NC STATE
UNIVERSITY
Designing new polar materials on a computer
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
NC STATE
UNIVERSITY
Properties of ferroelectric ceramics
Nature of polarization:
reduction of symmetry
see G. Saghi-Szabo et. al.
PRL 1998, PRB 1999,
also D. Vanderbilt
and K. Rabe
Lead Zirconate Titanate (PZT) ceramics
Representatives:
PbZrO 3 , PbTiO 3 , PbZrx Ti 1-x O3
Spontaneous polarization: up to 0.9
Piezoelectric const (stress): 5  10
C/m 2
C/m 2

Mechanical/Environmental properties:
Alternatives?
Very good pyro- and
piezoelectric properties!
Heavy, brittle, toxic!
NC STATE
UNIVERSITY
Nanotube primer
“Armchair”
“Zigzag”
BN nanotubes as possible pyro/piezoelectric materials:
excellent mechanical properties: light and flexible,
almost as strong as carbon nanotubes (Zhang and Crespi,
PRB 2000)

a2
NC STATE
UNIVERSITY
hexagonal BN

a1
chemically inert: proposed to be used as coatings
all insulators with no regard to chirality and constant
band-gap of around 5 eV
(n,0)
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
Zigzag NT ─ polar?
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

a2
NC STATE
UNIVERSITY
hexagonal BN

a1
( n, n )
(n,0)
intrinsically polar due to the polar nature of B-N bond
Armchair NT ─ non-polar
most of the BN nanotubes are non-centrosymmetric
(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
NC STATE
UNIVERSITY
Ferroelectric polymers
PVDF structural unit
β-PVDF
Representatives: polyvinylidene fluoride (PVDF),
PVDF copolymers, nylons, etc.
Spontaneous polarization: 0.1  0.2
Piezoelectric const (stress): up to 0.2
Mechanical/Environmental properties:
C/m 2
C/m 2

Weaker
than in PZT!
Light, flexible, non-toxic
Applications: sensors, transducers, hydrophone probes, sonar
NC STATE
UNIVERSITY
NC STATE
UNIVERSITY
Computing polarization
NC STATE
UNIVERSITY
A simple view on polarization
Finite macroscopic solid: (Ashcroft-Mermin)

P


  
   eZ l bl   r  (r )dr 
Vsample  sample l

sample
1

and  (r ) includes all
boundary charges.
Polarization is well defined but this definition cannot be used in realistic
calculations.
Periodic solid:
 1   
 1
 1
P   qi ri   eZ l bl   r  (r )dr
V i
V l
V cell
Ionic part:
Localized charges,
easy to compute
+
+
+
+
+
+
vs
+
+
Electronic part
Charges usually delocalized
ill-defined because charges
are delocalized
exception: “Clausius-Mossotti limit”
(R. M. Martin, PRB 1974)
Computing polarization in a periodic solid
NC STATE
UNIVERSITY
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.
Piezoelectric polarization:
P
P   e    
( xi  xi( 0 ) )

i xi
Spontaneous polarization:
 

P  P(polar )  P(nonpolar )
The scheme to compute polarization with MTP can be easily formulated
in the language of the density functional theory.
Berry phases and localized Wannier functions
 el 1   
Electronic part of the polarization P 
r  (r )dr

V cell


 2
 2
2e
 (r )  
 nk (r ) dk   2e  Wn (r )
3  
(2 ) n occ BZ
n occ
Bloch orbital


 nk (r )  unk (r )e
 el
2ie
P 
(2 ) 3

 
Wn (r )  V (2 )  nk (r )dk
Wannier
 function

ik r
n occ BZ
Computed by finite differences
on a fine k-point grid in the BZ


  el
Berry (electronic) phase   V G  P e
 
ion
“Ionic phase”    Z l G  bl
l

G : reciprocal lattice vector in direction α


3
BZ


 dk unk  k unk
el
NC STATE
UNIVERSITY


 
Polarization P  e R V ; R  G  1
Berry phases and localized Wannier functions
 el 1   
Electronic part of the polarization P 
r  (r )dr

V cell


 2
 2
2e
 (r )  
 nk (r ) dk   2e  Wn (r )
3  
(2 ) n occ BZ
n occ
Bloch orbital


 nk (r )  unk (r )e
 el
2ie
P 
(2 ) 3


ik r


 dk unk  k unk
n occ BZ
 el

el
P  e R V
(R. D. King-Smith &
D. Vanderbilt, PRB 1993)
NC STATE
UNIVERSITY
 
Wn (r )  V (2 )  nk (r )dk
Wannier
 function
3
BZ
 el


2e
2e
P    Wn r Wn    rn
V n occ
V n occ
Summation over WF centers
Dipole moment well defined!
WFs can be made localized by an
iterative technique
(Marzari & Vanderbilt, PRB 1997)
 el
In both cases P

is defined modulo 2eRl V
Summary for the theory section
NC STATE
UNIVERSITY
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
Piezoelectric properties of zigzag BN nanotubes
Born effective charges
Z* 
Piezoelectric constants
e33
V Pz
ec0 u
(w-GaN and w-ZnO data from
F. Bernardini, V. Fiorentini,
D. Vanderbilt, PRB 1997)
NC STATE
UNIVERSITY
Pz ec02 * du
 c0

Z
c
V
dc
Cell of volume V
c
u
c 0 , u 0 ─ equilibrium
parameters
Piezoelectric properties of zigzag BN nanotubes
Born effective charges
Z* 
Piezoelectric constants
e33
V Pz
ec0 u
(w-GaN and w-ZnO data from
F. Bernardini, V. Fiorentini,
D. Vanderbilt, PRB 1997)
NC STATE
UNIVERSITY
Pz

c
Pz ec02 * du
 c0

Z
c
V
dc
Piezoelectric properties of zigzag BN nanotubes
Born effective charges
Z* 
Piezoelectric constants
e33
V Pz
ec0 u
(w-GaN and w-ZnO data from
F. Bernardini, V. Fiorentini,
D. Vanderbilt, PRB 1997)
NC STATE
UNIVERSITY
Pz

u
Pz ec02 * du
 c0

Z
c
V
dc
Ionic phase in zigzag BN nanotubes
Ionic phase (modulo 2):
 ( )   Z
ion
z
( )
l

  ( )
Gz  bl
NC STATE
UNIVERSITY

l
Ionic polarization parallel to
the axis of the tube:
ec zion (n)
P (n) 
V
ion
z
BNNT
CNT
“virtual crystal” approximation
Carbon
Boron-Nitride
Ionic phase in zigzag BN nanotubes
Ionic phase:
 ( )   Z
ion
z
( )
l

  ( )
Gz  bl
NC STATE
UNIVERSITY

l
Ionic polarization parallel to
the axis of the tube:
ec zion (n)
P (n) 
V
ion
z
Ionic phase can be easily
unfolded:
 zion (n) 
n
 
3
Carbon
Boron-Nitride
NC STATE
UNIVERSITY
Electronic phase in zigzag BN nanotubes
J 1

el
( )
( )
Electronic phase (modulo 2):  z ( )  2Im ln   det u pk j u qk j 1

 j 0
( )
─
occupied
Bloch
states
u pk j






Axial electronic polarization:
el
ec


z ( n)
Pzel (n) 
V
Berry-phase calculations
provide no recipe for unfolding
the electronic phase!
Carbon
Boron-Nitride
Problems with electronic Berry phase
NC STATE
UNIVERSITY
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?
-orbital TB model
Electronic phase can not be unfolded;
can not unambiguously compute Pzel (n).
Have to switch to Wannier function
formalism to solve these problems.
(Kral & Mele, PRL 2002)
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
NC STATE
UNIVERSITY
NC STATE
UNIVERSITY
Wannier functions in C and BN nanotubes
Carbon
c
Boron-Nitride
c



N

B
0 1/12
1/3
7/12
5/6 1c
0 5/48 7/24
1/6
29/48 19/24 1c
2/3
NC STATE
UNIVERSITY
Unfolding the electronic phase
 el
 BN  C
2e
Pz (n)    (ri  ri )
V i
N
BN
B
0
½c
1c
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 ec
instead of 2ec V ): equivalent to phase
indetermination of !
V
2
n
BN
C
 (n)    ( zi zi )    
c i
3
el
z
can be folded into the 3 families of the Berry-phase
calculation:
C
0
½c
1c
(5,0):
-5/3
+2
+/3
(6,0):
-6/3
+1
-
(7,0):
-7/3
+2
-/3
(8,0):
-8/3
+3
+/3
NC STATE
UNIVERSITY
Total phase in zigzag nanotubes:
n n
 (n)   (n)   (n)    0
3 3
tot
z
ion
z
el
z
Zigzag nanotubes are not pyroelectric!
What about a more general case of chiral nanotubes?
 zion ( )  zel ( )  ztot ( )
(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)
Pztot  0.113 C/m 2
All wide BN nanotubes are not pyroelectric!
But breaking of the screw symmetry by bundling or
tot
2

P

0
.
01
C/m
z
deforming BNNTs makes them weakly pyroelectric.
Summary for the BN nanotubes
NC STATE
UNIVERSITY
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)
0.05-0.076 C/m 2
Polarization ( C/m 2)
Dipole summation with no interaction:
Mopsik and Broadhurst, JAP, 1975; Kakutani, J Polym Sci, 1970:
Purvis and Taylor, PRB 1982, JAP 1983:
Al-Jishi and Taylor, JAP 1985:
Carbeck, Lacks and Rutledge, J Chem Phys, 1995:
0.131
0.22
0.086
0.127
0.182
Which model is better? Ab Initio calculations can help!
What about copolymers?
Polarization in β-PVDF from the first principles
uniaxially oriented
non-poled PVDF – not polar
β-PVDF – polar
P  0.178 C/m 2
P  0.000 C/m 2
4.91 Å
Berry phase method
with DFT/GGA
NC STATE
UNIVERSITY
8.58 Å
P  0.178 C/m
2
crude estimate for 50%
crystalline sample:
P  0.09 C/m 2
experiment
0.05  0.076 C/m 2
NC STATE
UNIVERSITY
Polarization in PVDF copolymers
β-PVDF:
P  0.178 C/m 2
P(VDF/TrFE) 75/25 copolymer
P  0.150 C/m 2
P  0.132 C/m 2
Comparison with experiment:
very crude predictions for
73/27 P(VDF/TrFE) copolymer
projected to 100% crystallinity
P  0.120  0.160 C/m
P(VDF/TeFE) 75/25 copolymer
2
(Furukawa, IEEE Trans. 1989)
Comparison with experiment:
in 80/20 P(VDF/TeFE) copolymer
projected to 100% crystallinity
P  0.126  0.140 C/m 2
(Tasaka and Miyata, JAP 1985)
NC STATE
UNIVERSITY
Polar materials: the big picture
Material
class
Properties
Representatives
Polarization
2
(C/m )
Piezoelectric
2
const (C/m )
Pros
Cons
Good pyro- and
piezoelectric
properties
Heavy,
Brittle,
Toxic
Light,
Flexible
Pyro- and
piezoelectric
properties
weaker than in
PZT ceramics
Light,
Flexible; good
piezoelectric
properties
Expensive?
PbTiO 3
Lead Zirconate
Titanate (PZT)
ceramics
Polymers
BN nanotubes
PbZrO 3
PbZrx Ti 1-x O3
polyvinylidene fluoride
(PVDF),
PVDF copolymers
(5,0)-(13,0) BN nanotubes
up to 0.9
5-10
0.1-0.2
up to 0.2
Single NT:
Single NT:
0
0.25-0.4
Bundle:
Bundle:
~0.01
?
Conclusions
NC STATE
UNIVERSITY
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