1. North Carolina State University
Spontaneous polarization and piezoelectricity in 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 nanotube’s possible place among other pyro- and
piezoelectric materials?

2. I. Two main classes of industrial pyro/piezoelectrics
Material
class
Representatives
Properties
Pros
Cons
Polarization
( )
Piezoelectric const ( )
Lead Zirconate Titanate (PZT) ceramics
up to 0.9
5-10
Good pyro- and piezoelectric properties
Heavy,
Brittle,
Toxic
Polymers
polyvinylidene fluoride (PVDF),
PVDF copolymer with trifluoroethylene P(VDF/TrFE)
up to 0.12
Light,
Flexible
Pyro- and piezoelectric properties weaker than in PZT ceramics
Wurtzite oxides and nitrides
up to 0.1
up to 1.5

3. 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

4. Neat nanodevices that can be made out of pyro-
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

5. II. Polarization as a collection of dipoles
How was polarization computed before MTP?
Ashcroft-Mermin: Polarization is a collection of dipoles:
cell dipole moment
is ill-defined except for “Clausius-Mossotti limit” (R. M. Martin, PRB 1974). + vs
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).
We can formally fix this, summing over the whole sample:
and includes all
boundary charges.
This is still not a bulk property: depends on the shape of the sample.
Such definition can not be used in realistic calculations.

6. 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
• Piezo:
• Spontaneous:
– marks the state of the system along the adiabatic transition path.
System must stay insulating during the transition.
• In general:

7. II. Computing polarization: Wannier function connection
Electronic part of the polarization
Unitary transformation
Bloch orbital
Wannier function
Summation over WF centers
Dipole moment well defined!
Did not get rid of the multivalued nature of polarization:
defined modulo
because WF centers are defined modulo lattice vector
Calculations with Wannier functions: maximally localized Wannier functions
(Marzari & Vanderbilt, PRB 1997) obtained by minimization of the spread functional
Pros: the problem is reduced to Clausius-Mossotti case.
Cons: tedious to compute except in large cells (Γ-point approximates the whole BZ)

8. II. Berry phases It can be shown that (Blount, Sol. St. Phys. 13)
angular variable defined modulo
direction in which polarization is computed.
Recover polarization by
Phases: ionic
electronic
total
due to arbitrariness of the phases of
still defined modulo
in calculations

9. 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
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).

10. III. Nanotube primer
“Armchair”
“Zigzag”

11. III. Folding hexagonal BN into a nanotube
sheet of hexagonal BN
Armchair NT
Chiral NT
Zigzag NT

12. III. What should we expect from BNNTs polarization-wise?
Polarization as a collection of dipoles
Zigzag NT ─ polar
Chiral NT ─ somewhere in between
Armchair NT ─ nonpolar
(centrosymmetric)

13. III. Piezoelectric properties of zigzag BN nanotubes
Born effective charges
Piezoelectric constants
Cell of volume
─ equilibrium
parameters
(w-GaN and w-ZnO data from F. Bernardini, V. Fiorentini, D. Vanderbilt, PRB 1997)

14. 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)

15. 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)

16. III. Ionic phase in zigzag BN nanotubes
Carbon
Boron-Nitride
Ionic phase (modulo 2):
Ionic polarization parallel to
the axis of the tube:
BNNT
CNT
“virtual crystal” approximation

17. III. Ionic phase in zigzag BN nanotubes
Carbon
Boron-Nitride
Ionic phase (modulo 2):
Ionic polarization parallel to
the axis of the tube:
Ionic phase can be easily
unfolded:

18. III. Electronic phase in zigzag BN nanotubes
Electronic phase (modulo 2):
─ occupied Bloch states
Carbon
Boron-Nitride
Axial electronic polarization:
Berry-phase calculations provide no recipe for unfolding the electronic phase!

19. III. Problems with electronic Berry phase
3 families of behavior :  = /3, -, so that the polarization can be positive or negative depending on the nanotube index?
counterintuitive!
Previous model calculations find  = /3, 0. Are 0 and  related by a trivial phase?
Electronic phase can not be unfolded; can not unambiguously compute
(Kral & Mele, PRL 2002)
-orbital TB model
Have to switch to Wannier function formalism to solve these problems.

20. 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

21. III. Wannier functions in C and BN nanotubes
Carbon
Boron-Nitride c c    
0 5/48 7/ /48 19/24 1c
1/ /3 B N
0 1/ / / /6 1c

22. III. Unfolding the electronic phase
Electronic polarization is purely due to the -WF’s ( centers cancel out).
Electronic polarization is purely axial with an effective periodicity of ½c (i.e. defined modulo
instead of ): equivalent to phase indetermination of !
can be folded into the 3 families of the Berry-phase calculation: C ½c 1c B N BN
(5,0):
-5/3
+2
+/3
(6,0):
-6/3
+1 -
(7,0):
-7/3
-/3
(8,0):
-8/3
+3

23. Total phase in zigzag nanotubes:
Zigzag nanotubes are not pyroelectric!
What about a more general case of chiral nanotubes?

24. III. Extending to (n,m) nanotubes: example with ionic phase
Chiral vector
Translation vector
/ /2 2/ c B N
Dipole moment of one
hexagon along c:

25. 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
0 mod(π)
4,1
3.39 1
4,2
3.91
5,2
4.62 -1
8,2
6.78
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?

26. IV. BN nanotube’s place among other polar materials
Representatives
Properties
Lead Zirconate Titanate (PZT) ceramics
Polymers
polyvinylidene fluoride (PVDF),
PVDF copolymer with trifluoroethylene P(VDF/TrFE)
Material
class
Polarization
( )
Piezoelectric const ( )
up to 0.9
5-10
up to 0.12
Good pyro- and piezoelectric properties
Pros
Heavy,
Brittle,
Toxic
Pyro- and piezoelectric properties weaker than in PZT ceramics
Cons
Light,
Flexible
BN nanotubes
(5,0)-(13,0) BN nanotubes
Single NT:
Bundle: ?
Light,
Flexible; good piezoelectric properties
Expensive?

27. 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?

28. 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

29. II. Computing the electronic phase
“String” phase , contains information about the current flowing through the cell
due to arbitrariness of the phases of
still defined modulo
in calculations