Outline:

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Spontaneous polarization and piezoelectricityin
boron-nitride nanotubes
Serge Nakhmanson North Carolina State University
Outline I. Motivations Why BN nanotubes might
be an interesting pyro- and piezoelectric
material? II. Methodology How do we compute
polarization in semiconductors? 1.
Polarization as a collection of dipoles 2.
Modern theory of polarization (MTP), Wannier
functions and Berry phases III. Computations
piezo- and pyroelectric properties of BN
nanotubes IV. Conclusions BN nanotubes
possible place among other pyro- and
piezoelectric materials?
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I. Two main classes of industrial
pyro/piezoelectrics
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I. Properties of BN nanotubes
BN nanotubes as possible pyro- and piezoelectric
materials excellent mechanical properties
light and flexible, almost as strong as carbon
nanotubes (Zhang and Crespi, PRB
2000) chemically inert proposed to be used as
coatings all insulators with no regard to
chirality and constant band-gap of around 5
eV intrinsically polar due to the polar nature
of B-N bond most of the BN nanotubes are
non-centrosymmetric (i.e. do not have center of
inversion), which is required for the existence
of non-zero spontaneous polarization
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I. Applications

• Neat nanodevices that can be made out of pyro-
• and piezoelectric nanotubes
• actuators
• transducers
• strain and temperature sensors

Images from B. G. Demczyk et. al. APL 2001
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II. Polarization as a collection of dipoles
How was polarization computed before MTP?
Information about the charge transfer through the
surface of the cell is required to compute
polarization. Such cell dipole moment is not a
bulk property (cell shape dependent).
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II. Modern Theory of Polarization
References R. D. King-Smith D. Vanderbilt,
PRB 1993 R. Resta, RMP 1994
1) Polarization is a multivalued quantity (taking
on a lattice of values) and its absolute
value can not be computed.
2) Polarization derivatives are well defined and
can be computed
At zero external field
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II. Computing polarization Wannier function
connection
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II. Berry phases
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III. Software for polarization computations
Berry phases Massively parallel ab initio
real space LDA-DFT method with multigrid
acceleration (E.L. Briggs, D.J. Sullivan and J.
Bernholc, PRB 1996). Available at
http//nemo.physics.ncsu.edu/software/MGDFT-QMD/
Wannier functions Post-processing routine
for generation maximally localized Wannier
functions for entangled energy bands (Marzari and
Vanderbilt, PRB 1997 Souza, Marzari and
Vanderbilt, PRB 2001).
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III. Nanotube primer
Armchair
Zigzag
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III. Folding hexagonal BN into a nanotube
sheet of hexagonal BN
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III. What should we expect from BNNTs
polarization-wise?
Polarization as a collection of dipoles
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III. Piezoelectric properties of zigzag BN
nanotubes
Born effective charges
Piezoelectric constants
(w-GaN and w-ZnO data from F. Bernardini, V.
Fiorentini, D. Vanderbilt, PRB 1997)
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III. Piezoelectric properties of zigzag BN
nanotubes
Born effective charges
Piezoelectric constants
(w-GaN and w-ZnO data from F. Bernardini, V.
Fiorentini, D. Vanderbilt, PRB 1997)
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III. Piezoelectric properties of zigzag BN
nanotubes
Born effective charges
Piezoelectric constants
(w-GaN and w-ZnO data from F. Bernardini, V.
Fiorentini, D. Vanderbilt, PRB 1997)
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III. Ionic phase in zigzag BN nanotubes
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III. Ionic phase in zigzag BN nanotubes
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III. Electronic phase in zigzag BN nanotubes
Berry-phase calculations provide no recipe for
unfolding the electronic phase!
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III. Problems with electronic Berry phase
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III. Wannier functions in flat C and BN sheets
Carbon
Boron-Nitride
?
?
No spontaneous polarization in BN sheet due to
the presence of the three-fold symmetry axis
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III. Wannier functions in C and BN nanotubes
Carbon
Boron-Nitride
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III. Unfolding the electronic phase
(5,0) -5/3? 2? ?/3
(6,0) -6/3? 1? -?
(7,0) -7/3? 2? -?/3
(8,0) -8/3? 3? ?/3
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Total phase in zigzag nanotubes
Zigzag nanotubes are not pyroelectric! What
about a more general case of chiral nanotubes?
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III. Extending to (n,m) nanotubes example with
ionic phase
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III. General formula for polarization in BN
nanotubes
Chiral nanotubes
(n,m) R (bohr)
3,1 2.67 -1/3 0.113 -0.222
3,2 3.22 1/3 -1/3 0 mod(p)
4,1 3.39 1 1 0 mod(p)
4,2 3.91 -1/3 1/3 0 mod(p)
5,2 4.62 1 -1 0 mod(p)
8,2 6.78 0 1 0 mod(p)
All wide BN nanotubes are not pyroelectric! Is
the screw symmetry in BNNTs too strong to support
polarization? What happens when symmetry is
reduced? Or may the pseudo 1D character of BNNTs
be responsible for the absence of polarization?
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IV. BN nanotubes place among other polar
materials
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IV. Conclusions

• Materials Science
• Compared to wurtzite compounds and
  piezoelectric polymers, BN nanotubes are good
  piezoelectric materials that could be used for a
  variety of novel nanodevice applications
• Piezoelectric sensors
• Field effect devices and emitters
• Nano-Electro-Mechanical Systems (NEMS)
• Physics
• Quantum mechanical theory of polarization
  in BN nanotubes in terms of Berry phases and
  Wannier function centers BN nanotubes have no
  spontaneous polarization!
• Is it because the screw symmetry is too strong?
• What happens when the screw symmetry is broken
  bundles, multiwall
• nanotubes?
• Does the reduced dimensionality of BN nanotubes
  have anything to do with vanishing spontaneous
  polarization?

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Acknowledgments
NC State University group Jerry Bernholc
Marco Buongiorno Nardelli (also at ORNL)
Vincent Meunier (now at ORNL)
Wannier function code collaboration Arrigo
Calzolari (Universita di Modena, Italy) Nicola
Marzari (MIT) Ivo Souza (Rutgers)
Computational facilities DoD Supercomputing
Center NC Supercomputing Center
Funding NASA ONR
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II. Computing the electronic phase