Download Transport Measurement of Andreev Bound States in a Kondo

Transport Measurement of Andreev Bound
States in a Kondo-Correlated Quantum Dot
Phys. Rev. Lett. 110, 076803
arXiv:1209.4738
Rosa López
Institute of Interdisciplinary
Physics and Complex Systems
UIB
Experiment: B.-K. Kim, Y.-H. Ahn, J.-J. Kim, M.-H. Bae, N. Kim
Theory: R. Lopez, K. Kang, M.-S. Choi
Geneva 5th July 2013
Our story is about a Josephson-junction
Superconductor
Normal metal
Insulator
Quantum dot
...
Superconductor
Due to Andreev reflexion the current can flow without any
voltage drop!
-> Supercurrent
One-page summary
Usual Josephson phase-current relation
0-junction
But, under some conditions
pi-junction
This phase transition is followed by the crossing of the Andreev bound states!
Tunneling spectroscopy of the ABS and the 0-pi transition
Outline
1. Kondo effect
• Andreev bound states (ABS)
• 0-p transition
• Nonequilibrium transport measurement of ABS
Resistivity vs temperature (residual
resistance)
nonmagnetic impurity
metal
http://physics.info/condensed-matter/
Phonon scattering decreases with decreasing temperatures
Resistance minimum: due to static impurities, dependent on impurity concentration
Resistivity vs temperature: mathematical
formula
Residual resistance due to nonmagnetic impurities:
temperature independent
• Fermi liquid contribution (electron-electron scattering)
• Lattice vibration (phonon scattering)
What happens with magnetic impurities? (Bulk Kondo effect)
First observation:
Electrical resistivity of Au
de Hass et al., 1934
First theoretical explanation:
J. Kondo, 1964
Kondo temperature
Resistivity vs temperature: mathematical
formula
high-energy excitations also contribute!
Kondo’s explanation for the resistance minimum
As T goes to zero, this term diverges.
The so-called Kondo temperature TK is defined as the energy scale limiting the validity
of Kondo’s perturbation theory.
Kondo Hamiltonian and RG flow
Spin flip-flop scattering
For antiferromagnetic
isotropic model
ß-function
Hewson, The Kondo problem to heavy fermions (1993)
Kondo singlet
http://en.wikipedia.org/wiki/Kondo_effect
At low T, the impurity magnetic moment and one conduction electron moment bind
very strongly and make a singlet (nonmagnetic) state.
Asymptotic freedom
http://www.theory.caltech.edu/~preskill/Nobel2004_JP.pdf
Like the quark, at high energies the local moments inside metals are asymptotically free,
but at temperatures below TK they strongly interact with the surrounding electrons so that
they become confined at low energies.
P. Coleman (2006)
Mesoscopic Kondo effect
lead (contact,terminal)
L
C
R
central region
L. Kouwenhoven and L. Glazman, Physics Today (2001)
In a metal: scattering from impurities mixes electron waves with different momenta.
This momentum transfer increases the resistance.
In a quantum dot: all electrons have to travel through the device as there is no other
path. States belonging to the two opposite electrodes are mixed due to the Kondo
effect. Thus, this mixing increases the conductance.
In order to observe the Kondo effect, there
must be a well-defined local moment!
Anderson model
Noninteracting single particle
Hamiltonian
Conduction (lead) Hamiltonian
Dot Hamiltonian
Tunneling Hamiltonian
L
C
R
Formation of a local moment
Magnetization
vs
Coulomb interaccion
Spin flip and resonance level
Kondo resonance,
hallmark of the Kondo effect
Many spin-flip events give rise to the Kondo effect, and as a result an extra
resonance appears at the Fermi level.
Kondo effect in the unitary limit
W. G. van der wiel et al., Science (2000)
Question: In the presence of (magnetic) impurities,
what happens if we replace the normal leads by
the superconducting leads?
N
C
N
S
C
S
With nonmagnetic
impurities?
nonmagnetic impurity local (dot) density of state With N leads
Superconductor (SC)
Proximity induced gap
Bound states are induced, but they are
located at the gap edges!
Density of states (SC)
What happens in a SC with magnetic impurities?
Yu-Shiba-Rusinov (YSR) bound states:
Bound states emerging as a result of the exchange
coupling J
Pair creation (annihilation)
YSR bound state solution
YSR states
A. V. Balatsky et al., RMP (2006)
Andreev reflection and Andreev bound states (ABS)
At the NS interface, an electron produces a
Cooper pair in the superconductor and a
retroreflected hole in the normal region
N
S
S
electron and retroreflected hole, and vice
versa, make a complete loop so that
according to Bohr’s quantization rule we
have bound states.
Zagoskin, Quantum theory of many-body systems (1998)
The 0-p Transition
S
S
QD
M.-S. Choi et al., PRB (2004)
Two characteristic energy scales:
p-junction
• Kondo temperature
0-junction
• Superconducting gap
Ground state:
doublet
vs
singlet
Possible explanation for the p-junction
In order that electrons be in the canonical order, it
is necessary in the indicated step to permute the
order of the two electrons.
Canonical order
This exchange is responsible
for the negative sign due to fermion
anticommutation rule!
B. I. Spivak and S. A. Kivelson, PRB (1991)
Phase diagram
The energy flow goes to the strong fixed
point.
The flow is not attracted to the strong fixed
point.
Andreev bound
states
J. Bauer et al., JPCM (2007)
Importantly
The 0-p transition is always followed
by the crossing of the ABS!
J.S Lim and M-S Choi JPCM (2008) and J.
Bauer et al., JPCM (2007)
Large gap limit: effective single site Hamiltonian
Lead degrees of freedom
are integrated out so that
only the d-site is
considered.
Proximity effect
Bogoliubov transformation
A. Oguri et al., arXiv:1210.3260 (2012)
Why the 0-p transition is followed by the crossing of the
Andreev bound states?
Occupation and corresponding energy
Singlet
Doublet
Singlet
Phase boundary:
Possible excitations:
The excitations correspond to the positions of the ABS.
When EA=U/2, the ABS crosses the Fermi level and signal
the 0-p transition.
Critical current is measured
Vg is tuned to show the 0-p transition
What’s new in our experiment?
Previous experiments:
Up to date, ABS has been measured only in equilibrium.
Measurements of the ABS in the weak coupling regime.
• Either ABS or 0-p transition has been measured separately.
Our experiment:
First measurements of the ABS in the strong coupling regime.
We observe two different prototypes of the Kondo ridges depending
on the ratio
.
In nonequilibrium, simultaneous observation of ABS level crossing
and the 0-p transition has been achieved!
Physics gets complicated: What happens in a finite bias?
Multiple Andreev reflection (MAR)
AC Josephson effect
Time-dependent Hamiltonian
Bloch theory
Space periodic
H. Sambe, PRA (1973)
S.-I Chu and D. A. Telnov, PR (2004)
Floquet theory
Time periodic
MAR in a SNS junction
J. C. Cuevas and W. Belzig, PRL
(2003)
MAR in a S-QD-S
subpeaks
When we replace the normal region by a resonant level (quantum dot),
physics depends on the level position, onsite Coulomb interaction and
so on.
General condition:
The resonant level must align with the energetic
path.
M. R. Buitelaar et al., PRL (2003)
Andreev Transport is
probed
odd
odd
even
MAR peak
questionable ???
Our experiment
Phys. Rev. Lett. 110, 076803
arXiv:1209.4738
• Normal State
Kondo peaks appear in the odd valleys
Normal
• Superconducting State
Even valleys show peaks due to
quasiparticle cotunneling at Vsd=2D,-2D
toguether with a weak Andreev reflection at
at Vsd=D,-D
MAR (X)
Superconducting
(magnetic field x)
Odd valley show a rich subgap structure
for -D<Vsd<D
Two types of Kondo ridges: D and E
Depending on the Tk and D relative strength
Our experiment
Phys. Rev. Lett. 110, 076803
arXiv:1209.4738
More carefully
D valley Kondo dominant
D valley Anticrossing
E valley 0-p transition
E valley Crossing
Phys. Rev. Lett. 110, 076803
arXiv:1209.4738
Our experiment
Asymmetry Factor
• Right barrier much weaker coupled
to the CNT
• Right barrier works as a probe
of the ABS formed by the left barrier
and the CNT
Andreev Bound States
probed by the dI/dVsd
are
Our experiment
Phys. Rev. Lett. 110, 076803
arXiv:1209.4738
Andreev Bound States are probed by the dI/dVsd
The observed anticrossing (D)
and crossing (E) are indeed
an anticrossing and crossing of
the Andreev Bound States
Remember: an ABS anticrossing
signals a 0-p transition
Our experiment
Phys. Rev. Lett. 110, 076803
arXiv:1209.4738
Now we demonstrate the conection of the crossing of the
ABS with the 0-p transition
An ABS anticrossing
signals a 0-p transition:
The critical current
signal is high in the
0-junction behavior
whereas in the p-junction
the critical current
is drastically reduced
NRG result: J. S. Lim and R. Lopez’s
contribution
Ridge D
Ridge E
Actually, the left lead is in equilibrium with the
QD so that we can employ the NRG.
FYI: How to measure the critical current IC?
M. Tinkham, Introduction to superconductivity (1996)
Experiment
NRG
Conclusion
Gate tunable ABS are reported in I-V measurement in an Al-CNT-Al
Josephson junction.
The observed dI/dV shows the two distinct types of the Kondo ridges
associated with ABS.
ABS displays crossing (anti-crossing) behavior, which is the main
characteristics of the 0-p transition (0-junction) tuned by a gate voltage
applied to the QD.
This feature is also consistent with a measurement of the gatedependent critical current.
The experimental results are confirmed by a NRG calculation.
Thank you for your attention.
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