A coherent nanomechanical oscillator driven by single-electron tunnelling

TL;DR

This paper demonstrates that a single-electron transistor integrated with a nanomechanical resonator can produce coherent mechanical oscillations, exhibiting laser-like properties such as coherence, injection locking, and frequency narrowing.

Contribution

It provides experimental verification that strong electron tunnelling coupling can induce self-sustaining coherent mechanical oscillations in a nanomechanical system.

Findings

Confirmed coherent oscillations in a carbon nanotube transistor system.

Observed laser-like behaviors including injection locking.

Demonstrated frequency narrowing through feedback mechanisms.

Abstract

A single-electron transistor incorporated as part of a nanomechanical resonator represents an extreme limit of electron-phonon coupling. While it allows for fast and sensitive electromechanical measurements, it also introduces backaction forces from electron tunnelling which randomly perturb the mechanical state. Despite the stochastic nature of this backaction, under conditions of strong coupling it is predicted to create self-sustaining coherent mechanical oscillations. Here, we verify this prediction using time-resolved measurements of a vibrating carbon nanotube transistor. This electromechanical oscillator has intriguing similarities with a laser. The single-electron transistor, pumped by an electrical bias, acts as a gain medium while the resonator acts as a phonon cavity. Despite the unconventional operating principle, which does not involve stimulated emission, we confirm that the output is coherent, and demonstrate other laser behaviour including injection locking and frequency narrowing through feedback.

Equations

$$P(V^2)\propto e^{-V^2/V_{\text{T}}^2},$$

$$P(V^2)\propto e^{-(V^2-V_0^2{)}^2/{\sigma }_V^4}+A_{\mathrm{T}}{e}^{-V^2/V_{\text{T}}^2}$$

$$g^{(1)}(\tau )\equiv \frac{⟨V_{\text{I}}(t)V_{\text{I}}(t+\tau )⟩}{⟨V_{\text{I}}^2(t)⟩},$$

$$g^{(2)}(\tau )\equiv \frac{⟨V^2(t)V^2(t+\tau )⟩}{⟨V^2(t){⟩}^2}.$$

Full text

Backaction forces are an inescapable accompaniment to nanomechanical measurements. While their ultimate limit is set by quantum uncertainty Clerk2010 , in practical devices backaction often causes heating, damping, and dynamical instability even well before this limit is reached. Among the most sensitive measurement probe for nanomechanics is the single-electron transistor (SET), which transduces motion with a precision that can approach the standard quantum limit Schoelkopf1998a ; LaHaye2004 . However, the price is that the force exerted even by individual electrons modifies the mechanical dynamics. This results in strong electron-phonon coupling Mozyrsky2006 ; Steele2009 ; Lassagne2009 , leading to additional dissipation, frequency softening, nonlinearity, and cooling Naik2006 . Here, we show that the backaction force - due to stochastic single-electron tunnelling events - can also be harnessed to create a self-sustained oscillating state in a nanomechanical resonator. The resulting device is analogous to a laser, where the optical field is replaced by the mechanical displacement. In contrast to existing phonon lasers pumped by optical or mechanical drives Vahala2009 ; Grudinin2010 ; Mahboob2013 , this oscillator is driven by a constant electrical bias. The device exhibits several laser characteristics, detected via its electrical emission, including phase and amplitude coherence and injection locking. The resulting oscillator serves both as a novel on-chip phonon source and to explore the connection between the physics of backaction and of lasers.

To enter this regime of strong backaction, the SET, serving as a two-level system, must couple strongly to a mechanical resonator serving as a phonon cavity (Fig. 1a). As a high-quality resonator, we use a suspended carbon nanotube Sazonova2004 . Nanotubes have both low mass and high mechanical compliance, which are favourable for strong electron-phonon interaction Steele2009 ; Lassagne2009 ; Wen2018 ; DeBonis2018 ; Khivrich2019 . The selected nanotube is a narrow-gap semiconductor, allowing the SET to be defined in the nanotube itself using tunnel barriers at each end and a conducting segment near the middle. The two relevant SET states are the configurations with and without an excess electron. Flexural vibration of the nanotube modulates the electrical potential experienced by the SET, causing the current to depend on the displacement; at the same time, each added electron exerts a force that is larger than both quantum and thermal force fluctuations (see Supplementary Information).

The combination of these effects sets up an electromechanical feedback with rich predicted behaviour Armour2004 . If the SET’s energy splitting is resonant with the mechanical frequency, electrical excitations should be able to pump the resonator in a direct analogue of the micromaser Rodrigues2007 . More surprisingly, a laser-like instability is predicted even in a non-resonant situation, with complex dynamics that depend on level alignment and damping, and go beyond conventional laser behaviour Bennett2006 ; Usmani2007 . Previous experiments measuring time-average current through a nanotube have provided strong evidence for a threshold between resonance and oscillation Huttel2009 ; Steele2009 ; Eichler2011 . However, to test these predictions by fully characterizing the resulting states requires time-resolved displacement measurements Tsioutsios2017 ; Barnard2019 , which have not yet been possible in this regime of strong backaction.

Backaction turns a resonator into an oscillator

To explore these dynamic effects, we implemented an electromechanical circuit for measuring the nanotube’s vibrations directly Wen2018 (Fig. 1b). The carbon nanotube is stamped across metallic contact electrodes to give a vibrating segment of length $800$ nm Wu2010 , and is measured at a temperature of 25 mK. Voltages applied to five finger gates beneath the nanotube (labelled G1-G5) are used both to tune the electrical potential and to actuate vibrations by injecting an RF tone with drive power $P_{\text{D}}$. A voltage bias $V_{\text{DS}}$ is applied between the contacts to drive a current $I$. To configure the nanotube as an SET, the gate voltages are set to tune an electron tunnel barrier near each contact. The conductance thus depends strongly on the displacement, which allows sensitive electromechanical readout via the current through the nanotube. The radio-frequency (RF) part of the current is passed through an impedance transformer and then amplified, with the primary amplifier being an ultra-low-noise SQUID Schupp2018 . The resulting RF output voltage $V_{\text{out}}$ is therefore proportional to the instantaneous displacement and is a sensitive time-resolved record of the mechanical vibrations Wen2018 .

To identify signatures of electromechanical feedback, we first measure the DC conductance as a function of bias and DC gate voltage $V_{\text{G}}$ applied to gate G2 (Fig. 2a). Superimposed on the diamond pattern characteristic of single-electron charging are irregular sharp ridges of strongly positive or negative conductance as the nanotube switches between high and low-current states. Such features are associated with the onset of mechanical instability for bias exceeding a critical threshold Steele2009 ; Usmani2007 .

We detect the mechanical resonance by fixing the bias voltage and measuring the transmission of the drive tone to the RF amplifier input. When the drive frequency matches the mechanical resonance, the resulting motion relative to the gate electrodes changes the chemical potential of the SET, modulating the current at the drive frequency. This current, entering the impedance transformer, excites an RF output voltage $V_{\mathrm{out}}$, which is proportional to the nanotube’s displacement Wen2018 . The mechanical resonance therefore appears as a sharp peak in the electrical transmission from the drive to the output (Fig. 2b). The resonance frequency fluctuates quasiperiodically with gate voltage, which is a further indication of electromechanical coupling because the effective spring constant is softened when the SET is configured close to a Coulomb charge transition Steele2009 ; Lassagne2009 . From the peak width, the mechanical quality factor is $Q_{\text{M}}\approx 2.1\times 10^{3}$, with some gate voltage dependence because of electromechanical damping.

Mechanical oscillations, as distinct from a mechanical resonance, become evident when the output power spectrum is measured without driving (Fig. 2c). This undriven emission, plotted as a power spectral density $S$ referenced to the amplifier input, shows a peak whose frequency approximately follows the resonance of Fig. 2b. The peak is only present for some gate voltage settings, and is brightest close to the transport ridges of Fig. 2a. Furthermore, this peak strengthens with increasing bias (see Supplementary Information). For some gate voltage settings on the right of the graph, the peak switches between two or more frequencies, suggestive of dynamical bifurcation. All these observations imply that the observed emission is a result of self-excited mechanical oscillations driven by the DC bias across the device.

Mechanical coherence

With fast electromechanical readout, the coherence of this mechanical oscillator can be directly confirmed by measuring the output signal in real time. To do this, the signal is mixed with a local oscillator in a heterodyne circuit Liu2015Science ; Cassidy to generate records of the in-phase and quadrature voltages $V_{\text{I}}(t)$ and $V_{\text{Q}}(t)$ as a function of time $t$. The output record (Fig. 3a) shows clear sinusoidal oscillations. The onset of mechanical coherence is seen when the in-phase and quadrature time traces are represented as two-dimensional histograms for gate voltage settings above and below the oscillation threshold. Below threshold, the histogram is peaked near the origin, consistent with a band-limited but quasi-thermal source such as a randomly kicked resonator (Fig. 3b). However, above threshold the histogram has a ring shape, showing amplitude coherence characteristic of a laser-like oscillator (Fig. 3c). The ring diameter corresponds to an approximate phonon number $\bar{n}_{\text{p}}\sim 10^{5}$, i.e. an oscillation amplitude of $\sim 0.4$ nm, although there is a large uncertainty because of unknown device parameters (see Supplementary Information).

The clearest comparison to an ideal classically coherent source comes from a histogram of total output power, which is proportional to the number of phonons in the mode (Fig. 3d). Below threshold, the histogram follows the exponential distribution of completely incoherent quasi-thermal emission Liu2015Science . Above threshold, the histogram shifts to a distribution where the most probable state has a non-zero output power, as expected for a coherent source. It is approximately fitted by a Gaussian distribution, characteristic of a coherent oscillator in the limit of large phonon number $\bar{n}_{\text{p}}$. However, the distribution is slightly skewed, while its width, which for an ideal coherent state would be $\sim\sqrt{\bar{n}_{\text{p}}}$, is much larger than expected. Both the excess width and the skew indicate additional noise in the oscillator, presumably due to complex feedback between motion and electron tunnelling. The faint spot at the centre of Fig. 3c indicates bistability Usmani2007 ; Pistolesi , where the nanotube is either below threshold or has switched to a different frequency outside the measurement bandwidth. The weight of the spot shows that for this gate setting the device spends approximately 0.5 % of its time in such a state Liu2015Science .

While amplitude coherence is shown by the histogram, phase coherence is determined by plotting the autocorrelation of the demodulated signal $g^{(1)}(\tau)$ as a function of time interval $\tau$ (Fig. 3e). For these settings, the data are well fitted by an decaying sinusoid, reflecting the slow phase drift of the free-running oscillator. The envelope decay gives a phase coherence time $\tau_{\text{coh}}=99~{}\mu$s, i.e. a coherence linewidth of $\delta f_{\mathrm{coh}}=1/\pi\tau_{\text{coh}}=3$ kHz, approximately three times narrower than the resonance linewidth $f_{\text{M}}/Q_{\text{M}}$. Coherence is further confirmed by plotting the second-order correlation function $g^{(2)}(\tau)$, which shows chaotic quasi-thermal behaviour below threshold but nearly coherent behaviour above threshold FoxBook (Fig. 3f).

As the gate voltage is swept, the device switches between oscillating and non-oscillating states, and both the power and coherence time change (Fig. 4). By simultaneously measuring the RF and DC signals, the consequences for DC transport can be seen. Figure 4a shows current as a function of gate voltage over several periods of Coulomb blockade, while Fig. 4b shows the coherence time and emission power $P_{\text{total}}$ over the same range. The oscillator switches on and off approximately once per Coulomb period. Both the coherence time and the emitted power vary irregularly, but as expected most switches between oscillating and non-oscillating conditions coincide with abrupt current changes.

Stabilised oscillations

While the phase coherence time extracted from the autocorrelation characterizes the long-term oscillator stability, it is limited by slowly varying extrinsic effects such as charge noise or adsorbed atoms DeBonis2018 . To evaluate sensing schemes that rely on detecting mechanical frequency shifts, it important to identify the oscillator’s intrinsic linewidth if this slow variation could be eliminated, which may be much narrower. To measure the intrinsic linewidth, we employ two techniques from laser spectroscopy to stabilise the oscillator frequency.

First, we demonstrate that the oscillator can be locked to a stable but weak seed tone applied to the gate Stover1966 ; Liu2015PRA . This phenomenon of injection locking, previously demonstrated for trapped ions Knunz2010 and driven mechanical resonators Seitner2017 , arises because feedback amplifies small forces close to the operation frequency. In this measurement, the emission is monitored while the seed tone is applied at a nearby frequency $f_{\text{D}}$ (Fig. 5). As seen in Fig. 5a, b, for a range of $f_{\text{D}}$ settings near the free-running oscillator’s frequency and with sufficient drive power $P_{\text{D}}$, the broad emission line collapses onto the injection frequency. The locking events are accompanied by steps in the DC current (Fig. 5c, d).

The frequency range $\Delta f_{\text{D}}$ over which the oscillator is locked extends over many linewidths. Figure  5e shows the locking range as a function of injected power, confirming that a stronger injection tone has greater frequency pull. The data are well fitted by a power law of the form $f_{\text{D}}=AP_{\text{D}}^{\alpha}$, where $A$ and $\alpha$ are fit parameters. However, whereas the theory of conventional oscillators Adler1946 predicts an exponent $\alpha=0.5$, the data show a smaller exponent $\alpha\sim 0.3$, varying slightly with gate voltage but repeatable over two cooldowns. Another unexpected observation is a pair of spectral sidebands, whose frequency offset, surprisingly, depends on injection power (see Supplementary Information). These unexplained behaviours may be consequences of the stochastic nature of the current.

While injection locking clearly stabilises the oscillator’s state, it also contaminates the output spectrum with the high-frequency seed tone. An improved way to measure the oscillator’s intrinsic linewidth is to use feedback to cancel out slow frequency wander. This exploits the voltage tuning of the oscillator, and is analogous to Pound-Drever-Hall locking, used for sensitive laser measurements such as gravitational wave astronomy Drever1983 ; Abbott2009 . To implement this scheme (Fig. 6), the oscillator is incorporated into a phase-locked loop using error signal voltage fed to gate G1 (see Methods). Figure 6a shows dramatic frequency narrowing when the feedback is turned on. With optimised control parameters, the stabilised linewidth is $\delta f<2$ Hz (Fig. 6b), implying over $10^{8}$ coherent oscillations at the operating frequency of 230 MHz. This represents an upper limit on the intrinsic linewidth when slowly varying environmental perturbations are cancelled, and is limited by the spectral resolution. Similar to the Schawlow-Townes limit on a laser’s linewidth Schawlow1958 , the ultimate linewidth for an oscillator without stimulated emission is Wiseman1999 $\delta f_{\mathrm{ult}}=f_{\text{M}}/4\bar{n}_{\text{p}}Q_{\mathrm{M}}\sim 0.03$ Hz.

As expected, the feedback circuit succeeds in concentrating nearly the entire output into a narrow spectral line, provided that the oscillator’s free-running frequency is close to the target frequency (Fig. 6c, d) The stabilisation range is set by the maximum feedback voltage. However, feedback stabilises part of the emission even when this condition is not met, as seen by a weak spectral peak persisting beyond the expected voltage range (Fig. 6d). This indicates that the oscillator occasionally deviates by several linewidths from its central frequency. Feedback makes these excursions visible by temporarily capturing them.

Finally, we show how to generate a more complex output spectrum by exploiting the non-linear conductance of the SET. A time-varying drain-source voltage modulates the amplitude of the oscillator’s output, leading to regular sidebands spaced by the modulation frequency (Fig. 6e). Radio-frequency combs have been proposed for precise frequency comparison, as already used in optics Cundiff2003 , and this nanomechanical comb is an alternative to devices based on superconducting resonators Erickson2014 or Josephson junctions Solinas2015 ; Cassidy .

Conclusion

The dynamical instability explored here is an extreme consequence of invasive displacement measurement. For many kinds of nanomechanical sensing, it is a nuisance, because it means that same large bias necessary for precise measurement also strongly perturbs the displacement. However, when the aim is to detect a small frequency shift (e.g. for mass spectrometry Chaste2012 or some force-detected magnetic resonance schemes Stipe2001 ), introducing feedback directly into the sensing element can be beneficial. Clearly, the external frequency stabilization schemes described in the previous section are not directly useful for sensing because they render the oscillator insensitive both to the undesirable drift and to the desirable signal (unless they can be separated spectrally). However, even without applying external stabilization, the oscillation linewidth is narrower than the resonance linewidth, just as a laser’s emission is narrower than its cavity linewidth Bennett2006 , making small shifts easier to detect.

The similarities between SET nanomechanics and laser physics are intriguing Bennett2006 ; Rodrigues2007 . Like a laser, this device combines a pumped two-level system with a boson cavity, and shows phase and amplitude coherence as well as self-amplification. It differs from a conventional laser by not requiring degeneracy between the SET and the resonator, since there is no stimulated emission. A true phonon laser should emit directionally into a propagating sound wave Maryam2013 , which this experiment (like previous phonon laser realisations Vahala2009 ; Grudinin2010 ; Mahboob2013 ) does not test. However, to the extent that the key laser characteristic is output coherence Wiseman1997 , this experiment does indeed realise a phonon laser. It resembles unconventional lasers such as atom lasers that have coherent output statistics without stimulated emission Ottl2005 .

Further development from this device could replace the SET with a coherent two-level system such as a double quantum dot Brandes2003 , a superconducting SET Rodrigues2007 , or an electron spin Ohm2012 ; Palyi2012 . This would allow a phonon laser driven by conventional stimulated emission. Ultimately, superpositions might be transferred between the two-level system and the oscillator, allowing dynamic backaction to be studied in the fully quantum limit.