Superconducting Proximity Effect in Quantum-Dot Systems · 2013-10-04 · Michele Governale VU...
Transcript of Superconducting Proximity Effect in Quantum-Dot Systems · 2013-10-04 · Michele Governale VU...
Superconducting Proximity Effect in
Quantum-Dot Systems
Jurgen Konig
Institut fur Theoretische Physik
Universitat Duisburg-Essen
– p. 1
Transport through S-N and S-I-S Interfaces
BCS superconductivity: pair potential ∆ = g〈c−k↓ck↑〉
quasiparticles with gap |∆| in density of states
Cooper-pair condensate
subgap transport
Andreev reflection
∆
S N
V
Cooper pair breaks/forms
Josephson current
S S
transfer of Cooper pairs
Jjos(Φ) = Jc sinΦ– p. 2
QDs Coupled to Superconductors - Exps
carbon nanotube quantum dot
Buitelaar, Nussbaumer, Schonenberger, PRL ’02; Cleziou, Wernsdorfer, Bouchiat,
Ondarcuhu, Monthioux, Nature Nanotech. ’06; Jarillo-Herrero, van Dam,
Kouwenhoven, Nature ’06; Jorgensen, Grove-Rasmussen, Novotny, Flensberg,
Lindelof, PRL ’06; Hermann, Portier, Roche, Levy Yeyati, Kontos, Strunk, PRL ’10; Pillet,
Quay, Morfin, Bena, Levy Yeyati, Joyez, Nature Phys. ’10; . . .
quantum dot in InAs nanowire
van Dam, Nazarov, Bakkers, De Franceschi, Kouwenhoven, Nature ’06; Hofstetter,
Csonka, Nygard, Schonenberger, Nature ’10; . . .
InAs quantum dot with Al electrodes
Buizert, Oiwa, Shibata, Hirakawa, Tarucha, PRL ’07; Deacon, Tanaka, Oiwa, Sakano,
Yoshida, Shibata, Hirakawa, Tarucha, PRL ’10; . . .
quantum dot in graphene
Dirks, Hughes, Lal, Uchoa, Chen, Chialvo, Goldbart, Mason, Nature Phys. ’11– p. 3
QDs Coupled to Superconductors - Thy
Andreev and multiple Andreev reflections through QDs
Levy Yeyati, Cuevas, Lopez-Davalos, Martin-Rodero, PRB ’97; Fazio,
Raimondi, PRL ’98, PRL ’99; Kang, PRB ’98; Schwab, Raimondi, PRB ’99;
Johansson, Bratus, Shumeiko, Wendin, PRB ’99; Clerk, Ambegaokar,
Hershfield, PRB ’00; Cuevas, Levy Yeyati, Martin-Rodero, PRB ’01; . . .
transport in Kondo regime
Clerk, Ambegaokar, PRB ’00; Avishai, Golub, Zaikin, PRB ’03; Choi, Lee,
Kang, Belzig, PRB ’04; Siano, Egger, PRL ’04; Sellier, Kopp, Kroha, Barash, PRB
’05; Nussinov, Shnirman, Arovas, Balatsky, Zhu, PRB ’05; Bergeret, Levy Yeyati,
Martin-Rodero, PRB ’06; Lopez, Choi, Aguado, PRB ’07; Karrasch, Oguri,
Meden, PRB ’08; Karrasch, Meden, PRB ’09; . . .
Josephson current through QD with spin-orbit coupling
Dell’Anna, Zazunov, Egger, Martin, PRB ’07; Zazunov, Egger, Jonckheere,
Martin, PRL ’09; . . .
– p. 4
Nature 442, 667 (2006)
quantum dot in InAs nanowire
– p. 5
Nature 442, 667 (2006)
Josephson coupling due to
cotunneling
Jjos ∝ Γ2
π-state for odd occupation
theory:
van Dam et al., Nature ’06
Glazman, Matveev, JETP Lett. ’89
Rozhkov, Arovas, Guinea, PRB ’01
– p. 5
How to Establish a Josephson Coupling?
higher-order tunneling (cotunneling)
Jjos ∝ Γ2
(equilibrium) proximity effect in single-level QD
finite pair amplitude: 〈d↓d↑〉 6= 0
requirement: E0 ≈ E↑↓ < E↑ = E↓ ⇒ 0 < ǫ ≈ −U/2
impossible for large charging energy, U ≫ kBT,Γ
nonequilibrium proximity effect in single-level QD
finite pair amplitude: 〈d↓d↑〉 6= 0
third, normal electrode drives dot out of equilibrium
– p. 6
Model: Anderson Hamiltonian
ΓS ΓS
J
ΓN
L JR
µ N
−Φ/2Φ/2
N
QDE = 0 ǫ ǫ 2ǫ+ U
quantum dot: HD =∑
σ ǫd†σdσ + Un↑n↓
leads: Hη =∑
kσ ǫkc†ηkσcηkσ −
∑
k
(
∆ηc†ηk↑c
†η−k↓ +H.c.
)
superconducting leads: phase biased with ±iΦ/2
normal lead: voltage biased with µN
tunneling: Htunn,η = Vη
∑
kσ
(
c†ηkσdσ +H.c.)
⇒ charging energy, superconductivity, and nonequilibrium– p. 7
Diagrammatic Transport Theory
Pala, Governale, J.K., NJP ’07; Governale, Pala, J.K., PRB ’08
general idea: reduced density matrix for dot
– integrate out leads: contractions 〈c†ηkσcηkσ〉 and 〈cηkσc†ηkσ〉
– expand in tunnel coupling, treat interaction exactly
ρ0
A
0 00
0
0
0 000
L R L R L
LR
L
changes due to superconductivity:
– anomalous contractions: 〈c†ηkσc†η−k−σ〉 and 〈cη−k−σcηkσ〉
– finite pair amplitude on dot: 〈d↓d↑〉 6= 0
00 0
0
– p. 8
2S-dot-N in Nonequilibrium
ΓS ΓS
J
ΓN
L JR
µ N
−Φ/2Φ/2
N
QD
Governale, Pala, J.K., PRB ’08
|∆| → ∞: all orders in ΓS
first order in ΓN
isospin representation
Ix = Re〈d↓d↑〉; Iy = Im〈d↓d↑〉; Iz =〈d†↑d↑ + d†↓d↓〉 − 1
2
kinetic equation for isospin:dI
dt= A−R · I+ I×B
how to proximize the dot:
combined S-dot-N Andreev reflection: A(1)
Andreev reflection at S-dot interface: I×B(0)
– p. 9
Andreev bound states
without SC: resonances at ǫ and ǫ+ U
with SC: Andreev bound-state energies (for ∆ → ∞):
EA,γ′,γ = γ′U
2+ γ
√
(ǫ+ U/2)2 + Γ2Scos2(Φ/2) with γ, γ′ = ±
-1.5 -1 -0.5 0 0.5ε / U
-1.5
-1
-0.5
0
0.5
1
1.5
ΕΑ
,γ,γ
’ / U
EA,+,+
EA,+,-
EA,-,+
EA,-,-
four resonances
avoided crossing
at δ = 2ǫ+ U = 0
with gap 2ΓS| cos(Φ/2)|
– p. 10
– p. 11
– p. 12
Andreev Bound States for Finite ∆
Futterer, Swiebodzinski, Governale, J.K., PRB ’13
resummation scheme for finite ∆
exact for ∆ → ∞
very nice agreement with NRG
Martin Rodero, Levy Yeyati, JPCM ’12
much better than Hartree Fock
– p. 13
Josephson Current JL − JR
ΓS/U = 0.1
−1.5
−1
−0.5
0
0.5
1
1.5
−1.5 −1 −0.5 0 0.5
−4
−2
0
2
4
µN/U
hJjos/(2eΓN)
ǫ/U
(a)
ΓS/U = 0.5
−1.5
−1
−0.5
0
0.5
1
1.5
−20
−10
0
10
20
−1.5 −1 −0.5 0 0.5
µN/U
hJjos/(2eΓN)
ǫ/U
(a)
ΓS/U = 1
−1.5
−1
−0.5
0
0.5
1
1.5
−1.5 −1 −0.5 0 0.5
−1.5
−1
−0.5
0
0.5
1
1.5
−1.5 −1 −0.5 0 0.5
−40
0
40
80
µN/U
hJjos/(2eΓN)
ǫ/U
µN/U
hJjos/(2eΓN)
ǫ/U
(a)(a)
equilibrium Josephson current suppressed by charging
nonequilibrium Josephson current for finite µN
π-transition driven by ǫ or µN
Andreev bound-state energies:
EA,γ′,γ = γ′U2 + γ√
(ǫ+ U/2)2 + Γ2Scos2(Φ/2) with γ, γ′ = ±
Governale, Pala, J.K., PRB ’08– p. 14
Andreev Current JL + JR
ΓS/U = 0.1
−1.5
−1
−0.5
0
0.5
1
1.5
−1.5 −1 −0.5 0 0.5
−0.4
−0.2
0
0.2 0.4
µN/U
ǫ/U
hJand/(2eΓN) (b)
ΓS/U = 0.5
0
−0.4
−0.2
0.2
0.4
−1.5
−1
−0.5
0
0.5
1
1.5
−1.5 −1 −0.5 0 0.5
µN/U
ǫ/U
hJand/(2eΓN) (b)
ΓS/U = 1
−1.5
−1
−0.5
0
0.5
1
1.5
−1.5 −1 −0.5 0 0.5
−0.4
−0.2
0
0.2
0.4
µN/U
ǫ/U
hJand/(2eΓN) (b)
Andreev current probes nonequilibrium proximity effect
maximal around ε = −U/2
Andreev bound-state energies:
EA,γ′,γ = γ′U2 + γ√
(ǫ+ U/2)2 + Γ2Scos2(Φ/2) with γ, γ′ = ±
Governale, Pala, J.K., PRB ’08
– p. 15
Shot Noise Reveals Proximity Effect
Braggio, Governale, Pala, J.K., SSC ’11
ΓSΓNSN QD
off resonance (|δ| ≫ ΓS)
Cooper pair cotunneling
Poissonian transfer of 2e
Fano factor F = 2
on resonance (|δ| ≪ ΓS)
Cooper pair oscillations,
interrupted by SET
Poissonian transfer of e
Fano factor F = 1
a)
−1.5 −1 −0.5 0 0.5
−1.5
−1
−0.5
0
0.5
1.5
1
0.5
1
1.5
2
F 3
−1
0
1
F 2
ǫ/U
µ = 1.5Uµ = 0.5U
µ/U
hκ1/(eΓN )
b)
1
2
3
4
Skew
ness
Fano f
acto
r
Andreev current
– p. 16
Proximity by AC Driving
Moghaddam, Governale, J.K., PRB ’12
off resonant: |δ| ≫ ΓS ⇒ |Iz| =12
no bias voltage: µ = 0
ac driving: ΓS(t) = ΓS + δΓS cosΩt
⇒ B(t) acts on I ⇒ Rabi oscillations
DC current into N lead
resonance at Ω = δ
−4 −2 0 2 4
Ω/U
0.0
0.5
1.0
I N[(e/h)Γ
N]
δ/U = 3
δ/U = 2
δ/U = 1
– p. 17
Cooper Pair Splitter with Double Dots
double dot in InAs quantum wire
Hofstetter, Csonka, Nygard, Schonenberger, Nature ’09
Das, Ronen, Heiblum, Mahalu, Kretinin, Shtrikman, Nat. Comm. ’12
double dot in CNT
Hermann, Portier, Levy Yeyati, Kontos, Strunk, PRL ’10
→ spin-entangled electron pair in normal leads– p. 18
Nonlocal Proximity Effect in Double Dots
Eldridge, Pala, Governale, J.K., PRB ’10
ΓNL
ΓSL ΓSR
ΓNR
N RQD
S
N L QD
non-local pair amplitude:
〈dL↑dR↓ − dL↓dR↑〉 6= 0
negative bias µS < 0:
transport through singlet
Cooper pair splitter
positive bias µS > 0:
spin triplet occupied
triplet blockade
µ/U
δ/U
hIS/(eΓN )
−1.5
−1
−0.5
0
0.5
1
1.5
−2 −1 0 1 2
−1
0
0.6
– p. 19
Unconventional Superconductivity
pair amplitudes: Fiσ,i′σ′(t) = 〈Tciσ(t)ci′σ′(0)〉
symmetry: under exchange of spin/orbital/time
even-singlet: odd/even/even, BCS
even-triplet: even/odd/even, 3He, Sr2RuO4
odd-triplet: even/even/odd, FM-SC hybids
odd-singlet: odd/odd/odd, ?
even/odd - singlet/triplet pairing in double dots?
– p. 20
Unconventional SC in Double Dots
Weiss, Sothmann, Governale, J.K., in preparation
.
.
+µ −µ
Sµ = 0
QD
B
QD
B
S B
S
0
T
T
T
+
0
− ΓS
∆B
∆
large B field: T− occupied
supercond. lead: 0 ↔ S
inhom. B field: S ↔ T 0
noncol. B field: T 0 ↔ T−
triplet pairing detectable in transport– p. 21
Superconducting Proximity Effect in . . .
quantum dot + superconductor
0-π-transitions for Josephson current
local and non-local Andreev transport
double dot + superconductor
non-local Cooper pairs
triplet blockade
double dot + superconductor + magnetic field
non-local unconventional pairing
– p. 22
Collaborators
Alessandro Braggio Uni Genova, Italy
James Eldridge VU Wellington, New Zealand
Rosario Fazio SNS Pisa, Italy
David Futterer Uni Duisburg-Essen, Germany
Michele Governale VU Wellington, New Zealand
Bastian Hiltscher Uni Duisburg-Essen, Germany
Ali Moghaddam Uni Duisburg-Essen, Germany
Marco Pala CNRS Grenoble, France
Bjorn Sothmann Uni Geneva, Switzerland
Janine Splettstoesser RWTH Aachen, Germany
Jacek Swiebodzinski Uni Duisburg-Essen, Germany
Fabio Taddei SNS Pisa, Italy
Stephan Weiß Uni Duisburg-Essen, Germany– p. 23
extra slides
– p. 24
Proximity by AC Driving
Moghaddam, Governale, J.K., PRB ’12
off resonant: |δ| ≫ ΓS ⇒ |Iz| =12
no bias voltage: µ = 0
ac driving: ΓS(t) = ΓS + δΓS cosΩt
⇒ B(t) acts on I ⇒ Rabi oscillations
ac-induced proximity
resonances at Ω = δ±ΓS
−4 −2 0 2 4
Ω/U
−0.25
0.0
0.25
∆QD
δ/U = 3
δ/U = 2
δ/U = 1
– p. 25
Proximity by AC Driving
Moghaddam, Governale, J.K., PRB ’12
off resonant: |δ| ≫ ΓS ⇒ |Iz| =12
no bias voltage: µ = 0
ac driving: ΓS(t) = ΓS + δΓS cosΩt
⇒ B(t) acts on I ⇒ Rabi oscillations
DC current into N lead
resonance at Ω = δ
−4 −2 0 2 4
Ω/U
0.0
0.5
1.0
I N[(e/h)Γ
N]
δ/U = 3
δ/U = 2
δ/U = 1
– p. 25
Quantum Dot + FM + SC
Sothmann, Futterer, Governale, J.K., PRB ’10
ΓS
ΓL ΓR
S
+µ −µ
µ = 0
.
.
S
QDF F
non-collinear fm leads
spin accumulation on QD
fm leads → exchange field
current into superconductor (symmetric in µ)
very sensitive to exchange field
indication of unconventional pairing
– p. 26