Electron Theory of Ferroelectrics

Tamio Oguchi
ISIR, Osaka University

Electric Materials
pyroelectrics

piezoelectrics

dielectrics

ferroelectrics• Dielectrics: insulators• Piezoelectrics: non-polar/polar w. applied stress/strain• Pyroelectrics: polar, spontaneous polarization,
unswitchable• Ferroelectrics: polar, switchable → FERAM
2

Dielectric Constant and Polarization
condenser
f+ S‒‒‒‒‒‒‒‒+++++++DdD=
ChargeQ=|D| =0E

‒‒‒P‒
+ ‒‒
+ ‒‒
+ ‒‒
+ ‒

D=Surfacecharge density(C/m2 )

fSp‒

fElectric fieldVf|E| ==d⇥0V+f+++++++

Electric flux density

0E + P =Capacity

Q0S
C==Vd
Electric polarization
P = ⇥0 E0E

p relativefpermittivity

insulator
3

Susceptibility

Dielectric Constant and Polarization
f +b++
++
++
++
++
++
++
++

‒‒
‒
‒
‒
‒
‒
‒

P

A
V++++++++‒‒‒‒
‒‒
‒‒
‒‒
‒‒‒‒
‒‒

D=

0E + P =

|D| =
|E| =

C=

f +bf +b

⇥⇥00S
d0E

relative
permittivity=f⇥0⇤=1+⇥=

Dielectric constants and electric polarization
can be obtained by measuring charge transfer(electric current).
4f +fbElectric Dipole and Polarization
Electric dipolep = qxPolarization: dipole density

q +qpi‒1P =(C/m2 )

i+x

Localized ion model withpoint charges
e
P =
Zn Rn

‒

+

‒+

‒+

‒+

‒+

‒+

inversion operationr⇥ r⇤P ⇥ P

nno polarization with inversion

effective charge
5Paraelectrics and Ferroelectrics+

‒‒
++

‒+

‒+

‒+

‒+

‒E

‒++

• Ferroelectric phase- Spontaneous polarization- Non-volatile memory‒‒

P = ⇥0 E+P‒• Paraelectric phase

phase transition‒P‒

‒

+++‒‒

‒+++‒+

‒+‒+

‒+

‒E
order-disorder
6displaceElectron Theory of Ferroelectrics
Ferroelectrics
Spontaneous polarization
traditional interpretation:
polarization of ionic charges

PCrucial Factors → Necessity of electron theory
・Broken inversion symmetry
・Detailed response of electrons
7EBaTiO3
Paraelectric phase
P m3̄mFerroelectric phase
P 4mm403KPs
polarization
c/a ~ 1.01

c/a = 1.00

displacementsu(Ti) = 0.054Å
u(O1) = –0.097Å
u(O2) = –0.061Å

Ferroelectric Instability
Broken Inversion
Why？
8Born-Oppenheimer Potential•

BO Approximation⇤nNucleic system
22MnHen>nZn Zn e2+ Ee ({Rn })|Rn Rn |({Rn }) = E ({Rn })BO potential

Electron system({ri } ; {Rn }) = Ee ({Rn }) ({ri } ; {Rn })He =•⌃2n +⇤⇥

2i2m⌅2i +e2|rii>jrj |Zn e2|rR|ini,nexternal ﬁeld by nuclei

Density Functional Theory-One-electron problem in an effective field
9

Born-Oppenheimer Potential•

BO Potential (Total Energy)E({Rn }) = +•Atomic Forces → Structure Optimization, MD

Fn ({Rn }) =•n>nZn Zn e2+ Ee ({Rn })|Rn Rn |⌅n E({Rn })Force Constants → Phonon, Structure Stability

kn ,n ⇥ ({Rn }) =2

RnRn ⇥

E({Rn })

10

PhononCalculation
凍結フォノン近似•

Atomic Coordinates

Rn = Rl = Rl +lattice vector coordinates in cell••

Atomic Displacements(0)
Rlul = R lequilibrium
positionsHarmonic Approximation with wave vector Q andangular frequency

ul = u ei(Q·Rl•t)

Bloch Theorem

ul= ul eiQ·(RlRl )11Phonon
Calculation
凍結フォノン近似•

Atomic Forces=Flkl

ul,llk̄=k̄,(Q) =(Q)ul,kl,lk̄

,

eiQ·(Rll•

Equation of Motion
2M umass=12(Q)uRl )PhononCalculation
凍結フォノン近似•

Dynamical Matrix

D•,(Q) = (M M ) 1/2 k̄

(Q)Secular Equation

D•

,,

(Q)

2u

Eigenvalue2(Q)0 vibration mode2(Q) < 0

soft mode (instability)
13=0

BaTiO3
P 4mmionic model: Ba2+ Ti4+ O2-

PS =PspolarizationeZn un = 0.16C/m2
nexperiment: Ps = 0.27C/m2

c/a = 1.01
displacementsu(Ti) = 0.054Å
u(O1) = –0.097Å
u(O2) = –0.061Å

Polarization enhancement
Electron response
Why?

14

Macroscopic Polarization
R. Martin, Phys Rev B 9,1998 (1974)• Accurate electron density by first-principles••

calculations
Sensitive response of electron density spatiallydistributed due to lattice distortion and ionicdisplacements
Electron density: not sufficient to determine
polarization

P =

1P (r)d3 r· P (r) =cell

P =

1r (r) +1cellsurface
15(r)r[P (r) · dS]Theory of Macroscopic Polarization
R. Resta, Ferroelectrics 136, 51 (1992).

•Electric dipole is ill defined in a periodic charge
distribution unless decomposed into neutral,
localized charge segments.
➡

The absolute value of polarization P is not abulk property and a variation in P is actuallyobservable.

16

Theory of Macroscopic Polarization
R. Resta, Ferroelectrics 136, 51 (1992).

•Observation of polarization: current
by adiabatic variation of state⇥PJ( ) =⇥+d•AFirst-order polarization variation
P =⇤ 1J ( )d =0⇤ 10⇥P⇥⇥d⇤Pi e ⇤ ⇥⇥ik ( )|p|⇥jk ( )⇤⇥⇥ik ( )| ⇤ |⇥jk ( )⇤=⇥2kk⇤m
E ( ) E ( )⇤Vk,i=jij⇥pk
H( )⇥i ( ) =+ V ( ) ⇥ik ( ) = Eik ( )⇥ik ( )2m217

Theory of Macroscopic Polarization
King-Smith & Vanderbilt, PRB 47, 1651 (1993).

⇧occ. ⇥⌅kk⌃ie⇥ui ( ) ⇥ui ( )
⇥k⇥

⇥P=
⇥k,i

⌅

⇥uki ()⇥⇥uki (⇥k)⇧⇤⇥ik ( ) = eik·r uki ( )

P =

⇤ 1

⇥P⇥

0P

(⇥)

=

⇥d = P (1)occ.
⌅2ieuki (k,iBerry phase( )k =iP (0)⇤ ⇥ ⇤ k ⇥⇤u i ( ))⇤⇥kocc.

dki
18k( )
uikk( )ui

Calculation of Macroscopic Polarization• Variation in 1polarization
PP =d = P (1)

P (0)

0

P =P ion +P el• Localized charges
Pion=eion=nucleus + core electrons

Z u
Z =Zu

Ncore

ionic charge

atomic displacement
19Calculation of Macroscopic Polarization• Non-overlapping charges
e(0)elP =r n(1)ndraa
With boundary at zero density, integration can bemade as a cell integral like localized density.
• Charges with localized orbitals (Wannier ft.)n(a ) (r) = 2Pel=2ea(n ) (r)2nr̄n(1)r̄n(0)

r̄n( ) =2r a(n ) drn

center of gravity of Wannier ft.
20

Berry Phase
Berry, PRSL A 392, 45 (1984)Phase associated with adiabatic process originatingfrom geometric property of hamiltonian
Time-dependent Perturbation| (t) =i | ˙ (t)⇥ = H(t)| (t)⇥i ȧn (t) = En (t)an (t)m|m(t) am (t)H(t)|m(t)⇥ = Em (t)|m(t)⇥n(t)|ṁ(t) am (t)imAdiabatic Approximation (m=n only)an (t) = an (0) expi⇤ t0tn (t) = i⇥En (t )dt exp [i n (t)]n(t )|ṅ(t )⇥dt021

Berry Phase••Time dependence via a parameter x(t)A one-turn process in the parameter space withtime from 0 to TTn (T ) = in(x(t ))|ṅ(x(t ))⇥dt0=iC⇥n(x)|⇧x n(x)⇤ · dxCt=Tt=0x22

Berry Phase of Bloch Function
Zak, PRL 62, 2747(1989)

1D Bloch function in a vector potential⇤⌅⇥21e
˙V (x + a) = V (x)i (x, t) =pA(t) + V (x) (x, t)
2mc
⇤
⌅
⇥
21e
pA(t) + V (x) n (x, t) = ⇥n (t) n (x, t)
2mc

Generalized Bloch functionn (x, t) = eikxk(t) = kun (x, k(t))
Tn =ieA(t)c⇥n (k(t ))|⇥˙ n (k(t ))⇥dt0t=0a+

t=Tk=i+aaa23⇥n (k)|⇥kn (k)⇥dkBerry Phase and Wannier Function
1D Wannier Function
1ai (x Xl ) = ⇥eik(x Xl ) ui (x, k)
N k
1
ui (x, k) = ⇥e ik(x Xl ) ai (x Xl )N l
1⇥ui (x, k) = ⇥i(x Xl )e ik(x Xl ) ai (x⇥kNl

Berry Phase⌅ a ⇥⇤⇥ui (k)ui (k) dki =i⇥ka⇥⇤2=x|ai (x)|2 dx
center of gravitya24Xl )Calculation of Polarization with Berry PhaseElectric Polarization of 3D System
2ie(2 )3( )Pel =⇥n2e=(2⇥)3d3 k⇥u(n ) (k)|⇧k u(n ) (k)⇤d2 k( )

( )
0.32

Polarization (C/m2)

0.32




(k )

(k⌅ ) = Im

( )



limJ⇥⇤( )J⇤1( )

⇥det Sij (ks , ks+1 )s=0( )Sij (ks , ks+1 ) = ui (k , ks )|uj (k , ks+1 )

0.30

0.30
0.28
0.28
0.26
0.26
0.24
0.24
0.220.22
0.20
0.20

0.18
0.18
0.16
0.16



BaTiO300551010JJ15152020

Convergency with respect tonumber of k-points
25

Phase Problem of PolarizationeR
1.6 ⇥ 10 19 C · 4 ⇥ 10 8 cm2⇠=
100µC/cm⌦(4 ⇥ 10 8 )cm3

BiFeO3
PRB 71, 014113 (2005)

26

ZnO
Ishii, TO (unpublished) Pion = +14.7µC/cm2 
Pel = 19.6µC/cm2
Pion +

Pel =

4.9µC/cm2

Posternak

5µC/cm2

Vanderbilt

5.7µC/cm2

experiment

7 ± 2µC/cm2e **ΔP = Z u ⇒ Z = −1.99
Ω27

Born effective charges

Polarization of Ferroelectric BaTiO3••

Tetragonal（ionic model: 0.16 C/m2）-

0.31 C/m2 Ishii, TO-

0.28 C/m2 Zhong, Vanderbilt-

0.27 C/m2 experiment

Rhombohedral-

0.37 C/m2 Ishii, TO-

0.43 C/m2 Zhong, Vanderbilt-

0.33 C/m2 experiment28

Band Structure of Cubic BaTiO3ionic crystal
conduction bands
Ti 3dgaps
valence bands
O 2p
Ba 5p
O 2s
Ba 5s
29

Density of States of Cubic BaTiO3ionic crystal+strong hybridization
Ti-3dε (yz, zx, xy)
O-2pσ

O-2pπ
Ti-3dγ (3z2–r2, x2–y2)
no Ba orbitalaround gaps: Ba2+30

Electron Density of Tetragonal BaTiO3n(r) = n(r)natom (rR )Oreduction
increase
OTi

Ocharge transfer from Ti to O
→ ionic bonding
bond charge between Ti and O→ covalent bonding

31

Phonon Mode of Cubic BaTiO3
ω(cm–1)

BaTi

O1

O2

O2’

186i

-0.01

-0.62

1

0.51

0.51

01

1111165
180*

-0.66

10.78

0.97

0.97

277

000

1-1458482*-0.010.16

1-0.69-0.69*IR @395K

32

a = 4.00Å

soft mode
Ba2+ not joining
soft modePhonon Band of Perovskite Oxides: ABO3

Eigenvector:
red: A
green: B
blue: O

33

Phys. Rev. B 60, 836 (1999)

Energy Variation of BaTiO3 due to Displacement
24

BaTiO3 GGA
c/a=1.00

20

ENERGY (meV)16

c/a=1.011284

c/a=1.02

0c/a=1.03-4-8-12
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
DISPLACEMENT
34

displacementsu(Ti) = 0.054Å
u(O1) = –0.097Å
u(O2) = –0.061Å

Electronic Structure Variation of BaTiO3 due todisplacement
cubictetragonal

O2-pO1-p

Ti-ddisplacement→ Enhancement in hybridization→ band shift→ energy gain

broken lines: cubicsolid lines: tetragonal
35

Instability to Ferroelectric Phase
three-site model
H = H0 + Ht + Helaperovskite structure
TM1t1

TM2εdO†d (d1H0 =d1†+ d2d2 )Ht = t1†
d1

p + h.c.

+t2†d2p + h.c.

t2Hela =362x2

Instability to Ferroelectric Phase
Variation of hopping integral by O-ion displacement

t1 =t(1 + · x)t2 = +t(1electron-lattice coupling· x)

Energy variation up to 4th order of hopping t

E=8t2 2+32t4 2
3dd+x232t4 4 x4+324d

Negative coefficient of 2nd order term → instability

td

18t2 2d37>• large δ• small λInstability to Ferroelectric Phase
Equilibrium displacement
3dx2 =32t4 48t2 2dTo satisfy the instability condition,
appropriately large size of hopping integral t
compare to εd• large δTM1t1TM2εd• small λ• appropriate tt2ionic + covalent bondingO38

Instability to Ferroelectric Phase
Coupling constant δt(R)R(l+l +1)

Elastic constant λHt =2x2t(R0 ) 1

l+l +1xR0• kind of orbital• local structure• local structure: tolerance• Tolerance factor of perovskite structuret=

Ferroelectric instabilityt>1
rA + rOof B ion2(rB + rO ) t < 1 Antiferroelectric instabilityof A ion
39

Born Effective Chargee *ΔP = Z uΩu // <001>
1222
12BaTiO3A B O1 O2
2.75 7.16 -5.69 -2.11

SrTiO3

2.54 7.12 -5.66 -2.00

CaTiO3

2.58 7.08 -5.65 -2.00

KNbO3

1.14 9.23 -7.01 -1.68

NaNbO3

1.13 9.11 -7.01 -1.61

PbTiO33.90 7.06 -5.83 -2.56

PbZrO33.92 5.85 -4.81 -2.48

BaZrO32.73 6.03 -4.74 -2.01

Zhong, Vanderbilt, PRL 72, 3618 (1994).
40

Mechanism of Effective Charge
Enhancement

Ti-3dε

O2--Ti4+O-2pσ O-2pπ
Ti-3dγMovement of bond charge
due to variation inhybridization between
Ti-3d and O-2p orbitals

41

Role of Pb■ Lone pair

Repulsion of [6s]2 or [6sp]2
widely accepted since 70s especially in Chemistry■ Covalent bond with O-2p6s=O-2p

Cohen（1992-)

6p=O-2p

Miyazawa (2000-)42

Role of Pb: PbTiO3Pb-6s

Pb-6p

integrated DOSbonding

integrated DOSPb-s DOS

Pb-pDOS

anti-bondinganti-bonding
gap

gap

bonding

fully occupied Pb-6s
bonding and anti-bonding

occupied Pb-6p bonding
states only →covalent
43

Soft Mode in Cubic PbMO3 and BiMO344Summary• Electron Theory of Ferroelectrics- BO Potential- QM Representation of Polarization• Typical Ferroelectric BaTiO3- Ionic and Covalent Bonding- Ferroelectric Instability- Born Effective ChargeReferences• Reviews- 寺倉清之：固体物理 35, 620 (2000).
- R. Resta: Modelling Simul. Mater. Sci.
Eng. 11, R69 (2003).

- M. Dawber, K.M. Rabe, and J.F. Scott:Rev. Mod. Phys. 77, 1083 (2005).