edge channel interferometry in the quantum hall...

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Edge Channel Interferometryin the Quantum Hall Regime

Stefan Heun

NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore Piazza San Silvestro 12, 56127 Pisa, Italy

e-mail: [email protected]

Motivation

Interference phenomena

• Manifestation of wave nature of electrons

• Applications in solid-state quantum

information technology

2DES in the Quantum Hall regime

• Large electronic coherence length

• Edge channel chiral transport

• Solid state system to study interference

Microsoft Station Q

http://stationq.ucsb.edu/research.html

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Outline

• Introduction to Quantum Hall (QH) Physics

• Edge Channel Interferometry

• Basics of Scanning Gate Microscopy (SGM)

• SGM studies of samples in the QH regime

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Outline





Further reading:

Klaus von Klitzing: The quantized Hall effect, Nobel lecture, 1985 (http://nobelprize.org/nobel_prizes/physics/laureates/1985/klitzing.pdf)

R. J. Haug: Edge-state transport and its experimental consequences in high magnetic fields, Semicond. Sci. Technol. 8 (1993) 131.

http://nobelprize.org/nobel_prizes/physics/laureates/1985/klitzing.pdf

2 Dimensional Electron System

see also: Horst L. Stoermer, Nobel Lecture, December 8, 1998

Hall bar geometry

K. von Klitzing et al.: Physik Journal 4 (2005) No.6 p.37.

Measurement of longitudinal and transversal resistance

Drude model

vmBvEe

dt

vdmF

(=0; steady state)

zz

x

y

xy

x

xxx

Bne

B

j

E

constj

E

.

1

0

yxcy

xycx

jjE

jjE

0

0

Evnejm

ne

m

eBzc

,,

2

0with

Transverse current is zero:

0yj It follows:

Ashcroft / Mermin: Solid State Physics.

Hall effect


Drude model:

Rxy = proportional to B

Rxx = costant in B

Quantum Hall Effect


Behaviour observed at

low temperature in high

mobility samples in high

B ( ):

Plateaux in Rxy

Minima in Rxx

Quantum Hall Effect:

deviations from Drude

model observed around

B = nh / ne

n: filling factor

1c

The Nobel prize in Physics

1985: integer QHE 1998: fractional QHE

http://nobelprize.org/

Landau Levels

Single particle Hamiltonian for an electron in an external magnetic field:

Landau-Gauge: with

For the wavefunction can be written as:

It follows:

rVAepm

H

2

2

1

0,,0 BxA

zeBB

xeyxyiky ,

xxxmm

pxE c

x

2

0

22

2

1

2

0rV

R. J. Haug, Semicond. Sci. Technol. 8 (1993) 131.

Landau Levels

This is the equation of a harmonic oscillator with angular frequency

center of cyclotron orbit at and magnetic length

Drift velocity

meBc

2

0 Bykx

)(

25

TB

nm

eBB

xxxmm

pxE c

x

2

0

22

2

1

2

2

1 jE cj

Landau Levels0 yjy dkdEv


Landau Levels

Degeneracy of each Landau Level:

The number of filled Landau Levels for a given carrier concentration ns is then:

n: filling factor

Bh

eBn deg

eB

hn

n

n ss deg

n


Landau Levels

Yu/Cardona: Fundamentals of Semiconductors

Landau Levels

Quantum Hall droplets

Percolation

Hole droplets

Yu/Cardona: Fundamentals of Semiconductors

.

0

constxy

xx

increasesxy

xx

0

.

0

constxy

xx

Landau Levels

At the edges of the sample, a boundary confinement can be added by a small

V(x) which does not depend on y.

221 Bycyj kVjkE

B. I. Halperin, Phys. Rev. B 25, 2185 (1982).

Introduction to edge channels

xy x

see also: R. J. Haug, Semicond. Sci. Technol. 8 (1993) 131.


xy

edge: 1D “one-way” conductor, skipping orbits

bulk: insulatorx



xy

edge: 1D “one-way” conductor, skipping orbits

bulk: insulator

Quantization of energy levels (Landau levels)

Gap for excitation in the bulk

Transport via edge states

x


Edge states

compressibility:

incompressible:

sch n

sch n


incompressibilities

Example Semiconductor

n

VB

CB

n*

n* independent of B

Edge states

D. B. Chklovskii et al.: PRB 46 (1992) 4026.

compressibility:

incompressible:

sch n

sch n

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Non-interacting vs. interacting picture


• The self consistent potential due to e-e interactions modifies the edge structure

•For any realistic potential the density goes smoothly to zero.

•Alternating compressible and incompressible stripes arise at the sample edge

Incompressible stripes:•The electron density is constant•The potential has a jump

Compressible stripes:•The electron density has a jump•The potential is constant

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Non-interacting vs. interacting picture


• The self consistent potential due to e-e interactions modifies the edge structure

•For any realistic potential the density goes smoothly to zero.

•Alternating compressible and incompressible stripes arise at the sample edge

Incompressible stripes:•The electron density is constant•The potential has a jump

Compressible stripes:•The electron density has a jump•The potential is constant

Each of these edge channels carries

M. Büttiker, Phys. Rev. B 38, 9375 (1988)

VheheI 2

How to probe edge channels?

S. Roddaro et al., Phys. Rev. Lett. 90 (2003) 046805.

How to probe edge channels?

V

t

r 0

V

I(V)

Vg


Quantum Point Contact (QPC)

V

t

r 0

V

I(V)

Vg


we induce backscattering

by reducing this distance

Current conservation:

t + r = 1

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Conductance quantization in QPCs

1D confinement

In 1D systems the current is carried by a finite number of modes (arising from confined subbands) . Each mode contributes two quantum of conductance.

2e2/h

H. van Houten and C. Beenakker: Physics Today, July 1996, p. 22

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Outline





Further reading:

Yang Ji, Yanchui Chung, D. Sprinzak, M. Heiblum, D. Mahalu, and HadasShtrikman: An electronic Mach-Zehnder interferometer, Nature 422 (2003) 415.

Electronic coherence length

Thouless length LT is determined by

inelastic scattering processes (= inelastic

scattering length or quantum coherence

length).

LT2 = Din; D elastic diffusion constant

LT ~ T-1; LT ~ 1.4 mm @ 10 mK

W. Li et al.: Phys. Rev. Lett. 102 (2009) 216801.

Mach-Zehnder interferometer

Y. Ji et al., Nature 422 (2003) 415.


Y. Ji et al., Nature 422 (2003) 415.

V


Y. Ji et al., Nature 422 (2003) 415.

Two-particle Aharonov-Bohm

interferometer

Hanbury-Brown-Twiss interferometer

Topology limits complexity to max. 2

interferometers

I. Neder et al., Nature 448 (2007) 333.

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Motivation: electronic quantum interferometry

at the beam splitters the electrons are backscattered into the counter-propagating edge through two quantum point contacts (QPCs)

The state of the art of electronic quantum

interferometry

we induce backscattering by reducing this distance

B

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The edge picture in the QH effect

a new architecture for QH interferometry: the proposal of Giovannetti et al.

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The edge picture in the QH effect

in this architecture the beam splitters induce mixing between co-propagating channels

a new architecture for QH interferometry: the proposal of Giovannetti et al. coherent inter-channel mixing

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Motivation: a new architecture for QH interferometry

a simply connected QH interferometer: the proposal of Giovannetti et al.

Advantages:

•simply connected topology (no air bridges)

•very small F area, only a few flux quanta are involved

•the device is scalable: it is possible to put many devices in series

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Motivation: a new architecture for QH interferometry

a simply connected QH interferometer: the proposal of Giovannetti et al.

the only elusive parts are the beam mixers between co-propagating channels

coherent inter-channel mixing

Is it possible to study and image the microscopic details of the

inter-channel scattering?

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Outline





Further reading:

M. A. Topinka, B. J. LeRoy, S. E. J. Shaw, E. J. Heller, R. M. Westervelt, K. D. Maranowski, and A. C. Gossard: Imaging coherent electron flow from a quantum point contact, Science 289 (2000) 2323.

M. A. Topinka, B. J. LeRoy, R. M. Westervelt, S. E. J. Shaw, R. Fleischmann, E. J. Heller, K. D. Maranowski, and A. C. Gossard: Coherent branched flow in a two-dimensional electron gas, Nature 410 (2001) 183.

Scanning Gate Microscopy

AFM with conductive tip

Tip at negatively bias

(local gate - locally

depletes the 2DEG), no

current flows

SGM performed in

constant height mode

(10-50 nm above

surface), no strain

M. A. Topinka et al.:

Science 289 (2000) 2323.

Coherent branched flow of

electrons

M. A. Topinka et al., Nature 410 (2001) 183.

G: 0.00e2/h 0.25e2/h

Tip-induced backscattering

N. Paradiso et al., Physica E 42 (2010) 1038.

Coherent branched flow of

electrons

M. A. Topinka et al., Nature 410 (2001) 183.

G: 0.00e2/h 0.25e2/h

source-drain current

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The SGM @NEST lab in Pisa

• AFM non-optical detection scheme (tuning fork)

• With vibration and noise isolation system

• 3He insert (cold finger base temp. :300 mK)

• 9 T cryomagnet

Tip at negative bias (moveable gate locally depletes the 2DEG)

M. A. Topinka et al.: Science 289 (2000) 2323.Pioneering work by:

SGM performed in constant height mode (10-50 nm above surface), no strain

Setup:

AFM Head

• Positioner (range in xyz > 6 mm) allow to locate features on the sample with µm precision

• Scan range:

• 42µm x 42µm x 2µm @ RT

• 8.5µm x 8.5µm x 1.4µm @ 300 mK

• Temperature measurement (RuO2) close to sample

• Drift < 1 nm / h

• Temperature stability Delta T / T < 5% for hours, even at max. B field

• Noise in z at 300 mK: 1nm

Tuning Fork

Sample holder for transport

measurements

Base mounted on AFM scannerContact via pogo pins

Chip carrier

holds sample

Sample holder mounted on scanner

Tip – sample geometry

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Outline




• SGM studies of samples in the QH regimeFurther reading:

N. Paradiso, S. Heun, S. Roddaro, D. Venturelli, F. Taddei, V. Giovannetti, R. Fazio, G. Biasiol, L. Sorba, and F. Beltram: Spatially resolved analysis of edge-channel equilibration in quantum Hall circuits, Phys. Rev. B 83 (2011) 155305.

N. Paradiso, S. Heun, S. Roddaro, L. N. Pfeiffer, K. W. West, L. Sorba, G. Biasiol, and F. Beltram: Selective control of edge-channel trajectories by scanning gate microscopy, Physica E 42 (2010) 1038.

Coworkers

NEST, Pisa, Italy• Nicola Paradiso, Stefano Roddaro, Lucia

Sorba, and Fabio Beltram

• Davide Venturelli, Fabio Taddei, Vittorio Giovannetti, and Rosario Fazio

TASC, Trieste, Italy• Giorgio Biasiol

Bell Labs, Murray Hill (NJ), USA• L. N. Pfeiffer and K. W. West

Hall-bar samples

1900 m x 300 m


Hall-bar samples


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1D confinement


2e2/h

First we fix the mode number (QPC setpoint), then we start scanning the biased tip at a fixed height.

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1D confinement


First we fix the mode number (QPC setpoint), then we start scanning the biased tip at a fixed height.

source-drain current2DEG

The split gates define a constriction by depleting the 2DEG underneath

The biased tip creates a depletion spot that we use to backscatter the electrons passing through the constriction

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QPC at 3rd plateau3rd plateau

600nm600nm

�5.50

�

0.00

G = 0

G = 6 e2/h

600nm

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QPC at 2nd plateau2nd plateau

�5.50

�

0.00

G = 0

G = 6 e2/h

600nm

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QPC at 1st plateau1st plateau

�5.50

�

0.00

G = 0

G = 6 e2/h

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Branched flow and interference fringes

400nm100nm

Fringe periodicity: lF/2=20 nm

• QPC conductance G = 6 e2/h (3rd plateau)• Tip voltage Vtip = -5 V, height htip = 10 nm• see also M. A. Topinka et al., Nature 410 (2001) 183.

SGM in the Quantum Hall regime

Tip voltage Vtip = -5 V, height = 30 nm


Magnetoresistance


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Selective control of edge channel trajectories by SGM

• Bulk filling factor n=4

• B = 3.04 T

• 2 spin-degenerate edge channels

• gate-region filling factors g1 = g2 = 0

�2.90

�

0.00

4.0 e2/h

0.0 e2/h

4.0 e2/h

0.0 e2/h

600nm


SGM technique: we select individual channels from the edge of a quantized 2DEG, we send them to the constriction and make them backscatter with the biased SGM tip.

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How we probe incompressible stripes

-100 0 100 200 300 400 500 600 700 800

0

1

2

3

4

co

nd

uc

tan

ce

(e

2/h

)

tip position (nm)

tip position

Self-consistent potential

ħωc

Landau levels inside the constriction

tip induced potential

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-100 0 100 200 300 400 500 600 700 800

0

1

2

3

4

co

nd

uc

tan

ce

(e

2/h

)

tip position (nm)

tip position

backscattering

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-100 0 100 200 300 400 500 600 700 800

0

1

2

3

4

co

nd

uc

tan

ce

(e

2/h

)

tip position (nm)

tip position

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-100 0 100 200 300 400 500 600 700 800

0

1

2

3

4

co

nd

uc

tan

ce

(e

2/h

)

tip position (nm)

tip position

backscattering

Energy gap: ħω=5.7 meVPlateau width: 60 nmIncompr. stripe width: ≈30nm

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Asymmetrical gate bias

600nm 600nm 600nm


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Asymmetrical gate bias

600nm 600nm 600nm


We have selective control of edge-channel trajectories by scanning gate microscopy

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From QPCs to QH interferometry

at the beam splitters the electrons are backscattered into the counter-propagating edge through two QPCs

Present technology: beam mixers are obtained by means of QPCs

New architecture: beam splitters induce mixing between co-propagating edge channels

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Studying the inter-channel equilibration

d

Two co-propagating edge channels originating from two ohmic contacts at different potential

n=2

g=1

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Studying the inter-channel equilibration

devices with fixed interaction length d:elusive determination of the microscopic details of

the equilibration mechanisms

d

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The oppurtunity of the Scanning Gate Microscopy

Our technique allows to selectively control the channel trajectory

Our idea: exploit the mobile depletion spot induced by the SGM to continuously tune d

nbulk = 4two spin degenerate edges

SGM tip

n=0

n=2

n=4N. Paradiso et al., PRB 83 (2011) 115305.

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SGM tip

n=0

n=2

n=4N. Paradiso et al., PRB 83 (2011) 115305.


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SGM tip

n=0

n=2

n=4

Each channel carries 2e2/h units of conductance

N. Paradiso et al., PRB 83 (2011) 115305.


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Experimental setup

SEM micrograph of the device

source bias V=VAC + VDC

tip

Scheme of the electronic setup

edge-selector gate

reflected component

transmitted component

tip voltage=-10V

2DESmobility= 2.3x106cm2/Vse- density = 3.2x1011cm-2

depth= 55 nm

n=0

n=2

n=4


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Calibration step

Topography scan

Calibration SGM scan

[nm] 80

0

[e2/h] 4

0

GB


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Calibration step

the edges meet here

0 200 400 600 8001.0

1.5

2.0

2.5

3.0

dif

fere

nti

al

co

nd

uc

tan

ce

(e

2/h

)

position (nm)

0 1 2 3 4 5 60.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

position (m)

dif

fere

nti

al

co

nd

uc

tan

ce

(e

2/h

)

Topography scan


[nm] 80

0

[e2/h] 4

0

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Imaging the inter-channel equilibration

grounded

By grounding the upper contact an imbalance is established between the edges.

SGM map of the IB signal: direct imaging of the equilibration process.


Imaging the edge channel equilibration

Source bias: VAC=50V, VDC=0mV

4

0

[e2/h]

1

0

[e2/h]


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grounded

By grounding the upper contact an imbalance is established between the edges.

SGM map of the IB signal: direct imaging of the equilibration process.




4

0

[e2/h]

1

0

[e2/h]

Only the inner channel carries a non-zero current I = 2 e2/h V


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The profiles of GB(d) along the trajectory show a strict dependence on the local details


1

0

[e2/h]


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The profiles of GB(d) along the trajectory show a strict dependance on the local details

We can directly image the potential induced by the most important defects by means of a

scan at zero magnetic field

correlation found



SGM scan at zero magnetic field

1

0

[e2/h]


-1 0 1 2 3 4 5 6 7

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

dif

fere

nti

al

co

nd

uc

tan

ce

GB (

e2/h

)

position (m)

Experimental data

Tight binding simulations

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Pictorial model for the disorder potential

tip potential

“big impurities” potential

background potential

scattering centers

SGM map of the inter-channel equilibration in another device

1

0

[e2/h]

Simulations made by the theoretical group of Scuola Normale Superiore (Pisa, Italy)D. Venturelli, F. Taddei, V. Giovannetti and R.Fazio


-1 0 1 2 3 4 5 6 7

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

dif

fere

nti

al

co

nd

uc

tan

ce

GB (

e2/h

)

position (m)

Experimental data


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Pictorial model for the disorder potential

tip potential

“big impurities” potential

background potential

scattering centers

SGM map of the inter-channel equilibration in another device

1

0

[e2/h]

Simulations made by the theoretical group of Scuola Normale Superiore (Pisa, Italy)D. Venturelli, F. Taddei, V. Giovannetti and R.Fazio


Experimental realization of a beam mixer between co-propagating edge channels

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Next step: a simply connected MZI

Vin2=20V

Vin1=0V

Mixing (beam splitting)

Mixing (beam splitting)

If the electron mixing is coherent, it is possible to build an interferometer just

by adding another selector gate

d

BS1

BS2

Our idea to implement the Mach-Zehnder interferometer proposed by Giovannetti at al.

F

F

Vin1Vin2

Iout2

Iout1

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Nonlinear regime

The backscattered current is a function of the local imbalance V(x) that depends on the specific scattering process.

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Two mechanisms for the inter-channel scattering

For low bias the only relevant mechanism is the elastic scattering induced by impurities, which determines an ohmic behavior (linear I-V)

At high bias (Δμ≈ħωc) vertical transition with photon emission are enabled (threshold and saturation)

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Impact of the electron heating

Electron heating due to injection of hot carriers:

The relaxation of hot carriers induces a dramatic temperature increase. This is why the transition is smoothened and the threshold voltage reduced for high d

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Conclusions

We explore the use of Scanning Gate Microscopy to selectively control edge channel trajectories

We built size-tunable QH circuits to directly image the equilibration between imbalanced co-propagating edges

The comparison with the SGM scan at zero magnetic field reveals a correlation between the local potential and steps in the GB(d) curve

Shift of the threshold voltage for the onset of photon emission is explained by a simple model for the electron heating

Control of the edge channel trajectory allows us to study their structure

edge channel interferometry in the quantum hall...

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