Accessing nanotube bands via crossed electric and magnetic fields
We investigate the properties of conduction electrons in single-walled armchair carbon nanotubes in the presence of mutually orthogonal electric and magnetic fields transverse to the tube’s axis. We find that the fields give rise to an asymmetric dispersion in the right- and left-moving electrons along the tube as well as a band-dependent interaction. We predict that such a nanotube system would exhibit spin-band-charge separation and a band-dependant tunneling density of states. We show that in the quantum dot limit, the fields serve to completely tune the quantum states of electrons added to the nanotube. For each of the predicted effects, we provide examples and estimates that are relevant to experiment.
A highlight of graphene-based nanotubes is the presence of a band (valley) quantum degree of freedom which offers a new facet to a range of strongly correlated low-dimensional phenomena. In armchair nanotubes, the bands, which owe their existence to the two-atom basis of the underlying honeycomb lattice, form a pair of gapless linearly dispersing modes dresselhaus (). While a direct consequence of their presence is the measurable quantum conductance of , richer effects can arise, for instance, in field-induced orbital moments mceuen1 (), Coulomb-blockade profiles bockrath () and in Kondo physics kondo (). An ability to precisely control the band sector would not only shed light on these effects but also access other band-dependent physics. For instance, in the recently reported nanotube Mott phase MottBockrath (), the nature of this ordered phase could be probed by tuning inter-band interactions. In the Luttinger liquid phase typical of interacting one-dimensional systems, spin and charge sectors have been shown to decouple in quantum wires Yacoby05 (); nanotubes could extend such fractionalization into yet another sector. Recently, attention has turned towards “valleytronics” as a potential application for quantum information and quantum devices in the parent system of graphene valleytronics (); the ability to manipulate spin and band quantum states in nanotubes would offer an attractive alternative. However, save for some exceptions mceuen1 (), the band degree of freedom has thus far remained relatively resilient to any controlled coupling.
In this Letter, we explore the possibility of coupling to the band sector via applied transverse fields. Electric and magnetic fields individually have been shown to alter the low energy properties of conduction electrons bellucci (); novikov (). Here, building on our prior work us (), we find that the transverse field configuration shown in Fig. 1 in which both fields are present offers a powerful means of accessing the band sector. The fields break time-reversal, particle-hole and band symmetries, and in fact, render the dispersion asymmetric in the right- and left-moving electrons. Furthermore, the effective Coulomb interaction is modified from its field-free form to include couplings between the charge and band sectors. As a consequence, nanotubes in transverse fields could provide a realization of a tunable asymmetric Luttinger liquid trushin (); naon (); gil (). Most pertinently, the system displays spin-band-charge separation, acts as a band-selector for electrons tunneling into the nanotube and exhibits a variety of tunable quantum dot shell filling configurations.
The setup of interest, as shown in Fig. 1, consists of an armchair nanotube lying along the -axis subject to an applied magnetic field and electric field . For a tube of radius , where is the chiral index and nm is the carbon-carbon bond length, the effect of the fields can be characterized by the parameters and , where eV dresselhaus () is the electron hopping strength between neighboring sites. Near half-filling, the field-free low-energy electronic dispersion, which can be traced to bonding and anti-bonding states of the underlying graphene sublattices, consists of four degenerate gapless linear modes each having a Fermi velocity m/s. These modes correspond to right- and left-movers (indexed by ) at two Fermi points (indexed by ) which define the band degrees of freedom.
In this Hall-type setup, classically, each of the fields causes a spiralling motion of electrons, and thus for fixed kinetic energy, reduces their linear velocity along the tube’s axis. Moreover, the non-vanishing Poynting vector affects right- and left-movers differently. Indeed, a detailed perturbative band structure calculation us () for large enough tubes () shows that the right- and left-moving speeds are given by . The spectrum remains gapless gap (), but the two Fermi points reside at energies differing by , indicating a mild band-degeneracy breaking even at the band structure level. The corresponding low-energy non-interacting asymmetric Hamiltonian can be expressed as
where the band shift and Zeeman term are left implicit. The operator denotes the annihilation of a fermion and correspond to electronic spin components. For large but realizable tubes of nm and experimentally accessible field strengths T and V/nm, we find and , yielding a pronounced asymmetry in right- and left-moving velocities.
Transverse fields significantly alter not only the low-energy band structure but also the nature of Coulomb interactions within the tube. The usual forward scattering contribution, , remains and continues to dominate for larger tubes () kbf (). Here with is the net charge density and is the effective one-dimensional interaction egger (); us (), where is the charge of an electron, is the dielectric constant, and is the length of the tube. More interestingly, a second contribution becomes manifest due to the uneven distribution of charge in the circumferential direction induced by the presence of the applied fields. A careful accounting of circumferential modes shows that this additional effective Coulomb term along the tube has the form
where and is a numerical factor which depends on the details of the geometry of the underlying graphene lattice; for we have us (). Here, is the difference in charge density between the two bands. The presence of this interaction term has two origins: (i) the magnetic field couples to the crystal momentum, and thus the Fermi points, giving rise to the contribution involving the charge in the sector and (ii) the electric field couples to the net charge imbalance in bonding and anti-bonding states, or alternatively, to the density difference in the right- and left-movers in the sector. This interaction, which is key to the physics studied here, goes a step beyond the usual spin-charge-separating Coulomb term by directly coupling to the band degrees of freedom.
As with the field-free case, the asymmetric gapless modes and Coulomb interactions can be studied via bosonization fernandez (); naon (); trushin (); gil (), which renders the total Hamiltonian for the tube to be quadratic. The one-dimensional fermionic operators can be bosonized as
where the ’s are the so-called Klein factors which enforce anticommutation relations between different channels egger (). The chiral bosonic fields satisfy the commutation relations . The density associated with each sector takes the form . Given the asymmetric dispersion and current dependent interaction, we explicitly employ these chiral fields to ensure that this algebra is preserved. It is convenient to introduce a spin and channel decomposition for the chiral fields, . The total kinetic energy takes the bosonized form
and the total interaction term, , takes the form
As discussed in previous work on asymmetric bosonization fernandez (); naon (); gil (); trushin (), in diagonalizing the full Hamiltonian , care needs to be taken to ensure that the diagonal basis preserves the chiral commutation rules respected by the original fields, . Here, we circumvent unwieldy manipulations involving Bogoliubov transformations by employing the trick of converting left-handed fields to right-handed fields via the transformation , which then allows for standard rotations. The resultant plasmon modes for the coupled charge sectors move with four different velocities
where is the average velocity of the non-interacting fermions, the asymmetry, and is the standard Luttinger parameter. The spin sectors each retain their band structure velocities .
The existence of the two different plasmon velocities demonstrates that transverse fields can access the band sector, which is normally resilient to any bulk coupling. Thus far, momentum resolved bulk tunneling experiments on quantum wires Yacoby05 () have revealed two distinct plasmon dispersions corresponding to spin and charge modes; analogous experiments on nanotubes in transverse fields ought to unveil the three pairs of velocities predicted above. For example, for a tube nm and experimentally accessible field strengths T and V/nm, the plasmon velocities are given by , , , and for the charge sectors and and for the spin sectors. The observation of three distinct modes would reflect complete splintering of the nanotube electron into its spin, charge and band sectors.
The effects of transverse fields on the low-energy nanotube physics, particularly in the band degree of freedom, are prominently manifested in physical observables. The tunneling density of states, an ubiquitous, experimentally accessible quantity, retains its the power-law form characteristic of Luttinger liquids but exhibits a modified exponent. Using standard procedures giamarchi () that now account for field-dependent effects, we find that the tunneling exponent, to lowest order in , is given by
The first part of the exponent, also present in the field-free case, reflects the suppression in tunneling due to interactions (where is now tunable). The second part, which depends on the tunable coupling of , further suppresses or enhances tunneling depending on the sign of . As an estimate, for a nm tube in a T B-field and V/nm E-field with a field-free value of , we have distinctly different exponents and . The form of Eq. (7) reflects band (valley) selection; for example, a right-moving electron would preferentially tunnel into the Fermi point for the field configuration shown in Fig. 1
A physical consequence of transverse fields yielding such a band-dependent exponents would be the presence of two different power-law contributions to the non-Ohmic conductance of the nanotube yao (). On a related note, the presence of an impurity would distinguish these exponents; conductance properties would be sensitive to whether or not an electron impinging on the impurity switched band index. We note here that the related charge current, obtained from the continuity equation, has an atypical form, . Band-dependent effects similar to those in the density-of-states exponent would be manifest in any susceptibilities involving the sublattice degree of freedom. Tendencies towards different orderings, such as charge or spin density waves, which are governed by associated susceptibilities, would in turn reflect band dependence. For instance, charge density waves have contributions from operators , reflecting like () and staggered () band correlations at A and B sublattice sites. While these operators both come on an equal footing in the field-free case, we find that fields render more relevant in the renormalization group sense; fields can thus discriminate between two types of band-dependent ordering at the sublattice level.
In the quantum dot limit achieved by high resistance contacts or sufficiently low temperatures, we find that transverse fields enable controlled tuning of Coulomb blockade physics and nanotube quantum states. Here, as in previous treatments kbf (); fabrizio (); eggert (), we describe the dot as a finite-sized version of the net nanotube Hamiltonian and focus on the topological sectors as relevant for standard quantum dot experiments involving adiabatic tuning. In the presence of a gate voltage , the resultant Hamiltonian characterized by quantum numbers for each topological sector is given by
where , and is the interaction strength in a given mode. Here, , and . The term is proportional to , and accounts for the Zeeman splitting. The coupling between the and sectors arises due to the the interaction (Eq. (2)) and the no-current boundary conditions at the tube ends. The band splitting also depends on boundary conditions; in the absence of Fermi point mixing at the tube ends, it reduces to the band structure Fermi point mismatch us (). As typical parameter values, for a 1 m-tube of radius 3.39 nm, and magnetic and electric field strengths of 4.51 T and 0.026 V/nm, we find meV, , , , and .
By virtue of their field dependence, several parameters in can be varied to access a wide variety of shell filling configurations of the nanotube quantum dot in Coulomb blockade experiments. The configurations correspond to sets of electron occupation numbers which minimize the energy associated with Eq. (8) kbf (); us (). While the associated parameter space is too extensive for an exhaustive study, a few salient characteristics of shell-filling are as follows. (i) In actual experiments persistent (); bockrath (), the band degeneracy is often naturally broken due to physical attributes such as the confining potential created by the leads, yielding patterns such as in Fig. 2a. Here, the field dependence of enables controlled tuning of band degeneracy breaking as well as probing the extent of Fermi point mixing at the tube ends mccann (); us (). (ii) A four-fold periodicity has been observed in some experiments reflecting the band and spin degeneracy of the tube bockrath (). (iii) As a direct demonstration of band tuning, the parameter values quoted above yield the two-fold periodic shell-filling pattern in Fig. 2b. In particular, these values are chosen such that Eq. (8) is minimized by the condition , entirely restricting the tunneling into a given -point. We remark that this condition requires a fine-tuning of fields, and also that other field values can even render negative, resulting in an instability towards complete polarization into one band. (iv) Periodicity can also be entirely obliterated by choosing an irrational ratio between two of the relevant shell-filling parameters. As demonstrated above, transverse fields provide a precise means of preparing and manipulating the electronic spin and band quantum numbers of the nanotube quantum dot.
In conclusion, we have presented transverse fields as powerful probes to access and explore a rich range of physics in armchair SWNTs that directly addresses the band degree of freedom. We have found that these fields induce an unusual interaction that couples the charge and band sectors. We have predicted that signatures of these field effects will be apparent in a variety of measurements including those probing Luttinger liquid parameters and plasmon structure, tunneling density of states and impurity scattering. We have shown that transverse fields can be used for controlled manipulation of nanotube quantum dot states, making this proposed setup a potential building block for nanoscale quantum devices.
We would like to thank N. Mason and M. Stone for their perceptive comments. This work is supported by NSF, Grant No. DMR09-06521 (W. D. and S. V.), NSERC and MITACS (T.-C. W.), and UNLP and CONICET (V. F.)
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