Phase-field Modeling of Ferroelectric MaterialsPhase-field Modeling of Ferroelectric Materials Benjamin Völker, Marc Kamlah, Jie Wang COMSOL Conference 2009, October 14 – 16, Milano

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:


Technical applications

- NEMS



- NEMS - Memories

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



( , ) 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



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



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



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



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



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



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



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



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



image boundary


Approach:1) define reference nodes2) define Sij

image boundary

initial boundaryinitial boundary



image boundary



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:



only one DOF left

image boundary



image boundary

3) apply Sij to all other nodes:initial boundaryinitial 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



only one DOF left integration coupling variables


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:



P blProblem:- only one reference node- may coincide with domain wall

improvement: averaging over boundary p g g y integration of displacement over boundary



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



0xx


- model size: 16 nm x 24 nm



- stable configuration with 3 domains


- model size: 16 nm x 24 nm- stable configuration with 3 domains



stab e co gu at o t 3 do a s- different initial condition


- model size: 20 nm x 20 nm



- 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)



(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



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



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



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]



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



invitation to this conference: COMSOL

Documents

Phase-field Modeling of Ferroelectric MaterialsPhase-field Modeling of Ferroelectric Materials Benjamin Völker, Marc Kamlah, Jie Wang COMSOL Conference 2009, October 14 – 16, Milano