Magnetic field control of the Franck-Condon coupling of few-electron quantum states

TL;DR

This study investigates how magnetic fields influence the electron-vibration coupling in suspended carbon nanotubes, revealing valley-dependent effects and providing insights into quantum state control at cryogenic temperatures.

Contribution

It demonstrates magnetic field control over Franck-Condon coupling in nanotube quantum dots with well-defined electron states, highlighting valley-dependent effects.

Findings

Magnetic field induces evolution of Franck-Condon coupling.

Coupling depends on electronic valley states.

Model suggests wavefunction reshaping causes effects.

Abstract

Suspended carbon nanotubes display at cryogenic temperatures a distinct interaction between the quantized longitudinal vibration of the macromolecule and its embedded quantum dot, visible via Franck-Condon conductance side bands in transport spectroscopy. We present data on such side bands at known absolute number N=1 and N=2 of conduction band electrons and consequently well-defined electronic ground and excited states in a clean nanotube device. The interaction evolves only at a finite axial magnetic field and displays a distinct magnetic field dependence of the Franck-Condon coupling, different for different electronic base states and indicating a valley-dependent electron-vibron coupling. A tentative cause of these effects, reshaping of the electronic wavefunction by the magnetic field, is discussed and demonstrated in a model.

Figures

Equations

$$E_{\text{ev}}\propto \int \rho (x)\frac{\text{d}u}{\text{d}x}\text{d}x$$

$$\Delta ϵ=\frac{1}{1-(s_1/s_2)}e\Delta V_{\text{sd}}=f(B_{\parallel },V_{\text{g}})e\Delta V_{\text{sd}}.$$

$$\Delta {ϵ}_{\text{th}}=\frac{h}{L}\sqrt{\frac{Y}{\rho }},$$

$$\Delta {ϵ}_{\text{th}}\approx \frac{0.11\text{meV}}{L(\mu \text{m})}$$

$$\Delta I_n\propto \frac{e^{-g}{g}^n}{n!},n=0,1,2,\dots$$

$$G_{\text{th}}(V_{\text{sd}},V_{\text{sd}}^0)={\cosh }^{-2}\left(\frac{V_{\text{sd}}-V_{\text{sd}}^0}{\gamma }\right),$$

$$G(V_{\text{sd}},V_{\text{sd}}^0)=G_{\text{max}}\frac{{\sum }_{n=0}^9{P}_n{G}_{\text{th}}(V_{\text{sd}},V_{\text{sd}}^0+n\Delta V_{\text{sd}})}{{\sum }_{n=0}^9{P}_n{G}_{\text{th}}(V_{\text{sd}}^0,V_{\text{sd}}^0+n\Delta V_{\text{sd}})}.$$

$$\begin{gathered} {\hat{H}}_{\text{CNT}}=\underset{\text{“shell”}}{\underbrace{{\epsilon }_d{\hat{I}}_{\sigma }\otimes {\hat{I}}_{\tau }}}+\underset{\text{valley mixing}}{\underbrace{\frac{{\Delta }_{\text{KK’}}}{2}{\hat{I}}_{\sigma }\otimes {\hat{\tau }}_x}}+\underset{\text{spin-orbit interaction}}{\underbrace{\frac{{\Delta }_{\text{SO}}}{2}{\hat{\sigma }}_z\otimes {\hat{\tau }}_z}}+ \\ +\underset{\text{Zeeman effect}}{\underbrace{\frac{g_{\text{s}}{\mu }_{\text{B}}|\vec{B}|}{2}\left(\cos \phi {\hat{\sigma }}_z+\sin \phi {\hat{\sigma }}_x\right)\otimes {\hat{I}}_{\tau }}}+ \\ +\underset{\text{orbital angular moment}}{\underbrace{g_{\text{orb}}{\mu }_{\text{B}}|\vec{B}|\cos \phi {\hat{I}}_{\sigma }\otimes {\hat{\tau }}_z}} \end{gathered}$$

$$E_{\sigma \tau }(B_{\parallel })=\left(\sigma \frac{g_{\text{s}}{\mu }_{\text{B}}}{2}+\tau g_{\text{orb}}{\mu }_{\text{B}}\right)B_{\parallel }.$$

$$E_{\tau }(B_{\parallel })=\tau g_{\text{orb}}{\mu }_{\text{B}}{B}_{\parallel },\tau =\pm 1$$

$$\frac{\text{d}(\Delta E_{12}(B_{\parallel }))}{\text{d}{B}_{\parallel }}=\frac{\text{d}(E_2(B_{\parallel })-E_1(B_{\parallel }))}{\text{d}{B}_{\parallel }}.$$

$$e^{2i{\kappa }_{\parallel }L}\stackrel{!}{=}\frac{\tau {\kappa }_{\perp }+i{\kappa }_{\parallel }}{\tau {\kappa }_{\perp }-i{\kappa }_{\parallel }}.$$

$${\psi }_A(x,\varphi )=\sin (\pi \varphi x),x\in [0,1]$$

$${\psi }_B(x,\varphi )={\psi }_A(1-x,\varphi )$$

$$\rho (x,\varphi )=\frac{{\left|{\psi }_A(x,\varphi )\right|}^2+{\left|{\psi }_B(x,\varphi )\right|}^2}{\underset{0}{\overset{1}{\int }}{\left|{\psi }_A(x,\varphi )\right|}^2+{\left|{\psi }_B(x,\varphi )\right|}^2\text{d}x}$$

$$E_{\text{ev}}(\varphi )\propto \int \rho (x,\varphi ))\frac{\text{d}u}{\text{d}x}\text{d}x$$

Full text

I Introduction

Vibrational degrees of freedom, typically approximated at small deflection as harmonic oscillators, contribute in many ways to the fundamental properties of matter. The Franck-Condon principle Franck (1926); Condon (1926) relates vibrational wave functions in molecular physics to the intensity envelope of vibrational side bands in optical spectra. This principle, where an electronic transition is assumed to be instantaneous compared to the slow motion of the nuclei, also becomes directly visible in electronic low-temperature transport spectroscopy of single (macro)molecules Park et al. (2000); Braig and Flensberg (2003); LeRoy et al. (2004); Weig et al. (2004); Koch and von Oppen (2005); Izumida and Grifoni (2005); Sapmaz et al. (2006); Koch et al. (2006); Osorio et al. (2007); Burzurí et al. (2014). An experimental system where such Franck-Condon sidebands have been observed consistently is the longitudinal (stretching mode) vibration of a suspended single wall carbon nanotube quantum dot Sapmaz et al. (2006); Hüttel et al. (2008); Leturcq et al. (2009); Hüttel et al. (2009); Island et al. (2012); Ganzhorn et al. (2013); Jung et al. (2013); Weber et al. (2015). Results range from the nanotube length dependence of the vibration frequency Sapmaz et al. (2006), or thermal occupation of a vibration mode Leturcq et al. (2009), all the way to electronic pumping of nonequilibrium occupation Hüttel et al. (2009), spin-vibron coupling Ganzhorn et al. (2013) and a spin-dependent, electrostatically tunable electron-vibron coupling Weber et al. (2015).

Here, we present first observations of Franck-Condon sidebands at known absolute number $N_{\text{el}}=1$ and $N_{\text{el}}=2$ of conduction band electrons in the unperturbed carbon nanotube transport spectrum. The vibrational sidebands evolve only at finite axial magnetic field $B_{\parallel}$. The resulting millikelvin transport spectrum displays different sideband behaviour depending on the electronic base state; the data indicates a valley-dependent Franck-Condon electron-vibron coupling parameter Braig and Flensberg (2003). As a tentative mechanism for the observed phenomenon, reshaping of the electronic wavefunction by the magnetic field Margańska et al. (2019) is discussed and modelled; the model manages to capture the essential observed behaviour.

II Conduction side bands at finite field

A sketch of the measured device is depicted in Fig. 1(a). Following Cao et al. (2005), a carbon nanotube is grown over pre-defined rhenium contact electrodes and etched trenches. Subsequently the device is cooled down in a top-loading dilution refrigerator and characterized electronically at a base temperature of $T_{\text{mc}}\leq 30\,\text{mK}$, immersed into the diluted phase of the ${}^{3}\text{He}/{}^{4}\text{He}$ mixture. The length of the suspended nanotube segment is $L=700\,\text{nm}$. The device has already been characterized electronically in Margańska et al. (2019); Götz et al. (2018); Schmid et al. (2020); as can also be seen in the current trace at low bias of Fig. 1(b), it displays the behaviour of a small bandgap single-wall carbon nanotube, with transparent hole conduction and strong Coulomb blockade at low electron numbers. The first Coulomb oscillations exhibit very low current and require particular care to be resolved, see Fig. 1(c). Here, the opaque tunnel barriers are given by p-n junctions extended along the nanotube, between the electrostatically induced n-quantum dot and p-behaviour near the leads Park and McEuen (2001); Steele et al. (2009).

In the following we focus on the $1\leq N_{\text{el}}\leq 2$ transition, i.e., the second Coulomb oscillation at the electron side of the band gap. Fig. 1(d) and Fig. 1(e) display the stability diagram close to the corresponding degeneracy point, for (d) $B_{\parallel}=0$ and for (e) a magnetic field $B_{\parallel}=10\,\text{T}$ applied in parallel to the carbon nanotube axis. The overall conductance is very low; even so, the color scale in the figure has been cut off such as to focus on the substructure of the single electron tunneling (SET) regions. The strong black lines, corresponding to electronic excitations, shift with magnetic field; their detailed behaviour, as well as the negative differential conductance at $B_{\parallel}=0$, is topic of ongoing analysis. The figures display a clear qualitative difference: while the areas between the electronic excitation lines are featureless in Fig. 1(d), in Fig. 1(e) they display a multitude of fine, closely spaced conductance resonances (see the arrows in Fig. 1(e)). This also becomes visible in the comparison of the trace cuts of Fig. 1(f) ($B_{\parallel}=0\,\text{T}$) and Fig. 1(g) ($B_{\parallel}=10\,\text{T}$).

Figure 2(a) demonstrates the emergence of this phenomenon with increasing magnetic field. Here, we show the differential conductance $\text{d}I/\text{d}V_{\text{sd}}(B_{\parallel},V_{\text{sd}})$ as function of an applied field $B_{\parallel}$ parallel to the carbon nanotube axis and of the bias voltage $V_{\text{sd}}$. The gate voltage is kept constant and chosen such that we trace across the $N_{\text{el}}=1$ edge of the $1\leq N_{\text{el}}\leq 2$ single electron tunneling (SET) region, see the inset of Fig. 2(a) for a sketch and Fig. 7(a) in Appendix A for a larger-scale plot of the conductance at the degeneracy point. Here and later, we focus our evaluation on the low magnetic field region since it provides a better signal/noise ratio.

For $B_{\parallel}=0$, the SET region edge, visible as line of differential conductance, is located at approximately $V_{\text{sd}}=0.8\,\text{mV}$. Due to a shift in energy of the electronic states involved in transport, it rapidly moves to higher bias voltages until $B_{\parallel}\simeq 1.5\,\text{T}$ is reached. Here, the magnetic field induces a change in ground state, leading to a different energy dispersion. Soon afterwards, side bands of the differential conductance line emerge, see the arrows in Fig. 2(a). Figure 2(b) displays a larger parameter range than Fig. 2(a), though measured at reduced resolution. Still, the side bands of the conductance line become clearly visible as an asymmetric broadening of the main conductance line (towards higher bias voltages).

An example trace cut from Fig. 2(a), both smoothened for clarity (black line) and as raw numerical derivative of the current (red points), is shown in Fig. 2(c). A manual analysis of the relative peak positions in each such recorded trace $I(V_{\text{sd}})$ is given in Appendix A, see in particular Fig. 7(f). Its conclusion is that within the scatter the side bands are equidistant within each trace. The excitation energy is magnetic field independent for $B_{\parallel}\lesssim 6\,\text{T}$ 111At higher fields $B_{\parallel}\gtrsim 6\,\text{T}$ the extracted peak positions may indicate an increase in oscillator quantum, though the signal/noise ratio makes a quantitative analysis challenging here. and given by $\Delta\epsilon\simeq 50\,\mu\text{eV}$. This indicates a harmonic oscillator independent of the electronic spectrum. Given its energy scale, we can tentatively identify it with the longitudinal vibration of the carbon nanotube Sapmaz et al. (2006).

III Franck-Condon model

In multiple publications, the mechanism leading to vibrational harmonic side bands in transport spectroscopy has been identified as the Franck-Condon principle Braig and Flensberg (2003); Koch and von Oppen (2005); Sapmaz et al. (2006). As sketched in Fig. 2(d), the equilibrium position of the vibrational harmonic oscillator depends on the number of charges $N_{\text{el}}$ on the nanotube; the rate of single electron tunneling through its quantum dot is modified by the spatial overlap of the involved harmonic oscillator states $\left|\left.\left<{\Psi_{m}(N)\,}\right|{\Psi_{n}(N+1)}\right>\right|^{2}=\left|\left.\left<{\Psi_{m}(x)\,}\right|{\Psi_{n}(x+\Delta x)}\right>\right|^{2}$, with $m$ and $n$ as the vibrational quantum number at $N$ and $N+1$ electrons, respectively. $\Delta x$ is the displacement of the harmonic oscillator by the additional charge, cf. Fig. 2(d). The coupling strength is parametrized via the Franck-Condon coupling parameter $g=(\Delta x/x_{\text{zpf}})^{2}/2$, comparing $\Delta x$ with the characteristic length scale of the harmonic oscillator $x_{\text{zpf}}=\sqrt{\hbar/m\omega}$.

As sketched in Fig. 2(e), a finite value of $g$ leads to a redistribution of current between vibrational state channels: at the bias voltage corresponding to the bare electronic transition energy $N\longrightarrow N+1$ current is suppressed, but it increases whenever an additional vibration state becomes energetically available. A large number of extensions to this model has been developed to take into account specific details of transport spectra, see, e.g., Izumida and Grifoni (2005); Zazunov et al. (2006); Härtle et al. (2009); von Oppen et al. (2009); Mariani and von Oppen (2009); Hübener and Brandes (2009); Cavaliere et al. (2010); Maier et al. (2011); Härtle and Thoss (2011); Yar et al. (2011); Fang et al. (2011); Donarini et al. (2012); Zhang et al. (2012); Jovchev and Anders (2013); Simine and Segal (2014); Weber et al. (2015); McCaskey et al. (2015); Sowa et al. (2017); however, for now we focus our analysis on the simplest theoretical case, assuming a single harmonic oscillator mode and fast relaxation into the vibrational ground state. In this case, the current step heights or conductance peak amplitudes follow the Poisson formula Braig and Flensberg (2003); Koch et al. (2006), $P_{n}={(e^{-g}g^{n})}/{n!},\quad n=0,1,2,\dots$, at an energy $\Delta E_{\text{vib}}=n\hbar\omega$ from the bare electronic state transition supplied via the bias voltage. The resulting step function of the current, in absence of broadening effects, is sketched in Fig. 2(e) for the example of a large $g=2$.

At base temperature, with $25\,\text{mK}\,k_{\text{B}}\simeq 2\,\mu\text{eV}$, we expect the conductance lines to be lifetime-broadened, with a Lorentzian line shape. Thus, a sum of Lorentzians with amplitudes following the Poisson sequence and center points shifted by equidistant bias voltage offsets can be envisioned as model. In practice, it turns out that the Lorentzian line shape does not suit the measurement data well; this may indicate broadening effects beyond temperature and lifetime, as, e.g., fluctuating gate or bias voltages. Empirically, we choose the resonance shape of thermal broadening, $\propto\cosh^{-2}$, instead Beenakker (1991). Details of the fit procedure can be found in Appendix B.

The result of evaluating the Franck-Condon coupling parameter $g$ for each trace $\text{d}I/\text{d}V_{\text{sd}}(V_{\text{sd}})$ at fixed $B_{\parallel}$ is plotted in Fig. 2(f). It shows the resulting magnetic field dependence $g(B_{\parallel})$ for the $N_{\text{el}}=1$ $\longrightarrow$ $N_{\text{el}}=2$ ground state transition. The absence of side bands for $B_{\parallel}<1.5\,\text{T}$ corresponds to an absence of coupling, i.e., $g\sim 0$. For $B_{\parallel}\geq 1.5\,\text{T}$ the coupling increases monotonously, reaching a maximum value $g(B_{\parallel})\simeq 0.3$ at $2.5T\leq B_{\parallel}\leq 3\,\text{T}$. Subsequently, we observe a slow decrease and stabilization at $g(B_{\parallel})\simeq 0.2$.

IV Relation to electronic states

To our best knowledge, no similar observations of a magnetic field dependent electron-vibron coupling have been published so far. Its onset at an anticrossing of electronic states, see the dashed ellipsoid in Fig. 2(b), suggests a connection to the electronic quantum numbers. As opposed to previous reports on the longitudinal vibration mode Sapmaz et al. (2006); Hüttel et al. (2008); Leturcq et al. (2009); Hüttel et al. (2009); Ganzhorn et al. (2013); Weber et al. (2015), here we are characterizing a device where the nanotube has grown cleanly across pre-existing contacts, and no (accidental or intentional) strongly inhomogeneuous potential distorts the wave functions in the suspended macromolecule away from the metallic contacts Margańska et al. (2019). With this in mind, we have analyzed the vibrational side band behaviour of the electronic excited states in the transport spectrum.

A plot of the differential conductance at fixed gate voltage, as function of $B_{\parallel}$ and the bias voltage $V_{\text{sd}}$, now over a large bias range, is shown in Fig. 3(a). A preliminary evaluation of the data of Fig. 3(a), taking also into account the one-electron excitation spectrum of the device Margańska et al. (2019), reveals two energetically close shells with both intra- and inter-shell exchange interaction Margańska ; detailed modelling of the two electron transport spectrum will be the topic of a separate work. Here, we limit ourselves to a straightforward classification of conductance lines by magnetic field dispersion; see Appendix C for the details.

The dominant magnetic field dependence of the electron energies in a carbon nanotube in an axial field originates from the electronic orbital magnetic moment $\mu_{\text{orb}}$, see, e.g., Minot et al. (2004); Laird et al. (2015); Margańska et al. (2019), and Appendix C. Thus, both one- and two-electron quantum states become at large field angular momentum (and valley) eigenstates, and the slope of a conductance line in Fig. 3 indicates the angular momentum change when a second electron tunnels onto the quantum dot. If the state of the first electron remains unchanged, only the contribution of the second electron to the magnetic moment — parallel or antiparallel to the magnetic field — determines the magnetic field dispersion. From this we can classify the conductance lines of Fig. 3(a). As expected, in the low-bias region of the figure, two dominant slopes clearly emerge; we identify these with the addition of a K’-valley electron (downward slope) or a K-valley electron (upward slope), respectively Minot et al. (2004); Laird et al. (2015); Margańska et al. (2019).

Figures 3(b) and 3(c) enlarge the regions marked in 3(a) with dashed rectangles. Also here, the harmonic side bands are immediately visible. However, at a first glance, only the down-sloping spectral lines, where an electron is added in a K’-valley state, seem to exhibit electron-vibron coupling. Also the trace cuts at $B_{\parallel}=2.70\,\text{T}$ (Fig. 3(d)) and $B_{\parallel}=0.85\,\text{T}$ (Fig. 3(e)) demonstrate this, with the resonances accompanied by sidebands highlighted in green.

Extracting the Franck-Condon coupling parameter $g(B_{\parallel})$ for an exemplary excited state resonance (green arrowheads in Fig. 3(a)), using an identical fit procedure as for Fig. 2(f), we compare its evolution over the entire field range $0\,\text{T}\lesssim B\lesssim 3\,\text{T}$ with the two-electron ground state transition in Fig. 3(f). A finite $g$ persists to much lower magnetic field for the excited state transition; indeed, the plot shows a linear growth of $g(B_{\parallel})$ in this range with $g(B_{\parallel})\simeq 0.124\,\text{1/T}B_{\parallel}$ (gray solid line). The sudden onset of coupling for the ground state transition is thus consistent with a valley dependence and the change in ground state quantum numbers at $B_{\parallel}\simeq 1.5\,\text{T}$, see the structure marked with a dashed ellipsoid in Fig. 2(b).

We have performed a systematic analysis of the conductance resonances in Fig. 3(a), by dividing them into segments at each crossing or anticrossing and evaluating each segment; details can be found in Appendix D. The result is shown in Fig. 4. Fig. 4(a) displays symbols with their size representing the obtained $g$-parameter, superimposed on the line pattern of Fig. 3(a) at the parameters $(B_{\parallel},V_{\text{g}})$ where the evaluation has taken place. Fig. 4(b) plots the value of $g$ as function of the selected magnetic field, irrespective of bias voltage $V_{\text{sd}}$ and thus selected resonance line, with the fit function of Fig. 3(f) added. In both cases, red circles represent downsloping lines, black triangles upsloping lines. The result confirms our visual impression. In general the up-sloping lines, where an electron is added in a K-state, display no vibrational side bands, with the notable exception of the segment at ($3\,\text{T}$, $6\,\text{mV}$). The down-sloping lines, where an electron is added in a K’-state, show a $g$ growing with magnetic field, in most cases following the same linear behaviour as already observed in Fig. 3(f). One resonance line, also enlarged in Fig. 3(b), deviates with a large $g$.

It remains to clarify whether the observed effect is specific to the $1\leq N_{\text{el}}\leq 2$ region. Figure 5 displays the magnetic field dependence of a cut across the $0\leq N_{\text{el}}\leq 1$ ground state transtion. Also here, where an electron tunnels into a K’ state of the otherwise unoccupied conduction band, an onset of harmonic side bands can be visually identified, for both spin alignments. Given the significantly lower conductance here, a more detailed evaluation turns out to be challenging. We can however conclude that the electron-vibron coupling is already inherent to single electron phenomena, and not limited to two-electron states Weber et al. (2015). This indicates a selectivity directly related to the single-particle valley quantum number.

V Discussion

The transport spectra were measured with the carbon nanotube immersed into the ${}^{3}\text{He}/{}^{4}\text{He}$ mixture (D phase) of the dilution refrigerator. Its viscosity at base temperature, \eta\sim 10^{-5}\,\text{N s/m{}^{2}} Enss and Hunklinger (2005), is sufficiently high to mechanically dampen the transversal vibration mode Schmid et al. (2015a). As observed here, this does not affect the longitudinal mode; a likely explanation is that motion along the nanotube axis does not require any displacement of liquid.

In literature, Weber et al. Weber et al. (2015) have reported an electronic state dependent Franck-Condon coupling in a $N_{\text{el}}=4n+2$ quantum dot, switchable via a local gate potential deformation. They apply a magnetic field perpendicular to the nanotube, and see no effect of that field on the vibrational side bands. Due to valley mixing and the field direction, their data is in the regime of bonding and antibonding valley linear combinations. A different electron-vibron coupling of (valley-ground state) spin singlet and (valley-distributed) spin triplet states, as observed there, obviously can also indicate a valley-dependent effect. Whether this is related to our results still requires further analysis.

Models discussing strong electron-vibron coupling in carbon nanotube quantum dots typically assume an inhomogeneuous charge distribution relative to the vibration mode envelope, or more generally, a different localization of electron and vibron wave function, see, e.g., Sapmaz et al. (2006); Mariani and von Oppen (2009); Cavaliere et al. (2010); Donarini et al. (2012). This is consistent with the occurrence of Franck-Condon side bands in devices with local gates close to the nanotube Leturcq et al. (2009); Jung et al. (2013); Weber et al. (2015). In previous devices without such an electrode, a local potential may have been introduced via fabrication defects or impurities on the nanotube Sapmaz et al. (2006); Hüttel et al. (2008, 2009). These expanations do not lend themselves for the device presented here, displaying a highly regular one electron spectrum Margańska et al. (2019) and a regular addition spectrum over a large electron number range Schmid et al. (2015b); Götz et al. (2018). Nevertheless, the formation of the quantum dot via gate-induced p-n junctions, see, e.g., Park and McEuen (2001); Steele et al. (2009), leads to an electronic confinement more narrow than the mechanically active device length, and typically centered within the suspended nanotube segment.

Comparison with existing theoretical Mariani and von Oppen (2009); Donarini et al. (2012) and experimental works Weber et al. (2015) then suggests that the axial magnetic field modifies the Franck-Condon coupling $g$ by shifting the electronic wave function relative to the vibron envelope. A valley-dependent mechanism having this effect was recently proposed in Margańska et al. (2019), based on Akhmerov and Beenakker (2008); Castro Neto et al. (2009); Marganska et al. (2011). Essentially, the axial magnetic field introduces an Aharonov-Bohm phase around the nanotube. Due to the bipartite graphene lattice, the axial component $\kappa_{\parallel}$ and the circumferential component $\kappa_{\perp}$ of an electron state wave vector are coupled; one example solution is plotted in Fig. 6(a) following Margańska et al. (2019). The axial magnetic field thus also modifies the longitudinal boundary conditions and thereby the charge distribution along the carbon nanotube axis: the wave function envelope for each graphene sublattice is tuned from a $\lambda/4$ shape, with a finite value at one edge, near $B_{\parallel}=0$ to a $\lambda/2$ shape, as expected for a traditional “quantum box”, at large field; see the drawings in Fig. 6(a).

VI Variable coupling model

Based on the arguments brought forward in the previous section, we have constructed a toy model following Margańska et al. (2019); Donarini et al. (2012). Details can be found in Appendix E; our steps are summarized as follows: we simplify the electronic wave function amplitudes on the two graphene sublattices to be of the shape $\psi\propto\sin(\pi\nu x)$ over the length of the quantum dot, where $\phi\in[1,2]$ is used to continuously tune the wave function shape from one antinode to two antinodes and thus approximate the impact of the magnetic field. The total linear charge density $\rho(x)$ is obtained by summing up the charge densities $\propto|\psi|^{2}$ of the sublattices, one being the spatial mirror image of the other Margańska et al. (2019). The vibron is equally simplified as having a deflection amplitude $u(x)\propto\sin(\pi x)$.

Given the strong confinement of our electronic states and the low characteristic energy of our harmonic excitations, the quantum dot occupies likely a smaller part of the nanotube than the vibron. Fig. 6(b) shows both charge density and vibron envelope for a length ration $L_{d}/L_{v}=1/2$ and for a shift in center position $x_{d}-x_{v}=-0.625\,L_{v}$. The charge density is plotted as function of both the position and $\phi$, for the range $\phi\in[1.25,2]$ approximately covered by a K’ state when the axial magnetic field is tuned from zero to large values, cf. Fig. 6(a) and Margańska et al. (2019).

Following Donarini et al. (2012), we assume the interaction energy between linear charge density and deformation potential to be

[TABLE]

and the Franck-Condon parameter to be $g\propto|E_{\text{ev}}|^{2}$. The result for $g(\phi)$ is plotted in Fig. 6(c), see the red solid line. Recalling that $\phi=2$ here describes the asymptotic limit of large $B_{\parallel}$, the result qualitatively agrees with the behaviour of $g$ for a K’ state in our experiment, cf. Fig. 3(f) for the low-field and Fig 2(f) for the high-field behavior. For the same magnetic field range, a K state covers the range $\phi\in[1,1.25]$, with the large field limit $\phi=1$. Here, the coupling further decreases and remains small, see the dotted blue line.

Despite the many simplifications, this model describes the essential observations of the measurement. For a K’ state, the Franck-Condon coupling increases from near-zero with magnetic field, reaches a maximum, and then for large field becomes constant at smaller value. For a K state, the coupling remains low. The sudden onset of coupling for the ground state around $B_{\parallel}=2\,\text{T}$, Fig. 2(f), is due to a transition between these two cases. Limitations of the model become clear, however, when we look at the size ratio and placement of electron and vibron. In an ultraclean, suspended nanotube device, we expect the electronic system to be approximately centered in the suspended part. The vibron should cover at least the entire suspended part, which speaks against a spatial arrangement as shown in Fig. 6(b). A parameter search using our toy model has not yielded fundamentally different situations with suitable $g(B_{\parallel})$ behaviour so far. This shall likely require a more realistic and in particular also quantum mechanical treatment.

VII Conclusions

In conclusion, we demonstrate that vibrational sidebands emerge in the $0\leq N_{\text{el}}\leq 2$ transport spectrum of an ultraclean nanotube at a finite magnetic field parallel to the nanotube axis. The sidebands are equidistant, with an oscillator quantum field-independent for $B_{\parallel}\lesssim 3\,\text{T}$. Their evaluation results in a field-dependent Franck-Condon coupling parameter $g(B_{\parallel})$. Our data indicate that predominantly conductance lines corresponding to the addition of a K’ valley electron develop a finite coupling parameter $g$. A tentative mechanism for this can be a field-induced and valley-selective modification of the electronic wavefunction envelope Margańska et al. (2019). Following Donarini et al. (2012), we have developed a simplified classical model and are able to reproduce the essential behaviour of $g$. A realistic match of device and model parameters will require a more detailed theoretical treatment.

It has long been proposed that, similar to existing experiments using the transversal vibration Benyamini et al. (2014), also the longitudinal vibration coupling in a carbon nanotube Franck-Condon system can be controlled via changing the charge distribution along the nanotube Mariani and von Oppen (2009); Donarini et al. (2012). With our experimental results supporting this idea, the integration of the longitudinal vibration mode into quantum technological applications and its targeted manipulation as quantum harmonic oscillator Hüttel et al. (2009) now becomes an interesting challenge.

Acknowledgements.

We would like to thank M. Margańska, M. Grifoni, A. Donarini, Ch. Strunk, E. A. Laird, and P. Hakonen for insightful discussions, and Ch. Strunk and D. Weiss for the use of experimental facilities. The data has been recorded using the Lab::Measurement software package Reinhardt et al. (2019). The authors acknowledge financial support by the Deutsche Forschungsgemeinschaft (Emmy Noether grant Hu 1808/1, GRK 1570, SFB 689, SFB 1277) and by the Studienstiftung des deutschen Volkes.

Author contributions

The device was fabricated by P. L. S. and D. R. S; the low-temperature measurements were performed by all authors jointly. The resulting data was evaluated by P. L. S. and A. K. H.; the manuscript was written and the project was supervised by A. K. H.

Appendix A Harmonicity and excitation energy

Figures 7(a) and (b) display the conductance near the $1\leq N_{\text{el}}\leq 2$ degeneracy point, for $B_{\parallel}=0\,\text{T}$ and $B_{\parallel}=5\,\text{T}$ and over a large bias voltage range $|V_{\text{sd}}|\leq 10\,\text{mV}$. The magnetic field dependent measurements evaluated in Figs. 2 and 3 correspond to traces taken across the single electron tunneling region, in the parameter range indicated in Fig. 7(a) by a yellow marker. Part of such a trace $\text{d}I/\text{d}V_{\text{sd}}(V_{\text{sd}})$ for constant $B_{\parallel}$ and $V_{\text{g}}$ is shown schematically in Fig. 7(c). From it, the peak distances ${\Delta V_{\text{sd}}}_{1}$, ${\Delta V_{\text{sd}}}_{2}$, ${\Delta V_{\text{sd}}}_{3}$ as indicated in the drawing can be extracted. They correspond to excitation energies $\Delta\epsilon_{1}$, $\Delta\epsilon_{2}$, $\Delta\epsilon_{3}$.

In the stability diagrams, conductance lines ending at the $N_{\text{el}}=2$ Coulomb blockade region (i.e., for $V_{\text{sd}}>0$ with negative slope) correspond to 2-electron excitations; lines ending at the $N_{\text{el}}=1$ Coulomb blockade region (i.e., for $V_{\text{sd}}>0$ with negative slope) correspond to 1-electron excitations. In the evaluated parameter region, only lines with negative slope are visible. For them, the conversion from bias voltage distances $\Delta V_{\text{sd}}$ to excitation energies $\Delta\epsilon$, based on the capacitances in the quantum dot system, can be illustrated by the sketch of Fig. 7(b). Given the slopes $\Delta V_{\text{sd}}/\Delta V_{\text{g}}$ of the two edges of the single electron tunneling region in the stability diagram, $s_{1}<0$ and $s_{2}>0$, as indicated in the sketch, the conversion factor $f(B_{\parallel},V_{\text{g}})$ can be derived from elementary geometry as

[TABLE]

Here we indicate with $f(B_{\parallel},V_{\text{g}})$ that the factor can change with magnetic field and gate voltage due to modification of the electronic wave function shapes and thus the charge distributions and capacitances. The value of $f$, as extracted from stability diagrams at $B_{\parallel}=0,5,10\,\text{T}$, is plotted in Fig. 7(c), with a linear fit added. In the plotted range, $f$ varies by only approximately $10\%$.

The result of a manual evaluation of the data of Fig. 2(b), where peak positions have been read out from line plots, is shown in Fig. 7(d). Here, the red squares correspond to the peak distance of the first sideband relative to the base condutance resonance, the black dots to the one of the second sideband relative to the first sideband, and the blue triangles to the one of the third sideband relative to the second sideband. The left axis displays distance in bias voltage, the right axis the value converted into energy, using constant $f(5\,\text{T})$ for simplicity. Within the scatter, the three types of points lie in the same band of values, indicating equidistant quantum states and thus harmonic oscillator behaviour. The oscillator quantum is for $0\,\text{T}\leq B_{\parallel}\leq 6\,\text{T}$ approximately constant at $\Delta V_{\text{sd}}\simeq 0.15\,\text{mV}$ corresponding $\Delta\epsilon=\hbar\omega\simeq 50\,\mu\text{eV}$; this is the region predominantly discussed in the main text.

For larger $B_{\parallel}$, the data points seem to indicate a gradual increase in $\Delta\epsilon$. Given the decreasing signal to noise ratio and limited data for this field range it is unclear whether this is a real effect. Surprisingly, the stability diagram at $B_{\parallel}=10\,\text{T}$, Fig. 1(e), displays $\Delta V_{\text{sd}}\simeq 0.25\,\text{mV}$, which would confirm such an increase.

The theoretical value for the energy quantum of the carbon nanotube longitudinal vibration is given by Sapmaz et al. (2006)

[TABLE]

where $L$ is the nanotube length, $Y$ is Young’s modulus, and $\rho$ is the nanotube mass density. Assuming \rho=1.3\,\text{g/cm{}^{3}} and $Y=1\,\text{TPa}$, this results in Sapmaz et al. (2006)

[TABLE]

For our device, using the contact separation of L=0.7\,\text{\mum} as approximate value of the suspended nanotube segment length, we obtain $\Delta\epsilon_{\text{th}}\simeq 160\,\mu\text{eV}$. While there is a clear deviation, our measurement still lies at the edge of the typical scatter of oscillator quanta observed in experimental literature, see Fig. 8 for an overview.

Appendix B Franck-Condon model and fit procedure

In the context of single electron tunneling through a carbon nanotube, the Franck-Condon coupling parameter $g$ describes the spatial shift of the nanotube equilibrium position as harmonic oscillator when an additional electron is added to it. It is given by $g=({\Delta x_{0}}/{x_{\text{zpf}}})^{2}/2$, where $\Delta x_{0}$ is the shift in equilibrium position, $\Delta x_{0}=x_{0}(N+1)-x_{0}(N)$. In the denominator, $x_{\text{zpf}}=\sqrt{{\hbar}/({m\omega})}$ is the characteristic length scale of the harmonic oscillator, describing the wave function extension of the ground state and/or its zero point fluctuations. For $\hbar\omega=50\,\mu\text{eV}$ and $m=1.3\times 10^{-21}\,\text{kg}$ Schmid et al. (2012), we obtain $x_{\text{zpf}}=1.0\,\text{pm}$.

At finite temperature, a harmonic oscillator can both absorb and emit vibrons. Here, in the limit of low temperature and fast vibrational relaxation compared to the tunnel rates, we assume that for any single electron tunneling process we start out in the $N_{\text{el}}$ electron vibrational ground state. For each number of vibration quanta $n$, there is a distinct overlap $\left|\left.\left<{\Psi_{0}(N_{\text{el}})\,}\right|{\Psi_{n}(N_{\text{el}}+1)}\right>\right|^{2}$ of the $N_{\text{el}}$ electron vibrational ground state with an $N_{\text{el}}+1$ electron, $n$ vibron state. This leads to a series of equidistant steps in current $I(V_{\text{sd}})$ or peaks in differential conductance $\text{d}I/\text{d}V_{\text{sd}}(V_{\text{sd}})$, whenever sufficient energy for reaching the next vibrational state becomes available.

The contributions of the vibrational states and thus the current step heights are given for the limit of low temperature and fast vibrational relaxation by the Poisson formula Braig and Flensberg (2003); Koch et al. (2006),

[TABLE]

at an energy $\epsilon=n\hbar\omega$ from the bare electronic state transition.

The resulting step function of the current, in the idealized case of total absence of effects such as thermal or lifetime broadening, is shown in Fig. 9(a-c) for different values of $g$.

In a measurement, the steps will be broadenend due to finite temperature, finite lifetime of the involved quantum states, and additional mechanisms such as, e.g., potential fluctuations. As approximation we assume that this broadening equally affects all steps. Then, the differential conductance $G=\text{d}I/\text{d}V_{\text{sd}}$ exhibits a sequence of peaks, each corresponding to one step in current, with conductance peak heights proportional to the current step heights. Even though $G$ is in our measurement a derived quantity, it is both a better base for visualization of the phenomena, see the figures of this work, and a better base for numerical curve fitting than the measured current. To minimize the impact of postprocessing, we use the for fits the raw numerically differentiated conductance $\Delta I/\Delta V_{\text{sd}}$, without any smoothing or other numerical filtering applied.

As already mentioned in the main text, we find that our data is fitted well using the typical shape of a thermally broadened Coulomb oscillation,

[TABLE]

where $\gamma$ describes the peak width. For our fit model, we sum up a sequence of these peaks, weighted according to Eq. 5, cutting off at $n=10$ since subsequent terms are negligible for $g\lesssim 1$, and normalize such that the maximum conductance $G_{\text{max}}$ at the electronic base resonance ($n=0$) can be used as convenient fitting parameter. This results in

[TABLE]

Given the combination of large scatter in the data, see, e.g., Fig. 2(c), and the large number of free fit parameters $\left(V_{\text{sd}}^{0},G_{\text{max}},\gamma,g,\Delta V_{\text{sd}}=\hbar\omega/(ef)\right)$ the fit displays instabilities and large error bars. However, it turns out that the fit results for both $\gamma$ and $\Delta V_{\text{sd}}$ scatter in the evaluated regions around field-independent values (also validating our manual evaluation of Fig. 7(d)). As a consequence, we fix both parameters to an average value and re-run the fit with a reduced set of free parameters $\left(V_{\text{sd}}^{0},G_{\text{max}},g\right)$.

To give an insight into the results of this procedure, Figure 10 shows for part of the evaluation both the raw data and the best fit functions; these correspond to data traces of Fig. 2(b) and the fit prarameters plotted in Fig. 2(f) as blue squares.

Appendix C Classification of excited state resonances

In a magnetic field at an angle $\varphi$ to the carbon nanotube axis, the linearized single particle Hamiltonian of an electron in a carbon nanotube quantum dot close to the Dirac point is given by Schmid et al. (2015b); Margańska et al. (2019)

[TABLE]

Here, the Pauli matrices $\hat{\sigma}_{i}$ act on the spin part of the wave function, and the Pauli matrices $\hat{\tau}_{j}$ on the orbital/valley part of the wave function, respectively; the basis is given by spin and valley eigenstates in axial direction.

For a field alignment parallel to the nanotube ($\varphi=0$), already at moderate field (e.g., $B_{\parallel}\sim 0.5\,\text{T}$ Margańska et al. (2019)) the contribution of the orbital angular moment exceeds the valley mixing, and the eigenstates of the Hamiltonian become spin and valley eigenstates, with spin and orbital angular moment aligned parallel to the nanotube axis. With $\sigma=\pm 1$ and $\tau=\pm 1$ describing spin and orbital angular moment direction, the magnetic field dependent terms of Eq. 8 then lead to the energy contribution

[TABLE]

The dominant term is here the orbital contribution Minot et al. (2004); Laird et al. (2015); Margańska et al. (2019). This can be clearly observed, e.g., in Fig. 5, where at large field the two visible conductance resonances correspond to two single-particle states with equal orbital angular moment but opposite spin. As a consequence, we go one step further and classify our states only by orbital angular momentum direction (corresponding to K’ or K valley), i.e. by dominant magnetic field dispersion

[TABLE]

In the case of the second single electron tunneling region where the charge occupation is $1\leq N_{\text{el}}\leq 2$, resonance lines correspond to energy differences $\Delta E_{12}(B_{\parallel})=E_{2}(B_{\parallel})-E_{1}(B_{\parallel})$, and the dispersions in a magnetic field correspondingly to the orbital angular momentum change when a second electron tunnels onto the quantum dot, proportional to

[TABLE]

If the angular momentum of the first electron remains unchanged when the second eletron tunnels onto the dot, only two slopes corresponding to addition of a K’ or K electron remain possible. From this we can classify the conductance lines of Fig. 3(a). In the low-bias region of the figure, two dominant slopes clearly emerge; we identify these with the addition of a K’-valley electron (downward slope) or a K-valley electron (upward slope), respectively Minot et al. (2004); Laird et al. (2015); Margańska et al. (2019).

Appendix D Evaluating the sidebands of excited states

The data points of Fig. 4 are obtained as follows. We separate the conductance resonance pattern into line segments, each terminated by line crossings or anticrossings. On each segment that conforms to one of the two dominant slopes, we select a point $(B_{\parallel},V_{\text{sd}})$ near the high-$B_{\parallel}$ end of the segment; these points are the locations of the symbols in Fig. 4(a). The five traces $G(V_{\text{sd}})$ closest to this point in magnetic field are evaluated using the fit procedure as described above. To reduce the impact of outliers, the plotted values in Fig. 4 are then given by the median of the fit parameters $g$ from these five evaluations.

Appendix E Variable coupling model

One of the key results of Margańska et al. (2019), based on Akhmerov and Beenakker (2008); Castro Neto et al. (2009); Marganska et al. (2011), is that a magnetic field parallel to the axis of a carbon nanotube does not only modify the circumferential electronic wave function via an Aharonov-Bohm phase, but also the wave function in axial direction. While a typical one-dimensional “quantum box”, i.e., a potential well with hard walls of infinite height, forces the wave function of a trapped particle to zero at both ends, this construction is not possible for carbon nanotubes. With the exception of armchair nanotubes, there are no solutions that allow the electronic wave function to become zero on both ends for both graphene sublattices. Instead, the longitudinal quantization can be given by the condition that the wave function on each sublattice becomes zero at the end where that particular sublattice provides the majority of boundary atoms. With $L$ as the length of the nanotube segment, this introduces a so-called cross-quantization condition Margańska et al. (2019),

[TABLE]

The axial wave number $\kappa_{\parallel}$ and the circumferential wave number $\kappa_{\perp}$ of an electronic state are coupled; one example solution is sketched in Fig. 6(a) following Margańska et al. (2019). The axial magnetic field tunes the longitudinal profile and charge distribution of the electronic wave function from a $\lambda/4$ resonator like shape, with a finite value at one edge, near $B_{\parallel}=0$ all the way to a $\lambda/2$ resonator like shape at high field; see the schematic drawings in Fig. 6(a).

In Margańska et al. (2019), wave function envelopes have been calculated numerically using a tight-binding approach. Here, we approximate these results using a simple functional dependence: for a quantum dot of length 1, we write for sublattice A

[TABLE]

and accordingly for sublattice B

[TABLE]

The parameter $\phi$ describes the number of wave function antinodes along the quantum dot, i.e., in the range $x\in[0,1]$. Half-wavelength resonator solutions are given by $\phi=1,2,\dots$, quarter-wavelength resonator solutions are given by $\phi=0.5,1.5,2.5,\dots$. As can be seen in Fig. 6(a), increasing a magnetic field from zero continuously tunes $\phi$, for a K’ state from $\phi\approx 1.25$ towards $\phi=2$, for a K state from $\phi\approx 1.25$ towards $\phi=1$.

To obtain the total linear electron density, we now sum up the densities on the two sublattices, while normalizing such that a change in $\phi$ does not modify the total charge,

[TABLE]

The result is plotted in Fig. 11(a).

Reference Donarini et al. (2012) discusses the electron-vibron interaction of an interacting electron system using a Tomonaga–Luttinger model. From the interaction Hamiltonian used there, Eq. (12) in Donarini et al. (2012), coupling the electron density to the deformation potential, we take inspiration of our interaction energy,

[TABLE]

where $u(x)$ is the vibration envelope of the mechanical mode. In addition, also following Donarini et al. (2012) and its nomenclature, we allow for the vibron and the quantum dot to be shifted relative to each other and to be of different length; we parametrize this using length ratio $\lambda=L_{v}/L_{d}$ and relative position of the centres $\delta=(x_{v}-x_{d})/L_{d}$ of vibron and quantum dot. Fig. 11(b) plots for $\phi=1$ the interaction energy $\left|E_{\text{ev}}(\lambda,\delta)\right|$, reproducing basic features of the corresponding Tomonaga–Luttinger model plot, left panel of Fig. 5 in Donarini et al. (2012).

Given the comparatively small vibrational energy quantum in our device, cf. Fig. 8, and the strong electrostatic confinement of the quantum dot, we consider it likely that the quantum dot is smaller than the vibron envelope, i.e., $\lambda>1$. We are now interested in the magnetic field induced modification of the Franck-Condon parameter $g\propto E_{\text{ev}}^{2}$ Donarini et al. (2012), and thus plot in Fig. 11(c) this value as function of $\phi$ and $\delta$, for a fixed $\lambda=2$. The solid/dotted line corresponds to $\delta=1.25$, the value selected for the trace cut in Fig. 6(c).