Piezoelectric Nanotubes (!)

• Electrons on Carbon NT’s

• Heteropolar Nanotubes Pyroelectricity Piezoelectricity Photogalvanics

• Tubes as Optical Materials

….with Na Sai Charlie Kane Petr Kral

Carbon nanotube contacting platinum electrodes

Source DrainGate

“Long wavelength physics in the extended direction is controlled by the short distance physics in

the wrapped direction”

Some examples

• Semiconducting v. Conducting Carbon NT’s

• Pyrolectric and Piezoelectric Effects in III-V’s (this work)

• Structure Specific Near Infrared Fluorescence (in progress)

Graphene has a Critical Electronic State

Dispersion of a free particle in 2D..

…is replacedby an unconventional E(k) relation on thegraphene lattice

The (m,n) wrapping specifies a translation vector of the graphene lattice.

Rolling-up a graphene sheet

m=n mod(m-n,3) = ±1 mod(m-n,3) = 0, mn

Backscattering from elastic strains: bend and twist

Twist (but not bend) canbackscatter electronson an armchair tube.

this is responsible forthe T-linear observered resistivity.

Are Nanotubes Photogalvanic ?

Heteropolar NT’s of Boron Nitride

BN is the III-V homologto graphene. The B and N occupy different sublattices -- this lowers the symmetry and leads to new physical effects

Quantum Theory of Polarization(King-Smith & Vanderbilt, Phys. Rev. B47, 1651 (1993))

0 ( )HH H t

10

P PP dt dt

( )| ( ) ( , )2ieP d u u k

P is obtained from the geometric (Berry’s) phaseaccumulated by the u’s under adiabatic motion on a closed orbit in -space.

Nanotube Polarization as a Geometric Phase

† ( )( , , )

( )xFx

xF

v q iH q

v q i

Control parameters: qx, ,

with valence eigenstates that adiabatically follow

ˆexp( )2

v i 2 2 2

( )

2 ( )xx jF

j F

e ev p v e v

v q

sum over states and integrate over to obtain

2 2 2arctan( )x

se Wps W

The NT’s electric dipole moment that depends on its wrapping

0,3

FvR

The magnitude of the dipole issensitive to elastic strain (modulate

)

NT’s are molecular piezoelectrics, where P is sensitive to twist and stretch, so strain <=>voltage !

Pyroelectric v. Piezoelectric Effects

Geometrical (topological and discrete)

Strain (short range and continuously “tunable”)

(physics) (chemistry)

Piezoelectricity in a Heteropolar Sheet

elastic strain lowers the threefold symmetry of the BN sheetproducing an electric polarization

stretch twist

xxxxxx

Pe

xxx xyy yxy yyxe e e e

yyxy

xy

Pe

3m symmetry:

Elastic Strain on a Heteropolar Tube

stretch

twist

armchair zigzag

Calculated (N-TB) Piezo-Response of Nanotubes

zigzag armchair

N-TB

DFT

• electron response dominates• 1/R^2 finite size corrections

sheets

Chiral Tubes

• Chiral tubes have a wrapping vector high symmetry translation of the BN sheet (chiral angle ).

( ) ( ) ( )iaijk jb kc abcabc

e R R R e

11 sin(3 ) xxxe e 14 cos(3 ) yxye e stretch twist

• Electric Dipole couples to both stretch and torsion bilinear stretch-twist coupling!

• Low Symmetry Large Unit Cell, but …

Piezoelectricity of Chiral Tubes

mapped sheet response

From N-TB (calculations for (5,m) (6,m) (12,m) families)

tube

mapped sheet response

Size (R) Scaling of the Piezoelectric Constants of Chiral Tubes

Photogalvanic Effects inHeteropolar Tubes

C, BN NT’s are prototypes with many other compact meso-phases formed by folding lamellae

MoS2

Remska et al. Science 292, 479 (2001)

WS2

Single- anddouble- wall WS2

coat C-NT

and WS2 cones

Whitby et al. APL 79, 4574 (2001)

& many others: Tenne & Zettl, Topics. In Applied Physics 80, 81 (2001)

Physical Properties Control ofComposition AND Geometry

challenges opportunities

• control C&G in synthesis• structural sorting• assembly of networks and superstructures

• highly ordered(coherent) structures• access to quantum geometrical effects • phenomenology: systematics in “families”

Near-infrared Photoluminescence from Single-wall Carbon Nanotubes

Excitation (661 nm) Emission ( 850 nm)

Fluorescence Spectroscopy

• FS reveals electronic gap structure outside the conventional band model.

• The “ratio problem”

Gap Ratio < 2 (asymptote for large diameter tubes)

Hybridize e-h and 2e-2h excitations

1D + degeneracy from tube wrapping.

• The “deviations problem”

They are very large… with ± asymmetry

Curvature, Trig. Warping + Coul. Anisotropy(distinguished by scaling with R, n)

Long RangeInteraction

Short Range Interaction

Stick Boy and Match Girl

Stick Boy liked Match Girl

he liked her a lot.

He liked her cute figure,

he thought she was hot.

But could a flame ever burn

for a match and a stick?

It did quite literally;

he burned up pretty quick.

children’s poetry by Tim Burton (1997)