Longitudinal vibrations of suspended carbon nanotubes - - PowerPoint PPT Presentation

Longitudinal vibrations of suspended carbon nanotubes - Franck-Condon effect, cotunneling, and nonequilibrium Andreas K. H¨ uttel Kavli Institute for Nanoscience, Technische Universiteit Delft, Netherlands Current address: Institute for Experimental and Applied Physics, Universit¨ at Regensburg, Germany Workshop on Nano-Opto-Electro-Mechanical Systems Approaching the Quantum Regime, Abdus Salam International Centre for Theoretical Physics, Trieste 2010

Carbon, as we know it image source: Wikipedia

Carbon nanotubes: a more exciting form of carbon diamond fullerene�(C ) 60 nanotube graphite�/�graphene

Carbon nanotubes • different production methods; often: • use small catalyst particles • hot gas, with carbon feed (e.g. CH 4 ) • nucleation of tube structure • many different structures • single-wall, double-wall, multi-wall • zigzag, armchair, chiral (how the sheet is “wrapped together”) image source: Wikipedia

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/

Suspended carbon nanotube sample fabrication “the old way of doing things” catalyst�+�CVD�grown�nanotubes electrodes SW-CNT catalyst AFM�markers electrodes SW-CNT length L Au Cr SW-CNT SiO 2 + p doped�Si electrodes�as�etch�mask 500nm A. K. H¨ uttel et al. , New J. Phys. 10 , 095003 (2008)

Low-temperature transport measurements • Tunnel barriers between leads and nanotube • Low temperature k B T ≪ e 2 / C : formation of a quantum dot source dot drain N el. � S � D V SD I V g gate � V g � S � S � D � D � E � Dot dI stability�diagram:�������( V V , ) dV SD g SD CB SET V sd CB -�Coulomb�blockade “diamonds” CB SET -�single�electron�tunneling � E 0 Excited�states�visible�at�finite�bias! N N N+1 N+1 SET Spectroscopy�of�the�electronic�system V G A. K. H¨ uttel et al. , New J. Phys. 10 , 095003 (2008)

Vibration modes of carbon nanotubes • radial breathing mode(s) 10 RBM • stretching (longitudinal) mode: Energy (meV) h ν ∝ L − 1 1 h ν = 1100 ... 110 µ eV, ν = 270 ... 27GHz stretching (for 100nm ... 1 µ m) 0.1 • bending (transversal) mode: h ν ∝ L − 2 bending h ν = 10 ... 0 . 1 µ eV, 0.1K k B ν = 2 . 4GHz ... 24MHz 0.01 (for 100nm ... 1 µ m) h ν ∝ d , also tension-dependent 0 L (µm) 1 purely electronic excitations have different energy scale A. K. H¨ uttel et al. , New J. Phys. 10 , 095003 (2008)

The stretching mode – visible in electronic transport • Low-energy excitations • Equally spaced, ¯ h ω = 140 µ eV Log| I | • Identical for different charge states • Stretching mode as harmonic oscillator dI/dV L =1.2µm T =10mK red-pos, blue-neg S. Sapmaz et al. , PRL 96 , 026801 (2006)

Electron-vibron coupling, Franck-Condon physics � ℓ � � 2 p 2 g = λ 2 H = ˆ ¯ 2 m + 1 2 = 1 h x 2 + λ ˆ ˆ 2 m ω 2 0 ˆ ℓ 0 = x ℓ 0 m ω 0 2 new�equilibrium�position! � � eV eV Γ → Γ el |� Ψ after | Ψ before �| 2 � �� � P nm P n 0 = |� Ψ( x − ℓ ) | Ψ( x ) �| 2 = e − g g n n ! no effect for g < 0 . 1 additional steps in I ( V sd ) for g > 0 . 1 phonon blockade of transport for g > 1, V sd < g ¯ h ω 0 S. Braig and K. Flensberg, PRB 68 , 205324 (2003) M. C. Luffe et al. , PRB 77 , 125306 (2008), K. Flensberg, March Meeting 2006 slides

Vibrational excitations observed so far 10 prediction stretching mode prediction bending mode prediction box potential excitations in tunneling (published) excitations in cotunneling (published) excitations in tunneling excitations in cotunneling 1 Energy (meV) 0.1 0.01 100 1000 Length (nm) S. Sapmaz et al. , PRL 96 , 026801 (2006); A. K. H¨ uttel et al. , New J. Phys. 10 , 095003 (2008); A. K. H¨ uttel et al. , PRL 102 , 225501 (2009); R. Leturcq et al. , Nat. Phys. 5 , 327 (2009)

L = 250nm SC nanotube, few-hole regime 8 6 bias�voltage�V SD (mV) 4 N h =2 N h =1 2 d I d V sd 0 (nS) -2 4000 3000 2000 -4 1000 0 -6 -1000 -2000 -8 -100 0 100 200 300 400 500 gate�voltage�V g (mV) E gap = 0 . 2eV → d = 3 . 7nm , h ω RBM ≃ 7 . 8meV , ε ≃ 6 . 2meV ¯ maybe (0,46), length L = 250nm → expected ¯ h ω bend ≃ 0 . 002meV , ¯ h ω stretch ≃ 0 . 44meV bending lines → shifting potential minima, DQD-like properties A. K. H¨ uttel et al. , PRL 102 , 225501 (2009)

Stretching mode in SET and cotunneling (1 ≤ N h + ≤ 2) 3000 0 2000 N h =2 • excitations in SET, positive slope: 1000 -2 harmonic, ∆ ε = 0 . 42meV ≃ ¯ h ω stretch 0 V SD (mV) -1000 -4 • harmonic excitations in cotunneling! d I d V sd -6 (nS) -8 • Cotunnel-assisted sequential 150 200 250 V g (mV) tunneling, “CO-SET” (d) µ S µ S -4 µ D µ D V SD (mV) 1000 -6 µ dot µ dot 100 d I d V sd 10 cotunneling sequential�tunneling -8 (nS) 150 V g (mV) 200 A. K. H¨ uttel et al. , PRL 102 , 225501 (2009)

Reminder: cotunneling – second-order process 10000 example�data – 4 • current in Coulomb blockade: no�vibration�modes “several electrons tunneling at once” visible�here 1000 3 • two-electron processes: 2 100 • elastic: CB 1 10 V sd (mV) 0 1 -1 • inelastic (green arrow): -2 µ S -3 µ D -4 SET µ dot 1500 1600 V g (mV) unpublished data

Cotunnel-assisted sequential tunneling (CO-SET) 10000 example�data – • inelastic cotunneling, followed by a 4 no�vibration�modes tunnel-out process visible�here 1000 3 µ S µ S 2 100 µ D µ D 1 10 V sd (mV) µ dot µ dot 0 1 cotunneling sequential�tunneling -1 • requires energy storage -2 • this is the process we’ve seen -3 • requires energy storage: -4 tunnel-out must be faster than 1500 1600 relaxation V g (mV) unpublished data

Cotunnel-assisted sequential tunneling (CO-SET) B=0.1 T 2 δ eV bias P2 E+I+S P1 (c) B G2 E+I (b) 2 δ A E S E+I E+I+S D G1 δ / 2 α G G δ�α / 1.25 m � eV G • first observed and explained by Schleser et al. ∼ 2005 electronic excitations in GaAs/AlGaAs quantum dots R. Schleser et al. , PRL 94 , 206805 (2005)

Measurement detail (d) -4 V SD (mV) 1000 -6 100 d I d V sd 10 -8 (nS) 150 V g (mV) 200 • CO-SET current sets in at additional (electronic) excited state X • Tunnel rates coupling an 2h state to 1h ground state: small for 2h ground state, large for 2h excited state X • Real-time transport theory calculations, M. Leijnse & M. Wegewijs • Vibration mode is pumped by multiple inelasic cotunnel processes involving X (e.g. sequence ( 1 ) → ( 2 ) → ( 3 ) ) A. K. H¨ uttel et al. , PRL 102 , 225501 (2009); M. Leijnse & M. Wegewijs, PRB 78, 235424 (2008)

Numerical calculation ) 10 k T calculation 0 d I B N h =2 N h =1 ( d V sd 1 eV sd 1 eΓ ) ) ( 2 ħ ( measurement (2) 0.1 k T B X -200 0.01 (3) (4) F (1) -400 X B A -400 -200 0 ( k T ) α eV g B • CO-SET current sets in at additional (electronic) excited state X • Tunnel rates coupling an 2h state to 1h ground state: small for 2h ground state, large for 2h excited state X • Real-time transport theory calculations, M. Leijnse & M. Wegewijs • Vibration mode is pumped by multiple inelasic cotunnel processes involving X (e.g. sequence ( 1 ) → ( 2 ) → ( 3 ) ) A. K. H¨ uttel et al. , PRL 102 , 225501 (2009); M. Leijnse & M. Wegewijs, PRB 78, 235424 (2008)

CO-SET process requires energy storage, nonequilibrium µ S µ S µ D µ D µ dot µ dot cotunneling sequential�tunneling • Vibration mode must remain excited until tunnel-out • Vibrons are pumped as in a three-level laser! • Comparison of timescales & tunnelling rates − → first (weak) lower boundary for mechanical quality factor − → Q stretch � 30 • Known values for transversal CNT mode: Q bend,RT � 2000, Q bend,20mK � 150000 A. K. H¨ uttel et al. , PRL 102 , 225501 (2009); A. K. H¨ uttel et al. , Nano Lett. 9 , 2547 (2009)

Open question #1: Nature of the excited state X • simplest possibility: orbital excited state of the nanotube quantum dot • different orbital wavefunction • different tunnel couplings − → our model idea should work fine • alternative explanation: potential side minimum / double quantum dot • possible since the suspended nanotube is partially covered by the contacts • bending resonance lines in Coulomb diamonds: shifting potential minima − → our model idea should still work fine!

Some speculations about a “phaser” • idea: use analogy with the 3-level laser • current pumps vibration via the electronic excited state • use a double quantum dot to generate this level structure • feed a mechanical mode faster than it can decay, population inversion • ?