite spin orbit scattering rate, which leadsto a non-negligible contribution to the current from thesecond F p term in Eq. (40).The finite Zeeman energy modifies the current via thelast two F p terms in Eq. (40), leading to a correctionwhich becomes appreciable ( > D indistinguishable from thecase of large Zeeman splitting.The values of D ZSO for Samples D ZSO = 234 cm / s measured for Sample D = 260 ±
12 cm / s, consistent with the value measuredfor Sample VI. CONCLUSIONS
Motivated by a new and highly sensitive experimentaltechnique for measuring mesoscopic persistent currents,1we presented a theory of persistent currents in large, butnon-quantizing, magnetic fields. The theoretical resultsof this paper formed the basis for establishing the remark-able quantitative agreement between experiment and the-ory found in Ref. 4 and further refined in Sec.V. To reachthis agreement, we not only needed to take into accountthe large magnetic field, both for the single-particle andthe interaction contributions to the persistent current,but also spin effects.In addition to forming the basis for a quantitative com-parison with experiment, it is also worth emphasizingseveral theoretical conclusions from our results.(i) The magnetic field penetrating the ring leads toqualitative changes in the dependence of the persistentcurrent on the Aharonov-Bohm flux. At zero magneticfield, the persistent current is a periodic function of flux.Zero flux as well as integer and half-integer multiples ofthe flux quantum are special points where the persistentcurrent vanishes. At large magnetic fields, the persistentcurrent I ( φ ) is still a periodic function of flux, but thetypical magnitude (cid:104) I ( φ ) (cid:105) is no longer dependent on flux.(ii) Previous theoretical works have shown that thereare two principal contributions to mesoscopic persistentcurrents: a free-electron contribution and an interactioncontribution. In experiments, it is not always easy todisentangle these two contributions (especially for theeven harmonics of the persistent current). In fact, whilethe interaction contribution is expected to dominate theensemble-averaged persistent current, both of them con-tribute significantly in single- or few-ring experiments.We conclude from our results that the application of alarge magnetic field penetrating the ring strongly sup-presses the interaction contribution to the persistent cur-rent so that the technique of Ref. 4 provides direct accessto the free-electron contribution.(iii) One of the principal advantages of the experimen-tal technique of Ref. 4 is that unlike SQUID-based ap-proaches, it allows for measurements over a wide rangeof magnetic fields and thus of many oscillations of thepersistent current with flux. Our results for the auto-correlation function of the persistent current at differentmagnetic fields imply that averaging over magnetic fieldis equivalent to an ensemble average (ergodic hypothe-sis). One of the possibilities raised by this result is adirect measurement of the entire distribution function ofthe persistent current.The experimental technique of Ref. 4 has broughtmany additional experiments on persistent currents andrelated phenomena within experimental reach. Our ap-proach should be a valuable starting point for analyzingsuch future experiments. Acknowledgments
This work was supported in part by DOE grant DE-FG02-08ER46482 (LG), by DIP (FvO), as well as by NSFgrants 0706380 and 0653377 (JGEH). FvO and LG ac- knowledge the hospitality of KITP while part of this workhas been performed.
Appendix A: Arbitrary magnetic-field configurations
Within the toroidal-field model discussed and em-ployed in Secs. III B and IV, we could account for themagnetic field penetrating the ring by perturbation the-ory. This perturbative calculation was valid as long as R (cid:28) (cid:96) B . The toroidal-field model was special in that atthe surface of the ring, the vector potential A (cid:107) ( r ) associ-ated with the magnetic field penetrating the ring pointsparallel to the surface. As a result, the vector potentialdoes not enter into the boundary condition Eq. (18) forthe equation of the cooperon or the diffuson. Then, com-puting the perturbative shift of the eigenvectors by thein-plane magnetic field amounts to conventional pertur-bation theory as familiar from quantum mechanics.This is no longer the case for more general (and morerealistic) models of the in-plane field. Instead, the mag-netic field enters not only the diffuson or cooperon equa-tion, but also the boundary condition. In this appendix,we show how one can in principle reduce the resultinggeneralized problem of perturbation theory to the con-ventional case of quantum-mechanical perturbation the-ory.The equation for the cooperon or the diffuson is givenby − D (cid:2) ∇ − ie A ⊥ − ie A (cid:107) (cid:3) ψ = Eψ, (A1)with the appropriate choice of magnetic field. This equa-tion needs to be solved in conjunction with the boundarycondition ˆn · [ ∇ − ie A ⊥ − ie A (cid:107) ] ψ (cid:12)(cid:12) Σ = 0 (A2)valid at the surface Σ of the ring. We make the gaugechoice ∇ · A (cid:107) = 0. The basic observation is that wecan eliminate the vector potential A (cid:107) from the boundarycondition by the gauge transformation ψ ( r ) = e if ( r ) ψ ( r ) . (A3)The new function ψ ( r ) satisfies the modified diffusionequation − D (cid:2) ∇ − ie A ⊥ − ie A (cid:107) + i ∇ f (cid:3) ψ = Eψ (A4)with boundary condition ˆn · [ ∇ − ie A ⊥ − ie A (cid:107) + i ∇ f ] ψ (cid:12)(cid:12) Σ = 0 . (A5)If we choose the gauge transformation such that e ˆn · A (cid:107) (cid:12)(cid:12) Σ = ˆn · ∇ f | Σ (A6)combined with the gauge choice ∇ f = 0 , (A7)2we reduce the problem to a form which is amenable tostandard techniques of perturbation theory, namely Eq.(A4) combined with the boundary condition ˆn · [ ∇ − ie A ⊥ ] ψ | Σ = 0 . (A8) The principal technical difficulty consists in solving the“electrostatics” problem defined by Eqs. (A6) and (A7)to find the function f ( r ). F. Hund, Ann. Phys. (Leipzig) , 102 (1938). See, e.g., F. Bloch, Phys. Rev. , A787 (1965), ibid. ,415 (1968); M. Schick, ibid. , 401 (1968); L. Gunter andY. Imry, Solid State Commun. , 1391 (1969). M. Buttiker, Y. Imry, and R. Landauer, Phys. Lett. ,365 (1983). A. C. Bleszynski-Jayich, W. E. Shanks, B. Peaudecerf, E.Ginossar, F. von Oppen, L. Glazman, and J. G. E. Harris,Science , 272 (2009). B. L. Altshuler, Y. Gefen, and Y. Imry, Phys. Rev. Lett. , 88 (1991). A. Schmid, Phys. Rev. Lett. , 80 (1991). F. von Oppen and E. K. Riedel, Phys. Rev. Lett. , 84(1991). H. F. Cheung, E. K. Riedel, and Y. Gefen, Phys. Rev. Lett. , 587 (1989). E. K. Riedel and F. von Oppen, Phys. Rev. B , 15449(1993). V. Ambegaokar and U. Eckern, Phys. Rev. Lett. , 381(1990). B. L. Altshuler and A. G. Aronov, Solid State Commun. ,11 (1981). U. Eckern, Z. Phys. , 393 (1991). V. Ambegaokar and U. Eckern, Europhys. Lett. , 733(1990). F. von Oppen and E. K. Riedel, Phys. Rev. B , 3203(1992). N. C. Koshnick, H. Bluhm, M. E. Huber, and K. A. Moler,Science , 1440 (2007). L. P. L´evy, G. Dolan, J. Dunsmuir, and H. Bouchiat, Phys.Rev. Lett. , 2074 (1990). V. Chandrasekhar, R. A. Webb, M. J. Brady, M. B.Ketchen, W. J. Gallagher and A. Kleinsasser, Phys. Rev.Lett. , 3578 (1991). E. M. Q. Jariwala, P. Mohanty, M. B. Ketchen, and R. A.Webb, Phys. Rev. Lett. , 1594 (2001). H. Bluhm, N. C. Koshnick, J. A. Bert, M. E. Huber, andK. A. Moler, Phys. Rev. Lett. , 136802 (2009). D. Mailly, C. Chapelier, and A. Benoit, Phys. Rev. Lett. , 2020 (1993). L. P. L´evy, D. H. Reich, and L. Pfeiffer, K. West, PhysicaB , 204 (1993). F. von Oppen, Phys. Rev. B , 17151 (1994). D. Ullmo, K. Richter, and R. A. Jalabert, Phys. Rev. Lett. , 383 (1995). D. Ullmo, H. U. Baranger, K. Richter, F. von Oppen, andR. A. Jalabert, Phys. Rev. Lett. , 895 (1998). V. E. Kravtsov and V. I. Yudson, Phys. Rev. Lett. , 210(1993). A. G. Aronov and V. E. Kravtsov, Phys. Rev. B , 13409(1993). H. Bary-Soroker, O. Entin-Wohlman, and Y. Imry, Phys.Rev. Lett. , 057001 (2008). G. Schwiete and Y. Oreg, Phys. Rev. Lett. , 037001(2009). N. Byers and C. N. Yang, Phys. Rev. Lett. , 46 (1961). B. L. Altshuler and B. I. Shklovskii, Zh. Eksp. Teor. Fiz. , 220 (1986) [Sov. Phys. JETP , 127 (1986)]. A. G. Aronov and Yu. V. Sharvin, Rev. Mod. Phys. ,755 (1987). P. A. Lee, A. D. Stone, and H. Fukuyama, Phys. Rev. B , 1039 (1987). See, e.g., M. G. Vavilov and L. I. Glazman, Phys. Rev. B , 115310 (2003). A. C. Bleszynski-Jayich, W. E. Shanks, B. R. Ilic, and J.G. E. Harris, J. Vac. Sci. Technol. B26