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Nonequilibrium and quantum coherent phenomena in the electromechanics of suspended nanowires (Review Article)
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10.1063/1.3224725
/content/aip/journal/ltp/35/8/10.1063/1.3224725
http://aip.metastore.ingenta.com/content/aip/journal/ltp/35/8/10.1063/1.3224725
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Figures

Image of FIG. 1.
FIG. 1.

(Color online) Sketch of the system considered (a) and the equivalent circuit (b). Mechanical deviation of the wire away from the static configuration is described by the “displacement field” . An STM tip is put over the point along the suspended-nanowire axis, and the electron tunneling rate between the STM tip and the nanowire depends on the deflection of the midpoint of the nanowire. The tunneling rate between the nanowire and the electrodes is, however, constant. Adapted with permission from L. M. Jonsson et al., Nano Lett.5, 1165 (2005). © 2008, American Chemical Society.

Image of FIG. 2.
FIG. 2.

Threshold dissipation as a function of the junction resistances (Fig. 1b). The largest values of are obtained when the two junction resistances are of the same order of magnitude. Reprinted with permission from L. M. Jonsson et al., Nano Lett. 5, 1165 (2005).21 © 2005, American Chemical Society.

Image of FIG. 3.
FIG. 3.

(a) Amplitude of nanowire oscillations as a function of the bias voltage. The solid line shows a “soft” development of the instability, while the dotted line corresponds to a “hard” transition. (b) curves for the same “soft” (left) and “hard” (right) transitions. Insets show corresponding curves. Reprinted with permission from L. M. Jonsson et al., Nano Lett. 5, 1165 (2005).21 © 2005, American Chemical Society.

Image of FIG. 4.
FIG. 4.

(Color online) Threshold dissipation for the first and third transverse modes as a function of . Going from top to bottom, the electromechanical coupling is increased from the weak to the strong regime, (a), (b), and (c). When the electromechanical coupling is large the interaction between different modes is no longer negligible, which introduces a qualitatively new feature compared to the weak electromechanical regime. As can be seen, some values of allow for mode to be excited and mode to be suppressed, even though , i.e., . Reprinted with permission from L. M. Jonsson et al., New J. Phys. 9, 90 (2007).32 © 2007, Deutsche Physikalische Gesellschaft.

Image of FIG. 5.
FIG. 5.

(Color online) characteristics for the device of Fig. 1a in the weak coupling regime. For small voltages, the current is constant and depends only on the tunneling through the double static junction . Above the threshold voltage the current also depends on the vibration amplitude. Here the electromechanical instability occurs through the “hard” transition, characterized by the displayed hysteresis in the curve. Reprinted with permission from L. M. Jonsson et al., New J. Phys. 9, 90 (2007).32 © 2007, Deutsche Physikalische Gesellschaft.

Image of FIG. 6.
FIG. 6.

(Color online) Stationary points when two modes ( and ) are unstable. Two attractors, indicated by (○), corresponding to a finite amplitude of one mode while the other mode is suppressed, are shown. The stationary point marked with (x) is a repellor and the point indicated by (∗) is a saddle point. The thick lines are separatrices that trajectories cannot cross. The separatrix ensures that if , this inequality will hold for all times . Reprinted with permission from L. M. Jonsson et al., Phys. Rev. Lett. 100, 186802 (2008).35 © 2008, American Physical Society.

Image of FIG. 7.
FIG. 7.

(Color online) Numerical solution of equations (8) and (9) for the nanotube vibration amplitude as a function of time when three modes are unstable . (a) Weak electromechanical regime. Comparison with the approximate result, Eq. (10), is shown as dashed curves. The lower left panel shows the quasi-periodic oscillation of the nanotube center position just after the onset of the instability, while the lower right panel shows the regular vibrations that appear after all but the mode amplitudes have been suppressed (see text). (b) Strong coupling regime. The large amplitudes make an approximate analysis based on Eq. (10) invalid, but the phenomenon of selective excitation persists. Reprinted with permission from L. M. Jonsson et al., Phys. Rev. Lett.100, 186802 (2008).35 © 2008, American Physical Society.

Image of FIG. 8.
FIG. 8.

(Color online) (a) Regions of instability for dissipation modeled by a viscous term . (b) Regions of instability for internal dissipation given by the Zener model. The solid line and the dots that define the threshold dissipation curves are obtained respectively from the analytical and the numerical solutions of the linearized equations of motion. In both plots the shaded areas define the values of and for which only a single mode is unstable. Reprinted with permission from F. Santandrea.36

Image of FIG. 9.
FIG. 9.

Schematic diagram of the system considered in Sec. III A. A transverse magnetic field is applied to a suspended 1-dimensional model nanowire of length . If the wire is biased by a voltage , it carries a current, and the wire oscillates in response to the induced Lorentz force. Quantum fluctuations in the wire’s bending modes make the electrons propagate along an effectively two-dimensional wire. The magnetic-field-induced reduction of electron propagation in the elastic channel can be interpreted as an effect of destructive Aharonov-Bohm-type quantum interference between different paths of the tunneling electrons. Together with a blockade of some inelastic channels due to Pauli-principle restrictions (see text and the caption to Fig. 10) this leads to a finite magnetoresistance of the wire. The amplitude shown is greatly exaggerated. Reprinted with permission from G. Sonne, New J. Phys. 11, 073037 (2009).40 © 2009, Deutsche Physikalische Gesellschaft.

Image of FIG. 10.
FIG. 10.

Sketch of the different transmission channels available for electrons tunneling through the oscillating nanowire of Fig. 9. Electrons with energy tunneling from the left (source) to the right (drain) lead are transmitted in both elastic and inelastic tunneling channels (top image) with the corresponding energy exchange, ; . Due to Pauli-principle restrictions, some of the inelastic channels are affected by the electronic population in the drain lead (shaded region, lower image), which, together with a reduction of the tunneling rate in the elastic channel (see text and caption to Fig. 9), leads to a finite magnetoresistance of the wire. The Pauli-principle restrictions are important only if is close to the chemical potential in the drain lead. This is why the total current reduction saturates and becomes independent of both temperature and bias voltage for large enough bias voltage (see text).

Image of FIG. 11.
FIG. 11.

(Color online) (a) Magnetoconductance through a suspended SWNT system as a function of temperature and magnetic field . Reprinted with permission from R. I. Shekhter et al., Phys. Rev. Lett. 97, 156801 (2006).41 © 2006, American Physical Society. (b) Predicted offset current as a function of bias voltage for two different temperatures and two different vibrational frequencies. Note that the offset current (red line) does not extrapolate to the origin and scales as the square of the magnetic field, . Data shown in arbitrary units. Reprinted with permission from G. Sonne, New J. Phys. 11, 073037 (2009).40 © 2009, Deutsche Physikalische Gesellschaft.

Image of FIG. 12.
FIG. 12.

Time-averaged vibrational amplitude of the superconducting suspended nanotube as a function of driving voltage , clearly showing the onset of the parametric resonance at higher driving force . Reprinted with permission from G. Sonne et al., Phys. Rev. B 78, 144501 (2008).44 © 2008, American Physical Society.

Image of FIG. 13.
FIG. 13.

(Color online) (a) Time-averaged vibrational amplitude as a function of driving force for the second resonance peak, , clearly showing the saturation of the vibrational amplitude at . The inset shows the corresponding dimensionless dc current as a function of the magnetic field. (b) Predicted current bistability and hysteresis for two voltage pulses close to the parametric resonance. As shown, applying either a positive or negative voltage pulse in time can shift the system between the two stability points (1 and 2), which are separated by a finite current difference. Reprinted with permission from G. Sonne et al., Phys. Rev. B 78, 144501 (2008).44 © 2008, American Physical Society.

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2009-12-01
2014-04-17
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Scitation: Nonequilibrium and quantum coherent phenomena in the electromechanics of suspended nanowires (Review Article)
http://aip.metastore.ingenta.com/content/aip/journal/ltp/35/8/10.1063/1.3224725
10.1063/1.3224725
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