Post on 10-Mar-2020
ELECTRICAL TRANSPORT REGIMES
+ +++
+++++ le
φL
�
�
diffusive
ballistice
e
l L
L l
Inelastic scatteringCoulomb interactions
PhononsMagnetic moments
φLElastic scatteringGrain boundaries
EdgesImpurities
el
λλλλ
�
�
�
�
, , 3D
2
, 1
, , 0
F x y z
F z
F y z
F X y z
L L L
L D
L L D
L L L D
φ
φ
�
�
fully coherent
incoherent
L
L
L
Lconfinement
EXPERIMENTAL PLAYGROUND
80 (nm) 0.5
50meV 10eV
15�m 30nm
10 (�m) 101K
F
F
e
E
l
LT
φ
λ
<
( )3 210 mµ −Low density metals High density metals ( )10 310 mµ −
Epitaxial heterostructures Evaporated thin films
M. Heiblum et al, Weizmann Inst., IsraelN. Birge et F. Pierre, MSU, USA
TRANSPORT AS A SCATTERING PROBLEM
V
Fully Coherent Conductor
Landauer (1957)Büttiker (1992)
Iφ<<L L
Ideal lead
Lµε
1
)(εLf
Reservoir
Ve
ε
Rµ
1
)(εRf
Reservoir
Ideal lead
µ µ− =L R eV
SCATTERING MATRIX
( ) ( )( ) ( )
LL LR
RL RR
S S
S SS
� �� �=� �� �
L L
RR
b a
abS
� � � �=� � � �� � � �� �
� �
� �
( ),1
( )
( ),
L R
L R
L R N
a
a
a
� �� �= � �� �� �
��
,1
,
L
L
L N
b
b
b
� �� �= � �� �� �
��
Incoming modes
Outgoing modes
† † 1SS S S= =Current conservation
unitary
,1La
,2La
,3La
,1Ra
,2Ra
,3Ra
,1Lb
,2Lb
,3Lb
,1Rb
,2Rb
,3Rb
INDEPENDENT CONDUCTION CHANNELS
( ) ( ) ( )
( ) ( ) ( )
L R L R L R
L R L R L R
a U a
b V b
′ =
′ =
� �
� �
Unitary transformation
( ) ( )( ) ( )
† 1/ 2 1/ 2
† 1/ 2 1/ 2
0 00 0
LL LRL L
R RRL RR
S SV U iR TS
V U T iRS S
� �� � � � � �−� �′ = =� � � � � �� � −� �� � � �� �
With R and T real, diagonal, positive matrices
Collection of independent 1D problems
L L
R R
b aS
b a
� � � �′ ′� � � �′=� � � �′ ′� � � �
� �
� �
Th. Martin and R. Landauer, PRB 45, 1742 (92)
IDEAL 1D LEADS
2 2( ) / 2k k mε ± = �1
exp ( / )( )
εε
− �Li k x thv
If all states in energy range δεare occupied the current is
( )εδ δε δεε
= =dndL
v eI e
h
density of states in 1D
( )dn Ld hvε ε
=( )electron velocity
/for F F
kv eV
v ε εε
= ∂ ∂≈
�
�
Maximum rate of fermions emitted by a source into a 1D lead:1 ε=�dN dh
Rk
)( kε
δε
SINGLE CHANNEL SCATTERING MATRIX
( ) '( )( )
( ) '( )r t
St r
ε εε
ε ε� �
= � �� �
† † 1SS S S= =Current conservation
2 2 2 2
2 2 2 2
1
1
′ ′+ = + =
′ ′+ = + =
r t r t
r t r t
0
0
t r r t
t r r t
∗ ∗
∗ ∗
′ ′+ =′ ′+ =
2 2
2 2
transmission probability
= 1 reflection probability
t t
r r
τ
ρ τ
′= = =
′= = − =
r r’
t
t’
EXAMPLE: DELTA BARRIER
2 2
02 ( )2
ψ δ ψ εψ∂− + =∂
�U x
m x
( ) ( )
02
02
02
2( 0) ( 0) (0)
2
is continuous at 0
Derivative discontinuity t
a 0,
2L L R R R R
L L R R
L L R R R R
mU
mUika ikb ika ikb a b
a b a b
mUa b a
x
a
x
b b
ψ ψ ψ
ψ
′ ′+ − − =
� − + − + = +
� + = +
� − + − +
=
=
=
�
� �
( )0
0 0
0
0 0
22 2
22 2
iUiU iU
SiU
iU iU
εε ε
εε
ε ε
−� �� �+ +� �=� �−� �+ +� �
22
2mk
ε=�
ikx ikx ikx ikxL L R Ra e b e a e b e− −+ +
0 x
U0
Schrödinger eq.
SCATTERING STATES( ) ( )
( )
left
right
( )(e )1
)( , )
(
L L
R
ik x ik x
ik xF
L
e
vt
he
r
x
ε ε
εεε
εϕ
−� += �
( ) ( )
( )
right
left
( )((
( )
e )1
, )
R R
L
ik x ik x
ik xF
R
r
t
e
xhv
e
ε ε
ε
εε
ϕε
−
−
� + ′=
′
�
Complete orthonormal basis of current carrying states
rt
Li k xe
τ δεδ =L
eh
I
τ δεδ =R
eh
I
Ri k xe−
r’
t’
FILLING OF THE SCATTERING STATES BY THE RESERVOIRS
Lϕ Rϕ
r r’
t
t’Rk
)( kε
Lµ
Lk
)( kε
RµVe
Transport described using eigenstates in equilibrium with different reservoirs
[ ] [ ]
[ ]
( ) ( )( ) ( ) ( )
(( )
( )
) ) )( (
δ τ εε δ ε
ε
ε δ δε
ε τ ε
ε
ε
= − = −
= −�
RL L L
L
R R
R
f I ff I f
f
eI V
h
eV d fI
h
CONDUCTANCE from TRANSMISSION
[ ]( ) (( ))( )L Rfe
I V dh
fε εε τ ε= −�
2
0
( )lim ( )V
I e fG d
V hεε τ εε→
∂= =∂�
Current
Conductance
LANDAUER FORMULA
MULTICHANNEL2
ii
eG
hτ=
2
τ= eG
h2 (spin)×
2
0 38�S 1/ 26k�= � �e
Gh
If over the energy range between µL and µR( )τ ε τ=
[ ] ( )2
((( ) )) ετ ε τµτε µ= − = − =� L LR R
e e eI V d V
h h hf f
TRANSPORT from TRANSMISSION
{ττττ1111,..., ,..., ,..., ,..., ττττΝΝΝΝ}Normal state Superconducting
state
Conductance
Shot Noise
Josephsonsupercurrent
Shot NoiseCoulomb Blockade
…
(‘86)
(‘91)
(‘99)
(‘81)
(‘89)
(‘95)
(‘01)(‘01) Shapiro Steps
Current-Voltagecharacteristic
…
Thermopower
MESOSCOPIC PIN CODE
CONDUCTANCE QUANTIZATION
ττττ�=1 for all open channels
0<ττττN<1 for the last channel
( )2 22 2
1i N
e eG N
h hτ τ= = − +
van Wees; Wharam (1988)
T < 1 K
Counter-support
Insulating layer
Pushing rod
QUANTUM POINT CONTACTS IN METALS
0.0 0.2 0.4 0.6 0.8 1.00
2
4
6
8
��������������
�� ����������
0.0 0.1 0.2 compression (nm)
tunnel regime
Aluminum T = 50 mK
N state (H=0.2 T):
�� ��
�
-4 -2 0 2 4
-3
-2
-1
0
1
2
3 ∆ = 178 µVIc = 22.9 nA
I / I c
VJ / (∆ / e)
DC Josephsonsupercurrent peak
Multiple AndreevReflections
Counter-support
Insulating layer
Pushing rod
SUPERCONDUCTING ATOMIC CONTACTS
S state (H=0):
SUPERCONDUCTING RESERVOIRS
Ideal leadIdeal leads s
• New quasiparticles
• New scattering mechanism
NORMAL REFLECTION
)( kε
)( kε
An incident up-spin electron is reflectedas an up-spin electron of the same energy
ANDREEV REFLECTION
)( kε)( kε
S
An incident up-spin electron with energy µ+ε« drags » with it
a down-spin electron of energy µ−εto form a Cooper pair in the superconductor
leaving behind a hole in the spin-down band
Pairingpotential
ie φ∆ = ∆
COUPLING OF e���� AND h����
0 /Fvξ = ∆�
0ξ
S
0ξ
SN
N-S interface
Bulk S
BOGOLIUBOV-de GENNES EQUATION
Spin-up electrons ∂ =∂� N
ui H u
t
Spin-down holes ∗∂ = −∂� N
vi H v
t
Conjugated particles
† † †, ( ( ) )h e h ec cψ ψ ∗= =
( )N
N
H u ui
H v vt
µ
µ∗ ∗
− ∆� �� � � �∂=� �� � � �� �∆ − − ∂� � � �� ��
( ) /( )( )E
ik E x iEtu Ee e
v Eψ ± ± −� �
= � �� �
�
Propagating solutions of BdG equation
SUPERCONDUCTING QUASIPARTICLE STATESelectron-like and hole-like
( )
( )
2( ) 2
( )
22( )
1sgn( )
12 sgn( )
e h
ie h
e h
uE E E
v e
k m E E
φ
µ
= ± − ∆∆
= ± − ∆�
for or 0
( ) ( )1 0and
( ) ( )0 1e h
e h
E
u E u E
v E v E
∆ ∆ →
� � � �� � � �→ →� � � �� � � �
� � � �� � � �
�
-3 -2 -1 0 1 2 30
1
2
3
4
E ∆
( )( )n F
n E
n ξ ε=
Density of states
evanescent
ANDREEV REFLECTION AMPLITUDEie φ∆
x
( )1 0,
0 1e
e
ua E
vφ
� �� � � �+ = � �� � � �
� � � � � �
( ) ( )( )221, sgne
ie
va E E E E
u e φφ = = − − ∆∆
Matching at x=0
-3 -2 -1 0 1 2 3
2φ π+
φ π+
φ
E ∆-3 -2 -1 0 1 2 3
0.0
0.2
0.4
0.6
0.8
1.0
E ∆
( ),a E φ ( )( ),Arg a E φ
ALL TOGETHER NOW…
Normal + Andreev scattering
s s
s sRegion I
(N)Region II
(N)
MULTIPLE ANDREEV REFLECTIONS (MAR)
2E neV+
( )2 1E n eV+ +
E
E eV+
2E eV+
3E eV+
MAR + NORMAL SCATTERING
2E neV+
( )2 1E n eV+ +
( )2 1E n eV+ −
E
Scatterer
Region I(N)
Region II(N)
Superconductor (L)Superconductor (R)
SCATTERING STATES
2E neV+
( )2 1E n eV+ +
( )2 1E n eV+ −
nB2n na A
nA2n na B
1nD −
2 1 1n na C− −
1nC −
nC
2 1n na D+
nD
( )= +na a E neV
2 0
2 1
n n n n
n n n
B a A JS
C a D
δ
+
+� � � �=� � � �
� � � �
2
2 1 1
n n n
n n n
A a BS
D a C∗
+ −
� � � �=� � � �
� � � �
electrons
holes
1/ 22( ) 1 ( )� �= −� �
J E a E
a.c. JOSEPHSON CURRENTS
( ) ( )( ) ( )2 /1 2 0 2
N N N Ne e h h i E neV tik x ik x ik x ik x
n n n n n n nn
x a A J E e B e A e a B e eψ δ − +− −
∈
� �= + + + +� � �
( ) ( )( )2 1 /2 2 1 2 1
N N N Ne e h h i E n eV tik x ik x ik x ik x
n n n n n nn
x C e a D e a C e D e eψ − + +− −+ +
∈
� �= + + +� �
�
0 2eVω = �
( )( )( )( )
2 2 2
02 2( )
2tanh
12δ
∗− −
∗ ∗ ∗+ + +
� �� �+� � � �= − �� �� �+ + −� � � �� �� �
�
k k k k
k kn n k n n k n n kB
n
J E a A a Ae EI eV dE
a a A A B Bh k T
( ) ( ) ( )0 0, ik tk
k
I V t I V e δ ω+∞
+
=−∞
= Currents at all harmonics of
Josephson frequency
dc MAR CURRENT
( ) ( ) ( )( )2
2 2 20 0 0 0 2
2tanh 1
2 n n nnB
e EI eV dE J E a A A a A B
h k T∗� �� �� � = − + + + − �� �� �
� � � �� � �
� � � � ��
�
�
�
�
���τ ��� ���
���
���
������������∆��
�����∆��� ����� ! �"�#$%���"����������&�'(�� ���)� *�"�#$����"���++�����+����,�(���-�)� *�"�%.#���"�����������
MULTIPLE CHARGE TRANSFER PROCESSES
���
��
Blonder, Tinkham, Klapwijk (‘82)
�∆ ����∆ ����∆ ���
��
τ
S S
SUPERCONDUCTING ATOMIC CONTACTS IV
0 1 2 3 40
1
2
3
����������∆��
eV / ∆
τ3=0.09
τ2=0.11
τ1=0.55
exp 0 ( , )n
nI I V τ=
Number of channels fixed by number of valence orbitals
Fingerprint ofmesoscopic PIN code
0
5
10
15-60 -40 -20 0 20 40 60
{ 0.21, 0.20, 0.20 } { 0.40, 0.27, 0.03 } { 0.68, 0.25, 0.22 } { 0.996, 0.26 } Theory Poisson's Limit
�,�������/���������01��
�� ( nA )
-20 -10 0 10 20
������������/���������01��
Shot Noise in the Normal State
�
τ
�
Spectral density SI
Single channel ,� ��������/τ �
Multi-channel ,� ����������/ Στ�� ��Στ� �
Partition Noise {τι} known from the superconducting fingerprint
1 2 3 410-4
10-3
10-2
��2�����"�����"������3
��������
������∆
�,����������01��
SHOT-NOISE IN SUPERCONDUCTING STATE
�� � �
1 2 3 40
1
2
3
4
5
������∆
�45����
�����2�����"�����"������3
Effective Charge Q* = SI / 2eI
����6�$�.�� 7
$��' �"�6��8���99� �"�:��;�)��� �!�'��< =�� "�#$%��+"������������>��'��'(�� ���)� *������?��@ �����:����� ��������6�$�.�� 7
0 1 2 3 4 50.1
1
10 �2�����+"����&��3
�2�����"���&&"�����"������3���
�2���+�"����&������3
��6�$�.�� 7
Q *
/ e
eV / ∆
LARGE CHARGE QUANTA
SUMMARY
Scattering theory provides a rigorous framework to understand transport phenomena
Simple extension of equilibrium statistical mechanics:
( )
( )
ˆ
and eigenstates and eigenenergies of system
ˆ ,
scattering states
ε
ε
ε µ
= −
= −
nn
n
s
F
sss
A n A n f E
n
A s A s f
s
T
At equilibrium
Each scattering state at equilibrium with a different reservoir
Out of equilibrium
In superconducting mesoscopic systems the BdG equation can be used to construct scattering states in much the same way that the Schrödinger equation is used
in normal mesoscopic systems
� S. Datta : Electronic transport in Mesoscopic systems, Cambridge University Press(1997).
� C. Glattli, in « Quantum Entanglement and information processing », Les Houches Lecture, Session LXXIX, Elsevier (2004).
� S. Datta, P. Bagwell, and M.P. Anatram, « Scattering Theory of Transport for Mesoscopic Superconductors », Phys. Low-Dim. Struct., 3 pp1-58 (1996).
� M. Büttiker, Phys. Rev. B, 46 12485 (1992).
� Th. Martin and R. Landauer, Phys. Rev. B, 45 1742 (1992).
� G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B, 25 4515 (1982).
� C. Beenakker and H. van Houten, Quantum Transport in SemiconductorNanostructures, in `̀Solid State Physics'', Ehrenreich and Turnbull Eds., Academic Press(1991).
BIBLIOGRAPHY