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Landauer and inhomogeneous systems lution Although major advances in the understanding of d − 1 1 1 2 σeff = σ1 p1 − + σ2 p2 − electrical conductivity of disordered and heterogeneous d d ! d ! media were made by a number of his contemporaries, 1 1 2 Landauer’s approach was unique in this field: On one + σ1 p1 − + σ2 p2 − " d ! d !# hand, people like William Fuller Brown, Jr. [9] or Zvi  1/2 Hashin and Shmuel Shtrikman [10] confined themselves  d − 1 + 4 σ1σ2 . (2) to a discussion of systems where a classical physics d2 ) approach is valid, and described the local electrical response in terms of a position dependent conductiv- This approximation is still widely used, especially in ity. This lead, eventually, to concepts like percolation the context of a disordered microstructure, and is gen- threshold, which determines the macroscopic response erally known as the self-consistent (or symmetric) ef- of a metal/insulator mixture [11]. On the other hand, fective medium approximation (SEMA). This approx- people like Philip W. Anderson [12] and Neville F. Mott imation had actually already been discovered in 1935 [13, 14] focussed upon the effects of microscopic dis- by D. A. G. Bruggeman [21], who was then a high order on the detailed form of the quantum mechanical school teacher in the Netherlands. Landauer did not wave function. This lead, eventually, to concepts like know about this, (nor did the reviewer of his manuscript Anderson localization and Mott transition as governing at J. Appl. Phys.) and he therefore achieved this break- the macroscopic response of such a system. By con- through independently. This approximation can be con- trast, Landauer often tried to combine quantum consid- trasted with an earlier approximation, known as the erations with classical physics considerations. This is Clausius-Mossotti (CM) or Maxwell Garnett approxi- clearly evident in his work on electromigration [15, 16], mation. In the latter approximation, σeff satisfies a lin- but also in his classic paper which derived the famous ear algebraic equation which can be expressed in the “Landauer Formula” [17], where he used the classical following concise form: physics Einstein relation in order to derive the macro- σeff − σ2 σ1 − σ2 scopic conductivity of a one dimensional disordered = p1 . (3) σ + (d − 1)σ σ + (d − 1)σ system from the diffusion coefficient of a single elec- eff 2 1 2 tron. A similar combination of quantum and classical The CM result is non-symmetric in the two constituents: approaches can also be found in Landauer’s work on The σ2 constituent plays the role of host while the σ1 conductivity of cold-worked metals [18] and on Lorentz constituent plays the role of inclusions. It is easy to corrections to electrical conductivity [19]. generalize the CM result to any number of different in- clusion constituents that are embedded in one common host constituent. This is achieved by rewriting σ2 as 3. Classical inhomogeneous systems σhost, σ1, p1 as σi, pi, and summing the right hand side of the resulting equation over the different types of in- Rolf Landauer became interested in inhomogeneous clusions i. This leads to an equation that is still equiva- systems early on in his career. In 1952 he published a lent to a linear algebraic equation for σeff. paper entitled “The electrical resistance of binary metal- In contrast with CM, the SEMA result is symmetricin lic mixtures” [20], where he developed a simple ap- all of the constituents. When SEMA is extendedto more proximation for calculating that resistance, which is a than two constituents, the result for σeff then becomes macroscopic property of such mixtures. This approx- the solution of a polynomial equation with order equal imation yields the following equation for the macro- to the number of constituents. scopic scalar conductivity σeff of a multi-constituent, d- Interestingly, neither Bruggeman nor Landauer re- dimensional, isotropic composite medium in terms of alized, at first, that SEMA predicts the existence of 2 a conductivity threshold in the case where one of the These failures of SEMA are related to the fact that it constituents is a perfect insulator. This crucial conse- is an uncontrolled approximation which cannot be im- quence, which does not follow from the CM approxi- proved in any systematic fashion: SEMA is based on a mation, was first appreciated by others [22]. Somewhat simple, intuitive physical idea, namely, that when try- later it was realized that the conductivity threshold in ing to calculate the electric field and current in and near this system is associated with a percolation threshold a single spherical inclusion with conductivity σ1 or σ2 of the conducting constituent [23], which is a geomet- one can replace the rest of the heterogeneous system by ric property of the microstructure. This threshold is a a fictitious, uniform host with σeff as its uniform con- critical point, i.e., a singular point in the physical re- ductivity. The value of this initially unknown macro- sponse of the system as functionof the physical parame- scopic or “bulk effective” conductivity σeff is then found ters [11]. This point is characterized by the “percolation by imposing the self-consistency requirement that the threshold” pc = 1/d: When the volume fraction of the dipole current source, which is excited when an external conducting constituent pM is greater than pc, the macro- uniform electric field is imposed on any isolated spher- scopic conductivity σeff is nonzero, but it vanishes when ical inclusion in this fictitious uniform host, yields zero pM ≤ pc. For pM ≥ pc, σeff increases linearly with in- when summed over all the different inclusions in the creasing pM, starting from 0: system. While this approximation becomes exact when σ → σ or when the system is dilute, i.e., when ei- σ d 1 1 1 2 eff = p − , p ≥ p ≡ , (4) ther p ≪ 1 or p ≪ 1, it is impossible to estimate the − M M c 1 2 σM d 1 d ! d error when neither of these conditions is satisfied. The where σM is the scalar conductivity of the conducting failure of SEMA in predicting correct values for criti- constituent. There is thus no discontinuity in the func- cal exponents, like t, and its inability to include relevant tion σeff(pM), only a discontinuous slope at pM = pc. details of the microstructure, as in the case of the above Thevalue of pc = 1/d predicted by SEMA depends only mentioned swiss cheese model, have lead many scien- on the dimensionality. That and the linear dependence tists to abandon this approximation. Instead, they chose of σeff upon pM are leading characteristics of SEMA. to use techniques like renormalization group transfor- In practice, both characteristics are rather inaccurate: mation or brute force simulation of discrete, random Experiments on real continuum composites show that network models in order to study the macroscopic re- the value of pc depends on details of the microstructure sponse near a percolation threshold. (A review of the [24]. Only in the case of a two-dimensional (d = 2) different approaches to calculations of the macroscopic disordered composite medium where the microstruc- conductivity of a composite medium, including cases ture is symmetric in the two constituents is the value where the system is near a percolation threshold, can pc = 0.5, predicted by SEMA, correct. The linear form be found in Ref. [25].) However, extension of SEMA of σeff(pM), predicted by SEMA for pM ≥ pc, is also to the case where a strong magnetic field is applied to usually contradicted by experiments on real compos- a macroscopic mixture of two conductors with different ites and discrete network models, although the SEMA but comparable resistivities [28]–[36], or to a mixture prediction that σeff(pM) is continuous at pc is verified. of three constituents where one is a normal conductor In reality, the behavior of σeff(pM) for small positive while the other two are a perfect insulator and a per- values of pM − pc is well described by a power law fect conductor [37]–[39], have resulted in the discovery t σeff(pM) ∝ σM (pM − pc) , where the “critical exponent” of some new critical points which are unrelated to the t has values that depend on general properties of the mi- geometric percolation threshold. This demonstrates a crostructure but not on minute details—that is known as great advantage of SEMA and its extensions: Because “universality”. For example, in discrete network mod- they lead to closed form expressions for the elements of els with finite-range-correlated randomness, it is found the macroscopic resistivity tensor, or at least to closed that t ≈ 1.3 when d = 2, t ≈ 2.0 when d = 3, t = 0.5 form (though complicated) coupled equations which de- when d ≥ 6 [25]. In continuum composites, the value of termine those elements, it is much easier to identify a t also sometimes depends on details of the microstruc- mathematical singularity in those moduli, which signals ture, e.g., in the cases of “swiss cheese” and “inverse the existence of a critical point. swiss cheese” models of a conductor/insulator mixture The research by Landauer described above was done [26], and in the case of a singular distribution of conduc- while he worked at the NACA Lewis Laboratory, but by tances in a random resistor network [27]. In any case, the time Ref. [20] appeared in print, he had moved to in contrast with the SEMA prediction, t is never equal IBM. Shortly afterwards, Landauer got interested in the to 1. problem of magneto-transport in a macroscopically het- 3 turned out to exhibit surprising new features. While the exact results found by Landauer et al. in Refs. [40, 41] are valid for arbitrary field strength, other results have been found more recently which are asymptotically ex- act only in the strong field limit, i.e., when the Hall- to-Ohmic resistivity ratio in at least one constituent is much greater than 1 [49]–[52]. In 1957, Landauer wrote a pioneering paper on elec- tronic transport due to localized scatterers in a metal. In this article he pointed out that, besides the scattering of individual electrons by the impurity potential, another Figure 2: Landauer’s derivation of SEMA. This figure from Ref. [20] ff was captioned, “The shaded crystal of type 1 is surrounded by crystals important e ect must also be taken into consideration, of both types, which are imagined to be replaced by a single medium namely, the inhomogeneity of electron density induced of uniform conductivity.” Landauer made the further approximation of by that potential. That effect had previously been ig- replacing the shaded crystal by a sphere. Reprinted with permission nored. This article, which appeared in the first volume from The American Institute of Physics. of the IBM Journal of Research and Development [15], was not properly appreciated until it transpired that the erogeneous or composite medium. In a one page article, insights developed in it are extremely relevant for un- published in J. Appl. Phys. in 1956 [40] shortly after derstanding the phenomenon of electromigration—see the appearance of a 1955 IBM Technical Report [41], Ref. [53] for a detailed list of relevant references on this Landauer and two collaborators derived some basic re- topic. Because the availability of Ref. [15] was so lim- sults for the macroscopic Hall effect and electrical con- ited, the unusual step was taken of re-publishing it as ductivity of an inhomogeneous system constructed of a an article in the Journal of Mathematical Physics nearly conducting material that has a uniform local Hall resis- 40 years after the original publication—see Ref. [54]. (M) (M) tivity ρH and a uniform Ohmic resistivity ρ , except This is just one example of how far ahead of most other for the presence of non-conducting pores. (As shown physicists Landauer was in his scientific thinking and in Ref. [32], it is not even necessary for ρ(M) to be a insight: Until others caught up with his 1957 results, scalar quantity.) In particular, Landauer et al. obtained the original work had almost vanished into oblivion. exact results for the case where the microstructure has The phenomenon of electromigration was actually a cylindrical symmetry, i.e., when the microstructure de- major interest of Landauerfor much of his life. The sub- pends on only two out of three Cartesian coordinates: ject was of great practical importance to IBM, as a prin- If the magnetic field lies along the axis of cylindrical cipal mechanism for the failure of integrated circuits is (eff) symmetry, then the macroscopic Hall resistivity ρH is deterioration caused by electromigration of defects and the same as the Hall resistivity of the conducting con- impurities near junctions in the circuit. Landauer’s fo- (M) stituent ρH . If the magnetic field is perpendicular to cus was on the microscopic understanding of forces re- that axis, then sponsible for the motion of defects. An important early contribution to this field was the paper by Landauer and ρ(M) (eff) H Woo, “Driving force in electromigration”, published in ρH = , (5) 1 − pM 1974 [55]. The central idea of this paper was that the where pM is the volume fraction of the conducting con- inhomogeneity in the electron density near a defect or stituent. These early exact results were later extended impurity carries with it a change in the local conduc- by other groups who studied magneto-transport near a tivity. When an electric current is applied, this leads to percolation threshold [42]–[44]. Other studies involved formation of electric dipoles, which can exert a force on discrete network models which enabled the critical ex- the defect, in addition to forces resulting from the direct ponent for the weak (magnetic) field Hall effect of a transfer of momentum from an electron to the impurity percolating system to be evaluated with acceptable pre- during a scattering process. The issue confronting Lan- cision [45, 46], and scaling theories that discussed con- dauer and Woo was how to properly take this force into nections among different aspects of the critical behavior account. Landauer wrote a number of subsequent pa- [47, 48]. By extending SEMA to the case where the pers on the driving forces for electromigration, which local resistivity is no longer a scalar quantity, new criti- we will not have room to summarize here. However, the cal points were found, as already described above. The interested reader can find a review of Landauer’s contri- case of strong (magnetic) field magneto-transport has butions to the subject in an article by R. Sorbello, enti- 4 tled “Landauer fields in electron transport and electro- system, which he found to be migration”, published in 1998 [56]. e2 T When the first ETOPIM conference was convened, in G = . (6) Columbus, OH during 7–9 September 1977 [57], Rolf h 1 − T Landauer was asked to deliver the opening keynote ad- The formula is for spinless non-interacting electrons, in dress. In that talk, he presented an exhaustive review of the limit of zero temperature. The transmission prob- the development of theoretical treatments for the phys- ability is, in turn, related to the complex transmission ical properties of a composite medium up to that time. amplitude t, by T = |t|2. The article which summarizes that talk in the conference A decade after Landauer’s formulation, an alternative proceedings volume [58] is an invaluable historical re- relation between conductance and transmission proba- view, which also lists and discusses all the important bility was proposed by Economou and Soukoulis [60] articles in that field which were known at that time— and others. (See the discussion in Ref. [61], particularly altogether 163 references. pages 93–103.) For spinless electrons in one dimension, this relation is simply e2 4. Quantum systems Γ= T (7) h This formula was also generalized to the case where At an early stage of his career, Landauer became in- there can be several transverse channels for electrons in terested in the study of systems of non-interacting elec- the wires connected to the sample. In this case we have trons in a one-dimensional disordered potential, which [62] could be studied analytically or numerically with the computers of the time, and could shed light on more re- e2 Γ= |t |2 , (8) alistic three-dimensional systems which were then not h i j Xi j tractable. The work of Landauer and Helland, in 1953, was a pioneering work in this area [59]. However, where i and j label the channels in the left and right the most influential paper that Landauer wrote based leads respectively, and ti j is the matrix of transmission on the analysis of one-dimensional systems was his amplitudes. 1970 paper, “Electrical resistance of disordered one- For some time, there was much discussion about dimensional lattices” [17]. The 1970 paper was impor- which of the two quantities, G or Γ, is the “correct” tant because of its contribution to our understanding of definition of the electrical conductance. We now under- the phenomenon of localization in one-dimensional sys- stand that they are, in some sense, both correct, but refer tems, but even more significantly, it established a con- to different experiments [61, 63]. The quantity Γ should nection between electrical conductance and transmis- be thoughtof as a two-terminal conductance. If the sam- sion probabilities, that has been the basis for much fu- ple is connected by ideal wires to two large reservoirs, ture work on mesoscopic systems, often referred to as in equilibrium at voltages V1 and V2, and I is the current the Landauer formalism. through the sample, then What Landauer did in the 1970 paper was to study I statistical properties of the transmission matrix through Γ= . (9) V − V a one-dimensional region with a sequence of partially 1 2 reflecting barriers randomly spaced. As Landauer In contrast, G may be thought of as a four terminal con- noted, if the disordered region (let us call it the “sam- ductance. If one could attach an ideal voltage probe to ple”) is connected on either end by smooth wires to the leadson either side of the sample, which would mea- reservoirs at different chemical potentials, there will be sure the voltages V3 and V4 in the leads without drawing a net current through the sample determined by the po- any current from the leads and without disturbing them tential difference of the reservoirs and the transmission in any way, we would have probability T for an electron with energy close to the I Fermi energy, incident on the sample from either side. G = . (10) V − V (It is a consequence of the principle of detailed bal- 3 4 ance that the transmission probability will be the same Unfortunately, it is not entirely clear how one could whether the electron is incident from left or right.) Lan- construct an ideal voltage probe that would not disturb dauer used this result to define a conductance for the a mesoscopic system [61]. In fact, since the electrons 5 within the leads are not in thermal equilibrium, there his formula, Eq. (5) of Ref. [17], for the inverse of the may be some question how one should properly define conductance G of a sample consisting of two barriers in a voltage in the leads. Landauer had in mind that the series: voltage would be defined by the total density of left and e2 1 − T right moving electrons, as well as by the electrostatic = = hG T potential, which should be determined self-consistently. (1 − T ) + (1 − T ) + 2(1 − T )1/2(1 − T )1/2 cos φ On the other hand, it has proved relatively easy to fabri- 1 2 1 2 . cate mesoscopic systems with good connections to ex- T1T2 ternal reservoirs of known voltage, so the two-terminal (11) conductance Γ has proved to be an extremely useful Here T is the transmission of the system as a whole, T1 concept. Despite the difference between Γ and the con- and T2 are the transmissions of the individual barriers, ductance G that Landauer originally introduced, Lan- and the phase φ depends on the distance between barri- dauer deserves a great deal of credit for introducing the ers. That phase arises from the interference of contribu- idea that the conductance should be determined by the tions in which the particle is reflected multiple times by transmission probabilities. the barriers before finally emerging from one side or the In Landauer’s original paper, and much of the sub- other of the system. Formulas for three or more barri- sequent work, analysis was restricted to non-interacting ers can be obtained by iteration, adding one barrier at a electrons, or models where the Coulomb interaction is time. introduced only in the form of a self-consistent poten- To introduce the effects of disorder, Landauer consid- tial. However, we understand that the analysis is also ered a model of N barriers having identical individual applicable for interacting electron systems, provided transmission probabilities T1, but with random spacings that the temperature is low and the system sufficiently among them. In particular, Landauer assumed that the small so that electrons that enter the sample will leave phases φ between successive barriers could be treated it before suffering an inelastic collision. Since the time as independent random variables, uniformly distributed for inelastic collisions increases as the temperature is re- from 0 to 2π. This can be strictly justified when the duced, studies of these phenomena are generally carried variation in the distance between barriers is large on the out at very low temperature. scale of the wavelength of the electrons, but the final re- Landauer’s 1970 paper had importance separate from sults are actually much more general. Landauer showed the general question of conductance through meso- that the mean value of the resistance 1/G for his model scopic systems. The paper shed very important light is given by the formula on the issue of electron localization in one-dimensional e2 1 2 − T N systems. In previous work, by Mott and Twose, by Bor- = 1 − 1 . (12) land, and by others, it had been established that for non- *hG + 2  T1 !    interacting electrons in a disordered potential in one di- It follows that the mean value of the resistance will di- mension, in the limit of an infinite wire, the electron verge exponentially with the length of the system, un- eigenstates would all be localized, except for a possi- less T1 = 1, i.e., unless there is perfect transmission for ble set of measure zero [64, 65, 66]. What this meant the individual barriers. was that for each eigenstate, there would be a point on When the resistance of the sample is very large, it the line where the magnitude had a maximum, and on does not matter whether one considers the four-terminal either side of that point, the wave function would de- resistance 1/G or the two-terminal resistance 1/Γ. The crease exponentially, with a decay length that depended mean values of both quantities will diverge, at the same on the energy and the strength of the disorder, but would exponential rate, as N becomes large. By contrast, the remain finite in the limit of an infinite system. As a re- mean values of G and Γ are quite different. Though typ- sult of this, it was argued that the resistance of a long ical values of G will be exponentially small, as expected one-dimensional disordered system of non-interacting from the large value of hG−1i, the mean value of G, for electrons should increase exponentially with the length a sample of specified length N, will actually be infinite, L, in contrast to a classical wire, where the resistance as Landauer noted in his paper: is linear in L. Landauer was able to explain the expo- T nentially diverging resistance in terms of transmission hGi = = ∞ . (13) amplitudes and quantum mechanical interference in the 1 − T  wire. The reason for this can be seen by inspection of Eq. To understand Landauer’s argument, let us consider (11). The formula implies that when ∆T ≡ T1 − T2 6 and ∆φ ≡ φ − π go to zero, the value of G will di- verge as [(∆T)2 + (∆φ)2]−1. Let us divide our sequence of N barriers into two roughly equal halves, let T1 and T2 be the separate transmission probabilities of the two halves, and let φ be the phase accumulation in the space between the two halves. As the probability density for ∆T and ∆φ will, in general, be finite when the two vari- ables go to zero, the mean value of G will diverge loga- rithmically.

4.1. Applications An important application of the Landauer-type ap- proach to conductance resulted from the dramatic ex- perimental discovery in 1988 of quantized conductance steps in semiconductor devices with a narrow constric- tion [67, 68, 69]. In these devices, fabricated from two- dimensional electron systems in GaAs, the width of the constriction could be varied continuously by applying a negative bias to a pair of gate electrodes on the surface of the sample—see lower panel of Fig. 3. As the width of the constriction was varied, the conductance was not a linear function of the gate voltage, but was seen to exhibit a series of plateaus, with values Γ = 2Ne2/h , where N is a positive integer—see upper panel of Fig. 3. These observations could be understood using Lan- dauer’s ideas, if we assume that for electrons in a given channel of transverse motion, as soon as the constric- tion is wide enough to permit transmission of electrons at the Fermi energy, the transmission probability T is very close to unity. If the constriction is too narrow, then transmission in the given channel will be close to zero. Thus, the conductance will be close to an inte- ger times e2/h. The factor 2 appears because of the degeneracy due to electron spin. The sudden increase in T from complete reflection to complete transmission is very plausible in these systems because the control- ling gates are set back from the two-dimensional elec- Figure 3: Conductance steps in a two-dimensional electron system tron gas by a distance large compared to the Fermi wave with a constriction of variable width. Upper panel shows measured length. Thus the potential felt by the electrons should be conductances of two constrictions, QPC1 and QPC2, versus the volt- very smooth, and the the transmission problem reduces age applied to gate electrodes that control the widths of the con- strictions. Lower panel is a schematic of the experimental system. to the semiclassical problem of a particle incident upon Reprinted from Ref. [69] with permission from The American Insti- a barrier, where the transmission probability is either 0 tute of Physics. or 1, depending on whether the particle has enough en- ergy to get over the barrier. In the years since 1988, an enormous number of experimental and theoretical in- vestigations have been built on these experiments and their interpretation. Landauer’s ideas were also important in the under- standing of the resistance oscillations of a microscopic metal ring attached to two leads, as a function of the magnetic flux threading the ring. The key theoretical pa- per here was the work of B¨utikker, Imry, Landauer and 7 Pinhas, in 1985 [70]. This work was, in turn, closely to address fundamental issues of the nature of dissipa- tied to experiments carried out at IBM at that time [71], tion in small closed loops. which studied the magnetoresistance of a thin gold ring, Landauer’s approach to conductance was also the ba- approximately 800 nm in diameter, with a thickness of sis for important work on shot noise in mesoscopic sys- approximately 40 nm. Upon varying the magnetic field, tems. The formula for the current noise-power, per unit the experiments found oscillations in the resistance cor- frequency, in the limit of zero temperature, is given by responding to a fundamental period of the addition of [80, 81, 82, 83] one quantum of magnetic flux, Φ = h/e, through the 0 = 2/ − , hole in the ring. Fourier transform of the data showed S 2(e h)V Ti(1 Ti) (14) Xi strong peaks at frequencies corresponding to this fun- damental, and also at the first harmonic, corresponding where V is the applied voltage and Ti is the transmission to addition of half a flux quantum. As explained by probability for the ith channel. [Here, we have made a B¨uttiker et al. [70], the fundamental frequency in the unitary transformation on the channels in the two leads transmission amplitude arises from the quantum inter- so that the transmission matrix ti j is diagonal, and the 2 ference between paths in which an electron may travel conductance formula (8) becomes Γ = (e /h) i Ti.] from one contact to the other along either side of the The noise formula (14) is very widely used, andP has ring. The phase of this interference term depends on been the basis for much subsequent work. Landauer’s the precise location of scattering centers in the sam- own views on shot noise may be found in his article ple, and would be expected to vary randomly from one “Mesoscopic Noise: Common Sense View”, published sample to another. Thus oscillations at the fundamental in 1996 [4]. (See also the 1991 article by Landauer and period would be expected to vanish, or be greatly re- Martin [84].) duced, in an experiment where the signal was averaged Another interest of Landauer, related to mesoscopic over many different rings, However, oscillations corre- systems, was the concept of transit time in tunneling sponding to one-half flux quantum would not vanish on events. The reader is referred to Refs. [85, 86] for Lan- averaging, and thus would dominate an averaged mea- dauer’s views on this subject. surement. This analysis was consistent with the results of previous experiments and on multiple rings, and on 5. Biographical summary tubular samples, which may be thought of as many rings in parallel [72, 73, 74]. Rolf Landauer was born in Stuttgart, Germany, in Another geometry that interested Landauer was the 1927. He moved to the United States, with his family, in case of an isolated metal ring, with no electrical con- 1938, several years after the death of his father in 1935. tacts. Classically, the conductance of a closed metal ring His father, who had fought for Germany in World War I, can be measured by placing it in a time-varying exter- and had been severely wounded, was very patriotic, and nal magnetic field and measuring the magnetic moment did not want to leave the country. He strongly believed induced in the ring. The induced moment would be pro- that the Nazi antisemitism would pass. Landauer has portional to the current flowing around the ring, which said that were it not for the early death of his father, due in turn would be proportional to the time-derivative of in part to problems resulting from his war wounds, his the flux and the conductance of the ring. For a meso- family would have undoubtedly remained in Germany scopic wire at low temperatures, where the discrete until it was too late to leave. quantization of electronic levels becomes important, the Landauer’s family settled in New York City, where situation is more complicated. In this case, there can he went to high school, before entering Harvard Col- be a non-zero “persistent current”, in equilibrium in a lege at the age of 16. He graduated in two years, and dc magnetic field, which will be an oscillatory func- enlisted in the U. S. Navy, where he claimed to have tion of the flux through the loop. Such persistent cur- learned as much as he had learned in college. Eventu- rents are well known in superconductors, but they also ally, he returned to Harvard for graduate studies, where occur (with much smaller magnitudes) in normal metal he received his Ph.D. in 1950. After graduate school, loops. Landauer and his collaborators wrote a number Landauer worked for two years at the Lewis Labora- of important papers in the 1980s which discussed both tory of the National Advisory Committee for Aeronau- the existence and magnitude of persistent currents in a tics (NACA, later to be renamed NASA, acronym for dc magnetic field, and the behavior to be expected in a the National Aeronautics and Space Administration). In time-varying magnetic field [75, 76, 77, 78, 79]. Lan- 1952, he moved to IBM Laboratories in Poughkeepsie, dauer’s analysis of the latter problem also allowed him NY, (later to be renamed the IBM Thomas J. Watson 8 Research Center), where he continued to work until he [31] B. Ya. Balagurov, Fiz. Tverd. Tela 28, 3012 (1986) passed away in 1999. [Sov. Phys. Solid State 28, 1694 (1986)]. [32] D. J. Bergman, Y. M. Strelniker, Phys. Rev. B 60, 13016 (1999). At IBM, in addition to his research work, Landauer [33] D. J. Bergman, D. G. Stroud, Phys. Rev. B 62, 6603 (2000). held important management posts at various times. He [34] V. Guttal, D. Stroud, Phys. Rev. B 73, 085202 (2006). was responsible for much of IBM’s early work on large [35] R. Magier, D. J. Bergman, Phys. Rev. B 74, 094423 (2006). scale integration, and has been given credit for invent- [36] R. Magier, D. J. Bergman, Phys. Rev. B 77, 144406 (2008). [37] D. J. Bergman, Y. M. Strelniker, Phys. Rev. B 62, 14313 (2000). ing the term. Landauer was awarded the title of “IBM [38] D. J. Bergman, Phys. Rev. B 62, 13820 (2000). Fellow” in 1969. [39] D. J. Bergman, Phys. Rev. B 64, 024412 (2001). [40] H. J. Juretschke, R. Landauer, J. A. Swanson, J. Appl. Phys. 27, 838 (1956). Acknowledgments [41] R. Landauer, J. A. Swanson, IBM Technical Report (1955). [42] J. P. Straley, J. Phys. C 13, 4335 (1980). [43] D. J. Bergman, in Annals of the Israel Physical Society, vol. Research of BIH was supported in part by the Na- 5 Percolation Structures and Processes, eds. G. Deutscher, R. tional Science Foundation (grant DMR-09-06475). Re- Zallen, and J. Adler, pp. 297 (1983). search of DJB was supported by the Israel Science [44] D. Stroud, D.J. Bergman, Phys. Rev. B 30, 447 (1984). [45] D. J. Bergman, Y. Kantor, D. Stroud, I. Webman, Phys. Rev. Foundation (grant No. 585/06). Letters 50, 1512 (1983). [46] D. J. Bergman, E. Duering, M. Murat, J. Stat. Phys. 58, 1 (1990). [47] D. J. Bergman and D. Stroud, Phys. Rev. B (Rap. Comm.) 32, References 6097 (1985). [48] D. J. Bergman, Phil. Mag. B 56, 983 (1987). [1] G. Stix, Scientific American, September 1998, p. 32 [49] D. J. Bergman, Y. M. Strelniker, Phys. Rev. Lett. 80, 3356 [2] R. Landauer, Nature 365, 692 (1993). (1998). [3] R. Landauer, Philos. Trans. R. Soc. Lond. A 353, 367-376 [50] D. J. Bergman, Y. M. Strelniker, Phys. Rev. B 59, 2180 (1999). [51] D. J. Bergman, Y. M. Strelniker, Europhys. Lett. 45, 605 (1999). (1995). [52] D. J. Bergman, Y. M. Strelniker, R. Magier, this volume of Proc. [4] R. Landauer, Physica B 227, 156-160 (1996). [5] R. Landauer, Proceedings Workshop on Physics and Compu- ETOPIM8. tation, PhysComp’94. Dallas, Nov. 17-20, 1994 (IEEE Comp. [53] R. S. Sorbello, Solid State Physics 51, 147 (1997). Soc. Press, Los Alamitos, 1994, pp. 54-59). [54] R. Landauer, J. Math. Phys. 37, 5259 (1996); this is, essentially, [6] R. Landauer, In: Nanowires, eds. P.A. Serena and N. Garcia, a reprint of Ref. [15], the availability of which was too limited. (Kluwer Academic Publishers, Dordrecht, 1997) pp. 1-7. [55] R. Landauer and J.W.F. Woo, Phys. Rev. 10, 1266 (1974). [7] R. Landauer, Phys. Lett. A, Invited Comments Section 217, 188- [56] R. S. Sorbello, Superlattices and Microstructures 23, 711 (1998). 193 (1996). [8] R. 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Sciences Ad Hoc Committee on the Fundamentals of Amor- [69] J. R. Gao, B. J. van Wees, J. J. Kuipers, J. P. Heida, and T. M. phous Semiconductors (29 January 1971). Klapwijk, Appl. Phys. Lett. 64, 2529 (1994). [24] M. B. Heaney, Phys. Rev. B 17, 12477 (1995). [70] M. B¨uttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B 31, 6207 (1985). [Reprinted in Mesoscopic Physics, eds. A. [25] D. J. Bergman, D. Stroud, Solid State Physics 45, 147 (1992). Kawabata and K. Kawamura, The Physical Society of Japan, [26] B. I. Halperin, S. Feng, P. N. Sen, Phys. Rev. Lett. 54, 2391 (1985). Tokyo (1994), pp. 57-65.] [27] P. M. Kogut, J. P. Straley, J. Phys. C 12, 2151 (1979). [71] R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, [28] H. Stachowiak, Physica (Amsterdam) 45, 481 (1970). Phys. Rev. Lett. 54 2696 (1985). [29] M. H. Cohen, J. Jortner, Phys. Rev. Lett. 30, 696 (1973). [72] D. Yu. Sharvin and Yu. V. Sharvin, JETP Lett. 34, 272 (1981). [30] D. Stroud, Phys. Rev. B 12, 3368 (1975). [73] B. L. Al’tshuler et al., JETP Lett. 35, 588 (1982). 9 [74] B. Pannetier, J. Chaussy, R. Rammal and P. Gandit, Phys. Rev. Letters 53, 718 (1984). [75] M. B¨uttiker, Y. Imry and R. Landauer , Phys. Rev. 96A, 365 (1983). [76] R. Landauer and M. B¨uttiker, Phys. Rev. Lett. 54, 2049 (1985). [77] R. Landauer, Phys. Rev. B 33, 6497 (1986). [78] R. Landauer, Phys. Rev. Lett. 58, 2150 (1987). [79] R. Landauer, Phys. Rev. B 68, 217 (1987). [80] V. A. Khlus, Sov. Phys. JETP 66, 1243 (1987). [81] G. B. Lesovik, JETP Lett. 49, 594 (1989). [82] M. B¨uttiker, Phys. Rev. Lett. 65, 2901(1990). [83] B. Yurke and G. P. Kochanski, Phys. Rev. B. 41 8184 (1989). (1989) [84] R. Landauer and Th. Martin, Physica B 175, 167 (1991); Erra- tum, Physica B 182, 288 (1992). [85] R. Landauer and Th. Martin, Solid State Commun. 84, 115 (1992). [86] Th. Martin and R. Landauer, Phys. Rev. A 47, 2023 (1993).

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