Switchable Coupling of Vibrations to Two-Electron Carbon-Nanotube Quantum Dot States
P. Weber, H. L. Calvo, J. Bohle, K. Goß, C. Meyer, M. R. Wegewijs, C. Stampfer
SSwitchable coupling of vibrations to two-electroncarbon-nanotube quantum dot states
P. Weber, † , ‡ , ⊥ H. L. Calvo, ¶ , ‡ , ⊥ J. Bohle, ¶ , ‡ K. Goß, § , ‡ , @ C. Meyer, § , ‡ M. R.Wegewijs, § , ¶ , ‡ and C. Stampfer ∗ , † , § , ‡ nd Institute of Physics, RWTH Aachen University, 52056 Aachen, Germany,JARA – Fundamentals of Future Information Technology, Institute for Theory of StatisticalPhysics, RWTH Aachen University, 52074 Aachen, Germany, and Peter Grünberg Institute,Forschungszentrum Jülich, 52425 Jülich, Germany
E-mail: [email protected]
Abstract
We report transport measurements on a quantum dot in a partly suspended carbon nan-otube. Electrostatic tuning allows us to modify and even switch “on” and “off” the couplingto the quantized stretching vibration across several charge states. The magnetic-field depen-dence indicates that only the two-electron spin-triplet excited state couples to the mechanicalmotion, indicating mechanical coupling to both the valley degree of freedom and the exchangeinteraction, in contrast to standard models. ∗ To whom correspondence should be addressed † nd Institute of Physics, RWTH Aachen University, 52056 Aachen, Germany ‡ JARA – Fundamentals of Future Information Technology ¶ Institute for Theory of Statistical Physics, RWTH Aachen University, 52074 Aachen, Germany § Peter Grünberg Institute, Forschungszentrum Jülich, 52425 Jülich, Germany (cid:107)
ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels, Barcelona, Spain ⊥ Equal contribution
Instituto de Física Enrique Gaviola (IFEG-CONICET) and FaMAF, Universidad Nacional de Córdoba, CiudadUniversitaria, 5000 Córdoba, Argentina @ Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, Germany a r X i v : . [ c ond - m a t . m e s - h a ll ] J un arbon 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 spectro-scopic techniques and couple strongly to the excitations of different mechanical modes. The grow-ing interest in NEMS is fueled by the desire to accurately sense small masses and forces, ad-dress quantum-limited mechanical motion, and integrate such functionality into complex hybriddevices, leading to new applications. The central question is the strength of the coupling of elec-tronic states to the vibrational modes. Whereas molecular junctions display such modes also inelectrically gated transport measurements, carbon-nanotube (CNT) quantum dots allow for amuch more viable fabrication, higher mechanical Q-factors, and better tuneability as NEMS.
Also, the coupling to the bending mode can be combined with the spin-orbit (SO) interac-tion by making use of the recently demonstrated curvature-induced SO-coupling in CNTs. Whereas the frequency of the vibrational modes has been demonstrated to be tuneable, an-other desirable feature is the ability to switch “on” and “off” the electron-vibration coupling in thesame device, e.g., in envisioned quantum-information processing schemes.
This is also helpfulfor fundamental studies of systems in which mechanical motion is combined with other degrees offreedom, e.g, the spin and the valley. Recently, switchable coupling to a classical flexural mode of a CNT has been demonstrated. In this letter, we present a CNT quantum dot NEMS with a coupling of the electronic states to alongitudinal stretching vibration of about 200 GHz that can be turned “on” and “off”. We illustratethe advantage of this by transport measurements in the two-electron quantum-dot regime and findthat the well-known Anderson-Holstein scenario breaks down in an unexpected way: Differentspin states exhibit different coupling strengths to the vibrational mode.In Figure 1a we show a schematic of a typical suspended CNT quantum-dot device whosescanning electron microscope image is shown in Figure 1b. The CNT is electrically and mechan-ically connected to both source (s) and drain (d) contacts where the central electrode acts as asuspended, doubly clamped top gate (tg). The quantum dot is formed in the small band gap CNTby the electrostatic potentials of the top and back gate (bg), see Figures 1a and 2a, allowing for2 -60 -40 -20 0 20 40 60 80-8-4048 -15-10-50510 (c)(f)
Source DrainTop gate
N=1 C u rr en t ( p A ) DD +_ ST S D D- J/ J N=1 N=2 V ( m V ) s d V (mV) tg -4 -2 0 2 4 6V (mV) tg B ( T ) d I/ d V ( e / h ) s d2 A AB C D F B C D FE -5 0 5V (mV) tg V ( m V ) s d E Source DrainTop gateBack gate F CNT (b)(a)(d) (e) -8 Figure
1: Carbon-nanotube quantum dot characterization. (a) Schematic illustration of the cross-section of a partly suspended CNT connected to source and drain electrodes (5nm Cr, 50 nmAu). While the back-gate shifts the entire potential of the whole structure, the top-gate bridgeoverlaps with a 200 nm part of the CNT by an oxidized Cr layer, see Supporting Information.(b) Scanning electron-microscopy image of a partly suspended CNT sample. The CNT is justvisible and indicated by red arrows. (c) Source-drain current through the quantum dot at zeromagnetic field as function of the bias ( V sd ) and top-gate voltage ( V tg ), adjusting the back-gatevoltage V bg simultaneously to keep the average chemical potential in the leads constant: V bg = .
35 V − . × V tg . (d) dI / dV sd at zero magnetic field centered around the 1 ↔ V bg = .
25 V in the hole regime. We count the number ofelectrons relative to the last filled conduction band shell of the CNT as usual. The diagonal dashedlines marked A-F correspond to transitions between the N = D − ↔ S − ), B ( D − ↔ T ) and C ( D − ↔ S ) which are the most relevant ones forthe present discussion, see also Supporting Information. For panel (d) we extract ∆ = . J = . V sd = and enables a first characterization of the electronicproperties by the Coulomb and confinement energies. Importantly, the resulting estimates showthat the quantum dot formed in the CNT is comparable to or even larger in size than the top-gate,see Supporting Information.The key advantage of our device, in contrast to previous ones, is that we can first obtain de-tailed information about the electronic spectrum by measuring the differential conductance in agate voltage regime without signatures of vibrational excitations. For example, in the spectrumshown in Figure 1d the low-energy excitations indicated by dashed black lines can be assigned totransitions between states with electron number N = N = D − and D + , ob-tained by filling the (anti)bonding orbitals |±(cid:105) = ( | K (cid:105) ± | K (cid:48) (cid:105) ) / √ K and K (cid:48) valleys withone electron, which are split in energy by 2 ∆ due to the valley-mixing ∆ . For N = S − and S + (latter not shown) completely filling one of these orbitals, and a singlet S and a triplet T in which two different orbitals are filled. Here the labels of the many-body states S , D , T indicate the spin multiplicities (singlet, doublet, triplet), whereas the subscripts indicatethe relevant orbital polarizations. In the transport data of Figure 1d we identify a ground singlet( S − ), an excited triplet ( T ) and another singlet ( S ), split by the exchange energy J . The measuredmagnetic field transport spectroscopy in Figure 1f confirms this assignment: the slope of the linesA and C for transitions to S − and S , respectively, differs by the Zeeman spin splitting from theslope of line B for the transition to the triplet T . We note that for these parameters the singlet S + is the highest in energy in Figure 1e. It is not shown there nor discussed further below because this4 tg (mV)-10 -6 -2 2 6 1002468 04812 (b)(c) (d) (h) E ne r g y ( m e V ) -10 -6 -2 2 6 10-202468 0 5 10 15 20-20246810 -40 -20 0 20 40 (c) (1) N=0
N=2 N=4
V (mV) tg V ( m V ) s d V ( m V ) s d V (mV) tg (f) V (mV) tg A B C (e) -16 -12 -8 -4 0-202468 V (mV) tg V ( m V ) s d A B C (g) (2)
ONOFF D E ~ 0.8 meV vib (a) E ne r g y ( m e V ) D E ~ 0.8 meV vib -10 -6 -2 2 6 100246810 04812 V ( m V ) s d V ( m V ) s d V tg (mV) BB V ( V ) bg - d I/ d V ( / h ) s d - d I/ d V ( / h ) s d - d I/ d V ( / h ) s d - d I/ d V ( / h ) s d - d I/ d V ( / h ) s d LSM x x
Figure
2: Switching the coupling to the vibration “on” and“off”. (a) Schematic illustration of thequantum-dot tuning into a region with a longitudinal stretching mode (LSM). (b) Top- and back-gate voltage stability diagram recorded for V sd = V bg = .
41 V − . × V tg ] and (2) [ V bg = .
35 V − . × V tg ] indicate different regimes of electron-vibration coupling. (c) dI / dV sd measured along line (1) in panel (b) showing no effects of vibrations. (d) Measurementof dI / dV sd along line (2) in panel (b), where significant electron-vibration coupling is observed:the arrows indicate the vibrational sidebands introduced. Electronic lines A and B from panel (c)can still be identified, but C is commensurate with a vibrational sideband of B. [Note that thesame happens in the calculations in panel (g).] The blue markers indicate the end-points of theline (not shown for clarity) along which the measurements in Figure 3 are taken. (e) Similarmeasurement as in panel (d) but for a different relation of gate voltages ( V tg = .
45 V − . × V bg )showing vibrational excitations (arrows) with different gate-voltage slope, both in magnitude andsign. (f),(g) Calculated dI / dV sd corresponding to panel (c) and (d), respectively, see text. Theoverall conductance magnitude is adjusted through the coupling Γ , taking T = . ∆ E vib = ¯ h ω = . ± . ±
24 GHz. 5tate does not influence the measured transport in the considered regime. Our calculations belowdo, however, include the state S + and confirm that it has negligible influence.By independently tuning the top- and back gate voltages we can change the electrostatic con-finement of the quantum dot and thereby effectively operate a single quantum dot system whichcan be made sensitive to the vibrating part of the CNT, as illustrated in Figure 2a. The result-ing electronic stability diagram in Figure 2b, showing nearly parallel lines, indicates that we canindependently fix the electron number in the dot while modifying its shape, dimensions and posi-tion. 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 markedas (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 away that cannot be explained by a modification of the size-quantization energy on the quantumdot: for several subsequent charge states a dense spectrum of discrete excitation peaks appears,equally spaced by ¯ h ω = . ± . ≈
65 nm as in previous studies ). Furthermore, the predominance of the excitationlines with negative slope indicates that the quantum dot couples to only one of the two suspendedparts.
In Figure 2e we demonstrate that by tuning to a different voltage regime we are ableto make the other vibrating part dominate. Our system thus displays electrostatically tuneableelectron-vibration coupling.To illustrate how the switchable coupling to a quantized vibration can be exploited, we nowfocus on measurements for the N = ↔ ABCDE tg B ( T ) A B C D E D V ( m V ) s d (a) (b)(c) d I/ d V ( e / h ) s d2 tg A B C D E
Figure
3: Magnetic field evolution of the vibrational sidebands. (a) dI / dV sd as function of top-gatevoltage V tg along the line (not shown) connecting the blue markers Figure 2d. Each dI / dV sd linehas been measured at a different magnetic field B , which has been tuned from B = dI / dV sd lines are offset vertically by 0.015 e / h for clarity. Dashed lines indicatethe singlet (sloped) and triplet (vertical) transitions. (b) Bias-voltage spacings ( ∆ V sd ) of peaks A-Ein panel (a) as function of the magnetic field including additional intermediate line traces that arenot shown there. The dashed line for the peak-spacing A corresponds to g -factor 2. The data areoffset vertically by 1.5 mV for clarity. (c) Calculated dI / dV sd evolution corresponding to panel (a).7ion) which incorporate single-electron tunneling into both orbitals of the shell (with asymmetryparameter κ ) from both electrodes (with junction asymmetry parameter γ ). The electronic andvibrational states are assumed to relax with a phenomenological rate which exceeds the tunnelingrelaxation rates, taken for simplicity to be proportional to the energy change E in the transition: Γ rel ( E ) = Γ × ( E / . ) . The overall tunneling rate Γ merely sets the scale of the current andis 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, wehave investigated how the differential conductance measured along the line (not shown) connectingthe blue markers in Figure 2d evolves with a magnetic field B applied perpendicular to the CNT.The dominating features in Figure 3a are the vibrational sidebands of the lowest of the tripletexcitations T which in this presentation of the data appear as vertical lines. Strikingly, the groundstate singlet S − evolving with a slope has no vibrational sidebands as demonstrated by fits ofthe 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 Supporting Information for explicit attempts.Instead, in our modeling we must account for state-dependent Franck-Condon shifts resultingin the vibrational potentials plotted in Figure 4. To arrive at this, we start from a model accountingfor the observed set of accessible many-body transport states, which is restricted by Coulombblockade and bias voltage of a few mV to those shown in Figure 1e with electron numbers N = N = K - K (cid:48) shell: H el = ε N + ∆ ∑ τ = ± τ ∑ σ d † τσ d τσ − J S + · S − . (1)Here ε is mean level position controlled by V tg , ∆ is the subband or valley-mixing term and J is the exchange coupling between the spins in the two orbitals τ = ± with spin-operators S τ = ∑ σ , σ (cid:48) d † τσ σσσ σ , σ (cid:48) d τσ (cid:48) [ σ , σ (cid:48) are spin indices, σσσ is the usual vector of Pauli-matrices, and d † τσ cre-ates a spin- σ electron in orbital τ ]. To obtain a result as plotted in Figures 2f - 2g we first introduced8 Holstein coupling by allowing the level position ε to depend on Q , the dimensionless vibrationcoordinate normalized to the zero-point motion: we thus formally replace ε → ε + √ h ωλ ε Q .This results in the commonly assumed uniform vibration coupling with strength λ ε to all elec-tronic states with the same charge N , which is not observed here. The required state dependentelectron-vibration coupling is obtained by additionally accounting for a dependence of the otherparameters on the vibration coordinate, i.e., we formally replace ∆ → ∆ + √ h ωλ ∆ Q , where λ ∆ is a dimensionless one-electron valley-vibration coupling, and J → J + √ h ωλ J Q , where λ J is adimensionless coupling of the vibration to the two-electron exchange. Here many-body physicscomes in: when going from the singlet S − ground state to the triplet T , the Pauli principle forcesthe two electrons into different orbitals which can couple differently to the vibrational mode (dif-ference quantified by λ ∆ ). However, the coupling λ J is important as well: when allowing only for λ ∆ , the effective electronic excitation spectrum for fixed charge N (relative to which the vibrationexcitations are “counted”) becomes dependent on the vibrational couplings (polaronic renormal-ization). That experimentally no significant shift of the electronic excitations is found when turning“on” the couplings to the vibration requires the couplings λ J and λ ∆ to be comparable in magni-tude but opposite in sign. This results in an enhanced coupling of the triplet T over S − whilethe 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-specificcoupling, which we adjust to the experiment. Together this suffices to obtain results such as Fig-ures 2f - 2g that reproduce the main zero-field observations of Figures 2c - 2d. When the vibrationcouplings are “off’ in Figure 2f we estimate from Figure 2c the parameter values ∆ = . J = . γ = . κ = − .
3. When the vibrationcouplings are “on” in Figure 2g we use the same values for J and ∆ but nonzero vibration cou-plings λ ε = . λ ∆ = . λ J = − .
22 and frequency ¯ h ω = .
85 meV and we adjusted theasymmetries γ = − . κ = .
3. Despite the fact that there are several parameters, the experimentimposes strong restrictions, in particular, regarding the choice of vibrational couplings, excludinga simple Holstein mechanism ( λ ∆ = λ J = λ ε , λ ∆ , and λ J : (i) the observed T - S − splittingand (ii) S - T splitting (commensurate with 2¯ h ω ) should match energy expressions that dependon the vibrational couplings (polaron shift) and (iii) the vibrational-coupling of T is adjusted tonumerically reproduce the observed number of triplet vibrational sidebands. We note that in Fig-ures 2d - 2e, the higher vibrational sidebands become more intense at high bias. As expected, thisis not captured by our model since this may involve excitations beyond the lowest two electronicorbitals and energy-dependence of the tunnel barrier, neither of which we include here. We havefocused instead on the nontrivial interplay of vibrational and spin-excitations for N = N = D − and adding a second electron to the lowest orbital the lowest singlet state S − experiences onlya small horizontal shift of the vibrational potential minimum (both electrons in orbital |−(cid:105) havetheir coupling weakened by λ ∆ and there is no spin and therefore no exchange modification of thecoupling by λ J ). However, when adding the electron to the excited orbital, the coupling is notonly enhanced by λ ∆ , but also by a negative λ J when a spinfull triplet T is formed. This resultsin a large Franck-Condon shift of the potential minimum of T in Figure 4. The above horizontalshifts of the potential minima translate into suppressed vibrational sidebands for the singlet S − and a pronounced series of sidebands for the triplet T , respectively (Franck-Condon effect). Thepresence of the further electronic states and their quantized vibrational states in Figure 4, all ofwhich are included in our transport calculations, do not alter the above simple picture: Whereasthe excited singlet S + does not couple to transport, the role of S cannot be ascertained at zeromagnetic field because it is commensurate (within the line broadening) with one of the vibrationalsidebands of T .The field evolution in Figure 3c, calculated by adding a Zeeman term to equation ( ?? ), repro-duces the main observation of Figure 3a, namely, that the triplet maintains its vibrational sidebands(vertical) but the ground singlet S − (sloped) does not. However, to obtain this agreement with the10 E ne r g y ( ħ ω ) Figure
4: Vibrational potential energy surfaces underlying the quantum states of the vibratingCNT quantum dot included in the transport calculations using the same parameters as in Figures 2f- 2g and 3c. The excited triplet T and ground singlet S − have significantly different couplingsto the vibration, i.e., shifts of their potential minima relative to that of the one-electron groundstate D − strongly differ. Due to the weak SO coupling several avoided crossings can be seen. Themost important anticrossing is that of the T (blue) and S − (red) potential energy surfaces. Thiscan be understood directly from the SO operator as written in the text: it "flips" both the orbital( τ ) and spin index ( σ ) of an electron. In Figure 1e this implies that for N = T to S − ). The resulting admixture of T -components (blue) to S − (red) causes the latter toremain visible in the transport in Figure 3c with increasing the magnetic field when the tunnelingbecomes spin-selective due to the CNT leads. The remaining SO anticrossings are discussed in theSupporting Information, which for our parameters have negligible impact on transport.11easurements we are forced to further extend the above model. First, both the excited singlet( S ) as well as the Zeeman split-off states of the triplet ( T ) do not appear in the measurements.This we attribute to the fact that the source and drain leads of the quantum dot are not formedby metallic contacts but by small pieces of suspended CNT. Zeeman splitting of discrete states inthese CNT contacts may lead to spin-filtering which turns on with the magnetic field, developingfull strength at a few Tesla where g µ B B ≈ k B T . We phenomenologically account for this by a spin-dependence in tunneling to / from the electrodes which depends on B : ζ ( B ) = tanh ( g µ B B / k B T ) .Second, when only including this spin-filtering in the model, it suppresses the singlet groundstate S − (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 S − reappears(borrowing intensity from the triplet T , cf. also Figure 4), but, importantly, without reinstating theunobserved S and the Zeeman split-off states of T and their vibrational sidebands. This producesthe observed intensity pattern, which is impossible to achieve with simple commonly used models,see Supporting Information. Here, the spin-orbit coupling is included by adding to equation ( ?? )a term H SO = ∆ SO ∑ σ , τ d † τσ d − τ − σ with ∆ SO = . σ and orbitalindex τ to be flipped in the schematic Figure 1d, thereby coupling in particular T to S − , lending itintensity. Figures 2f - 2g and 3c are based on the inclusion of all these effects. However, we empha-size, that in the latter figure spin-filtering and spin-orbit coupling are needed exclusively to explainthe missing Zeeman lines, but do not lead to a qualitative change of the state-dependent coupling at B = S − and triplet T , respectively, and subsequently allowing us to study the vibrational sidebandsC-E.In conclusion, we have demonstrated switchable coupling of a quantized vibration of a carbonnanotube 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 vibrational12ransport sidebands not described by standard models. We showed that the interplay of intrinsiceffects on the carbon nanotube (Coulomb blockade, valley-index, spin-exchange) and experimen-tal details (junction, orbital, and spin asymmetries) can explain the observations. This, however,includes vibrational couplings that involve internal spin- and valley-degrees of freedom, bringingspin- and valley-tronics physics within range of NEMS. Author information
Corresponding author: [email protected]: The authors declare no competing financial interests.
Acknowledgement
We acknowledge F. Cavaliere for stimulating discussions, and S. Trellenkamp, J. Dauber for sup-port 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-1243and FOR912.
Supporting Information Available
Fabrication and experimental characterization of the quantum dot, the electrostatic control of thecoupling to vibrational modes and theoretical analysis of the electronic and vibrational quantumstates of the model and the transport calculations using master equations. This material is availablefree of charge via the Internet at http://pubs.acs.org/ . Notes and References (1) Moser, J.; Güttinger, J.; Eichler, A.; Esplandiu, M. J.; Liu, D. E.; Dykman, M. I.; Bachtold, A.
Nature Nanotech. , , 493. 132) Teufel, J. D.; Donner, T.; Castellanos-Beltran, M. A.; Harlow, J. W.; Lehnert, K. W. NatureNanotech. , , 820.(3) Viennot, J. J.; Delbecq, M. R.; Dartiailh, M. C.; Cottet, A.; Kontos, T. Phys. Rev. B , ,165404.(4) Schneider, B.; Etaki, S.; van der Zant, H.; Steele, G. Sci.Rep. , , 599.(5) Park, H.; Park, J.; Lim, A. K. L.; Anderson, E. H.; Alivisatos, A. P.; McEuen, P. L. Nature , , 57.(6) Pasupathy, A. N.; Park, J.; Chang, C.; Soldatov, A. V.; Lebedkin, S.; Bialczak, R. C.;Grose, J. E.; Donev, L. A. K.; Sethna, J. P.; Ralph, D. C.; McEuen, P. L. Nano Lett. , ,203.(7) Osorio, E. A.; O’Neill, K.; Stuhr-Hansen, N.; Nielsen, O. F.; Bjørnholm, T.; van der Zant, H.S. J. Adv. Mater. , , 281.(8) Sapmaz, S.; Jarillo-Herrero, P.; Blanter, Y.; Dekker, C.; van der Zant, H. S. J. Phys. Rev. Lett. , , 026801.(9) Steele, G. A.; Hüttel, A. K.; Witkamp, B.; Poot, M.; Meerwaldt, H. B.; Kouwenhoven, L. P.;van der Zant, H. S. J. Science , , 1103.(10) Leturcq, R.; Stampfer, C.; Inderbitzin, K.; Durrer, L.; Hierold, C.; Mariani, E.;Schultz, M. G.; von Oppen, F.; Ensslin, K. Nature Phys. , , 317.(11) Pei, F.; Laird, E. A.; Steele, G. A.; Kouwenhoven, L. P. Nature Nanotechnology , , 630.(12) Benyamini, A.; Hamo, A.; Viola Kusminsikiy, S.; von Oppen, F.; Ilani, S. Nature Phys. , , 151.(13) Moser, J.; Eichler, A.; Güttinger, J.; Dykman, M. I.; Bachtold, A. Nature Nanotech. , ,1007. 1414) Ohm, C.; Stampfer, C.; Splettstoesser, J.; Wegewijs, M. R. Appl. Phys. Lett. , ,143103.(15) Pa’lyi, A.; Struck, P. R.; Rudner, M.; Flensberg, K.; Burkard, G. Phys. Rev. Lett. , ,206811.(16) Kuemmeth, F.; Ilani, S.; Ralph, D. C.; McEuen, P. L. Nature , , 448.(17) Jespersen, T.; Grove-Rasmussen, K.; Paaske, J.; Muraki, K.; Fujisawa, T.; Nygård, J.;K.Flensberg, Nature Phys. , , 348.(18) Flensberg, K.; Marcus, C. M. Phys. Rev. B , , 195418.(19) Sazonova, V.; Yaish, Y.; Ustunel, H.; Roundy, D.; Arias, T. A.; McEuen, P. L. Nature , , 284.(20) Eichler, A.; del Álamo Ruiz, M.; Plaza, J. A.; Bachtold, A. Phys. Rev. Lett. , ,025503.(21) Barnard, A. W.; Sazonova, V.; van der Zande, A. M.; McEuen, P. L. PNAS , , 19093.(22) Palomaki, T. A.; Teufel, J. D.; Simmonds, R. W.; Lehnert, K. W. Science , , 710.(23) Ganzhorn, M.; Klyatskaya, S.; Ruben, M.; Wernsdorfer, W. Nature Nanotech. , , 165.(24) Pa’lyi, A.; Burkard, G. Phys. Rev. Lett. , , 086801.(25) Liang, W.; Bockrath, M.; Park, H.-K. Phys. Rev. Lett. , , 126801.(26) Buitelaar, M. R.; Nussbaumer, T.; Iqbal, M.; Schönenberger, C. Phys. Rev. Lett. , ,156801.(27) Sapmaz, S.; Jarillo-Herrero, P.; Kong, J.; Dekker, C.; Kouwenhoven, L. P.; van der Zant, H.S. J. Phys. Rev. B , , 153402. 1528) In this state two electrons fill the excited orbital causing the amplitude for a transition fromthe N = ground orbital to be strongly suppressed. Forthis 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 tunnelprocesses, neither of which are relevant here. This point is important since due to polaronicrenormalization discussed at the end of the letter S + may actually lie below S , but even thenit cannot be observed.(29) Cavaliere, F.; Mariani, E.; Leturcq, R.; Stampfer, C.; Sassetti, M. Phys. Rev. B , ,201303.(30) Donarini, A.; Yar, A.; Grifoni, M. New. J. Phys. , , 023045.16 upplementary Information for ‘Switchable coupling of vibrationsto two-electron carbon-nanotube quantum dot states’ P. Weber,
1, 2
H. L. Calvo,
3, 2
J. Bohle,
3, 2
K. Goß,
4, 2
C.Meyer,
4, 2
M. R. Wegewijs,
3, 4, 2 and C. Stampfer
1, 4, 2 nd Institute of Physics, RWTH Aachen University, 52056 Aachen, Germany JARA – Fundamentals of Future Information Technology ∗ Institute for Theory of Statistical Physics,RWTH Aachen, 52056 Aachen, Germany Peter Gr¨unberg Institut, Forschungszentrum J¨ulich, 52425 J¨ulich, Germany
CONTENTS
I. Experimental methods 2A. Fabrication 2B. Configuration of the quantum dot 2C. Electrostatic control of the coupling to vibrational modes 5D. Temperature dependence of vibrational sidebands 6E. Measurements in a magnetic field 7II. Theoretical modeling 7A. Model and eigenstates 71. Electronic model 72. Coupling to the vibration - beyond the Anderson-Holstein model 103. Spin-orbit interaction effects 11B. Master equations - tunneling and relaxation 131. Tunnel processes 132. Intrinsic relaxation 15C. Comparison with experiment 151. Zero magnetic field 152. Magnetic field spectroscopy 173. Influence of the various parameters and the problem of Anderson-Holstein coupling 19References 20In this Supplementary Information we provide a detailed description of the experimental meth-ods and additional measurements (Sec. I) and a precise formulation of the model and calculationsreported in the main text (Sec. II). In both sections, we provide an extensive discussion of claimsand results of the main article. Within the Supporting Information references are numbered as,e.g., equation (S-1) and Figure S-1, whereas regular numbers, e.g., equation (1) and Figure 1, referto the main article.
FIG. S-1. Fabrication work flow described in the text.
I. EXPERIMENTAL METHODSA. Fabrication
Devices were fabricated in a similar fashion as in Ref. [1] as outlined in Fig. S-1. The startingpoint is a highly doped silicon wafer covered by 290 nm silicon oxide (step a). Next, following therecipe described in Ref. [2] ferritin catalyst nanoparticles are dispersed on the substrate (step b)from which carbon nanotubes are grown by means of chemical vapor deposition (step c). For thesubsequent selection and localization of carbon nanotubes marker structures are evaporated in anelectron beam (e-beam) lithography step (step d,e). In a second e-beam lithography step metallicelectrodes and gate structures are deposited in a single evaporation (5nm Cr, 50nm Au) on selectedcarbon nanotubes (step f-h). Finally, diluted hydrofluoric acid (1% for 6min) is used to etch thesilicon oxide followed by critical point drying (step i). It is crucial that the central electrode iscompletely underetched so that the chromium layer oxidizes when exposed to environmental airto form the top-gate oxide.
B. Configuration of the quantum dot
In Fig. S-2 we show the back-gate voltage ( V bg ) characteristics of the investigated carbon nan-otube (CNT) at two different temperatures. The CNT is slightly p-doped and the charge neutralitypoint is found to be around 7 V. At gate voltages close to the charge neutrality point the current I sd is strongly suppressed, even for elevated temperatures T = 50 K. This is the typical character-istics of a CNT with a small semiconducting gap separating p- and n-conducting regions. For lowtemperatures the current within the semiconducting band gap, apart from a few small resonances,is pinched off. The band gap extends over a back-gate voltage range of ∆ V bg = 1 . α bg = 0 .
035 the energy splitting between valenceand conduction band is ∆ E gap ≈
50 meV. At T = 1 . V bg . Owing to the geometry of the device (see Figures 1a - 1b of the main text)different electrostatic potentials act on different parts of the nanotube depending on the proximityto the local top gate: The electrons just below the top gate experience electrostatic screeningof the back-gate voltage because the CNT is separated only by a few nanometers of oxide fromthe top-gate electrode while a combination of top gate and back-gate voltage is acting on thesuspended parts of the CNT. Therefore, we expect a bending of the CNT band structure along itsaxis.A device schematics together with an indication of the location and size of the quantum dotwith respect to the metallic leads and the top gate is given in Fig. S-3b. As deduced below, thequantum dot is formed close to the top gate and its lateral size is on the order of the top-gate V (V) bg I ( n A ) s d T ~ 50KT ~ 1.6K
FIG. S-2. Source-drain current I sd through the CNT as a function of back gate voltage V bg at small biasvoltage V sd = 0 . Top GateSource Quantum Dot DrainBack Gate source drainquantum dotmetallic leads tunneling barriers (a)
CNT-axis E C E V E F ElectronsHoles (b)
FIG. S-3. Formation scenario of the quantum dot. (a) Electrostatic bending of the CNT bands in closevicinity to the top gate induced by top and back gate voltages. (b) The upper illustration shows aschematic of the device. The semi-suspended CNT is connected to Cr/Au leads and a central top gate. Inthe suspended region longitudinal stretching modes (LSM) of vibration are indicated by the blue spiralssurrounding the CNT, while the red region below the top gate symbolizes the quantum dot. The lowerschematics describes the tunneling through the quantum dot. Electrostatically induced tunnel barriersseparate the CNT quantum dot electrically from the CNT leads. width of 200 nm. These parameters suggest, that the leads of the QD are not the metallic contactsbut the CNT itself and therefore the size of the QD can be tuned exclusively by the applied gatevoltages.In order to determine the electronic size of the quantum dot we analyse the addition energiesneeded to add the first and the second electron on the investigated electronic shell of the QD,respectively. The addition energy is defined as the change in electrochemical potential ∆ µ N whenadding the ( N + 1) charge to a quantum dot containing already N charges . It can be relatedto the charging energy E C , the quantum energy-level separation β , the valley degeneracy splitting2∆ and the exchange interaction J via∆ µ = E C + β −
2∆ (S-1)∆ µ = E C . (S-2)Additionally, we can infer J and ∆ from the energies of the first two excited states relative tothe two-electron ground state, E T − E S − = 2∆ − J/ E S − E S − = 2∆ + 3 J/
4, respectively[cf. equation (S-14), (S-45), and (S-46) below)]. From Fig. 2c of the main article [reproduced inFig. S-6b below] we extract ∆ µ = 9 . ± . µ = 3 . ± . µ = 1 . ± . d I/ d V ( e / h ) V (mV) tg -100 -50 0 500.0250.05 (a) (b) dI/dV(e /h) V (mV) tg V ( V ) bg FIG. S-4. (a) Charge stability map in the V tg - V bg -plane. The differential conductance is plotted forzero d.c. bias voltage V sd applied between source and drain. Resonances with a slope marked by theblue dashed lines are attributed to resonances off the central quantum dot. The Coulomb diamonds inFig. 1c of the main article are measured along the white continuous line and the Coulomb peaks in (b)are measured along the white dashed line ( V bg = 4 .
26 V − . × V tg ). µ = 2 . ± . β = 6 . , J = 1 . . . (S-3)Due to large level spacing β relative to J and ∆, the analysis of the transport spectrum involvingthe first few vibrational excitations – at the focus of the main article – only requires consideringelectron fillings of the first orbital shell (i.e., of the orbitals labeled τ = ± in the main article). Amore detailed fitting of the energy positions consistent with the above and including the vibrationsis given in Sec. II C 1. The expression for the quantization-induced level spacing β = hv F / L , with v F = 8 . × m / s allows to determine the quantum dot length L = 245 ±
25 nm, correspondingapproximately to the length below the top gate.For the determination of the position of the quantum dot we have evaluated the respective leverarms of both top gate and back-gate electrodes as well as the relative lever arms of the source anddrain leads. The analysis of the slopes of the edges of the Coulomb diamonds in Fig. 1d of themain article gives the following lever arms: α tg = 0 . ± .
04 and α s − α d = 0 . α rel = α tg /α bg = 15 . ± .
9, resulting in α bg = 0 . ± . V (V) tg dI/dV (e /h) sd V ( V ) bg FIG. S-5. Charge stability diagram in both gate voltages for finite d.c. bias voltage of V sd = 1 mV. Thedifferential conductance plot shows splitting of the lines belonging to the central quantum dot. The significantly larger lever arm to the top gate and the almost negligible difference betweenthe lever arms of source and drain confirm our assumption that the quantum dot is located inthe middle of the nanotube in close vicinity of the top gate and is as a result strongly screenedfrom the back gate. Additional lines with smaller slope in the charge-stability diagram indicatethe existence of additional quantum dots further away from the top gate.In Fig. S-4a we show a charge stability map as a function of both the top gate and the back-gatevoltage. Strikingly, the Coulomb-peak excitation lines with the steepest slope, which correspondto the central QD, appear in groups of four, reflecting the twofold orbital-degenerate bandstructureof high-quality CNTs. Further confirmation of the fourfold periodicity is provided in Fig. S-4bwhere we plot a Coulomb-peak measurement as function of V tg . These characteristics justify thetreatment of the CNT QD as an effective few-electron system.The CNT-QD is thus connected to the metallic electrodes by short CNT leads. To prove thatthe CNT-leads do not electrically influence our measurements on the central QD significantly, wehave measured the same diagram as in Fig. S-4a, but with a finite bias voltage V SD = 1 mV.Fig. S-5 shows that only the lines corresponding to the central QD split up into two lines, whilethe lines corresponding to the lateral parts are not affected. This broadening into ground andexcited states proves that the voltage drop occurs only at the tunnel barriers (marked in Fig. S-3b) between the CNT-QD and the CNT leads. The tunneling rates are estimated to be on theorder of Γ = 2 π ×
10 GHz, which corresponds to roughly 300-400 mK, well below the experimentaltemperature T = 1 . C. Electrostatic control of the coupling to vibrational modes
In this section we provide further experimental data on the tuning of the coupling of the QD tovibrational modes. In particular, in Fig. S-6 we provide an additional QD excitation spectrum tofurther illustrate our ability to continuously tune the electron-vibration coupling. Figures S-6a,S-6b and S-6d correspond to Figures 2b, 2c and 2d of the main article and show a pure electronicexcitation spectrum and a vibrational excitation spectrum, respectively. In addition, we show inFig. S-6c data for an intermediate state of the QD.In order to understand the switchable coupling of the QD to vibrational modes we compare thequantum dot size in the two extreme regimes. Unfortunately, when the vibrations are switched onwe can not assign the energy of the second excited state S in the two electron regime (becauseit is degenerate with a vibrational sideband within the experimental line width). Nevertheless,we can give bounds for the dot size. With ∆ µ = 9 . µ = 5 . µ = 0 . β = 4 . J/
4, which gives a larger quantum dotsize in the vibrational regime for 0 < J < . (b)(c)(d) 00.040.080.120.160.2-40 -20 0 20 404.24.34.44.5 V ( m V ) bg V (mV) tg V ( m V ) s d V ( m V ) s d V ( m V ) s d V (mV) tg V (mV) tg V (mV) tg -4048 -4048 -22610 dI/dV (e /h) dI/dV (e /h) dI/dV (e /h) dI/dV (e /h) (a) (b)(c) (d) Δμ Δμ Δμ Δμ μ μ μ FIG. S-6. (a) Differential conductance in the V tg - V bg -plane for small d.c. bias voltage. (b)-(d) Coulombdiamond charge stability diagrams in the V sd - V tg -plane for the effective charge states 0 − is fullfilled. If we choose, for example, J = 1 . D. Temperature dependence of vibrational sidebands
We also investigated temperature dependence of ground and excited states in the temperatureregime where Coulomb blockade peaks still could be resolved along the lines of Ref. [1]. In Fig. S-7a we show Coulomb diamonds measured in the very same region of gate voltage (same electronicstate) for different temperatures, 1.6 K, 2.5 K, 3.5 K and 5 K. As the temperature increases,the conductance peaks related to ground and excited states wash out, i.e., they broaden andthe conductance maximum decreases. In Fig. S-7b we show the temperature dependence of themaximum conductance G max for the electronic triplet excited state T (blue circles) and its firstvibrational replica, i.e. the emission side-band (red triangles). In the four panels in Fig. S-7athese excitations are marked by blue and red arrows, respectively. For the tunneling through theelectronic triplet T excited state and its vibrational emission peak, we observe a G max ∼ /k B T dependence (blue and red curves) that one expects for the derivative of the Fermi distribution inthe quantum Coulomb blockade regime .Unlike in Figs. 2c-2d in Ref. [1] we do not observe any extra vibrational- absorption conductancepeaks appearing with increasing temperature inside the Coulomb blockaded region, most likelybecause of weaker electron-vibron coupling strength / smaller tunneling rates. This prohibitsfurther investigation of the temperature dependence. V ( m V ) s d V (mV) tg V ( m V ) s d V (mV) tg V ( m V ) s d V (mV) tg V ( m V ) s d V (mV) tg - d I/ d V ( / h ) s d - d I/ d V ( / h ) s d - d I/ d V ( / h ) s d - d I/ d V ( / h ) s d G m a x ( e / h ) - (a) (b) FIG. S-7. (a) Coulomb diamond charge stability diagrams in the V sd - V tg -plane around the N = 1-2transition for T = 1 . T =2.5 K, 3.5 K and 5 K. (b) Conductance peak value as function of temperature for electronic triplet state(blue circles) and the first vibrational replica (red triangles). The blue and red curves are guides to theeye ∝ /k B T . E. Measurements in a magnetic field
The motivation for plotting the data as done in Fig. 3a in the main article is explained inSec. II C 2.
II. THEORETICAL MODELING
In this section we describe in detail the employed model, the method used for calculatingthe differential conductance of the CNT quantum dot, and the resulting understanding of thetransport measurements. The full Hamiltonian of the system under consideration reads as H = H qd + H tun + H res , where H qd describes the quantum dot states, including both their electronicand vibrational degrees of freedom, and H tun is the tunnel coupling Hamiltonian between the dotand the reservoirs described by H res . A. Model and eigenstates
1. Electronic model a. Carbon-nanotube quantum dot
We first set up an electronic model that accounts for themany-electron states observed in the experiment when the coupling to the CNT vibrations isswitched “off”. In Sec. II A 2 we then include the vibrations and their coupling to obtain our fullCNT quantum-dot Hamiltonian H qd . As mentioned at the end of Sec. I B we can restrict ourattention to a single orbital shell. We account for a significant valley-mixing ∆ > X σ ( d † Kσ d K σ + H.c.) , (S-4)where d † Kσ ( d K σ ) is a creation (annihilation) operator for an electron in valley K ( K ) with spin σ = {↑ , ↓} = { , − } whose quantization axis is chosen along the direction of the applied magneticfield. Due to the large splitting ∆, it is reasonable to use a basis of bonding ( τ = −
1) andantibonding ( τ = +1) combinations (BA) of the K and K valleys, i.e. d ( † ) τσ = ( d ( † ) Kσ + τ d ( † ) K σ ) / √ H el = εN − gµ B BS z + ∆ X τ τ N τ − J S + · S − + E C N ( N − . (S-5)In the first term, ε = − α tg V tg is the quantum-dot energy level, electrostatically controlled by thetop-gate, and N = P τσ d † τσ d τσ is the occupation number operator in the dot. Note that in theexperiment back- and top-gate are tuned simultaneously in a linearly dependent way and that V tg is taken as the independent parameter. Here N = 1 − N = 0 corresponds to the ‘empty dot’ state | i in which all lowershells are filled. The further number operators N τ = P σ d † τσ d τσ and N σ = P τ d † τσ d τσ count thenumber of electrons in orbital τ and with spin σ , respectively. The second term in equation (S-5)describes the spin Zeeman splitting due to magnetic field B applied perpendicular to the CNT axisand S z = P σ ( σ/ N σ is the operator of the spin component along the field B . The third term inthe electronic Hamiltonian (S-5) is a spin-exchange term with energy J . The spin operator S τ forelectrons in orbital τ = ± in the BA-basis is S τ = P σσ σ σσ d † τσ d τσ /
2, where σ is the vector ofPauli matrices. Finally, the last term is the charging energy E C accounting for Coulomb repulsion.The inter- and intra-valley electronic repulsion energies are assumed to be the same as is typicallyobserved in CNTs quantum-dot samples . Since we will restrict ourselves to the analysis of the1 ↔ E C and the level energy β [cf. equation (S-1)] canbe absorbed by a redefinition of the origin of top-gate voltage and therefore these do not need tobe included henceforth. Thus setting E C = 0 we obtain the Hamiltonian (1) of the main article,where we note that for brevity the Zeeman term was not written but only mentioned in the mainarticle.A convenient many-particle basis for N = 1 and N = 2 charge sectors is constructed by creatingspin-multiplets using these orbitals. In the following we will denote by | x, y i such electron fillings,where the first (second) slot x ( y ) corresponds to the ground orbital − (excited orbital +). Byfilling the empty dot state | i with one electron, we obtain two spin-doublets denoted by D ± inthe main article, with states | D σ − i = d †− σ | i = | σ, •i , (S-6) | D σ + i = d † + σ | i = |• , σ i . (S-7)where • denotes an empty ± orbital, respectively. The next step is to add a further electron tothese states. We therefore obtain two ‘localized’ singlet-fillings of the same orbital | S − i = d †−↓ d †−↑ | i = | ↓↑ , •i , (S-8) | S + i = d † + ↓ d † + ↑ | i = |• , ↓↑i . (S-9)By filling the empty dot with two electrons in different orbitals and diagonalizing the electronicHamiltonian of equation (S-5), we obtain a ‘delocalized’ singlet | S i = d † + ↓ d †−↑ − d † + ↑ d †−↓ √ | i = | ↑ , ↓i − | ↓ , ↑i√ , (S-10)and a triplet T of states | T m i with spin-projections m = 0 , ± | T i = d † + ↓ d †−↑ + d † + ↑ d †−↓ √ | i = | ↑ , ↓i + | ↓ , ↑i√ , (S-11) | T +0 i = d † + ↑ d †−↑ | i = | ↑ , ↑i , (S-12) | T − i = d † + ↓ d †−↓ | i = | ↓ , ↓i . (S-13)The labels of the many-body states S , D , T indicate the spin multiplicities (singlet, doublet,triplet), whereas the subscripts indicate the relevant orbital polarizations (signature of differenceof number of τ = ± electrons, respectively), which is important here.The above defined states are exact many-particle eigenstates of the model (S-5) whose corre-sponding eigenenergies are N = 1 : E D στ = τ ∆ − σgµ B B/ N = 2 : E S = 3 J/ E T m = − J/ − mgµ B BE S τ = 2 τ ∆ (S-14)For the moment we consider zero magnetic field B = 0 and can simplify the discussion by omittingthe spin projection indices σ in the doublets and m in the triplet. In Sec. II C we will return tothis notation when discussing the effect of a nonzero magnetic field. Clearly, in the N = 1 chargesector the ground state is the bonding doublet D − while for N = 2 the ground state is givenby the localized singlet S − in the expected regime of weak exchange energy relative to the valleymixing, J < b. Transport model
The source (s) and drain (d) leads are described as macroscopic reservoirsof noninteracting electrons through the Hamiltonian H res = X rkσ ( (cid:15) rk + µ r ) c † rkσ c rkσ , (S-15)where c † rkσ ( c rkσ ) creates (annihilates) an electron in lead r = { s , d } with spin σ = {↑ , ↓} and stateindex k . The eigenenergies of the leads are uniformly shifted by the bias voltage V sd such thatthe electro-chemical potentials read µ r = ± V sd / r = { s , d } , respectively. These reservoirs areassumed to be independently at equilibrium, characterized by a temperature T . We note that thematter of interactions in the CNT leads is a subtle one, but in the experiment we see no particulareffect indicating their importance. Rather, the fact that we have quantized (yet broadened) statesin the CNT leads seems to be important, resulting in an effective spin-dependence of the tunnelingrates, see below.The coupling between the dot and the leads is determined by the tunnel Hamiltonian H tun = X rkτσ t rτσ d † τσ c rkσ + H.c. , (S-16)with the tunnel amplitudes t rτσ assumed to be junction- ( r ), orbital- ( τ ) and spin-dependent ( σ ).Since the tunnel rates required below have the form 2 πρ ( t rτσ ) (by Fermi’s Golden Rule) thesedependencies are modeled using three asymmetry parameters κ , γ and ζ , respectively: t rτσ = √ (1 + τ κ )(1 + σζ ) t r , t r = 1 + rγ p γ s (Γ / πρ , (S-17)where r = ± corresponds to r = s/d . Here t r characterizes the tunneling through each junction r through the overall rates Γ r = 2 πρ ( t r ) and ρ is the density of states in the respective electrode. We let
Γ = Γ s + Γ d characterize the overall scale of the rates which merely sets the magnitudeof the current and is irrelevant to the relative strengths of the different excitations which are ofinterest here. The latter are controlled by the quantum dot electron-vibrational states and theparameters γ , κ , and ζ : • In equation (S-17) we include the usual junction asymmetry γ = ( t s − t d ) / ( t s + t d ) for thetunnel coupling to the source and drain leads through the last factor. Keeping Γ = Γ s + Γ d fixed, a little algebra shows that t r is given by the second equation in equation (S-17). Thisasymmetry is relevant for modeling the measured differential conductance, which manifestssome asymmetry between the intensities of positively and negatively sloped lines. We notethat any r -dependence in the density of states in the leads that we ignored above can beabsorbed into γ . • The first factor in equation (S-17) captures an orbital asymmetry κ = ( t r + σ − t r − σ ) / ( t r + σ + t r − σ )which is the same for all σ and r . This derives from the linear combinations of the valley-dependent tunnel amplitudes, i.e. t rKσ = ( t r + σ + t r − σ ) / √ t rK σ = ( t r + σ − t r − σ ) / √
2. Wenotice that for strictly symmetric couplings of the two orbitals, κ = t rK σ /t rKσ = 1, the tunnelHamiltonian is symmetric with respect to the interchange of the K and K valleys and cannot0induce transitions from the anti-symmetric N = 1 ground state D − into the symmetric state N = 2 ground state S − . This would cause a strong suppression of low-bias transport up toa voltage where the lowest excitation for either N = 1 or N = 2 becomes accessible. This isnot observed in the experiment and indicates that a definite orbital asymmetry is present,i.e., κ = ±
1. (A similar problem arises for κ = − • Finally, we introduced in addition a possible spin-dependence in the tunnel amplitudes through the parameter ζ = ( t rτ ↑ − t rτ ↓ ) / ( t rτ ↑ + t rτ ↓ ) which is the same for all τ and r . Belowwe turn on ζ only in a nonzero applied magnetic field B [cf. equation (S-59)]. As mentionedin the main article, this models a relevant aspect of the experiment, related to the fact thatwe have CNT leads, which we will discuss in Sec. II C when calculating the magnetic fieldevolution of the vibrational sideband lines observed in Fig. 3a of the main article.From the above Hamiltonian we calculate the tunnel matrix elements (TMEs) T rσηa ← b = X τ t rτσ h a | d ητσ | b i , (S-18)with the shorthand d ητσ = d † τσ , d τσ for η = ±
1. In Sec. II B we derive the explicit TMEs for theconsidered model after having considered the effect of the vibration, to which we turn now.
2. Coupling to the vibration - beyond the Anderson-Holstein model
The electronic model accounting for the many-electron states observed in the experiment is nowextended to deal with the case where the coupling to the vibrational stretching mode of the nan-otube is switched “on”. As discussed in the main article, the differential conductance of Fig. 2dtogether with its magnetic field evolution in Fig. 3a shows several vibrational sidebands associatedto the triplet state, but none for the ground singlet. This strong state-dependence in the couplingto the vibrational mode forces us to consider three different types of electron-vibration couplingswhich arise from the assumption of a linear dependence on the (dimensionless) mechanical dis-placement Q of the nanotube in the electronic parameters R = { ε, ∆ , J } of equation (S-5). In allcases, we assume R ( Q ) = R + λ R ~ ω √ Q . We therefore have, in addition to the standard Holsteincoupling λ ε to the number of particles N , a coupling λ ∆ which depends on the valley-mixing anda vibration-exchange coupling λ J . By plugging these into equation (S-5) we arrive to the fullHamiltonian of the CNT quantum dot H qd = H el + ~ ω h P + ( Q + √ i − ~ ω Λ , (S-19)which can be seen as a shifted quantum harmonic oscillator. The above electron-vibration cou-plings enter through the following operatorΛ = λ ε N + λ ∆ X τσ τ N τ − λ J S + · S − , (S-20)which shifts the harmonic potentials associated to each electronic state (horizontal shift) and italso introduces a polaronic shift in the energy (vertical shift). a. Adiabatic potentials To aid the intuition we consider the adiabatic potentials for this prob-lem, obtained by treating Q as a classical variable. These potentials were plotted in the main articlein Fig. 4. The adiabatic potentials associated to the electronic states | e i , where e = D ± , S ± , S , T ,can be characterized by polaron shift of its potential minimum Λ e = h e | Λ | e i , i.e.Λ D τ = λ ε + τ λ ∆ , Λ S τ = 2 λ ε + 2 τ λ ∆ , (S-21)Λ S = 2 λ ε + 34 λ J , Λ T = 2 λ ε − λ J . (S-22)Roughly speaking, the magnitude of these shifts determine whether none, several or many vi-brational sidebands will appear (see below). These expressions reveal that the polaronic shiftscorresponding to the various electronic states can indeed be different once one abandons the1simplifying assumption of the Anderson-Holstein model, that the electronic excitations are notaffected by a distortion (the vibration coordinate Q ):(i) Both the valley-vibration coupling λ ∆ and the exchange-vibration coupling λ J distinguish mul-tiplets with zero ( S , T ) and maximal orbital polarization ( D τ , S τ ).(ii) Also, the exchange-vibration coupling allows for a fine tuning of the coupling to the vibrationfor the S and T states. As we will show in Sec. II C, the horizontal shifts of the ground singlet S − and the triplet T can be tuned so they are qualitatively different, giving rise to contrastingFranck-Condon factors.Although the intuition is useful, in our calculations we account for the exact many-body eigen-states of the quantum-dot Hamiltonian in equation (S-19). These are obtained as a tensor productof the electronic states | e i and the quantum vibrational states | ν i . These electron-vibration eigen-states | e, ν i thus yield the following eigenenergies for e = D ± , S ± , S , T : E e ,ν = h e, ν | H qd | e, ν i = E e − ~ ω Λ + ~ ω (cid:18) ν + 12 (cid:19) . (S-23)The horizontal shift in equation (S-19), although not present in the above energies, enters as theequilibrium position for the vibrational states, and thereby it plays a crucial role in the transitionamplitudes when calculating the matrix elements of the tunnel Hamiltonian, as we will show inSec. II B.Finally, we note that in the limit λ ∆ = λ J = 0 the model reduces the standard Anderson-Holstein model. Here, the polaronic energy shifts can be absorbed into the gate voltage since theelectron-vibration coupling for transitions involving states differing by one electron is, in all cases,the same. b. State-dependent Franck-Condon shifts - the role of λ ∆ and λ J The tunnel amplitudes andthus the intensities of the lines in the differential conductance depend on the Franck-Condonamplitudes that can be calculated from the relative shifts between the harmonic potentials, i.e.Λ x,y = Λ x − Λ y where x, y are labels of the many-particle states differing by one electron. Ofparticular relevance to the experiment are the two relative oscillator shiftsΛ S − ,D − = λ ε − λ ∆ , (S-24)Λ T ,D − = λ ε + λ ∆ − λ J , (S-25)for transitions from the ground doublet D − to the ground singlet S − and to the triplet T ,respectively. Since λ ∆ is present in both transitions but with different sign, it can induce differentcoupling strengths for the singlet and the triplet. Additionally, λ J can be used to tune the couplingstrength of the triplet independently of the singlet. According to equations (S-57)-(S-23), thecouplings λ ∆ and λ J also induce vertical polaronic shifts (i.e., of the energy at the minimum)which are given by Λ S − = 4( λ ε − λ ∆ ) , (S-26)Λ T = (2 λ ε − λ J ) . (S-27)Here it is important to note that the triplet polaronic shift does not depend on λ ∆ , and inconsequence its coupling to the vibration can be modified through λ ∆ in equation (S-25), whereasthe polaronic shift in equation (S-27) remains unaffected. The coupling λ J can therefore be usedto tune the polaronic shift to the triplet without affecting the singlet S − state. It is thus indeedpossible with our model to describe the central observation in the experiment.
3. Spin-orbit interaction effects
In this section we consider the possible effects on the electronic properties of the many-bodystates of the CNT quantum dot due to a static curvature-enhanced spin-orbit (SO) coupling ∆ SO .There are two reasons for this: First, one may wonder how the above scenario is affected by ∆ SO ingeneral. Second, even when ∆ SO is too small as to produce a significant shift in the line positionsof the differential conductance, the SO mixing turns out to be crucial to give a nonvanishingamplitude to the ground singlet line in Fig. 3c.2When considering the nanotube axis oriented along the x -direction as we do here, this SOinteraction enters through the following Hamiltonian H SO = ∆ SO σ x ⊗ τ z , (S-28)where σ x and τ z are Pauli matrices in spin and valley subspaces, respectively. Hence, the spin-orbit interaction conserves the valley structure, but it couples different spins, yielding spin-flipprocesses. Written in the BA-basis, H SO = ∆ SO X τσ d † τσ d ¯ τ ¯ σ , (S-29)the spin-orbit term couples the Kramers doublets | D ↑− i ↔ | D ↓ + i and | D ↓− i ↔ | D ↑ + i . Here we usedthe compact notation ¯ σ = − σ and ¯ τ = − τ .Naively, one expects that for ∆ SO < ~ ω Λ T S − the state-dependent vibrational coupling willsurvive. To again develop some better intuition we discuss as before the adiabatic approximationwhere the mechanical displacement Q is considered as classical parameter. In the full numericalcalculations, however, we always exactly diagonalize the quantum-dot Hamiltonian model. a. One-electron states The adiabiatic Hamiltonian matrix in the N = 1 sector contains a SOmixing term between the doublets: H (1)ad = E D ↑− ( Q ) 0 0 ∆ SO E D ↓− ( Q ) ∆ SO
00 ∆ SO E D ↑ + ( Q ) 0∆ SO E D ↓ + ( Q ) , (S-30)where the electronic states are now associated to the following adiabatic potentials E x ( Q ) = E x − ~ ω Λ + ~ ω Q + √ x ) . (S-31)We therefore expect the SO interaction to couple the doublets at the points of intersection of theadiabatic potentials as Q is varied. For ∆ SO < ∆, the minimum of the ground doublet parabola isconserved, and one might expect little change for the line positions in the differential conductance.In Fig. S-8a we show the one-electron adiabatic potentials including a small spin-orbit coupling∆ SO = 0 . . E D τ . The importantconclusion to draw here is that in the parameter regime that we will discuss below in Sec. II Cthere is no significant change around the ground state potential energy minimum. b. Two-electron states - state dependent Franck-Condon shifts We now discuss the situationin the two-electron charge state. The form of the SO Hamilonian in equation (S-29) shows thatthis operator flips both the orbital and spin projection in one of the two electrons in the quantumdot. Therefore, the two states T and S with electrons with opposite spins in opposite orbitalsare not affected by this coupling due to Pauli’s exclusion principle. The other states S τ and T ± are indeed mixed, as revealed by the Hamiltonian matrix block: H (2)ad = E S − ( Q ) ∆ SO − ∆ SO SO E T ( Q ) 0 0 0 − ∆ SO E T ( Q ) 0 0 0 − ∆ SO E T − ( Q ) 0 ∆ SO E S ( Q ) 00 − ∆ SO SO E S + ( Q ) . (S-32)Again, we expect the spin-orbit coupling to be only important around the crossings of the adiabaticpotentials as Q is varied. The above matrix thus mixes the two-particle states as follows: (i) Thelocalized singlets S τ will be repelled by the triplets T ± . (ii) These triplet states slightly repel eachother due to a second order process (since there is no direct matrix element connecting them).It now depends on the strength of the spin-orbit coupling relative to the vibrational couplingenergy (shifts of the potential energy minima × ~ ω ) and the splitting between the “bare”electronicstates (vertical energies at the minima) whether the SO coupling has a negligible impact on our3 -2-101234-4 -3 -2 -1 0 1 2 3 E ne r g y Displacement -2-101234-4 -3 -2 -1 0 1 2 3 E ne r g y Displacement (a) (b)
FIG. S-8. Adiabatic potentials including spin-orbit interaction ∆ SO = 0 . . SO = 0 are shown in grey for comparison.For large Q we can associate each adiabatic potential to the marked electronic states. In both chargesectors, the relevant lower minima are slightly affected by ∆ SO . mechanism or not: the SO coupling only has a big effect at crossings of potential energies whichmay be far away from the relevant minima. For the estimated experimental parameters, theenergy difference between the triplet and the ground singlet is noticeable. Moreover, the adiabaticpotentials related to these states are strongly shifted, meaning that the SO coupling will not have astrong impact on the development of the minima of those if ∆ SO < ~ ω Λ T ,S − . This is corroboratedby Fig. S-8b where the two-electron adiabatic potentials are shown. This is in agreement withthe measurements. However, even in this case the SO coupling plays a role: it is required for theexplanation of the peak amplitudes in a magnetic field as we discuss in Sec. II C 2 b.For strong spin-orbit coupling, ∆ SO ≥ ~ ω Λ T ,S − , the situation is quite different: All involvedFranck-Condon shifts of the vibrational mode become approximately the same, in clear disagree-ment with the measurements. Roughly speaking, for strong spin-orbit coupling the states S − and T are strongly mixed and the difference in their coupling to the vibration is “averaged out”.We conclude two things: (i) The SO interaction (S-28)-(S-29) does not generate a state-dependentelectron-vibration coupling. (ii) When present and strong, the SO interaction rather tends toweaken it , merely renormalizing the vibration frequency. B. Master equations - tunneling and relaxation
In this section we describe the employed method for the calculation of the Coulomb-diamondstability diagrams shown in Figures 2f - 2g and Fig. 3c of the main article.
1. Tunnel processes
In the stationary limit and for weak couplings to the source and drain leads, the occupationsprobabilities p a in the dot obey the rate equations0 = X b = a ( W ab p b − W ba p a ) , (S-33)where W ab = W s ab + W d ab represents the probability per unit time for a state transition | b i → | a i in the quantum dot. Since we will restrict ourselves to lowest order contributions in Γ, the overall4scale of W ab , the above rates coincide with those obtained by Fermi’s Golden’s Rule, namely W ab = X r W rab = X rη Γ rηab f ηr ( E a − E b ) , (S-34)where f ηr ( x ) = [1 + exp( η ( x − µ r ) /k B T )] − is the Fermi distribution function for an electron( η = +1) or a hole ( η = −
1) and the sum runs over the reservoirs r = { s , d } . The tunnel current I r that flows out of electrode r is calculated through the standard master equation approach inthe single-electron tunneling regime (SET) I r = X ab ( N a − N b ) W rab p b , (S-35)where N a = h a | N | a i is the electron number in the quantum-dot state | a i . Like the state occupationprobabilities p a obtained from equation (S-33), the current thus also depends on the tunnelingrates Γ rηab . The tunnel rates Γ rηab are related to the tunnel matrix elements (TMEs) T rσηa ← b for anelectron with spin σ entering ( η = +1) or leaving ( η = −
1) the dot and the density of states inthe leads ρ . The latter is assumed to be constant in the model (wide-band limit) and henceΓ rηab = 2 π X σ | T rσηa ← b | ρ . (S-36)In equation (S-18) we only accounted for the tunnel matrix elements associated to the pure elec-tronic states | e i , i.e. the many-body eigenstates of equation (S-5) labeled by e = D ± , S ± , S , T .These now need to be extended to the electron-vibration states | e, ν i by adding the Frank-Condonoverlap F ν ,ν of the vibrational wave-functions involved in the tunnel event, i.e. T rσηa,ν ← b,ν = F ν ,ν T rσηa ← b . (S-37)where now a, b = D ± , S ± , S , T and T rσηa ← b is given by equation (S-18). The Franck-Condoncoefficient strongly depends on the horizontal shift λ between the adiabatic potentials of theelectronic states. If the dot is initially in the state | b, ν i , the probability of a transition to a finalstate | a, ν i shifted in λ is modulated by F ν ,ν = h ν | e i √ λP | ν i = e − λ / ( − λ ) ν − ν r ν ! ν ! L ν − νν ( λ ) , (S-38)for ν ≥ ν (replace ν ↔ ν for ν ≥ ν ) and L ij ( x ) is the associated Laguerre polynomial. Sincethe λ -shift strongly attenuates the transition probability between the involved electronic statesat different regimes of the bias, one of the key aspects of the model is that electronic transitionsto the triplet state allow a change in the number of vibrational quanta while this is exponentiallysuppressed for transitions to the ground singlet. We note that also in the presence of vibrations the overall tunneling rate
Γ – entering as an overall factor through equation (S-17), equation (S-18) and equation (S-37) – merely sets the scale of the current and is irrelevant to the relativemagnitude of the different excitations which is of interest here.We now proceed with the explicit calculation of the TMEs for the 1 ↔ N = 1 charge sector [the “discharg-ing” transitions involve the TMEs T rσ ¯ ηb ← a = ( T rσηa ← b ) ∗ ]. For a strong intrinsic relaxation (see below),the relevant transitions are those which begin from | D − , i , i.e., the ground doublet D − and novibration, ν = 0. For (charging) transitions to a doubly occupied state we use η = 1 and find thefollowing amplitudes T rσ S τ,ν ← D σ − , = F ν, (Λ S τ ,D − )¯ σt r − σ δ τ − δ σ ¯ σ , T rσ T ,ν ← D σ − , = F ν, (Λ T ,D − )¯ τ t r + σ δ σ ↑ δ σ ↑ , (S-39) T rσ T ,ν ← D σ − , = F ν, (Λ T ,D − )¯ τ t r + σ √ δ σ ¯ σ , T rσ T − ,ν ← D σ − , = F ν, (Λ T ,D − )¯ τ t r + σ δ σ ↓ δ σ ↓ , (S-40) T rσ S ,ν ← D σ − , = F ν, (Λ S ,D − )¯ σ t r + σ √ δ σ ¯ σ . (S-41)Here we again used the compact notation ¯ σ = − σ and ¯ τ = − τ . We note that the transition to the S + singlet from the ground doublet is forbidden in the SET regime since it would involve an orbital5flip process which is not present in the tunnel Hamiltonian. The same restriction also applies tothe S − ↔ D + transitions. However, as discussed in the previous section, a small spin-orbit termhybridizes the doublets and localized singlet states, making visible the resonance line associatedto this last transition (line E in Fig. 1d of the main article).
2. Intrinsic relaxation
In the measured stability diagrams of the main article, we note that electronic transitionsstarting from an excited state are strongly suppressed. In order to account for this effect in themost simple way we model the influence of an environmental bath by allowing for relaxationprocesses. Since the main source of excitations is the electron transport we neglect absorptionprocesses due to the bath by setting the bath temperature T b = 0. Furthermore, we allow in eachcharge sector for the relaxation of any energetically higher-lying state b into any lower-lying state a , independently of the spin or the orbital distribution of the involved states. The decay rates areassumed to be proportional to the energy difference between these states and is assumed to exceedthe tunneling relaxation rates: W rel ab = Γ rel ( E b − E a ) , Γ rel ( E ) = Γ × ( E/ . . (S-42)The relaxation rate matrix W rel is thus upper triangular and it is specified in units of Γ. Thecorresponding rates W rel ab are added to the golden rule rates of equation (S-34) and the masterequations are solved using these modified rates. This depends little on the details and has themain effect of preventing a very strong nonequilibrium state on the quantum dot, enhancing theground state occupation probability in each charge sector. C. Comparison with experiment
In this section we describe how we proceed in finding a unique parameter regime of the modelwhich is able to qualitatively explain the experimental data. The key effect of state-dependentvibrational coupling is discussed fully in the zero magnetic field case. The finite magnetic fieldexperiment brings in some complications due to spin-dependent tunneling and spin-orbit coupling,which are, however, not crucial for the central point of the article. Finally, we discuss the depen-dence on the parameters and identify the problem that a standard Anderson-Holstein model haswith explaining the observations.
1. Zero magnetic field a. Vibrations “off ”
As already outlined in the experimental Sec. I B we calculate the capaci-tive effect induced by the gates and the source and drain leads from Fig. S-9a. The left and rightresonance lines (red dashed lines in Fig. S-9a), related to the transition between the N = 1 and N = 2 ground states D − ↔ S − , are described by the following relations between the top-gate andthe source-drain voltages V ssd = − α tg α s − α d ) V tg = m s V tg , (S-43) V dsd = + 2 α tg − ( α s − α d ) V tg = m d V tg . (S-44)By extracting the slopes of these lines ( m s = − . m d = 0 .
9) we obtain the following valuesfor the lever arms: α tg = 0 . α s − α d = − .
1. We next extract the electronic parameters, thesubband splitting ∆ and the exchange energy J . To this end, we use the formulas [cf. equation (S-14), (S-45), and (S-46) below)] for the bias distances µ = 1 . µ = 2 . | e | = 1), measured in Fig. S-9a: µ = E T − E S − = 2∆ − J/ , (S-45) µ = E S − E S − = 2∆ + 3 J/ , (S-46)6 V s d ( m V ) V tg (mV)02468-7.5 -5 -2.5 0 2.5 5 7.5 (a) V s d ( m V ) V tg (mV)02468-7.5 -5 -2.5 0 2.5 5 7.5 (b) FIG. S-9. Resonance lines fitting for N = 1 ↔ µ and µ voltages are related, respectively, toelectronic transition energies E T − E S − and E S − E S − . (b) Vibrations switched “on” [zoom-in of Fig. S-6d and Fig. 2d in the main article]: The measured µ here corresponds to E T − E S − . In both panels,dashed lines are visible transitions described by the model: | D − , i ↔ | S − , i (red), | D − , i → | T , ν i with ν = 0 − | D − , i → | S , ν i with ν = 0 − | S − , i → | D − , i transition is shown in panel (b) in dashed magenta and isjust discernible in the measured differential conductance. We obtain ∆ = ( µ + 3 µ ) / . J = µ − µ = 1 . b. Vibrations “on” In the regime where the CNT quantum dot is coupled to the vibrationalmode, we aim to account for the following features observed in the experimental data of Fig. S-9b(Fig. 2d of the main article):1. The experiment shows that the two-electron ground state is given by the singlet S − and thefirst excited state is the triplet T . In addition, their line positions remain almost unchangedwhen turning “on” the vibrational coupling. In our model we thus include the base valuesof J and ∆, fixed by their fitting to the electronic spectrum in Fig. S-9a. When allow theseenergies to become sensitive to the vibration Q through λ ∆ and λ J we must thus imposethe severe restriction that these couplings do not give rise to a polaronic shift.2. The ground singlet S − transition is very weakly coupled to the vibrational mode, the onlyvisible lines of the | D − , ν i ↔ | S − , ν i transitions is the “zero-phonon” one, ν = ν = 0. Wethus require a small value for the horizontal shift between the N = 1 and N = 2 ground statepotential minima, i.e. | Λ S − ,D − | (cid:28)
1. In contrast, the triplet transition shows a whole seriesof vibrational sidebands and must therefore require a sizeable coupling, i.e., | Λ T ,D − | ’ . S transition appearsin Fig. S-9a, no transitions | D − , i ↔ | S , ν i for ν = 1 , , , ... can be distinguished in thestability diagram Fig. S-9b when they are turned “on”. The approximate relation J ≈ ~ ω suggests these transitions are in fact superimposed with the triplet sidebands: the energydifference between S and T , given by the exchange energy J , happens to be commensuratewith the vibration energy ~ ω to within the experimental thermal line broadenings. Werequire only J = n ~ ω and determine the best fitting integer n (confirming indeed that n = 2).In applying these constraints to the model parameters, we used energies obtained from lever armsthat we extracted independently for Fig. S-9b: in this case we obtain from the slopes ( m s = − . m d = 0 .
85) of the ground state lines, giving the following values: α tg = 0 .
46 and α s − α d = − . ~ ω = 0 .
85 meV, the fitted value of the mean level spacing∆ E vib obtained specifically for the upper panel in Fig. 2h of the main article.According to constraint (1) for fixed electronic parameters ∆ and J , we need to adjust theenergy difference between E T , and E S − , to the measured bias µ ’ . µ = E T , − E S − , = 2∆ − J ~ ω (Λ S − − Λ T ) . (S-47)For the following algebra it is convenient to denote the two free parameters by x = Λ S − and y = Λ T and introduce p = ( µ −
2∆ + J/ / ~ ω ’ − . y = p x − p . (S-48)The horizontal shift q = Λ T ,D − between the triplet and the doublet can be written as q = y − x/ y = x q . (S-49)From condition 2 we have | x | (cid:28) | q | ∼
1. Bearing in mind this restriction, we find a uniquesolution for x when requiring both the above two equations to hold simultaneously: x = 2 q ± r q + 34 p . (S-50)Notice here that we cannot take | q | < p | p | / ’ .
54 since this would imply no real-valuedsolution. On the other hand, we cannot increase | q | indefinitely either since otherwise the solutionfor x grows and will violate condition 2 [i.e., the sidebands associated to S − would become visible].For q we pick the value q = 0 . x = − .
04 and y = 0 . z = Λ S and equation (S-47) n ~ ω ∼ E S , − E T , = J − ~ ω ( z − y ) , (S-51)where n is an integer number. By using J = 1 . ~ ω = 0 .
85 meV, we obtain y + J/ ~ ω ’ .
15 and this implies that n cannot be larger than 2. As mentioned above, n = 2 yieldsapproximately the same line position for S as observed before in the pure electronic regime.Solving the above equation with n = 2 for z and using the polaronic shifts definitions in terms ofthe λ -parameters (equations (S-21) and (S-22)), we obtain the parameter values used in Figures 2dand 3c in the main article, λ ε = 0 . , λ ∆ = 0 . , λ J = − . . (S-52)We emphasize that the above procedure essentially determines a unique regime of parametersconsistent with the experimental results. This concerns their qualitative features: conditions 2and 3 only define a range of possible values for the polaronic shifts and therefore small deviationsof the above obtained values for the λ -parameters produce similar results. However, an Anderson-Holstein type model ( λ ∆ = λ J = 0 ) is certainly not consistent with the measurements, see alsoSec. II C 3. Thus, despite the fact that there are several parameters, the experiment imposes strongrestrictions, in particular, limiting the choice of vibrational couplings. Importantly, the transportparameters γ , κ and ζ adjust other aspects of the transport spectrum but do not generate or affectin an essential way the state-dependent vibrational coupling. In particular at B = 0 are not thatimportant.
2. Magnetic field spectroscopy
We now consider the predictions of the above described model for magnetic field evolution ofthe dI/dV sd -peak intensities shown in Fig. 3 of the main article.8 a. Zeeman effect on state-dependent vibrational side bands When applying a static magneticfield B , the many-electron states in the quantum dot experience a Zeeman shift. Since the fieldis perpendicular to the CNT axis the orbital splitting due to B can be neglected . The fielddependence of the doublet and triplet states with spin projection indices σ = { +1 , − } and m = { +1 , , − } , respectively, is E D σ ± ,ν = E (0) D ± ,ν − σgµ B B , (S-53) E T m ,ν = E (0) T ,ν − mgµ B B , (S-54)where E (0) x,ν denote the zero-field eigenenergies (S-23). The singlets energies are obviously inde-pendent of B . (Regarding the negligble importance of the singlet S + , see the remark in the mainarticle’s text and the footnote 28 there.) Since the N = 1 quantum dot ground state ( | D ↑− , i , ν = 0) evolves with − gµ B B/
2, the energy cost for adding one electron to the quantum dot dependsin all cases on the magnetic field strength, no matter which final state is reached: relative to thisstate, the addition energies for the N = 2 states accessible in the sequential tunneling regime (seeSec. II B) are ∆ E S − ,ν = ∆ E (0) S − ,ν + gµ B B , (S-55)∆ E T ,ν = ∆ E (0) T ,ν − gµ B B , (S-56)∆ E T ,ν = ∆ E (0) T ,ν + gµ B B , (S-57)∆ E S ,ν = ∆ E (0) S ,ν + gµ B B . (S-58)where ∆ E (0)x ,ν = E (0)x ,ν − E (0) D − , are the zero-field values. The ground-state singlet S − and the excitedtriplet T lines thus evolve with opposite slopes as function of B . Since the triplet T states havethe interesting vibrational side bands – both in the experiment and in our calculations – theirfield dependence is plotted such that the triplet peak positions stay fixed, both here in Fig. S-10and in Figures 3a - 3c of the main article. In the theoretical plots this is achieved by replacing V tg → V tg + gµ B B .In Fig. S-10a we show the magnetic-field evolution of the calculated differential conductancealong the line V sd = V tg + 5 mV for spin-independent tunnel barriers, i.e., ζ = 0 for all B (butincluding SO coupling). What the model accounts for at this level relates to the key observationmade in the measurements: the triplet T +0 state (vertical) show vibrational side bands, whereas the ground singlet state S − (left most sloped line) does not. Importantly, the other sloped lines should not be confused with vibrational side bands of the ground singlet S − : These are the transitionsinto the m = 0 component of the triplet, T , or the excited singlet S (the latter is shifted in about2 ~ ω with respect to the former, see above in Sec. II C 1) and their vibrational side bands. Thatthese lines should be there is clear from the theoretical model and its analysis [cf. equation (S-57)and equation (S-58)] but is also borne out by noting the difference between the distance A andthe following ones B-D [just as in the experimental data in Fig. 3a of the main article]. In theexperiment these Zeeman split-off triplet states in Fig. S-10a, as well as the singlet S are notobserved in Fig. 3a of the main article. b. Suppression of Zeeman split-off states In order to rationalize the observed absence of Zee-man split-off states in the experiment we now include a spin-dependence in the tunnel amplitudesthrough the parameter ζ [cf. equation (S-17)] which we take to be B -dependent: ζ ( B ) = 1 − e − gµ B B/k B T e − gµ B B/k B T , (S-59)This phenomenological function ensures that for gµ B B (cid:29) k B T we have ζ = 1, i.e., the tunnelamplitudes are fully polarized for spin-up carriers, see equation (S-17), whereas for gµ B B (cid:28) k B T we have ζ = 0. This spin-dependence is physically motivated by the fact that our quantum dotis contacted by CNT leads as shown in Fig. S-3b and explained in the main article [cf. Fig. 2b].When increasing the field by a few Tesla the strong spin-polarization in the tunnel amplitudessuppresses the passage of spin-down electrons through the dot and hence the T and S lines(together with their vibrational sidebands) are suppressed since the ground state in the N = 1sector is D ↑− . What remains are the T +0 excitation and its vibrational sidebands.9 B ( T ) V tg (mV)02468 0 1 2 3 4 5 6 7 A B C D dI/dV sd ( Γ ) B ( T ) V tg (mV)02468 0 1 2 3 4 5 6 7 dI/dV sd ( Γ ) (a) (b) FIG. S-10. Calculated magnetic field evolution for the plotted differential conductance of Fig. 2d ofthe main article: (a) Spin-independent tunnel amplitudes ( ζ = 0 for all B ). (b) Spin-dependent tunnelamplitudes according to equation (S-59). Further parameters [same as in Fig. 2g of the main article]:∆ = 0 . J = 1 . λ ε = 0 . λ ∆ = 0 . λ J = − . ~ ω = 0 .
85 meV, ∆ SO = 0 . κ = 0 . γ = − . T = 0 . However, the ground singlet S − is also suppressed when assuming this spin-dependence, sinceit requires an additional spin-down electron to fill the bonding orbital ( τ = − ), cf. equation (S-9).The presence of the ground singlet S − line, but not its vibrational sidebands is the key feature ofthe experiment. It is at this point that the spin-orbit coupling does have a decisive effect: wheneven a small spin-orbit coupling is included, the singlet S − reappears by borrowing intensity fromthe triplet T (see also Fig. 4), but without reinstating the unobserved S and the Zeeman split-offstates of T and their vibrational sidebands. The SO coupling mostly mixes the ground singlet S − and the triplet T states [cf. Sec. II A 3]. In Fig. S-10b [same data as in Fig. 3c of the mainarticle] we show the magnetic field evolution of the calculated differential conductance along theline V sd = V tg + 5 mV when we include the finite ζ of equation (S-59): indeed the suppressionof the T and S lines and their sidebands is maintained while only the ground S − has a clearintensity. This produces the observed intensity pattern of Fig. 3a of the main article. It is at thislast point that the Anderson-Holstein model fails, see Sec. II C 3. From this we infer that in our device an interplay of spin-filtering of the suspended CNT parts,functioning as tunnel junctions, and weak spin-orbit interaction in the CNT quantum dot may beresponsible for the suppression of Zeeman splitting.
As emphasized above it is not responsible forthe essential effect: The missing vibrational side bands while having a clear singlet ground state S − line.
3. Influence of the various parameters and the problem of Anderson-Holstein coupling
Having made the detailed comparison with the experiment we now outline the different influencethe various parameters on the differential conductance. We illustrate this for the case of anAnderson-Holstein type model ( λ ∆ = λ J = 0) and expose the problem it has in explaining thedata.In Figures S-11a - S-11c we plot the results obtained for our model, keeping all parameters asin the main article, except for γ which we vary and making the Anderson-Holstein approximationby setting λ ∆ = λ J = 0. Also, to obtain a similar number of visible vibrational sidebands wehave to choose a larger value of λ ε = 1 in this case. To find agreement with the experiment thelines with positive slope should first of all be suppressed: these are vibrational sidebands relatingto the | D − , ν i ↔ | S − , i where ν = 1 , , , .. quanta are excited for N = 1. This can be done byincreasing the junction asymmetry γ a lot. The best result obtainable this way, Fig. S-11c shouldnow be compared with Fig. 2g of the main article: as one adjusts γ to suppress the vibrationalsidebands relating to the S − (both with positive and negative slope) the ground state ν = 0transition also becomes suppressed, resulting in a severe current blockade at low bias V sd . ~ ω .In contrast, in Fig. 2g these lines are not present due to the state-dependent vibrational coupling0 V s d ( m V ) V tg (mV)-202468-10 -6 -2 2 6 10 dI/dV sd ( Γ ) V s d ( m V ) V tg (mV)-202468-10 -6 -2 2 6 10 dI/dV sd ( Γ ) V s d ( m V ) V tg (mV)-202468-10 -6 -2 2 6 10 dI/dV sd ( Γ ) (a) (b) (c) FIG. S-11. Differential conductance at B = 0 for the Anderson-Holstein model ( λ ∆ = λ J = 0) with λ ε = 1. The other parameters are the same as for the state-dependent vibrational coupling model, i.e.,∆ = 0 . J = 1 . ~ ω = 0 .
85 meV, κ = 0 . ζ = 0 . T = 0 . γ : (a) γ = 0, (b) γ = − . γ = − .
9. For completeness wehave also included here the SO coupling ∆ SO = 0 . λ ∆ , λ J = 0), although this has little effect at B = 0. λ ∆ and λ J which does not block the transport at low voltage. This allows for a much smallerasymmetry, leaving the ground-state transition fully visible.This problem becomes more pressing when we now consider the magnetic field evolution: while S − and its sidebands with negative V tg -slope are still present they are quasi-degenerate with thetriplet excitations at B = 0 in panel (c). Turning on the field B they will however split off fromthe triplet by the Zeeman effect. Also in this case one must assume a spin-dependent tunneling ζ [equation (S-59)] to find agreement with the experimental data of Fig. 3a where the Zeeman-splitoff states are not observed. As before, this also suppresses the S − transition, both the groundone ( ν = 0) and all vibrational sidebands ( ν > all S − transition are restored, i.e., including the vibrational side bands(this happens again by borrowing intensity from the T triplet). This disagrees with the keyexperimental observation that only the triplet transitions show vibrational side bands.The above discussion underlines the importance of the experimental advance reported in themain article: by being able to switch “on” and “off” the vibrational coupling as well as performingmagnetic field transport spectroscopy of the quantum dot states, we are able to identify electronicstates with different coupling to the vibration.Next, by varying orbital asymmetry in the tunneling κ one changes the magnitude of the S − transitions relative to the T and S transitions: when enhancing κ , the tunneling in the antibond-ing ( |−i ) orbital is enhanced: since S − has two electrons in that orbital, whereas T and S haveonly one, this enhances the former relative to the latter. As above, κ cannot be adjusted to findagreement with the experiment: as in the discussion of the γ dependence, it cannot suppress thevibrational side bands of S − relative to the S − transition: this requires state-dependent coupling. Thus the key problem in trying to use an Anderson-Holstein model to explain state-dependentvibration coupling lies in its basic assumption that all electronic states with the same charge coupleequally to the vibration.
It fails because it either does not at all show the D − → S − excitation(i.e., neither the ground ( ν = 0) or any vibrational sidebands ν = 0) or it does show it togetherwith all its vibrational sidebands. Which of the two is the case depends on parameters, but anapparent state-dependent vibrational coupling seems impossible to achieve. The sidebands for thesinglet S − , relative to the ground transition, are intense as those of the triplet T relative to itsground transition. Neither junction ( γ ), orbital ( κ ) nor spin-asymmetry ( ζ ) can get around thisfact. ∗ Equal contribution Leturcq, R. et al.
Franck-condon blockade in suspended carbon nanotube quantum dots.
Nature Phys. , 317 (2009). Durrer, L. et al.
Swnt growth by cvd on ferritin-based iron catalyst nanoparticles towards cnt sensors.
Sensors and Actuators B: Chemical , 485 (2008). Kouwenhoven, L. et al.
In Sohn, K., Kouwenhoven, L. & Sch¨on, G. (eds.)
Mesoscopic electron transport ,chap. Electron Transport in Quantum Dots, 105 (Kluwer, 1997). Kouwenhoven, L. P., Austing, D. G. & Tarucha, S. Few-electron quantum dots.
Rep. Prog. Phys. ,701 (2001). Sapmaz, S. et al.
Electronic excitation spectrum of metallic carbon nanotubes.
Phys. Rev. B ,153402 (2005). Flensberg, K.
Phys. Rev. B , 180516(R) (2010). Braig, S. & Flensberg, K. Vibrational sidebands and dissipative tunneling in molecular transistors.
Phys. Rev. B , 205324 (2003). Koch, J. & von Oppen, F. Franck-condon blockade and giant fano factors in transport through singlemolecules.
Phys. Rev. Lett. , 206804 (2005). Wegewijs, M. R. & Nowack, K. C. Nuclear wave function interference in single-molecule electrontransport.
New. J. Phys. , 239 (2005). Leijnse, M. & Wegewijs, M. R. Kinetic equations for transport through single-molecule transistors.
Phys. Rev. B , 235424 (2008). Flensberg, K. & Marcus, C. M. Bends in nanotubes allow electric spin control and coupling.
Phys.Rev. B81