Chaire de Physique Mésoscopique Michel Devoret Année 2007, Cours des 7 et 14 juin INTRODUCTION À LA PHYSIQUE MÉSOSCOPIQUE: ÉLECTRONS ET PHOTONS INTRODUCTION TO MESOSCOPIC PHYSICS: ELECTRONS AND PHOTONS Première Leçon / First Lecture This College de France document is for consultation only. Reproduction rights are reserved. 07-I-1
AIM OF THESE TWO LECTURES Discuss a selection of basic concepts of mesoscopic physics and contrast their treatment with that of atomic physics OUTLINE 1. General remarks on mesoscopic physics a) fundamental constants b) survival of quantum effects in macrosystems 2. What are "electrons"? a) example of mesoscopic resistor b) screening c) finite lifetime 3. What are "photons"? a) example of transmission line b) longitudinal and transverse modes c) 1-D Planck's law 4. Conclusion: quantum transport v.s. quantum optics part 1 part 2 07-I-2
General remarks Purpose: explain the novelty of mesoscopic phenomena by discussing status of variables and parameters 07-I-3
MACRO versus micro MACRO SYSTEM classical mechanics many particles, strongly coupled to environment 2 2 pθ kθ H = + +... 2M 2 can have engineered structural parameters micro system quantum mechanics few particles, almost isolated from environment 2 2 ˆ pˆ e H = +... 2m 4πε rˆ e 0 always "God-given" structural parameters 07-I-4
MESOSCOPIC PHENOMENA: Number of particles: macroscopic Collective degrees of freedom: quantum High-Q nanomechanical resonators @ low temperatures: clearly mesoscopic Superconducting magnets for MRI: clearly not mesoscopic Characteristic energies in mesoscopic phenomena involve not only fundamental constants, but also geometric dimensions adjusted by design and fabrication 07-I-5
REVIEW OF UNIVERSAL CONSTANTS S. I. Units : s, m, kg, A, K Classical constants Quantum constants speed of light impedance of vacuum c Z vac 377Ω charge quantum action quantum e electrical permittivity of vacuum magnetic permeability of vacuum ε 0 1 = cz Z µ 0 = c vac vac flux quantum resistance quantum h Φ 0 = 2e R K h = 26kΩ 2 e Boltzmann's constant gravitational constant k B G fine structure cst. α = Zvac 1 2R 137 K 07-I-6
UNIVERSAL AND MICROSCOPIC QUANTUM CONSTANTS Universal constants Microscopic constants charge of electron: e mass of electron: m e Planck's constant: flux quantum: Φ = 0 h 2e magn. mt. of electron: mass of proton: mass of neutron µ B m p m n resistance quantum: fine structure cst.: R K α = h = 26kΩ 2 e Zvac 1 2R 137 K magn. mt. of proton......... µ p 07-I-7
FORMS OF CHARACTERISTIC ENERGIES ω 20 GHz ev 80 µv 100 80 + + 60 40 - - kbt 1K 20 10 20 30 40 50 07-I-8
USUAL ENERGY SCALES OF MICROSCOPIC QUANTUM EFFECTS Atomic matter Bulk condensed matter E ~ ev Ry En = 2 n m c Ry 2 α 2 e 2 = = 13.6eV conduction band E gap ~ ev valence band E Debye ~ 10meV = sc E Debye e N 1 ( 0) V ~ mev IN MESOSCOPIC QUANTUM EFFECTS, NEW ENERGY SCALES EMERGE! 07-I-9
MESOSCOPIC PHYSICS CHALLENGES COMMON WISDOM ABOUT THE SUPPRESSION OF COLLECTIVE QUANTUM EFFECTS IN LARGE AND DIRTY SYSTEMS THERMAL ENERGY CAN BE BIGGER THAN SINGLE PARTICLE ENERGY LEVEL SPACINGS DISORDER DOES NOT FULLY SUPPRESS INTERFERENCES, IT IS DECOHERENCE WHICH IS IMPORTANT A SINGLE ELECTRON CAN COUNT LOCAL CIRCUIT THEORY WILL UNEXPECTEDLY FAIL 07-I-10
MESOSCOPIC PHYSICS CHALLENGES COMMON WISDOM ABOUT THE SUPPRESSION OF COLLECTIVE QUANTUM EFFECTS IN LARGE AND DIRTY SYSTEMS THERMAL ENERGY CAN BE BIGGER THAN SINGLE PARTICLE ENERGY LEVEL SPACINGS DISORDER DOES NOT FULLY SUPPRESS INTERFERENCES, IT IS DECOHERENCE WHICH IS IMPORTANT A SINGLE ELECTRON CAN COUNT LOCAL CIRCUIT THEORY WILL UNEXPECTEDLY FAIL 07-I-11
EXAMPLE OF COMMON WISDOM AT WORK: THE CROSSOVER TO DULONG-PETIT LAW C V Nk T 3 B 0 kt B ω Debye T quantum classical 07-I-12
2-DIMENSIONAL ELECTRON GAS SYSTEM contact electrode gate electrode z before charge transfer after charge transfer n-algaas i-gaas a 2DEG forms here! 07-I-13
PINCHING THE 2DEG INTO A SMALL WIRE source reservoir drain reservoir d small effective mass I V g V g v F ~ 1nm/ps V/2 V/2 Ballistic transport: L L what if F kt? e B v d 07-I-14
CONDUCTANCE QUANTIZATION G 2 / e h 12 d kt B v d F 10 8 van Wees et al. 1988: 6 4 2 0 G di = dv ev / k T 0 B V g waveguide effect! 07-I-15
UP TO WHAT SIZE CAN QUANTUM EFFECTS PERSIST IN PRESENCE OF DISORDER? weak disorder: 1/ e k F L e NON-TRIVIAL RESULT kt, B τ ϕ v L F e L Thouless energy Can be much larger than level spacing in box of size L! vk F F v ( Lk ) 2 L 2 F F e 07-I-16
BOOKS ON MESOSCOPIC PHYSICS S. Datta Y. Imry E. Akkermans and G. Montambaux H. Grabert and M. Devoret REQUISITES: Quantum mechanics Solid state physics Statistical mechanics Useful reference books on quantum statistical physics: Feynman; Schrieffer; Pines and Nozières 07-I-17
2. How mesoscopic physics models the "electron" Purpose: provide groundwork for Landauer's approach of transport phenomena 07-I-18
THE MESOSCOPIC RESISTOR + V _ quantum coherent wire I source reservoir L L ϕ drain reservoir Landauer reservoir: metallic electrode which injects into the quantum coherent region quasi-electron waves with well-defined chemical potential and temperature. It also accepts without reflection any quasi-electron waves and thoroughly recycles them. The Landauer reservoir is to Fermi waves what a black-body is to Bose waves. 07-I-19
Collection of independent channels t r t r µ + µ E ev f + f 07-I-19 bis
THE LANDAUER-BÜTTIKER FORMULA I = I I + e + I = f ( E) t ( E) 2 d E ± ± m h m f ± E ( ) = 1 E µ ± 1+ exp kt B µ µ = + ev schizophrenic! why does it work? 07-I-20
RESISTANCE DETERMINED BY ELASTIC TRANSMISSION COEFFICIENT! WHAT ABOUT JOULE HEATING? 07-I-21
QUESTION: What are the differences between a quasi-electron (a.k.a. dressed electron) and the usual (bare) electron? PARTICLE IDENTIFICATION CARD Last Name: Electron First First name: name: Bare Bare Address: Vacuum Genre: Genre: Fermion Fermion Occupation: Wave packet Lifetime: infinite Average energy: ω Average momentum: k Velocity: v=dω/dk Mass: dk/dv=m e Charge: -e Spin: S=1/2 Magnetic moment: µ g eb Sµ B ge α = 21 + + O 2 π 2 ( α ) 07-I-22
An example of a Feynman diagram involving the usual electron and photon of atomic physics which propagate in vacuum γ γ e - e - 07-I-22bis
THE "ELECTRON" OF MESOSCOPICS PARTICLE IDENTIFICATION CARD Last Name: Electron First name: Quasi Address: Metal Genre: Fermion Occupation: Wave packet Lifetime: finite, except @ k F Average energy: ω Average momentum: k Velocity: v=dω/dk Mass: dk/dv=m eff (k) Transverse charge: -e Longitudinal charge: 0 (q 0) Spin: S=1/2 Magnetic moment: g eff Sµ B 07-I-23
Definition of the longitudinal and transverse part of a field: F = F + F l F = 0 t F = 0 l t The longitudinal and transverse charges are the sources of the longitudinal and transverse parts of the electrical field, respectively. 07-I-23bis
SOLIDS - + - + + + + - + + + + - + - + + + van der Waals ionic insulators covalent + + + + + + metal + + + 07-I-24
REVIEW OF DRUDE MODEL E v x t τ v x e E ( ) = τ j = n( e) v x m j σ = = E ne m 2 τ 07-I-25
PROBLEM: ELECTRONS ARE FERMIONIC WAVES! 07-I-26
REVIEW OF FREE FERMI GAS MODEL Box volume V Nb of fermions N k y kf ( k ) a Total energy E K Length scale a 0 Energy scale Ry 2 4 me e = 4 πε ; Ry = 2 4πε 0 0 2 2 me e N k a n r k V = = ( π ) = = a a = r a 3 1 r 2 4π F 1 1.92 ; 2 s ; F 3 2 4π 3 0 s 0 F F g m E e C s 2 E K 3 F 3 2.22 = = E F = 2 N 5 2m e 5 rs v = k = v E N = Ry F Ry 0 k x 07-I-27
OTHER PROBLEM: ELECTRONS INTERACT STRONGLY! BUT TOO STRONG INTERACTION KILLS INTERACTION.. 07-I-28
Acknowledgements : D. Esteve, H. Pothier, D. Stone and C. Urbina