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 NakhmansonNorth 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 UNIVERSITYProperties of ferroelectric ceramics

Lead Zirconate Titanate (PZT) ceramics

Representatives:

Spontaneous polarization: up toPiezoelectric const (stress):

Mechanical/Environmental properties: Heavy, brittle, toxic!

Alternatives?

3x-1x33 OTiPbZr ,PbTiO ,PbZrO2C/m 9.02C/m 105 Very good pyro- and

piezoelectric properties!

Nature of polarization:reduction of symmetry

NC STATE UNIVERSITYBN 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

1a

2a

hexagonal BN

)0,(n

Zigzag NT ─ polar?

NC STATE UNIVERSITYBN 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

1a

2a

hexagonal BN

),( nn

Armchair NT ─ non-polar (centrosymmetric)

)0,(n

NC STATE UNIVERSITYFerroelectric 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

2C/m 2.01.0 2C/m 2.0 Weaker

than in PZT!

PVDF structural unit

NC STATE UNIVERSITYA simple view on polarization

Macroscopic solid:

and includes all boundary charges.

Polarization is well defined but this definition cannot be used in realistic calculations.

samplesample l

llsample

rdrrbeZV

P

)(1 )(r

Ionic part:Localized charges,easy to compute

Electronic partCharges usually delocalized

Periodic solid:

celll

lli

ii rdrrV

beZV

rqV

P

)(11

1

ill-defined because charges are delocalized

NC STATE UNIVERSITYComputing 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 1994

1) Polarization is a multivalued quantity and its absolute value cannot be computed.

Piezoelectric polarization:

)( )0(ii

i i

xxx

PeP

)nonpolar()polar( PPP

Spontaneous polarization:

The scheme to compute polarization with MTP can be easily formulatedin the language of the density functional theory.

NC STATE UNIVERSITYBerry phases and localized Wannier functions

occ

2

occ

2

3)(2 )(

)2(

2 )(

nn

n BZ

nk rWekdre

r

Wannier function

BZ

nkn kdrVrW

)()2()( 3 Bloch orbital

rkinknk erur

)()(

Electronic part of the polarization cell

el rdrrV

P

)(1

Computed by finite differenceson a fine k-point grid in the BZ

nkknkn BZ

el uukdie

P

occ 3)2(

2

Polarization 1 ; GRVReP

ePGV elel

Berry (electronic) phase

: reciprocal lattice vector in direction αG

ll

lion bGZ

“Ionic phase”

NC STATE UNIVERSITYBerry phases and localized Wannier functions

occ

2

occ

2

3)(2 )(

)2(

2 )(

nn

n BZ

nk rWekdre

r

Wannier function

BZ

nkn kdrVrW

)()2()( 3 Bloch orbital

rkinknk erur

)()(

Electronic part of the polarization cell

el rdrrV

P

)(1

nkknkn BZ

el uukdie

P

occ 3)2(

2

In both caseselP

is defined modulo VRe l

2

occ occ

22

nn

nnn

el rV

eWrW

V

eP

Summation over WF centersDipole moment well defined!

WFs can be made localized by an iterative technique

(Marzari & Vanderbilt, PRB 1997)(R. D. King-Smith &

D. Vanderbilt, PRB 1993)

VReP elel

NC STATE UNIVERSITYSummary 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 UNIVERSITYPiezoelectric properties of zigzag BN nanotubes

u

P

ec

VZ z

0

*dc

duZ

V

ec

c

Pce z *

20

033

(w-GaN and w-ZnO data from F. Bernardini, V. Fiorentini, D. Vanderbilt, PRB 1997)

Born effective charges Piezoelectric constants

c uCell of volumeV

00 ,uc ─ equilibrium parameters

NC STATE UNIVERSITYIonic phase in zigzag BN nanotubes

V

necnP

ionzion

z

)()(

Ionic polarization parallel to

the axis of the tube:

)()()( lzl

lionz bGZ

Ionic phase (modulo 2):

Carbon Boron-Nitride

“virtual crystal” approximation

BNNT CNT

NC STATE UNIVERSITYIonic phase in zigzag BN nanotubes

Ionic phase can be easily

unfolded:

3

)(n

nionz

V

necnP

ionzion

z

)()(

Ionic polarization parallel to

the axis of the tube:

)()()( lzl

lionz bGZ

Ionic phase:

Carbon Boron-Nitride

NC STATE UNIVERSITYElectronic phase in zigzag BN nanotubes

Berry-phase calculations provide no recipe for unfolding the electronic phase!

V

necnP

elzel

z

)()(

Axial electronic polarization:

Electronic phase (modulo 2):

)()(1

01

detln Im2)( jj qkpk

J

j

elz uu

)( jpku ─ occupied Bloch states

Carbon Boron-Nitride

NC STATE UNIVERSITYProblems 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 ).(nPelz

Have to switch to Wannier function formalism to solve these problems.

NC STATE UNIVERSITYWannier 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

NC STATE UNIVERSITYWannier 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 UNIVERSITYUnfolding 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

½c 1c0

B

N

BN

½c 1c0

i

Ci

BNi

elz rr

V

enP )(

2)(

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:

3

)(2

)(n

zzc

n Ci

i

BNi

elz

Vec2Vec

NC STATE UNIVERSITY

Total phase in zigzag nanotubes:

033

)()()( nn

nnn elz

ionz

totz

Zigzag nanotubes are not pyroelectric!

What about a more general case of 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(π)

4,1 3.39 1 1 0 mod(π)

4,2 3.91 -1/3 1/3 0 mod(π)

5,2 4.62 1 -1 0 mod(π)

8,2 6.78 0 1 0 mod(π)

)( totz)( el

z)( ionz

2C/m 113.0 totzP

All wide BN nanotubes are not pyroelectric!

But breaking of the screw symmetry by bundling ordeforming BNNTs makes them weakly pyroelectric.

2C/m 01.0 totzP

NC STATE UNIVERSITYSummary 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: 0.05-0.076

Empirical models (100% crystalline) Polarization ( )

Dipole summation with no interaction: 0.131Mopsik and Broadhurst, JAP, 1975; Kakutani, J Polym Sci, 1970: 0.22 Purvis and Taylor, PRB 1982, JAP 1983: 0.086Al-Jishi and Taylor, JAP 1985: 0.127Carbeck, Lacks and Rutledge, J Chem Phys, 1995: 0.182

2C/m

2C/m

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

2C/m 000.0P

uniaxially oriented non-poled PVDF – not polar

2C/m 178.0Pcrude estimate for 50%

crystalline sample:

2C/m 076.005.0

2C/m 09.0Pexperiment

2C/m 178.0PBerry phase method

with DFT/GGA

NC STATE UNIVERSITY

P(VDF/TeFE) 75/25 copolymerP(VDF/TrFE) 75/25 copolymer

Polarization in PVDF copolymers

2C/m 150.0P 2C/m 132.0P

2C/m 178.0Pβ-PVDF:

Comparison with experiment: very crude predictions for 73/27 P(VDF/TrFE) copolymer projected to 100% crystallinity

(Furukawa, IEEE Trans. 1989)

2C/m 160.0120.0 P

Comparison with experiment: in 80/20 P(VDF/TeFE) copolymer projected to 100% crystallinity

(Tasaka and Miyata, JAP 1985)

2C/m 140.0126.0 P

NC STATE UNIVERSITYPolar materials: the big picture

RepresentativesProperties

Lead Zirconate Titanate (PZT)

ceramics

Polymerspolyvinylidene fluoride

(PVDF),PVDF copolymers

Materialclass

3PbTiO

Polarization( )

Piezoelectric const ( )

2C/m 2C/m

up to 0.9 5-10

up to 0.20.1-0.2

3x-1x OTiPbZr3PbZrO

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:0.25-0.4Bundle:

?

Single NT:0

Bundle: ~0.01

Light,Flexible; good piezoelectric

properties

Expensive?

NC STATE UNIVERSITYConclusions

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