Carbon nanotubes as ultra-high quality factor mechanical resonators - PowerPoint PPT Presentation

Carbon nanotubes as ultra-high quality factor mechanical resonators — and much more! Andreas K. H¨ uttel Kavli Institute for Nanoscience, Technische Universiteit Delft, Netherlands Institute for Experimental and Applied Physics, Universit¨ at Regensburg, Germany Condensed Matter and Materials Physics (CMMP10) University of Warwick, Coventry, UK 14 December 2010

Carbon nanotubes: a more exciting (and not so flat) form of carbon diamond fullerene�(C ) 60 graphite�/�graphene nanotube

Mechanical properties of carbon nanotubes • stiffer than steel • resistant to damage from physical forces • very light F / A • Young’s modulus E = ∆ L / L : E CNT ≃ 1 . 2TPa, E steel ≃ 0 . 2TPa • Density: g g ρ CNT ≃ 1 . 3 ρ Al ≃ 2 . 7 cm 3 , cm 3 • (still) “material of dreams” http://www.pa.msu.edu/cmp/csc/ntproperties/

Doubly clamped nanotube resonators nanotube is suspended like a guitar or violin string low mass, high stiffness → high resonance frequency, large quantum effects single clean macromolecule → low dissipation???

Doubly clamped nanotube resonators 800nm nanotube is suspended like a guitar or violin string low mass, high stiffness → high resonance frequency, large quantum effects single clean macromolecule → low dissipation???

Vibration modes of carbon nanotubes • stretching (longitudinal) mode: h ν ∝ L − 1 h ν = 1100 ... 110 µ eV, 10 ν = 270 ... 27GHz RBM (for 100nm ... 1 µ m) Energy (meV) • bending (transversal) mode: 1 h ν ∝ L − 2 stretching h ν = 10 ... 0 . 1 µ eV, ν = 2 . 4GHz ... 24MHz (for 100nm ... 1 µ m) 0.1 h ν ∝ d , also tension-dependent bending 0.1K k B 0.01 stretching 0 L (µm) 1 bending RBM

Chip fabrication and measurement setup 400 nm V RF E (t) ~2cm source drain CNT V sd A V g gate 800 nm • First make chip (Pt electrodes, trench) • Then CVD-grow nanotubes across electrodes • Back gate connected to a gate voltage source V g • RF antenna suspended ∼ 2cm above chip • Dilution refrigerator ( T ≃ 20mK) • Only dc measurement G. A. Steele et al. , Nature Nanotech. 4 , 363 (2009); AKH et al. , Nano Lett. 9 , 2547 (2009)

Low-temperature transport: Coulomb blockade T � 20mK dilution refrigerator • Tunnel barriers between leads and nanotube • Low temperature k B T ≪ e 2 / C : formation of a quantum dot source dot Coulomb�blockade Single�electron drain tunneling � V g N el. � S � D V SD I � S � D � S � D V g gate d I V sd ≈ 0�(linear�response�regime) d V sd CB CB CB N-1 N N+1 el. el. el. SET SET SET 0 V g

Fixed V g and V SD , sweep of RF signal frequency 2 Q =140670 -17.8 dBm -64.5 dBm 88 I (pA) I (nA) 87 1 86 0 100 300 500 293.41 293.42 293.43 293.44 � (MHz) � (MHz) V RF • Sharp resonant structure in I dc ( ν ) E (t) ~2cm V sd • Very low driving power required A • High Q = ν / ∆ ν ( ∆ ν = FWHM) V g gate AKH et al. , Nano Lett. 9 , 2547 (2009)

V g dependence — this is really a mechanical resonance! � (MHz) 350 red: continuum beam model 300 � (MHz) 250 300 200 150 d I 250 d � -6 -5 -4 -3 -2 -1 0 V g (V) (pA/MHz) 1000 200 100 150 10 -6 -4 -2 V g (V) 0 larger | V g | − → increased tension − → higher frequency ν AKH et al. , Nano Lett. 9 , 2547 (2009)

Detection mechanism — mechanically induced averaging 2 • at resonant driving the nanotube I (nA) � I ac,eff V g position oscillates 1 • oscillating C g 0 − → fast averaging over I ( V g ) 0.1 � I (nA) 0 -0.1 • black line: dc measurement I ( V g ) • red line: this numerically averaged 0.1 � I (nA) 0 • blue: difference, effect of averaging -0.1 • red points: measured peak amplitude in I ( ν ) , for different values of V g -5.22 -5.21 -5.2 -5.19 V g (V) AKH et al. , Nano Lett. 9 , 2547 (2009)

Driving into nonlinear response... 80 mK, -70 dBm I (pA) • same temperature • same working point V g , V SD -62 dBm I (pA) I (pA) • low driving power: symmetric, “linear” response -56 dBm I (pA) • high driving power: asymmetric response, hysteresis -52.5 dBm Duffing-like oscillator I (pA) � (MHz) AKH et al. , Nano Lett. 9 , 2547 (2009)

Georg Duffing (1861 – 1944) and his oscillator • Driven mechanical oscillator with non-linear response • Response becomes bistable → large or small amplitude • Switching between branches Duffing differential equation: x + cx + bx 3 = F sin ω t m ¨ AKH et al. , Nano Lett. 9 , 2547 (2009)

Driving into nonlinear response... 80 mK, -70 dBm I (pA) • same temperature • same working point V g , V SD -62 dBm I (pA) I (pA) • low driving power: symmetric, “linear” response -56 dBm I (pA) • high driving power: asymmetric response, hysteresis -52.5 dBm Duffing-like oscillator I (pA) � (MHz) AKH et al. , Nano Lett. 9 , 2547 (2009)

... and then increasing the temperature -53 dBm, 20mK I (pA) • same driving power • same working point V g , V SD 80mK I (pA) • low temperature: asymmetric response, hysteresis 120mK Duffing-like oscillator I (pA) • high temperature: symmetric, “linear” response 160mK I (pA) peak broadening � (MHz) AKH et al. , Nano Lett. 9 , 2547 (2009)

Temperature dependence of Q (a) -66 dBm, 40mK I (pA) 5 Q=123578 10 -0.36 ~ T Q-factor (b) -50.5 dBm, 320mK I (pA) Q=59283 (c) -45 dBm, 1K I (pA) Q=23210 4 10 0.01 0.1 1 � (MHz) Temperature (K) Q ( T ) fits power law prediction for intrinsic dissipation in nanotube − → H. Jiang et al. , Phys. Rev. Lett. 93 , 185501 (2004) AKH et al. , Nano Lett. 9 , 2547 (2009)

V g dependence — this is really a mechanical resonance! � (MHz) 350 red: continuum beam model 300 � (MHz) 250 300 200 150 d I 250 d � -6 -5 -4 -3 -2 -1 0 V g (V) (pA/MHz) 1000 200 100 150 10 -6 -4 -2 V g (V) 0 larger | V g | − → increased tension − → higher frequency ν AKH et al. , Nano Lett. 9 , 2547 (2009)

Detailed ν ( V g ) : with current, frequency decreases I (nA) 8 dc current 0 140.0 � (MHz) N holes RF response N-1 holes 139.2 -0.90 -0.84 V g (V) “Coulomb blockade oscillations of mechanical resonance frequency” electrostatic contribution to spring constant G. A. Steele, AKH, et al. , Science 325 , 1103 (2009)

Model for ν ( V g ) – part I: “slope and steps” � (MHz) tension induced by a single elementary charge N holes N-1 holes V g (V) • Electrostatic force between tube and backgate: F dot = 1 d C g � 2 � V g − V dot 2 d z • Quantum dot voltage: V dot = C g V g + q dot , q dot ( q c ) = −| e |� N � ( q c ) , q c = C g V g C dot • Overall slope: continuous increase of voltage V g on gate • Steps: discrete change of V dot (single elementary charges!) G. A. Steele, AKH, et al. , Science 325 , 1103 (2009)

Model for ν ( V g ) – part II: “steps become dips” V sd = 0.5 mV V sd = 1.5 mV V sd = 2.5 mV 140.0 � (MHz) 139.5 -0.90 -0.88 -0.90 -0.88 -0.90 -0.88 V g (V) V g (V) V g (V) • q c = C g ( z ) V g is function of z • Electrostatic contribution to spring constant: � 2 � = V g ( V g − V dot ) � d C g 1 −| e | d � N � � k dot = − d F dot d z c dot d z d q c • Always negative, always decreasing frequency G. A. Steele, AKH, et al. , Science 325 , 1103 (2009)

Also mechanical Q and nonlinearity dominated by current ΔI sd (nA) 0.1 ΔI sd (nA) -0.1 -0.5 1.0 258 -60 dB -45 dB • Dissipation whenever charge can fluctuate � (MHz) • Q decreases on SET peaks 256 • Nonlinearity dominated -4.345 -4.335 -4.345 -4.335 V g (V) V g (V) by tunneling 1 nA Up Down f 0 Fit Q ~ 57000 • Switches between 20000 α < 0 ΔI sd weakening and 2900 α > 0 softening spring 90000 α < 0 -200 200 256 258 - 0 (kHz) � (MHz) � � G. A. Steele, AKH, et al. , Science 325 , 1103 (2009)

Interaction-induced nonlinearity α ( V g ) Frequency α < 0 α > 0 α < 0 Gate Voltage � 2 d 2 k dot α dot = − d 3 F d z 3 = d 2 � d C g d z 2 k dot ( q c ) = V 2 g d q 2 d z c The sign of α dot follows the sign of the curvature of k dot . G. A. Steele, AKH, et al. , Science 325 , 1103 (2009)

Self-excitation of the resonator 050 dI/dV ( S) dI/dV ( S) -10 5 A B D I sd 2 5 10 I sd (nA) sd (mV) sd (mV) V -4.93 g 5 V V -10 -2 0 -5.19 -5.17 -1.05 -0.95 V g (V) V g (V) E 284.5 C 10 nA f (MHz) I sd 283.5 0.5 sd (nA) V I V sd g -0.75 -4.935 -4.93 V g (V) G. A. Steele, AKH, et al. , Science 325 , 1103 (2009); Usmani et al. , PRB 75 , 195312 (2007)

What do we have so far? • Mechanical resonator, 120MHz � ν � 360MHz, Q � 150000 • Easy driving into nonlinear oscillator regime • Single-electron steps of the resonance frequency • Backaction of single electron tunneling on ν , Q , nonlinearity • Estimated motion amplitude at resonant driving ∼ 250pm compare thermal motion 6 . 5pm, zero-point motion 1 . 9pm u • Application as mass sensor: sensitivity 4 . 2 √ Hz • Without driving: mechanical thermal occupation n ≃ 1 . 2 AKH et al. , Nano Lett. 9 , 2547 (2009); G. A. Steele, AKH, et al. , Science 325 , 1103 (2009)

Higher frequency (I): higher vibration modes • higher harmonics visible too 1000 • dc current signal is smaller f (node(s) in nanotube motion, smaller (MHz) change in total capacitance) 500 • at high tension, integer frequency f (MHz) 1000 multiples 500 0 (expected for a string resonator) 1 2 3 n 0 200 300 N h+