THE KONDO EFFECT IN CARBON NANOTUBES Milena Grifoni University of Regensburg 700 nm Frauenchiemsee, October 5, 2015

TRANSPORT IN CARBON NANOTUBES: FROM FABRY-PEROT TO KONDO Milena Grifoni University of Regensburg Vg ms md 700 nm Frauenchiemsee, October 5, 2015

TRANSPORT IN CARBON NANOTUBES large transparencies one electron Kondo

TRANSPORT IN CARBON NANOTUBES Fabry-Pérot Coulomb blockade Kondo Courtesy of C. Strunk, A. Hüttel

TRANSPORT IN CARBON NANOTUBES Strong coupling Landauer-Büttiker + Green´s functions DM-NRG, Keldysh effective action Perturbative master equation for the reduced density matrix

KONDO EFFECT IN CNTs ?

CARBON NANOTUBES Quantization of : boundary condition k║ = k κ║ l ^ p 2 = boundary condition Ch·k κ║ κ K K’ τ = +1, -1 K´ = -K

Z-class valley preserving FINITE CNTs Reflection off-the boundaries  standing wave pattern, longitudinal quantization Zig-zag like Armchair like ΔKK' = D+- Z-class valley preserving A-class valley mixing Marganska et al., PRB 92 (2015)

Z-class valley preserving FINITE CNTs Zig-zag like Armchair like ΔKK' = D+- Z-class valley preserving A-class valley mixing With spin: SU(4) Z-class, broken SU(4) A-class but E+(D)=E _ (-D)

Z-class valley preserving FINITE CNTs Zig-zag like Armchair like ΔKK' = D+- Z-class valley preserving A-class valley mixing With spin-orbit: SU(4) broken for all classes but E1,2 (D)=E3,4 (-D)

CONJUGATION RELATIONS Valid also for finite B !!! T T C P C T Schmid et al., PRB 91, 155435 (2015) T, P antiunitary; C=PT -1

SU(4) KONDO PHENOMENA IN CNTs four-fold degeneracy of CNT shells allows for SU(4) Kondo when TK>>D parallel magnetic field lifts SU(4) symmetry Jarillo-Herrero et al. Nature 434, 484 (2005)

SU(4) KONDO PHENOMENA IN CNTs four-fold degeneracy of CNT shells allows for SU(4) Kondo when TK>>D ? Jarillo-Herrero et al. Nature 434, 484 (2005)

KONDO EFFECT WITH BROKEN SU(4) Nel 1 5 10 15 20 25 30 35 40 Nel 17 18 19 20 21 22 23

KONDO EFFECT WITH BROKEN SU(4) Nel = 21 2√∆SO2 + ∆KK'2 Vg (V) Vsd (mV)

MAGNETIC FIELD EVOLUTION No splitting of satellite peaks  P-transition is missing ! Schmid et al. PRB 91, 155435 (2015)

MAGNETIC FIELD EVOLUTION No splitting of satellite peaks  P-transition is missing ! Impose that conjugation relations are satisfied also at the level of the Keldysh effective action and hence at the level of the tunneling density of states Schmid et al. PRB 91, 155435 (2015)

MAGNETIC FIELD EVOLUTION No splitting of satellite peaks  P-transition is missing !

INELASTIC COTUNNELING … but P-line present in the weak coupling limit ? Jespersen et al., Nature Phys. 7, 348 (2011)

STRONG vs. WEAK COUPLING With Jean-Pierre Cleuziou @CEA Grenoble

STRONG vs. WEAK COUPLING

STRONG vs. WEAK COUPLING

STRONG vs. WEAK COUPLING

ANGULAR DEPENDENCE Ground state is a spin singlet!

AND THANKS TO GRK 1570 SFB 689 Andreas Hüttel Davide Mantellli Jean-Pierre Cleuziou (Grenoble) and the Regensburg team Alois Dirnaichner Magdalena Margańska Sergey Smirnov Daniel Schmid Christoph Strunk

CONJUGATION RELATIONS Valid also at finite B : E1,3(B)=E2,4(-B) T-conjugation E1,2(D(B))=E4,3(-D(B)) P-conjugation E3,4(D(-B))=E4,3(-D(-B)) C-conjugation T, P antiunitary; C=PT -1 Schmid et al., PRB 91, 155435 (2015)

CONJUGATION RELATIONS

+ PARALLEL MAGNETIC FIELD Aharonov-Bohm effect: changes the periodic boundary condition: Ch B  Aharonov-Bohm Ajiki, Ando J. Phys. Soc. Jpn 62 (1993) metallic semiconducting Zeeman effect: Zeeman term

STRONG vs. WEAK COUPLING Absence of P-transition is independent of direction of magnetic field !

FROM GRAPHENE TO CNTs a = 2.4 A° K K’ Dirac cones

Reflection off-the boundaries  standing wave pattern FINITE CNTs Reflection off-the boundaries  standing wave pattern Valley preserving Valley swapping

SYMMETRIES (8,2)

BUILDING A NANOTUBE: HELICAL CONSTRUCTION (8,2)

+ CURVATURE curvature spin-orbit coupling other terms in κ┴: determined by the boundary conditions T. Ando, J. Phys. Soc. Jpn 67 (2000)

UNIVERSALITY plot G(Vsd /TK) for different TK universality around Vsd = 0

TEMPERATURE DEPENDENCE experiment theory both main peak and satellites are washed out if T becomes comparable to the Kondo temperature

KELDYSH EFFECTIVE ACTION THEORY slave-boson technique + Keldysh field integral:

THANK YOU

PEAKS EVOLUTION B^ Nel = 21 B|| Nel = 17 one free parameter DSO/DKK' P-transitions T-transitions

TRANSMISSION IN PARALLEL B (6,3) (8,2) ΔSO Δ ΔKK‘ ΔKK' ≠ 0 ΔKK' = 0 Marganska et al., PRB 92, 075433 (2015)

SINGLE ELECTRON SPECTROSCOPY ∆∥ √∆SO2 + ∆KK'2 ∆KK' B sweep Experimental determination of

A-CLASS vs Z-CLASS A-class m=0 Z-class m=1, m=-1 Z-class m: quasi-angular momentum, distinguishes valley in Z-class only

A-CLASS vs Z-CLASS A-class m=0 Z-class m=1, m=-1 Z-class A-class: use odd/even eigenstates under U

Z-class valley preserving FINITE CNTs Z-class valley preserving A-class valley mixing Valley swapping

FINITE CNTs Dso=0 Zig-zag like Armchair like Valley preserving Valley swapping

(classical electrostatic) QUANTUM DOTS Vb e eVb=ms-md Dm source drain N-1 gate VG dot electrochemical potential addition energy charging energy (classical electrostatic) mean level spacing (quantum confinement) 1nm x 100nm dot: kBT273°K = 23 meV

QUANTUM DOTS b Transport condition: mN mN-1 blockade

QUANTUM DOTS b Transport condition in linear regime: mN mN mN-1 resonant tunneling mN mN-1 mN-1 kT >> G blockade

KONDO RESONANCE b Transport condition: mN+1 e N odd Kondo mN resonance kT < G, width ~ TK TK ~ Wexp(-pe/G), large EC

KONDO RESONANCE b Transport condition: mN+1 e N odd Kondo mN resonance kT < G, width ~ TK TK ~ Wexp(-pe/G), large EC

KONDO RESONANCE b Transport condition: mN+1 kT < G, width ~ TK Van der Wiel et al., Science 289, 2105 (2000) mN+1 kT < G, width ~ TK TK ~ Wexp(-pe/G), large EC

KONDO RESONANCE b Transport condition: kT < G, width ~ TK Van der Wiel et al., Science 289, 2105 (2000) kT < G, width ~ TK TK ~ Wexp(-pe/G), large EC

INFLUENCE OF MAGNETIC FIELD Bc B = 0 B = Bc see e.g. Kretinin et al., PRB 85, 201301 (2012) spin degeneracy of state on the QD lifted by Zeeman energy central resonance peak splits above Bc linear split in B at large fields

KONDO EFFECT s=1/2 LEVEL Low T, V High T ,V Fermi liquid behavior (Nozieres, Yosida and Yamada) High T ,V Perturbative regime (Anderson, Hamann) G universal cV/cT TK T,V Keldysh effective action theory  analytic tunneling DOS in the whole regime of parameters Smirnov and Grifoni, PRB 87, R121302 (2013)

KONDO EFFECT s=1/2 LEVEL Keldysh effective action theory  analytic tunneling DOS in the whole regime of parameters Smirnov and Grifoni, PRB 87, R121302 (2013)

FINITE CNTs With spin: SU(4) Z-class, broken SU(4) A-class but E+(D)=E _ (-D) Marganska et al., PRB 92, 075433 (2015)

FINITE CNTs With spin-orbit: SU(4) broken for all classes class but E3,4(D)=E1,2 (-D) Marganska et al., PRB 92, 075433 (2015)

KONDO EFFECT IN QUANTUM DOTS N even quantum dot N odd Van der Wiel et al., Science 289, 2105 (2000)

FROM GRAPHENE TO CNTs

BUILDING A NANOTUBE quantized (yellow lines) zigzag (n,0) θ E θ =0 k|| kx ky θ zigzag (n,0) metal k|| E θ =0 armchair (n,n) θ =π/6 l ^ p 2 = boundary condition Ch·k k|| E l ^ p 2 = boundary condition Ch·k semiconducting