Kondo effects in small-bandgap carbon nanotube quantum dots Patryk Florków, Damian Krychowski* and Stanisław Lipiński Full Research Paper Open Access Address: Beilstein J. Nanotechnol. 2020, 11, 1873–1890. Department of Theory of Nanostructures, Institute of Molecular https://doi.org/10.3762/bjnano.11.169 Physics, Polish Academy of Sciences, M. Smoluchowskiego 17,60-179 Poznań, Poland Received: 21 September 2020 Accepted: 03 December 2020 Email: Published: 23 December 2020 Damian Krychowski* - [email protected] Associate Editor: J. M. van Ruitenbeek * Corresponding author © 2020 Florków et al.; licensee Beilstein-Institut. Keywords: License and terms: see end of document. carbon nanotubes; Kondo effect; mesoscopic transport; quantum dots; valleytronics Abstract We study the magnetoconductance of small-bandgap carbon nanotube quantum dots in the presence of spin–orbit coupling in the strong-correlations regime. A finite-U slave-boson mean-field approach is used to study many-body effects. Different degeneracies are restored in a magnetic field and Kondo effects of different symmetries arise, including SU(3) effects of different types. Full spin–orbital degeneracy might be recovered at zero field and, correspondingly, the SU(4) Kondo effect sets in. We point out the possibility of the occurrence of electron–hole Kondo effects in slanting magnetic fields, which we predict to occur in magnetic fields with an orientation close to perpendicular. When the field approaches a transverse orientation a crossover from SU(2) or SU(3) symmetry into SU(4) is observed. Introduction Due to their remarkable electronic, transport, mechanical, and interference [13,14], Kondo physics [15-29], spintronic effects magnetic properties carbon nanotubes (CNTs) are of great [10,30-32], and induced superconductivity [33-35]. interest in molecular electronics and spintronics with potential applications, for example, as field-effect transistors, nano- In the present paper, we are interested in the effects of strong electromechanical devices, logic gates, spin valves, spin diodes, correlations. As electrons are confined in fewer dimensions, the and spin batteries [1-11]. CNTs are also interesting for funda- effects of interaction play a more fundamental role. Carbon mental science. Their study allows for the examination of many nanotubes are quasi one-dimensional systems, and the role of basic properties in ranges often not reachable in other systems. correlations further increases in quantum dots due to additional Many of the fundamental transport properties were observed in confinement. Of importance is also the low dielectric constant, nanotubes, including Coulomb blockade [12,13], Fabry–Perot which is especially low in suspended nanotubes [36]. 1873 Beilstein J. Nanotechnol. 2020, 11, 1873–1890. A single-walled carbon nanotube is a hollow cylinder formed of electric switching between different types of qubits (spin, graphene. A CNT can be either metallic or semiconducting, valley, or valley–spin) and their higher-dimensional equiva- depending on the way graphene is rolled up [37,38]. In the lents (qutrits [51,52], qudits [53,54]) in the same nanoscopic simple “zone folding” picture [39,40], the band structure of system. The storage capacity of three-state or four-state qudits CNTs is obtained from the band structure of graphene by is obviously higher than the capacity of a qubit. In the present imposing periodic conditions along the circumference. When paper, we are interested in transport properties. The regime of the closest quantization line misses the K point, a bandgap charge transport depends on the ratio between tunnel-induced appears. The bandgap depends on the minimum separation of broadening of dot energy levels and charging energy. For very the circular quantization lines from the Dirac points. The semi- weak dot–lead coupling and strong Coulomb interaction, the conducting gaps are of the order of a few hundred millielectron- electrons enter the dot one by one and Coulomb-blockade oscil- volts [9]. The nanotubes are characterized by the chiral vector lations of conductance are observed. For stronger coupling to C(n,m), where the integers n and m denote the number of unit electrodes, higher-order tunneling processes (i.e., cotunneling) vectors along two directions in the honeycomb crystal lattice of begin to play a decisive role. Also, many-body resonances form graphene. Zone folding theory predicts nanotubes to be metallic at the Fermi level, which enable transport in the valleys be- if n − m is a multiple of three. According to this prediction, one tween Coulomb peaks. In CNTs spin–orbit interaction plays an third of randomly selected nanotubes should be metallic. Exper- important role. The first experimental evidence showing the imentally, the fraction of nanotubes showing metallic behavior significance of this effect was the observation of splitting of a is very small (≤1% [41]). Even in nominally metallic tubes a spin–orbital quadruplet into two Kramers doublets recorded by narrow gap of the order of 10 meV is usually observed. These single-electron spectroscopy carried out on ultraclean nano- systems are sometimes called “nearly metallic” carbon nano- tubes [55]. Later, other experiments confirmed the fact that due tubes [42,43]. The reason for incorrect predictions of the zone to SO coupling spin and orbital degrees of freedom are not in- folding theory for small-diameter tubes is the neglect of the dependent even in the absence of magnetic fields [42,56-59]. curvature-induced breaking of the three-fold (C3) rotational SO coupling is an effect that emerges because of the curvature symmetry. When graphene is deformed into a nanotube, the of the CNTs [60]. Due to the curvature the inversion symmetry curvature causes the overlap matrix elements to depend on the is broken and hopping between p-orbitals of different parity direction [44]. The consequence of symmetry breaking is a shift from neighboring atoms is possible, which enhances SO cou- of the Dirac points in the reciprocal lattice away from the K and pling. Depending on the sign of SO coupling, this interaction K’ points. The broken symmetry also enhances the intrinsic introduces parallel or antiparallel alignments of spin and spin–orbit (SO) coupling in carbon nanotubes compared with angular momentum. The energy of SO coupling is comparable flat graphene. Apart from curvature [44-47], other perturba- to the energy scale of the Kondo effect. Therefore, taking this tions such as axial strain or twists can shift the dispersion cones perturbation into account is important when analyzing many- in CNTs and open the bandgap [44,48-50]. Unlike the quantiza- body effects in these systems. Several interesting papers have tion bandgaps, which depend on the inverse of the diameter, been devoted to the problem of interplay of Kondo effect and these narrow perturbation gaps are inversely proportional to the SO interaction [26,27,29,61,62]. In our previous paper [26], we square of the diameter and depend on the chiral angle. Small have pointed out the possibility of the occurrence of the SU(3) gaps are reflected in nonlinear dispersion curves and conse- Kondo effect in wide-bandgap nanotubes as a consequence of quent drastic changes of orbital effects induced by magnetic the threefold degeneracy induced by intervalley exchange or fields. The behavior in magnetic fields is distinctly different due to intershell mixing. We also announced in [26] the possi- than in wide-bandgap nanotubes. The field dependencies are de- bility of the occurrence of the SU(4) Kondo effect in narrow- termined not only by the response of orbital and spin magnetic bandgap nanotubes despite the presence of SO coupling. In the moments, as in the case of large gaps. They also crucially present paper, we show that the Kondo physics of narrow- depend on the value of the bandgap and the gate voltage. bandgap nanotubes is much richer. Apart from SU(2) Kondo Details of the band structure are decisive for the response on the resonances with effective spin, valley, or spin–valley fluctua- field. The degeneracy recovery lines plotted in the plane of tions, the emergence of an exotic SU(3) Kondo resonance is magnetic field and gate voltage are no longer parallel to the foreseen even without mixing between shells or valleys, but gate-voltage axis, like in wide-bandgap tubes. Instead, they simply due to the peculiarity of the band structure and a subtle depend on the gate voltage and they intersect at certain fields interplay of magnetic field, spin–orbit interaction, and changes with other similar lines, which leads to the appearance of higher of the bandgap. In the following we show how, for a given degeneracy in the system. The presence of different degeneracy nearly metallic nanotube, one can change the position of high- points and the gate dependence of degeneracy lines are interest- symmetry points by strain and a magnetic field. Our calcula- ing for quantum computing because they open the possibility of tions also show that in a quantum dot formed in a small- 1874 Beilstein J. Nanotechnol. 2020, 11, 1873–1890. bandgap nanotube electron and hole states can degenerate in (4) slanting magnetic fields. Based on this observation we antici- pate the possible occurrence of Kondo effects in which both where sz is the spin component along the nanotube axis and lx types of carriers take part. Apart from SU(2) Kondo lines, also is the Pauli matrix in the A–B graphene sublattice space. SU(3) Kondo points and SU(4) may appear for orientations of ΔZ = −δcos(3Θ)/D and ΔO = δ/D [63]. Various theoretical and the field close to perpendicular. experimental estimates differ not only in the reported values of the parameter δ but also often in the predictions of its sign. It Model and Formalism ranges from one tenth to a few meV·nm [42]. For wide-bandgap In our analysis we consider the low-energy and low-tempera- nanotubes, Eg ≫ ΔZ, Δo, the field dependence of single-particle ture ranges.