20-09-2010Dias 1
Jens PaaskeThe Niels Bohr Institute & Nano-Science Center
Cotunneling and Kondo effect in quantum dots
Part I/II
J. Paaske, NBI & NSC
Bad Honnef, September, 2010
20-09-2010Dias 2
J. Paaske, NBI
Part I
1. Basics of Coulomb blockade and quantum conductance- Quantum tunneling and classical charging
2. From Anderson model to cotunneling- Schrieffer-Wolff transformation
3. Elastic vs. inelastic cotunneling- Bias spectroscopy
4. Exchange cotunneling and basic Kondo effect - Signatures of Kondo effect
5. Tunneling renormalization of cotunneling thresholds- Ferromagnetic leads, quasi-degenerate systems
Part II
1. The nonequilibrium Kondo problem- What’s the problem?
2. Poor man’s scaling for nonequilibrium systems- Lineshapes for inelastic cotunneling?
3. The effect spin-orbit coupling- Source of bias-asymmetry and angular dependence of B-field
Lecture plan
20-09-2010Dias 3
J. Paaske, NBI
1. H. Bruus & K. Flensberg, “Many-Body Quantum Theory in Condensed Matter Physics”Oxford University Press (2004).
2. R. Hanson et al.: “Spins in few electron quantum dots”,Reviews of Modern Physics 79, 1217 (2007).
3. E. L. Wolf, “Principles of Electron Tunneling Spectroscopy”,Oxford University Press (1985).
4. J. Von Delft: “Kondo effect in metals and quantum dots”,lecture notes from The 4th Windsor Summer School on Condensed Matter Theory,Available at http://www.lancs.ac.uk/users/esqn/windsor07/programme.html
5. Articles cited along the way.
Suggested literature
20-09-2010Dias 5
Transistor Realizations …
In
J. Paaske, NBI
Molecular Transistor Realizations …
100 nm
1000 nm
10 nm
Present day Intel workhorse
Carbon nanotube (Delft)
B
E
C
Single molecule (NBI/NSC)
Bardeen, Brattain og Shockley, Bell Labs 1947
EC
B
20-09-2010Dias 6
J. Paaske, NBI
Basic (field effect) transistor setup
Current through the device (from source to drain)
- turns on (Logical 1)- turns off (Logical 0)
by adjusting electrical potential on the gate electrodes.
Vso
urce
drai
n
Vg
gate
I
? Field Effect Transistor
20-09-2010Dias 7
J. Paaske, NBI
Bias-spectroscopy of nanostructures
”Coulomb diamonds”
V
sour
ce
drai
n
Vg
gate
I
s
d
20-09-2010Dias 8
J. Paaske, NBI
Typical nanostructures of interestand many more ...
Heterostructure quantum dot(GaAs/AlGaAs)
Carbonnanotube
SemiconductorNano-wire
Organic molecule Metal complex
C60 Peapod
Single cell …
20-09-2010Dias 9
J. Paaske, NBI
Contacts of current interest
Normal metal (Au)
Ferromagnetic metal (Ni)
Superconducting metal (Ti/Al, Pd/Nb)
Various combinations: NDS, SDS, SDF, FDF, etc…
MAT
ERIA
LS
Mechanical break junctionsElectromigrationNAN
OG
APS
Electron Beam Lithography Au Nano-rods
New design!
2 nm gap
NSC®
20-09-2010Dias 10
J. Paaske, NBI & NSC
Charge conduction: Quantum tunneling + Classical charging
Source Drain
Filledstates
Potential-landscape:
V
sour
ce
drai
n
Vg
gate
I
Elctrostatic-landscape:
RS
CS CD
RD
CG
20-09-2010Dias 11
J. Paaske, NBI & NSC
The Harlequin diamond plot: Coulomb Blockade
Plotting conductance as a function of and
Chemical potential of dot or molecule:
0
N-1, N N, N+1
N-1 N N+1
C.B. C.B.C.B.C.B. C.B.
gives the slopes:
for .
Current thresholds:
0
D
S
I≠0
I=0
Addition energy:
20-09-2010Dias 12
J. Paaske, NBI
Steady state current (sequential tunneling)
Consider a single quantum level of energy
Occupations:
Source:
Drain:
Dot:
Tunneling rates:
Source and drain currents:
Steady state (nonequilibrium) occupation number of the level:
Steady state current:
Current is flowing only when the level lies within the ”bias-window”
D
S
20-09-2010Dias 13
J. Paaske, NBI
Bias dependence & level broadening
Steady state current:
But where is the voltage in Ohm’s law, ?
Tunneling broadens the quantum level and smears energy conservation:
,
This changes the current to:
,
Conductance through a single level cannot exceed the conductance quantum:
DS
Heisenberg!
20-09-2010Dias 14
I. Conductance through a single quantum level is limited by e2/h
II. Current is blocked by Coulomb-repulsion except for special resonant values of V and Vg .
III. Varying V and Vg leads to characteristic ’Coulomb-diamonds’ for the conductance: ”Single-electron transistor”.
J. Paaske, NBI
Summarizing:
InAs-wire based Quantum Dot,T. Sand Jespersen, NBI
20-09-2010Dias 16
J. Paaske, NBI
Cotunneling: Lifting Coulomb blockade by quantum fluctuations
0
N-1 N N+1
Charging cost:
Finite current:
Cotunneling rate (2.-order PT):
Spinful dot (odd occ.) (∞-order PT):
”Kondo-effect”:
20-09-2010Dias 17
J. Paaske, NBI
Inelastic Cotunneling: Bias spectroscopy
0
Excited state spectroscopy !!
Specific signatures:
• spin-flip transitions(Kondo-sharpened!)
• vibrationally assistedtransitions (sidebands!)
Extra contribution to the current:
20-09-2010Dias 18
J. Paaske, NBI
Kondo effect ( )
Hamiltonian:
(conduction (lead) electron s)(localized (dot) spin S)
(exchange amplitude J)
Transition probability in 3rd order perturbation-theory:
Perturbative Renormalization Group (Poor man’s scaling [PWAnderson, ’64]):
Universal scaling curve:
( Van der Wiel, Science 2000)
Interaction induced energy-scale !
Integrate down to relevant energy-scale:
Strong coupling regime: Landau Fermi Liquid Fixed Point [K.G. Wilson, ’71; P. Nozières, ’74]
J. Kondo, Prog. Theor. Phys. 32, 37 (1964)L. Glazman, M. Raikh, JETP Lett. 47, 452 (1988)T. K. Ng, P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988)
20-09-2010Dias 19
Observing a Kondo peak ...
(Liang et al., Nature 2002)
Spin is screened when lowering temperature!
Weak coupling:
- Doublet (S=1/2)
Strong coupling:
- Singlet (S=0)Binding energy TK ~4K
”Quark”
”Nucleon”
J. Paaske, NBI
20-09-2010Dias 20
J. Paaske, NBI
Dot/lead-Hamiltonian(2nd quantized many-body Hamiltonian)
Single-orbital Anderson modelD
S
Charge fluctuations are strongly suppressed!(Considered as a weak perturbation to Coulomb blockade)
Kondo-regime:
20-09-2010Dias 21
J. Paaske, NBI
Projecting out charge-fluctuations ( odd)The Schrieffer-Wolff transformation
Perform unitary transformation perturbatively:
Construct so as to cancel the tunneling term :
Satisfied with , where:
J. R. Schrieffer, P. A. Wolff, Phys. Rev. 149, 491 (1966).P.-O. Löwdin, J. Chem. Phys. 19, 1396 (1951).
20-09-2010Dias 22
J. Paaske, NBI
Effective exchange-cotunneling (Kondo) model
Finishing the Schrieffer-Wolff transformation:
With (exchange-)cotunneling amplitudes:
(AFM exchange coupling)
(Potential scattering)
J. Appelbaum, Phys. Rev. Lett. 17, 91 (1966).P. W. Anderson , Phys. Rev. Lett. 17, 95 (1966).
20-09-2010Dias 23
J. Paaske, NBI
Cotunneling current (2nd order PT, finite B-field)
Cotunneling-rates:
Nonequilibrium spin-occupation numbers:
for
20-09-2010Dias 24
J. Paaske, NBI
Cotunneling conductance (2nd, and 3rd order order PT, finite B-field)
M. R. Wegewijs, Y. Nazarov, arXiv: cond-mat/0103579J. Paaske, A. Rosch, P. Wölfle, Phys. Rev. B 69, 155330 (2004).V. N. Golovach, D. Loss, Phys. Rev. B 69, 245327 (2004).
20-09-2010Dias 25
J. Paaske, NBI & NSC
Inelastic cotunneling (typical experiments)
Osorio [OPV5]
Ralph [Charge-trap] Goldhaber-Gordon [GaAs/AlGaAs]
Cronenwet [GaAs/AlGaAs]
Schmid [GaAs/AlGaAs]
Zumbühl [GaAs/AlGaAs]
… Z z z z z z z z ... … z z z z z z z Z …
Kogan [GaAs/AlGaAs]
Osorio [Mn2+]
Nygård [CNT]
Babic [CNT]
20-09-2010Dias 26
J. Paaske, NBI
Contacting a single molecule (Electromigration: gold wire)
and a bit of chemistry...
(Herre van der Zant et al., TU-Delft)
20-09-2010Dias 28
J. Paaske, NBI
OligoPhenyleneVenylene5
• Chemical synthesis(Bjørnholm et al. NSC-Copenhagen)
• Low temperature bias-spectroscopy in electromigrated gold-junction(van der Zant et al., TU-Delft)
17_megah_lockin.dat
Vg (V)
Vb
(mV
)
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-80
-60
-40
-20
0
20
40
60
80
-4000
-2000
0
2000
4000
6000
8000
10000
12000dI/dV (nS)
The perfect void for inelastic cotunneling involving low-energy excitations!
Compare:
Molecule CNT-dot100 meV 5 meV
20-09-2010Dias 29
[Mn(terpy-O-(CH2)6-SAc)2)]2+
J. Paaske, NBI
• Chemical synthesis(Bjørnholm et al. NSC-Copenhagen)
SiO2
Al2O3 gate
AuPd
AuPd
Au
2 m
• Low temperature bias-spectroscopy inelectromigrated gold-junction(van der Zant et al., TU-Delft)
S=5/2 S=0
S=1S=1/2
N=5 N=6
S=1/2
Low-Spin
S=5/2
High-Spin
Electrical Spin Control !
20-09-2010Dias 30
J. Paaske, NBI
Spectroscopic fine-structure in carbon nanotubes:
Tunneling renormalization
20-09-2010Dias 31
Maria-Alm, Austria, January 2008
CNT Coulomb-blockade diamonds (bias-spectroscopy)
Adding 285 electrons,one by one...
88 odd-occupiedcharge states withzero-bias Kondo peak.
20-09-2010Dias 32
Maria-Alm, Austria, January 2008
The standard diamond
H He Li Be
B C N O
F Ne Na Mg
Shell-filling
20-09-2010Dias 33
Maria-Alm, Austria, January 2008
The standard diamond
Inelastic cotunneling
Elastic cotunneling (Kondo-peak)
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J. Paaske, NBI
N=1
Spin-polarized leads:
N=0N=2
N=1
Tunneling induced level-shifts in nanotube QD [Ni leads]
(tunneling out) (tunneling in)
Gate-dependent spin-splitting: ( )
(Bethe logarithms …)
J. Martinek et al., Phys. Rev. Lett. 91, 127203 (2003).J. Martinek et al., Phys. Rev. Lett. 72, 121302(R) (2005).M. Sindel et al., Phys. Rev. B 76, 045321 (2007).
20-09-2010Dias 37
J. Paaske, NBI
Gate-dependent exchange-field (tunneling induced ”Lamb-shift”)
0 21
Findings and prospects:
• Electrical spin-control (not via induction fields!)Allows for much faster switching (Spintronics)
• Extremely localized ’magnetic field’ of order 1T (even 70 Tesla !!!)Single electron spin control (Qubit initialization)
20-09-2010Dias 38
J. Paaske, NBI
N=1
Different tunneling-amplitudes to different orbitals:
Tunneling induced level-shifts in nanotube QD [Au leads]
N=0N=2
N=1
N=1
N=0N=2
N=1
20-09-2010Dias 40
Maria-Alm, Austria, January 2008
Tunneling-induced level shifts (2nd order PT)
tunneling out
tunneling in
Γ1 ¿ Γ2
Tunneling rate for orbital i=1,2 to lead =source, drain:
Energy of dot-state with i electrons in orbital 1 and j in orbital 2: ( )∝ Vg
20-09-2010Dias 41
J. Paaske, NBI
Strong coupling sub-gap structure … Unresolved ?!
Gate-dependent excitation energies
20-09-2010Dias 42
J. Paaske, NBI
Inelastic cotunneling in quantum dots and moleculeswith weakly broken degeneracies
Gate-dependent line-shapes
G. Begemann et al., Phys. Rev. B 82, 045316 (2010)
20-09-2010Dias 43
J. Paaske, NBI
Part I
1. Basics of Coulomb blockade and quantum conductance- Quantum tunneling and classical charging
2. From Anderson model to cotunneling- Schrieffer-Wolff transformation
3. Elastic vs. inelastic cotunneling- Bias spectroscopy
4. Exchange cotunneling and basic Kondo effect - Signatures of Kondo effect
5. Tunneling renormalization of cotunneling thresholds- Ferromagnetic leads, quasi-degenerate systems
Part II
1. The nonequilibrium Kondo problem- What’s the problem?
2. Poor man’s scaling for nonequilibrium systems- Lineshapes for inelastic cotunneling?
3. The effect spin-orbit coupling- Source of bias-asymmetry and angular dependence of B-field
Lecture plan