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