Mesoscopic Physics for Beginners · Mesoscopic Physics for Beginners . Gilles Montambaux ....

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Mesoscopic Physics for Beginners Gilles Montambaux Laboratoire de Physique des Solides Université Paris-Sud, Université Paris-Saclay Orsay µεσος GDR Physique Quantique Mésoscopique, Aussois, déc. 2015

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  • Mesoscopic Physics for Beginners Gilles Montambaux Laboratoire de Physique des Solides Université Paris-Sud, Université Paris-Saclay Orsay

    µεσος

    GDR Physique Quantique Mésoscopique, Aussois, déc. 2015

  • Mesoscopic physics = Phase coherence

    Breakdown of classical laws of electronic transport

    1 2R R R= +

    LRS

    ρ=

    1 2G G G= +

    SGL

    σ=

    1R 2R

    2G1G

    cf. Two path interferometer…

    1GR

    =

  • Phase coherence

    dimensionality disorder

    interactions

    The mesoscopic triangle

    H = p2

    2m+ V (~r) H = ¹hc~¾:~p + V (~r)

    Ã(~r) fÃA(~r); ÃB(~r)g

  • 4

    Domain of mesoscopic physics, deviations to classical transport Length scales, different regimes Conduction = transmission Landauer-Buttiker Quantization of conductance Universal conductance fluctuations Weak-localization What limits phase coherence ?

  • 5

    e F el v τ=

    Mean free path : distance between elastic collisions

    Phase coherence length

    el

    ( )L Tφ

    Interaction with an external degree of freedom (phonons, electrons, spin impurities… breaks phase coherence

    interference

    L Dφ ϕτ=

    Elastic collisions do not break phase coherence

    F elλ

  • 6

    Ohm’s law I GV= SGL

    σ=

    G conductance, σ conductivity

    2ene

    mτσ = Drude-Sommerfeld formula

    Validity ? Diffusive regime No quantum effects

    L Lφ>

    elL

  • 7

    int10

    22cosI I I I πφφ

    = ++

    R. Webb (IBM, 1985) THE founding experiment of mesoscopic physics

    1µm

    1I

    2II

    Classical physics Ohm’s law :

    2G1G

    φ

  • 8

    int10

    22cosI I I I φπφ

    = ++

    R. Webb (IBM, 1985) THE founding experiment of mesoscopic physics

    Interferences between electronic waves (cf. Young’s slits)

    1µm

    1I

    2II

    Aharonov-Bohm effect (1959) el L Lφ

  • 9

    int10

    22cosI I I I φπφ

    = ++

    R. Webb (IBM, 1985) THE founding experiment of mesoscopic physics

    Interferences between electronic waves (cf. Young’s slits)

    1µm

    1I

    2II

    Aharonov-Bohm effect (1959) el L Lφ

  • 10

    Reproducibles conductance fluctuations el L Lφ

  • 11

    Is the conductance of this Au atomic contact in any way related to the conductivity of gold ? NO new concepts, new tools

    What is conductance?

    SGL

    σ=

  • 12

    Quantization of the conductance (1988)

    W

    W

    G W∝

    Classically ballistic « Quantum Point Contact » QPC

  • 13

    2 2 22 2 int[ ]F

    e e WG Mh h λ

    = =

    2eh

    Quantum of conductance

    W

    W

    Quantization of the conductance (1988)

    « Quantum Point Contact » QPC ballistic

  • At which scale do we need new concepts ?

    14

    macroscopic 1nm 10 1000nm− 1 mµ

    nanoscopic

    Lφel

    Mean free path : distance between elastic collisions

    Phase coherence length

    mesoscopic

    ballistic diffusive

    el

    ( )L Tφ

  • 15

    What is conductance?

    Landauer-Büttiker : conductance = transmission

    metallic ring atomic contact nanotube

    2D gas graphene wire network

  • 16

    a b

    Analogies electronics - optics

    Aharonov-Bohm oscillations Young’s slits UCF Speckle Weak-localization Coherent backscattering

    Conductance – transmission coefficient

    … electron quantum optics…

  • 17

    1D wire

    Reservoir Contact Terminal

    1V 2V

    Hypothesis : coherent transport in the wire, dissipation in the reservoirs

    Lead

    scatterer

    Problem of 1D quantum mechanics

    T

    1 2( )I G V V= −

    I

    •A reservoir absorbs electrons and emits them at its own chemical potential and temperature. • No phase relation between ingoing and outcoming electrons in a reservoir. •The scatterer is elastic. •The resistance of the reservoirs is negligible.

  • 18

    1D wire

    1V 2V

    22eG Th

    =Landauer formula

    Without scatterer 22eGh

    =

    2

    1/(25812,807 )eh

    = Ω

    Conductance quantum

    T

    1 2( )I G V V= −

    I

  • 19

    1D wire

    1V 2V

    Very qualitatively…

    T

    I =charge

    time= e

    energy

    h= e

    e¢V

    h

    1 2( )I G V V= −

    I

    I =e2

    h¢V

    22eGh

    =

    time / 1T

    22eG Th

    =

    )

    )

  • 20

    The «multichannel » case

    Total current

    2

    1 22 ( )ab ab

    eI T V Vh

    = −

    Courant related to the transmission from a channel ‘b’ to a channel ‘a’

    2

    1 2,

    2 ( )aba b

    eI T V Vh

    = −∑

    1V 2V

    abT

    a b

    2

    ,

    2ab

    a b

    eG Th

    = ∑

    multichannel Landauer formula

  • 21

    The «multichannel » case

    2

    ,

    2ab

    a b

    eG Th

    = ∑

    1V 2V

    abT

    a b

  • 22

    2 2

    2 2 int[ ]/ 2F

    e e WG Mh h λ

    = =

    The conductance is proportional to the number of modes transmitted through the wave guide

    The «multichannel » case

    2

    ,

    2ab

    a b

    eG Th

    = ∑

    1V 2V

    abT

    ab abT δ=

  • 23

    Quantization of the conductance (1988)

    « Quantum Point Contact »

    2 2

    2 2 int[ ]/ 2F

    e e WG Mh h λ

    = =

  • 24

    4 vs. 2 terminals

    1V 2V

    2

    1 22 ( )eI V Vh

    = − ( ) ( )A BI V V −∞=

    AV BV

    for perfect sample, VA=VB

    2

    2 2eGh

    = 4G = ∞

    no potential drop in the wire :

    I

  • 25

    4 vs. 2 terminals

    1V 2V

    22 1( )I G V V= − 4 ( )A BI G V V= −

    AV BV

    with a scatterer VA=VB

    scatterer

    2

    2 2eG Th

    =

    I

    2

    4 2 1e TGh T

    =−

  • Potential profile ε

    1V

    2V

    T = 1 AV

    x

    BV

    T < 1 2VAV

    BV

    1V

    Ballistic

    One scatterer

    potential drop AT the contacts

    2

    2 2eGh

    =

    4G = ∞

    2

    4 2 1e TGh T

    =−

    2

    2 2eG Th

    =

    No dissipation in the wire

  • AVBV

    1V

    2VAVBV

    1V

    2VAV

    BV

    Diffusive regime

    Four terminal disorder + interferences

    2 1

    A

    B

    ENS,Paris

  • Landauer formula

    Conductance = Transmission

    2

    2 eG Th

    =

    Landauer-Büttiker formalism

    R. Landauer (1927-1999)

    M. Büttiker (1950-2013)

  • 29

    conductance = transmission

    ak

    bk

    2

    ,

    2ab

    a bG Te

    h= ∑

    b a

    analogy with optics

    abT

    optics - microwaves electronics

    b a

    ab abT Tδ = G Gδ

    b a

    fluctuations ~ average

    2eGh

    δ

    fluctuations

  • 30

    conductance = transmission

    ak

    bk

    2

    ,

    2ab

    a bG Te

    h= ∑

    b a

    analogy with optics

    abT

    optics - microwaves electronics b a

    In optics, you can mesure Tab , Ta , or T

    a abb

    T T= ∑,

    aba b

    T T= ∑abT

    In electronics, you can only mesure T

    ,ab

    a bT T= ∑

    The fluctuations of are much smaller than the fluctuations of ,

    aba b

    T T= ∑ abT

  • 31

    Weak-localization = first quantum correction to classical transport

    Phase coherence effect The negative correction is cancelled by a magnetic field B Negative magnetoresistance The characteristic field depends on temperature measures the phase coherence

    magnetoresistance of a Mg film Bergmann

    G = Gcl + ±G(B)

  • 2 *'

    , '( , ') ( , ') ( , ')j j j

    j j jG A r r A r r A r r∝ + ∑ ∑

    32

    Conductance = Transmission

    ( , ')jj

    A r r∑2

    ( , ')jj

    G A r r∝ ∑r r’

    j

    j’

    21 2I A A= +

    2 2 * *1 2 1 2 2 1I A A A A A A= + + +

    1

    2 S

  • 2 *'

    , '( , ') ( , ') ( , ')j j j

    j j jG A r r A r r A r r∝ + ∑ ∑

    33

    Classical transport: only paired identical trajectoires Aj Aj contribute If paired trajectories are different, the amplitudes Aj et Aj’ are different

    Classical term Interference term

    Quantum effects

    Disorder average

    Conductance = Transmission

    ???

    ( , ')jj

    A r r∑2

    ( , ')jj

    G A r r∝ ∑

    phases are uncorrelated all interference terms disappear in average

    r r’

    j

    j’

  • 34

    Quantum correction

    Classical conductance

    clG

    One loop and one crossing

    Weak-localization

    (0, )clP L∝

    G∆

    Z(¡~k)(¡d~l) =

    Z~kd~l

  • 35

    int ( )P t

    Quantum correction

    Classical conductance

    Opposite paired trajetories

    clG

    One loop and one crossing

    crossing

    = distribution of loops of time t = return probability

    Weak-localization

    (0, )clP L∝

    (0, )P L∝ ∆G∆

    i

    2

    nt ( )2eh

    G P t ∆ −

  • 36

    The return probability P(t) is larger at small d Phase coherence effects are more important in low dimension

    /2int ( ) (4 )

    d

    d

    LP tDtπ

    =

    2 2

    in

    ,

    t int2 ( ) ( )2

    e

    D

    D

    e eh h

    P dtG t P tφτ τ

    τ τ∆ − − = ∫

    Time spent in the sample Phase coherence time Elastic collision time eτ

    2

    DLD

    τ =

    2LD

    φφτ =

    volume explored after time t

    Weak-localization = return probability

  • 37

    /2e

    d

    dtt

    Gφτ

    τ

    ∆ ∝ ∫

    /2e

    d

    dtt

    φτ

    τ∫

    lne

    φττ

    eφτ τ− 1d =

    2d =

    L Dφ φτ=

    The measure of this quantum correction gives access to the phase coherence time (length)

    Weak-localization : importance of dimensionality

    ln Lφ

  • 38

    Oscillation of the WL correction with the flux (cf. oscillations Sharvin-Sharvin)

    Diffuson Cooperon

    Cooperon: in a field, time reversed trajectories acquire opposite phases

    φ φ02 φπ

    φ0

    2 φπφ

    0

    2 φπφ

    0

    4 φπφ

    Phase difference 02 2

    he

    φ= oscillations with period

    int ( ) ( )clP t P t= 04i

    eπ φ

    φ

    Weak-localization = phase coherence and magnetic field

    ( )clP t int ( )P t

  • 39

    Negative magnetoresistance Bergmann

    Diffuson Cooperon

    Cooperon: in a field, time reversed trajectories acquire opposite phases

    φ φ02 φπ

    φ0

    2 φπφ

    0

    2 φπφ

    0

    4 φπφ

    Phase difference 02 2

    he

    φ= oscillations with period

    int ( ) ( )clP t P t= 04i

    eπ φ

    φ

    Weak-localization = phase coherence and magnetic field

    ( )clP t int ( )P t

  • 40

    Diffuson (classical)

    Cooperon (quantum)

    ²

    Weak-localization = phase coherence

    ¿ ¿¿

    t¡ 2¿( )clP t int ( )P tLoop of time t

  • 41

    Suppression and revival of WL through control of time-reversal symmetry Vincent Josse et al. , Institut d’Optique, PRL 2015

    ¿ 6= t2

    t

    ¿ =t

    2« Suppression » « Revival »

  • 42 Magnetic impurities, e-e interaction, magnetic impurities

    Diffuson (classical)

    Cooperon (quantum)

    Phase coherence broken after a typical time Only trajectories of time contribute to the return probablity and to the WL

    t φτ<φτ

    /int ( ) ( )cl

    tP t P t e φτ−= 04ieπ φ

    φ

    ²

    Weak-localization = phase coherence

    Loop of time t

    ¿ ¿¿

    2t¡ ¿( )clP t int ( )P t

  • 43

    ( )tϕ+

    02

    ( )i

    i te eφπφ ϕ

    Random dephasing depends on the position of atoms, other electrons, magnetic impurities,…

    0

    2 ( )tφπ ϕφ

    +

    0

    2 ( )tφπ ϕφ

    − +

    0

    4 ( ) ( )t tφπ ϕ ϕφ

    + −

    04 ( )i i t

    eφπ ϕφ

    + ∆

    21 ( )( ) 2 /i t tt

    e ee φϕϕ τ− ∆ −∆

    Dephasing :

    Average on the trajectories and on the dynamics of external degrees of freedom

    Dephasing

  • 44

    0.1

    1

    10

    100

    0.001 0.01 0.1 1 10 100 T (K)

    L φ ( µ

    m)

    3Tφτ−∝

    e-ph interaction

    2/3Tφτ−∝

    3012

    3014

    3016

    3018

    3020

    3022

    3024

    -200 -150 -100 -50 0 50 100 150 200

    R +

    offs

    et (O

    hms)

    B (G)

    30mK

    60mK

    2000mK

    470mK

    10

    ( ) dBWL

    G B f φδφ

    =

    Magnetotransport gives access to the phase coherence length

    magnetic impurities

    ( )L Tφ

    2

    20

    ( ) dBL

    G B f φδφ

    =

    2/3 31( )

    AT B TTφτ

    = +

    e-phonon e-e

    e-e interaction

    quasi-1D wires

    Grenoble

  • https://users.lps.u-psud.fr/montambaux/X15-meso.htm

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