Electron quantum optics with edge states Objective of the mini-course : an introduction

Electron quantum optics with edge states Objective of the mini-course : an introduction

Create, manipulate and detect single electron wave pacquets

« à la manière de »

Quantum optics with single photon states in the 90’s

(Materials + Tools) + (Principles + Implementation)

First lecture : 2d electron materials & devices

o High-mobility semiconducting heterostructures o Quantum Hall Effect

o Electrons in graphene

o Electron reservoir and quantum transport o Scattering by a quantum point contact o The mesoscopic capacitor

Part 1 : 2d electron systems

Part 2 : 2d electron devices

Semiconductor Nanostructures, Thomas Ihn, Oxford University Press (2010)

Second lecture : quantum optics with edge states

o Electron optics principles o Single electron sources

o The mesoscopic capacitor source: average current and noise

o Partitioning single electrons

o Colliding indistinguishable electrons

o Decoherence and charge fractionalization

Part 3 : Introduction to electrons optics

Part 4 : Quantum optics experiments

Electron quantum optics in ballistic chiral conductors,

Bocquillon et al., Annalen der Physik 526, 1 (2014)

2D electron materials

Bernard Plaçais

placais@lpa.ens.fr

Gwendal Fève Jean-Marc Berroir

Princ i ple of 2D electron system

modulation doped heterostructures

GaAs AlGaAs

n doping (Si)

GaAs AlGaAs

+

Si⁺

+ +

Electron diffusion

+

equilibrium

Electric fied

counterbalancing diffusion

𝐸𝐸

conduction band profile quantum well

width ~ 10 nnm bound state Donor states

The AlGaAs/GaAS heterostructure

Schrödinger electrons : 𝜀𝜀 =

Effective mass : 𝑚𝑚

= 0,067 × 𝑚𝑚

E

GaAs AlGaAs

GaAs + Si

100 nm

2DEG

Al

Ga

As

∆=1.52+x.1.45eV GaAs

∆=1.52eV

Si

λ

~ 50 nm l

~ 10-20 µm

l

~ 15 µm

(T_~100 mK)

Molecular Beam Epitaxy

Laboratory of Photonics and Nanostructures,

CNRS-LPN, Marcoussis, France

o Density of states : ρ 𝜀𝜀 = 𝑔𝑔

𝑔𝑔

𝑚𝑚

⁄ 2πℏ

𝑔𝑔

= 2; 𝑔𝑔

= 1 (𝑒𝑒

ρ 𝜀𝜀

≈ 44 𝑓𝑓𝑓𝑓 𝜇𝜇𝑚𝑚

≡ 2.6 𝑛𝑛𝑚𝑚 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺)

o Electron density : 𝑛𝑛 ≈ 10

− 10

𝑐𝑐𝑚𝑚

(determined by doping) o Fermi-energy : 𝜀𝜀

=

𝑠𝑠𝑔𝑔_𝑣𝑣𝑚𝑚^∗

(≈ 10 𝑚𝑚𝑒𝑒𝑚𝑚 𝐺𝐺𝑎𝑎 3 10

𝑐𝑐𝑚𝑚

)

o Fermi wave lentgh: λ

=

(≈ 25 𝑛𝑛𝑚𝑚 𝐺𝐺𝑎𝑎 3 10

𝑐𝑐𝑚𝑚

) (>> metals !!) o Fermi velocity : 𝑣𝑣

=

(≈ 7 10

𝑚𝑚 𝐺𝐺

𝐺𝐺𝑎𝑎 3 10

𝑐𝑐𝑚𝑚

) (<< metals !!)

A (massive) 2d electron gas

ρ(ε)

E_C E₀ E_F E₁

ε

Conductivity / mobility of diffusive 2DEGs

o Good conductivity : 𝜎𝜎 = 𝑅𝑅

𝐿𝐿 𝑊𝑊 ⁄ (𝜎𝜎

≈ 1 − 10

𝑂𝑂𝑂𝑚𝑚𝐺𝐺)

o High mobility : 𝜇𝜇 𝑛𝑛 ≡ 𝑣𝑣

⁄ 𝐸𝐸 = 𝜎𝜎 𝑛𝑛𝑒𝑒 ⁄ 𝜇𝜇 ≈ 10

− 10

𝑐𝑐𝑚𝑚

𝑚𝑚

𝐺𝐺

o Diffusion const. : 𝜎𝜎 = 𝑒𝑒

ρ 𝜀𝜀

× 𝐷𝐷 𝜀𝜀

(𝐷𝐷 ≈ 10 − 10

𝑐𝑐𝑚𝑚

𝐺𝐺

)

o Diffusion length : 𝑙𝑙

= 2𝐷𝐷 𝑣𝑣 ⁄

(𝑙𝑙

≈ 30 𝑛𝑛𝑚𝑚 − 300𝜇𝜇𝑚𝑚 ‼)

High-mobility GaAs

N.D. Schlom, L.N. Pfeiffer, Nature Materials 9, 881–883 (2010)

Room temperature (Nokia phones, …)S

Ultra-clean limit (low-temp. physics)

Strong effect of temperature

(phonons)

Toward ballistic transport Shubnikov-de-Haas

𝑅𝑅

− 𝑚𝑚

⁄ 𝐼𝐼

−𝐵𝐵 𝑛𝑛𝑒𝑒 ⁄ 𝑅𝑅

− 𝑚𝑚

⁄ 𝐼𝐼

⁄ 𝑛𝑛𝑒𝑒𝜇𝜇

⁄ 𝜇𝜇 ≈

𝑚𝑚

𝐺𝐺

𝑛𝑛 ≈

Zeeman splitting

µ B = 𝑙𝑙_𝑒𝑒

𝑟𝑟_𝑐𝑐 ≈20

classical quantum

I

V

Ballistic transport Quantum Hall effect

𝑅𝑅

− 𝑚𝑚

⁄ 𝐼𝐼

−𝐵𝐵 𝑛𝑛𝑒𝑒 ⁄ 𝑅𝑅

− 𝑚𝑚

⁄ 𝐼𝐼

⁄ 𝑛𝑛𝑒𝑒𝜇𝜇

⁄ 𝜇𝜇 ≈

𝑚𝑚

𝐺𝐺

𝑛𝑛 ≈

Zeeman splitting QHE filling factor: 𝜈𝜈 = 𝑛𝑛 𝜑𝜑

⁄ 𝐵𝐵 (𝜑𝜑

= 𝑒𝑒 𝑂 ⁄ )

µ B = 𝑙𝑙_𝑒𝑒

𝑟𝑟_𝑐𝑐 ≈10

classical quantum integer fractional

I

V

Classical motion

B ^

cyclotron orbit

y

radius r_c cycloid

Quantum mechanical motion

Landau energy levels

2DEG under perpendicular magnetic field

Cyclotron frequency : 𝜔𝜔

𝑒𝑒𝐵𝐵 𝑚𝑚 ⁄

𝑣𝑣

⁄ 𝑟𝑟

Hamiltonian : 𝐻𝐻

𝑝𝑝

𝑒𝑒𝐺𝐺

𝑚𝑚(𝑦𝑦) …../…..

Flux quantum : 𝜑𝜑

𝑂 𝑒𝑒 ⁄ ; Magnetic length : 𝑙𝑙

𝜑𝜑

⁄

; Landau levels : 𝐸𝐸

𝑛𝑛

ℏ𝜔𝜔

Drift velocity : 𝑣𝑣

𝐸𝐸 𝐵𝐵 ⁄ (see T. Ihn) Edge channels follow equipotential lines

Edge states : ballistic transort

http://www.metas.ch/LesHouches/downloads/talks/22_Buttiker.pdf

Localized states

No back-scattering

extended states

back-scattering possible

Ballistic Diffusive

: 𝑅𝑅

𝑂𝑂 ; 𝑅𝑅

𝑂 ⁄

𝑅𝑅

≠ 𝑂𝑂 ; 𝑂 ⁄

𝑅𝑅

𝑂 ⁄

Edge channels follow equipotential lines !!

Graphene

2degs in semiconducting heterostructures

2degs in graphene

Electrons in 2D graphene

(K. Novoselov et al., Science 2004)

16

High-mobility graphene

Band structure of Graphene

(tight-binding model, Wallace 1947)

o Dirac electrons : 𝜀𝜀 = ℏ𝑣𝑣

𝑞𝑞 (linear dispersion relation) o Large Fermi velocity : 𝑣𝑣

= 10

𝑚𝑚𝐺𝐺

o Effective mass : 𝑚𝑚

= 0 (no band-gap) o 2-component spinnor ! : 𝜓𝜓 = 𝑒𝑒

1

𝑒𝑒

(chiral transport)

Schrödinger vs Dirac electrons

D ir ac point

Dirac electrons : Linear dispersion Mass-less fermions

Constant Fermi-velocity Electron/hole symmetry No band-gap !!

Wave function spinnor

Schrödinger electrons : parabolic dispersion

Massive fermions Electrons ≠ holes band-gap

Wave function scaler

Ballistic graphene

L. Wang et al., Science 342, 614 (2013)

Comparison with AlGaAs/GaAs 2DEGs

2013 : graphene /hBN

2005 : graphene /SiO2

Room temperature applications ?

Very weak effect of phonons

More 2D electron materials

o 1980’s : Si MOS-FETs

o 1990’s : III-V GaAS, InAs, InP, GaN, o 2010’s : LaAlO3-SrTiO3, ZnO,

o 2005’s : Bilayer graphene

o 2005’s : graphene

o 2010’s :Topological Insulators, Bi2Se3

o 20XX’s : New materials to be discovered …..

Schrödinger 2d electrons

Dirac 2d electrons

A. Tsukazaki , Nat. Mat. 9, 889 (2010)

B. D. Hsieh, Nature 460, 1101 (2009)

2d electron devices

o High-mobility semiconducting heterostructures o Quantum Hall Effect

o Electrons in graphene

o Electron reservoir and quantum transport o Scattering by a quantum point contact o The mesoscopic capacitor

Part 1 : 2d electron systems

Part 2 : 2d electron devices (GaAs only)

Semiconductor Nanostructures, Thomas Ihn, Oxford University Press (2010)

Example 1 : a Mach-Zehnder interferometer

Contacts (mustard yellow)

2D electron gas (blue)

Split gates (green)

D. Mailly, Laboratory of Photonics and Nanostuctures, CNRS-Marcousssis, France

Example 2 : our Hanbury Brown and Twiss interferometer

source contact Direct Current

reservoir

mesoscopic capacitor

split gates of the Quantum Point Contact

burried 2DEG

Optical image

SEM image SEM Image

Y. Jin, Laboratory of Photonics and Nanostuctures, CNRS-Marcousssis, France

Single Electron AC-Source Electron Beam Splitter

-V

B 

I

-V

contact black body for photons Fermi Dirac distribution

V e ε

) ( ε

f

f

( ε )

ε

µ

µ

0 1 1 0

Conductance quantization (one edge channel)

h V I e

=

conductance quantum : (𝑒𝑒

⁄ 𝑂 = 38,740459 𝜇𝜇𝜇𝜇)

+∞

∫

=

∆

0

2

S ( ω ) d ω

I

τ τ

ω I t I t e

d S ^{( ) = 2} ∫ ^{∆ ( ) ∆ ( +}

0 ) 0 ( ω = =

S no noise !

The Quantum Point Contact (QPC)

Metallic gate

Edge channels follow equipotential lines

B 

1 g

g V

V =

V

Metallic gate

Edge channels follow equipotential lines

V

1

2 g

g V

V <

B 

T ₂

The Quantum Point Contact (QPC)

Metallic gate

Edge channels follow equipotential lines

V

1 2

3 g g

g V V

V < <

T ₁

B

The Quantum Point Contact (QPC)

G (e2/h)

0 1 2 3

0 1 2 3

V QPC (V)

−1 −0,5 0

−1 −0,5 0

4

Vg

Vg

T ^B



∑

=

n

T

h G e

2

Tunable electronic beam splitter e

/ h h

e / 2

Vg

Vg

T

^B



Conductance quantization

T

=1/2

The transmission of a Quantum Point Contact (QPC)

M. Büttiker , Phys. Rev. B41, 7906 1990

See e.g. http://www2.le.ac.uk/departments/physics/people/mervynroy/lectures/tunnelling-results.pdf

Models of QPC transmission

𝐺𝐺𝐺𝐺𝑠𝑠𝑠𝑠𝑙𝑙𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑛𝑛𝑎𝑎 𝑏𝑏𝐺𝐺𝑟𝑟𝑟𝑟𝑝𝑝𝑒𝑒𝑟𝑟 ∶

V = 𝑚𝑚 =𝑚𝑚₀−¹₂ 𝑚𝑚𝜔𝜔_𝑥𝑥² 𝑥𝑥² +¹₂ 𝑚𝑚𝜔𝜔_𝑦𝑦² 𝑦𝑦² 𝑇𝑇_𝑛𝑛 = 1 +𝐸𝐸𝑥𝑥𝑝𝑝[(𝜖𝜖 − 𝑛𝑛+¹₂ ℏ𝜔𝜔_𝑦𝑦)/ℏ𝜔𝜔_𝑥𝑥] ⁻¹

𝑇𝑇_𝑛𝑛 = 1 +𝐸𝐸𝑥𝑥𝑝𝑝 − 𝜋𝜋 𝜔𝜔_𝑦𝑦/𝜔𝜔_𝑥𝑥 𝑦𝑦/𝑙𝑙_𝐵𝐵 ^{2 −1}

1𝐷𝐷 𝑏𝑏𝐺𝐺𝑟𝑟𝑟𝑟𝑝𝑝𝑒𝑒𝑟𝑟 ∶ 𝑇𝑇_𝑛𝑛 = 1 +_{4𝐸𝐸(𝑉𝑉}^𝑉𝑉02

0−𝜖𝜖)𝐺𝐺𝑝𝑝𝑛𝑛𝑂² (𝛼𝛼𝑏𝑏)] ⁻¹

QPC shot noise

conductance is transmission 𝐼𝐼 =

∫ 𝑇𝑇(𝐸𝐸) 𝑓𝑓

− 𝑓𝑓

𝑠𝑠𝐸𝐸 = T

V

transmission is stochastic => current is noisy 𝜇𝜇

= 2

∫ 𝑇𝑇 𝑓𝑓

1 − 𝑓𝑓

+ 𝑅𝑅 𝑓𝑓

1 − 𝑓𝑓

+ 𝑅𝑅𝑇𝑇 𝑓𝑓

− 𝑓𝑓

𝑠𝑠𝐸𝐸

equilibrium : temperature θ 𝜇𝜇

= 4

𝑇𝑇 ∫ 𝑘𝑘𝜃𝜃𝑓𝑓𝑓(𝐸𝐸)𝑠𝑠𝐸𝐸 = 4𝐺𝐺𝑘𝑘𝜃𝜃

transport : partition noise 𝜇𝜇

= 2e𝑇𝑇 1 − 𝑇𝑇

𝑚𝑚 = 2𝑒𝑒𝐼𝐼 1 − 𝑇𝑇

Several modes 𝐺𝐺 = 𝑒𝑒

𝑂 � 𝑇𝑇

; 𝜇𝜇

= 2𝑒𝑒

𝑂 � 𝑇𝑇

1 − 𝑇𝑇

𝑛𝑛

; 𝑓𝑓 = 𝜇𝜇

2𝑒𝑒𝐼𝐼 = ∑ 𝑇𝑇

1 − 𝑇𝑇

∑ 𝑇𝑇

0,5 1,0 1,5 2,0

0,0 0,1 0,2 0,3

∆ T

(K)

transmission G/G

-50 0 50

0 1 2

Vg (mV) G (2e2 /h)

Y. Blanter, M. Buttiker, Phys. Rep. 336, 1 (2000); Glattli’s group, e.g. PRL 90, 176803, (2003)

Exemples of QPC noise

PIB=f(Vbias), T=1/2, nu=3, B=2,75T

PIB (Vrms^2)

0 1e−09 2e−09 3e−09 4e−09 5e−09 6e−09 7e−09

0 1e−09 2e−09 3e−09 4e−09 5e−09 6e−09 7e−09

Vbias (µV)

−1 000 −500 0 500 1 000

−1 000 −500 0 500 1 000 Titre

PIB (Vrms^2)

0 5e−10 1e−09 1,5e−09

2e−09

Bruit

V(V)

0 2e−05 4e−05 6e−05

Bruit

Transmission

partition noise : 𝜇𝜇

∝ 𝑇𝑇 1 − 𝑇𝑇

partition noise Thermal noise

Thermal crossover 𝜇𝜇

= 2𝑒𝑒𝐼𝐼 tanh

𝑒𝑒𝑚𝑚

2𝑘𝑘𝜃𝜃

T

B

The mesoscopic capacitor : an on-demand single electron emitter

mesoscopic capacitor

split gates of the

Quantum Point Contact

Single Electron Source

The Mesoscopic capacitor

Quantum dot

VG

V

B

2 ε Φ π

= ∆

Christen-Büttiker PRL 70, 4114 (1993) Gabelli et al. Science 313, 499 (2006)

Numbers

∆≈ 2 𝐾𝐾; 𝐷𝐷 = 0 → 1 Quantum capacitance effect !

𝐶𝐶

= 𝑒𝑒

𝜌𝜌 𝜀𝜀

Quantization of the charge relaxation resistance 𝑅𝑅

= 𝑂

2𝑒𝑒

The charge relaxation resistance

Gabelli et al., Science 313, 499 (2006)

Gabelli et al., Rep. Prog. Phys. 75, 126504 (2012)

-0,74 -0,72 -0,85 -0,84 -0,83

2 4 6 8

Sample E1 C ω/2π = 1.085 GHz

R_q= h / 2e² A

Im(Z) (h/e2 )Re(Z) (h/e2 ) ^{Sample E3}

ω/2π = 1.2 GHz R_q= h / 2e²

D C_µ = 2.4 fF

V_G (V)

B C_µ = 1 fF

-0.05 0.00 0.05 0.10

-0.91 -0.90 -0.89

-0.02 0.00 0.02 0.04 -0.2 -0.1 0.0 0.1 0.2 0.3

f = 515 MHz

Conductance G (e2 /h)

V

(V)

f = 180 MHz f = 1.5 GHz

𝜌𝜌 𝜀𝜀

oscillations in the dot

𝑅𝑅

=

(coherent) 𝑅𝑅

=

(incoherent)

Raw admittance measurements

Next time :quantum optics with edge states

o Electron optics principles o Single electron sources

o The mesoscopic capacitor source : average current and noise

o Partitioning single electrons

o Colliding indistinguishable electrons

o Decoherence and charge fractionalization

Part 3 : Introduction to electrons optics

Part 4 : Quantum optics experiments

Electron quantum optics in ballistic chiral conductors,

Bocquillon et al., Annalen der Physik 526, 1 (2014)