Josephson effect in carbon nanotubes with spin-orbit coupling. Rosa López Interdisciplinary Institute for Cross-Disciplinary Physics and Complex Systems University of Balearic Islands IFISC (CSIC-UIB), Palma de Mallorca, Spain J. S. Lim (IFISC, UIB), R. López, R. Aguado (ICMM, CSIC): arxiv:114.513 Phys. Rev. Lett (211) in press, October 211
* 2 nm ORBITAL + SPIN Text W. Liang, M. Bockrath and H. Park, PRL, 88, 12681 (22) M. R. Buitelaar, A. Bachtold, T. Nussbaumer, M. Iqbal and C. Schönenberger, PRL, 88, 15681 (22) Jarillo-Herrero et al.,prl, 94, 15682, (25)
Periodic Boundary Conditions Nanotubes Sub-bands in 1D. K 1 K 2 Ethan Minot, Tuning the band structure of carbon nanotubes, PhD Thesis, Cornell 24.
Modelling Quantum Dot Carbon Nanotubes I Small bandgap nanotubes k = τ 3 k g K K k K and K are degenerate owing to time-reversal symmetry (isospin) τ 3 = ±1 H = v F (k τ 3 σ 1 + k τ σ 2 ) also becomes quantized due to the finite length (quantum dot).
Modelling Quantum Dot Carbon Nanotubes II µ orb = edν F /4 B t Large orbital moments couple to parallel magnetic fields H orb = τ 3 µ orb B = τ 3 ν F πb D/2Φ E k = τ 3 k g + Φ AB /DΦ Φ AB = B πd 2 H Z = 1 2 gµ BB τ σ s 3 B Determination of electron orbital magnetic moments in carbon nanotubes, E. D. Minot, Yuval Yaish, Vera Sazonova & Paul L. McEuen, Nature, 428, 536 (24) See News&Views Nanoscale physics: M. Chiao, Big moment for nanotubes (24)
SPIN KONDO EFFECT AND ISOSPIN KONDO EFFECT SU(4) SYMMETRY
SPIN KONDO ORBITAL KONDO SPIN + ORBITAL
Low temperature transport: evidence of SU(4) symmetry Zeeman orbital Experiment: Orbital Kondo effect in Carbon Nanotubes, Pablo Jarillo-Herrero, Jing Kong, Herre S.J. van der Zant, Cees Dekker, Leo P. Kouwenhoven, Silvano De Franceschi, Nature, 434, 484 (25). Theory: SU(4) Kondo effect in carbon nanotubes, Mahn-Soo Choi, Rosa López and Ramón Aguado, PRL, 95,6724 (25)
Carbon Nanotube with Spin-orbit coupling I The orbital motion of electrons also couples to a curvature-induced radial electric field. This creates an effective axial magnetic field which polarizes the spins along the NT axis and favors parallel alignment of the spin and orbital magnetic momenta or antiparallel depending on the sign of this spin-orbit coupling. As a result, the fourfold degeneracy breaks into two Kramers doublets (time-reversed electrons pairs). Spin-orbit interaction can be understood as an effective exchange field between Kramers pairs. Coupling of spin and orbital motion of electrons in carbon nanotubes, F. Kuemmeth, S. Ilani, D. C. Ralph & P.L.McEuen, Nature, 452, 448, 28. THEORY: T. Ando, J. Phys. Soc. Jpn. 69, 1757 (2). D. Huertas-hernando et al, Phys. Rev. B, 74, 155426 (26). D. V. Bulaev et al, Phys. Rev. B 77, 23531 (28) L. Chico et al, Phys. Rev. B, 79, 235423 (29). J. Jeong and H. Lee, Phys. Rev. B, 8, 7549 (29). W. Izumida, K. Sato and R. Saito, J. Phys. Soc. Jpn., 78, 7477 (29). B SO = v E
Carbon Nanotube with Spin-orbit coupling II The orbital motion of electrons also couples to a curvature-induced radial electric field. This creates an effective axial magnetic field which polarizes the spins along the NT axis and favors parallel alignment of the spin and orbital magnetic momenta or antiparallel depending on the sign of this spin-orbit coupling. As a result, SU(4) degeneracy breaks into two Kramers doublets (time-reversed electrons pairs). Spin-orbit interaction can be understood as an effective exchange field between these Kramers pairs (J-S Lim, R. López, G-L Giorgi, D. Sánchez Phys. Rev. B 83, 155325 (211)) H SO = 1 SOτ 3 σ 1 s 3 + SOτ 3 σ s 3 Coupling of spin and orbital motion of electrons in carbon nanotubes, F. Kuemmeth, S. Ilani, D. C. Ralph & P.L.McEuen, Nature, 452, 448, 28.
Nanotubes can be contacted with superconducting leads Pillet et al, Nature Physics, 6, 965, (21). J. P. Cleuziou et al, Nature Nanotech., 1, 53, (26)
Superconducting Contact Carbon Nanotube QD Superconducting Contact
Superconducting Contact Carbon Nanotube QD Superconducting Contact Andreev Bound states (ABS): entangled time-reversed electron-hole Kramers pairs. Recently measured in nanotubes and graphene Pillet et al, Nature Physics, 6, 965, 21 (nanotubes); T. Dirks, et al, Nature Physics, 6 February 211 (graphene)
Superconducting Contact Carbon Nanotube QD Superconducting Contact The Josephson current is mainly given by resonant tunneling of Cooper pairs through these bound states
Nanotubes can be contacted with superconducting leads Text Different curves correspond to different Vg Quantum supercurrent transistors in carbon nanotubes, Pablo Jarillo-Herrero, Jorden van Dam and Leo Kouwenhoven, Nature 439, 953 (26). For a review, see Hybrid superconductor quantum dot devices, Silvano de Franceschi et al, Nature Nanotechnology, 5, 73 (21)
As both phenomena, spin-orbit and the formation of Andreev bound states, are related to time-reversed Kramers pairs, it is interesting to address the following question: what happens to the Josephson effect in QD carbon nanotubes in the presence of spin-orbit?
Anderson-like hamiltonian + BCS leads H C = α=l/r,k,τ,s α,k,τ ξ k c αkτs c αkτs α e iφ α c αkτ c α k τ + h.c. H D = τ,s ε τs d τsd τs + U n τs n τ s H T = (τ,s)=(τ,s ) V α c αkτs d τs + h.c., α=l/r,k,τ,s
E/ FIG. 2. SO-mediated supercurrent reversal. a, Total (top) function of phase and different B (in Tesla) for Γ =.1 5 near the -π transition at B = B c =.52T. c, ABS vs. φ in π behavior. d, ABS versus V g for different B =,.5, for all V g <. The π transition is robust as V g is varied ( -5 Bc tization -1 in the longitudinal and perpendicular directi to QD -4 confinement -2 [15] and2the finite 4 diameter of t and the SO coupling. B [T] The levels can be approxim ε τ,σ = ε + στ SO + σ Z + τ orb, with Z = and orb = µ orb B (µ s and µ orb are the spin and
Calculation: Green s functions in Nambu space. The poles of the retarded Green s function give the Andreev bound states Det[G r d(ω) 1 ]=D + D = E 1(2) ε + ΓE 1(2) 2 E 2 1(2) E 1(2) + ε ± + ΓE 1(2) 2 E 2 1(2) Γ2 2 cos 2 (φ/2) 2 E 2 1(2) =
E 1(2) ε + ΓE 1(2) 2 E 2 1(2) E 1(2) + ε ± + ΓE 1(2) 2 E 2 1(2) Γ2 2 cos 2 (φ/2) 2 E 2 1(2) = 1 5 E/ E/ EF -5-1 Bc -4-2 2 4 B [T] -1 E2 E1 Bc B [T] 1 2µ orb B c = SO by studying the A
E 1(2) ε + ΓE 1(2) 2 E 2 1(2) E 1(2) + ε ± + ΓE 1(2) 2 E 2 1(2) Γ2 2 cos 2 (φ/2) 2 E 2 1(2) = 1 Each Kramer s doublet produces two ABS (four in total). E/ -1 E2 E1 Bc EF At B=Bc, the ABS corresponding to different Kramers doublets cross. After the crossing, the two ABS below EF belong to the same Kramers doublet. B [T] 1
Josephson current in terms of Green s functions (both discrete and continuum contribution calculated on the same footing) I = 2e dω ˆΣ< 2π Tr ˆσ 3 Ĝ a (ω)+ˆσ r Ĝ< (ω) = I dis + I con I dis Discrete Josephson current (resonant Cooper pairs) = eγ2 sin(φ) f(e 2 ) 2 E 2 ( 2 E2 2)D + (E 2) + f(e 1 ) 2 E 1 ( 2 E1 2)D (E 1)
I dis Discrete Josephson current (resonant Cooper pairs) = eγ2 sin(φ) E 2 f(e 2 ) 2 ( 2 E 2 2 )D + (E 2) + E 1 f(e 1 ) 2 ( 2 E 2 1 )D (E 1) I dis = 2e f(e ) E 1(2) 1(2) φ E 1(2) The discrete Josephson current is given by the derivative of the occupied (i. e. below EF) ABS with respect to phase
I dis = eγ2 sin(φ) f(e 2 ) 2 E 2 ( 2 E2 2)D + (E 2) + f(e 1 ) 2 E 1 ( 2 E1 2)D (E 1) Continuous Josephson current (quasiparticle states above the gap) I con = eγ2 π sin(φ) dω Θ( ω ) f(ω) 2 (ω 2 2 ) 1 D + (ω) + 1 D (ω)
Non-interacting regime,4 I J I J disc -,4,2 -pi transition: reversal of the supercurrent due to the combined effect of SO and external magnetic field. This is very unusual in a noninteracting system. I J cont -,2,5 -,5 1 2 B = B =.2 B =.4 B =.6
Non-interacting regime II,4 I J I J disc -,4,1 -pi transition: reversal of the supercurrent due to the combined effect of SO and external magnetic field. This is very unusual in a noninteracting system. I J cont -,1,6 -,6 1 2 B =.5 B =.51 B =.52 B =.53 B =.54
Non-interacting regime II I J,4 Magnetic field where spinpolarized orbital states become degenerate -,4,1 E/ 5 I J disc -,1-5 -1 Bc -4-2 2 4 B [T] I J cont,6 -,6 1 2 B =.5 B =.51 B =.52 B =.53 B =.54
1 1 E/6 c -1 B = B =.5 B =.6 EF B [T] After the crossing, the occupied ABS belong to the same Kramers doublet. Importantly, they have opposite derivative with respect to phase which gives discrete supercurrents of opposite sign. Only the continuous current (states above the gap) contributes. This reverses the sign of the supercurrent. 1-1 q/ 2 q/ 2 q/ 2 In standard QDs this happens in the cotunneling regime only, see Supercurrent reversal in quantum dots, J. van Dam et al, Nature 442, 667 (26).
Gate tunability 1 B =Bc d E/ -1-1 1 Vg/ -1 1-1 1-1 1 Vg/ Vg/ Vg/ At zero magnetic field, the SO splitted ABS show a diamondlike shape, similarly to spin-slit ABS due to Coulomb Blockade
Gate tunability 1 B =Bc d E/ -1-1 1 Vg/ -1 1-1 1-1 1 Vg/ Vg/ Vg/ At zero magnetic field, the SO splitted ABS show a diamondlike shape, similarly to spin-slit ABS due to Coulomb Blockade E. Vecino, A. Martín Rodero and A. L. Yeyati, Phys. Rev. B, 68,3515, 23
Cotunneling regime (fourth order perturbation theory)
Kondo regime (slave boson) I dis J = e 2 η=± sin(φ) [(1 + ηα) 2 + 1][(1 + ηα) 2 + cos 2 ( φ 2 )] α = SO 2T K,SU(4) Without SO we recover the results of: Zazunov, Levy-Yeyati and Egger, PRB 81, 1252 (21)
Kondo regime
I J I J disc,4 -,4,1 -,1,6 CONCLUSIONS The relatively small SO coupling in quantum dot carbon nanotubes induces a -pi transition in the Josephson current when an external magnetic field brings spin-polarized orbital levels to degeneracy. I J cont -,6 1 2 The transition is also tunable by a gate voltage. This is relevant in view of recent transport experiments in quantum dot carbon nanotubes. I c (2e 2 /h 1.5 1.5 -.5 -.5.5 1 1.5 V g / SO I dis/i c I( 1.8.6.4.2 SO = SO =.5TK, SU (4) SU (2).2.4.6.8 1 1.8.6.4.2 SU(4) SU(2) 1 2 3 4 5 SO /T K,SU(4) Cotunneling regime: the transition occurs even at zero magnetic field. Kondo regime: the Josephson current is always in the phase for both SU(4) and SU(2) symmetries. J. S. Lim, R. López, R. Aguado Phys. Rev. Lett (211) arxiv:114.513