Quantum Transport in Finite Disordered Electron Systems A Dissertation Presented by Branislav Nikoli´c to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Physics State University of New York at Stony Brook August 2000 State University of New York at Stony Brook The Graduate School Branislav Nikoli´c We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of the dissertation. Philip B. Allen, Professor, Department of Physics and Astronomy, Stony Brook Gerald E. Brown, Professor, Department of Physics and Astronomy, Stony Brook Vladimir J. Goldman, Professor, Department of Physics and Astronomy, Stony Brook Myron Strongin, Research Staff Member, Brookhaven National Laboratory, Upton This dissertation is accepted by the Graduate School. Graduate School ii Abstract of the Dissertation Quantum Transport in Finite Disordered Electron Systems by Branislav Nikoli´c Doctor of Philosophy in Physics State University of New York at Stony Brook 2000 The thesis presents a theoretical study of electron transport in various dis- ordered conductors. Both macroscopically homogeneous (nanoscale conductors and point contacts) and inhomogeneous (metal junctions, disordered interfaces, metallic multilayers, and granular metal films) samples have been studied using different mesoscopic as well as semiclassical (Bloch-Boltzmann and percolation in random resistor networks) transport formalisms. The main method employed is a real-space Green function technique and related Landauer-type or Kubo formula for the exact static quantum (zero temperature) conductance of a finite-size meso- scopic sample in a two-probe measuring geometry. The finite size of the sample makes is possible to treat the scattering on impurities exactly and thereby study all transport regimes. Special attention has been given to the transitional regions connecting diffusive, ballistic and localized transport regimes. Thorough analysis iii of the proper implementation of different formulas for the linear conductance has been provided. The thesis has three parts. In the first Chapter of Part I the quantum trans- port methods have been used to extract the bulk resistivity of a three-dimensional conductor, modeled by an Anderson model on an nanoscale lattice (composed of several thousands of atoms), from the linear scaling of disorder-averaged resis- tance with the length of the conductor. The deviations from the corresponding semiclassical Boltzmann theory have been investigated to show how quantum effects evolve eventually leading to the localization-delocalization transition in strongly disordered systems. The main result is discovery of a regime where semiclassical concepts, like mean free path, loose their meaning and quantum states carrying the current are “intrinsically diffusive”. Nevertheless, scaling of disorder-averaged resistance with the sample length is still approximately lin- ear and “quantum” resistivity can be extracted. Different mesoscopic effects, like fluctuations of transport coefficients, are explored in the regime of strong disorder where the concept of universality (independence on the sample size or the degree of disorder—within certain limits), introduced in the framework of perturbation theory, breaks down. The usual interpretation of a semiclassical limit of the disorder-averaged Landauer formula in terms of the sum of contact resistance and resistance of a disordered region was found to be violated even for low disorder. The “contact resistance” (i.e., the term independent of the sample length) diminishes with increasing disorder and eventually turns negative. The second Chapter of Part I investigates transport in metal junctions, strongly disordered interfaces and metallic multilayers. The Kubo formula in exact state representation fails to describe adequately the junction formed between two con- ductors of different disorder, to be contrasted with the mesoscopic methods (in iv the Landauer or Kubo linear response formulation) which take care of the finite- ness of a sample by attaching the ideal leads to it. Transmission properties of a single strongly disordered interface are computed. The conductance of different nanoscale metallic multilayers, composed of homogeneous disordered conductors coupled through disordered interfaces, is calculated. In the presence of clean conductors the multilayer conductance oscillates as a function of Fermi energy, even after disorder averaging. This stems from the size quantization caused by quantum interference effects of electron reflection from the strongly disordered interfaces. The effect is slowly destroyed by introducing disorder in the layer between the interfaces, while keeping the mean free path larger than the length of the that layer. If all components of the multilayer are disordered enough, the conductance oscillations are absent and applicability of the resistor model (mul- tilayer resistance understood as the sum of resistances of individual layers and interfaces) is analyzed. In Part II an atomic-scale quantum point contact was studied with the in- tention to investigate the effect of the attached leads on its conductance (i.e., the effect of “measuring apparatus” on the “result of measurement”, in the sense of quantum measurement theory). The practical merit of this study is for the analogous effects one has to be aware of when studying the disordered case. The transitional region between conductance quantization and resonant tunneling has been observed. The other problem of this Part is a classical point contact mod- eled as an orifice between two metallic half-spaces. The exact solution for the conductance is found by transforming the Boltzmann equation in the infinite space into an integral equation over the finite surface of the orifice. Such conduc- tance interpolates between the Sharvin (ballistic) conductance and the Maxwell (diffusive) conductance. It deviates by less than 11% from the na¨ıve interpolation v formula obtained by adding the corresponding resistances. The third Part is focused on the transport close to the metal-insulator tran- sition in disordered systems and effects which generate this transition in the non-interacting electron system. Eigenstate statistics are obtained by exact diag- onalization of the 3D Anderson Hamiltonians with either diagonal or off-diagonal disorder. Special attention has been given to the so-called pre-localized states which exhibit unusually high amplitudes of the wave function. The formation of such states should illustrate the quantum interference effects responsible for the localization-delocalization transition. The connection between the eigenstate statistics and quantum transport properties has been established showing that deviations (i.e., asymptotic tails of the corresponding distribution function in finite-size conductors) from the universal predictions of Random Matrix Theory are strongly dependent on the microscopic details of disorder. The mobility edge is located at the minimum energy at which exact quantum conductance is still non-zero. The second problem of Part III is a theoretical explanation of the infrared conductivity measurement on ultrathin quench-condensed Pb films. It was shown that quantum effects do not play as important a role as classical electromagnetic effects in a random network of resistors (grains in the film) and capacitors (ca- pacitively coupled grains). The experimental results exhibit scaling determined by the critical phenomena at the classical percolation transition point. vi Dedicated to the memory of my late grandfather Petronije Nikoli´c Contents List of Figures .................................... xiv Acknowledgements ................................. xv 1 INTRODUCTION ................................. 1 I Diffusive Transport Regime 19 2 Linear Transport Theories ............................ 20 2.1Introduction.................................... 20 2.2Ohm’slawandcurrentconservation....................... 24 2.3Semiclassicalformalism:Boltzmannequation.................. 32 2.4Quantumtransportformalisms......................... 36 2.4.1 Linearresponsetheory:Kuboformula................. 36 2.4.2 Scatteringapproach:Landauerformula................. 44 2.4.3 Non-equilibrium Green function formalism ............... 49 2.5 Quantum expressions for conductance: Real-space Green function technique 54 2.5.1 Latticemodelforthetwo-probemeasuringgeometry......... 54 2.5.2 Green function inside the disordered conductor . .......... 57 2.5.3 The Green function for an isolated semi-infinite ideal lead ...... 61 2.5.4 One-dimensionalexample:singleimpurityinacleanwire....... 63 viii 2.5.5 Equivalent quantum conductance formulas for the two-probe geometry 64 3 Residual Resistivity of a Metal between the Boltzmann Transport Regime and the Anderson Transition ............................. 70 3.1Introduction.................................... 70 3.2SemiclassicalResistivity............................. 73 3.3Quantumresistivity................................ 78 3.4 Conductance vs. Conductivity in mesoscopic physics....................................... 87 4 Quantum Transport in Disordered Macroscopically Inhomogeneous Con- ductors .......................................... 91 4.1Introduction.................................... 91 4.2Transportthroughdisorderedmetaljunctions................. 92 4.3Transportthroughstronglydisorderedinterfaces................ 105 4.4 Transport through metallic multilayers ..................... 109 II Ballistic Transport