SLIDE 1 Andreev and Majorana bound states in quantum dots
Alfredo Levy Yeyati
Chernogolovka 13/09/2012
In collaboration with: Alvaro Martín-Rodero, Bernd Braunecker (UAM) Reinhold Egger, Alex Zazunov, Roland Hützen (Dusseldorf) Exp results: Philippe Joyez (Saclay)
SLIDE 2 S
QW Effect of e-e interactions Charging energy in QD regime Andreev states spectroscopy in CNTs J.D. Pillet et al. Nature Phys. (2010) Search for Majorana states in semiconducting nanowires
- V. Mourik et al. Science (2012)
e h 2e
SLIDE 3
Outline Andreev states in QDs Majorana bound states in QDs
The superconducting Anderson model Experimental results – Fit by model calculations NRG vs mean field results The single charge Majorana transistor Transport properties: Known limits Weak blockade regime General Green functions formalism Zero band width limit Equation of motion method Master equation approach
Conclusions
SLIDE 4 QD regime: the superconducting Anderson model
. c . h c c e c c H
k k k / i k k , k k R , L
2
R L k , k k
H H . c . h d c t n n U n H
Single Level
L
R
U 0
2 2
R L R L
eV , , , U , ,
Equilibrium (V=0): Kondo vs Pairing
Energy scale : ~ Energy scale : ~ kb TK
crossover
K
T
p-junction behavior!
Review: A. Martín-Rodero & ALY, Adv. Phys. (2011)
SLIDE 5
Spectral properties: Andreev bound states
SLIDE 6
Experimental results: gate voltage dependence
SLIDE 7 R L k , k k
H H . c . h d c t n n U n H
ABS in SC Anderson model: Hartree Fock approximation
HF approx.
d d U n U
ind
ind ex
E
Minimal model Breaking spin symmetry
n n
(pheonomenological parameter)
0,0 0,5 1,0
0,00 0,25 0,50
0,00 0,25 0,50
0,0 0,5 1,0
0,0 0,5 1,0 1,5 2,0
0,0 0,5 1,0
0,5 1,0 1,5 2,0
0,00 0,25 0,50
E/
I/(4pe/h) I/(4pe/h) I/(4pe/h)
E/ E/
/p /p
2 Eex
5 . 25 0. Eex 75 0. Eex 50 1. Eex
- E. Vecino, A. Martín-Rodero, A. Levy Yeyati, PRB 68, 035105 (2003)
SLIDE 8 Fitting the experimental data: gate voltage dependence
Exp. Model
J.D. Pillet, Ch. Quay, C. Bena, A. Levy Yeyati and P. Joyez, Nature Phys. (2010)
SLIDE 9 ABS in SC Anderson model: Mean field vs “exact” results
- A. Martín-Rodero & ALY, J. Phys: Cond. Matter (2012)
SLIDE 10 Numerical Renomalization Group: basic ideas
D Logarithmic discretization Map into semi-infinite chain Iterative diagonalization Truncation: #states <
V
1
V
2
V
4
V
2 / N N
V
c
N
SLIDE 11 Dashed: HF, Full: NRG p phase: 4 ABSs
ABS: HF vs NRG results
300 4
c
N
SLIDE 12
Dashed: HF, Full: NRG p phase
ABS: HF vs NRG results
SLIDE 13
Symmetric case
Dashed: HF, Full: NRG
SLIDE 14 Quantum dots with Majonana bound states
- A. Zazunov, ALY & R. Egger, PRB (2011)
- R. Hützen, A. Zazunov, B. Braunecker, ALY & R. Egger, arXiv:1206.3912
SLIDE 15 Majorana generation: induced superconductivity in normal or topological semiconducting wires
Review: J. Alicea, arXiv:1202.1293
- R. Egger, A. Zazunov, and ALY, PRL (2010)
Helical states in TI nanowires
- V. Mourik et al. (Delft) Science (2012)
SLIDE 16 The Majorana Single Charge Transistor
“Non-local” fermion Cooper pairs number
gL gR
T E
R L c
, ,
,
SLIDE 17 Bolech & Demler, PRL (2007) “resonant Andreev reflection”
Known limits
“electron teleportation”
SLIDE 18
- A. Zazunov, A.L.Y. & R. Egger, PRB (2011)
Model Hamiltonian Equivalent representation: Cooper pairs + d fermion
) , ( ) 1 , ( N N
“normal” e tunneling
) 1 , 1 ( ) , ( N N
“anomalous” tunneling (Cooper pair splitting)
SLIDE 19 Weak blockade regime
Relevant dregree of freedom Keldysh path integral formulation
- A. Zazunov, A.L.Y. & R. Egger, PRB (2011)
Second order expansion equivalent to semi-classical Langevin equation Current in “P(E)” form
SLIDE 20
Keldysh GFs formulation: general current formula
Define Nambu spinors Exact current formula!
SLIDE 21 Linear conductance: evaluation within ZBWM
R
L
d
d
e i
SLIDE 22
Evaluation using EOM (Equation of Motion method)
Truncation Self-consistency
SLIDE 23
EOM results
e-spectral density “h”-spectral density
SLIDE 24
Crossover of peak conductance
SLIDE 25 Finite temperature: Master equation approach
Q Q+1 Q+2 Q-1 Q-2
) ( 1 , seq Q Q j
) ( 1 , seq Q Q j
) ( 2 , ' , AR Q Q j j
) ( , EC Q Q j
) ( 2 , ' , AR Q Q j j
SLIDE 26
Results from Master equation approach
CB oscillations Peak conductance
2 T
SLIDE 27
Finite voltage sideband peaks
SLIDE 28 Conclusions
* Validity of mean field (HFA): good agreement with NRG for
Andreev bound states in QDs Majorana Single Charge Transistor
* Crossover of peak conductance from 2e2 /h to e2/h as a function of Ec/ * Coulomb blockade oscillations and side band peaks in non-linear conductance * Work in progress: consequences for non-local transport (crossed Andreev) * Qualitative description of CNTs results using phenomenological models * Insight from several different methods (WB, ZBWM, EOM, ME)
K
T
