arXiv:0910.5172v1 [cond-mat.supr-con] 27 Oct 2009 ihafrewihcudsittepae∆ phase the connected shift the be could of which not force case can a the with experiment in The effect interference evi- Ahranov-Bohm unambiguous two-slit [16]. give the forces has that such [18] dence assume test - authors experimental field Some direct magnetic as any cross such [17]. not magnetic particles lines do that particles charged the seem though on even is act it can phenomenon forces this in observed hf 1]rte ae h rbe oeugn than urgent more problem the makes phase rather Ahranov-Bohm the [18] the explain of shift forces to with needed associated evidence delays experimental magnitude time can The of absence problems of solved. the case these of as interfer- considered the Both two-slit be in not the [1-3,14-16]. as in well experiment shift as ence call phase balance resistance Ahranov-Bohm force non-zero the the with question its loops of But in the limits formalism. the in quantum in observations recognized obtained universally [22-24] theories the agree- the in with observed is ment current in loops persistent observed superconductor The and current [2,7-9,19-21]. metal persistent normal the semiconductor, is nanostructures in tum charge atyas[71] eas ftepaesit∆ shift phase the the of in Because up bucked [17,18]. is years which last [14-16] debates provokes effect eslyrcgie nepeaino h hs gradient phase the of ▽ interpretation recognized versally ~ ainhsoyo unu hsc.Teiflec fthe of cente- influence the The potential vector in . magnetic attainment quantum remarkable of most history [13] the narian considered of is one and This as impotence fundamental [4-12]. has nanostructures effect metal observed and phenomena semiconductor quantum in mesoscopic The numerous apparent in becomes [2]. experiment first interference lines [3] two-slit the observed the field and for and magnetic [1] enclosed proposed any is effect cross flux Aharonov-Bohm not the do if state even particles quantum-mechanical particles on charged act of can flux formal- quantum magnetic recognized ism universally the to according ucinΨ= Ψ function ▽ ϕ n ftecneuneo h hrnvBh effect Aharonov-Bohm the of consequence the of One .Aaoo n .Bh aesoni 99[]that [1] 1959 in shown have Bohm D. and Aharonov Y. ϕ savlepootoa otecnnclmomentum canonical the to proportional value a as p = = q p u h o-oa oc-reqatmmomen- quantum force-free non-local the But . ~ = ▽ mv edbcueo h hrnvBh ffc.Ti eutcnass experiment. interference can experime two-slit the the result observed to of effect This contrast in Aharonov-Bohm a nanostructures the effect. flowing superconductor of Aharonov-Bohm current description for the dc needed the of of because evidence field unambiguous give current ϕ esrmnso h itePrsoclain tmeasurin at oscillations Little-Parks the of Measurements + | Ψ rnfripidi h Aharonov-Bohm the in implied transfer qA | exp usa cdm fSine,123 hroook,Moscow Chernogolovka, 142432 Sciences, of Academy Russian fapril ihtemass the with particle a of Introduction iϕ eut ietyfo h uni- the from directly results nttt fMcolcrnc ehooyadHg uiyM Purity High and Technology Microelectronics of Institute A ntephase the on hrnvBh ffcsi nanostructures in effects Aharonov-Bohm ..Groo,AV iuo,adVA Tulin V.A. and Nikulov, A.V. Gurtovoi, V.L. ϕ ϕ = ftewave the of m ϕ q Φ = n the and / q ~ Φ . / ~ evto ftepritn current persistent the of servation eas ftepsiiiyo cptnildifference resistance potential non-zero dc with in a halves effect of loop manifest Aharonov-Bohm possibility on more the the made of of be because case can problem the the In nanostructures it. solves ol enta tcnflwaanttefreo electric of force the against flow can field it that mean could fsg n antd ftemagnetization the of magnitude and change periodical sign The of in current [7,8,19-21,25]. equilibrium experiments circular direct many a as observed is rent s uecnutrnnsrcuecnitn ytmof radius system with We rings consisting aluminum investigation. nanostructure this superconductor for used use metal be and nature could Semiconductor paradoxical nanostructures effect. on Aharonov-Bohm information the new of experiment give with interference can two-slit together [17,18] the which of investigation investigation such the nanotechnology make of to progress allows The current. equilibrium lar eido antcfield magnetic of period 781]adted voltage dc the and [7,8,19] unu Φ quantum the ihlo perimeter loop with I a lcws rat-lcws ieto eedn on depending magnitude direction field anti-clockwise current magnetic or persistent clockwise the that has evidence unambiguous gives ihteperimeter the with opthe loop utn transition ducting amplitude tos antzto esrmnso lmnmrings aluminum section of fluctu- with measurements thermal the of ations. because observed is current sistent aervae at revealed have I eia rdcin[4.Ormaueet ftecritical the ∆ of shift measurements Our [24]. prediction retical p,A p,A is faloesol oeta h essetcur- persistent the that note should one all of First .EPRMNA VDNEO THE OF EVIDENCE EXPERIMENTAL 1. I ≈ E 1 = ESSETCRETA DIRECT A AS CURRENT PERSISTENT p ICLREULBIMCURRENT. EQUILIBRIUM CIRCULAR tlrsl bandrcnl o h case the for recently obtained result ntal ( urn uhlwrta h persistent the than lower much current g 200 = H I ) ▽ − nsmcnutr omlmtland metal normal semiconductor, in p,A nA I nA 0 ≈ anttefreo h celectric dc the of force the gainst p,A s m hta diinlfreis force additional an that ume 2 = au smc ihree bv supercon- above even higher much is value nsmcnutradnra ea loop metal normal and semiconductor in V at I 6000 = ≈ p,A fi a ecniee sadrc circu- direct a as considered be can it if T r π 700 T > T 2( ~ ≈ itit RUSSIA. District, = /q l T n 1 l c T ≈ nA aterials, − µm c nm nietelo ihtearea the with loop the inside nue yteetra current external the by induced eraig[] osntexceed not does [8], decreasing c 4 h oscillations the Φ H tradius at 7,where [7], 2 7 orsodn otetheo- the to corresponding [7] µm / H 0 V r Φ 10n-ie60-nm-thick) (110-nm-wide p h amplitude The . orsodn oteflux the to corresponding ≈ 0 siltos increasing oscillations, ) ∝ 1] nsuperconductor In [19]. 1 I µm p I r > R p 2,12]wt the with [20,21,25] . = nti case this in 0 6= > R l/ I n h per- the and 0 p 2 ( π H .Teob- The 0. ≈ ihthe with ) M 0 I . = p,A 5 µm SI V of S p , 2

because of a noise. Additional shielding has allowed us the measure R(H) at Iext = 2 nA ≪ Ip,A ≈ 100 nA, Fig.2. These results give unambiguous evidence of the dc current I = Ip − Iext/2 ≈ Ip flowing against the force of the dc electric field E = −▽ V ≈−RIext/πr because of the Aharonov-Bohm effect. The Aharonov-Bohm effect in this case can not be explained without an additional force in contrast to the result [18] obtained for the case

FIG. 1: The resistive transition R(T ) = V/Iext of a system of 110 aluminium rings connected in series with radius r ≈ 2 1 µm and half-ring sections sn = 4000 nm (200-nm-wide 2 20-nm-thick) and sn = 8000 nm (400-nm-wide 20-nm-thick) measured at different values of the measuring current Iext = 20 nA; 100 nA; 200 nA and the persistent current Ip = 0 at Φ = 0, Ip ≈ 100 nA at Φ = Φ0/2. The R(T ) shift −∆Tc ≈ 0.0025 K induced by the persistent current Ip with maximum value observed at Φ = Φ0/2 is large than the one −∆Tc ≈ 0.0015 K induced by the external current Iext = 100 nA and FIG. 2: The Little-Parks oscillations of the potential differ- smaller than −∆Tc ≈ 0.004 K induced by Iext = 200 nA. ence V = RIext measured on a system of 110 aluminium rings connected in series at low measuring current Iext = −3 nA; 0 nA; 2 nA; 3 nA and the temperature T ≈ 1.365 K and the persistent current have collaborated these results corresponding to the lower part of the resistive transition in order of value, Fig.1. R(T ). The resistance R = V/Iext oscillations in magnetic field R(H) do not depend on sign and magnitude of the mea- 2 suring current Iext and is consequence ∆R(H) ∝ Ip (H) of 2. MEASUREMENTS OF THE LITTLE-PARKS the persistent current oscillations Ip(H) with the amplitude OSCILLATIONS AT LOW MEASURING Ip,A ≈ 100 nA. The period of the oscillations H0 = 5.2 Oe CURRENT corresponds to the flux quantum inside the ring with the area S = πr2 = 4 µm2. In order to observe the Little-Parks oscillations R(H) [26] the ring resistance is found as the relation R = V/Iext of the potential difference V measured on ring halves on the two-slit interference experiment. to the measuring current Iext [25]. The dc electric filed E = −▽ V measured on both ring halves is directed from left to right or from right to left depending on the Acknowledgement Iext direction. The circular persistent current (and the total current I = Ip − Iext/2 at Iext < 2Ip ) is directed This work has been supported by a grant ”Possible ap- against the force of the dc electric field E = −▽ V in plications of new mesoscopic quantum effects for making one of the ring halves since the Ip 6= 0 is observed at of element basis of quantum computer, nanoelectronics a magnetic flux Φ 6= nΦ0 constant in time dΦ/dt = 0. and micro-system technic” of the Fundamental Research In order to observe the Little-Parks oscillations at low Program of ITCS department of RAS, the Russian Foun- measuring current Iext ≪ Ip we used a system with great dation of Basic Research grant 08-02-99042-r-ofi and a number of rings connected in series [25]. We could not grant of the Program ”Quantum Nanostructures” of the observe the R(H) oscillations at Iext < 50 nA in [25] Presidium of RAS.

[1] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). [4] Doing-In Chang et al., Nature Phys. 4, 205 (2008). [2] S. Olariu and I. I. Popescu, Rev. Mod. Phys. 57, 339 [5] F. Loder et al., Nature Phys. 4, 112 (2008). (1985) [6] R. Matsunaga et al., Phys. Rev. Lett. 101, 147404 (2008) [3] R. G. Chambers, Phys. Rev. Lett. 5, 3 (1960). [7] N. C. Koshnick et al., Science 318, 1440 (2007). 3

[8] N. A. J. M. Kleemans et al., Phys. Rev. Lett. 99, 146808 [19] B. Reulet, M. Ramin, H. Bouchiat, and D. Mailly, Phys. (2007) Rev. Lett. 75, 124 (1995); R. Deblock et al., Phys. Rev. [9] V. M. Fomin et al., Phys. Rev. B 76, 235320 (2007). Lett. 89, 206803 (2002). [10] V. L. Campo Jr et al., in Proceedings of 15th Interna- [20] V.L. Gurtovoi et al., in Proceedings of 16th International tional Symposium ”NANOSTRUCTURES: Physics and Symposium ”NANOSTRUCTURES: Physics and Tech- Technology” St Petersburg: Ioffe Institute, 2007 p. 236 nology” Vladivostok, Institute of Automation and Con- [11] O. A. Tkachenko et al., in Proceedings of 14th Interna- trol Processes, 2008, p.247. tional Symposium ”NANOSTRUCTURES: Physics and [21] S.V. Dubonos et al., in Proceedings of 15th International Technology” St Petersburg: Ioffe Institute, 2006, p. 250. Symposium ”NANOSTRUCTURES: Physics and Tech- [12] D. V. Nomokonov et al., in Proceedings of 13th Interna- nology” St Petersburg: Ioffe Institute, 2007 p. 60. tional Symposium ”NANOSTRUCTURES: Physics and [22] I. O. Kulik, Zh.Eksp.Teor.Fiz. 58, 2171 (1970); Pisma Technology” St Petersburg: Ioffe Institute, 2005, p. 197; Zh.Eksp.Teor.Fiz. 11, 407 (1970) (JETP Lett. 11, 275 O. A. Tkachenko et al., idid, p. 205 (1970)). [13] D. Kleppner and R. Jackiw, Science 289, 893 (2000) [23] F. von Oppen and E. K. Riedel, Phys. Rev.Lett. 66, 587 [14] M. Peshkin and A. Tonomura, The Aharonov-Bohm Ef- (1991). fect. Springer, New York, (1989). [24] F. von Oppen and E. K. Riedel, Phys. Rev. B 46, 3203 [15] M. Peshkin, Foun. Phys. 29, 481 (1999). (1992). [16] T. H. Boyer, Foun. Phys. 30, 893 (2000); Foun. Phys. [25] A. A. Burlakov et al., Pisma Zh.Eksp.Teor.Fiz. 86, 589 32, 41 (2002) (2007) (JETP Lett. 86, 517 (2007)). [17] A. Tonomura and F. Nori, Nature 452 298 (2008). [26] M. Tinkham, Introduction to . [18] A. Caprez, B. Barwick, and H. Batelaan, Phys. Rev. Lett. McGraw-Hill Book Company (1975). 99, 210401 (2007).