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 and National Large-scale Research Center of the Helmholtz Association www.kit.edu Presented at the COMSOL Conference 2009 Milan
28
Embed
Phase-field Modeling of Ferroelectric Materials · 2009. 12. 1. · Phase-field Modeling of Ferroelectric Materials Benjamin Völker, Marc Kamlah, Jie Wang COMSOL Conference 2009,
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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:
(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: