NC STATE UNIVERSITY Outline: I. Motivations: Why BN nanotubes might be an interesting pyro- and piezoelectric material? II. Methodology: How do we compute

NC 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?

Spontaneous polarization and piezoelectricityin boron-nitride nanotubes

Serge NakhmansonNorth Carolina State University

NC STATE UNIVERSITYI. Two main classes of industrial pyro/piezoelectrics

RepresentativesProperties

Lead Zirconate Titanate (PZT)

ceramics

Polymers

polyvinylidene fluoride (PVDF),

PVDF copolymer with trifluoroethylene

P(VDF/TrFE)

Materialclass

3PbTiO

Polarization( )

Piezoelectric const ( )

2C/m 2C/m

up to 0.9 5-10

up to 0.120.1-0.2

3x-1x OTiPbZr

3PbZrOGood pyro- and

piezoelectric properties

Pros

Heavy,Brittle,Toxic

Pyro- and piezoelectric

properties weaker than in PZT ceramics

Cons

Light,Flexible

Wurtzite oxides and nitrides up to 1.5up to 0.1

AlN-w

ZnO-w

NC STATE UNIVERSITYI. 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

NC STATE UNIVERSITYI. 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

NC STATE UNIVERSITYII. 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).

celll

lli

ii rdrrbeZV

rqV

P

)(1

1

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.

samplesample l

llsample

rdrrbeZV

P

)(1 )(r

+ +

+ +

+ +

+ +vs

Information about the charge transfer through the surface of the cellis required to compute polarization. Such cell dipole moment is not a bulk property (cell shape dependent).

NC STATE UNIVERSITYII. Modern Theory of Polarization

2) Polarization derivatives are well defined and can be computed:

piezosp PPP

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.

At zero external field

– marks the state of the system along the adiabatic transition path. System must stay insulating during the transition.

• In general: )()( 12 PPP

• Piezo:

)( )0(ii

i i

xxx

PeP

)nonpolar()polar( PPP

• Spontaneous:

NC STATE UNIVERSITYII. Computing polarization: Wannier function connection

Electronic part of the polarization cell

el rdrrV

P

)(1

)( )(

Calculations with Wannier functions: maximally localized Wannier functions(Marzari & Vanderbilt, PRB 1997) obtained by minimization of the spread functional

][2

occ

2n

nn rrΩ

Pros: the problem is reduced to Clausius-Mossotti case.

Cons: tedious to compute except in large cells (Γ-point approximates the whole BZ)

Unitary transformation :W

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

)()( )()(

occ

)(

occ

)()( 22)(

nn

nnn

el rV

eWrW

V

eP

Summation over WF centersDipole moment well defined!

Did not get rid of the multivalued nature of polarization: )(elP

defined modulo ,2 VRe l

because WF centers are defined modulo lattice vector .lR

NC STATE UNIVERSITYII. Berry phases

It can be shown that(Blount, Sol. St. Phys. 13)

)()(

occ 3

occ

)()(

)2(

nkknkn BZn

nn uukdiV

WrW

)()(

occ 3

)()(

)2(

2 )()()(

nkknk

n BZlll

elion uukdie

bZV

ePPP

Phases: ionic )()()(

l

ll

ion bGZ

electronic )()(

occ 3)2(

2 )(

nkknk

n BZ

el uukdiVG

total )()()( elion

Recover polarization by 1 ;)()( : GRVRePRα

ePGV )()(

Berry phase angular variable defined modulo .2

direction in which polarization is computed.

due to arbitrariness of the phases of)(elP

still defined modulo ,2 VRe l

).()( rnk

in calculations VReP lel

2

NC STATE UNIVERSITYIII. 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).

NC STATE UNIVERSITYIII. Nanotube primer

“Armchair” “Zigzag”

NC STATE UNIVERSITYIII. Folding hexagonal BN into a nanotube

1a

2a

sheet of hexagonal BN

)0,(n

Zigzag NT

),( nn

Armchair NT

),( mn

Chiral NT

NC STATE UNIVERSITYIII. What should we expect from BNNTs polarization-wise?

Polarization as a collection of dipoles

Armchair NT ─ nonpolar(centrosymmetric)

z

Zigzag NT ─ polar

z

Chiral NT ─ somewhere in between

NC STATE UNIVERSITYIII. Piezoelectric 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


u

P

ec

VZ z

0

*dc

duZ

V

ec

c

Pce z *

20

033



c

Pz


u

P

ec

VZ z

0

*dc

duZ

V

ec

c

Pce z *

20

033



u

Pz

NC STATE UNIVERSITYIII. Ionic 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 UNIVERSITYIII. Ionic 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 (modulo 2):


NC STATE UNIVERSITYIII. Electronic 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


NC STATE UNIVERSITYIII. 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 ).(nPel

z

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

NC STATE UNIVERSITYIII. Wannier functions in flat C and BN sheets


No spontaneous polarization in BN sheet due to the presence of the three-fold symmetry axis

NC STATE UNIVERSITYIII. 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


NC STATE UNIVERSITYIII. 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

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?

NC STATE UNIVERSITYIII. Extending to (n,m) nanotubes: example with ionic phase

0 1/6 1/2 2/3 1c

B

N 3

cedhex

Dipole moment of one hexagon along c:

)(3 21 aae

dhex

hC

1a

2a

T

21 amanCh

21

22a

D

mna

D

nmT

rr

Chiral vector

Translation vector

)(3

1),(),( mnmnP

Te

Vmn ion

zionz

)(6

),(

2

mnD

ec

VT

N

dTVT

NmnP

r

hex

hexhexion

z

NC STATE UNIVERSITYIII. General formula for polarization in BN nanotubes

033

),(),(),(

mnmn

mnmnmn elz

ionz

totz 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

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 responsiblefor the absence of polarization?

2C/m 113.0 totzP

NC STATE UNIVERSITYIV. BN nanotube’s place among other polar materials

RepresentativesProperties

Lead Zirconate Titanate (PZT)

ceramics

Polymers

polyvinylidene fluoride (PVDF),

PVDF copolymer with trifluoroethylene

P(VDF/TrFE)

Materialclass

3PbTiO

Polarization( )

Piezoelectric const ( )

2C/m 2C/m

up to 0.9 5-10

up to 0.120.1-0.2

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: ?

Light,Flexible; good piezoelectric

properties

Expensive?

NC STATE UNIVERSITYIV. 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?

NC STATE UNIVERSITYAcknowledgments

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

NC STATE UNIVERSITYII. Computing the electronic phase

k

G

A

A

el kkdVG

)()2(

2)( )(

3

Electronic phase

jk

1jk

)()(1

0

)()(

occ 0

)(

1detln Im lim

)(

jj mknk

J

jJ

nkknkn

G

uu

uudkik

)()( )()(

0rueru nk

riGnkJ

)()(kJ

“String” phase , contains information

about the current flowing through the cell

k

J

k

el kN

)(

2)( )(

due to arbitrariness of the phases of)(elP

still defined modulo ,2 VRe l

).()( rnk

in calculations VReP lel

2

Documents

NC STATE UNIVERSITY Outline: I. Motivations: Why BN nanotubes might be an interesting pyro- and piezoelectric material? II. Methodology: How do we compute