Dr. Max Mustermann
Referat Kommunikation & Marketing
Verwaltung
Shot noise of excited states
in a CNT quantum dot
Daniel Steininger
AG Strunk / Institut für Exp. und Angewandte Physik
FAKULTÄT FÜR PHYSIK
Sample setup:
Double Quantum Dot Layout:
source, drain, SC central contact, 2 sidegates
Operated as single quantum dot (QD)
Pd
-
Re
Pd
𝐶𝑑
𝐶𝑠
e-beam lithography
Metallization:
Sputter (Re)
Thermal (Pd)
S
𝑅𝑠
QD
𝑅𝑑
D
𝐶𝑔
Gate
𝑉𝑔
5µm
𝑉𝑏
Transport dominated by Coulomb Blockade:
𝜇 𝑁+1
𝜇 𝑁+1
𝜇 𝑁+1
𝜇𝑆
𝑉𝑏𝑖𝑎𝑠
𝜇𝑁
𝑉𝑔𝑎𝑡𝑒
𝜇𝑆
𝜇𝑆
𝑉𝑏𝑖𝑎𝑠
𝜇𝑁
𝜇𝐷
∗
𝜇𝑁
𝑉𝑏𝑖𝑎𝑠
𝜇𝑁
𝜇𝐷
𝜇𝐷
𝑉𝑔𝑎𝑡𝑒
𝑉𝑔𝑎𝑡𝑒
Condition for nonzero Conductance: 𝜇
𝑆
≳𝜇
𝑁+1
≳𝜇
𝐷
Coulomb Blockade:
N
N+1
Gate voltage
Without excited states:
„Coulomb Diamond“ pattern
N+2
E
N
N-1
N+1
Eadd
N-1
Bias voltage
Current
Coulomb peaks when state is aligned within the bias window
b
a
Gate voltage
Excited states included:
Additional steps in Current
Noise:
a)
𝐼(𝑡)
b)
𝐼
𝐼(𝑡)
𝐼
Average Current 𝐼 is the same for a) and b), while 𝐼 𝑡 is different.
𝐼𝛼 = −𝑒
𝑑
𝑑𝑡
𝑁𝛼 , where 𝑁𝛼 =
𝑑
𝑑𝑡
𝑑
𝑒2
𝑑𝑡
𝐼𝛼 = −𝑒
𝑆𝛼 =
𝑘
†
𝑐𝑘𝛼
𝑐𝑘𝛼 is the number of electrons in lead 𝛼.
𝑁𝛼
𝑁𝛼2 − 𝑁𝛼
time derivative of the average number of electrons
2
time derivative of variance of the number of electrons
Noise gives additional information which is discarded in standard
DC measurements
Sources of Noise:
Thermal Noise
𝑺𝑰 = 𝟒𝒌𝑩 𝑻𝑮
(Nyquist−Johnson formula)
𝑘𝐵 𝑇 ≫ 𝑒𝑉
1/f Noise
𝑺𝑰 ~ 𝟏/𝒇
low frequencies, strongly suppressed for 𝑓 ≳ 𝑘𝐻𝑧
Shot Noise
Consequence of charge quantization.
Electrons are randomly transmitted or reflected in the conductor.
→ Current fluctuations
For electrons passing a tunnel barrier with transmission probability 𝒕:
𝑡
𝑺𝑰 = 𝟐𝒆𝑰 𝟏 − 𝒕
𝑒𝑉 ≫ 𝑘𝐵 𝑇
1−𝑡
𝑡 → 0 transfer of electrons is completely random and is described by a Poissonian
distribution
𝑺𝑰 = 𝟐𝒆𝑰
(Schottky formula)
Sub-/Super Poissonian Noise:
We use the Fano factor
𝑺
𝑭≔
𝟐𝒆𝑰
to express deviations from the Poisson value SI = 2eI
Sub-poissonian (F < 1 ):
-Ballistic transport (no scattering), e.g. open channel in a QPC (𝑡 ≈ 1)
-Transport throught double barrier systems (QDs)
𝐹=
𝐹 = 0.5 for symmetric barriers
𝐹 = 1.0 for asymmetric coupling
Γ2𝐿 +Γ2𝑅
Γ𝐿 +Γ𝑅 2
0.5 ≤ 𝐹 ≤ 1.0
Super-poissonian (F > 1):
-Electron bunching due to cotunneling and/or blocking states (see later…)
Measurement Circuit:
300K
4.2K
20mK
-Dilution Cryo
-𝑓 ≈ 1.8𝑀𝐻𝑧
DMM1
I-V
π-filter
DC1
1Ω
100kΩ
100Ω
π-filter
100kΩ
1kΩ
~
~
100Hz
LI 1
Sample
Spectrum Analyzer
RLC-Circuit
Cryo-Amp
frequency-Splitter
Gain: 1.09
22nF
π-filter
2.0nF
10nF
2.2nF
1KΩ
50Ω
150Ω
~130 pF
10KΩ
high-frequencies
coax.
10MΩ
66uH
x1100
1KΩ
ATF - 34143
1.1nF
MITEQ – AU 1447
low-frequencies
22nF
Low frequencies (lock-in)
High frequencies (noise)
System calibration (in situ):
6x10
Thermal (equilibrium) noise of a known
Resistor (𝑅 = 50𝑘Ω).
-6
'500mK'
'450mK'
'400mK'
'350mK'
'300mK'
'200mK'
'175mK'
'150mK'
'125mK'
'100mK'
'075mK'
'050mK'
'025mK'
Vrms
5
4
Differences in peak
amplitude visible down to
T=20mK
3
2
1.6
1.7
1.8
f (Hz)
1.9
SV vs T:
Linear dependence: 𝑆𝑉 = 𝑉 2 = 4𝑘𝑏 𝑇𝑅
2.0
6
2.1x10
Sample Characterization:
Stability diagram:
Two different slopes of the Coulomb diamonds – Two CNTs?
Two sets of Coulomb diamonds:
90 meV
80 meV
~1𝜇𝑚
20 meV
5µm
10 meV
geometric length of the CNT ~ 1𝜇𝑚
Possible configuration:
D
S
ℎ𝑣𝐹
= 𝟖𝟒𝟎𝒏𝒎
4𝛿
ℎ𝑣𝐹
≈
= 𝟖𝟓𝒏𝒎
4𝛿
𝐿𝐶𝑁𝑇 ≈
𝐿𝐶𝑁𝑇
APL 78, 3693 (2001)
𝐿𝑄𝐷2
𝐿𝑄𝐷1
𝐿𝐶𝑁𝑇
2 CNTs in parallel
Stability Diagram:
Current:
dI/dV:
Excited states
∆𝐸 ≈ 1𝑚𝑒𝑉
What kind of excitations? Electronic or Vibronic?
Pro electronic: -CNT lies on a substrate
- ∆𝐸 fits 𝐿𝐶𝑁𝑇
Pro vibronic: - excitations are equidistant
- alternating pattern: pos./neg. dI/dV
Yar et al. PRB 84, 115432 (2011)
Comparison Franck-Condon model
𝑒 −𝑔 𝑔𝑛
𝑃𝑛 =
𝑛!
1 𝑥
𝑔=
2 𝑥0
𝒈 = 𝟏. 𝟗:
𝑃1 ≈ 0.284
𝑃2 ≈ 0.270
𝑃3 ≈ 0.171
𝑃3 ≈ 0.081
2
From experiment:
𝑃1 ≈ 0.284
𝑃2 ≈ 0.268
𝑃3 ≈ 0.175
𝑃3 ≈ 0.091
𝑃4 ≈ 0.91
𝑃3 ≈ 0.175
𝑃2 ≈ 0.268
𝑃1 ≈ 0.284
Step heights fit Franck-Condon model
for electron-phonon coupling 𝒈 = 𝟏. 𝟗
Sapmaz et al. PRL 96, 026801 (2006)
Noise Measurements:
300K
4.2K
20mK
DMM1
I-V
π-filter
DC1
1Ω
100kΩ
100Ω
π-filter
100kΩ
1kΩ
LI 1
~
Sample
Spectrum Analyzer
RLC-Circuit
f-Splitter
Cryo-Amp
22nF
π-filter
2.0nF
10nF
2.2
nF
10KΩ
1KΩ
50Ω
~130 pF
150Ω
coax.
10MΩ
66uH
x1100
1KΩ
ATF - 34143
1.1nF
MITEQ – AU 1447
22nF
Low frequencies (lock-in)
High frequencies (noise)
Data Processing:
Averaging time: t=10s
> Complete spectrum for every
data point (pixel)
> Remove distortions by cutting
> Do Lorentzian fit
> Extract amplitude and
convert to current noise 𝑆𝐼
Current
Current noise
Fano-Map:
- Pattern of different Fano factors
- Super Poissonian noise on excited states
- Enhanced Fano factors on NDC-areas
Modelling/Simulations required to explain this pattern
and distinguish different mechanisms (vibronic or electronic)
Origin of Super Poissonian Noise (F>1):
𝐼≠0
𝜏0
𝜇𝑆
𝜏0
𝜏0
𝑉𝑏𝑖𝑎𝑠
𝜏1 > 𝜏 0
𝐼=0
𝐼≠0
𝐼=0
𝐼≠0
…
…
𝜇𝐷
t
A state with longer lifetime 𝜏1 > 𝜏0 prevents electrons on higher states
from tunneling (blocking state)
Once the electron tunnels out of the dot, all electrons with higher
energy can tunnel out
Current flow is blocked again for 𝑡 = 𝜏1
Increase of noise, while average current remains constant
➔ Increase of Fano factor
Different gate regime:
DC Current:
Current Noise (SI):
dI/dV:
Fano Factor: 𝐹 = 𝑆𝐼 2𝑒|𝐼|
Very large Fano factors observed in this gate regime (𝐹~8)
Steps in Fano Factor:
1
2
3
1
2
3
Bias Voltage
SI vs Current:
F=10
F=1
1
F=0.5
F=10
F=1
2
F=0.5
F=10
3
F=1
F=0.5
1
2
3
Current
Summary:
•
•
Home built noise setup at mK-temperatures
 DC-/AC-/Noise-measurements simultaneously
 Very high resolution (10−29 𝐴2 /𝐻𝑧)
Plenty of additional information beyond standard DC transport:
 Shot Noise suppression / enhancement in Coulomb blockade regime
 Very high Fano factors on excited states
Outlook:
•
•
•
Modelling our experimental results
Repeat measurements with higher quality QDs (suspendended CNTs)
Use two amplifier chains to increase resolution (cross-correlations)
 2 amps already implemented, waiting for samples!
1.
2.
Thank you for
your attention!
Spectrum
Analyzer