Main

To explore the physics of few electrons, control of the number of interacting electrons is needed. This control has so far been obtained using semiconducting materials operated close to the bandgap edge3,4,5,6,7,14,15,16,17,18. In this letter, we show that nanotubes can be used to obtain the same kind of control. Owing to a robust shell structure, spin states can be identified in nanotube quantum dots even for high filling numbers19,20,21,22,23. Here, we show a CNT DQD exhibiting shell structure in both quantum dots10, which enables us to identify spin states of the DQD. This is in contrast to DQDs fabricated in other materials3,4,5,6,7,14,15, where the spin states in practice have to be determined by finding the absolute number of electrons, that is, the device has to be operated close to the bandgap edge.

The device analysed here, schematically shown in Fig. 1a, comprises a CNT contacted by titanium electrodes, and gated by three top-gate electrodes, G1, CG (centre gate) and G2, made of aluminium oxide and titanium 10,11,12. The device has two strongly coupled quantum dots in series, as confirmed by the observation of the so-called honeycomb pattern in plots of current (Isd) versus voltage applied to G1 (VG1) and G2 (VG2) (Figs 1b and 2a)1. The two dots have roughly equal charging energies and level spacings (see below), from which we infer that the tunnelling barrier between the two dots is located under, or close to the centre gate. As the room-temperature gate dependence of the device is metallic, the tunnelling barrier is most likely due to a defect under the centre gate formed during electron beam lithography. We have seen similar barriers in other devices fabricated by the same method. The number of electrons in dot 1 and dot 2 can be controlled by tuning VG1 and VG2, respectively. In the middle of each hexagon (white areas in Figs 1b and 2a), a fixed number of electrons are localized in each dot, and electron transport is suppressed by Coulomb blockade. Along the entire edge of the hexagons (blue lines), sequential tunnelling is allowed through molecular states formed in the DQD, indicating a strong coupling between the two dots. The height (width) of the hexagons corresponds to the energy required to add an extra electron in dot 1 (dot 2). In Fig. 1b, the width and height of the hexagons alternate in size in a regular pattern. The four hexagons marked with red numbers are distinctively larger than the other hexagons, with three smaller hexagons in between, indicating that each dot has four-fold degenerate levels due to spin and orbital degeneracy19,20,21. An eight-electron shell structure of the DQD can therefore be identified in this plot. Shell occupation numbers (N,M), where N (M) is the level occupation number in dot 1 (dot 2), are written on the honeycomb diagram.

Figure 1: Shell filling of a CNT DQD.
figure 1

a, Schematic diagram of the device consisting of a CNT contacted by titanium source and drain electrodes and gated with three top-gate electrodes, G1, CG (centre gate) and G2. Two quantum dots (QD1, QD2) are formed in series, one under G1 and one under G2. b, Surface plot of current (Isd) at constant bias (Vsd=0.5 mV) as a function of voltage applied to G1 (VG1) and G2 (VG2) with VCG=0 V and at T=0.35 K. Red numbers (N,M ) are shell occupation numbers for one eight-electron (DQD) shell. Two further eight-electron shells were observed in connection with this shell, one below, and one to the right.

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Figure 2: Four-electron DQD-shell structure.
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a, Surface plot of current (Isd) at constant bias (Vsd=0.2 mV) as a function of voltage applied to G1 (VG1) and G2 (VG2) with VCG=0 V and at T=0.35 K. The numbers (N,M) are shell occupation numbers for one four-electron shell. The charging energies and level spacings for the two dots are UC1∼3 meV and UC2∼3.5 meV, and ΔE1∼1.2 meV and ΔE2∼1.5 meV, respectively (see text). b, Black lines: schematic honeycomb diagram for a four-electron shell with strong tunnel coupling, and a small cross capacitance. Grey lines: same honeycomb diagram with negligible tunnel coupling. Dashed lines indicate where the line traces in Fig. 3 are measured.

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The honeycomb diagram in Fig. 2a is measured for the same device but in another gate region where a new pattern in the sizes of the hexagons is observed. The hexagons alternate in size between large and small owing to only spin degeneracy of the energy levels in each dot19,22, yielding a four-electron shell structure of the DQD. The charging energies (UC1, UC2) and level spacings (ΔE1, ΔE2) for the two dots can be extracted from the honeycomb pattern as schematically shown in Fig. 2b. The gate coupling of G1 (G2) to dot 1 (dot 2) is found from bias spectroscopy plots (not shown), and we find UC1∼3 meV and ΔE1∼1.2 meV for dot 1, and UC2∼3.5 meV and ΔE2∼1.5 meV for dot 2. As charging energy and level spacing are almost identical for the two dots, we deduce that the two dots are roughly equal in size. We have observed both four-electron and eight-electron shell structures in two different devices.

Herein, we will focus on a four-electron shell with level spacings and charging energies similar to the four-electron shell shown in Fig. 2a, except ΔE1∼1.9 meV. We show in the following that singlet and triplet states are formed in this four-electron shell. The singlet ground state between region (1,1) and (0,2) is in general a bonding state of the local singlet (S(02), both electrons in dot 2) and the non-local singlet (S(11), one electron in each dot):

The detuning (ɛ=E2−E1)-dependent parameters α and β determine the weight of each state, and E1 and E2 are the electrostatic potentials in dot 1 and dot 2, respectively. Similarly for the triplets

where −,0,+ denotes the spin magnetic moment in the z direction, Sz=−1,0,+1. In the following, we will show the existence of the singlet and triplet states on the basis of magnetic field spectroscopy on the four-electron shell.

In Fig. 3c, we analyse the magnetic field dependence of the width of hexagon (0,1), which involves 0, 1, and 2 electrons in dot 2. The chemical potential for these two Coulomb peaks is given by2: and , where is the chemical potential for adding an electron to charge state (01) given no electron in the DQD shell, and is the chemical potential for adding an electron in state SB given one electron charge state (01). These two Coulomb peaks are therefore expected to separate by g μBB, as shown in Fig. 3a. The height of hexagon (1,0) is similarly expected to separate by g μBB. The measured height of hexagon (1,0) and width of hexagon (0,1) versus magnetic field are in good agreement with theory, as shown in Fig. 3e.

Figure 3: Singlet and triplet states in a four-electron shell.
figure 3

a–d, Theoretical (a,b) and measured (c,d) magnetic field dependence of the chemical potentials, where EDQD=1/2(E1+E2). In a,c, the four-electron shell is filled from zero electrons (|00〉), to one spin-down electron in dot 2 (|↓2〉), to a singlet (SB). In b,d, the four-electron shell is filled from a spin-down electron in dot 2 (|↓2〉), to either a singlet (SB) or a triplet (TB−) (see text), to a three-particle state (|↓1↑2↓2〉). The line traces in c,d are extracted from honeycomb diagrams measured at B=0,1,2,…,7 T, where B is perpendicular to the nanotube. Each line in c,d is offset 0.3 nA and 0.7 nA, respectively, for clarity, and the left-most peak is centred at zero. e, Width of hexagon (01) (circles) (extracted from c) and height of hexagon (10) (squares) versus B. f, Size of hexagon (11) versus B at three values of detuning, as indicated in Fig. 2b. Squares are extracted from d. The solid lines in e,f are the Zeeman energy g μBB, with g=2 for nanotubes.

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We now analyse the size of hexagon (1,1), which involves 1, 2 and 3 electrons in the DQD shell. We show that by applying a magnetic field the two-electron ground state can be changed from SB to TB−, which is used to estimate the exchange energy (J) (energy separation between SB and TB0). Transport at the first Coulomb peak in Fig. 3d is through different chemical potentials at low and high magnetic field, given by2 at low magnetic field (g μBB<J), and at high magnetic field (g μBB>J). Similarly, transport at the second Coulomb peak in Fig. 3d is through at low magnetic field (g μBB<J) and through at high magnetic field (g μBB>J)2. Therefore, for increasing magnetic field, the gap between the two Coulomb peaks at the (1,1) to (0,2) anticrossing decreases when SB is ground state, and increases when TB− is ground state, as schematically shown in Fig. 3b. The measurement in Fig. 3d is in good agreement with Fig. 3b, with the bend (change of ground state from singlet to triplet) occurring at B∼3.2 T, that is, an exchange energy of J∼0.37 meV. The magnetic field dependence of the size of hexagon (1,1) at three constant values of detuning (Fig. 3f) shows that the exchange energy is minimum at the centre of hexagon (1,1), with the bend occurring at B∼2 T, that is, J∼0.23 meV. The exchange energy can also be estimated from the tunnel coupling strength (t). We estimate t∼0.32 meV from the curvature of the hexagons at the (1,1) to (0,2) anticrossing (see the Supplementary Information)11. This estimate of t yields a consistent estimate of the exchange energy (ref. 2) for large detuning (at ɛ=UC1, the centre of hexagon (11)), and at the anticrossing (see the Supplementary Information).

The anticrossing between (1,1) and (0,2) is analysed in Fig. 4. We find that transport is governed by elastic and inelastic cotunnelling via SB and TB−. The chemical potential for adding an electron to SB and TB− with E1+E2=0, that is, along the black dashed line in Fig. 4a, is given by (see the Supplementary Information):

We plot equations (1) and (2) in Fig. 4b with B=0 T (solid green and blue lines) and with B=6 T (dashed green and blue lines). We see that SB is ground state for B=0 T, and that the two chemical potentials cross at elevated magnetic field, indicated with a red arrow.

Figure 4: Singlet–triplet splitting probed by inelastic cotunnelling.
figure 4

a, Small section of the honeycomb diagram analysed in Fig. 3 with Vsd=50 μV at B=0 T (left) and B=6.5 T (right). The numbers (N,M) indicate electron occupation of the four-electron shell. b, Chemical potentials for the singlet bonding () and triplet bonding () with B=0 T (solid green and blue lines) and with B=6 T (dashed green and blue lines) calculated using equations (1) and (2). Onset of inelastic cotunnelling that excites electrons from singlet to triplet (black arrow marked B) and from triplet to singlet (black arrow marked C) occurs when the separation between and is equal to e Vsd. c, Schematic transport diagrams for elastic cotunnelling (A) and inelastic cotunnelling (B and C). d, Surface plot of current (Isd) at Vsd=0.2 mV versus magnetic field (B ), and detuning (ɛ) along the black dashed line in a. Onset of inelastic cotunnelling occurs along the white lines marked B and C, calculated using equation (3) with t=0.32 meV, ΔE2=1.5 meV, Vsd=0.2 mV and g=2. Dashed grey lines indicate where the two inelastic cotunnelling processes shown in b occur.

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At low magnetic field, one broad peak in conductance versus detuning between (1,1) and (0,2) is seen (Fig. 4a, white arrow marked A). This conductance peak is due to elastic cotunnelling via SB, schematically shown in Fig. 4c (marked A). As elastic cotunnelling via SB involves both S(11) and S(02), which have equal weight at ɛ=0, the elastic cotunnelling peak is centred around ɛ=0. At high magnetic field, the elastic cotunnelling via SB becomes suppressed because the ground state at ɛ=0 changes from SB to TB−. Figure 4d shows a surface plot of Isd versus ɛ and B along the black dashed line in Fig. 4a. The white vertical line marked A is the expected position of the elastic cotunnelling.

At high magnetic field, we observe two narrow peaks, marked B and C in Fig. 4a. These two narrow peaks are due to the onset of inelastic cotunnelling via SB and TB−, schematically shown in Fig. 4c marked B and C. Onset of inelastic cotunnelling via SB and TB− occurs when the energy separation between their chemical potentials becomes equal to the applied bias:

We have calculated the onset of inelastic cotunnelling in (ɛ,B)-space from these two conditions and plotted it as white lines marked B and C in Fig. 4d. Note that no fitting parameters are used in Fig. 4d; the parameters used, t=0.32 meV, ΔE2=1.5 meV, were found in the analysis above.

Methods

Fabrication and measurement set-up

The devices are made on a highly doped silicon substrate with a top layer of silicon dioxide. The CNTs are grown by chemical vapour deposition from islands of catalyst material and subsequently contacted by 50 nm titanium source and drain electrodes. Next, three narrow top-gate electrodes are fabricated between the source and drain electrodes, consisting of aluminium oxide and titanium12. A schematic diagram of the device together with the measurement set-up is shown in Fig. 1a. Source–drain voltage (Vsd) is applied to the source electrode and the drain electrode is connected through a current-to-voltage amplifier to ground. The three top-gate electrodes are named G1, CG (centre gate) and G2, starting from the source electrode. For the device reported here, we saw that G1 had a much lower gate coupling than G2 and CG (see Figs 1b and 2a), which we attribute to the G1 electrode being damaged somewhere, weakening its gate coupling. The gate coupling of G1 to dot 1 is αG1=2.9 meV V−1, and the gate coupling of G2 to dot 2 is αG2=400 meV V−1. The centre gate is kept at VCG=0 V for the measurements shown here. All data presented here are measured in a sorption pumped 3He cryostat at 350 mK.