Download Guard electrode - Princeton University

To Quantum Dots and Qubits
with electrons on helium
CEA-DRECAM
Service de Physique de l’Etat Condensé
CEA-Saclay
France
E. Rousseau
D. Ponarine
Y. Mukharsky
E.Varoquaux
O. Avenel
J.M. Richomme
Confine the electrons
• the exact quantity
of helium is added
to fill the two pools
and the ring.
Guard electrode
ring
L pool
R pool
• electrons are
created by a corona
discharge.
•Only electrons
above pools and
ring are mobile.
pool Guard electrode
550 nm
40 nm
trapping
0.00
-0.05
R pool
Gate
-V (Volt)
SET=0.1
-0.10
SET=0.2
X axis -0.15
Gate
R pool
-0.20
0.6
0.7
SET=0.35
0.8
x axis
• The electrostatic trap is created by the SET itself.
0.9
1.0
x10-5
Royal holloway
• Sample more or less similar to RHL one.
Counting Individual Electrons on liquid Helium .
G.Papageorgiou & al.
Applied physics letters 86, 153106 (2005)
Sample specificity: The Precious
• Ring diameter 3 mm, pyramidal SET island
Guard electrode
ring
L pool
R pool
Gate
ResR
Detection
Q+dq
Q
ResR
SET oscillations amplitude
1.0x10
-4
0.5
0.0
-0.5
SET island
•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 dq.
0.47
0.48
0.49
0.50
R pool potential
CVRe sR
f(
)
e
d
CVRe sR d q
f(
 )
e
e
0.51
Detection
0.3
-4
oscillations phase
SET oscillations amplitude
1.0x10
0.5
0.0
-0.5
0.2
0.1
0.0
-0.1
0.47
0.48
0.49
R pool potential
0.50
0.51
0.47
0.48
0.49
0.50
R pool potential
• We fit with a spline function a quiet part of the oscillations.
• The phase of the oscillations changes then by df.
Charging the ring
Drawing of the potential profile
guard
guard
Left and right pools
ring
One by one observation
-92
phase Q/e
-94
-96
-98
0.0
SET
ResR
-100
0.3
0.6
Right pool potential, V
0.9
1.2
One by one observation
-92.0
0.4e
-92.5
0.4e
-93.0
-93.5
Q/e
-94.0
-94.5
-95.0
0.6
0.7
0.8
0.9
ResR
last jumps are around ~0.4 e, value which is reproduced by
simulations.
1.0
Discharge curve
-90
-95
phase variation
-100
Zoom
-105
-110
-115
-120
-125
0.0
0.5
1.0
right pool potential
1.5
Discharge curve
phase variation
-92
-94
-96
-98
0.9
1.0
1.1
right pool potential
1.2
1.3
Discharge curve
-92
0
1
phase variation
2
-94
dq
5
4
3
6
8
-96
9
-98
0.9
1.0
1.1
right pool potential
1.2
1.3
Discharge curve
7
SET island charge Q/e
6
5
4
3
2
1
0
0
5
10
15
20
electron number
25
30
Density estimate
Electron sheet
h
R
qmax
Seff
SET island
• For a small number of electrons, density
can be considered as uniform.
A.A.Koulakov, B.I.Shklovskii condmat/9705030
• The electric field is supposed to be
constant on SET island.
• The electric charge on the SET island is
then given by:
Q nSeff
(1-cos(qmax))
=
e
2e0
1
N )
With qmax=arctan (
h pn
N=npR2 number of electron
n electronic density
Density estimate
7
Electronic density=(1.8 +/- 0.4)*1014 m-2
SET island charge Q/e
6
Seff =0.34 +/- 0.07 mm2
Sreal=0.35 mm2
5
4
3
2
1
0
0
5
10
15
20
electron number
25
30
• 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
Competition
between the symmetry of the trap
(parabolic trap)
 shell ordering
N=7
and the symmetry of the Wigner
crystal
 triangular lattice structure
N=19
macroscopic observation
camera
No observations with electrons on helium.
Addition Spectra
Current pA
20
Dm(1)
4
mright
mleft
10
gate
1
2
3
• charging energy:
energy required to add
Gate voltage, V one electron
• Dm(N) =m(N)-m(N-1)
~ e DVg
Addition spectra
phase variation
-92
Dm(2)
-94
Dm(1)
Dm(3)
-96
-98
0.9
1.0
1.1
right pool potential
1.2
1.3
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)
2
e

C

2
e
 D
C
D
Hund’s rule
“The numbers of
electrons on each
ring are not
universal and depend
on the type and
strength of
confinement
potential.”
For a parabolic trap
addition energy (eV)
0.008
0.006
0.004
0.002
0.000
0
5
10
15
20
number of electrons
Vladimir Bedanov and François Peeters
PRB 49, 2667
25
B.Partoens and F.Peeters
J.Phys:condens matter,9
5383 (1997)
No magic numbers when electrons form Wigner molecules.
Simulations needed to explain addition energy plot
Pour Vset=0.3 Vgate=0 Vground=-0.1
0.10
0.08
eV
0.06
0.04
0.02
0.00
0
2
4
6
nombre d'electron
8
10
chemical potential variation
0.12
Vset=Vgate=0.5 V
0.10
0.08
0.06
0.04
0.02
0.00
0
2
4
6
8
10
electron number
12
14
Monte Carlo simulations
• The model
potential 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.
-3.6
-3.8
energy (eV)
-4.0
-4.2
N=19
-4.4
N=20
N=21
-4.6
N=22
-4.8
-5.0
-0.1
0.0
0.1
0.2
reservoir right potential
0.3
0.4
Monte Carlo simulations
• The model
potential 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?
-0.15
potential profile
-0.20
d
-0.25
-0.30
-0.35
-0.40
0.5
0.6
0.7
0.8
0.9
x
1.0
-5
x10
Potential profile = one particule energy
One electron leaves the trap when
EN  EN 1  d
0.1
d (eV)
0.0
-0.1
E22-E21
-0.2
-0.3
-0.4
E2-E1
-0.2
0.0
0.2
0.4
0.6
reservoir right potential
0.8
1.0
Vset=0.5 Vgate=0.5
0.14
addition energy (eV)
0.12
simulation
experiment
0.10
0.08
0.06
0.04
0.02
0.00
0
5
10
number of electrons
15
addition spectra for different discharges
Addition energy Dmdot(N) (meV)
0.05
Clear magic
numbers!
Vset=0.1 Vgate=0.2
0.04
0.03
Cannot be explain
with M-C simulation
4
6
0.02
9
Positive impurity
increase locally the
density
0.01
0.00
2
4
6
8
10
12
14
electron number N
Structural transition in a finite classical two dimensional
system
G. A. Farias and F. M. Peeters
solid state communication, 100, 711 (1996)
One by one observation
Q/e
-95
-100
0.6
0.9
ResR, V
Probability of escape
0.00165
• If the barrier is
thin enough,
the electron can
tunnel.
barrier height
0.00160
0.00155
0.00150
0.00145
0.00140
8.15
8.20
8.25
8.30
ResR potential
8.35
8.40
x10
-6
Quantum state evidence?
1.0
Potential profile
|1>
Barrier
height
probability
0.8
0.6
0.4
0.2
|0>
0.0
position
0.80
0.85
0.90
0.95
1.00
1.05
pulse amplitude (arbitrary unit)
1.10
Existing to the first excited state
Potential felt by the electron
|1>
Oscillating E
|0>
I
SET
+e
Time
-e
Tunneling time
t=5.10-11 s
tmix=75 ms
measurement principles
Potentials profiles
?
Barrier
position
1. We choose a SET current
value
from 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.
Temperature estimate
p= 1-exp(-G0exp(-Eb/kbT))
1.0
probability of extraction
G0 =w0t/2p
0.8
w0/2p ~ 30 GHz trap frequency
t pulse length
0.6
0.4
Eb barrier high
0.2
Eb and w0 estimated from
simulations
0.0
6.50
6.55
6.60
6.65
pulse amplitude
6.70
6.75
Curve width with heating current
1.6
1.4
1.2
1.0
 
0.8
0.6
0.4
0.2
Fit with numerically calculated U
Multipler 33.61
v0=0.076, e0=.0052
0.0
-30
-20
-10
0
Ids
10
20
30
The pulse is applied during heating
probability of extraction
1.0
0.8
0.6
0.4
0.2
A0.5
0.0
6.50
6.55
6.60
6.65
pulse amplitude
6.70
6.75
4.1
Quantum tunneling
Thermal activation
A0.5
4.0
3.9
Crossover Temp ~ 250 mK
3.8
3.7
0
200
400
600
T, mK
800
1000
the frequency of the barrier
0.10
n0
energie (eV)
0.05
n0 ~ 30 GHz
(before the pulse)
0.00
Cross-over temperature
-0.05
-0.10
T
-0.15
-0.20
0.5
0.6
0.7
0.8
0.9
ResR potential
1.0
1.1
x10
-5
hn 0
~ 230 mK
2p k B
extraction curve
1.0
Vset=0.3
Vsetapp=0.2598
0.8
Only one
parameter is
adjusted
probability
Temp=0.235 K
0.6
Tcell=30 mK
0.4
 Vset.
0.2
0.0
0.644
0.646
0.648
0.650
0.652
0.654
ResR potential
Contact potential between niobium and aluminium
the potential on the SET is not really known
2.9
No change when
we heat the
electron!
2.8
A0.5
2.7
2.6
Need to probe
the coupling
between the
electron and its
environment
Ids 0
Ids 2.3
Ids 10
Ids 0
Ids 2.3
Ids 10
2.5
2.4
0
200
400 T, mK 600
800
1000
Coupling with environment
1. Heating for 300 ms
with 50 nAmps.
Potential profile
100 ms
SET
2. Wait between 2 and
100 ms.
Barrier
3. Applied the pulse on
reservoir right.
position
4. Fit the curve.
According to
position and width of
this curve get a
number proportional
to the temperature
Coupling with the environment
2000
t=0 high temperaure
Sep07
Sep09
Ekt
Even for t=2 ms, temp. is low.
4000
6000
8000
10000
12000
14000
16000
18000
20000
-20 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320
t ms
Heating pulse decay
Heating current
50 nAmps
time
~ 0.4 ms
0.1850
temperature a-u
0.1845
0.1840
0.1835
0.1830
0.1825
0.0
0.5
1.0
time in micro-second
1.5
0.1850
temperature a-u
0.1845
tau = 0.38 +/- 0.05 micro-s
0.1840
e
0.1835
t

tau
0.1830
0.1825
0.0
0.5
1.0
time in micro-second
1.5
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 t ~ 10 ms
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
Dynamical force?
|1>
|0>
I
SET
Two wave-guids:
~ 40 GHz  parrallel levels
~ 120 GHz  perpendicular levels
4x10
-7
d (eV)
2
0
-2
-4
-4
-2
0
2
4
reservoir right potential
x10
-7
Evolution with deformation
ellipsoïdal
Increase of the deformation
circular
Bias and IV characteristic
Guard electrode
Ids
ring
L pool
R pool
A
Uds
B
the SET is current bias
PreAmp
A-B
IV Characteristic
-4
6.0x10
-4
500 mK
sample 2h
4.0x10
-4
UDS, V
2.0x10
For C-B
oscillations
0.0
-4
-2.0x10
-4
-4.0x10
-4
-6.0x10
-4
-8.0x10
-5
-4
-3
-2
-1
0
IDS, nA
1
2
3
4
Measurement
• L-in technique: 125 Hz, 6-30 mV applied to the guard
Guard electrode
PreAmp
ring
L pool
R pool
L-in
6 mV
125
From L-in
C-B oscillations
From L-in
1.0x10
0.004934
C-B oscillations Amplitude
C-B oscillations Amplitude
Raw oscillations
0.004933
0.004932
0.004931
0.004930
-4
0.5
0.0
-0.5
-1.0
0.004929
0.07
0.08
0.09
0.10
VresR
0.07
0.08
0.09
VresR
Ids=2.6 nA
0.10
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 mm
a
• typical coulomb energy
• Typical kinetic energy (Fermi
energy)
Ec
• rs~2000 rscri~30
Coulombic interaction is
dominant
• Semiconductors qdots rs~2
EK
e2 1
4pe 0 a
2ma
2
chemical potential variation
0.07
Vset=Vgate=0.3 V
0.06
0.05
0.04
5
0.03
7
0.02
0.01
0.00
0
2
4
6
8
10
electron number
12
14