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In a polar semiconductor, a conduction electron deforms the surrounding lattice to form a polaron state14. The formation of this quasi-particle, by combining an electron and a cloud of lattice vibrations, or phonons, strongly influences the transport properties. The possibility for localization of strongly coupled polarons was suggested by Landau more than 70 years ago14. Recently, Koch et al. predicted that a related trapping of heavy polarons can occur in a quantum dot formed in a mechanically suspended nanostructure10,11. In such a nanoelectromechanical system, the vibrational modes of the nanostructure can be strongly affected by the presence of electrons in the quantum dot, as they deform the embedding medium. For strong electron–phonon coupling, the deformation effectively blocks electronic transport, termed Franck–Condon blockade. By analysing electronic transport through a suspended carbon nanotube (CNT) quantum dot over a wide range of electronic states, we are able to highlight generic features of vibron-assisted electronic transport, and unambiguously confirm the Franck–Condon blockade scenario.

Scanning electron microscope images and a scheme of our suspended CNT quantum dot device are shown in Fig. 1a–c. The CNT is electrically and mechanically connected to both source (S) and drain (D) contacts, and the central electrode acts as a suspended top-gate. A quantum dot in the CNT is formed between defects15, which are presumably created during the release process and act as local barriers. The double top- and back-gate configuration enables us to determine the location of the quantum dot below the top-gate (see Fig. 1c and Supplementary information). Although similar data were observed on a second sample with the same geometry, only data measured on the first sample are presented here.

Figure 1: Characterization of the suspended CNT quantum dot.
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a, Electron microscope micrograph in tilted view of the suspended CNT (white arrows) with the source (S) and drain (D) electrodes and the central top-gate (TG) electrode. b, Top view of the device. The scale bar in a and b is 200 nm. c, Scheme of the quantum dot (QD) device formed in the suspended CNT. As shown by the charge stability diagrams (see Supplementary information), a quantum dot is formed under the top-gate, separated from the other parts of the nanotube by tunnel barriers. Both the suspended top-gate and the underlying back-gate (BG) can be used to tune the electronic properties of the quantum dot and the leads (LL: left lead; RL: right lead). d, Source–drain current Isd through the quantum dot measured as a function of the bias voltage Vsd and the top-gate voltage Vtg, adjusting the back-gate voltage simultaneously to keep the average chemical potential in the leads constant (see Supplementary information): Vbg=−0.7576×Vtg+1.6784 V. Diamond-shape regions of suppressed current are characteristic of Coulomb blockade (so-called Coulomb diamonds), and the four-fold periodicity is characteristic of quantum dots formed in clean CNTs.

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The exceptional quality of our sample, which is crucial for this investigation, is revealed by the well-resolved observation of multiple four-fold degeneracies of electronic states in the large-scale presentation of our data (Fig. 1d). This stems from the combined spin and valley degeneracies in clean CNTs (refs 16, 17, 18), and enables us to fully characterize the electronic properties of our device, including the electronic confinement energy ΔEelec=6.8±2.5 meV (see Supplementary information).

At a finer energy scale (see Fig. 2), all of the probed Coulomb diamonds show several striking generic features. (1) Quasi-periodic lines running parallel to the edges of the Coulomb diamonds appear ubiquitously. We will see that they correspond to excited vibronic states. (2) For increasing temperatures, we observe the appearance of absorption sideband peaks within the Coulomb blockaded regions. (3) At the edge of the diamonds, there is a pronounced current suppression compared with the current magnitude for excited vibronic states. (4) Most diamonds show a significant apparent shift of their tip between positive and negative bias. (5) Finally, all of the probed diamonds show ubiquitous negative differential conductance appearing in between the excited vibronic states. We will show that most of these features are generic for transport-induced vibron excitations and are signatures of the strong electron–vibron coupling in our system.

Figure 2: Evidence and temperature dependence of vibron-assisted transport.
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a, Differential conductance dIsd/dVsd for a subset of the Coulomb diamonds shown in Fig. 1d, showing the quasi-periodic excited vibronic states (see dotted lines). The arrows point to electronic excited states, visible at higher energy. b, Energy spacing of the excited vibronic states for the whole gate voltage range shown in Fig. 1d. For each point, the energy spacing is averaged over half of a Coulomb diamond, corresponding to a given electronic state, with error bars determined as the standard deviation of the mean. The horizontal dashed line represents the average value over all points, ΔEvib=0.8±0.2 meV. This energy is compatible with the longitudinal stretching mode, for which ΔEvib=ℏvphq, with q=nπ/L for the nth vibron mode in the doubly clamped nanotube and L being the confinement length. Using the value of the stretching phonon group velocity in clean suspended CNTs, vst≈2.4×104 m s−1 (ref. 25), we deduce the characteristic length for the first vibron mode L≈65 nm, of the same order of magnitude as the suspended parts of the nanotube and in reasonable agreement with earlier experiments6. c–f, Coulomb diamonds measured in the same region of gate voltage (same electronic state) for different temperatures, 1.6 K (c), 3 K (d), 5 K (e) and 7 K (f). As the temperature increases, extra conductance peaks appear in the Coulomb blockaded regions. In e, the green uptriangle and red downtriangle mark the positions of conductance peaks corresponding to tunnelling through the ground state and the first excited vibronic state (emission peak), and the blue circle to the position of a conductance peak due to the absorption of a vibron (absorption peak). g, Maximum conductance Gmax for tunnelling through the ground state (green uptriangle), the emission peak (red downtriangle) and the absorption peak (blue circles), corresponding to points marked in e. For the tunnelling through the ground state and the emission peak, we use as a fit Gmax∝1/kBT (see green and red curves), as expected for the derivative of the Fermi distribution in the quantum Coulomb blockade regime26. For the absorption peak, we find a good fit with the product of the equilibrium population of the phonon states, given by the Bose distribution, 1/(exp(ℏω0/kBT)−1), and the derivative of the Fermi distribution, giving Gmax∝1/kBT×1/(exp(ℏω0/kBT)−1) (see blue curve). From the fit to the experimental data, we determine ℏω0=0.96±0.08 meV, which is consistent with the energy spacing of the excited vibronic states (ΔEvib=0.88±0.15 meV at this value of the gate voltage).

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Quasi-periodic excited states have been reported in previous experiments on suspended quantum dots. They have been interpreted as being due to the excitation of quantized phonons in the nanostructure, called vibrons6,7,8,9: when the bias voltage matches a multiple of the energy of a vibron, ℏω0, transport is enhanced by emission of a vibron. This interpretation is confirmed in our data because the energy spacing between these excited vibronic states ΔEvib=0.8±0.2 meV is much lower than the measured electronic level spacing. In addition, Fig. 2b shows that this energy is constant over a wide range of gate voltages and does not depend on the electronic structure of the quantum dot.

The bosonic origin of the excitations is further demonstrated by the temperature dependence of the data. As temperature is increased, we observe extra excited vibronic states appearing within the Coulomb blockade regime (Fig. 2c–f), similar to anti-Stokes resonances in Raman spectroscopy. This behaviour, not reported in previous experiments, is expected for excited vibronic states because higher vibrational states become populated at elevated temperature. As a consequence, electronic transport is enabled even in the Coulomb-blockade regime by absorption of a vibron. The quantitative analysis of the peak magnitude as a function of temperature confirms the bosonic origin of the absorption peak (Fig. 2g).

The very observation of vibrational sidebands is a signature of rather strong electron–vibron couplings in our CNT quantum dot. This is further highlighted by the current suppression at low bias detected in all of the probed diamonds.

Indeed, our data unambiguously confirm that this is a direct demonstration of the so-called Franck–Condon blockade10,11. When tunnelling onto the dot, the electron shifts the equilibrium coordinate of the vibron oscillator by an amount proportional to the electron–vibron coupling (Fig. 3a). The transition probability of this tunnelling process is proportional to the square of the overlap between the vibronic wavefunctions before and after tunnelling. In spectroscopy, this principle is well known as the Franck–Condon principle1,2 and explains the magnitude of peaks in absorption spectra. As the overlap of low-lying vibronic states is exponentially sensitive to this geometrical displacement, the ground-state-to-ground-state transition for vibrons is exponentially suppressed for strong electron–vibron coupling. This leads to the Franck–Condon blockade of the current at low bias. At the same time, the overlap between the vibronic ground state with highly excited vibronic states, of which the wavefunctions are widely spread in space, is still significant. This enables tunnelling while exciting vibrons, as long as the bias is large enough to compensate for the vibronic energy difference, causing the observed vibrational sidebands.

Figure 3: Theory and analysis of Franck–Condon blockade.
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a,b, Schematic description of the displacement of the vibron potential for strong (g≫1) (a) and weak (g≪1) (b) electron–vibron coupling g, with N (green curve) and N+1 electrons (blue curve) in the dot. The shift for g≫1 (a) suppresses the transitions between low-lying vibronic states and thus the current in the low-bias regime (Vsd1), causing the Franck–Condon blockade. In parallel, it opens transitions to excited vibronic states in the high-bias regime (Vsd2, see vertical double-headed arrows), yielding vibrational sidebands. In contrast, for g≪1 the shift is essentially absent, enabling ground-state-to-ground-state vibronic transitions (see vertical double-headed arrow). At the same time, the vibronic ground state before tunnelling is essentially orthogonal to all excited vibronic states after tunnelling. This suppresses the transition probabilities to highly excited vibronic states and hence the vibrational sidebands. c, Zoom into a part of the Coulomb diamonds (around Vtg=0.1570 V) showing the suppression of the low-bias transport and the apparent shift of the Coulomb diamond tip at positive bias with respect to that at a negative one. d, Simulated Coulomb diamonds in the same conditions as in the experiment (with kBT=0.15ℏω0, ΓR=10ΓL, g=3.3) for strongly asymmetric tunnel coupling between the leads and the dot, ΓR≫ΓL. e, Schematic diagrams of the rate-limiting processes through the opaque left barrier for the extremal lines (dotted in c and d) of the Coulomb diamond. For the extremal line with negative slope, the relevant process implies tunnelling through the left barrier with no vibronic excitation (i and iv). In the presence of strong vibronic relaxation, each electronic tunnelling event starts out of the vibronic ground state. Thus, these processes lead to a current proportional to ΓL|M0→0|2 involving the Franck–Condon overlap M0→0 between vibronic ground states, and the Franck–Condon suppression of vibronic transitions manifests itself also at large bias. The current at the extremal line with positive slope is also predominantly limited by tunnelling through the left junction, but as Vsd increases inelastic excitation of vibrons becomes energetically allowed (ii and iii). The strong vibronic relaxation implies that these processes are associated with vibronic transitions involving M0→Q, where Q=Int[e Vsd/ℏω0] is the maximum allowed vibronic quantum number (as long as M0→Q increases with Q). Because for strong electron–vibron coupling M0→Q≫M0→0, the resulting differential conductance peak proportional to ΓL|M0→Q|2 increases with the bias voltage. f, Differential conductance measured for Vtg=0.0678 V. Representative fits of the maxima with the Franck–Condon progression (equation (1)) enable us to extract the parameters g=3.5 (negative bias voltage, green squares) and g=5.0 (positive bias voltage, red circles). Peaks that deviate strongly from the progression at large bias voltage are due to electronic excited states. g, Same as in f, for Vtg=0.1570 V, giving g=3.0 (negative bias voltage, green squares) and g=4.5 (positive bias voltage, red circles).

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This interpretation provides a quantitative understanding of the experimental data. The apparent shift of the Coulomb diamond tips for positive and negative bias observed in many cases in our experiment (Fig. 3c) and others5, is explained by the asymmetric tunnel coupling of the quantum dot with source and drain in combination with strong vibrational relaxation (so that each electronic tunnelling event starts out of the vibronic ground state). The smaller tunnel coupling determines the current-limiting process, and energetic considerations (shown in Fig. 3e) imply that the edge of the diamond is due to ground-state-to-ground-state vibronic transitions for the diamond edge of one slope and due to ground-state-to-excited-state vibronic transitions for the other. As a result, the solution (see Fig. 3d) of the corresponding rate equations in Fig. 3e reproduces the experiment in Fig. 3c quite accurately.

To extract the electron–vibron coupling g despite the above asymmetry, we study the differential conductance at the gate voltage corresponding to the diamond tip (Fig. 3f,g). For equilibrated phonons, the Franck–Condon theory predicts differential conductance peaks following the progression:

with n being the difference in vibron quantum numbers and |M0→n| the overlap between vibronic states with 0 and n vibrons. By fitting the maxima of dIsd/dVsd with this expression (Fig. 3f,g), we find very good agreement for most measured diamonds. Our measurement not only shows a current suppression at low bias5,6 but rather matches the whole progression for excited vibronic states, proving global consistency with the Franck–Condon picture. We deduce a large value for the electron–vibron coupling parameter g. For the whole set of electronic states underlying the Coulomb diamonds of Fig. 1d, the average value is 〈g〉=3.3. This is about a factor of three larger than in previous measurements6. We also observe state-to-state fluctuations of g with a standard deviation of 0.9.

To obtain a theoretical understanding of the large electron–vibron coupling constant g in the quantum dot, we consider the intrinsic electron–phonon coupling of the underlying CNT. This turns out to be dominant compared with previously considered extrinsic electrostatic coupling mechanisms6,19. The coupling originates from (1) the modified hopping associated with changes in the C–C bond lengths and (2) the deformation potential due to local area variations; the deformation potential coupling constant gD≈30 eV is about an order of magnitude larger than the hopping-induced one20,21.

When combined with symmetry considerations, the existence of these two coupling mechanisms has several important consequences for the coupling strengths of the various CNT vibron modes. Longitudinal stretching modes (LSMs) and radial breathing modes couple most strongly. Their associated lattice deformations involve local area variations and produce a linear shift of the electronic energies, resulting in a linear deformation potential interaction. Twist modes also couple linearly, but they are area preserving and couple only through the weaker hopping-induced mechanism. Finally, bending modes have only a weak quadratic interaction because, for symmetry reasons, the electronic energy does not depend on the sign of the associated deformation of the CNT.

We can thus identify the relevant vibron mode probed in our data. Indeed, of the strongly coupled vibrons, only the LSM is in a frequency range compatible with our experimental observations (see Fig. 2b), as discussed previously6. The relevant dimensionless coupling constant g for the occurrence of the Franck–Condon blockade is given by (the square of) the shift of the equilibrium coordinate measured in units of the amplitude (oscillator length) of the vibronic quantum fluctuations. We compute this quantity in the framework of the effective Dirac Hamiltonian for the low-energy electronic properties of CNTs, combined with the theory of elasticity for the description of the vibron mode (see details in Supplementary information). Using literature values for the elastic constants of graphene21, this calculation yields a maximal coupling of

for the lowest LSM. An important consequence of our analysis is that the coupling constant is strongly sample dependent. It is inversely proportional to the CNT circumference , and proportional to the electronic matrix element of the deformation potential. The matrix element vanishes for the lowest longitudinal stretching mode when electrons and vibrons are confined to the same region of the CNT. It becomes maximal when the electronic wavefunction is sharply localized around a region of maximum strain, yielding the estimate in equation (2).

Atomic force microscope measurements indicate that our CNTs typically have a circumference of a few nanometres. Thus, our value for g given in equation (2) is consistent with the observation of the Franck–Condon blockade, which requires g>1. At the same time, the coupling constants extracted from experiment are slightly larger than our estimate. Here, the defects creating the electronic confinement could have a major role: first they help confine the wavefunction in a region of maximum strain; second they could also modify the vibrons and/or the deformation potential. Other observations also indicate that experiments exhibit more pronounced vibronic effects in transport than theory suggests: vibrational sidebands are accompanied by negative differential conductance, as already reported6, whereas theory predicts a step-like I–V characteristic. In addition, the experiment shows vibrational absorption sidebands within the Coulomb blockade diamonds, which are absent in theoretical simulations. We have checked that these intriguing enhancements are not related to asymmetric coupling of the dot to the leads (see Supplementary information). The underlying reasons are not understood at present.

Methods

Devices were fabricated from highly doped silicon wafer substrates covered by 200 nm silicon oxide. Single-walled CNTs (SWNTs) were grown randomly on the oxide substrate by chemical vapour deposition based on dispersed ferritin catalysts22. Electron beam lithography was used to pattern metallic electrodes and gate structures (2/30 nm Cr/Au) around (selected) individual SWNTs (Fig. 1b). Finally, diluted HF (4% for 5 min) etching followed by critical point drying completes the device fabrication (Fig. 1c). It is crucial that the Cr layer oxidizes after the etching and drying processes, as well as when exposed to environmental air, to form the top-gate oxide. For a more detailed process description, see refs 23, 24. Measurements were carried out in a variable temperature 4He cryostat at a base temperature of T=1.3 K. We have measured the two-terminal conductance through the SWNT device by applying a symmetric d.c. bias voltage Vsd while measuring the current through the quantum dot device with a resolution better than 10 fA. For differential conductance measurements, a small a.c. bias, Vac=48 μV was superimposed on Vsd and the differential conductance was measured with lock-in techniques. Measurements have been done in two different devices, with quantum dots formed at different locations along the CNT. All configurations show similar results as the one presented here.