prof. C. Strunk
prof. D. Weiss
Department of Applied Physics, Aalto University as visiting professor. This is a great chance to make contacts and contribute to projects there, as well as to learn techniques and push our own project planning ahead.
For Regensburg this means that I am currently not accepting any students for thesis projects anymore, and that it may be quite difficult to find time for oral exams (like those "Modulprüfungen").
Coulomb Blockade Spectroscopy of a MoS2 Nanotube" has been accepted for publication by pssRRL Rapid Research Letters.
Everybody is talking about novel semiconductor materials, and in particular the transition metal dichalcogenides (TMDCs), "layer materials" similar to graphene. With a chemical composition of TX2, where the transition metal T is, e.g., tungsten W or molybdenum Mo, and the chalcogenide X is, e.g., sulphur S or selenium Se, a wide range of interesting properties is expected.
What's by far not so well known is that many of these materials also form nanotubes, similar to carbon nanotubes in structure but with distinct properties inherited from the planar system. Here, we present first low temperature transport measurements on a quantum dot in a MoS2 nanotube. The metallic contacts to the nanotube still require a lot of improvements, but the nanotube between them acts as clean potential well for electrons.
Also, our measurements show possible traces of quantum confined behaviour. This is something that has not been achieved yet in planar, lithographically designed devices - since these have by their very geometric nature larger length scales. It means that via transport spectroscopy we can learn about the material properties and its suitability for quantum electronics devices.
A lot of complex physical phenomena have been predicted for MoS2, including spin filtering and intrinsic, possibly topologic superconductivity - a topic of high interest for the quantum computing community, where larger semiconductor nanowires are used at the moment. So this is the start of an exciting project!
"Coulomb Blockade Spectroscopy of a MoS2 Nanotube"
S. Reinhardt, L. Pirker, C. Bäuml, M. Remskar, and A. K. Hüttel
Physica Status Solidi RRL, doi:10.1002/pssr.201900251 (2019); arXiv:1904.05972 (PDF)
Nature Index 2019 Annual Tables have been published, and there is a valuable new addition: the tables now include a "normalized ranking", where the quality of a university's research output, and not its quantity counts. If we look at the world-wide natural sciences ranking, University of Regensburg is at spot 44, best of all universities in Germany, and in a similar ranking range as, e.g., University of Oxford, University of Tokyo, or University of California San Francisco! Cheers and congratulations!
When and where? Wednesdays, 12h - 14h, seminar room PHY 9.1.10. The next lecture is on 8 May 2019
See you next wednesday!
Regensburg University has published a press release (in German) on our recent Physical Review Letters "Editor's Suggestion" publication, "Shaping Electron Wave Functions in a Carbon Nanotube with a Parallel Magnetic Field". Read it on the university web page!
(A summary in English can be found in a previous blog post.)
PRL published: Shaping electron wave functions in a carbon nanotube with a parallel magnetic field (posted 2019-02-27)
Shaping electron wave functions in a carbon nanotube with a parallel magnetic field" has been published as Editor's Suggestion in Physical Review Letters.
When a physicist thinks of an electron confined to a one-dimensional object such as a carbon nanotube, the first idea that comes to mind is the ?particle in a box? from elementary quantum mechanics. A particle can behave as a wave, and in this model it is essentially a standing wave, reflected at two infinitely high, perfect barrier walls. The mathematical solutions for the wave function describing it are the well-known half-wavelength resonator solutions, with a fundamental mode where exactly half a wavelength fits between the walls, a node of the wave function at each wall and an antinode in the center.
In this publication, we show how wrong this first idea can be, and what impact that has. In a carbon nanotube as quasi one-dimensional system, an electron is not in free space, but confined to the lattice of carbon atoms which forms the nanotube walls. This hexagonal lattice, the same that also forms in planar form graphene, is called bipartite, since every elementary cell of the lattice contains two carbon atoms; one can imagine the nanotube wall as being built out of two sub-lattices, with one atom per cell each, that are shifted relative to each other. Surprisingly, the hexagonal bipartite lattice does not generally support the half-wavelength solutions mentioned above, where the electronic wave function becomes zero at the edges. In each sublattice, we can only force the wave function to zero at one end of the nanotube "box"; its value at the other end remains finite. This means that the wave function shape for each of the two sublattices is more similar to that of a quarter wavelength resonator, where one end displays a node, the other an antinode. The two sublattice wave functions are mirrored in shape to each other, with node and antinode swapping position.
When we now apply a magnetic field along the carbon nanotube, a magnetic flux enters the nanotube, and the boundary conditions for the electron wave function change via the Aharonov-Bohm effect. Astonishingly, its shape along the carbon nanotube can thereby be tuned between half-wavelength and quarter-wavelength behaviour. This means that the probability of the trapped electron to be near the contacts changes, and with it the tunnel current, leading to a very distinct behaviour of the electronic conductance. It turns out that our measurement and the corresponding calculations are agreeing very well. Thus, our work shows the impact of a non-trivial host crystal on the electronic behaviour, important for many novel types of material.
"Shaping electron wave functions in a carbon nanotube with a parallel magnetic field"
M. Marganska, D. R. Schmid, A. Dirnaichner, P. L. Stiller, Ch. Strunk, M. Grifoni, and A. K. Hüttel
Physical Review Letters 122, 086802 (2019), Editor's Suggestion; arXiv:1712.08545 (PDF, supplementary information)
Comp. Phys. Comm. published: "Lab::Measurement - a portable and extensible framework for controlling lab equipment and conducting measurements" (posted 2018-10-18)
"Lab::Measurement ? a portable and extensible framework for controlling lab equipment and conducting measurements", describing our measurement software package Lab::Measurement, has been published in Computer Physics Communications.
Lab::Measurement is a collection of object-oriented Perl 5 modules for controlling lab instruments, performing measurements, and recording and plotting the resultant data. Its operating system independent driver stack makes it possible to use nearly identical measurement scripts both on Linux and Windows. Foreground operation with live plotting and background operation for, e.g., process control are supported. For more details, please read our article, visit the Lab::Measurement homepage, or visit Lab::Measurement on CPAN!
"Lab::Measurement - a portable and extensible framework for controlling lab equipment and conducting measurements"
S. Reinhardt, C. Butschkow, S. Geissler, A. Dirnaichner, F. Olbrich, C. Lane, D. Schröer, and A. K. Hüttel
Comp. Phys. Comm. 234, 216 (2019); arXiv:1804.03321 (PDF)
PRL accepted: Nanomechanical characterization of the Kondo charge dynamics in a carbon nanotube (posted 2018-05-18)
Nanomechanical characterization of the Kondo charge dynamics in a carbon nanotube" has been accepted for publication by Physical Review Letters.
The Kondo effect is a many-body phenomenon at low temperature that results from a quantum state degeneracy, as, e.g., the one of spin states in absence of a magnetic field. In its simplest case, it makes a quantum dot, in our case a carbon nanotube with some trapped electrons on it, behave very different for an even and an odd number of electrons. At an even number of trapped electrons, no current can flow through the nanotube, since temperature and applied bias voltage are too low to charge it with one more elementary charge; this phenomenon is called Coulomb blockade. Strikingly, at odd electron number, when two degenerate quantum states in the nanotube are available, Coulomb blockade seems not to matter, and a large current can flow. Theory explains this by assuming that a localized electron couples to electrons in the contacts, forming a combined, delocalized singlet quantum state.
What carries the Kondo-enhanced current, and how does the electric charge now accumulate in the carbon nanotube? We use the vibration of the macromolecule to measure this. As also in the case of, e.g., a guitar string, the resonance frequency of a nanotube changes when you pull on it; in the case of the carbon nanotube this is sensitive enough to resolve fractions of the force caused by a single elementary charge. From the vibration frequency, as function of the electrostatic potential, we calculate the average number of electrons on the nanotube, and can then compare the odd and even number cases.
A surprising result of our evaluation is that the charge trapped on the nanotube behaves the same way in the even and odd occupation case, even though the current through it is completely different. Sequential tunneling of electrons can model the charge accumulation, and with it the mechanical behaviour. The large Kondo current is carried by virtual occupation of the nanotube alone, i.e., electrons tunneling on and immediately off again so they do not contribute to the charge on it.
"Nanomechanical Characterization of the Kondo Charge Dynamics in a Carbon Nanotube"
K. J. G. Götz, D. R. Schmid, F. J. Schupp, P. L. Stiller, Ch. Strunk, and A. K. Hüttel
Physical Review Letters 120, 246802 (2018); arXiv:1802.00522 (PDF, HTML, supplementary information)
Today's news is that we have submitted a manuscript for publication, describing Lab::Measurement and with it our approach towards fast, flexible, and platform-independent measuring with Perl! The manuscript mainly focuses on the new, Moose-based class hierarchy. We have uploaded it to arXiv as well; here is the (for now) full bibliographic information of the preprint:
"Lab::Measurement - a portable and extensible framework for controlling lab equipment and conducting measurements"If you're using Lab::Measurement in your lab, and this results in some nice publication, then we'd be very grateful for a citation of our work - for now the preprint, and later hopefully the accepted version.
FOSDEM 2018, the "Free and Open Source Developers' European Meeting", takes place 3-4 February at Universite Libre de Bruxelles, Campus Solbosch, Brussels - and our measurement control software Lab::Measurement will be presented there in the Perl devrooom! As all of FOSDEM, the talk will also be streamed live and archived; more details on this follow later. Here's the abstract:
Perl in the Physics Lab
Let's visit our university lab. We work on low-temperature nanophysics and transport spectroscopy, typically measuring current through experimental chip structures. That involves cooling and temperature control, dc voltage sources, multimeters, high-frequency sources, superconducting magnets, and a lot more fun equipment. A desktop computer controls the experiment and records and evaluates data.
linked PDF for the full talk announcement.
"Millikelvin transport experiments on carbon nanotubes - nanoelectromechanics, spectroscopy, and more"
PRL accepted: Secondary electron interference from trigonal warping in clean carbon nanotubes (posted 2016-09-16)
Secondary electron interference from trigonal warping in clean carbon nanotubes", was accepted for publication in Physical Review Letters.
Imagine a graphene "sheet" of carbon atoms rolled into a tube - and you get a carbon nanotube. Carbon nanotubes come in many variants, which influence strongly their electronic properties. They have different diameter, but also different "chiral angle", describing how the pattern of the carbon atoms twists around the tube axis. In our work, we show how to extract information on the nanotube structure from measurements of its conductance. At low temperature, electrons travel ballistically through a nanotube and are only scattered at its ends. For the quantum-mechanical electron wavefunction, metallic nanotubes act then analogous to an optical Fabry-Perot interferometer, i.e., a cavity with two semitransparent mirrors at either end, where a wave is partially reflected. Interference patterns are obtained by tuning the wavelength of the electrons; the current through the nanotube oscillates as a function of an applied gate voltage. The twisted graphene lattice then causes a distinct slow current modulation, which, as we show, allows a direct estimation of the chiral angle. This is an important step towards solving a highly nontrivial problem, namely identifying the precise
molecular structure of a nanotube from electronic measurements alone.
"Secondary electron interference from trigonal warping in clean carbon nanotubes"
A. Dirnaichner, M. del Valle, K. J. G. Götz, F. J. Schupp, N. Paradiso, M. Grifoni, Ch. Strunk, and A. K. Hüttel
accepted for publication in Physical Review Letters; arXiv:1602.03866 (PDF, supplemental information)
Nanotechnology accepted: Co-sputtered MoRe thin films for carbon nanotube growth-compatible superconducting coplanar resonators (posted 2016-02-04)
Co-sputtered MoRe thin films for carbon nanotube growth-compatible superconducting coplanar resonators" has just been accepted for publication in Nanotechnology.
For quite some time we have been working on techniques to combine ultra-clean carbon nanotubes and their regular electronic spectrum with superconducting material systems. One of our objectives is to perform high-frequency measurements on carbon nanotube nano-electromechanical systems at millikelvin temperatures. With this in mind we have established the fabrication and characterization of compatible superconducting coplanar resonators in our research group. A serious challenge here was that the high-temperature process of carbon nanotube growth destroys most metal films, or if not, at least lowers the critical temperature Tc of superconductors so much that they are not useful anymore.
In the present manuscript, we demonstrate deposition of a molybdenum-rhenium alloy of variable composition by simultaneous sputtering from two sources. We characterize the resulting thin films using x-ray photoelectron spectroscopy, and analyze the saturation of the surface layers with carbon during the nanotube growth process. Low-temperature dc measurements show that specifically an alloy of composition Mo20Re80 remains very stable during this process, with large critical currents and critical temperatures even rising to up to Tc~8K. We use this alloy to fabricate coplanar resonator structures and demonstrate even after a nanotube growth high temperature process resonant behaviour at Gigahertz frequencies with quality factors up to Q~5000. Observation of the temperature dependent behaviour shows that our devices are well described by Mattis-Bardeen theory, in combination with dissipation by two-level systems in the dielectric substrate.
"Co-sputtered MoRe thin films for carbon nanotube growth-compatible superconducting coplanar resonators"
K. J. G. Götz, S. Blien, P. L. Stiller, O. Vavra, T. Mayer, T. Huber, T. N. G. Meier, M. Kronseder, Ch. Strunk, and A. K. Hüttel
accepted for publication in Nanotechnology; arXiv:1510.00278 (PDF)
german research" magazine of the DFG includes an article about the work of our research group! This is a translation of a previous publication in the German language journal "Forschung" of the DFG. Enjoy!
"Carbon Nanotubes: Strong, Conductive and Defect-Free"
Carbon nanotubes are a fascinating material. In experiments at ultra-low temperatures, physicists make their different properties interact with one another - and in so doing find answers to fundamental questions.
Andreas K. Hüttel
german research 3/2015, 24-27 (2015) (PDF)
APL accepted: Liquid-induced damping of mechanical feedback effects in single electron tunneling through a suspended carbon nanotube (posted 2015-09-14)
One of the surprises that suspended, clean carbon nanotubes have in store is that they can start vibrating strongly at millikelvin temperatures without any applied radio-frequency driving signal. This was proposed theoretically several years ago by Usmani et al., as a strong feedback between the transversal vibration of the nanotube and the single electron tunneling through it. The effect was identified in measurements, and for example in a previous publication we have shown that damping induced by a magnetic field can suppress it.
Here, we demonstrate how one and the same device behaves distinctly different depending on the environment medium (or lack of the latter): we compare measurements made at the same temperature in a conventional dilution refrigerator, where the chip is placed into a vacuum chamber, and in a so-called top-loading dilution refrigerator, where the chip is inserted into the 3He/4He liquid of the mixing chamber. The overall electronic properties of the device do not change much, even though the thermal cycling could cause a lot of damage and has done so in the past for other devices. We can here even extract a rough estimate of the liquid helium dielectric constant by comparing the slightly shifted Coulomb oscillation positions of the two measurements.
However, a striking difference appears when looking at finite bias conductance and the mechanical feedback effects. In the viscous helium liquid, the resonator is damped and the vibrations are suppressed, and the unperturbed electronic transport spectrum emerges. Such an inert, liquid environment can thus be used to do transport spectroscopy at high transparency of the tunnel barriers and high applied bias voltages - parameter regions interesting for e.g. non-equilibrium Kondo phenomena, where otherwise mechanically-induced features would make data evaluation highly challenging.
"Liquid-induced damping of mechanical feedback effects in single electron tunneling through a suspended carbon nanotube"
D. R. Schmid, P. L. Stiller, Ch. Strunk, and A. K. Hüttel
Applied Physics Letters 107, 123110 (2015); arXiv:1407.2114 (PDF)
PRB accepted: Transport across a carbon nanotube quantum dot contacted with ferromagnetic leads: experiment and non-perturbative modeling (posted 2015-05-04)
When ferromagnetic materials are used as contacts for a carbon nanotube at low temperature, the current is strongly influenced by the direction of the contact magnetization via the so-called tunneling magnetoresistance (TMR). Since the nanotube contains a quantum dot, in addition its electronic energy levels play an important role; the TMR depends on the gate voltage value and can reach large negative and positive values. Here, in another fruitful joint experimental and theoretical effort, we present both measurements of the gate-dependent TMR across a "shell" of four Coulomb oscillations, and model them in the so-called "dressed second order" framework. The calculations nicely reproduce the characteristic oscillatory patterns of the TMR gate dependence.
"Transport across a carbon nanotube quantum dot contacted with ferromagnetic leads: experiment and non-perturbative modeling"
A. Dirnaichner, M. Grifoni, A. Prüfling, D. Steininger, A. K. Hüttel, and Ch. Strunk
Phys. Rev. B 91, 195402 (2015); arXiv:1502.02005 (PDF)
Broken SU(4) symmetry in a Kondo-correlated carbon nanotube" has been accepted for publication in Physical Review B.
This manuscript is the result of a joint experimental and theoretical effort. We demonstrate that there is a fundamental difference between cotunneling and the Kondo effect - a distinction that has been debated repeatedly in the past. In carbon nanotubes, the two graphene-derived Dirac points can lead to a two-fold valley degeneracy in addition to spin degeneracy; each orbital "shell" of a confined electronic system can be filled with four electrons. In most nanotubes, these degeneracies are broken by the spin-orbit interaction (due to the wall curvature) and by valley mixing (due to, as recently demonstrated, scattering at the nanotube boundaries). Using an externally applied magnetic field, the quantum states involved in equilibrium (i.e., elastic, zero-bias) and nonequilibrium (i.e., inelastic, finite bias) transitions can be identified. We show theoretically and experimentally that in the case of Kondo correlations, not all quantum state pairs contribute to Kondo-enhanced transport; some of these are forbidden by symmetries stemming from the carbon nanotube single particle Hamiltonian. This is distinctly different from the case of inelastic cotunneling (at higher temperatures and/or weaker quantum dot-lead coupling), where all transitions have been observed in the past.
"Broken SU(4) symmetry in a Kondo-correlated carbon nanotube"
D. R. Schmid, S. Smirnov, M. Marganska, A. Dirnaichner, P. L. Stiller, M. Grifoni, A. K. Hüttel, and Ch. Strunk
Phys. Rev. B 91, 155435 (2015) (PDF)
forschung" magazine of the DFG, published just a few days ago, includes an article about the work of our research group (in German)! Enjoy!
"Zugfest, leitend, defektfrei"
Kohlenstoff-Nanoröhren sind ein faszinierendes Material. In Experimenten bei ultratiefen Temperaturen versuchen Physiker, ihre verschiedenen Eigenschaften miteinander in Wechselwirkung zu bringen ? und so Antworten auf grundlegende Fragen zu finden.
Andreas K. Hüttel
forschung 4/2014, 10-13 (2014) (PDF)
NJP accepted: Thermally induced subgap features in the cotunneling spectroscopy of a carbon nanotube (posted 2014-11-10)
Thermally induced subgap features in the cotunneling spectroscopy of a carbon nanotube" has been accepted for publication by New Journal of Physics.
In a way, this work is directly building on our previous publication on thermally induced quasiparticles in niobium-carbon nanotube hybrid systes. As a contribution mainly from our theory colleagues, now the modelling of transport processes is enhanced and extended to cotunneling processes within Coulomb blockade. A generalized master equation based on the reduced density matrix approach in the charge conserved regime is derived, applicable to any strength of the intradot interaction and to finite values of the superconducting gap.
We show both theoretically and experimentally that also in cotunneling spectroscopy distinct thermal "replica lines" due to the finite quasiparticle occupation of the superconductor occur at higher temperature T~1K: the now possible transport processes lead to additional conductance both at zero bias and at finite voltage corresponding to an excitation energy; experiment and theoretical result match very well.
"Thermally induced subgap features in the cotunneling spectroscopy of a carbon nanotube"
S. Ratz, A. Donarini, D. Steininger, T. Geiger, A. Kumar, A. K. Hüttel, Ch. Strunk, and M. Grifoni
New J. Phys. 16, 123040 (2014), arXiv:1408.5000 (PDF)
Today we've received the good news that our Emmy Noether project on the electronic and nano-electromechanical properties of carbon nanotubes has been given a positive intermediate evaluation from the referees. This means funding for an additional period will be granted. Cheers!
PRB Rapid Comm. accepted: Sub-gap spectroscopy of thermally excited quasiparticles in a Nb contacted carbon nanotube quantum dot (posted 2014-05-25)
Sub-gap spectroscopy of thermally excited quasiparticles in a Nb contacted carbon nanotube quantum dot" was just accepted for publication by Physical Review B as a Rapid Communication.
Once again we visit the topic of a carbon nanotube quantum dot with superconducting contacts, and again we use niobium for these contacts. Only, this time the connection between the nanotube and the superconductor is pretty bad, i.e., very low electronic tunnel rates. In the end this means that the superconductor does not influence the localized electronic system very much. However, in the metallic contacts we still have a superconductor, meaning electrons pair up into Cooper pairs, and for free quasiparticles carrying only one electron charge an energy gap evolves (the so-called BCS density of states).
Since we're using niobium, we can see superconducting effects over a fairly large temperature range. If we increase the temperature enough, thermal quasiparticles are excited over this energy gap. This precisely is what we observe in our experiment, as additional discrete lines in the transport spectrum. A detailed theoretical analysis of single electron tunneling, in a close cooperation with the research group Prof. Dr. M. Grifoni, confirms our results very well, especially also the temperature dependence of the features visible in the measurements.
In addition there is an interesting bonus to be had here. The thermally activated processes lead to a distinct double-peak of the conductance at zero bias, and the relative height of the two maxima is controlled by the degeneracy of the quantum dot ground states involved in tunneling. This means that looking at the thermally activated current provides additional information to identify the carbon nanotube level spectrum, even if it is not immediately clear from the usual "Coulomb diamond spectroscopy".
"Sub-gap spectroscopy of thermally excited quasiparticles in a Nb contacted carbon nanotube quantum dot"
M. Gaass, S. Pfaller, T. Geiger, A. Donarini, M. Grifoni, A. K. Hüttel, and Ch. Strunk
Phys. Rev. B 89, 241405(R) (2014), arXiv:1403.4456 (PDF)
PRB accepted: Temperature dependence of Andreev spectra in a superconducting carbon nanotube quantum dot (posted 2014-01-08)
When you place a carbon nanotube at low temperature between contacts made from a superconducting metal, lots of interesting things happen. Strongly simplifying, currents in a superconductor are carried by Cooper pairs of two electrons each, while the localized electronic system in the carbon nanotube is normal-conducting and carries single electrons. One mechanism at a superconductor - normal conductor interface that mediates between these two types of charge transport is so-called Andreev reflection. An electron from the normal conductor enters the superconductor, at the same time a "missing electron", i.e. a "hole where an electron should be", is sent back into the normal conductor. The total charge passing through the interface is 2e, just right to form a Cooper pair. The superconductor-nanotube-superconductor system consistis of two such interfaces back to back; analogous to box potential, multiple reflections on both sides lead to the formation of bound quantum states within the nanotube, the so-called Andeev bound states (ABS).
So far, all other observations of ABS involved aluminum, which has a fairly low critical temperature and critical field. What is new in our work is that we use niobium as superconducting material, with higher critical temperature and larger energy gap. We can increase the temperature to over 1K and still see the superconductivity plus the ABS in the transport spectrum. This way, we can observe how thermal population of an excited Andreev state takes place. Additionally we observe a second pair of Andreev states in the larger superconducting energy gap, and a surprising multi-loop behaviour. All these effects are successfully modelled by calculations based on the superconducting Anderson model, in a collaboration with Alfredo Levy Yeyati and Alvaro Martin-Rodero from Universidad Autonoma de Madrid.
"Temperature dependence of Andreev spectra in a superconducting carbon nanotube quantum dot"
A. Kumar, M. Gaim, D. Steininger, A. Levy Yeyati, A. Martin-Rodero, A. K. Hüttel, and C. Strunk
Physical Review B 89, 075428 (2014), arXiv:1308.1020 (PDF)
SFB 689 "Spin phenomena in reduced dimensions" funding also renewed for another 4 years! (posted 2013-11-21)
SFB 689 "Spin phenomena in reduced dimensions" has now also officially received a renewal of funding for four more years. Time to celebrate - and then do a lot more of fascinating research!
research training group "Electronic Properties of Carbon Based Nanostructures" (GRK 1570) has been extended until 2018! The program focuses on the experimental and theoretical investigation of carbon-based nanostructures, i.e. devices based on graphene, carbon nanotubes, aromatic molecules or hybrids of those. A large number of projects is involved, with topics as various as e.g. the transport spectroscopy and analysis of electronic interactions in ultra-clean carbon nanotubes (our group), atomic force microscopy based research on forces in molecular electronics, or femtosecond plasmonics in graphene. More details on the research activities can be found on the projects web page, including direct links to the participating research groups. Thanks a lot to everyone who helped making this happen!
pss(b) accepted: Negative frequency tuning of a carbon nanotube nano-electromechanical resonator under tension (posted 2013-09-27)
The observed effect can be explained via so-called electrostatic softening of the vibration mode. Let us assume that the carbon nanotube is very close to the gate and vibrates towards and away from it. The capacitance between gate and nanotube varies within one oscillation cycle, and thereby the electrostatic force between these two obtains an additional position-dependent component. This can be seen as an electrodynamic contribution to the spring constant of the resonator; it is negative and thereby decreases the resonance frequency. We can estimate the size of this effect and obtain indeed consistent values for our sample geometry.
"Negative frequency tuning of a carbon nanotube nano-electromechanical resonator under tension"
P. L. Stiller, S. Kugler, D. R. Schmid, C. Strunk, and A. K. Hüttel
accepted for publication by physica status solidi (b), arXiv:1304.5092 (PDF)
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