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
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)
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”
In both caseselP
is defined modulo VRe l
2
(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