Phase-field Modeling of Ferroelectric Materials
Benjamin Völker, Marc Kamlah, Jie Wang
COMSOL Conference 2009, October 14 – 16, Milano
INSTITUTE FOR MATERIALS RESEARCH II
KIT – University of the State of Baden-Württemberg andNational Large-scale Research Center of the Helmholtz Association www.kit.edu
Presented at the COMSOL Conference 2009 Milan
Contents
theory of phase-field modeling of ferroelectric materials
parameter identification in free energy densityparameter identification in free energy density
finite element implementation:
PDE formPDE form
weak form
periodic boundary conditions:
electrical
mechanical
domain configurations
intrinsic and extrinsic contributions to small signal properties
Institute for Materials Research II 2 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
Technical applications
- NEMS
Institute for Materials Research II 3 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
- NEMS - Memories
Ferroelectric materials
Institute for Materials Research II 4 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
Phase-field modeling
polarization as continuous order parameter
fluctuating at the scale of domain dimensions
Total Helmholtz Free Energy densitygy y
,
, ,0
( , , , )
1ˆ ( , ) ( )( )2
i i j ij i
ijkl i j k l ij i i i i i
P P D
G P P P D P D P
System Free Energy
0
, ,0
21ˆ ˆ( ) ( , ) ( )( )
2Lan em
ijkl i j k l i ij i i i i iG P P P P D P D P
( ( ) ( ))P P P dV System Free Energy
equilibrium condition for order parameter
,( ( ), ( ))i i k i j kv
P P x P x dV
ˆ 0P Mi
temporal and spatial evolution: relaxation towards equilibrium:
, ,
: 0ii i i j j
P MinP P P
( )P t
Institute for Materials Research II 5 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
( , ) i ki
P x tP
Boundary value problem
field equations
balance of momentum
G i l,ij j i ib u
D q
Gaussian law
time dependent Ginzburg-Landau eqn.
boundary conditions
,
, ,
i i
ijkl k l ij jji
D q
G P PP
boundary conditions
ijij tn or iumechanical:
electrical: iinD or
potential relations
ii
polarization: 0, jji nP or iP
and
kinematicsij
ij
i
i DE
1
Institute for Materials Research II 6 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
and ijjiij uu ,,21
jiE ,
Structure of Helmholtz Free Energy density
(I)
free energy contains crystallographic and boundary information:
(II)
(III)
(IV)
ε: mechanical strainD: dielectric displacementP: polarization (order parameter,
ti d i ll )
(I) exchange energy: allows formation of domain walls with finite thickness(II) non-convex energy surface minima at spontaneous polarization states (Landau energy)
(IV) continuous on domain walls)
(II) non convex energy surface, minima at spontaneous polarization states (Landau energy)(III) adjustment of material properties (electromechanical coupling, elastic properties of
spontaneous polarized states)(IV) energy stored within free space occupied by material
Institute for Materials Research II 7 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
Adjustment of free energy density
Phenomenological approach:Phenomenological approach:
– fit to first order phase transition– based on experimental observationsp– requires κ(TC), PS(TC), TC, T0
however: ab-initio calculations can‘t yield this information [Devonshire1954]
Gi b L d th
this project: virtual material development (BMBF WING, COMFEM)new approach needed for multiscale simulation chain
b i iti / t i ti
piezoelectric coefficients dijk
dielectric permittivity κij
mechanical stiffness Cfree energy parameters
α
Ginzburg-Landau-theoryab-initio / atomistic
Aim:development
fmechanical stiffness Cijkl
spontaneous strain εS
spontaneous polarization PS
domain wall energy (90°/180°) γ90/180
αijklmnqijklcijklGijkl
of a new
adjustmentmethod
Institute for Materials Research II 8 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
domain wall energy (90 /180 ) γ90/180
domain wall thickness (90°/180°) ξ90/180
method
Adjustment of free energyadjustment for PbTiO : <111>
ab-initio/atomistic
(input)phase-field (adjusted)
adjustment for PbTiO3:
<110>
<111>
PS [C/m²] 0,880 0,880
ε11S -7,388E-03 -7,388E-03
ε33S 4 209E-02 4 209E-02
<100>
ε33 4,209E-02 4,209E-02
κ11 53,7 53,7
κ33 16,91 16,9
Px
C11 [N/m²] 2,86E+11 2,86E+11
C12 [N/m²] 1,17E+11 1,17E+11
C44 [N/m²] 6,70E+10 6,70E+10Py
ξ90 [nm] 0,478 0,478
γ90 [mJ/m²] 64 63.8
ξ180 [nm] 0,390 0,390 adjustment completely with ab-initio/atomistic results
Institute for Materials Research II 9 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
γ180 [mJ/m²] 156 156results
method works fine for PbTiO3
Finite element implementation
electric enthalpy: Legendre transformation
, ,( , , , ) ( , , , )ij i i i j ij i i i j i ih E P P D P P D E
potential relations
, ,j j j j
h hD h h , , ,
mechanical strain-displacement relations and electric potential
ijij ij
i
i
DE
i iP P
, ,i j i jP P
primary nodal variables for finite element formulation, , , ,( , , , ) ( , , , )ij i i i j i j j i i jh E P P h u P P
u P
physical principle: minimize System Free Energy WRT for given
, ,i iu P
( )i kP x ( ), ( )ij k i kx D x
Institute for Materials Research II 10 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
y gy g( )i k ( ), ( )ij k i k
Implementation in COMSOL (1)
general PDE form in COMSOL ,Fudue jjtatta ,,,
ijtti bhu
)(
, , ,( , , , ,...)i i t j jk u u u u
ii
ijij
tti
qEh
,
,,
)(
)(
boundary conditions in COMSOL or
iij
jitjij P
hPhP
,,
, )(
G 0 Rboundary conditions in COMSOL orGn ii
ijij
tnh
or iumechanical:
0 R
electrical:
ii
nEh
or
l i ti P0h
Institute for Materials Research II 11 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
polarization: or iP0,
j
ji
nP
Implementation in COMSOL (2)
weak form in COMSOL
principle of irt al ork for phase field theor eq ilibri m states
,Γ Γ n i i i iF dV dA
/ 0P t principle of virtual work for phase-field theory, equilibrium states
,ijij i i i i i ijkl k l j iV S
h hD E P dV t u G P n P dAP x P
/ 0iP t
expressing by means of electric enthalpy
h h h h
,
ijj j jV Si j i jP x P
,,
ij i i i jVij i i i j
h h h hE P P dVE P P
h h h
formulation of complex analytical expression to be entered in COMSOL by
,j i j j iS
ij j i j
h h hn u n n P dAE P
Institute for Materials Research II 12 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
symbolic algebra software
Importance of boundary conditions
short circuited boundary:
monodomain is lo est energ
0
monodomain is lowest energy state
open circuited boundary conditions: 0i iD n
flux closure, vortex
in this activity: bulk material behavior, far from free surface
Institute for Materials Research II 13 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
periodic boundary conditions, “elimination“ of the boundary
Periodic boundary conditions (1)
4
Periodic boundary conditions (part 1):polarization, electric potential
1 3 2 4==Px
Py
Px
Py
1 3
y
Φ + Vx
y
Φ + Vy
y2
x
2
Institute for Materials Research II 14 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
image boundary
Periodic boundary conditions (2)
Approach:1) define reference nodes2) define Sij
image boundary
initial boundaryinitial boundary
Institute for Materials Research II 15 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
image boundary
Periodic boundary conditions (2)
Approach:1) define reference nodes2) define Sij
image boundary
3) apply Sij to all other nodes:initial boundaryinitial boundary
- “new” node on image boundary:
- corresponding node on initial boundary:
periodic BC: for all nodes on image boundary:
Institute for Materials Research II 16 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
only one DOF left
image boundary
Periodic boundary conditions (2)
Approach:1) define reference nodes2) define Sij
image boundary
3) apply Sij to all other nodes:initial boundaryinitial boundary
- “new” node on image boundary:
- corresponding node on initial boundary:
P blperiodic BC: for all nodes on image boundary:
Problem:- only one reference node- may coincide with domain wall
improvement: averaging over boundary p g g y integration of displacement over boundary
Institute for Materials Research II 17 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
only one DOF left integration coupling variables
Periodic boundary conditions (2)
Ill t ti f i di b d iApproach:1) define reference nodes2) define Sij
Illustration of periodic boundaries:deformation plot (ux,uy) of domain structurecolor coding / arrows: polarization Py
3) apply Sij to all other nodes:
- “new” node on image boundary:
- corresponding node on initial boundary:
P blProblem:- only one reference node- may coincide with domain wall
improvement: averaging over boundary p g g y integration of displacement over boundary
Institute for Materials Research II 18 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
integration coupling variables
Mechanically predefined configurations
stable (90°) configuration: pinning mechanical displacement ui on corners of geometry
xx example: equal domain fractions
periodic boundary conditions
2xx yy
xu x
2xx yy
yu y
- periodic boundary conditions- minimum energy: 90°-domains
2
Py=Pspontx x00
x
y
uu
spon
yP P
0yy
Institute for Materials Research II 19 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
0xx
Mechanically predefined configurations
- model size: 16 nm x 24 nm
Institute for Materials Research II 20 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
- stable configuration with 3 domains
Mechanically predefined configurations
- model size: 16 nm x 24 nm- stable configuration with 3 domains
Institute for Materials Research II 21 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
stab e co gu at o t 3 do a s- different initial condition
Mechanically predefined configurations
- model size: 20 nm x 20 nm
Institute for Materials Research II 22 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
- different domain ratio
3D layer model (plain stress conditions)
x/y – directionas 2D-model:- periodic for Pi, ui, Φ
x
z z – direction- strain uzz free! (can move..)- Pz=0
x
y
model size: 16 nm x 24 nm(max model size: ~20 nm x 30 nm)
Institute for Materials Research II 23 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
(max. model size: ~20 nm x 30 nm)
Applied external loads
applied electric field
user defined angle α
applied stress
user defined angle α
tyy
ttyx
yPy α
Py α
x
Institute for Materials Research II 24 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
xR(α) * t * R(α)T
Contribution of reversible domain wall motion
0.85
0.90
13,5 0
0.75
0.80
Dx [C
/m²]
158 0
0.7108
0.890
0.00E+000 1.00E+008 2.00E+008 3.00E+008 4.00E+008
0.70
Ex [V/m]
0.7100
0.7102
0.7104
0.7106 C/m
²] x = 158 0 0.875
0.880
0.885
C/m
²]
x = 13,48 0
0.7092
0.7094
0.7096
0.7098
Dx [C
0.860
0.865
0.870Dx [C
Institute for Materials Research II 25 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
0 1x106 2x106 3x106 4x106
Ex [V/m]3.0x108 3.2x108 3.4x108 3.6x108 3.8x108 4.0x108
Ex [V/m]
Modeling defects in the phase-field model 1 Charged defect
bd i ( l ) t
1. Charged defect
subdomain (volume) terms boundary terms
2. “Fixed” polarization
+e.g.:Py = -Pdefect
approach for defects:- create geometric “point”
Throughout simulation:defect properties: constant
- create geometric point- define defect via boundary conditions- mesh geometry- solve
defects cannot switchdefects cannot move
Institute for Materials Research II 26 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
Investigation of 90° domain stacks
From PFM experiments: typical domain width ~100 200 nm [Fernandéz/Schneider TUHH]
PZT Zr/Ti 50/50 Nb 1 mol%
Example: 90°- stack, electrical loading (Y-direction)Domain wall (DW): 1) artificially completely fixed2) f
From PFM-experiments: typical domain width ~100-200 nm [Fernandéz/Schneider,TUHH]
2) free3) pinned by charged (line) defect
0.05
0.06
1) free DW 2) charge pinned DW 3) fixed DW
1 5 µm
Phase field model: 90° domain stack
0.02
0.03
0.04
Dy [C
/m²]
1) domain wall motion intrinsic + extrinsic
10 µm1.5 µm
200 nm
50 nm
0 0 2 0 106 4 0 106 6 0 106 8 0 106 1 0 107
0.00
0.01
D
3) intrinsic, domain wall fixed
2) domain wall bending
y
Py0-Pyspont Py
spont
- 2D, ~450k degrees of freedom- periodic boundary conditions
/
intrinsic/ extrinsic piezoelectric effect(reversible) DW moving / bending
0.0 2.0x106 4.0x106 6.0x106 8.0x106 1.0x107
applied electric field Ey [V/m]
Institute for Materials Research II 27 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
x- electrical / mechanical loading
Conclusionstheory of phase-field modeling of ferroelectric materials
parameter identification in free energy density
finite element implementationfinite element implementation
periodic boundary conditions
domain configurationsdomain configurations
intrinsic and extrinsic contributions to small signal properties
COMSOL: a powerful tool for nowadays mathematical physics
Acknowledgementsproject COMFEM: Federal Ministry of Education and Research, programme WING
Institute for Materials Research II 28 25.09.2009 B. Völker, M. Kamlah, J. Wang:
Phase-field Modeling of Ferroelectric Materials
invitation to this conference: COMSOL