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=

Lφ

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

Lφ

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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