Silvano De Franceschi Laboratorio Nazionale TASC INFM-CNR, Trieste, Italy Orbital Kondo effect in carbon nanotube quantum dots

‘Simple’ and controllable systems can be obtained in nanostructured materials. => quantum coherent electronics => fundamental quantum phenomena (spintronics, quantum computation, superconducting electronics…) (quantum coherent dynamics, entanglement, strongly correlated systems,…)

|  |  Spin ½ Kondo

|  |  Spin ½ Kondo T K ~ 10 K T K ~ 1 K Nygard et al., Nature (2000) Liang et al. & J. Park et al. Nature (2002) T K ~ K Goldhaber-Gordon et al., Nature (1998) Cronenwett et al., Science (1998) Schmid et al., Physica B (1998) Semiconductor dots Carbon nanotube dots Single-molecule dots

In a metal with magnetic impurties: In a quantum dot with spin 1/2  |  |  Spin ½ Kondo TKTK

|  |  Spin ½ Kondo Gate-voltage control: => Kondo effect in the unitary limit (G → 2e 2 /h) [Nature 405, 764 (2000)] [Science 289, 2105 (2000)] Magnetic-field control: => integer-spin Kondo effect at singlet-triplet degeneracy [ Phys. Rev. Lett. 88, (2002)] Bias-voltage control: [Phys. Rev. Lett. 89, (2002)] => Kondo effect out of equilibrium

+ + |  |  |+  |  + Spin ½ Kondo Orbital Kondo

|+,  |+,  | ,  | ,  |  |  |+  |  + Spin ½ Kondo  Orbital Kondo = SU(4) Kondo L. Borda et al., Phys. Rev. Lett. (2003). G. Zaránd et al., Solid State Comm. (2003). Theory proposals in 2DEG QDs Experiments in 2DEG QDs S. Sasaki et al., Phys. Rev. Lett. (2004)

k || E (k || ) ()() ()()   v   v Periodic boundary conditions:  Quantized momentum around circumference  One-dimensional subbands Orbital magnetic moment

Finite length L  discrete spectrum: 4-fold shell structure at B=0 (orbital+spin degeneracy) k || E (k || ) E  (1) E  (2) E  (3) E  (1) E  (2) E  (3) ()() ()() Nanotube quantum dot Gate V VGVG I SWNT   v   v

Orbital splitting k || E (k || ) E  (1) E  (2) E  (3) E  (1) E  (2) E  (3) ()() ()() B  0 Nanotube quantum dot Gate V VGVG I SWNT   v   v B B Prediction: Ajiki & Ando, J. Phys. Soc. Jpn (1993) discrete spectrum: 4-fold is lifted at B  0 (orbital splitting >> spin splitting) [Phys. Rev. Lett. 94, (2005)]

Kondo effect in a NT QD with 4-fold shell structure  Four-fold shell structure at B=0  Each shell has two orbitals with opposite orbital magnetic moment  Orbitals in different shells cross each other at high B B E |+,  | ,  | ,  |+,  gB0gB0 |+, > | , > |+, > B = 0 |+, > | , > |+, > B = B 0 gB0gB0 Intra-shell 4-fold degeneracy Inter-shell 2-fold degeneracy SU(4) Kondo Orbital Kondo [Nature 434, 484 (2005)]

B (T) n = 3n = 2 n = 1 half of 1st SHELL 2nd SHELL 3rd SHELL Linear conductance of a small-band-gap CNT QD V G (V) T=8K U U+ 

B (T)   v   v  orb  0.8 meV/T (>>  B = 0.06 meV/T) Consistent with theoretical predictions (Ajiki&Ando J.Phys.Soc. Jpn (1993)) and with recent experiments: Minot et al., Nature (2004); Zaric et al., Science (2004); Coskun et al., ibid. Orbital magnetic moment V G (V)

Orbital magnetic moment Colour scale x V g (mV) B (T) E. Minot et al. Nature 428, 536 (2004) V sd (mV) 0 V g (V) G (e 2 /h)  No 4-fold degeneracy  No link between spectrum & B-evolution of QD states They measured large orbital magnetic moments  orb = Dev F /4 ~ 0.7meV/T ~ 12  B Problems: Small band gap semic. nanotube

B (T)   v   v  orb  0.8 meV/T (>>  B = 0.06 meV/T) Consistent with theoretical predictions (Ajiki&Ando J.Phys.Soc. Jpn (1993)) and with recent experiments: Minot et al., Nature (2004); Zaric et al., Science (2004); Coskun et al., ibid. Orbital magnetic moment V G (V)

B(T) x20 IV I II III IV I II A BC D E F B1B1 C1C1 C2C2 D1D1 D2D2 E1E1 E2E2 F1F1 F2F2 G1G1 A’A’ B’B’ C’C’ D’D’ E’E’ F’F’ 0 1/ V G (V) 2.5 QD orbital & spin configuration k || E (k || ) E  (1) E  (2) E  (3) E  (1) E  (2) E  (3) k || E (k || ) E  (1) E  (2) E  (3) E  (1) E  (2) E  (3) k || E (k || ) E  (1) E  (2) E  (3) E  (1) E  (2) E  (3) k || E (k || ) E  (1) E  (2) E  (3) E  (1) E  (2) E  (3) k || E (k || ) E  (1) E  (2) E  (3) E  (1) E  (2) E  (3) k || E (k || ) E  (1) E  (2) E  (3) E  (1) E  (2) E  (3)

B(T) x20 IV I II III IV I II A BC D E F B1B1 C1C1 C2C2 D1D1 D2D2 E1E1 E2E2 F1F1 F2F2 G1G1 A’A’ B’B’ C’C’ D’D’ E’E’ F’F’ 0 1/ V G (V) 2.5 QD orbital & spin configuration B E

(AA ’ ): (CC ’ ): (DD ’ ): (EE ’ ): (FF ’ ): AA ’ (BB ’ ): BB 1 B1B’B1B’, CC 1 C2C’C2C’, C1C2C1C2,, DD 1 D2D’D2D’, D1D2D1D2, EE 1 E2E’E2E’, E1E2E1E2, FF 1 F2F’F2F’, F1F2F1F2 –  orb (2) – – g  B 1 2  orb (2) – – g  B 1 2 –  orb (2) + – g  B  orb (2) – – g  B 1 2 –  orb (1) – – g  B 1 2  orb (2) + – g  B 1 2 –  orb (1) – – g  B 1 2  orb (2) – – g  B 1 2 –  orb (1) – – g  B 1 2  orb (2) + – g  B 1 2 –  orb (1) + – g  B 1 2  orb (1) – – g  B 1 2 –  orb (1) + – g  B 1 2  orb (2) + – g  B 1 2 E  (3) E  (2) E  (1) E + (3) E + (2) E + (1) E B +,1 +,2 +,3 ,3 ,2 ,1     [Phys. Rev. Lett. 94, (2005)]

3 electrons 1 electron B(T) x20 IV I II III IV I II A BC D E F B1B1 C1C1 C2C2 D1D1 D2D2 E1E1 E2E2 F1F1 F2F2 G1G1 A’A’ B’B’ C’C’ D’D’ E’E’ F’F’ 0 1/ V G (V) 2.5 QD orbital & spin configuration B E Orbital crossing at B=3T

Orbital Kondo Effect V G (V) B (T) /2 III II IV I II III B = B 0  6T Orbital flip B E |+,  | ,  | ,  |+,  gB0gB0

B BB | ,  |+,  |+,  | ,  E B0B0 Orbital Kondo Effect B E |+,  | ,  | ,  |+,  gB0gB V G (V) B (T) /2 III II IV I II III B = B 0  6T Orbital flip at eV=  B B = B 0  6T 2gB02gB0 2B2B 2gB02gB0 2B2B

Low-impedance bipolar spin filter B VGVG I II III IV I II III Switch V G  switch filter polarity Orbital Kondo effect  low impedance

B(T) IV I II III IV I II G1G1 0 1/ V G (V) 2.5 Orbital+Spin Degeneracy => Strong Kondo (multilevel) V (mV) B = 0T I II III 0 IV V G (V)  Strong Kondo effect for 1 and 3 electrons in the shell  Strong triplet-singlet inelastic cotunneling peaks for 2 electrons in the shell [S. Sasaki, S. DF et al. Nature (2000)]

V (mV) V (mV) V G (V) B = 0T B = 1.5T I II III 0 IV Multiple finite B ! The Kondo resonance for 1 electron splits in 4 peaks

V (mV) B (T) Four-fold splitting  SU(4)-Kondo Zeeman splitting Orbital splitting [Theory: Choi, Lopez and Aguado, cond-mat/ ] I B dI/dV V

V G (V) V (mV) Inelastic cotunneling spectroscopy [PRL 86, 878 (2001)] Step in dI/dV at V=level spacing EE eV= +  E eV= -  E B (T) V (mV) B=0.7T dI/dV (e 2 /h) V (mV) dI/dV (e 2 /h) B=80mT 0123 B (T) 2gBB2gBB gBBgBB  4  orb B Zeeman, orbital, orbital + Zeeman

References Pablo Jarillo-Herrero Jing Kong Herre van der Zant Cees Dekker Leo Kouwenhoven Orbital Kondo effect [Nature 434, 484 (2005)] Magneto-transport spectroscopy [Phys. Rev. Lett. 94, (2005)] Collaborators