Outline: I. Motivations: Why do we need alternatives to ferroelectric ceramics?

1. Designing novel polar materials through computer simulations Serge Nakhmanson North Carolina State University Outline: I. Motivations: Why do we need alternatives to ferroelectric ceramics? II. Methodology: How do we compute polarization in periodic solids? III. Some alternatives studied in detail: 1. Boron-Nitride nanotubes 2. Ferroelectric polymers IV. Conclusions Acknowledgments: NC State University group: Jerry Bernholc Marco Buongiorno Nardelli Vincent Meunier (now at ORNL) Wannier functions collaboration: Arrigo Calzolari (U. di Modena) Nicola Marzari (MIT) Ivo Souza (Rutgers) Computational facilities: DoD Supercomputing Centers NC Supercomputing Center

2. Properties of ferroelectric ceramics Lead Zirconate Titanate (PZT) ceramics Representatives: Spontaneous polarization: up to Piezoelectric const (stress): Mechanical/Environmental properties: Heavy, brittle, toxic! Alternatives? Very good pyro- and piezoelectric properties! Nature of polarization: reduction of symmetry

3. BN nanotubes as possible pyro/piezoelectric materials: Zigzag NT ─ polar? hexagonal BN 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 Possible applications in nano-electro-mechanical devices: actuators, transducers, strain and temperature sensors

4. BN nanotubes as possible pyro/piezoelectric materials: Armchair NT ─ non-polar (centrosymmetric) hexagonal BN 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 Possible applications in nano-electro-mechanical devices: actuators, transducers, strain and temperature sensors

5. Ferroelectric polymers β-PVDF Representatives: polyvinylidene fluoride (PVDF), PVDF copolymers, nylons, etc. Spontaneous polarization: Piezoelectric const (stress): up to Mechanical/Environmental properties: Light, flexible, non-toxic Applications:sensors, transducers, hydrophone probes, sonar Weaker than in PZT! PVDF structural unit

6. A simple view on polarization Macroscopic solid: and includes all boundary charges. Polarization is well defined but this definition cannot be used in realistic calculations. Periodic solid: Ionic part: Localized charges, easy to compute Electronic part Charges usually delocalized ill-defined because charges are delocalized

7. Computing polarization in a periodic solid Piezoelectric polarization: Spontaneous polarization: Modern theory of polarization R. D. King-Smith & D. Vanderbilt, PRB 1993 R. Resta, RMP 1994 1) Polarization is a multivalued quantity and its absolute value cannot be computed. 2) Polarization derivatives are well defined and can be computed. The scheme to compute polarization with MTP can be easily formulated in the language of the density functional theory.

8. Berry phases and localized Wannier functions Electronic part of the polarization Wannier function Bloch orbital Summation over WF centers Dipole moment well defined! WFs can be made localized by an iterative technique (Marzari & Vanderbilt, PRB 1997) Berry (electronic) phase (R. D. King-Smith & D. Vanderbilt, PRB 1993) “Ionic phase” In both cases : reciprocal lattice vector in direction α is defined modulo Polarization Computed by finite differences on a fine k-point grid in the BZ

9. Summary for the theory section • In an infinite periodic solid polarization can be computed from the first principles with the help of Berry phases or localized Wannier functions • This method provides full description of polar properties of any insulator or semiconductor

10. Boron-Nitride Nanotubes

11. 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)

12. 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

13. Ionic phase in zigzag BN nanotubes Ionic polarization parallel to the axis of the tube: Ionic phase: Ionic phase can be easily unfolded: Boron-Nitride Carbon

14. 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!

15. 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)

16. 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

17. 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

18. 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

19. Total phase in zigzag nanotubes: Zigzag nanotubes are not pyroelectric! What about a more general case of chiral nanotubes? All wide BN nanotubes are not pyroelectric! But breaking of the screw symmetry by bundling or deforming BNNTs makes them weakly pyroelectric.

20. Summary for the BN nanotubes • Quantum mechanical theory of polarization in BN nanotubes in terms of Berry phases and Wannier function centers: individual BN nanotubes have no spontaneous polarization! • 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) • BN nanotubes can be made pyroelectric by deforming or bundling

21. Ferroelectric Polymers (work in progress)

22. “Dipole summation” models for polarization in PVDF Experimental polarization for approx. 50% crystalline samples: 0.05-0.076 Empirical models (100% crystalline)Polarization ( ) Dipole summation with no interaction: 0.131 Mopsik and Broadhurst, JAP, 1975; Kakutani, J Polym Sci, 1970: 0.22 Purvis and Taylor, PRB 1982, JAP 1983: 0.086 Al-Jishi and Taylor, JAP 1985: 0.127 Carbeck, Lacks and Rutledge, J Chem Phys, 1995: 0.182 Which model is better? Ab Initio calculations can help! What about copolymers?

23. Polarization in β-PVDF from the first principles uniaxially oriented non-poled PVDF – not polar Berry phase method with DFT/GGA 4.91 Å 8.58 Å crude estimate for 50% crystalline sample: experiment β-PVDF – polar

24. Polarization in PVDF copolymers P(VDF/TrFE) 75/25 copolymer P(VDF/TeFE) 75/25 copolymer Comparison with experiment: in 80/20 P(VDF/TeFE) copolymer projected to 100% crystallinity (Tasaka and Miyata, JAP 1985) Comparison with experiment: very crude predictions for 73/27 P(VDF/TrFE) copolymer projected to 100% crystallinity (Furukawa, IEEE Trans. 1989) β-PVDF:

25. Representatives 0.1-0.2 Properties Single NT: 0 Bundle: ~0.01 Single NT: 0.25-0.4 Bundle: ? up to 0.2 5-10 Polarization ( ) Piezoelectric const ( ) up to 0.9 Polar materials: the big picture 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 copolymers (5,0)-(13,0) BN nanotubes

26. Conclusions • Quantum mechanical theory of polarization in terms of Berry phases and Wannier function centers fully describes polar properties of any material • Polar boron-nitride nanotubes or ferroelectric polymers • are a good alternative/complement to ferroelectric ceramics: • Excellent mechanical properties, environmentally friendly • Polar properties still substantial • Numerous applications: sensors, actuators, transducers • Composites? • Methods for computing polarization can be used to study and predict • new materials with pre-designed polar properties