Outline: I. Motivations: Why BN nanotubes might be an interesting pyro- and

1. 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 nanotube’s possible place among other pyro- and piezoelectric materials?

2. Representatives 5-10 0.1-0.2 up to 0.12 up to 1.5 up to 0.9 Piezoelectric const ( ) up to 0.1 Properties Polarization ( ) I. Two main classes of industrial pyro/piezoelectrics Lead Zirconate Titanate (PZT) ceramics Good pyro- and piezoelectric properties Heavy, Brittle, Toxic Light, Flexible Material class Polymers Pros Pyro- and piezoelectric properties weaker than in PZT ceramics Cons Wurtzite oxides and nitrides polyvinylidene fluoride (PVDF), PVDF copolymer with trifluoroethylene P(VDF/TrFE)

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 insulatorswith no regard to chirality and constant band-gap of around 5 eV intrinsically polardue 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. 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 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 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. 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).

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

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 Berry phase direction in which polarization is computed. Recover polarization by Phases: ionic electronic total still defined modulo due to arbitrariness of the phases of 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 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).

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

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

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

13. III. Piezoelectric properties of zigzag BN nanotubes Cell of volume ─ equilibrium parameters Born effective charges Piezoelectric constants (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 Ionic polarization parallel to the axis of the tube: Ionic phase (modulo 2): BNNT CNT “virtual crystal” approximation Carbon Boron-Nitride

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

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

19. III. Problems with electronic Berry phase Problems: • 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 -orbital TB model Have to switch to Wannier function formalism to solve these problems. (Kral & Mele, PRL 2002)

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 c c   N   0 1/12 1/3 7/12 5/6 1c B 0 5/48 7/24 29/48 19/24 1c 1/6 2/3 Carbon Boron-Nitride

22. III. Unfolding the electronic phase N C BN • 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: B 0 ½c 1c 0 ½c 1c

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 Dipole moment of one hexagon along c: B N 0 1/6 1/2 2/3 1c Chiral vector Translation vector

25. III. General formula for polarization in BN nanotubes Chiral nanotubes: 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. Representatives 0.1-0.2 Properties Single NT: 0 Bundle: ? Single NT: 0.25-0.4 Bundle: ? up to 0.12 5-10 Polarization ( ) Piezoelectric const ( ) up to 0.9 IV. BN nanotube’s place among other polar materials Lead Zirconate Titanate (PZT) ceramics Light, Flexible Good pyro- and piezoelectric properties Light, Flexible; good piezoelectric properties Heavy, Brittle, Toxic Pros Pyro- and piezoelectric properties weaker than in PZT ceramics Cons BN nanotubes Polymers Material class Expensive? polyvinylidene fluoride (PVDF), PVDF copolymer with trifluoroethylene P(VDF/TrFE) (5,0)-(13,0) BN nanotubes

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 Electronic phase “String” phase , contains information about the current flowing through the cell still defined modulo due to arbitrariness of the phases of in calculations