Theory of Mesoscopic Systems

Transcription

1 Theory of Mesoscopic Systems Boris Altshuler Princeton University, Columbia University & NEC Laboratories America Lecture 1 01 June 2006

2 Meso: : intermediate between micro and macro Main Topics: Disordered conductors in d=3,2,1 and 0 dimensions Anderson localization Weak localization Sample to sample fluctuations Quantum chaos & spectral statistics Systems of interacting quantum particles Fermi liquid theory without translation invariance Decoherence Many body localization

3 ORIGINS E.P. Wigner, Conference on Neutron Physics by Time of Flight, November 1956 P.W. Anderson, Absence of Diffusion in Certain Random Lattices ; Phys.Rev., 1958, v.109, p.1492 L.D. Landau, Fermi-Liquid Theory Zh. Exp. Teor. Fiz.,1956, v.30, p.1058 J. Bardeen, L.N. Cooper & J. Schriffer, Theory of Superconductivity ; Phys.Rev., 1957, v.108, p.1175.

4

5 Einstein s Miraculous Year Six papers: 1. The light-quantum and the photoelectric effect. Completed March A new determination of molecular dimensions. Completed April 30. Published in1906 Ph.D. thesis. 3. Brownian Motion. Received by Annalen der Physik May 11. 4,5.The two papers on special relativity. Received June 30 and September Second paper on Brownian motion. Received December 19.

6 Einstein s Miraculous Year Six papers: 1. The light-quantum and the photoelectric effect. Completed March A new determination of molecular dimensions. Completed April 30. Published in1906 Ph.D. thesis. 3. Brownian Motion. Received by Annalen der Physik May 11. 4,5.The two papers on special relativity. Received June 30 and September Second paper on Brownian motion. Received December 19.

7 Einstein s Miraculous Year Diffusion and Brownian Motion: 2. A new determination of molecular dimensions. Completed April 30. Published in1906 Ph.D. thesis. 3. Brownian Motion. Received by Annalen der Physik May Second paper on Brownian motion. Received December 19. Are these papers indeed important enough to stay in the same line with the relativity and photons. Why Q:?

8 Einstein s Miraculous Year Six papers: 1. The light-quantum and the photoelectric effect. Completed March A new determination of molecular dimensions. Completed April 30. Published in1906 Ph.D. thesis. Nobel Prize By far the largest number of citations 3. Brownian Motion. Received by Annalen der Physik May 11. 4,5.The two papers on special relativity. Received June 30 and September Second paper on Brownian motion. Received December 19.

9 Brownian Motion - history Robert Brown ( ) The instrument with which Robert Brown studied Brownian Motion and which he used in his work on identifying the nucleus of the living cell. This instrument is preserved at the Linnean Society in London.

10 Brownian Motion - history Robert Brown, Phil.Mag. 4,161(1828); 6,161(1829) Random motion of particles suspended in water ( dust or soot deposited on all bodies in such quantities, especially in London ) Action of water molecules pushing against the suspended? object Giovanni Cantoni (Pavia). N.Cimento, 27,156(1867).

11 The Nobel Prize in Physics 1926 "for his work on the discontinuous structure of matter, and especially for his discovery of sedimentation equilibrium" Jean Baptiste Perrin France b d measurements on the Brownian movement showed that Einstein's theory was in perfect agreement with reality. Through these measurements a new determination of Avogadro's number was obtained. The Nobel Prize in Physics 1926 From the Presentation Speech by Professor C.W. Oseen, member of the Nobel Committee for Physics of The Royal Swedish Academy of Sciences on December 10, 1926

12 Brownian Motion - history Robert Brown, Phil.Mag. 4,161(1828); 6,161(1829) Random motion of particles suspended in water ( dust or soot deposited on all bodies in such quantities, especially in London ) Action of water molecules pushing against the suspend object Problems: 1. Each molecules is too light to change the momentum of the suspended particle. 2. Does Brownian motion violate the second law of thermodynamics? We see under our eyes now motion transformed into heat by friction, now heat changes inversely into motion. This is contrary to Carnot s principle. Jules Henri Poincaré ( ) H. Poincare, The fundamentals of Science, p.305, Scientific Press, NY, 1913

13 Problems: 1. Each molecules is too light to change the momentum of the suspended particle. 2. Does Brownian motion violate the second law of thermodynamics? 3. Do molecules exist as real objects and are the laws of mechanics applicable to them?

14 Kinetic theory S = k logw + const entropy probability k is Boltzmann constant Ludwig Boltzmann ρν, T ( ) = c 3 8πν h hν exp 1 kt 3 Max Planck

15 From Micro to Macro S = k logw + const From Macro to Micro

16 It is of great importance since it permits exact computation of Avogadro number. The great significance as a matter of principle is, however that one sees directly under the microscope part of the heat energy in the form of mechanical energy. Einstein, 1915

17 Brownian Motion - history Einstein was not the first to: 1. Attribute the Brownian motion to the action of water molecules pushing against the suspended object 2. Write down the diffusion equation 3. Saved Second law of Thermodynamics L. Szilard, Z. Phys, 53, 840(1929)

18 Brownian Motion - history Einstein was not the first to: 1. Attribute the Brownian motion to the action of water molecules pushing against the suspended object 2. Write down the diffusion equation 3. Saved Carnot s principle [L. Szilard, Z. Phys, 53, 840(1929)] Einstein was the first to: 1. Apply the diffusion equation to the probability 2. Derive the diffusion equation from the assumption that the process is markovian (before Markov) and take into account nonmarkovian effects 3. Derived the relation between diffusion const and viscosity (conductivity), i.e., connected fluctuations with dissipation By studying large molecules in solutions sugar in water or suspended particles Einstein made molecules visible

19 Diffusion Equation ρ D 2 ρ = t 0 Diffusion constant Einstein-Sutherland Relation for electric conductivity s dn σ = ed 2 ν ν dµ If electrons would be degenerate and form a classical ideal gas William Sutherland ( ) ν = 1 Tn tot

20 Einstein-Sutherland Relation for electric conductivity s metal n= n ( µ ) dn dn d dn = µ = ee dx dµ dx dµ No current Density of electrons Chemical potential Electric field ed dn dx = σ E dn σ = ed 2 ν ν dµ Conductivity Density of states

21 Diffusion Equation ρ D 2 ρ = t 0 Lessons from the Einstein s work: Universality: the equation is valid as long as the process is marcovian Can be applied to the probability and thus describes both fluctuations and dissipation There is a universal relation between the diffusion constant and the viscosity Studies of the diffusion processes brings information about micro scales.

22 What is a Mesoscopic System? Statistical description Can be effected by a microscopic system and the effect can be macroscopically detected Meso can serve as a microscope to study micro Brownian particle was the first mesoscopic device in use

23 Brownian particle was the first mesoscopic device in use First paper on Quantum Theory of Solid State (Specific heat) Annalen der Physik, 22, 180, 800 (1907) First paper on Mesoscopic Physics Annalen der Physik, 17, 549 (1905)

24 Finite size quantum physical systems Atoms Nuclei Molecules... Quantum Dots

25 e e e e e Quantum Dot 1. Disorder ( - impurities) 2. Complex geometry 3. e-e interactions

26 e e e e e 1. Disorder ( - impurities) 2. Complex geometry How to deal with disorder? Solve the Shrodinger equation exactly Start with plane waves, introduce the mean free path, and derive Boltzmann equation How to take quantum interference into account

27 Electrons in nanostructures Clean systems without boundaries: Electrons are characterized by their momenta or quasimomenta [ electronic wave functions are plane waves Physics is essentially local Interaction between electrons is often apparently not important In mesoscopic systems: Due to the scattering of the electrons off disorder (impurities) and/or boundaries the momentum is not a good quantum number Response to external perturbation is usually nonlocal Interaction between electrons is often crucial

28 Lesson 1: Beyond Markov chains: Anderson Localization and Magnetoresistance

29 R.A. Chentsov On the variation of electrical conductivity of tellurium in magnetic field at low temperatures, Zh. Exp. Theor. Fiz. v.18, , (1948).

30 Quantum particle in random quenched potential e

31 ORIGINS E.P. Wigner, Conference on Neutron Physics by Time of Flight, November 1956 P.W. Anderson, Absence of Diffusion in Certain Random Lattices ; Phys.Rev., 1958, v.109, p.1492 L.D. Landau, Fermi-Liquid Theory Zh. Exp. Teor. Fiz.,1956, v.30, p.1058 J. Bardeen, L.N. Cooper & J. Schriffer, Theory of Superconductivity ; Phys.Rev., 1957, v.108, p.1175.

32 Localization of single-particle wave-functions: extended Disorder localized

33 f = 3.04 GHz f = 7.33 GHz Anderson Insulator Anderson Metal

34 e Scattering centers, e.g., impurities Models of disorder: Randomly located impurities White noise potential Lattice models Anderson model Lifshits model

35 Anderson Model Lattice - tight binding model Onsite energies e i - random j i I ij I ij = Hopping matrix elements I ij -W < e i <W uniformly distributed I i and j are nearest neighbors 0 otherwise Anderson Transition I < I c Insulator All eigenstates are localized Localization length x I > I c Metal There appear states extended all over the whole system

36 Classical particle in a random potential Diffusion ρ - 2 D ρ = t 1 particle - random walk Density of the particles ρ Density fluctuations ρ(r,t) at a given point in space r and time t. Diffusion Equation D - Diffusion constant D = 2 l dτ l τ d 0 mean free path mean free time # of dimensions

37 Einstein - Sutherland Relation for electric conductivity s dn σ = ed 2 ν ν dµ Conductance G = σ L d - 2 G = e 2 h ( ν L d ) Dh L for a cubic sample of the size L ( ) g L ( ) g L = hd L d 1 ν L 2 = Thouless energy mean level spacing Dimensionless Thouless conductance

38 Energy scales (Thouless, 1972) 1. Mean level spacing L energy d 1 = 1/n L d d 1 L is the system size; d is the number of dimensions 2. Thouless energy E = hd/l 2 T D is the diffusion const E T has a meaning of the inverse diffusion time of the traveling through the system or the escape rate. (for open systems) g = E T / d 1 dimensionless Thouless conductance g = Gh/e 2

39 ( ) g L = hd L d 1 ν L 2 = Thouless energy mean level spacing Thouless conductance is Dimensionless Corrections to the diffusion come from the large distances (infrared corrections) Scaling theory of Localization (Abrahams, Anderson, Licciardello and Ramakrishnan, 1979). Universal description!

40 Scaling theory of Localization Abrahams, Anderson, Licciardello and Ramakrishnan g = E T / d Dimensionless Thouless conductance L = 2L = 4L = 8L... without quantum corrections E T L -2 d 1 L -d g = Gh/e 2 E T E T E T E T d 1 d 1 d 1 d 1 g g g g d( log g) d( log L) =β( g)

41 d( log g) d( log L) =β( g) β function Is universal, i.e., material independent But It depends on the global symmetries, e.g., it is different with and without T-invariance Limits: ( ) ( ) g >> 1 g L β g = d O Ł g ł d L ξ g << 1 g e β g» logg < 0 ( ) > 0 d > 2?? d = 2 < 0 d < 2

42 1-1 β(g) d log g β - function = β ( g) d log L unstable fixed point g c 1 3D 2D 1D g Metal insulator transition in 3D All states are localized for d=1,2

43 1-1 β(g) d log g β - function = β ( g) d log L unstable fixed point g c 1 3D 2D 1D g ( g) d 2 β = - + c d =?? c d g d -2 ( ) σ L g L = - cl cd d - 2 d 2 L c2 log Ł l ł d = 2

44 Questions: Why the scaling theory is correct? the corrections of the diffusion constant and conductance are negative? Why diffusion description fails at large scales?

45 Diffusion description fails at large scales Why? Einstein: there is no diffusion at too short scales there is memory, i.e., the process is not marcovian. r( t) dr dt = = Dt t 0? D 2t Does velocity diverge at No because at times shorter than mean free time process is not marcovian and there is no diffusion

46 Diffusion description fails at large scales Why? Einstein: there is no diffusion at too short scales there is memory, i.e., the process is not marcovian. Why there is memory at large distances in quantum case? Quantum corrections at large Thouless conductance weak localization Universal description

47 Quantum corrections c d ( g) d 2 β = - + =?? c d g d -2 ( ) σ L g L = - cl cd d - 2 d 2 L c2 log Ł l ł d = 2 Suggested homework: 1. Derive the equation for g(l) from this limit of the β-function 2. Suppose you know β(g) for some number of dimensions d. Let g at some size of the system L 0 be close to the critical value: g( L ) 0 = gc + δg; δg << 1 Estimate the localization length ξ (for δg<0) and the conductivity σ in the limit L fi (for δg>0)

48 WEAK LOCALIZATION ϕ = p r dr r Phase accumulated when traveling along the loop O j 1 = j 2 The particle can go around the loop in two directions Memory! Constructive interference probability to return to the origin gets enhanced diffusion constant gets reduced. Tendency towards localization b - function is negative for d=2

49 Diffusion Random walk Density fluctuations ρ(r,t) at a given point in space r and time t. ρ - 2 D ρ = t 0 Diffusion Equation D - Diffusion constant Mean squared distance from the original point at time t Ær t æ=dt ( ) 2 Probability to come back (to the element of the volume dv centered at the original point) ( ( ) ) P r t = = dv 0 dv d ( Dt) 2

50 O What is the probability P(t) the particle comes back in a time t? Probability to come back (to the element of the volume dv around the original point) ( ( ) ) P r t = = dv 0 dv d ( Dt) 2 Q: dv =? A: dv = D d -1 v dt F ( ) P t = t v dt - -D d 1 F» P( t ) ( ) d 2 max Dt g τ δ g

51 ( ) P t = -D d t v dt - 1 F ( ) d Dt τ 2 δ g g» ( ) P t max Q: t max =? A: t max min L 2 D, 1 ω,τ ϕ Decoherence in the leads AC conductivity Dephasing time (inelastic processes)

52 ( ) P t = -D d t v dt - 1 F ( ) d Dt τ 2 δ g g» ( ) P t max L 2 t = max D d = 2 h E T δ g v L» - D F log g D Dτ 2

53 ( ) P t = D d t v dt - 1 F ( ) d Dt τ 2 δ g g» P t ( ) max δ g Dv L 2Dv L» - F log = - F log g D Dτ D l 2 D v F 1 = πν g = ν Dh δ β g ( g) = - 2 log L π 2 π g l = - Universal!!!

54 Q: What does it mean d=2? A: Transverse dimension is much less than Dt max δ g = - 2 log L π l for a film with a thickness a much smaller than L,L ϕ β ( g) = - 2 π g

55 R.A. Chentsov On the variation of electrical conductivity of tellurium in magnetic field at low temperatures, Zh. Exp. Theor. Fiz. v.18, , (1948).

56 Magnetoresistance Φ O No magnetic field j 1 = j 2 O With magnetic field H j 1 - j 2 = 2*2p Φ/Φ 0

57 Length Scales Magnetic length L H = (hc/eh) 1/2 Dephasing length L j = (D t j ) 1/2 ( ) f H δ g H = d Ł L L ϕ ł Universal functions Magnetoresistance measurements allow to study inelastic collisions of electrons with phonons and other electrons

58 Weak Localization Negative Magnetoresistance Chentsov (1949) Aharonov-Bohm effect Theory B.A., Aronov & Spivak (1981) Experiment Sharvin & Sharvin (1981)

59 Aharonov-Bohm effect Theory B.A., Aronov & Spivak (1981) Experiment Sharvin & Sharvin (1981) Φ O With magnetic field H j 1 - j 2 = 2*2p Φ/Φ 0 Resistance is a periodic function of the magnetic flux with the period F o /2

60 Lesson 2: Brownian Particle as a mesoscopic system

61 Mesoscopic fluctuations

62 Resistance is a periodic function of the magnetic flux with the period F o

63 g( H) g Mesoscopic Fluctuations. g g 1 2 Properties of systems with identical set of macroscopic parameters but different realizations of disorder are different! Magnetoresistance... g >>1 - ensemble averaging H g( H) is sample -dependent

64 Before Einstein: Correct question would be: describe r( t) OK, maybe you can restrict yourself by r( t) r r º ß Einstein: Ør( 0) - r( t) ø 2 What is? r r ( 0 ) ( ) n غ - t øß =? r r Mesoscopic physics: Not only But also g( H) غ g H g H h ß ( ) - ( + ) ø 2

65 ensemble observables evolves as function of Interested in Brownian motion Set of brownian particles Position of each particle Time.. Statistics of r r t r ( t) Conductance fluctuations Set of small conductors Conductance of each sample.. g Magnetic field or any other external tunable parameter Statistics of H g( H) Example r r Ø º r t1 2 ß Ø ( ) ( ) 2 º g H1 - g H ø 2 ß ( ) - r( t ) ø 2

66 g 1 g 2 g H g 1 g 2 1 G 1 G 2 e 2 h Magnetoresistance ( ) g 1 H Statistics of the functions of g(h) are universal B.A.(1985); Lee & Stone (1985)

67 Statistics of random function(s) g(h) are universal! In particular, ( ) 2 δg 1 ( ) 2 δ g g L d-2 fi L 4-2d >> L -d g 2 Fluctuations are large and nonlocal

68 Waves in Random Media S 2 1 D W 1, W 2 probabilities W 1,2 = A 1,2 2 i A1,2 = A1,2 e ϕ A 1, A 2 probability amplitudes 1,2 Total probability W = A 1 + A 2 2 =W + W + 2Re( A A ) interference term: 2Re( A 1 A 2 )= 2 W 1 W 2 cos( ϕ 1 ϕ 2 )

69 W = A 1 + A 2 2 =W + W + 2Re( A A ) Re( A 1 A 2 )= 2 W 1 W 2 cos( ϕ 1 ϕ 2 ) A 1,2 = W 1,2 exp( iϕ ) 1,2 Phases ϕ 1,2 are random The interference term disappears after averaging 3. ϕ 1 ϕ 2 >> 2π cos( ϕ 1 ϕ 2 ) = 0 W = W 1 + W 2

70 S W = A 1 + A 2 2 =W 1 + W 2 + 2Re( A 1 A ) 2 Classical result for average probability: 2 1 D W = W 1 + W 2

71 S 2 1 D Consider now square of the probability W 2 = ( W +W ) 2 + 2W W Reason: cos( ϕ 1 ϕ 2 ) = 0 cos 2 ( ϕ 1 ϕ 2 ) =1 2 W 2 W 2

72 S 2 1 D W 2 W 2 CONCLUSIONS: 1. There are fluctuations! 2. Effect is nonlocal.

73 Now let us try to understand the effect of magnetic field. Consider the correlation function S 2 1 D ( ) ( + ) = ( ) ( + ) δϕ( ) δϕ W H W H h W H W H h 1 2 ( ) ( ( )) + 2W W cos H cos H + h δϕ ϕ1-ϕ2 cos( δϕ( H) ) cos( δϕ ( H + h) ) ( h) h ( area of the loop) F = for for h Φ 0 ( Φ( h) << Φ0 ) ( h) >> Φ0

74 Magnetoresistance g H ( ) g 1 F 0 H F Flux through the whole system

75 a R Φ πr 0 2 period of oscillations Φ 0 2πRa scale of aperiodic fluctuations

76 Quantum Chaos conductance (e 2 /h) 0.3 Marcus et al B(T) V g (mv) 0

77 Marcus et. al, 1998 Huibers, et al. PRL, (1998).

78 e e e e e 1. Disorder ( - impurities) 2. Complex geometry How to deal with disorder? Solve the Shrodinger equation exactly Make statistical analysis What if there in no disorder?