Carbon nanotube quantum dots: SU(4) Kondo and the Mixed ...gleb/kondo_talk_9.pdfCarbon nanotube...
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Carbon nanotube quantum dots:SU(4) Kondo and the Mixed Valence regimes
Gleb Finkelstein
Duke University
Alex Makarovski Thanks:F. Anders, H. Baranger, M. Galpin, L. Glazman, K. LeHur, J. Liu, D. Logan, G. Martins, K. Matveev, E. Novais, M. Pustilnik, D. Ullmo
Support: NSF DMR-0239748
Electronic band structure, sample
Coulomb blockade
Kondo effect / Mixed Valence
-apparent Kondo behavior in the Mixed Valence regime
Samples
doped Si
SiO2
nanotube
Vsource-drain Vgate
A
Single-wall CNT ~2 nm in diameter
Measurement: differential conductance = dI/dV (Vgate)
Graphene (2D graphite): semi-metal
kx
ky
E
k(2D vector)
Energy dispersion E(k)
k
E“Physical Properties of Carbon Nanotubes” “Physical Properties of Carbon Nanotubes” Saito, Dresselhaus and Dresselhaus 1998.1998.
k⊥
Nanotubes: quantization of quasi-momentum
k||
k⊥
k||
C= 2πR
r + C coincides with r =>k⊥= 2πN/C = N/Rk|| - arbitrary
Nanotubes: metallic and semiconducting
E
k
k- quasi-momentum along the length of the nanotube
metal
• Metallic or semiconducting • Two degenerate bands
k
semiconductor
k
E
k
Degenerate orbitals: ‘shells’
kE
k
Two degenerate subbands => degenerate orbitals
E
shell
W.J. Liang, M. Bockrath, and H. Park, PRL (2002)M. R. Buitelaar et al., PRL (2002)
Quantization along length
Electronic band structure, sample
Coulomb blockade
Kondo effect / Mixed Valence
-apparent Kondo behavior in the Mixed Valence regime
Ambipolar semiconducting nanotube
-8 -6 -4 -2 0 2 4 6 8 10 12 14 16 180.00.51.01.52.02.5
C
ondu
ctan
ce (e
2 /h)
Vgate (Volts)
T=1.5K
band gap
max: 4e2/h
P-type N-type
holes electrons
Quantum Dot
Leads and island formed within the same nanotube
M.S. Choi, R. Lopez and R. Aguado, PRL (2005)
4 degenerate levels in the dotare coupled one-to-one to 4 modes (↑, ↓, ↑, ↓) in the leads
The modes are not mixed by tunnelingTunneling Hamiltonian has
SU(4) symmetry
Nanotube quantum dot
Tunneling grows with Vgate
2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.00.0
0.5
1.0
Con
duct
ance
(e2 /h
)
V gate
Energy gap
Quantum Dot
Γ grows
2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.00.0
0.5
1.0
Con
duct
ance
(e2 /h
)
V gate
Groups of 4 peaks: orbital degeneracy
4 degenerate states:2 degenerate orbitals x 2 spin projections
Energy gap
E
-k0 k0
k||“shell”
Groups of 4 peaks: orbital degeneracy
E
-k0 k0
k||
7.0 8.0 9.0 10.00.0
0.5
1.0
Con
duct
ance
(e2 /h
)
V gate
EC+∆E
EC=e2/C – Charging energy
∆E – QM level spacing
EC
Conductance map: G(Vgate, Vsd)8 12 16 20 24
2 3 4 5 6 7 8 9 10-20-10
01020
Vsd
(mV
)
Vgate (V)
2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.00.0
0.5
1.0
Con
duct
ance
(e2 /h
)
V gate
E
B||
Nanotubes in Magnetic field
B||
B⊥
E
B⊥
Orbital + Zeeman ZeemanAndo, SST (2000)
0 1 2 3 4 5 6
9
10
11
12
Vga
te (V
)Magnetic field (Tesla)
Nanotube in B||
B||
B||
E
cond
ucta
nce
Orbital + Zeeman
B⊥: Zeeman splitting
B⊥
E
Sz = 1S=0,1
0 2 4 6 8
-3.4
-3.3
G
ate
volta
ge (V
)
Magnetic field (Tesla)
Addition energy = peak spacing in Vgate converted to energy
0 1 2 3 4 5 6 7 8 94.5
5.0
5.5
6.0
6.5
7.0
7.5
Ead
d (m
eV)
B (T)
Addition energies in B⊥
∆E
EC
Zeeman
5.0 6.00.0
0.5
Con
duct
ance
(e2 /h
)
V gate
1 2 3
Exchange is negligible
0 1 2 3 4 5 6 7 8 94.5
5.0
5.5
6.0
6.5
7.0
7.5
Ead
d (m
eV)
B (T)
Perpendicular magnetic field: Zeeman splitting
∆E
EC
Zeeman
Addition energy
B⊥
2
1, 3Exchange
Y. Oreg, K. Byczuk, and B.I. Halperin, PRL (2000)S-H. Ke, H.U. Baranger, and W. Yang, PRL (2003).
Electronic band structure, sample
Coulomb blockade
Kondo effect / Mixed Valence
-apparent Kondo behavior in the Mixed Valence regime
Kondo effect in Quantum Dots
G
Vgate
Conductance grows at low temperatures in valleys with a non-degenerate ground state
1e 2e 3e
Theory: Glazman and Raikh (1988)Ng and Lee (1988)
Experiment: Goldhaber-Gordon et al., (1998) Cronenwett et al., (1998)Schmid et al., (1998)
Kondo effect in Nanotubes
SU(4) theories for 1 electron:Double dots, dots with symmetries:D. Boese et al., PRB (2002)L. Borda et al., PRL (2003)K. Le Hur and P. Simon, PRB (2003)G. Zarand et al., SSC (2003)W. Izumida et al., J.P.Soc.Jpn. (1998)A. Levy Yeyati et al., PRL (1999)Nanotubes:M.S. Choi et al., PRL (2005)
1 electron SU(4) Kondo experiment:Quantum dots: S. Sasaki et al., PRL (2004)Nanotubes: P. Jarillo-Herrero et al., Nature (2005)
G
1e 2e 3eVgate
Kondo effect in Nanotubes
G
Vgate1e 2e 3e
2 electron SU(4) theoryM.R. Galpin, D.E. Logan and H.R. Krishnamurthy, PRL (2005)C.A. Busser and G.B. Martins, PRB (2007)
2-e Kondo in nanotubes – triplet or SU(4) ?W.J. Liang, M. Bockrath and H. Park, PRL (2002)B. Babic, T. Kontos and C. Schonenberger, PRB (2004)
Interactions ↑↓ = ↑↓ = ↑↓Exchange ↑↑ is small
Kondo effect in Nanotubes
G
Vgate1e 2e 3e
Friedel sum rule: Σ δi = πNKondo singlet: δ↑=δ↓=δ↑=δ↓1e δi = π/42e δi = π/2
G(2e) = 2G(1e)
This type of arguments may be found for the SU(2) case in Glazman and Pustilnikcond-mat/0501007
G=G0 Σ sin2(δi )
Temperature dependence of conductance
4 5 6 7 80.0
0.5
1.0
1.5
2.0
2.5
Vgate (Volts)
Temperature 15.0 K
Con
duct
ance
(e2 /h
)
1 2 3 1 2 3
barrier transparency grows
0/4 Ec, ∆ ~ 100K
Shell spacing
Temperature dependence of conductance
4 5 6 7 80.0
0.5
1.0
1.5
2.0
2.5
Vgate (Volts)
Temperature 1.3 K 3.3 K 6.4 K 10.4 K 15.0 K
Con
duct
ance
(e2 /h
)
Ec, ∆ ~ 100K
12
3 1
2 3
Growth of the signal in the valleys due to the Kondo effect
4 5 6 7 80.0
0.5
1.0
1.5
2.0
2.5
Vgate (Volts)
Temperature 1.3 K
Con
duct
ance
(e2 /h
)
Shape of the 4-electron clusters at low T and high transparency
Single-electron features are washed away
4e
4e
1
2 3
4 5 6 7 80.0
0.5
1.0
1.5
2.0
2.5
Vgate (Volts)
Temperature 1.3 K
Con
duct
ance
(e2 /h
)
Fabry- Perot (single-particle interference) does not work!
Shape of the 4-electron clusters at low T and high transparency
4 5 6 7 80.0
0.5
1.0
1.5
2.0
2.5
Vgate (Volts)
Temperature 1.3 K
Con
duct
ance
(e2 /h
)
G ~ sin2(πN/4) where N varies linearly with Vgate
N0e 4e
Shape of the 4-electron clusters at low T and high transparency
Electronic band structure, sample
Coulomb blockade
Kondo effect / Mixed Valence
-apparent Kondo behavior in the Mixed Valence regime
Phase shifts of scattered waves δ
Low temperature singlet:
δ↑ = δ↓ = δ↑ = δ↓
Friedel’s sum rule Σ δi = πN
N=CVg/e (open dot)
G=G0Σ sin2(δi)= 4G0sin2(πCVg/4e)
δi = πN/4
Characteristic energy scale T0~3 K << Ec, ∆, Γ (all ~100 K)
Evidently, the argument works even for non-integer N
Temperature dependence of conductance
4 5 6 7 80.0
0.5
1.0
1.5
2.0
2.5
Vgate (Volts)
Temperature 1.3 K 3.3 K 6.4 K 10.4 K 15.0 K
Con
duct
ance
(e2 /h
)
Ec, ∆ ~ 100K
12
3 1
2 3
4 5 6 7 8
-10-505
10
V
sd (m
V)
Vgate (Volts)
Spectroscopy at finite bias
E
Vgate
Tunneling from the weakly coupled lead probes the density of states in the Kondo / Mixed Valence system 1e 2e 3e
ΓL >> ΓRTunneling density of states
EF
2.25e2/h0Width of the resonance: 10 K ~ T0
T = 3.3 K
4 5 6 7 8
-10-505
10
V
sd (m
V)
Vgate (Volts)
Spectroscopy at finite bias
E
Vgate
Tunneling from the weakly coupled lead probes the density of states in the Kondo / Mixed Valence system 1e 2e 3e
ΓL >> ΓRTunneling density of states
EF
1.5e2/h0
T = 10 K
Electronic band structure, sample
Coulomb blockade
Kondo effect / Mixed Valence
Numerical Renormalization Group calculations
F. Anders, M. Galpin, D. Logan, and G.F., PRL (2008)
Leads and island formed within the same nanotube
4 degenerate levels (↑, ↓, ↑, ↓)in the dot are coupled one-to-one to 4 modes (↑, ↓, ↑, ↓) in the leads
The modes are not mixed by tunneling; t amplitude does not depend on α or σ Hamiltonian has SU(4) symmetry
two leads ν = L, R
spin σ, orbital α N number of electrons
SU(4) tunneling Hamiltonian
NRG calculations of the SU(4) tunneling Hamiltonian
F. Anders, M. Galpin, D. Logan, and G.F. PRL (2008)
Γ= 0.5, 1, 2 meVT= 15, 8.5, 5, 3, 1.7 K(color sequence reversed)
Γ= 1 meV
Γ= 0.5 meV Γ= 2 meV
Zero-bias conductance – numerical results
0 1 2 3 40.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0 4e2/ h sin2(πN/4) 15 K 8.5 K 5 K 3 K 1.7 K
N
σ (e
2 /h)
Ngate
Γ= 0.5 meV
Zero-bias conductance – numerical results
0 1 2 3 40.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0 4e2/ h sin2(πN/4) 15 K 8.5 K 5 K 3 K 1.7 K
N
σ (e
2 /h)
Ngate
Γ= 1 meV
Zero-bias conductance – numerical results
0 1 2 3 40.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0 4e2/ h sin2(πN/4) 15 K 8.5 K 5 K 3 K 1.7 K
N
σ (e
2 /h)
Ngate
Γ= 2 meV
Universal scaling curve G(T/T0) for 2 electrons
0.1 10.0
0.2
0.4
0.6
0.8
1.0
SU(2) SU(4)
shell II III IV
G(T
) / G
(0)
T/T0
4 5 6 7 80.0
0.5
1.0
1.5
2.0
2.5
Vgate (Volts)
Con
duct
ance
(e2 /h
)
2e SU(4) curve is more steep than inthe 1e SU(2) case
4 5 6 7 8
-10-505
10
Vsd
(mV
)
Vgate (Volts)
Electronic band structure, sample
Coulomb blockade
Kondo effect / Mixed Valence
-apparent Kondo behavior in the Mixed Valence regime
-Kondo in magnetic field
Kondo in B⊥: SU(4) Kondo suppressed
-4 -3 -2 -1 0 1 2 3 40.000.010.020.030.040.050.060.070.080.090.100.110.120.13
Con
duct
ance
(e2 /h
)
Vsd (mV)
0 T2 T4 T6 T8 T
2eE
B⊥
Non-degenerate ground stateKondo suppressed
Kondo in B⊥: SU(4) to orbital SU(2)
E
0 T2 T4 T6 T8 T
-4 -3 -2 -1 0 1 2 3 40.000.010.020.030.040.050.060.070.080.090.100.110.120.13
Con
duct
ance
(e2 /h
)
Vsd (mV)
1e
B⊥
E
B⊥
0 T2 T4 T6 T8 T
Degenerate ground state =>Orbital SU(2) Kondo
0 1 2 3 4 5 6
9
10
11
12
Vga
te (V
)Magnetic field (Tesla)
Nanotube in B||
B||
3e
2e
1eE
B||
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
-6
-4
-2
0
2
4
6
Vsd
(mV
)
Magnetic field (Tesla)
E E E
B||
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
-6
-4
-2
0
2
4
6
Vsd
(mV
)
Magnetic field (Tesla)
1e 2e 3e 2e singlet ground state => Kondo disappears
SU(2) spin Kondo still possible for 1e and 3e
Kondo in B||
-10 -5 0 5 100.0
0.5
1.0
1.5
2.0
2.5
3.0
Con
duct
ance
(e2 /h
)
Vsd (mV)-10 -5 0 5 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Con
duct
ance
(e2 /h
)
Vsd (mV)
3e2e
9T
0T
TKSU(2) < spin splitting singlet, non-degenerate
-10 -5 0 5 100.0
0.5
1.0
1.5
2.0
2.5
3.0
Con
duct
ance
(e2 /h
)
Vsd (mV)
1e
SU(4) => SU(2)SU(4) => no Kondo
TKSU(2) > spin splitting
SU(4) => no Kondo
-9 -6 -3 0 3 6 90.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
shell 1 shell 2
Con
duct
ance
(e2 /h
)
Vsd (mV)
1e
3e
2e
B|| =3.5 T
Kondo features in 2 successive shells
TKSU(2) > spin splittingSU(4) => SU(2)
singlet, non-degenerateSU(4) => no Kondo
TKSU(2) < spin splittingSU(4) => no Kondo
10.0 10.5 11.0 11.5 12.0
-10
-5
0
5
10
Vsd
(mV)
Vgate (Volts)
B|| = 3 T
1e 2e 3e 1e 2e 3e
Kondo features in 2 successive shells
The difference between 1e and 3e cases is persistent in 2 successive shells
B||
E
Spin-orbit
B||
E
B||
Without SO With SO
Large spin gap=> no Kondo
Small spin gap=> SU (2) Kondo
Kuemmeth et al., Nature (2008)
Summary
• Orbital degeneracy in nanotube Quantum Dots
• Kondo effect / Mixed Valence – a major modification of conductance at low T.
• Why does G ~ sin2(πN/4) work so well for noninteger N?
• What is the low energy scale?
• Lifting SU(4) symmetry by magnetic field:
B⊥ orbital SU(2) Kondo effect
B|| spin-orbit in 1e and 3e valleys
Determination of Γ
4 5 6 7 80.0
0.5
1.0
1.5
conductance at 15K 16 peak Lorentzian fit
Vgate (Volts)
Con
duct
ance
(e2 /h
)
0 2 4 6 8 10 12 14 160123456789
IVIIIIII
Γ (m
eV)
Peak Number
(E-E0)2-Γ21
10K
3.3K
Is the coupling of the two orbitals to the leads different?
10.0 10.5 11.0 11.5 12.00.0
0.2
0.4
0.6
0.8
1.0
1.2
3.5 T
765321
C
ondu
ctan
ce (e
2 /h)
Vgate (V)
The widths of the single-electron peaks at non-zero field for two orbitals in consecutive shells are about the same
What is the cause of the e-h asymmetry?
Spin-orbit
Positions of the resonances
T = 1.3 K
1.2e2/h0