E. Rousseau D. PonarineY. Mukharsky

E.VaroquauxO. AvenelJ.M. Richomme

CEA-DRECAMService de Physique de l’Etat CondenséCEA-SaclayFrance

To Quantum Dots and Qubitswith electrons on helium

L pool R pool

ring

Guard electrode

Confine the electrons

pool Guard electrode

550 nm40 nm

• the exact quantity of helium is added to fill the two pools and the ring.

• electrons are created by a corona discharge.

•Only electrons above pools and ring are mobile.

X axis

0.6 0.7 0.8 0.9 1.0x10-5

x axis

-0.20

-0.15

-0.10

-0.05

0.00

-V(V

olt)

SET=0.35

SET=0.2

SET=0.1

R pool

trapping

• The electrostatic trap is created by the SET itself.

Gate

R pool Gate

• Sample more or less similar to RHL one.

Counting Individual Electrons on liquid Helium .G.Papageorgiou & al. Applied physics letters 86, 153106 (2005)

Royal holloway

Sample specificity:• Ring diameter 3 m, pyramidal SET island

L pool R pool

ring

Guard electrode

The Precious

Gate

ResR

Detection

•The voltage across the SET is a periodic function of the island charge. •When an electron goes in or out of the trap the SET island charge change by q.

SET island

Q

0.47 0.48 0.49 0.50 0.51R pool potential

-0.5

0.0

0.5

1.0x10-4

SE

Tos

cill

atio

nsam

plit

ude

Q+q

Re( )sRCVf

e

Re( )sRCV qf

e e

ResR

0.47 0.48 0.49 0.50 0.51R pool potential

-0.5

0.0

0.5

1.0x10-4

SE

Tos

cill

atio

nsam

plit

ude

Detection

0.47 0.48 0.49 0.50R pool potential

-0.1

0.0

0.1

0.2

0.3

osci

llat

ions

phas

e

• We fit with a spline function a quiet part of the oscillations.• The phase of the oscillations changes then by

Charging the ring

Left and right pools

guardguard

Drawing of the potential profile

ring

0.0 0.3 0.6 0.9 1.2

-100

-98

-96

-94

-92

phase

Q/e

Right pool potential, V

One by one observation

ResR SET

0.6 0.7 0.8 0.9 1.0-95.0

-94.5

-94.0

-93.5

-93.0

-92.5

-92.0

ResR

Q/e

One by one observation

0.4e

0.4e

last jumps are around ~0.4 e, value which is reproduced by simulations.

Discharge curve

0.0 0.5 1.0 1.5right pool potential

-125

-120

-115

-110

-105

-100

-95

-90ph

ase

vari

atio

n

Zoom

Discharge curve

0.9 1.0 1.1 1.2 1.3right pool potential

-98

-96

-94

-92ph

ase

vari

atio

n

Discharge curve

0.9 1.0 1.1 1.2 1.3right pool potential

-98

-96

-94

-92ph

ase

vari

atio

n

12

5

34

0

6

89

q

Discharge curve

0 5 10 15 20 25 30electron number

0

1

2

3

4

5

6

7S

ET

isla

ndch

arge

Q/e

Density estimate• For a small number of electrons, density

can be considered as uniform. A.A.Koulakov, B.I.Shklovskii cond-mat/9705030

• The electric field is supposed to be constant on SET island.

• The electric charge on the SET island is then given by:

nSeff (1-cos(max))Qe = With max=arctan ( N

n1h

)

N=nR2 number of electronn electronic density

max

SET island

h

Electron sheetR

Seff

0 5 10 15 20 25 30

electron number

0

1

2

3

4

5

6

7S

ET

isla

ndch

arge

Q/e

Density estimate

Seff =0.34 +/- 0.07 m2

Electronic density=(1.8 +/- m-2

Sreal=0.35 m2

• The interaction between electrons is not screened by surrounding electrodes.

• From thermal fluctuations: n=1014 transition temperature Tc=2.5 K

• rs~2000 rscri~30

Coulombic interaction is dominant

• Electrons should form a Wigner crystal.

Wigner molecules

Competitionbetween the symmetry of the trap

(parabolic trap) shell ordering

and the symmetry of the Wigner crystal

triangular lattice structure

N=7

N=19

macroscopic observation

camera

No observations with electrons on helium.

Addition Spectra

Current pA

20

10

Gate voltage, V

1 2 3

4(1)

• charging energy: energy required to add

one electron

• (N) =(N)-(N-1) ~ e Vg

rightleft

gate

0.9 1.0 1.1 1.2 1.3right pool potential

-98

-96

-94

-92

phas

eva

riat

ion (1)(2)

(3)

Addition spectra

Liquid phase

• Average interaction between electrons Constant interaction model 1 particles model in a parabolic trap

• Confinement induce energy shell with magic numbers when a shell is completely filled: 2,6,12 (degeneracy due to spin)

2e

C

2e

C

Hund’s rule

Vladimir Bedanov and François PeetersPRB 49, 2667

0 5 10 15 20 25number of electrons

0.000

0.002

0.004

0.006

0.008

addi

tion

ener

gy(e

V)

“The numbers of electrons on each ring are not universal and depend on the type and strength of confinement potential.”

B.Partoens and F.PeetersJ.Phys:condens matter,9 5383 (1997)

For a parabolic trap

No magic numbers when electrons form Wigner molecules.

Simulations needed to explain addition energy plot

0 2 4 6 8 10nombre d'electron

0.00

0.02

0.04

0.06

0.08

0.10eV

Pour Vset=0.3 Vgate=0 Vground=-0.1

0 2 4 6 8 10 12 14electron number

0.00

0.02

0.04

0.06

0.08

0.10

0.12ch

emic

alpo

tent

ial

vari

atio

n Vset=Vgate=0.5 V

Monte Carlo simulations

• The modelpotential profile found with simulation simulation of the pyramidal island.

• Ground state of the configuration with N electrons is found with Monte-Carlo Simulations.

• Ground state energy vary linearly with reservoir right potential.

-0.1 0.0 0.1 0.2 0.3 0.4reservoir right potential

-5.0

-4.8

-4.6

-4.4

-4.2

-4.0

-3.8

-3.6en

ergy

(eV

)

N=19

N=20

N=21

N=22

Monte Carlo simulations

• The modelpotential profile found with simulation simulation of the pyramidal island.

• Ground state of the configuration with N electrons is found with Monte-Carlo Simulations.

• Ground state energy vary linearly with reservoir right potential.

• Criteria when the electron leaves the trap?

0.5 0.6 0.7 0.8 0.9 1.0x10-5

x

-0.40

-0.35

-0.30

-0.25

-0.20

-0.15

pote

ntia

lpr

ofil

e

Potential profile = one particule energy

1N NE E One electron leaves the trap when

-0.2 0.0 0.2 0.4 0.6 0.8 1.0reservoir right potential

-0.4

-0.3

-0.2

-0.1

0.0

0.1

(e

V)

E22-E21

E2-E1

0 5 10 15number of electrons

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14ad

diti

onen

ergy

(eV

)

experiment

simulation

Vset=0.5 Vgate=0.5

2 4 6 8 10 12 14

electron number N

0.00

0.01

0.02

0.03

0.04

0.05

Add

itio

nen

ergy

dot(N

)(m

eV)

addition spectra for different discharges

Vset=0.1 Vgate=0.2

4

6

9

Clear magic numbers!

Cannot be explain with M-C simulation

Positive impurity increase locally the density

Structural transition in a finite classical two dimensional system

G. A. Farias and F. M. Peeterssolid state communication, 100, 711 (1996)

0.6 0.9

-100

-95

ResR, V

Q

/eOne by one observation

Probability of escape

• If the barrier is thin enough, the electron can tunnel.

8.15 8.20 8.25 8.30 8.35 8.40x10-6

ResR potential

0.00140

0.00145

0.00150

0.00155

0.00160

0.00165

barr

ier

heig

ht

0.80 0.85 0.90 0.95 1.00 1.05 1.10

pulse amplitude (arbitrary unit)

0.0

0.2

0.4

0.6

0.8

1.0

prob

abil

ity

position

|0>

|1>

Potential profile

Barrier height

Quantum state evidence?

Existing to the first excited state

SETI

|0>

|1>Oscillating E

Time

Tunneling time=5.10-11 s

Potential felt by the electron

+e -e

mixs

measurement principles

1. We choose a SET current valuefrom 0 nAmp (no heating) to 50 nAmps.

2. We applied a voltage pulse to the right reservoir: the barrier height is reduce.

3. We check whether the electron is still in the trap or not.

4. 2000-3000 tries for one escape curve.

position

Potentials profiles

Barrier

?

6.50 6.55 6.60 6.65 6.70 6.75pulse amplitude

0.0

0.2

0.4

0.6

0.8

1.0

prob

abil

ity

ofex

trac

tion

p= 1-exp(-0exp(-Eb/kbT))

0 =

/2~ 30 GHz trap frequency pulse length

Eb barrier high

Eb and 0 estimated from simulations

Temperature estimate

-30 -20 -10 0 10 20 30

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Ids

Fit with numerically calculated UMultipler 33.61v

0=0.076,

0=.0052

The pulse is applied during heating

Curve width with heating current

6.50 6.55 6.60 6.65 6.70 6.75pulse amplitude

0.0

0.2

0.4

0.6

0.8

1.0pr

obab

ilit

yof

extr

acti

on

A0.5

0 200 400 600 800 10003.7

3.8

3.9

4.0

4.1

A

0.5

T, mK

Quantum tunnelingThermal activation

Crossover Temp ~ 250 mK

0.5 0.6 0.7 0.8 0.9 1.0 1.1x10-5

ResR potential

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

ener

gie

(eV

)

~ 30 GHz (before the pulse)

Cross-over temperature

~ 230 mK

the frequency of the barrier

0

2 B

hT

k

0.644 0.646 0.648 0.650 0.652 0.654

ResR potential

0.0

0.2

0.4

0.6

0.8

1.0

prob

abil

ity

extraction curve

Vsetapp=0.2598

Temp=0.235 K

Vset=0.3

Only one parameter is adjusted

Vset.

Contact potential between niobium and aluminiumthe potential on the SET is not really known

Tcell=30 mK

0 200 400 600 800 1000

2.4

2.5

2.6

2.7

2.8

2.9

Ids 0 Ids 2.3 Ids 10 Ids 0 Ids 2.3 Ids 10

A0.

5

T, mK

No change when we heat the electron!

Need to probe the coupling between the electron and its environment

1. Heating for 300 ms with 50 nAmps.

2. Wait between 2 and 100 s.

3. Applied the pulse on reservoir right.

4. Fit the curve.According to position and width of this curve get a number proportional to the temperature

Coupling with environment

position

Potential profile

Barrier

SET

100 s

-20 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320200001800016000140001200010000

8000

6000

4000

2000

E

kt

s

Sep07 Sep09

Coupling with the environment

t=0 high temperaure

Even for t=2 s, temp. is low.

Heating pulse decay

50 nAmps

~ 0.4 s

Heating current

time

0.0 0.5 1.0 1.5time in micro-second

0.1825

0.1830

0.1835

0.1840

0.1845

0.1850

tem

pera

ture

a-u

0.0 0.5 1.0 1.5time in micro-second

0.1825

0.1830

0.1835

0.1840

0.1845

0.1850

tem

pera

ture

a-u

tau = 0.38 +/- 0.05 micro-s

t

taue

What can we say?

• Either the coupling with the environment is huge

Where does it come from?

Coupling with phonons

• they induce modulations of the density. So they induce modulation of the image charge.

• This coupling depends highly with the electric field on the electron.

• In our case ~ 10 s

We maybe tried the smallest electric field as we could to keep the electron in the trap!

What can we say?

• Either the coupling with the environment is huge

• Or the picture of the heating procedure is too naïve

SETI

|0>

|1>

Dynamical force?

Two wave-guids:

~ 40 GHz parrallel levels

~ 120 GHz perpendicular levels

-4 -2 0 2 4x10-7

reservoir right potential

-4

-2

0

2

4x10-7

(e

V)

circular

ellipsoïdal

Evolution with deformation

Increase of the deformation

L pool R poolring

Guard electrode

Bias and IV characteristic

PreAmpIds A

B

A-B

the SET is current bias

Uds

IV Characteristic

For C-B oscillations

-5 -4 -3 -2 -1 0 1 2 3 4-8.0x10-4

-6.0x10-4

-4.0x10-4

-2.0x10-4

0.0

2.0x10-4

4.0x10-4

6.0x10-4

UD

S,

V

IDS, nA

500 mKsample 2h

Measurement • L-in technique: 125 Hz, 6-30 V applied to the guard

L pool R poolring

Guard electrodePreAmp

125

L-in

From L-in

6V

C-B oscillations

Raw oscillations From L-in

0.07 0.08 0.09 0.10VresR

0.004929

0.004930

0.004931

0.004932

0.004933

0.004934

C-B

osci

llat

ions

Am

plit

ude

0.07 0.08 0.09 0.10VresR

-1.0

-0.5

0.0

0.5

1.0x10-4

C-B

osci

llat

ions

Am

plit

ude

Ids=2.6 nA

Some properties of this kind of quantum dots

• Classical point of view: competition between thermal fluctuations and coulombian interaction

• for n=1014 transition temperature Tc=2.5 K

• Working temperature: from 50 mK to 1K

Gas parameter• Measure the competition between

quantum fluctuations and Coulombic interaction

• For n ~ 1014 m-2 a ~n-1/2 ~ 0.1 m

• typical coulomb energy

• Typical kinetic energy (Fermi energy)

• rs~2000 rscri~30Coulombic interaction is dominant

• Semiconductors qdots rs~2

2

0

1

4c

eE

a

a

22KEma

0 2 4 6 8 10 12 14

electron number

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07ch

emic

alpo

tent

ialva

riat

ion

5

7

Vset=Vgate=0.3 V