Conductance Viewed As Transmission

Conductance viewed as transmission

Yoseph Imry Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76 100, Israel

Rolf Landauer IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598

Electric current flow, in transport theory, has usually been viewed as the response to an applied electric field. Alternatively, current flow can be viewed as a consequence of the injection of carriers at contacts and their probability of reaching the other end. This approach has proven to be particularly useful for the small samples made by modern microelectronic techniques. The approach, some of its results, and related issues are described, but without an attempt to cover all the active subtopics in this field. [S0034-6861(99)00102-6]

CONTENTS essential to take the interfaces into account. Mesoscopic physics, concerned with samples that are intermediate in I. Conductance Calculated from Transmission S306 size between the atomic scale and the macroscopic one, II. Spatial Variation, Conductance Fluctuations, can now demonstrate in manufactured structures much Localization S308 of the quantum mechanics we associate with atoms and III. Electron Interactions S310 molecules. IV. Electron Coherence; Persistent Currents; Other When scattering by a randomly placed set of point Features S311 defects was under consideration, it quickly became cus- Acknowledgments S312 References S312 tomary in resistance calculations to evaluate the resistance after averaging over an ensemble of all possible defect placements. This removed the effects of I. CONDUCTANCE CALCULATED FROM TRANSMISSION quantum-mechanically coherent multiple scattering, which depends on the distance between the scatterers. Early quantum theories of electrical conduction were This approach also made the unwarranted assumption semiclassical. Electrons were accelerated according to that the variation of resistance between ensemble mem- Bloch’s theorem; this was balanced by back scattering bers was small. The approach made it impossible to ask due to phonons and lattice defects. Cross sections for about spatial variations of field and current within the scattering, and band structures, were calculated sample. Unfortunately, as a result, the very existence of quantum-mechanically, but the balancing process al- such questions, which distinguish between the ensemble lowed only for occupation probabilities, not permitting a average and the behavior of a particular sample, was totally coherent process. Also, in most instances, scatter- ignored. ers at separate locations were presumed to act incoher- Electron transport theory has typically viewed the ently. Totally quantum-mechanical theories stem from electric field as a cause and the current flow as a re- the 1950s, and have diverse sources. Particularly intense sponse. Circuit theory has had a broader approach, concern with the need for more quantum mechanical treating voltage sources and current sources on an equal approaches was manifested in Japan, and Kubo’s formu- footing. The approach to be emphasized in the following lation became the most widely accepted version. Quan- discussion is a generalization of the circuit theory alter- tum theory, as described by the Schro¨ dinger equation, is native: Transport is a result of the carrier flow incident a theory of conservative systems, and does not allow for on the sample boundaries. The voltage distribution dissipation. The Schro¨ dinger equation readily allows us within the sample results from the self-consistent pileup to calculate polarizability for atoms, molecules, or other of carriers. isolated systems that do not permit electrons to enter or The viewpoint stressed in this short note has been ex- leave. Kubo’s linear-response theory is essentially an explained in much more detail in books and review papers. tended theory of polarizability. Some supplementary We can cite only a few: Beenakker and van Houten handwaving is needed to calculate a dissipative effect (1991), Datta (1995), Ferry and Goodnick (1997), Imry such as conductance, for a sample with boundaries (1997). There are also many conference proceedings and where electrons enter and leave (Anderson, 1997). After special theme volumes related to this subject, e.g., Sohn all, no theory that ignores the interfaces of a sample to et al., 1997; Datta, 1998. the rest of its circuit can possibly calculate the resistance It seems obvious that the ease with which carriers of such a sample of limited extent. Modern microelec- penetrate through a sample should be closely related to tronics has provided the techniques for fabricating very its conductance. But this is a viewpoint that, with the small samples. These permit us to study conductance in exception of some highly specialized limiting cases, cases where the carriers have a totally quantum me- found slow acceptance. That the conductance of a single chanically coherent history within the sample, making it localized tunneling barrier, with a very small transmis-

fied in Eq. (1.2) occurs at the connections to the reservoir (Imry, 1986). Consider the left reservoir. Deep inside that reservoir there is a thermal equilibrium population. In the 1D channel only the right-moving electrons are present. The effective Fermi-level, or effective electrochemical potential, measures the level to which electrons are occupied. At point P in Fig. 1, deep FIG. 1. Two reservoirs on each side of a perfect tube, at dif- inside the left-hand reservoir, the electron distribution is ferent electrochemical potentials ␮1 and ␮2 . P is well inside that characteristic of thermal equilibrium with the Fermi reservoir 1; Q is in its entrance. level ␮1 . At point Q in Fig. 1, in the tapered part of the connection, the electron population shows some effect sion probability, is proportional to that probability was of the lowered density of electrons which have come out understood in the early 1930s (Landauer, 1994). In the of the right hand reservoir with electrochemical poten- case of a simple tunneling barrier it has always been tial ␮2 . Thus there is a potential difference between P apparent that the potential drop across the barrier is and Q. Along the ideal one-dimensional channel the localized to the immediate vicinity of the barrier and not electron population is equally controlled by both reser- distributed over a region of the order of the mean free voirs, and the electrochemical potential there must be path in the surrounding medium. The implicit accep- 1 2 (␮1ϩ␮2). Therefore the voltage drop specified by Eq. tance of a highly localized voltage drop across a single (1.2) is divided equally between the two tapered connec- barrier did not, however, readily lead to a broader ap- tors. The physics we have just discussed is essential. preciation of spatially inhomogeneous transport fields in Conductance can only be calculated after specifying the the presence of other types of scattering. The localized location where the potential is determined. The voltage voltage drop across a barrier has been demonstrated specification deep inside the reservoir and the geometri- with modern scanning tunneling microscopy (STM) cal spreading, are essential aspects of the derivation of probing methods (Briner et al., 1996). We return to spa- Eq. (1.2). Unfortunately, supposed derivations that ig- tial variations in Sec. II. nore these geometrical aspects are common in the litera- Figure 1 shows an ideal conducting channel with no ture. irregularities or scattering mechanisms along its length. If we insert an obstacle into the channel, which trans- A long perfect tube is tied to two large reservoirs via mits with probability T, the current will be reduced ac- adiabatically tapered nonreflecting connectors. Carriers cordingly, and we find approaching a reservoir pass into that reservoir with cer- 2 tainty. The reservoirs are the electronic equivalent of a Gϭ͑e /␲ប͒T. (1.3) radiative blackbody; the electrons coming out of a res- Note that it does not matter whether the T in Eq. (1.3) is ervoir are occupied according to the Fermi distribution determined by a single highly localized barrier or by a that characterizes the deep interior of that reservoir. As- more extended and complex potential profile. Expres- sume, initially, that the tube is narrow enough so that sions for the conductance with this same current, but only the lowest of the transverse eigenstates in the chan- with the potential measured within the narrow channel, nel has its energy below the Fermi level. That makes the on the two sides of the obstacle, also exist (Sec. 2.2 of channel effectively one-dimensional. Take the zero tem- Datta, 1995; Sec. 1.2.1 of Ferry and Goodnick, 1997, perature case and let the left reservoir be filled to up to Chap. 5 of Imry, 1997). If, in that case, the potential is level ␮1 , higher than that of the right-hand reservoir, averaged over a region long enough to remove interfer- ␮2 . Then in the range between ␮1 and ␮2 we have fully ence oscillations, then T in Eq. (1.3) is replaced by occupied states pouring from left to right. Thus the cur- T/(1ϪT). rent is The preceding discussion can easily be generalized to a channel that involves more than one transverse eigen- jϭϪ͑␮1Ϫ␮2͒ev͑dn/d␮͒, (1.1) state with energy below the Fermi level (Imry, 1986). In where dn/d␮ is the density of states (allowing for spin that case we utilize the transmission matrix t of the scat- degeneracy) and v is the velocity component along the tering obstacle, which specifies the transmitted wave tube at the Fermi surface. Now (␮1Ϫ␮2)ϭϪe(V1 functions relative to the incident wave, utilizing the ϪV2), where V is a voltage and e the magnitude of the transverse eigenstates of the channel as a basis. This electronic charge. Furthermore dn/d␮ϭ1/␲បv. There- yields fore the net current flow is given by Ϫ(e/␲ប)(␮1 Gϭ͑e2/␲ប͒Tr͑tt†͒. (1.4) Ϫ␮2). The resulting conductance is j In the particular case where we have N perfectly Gϭ ϭe2/␲ប. (1.2) transmitting channels this becomes V1ϪV2 GϭN͑e2/␲ប͒. (1.5) This is the conductance of an ideal one-dimensional conductor. The conditions along the uniform part of the One of the earliest and most significant experimental channel are the same; there is no potential drop there. verifications of this approach came from celebrated The potential drop associated with the resistance speci- studies of quantum point contacts (QPC). These are nar-

Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999 S308 Imry and Landauer: Conductance viewed as transmission row two-dimensional channels connecting wide reser- Equation (1.4) describes conduction as a function of voirs. The channel width can be controlled by externally quantum mechanically coherent transmission. Current applied gate voltages. As the conducting channel is wid- flow in the presence of a limited conductance is a dissi- ened, the number of transverse eigenstates below the pative process. Where are the dissipation and the irre- Fermi level increases. Conductance steps corresponding versibility (Sec. 2.3 of Datta, 1995)? They are in the res- to increasing values of N in Eq. (1.5) are clearly ob- ervoirs; carriers returning to them from the sample served. The original 1988 experiments were carried out eventually suffer inelastic collisions. These inelastic col- at Cambridge University and by a Delft-Philips collabo- lisions give the carriers, when they later again reach the ration (van Houten and Beenakker, 1996). transmissive sample, the occupation probability charac- The material of this section has been extended in teristic of the reservoir. The inelastic collisions in the many directions; we list only a few. Bu¨ ttiker (1986) de- reservoir also serve to eliminate any phase memory of scribes the widely used results when more than two res- the carrier’s earlier history. Thus the sample determines ervoirs are involved. Among a number of very diverse the size of the conductance, even though the irreversible attempts to describe ac behavior we cite only one (Bu¨ tt- process takes place elsewhere. For a narrow conductor, iker, 1993). In the ac case, however, the method of ap- attached to reservoirs which can serve as effective heat plying excitation to the sample matters. Moving the sinks, this means that the energy is released where it can Fermi level of reservoirs up and down is one possibility; easily be carried away and allows surprisingly large cur- applying an electric field through an incident electro- rents. Frank et al. (1998) pass current through a carbon magnetic field is another. The discussion of systems that nanotube, which would heat it to 20 000 K if the dissipa- consist of incoherent semiclassical scatterers occurs re- tion occurred along the tube. Such large currents and peatedly; we cite one with device relevance (Datta, As- the accompanying changes in the wave functions of the sad, and Lundstrom, in Datta (1998)). The extension to binding electrons may induce temporary atomic dis- nonvanishing temperatures is contained in many of our placements (Sec. 14 of Sorbello, 1997). broader citations. Nonlinearity has been treated repeat- edly. The correction for reservoirs of limited lateral ex- II. SPATIAL VARIATION, CONDUCTANCE tent has been described by Landauer (1989). It must be FLUCTUATIONS, LOCALIZATION stressed that the severe restrictions needed for the derivation of Eq. (1.4), i.e., the existence of ideal conducting We have already emphasized that ensemble members tubes on both sides of the sample joined smoothly to the differ and that transport fields are spatially inhomoge- reservoirs, are only conditions for that particular expres- neous. Spatial variations of current and field exist for sion. Transmission between reservoirs can be calculated two reasons. First of all geometry and preparation can under many other circumstances. Equation (1.4) has impose obvious patterns in space, as in a transistor or been applied to a wide variety of geometries. Many of scanning tunneling microscope. But a random arrange- the early experiments emphasized analogies to wave- ment of point scatterers can also provide inhomogeneity guide propagation. Transmission through cavities with with easy and hard paths through the sample. Why are classical chaotic motion has been studied extensively. spatial variations of interest? Calculating conductance Systems with superconducting interfaces and Andreev from Eq. (1.4) does not require an understanding of the reflections have been examined; see Chap. 7 of Imry spatial variations within the sample. But spatial varia- (1997). Three-dimensional narrow wires, resulting either tions are vital in other contexts. We can actually probe from an STM geometry or from mechanically pulling spatial distributions (Briner et al., 1996; Eriksson et al., wires, to or past their breaking point, have received ex- 1996). The notion, common in the middle 1980s, that tensive attention (Serena and Garcia, 1997). transport could only be examined by very invasive extra The preceding discussion assumes that we can ascribe conducting leads has been replaced by the awareness a transmission coefficient to electrons whose interac- that there is a growing set of minimally disturbing tions while in the reservoir are neglected. That does not probes, including, for example, electro-optic effects. prevent a Hartree approximation Coulomb interaction Spatial variations also matter in nonlinear transport. In along the conductor. Electron-electron interactions of that case the transport field itself is part of the field that almost any kind can exist within a sample, but that still determines transmission through the sample, and a self- permits us to discuss the transmission of uncorrelated consistent analysis invoking Poisson’s equation is electrons through that sample. needed. Datta et al. (1997), in an analysis of conduction Feeding current from reservoirs, with the carriers through an organic molecule caught in an STM configu- coming from each side characterized by a thermal equi- ration, illustrate this. Spatial variations matter in high- librium distribution, is only one possible way of driving a frequency behavior. In that case there can be capacitive sample. The exact distribution of arriving carriers, both shorting across the resistively hard parts of the sample. in real space and in momentum, matters. A sample does Spatial variations matter for electromigration (Sorbello, not really have a unique resistance, independent of the 1997). Electromigration is the motion of lattice defects way we attach to it. Wide reservoirs, connected to a nar- induced by electron transport. The moving defects rower sample, and emitting a thermal equilibrium distri- probe their local environment, not a volume average. In bution, are a good approximation to many real experi- that analysis care must be taken to include the spatial mental configurations. variation induced by that defect.

Rev. Mod. Phys., Vol. 71, No. 2, Centenary 1999 Imry and Landauer: Conductance viewed as transmission S309

We have alluded to the localized voltage drop across a Consider a one-dimensional ensemble of a fixed num- planar barrier. If, instead, we introduce a point scatterer, ber of identical localized scatterers, allowing for all pos- in the presence of a constant current flow, we can also sible relative phases between adjacent scatterers at the expect an increase in transport field related to the de- Fermi wavelength. (The total length of the chains cannot fect’s location. In fact, a planar barrier can be built from be held fixed.) This is a particularly simple case of dis- an array of point defects, and the two cases must show order. As already stated, this includes ensemble mem- related behavior. In the planar case the localized voltage bers that, over portions of the chain, simulate a forbid- drop arises from the pileup of incident electrons on one den band, resulting in an exponential decay of the wave side of the barrier and their removal from the other side, function. (The forbidden band can be associated with a allowing for self-consistent screening of these piled-up superlattice formed from the scatterers and in any case charges. A similar pileup at a point defect will generate represents only a physically suggestive way of pointing a dipole field, called the residual resistivity dipole to constructive interference in the buildup of reflec- (RRD), which has been studied for over four decades tions.) For an electron incident, say from the left, there (Sorbello, 1997; Zwerger, 1997; also see Ref. 19 of Lan- will be no regions providing a compensating exponential dauer’s introductory chapter in Serena and Garcia, increase to the right. An ensemble average of the result- 1997). A volume with incoherent point scatterers will ing resistance, rather than conductance, weights the generate a set of dipole fields, one dipole per scatterer. high-resistance ensemble members and can be shown to The resulting space-average field is that given by other increase exponentially with the number of obstacles elementary semiclassical theories; there is no new result (Sec. 5.3 of Imry, 1997). The problem of treating such a for the resistivity. We have only emphasized the strong highly dispersive ensemble (Azbel, 1983) was solved by spatial variation of the transport field. The current flow Anderson et al. (1980), who emphasized ln(1ϩgϪ1), pattern is also spatially nonuniform (Zwerger, 1997), where gϭG␲ប/e2, which behaves like a typical exten- representing the fact that the incident carrier flux, scat- sive quantity. The ensemble average of this quantity is tered by a localized defect, has to be carried around that proportional to the number of obstacles, and this quan- defect much as a current has to be carried around a tity also has a mean-squared deviation which scales lin- macroscopic cavity. early with length. The exponential decay, with length, of A review should not be confined to progress, but can transmission through a disordered array is a particularly also list questions. The spatial variation of the field in simple example of localization; electrons cannot propa- the presence of randomly placed point scatterers, pro- gate as effectively as classical diffusion would suggest. In viding coherent multiple scattering, is not understood. A two or three dimensions a carrier can detour around a set of dipole fields can still be expected, but the size of poorly transmissive region. As a result localization in each dipole can no longer depend only on the scattering higher dimensions is not as pronounced and is more action of a particular defect. The striking nonuniformity complex. of the potential drop along a coherent disordered one- Equation (1.4) tells us that the conductance can be dimensional array has been demonstrated (Maschke and considered to be a sum of contributions over the eigen- Schreiber, 1994). values of tt†. These represent, effectively, channels that An array of randomly placed point scatterers, acting transmit independently. The relative phase of what is incoherently, will allow for some variation in transmis- incident in different channels does not matter. The sion depending on the carrier’s path. In the presence of variation of transmission, which would depend on the coherence this variation is sensitive not only to density exact choice of path in a semiclassical discussion, is now fluctuations among the obstacles, but also to the exact represented by the distribution of eigenvalues of tt†. For relative phasing of scattered waves. At one extreme the a sample long enough so that conduction is controlled by random placement includes ensemble members that give many random elastic-scattering events, producing diffu- a periodic, or almost periodic, arrangement and cause sive carrier motion, but not long enough to exhibit local- the electron to see an allowed band, giving excellent ization, this distribution is bimodal (Beenakker, 1997). transmission. But the ensemble will also include mem- Most eigenvalues are very small, corresponding to chan- bers that in all or part of space simulate a forbidden gap, nels that transmit very poorly. There is, however, a clus- yielding exponentially small transmission. The relative ter of highly conducting channels, which transmit most phasing of scattered waves can be altered not only by of the current. Oakeshott and MacKinnon (1994) have changing atomic placement, but also by changing param- modeled the striking nonuniformity of current flow, eters, such as the Fermi energy or a magnetic field. The showing filamentary behavior, in a disordered block. resulting variations of conductance have been widely ob- This is a demonstration of the fact that the bimodal dis- served (Washburn and Webb, 1992; Sec. 5.2 of Ferry and tribution is related to the distribution in real space. The Goodnick, 1997) and are called universal conductance bimodal distribution, the strong variation between en- fluctuations (UCF). Relative phasing of waves, taking semble members, and the geometrical nonuniformity alternative paths, also plays a critical role in the ob- have also been persistent themes in the work of Pendry served oscillations in the conductance in a solid-state (e.g., Pendry et al., 1992), who stresses the importance of analog of the Aharonov-Bohm effect (Washburn and necklaces i.e., chains of sites that permit tunneling from Webb, 1992). one to the next, in the limit where localization matters.

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2 III. ELECTRON INTERACTIONS (EϪEF) Landau result. These changes are stronger in 1D and 2D, where the expressions obtained for the Both the thermodynamics and the transport proper- disorder-dominated e-e scattering rate diverge at nonva- ties of independent electrons propagating in a finite sys- nishing . However, physically meaningful scattering tem with random scatterers are relatively well under- rates areT finite. For example, Altshuler et al. (1982) stood. It is also well known that for the infinite found that the rate of dephasing of the relative phase homogeneous electron gas the Hartree term for the in- between two different paths is regular and goes as teractions cancels the ionic background. The remaining (with a small logarithmic prefactor) and 2/3 in 2D andT exchange-correlation contributions still play an impor- 1D respectively. These results are crucialT for numerous tant role, especially in soluble 1D models (Emery, 1979), mesoscopic interference situations and agree quantita- but a noninteracting model allows considerable progress tively with experiments at temperatures that are not too in most higher-dimensional situations. A partial justifi- low. cation for this, with modified parameters, is provided by A particular case of strong inhomogeneity occurs the Landau Fermi-liquid picture. In this description, the when the electrons are confined to a small spatial range, low-energy excitations of the interacting system are Fer- such as a lattice site or a small grain or ‘‘quantum dot’’ mion quasiparticles with a renormalized dispersion rela- (a two-dimensionally confined region of electrons). The tion and a finite lifetime due to collisions. This is valid as deviation from charge neutrality is accompanied by an long as the quasiparticle width is much smaller than its energy cost e2/2C, where C is an effective capacitance. excitation energy, which is the case for homogenous sys- This energy, when it is large compared to the thermal tems at low enough excitation energies. However, when energy, can prohibit double electronic occupancy for a strong inhomogeneities exist, a rich variety of new phe- hydrogenic impurity or exclude the electron transfer nomena opens up. Here we briefly consider the effects into or through a quantum dot. The latter phenomenon of disorder and finite size. Disorder turns out to enhance the effects of the inter- has been dubbed ‘‘the Coulomb blockade,’’ and it is rel- actions, as explained by Altshuler and Aronov (1985). evant to many experimental situations, including the op- This enhancement is not only in the Hartree term, but timistically named single-electron transistor (Chap. 4 of also in greatly modified exchange and correlation contri- Ferry and Goodnick, 1997). Most interestingly, the cor- butions. These effects become very strong for low di- relation embodied by this strong inhibition of electrons mensions or strong disorder. to populate certain locations often results in subtle and Singular behavior was found in the single-particle dramatic phenomena. Those include a Fermi-edge singu- density of states (DOS) near the Fermi energy. For dis- larity and the Kondo effect, both appearing (like the ordered 3D systems, the magnitude of this singularity Altshuler-Aronov singularities) at low energies, near the (Altshuler and Aronov, 1985) is determined by the ratio Fermi level. of Fermi wavelength to mean free path. That is small for The Fermi-edge singularity is well known from x-ray weak disorder, but increases markedly for stronger dis- absorption in metals. The attraction between the core order. The situation is much more interesting for effec- hole left by the photoexcitation and the conduction elec- tively 2D thin films and 1D narrow wires. There, when trons causes the absorption to diverge at its edge. carried to low excitation energies, these corrections di- Matveev and Larkin (1992) suggested that an analogous verge, respectively, like the log and the inverse square effect should exist for tunneling through a resonant im- root of EϪEF . Thus a more complete treatment, which purity (or a quantum dot) state in a small tunnel junc- is still lacking, is needed. These ‘‘zero-bias’’ DOS cor- tion. The hole left in that state by an electron tunneling rections should be ubiquitous. They have been observed out plays the same role as the above core hole. The experimentally and agree semiquantitatively with the interaction between this hole and the conduction elec- theory, as long as they are not too large. Direct trons increases the transmission amplitude near EF . The interaction-induced corrections to the conductivity in logarithmic divergence near threshold obtained from the the metallic regime were also predicted by Altshuler and simplest low-order perturbation theory is replaced in the Aronov and confirmed by numerous experiments. The full theory by a power-law singularity. Geim et al. (1994) 2D case, realized in semiconductor heterojunctions and observed this effect in impurity-assisted tunneling in thin metal films, is of special interest. The theory for through small resonant tunneling diodes. noninteracting electrons (Abrahams et al., 1979) pre- Another interesting near-Fermi-level structure is the dicts an insulating behavior as the temperature 0. Kondo resonance, due to the repulsion between the two The effect of electron-electron interactions is hardT→ to opposite-spin electrons on the same impurity (or quan- treat fully. The calculations by Finkelstein (1983) show a tum dot) state and their hybridization with the conduc- window of possible metallic behavior characterized by a tion electrons. Again, for a quantum dot connected to strong sensitivity to a parallel magnetic field. Several ex- two electrodes via tunnel junctions, this leads to a reso- perimental studies suggest a similar metallic behavior in nance in the zero-bias transmission. The splitting of this 2D (Lubkin, 1997), whose origin is still under debate. peak by a bias and its magnetic-field dependence were Schmid (1974) found the interaction-induced lifetime predicted by Meir et al. (1993). Very recently broadening strongly enhanced by the disorder, varying (Goldhaber-Gordon et al., 1998) this effect was ob- in 3D as (EϪE )3/2 at ϭ0 instead of the usual ballistic served with a high-quality quantum dot. F T

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IV. ELECTRON COHERENCE; PERSISTENT CURRENTS; There are now a number of experiments confirming OTHER FEATURES the existence of persistent currents in single mesoscopic rings and also in rather large ensembles of rings, as sum- It was pointed out in Sec. I that, in the absence of marized in Chap. 4, Sec. 2 of Imry (1997). In the latter inelastic scattering in an intervening sample, the conduc- case, the experiment measures the persistent current av- tance between reservoirs is determined by the elastic eraged over many rings. These rings differ through vary- scattering in that sample. In this case the ultimate source ing realizations of the random impurity potentials; an of irreversibility is in the inelastic scattering in the res- averaging over these differing realizations is effectively ervoirs. What happens if we eliminate the reservoirs and performed. The results of the single-ring experiments their inelastic scattering? We can do that by tying the agree roughly with the theory for noninteracting elec- leads to the sample to each other and creating a loop, trons, and it can be shown (see p. 75 of Imry, 1997) that considering the response of this quantum-mechanical the interactions do not change the order of magnitude of system to an external magnetic flux through the ring. the single-ring, sample-specific current. The situation is This will be done later in this section. very different for the ensemble-averaged current. The Inelastic scattering of the electron by other degrees of periodicity in the flux was experimentally found to be freedom of the ring acts like distributed coupling to ex- h/2e, rather than h/e, in accordance with the theory for ternal reservoirs. Inelastic scattering causes the electron the ensemble-averaged persistent current for noninter- waves to lose phase coherence; effects due to the interacting electrons. However, these theories underestimate ference between electron waves following alternative the magnitude of the measured current by more than paths are eliminated (Chap. 3 of Imry, 1997; Chap. 6 of two orders of magnitude. Introducing electron-electron Ferry and Goodnick, 1997). Because the eigenstates of a interactions perturbatively gives a result with the re- closed system are determined by periodicity and bound- quired period h/(2e), but still smaller than the experi- ary conditions, the energy levels of a bounded system ment by a factor of 5 to 10. This clearly goes in the right with some inelastic scattering are no longer sharp and direction, but there is still no definitive understanding of gradually lose their dependence on the periodicity con- the magnitude of the measured ensemble-averaged persistent currents. dition when the inelastic scattering increases. For our An interesting case in which persistent normal cur- ring, once the inelastic-scattering length exceeds the cir- rents may exist on the millimeter length scale is pro- cumference of the ring, the waves can respond to the vided by recent experiments on the magnetic response total set of boundary conditions. The ensuing flux sensi- in a proximity-effect system. Very-low-temperature tivity of the energy levels and their associated ‘‘persis- measurements of that response for a superconducting tent currents’’ are discussed below. The significant dif- cylinder with a normal-metal coating (Mota et al., 1994), ference between the effects of elastic and inelastic revealed an unexpected strong paramagnetic moment in scattering has been highlighted by recent research in dis- addition to the usual Meissner effect induced by the su- ordered and mesoscopic systems. Equilibrium and trans- perconductor in the proximity layer. This paramagnetic port experiments have determined the length the elec- moment is comparable to a diamagnetic moment. Whis- tron can propagate without losing phase coherence, pering gallery modes of the normal electrons, bouncing often in good agreement with theory. around the outer perimeter of the normal layer, qualita- That the ␲ electrons on a benzene-type ring molecule tively and speculatively explain this magnetic moment as have a large orbital magnetic response has been known due to unusually large normal persistent currents flow- for more than half a century. The explanation in terms ing near that surface (Bruder and Imry, 1998). of ‘‘ring currents’’ was advanced by Pauling in 1936. Or- Our subject has many more facets than we can dis- dinary metals exhibit a small but measurable orbital dia- cuss, or even list, in this short paper. We allude to only magnetism, a purely quantum-mechanical phenomenon. one subtopic. Noise measurements have developed into However, the prediction that a metallic ring structure in a surprisingly accurate probing method. Noise is more the usual mesoscopic size range can support an equilib- sensitive than the dc conductance to electron correla- rium circulating current in response to an external flux tions. Schoelkopf et al. (1997) have studied the fre- was greeted with skepticism 15 years ago. This applied quency dependence of noise in a current-carrying metal- particularly to the diffusive regime with repeated elastic lic conductor, with enough elastic scattering to give defect scattering for a carrier traversing the ring. When diffusive carrier motion, and find remarkable agreement most of the flux is through the ring’s opening, rather with the simplest independent-electron models. Rezni- than the conductor, the response is periodic in the flux, kov et al., in Datta (1998), discuss recent measurements with period ⌽0ϭh/e. This follows because such a flux ⌽ at the Weizmann Institute and by a CEA/Saclay-CNRS/ causes a phase change of 2␲⌽/⌽0 in the phase of the Bagneux collaboration, using shot-noise measurements eigenstates, upon taking an electron around the ring. to demonstrate the effective e/3 charge of the tunneling Therefore the energy levels depend periodically on the entity in a fractional-quantum-Hall-effect experiment. flux. This leads to a dependence of the thermodynamic Many alternative views of quantum transport have potential on the flux and hence, by thermodynamics, to been developed, e.g., the Keldysh formulation, adapt- an equilibrium current. Contrary to classical intuition, able to inelastic processes as treated in Chap. 8 of Datta elastic scattering alone does not cause current decay. (1995). Especially powerful is a block-scaling picture

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