Cooper pair splitting in electronic nanostructures
Andreas Baumgartner, Jens Schindele, Lukas Hofstetter, Szabolcs Csonka,
Samuel d‘Hollosy, Gabor Fabian and Christian Schönenberger
Department of Physics, University of Basel
Klingelbergstrasse 82, 4056 Basel, Switzerland
• Motivation:
electron entanglement in solids
• InAs quantum dot / superconductor hybrids
- Cooper pair splitting by Coulomb blockade
Outline
• Finite-bias spectroscopy on a Cooper pair splitter
• Carbon nanotube Cooper pair splitter:
- near-unity efficiency
• Summary
po
ster
Electron entanglement in solids
Two spatially separated particles are
entangled if their state can not be
prepared starting from a product state
using only local operations and classical
communication.
C.W.J. Beenakker, in "Quantum Computers, Algorithms and Chaos",
International School of Physics Enrico Fermi, vol. 162 (2005)
Our reference:
spin-singlet, maximally entangled:
→ Non-classical particle correlations
[ ]21212
1 ↑↓−↓↑=s
Electron entanglement in solids
Entanglement is
relevant in nature:
Gauger et al., Phys. Rev.
Avian (warm, wet!) compass:diluted dipolar-coupled
Ising magnet LiHo0.045Y0.955F4
susc
epti
bili
ty
Gauger et al., Phys. Rev.
Lett. 106, 040503 (2011)
C.H. Bennet and D.P. DiVincenzo,
Nature 404, 247 (2000)
Ghosh et al., Nature 425, 48 (2003)
Entanglement as a resource:
well-controled spatially separated particle pairs
(possibly all-electronic, on-chip)
e.g. in state teleportation:
temperature
↓↓
Entanglement in a 3D non-interacting electron gas
↓
wave function overlap (overlap ~ħ/pF= λF)and anti-symmetrization
→ „local singlet“
(Pauli principle, exchange interaction)
→ entanglement
→ decays with distance ~λF
↓↓
S. Oh, J. Kim, Phys. Rev. A 69, 054305 (2004)
↓
↓↓
Bosons: no Pauli principle!
enta
ngl
emen
t m
easu
re
↓↓
Entanglement in a superconductor
Superconductivity due to pairing of electrons (attractive electron-electron interaction via phonons)
Cooper pairs are singlets!
Bardeen, Cooper and Schrieffer:
BCS wave function:
( )Φ+=Φ ∏ ++
vacuum
↓↓
( ) 0Φ+=Φ ∏ +↓−
+↑
kkkkk aavu
„Size“ of a Cooper pair in aluminum:
~coherence length ξ0=1 µm
ξGL=140 nm (Ginzburg-Landau)→ nanostructures!
Cooper pairs
many electrons in the volume of a Cooper pair,
average electron spacing ~0.1 nm << ξ.
spin correlation between
two points due to many Cooper pairs
→ entanglement decays ~λF.
↓↓↓↓
Entanglement in a superconductor
↓↓↓↓
S. Oh and J. Kim, Phys. Rev. B 71, 144523 (2005)
entanglement measure
(concurrence)
At low temperatures only complete Cooper pairs
can be removed from the superconductor
↓↓↓
↓
Entanglement in a superconductor
not allowed: ↓
↓
allowed
Removing single electrons
NOT from the same Cooper pair is suppressed
→ two excitations in superconductor with energy ∆
(“any” excitations allowed in
non-interacting electron gas)
↓↓↓
(local) Andreev
reflection
energy conservation
~ momentum conservation
~ angular momentum conservationSN
↓↓ ↓
↑
x
E
↓
Transport mechanisms at N/S interfaces
x
SN1
↓↓ ↓
↑
x
N2(non-local)
Crossed Andreev
reflection
+ Multiple Andreev reflection, Andreev bound states, ...
Reverse process:
Cooper pair splitting
Al Pd
Experiments on metallic structures
A. Kleine, et al., EPL 87, 27011 (2009)
• Distance between contacts smaller than coherence length
→ nanometer scaled structures
• Other processes may dominate (elastic co-tunneling, charge imbalance, ...)
• Very limited control over crucial electron-electron interactions
(electrodynamic environment)
A. Kleine, et al., Nanotechnology 21, 274002 (2010)
I
Cooper pairs splitting with quantum dots
S
↓
↓ [ ])()()()( 1221 rrrr kkkk −+−+ ΨΨ+ΨΨ
conventional Cooper pair:
spin singlet
( )↑↓−↓↑⊗ ,,
I IQD1 QD2
mobile pairs of spatially
separated entangled electrons?
↓
↓ [ ])()()()( 12212211 rrrr NNNN ΨΨ+ΨΨ
( )↑↓−↓↑⊗ ,,N1 N2
split Cooper pair: I1 I2
correlated electrical currents!
General idea:
J. Torres and T. Martin, Eur. Phys. J. B 12, 319 (1999)
G.B. Lesovik et al., Eur. Phys. J. B 24, 287 (2001), …
With quantum dots:
e.g. Recher et al. Phys. Rev. B 63, 165314 (2001), …
QD1 QD2
N2N1
ΓN1 ΓN2
µN1 µN2
ΓS1 ΓS2
εD1 εD2
S
∆
−∆
initial
state
intermediate state: excitation in superconductor
suppressed ~1/∆
N2N1
µN1 µN2εD2
S−∆
local processes
N2N1 S
final state
N2N1
µN2εD1 εD2
S
∆
−∆
N2N1 S
doubly charged QD
intermediate state: suppressed ~1/U
N2N1
µN1 µN2εD2
S−∆
∆
Recher et al. Phys. Rev. B 63, 165314 (2001)
N2N1
ΓN1 ΓN2
µN1 µN2
ΓS1 ΓS2
εD1 εD2
S
∆
−∆
initial
state
intermediate state: NOT supressed by U or ∆
N2N1
µN1 µN2
S−∆
∆
non-local processes
N2N1 S
final state
N2N1
εD1 εD2
S
∆
−∆
N2N1 S
Elastic zero net current for µ1=µ2
co-tunneling : suppressed ~1/∆
N2N1
µN1 µN2
S−∆
Recher et al. Phys. Rev. B 63, 165314 (2001)
InAs nanowire
N1 N2S
QD2
QD1
InAs nanowire Cooper pair splitter
http://www.cityu.edu.hk/ieeeinec/abstract/samuelson.pdf
Overview: C.M. Lieber and Z.L. Wang,
MRS Bull. 32, 99-104 (2007).
Top gate 1 Top gate 2
InAs nanowire
N1 N2S
QD2
QD1
InAs nanowire Cooper pair splitter
• InAs NW: d ≈ 80nm
• superconductor (Ti/Al), w ≈ 200nm
• top gates with surface oxide, w ≈ 100nm
• Tbase ≈ 20 mK
• QDs: U ≈ 2-4meV
• gap feature: ∆ ≈ 160µV
• very weak cross capacitance
∆G1(∆Vg1) ≈ 1000 x ∆G1(∆Vg2)
L. Hofstetter, et al., Nature 461, 960-963 (2009)
Top gate 1 Top gate 2QD1
Measured quantity
Only Cooper pair splitting depends on QD1 and QD2
Measure G1(Vg2)
δG2 > 0
δR2<0
δUT < 0
δI = δU /R <0
R 200 Ω
V
R RI IU
R=200 Ω
Classical resistor model
‘resistive cross-talk’: conductance change
δG1 has opposite sign as the induced
conductance change on QD2, δG2.
δI1= δUT/R1 <0R1 R2I1 I2G1:=I1/V G2:=I2/V
0'' 11 <≈
V
IG
δδ
UT
GS (
G0)
VSG (mV)1.5 2.0 2.5 3.0
0.0
0.1
0.2
0.3
0.4
Vg1 (mV)Vg1 (mV)
zero bias
G1
(G0)
Measured quantity
Vg1 (mV)
∆G1
Result: correlated currents
∆G1 in the normal state (B > Bc):
• Negatively correlated signal : classical circuit
response (no fitting parameters)
∆G1 (Vg2) in the superconducting state:
• positively correlated non-local signal
• background, G1 ~ 0.15 G0
⇒ several % Cooper pair splitting!
L. Hofstetter, et al., Nature 461, 960-963 (2009)
N2N1
ΓN1 ΓN2
µN1
µN2
ΓS1 ΓS2
S
∆
−∆
-eUN2
Finite-bias spectroscopy
Similar experiments on metallic structures: J. Wei and V. Chandrasekhar, Nature Phys. 6, 494 (2010)
N2N1 S−∆
UN2I1
Measure G1(UN2)
µN1=µS
Local transport through QD1
independent of UN2
Qualitatively:
[Falci et al., EPL 54, 255 (2001)]
QD1: - ∆=130 µeV
- Γ=500 µeV > ∆ (state SB)→ sequential tunneling of CPs dominant
- state SB shows (split)
Kondo ridge at B>Bc (not shown)
QD2: - held at constant gate voltage
- open regime (not shown):
UN
1(m
V)
G1 (G0/10)
Quantum dot characteristics
- open regime (not shown):
→ D2(E) ≈ constant
Ug1 (mV)
Hofstetter et al., Phys. Rev. Lett. 107, 136801 (2011)
UN
1(m
V)
G1 (G0/10)
UN
2(m
V)
∆G1 (G0/100)
‚Non-local‘ experiments
Subtract local processes:For UN2 >>∆ non-local processes can be neglected(large density of states in S)
∆G1=G1(Ug1 , UN2 ) - G1(Ug1 , UN2=1mV)
Ug1 (mV)
UN
2(m
V)
Ug1 (mV)
G1 (G0/10)
Ug1 (mV)
Hofstetter et al., Phys. Rev. Lett. 107, 136801 (2011)
UN
1(m
V)
G1 (G0/10)
UN
2(m
V)
∆G1 (G0/100)
‚Non-local‘ experiments
Ug1 (mV)
UN
2(m
V)
Ug1 (mV)
G1 (G0/10)
Ug1 (mV)
Positive signal on resonances (pair splitting)
Negative signal slightly off resonance (EC)
asymmetric around resonances
simple model: different mechanisms probe DOS at
different energies
Hofstetter et al., Phys. Rev. Lett. 107, 136801 (2011)
UN
1(m
V)
G1 (G0/10)
UN
2(m
V)
∆G1 (G0/100)
‚Non-local‘ experiments
Ug1 (mV)
UN
2(m
V)
Ug1 (mV)
G1 (G0/10)
Ug1 (mV)
Ug1 (mV)
UN
2(m
V)
∆G1 (G0/1000)
Control
experiment:
Cooper pair splitting on carbon nanotubes
Strong QD1-QD2 tunnel coupling
limits efficiency to 50%
Device Charcteristics: - ∆ ≈ 130 µeV
- UQD1 = 5meV; UQD2 = 8meV
- typical Γ‘s from 120 to 500 µeV
- asymmetric coupling to S and N
contacts (ΓN/ΓS ≈ 10 to 100)
Herrmann et al., Phys. Rev. Lett. 104, 026801
(2010)
carbon nanotube
large Cooper pair splitting
Near-unity Cooper pair splitting efficiency
21
2
GG
Gs CPS
+=
CPS efficiency:
s=90%
21 GG +
1GGCPS ∆=Experiment:
UN
2(m
V)
∆G1 (G0/100)
Zero-bias
Cooper pair splitting
Finite-bias
Spectroscopy:identify competing
Summary and small-print
Ug1 (mV)
identify competing
processes
Up to 90%
Cooper pair splitting
efficiency
- What is the correct GCPS?
- What determines splitting efficiency?
- How to detect entaglement in
this system (optimal entangelment witness)?