The 2005 observation of a purely orbital Kondo effect in carbon nanotube quantum dots (Jarillo-Herrero et al., Nature 434, 484) implied that the orbital (valley) quantum number of the nanotube is conserved when electrons tunnel between the dot and the metallic leads — a conclusion the authors themselves could only justify by an unverified hypothesis: that electrons first tunnel into the nanotube section underneath the top-deposited metal contacts, dwell there, and only then enter the metal, so that the effectively CNT–CNT tunnel step conserves the orbital quantum number. The subsequent literature (theory and experiment on SU(4) Kondo, spin–orbit coupling, and intervalley mixing in nanotube dots, up to the 2015 Reviews of Modern Physics synthesis and 2020 numerical-renormalization-group work) does not derive or directly test the microscopic nature of this dot–lead coupling: later theory treats intervalley mixing as a phenomenological parameter, and the standard review still draws the two-step 'dwell' tunnelling path as a schematic assumption. The problem is to determine, theoretically or experimentally, how and why the orbital quantum number is conserved (or to what degree it is mixed) in tunnelling at metal–nanotube contacts, so that the observed orbital and SU(4) Kondo physics is microscopically explained and the finite-field split Kondo resonances can be compared in detail with theory.

Background

The Kondo effect is a many-body phenomenon in which a degenerate localized state is coherently screened by the Fermi sea of conduction electrons in the leads, producing a narrow resonance in the density of states at the Fermi level and a corresponding enhancement of the low-temperature conductance through a quantum dot. In the usual case the degeneracy is the electron spin (SU(2) Kondo effect), but any conserved two-valued degree of freedom of the localized state can play the role of a pseudospin and mediate a Kondo effect, provided the leads can exchange that quantum number with the dot. A single-wall carbon nanotube (CNT) is a rolled-up graphene sheet whose band structure derives from the two inequivalent valleys K and K' of graphene. For a wide class of chiralities (the wrapping angle of the tube) the two valleys are degenerate, and for each allowed longitudinal wave vector the two valleys correspond to electrons circulating clockwise or anticlockwise around the tube circumference. This is a twofold orbital degeneracy, characterized by an orbital magnetic moment coupling strongly to a magnetic field parallel to the tube axis. A CNT quantum dot (QD) is formed by depositing metal contacts on top of a nanotube; Coulomb blockade (the suppression of electron tunnelling by the electrostatic charging energy of a small conductor) confines electrons in the segment between the contacts, whose gate-voltage-tunable occupation shows a four-electron shell structure reflecting the simultaneous spin and valley degeneracy. In their 2005 experiment, Jarillo-Herrero, Kong, van der Zant, Dekker, Kouwenhoven and De Franceschi tuned a magnetic field so that spin-polarized states were brought into orbital degeneracy and observed a purely orbital Kondo effect (the valley pseudospin being screened), and, when spin and orbital degeneracies coexisted, an enhanced Kondo effect with a multiple (four-peak) splitting of the Kondo resonance in finite field, as predicted for the SU(4) symmetry group that acts on the four-component 'hyperspin' of spin and valley combined. Concurrent theory (Choi, López and Aguado, Phys. Rev. Lett. 95, 067204 (2005)) computed this four-peak splitting within an SU(4) Anderson impurity model (an Anderson model being the standard description of a dot level hybridized with lead continua, here with four degenerate flavors). The SU(4) interpretation rests on one load-bearing assumption: the valley quantum number must be exchanged between dot and leads, i.e. it must be conserved during tunnelling. This is not automatic. The metallic leads are disordered, three-dimensional Fermi seas with no valley index; direct tunnelling from the nanotube into a metal generically involves short-range (atomic-scale) scattering, which does not conserve the valley quantum number — the process known as intervalley (valley) mixing. The 2005 authors therefore proposed that in the actual device geometry electrons leaving the dot first enter the nanotube segment lying underneath the metal contacts, dwell there for a time, and then transfer into the metal; because a CNT–CNT (extended, valley-conserving) tunnel process is plausible for this intermediate step, the orbital quantum number could be effectively conserved. The paper itself presented this only as a hypothesis, and noted that the substructure of the outer peaks of the split Kondo resonance (inter-orbital cotunneling processes, i.e. second-order tunnelling events through virtual intermediate states with and without spin flip, which theory predicts as additional resolved components) was broadened beyond experimental resolution, precluding a detailed comparison with the predicted SU(4) splitting. Later work sharpened the context but not the mechanism: the discovery of spin–orbit coupling in nanotubes (Kuemmeth et al., Nature (2008)) split the fourfold degeneracy at zero field, and numerical-renormalization-group (NRG; Wilson's controlled iterative treatment of quantum impurity models) studies by Fang et al. (2008) and Galpin et al. (2010) reinterpreted the early Kondo data with spin–orbit and valley-mixing terms as adjustable parameters, while experiments on ultraclean tubes (Makarovski et al. (2007); Cleuziou et al. (2013)) and numerical-renormalization-group work (Teratani et al. (2020)) mapped SU(4)-to-SU(2) crossovers. The most recent authoritative synthesis (Laird et al., Rev. Mod. Phys. 87, 703 (2015)) still states that 'even if valley degeneracy exists in the nanotube, this does not ensure that the valley quantum number is also present in the leads', and depicts the two-step tunnelling path (into a nanotube lead segment, then into the metal) as an unverified schematic. Throughout this literature, intervalley mixing in tunnelling, disorder, and the lead segments is treated as a phenomenological parameter whose microscopic origin and magnitude at a metal–CNT top contact have not been derived.

Problem Statement

Determine, theoretically or experimentally, the microscopic nature of the dot–lead coupling in carbon nanotube quantum dots that enables the orbital (valley) quantum number to be conserved during tunnelling, as required for the observed orbital Kondo effect and the SU(4) spin–valley Kondo effect. Specifically: (i) establish whether, and to what degree, the orbital quantum number is conserved when an electron tunnels between a gated nanotube segment (the dot) and the metallic contacts deposited on top of it; (ii) identify the physical mechanism — in particular, verify or refute the hypothesis of Jarillo-Herrero et al. (Nature 434, 484 (2005)) that electrons first tunnel into the nanotube section underneath the contacts, dwell there, and then transfer into the metal through an effectively CNT–CNT (valley-conserving) process — or provide an alternative microscopic account of the contact tunnelling matrix elements, including the magnitude of contact-induced intervalley mixing relative to disorder- and dot-induced mixing; and (iii) show how this microscopic picture fixes the effective tunnelling amplitudes (including channel asymmetry and mixing) entering the SU(4) Anderson impurity model, so that the observed zero-field Kondo resonances, the field-induced multiple splitting of the Kondo resonance with its predicted but so-far-unresolved substructure of the outer peaks (broadened beyond experimental resolution), and the SU(4)-to-SU(2) crossovers can be compared quantitatively with theory.

References

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