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)?