Quantum Noise of a Carbon Nanotube Quantum Dot in the Kondo Regime
Exp : J. Basset, A.Yu. Kasumov, H. Bouchiat and R. DeblockLaboratoire de Physique des Solides – Orsay (France)
Theory : P. Simon (LPS), C.P. Moca and G. Zarand (Budapest)
Chernogolovka - June 2012
Kondo effect : - model system for electronic correlations- screening of a localized magnetic moment in a conductor
- nanophysics (quantum dots (Goldhaber-Gordon et al. Nature (1998), Cronenwett et al.
Science (1998); carbon nanotube (Nygard et al. Nature (2000))
Kondo effect on a single spinIn situ control of the parameters new situation (out of equilibrium, orbital Kondo effect)
Many body problem
Kondo effect and Mesoscopic physics
supra
normal
kondo
Increase of the resistance (T<TK)
3
Kondo effect in quantum dots
U : charging energy; 0: energy level; LR : coupling to the reservoirs
Under specific conditions:- Odd number of electrons in the dot- Intermediate transparency of the contacts- Temperature below Kondo temperature TK
Kondo effect : dynamical screening of the dot’s spin
L R
reservoir reservoir
Quantum dotVS
A
VG
gate
4
Kondo resonance in quantum dots
TK = (U )1/2 exp (-1/ Jeff )
- Transport through second order spin flip events
- Formation of a many body spin singlet (spin of the dot + conduction electrons)
- Peak in the DOS of the dot at the Fermi energy of the leads Kondo resonance
virtual
virtual
Heff = Jeff .S with Jeff = / U : DOS
5
2TK
T < TK
T > TK
TK
Signature of the Kondo effect on conductance
What about Kondo dynamics?
Increase of conductance at low
temperature
What about noise?
A
?
VS
VG
Out-of-equilibrium Kondo dynamics at frequencies h~kBTK?
TK = 1K, >20GHz
Noise detection in the Kondo regime
Low frequency regime (h << kBTK) and low bias voltage (eVsd < kBTK) :- semiconductor quantum dots (SU2) : Influence of Kondo correlations on the Fano factor
O. Zarchin et al. Phys. Rev. B (2008) Y. Yamauchi et al., Phys.Rev.Lett. (2011)
- carbon nanotube quantum dots : Signature of orbital and spin effect T. Delattre et al., Nature Phys. (2009)
Theoretical predictions
C.P. Moca et al., PRB (2011)
Signature of the Kondo effect on noise :
Logarithmic singularity at V=h/eRG calculation
eV=h=5kBTK
- RG calculations at high frequency h>kBTK and out-of-equilibrium
- Prediction of a logarithmic singularity at eV=heven when h>>kBTK
High frequency quantum noise detection at frequencies ≥ kBTK/h
Outline
- Introduction to noise measurement in the quantum regime
- Resonant coupling circuit and SIS detector :- Emission noise of a Josephson junction
- High frequency noise detection of a carbon nanotube in the Kondo regime
System Detector
Emission<0
>0Absorption
?
?
VS
VG
A
• What is electronic noise?
Introduction to electronic noise
I(t)=<I>+I(t)
Vsd
I(t)
Conducting system
•Why measure noise?
Electronic correlations, effective charge, characteristic energy scale, …
<I>
Noise in the quantum regime
• energy scales (eV, … and characteristic times • quantum noise : zero point fluctuations
System mesoscopic device
Detector Amplifier, Quantum dot, SIS junction,…
Emission<0
>0Absorption
h >> kBT, h > eV
Source Detector
Mesoscopic system S SI
Noise measurement in the quantum regime
Josephson Junction-
Carbon nanotube in the Kondo regime
Superconductor / Insulator / Superconductor (SIS) junction
Noise detection with SIS junction : Kouwenhoven’s group, Science (2003)
P.M. Billangeon et al.,PRL (2006)
Resonant Circuit
T=20 mK
0.0 0.2 0.4 0.6 0.8 1.0
0
10
20
30
40
50
IPAT
<0
IPAT
>0
h/e
h/e
ABSORPTION
I D
(nA
)
VD(mV)2/e
EMISSION
Quantum Noise Detection with a SIS Junction
EMISSION ABSORPTIONPhoto-assisted tunneling current
(PAT)
=30GHz
S SI
Ingold & Nazarov (1992)
Source/detector coupling with a resonant circuit
¼ wavelength resonant circuit as in T.Holst et al. PRL(1994)
Independent DC polarisations of the source and the detector
Coupling at eigen frequencies of the resonator (30 GHz and harmonics)
Coupling proportional to the quality factor (Q~10)
L=1mm
a=5µm
b=100µm
L=n/4
n odd integer
Source and detector coupled via the resonant circuit
Source
=
Josephson
Junction
• AC Josephson effect : « on-chip » calibration of the coupling
• Measurement of the quasi-particle high frequency noise
PAT current due to the tunneling of quasiparticlesIPAT measurement as a function of Vsource
Detector bias voltage (VD1 or VD2) : selection of frequency range
VD<2/e
Direct measurement of HF emission noise of a Josephson junction
DC current but No emission Noise while eVS<2+hi
Josephson junction emitting a photon
h=eVS-2
2Δ/e
Calculated (=0)
Theory at
Theory at & finite bandwidth
Quantum noise measurement with SIS detector
Detection of emission and absorption noise
Quantitative measurement of the HF Emission Noise of a Josephson Junction J. Basset et al. PRL 105,166801 (2010)
J. Basset et al. PRB 85, 085435 (2012)
Sensitivity : 2 fA² /Hz (1.5mK on 20k) at 28 GHz, 8 fA² /Hz (5.8mK on 20k) at 80 GHz
Powerful tool to measure HF noise of mesoscopic systems :
carbon nanotube quantum dot in the Kondo regime
What about emission noise?
A
?
VS
VG
Out-of-equilibrium Kondo dynamics at frequencies h~kBTK?
20
LL
VG
AVD
R
A
VS
R
source
draingate
NT
500nm
1m
junctions
Carbon nanotube coupled to the SIS detector
Detector biased for emission noise detection
21
-2 -1 0 1 2
0.2
0.4
0.6
0.8
1.0
1.2
dId
V(e
²/h
)
VS(mV)
Kondo effect in the measured carbon nanotube
Kondo ridge
Center of the ridge TK=1.4K =30GHz
2TK
VG=3.12V
Zero bias peak
What about noise?
Recent theoretical predictions
C.P. Moca et al., PRB (2011)
Signature of the Kondo effect on noise :
Logarithmic singularity at V=h/e
RG calculation
eV=h=5kBTK
- RG calculations at high frequency h>kBTK and out-of-equilibrium
- Prediction of a logarithmic singularity at eV=heven when h>>kBTK
-2 -1 0 1 2-4
-2
0
2
4
-4
-2
0
2
4
0
2
dS
I/dV
S(p
A².
Hz-1
.V-1)
VS (mV)
data
dS
I/dV
S(p
A².
Hz-1
.V-1)
dI/d
V (
e²/h
)
23
High frequency noise in the Kondo regime
30 GHzh~kBTK
2h1/e
2h3/e
No emission noise if |eVS| < h
Small singularity related to the Kondo resonance at
h~kBTK
h~kBTK :- Absence of emission noise if |eVS| < h - Singularity at |eVS| = h qualitatively consistent with
predictions
0 1-1 2-2VS (mV)
24
High frequency noise in the Kondo regime
30 GHzh~kBTK
80 GHzh~ 2.5 kBTK
ANY EXPLANATIONS??
Dynamics of the Kondo effect ? Not predicted by theory
C.P. Moca et al. PRB 10
2h1/e
2h3/e
-2 -1 0 1 2-4
-2
0
2
4
-4
-2
0
2
4
0
2
dSI/d
VS(p
A².
Hz-1
.V-1)
VS (mV)
data theory
dSI/d
VS(p
A².
Hz-1
.V-1)
dI/d
V (
e²/h
)
Singularity related to the Kondo resonance at h~kBTK
Qualitatively consistent but not quantitatively
Coll. with C.P.Moca, G.Zarand and P.Simon
- Theoretical comparison takes into account experimental data with no fitting parameter!
- Kondo temperature TK=1.4K TKRG=0.38K
- asymmetry a=0.67- U=2.5meV, =0.51meV
- Theoretical predictions approximately 2 times higher than experimental result
25
High frequency noise in the Kondo regime
30 GHzh~kBTK
80 GHzh~ 2.5 kBTK
Singularity related to the Kondo resonance at h~kBTK
Qualitatively consistent but not quantitatively
2h1/e
2h3/e
-2 -1 0 1 2-4
-2
0
2
4
-4
-2
0
2
4
0
2
dSI/d
VS(p
A².
Hz-1
.V-1)
VS (mV)
data theory
dSI/d
VS(p
A².
Hz-1
.V-1)
dI/d
V (
e²/h
)
No singularity at h~2.5 kBTK! Not consistent with theory
26
High frequency noise in the Kondo regime
30 GHzh~kBTK
80 GHzh~ 2.5 kBTK
Singularity related to the Kondo resonance at h~kBTK
Qualitatively consistent but not quantitatively
2h1/e
2h3/e
-2 -1 0 1 2-4
-2
0
2
4
-4
-2
0
2
4
0
2
dSI/d
VS(p
A².
Hz-1
.V-1)
VS (mV)
data theory
dSI/d
VS(p
A².
Hz-1
.V-1)
dI/d
V (
e²/h
)
No singularity at h~2.5 kBTK! Not consistent with theory
ANY EXPLANATIONS??
Decoherence at high VS ? Monreal et al. PRB 05
Van Roermund et al. PRB 10De Franceschi et al. PRL 02
Fit with additional spin decoherence rate
27
Decoherence due to voltage bias
- External decoherence rate
Form similar to the intrinsic rate (C.P. Moca et al. PRB 11) Consistent with the differential conductance Consistent with the noise power for both frequencies
Spin lifetime in the dot reduces with applied voltage bias VS
Coll. with C.P.Moca, G.Zarand and P.Simon
, : fitting parameters
28-2 -1 0 1 2-4
-2
0
2
4
-4
-2
0
2
4
dSI/d
VS(p
A².
Hz-1
.V-1)
VS (mV)
data theory with
decoherence theory
dSI/d
VS(p
A².
Hz-1
.V-1)
Single decoherence rate function reproduce the data
30 GHzh~kBTK
80 GHzh~ 2.5 kBTK
Fits OK using a single bias dependent spin decoherence
rate function
with =14, =0.15
29
-2 -1 0 1 2-4
-2
0
2
4
-4
-2
0
2
4
dS
I/dV
S(p
A².
Hz-1
.V-1)
VS (mV)
dS
I/dV
S(p
A².
Hz-1
.V-1)
Logarithmic singularity and decoherence effects
• eV increases Kondo peaks in the density of states (attached to the leads) split and vanish due to decoherence
• Decoherence already pointed out Exp. : De Franceschi et al. PRL 02, Leturcq et al. PRL 05
Th. : Monreal et al. PRB 05, Van Roermund et al. PRB 10
Many photons emitted at eV=h1
Few photons emitted at eV=h3
-2 -1 0 1 2-4
-2
0
2
4
-4
-2
0
2
4
0
2
data theory
dS
I/dV
S(p
A².
Hz-1
.V-1)
VS (mV)
data theory
dS
I/dV
S(p
A².
Hz-1
.V-1)
dI/d
V (
e²/h
)
• Real time Renormalization Group technique• Systematic expansion in the reservoir-system coupling
S. Andergassen et al., Nanotechnology (2010)
• Kondo system out of equilibrium
• Relaxation and decoherence included
Other theoretical approach
Good agreement with experiment with no fit parameter
S. Mülher and S. Andergassen
31
High frequency Fano like factor in the Kondo regime
30 GHz
80 GHz
• Subpoissonian Noise F1• F decreases when
conductance increases
Consistent with a highly
transmitted channel
0 1
N.B. : Energy independent transmission Fano factor
Conclusions
High frequency noise in the Kondo regime
• Singularity due to Kondo effect for h ~ kBTK
• No singularity for h ~ 2.5 kBTK
• Consistent with theory with decoherence due to the bias voltage
J. Basset et al. Phys. Rev. Lett. 108, 046802 (2012).
Quantum Noise of the resonant circuit
No source bias Detector only sensitive to Quantum Noise of the resonator at
equilibrium
Real part of the impedance seen by the detector
Low quality factor due to direct connection Even peaks due to finite value of impedance
mismatch
AC Josephson effect : calibration
IC : critical current
Dynamical Coulomb blockade Ingold et Nazarov (1992)
T~0.9K
Resonator
Resonator
Photons exchange
Resonator
T~0.02K
0.3 0.4 0.5 0.6 0.7 0.8 0.9
020406080
100120140160
Absorption
T=0.95K
T=0.88K
T=0.58K
80GHz 28GHz
dI D
/dV
D(µ
S)
VD(mV)
Emission
28GHz
T=0.02K
T~0.5K
S
SS I
SS I
SS I
Signal due to the resonant circuit Voltage Noise
Real part of the impedance of the resonant circuit.
Extracted from calibration.
• T=0 : No Emission noise but Absorption Zero Point fluctuations• T increases : Emission appears & Absorption increases
Crossover from Quantum to thermal Noise
crossover from quantum to thermal noise
1=28 GHz
38
-100 -50 0 50 100
0
1
2
3
4
+28GHz
2kBTR
Absorption
T=0.01K T=0.5K T=1K
SV
(a.u
.)
(GHz)
Emission
-28GHz
• T=0 : No Emission noise but Absorption Zero Point fluctuations
• T increases : Emission appears & Absorption increases
Thermal Noise
Noi
se @
T=0
(ZPF
)
Noise spectrum of a resistor R
Josephson junction as signal source2 operating modes : • Cooper pairs tunneling : AC Josepshon effect
• Quasiparticle tunneling :
and
Symmetric spectrum with peaks at frequencies
• Only absorption noise• Singularities at
• Emission and absorption noise • Singularity in emission at • Singularity in absorption at
emission absorption