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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=
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<
0he
φ =
2eG Gh
δ
φ
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<
0he
φ =φ
2eG Gh
δ
10
Reproducibles conductance fluctuations el L Lφ<
IGV
= ,ij
ij klkl
IG
V=
Exp. measures a conductance and not a conductivity Drude-Einstein conductivity provides an average description, valid if How to go beyond this average and describe interferences, fluctuations ?
L Lφ <
B(T )
2eGh
δ
Conductance depends not only on the system to be studied, but also on its connection to the outside world
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 / 1
T
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 2 eGh
= 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 2 eG Th
=
I
2
4 21
e TGh T
=−
Potential profile ε
1V
2V
T = 1 AV
x
BV
T < 1 2V
AVBV
1V
Ballistic
One scatterer
potential drop AT the contacts
2
2 2 eGh
=
4G = ∞
2
4 21
e TGh T
=−
2
2 2 eG Th
=
No dissipation in the wire
AVBV
1V
2VAV
BV
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 << average ??? Mailly,Sanquer
Maret
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 φ0
2 φπφ
0
2 φπφ
0
2 φπφ
−
0
4 φπφ
Phase difference 0
2 2he
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 φ0
2 φπφ
0
2 φπφ
0
2 φπφ
−
0
4 φπφ
Phase difference 0
2 2he
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= t
2
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−= 04i
eπ 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 tte ee φ
ϕϕ 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
( ) d
BWLG B f φδ
φ
=
Magnetotransport gives access to the phase coherence length
magnetic impurities
( )L Tφ
2
20
( ) d
BLG B f φδ
φ
=
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