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### Piezoelectric Nanotubes (!)

• Electrons on Carbon NT’s

• Heteropolar Nanotubes

Pyroelectricity

Piezoelectricity

Photogalvanics

• Tubes as Optical Materials

….with

Na Sai

Charlie Kane

Petr Kral

“Long wavelength physics in the extendeddirection 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 replaced

by an unconventional E(k) relation on the

graphene lattice

Rolling-up a graphene sheet

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

m=n

mod(m-n,3) = ±1

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

Backscattering from elastic strains: bend and twist

Twist (but not bend) can

backscatter electrons

on an armchair tube.

this is responsible for

the T-linear observered

resistivity.

Heteropolar NT’s of Boron Nitride

BN is the III-V homolog

to 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))

DP is obtained from the geometric (Berry’s) phase

accumulated by the u’s under adiabatic motion on a

closed orbit in t-space.

Nanotube Polarization as a Geometric Phase

Control parameters: qx, d, D

with valence eigenstates that adiabatically follow W

sum over states and integrate over D to obtain

The magnitude of the dipole issensitive to elastic strain (modulate d)

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 sheet

producing an electric polarization

stretch

twist

3m symmetry:

Calculated (N-TB) Piezo-Response

of Nanotubes

zigzag

armchair

sheets

N-TB

DFT

• p electron response dominates

• 1/R^2 finite size corrections

• Chiral tubes have a wrapping vector high

symmetry translation of the BN sheet (chiral angle q).

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

• Low Symmetry Large Unit Cell, but …

twist

stretch

Piezoelectricity of Chiral Tubes

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

mapped sheet response

mapped sheet

response

tube

Size (R) Scaling of the Piezoelectric

Constants of Chiral Tubes

C, BN NT’s are prototypes with many other compact

meso-phases formed by folding lamellae

MoS2

WS2

Single- and

double- 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)

Remska et al.

Science 292, 479 (2001)

Physical Properties Control of

Composition AND Geometry

challenges

opportunities

• highly ordered

(coherent) structures

• access to quantum

geometrical effects

• phenomenology:

systematics in “families”

• control C&G in

synthesis

• structural sorting

• assembly of networks

and superstructures

Near-infrared Photoluminescence from Single-wall Carbon Nanotubes

Excitation (661 nm)

Emission ( 850 nm)

• 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.

Long Range

Interaction

• The “deviations problem”

They are very large… with ± asymmetry

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

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.

for a match and a stick?

It did quite literally;

he burned up pretty quick.

children’s poetry by Tim Burton (1997)

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