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

graphite/graphene nanotube

Carbon nanotubes

• different production methods;
• ften:
• use small catalyst particles
• hot gas, with carbon feed

(e.g. CH4)

• 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
• Young’s modulus E =

F/A

∆L/L: ECNT ≃ 1.2TPa, Esteel ≃ 0.2TPa

• Density:

ρCNT ≃ 1.3

g cm3 ,

ρAl ≃ 2.7

g cm3

• (still) “material of dreams”

http://www.pa.msu.edu/cmp/csc/ntproperties/

Suspended carbon nanotube sample fabrication

“the old way of doing things”

AFMmarkers catalyst SW-CNT electrodes catalyst+CVDgrownnanotubes electrodes SW-CNT

electrodesasetchmask Au SW-CNT Cr 500nm length L SiO2 p dopedSi

+

• A. K. H¨

uttel et al., New J. Phys. 10, 095003 (2008)

Low-temperature transport measurements

• Tunnel barriers between leads and nanotube
• Low temperature kBT ≪ e2/C: formation of a quantum dot

N N+1

Vsd VG

E

N N+1

CB SET D Dot S D S

Vg

E

CB SET

• Coulombblockade “diamonds”
• singleelectrontunneling

CB SET Excitedstatesvisibleatfinitebias! Spectroscopyoftheelectronicsystem stabilitydiagram:( , ) V V

g SD

dI dVSD

D S

source dot gate

N el.

drain

Vg VSD I

• A. K. H¨

uttel et al., New J. Phys. 10, 095003 (2008)

Vibration modes of carbon nanotubes

0.01 0.1 1 10 1 stretching bending RBM 0.1K kB Energy (meV) L (µm)

• radial breathing mode(s)
• stretching (longitudinal) mode:

hν ∝ L−1

hν = 1100...110µeV,

ν = 270...27GHz

(for 100nm...1µm)

• bending (transversal) mode:

hν ∝ L−2

hν = 10...0.1µeV,

ν = 2.4GHz...24MHz

(for 100nm...1µm) hν ∝ d, also tension-dependent

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

Log|I|

red-pos, blue-neg

dI/dV

L T =1.2µm =10mK

• Low-energy excitations
• Equally spaced, ¯

hω = 140µeV

• Identical for different charge

states

• Stretching mode as harmonic
• scillator
• S. Sapmaz et al., PRL 96, 026801 (2006)

Electron-vibron coupling, Franck-Condon physics

ˆ

H = ˆ p2 2m + 1 2mω2

0ˆ

x2 +λˆ x g = λ 2 2 = 1 2

ℓ ℓ0 2 ℓ0 =

• ¯

h mω0

newequilibriumposition! eV

• eV
• Γ →Γel |Ψafter|Ψbefore|2
• Pnm

Pn0 = |Ψ(x −ℓ)|Ψ(x)|2

= e−ggn

n!

no effect for g < 0.1 additional steps in I(Vsd) for g > 0.1 phonon blockade of transport for g > 1, Vsd < g¯ hω0

• 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

0.01 0.1 1 10 100 1000 Energy (meV) Length (nm)

prediction stretching mode prediction bending mode prediction box potential excitations in tunneling (published) excitations in cotunneling (published) excitations in tunneling excitations in cotunneling

• 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

• 100

100 200 300 400 500 gatevoltageVg (mV)

• 8
• 6
• 4
• 2

2 4 6 8 biasvoltageVSD (mV)

• 2000
• 1000

1000 2000 3000 4000

Nh=2 Nh=1

dI dVsd (nS)

Egap = 0.2eV

→ d = 3.7nm, ¯

hωRBM ≃ 7.8meV, maybe (0,46),

ε ≃ 6.2meV

length L = 250nm

→ expected ¯

hωbend ≃ 0.002meV, ¯ hωstretch ≃ 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 ≤ Nh+ ≤ 2)

150 200

• 8
• 6
• 4

150 200 250 Vg (mV) Vg (mV)

• 8
• 6
• 4
• 2

VSD (mV)

• 1000

1000 2000 3000

Nh=2

VSD (mV)

10 100 1000

(d)

dI dVsd (nS) dI dVsd (nS)

• excitations in SET, positive slope:

harmonic, ∆ε = 0.42meV ≃ ¯ hωstretch

• harmonic excitations in cotunneling!
• Cotunnel-assisted sequential

tunneling, “CO-SET”

cotunneling

µD µdot µS µD µdot µS

sequentialtunneling

• A. K. H¨

uttel et al., PRL 102, 225501 (2009)

Reminder: cotunneling – second-order process

• current in Coulomb blockade:

“several electrons tunneling at once”

• two-electron processes:
• elastic:
• inelastic (green arrow):

µD µdot µS

1500 1600 Vg (mV)

• 4
• 3
• 2
• 1

1 2 3 4 Vsd (mV)

1 10 100 1000 10000

CB SET

exampledata – novibrationmodes visiblehere

unpublished data

Cotunnel-assisted sequential tunneling (CO-SET)

• inelastic cotunneling, followed by a

tunnel-out process

cotunneling

µD µdot µS µD µdot µS

sequentialtunneling

• requires energy storage
• this is the process we’ve seen
• requires energy storage:

tunnel-out must be faster than relaxation

1500 1600 Vg (mV)

• 4
• 3
• 2
• 1

1 2 3 4 Vsd (mV)

1 10 100 1000 10000

exampledata – novibrationmodes visiblehere

unpublished data

Cotunnel-assisted sequential tunneling (CO-SET)

B

2δ 2δ δα /

G

E

E+I E+I E+I+S E+I+S

eVG

δ/2αG

eVbias A

(c) (b)

P2 G2 G1 S D P1 1.25 m

• B=0.1 T
• 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

150 200

• 8
• 6
• 4

Vg (mV) VSD (mV)

10 100 1000

(d)

dI dVsd (nS)

• 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

• 400
• 200
• 400
• 200

0.01 0.1 1 10

eVsd αeVg ( ) k T

B

( ) k T

B

dI dVsd

eΓ k T

B

ħ (

(

) )

2

1 A B F

(3) (1) (2) (4)

X

Nh=2 Nh=1

calculation

X

measurement

• 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

cotunneling

µD µdot µS µD µdot µS

sequentialtunneling

• 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 − → Qstretch 30

• Known values for transversal CNT mode:

Qbend,RT 2000, Qbend,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

• ?

Open question #2: Frequency doubling

2 4 6 8 5 10 15 excitationlinenumber E(meV) SET pos.slope 1 2 3 1 2 3 4 E(meV) SET neg.slope 1 2 3 4 5 0 1 2 3 4 5 6 E(meV) CO-SET

ħω=0.425meV ħω=0.72meV? ħω=0.81meV

• pos. slope SET excitations: ¯

hω = 0.42meV

• CO-SET and neg. slope SET excitations: ¯

hω ≃ 0.8meV

• few-carrier system
• adding a hole redistributes the entire charge on the nanotube
• up to now, all measurements of the Franck-Condon sidebands were in

the metallic limit

Some speculations about 1D Wigner crystals

• few charge carrier limit
• one-dimensional chain of electrons or holes
• different charge number −

→ completely different charge distribution

• influence of charge distribution on the electrostatic forces inducing the

vibrations

• dynamic interaction?
• electronic system much faster than vibration

− → can regard mechanical oscillator fixed for each point in time

• ?

The team at Molecular Electronics & Devices, Delft and theory friends

Thanks!

Herre van der Zant Benoit Witkamp Hari Pathangi Menno Poot & Samir Etaki, Yaroslav Blanter, Fabio Pistolesi, Ivar Martin, Sami Sap- maz, Pablo Jarillo-Herrero, Raymond Schouten, Hidde Westra, ... Martin Leijnse Maarten Wegewijs

Meanwhile, things have moved on a bit...

... and me as well, back to Bavaria

• new research group at

Universit¨ at Regensburg

• spin injection and spin

transport in carbon nanotubes

• carbon nanotubes with

superconducting contacts

• and now (since funding has finally come):

carbon nanotubes as nano-electromechanical hybrid systems

• postdoc position available