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Electron–nuclear interaction in 13C nanotube double quantum dots

Abstract

For coherent electron spins, hyperfine coupling to nuclei in the host material can either be a dominant source of unwanted spin decoherence1,2,3 or, if controlled effectively, a resource enabling storage and retrieval of quantum information4,5,6,7. To investigate the effect of a controllable nuclear environment on the evolution of confined electron spins, we have fabricated and measured gate-defined double quantum dots with integrated charge sensors made from single-walled carbon nanotubes with a variable concentration of 13C (nuclear spin I=1/2) among the majority zero-nuclear-spin 12C atoms. We observe strong isotope effects in spin-blockaded transport, and from the magnetic field dependence estimate the hyperfine coupling in 13C nanotubes to be of the order of 100 μeV, two orders of magnitude larger than anticipated8,9. 13C-enhanced nanotubes are an interesting system for spin-based quantum information processing and memory: the 13C nuclei differ from those in the substrate, are naturally confined to one dimension, lack quadrupolar coupling and have a readily controllable concentration from less than one to 105 per electron.

Main

Techniques to prepare, manipulate and measure few-electron spin states in quantum dots have advanced considerably in recent years, with the leading progress in III–V semiconductor systems2,3,10,11. All stable isotopes of III–V semiconductors, such as GaAs, have non-zero nuclear spin, and the hyperfine coupling of electron spins to host nuclei is a dominant source of spin decoherence in these materials1,2,12,13. To eliminate this source of decoherence, group-IV semiconductors—various forms of carbon, silicon and silicon–germanium—which have predominantly zero nuclear spin, are being vigorously pursued as the basis of coherent spin electronic devices. Double quantum dots have recently been demonstrated in carbon nanotubes14,15,16, including the investigation of spin effects17,18.

The devices reported are based on single-walled carbon nanotubes grown by chemical vapour deposition using methane feedstock containing either 99% 13C (denoted 13C devices) or 99% 12C (denoted 12C devices; see the Methods section)19. The device design (Fig. 1a) uses two pairs of Pd contacts on the same nanotube; depletion by top-gates (blue, green and grey in Fig. 1a) forms a double dot between one pair of contacts and a single dot between the other. Devices are highly tunable, as demonstrated in Fig. 1, which shows that tuning the voltage on gate M (Fig. 1a) adjusts the tunnel rate between dots, enabling a crossover from large single-dot behaviour (Fig. 1b) to double-dot behaviour (Fig. 1c). Left and right tunnel barriers can be similarly tuned using the other gates shown in blue in Fig. 1a.

Figure 1: Nanotube double dot with integrated charge sensor.
figure 1

a, Scanning electron micrograph (with false colour) of a device similar to the measured 12C and 13C devices. The carbon nanotube (not visible) runs horizontally under the four Pd contacts (red). Top-gates (blue) create voltage-tunable tunnel barriers enabling the formation of a single or double quantum dot between contacts 1 and 2. Plunger gates L and R (green) control the occupancy of the double dot. A separate single dot contacted by Pd contacts 3 and 4 is controlled with gate plunger gate S (grey) and is capacitively coupled to the double dot by a coupling wire (orange). b, Current through the double dot, Idd, (colour scale) with the top-gates configured to form a large single dot. c, When carriers beneath the middle gate, M, are depleted, Idd shows typical double-dot transport behaviour, demarcating the honeycomb charge stability pattern. d, Within certain gate voltage ranges, honeycomb cells with larger addition energy and fourfold periodicity (outlined with dashed lines) indicate the filling of spin and orbital states in shells. Source–drain bias is −1.0 mV for bd.

A notable feature of nanotube quantum dots that is not shared by GaAs dots is that the energy required to add each subsequent electron, the addition energy, often shows shell-filling structure even in the many-electron regime18. An example of a shell-filling pattern, with larger addition energy every fourth electron in the right dot, is seen in Fig. 1d. We find, however, that evident shell filling is not necessary to observe spin blockade at finite bias. Figure 2a,b shows current through the double dot, Idd, as a function of gate voltages VR and VL for a weakly coupled, many-electron 13C double dot at +1 and −1 mV source–drain bias, respectively, in a range of dot occupancy that does not show shell structure in the addition spectrum of either dot. With a magnetic field B=200 mT applied along the tube axis, current flow is observed throughout the finite-bias triangles at positive bias, but is suppressed at negative bias for detuning below 0.8 meV, which presumably indicates where an excited state of the right dot enters the transport window.

Figure 2: Spin blockade in a 13C nanotube double dot.
figure 2

a, Current Idd (colour scale) at +1.0 mV source–drain bias, the non-spin-blockaded bias direction. Transport is dominated by resonant tunnelling through the ground state at the base of the finite-bias triangles and through an excited state at a detuning of 0.7 meV. bIdd (colour scale) at −1.0 mV source–drain bias, the spin-blockaded bias direction. Idd is suppressed except near the tips of the transport triangles. Suppressed transport for one bias direction is the signature of spin blockade. c, Charge-sensing signal, gs, (conductance of the sensing dot between contacts 3 and 4 in Fig. 1a), acquired simultaneously with a detects the time-averaged occupation of the right dot. d, Charge-sensing signal gs for −1.0 mV bias (blockade direction). In ad, dashed lines indicate allowed regions for current flow in the absence of blockade. e, Schematic diagram of spin- blockaded transport. Any spin may occupy the left dot, but only a spin singlet is allowed in the right dot, suppressing negative bias current once an electron enters the left dot and forms a triplet state. f, Current Idd near zero detuning (position marked by circles in a and b) as a function of magnetic field for positive bias (non-blockade, red trace) and negative bias (blockade, for two values of VM, purple and green traces). For VM=222, Idd was multiplied by 5.

Current rectification of this type is a hallmark of spin blockade10 (Fig. 2e): at positive bias, current flows freely as electrons of appropriate spin are drawn from the right lead to form the singlet ground state; at negative bias, current is blocked whenever a triplet state is formed between separated electrons, as the excess electron on the left can neither re-enter the left lead nor occupy the lowest orbital state on the right without flipping its spin. Spin blockade was identified in all four devices measured, two each of 12C and 13C. Spin blockade was occasionally found to follow a regular even–odd filling pattern, as seen in few-electron GaAs dots20, although no pattern was seen adjacent to the area in Fig. 2.

Electrostatic sensing of the double-dot charge state is provided by a gate-defined quantum dot formed on a separately contacted portion of the same nanotube. The sensing dot is capacitively coupled to the double dot by a 1 μm coupling wire21 (orange gate in Fig. 1a) but electrically isolated by a depletion gate between the Pd contacts. Charge sensor conductance gs as a function of VR and VL, acquired simultaneously with transport data in Fig. 2a,b, is shown in Fig. 2c,d. The location of the coupling wire makes gs sensitive to the occupancy of the right dot with no observable sensitivity to the left dot. Inside the positive-bias triangles (Fig. 2c), gs is intermediate in value between their bordering regions, indicating that the excess electron is rapidly shuttling between the dots as current flows through the double dot. In contrast, inside the negative-bias triangles (Fig. 2d), gs shows no excess electron on the right dot as a result of spin blockade.

The magnetic field dependence of spin blockade provides important information about electron spin relaxation mechanisms22,23. A first look at field dependence (Fig. 2f) for a 13C device shows that for negative bias (purple and green), spin-blockade leakage current is strongly peaked at B=0, whereas for positive bias (red), the unblockaded current does not depend on field. The peak in leakage current is shown for two values of VM, indicating that the width of the peak is independent of interdot tunnel coupling t. As discussed below, this field dependence can be understood in terms of hyperfine-mediated spin relaxation.

The striking difference in field dependence of spin-blockade leakage current between 12C and 13C devices is illustrated in Fig. 3a,b. These data show that for negative (spin-blockaded) bias, leakage current is a minimum at B=0 for the 12C device and a maximum at B=0 for the 13C device. In fourteen instances of spin blockade measured in four devices (two 13C and two 12C), we find that leakage current minima can occur at B=0 in both 12C and 13C devices, particularly for stronger interdot tunnelling. For weak interdot tunnelling, however, only the 13C devices show maxima of spin-blockade leakage at B=0, presumably because the width and height of this feature are strongly suppressed in 12C nanotubes. In all cases, the positive bias (non-spin-blockade) current shows no appreciable field dependence.

Figure 3: Contrasting magnetic field dependence of leakage current for 12C and 13C devices.
figure 3

a,b, Leakage current through spin blockade (colour scale) as a function of detuning and magnetic field, B, for 12C (a) and 13C (b) devices. The vertical axes in a and b are interdot detuning as indicated by the orange lines in c and d, respectively. In a, B was swept and detuning stepped, whereas in b, detuning was swept and B stepped. c,d, Current through the double dot for a 12C device (bias=−1.5 mV) and a 13C device (bias=−4 mV), respectively. e,f, Cuts along B at the detunings indicated by the black lines in a and b, respectively. The fit in e is a Lorentzian with a width of 30 mT, and the fit in f is to the theory of Jouravlev and Nazarov22, providing a measure of Bnuc=6.1 mT.

Figure 3e shows spin-blockade leakage current as a function of B at fixed detuning (the detuning value is shown as a black line in Fig. 3a), along with a best-fit Lorentzian, for the 12C device. The Lorentzian form was not motivated by theory, but seems to fit rather well. The width of the dip around B=0 decreases with decreasing interdot tunnelling (configuration Fig. 3e has t50 μeV, on the basis of charge-state transition width21), which may explain why it is not observed in the weakly coupled regime of Fig. 3b,f. We note that a similar zero-field dip in spin-blockade leakage current was recently reported in a double dot formed in an InAs nanowire24. There the dip was attributed to spin–orbit coupling, an effect that is also present in carbon nanotubes25.

Hyperfine coupling appears to the confined electrons as an effective local Zeeman field (the Overhauser field) that fluctuates in time independently in the two dots, driven by thermal excitation of nuclear spins. The difference in local Overhauser fields in the two dots will induce rapid mixing of all two-electron spin states whenever the applied field is less than the typical difference in fluctuating Overhauser fields (at higher fields, only the m=0 triplet can rapidly mix with the singlet). How hyperfine-mediated spin mixing translates to a field dependence of spin-blockade leakage current was investigated experimentally in GaAs devices23, with theory developed by Jouravlev and Nazarov22.

Field dependence of spin-blockade leakage current for a weakly coupled 13C double dot near zero detuning is shown Fig. 3f, along with a theoretical fit (equation (11) of ref. 22, with a constant background current added), from which we extract a root mean square amplitude of fluctuations of the local Overhauser fields, Bnuc=6.1 mT. We note that the width of the peak in Fig. 3f is independent of detuning (Fig. 3b), consistent with our interpretation that it is governed by Bnuc rather than t. Assuming Gaussian-distributed Overhauser fields and uniform coupling, Bnuc is related to the hyperfine coupling constant A by , where g is the electron g-factor and N is the number of 13C nuclei in each dot22. Taking N3–10×104 and g=2 (see Supplementary information), yields A1–2×10−4 eV, a value that is two orders of magnitude larger than predicted for carbon nanotubes8 or measured in fullerenes9.

Signatures of dynamic nuclear polarization provide further evidence of a strong hyperfine interaction in 13C double dots. Hysteresis in the spin-blockade leakage current near zero detuning is observed when the magnetic field is swept over a tesla-scale range, as shown in Fig. 4a. The data in Fig. 4a,b are from the same 13C device as in Fig. 3, but with the barriers tuned such that cotunnelling processes provide a significant contribution to the leakage current.

Figure 4: Hysteresis and fluctuations in leakage current.
figure 4

a, The spin-blockade leakage current for a 13C device measured for decreasing (increasing) magnetic field (sweep rate 0.4 mT s−1), shown in blue (red), after waiting at +1 T (−1 T) for 10 min. A 2 mV source–drain bias is applied at all times. Hysteresis is seen on a field scale >0.5 T for both sweep directions. b, Decay of leakage current over time measured by stopping a downward sweep at −0.25 T (marked by a black arrow in a). The fit is to an exponential decay with a time constant of 9 min. c, Dependence of leakage current on B near zero detuning in a second 13C device. The leakage current fluctuates over time at some values of B, but remains steady at others (insets).

We interpret the hysteresis in Fig. 4a as resulting from a net nuclear polarization induced by the electron spin flips required to circumvent spin blockade26. We speculate that this nuclear polarization generates an Overhauser field felt by the electron spins that opposes B once B passes through zero. The value of the coercive field, Bc0.6 T, the external field at which the two curves rejoin, places a lower bound for the hyperfine coefficient, Ag μBBc0.7×10−4 eV (equality corresponding to full polarization), independent of the value inferred from the width of the leakage current peak around zero field (Fig. 3f). If we instead use the value of A inferred from the current peak width (Fig. 3f), the size of Bc implies a 50% polarization for the data in Fig. 4a. Hysteresis is not observed for non-spin-blockaded transport in the 13C devices and is not observed in the 12C devices, suggesting that this effect cannot be attributed to sources such as the Fe catalyst particles or interaction with nuclei in the substrate or gate oxide.

Figure 4b shows that the induced nuclear polarization persists for 10 min, two orders of magnitude longer than similar processes in GaAs double dots27. The long relaxation time indicates that nuclear spin diffusion is extremely slow, owing both to the one-dimensional geometry of the nanotube and material mismatch between the nanotube and its surroundings. Field and occupancy dependence of relaxation were not measured.

Large fluctuations in Idd are seen at some values of magnetic field, but not at others (Fig. 4c), similar to behaviour observed in GaAs devices23. This presumably reflects an instability in nuclear polarization that can arise when polarization or depolarization rates themselves are polarization dependent26,28.

An important conclusion of this work is that the hyperfine coupling constant, A1–2×10−4 eV, in the 13C devices (for both electron and holes, see the Methods section) seems to be larger than anticipated8,9 and deserves further theoretical and experimental attention. It is possible that the substrate or gate oxide may enhance the degree of s-orbital content of conduction electrons, thus strengthening the contact hyperfine coupling. We also note that the one-dimensional character of charge carriers in 13C nanotubes may greatly enhance the effective electron–nuclear interaction29. Finally, the large value of A motivates the fabrication of isotopically enriched 12C nanotubes to reduce decoherence and the use of 13C tubes as a potential basis of electrically addressable quantum memory.

Methods

Carbon nanotubes are grown by chemical vapour deposition using methane feedstock and 5-nm-thick Fe catalyst islands on degenerately doped Si substrates with 1 μm thermal oxide. 12C devices are grown with methane containing natural abundance (1.1%) 13C; 13C devices are grown with 99% 13CH4 (Sigma-Aldrich). Nanotubes are located after growth using a scanning electron microscope, and catalyst islands, source and drain electrodes (15 nm Pd) and top-gates (30 nm Al) are patterned using electron-beam lithography. After contacting with Pd, samples are coated with a non-covalent functionalization layer combining NO2 and trimethylaluminium, followed by atomic layer deposition of a 30 nm Al2O3 top-gate insulator (Cambridge Nanotech Savannah atomic layer deposition system)30. Measurements were carried out in a dilution refrigerator with a base temperature of 30 mK and electron temperature of 120 mK, determined from the charge-sensing transition width21. The nanotubes presented in Figs 1 and 2 have small bandgaps (Eg25 meV); the 13C nanotube in Fig. 3b,d,f and the other 12C nanotube (data not shown) are large-gap semiconducting nanotubes. Charges occupying the dots and leads are electrons, except the data in Figs 3b,d,f and 4a,b, where the charge carriers are holes. No significant differences are seen between devices with electron and hole carriers.

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Acknowledgements

We thank M. Biercuk, K. Flensberg, L. Kouwenhoven, D. Loss, E. Rashba and O. Yazyev for discussions, and D. Reilly for experimental assistance. This work was supported in part by the National Science Foundation under grant no. NIRT 0210736 and the NSF-NNIN Program, ARO/iARPA, the Department of Defense, Harvard’s Center for Nanoscale Systems. H.O.H.C. acknowledges support from the NSF.

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Churchill, H., Bestwick, A., Harlow, J. et al. Electron–nuclear interaction in 13C nanotube double quantum dots. Nature Phys 5, 321–326 (2009). https://doi.org/10.1038/nphys1247

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