Switchable coupling of vibrations to two-electron carbon-nanotube quantum dot states
We report transport measurements on a quantum dot in a partly suspended carbon nanotube. Electrostatic tuning allows us to modify and even switch “on” and “off” the coupling to the quantized stretching vibration across several charge states. The magnetic-field dependence indicates that only the two-electron spin-triplet excited state couples to the mechanical motion, indicating mechanical coupling to both the valley degree of freedom and the exchange interaction, in contrast to standard models.
JARA – Fundamentals of Future Information Technology \altaffiliationICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels, Barcelona, Spain \altaffiliationEqual contribution \alsoaffiliationJARA – Fundamentals of Future Information Technology \altaffiliationInstituto de Física Enrique Gaviola (IFEG-CONICET) and FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina \altaffiliationEqual contribution \alsoaffiliationJARA – Fundamentals of Future Information Technology \alsoaffiliationJARA – Fundamentals of Future Information Technology \altaffiliationPhysikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, Germany \alsoaffiliationJARA – Fundamentals of Future Information Technology \alsoaffiliationInstitute for Theory of Statistical Physics, RWTH Aachen University, 52074 Aachen, Germany \alsoaffiliationJARA – Fundamentals of Future Information Technology \alsoaffiliationPeter Grünberg Institute, Forschungszentrum Jülich, 52425 Jülich, Germany \alsoaffiliationJARA – Fundamentals of Future Information Technology Carbon nanotubes are found to be an ideal playground for nano-electromechanical systems (NEMS) since their high-quality, quantum-confined electronic states are accessible by transport spectroscopic techniques and couple strongly to the excitations of different mechanical modes. The growing interest in NEMS is fueled by the desire to accurately sense small masses and forces 1, address quantum-limited mechanical motion 2, and integrate such functionality into complex hybrid devices 3, leading to new applications 4. The central question is the strength of the coupling of electronic states to the vibrational modes. Whereas molecular junctions display such modes also in electrically gated transport measurements 5, 6, 7, carbon-nanotube (CNT) quantum dots allow for a much more viable fabrication, higher mechanical Q-factors, and better tuneability as NEMS 8, 9, 10, 11, 12, 13. Also, the coupling to the bending mode can be combined 14, 15 with the spin-orbit (SO) interaction 16, 17 by making use of the recently demonstrated 11 curvature-induced SO-coupling in CNTs.18 Whereas the frequency of the vibrational modes has been demonstrated to be tuneable 19, 20, 21, another desirable feature is the ability to switch “on” and “off” the electron-vibration coupling in the same device, e.g., in envisioned quantum-information processing schemes 15, 22. This is also helpful for fundamental studies of systems in which mechanical motion is combined with other degrees of freedom, e.g, the spin 23 and the valley 24.
Recently, switchable coupling to a classical flexural mode of a CNT has been demonstrated 12. In this letter, we present a CNT quantum dot NEMS with a coupling of the electronic states to a longitudinal stretching vibration of about 200 GHz that can be turned “on” and “off”. We illustrate the advantage of this by transport measurements in the two-electron quantum-dot regime and find that the well-known Anderson-Holstein scenario breaks down in an unexpected way: Different spin states exhibit different coupling strengths to the vibrational mode.
In Figure 1a we show a schematic of a typical suspended CNT quantum-dot device whose scanning electron microscope image is shown in Figure 1b. The CNT is electrically and mechanically connected to both source (s) and drain (d) contacts where the central electrode acts as a suspended, doubly clamped top gate (tg). The quantum dot is formed in the small band gap CNT by the electrostatic potentials of the top and back gate (bg), see Figures 1a and 2a, allowing for electrostatic control of the size of the quantum dot in the range of 250 - 350 nm, see \textcolorblackSupporting Information. By changing the gate voltages we can modify the position and size of the dot with respect to the suspended vibrating region of the CNT, which is a crucial part of our experiment. The high quality of our CNT sample is revealed by the observation of well-resolved, multiple four-fold shell-structure of the electronic states in the stability diagram in Figure 1c measured at zero magnetic field and at a base temperature of 1.6 K. This shell-structure stems from the combined spin and valley degeneracies in clean CNTs 25, 26, 27, and enables a first characterization of the electronic properties by the Coulomb and confinement energies. Importantly, the resulting estimates show that the quantum dot formed in the CNT is comparable to or even larger in size than the top-gate, see \textcolorblackSupporting Information.
The key advantage of our device, in contrast to previous ones, is that we can first obtain detailed information about the electronic spectrum by measuring the differential conductance in a gate voltage regime without signatures of vibrational excitations. For example, in the spectrum shown in Figure 1d the low-energy excitations indicated by dashed black lines can be assigned to transitions between states with electron number and , respectively. These are indicated in the schematic in Figure 1e which shows for two spin doublets denoted and , obtained by filling the (anti)bonding orbitals of the and valleys with one electron, which are split in energy by due to the valley-mixing . For we have spin-singlets and (latter not shown) completely filling one of these orbitals, and a singlet and a triplet in which two different orbitals are filled. Here the labels of the many-body states , , indicate the spin multiplicities (singlet, doublet, triplet), whereas the subscripts indicate the relevant orbital polarizations. In the transport data of Figure 1d we identify a ground singlet (), an excited triplet () and another singlet (), split by the exchange energy . The measured magnetic field transport spectroscopy in Figure 1f confirms this assignment: the slope of the lines A and C for transitions to and , respectively, differs by the Zeeman spin splitting from the slope of line B for the transition to the triplet . We note that for these parameters the singlet is the highest in energy in Figure 1e. It is not shown there nor discussed further below because this state does not influence the measured transport in the considered regime. \bibnote[Note1] In this state two electrons fill the excited orbital causing the amplitude for a transition from the ground state with an electron in the ground orbital to be strongly suppressed. For this transition to happen, one must shuffle the lower electron up and add another electron, something which is only possible for very strong spin-orbit coupling or higher-order tunnel processes, neither of which are relevant here. This point is important since due to polaronic renormalization discussed at the end of the letter may actually lie below , but even then it cannot be observed. Our calculations below do, however, include the state and confirm that it has negligible influence.
By independently tuning the top- and back gate voltages we can change the electrostatic confinement of the quantum dot and thereby effectively operate a single quantum dot system which can be made sensitive to the vibrating part of the CNT, as illustrated in Figure 2a. The resulting electronic stability diagram in Figure 2b, showing nearly parallel lines, indicates that we can independently fix the electron number in the dot while modifying its shape, dimensions and position. When measuring the Coulomb diamonds along the lines indicated in Figure 2b one expects, electronically speaking, no qualitative difference. Indeed, along the initial working line marked as (1) in Figure 2b, the measurement in Figure 2c shows no indications of vibrations. However, when tuning to the working line (2), the excitation spectrum, shown in Figure 2d, changes in a way that cannot be explained by a modification of the size-quantization energy on the quantum dot: for several subsequent charge states a dense spectrum of discrete excitation peaks appears, equally spaced by as Figure 2h shows. This is the case across the entire electronic shell that we measure, see \textcolorblackSupporting Information. The spacing lies in the range expected for the high frequency of the longitudinal stretching mode (LSM) of the suspended parts of the CNT (length 65 nm as in previous studies 10, 8). Furthermore, the predominance of the excitation lines with negative slope indicates that the quantum dot couples to only one of the two suspended parts 29, 30. In Figure 2e we demonstrate that by tuning to a different voltage regime we are able to make the other vibrating part dominate. Our system thus displays electrostatically tuneable electron-vibration coupling.
To illustrate how the switchable coupling to a quantized vibration can be exploited, we now focus on measurements for the electron regime in Figures 2c - 2d. The corresponding calculations shown in Figures 2f - 2g are based on a quantum-dot model including the electronic states identified before in Figures 1d - 1e and coupling to a single vibrational mode. This model will be discussed in detail below, once we have presented all experimental data. Apart from this, the non-equilibrium transport is obtained from standard master equations (see \textcolorblackSupporting Information) which incorporate single-electron tunneling into both orbitals of the shell (with asymmetry parameter ) from both electrodes (with junction asymmetry parameter ). The electronic and vibrational states are assumed to relax with a phenomenological rate which exceeds the tunneling relaxation rates, taken for simplicity to be proportional to the energy change in the transition: . The overall tunneling rate merely sets the scale of the current and is irrelevant to the relative magnitude of the different excitations which is of interest here.
To experimentally identify the electronic states to which the vibrational excitations belong, we have investigated how the differential conductance measured along the line (not shown) connecting the blue markers in Figure 2d evolves with a magnetic field applied perpendicular to the CNT. The dominating features in Figure 3a are the vibrational sidebands of the lowest of the triplet excitations which in this presentation of the data appear as vertical lines. Strikingly, the ground state singlet evolving with a slope has no vibrational sidebands as demonstrated by fits of the difference of the peak position in Figure 3b. This can not be explained by an Anderson-Holstein model where all electronic states with the same charge couple equally to the vibration, see \textcolorblackSupporting Information for explicit attempts.
Instead, in our modeling we must account for state-dependent Franck-Condon shifts resulting in the vibrational potentials plotted in Figure 4. To arrive at this, we start from a model accounting for the observed set of accessible 28 many-body transport states, which is restricted by Coulomb blockade and bias voltage of a few mV to those shown in Figure 1e with electron numbers and and a single electronic - shell:
Here is mean level position controlled by , is the subband or valley-mixing term and is the exchange coupling between the spins in the two orbitals with spin-operators [ are spin indices, is the usual vector of Pauli-matrices, and creates a spin- electron in orbital ]. To obtain a result as plotted in Figures 2f - 2g we first introduced a Holstein coupling by allowing the level position to depend on , the dimensionless vibration coordinate normalized to the zero-point motion: we thus formally replace . This results in the commonly assumed uniform vibration coupling with strength to all electronic states with the same charge , which is not observed here. The required state dependent electron-vibration coupling is obtained by additionally accounting for a dependence of the other parameters on the vibration coordinate, i.e., we formally replace , where is a dimensionless one-electron valley-vibration coupling, and , where is a dimensionless coupling of the vibration to the two-electron exchange. Here many-body physics comes in: when going from the singlet ground state to the triplet , the Pauli principle forces the two electrons into different orbitals which can couple differently to the vibrational mode (difference quantified by ). However, the coupling is important as well: when allowing only for , the effective electronic excitation spectrum for fixed charge (relative to which the vibration excitations are “counted”) becomes dependent on the vibrational couplings (polaronic renormalization). That experimentally no significant shift of the electronic excitations is found when turning “on” the couplings to the vibration requires the couplings and to be comparable in magnitude but opposite in sign. This results in an enhanced coupling of the triplet over while the polaronic shifts that they induce cancel out, keeping the effective electronic excitations fixed. This thus leaves one free parameter, their magnitude, which controls the degree of state-specific coupling, which we adjust to the experiment. Together this suffices to obtain results such as Figures 2f - 2g that reproduce the main zero-field observations of Figures 2c - 2d. When the vibration couplings are “off’ in Figure 2f we estimate from Figure 2c the parameter values , , (similar to those in Figure 1d) and use , . When the vibration couplings are “on” in Figure 2g we use the same values for and but nonzero vibration couplings , , and frequency and we adjusted the asymmetries , . Despite the fact that there are several parameters, the experiment imposes strong restrictions, in particular, regarding the choice of vibrational couplings, excluding a simple Holstein mechanism (), see \textcolorblackSupporting Information. We arrive at the three electron-vibration couplings by imposing three experimental constraints after expressing the effective couplings of the electronic states in terms of , , and : (i) the observed - splitting and (ii) - splitting (commensurate with ) should match energy expressions that depend on the vibrational couplings (polaron shift) and (iii) the vibrational-coupling of is adjusted to numerically reproduce the observed number of triplet vibrational sidebands. We note that in Figures 2d - 2e, the higher vibrational sidebands become more intense at high bias. As expected, this is not captured by our model since this may involve excitations beyond the lowest two electronic orbitals and energy-dependence of the tunnel barrier, neither of which we include here. We have focused instead on the nontrivial interplay of vibrational and spin-excitations for and in the lowest sidebands.
The resulting physical mechanism is illustrated in Figure 4: when starting out from state and adding a second electron to the lowest orbital the lowest singlet state experiences only a small horizontal shift of the vibrational potential minimum (both electrons in orbital have their coupling weakened by and there is no spin and therefore no exchange modification of the coupling by ). However, when adding the electron to the excited orbital, the coupling is not only enhanced by , but also by a negative when a spinfull triplet is formed. This results in a large Franck-Condon shift of the potential minimum of in Figure 4. The above horizontal shifts of the potential minima translate into suppressed vibrational sidebands for the singlet and a pronounced series of sidebands for the triplet , respectively (Franck-Condon effect). The presence of the further electronic states and their quantized vibrational states in Figure 4, all of which are included in our transport calculations, do not alter the above simple picture: Whereas the excited singlet does not couple to transport 28, the role of cannot be ascertained at zero magnetic field because it is commensurate (within the line broadening) with one of the vibrational sidebands of .
The field evolution in Figure 3c, calculated by adding a Zeeman term to equation (1), reproduces the main observation of Figure 3a, namely, that the triplet maintains its vibrational sidebands (vertical) but the ground singlet (sloped) does not. However, to obtain this agreement with the measurements we are forced to further extend the above model. First, both the excited singlet () as well as the Zeeman split-off states of the triplet () do not appear in the measurements. This we attribute to the fact that the source and drain leads of the quantum dot are not formed by metallic contacts but by small pieces of suspended CNT. Zeeman splitting of discrete states in these CNT contacts may lead to spin-filtering which turns on with the magnetic field, developing full strength at a few Tesla where . We phenomenologically account for this by a spin-dependence in tunneling to / from the electrodes which depends on : . Second, when only including this spin-filtering in the model, it suppresses the singlet groundstate (without vibrational bands) which we do experimentally observe as excitation A in Figures 3a - 3b. However, when even a small spin-orbit (SO) coupling is included, the singlet reappears (borrowing intensity from the triplet , cf. also Figure 4), but, importantly, without reinstating the unobserved and the Zeeman split-off states of and their vibrational sidebands. This produces the observed intensity pattern, which is impossible to achieve with simple commonly used models, see \textcolorblackSupporting Information. Here, the spin-orbit coupling is included by adding to equation (1) a term with which allows both the spin and orbital index to be flipped in the schematic Figure 1d, thereby coupling in particular to , lending it intensity. Figures 2f - 2g and 3c are based on the inclusion of all these effects. However, we emphasize, that in the latter figure spin-filtering and spin-orbit coupling are needed exclusively to explain the missing Zeeman lines, but do not lead to a qualitative change of the state-dependent coupling at in Figure 2g, which is our main finding. The \textcolorblackSupporting Information explores the influence of the various parameters, confirming the necessity of including them. The key advantage of our tuneable setup is that we are able to first identify excitation A and B as relating to electronic singlet and triplet , respectively, and subsequently allowing us to study the vibrational sidebands C-E.
In conclusion, we have demonstrated switchable coupling of a quantized vibration of a carbon nanotube to its quantized electronic states. Using this advance we explored the two-electron regime – including the magnetic field dependence – and found indications of state-dependent vibrational transport sidebands not described by standard models. We showed that the interplay of intrinsic effects on the carbon nanotube (Coulomb blockade, valley-index, spin-exchange) and experimental details (junction, orbital, and spin asymmetries) can explain the observations. This, however, includes vibrational couplings that involve internal spin- and valley-degrees of freedom, bringing spin- and valley-tronics physics within range of NEMS.
Notes: The authors declare no competing financial interests.
We acknowledge F. Cavaliere for stimulating discussions, and S. Trellenkamp, J. Dauber for support with sample fabrication. We acknowledge support from the Helmholtz Nanoelectronic Facility (HNF) and financial support by the JARA Seed Fund and the DFG under Contract No. SPP-1243 and FOR912.
Fabrication and experimental characterization of the quantum dot, the electrostatic control of the coupling to vibrational modes and theoretical analysis of the electronic and vibrational quantum states of the model and the transport calculations using master equations.
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See pages 1-21 of supplementary-material.pdf