NC 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 Designing novel polar materials through computer simulations Serge Nakhmanson North Carolina State University 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

NC STATE UNIVERSITY 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

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

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

NC STATE UNIVERSITY 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

NC STATE UNIVERSITY 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. Ionic part: Localized charges, easy to compute Electronic part Charges usually delocalized Periodic solid: ill-defined because charges are delocalized

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

NC STATE UNIVERSITY Berry phases and localized Wannier functions Wannier functionBloch orbital Electronic part of the polarization Computed by finite differences on a fine k-point grid in the BZ Polarization Berry (electronic) phase : reciprocal lattice vector in direction α “Ionic phase”

NC STATE UNIVERSITY Berry phases and localized Wannier functions Wannier functionBloch orbital Electronic part of the polarization In both cases is defined modulo Summation over WF centers Dipole moment well defined! WFs can be made localized by an iterative technique (Marzari & Vanderbilt, PRB 1997) (R. D. King-Smith & D. Vanderbilt, PRB 1993)

NC STATE UNIVERSITY 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

NC STATE UNIVERSITY Boron-Nitride Nanotubes

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

NC STATE UNIVERSITY Ionic phase in zigzag BN nanotubes Ionic polarization parallel to the axis of the tube: Ionic phase (modulo 2  ): CarbonBoron-Nitride “virtual crystal” approximation BNNTCNT

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

NC STATE UNIVERSITY Electronic phase in zigzag BN nanotubes Berry-phase calculations provide no recipe for unfolding the electronic phase! Axial electronic polarization: Electronic phase (modulo 2  ): ─ occupied Bloch states CarbonBoron-Nitride

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

NC STATE UNIVERSITY Wannier functions in flat C and BN sheets   CarbonBoron-Nitride No spontaneous polarization in BN sheet due to the presence of the three-fold symmetry axis

NC STATE UNIVERSITY Wannier functions in C and BN nanotubes c   c  0 5/48 7/24 29/48 19/24 1c 1/6 2/3 B N 0 1/12 1/3 7/12 5/6 1c Carbon Boron-Nitride

NC STATE UNIVERSITY 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 C ½c1c0 B N BN ½c1c0 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:

NC STATE UNIVERSITY Total phase in zigzag nanotubes: Zigzag nanotubes are not pyroelectric! What about a more general case of chiral nanotubes? (n,m)R (bohr) 3, / ,23.221/3-1/30 mod(π) 4, mod(π) 4, /31/30 mod(π) 5, mod(π) 8, mod(π) All wide BN nanotubes are not pyroelectric! But breaking of the screw symmetry by bundling or deforming BNNTs makes them weakly pyroelectric.

NC STATE UNIVERSITY 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

NC STATE UNIVERSITY Ferroelectric Polymers (work in progress)

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

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

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

NC STATE UNIVERSITY Polar materials: the big picture Representatives Properties Lead Zirconate Titanate (PZT) ceramics Polymers polyvinylidene fluoride (PVDF), PVDF copolymers Material class Polarization ( ) Piezoelectric const ( ) up to up to 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: ? Single NT: 0 Bundle: ~0.01 Light, Flexible; good piezoelectric properties Expensive?

NC STATE UNIVERSITY 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