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ICE PAS demonstrate the haronov-Bohm effect_ Part of an electron beam passes through a toroidal mag

net (black ring at top) coated with niobium The other part ofthe beam passes outside the toroid Together the beams causean interference pattern (colored region at top). The background

interference pattern outside the ring results from interferenceamong electrons that do notgo tough the toroid The inter

ference pattern framed by the toroid is shifted with respect tothe backround even though the electrons were selded fromthe magnetic eld The shielding occurred because the niobi-

um was cooled below 9.1 degrees Kelvin and became superconducting. The shift, as predicted by Yar haronov and

David Bohm, is the result of interaction among the electronwves and the vector potential, which is present even in theabsence of the magnetic el When the niobium coating is

heated above 9.1 degrees K (bottom), it ceases to be superconducting, the magnetic eld contained within the toroidal mag

net changes and the interference patte shifts abruptly (col-ored region at bottom) Akira Tonomura and his colleaguescarried out ts experiment in 1986 at Hitac Ltd in Toko

SCIENIFIC AERIC Arl 1989 57

1989 SCIENTIFIC AMERICAN, INC

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other characteristics of the wave areconvenently described by a mathematical wave function. Consider, forinstance, an ocean wave the height ofwhich varies from one meter abovethe average surface to one meter below it and back. The wave can be described by a cosine function, sincethe value of the cosine changes from

+1 to -1 and back to +1 as its anglechanges from 0 to 180 to 360 degrees. The angle that corresponds tothe instantaneous height is called thephase angle.

The mathematical wave functionthat describes an electron wave is represented in terms of its mamumamplitude and phase angle. The amplitude of an electron wave describesa probabiliy, which is related to thefact that the position and velocity of aparticle can be determined to withinoy a certain degree of precision. Speically, the square of the mamumamplitude of the electrons wave function is the probability of nding theelectron at a particular location at aparticular time.

The phase angle of an electronswave function is especially useful forescribing the relation between twowaves. If two waves are "in phase ata particular location or time, the twowaves are in the same part of theircycle both have reached mamum orminimum amplitude. If two wavs are

"completely out of phase, one wavehas reached a mamum while theother is at a minimum. The phaseangle of an electron wave can also beeressed in terms of more intuitivephysical quantities. simple casesthe phase is related to the momentummultiplied by the distance the elec

tron wave has traveled and also to theenergy multiplied by the time.

These concepts provide an adequate explanation for the pattern yelded by the twoslit experiment. Since the particle generatoremits electrons having the same ener

gy and momentum, the electron wave

functions have the same phase at agiven distance from the generatoracondition known as coherence. s anelectron wave penetrates the two slits,it divides into partial waves. Since partial waves travel the same distance toeach slit, the partial wave emergingfrom the lefthand slit has the same

phase as the partial wave emergingfrom the righthand one. Thus at a

point on the lm that is equidistantfrom the two slits, the left and right

partial waves ll be in phase. Hencethe waves reinforce each other and

produce a bright band in the middle ofthe lm. It is also fair to say that thebright bands represent the fact thatelectrons have two times as muchchance of striking there as they do atan average point on the lm.

To the left of the bright band, however, the right partial wave must travela greater distance than the left partial

wave. Consequently, at some points tothe left, the two waves will be completely out of phase and wll canceleach other. Hence a dark band will

form because electrons have almostno chance of striking there). t a pointstill farther to the left on the lm, theight wave travels such a distance thatit is exactly one full cycle bend thelefthand wave. Once again the wavesare in phase and create another brighthigh probabili) band.

NIO UM

TORO MGNT and niobium f employed in the experiments of Tonomura aredepicted in the photograph at the left and in the illustration at the right. The magnet,

which is ve microns across, consists of an alloy containing 8 percent nckel and percent i silicon diode coating insulates the magnet from the obium.

58 April 8

order to obsee the haronovBohm effect, the twoslit interferenceexperiment must be altered slightly.Directly bend the plate and in between the slits is placed a very longsolenoid that has a magnetic eld inside it and absoluey no electric or

magnetic el outside it. en a beamof electrons now penetrates the wo

slits and goes around the solenoi, thelm records a new interference pattern. Compared wth the original patern, the new patte has shifted sothat preously bright regions will appear arker and dark regions appear righter. When the magnetic eldcontained in the solenoid is removedfrom the experiment, the interferencepattern retus to its original form.

ths new interference eerimentthe phases of te left and right partial waves apparently changed eventhough the magnetic eld was completely conned insie the solenoid.The change in phase of an electronwave function in a region where nomagnetic eld ests is one manfestation of the haronovBohm eect.

The eect reveaed that the phasechange of a wave function mustbe related to some physical entit present outside a conne magneic eld. haronov and Bohm derivefrom the fundamena equatios oquantum mechanics that the phase

change is due to an entiy hat estsanywhere in and around a mageiceld called the magnetic vector potential. lthough the vector potentia is avector eld in the sense that it has amagnitude an a irection at everpoint in space and can change withtime, the vector potential can be measured directly ony by obserng changes in phase of wave functions. Thephase shfts caused by the vector potential can account for all measurablemagnetic eects on charge paricles.

How did the vector potential act onthe phase of an electron in he twoslitexperiment? s he left and right partial waves traveled in the forcefreeregion near the solenoid, the vectorpotential changed the momentum ofthe left partial wave wit respect tothe right partial wave thout changing the netic energ. Since the phaseof a wave function is related to its

momentum, the left parial wavechanges phase in relation to the right

partial wave.The magnetic vector potential and

the haronovBohm eect have counterparts in electric interactions. Theyare the electric scalar potential andthe electrostatic haronovBohm effect. The electric scaar potential is not


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the hole exactly mismatched the reference, that is, the dark bands of oneested alongside the brght bands ofthe other. Whichever the case, thisgave a unique verication of the roleof the vector potental in changng the

phases of electron waves in a regionwhere no magnetic eld ests seeillustration on page ]

en electrons travel through avacuum, the haronovBohmeect can be obseed be

cause the phase of the electron wave

BEM

SOE

function remains well dened as thewave splits and interferes. The eects harder to observe in solids, becauseelectrons scatter o various imperfectons in the crystal lattice.

though every solid exhibits someform of scattering, techniques havebeen developed over the past decadethat reduce scattering to the point

where electrons travel much as theydo through a vacuum [see "Ballisticlectrons in Semiconductors, by ordehai Heiblum and Lester F. astman; , Februar, 1987]

ELECTROSTATIC Aharonov-Bohm eect can be obseed by splitting an electronbeam and directing it toward two hollow, metalic cylinders that shied electronsfrom eectric forces. electrons pass through the cylinders, a scalar potential difference (voltage) is applied between the cyiders. The interference pattern ob

seed on the screen is shifted by an amount directly related to the scalar potential.

LNGSLENID

-AGNETIFIELD

LNG_WIE

ETPTENTIAL

MAGNETIFIELD

tENT

VECTOR PO ( ) is compared with the magnetic eld (b )for a ong soenoid () and a long wire (g) Each line represents its respective ed at a given strength. The circulation of the vector potential ed arounda cue is equal to the magnetic eld multiplied by the area bound by that cure.

60 April 989

It was, however, the appreciation oftwo tpes of scattering in solid conductorsinelastic and elasticthatled to the rst discoveries of the haronovBohm eect and other quantuminterference eects in solid materials.

Inelastic scattering occurs when atoms that make up a solid conductorexchange energy with the electron.

Strictly speang, inelastic scatteringalters the wave functions of the atomsmaking up the solid, that is, scatternginduces a change in the quantum stateof the environment in whch the electron moves. For instance, the electroncan absorb energy from or give energyto the brations of atoms in a crystallattice. One key to reducing inelasticscattering is to limt the energy available for such nteractions. If enoughenergy is removed from the cstallattice and the electron system so that

they are essentially quiescent, inelastic scattering will be scarce. The wayone can remove this energy is to coolthe wre to low temperatures. t quiteattainable temperatures of a few degrees Kelvin, electrons in many metalscan move across several thousand atoms a distance of appromately onemicron) without undergoing inelasticscattering.

Cooling a solid conductor to lowtemperatures has another benecialeect. The range of energies withwch electrons travel through a solid

decreases as the temperature decreases. t temperatures low enough tomake inelastic scattering improbable,the range of energies is so narrowlydened that all electrons travelingthrough the re have eectively thesame energy. This makes all conducting electrons in he olid interfere inessentially the same way.

lastic scattering takes place whenan electron encounters a static potential, such as an impurity or a defectin the crystal lattice. static poten

tal changes the phase of the electronwave function in a welldened manner but not its total energy. lthough arandom distribution of static potentials in a solid lead to a randomchange in phase, the change ll be thesame for every electron that travelsthrough the solid at a particular energy. s temperature approaches absolute zero, it turns out that an electronwave should encounter only elasticscattering, which leads to a randombut constant phase change and doesnot obscure electroninterference ef

fects in a solid conductor. That wasthe key to obserng quantuminterference eects n solids.

In real experimental systems, however, solid conductors cooled to low


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temperatures still ehibit some degree of inelastic scattering that wllintroduce some' uncertanty in thephase of the electron wave function.As the size of the solid conductordecreases, the number of phase-randomzing events decreases. To obsee quantum interference the conductor must be suciently small to

essentially eliminate inelastic scattering. perments have sho that, although a metal wire 03 micron thck,.03 mcron wide and one mcron longcontains neary 100 milon atoms, thephase of an electron wave functontraveling through the wire wil typically be mantaned at temperatures below one degree K

In order o measure electron-interference eects n sold conductors,one must translate the mechanics

of electron waves into physcal quantities that can be measured easily. Wenan electron wave travels through asmal wire a low temperatures, part ofthe wave scatters from one end to theother whle other pars scatter back totheir pont of orgin. A measure of thediculty an eectron wave has traveling from one end of a wire to the otheris electrical resstance; conversey, ameasure of the ease with which thewave functon moves is the wires conductance. More than 2 years ago Rolfandauer of the M Thomas . Watson

Research Center in Yorktown Heights,N.Y., deveoped a theoretical frame

work expressing he conductance interms of the probability that an electron wave wll be transmitted throughthe wire. His work shows that theconductance is appromately proportional to the transmission probabiltydivided by a fundamental quantum

ut of resstance: 2,812 ohms. Thisvalue s equal to Plancks constantdded by the charge of an electronsquared.

One factor hat contributes to transmission probabilty and conductanceis wave-function interference. MarkusBttker, Landauer and one of us(mry) did theoretical work on metalic rings withou leads, which demonstrated that elastic scatterng did notdestroy quanum-interference eects.Then Yuval Gefen, Mark Ya. Azbel andone of us (Imy) predcted in 1984that, as a resul of the haronov-Bohmeect, the elecrical resistance of ametal ring would oscillate periodcallyas a magnetc eld appled o the cen

ter of the rng vared smoothly. Wenthe electron wave functions travelingin two dieren sections of the ringrenforce each other, the transmssionprobablty and us the conductance

should increase. en the electronwave functions cancel, the ransmission probability and the conductanceshould decrease. Hence the conductance or resistance of a wre shouldoscillate between these two extremes.In 1981Boris L. ltshuler, Arkady Aronov and Boris Spivak of the LenngradInstitute of Nuclear Physics made a

related predicton, and Yur V. Sharvinand his son at the Institute for Physica Problems n Moscow conrmed texpermentaly.

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One of us (Webb), working wih SeanWashburn, Cowin . Umbach and Robert B. aibotz of the Thomas . Watson Research Center rst demonstrated the haronov-Bohm eect in smallmetallic rings in 198 The group fabricated a gold ring on a silicon wafer.The ring had an inside diameter o.78 mcron and an outside diameter

of .86 micron. A current was appliedthrough an input lead attached to oneside of the loop and collected at anoutput lead on the opposite side of

CURRENT

020 025

AGNETI FIELD (TESLA)

G measures the haronovBohm eect in solid conductors p) Electron wavesenter from the left and scatter through the ring, which has been cooled to lowtemperatures vector potential eld due to a magnetic eld ) shifts thephase of the electron wave function and changes the ring's electrical resistance,

which is determined by measuring the voltage and the current. The haronovBohmeect accounts for the oscillation in the electrical resistance of the ring b)

Ap1989 61 1989 SCIENTIFIC AMERICAN, INC

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SWG D can be based on the electrostatic Aharonov-Bohm eect. antimony loop 8 micron on a side is lanked by two bars. By applying a potentialdierence (voltage) to either of the bars or to both, the wave functions of the elec

trons that travel throug the loop cange phase, so that the output voltage is altered.

the rng [see iusri precedigpge] Addtonal wres were attachednear he loop to each current lead tomeasure the voltage drop across therng The voltage dvded y the current elded the resstance of the rng

A magnetc eld appled perpendcularly created a magnetc vector potental that crculated n the plane of thesample

The workers oserved that the electrcal resstance of the rng oscllated prodcally as the magnetc eldncreased Ths agreed wth what sown aout the haronovBohm effect and potentals Eecron wavesthat traveled around the gold rng na clocse drecton nterfered wththe electron waves that traveled n the

opposte drecton As the magnetceld and the vecor potental) wasncreased the waves travelng clockwse shfted phase n relaton to thewaves travelng countercocse Asthe phase was shfted through a fulcycle y the vector potental the resstance of the rng luctuated. The average perod of oscllaton n termsof the magnetc eld was teslaThs quantty multpled y the average area enclosed y the rng yeldsa fundamenal quantummechancalvalue equal o Plancks constant dvd

ed y the charge of an electron aspredcted theoretcally

The magntude of he resstance oscllaton n ths case was qute small:aout percent of the total resstance

pri 989

of the rng Danel E Proer of YaleUnversty Supryo Data of PurdueUnversty and ther colleagues quckly conrmed the resstance oscllatons n other metals and n semconductors More recently experments

y numerous other groups have demonstrated osclatons as large as percent of the total resstance Furthermore oscllatons n the conductance of these samples are ndependent of the aerage resstance and areroughy equal to the charge of an electron squared dvded y Plancks constant Such unversaly that s resstance oscllatons ndependent of themateral and ts eastcscaterng mpurtes) was rst predcted y tshuler and shortly hereafter y Pat

rck A Lee of the Massachuses nsttute of Technology and A DouglasStone of Yale Unversty

The oservaton of the aronovBohm eect has opened up an entre new eld of research n whchthe quantum nature of he electronmovng n a sold can e studed nthe doman that les etween atomsand macroscopc ojects Such "mesoscopc systems whch are much larger than an atom or a molecule can emanpulated and measured y macroscopc means ye hey stll pla

y the rules of the game of mcroscopc physcs These systems dreclydsplay the unusual eects of quantum mechancs n for example ordnary electrca measurements s as

though one coud measure the resstance of the electrons ortng anaom These systems ll help to answer fundamental questons such ashow large a system must e to ehavemacroscopcally

nterestng n ther own rght theharonovBohm eect and quan

tum nterference may play a partcularly mportant role n the future ofelectroncs Snce the dscovery of thetransstor the dmensons of electronc deces have steadly decreased tothe pont where fewer than atoms make up the wdth of a re Atthe same tme the power per unt areathat computer chps dsspate n theform of heat has ncreased Unless

new deces are developed that perform relaly and consume less powera lmt on the numer of components

per chp wl e reached Ultmatelyths would lmt the operatng speedof electronc devces

Recent research on quantumnterference eects ndcates that newelectronc devces that dsspate extremely small amounts of power coulde developed expermental protote of one such devce has alreadyeen tested n a lowtemperature envronent The devce controls resstance and voltage y employng a potental to manpulate the wave characterstcs of an electron n the near

future as the sze of electronc components contnues to decrease devces could e constructed that mantan the quantummechancal ehav

or of electrons at much hgher temperatures We nd t remarkale thatthe haronovBohm eect and otherquantumnterference eects whchdeveloped from the astract foundatons of quanm mechan havefound ther way to experments ondotoearth samples

DIGSIGNFICNCE OF ECTROAGNETIC o-

TENL N QUANU HEORY. Y Aharonov and D. Bohm in e Physical Review, Vol. 115, No3, pags 485-491;August, 1959.

SINGLE VUEDNE OF WAVE UNCTON.E. Mrzbachr in American Joual ofPhysics, Vol 3, No. 4, pags 237247;April, 1962.

HE HYC OF OCOPIC SYTE

Yosph mr in Directions in CondensedMatter Physics, ditd by G. Grinstinand E. Maznko World Scintifc Publisng Co, 1986

QUANU NTERERENC UCTUATIONN ORDERED ETAL. ichard A.Wbb and San Washburn in PhysicsToday, Vol. 41, No. 12, pags 46-53;Dcmbr, 1988


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