We consider mesoscopic non-superconducting rings with an effective capacitance. We propose a Hamiltonian model describing magnetic flux in such rings. Next we incorporate dissipation and thermal fluctuations into
our kinetic model. We consider kinetics in limiting regimes of strong and weak coupling to thermal bath

Dajka, Jerzy

Machura, Łukasz

Rogoziński, S.

Łuczka, Jerzy

Acta Physica Polonica B

English

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Magnetic flux in mesoscopic rings : capacitance, inertia and kinetics  Author: Jerzy Dajka, S. Rogoziński, Łukasz Machura, Jerzy Łuczka  Citation style: Dajka Jerzy, Rogoziński S., Machura Łukasz, Łuczka Jerzy. (2007). Magnetic flux in mesoscopic rings : capacitance, inertia and kinetics. "Acta Physica Polonica B" (Vol. 38, no. 5 (2007), s. 1737-1744).   Vol. 38 (2007) ACTA PHYSICA POLONICA B No 5MAGNETIC FLUX IN MESOSCOPIC RINGS:CAPACITANCE, INERTIA AND KINETICS∗J. Dajka, S. Rogozinski, L. Machura, J. ŁuczkaInstitute of Physics, University of SilesiaUniwersytecka 4, 40-007 Katowice, Poland(Received October 20, 2006)We consider mesoscopic non-superconducting rings with an effectivecapacitance. We propose a Hamiltonian model describing magnetic flux insuch rings. Next we incorporate dissipation and thermal fluctuations intoour kinetic model. We consider kinetics in limiting regimes of strong andweak coupling to thermal bath.PACS numbers: 64.60.Cn, 05.10.Gg, 73.23.–b1. IntroductionIt is more than one decade after the first experiments [1] proving theexistence of the theoretically predicted [2, 3] persistent currents in normalmetal multiply connected samples.It has been expected [4] that currents in mesoscopic rings can flow evenin the absence of any driving. Such self-sustaining currents although theo-retically predicted have not been observed so far. One of the reasons is thatprecise experimental regimes for observation of such currents have not beensuggested.In our earlier work [5] we proposed the semi-phenomenological model ofnoisy dynamics of a magnetic flux produced by a current flowing in a meso-scopic ring or cylinder. In this model, the time evolution of the magneticflux is governed by a classical Langevin equation analogical to the one foran overdamped Brownian particle in a specific potential.In previous studies we have included both dissipation, related to theeffective resistance of the ring, and thermal equilibrium fluctuations but thepossible capacitance of the ring has been systematically neglected. The aimof this paper is to construct an extended version of the previous theory whichtakes into account effects related to the capacitance, which plays a role ofthe mass of a Brownian particle and is related to the inertia of the system.∗ Presented at the XIX Marian Smoluchowski Symposium on Statistical Physics,Kraków, Poland, May 14–17, 2006.(1737)1738 J. Dajka et al.First, in Section 2, we present the possible source of the capacitanceof the mesoscopic ring and we formulate the effective Hamiltonian of thesystem. Next, in Section 3, we include terms which consistently describedissipation processes. In Section 4, we consider characteristic time scalesin the system, which in the limiting cases allow to simplify the evolutionequation for the magnetic flux. We end with the summary in Section 5.2. Hamiltonian modelSmall disordered metallic rings and cylinders threaded by a magnetic fluxdisplay persistent, non-dissipative, currents run by coherent electrons. At fi-nite temperature T some of the electrons become ‘normal’, i.e. non-coherentand the amplitude of the persistent current decreases. The dissipative mo-tion of normal electrons is affected by thermal equilibrium Nyquist noise. Wemake an attempt to describe this situation by means of a two-fluid model,where normal and coherent electrons coexist.We consider the ring as a set of N one dimensional current channelsstacked along a certain axis. The coherent current as a function of magneticflux depends on the parity of the number of coherent electrons in a channel.Let p denote the probability of an even number of coherent electrons ina single current channel. The formula for the coherent current reads [6]Icoh(φ, T ) = pIeven(φ, T ) + (1− p)Iodd(φ, T ) ,withIeven(φ, T ) = Iodd(φ− φ0/2, T ) = I0∞∑n=1An(T ) sin(2npiφ/φ0) .The flux quantum φ0 = h/e and the maximal current I0 = heNeN/(2l2me),where Ne is the number of coherent electrons in a single channel of a circum-ference l and me stands for the electron mass. The temperature dependentamplitudes readAn(T ) =4TpiT ∗exp(−nT/T ∗)1− exp(−2nT/T ∗) cos(nkFl) ,where the characteristic temperature T ∗ is proportional to the energy gap∆F at the Fermi surface and kF denotes the Fermi momentum. The energygap ∆F in metals occurs due to finite small size of the system.We shall now construct an effective ‘Hamiltonian’ of the system. Thetotal energy of the system consists of three parts. The first is the effectivepotential related to the persistent current itself; the second is related to theenergy of the magnetic flux and the third is due to charging effects causedby a small but non-zero capacitance C of the system.Magnetic Flux in Mesoscopic Rings: Capacitance, Inertia and Kinetics 1739The total energy of the set of discrete energy levels carrying persistentcurrent can easily be related to this current at T = 0E(0)coh(φ) = −∫Icoh(φ, 0)dφ . (1)At non-zero temperature T , the energy levels become blurred but theyare still able to carry persistent currents as a suitable sum of single-levelcontributions weighted with the Fermi–Dirac probability distribution [6].In the following we apply this approach and define the effective potentialrelated to the persistent current by the relationEcoh(φ) = −∫Icoh(φ, T )dφ , (2)which reflects the well known fact that the persistent current is an equilib-rium and thermodynamic rather than transport phenomenon. This approachis applicable as long as the quantum part of the system which is responsiblefor the persistent currents remains in equilibrium i.e. the notion of discreteenergy levels makes sense. The high temperature limit does not violate theassumptions of the model due to vanishing amplitude of persistent currents.The approach used could be justified in a more elegant way applying themethods of thermofield dynamics [7].In the following we assume that the ring possesses also an effective capac-itance C. It was shown [8] that in the diffusive regime the energy associatedwith long-wavelength and low-energy charge fluctuations is determined byclassical charging energies of suitable defined capacitors. The flux depen-dence of these energies yields the contribution to the persistent current. Inother words, the ring behaves as it were a classical capacitor. The possibilitythat the charging energies could contribute to persistent current has beenmentioned by Imry and Altshuler [9]. They claimed that the local chargefluctuations in a globally neutral system might be the key to understandingproperties of persistent currents.For the above mentioned three parts of the energy, the effective Hamil-tonian takes the formH =C2(dφdt)2+12L (φ− φe)2 +Ecoh(φ), (3)where φe is the magnetic flux induced by an external magnetic field B and Lis a self-inductance of the system. Let us note that this Hamiltonian dependson temperature via Eq. (2). It is not an exception because all mean-fieldHamiltonians are temperature dependent. Moreover, it can be interpretedin terms of thermofield dynamics [7]. The corresponding equation of motionreads1740 J. Dajka et al.Cd2φdt2= − 1L(φ− φe) + Icoh(φ, T ) . (4)It describes a conservative system.3. Capacitive model with dissipationIn real systems a dissipative processes take place at non-zero tempera-ture. Therefore the resistance of the ring and thermal fluctuations shouldbe taken into account. For T > 0, there are coherent and dissipative partsof the total currentItot = Icoh + Idis . (5)The dissipative current Idis is determined by the Ohm’s law and Lenz’s rule,Idis = Idis(φ, T ) = − 1Rdφdt+√2kBTRΓ (t) , (6)where R is the resistance of the ring and kB denotes Boltzmann’s constant.We model the thermal Nyquist fluctuations Γ (t) of the Ohmic current bymeans of δ-correlated Gaussian white noise of zero average, i.e., 〈Γ (t)〉 = 0and 〈Γ (t)Γ (s)〉 = δ(t − s). The noise intensity D0 = 2kBT/R is chosen inaccordance with the fluctuation-dissipation theorem.Now, we generalize Eq. (4) by replacing Icoh by the total current Itot.We obtain then the basic evolution equationCd2φdt2+1Rdφdt= − 1L(φ− φe) + Icoh(φ, T ) +√2kBTRΓ (t). (7)By including the ‘inertial’ capacitive term we get an extended version of theequation proposed in [5].4. Overdamped and underdamped regimesIt is useful to work in dimensionless variables, because relations betweenscales of time, energy, currents, fluxes, etc., play a crucial role. The charac-teristic magnetic flux is determined in a natural way by the flux quantumφ0 = h/e. Accordingly, the flux is scaled as x = φ/φ0. Time can be scaledin several ways. In the following two examples we recognize some of thepossible time scales and corresponding energy scales of the system (7).Magnetic Flux in Mesoscopic Rings: Capacitance, Inertia and Kinetics 17414.1. Overdamped limitThe capacitance of ideally pure rings can safely be neglected and hencethe second order term in the equation of motion (7) for the magnetic fluxcan also be neglected. If the damping effects are dominating then the char-acteristic time τ0 can be obtained from the equation1Rdφdt= − 1L(φ− φe) , (8)by inserting the characteristic quantities, namely,1Rφ0τ0=φ0L , or τ0 =LR. (9)In the dimensionless units, for the rescaled flux x = φ/φ0 and rescaled timeτ = t/τ0, Eq. (7) can be rewritten in dimensionless form asMd2xdτ2+dxdτ= −dV (x)dx+√2D ξ(τ) . (10)The rescaled noise intensity is D = kBT/2ε0, the characteristic energyis ε0 = φ20/2L, the external flux scales as xe = φe/φ0. The generalizedpotential readsV (x) =12(x− xe)2 + F (x) , (11)withF (x) = α∞∑n=1An(T )2npi{p cos(2npix)+(1 −p) cos [2npi(x+1/2)]} , (12)where α = LI0/φ0. The rescaled zero-mean Gaussian white noise ξ(τ) hasthe same statistical properties as the thermal noise Γ (t). Eq. (10) corre-sponds to the evolution equation for the position x of a Brownian particleof mass M evolving in a potential V (x). The charging (inertial) effects arecharacterized by the rescaled ‘mass’M = CR2L =τLτ0, (13)which is the ratio of the two characteristic times τL = CR and τ0 = L/R.The Fokker–Planck equation for the probability distribution p(x, x˙, t)(with x˙ = dx/dτ) corresponding to (10) takes the form of a Kramers–Kleinequation for an inertial Brownian particle. Its stationary solution pst(x, x˙)is the Gibbs distribution describing the equilibrium state, namely,pst(x, x˙) ∝ exp{− 1D[Mx˙22+ V (x)]}. (14)1742 J. Dajka et al.When the charging effects can be neglected, i.e. when the ‘mass’ M ≪ 1we recover the overdamped (Smoluchowski) limit. In such a case Eq. (10)reduces to the formdxdτ= −dV (x)dx+√2D ξ(τ) . (15)The stationary distribution Pst(x) in the projected ‘coordinate’ space x canbe obtained from (14) by integration of the distribution p(x, x˙) over the‘velocity’ variable x˙Pst(x) ∝ exp [−V (x)/D] . (16)The Smoluchowski regime has been studied in [5].4.2. Underdamped limitIf the ‘inertial’ (charging) effects dominate, then the characteristic timeτ1 can be obtained from the relationCd2φdt2= − 1L(φ− φe) , (17)by inserting the characteristic quantitiesCφ0τ21=φ0L or τ21 = LC . (18)In the scaling τ = t/τ1, Eq. (7) assumes the formd2xdτ2+ ηdxdτ= −dV (x)dx+√2D1 ξ(τ) . (19)The rescaled friction coefficient η is the ratio of two characteristic times,namely,η =τ0τ1. (20)The rescaled noise intensity is D1 = L2kBT/φ20Rτ1. In the underdampedlimit, the friction coefficient η ≪ 1. Then Eq. (19) reduces to the formd2xdτ2= −dV (x)dx+√2D1 ξ(τ) . (21)In the noiseless case, when ξ(τ) = 0, this system is conservative.There are also other time scales and there is no unique prescription whichscaling is better. If there is an evident time scale separation (as e.g. M≪ 1or η ≪ 1), then one can eliminate the corresponding fast variables and getthe effective evolution equations for the slow variables. This method is well-known as the method of adiabatic elimination of fast variables and has beenexploited for decades. The above two equations (15) and (21) are examplesof this method.Magnetic Flux in Mesoscopic Rings: Capacitance, Inertia and Kinetics 17435. SummaryIn this paper we relate the effective capacitance of mesoscopic ring tothe inertial effects in the kinetics of magnetic flux. We show the possiblesource of such an effective capacitance. For such systems, we formulatethe Hamiltonian and the corresponding equation of motion. Contrary toprior studies, the proposed model includes also a second order term, whichdescribes the inertial effects.The kinetics of a real system under influence of dissipation and fluctu-ations is formulated. The proper form of the noise term ensures that theequilibrium conditions are satisfied. After identification of the characteristictime scales we consider the limiting cases of under- and overdamped systems.There is a natural but formal analogy between the model introduced inthis paper and that of noisy kinetics of the magnetic flux in a supercon-ducting ring with a Josephson junction [10]. In both cases one deals witha non-linear oscillator subjected to non-linear forces. Even at this formallevel there is a difference since in the case of normal metal rings the forceterm depends on temperature.It is known that Josephson junctions with a large charging energy canexhibit quantum mechanical properties: they can operate in the so-calledquantum Smoluchowski regime [11]. In the case of non-superconductingrings the limit of small capacity is natural and quantum effects are expectedto be more pronounce.J. D. thanks Joachim Ankerhold for useful discussions. The work sup-ported by the Polish Ministry of Science and Higher Education (MNiSzW)(grant N 202 131 32/3786).REFERENCES[1] L.P. Lèvy, G. Dolan, J. Dunsmuir, H. Bouchiat, Phys. Rev. Lett. 64 2074(1990); V. Chandrasekhar, R.A. Webb, M.J. Brady, M.B. Ketchen, W.J. Gal-lagher, A. Kleinsasser, Phys. Rev. Lett. 67, 3578 (1991); D. Mailly, C. Chape-lier, A. Benoit, Phys. Rev. Lett. 70, 2020 (1993); B. Reulet, M. Ramin,H. Bouchiat, D. Mailly, Phys. Rev. Lett. 75, 124 (1995); E.M.Q. Jariwala,P. Mohanty, M.B. Ketchen, R.A. Webb, Phys. Rev. Lett. 86, 1594 (2001);W. Rabaut, L. Saminadayar, D. Mailly, K. Hasselbach, A. Benoît, B. Etienne,Phys. Rev. Lett. 86, 3124 (2001); R. Deblock, R. Bel, B. Reulet, H. Bouchiat,D. Mailly, Phys. Rev. Lett. 89, 206803 (2002).[2] F. Hund, Ann. Phys. (Leipzig) 32, 102 (1938).[3] I.O. Kulik, JETP Lett. 11, 275 (1970).1744 J. Dajka et al.[4] D. Wohlleben, M. Esser, P. Freche, E. Zipper, M. Szopa, Phys. Rev. Lett.66, 3191 (1991); M. Lisowski, E. Zipper, M. Stebelski, Phys. Rev. D59, 8305(1999).[5] J. Dajka, J. Łuczka, M. Szopa, E. Zipper, Phys. Rev. B67, 073305 (2003);J. Dajka, M. Kostur, J. Łuczka, M. Szopa, E. Zipper, Acta Phys. Pol. B 34,3793 (2003).[6] H.F. Cheung, Y. Gefen, E.K. Riedel, W.H. Shih, Phys. Rev. B37, 6050 (1989).[7] H. Umezawa, H. Matsumoto, M. Tachiki, Thermo Field Dynamics and Con-densed States, North-Holland, Amsterdam 1982.[8] P. Kopietz, Phys. Rev. Lett. 70, 3123 (1993).[9] Y. Imry, B.L. Altshuler, in Nanostructures and Mesoscopic Phenoemena,P. Kirk, M.A. Reed, eds., Academic, San Diego, 1992.[10] A. Barone, G. Paterno, Physics and Applications of the Josephson Effect,Willey, New York 1982.[11] J. Ankerhold, Europhys. Lett. 61, 30 (2003).

Magnetic flux in mesoscopic rings : capacitance, inertia and kinetics

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Magnetic flux in mesoscopic rings : capacitance, inertia and kinetics

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