Transport Measurement of Andreev Bound States in a Kondo-Correlated Quantum Dot Experiment: B.-K. Kim, Y.-H. Ahn, J.-J. Kim, M.-H. Bae, N. Kim Theory: R. Lopez, K. Kang, M.-S. Choi Rosa López Institute of Interdisciplinary Physics and Complex Systems UIB Phys. Rev. Lett. 110, arXiv: Geneva 5 th July 2013

Our story is about a Josephson-junction Due to Andreev reflexion the current can flow without any voltage drop! -> Supercurrent Superconductor Normal metal Insulator Quantum dot... Superconductor

One-page summary Usual Josephson phase-current relation But, under some conditions This phase transition is followed by the crossing of the Andreev bound states! 0-junction pi-junction Tunneling spectroscopy of the ABS and the 0-pi transition

Outline 1. Kondo effect Andreev bound states (ABS) 0-  transition Nonequilibrium transport measurement of ABS

Resistivity vs temperature (residual resistance) Phonon scattering decreases with decreasing temperatures Resistance minimum: due to static impurities, dependent on impurity concentration metal nonmagnetic impurity

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 Kondo’s explanation for the resistance minimum As T goes to zero, this term diverges. The so-called Kondo temperature T K is defined as the energy scale limiting the validity of Kondo’s perturbation theory. high-energy excitations also contribute!

Kondo Hamiltonian and RG flow For antiferromagnetic isotropic model Hewson, The Kondo problem to heavy fermions (1993) ß-function Spin flip-flop scattering

Kondo singlet At low T, the impurity magnetic moment and one conduction electron moment bind very strongly and make a singlet (nonmagnetic) state.

Asymptotic freedom Like the quark, at high energies the local moments inside metals are asymptotically free, but at temperatures below T K they strongly interact with the surrounding electrons so that they become confined at low energies. P. Coleman (2006)

Mesoscopic Kondo effect 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. L. Kouwenhoven and L. Glazman, Physics Today (2001) LCR lead (contact,terminal) central region

In order to observe the Kondo effect, there must be a well-defined local moment!

Anderson model Conduction (lead) Hamiltonian Dot Hamiltonian Tunneling Hamiltonian Noninteracting single particle Hamiltonian LCR

Formation of a local moment Magnetization vs Coulomb interaccion

Spin flip and resonance level Many spin-flip events give rise to the Kondo effect, and as a result an extra resonance appears at the Fermi level. Kondo resonance, hallmark of the Kondo effect

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? NCN SCS

Density of states (SC) With nonmagnetic impurities? Superconductor (SC) nonmagnetic impurity local (dot) density of state Bound states are induced, but they are located at the gap edges! Proximity induced gap With N leads

What happens in a SC with magnetic impurities? A. V. Balatsky et al., RMP (2006) Yu-Shiba-Rusinov (YSR) bound states: Bound states emerging as a result of the exchange coupling J YSR bound state solution Pair creation (annihilation) YSR states -

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 Zagoskin, Quantum theory of many-body systems (1998) electron and retroreflected hole, and vice versa, make a complete loop so that according to Bohr’s quantization rule we have bound states. S N S

The 0-  Transition SS QD Two characteristic energy scales: Kondo temperature Superconducting gap 0-junction  -junction M.-S. Choi et al., PRB (2004) doubletsingletGround state:vs

Possible explanation for the  -junction B. I. Spivak and S. A. Kivelson, PRB (1991) In order that electrons be in the canonical order, it is necessary in the indicated step to permute the order of the two electrons. This exchange is responsible for the negative sign due to fermion anticommutation rule! Canonical order

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)

The 0-  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) Importantly

Large gap limit: effective single site Hamiltonian Bogoliubov transformation Lead degrees of freedom are integrated out so that only the d-site is considered. A. Oguri et al., arXiv: (2012) Proximity effect

Why the 0-  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 E A =U/2, the ABS crosses the Fermi level and signal the 0-  transition.

Critical current is measured V g is tuned to show the 0-  transition

What’s new in our experiment? Up to date, ABS has been measured only in equilibrium. Measurements of the ABS in the weak coupling regime. Either ABS or 0-  transition has been measured separately. Previous experiments: 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-  transition has been achieved!

Physics gets complicated: What happens in a finite bias? Multiple Andreev reflection (MAR) AC Josephson effect Time-dependent Hamiltonian Bloch theoryFloquet theory H. Sambe, PRA (1973) S.-I Chu and D. A. Telnov, PR (2004) Space periodicTime periodic

MAR in a SNS junction When we replace the normal region by a resonant level (quantum dot), physics depends on the level position, onsite Coulomb interaction and so on. MAR in a S-QD-S J. C. Cuevas and W. Belzig, PRL (2003) General condition: M. R. Buitelaar et al., PRL (2003) The resonant level must align with the energetic path. subpeaks

odd even odd MAR peak questionable ??? Andreev Transport is probed

Normal Superconducting Normal State Kondo peaks appear in the odd valleys Superconducting State Even valleys show peaks due to quasiparticle cotunneling at V sd =2  2  toguether with a weak Andreev reflection at at V sd =  Odd valley show a rich subgap structure for -  <V sd <  MAR (X) (magnetic field x) Our experiment Phys. Rev. Lett. 110, arXiv: Two types of Kondo ridges: D and E Depending on the T k and  relative strength

Our experiment Phys. Rev. Lett. 110, arXiv: More carefully D valley Anticrossing D valley Kondo dominant E valley Crossing E valley 0-  transition

Andreev Bound States are probed by the dI/dV sd Our experiment Phys. Rev. Lett. 110, arXiv: 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 are probed by the dI/dV sd Our experiment Phys. Rev. Lett. 110, arXiv: 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-  transition

Now we demonstrate the conection of the crossing of the ABS with the 0-  transition Our experiment Phys. Rev. Lett. 110, arXiv: An ABS anticrossing signals a 0-  transition: The critical current signal is high in the 0-junction behavior whereas in the  -junction the critical current is drastically reduced

NRG result: J. S. Lim and R. Lopez’s contribution Actually, the left lead is in equilibrium with the QD so that we can employ the NRG. Ridge D Ridge E

FYI: How to measure the critical current I C ? Experiment M. Tinkham, Introduction to superconductivity (1996) 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-  transition (0-junction) tuned by a gate voltage applied to the QD. This feature is also consistent with a measurement of the gate- dependent critical current. The experimental results are confirmed by a NRG calculation.

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