Heterogeneity and Disorder: Contributions of Rolf Landauer Bertrand I. Halperina, David J. Bergmanb aDepartment of Physics, Harvard University, Cambridge, MA 02138, USA bRaymond and Beverly Sackler School of Physics & Astronomy, Tel Aviv University, IL-69978 Tel Aviv, Israel Abstract Rolf Landauer made important contributions to many branches of science. Within the broad area of transport in disordered media, he wrote seminal papers on electrical conduction in macroscopically inhomogeneous materials, as well as fundamental analyses of electron transport in quantum mechanical systems with disorder on the atomic scale. We review here some of these contributions. Key words: transport, disorder, conductance, heterogeneous materials 1. Introduction Rolf Landauer was recognized for outstanding ac- complishments in many branches of science. In addi- tion to his work on transport in inhomogeneous sys- tems, which will be the focus of the present article, Lan- dauer wrote papers on noise and fluctuations, on nonlin- ear wave propagation and soliton formation, on ferro- electric instabilities and displacive soft modes, entropy production in systems out of equilibrium, philosophical principles of science and technology, and above all, on the physical limits to computation. His influence in this last area was of such a magnitude that he was the sub- ject of a “Profile” article by Gary Stix in the Septem- ber 1998 issue of Scientific American [1]. The article Figure 1: Photograph of Rolf Landauer. Printed with permission from was titled “Riding the Back of Electrons”, and subtitled his survivors. “Theoretician Rolf Landauer remains a defining figure in the physics of information.” During his lifetime, Landauer received many awards ence should be interpreted. He also delighted in chal- for his work including the Ballantine Medal of the lenging entrenched ideas and in forcing people to think Franklin Institute in 1992, the 1995 Buckley Prize of more carefully about the foundations of their work— the American Physical Society, the LVMH, Inc. Science see Fig. 1 for a typical appearance of Rolf Landauer in for Art Prize in 1997, the 1998 IEEE Edison Medal, an this mode. There is perhaps no better way to illustrate honorary doctorate from the Technion in 1991, and a this aspect of his character than to cite some of the titles arXiv:0910.0993v1 [cond-mat.mtrl-sci] 6 Oct 2009 Centennial Medal from Harvard University in 1993. He of articles that Landauer wrote in the last decade of his was elected to the National Academy of Science, the life: “Light faster than light” [2], “Is quantum mechan- National Academy of Engineering, and the American ics useful?” [3], “Mesoscopic noise: Common sense Academy of Arts and Sciences in the US, and to the Eu- view” [4], “Zig-zag path to understanding” [5], “Con- ropean Academy of Arts. ductance is transmission” [6], “The physical nature of Landauer’s influence on science and technology was information” [7], and “Fashions in science and technol- not limited to the importance of his scientific discover- ogy” [8]. ies and research. He had much to say about the conduct Due to limitations of space, our discussion of Lan- of research and about philosophical issues of how sci- dauer’s scientific contributions will be restricted to his Preprint submitted to Elsevier November 4, 2018 work on transport in inhomogeneoussystems, and some the constituent scalar conductivities σi and constituent closely related work on quantum mechanical effects in volume fractions pi, with i pi = 1: mesoscopic systems. However, we include at the end σ ff − σP of the article a brief summary of Landauer’s personal 0 = p e i . (1) i σ + (d − 1)σ history. Xi eff i In the case where there are just two components, this becomes a quadratic equation, which has the explicit so- 2. 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.