Quantum Interference Fluctuations in Disordered MetalsRichard A. Webb, and Sean Washburn

Citation: Physics Today 41, 12, 46 (1988); doi: 10.1063/1.881140View online: https://doi.org/10.1063/1.881140View Table of Contents: https://physicstoday.scitation.org/toc/pto/41/12Published by the American Institute of Physics

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QUANTUM INTERFERENCEFLUCTUATIONS INDISORDERED METALS

Phase coherence over thousands of lattice spacingsin disordered metals can produce quantum interference effectsin electrical resistance measured in very small devices.

Richard A. Webb and Sean Washburn

In statistical physics one is trained to think about theproperties of large ensembles of particles, and to calculatebulk properties by averaging over many microscopicconfigurations. Although the quantum mechanical prop-erties of the individual constituents of a macroscopicobject are important over some length scale (typically afew lattice spacings), they are usually not correlatedacross the whole object. We are, however, becomingacquainted with more and more disordered systems forwhich this effective length scale, at low temperatures, canbe 100-10 000 times the characteristic microscopic scale;the correlation can involve more than 10" particles. Suchphenomena occur in an intermediate "mesoscopic" regimethat lies between the microscopic world of atomic andmolecular orbitals and the thoroughly macroscopic worldwhere averages tell all. The wealth of novel quantumcoherence phenomena recently observed in this intermedi-ate size regime is the subject of this article.

We are mainly interested here in disordered systemsin which the motion of the electrons is diffusive. Inpractice this means that the mean free path / (typicallyabout 100 A) is much smaller than the sample length, or,more properly, than the separation L between the probesthat measure electrical resistance. In the theory, and inmany experiments, I is also smaller than the width orthickness of the wires that form the sample. The meanfree path, in turn, is much larger than the Fermiwavelength AF (about 1 A) of the electrons—the de Brogliewavelength corresponding to the energy of the Fermisurface. Therefore, treating the motion of the carriers

Richard Webb and Sean Washburn are experimentalphysicists at the IBM Thomas ). Watson Research Center,Yorktown Heights, New York.

semiclassically is a good approximation to the physics ofthe electrical resistance. The electrons move through thesample as nearly plane waves, colliding elastically withthe impurities, which sit at random places in the lattice.The resulting motion is a random walk among theimpurity sites. (See the article by Boris Al'tshuler andPatrick Lee on page 36.) Thermal fluctuations, phononcollisions and other such inelastic mechanisms alter thequantum state of an electron enough to rob it of its phasememory. As the temperature approaches zero, theseprocesses are diminished; inelastic scattering becomesrare.

Our IBM colleague Rolf Landauer has emphasizedthat elastic collisions with the impurities do not destroythe phase information contained in the electron's wave-function; they merely shift the phase by some fixedamount.' All electrons incident in the same state acquirethe same phase shift, and a time-reversed collision wouldrestore the original phase to the wavefunction. Inelasticcollisions, on the other hand, destroy the phase memory ofthe electrons irrevocably.

Thus the average diffusion length between inelasticcollisions is effectively the phase coherence length L^. Intypical metals Lj, can be several microns at temperaturesbelow 1 K. Therefore, with the aid of modern circuitlithography one can nowadays make resistance measure-ments at length scales where the electron retains its phasememory—where its wavefunction matters. With suchmeasurements one sees direct evidence of the interferenceof the electron wavefunctions. Each electron can bedescribed schematically by a wavefunction Ae1*, wherethe plane-wave phase 4>(t) is boosted or retarded determin-istically by elastic collisions with impurities. The electronhas a specific probability amplitude for every possible pathas it diffuses through the sample.

4 6 PHYSICS TODAY DECEMBER 1988 © 1986 Americon Insrirure of Physics

Shutters Faraday cages

Screen

Screen

Aharonov-Bohm magnetic (a) andelectrostatic (b) effects are counterintuitiveconsequences of quantum theory. Anelectron beam, split into two alternativepaths, can exhibit interference effects whenthe beams recombine. Surprisingly, therelative phase between the two pathsdepends on potentials whose derivativeforce fields are completely shielded fromthe electrons. In a the phase difference isdetermined by the magnetic flux <t> througha long solenoid threaded between thepaths. Changing <1> alters the interferencepattern at the screen. In b the two pathsare subjected to different electrostaticpotentials, but only while the choppedbeams are shielded within Faraday cages.Nonetheless, the potentials effect theinterference. Figure 1

The phases accumulated along different paths arerarely the same; partial waves from the different pathsinterfere with randomly displaced phases. Assume for themoment that the electron has only two possible pathsthrough the conductor. Then the superposition of the twowavefunctions yields the probability distribution

\Al+A \2=A-l2+ A2

2 - 4>2)In samples where measurements are made over distancesmuch larger than the phase-coherence length, the lastterm vanishes; its averaged value is zero. On the otherhand, at low-temperature in small samples with dimen-sions on the order of the phase coherence length, the crossterm has a specific, nonvanishing value that variesrandomly throughout the sample. Along the two alterna-tive paths of such a small, cold conductor, an electronwould acquire different phases from the different impuri-ties encountered on the way to a superposition measure-ment where the paths come together again. A pair ofvoltage probes placed at the two ends could, in principle,measure the resulting interference probabilities. Anothersample of the same size and impurity concentration willhave a resistance that differs from the first sample's by atmost ± |A,A2|. The averaged resistance from a largeensemble of such samples will of course yield the classicalvalue, because the third term in the superposition willhave averaged to zero.

The Aharonov-Bohm effectIn 1959 Yakir Aharonov and David Bohm formulated agedanken experiment that called attention to a surprising,counterintuitive consequence of quantum mechanics.They noted that the electromagnetic vector potential willaffect the phase of an electron wavefunction, withobservable consequences, even when the electron's path is

restricted to regions where the electric and magnetic fieldintensities vanish. The Aharonov-Bohm effect has beenexperimentally verified by several groups with electrontrajectories in vacuum.2 Its observation with electronstraveling inside small metal devices of various geometrieshas provided striking manifestations of quantum coher-ence over mesoscopic distances in disordered solids. (SeePHYSICS TODAY, January 1986, page 17.)

The Aharonov-Bohm mechanics provides a simplemeans of altering the phases of wavefunctions. These aredepicted schematically in figure 1. In figure la theelectrons are emitted from the source, and the beam issplit so that it encircles a magnetic flux. (In the strictestsense, the magnetic flux <t> must be completely shieldedfrom the electron paths,2 but in the experiments describedhere, this nicety is never attempted.) The equivalentexperiment for the electric potential is displayed in figurelb, where a voltage is turned on while the electrons are in-side field-free Faraday cages and the voltage is turned offagain before they emerge from the cages. The predictionof quantum theory is that the phase <f> of the wavefunctionis pushed up or down by the electromagnetic potentialsaccording to

I (Vdt-Ads)

where ds and dt are the space and time elements of theelectron's path. Simply ramping the potential causes thecurrent impinging at some point on the screen to oscillateperiodically as a function of the flux or time-integratedvoltage; increasing <1> or fVdt by hie completes anAharonov-Bohm cycle.

Classically, this interference effect cannot occur,because the electron never "sees" the fields E and B, and

PHYSICS TODAY DECEMBER 1988 4 7

MAGNETIC FIELD (T)

Electrical resistance of a tiny gold ring (electron micrograph, left), as a function of imposed magnetic field,measured by the authors6 at 0.06 kelvin. The fine Aharonov-Bohm oscillations (shown expanded) correspondto a flux periodicity of hie through the area encircled by the ring. Figure 2

so it does not experience the classical Lorentz force. TheseAharonov-Bohm effects have been studied in vacuum,3

where the electrons suffer no collisions of any kind; insmall metals whiskers, where transport is deep in theballistic regime4 (again, with no collisions); and in long,thin metal cylinders, where a magnetoresistance oscilla-tion of twice the Aharonov-Bohm frequency (A<t> = h/2e)occurs near zero magnetic field because of the incipientlocalization of the electron wavefunction in the metal.2

(See the article by Al'tshuler and Lee.) In the first twocases the absence of any scattering "guarantees" theoccurrence of Aharonov-Bohm effects. The third exam-ple, first observed in 1981 by Yuri Sharvin and his son atthe Institute for Physical Problems in Moscow, providedthe original demonstration that these effects are notdestroyed by elastic scattering. The observation of coher-ence effects from these relatively long cylinders is possiblebecause the interference from time-reversed motion alongpaths around the cylinder does not average to zero as thecylinder length exceeds L^,.

The simplest circuit analog of the Aharonov-Bohmgeometry is a loop of thin wire fed by two leads. Five yearsago theorists Markus Biittiker, Yoseph Imry and Lan-dauer at IBM and Yuval Gefen and Mark Azbel at theUniversity of Tel Aviv considered the possibility ofobserving oscillations of electrical resistance as a functionof the flux threading a tiny ring of this geometry.5 Theyconcluded that in this case, as distinguished from acylinder of macroscopic length, the "magnetoresistance"oscillations would exhibit a flux period of hie—twice theperiod for cylinders.

Such a loop of gold is shown in figure 2, together withthe data obtained by measuring its magnetoresistance inour laboratory. After taking some care to filter noise awayfrom the sample, we found that the resistance as afunction of the magnetic flux through the ring exhibitsoscillations that are periodic in the magnetic field. Ourresults and those of several other groups" have demon-strated that the period of oscillation is indeed A<t> = hie,where <t> is the flux through the area enclosed by theloop—just the period Aharonov and Bohm predicted fortheir gedanken experiment in vacua. We see then, in thisresistance measurement on a disordered metal loop of 1-micron diameter, the direct signature of the interference

of electron wavefunctions.There is a second quantum mechanical contribution

to the resistance of this loop. The periodic hie oscillationsare superimposed on a randomly fluctuating backgroundresistance. As the electrons diffuse through the wires thatform the ring, their trajectories randomly enclose a fluxthat pierces the metal. Douglas Stone (now at Yale) hasshown that these random fluxes also contribute toAharonov-Bohm interference.7 This interference appearsin figure 2 as random fluctuations in the resistance as afunction of the imposed magnetic field. These fluctuationstypically exhibit a much longer oscillation scale inmagnetic-field increment than does the primary, periodicAharonov-Bohm contribution. Naively speaking, thedifference in field scale reflects the different areasenclosing flux: For the periodic Aharonov-Bohm oscilla-tions, all of the area encircled by the loop contributes tothe flux, while for the random component, only the areacovered by metal contributes. Stone pointed out that theratio of the field scales is approximately the ratio of theseareas.

The total electrical resistance R(H) of a small two-leadmetal loop threaded by a flux <I> due to an imposedmagnetic field H can be written8

R(H) = RC+Ro+Rl cos f— + a,][h/e J

+ R-2 cos — + a2[•die

Rc is the classical resistance, which includes a magnetore-sistance term proportional to B2. Empirically, one findsthati?,, and an are random functions of if with oscillationscales proportional to the area of the metal, and, roughlyspeaking, inversely proportional to n. Al'tshuler, Lee andStone and others29 have proved that the amplitudes of thevarious terms are such that the conductance (l/R) exhibitsfluctuations with rms amplitude on the order of e2lh. Thisconductance fluctuation amplitude is "universal" inseveral loose senses of the word. For instance, the factorpreceding e2lh is weakly dependent on geometry, indepen-dent of length (so long as phase coherence is maintained)and independent of the average resistance. Any metalsample exhibits conductance fluctuations of about this

4 8 PHYSICS TODAY DECEMBER 1988

Failure of Ohm's law. For a sufficiently small, cold wire, theresistance depends on the voltage drop and is not

symmetrical under reversal of the current direction.Conductance fluctuations, deviating from Ohm's law, were

measured17 as a function of current in a 0.6-ftm length of Sbwire. Note the universal e2/h conductance fluctuation and

the asymmetry under current reversal. Figure 3 Oo-20 - 10 0

CURRENT10 20

amplitude. The reasons for this universality are quitesubtle and still under debate,109 but it has been thorough-ly verified in many experiments.211 The article byAl'tshuler and Lee on page 36 discusses the theory of theuniversal conductance fluctuation in some detail.

In the equation on page 47 for the interferenceamplitude, the absolute phase change does not enter—only the relative phases of the two paths. This in turn im-plies that the interference is not necessarily destroyed asthe magnetic field increases to rather strong values, wherea significant flux pierces the actual metal. The increasingmagnetic field intensity does not alter the character of theoscillations until the field reaches classically strong valueswhere Landau cyclotron orbits form, destroying thediffusive motion of the electrons.12 Figure 2 shows thatthe oscillations persist to rather large magnetic fieldwithout any attenuation whatsoever. They have, in fact,been observed at field intensities above 15 tesla withoutattenuation. Landau cyclotron orbits will not form in ourgold rings until one gets to about 200 tesla.

The quantum domainHow does a quantum system evolve into the more familiarclassical system? Experiments on these small phase-coherent systems have already provided a partial answer.Temperature, sample size, diffusion coefficient and detailsof how the measurement is made play a crucial role in de-termining whether or not quantum interference effectscontribute significantly to the electrical resistance in adisordered system. Phase randomization due to elasticscattering and thermal effects have historically beenthought to be the most devastating killers of coherence.But the gold-loop and cylinder experiments have demon-strated that elastic scattering is not destructive.

The question of thermal averaging arises becauseeven a device as small as the gold ring in figure 2 contains108 atoms; the average separation AE between its electronenergy levels is about 10 ~8 eV, which corresponds to atemperature of 10~4 K. When the thermal energy exceedssome characteristic energy of the system, the delicatequantum effects begin to disappear. One might naivelyexpect that the oscillations will decay like exp( - AE/kT).The surprising discoveries from the resistance fluctu-ations are that the spacing between energy levels is not thecharacteristic energy scale, and that the decay is muchslower. As Imry and others have pointed out, the energyscale that governs the physics is related to the Thouless en-ergy2.7.8

F *°if

determined from the coherence length and D, the diffusioncoefficient for elastic scattering. Once kTexceeds Ec, themagnitude of the quantum corrections to the resistance

decays slowly,378 like

For our gold ring, Ec Ik is only about 0.03 K, but because ofthe weak algebraic averaging, the quantum interferenceeffects are readily observable above 1 K. In semiconduc-tors, where EJk can be as high as 10 K, quantuminterference effects have been reported above 100 K.

This weak algebraic averaging of the quantuminterference also appears in the size dependence, whosecharacteristic length scale is Lt. When the length of thesample is increased beyond L^, the resistance fluctuationsgrow like (L/L^f2, but the average resistance growslinearly with L. Thus the relative magnitude of thequantum interference contributions decay as the square-root of the number of uncorrelated length segments. For asingle ring larger than L^, the loss of phase coherencereduces the oscillations severely; the oscillations decreaseexponentially with increasing path length L. The reasonis simply14 that the probability for an electron to reach theterminus with its phase coherence intact decreases likeexp( - L/Lj,).

Because all of these averaging mechanisms dependupon the coherence length, the temperature dependenceof L^ determines the observability of quantum interfer-ence effects. As the temperature increases, the probabili-ty that an electron will absorb and re-emit a phononincreases. For narrow wires at low temperature, it turnsout that Lt has only a weak power-law (nonexponential)dependence on temperature.

There are other ways to destroy the phase coherenceof electron wave functions; the dominant one at lowtemperatures is paramagnetic scattering.15 Because thephase shift acquired by scattering from an impurity with afluctuating spin depends on the instantaneous direction ofthat spin, the electron will in general acquire a time-dependent phase shift, and this will destroy the interfer-ence effects. In experiments where a low concentration ofparamagnetic impurities is adsorbed onto the surface of agold ring, the hie oscillations disappear at low fields. Butas the field is increased beyond some critical value, the fullamplitude of the resistance fluctuations is recovered. Thereason for this striking recovery is that when the magneticfield energy becomes larger than the thermal energy, theparamagnetic spins align with the field; classically theycease to fluctuate. Above this field intensity, everyelectron will acquire the same phase shift from theimpurity, which now acts like a stationary elastic scat-terer.

One obvious way to destroy quantum interferenceeffects in a disordered conductor is to pass too muchcurrent through it. Joule heating will raise the tempera-

PHYSICS TODAY DECEMBER 1988 4 9

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Electrostatic Aharonov-Bohm effect measured atIBM with an Sb loop 0.8 jam on a side, withcapacitor gates along two arms. (See micrograph a.)Plot of resistance change vs magnetic field (b) showsthat gate voltage (labeled) can shift oscillation phasesreproducibly. Wider field sweeps (c) show gradualchange of magnetoresistance pattern with increasingcapacitor voltage. Each trace is measured at 0.2volts higher than the one below. Figure 4

-0 .87 -0 .86 -0 .85

MAGNETIC FIELD (T)

ture. Perhaps not so apparent is the observation thatcurrents too small to cause significant Joule heating cannonetheless average away the interference terms. If thevoltage drop across a phase-coherent segment exceedsEc /e, there will be an averaging over uncorrelated energyregions, and the magnitude of the quantum interferenceeffects will be reduced by (Ec /eV)u2.

The current-voltage curveOhm's law fails when transport is phase coherent.Transmission of current across a random potential de-pends erratically on the average (net) slope of thepotential.16 Ohm's law, which prescribes a linear current-voltage curve, describes only the average behavior of thewire. Suppose a small conductor with a random impuritypotential is biased between two reservoirs of differingchemical potential—the terminals of a battery. This willsimply tilt the random potential distribution. Not only isthe effective resistance of such a biased impurity potentiala random function of the bias voltage; one cannot evenassume that R(I), the resistance as a function of current, isequal to R( — I). The resistance is not symmetric underexchange of the battery terminals because the potentialhas no mirror symmetry; a realistic three-dimensionalwire is not symmetric under inversion.

This nonlinearity in the I-V curve has been mea-sured. '7 A representative example of non-ohmic behaviorfound in an antimony wire at low temperatures, is shownin figure 3, where the deviation from Ohm's law is plotted.One sees that quantum interference causes the conduc-tance to fluctuate randomly (and reproducibly) through-out the range of current studied. Near zero current, thefluctuations are sharp, with rms amplitude e2/h. As thedrive current increases, the fluctuation amplitude de-creases slowly, and the current scale for a fluctuation

grows. The dependence of the amplitude and current scaleon the drive depends in a complicated way on certainaveraging processes and heating of the electrons by thecurrent, but the fluctuations are qualitatively in reasona-ble agreement with the theory.16 The startling predictionthat R(I) does not equal R ( — 1) is also borne out; there isno correlation between the fluctuations at positive andnegative currents.

As the drive current tends to zero, ohmic behaviorshould be recovered asymptotically—but only for currentsyielding voltage drops much smaller than kT/e. At atemperature of 0.01 K, one reaches the ohmic regime onlywhen the voltage drop across the sample is much less thana microvolt.

Electrostatic Aharonov-Bohm effectsThe electrical circuit analog of figure lb has been studiedto test the effects of a scalar potential on quantuminterference in disordered systems. The experimentaldevice, shown in figure 4, is a loop with capacitor probesalong its arms. A static voltage applied to the capacitorprobes will contribute an accumulated phase increment ofelfi$ Vdt to the electrons in the arms of the loop. Figure 4bshows that the phase of the hie oscillations from this loopcan be shifted by about n when the voltage applied to thecapacitor plates is changed by 0.75 V; but the amplitude ofthe oscillations is unaffected. Thus, at any constantmagnetic field, one can reproducibly change the outputvoltage from the ring by applying a potential to thecapacitor probes. One can, in fact, use the voltage to shiftthe position of the oscillation smoothly along the magneticfield axis.18 This is shown in figure 4c, which containsseveral magnetoresistance traces recorded with differentvoltages on the probes. Each trace is recorded at a voltage0.2 V higher than the trace below it. As one begins to in-

50 PHYSICS TODAY DECEMBER, 1968

crease the probe voltage, the oscillations shift towardsnegative field; but then, after the fourth trace, they shiftback toward positive field.

This is precisely what one expects if a global phasechange is added to all of the electrons in an arm of the loop.We expect that the time integral accumulating phase fromthe electric potential is cut off by the mean time betweeninelastic collisions. Using this time scale, and accountingnaively for the screening of the potential by the metal inthe loop, we find that the expected voltage required to shiftthe oscillations by v is on the order of one or two volts, inreasonable agreement with the observation.

A voltage on the capacitor plates affects the randomfluctuations in a similiar way. The net result of phasechanges caused by the capacitor voltage is to make Rn andan in the general equation for R(H) random functions ofthis voltage as well as of the magnetic field. In figure 4 wesee that the background resistance changes shape slightlyeach time voltage changes. The characteristic voltagerequired to cause fluctuations is about twice that needed toshift the oscillations. This is reminiscent of the situationfor the corresponding magnetic-field scales, which alsodiffer by a factor of about 2.

We are not studying strictly orthodox Aharonov-Bohm effects, because we make no effort to shield thecurrent from the electric or magnetic field. This meansthat there are also classical Lorentz forces at work on theelectrons in the loop. The electric field tends to push theelectrons toward or away from the edge of the wire. (Thesesame electrons screen the field.) Different paths amongthe impurities experience different potentials, causing theinterference pattern to change. Just how gradually itchanges with capacitor voltage remains an open question.Shifting so much as a single impurity atom19 by more thana Fermi wavelength will substantially affect the conduc-tance: it will cause a change of order e2/h. In this spirit webelieve that changing the electron paths within a screen-ing length of the wire surface by as little as a Fermiwavelength might also substantially alter the resistancefluctuation pattern.

Nonlocal quantum interferenceLong-range phase coherence in disordered systems hashad several unexpected experimental consequences. Oneof the most surprising is that the rms amplitude of the vol-tage fluctuations measured as a function of magnetic fieldin a four-terminal sample becomes independent oflength20 when the separation L between the voltageprobes is less than L6. The inset to figure 5 schematicallydisplays a four-wire sample configuration. Current isinjected at lead 1 and removed by lead 4, and the voltagedifference is measured between leads 2 and 3. Classically,the average resistance V23/Ii4 depends linearly on L. Theaverage voltage difference between any two points alongthe sample is given by L times the electric field. Thequantum-interference contributions to the voltage fluctu-ations with changing magnetic field, however, behave verydifferently. Figure 5 shows the results of a recentexperiment on an antimony wire, which has eight probeswith separations L ranging from 0.2 to 3.6 microns. Therms values of the voltage fluctuations are plotted as afunction of L. The length scale has been normalized to the11 [im correlation length, L,t.

The striking feature of these data is that voltagefluctuations do not vanish as L goes to zero. The quantuminterference contributions to the measured voltages re-main constant for all probe separations up to thecoherence length. For separations larger than a micron,the voltage fluctuations grow like (L/L,,,)"2, as expected.

This length independence comes about because the

Length dependence of voltage fluctuationamplitude. Inset is a schematic of a four-leadsample. Length L between the voltage probes(2 and 3) is scaled to L^, the coherencelength, and voltage is normalized to IR4,

2e2/h,where Rt is the resistance of a length Z.̂ and /is the current. Figure 5

voltage probes do not define the length scale for themeasurement.21 The effective length of the sample isreally determined by the properties of electrons them-selves. The electron is a wave whose phase coherenceextends over a distance L^. Therefore the electron existsin classically forbidden regions. In particular, the wavescan propagate coherently over a distance L^ into thecurrent and voltage probes. The application of a magneticfield alters the interference over the entire phase-coherentpath.

The microscopic details of the scattering impuritiesare different in each section of the sample and its leads.Because the electrons propagate into the probes, a finite"nonlocal" voltage is measured between leads 3 and 4when the current flows from lead 1 to lead 2—even thoughleads 3 and 4 are attached to the same point on theclassical current path. Recent theoretical analysis21'22 isconsistent with the general shape of the data curve infigure 5. This length independence clearly demonstratesthat there is a nonlocal relationship between electric fieldand current density in the sample due to quantuminterference over a distance of about 1 micron.

Another counterintuitive discovery was that themagnetoresistance R(H) measured in any four-terminalarrangement on a small wire does not remain the samewhen the magnetic field is reversed.23 Classically, ofcourse, R(H) should equal R( — H). Figure 6a dramatical-ly illustrates the asymmetric behavior found in a smallgold wire, which is typical of all four-probe measurementsmade on small wires and rings. R(—H) bears littleresemblance to R{ + H). The second trace in figure 6a isthe magnetoresistance obtained by interchanging thecurrent and voltage leads. At first glance, it seems to bearlittle resemblance to the upper curve. Upon closerinspection, however, one sees that the positive-field half ofthe second trace is very similar to the negative-field half ofthe first trace. All the other permutations of leads yielddifferent, asymmetric patterns of magnetoresistance fluc-tuation. These patterns are amazingly constant over time.After days of (gently) switching probes and currents, onecan return to the original configuration and find a patternof fluctuations that correlates with the original pattern tobetter than 95% (dashed curve with first trace in figure6a).

None of these observations violates any fundamentalsymmetries. In fact, as Biittiker has shown, the generalproperties displayed in figure 6a are consistent with theprinciple of reciprocity, which requires that the electrical

PHYSICS TODAY DECEMBER, 1988 51

- 4MAGNETIC FIELD (T)

resistance in a given measurement configuration andmagnetic field H be equal to the resistance at field — Hwhen the current and voltage leads are interchanged.23

= R231A-H)where

The Onsager symmetry relations for the local conductivitytensor may be more familiar, but they describe the localrelationship between current density and electric field.As we have seen, conductance is a nonlocal quantity, andtherefore the appropriate symmetries to consider arethose involving the total resistance R.

To demonstrate reciprocity in the data, one deter-mines the symmetric and antisymmetric parts of theresistance fluctuations.

RS(H) = 1/2\R14.23(H) + R2314(H)\RA{H)=\\Rl4,23(H)-R.13M(H)}

Figure 6b shows this decomposition of the data. To withinthe noise level of the experiment, Rs is perfectly symmet-ric with respect to magnetic field direction and i?A isperfectly antisymmetric. Applying this procedure toRl324 andi?24,i3 results in a similar Rs but a different RA.RA is not simply due to classical Hall resistance. Theantisymmetric part fluctuates randomly with an ampli-tude similar to the symmetric fluctuations. Once againthe conductance fluctuation is of order e2/h. The ampli-tude of the antisymmetric component is nearly constant,independent of L/L^, because it arises from nonlocaleffects occurring within a distance L^ of the junctions ofthe voltage probes. Excursions of electrons into thevoltage probes account for all of the fluctuations in RA .

Asymmetry under field reversal, a: Measurements on asmall gold wire23 show that conductance fluctuations changewhen magnetic field is reversed. These fluctuations arereproducible after days of measurement (dashed curve attop). Decomposing these measurements into symmetric (Cs)and antisymmetric (CA) parts, one sees (b) that thesecomponents do indeed exhibit the expected symmetry underfield reversal. Figure 6

Perhaps the clearest experimental evidence of nonlo-cal resistance is shown in figure 7. Two identical four-probe wires were fabricated at IBM, but on the secondsample a small ring was attached 0.2 fxm from the classicalcurrent path. Figure 7 shows the conductance of bothwires. Both exhibit random conductance fluctuationswith the same characteristic field scale, but the secondsample shows additional high-frequency "noise." TheFourier transforms of both data sets clearly show that thishigh-frequency noise is in fact an hie oscillation with thecharacteristic flux period equal to what is expected fromthe area of the ring.25 For this effect to be observable,some large fraction of the electrons must have encircledthe ring coherently.

The magnitude of the hie oscillations decays exponen-tially with distance from the classical current path, butthe characteristic length scale of this decay is not thegeometric dimension, but rather the quantum coherencelength Lj, .24 Thus the quantum interference measured inany small sample will have nonlocal contributions, andthe distance between voltage probes will not define thesample length.

The futureWe have tried to review some of the recent and unexpectedexperimental consequences of long-range phase coherencein disordered systems. Although many of the observationswere unexpected, they make sense in hindsight once weremember that the electron is a wave. Many of the aspectsof quantum transport we measure are similar to thephysics of electromagnetic radiation in waveguides andlight interference in disordered media. The concept of acoherence length thousands of times longer than thewavelength is familiar in these fields.

Long-range phase coherence in disordered systemsmay make it possible to observe Bloch oscillations andpersistent currents. Our study of these systems hasalready begun to teach us what it means to make ameasurement on a quantum system. Further progresswill come from extending the measurements to smallersample sizes and higher frequencies. The trend inteachnology has been to make smaller devices, pack themcloser together, and operate them at lower temperaturesand lower voltages. In the not-too-distant future theengineers will have to allow for quantum interferenceeffects in their efforts to optimize these circuits.

References1. R. Landauer, Philos. Mag. 21, 863 (1970).2. See the reviews by A. G. Aronov, Yu. V. Sharvin, Rev. Mod.

Phys. 59, 755 (1987); Y. Imry, in Directions in CondensedMatter Physics, G. Grinstein, E. Mazenko, eds., World Scien-tific, Singapore (1986); S. Washburn, R. A. Webb, Adv. Phys.35, 375 (1986).

3. A. Tonomura, N. Osakabe, T. Matsuda, T. Kawasaki, J. Endo,

5 2 PHY5IC5 TODAY DECEMBER 1968

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MAGNETIC FIELD (T)

- 1.4 - 1.2

0.02

tro

0.01

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Magnetoresistance of two similar wires(schematics on top), one of which has aring dangling outside the classical path,where it "should" have no effect. Infact the Fourier transforms of theconductance curves (bottom) show anadditional peak for the circuit with thering, indicating h/e Aharonov-Bohmoscillations corresponding to the areaenclosed by the ring. Figure 7

S. Yano, H. Yamada, Phys. Rev. Lett. 56, 792 (1986).4. N. B. Brandt, D. B. Gitsu, V. A. Dolma, Ya. G. Ponomarev,

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