Quantum physics in quantum dotsKlaus Ensslin

Solid State Physics

AFM nanolithography

Multi-terminal tunneling

Rings and dots

Time-resolved charge detection

Zürich

1970 1975 1980 1985 1990 1995 2000 2005 2010103

104

105

106

107

108

109Transistors per chip

Year

80786PentiumPro

Pentium80486

80386

80286

8086

8080

4004

?

micro nano

Moore‘s Law

gate length 100 nm

1985 1990 1995 2000 2010 2015 202010-1

100

101

102

103

104Electrons per device

2005

Year

(Transistors per chip)

(16M)

(4M)

(256M)

(1G)

(4G)

(16G)

(64M)

micro nano

Vanishing electrons

gate length 100 nm

Quantized charge

voltage U

Capacitance of a capacitor:

C =Q

U=

charge

voltage-Q

Q

Energy to charge the capacitor:

E = U dQ0

Q=

Q

C dQ

0

Q=

Q2

2C

Energy to put one electron (Q=e) on a capacitor with C = 1 nF

E =1.6 10 19 As( )

2

2 10 9 F=1.3 10 29 Joule = 8 10 9 eV

Equivalent to temperature T = 0.1 mK

10 nm

Size of a capacitor

-Q

Qcapacitance

C = 0

areaseparation

=

= 0

1 μm( )2

1 μm=10 16 F

1 μm

1 μm1 μm

equivalent to temperature T = 7 K

-> use nanotechnology to make a small capacitor

decoupled from its environment

direct patterning of AlGaAs/GaAs

high mobility two-dimensional electron gas (2DEG)below sample surface

Matsumoto et al., APL 68, 34 (1996)Held et al., APL 73, 262 (1998)

2DEG: W. Wegscheider Uni Regensburg

lateral resolution

1μm

0

1

2

3

4

600 800 1000 1200 1400

he

igh

t (n

m)

x(nm)

35 nm

Ti film

oxide line

writing speed 1μm/s

humidity 40 %

bias 8V

oxidation of GaAs - reproducibility

AFM galleryquantum dot

quantumpoint contacts

antidot lattice 4-terminal ring

rings, dots + qpc’s

1μm

ring + dots

3μm

Lithography on 8nm Ti top gates:

Martin Sigrist, Andreas Fuhrer

Double layer AFM lithography

Aharonov-Bohm effect

1

2

= 1 2 = geom. +

q

h

r A d

r l

conductance becomes a periodic function of magnetic flux

AFM defined quantum ring

300 nm

source

drain

QPCQPC

QPCQPC

plungerplunger

current flow KekuléBull. Soc. Chim. Fr. 3, 98 (1865)-> benzene

Aharonov & BohmPhys. Rev. 115, 485-491 (1959)-> magnetic flux

Büttiker, Imry, & LandauerPhys. Lett. 96A, 365-367 (1983)-> persistent currents

AB-oscillations in an open ring

At T=1.7K up to h/6e

Magnetoresistance Fourier-Spectrum

l (T) T 1, typical for e- - e- interaction

l (1.7K) 3μm ; l (100mK) = 60μm

electron rings on different scales

1 μm

Benzene ring: Ring accelerator :Large Electron Positron Collider at CERN in Geneva

0.5nm

1013

8.6km

Aharonov-Bohm effect: one flux quantum (h/e) through ring area

her2 = 5000 T

her2 = 7•10 23 T

Coulomb blockaded quantum ring

Ering (meV)0 0.4 0.8 1.2

0.02

0.01

0.00

source

drain

QPC QPC

QPC QPC

plungerplunger

T 100 mK

Coulomb blockade

kT << e /2C2

eU = E - E << e /2C2Fsource

Fdrain

-> no current transport

EF E

Fsource drain

e /2C2

discrete level between

EFsource

and EFdrain

-> coherent resonant tunneling

sourcedrain

EF

e /2C2

EF

disk: C = 4 0rr

r =100 nm

> C =100 aF

> e2 /2C = 600 μeV 7K

quantum ring

0

-0.2

-0.4

0.2

0.4

0.6

0.1 0.2 0.3V (V)plunger

B (

T)

h/e

Ering (meV)0 0.4 0.8 1.2

0.02

0.01

0.00

source

drain

QPC QPC

QPC QPC

plungerplunger

perfect 1D ring in a magnetic field

B = 0 - > H =h

2

2mr2

2

2

energies : El =h

2

2mr2 l2

wave functions : l ( ) =1

2eil

B 0 > Em,l =h

2

2mr2 (m l)2

m,l ( ) =1

2eil

El (m )

[h 2 2m*r02 ]

fixed N

m magnetic flux (h/e)

0.4

0

1.0

1.2E

(meV

)ri

ng

0.30.20 0.1B (T)

El (m )

[h 2 2m*r02 ]

fixedN

m magnetic flux (h/e)

13

energy spectrum

El =

h2l

m*r02 / 0 = I

l

l = 8; Il

22nA

Experiment:

imperfect ring

m ,l ( ) =1

2eil

perfect ring: -> probability density uniformly

spread over the ring

-> cannot explain oscillations of

Coulomb peak amplitude

m ,l* ( ) =

1sin m( 0 )( )

0

1

2

3

4

5

6

-1 -0.5 0 0.5 1 1.5 2 2.5 3

magnetic flux (h/e)

Em,l

(h2 2mr2)

0

imperfection at position = 0

symmetry breaking

asymmetric ringwith finite width

perfect ring

0 >

0 1 cos(2 )( )

energy levels and wave functions

flux quanta through ring

ener

gy

How to measure resistances

U

I

two-terminal

measurement of a

classical resistor

I

V four-terminal

measurement of a

classical resistor

-> elimination of contact

resistances

How to measure resistances

U

I

two-terminal

measurement of a

quantum dot

What about more than two terminals?

How to differentiate between contacts and quantum dot?

Quantum dot in the Coulomb blockade regime:

high impedance device

LG 1

LG 2

LG 3

LG 4

PG

lithographic size:600 450 nm2

electronic size:400 250 nm2

charging energy: EC

0.5 meV

mean level spacing: 35 eV

electronic temperature: kBT 10 eV

1

2

34

1 m

multi-terminal quantum dot

Renaud Leturcq & Davy Graf

Experimental set-up

VLG4

(V)

0.04

I/V

bia

s (e

2/h

)

lead 1

lead 2 lead 3

-0.24

0.02

0

-0.02

-0.04

0.04

0.02

0

-0.02

-0.04

0.02

0

-0.02

-0.22 -0.20 -0.18 -0.16

multi-terminal quantum dot

conductance matrix

VLG4

(V)

0.04

conduct

ance

Gij (

e2/h

)

lead 1

lead 2 lead 3

-0.24

0.02

0

-0.02

-0.04

0.04

0.02

0

-0.02

-0.04

0.02

0

-0.02

-0.22 -0.20 -0.18 -0.16

I1I2

I3

=

G11 G12 G13

G21 G22 G23

G31 G32 G33

V1

V2

V3

multi-terminal quantum dot

current conservation:

Ii = 0 Gij

i=1

3

= 0

V1 = V2 = V3 Ii = 0 Gij

j=1

3

= 0

sum rules

measurement set-upapply voltage to one terminalmeasure current in three terminals

Kirchhoff rules

G11 G12 G13

G21 G22 G23

G31 G32 G33

=

1

G1 + G2 + G3

G1(G2 + G3) G1G2 G1G3

G1G2 G2(G1 + G3) G2G3

G3G3 G3G2 G3(G1 + G2)

Gij: three-terminal conductanceGl: lead conductance

Gn =e2

4kT

1

nS

+1

nD

1

cosh 2 G VGn VG( )

2kT

sequentialtunneling:

VLG4

(V)

Lea

d c

on

du

ctan

ces

Gk (

e2/h

)

Tu

nn

elin

g r

ate ℏ

(eV

)

Weak coupling regime

⇒ independent fluctuations

strong overlap

weak overlap

Individual coupling to the leads

⇒ extend of the wave function in the dot

in the vicinity of the leads.

F

~5

0 n

individual tunnel couplings

D. Loss & D. DiVincenzo, PRA 57 (1998) 120

Spins in Coupled Quantum Dots for Quantum Computation

n.n. exchange local Zeeman

each dot has different g-factor

->individually addressable via ESR

magnetic field gradients

by current wire

spin as a qubit

one spin 1/2 particle is a natural qubit

singlet state: 1

2( )

triplet states: , 1

2+( ),

(entangled)

two spin 1/2 particles:

Spin coherence times have been shown to be much longer

than charge coherence times, up to 100 μs

Spin qubits in quantum dots

General qubit state: two-level system

Possible realizations employing quantum dots:

charge qubit spin qubit

Zeeman

= cos2

0 + ei sin2

1

S D

gate gate

gate

QPC

2 μm

detector

024

68

N N+1 N+2

I dot (

pA

)

Vgate

1.21.31.41.51.6

GQ

PC (

e2/h

)

Vgate

semi-circular dot with charge readout

Vgate (V)-0.1 -0.08 -0.06 -0.04 -0.02 0

4

5

6

dG

/dV

gate

(a.u

.)

Roland Schleser

Elisabeth Ruh

Thomas Ihn

See also Gardelis et al, PRB67, 073302 (2003), Elzermann et al. , Phys. Rev. B 67, 161308 (R), (2003)

time-resolved detector signal

time (s)

2 1064 8

close tunnel barriers -> electron transport one-by-one

pinch-off one tunnel barrier completely:

- one-off time is a measure for the tunnel rate on and off the quantum dot

- one-off probability is a measure for the state occupation -> Fermi distribution

dG

/dV

gate

(a.u

.)

EF

1

0

source drain

dot

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.40.5

0 0.2 0.4 0.6 0.8 1f (E)

E (m

eV)

Fermi-Dirac distribution

time (s)

2 106

0

1

0

1

0

1

0

1

0

1

0

1

fit: Fermi distribution

distribution extracted

from data

example sweeps

-> T ~ 150 mK

gate

vol

tage

Spectroscopy of

electronic states

source drainquantum

dot

EC

kBT

GS

GD

DE

VPG

(mV)

GS

D (

10

-3 e

2/h

) N

N+1

N-1

EC (+ )

kBT_ __ E C

source

drain

pg

Quantum point

contact as a charge

detector

VPG

(mV)

GS

D (

10

-3 e

2/h

) N

N+1

N-1

EC (+ )

kBT_ __ E C

source

drain

pg

VP

GQPC

2e2/h

M. Field et al., Phys. Rev. Lett. 70, 1311 (1993)

A few electron

quantum dot

source

drain

pg

M. Sigrist

Detection of

single electron transport• Quantum point contact

as a charge detector

• Low bias voltage on the

quantum dot

source drainquantum

dot

kBT

Te = 350 mK

Low bias - thermal noise

: effective dot-lead tunnel coupling

E: energy difference between Fermi level of the lead and

electrochemical potential of the dotR. Schleser et al., Appl. Phys. Lett. 85, 2005 (2004)

L. M. K. Vandersypen et al., Appl. Phys. Lett. 85, 4394 (2004)

Determination of the individual

tunneling rates

• Exponential distribution of waiting times

for independent events

• S=< in>, D=< out>

N

N+1

Measuring the current

by counting electrons

• Count number n of electrons entering the dot within a

time t0: I = e<n>/t

0

• Max. current = few fA (bandwidth = 30 kHz)

• BUT no absolute limitation for low current and noise

measurements

– we show here: I few aA, SI 10-35 A2/Hz

N

N+1

Histogram of current fluctuations

maximum: mean current

width: fluctuations, noise

Histogram of current fluctuations

• Poisson distribution for

asymmetric coupling

• Sub-Poisson distribution for

symmetric coupling

Theory: Hershfield et al., PRB 47, 1967 (1993)Bagrets & Nazarov, PRB 67, 085316 (2003)

Current fluctuations vs. asymmetry

• Reduction of the second and third

moments for symmetric coupling

Theory: Hershfield et al., PRB 47, 1967 (1993)Bagrets & Nazarov, PRB 67, 085316 (2003)

asymmetric

barriers

a=1

symmetric

barriers

a=0

Current fluctuations vs. asymmetry• Reduction of the second and third

moments for symmetric coupling

Theory: Hershfield et al., PRB 47, 1967 (1993)Bagrets & Nazarov, PRB 67, 085316 (2003)

width - noise asymmetry

Time-resolved electron transport

- small current level (< atto-Amperes)

- low noise levels (SI 10-35 A2/Hz)

- higher correlations in current are accessible

-> correlations, interactions and

entanglement in quantum dots

bandwidth 20 kHz

Aharonov-Bohm with cotunneling

Co-tunneling

– Electrons are injected

from the right lead

– They pass through either

the upper or lower arm

– The interference take

place in the left QD

WavesThe double slit experiment

double

slit

source

screen

Light

A. Tonomura et al.,

American Journal of Physics 57 117-120 (1989)

WavesThe double slit experiment

double

slit

source

screenParticles

Double slit experiment<-> Aharonov Bohm

Simon Gustavsson

Matthias Studer

huge visibility! >90%

little decoherence - > due to long dwell time in the collecting dot?

requires the couplings of upper and lower arm to be well symmetrized

1

-400 -200 0 200 4000

50

00

B-Field [mT]

counts

/ s

Aharonov-Bohm oscillations

AB amplitude stable below T=400mK

Destruction most likely due to thermal broadening

Temperature dependence

Future directions

• from quantum devices to quantum circuits

• non-equilibrium quantum mechanics

-> time dependent experiments, MHz - GHz

• detection of entanglement in solid state quantum systems

-> non-classical (microwave) radiation

• Combination of spatial and

temporal resolution

• novel quantum materials

graphene, nanowires

DDDD

QPC

thanksSimon Gustavsson

Thomas

Ihn

Martin SigristAndreas

Fuhrer

Renaud

Leturcq