^{1,a)}and Martin R. Galpin

^{1}

### Abstract

This paper provides a theoretical description of sequential tunneling transport and spectroscopy, in carbon nanotubequantum dots weakly tunnel coupled to metallic leads under a voltage bias. The effects of Coulomb blockade charging, spin-orbit fine structure, and orbital- and spin-Zeeman effects arising from coupling to applied magnetic fields are considered; and the dependence of the conductance upon applied gate voltage, bias voltage, and magnetic fields is determined. The work is motivated by recent experiments on ultraclean carbon nanotubedots [Kuemmeth *et al.*, Nature (London)452, 448 (2008)], to which comparison is made.

We are grateful to F. Anders, G. Finkelstein, F. Jayatilaka, A. Mitchell, D. Ralph, J. Wheater, and V. Zlatic for helpful discussions. We also thank the EPSRC for financial support, under Grant No. EP/D050952/1.

I. INTRODUCTION

II. BACKGROUND

III. ISOLATED CNTQUANTUM DOT

A. Classification of states

B. Parallel magnetic field

C. SO coupling

D. Net dot states

E. Ground states and zero-bias CB

F. Transverse magnetic field

IV. TRANSPORT AND SPECTROSCOPY

A. Equilibrium at finite bias

B. Finite-bias sequential tunneling spectroscopy

1. conductance spectra

2. conductance spectra

C. Finite-bias tunneling spectra: Field dependence

D. Transverse magnetic field

V. CONCLUDING REMARKS

### Key Topics

- Carbon nanotubes
- 46.0
- Ground states
- 28.0
- Tunneling
- 21.0
- Magnetic fields
- 20.0
- Lead
- 15.0

## Figures

Schematic of the CNT quantum dot device, showing the and leads and the degenerate pair of CNT orbitals (tunnel couplings are omitted for clarity). The chemical potential of the lead is , with a net voltage drop of magnitude ; the relative partitioning of which between the leads is determined by .

Schematic of the CNT quantum dot device, showing the and leads and the degenerate pair of CNT orbitals (tunnel couplings are omitted for clarity). The chemical potential of the lead is , with a net voltage drop of magnitude ; the relative partitioning of which between the leads is determined by .

Coulomb blockade staircase in the -plane, from Eqs. (25)–(28), choosing the ratio for purposes of illustration. With a small lifetime broadening (thickened lines) included as discussed in text (Sec. III), the figure equivalently shows the zero-bias conductance arising from sequential electron tunneling, as a function of and .

Coulomb blockade staircase in the -plane, from Eqs. (25)–(28), choosing the ratio for purposes of illustration. With a small lifetime broadening (thickened lines) included as discussed in text (Sec. III), the figure equivalently shows the zero-bias conductance arising from sequential electron tunneling, as a function of and .

Coulomb blockade staircase in the -plane, from Eq. (33). With a small lifetime broadening (thickened lines) included as discussed in text (Sec. III), the figure equivalently shows the zero-bias conductance arising from sequential electron tunneling, as a function of and . Note the very different field scale to Fig. 2.

Coulomb blockade staircase in the -plane, from Eq. (33). With a small lifetime broadening (thickened lines) included as discussed in text (Sec. III), the figure equivalently shows the zero-bias conductance arising from sequential electron tunneling, as a function of and . Note the very different field scale to Fig. 2.

Finite-bias equilibrium between the and ground states, in the -plane for (a) coupling to the -lead, (solid line) and (b) coupling instead to the -lead, (dashed line). A value of has been chosen [appropriate to experiment (Ref. 23), see Sec. ???].

Finite-bias equilibrium between the and ground states, in the -plane for (a) coupling to the -lead, (solid line) and (b) coupling instead to the -lead, (dashed line). A value of has been chosen [appropriate to experiment (Ref. 23), see Sec. ???].

Conductance resonances in the -plane, in the vicinity of the border (shown for a field ). As well as the weak resonance at , addition and removal resonances arise at (addition for and removal for ), , and , denoted by in the figure. A value of has been chosen, appropriate to experiment (Ref. 23) (see text). With and taken from experiment as in text, the axis extends to .

Conductance resonances in the -plane, in the vicinity of the border (shown for a field ). As well as the weak resonance at , addition and removal resonances arise at (addition for and removal for ), , and , denoted by in the figure. A value of has been chosen, appropriate to experiment (Ref. 23) (see text). With and taken from experiment as in text, the axis extends to .

Conductance resonances in the -plane, in the vicinity of the border (shown for ). Here there are addition resonances (to the ground state), removal resonances (from the ground state), and the single resonance (denoted ) at on the degeneracy line. The resonances denoted , , and correspond to , and [as given from Eq. (48)]. A value of has again been chosen; and with and taken from experiment as in text, the axis extends to .

Conductance resonances in the -plane, in the vicinity of the border (shown for ). Here there are addition resonances (to the ground state), removal resonances (from the ground state), and the single resonance (denoted ) at on the degeneracy line. The resonances denoted , , and correspond to , and [as given from Eq. (48)]. A value of has again been chosen; and with and taken from experiment as in text, the axis extends to .

Conductance resonances in the -plane, for a fixed in the vicinity of the border (cf. Fig. 6); showing the -dependence of the resonances , , , , and as given in text. With the experimental and (Ref. 23), the field . Agreement with the experimental conductance in the -plane [Fig. 2(c) of Ref. 23] is evident. Note that the vertical separation between the and resonances is for all , independent of ; enabling the proportionality constant between and [Eq. (46)] to be determined directly.

Conductance resonances in the -plane, for a fixed in the vicinity of the border (cf. Fig. 6); showing the -dependence of the resonances , , , , and as given in text. With the experimental and (Ref. 23), the field . Agreement with the experimental conductance in the -plane [Fig. 2(c) of Ref. 23] is evident. Note that the vertical separation between the and resonances is for all , independent of ; enabling the proportionality constant between and [Eq. (46)] to be determined directly.

Conductance resonances in the -plane, for a fixed in the vicinity of the border (cf. Fig. 6); showing the -dependence of the resonances , , , and as given in text. With and from experiment (Ref. 23), the field . Note that the vertical separation between the -resonance and the lowest energy resonance ( for and for ) is again the constant for all , independent of .

Conductance resonances in the -plane, for a fixed in the vicinity of the border (cf. Fig. 6); showing the -dependence of the resonances , , , and as given in text. With and from experiment (Ref. 23), the field . Note that the vertical separation between the -resonance and the lowest energy resonance ( for and for ) is again the constant for all , independent of .

(a) Conductance resonances in the -plane, in the vicinity of the border and for a fixed transverse field (cf. Fig. 5 for the parallel field case). In addition to the -resonance at , addition and removal resonances occur at (denoted , addition for and removal for ), and (addition resonance ). Shown for . (b) Corresponding conductance in the -plane (cf. Fig. 7 for a parallel field) for a constant . The resonances are given by Eq. (54). Note that the -scale extends up to . The vertical separation between the and resonances is again the constant for all .

(a) Conductance resonances in the -plane, in the vicinity of the border and for a fixed transverse field (cf. Fig. 5 for the parallel field case). In addition to the -resonance at , addition and removal resonances occur at (denoted , addition for and removal for ), and (addition resonance ). Shown for . (b) Corresponding conductance in the -plane (cf. Fig. 7 for a parallel field) for a constant . The resonances are given by Eq. (54). Note that the -scale extends up to . The vertical separation between the and resonances is again the constant for all .

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