Download Ferroelectrics from first principles Tips, tricks and pitfalls

Ferroelectrics from first principles
Tips, tricks and pitfalls
Nick Bristowe
School of Physical Sciences, University of Kent, Canterbury UK
CCP9 meeting, April 10th 2017
Ferroelectricity vs ferromagnetism
Botharetypesof“ferroics”
REVIEWSFerroelectricity – spontaneouspolarizationP (switchablewithanelectricfieldE)
Ferromagnetism
– spontaneousmagnetizationM (switchablewithamagneticfieldH)
REVIEWS
NATURE|Vol 442
NATURE|Vol 442|17
Classedbysymmetry:
Figure 2 | Time-reversal
and spatial-inversion
symmetrysymmetry
in ferroics.
may be
represented
by by
a positive
thatlies
liesasym
asy
Figure 2 | Time-reversal
and spatial-inversion
in ferroics.
may
be represented
a positivepoint
point charge
charge that
Eerenstein,
& Scott
Nature 442, 759
(2006)
a, Ferromagnets.
The local
magnetic
may
be represented
within
a crystallographic
unit
cellthat
thathas
has no net
There
a, Ferromagnets.
The local magnetic
moment
mmoment
mayMathur
bemrepresented
within
a crystallographic
unit
cell
netcharge.
charge.
Th
classically
by adynamically
charge that dynamically
traces as
an indicated
orbit, as indicated
dependence,
spatial
inversionreverses
reverses p.
classically by a charge
that
traces an orbit,
by the by the
dependence,
but but
spatial
inversion
p. c,c,Multiferroics
Multiferro
1.1 Oxides, interfaces and polarity
Ferroelectrics
(a)
1.1 Oxides, interfaces and polarity
P
up
Ferroelectrics
Pdown
(a)
Pup
e.g.
BaTiO3
PbTiO3
(b)
Figure 1.2:
f“Hysteresis”:
Pdown
Pup
(b) (c)
P
Figure 1.2:
Pdown
f
P
Pdown
Pup
Pup
P
(c)
P
εE
Two(ormore)stable
polarstates,
switchablewith
externalfield
Pup
ε
Pdown
Pdown
(a) Schematic of the o↵-centring for up and down ferroelectric
polarisation states. (b) The typical double well shape of the free energy as a
of polarisation.for
(c) up
Theand
hysteretic
of the polarisation
(a) Schematic offunction
the o↵-centring
downbehaviour
ferroelectric
with
Ferroelectrics are also…
Dielectrics
Piezoelectric
Pyroelectrics
Ferroelectrics
Ferroelectrics are also…
Dielectrics
Piezoelectric
Pyroelectrics
Ferroelectrics
Pinduced underE-field
Uses:insulators&
microwavedielectrics
Ferroelectrics are also…
Dielectrics
Piezoelectric
Pyroelectrics
Ferroelectrics
Pinduced underE-field
Uses:insulators&
microwavedielectrics
Pinduced understrain
Uses:Sparkplugs,
pressuresensors
Ferroelectrics are also…
Dielectrics
Piezoelectric
Pyroelectrics
Ferroelectrics
Pinduced underE-field
Uses:insulators&
microwavedielectrics
Pinduced understrain
Uses:Sparkplugs,
pressuresensors
SpontaneousP(T)
Uses:Tsensors
Ferroelectrics are also…
Dielectrics
Piezoelectric
Pyroelectrics
Ferroelectrics
Pinduced underE-field
Uses:insulators&
microwavedielectrics
Pinduced understrain
Uses:Sparkplugs,
pressuresensors
SpontaneousP(T)
Uses:Tsensors
SpontaneousandswitchableP
Uses:memorydevices,capacitors,high-kdielectrics,
optoelectronics,catalysis,“photoferroics”,“multiferroics”
Research Activity
Scopusresultforpaperspublishedwith“ferroelectric”mentionedinanyfield
Talk Outline
Highlightseveralbasicissuesforfirstprinciples
calculationsofferroelectrics
P
P
1.1 Oxides, interfaces and polarity
(a)
up
1.1 Oxides, interfaces and polarity
down
(a)
Pup
1)CalculatingthespontaneousPolarisation
(b)
(c)
f
P
Pup
(b)
2)Calculatingthehysteresisloop
Pup
Pdown
(c)
f
ε
P
Pdown
Pup
P
Pup
ε
P
Pdown
Pdown
Figure 1.2:
Figure 1.2:
Pdown
(a) Schematic of the o↵-centring for up and down ferroelectric
polarisation
states.ferroelectric
(b) The typical
(a) Schematic of the o↵-centring for up
and down
polarisation states. (b) The typical double well shape of the free energy as a
3)Determiningthetypeofferroelectric
double well shape of the free energy as a
function of polarisation. (c) The hysteretic behaviour of the polarisation with
applied electric field, E.
function of polarisation. (c) The hysteretic behaviour of the polarisation with
applied electric field, E.
short-range ionic Coulombic forces, the system would spontaneously polarise. The
dp hybridisation is intimately linked with the anomalously large Born e↵ective
charges, Z ⇤ , in ferroelectrics. The Born e↵ective charge is dynamical, and should
short-range ionic Coulombic forces, the system would spontaneously
polarise. The
be distinguished from static charges which are usually based on partitioning the
Multifunctional
magnetoelectrics
dp hybridisation is intimately
linked with the anomalously large Born e↵ective
4)Finitesizeeffects
full electron density in to contributions from di↵erent ions. Instead Z ⇤ relates the
change
in polarisation
upon
displacement of an atomic sub-lattice.
charges, Z ⇤ , in ferroelectrics. The Born e↵ective charge
is dynamical,
and
should
Z ⇤ is large in
(Generalized) Magnetoelectric: cross coupled ferroelectrics due to the charge transfers associated with bond formation upon B
be distinguished
from
staticand
charges
whichfields
are usually based on partitioning the
response to
electric
magnetic
cation displacements.
full electron density in to contributions from di↵erent ions.
Instead Z ⇤ relates the
Macroscopically, ferroelectric phase transitions can be described phenomenochange in polarisation upon displacement of an atomiclogically
sub-lattice.
Z ⇤ is large in
within thermodynamic
theory.
+
5)Multiferroics cation-displacements.
E
H
This is the Landau-Ginzburg-Devonshire
ferroelectrics due to the charge transfers associated with bond formation upon B
Polarization, P
Magnetization, M
P
M
Macroscopically, ferroelectric phase transitions can be described phenomeno-
7
1. Calculating P
1. Calculating P
1. Calculating P
1. Calculating P
1. Calculating P
1. Calculating P
1. Calculating P
1. Calculating P
Modern theory of polarisation
Defined in terms of Wannier centres
r r 2
" = $ w n ( r # R) i
occ
n,i
- +
+
- +
- +
z
!
P
P: Dipole moment per unit volume of point charges
DEFINED UP TO POLARISATION QUANTA
1. Calculating P
Modern theory of polarisation
Defined in terms of Wannier centres
r r 2
" = $ w n ( r # R) i
occ
n,i
- +
+
- +
- +
z
!
P (jumped by a polarisation quantum)
P: Dipole moment per unit volume of point charges
DEFINED UP TO POLARISATION QUANTA
1. Calculating P
For centrosymmetric bulk systems (or more generally for this,
any symmetry transforming z into -z), Pz should be invariant:
Pz = -Pz
Infinitesystem– notoneP,butasetofallowedPvalues
should
be invariant: which
is the case for the two values
Centrosymmetric
materialscanhave2setsbysymmetry:
{….. -2P0, -P0, 0, P0, 2P0 ……}
P0:quantum
{….. -3P0/2, -P0/2, P0/2, 3P0/2 ……}
Originof“polar”surfaces
Non-Centrosymmetric materials:
{……. -2P0+Ps , -P0+Ps , Ps , P0+Ps , 2P0+Ps ……}
Ferroelectric materials:
{……. -2P0-Ps , -P0-Ps , -Ps , P0-Ps , 2P0-Ps ……}
1. Calculating P
For centrosymmetric bulk systems (or more generally for this,
any symmetry transforming z into -z), Pz should be invariant:
Pz = -Pz
Infinitesystem– notoneP,butasetofallowedP
values
should
be invariant: which
is the case for the two values
Centrosymmetric
materialscanhave2setsbysymmetry:
{….. -2P0, -P0, 0, P0, 2P0 ……}
P0:quantum
{….. -3P0/2, -P0/2, P0/2, 3P0/2 ……}
Originof“polar”surfaces
Non-Centrosymmetric materials:
{……. -2P0+Ps , -P0+Ps , Ps , P0+Ps , 2P0+Ps ……}
Ferroelectric materials:
{……. -2P0-Ps , -P0-Ps , -Ps , P0-Ps , 2P0-Ps ……}
Whicharethe
relevantpairs?
2. Determining hysteresis
(b) a)WhatusPup- Pdown? (c)
P
f
Pdown
Figure 1.2:
Pup
Pup
ε
P
Pdown
(a) Schematic of the o↵-centring for up and down ferroelectric
P : {……. -2P +P , -P +P , P , P +P , 2P +P ……}
0
s
0
s
0
polarisation states. (b) The uptypical double well
shape
ofs thes free0 energy
as
as
function of polarisation. (c)
The
hysteretic
behaviour
of the polarisation with
Pdown
:{…….
-2P
0-Ps , -P0-Ps , -Ps , P0-Ps , 2P0-Ps ……}
applied electric field, E.
short-range ionic Coulombic forces, the system would spontaneously polarise. The
2. Determining hysteresis
2P0+Ps
P
2P0-Ps
P0+Ps
P0-Ps
Ps
Modeamplitude
-Ps
-P0+Ps
0-Ps
1.1-POxides,
interfa
-2P0+Ps
1.1 Oxides, interfaces and polarity
(a)
1.1 Oxides, interfaces and polarity
-2P0-Ps
(a)
Pup
Pdown
Pup
Pdown
2. Determining hysteresis
2P0+Ps
P
2P0-Ps
P0+Ps
P0-Ps
Ps
Modeamplitude
-Ps
-P0+Ps
0-Ps
1.1-POxides,
interfa
-2P0+Ps
1.1 Oxides, interfaces and polarity
1.1 Oxides, interfaces and polarity
-2P0-Ps
Connectthedots!
(a)
(a)
Pup
Pdown
Pup
Pdown
2. Determining hysteresis
2P0+Ps
P
2P0-Ps
P0+Ps
P0-Ps
Ps
Modeamplitude
-Ps
-P0+Ps
0-Ps
1.1-POxides,
interfa
-2P0+Ps
1.1 Oxides, interfaces and polarity
1.1 Oxides, interfaces and polarity
-2P0-Ps
Connectthedots!
(a)
(a)
Pup
Pdown
Pup
Pdown
2. Determining hysteresis
F
P
1.1 Oxides, interfa
1.1 Oxides, interfaces and polarity
(a)
1.1 Oxides, interfaces and polarity
(a)
Pup
Pdown
Pup
Pdown
2. Determining hysteresis
F=
aP2 +
F
bP4
E=
P
1.1 Oxides, interfa
1.1 Oxides, interfaces and polarity
(a)
1.1 Oxides, interfaces and polarity
(a)
Pup
Pdown
Pup
Pdown
2. Determining hysteresis
P
P
Ε
(a)
(b)
2. Determining hysteresis
P
P
Veryapproximate
estimateofhysteresis
Inreality,dominatedby
domainwallmotion
Ε
E
(a)
(b)
2. Determining hysteresis
P
P
Veryapproximate
estimateofhysteresis
Inreality,dominatedby
domainwallmotion
Ε
E
(a)
(b)
2. Determining hysteresis
P
P
Veryapproximate
estimateofhysteresis
Inreality,dominatedby
domainwallmotion
Ε
E
(a)
Requiresmesoscopicmodelling
Beyondscopefortoday
(b)
polarities
and, similarly,
can be computed
the mode th
ary conditions, which is particularly useful
in studying
ferroelectric
systems. from
Furthermore,
2.reformulation
Determining
hysteresis
eigenvector
and
the
of theand
magnetizations
citors suggests an appealing
in terms
ofknowledge
free charges
potentials,inwhi
duced
byan
displacing
individual
atment of stresses and strains. Using PbTiO
example,
we showatoms.
that our technique enabl
3 as
The magnetic response to an electric field is readily
variables within the density functionalFiniteFieldMethods
formalism.
odern theory of polarization1 P
has
in the theory of the ferroelectric
t could previously be inferred only
now be computed with quantum
rst-principles density functional
s focused on bulk ferroelectric
e balance between covalency and
ferroelectricity. Over time, these
the effects of external parameters
lds2,3 . Of particular note is the
od for performing calculations at
P4 . The ability to compute crystal
as a function of P provides an
nshire and related semiempirical
(a)
obtained from InthelinearP(E)regime,wecanuse
these expressions. Indeed, the equilibrium
value of un for applied
E and zero magnetic field is
“frozenphonon”approach
Sand
I C A Lthe
R E Berry-pha
VIEW LE
117201 (2008) energy PEHKSY P
v through PRL
the 101,
Kohn–Sham
X
1
d
¼ to the
pmoment
polarization The
P lattice
(ref.contributions
1).un(For
thepolarization
we fix the
latti
ni E i ; and magnetithus(5)
gainin
C i
zationwill
are be discussedn shortly.)
theof
compu
vectors; strains
The minimum
U
require sim
N
fixed from
D is which
given the
by induced magnetization
1 X
is
obtained
as
considered
P
¼
pd u
(3)
IR
latt;i
Ε
!0
ni n
sponse, wh
n¼1
NIR
X
X
1
@E
1
@U
@EKS
@P
Indeed, su
m
¼ " e.g.Iniguez
¼
!
E
þ
p
u
M jand
=
⌦ (D
·
= 0nj n playing a
elec;ij 4⇡P)
i
PRL101117201(2008)
!
@H
!
N
@v D 0 @v
@v
j
0 n¼1
i 1 X
polar crys
IR
M latt;j ¼
!0X
Nn¼1
IR
pm
nj un ;
(4)
X
X
1
1
m
Comparing respectively.
with
the
fixed-E
approach
2,3
¼GobeyondlinearP(E)regimebydefiningelectric
!elec;ij
E ithe
þ dielectric
pnj of ofrefs
pdninth
E i :IRin
pdn is
polarity
the
!0 from
Cn iBorn effecelectric enthalpy
given
by
mode, F
which
can
be
obtained
the
atomic
i is
n¼1
enthalpyfunctionalinDFTmethods
Then,
linear
we see that
tive charges and the mode eigenvector [10]. The pm
n cothe
mode-decomposed
lattice-mediated
part
efficients
are the magnetic analogue
of the dielectric
Fand,
(E,v)
Ecan
⌦ E ·P(v)
KS (v)
polarities
similarly,
be
from the mode
ME
response
is=given
bycomputed
eigenvector and the knowledge of the magnetizations inNIR individual atoms. X
NIR d m
ducedSouza,Iniguez,VanderbiltPRL89117602(2002)
by displacing
X
pni pnj
1
Umari
&Pasquarello
PRL89,157602(2002)
The
magnetic
to an
!latt;ij
¼ response
!latt;nij
¼electric field is readily
:
(b)
Ε
coupling t
included tin
which
(6)will
there
derivation
constant m
ofusing
thethe I
(
tural varia
dynamical
Results
(7) m
proposed
3. Types of ferroelectrics
DFThasplayedanimportantroleinunderstanding“origin”ofP
CohenNature358136(1992)
3. Types of ferroelectrics
HybridizationleadstoanomalousBorneffectivecharges
PH. GHOSEZ, X. GONZE, PH. LAMBIN, AND
6766
J.-P. MICHENAUD
TABLE I. Born effective charges for cubic BaTi03.
Shell model
Ions
1.63
7.51
2.9
6.7
—2.4
—4.8
Ba
Zo
—2.71
—3.72
'Reference 4.
"Reference 9.
Zhong et al. "
a„)(=4.00 A
a„ii=3.94 A
a„))=3.67 A
2.75
7. 16
—2. 11
—5.69
2.74
7.29
—2.13
—5.75
2.77
7.25
—2.15
—5.71
2.95
7.23
—2.28
—5.61
Ghosez etalPRB516765(1995)
of Zo ( —5.71) and Zo i
( —2. 15) refer, respectively, to a displacement of the oxygen
and
+7.25.
For
0,
the values
II
ion along the Ti-0 direction or perpendicular to it. Our results are in good agreement with those obtained by Zhong,
King-Smith, and Vanderbilt (also at the theoretical volume)
from finite difference of polarization and globally reproduce
the values deduced by Axe. The accordance with the shell
model is only qualitative and illustrates the limited precision
obtained within such an empirical description. We note however that this shell model was not designed to reproduce the
dielectric properties of BaTi03. The large value of Z~; (7.51)
orbitals is well known and was already pointed out from
and linear combinatight-binding models,
experiments,
tion of atomic orbitals calculations.
It was highlighted
more recently by Cohen from first principles as an essential
feature of ABO3 ferroelectric compounds. In this context, it
seemed realistic, following Harrison, to focus on 0 2p-Ti 3d
hybridization changes to explain intuitively large anomalous
contributions.
went bePosternak, Resta, and Baldereschi
yond this credible assumption. They showed for KNb03 that
the anomalous contributions disappear when the interaction
between the
2p and Nb 4d orbitals is artificially suppressed.
0
antipolar X5 mode contribution, in complete contrast to the
eigendisplacements (η) of the FE unstable mode for t
case of perovskite oxides [12].
of NaBF3 compounds, again comparing with BaTi
To understand the differences that we have found between
In all fluoride cases, we find a strong A-site cont
oxide and fluoride perovskites, we next calculate the Born
with a substantial contribution also from F∥ but a n
effective charges (Z ∗ ) of selected ABF3 cubic perovskites, and
contribution from the B-site ion. The dominance of th
HybridizationleadstoanomalousBorneffectivecharges
we compare them in Table II with the values for BaTiO3 . As
displacement can also be seen in Fig. 3, where w
expected for ionic compounds, the Born effective charges of
with red arrows the eigendisplacements for the FE
the fluoroperovskites are close to their nominal ionic charges,
NaMnF3 . The trend in A-site displacement magnitud
-P.
PH.
AND
MICHENAUD
PH. with
X.
LAMBIN,
J.
GHOSEZ,
GONZE,
the most anomalous value—a deviation of −0.71e for
the A- and B-site relative masses: the A-site cont
F∥ —resulting from a small but nonzero charge transfer from F
to the FE mode eigendisplacements for A = Li, Na
TABLE
Born
effective
for
cubic
I.
charges
BaTi03.
to B as the atoms move closer to each other. This is in striking
and Cs in ANiF3 are 0.347, 0.186, 0.138, 0.081, an
contrast to the oxides, where the
Born
effective
charges
are
respectively, while the contributions from Ni across
Shell model
et al. "
Ions
00 A
67 A
a„)(=4.
a„))=3.
a„ii=3.
strongly
anomalous.Zhong
For example,
the
corresponding
oxygen94 A series
are 0.003, −0.019, −0.053, −0.073, and −0.0
in63BaTiO3 is −3.86e,
indicating2.74
a much stronger
To explore
2.75
2.77
2.95 the origin of the unstable mode, we
2.9 anomaly 1.
Ba
charge transfer
and change of7.29
hybridization acthe
7.25 Table III 7.
23 on-site interatomic force constants (I
7.51
7. 16
6.7 dynamical
Since
key ingredient
—2.4 companying
—2.71the Ti-O displacement.
—2. 11
—2.a 13
—2.in15 the cubic
—2.NaBF
28 3 series. The on-site IFC gives the
ferroelectricity
—5.61 atom when it is displaced in the cryst
—4.8 explaining
—3.72
—5.in69perovskite oxides
—5.75 is the presence
—5.71 an individual
Zo
all the other atoms are kept fixed [27]. Even in high-s
phases with structural instabilities, the on-site IFCs ar
'Reference 4.
TABLE II. Born effective charges, Z ∗ (e), and eigendisplacements
Ghosez etalPRB516765(1995)
positive (indicating a restoring force) as a periodic c
"Reference 9.
of the FE unstable mode, η, in NaBF3 and BaTiO3 . X⊥ and X∥ indicate
3. Types of ferroelectrics
6766
the Born effective charge of the anion (F or O) when it is displaced
and
+7.25.
For
( —2. 15) refer,
is well known and was already pointed out from
parallel
perpendicular
the B-X orbitals
bond.
values of
5.71) and toZo
0, theAsopposedto“Geometricferroelectrics”withnominalcharges
Zo or( —
TABLE III. On-site IFCs (first four columns) and larg
i experiments,
and linear combinatight-binding models,
II
respectively, to a displacement of the oxygen
Z∗
ion along the Ti-0 direction or perpendicular to it. Our rethose obtained
A
B by X
X∥
sults are in good agreement with System
Zhong,
⊥
King-Smith, and Vanderbilt (also at the theoretical volume)
Nominal 1
2
−1
−1
from finite difference of polarization and globally reproduce
NaMnF3 1.17 2.21 −0.83 −1.72
with the shell
the values deduced by Axe. The accordance
NaVF3 1.18 1.99 −0.71 −1.75
model is only qualitative and illustrates the limited precision
NaZnF3 1.15 2.22 −0.84 −1.69
obtained within such an empirical description. We note howNaNiF3 1.15 2.05 −0.74 −1.73
the
to reproduce
ever that this shell model was not BaTiO
designed 2.75
7.37 −2.14 −5.86
3
dielectric properties of BaTi03. The large value of Z~; (7.51)
atomic IFCs (last three columns) of ABF3 (eV/Å2 ). X⊥ a
η
tion of atomic
orbitals calculations.
It was highlighted
defined as in Table II.
more recently by Cohen from first principles as an essential
A
B
X
X∥
of ABO3⊥ ferroelectric
feature
it
compounds.
FEmodeappearsduetoionicsizemismatch
System
A In this
B context,
X⊥
X∥
B − X∥ B − B∥′
seemed realistic, following Harrison, to focus on
2p-Ti 3d
0.174
−0.005 −0.020
0.17 large
13.14anomalous
0.76 13.35 −4.62
−2.77
to explainNaMnF
hybridization
intuitively
changes−0.088
3
0.181
−0.018 −0.019
−0.077Resta,
0.23 19.00 went
0.88 be17.29 −6.78
−1.96
NaVF
contributions.
and3 Baldereschi
Posternak,
0.177
−0.025assumption.
−0.084
0.43 for
12.63
1.22 that
13.81 −4.39
−3.21
NaZnF
this credible
They 3showed
KNb03
yond −0.005
0.186
−0.002 −0.025
−0.068 disappear
0.60 17.07
1.63 16.13 −5.69
−2.62
NaNiF3 when
the anomalous
contributions
the interaction
Garcia-CastroetalPRB89104107(2014)
0.001
8.27
13.09 6.74 sup11.70
2.71 −13.83
BaTiO
3
between0.098
the −0.071
and Nb 4d
orbitals
is artificially
2p −0.155
pressed.
0
0
temperature,
T,
3.
Types
of ferroelectrics
isotropy) in the absence of strain and
an external electric field, the free
f (T, P ) = a1 P 2 + a11 P 4 + a111 P 6 ,
ProperFerroelectrics
y of a ferroelectric can be expressed
as a function of polarisation, P , and
erature, T ,
Landaudescription
where the coefficients
are temperature dependent. Perhaps the m
2
6
f (T, P ) = a11,Ptakes
+ a11
P 4form,
+ a111aP
coefficient,
the
=
1 , ↵(T
T0 ), where ↵(1.2)
is a positiv
T0 is the phase transition temperature. Additionally the coeffici
the coefficients are temperature dependent. Perhaps the most important
mines the type of transition; first or second order. For a second or
cient, a1 , takes the form, a1 = ↵(T T0 ), where ↵ is a positive constant and
a11 is positive. When T > T0 , a1 is positive, and the free energy
the phase transition temperature. Additionally the coefficient, a11 , deterwith minimum at P = 0, and the material
therefore in the parae
Curie-Weisslaw
the type of transition; first or second order. For a second order transition,
instead T < T0 , the free energy is a double well (see figure 1.2)
positive. When T > T0 , a1 is positive, and p
the free energy is a single well
at P = ±PS , where PS =
a1 /(2a11 ) is the spontaneous pola
minimum at P = 0, and the material therefore in the paraelectric state. If
#$$ =
"R
#$$
=
Local
tions associated with the R point of the cubic first Brillouin
zone, respectively. We also compute the following quantities:
3. Types of ferroelectrics
!NZ!"2 2
&'u$( − 'u$(2),
V % ok BT
0.02
<u >=<u >=<u >
1
2
3
0.01
0.00
0.20
TriggeredFerroelectrics
(b)
!2"
N2
2
2
&'"R,
$( − '"R,$( ),
VkBT
R3c
Pm3m
I4/mcm
0.15
AFD motions (radians)
where “' (” denotes statistical averages and where u$ and
Landaudescription
"R,$ are the $ component of u and "R, respectively. N is the
number of sites in the supercell, while V is its volume. Z! is
0.10
the Born effective charge associated with the local mode, kB
< ! > =< ! >
1
R
2
R
2
4
2
2
is the Boltzmann constant, and %o is the permittivity of
0.05
vacuum. #$$ are the diagonal elements of the dielectric sus"R
23,24
<! >
ceptibility tensor,
while #$$ is the staggered
3
R
25
susceptibility associated with "Kisanonpolarmode,ais+ve
.
andc–ve
R,$
0.00
Figure 1!a" shows the 2predicted x-, y-, and z-Cartesian
350
If|cK |>a,PbecomesdoublewellwhenKiscondensedin
coordinates !'u1(, 'u2(, and 'u3(" of 'u( in BFO, while Fig.
(c)
Neithermodehastobe“primary”modeifcouplingcislarge
1!b" displays '"(R !with
'"1(R, '"2(R, and '"3(R denoting the
300
x-, y-, and z-Cartesian coordinates of '"(R", and Fig. 1!c"
250
reports the susceptibilities. Figures 1!a" and 1!b" indicate
that, as consistent with experiments !see Ref. 9 and refer200
ences therein", our numerical scheme predicts a R3c ground
<" >=<" >
11
22
state that exhibits a DoesnotfollowCurie-Weisslaw:
polarization pointing along the &111)
150
<"
>
direction !since 'u1( = 'u2( = 'u3( ! 0 at the lowest tempera33
100
tures" and a tilting of the oxygen octahedra about the &111)
axis !since '"1(R = '"2(R = '"3(R ! 0 at the lowest tempera50
tures". Figure 1 also indicates a first-order FE-to-paraelectric
transition from a rhombohedral R3c phase to a tetragonal
0
0
500
1000
1500
2000
I4 / mcm state that solely exhibits AFD motions about the
Temperature (K)
&001) axis, about TC * 1075 K as indicated by the nonzero
value of '"3(R above *1075 K in Fig. 1!b" and the vanishFIG. 1. Supercell average 'u( of the local mode vectors &panel
Holakovský,
Phys.1075
Status
ing of 'u( via the jump seen
in Fig. 1!a" around
K. Solidi B 56, 615 1973
!a"), AFD-related '"( quantity &panel !b"), and #$$ dielectric sus'"2(R 79 100105 2009 R
Figure 1!a" thus reveals that
'u1(, 'u2(,&'uBellaiche,
Kornev
3(, '"1(R, and PRB
ceptibilities &panel !c") as a function of temperature in BFO and as
are the OPs of this FE transition. The I4 / mcm phase exists
F = aP + bP + cP K
2000
AFD susceptibility
Dielectric susceptibility
1500
<"
<"
!
11
!
33
>=<"
!
>
22
>
1000
500
0
900
950
1000
1050
1100
1150
1200
3. Types of ferroelectrics
ImproperFerroelectrics
Landaudescription
1
1
2
E = A0 P + B0 P 4 + C12φ12φ22
2
4
1
1
+ A1φ12 + B1φ14 + C01φ12 P 2
2
4
1
1
+ A2φ22 + B2φ24 + C02φ22 P 2
2
4
+λφ1 φ2 P
Otheranharmonic couplingspossible
(Podd)
3. Types of ferroelectrics
ImproperFerroelectrics
Landaudescription
f1,2 µ (T-Tc)1/2
® P µ f1.f2 µ (T-Tc)
® c independent of T
Bousquet et al Nature 452 732 (2008)
A.P. Levanyuk and D.G. Sannikov, Sov. Phys. Usp. 17, 199 (1974)
4. Finite size effects
Ultrathin-films
miniaturizationoftransistors,developmentoflayer-by-layergrowthtechniques
4. Finite size effects
Ultrathin-films
miniaturizationoftransistors,developmentoflayer-by-layergrowthtechniques
Epitaxialstrain
4. Finite size effects
Ultrathin-films
“Strainengineering”
2 + csP2
F = aP2 + bP4 + KsPHYSICAL
REVIEW B 72, 144101 !2005"
Diéguez,Rabe&VanderbiltPRB 72 144101 (2005)
4. Finite size effects
Ultrathin-films
“Strainengineering”
2 + csP2
F = aP2 + bP4 + KsPHYSICAL
REVIEW B 72, 144101 !2005"
Diéguez,Rabe&VanderbiltPRB 72 144101 (2005)
4. Finite size effects
Opencircuitboundaryconditionsi.e.filmsinair
σ bound
! ! ρtotal
∇⋅E =
ε0
! !
= P ⋅ n = Pz
ε 0 ΔEz = σ free − ΔPz
FEinopen-circuitcreatesa“depolarizingfield”
F = aP2 + bP4 - E.P
E.P termdominatescurvatureofPtobecomepositive!
nd redox
nce of a
ernative
creening
4. Finite size effects
PHYSICAL REVIEW B 85, 024106 (2012)
Screeningmechanism1:Ionicadatoms
Metal
FE
Air
N. C. BRISTOWE et al.
FIG. 1. Schematic illustration of the conventional (left) and red
(center) mechanisms for ferroelectric screening in the absence o
top electrode. The presence of a biased tip can promote an alterna
Bristoweetal
redox mechanism
that provides an external circuit for the screen
electrons PRB
(right).85 024106 (2012)
enough to show bulklike features in the center, and 3 u
cells of BTO was experimentally shown to be thick enou
for ferroelectricity.1 We use a dipole correction to simul
open-circuit boundary conditions, enforcing zero macrosco
4. Finite size effects
Screeningmechanism2:Zenertunneling
Aguado-Puente,Bristoweetal
PRB92 03548(2015)
Yinetal
PRB92 115406(2015)
4. Finite size effects
Screeningmechanism3:Domains
w
d
Fluxclosure:
Aguado-Puente&Junquera PRB85 1841052012
in0 monodomain
Curves are calcu
parameters
obta
0
plot % = 1 V, re
-100
G (MJ/m )
3
G (MJ/m
)
3
◦ domain
phasesstructure,
(monodomain
andG
polydomain),
we
need
tothe
compare
width [53,54]
γ w/d,
propor180the
domain
in which
straight
ofwhere
the material,
elec = stripes
4.
Finite
size
effects
evolution
of direction
the
energy
ofbea thin film intothe
constant
can
calculated
to
all tionality
withthe
thethickness
same width
inbethe
perpendicular
thetwo
scenarios.
For the polydomain
phase,
we
assume
a
domaindifferent
wall,
have
an
out-of-plane
polarization
of
the
same
2
◦
8.416P
180
domain
structure,
in orientations.
which
straightFor
stripes
ofidealized
the material,
Screeningmechanism3:Domains
magnitude but
with alternating
such
(27)
γ =
all
with
the
same
width
in
the
direction
perpendicular
to
the
3
1/2
π ε0 [1
+ (εx εfound
] in tetragonal
version of the domain structures
typically
z)
domain wall, have an out-of-plane
polarization of the same
ferroelectric
◦ thin films, the energy per wunit of volume of the
for 180
stripebut
domains
[55]. The
width ofFor
domain
walls in a
magnitude
with alternating
orientations.
such idealized
polydomain phase can be expressed as
d
-100 where %
case
version
of the domain
structures typically
found
in tetragonal
typical
ferroelectric
is vanishingly
small and
remnant
electric -200
by surface red
!
ferroelectric
thin
films,
the
energy
per
unit
of
volume
of
the
fields in polydomain
configurations
decay
exponentially
away
G
+
G
=
U
+
,
(26)
poly
elec
nm. Interesting
polydomain phase can be expressed
as
w
from the surface and the domain wall, thus, except in the
0
2t
-200
predicts
that
! of a domain
where
the energy per
unit
of area
wall and
limit!ofis thicknesses
ofpoly
a few
G
+ Gelecwe
= U unit
+ cells,
, can approximate
(26)
polarization d
w is
the
domain
width.
The
electrostatic
energy
G
due
to
w
|P | ∼ PS throughout the film and U = U0 (PS elec
) as a constant.
0
2
crossover
betw
stray
fields
in
the
polydomain
configuration
is
proportional
to
FIG.
8.
(Color
onlin
where
isStrayfield:
theand
energy
per unit of area
a domain
wall and
Using
this !
result
differentiating
Eq. of
(26)
with respect
to w
thickness
the
thetodomain
width
[53,54]
G
=
γ
w/d,
where
the
proporin
monodomain
(black
w
is
the
domain
width.
The
electrostatic
energy
G
due
to
elec
elec
find the equilibrium domain width for a given thickness
of
formation
of
a
tionalitystray
constant
bepolydomain
calculated configuration
to be
are8.calculated
fo
fields can
in the
is proportional to CurvesFIG.
(Color
onli
the film,
one obtains the
well-known
Kittelwhere
law [53]
overobtained
the break
the domain width
[53,54]
G
=
γ
w/d,
the propor- parameters
fr
in monodomain
(blac
elec2
Kittellaw: 8.416P
tionality constant
be calculated
!d to be
plotCurves
% parameters,
= are
1 V,calculated
realistic
th
(27)
γ = can
2
3
1/2
(28) parameters obtained f
π ε0 [1w+ =
(εx εz ) . 2 ]
4.6 nm, well w
γ
8.416P
plot % = 1 V, realistic
(27)
= The
for 180◦ stripe domains γ[55].
width
of
domain
walls
in
a
grown%in=ferr
3 ε [1 + (ε ε )1/2 ]
π
case
where
C/
0
x
z
Substituting
this
expression
for
the
equilibrium
domain
typical ferroelectric
is vanishingly small and remnant electric
at which
◦
by surface
redoxthe
prot
for
180
stripe
domains
[55].
The
width
of
domain
walls
in
a
width
into
Eq.
(26)
leads
to
the
following
expression
for
the
fields in polydomain configurations decay exponentially away
case where % = C/
the atoms move from an sp2 environment towards sp3
bonding, resulting in polarization in the z direction [9].
In Table I, we report the lowest TO and LO phonon
Ferroelectrictypesthatareresistanttodepolarizingeffects
frequencies
and dielectric constants, as well as band gaps,
pP
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!
! 2
ΔZ
¼
ðZ
zz
zz Þm =N , and polarizations for a variety of
m
Hyperferroelectrics
ABC ferroelectrics; those with imaginary LO frequencies
are, by definition, hyperferroelectrics. The relatively small
LOisunstable(inadditiontoTO)
ProperferroelectricwhichlowP,andlarge
ΔZ!zz ≈ 3 and large ðϵ∞ Þzz ≈ 10–20 both contribute to the
Weakdepolarizingfields
weak depolarization fields in these materials [for reference,
cubic perovskites typically have ΔZ!zz ≈ 5 and ðϵ∞ Þzz ≈ 6].
Garrity,Rabe &VanderbiltPRL 112 127601 (2014)
Both the small effective charges and large dielectric
constants of ABC ferroelectrics are consequences of the
Improperferroelectrics
covalent bonding and resulting small band gaps of these
semiconductors. In Fig. 1(c), we show the phonon
Pdrivenbynon-polareffectivefield
dispersion for the hyperferroelectric LiBeSb, which is a
BalanceofenergyscalesdetermineswhetherPcanremaininopencircuit
previously synthesized material [23,24]. We see that the
lowest frequency phonon mode for q → 0 is unstable
regardless of the direction from which Γ is approached.
In order to investigate the electric equation of state of
hyperferroelectrics, we use a simple first-principles-based
Stengel,Fennie &Ghosez
094112
(2012)
model.PRB
We86first
define
a dimensionless polar internal degree
4. Finite size effects
22
mirror
plane onfor
thethe
central
metal
a complexof structure,
exponentially
the
2 layer. The
amplitude
this field,and
E d , decays
modesymmetry
, y, determined
sameTiO
tetragonal
cellrelaxation
geometry as
in has
depends
on (1) the in
screening
length of
metallic the
region
beyond
thepolarization
first RuO2 of
was
continued
until
thethe
maximum
force on case
the atoms
was smaller correlayer.
It
spreads
over a both monitor
the supercell.
Only
monodomain
was
considered,
metal
and
the
the
thin
film,
which
than 40 meVÅ21. The symmetry constraint prevents the appearance characteristic
distance that is related to the screening length of the
sponding to a situation that can be achieved experimentally9. In
of ferroelectricity, and the resulting structures correspond to the electrode. Each dipole is responsible for electrostatic potential drops
Fig. 2, the symbols show the evolution of the energy (with respect to
optimized paraelectric states. Starting from the previous relaxed in the same direction, so that the only way to preserve short-circuit
the referencetaken
paraelectric
phase)
when
freezing-in
displaceboundary conditions along the structure will be the appearance of a
configurations
as reference,
we then
searched
for theatomic
existence
y of increasing
amplitudes.
Calculations
have
been
donedepolarizing
for
electric field inside the ferroelectric film, illustrated in
ofments
a ferroelectric
instability,
moving BaTiO
atoms
continuously
3
differentthe
thicknesses
of the
BaTiO
layer.
2 points
Fig.a 3b for different values of the induced polarization. The
following
displacement
pattern
of 3the
bulkFigure
tetragonal
soft out
22
change
behaviourfor
at athe
critical
m ¼ 6as(26
For
of this field, E d , depends on (1) the screening length of
mode
, y,ofdetermined
same thickness
tetragonalaround
cell geometry
in Å).amplitude
the supercell. Only the monodomain case was considered, corre- the metal and the polarization of the thin film, which both monitor
sponding to a situation that can be achieved experimentally9. In
Fig. 2, the symbols show the evolution of the energy (with respect to
the reference paraelectric phase) when freezing-in atomic displacements y of increasing amplitudes. Calculations have been done for
different thicknesses of the BaTiO3 layer. Figure 2 points out a
effect of the depolarizing field for a realistic system and dem
change of behaviour at a critical thickness around m ¼ 6 (26 Å). For
4. Finite size effects
Capacitors
Closedcircuitwithrealelectrodes– imperfectscreening
letters to nature
its crucial role. First, it is responsible for the suppre
ferroelectricity at a finite size. Second, even well above th
thickness, the depolarizing field reduces the amplitud
spontaneous polarization, which only slowly approaches
value for films of increasing size (Fig. 2 inset).
The results reported here have been obtained for a de
insulating
Figure 2 Evolution of the energy as a function ofperfectly
the soft-mode distortion
y. First- BaTiO3 crystal, whereas typical ferr
principles results (symbols) and electrostatic model
results
(lines)
are
shown
for
capacitors are well-known
to exhibit high leakage currents
different thicknesses of the ferroelectric thin film: m ¼ 2 (circles, full line), m ¼ 4
monitored
Poole–Frenkel mechanism in the case
(upward-pointing triangles, dotted), m ¼ 6 (diamonds,
short-dashed), by
m ¼ the
8 (squares,
29
dot-dashed), and m ¼ 10 (downward-pointing triangles,
long-dashed).
The magnitude
oxide
electrodes
). Itof might therefore be questioned whethe
Figure 2 Evolution
of the energy
as a function
soft-mode
distortion
y. First- of BaTiO3 atoms
y is reported
as a percentage
of of
thethebulk
soft-mode
displacements
Figure 1 Structure of a typical ferroelectric capacitor. a, Schematic view of the system
redistribution
associated with the finite conductivity of th
principles results
and electrostatic
modelofresults
(lines)
are shown
for
considered here: a BaTiO3 film between SrRuO3 electrodes that are short-circuited.
(y ¼ 1(symbols)
corresponds
to the distortion
the bulk
tetragonal
ferroelectric
phase).The energy
different thicknesses
of the ferroelectric
film:asmreference.
¼ 2 (circles,
full line),
m ¼ could
4of the spontaneous
electric
further screen the depolarizing field and af
of the paraelectric
phase isthin
taken
Inset,
evolution
The whole structure is grown on top of a SrTiO3 substrate. The epitaxy is such that
(upward-pointing
triangles,
m ¼ 6 (diamonds,
short-dashed),
mdeduced
¼ 8 (squares,
polarization,
Ps,dotted),
SrTiO3 (001) k SrRuO3(001) k BaTiO3 (001)k SrRuO3(001). b, Atomic representation
of
with thickness.
For eachJunquera
thickness,
P
is
from
the
amplitude
of y current densities30 (J < 1025 A cm22,
&Ghosez
s
conclusions.
For usual
dot-dashed), and m ¼ 10 (downward-pointing triangles, long-dashed). The magnitude of
the related supercell that is simulated (for the case m ¼ 2).
at the minimum.
y is reported as a percentage of the bulk soft-mode
displacements
of BaTiO
Figure 1 Structure of a typical ferroelectric capacitor. a, Schematic view of the system
and
foratomsfields of the order of those of Fig. 3b), screenin
Nature423506(2003)
considered here: a BaTiO film between SrRuO electrodes that are short-circuited.
(y ¼ 1 corresponds to the distortion of the bulk tetragonal ferroelectric phase).The energy
© 2003 Nature Publishing Group
NATURE | VOL 422 | 3 APRIL 2003 | www.nature.com/nature
507
uncompensated surface
polarization charges ðjpol < 1026
of the paraelectric phase is taken as reference. Inset, evolution of the spontaneous
The whole structure is grown on top of a SrTiO substrate. The epitaxy is such that
polarization, P , with thickness. For each thickness, P is deduced from
SrTiO (001) k SrRuO (001) k BaTiO (001)k SrRuO (001). b, Atomic representation of
the amplitude
of y
would
require
a timescale of the order of 0.1 s. Compare
the related supercell that is simulated (for the case m ¼ 2).
at the minimum.
switching of the polarization (typically ,1029 s), this is a v
© 2003 Nature Publishing Group
NATURE | VOL 422 | 3 APRIL 2003 | www.nature.com/nature
507 can therefore conclude that this process cannot
process. We
Effectivescreeninglength
the film switching back to the paraelectric state below th
thickness. For m larger than the critical value and after a su
long time, the finite conductivity could allow the appea
additional screening charges. However, as it happens for
coming from the electrodes, the finite spread of this extr
3
3
3
3
3
3
3
3
s
s
M
5. Multiferroics
Multiferroic:materialcombiningtwoormoreferroic parameters
Ferromagnetic:
●
N
N
N
S
S
S
N
N
N
S
S
S
Ferroelectric:
M
P
Ferroelectric:
P
+
-
+
-
+
+
-
-
+
+
-
-
●
Hysteresis:
Multifunctional magnetoelectrics
(Generalized) Magnetoelectric: cross coupled
response to electric and magnetic fields
Non-volatile data-storage!
Magnetoelectric:
-
+
Polarization, P
E
H
Claude Ederer
First principles studies of multiferroic materials
P
M
Magnetization, M
Possibleapplications:
i.e. control of the magnetic M (electric P) phase with an applied
electric E (magneticRAM:electricwrite/magneticread
H) field
- Magnetoelectric
- 4-statememory
5. Multiferroics
Ferroelectricity as a lattice property
FE lattice distortion, Q, has identical symmetry properties as the
polarization, i.e. Q∝ P, involve small atomic distortions
Chemicalincompatibility
(Spaldin,J.Phys.Chem.B104 (2000))
Pb
FErequires“d0-ness”
Ti
O
FMrequirespartiallyfilledd
cubic-paraelectric
Commonroutetoavoidthisrule:
Space group Pm3m
P
+
–
Off-centering
drivenby
hybridisation
betweenemptyd
andoccupied2p
Cohen Nature 358, 136 (1992)
tetragonal-ferroelectric
Space group P4mm
- haveFEandFMactiveionsondifferentsiteeg BiFeO3EuTiO3
Cornell University!
School of Applied and Engineering Physics !
Problem:fewFEbuildingblockstostartfrom+weakcoupling?
fenniegroup.aep.cornell.edu!
42
5. Multiferroics
Ferroelectricity as a lattice property
FE lattice distortion, Q, has identical symmetry properties as the
polarization, i.e. Q∝ P, involve small atomic distortions
Chemicalincompatibility
(Spaldin,J.Phys.Chem.B104 (2000))
Pb
FErequires“d0-ness”
Ti
O
FMrequirespartiallyfilledd
cubic-paraelectric
Commonroutetoavoidthisrule:
Space group Pm3m
P
+
–
Off-centering
drivenby
hybridisation
betweenemptyd
andoccupied2p
Cohen Nature 358, 136 (1992)
tetragonal-ferroelectric
Space group P4mm
- haveFEandFMactiveionsondifferentsiteeg BiFeO3EuTiO3
Cornell University!
School of Applied and Engineering Physics !
Problem:fewFEbuildingblockstostartfrom+weakcoupling?
fenniegroup.aep.cornell.edu!
Tomorrow’stalk:UsingimproperFEtocreatemultiferroics
42