Main

Early work on spin transport in multiwall carbon nanotubes (MWNTs) with Co contacts showed that spins could propagate coherently over distances as long as 250 nm (ref. 12). The tunnel magnetoresistance (TMR)=(RAP−RP)/RP, defined as the relative difference between the resistances RAP and RP in the antiparallel and parallel magnetization configuration, was found to be positive and amounted to +4% in agreement with Jullière’s formula for tunnel junctions4,13. A negative TMR of about −30% was reported later for MWNTs contacted with similar Co contacts14. In these experiments, the nanotubes did not show quantum dot behaviour. It has been shown, however, that single-wall carbon nanotubes (SWNTs) and MWNTs contacted with non-ferromagnetic metals could behave as quantum dots and Fabry–Pérot resonators15,16,17,18,19, in which one can tune the position of discrete energy levels with a gate electrode. From this, one can expect to be able to tune the sign and the amplitude of the TMR in nanotubes, in a similar fashion as predicted originally for semiconductor heterostructures11.

In this letter, we report on TMR measurements of MWNTs and SWNTs that are contacted with ferromagnetic electrodes and capacitively coupled to a back-gate20. A typical sample geometry is shown in the inset of Fig. 1. As a result of resonant tunnelling, we observe a striking oscillatory amplitude and sign modulation of the TMR as a function of the gate voltage. We have studied and observed the TMR on nine samples (seven MWNTs and two SWNTs) with various tube lengths L between the ferromagnetic electrodes (see the Methods section). We present here results for one MWNT device and one SWNT device.

Figure 1: The TMR changes sign with Vg.
figure 1

Inset: SEM picture (Philips XL30-FEG) of a carbon nanotube contacted to ferromagnetic PdNi strips. The separation between the contacts along the nanotube amounts to L=400 nm. The magnetic field H was applied in plane. No qualitative difference has been seen for the field direction parallel and perpendicular to the long axis of the ferromagnetic electrodes. Main panel: Linear response resistance R as a function of H at temperature T=1.85 K for different gate voltages Vg. The blue (red) arrow indicates the up (down) magnetic field sweep direction, respectively. The observed amplitude and the sign of the TMR depend on Vg, but not on the high-field magnetoresistance, which is expressed by S denoting the percentage change of the magnetoresistance with magnetic field (see the Methods section). Note that we extract the maximum possible value for the TMR signal as indicated in the figure.

Full size image

We first discuss the results of the MWNT device. Figure 1 shows single traces of the linear response resistance R as a function of the magnetic field H at 1.85 K for two sweep directions and four different gate voltages Vg. For all cases, the characteristic hysteretic behaviour of a spin valve appears. On sweeping the magnetic field from −500 to 500 mT, the configuration becomes antiparallel between 0 and 100 mT, whereas it is always parallel for |H|>100 mT. At Vg=−3.1 V, for example, R increases from 49.7 to 51.5 kΩ when the sample switches from the parallel to the antiparallel configuration. This yields a normal positive TMR of +2.9%. In contrast, at Vg=−3.3 V, R switches from 30.5 kΩ in the parallel configuration to a smaller resistance of 29.5 kΩ in the antiparallel configuration, yielding an anomalous negative TMR of −3.5%. Therefore, the sign of the TMR changes with the gate voltage, demonstrating a gate-field-tunable magnetoresistance.

Figure 2a shows the variation of the TMR in a large Vg window of −5 to 2 V at temperature T=1.85 K. The TMR is observed to oscillate relatively regularly between −5 and +6% on a gate-voltage scale of ΔVgTMR=0.4–0.75 V. Two possible mechanisms may account for oscillations in spin transport: quantum interference11 and gate-field-induced spin precession by means of the Rashba spin–orbit interaction21 proposed in ref. 10. In the latter case, the spin–orbit interaction yields a spin precession that is reflected both in the TMR and the conductance. To lowest order, the spin precession would lead to a TMR∝cos(2mLβReVg/ħ2), where L is the length of the MWNT, m (e) the electron mass (charge) and βR the Rashba spin–orbit parameter. However, the measured magnitude of ΔVgTMR∼1 V for one TMR period requires a large βR∼10−12 m. Although a βR value of this order of magnitude has been reported in semiconductor heterostructures22, this value is unreasonably large for carbon nanotubes, owing to the small spin–orbit interaction of carbon leading to an electron g-factor close to 2 (refs 1924). We therefore conclude that the mechanism of TMR oscillations is quantum interference instead. To substantiate this, we next compare the TMR gate-voltage scale ΔVgTMR with the corresponding scale ΔVge for the addition of single electrons.

Figure 2: The TMR of the device shown in Fig. 1 oscillates with Vg.
figure 2

a, TMR as a function of Vg at T=1.85 K. The characteristic gate-voltage scale ΔVgTMR of the observed TMR modulation varies between 0.4 and 0.75 V. The bars reflect the error in deducing the TMR signal from R(B) curves, see Fig. 1. b,c, These data were measured in a different cryostat at 300 mK: b, nonlinear differential conductance dI/dV as a function of source–drain Vsd and Vg in a narrow Vg interval, corresponding to the addition of six electrons; and c, the linear conductance G over a much wider Vg interval.

Full size image

To resolve single-electron states, the same sample was measured at lower temperatures, that is at T=300 mK in a different cryostat that is not suitable for TMR studies. A measurement of the differential conductance dI/dV as a function of source–drain Vsd and gate voltage Vg at T=300 mK in a narrow Vg range is shown in Fig. 2b. It shows the diamond-like pattern characteristic for single-electron tunnelling in a quantum dot. The visible diamonds vary in size with single-electron addition energies ranging between 0.5 and 0.75 meV, in agreement with previous reports on MWNT quantum dots with non-ferromagnetic leads19. Comparing with VgTMR, we see that the average gate-voltage scale ΔVge=25 mV associated with one single particle level is much smaller than ΔVgTMR. The latter corresponds therefore to the addition of at least 16 electrons rather than one.

A gate-voltage scale that agrees with the TMR signal becomes visible if the linear conductance G at low temperatures is monitored over a wider gate-voltage range. This is shown in Fig. 2c. The single-electron conductance peaks are strongly modulated in amplitude, leading to a regular beating pattern with the proper gate-voltage scale of ΔVg≈0.4 V. Note, the absolute values of Vg cannot be compared with the TMR measurement in Fig. 1a, because the sample was thermally cycled.

Beatings in the amplitude of single-electron resonances are often observed in quantum dot structures and attributed to interference modulation owing to weak disorder. Indeed, as seen in the dI/dV plot of Fig. 2b, the diamonds do not alternate regularly. In the resonant tunnelling model, one expects each single-particle peak to contribute negatively to the TMR at sufficiently low temperatures. However, as we have measured the TMR at T=1.85 K, where the single-particle resonances are already strongly averaged out, the TMR is only sensitive to the average over these peaks, yielding a modulation that follows the envelope function of the single-electron peaks.

The final proof that interference of single-particle levels is that the physical origin for the observed TMR oscillation comes from measurements on SWNTs. Figure 3 shows the conductance G and the TMR of a SWNT device. The quantum dot behaviour is already observed at 1.85 K, whereas this was only evident at 0.3 K in the MWNT device. This is consistent with the higher energy scales (both single-electron charging energy and level spacing) for SWNTs as compared with MWNTs. As seen in the dI/dV plot of Fig. 3b, the typical single-electron addition energy amounts to ∼5 meV, whereas it was an order of magnitude smaller in the MWNT device.

Figure 3: The detailed evolution of the TMR signal for two single-particle resonances.
figure 3

Results for a SWNT device with a contact separation of L=500 nm measured at T=1.85 K. a, Measurement (▪) of the linear conductance G and the TMR around two resonances and theoretical fit (solid curves) using equation (1) with P=0.2, κ=0.32 meV and γL=0.014 (0.028) meV and γR=0.5 (0.85) meV for the left (right) resonance. The bars in a reflect the error in deducing the TMR signal from R(B) curves. b, Plot of the nonlinear differential conductance dI/dV as a function of source–drain Vsd and Vg. Typical R(B) curves for this F–SWNT–F are shown in Supplementary Information, Fig. S1, as are two representative R(B) curves for N–SWNT–F devices.

Full size image

In Fig. 3a, the variation of the linear conductance G and the TMR are simultaneously shown for two resonances. First, we observe that the TMR changes sign on each conductance resonance. Furthermore, we see that the line shape of the conductance resonances is symmetric, whereas that of the TMR dips is asymmetric. The jump in the G(Vg) data at Vg=4.325 V is not reproducible, but arises from background charge switching. This jump is absent in a second scan. The amplitude of the TMR ranges from −7 to +17%, which is a higher amplitude than for the MWNT samples. We think that this might be due to the higher charging energy in SWNTs25. Control experiments on two normal–SWNT–ferromagnetic (N–SWNT–F) devices yield an order of magnitude lower signal (see Methods and Supplementary Information), proving that the current in the F–tube–F devices is indeed spin polarized.

The transmission probability through a quantum dot near a resonance can be described by the Breit–Wigner formula. If the quantum dot is coupled to two continua with spin-dependent densities of states, the lifetime of an electron on the quantum dot becomes spin dependent. Therefore, the width of the resonance is different for carriers with up and down spins. In addition, the energy levels En of the carbon-nanotube quantum dot acquire a spin-dependent part caused by phases of the reflection amplitudes at the boundaries between the quantum dot and the ferromagnetic electrodes, which are spin dependent26,27,28. The spin-dependent Breit–Wigner transmission probability Tσ for electrons at energy E with spin orientation σ can conveniently be written as

where ΓL(R)σ=γL(R)(1+σPL(R)) denote the spin-dependent (σ=±1) coupling to the left (right) ferromagnetic lead, γL(R) is the bare coupling and E0σ is the spin-dependent energy level of the quantum dot. Note that the polarization in the leads PL(R) is measured relative to the spin-quantization axis.

The sign change of the TMR can be predicted with equation (1), provided that the couplings to the leads are asymmetric, see Fig. 4. Off resonance, that is if |E−E0|≫(ΓLσ+ΓRσ), Tσ is small and proportional to ΓLσΓRσ, yielding the normal positive TMR of +2P2/(1−P2) (we assume that |PL|=|PR|=P). On resonance, on the other hand, Tσ∝ΓLσ/ΓRσ (if, for example, ΓRσ≫ΓLσ), yielding an anomalous negative TMR of −2P2/(1+P2). This mechanism has already been suggested to explain an observed anomalous TMR in Ni/NiO/Co nanojunctions29. Unlike this earlier work, we are able to follow the conductance and the TMR by tuning the energy level E0 with the gate voltage Vg and comparing with the model. Whereas the negative TMR can be understood following this line of argument, the explicit shape and, in particular, the asymmetry in the TMR requires a spin-dependent energy level E0σ as we will now show. The eigenstate depends on the gate voltage and on the spin direction: E0σ=E0−εσ−αeVg, where α is a constant proportional to the gate capacitance, e the unit of charge and εσ the spin-dependent part of the energy level. In the limit of small spin polarization PL(R)≪1, one may use the ansatz εσ=κσ(PL+PR). We treat κ as a fitting parameter, which will be deduced from the experiment. κ determines the asymmetry of the TMR signal.

Figure 4: Schematics explaining the observed sign change of the TMR.
figure 4

a, The case for antiparallel magnetization. b, The case for parallel magnetization. The electric resistance of an asymmetric resonant-tunnelling junction (we assume ΓR≫ΓL) is proportional to the asymmetry A=ΓR/ΓL, where ΓR,L are the tunnelling rates to the right and left electrode, respectively. If ferromagnetic contacts are used, ΓR,L become spin-dependent. The rate is increased for the spin direction of the majority carriers, whereas it is decreased for the minority carriers. The two spin directions are colour-coded (red is up spin and blue is down spin). As the electric resistance is spin-dependent, the total resistance R is the parallel circuit of R↑ and R↓. Whereas R↑ and R↓ are equal in the parallel configuration, R↑ is smaller and R↓ is larger in the antiparallel configuration. Owing to the dominance of the smaller resistance in a parallel circuit, R is smaller in the antiparallel compared with the parallel case, corresponding to a negative TMR signal.

Full size image

The solid lines in Fig. 3 show fits to the measured conductance G and the TMR using equation (1). As the two resonances are well separated in energy, it is possible to fit them individually. In order to obtain the conductance at finite temperature, we convolved Tσ with the derivative of the Fermi–Dirac distribution at 1.85 K and summed over σ. The following parameters entered the fits: P=0.2 and κ=0.32 meV. γL,R differ for the two resonances: γL=0.014 (0.028) meV and γR=0.5 (0.85) meV for the left (right) resonance, respectively. Using these parameters, a very good agreement between theory and experiment is found. Convincing evidence for spin injection in a quantum dot is deduced from the observed asymmetric line shape of the TMR in the SWNT device. The spin imbalance expressed by εσ is substantial, amounting to as much as ±0.13 meV, which corresponds to an internal ‘exchange field’ of B=2.2 T.

Methods

Experimental

We have developed a reliable scheme to prepare transparent ferromagnetic contacts to MWNTs and SWNTs using the ferromagnetic alloy Pd1−xNix with x∼0.7 (ref. 20). Applying standard lithography techniques, a single nanotube is connected to two ferromagnetic PdNi electrodes that are further connected to bonding pads by means of Pd wires. We take advantage of the very good contacting properties of Pd to nanotubes30 and its giant paramagnetism. We have studied and observed the TMR on nine samples (seven MWNTs and two SWNTs) with various tube lengths L=0.4, 0.5, 0.8 and 1 μm between the ferromagnetic electrodes. Control samples with Pd and PdNi contacts were also fabricated, see below under ‘Control experiment’.

Stray-field effect

In order to rule out a simple stray-field effect, we compare the high-field magnetoresistance S (defined as a percentage change of the resistance per tesla) with the low-field hysteretic TMR signal. First, we see that the magnitude of the TMR signal of the curves shown in Fig. 1 is to a good approximation constant (to 3.4±0.4%), whereas the background (high-field) magnetoresistance may change by as much as an order of magnitude. Second, the sign change of the TMR from a positive value at Vg=−3.1 V to a negative value at Vg=−3.3 V is not accompanied by a change in the background S. In fact, all possible sign combinations of S and TMR have been observed. As the low-field TMR signal bears no correlation with the background magnetoresistance, we can exclude a stray-field effect from the contacts to the bulk nanotube as the source of the observed hysteretic signal.

Control experiment

In order to ensure that the measured TMR is caused by a coherent spin-polarized current, we have also analysed two devices with asymmetric contacts. One contact (F) is made from the ferromagnetic PdNi alloy of similar composition as was used in the F–SWNT–F devices and the other (N) from the non-ferromagnetic metal Pd. This yields an N–SWNT–F device, which ideally should show no hysteretic signal. On the basis of the noise signal of the resistance measurement, a hysteretic switching signal (if any) must be smaller than 1–1.5% (see Supplementary Information, Fig. S1). As this is up to ten times smaller than what we have observed in the F–SWNT–F device for similar conductances, any magnetic artefact arising from a single ferromagnetic contact alone must be small, proving that we have observed a spin effect in transport in the F–SWNT–F case.